Air Trenches for Dense Silica Integrated Optics
by
Milos Popovid
Submitted to the Department of Electrical Engineering and Computer Science
in partial fulfillment of the requirements for the degree of
Master of Science in Electrical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2002
@ Massachusetts
Institute of Technology 2002. All rights reserved.
Author..........
Department/of Electrical Engineering and Computer Science
February 4, 2001
Certified by.............
Hermann A. Haus
Institute Professor Emeritus
Thesis Supervisor
Accepted by..............
-
Arthur C. Smith
Chairman, Department Committee on Graduate Students
11;
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2
Air Trenches for Dense Silica Integrated Optics
by
Milo§ Popovid
Submitted to the Department of Electrical Engineering and Computer Science
on February 4, 2001, in partial fulfillment of the
requirements for the degree of
Master of Science in Electrical Engineering
Abstract
Air trench structures for reduced-size bends in low index contrast (e.g. silica) waveguides are
proposed. The proposed air trench bends (ATBs) improve on previous structures employing
trenches for sharp bending by the introduction of a new "cladding taper" which eliminates
junction loss between the low index contrast and air trench regions, and allows a smaller
radius (5-15pm) limited by bending loss to be used. This bend geometry is general in the
size-for-loss tradeoff, and offers an effective bend radius reduction dictated by the newly
introduced low index material in the trench; a reduction by a factor of 30-60 in effective
radius is predicted for silica integrated optics (20-250p-m in size). Design considerations
for bends and cladding tapers are presented. Bending, junction and substrate losses are
investigated. 2D FDTD/EIM simulations of complete ATBs in representative silica index
contrasts are presented. For proper account of substrate loss in the third dimension, an
air trench waveguide cross-sectional geometry is investigated that allows for arbitrarily low
substrate loss to be achieved at the expense of a deeper trench. The required trench depth,
for an acceptable substrate loss, is calculated. A simple, compact waveguide T-splitter using
ATBs is presented.
Thesis Supervisor: Hermann A. Haus
Title: Institute Professor Emeritus
3
4
Acknowledgments
I would like to thank Profs. H.A. Haus and E.P. Ippen for their mentorship over the course
of this thesis work, and all of my colleagues for making it a continuing thrill and challenge
to work in their midst.
In the context of the work presented in this thesis, I owe a debt of gratitude to my collaborators on the air trench bend project, Shoji Akiyama, Dr. Jirgen Michel and particularly
Dr. Kazumi Wada, whose suggestions initiated the project.
I am also grateful to my (former and current) officemates Christina Manolatou and
M. Jalal Khan, who were happy to bring me up to speed when I began my graduate
research. Christina's well-written FDTD simulation code, used throughout this thesis, has
saved me countless hours of programming and allowed me to concentrate on the project.
Jalal and I have spent hours debating all from waveguiding to world politics, sometimes in
an amusing combination but always worth the time. I would like to thank all of my other
colleagues (current and former) in the optics group for their part in introducing me to this
or their research when I first joined this research group, and for offering their advice when
needed. This includes Mike Watts, Dan Ripin, John Fini, Matt Grein, Leaf Jiang, Juliet
Gopinath, Pat Chou, Hanfei Shen, Jason Sickler, Laura Tiefenbruck, Pete Rakich, Aaron
Aguirre, Charles Yu and Poh-Boon Phua. I would also like to thank Tom Murphy for kindly
offering his advice to me, and for letting me use some of his code for dual boundary bends.
I have my sister and parents to thank for helping me get this far, and my wife Katherine
for patiently putting up with my late hours over the past two months of writing this thesis.
It's lucky this thesis is finished before her birthday.
Finally, I am very grateful for the support of an MIT Presidential Fellowship, a National
Science and Engineering Research Council (NSERC) of Canada Scholarship, and our grant
on the NPACI Cray T-90 supercomputer at SDSC for all FDTD simulations in this thesis.
5
6
Dedicated to my super grandparents in Yugoslavia, Kosara and Tomislav Jelenkovi6, and
Miroslava Risti.
Posvedeno mojim bakama i deki u Jugoslaviji, Kosari i Tomislavu Jelenkovi, i
Miroslavi Risti6.
7
8
9
Contents
1
2
Introduction: Integration of Optical Devices in Silica
15
. . . . . . . . . . . . . . . . . . . . . . .
16
. . . .
18
1.1
Silica in Integrated Optics [42],[28]
1.2
Previous Work on Dense Integration, Sharp Bends and Air Trenches
Analytic Tools for Planar Waveguide Structures
21
2.1
Modal Expansion of Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.2
Mode Solvers and FDTD
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2.1
Electromagnetic Theory on a Lattice [9],[74] . . . . . . . . . . . . . .
25
2.2.2
Finite-Difference Time-Domain (FDTD) [74],[63]
. . . . . . . . . . .
27
2.2.3
Vectorial 2D Cross-section Waveguide Mode Solver [51]
2.3
2.4
2.5
2.6
. . . . . . .
28
Effective Index Method [22],[65] . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.3.1
Standard Effective Index Method [22]
. . . . . . . . . . . . . . . . .
31
2.3.2
Perturbation-corrected Effective Index Method [11] . . . . . . . . . .
32
. . . . . . . . . . . . . .
34
2.4.1
Junction Scattering via Mode Expansion and Transfer Matrices . . .
35
2.4.2
Straight-Straight Waveguide Junction Loss
. . . . . . . . . . . . . .
37
2.4.3
Straight-Bent Waveguide Junction Loss [47] . . . . . . . . . . . . . .
38
Bend Loss in Step-Index Slab Waveguides . . . . . . . . . . . . . . . . . . .
40
2.5.1
Analytic Methods for Single and Dual Boundary Bends [47],[18],[16],[53]
41
2.5.2
Qualitative Dependencies of Bend Loss and the Leaky Mode Field
.
47
2.5.3
Optimal Design of Bends Terminating in Straight Waveguides . . . .
52
2.5.4
Optimal (Non-circular) Terminating Bend Geometry . . . . . . . . .
56
Junction Loss at Waveguide Interfaces [65],[47],[5]
Substrate Loss in Straight Leaky Waveguides
2.6.1
. . . . . . . . . . . . . . . . .
Substrate Loss of Layered Slab (1D) Waveguides
. . . . . . . . . . .
58
59
CONTENTS
10
2.6.2
Substrate Loss of 2D Cross-section Waveguides by the Equivalent
Source Method ........
3
67
Air Trench Bends: Design and Simulations
73
3.1
Limiting Factors of Integration Density in Silica . . . . . . . . . . . . . . . .
74
3.1.1
Index Contrast and Bend Loss in Integration Density
74
3.1.2
Scattering Loss and Fiber-to-Chip Coupling Limitations .......
78
3.2
Silica Waveguides and Optical Circuit Layout . . . . . . . . . . . . . . . . .
79
3.3
Air Trench Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
3.3.1
Cross-sectional Geometry for Strong Lateral Mode Confinement
. .
81
3.3.2
Bends in Air Trench Waveguides . . . . . . . . . . . . . . . . . . . .
85
3.3.3
The Cladding Taper: A Low-Loss Interface to the Silica Waveguide.
92
3.4
4
..............................
. . . . . . . .
Air Trench Bends for Silica PLCs . . . . . . . . . . . . . . . . . . . . . . . .
101
3.4.1
ATB Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
3.4.2
Numerical Simulation of ATBs . . . . . . . . . . . . . . . . . . . . .
103
3.4.3
Loss Mechanisms: Substrate and Scattering Loss . . . . . . . . . . .
114
3.4.4
Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
Beyond Air Trench Bends
121
4.1
Waveguide T-splitter using Two Air Trench Bends . . . . . . . . . . . . . .
121
4.2
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122
4.2.1
124
Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Reflection of Complex Waves at a Planar Boundary
127
11
List of Figures
1-1
Integrated optical components for which bend radius is an important parameter 17
1-2
Techniques for low-loss sharp bends on low and high index contrast platforms
2-1
Grid/field definitions for a discrete electromagnetic theory on a lattice (FDTD) 27
2-2
Electric field distribution of air trench waveguide quasi-TE and quasi-TM
19
fundamental modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
. . . . . .
31
2-3
Effective Index Method for buried, rib and air trench waveguides
2-4
Field distributions comparison from (2D) Effective Index Method and (3D)
vectorial mode solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2-5
Scattering at a junction: dielectric stacks; straight and bent slab interfaces
34
2-6
Single and dual boundary waveguide bend geometry and the radiation caustic 40
2-7
Index profile of a bend and the equivalent straight waveguide (conformal
m apping)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2-8
WKB and Airy function field distribution solutions for leaky mode of bend
45
2-9
FDTD simulations of single and dual boundary air-clad bends . . . . . . . .
49
. . . . . . . . . . . .
51
2-11 Single boundary bend loss vs. index contrast and radius . . . . . . . . . . .
53
2-12 Generic dual boundary bend dimensions; illustrated total loss optimization
54
2-13 Schematic of optimal bend geometry progression over a range of loss values
55
2-14 Leaky mode diagram of a layered ID waveguide . . . . . . . . . . . . . . . .
60
2-15 Substrate loss of layered (ID) leaky waveguide
. . . . . . . . . . . . . . . .
63
2-16 Equivalent current sheet configuration for layered 1D waveguide . . . . . . .
64
2-17 Equivalent current sheet configuration for air trench (2D) waveguide . . . .
68
2-18 Substrate loss results for straight, truncated air trench (2D) leaky waveguide
72
2-10 Leaky modes of single and dual boundary silica bends
12
LIST OF FIGURES
3-1
Crosstalk and bend loss effect on integration density: example circuit layout
76
3-2
Waveguide crosstalk and bend loss bounds on integration density . . . . . .
77
3-3
Polarization-independent integrated optical circuit topology . . . . . . . . .
81
3-4
Air trench waveguide cross-sectional geometry . . . . . . . . . . . . . . . . .
82
3-5
Air trench waveguide mode (dominant E-field component) contour plots for
exam ples A-C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
3-6
Idealized (infinite) and actual (finite-depth) air trench waveguide . . . . . .
85
3-7
Bend loss and whispering gallery mode width in the air trench waveguide
87
3-8
Straight ATW-straight silica waveguide junction loss vs. index contrast
3-9
FDTD simulations of dual boundary, finite-angle ATW bends . . . . . . . .
.
89
92
3-10 Conventional tapers and the problem of interfacing ATWs to silica waveguides 93
3-11 Cladding taper geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
3-12 Ray-optical estimate of required cladding taper angle for low loss . . . . . .
97
3-13 Guided power confinement of fundamental TE mode in slab waveguide . . .
99
3-14 Search algorithm for finding the optimal cladding taper, and example data.
100
3-15 FDTD electric field amplitude plots of cladding taper designs for case study
exam ples A-C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-16 Schematic of Air Trench Bend geometry and free parameters
3-17 Definition of the Total Size Box for silica and air trench bends
101
. . . . . . . .
103
. . . . . . .
104
3-18 FDTD field plots of TE mode, CW excited structures of examples C and B
108
3-19 Transmission and reflection spectra of ATB examples A-C . . . . . . . . . .
109
3-20 FDTD field plots of TE mode, CW excited structure of example A . . . . .
111
3-21 FDTD field plots of TE mode, CW excited ATBs without cladding tapers .
113
3-22 Fabrication process steps for Air Trench Bends . . . . . . . . . . . . . . . .
116
3-23 SEM of photoresist mask of ATB T-splitter trench on a plain Si wafer . . .
117
3-24 SEMs of two ATB waveguide and air trench masks and corresponding FDTD
sim ulations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
119
4-1
FDTD field plot and transmission/reflection spectra of air trench T-splitter
123
4-2
SEM of ATB-based waveguide T-splitter masks . . . . . . . . . . . . . . . .
124
A-1
Complex wave reflection at a planar boundary
128
. . . . . . . . . . . . . . . .
13
List of Tables
2.1
Effective Index Method and Vectorial Mode Solver Propagation Constants
.
33
2.2
Leakage Loss (dB/Mm) of ID Waveguides by Analytic and ECS Method .
.
62
3.1
Single-mode Silica Waveguide Square Core Width and Eff. Indices for Ex.
A-C
. . . . . .. .......
...........
.............
. ....
.
80
3.2
Air Trench Waveguide Core Width and Eff. Indices for Examples A-C . . .
83
3.3
Cladding Taper Dimensions and Loss for Examples A-C . . . . . . . . . . .
102
3.4
Air Trench Bend Design Dimensions for Ex. A-C . . . . . . . . . . . . . . .
105
3.5
Air Trench and Simple Silica Bend Sizes for Ex. A-C . . . . . . . . . . . . .
106
3.6
Air Trench Bend Loss Budget for Ex. A-C . . . . . . . . . . . . . . . . . . .
110
3.7
Air Trench Depth for 0.01dB Substrate Loss in Ex. A-C . . . . . . . . . . .
115
14
15
Chapter 1
Introduction: Integration of
Optical Devices in Silica
This thesis is concerned with the problem of increasing the integration density of silica
technology integrated optics. The basic principle investigated is the careful engineering of
etched air trenches introduced into the silica optical device geometry, to provide strong light
confinement (index contrast) for sharp bends with low loss and low reflection.
Integrated optics is a developing technology intended to drastically reduce the cost and
increase functionality of components for optical communication networks [52].
For wave-
length division multiplexed (WDM) optical networks, such components are optical crossconnect switches, wavelength multiplexers/demultiplexers, individual channel add/drop filters, and the associated lasers and photodetectors to interface the optical layer to the
endpoint electronics. A comprehensive treatment of optical (fiber) networks, and of the
potential role of integrated optics, is given in the book by Ramaswami and Sivarajan [52].
Integrated optics has its origins in 1969, in a series of papers by the AT&T Bell Labs
technical staff ([41],[33],[15],[32],[34]), proposing a "miniature form of laser beam circuitry"
[41]. In these papers, already the major issues in integrated optics were addressed: potential optical circuit geometries [41], modes of dielectric optical waveguides [15], coupling of
parallel waveguides in proximity [33], and radiation loss in curved waveguides [32],[34].
The last of these, bending loss, has been an area of active research over the last 30 years,
and is a major limiting factor in the achievable density of integration in low index contrast
(e.g. silica) integrated optics [28].
Introduction: Integration of Optical Devices in Silica
16
The sections that follow describe the problem being addressed in this thesis and the
previous work on the subject. Chapter 2 reviews the analysis of planar waveguide structures
in the context of the work in this thesis, and discusses junction, bend, and substrate leakage
losses of planar waveguides. In Chapter 3, the subject of this thesis - the Air Trench Bend
with cladding tapers - will be introduced, the design and simulation results presented,
along with SEM (scanning electromicrograph) images of some preliminary fabrication steps.
Chapter 4 briefly describes more complex structures based on the Air Trench Bend such as
a waveguide T-splitter, and discusses future directions for this work.
1.1
Silica in Integrated Optics [42],[28]
Since 1969, the field of integrated optics has expanded to include many major research areas,
including fixed and frequency-tunable integrated lasers, modulators, (semiconductor optical) amplifiers, wavelength-selective optical filters and multiplexers, and optical switches.
The platforms for integrated optics, in addition to silica which translated directly from
optical fibers, have grown to include electronically-active semiconductor materials such as
GaAs and InP. The range of refractive index contrasts in these platforms ranges from low
(e.g. silica) to very high (e.g. silicon-on-insulator, SOI), the latter offering extremely high
density of integration [30].
In this thesis, silica technology is considered. A review is given in [28],[42]. Low index
contrast silica bench technology - referred to as PLC (planar lightwave circuit) or SiOB
(silicon optical bench) - has gained widespread use in practice in the fabrication of passive
integrated optical components, by virtue of its use of well-tested IC industry manufacturing
processes and technology [42]. Large silica waveguide cross-sections offer low fiber-to-chip
coupling and propagation losses. A major drawback of SiOB technology is the relatively
large component size, where a critical factor is the minimum waveguide bending radius.
This radius is large - normally in the millimeters - in the low index contrasts (A = 0.25%1.5%) found in silica [42]. Low density of integration keeps production cost high and invites
yield problems.
In addition to bends required to route waveguide paths between components on a densely
integrated optical circuit, Fig. 1-1 shows several basic components of planar integrated
optical circuits whose size critically depends on the bend radius: a directional coupler-based
17
1.1 Silica in Integrated Optics [42],[28]
(a)
(b)
(c)
Figure 1-1: Some integrated optical circuit components for which bending radius is an important
factor for integration density:
(a) Mach-Zehnder interferometer from two cascaded
directional couplers, (b) second-order ring resonator channel add/drop filter, and (c)
the Arrayed Waveguide Grating multiplexer/demultiplexer.
Mach-Zehnder interferometer, a ring resonator channel dropping filter, and an Arrayed
Waveguide Grating (AWG) - an integrated wavelength multiplexer/demultiplexer based on
wave interference. Proposed in 1988 by Smit [56], the AWG, also known as the Waveguide
Grating Router (WGR) and Phased Array Grating (PHASAR), has become one of the most
practical and promising integrated optical components [67],[64],[13]. Primarily implemented
in silica, it is commercially available from, among others, Lucent Technologies, Hitachi and
NT&T/PIRI. One of the limiting factors in the size of the AWG is the minimum bend
radius of the "grating arm" waveguides. Keeping the bend loss within acceptable bounds
can limit the bend radius to over 5mm, while junction loss that results at straight-bent
waveguide interfaces can further push the minimum bend radius to over 10-15mm [8].
A technology that allows a drastic reduction in the bending radius would overcome one
of silica's major obstacles to attaining truly large-scale optical integration. We propose
a scheme using air trenches to provide locally enhanced lateral mode confinement.
Air
trenches have been investigated in the past in the context of bend loss (e.g. [73]), and have
been used to make compact bends in ridge waveguides (referred to as deep etching in that
context, [59],[47]).
For low index contrast buried waveguides (e.g. silica), large junction
losses occur at the interfaces between the straight silica waveguide and the air trench. We
introduce a new "cladding taper" to remove abrupt junction-induced mode mismatch and
Fresnel reflection in order to miniaturize optical waveguide bends while preserving low-loss
performance [49].
At the time of writing of this thesis, new results were published by den Besten et al.
on a low-loss (<4dB), compact (1mm x 1mm) 8x8 AWG in indium phosphide (InP) ridge
Introduction: Integration of Optical Devices in Silica
18
waveguides, by the use of deep etching [12]. This is precisely the type of application aimed
at in buried silica waveguides with the work in this thesis.
1.2
Previous Work on Dense Integration, Sharp Bends and
Air Trenches
Prior to delving into the design of small Air Trench Bends for buried silica waveguides, we
review the previous work on dense integration, sharp bends and air trenches.
While silica PLCs in general have low integration density with component sizes on the
order of centimeters [42], high index contrast platforms such as silicon-on-insulator (SOI)
offer very dense integration. Sharp, low loss 90 bends, on the order of a wavelength in
radius, have been proposed for SOI [31], shown in Fig. 1-2d.
Their operation is based
on a low-Q resonant cavity, but also relies on bend-guiding and corner mirror reflection
(like Fig. 1-2b). A detailed description of ultra-compact components in high index contrast
is given in [30].
However, high index contrast platforms pose challenges of fiber-to-chip
coupling loss due to mode shape mismatch and misalignment because waveguides are small
(e.g. -0.2pm).
They also have high scattering loss and sensitivity to other fabrication
defects and tolerances, and pose fabrication processing challenges. The case is similar with
high contrast, forward looking platforms like photonic crystal waveguides [40] (Fig. 1-2e).
In low index contrast, integration density is limited by, among other factors, the minimum bend radius for acceptable loss. An extensive literature exists on the analysis of bend
loss ([32],[18],[47],[16],[53]).
In bends which terminate in straight waveguides, there are ad-
ditional contributions to total loss from scattering at the straight-bent waveguide interfaces,
due to mode mismatch. To reduce this "junction loss", a lateral offset between the straight
and bent waveguides was proposed by Neumann [45], to align the laterally shifted modes
of the straight and bent waveguides. Pennings showed that, in addition, the widths of the
straight and bent waveguides should be unequal for optimum mode matching [47].
Numerous techniques for the reduction of loss in bent waveguides of small bend radii
have been proposed. In 1986, Korotky et al. proposed a "crowning" technique which formed
increased-index triangular prisms in the core of a LiNbO 3 diffused waveguide bend, in order
to steer light and reduce bend loss at small radii [24] (Fig. 1-2a). They achieved a three-fold
reduction of radius (to 5.5mm) in comparison to standard LiNbO 3 bends, with a loss of
1.2 Previous Work on Dense Integration, Sharp Bends and Air Trenches
19
(b)
(a)
(c)
(d)
(e)
Figure 1-2: Techniques for low-loss sharp bends in: low index contrast - (a) "crowning" in LiNbO 3
diffused waveguides [24], (b) corner mirrors [2],[59], (c) doubly-etched bends for ridge
waveguides (with waveguide cross-sections before and in the bend); and high index
contrast - simulations of a (d) low-Q resonant cavity bend for SOI waveguides [31],
and (e) 2D photonic crystal waveguide bend [40], both with high transmission efficiency.
~0.1dB.
More recently, several techniques for the better quality, lithographically etched waveguides have been proposed.
One of the first was the use of etched-wall corner mirrors,
proposed by Benson [2] (Fig. 1-2b). Such corners have not been able to yield losses much
lower than -1dB
[59], so attention turned back to the design of efficient curved waveguides.
The introduction of an asymmetric cladding or air trench into bent waveguides has been
studied by numerous authors (e.g. [73]) as a method to reduce bend loss by artificially
increasing the index contrast at the outer radius. Pennings described an extension of the
air trench to ridge waveguides [47]. Referred to as "doubly etched" (or deep-etched) bends,
these were fabricated by Spiekman et al. [59] achieving compact radii (30pm) with relatively low loss (0.6dB total loss, 0.2dB bend loss only) in InP/InGaAsP ridge waveguides.
Introduction: Integration of Optical Devices in Silica
20
Results on an Arrayed Waveguide Grating mux/demux using such ridge waveguide bends
in InP were recently published [12].
Although bend radius can be reduced, a significant loss in all air trench-aided bends
is the junction loss between the low index contrast waveguide and the air trench section,
due to mode mismatch and Fresnel reflection. This problem is particularly serious in silica
index contrasts, where junction loss is estimated to be limited to >0.1dB per junction, when
optimized.
In this thesis, an Air Trench Bend structure is proposed for silica buried waveguide PLCs
which uses air trenches to achieve tight bend radii, and eliminates abrupt junctions between
the trench and silica waveguides by employing a new "cladding taper". This taper ensures
an adiabatic transition can be achieved between the low and high index contrast regions.
The geometry proposed is general in the loss-for-size tradeoff, in that a set of dimensions
for this geometry can be found for any desired bend loss value with no fundamental lower
limits on the loss. Furthermore, the tapers allow the use of bend radii limited by bend loss
and taper-bend junction loss rather than the silica-air trench junction loss. As well, the
air trench waveguide is etched well below the core. It is shown that in this manner the
substrate leakage in the air trench region can be reduced to arbitrarily low values.
Chapter 2
Analytic Tools for
Planar Waveguide Structures
In this chapter, analytic tools are discussed for the analysis and design of the integrated
optical structures (air trench waveguides, bends, cladding tapers) presented in Chap 3.
Modal expansion of fields is reviewed for lengthwise-invariant dielectric waveguides, though
used also for bending waveguides in Sec. 2.5. The Finite-Difference Time-Domain method,
a popular numerical simulation tool used for the simulation of all complex structures in this
thesis, is briefly reviewed in terms of a discrete formulation of electromagnetic theory on a
lattice. A vectorial mode solver for 2D cross-section waveguides which also follows from this
discrete formulation, and is consistent with-FDTD, is described, and used for subsequent
waveguide design. All FDTD simulations, as well as bend and junction loss analyses in this
thesis are two-dimensional (2D). The Effective Index Method (EIM) is a commonly used and
very accurate method for reducing 3D (2D cross-section) planar waveguides to equivalent
2D (ID cross-section) structures. The standard EIM and a perturbative correction applied
for more accurate buried waveguide analysis are described.
