Photonic Signal Processing
Photonic Signal Processing
Second Edition
Le Nguyen Binh
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Library of Congress Cataloging‑in‑Publication Data
Names: Binh, Le Nguyen, author.
Title: Photonic signal processing / Le Nguyen Binh.
Description: Second edition. | Boca Raton : Taylor & Francis, a CRC title,
part of the Taylor & Francis imprint, a member of the Taylor & Francis
Group, the academic division of T&F Informa, plc, [2019] | Includes
bibliographical references.
Identifiers: LCCN 2018024823 (print) | LCCN 2018026380 (ebook) | ISBN
9780429436994 (eBook General) | ISBN 9780429792632 (Pdf) | ISBN
9780429792625 (ePub) | ISBN 9780429792618 (Mobipocket) | ISBN
9781498769938 (hardback : acid-free paper)
Subjects: LCSH: Optical communications. | Signal processing. | Photonics. |
Optoelectronic devices.
Classification: LCC TK5103.59 (ebook) | LCC TK5103.59 .B5243 2019 (print) |
DDC 621.36/5--dc23
LC record available at https://lccn.loc.gov/2018024823
Visit the Taylor & Francis Web site at
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Dedication
To the memory of my father
To my mother, Mrs. Nguyen Thi Huong
To Phuong Nguyen and Lam
Contents
Preface............................................................................................................................................xvii
Author .............................................................................................................................................xix
Chapter 1
Introduction ..................................................................................................................1
Acronyms .....................................................................................................................1
1.1 Preamble ............................................................................................................ 1
1.2 Introductory Remarks........................................................................................2
1.3 Organization of Chapters .................................................................................. 4
Chapter 2
Photonic Signal Processing Via Signal-Flow Graph.................................................... 7
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Introduction ....................................................................................................... 7
Incoherent Photonic Signal Processing ............................................................. 8
2.2.1 Fiber-Optic Delay Lines ..................................................................... 10
2.2.1.1 Fiber-Optic Directional Couplers ....................................... 10
2.2.2 Fiber-Optic and Semiconductor Amplifiers ....................................... 11
Coherent Integrated-Optic Signal Processing ................................................. 12
2.3.1 Integrated-Optic Delay Lines ............................................................. 15
2.3.2 Integrated-Optic Phase Shifters ......................................................... 16
2.3.3 Integrated-Optic Directional Couplers............................................... 16
2.3.4 Integrated-Optic Amplifiers ............................................................... 18
Remarks ........................................................................................................... 19
Signal-Flow Graph Approach and Photonic Circuits ......................................20
2.5.1 Introductory Remarks ........................................................................ 21
2.5.2 Signal-Flow Graph Theory ................................................................ 21
2.5.3 Definitions of SFG Elements .............................................................. 22
Rules of SFG.................................................................................................... 23
2.6.1 Rule 1: Transmission Rule.................................................................. 23
2.6.2 Rule 2: Addition Rule ......................................................................... 23
2.6.3 Rule 3: Product Rule ..........................................................................24
Mason’s Gain Formula ....................................................................................25
2.7.1 Analysis of an Incoherent Recursive Fiber-Optic Signal
Processor (RFOSP) ............................................................................25
2.7.2 SFG Representation of the Incoherent RFOSP ..................................25
2.7.3 Derivation of the Transfer Functions of the Incoherent RFOSP ........ 27
2.7.4 Stability Analysis of the Incoherent RFOSP...................................... 29
2.7.5 Design of the Incoherent RFOSP ....................................................... 29
2.7.6 Remarks.............................................................................................. 30
Optmason: A Program for Automatic Derivation of the Optical Transfer
Functions of Photonic Circuits from Their Connection Graphs ..................... 31
2.8.1 Overview ............................................................................................ 31
2.8.2 Using OPTMASON ........................................................................... 33
2.8.3 Contents of the Input File for above Examples .................................. 35
2.8.4 The OPTMASON Program Structure ............................................... 36
Appendix: Z-Transform ...................................................................................40
vii
viii
Contents
2.10
2.11
Chapter 3
Appendix: OPTMASON.PAS Program Listing............................................ 42
Appendix: Using “OPTIMASON” the Computer Aided Generator.............66
Bandpass Optical Filters by DSP Techniques ............................................................ 71
3.1
Optical Fixed Bandpass Filter ....................................................................... 71
3.1.1 Introductory Remarks ...................................................................... 71
3.1.2 Chebyshev Optical Filter Specification and Synthesis Algorithm ..... 71
3.1.3 Basic Characteristics of Chebyshev Lowpass Filters ....................... 72
3.1.3.1 Chebyshev-Type Optical Bandpass Filter
Specification .....................................................................72
3.1.3.2 Illustration of a Chebyshev Bandpass Optical Filter ........ 74
3.1.3.3 Optical Components for Chebyshev Filters ...................... 75
3.1.3.4 Realization of the Chebyshev Optical Bandpass Filters .......78
3.1.3.5 The COF1 ......................................................................... 78
3.1.3.6 Parallel Realization........................................................... 79
3.1.3.7 The COF2 ......................................................................... 81
3.1.3.8 Discussions ....................................................................... 83
3.1.4 Concluding Remarks ........................................................................ 83
3.2
Tunable Optical Bandpass Waveguide Filters ...............................................84
3.2.1 Introductory Remarks ......................................................................84
3.2.2 Transfer Function of IIR Digital Filters to Be Synthesized ............. 85
3.2.3 Basic Building Blocks of Tunable Optical Filters ............................ 86
3.2.3.1 Tunable Coupler ................................................................ 86
3.2.3.2 All-Pole Filter ................................................................... 87
3.2.3.3 All-Zero Filter................................................................... 89
3.2.4 Tunable Optical Filter....................................................................... 91
3.2.5 Synthesis of Tunable Optical Filters ................................................ 91
3.2.5.1 Design Equations for the Synthesis of Tunable
Optical Filters ................................................................... 91
3.2.5.2 Synthesis of Second-Order Butterworth Bandpass
and Bandstop Tunable Optical Filters...............................92
3.2.5.3 Designed Parameter Values of the Bandpass and
Bandstop Tunable Optical Filters .....................................92
3.2.5.4 Tuning Parameters of the Synthesized Bandpass
and Bandstop Tunable Optical Filters...............................94
3.2.6 Synthesis of Bandpass and Bandstop Tunable Optical Filters
with Variable Bandwidths and Fixed Center Frequencies ............... 98
3.2.6.1 Synthesis of Tunable Optical Filters with Fixed
Bandwidths and Tunable Center Frequencies .................. 98
3.2.6.2 Fabrication Tolerances of Filter Parameters ................... 100
3.2.7 Concluding Remarks ...................................................................... 104
References ................................................................................................................ 105
Chapter 4
Photonic Computing Processors .............................................................................. 107
4.1
Incoherent Fiber-Optic Systolic Array Processors ...................................... 107
4.1.1 Introduction .................................................................................... 107
4.1.2 Digital-Multiplication-by-Analog-Convolution Algorithm
and Its Extended Version................................................................ 109
4.1.2.1 Multiplication of Two Digital Numbers.......................... 109
ix
Contents
4.2
4.3
4.4
Chapter 5
4.1.2.2 High-Order Digital Multiplication.................................... 110
4.1.2.3 Sum of Products of Two Digital Numbers........................ 112
4.1.2.4 Two’s Complement Binary Arithmetic ............................. 112
4.1.3 Elemental Optical Signal Processors ............................................... 113
4.1.3.1 Optical Splitter and Combiner .......................................... 113
4.1.3.2 Binary Programmable Incoherent Fiber-Optic
Transversal Filter .............................................................. 114
4.1.4 Incoherent Fiber-Optic Systolic Array Processors for Digital
Matrix Multiplications ..................................................................... 116
4.1.4.1 Matrix–Vector Multiplication ........................................... 116
4.1.4.2 Matrix–Matrix Multiplication .......................................... 117
4.1.4.3 Cascaded Matrix Multiplication ....................................... 118
4.1.4.4 Performance Comparison ................................................. 121
4.1.4.5 Fiber-Optic Systolic Array Processors Using
Non-Binary Data............................................................... 122
4.1.4.6 High-Order Fiber-Optic Systolic Array Processors ......... 123
4.1.5 Remarks............................................................................................ 124
Programmable Incoherent Newton–Cotes Optical Integrator ...................... 125
4.2.1 Introductory Remarks ...................................................................... 125
4.2.2 Newton–Cotes Digital Integrators ................................................... 126
4.2.2.1 Transfer Function .............................................................. 126
4.2.2.2 Synthesis ........................................................................... 127
4.2.2.3 Design of a Programmable Optical Integrating
Processor........................................................................... 130
4.2.2.4 Analysis of the FIR Fiber-Optic Signal Processor ........... 132
4.2.2.5 Analysis of the IIR Fiber-Optic Signal Processor ............ 133
4.2.3 Section Remarks............................................................................... 143
Higher-Derivative FIR Optical Differentiators ............................................. 143
4.3.1 Introduction ...................................................................................... 144
4.3.2 Higher-Derivative FIR Digital Differentiators................................. 146
4.3.3 Synthesis of Higher-Derivative FIR Optical Differentiators ........... 147
4.3.4 Computed Differentiators of First and Higher Orders ..................... 150
4.3.4.1 First-Derivative Differentiators ........................................ 150
4.3.4.2 Second-Derivative Differentiators .................................... 152
4.3.4.3 Third-Derivative Differentiators....................................... 154
4.3.4.4 Fourth-Derivative Differentiator ...................................... 158
4.3.5 Remarks............................................................................................ 159
Appendix A: Generalized Theory of the Newton–Cotes Digital
Integrators ..................................................................................................... 160
4.4.1 Definition of Numerical Integration ................................................. 161
4.4.2 Newton’s Interpolating Polynomial .................................................. 162
4.4.3 General Form of the Newton–Cotes Closed Integration
Formulas .......................................................................................... 164
4.4.4 Generalized Theory of the Newton–Cotes Digital Integrators........ 164
Optical Dispersion Compensation and Gain Flattening .......................................... 167
5.1
5.2
Introductory Remarks.................................................................................... 167
Dispersion Compensation Using Optical Resonators .................................... 167
5.2.1 Signal-Flow Graph Application in Optical Resonators.................... 170
5.2.2 Stability Test ..................................................................................... 175
x
Contents
5.2.3
5.3
5.4
Chapter 6
Frequency and Impulse Responses .................................................. 176
5.2.3.1 Frequency Response ......................................................... 176
5.2.3.2 Impulse and Pulse Responses ........................................... 177
5.2.3.3 Cascade Networks ............................................................ 178
5.2.3.4 Circuits with Bi-directional Flow Path ............................. 178
5.2.3.5 Remarks ............................................................................ 178
5.2.4 Double-Coupler Double-Ring Circuit Under Temporal
Incoherent Condition ........................................................................ 178
5.2.4.1 Transfer Function of the DCDR Circuit ........................... 178
5.2.4.2 Circulating-Input Intensity Transfer Functions ................ 181
5.2.4.3 Analysis ............................................................................ 182
5.2.5 DCDR Under Coherence Operation ................................................. 199
5.2.5.1 Field Analysis of the DCDR Circuit ................................. 199
5.2.5.2 Output-Input Field Transfer Function ...............................200
5.2.5.3 Circulating to Input Field Transfer Functions ..................200
5.2.5.4 Resonance of the DCDR Circuit....................................... 201
5.2.5.5 Transient Response of the DCDR Circuit.........................204
5.2.6 DCDR Resonator as a Dispersion Equalizer: Group Delay
and Dispersion ..................................................................................209
Optical Eigenfilter as Dispersion Compensators ........................................... 220
5.3.1 Introductory Remarks ...................................................................... 220
5.3.2 Formulation and Design ................................................................... 222
5.3.2.1 Dispersive Optical Fiber Channel..................................... 222
5.3.2.2 Formulation of Optical Dispersion Eigencompensation ...... 223
5.3.2.3 Design and IM/DD System Performance .........................224
5.3.2.4 Performance Comparison of Eigenfilter and
Chebyshev Filter Techniques ............................................ 226
5.3.3 Synthesis of Optical Dispersion Eigencompensators ....................... 227
5.3.3.1 IM/DD Transmission System Model ................................ 228
5.3.3.2 Performance Comparison of Optical Dispersion
Eigencompensator and Chebyshev Optical Equalizer.........231
5.3.3.3 Eigencompensated System with Parameter Deviations
of the Optical Dispersion Eigencompensator ................... 234
5.3.3.4 Trade-Off Between Transmission Distance and
Eigenfilter Bandwidth ....................................................... 235
5.3.3.5 Compensation Power of Eigencompensating
Technique ....................................................................236
5.3.3.6 Remarks ............................................................................ 238
Photonic Functional Devices ......................................................................... 238
5.4.1 Preamble ........................................................................................... 238
5.4.2 Optical Dispersion Compensation Module (oDCM)........................ 239
5.4.3 Chromatic Dispersion Compensators ...............................................240
5.4.4 Optical Gain Equalizer .................................................................... 242
5.4.4.1 Introductory Remarks ....................................................... 242
5.4.4.2 Dynamic Gain Equalizer .................................................. 243
Optical Dispersion in Guided-Wave FIR and IIR Structures .................................. 245
6.1
6.2
Preamble/Introduction ...................................................................................246
Dispersion Mechanism in Fiber and Waveguide ........................................... 247
xi
Contents
6.3
6.4
6.5
6.6
6.7
6.8
6.9
Chapter 7
Micro-Ring Resonator (MRR) as an Optical Dispersion Compensator
(oDCM).......................................................................................................... 249
6.3.1 Why Resonator? ............................................................................... 249
6.3.2 Transfer Transmittance Function of the Thru Port (Notched
Resonant Filter) and Drop Port (Bandpass Filter) ............................ 250
6.3.2.1 Dispersion Characteristics and Dispersion
Compensation by MRR .................................................... 250
6.3.2.2 Dispersion Compensating of Multiple DWDM
Channels and Slope Dispersion Compensation ................ 251
6.3.3 Tunable Dispersion Compensator..................................................... 253
6.3.4 Length of Fiber Propagation and Dispersion Compensating
Module.............................................................................................. 254
6.3.5 Waveguide and Passive MRR Fabrication Technology for oDCM .....255
Active MRR................................................................................................... 255
6.4.1 Structure ........................................................................................... 255
oDCM by Fiber Bragg Grating...................................................................... 256
6.5.1 Motivation ........................................................................................ 256
6.5.2 Analytical Expression of Broadening (Fiber) and Compression
(TM-FBG) Factors ........................................................................... 257
6.5.2.1 Dispersion-Induced Pulse Broadening in Optical Fiber..... 257
6.5.2.2 Dispersion-Induced Pulse Broadening in FBG ................ 257
6.5.3 Design Cases ....................................................................................260
6.5.3.1 Design Case I: Finite Uniform Profile Grating ................260
6.5.3.2 Design Case II: Apodized Profile Grating ....................... 263
6.5.3.3 Remarks on FBG–oDCM .................................................264
FIR Discrete Wavelet Transform 2D Dispersion Compensating .................. 265
6.6.1 Introductory Remarks ...................................................................... 265
6.6.2 Analysis and Synthesis ..................................................................... 265
6.6.3 Design Procedures............................................................................ 268
6.6.4 Implementation................................................................................. 270
Concluding Remarks ..................................................................................... 274
Appendix: Dispersion Compensation a Historical View of Development
and Why MRR as DCM ................................................................................ 274
SFG and Mason Rules for Photonic Circuit Analysis ................................... 276
6.9.1 SFG and Mason Approach ............................................................... 276
6.9.2 The Gain Formula ............................................................................ 278
6.9.2.1 Procedure .......................................................................... 278
6.9.3 Derivation of Transfer Function of the Micro-Ring Resonator ....... 278
6.9.3.1 Single Ring ....................................................................... 278
6.9.3.2 MRR Incorporating MZDI Structure ...............................280
Photonic Ultra-Short Pulse Generators .................................................................... 283
7.1
Optical Dark-Soliton Generator and Detectors ............................................. 283
7.1.1 Introduction ...................................................................................... 283
7.1.2 Optical Fiber Propagation Model ..................................................... 285
7.1.3 Design and Performance of Optical Dark-Soliton Detectors ........... 286
7.1.4 Design of Optical Dark-Soliton Detectors ....................................... 286
7.1.5 Performance of the Optical Differentiator ....................................... 287
7.1.6 Performance of the Butterworth LPOF ............................................ 288
xii
Contents
7.1.7
7.2
7.3
7.4
Chapter 8
Design of the Optical Dark-Soliton Generator ................................. 289
7.1.7.1 Design of the Optical Integrator ....................................... 289
7.1.7.2 Design of an Optical Dark-Soliton Generator .................. 291
7.1.8 Performance of the Optical Dark-Soliton Generator and Detectors .... 293
7.1.8.1 Performance of the Optical Dark-Soliton Generator ........ 293
7.1.8.2 Performance of the Combined Optical Dark-Soliton
Generator and Optical Differentiator ............................... 295
7.1.8.3 Performance of the Combined Optical Dark-Soliton
Generator and Butterworth LPOF .................................... 295
7.1.9 Remarks............................................................................................ 297
Mode-Locked Ultra-Short Pulse Generators ................................................ 297
7.2.1 Introductory Remarks on Regenerative Mode-Locked Fiber
Laser Types ...................................................................................... 298
7.2.2 Ultra-High Repetition-Rate Fiber Mode-Locked Lasers .................302
7.2.2.1 Mode-Locking Techniques and Conditions for
Generation of Transform Limited Pulses from a
Mode-Locked Laser..........................................................302
7.2.3 MLL and MRLL Experimental Setup and Results.......................... 305
7.2.3.1 40 GHz Regenerative Mode-Locked Laser ......................307
7.2.3.2 Remarks ............................................................................309
7.2.4 Active Mode-Locked Fiber Ring Laser by Rational Harmonic
Detuning ........................................................................................... 311
7.2.4.1 Rational Harmonic Mode-Locking .................................. 311
7.2.4.2 Experiment........................................................................ 312
7.2.4.3 Phase Plane Analysis ........................................................ 313
7.2.4.4 Results and Discussion ..................................................... 316
7.2.4.5 Remarks ............................................................................ 319
Rep-Rate Multiplication Ring Laser Using Temporal Diffraction Effects ... 319
7.3.1 GVD Repetition Rate Multiplication Technique .............................. 320
7.3.2 Experiment Setup ............................................................................. 321
7.3.3 Phase Plane Analysis........................................................................ 322
7.3.3.1 Uniform Lasing Mode Amplitude Distribution ................ 322
7.3.3.2 Gaussian Lasing Mode Amplitude Distribution ............... 328
7.3.3.3 Effects of Filter Bandwidth .............................................. 328
7.3.3.4 Nonlinear Effects .............................................................. 328
7.3.3.5 Noise Effects ..................................................................... 328
7.3.4 Demonstration .................................................................................. 328
7.3.5 Remarks............................................................................................ 330
Multi-Wavelength Fiber Ring Lasers............................................................. 331
7.4.1 Theory .............................................................................................. 331
7.4.2 Experimental Results and Discussion .............................................. 333
7.4.3 Multi-wavelength Tunable Fiber Ring Lasers .................................. 336
7.4.4 Remarks............................................................................................ 338
Multi-Dimensional Photonic Processing by Discrete-Domain Approach ............... 341
8.1
Multi-Dimension (MULTI-D) PSP Design Techniques ................................ 341
8.1.1 An Overview of Photonic Signal Processing ................................... 341
8.1.1.1 Spatial and Temporal Approach ....................................... 342
8.1.1.2 Fiber-Optic or Integrated Optic Delay Line Approach .... 343
8.1.1.3 Motivation .........................................................................344
xiii
Contents
8.1.2
8.2
Multi-Dimensional Signal Processing..............................................344
8.1.2.1 Multi-Dimensional Signal ................................................344
8.1.2.2 Discrete Domain Signals .................................................. 345
8.1.2.3 Multi-Dimensional Discrete Signal Processing................346
8.1.2.4 Separability of 2-D Signals...............................................346
8.1.2.5 Separability of 2-D Signal Processing Operations ...........346
8.1.3 Filter Design Methods for 2-D PSP..................................................348
8.1.3.1 2-D Filter Specifications ................................................... 348
8.1.3.2 Mathematical Model of 2-D Discrete Photonic Systems .... 348
8.1.3.3 Filter Design Methods ...................................................... 352
8.1.3.4 Use of Matrix Decomposition .......................................... 352
8.1.4 Direct 2-D Filter Design Methods ................................................... 353
8.1.4.1 FIR and IIR Structures in 2-D Signal Processing ............ 353
8.1.4.2 Frequency Sampling Method............................................ 354
8.1.4.3 Windowing Method .......................................................... 356
8.1.4.4 McClellan Transformation Method .................................. 356
8.1.4.5 2-D Filter Design Using Transformation Method ............ 357
8.1.5 Concluding Remarks ........................................................................ 359
Decomposition Techniques and Implementation Using Fiber Optic
Delay Lines .................................................................................................... 359
8.2.1 Introductory Remarks ......................................................................360
8.2.2 Matrix Decomposition Methods ......................................................360
8.2.2.1 Single-Stage Singular Value Decomposition....................360
8.2.2.2 Multiple-Stage Singular Value Decomposition ................ 363
8.2.3 Iterative Singular Value Decomposition .......................................... 365
8.2.3.1 Iterative Singular Value Decomposition ........................... 365
8.2.3.2 A 2-D Filter Design Example Using Iterative Singular
Value Decomposition ........................................................ 366
8.2.4 Optimal Decomposition ................................................................... 367
8.2.4.1 Optimal Decomposition.................................................... 367
8.2.4.2 Other 2-D Filter Design Methods Based on Matrix
Decomposition .................................................................. 368
8.2.5 2-D Filter Order Reduction Using Balanced Approximation
Theory .............................................................................................. 369
8.2.5.1 Motivation for Lower Order Photonic Filters ................... 369
8.2.5.2 Description of 2-D System in State-Space Format........... 369
8.2.5.3 Balanced Approximation Method .................................... 369
8.2.5.4 Filter Order Reduction Using Balanced
Approximation: An Example............................................ 372
8.2.6 Fiber-Optic Delay Line Filters ......................................................... 374
8.2.7 Coherent and Incoherent Operation of Photonic Filters .................. 374
8.2.8 Using Optical Fibers to Realize Delayed Line Filter ....................... 375
8.2.8.1 Photonic Realization of Delay .......................................... 375
8.2.8.2 Photonic Realization of Tab Coefficients ......................... 376
8.2.8.3 Photonic Realization of Summer/Splitter ......................... 376
8.2.8.4 Graphical Representation of Photonic Circuits ................ 377
8.2.8.5 Remarks ............................................................................ 379
8.2.9 Concluding Remarks ........................................................................ 379
xiv
Contents
8.3
8.4
Realization ..................................................................................................... 380
8.3.1 Introductory Remarks ...................................................................... 380
8.3.2 Photonic Implementation of 2-D Filters ........................................... 381
8.3.2.1
Photonic Filter Structures ............................................... 381
8.3.2.2
Coherent System ............................................................. 381
8.3.2.3
2-D Direct Structure Filter ............................................. 381
8.3.2.4
2-D Separable Structure Filter ........................................ 383
8.3.2.5
Binary Tree Filter ........................................................... 384
8.3.2.6
Photonic Transversal Filter ............................................. 385
8.3.2.7
1-D Direct Structure Photonic Filter .............................. 388
8.3.2.8
Parallel Structure Filters ................................................. 389
8.3.2.9
Other 1-D Filter Structures ............................................. 391
8.3.2.10 Realization of Poles ........................................................ 392
8.3.2.11 Remarks .......................................................................... 393
8.3.3 Design Chart and Discussions.......................................................... 393
8.3.3.1
2-D Photonic Filter Design Flowchart ............................ 393
8.3.3.2
Examples of Photonic 2-D PSP Implementation ............ 393
8.3.3.3
Separable Implementation Using Matrix
Decomposition Methods ................................................. 396
8.3.3.4
Non-Separable Implementation Using Direct Methods .....399
8.3.3.5
Comparison of Matrix Decomposition Method Design
and Direct Method Design ..................................................402
Concluding Remarks .....................................................................................403
Chapter 9 Generation and Photonic Processing of Radio Waves, Tera-Waves and
Multi-Carrier Lightwaves ........................................................................................405
9.1
9.2
9.3
Introduction ...................................................................................................405
Generation of Tera-Hz Waves........................................................................407
9.2.1 Generation of Ultra-High Repetition Rate Pulse Trains.......................408
9.2.2 Necessity of Highly Nonlinear Optical Waveguide Section for
Tera-Hz Wave Ultra-High Speed Modulation ..................................409
Photonic Signal Processing of Radio Waves ................................................. 410
9.3.1 Generic Structures ............................................................................ 412
9.3.2 Polarization Dual-Mode Delay Processing Systems ........................ 413
9.3.2.1
Tunable Radio Wave Processing Systems Using
Differential Group Display Elements ............................. 413
9.3.2.2
Tunable Multi-Tap Radio Wave Filters Using Higher
Order Polarization Mode Dispersion Emulator .............. 416
9.3.3 Integrated Multi-Tap Delay Processing Systems.............................. 416
9.3.3.1
Dual Tunable RW Filters Using Sagnac Loop and
CFBGs ............................................................................ 418
9.3.3.2
Wavelength-Division Multiplexing (WDM) Multi-Tap
Tunable Radio Wave Filters ............................................ 420
9.3.3.3
Remarks .......................................................................... 423
9.3.4 Buffered Delay Processing Systems................................................. 423
9.3.5 Nonlinear Effects in Photonic Processing Systems of Radio
Waves.............................................................................................. 424
9.3.5.1
All Pass Interferometer as Radio Frequency Filter
Banks .............................................................................. 425
9.3.5.2
Integrated Radio Frequency and Photonic on Chip........ 428
xv
Contents
9.3.6
9.4
Remarks on the Photonic Signal Processing of Radio Waves ...... 430
9.3.6.1
Challenges and Uniqueness of Photonic
Processors ..................................................................430
9.3.6.2
Uniqueness of Tera-Hz Wave Generators ................... 432
Quantum Dot Solitonic Mode-Locked Comb Lasers .................................. 433
9.4.1
Structure and Quantum Optical Gain Waveguide ........................ 433
9.4.1.1
Quantum Dot Growth ................................................. 434
9.4.1.2
QD-BA and BU Structure........................................... 435
9.4.1.3
Lasing in Initial State ................................................. 436
9.4.1.4
Mode Locking and Comb Spectrum Generation ....... 437
9.4.1.5
Absorption Section ..................................................... 438
9.4.2
Performance .................................................................................. 438
9.4.2.1
Measurement Platform................................................ 438
9.4.2.2
Relative Intensity Noise ..............................................440
9.4.2.3
Linewidth of QD-MLL Generated Comb Laser.........440
9.4.3
Optical Frequency Comb in Multiple Radio Wave
Channelization ............................................................................. 442
9.4.4
Concluding Remarks on QD-MLL ............................................... 443
Chapter 10 Optical Devices for Photonic Signal Processing ...................................................... 445
10.1
10.2
10.3
Optical Fiber Communications.................................................................... 445
Photonic Signal Processors ..........................................................................446
10.2.1 Photonic Signal Processing ..........................................................446
10.2.2 Some Processor Optical Components ..........................................446
10.2.2.1 Optical Amplifiers....................................................... 447
10.2.2.2 Pumping Characteristics .............................................448
10.2.2.3 Gain Characteristics ....................................................449
10.2.3 Noise Considerations of EDFAs and Impact on System
Performance .................................................................................. 452
10.2.3.1 Noise Considerations................................................... 452
10.2.3.2 Fiber Bragg Gratings ................................................... 454
Optical Modulators ...................................................................................... 456
10.3.1 Introductory Remarks ................................................................... 456
10.3.2 Lithium Niobate Optical Modulators ........................................... 456
10.3.2.1 Optical-Diffused Channel Waveguides ...................... 457
10.3.2.2 Linear Electro-optic Effect .........................................468
10.3.3 Electro-absorption Modulators ..................................................... 472
10.3.3.1 Electro-absorption Effects .......................................... 472
10.3.3.2 Rib Channel Waveguides ............................................ 475
10.3.4 Operational Principles and Transfer Characteristics .................... 482
10.3.4.1 Electro-optic Mach–Zehnder Interferometric
Modulator .................................................................... 482
10.3.5 Modulation Characteristics and Transfer Function ...................... 485
10.3.5.1 Transfer Function ........................................................ 485
10.3.5.2 Extinction Ratio for Large Signal Operation .............. 487
10.3.5.3 Small Signal Operation ............................................... 488
10.3.5.4 DC Bias Stability and Linearization ........................... 488
xvi
Contents
10.3.6
Chirp in Modulators ..................................................................... 489
10.3.6.1 General Aspects .......................................................... 489
10.3.6.2 Modulation Chirp ........................................................ 490
10.3.7 Electro-optic Polymer Modulators ............................................... 492
10.3.8 Modulators for Photonic Signal Processing ................................. 494
10.4 Remarks ....................................................................................................... 495
References ................................................................................................................ 496
Index .............................................................................................................................................. 501
Preface
The evolution of optical networking is continuing to progress at tremendous pace. The transmission
rate has now reached 200Gbps per wavelength channel by employing discrete modulation format
over a limited bandwidth of only 35–45 GHz. The explosion of uses of data center clouds and
cloud radio access networking for 5G wireless networks have led to a high possibility of employing
photonic signal processing in optical networking for backhaul transporting of ultra-high capacity
information.
Photonic signal processing (PSP) is the art of manipulating photonic waves in the optical domain,
either in coherent or in-coherent states. This is very attractive as it has the potential to overcome
the electronic limits for processing ultra-wideband signals. Furthermore, PSP provides signal conditioning that can be integrated in-line with fiber optic systems in optic form or fiber form modules.
Several techniques have been proposed and reported for the implementation of the photonic
counter-parts of conventional electronic signal processing systems. Also, signal processing in the
photonic domain offers significant improvement of signal quality.
This book is written as the second edition of its first version published by CRC Press in 2007,1
to update and address the emerging techniques of processing and manipulating signals propagating
in optical domain. That means the pulses or signal envelopes are complex or modulating the optical
carriers. Naturally, the applications of such processing techniques in photonics are essential to illustrate their usefulness. The change of the transmission cable from coaxial and metallic waveguide
to flexible optically transparent glass fiber has allowed the processing of ultra-high-speed signals in
the microwave and millimeter-wave domain to the photonic domain in which the delay line can be
implemented in the fiber lines, which are very lightweight and space efficient. Previously found in
Chapter 2 of the first edition, a generic introduction addressing these fundamental understandings
of optical devices is now listed in the last chapter of this second edition.
We introduce the subjects of this second edition as follows. Chapter 2 gives a brief historical
perspective of PSP and introduces a number of photonic components essential for photonic processing systems, including, but not exclusively, the optical amplification devices, optical fibers, and
optical modulators. Chapter 2 illustrates the representation of photonic circuits using signal-flow
graph (SFG) techniques, which have been employed in electrical circuits since the 1960s. However,
these techniques are now adapted for photonic domain in which the transmittance of a photonic
subsystem determines the optical transfer function of a photonic subsystem. The coherence and
incoherence of photonic circuits are an important consideration as whether the field or the power of
the lightwaves should be used, as well as if the length of the photonic processors must be less than
that of the coherence length. Chapter 3 then illustrates the uses of SFG in the design of optical filters
of fixed and tunable passband.
Chapter 4 describes photonic signal processors, such as differentiators and integrators, leading
towards their applications for the generation of solitons and their uses in optically-amplified fiber
transmission systems. Chapter 5 illustrates the compensation of dispersion using photonic processors. Chapter 6 described the practical implementation of optical dispersion compensators in integrated optic technology. These compensators can offer highly low-cost modules for C- and L-band
optical transmission links, which can remove several limitations if they are using the O-band
spectral window.
1
Le Nguyen Binh, Photonic Signal Processing: Techniques and Applications, December 17, 2007 by CRC Press, ISBN
9781138746848.
xvii
xviii
Preface
Chapter 7 gives the design and implementation of generations of ultra-short pulse sequences, and
the dark solitons and their behavior under nonlinear instability conditions. Chapter 8 then gives a
multi-dimension PSP in which the sampling rates in such processor can have many different rates.
Chapter 9 then introduces the techniques for generation and processing in the photonic domain, of
radio waves, Tera-waves and multi-carrier lightwaves.
Many people have contributed, either directly or indirectly, to this book. Thanks are due to
Associate Professor John Ngo of Nanyang Technological University (NTU), Singapore, for the
works that he has conducted during his time as a PhD candidate member of my university research
group; Professor Shum Ping of the School of Electrical and Electronic Engineering (EEE) of NTU;
Dr. W.J. Lai and Miss Anh Le of Australia; my colleagues of Huawei Technologies, Dr. Sun Xu,
Bruce Liu Lei, Zhao Zhuang, Dr. Thomas Wang Tao, and Dr. Xie Chang Song are thanked for their
kind collaborations and helpful suggestions in developing integrated optical circuits and quantum
dot mode-locked lasers. I would like to thank the editors of CRC Press, Taylor and Francis, for their
encouragement. Last, but not least, I thank my wife Phuong, and my son Lam (LA, USA) for their
care and putting up with my busy writing schedule of this book besides my daily R&D activities
at the European Research Center of Huawei Technologies Munich, Germany, which certainly took
away a large amount of time that we could have spent together.
Le Nguyen Binh
Munich, Germany, Spring 2018
MATLAB® is a registered trademark of The MathWorks, Inc. For product information, please
contact:
The MathWorks, Inc.
3 Apple Hill Drive
Natick, MA 01760-2098 USA
Tel: 508 647 7000
Fax: 508-647-7001
E-mail: info@mathworks.com
Web: www.mathworks.com
Author
Le Nguyen Binh holds a BE (Summa Cum Laude) and PhD from the University of Western
Australia. He is currently a technical director at Huawei’s European Research Centre in Munich,
Germany, and has been awarded three Huawei Technologies Gold Medals for his work on advanced
optical communication technologies. He was previously the chair of Commission D (Electronics and
Photonics) of the National Committee for Radio Sciences of the Australian Academy of Sciences,
and a professorial fellow at Nanyang Technological University, Christian-Albrechts-Universität
zu Kiel, and various Australian universities. Widely published, Dr. Binh is the series editor of
Photonics and Optics for CRC Press.
xix
1
Introduction
ACRONYMS
Acronymic Terms
ASE
DC
DMAC
DSP
e-DSP
EDFA
E/O
FIR
FOSAP
GBaud
Gbps
HO-DMAC
IC
IIR
IM/DD
INCOI
OA
OEDC
O/E
PIC
PLC
PSP
RF
RW
SBS
S-DMAC
SOI
Tbps or Tera-bps
WDM Mux/DeMux
1.1
Full Terms
Amplified spontaneous emission
Directional couplers (optical)
Digital multiplication by analog convolution (algorithm)
Digital signal processing/processor
Electronic DSP
Erbium-doped fiber amplifiers
Electro-optic or electrical to optical conversion
Finite Impulse Response
Fiber-optic systolic array processors
Giga-Baud
Giga-bits/sec.
High order DMAC
Integrated circuits
Infinite impulse response
Intensity modulation/direct detection
Incoherent Newton–Cotes optical integrator
Optical amplification/amplifier
Optical dispersion eigencompensators
Optical to electrical conversion
Photonic integrated circuits
Planar Lightwave circuit
Photonic Signal Processing/processor
Radio frequency
Radio waves
Stimulated Brillouin scattering
Serial DMAC
Silicon on insulator
Tera-bits/sec.
Wavelength-division multi/demultiplexers
PREAMBLE
Optical networking has penetrated long-haul, metropolitan, and access networks. Furthermore, data
centers and cloud networking has quickly developed to carry information at a capacity reaching
several Tera-bits/s. The networking capacity of data centers has increased exponentially. Initially, the
transmission rate was low enough for multi-mode use as the transmission medium over ultra-short
distances of a few hundred meters to few kilometers. The bit rate has now reached 400 Gbps with
56 GBaud over single-mode fibers for the interconnection of clusters of servers. In connecting different
1
2
Photonic Signal Processing
sections of the sever stacks of large-scale data centers (DCs), a large-scale optical cross-connect
system via the processing or switching in the optical domain is preferred to minimize the latency
that electronic switching systems are currently facing. The fastest and highest switching capacity by
electronic system can only reach a total maximum capacity of not higher than one Tera-bps.
Data center networking (DCN) is highly important and inter-data center transmission links have
become highly critical with ultra-bit rate and increased capacity over short distances. Photonic
signal processing (PSP) can assist in the manipulation of optical channels in the optical domain.
The ultimate aim of this book is to show the many ways in photonic processing functions can be
implemented.
1.2 INTRODUCTORY REMARKS
The market demand for cost-effective transport of large amounts of information has accelerated
the pace for the widespread deployment of optical fiber communication systems, which have the
capacity to carry large amounts of information. These systems make use of optical signal processors as the basic functional all-optical device for signal processing and transmission. They are used
in many areas of telecommunications: undersea cables bridging the oceans, terrestrial cables connecting cities, trunk lines linking metropolitan areas, and subscriber loop systems serving customer
premises. Thus, there is an urgent need to design optical signal processors to meet the rapidly growing demands of advanced optical communication systems.
Digital signal processing (DSP) has been extensively employed in the coherent1,2 and incoherent3
optical transmission systems to increase the baud rate to 100 GBaud leading to several hundred
Giga-bits/s per wavelength channels. Although optical signal processing is an analogue technique
by its nature, DSP can be applied and transferred from the electrical domain into the photonic
domain as far as we can define the optical sampling feature so that the sampling z-transform can be
used. This is the main feature of the chapters described in this book.
In this book, as the second edition of the book Photonic Signal Processing: Techniques and
Applications (Optical Science and Engineering),4 two types of linear optical signal processors,
which make use of the photon nature of light for performing a wide variety of linear functions, are
considered. They are incoherent fiber-optic signal processors and coherent integrated-optic signal
processors, in which single-mode optical fibers and single-mode optical waveguides, respectively,
are employed as a delay-line medium for high-speed processing of broadband signals. The virtually unlimited bandwidth of these processors makes them attractive for processing radio frequency
(RF), microwave, and millimeter-wave signals directly in the optical domain. This direct processing
eliminates the need for inefficient and costly intermediate “opto-electronic-opto” conversions as in
conventional approaches, which eventually lead to electronic bottlenecks.
Incoherent fiber-optic signal processors employing optical fiber, fiber-optic directional couplers (DC), and erbium-doped fiber amplifiers (EDFAs) have been developed to perform various signal processing functions, such as convolution, correlation, analog matrix operations,
frequency filtering, pulse-train generation, data-rate transformation, and code generation. Coherent
integrated-optic signal processors employing silica-based waveguides on a silicon substrate have
M. Taylor, Phase estimation methods for optical coherent detection using digital signal processing, IEEE
J. LightwaveTechnol., 27(7), 901–914, 2009.
2 N. Stojanovic, F. Karinou, B. Mao, Chromatic dispersion estimation method for Nyquist and faster than Nyquist coherent optical systems, Optical Fiber Communications Conference and Exhibition (OFC), San Francisco, CA: OFC’2014,
PaperTh2A.19.
3 K. Kikuchi, Electronic polarization-division demultiplexing based on digital signal processing in intensity-modulation
direct-detection optical communication systems, Opt. Exp., 22(2), 1971–2001, 2014. doi:10.1364/OE.22.001971.
4 L. N. Binh, Photonic Signal Processing: Techniques and Applications (Optical Science and Engineering), 1st ed., Boca
Raton, FL: CRC Press, 2007.
1
Introduction
3
been demonstrated as optical switches, optical wavelength-division multi/demultiplexers (WDM
Mux/DeMux), optical frequency-division multi/demultiplexers, tunable optical filters, and optical
dispersion compensators.
Given the significant developments of this field, the overall aims of this book are, therefore,
(i) to describe the principal techniques for the design of new incoherent and coherent optical signal
processors and (ii) to demonstrate their applications in optical computing and optical communication systems. The specific aims of this investigation are described as follows.
The first aim of this book is to describe the design of new incoherent fiber-optic systolic array
processors (FOSAPs), which employ the digital multiplication by analog convolution (DMAC) algorithm, for performing real-valued digital matrix multiplications. The FOSAP multipliers are shown
to have massive pipeline capability and higher computational power than the digital electronic multipliers and other optical DMAC multipliers.
The second objective of this book is to design a new programmable incoherent Newton–Cotes
optical integrator (INCOI) for processing (or integrating) intensity-based signals in the time domain.
The new INCOI processor is synthesized using a generalized theory of the Newton–Cotes digital
integrators.
The third aim of this book is to present the design of the new higher-derivative finite impulse
response (FIR) coherent optical differentiators for processing (or differentiating) coherent signals in
the time domain. The new optical differentiators are synthesized using a theory of higher-derivative
FIR digital differentiators, which are commonly developed for electronic DSP.
Although digital integrators and differentiators have received considerable attention in the
field of digital signal processing, the proposed theories of optical integrators and differentiators are believed to be introduced, for the first time, to the area of optical signal processing. It
is hoped that these new theories form the basis for future development with a view to finding
potential applications for these new processors. One such application provides the fourth goal of
this study, which is to design new optical dark soliton generators and detectors by using the optical integrator and differentiator, respectively, as outlined above. This is because dark solitons,
which are difficult to generate and detect, have recently been predicted to offer better stability
than the well-known bright solitons against fiber loss, interactions between neighboring solitons,
and amplified noise-induced timing jitter. In addition, there are currently so few techniques for
dark soliton generation and detection—only three techniques for generating and one technique
for detecting dark solitons.
The fourth goal of this book is to present the design of new optical dispersion eigencompensators
(ODECs) for compensation of the combined effect of laser chirp and fiber dispersion at 1550 nm in
high-speed, long-haul intensity modulation/direct detection (IM/DD) optical communication systems. Although many optical dispersion compensators have been demonstrated to mitigate this detrimental effect, most of these devices are bandwidth-limited because of a lack of design flexibility.
It is found that the proposed ODECs perform better with dispersively chirped signals than with
dispersively chirp-free signals; an advantage which is not attainable with other techniques.
The fifth and final aim of this book is to systematically design new tunable optical filters with
the essential features of variable bandwidth and center frequency characteristics as well as general filtering characteristics, such as lowpass, highpass, bandpass, and bandstop characteristics.
Although several techniques have been developed to design tunable optical filters with variable
bandwidth and center frequency characteristics, such as bandpass and bandstop, there is still limited design flexibility in these methods; however, the proposed technique provides much greater
design flexibility.
The ultimate overall clarification in the chapters of this book is also to identify either the incoherence and coherence characteristics of the optical signals under operations, thus either the electromagnetic field or intensity of the optical waves or signals are to be employed in the PSP (Photonic
Signal Processing) systems. The implications are significant whether they are processed by coherent
or direct detection sub-systems. The design and applications of new incoherent and coherent optical
4
Photonic Signal Processing
signal processors reported in this book constitute a significant contribution to the fields of optical
computing and optical communications. Digital signal processing (DSP) in the electronic domain
by ASIC (Applied Specific Integrated Circuit) or FPGA (Field programmable gate array) can certainly be used as a hybrid co-processor to enhance the effectiveness of a PSP system. This kind of
hybrid processing technique is not presented in this edition of the book.
1.3 ORGANIZATION OF CHAPTERS
The chapters of this book are organized in the following order:
Chapter 2 presents the fundamental theory of incoherent fiber-optic signal processing and coherent integrated-optic signal processing, which provides the basis for Chapters 3–5 and Chapters 6–9,
respectively.
Chapter 3 gives a framework for the analysis of optical signal processors through the application of the signal-flow graph technique, which is developed as a mathematical tool applicable to all
chapters. The effectiveness of this approach can be demonstrated by applying it to the analysis and
design of an incoherent recursive fiber-optic signal processor.
Chapter 4 then describes structures and operations of incoherent FOSAPs employing the
DMAC, higher order DMAC (HO-DMAC) and serial S DMAC (S-DMAC) algorithms, which are
proposed for real-valued digital matrix multiplications. The performances of the FOSAP multipliers are also compared with those of the digital electronic multipliers and other optical DMAC
multipliers.
Chapter 5 describes a programmable INCOI processor, which is synthesized using a proposed
generalized theory of the Newton–Cotes digital integrators. The performance of the processor is
also analysed.
Chapter 6 gives a theory of higher-derivative FIR optical differentiators developed using a proposed theory of higher-derivative FIR digital differentiators. The performances of the optical differentiators are also analysed.
In Chapter 7, the trapezoidal optical integrator described in Chapter 5 is proposed as an optical
dark soliton generator and the first-order first-derivative optical differentiator outlined in Chapter 6
and a first-order Butterworth lowpass optical filter are proposed as optical dark-soliton detectors.
The performances of the dark soliton generator and detectors are also analyzed.
In Chapter 8, a digital eigenfilter approach is employed to design linear ODECs for the compensation of the combined effect of laser chirp and fiber dispersion at 1550 nm in high-speed long-haul
IM/DD lightwave systems. The performances of the ODECs are also compared with those of the
Chebyshev optical equalizers.
Chapter 9 presents a digital filter design method employed to systematically design tunable optical filters with variable bandwidth and center frequency characteristics, as well as lowpass, highpass,
bandpass and bandstop characteristics. The effectiveness of this technique is demonstrated with the
design of the second-order Butterworth lowpass, highpass, bandpass and bandstop tunable optical
filters with variable bandwidth and centre frequency characteristics. An experimental development
of the first-order Butterworth lowpass and highpass tunable fiber-optic filters is also described.
Comb lasers via the structure embedded with stack layers of quantum dots InAs integrated in a
buried heterostructure InGaAsP are also given as an example for a generation of multicarriers in
the optical domain. The integrated cavity can then generate soliton under injection of electrons and
formation of lightwaves and thence soliton waves till the saturation absorption layer reaches its
saturation level and Q-switching happens to thence generate ultra-short pulse train which can then
be transformed into wideband comb lines in the frequency domain.
Chapter 10 gives a number of principal optical devices required for photonic processing are
given. In the first edition of the book this chapter was described as Chapter 1. In this edition this
chapters updated and assigned as last chapter of the book and referred in other chapters, possibly
considerable as a combination of appendices of the chapters.
Introduction
5
Major conclusions drawn from the presentations and practical results reported in the chapters
and recommendations for future work are presented at the end of each chapter.
In this book, it is assumed that all optical systems operate in the C-band of silica fiber, which is
in the 1550 nm window, the operating wavelength of optical waves at which the signal loss is minimum. Optical signal losses can be compensated with EDFAs integrating optical isolators, and/or a
narrow-band optical filter may be required to suppress the stimulated Brillouin scattering (SBS) and
to minimize the amplified spontaneous emission (ASE), respectively. The effects of fiber polarization
and waveguide birefringence, which can be overcome in practice, on the performance of the proposed
optical signal processors are beyond the scope of this book and are therefore not considered.
2
Photonic Signal Processing
Via Signal-Flow Graph
The fields of photonic communications and integrated photonics have progressed significantly over
the last few decades. Spinning off from such developments, several related areas such as microwave
photonics, ultra-broadband optical fiber communications, photonic switching, intra- and inter-data center
communications, and others have been established. In these systems, photonic signals are propagating
and processed at the modulators, through the guided media, at the multiplexers and demultiplexers,
through optical amplifiers (Raman, rare-earth doped waveguide, or parametric amplification). Thus, the
necessity of processing of photonic waves has become important and requires fundamental approaches
for photonic circuit analysis, syntheses and design, and photonic signal processing.
Therefore, this chapter, as the first of the introductory chapters of photonic processors, presents
the fundamental principles and applications of this new field of photonic signal processing. We
state and define the significance of coherence and incoherence of lightwaves propagation through
photonic circuits and hence the representation of lightwaves and circuits either by the lightwave
intensity or by the photonic electromagnetic fields. The photonic circuits to be processed in coherent
or incoherent modes are described and followed by a number of photonic circuit elements. The
graphical representation of photonic circuits is described using the signal-flow graphs (SFG) and the
photonic Mason’s rules. Automatic generation for photonic transfer functions of the input and output
lightwave signals between any two nodes of a photonic circuit are presented.
Furthermore, this chapter gives a fundamental understanding of incoherent and coherent optical
signal processing, which provides the basis for later chapters. The advantages and disadvantages of
incoherent and coherent optical systems and means of overcoming their limitations are outlined in
Sub-sections 2.1–2.4. The characteristics of the fundamental components of incoherent fiber-optic
signal processors and coherent integrated-optic signal processors are described in Section 2.8.1
A Borland Pascal source code is included in Section 2.11 as a reference for readers who may want to
use it to generate photonic circuits and related transfer functions. Two appendices are given and applied
to the very basic z-transform and automatic SFG and transfer function from one node to the other.
2.1
INTRODUCTION
The phase of the optical signal is sensitive to environmental fluctuations, such as temperature and
pressure changes and acoustic vibrations as well as frequency fluctuations of the optical source. The
inherently high sensitivity of the optical phase to environmental effects has made it attractive for
sensor applications but unattractive for signal processing operations in which stability is essential.
Obviously, these effects can be obviated by discarding the optical phase using an incoherent light.
Incoherent optical signal processors require the coherence time of the optical source to be much
shorter than the basic time delay (or sampling period) to avoid any undesirable effects of optical
interference. Hence, incoherent systems use intensity variations on optical carriers for performing
signal processing operations. In incoherent systems, single-mode optical fibers can be used as a
promising delay-line medium for processing broadband signals because of the large bandwidth of
optical fibers. Typically, the basic delay-line length2 of an incoherent fiber-optic signal processor is
Details of the z-transform definitions and properties can be found in R.D Strum and D.E. Kirk, First Principles of
Discrete Systems and Digital Signal Processing, New York: Addison and Wesley, 1989.
2 The basic delay-line length has a delay corresponding to the basic time delay or sampling period of the system. This
allows the application of the z-transform, a common feature of DSP applicable in analog photonic processing domain.
1
7
8
Photonic Signal Processing
in the meter-order and is at least several orders of magnitude greater than the coherence length of
the optical source, depending on the frequency of operation. For this reason, changes in the basic
delay-line length, due to environmental effects and/or errors in cutting the fiber length, can be
tolerated without causing significant degradation of the system performance. Although incoherent
fiber-optic signal processors are stable and robust, they can only perform positive-valued signal
processing operations, but not bipolar or complex-valued signal processing operations. Hence, incoherent fiber-optic signal processors have limited applications. This serious limitation can clearly be
overcome with a coherent light if the instability can be found.
In contrast to the incoherent case, coherent optical signal processors require the coherence time
of the optical source to be much longer than the basic time delay to achieve coherent interference
of the delayed signals. Coherent systems are thus capable of performing complex-valued signal
processing operations because both the phase and amplitude of the optical signal are retained in
the processed information. As pointed out above, coherent systems cannot operate stably unless
the frequency fluctuations of the optical source and environmental effects can be prevented. The
frequency can be stabilized by using highly coherent semiconductor lasers that are commercially
available. The environmental effects can be suppressed by using integrated optical waveguides
(instead of optical fibers) as a comparatively small delay-line medium for broadband signal processing because of their large bandwidth. Coherent integrated-optic signal processors can operate
stably because the waveguide length, which is in the centimeter-order or millimeter-order, can be
accurately fabricated to the precision of the wavelength order and the phase of the optical signal can
be conveniently controlled to the precision of the wavelength order.
In this book, optical fibers and integrated optical waveguides have been considered as the delayline medium of choice for incoherent and coherent optical signal processing, respectively. The fundamental theories of both incoherent fiber-optic signal processing and coherent integrated-optic
signal processing are presented in the next sections.
2.2
INCOHERENT PHOTONIC SIGNAL PROCESSING
The potentially large bandwidth of optical fiber has made it an attractive delay-line medium for
incoherent processing of broadband signals. This section describes the theory of incoherent fiberoptic signal processing given in references [1] and [2].
In incoherent optical signal processors, the information signal (e.g., RF or microwave) to be
processed is modulated as intensity variations onto an optical carrier whose coherence time is much
shorter than the basic time delay in the system. The optical source can be a broad-linewidth semiconductor laser diode, which can be directly modulated at speeds up to several gigahertz. In the
time domain, the modulated wideband signals do not interfere with each other but are appropriately
delayed and incoherently combined at the system output. In the frequency domain, the frequency
response of the incoherent system depends on the interference of the modulation frequency (RF or
microwave) but not the optical carrier frequency. In other words, an incoherent system is incoherent
at the optical carrier frequency but coherent at the modulation frequency. Thus, the phase of the
optical carrier can be discarded, and the signals add on an intensity basis. As a result, incoherent
systems are stable and robust, but they can only perform positive-valued signal processing operations because intensity cannot be negative. Hence, they have limited applications. Using the theory
of positive systems, it has been shown that the impulse response of an incoherent system is real and
positive-valued.3
In addition, the magnitude of the frequency response of an incoherent system is maximum at
the origin of the frequency axis. Consequently, incoherent systems can only be designed to have a
limited number of lowpass characteristics but not highpass or bandpass characteristics.
3
B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, Fiber-optic lattice signal processing, Proc. IEEE, 72, 909–930, 1984.
Photonic Signal Processing Via Signal-Flow Graph
9
Incoherent fiber-optic signal processing was initiated by a research group at Stanford University
in the 1980s [3]4. Optical fibers and tunable fiber-optic directional couplers have been used in the
analysis, as well design and construction of a number of incoherent finite impulse response5 (FIR)
and infinite impulse response6 (IIR) fiber-optic signal processors that can perform a variety of
linear signal processing functions, which include convolution, correlation, analog matrix operations, frequency filtering, pulse-train generation, data-rate transformation, and code generation.
Considerable research effort has produced a number of new concepts, techniques, and applications
as a result of the advanced development of fiber-optic signal processors.7–8 Optical amplifiers, in
particular erbium-doped fiber amplifiers (EDFAs), have been used to overcome losses as well as to
provide greater flexibility in the analysis, synthesis, and construction of incoherent fiber-optic signal
processors for various filtering applications.9–10 The resulting amplified fiber-optic signal processors
have better performances and, hence, more applications than the unamplified processors.11,12,13,14,15
Adaptive techniques have also been proposed to provide dynamic weighting of the filter coefficients
as well as reconfiguration of the filter delays.16,17,18,19
The limitation of the incoherent (or positive) fiber-optic signal processors may be reduced by
using an electronic differential detection scheme, which can have negative filter coefficients but at
the expense of increased system complexity.20,21
It has been claimed that this synthesis technique can implement not only lowpass filters but also
highpass and bandpass filters. The performance of the synthesized filter can only approximate that of
the desired filter because the synthesis method is based on the least squares approach. Nevertheless,
impressive performances of the synthesized lowpass and highpass filters have been experimentally
K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, and H. J. Shaw, Photonic fiber delay-line
signal processing, IEEE Trans. Microw. Theory Tech., MTT-33, 193–210, 1985.
5 Note that FIR filters have no feedback loops and are also known as transversal, non-recursive or tapped delay-line filters.
6 Note that IIR filters have at least one feedback loops and are also known as recursive or recirculating delay-line filters.
All-pole and all-pass filters are special types of IIR filters.
7 C. C. Wang, High-frequency narrow-band single-mode fiber-optic transversal filters, J. Lightwave Technol., LT-5, 77–81,
1987.
8 E. C. Heyde, Theoretical methodology for describing active and passive recirculating delay line systems, Electron. Lett.,
31, 2038–2039, 1995.
9 B. Moslehi, Fiber-optic filters employing photonic amplifiers to provide design flexibility, Electron. Lett., 28, 226–228,
1992.
10 E. C. Heyde, Theoretical methodology for describing active and passive recirculating delay line systems, Electron. Lett.,
31, 2038–2039, 1995.
11 K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, and H. J. Shaw, Photonic fiber delay-line
signal processing, IEEE Trans. Microw. Theory Tech., MTT-33, 193–210, 1985.
12 C. C. Wang, High-frequency narrow-band single-mode fiber-optic transversal filters, J. Lightwave Technol., LT-5, 77–81,
1987.
13 R. I. MacDonald, Switched photonic delay-line signal processors, J. Lightwave Technol., LT-5, 856–861, 1987.
14 D. M. Gookin and M. H. Berry, Finite impulse response filter with large dynamic range and high sampling rate, Appl.
Opt., 29, 1061–1062, 1990.
15 A. Ghosh and S. Frank, Design and performance analysis of fiber-optic infinite impulse response filters, Appl. Opt., 31,
4700–4711, 1992.
16 B. Moslehi, K. K. Chau, and J. W. Goodman, Photonic amplifiers and liquidcrystal shutters applied to electrically reconfigurable fiber optic signal processors, Opt. Eng., 32, 974–981, 1993.
17 J. Capmany and J. Cascon, Photonic programmable transversal filters using fiber amplifiers, Electron. Lett., 28, 1245–
1246, 1992.
18 J. Capmany and J. Cascon, Discrete time fiber-optic signal processors using photonic amplifiers, J. Lightwave Technol.,
12, 106–117, 1994.
19 S. Sales, J. Capmany, J. Marti, and D. Pastor, Solutions to the synthesis problem of photonic delay line filters, Opt. Lett.,
20, 2438–2440, 1995.
20 J. Capmany, J. Cascon, J. L. Martin, S. Sales, D. Pastor, and J. Marti, Synthesis of fiber-optic delay line filters,
J. Lightwave Technol., 13, 2003–2012, 1994.
21 S. Sales, J. Capmany, J. Marti, and D. Pastor, Experimental demonstration of fiber-optic delay line filters with negative
coefficients, Electron. Lett., 31, 1095–1096, 1995.
4
10
Photonic Signal Processing
demonstrated [21]. However, the synthesis technique can only handle bipolar numbers but not complex numbers, which must be operated by coherent systems.
It is well known that the basic elements required for implementation of the FIR and IIR digital
signal processors are delays, adders and multipliers.22 As a result, the basic components required
for the realization of the FIR and IIR incoherent fiber-optic signal processors are: fiber-optic delay
lines, fiber-optic directional couplers, and fiber-optic (or semiconductor) amplifiers. These are
described in the following sections.
2.2.1 Fiber-Optic Delay lines
The low loss and broad bandwidth of optical fibers have made them an attractive delay-line medium
for incoherent processing of high-speed broadband signals directly in the optical domain. The
loss of optical fibers is about 0.5 dB km at 1300 nm and about 0.2 dB km at 1550 nm. The bandwidth–distance product of optical fibers is about 32 THz ⋅ km at 1300 nm and about 100 GHz ⋅ km at
1550 nm [26]. As a result, the time–bandwidth product of optical fibers exceeds 107 at 1300 nm and
exceeds 105 at 1550 nm, assuming a delay per unit length of 5 µs km.
2.2.1.1 Fiber-Optic Directional Couplers
One of the fundamental elements in incoherent fiber-optic signal processors is a fiber-optic directional coupler, which performs signal collection (or addition) or signal distribution (or tapping).
The [2 × 2] fiber-optic directional coupler (see Figure 2.1) is a symmetrical and reciprocal fourport device, which can be designed to have fixed or tunable coupling coefficient. The underlying
principle is based on the interaction of the evanescent fields between two parallel fiber cores placed
sufficiently close to each other.23,24
The coupler exhibits very little dependence on the state of polarization of the input fields, even
though the polarization effect can be easily overcome in practice by means of a fiber polarization
controller. An alternative and more promising approach is to use polarization maintaining fibers in
the construction of the couplers.
In the incoherent operating regime, the intensity transfer matrix of the 2 × 2 tunable fiber-optic
directional coupler can be described by [1]
 I3 
1 − κ
  = (γ ) 
 I4 
 κ
Port 1
κ   I1 
 
1 − κ   I2 
Tunable
Directional
Coupler
Port 2
FIGURE 2.1
(2.1)
Port 3
Port 4
Schematic diagram of a tunable fiber-optic directional coupler.
A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, Englewood Cliffs, NJ: Prentice-Hall, 1989.
S. K. Sheem and T. G. Giallorenzi, Single-mode fiber-photonic power divider: Encapsulated etching technique,
Opt. Lett., 4, 29–31, 1979.
24 M. J. F. Digonnet and H. J. Shaw, Analysis of a tunable single mode photonic fiber coupler, IEEE J. Quantum Electron.,
QE-18, 746–754, 1982.
22
23
Photonic Signal Processing Via Signal-Flow Graph
11
where:
{ I1, I2 } and { I3 , I4 } are the intensities at the input and output ports, respectively
κ (0 ≤ κ ≤ 1) is the cross-coupled intensity coefficient
γ (typically 0.95 < γ < 1) is the intensity transmission coefficient
Equation (2.1) means that, when the input port {2} is not excited, the signal intensity at the input port
is directly coupled to the output port {3} with an intensity coupling coefficient of γ (1− κ ) and cross
coupled to the output port {4} with an intensity coupling coefficient of γκ . Similarly, when the input
port {1} is not excited, the signal intensity at the input port {2} is directly coupled to the output port
{4} with an intensity coupling coefficient of γ (1− κ ) and cross coupled to the output port {3} with an
intensity coupling coefficient of γκ .
2.2.2 Fiber-Optic anD semicOnDuctOr ampliFiers
An optical amplifier is an important component in incoherent fiber-optic signal processors because
it can compensate for optical losses as well as providing design flexibility resulting in potential
applications.
Optical amplifiers can provide signal amplification directly in the optical domain. The operational principles, characteristics, and performances of the optical amplifiers described here have
been obtained from references [22] and25. The gain of an optical amplifier is generated by the processes of stimulated scattering induced by nonlinear scattering in an optical fiber, or by the stimulated emission caused by a population inversion in a lasing medium. The former process is utilized
by stimulated Raman scattering fiber amplifiers and stimulated Brillouin scattering fiber amplifiers,
which are of little interest in this investigation because they are based on nonlinear effects in fibers.
The latter process is employed by semiconductor laser amplifiers (SLAs) or rare-earth doped fiber
amplifiers.
Semiconductor lasers can be designed to act as amplifiers and hence the acronym SLAs. SLAs
can be categorized, according to biasing condition and structure, into three types: injection-locked
(IL), Fabry-Perot (FP), and travelling-wave (TW) SLAs. IL-SLA and FP-SLA, which are based on
resonance effects, require the biasing of the semiconductor laser above and below the lasing threshold, respectively. By contrast, TW-SLA, which exploits single-pass amplification, requires both
facets of the semiconductor laser to have antireflection coating. Considerable research attention was
initially paid to IL-SLA and FP-SLA with a view to improving the inferior anti-reflection coating
methods. However, TW-SLA has recently attracted the most attention because of its superior performance (saturation output, noise, and bandwidth, to mention a few) and considerable progress with
coating techniques. However, it is difficult to differentiate between FP-SLA and TW-SLA because
complete zero reflectivity cannot be easily achieved by actual antireflection coating techniques.
Thus, it is generally accepted that TW-SLA and FP-SLA have a reflectivity of less than 0.1%–1%
and more than 30%, respectively. For these reasons, the TW-SLAs have been chosen in this study
for application in incoherent fiber-optic signal processors.
The rare-earth doped fiber amplifiers make use of ions, such as erbium, neodymium, and praseodymium, as the gain medium to provide optical amplification. In recent years, EDFAs have been
the main subject of research simply because they operate near 1550 nm, the wavelength region in
which the fiber loss is minimum and hence the wavelength window of interest for next-generation
lightwave systems. For these reasons, EDFAs have also been considered in this investigation.
Because the TW-SLAs and EDFAs have been considered in this research, it is necessary to understand their characteristics and performances, which are summarized in Table 2.1 [25]. Compared
25
S. Shimada and H. Ishio (Eds.), Photonic Amplifiers and Their Applications, New York: John Wiley and Sons, 1994,
chapters 1–5.
12
Photonic Signal Processing
TABLE 2.1
Comparison of the Characteristics and Performances of TW-SLAs and EDFAs
TW-SLAs
Various wavelengths
15 ~ 20 dB
0 ~ 3 dBm
> 3 THz
Yes
6 ~ 9 dB
Large loss (9 ~ 10 dB)
Fast switching (< 1 ns)
Short carrier lifetime
Small, amplifier length of < 1 mm
Characteristics/Performances
EDFAs
Signal wavelength
Unsaturated gain
Saturation output power
Bandwidth
Polarization dependence
Noise figure
Fiber coupling loss
Switching speed
Currently 1550 nm band only
40 ~ 50 dB
10 ~ 20 dBm
1 ~ 4 THz
No
3 ~ 5 dB
Low loss (< 0.5 dB)
Slow switching ( 0.2 ~ 10 ms)
Long carrier lifetime
Size/length
Several meters to several 100 m of fiber
length
with TW-SLAs, the advantages of EDFAs are: higher unsaturated (or small-signal) gain, higher saturation output power, lower noise figure, lower fiber coupling loss, and polarization independence.
However, TW-SLAs generally have larger bandwidth and are more compact than EDFAs. The high
gain, high saturation output, large bandwidth, low noise, low fiber coupling loss, and polarization
independence of EDFAs make them an ideal choice for application in the incoherent fiber-optic signal processors. In addition to the enormous bandwidth and compactness, the relatively fast switching speed of TW-SLAs makes them more attractive than EDFAs for application in adaptive (or
programmable) incoherent fiber-optic signal processors, where the gains of the TW-SLAs can be
altered by varying the injection current sources driving the semiconductor lasers. Programmable
incoherent fiber-optic signal processors incorporating TW-SLAs must use polarization-preserving
fibers and couplers because of the polarization dependence of TW-SLAs.
In this book, EDFAs and TW-SLAs have thus been considered as the amplifiers of choice for
non-programmable and programmable incoherent fiber-optic signal processors, respectively.
2.3
COHERENT INTEGRATED-OPTIC SIGNAL PROCESSING
Integrated optical waveguides described in Section 2.3.1 can be used as an attractive delay-line
medium for coherent processing of high-speed broadband signals because of the high precision (and
hence stability) and large bandwidth of the waveguides.
In coherent optical signal processors, the information signal (RF, microwave or milli-meter-wave)
to be processed is impressed onto an optical carrier whose coherence time is much longer than the
basic time delay in the system. The optical source must be a frequency-stabilized highly coherent
semiconductor laser with a very narrow linewidth in order to suppress the frequency instability.
Thus, the optical source must be externally (rather than directly) modulated so that its high degree of
coherence can be maintained. This is because direct current modulation of the injection lasers causes
a dynamic shift of the peak emission wavelength resulting in broadening the spectral width [30]. In
addition, external modulation (e.g., using titanium-diffused lithium niobate (Ti:LiNbO3 waveguide
modulators) of the optical source permits high-speed modulation that is ideal for high-speed signal
processing. Thus, the use of external modulators allows the optical source to be optimized for
spectral quality as well as obtaining high modulation bandwidth. In the time domain, the modulated
signals constructively or destructively interfere with each other, depending on their relative phases.
In the frequency domain, the frequency response depends on the interference of the optical carrier
frequency. Thus, coherent optical signal processors using integrated-optic waveguides can stably
Photonic Signal Processing Via Signal-Flow Graph
13
perform high-speed complex-valued signal processing operations because both the phase and
amplitude of the optical carrier are retained in the processed information.
The term “integrated optics” was suggested by Miller in 1969 at a Bell Laboratory as the lightwave equivalent of “integrated electronics.”26,27 Since then research in integrated optics has begun
and gained momentum at about the same time as the development of low-loss optical fibers and
semiconductor lasers. The concept of integrated optics involves the use of thin-film and microfabrication technologies in the development of a large number of individually fabricated miniature
optical components, which may be integrated (or individually interconnected) onto a single chip
in a similar fashion to that which had taken place with integrated electronic circuits [26,27].28,29,30
Enormous progress has been made in this field with advances in material developments, design
techniques, fabrication processes, and component developments. Developments have now reached
the stage where integrated optical circuits can be realized to perform various all-optical signal
processing and switching functions. In fact, recent advances in lightwave technology and networks
have further accelerated the pace for the development of compact, rugged, stable and economical
integrated optical circuits for flexible processing and switching of high-speed broadband signals
directly in the optical domain.
The integrated optical circuit has several advantages over its counterpart, the integrated
electronic circuit, or over conventional bulk-optic systems consisting of relatively large discrete
components [31]. When compared to bulk-optic systems, integrated optical circuits share the same
advantages as those of the integrated electronic circuits such as smaller size and weight as well as
improved reliability and batch fabrication economy. However, the integrated optical circuit, which
uses a high carrier frequency for information processing, inherently has a higher processing speed
than the integrated electronic circuit. As with any other new technology, a high development cost
of integrated-optic technology (e.g., developing new fabrication technology) is initially required.
Nevertheless, the great potential of integrated optics will clearly justify the high development cost
in the long run.
Integrated optical circuits can be fabricated on several different materials, each with its own
particular features. The choice of a substrate material depends very much on the function to be performed by the circuit. Substrate materials commonly used are glass, LiNbO3, silicon (Si) , III–V31
semiconductors, such as gallium arsenide (GaAs) and indium phosphide (InP).32,33,34 For example,
the InGaAsP/InP material system has been used for the development of InGaAsP/InP 1.55-µm distributed feedback lasers because the InP substrate is capable of emitting light in the 1.3–1.6 µm
spectral region that is important for lightwave systems.35 The LiNbO3 dielectric material has been
widely used for the development of Ti:LiNbO3 waveguide modulators because of its linear electrooptic or Pockel’s effect, and that the optical waveguide can be easily formed by diffusing a thin
film of titanium into the LiNbO3 substrate [28–30]. The germanium, InGaAs/InP, and InGaAsP/InP
materials have been used for the fabrication of avalanche photodiodes because of their high absorption coefficients (or responsivity) in the 1.1–1.6 µm low-loss wavelength region [35].
S. E. Miller, Integrated optics: an introduction, Bell Syst. Tech. J., 48, 2059–2069, 1969.
T. Tamir (Ed.), Integrated Optics, New York: Springer-Verlag, 1975.
28 L. D. Hutcheson, Integrated Photonic Circuits and Components, New York: Marcel Dekker, 1987.
29 S. E. Miller and I. P. Kaminow (Eds.), Photonic Fiber Telecommunications II, San Diego, CA: Academic Press, 1988.
30 R. G. Hunsperger, Integrated Optics: Theory and Technology, New York: Springer-Verlag, third edition, 1991.
31 Note that compounds made of elements (e.g., gallium and arsenic) found in the third and fifth columns of the periodic
table are called III–V semiconductors.
32 M. J. F. Digonnet and H. J. Shaw, Analysis of a tunable single mode photonic fiber coupler, IEEE J. Quantum Electron.,
QE-18, 746–754, 1982.
33 S. Shimada and H. Ishio (Eds.), Photonic Amplifiers and Their Applications, New York: John Wiley and Sons, 1994,
chapters 1–5.
34 J. C. Cartledge and G. S. Burley, The effect of laser chirping on lightwave system performance, J. Lightwave Technol.,
7, 568–573, 1989.
35 J. M. Senior, Photonic Fiber Communications: Principles and Practice, London, UK: Prentice Hall, second edition, 1992.
26
27
14
Photonic Signal Processing
The fundamental component of any integrated optical circuit is the waveguide. Compared with
waveguides made of other materials, single-mode silica-based waveguides, which have almost the
same composition as that of single-mode optical fibers, are more compatible with optical fibers and
hence have lower fiber coupling loss.36,37,38 Two major processes have been used for the fabrication
of single-mode silica-based waveguides on planar silicon substrates: chemical vapor deposition and
flame hydrolysis39 deposition. The single-mode waveguide patterns are then defined by photolithographic pattern definition processes followed by reactive ion etching. The glass systems commonly
used for the silica-based waveguides are: phosphorous-doped silica (SiO2 − P2O5 ), which is formed
by chemical vapor deposition,40,41 and titanium-doped silica (SiO2 − TiO2 ) and germanium-doped
silica (SiO2 − GeO2 ), which are formed by flame hydrolysis deposition.42,43 It has been claimed by
research groups at NTT Japan laboratories that the combination of flame hydrolysis deposition and
reactive ion etching can produce low-loss silica-based waveguides, which are best matched to optical
fibers. A variety of passive integrated optical circuits, also known as planar lightwave circuits (PLCs),
using single-mode silica-based waveguides on silicon substrates fabricated by a combination of flame
hydrolysis deposition and reactive ion etching have been demonstrated as: splitters, low-speed optical
switches,44,45,46,47 optical wavelength-division multi/demultiplexers, optical frequency-division multi/
demultiplexers,48,49,50,51,52 tunable optical filters,53,54 and optical dispersion compensators [54].55
M. Kawachi, Silica waveguides on silicon and their application to integrated optic components, Opt. and Quantum
Electron., 22, 391–416, 1990.
37 M. Kawachi, Recent progress in planar lightwave circuits, 10th International Conference Integrated Opticsand Photonic
Fiber Communication, Hong Kong, Proceedings of the IOOC’95, Vol. 3, pp. 32–33, 1995.
38 C. H. Henry, Silica planar waveguides, Proceedings of the IREE, 19th Australian Conference Optical Fiber Technology,
Melbourne, Australia, pp. 326−328, 1994.
39 Flame hydrolysis is a method originally developed for fiber preform fabrication.
40 B. H. Verbeek, C. H. Henry, N. A. Olsson, K. J. Orlowsky, R. F. Kazarinov, and B. H. Johnson, Integrated four-channel
Mach-Zehnder multi/demultiplexer fabricated with phosphorous doped SiO2 waveguides on Si, J. Lightwave Technol., 6,
1011–1017, 1988.
41 L. Vivien, L. Pavesi, (Ed.), Handbook of Si photonics, CRC Press, Series of Optics and Optoelectronics, Boca Raton,
FL: CRC Press, 2012.
42 M. Kawachi, Silica waveguides on silicon and their application to integrated optic components, Opt. and Quantum
Electron., 22, 391–416, 1990.
43 M. Kawachi, Recent progress in planar lightwave circuits, 10th International Conf. Integrated Optics and Photonic
Fiber Commun., Hong Kong, Proc. IOOC’95, vol. 3, pp. 32–33, 1995.
44 T. Kitoh, N. Takato, K. Jinguji, M.Yasu, and M. Kawachi, Novel broad-band photonic switch using silica-based planar
lightwave circuit, IEEE Photon. Technol. Lett., 4, 735–737, 1992.
45 M. Okuno, K. Kato, Y. Ohmori, M. Kawachi, and T. Matsunaga, Improved 88 × integrated photonic matrix switch using
silica-based planar lightwave circuits, J. Lightwave Technol., 12, 1597–1606, 1994.
46 R. Nagase, A. Himeno, M. Okuno, K. Kato, K. Yukimatsu, and M. Kawachi, Silica-based 88 × photonic matrix switch
module with hybrid integrated driving circuits and its system application, J. Lightwave Technol., 12, 1631–1639, 1994.
47 Y. Hida, N. Takato, and K. Jinguji, Wavelength division multiplexer with passband and stopband for 1.3/1.55 µm using
silica-based planar lightwave circuit, Electron. Lett., 31, 1377–1379, 1995.
48 K. Sasayama, M. Okuno, and K. Habara, Photonic FDM multichannel selector using coherent photonictransversal filter,
J. Lightwave Technol., 12, 664–669, 1994.
49 K. Oda, S. Suzuki, H. Takahashi, and H. Toba, A photonicFDM distribution experiment using a high finesse waveguidetypedouble ring resonator, IEEE Photon. Technol. Lett., 6, 1031–1034, 1994.
50 Y. Xiao and S. He, An MMI-based demultiplexer with reduced cross-talk, Opt. Commun., 247(4), 335–339, 2005.
51 S. Suzuki, K. Oda, and Y. Hibino, Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz,
J. Lightwave Technol., 13, 1766–1771, 1995.
52 S. Suzuki, M. Yanagisawa, Y. Hibino, and K. Oda, High-density integrated planar lightwave circuits using SiO GeO
22− waveguides with a high-refractive index difference, J. Lightwave Technol., 12, 790–796, 1994.
53 E. Pawlowski, K. Takiguchi, M. Okuno, K. Sasayama, A. Himeno, K. Okamato, and Y. Ohmori, Variable bandwidth and
tunable centre frequency filter using transversal-form programmable photonic filter, Electron. Lett., 32, 113–114, 1996.
54 K. Okamoto, M. Ishii, Y. Hibino, and Y. Ohmori, Fabrication of variablebandwidth filters using arrayed-waveguide gratings, Electron. Lett., 31, 1592–1594, 1995.
55 R. Jones, J. Doylend, P. Ebrahimi, S. Ayotte, O. Raday, and O. Cohen., Silicon photonic tunable optical dispersion compensator, Opt. Express 15(24), 15836, 2007.
36
Photonic Signal Processing Via Signal-Flow Graph
15
Integrated optical circuits can be realized by two different approaches: hybrid integration, where
several devices are fabricated on different materials, with each optimized in a given material, and
combined on a common substrate, and monolithic integration, where all devices are fabricated
on a common substrate [32–35]. Although the monolithic integration is economically attractive
because mass production of the circuit can be achieved by automatic batch processing, there is
no single substrate material, which is ideal in all respects, as described above. In recent years,
hybrid integration has been considered as a practical and promising approach for combining many
desired functions on a common substrate, and silicon is an ideal substrate material [37,38,55–59].
The compatibility of silica-based waveguides on silicon with the optical fibers, the high thermal
conductivity of silicon (and hence a good heat sink), and the good mechanical stability of silicon make
silicon an attractive substrate not only for passive PLCs, but also as a platform (or motherboard) for
hybrid integration of opto-electronic devices. Hybrid integration platforms have been successfully
developed to enable the integration of silica-based waveguides and laser diode chips all on the
same silicon substrate. The present challenging task is to incorporate integrated-optic waveguide
amplifiers (e.g., using erbium-doped silica-based waveguides into the hybrid integration platform to
provide greater functionality for next-generation optical networks. In addition, the well-developed
silicon technology in the microelectronics industry can be applied to the mass production of hybrid
integration of opto-electronic devices, such as optical sources, optical amplifiers, and optical
functional circuits, all on a general-purpose silicon platform at a potentially low cost.
In this chapter, low-loss single-mode silica-based waveguides embedded on silicon substrates,
whose advantages have been described above, have been chosen as the integrated-optic technology
for the design of coherent integrated-optic signal processors. However, the methodology and
results are applicable to optical signal processors using other waveguide materials. Similar to the
incoherent fiber-optic signal processors described in Section 2.2, the basic components required
for the realization of FIR and IIR coherent integrated-optic signal processors are: integrated-optic
delay lines, integrated-optic phase shifters, integrated-optic directional couplers, and integratedoptic amplifiers. These are described in the following section.
2.3.1
integrateD-Optic Delay lines
The high precision and stability of the low-loss large-bandwidth single-mode silica-based waveguide on a silicon substrate have made the waveguide an attractive delay-line medium for processing high-speed broadband signals directly in the optical domain.
The minimum curvature (or bending) radius,56 the propagation loss of the waveguides and the
waveguide-fiber coupling loss are very important characteristics in designing and fabricating PLCs.
For high relative refractive index difference between the core and cladding ( ∆), the waveguides have
the advantage of having a small curvature radius but at the expense of having a large propagation
loss and a large fiber coupling loss. In recent years, the SiO2 − GeO2 waveguide has been preferred
to the SiO2 − TiO2 waveguide because the former glass system has a lower propagation loss. The
discussion here is thus focused on the SiO2 − GeO2 waveguides.
Typically, SiO2 − GeO2 waveguides with a core of 8 × 8 ~6 × 6 µm 2 and a low ∆ of 0.25% ~ 0.75%
have a minimum curvature radius of 25 ∼ 5 mm and a propagation loss of less than 0.1 dB cm [49].
However, high-∆ waveguides (∆ = 1.5%) with a core of 4.5 × 4.5 µm 2 have a minimum curvature
radius of 2 mm, a low propagation loss of 0.073 dB cm and a fiber coupling loss of 0.9 dB [49]. The
small curvature radius of high-∆ waveguides makes it possible to fabricate circuits with curvatures
and hence permits high-density integration of circuits. For example, ring resonators require a small
ring radius to have a large free spectral range. However, high-∆ waveguides are disadvantageous
in terms of poor coupling with conventional single-mode fibers because of the mismatch between
56
The minimum curvature radius of the waveguide is the radius above which the bending loss is negligible.
16
Photonic Signal Processing
their optical mode fields. This problem can be solved by using mode-field converters by means of a
thermally expanded core technique. For example, the coupling loss was reduced from 2.0 to 0.9 dB
when thermally expanded core waveguides were used.
The stress-induced birefringence of the waveguide, which is caused by the difference between the
thermal expansion coefficients of the silica glass layers and the silicon substrate, is unavoidable in
the fabrication of PLCs. The waveguide birefringence can be eliminated by using either polarization
mode converter with polyimide half wave-plate or the laser trimming method.
2.3.2
integrateD-Optic phase shiFters
A thermo-optic phase shifter (PS), which utilizes the thermo-optic effect to change the phase of the
optical carrier, is an important element in PLCs because it provides an extra degree of freedom in
circuit design.
The thermo-optic PS, which consists of a thin film heater placed on the silica waveguide, is based
on the temperature dependence of the refractive index of the waveguide. When an electric voltage is
applied to the thin film heater, the refractive index of the heated waveguide increases, thus changing
the optical path length by (dn dT) L∆T where dn dT = 1 × 10 −5 is the thermo-optic constant of silica
waveguide, L is the heated waveguide length and ∆T is the temperature increase. For example,
when a 5- mm long waveguide is heated by 30°C, the optical path length changes by 1.5 µm, which
corresponds to a phase shift of 2π for a 1.5 - µm lightwave.57
2.3.3 integrateD-Optic DirectiOnal cOuplers
One of the fundamental elements in PLCs is an integrated-optic waveguide directional coupler,
which performs signal collection (or addition) or signal distribution (or tapping). It is thus useful
to mathematically characterize both the non-tunable and tunable directional coupler. For analytical simplicity, the insertion loss, propagation delay and waveguide birefringence of the directional
coupler are not considered.
The electric-field transfer matrix of the lossless non-tunable waveguide directional coupler (the
left directional coupler (DC) of Figure 2.2) can be described by58
 E1   1 − k
 =
 E2   − j k
E1
DC
E2
k
− j k   E1 
 
1 − k   E2 
Phase Shifter
E1
E2
φ
(2.2)
DC
E3
k
E4
FIGURE 2.2 Schematic diagram of the symmetrical Mach−Zehnder interferometer, which is used as a tunable coupler (TC). DC represents the non-tunable directional coupler.
N. Takato, K. Jinguji, M. Yasu, H. Toba, and M. Kawachi, Silica-based single mode waveguides on silicon and their
application to guided-wave photonic interferometers, J. Lightwave Technol., 6, 1003–1010, 1988.
58 K. Oda, N. Takato, H. Toba, and K. Nosu, A wide-band guided-wave periodic multi/demultiplexer with a ring resonator
for photonic FDM transmission systems, J. Lightwave Technol., 6, 1016–1023, 1988.
57
17
Photonic Signal Processing Via Signal-Flow Graph
where {E1, E2 } and {E1, E2 } are, respectively, the electric-field amplitudes at the input and output ports
of the left DC, k (0 ≤ k ≤ 1) is the cross-coupled intensity coefficient, and j = −1. Equation (2.2)
means that, when the input port {2} is not excited (i.e., E2 = 0 ), the input lightwave E1 is directly
coupled to the output port {1} with an amplitude coupling coefficient of 1− k and cross coupled to
the output port {2} with an amplitude coupling coefficient of − j k . Similarly, when the input port
{1} is not excited (i.e., E1 = 0 ), the input lightwave E2 is directly coupled to the output port {2} with an
amplitude coupling coefficient of 1− k and cross coupled to the output port {1} with an amplitude
coupling coefficient of − j k . Note that the cross-coupled lightwave experiences a − π 2 phase shift.
The fixed coupling coefficient of the non-tunable DC restricts its application in some PLCs.
Another disadvantage is that it is difficult to fabricate the non-tunable DC with a precise coupling
coefficient. This problem can be overcome by using the symmetrical Mach−Zehnder interferometer
(see Figure 2.2), which can be designed to operate as a tunable coupler (TC) or an optical switch [57].
The TC consists of two identical non-tunable DCs interconnected by two waveguide arms of equal
length. A thermo-optic PS (see Section 2.3.2) placed on the upper arm induces a phase shift of ϕ .
When the input port {2} is not excited (i.e., E2 = 0 ) and using Eq. (2.2), the transfer functions are
given by
E3
= γ w  1 − k ⋅ exp( jϕ ) ⋅ 1 − k − j k ⋅ − j k 
E1 E2 =0
(2.3)
= γ w [(1 − k ) exp( jϕ ) − k ] ,
E4
= γ w  1 − k ⋅ exp( jϕ ) ⋅ − j k − j k ⋅ 1 − k 
E1 E2 =0
(2.4)
= − j γ w k (1 − k ) [1+ exp( jϕ ) ] ,
where:
exp( jϕ ) is the phase shift factor of the PS
γ w (typically γ w = 0.89 for an insertion loss59 of 0.5 dB) is the intensity transmission coefficient
of the waveguide TC
Similarly, when the input port {1} is not excited (i.e., E1 = 0), the transfer functions are given by
E3
E
= 4
E2 E1 0=
E1 E2 0
=
E4
= γ w  1 − k ⋅ 1 − k − j k ⋅ exp( jϕ ) ⋅ − j k 


E2 E1 =0
= γ w (1 − k ) − k exp( jϕ )  .
59
Note that the insertion loss including fiber coupling loss is 0.8 dB [64].
(2.5)
(2.6)
18
Photonic Signal Processing
By a simple algebraic manipulation of Eqs (2.3) through (2.6), the electric-field transfer matrix of
the TC, as proposed by Ngo et al. [66], is given by
 1 − K exp( jθ31 )
 E3 
  = γw 
 E4 
 K exp( jθ32 )
K exp( jθ32 )   E1 
 
1 − K exp( jθ 42 )   E2 
(2.7)
where:
K = 2k (1 − k )(1 + cosϕ )
(2.8)
0 ≤ K ≤ 4k (1 − k ) or 0.5 − 0.5 1 − K ≤ k ≤ 0.5 + 0.5 1 − K ,
(2.9)


sin ϕ
θ31 = tan −1 
,
 cos ϕ − k (1 − k ) 
(2.10)
 (1 + cosϕ ) 
θ32 = − tan −1 
,
 sin ϕ 
(2.11)


sin ϕ
θ 42 = tan −1 
.
ϕ
−
1
−
k
k
cos
(
)