We next discuss junction, bend and substrate loss. Focusing on 2D (ID cross-section)
waveguides, we consider junction scattering at interfaces between dissimilar straight waveguides, as well as at junctions between straight and bent waveguides. We then discuss bend
loss in circularly bent slab waveguides and single boundary bends under whispering gallery
operation. We review two very practical and insight-lending mode solution methods: WKB
analysis, and boundary matching with Airy function form modal field solutions. The de-
22
Analytic Tools for Planar Waveguide Structures
pendence of bend loss on index contrast, radius, wavelength and bent waveguide width is
reviewed. Finally, the optimization of bent waveguides which terminate in straight waveguides is described, considering bending and junction losses. From gained insight, an optimal
geometry (in terms of total size for a given loss and index contrast) for a bend terminating
in straight waveguides is suggested.
The treatment of the above structures in 2D by the Effective Index Method ignores
possible leakage loss to a substrate. Substrate loss of 3D (2D cross-section) waveguides is
treated last, by an equivalent source method with perturbation. In the analysis, a dyadic
Green's function is integrated over a source equivalent to an ideal, guiding waveguide, to
find the radiation field in the leakage loss region perturbed to represent the actual, leaky
waveguide. The method is also applied to a 2D (ID cross-section) example with an analytic
solution, for a validity check of the approximate method.
2.1
Modal Expansion of Fields
Analysis of electromagnetic field propagation in straight (e.g.
z-invariant in Cartesian
coordinates) structures is conveniently described in terms a set of guided and radiation
modes. Mode solution and field expansion in dielectric waveguides is well-covered in literature ([39],[43],[65],[58]).
We briefly review it here, starting with Maxwell's equations for
isotropic media in time-harmonic form (e.g. [23]),
V
x
V x E = iwpH
(2.1)
f
(2.2)
V - CE = p
(2.3)
H= -iWfE+
p - IP 0
with electric and magnetic field phasors, E and H, volumetric current density J and charge
density p, and the permittivity e and permeability /t all spatially dependent in general.
For p constant, propagation of the electric field is described by the vector wave equation
resulting from the curl equations (2.1) and (2.2),
V x V x E - k 2 = ipJ
with dispersion relation k 2
=
W2pE.
(2.4)
2.1 Modal Expansion of Fields
23
Eigenfunctions of the Source-Free Vector Wave Equation
Modes are the eigenfunctions E of the source-free electric field vector wave equation
In a z-invariant structure, E
=
(f
= 0).
E(x, y) and E(x, y, z) = E(x, y)eiz, where each mode is
p.
associated with a propagation constant (eigenvalue)
Using the simple z-dependence, the wave equation for modes of the structure can be
formulated in terms of two out of the six E and H field components.
In terms of the
transverse electric field components Ex(x, y) and E.(x, y) the vector wave equation is
22\&/
+2!3a1Ex
ax
VS+
2Ex
(Ey
E
Dc
1
aE+
--Ey
ax
ay
+ E
+3
Ey
+
E
&c\
)
2
EX(25
=#2E
(2.5)
2
(2.6)
-2
=
Ey
and the third component (Ez) is determined by Gauss' law (2.3), while the magnetic field
results from Faraday's law (2.1). Alternatively, a wave equation can be obtained in terms of
transverse magnetic field components (Hx, Hy), or the longitudinal electric and magnetic
fields (Ez, Hz). The wave equation can be written more compactly in operator form [72],
PX
FX
PYX
Ex 1
=0
vYY
Ey
2 EE~
(2.7)
Ey
The modes are found by solving for the eigenfunctions/values of this matrix equation. The
eigenvalue equation for 3 is set by the determinant of the matrix P - 321,
D(O)
=
PX
PXY
-2T
=o.
Pyx Pyy
We solve this determinantal equation numerically by discretizing the operators P as shown
in Sec. 2.2.3. The D(O) notation is used later in solving for leaky modes of straight and
bent structures (Sec 2.6.1,2.5).
In the one-dimensional (e.g. slab) waveguide case, &/&y = 0 and (2.5),(2.6) degenerate
to the familiar slab wave equations for the TM and TE polarizations, respectively,
2
+ k2
Ex +
IEx
2
+ k
=
EY
=
2
EX
/ 2 EY
(TM)
(2.8)
(TE)
(2.9)
The equations for the two polarizations are decoupled in the 1D waveguide. This is not so
in the general 2D waveguide, where (2.5),(2.6) must be solved simultaneously for modes.
24
Analytic Tools for Planar Waveguide Structures
Sometimes a semi-vectorial approximation is made to the wave equation set (2.5),(2.6)
to decouple them for two principal polarizations, termed quasi-TE and quasi-TM [61],[68].
Neglecting a field component contribution in each equation (Ey -+ 0 in (2.5), E -
0 in
(2.6)), the decoupled wave equations are
P..E.
=
PyyEy
=
#
(Eys
0)
quasi-TM
6(E
I ,
0)
quasi-TE.
This is usually a fairly good approximation as can be seen from the relative mode component
amplitudes plotted for the modes of a waveguide in Fig. 2-2.
Orthogonality of Modes [381
For z-invariant structures, an orthogonality relation can be defined for the guided and
radiation modes [38]. For forward guided modes in lossless waveguides it is given by
(Em
S
x H*)
where P is the guided power of mode m = n. For orthogonality relations among radiation
modes, or between guided and radiation modes, see e.g. [38]. Orthogonality is useful in the
projection of an arbitrary field distribution onto the basis set of modes. For an arbitrary
electric field distribution at z = zo in a lossless waveguide (satisfying Gauss' law, of course),
and its corresponding magnetic field from eqn. (2.1),
E(x, y,zo)
=
(an + bn)En(x,y)
n
N(x, y,zo)
=
(an - bn)Hn(X, y)
n
the individual mode amplitudes an for guided modes are given by
a
=
b
=
'
x
1 f
x
*) - da+
N*)
- da
I*
x H -da
f\E
Z
x
-
da.
2.2 Mode Solvers and FDTD
2.2
25
Mode Solvers and FDTD
Numerical methods are necessary for simulation of the dielectric optical structures considered in this thesis. Design and modeling of waveguides with a 2D cross-section requires an
accurate method for obtaining their modes.
It is known that idealized 1D cross-section slab waveguides have simple analytic mode
solutions (e.g.
[17],[23],[39]).
The eigenvalue equation for the propagation constants is
transcendental, but relatively easy to evaluate. General multilayer slab (ID cross-section)
waveguiding structures have fully decoupled determinantal equations for TE and TM modes,
conveniently reduced to the scalar wave equation (with a slight modification for TM), as
pointed out in Sec. 2.1. Multilayer ID waveguides have simple analytic forms for the mode
field solution in each layer, and a simple analytic determinantal equation. Solving for the
modes of this type of structure is not pursued here, but the solution follows a similar line
of reasoning to the mode solution for the 1D leaky waveguide presented in Sec. 2.6.1.
Waveguides of 2D cross-section do not in general have simple solutions, and much
literature exists on numerical methods for finding the modes.
They divide into scalar,
semi-vectorial (approximated for decoupled TE/TM equations) [61] and full-vector (most
accurate) methods [72]. Among the numerical approaches taken are circular harmonic expansion [15] or free-space mode matching, finite difference (e.g. [61],[72],[51])
and finite
element methods, and the spectral index method (e.g. [68]).
We use a full-vector mode solver based on an accurate and self-consistent finite-difference
formulation of Maxwell's equations [9], which obey a full set of discrete conservation laws.
This is exactly the Finite-Difference Time Domain algorithm ([74],[63]), commonly used for
simulation of microwave and optical wave propagation in structures of arbitrary dielectric
distribution. We use 2D FDTD for the simulation of all structures in this thesis.
2.2.1
Electromagnetic Theory on a Lattice [9],[74]
A discrete electromagnetic theory, defined on a lattice, has been formulated by Chew [9] in a
very compact and intuitive form. Replacing differential equations with difference equations,
it obeys a discrete law of charge conservation, and has an equivalent reciprocity theorem,
Poynting's theorem and uniqueness theorem.
This theory is of great use in numerical
modeling because it applies to a discretized grid and gives rise to no spurious solutions or
26
Analytic Tools for Planar Waveguide Structures
instabilities as guaranteed by its discrete conservation laws.
It is based on a discrete version of vector calculus with discrete forms of Gauss', Stokes'
and Green's theorems and Huygens' principle, and is described in detail in [9]. We quote
only the resulting discretized Maxwell's equations,
VxH
Jm2
=
2
M
V.B7
V
m
PM
=
(2.10)
with constitutive relations
B
m
1
fm
Dm
Bm±
Em B
1
where ft and I are the spatially dependent permeability and permittivity tensors. In the
above discrete formulation, integer vector m
=
(in,
p) is the spatial grid variable while
integer 1 is the temporal grid variable, with uniform spatial grid spacing A = Ax = Ay = Az
and time t = lAt (Fig. 2-1a). E, B, H and D are vector variables with spatial (m) and
temporal (1) dependence. Finally, peculiar to this discretization is the use of the tilde (~)
symbol on field quantities to indicate "fore-vectors" with :,
and i components defined
at grid points (m + 1/2, n, p), (m, n + 1/2, p) and (m, n, p + 1/2). The hat (^) similarly
denotes "back-vectors" with the components defined at (m - 1/2, n, p), (m, n - 1/2, p) and
(*, n, p - 1/2) (Fig. 2-1b). Likewise, tilde and hat define partial derivatives as forward and
backward differences, e.g. 5tFt = (Ft+l - Ft)/At. The del operator is defined in terms of
corresponding partial derivatives, e.g. V = s5&+
6y + 2z.
With this type of discretization, illustrated in Fig. 2-1, all derivatives are central differences, resulting in an O(A 2 ) numerical accuracy with respect to continuous Maxwell's
equations. The arrangement also makes possible the fulfillment of the discrete conservation
laws mentioned above.
This formulation is precisely the FDTD algorithm first proposed by K.S. Yee in 1966
[74]. We use FDTD for simulation of all structures studied in this thesis. We also use a
finite-difference vectorial mode solver for 2D cross-section waveguides, based on the above
discrete equations in time harmonic form, which is consistent with FDTD (based on [51]).
2.2 Mode Solvers and FDTD
27
27
2.2 Mode Solvers and FDTD
z
----
m,n,p+1)
Y
~-
-----------
m-
2n12p
12
(m-1 ,n+,p+ )
x
n+1,p)
(m+%n'-Mp+%)(m
S-+
(m+1,n,p) -
+
mi
,n+,p-2)
(a)
(m,n,p)
(b)
Figure 2-1: Electromagnetic theory on a lattice: definitions of (a) interleaved grids m and m +-1;
and (b) the corresponding electic field "fore-vector" (Em, after point m) and magnetic
field "back-vector" (Hm+.I, before point m
2.2.2
+ 1), associated with m and m + 1.
Finite-Difference Time-Domain (FDTD) [74],[63]
The Finite Difference Time Domain (FDTD) method is a popular, numerically rigorous
method for simulation of dielectric optical structures, particularly where high index contrast
is present and the Beam Propagation Method (BPM) [75] is not applicable [63],[74]. FDTD
can be used to simulate propagation in structures with arbitrary dielectric distributions.
Two-dimensional (2D) FDTD code, developed by C. Manolatou [30], is used for the
simulation of structures in this thesis (structures are reduced to 2D using the Effective
Index Method, Sec. 2.3). The edges of the computational domain are terminated using the
perfectly matched layer (PML) absorbing boundary condition [3], and a discretization of 1020 pixels per wavelength is used for accurate modeling. The details of the implementation,
including the time step At for stability of FDTD, sources, and measurement of spectral
response, are discussed in [30] and in the FDTD literature [63]. In our simulations, a pulse
spanning the spectrum of interest is launched into the optical structure, and the entire
spectral response with respect to the ports of interest is obtained in one simulation, which
is terminated after the launched pulse has left the computational domain, and all significant
residual electromagnetic power has decayed. The FDTD simulations in this thesis were
carried out on a Cray T-90 supercomputer, using 64-bit floating point precision.
Analytic Tools for Planar Waveguide Structures
28
2.2.3
Vectorial 2D Cross-section Waveguide Mode Solver [51]
A numerical finite-difference mode solver based on the discrete Maxwell's equations (2.10)
in time-harmonic form was described in [51]. Such a mode solver is consistent with FDTD,
and its modes are ideally suited to the excitation of waveguides in FDTD, with no residual
power. Such a mode solver was implemented in this thesis for the design of 2D crosssection waveguides. It is briefly described, and some example modal solutions of air trench
waveguides used later in this thesis are shown.
The discrete vector wave equation for modes of a z-invariant structure, resulting from
the time-harmonic form of (2.10), and given in terms of transverse field components is [51]
Mms .X
1 S X E-m
XH1 8
M+ IVS X
s/Lk
s
,
-V,
ms-8
EmEm - k 2 Enm
2
m jVs
1
p+1H
H
+ k Em= 0 (2.11)
_k
+Hs
-
k2 Hs
Z
1
with propagation constant kz, and s indicating transverse field vectors. These equations
can each be represented as a matrix eigenvalue problem for the modes of the form [51]
Ice -Es.
-ki Es
In what follows, we use the first (transverse electric field) form of the eigenvalue problem.
Slab (1D Cross-section) Waveguide Discrete Eigenvalue Equation
Consider first an arbitrary ID cross-section waveguide with dielectric profile Em =
(x), p
constant, and &, = 6, = 0. Eqn. (2.11) results in two decoupled eigenvalue equations, one
for each polarization, as expected in the 1D cross-section problem (compare to (2.8),(2.9)),
5x
EmE
+k 2Ex'
XOXEY+
where E
= -E
+ 9E
=
k2Ex"i
k2EL = k Ey
(TM)
(TE)
, and the y- and z- dependences have been removed
from the mode field in the equation (indices n,p). The equation above is. the discrete version
of the scalar (modified) wave equation for TE (TM) modes of 1D cross-section structures
in Sec. 2.1. Step-index ID waveguide mode solutions have a simple analytic form, and can
be solved more efficiently using other methods, but the equation above applies to arbitrary
dielectric distributions.
2.2 Mode Solvers and FDTD
29
2D Cross-section Waveguide Discrete Eigenvalue Equation
The 2D cross-section waveguide is of most interest in this thesis for the design of rectangular
silica and air trench waveguides (Chap. 3) because simple, exact analytic solutions do not
exist. In terms of the transverse electric field, eqn. (2.11) yields a matrix pair of eigenvalue
equations where the two transverse electric field components are coupled,
-
[-mo
-
x
Em-
k2 E
+ [Dy x
~
±
1 6Em Ex + [-6x285"8-B
.
Em
XIyEm
5y EmI6
Em
'Edm-
.
Elm'
k1 E
k 2Em
=
-k E
=
- k 2EYM
and can be represented as a matrix eigenvalue equation (compare (2.7))
prp~1.
Pix
P MY P&
Ek2
EY
-m
=-ki
EM
-l
EM
This matrix is of size (M x N) 2, where the computational domain is sized M x N. We
implement a perfect electric conductor (PEC) boundary condition at the edges of the copmutational domain. A number of methods can be used to find the eigenvalues, kz, and
eigenvectors (mode fields) of the above operator. An efficient algorithm is described in [51].
We simply use MATLAB's sparse matrix eigenvalue solver that uses the Arnoldi algorithm.
2D Mode Solutions of Example Silica and Air Trench Waveguides
We use this mode solver to find the guided modes of the waveguiding structures in this
thesis: buried, square core silica waveguides; and, ideal air trench waveguides (see Fig. 2-
3c, Sec. 3.3).
Fig. 2-2 shows the electric field distributions of the lowest order guided mode, of each
polarization, in an ideal air trench waveguide. The parameters of this waveguide are given
under Example C in Table 3.2. Contour plots of the fundamental dominant electric field
component for each polarization, for all three air trench waveguide examples used in this
thesis are given in Fig. 3-5.
Computed propagation constants are used to evaluate the
accuracy of the approximate Effective Index Method in Sec. 2.3 (see Table 2.1).
30
Analytic Tools for Planar Waveguide Structures
Quasi-TM
Quasi-TE
E
Ex (dominant)
0.02
0.01
1.5
-o
-0.01
15
=i 0.8
0.4-
-0.2
05
-0.02
-1
-
0.5
--
-0.5-
0
X (A )
y(
-
m
.1
0.5
-1.5
Ey(dominant)
0.8
1.5
-U0.4,,
RD0.2
LL
05.
-
-1
0.040.02
-0.02-1.
0
-0.6
-0.5
-0.04
y(
M)-1
0-1
X(pm)
--
0.5
0
-1.5
0
-
-.
0,5
X (Am)
Ez
Ez
0.15
0.1
0.05
d0
0.3
0.2
0.1
--
--
75 -0.05
- -AM
-0
-0.5
1.5
-
-
-0.11.
-
. -0.2-LL
-0.1
-0.3
-0.5
0
-0.15
-
-.
5-0.5
X (AM)
-0.4
y (Am
0
0.5
--
...--
0
--.
-0.5
1
0-5
-1.5
X (pM)
05-1.
Figure 2-2: Electric field component disributions for air trench example waveguide C (Table 3.2);
(a)-(c) Quasi-TE, and (d)-(f) Quasi-TM fundamental mode distributions.
uu~i~T
-
-
2.3 Effective Index Method [22],[65]
(a)
(b)
_______
31
(c)
(d)
Figure 2-3: Effective Index Method: (a) buried, (b) rib, and (c) air trench waveguides can be
reduced to (d) an equivalent slab waveguide with effective indices.
2.3
Effective Index Method [22],[65]
All integrated optics problems are physically posed in three dimensions.
Analytic tools
for 2D structures, however, are much simpler and thus more powerful. A very successful
technique used to reduce 3D waveguides to equivalent 2D (slab) waveguides, introduced by
Knox and Toulios [22], is known as the Effective Index Method. A detailed treatment is
given in [65]. The EIM is difficult to derive as an approximation to an exact mathematical
formulation, but has great intuitive appeal and has proven very accurate, particularly for
rib, ridge and even buried waveguides (e.g. [65],[76],[48],[39]).
The EIM simplifies the 3D waveguide to an equivalent slab waveguide by collapsing the
third ("vertical") dimension out of the way. This is done by solving the slab waveguide
problem in the vertical (y in Fig. 2-3)direction, and using the mode's effective index, 3/ko,
for the index of the equivalent slab in the appropriate region (here 1, 11 or III) [65],[39], [76].
For buried waveguides, this cannot be done in the cladding where no guided slab mode
exists in the cladding cross-section, so the lateral cladding index is not modified in the
equivalent waveguide. Better results are obtained by using a perturbative correction to the
Effective Index Method that yields effective indices for the lateral cladding.
We use the Effective Index Method in this form for all waveguides in this thesis, straight
and bending. However, for bent waveguides the correct formulation of the Effective Index
Method is slightly different, as pointed out in [47]. We disregard this in our calculations.
2.3.1
Standard Effective Index Method [22]
The original EIM was proposed for use in buried waveguides [22], where an effective index
N, is found for the core, but the cladding index is not modified, Ne1 = nci.
However, it is
Analytic Tools for Planar Waveguide Structures
32
in rib waveguides that it has found much use, and shown great accuracy.
We consider the fundamental guided mode. The rib waveguide in Fig. 2-3b is divided into
three lateral regions to correspond to the equivalent slab in Fig. 2-3d. The effective index of
the equivalent slab is then assigned the value of the modal index of the fundamental mode of
the slab waveguide in the vertical cross-section for the corresponding region. The problem
is thus reduced to 2D. The propagation constant of the rib mode can be approximated by
solving for that of the equivalent slab [65].
To stay consistent with quasi-TE and quasi-TM mode polarizations, the Effective Index
Method is applied to collapse the third (y) dimension using TM boundary conditions if the
equivalent waveguide is to be used in TE configuration, and vice versa.
2.3.2
Perturbation-corrected Effective Index Method [11]
The structures of interest in this thesis are buried (Fig. 2-3a) or air trench (Fig. 2-3c)
waveguides, which do not have guided slab modes in the y-section of the lateral cladding
regions, for purposes of the Effective Index Method.
Here, the core effective index is found by the regular EIM, while the cladding effective
index, Nd, is calculated as a perturbative correction of the former. The assumption made
is that the cladding region has a y-dependent field distribution identical to that in the core,
and only the perturbation of the propagation constant due to a different index distribution
in the cladding is calculated. The perturbatively calculated effective cladding index, Ni, is
Ne,
=
NcO - AN
with
2(Y)ID(Y)12dy
1 2NcO 0
Ari(y
=ni~y
J7(Y)2dy
-
rij(y)
=
rijII(y) - ni(y
where I(y) represents the field distribution from the effective index calculation in the core
region I. A more detailed treatment can be found in [11],[30].
For three example buried silica waveguides (Fig. 2-3a), in Table 2.1 we give comparison
of the perturbation-corrected Effective Index Method modal index solutions to the numerically exact modal index obtained using the vectorial mode solver from Sec. 2.2.3.
The
33
2.3 Effective Index Method [22],[65]
Table 2.1: Effective Index Method and Vectorial Mode Solver Propagation Constants
Property
Method
A
Polarization
Index Contrast (A)
Silica Guide
EIM
Modal Index*
Vectorial
Air Trench
Modal Index*
0.25
1.4618533
1.4619742
1.3820475
1.3781793
1.3454740
quasi-TE
both
quasi-TE
EIM
quasi-TE
Vectorial
Mode Solver quasi-TM
C
Example
0.68
1.4650879
1.4654191
1.3867043
1.3827550
1.3497562
6.62
1.5053808
1.5076706
1.4493810
1.4453893
1.4122388
units
%
*Straight waveguides with cross-sections as in Fig. 2-3a,c. Dimensions for examples A-C given in Tables 3.1,
3.2. Calculated indices at free-space wavelength of 1550nm.
x-dependent modal field distribution of the equivalent slab waveguide is compared to the
2D solution from the vectorial mode solver along the x-axis in Fig. 2-4, for the case of the
Air Trench Waveguide (Table 3.2) of example A (silica contrast of A = 0.25%). It shows
very good agreement. The equivalent waveguide is used in 2D FDTD simulations of the
structures in this thesis, and it is important that both the modal index and field distribution
resemble the original 3D waveguide.
EIM Slab Mode
2D Vector Mode
0.8 -
S0.6 -E
co
0.40.2 -
/
.,/...
llilllli
4
/
0
-3
-2
'
-..
A:.rerA
\
-1
0
1
Displacement (microns)
2
3
Figure 2-4: Lateral (x-axis) field distributions of the Air Trench Waveguide of example A (Table 3.2) using the 2D cross-section vectorial mode solver (dashed line), and that of the
equivalent slab waveguide obtained using the Effective Index Method (solid line).
Analytic Tools for Planar Waveguide Structures
Analytic Tools for PlanarWaveguide Structures
34
34
(a)
(b)
(c)
(d)
Figure 2-5: (a) Multi-junction dielectric stack with mode rays indicated, and (b) straight-straight,
(c) straight-bent and (d) bent-bent waveguide junctions. Forward and backward amplitudes of the nth mode in regions 1,2 are indicated in (b),(c).
2.4
Junction Loss at Waveguide Interfaces [65],[47],[5]
Junction analysis investigates scattering at an interface between two different structures,
for each of which a set of modes describes propagation relative to the interface. A simple
example is a (ID) dielectric stack, for which the modes in each layer are plane waves; more
complex junctions are shown in Fig. 2-5. A scattering matrix formalism is convenient, where
the solution is obtained by expanding the field distribution on either side of the junction
in terms of the its modes and matching the boundary conditions using this expansion. An
extensive literature is available on scattering and transfer matrix methods for analysis of
junctions (e.g. [65],[47],[5]). Concepts of use in this thesis are briefly summarized.
The full potential of planar junction scattering analysis is realized in its application to
lengthwise(z)-varying structures such as step-index waveguide tapers and directional couplers [39], and recently to more complex structures like photonic crystals [5]. By discretizing
the z-varying structure into a large number of short z-invariant sections, the behaviour of
the total structure can be obtained by cascading a set of transfer matrices, obtained in
terms of the Local Normal Modes of each section [39]. The scattering parameters of the
entire structure can be obtained at the end by converting the total transfer matrix to a
scattering matrix (e.g. [50]).
In this thesis, we are primarily concerned with single abrupt junctions between dissimilar
straight or straight and bent waveguides, in the context of optimizing the transfer of power
from a chosen source mode in one waveguide to a target mode in a second waveguide. Only
2.4 Junction Loss at Waveguide Interfaces [65],[47],[5]
35
two-dimensional (slab) waveguide problems are considered, with the Effective Index Method
used to reduce 3D problems to 2D (see Sec. 2.3).
2.4.1
Junction Scattering via Mode Expansion and Transfer Matrices
Consider the junction of the two z-invariant slab (layered, in general) waveguides in Fig. 25b. The TE modes of each waveguide are fully described by the electric field distribution
lein)eiz, a single scalar distribution for one vector component (
in Fig. 2-5b).
The set
of modes (these scalar distributions) is composed of guided and radiation modes, and is a
complete and orthogonal set. The transverse fields at each side of the interface z =
0+,-
are represented as a superposition of the modes of each structure,
Ei
Z(ain + bin)Iein)
=
n
-E
HiX=
An(ain-bin)Iein)
W-
n
in regions i = 1, 2, with forward and backward mode amplitudes ain and bin, respectively,
for the nth mode, and modal field Iein) = ein(x). By matching these fields at the boundary,
and making use of orthogonality relation (eimlein) = 6 mn, a transfer matrix to find mode
amplitudes in region 2 given those in region 1 is obtained (refer to Fig. 2-5b),
[
2
F
=
1
=T -
b2
I
b
where, for example, d 2 is a "vector" of amplitudes of all modes a2n, or, for a single mode
in region 2,
a2m
1
Fi1
Tmn -[=
=
b2 m
n
1
b
2(e2mle2m) n
T
mn
Tm1F
ain
T
T
bin
with matrix elements
Taa
Mn
=
T bb
mn =
T a
=
T ba =
\
(emein)
1 +
32m /e2len
1 -
Oin
(e2mlein)-
The 1D overlap integral above is defined as
(e2mlein)
=
Jesm()ein(x)dx.
36
Analytic Tools for Planar Waveguide Structures
Transfer matrices are handy because they can be cascaded easily with free-space propagation
and other junction transfer matrices to yield a total system transfer matrix. However, outgoing waves are not known at the outset, so a more desireable representation is a scattering
matrix which gives outgoing mode amplitudes in terms of given incoming mode amplitudes,
[
bim
= ESmn-* I
n
a2m
1
a1 n
Smn
smn
b2n
S2
mn
s22=
mn
n
an
bin
In terms of the above transfer matrices, the scattering matrix elements are
S"
mn
=
S12
=
T bb-1
mn
- [Tbb] -I. rba
S 21
mnmn Taa - Tab
mk - [Tbb]-1.Tba
k1
1n
Smn
=Tmank ' [T
kn-
with the understanding that repeated indices imply Einstein summation (i.e. matrix multiplication). This representation makes clear that the insertion loss between a source mode
n and target mode m in regions 1 and 2 respectively, S21,
is dependent not only on the
overlap integral of those two modes, but of all modes, due to the second term in the ex=21
pression for S
which involves the overlap integrals of all modes (if the T-matrices are not
diagonal).
The above notation does not make explicitly clear the symmetry of the scattering matrix
implied by reciprocity. Making use of the completeness property of the mode expansion
((e2mlelk)(elkle2n) =
6
mn, summed over k, for region 1) and of reciprocity, an explicitly
symmetric set of s-parameters is found by Pennings [47],
S
= -
n=
S2
(eikte2p)#2pq(e2qlelm) + #Ikm}
(e2k~e1p)1pq(e1qle2m) +
2kmJ -1
{(ek
e21)21r(e2rlein) + Likn
2(e2ke11)O1n
(2.12)
with the remaining two obtained by swapping indices 1,2 in the above expressions, and
the propagation constant matrix /9imn =
3
im 6 mn. Einstein summation is implied. These
expressions have a more clear connection to the approximate junction loss formulas we will
use.
Suppose that the set of modes (eigenfunctions) is identical for regions 1 and 2, while
their respective propagation constants may or may not be the same. The overlap integral
2.4 Junction Loss at Waveguide Interfaces [65],[47],[5]
37
matrices (eimlejn) in (2.12) are then diagonal because all overlap integrals are zero, except
for m = n when they are unity. In addition, the 3 matrix is diagonal by definition, so the
insertion loss s-parameter in this case simplifies to
S2 1
mm
2
3
(e2mlelm)
1m
(e2mleim)/31m(eimle2m) +
01M
- 2
m
/3 1m+/2m
02m
This is valid for a dielectric stack (Fig. 2-5a) where modes are plane wave, and a single
plane wave in one layer excites one and only one mode in the following layer because of
perfect mode overlap. It is also possible in slab waveguides with the right constraints, and
has been used to describe limits of fiber match to high index optical chips [44](p.103-4),
and to design extremely low-loss, high strength integrated optical Bragg reflectors [66] and
high-Q integrated Bragg resonators [70].
2.4.2
Straight-Straight Waveguide Junction Loss
It is of interest to us in this thesis to evaluate the insertion loss as defined in the context of
transfer of power from a source mode in region 1 to a chosen destination mode in region 2.
To avoid the necessity to compute a large (ideally infinite) number of modes, as required
by expr. (2.12), we make a simplification by assuming that the overlap integral of the
source mode with all modes in region 2 except the destination mode is very small. This
is consistent with a "low-loss" junction.
Then the reflection modes of region 1 are all
neglected in matching the boundary conditions, except the reflection into the source mode
=21
in the reverse direction. A simple formula for the S
S2 =
mn
2I3 1n 2)
\
1n
parameter results,
(e2mleln)
(2.13)
+ 02m ) (e2mle2m)
The net power carried in the +z direction in a mode is
Pik
=
(Iaik12
-|
bik 2) Oik (eikleik).
2wp-
Assuming no incident mode amplitude in region 2 (b 2n = 0), normalizing only to the incident
source mode power in region 1, and using (2.13) the insertion loss is
21
lqmn(ss) =
2
02m
mn k
1
#1n
+
/2mJ
2
(e2mlein) 2
(e2mle 2 m)(einlein)(
.
2.4
Variations of this formula are common in literature (e.g. [28],[18]). It is used for all straightstraight junctions in this thesis, and is valid for low loss. A comparison of junction loss
38
Analytic Tools for Planar Waveguide Structures
computed using this formula and using direct 2D FDTD simulations is given in Fig. 3-8,
and shows good agreement. The slight discrepancy between the two is attributed mostly to
error in the FDTD simulation due to a discretized waveguide width.
Fresnel Reflection and Mode Shape Mismatch
We observe from expr.
(2.14) that there are two independent mechanisms contributing
to junction loss. One is due to Fresnel reflection (a mismatch of the mode propagation
constants), and the other is due to the mode shape mismatch (measured by the overlap
integral in the numerator). The total junction loss may be due to one, the other, or both
of these contributing effects.
Normally, the former effect of Fresnel reflection can be suppressed by using similar refractive indices on the two sides of the junction, or index matching with a quarter-wave layer
at the expense of unwanted frequency depedence. Then, the insertion loss is determined by
the mode shape matching, governed by the geometry. Examples exist (including the plane
wave and slab examples mentioned above) where perfect mode shape matching is present
and the entire junction loss is due to Fresnel reflection ([44] (p.103-4),[66],[70]).
In the junctions encountered in this thesis, the core index on either side of the junction
is the same, so Fresnel reflection is subdominant to mode shape mismatch in determining
junction loss.
2.4.3
Straight-Bent Waveguide Junction Loss [47]
Thus far, this section has dealt with scattering from a junction of two straight guiding
structures. A type of junction of particular interest in this thesis is the junction between a
straight waveguide and a circularly bending waveguide. The junction loss analysis is slightly
different in this case.
In a circular bend, (leaky) modes are defined whose propagation is azimuthal, i.e. along
a constant radius curve. In cylindrical coordinates (p, q$), the field of a leaky mode is given
by [47],
EiYn
=
Hipn
=
(ainen
+ bine
An R (aineiin
in
I)
ein)
- bine-iinRo
ein)
where R is an arbitrarily chosen radius at which the propagation constant,
#,
is defined. In
2.4 Junction Loss at Waveguide Interfaces [65],[47],[5]
39
this thesis, this radius is usually chosen to be the central axis of the bent waveguide, or its
outer radius, to give the propagation constant a meaningful definition.
The modified insertion loss formula, akin to (2.14), for straight-bend junctions is
2
21
_
?lmn(BS) -
02m
i0
+6
(e2mlein) 2
21_
(e2mle2m)
(e2m ie2m)
(e2mle2m)(einlein)
(2.15)
with the bent waveguide mode overlap integral defined as
Roo =
(e2m I le2m)
P
o
0 e*
2(
R-enP~P
R
dp
Pep
This simplifies to the straight-straight junction loss formula (2.14) for R -+ oc, where
R/p --
1 over the range of p-values for which the integrand (modal field) significantly
contributes to the integral.
This formula is used in the optimization of straight-bent waveguide junction loss in later
sections; the total loss of a bend which terminates in straight waveguides may be dominated
by junction loss, in which case the optimization problem reduces to maximizing expr. (2.15).
Analytic Tools for Planar Waveguide Structures
Analytic Tools for Planar Waveguide Structures
40
40
(b)
(a)
Figure 2-6: Geometry of (a) dual, and (b) single boundary bends. The radiation caustic is indicated, beyond whose radius radiation loss is forced from the bend. The superimposed
leaky mode exhibits oscillatory (outward wave) behaviour beyond the caustic.
2.5
Bend Loss in Step-Index Slab Waveguides
Bend loss is a central issue in the problem addressed in this thesis. Of interest here are
planar integrated optical circuits, and thus planar waveguide bends. We address the twodimensional (2D) bent slab waveguide problem.
While waveguides of practical interest
(buried, rib, ridge, etc.) have a finite vertical extent for which a three-dimensional analysis
is appropriate (e.g. [55]), we reduce the dimensionality of the problem using the Effective
Index Method (Sec. 2.3).
This is commonly done in literature on waveguide bends and
is justified by the planar geometry [18],[47],[4],[25].
Although in practice the conventional
EIM is usually used, it has been pointed out that the Effective Index Method should be
differently applied for bent waveguides than for straight ones [47].
Bend loss is a fundamental radiation loss in bent step-index waveguides that occurs
beyond a critical radius, called the "radiation caustic" (Fig. 2-6, e.g. [47],[58]).
At this
radial distance, the velocity of the waveguide mode phase fronts reaches the speed of light
limit (e.g. [39], p.94). Outside this radius, the phase velocity is forbidden from going above
the speed of light, and radiation must occur.
Small radii imply large bend loss. It is the objective of the designer to minimize the
bend size for a given loss (or vice versa).
Extensive literature exists on the bend loss of step-index waveguides, beginning in 1969
and actively continuing into the present ([32],[18],[45],[16],[47],[39],[57],[25],[53],[4],[58]).
A
particularly useful work to this author was the Ph.D. dissertation of E.C.M. Pennings
2.5 Bend Loss in Step-Index Slab Waveguides
41
on bends in ridge waveguides [47], which comprehensively covers the subject and has an
excellent bibliography.
Several methods of analysis have been proposed for finding the
leaky modes of bent waveguides and hence the bending loss. The most general method to
find the modes and corresponding complex propagation constants is to solve the circular
bend problem directly by matching boundary conditions using a Bessel function expansion
[47].
This method, while theoretically exact, is computationally difficult to implement
and time consuming. The most practical methods, used in this thesis, apply a conformal
transformation and solve an equivalent ID eigenvalue problem to find the modes. They use
either global WKB analysis ([18],[47],[4]), or make a piecewise-linear local approximation
of the equivalent dielectric profile near the waveguide and patch the solution together from
Airy functions ([16],[53]).
In the first subsection that follows, we briefly describe the WKB and Airy function
analysis for the leaky modes, and refer the reader to the original works on these methods
for a detailed explanation. We use the subsections that follow to review the dependence
of bend loss and of the qualitative shape of the leaky mode on index contrast and radius,
the difference between single and dual boundary bends, and optimal design of bends which
terminate in straight waveguides. In the such terminated bends, additional contributions
to loss come from junction scattering due to mode mismatch between the straight and
bent waveguides. Finally, we speculate on the optimum (non-circular) bend geometry for
minimum size bends that terminate in straight waveguides.
2.5.1
Analytic Methods for Single and Dual Boundary Bends [47],[18],[16],[53]
Two types of circular bends are considered: single boundary (whispering gallery) and dual
boundary (waveguide) bends, adopting the naming convention from [47].
The latter is
simply a bent waveguide, with an inner and outer radius corresponding to the waveguide
walls (Fig. 2-6a). In the limit of infinite radius, the leaky mode of the dual boundary bend
(DBB) tends to a guided mode of the corresponding straight waveguide, for cases of practical
interest. A single boundary bend (SBB) is obtained by letting the inner radius of a dual
boundary bend go to zero (Fig. 2-6b). The bend still supports leaky modes, guided solely
by the outer radius. These are known as whispering gallery modes (WGM), first described
by Lord Rayleigh in the context of sound, and have been shown to be the minimum loss
limit of dual boundary bends of a given outer radius [47].
Analytic Tools for Planar Waveguide Structures
42
n(r)
n(u)
nco
nco
nci
ncl
Ro R1
0
-
UI1 U1II
UI
r
U
(b)
(a)
Figure 2-7: (a) Radial refractive index profile of single or dual boundary bend (inner and outer
radius, Ro,R 1 ), and (b) index profile of equivalent straight waveguide (1D problem
after conformal mapping) with solution region boundaries (ul-ulil) marked for the
SBB.
Conformal Transformation of a Bend to an Equivalent Straight Waveguide [18]
The circular bend problem is made simpler by applying a conformal transformation to convert the bend to an equivalent straight waveguide [18],[47]. Conformal transformations are
applicable to the two-dimensional Helmholtz (scalar wave) equation because the Laplacian
operator is invariant with respect to this kind of transformation [54]. Illustrated in Fig. 2-7,
the conformal mapping used is [53] (a nice explanation is given in [44]),
z = beaw
where z = x + iy represents real space, w = u + iv is the new domain, and a, b are scaling
parameters of the transformation. The result is a one-dimensional eigenvalue problem for
leaky modes along the u-axis, with the coordinate relationship (r = Vx
U=
2
+ y2)
1
r
In a
b
-
and a new refractive index profile for the equivalent straight waveguide
52(U) = (ab) 2 rn2 (r(u))e2 au
(2.16)
where n(r) = n(x, y) is the refractive index profile of the original bend.
Global WKB Analysis of Single Boundary Bends [47]
A meticulous WKB analysis of a single boundary bend after the conformal transformation
is given by Pennings [47].
The WKB field expressions, eigenvalue equation and validity
2.5 Bend Loss in Step-Index Slab Waveguides
43
condition are summarized in what follows, and the reader is referred to the original work
for a detailed description [47]. A treatment of global WKB analysis is given in the applied
mathematics text by Bender and Orszag [1].
With the definition 1/a = b = Rt (expr. (2.16)), the WKB leaky field is of the form [47]
exp
I
-
2c1 Cos
if -O0 < u < U
I(u')du'
I
Ukii (u')du'
-
11exp
+
+
(u
JI
exp +
2
C2exp
if ul < u < Ull
-i
/du
1 (u')du'
kiv(u')d'
--
if ull < U < Ulil
if UlIJ < U < 00
with solution region boundaries (Fig. 2-7b)
uj
=
Rt ln{ Re( )
uII
=
Rt In
U mlI =
R ln
R1 f
Rel o)
Lneiko Rt
and local u-dependent phase factors
Re(ti) > 0
;111- krn eu/Rt,
Re(kii) > 0
02ri2 1 e 2 u/Rt
kii
sk~v
kn
ie~e
-
7
Re(Kiii)
>
0
uR
Re(kiv) > 0.
Here, R 1 is the outer (single) boundary radius, and Rt is a scaling parameter normally set
equal to R 1 [47]; nco and nci are the core and cladding indices (Fig. 2-6a); ko = 27/Ao; and y
is the angular complex propagation constant of the leaky mode, with the actual propagation
constant /=
-y/r (at some radius r; we use here r = R1).
The eigenvalue equation for the leaky modes is found by matching the boundary conditions (field <1(u) and its derivative) at u = ull to determine one of the constants
c1,2
in
44
Analytic Tools for Planar Waveguide Structures
terms of the other. The resulting determinantal equation (of the form DWKB(Y)
=
0) is
complicated (see [47], p.69), but a simple, useful and accurate perturbative solution can be
found instead, starting from the simpler determinantal equation
DWKB(7o) -
sin
-
Ui
UI
kII(u')du - - - tan}-1
4
E(kIII(uII)
k 1 1(uII)
with real solution -yo. A simpler form of this determinantal equation is
j SUii kii(u')du' -
7T
_
kiii (uii)
- - tan-
vir = 0
with integer v indicating the mode number (v = 0 for lowest-order leaky mode). The loss
is then found by a Taylor series expansion of DWKB(-y) about Yo. The reader is referred to
[47](p. 71) for these expressions.
This WKB solution is valid approximately for [47]
N2
> n?(n?/4+
eff
2
k 2R1201 >z
n? - z Neeff
2
for i = "co" and i
=
"cl". It fails when the modal index Neff is near neo or
n.1 (i.e. near
the turning points), and for radius too small in comparison to the wavelength.
Plots of the field distribution of a leaky mode using this method are shown in Fig. 2-8.
They show the radiation away from the bend. The field is divergent in the figure at the
classical turning points.
Advantages of the WKB approach to the problem are a high accuracy for low loss
values and large radii and a fast and simple numerical implementation. The perturbative
eigenvalue equation for the leaky modes is a simple indicial equation where the leaky mode
to be calculated is explicitly selected by setting the value of the index v. The method works
well for bend losses less than 1dB/90', which is the case for all problems of interest in this
thesis.
The major disadvantage of the WKB method is the divergence of the modal field distribution near the classical turning points of the differential equation for the bend problem.
The field distribution is important for mode matching overlap integrals at straight-bend
junctions. For this, we use the Airy function solutions.
Global WKB Analysis of Dual Boundary Bends [18],[4]
Dual boundary bends are solved similarly using WKB analysis by Heiblum and Harris
[18], and by Berglund and Gopinath [4]. A correction to [18] is pointed out by Pennings
2.5 Bend Loss in Step-Index Slab Waveguides
45
R =300 sm
Loss =0.18335 dB/90'
1.4
1.2
1
:
1
a
0.8
0.8
0.6
0.6
0.4
CL
7
0.2
0
-WKB
--- Airy functions
0.2
395
410
415
400
405
Radial displacement (gm)
A
U-
0.4
0.2
0
0.2
-
420
295
(a)
300
305
310
Radial displacement (gm)
320
315
(b)
1.4
R1 =200 gm
1.4
1.2
Loss =2.8316 dB/90*
1.2
-~1
R =1 50 jim
Loss =8.588 dB/90
-
-~1
0.8
(D
0.6
~- CL
~-
0.4
0.8
0.6
I
0.4
;j
~I
~00.2
I I~I
-
-IIi
V1;
0
0
0.2
* * *
0.2
195
210
200
205
Radial displacement (pm)
215
145
155
150
160
Radial displacement (gm)
165
(d)
(c)
Figure 2-8: WKB (solid blue) and Airy function (dotted red) solutions for the leaky mode distribution of a single boundary bend (SBB), identical to the example in Table 5.1 in [47]
(n,, = 3.26106, nrj = 3.22000, A0
1300nm) for several radii (150-40Opm).
[47](p. 70).
Linearized-c Airy Function Solution for Dual Boundary Bends [53],[16]
Another approach to solving the bend loss problem makes use of the fact that bend radii
are large in comparison to the leaky mode size (i.e. waveguide width) in cases of practical
interest. The effective refractive index after the conformal mapping, which is exponentially
dependent on u, can be piecewise-linearized in the neighbourhood of the waveguide (i.e. the
outer radius, in the case of a single boundary bend).
Airy functions are the solutions to the wave equation in each region of linear dielectric
constant (i.e. ii 2 (u) linear in u). Rowland [53] sets up a direct complex eigenvalue problem
by matching boundary conditions, while Goyal et al. [16] use an indirect method where
Analytic Tools for Planar Waveguide Structures
46
the loss is evaluated by calculating the linewidth of a Lorentzian resonance characteristic
of the leaky mode.
The former is simpler, but the latter is more suited to numerical
implementation on a computer. We show the highlights of the first approach for its brevity,
and use Mathematica's arbitrary precision [71] to avoid numerical implementation issues.
Letting 1/a = R, b = p in expr. (2.16) for waveguide half-width p, the electric field
< 1)
distribution of a TE leaky mode is of the form [53] (valid for p/R
c1Ai(Z 2)
if
c2Ai(-Zi) + c3 Bi(-Zi)
if |1 < 1
c4(Bi(Z2) + iAi(Z2))
if
with the normalized spatial variable
axis radius, R
= (U
-
uR)/p, UR
<
-1
> 1
R ln(R/p). R is the waveguide
(Ro + R 1 )/2. Airy function arguments Z 1 ,2 are
Zi
U2 + A1
A 2 /3
=
1
-A2
Z2-W2
=
A 2/3
2
2n2k 2p3
R
with nO, nr
2n2k2p
3
1
A2
=
U
=
p~n20 k -
W
=
p
R
2
42 - n2k2
the core and cladding indices (Fig. 2-6b), and ko
equation is obtained by matching b( ) and its first derivative at
=
=
27r/Ao.
The eigenvalue
± 1, eliminating all but
one constant. The imaginary part of the complex propagation constant (eigenvalue) yields
the bend loss (Sec. 2.5.2). A detailed explanation of this solution is given by Rowland [53].
Linearized-c Airy Function Solution for Single Boundary Bends
The solution for dual boundary bends above can easily be simplified for single boundary
bends.
Because there is no explicit waveguide width here (Fig. 2-6a), a new choice of
normalization parameters (p, etc.) is necessary. This choice is arbitrary (because the answer
is the same regardless of the normalization), but for comparison, we use the approximate
waveguide width for whispering gallery operation found by Pennings (expr. (2.17)) to define
2.5 Bend Loss in Step-Index Slab Waveguides
47
the "waveguide" width p,
2p = WWGM-
The field distribution of leaky mode will be of the form
{
ciAi(-Zi)
if
<0
C2 (Bi(Z2) + iAi(Z 2 ))
if
> 0
with the same parameter definitions as in the previous section, and R defined to be the
outer (and only) boundary radius. The eigenvalue is obtained by matching the field and its
derivative at
= 0.
A comparison of the Airy function single boundary bend solution to the WKB solution
is shown in Fig. 2-8. The solutions match well except for the radiation field far from the
bend. There, a phase difference builds up because the Airy solution is based on a linearized
version of the exponentially-dependent index profile.
The Airy function representation of the field distribution is continuous and thus of use for
mode overlap integrals. The three (or two) piecewise-linear region approximation becomes
poor at small radii (p/R < 1 invalid), where the index is rapidly exponentially growing.
Such radii are of practical interest in high index contrast bends which may still have low
loss. Here, a division into multiple piecewise-linear layers, using the transfer matrix method
of Goyal et al. [16] is useful. The numerical implementation of the Airy method is more
involved than of the WKB method in the previous section.
2.5.2
Qualitative Dependencies of Bend Loss and the Leaky Mode Field
We now summarize the qualitative properties of bend loss in single (SBB) and dual boundary
bends (DBB). We quote some useful formulas from literature for the dependence of bend
loss on radius and index contrast, and for the minimum waveguide width of a DBB for
whispering gallery (SBB-like) operation. Throughout, we focus on the lowest-order leaky
mode of a bend, which is of practical value because it has the lowest bend loss and most
resembles the fundamental guided mode of a straight waveguide.
considered.
Only TE modes are
48
Analytic Tools for Planar Waveguide Structures
Properties of Single and Dual Boundary Bends
Two important differences exist between comparable (same outer radius) single and dual
boundary bends: bend loss, and the width and qualitative shape of the leaky mode field
distribution. The parameters of a (symmetric step-index, circular) dual boundary bend are
the core and cladding indices (n,,, ni)
and the outer and inner radii (R1, Ro). An alternate
set of parameters is nro, n,1, R 1, and w, where the waveguide width w - R1 - RO. In the
limit of RO -* 0, i.e. w large, the DBB approaches a SBB.
A single boundary bend supports leaky modes (whispering gallery modes, WGM) that
resemble the guided modes of a straight waveguide at small radii if they are low loss
(Sec. 2.5.1, Fig. 2-8).
The width of these modes is determined by the radius (R 1 ) and
the core index, and is very weakly dependent on the index contrast (eqn. 2.17) [47].
A
smaller radius implies a narrower leaky mode, approaching the width of the corresponding
single-mode waveguide mode in the same index contrast at small radii. At very small radii,
loss becomes high and the leaky modes lose their distinctness and cease to be meaningful.
By contrast, a straight waveguide mode width is determined by the waveguide width.
The fundamental mode width scales with the waveguide width above the cutoff of the second
guided mode, with a minimum width set by the index contrast.