(2.12)
In Eqs. (2.11) and (2.12), ( E3 , E4 ) are the output electric-field amplitudes of the TC, K is the
cross-coupled intensity coefficient of the TC, and θ nm is the effective phase shift from the input port
m to the output port n of the TC. The maximum value of K is K = 1, which only occurs at k = 0.5
according to Eq. (2.9). It is thus preferable to design both the non-tunable DCs with k ≅ 0.5 in order
to maximize the dynamic tuning range of the TC, which is 0 ≤ K ≤ 1 . Note that k = 0.5 results in
θ31 = θ32 = θ 42, which implies that the TC is a symmetrical and reciprocal device. Figure 2.3 shows
the effective intensity coupling coefficient and the effective phase shifts of the TC for k = 0.5 and
0 ≤ ϕ ≤ 2π . It can be seen that 0 ≤ K ≤ 1 and −π 2 ≤ θ31,θ32 ,θ 42 ≤ + π 2 for 0 ≤ ϕ ≤ 2π , and that the
same value of K occurs at two different values of ϕ because of the periodicity of K .
Note that it is difficult to precisely fabricate 3-dB (k = 0.5) non-tunable DCs, which are important for optical communication or sensor systems. However, this problem can be overcome by
using the TC, which can be made exactly 3 dB ( K = 0.5) provided that 0.1464 ≤ k ≤ 0.8536 (see
Eq. [2.9]).
The TC can behave as an optical switch when k = 0.5 and ϕ ∈(0, π ) [57]. When no electric power
is applied to the PS (ϕ = 0, off state), the input signals are cross-switched according to the paths:
(1 → 4 , 2 → 3). With electric power corresponding to a phase shift of ϕ = π is applied (on state),
the input signals are direct-switched according to the paths: (1 → 3, 2 → 4). Typically, the power
required for switching is about 0.5 W and the response time is about 1 ms.
The TC can operate stably against temperature variation because its operating condition is hardly
influenced by environmental temperature change. This is because it is the temperature difference
between the two waveguide arms, but not the absolute temperature of each arm, that is important
for tunable or switching operation.
2.3.4
integrateD-Optic ampliFiers
Integrated-optic waveguide amplifiers are important active elements for loss compensation as well
as for providing design flexibility of PLCs. The resulting amplified PLCs can perform functions,
which are otherwise not available with the unamplified PLCs.
Photonic Signal Processing Via Signal-Flow Graph
19
FIGURE 2.3 Output responses of the tunable coupler (TC) for k = 0.5 and 0 ≤ ϕ ≤ 2π . (a) Intensity coupling
coefficient K. (b) Effective phase shifts θ31 = θ32 = θ 42 .
A 50 cm long erbium-doped silica-based waveguide amplifier integrated with a wavelength-division
multiplexing (WDM) coupler has been successfully demonstrated with a gain of 27 dB, a low noise
figure (NF) of 5 dB, and a saturated output power of 4.4 dBm.60 In order to fully integrate amplifier
devices with other components on the same chip, the length of the amplifying waveguide must be
as short as possible. This can be achieved by increasing the doping level of erbium concentration
as much as possible. However, it is believed that there have been no reports to date of experimental
results of waveguide amplifiers integrated on PLCs. With recent success in PLC technology, it would
not be surprising that PLCs integrated with waveguide amplifiers (or semiconductor amplifiers)
become a reality in the next few years. In this research, PLCs using erbium-doped silica-based
waveguide amplifiers have thus been proposed as active functional optical devices for optical
communication systems.
2.4 REMARKS
The fundamental theories of incoherent fiber-optic signal processing and coherent integrated-optic
signal processing have been described. The major points that can be drawn from this chapter are
given below.
Incoherent Fiber-Optic Signal Processing
• Incoherent fiber-optic signal processors require the coherence time of the optical source to
be much shorter than the basic time delay in the system, and can be directly or externally
modulated.
• In incoherent fiber-optic signal processors, the low loss and large bandwidth of the singlemode optical fiber have made it an attractive delay-line medium for processing broadband
signals directly in the optical domain.
60
R. N. Ghosh, J. Shmulovich, C. F. Kane, M. R. X. de Barros, G. Nykolak, A. J. Bruce, and P. C. Becker, 8-mW threshold
Er3+ − doped planar waveguide amplifier, IEEE Photon. Technol. Lett., 8, 518–520, 1996.
20
Photonic Signal Processing
• The characteristics of the fundamental fiber-optic elements (such as fiber-optic delay lines,
fiber-optic directional couplers, and fiber-optic and semiconductor amplifiers) of the incoherent fiber-optic signal processors have been described.
• Although incoherent fiber-optic signal processors are stable and robust, they can only perform positive-valued signal processing operations and thus have limited applications. They
can process RF and microwave signals with speed ranging from hundreds of megahertz to
a few gigahertz because the basic filter length typically ranges from a few meters to tens of
meters depending on the frequency of operation.
• The fundamental theory of incoherent fiber-optic signal processing described in this chapter is used in the analysis and design of incoherent fiber-optic signal processors, which are
presented in Chapters 3 through 5.
Coherent Integrated-Optic Signal Processing
• Coherent integrated-optic signal processors require the coherence time of the optical source
to be much longer than the basic time delay in the system. The optical source must be a
frequency-stabilized highly coherent semiconductor laser, which must be externally modulated.
• In coherent integrated-optic signal processors, the high precision and stability of the lowloss large-bandwidth single-mode silica-based waveguide on a silicon substrate have made
the waveguide an attractive delay-line medium for high-speed processing of broadband
signals directly in the optical domain.
• The characteristics of the fundamental integrated-optic elements (such as integrated-optic
delay lines, integrated-optic phase shifters, integrated-optic directional couplers, and
integrated-optic amplifiers) of the coherent integrated-optic signal processors have been
described.
• Coherent integrated-optic signal processors can stably perform complex-valued signal processing operations and thus have potential applications in optical communication systems.
They can process millimeter-wave signals with speed in the range of tens of gigahertz
because the basic filter length typically ranges from a few millimeters to a few centimeters
depending on the frequency of operation.
• The fundamental theory of coherent integrated-optic signal processing described in this
chapter is used in the analysis and design of coherent integrated-optic signal processors,
which are presented in Chapters 6–9.
Note that the choice between an incoherent and a coherent optical signal processor depends on the
particular application.
2.5
SIGNAL-FLOW GRAPH APPROACH AND PHOTONIC CIRCUITS
The coherent and coherent aspects given in Sections 2.2 and 2.3 indicate that the transmittance of
an optical device can be considered as the transfer function of an optical circuit. The flow of either
the optical field or optical intensity can be represented by the input and output of a two-port optical
network via the transfer transmittance. This section thus gives a framework for the analysis of optical signal processing systems/circuits through application of the SFG technique, which is developed
as a mathematical tool for rest of the chapters. The limitations of other techniques for the analysis
of these processors are addressed in Section 2.5.1. Section 2.5 describes the fundamental theory
of the SFG approach whose effectiveness is then demonstrated by applying it to the analysis of an
incoherent recursive fiber-optic signal processor (RFOSP), as outlined in Section 2.7.1. Two designs
of the incoherent RFOSP and their applications as optical filters are also presented. The theory of
incoherent fiber-optic signal processing described in Sections 2.2 and 2.3 is employed in this section
where intensity-based signals are considered.
Photonic Signal Processing Via Signal-Flow Graph
2.5.1
21
intrODuctOry remarks
A linear time-invariant optical signal processor is often characterized by its transfer function(s).
The analysis and design of optical signal processors thus require the analysis of their transfer
functions, which have been obtained by the method of successive substitutions of simultaneous
equations61,62,63,64,65 and the transfer matrix method.66 However, these methods can be tedious, timeconsuming, and error-prone especially when dealing with a large-scale system, and do not provide a
clear picture of the mechanisms in which an input signal flows through (or reflects from) the system.
These problems can be overcome by the use of the SFG technique proposed in references [70]
and [71], which was applied to optics by Binh et al.67,68,69 References [67–69] were concerned with
the application of the SFG technique whose fundamental theory was, however, not given. In this
chapter, the fundamental theory as well as the application of the SFG method are presented in a
more comprehensive manner so that the underlying principles can be easily understood and applied.
2.5.2
signal-FlOw graph theOry
An SFG is defined as a network of directed branches, connected at nodes [68,71], and is simply a
pictorial representation of the simultaneous algebraic equations describing a system and graphically
displays the flow of signals through a system. The SFG method can be interpreted as a transformation
of either the method of successive substitutions of simultaneous equations or the transfer matrix
method to a topological approach. Thus, it may be said that “a graph is worth a thousand equations,”
which is analogous with the common saying that “a picture is worth a thousand words.” The SFG
technique has been widely used with great success in the diverse fields of electronics, and digital
signal processing and control systems since its development by Mason in the 1950s.70,71 In general,
the SFG theory can be applied to any linear time-invariant systems.
The advantages of the SFG technique over conventional methods are:
• It yields a pictorial representation of the flow of signals through the system, which enhances
an understanding of the system operation
• It provides an easy and systematic way of manipulating the variables of interests, which
allows graphical simulation of the system using a computer program72
• It enables solutions to be easily obtained by direct inspection of simple systems
• It permits the identification of the physical behavior and topological properties of a system
M. C. Vazquez, R. Civera, M. Lopez-Amo, and M. A. Muriel, Analysis of double-parallel amplified recirculating photonic-delay lines, Appl. Opt., 33, 1015–1021, 1994.
62 C. Vazquez, M. Lopez-Amo, M. A. Muriel, and J. Capmany, Performance parameters and applications of a modified
amplified recirculating delay line, Fiber Integrated Opt., 14, 347–358, 1995.
63 B. Vizoso, C. Vazquez, R. Civera, M. Lopez-Amo, and M. A. Muriel, Amplified fiber-optic recirculating delay lines,
J. Lightwave Technol., 12, 294–305, 1994.
64 B. Vizoso, I. R. Matias, M. Lopez-Amo, M. A. Muriel, and J. M. Lopez-Higuera, Design and application of double
amplified recirculating ring structure for hybrid fiber buses, Opt. and Quantum Electron., 27, 847–857, 1995.
65 E. C. Heyde and R. A. Minasian, A solution tothe synthesis problem of recirculating photonic delay line filters, IEEE
Photon. Technol. Lett., 6, 833–835, 1994.
66 K. Oda, N. Takato, and H. Toba, Awide-FSR waveguide double-ring resonator for photonic FDM transmission systems,
J. Lightwave Technol., 9, 728–736, 1991.
67 L. N. Binh, N. Q. Ngo, and S. F. Luk, Graphical representation and analysis of the Z-shaped double-coupler photonic
resonator, J. Lightwave Technol., 11, 1782–1792, 1993.
68 L. N. Binh, S. F. Luk, and N. Q. Ngo, Amplified double-coupler double-ring photonic resonators with negative photonic
gain, Appl. Opt., 34, 6086–6094, 1995.
69 L. N. Binh, X. T. Nguyen, and N. Q. Ngo, Realisation of Butterworth-type photonic filters using phase modulators and
3 3 × coupler ring resonators, IEE Proc.-Optoelectron., 143, 126–134, 1996.
70 S. J. Mason, Feedback theory-some properties of signal flow graphs, Proc. IRE, 41, 1144–1156, 1953.
71 S. J. Mason, Feedback theory-further properties of signal flow graphs, Proc. IRE, 44, 920–926, 1956.
72 L. P. A. Robichaud, M. Boisvert, and J. Robert, Signal Flow Graphs and Applications, Englewood Cliffs, NJ: PrenticeHall, 1962.
61
22
Photonic Signal Processing
In general, the SFG technique is used to solve a set of linear algebraic equations, which are described
by [68]
n
xj =
∑ t x , j = 2, 3…, n,
(2.13)
ij i
i =1
where:
x1, the only driving force in the system, is the independent variable, x2 , x3 ,…, n are the dependent
variables
tij is the transmittance
2.5.3
DeFinitiOns OF sFg elements
To understand the SFG technique, definitions of the fundamental elements of an SFG [74] are given
in Figure 2.4.
Definition 1: A node (•) represents a variable.
Definition 2: A branch is a directed path joining two nodes, its direction is indicated by an
arrow and its transmittance is specified by an attached symbol (or numeral) describing the
functional relation between the two nodes. For example, the symbol x1 → x2 represents a
branch that has a transmittance of t12.
Definition 3: A source node is a node at which all branches are directed outward. For
example, x2 is a source node.
Definition 4: A sink node is a node at which all branches are directed inward. For example,
x6 is a sink node.
Definition 5: A feedback loop is a closed path, which starts and terminates at the same node
such that the nodes can only be touched once per traversal. For example, x3 → x 4 → x3 is
a feedback loop.
Definition 6:A self-loop is a feedback loop consisting of a single branch. For example,
x2 → x2 is a self-loop.
Definition 7: Non-touching loops are separated loops, which have no node in common. For
example, x2 → x2 and x3 → x 4 → x3 are non-touching loops.
Definition 8: A loopgain is given by the product of all transmittances associated with a
feedback loop. For example, t34t43 is the loopgain of the feedback loop x3 → x 4 → x3.
t 34
x3
x1
t 12
t 23
t 45
t 25
x2
t 22
FIGURE 2.4 Example of a SFG.
x4
t 43
t 56
x5
x6
23
Photonic Signal Processing Via Signal-Flow Graph
Dummy
Node
t 12
x1
t 25
x2
t56
1
1
x1
x6
x5
x0
Dummy
Node
x7
t25
x2
t56
x6
x5
t 22
t 22
FIGURE 2.5
t 12
Example showing the equivalence of two SFGs.
Definition 9: A forward path is a path that has no feedback loop and consists of at least one
branch. For example, x1 → x2 → x5 → x6 and x1 → x2 → x3 → x4 → x5 → x6 are forward
paths.
Definition 10: A forward-path gain is given by the product of all transmittances associated
with a forward path. For example, t12t25t56 is the forward- path gain of the forward path
x1 → x2 → x5 → x6 .
For clarity, it is preferable to add an additional branch with a transmittance of unity to the source
node and to the sink node. Figure 2.5 shows an example of the equivalence of two graphs where
x0 = x1 the source node is and x7 = x6 is the sink node.
2.6
RULES OF SFG
The following basic rules, namely, the transmission, addition and product rules are frequently used
in SFG theory.73
2.6.1
rule 1: transmissiOn rule
The value of the variable denoted by a node is transmitted on every branch leaving that node. This
can be mathematically described by
=
x j tij=
xi , j 1, 2,, n
(2.14)
and graphically represented by Figure 2.6.
2.6.2
rule 2: aDDitiOn rule
The value of the variable denoted by a node is equal to the sum of all signals entering that node. This
can be mathematically described by
n
xj =
∑t x
ij i
(2.15)
i =1
and graphically represented by Figure 2.7.
73
J. J. Distefano, A. R. Stubberud, and I. J. Williams, Theory and Problems of Feedback and Control Systems, Singapore:
McGraw-Hill, Chapter 8, 1987.
24
Photonic Signal Processing
x1
t i1
t i2
xi
x2
t ik
xk
t in
xn
FIGURE 2.6
SFG representation of the transmission rule.
x1
t 1j
x2
t 2j
xj
t kj
xk
t nj
xn
FIGURE 2.7
2.6.3
SFG representation of the addition rule.
rule 3: prODuct rule
The effective transmittance of a branch is equal to the product of the transmittances of all branches
in cascade. This can be mathematically described by
xn = (t12t23 t( n −1)n ) x1
(2.16)
and graphically represented by Figure 2.8.
t 12
x1
t 12 t 23
t (n-1)n
x2
x n-1
xn
FIGURE 2.8 SFG representation of the product rule.
x1
t (n-1)n
xn
25
Photonic Signal Processing Via Signal-Flow Graph
2.7
MASON’S GAIN FORMULA
The transfer function H between the independent (or source) node j and the dependent (or sink)
node k in the SFG is determined using Mason’s gain formula [74]:
N
H=
∑
1
PD
i i
D i =1
(2.17)
where:
N = Total number of forward paths from node j to node k
Pi = ith forward-path gain of the forward path from node j to node k
Pmr = mth possible product of r non-touching loopgains
D = SFG determinant or characteristic function
= 1 − ( −1) r +1
= 1−
∑
m
∑∑ P
mr
m
r
Pm1 +
∑
m
Pm2 −
∑
Pm3 + ...
(2.18)
m
= 1 − (Sum of all loopgains) + (Sum of all gain-products of 2 non-touching loops) −
(sum of all gain-products of 3 non-touching loops) +...
Di = Cofactor of the ith forward path
= D; Evaluated with all loops touching Pi eliminated.
The application of Eq. (2.18) is considerably easier than it appears and is illustrated in Section 2.8.
2.7.1
analysis OF an incOherent recursive Fiber-Optic signal prOcessOr (rFOsp)
As an example, the SFG method is applied to the analysis of the incoherent RFOSP.
2.7.2
sFg representatiOn OF the incOherent rFOsp
Figure 2.9 shows the schematic diagram of the incoherent RFOSP, which consists of an optical fiber
loop interconnected by two tunable fiber-optic directional couplers DC1 and DC2. The couplers
are assumed to have the same intensity transmission coefficient γ and cross-coupled intensity coefficients κ1 and κ 2 . The signal intensities at the input and output ports are described by {I1, I8} and
{I2, I7}, respectively.
The optical transmittances of the lower (Λ1) and upper (Λ 2 ) halves of the ring are defined as
Λ1 = G exp( − jωT1 ),
(2.19)
and T1 = delay time of the optical path = equivalent guided propagation length L (2.20)
Λ 2 = exp( − jωT2 ),
(2.21)
G = G exp[ −2α ( L1 + L2 ) − 2(ε1 − ε 2 )].
(2.22)
26
Photonic Signal Processing
I7
I1
1
DC1
κ1
2
3
5
Loop Delay T
4
6
7
DC2
κ2
8
I2
I8
^
EDFA, G
FIGURE 2.9
the couplers.
Schematic diagram of the incoherent RFOSP. Numbers in circles denote the port numbers of
is the intensity gain of the
In Eqs. (2.20) and (2.21), G is the effective optical loopgain, G
EDFA, which has been described in Section 2.7.2, α is the amplitude attenuation coefficient of
the fiber, ε1 and ε 2 are the amplitude coefficients of the splice (or connector) losses of the lower
and upper halves of the ring, respectively, j = −1, ω is the angular modulation frequency, L1
and L2 are the fiber lengths of the lower and upper halves of the ring, respectively, and T1 and
T2 are the corresponding time delays of the lower and upper halves of the ring, respectively.
Note that the lengths of the lower and upper halves of the ring do not necessarily need to be the
same. It is the fiber loop length L = L1 + L2 or loop delay T = T1 + T2 that is important for signal
processing.
As a representative value, the exponential factor in Eq. (2.22), which represents the fiber loop
loss, is calculated using typical parameter values. The fiber loop length is assumed to be L = 100 m.
The fiber loss is assumed to be 0.2 dB/km at 1550 nm, and this results in α = 0.02303 km −1. Two
splices are required for the lower path because of the EDFA, whereas one splice is required for
the upper path. The splice loss is assumed to be 0.1 dB/splice, and this results in ε1 = 2 × 0.0115
and ε 2 = 0.0115. Using these values, the fiber loop loss is given by
exp[ −2α L − 2(ε1 + ε 2 )] = 0.93.
(2.23)
Using the SFG theory presented in Section 2.5 and the fiber-optic directional coupler defined in
Eq. (2.2), the resulting SFG representation of the incoherent RFOSP is shown in Figure 2.10.
FIGURE 2.10 SFG representation of the incoherent RFOSP. Numbers in circles denote optical nodes, which
correspond to the port numbers of the couplers.
Photonic Signal Processing Via Signal-Flow Graph
27
2.7.3 DerivatiOn OF the transFer FunctiOns OF the incOherent rFOsp
The intensity transfer function I2 /I1 of the incoherent RFOSP is derived in detail. Figure 2.10 shows
that there are two optical forward paths along which the input signal at node ① can flow to the output
node ②. Using Mason’s gain formula as defined in Eq. (2.17) where N = 2 , the two optical forwardpath gains can be written as follows:
Path 1:
Path 2:
①→②
P1 = γ (1− κ1 ),
(2.24)
P2 = γ 3κ12 (1 − κ 2 )Λ1Λ 2 .
(2.25)
①→④→⑥→⑤→③→②
There is only one optical loopgain that can be identified as follows:
Loop:
④→⑥→⑤→③→④
P11 = γ 2 (1 − κ1 )(1 − κ 2 )Λ1Λ 2 .
(2.26)
The SFG determinant is thus given by:
D = 1 − P11 = 1 − γ 2 (1 − κ1 )(1 − κ 2 )Λ1Λ 2 .
(2.27)
The cofactors of the forward paths are:
D1 = D because the forward path 1 does not touch the loop,
(2.28)
D2 = 1 because the forward path 2 touches the loop.
(2.29)
Inserting Eqs. (2.28) and (2.29) into Eqs. (2.26) and (2.27), the intensity transfer function I 2 / I1 of
the incoherent RFOSP can be written as
I2 P1 D1 + P2 D2 γ (1− κ1 )(1− zzero z −1 )
=
=
I1
D
1 − zpole z −1
(2.30)
where:
z = exp( jωT ) is the well-known z-transform parameter [25] and T = T1 + T2 is the basic time
delay (or sampling period) of the filter. Furthermore, the zero zzero and the system pole zpole
in the z-plane are given by
γ 2 (1 − 2κ1 )(1 − κ 2 )G
,
(1 − κ1 )
(2.31)
zpole = γ 2 (1 − κ1 )(1 − κ 2 )G.
(2.32)
zzero =
The intensity transfer function I 7 / I1 can be similarly derived but with less effort than
Eq. (2.30). This is because there is only one forward path from the input node ① to the output
node ⑦ [i.e., path ①→④→⑥→⑦], and this path also touches the loop. As a result, the cofactor of
this path is equal to unity. By inspection of Figure 2.10, the intensity transfer function I 7 / I1 of the
incoherent RFOSP can be written simply as
I7 γ 2κ1κ 2G exp(− jωΤ1 )
=
I1
1 − zpole z −1
(2.33)
28
Photonic Signal Processing
where the numerator corresponds to the optical forward-path gain. For analytical simplicity, the factor
exp(− jωΤ1 ) in Eq. (2.33), which represents the pure propagation delay and only introduces a linear phase
term to the phase response, is neglected because it does not alter the essential characteristics of the filter.
It is clear that Eq. (2.33) corresponds to the transfer function of an all-pole optical filter because
the zero is located at the origin. It can thus be stated that:
An optical signal processor will exhibit the characteristics of an all-pole optical filter
if the forward paths touch all optical loops in its SFG representation.
Based on the above statement, Figure 2.11a–c show other possible structures of the all-pole optical
filters with transfer functions Y2 /X1. Note that they all have only one forward path, which also
touches the optical loop. It is worth mentioning that a single-coupler fiber-optic filter having one
loop (or ring) cannot be used as an all-pole optical filter because there are two forward paths.
(a)
X1
Y2
Y1
X2
(b)
X1
Y2
Y1
X2
X1
Y1
(c)
Y2
X2
FIGURE 2.11 Schematic diagrams of other possible all-pole optical filters with transfer functions Y2 / X1
with the coupling (a) X2 is the main excitation port to the optical ring; (b) Y1 as the main tapping output of the
coupler involved Por X1 and Y1; and (c) Y1 is the main output port coupled out from the ring, and X2 is the main
excitation port into the optical ring.
29
Photonic Signal Processing Via Signal-Flow Graph
The above analysis shows the advantage of the SFG technique over conventional methods because
the intensity transfer functions of the incoherent RFOSP can be easily derived in a systematic
manner. In addition, the technique can identify the property of a particular optical system.
2.7.4
stability analysis OF the incOherent rFOsp
Stability is one of the most important requirements in the performance of amplified recursive optical
systems in which optical amplifiers are incorporated. To ensure stable operation of the incoherent
RFOSP, a first-order system, the system pole must be placed within the unit circle so that the
following condition holds:
G<
1
.
γ 2 (1 − κ1 )(1 − κ 2 )
(2.34)
The stability of first- and second-order amplified recursive optical systems can be easily determined
because explicit expressions for the system poles can always be obtained. However, it is mathematically
involved, if not impossible, to obtain explicit expressions describing the pole locations of thirdand higher-order amplified recursive optical systems, which can be constructed using additional
couplers and fiber loops. This difficulty can be overcome using Jury’s stability criteria, which has
been used with great success for testing the stability of linear time-invariant digital systems. Jury’s
stability test helps to obtain the necessary and sufficient conditions, which are usually functions of
the system parameters describing the system stability. Binh et al.74 have employed Jury’s stability
criteria to test the stability of incoherent recursive fiber-optic filters employing optical amplifiers.
2.7.5 Design OF the incOherent rFOsp
This section describes two designs of the incoherent RFOSP. Desired filtering characteristics of the
incoherent RFOSP can be obtained by appropriate placement of the pole-zero location in the z-plane.
In all photonic designs, a typical value γ = 0.95 is used. In all figures, the magnitude and phase
responses are shown over two frequency cycles that are normalized to ωΤ /(2π ). The time axis of
the impulse response represents the time normalized to the basic time delay T, and the pole and
zero locations in the pole-zero plot are represented by “x” and “o”, respectively.
For filter design purposes, it is useful to relate the zero (see Eq. [2.31]) and pole (see Eq. [2.32])
of the intensity transfer function I2 / I1 according to
zpole
(1 − κ1 )2
=
zzero (1 − 2κ1 )2
(2.35)
The intensity coupling coefficient of DC1 is determined from Eq. (2.33) to give
1/ 2
 z   z  z 
κ1 = 1 − pole  ± − pole 1 − pole  
 zzero   zzero  zzero  
(2.36)
zpole
z
≤ 0 or pole ≥ 1
zzero
zzero
(2.37)
where:
74
Le Nguyen Binh., Photonic Signal Processing, Boca Raton, FL: CRC Press, 2007.
30
Photonic Signal Processing
Equations (2.35) and (2.36) are used to determine the filter parameters required to satisfy a particular
prescribed magnitude response.
Design 1: Z pole = 0.99, zzero = −1 ⇒
zpole
= −0.99,
zzero
⇒ k1 = 0.5864, k2 = 0.50, G = 5.3
Note that the incoherent RFOSP cannot be designed to have a magnitude response that has
a maxima at ωΤ = π because the system pole is always positive. In addition, the intensity
transfer function I2 /I1 of the incoherent RFOSP cannot be designed to have a minima at
ωΤ = 2π because its zero is restricted by Eq. (2.37) to take a negative value. As pointed out in
Section 2.2, although incoherent fiber-optic signal processors are stable and robust, they can
only have certain filtering characteristics, and can only process positive-valued (or intensitybased) signals.
The above designs have shown that the EDFA can compensate for optical losses as well as
providing design flexibility in amplified fiber-optic signal processors. There is an extra degree
of freedom in manipulating the pole-zero pattern of amplified systems when compared with
unamplified systems, and this results in more possible applications of the amplified systems.
It is worth pointing out that the incoherent RFOSP described here has already been experimentally
demonstrated. The analytical and simulation results presented here, in particular Design 1, are
consistent with the experimental results given in reference.75
2.7.6
remarks
A framework for the analysis of optical signal processors through the use of the SFG method has
been presented. The effectiveness of this method has been demonstrated by applying it to derive
the intensity transfer functions of the incoherent RFOSP. The SFG technique described in this
chapter is an important mathematical tool in the analysis and design of optical signal processors
throughout this research.
The all-pole characteristics of the RFOSP are used for the design of an incoherent fiber-optic
integrator (see Chapter 5), the design of an optical dark-soliton generator (see Chapter 7), and the
design of tunable optical filters (see Chapter 9). This is why the RFOSP has been chosen as an
example for illustration of the effectiveness of the SFG method.
The new field of photonic signal processing is proposed. This section gives an illustration of how
a simple photonic circuit can be presented with the signal flow and processing the lightwaves with
the relationship between input ports and output ports. The photonic transfer functions can be then
derived, manually or automatically (see program listing and description in next Section 2.9), so that
their amplitude and phase characteristics can be obtained.
A number of photonic components have been illustrated and represented with the gain/
loss and propagation delay factors in forms of the z-transform parameters that would allow the
applications of popular digital signal processing techniques. The next chapters will present the
design and synthesis of several photonic processors in 1-D. Proposed techniques for multi-D PSP
systems have been reported [78]. This section therefore sets the initial steps for photonic signal
processing.
75
D. Pastor, S. Sales, J. Capmany, J. Marti, and J. Cascon, Amplified double coupler fiber-optic delay line filter, IEEE
Photon. Technol. Lett., 7, 75–77, 1995.
31
Photonic Signal Processing Via Signal-Flow Graph
2.8
2.8.1
OPTMASON: A PROGRAM FOR AUTOMATIC DERIVATION
OF THE OPTICAL TRANSFER FUNCTIONS OF PHOTONIC
CIRCUITS FROM THEIR CONNECTION GRAPHS
Overview
Determining the behavior of a given optical circuit has traditionally been done by solving the field
(vector) or intensity (scalar) equations for each optical component simultaneously. This is time
consuming and impractical for large circuits; provided some simple requirements are met, a better
method using an SFG approach may be used. (See Reference76)
The requirements to be satisfied are: (1) Linearity of all optical components; (2) Time invariance of all optical components; (3) Optical components must be lumped, that is not distributed.
In other words, we consider only lumped LTIV (linear time invariant) optical circuits. Therefore,
not considered here, are effects, such as backscatter of light along the length of an optical fiber
or saturation of an optical amplifier (the former is a linear distributed phenomenon, the latter is
a nonlinear process). Such circuits may be graphically represented as optical nodes connected
by links. Nodes represent points in the circuit and links the function of the optical components
connecting them.
For example, consider the following optical circuit constructed from two 2 × 2 optical couplers
and some lengths of optical fiber (Figure 2.12).
Let assuming that (I) no coupling between lightwaves propagating in opposite directions in fibers
or optical waveguide paths, and (ii) the fibers or waveguides are symmetrical in the transverse
plane. Then each fiber or waveguide path can be represented by an expression of the form t.z−L, with
t as the transmittance gain (possibly including also a fixed phase shifter), Lis the length of the fiber
or optical waveguide path, z is the z-transform or sampling transform parameter.77
z = e jβ z = e jωT
with β = ne k0 = ne 2π λ 0 = 2π ne f / c; T = is the delay time over the length z; ne is the effective
refractive index of the guided optical waves; β the propagation constant of the guided fundamental
mode (single mode propagation assumed).
X
k1
Fiber 3
Y
Fiber 2
Fiber 1
k2
FIGURE 2.12 Optical structure of an optical feedback resonator. Fiber or integrated optical waveguides
can be used as the optical circuit interconnections. Optical couplers k1 and k2 are the cross coupling for the
photonic circuit.
L.N. Binh, N.Q. Ngo, and S.F. Luk, Graphical representation and analysis of the Z-shaped doublecoupler optical resonator, IEEE Journal of Lightwave Technology, 1993.
77 This is the principal definition and well related to the physical system. The delay time acts as the sampling function in
the digital domain.
76
32
Photonic Signal Processing
X
1
3
C1
k1
2
k1 4
C1
t3.z-L3
5
Y
7
C2
k2
t2.z-L2
t1.z-L1
6
k2 8
C2
FIGURE 2.13 Nodes and optical transmission path of a photonic circuit.
The coupling constants of the various ports of the ith 2 × 2 optical coupler are as follows: coupling factor Ci = (1−|ki|) from a port to the port directly opposite coupling factor ki from a port to the
port diagonally opposite. ki can be imaginary (= j |ki |) if coherent light is being considered, but real
if only light intensities are used. The couplers are assumed to be symmetrical.
The circuit may then be depicted as follows (Figure 2.13).
This could be called a “photonic connection graph.” The nodes are represented by dark circles,
and the links by lines. Note that this is NOT an SFG: the links depicted here are bi-directional. To
create a SFG from the optical connection graph, it is necessary to double each node and each link to
create separate nodes and links for each direction of light propagation. For example, (Figure 2.14)
Once a SFG has been obtained by this doubling method, Mason’s rule may be applied to calculate
the transfer function between a source node and a sink node (nodes that have only outputs and only
inputs respectively; these nodes should not be “doubled”). We take X and Y above to be the source
and sink respectively. The Mason’s rule for optical circuits are re-iterated here in the following
definitions (see also Section 2.7):
N
M
N
M
N’
M’
FIGURE 2.14
Node and links of bidirectional photonic paths.
33
Photonic Signal Processing Via Signal-Flow Graph
• A forward path is a connected sequence of directed links going from one node to another
(along the link directions), encountering no node more than once.
• A loop is a forward path that begins and ends on the same node.
• The loopgain or path gain of a loop or path is the product of all the links along that loop or
path (NB: links are labeled with their values; unlabeled links have a default value of unity).
• Two loops or paths are said to be non-touching if they share no nodes in common.
• Then Mason’s rule states that the transfer function T from node X to node Y is given by:
T=
Y 1
=
X ∆
∑G ∆
k
k
(2.38)
k
In which
• ∆ = 1 − (sum of all individual loopgains) + (sum of products of loopgains for all possible
pairs of non-touching loops) − (sum of products of loopgains for all possible triples of
non-touching loops) + (sum of products of loopgains for all possible quadruples of nontouching loops) −…etc.
• The summation is over all forward paths from X to Y;
• Gk is the path gain of the kth forward path from X to Y.
• ∆k is the same as ∆, but calculated using ONLY those loops not touching said path.
2.8.2 using OptmasOn
OPTMASON, whose source codes are given in Section 2.11, is a Borland Pascal program that
accepts a text file description of an optical circuit (in “optical connection graph” form) and
generates its transfer function (from source to sink as specified in the input file). Internally, the
optical connection graph is converted into a signal flow graph, and then Mason’s rule is applied.
The resulting expression is simplified by nested grouping-of-terms.
OPTMASON is started from the DOS command line by typing:
optmason input_file [output_file] [-d]
here “input_file” and “output_file” are the corresponding filenames, and “d” is the number of
decimal places to display real numbers to in the output; if omitted, “d” defaults to 3. If the output
filename is omitted, output is to the screen (actually to DOS’s standard output file, to enable
redirection). If the input and output filenames are the same, the input file is not overwritten, but is
instead appended (i.e., OPTMASON’s output is added to the end of it).
The input text file format for OPTMASON is as follows:
$INPUT = nodename
$OUTPUT = nodename
nodename: nodename, nodename,…; nodename, nodename,…
nodename: nodename, nodename,…; nodename, nodename,…
nodename: nodename, nodename,…; nodename, nodename,…
$TRANSMITTANCES
[*]nodename, nodename = expression
[*]nodename, nodename = expression
[*]nodename, nodename = expression
Here nodename is a label for a node of the optical connection graph. It may be any string of up
to 15 characters, but not containing any of the characters “;:,=$*” (double quotes are OK though).
“expression” is a mathematical expression (see below).
34
Photonic Signal Processing
The first line ($INPUT=…) identifies the input (source) node.
The second line ($OUTPUT=…) identifies the output (sink) node.
The input and output nodes may be the same node.
The next section defines the geometry of the optical connection graph. A line of the form
nodename: nodename, nodename,…; nodename, nodename,…
is required for each node in the graph, except the output node. (If a line for the output node is
included, it will be ignored, unless that node is also the input node.)
Each such line begins with the name of the node being defined, followed by a colon “:”. The
remainder of the line is a list of all the other nodes that it connects to. Since light may travel
independently in two directions through a node in an optical connection graph, connections on
either “side” (optically speaking) are separated by a semicolon “;”. The definition for the input
node and any node at the free end of an optical fiber (reflection point) will only include one “side”
of this list.
Since links in an optical connection graph are bi-directional, a link from n to m in the definition
of a node n must be matched by a corresponding link from m to n in the definition of node m;
links to the output node are an exception. Note that only a single link may join any two nodes. For
multiple optical paths between two nodes, intermediate nodes must be inserted.
The geometry definition section is terminated by the “$TRANSMITTANCES” line.
The section that follows this line defines the values of the links in the optical connection graph.
Since all links have a default value of unity in both directions, only the values of links that differ
from this need be defined. The format for specifying the value of a link between two nodes n and
m (value given by “expression”) is:
n, m = expression
To define the value of the link in the direction n → m only, place an asterix “*” at the start of the
line. Note that if a link value is defined twice, the second definition replaces the first (both directions
are treated independently).
To define the reflection coefficient at a node r, simply write:
r, r = expression
An asterix “*” is optional and has no effect. Reflection coefficients may ONLY be defined at nodes
that are optically single-sided (e.g., cut-end of an optical fiber), and may not be defined for the input
or output nodes.
Some other things to note about the input file format are:
• The input is case-sensitive for node names and variables within expressions.
• Whitespace (spaces, tabs, and blank lines) are ignored or filtered out.
• The start of the file is ignored up to the line starting with “$INPUT=” so it may be used for
a description of the file contents.
• Any line beginning with a semicolon “;” is treated as a comment and ignored.
• Input lines will be truncated beyond 255 characters.
“expression”s have the following form:
magnitude {mag_expr} ^power < angle {angle_expr} z ^power {power_expr}
magnitude, angle, and power are real numbers (scientific notation, e.g., -1.2e+7 is permitted).
35
Photonic Signal Processing Via Signal-Flow Graph
“z” may be uppercase or lowercase (the z-transform parameter).
mag_expr, angle_expr, power_expr may be any strings at all provided they do not contain “}”. They
are intended to be variable names or entire subexpressions.
Any part or parts of the above expression format may be omitted, provided that a meaningful
expression results. The following are examples of valid expressions and their corresponding
mathematical meaning:
6
=6
2.96E-9{a}^2
= 2.96 × 10−9 a2
< {pi.beta}
= exp(j(pi.beta))
−3 < 1.2
= −3exp(j1.2)
Z^–4{L}
= z−4L
−5.2{x*y/z}^3 < 6.4{beta.pi} z^{L+n}
= −0.2(x*y/z)3exp(j.6.4(beta.pi))z(L+n)
Note that although parts of the expression format may be omitted, the ordering of the expression
components must be strictly adhered to. Also:
• The first “^power” is a power of “mag_expr” and may only be present if “mag_expr” is
also.
• The second “power” and “power_expr” apply to the “z”, so can only be present if it is
also.
• A single “-” sign is not a number, and cannot precede any of the {} brackets on its own. Use
-1{…} to obtain the same effect.
• Expressions in {} brackets are not simplified internally; they are treated as single variables.
However when displaying the output, OPTMASON distinguishes two classes of {} expression:
1. A string inside {} brackets is treated as a single variable if it is composed only
of letters, numbers, and the characters “~#@$%?”, and if it does not begin with
a digit (0–9). It may also end with a string of single quotes “ ‘ ”. As a single
variable, it will appear in summations, products, and power expressions without
brackets around it. So for correct output, products of two variables contained
within {} should include a “.” or “*” symbol. e.g. if t2 x is a product, entering
{t2x} may result in the output containing (say) t2x^3, which looks like t2 x3 when
what is desired is (t2 x)3.
2. Any other string inside {} brackets will appear bracketed in the output. In particular,
any string containing any of the mathematical symbols “[]()+-*./^<>=!” will appear
bracketed (inside square brackets) in the output except if it appears on its own (without
any multiplying terms) in a summation.
2.8.3
cOntents OF the input File FOr abOve examples
This file is an example of an input file for the sample network included
in the OPTMASON documentation.
The next two lines define the names of the input (source) and output
(sink)
nodes:
$INPUT = X
$OUTPUT = Y
; The transfer function calculated by OPTMASON will be T = Y/X
; Here is a description of the network geometry:
X: 1
36
Photonic Signal Processing
1: X; 2,4
2: 1,3; 8
3: 6; 2,4
4: 1,3; 5
5: 4; 6,8
6: 5,7; 3
7: Y; 6,8
8: 5,7; 2
; not necessary, but included for completeness:
Y: 7
$TRANSMITTANCES
; Internal optical coupler transmittances:
1,2 = {C1}
3,4 = {C1}
1,4 = {k1}
2,3 = {k1}
5,6 = {C2}
7,8 = {C2}
5,8 = {k2}
6,7 = {k2}
; We ignore the links from X and Y because they don’t contribute to the
; magnitude response of the system, or affect its pole and zero
positions.
;...so they default to values of 1.
; The three main optical fiber transmittances:
2,8 = {t1} z^-1{L1}
4,5 = {t2} z^-1{L2}
3,6 = {t3} z^-1{L3}
; Thats all we need! (simple, isn’t it?)
Here is OPTMASON’s output:
T = numerator/denominator
numeratorá=á((k1^2+C1^2)*k2^2+C2^2*k1^2-2*C1^2*C2^2)*t1*t2*t3*z^(-L3-L2L1)+((C1*C2*k1^2-C1^3*C2)*k2^2-C1*C2^3*k1^2+C1^3*C2^3)*t1*t2^2*t3^
2*z^(-2*L3-2*L2-L1)+C1*C2*t1*z^-L1
denominator =
-2*C1*C2*t2*t3*z^(-L3-L2)+C1^2*C2^2*t2^2*t3^2*z^(-2*L3-2*L2)+1
2.8.4
the OPTMASON prOgram structure
OPTMASON is written in Borland Pascal (version 7.0). The source code is listed in the appendix
of Section 2.11. It performs the following basic steps, which can be referred to in order to produce
the output:
1. The input file lines $INPUT=… and $OUTPUT=… lines are read, and corresponding
input and output (source and sink) nodes are created.
2. The geometry description section is processed a line at a time. Each new node mentioned in
this section actually causes the creation of two nodes (of the same name) in OPTMASON’s
data structure. Links listed before the semicolon “;” head out from the first of these, and links
listed after the semicolon head out from the second. (NB: internally, all links associated
with a node are outgoing… that’s why the output node doesn’t need a description line). The
destination of a link is always to the first node with the correct name. Links to the input
node are created to ensure correct operation when the input and output nodes are the same
(see [3] below). This stage is complete at the end of the file or when a line starting with
“$TRANSMITTANCES” is encountered.
Photonic Signal Processing Via Signal-Flow Graph
3. The link structure is checked and adjusted: It is verified that every node (except the sink)
has at least one outgoing link, that for each link n → m there is a reciprocal link m → n,
and that there is at most one link between any two nodes. The following adjustment is also
carried out for each link n → m: If the two nodes with the name m are denoted m1 and
m2 then the link is always n → m1 at first. If the reciprocal link m → n is of the form m2
→ n then no action need be taken; but if it is of the form m1 → n, then the n → m link is
adjusted to be n → m2. This eliminates all link loops of the form A → B → A, and results
in the correct signal flow graph corresponding to the optical connection graph. This procedure also correctly handles the case where the input and output nodes are the same. (In
step (1) it is ensured that the output node precedes the input node in the list of nodes, and
so is treated as the first node of that name if the two are the same). If they are not the same,
some “phantom” links to the input node will remain and should be ignored.
4. The link value description section is parsed one line at a time, and the link values are set
(all are already initialized to 1 at link creation). When a link from a node to itself (i.e., a
reflection coefficient) is encountered, it is first checked that one of the two corresponding
nodes with that name has NO outputs, then a new link with the desired value is created
from that node to its pair node. A check is also done to see whether this link already exists,
and if so the value is overwritten rather than a new link created.
5. A complete depth-first search through the signal flow graph is performed recursively,
starting at the input node. When the output node is reached, a new path has been found.
An entry in the list of paths is created for it and the path gain is computed. Associated
with each node is a list of the paths and the loops that it touches (the “touchpath” and
“touchloop” lists respectively). The new path is added to the “touchpath” list of each node
along it. Associated with each path is the list of loops that it touches, and to the new path’s
“touchloop” list is added the contents of each “touchloop” list on each node along the
path, checking to ensure that duplicate entries are not made. (NB a path touches any loop
that any one of its nodes touches). Similarly, when the search encounters a node that it has
been to before, then a loop has been found. (The last-taken link out of each node is stored
during the search, so if this is not nil then the node has been previously traversed. NB this
information is also used to follow loops and paths when computing loop/path gain, without
disrupting the recursive nature of the search procedure). Associated with each loop is a
“touchloop” and a circularly-linked list of nodes in the loop. When a loop has been found,
this node list for each loop in the “touchloop” list of the current node is checked, to see
if that loop has actually been found before. If not, a new entry for it in the list of loops
is created and placed at the start of this list. The loopgain of the new loop is computed
(actually (−loopgain), since this is what Mason’s rule uses). Also the following is done:
(i) A circular node-list for the loop is created; (ii) the loop is added to the “touchloop”
list of each node along it; (iii) the loop is added to the “touchloop” list of each path in
the “touchpath” lists of all the nodes along it; (iv) the “touchloop” lists of each of the
nodes along the new loop are amalgamated (removing duplicate entries) and stored as the
“touchloop” list for this loop. Naturally when performing the computations for a new loop
or path, they are all done together on a node-by-node basis rather than being completed
sequentially as the above descriptions imply. It would be possible to speed up the operation
of this stage of OPTMASON by storing the “touchloop” lists in sorted form (or even
in binary tree form) to make searching for duplicate entries and list amalgamation more
efficient. The lists actually contain pointers, but these could be converted to “longints” for
use as sorting keys.
6. When the search is complete, the result is a list of all paths and an ordered list of all loops
in the signal flow graph, with their associated path gains and (−loopgains). Also associated
with each path is a list of the loops it touches, and associated with each loop is a list of the
paths it touches, and a list of the loops below it in the loop-list that it touches. (That is why
37
38
Photonic Signal Processing
the loop list order is important; the loops are in no particular order, per se). Since this is all
the information Mason’s rule requires, all the data associated with each node and link is
now deleted to free memory space for further calculation. The node lists for each loop are
also deleted.
7. The Mason’s rule denominator ∆ is calculated. To explain how this is done, it must first
be observed that the rule for the calculation of ∆ is given earlier and can be expressed
as: ∆ = 1 + sum of all possible products of (−loopgain) of all non-touching loops. The
list of loops can be regarded as a tree structure: the root node is not a loop, but its branch
nodes are each of the loops, and every other node in tree is a loop. Further down the tree,
a node has as its branch nodes all loops below it in the list (of loops) that do not touch
it or any node above it in the tree. Each possible descent through the tree terminating on
an arbitrary node (i.e., not necessarily a leaf node) then represents a unique combination
of non-touching loops, and the set descents to all possible nodes in the tree represents
the set of all possible combinations of non-touching loops. ∆ is calculated, therefore, by
a recursive depth-first tree search, which at each node adds to delta the running product
of that node’s (−loopgain) times the (−loopgain) product of the nodes earlier in the tree,
before proceeding to search the branches. Associated with each loop is the number of
times it has been “deactivated” or touched by loops along the current descent path. The
search works by simply searching all loops below the current tree-node in the loop-list that
have not been deactivated (touched) yet.
8. The Mason’s rule numerator is calculated. This involves stepping through each path in
turn and adding to the numerator the path gain times ∆k for that (the kth) path. ∆k is
calculated the same way ∆ is, but the loops touching the current (kth) path are first given a
“deactivation level” of 1 so they will not be included.
9. The results are outputted (in symbolic form) either to the screen (DOS standard re-directable
output) or to the specified output file (appending it if it is the same as the input file). All
dynamic variables are then disposed of before the program completes execution.
The above functions account for approximately 2/3 of the OPTMASON listing. The other 1/3 of the
program is devoted to manipulating the symbolic expressions that result from calculations on link
values and on path and loopgains.
OPTMASON expressions are stored in a hierarchical tree-like structure, and all operations on
them are performed recursively. The structure corresponding to an expression is built up out of
records of type “expression_type”. A “format” field in each of these records identifies it as belonging
to one of four possible types:
• A “zterm”: a term of the form znum*{expr}. If “expr” is absent (nil pointer) then it just represents
znum.
• An “anglterm”: a term of the form exp(j.num.{expr}). If expr is absent, just exp(j.num).
• A “magterm”: a term of the form {expr}num.
• A “magval”: a real number, = num.
In all the above cases, “num” is a double-precision real number, and “expr” is a string representing
a variable or a mathematical expression; it is what appears inside {} brackets in the input file.
No processing is performed on the contents of expr. (Actually expr^, since expr is a pointer to
the string).
Apart from the “format”, “num”, and “expr” fields, expression_type also has “sumterm” and
“prodterm” field, which point to other expression records, or are nil. They define the structure
of a mathematical expression. This is best illustrated graphically: horizontal arrows represent the
sumterm pointers, vertical arrows the prodterm pointers, and the absence of an arrow indicates that
the pointer is nil. Each letter below represents a single term, such as single expression_type record.
Photonic Signal Processing Via Signal-Flow Graph
39
This entire structure corresponds to the expression:
(o.i.d+(j+k)e+f)a + b + (g+(l+p.m+n)h)c
A Boolean function precedes(x, y) and is used to determine if the term x “precedes” the term y.
Order of precedence is as follows: zterm < anglterm < magterm < magval. If x and y are of the
same format, then ASCII comparison of {expr}x and {expr}y is used to decide whether x precedes y,
and if these are equal, numx is compared to numy. The precedes () function always returns “true” if
x=y, or if they are both “magval”s (so it should really be called “equal_or_precedes”).
The expression tree for an expression is ordered so that if a term x contains a link to a term y
(either sumterm or prodterm link), then precedes(x, y) will be true. This has the effect of bringing
“zterm”s outside of brackets, and summing “magval”s and “magterm”s together inside brackets—
which hopefully results in the most compact and legible output.
Thus, “magval”s are the leaves of the expression tree. A nil pointer in an expression or to an
expression is treated as a zero. Thus, if a prodterm pointer is nil, the terms it multiplies get canceled.
The exception is on “magval” terms, which must therefore terminate every path down through the
tree; a nil prodterm pointer is required on “magval” type terms.
Addition of one expression to another is performed as follows: Each term in the top-level sum
list (i.e., root, root^.sumterm, root^.sumterm^.sumterm… etc.) of the expression being added is
compared with each successive term of the top-level sum list of the expression being added to, until
the term being added is found to precede () the term it is compared to. If they are addable—which
occurs if they are equal or both magvals—then if they are magvals the value of the term being
added is just added; otherwise, the addition procedure is called recursively to add their prod term
subexpressions (since ax + bx = (a + b)x). If they are not addable, then the term being added is simply
inserted into the top-level sum list of the expression being added to, along with the subexpression
pointed to by its prod term… i.e., along with anything multiplying it.
Multiplication is performed in a similar manner: to create a new expression that is the product of two others, the product expression is set to zero, then every term in the top-level sum
list of the first is compared with every term in the top-level sum list of the other. If a pair of
such terms can be multiplied directly (i.e., if they are both “magval”s or if they are of the same
format and have the same {expr}) then this is done, and both their prod term subexpressions are
multiplied (calling the multiply procedure recursively). Otherwise the prod term subexpression
of the “precede ()”ing term is multiplied (recursively) by the other term and its product subexpression. In either case, the result is added to the final product (using the addition procedure
described above).
Due to the regular sorted (by a precedes () function) way in which expressions are stored and
manipulated, they are automatically simplified (to avoid running out of memory) by grouping of
terms at every stage in the calculation. This is not only efficient, but almost certainly necessary
for handling the huge sum-of-product type expressions that might result from calculation of the
Mason’s rule ∆s for large graphs.
40
Photonic Signal Processing
Finally, the procedure used to output expressions works in a similar manner as those used
to manipulate them: Each top-level sum term is handled in turn, first outputting the prod term
subexpression (by calling the output routine recursively), then outputting the term it multiplies,
and a “+” if more sum terms remain. At least, that’s how it works in principle: In practice, there
are exceptions and modifications. In particular, when “+” needs to be output, it is not done right
away, but a plus waiting flag is set instead. Then, when the next item is outputted, the waiting “+”
is outputted first, unless then next item to be outputted begins with a minus sign. This eliminates
output of the form x ± 6y. Similarly, checks are made for multiplication by ± 1 to avoid outputting
the numeral. “prod term” subexpressions involving sums are bracketed, and those that do not
are not. An {expr} part of a term that is treated as a single variable (see above) will not have
brackets around it, otherwise it will be enclosed in [] brackets, except when it appears in a sum
nothing except +1 multiplying it. “prod term” subexpressions are in general outputted recursively,
but where products of multiple single z terms or single angl terms occur, the entire product is
displayed within a single z^(…+…) or exp(j(…+…)) to make the output easier to read; recursion is
not used in this case, and the product list is stepped through manually until an incompatible term
is found (i.e., that cannot be put within the same brackets).
The Borland C program is listed in the Appendix of Section 2.11.
2.9
APPENDIX: Z-TRANSFORM
A z-transform is commonly used in digital signal processing (DSP) [3]. It can also be called
discrete-time signal processing, and it is employed for the analysis of the photonic circuit in this
work. The discrete-time signal can be considered an equally spaced sampling of a continuous-time
signal. The z-transform of a sequence x[n] is defined as
∞
X ( z) =
∑ x[n]z .
−n
(2.39)
n =−∞
where x[n] can be interpreted as the nth term of the sequence of numbers, which describe the
discrete-time signal in the time domain. The ratio between the output transform Y [ z] and the input
transform U[ z ] of a system is called the transfer function H[ z ], which is given as:
H[ z] =
Y[ z]
U[ z ]
(2.40)
Rearranging Equation (2.40), it gives:
H [ z ]* U [ z ] = Y [ z ]
(2.41)
This shows that in the z-domain multiplication of the input with the transfer function produces the
output. The transfer function given in Eq. 2.40 is related to the corresponding time domain sequence
h[n], which is usually called the impulse response of the system, by Eq. 2.39. The corresponding
operation in the time-domain for Eq. 2.40 is
∞
y[n] =
∑ u[r ]h[n − r ]
(2.42)
r =−∞
Equation 2.41 is called the convolution sum and its operation can be expressed as
=
y[n] u=
[n]* h[n] h[n]* u[n]
(2.43)
41
Photonic Signal Processing Via Signal-Flow Graph
where the symbol * denotes convolution. It is to be noted that the convolution operation is
commutative. There are two important properties of convolution. The convolution of two functions
in the time domain corresponds to the multiplication of their z-transformations in the z-domain.
On the other hand, the multiplication of two functions in the time domain corresponds to their
convolution in the z-domain.
B
A
∑
∑
Defining NUM ( z ) =
bi z −i and DEN ( z) =
ai z −i , the z-domain transfer function can be writi =0
i =0
ten as
B
∑b z
NUM ( z)
i
H (z) =
DEN ( z)
−i
= i =A0
∑a z
i
(2.44)
−i
i =0
To change Equation (2.43) into product form, it can be rewritten as
B
∏ (z − q )
k
b
H ( z ) = 0 z A− B kA=1
a0
∏ (z − p )
(2.45)
j
j =1
It is assumed that H ( z ) has been expressed in the irreducible form. The values p j are called the
poles of H ( z) such that H(pj) = * H ( p j ) = ∞ . The values qk are called the zeroes of H(z) for which
H(qk) = 0. Also, there is a pole at z = 0 of the multiplicity of (B*A) if B > A. If A > B, there would
be a zero at z = 0 of the multiplicity of (A–B).
As it has been found that the transfer functions can be expressed in z-domain, thus the transfer
characteristics are dependent on the zero-pole patterns [4] of these transfer functions in the z-plane.
The magnitude-frequency response at a particular frequency as the operating point moving on
the unit-circle, *z* = 1, of the z-plane is given by
B
H ( z ) = H (e
jωτ
b0
)=
a0
∏l
zk
k =1
A
∏l
(2.46)
pj
j =1
where:
z = e jωτ , ω is the angular frequency in radians per second
τ is the sampling period of signals in seconds
lzk and l pj are the lengths from the operating point to the position of the kth zero and the jth pole
of the transfer function respectively.
The corresponding phase-frequency response at a particular operating frequency (wavelength) is
given by
B
arg( H ( z)) =
∑
k =1
A
ϕ zk −
∑ϕ + ( A − B)ωτ
pj
j =1
(2.47)
42
Photonic Signal Processing
where ϕzk and ϕpj are the phase angles of the zeroes and poles respectively formed by the horizontal
real axis and the lines connecting the poles and zeroes to the operating point in the z-plane.
Thus, from Eqs. 2.45 and 2.46 we can tailor the magnitude-frequency response by adjusting
the pole and zero patterns of the transfer functions. To obtain a maximum magnitude at a
A
particular operating wavelength (frequency), we require a pole or a very small value of ∏ j =1 l pj
at that wavelength. Similarly, in order to obtain a minimum at a particular wavelength, a zero
B
or an infinitesimal value of ∏ k =1 lzk is required at that wavelength. There are some relationships
between the positions of poles in the z-plane to those correspondingly in the s-plane (the continuous
frequency domain). One basic property of this relationship is, when recalling that z = e sτ where
s = jω , a pole position moves on the imaginary axis of the s-plane, it would move along the unitcircle of the z-plane. In this case, we would have marginal stability and a lossless system. When
a pole moves on the imaginary axis towards the left half of the s-plane, the corresponding pole
moves inside the unit circle in the z-plane. The system would then become lossy and stable. If
one of the system poles lies outside the unit circle in the z-plane, the system becomes unstable.
Its temporal response would increase with time. In general, the system would be stable if all the
system poles lie inside or on the unit circle in the z-plane. Stability plays an important role in
design of photonic circuits. Stability test for the photonic circuit is already introduced in Section 2.7.
2.10 APPENDIX: OPTMASON.PAS PROGRAM LISTING
{ OPTMASON.PAS
{ Written in 1996 by Dean Trower and Le Binh
{ for Photonic Signal Processing
{
{ This program uses Mason’s rule for signal flow graphs to generate
{ the optical transfer function (in symbolic form) of an arbitrary
{ lumped, linear time-invariant optical circuit, as specified in the
{ input file (file name is a command line parameter).
{
{ See the file OPTMASON.DOC for more information given in this section}
type term_type = (zterm, anglterm, magterm, magval);
link_ptr = ^link_type;
node_ptr = ^node_type;
path_ptr = ^path_type;
loop_ptr = ^loop_type;
looplist_ptr = ^looplist_type;
pathlist_ptr = ^pathlist_type;
nodelist_ptr = ^nodelist_type;
expression_ptr = ^expression_type;
link_type = record {record for a signal flow graph link}
dest:node_ptr; {destination node}
val:expression_ptr; {value}
next:link_ptr;
end;
expression_type = record {record for a single term in an
expression}
format:term_type;
sumterm:expression_ptr;
prodterm:expression_ptr;
num:double;
expr:^string;
end;
Photonic Signal Processing Via Signal-Flow Graph
43
looplist_type = record {record for a single entry in a list of
loops}
loop:loop_ptr;
next:looplist_ptr;
end;
pathlist_type = record {record for a single entry in a list of
paths}
path:path_ptr;
next:pathlist_ptr;
end;
nodelist_type = record {record for a single entry in a list of nodes}
node:node_ptr;
next:nodelist_ptr;
end;
node_type = record {record for a node of the s.f. graph}
name:string[15];
firstlink:link_ptr; {linked list of OUTPUT links}
next:node_ptr; {i.e. next node in s.f. graph}
link_taken:link_ptr; {used during search for loops/
paths}
touchloop:looplist_ptr; {list of loops containing
node}
touchpath:pathlist_ptr; {list of paths containing
node}
end;
path_type = record
pathgain:expression_ptr;
touchloop:looplist_ptr; {list of loops touching path}
next:path_ptr;
end;
loop_type = record
loopgain:expression_ptr;
touchloop:looplist_ptr; {!!!loops later in list only}
nodelist:nodelist_ptr; {circularly linked list of
nodes}
deactivated:word; {used for calc. of Mason rule
deltas}
next:loop_ptr;
end;
var firstnode, sourcenode, sinknode:node_ptr;
firstpath:path_ptr;
firstloop:loop_ptr;
numerator, denominator:expression_ptr;
f:text;
outfile_name:string;
precparam:string; {stores the -d command line parameter}
line:longint; {counts lines in the input file}
prec:byte; {no. of digits to display reals to in the output}
errpos:integer; {used while parsing input and cmnd line params}
heap_state:pointer; {so it can be restored when program ends}
{the following variables aren’t really global,}
path:path_ptr; {but they are here to prevent them being created}
loop:loop_ptr; {many times during recursion.}
node:node_ptr;
pathentry:pathlist_ptr;
loopentry, loopentry2:looplist_ptr;
44
Photonic Signal Processing
nodeentry:nodelist_ptr;
found_duplicate, plus_waiting, addbracket:boolean;
tmp_exp, mult_result:expression_ptr;
function singleterm(s:string):boolean;
{
determines if an "expr" string should be treated as a single variable}
var i:integer;
b:boolean;
begin
b:=true;
i:=1;
while b and (i<=length(s)) do
begin
b:=(pos(upcase(s[i]),'~#@$%?_ABCDEFGHIJKLMNOPQRSTUVWXYZ')>0) or
(pos(copy(s, i,length(s)-i+1),’’’’’’’’’’’’’’’’’’’’’’’’)>0) or
((i>1) and (pos(s[i],'0123456789')>0));
inc(i);
end;
singleterm:=b;
end;
procedure writeexpression(var f:text; e:expression_ptr);
{
recursively procedure to display an expression (or output it to file f)}
var t, q:expression_ptr;
begin
if e=nil then write(f,'0');
t:=e;
while t<>nil do {step through each root-level sumterm}
begin
{skip over sums of angles inside single exp()}
q:=t; {and sums inside brackets in z^(***)... done later}
if (t^.format=anglterm) or (t^.format=zterm) then
while (q^.prodterm<>nil) and (q^.prodterm^.sumterm=nil)
and (q^.prodterm^.format=t^.format) do q:=q^.prodterm;
if q^.prodterm<>nil then
if q^.prodterm^.sumterm<>nil then
begin
if plus_waiting then write(f,'+');
plus_waiting:=false;
write(f,'(');
writeexpression(f, q^.prodterm);
write(f,')*');
end
else
begin
if not((q^.prodterm^.format=magval) and {don’t write "1*"}
(abs(q^.prodterm^.num)=1)) then {or "-1*"}
begin
writeexpression(f, q^.prodterm);
write(f,'*');
end
else if q^.prodterm^.num=-1 then write(f,'-')
else if plus_waiting then write(f,'+');
plus_waiting:=false;
end;
if not((t^.format=magval) and (t^.num<0)) and plus_waiting
then write(f,'+'); {don’t write +-number for negative nums}
Photonic Signal Processing Via Signal-Flow Graph
45
plus_waiting:=false;
case t^.format of
magval: if t^.num=round(t^.num) then write(f, round(t^.num))
else write(f, t^.num:0:prec);
magterm: if t^.num<>1 then
begin
{ie must display the ^...}
if singleterm(t^.expr^)
then write(f, t^.expr^,'^')
else write(f,'[',t^.expr^,']^');
if t^.num=round(t^.num) then write(f, round(t^.num))
else write(f, t^.num:0:prec);
end
else if singleterm(t^.expr^) or
(not((t=e) and (t^.sumterm=nil)) and
((t^.prodterm=nil) or
((t^.prodterm^.sumterm=nil) and
(t^.prodterm^.format=magval) and
(t^.prodterm^.num=1))))
then write(f, t^.expr^)
else write(f,'[',t^.expr^,']');
anglterm: begin
write(f,'exp(j*');
if ((t^.prodterm=nil) or (t^.prodterm^.
format<>anglterm))
and
((t^.expr=nil) or singleterm(t^.expr^) or (t^.
num<>1))
then addbracket:=false
else
begin
addbracket:=true;
write(f,'(');
end;
q:=t;
{display an entire list of single-anglterm}
repeat
{products inside the brackets}
if (q<>t) and (q^.num>=0) then write(f,'+');
if (q^.num<>1) or (q^.expr=nil) then
begin
if q^.num=-1 then write(f,'-')
else if q^.num=round(q^.num)
then write(f, round(q^.num))
else write(f, q^.num:0:prec);
if (q^.expr<>nil) and (q^.num<>-1)
then write(f,'*');
end;
if q^.expr<>nil then
if singleterm(q^.expr^) or (q^.num=1)
then write(f, q^.expr^)
else write(f,'[',q^.expr^,']');
q:=q^.prodterm;
until (q=nil) or (q^.sumterm<>nil) or
(q^.format<>anglterm);
if addbracket then write(f,'))')
else write(f,')');
end;
46
Photonic Signal Processing
zterm: begin
write(f,'z');
if ((t^.prodterm=nil) or (t^.prodterm^.format<>zterm)) and
((t^.expr=nil) or (abs(t^.num)=1))
then addbracket:=false else addbracket:=true;
if addbracket or (t^.expr<>nil) or (t^.num<>1) then
begin
if addbracket then write(f,'^(') else
write(f,'^');
q:=t;
{display an entire list of
single-zterm}
repeat {products using the same z^(...)}
if (q<>t) and (q^.num>=0) then write(f,'+');
if (q^.num<>1) or (q^.expr=nil) then
begin
if q^.num=-1 then write(f,'-')
else if q^.num=round(q^.num)
then write(f, round(q^.num))
else write(f, q^.num:0:prec);
if (q^.expr<>nil) and (q^.num<>-1)
then write(f,'*');
end;
if q^.expr<>nil then
if singleterm(q^.expr^) or (q^.num=1)
then write(f, q^.expr^)
else write(f,'[',q^.expr^,']');
q:=q^.prodterm;
until (q=nil) or (q^.sumterm<>nil) or
(q^.format<>zterm);
if addbracket then write(f,')');
end;
end;
end;
t:=t^.sumterm;
if t<>nil then plus_waiting:=true; {next term to be added}
end;
end;
procedure dispose_expression(var e:expression_ptr);
{ deletes all the memory allocated to an expression and sets it to nil
(=0)}
begin
if e<>nil then
begin
dispose_expression(e^.prodterm);
dispose_expression(e^.sumterm);
if (e^.format<>magval) and (e^.expr<>nil)
then freemem(e^.expr, length(e^.expr^)+1);
dispose(e);
e:=nil;
end;
end;
procedure copy_expression(var exp1:expression_ptr; exp2:expression_ptr);
{ recursively copys the contents of exp2 to exp1; new exp1 pointer created }
begin
if exp2=nil then exp1:=nil else
Photonic Signal Processing Via Signal-Flow Graph
47
begin
new(exp1);
exp1^:=exp2^;
if (exp2^.format<>magval) and (exp2^.expr<>nil) then
begin
getmem(exp1^.expr, length(exp2^.expr^)+1);
exp1^.expr^:=exp2^.expr^;
end
else exp1^.expr:=nil;
copy_expression(exp1^.prodterm, exp2^.prodterm);
copy_expression(exp1^.sumterm, exp2^.sumterm);
end;
end;
procedure new_scalar(var scalar_expr:expression_ptr; val:double);
{ creates a new expression pointed to by scalar_expr, with value val }
var e:expression_ptr;
begin
if val=0 then scalar_expr:=nil else
begin
new(e);
scalar_expr:=e;
e^.format:=magval;
e^.num:=val;
e^.expr:=nil;
e^.sumterm:=nil;
e^.prodterm:=nil;
end;
end;
procedure new_link_val(var link_expr:expression_ptr;
mag:double; mag_str:string; mag_pwr:double;
angl:double; angl_str:string;
zcoeff:double; z_str:string);
{ creates a new expression pointed to by link_expr, of the form
}
{ mag*[mag_str]^mag_pwr * exp(j*(angl*[angl_str])) * z^(zcoeff*[z_str])}
{
}
{ NB this procedure creates the correct expression tree for precedes() }
{ as it is now; but if precedes() is altered w.r.t. the precedence of}
{ the different term formats, this procedure MUST be modified also. }
var e:expression_ptr;
begin
if mag=0 then link_expr:=nil else {anything*0 = 0}
begin
new(e);
link_expr:=e;
if zcoeff<>0 then {include a zterm if present}
begin
e^.format:=zterm;
e^.num:=zcoeff;
if z_str=''then e^.expr:=nil else
begin
getmem(e^.expr, length(z_str)+1);
e^.expr^:=z_str;
end;
48
Photonic Signal Processing
e^.sumterm:=nil;
new(e^.prodterm);
e:=e^.prodterm;
end;
if angl<>0 then
{include an anglterm if present}
begin
e^.format:=anglterm;
e^.num:=angl;
if angl_str=''then e^.expr:=nil else
begin
getmem(e^.expr, length(angl_str)+1);
e^.expr^:=angl_str;
end;
e^.sumterm:=nil;
new(e^.prodterm);
e:=e^.prodterm;
end;
if (mag_str<>'') and (mag_pwr<>0) then {include a magterm if
present}
begin
e^.format:=magterm;
getmem(e^.expr, length(mag_str)+1);
e^.expr^:=mag_str;
e^.num:=mag_pwr;
e^.sumterm:=nil;
new(e^.prodterm);
e:=e^.prodterm;
end;
e^.format:=magval; {the magval term is compulsory for}
e^.num:=mag; {nonzero expressions.}
e^.expr:=nil;
e^.prodterm:=nil;
e^.sumterm:=nil;
end;
end;
function precedes(x, y:expression_ptr):boolean;
{ establishes an order-of-precedence among expression components}
begin
if y=nil then precedes:=true else if x=nil then precedes:=false
else if (x^.format<>y^.format) then precedes:=(x^.format<y^.format)
else if x^.format=magval then precedes:=true
else if (x^.expr=nil) xor (y^.expr=nil) then precedes:=(y^.expr=nil)
else if (x^.expr=nil) or (x^.expr^=y^.expr^)
then precedes:=(x^.num>=y^.num)
else precedes:=(x^.expr^>y^.expr^);
end;
function can_add(x, y:expression_ptr):boolean;
{ true if the expression components pointed to by x, y are directly
addable }
{ they are addable if they are equal or if they are both magvals.
}
begin
if ((x=nil) xor (y=nil)) or (x^.format<>y^.format) then can_add:=false
else if (x=nil) or (x^.format=magval) then can_add:=true
Photonic Signal Processing Via Signal-Flow Graph
49
else if (x^.num<>y^.num) or
((x^.expr=nil) xor (y^.expr=nil)) or
((x^.expr<>nil) and (x^.expr^<>y^.expr^))
then can_add:=false else can_add:=true;
end;
procedure add(var e1:expression_ptr; e2:expression_ptr);
{ computes e1 <- e1 + e2}
{ NB must not have e1,e2 pointing to same expression}
var temp, last:expression_ptr;
begin
while e2<>nil do {e2 will step through successive top-level sumterms}
if e1=nil then {if e1 = 0, just set e1:= e2}
begin
copy_expression(e1,e2);
e2:=nil;
end
else
begin
temp:=e1; {step through top-level sumterms of e1}
while precedes(temp, e2) and not(can_add(e2,temp)) do
begin
last:=temp;
temp:=temp^.sumterm;
end;
if not(precedes(temp, e2)) then {couldn’t find one to add to,}
begin { so must insert the e2 term}
tmp_exp:=e2^.sumterm; {TEMPORARILY disconnect the remaining}
e2^.sumterm:=nil;
{sumterms so that only e2 and its}
if temp=e1 then
{prodterms get inserted.}
begin
{insert at start of list}
copy_expression(e1,e2);
e1^.sumterm:=temp;
end
else
begin
{insert not at start of list}
copy_expression(last^.sumterm, e2);
last^.sumterm^.sumterm:=temp;
end;
e2^.sumterm:=tmp_exp; {restore remaining sumterms}
end
else if e2^.prodterm<>nil then {can add directly by adding}
begin
{prodterm subexpressions}
add(temp^.prodterm, e2^.prodterm);
if (temp^.prodterm=nil) then
{check for a zero after the}
begin
{addition and if there, get}
if temp=e1 then e1:=e1^.sumterm {rid of it}
else last^.sumterm:=temp^.sumterm;
dispose_expression(temp);
end;
end
else if temp^.format=magval then {can add directly by adding
nums }
begin
temp^.num:=temp^.num + e2^.num;
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Photonic Signal Processing
if temp^.num=0 then {check for a zero after addition and}
begin
{if there, get rid of it.}
if temp=e1 then e1:=e1^.sumterm
else last^.sumterm:=temp^.sumterm;
dispose_expression(temp);
end;
end;
e2:=e2^.sumterm; {do next term}
end;
end;
procedure copy_term(var x:expression_ptr; y:expression_ptr);
{copies a single expression term x <- y, excluding sumterm/prodterm
links}
begin
new(x);
x^:=y^;
x^.sumterm:=nil;
x^.prodterm:=nil;
if (y^.format<>magval) and (y^.expr<>nil) then
begin
getmem(x^.expr, length(y^.expr^)+1);
x^.expr^:=y^.expr^;
end
else x^.expr:=nil;
end;
function can_multiply(x, y:expression_ptr):boolean;
{ true if the components x, y can multiply to a single component}
begin
if (x^.format<>y^.format) then can_multiply:=false else
if x^.format=magval then can_multiply:=true
else if ((x^.expr=nil) and (y^.expr=nil)) or
((x^.expr<>nil) and (y^.expr<>nil) and (x^.expr^=y^.expr^))
then can_multiply:=true else can_multiply:=false;
end;
procedure multiply(var product:expression_ptr; e1,e2:expression_ptr);
{ computes product <- e1 * e2}
var partprod, t1,t2,tmp_ptr:expression_ptr;
begin
product:=nil; {start by setting product = 0}
t1:=e1;
while t1<>nil do {t1 steps through top-level sumterms of e1}
begin
t2:=e2;
while t2<>nil do {t2 steps through top-level sumterms of e2}
begin
if can_multiply(t1,t2) then {multiply t1,t2 directly, then}
begin {multiply their prodterms}
copy_term(partprod, t1);
if t1^.format=magval then partprod^.num:=t1^.num*t2^.num
else partprod^.num:=t1^.num+t2^.num;
if partprod^.num=0 then {get rid of zero terms}
begin
dispose_expression(partprod);
if t1^.format=magval
Photonic Signal Processing Via Signal-Flow Graph
then partprod:=nil
else multiply(partprod,t1^.prodterm, t2^.prodterm);
end
else multiply(partprod^.prodterm, t1^.prodterm, t2^.
prodterm);
end
else if precedes(t1,t2) then {t1 precedes t2, so}
begin
{partprod = t1.(t1_prodterm*t2)}
copy_term(partprod, t1);
tmp_ptr:=t2^.sumterm; {temporarily disconnect}
t2^.sumterm:=nil; {remaining sumterms}
multiply(partprod^.prodterm, t1^.prodterm, t2);
t2^.sumterm:=tmp_ptr;
end
else
begin {same as above, but for t2 preceding t1}
copy_term(partprod, t2);
tmp_ptr:=t1^.sumterm;
t1^.sumterm:=nil;
multiply(partprod^.prodterm, t1,t2^.prodterm);
t1^.sumterm:=tmp_ptr;
end;
add(product, partprod); {add the product of the two}
dispose_expression(partprod); {terms to the final product}
t2:=t2^.sumterm; {get next sumterm from e2}
end;
t1:=t1^.sumterm; {get next sumterm from e1}
end;
end;
procedure killwhitespace(var s:string);
{ removes spaces and tabs from each line of the input file.}
{ also removes lines starting with ';' - i.e. comments.}
var i:byte;
begin
i:=0;
while i<length(s) do
if (s[i+1]='') or (ord(s[i+1])=9)
then delete(s, i+1,1)
else inc(i);
if (length(s)>0) and (s[1]=';') then s:='';
end;
procedure error(i:byte; s:string);
{ displays an error message on the screen, cleans up, and exits the
program}
begin
case i of
0: writeln('ERROR OPENING FILE',s);
1: writeln('INVALID INPUT FILE FORMAT: NO "$INPUT="LINE FOUND');
2: begin
writeln('ERROR LINE ',line,': MAXIMUM LENGTH OF NODE NAME IS
15 CHARACTERS:');
writeln('"',s,'"IS TOO LONG.');
end;
3: writeln('ERROR LINE ',line,': INPUT NODE NAME NOT SPECIFIED');
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Photonic Signal Processing
4: writeln('INVALID INPUT FILE FORMAT: "$OUTPUT="LINE MUST FOLLOW
"$INPUT="LINE');
5: writeln('ERROR LINE ',line,': OUTPUT NODE NAME NOT SPECIFIED');
6: begin
writeln('SYNTAX ERROR IN LINE ',line,' (LINE DISPLAYED UP TO
ERROR):');
writeln(s);
end;
7: writeln('ERROR LINE ',line,': NODE CANNOT CONNECT TO ITSELF
(',s,')');
8: writeln('INVALID FILE FORMAT: "$TRANSMITTANCES" LINE MUST FOLLOW
LINKS LIST');
9: writeln('ERROR: NO LINKS DEFINED FOR NODE ',s);
10: writeln('ERROR: RECIPROCAL LINK FOR LINK ',s,' MISSING');
11: writeln('ERROR: MULTIPLE LINKS ',s,' NOT PERMITTED');
12: writeln('ERROR LINE ',line,': NODE NOT FOUND (',s,')');
13: writeln('ERROR LINE ',line,': LINK NOT FOUND (',s,')');
14: writeln('ERROR LINE ',line,': INPUT NODE LINK DESCRIPTION MAY NOT
CONTAIN ";"');
end;
close(f); {close input/output file if open}
release(heap_state); {release all dynamic variables}
halt; {ABORT PROGRAM!!!}
end;
procedure findnode(var node:node_ptr; nm:string);
{ returns a pointer to the FIRST node in the list with name 'nm'}
{ if no node is found, TWO new nodes are created with this name,}
{ and are added to the START of the node list.}
var i:byte;
begin
if length(nm)>15 then error(2,nm);
node:=firstnode;
while (node<>nil) and (node^.name<>nm) do node:=node^.next;
if node=nil then
for i:=1 to 2 do
begin
new(node);
node^.next:=firstnode;
firstnode:=node;
node^.name:=nm;
node^.firstlink:=nil;
node^.link_taken:=nil;
node^.touchloop:=nil;
node^.touchpath:=nil;
end;
end;
procedure getsourcesink;
{ parses the first 2 meaningful lines of the input file
var s:string;
i, l:byte;
begin
line:=0;
repeat {find the $INPUT line}
readln(f, s);
}
Photonic Signal Processing Via Signal-Flow Graph
inc(line);
killwhitespace(s);
until eof(f) or (copy(s,1,7)='$INPUT=');
if eof(f) then error(1,'');
i:=8; {get the input node name}
l:=0;
while (i+l<=length(s)) and (pos(s[i+l],';:,=$*')=0) do inc(l);
if i+l<=length(s) then error(6,copy(s,1,i+l))
else if l=0 then error(3,'')
else if l>15 then error(2,copy(s, i,l));
new(firstnode); {create a node for it}
firstnode^.next:=nil;
firstnode^.name:=copy(s, i,l);
firstnode^.firstlink:=nil;
firstnode^.link_taken:=nil;
firstnode^.touchloop:=nil;
firstnode^.touchpath:=nil;
sourcenode:=firstnode;
repeat {find the $OUTPUT line}
readln(f, s);
inc(line);
killwhitespace(s);
until eof(f) or (s<>'');
if copy(s,1,8)<>'$OUTPUT='then error(4,'');
i:=9; {get the output node name}
l:=0;
while (i+l<=length(s)) and (pos(s[i+l],';:,=$*')=0) do inc(l);
if i+l<=length(s) then error(6,copy(s,1,i+l))
else if l=0 then error(5,'')
else if l>15 then error(2,copy(s, i,l));
new(firstnode^.next); {create a node for it, at the START}
node:=firstnode^.next; {of the node list (which then contains}
node^.next:=nil; {exactly two nodes)}
node^.name:=copy(s, i,l);
node^.firstlink:=nil; {NB names of sourcenode and sinknode}
node^.link_taken:=nil; {may be the same.}
node^.touchloop:=nil;
node^.touchpath:=nil;
sinknode:=node;
end;
procedure getlinks;
{ parses the geometry description section of the input file }
var outputnode:node_ptr;
link:link_ptr;
s, nm:string;
i, l,j:byte;
begin
repeat {get a line}
readln(f, s);
inc(line);
killwhitespace(s);
if (s<>'') and (s[1]<>'$') then {skip blank lines/$TRANSMITTANCES
line}
begin
i:=1; {get the name of the node that the line defines}
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Photonic Signal Processing
l:=0;
while (i+l<length(s)) and (pos(s[i+l],';:,=$*')=0) do inc(l);
if (l=0) or (s[i+l]<>':') or (i+l=length(s))
then error(6,copy(s,1,i+l));
if l>15 then error(2,copy(s, i,l));
nm:=copy(s, i,l);
findnode(node, nm);
{find the node or create it}
if node<>sinknode then
{ignore geometry defn. for output
node}
repeat {NB if input=output name, the INPUT}
i:=i+l+1;
{node will be found.}
l:=0;
{find each node name in geometry
list in turn}
while (i+l<=length(s)) and (pos(s[i+l],';:,=$*')=0) do inc(l);
if ((l=0) and not((i+l<=length(s)) and (s[i+l]=';'))) or
((i+l<=length(s)) and (pos(s[i+l],':=$*')>0)) or
((i+l<length(s)) and (s[i+l]=';') and
((node^.next=nil) or (node^.name<>node^.next^.name)))
then error(6,copy(s,1,i+l)); {syntax check}
if l>15 then error(2,copy(s, i,l));
nm:=copy(s, i,l);
if nm=node^.name then error(7,nm);
findnode(outputnode, nm);
{find the listed node, and}
new(link);
{create a link (value=1) to it}
link^.next:=node^.firstlink;
node^.firstlink:=link;
link^.dest:=outputnode;
new_scalar(link^.val,1); {switch to 2nd node of same name}
if (i+l<length(s)) and (s[i+l]=';') then {when ';' found}
if node=sourcenode then error(14,'')
else node:=node^.next;
until (i+l>length(s)) or ((i+l=length(s)) and (s[i+l]=';'));
end;
{ignore ';' if it is the last thing on the line}
until (s[1]='$') or eof(f);
if not(eof(f)) and (copy(s,1,15)<>'$TRANSMITTANCES') then error(8,'');
end;
{terminate at end of file or at "$TRANSMITTANCES" line}
procedure correctlinks;
{ checks that (1) outgoing links exist, (2) reciprocal links exist, and}
{ (3) that no multiple links exist. Shifts link destinations from a node}
{ to its mirror node (i.e. same point in optical cct, opposite direction}
{ of light travel) as required so that no A -> B -> A type links exist.}
var destnode:node_ptr;
link, otherlink:link_ptr;
begin
node:=firstnode;
while node<>nil do {step through each node in turn}
begin
link:=node^.firstlink;
if link<>nil then
while link<>nil do {step through each link on that node in turn}
begin
destnode:=link^.dest;
if (destnode<>sinknode) then
{check for reciprocal link}
begin
{except from sink node}
otherlink:=destnode^.firstlink;
Photonic Signal Processing Via Signal-Flow Graph
55
while (otherlink<>nil) and (otherlink^.dest^.
name<>node^.name)
do otherlink:=otherlink^.next;
if destnode<>sourcenode then {if reciprocal link found}
if otherlink<>nil then link^.dest:=destnode^.next
else {on first node of pair, shift original link to}
begin {2nd...otherwise continue check on 2nd
same-name}
otherlink:=destnode^.next^.firstlink; {node}
while (otherlink<>nil) and
(otherlink^.dest^.name<>node^.name)
do otherlink:=otherlink^.next;
if otherlink=nil
then error(10,node^.name+' -> '+destnode^.name);
end {err if no reciprocal link found}
else {link goes to source node => error unless names of}
if otherlink=nil {source, sink are the same - then
just}
then error(10,node^.name+' -> '+destnode^.name)
else if sourcenode^.name=sinknode^.name {shift it to}
then link^.dest:=sinknode; {the sink}
end;
otherlink:=link^.next; {check for 2nd link to same node}
while (otherlink<>nil) and
(otherlink^.dest^.name<>link^.dest^.name)
do otherlink:=otherlink^.next;
if (otherlink=nil) and (node^.next<>nil) {may need to
continue}
and (node^.next^.name=node^.name) {search on 2nd same-}
then otherlink:=node^.next^.firstlink; {name node}
while (otherlink<>nil) and
(otherlink^.dest^.name<>link^.dest^.name)
do otherlink:=otherlink^.next;
if (otherlink<>nil) then {error if duplicate link found}
error(11,node^.name+' <→ '+link^.dest^.name);
link:=link^.next; {do next link}
end
else if (node=sourcenode) or ((node^.next<>nil) and
(node^.next^.firstlink=nil) and (node^.next^.name=node^.
name))
then error(9,node^.name); {no links from sourcenode or from}
node:=node^.next; {a pair of same-name node => err!}
end; {check next node...}
end;
procedure getlinkvals;
{read the input file to obtain expressions for links (=1 if not listed)}
var s:string;
node1,node2,tmp_node:node_ptr;
link:link_ptr;
nm1,nm2:string[15];
oneway:boolean;
i, l:byte;
mag_val, angl_num, mag_pwr, zcoeff:double;
mag_expr, angl_expr, z_expr:string;
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Photonic Signal Processing
begin
repeat {get each remaining line of the input file in turn}
readln(f, s);
inc(line);
killwhitespace(s);
if s<>''then {ignore blanks, comments}
begin
i:=1;
l:=0;
if s[1]='*' then
{check for * at start of line, denoting}
begin {a one-way only link assignment.}
oneway:=true;
i:=2;
if length(s)=1 then error(6,s);
end
else oneway:=false;
{get first node name and the comma}
while (i+l<length(s)) and (pos(s[i+l],';:,=$*')=0) do inc(l);
if (s[i+l]<>',') or (length(s)=i+l) or (l=0)
then error(6,copy(s,1,i+l)) else
if l>15 then error(2,copy(s, i,l)) else nm1:=copy(s, i,l);
i:=i+l+1;
l:=0;
{get second node name}
while (i+l<length(s)) and (pos(s[i+l],';:,=$*')=0) do inc(l);
if (s[i+l]<>'=') or (length(s)=i+l) or (l=0)
then error(6,copy(s,1,i+l)) else
if l>15 then error(2,copy(s, i,l)) else nm2:=copy(s, i,l);
i:=i+l+1;
l:=0;
node1:=firstnode; {find both nodes, check that they exist}
node2:=node1;
while (node1<>nil) and (node1^.name<>nm1) do
begin
if (node2^.name<>nm2) then node2:=node1;
node1:=node1^.next;
end;
while (node2<>nil) and (node2^.name<>nm2) do node2:=node2^.next;
if (node1=nil) then error(12,nm1);
if (node2=nil) then error(12,nm2);
mag_val:=1; {set defaults for any part of the}
mag_pwr:=0; {expression not included in the input}
angl_num:=0;
zcoeff:=0;
mag_expr:='';
angl_expr:='';
z_expr:=''; {(1) look for a number}
while (i+l<=length(s)) and (pos(s[i+l],'+-0123456789eE.')>0) do inc(l);
if l>0 then
begin
val(copy(s, i,l),mag_val, errpos);
if errpos>0 then error(6,copy(s,1,i+errpos));
i:=i+l;
l:=0;
end;
{(2) look for a magterm expr}
if (i<length(s)) and (s[i]='{') then
begin
inc(i);
Photonic Signal Processing Via Signal-Flow Graph
57
while (i+l<length(s)) and (s[i+l]<>'}') do inc(l);
if s[i+l]<>'}' then error(6,s);
mag_expr:=copy(s, i,l);
mag_pwr:=1;
i:=i+l+1;
l:=0; {(3) if found one, look for associated ^num}
if (i<length(s)) and (s[i]='^') then
begin
inc(i);
while (i+l<=length(s)) and
(pos(s[i+l],'+-0123456789eE.')>0)
do inc(l);
if l=0 then error(6,copy(s,1,i));
val(copy(s, i,l),mag_pwr, errpos);
if errpos>0 then error(6,copy(s,1,i+errpos));
i:=i+l+1;
l:=0;
end;
end;
{(4) look for an angle term identifier
'<'}
if (i<length(s)) and (s[i]='<') then
begin
inc(i); {(5) if found '<'look for a number}
while (i+l<=length(s)) and (pos(s[i+l],'+-0123456789eE.')>0)
do inc(l);
if l>0 then
begin
val(copy(s, i,l),angl_num, errpos);
if errpos>0 then error(6,copy(s,1,i+errpos));
i:=i+l;
l:=0;
end
{(6)... then look for an anglterm expr}
else if (i<length(s)) and (s[i]='{') then angl_num:=1;
if (i<length(s)) and (s[i]='{') then
begin
inc(i);
while (i+l<length(s)) and (s[i+l]<>'}') do inc(l);
if s[i+l]<>'}' then error(6,s);
angl_expr:=copy(s, i,l);
i:=i+l+1;
l:=0;
end;
end;
{(7) look for a zterm identifier 'z' or 'Z'}
if ((i<=length(s)) and (upcase(s[i])<>'Z')) then
error(6,copy(s,1,i))
else if i=length(s) then zcoeff:=1 else if i<length(s) then
begin
{if it's not the last char. on the line, it must}
inc(i); {be followed by ^something}
if s[i]<>'^' then error(6,copy(s,1,i)) else
if i=length(s) then error(6,s) else inc(i);
while (i+l<=length(s)) and (pos(s[i+l],'+-0123456789eE.')>0)
do inc(l); {(8) look for a number (zterm.num field)}
if l>0 then
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Photonic Signal Processing
begin
val(copy(s, i,l),zcoeff, errpos);
if errpos>0 then error(6,copy(s,1,i+errpos));
i:=i+l;
l:=0;
end {(9) look for a zterm expr}
else if (i<length(s)) and (s[i]='{') then zcoeff:=1;
if i=length(s) then error(6,s);
if (i<length(s)) and (s[i]='{') then
begin
inc(i);
while (i+l<length(s)) and (s[i+l]<>'}') do inc(l);
if s[i+l]<>'}' then error(6,s);
if i+l<length(s) then error(6,copy(s,1,i+l+1));
z_expr:=copy(s, i,l);
end;
end;
if node1<>node2 then {is it a reflection coefficient?}
begin {NO: then if it involves source or sink, may have}
if sourcenode^.name<>sinknode^.name then {to swap}
begin {if oneway is set, order is already explicit}
if not(oneway) and (node1=sinknode) or
(node2=sourcenode)
then begin
tmp_node:=node1; {swap nodes so that oneway}
node1:=node2; {direction is appropriate,}
node2:=tmp_node; {and set oneway. (since}
oneway:=true; {source/sink only have one}
end; {way links)}
if (node1=sourcenode) or (node2=sinknode) then
oneway:=true;
end;
{find the desired link}
link:=node1^.firstlink;
while (link<>nil) and (link^.dest^.name<>nm2) do
link:=link^.next;
if (link=nil) and (node1^.next<>nil) and (node1^.next^.
name=nm1)
then link:=node1^.next^.firstlink;
while (link<>nil) and (link^.dest^.name<>nm2) do
link:=link^.next;
if link=nil then error(13,nm1+' -> '+nm2) else {link
absent?}
begin {set its value}
dispose_expression(link^.val);
new_link_val(link^.val,
mag_val, mag_expr, mag_pwr,
angl_num, angl_expr,
zcoeff, z_expr);
end;
if not(oneway) then {if setting value for both directions,}
begin
{find the reciprocal (other way) link}
link:=node2^.firstlink;
while (link<>nil) and (link^.dest^.name<>nm1)
do link:=link^.next;
if (link=nil) and (node2^.next<>nil) and (node2^.next^.
name=nm2)
Photonic Signal Processing Via Signal-Flow Graph
59
then link:=node2^.next^.firstlink;
while (link<>nil) and (link^.dest^.name<>nm1)
do link:=link^.next;
if link=nil then error(13,nm2+' -> '+nm1) else {absent?}
begin
{set its value}
dispose_expression(link^.val);
new_link_val(link^.val,
mag_val, mag_expr, mag_pwr,
angl_num, angl_expr,
zcoeff, z_expr);
end;
end;
end
else {from earlier: names are the same so its a reflection
coeff}
begin
if (node1=sourcenode) or (node1=sinknode)
then error(13,nm1+' -> '+nm1); {source or sink not
allowed}
node2:=node1^.next; {get node and its same-name
counterpart}
if (node1^.firstlink<>nil) and {reflection already set?}
(node1^.firstlink^.dest=node2) then
begin {if so, replace it}
dispose_expression(node1^.firstlink^.val);
new_link_val(node1^.firstlink^.val,
mag_val, mag_expr, mag_pwr,
angl_num, angl_expr,
zcoeff, z_expr);
end
else if (node2^.firstlink<>nil) and {already set, other
way?}
(node2^.firstlink^.dest=node1) then
begin
{replace existing link}
dispose_expression(node2^.firstlink^.val);
new_link_val(node2^.firstlink^.val,
mag_val, mag_expr, mag_pwr,
angl_num, angl_expr,
zcoeff, z_expr);
end
else if (node1^.firstlink=nil) and (node2^.
firstlink<>nil) then
begin {create new link on 1st node and set value}
new(link); {if the 1st node is the input node for
the}
link^.next:=nil; {same-name pair}
node1^.firstlink:=link;
link^.dest:=node2;
new_link_val(link^.val,
mag_val, mag_expr, mag_pwr,
angl_num, angl_expr,
zcoeff, z_expr);
end
else if (node2^.firstlink=nil) and (node1^.
firstlink<>nil) then
begin {otherwise create new link on 2nd node...}
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Photonic Signal Processing
new(link);
link^.next:=nil;
node2^.firstlink:=link;
link^.dest:=node1;
new_link_val(link^.val,
mag_val, mag_expr, mag_pwr,
angl_num, angl_expr,
zcoeff, z_expr);
end
else error(13,nm1+' -> '+nm1); {neither node was the
input}
end; {of a cut-off fiber end. So reflections aren’t
allowed!!!}
end;
until eof(f); {keep getting lines until end of input file}
end;
procedure searchfrom(root:node_ptr);
{
performs a recursive search from "root" node to find loops and
paths }
begin
if root^.link_taken<>nil then {hit a node already passed: loop found}
begin
found_duplicate:=false;
{CHECK IF LOOP ALREADY EXISTS:}
loopentry:=root^.touchloop; {compare with existing loops at node}
while not(found_duplicate) and (loopentry<>nil) do {check each one}
begin
{find the current node in its node list}
nodeentry:=loopentry^.loop^.nodelist;
while nodeentry^.node<>root do nodeentry:=nodeentry^.next;
node:=root;
repeat
{follow both loops a step at a time until we cycle, or}
node:=node^.link_taken^.dest; {we get to 2 different nodes}
nodeentry:=nodeentry^.next;
until (node=root) or (node<>nodeentry^.node); {if we cycled, loop}
if node=nodeentry^.node then found_duplicate:=true; {isn’t new}
loopentry:=loopentry^.next; {check next loop at node}
end;
if not(found_duplicate) then {CREATE NEW LOOP:}
begin
new(loop);
{create loop entry}
loop^.next:=firstloop;
firstloop:=loop;
loop^.touchloop:=nil;
new_scalar(loop^.loopgain,-1);
{-loopgain actually
stored}
loop^.nodelist:=nil;
{will * one link at a time}
loop^.deactivated:=0;
{activate loop- used later}
nodeentry:=nil;
node:=root;
{along each node on the loop do 5 things:}
repeat with node^ do
begin
new(loopentry); {(1) add loop to list of loops at node}
loopentry^.loop:=loop;
loopentry^.next:=touchloop;
touchloop:=loopentry;
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61
if nodeentry=nil then {(2) add node to nodelist for loop}
begin
new(loop^.nodelist);
nodeentry:=loop^.nodelist;
end
else
begin
new(nodeentry^.next);
nodeentry:=nodeentry^.next;
end;
nodeentry^.node:=node;
loopentry:=touchloop; {(3) add all loops at node to}
while loopentry<>nil do {touchloop list for this loop}
begin {(but check for duplicates first)}
loopentry2:=loop^.touchloop;
while (loopentry2<>nil) and
(loopentry2^.loop<>loopentry^.loop)
do loopentry2:=loopentry2^.next;
if loopentry2=nil then {add loop to list}
begin
new(loopentry2);
loopentry2^.loop:=loopentry^.loop;
loopentry2^.next:=loop^.touchloop;
loop^.touchloop:=loopentry2;
end;
loopentry:=loopentry^.next;
end;
pathentry:=touchpath; {(4) add this loop to touchloop}
while pathentry<>nil do {list of all paths at node}
begin
loopentry2:=pathentry^.path^.touchloop;
while (loopentry2<>nil) and (loopentry2^.loop<>loop)
do loopentry2:=loopentry2^.next;
if loopentry2=nil then {check for existing entry}
begin
{none, so add to list}
new(loopentry2);
loopentry2^.loop:=loop;
loopentry2^.next:=pathentry^.path^.touchloop;
pathentry^.path^.touchloop:=loopentry2;
end;
pathentry:=pathentry^.next;
end;
{(5) multiply loopgain by value of outgoing link}
multiply(mult_result, loop^.loopgain, link_taken^.val);
dispose_expression(loop^.loopgain);
loop^.loopgain:=mult_result;
node:=link_taken^.dest
end;
until node=root; {cycle through nodes until back at the start}
nodeentry^.next:=loop^.nodelist;
{nodelist itself
should loop}
end;
{i.e. it’s circularly linked}
end
else if root=sinknode then {REACHED SINKNODE: NEW PATH FOUND}
begin
new(path); {create path entry}
path^.next:=firstpath;
62
Photonic Signal Processing
firstpath:=path;
path^.touchloop:=nil;
new_scalar(path^.pathgain,1);
node:=sourcenode; {along each node on the path do 3 things:}
repeat with node^ do
begin
new(pathentry); {(1) add path to list of paths at node}
pathentry^.path:=path;
pathentry^.next:=touchpath;
touchpath:=pathentry;
loopentry:=touchloop; {(2) add all loops at node to loop}
while loopentry<>nil do {list for this path (but check}
begin {for duplicates first)}
loopentry2:=path^.touchloop;
while (loopentry2<>nil) and
(loopentry2^.loop<>loopentry^.loop)
do loopentry2:=loopentry2^.next;
if loopentry2=nil then
{add loop to list}
begin
new(loopentry2);
loopentry2^.loop:=loopentry^.loop;
loopentry2^.next:=path^.touchloop;
path^.touchloop:=loopentry2;
end;
loopentry:=loopentry^.next;
end;
if link_taken=nil then node:=nil else {(3) multiply pathgain by}
begin
{value of outgoing link}
multiply(mult_result, path^.pathgain, link_taken^.val);
dispose_expression(path^.pathgain);
path^.pathgain:=mult_result;
node:=link_taken^.dest
end;
end;
until node=nil; {step through nodes on the path until none remain}
end
else with root^ do {continue depth-first search}
begin
link_taken:=firstlink; {mark the way we went}
while link_taken<>nil do
begin {don’t follow 'phantom' links TO the source node}
if link_taken^.dest<>sourcenode then searchfrom(link_taken^.
dest);
link_taken:=link_taken^.next;
{go a different way...}
end;
end;
end;
procedure deletenodedata;
{ frees memory used for storing node information (no longer needed) }
var nextentry:pointer;
link:link_ptr;
begin
loop:=firstloop; {remove each loop’s nodelist}
while loop<>nil do
begin
nodeentry:=loop^.nodelist;
Photonic Signal Processing Via Signal-Flow Graph
while nodeentry<>nil do
begin
nextentry:=nodeentry^.next;
if nextentry=loop^.nodelist then nextentry:=nil;
dispose(nodeentry);
nodeentry:=nextentry;
end;
loop^.nodelist:=nil;
loop:=loop^.next;
end;
node:=firstnode; {remove all data for each node:}
while node<>nil do
begin
pathentry:=node^.touchpath; {remove the touchpath list}
while pathentry<>nil do
begin
nextentry:=pathentry^.next;
dispose(pathentry);
pathentry:=nextentry;
end;
loopentry:=node^.touchloop; {remove the touchloop list}
while loopentry<>nil do
begin
nextentry:=loopentry^.next;
dispose(loopentry);
loopentry:=nextentry;
end;
link:=node^.firstlink; {remove the links}
while link<>nil do
begin
nextentry:=link^.next;
dispose_expression(link^.val);
dispose(link);
link:=nextentry;
end;
nextentry:=node^.next; {remove the node itself}
dispose(node);
node:=nextentry;
end;
end;
procedure addloopcombination(var delta:expression_ptr;
running_product:expression_ptr;
current_loop:loop_ptr);
{ recursively adds to delta the sum of all possible products of}
{ nontouching loop gains, down the list from 'current_loop'. }
{ 'running product' contains the product of those loops already }
{ used... i.e. the ones up the list from current_loop that were }
{ taken on the way to getting to current_loop. }
var tempexpression:expression_ptr;
begin
while current_loop<>nil do {step through all possible branches}
begin {and descend to them in turn if active}
if current_loop^.deactivated=0 then {skip deactivated ones, i.e.}
begin
{those already 'touched'}
multiply(tempexpression, current_loop^.loopgain, running_product);
63
64
Photonic Signal Processing
add(delta, tempexpression); {add to delta the tree descent so
far}
loopentry:=current_loop^.touchloop; {deactivate touching loops}
while loopentry<>nil do
{further down the list}
begin
inc(loopentry^.loop^.deactivated);
loopentry:=loopentry^.next;
end; {recursively add to delta all terms downbranch}
addloopcombination(delta, tempexpression, current_loop^.next);
dispose_expression(tempexpression);
loopentry:=current_loop^.touchloop; {reactivate loops}
while loopentry<>nil do
{(or clear deactivations}
begin
{set by current loop)}
dec(loopentry^.loop^.deactivated);
loopentry:=loopentry^.next;
end;
end;
current_loop:=current_loop^.next; {do next branch}
end;
end;
procedure calcdelta(var delta:expression_ptr);
{ calculates the Mason rule delta, using all loops not already
deactivated}
var running_product:expression_ptr;
begin
new_scalar(delta,1);
new_scalar(running_product,1);
addloopcombination(delta, running_product, firstloop);
dispose_expression(running_product);
end;
procedure calctransferfunction;
{ calculates the numerator and denominator of the signal flow}
{ graph transfer function using the Mason’s rule equation}
var delta:expression_ptr;
begin
calcdelta(denominator); {calculate denominator = delta}
new_scalar(numerator,0); {set numerator = 0}
path:=firstpath;
while path<>nil do {for each path in turn:}
begin
loopentry:=path^.touchloop; {deactivate each loop that touches it}
while loopentry<>nil do
begin
loopentry^.loop^.deactivated:=1;
loopentry:=loopentry^.next;
end;
calcdelta(delta);
{find delta for this reduced loop set}
multiply(mult_result, delta, path^.pathgain);
dispose_expression(delta);
dispose_expression(path^.pathgain);
add(numerator, mult_result); {numerator <- numerator +
delta*pathgain}
dispose_expression(mult_result);
loopentry:=path^.touchloop; {reactivate all the loops}
Photonic Signal Processing Via Signal-Flow Graph
65
while loopentry<>nil do {deactivated by current path}
begin
loopentry^.loop^.deactivated:=0;
loopentry:=loopentry^.next;
end;
path:=path^.next; {do next path}
end;
end;
{ ########################## MAIN PROGRAM ###############################}
begin
writeln;
writeln;
prec:=3; {default to 3 decimal places in the output}
errpos:=0;
outfile_name:=paramstr(2);
precparam:=paramstr(3);
if outfile_name[1]='-' then {if outfile is omitted, -d may be the}
begin {second command line parameter}
precparam:=outfile_name;
outfile_name:='';
end;
if precparam<>''then {set precision if parameter present}
val(copy(precparam,2,length(precparam)-1),prec, errpos);
if (paramcount=0) or (paramcount>3) or
((paramcount=3) and (precparam[1]<>'-')) or (errpos<>0) then
begin {if the parameters weren’t right, show instructions and quit}
writeln('OptMason generates the transfer function of an optical
network using Mason''s');
writeln('rule for signal-flow graphs.');
writeln;
writeln('Written by Dean Trower, 1996. This program may be freely
distributed.');
writeln;
writeln;
writeln('USAGE: optmason input_file [output_file] [-d]');
writeln;
writeln('d is the number of decimal places that numbers are
displayed to in the output.');
writeln('If the output file is omitted, output is to the screen
(or standard output).');
writeln('If the output file has the same name as the input file,
the output is appended');
writeln('to the input file.');
writeln;
halt;
end;
mark(heap_state); {record the state of memory, to restore it when done}
firstnode:=nil; {init lists and miscellaneous}
firstpath:=nil;
firstloop:=nil;
plus_waiting:=false;
assign(f, paramstr(1)); {attempt to open input file}
reset(f);
if IOResult<>0 then error(0,paramstr(1));
getsourcesink; {parse input, output definitions}
66
Photonic Signal Processing
getlinks; {parse geometry description}
correctlinks; {create signal flow graph geometry, and check}
getlinkvals; {parse transmittance expressions}
close(f); {close the input file}
writeln('SIGNAL FLOW GRAPH CREATED'); {this always goes to the SCREEN}
searchfrom(sourcenode); {recursively find all loops and paths}
deletenodedata; {get rid signal flow graph, now no longer necessary}
writeln('PATH AND LOOP INFORMATION COMPUTED'); {displays to SCREEN}
calctransferfunction; {apply Mason’s rule to loop and path info}
writeln('TRANSFER FUNCTION T COMPUTED'); {displays to SCREEN}
assign(f, outfile_name); {create or append the desired output file}
if paramstr(1)=outfile_name then append(f) else rewrite(f);
if IOresult<>0 then error(0,outfile_name);
writeln(f); {NB a file name of ''get assigned to DOS standard output}
writeln(f,'T = numerator/denominator'); {output the results}
writeln(f);
write(f,'numerator = ');
writeexpression(f, numerator);
writeln(f);
writeln(f);
write(f,'denominator = ');
writeexpression(f, denominator);
writeln(f);
close(f); {close the output file}
release(heap_state); {deallocate all memory that got used}
end.
2.11
APPENDIX: USING “OPTIMASON” THE COMPUTER AIDED GENERATOR
OPTMASON (Program listed in Section 2.9) is started from the DOS command line by typing:
“optmason input_file [output_file] [-d]” where “input_file” and “output_file” are the corresponding
filenames, and “d” is the number of decimal places to display real numbers to in the output; if omitted, “d” defaults to 3. If the output filename is omitted, output is to the screen (actually to DOS’s
standard output file, to enable redirection). If the input and output filenames are the same, the input
file is not overwritten, but is instead appended (i.e., OPTMASON’s output is added to the end of it).
The input text file format for OPTMASON is as follows:
$INPUT = nodename
$OUTPUT = nodename
nodename: nodename, nodename,…; nodename, nodename,…
nodename: nodename, nodename,…; nodename, nodename,…
nodename: nodename, nodename,…; nodename, nodename,…
$TRANSMITTANCES
[*]nodename, nodename = expression
[*]nodename, nodename = expression
[*]nodename, nodename = expression
Here nodename is a label for a node of the photonic connection graph. It may be any string of up
to 15 characters, but not containing any of the characters “;:,=$*” (double quotes are OK though).
Expression is a mathematical expression (see below).
The first line ($INPUT=…) identifies the input (or source) node.
The second line ($OUTPUT=…) identifies the output (or sink) node.
Photonic Signal Processing Via Signal-Flow Graph
67
The input and output nodes may be the same node.
The next section defines the geometry of the photonic connection graph.
A line of the form nodename: nodename, nodename,…; nodename, nodename,…is required for
each node in the graph exceptthe output node. (If a line for the output node is included, it will be
ignored, unless that node is also the input node.)
Each such line begins with the name of the node being defined, followed by a colon “:”. The
remainder of the line is a list of all the other nodes that it connects to. Since light may travel
independently in bi-directions through a node in a Photonic connection graph, connections on either
“side” (photonically speaking) are separated by a semicolon “;”. The definition for the input node
and any node at the free end of a photonic waveguide(reflection point) will only include one “side”
of this list.
Since links in a photonic connection graph are bi-directional, a link from n to min the definition
of a node n must be matched by a corresponding link from m to n in the definition of node m; links to
the output node are an exception. Note that only a single link may join any two nodes. For multiple
photonic paths between two nodes, intermediate nodes must be inserted.
The geometry definition section is terminated by the “$TRANSMITTANCES” line.
The section following this line defines the values of the links in the photonic connection graph.
Since all links have a default value of unity in both directions, only the values of links that differ
from this need be defined. The format for specifying the value of a link between two nodes n and m
(value given by “expression”) is: “n, m = expression”.
To define the value of the link in the direction n only, place an asterix “*” at the start of the line.
Note that if a link value is defined twice, the second definition replaces the first (both directions are
treated independently).
To define the reflection coefficient at a node r, simply write: “r, r = expression”
An asterix “*” is optional and has no effect. Reflection coefficients may ONLY be defined at
nodes that are photonically single-sided (e.g., cut-end of a photonic fiber/waveguide), and may not
be defined for the input or output nodes.
Some other things to note about the input file format are:
• The input is case-sensitive for node names and variables within expressions.
• Whitespace (spaces, tabs, and blank lines) are ignored or filtered out.
• The start of the file is ignored up to the line starting with “$INPUT=” so it may be used for
a description of the file contents.
• Any line beginning with a semicolon “;” is treated as a comment and ignored.
• Input lines are truncated beyond 255 characters.
"expression"s have the following form:
magnitude {mag_expr} ^power <angle{angle_expr} z ^power
{power_expr}
magnitude, angle, andpowerare real numbers (scientific notation, e.g.
-1.2e+7 is permitted).
"z" may be uppercase or lowercase (the z-transform parameter). mag_expr,
angle_expr, power_exprmay be any strings atall provided they do not
contain "}". They are intended to be variable names or entire
subexpressions.
Any part or parts of the above expression format may be omitted, provided that a meaningful
expression is the result. The following are examples of valid expressions and their corresponding
mathematical meaning:
6 = 6
2.96E-9{a}^2 = 2.96 × 10-9a^2
68
Photonic Signal Processing
< {pi.beta} = exp(j(pi.beta))
-3 < 1.2 = -3exp(j1.2)
Z^-4{L} = z^-4L
-5.2{x*y/z}^3 < 6.4{beta.pi} z^{L+n} =-.2(x*y/z)^3exp(j.6.4(beta.pi))
z^(L+n)
Note that although parts of the expression format may be omitted, the ordering of the expression
components must be strictly adhered to. Also:
• The first “^power” is a power of “mag_expr” and may only be present if “mag_expr” is also.
• The second “power” and “power_expr” apply to the “z”, so can only be present if it is also.
• A single “-” sign is not a number, and cannot precede any of the {} brackets on its own. Use
-1{…} to obtain the same effect.
• Expressions in {} brackets are not simplified internally; they are treated as single variables.
However, when displaying the output, OPTMASON distinguishes two classes of {}
expression:
1. A string inside {} brackets is treated as a single variable if it is composed only of letters,
numbers, and the characters “~#@$%?”, and if it does not begin with a digit (0–9). It may
also end with a string of single quotes “ ‘ ”.
As a single variable, it will appear in summations, products, and power expressions without
brackets around it. So for correct output, products of two variables contained within {}
should include a “.” or “*” symbol. For example, if t2x is a product, entering {t2x} may
result in the output containing (say) t2x^3, which looks like t2 × 3 when what is desired is
(t2 x) 3.
2. Any other string inside {} brackets will appear bracketed in the output. In particular,
any string containing any of the mathematical symbols “[]()+-*./^<>=!” will appear
bracketed (inside square brackets) in the output, except if it appears on its own (without
any multiplying terms) in a summation.
Here are the contents of the input file for the example given earlier:
• This file is an example of an input file for the sample network included in the OPTMASON
documentation given in the Section 2.7.
• The next two lines define the names of the input (source) and output (sink) nodes:
$INPUT = X
$OUTPUT = Y
; The transfer function calculated by OPTMASON will be T = Y/X
; Here is a description of the network geometry:
X: 1
1: X; 2,4
2: 1,3; 8
3: 6; 2,4
4: 1,3; 5
5: 4; 6,8
6: 5,7; 3
7: Y; 6,8
8: 5,7; 2
; not necessary, but included for completeness:
Y: 7
$TRANSMITTANCES
; Internal photonic coupler transmittances:
Photonic Signal Processing Via Signal-Flow Graph
69
1,2 = {C1}; 3,4 = {C1}; 1,4 = {k1};2,3 = {k1}; 5,6 = {C2};
7,8 = {C2}; 5,8 = {k2}; 6,7 = {k2};
; We ignore the links from X and Y because they do not contribute to the
magnitude response of the system, or affect its pole and zero positions.
;...so they default to values of 1.
; The three main photonic fiber transmittances:
2,8 = {t1} z^-1{L1}; 4,5 = {t2} z^-1{L2};3,6 = {t3} z^-1{L3}; That’s all
we need! (simple, isn’t it?).
Here is OPTMASON’s output:
T = numerator/denominator
Numerator =((k1^2+C1^2)*k2^2+C2^2*k1^2-2*C1^2*C2^2)*t1*t2*t3*z
^(-L3-L2-L1)+((C1*C2*k1^2-C1^3*C2)*k2^2-C1*C2^3*k1^2+C1^3*C2^3)
*t1*t2^2*t3^2*z^(-2*L3-2*L2-L1)+C1*C2*t1*z^-L1
denominator =
-2*C1*C2*t2*t3*z^(-L3-L2)+C1^2*C2^2*t2^2*t3^2*z^(-2*L3-2*L2)+1
3
Bandpass Optical Filters
by DSP Techniques
Optical filters play a significant part in both the optical signal processors and communications
systems. A systematic procedure for synthesizing bandpass optical filters of the Chebyshev type is
described. Proposed structures for these filters in cascade or parallel forms are given using optical
resonators exhibiting a single zero and single pole transfer function in the z-domain, and a quasi-allpole and all-zero optical circuits are described for implementation of the synthesized filter functions.
3.1
3.1.1
OPTICAL FIXED BANDPASS FILTER
Introductory remarks
Recently, optical filters and equalizers are attracting great interest due to the extension of the repeaterless transmission distance and their applications in wavelength division multiplexing (WDM) systems and networks [1]. However, only Butterworth-type optical filters have been considered [2,3].
In filter design, the Chebyshev types are also very important because they would generate a better filter passband and much improved stability of the filtering systems. In practice, it is very often
that the filter characteristics are specified, and the designer must tailor the optical filters accordingly. Several works have been published on the analysis of a certain type of optical configurations
and, from the characteristics obtained, some filters are proposed [4,5]. Thus, there is an urgent need
to develop a systematic procedure to synthesize optical filters from practical devices, which are
available in research laboratories and implemented devices.
This chapter describes a synthesis for optical filters based on the digital filter technique following
that of a Chebyshev filter type. The background about the program and an algorithm for synthesizing the filters are given in Section 3.2. The transfer functions for these filters, which are derived and
described in Section 3.1.2, are expressed in cascade and parallel forms as the fundamental structures for implementation of the filters. Essential optical components are analyzed in Section 3.1.3.3,
which describes two optical resonators and an optical interferometer. In one resonator, a 3 × 3 optical
directional coupler is employed with two feedback back paths: one direct connection and the other
with single or multiple order optical delay to obtain an optical transfer function having non-zero
roots in both its numerator and denominator. The other resonator employs two [2 × 2] optical directional couplers and only one feedback path [1] to obtain an optical transfer having non-zero roots in
the denominator. Optical implementation for the Chebyshev filters are proposed in Section 3.1.3.4.
Conclusions and properties of the Chebyshev filters are given in the last section.
3.1.2
chebyshev optIcal FIlter specIFIcatIon and synthesIs algorIthm
Optical filters can have the characteristics of all basic filter characteristics of several types, such as
lowpass, highpass, and bandpass. The lowpass filter is the fundamental filter, and design algorithm
is based on the design of this filter plus a further transformation, such as bilinear transformation
[6], [7]. In this section, the lowpass type of Chebyshev filters is given then followed by an algorithm to develop bandpass optical filters.
71
72
Photonic Signal Processing
3.1.3
basic characteristics OF chebyshev lOwpass Filters
Bandpass Chebyshev optical filters can be designed by transforming the characteristics of a lowpass
type, which is given by [5]:
2
H (ω 2 ) =
1
1 + ε Cn2 (ω )
(3.1)
2
where ω is the optical angular frequency, Cn (ω ) is the nth order Chebyshev polynomial, and ε < 1
is a real constant and specifies the amplitude of the ripple of the transfer function in the passband.
Thus the nth order Chebyshev polynomial can be obtained as [5]
and
Cn (ω ) = cos( n.cos −1 ω )
if |ω| < 1
(3.2)
Cn (ω ) = cosh( n.cos −1 ω )
if |ω| > 1
(3.3)
3.1.3.1 Chebyshev-Type Optical Bandpass Filter Specification
A typical specification for photonic bandpass filter in the wavelength domain is shown in
Figure 3.1 where λc is the center wavelength, λl is the lower wavelength, λu upper wavelength and λst
the stopband wavelength. Rp is the ripple of the pass band, which is normally specified in dB. Rp
corresponds to the parameter ε of (1) and Rs (in dB) is the overall transfer magnitude of the transfer
function. Once the filter characteristics are specified, the following steps are used to obtain the filter
transfer function in the z-domain so that the filter hardware can be implemented. The z-domain is
used for convenience in analysis and synthesis. Our algorithm is applicable in both analog and digital domain. The z parameter is defined as usual as z = e jωT = e j β L where T is the sampling period,
which is the delay time in an optical loop or path L, and β is the propagation constant of the guided
lightwaves, and L is the optical delay length corresponding to T .
• Step 1: Converting the optical wavelengths to frequency domain from the above specification. This can be achieved by using f = c / neλ with ne as the effective refractive index of
the guided medium (n ≈ 1.448 for optical silica fiber and 2.57 for Si waveguide in Silicon
on insulator platform) and c is the speed of light in vacuum. This would make the optical
frequencies, f c , f u , f l , f st , correspond to the optical wavelengths at the center, upper, lower,
and stop band of the filter, as specified above.
• Step 2: Choosing a sampling frequency to normalize and compute the analog pre-warped
frequency. The Nyquist sampling rate is 2 f c for the narrow bandpass filter, then choosing
FIGURE 3.1 Bandpass optical filter specification.
73
Bandpass Optical Filters by DSP Techniques
sampling rate: f s = 4 f c to normalize for desired digital filter for each specified frequency
by using θ = 2π f / f s, thus the pre warped analog frequencies θ c ,θ l ,θ u ,θ st can be calculated.
• Step 3: Converting pre-warped bandpass frequencies to equivalent lowpass by using
ωLp =
ω Bp 2 − ω0 2
Bω Bp
(3.4)
where the ωLP and ωBP denote the optical frequency for the lowpass and bandpass filters
respectively. B is the optical 3-dB bandwidth of the transfer function, and ωo is the center
optical frequency of the filter. The bandpass ripple is Rp (in dB), and it thus corresponds to
−10 log10 1 +1ε = Rp. At the specified stopband, if the maximum gain or loss is Rs (dB), we have
2
1/ 2
  Rs  
 10 10 − 1 
Rs
−
1
2


≤ 10 10 ⇒ Cn(ω ) =  
H ( jω ) =

1 + ε 2Cn2 (ω )
ε2






(3.5)
Since ω = ωstLp ⟩1, from (3.1) we have
(
)
⇒ Cn(ω ) = Cosh nCosh −1(ω ) ⇒ n >=
Cosh −1 [Cn(ω ) ]
Cosh −1(ω ) ω =ω
(3.6)
stLp
Eq. (3.5) thus gives the order of the filter, which is then chosen to be upper nearest
integer.
• Step 4: Obtaining the lowpass prototype transfer function corresponding to the defined filter
order n above. This transfer function is then normalized, such that H ( j0 ) = 1. Transforming
this lowpass transfer function back to the bandpass filter by using s = s +Bsω0 where s = jω is
the Laplace variable. We then obtain an 2n order Chebyshev analog bandpass filter transfer
function
2
H Bp ( s) = H Lp ( s) s= s2 +ω02
2
(3.7)
Bs
• Step 5: Applying the bilinear transformation to obtain required digital filter transfer
function
H Bp ( z ) = H Lp ( z ) s= z −1
(3.8)
z +1
where the z variable denotes the z-transform variable of the transfer function. Following
this process, converting the digital response to the optical wave length domain can be
conducted.
• Step 6: From this transfer function, the optical system consists of components that are
implementable in fiber or integrated optic structures, in particular resonators, interferometers, and others [2].
74
Photonic Signal Processing
3.1.3.2 Illustration of a Chebyshev Bandpass Optical Filter
For sharper roll-off attenuation at stop band frequencies, one must require a higher order bandpass
filter or more hardware devices in implementation. In this chapter, an example is given for synthesizing a sixth-order Chebyshev bandpass filter (i.e., a third-order lowpass is designed in the first
step) for an arbitrary center frequency in the useful optical frequency range. In present silica-based
optical fiber communications, the second and third windows at 1300 and 1550 nm, respectively, are
the working spectral regions for optical communications using silica fibers. Supposing that we want
to synthesize an optical Chebyshev bandpass filter with the following specifications:
Centre wave-length:
Lower cut-off wave length:
Upper cut-off wave length:
Lower stop band wavelength:
1310 nm
1308 nm
1312 nm
1302 nm
with a passband ripple of 1.0 dB and a −40 dB stopband.
It is can be shown easily by using Eqs. 3.1 through 3.9 that for a sixth order Chebyshev bandpass
filter the required transfer function is given by:
H BP ( z −1 ) =
−6.792372e − 9(1 − z −2 )3
1 + 3.004723z −2 + 3.009475 z −4 + 1.0004752 z −6
(3.9)
The magnitude and phase responses of the synthesized Chebyshev bandpass filter, according to
the above stringent specification, is shown in Figure 3.2a and b, and the poles and zeros positions
are plotted in the z-plane as shown in Figure 3.2c. This function can be decomposed into a sum of
fractions or a multiplier of a number of subsystems, which exhibit only a single root in the numerator or denominator. However, if the roots are complex conjugate then the order of the subsystem
(a)
(b)
(c)
(d)
FIGURE 3.2 Responses of the Chebyshev bandpass filter (a) and (b) magnitude and phase responses as
a function of optical wavelength (c) poles and zeros positions in the z-plane (d) Impulse response at center
wavelength of 1550 nm, passband of 0.4 nm and 20 dB roll of with 1 dB ripple in the passband. (a) and (d) are
assigned clockwise from top to bottom.
75
Bandpass Optical Filters by DSP Techniques
can be quadratic. The partitioning of the transfer function is described in Section 3.1.3.3. In the
Section 3.1.3.3, essential optical components for implementing this transfer function are given.
3.1.3.3 Optical Components for Chebyshev Filters
Chebyshev filters can be implemented by using the single pole single zero resonator (SPSZR), which
is formed by using a [3 × 3] optical directional coupler with a planar cross section and two optical
feedback paths connecting two outputs to two inputs of the coupler. This type of resonator has been
described in detail in another article [3], where we outline very briefly its main characteristics for
the sake of clarity. Further, an all-pole and an all-zero optical circuits (APOC and AZOC), which
are required for implementation of the filters, are described.
3.1.3.3.1 The SPSZR
For a planar [3 × 3] optical directional coupler, whose schematic diagram and its signal-flow graph
are shown in Figure 3.3a and b, respectively, with a direct (or the order of the delay path is zero)
shunt feedback from output port 3 to input port 3, the output-input transfer function is given as:
1
1
1
j φ +φ
jφ 1k −1
jφ 2 k
+
t1k t2 k e ( 1k 2 k ) z −1
E1( d ) − 2 t1k e z − 2 t2 k e
2
=
1
1
j
j
E1(0)
(1 −
t2 k e φ 2 k )(1 −
t1k e φ 1k z −1 )
2
2
(3.10)
where E1( d ) and E1(0) are the optical fields of the lightwaves at the output and input ports, respectively. The t1k and t2 k are the intensity transmission coefficients of paths 1 and 2, respectively. The
φ1k ,φ2 k are the incorporated optical phase modulation in corresponding paths. The coefficients
xij ( i , j = 1, 2, 3) in Figure 3.3b are the coupling coefficient of the [3 × 3] coupler matrix. It is assumed
that the [3 × 3] coupler has a planar cross-section [3] with a coupling length d and a factor kd = π2 .
It can be easily seen that Eq. 3.10 has only one pole and one zero. Therefore, the pole and zero
can be independently adjusted by tuning the coefficients tik and _ φik , that is, optical attenuators/
amplifiers or phase modulators placed in-line in the feedback paths. In designing optical Chebyshev
filters, the transfer function (3.10) can provide an arbitrary pair of pole-zero denoted by ( a, b ) given
by the roots of its numerator and denominator as:
a=
1
t1k e jφ1k
2
or
t1k e jφ1k = 2a
(3.11)
Alternatively,
t2 k e jφ2 k = −
ab −1
0.5 − ab −1
(3.12)
with this set of chosen parameters, the transfer function in (3.10) becomes
ab −1
(1 − bz −1 )
−1
E1( d )
= 1 − 2ab−1
ab
E1(0)
(1 −
)(1 − az −1 )
1 − 2ab −1
(3.13)
Again, Eq. 3.13 clearly demonstrates that the resonator would exhibit only one pole and one zero
which can be independently adjusted with each other. The gain of the transfer function is dependent
on these values of the pole and zero. However, this amplitude gain or loss can be compensated by
an in-line optical amplifier.
76
Photonic Signal Processing
FIGURE 3.3 The 3 × 3 optical coupler and optical feedback paths as the SPSZR (a) schematic diagram and
(b) Graphical signal-flow representation.
The circuit of Figure 3.3a can be observed to exhibit only one pole due to the fact that there is
only one loop with a single delay line in the graph. According to Mason’s rule the number of poles is
the roots of the graph determinant. The graph determinant order is the order of delay of the optical
loop. Thus, there is only one delay in the loop and there must be only one pole.
The number of zeros of the resonator depends on the number of non-touching loops of the optical circuit. In this case, there are two loops in this resonator and they are non-touching; however,
only one unit delay in one loop, thus, ensuring that the order of the numerator is one. It is therefore
concluded that this identification of the double feedback optical resonator leads to implementation
of an optical transfer function having a pole and zero pair. We can thus name this type of optical
resonance circuit as the single-pole single zero resonator (SPSZR).
Since the SPSZR is the core component for designing Chebyshev optical filters, it is necessary to
closely examine the feasibility of a direct optical feedback from the output to the input of the [3 × 3]
optical coupler. This type of delay is termed the delay-free feedback and, thus, a delay-free loop is
77
Bandpass Optical Filters by DSP Techniques
formed at the upper part of Figure 3.3a and b [8]. It is stated in Reference [8] that the necessary and
sufficient condition so the signal-flow graph of the structure can be computable for a digital filter
is that there is no delay-free loop. We must make it very clear here that the direct connecting shunt
feedback path from the output port 3 to the input port 3 is extremely smaller than the delay of the
other loop. Furthermore, the direct connection loop is not operating under resonance at the operating wavelength. Thus, it is reasonable to assume that this loop does not have the same meaning as
the delay-free loop defined in Reference [8]. This ensures that our derivation for the transfer function of Eq. 3.10 is valid. In practice, the delay path of the lower loop is much greater than that of the
direct loop and that of the coupling length of the 3 × 3 directional coupler.
3.1.3.3.2 The APOC
Besides the SPSZR optical circuit described above, there must be optical circuits or components
that exhibit pole (or higher order multiple poles) characteristics so that it could form a set of optical
components with the SPSZR to simulate the filter structure. The APOC is in fact an optical resonator using two 2 × 2 optical couplers with an optical feedback from the output of the second coupler
to the input of the first coupler [2]. The schematic diagram of the APOC is given in Figure 3.4. The
transmission coefficients are denoted as tip ( i = 1, 2 ) with p and z denoting the all-pole or all-zero,
and the phases of each optical paths as γ.
The transfer function of the APOC with a first-order delay in the feedback path can be obtained
by using the graphical method given in Chapter 2 as:
H ap (z −1 ) =
(1 − k1 )(1 − k2 )t1z e jϕ1
E7
=
E1 1 + t1zt2z k1k2 e j (ϕ21 +ϕ11 ) z −1
(3.14)
Thus the transfer function has a zero at origin and a pole at
z = − t1zt2 z k1k2 e ( 1
j φ +φ 2 )
(3.15)
This APOC is a quasi-all-pole optical circuit, that is, the zero at the origin can be cancelled by
another optical circuit, which would have a finite zero and a pole at the origin such as the AZOC to
be considered next.
FIGURE 3.4 Optical resonance loop to obtain a quasi all-pole optical circuit (APOC).
78
Photonic Signal Processing
3.1.3.3.3 The AZOC
The AZOC is in fact an optical interferometer, which has been studied in detail in [1,2]. The schematic diagram of the AZOC is shown in Figure 3.6. The transmission coefficients are denoted as tip
or tiz (i = 1,2) with the subscript z denoting the all-zero type, and the phases of each optical paths
as γ ,φ . The transfer functions of the AZOC and its zero position are given in Eqs. 3.16 and 3.17,
respectively
H az ( z −1 ) =
E7 −1
( z ) = (1 − k1 )(1 − k2 )t1 e jφ1 − k1k2t2 e jφ2 z −1
E1
(3.16)
which has one pole at the origin, p = 0 and one zero at
z=
k1k2t2
e j (φ1 −φ2 )
(1 − k1 )(1 − k2 )t1
(3.17)
Although the transfer function (3.16) of the AZOC contains a pole, it is at the origin in the z-plane.
The positions of the zero can be changed to suit the needs for the design of optical systems. It is,
thus, a quasi-all-zero optical circuit.
It is interesting to note that if an APOC and an AZOC of the same order are cascaded then the
overall transfer function exhibit a numerator and denominator of the same pole and zero order
because the poles and zeros at the origin cancel each other. This is considered for implementing the
Chebyshev filters in the next section.
3.1.3.4 Realization of the Chebyshev Optical Bandpass Filters
In realizing the optical filters of Chebyshev types, there are two structures that can perform the
same filtering functions, namely, the cascaded and the parallel types. The differences between these
two types are that one is in tandem and one in parallel combination of each modular optical block.
Ideally, each modular optical block must have a single pole and single zero in its transfer function [8]. Two designs for the Chebyshev filters are considered in this section. One uses a combination of the SPSZRs and the APOC, called Chebyshev Optical Filter-type 1 (COF1), and the other
combining the APOCs and AZOCs, called Chebyshev Optical Filter type 2 (COF2).
3.1.3.5 The COF1
3.1.3.5.1 Cascaded form Chebyshev Filters
From the transfer function obtained above, the design system should provide the following system
of poles and zeros:
3.1.3.5.1.1 Five Zeros at
z1 = −1.00567002 + j 0.0000,
z2 = −0.99716505 + j 0.0048946,
z3 = −0.99761605 − j 0.0048946,
z4 = 1.00339252 + j 0.0000,
z5 = 0.99830373 + j 0.00294813,
z6 = 0.99830373 − j 0.00294813
and
79
Bandpass Optical Filters by DSP Techniques
3.1.3.5.1.2 6 Poles at
p1 = −0.00000367 + j1.00118581,
p2 = −0.00000367 − j1.00118581,
p3 = 0.00231434 + j1.00059005,
p4 = 0.00231434 − j1.00059005,
p5 = −0.00232166 + j1.00059003,
p6 = −0.00232166 − j1.00059003
These poles are quite close to the unit circle, thus the system is marginally stable. However, there
are also an equal number of zeros on the unit circle and clustered around the poles. This would
generate a stable system as we can observe from the optical response in Figure 3.2a. The poles and
zeros of the systems are plotted in the z-plane, as shown in Figure 3.2c. The phase of the systems
shown in Figure 3.2b indicate a quasi-linear phase inside the optical passband. The poles and zeros
are complex pole pairs.
The filter with the above system of poles and zeros can be implemented by cascading six SPSZR
with the chosen parameters shown in Table 3.1.
The filter system thus has the following transfer function:
H BP ( z −1 ) =
0.125(1 − z −2 )3
1 + 3.004723z + 3.009475 z −4 + 1.004752 z −6
(3.18)
−2
3.1.3.6 Parallel Realization
For parallel realization the transfer function of the Chebyshev bandpass filter (3.18) can be
expressed as
H BP ( z −1 ) = −6.7923e − 9 +
8.69874e − 9(1 − 272.8e3z −1 )
(1 − p1z −1 )(1 − p2 z −1 )
−3.00551e − 4(1 + 3.9499 z −1 ) 3.00551e − 4(1 + 3.9477 z −1 )
+
+
(1 − p3 z −1 )(1 − p4 z −1 )
(1 − p5 z −1 )(1 − p6 z −1 )
(3.19)
The system can be implemented by a parallel realization as shown in Figure 3.5. The sub-systems
H1, H 2 , H 3 and H 4 are implemented as followed:
TABLE 3.1
Chosen Parameters for Chebyshev Bandpass Optical Filter
SPSZR1
SPSZR2
SPSZR3
SPSZR4
SPSZR5
SPSZR6
t11 = 4.009492
t12 = 4.009492
t13 = 4.004721
t14 = 4.004721
t15 = 4.004743
t16 = 4.004743
t21 = 0.800381366
t22 = 0.800381366
t23 = 0.798711128
t24 = 0.798711128
t25 = 0.801677094
t26 = 0.801677094
ϕ11 = 1.570799992
ϕ12 = −1.570799992
ϕ13 = 1.568483356
ϕ14 = −1.568483356
ϕ14 = 1.573116614
ϕ14 = −1.573116614
ϕ21 = 0.463174471
ϕ22 = 0.463174471
ϕ23 = 0.462948918
ϕ24 = 0.462948918
ϕ25 = 0.463873612
ϕ26 = 0.463873612
p = p1
p = p2
p = p3
p = p4
p = p5
p = p6
z = −1
z=1
z = −1
z=1
z = −1
z=1
80
Photonic Signal Processing
FIGURE 3.5
Hardware implementation diagram of Chebyshev filter.
3.1.3.6.1 Sub-system H1
The sub-system H1 is simply implemented by cascading one optical device with gain 6.7923e-9,
and an optical phase modulator with a phase of π. H1 contributes to the system gain factor
H1 = −6.7923e-9.
3.1.3.6.2 Sub-system H2
This sub-system H 2 should provide a system of poles and zeros as follows: 2 poles at
=
p p=
p2
1; p
and 2 zeros z1 = 272.8e +3 and z2 = 0. This sub-system can be implemented by cascading a SPSZR
and an APOC. The chosen parameters for these elements are shown in Table 3.2.
The two optical components in Table 3.2 are cascaded with an optical device with gain
0.0122456e-06 and an optical phase modular with phase −0.7848 rads.
3.1.3.6.3 Sub-system H3
This sub-system H3 should provide a system of poles and zeros as follows: 2 poles at p = p3 and
p = p4 , and 2 zeros at z = 3.9499 and z = 0. This sub-system can be implemented by cascading a
SPSZR and an APOC. The chosen parameters for these elements are shown in Table 3.3.
The two optical devices above are cascaded with an optical device with gain 1.254e-04 and an
optical phase modular with phase −0.37841 rads.
3.1.3.6.4 Sub-system H 4
This sub-system H 4 should provide a system of poles and zeros as follows: 2 poles at p = p5
and p = p6, and 2 zeros at z = −3.9477 and z = 0. This sub-system can be implemented by cascading
TABLE 3.2
Chosen Parameters for Subsystem H2
SPSZR1
t11 = 4.009492
t21 = 0.0135526
φ 11 = 1.570799999
φ 21 = 4.712404286
p = p1
z = 272.8e03
APOC1
k=
k=
0.5
11
21
t11
= t21
= 2.00237
φ 11 = 0
φ 21 = −1.57079999
p = p2
z=0
ϕ12 = 1.568483356
ϕ12 = 0
ϕ22 = 1.099537311
ϕ22 = −1.568483356
p = p3
p = p4
z = −3.9499
z=0
TABLE 3.3
Chosen Parameters for Subsystem H3
SPSZR2
APOC2
t12 = 4.004721
k12 = k22 = 0.5
t22 = 0.2009535
t12 = t22 = 2.00118
81
Bandpass Optical Filters by DSP Techniques
TABLE 3.4
Chosen Parameters for Subsystem H4
SPSZR3
t13 = 4.004743
t23 = 0.1543989
φ 13 = 1.57311662
φ 23 = 1.205039363
p = p5
z = −3.9477
APOC3
k=
k=
0.5
13
23
t13
= t23
= 2.0005926
φ 13 = 0
φ 23 = −1.57311662
p = p6
z=0
a SPSZR and an AZOC as described in Reference [1]. The chosen parameters for these elements
are shown in Table 3.4.
The two elements above are cascaded with an optical device with gain 1.152e-04 and an optical
phase modular with a phase −0.379212.
3.1.3.7 The COF2
3.1.3.7.1 Cascading Realization
The Chebyshev bandpass transfer function with the numbers of poles and zeros given in Section
4.1.1 above can be realized in the cascade configuration from the combination of all-poles and allzeros basic elements. The overall system transfer function of Eq. (3.18) can be expressed in the more
general form that separates the factors containing the poles and zeros as:
m
H BP ( z ) =
∏ H ( z )H ( z )
azi
−1
api
−1
(3.20)
i =1
where each pole and each zero are realized from one basic element of the AZOC or APOC. The
functions H azi ( z −1 ) and H api ( z −1 ) are the first order function of z−1 numerator and denominator. The
schematic diagram showing the cascade realization for this transfer function is shown in Figure 3.6.
FIGURE 3.6 Schematic diagram showing Tandem all-pole and all-zero subsystems using integrated photonic circuits.
82
Photonic Signal Processing
TABLE 3.5
Chosen System Parameters for Cascade Realization
Subsystem
Coupling
Coefficient
Transmission
Coefficient
Delay Order
Poles or Zeros
H az1
b11 = b21 = 0.5
t11 = t21 = 1
Φ11 = 0
Φ21 = 0
2
z = ±1
H az2
b12 = b22 = 0.5
t12 = t22 = 1
Φ12 = 0
Φ22 = 0
2
z = ±1
H az3
b13 = b21 = 0.5
t13 = t23 = 1
Φ13 = 0
Φ23 = 0
2
z = ±1
H ap1
a11 = a21 = 0.5
Φ11 = 0
Φ21 = −1.5684760
1
p1
H ap2
a12 = a22 = 0.5
Φ12 = 0
Φ22 = 1.56847603
1
p2
H ap3
a13 = a23 = 0.5
Φ13 = 0
Φ23 = −1.5731092
1
p3
H ap4
a14 = a24 = 0.5
Φ14 = 0
Φ24 = 1.57310929
1
p4
H ap5
a15 = a25 = 0.5
Φ15 = 0
Φ25 = −1.5707926
1
p5
H ap6
a16 = a26 = 0.5
t11 = 1;
t21 = 4.004743245
t12 = 1
t22 = 4.004743245
t13 = 1;
t23 = 4.004743245
t14 = 1;
t24 = 4.004743245
t15 = 1;
t25 = 4.009492105
t16 = 1;
t26 = 4.009492105
Φ16 = 0
Φ26 = 1.57079266
1
p6
Phase Shift Modulator
It is noted that if the poles or zeros appear in conjugate pairs of either real or imaginary, then
the numbers of subsystems can be reduced by using more delay coefficients (d ≥ 2) for higher order
subsystem realization.
From Eqs. 3.13 and 3.15, we can obtain the required parameters with some arbitrary chosen
values for each AZOC and APOC subsystems in the cascaded configuration in Figure 3.6. They
are showed in Table 3.4 with parameters X pq that indicate p couplers and q subsystems. It is noted
that the accuracy of parameters in the table is necessary and should not be rounded off. Additional
amplifier gain may also be required to achieve a unity gain response (Table 3.5).
3.1.3.7.2 Parallel Realization
Using the transfer function for the Chebyshev bandpass filter given by (3.19), the system can be
implemented by a parallel realization as shown in Figure 3.5. This structures for the sub-systems
H1 to H 4 are given as follows:
3.1.3.7.2.1 Sub-system H1 Sub-system H1 is simply implemented by cascading one optical
device with gain 6.7923e-9, and an optical phase modulator with phase biased at π. H1 contributes
to the system H1 = −6.7923e−9.
3.1.3.7.2.2 Sub-system H2 This sub-system H 2 should provide a system of poles and zeros of
2 poles at p = p1 and p = p2 and 2 zeros at z = 272.8e03 and z = 0. This sub-system can be implemented by cascading two APOCs and one AZOC with the chosen parameters shown in the Table 3.6.
An optical device with gain 1.5011e+6 is required to cascade with these optical components.
3.1.3.7.2.3 Sub-system H3 This sub-system H 3 should provide a system of poles and zeros as
follows: 2 poles at p = p3 and p = p4, and 2 zeros at z = −3.9499 and z = 0. This sub-system can
be implemented by cascading two APOCs and one AZOC with the chosen parameters shown in
Table 3.7. These elements are cascaded with an optical device with gain 1.7882e-3 and an optical
phase modulator with phase π.
83
Bandpass Optical Filters by DSP Techniques
TABLE 3.6
Chosen Parameters for Subsytem H2
H az11
b11 = b21 = 0.9996335
t21 = 1000; t11 = 0.1
ϕ11 = 0
ϕ21 = 0
z = 272.8e3
H ap11
a11 = a21 = 0.5
t11 = t21 = 2.00237
ϕ11 = 0
ϕ21 = 1.57079999
p = p1
H ap21
a12 = a22 = 0.5
t12 = t22 = 2.00237
ϕ12 = 0
ϕ22 = −1.57079999
p = p2
ϕ21 = 0
z = −3.9499
TABLE 3.7
Chosen Parameters for Subsytem H 3
H az12
b11 = b21 = 0.66385
t21 = 4; t11 = 1
ϕ11 = π
H ap12
a11 = a21 = 0.5
t11 = t21 = 2.00237
ϕ11 = 0
ϕ21 = 1.56848336
p = p3
H ap22
a12 = a22 = 0.5
t12 = t22 = 2.00118
ϕ12 = 0
ϕ22 = −1.56848336
p = p4
TABLE 3.8
Chosen Parameters for Subsystem H4
H az13
b11 = b21 = 0.663735
t21 = 4; t11 = 1
ϕ11 = π
ϕ21 = 0
z = −3.9477
H ap13
a11 = a21 = 0.5
t11 = t21 = 2.0005926
ϕ11 = 0
ϕ21 = 1.57311662
p = p5
H ap23
a12 = a22 = 0.5
t12 = t22 = 2.0005926
ϕ12 = 0
ϕ22 = −1.57311662
p = p6
3.1.3.7.2.4 Sub-system H 4 Similarly, the sub-system H 4 should provide a system of poles and
zeros as follows: 2 poles at p = p5 and p = p6, 2 zeros at z = −3.9477 and z = 0. Again this subsystem
can be implemented by cascading 2 all-pole and 1 all-zero subsystems with the chosen parameters
shown in Table 3.8. These elements are to be cascaded with an optical device with gain 1.787841e-3.
3.1.3.8 Discussions
Although the use of the 3 × 3 optical directional coupler to form the SPSZR would reduce the
required number of the couplers for structuring the filters type COF1, the direct feedback of the
optical path delay restrict the operation frequency of the filter far below the resonance frequency
range of the delay-free loop. On the other hand, the COF2 employs the all-poles and all-zeros the
optical circuits allowing for better stability, as well as a much wider range of operating frequency or
much narrow range of the optical bandpass filters. These filters can be implemented in optical fiber
or integrated optical configuration. A much better technology would be the use of in-line fiber gratings in forward or reflection modes. We are currently implementing a number of optical filters, such
the Butterworth or Chebyshev types, using UV written fiber gratings or photorefractive gratings in
lithium niobate diffused optical waveguides.
3.1.4
cOncluDing remarks
We have demonstrated the synthesizing process for a narrow Chebyshev bandpass filter according
to a given set of specifications. In practice, the hardware implementations of these kind of filters
should be very accurate due to the effects of operational parameters of optical components and the
phase modulators used in the network. Because the filters have a very narrow bandpass centered at a
very high frequency region, they are highly sensitive to minute fluctuation of these optical elements.
84
Photonic Signal Processing
Care should be considered with these filters implementations. On the other hand, this filter can be
used as a sensitive optical sensor.
The employment of the SPSZRs as well as the two quasi all-pole and all-zero optical circuits has
been proven to be compact for the implementation of the filters and the reduction of the required
number of the optical couplers. The quasi-all-pole and all-zero optical circuits are also used and
integrated into the optical systems for signal processing. We believe that these optical components
described in this chapter would find many applications in optical communications and signal processing networks.
However, it is believed that the equalization of optical signals in high-speed data transmission
can be implemented using fiber-optic resonators with some specific configuration to be feasible in
the near future [9].
3.2
TUNABLE OPTICAL BANDPASS WAVEGUIDE FILTERS
This section presents a highly versatile synthesis method for the design of a variety of tunable optical waveguide filters with independently variable bandwidths and tunable center frequencies and
arbitrary infinite impulse response (IIR) characteristics. The synthesized Mth-order tunable optical
filter consists of the concatenation of M all-pole filters (APFs) with M all-zero filters (AZFs). The
bandwidth and center frequency of the designed tunable optical filter can be independently tuned
by applying electric power to thin-film heaters loaded on the waveguides of the both APFs and
AZFs. One unique advantage of the proposed synthesis technique is that the poles and zeros of the
filter can be adjusted independently of each other to enable the design of tunable optical filters with
arbitrary IIR characteristics. By means of computer simulation, the effectiveness of the synthesis
method is demonstrated with the design of the second-order Butterworth bandpass and bandstop
tunable optical filters with variable bandwidths and tunable center frequencies. To study the effects
of fabrication tolerances on the filter performances, the maximum allowable deviations of the filter
parameters from their designed values are also determined. The proposed synthesis method is general and flexible enough to enable the design of a variety of tunable optical filters with arbitrary IIR
characteristics, which include the Chebyshev and elliptic filter types.
3.2.1
intrODuctOry remarks
Optical filters with certain spectral characteristics are important devices in a wide variety of applications. For example, in wavelength division multiplexing (WDM) networks, optical filters have
been used as optical demultiplexers, optical add-drop multiplexers, optical dispersion compensators, and optical gain equalizers. Optical filters are normally designed using complex electromagnetic models, where the fields are determined in the frequency or time domain in a cumbersome
manner. As a simpler alternative to filter design, we have previously proposed the use of digital
signal processing technique for the design or synthesis of fixed or non-tunable optical filters using
interferometers and ring resonators as the basic building blocks [10]. Optical filter design based on
the digital signal processing techniques provide unique advantages, such as systematic and flexible
approach, scalable to larger and more complex filters, and great insight into the properties of the
filters. This is because the signal processing techniques that have been well-established over the last
several decades can provide a readily available mathematical framework for the design of complex
optical filters. In fact, in recent years, there has been an increasing number of researchers adopting
the digital signal processing techniques for the design and analysis of optical filters, for example,
see references [11–13]. In analog to digital filters [5,14], there are two types of optical filter architectures, namely, finite impulse response (FIR) and infinite impulse response (IIR) [10]. FIR filters
have only feed-forward paths (i.e., no loops) and are also known as transversal, tapped delay-line,
and non-recursive filters. IIR filters have feedback paths (or loops) and are also known as recirculating delay-line and recursive filters. It is well-known in the field of digital signal processing that the
85
Bandpass Optical Filters by DSP Techniques
IIR filters, having the same filter order (or complexity) as that of the FIR filters, can provide flatter
passband, sharper roll-off, and greater stopband rejection than the FIR filters due to the feedback
effect [14]. From a practical implementation point of view, it is therefore more advantageous to
design (or synthesize) optical filters based on the IIR topology than the FIR topology. However,
the IIR filters having a rational transfer function are generally more difficult to design than the FIR
filters that only have a polynomial transfer function. Although we have previously employed the
digital signal processing technique to design the non-tunable optical filters with arbitrary IIR characteristics [10], the capability of this technique to enable both the bandwidth and center frequency
of the filter to be independently tuned has not yet been demonstrated.
Tunable optical filters are mostly based on fiber Fabry-Perot (FP) interferometer [15], waveguide
ring resonator [16], and fiber grating [17]. Tunable optical filters are also important in a wide variety
of applications. In a frequency-division multiple access (FDMA) network, a tunable optical filter is
required as an optical demultiplexer at each receiver to select one or more desired frequency channels (or users) at any optical frequency [16]. In a reconfigurable WDM network, tunable optical
filters are required to provide dynamic selection of the wavelength channels [18]. In an optical sensor system, a tunable optical filter is needed to provide interrogation of several fiber grating sensors
on a single strain of optical fiber [19]. In a tunable fiber laser system, a tunable optical filter can
be used to select the resonance wavelength in the laser cavity to extract the desired output lasing
wavelength [20]. In a soliton transmission system, a tunable optical filter can be used as a slidingfrequency guiding filter to reduce the amplified noise-induced timing jitter [21]. A tunable optical
filter can also be used for spectral filtering in the generation of ultrashort pulses [13,22]. It is noted
that, in these tunable optical filters, only the center frequencies (but not the bandwidths) of the filters
can be tuned. That is, both the bandwidths and center frequencies of these tunable optical filters
cannot be simultaneously tuned. There was no previous report of a synthesis method that can be
used for the design of a variety of tunable optical filters with variable bandwidths and tunable center
frequencies and arbitrary IIR characteristics.
In this section, an effective and versatile synthesis method based on the digital signal processing technique is proposed for the design of tunable optical waveguide filters with variable bandwidths and tunable center frequencies and arbitrary IIR characteristics. Section 3.2.2
describes the transfer function of an IIR digital filter to be synthesized using optical waveguide
components. Section 3.2.3 presents the basic building blocks, namely, APF and the AZF of
the tunable optical filter. Section 3.2.4 describes the proposed synthesis method of the tunable optical filters using the digital signal processing technique, and the design of second-order
Butterworth bandpass and bandstop tunable optical filters. Concluding remarks are given in
Sections 3.2.5 and 3.2.6.
3.2.2
transFer FunctiOn OF iir Digital Filters tO be synthesizeD
For clarity, variables with a cap (e.g., Hˆ ( z )) are associated with digital filters while the corresponding variables without a cap (e.g., H ( z )) are associated with optical filters. The transfer function of
an Mth-order IIR digital filter to be synthesized can be expressed in a rational form as
Hˆ ( z ) = Aˆ
M
( z − zˆ k )
( z −zˆ1 ) ( z −zˆ 2 )
( z − zˆM )
∏ ( z − pˆ ) = Aˆ ( z − pˆ ) ( z − pˆ ) ( z − pˆ )
k =1
k
1
2
(3.21)
M
where A is a constant and z is the z-transform parameter [5]. Furthermore, pk and zk are the kth pole
and kth zero in the z-plane and they can be expressed in phasor forms as
(
)
pk = pk e jarg( pk ), 0 ≤ pk < 1
(3.22)
86
Photonic Signal Processing
zk = zk e jarg(z )
k
(3.23)
where j = −1 and arg denotes the argument of a complex number. The filter stability requires all
the poles to be located inside the unit circle as described by the condition given in Eq. 3.22. It can
be seen from Eq. 3.21 that the Mth-order IIR digital filter simply consists of the concatenation of
M all-pole filters (with each filter transfer function given by 1 ( z − pk )) with M all-zero filters (with
each filter transfer function given by ( z − z k )). Hence, an Mth-order IIR optical filter with a transfer
function in the form of Eq. 3.21 can be synthesized from its digital counterpart (i.e., the Mth-order
IIR digital filter). This is presented in Section 3.2.4.
3.2.3
basic builDing blOcks OF tunable Optical Filters
This section describes the basic building blocks, namely, the first-order all-pole optical filter
(FOAPOF) and the first-order all-zero optical filter (FOAZOF) of the tunable optical filter.
3.2.3.1 Tunable Coupler
In this section, we describe the characteristics of a tunable coupler (TC), which is a fundamental
component in the APF as described in Section 3.2.3.2. A TC is also one of the fundamental components in many integrated optical circuits because it can provide tuning or switching functions. The
coupling coefficient of the TC can be varied from zero to unity, a feature which is not available with
a directional coupler (DC) with a fixed coupling coefficient, and can therefore provide great flexibility in the design of integrated optical circuits. The kth-stage TC is shown schematically on the left
side of Figure 3.7. It is a symmetrical Mach−Zehnder interferometer, which consists of two identical
directional couplers (DCs) (with each having an intensity cross-coupling coefficient of dk ) interconnected by two waveguide arms of equal length, and a phase shifter (PS) (with a phase shift of ϕk )
loaded on one of the arms. The PS, which is used to tune the TC’s coupling coefficient to a desired
value as shown below, is a thin-film heater loaded on the waveguide and utilizes the thermo-optic
effect to change the phase of the optical carrier. When an electric voltage is applied to the thin-film
heater, the optical path length of the heated waveguide would change because of the temperature
dependence of the refractive index. For instance, a change in the optical path length of the heated
waveguide by 1.55 µm will correspond to a change in the phase of a 1.55 µm optical carrier by 2π .
The TC is stable against temperature variation because it is the temperature difference between
FIGURE 3.7 Schematic diagram of the kth-stage all-pole filter (APF). The left side of the figure shows the
schematic of the tunable coupler (TC).
87
Bandpass Optical Filters by DSP Techniques
the two waveguide arms, not the absolute temperature of each arm, that is important for tuning or
switching operation. The transfer matrix of the kth TC (in the electric-field amplitude domain) can
be shown to be given by
 1 − ak e jθ13,k
 E3 
−α w L24 ,k − jωT24 ,k
=
e
e

E 
 ak e jθ14 ,k
 4
ak e jθ23,k   E1 
  ,
1 − ak e jθ24 ,k   E2 
(3.24)
where:
ak = 2dk (1 − dk )(1 + cos ϕk ),
(3.25)
0 ≤ ak ≤ 4dk (1 − dk ) or 0 ≤ ak ≤ 1 for dk = 0.5,
(3.26)


ak
− 1 or ϕk = cos −1(2ak − 1) for dk = 0.5, (0 ≤ ϕk ≤ π )
ϕk = cos −1 
 2dk (1 − dk ) 
(3.27)


sin ϕk
θ13,k = tan −1 
 , ( − π 2 ≤ θ13,k ≤ π 2 ) ,
ϕ
−
d
1
−
d
cos
(
)
k
k
k 

(3.28)
 1 + cos ϕk 
θ14,k = θ 23,k = − tan −1 
 , ( − π 2 ≤ θ14,k ,θ 23,k ≤ π 2 ) ,
 sin ϕk 
(3.29)


sin ϕk
θ 24,k = tan −1 
 , ( − π 2 ≤ θ 24,k ≤ π 2 ) ,
ϕ
−
1
−
d
d
cos
(
)
k
k
k 

(3.30)
The variables in Eqs. 3.4 through 3.10 are defined as follows. ( E1, E2 ) and ( E3 , E4 ) are the electricfield amplitudes at the input ports (1, 2) and output ports (3, 4) of the TC, respectively. e −α w L24,k is the
propagation loss of the TC, where α w is the amplitude waveguide loss and L24,k (whose corresponding time delay is T24,k ) is the waveguide length of the TC from the input port 2 to the output port 4.
It is valid to assume that L=
L=
L23,k = L24,k (and hence T=
T=
T23,k = T24,k ) are due to
13,k
14 ,k
13,k
14 ,k
the small dimensions of the waveguide lengths. e − jωT24,k is the propagation delay of the TC, where ω
is the relative optical frequency (radial) and T=
T=
T23,k = T24,k is the time delay of the TC.
13,k
14 ,k
ak is the intensity cross-coupling coefficient of the TC (i.e., from port 1 to port 4 or from port 2
to port 3). From Eqs. 3.5 and 3.6, the maximum tuning range of the TC (i.e., 0 ≤ ak ≤ 1) can be
achieved by using 3-dB DCs (i.e., dk = 0.5 ) and by tuning the required phase of the phase shifter in
the range of 0 ≤ ϕk ≤ π according to Eq. 3.7. For this reason, dk = 0.5 is chosen for filter synthesis as
described in Section 3.2.4. In Eqs. 3.8 through 3.10, θ13,k , θ14,k = θ 23,k , and θ 24,k are the output phases
of the TC. It is noted that θ13,k = θ14,k = θ 23,k = θ 24,k when dk = 0.5. This implies that the TC is a symmetrical and reciprocal device.
3.2.3.2 All-Pole Filter
Figure 3.7 shows the schematic diagram of the kth-stage all-pole filter (APF). It consists of a waveguide loop interconnected by a TC (as described in Section 3.2.3.1 above) and a DC with an intensity cross-coupling coupling coefficient of ck. A thermo-optic phase shifter (PS) (as described in
Section 3.2.3.1 above) with a phase shift of φk is loaded on the lower section of the loop, and an
amorphous-silicon (a-Si) film is loaded on the upper section of the loop. The TC and DC are used to
provide the required amplitude (or magnitude) of the filter pole, and the PS to provide the required
phase (or angle) of the filter pole. By laser trimming the a-Si film, the stress-induced waveguide
88
Photonic Signal Processing
birefringence can be eliminated [14]. Using the signal-flow graph method [1], the transfer function
of the kth-stage APF (from the input port 1 to the output port 8) is simply given by
H ap,k (ω ) =
forward path gain
1 − loop gain
(3.31)
where the subscript ap denotes all pole. The forward path gain is given by the product of all transmittances associated with a forward path that connects the input port (or node) {1} and the output
port (or node) {8} through nodes {4} and {5}. The loop gain is given by the product of all transmittances associated with a feedback loop that connects nodes {2} → {4} → {5} → {7} → {2}. Using
the TC transfer matrix as defined in Eq. 3.4, the forward path gain is given by
forward path gain ({1} → {4} → {5} → {8})
= e −α w L14 ,k × e − jωT14 ,k × ak e jθ14 ,k  × e −α w L45,k × e − jωT45,k × e jϕk 
(3.32)
× e −α w L58,k × e − jωT58,k × ck e − jπ 2 
where the first term in brackets is the transmittance of path {1} → {4}, the second term in brackets
is the transmittance of path {4} → {5}, and the third term in brackets is the transmittance of path
{5} → {8}. Similarly, the loop gain is given by
loop gain ({2} → {4} → {5} → {7} → {2})
= e −α w L24 ,k × e − jωT24 ,k × 1 − ak e jθ24 ,k  × e −α w L45,k × e − jωT45,k × e jϕk 
(3.33)
× e −α w L57,k × e − jωT57,k × 1 − ck  × e −α w L72,k × e − jωT72,k 
It is useful to define the loop length L, its corresponding loop delay T, and the z-transform parameter z as
L = L24,k + L45,k + L57,k + L72,k ,
(3.34)
T = T24,k + T45,k + T57,k + T72,k ,
(3.35)
z = e jωT .
(3.36)
Inserting Eqs. 3.32 through 3.36 into Eq. 3.21 and using L14,k = L24,k and L58,k = L57,k, the z-transform
transfer function of the kth-stage APF becomes
H ap,k ( z ) =
Aap,k e j (θ23,k +φk −π 2)e − jω (T24 ,k +T45,k +T57,k ) z
,
z − pk
(3.37)
where the amplitude Aap,k and the pole location pk in the z-plane are given by
Aap,k = ak ck e −α w ( L24 ,k + L45,k + L57,k ) ,
(3.38)
pk = (1 − ak )(1 − ck ) e −α w Le j (θ24 ,k +φk ) .
(3.39)
Bandpass Optical Filters by DSP Techniques
89
where e −α w L is the transmission factor of the loop. It is useful to express Eq. 3.39 in the phasor
form as
pk = pk e jarg( pk ) , ( 0 ≤ pk < 1) ,
(3.40)
pk = (1 − ak )(1 − ck ) e −α w L ,
(3.41)
arg( pk ) = θ 24,k + φk .
(3.42)
where:
3.2.3.3 All-Zero Filter
Figure 3.8 shows the schematic diagram of the kth-stage all-zero filter (AZF). It is an asymmetrical Mach−Zehnder interferometer, which consists of one tunable coupler (TC) (as described
in Section 3.1.3.3) and one directional coupler (DC) (with an intensity cross-coupling coefficient
of bk ) interconnected by two waveguide arms with a differential time delay of T. The TC and
DC are used to provide the required amplitude (or magnitude) of the filter zero. The thermooptic phase shifter (PS) (as described in Section 3.1.3.3 above) with a phase shift of Ψ k loaded
on the right arm is used to provide the required phase (or angle) of the filter zero. The a-Si film
(as described in Section 3.2.3.2 above) is used to eliminate any polarization dependence of the
AZF. Note that the TC uses two 3-dB DCs because, according to Eqs. 3.24 and 3.25, this will
allow the TC to have a maximum tuning range of the intensity cross-coupling coefficient, gk
(i.e., 0 ≤ gk ≤ 1), which is given by
gk = (1 + cos Φ k ) 2 ; ( 0 ≤ gk ≤ 1)
FIGURE 3.8 Schematic diagram of the kth-stage all-zero filter (AZF).
(3.43)
90
Photonic Signal Processing
From Eq. 3.43, the phase shift, Φ k , of the phase shifter (PS) loaded on one arm of the TC is given by
Φ k = cos −1(2 gk − 1); ( 0 ≤ Φ k ≤ π )
(3.44)
From Eq. 3.39, the output phase of the TC is given by
 sin Φ k 
ξ k = tan −1 
 , ( − π 2 ≤ ξk ≤ π 2)
 cos Φ k − 1 
(3.45)
Using the signal-flow graph method [8], the transfer function of the kth-stage AZF (from the input
port to the output port) is simply given by
H az ,k (ω ) = e −α w L3,k  × e − jωT3,k  ×  1 − gk e jξk  ×  1 − bk 
+ e −α w L4 ,k  × e − jωT4 ,k  ×  gk e jξk  × e jΨ k  ×  j bk 
(3.46)
where 1− gk e jξk is the direct-coupling coefficient of the TC, gk e jξk is the cross-coupling
coefficient of the TC, az denotes all zero, L3,k is the length of the left waveguide arm from the
input to the output port, and L4,k is the length of the right waveguide arm from the input to
the output port. It is useful to define the differential length L and its corresponding differential
delay T as
L = L4,k − L3,k
(3.47)
T = T4,k − T3,k .
(3.48)
Substituting Eqs. 3.35, 3.36, and 3.47, into Eq. 3.45, the z-transform transfer function of the kth-stage
AZF becomes
H az ,k ( z ) = Aaz ,k e − jωT3,k z −1( z − zk )
(3.49)
where the amplitude and the zero location in the z-plane are given by
Aaz ,k = [(1 − gk )(1 − bk ) ] e −α w L3,k
12
(3.50)
12
 −α w L j (Ψ k +3π 2)

gk bk
e
zk = 
.
 e
g
b
(
1
)(
1
)
−
−
k
k 

(3.51)
It is useful to express Eqs. 3.50 and 3.51 in the phasor form as
zk = zk e jarg( zk )
(3.52)
where
12
 −α w L

gk bk
zk = 
 e
(
1
g
)(
1
b
)
−
−
k
k 

(3.53)
arg( zk ) = Ψ k + 3π 2 .
(3.54)
91
Bandpass Optical Filters by DSP Techniques
3.2.4
tunable Optical Filter
The z-transform transfer function of the Mth-order tunable optical filter, which is the transfer function of the concatenation of M APFs with M AZFs, is given by
 M
  M

H ( z) = 
H ap,k ( z )  × 
H az ,k ( z )  .
 k =1
  k =1

∏
∏
(3.55)
Substituting Eq. 3.36 and 3.38 into Eq. 3.54, the z-transform transfer function of the Mth-order tunable optical filter becomes
H ( z) = e
M
M

Mπ
j  ∑ (θ23,k +ϕk +ξ k )−
−ω ∑ (T24 ,k +T45 ,k +T57 ,k +T3,k ) 
2
k =1
 k =1

 M ( z − zk ) 
× A

 k =1 ( z − pk ) 
∏
(3.56)
where the amplitude A is given by
12
M
A=e
− Mα w L
×
∏[a c (1 − g )(1 − b )] .
k k
k
k
(3.57)
k =1
It should be noted in Eq. 3.56 that e −α w L = e −α w ( L24 ,k + L45,k + L57,k + L3,k ) and L72,k = L3,k have been assumed
for analytical simplicity. It can be seen from Eq. 3.55 that the Mth-order tunable optical filter has the
unique advantage in that its poles and zeros can be adjusted independently of each other, enabling
a particular pole-zero pattern to be easily obtained to allow the design of a variety of filters with
arbitrary IIR characteristics.
3.2.5
synthesis OF tunable Optical Filters
3.2.5.1 Design Equations for the Synthesis of Tunable Optical Filters
For analytical simplicity, the exponential factor in Eq. 3.55, which represents a linear phase term,
is neglected here because it has no effect on the magnitude response of the filter. The synthesis of a
tunable optical filter from the characteristics of a digital filter requires the second factor of Eq. 3.56
to be equal to Eq. 3.21 such that the following equations hold
A = A
(3.58)
pk = pk
(3.59)
zk = zk .
(3.60)
Substituting Eq. 3.22 and 3.40 through 3.42 into Eq. 3.59 results in
2
pk
ak = 1 −
,
(1 − ck )e −2α w L
arg( pk ) − θ 24,k ,

φk = 
arg( pk ) − θ 24,k + 2π ,
pk < 1 − ck e −α w L
arg( pk ) − θ 24,k ≥ 0
arg( pk ) − θ 24,k < 0
(3.61)
(3.62)
92
Photonic Signal Processing
It is clear from the condition (i.e., pk < 1 − ck e −α w L ) given in Eq. 3.61 that the value ck (which can
be fixed at the design stage) of the directional coupler (DC) must be designed to be as small as possible and that the transmission factor of the loop, e −α w L , of the all-pole filter must be designed to
be as large as possible (by having a small waveguide propagation loss α w ) in order to achieve the
maximum allowable value of the digital filter pole, pk , and hence the maximum tuning range of the
filter bandwidth. This can be achieved by varying the coupling coefficient ak of the tunable coupler
(TC) (see Eq. 3.61), which requires tuning of the phase shift φk (see Eq. 3.27) of the TC. It is noted
that, for practical purposes, a full-cycle phase shift of 2π has been added, without affecting the filter
performance, to the second equation of Eq. 3.62 so that φk takes a positive value (i.e., 0 ≤ φk ≤ 2π ),
which can be realized by applying a positive voltage to the thin film heater. Substituting Eqs. 3.23
and 3.52 through 3.54 into Eq. 3.60 results in
gk =
1
; 0 < gk ≤ 1
−2 −2α L
 bk 
w
1+ 
z
e
 k
 1 − bk 

arg( zk ) − 3π 2 ,
Ψk = 
arg( zk ) − 3π 2 + 2π ,
arg( zk ) − 3π 2 ≥ 0
arg( zk ) − 3π 2 < 0
(3.63)
(3.64)
The filter bandwidth can be varied by changing the magnitude of the filter pole(s) pk , and this can
be achieved by varying the coupling coefficient of the TC, ak (see Eq. 3.61 and hence tuning the
phase shift of the phase shifter ϕk (see Eq. 3.27 of the TC. It can be seen from the condition given
in Eq. 3.61 that the maximum value of the filter pole and hence the maximum achievable filter
bandwidth is limited by the transmission factor of the loop e −α w L . To change the center frequency of
the filter without affecting its bandwidth, a phase shift of δ 0 (0 < δ 0 < 2π ) must be added to Eqs. 3.62
and 3.64 (i.e., to the phase shifters of both the APFs and the AZFs) to give
arg( pk ) − θ 24,k + δ 0 ,

φk = 
arg( pk ) − θ 24,k + δ 0 + 2π ,
arg( pk ) − θ 24,k + δ 0 ≥ 0
arg( zk ) − 3π 2 + δ 0 ,

Ψk = 
arg( zk ) − 3π 2 + δ 0 + 2π ,
arg( zk ) − 3π 2 + δ 0 ≥ 0
arg( pk ) − θ 24,k + δ 0 < 0
arg( zk ) − 3π 2 + δ 0 < 0
(3.65)
(3.66)
In summary, the design equations for the synthesis of the tunable optical filter are Eqs. 3.27, 3.30,
3.57, and 3.61 through 3.66.
3.2.5.2
Synthesis of Second-Order Butterworth Bandpass and Bandstop
Tunable Optical Filters
To demonstrate the effectiveness of the design method, this section describes a practical design
example of the second-order (M = 2) Butterworth bandpass and bandstop tunable optical filters
with variable bandwidths and tunable center frequencies characteristics.
3.2.5.3 Designed Parameter Values of the Bandpass and Bandstop Tunable Optical Filters
This section presents a practical design example of the tunable optical filters with variable
bandwidths and tunable center frequencies using the proposed digital filter synthesis method.
The propagation loss of the waveguide is assumed to be typically 0.1 dB cm and this gives
α w = 0.1 dB cm 8.686 = 0.01151 cm −1. As a case study, the loop length L is assumed to be L = 4 cm,
93
Bandpass Optical Filters by DSP Techniques
which results in the loop delay of T = 200 ps assuming the effective refractive index of the waveguide is about 1.5, and hence the free spectral range (FSR) of the filter is FSR
= 1=
T 5 GHz . Using
α w = 0.01151 cm −1, and L = 4 cm, e −α w L = 0.955 . By choosing c = 0.25 and using e −α w L = 0.955 , the
coupling coefficient of the TC, ak , in the APF defined in Eq. 3.41 becomes
2
ak = 1 − 1.462 pk , pk < 0.827
(3.67)
The value ck = 0.25 is chosen here so that the pole value is pk < 0.827, which is adequate to
provide a sufficient tuning range of the filter bandwidth as will be shown later. As described in
Section 3.2.3.1, the maximum tuning range of the coupling coefficient of the TC (i.e., 0 ≤ ak ≤ 1 )
can be achieved by using 3-dB DCs (i.e., dk = 0.5 ) and by tuning the required PS phase in the range
0 ≤ ϕk ≤ π according to Eq. 3.27. Thus for dk = 0.5 , Eq. 3.27 becomes
ϕk = cos −1(2ak − 1) ( 0 ≤ ϕk ≤ π )
(3.68)
Let θ13,k = θ14,k = θ 23,k = θ 24,k = θ k and dk = 0.5 , Eq. 3.10 becomes
 sin ϕk 
θ k = tan −1 
 . ( − π 2 ≤ θk ≤ π 2)
 cos ϕk − 1 
(3.69)
Using e −α w L = 0.955 and putting bk = 0.5 (i.e., the DC is 3 dB in the TC), the coupling coefficient of
the TC, gk , in the AZF defined in Eq. 3.43 becomes
gk =
1
1 + (0.955)2 zk
−2
(
; zk ≥ 0
)
(3.70)
where 0 ≤ gk ≤ 1. Note from Eq. 3.70 that zk ≥ 0 means that the zeros can be located anywhere in the
z-plane, and this unique feature can provide great flexibility in the design. This is because the proposed
method can generally be applied to the design of a variety of tunable filters, such as the Butterworth,
Chebyshev and elliptic filter types. In particular, the Chebyshev and elliptic filters can have zero locations outside the unit circle in the z-plane. The maximum tuning range of the coupling coefficient of
the TC (i.e., 0 ≤ gk ≤ 1 ) in the AZF defined in Eq. 3.70 can be achieved by tuning the phase shift of the
phase shifter (PS) loaded on one arm of the TC in the range of 0 ≤ Φ k ≤ π , which is given as
Φ k = cos −1(2 gk − 1); ( 0 ≤ Φ k ≤ π )
(3.71)
Using M = 2, e −α w L = 0.955, ck = 0.25 and bk = 0.5, Eq. 3.37 becomes
A = 0.114 [ a1a2 (1 − g1 )(1 − g2 ) ] .
12
(3.72)
In summary, Eqs. 3.65 through 3.72 are used for the synthesis of the second-order Butterworth
bandpass and bandstop tunable optical filters. In all frequency responses, the transmission (or
magnitude) responses are plotted over the normalized FSR of ωT (2π ) = f T , where f is the relative optical frequency. The actual FSR is given by FSR = 1 T (which is 5 GHz in this design
example). Note that the transmission responses are periodic with a normalized FSR of f T = 1 or
with an FSR of 1 T .
Table 3.9 shows the computed designed parameter values of the second-order Butterworth bandpass
and bandstop digital filters with variable normalized 3-dB bandwidths (i.e., ∆ωT (2π ) = ∆f T , where
0.1 ≤ ∆f T ≤ 0.9). For a particular bandwidth, both the bandpass and bandstop filters have the same
values of p1 = p2 , arg( p1), arg( p2 ) = −arg( p1 ) and z1 = z2 . Note that the bandpass digital filter has
94
Photonic Signal Processing
TABLE 3.9
Designed Parameter Values of the Second-Order Butterworth Bandpass and Bandstop
Digital Filters with Variable Normalized 3-dB Bandwidths of 0.1 ≤ ∆f T ≤ 0.9
Poles of H ap,k (z ), k = 1,2.
Filter Type
Bandpass
Bandstop
Zeros of H az,k (z ), k = 1,2.
Normalized
Bandwidth ∆f T
A
p 1 = p 2
arg(p 1)
(radian)
arg(p 2 ) = − arg(p 1)
(radian)
z 1 = z 2
arg(z 1) = − arg( z 2 )
(radian)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.0201
0.0675
0.1311
0.2066
0.2929
0.3913
0.5050
0.6389
0.8006
0.8006
0.6389
0.5050
0.3913
0.2929
0.2066
0.1311
0.0675
0.0201
0.8008
0.6425
0.5217
0.4425
0.4142
0.4425
0.5217
0.6425
0.8008
0.8008
0.6425
0.5217
0.4425
0.4142
0.4425
0.5217
0.6425
0.8008
−2.9158
−2.6670
−2.3698
−2.0015
−1.5708
−1.1401
−0.7718
−0.4746
−0.2258
−2.9158
−2.6670
−2.3698
−2.0015
−1.5708
−1.1401
−0.7718
−0.4746
−0.2258
2.9158
2.6670
2.3698
2.0015
1.5708
1.1401
0.7718
0.4746
0.2258
2.9158
2.6670
2.3698
2.0015
1.5708
1.1401
0.7718
0.4746
0.2258
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
π
π
π
π
π
π
π
π
π
Note: For each bandwidth, both the bandpass and bandstop filters have the same poles which occur in complex-conjugate
pairs. The zeros are located exactly on the unit circle.
arg
=
( z1 ) arg
=
( z2 ) 0 while the bandstop digital filter has arg( z1 ) = arg( z2 ) = π . Using Eqs. 3.45 through
3.52, Table 3.10 shows the computed designed parameter values of the corresponding second-order
Butterworth bandpass and bandstop tunable optical filters with variable normalized 3-dB bandwidths
of 0.1 ≤ ∆f T ≤ 0.9 and fixed normalized center frequencies which are chosen to be f T = 0.5 in this
design example (i.e., δ 0 = 0). For a particular bandwidth, both the bandpass and bandstop tunable optical filters have the same parameter values of a1 = a2 , ϕ1 = ϕ2 , θ1 = θ 2 and φ1, φ2 of H ap,k ( z ) for k = 1, 2
and the same parameter values of g1 = g2 and Φ1 = Φ 2 of H az ,k ( z ) for k = 1, 2 . Note that the bandpass
tunable optical filter has Ψ1 = Ψ 2 = π /2 while the bandstop tunable optical filter has Ψ1 = Ψ 2 = 3π / 2.
3.2.5.4 Tuning Parameters of the Synthesized Bandpass and Bandstop
Tunable Optical Filters
For filter implementation, it is important to know the range of values of the tuning parameters
required for various filter bandwidths. Figure 3.9 shows the characteristics of the tuning parameters
of the bandpass and bandstop tunable optical filters with variable normalized 3-dB bandwidths and
fixed normalized center frequencies of f T = 0.5 (i.e., δ 0 = 0). The values of these parameters are
obtained from Table 3.10. Figure 3.9a shows the intensity coupling coefficients of APFs (i.e., a1 = a2 ,
c=
c=
0.25 and d=
d=
0.5 ) and the AZFs (i.e., b=
b=
0.5 and g=
g=
0.523 ) versus the nor1
2
1
2
1
2
1
2
malized filter bandwidth for both the bandpass and bandstop tunable filters. The required values of
c=
0.25 , d=
d=
0.5 , b=
b=
0.5 , and g=
the coupling coefficients of c=
g=
0.523 are fixed
1
2
1
2
1
2
1
2
over the bandwidth range. To tune the filter bandwidth, the coupling coefficients a1 = a2 of the tunable couplers (TCs) of the all-pole filters (APFs) must be varied by tuning the phase shifts ϕ1 = ϕ2 of
0.6021
0.7137
0.7492
38.81 × 10 −3
−3
0.4
0.5
0.0625
21.56 × 10 −3
0.8
3.39 × 10
0.3965
32.74 × 10 −3
0.7
0.9
0.6021
38.81 × 10
0.6
−3
0.7137
−3
40.74 × 10
0.3965
21.56 × 10
32.74 × 10 −3
0.3
0.0625
a1 = a2
0.2
Bandpass
−3
−3
3.39 × 10
0.1
Filter Type
Amplitude A
Normalized
Bandwidth
∆f T
2.6363
1.7793
1.3651
1.1291
1.0491
1.1291
1.3651
1.7790
2.6363
ϕ 1 = ϕ 2 (radian)
−0.2526
−0.6811
−0.8882
−1.0063
−1.0463
−1.0063
−0.8882
−0.6811
−0.2526
θ 1 = θ 2 (radian)
0.0268
0.2065
0.1164
6.1493
5.7586
5.2879
4.8016
4.2973
3.6200
φ 1 (radian)
Parameters of H ap,k (z ), k = 1,2.
0.25 and d=1 = d==
0.5)
(c=1 = c==
2
2
0.4784
1.1557
1.6600
2.1464
2.6171
3.0078
3.2580
3.3481
3.1684
φ 2 (radian)
0.5230
0.5230
0.5230
0.5230
0.5230
0.5230
0.5230
0.5230
0.5230
g1 = g2
1.5248
1.5248
1.5248
1.5248
1.5248
1.5248
1.5248
1.5248
1.5248
Φ 1 = Φ 2 (radian)
π /2
(Continued )
π /2
π /2
π /2
π /2
π /2
π /2
π /2
π /2
Ψ1 = Ψ 2 (radian)
Parameters of H az,k (z ), k = 1,2.
0.5)
(b=1 = b==
2
TABLE 3.10
Designed Parameter Values of the Second-Order Butterworth Bandpass and Bandstop Tunable Optical Filters with Variable Normalized
3-dB Bandwidths of 0.1 ≤ ∆f T ≤ 0.9 and Fixed Normalized Center Frequencies at f T = 0.5 (i.e., δ 0 = 0)
Bandpass Optical Filters by DSP Techniques
95
0.6021
0.7137
0.7492
38.81 × 10 −3
−3
0.4
0.5
2.6363
1.7793
1.3651
1.1291
1.0491
1.1291
1.3651
1.7793
2.6363
ϕ 1 = ϕ 2 (radian)
−0.2526
−0.6811
−0.8882
−1.0063
−1.0463
−1.0063
−0.8882
−0.6811
−0.2526
θ 1 = θ 2 (radian)
0.0268
0.2065
0.1164
6.1493
5.7586
5.2879
4.8016
4.2973
3.6200
φ 1 (radian)
Parameters of H ap,k (z ), k = 1,2.
(c=1 = c==
0.25 and d=1 = d==
0.5)
2
2
0.4784
1.1557
1.6600
2.1464
2.6171
3.0078
3.2580
3.3481
3.1684
φ 2 (radian)
0.5230
0.5230
0.5230
0.5230
0.5230
0.5230
0.5230
0.5230
0.5230
g1 = g2
1.5248
1.5248
1.5248
1.5248
1.5248
1.5248
1.5248
1.5248
1.5248
Φ 1 = Φ 2 (radian)
3π / 2
3π / 2
3π / 2
3π / 2
3π / 2
3π / 2
3π / 2
3π / 2
3π / 2
Ψ1 = Ψ 2 (radian)
Parameters of H az,k (z ), k = 1,2.
(b=1 = b==
0.5)
2
Note: For each bandwidth, both the bandpass and bandstop filters have the same pole values (see Table 3.9) and hence the same parameters of APFs (i.e., the same coupling coefficients of
a1 = a2 of the TCs and the same phase shifts of ϕ 1 = ϕ2 , φ 1 and φ 2), and the same zero values (see Table 3.9) and hence the same parameters of AZFs (i.e., the same coupling coefficients
of g1 = g2 of the TCs and the same phase shifts of Φ1 = Φ 2 ). For each bandwidth, there is a phase difference of π between the zeros of the bandpass and bandstop filters, that is, the
bandpass filters have Ψ1 = Ψ 2 = π / 2 and the bandstop filters to have Ψ1 = Ψ 2 = 3π / 2.
0.0625
21.56 × 10 −3
0.8
3.39 × 10
0.3965
32.74 × 10 −3
0.7
0.9
0.6021
38.81 × 10
0.6
−3
0.7137
−3
40.74 × 10
0.3965
21.56 × 10
32.74 × 10 −3
0.3
0.0625
a1 = a2
0.2
Bandstop
−3
−3
3.39 × 10
0.1
Filter Type
Amplitude A
Normalized
Bandwidth
∆f T
TABLE 3.10 (Continued)
Designed Parameter Values of the Second-Order Butterworth Bandpass and Bandstop Tunable Optical Filters with Variable Normalized
3-dB Bandwidths of 0.1 ≤ ∆f T ≤ 0.9 and Fixed Normalized Center Frequencies at f T = 0.5 (i.e., δ 0 = 0)
96
Photonic Signal Processing
Bandpass Optical Filters by DSP Techniques
97
FIGURE 3.9 Characteristics of the tuning parameters versus the normalized 3-dB bandwidth of the bandpass and bandstop tunable optical filters with variable normalized 3-dB bandwidths and fixed normalized
center frequencies at f T = 0.5 (i.e., δ 0 = 0). (a) Intensity coupling coefficients of the all-pole filters and the allzero filters of both the bandpass and bandstop tunable filters. (b) Optical phase shifts of the all-pole filters and
the all-zero filters of both the bandpass and bandstop tunable filters.
the TCs (see the dotted-dotted curve in Figure 3.9b). Note that Figure 3.9b also shows that the phase
shifts φ1 (dashed curve) and φ2 (dotted-dashed curve) of the APFs of both the bandpass and bandstop
tunable filters must also be tuned in order to tune the filter bandwidth. However, both the bandpass
and bandstop tunable filters require the same phase shifts of Φ1 = Φ 2 = 1.5248 and of the all-zero
filters (AZFs). The bandpass and bandstop tunable filters require the phase shifts Ψ1 = Ψ 2 = π /2
and Ψ1 = Ψ 2 = 3π /2 (see the solid curves) of the AZFs, respectively. The tuning parameters of the
filter can be summarized as follows.
98
Photonic Signal Processing
• For a particular filter bandwidth (i.e., for a particular set of phase shift values of ϕ1 = ϕ2, φ1,
and φ2 of the APFs and Φ1 = Φ 2 of the AZFs), the phase shifts Ψ1 = Ψ 2 of the AZFs determine the transmission response characteristics of the filter. That is, a bandpass filter with
a certain bandwidth can be tuned to a bandstop filter without changing the bandwidth by
tuning Ψ1 = Ψ 2 = π / 2 to Ψ1 = Ψ 2 = 3π / 2 of the AZFs and vice versa.
• For a particular filter characteristics (i.e., bandpass where Ψ1 = Ψ 2 = π / 2 or bandstop
where Ψ1 = Ψ 2 = 3π / 2), fixing the phase shifts Φ1 = Φ 2 = 1.5248 (of the TCs) of the AZFs
and tuning the phase shifts ϕ1 = ϕ2 (of the TCs), and φ1 and φ2 of the APFs will change the
locations of the poles and hence changing the bandwidth of the filter.
3.2.6
synthesis OF banDpass anD banDstOp tunable Optical Filters
with variable banDwiDths anD FixeD center Frequencies
Using the designed parameter values shown in Table 3.10 and Figures 3.9 and 3.10a, b show, respectively, the transmission responses of the bandpass and bandstop tunable optical filters with variable bandwidths (normalized) and fixed center frequencies (normalized) at f T = 0.5 (i.e., δ 0 = 0 in
Eqs. 3.65 and 3.66. The 3-dB bandwidth (normalized) of each filter characteristic (i.e., bandpass or
bandstop) can be varied from ∆f T = 0.2 to ∆f T = 0.4 and to ∆f T = 0.6 by tuning the phase shifters
of the APFs (i.e., ϕ1 = ϕ2, φ1, and φ2 of H ap,1( z ) and H ap,2 ( z ), and Φ1 = Φ 2 = 1.5248 of H az,1( z ) and
H az,2 ( z ), see Figure 3.9b and Table 3.9) and by keeping the phase shifters of the AZFs unchanged
(i.e., Ψ1 = Ψ 2 = π / 2 for bandpass and Ψ1 = Ψ 2 = 3π / 2 for bandstop, see Figure 3.9b and Table 3.10).
Note that the center frequencies (normalized) here are designed to be at f T = 0.5 for both the bandpass and bandstop tunable filters.
Synthesis of Tunable Optical Filters with Fixed Bandwidths
and Tunable Center Frequencies
As a design example, the tunable bandpass filter with a normalized bandwidth of ∆f T = 0.2 and a
normalized center frequency of f T = 0.5 is employed as a reference filter (see the solid-line curve
shown in Figure 3.10a where δ 0 = 0 ), and similarly the tunable bandstop filter with a normalized bandwidth of ∆f T = 0.2 and a normalized center frequency of f T = 0.5 is used as a reference filter (see the solid-line curve shown in Figure 3.10b where δ 0 = 0 ). The center frequencies
of these two types of filters can be tuned, without affecting their bandwidths (i.e., the normalized bandwidths will remain unchanged at ∆f T = 0.2), to within one FSR by adding the same
phase shift values of δ 0 (0 < δ 0 < 2π ) to the phase shifters of both the APFs (i.e., φ1 and φ 2 in
Eq. 3.65) and the AZFs (i.e., Ψ1 and Ψ 2 in Eq. 3.66). In doing so, Figure 3.11a and b show, respectively, the transmission responses of the designed bandpass and bandstop tunable optical filters
with fixed normalized 3-dB bandwidths of ∆f T = 0.2 and variable center frequencies. In both
Figure 3.11a and 5b, the normalized center frequency can be varied to within one normalized
FSR of between 0 and 1 from f T = 0.2 (where δ 0 = −0.6π , see the dashed-line curves) to f T = 0.5
(where δ 0 = 0 , see the solid-line curves) and to f T = 0.8 (where δ 0 = +0.6π , see the dashed-dotted
curves). Note that the designed parameter values here are exactly the same as those shown in
Table 3.10 and Figure 3.9, except that appropriate phase shift values of δ 0 (i.e., δ 0 = −0.6π , 0 and
+0.6π in this design example) have been added to the phase shifters of the APFs (i.e., φ1 and
φ2 in Eq. 3.65) and to the phase shifters of the AZFs (i.e., Ψ1 and Ψ 2 in Eq. 3.66). That is, in
Figure 3.11a, for the bandpass tunable filters with a fixed normalized 3-dB bandwidth of ∆f T = 0.2,
the phase shift values of φ 1, φ 2 , Ψ1, and Ψ 2 shown in Table 3.10 and Figure 3.9 have been changed
to φ 1 = 4.2973 − 0.6π = 2.4123, φ 2 = 3.3481 − 0.6π = 1.4631 and Ψ1 = Ψ 2 = π / 2 − 0.6π = −0.1π ,
respectively; for the new normalized center frequency of fT = 0.2 (see the dashed-line curve), to
φ 1 = 4.2973, φ 2 = 3.3481, and Ψ1 = Ψ 2 = π / 2, respectively; for the new normalized center frequency
3.2.6.1
Bandpass Optical Filters by DSP Techniques
99
FIGURE 3.10 Transmission responses of the designed bandpass and bandstop tunable optical filters with
variable normalized 3-dB bandwidths of ∆f T = 0.2, ∆f T = 0.4 and ∆f T = 0.6, and fixed normalized center
frequencies at f T = 0.5 (i.e., δ 0 = 0). (a) Bandpass responses. (b) Bandstop responses.
of f T = 0.5 (see the solid-line curve), and to φ 1 = 4.2973 + 0.6π = 6.1823, φ 2 = 3.3481 + 0.6π = 5.2331,
and Ψ1 = Ψ 2 = π / 2 + 0.6π = 1.1π , respectively; for the new normalized center frequency of
f T = 0.8 (see the dashed-dotted curve). Similarly, in Figure 3.11b, for the bandstop tunable filters with a fixed normalized 3-dB bandwidth of ∆f T = 0.2, the phase shift values of φ 1, φ 2 , Ψ1,
and Ψ 2 shown in Table 3.10 and Figure 3.9 have been changed to φ 1 = 4.2973 − 0.6π = 2.4123,
φ 2 = 3.3481 − 0.6π = 1.4631, and Ψ1 = Ψ 2 = 3π / 2 − 0.6π = 0.9π , respectively; for the new normalized center frequency of f T = 0.2 (see the dashed-line curve), to φ 1 = 4.2973, φ 2 = 3.3481,
and Ψ1 = Ψ 2 = 3π / 2, respectively; for the new normalized center frequency of fT = 0.5
100
Photonic Signal Processing
FIGURE 3.11 Transmission responses of the designed bandpass and bandstop tunable optical filters with
a fixed normalized 3-dB bandwidth of ∆f T = 0.2 and tunable normalized center frequencies of f T = 0.2
(i.e., δ 0 = −0.6π ), f T = 0.5 (i.e., δ 0 = 0) and f T = 0.8 (i.e., δ 0 = +0.6π ). (a) Bandpass responses. (b) Bandstop
responses.
(see the solid-line curve) and to φ 1 = 4.2973 + 0.6π = 6.1823, φ 2 = 3.3481 + 0.6π = 5.2331, and
Ψ1 = Ψ 2 = 3π / 2 + 0.6π = 2.1π , respectively; for the new normalized center frequency of f T = 0.8
(see the dashed-dotted curve).
3.2.6.2 Fabrication Tolerances of Filter Parameters
It is difficult to fabricate filter parameter values that do not deviate from their designed values. So,
it is important to determine the allowable range of the parameter errors for fabrication purposes.
The essential parameters that are considered to have fabrication errors are the phase shifts of the
Bandpass Optical Filters by DSP Techniques
101
various phase shifters (i.e., ϕ1, ϕ2 , φ1 , φ2 , Φ1, Φ 2, Ψ1, and Ψ 2 ) and the coupling coefficients of the
directional couplers (DCs) (i.e., b1, b2, c1, c2, d1, and d2). It is convenient to consider the coupling
angles rather than the coupling coefficients as parameter errors because the phase shifts are considered as parameter errors. The intensity coupling coefficient of the directional coupler y (where
=
y b=
1, 2) relates to the coupling angle x (where x is the coupling angle of bk , ck , dk ;
k , ck , dk ; k
k = 1, 2 ) by y = sin 2 x . Thus, the change in the intensity coupling coefficient ∆y (i.e., due to fabrication error) relates to the change in the coupling angle ∆x by
∆y = 2 y(1 − y ) ∆x
(3.73)
The phase shift errors of the phase shifters (i.e., ∆ϕ1, ∆ϕ2, ∆φ1, ∆φ2, ∆Φ1, ∆Φ 2, ∆Ψ1, and ∆Ψ 2 )
and the coupling angle errors of the directional couplers (i.e., ∆x ) are considered to be ±0.001π ,
±0.01π , and ±0.1π . Figure 3.12a–c shows the transmission responses of the designed bandpass
tunable optical filter (which has a normalized 3-dB bandwidth of ∆f T = 0.4 and a normalized
center frequency of f T = 0.5 in this example) with the phase shift errors and the coupling angle
errors of ±0.001π , ±0.01π , and ±0.1π , respectively. It can be seen from these figures that a fabrication accuracy of better than ±0.001π is required for the actual transmission response to be very
close to the ideal one. However, Figure 3.12b shows that the transmission response is still sufficiently practical even with the random phase errors of ±0.01π . The same conclusion can also be
made on the bandstop filter (see Figure 3.13a–c), because the stopband in the transmission response
(see Figure 3.13b which corresponds to the random phase errors of ±0.01π ) is greater than 40 dB.
Similar results are also obtained for the designed bandpass and bandstop tunable filters with other
bandwidth values. Thus, the allowable phase shift errors of the phase shifters and the coupling
angle errors of the directional couplers must be less than ±0.01π , which is used as a basis to determine if the designed filters could be fabricated to such an accuracy. This is described as follows.
Using the coupling angle errors of ±0.01π in Eq. 3.53, the allowable errors in the intensity coupling
FIGURE 3.12 Transmission responses of the designed bandpass tunable optical filter with phase shift errors
of the phase shifters and coupling angle errors of the directional couplers. (a) ±0.001π ,
(Continued)
102
Photonic Signal Processing
FIGURE 3.12 (Continued) Transmission responses of the designed bandpass tunable optical filter with
phase shift errors of the phase shifters and coupling angle errors of the directional couplers. (b) ±0.01π , and
(c) ±0.1π . This particular filter design has a normalized 3-dB bandwidth of ∆f T = 0.4 and a normalized center
frequency of f T = 0.5.
Bandpass Optical Filters by DSP Techniques
103
FIGURE 3.13 Transmission responses of the designed bandstop tunable optical filter with phase shift
errors of the phase shifters and coupling angle errors of the directional couplers. (a) ±0.001π , (b) ±0.01π , and
(c) ±0.1π . This particular filter design has a normalized 3-dB bandwidth of ∆f T = 0.4 and a normalized center
frequency of f T = 0.5.
104
Photonic Signal Processing
coefficients are ∆b1 b1 = ∆b2 b2 = ±0.063 (or ± 6.3%), ∆c1 c1 = ∆c2 c2 = ±0.109 (or ± 10.9%), and
∆d1 d1 = ∆d2 d2 = ±0.063 (or ± 6.3%). Thus, the directional couplers must be fabricated with an
accuracy of better than 6% and such an accuracy can be achieved in practice [14].
As discussed in Section 3.2.3.1, when an electric voltage is applied to the thin-film heater, the
effective refractive index neff of the heated waveguide increases and thus changes the optical path
length neff × l by
∆( neff × l ) = ∆Temp
dneff
l
dTemp
(3.74)
where Temp is the temperature, ∆Temp is the temperature change in Celsius degrees,
dneff dTemp = 1× 10 −5 °C is a typical thermo-optic constant of the silica waveguide, and l is the
length of the heated waveguide (or the phase shifter). Furthermore, the change in the optical path
length ∆( neff × l ) relates to the change in the phase shift of the thin-film heater ∆Θ by
∆( neff × l ) =
∆Θ
λ
2π
(3.75)
where λ is the wavelength of the optical carrier. When Eq. 3.75 is substituted into Eq. 3.74, the temperature change ∆Temp relates to the phase shift change ∆Θ by
∆Temp =
λ
∆Θ
dneff
2π
l
dTemp
(3.76)
For a λ = 1.55 µ m lightwave, a typical l = 4 mm of the thin-film heaters, and the allowable phase
errors of ∆Θ = ±0.01π as described above, the allowable temperature change of the filter must be
less than ∆Temp = ±0.19°C according to Eq. 3.76. Thus, the fabricated filter must be stabilized to
within ±0.19°C to ensure that its transmission response is not greatly deteriorated. Such a temperature stability of the filter can be achieved in practice using a Peltier device, which could stabilize the
device temperature to within ±0.1°C [23].
We now discuss means of overcoming the fabrication errors of the waveguide lengths. The
error in the waveguide loop length L of the APF (see Eq. 3.34) and the error in the differential
waveguide length L of the AZF (see Eq. 3.47) could have an effect on the filter performance,
especially on the FSR. Although the phase shifters are mainly used to change the phase of the
optical carrier by means of the thermo-optic effect, as described above, they can also be used
to compensate for the waveguide length error because of the change in the optical path length
when the phase shifters are heated. Thus, the phase shift of the phase shifter φk of the all-pole
filter can also be used to accurately adjust the waveguide loop length to the designed value of
L (see Figure 3.7), and similarly the phase shift of the phase shifter ψ k of the all-zero filter can
also be used to accurately adjust the differential waveguide length to the designed value of L (see
Figure 3.8). Nevertheless, it has been found that the waveguide length could be accurately fabricated to within 1% accuracy [23].
3.2.7 concludIng remarks
An effective synthesis method has been developed for the design of a variety of tunable optical filters with independently variable bandwidths and tunable center frequencies and arbitrary IIR characteristics. The synthesized Mth-order tunable optical filter consists of the cascade of M all-pole
filters (APFs) with M all-zero filters (AZFs). The bandwidth and center frequency of the designed
tunable optical filter can be independently tuned by applying electric power to thin-film heaters
Bandpass Optical Filters by DSP Techniques
105
loaded on the waveguides of the both the APFs and AZFs. The synthesis method has the unique
advantage in that the poles and zeros of the tunable optical filter can be adjusted independently of
each other and can thus be used to design tunable optical filters with arbitrary IIR characteristics.
By means of computer simulation, the effectiveness of the synthesis method has been demonstrated
with the design of the second-order Butterworth bandpass and bandstop tunable optical filters with
variable bandwidths and tunable center frequencies. In this design, for a fixed center frequency,
the filter bandwidth can be tuned by tuning the phase shifts of the phase shifters of the APFs and
by keeping the parameters of the AZFs unchanged; and for a fixed bandwidth, the filter center
frequency can be tuned, to within one FSR, by changing the phase shifts of the phase shifters of
both the APFs and AZFs. In terms of fabrication tolerances, the allowable values of the phase shift
errors of the phase shifters and the coupling angle errors of the directional couplers have also been
determined. In addition to the designed Butterworth filters as shown in this paper, the synthesis
method is general and flexible enough to enable the design of a variety of tunable optical filters with
variable bandwidths and tunable center frequencies and arbitrary IIR characteristics which include
the Chebyshev and elliptic filter types.
REFERENCES
1. E. M. Dowling and D. L. MacFarlane, Lightwave lattice filters for optically multiplexed communications systems, J. Lightwave Technol., 12, 471–486, 1994.
2. N. Q. Ngo and L. N. Binh, Novel realization of monotonic Butterworth-type low pass, highpass and bandpass filters using phase-modulated fiber-optic interferometers and ring resonators, IEEE J. Lightwave
Technol., 12, 827–841, 1994.
3. L. N. Binh, X. T. Nguyen, and N. Q. Ngo, Realization of Butterworth-type optical filters using 3 × 3
optical directional couplers, IEE Part J. Optoelectronics, 143(2), 126–134, 1996.
4. Y. H. Ja, Analysis of optical fiber loop resonators with a collinear 3 × 3 fiber coupler, Applied Opt., 33,
6402–6411, 1994.
5. Y. H. Ja and X. Dai, Butterworth-like filters using an S-shape two coupler optical fiber ring resonator,
Micorw. Opt. Tech. Lett., 6, 376–378, 1993.
6. R. D. Strum and D. E. Kirk, First Principles of Discrete Systems and Digital Signal Processing,
New York: Addison-Wesley, 1989, pp. 676–685.
7. S. K. Mitra and J. F. Kaiser, (Ed.), Handbook of Digital Signal Processing, New York: John Wiley &
Sons, 1993, p. 321.
8. S. K. Mitra and J. F. Kaiser, (Ed.), Handbook of Digital Signal Processing, New York: John Wiley &
Sons, 1993, pp. 137–142.
9. N. Q. Ngo, L. N. Binh, and X. Dai, Eigenfilter approach for designing FIR all-pass optical dispersion
compensators for high speed long-haul systems, Proceedings of the Australian Conference on Optical
Fibre Technology, Melbourne, Australia: ACOFT, December. 1994, pp. 355–358.
10. N. Q. Ngo and L. N. Binh, Novel realization of monotonic Butterworth-type lowpass, highpass and bandpass optical filters using phase-modulated fiber-optic interferometers and ring resonators, J. Lightwave
Technol., 12, 827–841, 1994.
11. K. Jinguji, Synthesis of coherent two-port optical delay-line circuit with ring waveguides, J. Lightwave
Technol., 14, 1882–1898, 1996.
12. C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach,
New York: John Wiley & Sons, 1999.
13. A. Rostami and G. Rostami, All-optical implementation of tunable lowpass, highpass, and bandpass
optical filters using ring resonators, J. Lightwave Technol., 23, 446–460, 2005.
14. A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, Englewood Cliffs, NJ: Prentice
Hall, 1989.
15. A. A. M. Saleh and J. Stone, Two-stage Fabry-Perot filters as demultiplexers in optical FDMA LAN’s,
J. Lightwave Technol., 7, 323–330, 1989.
16. K. Oda, S. Suzuki, H. Takahashi, and H. Toba, An optical FDM distribution experiment using a high
finesse waveguide-type double ring resonator, IEEE Photonics Technol. Letts., 6, 1031–1034, 1994.
17. S. Y. Li, N. Q. Ngo, S. C. Tjin, P. Shum, and J. Zhang, Thermally tunable narrow bandpass filter based
on linearly chirped fiber Bragg grating, Opt. Lett., 29(1), 29−31, 2004.
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Photonic Signal Processing
18. D. Sadot and E. Boimovich, Tunable optical filter for dense WDM networks, IEEE Comm. Magaz., 36,
50–55, 1998.
19. M. G. Xu, H. Geiger, and J. P. Dakin, Interrogation of fiber-optic interferometer sensors using acoustooptic tunable filter, Electron. Lett., 31, 1487–1488, 1995.
20. S. H. Yun, D. J. Richardson, D. O. Culverhouse, and B. Y. Kim, Wavelength-swept fiber laser with frequency shifted feedback and resonantly swept intra-cavity acoustooptic tunable filter, IEEE J. Sel. Top.
Quantum Electron., 3, 1087–1096, 1997.
21. P. V. Mamyshev and L. F. Mollenauer, Stability of soliton propagation with sliding-frequency guiding
filters, Opt. Lett., 19, 2083–2085, 1994.
22. K. Tamura, E. P. Ippen, and H. A. Haus, Optimization of filtering in soliton fiber lasers, IEEE Photonics
Technol. Lett., 6, 1433–1435, 1994.
23. S. Suzuki, M. Yanagisawa, Y. Hibino, and K. Oda, High-density integrated planar lightwave circuits
using waveguides with a high-refractive index difference, J. Lightwave Technol., 12, 790–796, 1994.
4
Photonic Computing
Processors
In this chapter, incoherent fiber-optic systolic array processors (FOSAPs), which employ a
digital-multiplication-by-analog-convolution (DMAC) algorithm and the extension of the
DMAC algorithm, are proposed for real-valued digital matrix computations. The important
role of optics in optical computing and a variety of existing optical architectures using the
DMAC algorithm are described in Section 4.1.2. Section 4.1.3 presents mathematical formulations of the DMAC algorithm and the two’s complement binary (TCB) arithmetic, while
the next section outlines the operational principles of the elemental processors of the FOSAP
architectures. The performances of the FOSAP multipliers (Section 4.1.4) are compared
with those of digital electronic multipliers and other optical DMAC multipliers. Means of
overcoming the limitation of the FOSAP architectures are discussed in Section 4.1.5. The
theoretical aspects of incoherent fiber-optic signal processing described in Chapter 2 are
applied in this chapter where intensity-based signals are considered.
Furthermore, the design of a programmable incoherent Newton–Cotes optical integrator
(INCOI) is described. The definition, existing design techniques, and application of digital integrators are described in Section 4.2. A generalized theory of the Newton–Cotes digital integrators, whose derivation is given in Section 4.4, and their magnitude and impulse responses are
given in Section 4.2. Based on this theory, algorithms for the synthesis of the INCOI processor are
proposed. These algorithms are then used in the design of the programmable INCOI processor,
which essentially consists of a microprocessor, fiber-optic architectures, optical switches, optical,
and semiconductor amplifiers. Several types of input pulse sequences are chosen as examples
for illustration of the incoherent processing accuracy of the programmable INCOI processor.
In addition, the incoherent recursive fiber-optic signal processor (RFOSP) is used in the design
of the programmable INCOI processor. The theory of incoherent fiber-optic signal processing
described in Chapter 2 is employed in this chapter where intensity-based signals are considered.
Section 4.3 then outlines the theoretical development for optical differentiating processors and
their implementation.
4.1 INCOHERENT FIBER-OPTIC SYSTOLIC ARRAY PROCESSORS
4.1.1
IntroductIon
In a digital electronic computer, a co-processor is an important special-purpose processor, which is
mainly responsible for performing specific high-speed arithmetic operations. A high-performance
co-processor is necessary for performing massive computational tasks, such as image processing,
pattern recognition, and signal processing problems, which would otherwise be performed by very
computationally expensive computer software. There has been considerable research interest in
developing optical co-processors for general-purpose digital electronic computers because of the
massive parallelism, high processing speed, and high spatial- and temporal-bandwidth of optics
compared with electronics.
107
108
Photonic Signal Processing
Optical systolic array processors1,2 have been proposed as optical co-processors. Initially, they
were proposed for performing analog matrix–vector and matrix–matrix operations using acoustooptic architectures3 and fiber-optic architectures.4,5 However, the main drawback of these analog
optical processors is their restriction to operating in the low accuracy range (13–16 bits). The accuracy is limited primarily by the linear dynamic range of the devices used in the system. For example,
an erbium-doped fiber amplifier (EDFA) can provide an intensity gain of 40–50 dB, which translates
to 13–16 bits of accuracy.
Various algorithms have been described that enable optical systolic array processors to compute with high (or digital) accuracy techniques, such as residue arithmetic,6 modified signed-digit
number representation,7 redundant number representation,8 and symbolic substitution.9 The most
commonly used technique is the DMAC algorithm,10 which is employed in this chapter, because
convolution can be easily performed in optics.
Whitehouse and Speiser11 proposed that the digital multiplication of two binary numbers (originally known as the Swartzlander multiplier12) is equivalent to the analog convolution of these numbers, provided that the analog result of the digital product is represented in the mixed-binary format.
This digital-multiplication-by-analog-convolution technique is known as the DMAC algorithm.13 The
mixed-binary representation of the digital product can be converted to the standard binary representation of the digital product by an analog-to-digital converter (ADC) and a shift-and-add (S/A) circuit in
the post-processing unit. This basic idea has been applied to digital matrix–vector and matrix–matrix
operations using time-integrating (TI) and space-integrating (SI) acousto-optic architectures,14,15
magneto-optic spatial light modulators,16 nonlinear optical devices,17,18 and logic counters.19
In this chapter, incoherent FOSAPs employing the DMAC algorithm are proposed for realvalued digital matrix multiplications. Most of the work presented here has been described by Ngo
and Binh.19,20
Although there are various architectures of optical systolic array processors, they all have one general feature in that
the input data flows in a pulsating fashion, and hence the name “systolic”, through one-dimensional or two-dimensional
identical array processing elements where computation is performed on the data currently present.
2 H. J. Caufield, W. T. Rhodes, M. J. Foster, and S. Horvitz, Optical implementation of systolic array processing,
Opt. Commun., 40, 86–90, 1981.
3 R. A. Athale and J. N. Lee, Optical processing using outer-product concepts, Proc. IEEE, 72, 931–941, 1984.
4 B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, Fiber-optic lattice signal processing, Proc. IEEE, 72, 909–930, 1984.
5 K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, and H. J. Shaw, Optical fiber delay-line
signal processing, IEEE Trans. Microw. Theory Tech., MTT-33, 193–210, 1985.
6 A. Goutzoulis, E. Malarkey, D. K. Davies, J. Bradley, and P. R. Beaudet, Optical processing with residue LED/LD
lookup tables, Appl. Opt., 25, 3097–3112, 1986.
7 B. Drake, R. Bocker, M. Lasher, R. Patterson, and W. Miceli, Photonic computing using the modified signed-digit number representation, Opt. Eng., 25, 38–43, 1986.
8 R. A. Athale, Highly redundant number representation for medium accuracy optical computing, Appl. Opt., 25, 3122–3127, 1986.
9 K. H. Brenner, Programmable optical processor based on symbolic substitution, Appl. Opt., 27, 1687–1691, 1988.
10 H. J. Whitehouse and J. M. Speiser, Linear signal processing architectures, in Aspects of Signal Processing. Part 2,
G. Tacconi (Ed.). NATO Advanced Study Institute, Boston, MA, 1976. pp. 669–702.
11 Ibid.
12 E. E. Swartzlander, The quasi-serial multiplier, IEEE Trans. Comp., C-22, 317–321, 1973.
13 D. Psaltis, D. Casasent, D. Neft, and M. Carlotto, Accurate numerical computation by optical convolution, in 1980
International Optical Computing Conference II, W. T. Rhodes (Ed.). Proc Soc Photo-Opt Instrum Eng, 232, 151–156, 1980.
14 Ibid.
15 E. J. Baranoski and D. P. Casasent, High accuracy optical processors: a new performance comparison, Appl. Opt., 28,
5351–5357, 1989.
16 F. T. S. Yu and M. F. Cao, Digital optical matrix multiplication based on a systolic outer-product method, Opt. Eng., 26,
1229–1233, 1987.
17 G. Eichmann, Y. Li, P. P. Ho, and R. R. Alfano, Digital optical isochronous array processing, Appl. Opt., 26, 2726–2733, 1987.
18 Y. Li, G. Eichmann, and R. R. Alfano, Fast parallel optical digital multiplication, Opt. Commun., 64, 99–104, 1987.
19 Y. Li, B. Ha, and G. Eichmann, Fast digital optical multiplication using an array of binary symmetric logic counters,
Appl. Opt., 30, 531–539, 1991.
20 N. Q. Ngo and L. N. Binh, Fibre-optic array processors for algebra computations, Proc. IREE, 18th Australian Conf.
Opt. Fibre Technol., Wollongong, 356–359, 1993.
1
109
Photonic Computing Processors
4.1.2
Digital-multiplicatiOn-by-analOg-cOnvOlutiOn
algOrithm anD its extenDeD versiOn
The DMAC algorithm, its extended version, and the TCB arithmetic are described in this section.
For the sake of hardware simplicity, the binary words are assumed to be of the same length, although
the DMAC algorithm is generally valid for binary data of any length. Unless otherwise stated,
variables with brackets denote binary sequences. For example, f represents the analog value whose
binary sequence is denoted by { f }.
4.1.2.1 Multiplication of Two Digital Numbers
Multiplication is a standard operation in matrix computations. It is shown here that the digital multiplication of two binary numbers can be determined by performing the analog convolution of these
binary numbers followed by the function of the postprocessor.
The standard binary representation of two positive integers f and g are given by the sequences
{ f } = { f n −1 f1 f 0} and {g} = {gn −1 g1 g0}, where n is the number of bits in a binary number,
fi ,gi ∈ (0,1) for 0 ≤ i ≤ n − 1, f 0 and g0 are the least significant bits (LSBs), and f n−1 and gn−1 are
the most significant bits (MSBs). The mathematical descriptions of these positive n-bit words
are given by
n −1
f = ∑ fi 2i ,
i=0
(4.1)
n −1
j
g = ∑ g j2 ,
j=0
(4.2)
The product of these binary numbers is given by13
f ⋅g =
2n − 2 k
∑ 2 yk ,
k =0
(4.3)
where
k
yk = ∑ fi gk − i ,(0 ≤ k ≤ 2n − 2),
i=0
(4.4)
g=
0 for i < 0 or i > n −1.
with f=
i
i
Eq. (4.4) is easily recognized as the discrete convolution of two sequences, such as { f } ∗ {g} where
* designates the convolution operation. The analog result of the discrete convolution is in the mixedbinary format. Note that the digital multiplication of two binary numbers without carries, that is
{f}·{g}, is also in the mixed-binary format. Furthermore, the kth analog value yk of the convolution
result represents the sum of partial products (without carries) in the kth column of the product {f}·{g}.
This process of digital multiplication by analog convolution is known as the DMAC algorithm.
Evaluation of Eq. (4.3) requires an optical convolver, an optical detector, an ADC, and a S/A
circuit. At a particular discrete time, k, an ADC21 is used to convert the kth analog value yk into its
binary representation, which is then up-shifted to the left by k bits by a shift register. This process is
equivalent to evaluating the partial product 2k yk of Eq. (4.3). The last step for evaluation of Eq. (4.3)
requires the addition of the partial products (with carries) by a binary adder.
21
The ADC requires log 2 n bits accuracy because n is the maximum analog value as a result of the convolution of two n-bit
words.
110
Photonic Signal Processing
The DMAC algorithm is best illustrated by means of a numerical example of multiplying two
positive n-bit words whose standard binary representations are { f } = {110} and { g} = {011}, which
correspond to the analog values f = 6 and g = 3. Figure 4.1a and b show the discrete-time representation of these sequences, where f [i ] is the impulse response of the optical convolver, g [i ]
represents the modulated laser pulses (rectangular profile assumed) of unity height to be launched
into the convolver, T p is the pulse width, and T is the bit period of the input pulse sequence or the
sampling period of the optical convolver. The convolver pulse response, which is simply the discrete convolution of its impulse response f [i ] with the input pulse sequence g [i ] is given, from
Eq. (4.4), as
k
y[k ] = ∑ f [i ]g[k − i ], (0 ≤ k ≤ 2n − 2).
i=0
(4.5)
Eq. (4.5) can be evaluated graphically as shown in Figure 4.1c–g for n = 3. In Figure 4.1c, g [i ]
is folded about i = 0 to become g [ −i ] and is slid past (with the LSB first) the digits of f [i ]. The
sequence g [ k − i ] is simply the folded sequence of g[i] shifted to the right by k units of delay T , as
shown in Figure 4.1c–g for k = 0, 1, 2, 3, 4, respectively. The pulse shown by the broken curve corresponds to the LSB of the next word. t0 , t1 denote the starting times of the LSB of the first and second
words, respectively. The word strobe period Tw = t j +1 − t j is the time separation between the LSB
of the previous word and the LSB of the next word and is given by Tw = (2n − 1)T . In order to avoid
overflow, n −1 zeros are required for padding between the words. The analog result y[k] is shown in
Figure 4.1h, where Tconv is the propagation delay of the optical convolver, which has been ignored
during the convolution process in Figure 4.1c–g for the sake of clarity. These mixed-binary pulses
are converted into their binary representations by a 2-bit ADC and shifted and added by a 2-bit S/A
circuit in the post-processing unit, as shown in Figure 4.1. (i). The standard binary representation
{010010} corresponds to the integer 18.
4.1.2.2 High-Order Digital Multiplication
High-order multiplication is required for high-order matrix computations. It is shown here that the
digital multiplication of N̂ binary numbers can be determined by performing the analog convolution
of these N̂ binary numbers followed by the function of the postprocessor. This process of “HighOrder Digital-Multiplication-by-Analog-Convolution” is referred to as the “HO-DMAC” algorithm.
The product of N̂ positive n-bit words is defined as
PN = f (1)f (2) f ( Nˆ − 1)f ( Nˆ )

 n −1
n − 1
 n − 1

  n −1
=  ∑ f (1,a)2a   ∑ f (2,b)2b   ∑ f ( Nˆ − 1,α )2α   ∑ f ( Nˆ ,β )2 β  ,

 a=0
  b=0

α =0
  β =0

(4.6)
where f (1, a), f (2, b),, f ( Nˆ − 1,α ), f ( Nˆ , β ) ∈ (0,1). Substitution of γ = a + b + + α + β (or
β = γ − a − b − − α ) into Eq. (4.6) results in
PN =
N (n − 1)
γ
∑ 2 yγ ,
γ =0
(4.7)
where
γ
γ
γ
yγ = ∑ ∑ ∑ f (1,a)f (2,b) f ( Nˆ − 1,α )f ( Nˆ ,γ − a − b − − α ) ,
a = 0b = 0 α = 0
(4.8)
Photonic Computing Processors
111
FIGURE 4.1 Graphical illustration of the DMAC technique. (a) The input pulse sequence g[i ]. (b) The
convolver impulse response f [i ]. (c)–(g) The convolution operation. (h) The convolution output y[k ] is in the
mixed-binary format. (i) The analog output is operated by the post-processing unit to obtain the expected
standard binary representation of the decimal number 18.
for 0 ≤ γ ≤ N̂ (n − 1), and f (1,a) = f (2,a) = = f ( Nˆ − 1,a) = f ( Nˆ ,a) = 0 for a < 0 or a > n −1.
Eq. (4.8) can be recognized as the discrete convolution of N binary sequences
yγ = { f (1)} ∗ { f (2)}{ f ( Nˆ − 1) } ∗ { f (Nˆ )},
(4.9)
The order of performing the convolution in Eq. (4.9) is unimportant because convolution is commutative and associative. In general, there are N̂ ( n − 1) + 1 mixed-binary points as a result of the convolution of N̂ n-bit words, the word cycle is Tw = [ Nˆ ( n − 1) + 1]T , and the number of zeros required
112
Photonic Signal Processing
for padding is ( N̂ − 1)( n − 1). These mixed-binary points can be converted to the standard binary
representation by the postprocessor, as described in Section 4.2.1.
4.1.2.3 Sum of Products of Two Digital Numbers
The sum of products of two digital numbers, which is a vector inner-product operation, is also a
standard operation in matrix computations. It is shown here that the digital summation of products
of two binary numbers can be determined by performing the analog summation of convolutions
of these binary numbers followed by the function of the postprocessor. This process of “Sum of
Digital-Multiplication-by-Analog-Convolution” is referred to as the “S-DMAC” algorithm.
The standard binary representations of the pth positive integers f ( p) and g ( p) are given by
n −1
n −1
j
f ( p) = ∑ fi (p)2i , g (p) = ∑ g j (p)2 ,
i=0
j =0
(4.10)
where fi (p), g j (p) ∈(0,1). The sum of products of these positive n-bit words is given by
ˆ
M
SM̂ = ∑ f ( p)g ( p),
p =1
(4.11)
where M̂ is the number of products of such two n-bit words. Eq. (4.11) can be shown to be94
SM̂ =
ˆ

2n − 2 k  M
∑ 2  ∑ yk (p)  ,
k =0
 p = 1

(4.12)
where
k
yk (p) = ∑ fi (p)gk − i (p), (0 ≤ k ≤ 2n − 2).
i=0
(4.13)
Eq. (4.13) can be recognized as the discrete convolution of two binary numbers. Evaluating Eq. (4.12)
requires M̂ optical convolvers, an optical combiner, an optical detector, an ADC, and a S/A circuit.
At a particular discrete time k, the term in brackets in Eq. (4.12), which represents the sum of M̂
, can be performed by an optical combiner. The analog result
analog values of yk (p) for 1 ≤ p ≤ M
of this summation is then passed to the ADC for digitization, followed by the S/A circuit to obtain
the standard binary representation. The advantage of the S-DMAC algorithm lies in the fact that a
combination of product and summation operations can be performed.
4.1.2.4 Two’s Complement Binary Arithmetic
The TCB representation is a powerful encoding scheme that permits both positive and negative
numbers to be represented in binary form. The encoding scheme requires a sign-bit (SB) to be
attached to the leftmost bit of the binary number, i.e., SB = 0 for positive numbers and SB = 1 for
negative numbers. For example, the TCB sequence of the positive number22 +5.5 is {0 1 0 1.1}, and
the negative number −13.375 is {1 0 0 1 0 _ 1 0 1}.
Based on the DMAC algorithm as described in Section 4.2.1, the multiplication of two real
numbers using the TCB arithmetic requires the input numbers to be represented by the same
number of bits as the output number. For example, the TCB representation of the product (+5.5)
(−13.375) = −73.5625 is {101101100111}, which is a 12-bit word. The 12-bit TCB sequence
22
Implementation of the radix (or decimal) point shifting is not discussed here because it is not a hardware issue.
113
Photonic Computing Processors
of the input numbers can thus be obtained by inserting seven zeros to the left of the SB of
+5.5 to become {0 0 0 0 0 0 0 0 1 0 11} and four ones to the left of the SB of −13.375 to give
{1 1 1 1 1 0 0 1 01 0 1}. The discrete convolution of these two sequences results in the mixedbinary sequence {1 1 1 21 2 0 2 2 2 3 3 2 1 1 0 0 0 0 0 0 0 0} in which the last 13 analog bits
are discarded. The standard TCB format of the first 12 chosen analog bits (the boldface bits) is
{1 0 1 1 0 1 1 00 1 1 1} after the ADC and S/A operations, which is expected for the negative number
−73.5625. The TCB arithmetic is not only applicable to the DMAC algorithm, but can also be used
in conjunction with the HO-DMAC (see Section 4.2.2) and S-DMAC (see Section 4.2.3) algorithms.
The main disadvantage of any optical DMAC processor employing the TCB arithmetic is the
reduction in the pre-processing speed, and this can be shown as follows. In the following discussion,
primes are used to denote the TCB variables. The convolution of N̂ n′-bit TCB numbers generates
[ N̂ ( n′ − 1) + 1] mixed-binary points, in which only the first n′ analog bits are useful for decoding
into the binary format, and the word cycle is Tw′ = [ Nˆ ( n′ − 1) + 1]T . If the first useful n′ analog bits
of the TCB convolution are equal to the [ N̂ ( n − 1) + 1] analog bits of the unsigned convolution (see
Section 4.2.2), then the following relationship is obtained:
ˆ w + (1 − Nˆ )T ,
Tw′ = NT
(4.14)
in which the first term is dominant for large word length. Thus, the processing power of the optical
DMAC processor incorporating the TCB arithmetic is approximately reduced by a factor of N̂ (i.e.,
ˆ w where N̂ is the number of integers (or matrices) to be multiplied) as compared with its
Tw′ ≈ NT
unsigned counterpart. Optical DMAC processors based on the TCB arithmetic also suffer from an
increase in the resolution bits of the ADC and the S/A circuit because all the bits representing the
output number are not fully utilized. This has the effect of reducing the processing speed because
a higher-resolution ADC operates at a much lower speed. Nevertheless, optical DMAC processors
incorporating the TCB representations can operate on real numbers.
4.1.3
elemental Optical signal prOcessOrs
This section describes three elemental optical signal processors, namely, an optical splitter, an optical combiner, and a binary programmable incoherent fiber-optic transversal filter. These processors
are the building blocks of the FOSAP matrix multipliers, which are to be outlined in Section 4.4.
It is assumed that the optically encoded signals to be processed are modulated onto an optical
carrier whose coherence time T p is very short compared to the basic time delay T of the incoherent
fiber-optic transversal filter, i.e., T p << T . In other words, incoherent fiber-optic signal processing
(see Section 4.2.2) is considered here where signals add on an intensity basis. It is further assumed
that the modulated laser pulse has a pulse width Tp and a height of one unit.
4.1.3.1 Optical Splitter and Combiner
Optical splitter and combiner are useful for signal distribution and collection, respectively. The 1xn
optical splitter (see Figure 4.2) and the nx1 optical combiner (see Figure 4.3) can be constructed
from n−1, 3-dB fiber-optic directional couplers arranged in a binary tree structure. The 1xn splitter
distributes the incoming signal intensity evenly to n output ports, while the nx1 combiner collects
signal intensities from n input ports into a single output port.
The signal intensity coming into the splitter will experience a 3-dB coupling loss at every stage
of the tree. This results in a common intensity loss of 10 log (n) (dB) at each of the n output ports.
An EDFA, which has been described in Section 4.2.2, with an intensity gain Gsplit is incorporated at
the start of the tree to compensate for such a loss. Thus, the signal intensity at each of the output ports
is equal to the incoming signal intensity provided that Gsplit = 10log (n) (dB). Likewise, the incoming
signal intensity at each of the n input ports of the combiner will also experience a common intensity
114
Photonic Signal Processing
FIGURE 4.2 1xn Optical splitter (a) Schematic diagram. (b) Block diagram.
FIGURE 4.3
Optical combiner. (a) Schematic diagram. (b) Block diagram.
loss of 10log (n) (dB) at the end of the tree. The signal intensity of each input port can be recovered at
the output port, provided that an EDFA of intensity gain Gcomb = 10log (n) (dB) is incorporated at the
output of the device. For a typical 30-dB gain of the EDFA, the value n can be as large as 1000. Note
that the propagation delays of the splitter (Tsplit) and combiner (Tcomb) are mainly due to the amplifier
lengths of the EDFAs.
4.1.3.2 Binary Programmable Incoherent Fiber-Optic Transversal Filter
An optical transversal filter (or optical convolver) is useful for performing the convolution operation. A binary programmable incoherent fiber-optic transversal filter has the advantage of being
adaptive, enabling an arbitrary binary intensity impulse response to be obtained by means of external optical and/or electronic control.
Figure 4.4a and b shows the schematic and block diagrams of the binary programmable incoherent fiber-optic transversal filter, which essentially consists of an optical splitter, an optical combiner, optical switches, and fiber delay lines.23 The binary-code selector is word-parallel loaded
into the optical convolver and used to set the binary impulse response of the optical convolver by
23
P. R. Prucnal, M. A. Santoro, and S. K. Sehgal, Ultrafast all-optical synchronous multiple access fiber networks, IEEE J.
Selected Areas Commun., SAC-4, 1484–1493, 1986.
115
Photonic Computing Processors
FIGURE 4.4 Binary programmable incoherent fiber-optic transversal filter (or optical convolver). (a)
Schematic diagram. (b) Block diagram.
simultaneously controlling the binary states of the 1 × 2 (or 2 × 2) optical switches. The binary
states of the optical switches correspond to the code word bi = {sn −1 sn − 2 s0}, where si ∈(0,1),
s0 and sn−1 designate the LSB and MSB, respectively, and n is the number of bits in a binary word bi.
Each optical switch permits the signal intensity from the optical splitter to either connect (binary
state si = 1) or bypass (binary state si = 0) a particular fiber delay path. The optical combiner collects the signal intensities according to the binary states of the optical switches. The optical splitter
and combiner are those of Figures 4.2 and 4.3, except that no EDFAs are used. Instead, an EDFA of
intensity gain Gconv is placed at the start of the splitter to compensate for the losses24 of the splitter
and combiner. That is, the signal intensity at the output of the optical convolver is the same as the
signal intensity at its input provided that Gconv = 20 log n (dB). The convolver impulse response is
given by, provided that Gconv = 20 log n (dB),
n −1
h(t ) = ∑ si δ ( t − iT − Tconv ) ,
i=0
(4.15)
where δ (t ) is the delta function, Tconv the filter propagation delay, which is mainly due to the amplifier length of the EDFA, T the time delay difference between the fiber delay lines or the sampling
period of the filter, and 1/ T is the sampling frequency of the filter.
Another form of the programmable fiber-optic transversal filter is in the forward-flow bus structure,10 which has a common intensity loss of 3( n + 1) dB and requires 2n couplers. This bus structure
has loss that varies linearly with the number of bits n, but the loss of the proposed binary structure,
which requires 2 x( n − 1) couplers, varies as the logarithm of n. As a result, the programmable fiberoptic transversal filter considered here has less loss and fewer couplers required.
24
Other minor losses associated with the fiber delay lines, such as the insertion loss of the optical switch and connector
(or splice) loss, are not considered here for the sake of analytical simplicity.
116
4.1.4
Photonic Signal Processing
incOherent Fiber-Optic systOlic array prOcessOrs FOr
Digital matrix multiplicatiOns
This section describes the incoherent FOSAP architectures for computation of positive-valued digital matrix–vector, matrix–matrix and triple-matrix products by using the DMAC, HO-DMAC and
S-DMAC algorithms.
4.1.4.1 Matrix–Vector Multiplication
A high-accuracy FOSAP matrix–vector multiplier based on the S-DMAC algorithm and the vector
inner-product operations is described here.
The matrix–vector product of a MxN matrix A and a Nx1 column vector B results in a Mx1 column vector C, which is given by
 c1   a11
c  a
 2  =  21
     
 c M   aM 1
a12
a22
aM 2
a1N   b1   a11b1 + a12b2 + … + a1N bN 
a2 N   b2   a21b1 + a22b2 + … + a2 N bN 
=
,
    
  

aMN  bN   aM 1b1 + aM 2b2 + … + aMN bN 
(4.16)
where the words of vector C are given by
N
ci = ∑ aij b j
j =1
(i = 1, 2,, M ),
(4.17)
with aij and b j being the n-bit words of matrix A and vector B, respectively. Note that the matrix–
vector product can be decomposed into parallel vector inner-product operations, as shown in
Eq. (4.16).
Figure 4.5 shows the block diagram of the high-accuracy FOSAP matrix–vector multiplier based
on the S-DMAC algorithm and the vector inner-product operations. At time t0, the N n-bit binary
words (b1 b2 bN ) of vector B are word-parallel loaded into the corresponding N optical convolvers
by selection of the appropriate optical switches. This process is equivalent to setting the desired
binary impulse responses of the optical convolvers, as described in Section 4.3.2. At the same time,
the LSBs of the N n-bit words (a11 a12 a1N ) on the first row of matrix A are bit-serial fed into the
appropriate inputs of the N optical convolvers. The N analog output signals of the optical convolvers
are then passed to the Nx1 optical combiner, where analog summations begin. The resulting analog
optical signal is then passed to an optical detector and converted into its binary representation by a
log2 Nn-bit ADC25 and a log2 Nn-bit S/A circuit. At a later bit time t0 + T , the second bits of the same
words (a11 a12 a1N ) are bit-serial loaded into the optical convolvers, and the analog signals are then
combined by the optical combiner. The detected analog signal is converted into its binary format
by an ADC, which is then upshifted by one position to the left and binary added to the previous
log2 Nn binary bits by the S/A circuit to form a new binary sequence. This process continues until
the computation of the first row of matrix A is completed, which takes one-word cycle Tw where
Tw = (2n − 1)T . At the instant t1 + T , the first element c1 of the column vector C is computed and
stored in an output buffer. At the same time, the LSBs of the N words (a21 a22 a2 N ) on the second
row of the matrix A are convolved with the LSBs of the N optical convolvers. This computation
process is repeated as before, and the second element c2 of the vector C is available after two-word
cycles 2Tw and stored in the output buffer. There are n −1 zeros for padding between the words to
avoid overflow. The computation time of the high-accuracy FOSAP matrix–vector multiplier, which
takes M word cycles, is thus, provided that TADC < T , given by
25
It is assumed that the coversion time of the ADC (TADC) is smaller than the bit time T.
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Photonic Computing Processors
FIGURE 4.5 The high-accuracy FOSAP matrix–vector multiplier based on the S-DMAC algorithm and the
vector inner-product operations.
TMV = (2n − 1)MT + Tconv + Tcomb ,
(4.18)
4.1.4.2 Matrix–Matrix Multiplication
A high-accuracy FOSAP matrix–matrix multiplier based on the S-DMAC algorithm and the vector
outer-product operations is described here.
The matrix–matrix product of a MxN matrix A and a NxP matrix B results in a MxP matrix C,
which is given by
 c11
c
 21
 
 cM 1
c12
c22
cM 2
c1P   a11
c2 P   a21
=
   
cMP   aM 1
a12
a22
aM 2
a1N   b11
a2 N   b21
 
aMN  bN 1
b12
b22
bN 2
b1P 
b2 P 
,


bNP 
(4.19)
where the words of matrix C are given by
N
cij = ∑ aik bkj ,
k =1
for i = 1, 2, , M and j = 1, 2, , P
(4.20)
The n-bit words aik and bkj are the elements of matrices A and B, respectively. The matrix–matrix
product can be computed by successive vector outer-product operations, as in the following
illustration:
 c11
c
 21
 c31
c12
c22
c32
c13   a11 
c23  =  a21  ⋅ [ b11
c33   a31 
b12
 a12 
b13 ] +  a22  ⋅ [ b21
 a32 
b22
 a13 
b23 ] +  a23  ⋅ [ b31
 a33 
b32
b33 ]
(4.21)
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Photonic Signal Processing
Note that the vector outer-product operations, for example, the first term on the right of Eq. (4.21),
is defined as
 a11 
a  ⋅ b
 21  [ 11
 a31 
b12
 a11b11
b13 ] =  a21b11
 a31b11
a11b12
a21b12
a31b12
a11b13 
a21b13  .
a31b13 
(4.22)
Figure 4.5 shows the block diagram of the high-accuracy FOSAP matrix–matrix multiplier based
on the S-DMAC algorithm and the vector outer-product operations. At the instant t0 , all the n-bit
binary words (b11 bNP ) of matrix B are word-parallel loaded into the optical convolvers by selection of the appropriate optical switches. At the same time, the N n-bit words ( a11a12 a1N ) on the
first row of matrix A are bit-serial fed (with the LSBs first) into the N 1xP optical splitters. The LSBs
of these words are then bit-serial fed into the inputs of the appropriate optical convolvers, where
convolution computations begin. The analog output signals of the optical convolvers are then passed
to the PNx1 optical combiners, where analog additions begin. The P postprocessors, as shown in
Figure 4.5, perform the ADC and S/A operations to yield the standard binary sequences for the first
row of matrix C, i.e., (c11c12 c1P ) , which takes one-word cycle Tw, where Tw = (2n − 1)T . At time
t1 + T , the LSBs of the N n-bit elements ( a21a22 a2 N ) on the second row of matrix A are sequentially
fed into the N 1xP optical splitters, and the process repeats as before until time t M −1, which takes M
word cycles.
H m2 ( z=
; p 2=
; m 2, 4, 5) . (c) Simpson’s 3/8 integrator H 33 ( z ), Boole’s integrator H 44 ( z ), and the
fifth-order integrator H 55 ( z ).
The computation time of the high-accuracy FOSAP matrix–matrix multiplier, which takes M
word cycles, is thus given by
TMM = (2n − 1) MT + Tsplit + Tconv + Tcomb ,
(4.23)
provided that TADC < T . Both the ADC and the S/A circuit in each post-processor require log2 Nn
resolution bits, the same number as that required by the FOSAP matrix–vector multiplier of
Figure 4.5. The pre-processing time, as shown by the first term of Eq. (4.23), is exactly the same
as the first term of Eq. (4.18), and this clearly shows the massive parallel-processing capability of
the high-accuracy FOSAP matrix multiplier. Figures 4.6 and 4.7 show the magnitude and impulse
responses respectively of the integrator.
4.1.4.3 Cascaded Matrix Multiplication
A high-accuracy FOSAP triple-matrix multiplier based on the S-DMAC and HO-DMAC algorithms, and the vector outer-product operations is described here.
The triple-matrix product of a MxN matrix A, a NxP matrix B, and a PxQ matrix C results in a
MxQ matrix D, which is given by
 d11 d12
d
 21 d22
 
dM1 dM 2
d1Q   a11 a12
d2Q   a21 a22
=
   
d MQ   aM 1 aM 2
a1N 
a2 N 


aMN 
 b11 b12
b
 21 b22
 
bN 1 bN 2
b1P   c11 c12
b2 P   c21 c22
×
   
bNP  cP1 cP 2
c1Q 
c2Q 
(4.24)


cPQ 
119
Photonic Computing Processors
(a)
(b)
(c)
FIGURE 4.6 Magnitude responses of several families of the Newton–Cotes digital integrators. Here, the
normalized frequency corresponds to ωT (2π ), Hmp, denoted by H mp ( z ), and the sampling period is assumed
to be T = 1 without loss of generality. (a) Trapezoidal family H m1 ( z=
; p 1;=
m 1, 2, 3, 4, 5). (b) Simpson’s
1/3 family. H m2 ( z=
; p 2=
; m 2, 4, 5).
) (c) Simpson’s 3/8 integrator H33(z), Boole’s integrator H44(z), and the fifthorder integrator H55(z).
where the words of matrix D are given by
P
dil =
N
∑ ∑ aik bkj c , for i = 1,2,,M and l = 1,2,,Q.
jl
(4.25)
j =1 k =1
The n-bit binary words aik , bkj , and c jl are the elements of matrices A, B and C, respectively.
Figure 4.8 shows the block diagram of the high-accuracy FOSAP triple-matrix multiplier based
on the S-DMAC and HO-DMAC algorithms, and the vector outer-product operations. The operating
120
Photonic Signal Processing
(a)
(b)
(c)
FIGURE 4.7 Impulse responses of several families of the Newton–Cotes digital integrators. Here, Hmp
means H mp ( z ), time n means nT , and the sampling period T is assumed to be T = 1 without loss of generality.
(a) Trapezoidal family H m1 ( z=
; p 1;=
m 1, 2, 3, 4, 5). (b) Simpson’s 1/3 family H m2 ( z=
; p 2=
; m 2, 4, 5) .
(c) Simpson’s 3/8 integrator H 33 ( z ), Boole’s integrator H 44 ( z ), and the fifth-order integrator H 55 ( z ).
principle of the FOSAP triple-matrix multiplier is very similar to that of the FOSAP matrix–matrix
multiplier. The analog outputs of the P Nx1 optical combiners are due to the convolution of the
elements of matrix A with the elements of matrix B, i.e., A*B. At this point, the pre-processing
architecture is exactly the same as that of the FOSAP matrix–matrix multiplier of Figure 4.8. The P
analog outputs of A*B are sequentially fed to the P 1xQ optical splitters and then to the appropriate
optical convolvers, whose binary impulse responses correspond to the words of matrix C. The Q
Px1 optical combiners perform the analog summations to generate the Q analog outputs of (A*B)*C,
which are then fed to the postprocessors for digitization to yield the standard binary sequences of
the elements of matrix D.
The word cycle Tw is given by Tw = (3n − 2)T , and the number of zeros required for padding between
the words is 2(nx1). The largest analog value of the convolution of 3 n-bit words (i.e., aik ∗ bkj ∗ c jl )
in which all binary bits have logic one is given by 0.75n2 when n is even and 0.25(3n2 + 1) when
Photonic Computing Processors
121
FIGURE 4.8 The high-accuracy FOSAP matrix–matrix multiplier based on the S-DMAC algorithm and the
vector outer-product operations.
n is odd. Thus, the largest analog bit of dil in Eq. (4.24) is given by 0.25NP (3n2 + 1) , where N and
P are the number of summations performed by the Nx1 and Px1 optical combiners to generate
(A*B) and (A*B)*C, respectively. Thus, both the ADC and S/A circuit in the postprocessor require
log2 [0.25NP (3n2 + 1)] resolution bits. The computation time of the high-accuracy FOSAP triplematrix multiplier, which takes M word cycles, is thus given by
TMMM = (3n − 2) MT + 2Tsplit1 + 2Tconv + Tcomb1 + Tsplit2 ,
(4.26)
provided that TADC < T and Tsplit1 = Tcomb2.
The major advantage of the S-DMAC and HO-DMAC algorithms lies in the fact that a combination of convolution and addition operations can be optically performed in the pre-processing unit.
Higher-order matrix operations can be performed by cascading the basic building blocks, as highlighted in Figure 4.8, where each block corresponds to the elements of a matrix.
4.1.4.4 Performance Comparison
In this section, the performance of the FOSAP architecture using non-binary data is described. The
performances of the FOSAP matrix–vector and matrix–matrix multipliers are compared with those
of the digital electronic multipliers and other optical multipliers. Also described is the performance
of the high-order FOSAP architecture. For analytical simplicity, the propagation delays Tsplit, Tcomb,
122
Photonic Signal Processing
and Tconv of the FOSAP architectures are assumed to be negligible compared with the word cycles
because the matrix dimensions considered here are large.
4.1.4.5 Fiber-Optic Systolic Array Processors Using Non-Binary Data
Using the Psaltis-Athale performance ratio as a performance measure, the performance of the
FOSAP matrix multiplier, in which the FOSAP matrix–vector multiplier of Figure 4.8 is chosen as
an example, is described here for non-binary encoded data.
The number of bits n required for representing a base-b m-bit number is given by
n ≥ logb 2m.
(4.27)
The Psaltis-Athale performance ratio R is defined as the number of multiplications per ADC operation, and can be obtained by dividing the number of operations26 per second by the number of ADC
operations per second (the number of ADCs times the clock rate)23:
R = Number of Multiplications per ADC Operation =
Number of Opeerations Per Second
. (4.28)
Number of ADCs × Clock Rate
The Psaltis-Athale performance ratio provides a direct comparative estimate of an optical implementation versus an electronic implementation. The performance ratio R is independent of the clock
rate and must exceed unity to keep the complexity of the electronics to a minimum. As a result,
optical DMAC processors are only superior to the digital electronic processors if R > 1. If, for
example, R = 1 then only one binary multiplication is being performed by the optical system per
ADC operation. Because it is about equally difficult to perform multiplications and ADC operations
electronically, this example shows that optics offers no advantage over electronics.
The maximum value of the matrix–vector convolution, where the elements are base-b n-bit numbers, is given by Nn(b −1)2, which must not exceed the dynamic range of the ADC of 2 N ADC where
N ADC is the ADC resolution bits. Thus, the matrix dimension N of the FOSAP matrix–vector multiplier of Figure 4.8 is limited by the dynamic range of the ADC according to
N≤
2 N ADC
.
n(b − 1)2
(4.29)
The FOSAP matrix–vector multiplier, which performs MN operations in time MTw, has the processing speed given by
MOPS =
MN
N
,
=
MTw (2n − 1)T
(4.30)
where MOPS stands for mega operations per second and 1/T is the clock rate or ADC speed in
megahertz. The performance ratio of the FOSAP matrix–vector multiplier is thus given, from
Eq. (4.28), as
R=
MOPS
N
=
.
1× 1 T
(2n − 1)
(4.31)
Table 4.1 shows the operational parameters for various case studies of the FOSAP matrix–vector
multiplier of Figure 4.8 using Eqs. 4.27 through 4.31. Cases 1–3, where the resolution bits NADC and
26
Here, one operation is considered to be equivalent to one multiplication and one addition.
123
Photonic Computing Processors
TABLE 4.1
Operational Parameters for Various Case Studies of the Fosap Matrix–Vector
Multiplier for 32-bit (m = 32) Multiplications
Case
b
n
N
NADC
1/T (MHz)
MOPS
R
Remarks
1
2
3
4
5
6
2
4
8
2
4
8
32
16
11
32
16
11
128
28
7
128
128
128
12
12
12
12
16
18
100
100
100
100
20
0.20
203.2
90.3
33.3
203.2
82.6
1.22
2.032
0.903
0.333
2.032
4.13
6.1
Constant clock rate 1 T , constant
ADC bits N ADC
Constant matrix dimension N
Note: b is the base used; n, the digits of accuracy; N, the matrix dimension; NADC, the ADC resolution bits;
1/T, the ADC speed or clock rate; MOPS, mega operations per second; and R, the Psaltis-Athale
performance ratio.
speed 1/T of the 12-bit 100-MHz ADC are fixed, show that increasing the base b results in decreasing the values of n and N. Consequently, the values of MOPS and R are significantly reduced,
and the FOSAP matrix–vector multiplier performs best with binary-encoded data (case 1). Cases
4–6 correspond to a fixed matrix dimension N = 128. Increasing the base b results in increasing NADC, which greatly reduces the processing speed of MOPS because higher-resolution ADCs
operate at much lower speeds. However, the FOSAP matrix–vector multiplier still outperforms its
digital electronic counterpart because of the large value of R but at the expense of lower accuracy
of n. Hence, the overall performance of the FOSAP matrix–vector multiplier is best achieved with
binary-encoded data because of the desirable values of MOPS and R, as shown in Case 4, which
is in fact Case 1.
Similarly, the FOSAP matrix–matrix multiplier of Figure 4.8 also has the same operational
parameters as those in Table 4.1 except that it has P times the values of the MOPSs. The analysis
here also applies to higher-order FOSAP matrix multipliers, and it can be deduced that the FOSAP
matrix multipliers perform better with binary-encoded data.
4.1.4.6 High-Order Fiber-Optic Systolic Array Processors
From Section 4.1.4.2, the FOSAP matrix–matrix multiplier achieves the performance ratio
R = 2.032, which indicates its superiority over other non-fiber and digital electronic processors.
In this section, the performance of the positive-valued high-order FOSAP matrix multiplier is
compared with that of its digital electronic counterpart. For analytical simplicity, the matrix is
assumed to be square and has dimension M, and its elements are n-bit words.
The high-order FOSAP matrix multiplier requires M word cycles to perform the product of x
MxM matrices, and its processing time (or the number of clock cycles) is thus given by
Tx = M [ x( n − 1) + 1]T .
(4.32)
The number of operations (NO) involved in the product of x MxM matrices is
NO = ( x − 1) M 3 .
(4.33)
The number of operations per second (NOPS) performed by the high-order FOSAP matrix multiplier is thus given by
NOPS =
( x − 1) M 2
NO
.
=
Tx
[ x( n − 1) + 1]T
(4.34)
124
Photonic Signal Processing
The number of ADCs required by the high-order FOSAP matrix multiplier is always M . Using
Eqs. 4.32 and 4.27, the Psaltis-Athale performance ratio of the positive-valued high-order FOSAP
matrix multiplier is thus given by
R=
NOPS
( x − 1) M
 M   x −1 
=
≈  

M × 1 T [ x( n − 1) + 1]  n   x 
(4.35)
which is greater than unity if
 x 
M > n
.
 x −1 
(4.36)
The performance of the ratio R in Eq. (4.35) is expected to exceed unity because the matrix dimension M is, for many practical high-order matrix operations, usually a few orders of magnitude larger
than the word length n. This shows the superiority of the high-order FOSAP matrix multiplier over
its digital electronic counterpart. Thus, the high-order FOSAP matrix multiplier may be used to perform various linear algebra operations, such as solutions of algebraic equations, 2-D mathematical
transform, matrix-inversions, and pattern recognition, which require high-order matrix operations.
4.1.5
remarks
The FOSAP matrix multipliers may be used to perform several linear algebra operations, which
require basic matrix–vector and matrix–matrix operations, for advanced signal processing tasks
such as pattern recognition and image processing. For example, the following linear algebra operations involve matrix operations: LU factorization; QR factorization; singular value decomposition;
solution of simultaneous algebraic and differential equations; least squares solution; matrix inversion; and solutions to eigenvalue problems.
The main limitation of any optical DMAC processor is the slow processing speed of the electronic post-processor in which the ADC is often the slowest component with a bit-time limit of T .
This limitation means that the overall performance of any optical DMAC system is highly compromised. It has been predicted that an electronic 8-bit ADC can operate at 1.5 GHz, and that
future development of a 6-bit ADC at 6 GHz is feasible.27 Electro-optic 2-bit and 4-bit ADCs were
experimentally demonstrated to operate in the gigahertz range.28 However, future development of
high-speed and high-resolution optical ADCs, with speed in the gigahertz range and resolution bits
greater than 12, will enable the proposed FOSAP architectures to process more information at a
faster rate than the digital electronic architectures.
• Using the DMAC, HO-DMAC and S-DMAC algorithms, the incoherent FOSAP matrix
multipliers have been designed using optical splitters, optical combiners, programmable
fiber-optic transversal filters and postprocessors consisting of optical detectors, ADCs and
S/A circuits.
• The FOSAP multipliers, which perform best with binary-encoded data, can perform
real-valued digital matrix–vector, matrix–matrix, triple-matrix, and higher-order matrix
computations.
• The positive-valued FOSAP matrix–vector and matrix–matrix multipliers have higher
computational power than the digital electronic multipliers and other optical DMAC multipliers, showing their massive parallel-processing capability.
27
28
C. A. Liechti, High speed transistors: Directions for the 1990s, Microwave J., 30, 165–177, 1989.
R. A. Becker, C. E. Woodward, F. J. Leonberger, and R. C. Williamson, Wide-band electrooptic guided-wave analog-todigital converters, Proc. IEEE, 72, 802–819, 1984.
Photonic Computing Processors
125
• The computational power of any real-valued optical DMAC system incorporating TCB
arithmetic is reduced by a factor approximately equal to the number of matrices to be multiplied compared with the positive-valued optical DMAC systems.
• The positive-valued high-order FOSAP multipliers have higher computational power than
the digital electronic multipliers when the matrix dimension is greater than the word length
by a few orders of magnitude, which is often the case for many signal processing applications. However, the real-valued high-order FOSAP multipliers using TCB arithmetic are
unlikely to offer any advantage over the digital electronic multipliers.
• The FOSAP architectures have a number of advantages including: massive pipeline capability; high computational power; modularity (construction of a larger architecture from several
smaller architectures); and size scalability (the size of the architecture can be increased with
nominal changes in the existing architecture with a comparable increase in performance).
• The FOSAP multipliers may be used as high-performance co-processors in a generalpurpose digital electronic computer to perform various linear algebra operations, such as
those in pattern recognition and image processing.
The optical transversal filter structure described in this Section will be used in the design of incoherent
optical integrators in Section 4.2 and in the design of coherent optical differentiators in Section 4.3.
4.2
PROGRAMMABLE INCOHERENT NEWTON–COTES
OPTICAL INTEGRATOR
In this section, the design of a programmable INCOI is described. The definition, existing design techniques and application of digital integrators are described in Section 4.2.2. A generalized theory of
the Newton–Cotes digital integrators, whose derivation is given in Appendix A, and their magnitude
and impulse responses are given in Section 4.1.2.1. Based on this theory, algorithms for the synthesis
of the INCOI processor are proposed in Section 4.2.2.3. These algorithms are then used in the design
of the programmable INCOI processor, which essentially consists of a microprocessor, fiber-optic
architectures, optical switches, and optical and semiconductor amplifiers. Several types of input pulse
sequences are chosen as examples for illustration of the incoherent processing accuracy of the programmable INCOI processor. In addition, the incoherent RFOSP described in Section 4.1.3 is used in
the design of the programmable INCOI processor. The theory of incoherent fiber-optic signal processing described in Chapter 2 is employed here where intensity-based signals are considered.
4.2.1 Introductory rEMarks
While digital signal processing was originally used in the 1950s as a technique for simulating
continuous-time systems using discrete-time computations, it has since become a field of study in
its own right. One example is a discrete-time (or digital) integrator, which can be used to simulate
the behavior of a continuous-time (or analog) integrator. A digital integrator forms a fundamental
part of many practical signal processing systems because the time integral of signals is sometimes
required for further use or analysis. For example, digital integrators have been used in the design
of compensators for control systems and for measuring the cardiac output, the volume of blood
pumped by the heart per unit time.
A digital integrator is a processor whose output pulse sequence is obtained by approximating the
integral of a continuous-time signal from the samples of that signal. A continuous-time signal x(t ),
whose values are known at the discrete time t = nT for n = 0,1, 2, , where T is the period between
successive samples, can be integrated by a digital integrator. The frequency response of an ideal
digital integrator is given by29
29
W. J. Tompkins and J. G. Webster, (Eds.), Design of Microcomputers-based Medical Instrumentation, Englewood
Cliffs, NJ: Prentice-Hall, 1981.
126
Photonic Signal Processing
 1
0 ≤ ωT (2π ) ≤ 1 2 ,
 jωT ,
H I (ω ) = 
1

, 1 2 < ωT (2π ) ≤ 1,
 j (2π − ωT )
(4.37)
where j = −1, ω is the angular frequency, and T is the sampling period of the integrator.
Digital integrators may be designed by using one of the many classical numerical integration techniques such as the Newton–Cotes, Lagrange, Romberg and Gauss-Legendre formulas.29,30,31,32,33,34
Of these, the Newton–Cotes integration scheme has been extensively used. For example, the wellknown trapezoidal, Simpson’s 1/3, Simpson’s 3/8 and Boole’s integrators all belong to the family of
the Newton–Cotes digital integrators.30–34 The underlying principle of the Newton–Cotes integration
scheme is to fit a continuous-time interpolation polynomial x(t ) to a given input pulse sequence f ( nT )
where f ( nT ) = x( nT ). The continuous-time interpolation polynomial x(t ) is then integrated resulting
t
in y(t ) = ∫0 x(t )dt . Sampling the integrated continuous-time polynomial y(t ) by a digital integrator
at the sampling period yields the output pulse sequence y( nT ). Thus, the output pulse sequence of a
digital integrator effectively approximates the integral of a continuous-time signal according to
nT
y( nT ) =
∫ x(t )dt.
(4.38)
0
The magnitude responses of the Newton–Cotes digital integrators generally approximate the ideal
magnitude response reasonably well over the lower frequency band of 0 ≤ ωT (2π ) ≤ 0.2. The
Newton–Cotes digital integrators may thus be referred to as narrow-band integrators.
Unlike the well-known Newton–Cotes digital integrators, the concept of optical integration is
still new in the area of optical signal processing. In this section, a programmable INCOI processor
is described. Most of the work presented here has been described by Ngo and Binh.35 The derivation
of a generalized theory of the Newton–Cotes digital integrators, which was not given in reference,36
is shown in Appendix A.
4.2.2 newtOn–cOtes Digital integratOrs
4.2.2.1 Transfer Function
The transfer function of the pth-order Newton–Cotes digital integrators can be generally expressed as37
m
H mp ( z ) =
∑
T
C k ( p) ∆ k D ( z )
1 − z − p k =0
(4.39)
T
C0 ( p) + C1( p)∆D( z ) + C2 ( p)∆ 2 D( z ) + + Cm ( p)∆ m D( z ) 
=
1− z− p 
G. F. Franklin, J. D. Powell, and M. L. Workman, Digital Control of Dynamic Systems, 2nd., Reading, MA: Addison–
Wesley, 1990.
31 R. Vich, Z Transform Theory and Applications, Norwell, MA: Kluwer Academic Publishers, 1987.
32 R. Pintelon and J. Schoukens, Real-time integration and differentiation of analog signals by means of digital filtering,
IEEE Trans. Intrum. Meas., 39, 923–927, 1990.
33 M. Abramowitz and I. A. Segun, Handbook of Mathematical Function, New York: Dover Publications, 1964.
34 S. C. Chapra and R. P. Canale, Numerical Methods for Engineers, 2nd ed., Singapore: McGraw-Hill, 1989.
35 N. Q. Ngo and L. N. Binh, Programmable incoherent Newton–Cotes optical integrator, Opt. Commun., 119, 390–402, 1995.
36 A. Ehrhardt, M. Eiselt, G. Großkoptf, L. Küller, R. Ludwig, W. Pieper, R. Schnabel, and G. Weber, Semiconductor laser
amplifier as optical switching gate, J. Lightwave Technol., 11, 1287–1295, 1993.
37 The derivations of Eqs. (4.39) through (4.41) are given in Appendix A.
30
127
Photonic Computing Processors
where the kth coefficient is given by38
p
η 
Ck ( p) =   dη ,
k 
0
∫
(4.40)
∆ k D( z ) = ( −1) k (1 − z −1 ) k ,
(4.41)
and the kth difference equation is given by
where k = 0,1, , m, 1 ≤ p ≤ m, and z = e jωT is the z-transform parameter.25 Note that these digital
integrators are marginally stable because of p poles on the unit circle in the z-plane.
Eq. (4.39) can be generally expressed as the product of the transfer function of the FIR (finite
impulse response) filter and the transfer function of the IIR (infinite impulse response) filter
according to
m
T⋅
H mp ( z ) =
∑b z
k
−k
k =0
1 − az − p
(4.42)
where a = 1 is the pole value, bk is the tap coefficient of the FIR filter, bk = bm − k is positive for m = p ,
and bk is real for m > p . The transfer functions of several families of the Newton–Cotes digital
integrators are tabulated in Table 4.1. Note that H11( z ), H 22 ( z ), H 33 ( z ) and H 44 ( z ) are, respectively,
the well-known transfer functions of the trapezoidal, Simpson’s 1/3, Simpson’s 3/8 and Boole’s
integrators.
Figure 4.9 shows a block diagram of programmable INCOI processor. Figure 4.10a–c show the
magnitude responses of several families of the Newton–Cotes digital integrators. Figure 4.10a
shows that the magnitude response of the trapezoidal integrator H 31( z ) approximates that of the
ideal integrator much better than other integrators of the same or a different family. This does not
necessarily mean that the trapezoidal integrator is superior to other Newton–Cotes integrators
because the time-domain performance, which is described in Section 4.4, needs to be considered.
Figure 4.10a–c show that the impulse responses of several families of the Newton–Cotes digital
integrators are real and positive and hence characterize incoherent systems (see Section 4.2.2.2). As
a result, incoherent Newton–Cotes optical integrators can be synthesized from the Newton–Cotes
digital integrators.
4.2.2.2 Synthesis
One common approach to the optical synthesis of Eq. (4.42) is to cascade the FIR fiber-optic signal
processor (FOSP) with the IIR FOSP. This approach can only be used for the synthesis of incoherent
FOSPs comprising positive tap coefficients, e.g., the tap coefficients bks of H mp ( z ) m = p are all positive
(see Tables 4.1 and 4.2). However, such a technique cannot be used for the case H mp ( z ) m > p where
some of the negative tap coefficients cannot be optically implemented with incoherent FOSPs. An
alternative optical synthesis method, which enables the negative tap coefficients to be implemented
with incoherent FOSPs, is described here. In the following analysis, the transfer function H mp ( z )
and parameters a and bk are associated with any generic digital filters, while Ĥ mp ( z ), â and b̂k are
the variables of their optical counterparts.
38
The binomial coefficient is defined as 
η!
η  η (η − 1) (η − ( k − 1))
.
=
=
k
k!
(η − k )! k !
 
128
Photonic Signal Processing
FIGURE 4.9 Block diagram of the proposed programmable INCOI processor.
When p incoherent IIR FOSPs are incorporated into p higher-order delay lines of an incoherent
FIR FOSP, the overall transfer function of the pth-order INCOI can be described by
m
 bˆ k z − k 
Hˆ mp ( z ) = T ⋅
(4.43)
 ˆ − p ,
1 − az 
k =0 
∑
0,
where a = 
a = 1,
for k = 0,1,,m − p,
for k = m − ( p − 1),, m.
Eq. (4.43) can be rewritten as
m
T⋅
Hˆ mp ( z ) =
∑ ( bˆ − bˆ aˆ ) z
k
k =0
k− p
ˆ −p
1 − az
−k
,( bˆ i = 0 for i < 0).
(4.44)
The synthesis technique requires of Eq. (4.44) requires the following necessary and sufficient
conditions:
â= a= 1,
(4.45)
bk = bk > 0,( k = 0,1,, p − 1)
(4.46)
129
Photonic Computing Processors
FIGURE 4.10 Graphical illustration of the performance of the programmable trapezoidal INCOI processor
Ĥ11 ( z ) in processing a rectangular input pulse sequence. (a) TW-SLA gain profile of G0 [n]. (b) TW-SLA
gain profile of G1[n]. (c) Rectangular input pulse (rectangular profile assumed) sequence x[n] = 1 for n ≥ 0.
(d) Output pulse sequence y[n].
k
b̂k =
∑b
k − pq
a q > 0, ( k = p, p + 1,, m)
(4.47)
q=0
where bi = 0 for i < 0. The advantage of this synthesis method is evident from Eqs. (4.45) through
(4.47), where the new optical tap coefficient ( bˆ k − bˆ k − p aˆ ) can be made positive or negative depending on the design requirements. Table 4.1 shows that the computed optical tap coefficients are
positive, showing the effectiveness of the synthesis method. As a result, the requirement for the
synthesis of the incoherent INCOI processor is met. Thus, the optical synthesis technique described
here provides greater design flexibility than the conventional approach.
130
Photonic Signal Processing
TABLE 4.2
Digital Tap Coefficients of Several Families of the Newton–Cotes
Digital Integrators, as Computed From Eqs. (4.43)–(4.47), with
Transfer Functions Expressed in the form of Eq. (4.39 or 4.42)
H mp ( z)
b0
b1
b2
b3
b4
b5
2T H11 ( z )
1
5
9
251
475
1
8
19
646
1427
0
−1
−5
−264
−798
0
0
1
106
482
0
0
0
−19
−173
0
0
0
0
27
1
29
28
4
124
129
1
24
14
0
4
14
0
−1
−6
0
0
1
Filter Family
Trapezoidal
−1
−1
12T H 21 ( z )
24T −1 H 31 ( z )
720T −1 H 41 ( z )
1440T −1 H 51 ( z )
Simpson’s 1/3
3T −1 H 22 ( z ) =

90T −1 H 42 ( z )
90T −1 H 52 ( z ) 
Simpson’s 3/8
Boole
Fifth-order
(8 3)T −1 H 33 ( z )
−1
( 45 2)T H 44 ( z )
−1
288T H 55 ( z )
1
3
3
1
0
0
7
32
12
32
7
0
95
375
250
250
375
95
4.2.2.3 Design of a Programmable Optical Integrating Processor
Based on the optical synthesis technique described in the previous section, the design of a programmable INCOI processor is outlined in this section.
4.2.2.3.1 Adaptive Algorithm for the INCOI Processor
One possible approach for overcoming the marginal stability of the INCOI processor is to replace
the IIR filter, as described by Eq. (4.42), by another FIR filter knowing that
∞
∑
1
=
( az − p )i .
1 − az − p i = 0
(4.48)
This approach is not practically feasible because of the infinite number of taps involved in the FIR
filter. However, if the order of the INCOI processor is low (e.g., p = 1) and the duration of the input
pulse sequence to be processed is short (e.g., a 20-tap INCOI processor is sufficient to process a
signal with a duration of 20 sampling intervals), then such a technique can be useful for this specific
requirement. The drawback of this method is that only the positive tap coefficients of the two cascaded FIR filters can be optically implemented with incoherent FOSPs. As a result, the technique is
restricted to the INCOI processor with m = p.
An adaptive algorithm for overcoming the marginal stability of the INCOI processor is described
here. The INCOI impulse response ĥmp [n] is given by the inverse z-transform of Eq. (4.48) as
T −1 hˆ mp [n] = bˆ 0δ [n] + + bˆ m− pδ [n − ( m − p)] +
m
∑
k = m−( p −1)
bˆ k aˆ ( n−k ) p
∞
∑δ [n − (k + pq)] (4.49)
q =0
131
Photonic Computing Processors
where the unit-sample sequence is defined as25
1,
δ [n] = 
0,
for n = 0,
(4.50)
for n ≠ 0,
and the delayed unit-sample sequence is defined as
1,
δ [n − i ] = 
0,
for n = i ,
(4.51)
for n ≠ i ,
where n is the discrete-time index, i.e., n = 1 means one unit-time delay T. Optical implementation
of the positive tap coefficients in Eq. (4.49), requires adaptive control of the optical gain values such
that
T −1 hˆ mp [n] = G0 [0] + + Gm − p [m − p] +
m
∑ G [n]aˆ
k
( n−k ) p
,
(4.52)
k = m − ( p −1)
where the time-variant optical gains are given by
Gk [n] n=k = bˆ k ,( k = 0,1,,m − p),
Gk [n] = bˆ k aˆ −( n−k ) p
(4.53)
∞
∑ δ[n − (k + pq)],(k = m − ( p − 1),, m)
(4.54)
q =0
where 0 < â < 1. The summation term in Eqs. (4.49) and (4.54) shows that the pole value â is not
strictly required to be â = 1 but can take any value in the range of 0 < â < 1 while still maintaining
the characteristics and overcoming the marginal stability of the INCOI processor. The dynamic
range of the time-variant optical gain in Eq. (4.54) can be increased by setting the pole value a as
close to unity as possible (e.g., â = 0.95) but not so close to unity that the original problem of marginal stability reappears.
4.2.2.3.2 Analysis of the Programmable INCOI Processor
The traveling-wave semiconductor laser amplifiers (TW-SLAs) are considered to be the optical
gain elements in the programmable INCOI processor because of their fast switching speed
(see Section 4.2.2). The TW-SLA can operate at either in the 1300 nm O-band or 1550 nm C-band.
The TW-SLA gain depends on both the injection current and the injected light power; it increases
with increasing injection current but decreases with increasing injected power. Thus, the injected
power level must be chosen to within the range over which the TW-SLA gate can be driven into
saturation where the gain does not differ significantly. With this fixed level of input power at a particular operating wavelength, the required TW-SLA gain can be obtained from the injection current
source driving the gate. The effects of polarization sensitivity, arising from different transverse
electric (TE) and transverse magnetic (TM) mode confinement factors, and the amplifier spontaneous emission (ASE) noise of the TW-SLAs are ignored here because they can be easily overcome
in practice. A polarization insensitive TW-SLA has been demonstrated to be capable of achieving
an effective gain of up to 20 dB, with less than 1 dB polarization sensitivity, less than 1 dB spectral
gain ripple, and a 3 dB bandwidth of 55 nm. The ASE noise (~7 dB noise figure) can be minimized
by means of a tunable optical filter with sufficient narrow bandwidth.
Figure 4.9 shows a block diagram of the proposed microprocessor ( µ P ) controlled INCOI processor. The following discussion focuses on the µ P-controlled TW-SLAs. The software-controlled
132
Photonic Signal Processing
µ P chip is used to control the injection current sources driving the TW-SLAs to provide the required
time-variant optical gains according to Eq. (4.53).
Figure 4.10 shows a graphical illustration of the performance of the programmable trapezoidal INCOI processor Ĥ11( z ) in processing a rectangular input pulse sequence, x[n] = 1 for n ≥ 0.
Figure 4.10a and b show, respectively, the profiles of the time-variant TW-SLA gains G0 [n] and
G1[n]. From Eq. (4.53), the time-variant TW-SLA gain Gk [n] n=k is only active at the appropriate discrete time n. This is shown in Figure 4.10a where G0 [n] n=0 = 0.5 is only applied at time n = 0. It is
assumed that G0 [n] has already reached its steady-state value when the first pulse of the input pulse
sequence arrives. Figure 4.10b shows that the time-variant TW-SLA gain G1[n] is active at every
clock period Tµ of the µ P and takes TSLA, the switching time of the TW-SLA, to reach its steadystate and thereby provides the required gain G1[n] according to Eq. (4.53).
From Figure 4.10a–c, the following timing requirements must be met in order to obtain optimum
performance of the programmable INCOI processor:
τ w << τ ,
(4.55)
τ = qTSLA ,
(4.56)
T = qτ , q = 1, 2,
(4.57)
Tµ = T .
(4.58)
Eq. (4.55) indicates that the full-width half-maximum (FWHM) of the input pulse Tw must be
very much less than the bit period T of the input pulse sequence in order to satisfy the incoherent
requirement of the INCOI processor. Eq. (4.56) requires τ to be a multiple integer of TSLA,
e.g., Figure 4.10 shows the case where τ = TSLA . Eq. (4.57) shows that the multiple integer of τ
must be equal to the unit-time delay T, e.g., Figure 4.10 shows the case where T = 3τ. Eq. (4.58)
requires the clock period Tµ of the µ P to be equal to the unit-time delay T of the INCOI processor
in order to achieve system synchronization. The conditions described by Eqs. (4.55) and (4.58)
are necessary for obtaining high processing accuracy. The deviation of the unit-time delay from
below (−∆T) or above (+∆T) its nominal value T by /∆T/, as shown in Figure 4.10b, would not
cause significant performance degradation on the programmable INCOI processor provided that
the following condition is satisfied:
∆T ≤
τw
.
2
(4.59)
The output pulse sequence is shown in Figure 4.10d. Note that the speed (1 T ) of the programmable
INCOI processor is ultimately limited by the µ P speed (1 Tµ ) , the clock rate at which the µ P
sequentially fetches and executes one instruction after another until a halt instruction is processed.
Commercially available 32-bit µ P s are capable of achieving a speed of 60 MHz.
4.2.2.4 Analysis of the FIR Fiber-Optic Signal Processor
The delay and loss of the FIR FOSP block, as highlighted in Figures 4.9 and 4.10, are analyzed. The
insertion loss of an 1× ( m + 1) optical splitter ( Los ) or an ( m + 1) × 1 optical combiner ( Loc ) are given by
Los = Loc =
( Ls/c Ldc )log2 ( m +1)
,
( m + 1) Ls/c
(4.60)
Where Ls/c is the splice (or connector) loss and Ldc is the insertion loss of the 3-dB fiber-optic directional coupler (DC). Note that the structures of the optical splitter and combiner have already been
described in Chapter 2.
133
Photonic Computing Processors
4.2.2.4.1 Delay Analysis of the FIR Fiber-Optic Signal Processor
The delay analysis considered here is associated only with the optical components inside the FIR
FOSP block of Figure 4.9. In the following analysis, it is assumed that the cross talks of the optical
switches are acceptably low, e.g., < −50 dB. It is presumed that the µ P -controlled optical switches
S1 Sm , Sk ∈ (0,1) , of the FIR FOSP are in the bar state so that the optical intensity signals from
the 1× ( m + 1) optical splitter (OS1) are routed directly to the ( m + 1) × 1 optical combiner (OC1). The
FIR FOSP requires the differential delay between neighboring fiber delay lines to be exactly one
unit-time delay T, i.e.,
Tk = T0 + kT ,( k = 0,1, , m),
(4.61)
where Tk is the delay of the kth fiber path between the output of the OS1 and the input of the OC1.
4.2.2.4.2 Loss Analysis of the FIR Fiber-Optic Signal Processor
The loss analysis considered here is associated only with the optical components inside the FIR
FOSP block. An optical intensity signal coming into the input port (IN) of the FIR FOSP will experience an intensity loss LFIR at its output port (OUT) according to
LFIR = Ls/c 7 ⋅ Los1 ⋅ Lsw ⋅ (0.5Ldc ) ⋅ Loc1,
(4.62)
where Lsw is the insertion loss of the optical switch, and Los1 and Loc1 are, respectively, the insertion
losses of the OS1 and the OC1 (see Eq. 4.60). The loss in Eq. (4.62) can be compensated by a timeinvariant (or fixed) TW-SLA gain GFIR placed at the front of the FIR FOSP such that
GFIR =
2( m + 1)2 T
L5s/c Ldc Lsw ( Ls/c Ldc )log2 ( m+1)
2
.
(4.63)
Eq. (4.63) has been obtained by assuming that all the fiber paths of the FIR FOSP have the same
intensity loss. However, the loss of the upper fiber path (where k = 0) is less than those on the
lower fiber paths (where k = 0,1, 2 …) because of the absence of one splice (or connector), one optical switch and one 3-dB DC. The absence of this loss must be incorporated into the time-variant
TW-SLA gain G0 [0] so that every fiber path of the FIR FOSP has the same intensity loss. As a
result, the new G0 [0] in Eq. (4.55) now becomes
G0 [0] = (0.5Ls/c Lsw Ldc ) bˆ 0 .
(4.64)
4.2.2.5 Analysis of the IIR Fiber-Optic Signal Processor
The programmable fiber loop length (PFLL), the delay and loss of the IIR FOSP block, as highlighted in Figure 4.9, are described here. From Eq. (4.42), p incoherent IIR FOSPs are required
to be incorporated into the p higher-order delay lines of the FIR FOSP. This requirement can be
alternatively achieved, as shown inside the IIR FOSP block of Figure 4.9, with a mx1 optical combiner (OC2), a 1xm optical splitter (OS2) and only one (instead of p) IIR FOSP. It is clear that this
IIR FOSP block is structurally simpler and yet performs the same function as if p IIR FOSPs are
introduced into the p higher-order fiber paths of the FIR FOSP. Note that the IIR FOSP, which is
an all-pole filter, has already been described in Chapter 2 where it was referred to as the incoherent
RFOSP.
4.2.2.5.1 Programmable Fiber Loop Length
The PFLL is schematically shown and highlighted inside the IIR FOSP block (see Figure 4.9). The
intensity signal coming into the PFLL is optionally routed through N fiber segments whose lengths
134
Photonic Signal Processing
are arranged in a binary sequence of lengths giving delays of 2i T (i = 0,1, , N − 1) so that 2 N − 1
different delays with a resolution of T can be selected. Selection of the appropriate fiber segments
is by the µP-controlled optical switches, which permit the signal either to connect or bypass a particular fiber segment according to the state pi ∈ (0,1). The PFLL requires N optical switches and one
3-dB DC. For example, Figures 4.9 and 4.10 show the case where N = 4. The required fiber loop
delay T p,IIR of the pth-order IIR FOSP is given by
T p,IIR = pT ,(1 ≤ p ≤ m),
(4.65)
as required by the loop delay z − p . The integer value p in Eq. (4.65) takes the binary representation
of the N fiber segments as
N −1
p=
∑ p 2 , p ∈ (0,1),
i
i
i
(4.66)
i =0
which lies in the range of 1 ≤ p ≤ 2 N − 1. Thus, the number of optical switches required to achieve p
unit-time delays is given by
N ≥ log2 ( p + 1)
(4.67)
which is relatively small because of its logarithmic relation.
4.2.2.5.2 Delay Analysis of the IIR Fiber-Optic Signal Processor
The delay analysis considered here is associated only with the optical components inside the IIR
FOSP block of Figure 4.9. It is presumed that the µP-controlled optical switches S1 Sm of the
FIR FOSP are now in the cross state so that the intensity signals coming into these switches are
routed through the OC2 (See Figure 4.9). Selection of the states of the switches is governed by Eq.
(4.66). There are m + 1 possible paths through which the signal at the input port can propagate
to the output port. Each of these paths introduces a delay Dk between the input and output ports
according to
D0 = τ FIR + 2Tos1 + T0 ,
(4.68)
Dk = τ FIR + 2Tos1 + 2Tos2 + τ 1 + τ 2 + τ 3 + Tk1 + Tk 2 ,( k = 1, 2,, m)
(4.69)
where Tos1 = Toc1 and Tos2 = Toc2 have been used. The parameters of Eq. (4.69) are defined as follows:
Tos1 is the propagation delay of the OS1;
Toc1 is the propagation delay of the OC1;
Tos2 is the propagation delay of the OS2;
Toc2 is the propagation delay of the OC2;
τ FIR is the delay of the fiber path between the input port IN and the input of the OS1;
τ 1 is the delay of the fiber path between the output of the OC2 and the input of the IIR FOSP via
the TW-SLA with gain GR1( N );
τ 2 is the delay of the fiber path between the two 3-dB DCs via the TW-SLA with gain GR2 ( N );
τ 3 is the delay of the fiber path between the output of the IIR FOSP and the input of the OS2;
Tk1 is the delay of the kth fiber path between the output of the OS1 and the input of the OC2 via
the kth optical switch;
Tk 2 is the delay of the kth fiber path between the output of the OS2 and the input of the OC1 via
the 3-dB DC and the kth time-variant TW-SLA with gain Gk [n]. The delay D0 in Eq. (4.68)
is the pure propagation delay of the INCOI processor.
135
Photonic Computing Processors
Again, the FIR FOSP requires the differential delay between neighboring fiber delay lines to be
exactly one unit-time delay T, i.e.,
Dk − D0 = kT ,( k = 0,1, , m),
(4.70)
Tk = 2Tos2 + τ 1 + τ 2 + τ 3 + Tk1 + Tk 2 ,( k = 1, 2,, m)
(4.71)
or
when Eqs. (4.61) and (4.68) are substituted into (4.71). The µ P -controlled optical switches S1 Sm
must be active at the instant when the first pulse of the input pulse sequence arrives. This timing
requirement can be met by having
τ FIR + Tos1 = Tsw + Tµ ,
(4.72)
where Tsw is the switching time of the optical switches.
4.2.2.5.3 Loss Analysis of the IIR Fiber-Optic Signal Processor
The loss analysis considered here is associated only with the optical components inside the IIR
FOSP block (Figure 4.9). The loop gain Gloop (ω ) of the IIR FOSP is given by
Gloop (ω ) = (0.5Ldc )3 ⋅ Ls/c N + 4 ⋅ LNsw ⋅ GR2 ( N ) ⋅ exp[− jω ( pT + τ 2 + τ 4 + τ 6 )],
(4.73)
where τ 4 is the delay of the fiber path between the 3-dB DC, and the optical switch p0 and τ 6 is
the delay of the fiber path between the 3-dB DC of the PFLL and another 3-dB DC. In the PFLL,
the effect of the delay of the lower fiber line between neighboring optical switches, such as τ5 has
been ignored in Eq. (4.73) because the states of the optical switches are not known in advance to
the designer. Comparison of Eqs. (4.61) and (4.65) requires τ 2 = τ 4 = τ 6 = 0, which is not practically
attainable. However, the effects of the delays τ 2, τ 4 , τ 6, and τ 5 can be ignored without much deterioration of the performance of the programmable INCOI processor if they can be made sufficiently
small compared to T, for example (τ 2,τ 4 ,τ 6,τ 5) << T . The absolute value of the loop gain in Eq. (4.73)
is required to be equal to â according to Eq. (4.42) such that
GR2 ( N ) =
8aˆ
L L L
3 N N +4
dc sw s/c
,(0 < aˆ < 1).
(4.74)
In the transfer function given by Eqs. (4.42), (4.46), and (4.48), the numerator of the IIR FOSP is
equal to unity, and this requires the forward-path gain of the kth fiber path between the input of the
OC2 and the output of the OS2 to be equal to unity, i.e.,
L2os2 ⋅ L6s/c ⋅ (0.5Ldc )2 ⋅ GR1( N ) ⋅ GR 2 ( N ) = 1,
(4.75)
or
GR1( N ) =
when Los 2 = Loc 2 is used.
Ldc LNsw LNs/c−2
,
ˆ 2os2
2 aL
(4.76)
136
Photonic Signal Processing
4.2.2.5.4 Timing Requirements for Multiple Input Pulse Sequences
The timing requirements for sequential processing of multiple input pulse sequences are described
here. The INCOI processor time, TINCOI , the time required to process the input pulse sequence of
duration, Tpulse , is given by
TINCOI = 2Tpulse − T ,
(4.77)
where Tpulse >> T . The next timing requirement assumes that each input pulse sequence is processed
by a different family of the INCOI processor. The pulse strobe period, Tpsp, defined as the time
separation between the first pulse of the present input pulse sequence and the first pulse of the next
input pulse sequence, and the programmability period, Tpr , defined as the time separation between
the setup of two consecutive programs, must be equal to the INCOI processor time, i.e.
Tpsp
= T=
TINCOI .
pr
(4.78)
4.2.2.5.5 Remarks
This section describes the processing accuracy of the ideal and non-ideal INCOI processor in the
time domain. For analytical simplicity, the timer instant n means nT and the sampling period is
assumed to be T = 1.
The processing accuracy of the INCOI processor is evaluated as
Error Response =
True Integral − Output Pulse Sequence
× 100 %
True Integral
(4.79)
where the True Integral corresponds to the true integral of the input pulse sequence x [ n], and the
Output Pulse Sequence corresponds to the convolution of the impulse response h mp [n] of the INCOI
processor with the input pulse sequence x [ n].
There are two major error sources in the implementation of the programmable INCOI processor.
The first arises from the deviation of the optical tap coefficient, that is ∆ b̂k , caused by the inaccuracy
in the time-variant TW-SLA gain Gk [n], from its nominal value b̂k . However, the effect of the deviation of the fiber loop gain value (i.e., ∆ â), caused by the inaccuracy in the TW-SLA gain GR2 [n],
from its nominal value â can easily be included in Gk [n]. The second error source is due to the deviation of the unit-time delay from its nominal value T, which is ∆T, caused by the inaccurate cutting
of fiber lengths. The effects of the detector noise and crosstalks of the optical switches are assumed
to be negligible here.
The performance of the trapezoidal INCOI processor Ĥ11( z ) in processing the linear input pulse
sequence is now considered in detail. The basic time delay of the INCOI processor is defined as
T1 − T0 = T , which is used as a basis for the higher-order delays (i.e., Tk − T0 = kT , k = 2, 3, , m) of
the FIR FOSP for m ≥ 2 and for the fiber loop delay T p,IIR = pT of the IIR FOSP. Thus, the trapezoidal integrator requires the nominal value of the fiber loop delay T1,IIR of the IIR FOSP to be exactlyT
(i.e., T1,IIR = T ) and any deviation from this nominal value is denoted as ∆T1,IIR .
Table 4.3 shows three cases of the ±5% deviation of the fiber loop delay ∆T1,IIR T1,IIR and their corresponding ±5% deviations of all the possible sets of the optical tap coefficients {∆ bˆ 0 bˆ 0 , ∆ bˆ 1 bˆ 1 }.
The percent relative errors (REs) at n = 20 are also shown for each case.
Figures 4.11 through 4.13 show the percentage error responses of the trapezoidal integrator for
the three cases. Curves Ai , Bi and Ci (i = 1, , 9) show the error responses of cases A, B, and
C with the corresponding ith set of optical tap coefficients. The solid curve A1 in Figure 4.11a
shows the performance of the ideal trapezoidal integrator, where there are no deviations of the fiber
loop length (i.e., ∆T1,IIR = 0 ) and the optical tap coefficients (i.e., ∆b 0 = ∆b 1 = 0) from their nominal
values. Curve A1 shows that the trapezoidal integrator has zero processing error at the sampling
137
Photonic Computing Processors
TABLE 4.3
Optical Tap Coefficients of Several Families of the Synthesized INCOI
Processor as Computed from Eqs. (3.44b) and (3.44c)
INCOI Family
H mp (z )
b 0
b 1
Trapezoidal
T −1 H 11 ( z )
T −1 H 21 ( z )
0.50
1.0
0
0
0
0
0.4167
1.0833
1.0
0
0
0
T −1 H 31 ( z )
T −1Hˆ 41 ( z )
0.375
1.1667
0.9583
1.0
0
0
0.3486
1.2458
0.8792
1.0264
1.0
0
T Hˆ 51 ( z )
0.3299
1.3208
0.7667
1.1014
0.9812
1.0
T Hˆ 22 ( z ) =
T −1Hˆ 42 ( z )
0.3333
1.3333
0.6667
0
0
0
0.3222
1.3778
0.5889
1.4222
0.5778
0
T −1Hˆ 52 ( z )
0.3111
1.4333
0.4667
1.5889
0.40
1.6
Simpson’s 3/8
T Hˆ 33 ( z )
0.375
1.125
1.125
0.50
0
0
Boole
T Hˆ 44 ( z )
0.3111
1.4222
0.5333
1.4222
0.6222
0
Fifth-order
T Hˆ 55 ( z )
0.3299
1.3021
0.8681
0.8681
1.3021
0.6597
−1
Simpson’s 1/3
−1
−1
−1
−1
b 2
b 3
b 4
b 5
FIGURE 4.11 Relative error response (%) of the trapezoidal INCOI processor H 11 ( z) in processing the linear
input pulse sequence x[n] = n for n ≥ 0. (a) Cases A1–A4, and (b) Cases A5–A9 correspond to the conditions
given in Table 4.3. They are given in the Appendix.
138
Photonic Signal Processing
FIGURE 4.12 Relative error response (%) of the trapezoidal INCOI processor H 11 ( z) in processing the linear
input pulse sequence x[n] = n for n ≥ 0. (a) Cases B1–B4, and (b) Cases B5–B9 correspond to the conditions
given in Table 4.3.
FIGURE 4.13 Relative error response (%) of the trapezoidal INCOI processor H 11 ( z) in processing the linear
input pulse sequence x[n] = n for n ≥ 0. (a) Cases C1–C4, and (b) Cases C5–C9 correspond to the conditions
given in Table 4.3.
Photonic Computing Processors
139
FIGURE 4.14 Relative error response (%) of the trapezoidal INCOI processor Ĥ11 ( z ) in processing the linear
input pulse sequence x[n] = n for n ≥ 0. (a) Cases A1–A4 (See Appendix), and (b) Cases A5–A9 correspond to
the conditions given in Table 4.3.
instants (i.e., n = 0,1, 2,) in processing the linear input pulse sequence. This is as expected from
the numerical integration scheme where the trapezoidal rule is equivalent to approximating the area
of the trapezoid under the straight line.42
However, Figures 4.14 through 4.16 show that the processing errors are large between the sampling instants and eventually reach their steady states after n = 20 sampling intervals, beyond which
they do not differ significantly. The steady-state errors of Figures 4.15, and 4.16 are tabulated in
Table 4.3, which shows that the cases {A1,A4,A7}, {B1,B4,B7} and {C1,C4,C7}, where ∆ b̂1 = 0 ,
have smaller processing errors than other sets belonging to the same cases. Thus, the performance
of the trapezoidal integrator is greatly affected by large deviations of b̂1. In addition, the absolute
values of the steady-state errors for all cases in Tables 4.1 and 4.2 are less than 6%. Thus, the ±5%
parameter deviation may be considered as an acceptable upper bound value in the implementation
of the programmable INCOI processor.
Next, the performances of several families of the ideal INCOI processor are analyzed for several
types of input pulse sequence. It was found that the ideal Simpson’s 1/3 integrator H 22 ( z ) has zero
processing error of the parabolic input pulse sequence (i.e., x[n] = n2, n ≥ 0) at the sampling instants.
This is also expected because the numerical integration scheme was employed.39
From Table 4.4, the Simpson’s 1/3 integrator Ĥ 22 ( z ) has higher processing accuracy of the cubic
input pulse sequence (i.e., x[n] = n3 , n ≥ 0) than other families of the INCOI processor, and the error
rapidly decreases with time and eventually converges to zero in the steady state. For the fourth-order
polynomial input pulse sequence (i.e., x[n] = n4, n ≥ 0), Table 4.5 shows that the integrator H 52 ( z ) outperforms other families of the INCOI processor and has negligibly small processing error for n ≥ 3.
39
S. C. Chapra and R. P. Canale, Numerical Methods for Engineers, 2nd ed., Singapore: McGraw-Hill, 1989.
140
Photonic Signal Processing
FIGURE 4.15 Relative error response (%) of the trapezoidal INCOI processor Ĥ11 ( z ) in processing the linear
input pulse sequence x[n] = n for n ≥ 0. (a) Cases B1–B4, and (b) Cases B5–B9 correspond to the conditions
given in Table 4.3.
FIGURE 4.16 Relative error response (%) of the trapezoidal INCOI processor Ĥ11 ( z ) in processing the linear
input pulse sequence x[n] = n for n ≥ 0. (a) Cases C1–C4, and (b) Cases C5–C9 correspond to the conditions
given in Table 4.3.
141
Photonic Computing Processors
TABLE 4.4
Relative Errors (REs) (%) at n = 20 of the Trapezoidal INCOI Processor H 11(z ), with ± 5%
%
Deviations of the Optical Tap Coefficients and ± 5%
% Deviation of the Fiber Loop Delay, in
Processing the Linear Input Pulse Sequence x[n] = n for n ≥ 0
Case A
∆T1,IIR
=0
T1,IIR
Case B
Case C
∆T1,IIR
= + 0.05
T1,IIR
∆T1,IIR
= − 0.05
T1,IIR
Set
∆ b 0
b 0
∆ b 1
b 1
RE (%)
∆ b 0
b 0
∆ b 1
b 1
RE (%)
∆ b 0
b 0
∆ b 1
b 1
RE (%)
1
2
3
4
5
6
7
8
9
0
0
0
−0.05
+0.05
+0.05
−0.05
−0.05
−0.05
0
+0.05
−0.05
0
+0.05
−0.05
0
+0.05
−0.05
0
−4.75
+4.75
−0.25
−0.5
+4.5
+0.25
−4.5
+5.0
0
0
0
+0.05
+0.05
−0.05
−0.05
−0.05
−0.05
0
+0.05
−0.05
0
+0.05
−0.05
0
+0.05
−0.05
+0.463
−4.625
+5.19
+0.123
−4.514
+4.491
+0.712
−4.016
5.439
0
0
0
+0.05
+0.05
+0.05
−0.05
−0.05
−0.05
0
+0.05
−0.05
0
+0.05
−0.05
0
+0.05
−0.05
−0.488
−5.261
+4.286
−0.738
−5.512
+4.036
−0.237
−5.011
+4.537
TABLE 4.5
Relative Errors (%) of Several Families of the Ideal INCOI Processor, at the Sampling Instants, in
Processing the Cubic Input Pulse Sequence x[n] = n3 for n ≥ 0. Error at Time n = 0 Is Undefined
Time, n
Ĥ mp ( z )
1
2
3
4
H 11 ( z )
H 21 ( z )
−100
−25
−11.1
−6.25
−4.0
−2.5E−03
H 31 ( z )
H 41 ( z )
−66.7
−10.4
−3.29
−1.43
−0.75
−1.25E−05
−50
−4.17
−0.82
−0.26
−0.107
−4.17E−08
−39.4
−0.87
−0.041
−0.013
−0.0053
−2.08E−09
−31.9
1.007
0.051
−0.013
−0.0053
−2.08E−09
H 22 ( z ) =
H 42 ( z )
H 52 ( z )
−33.3
−28.9
−24.4
0
1.11
1.94
−0.41
−0.302
−0.41
1.11E–14
0.0694
0.104
−0.053
−0.0391
−0.053
0
1.11E−08
1.67E−08
H 33 ( z )
−50
−3.13
0
−0.195
−0.08
−3.13E−08
H 44 ( z )
−24.4
2.22
−0.302
0
−0.0391
0
H 55 ( z )
−31.9
1.48
0.29
−0.125
−1.82E−14
0
H 51 ( z )
5
200
It was also found that the overall performance of the ideal Boole’s integrator H 44 ( z ) is superior to
other families of the INCOI processor in processing the rth-order polynomial input pulse sequence
(i.e., x[n] = nr , r ≥ 5 for n ≥ 0). From Tables 4.4 and 4.5, as the order of the polynomial input pulse
sequence is increased (i.e., r ≥ 2), the higher-order trapezoidal integrator H m1( z ) m≥2 outperforms
the lowest-order trapezoidal integrator H 11( z ). This shows the advantage of using a higher-order
trapezoidal integrator.
142
Photonic Signal Processing
For the exponential input pulse sequence, that is x[n] = exp( − n w ) for n ≥ 0, with the FWHM
pulse width w = 50. Table 4.6 shows that the ideal trapezoidal integrator H 11( z ) has the smallest
processing error among other families of the INCOI processor, and its steady-state error converges
to −1.006% at n = 300. Its processing error was also found to decrease with increasing pulse width,
for example, when w = 100 the steady-state error was reduced to −0.527% at n = 300, which is about
half of that for the case w = 50 (Table 4.7).
TABLE 4.6
Relative Errors (%) of Several Families of the Ideal INCOI Processor, at the Sampling Instants,
in Processing the Fourth-order Polynomial Input Pulse Sequence x[n] = n4 for n ≥ 0. Error at
Time n = 0 is Undefined
Time, n
Ĥ mp ( z )
1
2
3
4
5
200
Ĥ11 ( z )
−150
−40.6
−18.3
−10.4
−6.64
−4.2E–03
−108
−21.1
−7.17
−3.23
−1.72
−3.1E–05
−87.5
−11.9
−2.88
−0.99
−0.427
−1.97E–07
Ĥ 21 ( z )
Ĥ 31 ( z )
−74.3
−6.62
−0.926
−0.219
−0.072
−7.02E–10
Ĥ 51 ( z )
−64.9
−3.1
−0.0386
1.39E–14
1.82E–14
−7.51E–13
Ĥ 22 ( z )
−66.7
−4.17
−0.823
−0.26
−0.107
−4.17E–08
Ĥ 41 ( z )
−61.1
−2.08
−0.274
−0.065
−0.0213
−2.08E–10
Ĥ 52 ( z )
−55.6
−0.174
0
0
0
−2.03E–13
Ĥ 33 ( z )
−87.5
−11.3
−1.85
−0.525
−0.26
−9.39E–08
Ĥ 44 ( z )
−55.6
0
0.229
−1.39E–14
−0.0178
−1.91E–13
Ĥ 55 ( z )
−64.9
−2.81
0.37
0.0634
0
0
Ĥ 42 ( z )
TABLE 4.7
Relative Errors (%) of Several Families of the Ideal INCOI Processor, at the Sampling Instants,
in Processing the Exponential Input Pulse Sequence x[n] = exp(− n w ) for n ≥ 0 Where
w = 50. Error at Time n = 0 is Undefined
Time, n
Ĥ mp ( z )
1
2
3
4
5
300
Ĥ11 ( z )
−50.51
−25.51
−17.18
−13.01
−10.51
−1.006
−50.67
−25.59
−17.22
−13.05
−10.54
−1.006
−54.96
−25.59
−17.23
−13.05
−10.54
−1.006
Ĥ 21 ( z )
Ĥ 31 ( z )
−60.35
−24.22
−17.23
−13.05
−10.54
−1.006
Ĥ 51 ( z )
−66.06
−21.31
−17.89
−13.05
−10.54
−1.006
Ĥ 22 ( z )=
−67.67
−17.00
−23.01
−8.67
−14.08
−0.668
Ĥ 41 ( z )
−71.06
−14.71
−24.55
−7.503
−15.02
−0.578
Ĥ 52 ( z )
−75.57
−10.71
−27.63
−5.168
−16.91
−0.398
Ĥ 33 ( z )
−50.76
−32.01
−12.89
−13.07
−13.19
−0.752
Ĥ 44 ( z )
−74.45
−13.56
−23.79
−8.093
−15.49
−0.624
Ĥ 55 ( z )
−64.17
−25.54
−12.67
−17.52
−6.933
−0.661
Ĥ 42 ( z )
Photonic Computing Processors
143
The processing errors of the families of the INCOI processor are large over the initial time interval but eventually converge to a negligibly small, if not zero, value in the steady state. Note that, if
the deviation of the basic time delay satisfies the condition at which the performance of the INCOI
processor is not greatly affected by this factor. It is worth mentioning that the rectangular integrator,
whose transfer function is given by Ĥ R ( z ) = z −1 / (1 − z −1 ), has zero processing error of the constant
(or rectangular) input pulse sequence, such as that shown in Figure 5.4c, at the sampling instants.
4.2.3 sectiOn remarks
• A generalized theory of the Newton–Cotes digital integrators has been developed based
on a programmable INCOI processor that has been designed using a microprocessor, fiberoptic architectures, optical switches, and optical and semiconductor amplifiers.
• The pth-order programmable INCOI processor has high processing accuracy of the pthorder polynomial input pulse sequence (i.e., x[n] = n p , p ≥ 1), while the trapezoidal processor, Ĥ11( z ), outperforms other higher-order processors in processing the exponential
pulse (i.e., x[n] = exp( − n w ) , n ≥ 0). In general, the incoherent processing accuracy of a
particular processor order depends very much on the type of input pulse.
• Optical integration is a new concept with many potential applications. One example is to
apply the trapezoidal optical integrator described here to the design of an optical darksoliton generator, which is outlined in Section 4.3.
4.3
HIGHER-DERIVATIVE FIR OPTICAL DIFFERENTIATORS
In this section, a theory of higher-derivative FIR (finite impulse response) optical differentiators
is proposed (see Figure 4.17). Section 4.3.1 describes underlying principle of digital differentiators
and their design techniques and applications. Section 4.3.3 presents a theory of the qth-order pthderivative FIR digital differentiator whose derivation is given in Section 4.4. Section 4.3.3 describes
the synthesis of a qth-order pth-derivative FIR optical differentiator using an optical transversal
filter structure, as described in Sections 3.2 and 3.3, which consists of integrated-optic components.
The theory of coherent integrated-optic signal processing described in Chapter 2 is employed in this
chapter where electric-field amplitude signals are considered.
FIGURE 4.17 Schematic diagram of the proposed (q + 1)-tap FIR coherent optical filter used to synthesize
the qth-order pth-derivative FIR optical differentiator. TC: tunable coupler, PS: phase shifter, and PC: polarization controller.
144
4.3.1
Photonic Signal Processing
intrODuctiOn
Section 4.2 has established that a digital integrator can be used to simulate the behavior of an analog integrator. In this chapter, a digital differentiator is also used to simulate the behavior of an analog differentiator.
Like the digital integrator, a digital differentiator also forms an integral part of many practical signal processing systems because the time derivative of signals is sometimes required for further use or analysis. Digital
differentiators are useful in modifying the shape of the signal; they can be used to find positive-going or
negative-going slopes, maxima or minima, or point of greatest slope, where the differentiated output would
be positive or negative, zero, or maximum, respectively. First-derivative and second-derivative digital differentiators have been used in the design of compensators for control systems,40 for monitoring electrocardiograph (ECG) signals,41 in the study of velocity and acceleration in human locomotion,42 in the analysis of
radar signals in radar systems,43 and for the calculation of geometric moments in optical systems.44
A digital differentiator is a processor whose output pulse sequence is obtained by approximating the derivative of a continuous-time signal from the samples of that signal. A continuous-time
signal x(t), whose values are known at the discrete time t = nT for n = 0,1,2… where T > 0 is the
period between successive samples, can be differentiated by a digital differentiator. The frequency
response of an ideal pth-derivative digital differentiator is given by45
p
0 ≤ ωT (2π ) ≤ 1 2 ,
( jωT ) ,
H d (ω ) = 
p
 [ j (2π − ωT )] , 1 2 < ωT (2π ) ≤ 1,
(4.80)
where j = −1, p = 1, 2, , ω is the angular frequency and T is the sampling period of the differentiator. The output pulse sequence of the ideal pth-derivative digital differentiator y ( p ) ( nT ) approximates the true pth-derivative of the continuous-time signal x(t ) according to
y ( p ) ( nT ) =
d p x (t )
.
dt p t = nT
(4.81)
Digital differentiators can be designed in two ways for formulation in either the time or the frequency domains approach as follows.
In the time-domain approach, first-derivative FIR digital differentiators can be designed by using
one of the many classical numerical differentiation algorithms such as the Newton, Bessel, Everett,
Stirling, and Lagrange formulas.46,47,48,49,50 The underlying principle of these numerical differentiation
algorithms is to fit a continuous-time interpolation polynomial x(t) to a given input pulse sequence
f(nT), where f(nT) = x(nT), which is then differentiated to give y(t ) = dx(t ) / dt . Sampling the differentiated continuous-time polynomial y(t ) (by a digital differentiator) at the discrete time t = nT
G. F. Franklin, J. D. Powell, and M. L. Workman, Digital Control of Dynamic Systems, 2nd ed., Reading, MA: Addison–
Wesley, 1990
41 W. J. Tompkins and J. G. Webster, (Eds.), Design of Microcomputers-based Medical Instrumentation, Englewood
Cliffs, NJ: Prentice-Hall, 1981.
42 S. Usui and I. Amidror, Digital low-pass differentiation for biological signal processing, IEEE Trans. Biomed. Eng.,
BME-29, 686–693, 1982.
43 M. I. Skolnik, Introduction to Radar Systems, 2nd ed., Boston, MA: McGraw-Hill, 1980.
44 B. V. K. Vijaya Kumar and C. A. Rahenkamp, Calculation of geometric moments using Fourier plane intensities, Appl.
Opt., 25, 997–1007, 1986.
45 L. R. Rabiner and R. W. Schafer, On the behavior of minimax relative error FIR digital differentiators, Bell Syst. Tech.
J., 53, 333–360, 1974.
46 W. J. Tompkins and J. G. Webster, (Eds.), Design of Microcomputers-based Medical Instrumentation, Englewood
Cliffs, NJ: Prentice-Hall, 1981.
47 R. Vich, Z Transform Theory and Applications, Norwell, MA: Kluwer Academic Publishers, 1987.
48 R. Pintelon and J. Schoukens, Real-time integration and differentiation of analog signals by means of digital filtering,
IEEE Trans. Intrum. Meas., 39, 923–927, 1990.
49 M. Abramowitz and I. A. Segun, Handbook of Mathematical Function, New York: Dover Publications, 1964.
50 S. C. Chapra and R. P. Canale, Numerical Methods for Engineers, 2nd ed., Singapore: McGraw-Hill, 1989.
40
145
Photonic Computing Processors
yields the output pulse sequence y( nT ) , which effectively approximates the true derivative of a
continuous-time signal, that is
y( nT ) =
dx(t )
.
dt t = nT
(4.82)
The magnitude responses of these digital differentiators generally approximate the ideal magnitude
response reasonably well over the lower frequency band 0 ≤ ωT (2π ) ≤ 1 4. Thus, these differentiators are normally referred to as narrow-band digital differentiators.
In the frequency-domain approach, first-derivative and higher-derivative FIR digital differentiators satisfying prescribed specifications of the ideal frequency response can be designed by
using the minimum method,51,52 the Fourier series method in conjunction with the Kaiser window function,53,54 the Fourier series method in conjunction with accuracy constraints,55,56 the
eigenfilter method,57,58 and the least-squares methods.59,60 Because of the constraints imposed on
the frequency responses of these digital differentiators, they are normally referred to as frequencyselective differentiating filters that, in addition to performing the function of signal differentiation,
are capable of passing as well as rejecting certain frequency components of the signal. That is, they
can be designed to have a narrow-band,61,62 mid-band,63,64 or wide-band,65,66,67,68,69,70,71,72 magnitude
L. R. Rabiner and R. W. Schafer, On the behavior of minimax relative error FIR digital differentiators, Bell Syst. Tech. J.,
53, 333–360, 1974.
52 C. A. Rahenkamp and B. V. K. Vijaya Kumar, Modifications to the McClellan, Parks, and Rabiner computer program for
designing higher order differentiating FIR filters, IEEE Trans. Acoust. Speech Signal Process., ASSP-34, 1671–1674, 1986.
53 A. Antoniou, Design of digital differentiators satisfying prescribed specifications, Proc. IEE, 127, pt. E, 24–30, 1980.
54 A. Antoniou and C. Charalambous, Improved design method for Kaiser differentiators and comparison with equiripple
method, Proc. IEE, 128, pt. E, 190–196, 1981.
55 B. Kumar and S. C. Dutta Roy, Design of digital differentiators for low frequencies, Proc. IEEE, 76, 287–289, 1988.
56 B. Kumar and S. C. Dutta Roy, Maximally linear FIR digital differentiators for high frequencies, IEEE Trans. Circuits
and Syst., CAS-36, 890–893, 1989.
57 S. C. Pei and J. J. Shyu, Design of FIR Hilbert transformers and differentiators by eigenfilter, IEEE Trans. Circuits and
Syst., 35, 1457–1461, 1988.
58 S. C. Pei and J. J. Shyu, Eigenfilter design of higher order digital differentiators, IEEE Trans. Acoust. Speech Signal
Process., 37, 505–511, 1989.
59 K. Sasayama, M. Okuno, and K. Habara, Coherent optical transversal filter using silica-based waveguides for highspeed signal processing, J. Light. Technol., 9, 1225–1230, 1991.
60 S. Sunder and R. P. Ramachandran, Least-squares design of higher order nonrecursive differentiators, IEEE Trans.
Signal Process., 42, 956–961, 1994.
61 A narrow-band, mid-band, or wide-band digital differentiator has a magnitude response that closely matches the ideal magnitude response over the frequency band of 0 ≤ ωT ( 2π ) ≤ 1 4 , 1 8 ≤ ωT ( 2π ) ≤ 3 8 or 0 ≤ ωT ( 2π ) ≤ 1 2, respectively.
62 R. R. R. Reddy, B. Kumar, and S. C. Dutta Roy, Design of efficient second and higher order FIR digital differentiators
for low frequencies, Signal Process., 20, 219–225, 1990.
63 B. Kumar and S. C. Dutta Roy, Maximally linear FIR digital differentiators for midband frequencies, Int. J. Circ. Theor.
App., 17, 21–27, 1989.
64 B. Kumar and S. C. Dutta Roy, Design of efficient FIR digital differentiators and Hilbert transformers for midband
frequency ranges, Int. J. Circ. Theor. App., 17, 483–488, 1989.
65 G. P. Agrawal, Fiber-Optic Communication Systems, Hoboken, NJ: John Wiley & Sons, 1992.
66 L. R. Rabiner and R. W. Schafer, On the behavior of minimax relative error FIR digital differentiators, Bell Syst. Tech.
J., 53, 333–360, 1974.
67 C. A. Rahenkamp and B. V. K. Vijaya Kumar, Modifications to the McClellan, Parks, and Rabiner computer program for
designing higher order differentiating FIR filters, IEEE Trans. Acoust. Speech Signal Process., ASSP-34, 1671–1674, 1986.
69 S. C. Pei and J. J. Shyu, Design of FIR Hilbert transformers and differentiators by eigenfilter, IEEE Trans. Circuits Syst.,
35, 1457–1461, 1988.
71 S. Sunder, W. S. Lu, A. Antoniou, and Y. Su, Design of digital differentiators satisfying prescribed specifications using
optimisation techniques, Proc. IEE, 138, pt. G, 315–320, 1991.
68 A. Antoniou and C. Charalambous, Improved design method for Kaiser differentiators and comparison with equiripple
method, Proc. IEE, 128, pt. E, 190–196, 1981.
70 S. C. Pei and J. J. Shyu, Eigenfilter design of higher order digital differentiators, IEEE Trans. Acoust. Speech Signal
Process., 37, 505–511, 1989.
72 B. Kumar and S. C. Dutta Roy, Maximally linear FIR digital differentiators for high frequencies, IEEE Trans. Circuits
Syst., CAS-36, 890–893, 1989.
51
146
Photonic Signal Processing
frequency response, depending on the application. Although impressive frequency-domain performances can generally be achieved by these frequency-selective differentiating filters, the required
filter order is generally very high (e.g., 30–40 taps are common).
In both the time-domain and frequency-domain approaches, the performances of the digital differentiators have usually been evaluated in the frequency domain by using the ideal frequency response (usually magnitude but not phase response) as a basis. This is because the differentiating accuracy of these
digital differentiators in the time domain is difficult to assess as this would depend on a specific application, and hence the actual shape of the signal. It is obvious that a good digital differentiator must be
capable of achieving high differentiation accuracy in the time domain while still able to reject unwanted
frequency components. For example, the high frequency components of a digital signal are often corrupted with wide-band noise. A narrow-band digital differentiator would be useful in this case.73,74,75,76
Unlike the digital differentiators, which have been studied for some time, optical differentiation is still
a new concept in the area of optical signal processing. Although a 3-tap coherent optical transversal (or
FIR) filter using silica-based waveguides integrated on a silicon substrate has been experimentally demonstrated as a second-derivative optical differentiator, no theoretical background was given.77 In addition,
a fiber-optic ring resonator has been claimed as a first-derivative optical differentiator under the resonance condition, but it is not a true differentiator and hence would suffer from low processing accuracy.78
In this section, a theory of higher-derivative FIR optical differentiators using integrated-optic
structures is described. Most of the work presented here has been described by Ngo and Binh.79 The
derivation of a theory of higher-derivative FIR digital differentiators, which was not presented in
reference,79 is given in Appendix A.
4.3.2
higher-Derivative Fir Digital DiFFerentiatOrs
The transfer function of the qth-order pth-derivative FIR digital differentiator [i.e., H q( p ) ( z )] can be
generally expressed as80
 −a11