A dual boundary bend is obtained by considering a straight waveguide, and deforming
it slightly into a large radius, low loss bend. The guided modes of the straight waveguide
become the leaky modes of the bend. The lowest order (fundamental) mode of the waveguide
becomes the lowest loss leaky mode of the bend. At large radii (R 1 -* 00) the leaky mode
width is determined by the waveguide width. At small radii compared to the waveguide
width, the mode width may be limited by the outer bend radius (R 1 ) instead (whispering
gallery operation). In this case, the inner radius has weak influence on the mode and the
bend behaves as a single boundary bend.
Fig. 2-9 shows some FDTD simulations of dual boundary bends of varying bend width
(a-f), showing effectively single boundary operation past a certain width (d-f); and, single
boundary bends of varying radii (g-1) with large (g), tolerable (h-j) and small (k-1) bending
loss. The bends in Fig. 2-9k-l do not have circularly propagating light in the bend because
multiple leaky modes are excited by the suboptimally placed waveguide. Their interference
causes large junction loss at the output in Fig. 2-91.
it
U~L~U
-~
-
--
2.5 Bend Loss in Step-Index Slab Waveguides
49
(a)
(D)
(C)
(d)
(e)
(1)
(g)
(h)
(i)
(j)
(k)
(1)
Figure 2-9: FDTD field plots of 1.481:1.0 index bends with R 1 flush with 0.7pm straight waveguides. (a)-(f) DBBs of varied bend width showing effective single boundary operation
in (d)-(f); (g)-(l) SBBs of varied radii: 2, 4, 6, 8, 11 and 14 waveguide widths. High
bend loss is seen in (g),(h), while significant junction loss occurs in (k),(l) due to excitation of multiple leaky modes, and interference phase misalignment at the output.
Analytic Tools for Planar Waveguide Structures
50
Qualitative Features of the Leaky Mode Distribution
Given n,,, nei and R 1 , it has been shown that the minimum bend loss results for a single
boundary bend, i.e.
for Ro -- 0.
An empirical formula found by Pennings gives the
minimum width w required for a dual boundary bend to act as a single boundary bend [47]:
W
3.663
> WWGM - 3.6
koric0
o'kncoR1.
(2.17)
In applications where bend loss is the only consideration, an SBB gives the minimum size
bend. It is useful for continuous bends such as ring (i.e. disk) resonators. A width w as
calculated in (2.17) is useful because it guarantees WGM operation (minimum loss), but
works to suppress (increase the loss of) higher-order leaky modes which are not of interest.
The field distribution shape of a leaky mode of a bend has several qualitative differences
from the corresponding guided mode of a straight waveguide. In addition to having an
oscillatory rather than evanescent field past the radiation caustic, a leaky bend mode has
an asymmetric shape unlike the centrosymmetric (or anti-symmetric) guided modes of a
slab. This mode asymmetry is greater for single boundary bends (i.e. in whispering gallery
modes), and for larger radii and index contrasts. The asymmetry works against good mode
matching for excitation of the leaky mode by a straight waveguide. This issue is encountered
in subsection 2.5.3.
A comparison of the fundamental (respectively guided and leaky) modes of straight
waveguides and the corresponding single and dual boundary bends is given in Fig. 2-10.
The modes are shown on the same length scale, for several radii in a silica index contrast
(1.46:1.45). Loss of both types of bend is indicated. The field amplitudes are offset vertically
from each other for clarity.
Bend Loss Dependence on Index Contrast and Radius
Bend loss has simple scaling behaviour with index contrast and radius. We consider single
boundary bends because they have minimum loss (and one less free parameter).
Bend loss a (in dB/unit angle) is well-known to be exponentially dependent on the bend
radius (adB oc e-AR1), given a fixed index profile [21],[47]. Another very useful empirical
formula found in [47] is for single boundary bend radius as a function of core index, index
2.5 Bend Loss in Step-Index Slab Waveguides
R1 = 45000 gm
1.2
51
Boundary Bend Mode
Straight Waveguide Mode
Dual Boundary Bend Mode
-
1.2
1
1
0.8
0.8
0.6
0.6
-Single
R, =2685 gm
0.4
0.4
0.2
0.2
DBB: 2e-59
SBB1: 1le-1l11
0
-40
Loss (dB/90')'
Loss (dB/90):
0 DBB:0.f107
SBB:0.O1 0
h
-40
-20
0
20
40
I
-40
60
R,=5000 gm
1.2
-20
0
20
40
60
R=2195 pm
1.2
1
0.8
-
-
0.8
:3
E
U_
0.6
0.6
0.4
0.4
-------
-
0.2
Loss (dB/90'):
------
-0.2 '
-40
-20
0
20
40
0
-0.2
60
R =3153 m
1.2
-4(
-
0.8
0.8
-
0.4
0.2
-20
0
20
40
60
0.6
0.4
Loss (dB/90'):
.- - -....
..... -------------- --I ....
DBB:0.028
S131:0.0011
0
-4
SBB:0.101
R, =1670 gm
1
I
DBB:0.414
1.2
1
0.6
-
0.2
DBB:0.0001
SBB:7e-8
0
-
Loss (dB/90'):
-40
-20
0
20
40
Loss (dB/90):
0.20
60
-0.2
-40
DBB:1.599
SBB:11.007
-20
0
20
40
60
Displacement from outer radius, r - R1 (gm)
Figure 2-10: Lowest-order leaky modes (Airy solution) of single (blue solid) and dual (red dashdot) boundary bends compared with straight waveguide fundamental mode (green
dotted). Plots given over a range of radii, from lossless to
1dB/90; indices 1.46:1.45.
Analytic Tools for Planar Waveguide Structures
52
contrast, wavelength and bend loss:
koRinco
1
-
n
1
=
(2.18)
C(a)
with
C(a) = 13.7 - 3.72 logo (
(2.19)
1 dB/90-
where the denominator in the logarithm denotes the units. The formula is plotted in Fig. 211c along with WKB bend loss calculations from Sec. 2.5.1 as a set of constant-loss curves
of bend radius vs. index contrast, and shows good agreement with some deviation at high
index contrasts. Figs. 2-11a-b show bend loss vs. radius for a set of index contrasts on a
log-radius scale, and the same for a single index contrast of 1.46:1.45 (A = 0.7%, also used
in Fig. 2-10) to make clear the exponential dependence of adB on radius.
2.5.3
Optimal Design of Bends Terminating in Straight Waveguides
The analysis of 2D circular bends from the previous subsections fully describes the problem
of continuous circular bends, such as standalone ring and disk resonators (e.g. see [10]). In
other applications of interest bends of a finite sector angle terminate in straight waveguides,
or other bends (of different radii, possibly bending in the opposite direction). The former
case is considered here, because 90 bends terminating in straight waveguides are treated
later in this thesis. The latter case of bends terminating in bends occurs for example in
S-bends used for directional couplers; their design is a simple extension of the case for a
bend terminating in straight waveguides. It is considered in [47].
For minimum loss of the lowest-order leaky mode of a standalone ring resonator, given
an index system (nc0 , nc) and outer radius R 1 , the optimal structure is a single boundary
bend in whispering gallery operation (see Sec. 2.5.2), with inner radius Ro -
0 (a disk
resonator).
A circular bend terminating in straight waveguides has a larger number of parameters nc, ncd, R 1 , Ro as before, plus the straight waveguide width (we, adopting identical input
and output waveguides by symmetry) and the alignment offset Aw between the straight
waveguide and the bend (Fig. 2-12a). An angular offset parameter is not considered because
it offers no benefit. In addition to bending loss, in a terminating bend contributions to total
loss come from scattering at the interfaces between the straight and bent waveguides (see
53
2.5 Bend Loss in Step-Index Slab Waveguides
Single Boundary Bend Loss vs. Radius for Several A's (log-log) 1n Single Boundary Bend Loss vs. Radius for A = 0.7% (semilog)
I
i
s' '
e
10n
,
1010-2
102
1 -4
10
10
U)
106
10
0.25%
0.7%
7%
-1"
0-8
8
(
40%
10' 10
10
10
102
10
10
105
-
1000
2000
Bend Radius, R, (Am)
3000
4000
Bend Radius, R1 (m)
5000
6000
(b)
(a)
SBB Constant Loss Curves of Bend Radius vs. Index Contrast
-
- - -
101
0.001 dB/90*
0.01 dB/90'
0.1
dB/90'
1 dB/90'
0
C
C
0
V\\
"\N
102
10 3
104
Normalized bend radius, k0 R1 n
%
10
106
(c)
Figure 2-11: (a)-(b) Single boundary bend loss vs. radius for several index contrast values, computed by WKB for nej = 1.45; (c) comparison of WKB computed bend loss (blue
lines, see legend) and the normalized empirical formula (red, finely dotted lines) from
[47], valid for all index contrasts and radii, for losses a < 0.1 dB/90 (see [47], p.108).
Analytic Tools for Planar Waveguide Structures
54
bend loss
junction loss
dominates
dominates
E
(whispering
gallery limit)
Bn
C0
Ws w
Gei
(a)
A
-
Oge oWav
(b)
-ii
traghtwaveguide
R
Figure 2-12: (a) Parameters of a generic Dual Boundary Bend (DBB), and (b) an illustration of
minimum achievable loss for a given outer radius R1 (and index contrast) for single
and dual boundary bends terminating at single-mode straight waveguides; SBBs are
not general in the size-loss tradeoff.
Sec. 2.4). The optimization problem then involves choosing the free parameters (of which
there are now more) to produce the smallest possible terminated bend for a given total loss.
This may no longer be a single boundary bend.
Design techniques to minimize bend loss were presented in the previous subsections.
To minimize junction loss, several simple techniques have also been proposed. For joining
bends and waveguides of equal width, the introduction of a non-zero offset (Aw) between the
bent and straight waveguide was proposed by Neumann ([45], [47]). It compensates for the
lateral offset of the DBB bend mode with respect to the corresponding straight waveguide
mode (slightly noticeable in Fig. 2-10). Pennings suggested ([57], [47]) that junction loss
can further be reduced by optimizing the waveguide width to best match the widths of the
leaky mode of the bend and the guided mode of the straight waveguide.
It is to be noted that a single boundary bend, while in possession of minimum bending
loss, can have large junction loss. While the width and offset of the straight waveguide
can be optimized to match the leaky mode (Fig. 2-10), the inherent asymmetry of the field
distribution means that the junction loss is enhanced at large radii. Thus, a larger radius
results in lower bend loss and larger junction loss. The single boundary bend is thus not
general in the loss-for-size tradeoff in the terminated bend configuration, because arbitrarily
low losses cannot be achieved for a given index contrast (illustrated in Fig. 2-12b).
Dual boundary terminated bends with equal width straight and bent waveguides (and
2.5 Bend Loss in Step-Index Slab Waveguides
2.5 Bend Loss in Step-Index Slab Waveguides
55
55
(decreasing loss)
straight waveguide (fossless) limit
whispering gallery (SBB) limit
Figure 2-13: Illustration of the progression of the optimal (in terms of size) bend geometry as a
function of total loss: from SBB-like (whispering gallery operation) at small radii
when higher loss is tolerable to a straight waveguide-like DBB in the low loss limit.
no offset), as is clear from Fig. 2-10, do offer a general tradeoff of size for loss, because both
their bend loss and junction loss decrease with increasing radius. This is evident because the
leaky mode of the bend approaches the shape of the guided mode of the straight waveguide,
and in the limit of infinite radius, the loss is zero. However, the equal-width/no-offset dual
boundary bend does not have minimum loss for a given size (or vice versa).
An optimal circular terminated bend is a dual boundary bend with the free parameters
chosen to minimize loss. It will look more like a single boundary bend (whispering gallery
operation) at small radii where bend loss is critical, but will resemble the equal-widths, nooffset dual boundary bend at large radii where junction loss dominates (becoming a straight
waveguide in the limit of R 1 -* oc, see illustration Fig. 2-12b).
Fig. 2-13 illustrates this
progression of the optimal terminated bend.
Practical Numerical Optimization
The optimization problem is now formulated more exactly. In this thesis, bends which
terminate in single mode waveguides are of interest, and thus the straight waveguide width
w, is fixed, in addition to the chosen core and cladding index, and the desired loss (Ld). w,
is typically set for the waveguide to be at the cutoff of the second guided mode.
The remaining free parameters are then the offset Aw and outer radius R 1 (for single
boundary bends), and also the inner radius Ro (for dual boundary bends).
We replace
Ro by the equivalent parameter of bend width, Wb = R1 - Ro. The optimization problem
56
Analytic Tools for Planar Waveguide Structures
minimizes loss (Lmin) over the possible values of Aw and
Wb,
given an outer radius R 1 . The
minimum R 1 is sought for which the desired loss is still within the optimization space of loss
over the range of possible Aw,
Wb
values; i.e. is at the edge of the design space, Lmin
= Ld.
Reflection and Excitation of Multiple Leaky Modes
In optimizing the terminating circular bend, we have used insertion loss as the minimization
goal (or alternatively size given the insertion loss). However, junction scattering also gives
rise to reflection into the backward guided mode of the straight waveguide. This reflection or
return loss is an important consideration in integrated optical circuits, often required to be
greater than 30dB. If it (rather than the insertion loss) becomes the limiting specification,
junction loss will need to be reduced to meet this requirement. This is not normally the case
for low loss. In ideal waveguides, bend loss itself does not contribute to the excitation of
contradirectional modes. In fabricated ring resonators, such an effect is seen due to surface
roughness which couples the clockwise and counter-clockwise leaky modes.
In our consideration of junction loss, we concentrate on optimizing the shape match (i.e.
corresponding overlap integral) between the straight waveguide's fundamental guided mode
and the lowest-order leaky mode of the bend that it is intended to excite. In some (strange)
situations, it may be that a larger total amount of power would be transmitted through the
junction without loss if a larger number of the bend's leaky modes were employed (e.g. the
first and second). There are two drawbacks. First, higher-order leaky modes have greater
bend loss, so the proportion of power coupled to each mode at the input and extracted
at the output would be different.
Secondly, because the propagation constants of these
modes are different, the output straight waveguide would be able to extract the guided
power efficently only when the two leaky modes are in-phase (with respect to their phase
difference at the input junction, e.g. see Fig. 2-91). This would limit the bend to a discrete
set of possible bend angles. There would be wavelength-dependent loss in the bend, an
undesireable quality in conventional applications.
2.5.4
Optimal (Non-circular) Terminating Bend Geometry
In optimizing circular terminating bends, we have considered two types of loss: bending
loss, which is a distributed loss and depends on propagation distance; and junction loss,
which is a fixed loss incurred at a point of discontinuous radius (and other parameters)
2.5 Bend Loss in Step-Index Slab Waveguides
57
along the waveguide path, such that the chosen (used) modes of the structure on either
side of the junction differ in shape and have non-unity overlap. A finite loss results from
excitation of unused modes.
Junction loss can be made distributed by making the change in radius (and other parameters) infinitesimal, thereby making the junction loss infinitesimal. A succession of such
junctions to achieve the desired change in radius would result in a distributed junction loss.
This suggests a more general geometry for a bend that terminates in straight waveguides,
namely one with multiple consecutive circular sectors of different radius between the two
straight waveguides. In the limit of an infinite number of such sections, varying the radius
from infinite to a finite minimum and back to infinite (straight), the junction loss (and
bend loss) would be distributed. However, such a structure has a growing number of free
parameters with the number of sections of constant radius, and is difficult to optimize by
brute force. Methods to find the optimal bend of this kind are not pursued further here.
Analytic Tools for Planar Waveguide Structures
58
2.6
Substrate Loss in Straight Leaky Waveguides
The air trench waveguides that will be used in this thesis are inherently leaky. For proper
design, it is of interest to calculate their per-unit-length leakage loss, which we will henceforth refer to as substrate (or bulk cladding) loss, after the material region to which lost
power is radiated in our structures. We consider a class of leaky waveguides whose crosssectional refractive index configuration can be represented as the superposition of that of a
waveguide with perfectly guided modes and an index perturbation far from the core which
causes leakage, by virtue of resulting in a refractive index higher than the modal index of
the waveguide mode.
In the case of practical interest where substrate loss is small, the
shape of the modal field distribution near the core (and the modal index) exhibits little
change from the unperturbed case. In the region of leakage, however, the field will change
from evanescent to oscillatory. The analysis is valid for any order leaky mode, but in our
later calculations, we concentrate on the fundamental mode.
In the subsections that follow, we develop an analysis that takes advantage of the uniform
index distribution of the leakage region in our cases of interest. Assuming that the field
distribution near the core is unaffected by the index perturbation, we define an equivalent
current sheet [23] at the boundary to the leakage region, using the unperturbed mode,
and integrate a Green's function over this source in the now perturbed, leaky region to
calculate radiation. The radiation field due to an arbitrary vector source distribution can
be conveniently obtained in homogeneous media by the method of dyadic Green's functions.
A detailed treatment is given by Kong [23]. This method is attractive for its conceptual
simplicity and direct physical interpretation, and for its simple extension to 2D cross-section
waveguides by utilizing the Green's function of appropriate dimensionality.
A more direct way, not pursued here, to evaluate the substrate loss is using a numerical modesolver with an absorbing boundary condition at the edges of the computational
domain to solve for the leaky mode. The imaginary part of the complex propagation constant represents the loss.
A finite difference mode solver (such as in Sec. 2.2.3) with a
perfectly-matched layer (PML) absorbing boundary condition [3] was demonstrated for a
1D cross-section leaky waveguide by Huang et al. [19]. An absorbing boundary condition is
needed to sufficiently attenuate the outward-propagating leaky wave before the computational domain is terminated by a conducting (zero E-field) boundary condition. This method
2.6 Substrate Loss in Straight Leaky Waveguides
59
is conceptually simple and can handle more general leaky structures, but its implementation can be difficult for low values of loss where the matrix problem is badly scaled, and
the numerical accuracy is limited by the computer's floating-point precision. For low index
contrast structures, the computational domain can be exceedingly large. Extension of this
approach to 2D cross-sections, which are of interest to us, could become computationally
taxing.
2.6.1
Substrate Loss of Layered Slab (D)
Waveguides
Prior to developing the dyadic Green's function method for leaky waveguides with 2D
cross-sections, we consider the ID problem.
The 1D cross-section leaky slab waveguide
composed of multiple layers of piecewise constant refractive index has an analytic eigenvalue
equation for leaky modes. We will use its exact solution for comparison in the 1D case, to
validate the approximations made in the Green's function method. We will also investigate
compensation of the equivalent current sheet in the Green's function approach for Fresnel
reflection at the interface with the leaky region. We will find that with compensation for
reflection, the Green's function method is an excellent approximation, and even without
gives a accurate figure of loss.
For the 1D analysis and comparison, we consider a simple, symmetric five-layer leaky
waveguide shown in Fig. 2-14b, a perturbation of the perfectly guiding slab in Fig. 2-14a.
Refractive indices ni, n2 and n3 are chosen in the core, cladding and leaky regions, with
interfaces at x, and X2, cladding thickness Ax
x2
- x1, and free space wavelength A0 .
Leaky modes of this structure have a decay constant in the propagation direction representative of the leakage loss. As a result they must, in lossless media, have an exponentially growing outward wave in the leaky region (Fig. 2-14b), which makes leaky modes
non-normalizable [39].
The growing nature of the outward wave with distance from the
core can be interpreted as time-delayed power lost from the core at progressively earlier
points in space along the propagation axis, where the mode carried more power (Fig. 2-14b,
inset). For this reason, we define lengthwise modal leakage loss by the ratio of outwardradiated power - per unit length - at the point of departure (x = X)
and the guided
power (-x4 < x < x-). Alternatively, a point of reference for outward radiation could be
chosen further from the core, but in this case would need to be normalized to the carried
modal power from an appropriate, earlier point along the propagation axis. The loss value
Analytic Tools for Planar Waveguide Structures
Analytic Tools for Planar Waveguide Structures
60
60
(a)
(b)
Figure 2-14: (a) Lowest-order TE guided mode of the slab waveguide, and (b) real part (solid),
magnitude (dotted) of the lowest-order leaky mode of a five-layer leaky waveguide.
is independent of the choice.
There is a spatial "cone" of validity of the solution based on the excitation point of the
mode (Fig. 2-14b, inset). That is, the range of x-values (here, about x = 0) for which the
solution is valid is greater after a larger propagation distance from the point of excitation.
Leaky modes are mutually non-orthogonal, and one cannot excite only a leaky mode by a
finite source, so the non-normalizability of the modes is not a physical issue. A discussion
can be found in Marcuse [39](p.31-43).
Exact Analytic Solution of the 1D Leaky Mode Problem
In the 1D case, waveguide modes are solutions of the scalar wave equation. We consider the
TE case. Taking advantage of the even symmetry of the structure and the sought solution,
we set the origin at x = 0, and postulate the form of the electric field (Sy) solution by
analytic extension from the guiding slab (real
Re{3} < ki, k3 , Re{/3} > k 2 , ki =
#)
case, assuming the leaky mode conditions
-ni (lossless media):
AO
A1 cos(kixx)
< (x)
=
A2
e-a2x(x-x)
if 0 < x < x 1
+
B 2 e+a2x(x-x2)
(2.20)
X2
if x > X2
A 3 eik3x(x-x2)
chosen for good numerical scaling, where kix
if X1 < X
=
-
12, ci
=
# 2 -kr,
and
1
is the
complex propagation constant of the leaky mode. In general, the cosines and exponentials in
2.6 Substrate Loss in Straight Leaky Waveguides
61
the above solution will have complex arguments. For low loss, the imaginary components of
i3, ki_, a2x in x < X2 will be small, and the field will resemble its familiar functional forms.
It is also clear that the field must exponentially grow in x > x2 for an outward propagating
leaky wave, if the mode propagates and attenuates along +z, because
(#R/I
=
/32
+ k2e= k2 is real
-k3xRk3xI)-
The modal eigenvalue equation is found by matching the tangential electric and magnetic
fields, here the scalar field amplitude <(x) and its first derivative, at x1 and X2.
These
boundary conditions yield a set of four linear equations, representable as a single 4x4
matrix equation:
--
-C
0
A1
a2x
-a2xC
0
A2
0
C
1
-1
B2
0
-a2xC
a2x
A
-1
B
=
0
(2.21)
-iD __ A 3 J
with the definitions
A
cos(kixxi)
B
-kil
sin(kixxi)
C
e-2,
a
AX
D
k3 x.
=
X2
-
X1
Vectors corresponding to existing modes comprise the nullspace of matrix M.
For the
matrix to have a non-empty nullspace, its determinant must be zero [62], so the eigenvalues
(propagation constants) of the leaky modes are given by the roots of the determinantal
equation D(,3) = detIMI = 0, with
D(13) = (a2x - iD)(a 2xA + B) - (a2x + iD)(z2xA - B)C 2
(2.22)
D(3) is a transcendental function of #, but a complex root finding package, such as FindRoot
in Mathematica [71], can easily solve for the complex propagation constant, /3, of a leaky
mode. For low losses, always true in the cases of interest to us, a perturbative solution gives
excellent results.
When the leakage region is pushed away from the core to infinity (Ax -- oc), C -* 0 in
eqns. (2.21) and (2.22), all elements of the off-diagonal 2x2 sub-matrices of M go to zero,
and the eigenvalue equation (2.22) degenerates to the well known one for the TE guided
modes of a slab (see, e.g. [17]). When the leakage region is far from the core, the complex
62
Analytic Tools for Planar Waveguide Structures
Table 2.2: Leakage Loss (dB/pm) of ID Waveguides by Analytic and ECS Method
Analytic Method
Exact
Perturbative
0.260298905
1.147679535
0.420134537
1.100932465
0.613103225
0.840265690
0.540820423
0.633875464
0.334980594
0.353687145
X2
2x1
0.74
0.75
0.8125
0.875
1
Equivalent Current Source Method
Uncompensated
Reflection-comrpensated
2.266922771
1.435280986
2.155247733
1.364574978
1.571719882
0.995119768
1.146180715
0.725693617
0.609549524
0.38593059
1.5
2
2.5
0.0304060142
0.00246577041
0.000197544070
0.0304762610
0.00246618649
0.000197546713
0.0487565083
0.00389992447
0.000311946269
0.0308697280
0.00246920076
0.000197505867
3.5
4.5
1.26412781 E-06
8.08795896E-09
1.26412792E-06
8.08795896E-09
1.99584584E-06
1.27695088E-08
1.26365116E-06
8.08489529E-09
# will approach the real
#0
of the TE slab. A first-order Taylor series expansion of D(O)
about /0 yields a complex correction to the propagation constant that accounts for loss,
D (0) = D (0) + --D
()3 - )3o) + --- =0
-D(0o)
50 = 0 - 0
0
resulting in 3 = /0 + 6/,
DO(O) = D(O)|c=0.
solution of
/
a a3
D1 3
where io is a root of the guided slab eigenvalue determinant
This calculation is in excellent agreement with an exact numerical
for |631 < 10|, which is valid for loss values of practical interest.