 −a21
 
 −aM 1
Where a pq =
a12
a22
aM 2
−a13
−a23
− aM 3
( −1) M a1M   TH M(1) ( z )   z −1 − 1 

 

( −1) M a2 M   T 2 H M( 2) ( z )   z −2 − 1 
≅

  

 

( −1) M aMM  T M H M( M ) ( z )   z − M − 1
pq
, p, q = 1, 2, , M ,
q!
(4.83)
(4.84)
and z = e jωT is the z-transform parameter. The pulse response of the qth-order pth-derivative FIR
digital differentiator is defined as
yq( p ) ( nT ) = x( nT ) ∗ hq( p ) ( nT ),
(4.85)
W. J. Tompkins and J. G. Webster, (Eds.), Design of Microcomputers-based Medical Instrumentation, Englewood
Cliffs, NJ: Prentice-Hall, 1981.
74 S. Usui and I. Amidror, Digital low-pass differentiation for biological signal processing, IEEE Trans. Biomed. Eng.,
BME-29, 686–693, 1982.
75 M. I. Skolnik, Introduction to Radar Systems, 2nd ed., Boston, MA: McGraw-Hill, 1980.
76 B. V. K. Vijaya Kumar and C. A. Rahenkamp, Calculation of geometric moments using Fourier plane intensities, Appl.
Opt., 25, 997–1007.
77 K. Sasayama, M. Okuno, and K. Habara, Coherent optical transversal filter using silica-based waveguides for highspeed signal processing, J. Light. Technol., 9, 1225–1230, 1991.
78 G. S. Pandian and F. E. Seraji, Optical pulse response of a fiber ring resonator, Proc. IEE, 42, part J, 235–239, 1991.
79 N. Q. Ngo and L. N. Binh, Theory of a FIR optical digital differentiator, Fiber and Integrated Optics, 14, 359–385, 1995.
80 The derivation of Eq. (6.4) is given in Appendix B where Eq. (B.12) corresponds to Eq. (6.4).
73
147
Photonic Computing Processors
which approximates the true pth-derivative of the input pulse sequence x( nT ) as
yq( p ) ( nT ) ≅
d p x (t )
dt p t = nT
(4.86)
where hq( p ) ( nT ) is the impulse response of H q( p ) ( z ).
The transfer function of the qth-order pth-derivative FIR digital differentiator takes the general form of
q
H
( p)
q
∑
b 
( z ) =  max
b( k ) z − k , p, q = 1, 2, , M ,
p
 T  k = 0
(4.87)
where −1 ≤ b( k ) ≤ 1 is the normalized tap coefficient and bmax ≥ 1 is the normalization factor. The
transfer functions of several families of the digital differentiators, as computed from Eq. (4.82)
for M = 4, are tabulated in Table 4.8. Note that the signs of the tap coefficients alternate such that
even are positive and odd negative. For the special case where q = p , the transfer function of the
qth-order pth-derivative FIR digital differentiator is generally given by
T p H q( p ) ( z )
4.3.3
q= p
(
= 1 − z −1
).
p
(4.88)
synthesis OF higher-Derivative Fir Optical DiFFerentiatOrs
The characteristics of the higher-derivative FIR digital differentiators and the planar lightwave circuit (PLC) technology outlined in Section 4.2.3 are used to synthesize higher-derivative FIR optical
differentiators.
Coherent integrated-optic signal processing of electric-field amplitude signals is considered here.
The unmodulated signal of the optical source is assumed to be externally modulated by an optical
intensity modulator, which minimizes laser chirp as well as permitting high-speed modulation and
hence high-speed signal processing. It is also assumed that the optically encoded signals to be processed by the optical differentiator are modulated onto the optical carrier whose coherence time is
TABLE 4.8
Normalized Tap Coefficients and Normalization Factor, as Computed from Eq. (6.4) for
M = 4, of the Several Families of the FIR Digital Differentiators with Transfer Functions
Expressed in the Form of Eq. (3.77)
T p Hq( p ) ( z)
(1)
1
(1)
2
(1)
3
TH ( z )
TH ( z )
TH ( z )
TH 4(1) ( z )
2
( 2)
2
( 2)
3
( 2)
4
bmax
b(0)
1
2
3
4
1
0.75
0.6111
0.5208
2
5
0.5
0.4
b(1)
b(2)
b(3)
b(4)
−1
−1
−1
−1
0
0.25
0.5
0.75
0
0
−0.1111
0
0
0
0.0625
0.5
0.8
1
−0.2
−0.4912
0
0
0.0965
−0.3333
9.5
0.3070
−1
−1
−0.9123
T 3 H 3( 3) ( z )
T 3 H 4( 3) ( z )
3
12
0.3333
0.2083
−1
−0.75
1
1
−0.3333
−0.5833
0
0.125
T 4 H 4( 4 ) ( z )
6
0.1667
−0.6667
1
−0.6667
0.1667
T H
T 2H
T 2H
( z)
( z)
( z)
0
148
Photonic Signal Processing
much longer than the sampling period T of the optical differentiator. As a result, the pulse response
of the optical differentiator depends on the coherent interference of the delayed signals.
Figure 4.17 shows the schematic diagram of the proposed ( q +1) tap FIR coherent optical filter,81
which is used to synthesize the qth-order pth-derivative FIR optical differentiator. The FIR coherent optical filter essentially consists of a lx( q +1) optical splitter, an ( q +1) xl optical combiner, and
( q +1) waveguide delay lines, into each of which a tunable coupler (TC) and a phase shifter (PS) are
incorporated. A coherent optical signal coming into the optical splitter will be evenly distributed to
( q +1) signals, which are then appropriately delayed by the delay lines and weighted by the TCs and
PSs. These signals are then coherently collected by the optical combiner to generate the differentiated
optical signal.
The FIR coherent optical filter can be constructed using the PLC technology, namely, silicabased waveguides embedded on a silicon substrate as described in Section 4.2.3. The optical splitter
and optical combiner can be developed using 3-dB silica-based waveguide directional couplers
(DCs), except that no erbium-doped fiber amplifiers (EDFAs) are used here. In each delay line, the
PS following the TC is a waveguide with a thin-film heater deposited on it and utilizes the thermooptic effect to induce a carrier phase change of φ ( k ). The TC is a symmetrical Mach−Zehnder
interferometer (see the inset of Figure 4.17 that consists of two 3-dB DCs), two equal waveguide
arms, and a thin-film heater, with a carrier phase change of φ(κ), attached to one of the arms for
controlling the output amplitude (see Section 4.3).
Neglecting the insertion loss of the 3-dB DCs, the propagation delay and waveguide birefringence of the TC, the kth TC transfer function, which corresponds to the transfer function E3 / E1 as
the output port 3 over the input port 1 fields of the coupler, is given by
C ( k ) = C ( k ) exp ( j ∠C ( k ) ) = 0.5 exp ( jϕ ( k ) ) − 1
(4.89)
where ∠C ( k ) denotes the argument of C ( k ),
C ( k ) = 0.5 − 0.5 cos (ϕ ( k ) )
(4.90)
ϕ ( k ) = cos −1 1 − 2 C ( k )  ,


(4.91)
or
2
and ∠C ( k ) = tan −1 sin(ϕ ( k )) ( cos(ϕ ( k )) − 1) 
for k = 0,1, , q
(4.92)
Eq. (4.90) indicates that a desired TC amplitude can be obtained by choosing an appropriate PS
phase according to Eq. (4.91), and this results in the TC phase as given by Eq. (4.92). The amplitude
and phase of the TC can be changed from 0 to 1 and from − π 2 to + π 2, respectively, when ϕ ( k )
is varied from 0 to 2π .
Neglecting the propagation delay and waveguide birefringence, the transfer function of the
( q +1) -tap FIR coherent optical filter is given by
( p)
H q ( z ) = lpath ⋅ ( q + 1) −1 ⋅ G ⋅
q
∑ (−1) ⋅ C (k ) exp( j∠C (k )) ⋅ exp( jφ(k )) ⋅z
k
−k
(4.93)
k =0
81
The FIR coherent optical filter described here has a similar structure to the incoherent fiber-optic transversal filter (see
Figure 4.4) outlined in previous section.
149
Photonic Computing Processors
where lpath is the intensity path loss that takes into account all the losses associated with each delay
line. Such losses include the loss of the straight and bend waveguides, the insertion loss of the
3-dB DCs in the splitter and combiner, and the insertion loss of the TC. Additionally, ( q + 1) −1 is
the coupling loss of the splitter and combiner as a result of a 3-dB coupling loss at each stage of the
structure, G is the intensity gain of the EDFA, and ( −1) k = exp( jkπ ) is the phase shift factor due to
the π /2 cross-coupled phase shift of the 3-dB DCs in the splitter and combiner.
Synthesis of the qth-order pth-derivative FIR optical differentiator requires the equality of Eqs.
(4.93) and (4.88), i.e., Ĥ q( p ) ( z ) = H q( p ) ( z ), such that
G=
2
bmax
( q + 1)2
,
lpathT 2 p
(4.94)
C ( k ) = b( k ) ,
(4.95)
φ ( k ) = ∠b( k ) − ∠( −1) k − ∠C ( k ),
(4.96)
φ ( k ) = −∠C ( k ), ∠b( k )= ∠( −1) k ,
(4.97)
where ∠b( k )=∠( −1) k = 0 for even k or ∠b( k )=∠( −1) k = π for odd k . Eq. (4.94) shows that the
required gain G is dominated by the small value of the sampling period T, and hence several
EDFAs in cascade may be required at both the input and output of the optical differentiator,
depending on the application. In each kth delay line, Eq. (4.95) shows that the amplitude of the
digital coefficient [i.e., 0 ≤ ωT (2π ) ≤ 1 2] can be optically implemented by the TC amplitude
[i.e., C ( k ) ], and Eq. (4.97) shows that the PS must provide a phase shift [i.e., φ ( k )] opposite to the
TC phase [i.e., ∠C ( k )]. Because of the temperature dependence of the refractive index change of
the silica waveguide, the PS can also be used to compensate for the optical path-length difference
resulting from imperfect fabrication of the waveguide length. Note that if, for each delay line, a
non-tunable DC is used instead of the TC, then the PS is not required. However, it is difficult to
practically fabricate a non-tunable DC with a precise coupling coefficient as described in Section
4.2.3.3. Thus, it is preferable to use the TC, which, in addition to implementing the digital coefficient, can accommodate for the deviations of the coupling coefficients of the 3-dB DCs in the
splitter and combiner as a result of fabrication errors.
Because the temperature of the silicon substrate can be maintained to within a small fraction of
a degree to stabilize the refractive index of the waveguides,82 the PS can provide a very accurate
phase shift. The temperature stability of the waveguides means that the optical differentiator can
operate stably. Since the control of the TC amplitude and PS phase in a particular delay line is
independent of those in other delay lines, the amplitude and phase of the digital coefficient can be
optically implemented with high accuracy, showing the advantage of the proposed filter structure.
The polarization controller (PC) placed at the input of the optical differentiator is used to counter
any birefringence induced in the fiber, while the PC placed at its output is used to counter any
waveguide birefringence arising from the optical differentiator. Alternatively, the waveguide birefringence may be overcome by inserting the polyamide half waveplates, acting as TE/TM mode
converter, into the delay lines.
82
K. Sasayama, M. Okuno, and K. Habara, Coherent optical transversal filter using silica-based waveguides for highspeed signal processing, J. Light. Technol., 9, 1225–1230, 1991.
150
4.3.4
Photonic Signal Processing
cOmputeD DiFFerentiatOrs OF First anD higher OrDers
The proposed qth-order pth-derivative FIR optical differentiator, Ĥ q( p ) ( z; p, q = 1, 2, 3, 4) , described
in Section 4.3.2 is now analyzed. For analytical simplicity, the following assumptions are used
in all figures: the discrete-time index m means m = nT , the normalized optical frequency means
ωT (2π ), the normalized time t T means m, Hpq means Ĥ q( p ) ( z ), the magnitude response corresponds to Ĥ q( p ) ( z ) , and the sampling period T is set to unity.
The processing accuracy of the qth-order pth-derivative FIR optical differentiator is evaluated
by means of the
Error Response =
True Derivative − Pulse Response
× 100 %
True Derivative
(4.98)
where True Derivative corresponds to the true derivative of the input pulse sequence x[m] and Pulse
Response corresponds to the pulse response of the differentiator. To characterize the performance
of the differentiator, the pulse response is defined as the amplitude response at the output of the
optical differentiator prior to detection by an optical detector.
4.3.4.1 First-Derivative Differentiators
This section analyses the performances of the qth-order first-derivative differentiators,
(1)
H q ( z; q = 1, 2, 3, 4) .
Figure 4.18a shows the magnitude responses of the qth-order and ideal differentiators. The magnitude response increases with increasing filter order q. The magnitudes are zero at ωT (2π ) = 0
FIGURE 4.18 Magnitude responses of the optical differentiators. (a) qth-order first-derivative differentiators Ĥ q(1) ( z; q = 1, 2, 3, 4) . (b) qth-order second-derivative differentiators Ĥ q( 2) ( z; q = 2, 3, 4). (c) qth-order thirdderivative differentiators Ĥ q( 3) ( z; q = 3, 4). (d) Fourth-order fourth-derivative differentiator Ĥ 4( 4 ) ( z ).
Photonic Computing Processors
151
FIGURE 4.19 Error responses of the qth-order first-derivative differentiators Ĥ q(1) ( z; q = 1, 2, 3, 4) when processing the qth-order polynomial input pulse. (a) First-order polynomial input pulse: x[m] = m. (b) Secondorder polynomial input pulse: x[m] = m2 . (c) Third-order polynomial input pulse: x[m] = m3. (d) Fourth-order
polynomial input pulse: x[m] = m4.
and ωT (2π ) = 1, and maximum at ωT (2π ) = 0.5 because there are at least one zero on the unit
circle in the z-plane. Similar accounts can be made for the magnitude responses of the qth-order
second-derivative, third-derivative and fourth-derivative differentiators, which are shown in
Figure 4.19b–d, respectively.
Figure 4.20a–d show the differentiator pulse responses when processing the Gaussian input
pulse with various pulse widths83 w, where 2w is the temporal full-width of the intensity pulse at
the 1 exp(1) points. The pulse responses closely resemble the true derivatives and the processing
accuracy increases with increasing value of 2w.
Figure 4.21a and b show the differentiator error responses when processing the Gaussian input
pulse with pulse widths w = 5 and w = 20, respectively. For w = 5, Figure 4.22a shows that the
second-order, third-order and fourth-order differentiators have lower processing accuracy than the
first-order differentiator for time m > 25. However, for a larger pulse width, w = 20, Figures 4.23
and 4.24b shows that the second-order, third-order and fourth-order differentiators have higher processing accuracy than the first-order differentiator, but at the expense of having lower processing
accuracy over the initial time interval [see the enlarged curves in Figure 4.21c]. A higher-order
differentiator requires more hardware components than a lower-order differentiator. Thus, the firstorder differentiator is considered as the optimum filter for processing a Gaussian pulse because of
its structural simplicity and impressive performance.
83
The pulse width w corresponds to the normalized pulse width, i.e., w means wT.
152
Photonic Signal Processing
FIGURE 4.20 Pulse responses of the first-order first-derivative differentiator Ĥ1(1) ( z ) when processing
the gaussian input pulse (i.e., x[m] = exp[− m2 (2w 2 )]) With various pulse widths w. (A) w = 5. (B) w = 10.
(C) w = 15. (D) w = 20.
Figures 4.24 and 4.25 show the differentiator error responses when processing the qth-order
polynomial input pulse x[m] = mq. The first-order, second-order, third-order and fourth-order differentiators have higher processing accuracy of the first-order, second-order, third-order and fourthorder input pulses, respectively, than other differentiator orders. Thus, for x[m] = mq, the qth-order
first-derivative differentiator has large processing error over the initial time interval 0 ≤ m < q but
has zero processing error over the longer time interval m ≥ q. This is because the differentiators
have been designed from the polynomial perspective (see Appendix A).
4.3.4.2 Second-Derivative Differentiators
This section analyses the performances of the qth-order second-derivative differentiators,
Ĥ q( 2) ( z; q = 2, 3, 4), whose magnitude responses were shown in Figure 4.23b.
Figure 4.25a–d show the differentiator error responses when processing the qth-order polynomial input pulse. Figures 4.24 through 4.27 show respectively that the second-order, third-order
153
Photonic Computing Processors
(a)
(b)
(c)
FIGURE 4.21 Error responses of the qth-order first-derivative differentiators Ĥ q(1) ( z; q = 1, 2, 3, 4) when processing the Gaussian input pulse with two different pulse widths. (a) w = 5. (b), (c) w = 20 with different time
scales.
and fourth-order differentiators have higher processing accuracy of the second-order, third-order
and fourth-order pulses, respectively, than other differentiator orders. However, Figure 4.28d
shows that the fourth-order differentiator has higher processing accuracy of the fifth-order pulse
than the second-order and third-order differentiators. Thus, for x[m] = mq, the qth-order secondderivative differentiator has large processing error over the initial time interval 0 ≤ m < q but has
zero processing error over the longer time interval m ≥ q.
Figure 4.29 show the differentiator pulse responses when processing the Gaussian input pulse
with various pulse widths w. The pulse responses closely resemble the true derivatives and the
processing accuracy increases with increasing value of w. The second-order second-derivative differentiator analyzed here can be experimentally demonstrated where a square-type input pulse can
be processed.
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Photonic Signal Processing
FIGURE 4.22 Error responses of the qth-order second-derivative differentiators Ĥ q( 2) ( z; q = 2, 3, 4) when
processing the qth-order polynomial input pulse. (a) Second-order polynomial input pulse: x[m] = m2 . (b)
Third-order polynomial input pulse: x[m] = m3 . (c) Fourth-order polynomial input pulse: x[m] = m4. (d) Fifthorder polynomial input pulse:x[m] = m5 .
Figure 4.21a and b show the differentiator error responses when processing the Gaussian pulse
with pulse widths w = 5 and w = 20, respectively. For w = 5, Figure 4.26a shows that the third-order
and fourth-order differentiators have lower processing accuracy than the second-order differentiator
for time m > 25. However, for a larger pulse width, w = 20, Figure 4.28b shows that the third-order
and fourth-order differentiators have higher processing accuracy than the second-order differentiator, but at the expense of having lower processing accuracy over the initial time interval [see the
enlarged curves in Figure 4.23c].
4.3.4.3 Third-Derivative Differentiators
This section analyses the performances of the qth-order third-derivative differentiators,
Ĥ q( 3) ( z; q = 3, 4), whose magnitude responses were shown in Figure 4.25c.
Figure 4.25a and b show the differentiator error responses when processing the qth-order polynomial input pulse. Figure 4.25a shows that the third-order differentiator has higher processing
Photonic Computing Processors
155
FIGURE 4.23 Pulse responses of the second-order second-derivative differentiator Ĥ 2( 2) ( z ) when processing
: (a ) w 5=
. ( b) w 10
=
. (c) w 15
=
. (d ) w 20.3.
the Gaussian input pulse with various pulse widths w=
accuracy of the third-order pulse than the fourth-order differentiator. However, Figure 4.25b shows
that the fourth-order differentiator has higher processing accuracy of the fourth-order pulse than
the third-order differentiator. Thus, for x[m] = mq, the qth-order third-derivative differentiator has
large processing error over the initial time interval 0 ≤ m < q but has zero processing error over the
longer time interval m ≥ q.
Figure 4.26a–d show the differentiator pulse responses when processing the Gaussian input
pulse with various pulse widths w. The pulse responses closely resemble the true derivatives and the
processing accuracy increases with increasing value of w. Figure 4.26a and b show the differentiator error responses when processing the Gaussian input pulse with pulse widths
=
w 5=
and w 20,
respectively. For w = 5, Figure 4.26a shows that the fourth-order differentiator has lower processing accuracy than the third-order differentiator for time m > 35. However, for a larger pulse width
w = 20, Figure 4.28b shows that the fourth-order differentiator has higher processing accuracy than
the third-order differentiator, but at the expense of having lower processing accuracy over the initial
time interval [see the enlarged curves in Figure 4.26c].
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Photonic Signal Processing
FIGURE 4.24 Error responses of the qth-order second-derivative differentiators Ĥ q( 2) ( z; q = 2, 3, 4) when processing the Gaussian input pulse with two different pulse widths. (a) w = 5. (b), (c) w = 20 with different time scales.
Photonic Computing Processors
157
FIGURE 4.25 Error responses of the qth-order third-derivative differentiators Ĥ q( 3) ( z; q = 3, 4) when processing the qth-order pulse. (a) Third-order polynomial input pulse: x[m] = m3. (b) Fourth-order polynomial
input pulse: x[m] = m4.
FIGURE 4.26 Pulse responses of the third-order third-derivative differentiator Ĥ 3( 3) ( z ) when processing the
Gaussian input pulse with various pulse widths w.
=
(a) w 5=
. ( b) w 10=
. (c) w 15
=
. (d) w 20.
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Photonic Signal Processing
q ( z; q = 3, 4) when proFIGURE 4.27 Error responses of the qth-order third-derivative differentiators H
cessing the Gaussian input pulse with two different pulse widths. (a) w = 5. (b), (c) w = 20 with different
time scales.
( 3)
4.3.4.4 Fourth-Derivative Differentiator
This section analyes the performance of the fourth-order fourth-derivative differentiator Ĥ 4( 4 ) ( z )
whose magnitude response was shown in Figure 4.28d.
Figure 4.28a–d show the pulse and error responses of the differentiator when processing the
fourth-order and fifth-order polynomial input pulses. For the fourth-order pulse, Figure 4.28a and
b show that the fourth-order differentiator has large processing error over the initial time interval
0 ≤ m < 4 but has zero processing error over the longer time interval m ≥ 4. For the fifth-order
pulse, Figure 4.29c and d show that the processing error never converges to zero. This is because
the order of the differentiator is lower than the order of the input pulse. However, a fifth-order
differentiator, which is not considered here, is expected to improve the processing accuracy of the
fifth-order pulse.
Figure 4.29a–d show the differentiator pulse responses when processing the Gaussian input pulse
with various pulse widths w. The pulse responses closely resemble the true derivatives and the processing accuracy increases with increasing value of w.
159
Photonic Computing Processors
(4)
FIGURE 4.28 Pulse and error responses of the fourth-order fourth-derivative differentiator H 4 ( z ) when
processing the fourth-order and fifth-order polynomial input pulses. (a), (b) Fourth-order polynomial input
pulse: x[ m ] = m 4 . (c), (d) Fifth-order polynomial input pulse: x[ m ] = m 5 .
4.3.5
remarks
• A theory of the qth-order pth-derivative FIR digital differentiator has been proposed,
based on which the qth-order pth-derivative FIR optical differentiator has been synthesized using integrated-optic components.
• For a qth-order polynomial input pulse (i.e., x[m] = mq, q ≥ 1), the qth-order pth-derivative
FIR optical differentiator, Ĥ q( p ) ( z ), which has higher processing accuracy than other differentiator orders, has large processing error over the initial time interval 0 ≤ m < q but has
zero processing error over the longer time interval m ≥ q.
• For a Gaussian input pulse (i.e., x[m] = exp[− m2 (2w 2 )]), the qth-order pth-derivative
differentiator, Ĥ q( p ) ( z; q = p) , whose processing accuracy increases with increasing
pulse width w, is the optimum filter when compared with the higher-order pth-derivative
160
Photonic Signal Processing
FIGURE 4.29 Pulse responses of the fourth-order fourth-derivative differentiator Ĥ 4( 4 ) ( z ) when processing
the Gaussian input pulse with various pulse widths w.
=
(a ) w 5=
. ( b) w 10
=
. (c) w 15
=
. (d ) w 20.
differentiator, Ĥ q( p ) ( z; q > p) , because of its structural simplicity and impressive performance. The results for the Gaussian pulse are similar to those for the exponential pulse
(i.e., x[m] = exp( − m w ), m ≥ 0).
• In general, regardless of the type of input pulse, the optical differentiators have large processing error over the initial time interval corresponding to q sampling periods but their
processing errors reduce significantly over a longer time interval.
• Optical differentiation is still a new research area with many potential applications.
One example is to apply the first-order first-derivative FIR optical differentiator described
here to the design of an optical dark-soliton detector, which is outlined in Section 7.1
of Chapter 7.
4.4 APPENDIX A: GENERALIZED THEORY OF THE
NEWTON–COTES DIGITAL INTEGRATORS
In this appendix, a classical numerical integration scheme together with the digital signal processing
technique are employed to develop a generalized theory of the Newton–Cotes digital integrators,
which is believed to be described for the first time. This theory has been used in the synthesis of
the programmable incoherent Newton–Cotes optical integrator (INCOI) as described in Chapter 5.
A definition of numerical integration is first given and used as a basis in the derivation process. The
Newton’s interpolating polynomial is then described, and based on which a general form of the
Newton–Cotes closed integration formula is derived. Finally, a generalized theory of the Newton–
Cotes digital integrators is obtained.
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Photonic Computing Processors
4.4.1
DeFinitiOn OF numerical integratiOn
It is assumed that a continuous-time signal x(t) is given and that its integral
t
∫ x(t )dt
y (t ) =
(4.99)
0
is to be determined from a sequence of samples of the continuous-time signal x(t) at the discrete
time t = t n where
=
t n nT
=
, n 0,1, 2,
(4.100)
with T > 0 being the period between successive samples. Intuitively, the integral y(t ) cannot be
obtained for all t, but only for t = t n . Thus, Eq. (4.99) can be written as
tn
yn = y (t n ) =
∫ x(t )dt.
(4.101)
0
To simplify the numerical integration algorithm, the integration interval [ 0,t n ] is divided into a
number of equal segments, each with a step size of T . The underlying principle of the numerical
integration algorithm is shown in Figure 4.30.
From Figure 4.30, the integral in Eq. (4.101) can be divided into two integrals as
t n− p
yn =
∫
tn
x(t )dt +
∫ x(t )dt = y + i
n− p
p
(4.102)
t n− p
0
where the partial integral i p, which represents the area of the hatched region of Figure 4.30, is
given by
tn
ip =
∫ x(t )dt.
t n− p
FIGURE 4.30 Graphical illustration of the numerical integration technique.
(4.103)
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Photonic Signal Processing
The z-transform of Eq. (4.102) is given by
Y ( z ) = z − pY ( z ) + I p ( z )
(4.104)
 1 
I p ( z ),
Y ( z) = 
−p
 1 − z 
(4.105)
or
where Y ( z ) = Z { yn } and I p ( z ) = Z {i p } with Z {}
. being the z-transform of {}
. . In Eq. (4.105), the
z-transform parameter is defined as z = exp( jωT ) where j = −1, ω is the angular frequency, and T
is the sampling period of the integrator. The z-transform of the partial integral I p ( z ) is to be determined in Sections 4.4.2 and 4.4.3.
4.4.2
newtOn’s interpOlating pOlynOmial
For analytical simplicity, the discrete-time variables in Figure 4.30 are re-defined as
t k = t n − p , k = n − p,
(4.106)
tk + p = tn ,
(4.107)
tk + m = tm .
(4.108)
Using Eqs. (4.106) and (4.107), Eq. (4.105) and taking the inverse transform we have
tk + p
ip =
∫ x(t )dt.
(4.109)
tk
For the time interval [t k , t k + m ] as shown in Figure 4.30, the curve x(t ) can be approximated by the
mth-order Newton’s interpolating polynomial, which passes through m +1 data points, as84
x (t ) = x (t k ) +
∆x(t k )
∆ 2 x (t k )
(t − t k )(t − t k +1 )
(t − t k ) +
T
2 !T 2
∆ m x (t k )
+ +
(t − t k )(t − t k +1 ) (t − t k + m −1 ),
m!T m
(4.110)
where the ith discrete-time variable is given by
t k + i = t k + iT , i = 0,1, , m − 1,
(4.111)
and the qth forward difference equation is given by85
q
q
∆ x (t k ) =
q
∑ (−1)  i  x(t
i
k + q −i
), q = 0,1, , m.
(4.112)
i =0
84
85
N. Q. Ngo and L. N. Binh, Programmable incoherent Newton–Cotes optical integrator, Opt. Commun., 119, 390–402, 1995.
M. Abramowitz and I. A. Segun, Handbook of Mathematical function, New York: Dover Publications, 1964.
163
Photonic Computing Processors
The binomial coefficient in Eq. (4.112) is defined as
 q  q( q − 1) ( q − (i − 1) )
q!
=
 =
( q − i )!i !
i!
i
(4.113)
q q
  =   = 1.
0 q
(4.114)
with
Eq. 4.112 can be written in a recursive form as85
∆ 0 x(t k ) = x(t k ),
(4.115)
∆1x(t k ) = ∆x(t k ) = x(t k +1 ) − x(t k ),
(4.116)
∆ q x(t k ) = ∆ q −1x(t k +1 ) − ∆ q −1x(t k ), q = 2, , m.
(4.117)
Eqs. (4.110) and (4.117) can be simply expressed as
q −1

∆ q x(t k ) 
x (t ) = x (t k ) +
(
t
−
t
)
+
k
i


q !T q  i = 0
q =1


m
∑
∏
(4.118)
which can be further simplified by defining a new quantity
η=
t − tk
.
T
(4.119)
Substituting Eq. (4.119) into Eq. (4.111), the following equation is obtained:
t − t k +i = T (η − i ), i = 0,1,, m − 1.
(4.120)
Substituting Eq. (4.120) into Eq. (4.118) results in
q −1
q −1
m


∆ q x(t k ) 
∆ q x(t k ) 
=
T
(
η
−
i
)
x
(
t
)
+
(η − i ) 
[
]
k


q 
q !  i =0
q!T  i =0


q =1
q =1

m
x (t ) = x (t k ) +
∑
∑
∏
∏
(4.121)
η (η − 1)(η − ( m − 1) )
η (η − 1)
+ ∆ m x (t k )
= x(t k ) + ∆x(t k )η + ∆ 2 x(t k )
,
2!
m!
which can be further simplified to
m
x (t ) = x (t k ) +
m
η  q
η  q
 ∆ x(t k ) =
 ∆ x(t k ).
q
q
q =1  
q =0  
∑
∑
(4.122)
Thus, for the time interval [t k , t k + m ], the mth-order Newton’s interpolating polynomial of the curve
x(t ) can be simply described by Eq. (4.122).
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Photonic Signal Processing
4.4.3 general FOrm OF the newtOn–cOtes clOseD integratiOn FOrmulas
Substituting Eq. (4.122) into Eq. (4.109) results in
tk + p
ip =
 m η 

q

 ∆ x(t k )  dt .


q
tk  q =0
∫ ∑
(4.123)
From Eq. (4.123), dt = Tdη and the limits of integration are changed from t = t k to η = 0 and from
t = t k + p to η = p. Substituting these parameters into Eq. (4.123) results in

 p η 
   dη ⋅ ∆ q x(t k ) 
=T
 q

q =0  0  

p
 m η 

q
ip = 
 ∆ x(t k )  ⋅ Tdη


q
0  q =0
m
∫∑
∑∫
(4.124)
∑ C ( p)∆ x(t ),
(4.125)
which can be rearranged to give
m
ip = T
q
q
k
q=0
where the qth coefficient Cq ( p) is given by
p
η 
Cq ( p) =   dη , q = 0,1,, m.
q
0
∫
(4.126)
The qth forward difference equation, as described by Eq. (4.118), can be further simplified by substituting u = q − i or i = q − u into Eq. (4.121) to give
q
∆ q x(t k ) = ( −1) q
q
q
 q 
q
( −1) − u 
( −1)u   x(t k +u )
 x(t k +u ) = ( −1)
u
q −u
u =0
u =0
∑
∑
(4.127)
u ∈ integer
(4.128)
Where the following equations are used
( −1) − u = ( −1)u , with
 q  q

 =  ,
q − u u
(4.129)
Thus, the general form of the Newton–Cotes closed integration formulas can be simply described
by three closed-form formulas, as given in Eqs. (4.127) through (4.129).
4.4.4 generalizeD theOry OF the newtOn–cOtes Digital integratOrs
Taking the z-transform of Eq. (4.125) leads to
m
I p ( z) = T
∑ C ( p) ∆ X ( z )
q
q=0
q
(4.130)
165
Photonic Computing Processors
{
}
where ∆ q X ( z ) = Z ∆ q x(t k ) is the z-transform of Eq. (4.127), which is given by
q
∆ q X ( z ) = ( −1) q
q
∑ (−1)  u  ⋅ X ( z) z
u
−u
(4.131)
u =0
where X ( z ) z − u = Z { x(t k + u )}. Eq. (4.131) can be rearranged to give
q
q
∆q X ( z)
q( q − 1) −2
q!
( −1)u   z − u =1 − qz −1 +
z + + ( −1) r
=
z − r + ( −1) q z − q
q
u
 X ( z )( −1)  u = 0
!
q
r
r
2
(
−
)!
!
 
∑
(4.132)
Note that Eq. (4.132) can be recognized as86
∆q X ( z)
= (1 − z −1 ) q
 X ( z )( −1) q 
(4.133)
∆ q X ( z ) = ∆ q D( z ) ⋅ X ( z ),
(4.134)
∆ q D( z ) = ( −1) q (1 − z −1 ) q , q = 0,1, , m.
(4.135)
or
where
Substituting Eq. (4.135) into Eq. (4.130) gives
m
I p ( z ) = X ( z )T
∑ C ( p)∆ D( z).
q
q
(4.136)
q=0
Substituting Eq. (4.136) into Eq. (4.105), the pth-order transfer function of the Newton–Cotes digital
integrators can be generally described by
m
H mp ( z ) =
=
∑
Y ( z)
T
=
C q ( p) ∆ q D ( z )
X ( z ) 1 − z − p q =0
(4.137)
T
C0 ( p) + C1( p)∆D( z ) + C2 ( p)∆ 2 D( z ) + + Cm ( p)∆ m D( z ) 
1− z− p 
where the qth coefficient is given, from Eq. (4.126), as
p
η 
Cq ( p) =   dη ,
q
0
∫
86
(4.138)
S. Usui and I. Amidror, Digital low-pass differentiation for biological signal processing, IEEE Trans. Biomed. Eng.,
BME-29, 686–693, 1982.
166
Photonic Signal Processing
and the qth difference equation is given, from Eq. (4.138), as
∆ q D( z ) = ( −1) q (1 − z −1 ) q, for q = 0,1, , m and 1 ≤ p ≤ m
(4.139)
In summary, a generalized theory of the Newton–Cotes digital integrators has been derived, and
Eqs. (4.137) through (4.139), which are referred to in Section 4.2.
5
Optical Dispersion
Compensation and
Gain Flattening
5.1 INTRODUCTORY REMARKS
This chapter introduces the compensation of dispersion effects in the transmission of
lightwave-modulated signals is a very critical task in long haul, ultra-high-speed optical
communications.
Fast, reliable, and error-free 100G/200G/400G backbone and metro networks are a key ingredient to deliver bandwidth-hungry services, such 4G/5G mobile connectivity, fiber to the home
(FTTH), and Remote PHY* (for cable operators). Optical functional devices, in such optical transmission systems, mainly are composed of two types: (i) Firstly, dispersion compensators in order
to compensate for the dispersion broadening of transmitted pulse sequence; and (ii) secondly, the
gain equalizer to equalize the gain over the operating spectral region so that the unevenness of the
optical attenuation of fiber and that of the optical amplification can be flattened. Thus even optical
powers of all channels at different wavelengths can be achieved over the transmission distance.
Variable chromatic dispersion compensators are becoming increasingly important in high
speed optical transmission systems with bit rates of 25 or 56Gbaud or more, where it is essential
to compensate adaptively for the various dispersions of installed fibers and the dispersion change
caused by changes in the environmental temperature or path differences in optical networks.
A number of compensation and equalization techniques have been reported. However, this chapter
describes only the techniques employing optical filters and resonators, which can be designed and
implemented using photonic signal processing methodology. The generation of highly dispersive effects
in resonators that are operated under resonance and eigenfiltering are given. Photonic functional devices
are described including photonic dispersion compensator, gain equalizers using lattice filters.
5.2
DISPERSION COMPENSATION USING OPTICAL RESONATORS
In recent years, there have been growing interests in studying the characteristics of the all-fiber
photonic circuit components, especially the re-circulating delay line1,2,3 and optical resonator.4,5,6,7,8
It has been found that the photonic circuits have great potential in many applications, such as
https://www.cisco.com/c/en/us/solutions/collateral/service-provider/converged-cable-access-platform-ccap-solution/
white-paper-c11-732260.html
1 S. J. Mason, Feedback theory–Some properties of signal flow graphs, Proc. IRE, 1144–1156, 1953.
2 S. J. Mason, Feedback theory-Further properties of signal-flow graphs, Proc. IRE, 44, 920–926, 1956.
3 A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing, Upper Saddle River, NJ: Prentice-Hall, 1989.
4 B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, Fiber lattice optic signal processing, Proc. IEEE, 72,
909–930, 1984.
5 B. Moslehi, Fiber-optic filters employing optical amplifiers to provide design flexibility, Elect. Lett., 28, 226–228, 1992.
6 B. Moslehi, J. W. Goodman, Novel amplified fiber-optic recirculating delay line processor, J. Light. Technol., 10,
1142–1147, 1992.
7 B. C. Kuo, Digital Control Systems, Chapter 5, pp. 267–296, Hongkong: CBS Pub. Asia, 1987.
8 Y. H. Ja, Single-mode optical fiber ring and loop resonators using degenerate two-wave mixing, Appl. Opt., 30,
2424–2426, 1991.
*
167
168
Photonic Signal Processing
communications,9,10 sensing devices,11,12 and signal processing devices.13 The progress in fiber-optic
technology also makes the implementation of these circuits easier. For instance, due to the development in optical amplifier technology,14 active optical devices1,3 are a possibility. As the trends of
integrated photonic circuits progress, it is expected that the circuits analyzed may consist of more
and more elements in them. This surely causes difficulties in the examination of the circuits. Thus,
an efficient method is needed to facilitate these problems.
As far as the examination of the circuit behavior and its design are concerned, the analysis of
the circuit involves the determination of the circuit’s transfer function. This is usually achieved by
simultaneously solving a set of linear field or intensity equations. However, this method becomes
error-prone and time-consuming as the complexities of the circuit increase due to the increasing
numbers of components and interconnections in the optical circuit network. It makes the formulation of the transfer functions for the circuit and the analysis on the circuits’ characteristics more
difficult and tedious. Although a scattering matrix has been used in describing the circuit characteristics15,16 and is shown to be a more systematic approach to the problem, a simple method of determining the circuit’s transfer functions from the set of governing equations has not been proposed,
to the best of our knowledge. This restricts our analysis to simple circuits only.
The new method employs the use of signal-flow graph theory17,18 in optical circuits. The signalflow graph theory was first introduced 80 years ago by S. J. Mason.1,2 The theory has been applied
in electrical and electronic circuits for a long time already. This is the first time, to the best of
our knowledge, when the signal-flow graph theory is being applied during the analysis of optical
circuits. The unique feature of our work is that the optical circuits can be represented in form of a
signal-flow graph (SFG), and their circuits’ transfer functions can then be determined systematically
using the corresponding manipulation rules. The analysis of optical circuits, which employs the new
method, has been published recently by our group.19,20,21,22 The new method is found to be much
more efficient than the conventional method. The main advantage of our method over the conventional one is that more simple and systematic procedures are used in deriving the circuits’ transfer
functions. This certainly lowers the possibility of error. Moreover, the graphical representation of
the optical circuits also allows easier examination of recirculating or resonating loops in the circuit.
Thus, the locations of loops in the circuits can be identified with lesser effort and the mechanism
L. N. Binh, N. Q. Ngo, and S. F. Luk, Graphical representation and analysis of the z-shaped double-coupler optical
resonator, J. Light.Technol., 11, 1782–1792, 1993.
10 Y. H. Ja, Generalized theory of optical fiber loop and ring resonators with multiple couplers: 1: Circulating and output
fields and 2: General characteristics, Appl. Opt., 29, 3517–3529.
11 Y. H. Ja, Optical fiber loop resonators with double couplers, Opt. Commun., 75, 239–245, 1990.
12 Y. H. Ja, A double-coupler optical fiber ring-loop resonator with degenerate two-wave mixing, Opt. Commun., 81, 113–
120, 1991.
13 Y. H. Ja, On the configurations of double optical fiber loop or ring resonator with double couplers, J. Opt. Commun., 12,
29–32, 1991.
14 B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, Fiber lattice optic signal processing, Proc. IEEE, 72, 909–930, 1984.
15 J. M. Vigoureux and F. Raba, A model for the optical transistor and optical switching, J. Mod. Opt., 38, 2521, 1991.
16 B. Moslehi, J. W. Goodman, Novel amplified fiber-optic recirculating delay line processor, IEEE J. Light. Technol., 10,
1142–1147, 1992.
17 D. Marcuse, Pulse distortion in single-mode fibers, Appl. Opt., 19, 1653–1660, 1980.
18 D. Marcuse, Selected topics in the theory of telecommunications fibers, Optical Fiber Telecommunications II, Chapter 3,
Boston, MA: Academic Press, 1988.
19 B. E. A. Saleh and M. I. Irshid, Transmission of pulse sequences through mono-mode fibers, Applied Optics, 21, 4219–
4222, 1982.
20 B. E. A. Saleh and M. I. Irshid, Coherence and inter-symbol interference in digital fiber optic communication systems,
IEEE J. Quantum Electron., QE-18, 944–951, 1982.
21 K. Joergensen, Transmission of Gaussian pulses through monomode dielectric optical waveguides, Appl. Opt., 16,
22–23, 1977.
22 K. Joergensen, Gaussian pulse transmission through monomode fibers, accounting for source linewidth, Appl. Opt., 17,
2412–2415, 1978.
9
Optical Dispersion Compensation and Gain Flattening
169
of resonance can then be studied easily. Furthermore, our new method incorporates a stability test,
which is useful in the assessment of the photonic circuit operation. The stability of the photonic
circuit should be an important part in the design. However, very few studies have been conducted
on aspects of stable and unstable states of the photonic processing unit. The stability test should
prove to be an important tool in photonic circuit design in the future. The result is significant as a
systematic method is presented to the analysis of photonic circuit. The new method should lead to
a better understanding of the performance of various photonic circuits and may initiate studies on
complex photonic circuits on a large scale.
Passive resonators have been studied for some simple photonic circuit configurations.4,5,6,7,8
However, optical amplifiers can be inserted in the fiber paths of the circuit to enhance the circuit’s
performance. It has been shown3 that optical amplifiers can provide flexibility in the design of the
photonic circuit, not only just as compensators for losses in fiber paths. An amplified double-coupler
double-ring (DCDR) photonic circuit is studied in detail in this work. The passive operation of this
circuit has been examined in the case of a resonator23,24 but its use as re-circulating delay line and
the circuit’s temporal response have never been studied. The development of the signal-flow graph
theory in optical circuits allows us to carry out the generalized study of the DCDR circuit easily
and efficiently. We can identify the recirculating or resonant loops in the circuit immediately once
its signal-flow graph has been drawn. This would significantly increase our understanding on the
circuit performance, particularly the effects from the physical structure of the circuit. The special
feature of the DCDR circuit is that it consists of two loops, which share one common fiber path in
the circuit. This special feature produces some useful results, like the application as an adder, which
are not realizable in other simple configurations.
The DCDR circuit, like most of the optical circuits, is discrete in nature and can be treated as a
discrete-time signal system. The discrete behavior of the all-fiber optical circuit is due to the delays
in the fiber paths of the circuit. Hence, z-transform is used for the analysis of discrete-time signal
processing in our work. The DCDR circuit system will be represented by transfer functions in the
z-domain. The transfer function will contain poles and zeros on the z-plane, which are the roots of
the denominator and numerator of the transfer function, respectively. The report will show that the
poles and zeros locations, or distributions, of the transfer functions determine the characteristics
of the circuit, for instance, the filtering properties. Thus, the circuit responses will be explained in
terms of the poles and zeros values of the transfer functions. In this chapter, the circuit is examined
under two conditions—illuminated by a temporal incoherent source and a coherent source with
finite linewidth. In the former case, the analysis is carried out on the intensity-basis and the circuit
is operating as a re-circulating delay line. In the latter case, the use of the circuit as a resonator
is investigated together with the transient response of the circuit. The computation of transient
response is generalized for optical circuits.
Investigation is also carried out on the possibility of employing the DCDR as a fiber dispersion
equalizer. The use of optical resonators as dispersion equalizers were studied recently,9,10 but they
were confined to simple resonators, which may limit the design flexibility. In this work, the DCDR
circuit is shown to give more freedom in the design as an equalizer.
Section 5.2 is organized as follows. In Section 5.2.3, the graphical technique is used to derive
the transfer function of the resonator in association with the Mason’s gain rules. SFG theory and
Mason’s rule17,18 are found to be applicable in the analysis of photonic circuits. The unique feature
of this method is shown and the advantages of our method over the conventional ones are presented.
Section 5.2.4 discusses the DCDR circuit operating under a temporally incoherent source. The
circuit is considered on an intensity-basis and it is treated as re-circulating delay lines. Several
C. Lin and D. Marcuse, Optimum optical pulse width for high bandwidth single-mode fiber transmission, Electron. Lett.,
17, 54–55, 1981.
24 D. Marcuse and C. Lin, Low dispersion single-mode fiber transmission–The question of practical versus theoretical
maximum transmission bandwidth, IEEE J. Quantum Electron., QE-17, 869–878, 1981.
23
170
Photonic Signal Processing
operation modes of the circuit are displayed including passive, active, and active with negative optical gain. Both the frequency and temporal responses are studied. Possible applications of the circuit
in signal processing are suggested. Procedures in the design of the circuit are proposed for the case
with negative optical gain acting as an illustration of a general design procedure. The DCDR circuit
under a coherent source is studied with emphasis on the resonance effects and the transient response
of the circuit. The resonance of the DCDR circuit is displayed and its use in certain signal processing applications are mentioned. The effects of the source coherence and the input pulse shape on the
transient response of the circuit are studied in detail. Also, algorithms for computations of transient
responses for general photonic circuits are given.
In Section 5.2.5, one of the applications of the DCDR circuit, a fiber dispersion equalizer, is
examined specifically. The equalization achieved by the circuit under a different circuit’s parameters is demonstrated. The capability of the DCDR circuit as a dispersion equalizer is shown with
different operating points.
5.2.1
signal-FlOw graph applicatiOn in Optical resOnatOrs
The mathematical tools that are used in our analysis are introduced here. They include the z-transform
and SFG theory. Z-transform of discrete-time signal processing is useful for the analysis of photonic
circuits, especially for those consisting of delay elements. The SFG theory enables us to examine
and efficiently investigate the characteristics of photonic circuits. The graph reduction rule and
Mason’s rule1,2 are associated with the SFG theory and included. Stability of the circuit is usually an
important criterion of evaluating the circuit performance. Therefore, the stability test for the system
is also introduced.
For the sake of simplicity, a simple photonic circuit—a single-loop resonator circuit—is used as an
example to illustrate the manipulation of the signal-flow graph theory. The z-transform techniques are
used in digital signal processing3 (also called discrete-time signal processing) and employed for the
analysis of the photonic circuit in this work. The discrete-time signal can be considered as an equally
spaced sampling of a continuous-time signal. The z-transform of a sequence x[n] is defined as
∞
X ( z) =
∑ x[n]z
−n
(5.1)
n =−∞
x[n] can be interpreted as the nth term of the sequence of numbers that describe the discrete-time
signal in the time domain. The ratio between the output transform Y ( z ) and the input transform
U ( z ) of a system is called the transfer function H ( z), which is given as:
H ( z) =
Y ( z)
U ( z)
(5.2)
Rearranging (5.2) gives:
Y ( z) = H ( z) ⋅U ( z)
(5.3)
This shows that in the z-domain multiplication of the input with the transfer function produces
the output. The transfer function given in Eq. (5.2) is related to the corresponding time domain
sequence h[n], indeed it is usually defined as the impulse response of the system. The corresponding
operation in the time-domain of Eq. (5.3) is
∞
y[n] =
∑ u[r]h[n − r],
r =−∞
(5.4)
171
Optical Dispersion Compensation and Gain Flattening
which is the convolution sum that can be expressed as
=
y[n] u=
[n] * h[n] h[n] * u[n].
(5.5)
where the symbol * denotes the convolution. It is to be noted that the convolution operation is
commutative. There are two important properties of convolution. The convolution of two functions in the time domain corresponds to the multiplication of their z-transforms. On the other
hand, the multiplication of two functions in the time domain corresponds to their convolution in
the z-domain.
Now writing the numerator and denominator NUM ( z ) as ∑ iB=0 bi z −i and DEN ( z ) as ∑ iA=0 ai z −i ,
H ( z ), the z-domain transfer function can then be expressed as
B
∑b z
NUM ( z )
−i
i
H ( z) =
DEM ( z )
= i =A0
∑a z
i
(5.6)
−i
i =0
which can also be rewritten into product form as
B
b
H ( z ) = 0 z A− B
a0
∏(z − q )
k
k =1
A
∏(z − p )
(5.7)
j
j =1
It is assumed that H ( z) has been expressed in the irreducible form. The values p j are called the
poles of H ( z ) such that H ( p j ) = ∞.; similarly the values qk are the zeroes of H ( z ) for which
H ( qk ) = 0. Also, there is a pole at z = 0 of the multiplicity of ( BA) if B > A. If A > B , there would
be a zero at z = 0 of the multiplicity of ( A − B). As it has been found that the transfer functions can
be expressed in z-domain, thus the transfer characteristics are dependent on the zero-pole patterns4
of these transfer functions in the z-plane. The magnitude-frequency response at a particular frequency as the operating point moving on the unit-circle, z = 1, of the z-plane is given by
B
H ( z) = H ( e
jωτ
b0
)=
a0
∏l
zk
k =1
A
∏
(5.8)
l pj
j =1
where z = ejωτ, ω is the angular frequency in radians per second and τ is the sampling period of
signals in seconds. lzk and l pj are the lengths from the operating point to the position of the kth zero
and the jth pole of the transfer function respectively. The corresponding phase-frequency response
at a particular operating frequency (wavelength) is given by
B
arg( H ( z )) =
∑
k =1
A
φzk −
∑φ + ( A − B)ωτ
pj
(5.9)
j =1
where ϕzk and ϕpj are the phase angles of the zeroes and poles respectively formed by the horizontal
real axis and the lines connecting the poles and zeroes to the operating point in the z-plane.
172
Photonic Signal Processing
Thus, from Eqs. (5.8) and (5.9) we can design the magnitude-frequency response by adjusting
the pole and zero patterns of the transfer functions. To obtain a maximum magnitude at a particular
A
operating wavelength (frequency), we require a pole or a very small value of ∏ j =1 l pj at that wavelength. Similarly, in order to obtain a minimum at a particular wavelength, a zero or an infinitesimal
B
value of ∏ k =1 lzk is required at that wavelength.
There are some relationships between the positions of poles in the z-plane to those correspondingly in the s-plane (the continuous frequency domain). One basic property of this relationship is that,
recall that z = e sτ where s = jω when a pole position moves on the imaginary axis of the s-plane, it
would move along the unit-circle of the z-plane. In this case, we would have marginal stability and
a lossless system. When a pole moves on the imaginary axis towards the left-half of the s-plane, the
corresponding pole moves inside the unit circle in the z-plane. The system would thus become lossy
and stable. If one of the system poles lies outside the unit circle in the z-plane, the system becomes
unstable. Its temporal response would increase with time. In general, the system would be stable if
all the system poles lie inside or on the unit circle in the z-plane. Stability plays an important role in
design of photonic circuits. Stability test for the photonic circuit is introduced in Section 5.2.4.
The SFG has been well known in the analyses and formulation of massive electrical and electronic circuits. We employ extensively this technique in the analysis of optical circuits here. As
mentioned in Chapter 2, SFG is a graphical diagram that comprises directed branches and nodes, in
other words, a graph with nodes linked up by directed branches in some way. A branch represents
the relationships between two nodes and it is usually associated with a transmittance or transfer
function. The signal flows through the branch in a direction indicated by the arrowhead. The transmittance denotes the functional operation that the signal would undergo as it travels through the
branch. In general, that functional operation can be linear or non-linear. The rule of engagement of
the transmission path is described in Chapter 2.
The nodes in the SFG represent the circuit variables. In photonic circuits they are usually optical
fields, for coherence case and intensities for incoherence. The value of a node variable is taken to
be the sum of all incoming signals entering the node. This node value or signal travels along each
outgoing branch connected to it. There are two special kinds of nodes in SFG, source and sink.
A source is a node with outgoing branches connected to it. Likewise, a sink is the opposite, it is a
node with incoming branches only.
There are several terms that are frequently used for describing SFG. For instance, a feed-forward
path from node a to node b is a sequence of nodes and branches that the signal passes through from
node a to b, in which all nodes are traversed only once. A feedback loop is a path that starts and
terminates at the same node such that no node is traversed more than once. A feedback loop that
contains only a single node is called a self-loop.
Thus, the SFG is a graphical illustration of the relationships among the variables in a photonic
circuit. In the following presentation, a single loop resonator circuit is used as a simple example
to illustrate different aspects of the SFG theory and associated rules. A schematic diagram of the
single loop resonator is shown in Figure 5.1.
Input
1
coupler
3
Output
2
4
T
FIGURE 5.1
The schematic diagram of the single loop resonator.
173
Optical Dispersion Compensation and Gain Flattening
A single loop resonator comprises a 2 × 2 directional coupler and a fiber path as shown in
Figure 5.1. The fiber path is connected in such a way that one of the output ports of the coupler is
joined to one of the coupler’s input ports through the path. The excess power loss of the coupler is
assumed to be zero, i.e. the coupler is lossless. This assumption of the coupler is valid for the whole
scope of our analysis. The power coupling coefficient of the coupler is given as k. T ( z ) represents
the transmittance of the fiber path, it is given as:
T ( z ) = t aGz −1 .
(5.10)
where t a is the passive transmission coefficient of the fiber path (3)–(2), which is related to the fiber’s
loss. If ta is equal to its maximum value of 1, the fiber is lossless. G is the optical intensity gain factor of the fiber optical amplifiers (if any) inserted in the path. G equals to one corresponds to the
case where there is no amplifier in the feedback fiber path, i.e. passive operation of the circuit. By
using the z-transform representation, the delay element in the photonic circuit with a unit delay can
be represented by z −1. The unit delay or the basic time delay is defined as the time required for the
optical waves to travel along a length l of optical fiber such that it is equal to the sampling period
of the input optical signal. In this case, z −1 is used to show that the signal at port 2 of the circuit is
a unit-delayed version of the signal at port 3, apart from the scaling of the signal’s magnitude. This
represents the discrete nature of the signals in the circuit.
As the circuit is taken to be an example of the SFG illustration only, we further assume that the
source is temporally incoherent so as to simplify the situation. By this assumption, it means that the
source coherence length is much shorter than the shortest delay path in the circuit so that the phase
change can be ignored. Intensity rather than field amplitude is thus considered. The directional coupler has a direct coupling of (1− k ) and cross coupling of k of the intensity at the input port. Further
discussion of the temporal incoherent source is given in Chapter 2. The SFG in terms of intensity
for the single loop resonator circuit is shown in Figure 5.2.
It can be seen that the single loop in the circuit and the direction of signal flow in the loop can be
identified easily in the SFG in Figure 5.2. In all-fiber photonic circuits, the directional couplers and
the delay lines usually form the basic building blocks. From Figure 5.2, it can be seen that the SFGs
of these two elements are simple.
After obtaining the SFG of the photonic circuit, the derivation of the transfer functions between
node variables is very straight forward. There are generally two methods to achieve this objective,
the graphical reduction of the SFG and Mason’s Rule.
T
3
k
1-k
1-k
1
2
k
FIGURE 5.2 SFG of the single loop resonator circuit.
4
174
Photonic Signal Processing
If the SFG reduction rules (iii), (iv), and (i) are applied in succession to the SFG in Figure 5.2, the
output-input intensity transfer function of the single loop circuit is obtained to be:
k + (1 − 2k )T
1 − kT
H14 =
(5.11)
It is not necessary to reduce the SFG to find the desired transfer functions of the circuit. Mason’s
Rule1,2 can be applied to give the transfer functions directly. Considering the independent input node
a and dependent output node b of an arbitrary photonic graphical network, which are its input and
any port. Let the transfer function H ab for the SFG be defined as the overall transmittance between
the two nodes a and b, then Mason’s rule can be stated for the circuit as1,2:
N
∑ (F ) (∆ )
ab q
H ab =
ab q
q =1
∆
,
(5.12)
where N is the total number of optical transmission paths from node a to node b, ( Fab ) q is the optical transmittance of the qth path from node a to node b in the SFG, and ∆ is given by
∆ = 1−
∑T + ∑T T − ∑ T T T +
lu
u
lu lv
u ,v
lu lv lw
(5.13)
u ,v ,w
with Tlu is the loop transmittance and only products of non-touching loops are included in the
∆ expression. Thus Eq. (5.13) can be written in plain word as:
∆ = 1 − ( sum of all loop transmittances )
+ (sum of products of all loop transmittances of 2 non-touching optical loops)
− ( sum of products of all loop transmittances of 3 non-touching loops ) +
with ∆ is defined as the graph determinant. Therefore, it follows that ( ∆ ab ) q is the cofactor of
the qth forward transmission path, which is equal to ∆ after all loops touching the qth path have
been excluded. Two optical loops are considered as non-touching if they do not share any common node. However, one criterion of using the Mason’s rule is needed to be satisfied. The SFG
should be able to be represented in the planar form where there are no crossings between the
signal-flow paths.
From the SFG of the single loop photonic circuit, it can be observed that there is only one optical feedback loop in the circuit that is (3)(2)(3). Thereafter, the loops or the paths in the circuit are
given by sequences of the node numbers the loop or path follows. Its output-input intensity transfer
function can be determined using Mason’s rule of Eq. (5.13). Applying Mason’s Rule, H14 ( z ) of the
single loop resonator circuit can then be determined by:
2
∑ (F ) (∆ )
14 q
H14 =
14 q
q =1
∆
(5.14)
Firstly, the loop transmittance of the single loop circuit is given as:
Tl1 = kT
(5.15)
175
Optical Dispersion Compensation and Gain Flattening
Thus, ∆ can be obtained by as
∆ = 1 − Tl1 ⇒ ∆ = 1 − kT .
(5.16)
There are two forward paths going from node 1 to node 4, they are: (i) the direct path (1)(4), which
has a transmittance of k, that is ( F14 )1 = k , and ( ∆14 )1 =1 − kT , and (ii) path (1)(3)(2)(4) with the
transmittance of ( F14 )2 = (1 − k )2 T and ( ∆14 )2 = 1. Substituting (5)–(8) into (4) yields
H14 =
k + (1 − 2k )T
1 − kT
(5.17)
which is the same as the expression obtained in Eq. (5.15) that used the graph reduction rules.
5.2.2
stability test
Before computing the response of the circuit, it is sometimes useful to consider the system’s stability
in the design. Stability consideration of the photonic circuit is very important as it is directly related
to the performance of the circuit especially in the time-domain applications. Since the photonic
circuit can be modeled under the discrete signaling, the Jury’s stability test usually applied for
digital control system25 can be employed. Consider a characteristic equation in the form of
F ( z ) = an z n + an−1z n−1 +...... + a2 z 2 + a1z + a0 = 0
(5.18)
This characteristic equation is defined as the equation obtained by equating the denominator of
the transfer function to zero. For n = 2, that is a second-order system, the necessary and sufficient
conditions for a stable operation of the circuit requires
F (1) > 0, and F ( −1) > 0 and a0 < a2
(5.19)
That is all the roots of the equation must lie inside the unit circle in the z-plane. In this example,
n = 1, the stability criterion becomes F(1) > 0 and F( −1) < 0. Now consider Eq. (5.10), the characteristic equation of the single loop circuit is
F (z ) = z − kt aG = 0
(5.20)
Thus, the stability criterion is then given by
1 − kt aG > 0 ⇒ kt aG < 1
(5.21)
The expression on the LHS of the inequality in Eq. (5.21) is in fact the “pole” value of the circuit.
If the pole value is less than one, the pole will then lie inside the unit circle in the z-plane and
stability is achieved. Therefore, it is clearly shown that the Jury’s stability test criterion gives the
condition where the poles of the system will lie inside the unit circle in the z-plane.
25
B. C. Kuo, Digital Control Systems, Chapter 5, Asia, Hong Kong: CBS Publications, 1987. pp. 267–296.
176
Photonic Signal Processing
5.2.3 Frequency anD impulse respOnses
5.2.3.1 Frequency Response
The magnitude and phase responses of the photonic circuit can be obtained by repeating computations at different values of frequency along the unit circle, i.e. z = 1. The frequency responses of
H14 ( z ) of the single loop resonator circuit is plotted against ωτ , as shown in Figure 5.3, together
with a plot showing the pole and zero positions. The ω is the angular frequency of the source, and
τ is the unit time delay of the fiber path in the circuit. The parameters used in this example are
=
k 0=
.4142, t a 1 and G = 2.414 , which gives a pole at +1 and a zero at −1. From the magnitude
plot, we can see that the response has minima when the frequency passing a pole and maxima for
zeroes. When passing through a pole, the phase changes from −1.57rad ( −90°) to 1.57rad (90°) ,
likewise, when traveling past a zero, the phase changes from 1.57rad (90°) to −1.57rad ( −90°). At
other frequencies, the phase remains constant at either −1.57rad or 1.57rad. This can be explained
by examining the phase response expression for the circuit. The phase response of the single loop
circuit is given by
arg ( H14 ( z ) ) = φz1 − φ p1
(5.22)
Consider geometrically in the z-plane, from Figure 5.4, the difference between φz1 and φ p1 is always
90°. Thus (φz1,φ p1 ), depending on the position of the operating point, takes the value of either +90°
or −90°.
FIGURE 5.3 Frequency response for the single loop circuit together with the pole-zero positions with pole
at +1 and zero at −1 (pole: x, zero: o).
177
Optical Dispersion Compensation and Gain Flattening
z
-1
φz1
φp1
+1
FIGURE 5.4 Representing the phase response obtained in Figure 5.3.
5.2.3.2 Impulse and Pulse Responses
The impulse response and the pulse response of the feedback resonator given for Figure 5.1
are shown in Figure 5.5. The pulse input can be obtained by launching a sequence of “111” into
the circuit, the “1’s” are delayed from each other by a unit time delay. It can be seen that the impulse
response has a steady state value, that means the system is undamped or loseless. From Eq. (5.12) the
magnitude of the loop transmittance has a pole at +1. Therefore, the signal maintains its magnitude
in each circulation inside the circuit loop, the signals cross-coupled to the circuit’s output after every
circulation are thus remaining constant. Using the stability criterion (5.20), kt aG or the pole value
equates to unity. Thus, a marginally stable condition is achieved, that is the pole lies exactly on or
very close to the unit circle in the z-plane. If the magnitude of pole value is larger than 1, the system
becomes unstable and the impulse response will then be increasing as time proceeds.
FIGURE 5.5 Impulse response and pulse response for the single loop circuit with the same circuit parameters as in Figure 5.3.
178
Photonic Signal Processing
5.2.3.3 Cascade Networks
This photonic network theory can now be extended to a network of interconnected basic photonic
circuits. In fact, we have so far treated the photonic circuits as elements of a signal processing system, in other words, we have systematized the circuit. Thus, the system theory can be applied in the
manipulation of the circuit. For instance, there are two photonic circuits with output-input transfer
functions H a and H b respectively. If the output of the first circuit is connected to the input of the
second circuit, for example, the two circuits are connected in cascade, then the overall output-input
transfer function of the system will then be equal to H a H b, the product of the two transfer functions.
In general, if we connect n photonic circuits in cascade, the overall transfer function of the system
can be easily determined as the product of all individual transfer functions.
5.2.3.4 Circuits with Bi-directional Flow Path
If in a photonic circuit, the signal travels in both directions through a fiber path, for instance one
signal is transmitted and the other one is reflected, the circuit can be represented by two SFGs. If the
circuit is linear, the circuit transfer function can be obtained by the superposition of the two transfer
functions as obtained individually from each SFG,9 the single ring resonator with two-wave mixing,
or reference, the z-shaped double-coupler resonator.
5.2.3.5 Remarks
The tools for analyzing general photonic circuit are introduced here. The application of z-transform
in the analysis makes both the manipulation and understanding easily. We have developed the application of the powerful signal flow graph theory in the study of photonic circuits. This provides an
examination of the circuit characteristics visually and it effectively reduces the effort of determining the transfer functions of the circuit. The techniques of handling the signal flow graph are demonstrated and include the Mason’s rule and the graphical reduction rule. Together with the stability
test provided, the circuit behavior can be examined thoroughly.
5.2.4
DOuble-cOupler DOuble-ring circuit unDer tempOral incOherent cOnDitiOn
Resonator circuits with double or multiple couplers have been studied.9–12 In this section, a DCDR
circuit is analyzed. The use of this circuit as a resonator is studied9,10 but its use as recirculating delay
lines had not been examined. Here, temporally incoherent of the signal is assumed. Therefore, with
this assumption, the circuit is operated as recirculating delay line instead of resonator. Incoherent
processing is considered by using the intensities. A detailed study of the circuit is carried out for
different sets of circuit parameters including the case where negative optical gain is provided by
the optical amplifier in the circuit. The frequency response and impulse response of the circuit are
given. From the studies, several applications of the DCDR circuit are illustrated.
5.2.4.1 Transfer Function of the DCDR Circuit
The schematic diagram of the DCDR circuit is shown in Figure 5.6 consisting of two 2 × 2 directional couplers interconnected with three optical fiber forward and feedback paths. The fiber path
(3)(6) and (4)(5) are referred as the forward paths of the circuit while path (7)(2) is called the feedback path of the circuit. The circuit can be considered as a two-port device, which incorporates
one input port and one output port as visualized Figure 5.6b. In this latter arrangement, the circuit
can be viewed as a cascade form of two couplers with an overall feedback loop. The k1 & k2 are the
power coupling coefficients of the two couplers 1 and 2, respectively. T1 , T2 , and T3 denote the transmission functions of the forward paths (3)(6) and (4)(5), and the feedback path (7)(2), respectively.
The optical transmittances can be represented by
Ti = t aiGi z − mi for.....i = 1, 2, 3.
(5.23)
179
Optical Dispersion Compensation and Gain Flattening
Input
1
coupler 1
2
(a)
FIGURE 5.6
T2
7
8
coupler 2
1
3
T1
6
8
2
4
T2
5
7
Output
4
T3
Output
coupler 1
Input
3
T1
5
coupler 2
6
T3
(b)
(a) The schematic diagram of the DCDR circuit, (b) the same circuit in a different arrangement.
T
1
3
k1
1-k1
T3
k1
k2
7
6
1-k2
1-k1
1
FIGURE 5.7
2
4
T2
5
1-k2
k2
8
The SFG of the DCDR circuit.
i = 1, 2, 3 corresponds to the three paths as shown in Figure 5.6. tai is the transmission coefficient
of the ith optical waveguide path as the same parameter t a defined in Chapter 2. Gi is the optical
intensity gain factor, which is provided by the waveguide/fiber optical amplifiers incorporated in
the paths and mi is the order of the delay path. The SFG of the DCDR circuit is shown in Figure 5.7.
The SFG of the DCDR circuit is represented in planar form where, for instance, there is no crossing
of optical paths in the graph. The node variable in the SFG represents the optical intensity at that
point of the DCDR circuit. From the SFG of the DCDR circuit, it can be easily seen that there are
two optical closed loops in the circuit, which can be called feedback loops or the recirculating loops
of the circuit. One of the loops is the loop connecting nodes numbered (2), (3), ( 6) and (7), and
the other one through nodes (2), ( 4), (5) and (7). Furthermore, these two loops share one common
path, the path (7)(2). This demonstrates one of the advantages of representing the photonic circuit
with an SFG as we can identify the loops in the circuit without any difficulty. The SFG gives us
an insight into the circuit properties. The next step in the analysis is to derive the optical transfer
functions of the circuit from the SFG.
First of all, the output-input intensity transfer function of the DCDR circuit can be obtained by
using either the graph reduction rules or Mason’s rule as introduced in Chapter 2. In deriving the
optical transfer functions between nodes a and b , the subscript ab is omitted for the sake of clarity.
180
Photonic Signal Processing
By applying Mason’s rule for the SFG of the DCDR photonic circuit, the output-input intensity
transfer function is given by:
4
∑
Fq ∆ q
I8 q = 1
H18 = =
I1
∆
(5.24)
where I 8 and I1 are the output and input intensity respectively of the DCDR circuit. The path and
loop transmittances can be obtained as follows.
1. Two loop transmittances
a. Loop 1: Loop 1 is formed by nodes (2), (3), ( 6) and (7), thus Loop 1 can be written as
(2 )(3)(6)(7)(2) . The loop optical transmittance of Loop 1 is
Tl1 = k1k2T1T3 .
(5.25)
b. Loop 2: Similarly the optical transmittance of Loop 2, which is (2)( 4)(5)(7)(2) and is
given by
Tl 2 = (1 − k1 )(1 − k2 )T2T3
(5.26)
2. Forward path transmittances
From the graphical signal flow diagram, it can be observed that there are four optical
forward paths connecting nodes 1 and 8. The four forward paths and its related
transmittances are:
Path 1: (1)(3)(6)(8)
F1 = (1 − k1 )(1 − k2 )T1
and... ∆1 = 1 − Tl 2 .
(5.27)
Path 2: (1)(3)(6)(7)(2)(4)(5)(8)
F2 = (1 − k1 ) k22T1T2T3
2
and ∆ 2 = 1
(5.28)
∆2 is equal to unity due to its forward path touching both optical loops.
Path 3: (1)(4)(5)(7)(2)(3)(6)(8)
F3 = k12 (1 − k2 ) 2 T1T2T3 , and ∆ 3 = 1.
(5.29)
Similar to ∆ 2 , ∆ 3 is equal to unity due to the touching of two loops of the forward path.
Path 4: (1)(4)(5)(8)
F4 = k1 k 2 T2 , and ∆ 4 = 1 − Tl 1 .
(5.30)
The loop determinant Δ in the denominator is then given by
∆ = 1 − Tl1 − Tl 2
(5.31)
Therefore, the optical transfer function I8/I1 as H18 can be obtained as
H18 =
(1 − k1 )(1 − k2 )T1 + k1k2T2 − (1 − 2k1 )(1 − 2k2 )T1T2T3
,
DEN
DEN = 1 − k1k2T1T3 − (1 − k1 )(1 − k2 )T2T3 .
(5.32)
(5.33)
181
Optical Dispersion Compensation and Gain Flattening
(1-k2)T1
8
(1-k2)T1
k2T2
8
k2T2
k1(1-k2)T2T3
1
1-k1
3
(a)
k2T1
7
(1-k1)T3
d1
1-k1
d1
(1-k2)T2
k1T3
1
4
3
(b)
k1
(1-k1)k2T1T3
d2
4
k1
d2
FIGURE 5.8 The reduced signal flow graphs for the SFG of the DCDR circuit. (a) First reduction, and
(b) Further reduction. d1 = 1 − k1k2T1T3 and d2 = 1 − (1 − k1 )(1 − k2 )T2T3.
Alternatively, the transfer functions of the DCDR circuit can be found by performing a graph
reduction of the SFG in Figures 5.7 and 5.8a and b. After reduction to a certain stage, we can
see that the transfer function H18 obtained from Figure 5.8b is the same as that given in (5.32). In
Figure 5.8b, this can be performed by summing up the transmittances going to port 8, from path (1)
(3)(8) and path (1)(4)(8).
5.2.4.2 Circulating-Input Intensity Transfer Functions
The same procedure as given in the above Section 5.2.4.1 can be used to get the circulatinginput intensity transfer functions of the DCDR circuit. To obtain the transfer function H13 ( z ), the
circulating intensity I3 with respect to the input intensity, we consider the SFG of the DCDR circuit
again. There are two optical forward paths traveling from node 1 to node 3, they are the direct path
(1)(3) and path (1)(4)(5)(7)(2)(3). Their transmittances are given by:
Path 1: (1)(3) :
F1 = 1 − k1, ∆1 = 1 − Tl 2
(5.34)
F2 = k12 (1 − k2 )T2T3 , and ∆ 2 = 1
(5.35)
Path 2: (1)( 4)(5)(7)( 2)(3) :
H13 ( z ) is then given as
2
∑ F∆
i
H13 ( z ) =
i =1
i
∆
(5.36)
By substituting (5.34) and (5.35) into (5.36) we have
H13 ( z ) =
(1 − k1 ) − (1 − 2k1 )(1 − k2 )T2T3
,
DEN ( z )
Similarly, the transfer functions H14 ( z ) and H17 ( z ) can be obtained as
(5.37)
182
Photonic Signal Processing
I1
k + (1 − 2k1 )k2T1T3
= 1
,
I4
DEN
(5.38)
I7
(1 − k1 )k2T1 + k1 (1 − k2 )T2
=
.
I1
DEN
(5.39)
H14 ( z ) =
H17 ( z ) =
5.2.4.3 Analysis
In this section, the behavior of the DCDR circuit under incoherent source condition is examined.
The responses of the circuit with different circuit parameters are studied. The responses investigated
including both frequency response and time response. The frequency responses are magnitude
response and phase response, whereas the time responses are impulse and pulse responses of the
circuit. The relationships between the pole-zero positions of the transfer functions and the responses
of the circuit are considered. This Section mainly focuses on the output response rather than the
circulating intensity response as the former one can be easily obtained.
5.2.4.3.1 Case 1: DCDR Resonance with Unity Delay in Each Transmission Path
In the first case the delay introduced by the interconnection between couplers in feedforward
and feedback paths is equal to the sampling period of the signal entering the circuit, that means
m
=1 m=
m3 = 1. The denominator DEN of the transfer functions derived in Section 5.2.4.2 becomes:
2
DEN ( z ) = 1 − k1k2t a1t a3G1G3 z −2 − (1 − k1 )(1 − k2 )t a 2t a3G2G3 z −2 ,
(5.40)
t a=
t a3 = 1, we
For simplicity, assuming that the fiber transmission paths are lossless, that is t=
a1
2
would take this assumption throughout this Section unless otherwise specified. Thus, we have
DEN ( z ) = 1 − k1k2G1G3 z −2 − (1 − k1 )(1 − k2 )G2G3 z −2
(5.41)
Rearranging Eq. (5.41) in the form and setting to zero in order to find the roots, and hence then poles
of the transfer function
F ( z ) = z 2 − k1k2G1G3 − (1 − k1 )(1 − k2 )G2G3 = 0
(5.42)
So, with a2 = 1 and a0 = −k1k2G1G3 − (1 − k1 )(1 − k2 )G2G3
Applying Jury’s stability test (5.40) through (5.42), a stable operation of the circuit requires
F (1) > 0 ⇒ 1 − k1k2G1G3 − (1 − k1 )(1 − k2 )G2G3 > 0,
(5.43)
F ( −1) > 0 ⇒ 1 − k1k2G1G3 − (1 − k1 )(1 − k2 )G2G3 > 0
(5.44)
and
a0 < a2 ⇒ −k1k2G1G3 − (1 − k1 )(1 − k2 )G2G3 < 1
(5.45)
After some simple algebra, the stability conditions becomes
k1k2G1G3 + (1 − k1 )(1 − k2 )G2G3 < 1
(5.46)
If we consider the system stability in terms of the system poles, the necessary and sufficient
condition for the optical networks to be stable is that all the poles of the system must lie on and/or
inside the unit circle of the z-plane. In another word, the magnitude of these poles must be less than
or equal to one. Thus, the poles magnitudes given by the characteristics Eq. (5.46) would result in:
k1k2G1G3 + (1 − k1 )(1 − k2 )G2G3 < 1
(5.47)
183
Optical Dispersion Compensation and Gain Flattening
It is shown once again that the stability test result is strongly linked to the pole positions in the
z-plane. The system poles can be made imaginary when the expression inside the absolute sign on
the left-hand-side of (5.47) is set negative. This case would be treated later in the analysis.
Several scenarios of Case 1 would now be studied depending on several parameters of the DCDR
circuit.
5.2.4.3.2 Passive DCDR Circuit: Case 1(a)
Operating the DCDR circuit under passive condition, that is when there is no optical amplification
G=
G3 = 1. The stability condition in (5.47) becomes
in the circuit implying G=
1
2
k1k2 + (1 − k1 )(1 − k2 ) < 1
(5.48)
The transfer function H18, Eq. (5.38), becomes
H18 =
[(1 − k1 )(1 − k2 ) + k1k2 ] z −1 − (1 − 2k1 )(1 − 2k2 )z −3
1 − k1k2 z −2 − (1 − k1 )(1 − k2 ) z −2
(5.49)
where the zeroes are at:
z z (1, 2) = ±
(1 − 2k1 )(1 − 2k2 )
(1 − k1 )(1 − k2 ) + k1k2
(5.50)
Let y be the expression of the LHS of (5.48), the plot of y as a function of the coupling coefficients
k1 and k2 is shown in Figure 5.9. As the ranges of k1 and k2 fall within [0.5 − 1] only, the stability
inequality (5.48) is always satisfied. This can also be observed from Figure 5.9. It implies that the
passive circuit is always stable. Recall that y is also the square of the system poles’ magnitude,
k=
0 or k=
k=
1. In either of these two circumstances
y attains its maximum value of 1 when k=
1
2
1
2
the circuit reduces to a single straight through optical path and contains no feedback loop. The system poles can never be positioned on the unit circle of the z-plane.
1
k2 = 0
0.9
0.1
0.8
0.2
0.7
0.3
0.4
0.6
0.5
y 0.5
0.6
0.4
0.7
0.8
0.3
0.9
0.2
1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
k1
0.6
0.7
0.8
0.9
1
FIGURE 5.9 Variation of y (LHS of Eq. (5.48) against the coupling coefficient k1 with k2 as a variable.
184
Photonic Signal Processing
Examining the transfer function H18, it is noted that the zeros of the output-input intensity transfer
function for the passive DCDR circuit can be either purely real or purely imaginary depending on
the values of k1 and k2.
The output-input frequency responses and impulse responses of the passive DCDR circuit have
been computed using different values of k1 and k2, they are shown in Figure 5.10. The corresponding
responses are given in Figures 5.10 through 5.12.
FIGURE 5.10 From left to right in anticlockwise, frequency responses of the photonic circuit amplitude and
phase, the sampled impulse response and the pole-zero plot for the passive DCDR circuit whose parameters
are tabulated in Table 5.1 (correspondent row of Figure 5.10).
FIGURE 5.11 From left to right and anticlockwise, frequency responses of the photonic circuit amplitude
and phase, the sampled impulse response and the pole-zero plot for the passive DCDR circuit whose parameters are tabulated in Table 5.1 (correspondent row of Figure 5.11).
Optical Dispersion Compensation and Gain Flattening
185
FIGURE 5.12 From left to right and anticlockwise, frequency responses of the photonic circuit amplitude
and phase, the sampled impulse response and the pole-zero plot for the passive DCDR circuit whose parameters are tabulated in Table 5.1 (correspondent row of Figure 5.12).
Figure 5.10 shows that the variation in the magnitude plot is a minute and less than 1 dB. This
can be clearly observed that the zero and pole pairs are very close to each other and their effects
counteract each other. Therefore, the circuit resembles a single pole system, which has the pole
at the center of the unit circle in the z-plane, and the magnitude plot is quite uniform with respect to
the frequency. The output-input impulse response is mainly an impulse delayed by one-unit delay
time from the input, which is similar to the response of a unit delay line with loss. It is noted that
there are quasi-linear portions in the phase response, which can be useful.
The output-input frequency response in Figure 3.6 shows quite sharp dips. Its impulse response shows
a maximum after every circulation in the loop and then decays. In this case, the zeros are purely imaginary,
and the poles are purely real. Thus, they would not cancel each other as in the case in Figure 5.10.
In Figure 5.12, where k=
k=
0.5, it gives the zeros at 0, which is at the center of the unit circle in
1
2
the z-plane. As seen in (5.48) that if any one of k1 and k2 equal to 0.5, then the zeros would be 0 (at the
origin in z-plane). Also, it is shown from Figure 3.4 that y (hence the square of system poles’ magnitude) remains constant with different k1 for k2 = 0.5. It is also true when k1 = 0.5, then y is constant when
we change k2. In other words, if we set any one of k1 or k2 equal to 0.5, the poles and zeros patterns of
the output-input transfer function stays the same when we change the other k. As the output response
of the DCDR circuit depends solely on the poles and zeros patterns of the output-input transfer function,
the response would stay the same in this case. Moreover, it can be seen from the impulse response in
Figure 5.12 that it is a lossy system. It is also found that in the passive circuit the two couplers exhibit
k=
K is equal to that of k1 = k2 = 1 − K .
symmetrical behavior. For instance, the response of k=
1
2
5.2.4.3.3 Case: Active DCDR Circuit with Unit Delay in Each Path
5.2.4.3.3.1 G1 > 1. One Optical Amplifier in Forward Path We investigate here the case
where only one optical amplifier is inserted in the DCDR circuit and it is inserted in the path connecting between port 3 and 6, one of the two forward paths. The output-input intensity transfer
function H18 ( z ) simplifies to:
H18 ( z ) =
[(1 − k1)(1 − k2 )G1 + k1k2 ] z −1 − (1 − 2k1 )(1 − 2k2 )G1z −3
1 − [ k1k2G1 + (1 − k1 )(1 − k2 ) ] z −2
(5.51)
186
Photonic Signal Processing
the zeroes of H18 are thus located at
(1 − 2k1 )(1 − 2k2 )G1
(1 − k1 )(1 − k2 )G1 + k1k2
(5.52)
| k1 k2G1 + (1 − k1 )(1 − k2 ) | < 1
(5.53)
z z ( 1, 2 ) =
The stability condition now becomes
The pole values of the output intensity transfer function H18 ( z ) and its zero values are plotted
against the optical gain G1 with k1 and k2 as parameters are shown in Figure 5.13a–c with different
values of k1 and k2 as indicated.
To get a pole value close to one and just inside the unit circle, the value of G1 needs to be 1.2,
5.2, and 3 respectively in the cases shown in Figure 5.13a–c. These give us a sharper roll-off in frequency response than the passive counterpart displayed in Figures 5.14 through 5.16. The data for
the three figures are listed in Table 5.1.
Comparing Figures 5.13 and 5.14 with Figure 5.10, it is observed that the introduction of optical
amplifier in the DCDR circuit enhance the performance especially the frequency response. As the
pole pairs are pushed very close to the unit circle in the z-plane, it results in sharper response in
the magnitude plot. Recall that the pole-zero pattern in the z-plane would play a major part in the
sharpness of the resonance peak of photonic circuits. To get a sharper maximum in this particular
situation, it is required that the distance between the pole and the unit circle must be less than that
between the zero and the unit circle (refer to Eq. (2.2–2.8)). The effect of this active mode operation
can be noted by realizing that the peaks in the magnitude plot have values greater than 0 dB, while
in the passive operation the corresponding peaks can only come up to 0 dB. This operation can be
used as a bandpass filter with narrow passband.
FIGURE 5.13 Plot of absolute magnitudes of the pole and zero of H18 against G1 in case 1(b)(i).
(a ) k1 = 0.9 and k2 = 0.9, ( b) k1 = 0.2; k2 = 0.8, (c) k1 = 0.5; k2 = 0.5.
Optical Dispersion Compensation and Gain Flattening
187
FIGURE 5.14 From left to right and anticlockwise, frequency responses of the photonic circuit amplitude
and phase, the sampled impulse response and the pole-zero plot for the passive DCDR circuit whose parameters are tabulated in Table 5.2 (correspondent row of Figure 5.14).
FIGURE 5.15 From left to right and anticlockwise, frequency responses of the photonic circuit amplitude
and phase, the sampled impulse response and the pole-zero plot for the passive DCDR circuit whose parameters are tabulated in Table 5.2 (correspondent row of Figure 5.15).
188
Photonic Signal Processing
FIGURE 5.16 From left to right and anticlockwise, frequency responses of the photonic circuit amplitude
and phase, the sampled impulse response and the pole-zero plot for the passive DCDR circuit whose parameters are tabulated in Table 5.2 (correspondent row of Figure 5.16).
TABLE 5.1
Parameters Used in the Analysis of the Passive DCDR Response with the Corresponding
Poles and Zeros of the Output-Input Intensity Transfer Function H18
Figure 5.10
Figure 5.11
Figure 5.12
k1
k2
Poles
Zeros
0.9
0.2
0.5
0.9
0.8
0.5
0, ±0.905539
0, ±0.565685
0, ±0.707107
±0.883452
±j1.06066
0, 0
TABLE 5.2
Circuit Parameters Used in the Analysis of the DCDR Response in Case 1(b)(i) with the
Corresponding Poles and Zeros of H18 (z )
Figure 5.14
Figure 5.15
Figure 5.16
k1
k2
G1
Poles
Zeros
0.9
0.2
0.5
0.9
0.8
0.5
1.2
5.2
3.0
0, ±0.990959
0, ±0.995992
0, ±1
±0.966595
±j1.373716
0, 0
Inspecting Figure 5.15, it is noticed that the large amplifier gain has changed the response
dramatically as compared to Figure 5.11, since its presence changes the pole-zero pattern greatly.
In Figure 5.16 we have a system with a pole at 0, a pole pair at ±1 and zeros at 0 realized by the
DCDR circuit. This yields a magnitude plot with very sharp peaks. The lossless system also produces
an infinite constant-value impulse response. The impulse response here can be applied to generate
continuous sequences of “1” pulses in signal processing system. In other words, a single impulse
Optical Dispersion Compensation and Gain Flattening
189
fed into the circuit can trigger a continuous stream of impulses of the same magnitude at the output.
The causes of “0” in between the ones in the impulse response can be understood by inspecting
the geometry of the circuit. As can be seen, the consecutive output signals can only be detected
every two sampling times, which, in this case, is the round loop time. Thus, if we want to get a
stream of “1” signals, we need to acquire the output every two sampling times. If not, we would get
’0 ’’1’’0 ’’1’’0 ’’1’....
Another application is found with the impulse response generated by the DCDR circuit in the situation of Figure 5.16. We examine the pulse response as shown in Figure 5.17a–f for input sequences
consisting of “1” and “0.” For Figure 5.17a, the steady state value of the pulse response is oscillating
between 1 and 0 for an input stream with one “1.” For Figure 5.17b, the steady state value of the
pulse response is oscillating between 1 and 1 for an input stream with two “1.” In Figure 5.17c, the
steady state value of the pulse response is oscillating between 2 and 1 for an input stream with three
“1.” In Figure 5.17d, the steady state value of the pulse response is oscillating between 1 and 2 for an
input stream with three “1.” Following the above pattern, we can observe that from the steady state
magnitude of the pulse response, the number of “1” in the input stream can be detected. It is shown
in Figure 5.17e and f for longer input streams. If we look at the stream as sequence of two-digit
binary numbers, from the response we can indeed determine the occurrence of “1” in a certain digit
position. For instance, in Figure 5.17d, the input streams are “1 1” and “0 1.” The output stream in
this case shows one in the first digit position and two in the second digit position, which corresponds
to the occurrence of “1” at that positions. The response, in fact, counts the numbers of “1” at the two
digit positions, and this may be used as an adder. This ability of counting arises from the geometry
of the circuit and the orders of delay in each path. In Figure 5.17g, m
=
m=
3, the circuit can count
1
2
numbers of “1” in four-digit numbers as expected.
FIGURE 5.17 Pulse response of the DCDR circuit for different input sequence with k1 = k2 = 0.5, G1 = 3,
G2 = G3 = 1 and m1 = m2 = m3 = 1 except (g), which has m1 = m2 = 3 and m3 = 1. Input sequences for each
Figure 5.17: ( a) [0 1], (b) [1 1], (c) [111], ( d ) [1101]
(Continued )
190
Photonic Signal Processing
FIGURE 5.17 (Continued) Pulse response of the DCDR circuit for different input sequence with
k1 = k2 = 0.5, G1 = 3, G2 = G3 = 1 and m1 = m2 = m3 = 1 except (g), which has m1 = m2 = 3 and m3 = 1. Input
sequences for each Figure 5.17: (e) [1 0 1 0 1 1 0 1], ( f ) [1 1 1 0 1 0 1 1 0 1], ( g ) [1 0 1 0 1 1 0 1].
5.2.4.3.3.2 G2 > 1. Optical Amplifier in the Other Feed Forward Path This case would be
similar to Case 1(b)(i) where only one optical amplifier is placed in the other feedforward path. The
output-input intensity H18 becomes
H 18 ( z ) =
[(1 − k1)(1 − k2 ) + k1k2G2 ] z −1 − (1 − 2k1)(1 − 2k2 )G2 z −3
1 − [ k1k2 + (1 − k1 )(1 − k2 )G2 ] z −2
(5.54)
(1 − 2k1 )(1 − 2k2 )G2
(1 − k1 )(1 − k2 )+k1k2G2
(5.55)
with the zeros at:
z z ( 1, 2 ) = ±
The characteristic equation remains similar to those equations given in Section 5.2.2 except that the
coupling coefficients are interchanged and G2 is placed appropriately. Applying the Jury’s stability
test again we obtain the stability condition as:
| k1k2 + (1 − k1 )(1 − k2 )G2 | < 1
(5.56)
The responses of the DCDR circuit in this case are similar to Case 1(b)(i), which has the amplifier
inserted in the other forward path of the circuit. To illustrate the duality of the two cases we try
k=1 k=
0.1=
, G1 1,=
G2 1.2 and G3 = 1. The circuit responses are shown to be the same of that in
2
Figure 5.14 for case 1(b)(i) where k=
k=
0=
.9, G1 1=
.2, G2 G3 = 1, since the values of poles and
1
2
zeros are the same for these two sets of parameters.
Optical Dispersion Compensation and Gain Flattening
191
5.2.4.3.4 G3 > 1. Optical Amplifier in the Feedback Path
The output intensity transfer function H18 now becomes:
H 18 ( z ) =
[(1 − k1 )(1 − k2 ) + k1k2 ] z −1 − (1 − 2k1 )(1 − 2k2 )G3 z −3
1 − [ k1k2 + (1 − k1 )(1 − k2 ) ] G3 z −2
(5.57)
(1 − 2k1 )(1 − 2k2 )G3
(1 − k1 )(1 − k2 ) + k1k2
(5.58)
with the zeros at:
z z1,2 = ±
The stability condition follows immediately that
| k1 k2G3 + (1 − k1 )(1 − k2 )G3 | < 1
(5.59)
0.5, G=1 G=
In the special case of k=1 k=
2
2 1 and G3 = 2, the output intensity responses are plotted in
Figure 5.18. The frequency response of the DCDR circuit is the same as that in Figure 5.16 where the
same k values are used there. However, with the optical amplifier situated at the feedback path rather
than the feed forward path of the circuit, the optical gain required in the former case (to have the
same pole and zero patterns and satisfying the stability condition) is less than that in the latter one.
This makes one think to put the amplifier in the feedback path instead of putting it in the feed forward
path. Nevertheless, the amplitude of the impulse response is smaller in this case where we put the
amplifier in the feedback path. This is because the gain we used in Figure 5.18 is smaller than that
in Figure 5.16 and the output is obtained immediately after the amplifier for the case in Figure 5.16.
The three cases in case 1(b) have been studied where only one optical amplifier is available
for the DCDR circuit. In fact, this is very important in practice when several DCDR circuits are
integrated to form a network then the least number of optical amplifiers in the loop is required.
Therefore, the most important question remained to be addressed is where should we place the
FIGURE 5.18 Frequency response, impulse response and pole-zero plot for the active DCDR circuit
(case 1(b)(iii)) with k1 = k2 = 0.5, G1 = G2 = 1 and G3 = 2.
192
Photonic Signal Processing
optical amplifier in the DCDR circuit? Should it be- in the feed forward or feedback path? The
answer depends on the specific applications.
Considering the case where the DCDR circuit is applicable as a filter then the optical amplifier
should be placed in the feedback path. As stated above, the amplifier in the feedback path requires
less optical gain than that in the feed forward path to achieve the same performance. From another
point of view, we can say that with an amplifier of a particular gain, better performance is achieved
(as far as the use as a filter is concerned) if we put it in the feedback path (providing that the stability
criterion is met).
If we are interested in the impulse response for applications in photonic digital processing
systems, the position of the optical amplifier is very important. Also, depending on the output
ports where signals are tapped, the optical amplifier must be closed to that port. Thus, both cases
where the optical amplifier is placed in feed forward or feedback paths can be used with appropriate
applications.
5.2.4.3.5 DCDR Circuit with Multiple Delays: Case 2
Different combinations of delay orders in the two feed forward paths and the feedback path would
result in different circuit performances. One useful point to note here is that we can obtain poles,
which are evenly and equally spaced around the circle in z-plane by certain choice of delay orders
m1 , m2 , and m3.
Inspecting the denominator of the circuit’s transfer function, which is DEN in Eq. (5.33), if
m1 = m2 , DEN will always follow:
1− az − b
(5.60)
where a is any real number and b is any positive integer. Thus, we would have multiple poles
for the system, and the same stability criterion as outlined in Section 5.2, can be applied as
well. The magnitude frequency responses with m1 = m2 , which equals to 2 and 3, respectively,
are shown in Figures 5.19 and 5.20. The same k and G values used in Figure 5.16 are used
here. In Figure 5.18, the poles’ value is located at 0, 0, −0.5 ± j 0.866 and 1. In Figure 5.20, the
poles are located at 0, 0, 0,± j and ± 1. An application of this feature is that optical frequencies or
FIGURE 5.19 Frequency response, impulse response and pole-zero plot for the active DCDR circuit (case 2)
with k1 = k2 = 0.5, G1 = 3, G2 = G3 = 1 and m1 = m2 = 2, m3 = 1.
Optical Dispersion Compensation and Gain Flattening
193
FIGURE 5.20 Frequency response, impulse response and pole-zero plot for the active DCDR circuit (case 2)
with k1 = k2 = 0.5, G1 = 3, G2 = G3 = 1 and m1 = m2 = 3, m3 = 1.
wavelengths of input optical signal can be filtered at equal interval, for example as a group of
optical carriers at equal intervals.
It is expected that if the delays in the two forward paths are different, interesting responses would
result. In this case, the intensities in the two forward paths do not always add up at coupler 2 at the
same time. A result with=
m1 2=
, m2 1, m3 = 1 is shown in Figure 5.21. The conjugate pole pairs are
now located well inside the unit circle leading to a lossy system and an oscillating time response.
The appearance of the impulse response confirms this point.
FIGURE 5.21 Frequency response, impulse response and pole-zero plot for the active DCDR circuit (Case 2)
with k=1 k=
0.5=
, G1 3=
,=
G2 G3 1=
and m1 2=
, m2 1=
, m3 1.
2
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Photonic Signal Processing
5.2.4.3.6 Special Case with Negative Optical Gain: Case 3
5.2.4.3.6.1 Purely Real or Purely Imaginary Poles, m1 = m2 = m3 = 1 In this section, an optical
amplifier with a negative optical small-signal gain is used, this would allow much greater flexibility
in tuning the peaks of the magnitude plot corresponding to adjusting the pole-zero pattern of the
optical transfer function of the DCDR circuit. The negative optical gain factor can be achieved by
using an optical transistor as described in reference. Alternatively, an optical amplifier incorporated
with a π-phase shifter such as a LiNbO3 integrated optic phase modulator would allow a negative
optical gain.
It is mentioned in Section 5.2.4.2 that DEN is of the form as in Eq. (3.3–19)
if m1 m=
=
2 . If m1
m=
m=
1, DEN is of the form:
2
3
DEN = 1 − az −2
(5.61)
With a = k1k2G1G3 + (1 − k1 )(1 − k2 )G2G3 . The optical system poles would be located at 0, π , 2π ...
if a > 0 and at π / 2, 3π / 2, ... if a < 0. As it can be easily seen, the characteristic equation of the
photonic DCDR circuit would result in a quadratic equation with roots as purely real or purely
complex conjugates in this situation. To achieve the latter case, it requires amplifiers in the
DCDR circuit with negative gain.
k=
0.1 and the gain factors
Firstly, we examine a typical example in the above case where k=
1
2
for the three optical paths are G3 = 1 and G2 = −1.2, −1.25 and − 1.3, respectively, with the gain G1 as
a variable. Figure 3.17a shows the magnitudes of the poles (solid line) and that of the zeroes (dashed
line) of H18 ( z ) as a function of gain G1 for G3 = 1 and G2 = −1.2. For stability, the values of the poles
and zeroes would never reach unity for this particular circumstance. In fact, it can reach closed to
unity, however both zero and pole approach unity leading to the cancellation of their effects.
When the value of the negative gain G2 is increased to −1.25 and −1.3, the values of the pole and
zero are plotted in Figure 5.22b and c respectively. Figure 5.22b exhibits a similar characteristic as
in Figure 5.22a except that the pole and zero can now reach unity.
When G2 = −1.3 the pole and zero at unity are separated as it can be observed in Figure 5.22c.
Thus, we can select G1 = 5.3 and G3 = 1, G2 = −1.3 for the pole equals to unity corresponding to a
FIGURE 5.22 Values of poles (solid line) and zeroes (dashed line) versus gain G1 for DCDR circuit with
k1 = k2 = 0.1, G3 = 1 and (a) G2 = −1.2, (b) G2 = −1.25, (c) G2 = −1.3.
Optical Dispersion Compensation and Gain Flattening
195
FIGURE 5.23 Magnitude frequency response of H 18 ( z ), H 18 ( z ) with k1 = k2 = 0.1, G1 = 2, G2 = −1.25,
G3 = 1 and m1 = m2 = m3 = 1 .
zero value of 1.015. It is also noted here that for two identical couplers the value of zero is always
greater than that of the pole. Thus, the amplitudes of the frequency response always have minima
according to the analysis of transfer function in the z-domain. Hence, the DCDR with negative gain
in one of the forward or feedback paths can only be used as an optical notch filter at output port 8.
For the case where G2 = −1.25 and G1 = 2, the DCDR circuit has altogether three system poles:
one at the origin, z p (1) = 0 and one complex pole pair at z p( 2,3) = ± j 0.996243. The transfer function
H18 ( z ) has two zeroes at the location z z(1,2) = ± j 0.997663. Clearly the zeroes follow closely the complex pole pair. Figure 5.23 shows the magnitude response of this optical transfer function.
The relation in Eq. (2.2–8) allows us to design photonic circuit with adjustable response by
appropriately positioning the poles and zeroes of the optical transfer function. This fact is indeed
the novel feature of our analysis using the SFG technique and z-transform. It is also determined that
when G1 is increased from 2 to 5 in the case for Figure 5.23 the magnitude H18 ( z ) changes from a
ratio of 2336 / 3757 to 4655 / 18929.
Figure 5.24 shows the magnitude and phase responses together with the pole-zero position in
the z-plane of the circulating transfer function H 14 ( z ) with the same circuit parameters as used in
Figure 5.23. This H 14 ( z ) has maxima at an optical frequency corresponding to ωτ equals to odd
multiple order of π/2. At these values, the transfer functions H 18 ( z ) and H 13 ( z ) have minima as it can
be observed from Eqs. (5.32), (5.49) and (5.57). Thus, the circuit performs a kind of quasi-resonance.
Since it is assumed that the source is temporal incoherent here, thus resonance effect should
not occur in the circuit. This is because the beams traveling inside the circuit always add up
constructively. However, with the application of negative optical gain in one of the paths, destructive
interference can occur, which results in a behavior similar to resonance in the coherent case. The
resonance condition for the DCDR circuit under coherent source will be determined in Chapter 7.
H13 denotes the circulating transfer function in one of the two optical loops. Thus, we can conclude
that at this resonance the energy is stored in one of the optical loops only for circuits with two
optical loops that share one common path. In addition, with this case of G2 = −1.25 and G3 = 1 and
G1 = 2, the loop (3)(6)(7)(2)(3) has a positive transmittance value while the other loop (4)(5)(7)(2)
(4) would have a negative loop transmittance. The negative gain in the forward path (4)(5) would
interfere with the optical waves in the forward paths (3)(6) in a destructive manner and hence can
generate a depletion of the output at a destructive interference. Hence the optical energy is stored
only in the loop (4)(5)(7)(2)(4). This helps explaining the quasi-resonant effect mentioned earlier.
196
FIGURE 5.24
Photonic Signal Processing
The responses of H 14 ( z ) with k1 = k2 = 0.1, G1 = 2, G2 = −1.25, G3 = 1 and m1 = m2 = m3 = 1.
This finding is a significant development by applying the SFG technique in optical resonators in
particular and photonic circuits in general.
=
m3 1=
and either m1 0 or m2 = 0
5.2.4.3.7
Next, we consider another situation where we have direct connection in either one of the forward
paths (3)(6) or (4)(5). m3 is equal to 1. The direct connection would convert the DEN to a quadratic
equation of the form:
DEN = a + bz −1 + cz −2 ,
(5.62)
The values of the poles are then given by
z p( 1, 2 ) =
−b ±
b2 − 4ac
,
2a
(5.63)
where a, b, and c are three constants whose values depend on the photonic components and
m1 0=
, m2 1, m3 = 1. The
correspond with those of Eq. (5.33). We consider here the case of=
expression of DEN in terms of the circuit parameters:
1 − k1k2G1G3 z −1 − (1 − k1 )(1 − k2 )G2G3 z −2 = 0
(5.64)
By applying the Jury’s stability test for digital systems again, with
a2 = 1, a1 = −k1k2G1G3 , a0 = (1 − k1 )(1 − k2 )G2G3
(5.65)
Optical Dispersion Compensation and Gain Flattening
197
The stability criteria requires
1 − k1k2G1G3 − (1 − k1 )(1 − k2 )G2G3 > 0
1 + k1k2G1G3 − (1 − k1 )(1 − k2 )G2G3 > 0
(5.66)
(1 − k1 )(1 − k2 )G2G3 < 1
Rearranging Eq. (5.66), we obtain
k1k2G1G3 + (1 − k1 )(1 − k2 )G2G3 < 1
(1 − k1 )(1 − k2 )G2G3 − k1k2G1G3 < 1
(5.67)
(1 − k1 )(1 − k2 )G2G3 < 1,
To obtain a complex conjugate pole pair it then requires from Eq. (5.67) that
(k1k2G1G3 )2 + 4(1 − k1 )(1 − k2 )G2G3 < 0,
(5.68)
In order to satisfy Eq. (5.68), the amplifier gain G2 or G3 must take sufficient negative values if the
couplers are passive i.e. k1, k2 < 1. There are several combinations of photonic circuit parameters
of the DCDR resonators to satisfy this equation. A procedure can be established as follows: select
a combination of four out of five of k1, k2 , G1, G2 and G3 , plot Eq. (5.68) against the fifth parameter
as a variable, then find a range of operation so that this inequality is satisfied, and finally choose
values in this range to design the photonic circuit and re-check the stability condition.
An example is given here to illustrate the above procedure where k1 = k2 = 0.3, G1 = 2, G2 = −1 are
chosen. Figure 5.25 shows the L.H.S. of Eq. (5.68) as solid line and the other three lines represent
the L.H.S. of Eq. (5.68), which are the stability expressions. The optical gain G3 must be chosen so
that it satisfies Eq. (5.68) as well as the stability conditions. It follows that the maximum value of
G3, where the system remains stable, is about 2.
FIGURE 5.25 Selecting optical gain parameters for the design of a DCDR circuit when=
m1 0=
, m2 m3 = 1.
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Photonic Signal Processing
FIGURE 5.26 Frequency and discrete-time responses for H18(z) with k1 = k2 = 0.3, G1 = 2, G2 = −1, G3 = 2
and m1 = 0, m2 = m3 = 1. Poles : 0.1800 + j 0.9734, Zeroes : 0.0459 + j 0.8068.
This analytical examination of the stability of the photonic circuits is unique when a z-transform
method is employed. Therefore, flexibility in design is achieved. The frequency response of the
intensity transfer function H18 ( z ) is shown in Figure 5.26 with the circuit parameters selected as
given above. From the pole-zero pattern plotted, we can see that the poles and zeroes of the optical
transfer function H18 ( z ) are complex in nature. It is noted here that the poles and zeroes are complex
conjugates and are not purely real or purely imaginary. This would allow the peaks of the magnitude plot designed to be closely spaced together. An application of this feature is that two or several
wavelengths can be demultiplexed as we can observe from Figure 5.26. A phase change of occurs
at each pole position. The periodic variation of the output impulse response of the DCDR shows the
circulating of the impulse delay in the re-circulating loops.
Another example of the operation of the DCDR circuit is to choose the poles closely spaced.
The pole and zero patterns of a DCDR with m1 = 0, m2 = m3 = 1 and k=
k=
0=
.5, G1 1=
.9, G2 1
1
2
and G3 = −4 is shown in Figure 5.27. The zeroes of the system are now located at the origin and
a real negative of −0.5263. The magnitude response shows the feature of demultiplexing of this
DCDR configuration. Furthermore, the impulse response shows the variation of the output pulses
with certain pattern different from that in Figure 5.26.
5.2.4.3.8 Remarks
In this section, the DCDR circuit is studied under temporal incoherent source for different operation
modes. It is found that for active operation, the locations of the amplifiers in the circuit depend on
the application required. Greater design flexibility is found with presence of negative optical gain in
the circuit. It is discovered that a quasi-resonance behavior is possible with the insertion of negative
optical gain in the delay line. Procedures in design of the DCDR circuit are shown in the special
case of negative optical gain but in general, this can be applied to other situations. Applications of
the circuit in signal processing such as filters, counters, and adders are realized.
One point is needed to be clarified before proceeding to the next chapter. It is more appropriate to
call the DCDR photonic circuit a re-circulating delay line rather than a resonator when it is operating at
the incoherent source situation. Although the output intensity response can have minimums for certain
Optical Dispersion Compensation and Gain Flattening
199
FIGURE 5.27 (a) Amplitude and (b) phase frequency response and (c) discrete-time impulse response
and (d) pole-zero pattern for H18 ( z ) with k1 = k2 = 0.5, G1 = 1.9, G2 = 1, G3 = −4 and m1 = 0, m2 = m3 = 1. Poles:
0.9500 ± j 0.3123, Zeroes : 0, −0.5263.
circuit’s parameters (negative optical gain), the circulating intensity responses do not have maximums at
the same condition. It would be shown in the next chapter that for a resonator to have resonance, there are
generally two constraints it need to be met. They are the constraints on the circuit’s parameters and the
other one is the constraint on the operating point of the resonator. The latter one is usually related to the
phase change the signal experiences when it travels along a loop in the circuit. For the source incoherent
condition, we have ignored the effect of phase that could have on the signal. The parameter constraint can
be easily met but due to the ignorance of the phase in the circuit, the phase constraint cannot be met. The
circuit is also very stable due to the ignorance of the phase effect on the circuit’s performance.
5.2.5 DcDr unDer cOherence OperatiOn
In the previous sections of this chapter, the DCDR circuit is studied under incoherent source
condition. In order to study the resonance effect of the DCDR circuit, the analysis is required to be
carried out on the field-basis. Therefore, in this chapter, the circuit excited by coherent source with
finite linewidth will be considered. The DCDR circuit is considered as a resonator here while in the
previous chapter it is treated as re-circulating delay line.
Previous studies on the effect of source coherence on the performance of resonator circuit, but
no one had ever performed the analysis on DCDR circuit or more complicated photonic circuits.
Under the current trends of integrated photonics circuits based on Silicon, Si on insulator (SOI)
technology photonics circuits will be complex, and the computer aided circuit design will become
useful. This mainly due to the complicated manipulations involved. However, with the help of our
newly developed SFG theory in optical circuits, these difficulties can be overcome easily. In this
chapter, an algorithm is derived to compute the response systematically. The temporal transient
response of the circuit is shown with source coherence.
5.2.5.1 Field Analysis of the DCDR Circuit
Before considering the case for source with finite linewidth, we first derive the transfer functions of
the DCDR circuit in terms of field rather than intensity as in the previous chapter. In this section, we
derive the field version of the transfer functions for the DCDR circuit. The analysis in this Section is
200
Photonic Signal Processing
different from the preceding intensity-basis analysis in the sense that the phase change encountered
by the signal in the circuit would be taken into consideration, thus interferometric effects can take
place. Therefore, the resonance effect of the DCDR circuit can be investigated.
Firstly, we define the DCDR circuit parameters with lightwaves represented in the field
amplitude. The subscript f distinguishes the nature of the variables as field variables from those
in terms of intensity. The circuit parameters are defined as: k pcf = − j k p where p = 1,2 as the
field cross-coupling coefficient for the two couplers, k pdf = 1 − k p where p = 1, 2 is the field
direct-coupling coefficient for the two couplers 1 and 2, and Tif = tai Gi z − mi for i = 1, 2 and 3. The
−j term in the k pcf expression above accounts for the −π/2 phase shift induced by the coupler
during the cross-coupling.
5.2.5.2 Output-Input Field Transfer Function
The output-input field transfer function is obtained by carrying out the same procedure as introduced
in Section 5.2.3. The output-input field transfer function of the DCDR circuit, by using Mason’s rule
given in Chapter 2, is given by
4
∑
Fqf ∆ qf
E8 q = 1
H18 f ( z)
=
∆f
E1
(5.69)
E8 and E1 are the output and input field amplitude respectively of the DCDR circuit. The loop
transmittances and the forward path transmittances are stated as follows:
1. Loop transmittances
a. Loop 1: (2)(3)(6)(7)(2): The loop optical transmittance of Loop 1 is
b. Loop 2: (2)(4)(5)(7)(2) The optical transmittance of Loop 2 is Tl 2 f = k1df k2 df T2 f T3 f
2. Forward path transmittances
The four forward paths and their related transmittances are:
Path 1: (1)(3)(6)(8) F1 f = k1df k2 df T1 f and ∆1 f = 1 − Tl 2 f
Path 2: (1)(3)(6)(7)(2)(4)(5)(8) we have F2 f = k1df 2k2cf 2T1 f T2 f T3 f and ∆ 2 f = 1 where ∆ 2 f is equal
to unity due to its forward path touching both optical loops.
Path 3: (1)(4)(5)(7)(2)(3)(6)(8) F3 f = k1cf 2k2 df 2T1 f T2 f T3 f , and ∆ 3 f = 1.
∆ 3 f is equal to unity due to the touching of two loops of the forward path.
Path 4: (1)(4)(5)(8) F4 f = k1cf k2cf T2 f , and ∆ 4 f = 1 − Tl1 f
Furthermore, the loop determinant ∆ f is given by ∆ f = 1 − Tl1 f − Tl 2 f
Hence, the output-input field transfer function, in terms of k1, k2 , T1 f , T2 f and T3 f can be expressed as:
H18 f ( z ) =
(1 − k1 )(1 − k2 )T1 f − k1k2 T2 f − T1 f T2 f T3 f
1 + k1k2 T1 f T3 f − (1 − k1 )(1 − k2 )T2 f T3 f
(5.70)
5.2.5.3 Circulating to Input Field Transfer Functions
Similarly, the circulating-input field transfer functions can be derived and is given as follows:
H13 f ( z ) =
(1 − k1 ) − (1 − k2 )T2 f T3 f
DEN f
(5.71)
201
Optical Dispersion Compensation and Gain Flattening
− j k1 − j k2 T1 f T3 f
DEN f
(5.72)
− j k1 (1 − k2 )T2 f − j (1 − k1 )k2 T1 f
DEN f
(5.73)
H14 f ( z ) =
H17 f ( z ) =
where DEN f = 1 + k1k2 T1 f T3 f − (1 − k1 )(1 − k2 )T2 f T3 f .
5.2.5.4 Resonance of the DCDR Circuit
The usual definition of optical resonance in a photonic circuit is that at a particular frequency or
wavelength the optical output takes a minimum value while the optical energy is circulating in the
loops of the photonic circuit. Thus, the resonant condition can be found by setting the output transfer
function to zero or effectively finding the zeroes of the output-input optical transfer function. In this
case, the DCDR circuit can be referred as a resonator. Considering (5.70), if ta1 = ta2 = ta3 = ta ,
k1 = k2 = k, G1 = G2 = G3 = 1 and m1 = m2 = m3 = 1, the equation simplifies to:
H18 f ( z ) =
(1 − 2k ) t a z −1 − t a t a z −3
1 − (1 − 2k )t a z −2
(5.74)
Rearranging, Eq. (5.74) gives
H18 f ( z ) =
(1 − 2k ) t a z 2 − t a t a
(5.75)
z 3 − (1 − 2k )t a z
Setting the numerator in Eq. (5.75) to zero resulting in two resonance conditions: (i) if
z 2 = −1, k = (1 + ta ) / 2; and (ii) if z 2 = 1; k = (1 − t a ) / 2; z 2 = −1 means z = ± j , which can be
interpreted as ωτ = nπ − π 2 ... with n = 1, 2... Recall that ωτ is the phase change through a fiber
path in the circuit with unit delay time. Similarly, z 2 = 1, which means z = ±1, can be interpreted as
ωτ = n, n = 1, 2.... As ta is close to 1 for low-loss fibers, k for resonance would be close to 1 and 0,
respectively, for the two resonant conditions.
By inspecting the SFG of the DCDR circuit in Figure 5.28, it can be seen that there are two
touching loops. They are Loop 1, which is (2)(3)(6)(7)(2), and Loop 2, which is (2)(4)(5)(7)(2), with
their loop transmittances. The frequency response of the output-input intensity transfer function and
the circulating-input intensity transfer functions under the resonant conditions given in Eqs. (5.72)
and (5.75) are plotted in Figures 5.29 and 5.30 respectively. The instantaneous optical intensities are
T1f
3
k1df
k1cf
2
k3f
7
k1cf
6
k2df
k2df
k1df
1
k2cf
4
T2f
5
k2cf
FIGURE 5.28 The SFG of the DCDR circuit in terms of the field variables.
8
202
Photonic Signal Processing
FIGURE 5.29 Frequency response of the output-input intensity transfer function and the circulating-input
intensity transfer functions of the DCDR resonator under the resonant conditions of ta = 0.99 and k = 0.995,
which satisfy the condition listed in Eq. (4.2–4.16).
FIGURE 5.30 Frequency response of the output-input intensity transfer function and the circulating-input
intensity transfer functions of the DCDR resonator under the resonant conditions of t a = 0.99 and k = 0.005 ,
which satisfy the condition listed in Eq. (4.2–4.17).
obtained by taking the square of the modulus of the corresponding field amplitudes. The vertical scale
of the magnitude plot is shown with the absolute magnitude rather than dB as used in the previous
chapters. It is found that this scale would give a better illustration of the resonance effect of the circuit.
In Figure 5.29, there are maximums for the circulating intensity I3 at ωτ = π / 2, 3π / 2, 5π / 2 and
7π / 2 rad. and correspondingly there are minimums for the output intensity I8 and circulating
intensity I4 at these positions. This shows the resonance of Loop 1 of the DCDR resonator. Indeed,
the resonance criterion on the phase change for the signal in the circuit is that the round-loop phase
Optical Dispersion Compensation and Gain Flattening
203
change of the resonant loop needed to be 2nπ where n is an integer. Looking at the criterion on ωτ
(the phase change per fiber path with unit delay) for resonance of Loop 1 in Eq. (5.76), it shows the
above point. The round-loop phase change for Loop 1, in this case, is 2(nπ − π /2), plus the phase
change encountered across the couplers that is π /2 per coupler. It is obvious that this sum adds
up to the value of 2nπ . This is another way of looking at the resonance condition, and it can be
determined by examining the circuit’s signal flow graph alone. The resonance of Loop 2 of the
DCDR resonator is shown in Figure 5.30. There are maximums for the circulating intensity I4
at ωτ = π , 2π , 3π and 4π and correspondingly there are minimums for the output intensity I8 and
circulating intensity I4 at these positions.
It is also found that the sharpness of the resonance depends on the circuit parameters. In Figure 5.31,
resonance behavior of the DCDR resonator is shown with t a = 0.8 and k = 0.1, which corresponds to
the resonance of Loop 2 of the resonator. A smaller value of ta is used here, and it indicates the loss in
the fiber path is larger. It is clearly seen that the maximum value of the circulating intensity I4 is much
smaller than the one in Figure 5.30. Thus, in this case, it can be stated that low-loss fibers should be
used in building up the resonator in order to have sharp resonant peaks.
In the above results, it is observed that the energy can be stored in either of the two loops of the
DCDR resonator depending on the circuit parameters. It is also noticed from Figures 5.29 and 5.30
that H17 f has maximums in both cases.
Next, we examine the stability of the circuit under the above resonant condition. Applying Jury’s
stability test to H18 f results in the stability condition of
(1 − 2k )ta < 1
(5.76)
From (5.76), it can be seen that a stable operation of the passive DCDR resonator is always fulfilled
provided that 0 < k < 1 and 0 < t a < 1. If the DCDR resonator is operated in a different situation as
mentioned above, for instance G1 > 1 or k1 ≠ k2, there would be other sets of resonant conditions for
the resonator.
FIGURE 5.31 Frequency response of the output-input intensity transfer function and the circulating-input
intensity transfer functions of the DCDR resonator under the resonant conditions of ta = 0.8 and k = 0.1, which
satisfy the condition listed in Eq. (4.2–4.17).
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Photonic Signal Processing
5.2.5.5 Transient Response of the DCDR Circuit
5.2.5.5.1 Effects of Source Coherence
In Section 5.2.4.3, we study the response of the DCDR circuit to a monochromatic light source,
i.e. single frequency, which can be called the steady-state response of the circuit. But, practically,
a purely monochromatic source is not available. The best we can have is a single frequency source
with finite linewidth. In the following analysis, the effect of the source coherence is taken into
account in evaluating the transient response of the DCDR circuit.
We start the analysis by considering the electric field of the input light source, which is expressed
in the form:
E1(t ) = Es (t ) exp ( j (ωot + φ (t ))
(5.77)
where Es (t ) is the amplitude of the electric field at time t, ω0 is the frequency of the source, and
φ (t ) is the time-dependent phase that represents the phase fluctuation. This is often called the phase
noise in optical fiber systems. This phase fluctuation is the cause of broadening of the optical source
spectrum which results in finite linewidth of the source spectrum. If the source amplitude is time
invariant and recalling some basic properties of the z-transform, the output-input field transfer
function of the circuit H18 f = H ( z ) can be stated in the form:
H18 f ( z ) = ... + h[1]z1 + h[0] + h[1]z −1 + h[2]z −2 + h[3]z −3 + ...
(5.78)
An inverse z-transform manipulation would convert H18f from the z-domain to the time domain. For
t a=
t a3 = 1,
m1 = m2 = m3 = 1, that is, there is unity delay in each path of the DCDR circuit and t=
a1
2
the expansion of Eq. (5.78):
h[n] = 0 for all n < 0,
h[0] = 0,
h[1] = ( (1 − k1 )(1 − k2 )G1 − k1k2G2 ),
h[2] = 0 ,
h[3] = −  k1k2 (1 − k1 )(1 − k2 )G3 (G1 + G2 ) + (k1 + k2 − 2k1k2 ) G1G2G3  ,


Hence, h[n] represents the impulse response of the system at time index n. Although h[n] is
=
m=
m3 = 1 and t=
t a3 = 1 for simplicity, the
calculated for the restricted condition m
1
2
a1 t a=
2
following analysis is applicable to the general situation. To compute the output of the resonator for
an arbitrary input sequence (or pulse shape) x[n], we perform the convolution in the time domain
between h[n] and x[n], i.e. the output y[n] is given by:
y[n] = h[n]* x[n]
(5.79)
where * represents the convolution operation. This is in fact the theory of operation to get the pulse
response of the system.
Recall that the basic delay time of the circuit is denoted as τ and it is the sampling time as well.
At time t n = n, the time-averaged output intensity I ( tn ) of the DCDR circuit is
I8 ( tn ) = E8 ( tn ) E8* ( tn )
(5.80)
where E8* ( tn ) is the complex conjugate of E8 ( tn ). The angular brackets denote ensemble average. From a different point of view, E8 ( tn ), which is the output at time tn = n , can be considered
205
Optical Dispersion Compensation and Gain Flattening
as the nth sampled output and, similarly, the input E1 ( tn ) at time tn = n can be regarded as the
nth sampled input into the circuit. So, the former term is in fact y[n] and the latter term corresponds to x[n] in Eq. (5.79). We can see the relationship between the discrete-time signal
representation and the signals in the photonic circuit. Next, the effect of source coherence is
considered.
In order to include the source coherence contribution to our analysis, the phase fluctuation
term e( jφ ( t )) of the light source as given in Eq. (5.80) is needed to be examined. This time-varying
phase fluctuates randomly and is statistical in nature, thus statistical method is used to handle it.
Traditionally, it is treated as random signal or process, and it is best described by its correlation
function. In general, we consider the autocorrelation function R[( p − s )τ ] of the input optical wave
field and it is given as
R[( p − s)τ ] =
E1 (t − pτ ) E1* (t − sτ )
(5.81)
 Es (t − pτ ) Es * (t − sτ ) 


where p and s are integers. From Eqs. (5.80), (5.81) becomes:
R[( p − s)τ ] = expj[(s − p)ωoτ ] exp[ jφ (t − pτ )  exp  jφ (t − sτ )]
(5.82)
If the spectrum broadening of the laser due to the random phase fluctuation ϕ(t) is of the Lorentzian
form, the phase function exp[ jφ (t )] would have the following correlation
exp [ jφ (t − pτ ) ]exp[ − jφ (t − sτ ) ] = exp [( − s − p)∆ωτ ]
(5.83)
Where ∆ω 2π is the half-width at half-maximum of the Lorentzian spectrum. We also note that
e −∆ωτ is the Fourier transform of the Lorentzian spectrum. It is related to the autocorrelation function for the phase. Since the coherence time τ c of the light source is equal to26:
τc =
1
2∆ω
(5.84)
Thus, this can be rewritten as
exp[ jφ (t − pτ )  exp  − jφ (t − sτ )] = exp (− s − p )τ / τ c 
(5.85)
Hence, Eq. (5.82) can be rewritten as
R[( p − s)τ ] = exp j[(s − p)ωoτ ]exp(− s − p τ / 2τ c ) = exp j[(s − p)ωoτ ]D
with
− s− p
(5.86)
D = eτ /2τ c
This gives an expression for the phase autocorrelation function in terms of the source
coherence time. When computing I8 ( tn ) in Eq. (5.83), products of the input field such as
< E1 ( t n ) E1 * ( t n ) >, < E1 ( t n ) E1 * (t n − τ) >,... , E1(t n − pτ) E1 * (t n − sτ) ,... etc. are involved because
E8 (t ) is related to E1(t ) via the sampled relation of Eq. (5.79). By using Eqs. (5.86), (5.80), and (5.85),
the values of the above products and I 8 ( t n ) can be found.
The general algorithms of computing the transient response of the photonic circuit are derived
from the following steps: (i) From the transfer function H ( z ) of the optical circuit, obtain the
26
L. N. Binh, Advanced Digital Optical Communications, 2nd ed., Boca Raton, FL: CRC Press, 2017.
206
Photonic Signal Processing
impulse response or indeed the sequence h[n] (Eq. 5.78). (ii) Convolute the impulse response
obtained in step 1 with the input sequence to obtain the output of the system for the pulse input
(Eq. 5.77). (iii) Obtain the expression for the phase fluctuation of the input, for example, the correlation between values of phases at different times. (iv) Compute the output intensity of the resonator
by using the result obtained in step 2 in combination with expression in step 3.
We can compute the transient response of any photonic circuit with transfer function H ( z ). The
results for the DCDR circuit are presented in the following section.
It can be seen from Eq. (5.79) that the transient response of the circuit mainly depends
on the ratio of the basic time delay to the source coherence time. For the two extreme cases,
monochromatic source (very long coherence time, τ c >> τ ) and temporal incoherent source (very
short coherence time, τ c << τ ), the value of D is equal to 1 and 0 respectively. Hence the range
of D is within [0, 1]. In general, the temporal response with monochromatic source is called the
steady state response. The transient response corresponds to the case where the source contains
finite linewidth.
The transient responses are shown in Figures 5.32 and 5.33 for the passive DCDR circuit with unit
delay in each path. In Figure 5.32, the circuit parameters satisfying the resonant condition Eq. (5.73) is
used with ta = 0.99, k = 0.995 and ωoτ is chosen to be π / 2. The input pulse is shown in Figure 5.32a
where the output intensity response is shown in Figure 5.32b–d with different degrees of source
coherence. It is interesting to look at the incoherent case in Figure 5.32 d in which the output is constant. In Figure 5.33, the circuit response to other input sequence (or in other words, the pulse shape)
is examined. It is found in Figure 5.33 that the responses for different source coherence are similar.
FIGURE 5.32 Transient response of the passive DCDR circuit with t a = 0.99, k = 0.995 and ωoτ = π /2.
(a) Input pulse; (b) Output intensity pulse with τ/τc = 0.001 (monochromatic); (c) Output intensity pulse with
τ /τ c = 0.2; (d) Output intensity pulse with τ /τ c = 500 (temporal incoherent).
Optical Dispersion Compensation and Gain Flattening
207
FIGURE 5.33 Transient response of the passive DCDR circuit with ta = 0.99, k = 0.995 and ωoτ = π/2.
(a) Input pulse; (b) Output intensity pulse with τ /τ c = 0.001 (monochromatic); (c) Output intensity pulse with
τ/τc = 0.2; (d) Output intensity pulse with τ /τ c = 500 (temporal incoherent).
In Figure 5.34, the circuit responses are computed with the same input pulse shape as in
Figure 5.33a, but the circuit parameters here are different. The values of t a and k still satisfy
the resonant condition Eq. (5.75) but non-resonant values of ωoτ, have been used in (b) and (d).
Also, effects of different degree of source coherence on the response are compared. It can be
seen in Figure 5.34a and b that the responses oscillate. The output in (a) satisfies the resonant
condition and oscillates more than that in (b). This is because at resonant condition there are
many circulating fields in the circuit interacting with each other. Consider the geometry of our
DCDR circuit, which has two forward paths and signals that add up at coupler 2 after each circulation. There may not always have constructive interference occurred due to the relative magnitudes of the two beams and their “signs.” The “signs” are determined by the phase change the
signal experienced when traveling. For instance, after traveling along a fiber path whose ωoτ is
denoted by π /2 , this is equivalent to multiplying the complex magnitude of the signal by j that
represents the phase. In the case of (a), it may suggest that there are alternating constructive and
destructive interference at each circulation that results the oscillations in the output response. In
(b), there are fewer circulating fields taking part in the interaction as most of them travel only
once inside the circuit and are out to the output directly. This explains why the average output
intensity is larger in (b). The same reasoning can be applied to explain the difference between
Figures 5.33c and 5.34a. In the former one, since the value of k is close to 1, most of the beams
208
Photonic Signal Processing
FIGURE 5.34 Transient response of the passive DCDR circuit with ta = 0.8 and k = 0.9. The input pulse
shape is the same as in Figure 4.6a. (a) Output intensity pulse with ωoτ = π/2 and τ/τc = 0.2, (b) Output intensity pulse with ωoτ = π and τ/τc = 0.2, (c) Output intensity pulse with ωoτ = π/2 and τ/τc = 10, (d) Output
intensity pulse with ωoτ = π and τ/τc = 10.
are coupled out of the circuit after they have travelled only one path in the circuit leaving less
circulating beams to interact with incoming beams. Thus, the former one does not have oscillation in the output response.
When we increase the value of τ /τ c = 10 (i.e., decrease in the source coherence) in Figure 5.34c
and d, the oscillations cease. In this case, the phase change in each fiber path becomes less pronounced
since the signals inside the circuit are more incoherent. As a consequence of this, the interference
around the path each time would be of the same sign (like a recirculating delay line). Therefore, the
output responses are more stable than that in (a) and (b).
The effect of source coherence is again shown with an input pulse of the shape as given in
Figure 5.35a. The most coherent one has the largest oscillation, and, interestingly the response has
a large oscillation even at a time after the input pulse ceases completely. Generally, a less coherent
source results in a less oscillating output response.
As the above procedures of computing transient response of the resonator is “programmed”
already, so we can compute the response of other photonic circuit configurations as well. The
resonant conditions of the DCDR are found, and the corresponding behavior is examined. For the
transient response, we have found that the response is oscillatory for a source with higher degree of
coherence. The results are explained in terms of the interferences among circulating beams in the
circuit. The procedures of computation of the transient response are programmed and provide the
possibility of analyzing other photonic circuits.
Optical Dispersion Compensation and Gain Flattening
209
FIGURE 5.35 Transient response of the passive DCDR circuit with ta = 0.8, k = 0.9 and ωτ = π /2. (a) Input
pulse, (b) Output intensity pulse with τ/τc = 0.001, (c) Output intensity pulse with τ /τ c = 0.2, (d) Output intensity pulse with τ /τ c = 10.
5.2.6 DcDr resOnatOr as a DispersiOn equalizer: grOup Delay anD DispersiOn
The performance of optical communication systems can be limited by several effects. One of these
effects is the pulse distortion due to fiber chromatic dispersion.
The study of pulse propagation in dispersive media is important in many applications, including the transmission of optical pulses through the optical fibers27 used in optical communication
systems.28 In the fiber-optic transmission system, compensators or dispersion equalizers29,30,31,32 are
required to compensate for the distortion resulted during the signal transmission in the fiber. This
ensures that the signal distortion at the end of the transmission link is kept to the minimum so that
the signal is received without error. In this chapter, the possibility of employing the DCDR resonator
as a dispersion equalizer is investigated.
In the first part of this section, the equations governing the optical pulse transmission in single
mode fiber are derived. These equations will include the effects of source coherence and the modulation signal bandwidth on the pulse broadening behavior in the fiber. The application of DCDR
circuit as the dispersion equalizer is studied in the second part of the chapter.
Since the optical fiber is a dispersive medium, an optical pulse broadens as it travels along a fiber.
If several pulses are transmitted through the fiber, the broadening of the optical pulse will cause the
B. Moslehi and J. W. Goodman, Novel amplified fiber-optic recirculating delay line processor, J. Light. Technol., 10(8),
1142–1147, 1992.
28 L. N. Binh, N. Q. Ngo, and S. F. Luk, Graphical representation and analysis of the z-shaped double-coupler optical resonator, IEEE J. Light. Technol., 11(11), 1782–1792, 1993.
29 Y. H. Ja, Generalized theory of optical fiber loop and ring resonators with multiple couplers: 1: Circulating and output
fields and 2: General characteristics, Appl. Opt., 29, 3517–3529, 1990.
30 Y. H. Ja, Optical fiber loop resonators with double couplers, Opt. Commun., 75, 239–245, 1990.
31 Y. H. Ja, A double-coupler optical fiber ring-loop resonator with degenerate two-wave mixing, Opt. Commun., 81,
113–120, 1991.
32 Y. H. Ja, On the configurations of double optical fiber loop or ring resonator with double couplers, J. Opt. Commun., 12,
29–32, 1991.
27
210
Photonic Signal Processing
overlapping among the pulses after traveling a long distance. This interference between adjacent
pulses is called the inter-symbol interference (ISI). The ISI will affect the signals received at the
other end of the fiber causing errors in the receiver. The fiber dispersion thus limits the informationcarrying capacity of the fiber system as adjacent signal pulses cannot be transmitted too close to
each other hence lowering the rate of transmission.
There are two types of dispersion mechanisms of pulse spreading in fibers—intra-modal
dispersion and intermodal dispersion. Intra-modal dispersion, sometimes called chromatic
dispersion, is the pulse spreading that occurs within a single mode. The inter-modal dispersion is
the dispersion between the modes. Only chromatic dispersion will be considered in the following
analysis as pulse transmission in single mode fiber is concerned. Chromatic dispersion is mainly
due to the wavelength dependence of the refractive index of the fiber’s core material. Hence, the
chromatic dispersion can be represented by the wavelength dependence of the propagation constant.
The main aim of this Section is to obtain the signal at the output of the fiber link so that design of
equalizer can be performed. As this Section is mainly devoted to the study of equalization of fiber
dispersion by the DCDR circuit, the discussion on pulse propagation will concentrate on typical
case rather than general one. In this instance, only Gaussian pulse will be considered.
The expression for the signal at the output of a fiber link is derived. This output will be fed into
the DCDR circuit at the end of the fiber link for dispersion equalization. We start with the signal
entering the fiber, that is, the light source. We assume that the light source emits an optical signal
with Gaussian spectrum centered at ω = ω0, which takes the form
S(ω ) =
2π
exp[(ω − ωo )2 / 2Ws 2 ]
Ws
(
)
(5.87)
where WS is the spectral width at the 1 / e points in units of angular frequency, ω is the angular
optical frequency, and ωo is the center angular frequency of the source. The corresponding temporal
(time domain) function s(t ) by taking the Fourier transform on Eq. (5.87) is
s (t ) = e
 − t 2 Ws2 


 2  jω t
o


e
(5.88)
The signal needs to be modulated to get the light pulse. The modulation of the light source signal can
be done either directly or externally. In direct modulation, one would modulate the driving current
of the source in order to modulate the source output directly. In external modulation, the continuous
optical signal from the source is modulated using an optical modulator following the source.
Suppose the source is amplitude modulated with a pulse of Gaussian shape of the form:
Am (t ) = e
 −t2 


 2τ 2 
 m
(5.89)
where τm is the 1/e half width. Thus, the light signal, or the electric field at the input of the fiber,
Eif (t ) can be expressed as
Eif (t ) = Am (t ) s(t )
(5.90)
that is
Eif (t ) = e
 −t2  1


 2 + Ws2  
 2  τ m
  jωo t
The electric field at the end of the fiber of length L is
e
(5.91)
Optical Dispersion Compensation and Gain Flattening
∫
Eof (t ) = Εif (ω ) exp[ j (ω t − β L)]dω
211
(5.92)
where Εif (ω ) is the Fourier transform of Εif (t ), β is the propagation constant of the guided mode in
the fiber or optical; waveguide. Low-loss fiber is assumed to be used so that attenuation of the fiber
can be neglected. The dispersion effect of the fiber can be represented by the fact that is a function
of frequency. By Taylor’s series expansion of β about ω = ωo, β can be approximated by
β (ω ) = β (ωo ) + β ′(ωo )(ω − ωo ) +
β ′′(ωo )
(ω − ωo )2 + ...
2
(5.93)
For simplicity, we use β 0 to represent β (ω0 ) thereafter. In the following analysis, only the terms up to
the second derivative of the propagation constant are included. The higher-order derivative terms in
the Taylor series expansion of the propagation constant are neglected. Moreover, the first two terms in
Eq. (5.93) represent the pure delay of the carrier and the envelope respectively and they do not contribute to the dispersion. Therefore, these two terms are ignored in the following analysis for clarity. After
these simplifications, the effective propagation constant used in our analysis can be represented by
β (ω ) =
β o′′
(ω − ωo )2
2
(5.94)
The value of the second derivative of the propagation constant is a measure of the so-called first
order dispersion.33,34 For a silica fiber, the wavelength at which this first order dispersion vanishes is
about 1300 nm. At wavelength other than the zero-dispersion wavelength, the first-order dispersion
is non-zero and it accounts for the broadening of optical pulse traveling in the fiber. Also, the
values of higher order of derivatives, which correspond to second- and higher-order dispersion are
negligible at wavelength other than the zero-dispersion wavelength.
Using (5.90) to (5.94) the output power at the fiber end can be obtained as
I of ( t ) =


−(t / τ m )2
exp 

2
 1 + 4 D (1 + V 2 ) 
[1 + 4 D (1 + V 2 )]1/ 2
1
2
(5.95)
with
V = t mWs ,
(5.96)
β ′′L
D = o 2 ,
2τ m
(5.97)
Eq. (5.95) shows that the Gaussian shape pulse remains Gaussian after traveling through the fiber. It
can be stated that the output power at the fiber end has a pulse width τo, which is given by:
τ o2 = τ m2 [1 + 4 Dˆ 2 (1 + V 2 )]
(5.98)
This equation shows that a larger value of V would produce a wider spreading pulse width τ 0 at the
fiber output end. Naturally one can observe that Eq. (5.98) indicates for an incoherent source (large
Ws) signals broadening becoming more than the case with a coherent source (small Ws). Moreover,
the broadening of pulses depends on the value of D , hence the values of β o′′ and L.
33
34
S. J. Mason, Feedback theory—Some properties of signal flow graphs, Proc. IRE, 41, 1144–1156, 1953.
S. J. Mason, Feedback theory—Further properties of signal-flow graphs, Proc. IRE, 44, 920–926, 1956.
212
Photonic Signal Processing
Without loss of generality, the source is assumed to be monochromatic in the following analysis,
i.e. Ws = 0. This makes V = 0. The electric field at the fiber output for a monochromatic light source
is given by
Eof (t ) =
 −(t / τ 2 ) 
 − j D (t / τ 2 ) 
1
m
m




exp(
j
ω
t
)
exp
ex
p
o
2
2 1/ 4
2



(1 + 4 D )
 2(1 + 4 D ) 
 (1 + 4 D ) 
(5.99)
Expressing Eq. (5.99) in terms of the physical parameters, resulting:
Eof (t ) =
τm
τ m4 + ( β o′′L)2 


1/ 4
 −t 2 (τ m2 + j β o′′L) 
exp( jωot )exp 
4
2 
 2[τ m + ( β o′′L) ] 
(5.100)
which is the same expression as obtained in reference35 and it is the field equation required for the
subsequent analysis. This would be the field at the input of our equalizer.
We can now investigate the application of the DCDR circuit as an equalizer for single mode fiber
dispersion. The principle of the fiber dispersion equalizer lies in the fact that the equalizer has a group
delay in the opposite sign to that introduced by the fiber. Thus, the group delay induced on the signal
after the signal travels through the fiber would be partially or completely cancelled by that in the equalizer. Recall that the magnitude of the phase change induced by the fiber on the signal after the signal
travels length L of fiber is β L. Thus, the group delay time τ f induced by the fiber can be expressed as:
τ f = β 0′′(ω − ω0 ) L
(5.101)
which is obtained by differentiating β L with respect to ω making use of Eq. (5.101). The chromatic
dispersion induced by the fiber link of length L is thus equal to
dτ f
= β 0′′L
dω
(5.102)
Ideally, we need to use the linear portion (if available) of the equalizer group delay that has the
suitable slope, both the magnitude and the sign, to counteract the fiber dispersion. In addition, that
portion should be positioned to the center frequency of the signal. This point is called the operating
point of the equalizer thereafter.
Firstly, we consider the pulse at the fiber link output whose field equation is given in Eq. (5.100).
The following parameters are used in our analysis. The input pulse into the fiber is a Gaussian shaped
pulse with temporal half width at the 1/e points and τm equals to 50 ps. The working wavelength of the
system, λo, is taken to be 1550 nm in the following analysis. As we only concentrate on the dispersive
effect of the fiber, the attenuation effect of the fiber is ignored. Therefore, the fiber is assumed to be
loseless. As the operating wavelength of the system is 1550 nm at which the loss of standard fiber is
lowest, the above assumption is valid. Actually, the attenuation introduced by the fiber can be easily
compensated by an amplifier or repeater at the end of the fiber length, but this is not the main point
in our analysis. The fiber dispersion value is represented by a parameter D, which is 17 ps/nm/km for
conventional silica fiber. The dispersion caused by fiber is given by β 0′′L (Eq. 5.102), which is
β 0′′L =
35
Dλo2 L
2π c
(5.103)
Y. H. Ja, Generalized theory of optical fiber loop and ring resonators with multiple couplers: 1: Circulating and output
fields and 2: General characteristics, Appl. Opt., 29, 3517–3529, 1990.
Optical Dispersion Compensation and Gain Flattening
213
FIGURE 5.36 The intensities of the optical pulses at the input of the fiber (dotted line) and at the fiber output
(solid line) with the above fiber parameters and operating conditions.
where c is the speed of light in vacuum. The transmission fiber path length L is taken to be 200 km
unless otherwise stated. Using the given parameters, the value of β o′′L is 4.3335e-21. The intensities
of the optical pulses at the input of the fiber and at the fiber output with the above fiber parameters
and operating conditions are plotted in Figure 5.36. The dispersive properties of the fiber can be
clearly observed when the difference between the pulse widths of the two pulses is compared.
Obviously, the optical pulse broadens as it travels along the fiber.
The group delay and the dispersion of the DCDR resonator can be evaluated numerically
using programs written in MATLAB® commands. Recall that τ is the basic delay time of the DCDR
circuit and it is set to the value of 20 ps, which is equal to τm/2.5. The normalized group delay and
dispersion with respect to frequency is obtained for the case with k=1 k=
0.1, G
=
G=
G3 = 1 and
2
1
2
=
m1 m
=
m3 = 1.
2
Since the group delay and dispersion values are normalized, the group delay value read on the
axis needed to be multiplied by τ to get the absolute value, it need to be multiplied by τ 2 for the
dispersion. It is found that the group delay of the equalizer is always negative while its dispersion has
both positive and negative values. The negative portion of the equalizer dispersion can be employed
to cancel the effect of fiber dispersion.
The task of the DCDR circuit is to compensate the dispersion created by the fiber path. Let the
ratio of equalizer dispersion to the fiber dispersion be R, the ideal value of R would then be −1. We
also need to choose the operating point of the equalizer so that we can get the desired equalizer
dispersion to perform the equalization. The operating point of the equalizer, φo is actually the
operating ω/τ of the DCDR circuit. The operating point can be positioned by changing the length
of the fiber path in the DCDR circuit using piezoelectric effect. In order to get better equalization,
the equalizer dispersion should vary as small as possible with ωτ in the vicinity of φo . This is due
to the fact that we have assumed a uniform fiber dispersion in the vicinity of the source operating
wavelength. The value of φo can be chosen from the resonator dispersion in
The comparison between the resonator input intensity and the resonator output intensity in
Figure 5.27 shows that the broadening of the former pulse has been compensated to a great extent
by the resonator. The resonator cannot give full equalization because its dispersion is not constant
with frequency. For that reason, we attempt to design parameters giving the best rather than 100%
equalization for a given pulse. The loss in the magnitude of the equalized pulse can be compensated
by an amplifier inserted after the resonator (Figure 5.37).
214
Photonic Signal Processing
FIGURE 5.37 The intensities of the optical pulses at the input of the fiber (dashed ––), at the fiber output
(i.e., resonator input) (dash dot –.–) and at the resonator output (solid line) with k=1 k=
0.1 , t=
t a=
t a3 = 1,
a1
2
2
G=
=
G=
and m1 m=
m=
1
2 G 3 1=
2
3 1, operating point of the resonator ωo = 2.8225 rad (161.7 °).
TABLE 5.3
Operating Points of the Equalizer
together with the Value of R for
Figure 5.4a–e
Figure 5.4a
Figure 5.4b
Figure 5.4c
Figure 5.4d
Figure 5.4e
ϕo (rad)
R
2.5035
2.6814
2.9391
3.0005
3.1355
−0.1473
−0.3758
−2.9215
−5.5526
−1.6116
The effects of different operating points on the equalization are shown in Figure 5.38. The
operating points used for the plots are given in Table 5.3.
The operating points for the plots in Figure 5.38 are all in the proximity of ϕo = 2.8225 rad, i.e.
the operating point used in Figure 5.37. In this range of ϕo, the resonator’s dispersion is negative,
which is appropriate for its use as an equalizer. It can be observed that when we move from the
operating point ϕo = 2.5035 to ϕo = 3.1355, a trailing tail of the signal is building up. It becomes
significant as compared to the original signal thus causing other distortion effects. Therefore, ϕo in
Figure 5.38c–e are not good operating points of the equalizer in this case.
To obtain a more visual presentation on the effect of broadened pulses, we consider a sequence
of pulses. Figure 5.39b shows the overlapping between two adjacent pulses separated 6τ m in time
at the end of the fiber link after undergoing dispersion. Figure 5.39a shows that the two pulses are
originally with no interference between them at the transmitted end. If the sampling time of the
detection system is 6τm in this situation, the fiber dispersion would not cause any problem to the
detection. But, if the pulses are getting closer to each other, then, for instance, at a higher transmission rate, the ISI would limit the capacity of the system.
215
normalized intensity
(a)
(c)
1
0.9
0.8
0.8
0.7
0.7
normalized intensity
1
0.9
0.6
0.5
0.4
0.3
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
-8
-6
-4
-2
0
2
4
6
8
× τm time
(b)
0
-8
1
1
0.9
0.9
0.8
0.8
0.7
0.7
normalized intensity
normalized intensity
Optical Dispersion Compensation and Gain Flattening
0.6
0.5
0.4
0.3
0
2
4
6
8
× τm time
(d)
-6
-4
-2
0
4
6
2
4
6
8
× τm time
0.3
0.1
-2
0
0.4
0.2
-4
-2
0.5
0.1
-6
-4
0.6
0.2
0
-8
-6
0
-8
2
4
6
8
× τm time
1
0.9
normalized intensity
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(e)
0
-8
-6
-4
-2
0
× τm time
2
8
FIGURE 5.38 The intensities of the optical pulses at the input of the fiber (dashed – –), at the fiber output (i.e., resonator input) (dash dot –.–) and at the resonator output (solid line) with k1 = k2 = 0.1,
t=
ta=
ta3 = 1, G1 = G2 = G3 = 1 and m1 = m2 = m3 = 1 for different operating points of the resonator with
a1
2
(a) φo = 2.5035 rad, ( b) φo = 2.6814 rad, (c) φo = 2.9391 rad, (d) φo = 3.0005 rad, and (e) φo = 3.1355 rad.
216
Photonic Signal Processing
0.6
1
0.9
0.5
normalized intensity
normalized intensity
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.4
0.3
0.2
0.1
0.1
0
-12 -10 -8
(a)
-6
-4 -2
0
× τm time
2
0
-12 -10 -8
4
(b)
-6
-4 -2 0
× τm time
2
4
FIGURE 5.39 Two adjacent pulses (a) at the input of the fiber link and (b) at the end of the fiber link before
the equalizer. The two pulses are separated 6τm in time.
(a)
1
1
0.9
0.9
0.8
0.8
0.7
0.7
normalized intensity
normalized intensity
An eye pattern diagram is usually used to visually assess the system performance in a digital communication system. Initially, random data signal patterns are generated. For a 3-bit-long
sequence, for instance, we could have ′111′, ′ 011′, ′110′..., and so on When these combinations are
superimposed simultaneously, an eye pattern diagram is formed. The eye pattern diagrams for the
intensities before and after the equalizer corresponding to the above case are shown respectively in
Figure 5.40a and b respectively. Recall that the operating point of the equalizer here is 2.8225 rad.
It is noted in Figure 5.40b that the eye opening of the diagram is greater than that in Figure 5.40a,
thus the equalizer indeed improves the quality of the received signals. In Figure 5.41, the eye
diagrams for two different transmission rates are shown. In Figure 5.41a, the pulses are pushed
closer together, and the time between their adjacent peaks is 4τ m. The corresponding time is 8τ m in
Figure 5.41b. It is obvious that the eye closure in Figure 5.41a is greater than that in Figure 5.41b. This
means that it is more difficult to detect signal correctly in the case of Figure 5.41a. By comparing the
eye diagrams in Figures 5.40 and 5.41, we can state that the minimum spacing between transmitted
pulses is about 6τ m in this instance. This, in turn, is a way of determining the maximum transmission
rate of a transmission channel.
0.6
0.5
0.4
0.3
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
-10 -8 -6 -4 -2 0 2
× τm time
4
6
8
10
(b)
0
-10 -8 -6 -4 -2 0 2
× τm time
4
6
8
10
FIGURE 5.40 (a) The eye pattern diagram before the equalizer, (b) the eye pattern diagram after the equalizer.
The parameters used in the equalizer is the same as that used in Figure 5.38. Pulses are separated by 6τm in time.
217
0.6
0.6
0.5
0.5
0.4
0.4
normalized intensity
normalized intensity
Optical Dispersion Compensation and Gain Flattening
0.3
0.2
(a)
0.2
0.1
0.1
0
-8
0.3
-6
-4
-2
0 2 4
× τm time
6
8
10
0
-10
(b)
-5
0
5
× τm time
10
FIGURE 5.41 The eye pattern diagrams after the equalizer. The parameters used in the equalizer is the
same as that used in Figure 5.3. (a) Pulses are separated by 4τm in time, (b) pulses are separated by 8τm in time.
In Figure 5.42, we have shown that with ϕo = 2.5035 rad and R = −0.1473 (first row in Table 5.3),
the result is even better than that in Figure 5.6 with ϕo = 2.8225 rad and R = −1.0065. These results
show that the DCDR resonator can be used as an equalizer even in the passive operation.
Since the dispersion of equalizer is proportional to τ2, it is expected that a larger value of τ would
result in a greater magnitude of the equalizer dispersion. In Figure 5.43, two values of τ are used
with the same operating point ϕo = 2.5035 rad. It can be observed that a better compensation is
obtained by the equalizer with a larger τ.
An active mode operation of the equalizer is shown in Figure 5.45. The parameters used are
k=
k=
0.1, t=
ta=
ta3 = 1, G1 = 1.9, G=
G=3 1 and m
=
m=
m3 = 1. The operating point of
1
2
a1
2
2
1
2
the resonator is located at 2.8225 rad, which is the same as that in Figure 5.3. The value of R in this
case is −1.0074. By comparing between Figures 5.3 and 5.45, it is obvious that the equalized pulse in
the latter graph has a larger magnitude. Thus, the amplifier in the DCDR circuit compensates for the
loss in magnitude as well. This is a useful feature in our amplified DCDR circuit as the dispersion
1
0.9
normalized intensity
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-10 -8
-6
-4
-2 0 2
× τm time
4
6
8
10
FIGURE 5.42 The eye pattern diagrams after the equalizer. The parameters used in the equalizer is the
same as that used in Figure 5.41, except that ϕo = 2.5035 rad. Pulses are separated by 6τm in time.
218
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.5
0.4
0.3
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
-8
(a)
normalized intensity
normalized intensity
Photonic Signal Processing
-6
-4
-2
0
2
× τm time
4
6
0
-8
8
-6
-4
(b)
-2
0
2
× τm time
4
6
8
FIGURE 5.43 The intensities of the optical pulses at the input of the fiber (dashed ––), at the fiber output
(i.e., resonator input) (dashdot ––) and at the resonator output (solid line) with k1 = k2 = 0.1, t=
t a=
t a3 = 1,
a1
2
G1 = G2 = G3 = 1 and m1 = m2 = m3 = 1, operating point of the resonator ϕo = 2.5035 rad. (a) τ = 15 ps,
(b) τ = 25 ps.
1
0.9
normalized intensity
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-8
-6
-4
-2
0
2
4
× τm time
6
8
10
12
FIGURE 5.44 The intensities of the optical pulses at the input of the fiber (dashed ––), at the fiber
output (i.e., resonator input) (dash dot –.–) and at the resonator output (solid line) with k=
k=2 0.1,
1
G1 = 1.9, G2 = G3 = 1 and m1 = m2 = m3 = 1, operating point of the resonator ϕo = 2.8225 rad.
compensation and the compensation for the pulse magnitude can be done with a single device. In other
words, in addition to the dispersion equalization, the DCDR circuit can be tailored to compensate for
the attenuation in the pulse amplitude. This would be an economical way of equalizer design.
The eye diagram for the equalized pulse in Figures 5.44 and 5.37 is given in Figure 5.45. The eye
diagram before the equalizer is the same as that in Figure 5.40a. The eye diagram in Figure 5.45 is
to be compared with that in Figure 5.40b where we have passive mode operation. In both cases, the
equalizers have performed considerable amount of dispersion equalization.
219
Optical Dispersion Compensation and Gain Flattening
1.2
normalized intensity
1
0.8
0.6
0.4
0.2
0
-10
-5
0
× τm time
5
10
FIGURE 5.45 The eye pattern diagram after the equalizer. The parameters used in the equalizer is the same
as that used in Figure 5.10. Pulses are separated by 6τm in time.
The programs have been tested for a single loop resonator using the same parameters as given in
Pandian paper,36 and after value-by-value comparison, our results are consistent with theirs. Thus,
the programs can be used with great confidence. One advantage of our programs is that they are
very generalized, and they can handle other circuit configurations provided that the input-output
field transfer functions of the circuits are given.
The application of using the DCDR circuit as a dispersion equalizer is demonstrated. The
effects of working the equalizer at several operating points are shown. It is found that the DCDR
circuit can provide a reasonable amount of chromatic dispersion equalization. Active operation of
the DCDR circuit can provide compensation for the pulse amplitude in addition to the dispersion
compensation. Therefore, theoretically, the circuit can be employed as a dispersion equalizer in
fiber-optical communication systems.
The resonant behavior of the circuit operating as a resonator is observed. It is discovered that by
proper choice of circuit parameters, the sharpness of the resonant peak can be adjusted. The transient
responses of the DCDR circuit with different source coherences and different input pulse shapes are
examined. The results show that responses of the circuit are oscillatory for highly coherent source.
Moreover, the procedures of computation of transient response are generalized and can be applied
to general photonic circuits.
It is shown to be possible to employ the DCDR circuit as a fiber dispersion equalizer. The
circuit provides larger flexibility in equalizer design than simple configurations. One significant
consequence is that the DCDR circuit could provide compensation for the pulse amplitude in
addition to dispersion compensation.
In general, our novel graphical representation of optical circuits can tackle non-linear optical elements.
In other words, the transmittances in the signal flow graph of the circuit can be non-linear in nature. The use
of our technique may lead to easier understanding of the performance of non-linear optical circuits. Over
the last decade, there were advancing research interests in the optical fiber communication systems. The
use of an all-fiber photonic circuit as an optical equalizer for the fiber dispersion should be a considerable
step in going towards an all-optical communication system. Further research on this topic could be the
investigations of possibilities of using other photonic circuits as the optical-based dispersion equalizer.
36
Y. H. Ja, Generalized theory of optical fiber loop and ring resonators with multiple couplers: 1: Circulating and output
fields and 2: General characteristics, Appl. Opt., 29, 3517–3529, 1990.
220
Photonic Signal Processing
Considerations may cover different signal modulation methods, various signal pulse shapes and different
detection techniques. We expect that the novel graphical representation and the associated manipulation
rules of the photonic circuits would play an important role in future optical network analysis.
5.3 OPTICAL EIGENFILTER AS DISPERSION COMPENSATORS
In this section, a digital eigenfilter approach is employed to design linear optical dispersion
eigencompensators (ODECs) for compensation of the combined effect of laser chirp and fiber dispersion
at 1550 nm in high-speed long-haul intensity modulation/direct detection (IM/DD) lightwave systems
[48]. Various techniques developed to mitigate this detrimental effect are reviewed in Section 5.3.3.1 that
describes the formulation of the dispersive fiber channel, which is used as a basis for designing ODECs and
compares the performance of the proposed eigencompensating method with the Chebyshev technique in
the frequency domain. A procedure for the synthesis of the ODECs using the PLC technology described
in Section 3.3 is outlined in Sections 5.3.2.2 to 5.3.2.4. Section 5.3.3.1 describes the transmission system
model used to simulate the performance of the ODEC in an IM/DD optical communication system and
compares the performance of the ODEC with that of the Chebyshev equalizer in the time domain. The
reasons for comparing the eigenfilter approach with the Chebyshev technique are given in 5.3.2.4. The
theory of coherent integrated-optic signal processing described in Section 2.3 of Chapter 2, is employed
in this chapter where electric-field amplitude signals are considered.
5.3.1 introductory remarkS
The advent of the erbium-doped fiber amplifier (EDFA) has created a need for optical dispersion
compensators to be developed to solve the important problem of fiber chromatic dispersion in optical
communication systems operating near the minimum-loss of 1550 nm wavelength. The combined
effect of laser chirp and fiber dispersion, which results in significant pulse broadening, is the critical
factor limiting the maximum bit rate-distance in high-speed long-haul IM/DD systems. To exploit the
potential benefits of the EDFA for long-distance transmission at 1550 nm, this detrimental effect must
be minimized either by using dispersion-shifted fibers (DSFs) with zero-dispersion wavelength near
1550 nm or by using a 1300-nm optimized standard fibers with some means of dispersion compensation.
The use of DSFs to upgrade the already-embedded fiber network is clearly not an economical way to
enhance network performance. The employment of DSFs in conjunction with EDFAs as repeaters
is seen to be a flexible and cost-effective choice of technology for the generation of new ultra-long,
ultra-high-speed transoceanic systems. However, this technology still suffers from residual dispersion
at the end of the link because each DSF span has its own zero-dispersion wavelength. In fact, it has
been proposed that operating exactly at the zero-dispersion wavelength is undesirable because of the
strong build-up of noise due to the four-wave mixing between the signal and the amplified noise.37
The residual dispersion can be dealt with by placing an adjustable optical dispersion compensator at
the receiver end. Thus, dispersion compensation will remain a key technological issue for both the
upgrading of existing terrestrial fiber networks and the installation of new transoceanic systems for
years to come.Several nonlinear and linear optical dispersion-compensating techniques have been
demonstrated to mitigate the primary effect of fiber dispersion on lightwave systems at 1550 nm. The
nonlinear methods, such as bright-soliton transmission38 or dark-soliton transmission as described
in Chapter 7 and optical phase conjugation39 using DSFs and EDFAs, have been demonstrated in
ultra-long transmission systems by means of nonlinear optical effects in fibers. The linear techniques,
N. Henmi, T. Saito, and T. Ishida, Prechirp technique as a linear dispersion compensation for ultrahigh-speed long-span
intensity modulation directed detection optical communication systems, J. Light. Technol., 12, 1716–1729, 1999.
38 W. Zhao and E. Bourkoff, Propagation properties of dark solitons, Opt. Lett., 14, 703–705, 1989.
39 N. Henmi, T. Saito, and T. Ishida, Prechirp technique as a linear dispersion compensation for ultrahigh-speed long-span
intensity modulation directed detection optical communication systems, J. Light. Technol., 12, 1716–1729, 1994.
37
Optical Dispersion Compensation and Gain Flattening
221
the subject of this chapter, require the optical power level in the transmission fiber to be below that
required for the onset of the nonlinear optical effects, and they are briefly described below.
Wedding et al. demonstrated a dispersion-supported transmission technique capable of increasing
the bit rate-distance of a conventional direct-modulation system four-fold.40 This method is based
on the interferometric conversion of the optical frequency modulation generated by a directly
modulated laser to an amplitude modulation by means of fiber dispersion. Despite the simplicity
and low cost of this scheme, it still imposes a limit on the achievable bit rate-distance.
Henmi et al. reported pre-chirp and modified pre-chirp techniques for improving the
dispersion limitation of standard external-modulation systems.41 The pre-chirp scheme is based
on a combination of direct frequency modulation of the laser and external amplitude modulation
for extending the dispersion-limited distance to nearly double. The modified pre-chirp technique
is based on time division multiplexing of two independent pre-chirp RZ (return-to-zero) signals
by utilizing polarization multiplexing techniques for increasing the dispersion-limited distance
to nearly four times. Although these techniques are simple and cheap to implement they are not
applicable to direct-modulation systems, and still set a limit on the achievable bit rate-distance.
Poole et al. developed a broadband fiber-based compensating technique capable of achieving
negative chromatic dispersion as large as −770 ps nm km at 1555 nm.42 The fiber compensator
was designed to operate in a higher-order mode near its cut off wavelength at which a large negative
dispersion could be obtained to counteract its positive counterpart in a standard fiber. Despite
the broadband capability of this device, its insertion loss is high due to the high attenuation loss
(~5 dB km ) and the long fiber length required. In order to achieve effective dispersion compensation,
this device requires high-efficiency mode converters to excite the desired higher-order mode.
Onishi et al.43 designed a dispersion compensating fiber with a negative chromatic dispersion of
−80 ps nm km at 1550 nm for countering its positive counterpart in a conventional fiber. This technique is based on optimization of the refractive index profile of the fiber core and special fluorine
doping in the cladding. The low negative dispersion and relatively high loss (0.32 dB km ) of the
compensating fiber reflect its high cost for system implementation.
Linearly chirped fiber Bragg grating has been demonstrated as an all-fiber dispersion compensator.44 The spectral components of the dispersive input pulse are reflected at different locations
along the grating length and hence experience different group velocities. It is this frequency dependence of the group velocity of the compensator that leads to dispersion compensation. Long grating
length, which results in a reduction of the filter bandwidth, is necessary for the compensation of a
long dispersive fiber. The grating chirp must also be carefully designed to obtain the desired frequency responses. Although this device is compact and has low insertion loss, it is limited to singlewavelength applications because of its aperiodic frequency characteristics. Also, this compensator
has limited design flexibility resulting in bandwidth limitation.
Cimini et al. proposed a reflective fiber Fabry-Perot filter for dispersion compensation.45 In
order to avoid degradation of the filter performance, its mirror reflectivity must be accurately
designed. Despite the structural simplicity of the filter, it is inherently bandwidth-limited because
C. D. Poole, J. M. Wiesenfeld, D. J. Di Giovanni, and A. M. Vengsarkar, Optical fiber-based dispersion compensation
using higher order modes near cutoff, IEEE J. Light. Technol., 12, 1756–1768, 1994.
41 M. Onishi, Y. Koyano, M. Shigematsu, H. Kanamori, and M. Nishimura, Dispersion compensating fiber with a high
figure of merit of 250 ps/nm/dB, Electron. Lett., 30, 161–164, 1994.
42 P. A. Krug, T. Stephens, G. Yoffe, F. Ouellette, P. Hill, and G. Dhosi, Dispersion compensation over 270 km at 10 Gbit/s
using an offset-core chirped fiber Bragg grating, Electron. Lett., 31, 1091–1093, 1995.
43 L. J. Cimini, Jr., L. J. Greenstein, and A. A. M. Saleh, Optical equalization to combat the effects of laser chirp and fiber
dispersion, IEEE J. Light. Technol., 8, 649–659, 1990.
44 A. H. Gnauck, C. R. Giles, L. J. Cimini, Jr., J. Stone, L. W. Stulz, S. K. Korotky, and J. J. Veselka, 8-Gb/s-130 km transmission experiment using Er-doped fiber preamplifier and optical dispersion equalization, IEEE Photon. Technol. Lett.,
3, 1147–1149, 1991.
45 T. Ozeki, Optical equalizers, Opt. Lett., 17, 375–377, 1992.
40
222
Photonic Signal Processing
its group-delay response is only linear over a narrow optical bandwidth as a result of its limited
design flexibility. Nevertheless, impressive performance of the Fabry-Perot equalizer has been
experimentally demonstrated in an 8-Gbit/s 130-km external-modulation system.46
Ozeki proposed a non-recursive Chebyshev optical equalizer for dispersion compensation.47 A
dispersive waveform, which has higher and lower frequency components at the front and end of the
pulse, is passed through the equalizer that slows down and speeds up the higher and lower frequency
components of the pulse. In this way, the dispersive pulse is compressed by the equalizer resulting in
dispersion compensation. The Chebyshev equalizers using the PLCs45 and birefringent crystals48 have
been demonstrated with impressive results for dispersion compensation in external-modulation systems.
Most of the linear dispersion-compensating techniques,49,50,51 excluding40,44,42 described above, were
only applied to externally modulated systems for the purpose of increasing data rate and/or transmission
distance. When these techniques are used in direct-modulation systems the transmission speed and/or
distance are expected to be reduced. Thus, they are only effective in the absence of laser chirp.
In this chapter, a digital eigen-filter approach is applied to the design of linear optical dispersion
compensators for overcoming the combined effect of laser chirp and fiber dispersion at 1550 nm
in high-speed long-haul IM/DD lightwave systems. This technique is called optical dispersion
eigencompensating and, hence, the acronym for optical dispersion eigencompensators is ODECs,
because it is adopted from the digital eigenfilter technique. Most of the work presented here has
been described by Ngo et al.52,53
The fiber loss is not considered so that the underlying principle of the eigencompensating
technique can be demonstrated.
5.3.2 FOrmulatiOn anD Design
5.3.2.1 Dispersive Optical Fiber Channel
Neglecting the nonlinearity and loss of the optical fiber, the frequency response of the standard
fiber, containing only the quadratic phase response, is given by
2
H f (ω ) = exp  − jDˆ (ωT − ωcT ) 


(5.104)
where the dimensionless first-order fiber chromatic dispersion parameter is defined as
DLλ
.
Dˆ =
4π cT 2
2
(5.105)
In these equations, j = −1, ω is the angular optical frequency, ωc the central angular frequency
of the optical source, D ( −17ps/nm/km) the chromatic dispersion at the operating wavelength
M. Sharma, H. Ibe, and T. Ozeki, Optical circuits for equalizing group delay dispersion of optical fibers, IEEE J. Light.
Technol., 12, 1769–1775, 1994.
47 S. C. Pei and J. J. Shyu, Eigen-approach for designing FIR filters and all-pass phase equalizers with prescribed magnitude and phase response, IEEE Trans. Circuits and Syst., 39, 137–146, 1992.
48 N. Q. Ngo, L. N. Binh, and X. Dai, Eigenfilter approach for designing FIR all-pass optical dispersion compensators for
high-speed long-haul systems, Proc. IREE, 19th Australian Conf. Opt. Fibre Technol., Melbourne, pp. 355–358, 1994.
49 M. Onishi, Y. Koyano, M. Shigematsu, H. Kanamori, and M. Nishimura, Dispersion compensating fiber with a high
figure of merit of 250 ps/nm/dB, Electron. Lett., 30, 161–164, 1994.
50 B. Wedding, B. Franz, and B. Junginger, 10-Gb=s optical transmission up to 253 km via standard single-mode fiber
using the method of dispersion-supported transmission, J. Light. Technol., 12, 1730–1737, 1994.
51 N. Henmi, T. Saito, and T. Ishida, Prechirp technique as a linear dispersion compensation for ultrahigh-speed long-span
intensity modulation directed detection optical communication systems, J. Light. Technol., 12, 1716–1729, 1994.
52 N. Q. Ngo, L. N. Binh, and X. Dai, Optical dispersion eigencompensators for high-speed long-haul IM/DD lightwave
systems: Computer simulation, J. Light. Technol., 14, 2097–2107, 1996.
53 A. P. Clark, Equalizers for Digital Modems, London, UK: Pentech, 1985.
46
223
Optical Dispersion Compensation and Gain Flattening
λ (1550 nm ) of a 1300-nm optimized fiber, L the fiber length, c the speed of light in free space, and
T is the sampling period of the eigenfilter (or the Chebyshev filter). In Eqs. (5.104) and (5.105), the
constant and linear phase terms have been neglected since they do not contribute to pulse distortion,
and the cubic and higher-order phase terms have also been ignored as insignificant at 1550 nm.
Full equalization of the dispersive fiber channel can be achieved by means of an optical dispersion
compensator whose frequency response is opposite to that of the fiber, that is
2
F (ω ) = exp  + jDˆ (ωT − ωcT )  .


(5.106)
5.3.2.2 Formulation of Optical Dispersion Eigencompensation
The eigenfilter technique54 is based on the finite-impulse response (FIR) filter method, which has
been widely used for equalization of phase distortions in digital communication channels.55 This
scheme is used to design the ODECs with approximately unity magnitude response and prescribed
phase response that is opposite to that of the fiber. The eigenfilter technique is a least squares approach
used to minimize some measure of the difference between the phase response of the ODEC and the
desired fiber phase response. It is based on the formulation of minimizing a quadratic measure of the
frequency band errors in the magnitude and phase responses. The coefficients of the FIR ODEC can
be obtained from computation of the eigenvectors that correspond to the smallest eigenvalues of the
real, symmetric and positive-definite matrices. In this way, the magnitude and phase responses of the
fiber are simultaneously equalized resulting in very effective dispersion compensation.
The frequency response Heq (ω ) of a N-tap FIR filter with tap coefficients a(n), n = 0,1,, N − 1,
is given by56
N −1
exp( jM ωT ) H eq (ω ) = exp( jM ωT )
∑ a(n) exp(− jnωT )
(5.107)
n=0
where M = ( N − 1) 2 with N being an odd integer. The linear phase factor exp( jMω T ) has been
included on both sides of Eq. (5.107) to facilitate the eigenfilter technique. The eigenfilter method
requires the real and imaginary parts of Eq. (5.107) to approximate the real and imaginary parts,
respectively, of the desired eigenfilter frequency response
F (ω ) = exp ( jφ̂ (ω ) )
(5.108)
where φ̂ (ω ) = Dˆ (ωT − ωcT ) is the desired phase function of the eigenfilter. The eigenfilter algorithm
requires Eq. (5.108) to be an even function about ω T = π 2; thus ωcT = π 2 must be chosen in all
designs. By separately minimizing the real and imaginary parts of the error functions, the elements
of matrix QR are given by55
2
 cos 2m(ωoT )

ˆ
 cos 2n(ωoT )

 cos φˆ (ω ) cos φ (ω ) − 
ˆ
o
cos φ (ω ) − cos 2n(ωT )  ⋅ 
qR ( n, m; ω ) = 
 d (ωT )
(5.109)
 cos φˆ (ωo )
 

0
cos 2m(ωT )

π
∫
for _ 0 ≤ n, m ≤ I
A. P. Clark, Equalizers for Digital Modems, London, UK: Pentech, 1985.
J. C. Cartledge and A. F. Elrefaie, Effect of chirping-induced waveform distortion on the performance of direct detection
receivers using traveling-wave semiconductor optical preamplifiers, J. Light. Technol., 9, 209–219, 1991.
56 T. Q. Nguyen, T. I. Laakso, and R. D. Koilpillai, Eigenfilter approach for the design of allpass filters approximating a
given phase response, IEEE Trans. Signal Process., 42, 2257–2263, 1994.
54
55
224
Photonic Signal Processing
and the elements of matrix QI are given by
π  sin(2n − 1)(ωoT )
  sin(2m − 1)(ωoT )

sin φˆ (ωo )  
sin φˆ (ω ) 

ˆ
ˆ
sin φ (ωo )
qI (n, m;ω ) =  sin φ (ωo )
x
 d (ωT )
 

0  − sin(2n − 1)(ωT )

  − sin(2m − 1)(ωT )

∫
(5.110)
with 1 ≤ n, m ≤ J
The reference frequency ωo is arbitrary except that cos φ̂ (ωo ) ≠ 0 and sin φ̂ (ωo ) ≠ 0, for example
ωoT = 0.32π . The eigenvectors of matrices QR and QI , that correspond to the smallest eigenvalues,
are the coefficients u(n) and v(n), respectively, which relate to the tap coefficients according to
 a( M ),
u (n ) = 
2a( M − 2n),
v(n) = 2a( M − 2n + 1),
n=0
n = 1,, I
(5.111)
n = 1,, J
(5.112)
where I = M 2 and J = M 2 for even M or I = ( M − 1) 2 and J = ( M + 1) 2 for odd M. Thus, the
eigenfilter technique requires the group-delay response of exp( jMω T ) Heq (ω ) to approximate the
desired filter group-delay response
G(ω ) = −
dφ̂ (ω )
ˆ (ωT − ω T ) .
= −2 DT
c
dω
(5.113)
Consequently, the group-delay response of the eigenfilter Heq (ω ) approximates the total desired
group-delay response
ˆ (ωT − ω T )
Gtotal (ω ) = MT − 2 DT
c
(5.114)
where the constant group-delay factor MT, which is the pure propagation delay of the eigenfilter, has
no effect on channel equalization.
5.3.2.3 Design and IM/DD System Performance
To show the effectiveness of the eigenfilter technique, one particular design is considered. In this
example, D̂ = −8 π , which corresponds to the typical values L = 100 km and T = 20.63 ps according
to Eqs. (5.105), (5.107), (5.114) and N = 21 is chosen for reasons to be given shortly. Using the
eigenfilter approach described in Section 5.3, Table 5.4 shows that the computed tap coefficients
of the ODEC are bipolar and less than unity. Note that the coefficients have even or odd symmetry
about the center point a(10).
Figure 5.46a shows that the ODEC magnitude response lies within 1.5 dB over the Nyquist
bandwidth (or maximum filter-bandwidth), given by ∆fmax = 1 (2T ) (or 24.24 GHz in this case),
while Figure 8.1b shows that the ODEC group-delay response approximates the desired group-delay
response reasonably well over the Nyquist bandwidth. An ODEC dispersion of 1710 ps/nm is achieved
over an operating bandwidth of ∆fop = 18.42 GHz (or ∆fop = 0.76∆fmax), the frequency interval over
which the ODEC group-delay response is linear to within 10%, which is the criteria used for choosing
225
Optical Dispersion Compensation and Gain Flattening
TABLE 5.4
?
For 0 ≤ n, m ≤ I Computed Tap Coefficients of the ODEC for D = −8 π
(L = 100 km, T = 20.63 ps), and N = 21
n
0
1
2
3
4
5
6
a(n)
−0.0423
+0.0708
+0.1320
+0.1917
+0.3058
+0.3310
+0.2641
n
7
+0.0423
8
−0.3585
9
−0.2308
10
+0.2642
11
+0.2308
12
−0.3585
13
+0.0347
14
+0.2461
15
−0.3311
16
+0.3508
17
−0.1912
18
+0.1320
19
−0.0718
20
+0.0423
a(n)
n
a(n)
N = 21. The maximum bandwidth of the ODEC can be obtained by increasing the number of taps,
which also results in improving the accuracy of the frequency responses. For example, the maximum
filter-bandwidth of ∆fop = ∆fmax = 24.24 GHz can be achieved with a 31-tap ODEC.
It is approximated from Table 5.5 that N increases linearly with D according to
N ≅ 6.7 Dˆ + 4.3 (13 ≤ N ≤ 47).
(5.115)
The relation between the fiber and ODEC parameters is given, from (5.105) and (5.115), as
 π c  ( N − 4.3)
L=
⋅
2
2
 6.7 D λ  ∆fmax
(5.116)
FIGURE 5.46 Frequency responses of the ODEC (N = 21) with transfer function exp( jM ωT ) H eq (ω) and
the Chebyshev equalizer (N = 12) for D̂ = −8 π (L = 100 km, T = 20.63 ps, ∆f max = 24.24 GHz ). (a) Magnitude
response, (b) Group-delay response, over the maximum filter-bandwidth.
226
Photonic Signal Processing
TABLE 5.5
Number of Taps N Required for Various Values of the
Dispersion Parameter D̂ for ∆fop ≥ 0.6∆fmax
D̂
N
−4 π
13
−8 π
21
−4
31
−16 π
39
−20 π
47
FIGURE 5.47 Plot showing the relation of the transmission distance L, filter-order N and maximum filterbandwidth ∆f max of the ODEC.
and is graphically shown in Figure 5.47. For a given value of N (and hence D̂), there is a trade-off
between the transmission distance L and the maximum filter-bandwidth ∆fmax. For N = 21, for
example, increasing L from 100 to 200 km reduces ∆fmax from 24.24 to 17.4 GHz. For a given value
of ∆fmax, L increases linearly with N. For ∆fmax = 24.24 GHz, for example, L can be increased from
100 to 200 km by increasing N from 21 to 39. Thus, with a sufficient number of taps, the ODEC is
capable of achieving high-bit-rate long-distance fiber channel equalization.
Note that, for any linear optical dispersion-compensating technique, there is always a compromise
between the compensated distance and bandwidth of the compensator. This is because Eqs. (5.114)
and (5.115) are generally formulated and, in particular, the desired filter frequency response in
Eq. (5.105) can be used as a basis for designing optical dispersion compensators.
The eigenfilter technique has also been extended to designing IIR (infinite-impulse response) or
recursive digital all-pass filters that approximate a desired frequency response in the least squares
sense.57 The IIR eigenfilter algorithm may be adapted to designing IIR optical equalizers with a
phase response that approximates the desired phase function given in Eq. (5.105). One possible
optical synthesis of the IIR digital filters is to cascade the basic all-pole and all-zero optical filters
to obtain the desired filtering characteristics as described in Chapter 3.
5.3.2.4 Performance Comparison of Eigenfilter and Chebyshev Filter Techniques
The performance of the ODEC is compared with that of the Chebyshev optical equalizer in the
frequency domain. The parameters of the Chebyshev equalizer using birefringent crystals are
chosen as follows: filter-order N = 12, optical axis rotation between adjacent birefringent crystals
Θ = π 4, and sampling period T = 20.63 ps. From Figure 5.6, the frequency responses of the ODEC
compare favorably with those of the Chebyshev equalizer. For this particular Chebyshev filter
57
A. F. Elrefaie, R. E. Wagner, D. A. Atlas, and D. G. Daut, Chromatic dispersion limitations in coherent lightwave transmission systems, J. Light. Technol., 6, 704–709, 1988.
227
Optical Dispersion Compensation and Gain Flattening
design, optimum performances are achieved in both the frequency domain and the time domain
given in Section 5.3). Increasing the order of the Chebyshev filter only deteriorates its performance
instead of improving it. Unlike the eigenfilter technique, the Chebyshev technique is not systematic
in the sense that its formulation does not approximate the desired magnitude and phase responses
simultaneously. As a result, the Chebyshev equalizer is bandwidth-limited compared with the
eigenfilter technique.
5.3.3
synthesis OF Optical DispersiOn eigencOmpensatOrs
The ODECs can be synthesized using a P-tap FIR coherent optical filter (see Figure 5.48), which
has been used in the synthesis of the higher-derivative FIR optical differentiators as described in
Chapter 4. The FIR filter essentially consists of a 1xP optical splitter, a Px1 optical combiner, and
P waveguide delay lines, into each of which a tunable coupler (TC) and a phase shifter (PS) are
incorporated. A dispersive signal entering the splitter will be evenly distributed to P signals, which
are then appropriately delayed by the delay lines and weighted by the TCs and PSs. These signals
are then coherently collected by the combiner to generate the compensated signal.
The ODECs can be constructed using the PLC technology, namely, silica-based waveguides
embedded on a silicon substrate. In each kth delay line, the PS following the TC is a waveguide
with a thin-film heater attached and uses the thermo-optic effect to induce a carrier phase change
of φ (k ). The PS can use the temperature dependence of the refractive index of the silica waveguide
to compensate for any optical path-length differences of the waveguides. The TC is a symmetrical
Mach−Zehnder interferometer (see the inset of Figure 5.48). It consists of two 3-dB directional
couplers (DCs), two equal waveguide arms, and a thin-film heater, with a carrier phase change of
ϕ (k ) , attached to one of the arms to control the output amplitude.
Neglecting the insertion loss of the 3-dB DCs, the propagation delay and waveguide birefringence of the TC, the kth TC transfer function, is given by
C ( k ) = C ( k ) exp ( j ∠C ( k ) ) = 0.5 exp ( jϕ ( k ) ) − 1
(5.117)
where ∠C(k ) denotes the argument of C(k ),
P-tap FIR Coherent Optical Filter
Symmetrical Mach-Zehnder Interferometer
IN
3-dB
Directional
Coupler
Thin-Film
Heater
C(0)
OUT
PS
TC
ϕ(0)
T
PC
EDFA
φ(0)
PS
TC
PS
C(1)
φ(1)
Px1
1xP
Optical
Splitter
Thin-Film
Heater
(P−2)T
(P−1)T
C(P−2)
φ(P−2)
TC
PS
TC
C(P−1)
PC
Optical
Combiner
PS
φ(P−1)
FIGURE 5.48 Schematic diagram of the P-tap FIR coherent optical filter used to synthesize the ODECs
using the PLC technology. TC: tunable coupler, PS: phase shifter, and PC: polarization controller.
228
Photonic Signal Processing
C(k ) = 0.5 − 0.5 cos (ϕ (k ) )
(5.118)
ϕ ( k ) = cos −1 1 − 2 C ( k )  ,


2
(5.119)
and
(
)
∠C(k ) = tan −1 sin (ϕ (k ) ) cos (ϕ (k ) ) − 1 


(5.120)
for k = 0,1,, P − 1. Eq. (5.118) shows that a desired TC amplitude can be obtained by choosing an
appropriate PS phase ϕ ( k ) according to Eq. (5.119), and this results in the TC phase as given by
Eq. (5.120). The amplitude and phase of the TC can be changed from 0 to 1 and from −π/2 to +π/2,
respectively, when ϕ (k ) is varied from 0 to 2π.
Assuming an isotropic and neglecting the propagation delay and waveguide anisotropy, the
transfer function of the synthesized P-tap ODEC is given by
P −1
Heq (ω ) = lpath ⋅ P −1 ⋅ G ⋅
∑ (−1) ⋅ C(k) exp ( j∠C(k)) ⋅ exp ( jφ (k)) ⋅ exp ( − jkωT ) (5.121)
k
k =0
where lpath is the intensity path loss, which considers all the losses associated with each delay line
such as the losses of the straight and bend waveguides, the insertion loss of the 3-dB DCs in the
splitter and combiner, and the insertion loss of the TC. The P −1 is the coupling loss factor as a result
of a 3-dB coupling loss at each stage of the splitter and combiner, G is the intensity gain of the
EDFA, (−1)k = exp( jkπ ) is the phase shift factor due to the π 2 cross-coupled phase shift of the
3-dB DCs in the splitter and combiner, and T, the sampling period of the ODEC, is the differential
delay between neighboring delay lines.
Optical synthesis of the digital coefficients requires the equality of the amplitude and phase of
Eq. (5.121) such that
G=
P2
, C ( k ) = a( k ) and φ ( k ) = ∠a( k ) − ∠( −1) k − ∠C ( k )
lpath
(5.122)
where ∠a(k ), ∠(−1)k ∈ (0, π ). Thus, the insertion loss of the ODEC can be overcome by the EDFA
with gain as given in Eq. (5.122). In each kth delay line, Eq. (5.122) shows that the amplitude of the
digital coefficient can be optically implemented by the TC amplitude, and the PS must provide a
phase shift according to Eq. (5.122). Note that the TC amplitude can also incorporate, in addition
to a(k ), the deviations of the coupling coefficients of the couplers from the desired 3-dB values as
a result of fabrication errors.
5.3.3.1 IM/DD Transmission System Model
Figure 5.49 shows a block diagram of a 1550-nm IM/DD lightwave system with the ODEC. The
optical source is assumed to be a single-longitudinal mode semiconductor laser, which is directly
modulated by an 8-Gbit/s non-return-to-zero (NRZ) (or RZ with 50% duty cycle) current pulse with
a pseudorandom 6-bit pattern of length 64.
The injected laser current ic (t ) is a pseudorandom digital pulse waveform
∞
ic (t ) = Ibias +
∑ A i (t − kT )
k p
k =−∞
b
(5.123)
229
Optical Dispersion Compensation and Gain Flattening
Hf(ω)
LASER
El(ω)
Ef(ω)
STANDARD OPTICAL FIBRE
Heq(ω)
ODEC
Heq(ω)
OPTICAL
DETECTOR
id(t)
ic(t)
FILTER
Hr(ω) FILTER
ir(t)
Ak
DATA
SIGNAL
EYE
DIAGRAM
FIGURE 5.49 Block diagram of a 1550-nm IM/DD optical communication system with the ODEC.
where Ibias is the bias current, Ak ∈(0,1) is the digital sequence, ip (t ) is the current pulse shape, and
Tb is the bit period that is the inverse of the bit rate (or 1 Tb = 8 Gbit s). If a binary ONE (ZERO) is
transmitted during the kth time interval, then Ak = 1 (Ak = 0). The current pulse shape is of the form
0,

i p (t ) =  I m [1 − exp( −t t r )],
 I exp ( −(t − T ′) t ) ,
b
r
 m
t <0
0 ≤ t ≤ Tb′
t > Tb′
(5.124)
where Tb′ = Tb for the NRZ encoding format and Tb′ = Tb 2 for the RZ encoding format, Im is the peak
modulation current and tr is the pulse rise and fall times (approximately 15%–85%). Note that the
injected laser current ic (t ) is assumed to be the response of a first-order electrical filter when subject
to an input data signal Ak.
The optical field amplitude and phase of the semiconductor laser, in response to the injected current ic (t ), are determined by numerically solving the large-signal laser rate equations describing the
nonlinear modulation dynamics of the device. A fourth-fifth order Runge-Kutta algorithm is used
to integrate the coupled set of first-order differential equations for the photon density s(t ), carrier
density n(t ), and optical phase φm (t ):
n(t ) − n0 1 
ds(t ) 
βΓn(t )
−  s(t ) +
=  Γa0 vg
,
dt
1 + ε s(t ) τ p 
τn

(5.125)
dn(t ) ic (t ) n(t )
n(t ) − n0
=
−
− a0 vg
s(t ),
dt
qVa τ n
1 + ε s(t )
(5.126)
dφm 1 
1 
= α  Γa0 vg ( n(t ) − n0 ) −  ,
dt
2 
τp 
(5.127)
where Γ is the optical confinement factor, a0 is the gain coefficient, vg is the group velocity, n0
is the carrier density at transparency, ε–the gain compression factor, τ p is the photon lifetime,
230
Photonic Signal Processing
β- the fraction of spontaneous emission coupled into the lasing mode, τ n is the carrier lifetime,
q is the electron charge, Va is the active layer volume, α is the linewidth enhancement factor or
laser chirp parameter. The time variations of the optical power and laser chirp are, respectively,
given by
m(t ) = 0.5s(t )Vaη0 hf c (Γτ p ),
∆v(t ) =
1 dφm (t )
,
2π dt
(5.128)
(5.129)
where hfc is the photon energy at the carrier frequency fc and η0 is the total differential quantum
efficiency. The time-dependent electric-field amplitude at the laser output is given by
el (t ) = m(t ) exp ( jφm (t ) )
(5.130)
which can be obtained by numerically solving Eqs. (5.123) through (5.129).
The frequency-dependent electric-field amplitude at the output of the dispersive fiber channel is
given by
E f (ω ) = El (ω ) H f (ω )
(5.131)
where El (ω ) = FFT {el (t )} with FFT denoting fast Fourier transform. The frequency-dependent
electric-field amplitude at the output of the ODEC is given by
Eeq (ω ) = E f (ω ) H eq (ω ).
(5.132)
The time-dependent optical intensity at the output of the optical detector is given by
id (t ) = IFFT {Eeq (ω )}
2
(5.133)
where IFFT denotes the inverse FFT operation. The time-dependent filtered signal at the output of
the baseband receiver filter is given by
ir (t ) = IFFT {FFT {id (t )} H r (ω )}.
(5.134)
The receiver filter is a third-order Butterworth electrical lowpass filter whose transfer function is
given by
Hr (ω ) =
1
(S + 1)(S + 0.5 − j 0.866)(S + 0.5 + j 0.866)
(5.135)
where S = j ω ωr is the S-transform parameter with ωr being the 3-dB angular bandwidth. The
3-dB bandwidth of the receiver filter is ωr (2π ) = 0.65 Tb = 5.2 GHz for the NRZ signal or
ωr (2π ) = 0.55 Tb = 4.4 GHz for the RZ signal.
The third derivative of the fiber propagation constant, typically β3 = 0.13/km [5.5.26], was also
included in the simulation model even though it has been neglected for the purpose of facilitating the
eigenfilter design. However, it was found to have negligible effect on the system performance. The
parameter values used to obtain the numerical results are given in Table 5.6.
231
Optical Dispersion Compensation and Gain Flattening
TABLE 5.6
List of Parameters
Symbol
Parameter Value
I th (threshold current)
I m = I th
I bias = 1.05I th
29.8 mA
29.8 mA
Γ
vg
31.3 mA
0.4
7.5 ×109 cm s
2.5 × 10 −16 cm 2
1.0 × 1018 cm −3
6.0 × 10 −17 cm 3
a0
n0
ε
τp
0.9 ps
β
τn
Va
η0
Q
2.0 × 10 −4
1.0 ns
0.75 × 10 −10 cm 3
0.13
1.6022 × 10 −19 C
6.6262 × 10 −34 Js
H
λ
D
1550 nm
−17 ps nm km
Tb
t r = 0.15Tb
α
125 ps
18.75 ps
0 ≤α ≤ 6
5.3.3.2
Performance Comparison of Optical Dispersion Eigencompensator
and Chebyshev Optical Equalizer
The performances of the ODEC and Chebyshev equalizer are assessed in the time domain by means of
the dispersion-induced optical power penalty PD (in dB) at the baseband filter output, which is defined as
 y
PD = 10 log  
x
(5.136)
where x is the ideal eye-opening at the receiver baseband filter output with no fiber in place, and y
is the eye-opening of a particular system under study.
Figure 5.50 shows the NRZ eye diagrams of an 8-Gbit/s 100-km 1550-nm uncompensated and
compensated external-modulation system, in which the laser chirp factor is α = 0 and the source
spectrum is about 6 GHz. The ODEC and the Chebyshev equalizer improve the dispersion-induced
optical power penalty from PD = −3.2 dB to PD = −0.18 dB and to PD = −0.77 dB, respectively. This
shows a 3.02 dB power improvement provided by the ODEC as compared with 2.43 dB by the
Chebyshev equalizer. The superiority of the eigenfilter technique to the Chebyshev technique is
as expected because the 18-GHz bandwidth provided by the ODEC is more than enough to cover
the whole signal spectrum of 6 GHz, while the linear region of the group-delay response of the
Chebyshev equalizer is less than 5 GHz. Furthermore, the Chebyshev equalizer can only provide
partial dispersion compensation because its dispersion (or the gradient of the linear region of its
group-delay response) is not closely matched to the desired filter dispersion. This explains why
the Chebyshev-compensated eye diagram in Figure 5.50 dis slightly closed although detectable
232
Photonic Signal Processing
FIGURE 5.50 NRZ eye diagrams of an 8-Gbit/s 1550-nm IM/DD lightwave system for D̂ = −8 π and
α = 0. (a) Uncompensated link (L = 0 km). (b) Uncompensated link (L = 100 km). (c) Compensated link
(L = 100 km) using the ODEC with T = 20.63 ps and N = 21. (d) Compensated link (L = 100 km) using the
Chebyshev equalizer with T = 20.63 ps, Θ = π/4 and N = 12.
when compared with the eigencompensated eye diagram in Figure 5.50c, which resembles the ideal
eye diagram in Figure 5.50a. The fact that the ODEC still experiences a penalty of −0.18dB, even
though its bandwidth covers the whole signal spectrum, may be due to a very small error in the
group-delay response, resulting in a very small uncompensated residual dispersion.
This case study shows that, for an ideal chirp-free system, the ODEC outperforms the Chebyshev
equalizer by 0.59 dB in the time domain.
Figure 5.50 shows the NRZ eye diagrams of an 8-Gbit/s 100-km 1550-nm uncompensated and
compensated direct-modulation system, in which the laser chirp parameter is α = 6 and the signal
spectrum is about 18 GHz. The 18 GHz bandwidth of the chirped signal spectrum clearly shows
the effect of the laser chirp on the broadening of the optical source spectrum when compared
with the 6 GHz bandwidth of the chirp-free signal spectrum. Figure 5.50 shows that the ODEC
and the Chebyshev equalizer improve the dispersion penalty from −10.9 dB (Figure 5.51b) to
+0.72 dB (Figure 5.51c) and to −7.9 dB (Figure 5.51d), respectively. This shows that the ODEC
performs significantly better than the Chebyshev equalizer in compensation of the dispersively
chirped signals. Thus, for a chirped system, the ODEC significantly outperforms the Chebyshev
equalizer by 8.62 dB. The Chebyshev-compensated eye diagram (Figure 5.51d) is significantly
distorted and almost closed as would be expected from its narrow bandwidth and poor groupdelay characteristics. The eigencompensated eye-opening (Figure 5.52c) is slightly wider than the
ideal eye-opening (Figure 5.52a), resulting in a positive dispersion penalty of +0.72 dB. Compared
with the eigencompensated chirp-free system with a penalty of −0.18 dB (Figure 5.52c), this case
study shows that the ODEC performs better with the dispersively chirped signals than with the
dispersively chirp-free signals, provided that the chirped signal spectrum lies within the eigenfilter
bandwidth. It is believed that the eigencompensating technique is the first linear optical dispersioncompensating scheme that actually “likes” to work with chirped signals, instead of “hates” as in
other techniques.40–42
Optical Dispersion Compensation and Gain Flattening
233
FIGURE 5.51 Eye diagrams of the transmitted sequences with the use of the eigencompensator whose
conditions are given as in Figure 5.50, except with the factor α = 6 (a) Uncompensated link (L = 0 km). (b)
Uncompensated link (L = 100 km). (c) Compensated link (L = 100 km) using the ODEC with T = 20.63 ps and
N = 21. (d) Compensated link (L = 100 km) using Chebyshev equalizer with T = 20.63 ps, Θ = π/4 and N = 12.
FIGURE 5.52 Dependence of the dispersion-induced ( D̂ = −8 π ) optical power penalty on the laser chirp
parameter for the uncompensated link (L = 0 km), the uncompensated link (L = 100 km), the compensated
link (L = 100 km) using the ODEC with T = 20.63 ps and N = 21, and the compensated link (L = 100 km)
using the Chebyshev equalizer with T = 20.63 ps, Θ = π 4 and N = 12. (a) An 8-Gbit/s 1550-nm system with
the NRZ signal. (b) An 8-Gbit/s 1550-nm system with the RZ signal.
Figure 5.50a and b show the dispersion penalties of an 8-Gbit/s 100-km 1550-nm uncompensated
and compensated system using the NRZ and RZ signals, respectively. In these Figure s, α varies
from 0 to 6 in step of 0.5, and this leads to varying the signal spectrum from 6 to 18 GHz for the NRZ
signal and from 8 to 20 GHz for the RZ signal. There is an optimum value of the chirp parameter
where the dispersion penalty of the uncompensated link is minimum. The uncompensated link
234
Photonic Signal Processing
experiences a minimum penalty of −0.08 dB for the NRZ signal with α = 1 (which also agrees with
the experimental result that the chirp parameter must be α < 1 to achieve the longest transmission
distance) and of −0.06 dB for the RZ signal with α = 2. Some means of dispersion compensation are
obviously required for large departures from the optimum chirp value.
For α = 0, the Chebyshev equalizer improves the penalties from −3.2 dB to −0.77 dB for the NRZ
signal and from −3.2 dB to −1.8 dB for the RZ signal. However, the performance of the Chebyshev
equalizer deteriorates significantly for α ≥ 1 and, in most cases, becomes worse than that of the
uncompensated link. This study shows that the interaction of the poor filter group-delay characteristics, laser chirp, and fiber dispersion further degrades the uncompensated eye diagram, instead
of improving it. Note that the Chebyshev equalizer with Θ = π 4 and N = 12 was found to yield
optimum performance for this particular system and increasing (or decreasing) its order N (for the
same Θ = π 4) would, instead of improving, further degrade the eye diagram.
The ODEC has the capability to achieve a significant re-opening of an eye diagram that would
otherwise be closed. For the NRZ (with α ≥ 1) and the RZ (with α ≥ 2) signals, the dispersion penalties are positive for the eigencompensated system, showing that the ODEC can re-open the receiver
eye further than the ideal eye-opening. Thus, the ODEC performs better with dispersively chirped
signals and can hence be used in external-modulation high-speed systems without sacrificing for the
data rate and transmission distance. This is because an external optical modulator can be designed to
have a residual chirp, and, in fact, it is preferable to have a non-zero chirp value to achieve the longest
transmission distance. This advantage is clearly due to the sufficient bandwidth and high-accuracy
group-delay characteristics of the ODEC. From this analysis, the ODEC can significantly and simultaneously compensate for the laser chirp and fiber dispersion and can, hence, re-open the receiver eye
further than the ideal eye-opening, showing the phenomenon of optical power enhancement.
5.3.3.3
Eigencompensated System with Parameter Deviations
of the Optical Dispersion Eigencompensator
The combined effect of the deviation of three ODEC parameters on the performance of the
eigencompensated system can now be considered. Deviation of filter parameters will arise in
practice due fabrication error of device parameters. The deviation of the ODEC parameters to be
considered is the TC amplitude [see Eq. (5.126)], the PS phase following the TC [see Eq. (5.127)],
and the waveguide delay line.
Figure 5.53a and b show the dispersion penalties of an 8-Gbit/s 100-km 1550-nm eigen compensated link (D = −8 π , α = 6) using the non-ideal ODEC for the NRZ and RZ signals, respectively.
The legend boxes represent, respectively, the percentage deviations of the TC amplitude and PS phase
from below or above their nominal values. The TC amplitude and PS phase deviations are randomly
chosen to be (0%, 0%) , ( ±5%, ± 5%) and ( ±10%, ± 10%) , and the delay-line deviation is randomly
varied from 0 to ±5%.
The dispersion penalties of the non-ideal ODEC vary from −2.5 dB to +0.72 dB for the NRZ
signal and from −2 dB to +1.2 dB for the RZ signal. For the 0% delay-line deviation, Figure 5.53a
indicates that the power enhancement is reduced from the ideal case of +0.72 dB to +0.43 dB for
the ( ±5%, ± 5%) case and to +0.39 dB for the ( ±10%, ± 10%) case. For the ±5% delay-line deviation,
the power penalty of the (0%, 0%) case, i.e., −2.5 dB, is surprisingly more severe than those of the
( ±5%, ± 5%) case, i.e., −1.1 dB, and the ( ±10%, ± 10%) case, i.e., −1.9 dB. The randomness of the
parameter deviations makes it difficult to provide a reasonable explanation for this discrepancy.
Similar explanations can be made for the case of the RZ signal as shown in Figure 53b. For a
1-dB penalty, the eigencompensated performance allows a ±2% delay-line deviation for the NRZ
signal and a ±4% delay-line deviation for the RZ signal, and the TC amplitude and PS phase
deviations of up to ( ±10%, ± 10%) for both cases. It has been claimed that the waveguide delay line
could be accurately fabricated to within 1% deviation. The effect of the delay-line deviation on the
performance of the non-ideal ODEC can thus be neglected, and the TC amplitude and PS phase
deviations may be allowed to more than ( ±10%, ± 10%) with less than 1-dB penalty.
Optical Dispersion Compensation and Gain Flattening
235
FIGURE 5.53 Dependence of dispersion-induced optical power penalty on delay-line deviation of the ODEC
for an 8-Gbit/s 100-km 1550-nm direct-modulation system ( D̂ = −8 π ,α = 6). The curves correspond to
the non-ideal ODEC (T = 20.63 ps, N = 21) with the TC amplitude and PS phase deviations of (0%, 0%) ,
( ±5%, ± 5%) and ( ±10%, ± 10%) , respectively. (a) NRZ signal. (b) RZ signal.
5.3.3.4 Trade-Off Between Transmission Distance and Eigenfilter Bandwidth
Section 5.3.3.3 has shown that there is a compromise between the eigencompensated distance and
the eigenfilter bandwidth for a fixed number of taps N. Such an effect on the eigencompensated
performance is now investigated.
Figure 5.54 shows the variation of the bandwidth-limited optical power penalty with the
transmission distance for an 8-Gbit/s 1550-nm eigencompensated system (N = 21, D̂ = −8 π ) for
both the NRZ and RZ signals with α = 0 and α = 6. As expected from Eq. (5.116), increasing
the distance from L = 100 km to L = 1000 km increases the power penalty since the eigenfilter
bandwidth is reduced from ∆fmax = 24.24 GHz (T = 20.63 ps) to ∆f max = 7.67 GHz (T = 65.3 ps). For
L > 100 km, the vertical amplitudes of the ideal and eigencompensated eye diagrams, which are still
widely open, are smaller than those of the 100-km case. This can be explained in terms of either a
reduction in bandwidth ∆fmax or an increase in the sampling period T of the whole system because
of the interrelation between ∆fmax and T. The sampling period to be chosen must be sufficiently
small enough to capture most of the encoded information for transmission, or the compensator
bandwidth must be designed to be sufficiently large enough to cover the signal spectrum. Thus, the
ideal eye diagram of the 100-km case is used as a basis for computing the bandwidth-limited (but
not dispersion-limited) optical power penalty for the L > 100 km cases.
For distances up to 200 km for the NRZ signal and up to 400 km for the RZ signal, the
eigencompensated system performs significantly better with chirped than with chirp-free signals.
For longer distances, the eigencompensated performance shows little difference for both cases
because of the reduced eigenfilter bandwidth. For a 1-dB bandwidth-limited penalty, error-free
transmission may still be possible for this particular eigencompensated system with distances
up to 600 km for both the NRZ and RZ chirped signals. The peaks and dips of these curves
are difficult to explain because of the complicated effect of the interaction of the laser chirp,
fiber dispersion, and eigenfilter group delay. However, they provide useful information about the
optimum eigencompensated performance for a particular transmission distance, which can be
greatly increased without any reduction in bandwidth, by increasing the eigenfilter tap according
to Eq. (5.116).
236
Photonic Signal Processing
FIGURE 5.54 Dependence of bandwidth-limited optical power penalty on transmission distance of an 8-Gbit/s
1550-nm IM/DD system with ODEC (N = 21, D̂ = −8 π ) for α = 0 and α = 6. (a) NRZ signal. (b) RZ signal.
5.3.3.5 Compensation Power of Eigencompensating Technique
The performance of the eigencompensated system is now characterized by means of the compensation
power (CP) of the ODEC. Section 5.3.3.4 has shown that, for a 1-dB bandwidth-limited penalty, the
maximum eigencompensated distance is about 600 km for both the NRZ and RZ chirped signals.
= −8 π , L = 600 km, T = 50.56 ps, and ∆fmax = 9.89 GHz = 1.24 B, where B
In this case, N = 21, D
denotes the bit rate. For a 1-dB bandwidth-limited penalty, the product of the square of the bit rate
and the eigencompensated distance for both the NRZ and RZ chirped signals is thus given by
 c 
B2 L ≤ 0.6( N − 4.3) 
 2 D λ 2 


(5.137)
when ∆fmax ≥ 1.24 B has been substituted into Eq. (5.137). For an uncompensated system, the B2 L
value is given by
B2 L ≤
π
2(1 + α )

2 12
 c 

2 
+ 2α   2 D λ 
(5.138)
which decreases with increasing chirp parameter α. For α = 0, Eq. (5.138) is almost the same and a
factor of π/2, where a 1-dB penalty was used as a criterion in both references. Eqs. (5.137) and (5.138)
are graphically illustrated in Figure 5.55, where the dispersion limits for both α = 0 and α = 6 are
shown to be significantly improved by the ODEC. For example, for B = 8 Gbit/s, the uncompensated
distance is limited to only 7.5 km for α = 6 but is extended to 90 km for α = 0. However, the
presence of the ODEC significantly extends the transmission distance to 575 km for N = 21 and
B = 8 Gb/s.
Optical Dispersion Compensation and Gain Flattening
237
FIGURE 5.55 Dispersion limits for the uncompensated transmission using external modulation (α = 0) and
direct modulation (α = 6), and dispersion improvements for the eigencompensated transmissions using direct
modulation (α = 6).
The overall performance of the eigencompensated system can be characterized by the
compensation power of the ODEC, which is a measure of the increase in the B2 L value and is
defined as the ratio of Eq. (5.138) to the dispersion factor given in Eq. (5.105) as
CP = 0.19( N − 4.3) 2(1 + α 2 )1 2 + 2α  .
(5.139)
Figure 5.56 shows that the CP value increases with the eigenfilter order as well as with the laser
chirp parameter. The above example gives CP = 77 for α = 6, showing that the B2 L value of the
eigencompensated direct-modulation system is improved by a factor of 77.
FIGURE 5.56 Compensation power (CP) of the ODEC for various values of the chirp parameter and filter order.
238
Photonic Signal Processing
The reasons for comparing the eigenfilter technique with the Chebyshev technique are two-fold.
First, the Chebyshev technique58,59 has a greater design flexibility than those of the chirped fiber
Bragg grating60 (5.54) and the Fabry-Perot equalizer61,62 in modifying the filter frequency response.
Second, the ODEC and the Chebyshev equalizer have common features, such as linearity, nonrecursiveness, and periodic frequency response, and they both can be implemented using the same
PLC technology. As briefly described in Section 5.3.1, there are advantages as well as disadvantages
associated with various techniques, so that it is difficult to make a fair comparison between them.
However, the unique feature of the ODEC is its ability to perform better with dispersively chirped
signals than with dispersively chirp-free signals, an advantage that has not yet been claimed by
other methods. In addition, the ODEC is compact, capable of operating stably and can perform
high-speed signal compensation because of its integrated-optic form. Further progress in the PLC
technology would make the ODEC even more attractive.
5.3.3.6 Remarks
• An effective digital eigenfilter approach has been employed to design linear ODECs for
compensation of the combined effect of laser chirp and fiber dispersion at 1550 nm in highbit-rate long-distance IM/DD lightwave systems.
• The ODECs, which have been synthesized using an integrated-optic transversal filter, are
very effective in equalization of a dispersively chirped optical communication channel.
• In an 8-Gbit/s 100-km 1550-nm system, the performance of the ODEC is more impressive
than that of the Chebyshev equalizer in both the frequency and time domains. The ODEC
slightly outperforms the Chebyshev equalizer for external-modulation transmission but
significantly outperforms the Chebyshev equalizer for direct-modulation transmission.
• For a 1-dB power penalty, a 21-tap ODEC may provide error-free transmission of an
8-Gbit/s 1550-nm direct-modulation system for distances up to 600 km for both the NRZ
and RZ chirped signals.
• The ODEC has a large value of compensation power, which increases with the eigenfilter
order and also with the laser chirp parameter.
• The combined effect of the laser chirp, fiber dispersion, and ODEC group delay can
re-open the receiver data eye further than that of the ideal eye-opening, resulting in the
phenomenon of optical power enhancement, a feature that is not available with other
compensation techniques.
5.4
5.4.1
PHOTONIC FUNCTIONAL DEVICES
preamble
Photonic functional devices based on a silica planar lightwave circuit. First, lattice-form optical
devices are described for chromatic dispersion slope compensation, and the dynamic equalization
of chromatic and polarization-mode dispersion and gain non-uniformity, in high-speed wavelength
division multiplexing transmissions.
S. C. Pei and J. J. Shyu, Eigen-approach for designing, FIR filters and all-pass phase equalizers with prescribed magnitude
and phase response, IEEE Trans. Circuits Syst.,39, 137–146, 1992.
59 N. Q. Ngo, L. N. Binh, and X. Dai, Eigenfilter approach for designing FIR all-pass optical dispersion compensators
for high-speed long-haul systems, Proc. IREE, 19th Australian Conference on Optical Fibre Technology, Melbourne,
pp. 355–358, 1994.
60 G. P. Agrawal, Nonlinear fiber optics, Boston, MA: Academic Press, 1989.
61 N. Sugimoto, H. Terui, A. Tate, Y. Katoh, Y. Yamada, A. Sugita, A. Shibukawa, and Y. Inoue, A hybrid integrated waveguide isolator on a silica-based planar lightwave circuit, J. Light. Technol., 14, 2537–2546, 1996.
62 D. Marcuse, Single-channel operation in very long nonlinear fibers with optical amplifiers at zero dispersion, J. Light.
Technol., 9, 356–361, 1991.
58
239
Optical Dispersion Compensation and Gain Flattening
Advanced optical systems require highly functional optical devices in order to exceed the
electrical speed limit. If we are to realize high-speed WDM links, we must develop compensators
to deal with the undesirable characteristics of optical fibers, namely, chromatic and polarizationmode dispersion and gain non-uniformity.
Lattice-form filters, which comprise cascades of alternating symmetrical and asymmetrical
Mach Zehnder interferometers (MZIs), are suitable for realizing these compensators, because they
can achieve various characteristics adaptively and flexibly.
Photonic Lightwave Circuits (PLC) based circuits are fabricated on a silica on silicon substrate
by a combination of flame hydrolysis deposition and reactive ion etching. The typical waveguide
bending radius is around 2 to 25 mm, and large-scale integrated circuit chips can be as large
as several centimeters square. This fabrication process can provide a uniform refractive index
and core geometry, and the propagation loss throughout a large wafer is low (about 0.01 dB/cm).
Alternatively Si integrated photonics can also offer the designed lattice filters, that these devices
can be integrated with electronic circuits and active modulator components. Figure 5.57 shows
the cross Section of PLC waveguide. Table 5.7 tabulates the properties of PLC.
5.4.2
Optical DispersiOn cOmpensatiOn mODule (ODcm)
Figure 5.58 shows the basic configuration of a lattice-form optical filter, which comprises a cascade
of alternating n + 1 symmetrical and n asymmetrical MZIs (n: natural number). The symmetrical
and asymmetrical MZIs function as tunable couplers and delay applying parts, respectively. Such
filters are attractive because they can realize several kinds of dynamic compensator for chromatic
and polarization-mode dispersion and gain non-uniformity. Since they confine lights in the MZIs
without radiation, their properties are superior to those of transversal form devices in that their
losses and loss variations are intrinsically smaller. The transfer functions of the lattice filters are
given by a Fourier series with terms identical to the symmetrical MZIs,
FIGURE 5.57 (a) Structure of PLC waveguide and progress of PLC technology, SiON material low-loss is
also the technology similar to silica on silicon; (b) Progresses of PLC.
TABLE 5.7
Properties of PLC
Index difference (%)
Core size (µm)
Loss (dB/point)
Coupling loss (dB/point)a
Bending radius (mm)b
a
b
b
Low-Δ
Medium-Δ
High-Δ
Super high-Δ
0.3
8×8
<0.01
<0.1
25
0.45
7×7
0.02
0.1
15
0.75
6×6
0.04
0.4
5
1.5∼2.0
4.5 × 4.5∼3 × 3
0.07
2.0 (∼0.7c)
2
Connection with standard single-mode fiber
Bending radium where bending loss in a 90° arc is 0.1 dB
When PLC spot-size converter is used
240
Photonic Signal Processing
FIGURE 5.58
Structure of lattice form oDCM.
n +1
T ( z) =
∑a z
i
−i
(5.140)
i =1
where z = e j 2π fne ∆L / c0 acting as the sampling variable as defined in Chapter 2 and ai is the complex
expansion coefficient of the series (j: imaginary unit, ne: effective refractive index of the waveguide,
∆L is length difference between two arms of the asymmetrical MZI, f denotes the operating
frequency, c0 is the light speed in vacuum). By varying the amplitude level ai , we can obtain
arbitrary and flexible filter characteristics whose performance limits are determined by the number
n, the number of cascaded MZDI structures. Thus, the order of each MZDI is determined by the
length difference between the two paths. The length difference must be taken in multiple number of
a basic length difference to simplify the analytical solution.
The problem with the lattice filters lies in the difficulty of tuning the MZI phase parameters.
Recently, we proposed a novel method for tuning the phases in lattice filters accurately to obtain
desired characteristics without the need for monitor couplers, thus reducing both device size and loss.
5.4.3
chrOmatic DispersiOn cOmpensatOrs
Variable chromatic dispersion compensators are63 becoming increasingly important in high speed
optical transmission systems with bit rates of 25 or 56 Gbaud or more, where it is essential to compensate adaptively for the various dispersions of installed fibers and the dispersion change caused
by changes in the environmental temperature or path differences in optical networks. The allowable
dispersion limit is inversely proportional to the square of the bit rate. We have realized the first variable dispersion compensator with a lattice configuration and recently improved its characteristics by
adopting the adjustment method shown in Figure 5.58. Figure 5.59a and b, respectively, show the
power transmittance and the relative delay time versus the operating wavelength.
The fiber dispersion slope, which generates a different dispersion in each wavelength channel, is
also one of the factors that degrade the quality of high-speed WDM transmission systems. However,
it has been difficult to compensate stably for the dispersion slope of dispersion shifted fiber (DSF)
and non-zero dispersion shifted fiber (NZ-DSF), which are indispensable if we are to achieve large
capacity transmissions. The lattice configuration is used to realize slope compensators for N × 20
and 40 Gbit/s transmissions, whose configuration and parameters are shown in Figure 5.60. An
array of lattice compensators was integrated on one silica waveguide wafer with a ∆ of 1.5% in
order to integrate all the equalizers monolithically. The pre-dispersion compensated wavelength
63
K. Takiguchi, Photonic functional devices based on a silica planar lightwave circuit, NTT Japan, Technical report, 1999.
Optical Dispersion Compensation and Gain Flattening
FIGURE 5.59
Transmittance and relative delay parameters versus frequency/wavelength.
FIGURE 5.60
Configuration and parameters of dispersion slope compensator.
241
components on the International Telecommunication Union (ITU) grids are combined by a
conventional multiplexing AWG and transmitted into the fibers. Each compensator is composed
of five asymmetrical MZIs cascaded in series, and each has a different compensation value.
Figure 5.61a and b, respectively, show the relative delay time obtained with the compensators for
16 × 20 Gbit/s, 640 km and 8 × 40 Gbps, 320 km DSFs, respectively. The fiber-to-fiber losses
were about 7 dB. The waveguide phase shifts are trimmed using local heating and quenching
with a high electrical power to decrease the electrical offset power, and reduced the total power
of a few tens of watts by over 80%. We were able to construct two other configurations flexibly,
namely, two AWGs interleaved with a compensator array or a compensator array installed behind a
demultiplexing AWG, for in-line or post dispersion compensation, respectively. In addition, we have
been investigating a slope compensator within a single signal bandwidth for ultra-high-speed time
division multiplexing (TDM) transmissions of more than 100 Gbit/s.
242
Photonic Signal Processing
FIGURE 5.61 Measured relative delay time of dispersion slope compensators. (Extracted from Suzuki, K.
et al., Electron. Lett., 38, 1030–1031, 2002.)
5.4.4
Optical gain equalizer
5.4.4.1 Introductory Remarks
Gain equalization is a technique for making the gain spectrum of an optical amplifier device
flatter over a certain optical frequency range. The optical gain e.g. from a fiber amplifier has some
dependence on the wavelength, which can be disturbing. For example, in optical fiber communications
employing wavelength division multiplexing (WDM), the wavelength dependence of the gain can
unbalance the powers in the transmitted channels. Therefore, it is common to apply methods for
gain equalization, also termed as gain flattening. There are various technological options:
• For a given kind of active fiber, the shape of the gain spectrum can be optimized by
adjusting the average inversion level of the laser-active ions. This can be done e.g. by
varying the pump power or the length of the active fiber.
• The material composition of the fiber core can be optimized. For example, silica fibers can be
optimized with various co-dopants, and fluoride fibers can offer a fairly flat gain spectrum.64
• Amplifier fibers of different glass compositions can be combined in an amplifier chain to
obtain a wideband hybrid amplifier. Such a device may also contain a Raman amplifier.
• Another approach is a split-band amplifier, where a wavelength-dependent splitter
distributes the signal content over two or more different fiber amplifiers, and another
wavelength-dependent fiber coupler serves to recombine the spectral components.65
• Optical filters (gain flattening filters)66 can be used, which have higher losses in wavelength
regions where the gain is higher. Such filters are often based on fiber Bragg gratings (longperiod gratings, slanted gratings or chirped gratings), although various other types of filters
have been demonstrated.
The optimization of a multi-stage amplifier with flattening filters is a complex task, since it is not
obvious, for instance, which combination of amplifiers and filters gives the best results in terms of
noise figure and power efficiency. A typical solution for a two-stage amplifier based on doped silica
fibers would include an optical filter between the two stages.
B. Clesca et al., 1.5 µm fluoride-based fiber amplifiers for wideband multichannel transport networks, Opt. Fiber
Technol., 1, 135, 1995.
65 N. Park et al., High-power Er–Yb-doped fiber amplifier with multichannel gain flatness within 0.2 dB over 14 nm, IEEE
Photon. Technol. Lett., 8, 1148, 1996.
66 K. Inoue et al., Tunable gain-equalization using a Mach–Zehnder optical filter in multistage amplifiers, IEEE Photon.
Technol. Lett., 3, 718, 1991.
64
Optical Dispersion Compensation and Gain Flattening
243
The gain spectrum of a Raman amplifier can be flattened by using multiple pump beams with
well-balanced pump power level.67
5.4.4.2 Dynamic Gain Equalizer
A dynamic gain equalizer is needed to flatten the various gain profiles of optical fiber amplifiers,
whose gain spectra may change due to environmental fluctuations or the add/drop of WDM signals.
We have developed a dynamic gain equalizer whose configuration is shown in Figure 5.62. The
polarization properties were eliminated by the polarization diversity technique. The fiber-to-fiber
loss including the diversity device was 9.0 dB. The equalizer comprises ten asymmetrical MZIs
with one tunable coupler on either side whose free spectral ranges (FSRs) increase in 8 nm steps
from 8 to 80 nm. One output port in each asymmetrical MZI is connected to one input port in the
next-stage MZI in series. Although the device in Figure 5.62 is not completely lattice-form and its
loss properties are somewhat inferior to the configuration in Figure 5.57, it is easier to adjust and thus
obtain the desired characteristics. Figure 5.63 shows the un-equalized (raw) and equalized spectra.68
FIGURE 5.62
Configuration of dynamic gain equalizer.
FIGURE 5.63
Raw and equalized spectra of EDTFA ASE.
Y. Emori et al., 100 nm bandwidth flat-gain Raman amplifiers pumped and gain-equalized by 12-wavelength-channel
WDM laser diode unit, Electron. Lett., 35, 1355, 1999.
68 K. Suzuki, T. Kitoh, S. Suzuki, Y. Inoue, Y. Hibino, T. Shibata, A. Mori, M. Shimizu, PLC-based dynamic gain equaliser
consisting of integrated Mach-Zehnder interferometers with C- and L-band equalising range, Electron. Lett., 38,
1030–1031, 2002.
67
6
Optical Dispersion
in Guided-Wave FIR
and IIR Structures
Following Chapter 5 on the more theoretical design of optical filter via the use of digital signal
processing (DSP) technique, this chapter gives a brief description of the dispersion phenomena of
lightwaves propagation in fiber and waveguide integrated structures and real practical experimental
implementation of the filters whose dispersion factor can counter that of signal broadening over a
long length of fibers. These dispersion compensators in integrated structures are highly compact
and produced at a very low cost, especially in the Si on Insulator (SOI) platform. The use of microring resonator (MRR) offers simultaneous compensation of several channels in the same device/
module due to its repeatable characteristics or effectively the free spectral range (FSR).
The total dispersion of an optical guiding wave structure operating in the linear region is contributed by material refractive index as a function of wavelength and waveguide structure which
confines the waves, thus tighten the phase velocity of optical waves of different frequencies or wavelength. It is the waveguide dispersion factor that contributes significantly to the dispersive effects
on the modulated signals whose bandwidth is finite. Hence, several spectral components travel at
different phase velocities leading to different arrival times resulting in a dispersion or broadening
of modulated lightwaves.
Thus, photonic integrated circuits (PIC) can play a significant role in the dispersive effects,
which are counter those of the fiber, thence the compensation or equalization of transmitted optical
pulses. In order to obtain high dispersive effects in guided structure the second order differentiation
of the phase with respect to the optical frequency. Similar to that employed in electrical circuits,
an optical transfer function or transmittance of a photonic integrated circuit (PIC) consists of the
amplitude and phase parts. The phase changes significantly when the PIC exhibits a resonance,
i.e., the complex pole pairs of the transfer function (TF) if we can obtain by using the sampling
z-transform described in Chapters 2 and 3.
In this chapter, we give brief designs of the MRR, which can act as a dispersion compensator in
the optical domain. The transfer function of both amplitude and phases are obtained, and the dispersion of the MRR estimated. It is found that a single MRR can perform best at compensating 10 km
of standard single-mode optical fibers (SSMF), which operates at the through port, or a notched
filter with the dispersion factor in the opposite or normal dispersion to the SSMF. Notably, with the
periodic resonant characteristics of the MRR, by designing this to be equal to the spectral spacing
of the wavelength-division multiplexing (WDM) channels, such as 50/100 GHz, an MRR PIC can
be designed to simultaneously compensate for WDM dispersive channels.
Fiber Bragg gratings (FBG), whose profile is chirped or apodized, can also offer a dispersive
device much better than a finite uniform index profile FBG. The dispersion compensation of these
FBGs can be equivalent to more than 100 km SSMF. However, an FBG cannot be integrated with
a Silicon-based transmitter or receiver. It is given here for reference only, and we believe that the
solutions should be strongly based on an MRR PIC system.
Thence, the IIR (infinite impulse response) structure consisting mainly of Mach–Zehnder
interferometers in integrated structures are described and related to the half-band filter (HBF) or
alternatively the wavelet filters (WP) based on Wavelet Packet Transform (WPT). These types of
filters offer the dispersion of the same sign over the whole band of modulated channels. They consist
of all interferometers so they are considered the most stable.
245
246
6.1
Photonic Signal Processing
PREAMBLE/INTRODUCTION
This short section attempts to give a brief explanation on the dispersion mechanism of lightwaves
propagating in guided optical waveguides, including optical fiber, especially SSMF, and the Si-based
ridged waveguide. Dispersion, large or small, relies mainly on the tightness of lightwaves by the
waveguide, either when the modulated channel signals travel through the weakly guiding fiber or
strongly guided waves in an integrated photonic structure. Then the dispersion mechanism in optical resonators, especially the MRR based on a Si-waveguide can be described.
To begin with, this chapter gives a brief account of dispersion in SSMF then dispersion in MRR.
Additionally, the chapter will discuss the dispersion compensating mechanism and parameters of
the MRR, which can offer the dispersion and dispersion slope compensation of SSMF with length
of 10–80 km, as well as the tunability and switched dispersion compensation of multiple number
of 10 km length. Thence, structures employing Finite Impulse Response (FIR) structures, which
can offer a more stable and a fuller band dispersion factor, are given with a synthesis and design
techniques using HBF or wavelet transform. The structure employs delay interferometers of Mach–
Zehnder type or MZDI (Mach–Zehnder Delay Interferometers), which are stable. Implementation
of the dispersion compensators of both types is given.
System applications of such dispersion compensators are dealt in1. References are given and
listed in the footnote given here for oDCM using MRR2,3,4,5,6,7,8,9,10 and FBG as oDCM.11,12,13,14
M. Seifouri, S. Olyaee, M. Dekamin, and R. Karami, Dispersion compensation in optical transmission systems using
high negative dispersion chalcogenide/silica hybrid microstructured optical fiber, Opt. Rev., 24, 318–324, 2017.
2 S. Dilwali and G. S. Pandian, Pulse response of a fiber dispersion equalizing scheme based on an optical resonator , IEEE
Photonic Tech. Lett., 4, 942, 1992.
3 G. S. Pandian, F. E. Seraji, Optical pulse response of a fibre ring resonator, IEE Proc-J., 138, 235, 1991.
4 K. Takiguchi, S. Suzuki, and Y. Ohmori, Planar lightwave circuit optical dispersion equalizer, IEEE Photonic Tech., 6,
86, 1994.
5 C. K. Madsen and G. Lenz, Optical all-pass filters for phase response design with applications for dispersion compensation, IEEE Photonic Tech L., 10, 994, 1998.
6 C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, T. N. Nielsen, L. E. Adam, and I. Brenner, An all
pass filter dispersion compensator using planar waveguide ring resonators, CLEO, FE6-1/99.
7 P. Chindachaichuay, P. Yupapin, Multi-stage ring resonator all-pass filters for dispersion compensation, Opt. Appl., 39,
2009.
8 C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Capuzzo, L. T. Gomez, T. N. Nielsen, and I. Brener, Multistage dispersion
compensator using ring resonators, Opt. Lett., 24, 1555, 1999.
9 C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, Integrated all-pass filters for tunable
dispersion and dispersion slope compensation, IEEE Photonic Tech L., 11, 1623, 1999.
10 G. Lenz and C. K. Madsen, General optical all-pass filter structures for dispersion control in WDM systems, IEEE J.
Lightw. Technol., 7, 1248, 1999.
11 I. Riant, S. Gurib, J. Gourhant, P. Sansonetti, C. Bungarzeanu, and R. Kashyap, Chirped fiber Bragg gratings for WDM
chromatic dispersion compensation in multispan 10-Gb/s transmission, IEEE J. Sel. Top. Quantum Electron, 5, 1312,
1999.
12 I. Navruz, and A. Altuncu, Optimization based synthesis methods to design multi-channel sampled fiber Bragg gratings
with phase-shift, ELECO, pp. 213–217. Budapest 2009.
13 L. M. Rio de Sousa Ramosand Rui, P. M. Alves Ramos, Characterization of fiber Bragg grating for dispersion compensation, Instituto de Engenharia de Sistemas e Computadores do Porto, Faculdade de Engenharia da Universidade do
Porto, Licenciatura em Engenharia Electrotecnica e de Computadores. (Thesis)
14 N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, Fiber Bragg gratings for dispersion compensation in transmission:
Theoretical model and design criteria for nearly ideal pulse recompression, IEEE J. Lightw. Technol., 15, 1303, 1997.
1
Optical Dispersion in Guided-Wave FIR and IIR Structures
6.2
247
DISPERSION MECHANISM IN FIBER AND WAVEGUIDE
In general, the dispersion of an optical pulse happens when the components of its modulated spectrum are transmitted and propagated in a medium with different velocities—the phase velocity.
Thus, they arrive at different times even though they were transmitted at the same time, as shown
in Figure 6.1.
Now in a SSMF, the lightwaves are guided in a weakly guided manner so that they can travel (in
a single-guided mode) without much loss (0.15 dB/km at 1550 nm). If operating in a linear regime
or below the nonlinear threshold, the dispersion occurs in SSMF due to:
• Material dispersion: The refractive index of the core and cladding of the fiber are a function of wavelength n(λ ). Thus, the different optical frequency components of an optical
modulated pulse sequence are dispersed due to the material (e.g., Ge2O-doped Silica). One
can change the material dispersion by designing the composition of the materials of the
core and cladding. But this can change only a small amount.
• Waveguide dispersion: The optical waveguide or optical fiber (if the core is circular) can
influence the travelling velocity of the guided mode whose propagation phase velocity,
normally called the propagation constant β (λ ), which varies with respect to the operating
wavelength of the lightwaves. Note that an optical modulated pulse does have its components of [ λi , λi + ∆λi , λi + 2∆λ .....λi + N ∆λi ] ; with..∆λi << λi channels are traveling at different velocities which are strongly influenced by the structure of core and cladding. Indeed,
this is β (λ ) and it varies with the fiber parameters:
∆n,core− cladding_index_difference; r = core radius − circular_for_SSMF.
For SSMF, the refractive index difference is very small (less than 0.3%), and this why it
is called a weakly guided phenomena. The reflection loss at the core-cladding interface is
small so as the lightwaves can propagate with very small loss.
FIGURE 6.1 Pulse input → dispersive pulse of color feature after propagating through in an optical fiber.
If the propagation medium has an opposite dispersion factor, then the pulse is also dispersed with reversed
“color” components.
248
Photonic Signal Processing
• Dispersion factor: Dispersion factor of an optical waveguide can be obtained by taking
the derivative of the material n(λ ) or waveguide propagation constant β (λ ) with respect
to wavelength (or optical frequency) to give the group velocity delay (GVD) or the delay
variation of each frequency component and their gain of difference. By taking the derivative of this GVD, it gives the dispersion factor or the pulse dispersive feature. This depends
on the bandwidth of the modulate pulse and the length of the waveguide (fiber length).
Figure 6.2 shows (a) the material as a function of optical frequency (wavelength); (b) GVD
versus wavelength due to material (light grey) and waveguide (dark grey); and (c) total
broadening/dispersion factor.
Inflection point --> dispersion
zero @1290 nm -1310 nm region
depending on
(a)
(b)
(c)
FIGURE 6.2 Dispersion of SSMF optical waveguide (a) refractive index of Silica as a function of material
→ material dispersion (b) GVD due to material contribution and that of waveguide; and (c) dispersion factor
curve versus wavelength of Ge2O:SiO2 core SSMF.
249
Optical Dispersion in Guided-Wave FIR and IIR Structures
6.3
6.3.1
MICRO-RING RESONATOR (MRR) AS AN OPTICAL
DISPERSION COMPENSATOR (oDCM)
why resOnatOr?
A resonator structure (whether electrical or optical circuits) has a resonant frequency at which
the amplitude is peaking, as shown in Figure 6.3 (light grey curve), and of an optical MRR shown
in Figure 6.4. At the two 3dB points on both sides of the resonant frequency, the phase of the
lightwaves passing through the resonant structure changes significantly, about 180° of arc. Due to
this large phase change15 over a narrow optical frequency band around the resonant frequency, GVD
Passband resonant filter
Notched resonant filter
FSR
(a)
(b)
FIGURE 6.3 (a) Resonant transfer transmittance as a function of optical frequency. This is for MRR.
Blue = notched filter (Dip at resonant) and Red = resonant passband filter (Peak at resonant) (b) resonant
notched filter (single ring MRR) of 30 GHz = FSR; κ = 0.5; cross-port output, loop length = 400 µm.
15
Similarly in electrical and electronic circuits a resonant electrical circuit has a transfer function whose poles are
complex conjugates at which the phases change significantly. In optical circuit such as MRR there a also two complex poles of the z-transform transfer function whose z parameter (sampling variable) can be represented by
z −1 = e jωt = e j β z ; phase = ωτ = β L. β is the propagation constant of the guided lightwave; L is the length of the resonator;
τ is the propagation time/delay of the lightwaves in the guided resonator.
250
Photonic Signal Processing
(a)
(b)
(c)
FIGURE 6.4 MRR structure (a) wave propagating through to (b) DROP (resonant drop) and THRU (output)
ports; (c) MRR Si-waveguide, highly/strongly guided mode.
is large and thus the dispersion factor (derivative of the GVD with respect to the wavelength) is
large and can equal some 10’s km of SSMF. The lightwaves in an integrated optical MRR are
strongly guided and well confined in the waveguide, as shown in Figure 6.4c. This tight guiding of the lightwaves allows waveguide dispersion, since the propagation constant of the guided
lightwaves is a function of the wavelength (as mention in the fiber dispersion section).
Thus, resonant structures offer dispersive properties and can be designed to compensate the
dispersion of lightwave pulse sequence after propagating through a long length of optical SSMF.
6.3.2
transFer transmittance FunctiOn OF the thru pOrt (nOtcheD
resOnant Filter) anD DrOp pOrt (banDpass Filter)
6.3.2.1 Dispersion Characteristics and Dispersion Compensation by MRR
Now then, if we observe the transmittance (Power) of the thru and drop ports of the MRR, as given
in Figure 6.3, we can see that for the notched resonant filter (lightwaves out at THRU port) the
resonant dip allows us to imagine that the dispersion is negative (due to sign definition of the GVD –
this becomes positive). Similarly, the dispersion of the lightwaves at the drop port is a passband
transmittance giving a positive dispersion characteristic. Thus, using either of these ports, one can
compensate for a negative or positive dispersive pulse sequence. It is noted here that positive or
negative dispersive pulses are all dispersive pulses but opposite in the “color” sequential frequencies
contained inside the pulse spectrum (see Figure 6.1).
We can obtain the resonant characteristics of an MRR, which is dependent on the guided propagation constant (hence the phase velocity) of the lightwaves in the optical waveguiding path of
the resonator. The resonant condition is that the total phase of the lightwaves traveling around the
ring must be a multiple number of 2π . Thus, we can obtain the GVD and dispersion as a function
of the optical frequency or normalized parameter ωτ , as shown in Figure 6.5. The free spectral
251
Optical Dispersion in Guided-Wave FIR and IIR Structures
passband
FIGURE 6.5 GVD and dispersion as function of phase of guided lightwaves ωτ or angular optical frequency
(by keeping constant delay tie or loop length) obtained at the thru port of a MRR. τ ≈ 10 ps for a 1000 µm
resonant loop length at 1550nm for normal Si ridge waveguide. ωτ = β L = loop phase or sampling period in
z-transform.
range (FSR) is defined as the spectral distance between two resonant peaks, as shown in Figure 6.1.
That is, the resonance occurs for lightwave channels that are spaced with the next channel equals
to the FSR.
6.3.2.2
Dispersion Compensating of Multiple DWDM Channels
and Slope Dispersion Compensation
The FSR can be designed to equate that of the wavelength spacing of the WDM channels so that the resonance occurs at the wavelengths of the WDM channels, hence dispersion compensation can be implemented. We can imagine the multiple dispersion curve of an MRR is shown in Figures 6.5 and 6.6.
Thus, using only a single MRR, this allows the compensation of the WDM channels whose
spacing is equal to the FSR of the MRR. The GVD and dispersion factor Γ(ω ), d Γ(ω ) , respectively,
dω
are given by
2
 κ + t + 2 tκ sin (ωτ ) 
Ec
=
 ...assuming..lossless..coupler
Ein
 1 + κ t + 2 tκ sin (ωτ ) 

 tκ + tκ sin (ωτ ) 
t cos (ωτ ) 
E
−1
Phase(ω ) − of − TF = c = − tan −1 
 − tan  −