Figure 2-15 shows leakage loss (in dB/m) vs. displacement of the leaky region from
the core, Ax, calculated for two examples with ni
=
1.5, n 2 = 1.0, n 3 = 1.4 (silica) or
3.0 (Silicon), Ao = 1550nm and slab halfwidth x, chosen at the cutoff of the second guided
mode. Table 2.2 gives a comparison of the exact (complex eigenvalue) and perturbative
evaluation of the leaky mode propagation constant.
Equivalent Source-Green's Function Solution of ID Substrate Loss
The 1D leakage loss problem solved exactly above is now treated by the approximate equivalent source method. The waveguide is now centered at an arbitrary xO.
We define an
equivalent current sheet at X2 in Fig. 2-16a to represent the field in x > x2, then introduce
an index perturbation to make the region leaky (Fig. 2-16b) and evaluate the leakage loss
using a Green's function (of the driven vector wave equation (2.4)) method to compute the
radiation field in the leaky region. This approach is more involved, but is easily extended
to the 2D cross-section case, where an exact solution may not exist.
2.6 Substrate Loss in Straight Leaky Waveguides
63
Leaky slab waveguide loss per unit length (Silica substrate, n
I
I
I
=
1.4)
II
I
Uncompensated ECSM (1 D Green fcn)
-.' . ..Reflection-compensated
..-...-...
ECSM
O Exact analytic solution
-
10 0
10-2
10
~0
-4
10 -
0.
5
1
1.5
2
2.5
3
3.5
4
4.5
Interface distance from core axis (core widths)
(a)
Leaky slab waveguide loss per unit length (Silicon substrate, n3 = 3)
''0Q
100
--
0
'".
Uncompensated ECSM (1DGreen fcn)
Reflection-compensated ECSM
Exact analytic solution
E10 -4
cj)
0
'0
10-
IiI
10 -8
0.5
1
3.5
4
1.5
2
2.5
3
Interface distance from core axis (core widths, x2/(2x,))
4.5
(b)
Figure 2-15: Substrate loss using the analytic and equivalent current source (ECSM) methods, for
an example with ni = 1.5,
n2 = 1.0, AO = 1550nm at the cutoff of the second TE
mode. (a) Silica bulk cladding (n3 = 1.4) or (b) Silicon substrate (n 3 = 3) as leaky
region.
Analytic Tools for Planar Waveguide Structures
Analytic Tools for Planar Waveguide Structures
64
64
(a)
(b)
Figure 2-16: Equivalent current sheet configuration for (a) the unperturbed (guiding) waveguide,
and (b) the leaky waveguide after the introduced index perturbation in x
> X2.
Modes are superimposed for reference.
Consider first the unperturbed slab waveguide from Fig. 2-14a (reproduced in Fig. 216a). If we focus on a region of interest x >
we can define an equivalent current sheet at
x2,
x 2 to replace the (source) fields near the core (x
(J) and magnetic
(M)
surface currents at
X2
<
x 2 ). A range of combinations of electric
(in Fig. 2-16a) can reproduce the identical
modal field pattern to the right of the interface [23].
For simplicity, we set M = 0 and
choose an electric surface current alone, defined as
Ko = n
x
3'R(x2)
= -
27Lz(x2)
=
2 a Y
i--
(A/m)
(2.23)
In the unperturbed waveguide, this equivalent source will reproduce the evanescent field of
the guided mode in the region x > x 2 . Keeping the same current sheet at x = x2, but
modifying the region x >
x2
to represent the leaky waveguide implies the assumption that
this index perturbation has a small effect on the source mode in the core. With source K 0 to
the left of the leaky region (still in n 2 index), some Fresnel reflection of the evanescent wave
in region x1
x <
X2
is expected at the boundary. The reflected wave will be exponentially
decaying toward the source (B 2 in eqn. (2.20)), so moving the equivalent current sheet
slightly further to the left of
X2
exponentially diminishes its influence on the sheet itself.
Thus we can use the same equivalent source in the presence of reflection, and account for
the reflection independently.
First, we ignore reflection, move the current sheet without compensation across the
boundary to x4 (inside index
n 3 ),
and use the Green's function method to find the radiation
2.6 Substrate Loss in Straight Leaky Waveguides
65
field in the homogeneous leaky region. We now develop the 1D Green's function method.
The wave equation driving term is the volume current source (with &/&y = 0),
J(i) = f(x, z) =()ez
=
(A/iM2 )
Ko6(x - X2)ei
(2.24)
because the equivalent source has the same z-dependence as the modal field used to generate
it. The radiation loss then results from the far-field solution to the driven vector wave
equation (compare eqn. (2.4)),
(V
(x, z) = iwpkoeilz3(x
x V x -k 2)
-
(2.25)
x2)
Noting that E(x, z) is a vector Green's function of the wave equation, we simplify it by
letting [23],
E(x, z)
+
VV - g i(x, z)
(2.26)
where I is the unit dyad (matrix). Eqn. (2.25) is reduced to the scalar wave equation (in
three vector components)
(V 2 + k 2 )4i(x, z)
=
-iwpkoei'3 z6(X - X2)
(2.27)
Another definition,
(2.28)
g1(X, z) = iWPKOei3zg1(X - X2)
reduces eqn. (2.27) further to the scalar wave equation
(2 + k
with k
-
)g()
=
-6W()
/32 , whose solution is the 1D scalar Green's function [23],
91()
(2.29)
= i2ikxIxI
Substituting (2.29), (2.28) into (2.26), noting that here V = ik =
ikx +
i3 and setting
the origin at the source, X2 = 0, for simplicity, we find the radiated electric field for x > X2,
$(xz )=
-x
-Jx
Xk -2
Jz)
k2zJY
+ 9 Jy +
Jz -x
kk2z
Jx)
k2Je
ei(,3z+kx x)
(.0
The magnetic field follows from Faraday's law (2.1),
1k
z)
=
-(X,
kV
=
X E =
OJ, -y#
)
(x
J
Jz) - kx
JzJz
kx Jyj eN(23
Analytic Tools for Planar Waveguide Structures
66
The only x- and z-dependence of (2.30) and (2.31) is the exponential propagation factor.
With no approximations, we find the time-averaged, far-field radiated power density,
(S)
Re{Ex H*
=
81 kx1 2
kTJ2-_
+
[Xsk*JY2
2i Im{kx*} kxJ,
k~+y k2
) J* +
,J*
kJ -
+2(T*tJY|2 +
+
Jx
(2.32)
e2
Jx
Returning to the problem in Fig. 2-16, we have k = k3, kx = k3x, and from eqns. (2.23) and
(2.24), Jx = Jz = 0. With these simplifications, the power density radiated in the lateral
(x) direction is
azay
)X = Re
=
W
_8k3
e-2(z-k3xX)
J(.
12
(W/m 2 )
(2.33)
This is the power lost in the direction normal to the core (x), per unit waveguide length
(z) and height in the invariant (y) slab direction. To obtain the loss, this radiated power is
normalized by the power guided in the core (only in xo < X
O-guided
Re{/*}
Oy
2wp
Loss coefficient a (P oc e-
2a
2
az)
z
X
2
I
2
<
dx
x 2 by symmetry),
(W/m)
(2.34)
1
can be approximated as the ratio loss per unit length,
y
__2P____/_z
(Pguided/09y
= Re
(wop)
2
____
X2
4ksxO*
(Np/m)
(2.35)
X2|Iy|12 dx
resulting in a dB loss L = -20 log1 0 e_,(,0-6 m) (dB/pm). The term exponentially blowing
up in x was removed by evaluating radiation at the point of departure, near x
= X2 =
0.
Figure 2-15 shows a calculation of loss and a comparison to the exact solution. The loss
is slightly overestimated by this method because it does not account for Fresnel reflection
at
X2
introduced by the index perturbation used to add the leaky region to the guided
slab. Fresnel reflection (of the evanescent wave at x-) reduces the effective current sheet
"translated" across to x.
The loss calculated here is a good upperbound, particularly
when the index contrast between the cladding and the leaky substrate region is low. We
now proceed to correct for the Fresnel reflection and obtain a better result with this method.
2.6 Substrate Loss in Straight Leaky Waveguides
67
Fresnel Reflection-Corrected ID Green's Function Solution
We can calculate the field reflection coefficient at boundary x 2 by matching the evanescent
waves in region 2 to the leaky wave in region 3 of Fig. 2-16. We use this reflection coefficient,
calculated in Appendix A,
I-'~F
exp 2i tan- 1C4
k 3 cos
3
}
where 03 = tan- 1 (/R/k3), to translate the tangential magnetic field, which generates the
equivalent current sheet, across the boundary at x2:
N2(42+) = (1 - I-)Hz(22)Repeating the radiation loss calculation with this new value of magnetic field generating
the equivalent current source in eqn. (2.23) yields an excellent match to the exact solution
in Fig. 2-15 (see Table 2.2).
2.6.2
Substrate Loss of 2D Cross-section Waveguides
by the Equivalent Source Method
Having validated the equivalent source calculation in the ID case against the exact solution,
we now proceed to analyze a leaky waveguide with a 2D cross-section. The particular case
of interest in this thesis is one where the leaky region is of constant refractive index.
The ideal, guiding air trench waveguide in Fig. 2-17a becomes leaky in a practical
realization where its depth is truncated below and above the core (Fig. 2-17b). Leakage
occurs only to the substrate, when the substrate index is higher than the modal index,
#/ko.
The air or trench material above the waveguide is always guiding, as its index is always
lower than the modal index in this configuration.
Analogously to the 1D case, focusing on a region of interest y < -h, we define an
equivalent current sheet,
f(x, y, z) = f x (2HT)
=
,Ux+ Jz = K(x, z)6(y+h) = K(x)eiWz3(y+h)
(A/M 2 ) (2.36)
where the generating magnetic field, R comes from the guided mode of the unperturbed
structure, obtained for this structure from a numerical modesolver (Sec. 2.2.3).
Because
the source here is a function of x, reflection compensation is not trivial, as in the ID case,
and would require a decomposition of the radiated field into a full set of radiation modes on
-
68
bf
L
Analytic Tools for Planar Waveguide Structures
y
z
X
region of
no interest
current sheet
K(x,z)
region of
interest
truncated
depth
o
X : -O
current sheet
K(x,z)
h
n.
3n
z
region of
o interest
n3
(a)
(b)
Figure 2-17: Equivalent current sheet configuration for (a) the unperturbed (guiding) air trench
waveguide, and (b) the leaky waveguide after truncating the air trench at a finite
depth above and below the core. Guided mode in (a) is superimposed for reference.
both sides of the boundary y = -h. This is a complicated mode-matching matrix problem,
and we only pursue an uncompensated solution to obtain a good upperbound on the leakage
loss.
Green's Function Formulation
The radiation field solution in the leaky region follows similarly to the 1D case, by integrating
a dyadic Green's function over the source J, in this case a planar distribution at y = -h.
The dyadic Green's function of the vector wave equation is defined as the solution to [23]
V x V x -k2)G(,
F')
=
F')
6(+-
(2.37)
where primed coordinates describe the source distribution and the unprimed coordinates
describe the observation point. A comparison with the driven vector wave equation (2.4)
yields the solution for the electric field [23],
E(x, y, z)
=
iwpU
= iWp
JJ
J
dx'dz' G(x, x', y, -h, z, z') - K(x')ei/z'
dx' U(x, x', y, -h, z, 3). -K(x')
(2.38)
where G is the spatial Fourier transform of G with respect to z'. Following [23] we let
(similarly to, but not the same as eqn. (2.26))
G(r, rf') =
+ k2 VV] g3(,
')
(2.39)
2.6 Substrate Loss in Straight Leaky Waveguides
69
and simplify eqn. (2.37) to the scalar wave equation (with 3D scalar Green's function 93)
(V2 + k 2 )93 (Fl
') = -6(
, ')
Exploiting the symmetry of our problem, a further simplification can be made by removing
the z-dependence of the source (from eqn. (2.39)),
G(x,',y, y', z,)
= i+
1 V
dz'93 (r, r')e iz'
k2VV]
4
+ k2VVg 3 (x, x', y, y', z, f3)
where j3, the Fourier transform of 93 with respect to z', is a solution to
(V 2 + k 2 )j 3 (x, x', y, y' Z,
)
-S(x - x')3(y - y')eiIz.
A further definition
rG3(e
dimensions,
', y,
ati
te
Y
w
ea)eisz
reduces the Green's function to two dimensions, satisfying the scalar wave equation
(v
+ (k 2 _ /32))g2(x, x', y, y')
(vT+
for kP =
/k
2
-
=
k) g2( ,P')
-6(x - x')J(y - y')
6VI~ p'|
i 'I
26(,p-
2. In polar coordinates, this is the usual 2D scalar Green's function [23],
92(p, p') =
4 Ho(')(kP p - p'|)
where HOM1 is the zeroth-order Hankel function of the first kind [7].
The Radiation Field
The electric field in eqn. (2.38), due to the equivalent source J, is simplified by the 2D
Green's function to
E
=
iWp i+
esZ
_VV
Jdx'
g2(x, x', y, -h)K(x')
(2.40)
To find the radiation field, we take the leading asymptotic (far-field) form of the cylindrical
Green's function in eqn. (2.40) (kp1p5--
1 irkPIP-2
92(,~'- ~'
9(A")-4
2
P'
p'
>> 1 where I -,- "'I ~ p - V'. p)
i(kpI6)'I-7/ 4 )
2
r
4 irkppei(kpp-7r/4) -ikp.j3'
Analytic Tools for Planar Waveguide Structures
70
as detailed in [23]. For simplicity of further analysis, we move the coordinate origin to the
plane of the equivalent source to (x
=
0, y
For our purposes this can be accomplished
-h).
=
by letting h = 0 with suitable redefinition of the mode coordinates. The expression 2.40 for
electric field becomes
4
+IeVV]k2p7r/
r '
E
+
iWP
vv
2
)eioz
i(kprx'
(2.41)
k(x)e-ikp-p'
f(4)
h(p,z)
where we have defined a propagation factor h(p, z), and a vector current moment f().
vector current moment is only a function of the angle of observation,
retardation factor in the integral, for "5'= : x'+ y' and ,
-ik(xcos
ikp-
Only
#-dependence
=
#,
The
because the phase
cos # + 9 sin q, expands as
4+y'sin q )
remains after integration over x', as y'= -h is fixed (see eqn. (2.40)).
The del operator above operates on the unprimed (observer) variables.
We find its
far-field form by considering (2.41) and keeping only the leading asymptotic (p-2) terms,
a
-18(9
+
+6
~ikp + i/
az
ap
p ao
V
This simplifies the dyadic operator in (2.41) to a multiplicative factor (using k 2+02
The resulting radiation electric field (with f=
E ~ iwp h(p, z) [
fpf +
(Of, - kpfz) +fo
-
2),
qfp + ;f,) is,
-
p-
L
kpfz)]
The magnetic field follows similarly as (2.31) does from (2.30), resulting in a radiation power
density (compare with (2.32))
(L)
=
(k*If2+
h(p,z) 2 Re
+
I{kp,3*} /
-i
Imk2i
+
kpffz -fP)
O*|fO|2+
(kpfz
kpfzz
-
3kp) f +
-
3fp
2)
where the only p- and z-dependence of (S) is in the h(p, z) propagation factor,
h(p, z)
=
-
1
I
87rjkP p
e 2kpIpe-20z.
2.6 Substrate Loss in Straight Leaky Waveguides
71
The exponential decay in z and growth in p are consequences of the leaky mode formulation.
We will evaluate the radiated power using the far-field result above, but integrating near
the point of leakage (current source plane) where the radiated field is not time-delayed with
respect to its point of departure and the exponential growth in p is thus absent. Recall that
for evaluating the vector current moment in the above expressions, we have set h = 0.
Extracting the Loss Coefficient
We evaluate loss similarly to the ID case, by finding the ratio between the power lost
radially per unit waveguide length, to the guided power above the leaky region (compare
Radiated power per unit length is found by integrating the
expr. (2.33), (2.34), (2.35)).
radial power density due to leakage over the lower half-plane (90' <
#
< 180') at any
constant radius R, with the implicit understanding of allowing R -+ 0 in order to eliminate
the exponential growth in p:
dPos
dz
=
lim
R-0
=
f
Rd#((S) . )
r
(W/m)
p=
e -2/z j
dol6
ij
(Re{kp}If 4 J2 + Re
kpfz
-
fp 12)
(2.42)
The "guided" power below has the same z-dependence, and thus the ratio which gives the
loss coefficient will be independent of z. Integrated over the cross-section above the leaky
region (denoted A), the guided power (from guided mode fields S, H) is
Pguided =
(S) -ida =
Re{8
x
*
da
(2.43)
(W)
To evaluate the guided power (as well as the equivalent current sheet), we assume that the
leakage does not change the mode shape significantly, and we use the mode for the original
(infinite) guided structure, obtained from a 2D cross-section vectorial mode solver (see Sec.
2.2.3) to evaluate expr. (2.43). The same mode solution is used in defining the equivalent
current sheet, required for the evaluation of the vector current moment, f(#).
Noting that the equivalent current sheet derived from the unperturbed mode solution
has only , and
components (expr. (2.36)), K, = 2Hz and Kz = -2H,,
the vector current
moment is then (with each component evaluated and integral as defined in expr. (2.41)),
(#)
= 4Ux +
fz =
x
cos 0 - O fX sin
#
+
(2.44)
(fz4
72
72
Analytic Tools for Planar Waveguide Structures
Analytic Tools for Planar Waveguide Structures
Air trench waveguide leakage loss to silica bulk cladding (n = n2 = 1.46) vs. trench depth
10-
A (A = 0.25%)
B (A = 0.7%)
C (A = 7% )
0-2
10
10
a 10
4
CO
CD
0
-j
105
"I
0-6
10-7 0. 5
1
1.5
Interface distance, h, from core axis (in core heights, WL)
2
Figure 2-18: Substrate loss using the equivalent current source method (ECSM), for air trench
waveguide examples from Section 2.2.3, 3.3.1 (Table 3.2), varying in index contrast
A = 0.25% -*
7%. Silica bulk cladding (n4 = n2 = 1.46, Fig. 2-17) only is considered
in the leaky region.
The unperturbed mode solution, and expressions (2.41), (2.44), (2.42) and (2.43) are sufficient to evaluate, for small loss values, the loss coefficient,
1 dPoss/dz
2 Pguided
(Np/m)
Calculated Results
The method above was used to calculate substrate loss from air trench waveguides used in
this thesis. The waveguides used are those for which vectorial mode solutions are obtained
as examples in Section 2.2.3. Figure 2-18 gives a plot of the leakage loss as a function of air
trench depth below the core axis, in units of core height, for three waveguides.
Chapter 3
Air Trench Bends:
Design and Simulations
The present chapter contains the main work in this thesis, the design and analysis of Air
Trench Bends (ATBs) that allow denser silica integrated optical circuits. The intent here
is to give an efficient exposition of the problem at hand, and for results which involve a
significant amount of mathematics or background theory, the reader is referred to Chapter 2.
Here, data relevant to the discussion is included from Chapter 2 and referenced as needed.
Air Trench Bends are proposed as a theoretically and technologically simple way of
dramatically reducing waveguide bend radius, and increasing the density of integration in
silica PLCs (planar lightwave circuits). High density silica PLCs are desirable because of
their efficient fiber-to-chip coupling, low scattering losses, and reliance on widely-used silica
bench technology. As part of the ATB design, a new "cladding taper" is introduced to
minimize junction losses [49]. Other applications of this new kind of taper are described in
Chapter 4.
First, a brief discussion of the limiting factors of the density of integration in silica
PLCs is given to justify our concentration on the waveguide bend problem (Sec. 3.1). In
the sections that follow, the waveguide cross-section and circuit layout is discussed (Sec. 3.2),
the design of the straight and bent air trench waveguide (Sec. 3.3), and that of the cladding
tapers for low junction loss (Sec. 3.3.3). These components are put together and optimized
to obtain compact Air Trench Bend designs for several silica PLC index contrast scenarios
(Sec. 3.4).
Air Trench Bends: Design and Simulations
74
Throughout this chapter, three case studies with different index contrasts - A ~ 0.25%,
0.7% and 7% - are followed, and referred to as examples A, B and C, respectively. The
cladding is chosen as silica (nci = 1.46) for all three cases. Example A represents a fiber-like
(low-A) silica PLC with low-loss fiber-to-chip coupling; B is a high-A silica example [42];
and C is above normal silica PLC A's but is useful for simulations (because of its small size
in wavelengths) and helpful for comparison. These are realizeable in the silicon oxynitride
(SiOxNy) material system (see, e.g., [6]).
Limiting Factors of Integration Density in Silica
3.1
It is generally recognized that the integration density of conventional (buried channel, ridge
waveguide, etc.)
integrated optical circuits is dependent on the index contrast between
the core and cladding regions. This has motivated the push toward high index contrast
(HIC) platforms, such as SOI (silicon-on-insulator), which can achieve very high integration
densities (e.g. [31]). HIC platforms are hindered by the difficulty of fiber-to-chip coupling
and by fabrication defects, which manifest themselves as scattering loss.
Density of integration is limited by several factors related to index contrast, which
defines not only the size of waveguides, but also the minimum spacing between them.
Some of these factors are the waveguide bending loss, scattering loss and fiber-to-chip
coupling requirements. The latter two in direct tradeoff with increasing index contrast. We
concentrate on the basic index contrast limitations of waveguide confinement and coupling,
and those imposed by bend loss, and illustrate their relative importance by a simple analytic
study. Scattering loss and fiber-to-chip coupling are briefly discussed afterward.
3.1.1
Index Contrast and Bend Loss in Integration Density
The inherent limitation imposed by index contrast on the integration density is the dictated
size of waveguides for single mode operation, and the minimum relative spacing between
parallel waveguides that keeps direct-coupling crosstalk within acceptable bounds. In the
case of buried channel waveguides, the aspect ratio of the rectangular core - under a given
index contrast and with the single-mode condition - strongly affects side-coupling distance.
Before going further, we define a measure of index contrast. In general, in the literature,
no exact boundary between low and high index contrast is commonly accepted. In silica,
3.1 Limiting Factors of Integration Density in Silica
75
the index contrast parameter A is commonly used (e.g. [28],[42]),
no - n.1
neo + ncl
where nol,
An
A
nci are respectively the core and cladding indices. A represents the fractional
index difference with respect to the average index and can be expressed as a percentage.
The "average index" h in the definition of A retains a sensible physical meaning only where
An
<
ii (to call A a percentage difference), which is essentially the commonly accepted
range of index contrast for silica PLCs, 0 < A < 1.5% [42].
Bend loss is normally dominant over waveguide coupling crosstalk in limiting integration
density. Because there are many possible geometries of integrated optical circuits, rather
than pursuing a general argument, we illustrate this with a simplistic analytic example.
We consider a 2D integrated optical circuit of N x N slab waveguides in a grid arrangement (Fig. 3-1) as, for example, in an N x N WDM multiplexer [29]. With a regular spacing
d between all waveguides (of width 2w), the size of the circuit is calculated for an acceptable
loss or crosstalk after a single traversal of the circuit. Because slab waveguides are used, an
analytic expression for the normalized circuit size (in units of wavelength) can be obtained
as a function of index contrast A and grid size N, given a loss or crosstalk specification.
Bend loss limitation of integration density in long, narrow geometry optical circuits (e.g.
Mach-Zehnder filters) is discussed in [28].