Ein
cos (ωτ )
 t + 2 κ sin (ωτ ) 


E
d c
 κ + tκ sin (ωτ )
Ein
1 + tκ sin (ωτ ) 
GVD :→ Γ(ω ) = −
=τ 
−

dω
 κ + t + 2 tκ sin (ωτ ) 1 + κ t + 2 tκ sin (ωτ ) 
Transfer-function → TF (ω ) =
Dispersion-facttor →


t −κ
Γ(ω )
1 − tκ

= τ 2 tκ cos (ωτ ) 
+
 κ + t + 2 tκ sin (ωτ ) 2 1 + κ t + 2 tκ sin (ωτ ) 2 
dω


(6.1)
(
) (
)
252
Photonic Signal Processing
(a)
(b)
FIGURE 6.6 Dispersion GVD as a function of wavelength of a SINGLE Si-based waveguide MRR of a
dispersion factor of 800 ps/nm over 0.1 nm band or 12.5 GHz (a) Single channel (b) Multiple channel.
With κ ,t, the power cross coupling coefficient of the coupler and the transmittance of the loop,
respectively. The coupler coefficient is a coupling parameter of a [2 × 2] coupling matrix, in general.
Later we will see an effective coupling coefficient κ C representing a [2 × 2] MZDI with a phase
shifter to tune the coupling coefficient of the resonator. ωτ = β L is the total phase of the lightwaves
after circulating through the complete length L, and β is the propagation constant of the guided
waves by which the effective refractive index of the guided mode can be derived. t is the transmittance of the loop L, and κ is the intensity coupling coefficient of the coupler.
The output power is depleted at a minimum (notched filtering of the THRU port) when
t = κ ; and ωτ = 2qπ −
π
2
(6.2)
This leads to the periodic filtering of the MRR. The spectral spacing between the two-filter central
frequency/wavelength is defined as the FSR. This allows simultaneous dispersion compensation of
WDM channels. By determining the channel spacing of the channels, we can determine the length
of the resonant loop. Thus, the FSR can be estimated by
FSR =
2n L
1
c
→τ =
= g
2ng L
FSR
c
(6.3)
Slope dispersion exists in a DM channel propagating through SSMF. This is well known and can
be dealt with by designing slightly different MMR with a combination of a MZDI, whose length
Optical Dispersion in Guided-Wave FIR and IIR Structures
253
(a)
(b)
FIGURE 6.7 Modified MRR by incorporating a MZDI with phase control (a) combined MZDI and MRR;
(b) combined MZDI+MRR represented by a single loop ring resonator in which the MZDI is simplified by a
[2 × 2] port optical network with ϕ m the tuning phase applied to the MZDI to tune the resonant curve. The ring
is in resonant by tuning the phase Φm of the ring, the phase shifter denoted by PS.
difference of the two arms of the MZ interferometer is much smaller than that of the resonator loop
as shown in Figure 6.7.
6.3.3
tunable DispersiOn cOmpensatOr
By observing the GVD and dispersion factor of a single ring operating under different coupling
coefficient of the coupler inside the MRR and the transmission coefficient (transmittance, t) of the
loop of the MRR, we can see that the dispersion can be tuned by changing the coupling coefficient
from 0.575 to 0.275. This can be done easily by tuning the phase of the MZDI. This coupling coefficient κ C is given by
∆L
1

κ C = 1 − 4κ (1 − κ ) cos2  2π ng
+ Φm 
λ
2

(6.4)
where the waveguide coupling coefficient is κ , ∆L is the length difference of the two arms of
the MZDI structure, ng is the effective index of the guided wave or β k0 (k0 is the wave number
in vacuum), Φ m is the phase shift applied to the electrode of an MZϕDI arm, and λ is the central
wavelength of the lightwave channel. Thus, we can see that the coupling coefficient of a [2 × 2]
equivalent coupler of a MZϕDI (phase tune MZDI) can be tuned by tuning the phase shifter, and
the dispersion factor can be tuned to match the length of the SSMF.
The FSR can be designed to be 100 GHz with a resonant loop length of 3 mm. It is to be optimized by detailed design. In particular, the transmittance of the waveguide must be as high as
possible or the loss needs to be within 5%–10%, which is achievable by selecting Si-waveguide
fabrication platform.
Indeed, the resonance peak of the modified MRR (MZDI-MRR) shown in Figure 6.7 can be
tuned so that the resonance transmittance can be slightly variable from one FSR location to the other.
254
Photonic Signal Processing
6.3.4
length OF Fiber prOpagatiOn anD DispersiOn cOmpensating mODule
It is expected that a short length unit (in 10 km length of SSMF) can be compensated by one
single MRR. Longer length SSMF can be designed using single MRR. Currently, we can see that
the loss of a single MRR can be about 0.8 dB. Thus, longer SSMF length can be compensated by
cascading a single ring MRR, as shown in Figure 6.8. Two MRRs can also be possible by designing another modified MRR to minimize the total size of the optical dispersion compensating
module (oDCM).
FSR =100GHz;
D(1550nm) = 40x17ps/nm=680 ps/nm
κ1=0.757; κ2 =0.843 κ3 = 0.608; κ4 =0.5; φ1 = −0.772 rad; φ2=0.090; φ3=−1.393; φ4 = 0.01 rad.
4xMRR 620ps/nm operang under the above condion of phase and coupling coefficient (Power).
This dispersion factor can be tuned to obtain exact dispersion compensang.4xMRRs loss ~ 3.2dB
assuming Si /SOI @ 1.5dB/MRR
Passband of each MRR is 50GHz for 56GBaud channels /MRR – each MRR is to compensate for 10Km
SSMF. Note the power coupling coefficients can be modified by using MZφDI structure [2x2] port
coupling to improve and tune the dispersion factor, i.e. tuning the GVD slope.
Core size area = 3.5 µm2; coupler length = 480 µm; radius of curvature~ 1000 µm; loop length =
3000/2.72 ~ 1000 µm (for 100GHz FSR). Wide width Si waveguide technology. 2.72 = ng effective
(a)
refractive index of lightwaves propagating in the Si-SOI waveguide.
(b)
FIGURE 6.8 Cascade of 4 MMR to dispersion compensating of 40 km SSMF in the C-band. Coupler is an
equivalent MZϕDI (a) no scalable with possible designed parameters of MRR; (b) scalable via use of optical
switches to connect desired MRR for compensating appropriate SSMF length. ∆φm is the phase shift to be
applied to the heating electrode of the loop of the MRR so that resonant frequency can be tuned. Each MRR
is assumed to compensate for a dispersion amount of 10 km SSMF or 170 ps/nm.
255
Optical Dispersion in Guided-Wave FIR and IIR Structures
6.3.5
waveguiDe anD passive mrr FabricatiOn technOlOgy FOr ODcm
We must select the fabrication technology to implement the prototype and eventually the production
of the oDCM so that it can satisfy:
1. CMOS Compatible technology to achieve low cost mass production.
2. Wider waveguide Si size to reduce loss in order to achieve accurate transmittance of the
MRR loop.
3. Coupling from fiber to waveguide with low loss.
4. Precision in fabrication of coupler sections so that the coupling coefficient is achievable
within tolerance.
5. Waveguide with tightly confined guided mode and minimum propagation loss with tight
radius of curvature. FSR = 100 GHz leads to a loop length of about 3 mm. Thus, small
radius of the curvature should be considered. SOI Si-integrated photonic CMOS compatible technology should offer the high possibility for low cost production, for example 3 or
5 µm Si ridged waveguide width.
6. Consideration for integrating these oDCM in Si with low cost DFB lasers on SOI integrated photonic module at the transmitter side or receiver side.
7. Different modules of oDCM can be implemented with scalable, on-scalable, or dispersion
slope compensation depending on the application scenarios of oDCM placed in optical
transmission in metro or access networks.
8. Finally, before producing prototype we must select which laboratory to engage into prototype. I prefer more a commercial company such as VTT. UPV can be considered as an
associate in this process to ensure correctness in design flexibility. We will do detailed
designs so that the intellectual property (IP) is kept within Huawei and share the IP with
other partners.
6.4
ACTIVE MRR
6.4.1
structure
An active MRR can be considered in which a gain medium is inserted in the loop to equalize the
transmission loss of the loop. Normally, an semiconductor amplifier (SOA) is integrated to obtain
the gain if erbium-doped amplification cannot be used for C-band operation. Such an active MRR is
shown in Figure 6.9. A gain section is used to compensate for the loss of loop transmission.
Phase to tune resonance condition
G = SOA/EDFA gain
section
FIGURE 6.9
Active MRR, Gain section is incorporated.
256
6.5
6.5.1
Photonic Signal Processing
oDCM BY FIBER BRAGG GRATING
mOtivatiOn
Fiber gratings are considered key components in optical communication links as filters,16 gain
flatteners,17 and dispersion compensators.18 Fiber Bragg gratings (FBG’s) are very attractive components because, as well as being passive, linear, and compact, they possess strong dispersion in
both reflection and transmission. In reflection, the dispersion arises when the edge of the band gap
varies with the axial position along the grating, such as in linearly chirped or ramped gratings.18,19
Different wavelengths in a dispersed pulse are reflected at different positions in the grating, leading
to different optical path lengths and, thus, providing the possibility of compensating for dispersion
in long-haul fiber links.20,21 We note that this thinking was proposed in the late 1990s when no
DSP-receivers were available. Under current long-haul transmission, coherent transmission with
DSP assisted processing to compensate for dispersion has been extensively exploited. However, in
metro and access optical networks, the employment of the DSP consumes lots of power and has
high costs. Thus, FBG, as a dispersion compensator, can be considered very attractive as a plug-in
component. It can be placed at the end of a SSMF so that it can be considered as a dispersion less
SSMF length cable.
In order to minimize the cost of the oDCM, the transmission mode FBG (TM-FBG) is to be
considered in which no circulator is required in contrast to that of the reflection-mode FBG. The
TM-FBG in such an optical transmission link is shown in Figure 6.10 in which the pulse is broadened after propagating through an SSMF and then compressed through the TM-FBG.
FIGURE 6.10
Pulse broadening via fiber transmission and compression via FBG in transmission mode.
G. Meltz, W. W. Morey, and W. H. Glenn, Formation of Bragg gratings in optical fibers by a transverse holographic
method, Opt. Lett., 14, 823, 1989.
17 A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, and P. J. Lemaire, Long-period fiber-grating-based gain equalizers,
Opt. Lett., 21, 336, 1996.
18 F. Ouellette, Dispersion cancellation using linearly chirped Bragg grating filters in optical waveguides, Opt. Lett., 12,
847, 1987.
19 T. Stephens, P. A. Krug, Z. Brodzeli, G. Dhosi, F. Ouellette, and L. Poladian, 257 km transmission at 10 Gbit/s in nondispersion-shifted fibre using an unchirped fibre Bragg grating dispersion compensator, Electron. Lett., 32, 1599, 1996.
20 P. A. Krug, T. D. Stephens, G. Dhosi, G. Yoffe, F. Ouellette, and P. C. Hill, Dispersion compensation over 270 km at
10 Gbit/s using an offset-core chirped fibre Bragg grating, Electron. Lett., 31, 1091, 1995.
21 W. H. Loh, R. I. Laming, N. Robinson, A. Cavaciuiti, F. Vaninetti, C. J. Anderson, M. N. Zervas, and M. J. Cole,
Dispersion compensation over distances in excess of 500 km for 10-Gb/s systems using chirped fiber gratings, IEEE
Photon. Technol. Lett., 8, 944, 1996.
16
Optical Dispersion in Guided-Wave FIR and IIR Structures
6.5.2
257
analytical expressiOn OF brOaDening (Fiber)
anD cOmpressiOn (tm-Fbg) FactOrs22
6.5.2.1 Dispersion-Induced Pulse Broadening in Optical Fiber
First consider the problem of optimizing a uniform grating for compression of an initially transform-limited pulse of 1/e-width, broadened by propagation through optical fiber of length with dispersion, as illustrated in Figure 6.10. We only consider the second-order fiber dispersion, ignoring
higher order fiber dispersion. We also ignore any losses and nonlinear effects in both the fiber and
the grating.
A Gaussian pulse maintains its shape with propagation through the fiber, but its width increases
due to group velocity dispersion. The 1/e-width after propagating a distance is related to the initial
width by the relation23
( L β ) = pulse_thru_SSMF
τ = τ 1+
f
2
f
f
0
2
τ 04
(6.5)
and the pulse becomes chirped, so that the frequency changes linearly across the pulse with chirp
parameter. The subscript or superscript f indicates the parameter of the “fiber – SSMF.”
αf =
Lf
2

 τ2  
β  L2f +  0f  

 β 2  
(6.6)
F
2
6.5.2.2 Dispersion-Induced Pulse Broadening in FBG
Now consider the propagation of this chirped Gaussian pulse in a FBG. In the grating, cubic dispersion is significant and cannot be ignored. To describe propagation through a Bragg grating,
there needs to be consideration for the dispersion relation, which is the relationship between the
frequency detuning parameter12
δ=
ne
(ω − ωB )
c
(6.7)
and the effective propagation constant γ = β g − β B, which is the non-matching wave number
between the effective propagation constant of the guided waves in the FBG and that of the fiber
grating at Bragg resonance; ω is the optical angular frequency and ωB is the resonant Bragg angular
frequency. ne denotes the effective refractive index of the guided mode as it propagates through the
fiber or the fiber grating.
The dispersion relation is obtained by inserting plane wave solutions into the coupled-mode
equations, yielding24
δ 2 = γ 2 +κ2
(6.8)
N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, Fiber Bragg gratings for dispersion compensation in transmission:
Theoretical model and design criteria for nearly ideal pulse recompression, IEEE J. Lightw Technol., 15, 1303, 1997.
23 G. P. Agrawal, Nonlinear Fiber Optics. New York: Academic, 1995.
24 J. E. Sipe, L. Poladian, and M. de Sterke, Propagation through nonuniform grating structures, J. Opt Soc. Amer. A, 11,
1307, 1994.
22
258
Photonic Signal Processing
Operating region
for oDCM-FBG
Periodic medium
(Continuous)dispersive
Band gap
No-oDCM κ
-dependent
δ0
INFINITE Uniform
medium (dotted) –
no dispersion
FIGURE 6.11 Photonic band gap for infinite uniform Bragg grating. Operating region over which the slope
is nonlinear for dispersive operating.
πη∆n
in which κ = λ is the coupling coefficient, defined by the FBG refractive index modulation, ∆n,
B
and the fraction of energy contained in the FBG fiber core, η , and λB is the Bragg wavelength.
The dispersion curve of (6.8) can be illustrated in Figure 6.11, for both a uniform medium
(dashed line) and a periodic medium (solid line). Recall that the group velocity dispersion is given
by the curvature of the dispersion relation. Therefore, by observing the curves given in Figure 6.11,
for the uniform medium the slope is constant, and thus the dispersion is negligible. By introducing
a grating, the dispersion relation is modified, with the most noticeable feature being the “photonic
band gap” in the dispersion relation that corresponds to the range of detuning −κ < δ < +κ . Over this
detuning range the guided lightwaves can not propagate through the grating but strongly reflected.
Optical pulses detuned outside of the gap δ > κ can propagate, but at velocities that can be substantially less than the speed of light in the uniform medium. The reduced velocity can be understood in terms of the multiple reflections at the grating periodic variations with a resulting path length
increase. The group velocity of the modulated signals propagating through the FBG, v g = nce 1 − κ 22 ,
δ
vanishes at the band edge and asymptotically approaches c ne far from the Bragg resonance. The spectral range over which this occurs is roughly equal to the bandwidth of the passband of the grating.
For the fiber gratings under our considerations, for metro and access networks (e.g., 25GBaud signals), this is of the order of a few nanometers. This extreme variation in the group velocity over such
a small range of wavelengths leads to strong group velocity dispersion. On the upper branch of the
band gap, the sign of the dispersion is positive corresponding to anomalous dispersion. Similarly,
we can state that the normal dispersion exhibited by the grating is located on the lower side of
the photonic band gap, which can be used to compensate for anomalous dispersion effects caused
by fiber propagation at C-band or upper O-band of the communication wavelength windows.
The effects of this grating dispersion can be accounted for mathematically by expanding the
propagating constant β g through the FBG (the superscript/subscript g stands for the FBG) in a
Taylor series about δ 025 as
2
β g (δ ) = β 0g +
3
c g
1 c 
1 c 
β1 (δ − δ 0 ) +   β 2g (δ − δ 0 )2 +   β 3g (δ − δ 0 )3 + ...
n
2n
6n
(6.9)
in which β ng is the nth derivative of β g with respect to evaluated at δ 0 . We include only the terms to
the third order of the Taylors series expansion.
25
E. Peral and J. Capmany, Generalized bloch wave analysis for fiber and waveguide gratings, IEEE J. Lightw Technol.,
15(8) 1295–1302, 1997.
259
Optical Dispersion in Guided-Wave FIR and IIR Structures
The term β 2g as the (quadratic) group velocity dispersion and the cubic dispersion, or dispersion
slope of the FBG, can be witten by
2
κ2
n 
β 2g = −  e 
3 / 2 .sign(δ )
 c  δ 2 −κ2
(
)
3
κ 2δ
n 
β3g = −3  e 
5/ 2
 c  δ 2 −κ2
(
(6.10)
)
Thus, for FBG as an indeal oDCM then we must have
LFBG β 2g = LSSMF β 2F
(6.11)
where β 2 is the quadratic dispersion of the FBG given by (6.10) and β 2F is the quadratic dispersion
of the fiber. An ideal dispersion compensator recompresses the fiber dispersion-broadened pulse to
its transform-limited pulse width. Of course, the cubic dispersion of the grating acts to diminish the
efficiency of compression and to distort the pulse, and thus must be as small as possible for the grating to approach ideal behavior, that a constant second order dispersion or matching to the dispersion
slope of the fiber when multiple λ channels are to be simultaneously compensated.
We can now model the grating structure by a highly dispersive uniform medium with the quadratic dispersion and cubic dispersion. Thus the wave equation can be employed instead of solving coupled-mode equations to study pulse propagation in the grating. As we can neglect fourth
and higher order dispersion, the slowly varying amplitude of the pulse envelope satisfies the
Schroedinger equation
g
 ∂
β 2g ∂ 2 β3g ∂ 3 
−
 +j
 E ( z ,τ ) = 0
2 ! ∂τ 2 3! ∂τ 3 
 ∂z
(6.12)
where z is the axial position along the grating and j = −1, and the sign “!” indicates the factorial.
Following Marcuse26 and Akhmanov,27 the compression ratio C, defined as the ratio of the input
to output (root mean square) RMS pulse widths, can be found analytically for the case including
quadratic and cubic dispersion
2
2
g

2
 β 2g z 

2 4  β3 z  
g


C =  1 − α f β 2 z +  2  + 1 + α f τ f   f  
 2τ 3  
 τf 

(
)
−1/ 2
(6.13)
We note that the RMS pulse width gives a good representation of the actual pulse width if the
pulse shape is nearly Gaussian, but for non-Gaussian pulses with long, asymmetric leading or
trailing edges, the RMS width can be much larger than the full-width at half maximum (FWHM)
width. It is found that the compression ratio reaches its maximum at zopt = Lopt along the FBG
given by
26
27
D. Marcuse, Pulse distortion in single-mode fibers. 3: Chirped pulses, Appl. Opt., 20(20), 3573–3579, 1981.
S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses. New York: AIP, 1992.
260
Photonic Signal Processing
L f β 2f
β 2g
Lopt =
2


 βg 
τ 04

1 +  3g  + 
2
 2β 2 τ 0   τ 04 +  L f β 2f  


(6.14)
And the coupling factr Copt equals to
1/ 2
2

 L f β 2f  

 2  
τ0  

Copt = 1 + 
2
 β3g  

 1 +  2β gτ  
 2 0 

(6.15)
Note that the optimum length of the grating depends on detuning δ . If we now fix z = Lopt , then
from (6.5), (6.6) and (6.13) the compression ratio C at optimum dispersion of the grating can be
seen varying as
2
2

2
 β 2g Lg    β3g   

g
C =  1 − β 2 Lgα f +  2  + 1 +  g   
 τ f    2β 2 τ 0   

(
6.5.3
)
−1/ 2
(6.16)
Design cases
6.5.3.1 Design Case I: Finite Uniform Profile Grating
The propagation constant of the guided waves through the FBG can be estimated as given in
Figure 6.12 for different tuning parameters of the grating from the resonant position.
FIGURE 6.12 Estimation of the FBG propagating constant versus the grating coupling parameter with the
detuning factor as a parameter. FBG parameters: ∆δ = 5 cm −1(Solid ); 10 cm −1(long-dash); 20 cm −1(Short-dash).
Optical Dispersion in Guided-Wave FIR and IIR Structures
261
FIGURE 6.13 The calculated group delay for a uniform FBG (Lg =1 cm and coupling coefficient κ = 5 cm−1)
in transmission (transmittance- solid line) and the delay in transmission (dashed line), ignoring end effects as
a function of detuning parameter ∆δ after Ref.[22].
FIGURE 6.14 The calculated delay in transmission (solid line) for an apodized FBG (Lg = 1 cm and its
coupling coefficient κ = 5 cm−1) and transmission spectrum (transmittance dashed line) as a function of
detuning parameter ∆δ.
The transmission and group delay of the guided wave channel propagating through an FBG of
finite uniform and apodized FBG are given in Figures 6.13 and 6.14, respectively. It is observed
from these displays that apodization can effectively removes the side-lobes in the reflection spectrum of a uniform grating by allowing the index modulation to vanish smoothly at each end. This
can largely remove the Fabry–Perot reflections, which occur at the boundary between the grating
and the surrounding end medium, thus suppressing the side-lobes and the oscillations in the delay
spectrum.
Thus, apodized-profile FBG should be used or, alternatively, chirped (one-sided apodization) can
be relaced in order for transmission oDCM by the FBG. In the limit that is ∆δ << κ , then we can be
approximated by
262
Photonic Signal Processing
2
κ 1/ 2
n 
β 2g ≈ −  e 
3/ 2
 c  ( 2∆δ )
or_alternatively
1n 
∆δ =  e 
2 c 
4/3
κ
1/ 3
 Lg 
 f 
 β2 L f 
2/3
(6.17)
by using the condition for complete compensating condition between the fiber and the FBG.
Example: Consider a uniform grating with κ = 50 cm −1 and length Lg = 10 cm, for compensating
dispersion in a fiber link of length L f = 100 km. We consider a pulse with 40 ps (or 25 Gbps) at 1550 nm
where the dispersion is −20 ps2 /km. Solving the optimum FoM M3 numerically gives an optimum
detuning offset of 9.09 cm−1, corresponding to Region II in Figure 6.15. This suggests that, although
quadratic dispersion dominates cubic dispersion, the higher order effects are not negligible in this
case. We now model the grating characteristics numerically. Shown in Figure 6.16 are the results of
recompression via transmission through the grating designed above. The transform-limited pulse is
broadened to 144 ps (FWHM) upon propagation through the dispersive fiber and compressed back
to approximately 46 ps upon transmission through the 10-cmlong grating. The pulse is retarded
in time by approximately 900 ps, indicating that the pulse propagates along the length of the grating at an average velocity of 53% of the speed of light in the uniform medium. The reduced peak
intensity is associated with out of band reflections and residual cubic dispersion; note that the peak
intensity of the compressed pulse is approximately 40% of the transform-limited pulse. Figure 6.17a
and b show, respectively, the compression ratio and peak intensity versus the central detuning of
the initial pulse. Note that the compression ratio peaks at δ opt = 59.09 cm −1, consistent with our
analytical results. For detunings close to the band gap, dispersion orders higher than two become
large and reduce the efficiency of compression. For detunings, the group velocity dispersion is too
small, so that the grating length must be increased (for fixed) to satisfy condition (6.8). Of interest,
the bandwidth corresponding to compression ratios are greater than two is 5 cm−1. The bandwidth
can always be improved by either making the grating longer or stronger.
The optimal compression ratio versus M3 for L f = 100 km and τ 0 = 24 ps. The figure of merit
βg
3n 1
.
M3 is given by (when ∆δ << κ ) M 3 = g3 ≈ e
β 2 τ 0 2τ 0 c ∆δ
FIGURE 6.15
Optical Dispersion in Guided-Wave FIR and IIR Structures
263
FIGURE 6.16 Initial, broadened and recompressed pulses after transmission through 100 km of fiber and
10 cm uniform grating with κ = 50 cm−1 and ∆δ = 9:09 cm−1 and recompressed pulse (after FBG) superimposed on transform limited pulse.
FIGURE 6.17 (a) The compression ratio as a function of for a 40-ps pulse broadened to 144 ps by propagation through 100 km of fiber and recompressed via transmission through a 10-cm long uniform (triangles) and
apodized (squares) grating with κ = 50 cm−1, compared with theoretical curve (circles). (b) The peak intensity
of recompressed pulse normalized to the intensity of initial transform-limited pulse as a function of detuning
parameter δ . (Extracted from Litchinitser, N.M. et al., IEEE J. Lightw Technol., 15, 1303, 1997.)
6.5.3.2 Design Case II: Apodized Profile Grating
Now we consider the same fiber system as in Example I (τ 0FWHM = 40 ps(25G );
τ FWHM
= 144 ps at 100 km SSMF), but increasing the grating parameters to κ = 100 cm −1; Lg = 20 cm ).
f
Such strong gratings have been fabricated in laboratories, exhibiting index modulations as large as
0.0128). For this grating, the calculated value of M 3 = 0.17, corresponds to Region III in Figure 6.15
28
V. Mizrahi et al., Ultraviolet laser fabrication of ultrastrong optical fiber gratings and of germania‐doped channel waveguides, Appl. Phys. Lett., 63, 1727, 1993.
264
FIGURE 6.18
Photonic Signal Processing
The ratio of the transform-limited pulse width to the recompressed pulse width for Example II.
FIGURE 6.19 Compression ratio versus detuning parameter δ for Example II: numerical results (squares)
and theoretical values (circles).
where nearly perfect compensation can be expected. This near-ideal behavior is demonstrated in
Figure 6.18, where we plot the ratio of the transform-limited pulse width to the recompressed pulse
width predicted theoretically for this case. Here, we observe that the ratio approaches unity at the
optimum detuning, δ = 117.37cm −1. Figure 6.19 compares the theoretical and numerical results for
the compression ratio. Here we note that the bandwidth corresponding to compression ratios which
are greater than two is approximately 1.6 times greater than that for the weaker, shorter grating
of Example I. If we now apodize this 20-cm long grating (κ = 100 cm −1 ), it is found that the peak
intensity increases to 95% of the initial peak intensity, so that the insertion losses in the system are
less than 0.25 dB. This indicates that almost perfect recompression can be obtained using longer and
stronger apodized gratings in transmission system.22
6.5.3.3 Remarks on FBG–oDCM
FBG can operate in transmission mode to compensate for dispersion over long distance but it is
not being possible to be integrated with Si-based transmitter or receiver platform. Furthermore, the
Optical Dispersion in Guided-Wave FIR and IIR Structures
265
FBG must be tuned by stretching mechanically, thus the aging effect is very serious. We thus should
preserve the solution of FBG for long haul coarse DCM only and possibly integrated with MRR
oDCM module for fine tuning.
6.6
FIR DISCRETE WAVELET TRANSFORM 2D
DISPERSION COMPENSATING
6.6.1
intrODuctOry remarks
In Chapter 8 we will give a detailed design of photonic signal processing in multi-dimensional
discrete transform. This section gives a practical design of dispersion compensation using 2D discrete transform, the discrete wavelet transform (DWT), which is commonly applied to wavelets for
multiband filtering, because the wavelet filter can be a type of half-band filtering device (HBF).29
The motivation to introduce this section is
• Highly stable and linear group delay, thus constant broadening factor over wideband signals are critical for ultra-high-speed channels over short reach for inter or intra-data center
links.
• IIR (Infinite Impulse Response) filters type are important to offer high stability. This type
of filters can be straight forward implemented by cascading of MZDI of only two delay
paths.
• Flat-band, low-loss, constant dispersion over the whole passband filters are required for
dispersion compensation.
• Few channels are required for Tera-bits/s channel Bandwidth (DMT – 35 GHz for
200 Gbps/channel.
• Highly stable and linear group delay, thus constant broadening factor over wideband signals are critical for ultra-high-speed channels over short reach for inter- or intra-datacenter
links.
• IIR (Infinite Impulse Response) filters type are important because they offer high stability.
This type of filter can be easily implemented by the cascading of MZDI of only two delay
paths.
• Flat-band, low loss, constant dispersion over the whole passband filters are required for
dispersion compensation.
• Few channels are required for Tera-bits/s channel Bandwidth (DMT – 35 GHz for
200 Gbps/channel, especially for short haul links such as in DCN (data-center networks,
DCI (DC interconnect).
6.6.2
analysis anD synthesis
The general schemes for DWT and WP (wavelet packets) decomposition of a digital optical signal
has been described in,30 considering chains or trees of wavelet filters. The aim of this section is
to furnish the synthesis method of optical wavelet filters using planar lightwave circuits (PLC) or
Si-on-insulator (SOI) photonics two-port lattice-form architectures. We can show that wavelet filters
(WF) can be realized according to the synthesis procedure of power half-band (HB) filters proposed
by Jinguji et al.,29; however, since wavelet filters are quadrature mirror filters (QMFs), their design
guidelines are simplified and the circuit parameters can be evaluated directly from the knowledge
of the filter coefficients.
29
30
K. Jinguji and M. Oguma, Optical half-band filters, IEEE J. Lightw. Technol., 18, 252–259, 2000.
G. Cincotti, Fiber wavelet filters, IEEE J. Quantum Electron., 38, 1420–1427, 2002.
266
Photonic Signal Processing
Given an input sampled sequence s( n) representing the discrete time transfer function, it can
be decomposed into two parts, a recursive discrete convolution with a lowpass h[k ] and a highpass
g[k ] filter. Notably, the delay time difference represents the sampling time in the optical domain
equivalent to that of the switching or sampling in the electronic domain. Thus, for one-unit delay,
it is considered a fundamental sampling rate, twice the delay time is considered half of the sampling rate, and one is half of the rate of the other (or by a factor of 2). The sampling and detailed
coefficients of the system are given by31
c 1[n] =
∑ h[2n − k ]s[k ];
scaling_coefficients
∑ g[2n − k ]s[k ];
details_coefficients
k
d 1[n] =
(6.18)
k
The coefficients at the resolution of 2l (l > 1) can be iteratively regenerated from the scaling coefficients at resolution 2l −1 as
cl [n] =
∑ h[2n − k ]c [k ]
l −1
k
dl [n] =
∑ g[2k − n]c [k ]
(6.19)
l −1
k
Thus, in the recovery of the input sequence, an inverse filtering process can be employed to obtain
from the distorted sequence as
s′[n] =
∑ h[2k − n]c [k ] + g[2k − n]d [k ]
1
1
(6.20)
k
The scaling and detail coefficients are an orthogonal projection of the sequence s[n] onto two complementary sub-spaces Vl ,Wl, which are a spanned and scaled version of the scaling and wavelet
functions given as
ϕl ,k = 2− l / 2ϕ (2− l t − k ); scaling_function
ψ l ,k = 2− l / 2ψ (2− l t − k ); wavelet_function
(6.21)
These two functions satisfy the dilation relation
ϕ l (t ) = 2
∑ h[k ] .ψ [2t − kτ ]
k
ψ (t ) = 2
∑ g[k ] . ϕ[2t − kτ ]
(6.22)
k
where τ = 1 FSR ; FSR = free spectral range, that is the repeatable spectral window of the filter.
31
S. Mallat, A theory for multi-resolution signal decomposition: The Wavelet representation, IEEE Trans. Pattern Anal
Mach Intell., 11, 674–693, 1969.
267
Optical Dispersion in Guided-Wave FIR and IIR Structures
In the corresponding frequency domain, we have the filters transfer functions H (ω ) − and − G(ω )
given as
H ( z) =
1
( H1( z ) z −1 + G*1( z ) z −2( N −1)
2
G( z ) =
1
( H1( z ) z −2( N −1) − G*1( z ) z −1
2
(6.23)
So from (6.20), the composite transfer function S ( z ) and the S1( z ) in the sampling z-transform
domain can be given as
 H ( z)
S ( z) = 
 − jG( z )
− jG* ( z ) z −2( N −1) 
 ; and
H * ( z ) z −2( N −1) 
 H1 ( z )
S1( z ) = 
 jG( z )
jG*1( z ) z −2( N −1) 

H1* ( z ) z −2( N −1) 
(6.24)
With
(
)
(
)
H1 ( z ) =
1
H ( z ) z + G* ( z ) z −2( N −1)
2
G1( z ) =
1
H * ( z ) z −2( N −1) − G( z ) z
2
(6.25)
With j as the complex number. The quadrature mirror functions and half-band filters H (ω ); G(ω )
are interpreted as finite impulse response (FIR) filters as
M −1

1

π
jωτ
*
h[k ]e − jkωτ 
 G(ω ) = e H  ω + τ 
2 k =0




→
 sa
2
atisfy
M −1
1
 H (ω ) 2 + G(ω + π ) = 1

G(ω ) =
g[k ]e − jkωτ 

τ
2 k =0

H (ω ) =
∑
∑
(6.26)
Which satisfy the energy conservation conditions when lossless
2
2
H (ω ) + G(ω ) = 1
(6.27)
In the following, the synthesis procedure of a wavelet filter, using the SOI technology. An arbitrary FIR digital filter of length M can be synthesized in a two-port lattice-form configuration
using M guided optical delay lines (ODL), M phase shifters (PS), and M +1 directional couplers
(DC) as given in Chapter 10. If the FIR (finite impulse response) filter satisfies the HBF property,
the optical realization has about 1/2 elements of a conventional FIR filter. Jinguji et al.29 shows
that an HB filter of length M = 2 N can be realized using a Mach–Zehnder interferometer (MZI)
with a differential path delay τ and ( N −1) MZDIs with delay 2τ , as shown in Figure 6.21a.32
The first directional coupler is a 3-dB coupler, and the value of phase shifter in the first MZI
is θ 0 = π 4 and the value of the phase shift is ϕ0 = 0 . The remaining circuit parameters can be
calculated applying a recursive algorithm on the transfer functions and of H1( z ), G1( z ) of S1( z )
where S1( z ) is the transfer matrix given by the product of the transfer matrices of MZDIs with
32
G. Cincotti, Full optical encoders/decoders for photonic IP routers, IEEE J. Lightw. Technol., 22, 337–342, 2004.
268
Photonic Signal Processing
delay 2τ as depicted in Figure 6.21b. In this section, it can be shown that the synthesis procedure
of QMF is immediate and that the circuit parameters are directly related to the filter coefficients.33
We note here that the coupler can be a directional coupler type or multi-mode interference (MMI)
coupler whose coupling characteristics are different. H ( z ) can be the lowpass and G(z) is highpass
filter type.
The DWT or WP decomposition of a digital optical signal is performed with chains or trees of
wavelet filters with unit time delays that increase in a logarithmic way; the unit delay τ of the head
filter coincides with the bit period of the incoming signal. The time-frequency resolution of a WP
decomposition depends only on the decomposition level and the wavelet filters, that is, the selection
and the number of the filter stages in each chain.30,33
Since wavelet filters are HB filters, the design guidelines of Jinguji et al.29 and Cincotti30,32 can
be used. The bar and cross transfer functions of the devices are the lowpass and highpass filters,
respectively, so we can write the total transfer matrix and the matrix in the form of Eq. (6.24). So
the transfer functions can be interested as given in Eq. (6.23). Thus, using the properties of the QMF
that H ( − z ) = −G* ( z ) z −( 2 n −1) the transfer functions of the fundamental sampling of Eq. (6.25) can be
rewritten as
H1 ( z ) =
1
( H ( z ) − H ( − z )) z
2
G1( z ) =
1
1 −2( N −1)
(G( − z ) − G( z )) z =
z
( H ( z ) − H ( − z ))
2
2
(6.28)
The subscript (*) indicates the para-Hermitian conjugation of H * ( z ) = H * (1 z ). The function H ( z )
can be split into even and odd terms as
H ( z) =
N −1
N −1

1 
h[2k ]z −2 k +
h[2k + 1]z −( 2 k +1) 


2  k = 0
k =0

∑
∑
N −1

 N −1
⇒ H1 ( z ) = 
h[2k + 1]z −2 k  ; G1( z ) = z −2( N −1)
h*[2k ]z + ( 2 k )


k =0
 k =0

∑
∑
(6.29)
To illustrate the wavelet of order M = 2, 4, and 6, Ref.33 has illustrated in Figure 6.20 these wavelets
in an optical domain.
6.6.3
Design prOceDures
For M = 4, the transfer function H1( z ) and G1( z ) in the second order delay is given as
H1( z ) = h[1] + h[3]z −2 
=
G1( z ) = h[2] + h[0]z −2 
h[2] h[1]
5π
⇒ θ1 =
=
h[3] h[0]
3
h[0] h[1]
π
tan θ 2 = −
=
⇒ θ1 =
h[3] h[2]
12
tan θ1 = −
(6.30)
With the phase shift at the coupler denoted as θ1 and θ 2 , the coupler of second order delay times is
shown in Figure 6.21.
33
M. S. Moreolo, G. Cincotti, and A. Neri, Synthesis of optical wavelet filters, IEEE Photonic Tech. L., 16, 1679.
269
Optical Dispersion in Guided-Wave FIR and IIR Structures
MMI
∆L1
MMI
2x2
π/4……………π/4
(a)
MMI
∆L1
∆L1
2∆L1
MMI
2x2
FIGURE 6.20
MMI
2x2
2x2
π/4……………5π/3…………………….π/12
MMI
(c)
2∆L1
MMI
2x2
(b)
H(z)
−jG(z)
2x2
MMI
2∆L1
MMI
2x2
2x2
2x2
π/4……………0.432π…………0.385π…………..0.067π
Optical circuits using MZDI for (a) M = 2 or Haar wavelet (b) M = 4 and (c) M = 6.
PS = ∆φ1
(a)
Ei
6 ]
∆φ0
MMI
2x2
MZDI0
3dB
∆L1
∆L2 + ∆L1
MMI
2x2
∆φN
Eo
∆L2 + ∆L1
2τ
2τ
MZDI2
MZDIN
θ1
θ2..........θ3..................................
...................
.......θN
MZDI0
3dB
(b)
τ
θ0
S1(z)
FIGURE 6.21 Layout of an optical compact-support Quadrature Mirror Filter: (a) Circuit configuration.
(b) Schematic circuit configuration. Legends: t = 1/FSR – FSR = free spectral range, φ0; θ = phase shift
between optical paths (PS) of asymmetric MZDI besides the phase difference as the sampling variable of unit
one or two and at the cross-coupling port of the coupler. H ( f ), G ( f ) are the real part and magnitude of the
complex part of the QMF transfer function. We note that the quadrature feature is already existed in couplers
either directional or MMI types.
The transfer function of the MZDI cascade can be interpreted as
E
sin ϕ sin N θ 

TF ( f ) = o ( f ) = − je − jN φ  cos( N θ + j
 = TF ( f ) TF ( f )
Ei
2 sin θ 

∆L2 − ∆L1
∆L + ∆L1
 cos ϕ 
ϕ=β
;φ = β 2
;θ = cos −1 

2
2
 2 
β = effective__prop_constant_of_guided-waves
2
⇒ TF ( f ) and TF ( f ) angle-phase( f )
where
c
c
; ∆L1 = ( m − 0.5)
; l , m = 2, 4, 6...2n = even_intergers
2ne f
2ne f
f 0 = center_operating_frequency_of_lightwaves; ne = effective_ref_index_guided-waves
c = light-vacum-velocity
W = channel_bandwidth ⇒ edges = lower_freq f L = f 0 − W / 2 and upper f H = f 0 + W / 2
∆L1 = (l + 0.5)
(6.31)
270
Photonic Signal Processing
The transmittance of the cascaded MZDI structure can be written as
2
T ( f ) = H ( f ) = cos2 N θ +
sin 2 ϕ sin 2 N θ
2 sin 2 θ
(6.32)
The phase and group delay of the cascaded MZDI HBF structure are given by
τ( f ) = −
1 d
1
H ′( f )
(arg( H ( f )) = −
ℑm
2π df
2π
H( f )
 2 sin θ cos ϕ cosθ sinψ sin ϕ 
N sinψ sin ϕ + sin N θ cos N θ 
−

sin θ
sin θ
1

 ne ( ∆L1 + ∆L2 )
τ( f ) = −
2
2 sin 2 θ cos2 N θ + sinϕ sin 2 N θ
c
(6.33)
(6.34)
N is the number of stages; ne is the effective ref-index of the guided mode along all paths of the
structure.
6.6.4
implementatiOn
The interleaved and flat top of the passband of HBF has been taken into account and designed by
VTT Finland.34 However, the dispersion or phase behavior of the filters is not considered and demonstrated. Figure 6.22a shows the flattened layout of the FIR HBF using 3 um Si on insulator technology incorporating MMI structures. Thus, the phase shift with the MMI is specifically designed
along with another phase shifter by thermal tuning along the path of the MZDI. The frequency
response of the filters at the output and its complementary ports in Figure 6.22a lower part indicates
the flat top and interleaved features of the filter. Two MMI couplers, at the input and output ports of
the fourth order filter, are seen clearly with the shift in each port of the MMI before interconnecting
in cascade with the next MZDI. Figure 6.22b depicts the implemented structure on the 3 um Si on
insulator. Figure 6.22c shows the implemented structure in a compact form, or spiral, with appropriate delay paths for MZDIs, and, hence, the frequency response of the filters with their passband
are quite flat.
Figure 6.23 shows the design and performance of the HBF filter structure of Ref.29 and, thence,
the simulated transmittance and group delay as function of frequency of cascade MZDI structure
with Figure 6.23a depicting structure with fundamental MZDI (single delay path in cascade with
double delay path; Figure 6.23b depicting the simulated transmittance of the output port and its
orthogonal and complementary port; and Figure 6.23c showing the group delay, hence, constant
dispersion factor.
It is noted that the linear group delay depicted in Figure 6.23 leads to a constant dispersion
over the whole band of the filter. Thus, unlike the FIR structure described in Section 6.6.1, we
can obtain only either positive or negative dispersion factors; the HBF type by cascade MZDIs
can offer only a dispersion factor of either positive or negative sign. However with either negative or positive dispersion property of the over whole passband of the filter can be highly unique
of such filter and dispersion compensation as one does not have to control precisely the exact
wavelength position of the transmitted channel. Thence, the complexity of control of dispersion
is the unique property of the FIR cascade MZDI for dispersion compensating device.
34
M. Cherchi, M. Harjanne, S. Ylinen, M. Kapulainen, T. Vehmas, and T. Aalto, Flat-top MZI filters: A novel robust
design based on MMI splitters, Proc. SPIE., 9752, 9 pages, 2016.
Optical Dispersion in Guided-Wave FIR and IIR Structures
271
(a)
(b)
(c)
FIGURE 6.22 (a) Cascade MZDI structure using MMI (2 × 2 and delay paths; simulated transmittance
(b) SOI 3 um waveguide structures and transmittance [courtesy of VTT Findland] and (c) spiral structure for
long MZDI structures with polarization independent curvature.
Figure 6.24 gives another MZDI structure for evaluating the group delay and, thence,
the dispersion as a function of wavelength, depicted in Figures 6.25 through 6.27, respectively.
In Figure 6.26, the dispersion is plotted over the entire region of 100 GHz spacing or the FSR of
the device. There are two anti-symmetrical sections of the dispersion factors that are in opposite
side. These two regions are separated by a band, which can be a guard band so no overlapping can
occur unlike the one cascade MRR given in Sections 6.3 and 6.4 where the dispersion crosses the
zero point. The dispersion obtained by an all-pass filter of the MZDI type is much flatter than that
from the MRR cascade structure as these photonic circuits can only have “zeros,” the interference
nature of optics.
272
Photonic Signal Processing
FIGURE 6.23 Simulated transmittance and group delay as function of frequency of cascade MZDI structure
(a) structure with fundamental MZDI (single delay path in cascade with double delay path (b) transmittance
and (c) group delay of the structure.
FIGURE 6.24 Dispersion design of all-pass filter by cascading MZDI.
273
Optical Dispersion in Guided-Wave FIR and IIR Structures
FIGURE 6.25 Group Delay versus wavelength of Cascade MZDI depicted in Figure 6.24.
FIGURE 6.26 Dispersion versus wavelength of Cascade MZDI given in Figure 6.24.
~80pm (20GHz),
D>100ps/nm
~80pm (10GHz),
D~400ps/nm
FIGURE 6.27 Dispersion factor as a function of wavelength.
274
Photonic Signal Processing
Further we observe that (i) Increase the quantities of MZI: larger dispersion compensation,
smaller bandwidth, flatter; (ii) For DWDM with 100 GHz spacing: the largest bandwidth is about
20 GHz, with dispersion compensation larger than 100 ps/nm.
In conclusion, the MZDI structures in cascade are more stable and adjustable as compared
with cascaded MRR structure. The biasing is also much better in the operation as the dispersion
compensator.
6.7
CONCLUDING REMARKS
In brief, we believe that this chapter has described the design and application of dispersion compensating structures employing IIR and FIR design techniques in integrated optic structures. Also,
their applications in optical channels of transmission links employed in metro or short haul transmission can be implemented using MRR and scalable DC by cascading of MRR as IIR filter structures, described in this document, should offer products for telecommunication network owners
and equipment manufacturers in short haul or metro optical transmission systems and networks.
Furthermore, the FIR structure employing the HBF techniques or the wavelet transform (WT)
design procedure offer a full band dispersion factor in contrast to only the half band of the IIR using
MRR. These FIR structures can be tuned via the phase shifted embedded in the MZDI in cascade
for from negative to positive dispersion and vice versa.
6.8 APPENDIX: DISPERSION COMPENSATION A HISTORICAL
VIEW OF DEVELOPMENT AND WHY MRR AS DCM
It is well known that since the invention of guiding lightwaves in dielectric waveguide in 1966 by
Kao and Hockham,35 optical transmission in the 1970s was through multi-mode fibers. Then, in
the 1980s, single mode optical fibers were widely recognized as the low dispersion and low loss
transmission medium, as the multi-mode interference could no longer be the severe effect. Optical
transmission was then installed by single mode with a span distance of 40 km with repeaters due to
limited optical power. Dispersion in SMF was studied but not critically considered until the invention of optical amplifiers EDFA in late 1989. Since then, the fiber loss was not that critical and many
reasonably lossy devices were considered to be revived such as ultra-high speed LiNbO3 optical
modulators. The dispersion in a single-mode optical fiber was then dealt with for long-haul transmission systems with span lengths of 80–100 km, in which dispersion compensating fiber (DCF)
was exploited.
However, the loss of DCF module to compensate for 80 km SMF is high, about 12 dB, due to its
smaller core diameter so as to achieve high waveguide. Thus, two EDFAs are used instead of one to compensate for the transmission fiber and one for the DCF module, which is about 20 km for every 80 km.
Dispersion shifted fiber (DSF) was also considered and designed for zero dispersion at 1550 nm
and low dispersion-factor in the C-band, but four-wave mixing (FWM) was creating severe cross
talks in adjacent channels as the phase matching for such four-wave interaction in non-dispersive
medium. Then, nonzero-DSF was designed so that there was some dispersion in the transmission
band so that no FWM, that is, the zero dispersion wavelength is place around 1510 nm and the
dispersion factor is only about 3 ps/nm/km at 1550 nm to a maximum of 6 ps/nm/km at the upper
limit of the C-band.
The operating bit rate in commercially installed transmission systems was only t 10 GBps
DWDM of 100 GHz spacing. R&D activities were carried out in several laboratories around the
world on dispersion compensation, especially for transmission bit rate at 40 Gbps NRZ and 80 Gbps
NRZ OTDM (optical time division multiplexing).
35
K. C. Kao and G. A. Hockham, Dielectric-fibre surface waveguides for optical frequencies, Proc. IEE, 113, 1151, 1966.
Optical Dispersion in Guided-Wave FIR and IIR Structures
275
Compact dispersion compensating module was developed firstly using optical fiber36 for an FSR
of 30 GHz and a 3-dB bandwidth of about 7.5 GHz. This is sufficient for 10 Gbps NRZ compensation in coherent transmission system. This was made by a polished fiber coupler and the looping the
fiber through the coupler of a micro-ring resonator (MRR).
The compactness and accurate length of an MRR can only be achieved in integrated optics.
We have seen several MRRs developed in integrated SiP platforms by Bell Labs.37,38
Note that the z-transform was also attracting a lot of interest in the design of dispersion compensator37,39 due to the fact that the lightwaves circulating through a resonator with repeatability of
FSR can be considered as sampling parameter z.
The design and demonstration of MRR by Bell Labs37,38 was only for 10 Gbps with a FSR of
25 GHz in the late 1999. Since then we have seen not much R&D on MRR as dispersion compensator due to the following reasons: The dot.com market collapse and the privatization of US
telecom industries. Bell Labs and Belcore were split and belong to AT&T and Lucent technologies.
The R&D in DCM was then abandoned.
The 10 Gbps was then upgraded to 28 Gbps NRZ to 100 GBps DP-DPSK coherent with DSP.
Dispersion compensation could be done via DSP algorithms. Note that the bandwidth of the baud
rate 28 GBps is only about 14 GHz. Thus the DCM has been ignored until the Metro low cost for
ultra-high capacity demands no–DSP and only FEC at Tx and Rx. Thus fibers SSMF terminated
with compact DCM module attract Huawei attention, thus this study and project. The cost of MRR
would be only a few USD with a fabrication cost of only few cents if Si-Integrate photonic (Si-IP)
CMOS-compatible technology is used.
Over the last 10 years Si-IP has been fast developed and several R&D works on MRR were published but none on DCM-MRRs no commercial companies have paid attention to low cost metro in
the global competition. Thus this is urgent for Huawei to enter into this direction for MRRR dispersion compensating module to make Huawei product more flexible. Flexibility in term of spectral
usage in the C-band rather than limits faced by O-band as EDFA can be used in critical system
reach and power.
It is noted that only FEC is allowable and no DSP is to be used in these metro and access application scenarios. The baud rate can reach 56GBaud and higher for 400 G, 800 G and 1,6 Tbps module.
Furthermore the bandwidth of the signal channels is no longer narrow in the 14 GHz but must be
at least 20 GHz to pass all bands of signals. Thus different design and considerations must be made
to demonstrate. Hence new classes of DCMs can be considered.
The uses of Si-IP allow us to integrate DCM into transceiver module either by pre-emphasis or
pre-receiving Ge-PD on Si waveguide. Thus the cost would be most nil in this integration.
Note that other DC device technologies were also investigated by many other research groups in
the following:
• FBG under long and short period with dispersion compensation of some tens of kms of
SSMF. This can also be considered as a competitive solution for 10–20 km with two FBGs
and a circulator – one for fine and one coarse compensating to tune exact compensating
of dispersion.
• Phase modulation of individual frequency components via a planar demultiplexing section
and reflection via LCOS as phase manipulating mirrors.
C. Caspar and E. J. Bachus, Fiber-optic Micro-Ring-Resonator with 2mm Diameter, Elect. Lett., 25, 1560, 1989.
G. Lenz and C. K. Madsen, General optical all-pass filter structures for dispersion control in WDM systems, IEEE
J. Lightw. Technol., 17, 1248, 1999.
38 C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, Integrated all-pass filters for tunable
dispersion and dispersion slope compensation, IEEE Photonic Tech. Lett., 11, 1623, 1999.
39 L. N. Binh, Photonic Signal Processing: Techniques and Applications, Boca Raton, FL: CRC Press, 2007.
36
37
276
6.9
6.9.1
Photonic Signal Processing
SFG AND MASON RULES FOR PHOTONIC CIRCUIT ANALYSIS
sFg anD masOn apprOach
Mason’s gain formula (MGF) is a method for finding the transfer function of a linear signalflow graph (SFG). The formula was derived by Samuel Jefferson Mason,40 whom it is also named
after. MGF is an alternate method to finding the transfer function algebraically by labeling each
signal, writing down the equation for how that signal depends on other signals, and then solving the
multiple equations for the output signal in terms of the input signal. Originally Shanon in 1948 has
developed this approach. MGF provides a step by step method to obtain the transfer function from
a SFG. Often, MGF can be determined by inspection of the SFG. The method can easily handle
SFGs with many variables and loops including loops with inner loops. MGF comes up often in the
context of control systems and digital filters because control systems and digital filters are often represented by SFGs.41 A typical SFG for an optical resonator with an optical feedback path is shown
in Figure 6.28.
Binh42 has applied this method into the derivations of transfer functions for photonic circuits,
especially integrated photonic circuits whether in planar or 3D waveguide structures. This method
is described in Chapter 2. In this section, a brief outline on this SFG (Signal flow graph) technique
is given to present the optical guided waves propagation in steady state and its transmittance in
optical fields and the transfer function in field term. Only the field terms are applied as it follows
all rules of algebraic linear functions. Thus the gain or loss are in field term note that commonly
the intensity or power terms (in dB) are normally measured and specified. Its corresponding
linear parameters must be obtained from the power term to apply into these SFG and Mason gain
formula. The Mason formula and procedure are used to simplify the derivation of the transfer
functions from any port input to output port, both in optical field or intensity transfer function
(Figure 6.29).
(a)
(b)
FIGURE 6.28 (a) Optical resonator (b) SFG representing the optical waves transmission paths and
connection j = −1.
S. J. Mason, Feedback theory-Further properties of signal flow graphs, Proc. IRE, 44, 920–926. doi:10.1109/
jrproc.1956.275147.
41 B. C. Kuo, Automatic Control Systems, 2nd ed., Englewood Cliffs, NJ: Prentice-Hall, 1967. pp. 59–60.
42 L. N. Binh, Photonic Signal Processing, Boca Raton, FL: CRC Press, 2007.
40
Optical Dispersion in Guided-Wave FIR and IIR Structures
277
t = intensity – transmittance_over_L_waveguide_length
FIGURE 6.29 SFG and transfer function: Coupler represented by a flat-square with flow directions for thru
and cross coupling; delay length L/2 × 2 then this equals to the loop length L of the MRR incorporated in the
MZDI (a) layout by VTT optical circuit (b) SFG representation with gain /loss transmittance in field term.
κ C , κ1, κ 2 = intensity coupling coefficient of the MRR coupler, the coupler 1 (MZDI); and coupler 2 (MZDI),
respectively.
278
6.9.2
Photonic Signal Processing
the gain FOrmula
In brief, the gain transfer function formula can be given by
N
∑
Gk ∆ k
NUM ( z )
Eout
k =1
z
=
; z = e − j β L = e − jωτ = sampling − variable
( )=
DEN ( z )
Ein
∆
∆ = 1−
∑ L +∑ L L −∑ L L L +.... + (−1) ∑
i
i
j
i
j k
m
∆ = graph − determinant
(6.35)
where
ωτ = β L;or j β L = jωτ = phase-after-propagation_thru_optical-waveguuide_
of_length_L_with_a_time _ τ
We summarize the denotations which are described in Chapter 2 with more details, of the following;
Δ is the determinant of the graph; Ein is the input-node field variable of the optical waves; Eout is
the output-node field variable of theoutput lightwaves; G is the complete field gain transmittance
between Ein and Eout optical waves; N is the total number of forward paths between Ein and Eout; Gk is
transmittance
loop field gain
the path field
gain of the kth forward path between Ein and Eout ; Li is the
transmittance of each closed loop in the system; LiLj is the product of the loop transmittance gains
of any two non-touching loops (no common nodes); LiLjLk is the product of the loop transmittance
gains of any three pair-wise non-touching loops Δk is the co-factor value of Δ for the kth forward
path, with the loops touching the kth forward path removed.
In which we can make the following definitions: A Path is a continuous set of transmission
branches traversed in the direction that they indicate, i.e., the optical fields are directed in the direction of the lightwaves transmitting; A forward path: A path from an input node to an output node
in which no node is touched more than once; A Loop is a path that originates and ends on the same
node in which no node is touched more than once. A Non-touching loop is a loop through which
no transmission paths are recorded. Thus, it is preferred that a graph is laid out in such a way that
it is planar, and the loop has no path passing through; Path transmittance gain: the product of the
field transmittances of all the branches in the path; Loop transmittance gain: the product of the field
transmittances of all the branches in the loop.
6.9.2.1 Procedure
To use this technique, we can do the following: (i) Make a list of all forward paths and their gains,
and label these Gk; (ii) Make a list of all the optical loops and their transmittance gains, and label
these optical Li (for i loops); (iii) Make a list of all pairs of non-touching loops, and the products
of their gains (LiLj); (iv) Make a list of all pair-wise non-touching loops taken three at a time (LiLjLk),
then four at a time, and so forth, until there are no more; (v) Compute the determinant Δ and cofactors Δk; then finally (vi) Apply the transmittance gain formula given in Eq. (6.35).
6.9.3
DerivatiOn OF transFer FunctiOn OF the micrO-ring resOnatOr
6.9.3.1 Single Ring
The transfer functions are derived by using Mason’s gain formulation and procedures described in
Chapter 2. The determinant of the graph (resonant optical circuit) with only one non-touching loop
and, hence, its transmittance gain and its determinant
279
Optical Dispersion in Guided-Wave FIR and IIR Structures
∆ = 1 − e −α Le − j β L 1 − κ C = DEN ( z )
with z = e − j β L = e − jωτ
(6.36)
With κ C as the effective coupling coefficient of the [2 × 2] optical coupler; α , β are the linear
attenuation constant and phase propagation constant of the guided optical feedback path. The cross
coupling is expected to be 100% and, thus, a pi/4 phase shift.
The optical paths are obtained as follows:
Path 1: for nodes input (Node 2) to output (Node 4): Node 2-3-1-4. This loop is touching the
loop. So, removing this loop leaves the cofactor of ∆1 = 1. The gain transmittance of this
path is
G1 = j κ C .e j β Le −α L . j κ C = −κ C e j β Le −α L
(6.37)
Path 2: Node 2- to node 4. Not touching the loop (only one loop) in the circuit. This loop is not
touching the loop. 1-3-1 thus the gain transmittance and its co-factor are given by
G2 = 1 − κ C
∆ 2 = ∆ = 1 − 1 − κ C e j β Le −α L
(6.38)
Thus, the transfer function is given by
N
∑
Gk ∆ k
j β L −α L
j β L −α L
NUM ( z ) 1.κ C e e + 1 − κ C 1 − 1 − κ C e e
Eout
k =1
z
(
)
=
=
=
DEN ( z )
Ein
∆
1 − e −α Le − j β L 1 − κ C
(
(
)
)
1.κ C t 1/ 2 z + 1 − κ C 1 − 1 − κ C t 1/ 2 z
Eout
(
z
)
;
=
Ein
1 − t 1/ 2 z 1 − κ C
with : ..z = e − j β L = e − jωτ = sampling-variabble; t1/ 2 = e −α L
(6.39)
Thus, the transfer function can be written in normal expression to expose pole-Zero pattern as
Eout
E4
( z) = =
Ein
E2
(
1.κ C t 1/ 2 z + 1 − κ C 1 − 1 − κ C t 1/ 2 z
κ − 1 − κC
= C
1 − κC
1 − t 1/ 2 z 1 − κ C
) = κ t − 1− κ t
C
1/ 2
C
t 1/ 2 1 − κ C
1/ 2
z−
− 1 − κC
κ C t − 1 − κ C t 1/ 2
1
z − 1/ 2
t
1 − κC
1/ 2
− 1 − κC
z − 1/ 2
t κ C − t 1/ 2 1 − κ C
;
1
z − 1/ 2
1 − κC
t
Where : ...t 1/ 2 = e −α L ; z = e − j β L = e − jωτ ; κ C = effective-coupling-cofficient-of _[2 × 2] coupler
(6.40)
280
Photonic Signal Processing
6.9.3.2 MRR Incorporating MZDI Structure
The transfer function between input and output ports E1 and E7 can be obtained by:
Transfer function from 1 to 7:
DEN(z): one non-touching loop --> graph determinant
∆ = 1 − loop transmittance = 1 − t 1/ 2 1 − κ C z −1
(6.41)
NUM(z): gain transmittance (Field) paths
Path 1: 1-3-10-12-5-7 no touching the loop thus
∆1 = ∆; path1 .field − transmittance = pt1 = 1 − κ1 1 − κ C 1 − κ 2
(6.42)
Path 2: 1-4-6-7: no touching the loop so
∆ 2 = ∆; path 2 .field − transmittance = pt2 = j κ1 t 1/ 2 z −1 j κ 2 = − κ1 κ 2 t 1/ 2 z −1
(6.43)
Path 3: 1-3-10-11-9-12-5-7: touching the loop so ∆ 3 = 1; path3.field_transmittance =
pt3 = 1 − κ1 j κ C t 1/ 2 z −1 j κ C 1 − κ 2 = − 1 − κ1 κ C κ C 1 − κ 2 .t 1/ 2 z −1
(6.44)
Thus, the transfer function can be written by combining Eqs. (6.38) through (6.41) giving
E7out ∆1. pt1 + ∆ 2 . pt2 + ∆3 . pt3
=
E1in
∆
=
=
=
=
(1 − t
1 − κC z −1
1/ 2
) ( 1 − κ 1 − κ 1 − κ ) + (1 − t
C
1
2
1− t
(1− t
1/ 2
1 − κC z −1
C
2
)
)(
)
1 − κC z −1 − κ1 κ2 t 1/ 2 z −1 + − 1 − κ1 κC 1 − κ2 .t 1/ 2 z −1
1 − κC z −1
) ( (1 − κ ) (1 − κ ) (1 − κ ) ) + (1 − t
1
)(
1/ 2
1/ 2
1/ 2
1 − κC z −1 − κ1κ2 t 1/ 2 z −1 −
(1 − κ1 ) (1 − κ2 ) κC .t1/2 z −1
1 − t 1/ 2 1 − κC z −1
( (1 − κ ) (1 − κ )(1 − κ ) − (1 − κ ) (1 − κ )(1 − κ )t z ) + (− κ κ t z + − κ κ t 1 − κ z ) − (1 − κ ) (1 − κ ) κ .t z
1
C
2
C
1
2
1/ 2 −1
1 2
1/ 2 −1
1 2
C
−2
1
2
C
1/ 2 −1
1 − t 1/ 2 1 − κC z −1
(1 − κ1 ) (1 − κC ) (1 − κ2 ) +  κ1κ2 t1/2 − (1 − κC ) (1 − κ1 ) (1 − κ2 ) t1/2 + (1 − κ1 ) (1 − κ2 ) κC .t1/2  z −1 − ( κ1κ2 t 1 − κC z −2 )
1 − t 1/ 2 1 − κC z −1
(6.45)
If κ1 = κ1 = 0.5 = 3dB_coupler then we can have
E7out
=
E1in
(1 − κ1 ) (1 − κC ) (1 − κ2 ) +  κ1κ2 t1/2 − (1 − κC ) (1 − κ1 ) (1 − κ2 ) t1/2 + (1 − κ1 ) (1 − κ2 ) κC .t1/2  z −1 − ( κ1κ2 t 1 − κC z −2 )
E7out 0.5 (1 − κC ) + (0.5t
=
E1in
1 − t 1/ 2 1 − κC z −1
1/ 2
− (1 − κC ) 0.5t 1/ 2 + 0.5κC .t 1/ 2 ) z −1 − 0.5 1 − κC z −2
1 − t 1/ 2 1 − κC z −1
=
(6.46)
Optical Dispersion in Guided-Wave FIR and IIR Structures
281
Now, if we consider the low loss transmission t = 1.0– and a variable coupling coefficient, then we
have the following:
• Resonance frequency (pole) at
1 − t 1/ 2 1 − κ C z −1 = 0 → z p 2 = t 1/ 2 1 − κ C _ and _ z p1 = 0
(6.47)
• Zeroes are located at:
0.5 (1 − κ C ) + (0.5t 1/ 2 − (1 − κ C ) 0.5t 1/ 2 + 0.5κ C .t 1/ 2 ) z −1 − 0.5 1 − κ C z −2 = 0 →
0.5 (1 − κ C ) z +2 + (0.5t 1/ 2 − (1 − κ C ) 0.5t 1/ 2 + 0.5κ C .t 1/ 2 ) z +1 − 0.5 1 − κ C  / z 2 = 0


z p1=........... and z p 2=........... and_a doublee poles at origin
(6.48)
So, we have one z = 0 cancelled by one divided z of the denominator, and we have one additional
pole at the origin. Thus, the optical circuit MZDI-MRR offers one real pole at … and one pole at
the origin.
And two real=
zeroes z p1 ...........
and z p 2 ...........
=
7
Photonic Ultra-Short
Pulse Generators
Ultra-short pulse generators are lightwave sources that emit short pulses of high intensity. This
means the energy has been concentrated into a very short time and periodically distributed along
the pulses of the sequence. The peak power of these pulses reaches the nonlinear threshold of
several materials that are used as the interaction and guided media for photonic signal processing.
This chapter presents a number of important and emerging light sources of very narrow pulse
width sequences considered for application in soliton communications and soliton logics. Photonic
generators for bright and dark solitons as well as bound solitons are given.
Mode-locked fiber lasers have become well known over the last decades and are emerging as
important sources for advanced lightwave communications technology. Practical implementations
of these types of lasers will be subsequently discussed after the sections on solitons.
7.1
OPTICAL DARK-SOLITON GENERATOR AND DETECTORS
In this section, the trapezoidal optical integrator described in Chapter 3 is proposed as an optical
dark-soliton generator, and the first-order first-derivative optical differentiator outlined in Chapter 6
and a first-order Butterworth Lowpass optical filter (LPOF) are proposed as optical dark-soliton
detectors. A brief review of solitons in optical fibers is presented in Section 7.1. The nonlinear
Schroedinger equation describing soliton propagation in a lossless optical fiber and the parameters
of the dispersion shifted fiber and laser source used are given in Section 7.2. The design and performance of the optical dark-soliton detectors are first investigated in Section 7.3 so that they can
be characterized. The design (Section 7.4) and performance (Section 7.5) of the optical dark-soliton
generator are then outlined. The performances of the combined dark-soliton generator and detectors are also described in Section 7.5. The theory of coherent integrated-optic signal processing
described in Section 7.2.3 is employed in this chapter where electric-field amplitude signals are
considered.
7.1.1
intrODuctiOn
In the near future, optical solitons are believed to be promising candidates as information carriers in ultra-long-distance and/or ultra-high-speed repeater-less, optically-amplified communication
systems. The history of solitons in optical fibers began almost a quarter of a century ago when
Hasegawa and Tappert1,2 proposed that optical solitons can propagate without distortion over an
infinitely long distance in a lossless single-mode optical fiber through the exact balancing of the
inherent effects of group velocity dispersion (GVD) and self-phase modulation: bright solitons exist
in the negative (or anomalous) GVD regime, while dark solitons appear in the positive (or normal)
GVD regime. The pioneering work of Hasegawa and Tappert has led to the field of optical soliton
communications, which has been under intensive worldwide investigation in the last decade.
A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers: I.
Anomalous dispersion, Appl. Phys. Lett., 23, 142–144, 1973.
2 A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers: II.
Normal dispersion, Appl. Phys. Lett., 23, 171–173, 1973.
1
283
284
Photonic Signal Processing
The practical importance of bright solitons in optical fibers was not realized until 1980 when
Mollenauer et al.3 reported the first experimental verification of the existence of a bright soliton over
a 700-m standard single-mode optical fiber. Several years after the prediction made by Hasegawa4
that practical propagation of bright solitons over long distances could be made possible by
periodically compensating for the fiber loss through the use of Raman amplifier gain, Mollenauer
and Smith5 exploited this idea and presented the first bright-soliton transmission experiment ever
carried out over 4000 km. One year later, Nakazawa et al.6 reported the first soliton transmission
experiment that used erbium-doped fiber amplifier (EDFA) as an optical repeater for the 50-km
fiber link. Recent technological advances have generated many successful ultra-high-speed and/or
ultra-long-reach transmission experiments in which EDFAs were used as optical repeaters; see, for
example, the recent papers by Nakazawa7 and Aubin et al.8
Dark solitons have been predicted to offer better stability than bright solitons against fiber
loss,9 and improved interactions between neighboring solitons10 and amplified noise-induced
timing jitter.11,12 However, Dark-soliton propagation experiments are much more difficult to
implement than bright-soliton propagation experiments because of the difficulty of generating
and detecting dark solitons. In 1987, Emplit et al.13 experimentally studied the propagation of
odd-symmetry optical dark pulses that were generated using amplitude and phase filtering techniques. However, dark pulse propagation was not convincingly observed in their experiment
because the fiber length was shorter than the soliton period and the low resolution in their pulseshaping and pulse-measurement apparatus. Nevertheless, it was the first experimental propagation of dark pulses in optical fiber. One year later, Krökel et al.14 experimentally demonstrated
the evolution of an even-symmetry dark pulse into a pair of low contrast dark pulses. In the same
year, Weiner et al.15 successfully developed a technique for synthesizing femtosecond optical
pulses with almost arbitrary shape and phase. They also presented a more convincing experimental observation of soliton-like propagation of odd-symmetry dark pulse superimposed upon
a broader Gaussian background pulse.
L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Experimental observation of picosecond pulse narrowing and solitons
in optical fibers, Phys. Rev. Lett., 45, 1095–1098, 1980.
4 A. Hasegawa, Numerical study of optical soliton transmission amplified periodically by the stimulated Raman process,
Appl. Opt., 23, 3302–3309, 1984.
5 L. F. Mollenauer and K. Smith, Demonstration of soliton transmission over more than 4000 km in fiber with loss periodically compensated by Raman gain, Opt. Lett., 13, 675–677, 1988.
6 M. Nakazawa, Y. Kimura, and K. Suzuki, Soliton amplification and transmission with Er3+-doped fibre repeater
pumped by GainAsP laser diode, Electron. Lett., 25, 199–200, 1989.
7 M. Nakazawa, Ultrahigh speed optical soliton communication and related technology, IOOC’95 Technical Digest
Series, Hong Kong, 4, 96–98, 1995.
8 G. Aubin, E. Jeanney, T. Montalant, J. Moulu, F. Pirio, J.-B. Thomine, and F. Devaux, Electro absorption modulator for
a 20 Gbit/s soliton transmission experiment over 1 Million km with a 140 km amplifier span, IOOC’95 Technical Digest
Series, Hong Kong, 5, 29–30, Post-deadline paper PD2–5, 1995.
9 W. Zhao and E. Bourkoff, Propagation properties of dark solitons, Opt. Lett., 14, 703–705, 1989.
10 W. Zhao and E. Bourkoff, Interactions between dark solitons, Opt. Lett., 14, 1371–1373, 1989.
11 J. P. Hamaide, P. Emplit, and M. Haelterman, Dark-soliton jitter in amplified optical transmission systems, Opt. Lett., 16,
1578–1580, 1991.
12 Y. S. Kivshar, M. Haelterman, Ph. Emplit, and J. P. Hamaide, Gordon-Haus effect on dark solitons, Opt. Lett., 19, 19–21,
1994.
13 P. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, and A. Barthelemy, Picosecond steps and dark pulses through nonlinear single mode fibers, Opt. Commun., 62, 374–379, 1987.
14 D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, Dark-pulse propagation in optical fibers, Phys. Rev. Lett., 60,
29–32, 1988.
15 A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson,
Experimental observation of the fundamental dark soliton in optical, fibers, Phys. Rev. Lett., 61, 2445–2448, 1988.
3
Photonic Ultra-Short Pulse Generators
285
Unlike previous schemes for generating dark pulses with a finite background,16,17 techniques for
generating dark pulses with a CW (continuous wave) background (or dark solitons) have also been
proposed and experimentally demonstrated.18,19 Richardson et al.16 reported the first experimental
demonstration of the generation of a 100-GHz dark-soliton train by means of nonlinear conversion of a high-intensity beat signal in a positive GVD decreasing fiber. They also confirmed the
stability of the generated dark-soliton train by propagating it through a 2-km length of positive
GVD shifted fiber. Zhao and Bourkoff17 proposed the use of an electro-optic intensity modulator
driven by square pulses to generate dark solitons. Recently, Nakazawa and Suzuki18 modified this
scheme by using a logic AND circuit and a T-flip-flop circuit to perform data conversion to obtain
a pseudo-random dark-soliton train. They also developed the first dark-soliton detection scheme,
based on a one-bit-shifting technique with a Mach–Zehnder interferometer, which converts a darksoliton signal into a modified return-to-zero (RZ) signal and into an inverted non-return-to-zero
(NRZ) signal. Using the developed dark-soliton generation and detection schemes, Nakazawa and
Suzuki18 conducted for the first time a dark-soliton transmission experiment over 1200 km.
In this chapter, the trapezoidal optical integrator is proposed as an optical dark-soliton generator,
and the first-order first-derivative optical differentiator and the first-order Butterworth LPOF are
proposed as optical dark-soliton detectors. Most of the work presented here has been described by
Ngo et al.20 The effect of fiber loss on dark-soliton propagation is not considered here, so that the
underlying principles of dark-soliton generation and detection can be demonstrated.
7.1.2
Optical Fiber prOpagatiOn mODel
To demonstrate the effectiveness of the proposed optical dark-soliton generator and detectors, the
nonlinear Schroedinger equation describing pulse propagation in a lossless single-mode optical
fiber
j
∂ A β2 ∂ 2 A
2
−
+ξ A A = 0
2
∂Z 2 ∂t
(7.1)
can be numerically solved using the split-step Fourier method.21 In Eq. (7.1), j = −1 ,
A( Z , t ) = P0 U ( Z , t ), P0 is the peak power of the incident pulse with normalized amplitude U ( Z , t ), Z
the distance of propagation, t the retarded time,22 β 2 = −Dλ 2 (2π c) the GVD parameter, D the fiber
dispersion parameter, λ the operating wavelength, c the speed of light in vacuum, ξ = 2π n2 ( Aeff λ )
the fiber nonlinear coefficient, n2 the nonlinear refractive index, and Aeff is the effective core area.
It is useful to write the peak power and soliton period as P0 = N 2 β 2 (ξ T02 ) and Z0 = π T02 (2 β 2 ),
respectively, where the integer N is the soliton order and T0 the initial pulse width. The dispersionshifted fiber, with zero-dispersion wavelength at 1550 nm, and soliton source are assumed to have
the following typical parameters: β 2 = +1.27 ps2 km (for D = −1.0 ps nm km), ξ = 3.2 W −1km −1
(for Aeff = 40 µm 2 and n2 = 3.2 × 10 −20 m 2 W ), P0 = 0.494 mW (for N = 1 and T0 = 28.4 ps) and
Z0 = 995 km.
D. J. Richardson, R. P. Chamberlin, L. Dong, and D. N. Payne, Experimental demonstration of 100 GHz dark soliton
generation and propagation using a dispersion decreasing fibre, Electron. Lett., 30, 1326–1327, 1994.
17 W. Zhao and E. Bourkoff, Generation of dark solitons under a cw background using waveguide electro-optic modulators,
Opt. Lett., 15, 405–407, 1990.
18 M. Nakazawa and K. Suzuki, Generation of a pseudorandom dark soliton data train and its coherent detection by onebit-shifting with a Mach–Zehnder interferometer, Electron. Lett., 31, 1084–1085, 1995.
19 M. Nakazawa and K. Suzuki, 10 Gbit/s pseudorandom dark soliton data transmission over 1200 km, Electron. Lett., 31,
1076–1077, 1995.
20 N. Q. Ngo, L. N. Binh, and X. Dai, Optical dark-soliton generators and detectors, Opt,. Commun., 132, 389–402, 1996.
21 G. P. Agrawal, Nonlinear Fiber Optics, Boston, MA: Academic Press, 1989.
22 The retarded time is defined as the normalized time measured in a frame of reference moving with the pulse at the group
velocity, i.e., t = t − Z vg where t is the actual time and vg the group velocity.
16
286
7.1.3
Photonic Signal Processing
Design anD perFOrmance OF Optical Dark-sOlitOn DetectOrs
The optical differentiator and Butterworth LPOF are now designed as optical dark-soliton detectors.
The fundamental dark-soliton normalized pulse at the fiber input is given by10
 tanh(t T0 + q0 ),
U (0, t ) = 
− tanh(t T0 − q0 ),
− ∞ < t T0 < 0
0 ≤ t T0 < ∞
(7.2)
where q0 is a constant, Tb = 2q0T0 the bit time and
=
Tw1 1=
.76T0 50 ps is the full width half mark
(FWHM) of the soliton pulse. It is considered that Tb = 200 ps (for q0 = 3.52) corresponds to a bit
rate of 1 Tb = 5 Gbit s.
7.1.4
Design OF Optical Dark-sOlitOn DetectOrs
By definition,23 the derivative of a fundamental dark-soliton pulse with amplitude tanh(t ) is given
by a “bright-squared”24 soliton pulse with a temporal amplitude function sech 2 (t ) given by:
d [ tanh(t ) ]
= sech 2 (t )
dt
(7.3)
The design of an optical differentiator, which can perform the above derivative function, and a firstorder Butterworth LPOF are now described as optical dark-soliton detectors. Figure 7.1 shows the
schematic diagram of the asymmetrical Mach–Zehnder interferometer (AMZI) used for dark-soliton
detection. The AMZI, which can be implemented using planar lightwave circuit (PLC) technology,
namely, silica-based waveguides embedded on a silicon substrate as described in Section 7.2.3, consists of two directional couplers, DC1 and DC2, with cross-coupled intensity coefficients k1 and k2 ,
which are interconnected by two unequal waveguide arms with a differential time delay of Td .
Neglecting the propagation delay, insertion loss and waveguide birefringence,25 the transfer
functions of the AMZI are given by26
E3
12
= exp( − jωTu ) [(1 − k1 )(1 − k2 ) ] 1 − z31z −1 ,
E1
(7.4)
E4
12
= exp[− j (ωTu + π 2)][(1 − k1 )k2 ] 1 − z41z −1 ,
E1
(7.5)
H 31( z ) =
H 41( z ) =
E1
Dark
soliton
signal
{
}
{
DC1
k1
Td
}
DC2
E3 (RZ signal)
k2
E4 (NRZ signal)
FIGURE 7.1 Schematic diagram of the asymmetrical Mach–Zehnder interferometer (AMZI) used for darksoliton detection. All lines are integrated optical waveguides using planar lightwave technology.
M. Abramowitz and I. A. Segun, Handbook of Mathematical Function, New York: Dover Publications, 1964.
It is well known that the electric-field amplitude of a bright-soliton pulse is given by a secant pulse sech(t ) . Here, the
2
electric-field amplitude of a secant-squared pulse sech (t ) is referred to as a “bright-squared” soliton pulse. Thus, the
electric-field amplitude of the “bright-squared” soliton pulse is the square of that of the bright-soliton pulse.
25 The waveguide birefringence of the AMZI can be eliminated by a fiber polarisation controller or by inserting polyimide
half waveplates into the gap of the waveguide paths.63
26 These transfer functions have been obtained by using the waveguide directional coupler defined in Eq. (2.2) and the
signal-flow graph technique described in Chapter 3.
23
24
287
Photonic Ultra-Short Pulse Generators
whose zeros in the z-plane are given by
12