Limitation due to Parallel Waveguide Coupling Crosstalk
In the case of parallel waveguide coupling, crosstalk in the adjacent waveguide becomes
For example, a crosstalk
significant before loss in the source waveguide is of concern.
suppression of -30dB corresponds to a loss of -0.004dB. We choose a crosstalk of < -30dB
after a propagation distance of two circuit lengths as the target. The expression for the
crosstalk results from coupling of modes in space (e.g. [17]), the geometry of the hypothetical
circuit described above (Fig. 3-la), the simple eigenvalue solution and functional form of
the lowest-order slab TE mode, and from the exactly soluble overlap integrals for the spatial
coupling constant. The adjacent waveguide crosstalk is
d
X21(V, -- , A,
W
-sny2l
-sn(e(Q
N) = sin2(1K21INd) = sin2
"w) d
3w
-N
W
f(V)
(3.1)
Air TErench Bends: Design and Simulations
76
Air Trench Bends: Design and Simulations
76
:-
N waveguides
-
-N
(a)
waveguides
(b)
-
>
Figure 3-1: N x N slab waveguide grid optical circuit for comparison of integration density bounds
due to (a) parallel waveguide coupling crosstalk and (b) bend loss.
with
cos (kxw ) [axw(e2axw - 1) cos( kxw) + kxw(e 2 axw + 1) sin(kxw)]
2=1 + e- 2axw +sin(2kxw)
f(V)
czw
2kxw/
where crosstalk X21 is only a function of four normalized parameters, because the slab
waveguide parameters czw and kxw are determined by the normalized frequency, V =
kow
n2
_ rc2 (e.g. [20]). Single mode slab waveguides at the cutoff of the second mode, in
TB polarization, are used (V = ir/2) to maximize mode confinement for small side-coupling,
but avoid multimode support. Thus, X21 is a function of three remaining normalized variables
-
d/w, A and N.
The dimension of the optical circuit is computed by making an approximation valid for
all silica index contrasts, V =k w 2 2iAn
2
2A(kown)
(i.e. )
~
c
ncj). Then, the
normalized device size, z/A (where average material wavelength A = Ao/h) is
z
dw
/d
VN
where d/w is implicitly defined in (3.1) in terms of chosen parameters V, X21, A and N.
These lower bounds on the (a) total and (b) per-waveguide (of N) normalized device size
are plotted in thin line in Fig. 3-2. Fig. 3-2a also shows that silica PLCs are fundamentally
limited to about a 1000 x 1000 waveguide grid on a 1cm x 1cm chip, under the assumptions
made here.
3.1 Limiting Factors of Integration Density in Silica
77
Parallel Coupling and Bend Loss Lower Bounds on NxN Waveguide PLC Normalized Total Size
10
-
-11
-
-
- -
106
---
N
=104
7
Bend Loss
LowerBoun
-
---.....
.........
105
N=101
2
N= 10
N= 103
n d%
1a 10
E
0
10-
N
E
U5
0
-j
Q_ 103
E
E
Parallel Waveguide
Coupling Lower Bounds
0.01
101L
0.01
0.0 01
0.1
Index contrast, A
(a)
104
Parallel Coupling and Bend Loss Lower Bounds on Normalized PLC Size Per Unit Waveguide
.
.
. . . . . .
.
.
.
-N =10
-
N = 103
N=10
Bend Loss
Lower Bound
(identical for all N)
10
.0
E51012
0
-J
Fn 1
100
0.001
Parallel Waveguide
--
Coupling Lower Bounds
-
-
0.1
0.01
Index contrast, A
(b)
Figure 3-2: Lower bounds on the N x N slab waveguide optical circuit size due to parallel waveguide
coupling crosstalk (thin lines) and bend loss (thick lines). Minimum circuit size plotted
against A and N, normalized to (a) material wavelength, and (b) material wavelength
and N. Coupling crosstalk: -30dB, bend loss:
0.01dB/90*. Chip-size upper bound on
device size is indicated for silica at AO = 1.5pm.
78
Air Trench Bends: Design and Simulations
Limitation due to Bend Loss
Next, we consider bend loss limitation of integration density, and disregard the waveguide
coupling, but assume that a single bend with radius R
=
d (Fig. 3-1b) is required along any
traversed path. Bending loss only is considered and junction loss ignored.
The empirical fit formula (2.18) (Sec. 2.5.2 and [47]) for required radius as a function of
index contrast, loss and wavelength that is valid for low index contrast is used, expressed
in the present variables,
R
R(A,
R
=
A A , Y) =
Rneo
O
c0
C(-Y)
=2
27r (1 - '
1.54
C(-Y)
C(
27r (2A)
.54
with C(-y) given in (2.19), where we approximate 1 - ni /n 2 by our 2A, valid for low index
contrast, and we replace Ace by the average material wavelength A as no '
i. The loss, y,
in this formula is in decibels, dB/90 . Also, here R - d.
For comparison with coupling crosstalk, we choose a reasonable bend loss to be 0.01
dB/90'. The lower bound due to bend loss for optical circuit size vs. A and grid size N is
overlaid in Fig. 3-2 in thick line. Clearly, bend loss presents a much more restrictive lower
bound on the PLC size, for the adopted scenario. A better bend design would allow us to
approach the lower bound set by coupling crosstalk in Fig. 3-2.
3.1.2
Scattering Loss and Fiber-to-Chip Coupling Limitations
Scanning Fig. 3-2, we observe that the device size is most easily decreased by increasing the
index contrast, particularly if the dominant bound is the bending loss, as it is here. However,
high index contrast (HIC) waveguides have much higher scattering losses than their low
index contrast counterparts. Because HIC waveguides are small, surface defects are large
in relative comparison, and their scattering is enhanced by the high index contrast. There
will be a tradeoff between increasing index contrast to improve the density of integration
and keeping scattering losses within acceptable bounds. A study of bounds due to surface
scattering loss is not pursued, but much work exists on its modeling (e.g. [35, 36, 46, 26, 27]).
If a better bend design allowed us to approach the bound imposed by waveguide coupling,
the incentive to go to high index contrast would be diminished, due to the much gentler
slope of the coupling bounds in Fig. 3-2. This is the intended purpose of the Air Trench
Bends in this thesis.
3.2 Silica Waveguides and Optical Circuit Layout
79
Another important, though indirect, bound on integration density is imposed by the size
of the single mode fiber, with a core-and-cladding diameter of 125pum, which couples optical
signal into and out of the PLC. Clearly, for a given PLC size, the number of input and
output waveguides is limited to the number of fibers that can be butt-coupled side-by-side
to the chip perimeter, proportional to N. Nevertheless, the total integration density on the
chip including waveguides that do not lead to fibers can still be that of Fig. 3-2.
3.2
Silica Waveguides and Optical Circuit Layout
In the sections that follow, we will propose to increase the integration density of silica PLCs
by introducing Air Trench Bends. We first choose the silica PLC waveguide parameters.
For buried channel (rectangular core) waveguides, the design parameters are the core size
and aspect ratio (equivalently, width and height), given an index contrast and wavelength.
The size of the waveguide core is chosen as large as possible for strong confinement (low
side-coupling and scattering loss) and good coupling to fiber, while keeping the guide small
enough to be single-moded in each polarization. Multimode waveguides have the problem
of modal dispersion [20].
Low Index Contrast Rectangular Silica Waveguides
For design of all rectangular silica waveguides in this thesis, instead of the Effective Index
Method, we use the (full 2D cross-section) numerical modal solutions calculated by Goell
[15] (also in [33],[39]). Goell's normalized dispersion curves are valid for nc/nci < 1.05, or
A < 5%.
We choose a square (1:1 aspect ratio) core for good match to the fiber mode. From [15],
a square waveguide of core width d at the cutoff of the second guided mode has V-number
Vsquare _ 2d
n2-
Ao
c
7r
n
1.38
(3.2)
compared to a slab waveguide with core width d = 2w, and V defined for the halfwidth,
2
Vslab
7r
2w
= 2A
2
2
no - n=1.
Hence, in silica waveguides with A < 5%, dsquare/dslab
(3.3)
=
1.38.
A square cross-section
waveguide can be 38% wider than a slab guide and still be single-moded (per polarization).
Air Trench Bends: Design and Simulations
80
We obtain silica waveguide dimensions from (3.2). For our case study examples, they are
given in Table 3.1. To allow for core width and refractive index tolerances in fabrication, the
waveguides chosen and used from here on are slightly smaller than the largest single-mode
waveguides, as shown in the table. Effective indices for 2D analysis of structures using these
waveguides are also in Table 3.1 (see Sec. 2.3).
Table 3.1: Single-mode Silica Waveguide Square Core Width and Eff. Indices for Ex. A-C
Property
A
Index Contrast (A)
Core Index (n0o)
Cladding Index (ncl)
Ideal Square Core Width
Actual Core Width (allowing for fab tolerances)
Core TM Effective Index (Nco)
Cladding TM Effective Index (Nc1)
TE Modal Index from EIM (nmode, see Table 2.1)
0.25
1.46365
1.46
10.35
10.05
1.462736
1.459409
1.461853
Example
B
0.68
1.47
1.46
6.25
6.05
1.467495
1.458381
1.465088
units
unt
6.62 %
1.56
1.46
1.95 pm
1.75 ptm
1.531297
1.442006
1.505381
C
Polarization-Independent Optical Circuit
By symmetry, a square waveguide has degenerate polarization modes, making it ideally polarization independent. In high index contrast, the large scattering due to surface roughness
can couple the two degenerate polarization modes and introduce polarization crosstalk, so
dual-polarization operation is better avoided.
Here, in low index contrast, this is not a
concern. However, polarization dependence is still an issue in low-A waveguide bends, resonators, and will be (as we later show) in the Air Trench Bends. Much labour is expended in
literature to compensate and design polarization independent devices. Polarization dependence is also introduced in the fabrication process, by imperfect core aspect ratios, slanted
waveguide walls, strain in the material and surface defects.
We abandon attempts to create a polarization independent Air Trench Bend, and instead
propose using a polarization-independent optical circuit topology that uses polarizationdependent elements (Fig. 3-3).
The optical circuit is duplicated in area, and requires a
polarization beam splitter and polarization rotator, but does not require any polarizationindependent components. The incoming signal polarizations are split, one is rotated, and
both are processed by identical optical circuits side-by-side. Then, the other polarization is
3.3 Air Trench Waveguides
81
TE
optical
circuit
TE
output
TM
90'
C 90'
TM
input
TE
optical
circuit
TE
Figure 3-3: Proposed polarization-independent optical circuit topology that uses polarizationdependent components. Uses two polarization beam splitters, polarization rotators
and two identical circuits (designed for the same polarization, e.g. TE-like).
rotated and they are recombined.
We argue that the chip area savings that will be made possible by using Air Trench
Bends will more than compensate for the doubling of the circuit (Fig. 3-3).
3.3
Air Trench Waveguides
In this section, in pursuit of a sharp waveguide bend for silica PLCs, we design a waveguide
with strong lateral mode confinement, by virtue of which it will be able to make sharp turns
with low loss. We then investigate its mode shape, bend loss and junction loss properties
in two dimensions (using effective indices from the EIM). We first design bends to have low
junction loss at interfaces to straight Air Trench Waveguides.
Then, we design cladding
tapers to allow these sharp bends to interface with low index contrast input silica waveguides. Putting together these bends and cladding tapers, and optimizing their parameters
for minimum size Air Trench Bends (given a total loss budget) is the subject of Sec. 3.4.
3.3.1
Cross-sectional Geometry for Strong Lateral Mode Confinement
Our first step to a sharp waveguide bend is a waveguide with strong lateral confinement.
This translates to high index contrast (HIC), and indeed bend radii on the order of the
wavelength have been reported for HIC systems such as silicon-on-insulator (SOI) [31].
However, HIC waveguides are very small in both dimensions. Coupling one to a silica PLC
waveguide would require horizontal and vertical tapering, and would complicate fabrication.
Air Trench Bends: Design and Simulations
82
trench
Sair
cocore
cladding
(a)
air trench
core
air trench
core/a
cladding
claddingg
(b)
(c)
(d)
Figure 3-4: Designing a waveguide with strong lateral confinement from a weakly-guided silica
waveguide in (a). Air Trench Waveguides with (b) trenches only to the sides of the
core, (c) trenches extended vertically to confine the evanescent tails of the mode above
and below the core, and (d) infinite trenches, where the ATW has a guided mode.
Furthermore, because we are concerned with planar structures, strong confinement in the
vertical dimension is not as critical as that in the horizontal plane in which sharp bends are
to be made.
A simple solution compatible with planar fabrication is to introduce air (or very low
index material) trenches only to the sides of the core region. The cartoon in Fig. 3-4 shows
a silica waveguide and a progression of Air Trench Waveguide designs. The introduction
of an air trench to either side of the core increases the lateral index contrast and modal
confinement. The core (keeping its height constant) must be narrower laterally than in the
silica waveguide to remain single-moded (Fig. 3-4)b. The mode remains (even more) weakly
confined in the vertical dimension. In bending, the laterally weakly-confined evanescent
tails of the mode above and below the core will radiate to a much greater degree than the
well-confined modal field in the core. Lateral confinement of the evanescent tails can be
improved by allowing the air trenches to extend further above and below the core (Fig. 3-4c),
consistent with planar fabrication. In the limit of allowing the trenches to extend infinitely
above and below (Fig. 3-4d), the issue is averted and bend loss is entirely due to the stronger
lateral index confinement. This loss and design of bend radii is pursued in Sec. 3.3.2.
Guided Modes in the Air Trench Waveguide
Another strong reason for extending the air trench above and below the core is that the
single-mode Air Trench Waveguide, for high enough lateral index contrast, will only have
a guided fundamental mode if the air trench is vertically infinite in extent. Otherwise, the
fundamental mode will be leaky. A brief explanation follows.
3.3 Air T'rench Waveguides
83
Due to the strong lateral confinement, the modal index of the fundamental mode may
be lower than the cladding index (see Table 3.2). In 1D multi-layer slab waveguides, we saw
that this implies an oscillatory, i.e. leaky, field in the cladding (Sec. 2.6.1). Here, this would
translate to an oscillatory, rather than an evanescent, field in the cladding "posts" above
and below the core in Fig. 3-4d, implying a leaky mode. In fact, this is not necessarily the
case, and the field in these regions can still be evanescent.
This is because a 2D cross-section waveguide has an extra degree of freedom for solutions
of the wave equation in each region of constant index. In a constant-index region
of a 1D waveguide, k2
=
k
2
-/32
=
(n2egi
-
n2ode)k2. If nmode <
rregion,
(nregion)
kx must be real,
and the field is oscillatory. However, in a cladding post of the 2D cross-section waveguide
in Fig. 3-4d, k2 + k2
=
k2 _
2
. If k2 + k > 0 (nmode < nregion), but k2 > k2 _
then ky
32,
can still be imaginary and the field evanescent in the vertical direction away from the core.
This is exactly the effect of the strong lateral confinement provided by the air trench. It
makes the lateral field fast-oscillating (large kx) leading to a guided mode.
For our case study examples A-C, we have used silica waveguides from Table 3.1 and
constructed Air Trench Waveguides by keeping the core height fixed and varying the width
to arrive at ATWs at the cutoff of the second mode. Because the index contrast between
the core and cladding for all our examples is negligible in comparison to the air trench, an
approximate width for single-mode operation can be obtained by assuming a slab waveguide
with silica (n = 1.46) core and air cladding, and using expr. (3.3). This yields an estimate of
0.75pm. Because of some vertical mode confinement in our ATW, its maximum single-mode
width will be larger. We choose a width of 1.05tpm for all examples to make the ATW more
Table 3.2: Air Trench Waveguide Core Width and Eff. Indices for Examples A-C
Property
A
units
Example
%
0.25
0.68
6.62
1.46365
1.47
1.56
Cladding Index (n01)
1.46
1.46
1.46
Trench Index (ntr)
Approximate Ideal Air Trench Core Width (Only)
Actual Air Trench Width (allowing for fab tolerances)
1.00
0.75
1.05
1.00
0.75
1.05
1.00
0.75 gm
1.05 pm
1.462736
1.072599
1.467495
1.072826
1.531297
1.072515
1.382048
1.386704
1.449381
Index Contrast (A)
Core Index (n0O)
Core TM Effective Index (N00 )
Cladding TM Effective Index (N01)
TE Modal Index from EIM (nmode, see Table 2.1)
Air Trench Bends: Design and Simulations
84
Air Trench Bends: Design and Simulations
84
)(
q-TE q-TM
(a)
q-TE
q-TM
q-TE
(b)
X :
Z
q-TM
(c)
Figure 3-5: Contour plots of the dominant electric field component of the fundamental quasi-TE
and quasi-TM modes of Air Trench Waveguides at the cutoff of the second quasi-TE
mode. (a)-(c) correspond to ATWs for case study examples A-C.
easily fabricable. The ATW is then weakly multimode, but the second mode is so weakly
guided that in practice it will experience large loss and not affect performance. The ATW
parameters are in Table 3.2, and contour plots of the dominant electric field component of
the TE-like (E.) and TM-like (E.) modes are shown in Fig. 3-5. The quasi-TE/TM mode
propagation constants in Table 2.1, and the fields in the mode contour plots suggest that
optical components made of Air Trench Waveguides will be polarization dependent. This
is inherent due to the ATW's cross-sectional asymmetry along the principal polarization
directions, x and y. It is for this reason that we will only use the better confined quasi-TE
polarization, and rely on an optical circuit layout as in Fig. 3-3.
Finite Width and Height Air Trenches
In the realization of an Air Trench Waveguide, the air trench must be truncated at a finite
height above and below the core (Fig. 3-6b).
Now, the bulk cladding below the trench
cannot provide a large lateral confinement (k.).
Thus, in the bulk cladding region, the
mode is leaky for nmode < nregion. The substrate or bulk cladding loss due to the finite
trench height must be designed to be small. In all design and analysis of bends, we assume
infinite trenches. The substrate loss is calculated as a correction according to the method
in Sec. 2.6.2.
A finite lateral width of the total air trench cross-section (Fig. 3-6b) is also desirable to
I
-
-
-
-
- -------
85
3.3 Air Trench Waveguides
Idealized Infinite Trench Cross-section
Actual Finite Trench Cross-section
air
air
trench
cldng
ridgesg
core
cor
trench
cladding core
trench
rga
depth
cladding
ze
(a)
bulk cladding (substrate) loss
(b)
Figure 3-6: Schematic of (a) an idealized Air Trench Waveguide, free of substrate loss, and (b) a
truncated, finite-depth Air Trench Waveguide, susceptible to leakage loss.
make the trench least intrusive to the surrounding components on the optical chip. This
width is easily chosen to be only several core widths away from the core, because the mode
is well confined laterally. For further analysis and design in two dimensions, we use the
ideal ATW, and treat it as a slab waveguide with the effective indices from Table 3.2.
3.3.2
Bends in Air Trench Waveguides
We now investigate the circular bending properties of Air Trench Waveguides with strong
lateral mode confinement for the purpose of designing sharp bends in silica. In continuous
bends, relevant to disk or ring resonators, only radiation loss due to a finite bend radius
is of concern. In finite-angle or "pie-slice" bends (e.g. 90') which terminate in straight
waveguides, we must also consider junction loss due to mode mismatch (including Fresnel
reflection) at the interfaces between the bent and straight waveguides.
Keeping in mind that single boundary (whispering gallery) bends have the minimum
bending loss for a given radius and index contrast (Sec. 2.5, e.g. [47],[57]), we calculate
the minimum bend radii vs. index contrast for a prescribed loss. This yields a bound on
achievable bend radius using ATWs.
Next, we consider junction loss in order to make low loss finite-angle ATW bends. We
show that junction-matching directly to a large silica waveguide mode can be low-loss,
but requires extremely large bend radius for good mode match and does not permit the
benefits of sharp bending offered by ATWs. Instead, we design ATW bends for good match
to straight ATW's, and defer the problem of mode matching to the silica waveguide to
Sec. 3.3.3.
86
Air Trench Bends: Design and Simulations
Minimum Bend Radius of Air Trench Waveguides
To establish a lower bound on the achievable bend radius in Air Trench Waveguides, given
a bend loss criterion, we consider the lowest-order whispering gallery mode (WGM) of a
single boundary bend (see Sec. 2.5).
The effective lateral index contrast is roughly the same in all ATW cases under our
consideration (A-C, see effective indices in Table. 3.2). We only show plots for a generic
slab waveguide case with indices 1.47:1.07, relevant to the 2D analysis of all three of our
examples. Figure 3-7a shows bend loss vs. radius for this case. Owing to the exponential
dependence of the bend loss value (in dB) on radius (Figs. 3-7a, 2-11b), the relative change
in radius for various loss values of interest to us (e.g. 0.001dB/90' to 0.1dB/90') is not large
(see Fig. 2-11a). We choose a target loss equivalent to 98% transmission (~0.088dB/90')
for further design. For all loss values of interest, the minimum achievable bend radius is of
the order of 5 -10pm in this index contrast (silica-to-air).
The full width at half-maximum (FWHM) field of the whispering gallery mode is plotted
against a larger range of radii (to be of use later) in Fig. 3-7b, with that of the straight slab
waveguide fundamental mode at the cutoff of the second TE mode included for reference.
The approximate minimum width of the bent waveguide to achieve single boundary, i.e.
whispering gallery, operation is also plotted (expr. (2.17)).
radius in single boundary bends.
The mode width depends on
The mode becomes more compressed at smaller bend
radii, until it passes a critical radius determined by the index contrast, beyond which the
bend loss is excessive and the mode expands radially outward (Fig. 3-7b). The plot shows
that the mode width is very insensitive to radius, or, conversely, that the bend radius is
extremely sensitive to modal width (Sec. 2.5).
Sources of Junction Loss in Finite-Angle ATW Bends
While not of particular interest in the analysis of bending loss, the modal width and qualitative shape are of direct consequence to junction loss in bends which terminate at straight
waveguides (Secs. 2.4.3, 2.5.3). In a junction of two dissimilar waveguides, the objective is
to match a mode of the one waveguide as efficiently as possible to a mode of the other. Here,
the modes of interest are the fundamental mode of the straight waveguide and the lowestorder leaky mode of the bend. Expressions for junction loss at straight-to-straight and
3.3 Air Trench Waveguides
87
Single Boundary Bend Loss vs. Radius for n c:n -1.47:1.07
100
10 1
1010-2
-4
10
0
4
8
10 ~
(D 10
6 10
1
10
12
-6
107
10-8
10-0
4
8
6
14
16
18
20
Bend Radius (pm)
(a)
Single Boundary Bend Whispering Gallery Mode Width (FWHM) and Waveguide Width for WGM Operation
2
[
.
. . , ,,
,
.
. . . . I
.
10
-
.
.
. . . ,I
I
.
. I
Straight Slab Mode Width for V =
-
. . .
Whispering Gallery Mode Width (FWHM)
Min. Waveguide Width for WGM Operation
n/2 (FWHM)
.
-
0'1
0,
a)
'0
0
-o
100
- . ..
.. . .
10
. . .. .. .. . .
10 2
Bend Radius (pm)
10
. . .. . . -..
104
(b)
Figure 3-7: For a single boundary bend with effective indices 1.47 and 1.07: (a) bend loss vs.
radius (by WKB method), and (b) whispering gallery mode width and minimum bent
waveguide width to be in the whispering gallery regime vs. radius.
88
Air Trench Bends: Design and Simulations
straight-to-bent slab waveguide interfaces, found in Sec. 2.4, are used for the calculations
here.
Two mechanisms of mismatch result in junction loss. They are evident in expr. (2.14)
and (2.15) in the overlap integral, where loss is due to mode shape mismatch, and in the
difference of the propagation constants, where it is due in effect to Fresnel reflection resulting
from differing refractive indices on the two sides of the interface (Sec. 2.4.2, 2.4.3). In our
junctions, the core index on either side of the junction is the same, so Fresnel reflection is
subdominant to mode shape mismatch in determining junction loss.
Junction Loss: ATW Bend Interfaced with Straight Silica Waveguides
Finite-angle ATW bends can be interfaced with straight Air Trench or silica waveguides. A
low-loss bend for a silica PLC would preferably be interfaced directly with a silica waveguide
and optimized for low junction loss. We demonstrate here that this is not a viable solution.
Using this approach, the loss has a lower bound of about 0.1dB per junction, and even so
the radius required for the optimum loss is far too large to be of use in reducing total bend
size.
Consider first a straight-to-straight waveguide interface, with a fixed-width PLC silica
waveguide on one side (Table 3.1), and a variable-width ATW on the other, to be optimized
for minimum junction loss.
The minimum loss, and corresponding optimal Air Trench
Waveguide width are given in Fig. 3-8 as a function of index contrast, assuming a silica
cladding (nct = 1.46) and air in the trench (ntr = 1.0).
In lower index contrast, surprisingly low loss is achievable (considering the silica-trench
index difference), around
0.1dB per junction. Nevertheless, in designing a bend, lower
overall losses may be desired and this limitation is a problem. It is interesting to note that
the optimal core width of the ATW is on the order of twice the size of the silica core for
low-A cases. The air trench waveguide is multimode.