k1k2
z31 = 
 ,
 (1 − k1 )(1 − k2 ) 
(7.6)
12
 k (1 − k2 ) 
z41 = −  1
 ,
 (1 − k1 )k2 
(7.7)
where E1 and (E3, E4 ) are the electric-field amplitudes at the input and output ports, respectively,
z = exp( jωTd ) is the z-transform parameter, ω the angular optical frequency, and Td = T − Tu (the
sampling period of the AMZI) is the differential time delay between the lower arm (with delay T)
and the upper arm (with delay Tu ). For k=
k=
0.5 and hence z31 = 1, H 31( z ) corresponds to the
1
2
transfer function of the first-order first-derivative optical differentiator as described in Chapter 6.
Note that H 31( z ) is also the transfer function of the first-order Butterworth highpass optical filter
with a 3-dB cut-off frequency at ωTd = π 2. While, for k=
k=
0.5 and hence z41 = −1, H 41( z )
1
2
corresponds to the transfer function of the first-order Butterworth lowpass optical filter with a 3-dB
cut-off frequency also at ωTd = π 2.
7.1.5
perFOrmance OF the Optical DiFFerentiatOr
The sampling period of the AMZI is chosen to be equal to the bit period, i.e. Td = Tb , for reasons
to be given below. Figure 7.2 shows the 5 Gb/s bit fundamental dark-soliton signals at Z = 0 and
Z = 100 Z0. Compared with the input dark-soliton signal at Z = 0, the initial separation of the darksoliton signal at Z = 100 Z0 is increased by 2.5% as a result of the repulsive force between neighboring
dark solitons at a long distance.10
The propagated dark-soliton signal at Z = 100 Z0 is detected by the optical differentiator and
the resulting RZ signals are shown in Figure 7.2 for 0.8 ≤ z31 ≤ 1.2. The solid curve corresponds
to the performance of the ideal differentiator, which requires k=
k=
0.5 and hence z31 = 1. Such
1
2
a requirement is often difficult to achieve in practice because of the difficulty of fabricating directional couplers with the very precise values of the coupling coefficients. From Figure 7.2, as z31
deviates from its nominal value, the power levels of the space signals are raised from the otherwise zero values to P ≤ 0.0125, and this results in decreasing the otherwise infinite mark to space
FIGURE 7.2 5 Gb/s bit fundamental dark-soliton signals at Z = 0 and Z = 100 Z 0 .
288
Photonic Signal Processing
ratio (MTSR) to MTSR ≥ 80. Surprisingly, the middle portion of the adjacent mark signals is not
affected by this variation in z31. For MTSR ≥ 80, the allowable values of the coupling coefficients
of the AMZI lie in the range of 0.45 ≤ k1, k2 ≤ 0.55 , which is obtained by imposing the condition
0.8 ≤ z31 ≤ 1.2 on Eq. (7.6). Such variation of the coupling coefficients can be easily tolerated using
the PLC technology.
Note that the detected RZ signals are of the square-type intensity pulse shape rather than the
sech 4 (t ) shape expected from the derivative of a tanh(t) amplitude pulse shape. This is because the
chosen sampling period is large compared with the dark-soliton pulse width (i.e., Td = 4Tw1). When
=
Td 0=
.05Tw1 2.5 ps, the RZ signals have the expected sech 4 (t ) intensity pulse shape, showing the
high processing accuracy of the differentiator. However, the drawbacks of using Td << 4Tw1 are that
the performance of the differentiator significantly deteriorates through the large variation of z31 and
that a small differential length of interferometer arms is required, which then requires very high
fabrication accuracy.
It is clear from Figure 7.3 that the RZ signals can be detectable, and that high processing accuracy of the differentiator is not necessary as far as dark-soliton detection is concerned.
The bit time of the dark-soliton signal must be sufficiently large to minimize the effect of darksoliton interactions, especially at very long distances. Thus, the sampling period of the AMZI, Td ,
the bit period of the dark-soliton signal, Tb , and the dark-soliton pulse width, Tw1, must be related to
each other in such a way that the optical bandwidth of the AMZI [i.e., 1 (2Td )] is sufficiently large to
accommodate the optical spectrum of the input dark-soliton signal. It was found that T=
T=
4Tw1
d
b
satisfies this bandwidth requirement. In this study, for example, the 2.5 GHz bandwidth of the
AMZI is wide enough to accommodate the 1 GHz spectral width of the dark-soliton spectrum at
Z = 100 Z0.
7.1.6
perFOrmance OF the butterwOrth lpOF
The propagated dark-soliton signal at Z = 100 Z0, as shown in Figure 7.2, is detected by the
Butterworth LPOF, resulting in the inverted NRZ signals as shown in Figure 7.4 for −1.2 ≤ z41 ≤ −0.8.
The solid curve corresponds to the performance of the ideal LPOF, which requires k=
k=
0.5 and
1
2
FIGURE 7.3 Detected 5 Gb/s bit RZ signals at the output of the optical differentiator with various zero loca2
tions. The actual optical power is given by A( Z , t ) = (1 − k1 )(1 − k2 ) PCP0 where P and C are constant factors.
289
Photonic Ultra-Short Pulse Generators
FIGURE 7.4 Detected 5 Gbit s bit inverted NRZ signals at the output of the Butterworth lowpass optical
2
filter with various zero locations. The actual optical power is given by A( Z , t ) = (1 − k1 )k2 PCP0.
z41 = −1. As z41 deviates from its nominal value, the power levels of the space signals are increased
from the otherwise zero values to P ≤ 0.0125 that gives MTSR ≥ 80. For MTSR ≥ 80, the allowable
values of the coupling coefficients of the AMZI are also in the range of 0.45 ≤ k1, k2 ≤ 0.55.
For both cases of the differentiator and LPF, the variations in z31 and z41 have more pronounced
effects on the space signals than the mark signals. Compared with the propagated dark-soliton signal at Z = 100 Z0 (see Figure 7.2), Figures 7.3 and 7.4 show that the detected signals are shifted by a
1/2-bit to the right of the time axis. The simulation results for both cases are also consistent with the
experimental results of Nakazawa and Suzuki.19
The proposed optical differentiator is based on the concept of optical differentiation, while the
AMZI described by Nakazawa and Suzuki18 was based on the one-bit-shifting technique. However,
these two completely different concepts can be implemented by the AMZI. The LPF is based on the
experimental investigations of Nakazawa and Suzuki.18 Thus, the simulation results presented here
provide the groundwork for the design of optical dark-soliton detectors since no theoretical results
are given in reference.
7.1.7
Design OF the Optical Dark-sOlitOn generatOr
7.1.7.1 Design of the Optical Integrator
By definition,21 the integral of a “bright-squared” soliton pulse with amplitude sech 2 (t ) is given by
a fundamental dark-soliton pulse with its amplitude following the shape of tanh(t ) :
t
∫ sech (t′)dt′ = tanh(t )
2
(7.8)
−∞
The design of an optical integrator, which can perform the above integral function, is now described.
Figure 7.5 shows the schematic diagram of the all-pole optical filter using the PLC technology, which
is designed to be used as the optical integrator as described in Chapter 5. Note that the physical
structure is exactly the same as that of the incoherent recursive fiber-optic signal processor in which
fiber or integrated optic components can be used. The filter consists of an active optical waveguide
loop formed by two directional couplers, DC3 and DC4, which are interconnected by waveguides.
290
Photonic Signal Processing
Input
Output
DC3
k3
DC4
k4
Loop Delay
TI
EDWA,Gw
FIGURE 7.5 Schematic diagram of the all-pole optical filter using the PLC technology, which is used as the
optical integrator.
A portion of the input signal is coupled by DC3 to the lower active waveguide path where it is
amplified by an erbium-doped waveguide amplifier (EDWA) of intensity gain Gw , as described in
Chapter 2, Section 2.3. A portion of the amplified signal is coupled to the output port by DC4 while
the rest is coupled to the upper path. A portion of the signal in the upper path is then coupled to the
lower path where it is further amplified by the EDWA and the process continues. Optical signals
can be maintained circulating in the loop for as long as the losses are compensated by the EDWA.
Following the EDWA, an integrated waveguide isolator21 and a narrow-band optical filter may be
required to suppress the stimulated Brillouin scattering (SBS) and to minimize the amplified spontaneous emission (ASE), respectively.
Neglecting the propagation delay, insertion loss and waveguide birefringence, the electric-field
transfer function of the all-pole optical integrator is given by27
− [ k3k4Gw ]
12
H ( z) =
1 − [(1 − k3 )(1 − k4 )Gw ] z −1
12
(7.9)
which has one zero at the origin and one pole in the z-plane given by
pole = [(1 − k3 )(1 − k4 )Gw ]
12
(7.10)
where z = exp( jωTI ) is the z-transform parameter, TI the optical loop delay or sampling period of
the filter, and k3 and k4 are, respectively, the cross-coupled intensity coefficients of DC3 and DC.
As described in Chapter 5, the optical integrator requires the pole location to be exactly on the unit
circle in the z-plane, i.e. pole = 1. A convenient choice to accomplish such a requirement is to choose
k=
k=
0.5 and Gw = 4, and this results in the transfer function of the optical integrator as given by
3
4
−H ( z) =
1
.
1 − z −1
(7.11)
Note that the excess losses of DC3 and DC4 and the losses of the straight and bend waveguides of
the optical loop, which are not taken into account here, can also be compensated by the EDWA.
Also note that the transfer function of the optical integrator is the reciprocal of that of the optical
k=
0.5 is used. This is as expected from the
differentiator as described by Eq. (3.4) when k=
1
2
inverse operation between the integral and derivative functions.
27
This transfer function has been derived by using the waveguide directional coupler defined in Eq. (2.2) and the signalflow graph method described in Chapter 3.
291
Photonic Ultra-Short Pulse Generators
The un-modulated (or continuous wave) signal of the soliton source is assumed to be externally
modulated by an ideal optical intensity modulator. The modulated signal to be processed by the
optical integrator is assumed to be of the normalized form
x (t ) =
∑ (−1) a S (t T − 2nq )
n
n
12
0
(7.12)
0
n
where an ∈(0,1) is the digital sequence and the intensity pulse shape
S (t ) = sech 4 (t ).
(7.13)
The processed signal at the output of the optical integrator is given, by the convolution of the modulated signal with the impulse response of the optical integrator ( −h(t )) , as
y(t ) = x(t ) ∗ ( −h(t )) ≅
∑ (−1) a tanh(t T − 2nq ) + 1
n
n
0
0
(7.14)
n
where * denotes the convolution operation and ( −h(t )) is given by the inverse z-transform of −H ( z ).
The integrated signal in Eq. (7.14) is a positive function because of the positive nature of the input
pulse S (t ) and of the positive impulse response ( −h(t )) , as described in Chapter 5. Thus, the amplitude level of this integrated signal must be lowered by an amount equal to a CW signal of unity
amplitude in order to obtain the required fundamental dark-soliton signal. As a result, the fundamental dark-soliton signal yI (t ) can be generated according to
yI (t ) ≅ y(t ) − ymax 2 ≅
∑ (−1) a tanh(t T − 2nq )
n
n
0
0
(7.15)
n
where
=
ymax max[
=
y(t )] 2 is the maximum amplitude of y(t ).
7.1.7.2 Design of an Optical Dark-Soliton Generator
The optical dark-soliton generator is now designed based on the mathematical formulations given
in Eqs. (7.12) through (7.15). Figure 7.6 shows the schematic diagram of the proposed optical darksoliton generation scheme using PLC technology. The unmodulated light signal of the laser source
is equally split by a 3-dB waveguide directional coupler into the upper and lower waveguide paths
that are of equal length. The processed modulated signal at the end of the upper path and the
unmodulated signal at the end of the lower path are synchronously combined by another waveguide
3-dB directional coupler and then amplified by the EDFA, resulting in the required fundamental
dark-soliton signal at the fiber input. The EDFA of intensity gain Gf is used as a booster amplifier to
provide sufficient power to the fiber input in order to enable dark-soliton propagation. The requirement that the signals in the upper and lower paths be synchronously combined can be achieved by
having both paths equal in length. In the upper waveguide arm, the optical intensity modulator can
be of a Mach–Zehnder type and the optical integrator, which can be implemented using integrated
photonics, based on LInBO3 or Si on insulators.28 In the lower waveguide arm, the optical tunable coupler, having a transfer function29 K exp( jθ32 ) with K being the cross-coupled intensity
coefficient and θ32 the phase shift, and the optical phase shifter with a phase shift φ , can also be
implemented using the PLC technology. The characteristics of the phase shifter and tunable coupler
have been described in Sections 2.3.2 and 2.3.3 in Chapter 2. Thus, the optical integrator, tunable
coupler, phase shifter, 3-dB directional couplers, and the waveguide paths can all be implemented
using the same PLC technology.
L. N. Binh, Guided wave integrated photonics: Fundamentals and applications with MATLAB®, August 23, 2011 by
CRC Press Boca Raton, FL.
29 This transfer function corresponds to the transfer function E4
in Eq. (2.7).
E
28
1
292
Photonic Signal Processing
Optical Integrator
Optical
Intensity
Modulator
Optical waveguide
DC3
k3
Electrical
Signal
Generator
LASER
Optical Fiber
TI
Connector/Joint
DC4
k4
EDWA,Gw
EDFA
h(t)
Gf
3-dB
Directional
Coupler
3-dB
Directional
Coupler
Fiber Input
A (Z = 0, t)
Phase
Shifter
Tunable
Coupler
ϕ
(K, θ32)
FIGURE 7.6
Schematic diagram of the proposed optical dark-soliton generator using the PLC technology.
The fundamental dark-soliton signal at the fiber input is given by
A( Z = 0, t ) = − j Gf Ps 4  α u x(t ) ∗ ( −h(t )) − α K exp ( j (θ32 + φ ) ) 
(7.16)
where Ps is the peak power of the laser source and α u and α are, respectively, the intensity losses
of the upper and lower waveguide paths that include all possible optical losses, such as the connector loss between the fiber and waveguide, the straight and bend waveguide losses, and the insertion losses of the optical intensity modulator, the optical integrator, and the tunable coupler. The
first and second terms in Eq. (7.16) represent the processed modulated signal and the unmodulated
signal in the upper and lower paths, respectively. The generation of the fundamental dark-soliton
signal requires the terms inside the square brackets in Eq. (7.16). Figures 7.12 and 7.13 show the
composition of an mode-locked ring laser (MLRL) with and without feedback loops used in this
study, respectively. It consists principally for a non-feedback ring, an optical close loop with an
optical gain medium, an optical modulator (intensity or phase type), an optical fibers coupler, and
associated optics. An opto-electronically (O/E) feedback loop detecting repetition-rate signal and
generating RF sinusoidal waves to electro-optically drive the intensity modulator is necessary for
the regenerative configuration as shown in Figures 7.12 and 7.13 show the composition of a MLRL
without and with feedback loop used in this study respectively. It consists principally, for a nonfeedback ring, an optical close loop with an optical gain medium, an optical modulator (intensity
or phase type) an optical fibers coupler and associated optics. An O/E feedback loop can detect the
repetition-rate of the circulating signal. An RF sinusoidal waves can then be generated to electrooptically drive the intensity modulator for the regenerative configuration as shown in Figure 7.13.
Eq. (7.16) can be set to be equal to the respective terms in Eq. (7.15) such that
α K exp ( j (θ32 + φ ) ) = max  α u x(t ) ∗ ( −h(t ))  2 ,
(7.17)
αu
,
α
(7.18)
φ = −θ32 .
(7.19)
from which
K=
Photonic Ultra-Short Pulse Generators
293
The upper path is probably more lossy than the lower one, i.e., α u < α , because of the relatively
high loss incurred by the optical intensity modulator. The required value of K is expected to be
in the range of 0 < K < 1. The tunable coupler can be used as a variable optical attenuator, which
adjusts the power level in the lower waveguide path so that both the upper and lower paths have the
same power level, while the phase shifter is used to equalize the resulting phase shift of the tunable coupler. The phase shifter can also be used to adjust the length of the lower path to the precision of the wavelength order because of the temperature dependence of the refractive index of the
silica waveguide. The phase shifter can thus provide flexible control of the waveguide length and
hence synchronization of the system. The temperature of the silicon substrates can be maintained
to within a small fraction of a degree to stabilize the refractive index of the waveguides, and this
stability enables the optical dark-soliton generation scheme to efficiently and stably generate highspeed dark-soliton signals.
The following practical measures must be considered for stable operation. The CW light of the
laser source must be in one polarization state; easily achieved by means of a fiber polarization controller (PC) placed at the laser output. A fiber polarization controller is also required at the fiber
input because of the presence of the waveguide birefringence.
7.1.8
perFOrmance OF the Optical Dark-sOlitOn generatOr anD DetectOrs
It is considered that the input amplitude pulse to the optical integrator is a “bright-squared” soliton
pulse pair (normalized):
U (0, t ) = −sech 2 (t T0 − q0 ) + sech 2 (t T0 + q0 )
(7.20)
which has a FWHM of
=
Tw2 1=
.21T0 34.4 ps. The bit time Tb = 2q0T0 is chosen to be Tb = 200 ps
(for q0 = 3.52), the same as that used for the dark-soliton detectors in Section 7.1.4. The sampling
period of the optical integrator is considered to be TI = 25 ps, which corresponds to an optical bandwidth of 20 GHz.
7.1.8.1 Performance of the Optical Dark-Soliton Generator
Figure 7.7 shows the ideal, generated and propagated 5 Gb/s bit dark-soliton signals. The solid curve
corresponds to the ideal dark-soliton signal whose amplitude pulse is described by Eqs. (7.1) and
(7.2). The “bright-squared” soliton pulse, as described by Eq. (7.20), is processed by the ideal optical dark-soliton generator in which the optical integrator has the desired pole location pole = 1. This
results in the generated dark-soliton signal at Z = 0 (dashed curve), which is slightly shifted to the
left of the time axis compared with the ideal dark-soliton signal. This is because of the low processing accuracy of the integrator as a result of its relatively large sampling period, which occupies a
large fraction of the pulse width, i.e., TI = 0.73Tw 2. It was found that when the sampling period is
reduced by ten-fold to TI = 0.073Tw 2, the generated dark-soliton signal has almost the same shape as
the ideal dark-soliton signal, showing the high processing accuracy of the integrator. As discussed
in Section 7.1.5 for the case of the differentiator, the disadvantages associated with using a small
sampling period are that the performance of the integrator significantly deteriorates when the pole
position deviates from its nominal value and that high fabrication accuracy of the loop length of the
integrator is required. The sampling period of the integrator is therefore chosen so that its optical
bandwidth is large enough to accommodate the optical spectrum of the input “bright-squared” soliton signal. In this study, for example, the 20 GHz bandwidth of the integrator is sufficient to cover
the 20 GHz spectral width of the input “bright-squared” soliton signal.
Figure 7.7 shows that the signal at a very large distance of Z = 100 Z0 still preserves the characteristics of the generated signal. This shows the effectiveness of the proposed scheme to generate high-quality dark solitons, which can propagate very stably over very long distances. The low
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Photonic Signal Processing
FIGURE 7.7 5 Gb/s bit dark-soliton signals, showing the ideal signal, the signal at the output of the darksoliton generator, and the signal at a large distance of Z = 100 Z 0 .
background noise of the propagated signal is due to the low processing accuracy of the integrator.
The background noise is reduced significantly when the sampling period is reduced ten-fold but,
as discussed above, the performance of the integrator is very sensitive to the variation of its pole
location.
Figure 7.8a shows the generated dark-soliton signals at Z = 0 for various pole values of the
integrator. The quality of the signals deteriorates as the pole of the integrator moves away from it
within the unit circle. As expected, the quality of the corresponding signals at Z = 100 Z0, as shown
in Figure 7.8b, also deteriorates with pole variation.
Figure 7.9a and b show the evolution of the generated dark-soliton signals over 100 soliton periods for the pole values 0.994 and 1, respectively. From Figure 7.9a, the background noise fluctuates
FIGURE 7.8 5 Gb/s bit generated and propagated dark-soliton signals for various pole values of the optical
integrator. (a) Generated dark-soliton signals at Z = 0. (b) The corresponding dark-soliton signals of Figure 7.8a
at Z = 100 Z 0 .
Photonic Ultra-Short Pulse Generators
295
FIGURE 7.9 Evolution of the 5 Gb/s bit generated dark-soliton signals over 100 soliton periods for two different pole values of the optical integrator. (a) Pole = 0.99 and (b) Pole = 1.
about the unity along the propagation distance, because the generated signal does not have the exact
shape of the fundamental signal but still has the essential characteristics of the fundamental signal.
Figure 7.9b shows that, when the integrator has the desired pole value, the noise reduces significantly
with distance, and the propagated signal has almost the same shape as that of the fundamental signal.
Thus, to generate stable high-quality dark-soliton signals, the integrator must operate just within the
unit circle. If operated outside, the integrator becomes unstable.
7.1.8.2 Performance of the Combined Optical Dark-Soliton
Generator and Optical Differentiator
The individual performances of the optical dark-soliton generator and optical differentiator have
been separately analyzed in Section 7.1.3. The overall performance of the combined generator and
differentiator is now described.
The propagated signals at Z = 100 Z0, as shown in Figure 7.8b, are detected by the ideal
differentiator having the zero location z31 = 1, and the resulting RZ signals are shown in Figure 7.10a.
The solid curve corresponds to the ideal performance of the combined generator and differentiator,
and has an infinite value of MTSR. The pole variation of the integrator has more pronounced effects
on the space signals than the mark signals. Increasing the variation of the pole position can be
achieved by raising the pulse amplitude. Otherwise the “zero” power levels can increase to some
finite power levels or noises.
Figure 7.10b shows the detected RZ signals using the same conditions as those in Figure 7.10a,
except that the zero location of the differentiator is now at z31 = 0.8. The mark signals in Figure 7.10b
are identical to the corresponding mark signals in Figure 7.10a, while the power levels of the space signals in Figure 7.10b are relatively higher than those of the corresponding space signals in Figure 7.10a.
Similar results were also obtained for the differentiator having zero values z31 = 0.9,1.1.. or, 1.2 as
would be expected from the results obtained in Section 7.1.5 (see Figure 7.3). Thus, the variation of
the differentiator zero has more effect on the space signals than the mark signals, and this result in
decreasing the MTSR. These results are also consistent with those discussed in Section 7.1.5 where
an ideal dark-soliton signal was detected by the differentiator.
7.1.8.3 Performance of the Combined Optical Dark-Soliton
Generator and Butterworth LPOF
The individual performances of the optical dark-soliton generator and Butterworth LPOF (lowpass
optical filter) have been separately analyzed in Sections 7.1.8 and 7.1.6, respectively. The overall
performance of the combined generator and LPF is now described.
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Photonic Signal Processing
FIGURE 7.10 Detected 5 Gb/s bit RZ signals at the output of the optical differentiator, having two different
zero locations, as a result of processing the corresponding propagated dark-soliton signals as shown in
2
Figure 8b. The actual optical power is given by A( Z , t ) = (1 − k1 )(1 − k2 ) PCP0 . (a) Zero location at z31 = 1.
(b) Zero location at z31 = 0.8 .
The propagated signals at Z = 100 Z0, as shown in Figure 7.8b, are detected by the ideal LPF having the zero location z41 = −1, and the resulting inverted NRZ signals are shown in Figure 7.11a. The
solid curve corresponds to the ideal performance of the combined generator and LPF. Figure 7.11a
shows that the pole deviation of the integrator affects both the mark and space signals. Increasing
the variation of the pole value results raises the otherwise zero power levels of the space signals to
some finite values and hence results in reducing the MTSRs.
Figure 7.11b shows the inverted NRZ signals using the same conditions as those in Figure 7.11a
except that the zero location of the LPF is now at z41 = −0.8. The mark signals in Figure 7.11b resemble those in Figure 7.11a, while the power levels of the space signals in Figure 7.11b are relatively
higher than those of the corresponding space signals in Figure 7.11a.
Similar results were also obtained for the LPOF having zero values z41 = −0.9, −1.1, −1.2 as
expected from the results obtained in Section 7.1.6 (see Figure 7.4). Thus, the zero variation of the
FIGURE 7.11 Detected 5 Gb/s bit inverted NRZ signals at the output of the Butterworth LPOF, having
two different zero locations, as a result of processing the corresponding propagated dark-soliton signals as
2
shown in Figure 5.8b. The actual optical power is given by A( Z , t ) = (1 − k1 )k2 PCP0. (a) Zero location z41 = −1.
(b) Zero location z41 = −0.8.
Photonic Ultra-Short Pulse Generators
297
LPOF has more pronounced effects on the space signals than the mark signals and hence decreases
the MTSR. These results also agree with those discussed in Section 7.1.6 where an ideal dark-soliton
signal was detected by the LPF.
In summary, for both cases of the combined generator and differentiator and the combined generator and LPF, it has been found that the variation of the zero location has more pronounced effects
on the space signals than the mark signals.
7.1.9
remarks
• An optical dark-soliton generator and detectors have been designed using integrated-optic
structures to achieve stable generation and efficient detection of high-speed dark-soliton
signals.
• The optical dark-soliton generator, in which the trapezoidal optical integrator is incorporated, can convert a “bright-squared” soliton signal into a fundamental dark-soliton signal.
The generator can generate very stable dark-soliton signals when the optical integrator is
kept operating just within (i.e., less than 1% variation) the unit circle in the z-plane.
• The first-order first-derivative optical differentiator and the first-order Butterworth LPOF,
both of which have been designed using the AMZI, can convert a fundamental dark-soliton
signal into a conventional RZ signal and into an inverted NRZ signal, respectively. For optimum performance, the sampling period of the AMZI, Td , relates to the pulse width, Tw1,
and the bit period, Tb , according to=
Td 4=
Tw1 Tb. For up to 10% variation of the coupling
coefficients of the AMZI, dark-soliton signals can still be detected with high quality with a
MTSR ≥ 80.
• Based on the excellent performance of the generator, impressive results for the combined
generator and differentiator and the combined generator and LPF are obtained.
The AMZI and the all-pole optical filter described here are used in the design of tunable optical
filters in Chapter 4.
7.2 MODE-LOCKED ULTRA-SHORT PULSE GENERATORS
This section gives a detailed account of the design, construction, and characterization of photonic
fiber ring lasers: the harmonic and regenerative mode-locked type for 10 and 40 G-pulse/sec, harmonic detuning type for up to 200 G-pulses/sec, the harmonic repetition multiplication via temporal
diffraction, and the multi-wavelength type.
Mode-locked (ML) laser structures consist of in-line optical fiber amplifiers, guided-wave
optical intensity Mach–Zehnder interferometric modulator (MZIM), and associate optics to
form the structure of a ring resonator for the generation of photonic pulse trains of several GHz
repetition-rate with pulse duration in order of ps or sub-ps. This rate has been extended further to
a few hundred Giga-pulses/sec by using the temporal diffraction Talbot effect that eliminates the
bandwidth limitation of the optical modulator incorporated in the laser ring.
A mode-locked laser operating at 10 GHz repetition rate has been designed, constructed, and
tested. The laser generates optical pulse train of 5 ps pulse width when the modulator is biased at the
phase quadrature quiescent region. An experimental set-up of a 40 GHz repetition rate mode-locked
laser has also been demonstrated. We have achieved long term stability of amplitude and phase
noise. This indicates that the optical pulse source can produce an error-free pseudo random binary
sequence (PRBS) pattern in a self-locking mode for more than 20 hours, the most stable photonic
fiber ring laser reported to date.
The repetition-rate of the mode-locked fiber ring laser is demonstrated up to 200 G-pulses/
sec using a harmonic detuning mechanism in a ring laser. In this system, we investigate the system behavior of rational harmonic mode-locking in the fiber ring laser using phase plane technique. Furthermore, we examine the harmonic distortion contribution to this system performance.
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Photonic Signal Processing
We also demonstrate 660x and 1230x repetition rate multiplications on 100 MHz pulse train generated from an active harmonically mode-locked fiber ring laser, hence, achieving 66 and 123 GHz
pulse operations, which is the highest rational harmonic order reported to date.
The system behavior of the GVD repetition rate multiplication in optical communication systems
is also demonstrated. The stability and the transient response of the multiplied pulses are studied
using the phase plane technique of nonlinear control engineering. We also demonstrated four times
repetition rate multiplication (Section 7.3) on 10 G-pulse/sec pulse train generated from the active
harmonically mode-locked fiber ring laser, hence achieving 40 G-pulses/sec pulse train by using fiber
GVD effect. It has been found that the stability of the GVD multiplied pulse train, based on the phase
plane analysis is hardly achievable even under the perfect multiplication conditions. Furthermore,
uneven pulse amplitude distribution is observed in the multiplied pulse train. In addition to that, the
influences of the filter bandwidth in the laser cavity, nonlinear effect, and the noise performance are
also studied in our analyses.
Finally, we present the theoretical development and demonstration of a multi-wavelength optically amplifier fiber ring laser using an all-polarization-maintaining fiber (PMF) Sagnac loop. The
Sagnac loop simply consists of a PMF coupler and a segment of stress-induced PMF with a singlepolarization coupling point in the loop. The Sagnac loop is shown to be a stable comb filter with
an equal frequency period, which determines the possible output power spectrum of the fiber ring
laser. The number of output lasing wavelengths is obtained by adjusting the polarization state of the
light in the unidirectional ring cavity by means of a polarization controller.
Multi-wavelength ultra-short-width, ultra-high rep-rate photonic generators will be proposed
(Section 7.4) and reported in the near future, by combining the principles of the above photonic
fiber ring lasers.
7.2.1
intrODuctOry remarks On regenerative mODe-lOckeD Fiber laser types
Generation of ultra-short optical pulses with multiple gigabits repetition rate is critical for ultra-high bit
rate optical communications, particularly for the next generation of terabits/sec optical fibers systems.
As the demand for the bandwidth of the optical communication systems increases, the generation of
short pulses with ultra-high repetition rate becomes increasingly important in the coming decades.
The mode-locked fibers laser offers a potential source of such pulse train. Although the generation of ultra-short pulses by mode locking of a multi-modal ring laser is well known, the applications
of such short pulse trains in multi-gigabits/sec optical communications challenges its designers on
its stability and spectral properties. Recent reports on the generation of short pulse trains at repetition rates in order of 40 Gb/s, possibly higher in the near future,23 motivates us to design and
experiment with these sources in order to evaluate whether they can be employed in practical optical
communications systems.
Further, the interest of multiplexed transmission at 160 Gb/s and higher in the foreseeable future
requires us to experiment with optical pulse source having a short pulse duration and high repetition
rates. This section describes laboratory experiments of a mode-locked fibers ring laser (MLFRL),
initially with a repetition rate of 10 GHz and preliminary results of higher multiple repetition rates
up to 40 GHz. The mode-locked ring lasers reported here adopt an active mode locking scheme
whereby partial optical power of the output optical waves is detected, filtered, and a clock signal is
recovered at the desired repetition rate. It is then used as a RF drive signal to the intensity modulator incorporated in the ring laser. A brief description on the principle of operation of the MLFRL
is given in the next section followed by a description of the mode-locked laser experimental set up
and characterization.
Active mode-locked fiber lasers remain a potential candidate for the generation of such pulse
trains. However, the pulse repetition rate is often limited by the bandwidth of the modulator used or
the radio frequency (RF) oscillator that generates the modulation signal. Hence, some techniques
have been proposed to increase the repetition frequency of the generated pulse trains. Rational
Photonic Ultra-Short Pulse Generators
299
harmonic mode-locking is widely used to increase the system repetition frequency.30,31,32 A 40 GHz
repetition frequency has been obtained with a fourth order rational harmonic mode locking at
10 GHz base band modulation frequency.4,25,33 has reported 22nd order rational harmonic detuning in the active mode-locked fiber laser, with 1 GHz base frequency, leading to 22 GHz pulse
operation. This technique is simple and achieved by applying a slight deviated frequency from
the multiple of fundamental cavity frequency. Nevertheless, it is well known that it suffers from
inherent pulse amplitude instability as well as poor long-term stability. Therefore, pulse amplitude
equalization techniques are often applied to achieve better system performance.34,35
Other than this rational harmonic detuning, there are some other techniques that have been
reported and used to achieve the same objective. A fractional temporal Talbot-based repetition rate
multiplication technique36,37 uses the interference effect between the dispersed pulses to achieve the
repetition rate multiplication. The essential element of this technique is the dispersive medium, such
as linearly chirped fiber grating (LCFG) and dispersive fiber.38,39,40 Intra-cavity optical filtering41
uses modulators and a high finesse Fabry-Perot filter (FFP) within the laser cavity to achieve higher
repetition rate by filtering out certain lasing modes in the mode-locked laser. Other techniques used
in repetition rate multiplication include higher order FM mode-locking,42 optical time domain multiplexing,43 and others.
The stability of high repetition rate pulse train generation is one of the main concerns for practical
multi-Giga bits/sec optical communications system. Qualitatively, a laser pulse source is considered
as stable if it is operating at a state where any perturbations or deviations from this operating point
is not increased but suppressed. Conventionally the stability analyses of such laser systems are based
on the linear behavior of the laser in which we can analytically analyze the system behavior in both
time and frequency domains. However, when the mode-locked fiber laser is operating under nonlinear
regime, none of these standard approaches can be used, since direct solution of nonlinear different
equation is generally impossible, and frequency domain transformation is not applicable. Some inherent nonlinearities in the fiber laser may affect its stability and performance, such as the saturation of
K. Kuroda and H. Takakura, Mode-locked ring laser with output pulse width of 0.4 ps, IEEE Trans. Inst. Meas., 48,
1018–1022, 1999.
31 G. Zhu, H. Chen, and N. Dutta, Time domain analysis of a rational harmonic mode-locked ring fiber laser, J. Appl. Phys.,
90, 2143–2147, 2001.
32 C. Wu and N. K. Dutta, High repetition rate optical pulse generation using a rational harmonic mode-locked fiber laser,
IEEE J. Quantum Electron., 36, 145–150, 2000.
33 K. K. Gupta, N. Onodera, K. S. Abedin, and M. Hyodo, Pulse repetition frequency multiplication via intracavity optical
filtering in AM mode-locked fiber ring lasers, IEEE Photon. Technol. Lett., 14, 284–286, 2002.
34 K. S. Abedin, N. Onodera, and M. Hyodo, Higher order FM mode-locking for pulse repetition-rate enhancement in
actively mode-locked lasers: Theory and experiment, IEEE J. Quantum Electron., 35, 875–890, 1999.
35 Y. Shiquan, L. Zhaohui, Z. Chunliu, D. Xiaoyi, Y. Shuzhong, K. Guiyun, and Z. Qida,, Pulse-amplitude equalization in
a rational harmonic mode-locked fiber ring laser by using modulator as both mode locker and equalizer, IEEE Photon.
Technol. Lett., 15, 389–391, 2003.
36 J. Azana and M. A. Muriel, Technique for multiplying the repetition rates of periodic trains of pulses by means of a
temporal self-imaging effect in chirped fiber gratings, Opt Lett., 24, 1672–1674, 1999.
37 S. Atkins and B. Fischer, All optical pulse rate multiplication using fractional Talbot effect and field-to-intensity conversion with cross gain modulation, IEEE Photon. Technol. Lett., 15, 132–134, 2003.
38 D. A. Chestnut, C. J. S. de Matos, and J. R. Taylor, 4x Repetition rate multiplication and Raman compression of pulses
in the same optical fiber, Opt. Lett., 27, 1262–1264, 2002.
39 S. Arahira, S. Kutsuzawa, Y. Matsui, D. Kunimatsu, and Y. Ogawa, Repetition frequency multiplication of mode-locked
using fiber dispersion, J. Light. Technol., 16, 405–410, 1998.
40 W. J. Lai, P. Shum, and L. N. Binh, Stability and transient analyses of temporal Talbot effect-based repetition-rate multiplication mode-locked laser systems, IEEE Photon. Technol. Lett., 16, 437–439, 2004.
41 K. K. Gupta, N. Onodera, K. S. Abedin, and M. Hyodo, Pulse repetition frequency multiplication via intra-cavity optical
filtering in AM mode-locked fiber ring lasers, IEEE Photon. Technol. Lett., 14, 284–286, 2002.
42 K. S. Abedin, N. Onodera, and M. Hyodo, Higher order FM mode-locking for pulse repetition-rate enhancement in
actively mode-locked lasers: Theory and experiment, IEEE J. Quantum Electron., 35, 875–890, 1999.
43 W. Daoping, Z. Yucheng, L. Tangjun, and J. Shuisheng, 20 Gb/s optical time division multiplexing signal generation by
fiber coupler loop-connecting configuration, presented at 4th Optoelectronics and Communications Conference, 1999.
30
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Photonic Signal Processing
the embedded gain medium, non-quadrature biasing of the modulator, nonlinearities in the fiber, and
so forth. Hence, nonlinear stability approach should be used in any laser stability analysis.
In Section 7.2.2, we focus on the stability and transient analyses of the rational harmonic modelocking in the fiber ring laser system using phase plane method, which is commonly used in nonlinear
control system. This technique has been previously used in Ref.40 to study the system performance
of the fractional temporal Talbot repetition rate multiplication systems. It has been shown that it is
an attractive tool in system behavior analysis. However, it has not been used in the rational harmonic
mode-locking fiber laser system. In Section 7.2.4.1, the rational harmonic detuning technique is briefly
discussed. Section 7.2.4.2 describes the experimental setup for the repetition rate multiplication used.
Section 7.2.4.3 investigates the dynamic behavior of the phase plane of the fiber laser system, followed
by some simulation results. Section 7.2.4.4 presents and discusses the results obtained from the experiment and simulation. Finally, some concluding remarks and possible future developments for this type
of ring laser are given. Rational harmonic detuning44 is achieved by applying a slight deviated frequency
from the multiple of fundamental cavity frequency. 40 Ghz repetition frequency has been obtained by
Ref.32 using 10 GHz base band modulation frequency with 4th order rational harmonic mode locking.
This technique is simple in nature. However, this technique suffers from inherent pulse amplitude instability, which includes both amplitude noise and inequality in pulse amplitude, furthermore, it gives poor
long-term stability. Hence, pulse amplitude equalization techniques are often applied to achieve better
system performance.31,33,35 Fractional temporal Talbot based repetition rate multiplication technique8
uses the interference effect between the dispersed pulses to achieve the repetition rate multiplication.
The essential element of this technique is the dispersive medium, such as LCFG36,44 and single-mode
fiber.30,31 This technique will be discussed further in Section 7.2.2. Intra-cavity optical filtering42,45 uses
modulators and a high finesse FFP within the laser cavity to achieve higher repetition rate by filtering
out certain lasing modes in the mode-locked laser. Other techniques used in repetition rate multiplication include higher order FM mode-locking, optical time domain multiplexing, etc.
Although Talbot based repetition rate multiplication systems are based on the linear behavior of
the laser, there are still some inherent nonlinearities affecting its stability, such as the saturation of
the embedded gain medium, non-quadrature biasing of the modulator, nonlinearities in the fiber,
and others; hence, nonlinear stability approach must be adopted.
In Section 7.3, we focus on the stability and transient analyses of the GVD multiplied pulse train
using the phase plane analysis of nonlinear control analytical technique. This was the first time
that the phase plane analysis is being used to study the stability and transient performances of the
GVD repetition rate multiplication systems. In Section 7.3.1, the GVD repetition rate multiplication technique is briefly given. Section 7.3.2 describes the experimental setup for the repetition rate
multiplication. Section 7.3.3 investigates the dynamic behavior of the phase plane of GVD multiplication system, followed by some simulation results. Section 7.3.4 presents and discusses the results
obtained from the experiment. Finally, some concluding remarks and possible future developments
for this type of lasers are given.
Due to the enormous practical applications in components testing and DWDM optical networks and considering that the complexity and cost of optical sources would increase as the number of wavelength channels increases, multi-wavelength optical sources capable of generating
a large number of wavelengths have been an area of strong and continuing worldwide research
interest. Erbium-doped fiber lasers operating in the 1550-nm window are widely used in WDM
systems, fiber-optic sensor systems, and photonic true-time delay beam forming systems.46,47
D. L. A. Seixasn and M. C. R. Carvalho, 50 GHz fiber ring laser using rational harmonic mode-locking, presented at
Microwave and Optoelectronics Conference, 2001.
45 K. S. Abedin, N. Onodera, and M. Hyodo, Higher order FM mode-locking for pulse repetition-rate enhancement in
actively mode-locked lasers: Theory and experiment, IEEE J. Quantum Electron., 35, 875–890, 1999.
46 R. Y. Kim, Fiber lasers and their applications, presented at Laser and Electro-Optics, CLEO Pacific Rim’95, 1995.
47 H. Zmuda, R. A. Soref, P. Payson, S. Johns, and E. N. Toughlian, Photonic beam former for phased array antennas using
a fiber grating prism, IEEE Photon. Technol. Lett., 9, 241–243, 1997.
44
Photonic Ultra-Short Pulse Generators
301
Among these different types of fiber lasers, unidirectional traveling-wave ring lasers have been
studied extensively in recent years due to several advantages, such as elimination of back scattering and spatial hole-burning effects. Several types of optical filters, such as fiber Bragg gratings
(FBGs),33,34 Fabry-Perot filters,48 multi-mode filters,49 and high-birefringence fibers,50 have been
used in the construction of multi-wavelength fiber ring lasers. Multi-wavelength fiber lasers using
FBGs as wavelength-selective filters will require a large number of individual FBGs to be written
on one single segment of fiber depending on the number of lasing wavelengths to be generated.
This type of fiber laser has a high insertion loss and is costly due to a large number of phase masks
required to fabricate the FBGs.
The Sagnac loop simply consists of a PMF coupler and a segment of stress-induced PMF with a
single-polarization coupling point in the loop. The Sagnac loop is shown to be a stable comb filter
with equal frequency period, which determines the possible output power spectrum of the fiber ring
laser. The number of output lasing wavelengths is obtained by adjusting the polarization state of the
light in the unidirectional ring cavity by means of a polarization controller.
Sagnac fiber interferometers have several applications, such as gyroscopes, magnetic field
sensors, and secure optical communications. A Sagnac loop filter simply consists of a fiber coupler
whose two ends are spliced to the two ends of a segment of optical fiber (see Figure 7.49a). The
Sagnac loop can function as a reflector or periodic filter depending on the birefringence with the
fiber loop. The Sagnac loop is simply a reflector when there is no birefringence in the fiber loop.
Due to the frequency dependence of the coupler, the coupler acts as a reflector when its coupling
coefficient is 50% and as a partial reflector for other values of coupling coefficient. The Sagnac
loop as a reflector has been used in the fabrication of fiber laser with a linear cavity.51 Also the
Sagnac loop can function as a periodic filter when there is some birefringence induced in the
loop.52,53 Compared with a Mach–Zehnder interferometer, the Sagnac loop filter is more robust
against environmental changes because it is a two-beam interferometer with one common path.
However, the Sagnac filter may not operate stably when only standard single-mode fibers are used
to form the coupler and the loop. This is because the small fiber core imperfection, external stress,
bending, twisting, and temperature variation may cause the spliced sections of the coupler and
the loop to behave as a birefringent media whose birefringence changes randomly with time. It
should be noted that references54 and53 do not provide a complete theoretical basis of the effect of
the linear birefringence on the performance of the Sagnac filter, which is presented in this section.
This kind of birefringent loop filter has been placed inside the linear cavity of a fiber laser with
a polarization controller inside the loop, because the coupler was made of standard single-mode
fiber (SSMF).55
In Section 7.4, we present a theoretical analysis and implementation of a multi-wavelength fiber
ring laser (MWFRL) with a Sagnac PMF loop filter. It is shown that when a single-polarization
D. Wei, T. Li, Y. Zhao, and S. Jian, Multi-wavelength erbium-doped fiber ring laser with overlap-written fiber Bragg
gratings, Opt. Lett., 25, 1150–1152, 2000.
49 S. K. Kim, M. J. Chu, and J. H. Lee, Wideband multi-wavelength erbium-doped fiber ring laser with frequency shifted
feedback, Opt. Commun., 190, 291–302, 2001.
50 R. M. Sova, C. S. Kim, and J. U. Kang, Tunable dual-wavelength all-PM fiber ring laser, IEEE Photon. Technol. Lett.,
14, 287–289, 2002.
51 I. D. Miller, D. B. Mortimore, P. Urquhart et al., A Nd3+:doped cw fiber laser using all fiber reflectors, Appl. Opt., 26,
2197–2201, 1987.
52 X. Fang and R. O. Claus, Polarization-independent all-fiber wavelength-division multiplexer based on a Sagnac interferometer, Opt. Lett., 20, 2146–2148, 1995.
53 X. Fang, H. Ji, C. T. Aleen, A compound high-order polarization-independent birefringence filter, IEEE Photon.
Technol. Lett., 19, 458–460, 1997.
54 X. Fang and R. O. Claus, Polarization-independent all-fiber wavelength-division multiplexer based on a Sagnac interferometer, Opt. Lett., 20, 2146–2148, 1995.
55 X. P. Dong, Li, S., K. S. Chiang et al., Multi-wavelength erbium-doped fiber laser based on a high-birefringence fiber
loop, Electron. Lett., 36, 1609–1610, 2000.
48
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coupling point is formed by the stress-induced PMF loop, the fiber laser can provide stable generation of multiple lasing wavelengths with equal wavelength spacing. Section 7.4.1 presents the theory
of the Sagnac PMF loop filter, which consists of a PMF coupler (instead of a standard single-mode
fiber coupler used in previous works as described above) and a segment of PMF. Section 7.4.2
presents the experimental results and discussion.
7.2.2
ultra-hIgh repetItIon-rate FIber Mode-locked lasers
This section gives a detailed account of the design, construction and characterization of a modelocked (ML) fibers ring laser. The ML laser structure employs in-line optical fibers amplifiers,
a guided-wave optical intensity Mach–Zehnder interferometric modulator (MZIM), and associate
optics to form a ring resonator structure generating optical pulse trains of several GHz repetition
rate with pulse duration in order of pico-seconds. Long term stability of amplitude and phase noise
has been achieved indicating that the optical pulse source can produce an error-free PRBS pattern
in a self-locking mode for more than 20 hours. A mode-locked laser operating at 10 GHz repetition
rate has been designed, constructed, tested, and packaged. The laser generates an optical pulse train
of 5 ps pulse width when the modulator is biased at the phase quadrature quiescent region. A preliminary experiment of a 40 GHz repetition rate mode-locked laser has also been demonstrated.
Although it is still unstable in long term without an O/E feedback loop, optical pulse trains have
been observed.
7.2.2.1 Mode-Locking Techniques and Conditions for Generation
of Transform Limited Pulses from a Mode-Locked Laser
7.2.2.1.1 Schematic Structure of MLRL
Figures 7.12 and 7.13 show the composition of a MLRL without and with feedback loop used in
this study respectively. It consists principally, for a non-feedback ring, an optical close loop with an
optical gain medium, an optical modulator (intensity or phase type) an optical fibers coupler and
associated optics. An O/E feedback loop detecting and repetition-rate signal and generating RF
Frequency tuning
to loop resonance frequency
Word (Packet)
H - PL
generator - CLK
Fiber connection
delay T
clk
RF Amp
RF CW signal
Re-circulating
EDFA
loop comp
Loop
MZI
DC bias
PC
OF
2
1
3 90%
10:90
Analog CRO
HS-30
4 10%
Power meter
High Speed
Digital
Sampling
CRO
FIGURE 7.12 Schematic arrangement of a mode-locked ring laser without the active feedback control.
303
Photonic Ultra-Short Pulse Generators
10 GHz repetition rate - ps pulse width EFA-MLL
Basic fiber 2-STAGE EDFA ring
-10 or -25 dB
RF dir.coupler
Fiber connection
delay T
10GHz BP Amp
DC bias
Adjustable
phase shifter
RF Amp
EDFA Section 1
10GHz BPF
clock recovery
Loop
MZI
Re-circulating
port
3nm OF
Linear amplifier
PC
2
3
90%
1 10:90 4 10%
FC
SYNC
LD det.
EDFA Section 2
Opt.att
3dB FC
OUTPUT
PULSES
HP- det
High Speed
Digital
Sampling
CRO
FIGURE 7.13 Schematic arrangement of a mode-locked ring laser with an O/E-RF electronic active
feedback loop.
sinusoidal waves to electro-optically drive the intensity modulator is necessary for the regenerative
configuration as shown in Figure 7.13.
7.2.2.1.2 Mode-Locking Conditions
The basic conditions for MLRL to operate in pulse oscillation are:
7.2.2.1.2.1 For Non-feedback Optical Mode-Locking
Condition 1: The total optical loopgain must be greater than unity when the modulator is
ON-state, i.e. when the optical waves transmitting through the MZIM is propagating in
phase;56
Condition 2: The optical lightwaves must be depleted when the optical modulator is in the
OFF-state, i.e. when the lightwaves of the two branches of the MZIM is out of phase or in
destructive interference mode;57
Condition 3: The frequency repetition rate at a locking state must be a multiple number of the
fundamental ring resonant frequency.58
7.2.2.1.2.2 For Optical-RF Feedback Mode Locking—Regenerative Mode-Locking
Condition 4: Under an O/E-RF feedback to control modulation of the intensity modulator the optical noise at the output of the laser must be significantly greater than that of the electronic noise for
the start-up of the mode locking and lasing. In other words the loopgain of the optical-electronic
feedback loop must be greater than unity.
Thus, it is necessary that the EDF amplifiers are operated in saturation mode, and the total average optical power of the lightwaves circulating in the loop must be sufficiently adequate for the
J. Azana and M. A. Muriel, Technique for multiplying the repetition rates of periodic trains of pulses by means of a
temporal self-imaging effect in chirped fiber gratings, Opt. Lett., 24, 1672–1674, 1999.
57 D. Jones, H. Haus, and E. Ippen, Sub-picosecond solitons in an actively mode-locked fiber laser, Opt. Lett., 1818–1820,
1996.
58 X. Zhang, M. Karlson, and P. Andrekson, Design guideline for actively mode-locked fiber ring lasers, IEEE Photon.
Tech. Lett., 1103–1105, 1998.
56
304
Photonic Signal Processing
detection at the photo-detector and electronic preamplifier. Under this condition the optical quantum shot noise dominates the electronic shot noise.
7.2.2.1.3
Factors Influencing the Design and Performance of Mode-Locking
and Generation of Optical Pulse Train
The locking frequency is a multiple of the fundamental harmonic frequency of the ring defined as
the inverse of the traveling time around the loop and is given by
f RF =
Nc
neff L
(7.21)
where:
f RF is the RF frequency required for locking and the required generation rate
N is an integer and indicates mode number order, c is the velocity of light in vacuum
neff is the effective index of the guided propagating mode
L is the loop length including that of the optical amplifiers.
Under the requirement of the OC-192 standard bit rate, the locking frequency must be in the region
of 9.95 G-pulses/sec or 50 GHz channel spacing for C-band grid if used as frequency comb derived
from ultra-short pulse spectrum.59,60 The laser must be locked to a very high order of the fundamental loop frequency that is in the region of 1 to 10 MHz depending on the total ring length. For an
optical ring of length about 30 meters and a pulse repetition rate of 10 GHz, the locking occurs on
approximately the 1400th harmonic mode.
It is noted also that the effective refractive index n can be varied in different sections of the
optical components forming the laser ring. Furthermore, the two polarized states of propagating
lightwaves in the ring, if the fibers is not a polarization maintaining type, would form two simultaneously propagating rings, and they could interfere or hop between these dual polarized rings.
The pulse width, denoted as ∆τ of the generated optical pulse trains, can be found to be given by61
1/ 4
α G 
∆τ = 0.45  t t 
 ∆m 
1
( f RF ∆υ )1/ 2
(7.22)
with ατΓ τ is the round trip gain coefficient as a product of all the loss and gain coefficients of all
optical components including their corresponding fluctuation factor, ∆m is the modulation index and
∆ν is the overall optical bandwidth (in units of Hz) of the laser.
Hence, the modulation index and the bandwidth of the optical filter influence the generated pulse
width of the pulse train. However, the optical characteristics of the optical filters and optical gain
must be flattened over the optical bandwidth of the transform limit for which a transform limited
pulse must satisfy, for a sech2 pulse intensity profile, the relationship
∆τ∆υ = 0.315
(7.23)
Similarly, for Gaussian pulse shape the constant becomes 0.441. The fluctuation of the gain or loss
coefficients over the optical flattened region can also influence the generated optical pulse width
and mode locking condition.
R. P. Scott, N. K. Fontaine, J. P. Heritage, B. H. Kolner and S. J. B. Yoo, 3.5-THz wide, 175 mode optical comb source,
Opt Fiber Conf. 2007, Paper OWJ3, Annaheim, CA.
60 D. J. Geisler, N. K. Fontaine, R. P. Scott, T. He, K. Okamoto, J. P. Heritage, and S. J. Ben Yoo, 3 b/s/Hz 1.2 Tb/s packet
generation using optical arbitrary waveform generation based optical transmitter, OFC 2009, paper JTh8.
61 X. Zhang, M. Karlson, and P. Andrekson, Design guideline for actively mode-locked fiber ring lasers,IEEE Photon.
Tech. Lett., 1103–1105, 1998.
59
305
Photonic Ultra-Short Pulse Generators
For regenerative mode-locking case, as illustrated in Figure 7.13, the optical output intensity is split
and O/E detected, we must consider the sensitivity and noises generated at the photo-detector (PD).
Two major sources of noises are generated at the input of the PD, firstly the optical quantum shot
noises generated by the detection of the optical pulse trains and secondly the random thermal electronic noises of the small signal electronic amplifier which follows that detector. Usually the electronic
amplifier would have a 50 Ω equivalent input resistance R referred to the input of the optical preamplifier as evaluated at the operating repetition frequency, this gives a thermal noise spectral density of
SR =
4kT
⋅ A 2 / Hz
R
(7.24)
with k the Boltzmann’s constant. This equals to 3.312 × 10 −22 A2/Hz at 300 K. Depending on the
electronic bandwidth Be of the electronic pre-amplifier, i.e. wideband or narrow band type, the total
2
equivalent electronic noise (square of noise “current”) is given by, iNT
= S R Be. Under the worst case,
when a wide-band amplifier of a 3-dB electrical bandwidth of 10 GHz, the equivalent electronic
noise at the input of the electronic amplifier is 3.312 × 10 −11 A2, i.e. an equivalent noise current of
5.755 µA is present at the input of the “clock” recovery circuit. If a narrow bandpass amplifier of
50 MHz 3-dB bandwidth centered at 10 GHz is employed, this equivalent electronic noise current
is 0.181 µA.
Now, considering the total quantum shot noise generated at the input of the “clock” recovery
circuit, suppose that a 1.0 mW (or 0 dBm) average optical power is generated at the output of the
MLRL, then a quantum shot noise62 of approximately 2.56 × 10 −22 A2/Hz (i.e., an equivalent electronic noise current of 16 nA) is present at the input of the clock recovery circuit. This quantum shot
noise current is substantially smaller than that of the electronic noise.
In order for the detected signal at the optical receiver incorporated in the “clock” recovery circuit
to generate a high signal-to-noise ratio, the optical average power of the generated pulse trains must
be high, at least at a ratio of 10. We estimate that this optical power must be at least 0 dBm at the PD
in order for the MLRL to lock efficiently to generate a stable pulse train.
Given that a 10% fiber coupler is used at the optical output and an estimate optical loss of about
12 dB, which is due to coupling, connector loss, and attenuation of all optical components employed
in the ring, the total optical power generated by the amplifiers must be about 30 dBm. To achieve this,
we employ two EDF amplifiers of 16.5 dBm output power each positioned before and after the optical coupler. One is used to compensate for the optical losses, and the other for generating sufficient
optical gain and power to dominate the electronic noise in the regenerative loop.
7.2.3
mll anD mrll experimental setup anD results
The experimental setups for MLL and RMLL are shown, again, in Figures 7.12 and 7.13. Associate
equipment used for monitoring of the mode locking and measurement of the lasers are also included.
However, we note the following:
In order to lock the lasing mode of the MLL to a certain repetition rate or multiple harmonic of
the fundamental ring frequency, a synthesizer is required to generate the required sinusoidal waves
for modulating the optical intensity modulator and tuned to a harmonic of the cavity fundamental
frequency.
A signal must be created for the purpose of triggering the digital oscilloscope to observe the
locking of the detected optical pulse train. For the HP-54118A amplitude of this signal must
be >200 mV. This is also critical for the RMLL set up as the RF signal detected and phase locked
via the clock recovery circuitry must be spit to generate this triggering signal.
62
By using the relationship of the quantum noise spectral density of 2qRPav with Pav the average optical power, q the electronic
charge and R the responsivity of the detector.
306
Photonic Signal Processing
Typical experimental procedures are: (i) After the connection of all optical components with
the ring path broken, ideally at the output of the fibers coupler, a CW optical source can be used
to inject optical waves at a specific wavelength to monitor the optical loss of the ring; (ii) Close
the optical ring and monitor the average optical power at the output of the 90:10 fibers coupler and
hence estimate the optical power available at the PD is about −3 dBm after a 50:50 fibers coupler;
(iii) Determine whether an optical amplifier is required for detecting the optical pulse train or
whether this optical power is sufficient for O/E RF feedback condition as stated above; (iv) Set the
biasing condition and hence the bias voltage of the optical modulator; (v) Tune the synthesizer or the
electrical phase to synchronize the generation and locking of the optical pulse train.
One could observe the following: (i) The optical pulse train generated at the output of the MLL
or RMML. Experimental set up is shown in Figure 7.14; (ii) Synthesized modulating sinusoidal
waveforms can be monitored as shown in Figures 7.16 through 7.18. Figure 7.16 illustrates the
mode locking of an MLL operating at around 2 GHz repetition rate with the modulator driven from
a pattern generator.
Figures 7.17 and 7.18 show the sinusoidal waveforms generating when the MLRL is operating
at the self-mode-locking state. (iii) The interference of other super-modes of the MLL without RF
feedback for self-locking is indicated in Figure 7.16.; (iv) Observed optical spectrum (not available
in electronic form); (v) Electrical spectrum of the generated pulse trains was observed showing a
70-dB super-mode suppression under the locked state of the regenerative MLRL; (vi) Figures 7.17
through 7.19 show that the regenerative MLRL can be operating under the cases when the modulator
is biased either at the positive or at the negative going slope of the optical transfer characteristics of
the Mach–Zehnder modulator.
Optical pulse width is measured using an optical auto-correlator (slow or fast scan mode). Typical
pulse width obtained with the slow scan auto-correlator is shown in Figure 7.21. Minimum pulse
duration obtained was 5 ps with a time-bandwidth product of about 3.8 showing that the generated
pulse is near transform limited.
The bit error rate (BER) measurement can be used to monitor the stability of the regenerative
MLRL. The BER error detector was then programmed to detect all “1” at the decision level at
a tuned amplitude level and phase delay. The clock source used is produced by the laser itself.
This set up is shown in Figure 7.15 with an error-free has been achieved for over 20 hours. The
O/E detected waveform of the output pulse train for testing the BER is shown in Figure 7.20.
Clock signals generated from RMLL
Trig input
OPTICAL PULSE
TRAINS GENERATED
FROM
R-MLL
Opt. Att.
HP-34GHz
If req. dep. pin DECTECTOR
on detector
or
Fermionics HSD-30
HP-54118A
0.5 - 18 GHz
Output trig
HP-54123A (DC-34
GHz)
input
Channel inputs
HP-54118A
sampling head and display unit
FIGURE 7.14 Experimental set up for monitoring the locking of the photonic pulse train.
307
Photonic Ultra-Short Pulse Generators
BER MEASUREMENT OF RMLL-EDF
FIGURE 4
BER circuitry and equipment set-up
Optical signals
from output port of RMLL
To CLK recovery and feedback to drive MZIM
3 dB FC
PIN bias (9V)
HBT TI amp supply (11.5V)
DC blocking
cap
Linear power amplifier to provide
> 0.2 V signals for BER input port
RMLL Clock
delay
Set @ ALL ‘1’
16-bit length periodic
Nortel PP-10G
500 Ω TI optical receiver
RF amp AL-7
MA Ltd
BER detector
ANRITSU MP1764A
RF amp
ERA
10GHz RZ sinusoidal waveform
Data IN
CLK
Clk freq: 9.947303 GHz var. 100 MHz
FIGURE 7.15 Experimental set up for monitoring the BER of the photonic pulse train.
After 20 hours operation, the recorded waveform is obtained under infinite persistence mode
of the digital oscilloscope.
A drift of clock frequency of about 20 kHz over one hour in an open laboratory environment is
observed. This is acceptable for a 10 GHz repetition rate.
The “clock” recovered waveforms were also monitored at the initial locked state and after the
long-term test, as shown in Figures 7.17 and 7.18, respectively. Figure 7.18 was obtained under the
infinite persistence mode of the digital oscilloscope.
We note the following factors which are related to the above measurements (Figures 7.16
through 7.18):
• All the above measurements have been conducted with two distributed optical amplifiers
(GTi EDF optical amplifiers) driven at 180 mA and with a specified output optical power
of 16.5 dBm.
• Optical pulse trains are detected with 34 GHz 3 dB bandwidth HP pin detector directly
coupled to the digital oscilloscope without using any optical pre-amplifier (Figures 7.19
through 7.21).
7.2.3.1 40 GHz Regenerative Mode-Locked Laser
Following our initial success of the construction and testing of a regenerative MLRL at 40 GHz
repetition rate, a regenerative mode-locked laser was constructed. The schematic arrangement of the
40 G regenerative MLRL is shown in Figure 7.22. Initial observation of the locking and generation
of the laser has been observed and progress of this laser design and experiments will be reported in
the near future.
308
Photonic Signal Processing
Modulation frequency = 2,143,500 kHz
SCOPE 2
2 GHz ML_EDFA_L Competition from other supermodes
Generated -Detected Laser pulses
Neg CLK PULSES
0.000501
Generated output pulses
0.000401
@ coupler port 4
0.000301
0.000201
Clock for modulating
MZI Modulator in-loop
0.000101
9.99996E-07
00
00
8.
50
19
8.
00
00
00
30
19
8.
00
00
00
8.
10
19
00
90
18
18
70
8.
00
8.
50
00
00
00
00
18
8.
30
18
10
8.
00
00
00
00
18
00
8.
90
17
8.
70
17
17
50
8.
00
00
00
-9.9E-05
Time (picoseconds)
FIGURE 7.16 Detected pulse train at the MLRL output tested at a multiple frequency in the range of 2 GHz
repetition frequency.
31.375
Regenerative mode-locked generated pulse trains:
10 G rep rates - detected by 34 GHz
(3dB BW HP-detector into 50 Ω)
SCOPE 3
21.375
Generated pulse trains
100 ps spacing - pulse
width ~ 12 ps - limited
by PD bandwidth
11.375
1.375
-8.625
-18.625
Clock recovery signals
for driving MZIM
-28.625
-38.625
10
00
9.
69
16
16
59
9.
00
10
10
16
49
9.
00
10
16
39
9.
00
10
00
9.
29
16
16
19
9.
00
10
-48.625
Time (picoseconds)
FIGURE 7.17 Output pulse trains of the regenerative MLRL and the RF signals as recovered for modulating
the MZIM for self-locking.
309
Photonic Ultra-Short Pulse Generators
43
RML laser with V(MZIM bias)=1.9V
(more sensitive to competition of supermodes than
biased @ 0 V - maximum transmission bias)
SCOPE 4
Generated pulse train
33
23
13
3
-7
Clocked regen for
modulating MZIM
-17
-27
67
66
01
0.
01
0.
66
65
01
0.
01
65
0.
0.
01
64
01
64
0.
01
63
0.
63
01
0.
01
62
0.
01
0.
0.
01
62
-37
FIGURE 7.18 Detected output pulse trains of the regenerative MLRL and recovered clock signal when the
MZIM is biased at a negative going slope of the operating characteristics of the modulator.
RMLLL-09-May-2000 V(bias)= 9.34 Volts
09-May-2000 Regen MLL
RF pulses to drive MZIM
Generated optical output pulses
43
10G ps-pulses generated
clk & RF signals to MZIM
33
23
13
3
-7
-17
-27
0
.6
99
0
16
49
.6
16
0
99
.5
16
16
.5
49
0
0
99
.4
16
0
49
.4
16
.3
99
0
0
16
.3
49
16
.2
99
0
16
0
.2
49
16
16
.1
99
0
-37
Time (nanoseconds)
FIGURE 7.19 Output pulse trains and clock recovered signals of the 10 G regenerative MLRL when the
modulator is biased at the positive going slope of the modulator operating transfer curve.
7.2.3.2 Remarks
We have successfully constructed a mode-locked laser operating under an open loop condition and
with O/E RF feedback providing regenerative mode locking. The O/E feedback can certainly provide a self-locking mechanism under the condition when the polarization characteristics of the ring
laser are manageable. This is done by ensuring that all fibers have a path under constant operating
condition. The regenerative MLRL can self-lock even under the DC drifting effect of the modulator
310
Photonic Signal Processing
BER measurement
BER=0 x 10-15 measured for over 20 hours - RF (CLK) frequency
varied from 9.954 to 9.952 GHz gradually over measurement period
BER measurement - clock signal - infinite persistence
non- pulses
clock RF signals while BER measurement
280
180
80
-20
-120
-220
0
0
99
.6
16
0
.6
16
99
.5
16
49
0
0
49
.5
16
0
99
.4
16
0
16
.4
49
0
99
.3
16
0
16
.3
49
0
99
16
.2
49
99
.1
16
16
.2
0
-320
Time (nanoseconds)
FIGURE 7.20 BER measurement – O/E detected signals from the generated output pulse trains for BER test
set measurement. The waveform is obtained after 20 hours persistence.
Auto-correlated pulse
V(bias) = 1.55 volt - phase quadrature neg slop
FWHM ∆τ = 4.48 ps
autocorrelator set thumbwheel 6 and 0 (100ps range @10ps/s delay rate)
FIGURE 7.21 Auto-correlation trace of output pulse trains of 9.95 GHz regenerative MLRL.
bias voltage (over 20 hours).63 The generated pulse trains of 5 ps duration can be, without difficulty,
compressed further to less than 3 ps for 160 Gb/s optical communication systems.
The regenerative MLRL can be an important source for all-optical switching of an optical packet
switching system. We recommend the following for successful construction and operation of the regenerative MLRL: (i) Eliminating polarization drift through the use of Faraday mirror or all polarization
63
Typically the DC bias voltage of a LiNbO3 intensity modulator is drifted by 1.5 volts after 15 hours of continuous operation.
311
Photonic Ultra-Short Pulse Generators
FIGURE 3
40 GHz repetition rate - ps pulse width EFA-MLL
SCC 40 GHz MZ modulator
Basic fiber 2-STAGE EDFA ring
Fiber connection
delay T
-10 or -25 dB
RF dir.coupler
40 GHz BP Amp
DC bias
Adjustable
phase shifter
RF BPF Amp
40 GHz BPF
EDFA Section 1
Loop
MZI
Re-circulating
port
Linear BP amplifier
3nm OF
PC
2
EDFA Section 2
NTT 40 GHz
det.
90%
3
Opt.att
10%
FIGURE 7.22
SYNC
HP-det
High Speed
Digital
Sampling
Set up of a 40 G regenerative MLRL.
maintaining (PM) optical components, for example polarized Er-doped fiber amplifiers, PM fibers at
the input and output ports of the intensity modulator; (ii) Stabilizing the ring cavity length with appropriate packaging and via piezo/thermal control to improve long term frequency drift; (iii) Controlling
and automatic tuning of the DC bias voltage of the intensity modulator; (iv) Developing electronic
RF ‘clock’ recovery circuit for regenerative MLRL operating at 40 GHz repetition rate together with
appropriate polarization control strategy; (v) Studying the dependence of the optical power circulating
in the ring laser by varying the output average optical power of the optical amplifiers under different
pump power conditions; and (vi) Incorporating a phase modulator, in lieu of the intensity modulator, to
reduce the complexity of polarization dependence of the optical waves propagating in the ring cavity,
thus minimizing the bias drift problem of the intensity modulator.
7.2.4
active mODe-lOckeD Fiber ring laser by ratiOnal harmOnic Detuning
In this section, we investigate the system behavior of rational harmonic mode-locking in the fiber
ring laser using phase plane technique of the nonlinear control engineering. Furthermore, we examine the harmonic distortion contribution to this system performance. We also demonstrate 660x and
1230x repetition rate multiplications on 100 MHz pulse train generated from an active harmonically
mode-locked fiber ring laser, hence achieving 66 and 123 GHz pulse operations by using rational
harmonic detuning, which is the highest rational harmonic order reported to date.
7.2.4.1 Rational Harmonic Mode-Locking
In an active harmonically mode-locking fiber ring laser, the repetition frequency of the generated
pulses is determined by the modulation frequency of the modulator, fm = qfc, where q is the qth harmonic of the fundamental cavity frequency, fc, which is determined by the cavity length of the laser,
fc = c/nL, where c is the speed of light, n is the refractive index of the fiber, and L is the cavity length.
Typically, fc is in the range of kHz or MHz. Hence, in order to generate GHz pulse train, mode-locking
is normally performed by modulation in the states of q >> 1, i.e. q pulses circulating within the
cavity, which is known as harmonic mode-locking. By applying a slight deviation or a fraction of the
fundamental cavity frequency, ∆f = fc/m, where m is the integer, the modulation frequency becomes
312
Photonic Signal Processing
f m = qf c ±
fc
m
(7.25)
This leads to an m-times increase in the system repetition rate, fr = mf m, where fr is the repetition
frequency of the system. When the modulation frequency is detuned by an m fraction, the contributions of the detuned neighboring modes are weakened, only every mth lasing mode oscillates
in phase and the oscillation waveform maximums accumulate, hence achieving in m times higher
repetition frequency. However, the small but not negligible detuned neighboring modes affect the
resultant pulse train, which leads to uneven pulse amplitude distribution and poor long term stability. This is considered as harmonic distortion in our modeling, and it depends on the laser linewidth and amount of detuned, i.e. fraction m. The amount of the allowable detunes or rather the
obtainable increase in the system repetition rate by this technique is very limited by the amount
harmonic distortion. When the amount of frequency detuned is too small relative to the modulation frequency, i.e. large m, contributions of the neighboring lasing modes become prominent,
thus reducing the repetition rate multiplication capability significantly. Another words, no repetition
frequency multiplication is achieved when the detuned frequency is unnoticeably small. Often the
case, it is considered as the system noise due to improper modulation frequency tuning. In addition,
the pulse amplitude fluctuation is also determined by this harmonic distortion.
7.2.4.2 Experiment
The experimental setup of the active harmonically mode-locked fiber ring laser is shown in
Figure 7.23. The principal element of the laser is an optical close loop with an optical gain medium,
a Mach–Zehnder amplitude modulator (MZM), an optical polarization controller (PC), an optical
bandpass filter (BPF), optical couplers, and other associated optics.
The gain medium used in our fiber laser system is an erbium doped fiber amplifier (EDFA) with
a saturation power of 16 dBm. A polarized, independent optical isolator is used to ensure unidirectional lightwave propagation as well as to eliminate back reflections from the fiber splices and
optical connectors. A free space filter with a 3-dB bandwidth of 4 nm at 1555 nm is inserted into the
cavity to select the operating wavelength of the generated signal and to reduce the noise in the system. In addition, it is responsible for the longitudinal modes selection in the mode-locking process.
The birefringence of the fiber is compensated by a polarization controller, which is also used for
the polarization alignment of the linearly polarized lightwave before entering the planar structure
modulator for better output efficiency. Pulse operation is achieved by introducing an asymmetric
coplanar traveling wave 10 Gb/s Ti:LiNbO3 Mach–Zehnder amplitude modulator into the cavity
with a half-wave voltage, Vπ of 5.8 V and an insertion loss of ≤7 dB. The modulator is DC biased
FIGURE 7.23 Schematic diagram of an active mode-locked fiber ring laser.
Photonic Ultra-Short Pulse Generators
313
near the quadrature point and not more than the Vπ such that it operates around the linear region of
its characteristic curve. The modulator is driven by a 100 MHz, 100 ps step recovery diode (SRD),
which is in turn driven by a RF amplifier (RFA)—a RF signal generator. The modulating signal
generated by the step recovery diode is a ~1% duty cycle Gaussian pulse train. The output coupling
of the laser is optimized using a 10/90 coupler. 90% of the optical field power is coupled back into
the cavity ring loop, while the remaining portion is taken out as the output of the laser and analyzed.
7.2.4.3 Phase Plane Analysis
Nonlinear system frequently has more than one equilibrium point. It can also oscillate at fixed
amplitude and fixed period without external excitation. This oscillation is called the limit cycle.
However, limit cycles in nonlinear systems are different from linear oscillations. First, the amplitude of self-sustained excitation is independent of the initial condition, while the oscillation of a
marginally stable linear system has its amplitude determined by the initial conditions. Second, marginally stable linear systems are very sensitive to changes, while limit cycles are not easily affected
by parameter changes.53,64
Phase plane analysis is a graphical method of studying second-order nonlinear systems. The
result is a family of system motion trajectories on a two-dimensional plane, which allows us to visually observe the motion patterns of the system. Nonlinear systems can display more complicated
patterns in the phase plane, such as multiple equilibrium points and limit cycles. In the phase plane,
a limit cycle is defined as an isolated closed curve. The trajectory has to be both closed, indicating
the periodic nature of the motion, and isolated, indicating the limiting nature of the cycle.53
The system modeling of the rational harmonic mode-locked fiber ring laser system is done based
on the following assumptions: (i) detuned frequency is perfectly adjusted according to the fraction
number required, (ii) small harmonic distortion, (iii) no fiber nonlinearity is included in the analysis, (iv) no other noise sources are involved in the system, and (v) Gaussian lasing mode amplitude
distribution analysis.
The phase plane of a perfect 10 GHz mode-locked pulse train without any frequency detune is
shown Figure 7.24 and the corresponding pulse train is shown in Figure 7.25a. The shape of the
FIGURE 7.24 Phase plane of a 10 GHz mode-locked pulse train. (solid line – real part of the energy, dotted
line – imaginary part of the energy, x-axes – E(t) and y-axes – E′(t)).
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J. J. E. Slotine and W. Li, Applied Nonlinear Control, Englewood Cliffs, NJ: Prentice Hall, 1991.
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Photonic Signal Processing
FIGURE 7.25 Normalized pulse propagation of original pulse (a); detuning fraction of 4, with 0% (b) 5%
(c) harmonic distortion noise.
phase plane exposes the phase between the displacement and its derivative. From the phase plane
obtained, one can easily observe that the origin is a stable node and the limit cycle around that vicinity is a stable limit cycle, hence leading to stable system trajectory. 4x multiplication pulse trains,
i.e. m = 4, without and with 5% harmonic distortion are shown in Figure 7.25b and c. Their corresponding phase planes are shown in Figure 7.26a and b. For the case of zero harmonic distortion,
FIGURE 7.26 Phase plane of detuned pulse train, m = 4, 0% harmonic distortion (a), and 5% harmonic distortion (b); (solid line – real part of the energy, dotted line – imaginary part of the energy, x-axes – E(t) and
y-axes – E′(t)).
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Photonic Ultra-Short Pulse Generators
1.2
% Fluctuation
1
0.8
0.6
0.4
0.2
0
0
0.16
0.32
0.48
0.64
0.8
0.96
% Harmonic Distortion
m=2
m=4
m=8
FIGURE 7.27 Relationship between the amplitude fluctuation and the percentage harmonic distortion
(diamond – m = 2, square – m = 4, triangle – m = 8).
which is the ideal case, the generated pulse train is perfectly multiplied with equal amplitude and
the phase plane has stable symmetry periodic trajectories around the origin, as well. However, for
the practical case, with 5% harmonic distortion, it is obvious that the pulse amplitude is unevenly
distributed, which can be easily verified with the experimental results obtained in.32 Its corresponding phase plane shows more complex asymmetry system trajectories (Figures 7.27 and 7.29).
One may naively think that the detuning fraction, m, could be increased to a very large number,
so a very small frequency deviated, ∆f, so as to obtain a very high repetition frequency. This is only
true in the ideal world, if no harmonic distortion is present in the system. However, this is unreasonable for a practical mode-locked laser system.
We define the percentage fluctuation, %F as follows:
%F =
Emax − Emin
×100%
Emax
(7.26)
where Emax and Emin are the maximum and minimum peak amplitude of the generated pulse train. For
any practical mode-locked laser system, fluctuations above 50% should be considered as poor laser
system design. Therefore, this is one of the limiting factors in a rational harmonic mode-locking
fiber laser system. The relationships between the percentage fluctuation and harmonic distortion for
three multipliers (m = 2, 4 and 8) are shown in Figure 7.26. Thus, the obtainable rational harmonic
mode-locking is very much limited by the harmonic distortion of the system. For 100% fluctuation,
it means no repetition rate multiplication, but with additional noise components; a typical pulse train
and its corresponding phase plane are shown in Figure 7.28. (lower plot) and Figure 7.29 with m = 8
and 20% harmonic distortion. The asymmetric trajectories of the phase graph explain the amplitude
unevenness of the pulse train. Furthermore, it shows a more complex pulse formation system. Thus,
it is clear that for any harmonic mode-locked laser system, the small side pulses generated are
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Photonic Signal Processing
FIGURE 7.28 10 GHz pulse train (upper plot), pulse train with m = 8 and 20% harmonic distortion (lower plot).
FIGURE 7.29 Phase plane of the pulse train with m = 8 and 20% harmonic distortion.
largely due to improper or not exact tuning of the modulation frequency of the system. An experiment result is depicted in Figure 7.30 for a comparison.
7.2.4.4 Results and Discussion
By careful adjustment of the modulation frequency, polarization, gain level and other parameters
of the fiber ring laser, we manage to obtain the 660th and 1230th order of rational harmonic detuning in the mode-locked fiber ring laser with base frequency of 100 MHz, hence achieving 66 and
123 GHz repetition frequency pulse operation. The auto-correlation traces and optical spectrums of
the pulse operations are shown in Figure 7.31. With Gaussian pulse assumption, the obtained pulse
widths of the operations are 2.5456 ps and 2.2853 ps, respectively. For the 100 MHz pulse operation, i.e. without any frequency detune, the generated pulse width is about 91 ps. Thus, not only do
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Photonic Ultra-Short Pulse Generators
FIGURE 7.30 Autocorrelation trace (a) and optical spectrum (b) of slight frequency detune in the modelocked fiber ring laser.
we achieve an increase in the pulse repetition frequency, but also a decrease in the generated pulse
widths. This pulse narrowing effect is partly due to the self-phase modulation effect of the system,
as observed in the optical spectrums. Another reason for this pulse shortening is stated by Haus
in40,65 where the pulse width is inversely proportional to the modulation frequency, as follows:
τ4 =
2g
M ωm2 ω g2
(7.27)
where τ is the pulse width of the mode-locked pulse, ωm is the modulation frequency, g is the gain
coefficient, M is the modulation index, and ωg is the gain bandwidth of the system. In addition, the
duty cycle of our Gaussian modulation signal is ~1%, which is much less than 50% and leads to a
narrow pulse width, as well. Besides the uneven pulse amplitude distribution, a high level of pedestal noise is also observed in the obtained results.
For 66 GHz pulse operation, a 4-nm bandwidth filter is used in the setup, but it is removed for
the 123 GHz operation. It is done to allow more modes to be locked during the operation, thus, to
achieve better pulse quality. In contrast, this increases the level of difficulty significantly in the
system tuning and adjustment. As a result, the operation is very much determined by the gain bandwidth of the EDFA used in the laser setup.
The simulated phase planes for the above pulse operation are shown in Figure 7.32. They are
simulated based on the 100 MHz base frequency, 10 round trips condition and 0.001% of harmonic
distortion contribution. There is no stable limit cycle in the phase graphs obtained; hence, the system stability is hardly achievable, which is a known fact in the rational harmonic mode-locking.
Asymmetric system trajectories are observed in the phase planes of the pulse operations. This
reflects the unevenness of the amplitude of the pulses generated. Furthermore, more complex pulse
formation process is also revealed in the phase graphs obtained.
65
H. Zmuda, R. A. Soref, P. Payson, S. Johns, and E. N. Toughlian, Photonic beam former for phased array antennas using
a fiber grating prism, IEEE Photon. Technol. Lett., 9, 241–243, 1997.
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Photonic Signal Processing
FIGURE 7.31 Autocorrelation traces of 66 GHz (a) and 123 GHz (c) pulse operation; optical spectrums of
66 GHz (b) and 123 GHz (d).
(a)
(b)
FIGURE 7.32 Phase plane of the 66 GHz (a) and 123 GHz (b) pulse train with 0.001% harmonic distortion noise.
Photonic Ultra-Short Pulse Generators
319
By a very small amount of frequency deviation, or improper modulation frequency tuning in
the general context, we obtain a pulse train with ~100 MHz with small side pulses in between, as
shown in Figure 7.10. It is rather similar to the Figure 7.6 (lower plot) shown in the earlier section
despite the level of pedestal noise in the actual case. This is mainly because we do not consider other
sources of noise in our modeling, except the harmonic distortion.
7.2.4.5 Remarks
We have demonstrated 660th and 1230th order of rational harmonic mode locking from a base modulation frequency of 100 MHz in the erbium doped fiber ring laser, hence achieving 66 and 123 GHz
pulse repetition frequency. To the best of our knowledge, this is the highest rational harmonic order
obtained to date. Besides the repetition rate multiplication, we also obtain high pulse compression
factor in the system, ~35x and 40x relative to the non-multiplied laser system.
In addition, we use phase plane analysis to study the laser system behavior. From the analysis
model, the amplitude stability of the detuned pulse train can only be achieved under negligible or
no harmonic distortion condition, which is the ideal situation. The phase plane analysis also reveals
the pulse forming complexity of the laser system.
7.3 REP-RATE MULTIPLICATION RING LASER USING
TEMPORAL DIFFRACTION EFFECTS
The pulse repetition rate of a mode-locked ring laser is usually limited by the bandwidth of the
intra-cavity modulator. Hence, a number of techniques have to be used to increase the repetition
frequency of the generated pulse train. Rational harmonic detuning32,44 is achieved by applying a
slight deviated frequency from the multiple of fundamental cavity frequency. A 40 Ghz repetition
frequency has been obtained by32 using a 10 GHz base band modulation frequency with fourth order
rational harmonic mode locking. This technique is simple in nature. However, this technique suffers
from an inherent pulse amplitude instability, which includes both amplitude noise and inequality in pulse amplitude, and, furthermore, it gives poor long-term stability. Hence, pulse amplitude
equalization techniques are often applied to achieve better system performance.31,33,34 Fractional
temporal Talbot-based repetition rate multiplication technique uses the interference effect between
the dispersed pulses to achieve the repetition rate multiplication. The essential element of this technique is the dispersive medium, such as linearly chirped fiber grating (LCFG)56,38 and single mode
fiber.30,31 This technique will be discussed further in Section 7.3.1. Intra-cavity optical filtering42
uses modulators and a high finesse Fabry-Perot filter (FFP) within the laser cavity to achieve higher
repetition rate by filtering out certain lasing modes in the mode-locked laser. Other techniques used
in repetition rate multiplication include higher order FM mode-locking,42 optical time domain multiplexing, and others.
The stability of high repetition rate pulse train generated is one of the main concerns for practical
multi-Giga bits/sec optical communications system. Qualitatively, a laser pulse source is considered
stable if it is operating at a state where any perturbations or deviations from this operating point is
not increased but suppressed. Conventionally the stability analyses of such laser systems are based
on the linear behavior of the laser in which we can analyze the system behavior in both time and frequency domains. However, when the mode-locked fiber laser is operating under nonlinear regime,
none of these standard approaches can be used, since direct solution of nonlinear different equation
is generally impossible, hence frequency domain transformation is not applicable. Although Talbotbased repetition rate multiplication systems are based on the linear behavior of the laser, there are
still some inherent nonlinearities affecting its stability, such as the saturation of the embedded gain
medium, non-quadrature biasing of the modulator, nonlinearities in the fiber, and so on. Hence, a
nonlinear stability approach must be adopted.
We investigate the stability and transient analyses of GVD multiplied pulse train using the phase
plane analysis of nonlinear control analytical technique.31 This was the first time that the phase
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Photonic Signal Processing
plane analysis of modern control engineering is being used to study the stability and transient performances of the GVD repetition rate multiplication systems.
The stability and the transient response of the multiplied pulses are studied using the phase
plane technique of nonlinear control engineering. We also demonstrated four times the repetition
rate multiplication on 10 Gb/s pulse train generated from the active harmonically mode-locked
fiber ring laser, hence achieving 40 Gb/s pulse train by using fiber GVD effect. It has been found
that the stability of the GVD multiplied pulse train, based on the phase plane analysis is hardly
achievable even under the perfect multiplication conditions. Furthermore, uneven pulse amplitude distribution is observed in the multiplied pulse train. In addition to that, the influences of the
filter bandwidth in the laser cavity, nonlinear effect and the noise performance are also studied
in our analyses.
In Section 7.3.1, the GVD repetition rate multiplication technique is briefly given. Section 7.3.2
describes the experimental setup for the repetition rate multiplication. Section 7.3.4 investigates the
dynamic behavior of the phase plane of GVD multiplication system, followed by simulation and
experimental results. Finally, some concluding remarks and possible future developments are given.
7.3.1
gvD repetitiOn rate multiplicatiOn technique
When a pulse train is transmitted through an optical fiber, the phase shift of kth individual lasing
mode due to group velocity dispersion (GVD) is
ϕk =
πλ 2 Dzk 2 f r2
c
(7.28)
where:
λ is the center wavelength of the mode-locked pulses
D is the fiber’s GVD factor
z is the fiber length
fr is the repetition frequency
c is the speed of light in vacuum
This phase shift induces pulse broadening and distortion. At Talbot distance, zT = 2/∆λ fr /D/ 36 the
initial pulse shape is restored, where ∆λ = fr λ2 /c is the spacing between Fourier-transformed spectrum of the pulse train. When the fiber length is equal to zT /(2m), (where m = 2,3,4, …), every mth
lasing modes oscillates in phase and the oscillation waveform maximums accumulate. However,
when the phases of other modes become mismatched, this weakens their contributions to pulse
waveform formation. This leads to the generation of a pulse train with a multiplied repetition frequency with m-times. The pulse duration does not change that much, even after the multiplication, because every mth lasing mode dominates in pulse waveform formation of m-times multiplied
pulses. The pulse waveform therefore becomes identical to that generated from the mode-locked
laser, with the same spectral property. The optical spectrum does not change after the multiplication
process, because this technique utilizes only the change of phase relationship between lasing modes
and does not use the fiber’s nonlinearity.
The effect of higher order dispersion might degrade the quality of the multiplied pulses, including pulse broadening, appearance of pulse wings, and pulse-to-pulse intensity fluctuation. In this
case, any dispersive media to compensate the fiber’s higher order dispersion would be required in
order to complete the multiplication process. To achieve higher multiplications, the input pulses
must have a broad spectrum and the fractional Talbot length must be very precise in order to receive
high quality pulses. If the average power of the pulse train induces the nonlinear suppression and
experience anomalous dispersion along the fiber, solitonic action would occur and prevent the linear
Talbot effect from occurring.
Photonic Ultra-Short Pulse Generators
321
The highest repetition rate obtainable is limited by the duration of the individual pulses, as
pulses start to overlap when the pulse duration becomes comparable to the pulse train period,
i.e. mmax = ∆T/∆t, where ∆T is the pulse train period and ∆t is the pulse duration.
7.3.2
experiment setup
GVD repetition rate multiplication is used to achieve 40 Gb/s operation. The input to the GVD multiplier is a 10.217993 Gb/s laser pulse source, obtained from active harmonically mode-locked fiber
ring laser, operating at 1550.2 nm.
The principle element of the active harmonically mode-locked fiber ring laser is an optical closed
loop with an optical gain medium (i.e., Erbium doped fiber with 980 nm pump source), an optical
10 GHz amplitude modulator, optical bandpass filter, optical fiber couplers, and other associated
optics. The schematic construction of the active mode-locked fiber ring laser is shown in Figure 7.33.
The active mode-locked fiber laser design is based on a fiber ring cavity where the 25 meter EDF
with Er3+ ion concentration of 7 × 1024 ions/m3 is pumped by two diode lasers at 980 nm: SDLO-278000-300 and CosetK1116 with maximum forward pump power of 280 mW and backward pump
power of 120 mW. The pump lights are coupled into the cavity by the 980/1550 nm WDM couplers;
with insertion loss for 980 and 1550 nm signals are about 0.48 and 0.35 dB, respectively. A polarized, independent optical isolator ensures the unidirectional lasing. The birefringence of the fiber is
compensated by a polarization controller (PC). A tunable FP filter with a 3-dB bandwidth of 1 nm
and a wavelength tuning range from 1530 to 1560 nm is inserted into the cavity to select the center
wavelength of the generated signal, as well as to reduce the noise in the system. In addition, it is
used for the longitudinal modes selection in the mode-locking process. Pulse operation is achieved
by introducing a JDS Uniphase 10 Gb/s (8 GHz 3 dB bandwidth) Ti:LiNbO3 Mach–Zehnder amplitude modulator into the cavity with half wave voltage, Vπ of 5.8 V. The modulator is DC biased near
the quadrature point and not more than the Vπ such that it operates on the linear region of its characteristic curve and driven by the sinusoidal signal derived from an Anritsu 68347C Synthesizer
Signal Generator. The modulating depth should be less than the unity to avoid signal distortion.
FIGURE 7.33 Schematic diagram for active mode-locked fiber ring laser.
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Photonic Signal Processing
FIGURE 7.34
Experiment setup for GVD repetition rate multiplication system.
The modulator has an insertion loss of ≤7dB. The output coupling of the laser is optimized using
a 10/90 coupler. Additionally, 90% of the optical field power is coupled back into the cavity ring
loop, while the remaining portion is taken out as the output of the laser and is analyzed using a New
Focus 1014B 40 GHz photo-detector, Ando AQ6317B Optical Spectrum Analyzer, Textronix CSA
8000 80E01 50 GHz Communications Signal Analyzer or Agilent E4407B RF Spectrum Analyzer.
One rim of about 3.042 km of dispersion compensating fiber (DCF), with a dispersion value of
−98 ps/nm/km, was used in the experiment; the schematic of the experimental setup is shown in
Figure 7.34. The variable optical attenuator used in the setup is to reduce the optical power of the
pulse train generated by the mode-locked fiber ring laser, hence to remove the nonlinear effect of the
pulse. A DCF length for 4x multiplication factor on the ~10 GHz signal is required and estimated
to be 3.048173 km. The output of the multiplier (i.e., at the end of DCF) is then observed using
Textronix CSA 8000 80E01 50 GHz Communications Signal Analyzer.
7.3.3
phase plane analysis
A nonlinear system frequently has more than one equilibrium point. It can also oscillate at a fixed amplitude and fixed period without external excitation. This oscillation is called the limit cycle. However,
limit cycles in nonlinear systems are different from linear oscillations. First, the amplitude of selfsustained excitation is independent of the initial condition, while the oscillation of a marginally stable
linear system has its amplitude determined by the initial conditions. Second, marginally stable linear
systems are very sensitive to changes, while limit cycles are not easily affected by parameter changes.
Phase plane analysis is a graphical method of studying second-order nonlinear systems. The
result is a family of system motion trajectories on a two-dimensional plane, which allows us to visually observe the motion patterns of the system. Nonlinear systems can display more complicated
patterns in the phase plane, such as multiple equilibrium points and limit cycles. In the phase plane,
a limit cycle is defined as an isolated closed curve. The trajectory has to be both closed, indicating
the periodic nature of the motion, and isolated, indicating the limiting nature of the cycle.
The system modeling for the GVD multiplier is done based on the following assumptions:
(i) perfect output pulse from the mode-locked fiber ring laser without any timing jitter, (ii) the
multiplication is achieved under ideal conditions (i.e., exact fiber length for a certain dispersion
value), (iii) no fiber nonlinearity is included in the analysis of the multiplied pulse, (iv) no other noise
sources are involved in the system, and (v) uniform or Gaussian lasing mode amplitude distribution.
7.3.3.1 Uniform Lasing Mode Amplitude Distribution
Uniform lasing mode amplitude distribution is assumed at the first instance, i.e. ideal mode-locking
condition. The simulation is done based on the 10 Gb/s pulse train, centered at 1550 nm, with fiber
dispersion value of −98 ps/km/nm, and a 1-nm flat-top passband filter is used in the cavity of modelocked fiber laser. The estimated Talbot distance is 25.484 km.
The original pulse (direct from the mode-locked laser) propagation behavior and its phase plane
are shown in Figures 7.35a and 7.36a. From the phase plane obtained, one can observe that the
Photonic Ultra-Short Pulse Generators
323
FIGURE 7.35 Pulse propagation of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x
multiplication with 1 nm filter bandwidth and equal lasing mode amplitude analysis.
FIGURE 7.36 Phase plane of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x multiplication with 1 nm filter bandwidth and equal lasing mode amplitude analysis; (solid line – real part of the
energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E′(t)).
origin is a stable node and the limit cycle around that vicinity is a stable limit cycle. This agrees
very well to our first assumption: ideal pulse train is at the input of the multiplier. Also, we present
the pulse propagation behavior and phase plane for 2×, 4× and 8× GVD multiplication system in
Figures 7.33 and 7.34. The shape of the phase graph exposes the phase between the displacement
and its derivative (Figures 7.35 through 7.44).
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Photonic Signal Processing
As the multiplication factor increases, the system trajectories are moving away from the origin.
As for the 4× and 8× multiplications, there is neither a stable limit cycle nor stable node on the phase
planes even with the ideal multiplication parameters. Here, we see the system trajectories spiral out
FIGURE 7.37 Pulse propagation of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x
multiplication with 1 nm filter bandwidth and Gaussian lasing mode amplitude analysis.
FIGURE 7.38 Phase plane of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x multiplication with 1 nm filter bandwidth and Gaussian lasing mode amplitude analysis; (solid line – real part of
the energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E′(t)).
Photonic Ultra-Short Pulse Generators
325
FIGURE 7.39 Pulse propagation of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x
multiplication with 3 nm filter bandwidth and Gaussian lasing mode amplitude analysis.
FIGURE 7.40 Phase plane of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x multiplication with 3 nm filter bandwidth and Gaussian lasing mode amplitude analysis; (solid line – real part of
the energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E′(t)).
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Photonic Signal Processing
FIGURE 7.41 Pulse propagation of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x
multiplication with 3 nm filter bandwidth, Gaussian lasing mode amplitude analysis and input power = 1 W.
FIGURE 7.42 Phase plane of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x multiplication with 3 nm filter bandwidth, Gaussian lasing mode amplitude analysis and input power = 1 W; (solid
line – real part of the energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E′(t)).
Photonic Ultra-Short Pulse Generators
327
FIGURE 7.43 Pulse propagation of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x
multiplication with 3 nm filter bandwidth, Gaussian lasing mode amplitude analysis and 0 dB signal to noise ratio.
FIGURE 7.44 Phase plane of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x multiplication with 3 nm filter bandwidth, Gaussian lasing mode amplitude analysis and 0 dB signal to noise ratio;
(solid line – real part of the energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E′(t)).
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Photonic Signal Processing
to an outer radius and back to an inner radius again. The change in the radius of the spiral is the
transient response of the system. Hence, with the increase in the multiplication factor, the system
trajectories become more sophisticated. Although GVD repetition rate multiplication uses only the
phase change effect in multiplication process, the inherent nonlinearities still affect its stability indirectly. Despite the reduction in the pulse amplitude, we observe uneven pulse amplitude distribution
in the multiplied pulse train. The percentage of unevenness increases with the multiplication factor
in the system.
7.3.3.2 Gaussian Lasing Mode Amplitude Distribution
In this set of the simulation models, the practical filter used in the system. It gives us a better insight
on the GVD repetition rate multiplication system behavior. The parameters used in the simulation
are exactly the same except the filter of the laser has been changed to 1 nm (125 GHz @ 1550 nm)
Gaussian-profile passband filter. The spirals of the system trajectories and uneven pulse amplitude
distribution are more severe than those in the uniform lasing mode amplitude analysis.
7.3.3.3 Effects of Filter Bandwidth
Filter bandwidth used in the mode-locked fiber ring laser will also affect the system stability of the
GVD repetition rate multiplication system. The analysis done above is based on 1 nm filter bandwidth. The number of modes locked in the laser system increases with the bandwidth of the filter
used, which gives us a better quality of the mode-locked pulse train. The simulation results shown
below are based on the Gaussian lasing mode amplitude distribution, a 3-nm filter bandwidth used
in the laser cavity, and other parameters remain unchanged.
With wider filter bandwidth, the pulse width and the percentage pulse amplitude fluctuation
decreases. This suggests a better stability condition. Instead of spiraling away from the origin, the
system trajectories move inward to the stable node. However, this leads to a more complex pulse
formation system.
7.3.3.4 Nonlinear Effects
When the input power of the pulse train enters the nonlinear region, the GVD multiplier loses its
multiplication capability as predicted. The additional nonlinear phase shift due to the high input
power is added to the total pulse phase shift and destroys the phase change condition of the lasing
modes required by the multiplication condition. Furthermore, this additional nonlinear phase shift
also changes the pulse shape and the phase plane of the multiplied pulses.
7.3.3.5 Noise Effects
The above simulations are all based on the noiseless situation. However, in the practical optical
communication systems, noises are always sources of nuisance which can cause system instability,
therefore it must be taken into the consideration for the system stability studies.
Since the optical intensity of the m-times multiplied pulse is m-times less than the original pulse,
it is more vulnerable to noise. The signal is difficult to differentiate from the noise within the system
if the power of multiplied pulse is too small. The phase plane the multiplied pulse is distorted due
to the presence of the noise, which leads to poor stability performance.
7.3.4
DemOnstratiOn
The obtained 10 GHz output pulse train from the mode-locked fiber ring laser is shown in
Figure 7.45. Its spectrum is shown in Figure 7.46. This output was then used as the input to the
dispersion compensating fiber, which acts as the GVD multiplier in our experiment. The obtained
times multiplication by the GVD effect and its spectrum are shown in Figures 7.47 and 7.48.
Photonic Ultra-Short Pulse Generators
FIGURE 7.45
10 GHz pulse train from mode-locked fiber ring laser (100 ps/div, 50 mV/div).
FIGURE 7.46
10 GHz pulse spectrum from mode-locked fiber ring laser.
329
The spectrums for both cases (original and multiplied pulse) are exactly the same since this repetition rate multiplication technique utilizes only the change of phase relationship between lasing
modes and does not use fiber’s nonlinearity.
The multiplied pulse suffers an amplitude reduction in the output pulse train; however, the pulse
characteristics should remain the same. The instability of the multiplied pulse train is mainly due
to the slight deviation from the required DCF length (0.2% deviation). Another reason for the pulse
instability, which derived from our analysis; is the divergence of the pulse energy variation in the
vicinity around the origin, as the multiplication factor gets higher. The pulse amplitude decreases
with the increase in multiplication factor, as the fact of energy conservation, when it reaches certain energy level, is indistinguishable from the noise level in the system, and the whole system will
become unstable and noisy.
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Photonic Signal Processing
FIGURE 7.47 40 GHz multiplied pulse train (20 ps/div, 1 mV/div).
FIGURE 7.48
7.3.5
40 GHz pulse spectrum from GVD multiplier.
remarks
We have demonstrated 4× repetition rate multiplication by using fiber GVD effect; hence, 40 GHz
pulse train is obtained from 10 GHz mode-locked fiber laser source. However, its stability is of
great concern for practical use in the optical communications systems. Although the GVD repetition rate multiplication technique is linear in nature, the inherent nonlinear effects in such system
may disturb the stability of the system. Hence, any linear approach may not be suitable in deriving
the system stability. Stability analysis for this multiplied pulse train has been studied by using the
nonlinear control stability theory, which is the first time, to the best of our knowledge, that phase
plane analysis is being used to study the transient and stability performance of the GVD repetition
rate multiplication system. Surprisingly, from the analysis model, the stability of the multiplied
pulse train can hardly be achieved even under perfect multiplication conditions. Furthermore, we
observed uneven pulse amplitude distribution in the GVD multiplied pulse train, which is due to the
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Photonic Ultra-Short Pulse Generators
energy variations between the pulses that cause some energy beating between them. Another possibility is the divergence of the pulse energy variation in the vicinity around the equilibrium point
that leads to instability.
The pulse amplitude fluctuation increases with the multiplication factor. Also, with wider filter
bandwidth used in the laser cavity, better stability condition can be achieved. The nonlinear phase
shift and noises in the system challenge the system stability of the multiplied pulses. They not only
change the pulse shape of the multiplied pulses, they also distort the phase plane of the system. Hence,
the system stability is greatly affected by the self-phase modulation (SPM) as well as the system noises.
This stability analysis model can further be extended to include some system nonlinearities, such
as the gain saturation effect, non-quadrature biasing of the modulator, fiber nonlinearities, and so
on. The chaotic behavior of the system may also be studied by applying different initial phase and
injected energy conditions to the model.
7.4
MULTI-WAVELENGTH FIBER RING LASERS
This section presents the theoretical development and demonstration of a multi-wavelength erbiumdoped fiber ring laser with an all-polarization-maintaining fiber (PMF) Sagnac loop. The Sagnac loop
simply consists of a PMF coupler and a segment of stress-induced PMF, with a single-polarization
coupling point in the loop. The Sagnac loop is shown to be a stable comb filter with equal frequency
period which determines the possible output power spectrum of the fiber ring laser. The number of output lasing wavelengths is obtained by adjusting the polarization state of the light in the unidirectional
ring cavity by means of a polarization controller. This section is organized as follows: Section 7.4.1
presents the theory of the Sagnac PMF loop filter, which consists of a PMF coupler (instead of a standard single-mode fiber coupler used in previous works as described above) and a segment of PMF.
Section 7.4.2 presents the experimental results and discussion. Concluding remarks are given.
7.4.1
theory
In this section, we present a theoretical analysis of the all-PMF Sagnac loop, which is the key component in the fiber ring laser. We consider the simplest case (see Figure 7.49a), in which only one
polarization mode-coupling point exists. That is, the loop filter is constructed by splicing the two
pigtails (with lengths l1 and l2) of the PMF coupler with a phase difference θ along a certain principal
axis at the spliced point. The input light is equally split into two waves by the 3-dB PMF coupler, and
the two counter-propagating waves are recombined at the coupler output port after traveling through
the loop. The electric components, Eix (ω ) and Eiy (ω ) , of the input light, Ein (ω ) , can be defined as
 Eix (ω ) 
(7.29)
Ein (ω ) = 

 Eiy (ω ) 
where ω is the angular optical frequency. The PMF with length l can be considered as an ideal
waveguide with linear birefringence, which is described by the Jones propagation matrix as66
 exp ( j ∆β (ω )l/2 )
J PMF (ω , l ) = 
0


0

exp ( − j ∆β (ω )l / 2 ) 
(7.30)
where j = −1 and ∆β (ω ) = β x (ω ) − β y (ω ) is the difference between the two propagation constants of a high-birefringence fiber, which supports two linearly orthogonal fundamental modes
66
D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, Dark-pulse propagation in optical fibers, Phys. Rev. Lett., 60,
29–32, 1988.
332
Photonic Signal Processing
y1
y1
x1
l1
Fre
y2
X2
Fnon-re
y1 y2
l2 x2
X1
x1
θ
(a)
(b)
FIGURE 7.49 (a) Schematic diagram of the proposed all-PMF Sagnac loop filter with a single coupling
point. (b) Representation of the coordinates of the single coupling point.
(i.e., HEx11 and HEy11). It should be noted that a common average phase shift of exp( j β (ω )l ) is omitted in (7.10), because the Sagnac interferometer cannot distinguish the common phase term for the
clockwise wave (CW) and counterclockwise wave (CCW). The transfer matrix of the coordinate
(i.e., Θ(θ ) ) of the polarization mode-coupling point at the principal axes with a phase difference
of θ (see Figure 7.49b) is given as
 cos θ
Θ (θ ) = 
 sin θ
sin θ 

− cosθ 
(7.31)
The 3-dB PMF coupler is assumed to be ideal so that polarization coupling, polarization-dependent
loss, and frequency dependence of the coupler are negligible (i.e., the coupling ratio is 50% in the
operating wavelength range). The CW and CCW will experience the same phase shift and a 3-dB
loss through the coupler, thus there is no phase difference at the reciprocal port. Hence, the CW’s
Jones matrix for the Sagnac loop is given by
Gcw (ω ) =
1
J PMF (ω , l1 )Θ(θ ) J PMF (ω , l2 )
2
(7.32)
where J PMF (ω , l1 ) and J PMF (ω , l2 ) are defined in (7.30). The CCW’s Jones matrix is simply the transpose of the CW’s Jones matrix and is given by
T
Gccw (ω ) = Gcw
(ω )
(7.33)
The electric components of the light at the output ports are given by
 Eox (ω ) 

 Eoy (ω ) 
Eout (ω ) = 
(7.34)
The relationship between the electric components at the input and output ports is given by
Eout (ω ) = [Gccw (ω ) + Gcw (ω ) ] Ein (ω )
(7.35)
Using Eqs. (7.29) through (7.32), the intensity transfer function, Fre, for the reciprocal port can be
derived as
Fre = 1 − sin 2 θ ⋅ sin 2 ( ∆β ⋅ ∆l/2 )
(7.36)
333
Photonic Ultra-Short Pulse Generators
where ∆l = l2 − l1 is the difference between the length of the two PMF segments in the loop. It is
noted that (7.16) is independent of the polarization state of the input light due to the fact that the
interference terms of the x-component and y-component of the light cancel out with each other at
the output port. From (7.36), it can be shown that when θ ≠ 0 or θ ≠ π the spectral peaks of the
reflection spectrum will have maximum intensity at frequencies according to
∆β (ωm ) ⋅ ∆l = 2π m ( m = 1, 2, ⋅⋅⋅)
(7.37)
Note that light with frequency ωm will disappear at the non-reciprocal port for the case of θ = π 2
because the transfer function of the non-reciprocal port is Fnon−re = 1 − Fre. There are two kinds of highbirefringence fibers, namely, stress-induced birefringent fiber and geometry-induced birefringent fiber,
where the former one has greater birefringence. Here, we only consider the stress-induced birefringent
fiber, where the effective index of each polarization is influenced by stress alone. Thus, the modal birefringence B is independent of wavelength over a particular wavelength range, and is given by
∆β (ωm ) =
2π
B ( m = 0, 1, 2…….)
λ
(7.38)
Using (7.17) and (7.18), the spectral peaks of the reflection spectrum will have maximum intensity
at frequencies f m given by
fm =
mc
( m = 0,1, 2…….)
B∆l
(7.39)
where c is the speed of the light in vacuum. From (7.39), the Sagnac loop is a comb filter whose
spectral peaks are separated by frequency spacing given by
f m+1 − f m =
c
B∆l
(7.40)
It is noted that although the frequency-dependent intensity transfer function of the loop filter is
independent of the state of polarization of the input light, the polarization state of the output light
generally depends on the polarization state and frequency of the input light.
7.4.2
experimental results anD DiscussiOn
This section presents the experimental verification of the theoretical analysis of the all-PM Sagnac
loop filter described in Section 7.4.1 and the experimental results of the fiber ring laser. Figure 7.52
shows a typical reflective spectrum of the all-PMF Sagnac loop filter. The loop filter was constructed by splicing the two pigtails (with lengths l1 and l2) of the PMF coupler in 0° and 90° with
respect to their principal axes to form a single coupling point in the loop (i.e., a phase difference
of θ = 90°). The loop filter is highly stable, as expected, because all the components used are allPM components. From Figure 7.50, the frequency period of the filter is 0.35 nm, which agrees well
with the theoretical value as predicted by 20 when the following parameter values, B = 5.2e-4 and
∆l = 13.2 m (in the 1550-nm window) are substituted into the equation.
Figure 7.51 shows the schematic diagram of the proposed unidirectional fiber ring laser with
the all-PMF Sagnac loop filter. It consists of a 15-m long of Er3+ silica fiber doped with ~200 ppm
of erbium. The erbium-doped fiber has a numerical aperture (NA) of 0.21, a cut-off wavelength of
920 nm, and an absorption coefficient of 12 dB/m at 980 nm. To increase the optical pump efficiency, the erbium-doped fiber is pumped by a 980-nm laser diode (LD), which generates 70 mW
power in both directions in the ring cavity through the two 980/1550-nm WDM couplers. A polarized, independent fiber isolator is used to provide unidirectional ring oscillation so as to avoid spatial hole burning in the gain medium. The coupler-2 is used as the output coupler for the fiber laser
334
FIGURE 7.50
Photonic Signal Processing
A typical reflective spectrum of the all-PMF Sagnac loop filter with a single coupling point.
PC
WDM2
980nm coupler
Er3+
doped
fiber
980nm LD
Isolator
WDM1
Output
Coupler-2
PMF
coupler
coupling
point
FIGURE 7.51 Schematic of the proposed unidirectional fiber ring laser using the all-PMF Sagnac loop as a
stable periodic filter.
and also to direct the light wave to the Sagnac loop filter. The periodic spectral peaks of the Sagnac
filter will determine the lasing wavelengths. A polarization controller (PC) is used in the cavity to
adjust the polarization state to obtain several lasing wavelengths at the output port.
Figure 7.52a and b show the experimental results of the output lasing wavelengths of the fiber ring
laser under different polarization conditions by adjusting the PC in the cavity. It can be seen that the
wavelength spacing is 1.0 nm, which is defined by the 1.0 nm frequency period of the Sagnac loop
filter with parameter values of B = 5.2e^-4 and ∆l = 4.5 m. Figure 7.53 shows the output spectra of
the lasing wavelengths of the fiber ring laser under a particular polarization condition in the cavity,
where the wavelength spacing is 0.50 nm, which is defined by the 0.50 nm frequency period of the
Sagnac loop filter with parameter values of B = 5.2e-4 and ∆l = 9.0 m. It should be noted that the
number of output lasing wavelengths of the proposed fiber ring laser could be greatly increased by
overcoming the large homogeneous broadening of the gain medium of the erbium-doped fiber at
335
0.1
0.1
0.01
0.01
Output (dBm)
Output (dBm)
Photonic Ultra-Short Pulse Generators
1E-3
1E-4
1E-4
1556
(a)
1E-3
1558
1560
1562
Wavlength (nm)
1564
1556
(b)
1558
1560
1562
Wavlength (nm)
1564
FIGURE 7.52 (a), (b) Typical output lasing wavelengths of the fiber ring laser under different polarization
conditions of the PC in the ring cavity. Wavelength spacing is 1.0 nm.
Output (dBm)
1
0.1
0.01
1E-3
1E-4
1552
1554
1556
1558
Wavlength (nm)
1560
1562
FIGURE 7.53 Typical output lasing wavelength of the fiber ring laser under a particular polarization condition of the PC in the ring cavity. Wavelength spacing is 0.50 nm.
room temperature.67 This problem can be overcome by cooling the erbium-doped fiber to 77 K, but
this technique is probably not suitable for practical applications.33,68 A more practical approach is to
use an acousto-optic frequency shifter in the ring cavity to prevent the steady-state laser oscillation
in order to generate a larger number of stable lasing wavelengths.34
In this section we have demonstrated an Er-doped fiber ring laser using an all-polarizationmaintaining-fiber (PMF) Sagnac loop filter for multi-wavelength operation. The theoretical analysis
and experimental results of the all-PMF Sagnac loop as a stable comb filter have been presented.
The Sagnac loop filter is a simple and all-fiber device that consists of a PMF coupler and a segment of stress-induced PMF to form the loop. The number of output lasing wavelengths has been
obtained by adjusting the polarization state of the light in the ring cavity using a polarization controller. The wavelength spacing is determined by the frequency period of the comb filter with equal
frequency interval.
S. Arahira, S. Kutsuzawa, Y. Matsui, D. Kunimatsu, and Y. Ogawa, Repetition frequency multiplication of mode-locked
using fiber dispersion J. Light. Technol., 16, 405–410, 1998.
68 W. J. Lai, P. Shum, and L. N. Binh, Stability and transient analyses of temporal Talbot effect-based repetition-rate multiplication mode-locked laser systems, IEEE Photon. Technol. Lett., 16, 437–439, 2004.
67
336
7.4.3
Photonic Signal Processing
multi-wavelength tunable Fiber ring lasers
Furthermore, tunable EDFR lasers57–61,69,70,71,72,73 with a continuous tuning range of 15.5 nm (i.e., from
1546.8 to 1562.3 nm) can be demonstrated. The laser output power is ~9 dBm and the power variation is
<2 dB over the tuning range. The laser is tuned using our recently developed tunable narrow-bandpass
filter, which is based on a phase-shifted linearly chirped FBG (LCFBG). Heating the LCFBG at a
small contact point using a resistance wire would introduce a temporary phase-shift into the LCFBG,
and hence creating a very narrow passband peak within the stopband of the LCFBG operating in the
transmission mode. Scanning the resistance wire along the LCFBG, which is controlled by a linear
travel stage, the narrow passband peak will shift across the stopband of LCFBG. This tuning technique overcomes the limited tuning range of FBG due to the failure feature of silica fiber.
Figure 7.54 shows the schematic of the proposed tunable EDFRL. The laser consists of 25 m of
erbium-doped fiber (EDF) pumped by a 980-nm laser diode (LD), a tunable phase-shifted LCFBG,
an optical circulator, a LCFBG1, a 2:1 fiber coupler, a polarization controller (PC), and an isolator.
The mode field diameter of the EDF is 6.6 µm and its cladding diameter is 80 µm.
Our recently developed tunable phase-shifted LCFBG is used as a tunable bandpass filter (TBF)
in the laser cavity. The length of the LCFBG is 6 cm and the chirp rate is 2.25 nm/cm. The stopband
of the LCFBG is 21 nm (i.e., from 1544 to 1565 nm). The LCFBG was inscribed into a hydrogenloaded germanium-doped cladding-mode suppression fiber using the UV scanning beam technique.
The cladding mode loss, which can cause an undesirable effect on the filter characteristics, is eliminated using this cladding-mode suppression fiber. A resistance wire is used as a thermal head to heat
a small contact point of LCFBG. A thermally-induced temporary phase shift will be introduced into
the LCFBG, and, hence, a narrow passband peak will be created in the stopband of the LCFBG.
When the wire heater is scanned along the LCFBG using a linear travel stage, the center wavelength
of the narrow passband peak will shift according to the position of the resistance wire. As a result,
an all-fiber TBF can be achieved. To eliminate the broadened spectral disturbance due to thermal
EDF
Coupler
980/1550
WDM
LCFBG1
980 nm
Pump LD
PC
Isolator
Tunable
phase-shifted
LCFBG
66.7%
33.3%
Output
Circulator
Tunable bandpass filter
FIGURE 7.54 Experimental configuration of the proposed tunable fiber ring laser.
S. Yamashita and M. Nishihara, Widely tunable erbium-doped fiber ring laser covering both C-band and L-band, IEEE
J. Select Topics Quantum Electron., 7, 41–43, 2001.
70 M. Y. Jeon, H. K. Lee, K. H. Kim, E. H. Lee, S. H. Yun, B. Y. Kim, and Y. W. Koh, An electronically wavelength-tunable
mode-locked fiber laser using an all-fiber acousto-optic tunable filter, IEEE Photon. Technol. Lett., 8, 1618–1620, 1996.
71 Y. T. Chieng and R. A. Minasian, Tunable erbium-doped fiber laser with a reflection Mach–Zehnder interferometer,
IEEE Photon. Technol. Lett., 6, AQ36 153–156, 1994.
72 S. K. Kim, G. Stewart, and B. Culshaw, Mode-hop-free single-longitudinal-mode erbium doped fiber laser frequency
scanned with afiber ring resonator, Appl. Opt., 38, 5154–5157, 1999.
73 Y. W. Song, S. A. Havstad, D. Starodubov, Y. Xie, A. E. Willner, and J. Feinberg, 40-nm-wide tunablefiber ring laser with
single-mode operation using a highly stretchable FBG, IEEE Photon. Technol. Lett., 13, 1167–1169, 2001.
69
Photonic Ultra-Short Pulse Generators
337
conduction, a heat sink beside the heating section is used to cool the LCFBG. It should be noted
that, in this experiment, an identical LCFBG1 operating in the reflection mode and a circulator
are needed in the cavity to suppress the transmission outside the stopband of the LCFBG. This is
to eliminate the unwanted laser output outside the stopband of the LCFBG to achieve good laser
performance. The measured spectrum of the TBF (with LCFBG1 and the circulator) is shown in
Figure 7.55. The 3-dB bandwidth of the TBF is smaller than 0.01 nm, which is only limited by the
0.01 nm resolution of the optical spectrum analyzer. The total insertion loss is ~8 dB. However, the
LCFBG1 and the optical circulator will not be needed when the stopband of the LCFBG can cover
the whole gain bandwidth of EDF, and this will reduce the insertion loss and cost of the laser.
The optical spectrum of the tunable fiber ring laser over the tuning range of 15.5 nm is shown in
Figure 7.56. The power of the pump LD is ~98 mW. The laser output is ~9 dBm. When scanning the
FIGURE 7.55 Measured transmission spectrum of TBF over the tuning range.
FIGURE 7.56 Output power of the tunable laser over the 15.5 nm tuning range.
338
Photonic Signal Processing
FIGURE 7.57 Stability of laser output spectrum (10 min/scan).
resistance wire along the LCFBG over 50 mm, the lasing wavelength was shifted from 1546.8 to
1562.3 nm (or 15.5 nm range). It is observed that there is no spectral distortion of the lasing wavelength over the tuning range and that the output power is relatively constant with a power variation
of <2 dB. The side-mode suppression ratio is greater than 30 dB. Figure 7.57 shows the output
spectrum of the fiber ring laser and its stability with time. The time interval between the scans
is 10 minutes. It can be seen that the laser output is very stable with time. The laser linewidth is
measured using a scanning F-P interferometer with 10 GHz free spectral range and 50 MHz resolution. The measured linewidth is ~6.5 MHz, which is limited by the resolution of the scanning F-P
interferometer.
The resolution of the travel stage to control the resistance wire is 0.01 mm, which corresponds
to a tuning step of 3 pm of the laser. The tuning step can be made smaller using a travel stage with
higher resolution and/or LCFBG with smaller chirp rate. It should be noted that the tunable range of
the fiber ring laser is limited only by the stopband width of the LCFBG. Using LCFBG with longer
length and/or larger chirp rate, the laser tuning range can be increased to cover the whole C-band.
The fiber ring laser can also operate in the S-band and L-band using LCFBG with appropriate
stopband.
7.4.4
remarks
This chapter described a mode-locked laser operating under the open loop condition and with O/E
RF feedback providing regenerative mode locking. The O/E feedback can certainly provide a selflocking mechanism under the condition that the polarization characteristics of the ring laser are
manageable. The regenerative MLRL can self-lock even under the DC drifting effect of the modulator bias voltage (over 20 hours).74 The generated pulse trains of 5 ps duration can be, with minimum
difficulty, compressed further to less than 3 ps for 160 Gb/s optical communication systems.
The temporal Talbot phenomena is also given by demonstrating the 660th and 1230th order of
rational harmonic mode locking from a base modulation frequency of 100 MHz in the optically
74
Typically the DC bias voltage of a LiNbO3 intensity modulator is drifted by 1.5 volts after 15 hours of continuous
operation.
Photonic Ultra-Short Pulse Generators
339
amplified fiber ring laser, hence achieving 66 and 123 GHz pulse repetition frequency. Besides the
repetition rate multiplication, we also obtain high pulse compression factor in the system, ~35x and
40x relative to the non-multiplied laser system. In addition, we use phase plane analysis to study
the laser system behavior. From the analysis model, the amplitude stability of the detuned pulse
train can only be achieved under negligible or no harmonic distortion condition, which is the ideal
situation. The phase plane analysis also reveals the pulse forming complexity of the laser system.
We have demonstrated times repetition rate multiplication by using fiber GVD effect; hence,
40 GHz pulse train is obtained from 10 GHz mode-locked fiber laser source. Stability analysis for
this multiplied pulse train has been studied by using the nonlinear control stability theory, that
phase plane analysis is being used to study the transient and stability performance of the GVD repetition rate multiplication system. Surprisingly, from the analysis model, the stability of the multiplied pulse train can hardly be achieved even under perfect multiplication conditions. Furthermore,
we observed uneven pulse amplitude distribution in the GVD multiplied pulse train, which is due
to the energy variations between the pulses that cause some energy beating between them. Another
possibility is the divergence of the pulse energy variation in the vicinity around the equilibrium
point that leads to instability.
The pulse amplitude fluctuation increases with the multiplication factor. Also, with wider filter
bandwidth used in the laser cavity, better stability condition can be achieved. The nonlinear phase
shift and noises in the system challenge the system stability of the multiplied pulses. They not only
change the pulse shape of the multiplied pulses, they also distort the phase plane of the system.
Hence, the system stability is greatly affected by the self-phase modulation as well as the system
noises. This stability analysis model can further be extended to include some system nonlinearities,
such as the gain saturation effect, non-quadrature biasing of the modulator, fiber nonlinearities,
etc. The chaotic behavior of the system may also be studied by applying different initial phase and
injected energy conditions to the model.
In Section 7.4, we have demonstrated an erbium-doped fiber ring laser using an all-polarizationmaintaining-fiber (PMF) Sagnac loop filter for multi-wavelength operation. The theoretical analysis
and experimental results of the all-PMF (polarization maintaining fiber) Sagnac loop as a stable
comb filter have been presented. The Sagnac loop filter is a simple and all-fiber device that consists
of a PMF coupler and a segment of stress-induced PMF to form the loop. The number of output lasing wavelengths has been obtained by adjusting the polarization state of the light in the ring cavity
using a polarization controller. The wavelength spacing is determined by the frequency period of
the comb filter with equal frequency interval.
We are currently pursuing the design and demonstration of multi-wavelength mode-locked lasers
to generate ultra-short and ultra-high rep-rate pulse sequences by employing a multi-spectral filter
de-multiplexers and multiplexers within the photonic fiber ring.
8
Multi-Dimensional Photonic
Processing by DiscreteDomain Approach
8.1 MULTI-DIMENSION (MULTI-D) PSP DESIGN TECHNIQUES
Information transport infrastructures are currently based and directionally developed towards
super-dense wavelength division multiplexing with provisional routing and switching in spatial
(e.g., routing to different sub-network elements) and spectral (e.g., multiplexing/demultiplexing,
filtering, adding/dropping of wavelength channels) domain. The processing of photonic signals is
now becoming very important in these diverse domains.
This section has sought to integrate the fields of discrete signal processing and fiber-optic signal
processing, integrated photonics and/or possibly nano-photonics to establish a methodology based
on which physical systems can be implemented. Because fiber-optics is essentially one-dimensional
planar medium, the methodology has been proposed to implement 2-D signal processing using 1-D
sources and processors. A number of 2-D filter design algorithms are implemented. These algorithms
are applicable to photonic filters that perform 2-D processing. The developed 2-D filter design methods are generic allowing the proposals of several photonic signal processing (PSP) architectures in
Sections 8.1.3, 8.1.4 and 8.2.5 to enable efficient coherent lightwave signal processing.
8.1.1
an Overview OF phOtOnic signal prOcessing
In recent years, there has been a notable increase in the number of applications that require an
extremely fast signal processing speed that cannot be met by the current all-electronic technology.
Photonic signal processing (PSP) opens the possibilities for meeting the demands of such highspeed processing by exploiting the ultra-high bandwidth capability of lightwave signals with specific applications in the field of photonic communications and fiber optic sensor networks.
The field of signal processing is concerned with the conditioning of a signal to fit certain required
characteristics, such as bandwidth, amplitude, and phase. Conventional techniques of signal processing
make direct or indirect use of electronics. For example, frequency filtering, a most important signal
processing procedure, can be performed through direct electronic means, such as tunable IC filters,
or indirectly by digitizing the input for subsequent processing by computers or special purpose digital
signal processing chips. Although a high performance can be obtained using either of the techniques,
electronic methods suffer from physical limitations that govern the maximum processing speed. The
demands for high performance beyond that is achievable by electronic means have been increasing
recently due to the increase in computationally demanding real-time processing applications.
Using lightwaves instead of electronic signals as the information carrier in signal processing
is an appealing concept. The full potential of the technology has been accelerated in recent years
due to the invention and discovery of photonic crystals. Several important advances have been
made in utilizing light as the information carrier including real-time spatial-light modulators and
electro-optic devices, micro-ring resonators, photonic crystal fibers, guided-wave crystal photonics,
and super-prisms. Another incentive for using light as the information carrier is the superiority of
341
342
Photonic Signal Processing
fiber-optic communication systems, which offer the wide bandwidth properties of photonic fiber
medium. To fully exploit the capability of photonic systems, PSP is very essential.
The field of photonic signal processing (PSP) can be divided into two distinctive approaches
which are outlined in the following sections.
8.1.1.1 Spatial and Temporal Approach
The first use of lightwaves for signal processing applications was developed as early as 1968
when an “integrated photonic correlator”1,2 consisting of spatial light modulators and lenses in a
planar waveguide was suggested. Further developments along this line were made, and several
experimental devices including acousto-optic spectrum analyzer, a time-integrating acousto-optic
correlator, a hybrid electro-optic/acousto-optic vector multiplier, a high-speed electro-optic analogto-digital converter, and several fiber delay-line processors2 were demonstrated.
The advantage of the spatial and temporal approach over the conventional electronic approach can
be seen in light of the fact that lenses, which are 2-D devices, have Fourier transform properties and
can, therefore, act as a massively parallel Fourier transform processor. Taking advantage of the massive
parallelism can mean the removal of the Von-Neumann bottleneck of present-day digital computers.
Although an all-photonic computer does not seem feasible in the near future, a hybrid photonicelectronic computer offering ultra-high-speed processing capability could be realized by combining
photonic information processing for some specific functions and electronics for general operation.3
The drawback with the spatial and temporal approach is the fact that the signal processing is
performed in an analogue manner. As shown in Figure 8.1, lightwaves carrying different signals must
travel through different media suffering acoustic diffraction resulting in crosstalk.2 It is interesting to
note that using holographic techniques, several layers of neural nets can be implemented, with each
layer in parallel format, making spatial and temporal approach a suitable technique for neural network
implementation. Although this technique may be useful in the implementation of an opto-electronic
computer, the approach is not suitable for signals that have been transmitted through photonic fiber
communication networks. Such signals are sequentially linear, and, in order to be processed by a
2D spatial
modulator
2D spatial
modulator
2D photodetector
array
Input pattern
Reference pattern
Output
Lighwave input
FIGURE 8.1
diffractors.
Spatial Fourier optical signal processor. The photonic active components are the acousto-optic
N. Q. Ngo and L. N. Binh, Programmable incoherent Newton-Cotes optical integrator, Opt. Commun., 119, 390–402,
1995.
2 R. T. Robert, A. K. Kar1, and J. Allington-Smith, Ultrafast laser inscription: An enabling technology for astrophotonics,
Opt. Exp., 17(3), 1964–2969, 2009.
3 H. F. Taylor, Application of guided-wave optics in signal processing and sensing, Proc. IEEE, 75(11), 1524–1535, 1987.
1
343
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
spatial and temporal processor, a conversion into a suitable 2-D format using demultiplexing devices
and laser arrays will be required. The following section introduces a technique that is ideal for
lightwave signals from guided media, such as photonic fibers and photonic crystals.
The spatial structures can be translated in fiber and integrated photonic forms using planar
lightwave circuit (PLC) using silica-on-silicon technology, for example the array waveguide filters
acting as wavelength muxes and demuxes and spatial separators.
8.1.1.2 Fiber-Optic or Integrated Optic Delay Line Approach
Guided-wave photonics and fiber optics provide alternative architectures for PSP compared to the
classic spatial or time-integrating architecture introduced in Section 8.1.1.1. The main advantage of
guided-wave systems over spatial and temporal systems is the wide bandwidth property available
with photonic fiber transmission medium. For example, a silica fiber with a nominal 5 µs delay can
store 1 GHz bandwidth signals for time periods less than one millisecond.3 Another advantage of
guided-wave optics can be the elimination of acoustic diffraction. However, since photonic fiber
is essentially a 1-D medium (signal propagates along one axis—that of the fiber), this architecture
sacrifices the 2-D nature of light that is utilized in time and space integrating architectures. In
effect, in guided-wave systems, the advantage of massive 2-D parallel processing capability of light
is sacrificed for the wide bandwidth of guided-wave optics that enables high-speed processing.
Despite this limitation, which confines the use of fiber-optic technology to signals from guided
lightwave transmission medium, the simple fact that the current major usage of photonic systems is
in communication systems makes the technology useful as it presents the possibilities of removing
the bottleneck caused by opto-electronic conversion and, therefore, ensuring full utilization of fiber
bandwidth. So far various uses have been found for fiber-optic signal processors as frequency filters,
matched filters, correlators, and waveform and sequence generators.4,5,6,7,8,9,10
Figure 8.2 shows one possible configuration of a fiber-optic processor. Although filter coefficients
were realized using reflectors in Figure 8.2, other in-line components, such as photonic attenuator/
amplifier, can also be used for implementing filter coefficients. It is evident that the operation of fiberoptic delay line filters is similar to that of digital filters. In fact, the correct term to describe the fiberoptic signal processing would be “discrete-time PSP” rather than digital signal processing as the range
optical coupler
optical fiber
reflector
FBG
Input lightwave
Output
FIGURE 8.2 Fiber-optic delay line processor. The coupler can be replaced by a3-port optical circulator. The
reflector can be a fiber Bragg grating (FBG).
C. K. Marsden and J. H. Zhao, Optical Filter Analysis and Design: A Digital Signal Processing Approach, New York:
John Wiley & Sons, 1999.
5 J. Capmany and J. Cascon, Optical programmable transversal filters using fiber amplifiers, Electron. Lett., 28, 1245–
1246, 1992.
6 B. Moslehi, Fiber-optic filters employing optical amplifiers to provide design flexibility, Electron. Lett., 28, 226–228,
1992.
7 E. Heyde and R. A. Minasian, Photonic signal processing of microwave signals using an active fiber Bragg grating pair,
IEEE Trans. Microw. Theory Tech., 45, 1463–1466, 1997.
8 F. T. S. Yu and I. C. Khoo, Principles of Optical Engineering, Section 9, New York: John Wiley, 1990.
9 L. N. Binh, N. Q. Ngo, and S. F. Luk, Graphical representation and analysis of the Z-shaped double-coupler optical
resonator, IEEE J. Lightw. Technol., 11, 1782–1792, 1993.
10 J. S. Lim, Two-Dimensional Signal and Image Processing, Englewood Cliffs, NJ: Prentice Hall, 1990.
4
344
Photonic Signal Processing
of the input or output signal is not digital at all. In any case, the discrete-time property makes it possible
to apply the well-developed z-transform techniques to filter design. In Section 8.1.2, the application of
the z-transform techniques for analysis and design of fiber-optic systems is discussed in detail.
8.1.1.3 Motivation
The demand for multi-dimensional photonic signal processing (M-ary PSP) can be attributed to
various factors due to the growing feasibility of high-capacity digital transmission networks capable
of transmitting ultra-high bit rate and time division multiplexing up to 160 Gb/s as well as fiber
optical sensor networks.
A problem with the implementation of such systems is the lack of devices with the capability of
processing an enormous amount of data associated with multi-dimensional signals. With photonic
transmission networks becoming the transport infrastructure, PSP technique has become increasingly
more desirable compared to O/E and E/O conversion techniques. As discussed in Section 8.1.1.2, fiberoptic signal processing systems are ideal for such processing demands for several reasons: all-optical
(or photonic) processing of photonic information of optical communication systems are possible
using fiber-optic signal processing; 2-D signals usually require much higher bandwidth than 1-D
signals and therefore must be processed by a high bandwidth system to allow real-time performance;
and it is likely that future telecommunication networks would be all fiber-optic (Table 8.1).
8.1.2
multi-DimensiOnal signal prOcessing
Multi-dimensional signal processing enables processing of signals that depend on more than one
co-ordinate. Although many concepts of multi-dimensional signal processing are straightforward
extensions of 1-D signal processing theory, there are also significant differences that need to
be clarified, particularly when referred to photonics. Discussions of Multi-dimensional signal
processing in this paper is limited to 2-D signal processing applicable to photonics that is by far the
most important class of multi-dimensional signal processing.
8.1.2.1 Multi-Dimensional Signal
One may define multi-dimensional signals as signals whose values at a certain instance of time,
space, or other coordinates depend on more than one variable. In 2-D signal processing, each of
the properties depends on both x and y direction and therefore the concepts of spatial signal, and
therefore spatial frequency must be introduced. Spatial frequency does not depend on time, but rather
depends on the spatial variations of the 2-D signal. There are two distinct spatial frequencies: one in
x direction and one in y direction. 2-D signals form the most important class of multi-dimensional
TABLE 8.1
Outline of Two Different Approaches to PSP
Spatial and Temporal
Fiber-Optic/Integrated Optic
Time mode
Flexibility
Unguided
Lenses, light modulators, mirrors, masks, LED
or laser arrays, slits
Continuous-time
Hard to change configuration once developed
Analysis method
Accuracy
Cross-talk
Major use
Parallel processing capability
Difficult (some Fourier transforms)
Low
Yes
Photonic computing
Massive parallel processing
Guided
Lasers or LED’s, optical fibers, optical
amplifiers (OA), attenuators, reflectors
Discrete-time
Easy to adjust the function using
different tab values
Well known z-transform method
High
No
Communication signal processing
Limited parallel processing
Principle operating mode
Components used
345
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
signals, and methods developed for 2-D signal can be generalized to signals of larger dimensions.
This chapter concentrates on developing filter design methods for two-dimensional signals.
8.1.2.2 Discrete Domain Signals
A signal domain can be either continuous or discrete. For digital signal processing purposes however,
it is convenient to “sample” continuous domain signals at a discrete interval so that in effect it has a
discrete domain. In 1-D, signal to be processed or stored in a sequential manner can be sampled at
discrete intervals of time or direction. Put into an equation form, 1-D signal can be represented by
a train of scaled impulses as
∞
a(t ) = lim
∆ →0
∑ a(k∆)δ (t − k∆)∆
(8.1)
k =−∞
where ∆ is the sampling period and n denotes the sequence number. The sampled signal can be
infinite in extent and reflects this accordingly. If ∆ is infinitely short, then above expression reduces
to the representation of a continuous signal as expected.
In 2-D, a natural extension to (8.1) can be made as s
∞
a( x, y) = lim lim
∆1 →0 ∆2 →0
a(k1∆1, k2 ∆ 2 )
∞
∑ ∑ δ ( x − k ∆ , y − k ∆ )∆ ∆
k1 =−∞ k2 =−∞
1 1
2
2
1
(8.2)
2
There is an important difference between sampling of 1-D signals and 2-D signals in practice. Assuming
there is only one sampling device, 1-D signal such as the one shown in Figure 8.3a can be sampled by
taking values at discrete intervals. If the signal duration is infinite in extent, no truncation is needed
as the transfer function defines the limit even if the signal is not periodic. For 2-D signals with infinite
duration, this is not the case. As seen in Figure 8.3b, if the 2-D signal was sampled infinitely in one
dimension, the part of the 2-D signal that extends in the other dimension will never be sampled. For
2-D signals, there is always a predefined limit on how many samples are taken in each dimension.
After reaching the limit in one dimension, the coordinate on the other dimension is incremented by one
sampling period, and the sampling process continues until the number of samples in the first dimension
again reaches the limit. The process is repeated until the pre-defined number of samples in the second
dimension is reached. The consequence is that the process gives a train of sampled 2-D signals stretched
out in 1-D as shown in Figure 8.3c. For 1-D discrete time processing of 2-D signals, the signal must be
y
0 1
2 3
4 ......
42 43 44 45 46 47 48
35 36 37 38 39 40 41
28 29 30 31 32 3 34
21 22 23 24 25 26 27
14 15 16 17 18 19 20
7 8 9 10 11 12 13
0 1 2 3 4 5 6
t
(b)
(a)
0 1 2 3 4 5 6 7 8 9 . . . . . .48
(c)
x
t
FIGURE 8.3 (a) Infinite extent 1-D signal. (b) 2-D signal with finite predefined limit of 7 × 7 (each index refer to
the crossing at the bottom-left corner of the grid it belongs to) and (c) The signal in (b) fed into 1-D signal processor.
346
Photonic Signal Processing
sampled in this way so that the processor can implement the delays z1−1 and z2−1 using only 1-D delay
photonic element. The limiting of sample space is similar to windowing or the truncation performed on
1-D signals for some signal processing operations, such as discrete Fourier transform (DFT).
Discrete space form of a 2-D signal with predefined limits can be expressed by
n1
a[n1, n2 ] =
n2
∑ ∑ a[k , k ]δ [n − k , n − k ]
1
2
1
1
2
(8.3)
2
k1 =0 k2 =0
Using the above form of coding for 2-D signals in a linear sequence, 2-D signal processing using the
1-D medium, such as optical fiber, can be made possible.
8.1.2.3 Multi-Dimensional Discrete Signal Processing
Having made a reasonable compromise in the size of the predefined limit, i.e., truncation window
size, the Nyquist rate can be applied to 2-D signals to determine the sampling rate. In 1-D, the
Nyquist rate is twice the highest frequency component of the sampled signal and defines the
sampling rate necessary to preserve the entire bandwidth of the signal.
In 2-D, the direction in which the Nyquist rate is applied must be made clear as sampling in 1-D
at the Nyquist rate may not guarantee the preservation of the 2-D signal if the signal varies faster
with respect to the other dimension. To preserve the entire 2-D signal bandwidth, sampling must be
performed at twice the highest spatial frequency component of the 2-D signal in any direction in the
sampled space. For example, consider a signal that has a 20 GSamples/s (Ga/s) component in n1-axis
but has a 60 GSa/s component at 70° from n1-axis. In this case, the sampling rate of 40 GSa/s in both
dimensions is not adequate as the signal has a frequency component of 60 GSa/s × sin(70°) = 56 GSa/s
along n2-axis. Since the sampling rates in both dimensions are usually kept the same, the sampling
rate of 56 × 2 = 112 GSa/s in both dimensions will preserve the entire 2-D signal bandwidth.
In discrete-time signal processing, the term normalized frequency is used to describe a frequency
independent of the system sampling frequency. The concept is applied in 2-D processing with a
straightforward extension to spatial frequency.
8.1.2.4 Separability of 2-D Signals
A 2-D sequence is separable if it can be represented by a product of two 1-D sequences, as shown
in (8.1.2.5). Separable sequences form an important and special, but limited class of 2-D sequences.
Many results in 1-D theory have a simple extension for separable 2-D sequences whereas for
non-separable sequences such extensions often do not exist. If a 2-D sequence is separable, the
separability can be exploited to reduce the processing requirements resulting in considerably less
amount of computation. Unfortunately, most 2-D sequences are not separable.
s[n1, n2 ] = f [n1 ]g[n2 ]
(8.4)
An example of a separable sequence is the unit sample sequence δ(n1, n2) as shown in Eq. (8.5a).10 Other
examples of separable sequences include the unit step sequence u(n1, n2) and the example in Eq. (8.5b).
 δ [n1, n2 ] = δ [n1 ]δ [n2 ]
an1 bn2 + bn1 +n2 =  n1
a + bn1 bn2

(
)
(a )
( b)
(8.5)
8.1.2.5 Separability of 2-D Signal Processing Operations
Similar to 2-D sequences, a 2-D signal processing operation can be classified as separable or nonseparable. The consequence of an operation being separable is that the operation yields the correct
answer when it is performed in two independent cascade stages with each stage performing the
operation with respect to only one of the independent variables. The situation is illustrated in Figure 8.4.
347
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
2D Input
2D Input
Signal processing
operation
S. P.
Operation
in n1
2D Output
S. P.
Operation
in n2
2D Output
FIGURE 8.4 A separable 2-D signal processing operation.
An example of a 2-D separable signal processing operation is double integration. A double integration procedure can be expressed as
∞
F(n1, n2 ) =
∞
∫ ∫ f (n , n )dn dn
1
2
1
2
(8.6)
n2 =−∞ n1 =−∞
The 2-D sequence f ( n1, n2 ) is integrated with respect to n1 first, and then with respect to n2. The two
procedures can be put in cascade and thus double integration operation is classified as a separable
operation. Note that the separability of the 2-D input sequence f ( n1, n2 ) is not a pre-requisite for the
success of the operation. In addition, separable signal processing operations have separable impulse
responses. The 2-D signal processing can be performed by convoluting the 2-D input with the 1-D
filter impulse response in one dimension, and the operation can be completed by convoluting the
result of the first convolution with the filter impulse response in the other dimension. It is therefore
clear that the operations can be performed using a cascade stage of two filters. As with the case of
separable sequences, separable operations form a special class of 2-D signal processing operations.
Most signal processing operations are not separable.
In discrete domain, separable operations can be expressed in terms of a product of two z-transform
transfer functions. For example, the double integration (in Eq. 8.6) using Simpson’s rule for digital
integration.11
H ( z1, z2 ) =
T1  1 + z1−1  T2  1 + z2−1 

⋅ 

2  1 − z1−1  2  1 − z2−1 
(8.7)
It is clear that H ( z1, z2 ) is separable. For other functions, the separability is often not the case.
A circularly symmetric 2-D digital lowpass filter cannot be separated into a product of two
functions, each dealing with only one kind of delays ( z1 or z2 ). In such cases, a way of dealing with
non-separability must be found as cascade stages will no longer work. It is the non-separability of
most 2-D signal processing functions that makes implementation of 2-D filters a difficult task.
H ( z1, z2 ) = 1 + 0.9z1−1 + 0.9z2−1 + 0.8z1−1z2−1 − 0.1z1−2 − 0.05z1−1z2−22 − 0.05z1−2 z2−1
−0.1z2−2 + 0.1z1−3 + 0.1z1−3 z2−1 + 0.07z1−3 z2−2 + 0.07z1−2 z2−3 − 0.05z1−3 z2−3
(8.8)
Another advantage of having a separable implementation is the issue of stability. The stability analysis
of non-separable filters is very difficult and there are no known simple methods of checking the
348
Photonic Signal Processing
stability of 2-D filters directly from the transfer function or from pole-zero plots as is the case with 1-D
systems.11 However, with separable filters, if 1-D sub-sections are stable, then the overall stability is
guaranteed. Stability of 1-D filters can be guaranteed by having all system poles inside the unit circle.
8.1.3 Filter Design methODs FOr 2-D psp
In Section 8.1.2, the concepts of 2-D signal processing have been introduced. Out of many possible
mathematical models for 2-D systems, the model best suited to fiber-optic signal processing must be
found. In this section, two different mathematical models of representing 2-D systems are presented
and a brief introduction to 2-D filter design methods is given.
8.1.3.1 2-D Filter Specifications
To specify a filter, two approaches can be adopted. One approach specifies a filter in mathematical
form by specifying the transfer function or the state-space equations of the filter. This method
can specify the exact behavior of the filter. The other approach specifies a filter by its transfer
characteristics of magnitude and phase response or impulse response of the filter. This later approach
is more intuitive than the former because it is easy to see how the filter would behave in practical
implementation. However, the accuracy of the filter then depends on the accuracy of the specification,
which can sometimes be inadequate. In any case, the later approach must go through the mathematical
description before implementation. Developing a method for designing and implementing a filter
from its dynamic characteristics therefore encompasses the mathematical description.
The method developed in this section assumes the spatial frequency responses of the filter to
be specified. The 2-D photonic filter design process can be as follows: specification of magnitude
or impulse response of the desired 2-D filter; development of transfer function or state-space
description of the 2-D filter; development of signal flow diagram of the 2-D filter; and development
of photonic implementation of the 2-D filter.
To specify a filter using its frequency response, both magnitude and the phase responses need
to be supplied. However, designing a filter with a certain phase response is a very difficult task. In
many cases of interest, a condition of linear phase is all that is required and, in this chapter, the
condition is adhered to. The reason for requiring a linear phase can be explained by the Fourier
transform of a 1-D linear phase filter as
V ( f )e − jϕω ⇔ v(t − ϕ )
(8.9)
In Eq. (8.9), the phase φ(ω) is proportional to frequency. The Fourier transform (FT) of the linear
phase on the right of ⇔ shows that a linear phase corresponds to pure time delay. The result is extendible to 2-D simply by substituting frequency by spatial frequency and time delay by spatial delay.
A non-linear phase response leads to non-uniform delays and, thus, inter-symbol interference (ISI).
8.1.3.2 Mathematical Model of 2-D Discrete Photonic Systems
8.1.3.2.1 Transfer Function Description
2-D transfer function description of the filter can also be explained by using a 2-D difference
equation. As in 1-D, 2-D transfer functions can readily be turned into 2-D difference equations
H ( z1, z2 ) =
T1  1 + z1−1  T2  1 + z2−1 

⋅ 

2  1 − z1−1  2  1 − z2−1 
The equivalent difference equation is given by
11
For more detailed discussion on stability checking using position of poles, refer to Section 8.4.25
(8.10)
349
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
y(n1, n2 ) =
T1T2
[ x (n1, n2 ) + x (n1 − 1, n2 ) + x (n1, n2 − 1)
4
+ x (n1 − 1, n2 − 1)] + y(n1 − 1, n2 )
(8.11)
+ y(n1, n2 − 1) − y(n1 − 1, n2 − 1)
Figure 8.5 illustrates the sample points that are summed in y ( n1, n2 ) of (8.10). Because we are dealing with
spatial delay and not time delay, the actual implementation of delay depends on the signal transmission
format. If all points of the 2-D signal are transmitted in parallel, then the delays need not be time delay
raising the possibility of parallel processing similar to that of spatial and temporal architecture.
A 1-D integrator transfer function amounts to just the first half or the second half of H ( z1, z2 ) in
(8.10) and can readily be turned into a signal-flow graph (SFG) as shown in Figure 8.6 and described
in Chapter 2.
The SFG of 2-D version of the trapezoidal integrator transfer function H ( z ) is shown in Figure 8.7.
A notable difference is the presence of two different delay elements. In spatial terms, one represents
the vertical delay and the other represents the horizontal delay.
Whether an SFG can readily be turned into photonic domain depends on its structure. A major
obstacle that prevents a direct translation of SFGs into photonic circuits is the number of complex
interconnections. Too many complex interconnections can result in the loss of modularity making
the photonic implementation of the transfer function difficult.
Input array
FIGURE 8.5
Output array
x31
x32
x33
y31
y32
y33
x21
x22
x23
y21
y22
y23
x11
x12
x13
y11
y12
y13
Illustration of the difference Eq. (8.10).
1
Input
1
Output
z-1
-1
FIGURE 8.6
1
1-D trapezoidal integrator signal flow diagram.
1
1
-1
z1-1
1
z2-1
Input
T1/2
1
1
z1-1
-1
FIGURE 8.7
Current
output
2-D trapezoidal integrator SFG diagram.
1
T2/2
Output
350
Photonic Signal Processing
Transfer functions can be manipulated into potentially useful forms for photonic implementation.
One such manipulation technique is called continued fraction expansion realization (CFER) as
described below. Although the method results in reduced number of filter components, not all transfer
function can be expanded easily. A method of designing a filter transfer function that can be expanded
using continued fraction expansion is therefore required. Such a design method does not exist currently
and, as a consequence, the use of CFER is confined to only transfer functions that are expandable.
8.1.3.2.1.1 Continued Fraction Expansion Given a transfer function, the numerator of the transfer function is recursively long-divided by the denominator until the remainder is only a simple fraction. A possible form for a fraction that has been expanded using continued fraction expansion as
H ( z1, z2 ) = C1 +
1
A1z1 +
1
C2 +
(8.12)
1
B1z2 +
1
It is intuitively obvious that such expansions do not exist for all polynomial fractions. A method of
checking the existence of such expansion is given in references.12,13,14
8.1.3.2.2 State-Space Equation Description
State-space description can be seen as an alternative description method to the transfer function
description. The advantages offered by state-space description include the notion of observability and
controllability. Although such concepts are useful in 2-D dynamic control system, the applications
of the concepts are not obvious in 2-D signal processing. At best, the main advantage of using the
state-space approach can be stated as the established techniques of 1-D state-space theory, such as
algorithms to manipulate state-space matrices to obtain a reduced order system (see Section 8.2).
8.1.3.2.2.1 An Algorithm for Conversion of a 2-D Transfer Function into 2-D State-Space
Equation In Antoniou and Lu,16 a 2-D state-space description is formulated from a 2-D FIR
transfer function description using the following method. Ref.15 has given a more generalized case
of transfer functions of 2-D IIR filters.
8.1.3.2.3 Formulation of State-Space Equations from Transfer Function16
A 2-D FIR transfer function can be expressed by
N1
H ( z1, z2 ) =
N2
∑ ∑ h (n , n ) z z
1
2
− n1 − n2
1
2
(8.13)
n1 =0 n2 =0
L. N. Binh, Generalized analysis of a double-coupler doubly-resonant optical circuit, IEE Proc. Optoelectron., 142,
296–304, 1995.
13 R. Mersereau, W. F. G. Mecklenbrauker, and T. F. Quatieri, Jr., McClellan transformations for two-dimensional digital
filtering: I-design, IEEE Trans. Circ. Syst., CAS-23(7), 405–422, 1976.
14 Z.-J. Mou, Efficient 2-D and multi-D symmetric FIR filter structures, IEEE Trans. Signal Process., 42(3), 1994.
15 S. Kung, B. C. Levy, M. Morf, and T. Kailath, New results in 2-D systems theory, Part II: 2-D state-space modelsrealization and the notions of controllability, observability, and minimality, Proc. IEEE, 65(6), 945–961, 1977.
16 A. Antoniou and W. Lu, Design of two-dimensional digital filters by using the singular value decomposition, IEEE
Trans. Circ. Syst., CAS-34(10), 1191–1198, 1987.
12
351
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
A state-space form can be expressed as
 x h (i + 1, j )   A1
 v
=
 x (i , j + 1)  A 3
y(i , j ) = [ c1
A 2   x h (i , j )   b1 
u(i , j ) ≡ Ax + b

+
A 4   x v (i , j )   b2 
 x (i , j ) 
c2 ]  v
 + du(i , j ) ≡ cx + du
 x (i , j ) 
(a)
(8.14)
h
0
A1 = 
0
0
A4 = 
0
A 3 = 0,
b =  h10
c = 1
 h1N2

A2 =   hN1N2
I N1 −1 
,
0 
0
hN1 0
0
0
h0 N2
(b)
h11 


hN11 
…
I N2 −1 

0 
(8.15)
1
0
T
h01 
d = h00
8.1.3.2.3.1 An Algorithm to Convert a 2-D State-Space Equations into a 2-D Transfer
Function For the state-space approach to be useful, there must be a method of converting a 2-D
state-space equation into a 2-D transfer function form. An algorithm to perform such a task could
not be found in standard text books of digital signal processing. Therefore, it had to be devised
independently. The algorithm given in this section performs conversion from a 2-D state-space
equation specified in (8.16) to a 2-D transfer function description.
By re-arranging (8.13), we can obtain the transfer function form of the same system with input
denoted by x and output denoted by y as
 z1I m
H ( z1, z2 ) = C 