Consider now an ATW bend interfaced with the straight silica waveguide of example
A in Table 3.1.
The FWHM mode width of this silica waveguide is about 10pm.
For
good mode match, the leaky mode of the ATW bend must be of the same width. From
Fig. 3-7b (see also Fig. 2-10, remembering that mode width is very weakly dependent on
A), the radius of a single boundary bend must be on the order of millimeters to meet this
requirement, becoming of no use in reduction of bend size (compare to radii of conventional
3.3 Air Trench Waveguides
89
Straight Air Trench-Straight Silica Waveguide Interface: Optimal Junction Loss vs. Index Contrast
-0.01
optimized free parameter)
-0.02
ntr
-.
-0.03
1
(fixed)
--
-0.04
-0
-0.05
0
-J
a
.2
-0.06
-0.07
-E
-0.08
-0.09
qn. (2.14)
FDTqD
0 2D
0.
--
10
10-1
10-2
10
Silica waveguide index contrast, A
(a)
Straight-Straight Waveguide Interface: Optimal Air Trench Waveguide Width vs. Index Contrast
1.9
. 1.8
CU
~1.7
LV
t-
1.6
FZ 1.6
1.5
5D
0 1.5
1.2'
10
3
2
10-1
10
Silica waveguide index contrast, A
100
(b)
Figure 3-8: Optimal junction loss at a straight ATW-straight silica waveguide interface. Claddings
1.0, 1.46 respectively, common core index from silica waveguide index contrast, A. (a)
Optimal junction loss, (b) ATW (normalized) core width for optimal loss.
Air Trench Bends: Design and Simulations
90
silica bends in Sec. 2.5, Fig. 2-10). Furthermore, because it is virtually lossless (out of range
in Fig. 3-7a), a much smaller radius could have been used, were it not for junction loss.
It is possible to use a smaller radius, and to expand the mode width by contracting
(instead of overmoding) the waveguide width into weak guiding as in a straight waveguide
far below cutoff of the second guided mode. An inner boundary added to the bend could
be used to squeeze out the mode. However, the mode would not begin expanding in width
before first compressing to the order of 1pm, the single-mode waveguide width, and the
qualitative shape of the leaky mode would be far different from the input silica waveguide
mode. Furthermore, introducing this weak guiding would bring bend loss quickly back into
the game.
Junction Loss: ATW Bend Interfaced with Straight Air Trench Waveguides
To obtain a small bend in silica, it is necessary to be able to use the well confined leaky
mode of a small, but still low-loss, ATW bend to traverse the 90 turn, but to find a sizeand loss-efficient way of coupling a straight silica waveguide mode into and out of this turn.
The leaky mode of the smallest ATW bends with acceptable loss (5-10pim in Fig. 3-7a) is of
a similar size and shape to the straight ATW fundamental mode at the cutoff of the second
mode (Fig. 3-7b).
For this reason, we interface the ATW bend to straight ATWs, and
optimize the junction loss. We discuss a new taper for efficiently interfacing the straight
ATW mode to the silica waveguide mode in Sec. 3.3.3.
The size of the finite-angle bend consisting of an ATW bend and input/output interfaces
to straight ATWs is now optimized for a given overall loss. The overall loss is the sum of
the bend loss and the two interface junction losses.
The junction loss here, where the bent and straight waveguides have the same core and
trench indices, is due to shape mismatch (width and qualitative shape) and lateral misalignment of the modes. The optimization of this standard bend is described in Sec. 2.5.3.
Even for optimal mode width and lateral offset, the junction loss may be non-zero due
to qualitative mode shape differences. As discussed in Sec. 2.5, high index contrast single
boundary bends where the radius is roughly of the order of the wavelength have a leaky
mode that may have negligible loss due to the high index contrast, but is asymmetrically
deformed in shape in comparison to the straight slab fundamental mode, which is symmetric
about the waveguide axis. No mode width or lateral offset can improve the mode match
3.3 Air TIrench Waveguides
91
past a certain maximum.
The remedy is in increasing the single boundary bend radius to allow room in the loss
budget, then introducing an inner wall to the bend to push the now-widened mode back to
the correct width, now with less asymmetry (Sec. 2.5.3). With the new degree of freedom,
in the limit of infinite radius, we obtain the straight slab waveguide and no junction loss.
In optimizing for a given finite overall loss, the radius is increased and the bent waveguide
width
(w),
straight waveguide width (w,) and mutual offset (Aw) optimized for minimum
loss until the loss spec is met,
min
Loss(w, Wb, AW)
{Ws,WbAW}
R
< LossSpec
-
for the smallest possible R. Because there are now three free parameters, the optimization
space is large. However, the bent waveguide width is limited to be smaller than the minimum width for whispering gallery operation (Fig. 3-7) and larger than single-mode straight
waveguide width; and the straight ATW width, due to equal index contrast, will be near
the bent waveguide width. A small increase in radius can quickly decrease the loss, so the
right order of magnitude size reduction can be obtained in a few manual steps.
Instead of optimization examples the parameters for which will only become meaningful
in the context of the full Air Trench Bend (Sec. 3.4), we show an FDTD simulation of
a bend optimized using all three parameters for illustration (Fig. 3-9c). Field plots of a
similar dual boundary bend with no offset and bend width chosen for dual and effectively
single boundary operation are shown in Fig. 3-9a-b, respectively.
Return Loss in Finite-Angle ATW Bends
As described in Sec. 2.5.3, transmission efficiency (insertion loss) is normally the only consideration in optimizing circular waveguide bends because ideal, non-terminating bends
have no contradirectional mode coupling or scattering. In practical bends, return loss, or
reflection into the backward-traveling guided mode, is an important issue and must be considered in the finite-angle bends discussed and the fully-assembled Air Trench Bends to
come.
We adopt a figure of -30dB
suppression of the signal to the reverse direction to be
sufficient for practical applications. Because of the matched core indices, in the present
structures scattering into the reverse guided mode is very weak. The greater restriction is
Air Trench Bends: Design and Simulations
92
Air Trench Bends: Design and Simulations
92
(a)
(b)
(c)
Figure 3-9: FDTD simulations showing dual boundary, finite-angle ATW bends in (a) dual boundary, and (b) single boundary operation with offset or width optimization, and (c) a
bend optimized for radius, waveguide widths and offset.
the insertion loss, and when met for the low loss values of interest it normally ensures that
the return loss specification is also met.
3.3.3
The Cladding Taper: A Low-Loss Interface to the Silica Waveguide
A size-optimized ATW bend (given a target loss), as designed in Sec. 3.3.2 (Fig. 3-9),
interfaces to straight ATWs. For this bend to be of use in increasing the integration density
in silica PLCs, a size- and loss-economic method is needed to couple the tightly confined
mode of the straight ATW to the large guided mode of the straight silica waveguide. A
"cladding taper" is proposed. The geometry of this new kind of taper is first justified; then,
its practical design and optimization are described. Two such tapers and an ATW bend,
jointly optimized, comprise the complete Air Trench Bend for silica PLCs (Sec. 3.4).
Geometry of the Cladding Taper
Conventional adiabatic tapers (here referred to as core tapers) are commonly used to interface (with low loss) waveguides of identical core and cladding index, but of different width
and thus different level of confinement of the fundamental mode [37] (Fig. 3-10a).
The
problem addressed here concerns an interface between the air trench and silica waveguides
(Fig. 3-10b).
It is slightly different in that the two waveguides of different width have
the same relative level of mode confinement, V-number (each is at the cutoff of its second
3.3 Air Trench Waveguides
93
air
trench
cladding
Scladding
(a)
(b)
Figure 3-10: (a) Conventional core width taper with identical indices but weakly confined mode
in narrow waveguide, and (b) a needed interface for air trench and silica waveguides,
both well confined. The large waveguide is single-mode in both cases.
guided mode), but different cladding indices (air and silica, respectively). They share the
same core index. For low loss, an adiabatic structure to match the two is desirable, but
such a structure must deal with the change of cladding index.
A lengthwise-graded index profile is not a practical solution. A simple step-index solution is the combination of an optimized abrupt junction that deals with the index change
in the cladding with a conventional core taper to modify the core width (Fig. 3-11a). This
type of structure was used by Spiekman et al. [59, 60] in exactly the scenario relevant to
this thesis, for matching air trench bends to InGaAsP:InP (multimode) ridge waveguides
(see also Fig. 1-2c). A junction loss of <0.1dB was quoted for matching to multimode ridge
waveguides. Single-mode waveguides would force higher losses at optimized junctions.
For single-mode, buried silica waveguide PLCs of interest in this thesis, optimal junction
loss values computed in Fig. 3-8 give a lower bound on the loss of the matching structure
of ~0.1dB.
A better taper design avoids abrupt junctions, and instead provides an adiabatic taper
of the core width and cladding index by the (still step-index) structure in Fig. 3-11b. Here
both the core width and the cladding index are changed in incremental perturbations of the
cross-sectional geometry. Using this method, arbitrarily low transition losses are achievable
at the cost of taper length. Our goal is to design the minimum length taper for a given
transition loss budget.
Before proceeding, a further simplification of the geometry of the new taper is introduced
out of consideration for fabrication. Because the air trench is etched using a second mask
Air Trench Bends: Design and Simulations
94
94
Air Trench Bends: Design and Simulations
lir
(a)
(b)
(c)
(d)
Figure 3-11: Geometries for the cladding taper, the interface between the air trench and silica
waveguide. (a) Abrupt junction followed by a conventional taper, (b) core and index
taper that eliminates the abrupt junction, (c) the simplified cladding taper with a
constant width core, and two constant taper rates for fabricability, and (d) a cartoon
of the ideal finite length, three-index taper shape.
after the core has been formed, misalignment between the core and air trench in Fig. 311b can result in poor performance. Instead, we use a constant-width core waveguide at
a relatively small loss penalty (Fig. 3-11c). For purposes of optimization, we have chosen
two regions of piecewise-linear tapering due to qualitative differences between the cladding
taper and the core taper (Fig. 3-11c).
Optimal tapers will not in fact be linear, because the linear tapers provide an adiabatic
modification of the mode width, but force a fast change of phase front curvature.
The
optimal tapers corresponding to those in Fig. 3-11a-c will have a more complex shape and
are illustrated in Fig. 3-11d. The number of degrees of freedom in the optimization problem
is large for such a taper without a new physical insight as guidance, and we do not pursue
it.
Optimization of a Simple Cladding Taper Geometry
We focus on the geometry of Fig. 3-11c in this thesis. Although it can achieve arbitrarily
low loss for large lengths, this taper is suboptimal in the sense that it will not have the
minimum achievable length using the three available refractive indices, given the input and
output waveguide. However, the achievable lengths with this taper will still permit dramatic
reduction in bend size, so it will be sufficient for our purposes.
3.3 Air Trench Waveguides
95
Given the effective refractive indices (Nco,Ned,Ntr) and dimensions (Fig. 3-11c) of the
air trench and silica waveguides (wa,ws), the three free parameters in the new taper are:
the width of the air taper "mouth" at the silica waveguide end (win), the length of the
cladding taper section (Lei), and the length of the core taper section (L,,). These are to be
optimized for minimum total taper length (Ltet = Ldi + L, 0 ) given a target insertion loss.
Because the taper is adiabatic, reflection is negligible and return loss is not of concern.
Given a total taper length Ltot, the optimization of the taper is then effectively subject
to two free parameters,
min
{LJQ.WaLei
Loss Leo ,Wa
<LossSpec.
Lt~t
The goal is to find a solution to the above optimization problem for the smallest possible
Ltot.-
Numerical Modeling of the Cladding Taper
To optimize the taper a method to calculate the loss of a given taper is necessary. Several
methods can practically model the proposed cladding taper. Among them are the transfer
matrix method using expansion into local normal modes (Sec. 2.4), and fully numerical
simulation tools such as the beam propagation method (BPM) and finite difference-time
domain (FDTD). The latter two are accurate, but slow. We use FDTD because high index
contrast is present in part of the structure, so BPM is not a favourable option.
As with bends, the taper loss is sensitive to length, so the order of length needed to
meet a loss spec can be reached in a few crude optimization steps. We use beam diffraction
and ray optics for an estimate of the taper angle (length) required, and then optimize the
taper further about this point manually using FDTD simulations.
Taper Length Estimate by Gaussian Beam Diffraction Angle
An order of magnitude for the length of the simple taper in Fig. 3-11a can be obtained
by considering the diffraction angle for a Gaussian beam in a uniform medium with the
core index. For the small air trench waveguide launching the beam, the beam waist (1/e
full width wo) is approximately equal to the waveguide mode width, i.e. the core width
(Wa).
Such an incident beam will in turn have a good match to the waveguide, both in
field distribution and phase front. Gaussian beams are strictly valid for beam waists much
Air Trench Bends: Design and Simulations
96
larger than the wavelength, wo
> A,,, which is not the case here for A,, = 1.55pm/ni,
and
wo - 1pm, where they are of the same order. Nonetheless, the diffraction angle ([17]; wo,
0, definitions differ),
0, = 2 tan_1
2Aco
(7FWo
gives a beam width of -60' for flat phase front matching to the small waveguide, without
any help from the cladding taper in guiding. Although the formula in [17] is computed for
the Gaussian beam of 2D cross-section, the same expression applies to a "slab" beam with
only one transverse dimension, because the paraxial wave equation is separable in xy, with
a product solution. This diffraction angle likely yields an underestimate for the taper length
required. The beam will have a curved phase front at the other end of the taper, where it
is to be matched to the plane phase front of the silica waveguide mode. A longer taper is
needed to accomodate this constraint with low loss.
A longer taper is helped by the guiding of the core and the guiding and phase front
shaping due to the air trench taper to match to the modes on both sides. These effects are
not studied separately in detail, but are compensated for manually with the feedback from
direct FDTD simulation, given the above taper angle as a starting point.
Taper Length Estimate by Ray Optics
Another estimate of the required taper length may be obtained from a simple ray optics
model.
The fundamental guided mode of the air trench and silica slab waveguides can
each be described in terms of plane waves in the core region, with the Goos-Hiinschen shift
describing effective metallic mirrors for these plane waves. The Goos-Hiinschen shift is
small at the silica-air interfaces of the air trench waveguide and the air trench taper, so the
mirrors can be placed at those interfaces. We attribute a ray to each plane wave, and search
for the taper angle that will guide a silica waveguide input ray into an air trench waveguide
output ray. One reflection from the taper wall is sufficient. The ray angles of the guided
modes with respect to a normal of the waveguide wall are,
Gi
= tan_
= tan-,(,)
kx
/2
(
\|
Nmode,i
:- N
-~
-
-
-
---
~-------~-.
-,
97
3.3 Air 'irench Waveguides
ATW guided ray
ot
Goos-Hinschen Mirror"
Figure 3-12: A cladding taper with a single taper rate, shown with a ray optical description of the
air trench and silica guided modes. The Goos-Hinschen shift (dashed line) indicates
interfaces where the guided rays are effectively reflected. From the indicated angles,
the taper angle Ot is found that guides the k-vectors of one guided mode into those
of the other upon one reflection from the taper interface.
for i = s, a for the silica and air trench waveguide, respectively. The single-reflection taper
angle, Ot, as defined in Fig. 3-12, is then estimated as
- a
Ot = 90* _ Os
2
Using the modal solutions and effective indices computed in Tables 3.1 and 3.2, the cladding
taper angle estimated for case study examples A, B and C is on the order of 80 (81 , 82 and
860 , respectively). We use these taper angles as a starting point in numerical optimization.
Mouth Opening of the Cladding Taper
A second parameter, in addition to the taper angle, that is needed to determine a finite
starting taper length for optimization is the width of the cladding taper at the interface to
the silica waveguide. Because the taper begins abruptly there, a junction loss is incurred.
A taper mouth that is too narrow incurs large junction loss, while one that is too wide
minimizes junction loss but sharpens the taper angle (given a fixed length) increasing the
associated taper radiation loss. A taper mouth width is chosen to encompass most of the
guided power, such that little is left outside to be reflected at the trench taper-silica guide
junction.
Figure 3-13 shows the fraction of guided power contained within a given lateral distance
-
Air Trench Bends: Design and Simulations
98
from the waveguide axis. A normalized plot can be given for a slab waveguide with a
normalized frequency V. Our silica waveguides are at the cutoff of the second guided mode.
For slabs, this corresponds to V = ir/2 (eqn. (3.3)). In our square cross-section waveguides,
due to the effective index values in Table 3.1, we have Veff ~ 0.647r for case study examples
A and B (Fig. 3-13a), and Veff ~~0.587r for example C (Fig. 3-13b). Indicator lines show
99%, 99.5%, 99.9% and 99.99% power containing widths. Using a taper wide enough to
contain 99.99% is more than sufficient to eliminate junction loss in the context of a sought
0.1dB/90 total loss of the fully-assembled Air Trench Bend.
The taper mouth widths needed to contain 99.99% of the guided power for examples A,
B and C are on the order of 30pm, 18pm and 6pm, respectively, using the silica waveguide
widths in Table 3.1.
The guided power contained in a given width of the cross-section is also used to find a
practical width of the air trench in the bend region, such that additional loss introduced is
negligible.
Numerical Simulations of Cladding Tapers
With the taper angle and mouth-opening width on the silica waveguide end estimated, a set
of successive numerical simulations is used to optimize the angle and mouth opening first,
and then given a total length, for the relative lengths (angles) of the two taper sections (see
Fig. 3-11c).
The iterative algorithm used manually to optimize the taper length is briefly described
and illustrated in Fig. 3-14a. A starting point for the taper is chosen from the ray optical
taper angle estimate (setting the core and cladding taper sections to initially have an equal
angle, Figs. 3-11c, 3-12) and a mouth opening selected to contain a sufficient proportion of
the input mode that junction loss is limited to being much lower than the total loss budget
for the taper. The taper is optimized with respect to the total length and the ratio of the
core and cladding taper lengths within it (Leo : Lci).
An iteration i of the optimization consists of two steps (ia and ib). In the first step
(ia), the total length of the taper is reduced along a constant Lco : LdJ ratio line (the taper
is scaled in proportion in the lengthwise dimension) to meet the loss budget value. Each
step here involves a couple of FDTD simulations, but relatively few steps are required as
illustrated by an example in Fig. 3-14b. At the budget loss, the taper is then optimized
3.3 Air Trench Waveguides
99
99
3.3 Air Trench Waveguides
Guided Mode Power Confinement in Normalized Slab Waveguide for V = 0.647c
1
0.9
99%
-
:99.9%
:99.99%
0.8-
0
0
S0.4
0.2
035
0.
D
-1
core halfwidth
d = ws/2
El
01
0
0
0.5
1.5
2
2.5
3
3.5
4
Normalized Distance from Waveguide Axis, x/d (unitless)
(a)
4.5
5
Guided Mode Power Confinement in Normalized Slab Waveguide for V = 0.58n
0.9
:99%
:99.9%
:99.99%
E 0.8
0
0.70. 0
.
-
W0.6 -
0.250.J-
El
$E0.120
0
0.5
4
1.5
2
2.5
3
3.5
Normalized Distance from Waveguide Axis, x/d (unitless)
4.5
5
(b)
Figure 3-13: Fraction of guided power contained within a distance from the core axis. Plotted for
the fundamental TE mode of a slab waveguide in normalized parameters, for case
study examples (a) A and B (V
0.647r), and (b) C (V
0.587r).
Air Trench Bends: Design and Simulations
100
+
Lcl
I
L/
\J
P4
*
Equal-Angle Line
o, = Oci= Ot
Constant Ratio Lines
i
(Lco/Lcl)
C
-L
Ray Optics
AV,
se9
Estimate
go.
to
0 tC al
onstant Length Lines (Ltot
Core taper length
=
Lco + Lcl)
L_
(a)
Reducing Ltot along Equal-Angle Line to reach Loss Spec
100
Optimization of L /LCI ratio over a constant Lt0t line
100
)
n 0 = 1.49, nC1= 1.48, n, = 1.0
98- (NCO = 1.48685, N = 1.4781, N, = 1.0778)
w = 0.7pm, ws =5pm
96 - w =22sm
C
0
wa
C
CL
99.5
0U.
C
0
94-
w
92
19
E 98.5
90
a98
CP
88
Case Study Example B
nco = 1.47, n =1.46, n = 1.0
(waveguide parameters in Tables)
wm
86 -
75
77
76
78
79
80
77
78
79 80
Taper Angle, Ot = 0
(b)
83
84
8
81
82
83
= O8I (degrees)
84
85
81
82
97.5 -
0.2
=16.051im
0.5
0.6
0.3
0.4
Core Taper:Cladding Taper Length Ratio, LO/Ld
0.7
(c)
Figure 3-14: (a) Search algorithm for optimal taper lengths L,,, Lcj given a loss budget and taper
mouth opening, win; (b) data of an optimization step la for an example similar to
our Case Study Example B, and (c) step lb for the actual Example B.
3.4 Air Trench Bends for Silica PLCs
101
(a)
(b)
(c)
Figure 3-15: Electric field amplitude from FDTD simulations (2D, using EIM) of cladding tapers
designed for case study examples A-C in (a)-(c), respectively.
over the L,, : Ld ratio for minimum loss (step
ib),
along a constant length (Ltet) line. The
room gained in the loss budget can be used in successive iterations.
One iteration was
sufficient for the tapers designed here. At the end, the taper is truncated in length at the
silica waveguide end until the taper width begins to impact loss. In this manner, the mouth
opening is minimized.
Electric field amplitude plots from FDTD simulations of the tapers designed for case
study examples A-C are shown in Fig. 3-15, with dimensions and transmission loss values
in Table 3.3.
The angles are similar to the ray angle estimate.
Comparing the length
of these tapers and the ATW bend radii in the previous section to the millimeter radii
of conventional silica bends (Sec. 2.5) gives indication that this approach will result in
significant size reduction.
3.4
Air Trench Bends for Silica PLCs
In the previous sections, air trench waveguides with strong lateral mode confinement were
introduced, and low-loss ATW bends and transitions to silica waveguides (cladding tapers)
were designed.
In this section, we put together these components to create Air Trench
Bends (ATBs) for silica PLCs.
ATBs make use of a third low index material (air) to
provide compact, low-loss finite-angle bends for silica waveguides.
As with conventional
dual-boundary silica waveguide bends (see Sec. 2.5), ATBs are general in that they can be
Air Trench Bends: Design and Simulations
102
Air Trench Bends: Design and Simulations
102
Table 3.3: Cladding Taper Dimensions and Loss for Examples A-C
Property
Core Taper Length (LCO)
Cladding Taper Length (Li)
Taper Mouth Opening (wm)
Core Taper Angle (OCO)
Cladding Taper Angle (Od)
Total Taper Length (Ltot)
Throughput Efficiency/Insertion Loss
A
Example
C
units
pum
gm
Rm
deg
deg
gm
70.20
155.00
25.15
86.33
87.21
225.20
19.00
60.00
16.05
82.50
85.24
79.00
2.05
7.90
5.95
80.31
75.11
9.95
-0.022
-0.0187
-0.089
d
designed for arbitrarily low loss as a trade-off against size, or effective radius. However,
in ATBs the third material can be used to advantage to achieve equal insertion loss to
conventional bends, at much smaller sizes.
The trade-offs for this luxury are enhanced
scattering loss, substrate leakage loss and stronger polarization dependence.
3.4.1
ATB Geometry
Consisting of a planar air trench waveguide structure in cross-section, the ATB geometry
in the plane is illustrated in Fig. 3-16a. The ATB consists of an ATW bend, in this case
chosen to be 90', and two cladding tapers to interface the ATWs to the silica waveguides.
The square shape of the air trench region surrounding the bend is such only for simplicity
of generating the structure in the numerical FDTD simulation code. The air trench regions
need be no wider on either side of the bent waveguide than 2-3 core widths of the bend.
The ATBs are directly built from concatenated tapers and bends from the previous
section. We choose for the ATB loss budget 98% transmission (~0.1dB insertion loss) and
<-30dB reflection.
A comparison of the ATW bend and taper sizes to silica bends has
shown that a large size reduction is possible. Furthermore, the ATW bend region is much
smaller than the cladding taper in all cases of interest in silica (our examples A and B, but
not C, which has A ~ 7%), and thus contributes little to the overall size of the ATB.
Instead of optimizing for the minimum possible ATB size, we simply increase the bend
radius to meet the loss budget. As bend loss (in dB) depends exponentially on radius (e.g.