−1


 − A B
z2 In 

(8.16)
What makes the implementation of the above equation difficult, is the presence of matrix inverse. Because
the matrix inside the bracket in (8.16) describes a 2-D system, the determinant of the matrix contains two
independent variables z1 and z2 and cannot therefore be solved by simply obtaining the eigenvalues of the
matrix and cross-multiplying to get coefficients of variables as in 1-D determinant calculation.
8.1.3.2.3.2
An Algorithm to Obtain the Characteristic Polynomial of a Matrix Describing a
2-D System
1. Let A be the matrix of size m × n describing a 2-D system and Z be a zero matrix of size m × n.
2. Let z1 = 0, z2 = 0.
3. Let A’ be a matrix formed by eliminating m-z1 rows and n-z2 columns from matrix A.
4. Let Zz1,z2 = Zz1,z2 + ∆(A’). ∆ is 1-D determinant operator.
5. Repeat step 3 and 4 until all combinations of z1 rows and z2 columns are tried.
6. Repeat step 3 to 5 with different value of z1 and z2 until all coefficients of the characteristic
polynomial Z are found.
7. Reverse the signs of elements of Z whose indices sum to an odd number. The resulting
matrix Z contains the coefficients of 2-D characteristic polynomial in format as.
352
Photonic Signal Processing
N1
H ( z1, z2 ) =
N2
∑ ∑ h(n , n ) z z
1
− n1 − n2
1
2
2
n1 =0 n2 =0
is expressed in maatrix form as
 h00
h
10
Z=
 
 hm0
h01
h11
hm1
h0 n 
h1n 


hmn 
(8.17)
For the 2-D transfer function description, denominator and the numerator can be calculated by
h a = det(A)
h b = det(A − B × C) − det(A) × ( D − 1)
(8.18)
8.1.3.3 Filter Design Methods
8.1.3.3.1 Direct Design Methods
Direct design methods include window method, frequency sampling method, transformation
method for FIR filter implementations, and impulse response method for IIR filters. All of the design
methods listed here are similar to the 1-D methods of the same name involving some extensions of
the concepts into 2-D and they all result in non-separable transfer functions.
The general format of non-separable filter transfer functions generated by FIR filter design
methods is given in Section 8.1.2, and the most general form of 2-D FIR filter signal flow diagram is
shown in Figure 8.8. In Section 8.1.4, the details of the algorithms and the implementations of 2-D
direct design of FIR filters are discussed. Direct design methods for IIR filters are also discussed.
8.1.3.4 Use of Matrix Decomposition
Mitra et al.17 introduces a method where matrix decomposition is used to separate a non-separable
function into a cascade of two separable filter stages each one involving only one set of delay elements.
A drawback with Mitra’s method is that there is no general structure for filter implementation.
Input
z2-1
h30
h31
h32
h33
z2-1
h20
h21
h22
h23
z2-1
h10
h11
h12
h13
h00
h01
h02
h03
z1-1
z1-1
z1-1
Output
FIGURE 8.8 General form of FIR filter signal flow diagram.18
S. Mitra, A. Sagar, and N. Pendergrass, Realizations of two-dimensional recursive digital filters, IEEE Trans. Circ. Syst.,
Cas-22(3), 177–184, 1975.
18 Z.-J. Mou, Efficient 2-D and multi-D symmetric FIR filter structures, IEEE Trans. Signal Process., 42(3), 1994.
17
353
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
−
2
z2-1
+
1
z2-1
+
z2-1
+
FIGURE 8.9 A subsection of filter in Antoniou and Lu. (From Antoniou, A., and Lu, W., IEEE Trans. Circ.
Syst., CAS-34, 1191–1198, 1987.)
The filter structure is therefore heavily dependent on the transfer function, and this lack of generality
of filter design limits the usefulness of the method in photonic filter implementation where the final
product corresponds closely to the SFG representation of the filter transfer function. Figure 8.9
shows a part of the filter implementation.
Another approach, which uses matrix decomposition, is decomposition of 2-D magnitude
specification into the sum of products of 1-D magnitude specifications. Because 2-D magnitude
specification becomes a set of two 1-D magnitude specifications, the design process of a 2-D filter
reduces to a set of 1-D filter designs for which the established design methods are aplenty. In
addition, the resulting 2-D filter is separable. Section 8.2 discusses a number of 2-D filter design
algorithms based on matrix decomposition of 2-D magnitude specification.
8.1.4
Direct 2-D Filter Design methODs
As the first of two streams of 2-D filter design methods, direct 2-D filter design methods are
introduced. This class of 2-D filter design methods does not use matrix decomposition and instead
uses 2-D magnitude specifications directly to produce non-separable designs.
8.1.4.1 FIR and IIR Structures in 2-D Signal Processing
A digital filter can be divided into two broad classes, FIR (Finite Impulse Response) and IIR (Infinite
Impulse Response). FIR filters only use feed forward structure and, therefore, are non-recursive. Whereas
IIR filters use feedback as well as feed forward structure and, therefore, are recursive. The impulse
response of an IIR filter is infinite in duration, therefore it has the name “infinite impulse response filter.”
Given a 2-D frequency response specification, one can either try to formulate a FIR filter transfer
function or an IIR filter transfer function. There are several factors that must be considered when
deciding which structure to implement for a given frequency response specification: (i) Linearity of
the phase response of the filter; (ii) Stability of the filter; and (iii) Order of the implemented filter.
Linearity of the designed filter’s phase response is very important as explained previously in
Section 8.1.3.1. Linear phase FIR filters are very easy to design as the condition for the linear phase
is simply a symmetric impulse response, which, in turn, is guaranteed if the magnitude response
of the 2-D filter is symmetric about the two axes. For 2-D IIR filters, phase linearity is much
more difficult to guarantee. Often IIR filters are specified only with magnitude characteristics, and
the phase response is generally accepted for what it is; non-linear. The lack of control over phase
response of IIR filters limits its usefulness in many applications.10
Stability is a very important issue in designing of any dynamic system. This requires no explanations.
The advantage of 2-D FIR filter over 2-D IIR filter regarding the issue of stability is that for 2-D FIR
filters stability is inherent in its definition. Since the impulse response of the FIR filter is finite in
354
Photonic Signal Processing
duration, bounded input results in bounded output and the filter is, therefore, always stable. Although
2-D IIR filters can be designed to be stable, as mentioned in Section 8.1.2.5, there is no simple algorithm
for checking the condition for stability of a 2-D IIR filter. A mathematical theory involving complex
cepstrum to check for the stability condition of 2-D filters is quite involved, and most algorithms for
checking the 2-D stability simply repeat 1-D stability condition over the 2-D space many times over
which can be computationally very inefficient.10 As a consequence of the lack of usable algorithms or
a simple method for stability testing, there is no known method of designing stable 2-D IIR filters.10 In
practice, 2-D FIR filters are, therefore, much more preferred to 2-D IIR filters.
Order of the filter refers to the number of delay elements in the numerator (or the denominator if
the filter is IIR). Order of the filter has a direct consequence in the final implementation of the filter
as the determining factor of the number of processing elements. Higher order filters require more
processing elements than lower order filters making low order filters more desirable. One distinctive
advantage of IIR filters over FIR filters is that the order of the filter required for a given magnitude
specification is smaller. FIR filters can sometimes require an excessive order (the definition of
“excessive” depends on the implementation medium; for example, in software implementation of
a digital filter, a 1000th order filter might be quite acceptable but with fiber-optic delay line filters,
the maximum order is well below 50). For example, an integrator, which has an infinite impulse
response by nature, can be described using a first order filter within 12.5% error,11 whereas achieving
the same error with 1st order FIR filters would be impossible. In addition, 2-D filters usually require
much higher order filters than 1-D filters of similar transition band requirements, so it is important
that the 2-D filter design methods keep the order of the filter to an acceptable level.
8.1.4.2 Frequency Sampling Method
As the first of the direct 2-D filter design methods, 2-D frequency sampling method produces an FIR
filter with the minimum of fuss. Using the fact that the transfer function of a FIR filter is the same as
the impulse response of the filter, the 2-D frequency sampling method takes discrete Fourier transform of even-spaced samples of 2-D frequency response and uses the result as the coefficients for 2-D
transfer function. It is noted that the procedure is identical to that of 1-D frequency sampling method.
It is observed in Lim10 that the filter designed using frequency sampling method is not optimal as far
as number of delay elements is concerned. Also, the frequency response of the filter is controlled only
by the sampling rate of the frequency sampling and the nature of the frequency samples. For example,
increasing the frequency sampling rate will increase the number of discrete sampling points of the
impulse response and, hence, will result in a filter with a better frequency response but with a larger
order. It is also found that frequency response can be improved considerably, especially around the
transition band, if the ideal frequency response takes account of the transition frequency values.
8.1.4.2.1 A Filter Design Example Using Frequency Sampling Method
Design Aim: A lowpass filter with normalized cut off frequency of 0.5 in both dimensions
Method: Frequency sampling method
Program used: FIR2-DFS.m
Result: The magnitude response of the designed filter is shown below. Filter is of 33 × 33rd
order and the error is 3.63% (Figure 8.10)
The filter error is calculated using Eq. (8.19) where 2N1 + 1 and 2N2 + 1 are the order of the filter in
n1 and n2 dimension, respectively. Hd is the ideal frequency response, Hf is the actual filter response,
and Ω1 and Ω2 are the frequency sampling rates.
N1
N2
∑ ∑ H [n Ω , n Ω ] − H [n Ω , n Ω ]
d
e=
n1 =− N1 n2 =− N2
1
1
2
2
(2 N1 − 1)(2 N2 − 1)
f
1
1
2
2
(8.19)
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
355
1
0.8
0.6
0.4
0.2
0
2
0
-2
-2
0
2
FIGURE 8.10 Magnitude response of a 2D filter designed using frequency sampling method.
The magnitude response shown below is obtained by incorporating the transition band values into
the ideal frequency response parameter. It can be seen that the ripple in the transition band has
disappeared and the error is found to be only 2.74%, which is nearly 1% less than that obtained
without any transition band consideration (Figure 8.11).
The response shown below is obtained with a filter of order of 20 × 20. The error is found to be
5.85% which compares unfavorably with 3.63% obtained with the first design. Clearly lower order
filters result in considerably worse performance (Figure 8.12).
1
0.8
0.6
0.4
0.2
0
2
2
0
-2
FIGURE 8.11
-2
0
Magnitude response of the filter designed with transition band consideration.
1
0.8
0.6
0.4
0.2
0
2
0
-2
-2
FIGURE 8.12 Magnitude response of filter of order 20 × 20.
0
2
356
Photonic Signal Processing
8.1.4.3 Windowing Method
The window method for 2-D filters uses a 2-D window instead of 1-D window to achieve a finite
impulse response sequence in 2-D. As with a 1-D windowing method, the 2-D windowing method
begins by performing a Fourier transform on the desired frequency response expression. The
impulse response of the filter in 2-D is then multiplied by the expression for window. The purpose
of multiplying by a windowing function is to reduce the effect of sharp transitions in the transition
band and to make the impulse response finite through truncation. The windowing function is
chosen so that the frequency response is least affected, and the impulse response is as short as
possible. It is noted in Lim10 that the performances of window method and the frequency sampling
method are similar, and an example is therefore omitted.
8.1.4.4 McClellan Transformation Method
The McClellan transformation method takes an entirely different approach to the design process of
2-D filters. The idea is to transform a 1-D FIR filter into a 2-D filter with the desired characteristic.
The 1-D filter can be designed using any 1-D filter design method so that its frequency response is
a cross-section of the desired 2-D filter response (see Figure 8.13).
Given the transfer function of a 1-D FIR filter, each coefficient is multiplied by the transformation
function T, which is a function of ω1 and ω2. The resulting transfer function is also a function of ω1
and ω2 and describes a 2-D filter with the desired magnitude response.
N
H (ω1, ω2 ) = h(0) +
∑ 2h(n) ⋅[T (ω ,ω )]
1
2
n
n =1
with T (ω1, ω2 ) =
(8.20)
∑ ∑ t (n , n ) ⋅ e
1
2
− jω1n1
⋅ e − jω2n2
( n1 ,n2 ) ∈RT
In (8.20), RT is the region of support of t(n1,n2), which describes the transformation function.
It should be noted that using different transformation functions, many 2-D filters can be designed
from a single 1-D filter. It is also important to note that as long as the 1-D filter is a linear phase filter,
the transformed 2-D filter is also a linear phase filter as long as phase of the transformation function
T is linear (i.e., the transfer function is symmetric about the zero delay point) since multiplying a
linear phase function by another linear phase function does not affect the linearity of the phase of
the resultant function.
0.5π
0.5π
0.5π
ω2
ω1
0.5π
0
0.5π
0.5π
FIGURE 8.13 Desired 2-D filter magnitude specification and the required 1-D filter specification.
ω2
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
357
8.1.4.5 2-D Filter Design Using Transformation Method
Design Aim: 2-D lowpass filter with frequency cut off at 0.5 in both dimensions.
Method: McClellan transformation method. The filter order is 13 × 13 and the transformation
function used is given by
1 1
1
1
1
T (ω1, ω2 ) = − + e − jω1 + e − jω2 + e jω1 + e jω2
2 4
4
4
4
1
1
1
1
+ e − jω1 ⋅ e − jω2 + e jω1 ⋅ e jω2 + e jω1 ⋅ e − jω2 + e − jω1 ⋅ e jω2
8
8
8
8
(8.21)
Graphically, the above transformation function can be expressed as in Figure 8.14.
Figure 8.15 shows the frequency response of the designed filter. Clearly, the
performance of the filter designed using the transformation method is somewhat worse
than that designed using frequency sampling method. A factor, which should be taken
into consideration, is the order of the filter. The order of the filter in is only 13 × 13.
The reason for such a large difference lies with the coefficients of the resulting filter
of transformation method. In Figure 8.15b, the coefficients of the filter as originally
designed by the transformation method is shown as the form of the impulse response
of the filter. Clearly, it is a 33 × 33 order filter; however, the surrounding 10 rows and
columns do not contribute to the filter response at all and, therefore, can be removed
without affecting the response of the filter and the remaining coefficients constitute a
13 × 13 order filter.
MATLAB® program used: FIR2-DTF.m.
Result: The error calculated = 13.71%.
The transform function has a large bearing on the eventual filter transfer function. The filter shown
in Figure 8.16 is of the same order as the filter shown in Figure 8.13.
n2
1/8
1/4
1/8
1/4
-1/2
1/4
1/8
1/8
1/8
n1
FIGURE 8.14 Transformation sequence used. (From Lim, J.S., Two-Dimensional Signal and Image
Processing, Englewood Cliffs, NJ, Prentice Hall, 1990.)
358
Photonic Signal Processing
1
0.8
0.6
0.4
0.2
0
2
0
-2
2
0
-2
(a)
0.2
0.15
0.1
0.05
0
30
20
10
10
00
30
20
(b)
FIGURE 8.15 (a) Frequency response of a filter of order 13 × 13 designed using transformation method and
(b) impulse response (filter coefficients) of the filter designed by transformation method.
1
0.8
0.6
0.4
0.2
0
2
2
0
-2
FIGURE 8.16
-2
0
2-D filter designed using a different transform function.
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
359
Figure 8.15a and the 1-D prototype function is identical. However, the transformation function is now a 7 × 7 function.19 The transition band is much more distinct, whereas the stopband is
not as well attenuated as the filter response in Figure 8.15a. The overall error of the filter, whose
frequency response is shown in Figure 8.16, is found to be 12.84%. Clearly, the good performance in passband and the transition band is offset by the poor stopband attenuation. Finally, it
is noted that the resulting transfer function is a non-separable FIR transfer function.
8.1.5
cOncluDing remarks
The objective of this section of the chapter is to explore possible ways of realizing a 2-D signal
processing system using fiber-optic signal processing architecture. This section describes a
general technique for designing 2-D filters. Numerous examples of utilization of the technique
are given. Although the discussion is focused on fiber-optic systems, the design procedure for 2-D
filters are just as applicable to any other signal processing architectures. For example, the 2-D
filter order reduction method given in Section 8.4 can be used to simplify 2-D lightwaves systems,
which may or may not be fiber-optic systems.
The design of 2-D filters is classified into two different classes. One class used matrix
decomposition to reduce the design of 2-D filters into a set of 1-D filter design procedures.
The other class uses direct extensions of 1-D filter design methods. It is found that neither has a
distinctive superiority over another, and that the designer has to choose what is the best for the
particular application, most likely by designing both and comparing the performances. All the
design procedures are implemented using the MATLAB programming language.
The next section will describe the techniques of matrix decomposition methods of these
techniques, the multiple stage singular value decomposition method performs the best whereas for
direct methods, frequency sampling method produced filters with smallest errors. A 2-D Filter order
reduction method is applied to make fiber- and integrated optic signal processing more feasible.
The technology allows the filter designer to produce filters of orders that are implementable in
practice without sacrifices in performance. Section 8.2.3.2 will deal with different possible filter
structures are proposed and illustrated for photonic implementation of 2-D filters. Filter structures
for FIR and IIR filters are also shown and examples are given in Section 8.2.4.
8.2
DECOMPOSITION TECHNIQUES AND IMPLEMENTATION
USING FIBER OPTIC DELAY LINES
Ultra-high bandwidth properties of fiber-optic signal processing systems could provide the necessary processing power for computationally demanding 2-D signal processing applications. The
techniques of fiber-optic signal processing so far have not been applied to the area of 2-D signal
processing. Furthermore, the matured fields of integrated optics and integrated photonics as well
as recent developments of nano-photonics allow innovative structures for processing of lightwaves
in photonic domain.
This section has sought to integrate the fields of discrete signal processing and fiber-optic
signal processing, integrated photonics and/or possibly nano-photonics to establish a methodology
based on which physical systems can be implemented. Several photonic signal processing (PSP)
architectures are proposed to enable efficient coherent lightwave signal processing. Although the
structures are originally developed for 2-D processing, they are also applicable for 1-D structures.
Using a combination of one-dimensional filter structures, 2-D fiber-optic filters can be constructed.
The relationship between the fiber-optic model and the mathematical model has been linked to allow
quick implementation. Using the developed methodologies, multi-dimensional coherent photonic
signal processors can be designed.
19
Ref.13 discusses methods of designing such transformation functions in great detail.
360
Photonic Signal Processing
8.2.1 intrODuctOry remarks
This chapter can be characterized in three parts. Section 8.2.1 outlines the motivation of this
chapter and the fundamental theory for multi-dimension signal processing applicable in photonic
domain. Section 8.2 and Section 8.3 focus on the theoretical fundamentals of multidimensional
PSP and the proposed realization.
The demand for multi-dimensional photonic signal processing (M-D PSP) can be attributed to
various factors due the growing feasibility of high-capacity digital transmission networks capable of transmitting ultra-high bit rate and time division multiplexing up to 160 Gb/s and even
200 Gbps by Discrete multi-carrier modulation (DMT) as well as fiber optical sensor networks.
A problem with the implementation of such systems is the lack of devices that can process an
enormous amount of data associated with multi-dimensional signals. With photonic transmission
networks becoming the transport infrastructure, the PSP technique has become increasingly
more desirable compared to O/E and E/O conversion techniques. As introduced, fiber-optic
signal processing systems are ideal for such processing demands for several reasons: all-optical
(or photonic) processing of photonic information of optical communication systems are possible
using fiber-optic signal processing; 2-D signals usually require much higher bandwidth than
1-D signals and therefore must be processed by a high bandwidth system to allow real-time
performance; it is likely that future telecommunication networks would be all fiber-optic.
This section outlines a number of techniques for multi-dimension signal processing, which
can be implemented in photonic domain and most importantly they must be simplified so that the
optical lightwave paths are minimized and thus minimum losses occur in the photonic processors.
This is very critical as lightwave propagation is involved with the overall transmittance of the
photonic circuit.1–5 Thus, if loss is high, then optical amplification is required. If this amplification
is implemented on the same optical integrated circuit, then they would occupy a large area of
the circuit. Therefore, we propose a number of techniques, such as the matrix decomposition
methods6,7 described in Section 8.2. We then present the methods for reduction for the design of
2-D optical filters in Section 8.3. Finally, in Section 8.4, we present an implementation of 2-D
optical filters using fiber optic delay lines and give some concluding remarks.
8.2.2 matrix DecOmpOsitiOn methODs
As the second of the two streams of 2-D filter design methods, matrix decomposition methods are
introduced. Matrix decomposition methods result in a set of separable 1-D magnitude responses,
which can be implemented using any 1-D filter design methods. Using either this approach or direct
approach of Section 8.1,1 a transfer function of the desired 2-D filter can be obtained and implemented by the photonic implementation methods.
8.2.2.1 Single-Stage Singular Value Decomposition
In Section 8.1, the application of matrix decomposition to 2-D filter design is briefly introduced.
Matrix decomposition is a mathematical procedure where a matrix is split into a sum of products
of vectors as
n
H=
∑λ u v
1/ 2
i i
(8.22)
i =1
where λI is ith eigenvalue of H and ui and vi are the decomposed vectors. An example of matrix
decomposition is the well-known lower-upper (LU) decomposition, which splits a matrix into
a lower triangular matrix and an upper triangular matrix. The LU decomposition can be used to
express a matrix as sum of products of vectors.
361
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
8.2.2.1.1 LU Decomposition
1
A =  4
 7
2
5
8
3
6 
9 
0
= P • L • U = 0
1
 0.1429 
= 0.5714  [ 7
 1 
1
=  4
 7
1
0
0
8
0  1
1   0.1429
0  0.5714
0
1
0.5000
0  7
0   0
1   0
1 
9] + 0.5 [ 0
 0 
0.8517
1.1743]
1.4867  0
5.2429  + 0
9  0
1.1432
4.5712
8
8
0.8517
0
1.1743  1
0.5871 =  4
0   7
0.8517
0.4258
0
9 
1.1743
0 
3
6 
9 
2
5
8
3
A=
∑ P•L •U
xi
(8.23)
ix
i =1, x =1,2,3
A decomposition method particularly suited to 2-D filter design is the singular value decomposition
(SVD). The SVD reduces a 2-D matrix into two matrices U and V, and a diagonal matrix S of
singular values of the original matrix. Singular values are related to the eigenvalues of the matrix
and the result has the form
A = U•S•B
(8.24)
N
=
∑
U xiSii B xi
i =1, x =1,2.. N
The unique feature of SVD is that the matrix “power” is distributed to the singular values of the
matrix in decreasing order of the position of the singular value in the matrix S starting from the
top-left corner. Consequently, to approximate the matrix A by the product of just one set of two vectors Uxi and Bxi, the best approximation will be made by the product of first set of vectors that result
from SVD of the matrix. Mathematically, the property can be described as shown in Eq. (8.3). As
1 2
an example, if we wanted to approximate the matrix 3 4 by the product of a set of vectors, SVD
0.5760
0.8174
0.4046
0.9145 
would be performed on the matrix resulting in U = 
, V = 
, and
−0.8174 0.5760
0.9145 −0.4046
5
.
4650
0
 . U = 0.4046 0.9145 , S = 5.4650, and V = [ 0.5760 0.8174 ] form the first set
S=
[
] 11
x1
x1
 0
0.3660
1.2736 1.8072 
. To obtain a better
of vectors. The resulting approximation would then be U x1S11VxT1 = 
2.8790 4.0853
1 2 
, the product of the second set of vectors could be added.
approximation of 
3 4 
N
N
A−
∑
i =1, x =1, 2.. N
U xiSii BTxi =
min
U xi , Uii , U xi
A−
∑ Uˆ S S
xi
ii
xi
i =1, x =1, 2.. N
where Û xi and B̂xi are subsets of U xi and B xi , and Ŝ ii is the corresponding singular value.
(8.25)
362
Photonic Signal Processing
In Yu and Khoo,8 the SVD is used to decompose a 2-D magnitude specification matrix into
two 1-D magnitude specification. The result is a design procedure in which a 2-D filter design
becomes a set of 1-D filter design. The matrix decomposition methods have several advantages
over the direct methods of Section 8.11 as follows: (i) The resulting 1-D magnitude specifications
can be met by any of the standard algorithms for a 1-D filter design, such as least-squares method
or Parks-McClellan algorithms available in many computer simulation packages; (ii) As long
as 1-D filter sections are stable, the overall 2-D filter is also stable. The stability of final 2-D
design can therefore be guaranteed easily without the need for heavy mathematical analysis or
computationally expensive algorithms involved with 2-D filter designs; (iii) The filter designer is
given the flexibility to decide how many sets of 1-D filter sections are included in the system; and
(iv) The resulting 2-D filter is parallel in structure and therefore does not introduce unnecessary
processing delays.
All 2-D filters designed using matrix decomposition can be described by (8.26) where Fi(z1)
and Gi(z2) are 1-D filter sections and K is the number of singular values included in the system.
K
H ( z1, z2 ) =
∑ F (z ) ⋅ G (z )
i
1
i
(8.26)
2
i =1
A simple example of a 2-D filter design using only one filter section as in Thomson et al.2 is given
below. The example chosen is deliberately simple to show the fundamental concepts involved in 2-D
filter design using matrix decomposition methods.
8.2.2.1.2 A 2-D Filter Design Using SVD with Single Parallel Section
Design Aim: A lowpass filter with normalized cut-off frequency of 0.5 in both dimensions
Method: Single-stage singular value decomposition
Program used: SVDFIR2-D.M
Result: With single stage, the error of frequency response of the designed filter is 6.663%.
Although this is quite low, it may not be acceptable in some cases for which an extension of
single-stage singular value decomposition may be employed as shown in Section 8.2.2.2
(Figure 8.17)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
2
0
(a)
-2
0
-2
2
2
0
(b)
-2
0
2
-2
FIGURE 8.17 (a) Magnitude specification of lowpass filter and (b) magnitude response of 15 × 15 2-D filter
designed using single stage singular value decomposition.
363
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
TABLE 8.2
Coefficients of the FIR Filter Designed Using Single-Stage
Singular Value Decomposition
Coefficient Order
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
2D
input
1D filter in z1
domain with
coefficients b1
b1
b2
−0.0007
0.0010
0.0025
−0.0090
−0.0273
0.0197
0.1837
0.3553
0.3553
0.1837
0.0197
−0.0273
−0.0090
0.0025
0.0010
−0.0007
−0.0007
0.0010
0.0025
−0.0090
−0.0273
0.0197
0.1837
0.3553
0.3553
0.1837
0.0197
−0.0273
−0.0090
0.0025
0.0010
−0.0007
1D filter in z2
domain with
coefficients b2
2D
output
FIGURE 8.18 Separable implementation of 2-D filter using single-stage singular value decomposition.
The 1-D filters designed are FIR filters, and the actual filter coefficients are given in a table format as
shown below. It is noted that the first and the second filter sections are identical since the frequency
specification is symmetric about the origin. Consequently, for symmetric filters, only one 1-D filter
needs to be designed to complete the design for a 2-D filter implying significant simplification in
the filter design procedure.
The filter designed in Table 8.2, when implemented takes on the structure shown in Figure 8.18.
In the case of a multiple-stage implementation, several of the structure shown below would be
connected in parallel to form the 2-D filter.
8.2.2.2 Multiple-Stage Singular Value Decomposition
Multiple-stage SVD takes the leap forward from single-stage SVD method and includes stages
that belong to the second largest singular values and smaller. Depending on the relative magnitude of singular values, the inclusion of extra stages can result in a sizable reduction in error, or
sometimes it has no effect at all. The sampled design given below shows a case where inclusion of
multiple stages results in more than 30% reduction in the error of the single stage implementation
(Figure 8.19).
364
Photonic Signal Processing
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
2
(a)
0
-2
-2
0
2
2
0
(b)
-2
-2
0
2
FIGURE 8.19 (a) Magnitude specification of 90° fan filter and (b) magnitude response of 32 × 32 2-D filter
designed using multiple-stage singular value decomposition.
8.2.2.2.1 A 2-D Filter Design Using SVD with Multiple Parallel Sections
Design Aim: A 90° fan filter of order of 32 × 32
Method: Multiple stage singular value decomposition with 1-D FIR filters
Program used: SVDFIR2-D.m
Result: The error of frequency response of the filter is given as 18.65%20
The filter is obtained after six parallel stages. Figure 8.20 shows the error and magnitudes of the
singular values against the number of included parallel stages. Figure 8.20 appears to show that
there is roughly a linear relationship between the error curve and the singular values curve. The
relationship is actually subtler than this. A little thought will reveal that greater the gradient of
singular value curve is, the flatter the error curve will be. This is because if there is a large difference
between two singular values, adding the stage which belongs to the smaller singular value will have
little effect on the overall performance. As a rule of thumb, if the singular value of a parallel stage
is less than one-tenth of the first singular value, then it is probably not worth including.
With the current computer technology, calculations for around ten 32nd order 1-D filters can be
done virtually in real time and, therefore, the SVD method is practical even for adaptive filtering.
The resulting filter structure is shown in Figure 8.21.
35
Errors(%)
Singular values
30
25
20
15
10
5
0
1
2
3
4
5
6
FIGURE 8.20 Errors and magnitudes of singular values.
20
Although this value is quite large compared to the single digit figure we have been obtaining so far, it should be kept
in mind that the error largely depends on the specification. Therefore it is only meaningful to compare error between
different implementations of the same magnitude specification.
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
1D filter section
F1 (z1)
1D filter section
G1 (z2)
1D filter section
F1 (z1)
1D filter section
G2 (z2)
1D filter section
F6 (z1)
1D filter section
G6 (z2)
365
FIGURE 8.21 Structure of 2-D 90° fan filter.
8.2.3 iterative singular value DecOmpOsitiOn
There are many variations on the theme of matrix decomposition, specifically the SVD. The iterative
singular value decomposition (ISVD)21 is devised for avoiding “negative” magnitude definitions that
arise from the plain SVD procedure of the previous section. By keeping the 1-D magnitude positive,
the chapter claims that the 1-D filter design procedure becomes less intricate.
8.2.3.1 Iterative Singular Value Decomposition19
1. Let the 2-D magnitude specification be A. Let A+1 = A and A−1 = 0.
2. Perform singular value decomposition on A+. λ1i are the singular values of A+1. By the
definition of SVD given in Mitra et al.,18 λ11 is larger than any other λs.
A1+ =
r1
∑λ u v
t
1i 1i 1i
i =1
(8.27)
r1
=
∑u λ ⋅λ v
1/ 2
1i 1i
1/ 2 t
1i
1i
i =1
3. Because of Perron’s result on non-negative matrices,21 the vectors u11 and v11 are also non1/ 2
1/ 2 t
negative. It is then possible to estimate A+ by u11λ11
⋅ λ11
v11 by assigning the first of the
pair as F+1, and G+1 gives the first non-negative 1-D magnitude specifications. F1 and G1 are
assigned F+1 and G+1
S1 = 1
F1 = F1+
(8.28)
G1 = G1+
4. A2 can now be calculated using Eq. (8.27). This matrix can now be separated into A 2+ and
A 2− sum of which make up the error matrix A2.
A 2 = A − S1F1G1
= A 2+ + A 2−
21
(8.29)
F. T. S. Yu and I. C. Khoo, Principles of Optical Engineering, Section 9, 3rd ed., Oxford, UK: Pergamon Press, 1999.
366
Photonic Signal Processing
where
A 2+ ( m, n) =
A −2 ( m, n) =
0
if A 2 ( m, n) ≥ 0
if A 2 ( m, n) < 0
A 2 ( m, n)
0
iff A 2 ( m, n) < 0
if A 2 ( m, n) ≥ 0
A 2 ( m, n)
5. Singular value decomposition is performed on both matrices resulting in two sets of vectors
S2 , F2+ , G 2+ and S2 , F2− , G 2− . S2 is 1 for the vectors resulting from decomposition of A+, and −1
for the vectors from A−.
6. Euclidean norms are calculated for resulting error matrix defined in Eq. (8.27). The same
operation is performed with S2 , F2− , G 2− in place of S2 , F2+ , G 2+ in Eq. (8.29) with E2− as the result.
2
E 2+ = A −
∑S F G
+
i i
+
i
i =1
(8.30)
1/ 2
 M N

E2+ = 
[ E2 ( m, n)]2 
2
 m =0 n =0

∑∑
7. E2+ and E2− are compared. Since smaller error means closer approximation to the original
matrix, the set of vectors that results in the lower Euclidean norm is chosen as F2 and G2 .
8. A3 is assigned the error matrix A 2 − A + or A 2 − A − depending on whether E2+ is greater or
smaller than E2− . Steps 4, 5, 6 and 7 are then repeated with appropriate substitution.
The procedure is repeated until a satisfactory approximation of the original matrix is obtained.
Compared to a plain singular value decomposition algorithm, ISVD algorithm converges more
slowly because adding an extra stage does less to compensate for the error than plain SVD algorithm
since only a part of the error is compensated. An example of a filter designed using the iterative
singular-value decomposition is given as
8.2.3.2 A 2-D Filter Design Example Using Iterative Singular Value Decomposition
Design aim: A bandpass filter of order of 32 × 32 with normalized passband between 0.3
and 0.6
Method: Iterative singular value decomposition algorithm Program used: ISVDFIR2-D.m
(Figure 8.22)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
2
0
(a)
-2
-2
0
2
2
0
(b)
-2
-2
0
2
FIGURE 8.22 Iterative singular value decomposition (a) Ideal magnitude response and (b) actual filter
magnitude response.
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
367
The filter error is reasonably low at 9.88% after seven approximation stages. Overall, the filter
requires seven 1-D FIR filter design stages as the magnitude specification is symmetric about the
two axes. Because of the complexity of the magnitude specification, the designed filter does not
perform as well as one might expect. This can be corrected to some extent using better 1-D filter
design procedures such as the Parks-McClellan algorithm.
8.2.4 Optimal DecOmpOsitiOn
Optimal decomposition is an improvement on ISVD, which is based on the optimization of the 1-D
magnitude vectors so that the Euclidean error is minimized. The error is found to be 9.68%, which
is only slightly better than that of ISVD.
8.2.4.1 Optimal Decomposition22
In Eq. (8.30), the definition of the Euclidean norm is defined. In the optimal decomposition, the
objective is to minimize the value of this error estimate to provide the best set of vectors that will
make up the original specification matrix.
Continuing with the constraint that magnitude vectors must all be positive, we then perform
exponential mapping to Fi and Gi .
Fi = e xi 0
e xil
Gi = e yi 0
yil
e
⋅⋅⋅
e xiM 
⋅⋅⋅
yiM
e


(8.31)
The purpose of exponential transformation is so that the optimizing variables xij and yij are
not constrained to be positive. However, the condition of positive magnitude is retained as
all values of Fi and Gi will be positive no matter what the values of xij and yij are. Non-linear
optimizing technique must be applied since this problem is non-linear. Numerous algorithms
exist for non-linear optimization of several variables and any technique can be used to obtain
the answer.
Choosing bad starting points for optimization routines results in local minima or bad convergence
points. It is recommended that iterative singular value decomposition algorithm be used to provide
the initial points for optimization.
8.2.4.1.1 2-D Filter Design Example Using Optimal Decomposition
1. Design aim: A 2-D bandpass filter with normalized passband frequencies of 0.33 and 0.66
in both dimensions
2. Method: Optimal decomposition22
3. Program used: ODFIR2-D.m
4. Result: The error is 9.65% compared to 9.88% for ISVD algorithm after seven stages.
The number of filter designs required is seven (same as ISVD); however, each filter
stage requires a great deal more computational effort than the ISVD method as it
requires non-linear optimization to be performed on quite a large number of variables
(Figure 8.23)
Due to the computational constraints, full optimization is not performed. Even then the optimization routine took a very long time to perform and the reason is attributed to the number of
variables to be optimized being so large (20–30 variables, depending on the order of the filter
transfer function).
22
T. Deng and M. Kawamata, Frequency-domain design of 2-D digital filters using the iterative singular value
decomposition, IEEE Trans. Circ. Syst., 38(10), 1225–1228, 1991.
368
Photonic Signal Processing
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
2
0
(a)
-2
-2
0
2
2
0
-2
(b)
0
-2
2
FIGURE 8.23 Optimal decomposition (a) Ideal magnitude response and (b) actual filter magnitude response.
8.2.4.2 Other 2-D Filter Design Methods Based on Matrix Decomposition
There are many other 2-D filter design methods that are based on the idea of matrix decomposition. Thus far, all the methods discussed decompose the 2-D magnitude specification into a set
of two 1-D magnitude specifications so that the 2-D filter design procedure is essentially reduced
to that of 1-D. It is shown that, by using this approach, the design problem is reduced significantly. But, since the approach produces only an approximation to the 2-D transfer function, the
methods based on magnitude decomposition do not perform as well as the direct methods.
One notable 2-D filter design method uses matrix decomposition, but it is s not based on
magnitude decomposition is by Shaw and Mistra.23 In this approach, it is assumed that the 2-D
transfer function is already obtained using some 2-D filter design.
A 2-D transfer function can be represented by matrices as shown in Eq. (8.32).
H ( z1, z2 ) =
hb ( z1, z2 )
ha ( z1, z2 )
 hb (nb1 − 1, mb1 − 1)

z Tb1 
 hb (1, mb1 − 1)

 hb (0, mb1 − 1)
=
 ha (na1 − 1, ma1 − 1)

z Ta1 
 ha (1, ma1 − 1)

 ha (0, ma1 − 1)
23
hb (nb1 − 1, mb1 − 2)
hb (1, mb1 − 2)
hb (0, mb1 − 2)
ha (na1 − 1, ma1 − 2)
ha (1, ma1 − 2)
ha (0, ma1 − 2)
…
…
…
…
hb (nb1 − 1, 0) 

 z b2
hb (1, 0) 

hb (0, 0) 
ha (na1 − 1, 0) 

 z a2
ha (1, 0) 

ha (0, 0) 
z b1 ∆  z1bm1 −1 z1bm1 −2
z1
1
z b 2 ∆  z2bn1 −1 z2bn1 −2
z2
1
z a1 ∆  z1am1 −1 z1am1 −2
z1
1
z a 2 ∆  z2an1 −1 z2an1 −2
z2
1
(8.32)
T
T
T
T
A. K. Shaw and P. Mitra, An exact realization of 2-D IIR filters using separable 1-D modules, IEEE Trans. Circ. Syst.,
37, 1990.
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
369
In Shaw’s method, the decomposition is performed on 2-D transfer function matrices. The method
can result in efficient filters in terms of the required elements, however it results in 1-D filter
sections with different orders and thus does not offer the modularity of the other decomposition
methods.23
Other methods exist for yet more efficient filter design and a 2-D filter order reduction method is
described in Section 8.1.1
8.2.5 2-D Filter OrDer reDuctiOn using balanceD apprOximatiOn theOry
Keeping the filter order to the minimum is important for photonic circuit implementation as coupling
losses of higher order filters may render the actual implementation impossible. To achieve lower
order filters with good performance, the balanced approximation method used in control systems
theory is applied to the order reduction of 2-D digital filters.
8.2.5.1 Motivation for Lower Order Photonic Filters
For an adequate filter performance, that is satisfying the roll-off factor and the passband ripple,
the order of the filter must be appropriately chosen. In general, increasing the order of the filter
can significantly reduce the error. However, higher filter order directly translates to extra filter components, noise, and higher attenuation which are unacceptable in many cases including
fiber-optic implementations. It is therefore critical that the order of the filter remains low without
sacrificing overall performance measures such as error response.
The motivation for keeping filter order low is greater for fiber-optic systems than in other filter
implementations. For this reason, higher order filters cause large coupling losses that must be
compensated by a pre-amplifier, which, in turn, introduces noise when the amplification factor
is large. It is generally accepted that filters with an order greater than 16 start becoming difficult
to realize in practice with the current technology. However, as we have seen in Sections 8.2.2 and
8.2.5, 2-D filters with orders of around 30 × 30 are quite common.
Balanced approximation, derived from control theory, is a model reduction technique for 1-D
systems. As 2-D filters usually have high orders, application of the filter order reduction method to
2-D filters may prove to be very rewarding especially for fiber-optic filters, which must have low
orders for feasibility.
8.2.5.2 Description of 2-D System in State-Space Format
As balanced approximation is originally developed for model reduction of dynamic systems, it uses
the state-space model of digital systems. The implication is that 2-D systems, which we have been
representing using transfer functions must now be represented in state-space format.
The representation of 2-D systems in a state-space format has been a topic of research for a
number of years and several models have been proposed.22,23 It is noted in Deng and Kawamata22
that the model in Shaw and Mistra23 (see Box 3.2) is the most general and the model proposed in
Mason’s rule33 can actually be embedded into the model in Shaw and Mistra.23
Although converting from a 2-D transfer function description to a 2-D state-space description
involves only plug-in formulae, converting from 2-D state-space description to 2-D transfer function
is much more involved and a novel algorithm is described in Section 8.1 (Part I).1 Using the two
algorithms, the balanced approximation method can be applied to 2-D systems.
8.2.5.3 Balanced Approximation Method
Using a known 2-D filter transfer function, the balanced approximation method (BAM) finds
the balancing transformation matrix T, which “balances” the system. The order reduction is
subsequently performed by removing states that do not contribute substantially to the system
behavior.
370
Photonic Signal Processing
The first task is to find the generalized reachability and observability that Gramians defined as
K=
1
(2π j )2
∫ ∫
f ( z1, z2 ) f * ( z1, z2 )
dz1 dz2
z1 z2
W=
1
(2π j )2
∫ ∫
g * ( z1, z2 ) g ( z1, z2 )
dz1 dz2
z1 z2
| z1|=1
| z1|=1
| z2 |=1
| z2 |=1
where
−1
f ( z1, z2 ) =  I( z1, z2 ) − A  b
g( z1, z2 ) = c  I( z1, z2 ) − A 
(8.33)
−1
Fortunately, the double integrations do not have to be solved directly and can be partially solved (as
distinct from partial integration) using the Lyapunov approach. If K11 denotes the upper-left, upperblock of K and K22 denotes the lower-right block of K, then K11 and K22 can be obtained. The same
notations apply to the observability Gramian, W.
N1 −1
K11 =
∑
N2 −1
A1i P(A iT )i
W22 =
i =0
∑ A Q( A )
i
4
i =0
where
where
P = H bH Tb
 h10

Hb =   hN1 0
T i
4
Q = H Tc H b
…
h1N2 
 h10


 Hc =   hN1 0
hN1N2 
W11 = I N1
…
(8.34)
h1N2 


hN1N2 
K22 = I N2
A system is said to be balanced if its Gramians satisfy the following condition where σij are the
Hankel singular value of the system.
σ 11

0
K11 = W11 = 
 
 0
0
σ 12
0
0 


0 

σ 1N1 
σ 21

0
K22 = W22 = 
 
 0
0
σ 22
0 


0 

σ 2 N2 
0
(8.35)
To balance a system, the similarity transform T that will achieve the above condition must be found.
Applying the balancing similarity transform T to the subsections of the Gramians will result in the
condition satisfied.
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
K = T −1KT −T
371
(8.36)
W = T TWT
The balancing transformation T can be found by applying the algorithm described here.
8.2.5.3.1 Determination of the Balancing Transformation T
1. Cholesky factorization of K11: The resulting lower triangular matrix is assigned Lc.
2. Formation of LTc W11 Lc .
T
T
3. Symmetric eigenvalue/eigenvector problem, U11
(LT11c W11L11c )U11 = Λ11
.
−1/ 2
4. Formation of T11: T11 = LcU1 L11 .
The same procedure with appropriate subscript substitutions can be used to find T22. Once both T11
and T22 are found, the overall transformation matrix T can be found by performing an operation
denoted by ⊕ in Yu and Khoo.21
T = T11 ⊕ T22
T11
=
0
(8.37)
0 

T22 
Using the balancing matrix T, a balanced realization of the system can be found by similarity
transformation of state-space matrices as shown in Eq. (8.37).
ˆ = T−1AT
A
ˆ = T−1b
b
(8.38)
cˆ = cT
dˆ = d
By observing how many significant Hankel singular values exist, one can make the decision on how
many states should be preserved thereby determining the order r1 and r2. The state-space matrices
can then be partitioned using the following scheme.
A1r

ˆ
A2   *
=
ˆ   0
A
4 

 0
ˆ
ˆ =  A1
A
ˆ
A 3
A2r
*
A 4r
*
*
*
0
0
*

*
*

*
 b1r 
 
ˆ
ˆ =  b1 =  * 
b
ˆ   
b2 r
 b 2
 
 * 
cˆ =  cˆ 1
|
cˆ 2  =  c1r
(8.39)
*
c 2r
*
A1r is a [r1 × r1] matrix if r1 is greater than r2 and a r1 × r2  matrix if r2 is greater than r1. On the
other hand, A2r is a [ r2 × r1 ] matrix if r1 is greater than r2 and a r2 × r2  matrix if r2 is greater than r1.
372
Photonic Signal Processing
The dimensions of A3r and A4r are the same of that of A1r and A2r, respectively. Finally, the reduced
system is obtained by forming a new system matrices by
(8.40)
The resulting system, which is described by the matrices Ar , br , and cr , is of order of [r1 × r2]. From
the new 2-D state-space description of the reduced system, one can obtain the 2-D system transfer
function of lower order.
8.2.5.4 Filter Order Reduction Using Balanced Approximation: An Example
In this section, a 15 × 15 order bandpass filter is designed using optimal decomposition method, and
the balanced approximation method is applied to reduce the filter order.
8.2.5.4.1 Application of Balanced Approximation Method for 2-D Filter Order Reduction
Design Aim: 2-D bandpass filter with normalized passband frequency between 0.33 and 0.66
with lowest order acceptable
Method Used: Optimal decomposition for filter design, and balanced approximation for order
reduction
Programs Used: ODFIR2-D.m, BA.m
Results: Using optimal decomposition method, a filter with specifications shown below
is designed. The error is approximated at 11% after 6 stages of approximations
(Figure 8.24)
Balanced approximation is then applied to the filter design. To apply the order reduction; however,
a new reduced order had to be chosen and the choice is made based on the Hankel singular values
of the system plotted in Figure 8.25. Clearly, it seems reasonable to retain only up to the 10th order
in both dimensions since from 11th order onwards, the Hankel singular values become very small
indeed (see Figure 8.26a and b).
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
2
0
(a)
2
2
-2
-2
2
0
0
(b)
-2
0
-2
FIGURE 8.24 (a) Ideal magnitude response of the filter and (b) actual filter response of the 16 × 16 order
filter.
373
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
0.6
0.4
0.2
0
(a)
0
5
10
15
5
10
15
0.6
0.4
0.2
0
(b)
0
FIGURE 8.25 Hankel singular values of the filter (a) Hankel singular values of N1 dimension and (b) Hankel
singular values of N2 dimension
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
2
0
(a)
FIGURE 8.26
of the filter.
-2
-2
0
2
2
0
(b)
-2
-2
0
2
Reduced order (10 × 10) filter (a) ideal characteristic of the filter and (b) actual characteristic
Choosing the new order of the filter as 10 × 10, the BAM is applied with the following excellent
results.
As the plots of magnitude characteristic shows, there is hardly any difference between the
original design and the reduced order design. The error estimate of 11.46% compared to 11% of the
original 15 × 15 order design confirms this point and shows that balanced approximation indeed
produces filters of significantly lower order with very little sacrifice in performance.
The result of the application of BAM to a 2-D filter transfer function can be summarized as
follows: (i) Reduced order filter; (ii) Usually IIR structure; (iii) If the original filter has a separable denominator, then the reduced filter also has a separable denominator allowing a separable
implementation; and (iv) Little sacrifice in performance (magnitude error and phase linearity).
The phase remains nearly linear for the resulting IIR structure, as well, which is a feature difficult to achieve with other 2-D IIR filter design methods. The proof of the linearity is given in
Capmany and Cascon.5
All in all, the BAM provides an excellent method of reducing filter order to a realizable level
without a large deterioration in performance and should therefore be given a consideration before
implementation of 2-D filters.
374
Photonic Signal Processing
8.2.6 Fiber-Optic Delay line Filters
As introduced in Section 8.1,1 fiber optic delay line architecture is an alternative architecture to
spatial and temporal architecture. The fiber optic delay line architecture used in this section to
implement 2-D filters is described in further detail with a mathematical analysis.
8.2.7 cOherent anD incOherent OperatiOn OF phOtOnic Filters
When the advances in laser technology first made guided-wave photonic systems possible, most
pioneering photonic systems used multi-mode propagation of light as the main mode of signal
transmission. However, with the advent of lasers with narrower line-widths, it has become possible to
operate lightwave systems in single mode resulting in greater bandwidth-distance product. Single mode
systems are becoming increasingly popular, and the trend towards single mode systems is set to continue.
Aside from the mode of propagation, another factor that determines the characteristic of a lightwave system is whether the system is operating in coherent or in incoherent modes. The differences
between the two operations can be summarized as follows: in a coherent system, the light source
can be regarded as operating in a single wavelength (although in reality, no matter how small the
linewidth is, the emitted lightwave is certain to contain more than one wavelength component). The
use of coherent light as the signal carrier simply means that the phase, as well as the amplitude
of the lightwaves, must be regarded as a part of the information being carried by the lightwave.
In incoherent systems, the information is carried only by the intensity of the lightwave. One may
therefore consider incoherent systems as the amplitude modulated system with intensity modulation
instead of amplitude modulation. It is obvious that negative range cannot be expressed by intensitybased incoherent systems unless one biases the light intensity to a predefined level. The receiver can
then detect negative range by comparing the received intensity value to the predefined level.
The differences between coherent systems and incoherent systems are shown in Table 8.3. For
signal processing purposes, incoherent operation implies that the modulating frequency of the source
must be much lower than the photonic frequency implying that the full bandwidth of the laser cannot
be used. Coherent operation on the other hand allows utilization of the full bandwidth of the system
resulting in greater processing speed. However, because coherent systems tend to be more vulnerable to environmental effects such as phase jitter, some shielding must be used to reduce the adverse
effects to a negligible level.24 Another important advantage the coherent systems have over incoherent systems is the flexibility in system design. Incoherent systems fall into the category of so called
positive systems and have restrictions on quantities, such as number and positions of system poles and
zeros.25 In spite of such constraints, most of the research works reported so far on fiber-optic delay
TABLE 8.3
Differences Between Coherent and Incoherent Lightwave System
Information carrier
Bandwidth
Required linewidth of the source
Negative range
24
25
Coherent System
Incoherent System
Amplitude, phase
Very wide
Very narrow
Amplitude and phase can combine to
express a negative value
Intensity
Wide
Narrow
Predetermined bias value is necessary
B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, Fiber-optic lattice signal processing, Proc. IEEE, 72(7), 1984.
L. N. Binh, Photonic Signal Processing: Techniques and Applications, Boca Raton, FL: CRC Press, 2009.
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
375
line signal processing have been using incoherent systems.26,27,28,29,30 The reasons for avoiding coherent
systems have been that “coherent systems are more difficult to implement in practice and are usually
more complicated than incoherent systems because of the stringent requirements on the stability of
the source and photonic delay paths.”4 In the future, it is likely that lasers capable of coherent operation over longer distances as well as better techniques for controlling the delay paths will be available.
Coherent systems, thus, may yet represent the possibility for full bandwidth all-photonic processing.
8.2.8 using Optical Fibers tO realize DelayeD line Filter
Three main components are required in most forms of discrete-time filters: delay, coefficient, and
summer/splitter. To illustrate how the components are realized in photonic domain, discrete-time
tab filter shown in Figure 8.27 is used as an example. As the signal-flow diagram for a discrete-time
tab filter is general, the photonic components used to realize a discrete-time tab-filter can be used in
other filter structures.
8.2.8.1 Photonic Realization of Delay
In fiber-optic delay line filters, photonic fibers are used as delay elements as signal propagation time
can be controlled using the length of the fiber. The transfer function of optical fiber, ignoring the
fiber signal dispersion and the fiber intensity loss, can be expressed mathematically by.
H (ω ) = e − jβ L
(8.41)
where L is the length of the delay lines, β is the propagation constant of the guided fundamental
mode. The propagation constant β is defined by β = ωneff /c where neff is the effective refractive
index of the guided mode in the fiber or optical planar channel waveguide, ω is the operating optical
frequency in radians, and c is the speed of light. The inverse of the time delay T is neff f/c equals to
the sampling frequency of the filter. Choosing a reference length of the optical delay as L d, if L is an
integer multiple of L d the transfer function can be expressed as
H (ω ) = e
− jd ( L
2π fneff
z − d = e − jd (ωT )
h0
Input
z-1
h0
z-1
h0
z-1
c
)
(a )
(8.42)
( b)
Output
h0
FIGURE 8.27 Signal flow diagram of discrete-time tab filter, the unit delay is the traveling time of lightwaves
over a distance equivalent to the unit sampling time. The coefficients h’s are the transmittances over the
specific path.
See for example, B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, Fiber-optic lattice signal processing, Proc. IEEE,
72(7), 1984.
27 J. Capmany and J. Cascon, Optical programmable transversal filters using fiber amplifiers, Electron. Lett., 28, 1245–1246, 1992.
28 B. Moslehi, Fiber-optic filters employing optical amplifiers to provide design flexibility, Electron. Lett., 28, 226–228, 1992.
29 E. Heyde and R. A. Minasian, Photonic signal processing of microwave signals using an active fiber Bragg grating pair,
IEEE Trans. Microw. Theory Tech., 1463–1466, 1997.
30 C. K. Marsden and J. H. Zhao, Optical Filter Analysis and Design: A Digital Signal Processing Approac, New York:
John Wiley, 1999.
26
376
Photonic Signal Processing
Upon comparing (8.42a) and (8.41), it can be observed that the two equations are very similar and, in
fact, if ωT in (8.42b) is replaced by (2πf)(neff L d /c), then the two equations are identical. It is therefore
clear that fiber can act as a delay whose length is controlled by neff and L.
Optical fiber has several properties that enable it to be an ideal delay line medium: flexibility, which
enables a relatively compact implementation of the system, the accuracy of time interval between
tabs that can be produced, and insensitivity to electromagnetic interference, which is useful when
used in electro-magnetic environments—quite often the case with signal processing equipment.
8.2.8.2 Photonic Realization of Tab Coefficients
General form of a feed forward transfer function in z-domain can be expressed as.
n
H (z) =
∑h z
d
−d
(8.43)
d =0
There are several ways to realize the coefficients magnitude hd , such as optical amplifiers (OAs)/
attenuators31 (earlier methods of achieving filter coefficients included reflectors, radiation due to bending, and evanescent coupling by polishing the cladding down very close to the fiber core32). However,
the negative sign can be difficult to realize as it represents a negative intensity in an incoherent system!
The inability to represent negative quantities effectively is a major limitation of incoherent systems.
On coherent systems, a multiplication by a negative coefficient represents a phase shift of 180°.
8.2.8.3 Photonic Realization of Summer/Splitter
In photonic domain, summing/splitting of signals can be performed by optical couplers (see
Figure 8.28).
 E3   1 − k1
 =
 E4   − j k2
− j k1   E1 
 
1 − k2   E2 
(8.44)
Splitting of signal can be performed by using just one of the input terminals ( E3 , E4 ) and both
output terminals ( E1, E2 )—see Figure 8.29a. Summing can be achieved by using just one output
signal port and both input ports—see Figure 8.29b.
When using an optical coupler as a summer/splitter, there are two undesirable properties
that must be taken into account. First, as can be seen from the transfer matrix, when a photonic
signal goes through a coupler the signal amplitude (and therefore intensity) is attenuated by the
1
coupling factor of the optical coupler, which for a half intensity splitter is 2 . A photonic filter
is likely to have cascaded stages of optical couplers and the combined coupling coefficients
cause quite a substantial attenuation of the original input. OAs are therefore usually necessary to
E1
E2
√1-k1
-j√k2
-j√k1
√1-k2
E3
E4
FIGURE 8.28 Schematic diagram of an optical coupler.
B. Moslehi, Fiber-optic filters employing optical amplifiers to provide design flexibility, Electron. Lett., 28, 226–228,
1992.
32 F. T. S. Yu and I. C. Khoo, Principles of Optical Engineering, Section 9, New York: John Wiley, 1990.
31
377
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
√1-k1
E1
-j√k1
E3
E1
E4
E2
√1-k2
(a)
-j√k2
E3
(b)
FIGURE 8.29 (a) Optical coupler as a splitter and (b) optical coupler as a summer.
compensate for the amplitude attenuation.6 Second, the problem arising from the use of optical
couplers as splitter/summer is the phase shift of −90° associated with cross-coupling of photonic
signals. The phase shift is not an issue for concern in an intensity-based system (incoherent
system); however, it can cause difficulty in coherent systems, especially when coupler is being
used as a summer as the signals that are being added must be in the same phase at the output
of the coupler. To illustrate this problem, consider adding two signals, E1 and E2, which are in
phase before they enter the coupler Eq. (8.44). At the output of the coupler, only one output is
cross-coupled and therefore phase shifted whereas the other output retains the phase of the input
signal. The added signal is therefore an inaccurate representation of the summing operation.
 Eout   1 − k1

=

 
− j k1   E1 
 
  E2 
(8.45)
Eout = 1 − k1 E1 − j k1 E2
However, the phase shifting property of couplers, if manipulated well, can act be used as perfect
phase-shifters necessary in implementing negative coefficient taps. It is conceivable that with the
right choice of input and output terminals, a coherent signal processing system that represents
negativity without biasing (as in incoherent systems) is feasible.
8.2.8.4 Graphical Representation of Photonic Circuits
A photonic circuit can be translated directly into a signal-flow diagram (SFG) as the elements in a
photonic circuit and the elements in its SFG have a direct one-to-one correspondence. To effectively
utilize the SFG representation in analyzing photonic circuits, the well-known Mason’s rule,9,33 given
in Chapter 2 for analyzing the SFGs, is applied to the photonic circuits. The key to the application of
the rule is the planar SFG representation of optical coupler as shown in Figure 8.30.5
Photonic components other than couplers, such as fiber delay lines and amplifier/attenuators, have
a straightforward representation in the SFG. Using the above representation for optical couplers,
photonic circuits can be analyzed systematically. The result is a very powerful technique that enables
a systematic mathematical analysis of photonic lumped circuits. Using this technique, the z-transfer
1
E1
-j√k2
4
√1-k1
E3
E4
√1-k2
3
-j√k2
2
E2
FIGURE 8.30 Graphical representation of optical coupler.
33
Mason’s rule can be found in many digital signal processing textbooks. The rule is applied without modifications to SFG
representations of optical circuits.
378
Photonic Signal Processing
function of a system from any one node to another (instead of just from one preset input node to a preset
output node) can be calculated allowing the system designer more degrees of freedom in designing
and using photonic circuits. Once the transfer function in z-domain has been obtained, as z-transform
theory is very well developed, one can simply apply the conventional analysis to the photonic circuits.
Alternative to this method of analyzing photonic circuit is the matrix-method, which attempts
to analyze photonic circuit by direct manipulation of the coupler transfer matrix. The disadvantage
of such approach is that when the photonic circuit consists of more than a few photonic elements,
it becomes extremely difficult to recognize what effect each element is having on the overall
function of the system. A graphical approach allows direct manipulation of the photonic circuit as
the correlation between an SFG and the photonic circuit it represents is very high.
The graphical method is best suited to analyzing a lumped photonic system, most likely to confirm
the operations of a photonic circuit or to find new functions of a photonic circuit configuration. For
further discussion on the uses of the graphical method, see Section 8.1.1
8.2.8.4.1
Graphical Method of Analysis of Double- Coupler Feedback
Photonic Resonator
Double coupler feedback photonic resonator (DCFBOR) is a configuration that results in one optical
energy storage element through feedback and one interferometer through different path lengths in
the feed forward path. The resulting transfer function contains one pole and one zero at the origin,
and, therefore, the configuration can be used to realize all pole IIR filters.
Method: Graphical technique for photonic circuit analysis5
Result: The photonic circuit of double coupler feedback optical resonator is shown in
Figure 8.31. The SFG of DCFBOR is shown in Figure 8.32
t11ejτ 11
E1
1
2
a11
3
5
4
6
E4
E6
a21
7
8
E7
t21ejτ 21 z−1
FIGURE 8.31 Schematic diagram of DCFBOR.
E1
E4
(1− γ )(1− a11)
1
3
− j (1 −γ ) a11
− j (1 −γ ) a11
4
2
(1− γ )(1− a11)
t11ejφ11
(1− γ )(1− a21)
5
− j (1 − γ )a21
8
t21e
jφ211 −1
z
E7
7
− j (1 − γ )a21
6
(1− γ )(1− a21)
FIGURE 8.32 Signal flow diagram representation of DCFBOR.
E6
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
379
The details of application of Mason’s rule of determining the signal flow diagram transfer functions
can be found in Capmany and Cascon.5 The resulting transfer function, as expected, has one zero at
the origin and one pole at a location in z-plane determined by the circuit parameters.
H (z) =
(1 − γ ) (1 − a11 )(1 − a21 )t11 e jϕ11
E7
=
E1 1 + (1 − γ ) a11a21t11t11 e j (ϕ11 +ϕ21 ) z −1
(8.46)
In conclusion, the graphical technique presents a previously unavailable systematic method
of analysing photonic circuits. The greatest potential will be realized when the technique is
implemented in a software form as the technique can be time-consuming to apply manually if there
are more than two feedback loops.
8.2.8.5 Remarks
In this section we have described: (i) The differences between coherent and incoherent operation
of lightwave systems. (ii) The architecture and components of delayed line filters; (iii) Table 8.4
showing components that make up a photonic digital filter with corresponding element in an
SFG. (iv) A method of representing photonic circuits in signal flow diagram is introduced and its
advantages are outlined. It is stated that, by using Mason’s rule, the transfer function of a photonic
circuit can be obtained directly from signal-flow diagrams. An example of application of the graphical method is also given.
8.2.9
cOncluDing remarks
The objective of this Section 8.2 (Part II) is to explore possible ways of realizing a 2-D signal
processing system using fiber-optic signal processing architecture. A general technique for designing
a 2-D filter is illustrated and numerous examples of utilization of the technique are given. Although
the discussion is focused on fiber-optic systems, the design procedure for 2-D filters are just as
applicable to any other signal processing architectures. For example, the 2-D filter order reduction
method given in Section 8.2.5 can be used to simplify 2-D lightwaves systems which may or may
not be fiber-optic systems.
The design of 2-D filters is classified into two different classes. One class used matrix decomposition
to reduce the design of 2-D filters into a set of 1-D filter design procedures. The other class used
direct extensions of 1-D filter design methods. It is found that neither has a distinctive superiority
over another and that the designer has to choose what is the best for the particular application,
most likely by designing both and comparing the performances. All the design procedures are
implemented using the MATLAB programming language.
Among the matrix decomposition methods, the multiple stage singular value decomposition
method of Section 8.2.5.2 performed the best whereas, for direct methods, frequency sampling
method produced filters with the smallest errors. However, the result should be taken with caution
as there are many factors to be considered before declaring one method superior over another. The
differences between the various methods are outlined.
TABLE 8.4
Comparison of Photonic and Signal Flow Diagram Elements
Photonic Implementation
Unit length fiber
Optical amplifier, optical attenuator, etc.
Coupler
Signal Flow Diagram Element
Delay line
Multiplicative coefficient
Summing/splitting point
380
Photonic Signal Processing
A 2-D Filter order reduction method is applied to make fiber- and integrated-optic signal
processing more feasible. The technology allows the filter designer to produce filters of orders that
are implementable in practice without sacrifices in performance.
Different possible filter structures are proposed and illustrated for photonic implementation of
2-D filters. Most of the filter structures discussed can be used in 1-D coherent fiber-optic signal
processing and are not limited to 2-D coherent fiber-optic signal processing. Some of the proposed
structures, such as a transversal structure, are extremely efficient in the number of components used
to achieve a certain performance requirement. To make the efficient structures possible, the phase
shifting property of optical couplers when the incoming lightwaves is cross-coupled is utilized.
Filter structures for FIR and IIR filters are also shown and examples are given in Section 8.1.4.
It is evident that the fiber-optic signal processing technology presents a new direction in the usage
of optical fiber, lasers, and photonics technologies, which are evolving very fast. In34 an incoherent
signal processing system operating at 100 MHz is demonstrated. The authors note that raising this
capability to over 10 GHz is a relatively straightforward procedure involving shorter fiber lengths
and lasers and detectors with faster rise and fall time. They also note that conventional digital
signal processing and analog signal processing techniques are limited in their usefulness for signal
bandwidths exceeding one or two GHz. Current research efforts on fiber-optic signal processing
on lightwaves of the millimeter wavelength region will allow signal processing at bandwidths of
up to 100 GHz, even to THz region, if parametric amplification is employed. The field of 2-D
signal processing which requires ultra-fast processing capability has a great deal to gain from the
usage of the high-speed processing capability of fiber-optic architectures. Particularly with the fast
pace of research and inventions of photonic circuits reaching the nano-scale employing photonic
crystal wave guiding techniques, will allow multi-dimensional processing in the photonic domain
flourishing in the near future.
8.3 realizatiOn
This section of the chapter on multi-dimension photonic signal processing proposes techniques
to integrate the fields of discrete signal processing and fiber-optic signal processing, integrated
photonics and/or possibly nano-photonics to establish a methodology based on physical systems
that can be implemented. Using a combination of 1-D filter structures, 2-D fiber-optic filters can
be constructed. The relationship between the fiber-optic model and the mathematical model has
been linked to allow quick implementation. Using the developed methodologies, multi-dimensional
coherent photonic signal processors can be designed.
The technologies to support physical realization of such systems are not yet matured, therefore it
is difficult to estimate the exact capabilities of such systems. However, theoretically, the architectures
provide the potential for all-optical signal processing and it is envisaged that the processing
bandwidth can reach the Tera-Hz region. This section proposes a number of multi-dimensional
structures that would be realizable in the near future.
8.3.1
intrODuctOry remarks
In recent years, there has been a notable increase in the number of applications that require an
extremely fast signal processing speed that cannot be met by current all-electronic technology.
Photonic signal processing (PSP) opens the possibilities for meeting the demands of such highspeed processing by exploiting the ultra-high bandwidth capability of lightwave signals with specific
applications in the field of photonic communications and fiber optic sensor networks.
34
T. Deng and M. Kawamata, Frequency-domain design of 2-D digital filters using the iterative singular value
decomposition, IEEE Trans. Circ. Syst., 38(10), 1225–1228, 1991.
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
381
The demand for multi-dimensional photonic signal processing (M-D PSP) can be attributed to
various factors due the growing feasibility of high-capacity digital transmission networks capable of
transmitting ultra-high bit rate and time division multiplexing up to 160 Gb/s as well as fiber optical
sensor networks.
A problem with the implementation of such systems is the lack of devices that can process an
enormous amount of data associated with multi-dimensional signals. With photonic transmission
networks becoming the transport infrastructure, PSP technique has become increasingly more
desirable compared to O/E and E/O conversion techniques. As discussed in Section 8.1,1 fiber-optic
signal processing systems are ideal for such processing demands for several reasons: all-optical (or
photonic) processing of photonic information of optical communication systems are possible using
fiber-optic signal processing; 2-D signals usually require much higher bandwidth than 1-D signals
and therefore must be processed by a high bandwidth system to allow real-time performance. It is
likely that future telecommunication networks will be all fiber-optic.
Section 8.1 gave an introduction to multi-dimension signal processing and the fundamentals of 2D
processing with mathematical representation and optical signal-flow graphs so that the implementation
using photonic structures can be carried out. Section 8.2 dealt with methods of decomposition and
related photonic structures so as to simplify the practical implementation in photonic domain. This
section gives some examples on the implementation of multi-dimension photonic signal processors.
General concluding remarks are given to summarize all areas of this chapter.
8.3.2
phOtOnic implementatiOn OF 2-D Filters
In Section 8.11 the fiber-optic signal processing technique was introduced. In this section, various
structures of fiber-optic signal processing systems, some of them novel, are shown. Using these
structures along with the 2-D filter design methods given in Section 8.2,2 2-D filters can be
implemented in photonic domain.
8.3.2.1 Photonic Filter Structures
In Section 8.2,2 various methods of developing 2-D transfer function of the filter with desired
characteristics have been developed. To implement a 2-D transfer function in photonic domain, as
with other implementations of digital filtering systems, one must consider what kind of structure the
fiber-optic filter should have to reduce error and, at the same time, be economical.
Many of the structures used here are similar to 1-D fiber-optic filter structures and simply require
cascading of the structures to construct a 2-D fiber-optic filter. However, for transfer functions not
derived using matrix decomposition methods, novel structures must be devised to accommodate
the requirements of non-separable transfer functions. Possible filter structures include binary tree
structure, direct structure, lattice structure, parallel structure, and transversal structure. All, of
which, are described in the following sections.
8.3.2.2 Coherent System
The filters implemented in this paper assume coherent operation of the laser. As explained in
Section 8.2,2 this implies that the phase of incoming signal cannot be discarded and the constructive
and destructive interference of signals from different paths combining must be taken into account
when designing the filter. Although the restriction is somewhat harsh, it is noted in Section 8.22 that
there are important advantages of using coherent operation over incoherent operation. Furthermore,
unless specifically noted otherwise, all filters implement 2-D FIR structures. The implementation
methods for IIR filters are discussed in Section 8.3.2.10.
8.3.2.3 2-D Direct Structure Filter
We first consider a filter structure that is suited to direct design methods of Section 8.2,2 i.e., no
matrix decomposition. None of binary tree structure, transversal filter structure, or other 1-D
382
Photonic Signal Processing
structures, which are introduced later this section, are suitable for implementation of 2-D filters
designed without using matrix decomposition. The 2-D frequency sampling method, for example,
does not generate a set of separable transfer functions and therefore cannot be implemented using
1-D structures. In such cases, two options exist for photonic implementation: (i) Break down the
2-D transfer function into a set of 1-D transfer functions by decomposing the filter transfer function
using singular value decomposition; (ii) Use a 2-D direct filter structure.
2-D direct structure filter is derived from the signal-flow diagram shown in of Section 8.2.2
Translating the signal-flow diagram into photonic domain is a relatively easy task in this case and
the result is shown in Figure 8.33.
In Figure 8.33, each box labeled Amn represents a coefficient module, which contains a one
photonic attenuator (or some form of attenuation mechanism) and possibly a phase modulator. The
coupler ports are arranged so that the number of cross coupling a signal path contains is four for
all signal paths therefore ensuring that the phase of the signal entering each coefficient module is
consistent over the whole network. The negative coefficients can then be realized by including a
180° phase shifter in each negative coefficient module.
It should be noted that there are two different kinds of delays. The implementation of the delays
is quite simple if we are considering signal input of the form described in of Section 8.2.2 For
the 2-D sequence of Section 8.2,2 z1−1 can be implemented as just one-unit delay, and z2−1 can be
implemented as 7 unit delays as this is the number elements in a row of the signal.
The quantities of photonic elements required for 2-D direct photonic implementation are given by
C2 D = 2 N1 N2 + N1 + N2
NPM 2 D = NOAt 2 D = N1 × N2
(8.47)
NOA 2 D = 1
z2-1
z2-1
z2-1
A30
A31
A32
A33
A20
A21
A22
A23
A10
A11
A12
A13
A00
A01
A02
A03
z1-1
z1-1
z1-1
FIGURE 8.33 Fiber/integrated optic implementation of direct structure.
383
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
where C2− D is the number of optical couplers required; NPM2− D is the number of optical phase
modulators required; NOAt 2− D is the number of optical attenuators required; N1 is the filter order in
n1 dimension; N 2 is the filter order in n2 dimension.
8.3.2.4 2-D Separable Structure Filter
In the previous section, it is made clear that matrix decomposition methods of Section 8.22 can be
implemented in the photonic domain by cascading of 1-D fiber-optic structures since matrix decomposition methods basically generate sets of 1-D transfer functions. The 1-D structures however
must be combined in a way so that the filter performs 2-D signal processing operation as intended.
Essentially, the combined structure must implement a product of sum of 1-D transfer functions.
Figure 8.34 shows a fiber-optic structure that implements the required function.
Any 1-D filter implementation can be substituted for the filter banks if it is modular (the signal
in and the signal out must have the same zero reference point). One can therefore regard the above
structure as the general structure for a separable 2-D photonic filter. Specifically, Figure 8.34 implements a 2-D filter generated by matrix decomposition with eight significant singular values. The
order of the resultant 2-D filter is defined by the order of 1-D filters in the filter banks. Even though
the diagram does not make any distinctions, it should be kept in mind that the length of the delays
z1−1 and z2 −1 are also different for the two filter banks in each parallel branch.
In the following sections, a number of 1-D sub-structures that can be used to implement the filter
banks are described. It is up to the filter designer to choose which structure is most suitable for the
particular application, as any one of the structures can be substituted into the filter banks.
The number of photonic elements required for the separable 2-D structure is given by
log2 Ns
C=
∑2
i +1
i =0
NPM =
Ns
2
(8.48)
NOA = 2
filter bank
filter bank
filter bank
filter bank
filter bank
filter bank
filter bank
Phase
modulator
filter bank
filter bank
Phase
modulator
filter bank
OA
OA
filter bank
filter bank
filter bank
filter bank
filter bank
FIGURE 8.34
Phase
modulator
2-D fiber-optic filter structure.
Phase
modulator
filter bank
384
Photonic Signal Processing
where C is the number of optical couplers required; NPM is the number of optical phase modulators
required (maximum); NOA is the number of OAs required (maximum); Ns is the number of parallel
stages included.
8.3.2.5 Binary Tree Filter
A 1-D FIR filter transfer function can be implemented in fiber-optic format by arranging filter elements in binary tree structure, as shown in Figure 8.35. The particular filter shown in the Figure is
a 1-D 3rd order FIR filter. The extensions to higher order filters are obvious.
The transfer function of the filter implemented with the structure can be expressed as
k
Haz ( z ) =
∑
Eo,az
= (1 − γ )log2 ( k +1) (k + 1)−1 A (−1)d exp( jϕd ) ad z − d
Eo,az
d =0
(8.49)
(1 − γ )log2 ( k +1) = common excesslossfctor
(k + 1)−1 = common 3dBloss faactor of the couplers
where
(8.50)
A, a d = int ensity attenuation of optical attenuuators
ϕd = optical phaseshift of the optical phase modulator
d = delay order
Because the system is not intensity based, the signal is assumed to have the form Eoe j (ωot +φ ( t )+ε ),
which includes the optical phase ϕ. Negative coefficients of the transfer function can therefore be
achieved by shifting the phase of the signal by 180° through phase modulators.3 With appropriate
manipulation of the optical coupler characteristic as a perfect −90° phase shifter, the use of optical
modulators to achieve negative coefficients can be reduced or even eliminated as with transversal
filter structure, shown in the next section.
The advantage of the binary structure is that the number of 3-dB splitting stages the signal must
travel through is the minimum achievable. The loss due to splitting is thus at a minimum and the
effect of noise due to splitting is also at minimum. The structure is used in Taylor3 to implement a
fiber-optic integrator in which the input and output terminals of couplers are selected, so that the
3dB optical
coupler
ejϕ
0
a0
OPM
OAt
3dB optical
coupler
z-1
A
OPM
OAt
ejϕ
a1
1
A
OA
OA
3dB optical
coupler
ejϕ
2
3dB optical
coupler
z-2
OPM
a2
OAt
3dB optical
coupler
3dB optical
coupler
z-3
OPM
OAt
e
a3
jϕ3
FIGURE 8.35
Binary tree structure. (From Taylor, H. F., Proc. IEEE, 75, 1524–1535, 1987.)
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
385
sign alternation in the numerator that is the characteristic of Newton’s family of digital integrators
is achieved without phase modulators. However, this (alternating sign) is not true for an arbitrary
filter transfer function, and most filter transfer functions would necessitate the use of optical phase
modulators (OPMs) as shown in Figure 8.35.
The number of photonic elements required for the binary tree structure is given by
log2 N
CBT =
∑2
i +1
i =0
NPMBT = N
(8.51)
NAt BT = N
NABT = 1
where CBT is the number of optical couplers required; NPMBT is the number of optical phase
modulators required (maximum); NAt BT is the number of optical attenuators required; N A BT is the
number of OAs required (minimum) and N is the order of the 1-D filter sections
8.3.2.6 Photonic Transversal Filter
In the previous section, it is shown that the binary tree structure requires phase modulators, except
in cases when the signs are alternating or when there is equal number of positive coefficients
as negative coefficients in the transfer function. Such cases are rare, and in most cases phase
modulators will be needed to achieve the negativity. In this section, a structure that does not
require any phase modulators is proposed. This structure, based on the transversal filter structure
in Refs.7–9 (shown in Figure 8.36a), achieves 180° phase shift required by negative coefficients
through appropriate arrangement of coupler ports. The signal paths for positive coefficients
contain four cross couplings resulting in –j × –j × –j × –j = 1, i.e., no phase shift, and all
signal paths for negative coefficients contain two cross couplings resulting in –j × –j = –1, i.e.,
180° phase shift. Shown in Figure 8.36b is the proposed transversal filter structure that does not
require phase modulators to achieve negative coefficients. The phase of the signal at various
points of the filter is shown in gray scale.
The proposed structure is particularly suited to the transfer functions generated by matrix
decomposition as it can implement a 1-D FIR transfer function with very little modifications
(needed for the coupling attenuations) to its filter coefficients. How the additional sections can be
accommodated to form a 2-D filter is shown later in this section.
The advantages of the proposed filter structure over the simple transversal filter structure
in Heyde and Minasian7 are threefold. First, as mentioned previously, is the fact that no phase
modulators are required to achieve negative coefficient. The second advantage is that no signal
biasing or differential detection is required. The implication of the second advantage is that the
filter structure is modular and can therefore be connected in parallel or in series without any
modification to the filter module. Another advantage of the proposed structure is that the signal
goes through less couplers than the implementation of Heyde and Minasian.7 This property can
be observed if one considers the fact that the signal only goes through in average N/2 couplers for
the proposed structure (since the coefficients are split into two groups), whereas for the traditional
transversal filter the number of couplers the signal must travel through is always N, where N is
the number of coefficients of the filter transfer function, which may or may not be the same as the
order of the filter.
The proposed structure can also be used for adaptive filtering as only the coefficients of the optical
amplifier/attenuator need to be changed to modify the filtering operation. However, for adaptive
operation, the filter coefficients that are zero and therefore not included in the original structure may
386
Photonic Signal Processing
z-1
z-1
Incoherent
lightwave
input
OA
+
PM
-j
(a)
OA
+
PM
OA
+
PM
Filter
output
1
z-1
OA
-ve
0
Coherent
lightwave
input
(b)
1
1
-j
-j
z-1
OA
-ve
1
OA
-ve
N-ve
1
1
j
-1
OA
+ve
0
z-1
Negative subsection
1-1
j
OA
+ve
N+ve
OA
+ve
1
Filter
output
Positive subsection
z-1
FIGURE 8.36 (a) The original transversal structure. (From Heyde, E., and Minasian, R. A., IEEE Trans.
Microw. Theory Tech., 45, 1463–1466, 1997.) (b) The proposed structure.
have to be included in case the adapting algorithm changes them to non-zero values. The situation is
slightly disadvantageous for the proposed structure compared to the simple transversal structure of
Heyde and Minasian,7 because the number of positive and negative coefficients (N−ve and N+ve) of an
Nth order filter can range from 0 to N + 1 and both sets of coefficients need to be fully implemented.
Fortunately, the situation is not as bad as one might expect. At first, it appears that the number of
optical attenuators (OAs) needed is 2N, since both negative and positive sections should implement
the full order of the filter transfer function. For the proposed structure, it can be shown that the
number of OAs required is approximately [(N + 1) × 1.5* (N + 1)] components are required for the
simple transversal filter structure of Heyde and Minasian7 for adaptive operation). The reason for
requiring only (N + 1) × 1.5 instead of 2N can be explained as follows.
Assume that we can exchange the input connections to two parallel sections using optical
switches so that the signs of the two parallel sections can be reversed. First consider the case where
the number of positive coefficients and negative coefficients of the filter transform function are
exactly the same, i.e., N+ve = N−ve = (N + 1)/2. Clearly (N + 1)/2 OAs are needed for each parallel
subsection and therefore total of N + 1 OAs are required. Now consider the case where one sign is
completely dominant over the other, i.e., N+ve = N + 1 and N –ve = 0, or N–ve = N + 1 and N+ve = 0. In
this case, N + 1 OAs are necessary in at least one of the parallel subsections even though the other
subsection does not need any. To accommodate both situations with one filter structure, it is clear
that (N + 1) + (N + 1)/2 = 1.5(N + 1) OAs are necessary. The result for optical attenuators can be
extended to phase modulators and couplers with a few simple modifications.
In the case where adaptive filtering is not required, the total number of components required is
minimized by the use of the proposed structure. For 1-D digital filtering purposes, this structure is
also economical, modular, and easy to implement. Example 8.1 is a simple exemplar of a fiber-optic
filter realized from a transfer function using the parallel structure.
387
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
8.3.2.6.1 Photonic Filter Implementation Using the Proposed Transversal Structure
Design Aim: A sample implementation of a simple transfer function.
Method: The proposed transversal structure. Fiber losses and other non-linear effects are ignored.
Results:
H ( z −1 ) = 1 + 0.2 z −1 − 0.3z −2 − 0.2 z −3 + 0.4 z −4 + 0.6 z −5 − 0.5 z −6
(8.52)
The first step is to divide the transfer function into two parts, namely the positive coefficients and the
negative coefficients denoted by k+ and k−. It is assumed that all the coefficients are realized using
optical attenuators and not amplifiers (optical attenuator is known to produce less noise than does
the optical amplifier).
An observation of the filter structure shown previously in Figure 8.36 will reveal that all streams
of lightwaves in each parallel section pass through the same number of couplers and therefore
suffers the same attenuation from coupling. In this example, the positive coefficients suffer (√2)7
attenuation, whereas the negative coefficients suffer a (√2)6 attenuation. Optical preamplifier gain is
therefore set at (√2)7 and the negative coefficient attenuators are set at h−n ÷ √2 where h−n denotes the
negative coefficient values. This factor of √2 compensates for the effect of the extra coupler positive
coefficient signals go through. Optical preamplifier is set at (√2)k where k is the greater of k+ and
k− to compensate for the coupling losses (Figure 8.37).
Because the binary tree splitting stages of Figure 8.35 add further attenuation to the lightwave,
when the transversal structure is incorporated into a 2-D system, the filter amplifier and the
attenuator settings must be adjusted accordingly. The number of optical elements required for the
transversal structure is given by
CTF = 2 N + 2
NPMTF = 0
(8.53)
NAtTF = N
NATF = 1
z-2
Coherent
lightwave
input
z-1
z-3
0.2121
0.1414
0.3536
Filter
output
11.3137
z-1
Optical attenuator
Optical amplifier
0.2000
0.4000
z-3
z-1
0.4000
Optical coupler
z-1
Unitdelay line
FIGURE 8.37 Fiber optic implementation of (8.52) using the proposed transversal filter structure.
388
Photonic Signal Processing
where CTF is the number of optical couplers required; NPMTF is the number of optical phase
modulators required; NAtTF is the number of optical attenuators required; NATF is the number of OAs
required (minimum) and N is the order of the 1-D filter sections
8.3.2.7 1-D Direct Structure Photonic Filter
A 1-D direct structure provides the classic alternative to the binary tree structure and transversal
filter structure for implementation of 1-D filters. Unlike the structures mentioned, a direct structure
can also implement IIR filters very easily. The direct structure signal-flow diagram can be found in
most signal processing books and is quite straightforward. However, photonic implementation of
the direct structure has not been developed, as yet, and this section shows an implementation of the
structure using fiber-optic elements.
L
∑b z
k
H (z) =
−k
k =0
N
1−
∑a z
k
(8.54)
−k
k =1
This transfer function can be represented in signal flow diagram format, as shown in Figure 8.38,
and the photonic implementation is shown in Figure 8.39.
As with the 2-D direct structure, each signal path contains four cross-couplings and suffers no
overall phase shift (− j × − j × − j × − j = 1). The boxes in the horizontal branches represent coefficient module identical to that in 2-D direct structure and the boxes in the vertical branches represent
the delay elements. The structure is optimal in the sense that the number of delay elements is minimal and can be used to implement 1-D IIR functions in the photonic domain. The couplers used in
the middle section of the implementation represent either 3 × 3 couplers or a cascade arrangement
of two 2 × 2 couplers shown in Figure 8.40.
w (n)
y (n)
x (n)
-1
z
b1
a1
z-1
a2
b2
bL-1
z-1
bL
z-1
aN-1
z-1
aN
FIGURE 8.38 Signal flow diagram of 1-D direct structure.
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
a′1
z-1
389
b′1
a′2
b′2
a′N
b′3
FIGURE 8.39 Photonic implementation of direct 1-D structure.
FIGURE 8.40
An arrangement of two 2 × 2 couplers to form a 3 × 3 coupler.
The number of photonic elements required for the direct structure is given by
CDF = 3 N + Nnum + 3
NPMDF = NAtDF = Nnum + Nden
(8.55)
NADF = 1
where CDF is the number of optical couplers required; NPMDF is the number of optical phase
modulators required; NAtDF is the number of optical attenuators required; NADF is the number of
OAs required; Nnum is the order of the 1-D filter numerator and Nden is the order of the 1-D filter
section denominator.
8.3.2.8 Parallel Structure Filters
Parallel structure is a simple variation on the theme on the direct structure. This structure is slightly
different from the direct structure in that signals pass through less couplers, and the signals are not
attenuated as much.
Basically, the structure can be thought of as a parallel arrangement of 2nd order direct structures.
The signals are split into several branches at the beginning, and, after the signals have traveled
through the parallel sub-structures, they are merged back into one signal path. To implement the
parallel structure, the filter transfer function must be factorized into a sum of several sub-transfer
functions. If each sub-transfer function is in the form of 2nd order substructure, it is then possible to
turn the sub-transfer function into a signal-flow diagram form, which can be realized in the optical
domain.
390
Photonic Signal Processing
8.3.2.8.1 A Filter Design Using Parallel Structure Filter
Design Aim: A 1-D lowpass filter of 7th order with normalized cut-off frequency at 0.3
Method: Parallel factorization
Program used: PARALLEL.m
Results: The transfer function designed using Chebyshev approximation routine in MATLAB
simulation package is decomposed as shown below in signal-flow diagram format. It is found
that the performances of the two implementations are nearly identical (see Figure 8.41).
For photonic implementations of parallel structures, the splitting of the signal at the beginning of
the structure can be performed using a binary tree of couplers, in which case the amplitude and
the phase change caused by cross-coupling must be considered. The task is very similar to that
performed for binary filter structure. Essentially, the phase changes can be corrected by including
a 180° shift every second parallel branch, and amplitude compensation is equal to 2 raised to log2P
where P is the number of parallel stages (which must be a power of 2).
The performance of the parallel structure is similar to that of the direct structure, as seen in
Figure 8.41, and the number of delays as well as the number of amplifiers/attenuators required
are also the same. However, the number of summer/splitters in parallel structure is significantly
increased, it will be shown that the number of optical couplers needed for the implementation of
parallel structure is also very much larger than that required for other 1-D structures.
The number of photonic elements required for the parallel structure is given by
w(n) 0.0304
x(n)
z -1
y(n)
0.1218
-
13834
1.4721
z -1
0.1827
z -1
08012
(a)
0.2286
0.1218
z -1
0.0304
02960
z-1
04830
0.1089
z-1
0.7194
0.4599
x(n)
y(n)
z-1
09004
0.3177
(b)
00155
z-1
0.0304
FIGURE 8.41 Parallel structure example signal flow diagram: (a) Direct structure and (b) parallel structure.
391
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
N
CPF = den × 7 +
2
NPMPF = 2 Nden +
N
log2 den
2
∑2
i +1
i =0
Nden
2
2
log2
(8.56)
NAtPF = 2 Nden
NAPF = 1
where CPF is the number of optical couplers required; NPMPF is the number of optical phase
modulators required; NAtPF is the number of optical attenuators required; NAPF is the number of OAs
required (minimum); Nnum is the order of the 1-D filter numerator and Nden is the order of the 1-D
filter section denominator.
8.3.2.9 Other 1-D Filter Structures
Several other filter structures exist for filter realization in the optical domain. The lattice form,
shown in Figure 8.42, is a structure particularly suited to incoherent systems since each section of
the signal-flow diagram is identical to the optical coupler signal flow diagram shown in Figure 8.41a.
The coefficients k can be implemented using optical amplifier/attenuators. The lattice structure is
not suitable for coherent signal processing as coherent systems must take phase into account and,
since the lattice structure assumes that the cross coupling does not cause any phase shift, it will
be inefficient and difficult to implement coherent systems using the lattice structure (Figure 8.43).
y1(z)
X(z)
yM-1(z)
k1
z-1
yM(z)
kM
z-1
w1(z)
wM(z)
wM-1(z)
FIGURE 8.42 Lattice form signal flow diagram.
1.2
Direct form
Parallel form
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
FIGURE 8.43 Frequency response of direct and parallel structures.
3
3.5
392
Photonic Signal Processing
8.3.2.10 Realization of Poles
In discussing feed forward structures, such as the binary tree structure or the direct structure, the
realization of the denominator part of filter transfer functions are ignored. This can be justified in the
case of finite impulse response filters (FIR) where the denominator is simply 1. In cases of IIR filters,
the denominator must be incorporated into the system as in 1-D direct structure. An alternative is to
form a pure pole filter by using a feed forward structure in a feedback loop as shown in Figure 8.44.
Assuming that the denominator of transfer function can be written as
p( z ) = K + f ( z )
(8.57)
where f ( z) has no constant terms, applying the feedback structure results in the following transfer
function:
H p (z) =
1+
1
K
1
K
(8.58)
f (z)
It is therefore possible to use the feed forward structure, such as transversal structure, without
modifications in feedback branch to form the denominator of an IIR filter transfer function. The overall
filter structure incorporating numerator q( z ) and denominator K + f ( z ) is shown in Figure 8.45.
Note that q( z ) and f ( z ) can be replaced by q( z1, z2 ) and f ( z1, z2 ). The structure in Figure 8.45 can
therefore be used to implement a subsection of separable filters of Section 8.22 or a complete 2-D
direct filter implementation. We implement a case of the former as follows.
8.3.2.10.1 A 2-D Recursive Filter Subsection Realization
Design Aim: The recursive transfer function given in Eq. (8.58).
H ( z −1 ) =
1 + 0.2 z −1 − 0.3z −2
1 − 0.2 z + 0.4 z −4 + 0.6z −5 − 0.5z −6
−3
(8.59)
The filter subsection can be implemented in photonic domain as shown in Figure 8.46.
Finally, when the SFG is ready, the graphical technique introduced in Section 8.22 can be used
to check the validity of the design. However, applying the technique by hand to photonic circuits
1/K
f(z)
FIGURE 8.44 Denominator realization using feed forward structure.
q(z)
+
1/K
f(z)
FIGURE 8.45 IIR filter structure using FIR subsections.
393
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
z-1
z-1
0
Coherent
lightwave
input
0.212
2
5.657
1
0.2
0.5
z-3
0.2
z-1
5.6569
0.6
z-1 Unit delay line
Optical attenuator
Filter
output
z-3
0.4
Optical coupler
z-1
z-4
Optical amplifier
FIGURE 8.46 A 1-D IIR filter subsection example.
shown in this section would be time-consuming and error prone as there are too many photonic elements to consider. In fact, applying any graphical technique by hand to the photonic circuits in this
section would be impractical. Due to the impracticality of performing a graphical analysis by hand,
the graphical analysis of the photonic circuits is omitted here.
8.3.2.11 Remarks
In this Section (i) The significance of using coherent systems for PSP is stated; (ii) 2-D direct
structure illustrated; (iii) Various matrix decomposition realizations using 1-D filter structures
shown; and (iv) Realizations of IIR filters using FIR filter sections shown.
8.3.3 Design chart anD DiscussiOns
The design of 2-D photonic filter can be summarized in this section. Another example of such
design is also illustrated and thence some concluding remarks are given.
8.3.3.1 2-D Photonic Filter Design Flowchart
In designing and implementing a 2-D filter, several design decisions must be made. The following
flowchart illustrates the possible paths that may be taken when designing and implementing a 2-D
filter in photonic domain. Tables in the chart show various design methods any one of which may
be substituted for another within the same table. For example, for separable design methods path
(the branch on the right side of the flowchart), any one of single stage SVD, multistage SVD,
iterative SVD, or OD may be used to calculate the appropriate 2-D transfer function. The numbers
inside brackets show the relevant section which to be referred to. Obviously, if a design result is not
satisfactory, some of the procedures must be repeated to find a better solution. The iteration paths
are not shown in the flowchart as they are reasonably obvious.
8.3.3.2 Examples of Photonic 2-D PSP Implementation
In this section, 2-D filter implementations of the same frequency specification are performed from
scratch first using a matrix decomposition method, and then a direct method. The design methods
used in the example are by no means “ideal.” As shown in the flowchart shown in Figure 8.47, there
are many possible paths and ultimately, the choices lie with the filter designer.
8.3.3.2.1 Specification
The filter to be designed is a lowpass filter with normalized spatial cut off frequency of 0.7 in x and
y directions (see Figures 8.47 and 8.48). The overall filter magnitude response error is to be less
than 10%. The 2-D signal is transmitted coherently through an optical fiber medium using linear
394
Photonic Signal Processing
2D Magnitude
Specification (3.1)
Direct design methods
Separable design methods
Choose a design
approach (3.3)
Single stage
SVD(5.1)
Frequency sampling
method (4.2)
Multistage SVD (5.2)
Window method
(4.3)
Iterative SVD (5.3)
McClellan
transformation (4.4)
Filter order
acceptable? (6.1)
YES
Require a separable
filter?
OD (5.4)
NO
Filter order
acceptable? (6.1)
NO
2D Balanced
approximation
(6.3)
YES
n=1
Do below for zn
YES
SVD
(6.2)
FIR or
IIR1
NO
IIR
FIR
or (4.1)
IIR
FIR
IIR
Binary
(7.5)
Transveral
(7.6)
Direct
(7.7)
Parallel
(7.8)
Separate into
NUM DEN
numerator and
denominator
2D Direct
Structure (7.3)
Separate into
numerator and
denominator
DEN
2D Direct
Structure (7.3)
FIR
NUM
Binary
(7.5)
Transveral
(7.6)
Direct
(7.7)
Parallel
(7.8)
Merge using the IIR
structure in (7.10)
Merge using the IIR
structure in (7.10)
n=2
YES
NO
n=1
Combine structures for
z1 and z2 (7.4)
The final
filter design
1. 1D FIR and IIR filter discussion can be
found in many signal processing textbooks
FIGURE 8.47 Flowchart of the design of multi-dimension photonic filters.
395
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
1
0.8
0.6
0.4
0.2
0
1
0.5
0
-0.5
-1 -1
-0.5
0.5
0
1
FIGURE 8.48 Magnitude specification.
sequencing of a 256 × 256 pixel frame starting from x = 0 and y = 0. The frequency response of the
filter is to be circularly symmetric. The order of the filter should be sufficiently low to enable it to
be implemented in photonic domain. The phase response of the filter should be linear. The filter is
intended to be a noise remover as most noise components are in the high frequency band.
8.3.3.2.2 Choice of a Design Methodology
As the first decision to be made, a choice between direct design methods and separable design methods must be made. The following table shows the differences between separable and direct design
methods. Although the two methodologies result in very different designs, there are no apparent
advantages to be gained from preferring one structure over another. There are multitude of factors
contributing to the actual performance and the economy of the designed filter and the best design
will be obtained by using both methodologies and comparing various statistics to find out which
design method gives better results for the particular application (Table 8.5).
TABLE 8.5
Comparison of Direct Structure and Separable Structure
Property
Number of parallel stages
Filter structure
Filter design procedure
Coherent processing
Performance
Fiber-optic implementation
Direct Design Methods
1
FIR or IIR
One 2-D filter design
Yes
Depends on the order of the filter and
the 2-D filter design method
Direct structure only
Separable Design Methods
≥1
FIR or IIR
Many 1-D filter designs
Yes
Depends on number of additional stages
and the 1-D filter design method
Binary, parallel, transversal, direct
396
Photonic Signal Processing
8.3.3.3 Separable Implementation Using Matrix Decomposition Methods
8.3.3.3.1 Choice of a Decomposition Method
There are four decomposition methods to consider. All methods are tried to compare their performances.
The comparison is made based on the least number of parallel stages that will satisfy the specified 10%
overall error. Also, the 1-D filter design methods used are identical for the four different decomposition
methods (note that perhaps this is not quite fair on methods such as ISVD algorithm or OD algorithm
which put their strengths on making filter design task easier by keeping the 1-D magnitudes all positive).
As can be seen in the table, the best performance can be obtained by multiple stage SVD algorithm that,
with two parallel stages, results in a filter error of 8.274% (Table 8.6).
As multiple stage SVD algorithm produces four sets of 1-D magnitude responses, there are four
1-D filters to be implemented and their coefficients are shown in Table 8.7. The 1-D filter design
method used is least-squares algorithm.
The filters shown above are symmetric about the centre (which is formed by the coefficients
for 7th order and 8th order), therefore the phase response is guaranteed to be linear. The repeated
coefficients also raise the possibility of saving of components through exploitation of symmetry.
A filter order of 15, although slightly too high for photonic implementation, will be retained for the
example purpose. If the filter order is to be reduced at this stage, then one would merge the separable
expressions into a single expression, then use the balanced approximation method illustrated in
Section 8.1.2 Such a procedure would result in an IIR filter design of reduced order.
8.3.3.3.2 Photonic Implementation of the Separable 2-D Filter
Because we are dealing with 2-D separable structure, we need to decide which 1-D photonic filter
structure should be used to implement the four 1-D filter stages. Table 8.8 is a comparison of the
structures on the number of components required.
In Table 8.8, the number of splitting stages per OA is used to indicate how many signal splitting stages the optical pre-amplifier must compensate for. If 30 stages must be compensated
for as in the direct structure, the amplification factor would need to be (√2)30 = 32768 which
is clearly unacceptable for OAs. A solution such as including several OAs in cascade would be
required in such situations provided the accumulated ASE noises do not surpass the signal level.
TABLE 8.6
Comparison of Various Decomposition Methods
Single Stage SVD
Number of parallel stages
Resulting design
1
FIR (or IIR)
Computational requirements Minimal
Major claim for advantage
Simple and quick
Major disadvantage
Filter error (%)
Order
Total no of multiplications
required
35
35
Multiple Stage SVD
Optimal
Decomposition
2
FIR (or IIR)
2
FIR (or IIR)
2
FIR (or IIR)
Intermediate
Accurate
Intermediate
Easy 1-D filter design
Very heavy
Accurate and easy 1-D
filter design
Not accurate enough
and heavy
computational load
Too rough (large error) Phase must be
considered when
designing 1-D filter
sections
10.7262
8.2740
16
16
32
64
() contains the possible alternatives.
Iterative SVD
Not accurate enough
9.9603
16
64
9.8521
16
64
z1 stage 1
z1 stage 2
z2 stage 1
z2 stage 2
Order
0.0004
0.0022
−0.0004
−0.0025
0
2
−0.0049
−0.0074
−0.0167
0.0130
1
−0.0004
0.0055
0.0028
0.0080
3
0.0114
0.0037
−0.0246
−0.0165
TABLE 8.7
Coefficients of the Designed Filter
4
0.0083
0.0406
0.0600
−0.0338
5
−0.0704
−0.0841
0.0849
−0.0109
6
0.0819
−0.1752
0.0071
0.0028
7
0.5015
0.0082
0.0139
0.0050
8
0.5015
0.0082
0.0139
0.0050
9
0.0819
−0.1752
0.0071
0.0028
10
−0.0704
−0.0841
0.0849
−0.0109
11
0.0083
0.0406
0.0600
−0.0338
12
0.0114
0.0037
−0.0246
−0.0165
13
−0.0049
−0.0074
−0.0167
0.0130
14
−0.0004
0.0055
0.0028
0.0080
15
0.0004
0.0022
−0.0004
−0.0025
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
397
398
Photonic Signal Processing
TABLE 8.8
Comparison of Different Fiber-Optic Filter Structures
Binary
No of splitting stages per
amplifier
Total no. of optical phase
modulators
Total no. of optical attenuators
Total no. of optical couplers
Total no. of OAs
Total no. of components
Remarks
Transversal
Direct
Parallel
10
19
30
10
34
0
64
96
64
126
4
228
Low attenuation
64
64
142
136
4
4
210
268
No phase modulators Simple, but inefficient
required
96
184
4
380
Inefficient, but
low attenuation
For the example, transversal filter structure is chosen as the implementation as it appears to
be the most efficient structure. Below diagram shows the schematic diagram for the photonic
implementation.
Because the filter is circularly symmetric, the filter structure and attenuator/amplifier coefficients
are the same for the second dimension except for the delay which must be replaced by z2−1. Physically,
this delay would correspond to (z1−1)255 as one-line delay is same as the entire row of pixels. For the
bottom parallel section of Figure 8.49, the positive subsections and the negative subsections of the two
1-D subsections will need to be exchanged. This is because the binary splitting stages at the start and
end of the structure shown in Figure 8.50 introduce another phase shift of 180° for the lower stage.
Exchanging of subsections can compensate for the phase shift. The filter coefficients are related
to the attenuator settings by
a+ i =
hi
k +2
2
where
, a−i = −hi
(8.60)
k = No of +ve terms − No of − ve terms
z-1
Unit delay line in z1 dimension
Optical coupler
Optical attenuator
z-1
z-1
z-3
Optical amplifier
z-5
z-3
z-1
0.0001 0.0012 0.0176 0.0176 0.0012 0.0001
Negative subsection
181
0.0001 0.0012 0.0176 0.0176 0.0012 0.0001 0.0004 0.0049 0.0704 0.0704
z-3
FIGURE 8.49 1-D filter stage 1.
z-1
z-2
z-1
z-1
z-1
z-2
z-1
z-3
Positive subsection
399
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
FIGURE 8.50
1D
Filter stage 1
in z1 domain
1D
Filter stage 1
in z2 domain
1D
Filter stage 2
in z1 domain
1D
Filter stage 2
in z2 domain
Schematic diagram of the separable 2-D filter.
z-2
z-3
z-1
z-3
z-1
z-3
Negative subsection
0.0019 0.0160 0.0438 0.0438 0.0160 0.0019
181
0.0022 0.0055 0.0037 0.0406 0.0082 0.0082 0.0406 0.0037 0.0055 0.0022
-1
-2
z
z
-1
z
-3
z
-1
-3
z
z
-1
z
-2
Positive
-3
z
z
FIGURE 8.51 1-D filter stage 2.
1
0.5
0
1
FIGURE 8.52
0.5
0
-0.5
-1 -1
-0.5
0
0.5
1
Magnitude response of the designed 2-D filter.
where the number of positive terms is greater than the number of negative terms in the transfer
function. a+i and a−i can be exchanged and multiplied by −1 to obtain the settings if the reverse is
true. The magnitude response of the filter is plotted in Figures 8.51 and 8.52.
8.3.3.4 Non-Separable Implementation Using Direct Methods
8.3.3.4.1 Choice of a Design Method
There are two methods to choose from: frequency sampling method and McClellan transformation
method. The designs results are shown in Table 8.9.
400
Photonic Signal Processing
TABLE 8.9
Comparison of Direct Design Methods
Property
Frequency Sampling Method
McClellan Transformation Method
Resulting design
Requirements
FIR
Magnitude specification
Remarks
The filter accuracy depends heavily on the
accuracy of frequency sampling
20 × 20
7.61
Filter order
Magnitude spec.
error (%)
FIR or IIR
1-D magnitude specifications and a transformation
function
Better transformation functions may result in much
improved error
12 × 12
22.68%
The poor performance of McClellan transformation method can be attributed to its low attenuation
in the stopband. It is envisaged that with a better transformation function, McClellan transformation
method will perform better. For the current design task however, frequency sampling method is
chosen. The resultant filter coefficients are not shown due to the number of coefficients being too
large to show here.
8.3.3.4.2 Reduction of Filter Order Using Balanced Approximation Method
The filter design in the current form requires a 20 × 20 order filter. This implies that for some signal
paths, the signal must go through 20 + 20 = 40 power splitting stages. A huge pre-amplification
factor will therefore be required. A solution to this problem is to use the filter order reduction
method of Section 8.2. In this case, the reduced filter is 10 × 10 resulting in 11 + 11 = 22 splitting
stages which is still large, but again for example purpose, is retained (Tables 8.10 and 8.11).
The filter error is found to be 6.57%, which is actually less than the original design! It has therefore
been shown that the application of filter order reduction did not result in any notable degradation
in the performance. As can be deduced from the fact that there are two sets of filter coefficients,
the resulting design is an infinite impulse response design. Another factor which must be taken into
account is the fact that neither numerator nor the denominator is symmetrical. No components are
duplicated therefore removing the possibility for saving of components by exploitation of symmetry.
TABLE 8.10
Denominator Coefficients
Order
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0.0015
−0.0045
0.0128
−0.0273
0.0370
−0.0437
0.0520
−0.0531
0.0469
−0.0291
0.0117
−0.0046
0.0098
−0.0325
0.0852
−0.1279
0.1660
−0.2103
0.2156
−0.1903
0.1171
−0.0487
0.0120
−0.0251
0.0702
−0.1757
0.2614
−0.3410
0.4309
−0.4349
0.3841
−0.2344
0.1017
−0.0239
0.0628
−0.1456
0.2946
−0.3985
0.4815
−0.5751
0.5640
−0.4995
0.3020
−0.1362
0.0290
−0.0895
0.1945
−0.3387
0.4165
−0.4550
0.5001
−0.4755
0.4244
−0.2527
0.1171
−0.0258
0.0991
−0.2091
0.3073
−0.3249
0.2844
−0.2461
0.2151
−0.1944
0.1154
−0.0529
0.0246
−0.1155
0.2341
−0.2844
0.2366
−0.1124
−0.0139
0.0571
−0.0508
0.0245
−0.0127
−0.0250
0.1212
−0.2318
0.2461
−0.1628
0.0031
0.1646
−0.2204
0.1957
−0.1134
0.0468
0.0262
−0.1174
0.2181
−0.2319
0.1567
−0.0112
−0.1466
0.2045
−0.1731
0.1145
−0.0388
−0.0197
0.0816
−0.1479
0.1602
−0.1155
0.0317
0.0590
−0.1033
0.0941
−0.0534
0.0156
0.0092
−0.0373
0.0706
−0.0844
0.0711
−0.0394
0.0059
0.0123
−0.0124
0.0051
0.0003
401
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
TABLE 8.11
Numerator Coefficients
Order
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
1.0000 −3.3531
6.6740 −9.7374 11.1831 −10.3924
7.8531 −4.7539
2.2164 −0.7249 0.1283
9.8116 −19.5290 28.4926 −32.7230 30.4093 −22.9790 13.9103 −6.4854 2.1211 −0.3753
−2.9261
5.4318 −18.2134 36.2521 −52.8915 60.7444 −56.4494 42.6564 −25.8220 12.0390 −3.9375 0.6967
−7.5706 25.3851 −50.5266 73.7179 −84.6629 78.6767 −59.4526 35.9896 −16.7795 5.4880 −0.9710
8.4280 −28.2599 56.2486 −82.0662 94.2508 −87.5866 66.1854 −40.0654 18.6797 −6.1095 1.0809
−7.6733 25.7295 −51.2120 74.7179 −85.8115 79.7440 −60.2591 36.4779 −17.0071 5.5624 −0.9841
5.7326 −19.2220 38.2595 −55.8202 64.1080 −59.5752 45.0184 −27.2519 12.7057 −4.1556 0.7352
−3.4579 11.5948 −23.0783 33.6710 −38.6703 35.9360 −27.1553 16.4385 −7.6641 2.5067 −0.4435
1.6190 −5.4287 10.8053 −15.7649 18.1056 −16.8254 12.7142 −7.6965
3.5884 −1.1736 0.2076
1.7992 −3.5812
5.2250 −6.0008
5.5765 −4.2139
2.5509 −1.1893 0.3890 −0.0688
−0.5366
0.0965 −0.3236
0.6441 −0.9398
1.0793 −1.0030
0.7579 −0.4588
0.2139 −0.0700 0.0124
8.3.3.4.3 Photonic Implementation of Non-separable Filters
As the filter structure is in form of IIR filter, the structure proposed in Section 8.3.2 must be employed
in realizing the fiber-optic filter. The filter shown in Figure 8.33 will form the subsections of the IIR
structure shown in Figure 8.46. The actual filter schematic diagram is omitted as the diagram will be too
cluttered to make any clear statement of the structure of the filter. The attenuator settings can be calculated
by considering how many splitting stages the signal being attenuated by the coefficient module must go
through. The relationship between the attenuator settings aij and the filter coefficients hij is given by
aij =
h
2
ij
ki + 2 j + 2
(8.61)
where ki is the order of the filter in n1 dimension. The filter statistics are given in Table 8.12. The
filter is shown to be very inefficient compared to the separable implementations as the number of
components required is very much higher. Clearly, the trade-off for the good error response is the
large number of components required to implement the design in practice.
The filter magnitude response is shown below in Figure 8.53. Although the passband contains
some irregularities, the phase response of IIR filters designed using balanced approximations
remains approximately linear (refer to Sections 8.2.4 and 8.1.4.5) therefore satisfying the phase
requirement of the specification.
TABLE 8.12
Direct Structure Filter Statistics
Factor
2-D Direct Form
No of splitting stages per amplifier
No. of OPMs
No. of optical attenuators
No. of optical couplers
No. of OAs
Total no. of components
Remarks
33 (worst case)
242
242
528
2
1014
Very small error
402
Photonic Signal Processing
1
0.5
0
1
0.5
0
-0.5
-1 -1
-0.5
0
0.5
1
FIGURE 8.53 Normalized filter magnitude response.
8.3.3.5 Comparison of Matrix Decomposition Method Design and Direct Method Design
Although we have started out with the exactly the same specification, the two design results are strikingly different. Direct method results in a filter with over 1000 elements whereas matrix decomposition method results in a filter with around 200 elements. The error is also different. Various factors
such as performance and economy must be taken into account when deciding which structure to
implement. The table below shows the comparison of the two methods using the results obtained for
this example. For the example, the better implementation technique would be matrix decomposition
method as it offers a filter of reasonable performance with smaller number of components (Table 8.13).
8.3.4
POSSiBle AreAS Of APPlicAtiOnS
The fiber-optic and integrated photonic signal processing has found applications in areas such
as optical matrix multiplications, convolutions and broadband signal processing, time division
multiplexing in the femtosecond time resolution, high order optical correlation, higher order
spectrum analysis for amplitude and phase, spatial and time photonic Fourier transformation.
Because of the wideband properties of optical fibers and integrated photonic waveguides, PSP
technique makes a natural processing architecture for fiber transmission systems carrying wide
bandwidth data. An application of 2-D fiber-optic signal processing system would be the signal
processing of ultra-fast time division multiplexing (TDM) information that is the photonic packet
switching technique. Novel time division coding used as photonic headers is also another potential
application of M-D PSP. 2-D fiber-optic signal processing architecture can be therefore a natural
candidate for the processing of ultra-fast signals transmitted over optical guided media.
TABLE 8.13
Comparison of Direct Method and Matrix Decomposition Method
Property
No. of splitting stages per OA
No. of OAs
No. of attenuators
No. of couplers
No. of phase modulators
Total no. of components
Error (%)
Remark
Direct Design Method
Matrix Decomposition Method
33 (worst case)
2
242
528
242
1014
6.57
Very good transition band and low error
19
4
64
142
64
210
8.27
Economical
Multi-Dimensional Photonic Processing by Discrete-Domain Approach
403
Further capabilities of 2-D fiber-optic signal processing can be explored by considering
multiplexing of signals by some form of wavelength division multiplexing. By transmitting the
entire frame of the ultra-fast signals at once, it becomes possible to process the signal without using
delays and by just using optical attenuators and OAs. However, there are many practicalities to be
overcome in realizing such as system and further works is required.
8.4
CONCLUDING REMARKS
The objective of the research presented in this chapter is to explore possible ways of realizing a 2-D
signal processing system using fiber-optic signal processing architecture. A general technique for
designing a 2-D filter is illustrated and numerous examples of utilization of the technique are given.
Although the discussion is focused on fiber-optic systems, the design procedure for 2-D filters are
just as applicable to any other signal processing architectures. For example, the 2-D filter order
reduction method given in Sections 8.1 and 8.21,2 can be used to simplify 2-D lightwaves systems
which may or may not be fiber-optic systems.
The design of 2-D filters is classified into two different classes. One class used matrix decomposition to reduce the design of 2-D filters into a set of 1-D filter design procedures. The other class used
direct extensions of 1-D filter design methods. It is found that neither has a distinctive superiority
over another and that the designer has to choose what is the best for the particular application,
most likely by designing both and comparing the performances. All the design procedures are
implemented using the MATLAB programming language.
Among the matrix decomposition methods, the multiple stage singular value decomposition method
of Section 8.236,37 performed the best whereas for direct methods, frequency sampling method produced
filters with smallest errors. However, the result should be taken with caution as there are many factors
to be considered before declaring one method superior over another. The differences between the various methods are outlined in Tables 8.6 and 8.9. A 2-D Filter order reduction method is applied to make
fiber- and integrated optic signal processing more feasible. The technology allows the filter designer to
produce filters of orders that are implementable in practice without sacrifices in performance.
Different possible filter structures are proposed and illustrated for photonic implementation of
2-D filters. Most of the filter structures discussed can be used in 1-D coherent fiber-optic signal
processing and are not limited to 2-D coherent fiber-optic signal processing. Some of the proposed
structures, such as transversal structure, are extremely efficient in the number of components used
to achieve a certain performance requirement. To make the efficient structures possible, the phase
shifting property of optical couplers when the incoming lightwaves is cross-coupled is utilized.
Filter structures of FIR and IIR types are also shown and examples are given. It is evident that
the fiber-optic signal processing technology presents a new direction in the usage of optical fiber,
lasers, and photonics technologies which are evolving very fast. In38,39,40,41,42,43 an incoherent signal
L. N. Binh, Multi-dimensional photonic processing: A discrete-domain approach: Part I, Int. J. Comput. Sci. Netw.
Security, 9(5), 97–109, 2009.
37 L. N. Binh, Multi-dimensional photonic processing: A discrete-domain approach: Part II, Int. J. Comput. Sci. Netw.
Security, 9(5), 110–123, 2009.
38 C. K. Marsden and J. H. Zhao, Optical Filter Analysis and Design: A Digital Signal Processing Approach, New York:
John Wiley, 1999.
39 J. Capmany, and J. Cascon, Optical programmable transversal filters using fiber amplifiers, Electron. Lett., 28, 1245–
1246, 1992.
40 W. Lu, H. Wang, and A. Antoniou, Design of two-dimensional digital filters using singular-value decomposition and
balanced approximation method, IEEE Trans. Signal Process., 39(10), 2253–2262, 1991.
41 L. N. Binh and D. Trowser, “OPTMASON”: A program for Automatic Derivation of the optical transfer functions of
photonic circuits from their connection graphs, Report No. MECSE-34-2004, 2004.
42 L. N. Binh, Photonic Signal Processing, Boca Raton, FL: CRC Press, 2009.
43 See for example, B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, Fiber-optic lattice signal processing, Proc. IEEE,
72(7), 1984.
36
404
Photonic Signal Processing
processing system operating at 100 MHz is demonstrated. The authors note that the raising this
capability to over 10 GHz is a relatively straightforward procedure involving shorter fiber lengths
and lasers and detectors with faster rise and fall time. They also note that conventional digital
signal processing and analog signal processing techniques are limited in their usefulness for signal
bandwidths exceeding one or two GHz. Current research efforts on fiber-optic signal processing
on lightwaves of the millimeter wavelength region will allow signal processing at bandwidths of
up to 100 GHz even to THz region if parametric amplification is employed. The field of 2-D signal
processing which requires ultra-fast processing capability has a great deal to gain from the usage
of the high-speed processing capability of fiber-optic architectures. In particular, with the fast
pace of research and inventions of photonic circuits reaching the nano-scale, employing photonic
crystal wave guiding techniques will allow multi-dimensional processing in the photonic domain
flourishing in the near future.
Furthermore, advanced modulation formats for 100 Gb/s such as quadrature amplitude
modulation (QAM), quadrature differential phase shift keying (QPSK) schemes are 2-D phase
coding on information signals. The proposed multi-dimension optical signal processing may offer
significant advantages for demodulation at the front of the optical receivers.
9
Generation and Photonic
Processing of Radio
Waves, Tera-Waves and
Multi-Carrier Lightwaves
By photonic processing of radio waves (RW), we express the techniques in which the processing
of RW can be implemented in the photonic domain. Thus a RW photonic processor is a subsystem
that can perform all necessary processing and manipulating functions as those of a radio wave
processing or transmission link, but also take advantage of the low loss, ultra-wide bandwidth,
re-configurability, and immunity to electromagnetic (EM) radiations. At the input and output stages,
radio waves, whose property can be of random manner, are transformed from the electronic to
photonic domain via an optical modulation scheme and vice versa. This chapter describes the main
principles of several PSP (photonic signal processing) systems that enable the processing of RW in
the frequency regions of radio wave, microwave, mm-wave or Tera-Hz waves in all-optical (photonic) domains. Generic structures of photonic signal processing of radio waves (PSPRW) are given
incorporating electronic to optical converter, photonic signal processing systems and then optical
to electronic conversion to obtain processed radio waves. Based on these structures, different types
of processing functions, such as filtering, correlating, and integrating, can be established based on
these photonic radio frequency configurations. Differential group delay (polarization maintaining
fibers) and polarization mode dispersion/rotating photonic elements (linear and nonlinear chirped
gratings) can be incorporated for performing several photonic processing functions, such as parallel
delayed signals for tapped delay as in discrete processing. The challenges and novelty of the proposed systems, processors, and generators of radio waves are identified and stated in this chapter.
This will allow for the setting of a research program and the planning of experimental works for this
emerging technological development.
9.1
INTRODUCTION
Over the last few years, the field of radio wave photonics has attracted considerable attention due
to the unique properties that photonic devices and systems bring to the generation, transportation,
processing, and detection of radio waves, millimeter wave, and emerging Tera-Hz wave (radio wave,
mm-wave, T-wave) signals. The term “radio waves” is used for generic purpose instead of the popular term “microwave” in microwave photonics. This term designates either microwave or RF, mmwave or Tera-Hz waves, which can be used interchangeably, especially when processing of weak
signals is under in electro-magnetic (EM) radiation noisy environments such as in radio astronomy,
radar signal identification etc. By PSPRW, we mean a photonic sub-system that can perform equivalent tasks to those of an ordinary microwave processor within a radio wave system or transmission
link, but taking advantages of the photonic domain of the low loss, ultra-high bandwidth and immunity to electro-magnetic radiation (EMR) and re-configurability.
Such processing of RW would commonly occur in the processing of random RW in which a specific frequency RW and its associated information are embedded. The processing of such signals is
difficult and bulky in an electrical domain.
The traditional approach toward RW signal processing is that an RW signal originated at an RF
(Radio Frequency) source, or came from an antenna, is fed to an RF circuit that performs the signal
405
406
Photonic Signal Processing
processing tasks either at the RF signal level or at an IF (intermediate frequency) band after a downconversion operation. In any case, the RF circuit is capable of performing the signal processing tasks
for which it has been designed only within a specified (often reduced) spectral band. However, this
approach results in poor flexibility since changing the band of the signals to be processed requires
the design of a novel RF circuit and the use of different hardware technologies. Furthermore, even
if the RF carrier is not changed, the nature of the modulating signal might require more bandwidth
or sampling speed from the processor. These drawbacks are often termed in the literature as the
electronic bottleneck.1,2,3,4 Also, the emerging Tera-Hz EM waves would require the signal processing functions performed in the photonic domain.
Thus, it is of very important that photonic signal processing be exploited for applications in
many areas, such as broadband wireless access networks, wireless sensor networks, radar and
satellite communication systems, and radio astronomical systems to overcome several bottleneck
problems facing in the electronic domain. The PSPRW has many advantages: has optical delay
lines with very low loss (independent of the RW signal frequency), provide very high-time bandwidth products, are immune to electromagnetic interference (EMI), are lightweight, can provide
very short delays resulting in very high-speed sampling frequencies (over 100 GHz in comparison with a few GHz with the available electronic technology), and finally, but not less important,
provides the possibility of spatial and wavelength parallelism using multi-wavelength multiplexing techniques. In particular, the PSPRW approach is of interest in radio over fiber systems, for
both channel rejection and channel selection applications.5 The ability to reject/select RW signals
directly in the optical domain, without OE (optical to electrical) and EO (electrical to optical)
conversions, is a unique characteristic of these photonic processors. Another example of their
application is in noise suppression and channel interference mitigation in the front-end stage after
the receiving antenna of a wireless base station prior to a highly selective surface acoustic waves
(SAW) filter.6
Having identified the role of photonic processing of radio waves, we need to address the generation of RW, in particular the Tera-Hz waves from the higher spectral region of lightwaves.
The generation of ultra-high speed photonic pulse trains is very important in the areas of ultrahigh bit rate optical communications, soliton science, and optical switching and optical ultra-fast
analog-to-digital conversion applications.7,8 Furthermore, if one can generate optical pulse trains
of a repetition of few tens of THz, then they can be employed to count the frequency oscillation
of lightwaves, provided that these pulse trains exhibit very low noise jitter. In these applications,
the timing jitter of pulsed signals can be a limiting factor for obtainable communication speed or
sampling rate. These generation techniques are described in the next section.
Section 9.3 then describes detailed photonic systems for processing of radio waves. Several
novel structures are designed incorporating techniques developed for mitigation of polarization
H. A. Haus, Mode-locking of lasers, IEEE J. Sel. Top. Quant. Electron., 6, 1173–1185, 2000.
W. J. Lai, P. Shum, and L.N. Binh, Stability and transient analyses of temporal talbot effect-based repetition-rate multiplication mode-locked laser systems, IEEE Photonics Tech. Lett., 16, 437–439, 2004.
3 G. R. Lin, Y. C. Chang, and J. R. Wu, Rational harmonic mode-locking of erbium-doped fiber laser at 40 GHz using a
loss-modulated fabry-perot laser diode, IEEE Photonics Tech. Lett., 16, 1810–1812, 2004.
4 K. K. Gupta, N. Onodera, K. S. Abedin, and M. Hyodo, Pulse repetition frequency multiplication via intracavity optical
filtering in AM mode-locked fiber ring lasers, IEEE Photonics Tech. Lett., 14, 284–286, 2002.
5 J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P-O. Hedekvist, Fibre-based optical parametric amplifiers and their
amplications, IEEE J. Sel. Top. Quantum Electron., 8, 506–519, 2002.
6 W. J. Lai, P. Shum, L. N. Binh, and M. Karlsson, Phase plane analysis of rational harmonic mode-locking in an erbium
doped fiber ring laser, IEEE J. Quantum Electron., Paper No. JQE-130067-2004.R1, 2004.
7 L. N. Binh, Ultra-stable fibre mode-locked lasers at 10 and 40 GHz repetition rate, Presented at Proceedings of the
Photonics West, San Jose, CA, 2003.
8 L. N. Binh, and Chang, K-F., A platform for multi-channel transmission systems frequency domain using Volterra series
transfer function, Presented at Proceedings of the First International Conferenceof the Optical Communication and
Networks, Singapore, 2002.
1
2
Generation and Photonic Processing of Radio Waves
407
mode dispersion (PMD) in long-haul fiber optic communications. The incorporation of the
differential delays of the polarized mode will overcome the coherence problems in photonic
processors. The employment of optical buffers allows the temporal shift and, hence, the tenability of the passband of the RW filters. They will be described in this chapter. Finally, the
novelty and challenges of the RW photonics implementation and characterization are stated.
9.2
GENERATION OF TERA-HZ WAVES
Recently, actively mode-locked erbium-doped fiber lasers (EDFL) have been demonstrated to be
able to generate stable, short pulses at 40 GHz and lower.9 This kind of device is particularly attractive not only because of the all-fiber approach, but also because of its compactness and environmental stability. In an actively mode-locked laser, the pulse-to-pulse jitter is ultimately limited by the
phase noise of the mode-locking microwave source (typically electronic frequency synthesizers).
On the other hand, microwave photonic oscillators using fiber delay lines have been shown capable
of generating microwave frequency with extremely low phase noise. Therefore, one can take advantage of the low phase noise of a microwave driven optical modulator in association with the nonlinear parametric conversion in the fiber ring for ultra-low jitter pulse generation. In this chapter, we
will describe the development of a new low-jitter pulse generator by combing the two technologies.
In this approach, the optical oscillator (the mode-locked EDFL) and the microwave oscillator can be
coupled through a common Mach−Zehnder (MZ) optical modulator as an integrated pulse generator via the atomic parametric conversion processes, named the all-photonic atomic clock.
The novel technique proposed in this application is the generation of repetitive pulse sequences
from the locking into one of the longitudinal mode of a ring laser whether it is fiber-based, fiber, or
semiconductor-based structures.10,11,12,13,14 A fundamental schematic diagram of mode-locked lasers
is shown in Figure 9.1, which consists of a nonlinear waveguide section, an amplifying device to
compensate for the energy loss, a tuning section to generate the locking condition of the lightwave
energy to a particular harmonic, an input and output coupling section for tapping the laser source,
and an optical modulator to generate the repetition rate in association with the locking mechanism.
All these optical components are interconnected by a ring of single-mode optical fiber. The length
of the optical waveguide is used to determine the frequency spacing between adjacent longitudinal
modes of the ring laser.
The generated pulse repetition frequency by the above convention technique is often limited
by the operating frequency of the optical modulator incorporated within the ring resonator. Some
techniques have been proposed to increase the repetition rate of the mode-locked system using
temporal Talbot effects (temporal diffraction15), rational harmonic detuning [3], optical division
multiplexing, [4] and others. This section proposes novel techniques for the generation of extremely
K. S. J. Nishizawa, Widely frequency-tunable terahertz wave generation and spectroscopic application, Int. J. Infrared
Millim. Waves, 26(7), 937–952, 2005.
10 W. J. Lai, P. Shum, L. N. Binh, and M. Karlsson, Phase plane analysis of rational harmonic mode-locking in an erbium
doped fiber ring laser, IEEE J. Quantum Electron., Paper No. JQE-130067-2004.R1, 2004.
11 L. N. Binh, Ultra-stable fibre mode-locked lasers at 10 GHz and 40 GHz repetition rate, Presented at Proceedings of the
Photonics West, San Jose, CA, 2003.
12 L. N. Binh, and Chang, K-F., A platform for multi-channel transmission systems frequency domain using Volterra series
transfer function, Presented at Proceedings of the First International Conference on Optical Communications and
Networks, Singapore, 2002.
13 H. Q. Lam, and L. N. Binh, Generation and propagation of pulses in mode-locked fiber lasers using finite-difference
time-difference method, Clayton, Melbourne: Monash University, 2004.
14 H. Q. Lam, L. N. Binh, J. Q. N. Ngo, and P. Shum, Locking of ultra-high peak pulses in harmonically mode-locked
Erbium-doped fiber ring lasers, Opt. Eng., 44(12), 124201, 2005.
15 W. J. Lai, P. Shum, L. N. Binh, and M. Karlsson, Phase-plane analysis of rational harmonic mode-locking in an erbiumdoped fiber ring laser; IEEE J. Quantum Electron., 41(3), 426–433, 2005.
9
408
Photonic Signal Processing
FIGURE 9.1 Schematic diagram of a generic mode-locked laser for generation of multi-GHz repetition rate
pulse sequence. Note the optical modulator is incorporated in the ring → the repetition rate is limited by the
bandwidth of the modulator.
high repetition rate pulse sequence in the THz range by using the nonlinear parametric amplification and the degenerate four-wave mixing (FWM) phenomenon in a special optical waveguide and
the mechanism of rational frequency detuning as well as the interference between the modulated
pump signals via an optical modulator.
9.2.1
generatiOn OF ultra-High repetitiOn rate pulse trains
The novelty of our system is the use of parametric amplification in a section of special optical
waveguide, which can be a highly nonlinear dispersion of flattened and shifted fibers [5] or an
integrated optical waveguide. Parametric amplification is achieved by mixing four waves at three
different frequencies based on the nonlinear intensity-dependent change of the waveguide refractive index.
Under a perfect phase matching condition, a signal at the frequency fs and a pump source at f p
will mix and modulate the refractive index of the nonlinear optical waveguide section in such a way
that a third lightwave, also at f p, will create the sidebands at f p ± ( fs − f p ) and result in signal gain
and the generation of an idler.
An active ring laser, whose repetition frequency is generated by locking at a harmonic, is determined by the modulation frequency of the modulator. The optical modulator is shown in both
Figures 9.1 and 9.2. The modulation frequency fm is fm = qfc with fc as the fundamental resonant
frequency of the waveguide ring determined by the length of the laser cavity (the ring length), and
q is the order of the harmonics of the cavity longitudinal modes.
By applying a small deviation of the modulation frequency ∆f = f c /N (N = a large integer), the
modulation frequency becomes fm = qfc + fc / N . This leads to an N-times increase of the repetition
rate with the repetition rate, expressed as fR = Nfm.
In conventional high-repetition rate mode-locked lasers,16 the modulation of the circulating
lightwave for mode locking is implemented by an optical Mach–Zehnder modulator (MZM) type
placed within the cavity. In our proposed system, we will modulate the pump signal for mode
16
L. N. Binh and N. Quoc Ngo, Ultra-Fast Fiber Lasers: Principles and Applications with MATLAB® Models, Boca
Raton, FL: CRC Press, 2010.
Generation and Photonic Processing of Radio Waves
409
FIGURE 9.2 Proposed mode-locked laser using parametric amplification for generation of sidebands and
hence Tera-Hz repetitive pulse sequence.
locking. This locking, in association with the four-wave-mixing effect (or four photon interaction)
under a phase matching condition, will transfer a low modulation rate into an extremely high
repetition rate by the detuning of the locking frequency. Figure 9.2 shows the schematic diagram
of the proposed laser system and details of the experimental set-up as shown in the next section.
The modulation of the pump source can be implemented via a step recovery diode (SRD), which
would produce a very short pulse and low repetition rate to modulate the amplitude of the pump
lightwave via the optical modulator of a moderately wide bandwidth. This is the principal novelty
of our scheme. That is how the bandwidth limitation of the modulator in conventional modelocked lasers can be overcome. Hence, it is possible to push the repetition rate to the extremely
high region—in the few tens of THz range. An optical filter is placed at the output of the parametric amplifier to filter the generated idler wave from the four-wave mixing process. This generation
scheme was the world first at the time.
9.2.2
necessity OF highly nOnlinear Optical waveguiDe sectiOn FOr tera-hz
wave ultra-High speeD mODulatiOn
Parametric amplification is achieved by the interaction of the four photons of the signals and
the pump waves. The non-degenerative process starts with two photonic waves at a different
frequency that co-propagate through the guided waveguide. As they propagate, they are beating
with each other. The intensity modulated group wave at the different frequency will modulate the
refractive index (RI) of the guided medium via the self-phase modulation (SPM) effect (e.g., the
Kerr’s effect).
When a third wave is injected, it will become phase modulated at this beating frequency due to
the modulated RI. There will then be a phase-modulation to amplitude conversion, and sidebands
would develop with amplitudes proportional to the signal amplitude. This happens for both the signal and the pump lightwaves. This process will enhance the gain of the signals, hence parametric
amplification. This process is not possible if there is no nonlinear change of the refractive index of
the guided medium. Therefore a highly nonlinear optical waveguide, either in form of an optical
fiber or an integrated photonic waveguide having highly nonlinear core materials such as polymers
or doped silica or diffused LiNbO3 will be employed. The optical waveguide must have zero dispersion characteristics at the signal frequency for phase matching.
410
Photonic Signal Processing
In the initial stage of this proposed project we will use highly nonlinear optical fibers with a
nonlinear coefficient of around 10 (W.km)−1 with a zero dispersion in the region 1540–1550 nm and
a dispersion slope of about 0.035 ps/(nm)2/km. These fibers are commercially available.
9.3
PHOTONIC SIGNAL PROCESSING OF RADIO WAVES
In general, a network analyzer in an electrical domain is used to generate radio frequency signals,
which are then fed into an optical modulator, an electro-optic modulator (EOM). A light source
is coupled to the optical modulator. Thus, the RW waves are converted to the optical domain in
which the processing or filtering is done in the optical domain. Thence, the conversion from the
optical to electrical domain is implemented via the photodetector (PD) and electronic amplifier
in cascade with an automatic gain control (AGC) or limiting amplifier. We assume that the bandwidth of all components are wide enough and the system is operating in the mid-band region.
The ultimate aim of an RW PSP system is to identify the unknown randomness nature of the RW
and, thus, clarify the signatures of the RW, which is commonly known as a possible “enemy”
frequency to be identified.
Any processors implemented using discrete time-delays for PSPRW17,18,19,20,21,22 can provide a
system function for processing the RW signal. A PSP system in which two nodes, ingress and
egress, can be represented are related by a transfer function H ( z )23 given by
n
∑a z
r
H ( z) =
r =0
−r
m


1 −
bj z− j 


j =0


∑
(9.1)
where z = e jωT = e j β L is the z-transform with T is the sampling period, β is the propagation
constant of the lightwaves in guided media, and L is the propagation distance corresponding to
the sampling period. For example, a ring resonator, whose loop length is L, acts as a sampling
of optical signals when circulating around the loop. It is noted here again that photonic signal
processing is analog in nature as there is not optical sampling and digitalization here. However,
the digital processing technique can be applied with the sampling defined as the delay of the
lightwaves propagating through a length of optical waveguides, such as that in a loop resonance length or that of a different length of an interferometer. The transfer function exhibits the property of the processing in the optical domain, and translates to electrical domain
when the optical to electrical conversion process is performed. Thus, the optical domain can
process extremely high frequency via the conversion and processed by optical components.
Thus z −1 represents the basic time delay between samples, and ar , b j are the filter coefficients
K. Wilner and A. P. V. den Heuvel, Fiber-optic delay lines for microwave signal processing, Proc. IEEE, 64(5), 805–807,
1976.
18 K. Jackson, S. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, and H. J. Shaw, Optical fiber delay-line signal
processing, IEEE Trans. Microw. Theory Tech., MTT-33(3), 193–204, 1985.
19 M. Tur, J. W. Goodman, B. Moslehi, J. E. Bowers, and H. J. Shaw, Fiber-optic signal processor with applications to
matrix-vector multiplication and lattice filtering, Opt. Lett., 7(9), 463–465, 1982.
20 S. A. Newton, R. S. Howland, K. P. Jackson, and H. J. Shaw, High speed pulse train generation using single-mode fiber
recirculating delay lines, Electron. Lett., 19, 756–758, 1983.
21 B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, Fiber-optic lattice signal processing, Proc. IEEE, 72(7), 909–930, 1984.
22 R. A. Minasian, Photonic signal processing of high-speed signals using fiber gratings, Opt. Fiber Technol., 91–108, 2000.
23 L. N. Binh, Photonic Signal Processing: Principles and Applications, Boca Raton, FL: CRC Press, 2007.
17
Generation and Photonic Processing of Radio Waves
411
that are implemented by optical components. If bk = 0 for all k, the filter is non-recursive and
is also known as transversal filter. For this case, the transfer function has periodic spectral
characteristics; the frequency period is known as free spectral range (FSR), which is inversely
proportional to the basic time delay T.
The simplest technique for realizing photonic filters is based on using a single laser and a single photodetector with multiple optical delay paths [23,19,24,25,26,27]. As shown in Figure 9.3, multiple
optical signal components carrying the RW signal are mixed at the detector end of the structure.
In the detection process, the different taps (or optical contributions) can be mixed according to
two different regimes. One is the coherent regime, which applies when a single optical source is
employed, and its “coherence time” (τ c ) is much longer than the time delay T between adjacent
samples or taps of the optical filter. Under a coherent regime operation, any slight change in the
propagation characteristics of any part of the optical processor (length of any delay line, refractive
index changes due to environment, and so on) drastically affects the filter response. In comparison, in the incoherent regime, the optical phase relationship between the filter taps is completely
random due to the limited source coherence time. For this to happen when a single optical source
is employed, τ c << T and the optical power at the PD input is the sum of the optical powers of the
filter samples. In this case, the filter structure is free from environmental effects and, thus, very
stable and its performance is quite repeatable.
For incoherent processing, the coherence length (thence time) of the light source is assumed
smaller than the minimum delay time of the processor. There are two problems with this approach:
(i) First, normal narrow-linewidth telecommunications lasers cannot be used, in general; and (ii),
more important, the incoherent approach produces an excessive amount of phase noise at the output
due to the optical interference of the delayed optical signals.
FIGURE 9.3 A classic photonic sign