Fig. 2-11), a small change in the bend radius will allow the loss budget to be met. The
3.4 Air Trench Bends for Silica PLCs
103
(a)
(b)
(c)
(d)
Figure 3-16: Air Trench Bend structure top-view schematic: (a) labeled, and (b) dimensioned
ATB plan geometry; and, cross-section geometry in the (c) silica waveguide, and (d)
air trench waveguide regions.
cladding tapers in Sec. 3.3.3 were designed to each have a throughput efficiency >99%, so
that the loss budget for the ATB can always be met by modifying the small bend only.
Because there are many free parameters in the complete Air Trench Bend, they have
been renamed from previous sections as defined in Fig. 3-16b.
3.4.2
Numerical Simulation of ATBs
Using effective indices from Tables 3.1 and 3.2, two-dimensional FDTD simulations were
done of the structures for case study examples A-C. A discretization of 20 points per wavelength was used, and a 50fs Gaussian pulse launched to measure the transmission and
reflection spectrum. The fields passing through the input and output waveguides were fil-
104
Air Trench Bends: Design and Simulations
Air Trench Bends: Design and Simulations
104
(a)
(b)
Figure 3-17: Schematic definition of a "total size box" (dash-dot), that contains input/output
ports to silica waveguides encompassing 99.99% of the guided power, for (a) simple
silica bends, and (b) Air Trench Bends.
tered for the fundamental mode amplitude (using the overlap integral) to yield insertion
loss and return loss values.
Example B and C were simulated directly using the structure from Fig. 3-16a-b. Simulating Example A in the same manner would require a large computational domain due
to the low index contrast of 0.25% and thus large size of the component in units of wavelength. We fold Example A into a 180 bend for simulation purposes. The performance of
this bend is representative of (and a lower bound for) its 90* counterpart, because the bends
in the lower index contrast examples have been enlarged to carry less loss than the tapers.
Also, the bending loss itself is only a portion of the total ATW bend loss, which includes
two straight-bent waveguide junctions. It is only the bending loss that is doubled in the
180 bend.
Case Study ATB Simulations
Air Trench Bends for the three example index contrast systems (A-C) were put through a
coarse manual optimization via a small number of successive 2D FDTD simulations. The
resulting dimensions of the total structure are given in Table 3.4, as defined in Fig. 3-16b.
For comparison with simple silica waveguide bends, the total size of a bend is defined by a
box which encompasses the entire bend structure and a chosen fraction of the guided power
of the input and output waveguide modes (Fig. 3-17).
3.4 Air Trench Bends for Silica PLCs
105
Table 3.4: Air Trench Bend Design Dimensions for Ex. A-C
Property
Contrast (A)
Index
Core
Cladding
A
Example
B
C
0.25
0.68
6.62
1.46365
1.46
1.47
1.46
1.56
1.46
units
unt
%
R1
Inner Radius
14.20
8.10
6.10 pam
R2
Outer Radius
15.35
9.25
7.25 lam
WL
WH
Big Waveguide Width
Small Waveguide Width
10.05
1.05
6.05
1.05
1.75 gm
1.05 pam
Bend Offset
Hi-A Air Triangle Length
Hi-A Air Triangle Width
Lo-A Air Triangle Length
Lo-A Air Triangle Width
Air Square Edge Length
Air Buffer Width
0.00
0.10
0.10 pm
70.20
4.50
155.00
7.55
98.60
1.00
19.00
2.50
60.00
5.00
36.85
1.00
2.05
0.35
7.90
2.10
12.85
1.00
Total Structure Edge Size
252.60
95.85
19.75 jim
AWB
o LHA
SLHB
E LLA
0
LLB
Ls
LA
L
pm
pam
lam
pam
jim
gm
Here, the fraction of mode power chosen to be contained by the size total size box
is 99.99%, and is relevant to the definition of size of a bend. Clearly, by this definition,
the bend size is at least as large as the mode size. This lower bound is approached by a
plane, perfectly reflecting mirror, which is a viable solution for a corner bend in very low
index contrasts, where the mode k-vectors are paraxial such that acceptable losses can be
achieved. Corner mirrors were briefly described in Chap. 1 (Fig. 1-2a). Their geometry is
not general in the size-for-loss tradeoff, and their performance makes them not viable in
silica.
The example structures A-C will be discussed in the reverse order, for that is the order
in which they were designed, going from higher to lower index contrast.
Example C: Very High Index Contrast
Example C, with an index contrast of ~7%, is outside the accepted range of silica index
contrasts (0.25% < A < 1.5%) [42]. It was simulated first because of its small size relative
to the wavelength, leading to a small computational domain and fast simulation times in
FDTD.
In this example, unlike the other two (A,B), the air trench waveguide bend and the
Air Trench Bends: Design and Simulations
106
cladding taper are of comparable size (Fig. 3-18a). The bend and taper have strong dependence of loss on size. For demonstration of the order of magnitude of size reduction in
comparison to a simple silica bend in the same index contrast (7%), a design is easily found
manually in a few iterations. The optimal structure dimensions from joint optimization of
the parameters will only bring small additional gains. This avenue isn't pursued because
the effort of optimizing the structure is not justified by the small further gains to be made.
Furthermore, the numerical optimization is brute force (although local about sensible start
values for the parameters), and provides no additional insight into the physical workings of
the structure.
Designed for 98% transmission, the size, transmission efficiency and comparison to the
radius of a simple silica bend of example C is given in Table 3.5. A reduction by a factor of
4 is achieved for the same transmission efficiency, with an increase of lateral index contrast
in the bend region from 1.56:1.46 to 1.56:1.0. A 20ptm total bend size results.
The FDTD electric field plot of a CW (TE mode) excitation of the structure at the freespace wavelength of 1550nm is shown in Fig. 3-18a. Absence of significant field strength
anywhere except in the waveguide path is a qualitative indicator of the low loss of the
structure. The mouth of the cladding taper is made wide enough for the junction loss at
the entrance to the structure to be a negligible contributor to the total loss. The smooth
continuity of the mode field between the taper and the bend section is also indicative of
a satisfactory offset of the bend with respect to the taper waveguide. An incorrect offset
between the leaky mode of the bend and the field entering from the taper would cause a
multitude of leaky modes to be excited in the bend. All higher order leaky modes have much
higher bend loss than the lowest order mode, and thus such a configuration is suboptimal.
Table 3.5: Air Trench and Simple Silica Bend Sizes for Ex. A-C
Property
Index
Contrast (A)
Regular Bend
Radius
Total Size
Air Trench Bend Radius
Total Size
Size Reduction by Length
by Area
A
0.25
15700
15708
15.35
252.60
62
3867
Example
0.68
2565
2570
9.25
95.85
27
719
C
units
6.62
76
78
7.25
19.75
4
16
%
pam
am
lam
4m
3.4 Air Trench Bends for Silica PLCs
107
Furthermore, use of multimode interference leads to wavelength dependence.
Spectra of transmitted and back-scattered power normalized to the input power are
shown in Fig. 3-19 (solid line). The spectra show that the bend is wavelength insensitive
across the C-band communications window (1530-1570nm), not unexpected because we have
avoided the use of any feed-forward (multipath) or feedback (resonant) mode interference in
the bend design. The power scattered back through the input plane is more than 30dB below
the input power. The majority of the back-scattered power is in radiation modes rather
than the reverse guided mode. At 1550nm, an indicator (square) shows the transmission
and reflection into the fundamental waveguide mode. The slight wavelength variation of
backscattered power can be attributed to Fabry-Perot reflections at the bend-taper offsets,
due to remaining mode mismatch.
Example B: High-A Silica
Example B is in a high-A silica material system, which is becoming more commonly used in
integrated optics in industry [42]. With an index contrast of 0.7%, the 6pm silica waveguide
is similar to the fiber core diameter of ~9pm. The fiber-chip coupling loss in this example
will be tolerable in contrast to high index contrast systems, which suffer from large coupling
losses.
In this example and the following, the cladding taper dominates the size of the air trench
bend. This is because the lateral index contrast in the air trench is roughly the same for
all examples, and thus the radius hardly changes. However, a lower index contrast requires
a longer taper for a low-loss transition from the silica to the air trench waveguide. For this
reason, the bend loss is made small by increasing the bend radius, and the majority of the
losses are placed on the tapers to minimize their length.
In low index contrasts, the size of the Air Trench Bend is limited not by the bend radius,
but by the taper length required for low-loss, adiabatic transition, and by the taper-bend
junction loss.
In this example, an ATB of -100pm in size can more than meet the 98% transmission
budget (Fig. 3-19, Table 3.6). In comparison, a simple silica bend of the same index contrast
must have a 3mm radius (Table 3.5, Fig. 2-11). Clearly, in lower index contrasts, there is
more to gain by using Air Trench Bends. Even though tapers grow larger in lower index
contrast, simple silica bends do so at a faster rate.
108
108
Air Trench Bends: Design and Simulations
Air Trench Bends: Design and Simulations
(a)
(b)
Figure 3-18: 2D FDTD simulation field plots. CW-excited (TE mode) electric field amplitude for
structures of (a) example C (A = 7%), and (b) example B (A = 0.7%). The total size
box and data marker/line type for the transmission plots in Fig. 3-19 are indicated.
3.4 Air Trench Bends for Silica PLCs
109
109
3.4 Air Trench Bends for Silica PLCs
Transmission
100
-0
98-
01
0
96-
0
C)
0
C',
94-
Cn
C,
-63--
92-
-
0
'
'
1535
1540
0' I
1530
'
'
A (A = 7%)
B (A = 0.7%)
-A--
C(A = O.25%)
'
'
'
1560
1565
1545 1550 1555
Wavelength (nm)
1570
(a)
Back-scattering
-El-&-
-25
---
A (A =7%)
B (A= 0.7%)
C (A= 0.25%)
-
-3
CO
Cn
-
-35
-
-
NN
C.)
N
N
/
-40
1530
0
I
I
1535
1540
I-I
Reflection to source
mode
I _ __ j
1545 1550 1555
Wavelength (nm)
1560
I
1565
1570
(b)
Figure 3-19: (a) Transmitted and (b) backscattered power spectra, and transmission and reflection
into the fundamental silica waveguide mode at the central wavelength of 1550nm for
examples A-C.
Air Trench Bends: Design and Simulations
110
Table 3.6: Air Trench Bend Loss Budget for Ex. A-C
Property
A
Index Contrast (A)
Bending Loss
Junction Loss
Taper Loss
Total Loss*
Transmission
units
Example
0.25
0.68
6.62
0.0020
0.0090
0.0232
0.074
98.30
0.0024
0.0355
0.0187
0.061
98.60
0.0023
0.0394
0.0089
0.052
98.80
%
dB/90'
dB/jctn
dB/taper
dB
%
*Total loss is due to 2 tapers, 2 junctions and 90 of bending. Junction loss at the low index waveguide to
taper interface is deemed negligible and ignored.
Example A: Fiber-like Silica
The last example has an index contrast identical to common optical fibers, A = 0.25%,
and is considered low-A silica integrated optics [42]. The silica waveguide will have ideally
a perfect match to the fiber mode, resulting in very low loss fiber-chip coupling. Simple
silica bends in this index contrast are on the order of 15mm for 98% transmission through
a 90' bend (Fig. 2-11).
A difficulty arises with simulations as the index contrast is lowered. Namely, the structure becomes large relative to the wavelength, and the computational domain for the FDTD
simulation becomes too large to simulate this structure in a reasonable amount of time. To
reduce the computational requirements, we use the fact that the taper is by far the predominant contributor to bend size. We increase the air trench bend radius to 15pam where it has
very little contribution to bend loss (Table 3.6). We then fold the 90' bend into a 180' bend,
where the only additional loss incurred will be the extra 90 of bending loss, which here is
negligible.
The field plot of the FDTD simulation used to calculate the bend performance is given
in Fig. 3-20a. The discretization here was 20 points per wavelength, as with the previous examples. Fig. 3-20b shows what the bend actually looks like, where a very coarse
(inaccurate) discretization was used to obtain the snapshot with CW excitation.
The Air Trench Bend in the lowest index contrast (A = 0.25%) used in practice in silica
integrated optics is
25 0
ptm in total size. In comparison to the simple silica bend radius of
15mm, the ATB offers a significant size reduction.
3.4 Air TJIrench Bends for Silica PLCs
ill
111
3.4 Air Trench Bends for Silica PLCs
(a)
(b)
Figure 3-20: 2D FDTD simulation field plots. CW-excited (TE mode) electric field amplitude for
structure of example A (A = 0.25%). (a) Folded structure used to simulate insertion
loss and reflection, and (b) coarse-discretized simulation of the actual structure to
scale, for display purpose only.
Air Trench Bends: Design and Simulations
112
The price paid for the reduction in bend size, in addition to an extra fabrication step,
is difficulty in fabrication. For lower index contrast examples, where the gain in terms of
size reduction is larger, the air trench waveguide aspect ratio grows as well. Air trenches
must be etched to a large depth in order to avoid significant substrate loss. This loss and
the required trench depth to avoid it are described in Sec. 3.4.3.
ATBs without Cladding Tapers
Before considering additional loss mechanisms in Air Trench Bends, we show an FDTD
simulation of the structure in example B without the benefit of the cladding taper to
illustrate its role in keeping losses low.
Fig. 3-21a shows example B with the cladding taper part of the compound taper removed, and yields a transmission (insertion) loss of 1.5dB, and a reflection suppression of
over 40dB. The high insertion loss is due to the abrupt junction loss rather than the angle of
the core taper, and a longer core taper would not significantly reduce the loss. The reason
for a still good suppression of reflection is that there is only a cladding index discontinuity
at the junction, while the core index is continuous. While scattering into unwanted modes is
present at the junction and results in the loss observed, the lost power is scattered primarily
into radiation modes rather than into the reverse guided mode.
A more fair comparison of the cladding taper is to a regular core taper that follows an
optimized abrupt junction, as in Fig. 3-11a. A simulation of a bend with a taper of the
latter kind, for the same total size, is shown in Fig. 3-21b. The insertion loss is reduced
to 0.5dB. Interestingly, the loss is far above the ~0.2dB loss lower bound imposed by the
two abrupt junctions.
Observing the multimode behaviour in the taper, it appears that
the taper is not gradual enough to be adiabatic, even though its length is identical to the
cladding taper in Fig. 3-18b, whose loss indicates that its transition is adiabatic.
This is an indicator that the cladding taper, in addition to eliminating the abrupt
junction for all practical purposes, also curves the phase front of the incoming mode. The
lateral distribution of index, although in steps, resembles a parabolic index profile at some
stage in the taper. Air is gradually introduced and becomes narrower as the taper progresses,
at some stage allowing significant focusing.
_____________________________
I~
~
-- -
-
---
~---.-----
-~
113
3.4 Air Trench Bends for Silica PLCs
(a)
(b)
Figure 3-21: FDTD simulations of the Air Trench Bend of example B without cladding tapers;
for comparison, the simulations shown are using (a) removed cladding tapers leaving
an abrupt junction, and (b) an optimized abrupt junction followed by a conventional
taper.
114
Air Trench Bends: Design and Simulations
3.4.3
Loss Mechanisms: Substrate and Scattering Loss
Substrate loss resulting from the truncation of the air trench depth (Fig. 3-6b) is assumed
small, in order to leave it out of the optimization problem of the previous sections. It is
calculated now as a correction. All ATB examples (A-C) were designed with a target loss
of 98% (approx. 0.1dB) in mind. A trench depth here is sought for each example to keep
substrate loss under 0.01dB.
An approximate calculation of substrate loss for the truncated air trench waveguide
used here was shown in Sec. 2.6.2. We refer the reader to Fig. 2-18. It shows substrate
loss vs. normalized trench depth for all three case study examples considered (A-C). The
assumptions made in the calculation are that the substrate is silica (n = 1.46), and (only
for purposes of the required depth in Table 3.7) that the finite trench extent is symmetric
above and below the waveguide axis. In practice, this assumption is too strict. The trench
can be shorter above the waveguide, as there is no leakage loss to the top. The top must
be sufficiently far from the core axis to avoid pushing the mode out of the core toward the
substrate.
It is interesting that the leakage loss curves normalized to the core height are roughly the
same for all three examples, although their index contrast varies from 0.25% to 7%. This
is because the silica waveguide of each example is at the cutoff of its own second guided
mode. Thus, their field distributions are similar when each is considered on the scale of its
own core size.
To have a substrate loss of 0.01dB for each example, the loss per unit length is found
by using an effective propagation length for substrate leakage. We choose this length to
be two edge lengths of the region which contains only the bend and the core taper. That
is, because the leaky mode in the air trench waveguide has a significantly larger vertical
extent than the silica waveguide mode, we assume that the substrate loss is significant only
where the core is surrounded by air. Thus the effective propagation length used is 2LS (see
Fig. 3-16b, Table 3.4).
Using this effective propagation length for substrate loss and Fig. 2-18, the required
trench depths are estimated for examples A-C, and given in Table 3.7. Not surprisingly, the
required trench depth for the same loss is larger for lower index contrast silica examples.
The ability to fabricate trenches with high aspect ratios will place a lower limit on the index
3.4 Air Trench Bends for Silica PLCs
115
115
3.4 Air Trench Bends for Silica PLCs
Table 3.7: Air Trench Depth for 0.01dB Substrate Loss in Ex. A-C
Property
Index Contrast (A)
Target Substrate Loss
Effective Propagation Length (2 Ls)a
Target Leakage Loss per Unit Length
Normalized Trench Interface Displacement (d/w)b
Core Height (w)c
Full Trench Depth (2d)
a see Table 3.4;
A
0.25
0.01
197.2
0.00005
1.39
10.05
27.9
units
Example
0.68
0.01
74.6
0.00013
1.29
6.05
15.6
6.62
%
dB
0.01
gm
25.7
0.00039 dB/gm
1.17
1.75
gm
4m
4.1
see Fig. 2-18; ' see Table 3.1
b
contrast range in which air trenches of this kind will be useful.
Finally, depending on the difficulty of fabrication, it may be desirable to allow the
majority of the loss budget for an Air Trench Bend to be attributed to substrate loss to
minimize the air trench depth. The remaining components can be made larger to meet a
much lower loss. A significant size reduction in comparison to silica bends should still be
realizable, since the dependence of the loss on component length is strong.
3.4.4
Fabrication
This section on the fabrication of Air Trench Bends is included for completeness. This work
was not part of the author's work, but is due to Shoji Akiyama.
Process for Fabrication of Air Trench Bend Structures
The first step in the fabrication of the present ATB structures is the formation of a cladding
layer (Fig. 3-22a).
There are two choices for the fabrication of this layer: (a) Thermal
oxidation and (b) Plasma Enhanced Chemical Vapor Deposition (PECVD). We started
with the former process because it gives the best quality oxide film. PECVD will be used
later, because it produces a thick oxide layer in a short time, but post-annealing will be
necessary to remove grain boundaries.
For the core layer (in Fig. 3-22b) we have chosen silicon oxynitride (SiON). The refractive index of this material can be varied from 1.46 to 2.0 at the 1.55pm wavelength[6], by
changing the composition of the source-gases such as SiH 4 , NH 3 and N 2 0. The N-H bonds,
which have been reported to have strong absorption bands around 1.5pim[14], are removed
Air Trench Bends: Design and Simulations
116
Air Trench Bends: Design and Simulations
116
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 3-22: Fabrication steps: (a) forming the under-cladding, (b) forming the core layer, (c)
patterning the waveguide, (d) forming the top cladding, (e) surface smoothing, (f)
deposition of nitride mask, (g) nitride etching, (h) making the air trench.
by annealing.
The waveguide is patterned (Fig. 3-22c) by an i-line stepper followed by a high density
plasma dielectric etch (HDPDE). The top cladding (Fig. 3-22d) is formed using PECVD.
The quality of the film is less critical than that of the waveguide and therefore the faster
process of PECVD is used. Post-annealing is done to remove grain boundaries. Chemical
mechanical polishing (CMP) is used to produce a smooth surface (Fig. 3-22e).
Next comes the fabrication of the trench.
A silicon nitride layer (~1500A thick) is
deposited first on the top cladding using PECVD (Fig. 3-22f).
This layer serves for the
etching mask. The i-line stepper defines the trench pattern (Fig. 3-22g), which is then
etched by a plasma etcher. The trench pattern must be aligned carefully with the underlying
waveguide; otherwise the cross-section of the waveguide is deformed.
Once the mask is
fabricated, the air trench is formed using HDPDE (Fig. 3-22h).
Progress in the Fabrication of Test Structures
At the time of writing of this thesis, the first steps are being taken toward the fabrication
of test Air Trench Bend devices. Measurements of loss will be performed to evaluate the
design presented in the thesis. The first test of placing a p-type photoresist mask onto a
silicon wafer is shown in the SEM (scanning electromicrograph) in Fig. 3-23. It shows the
pattern of a waveguide T-splitter made from two Air Trench Bends.
3.4 Air Trench Bends for Silica PLCs
3.4 Air Trench Bends for Silica PLCs
117
117
Figure 3-23: SEM of the first photoresist mask test of an ATB T-splitter structure on a plain Si
wafer; exposure time: 200ms.
Air Trench Bends: Design and Simulations
118
A set of masks is shown in Fig. 3-24 for two Air Trench Bends of older design than
the examples shown in this thesis (see [69]).
Fig. 3-24a-c shows the low index contrast
waveguide mask, the air trench mask, and the corresponding FDTD simulation, in that
order, for a bend in 2.0:1.9 index contrast. With a edge length of 20Qm, it is predicted to
have a transmission efficiency of 99% and -33dB reflection. For comparison, a conventional
bend without an air trench would need to have a radius of 230Qpm for the 98% bend loss
through a 90 turn.
Fig. 3-24d-f shows the same features for an Air Trench Bend purposely designed to have
a radius and taper lengths that are too small. It is one of a large set of varying bend sizes
to be fabricated, for comparison in testing. An FDTD simulation shows its expected poor
performance.
119
3.4 Air T7rench Bends for Silica PLCs
119
3.4 Air Trench Bends for Silica PLGs
(a)
(d)
(b)
(e)
(c)
(f)
Figure 3-24: SEMs of ATBs of two different sizes, (a)-(c) and (d)-(f), in 2.0:1.9 index contrast;
(a),(d) low index contrast waveguide masks, (b),(e) air trench masks, and (c),(f)
corresponding FDTD simulations [69].
120
121
Chapter 4
Beyond Air Trench Bends:
Applications and Other Ideas
The purpose of this short chapter is to present some simulation results for a simple compound device - a waveguide T-splitter - based on the Air Trench Bends presented in Chap. 3,
summarize the thesis results in a concluding section, and suggest some directions for future
work and other potential applications for ATBs and cladding tapers.
4.1
Waveguide T-splitter using Two Air Trench Bends
As a simple application of the Air Trench Bend, a 90 waveguide T-splitter is designed
using two back-to-back ATBs. The T-splitter's geometry is that of an ATB symmetrically
reflected about its input waveguide axis, with one exception.
The taper-to-bend lateral
offset is larger at the input side of the splitter.
The purpose of the lateral offset in bends is to align the guided mode of the straight
waveguide with the leaky mode of the bend for optimal mode match. In the case of the
T-splitter, a larger offset is needed at the input to aid in the splitting of the input mode
into two output waveguides. Wider offsets create a small wide-angle diffraction region in
the splitting section (Fig. 4-1a, inset), somewhat alike to a multimode interference coupler
(MMI).
With the use of this small modification of the geometry, a waveguide T-splitter is designed (based on the ATB of example B in Chap. 3) with 50:50 splitting ratio and 95% overall
transmission efficiency (at 1550nm) from the source guided mode at the input to the two
122
Beyond Air Trench Bends
output waveguide target modes. A reflection to the source of less than -40dB is predicted.
In addition to the FDTD (CW-excited) field plot, transmitted and back-scattered power
ratio (through the output and input waveguide planes, respectively) spectra are given in
Fig. 4-1. The transmission (insertion loss) and reflection (return loss) figures are indicated
at 1550nm with a circle marker.
The compact (180 x 100pm), wavelength-independent T-splitter demonstrates the utility
of the Air Trench Bend for increasing integration density, in this case on a A = 0.7% silica
platform. Several issues of practical importance are not considered in this design, such
as the fabricability of the sharp wall between the bends in the splitting section, and the
sensitivity of the splitting ratio to geometrical irregularities in the practical realization of
the device. A better, more realistic design is left as a future exercise.
T-splitters are among the test structures being fabricated, and some masks for the air
trench, readied at the time of this writing, are shown in Fig. 4-2.
4.2
Concluding Remarks
In this thesis, a new bend geometry incorporating an air trench was proposed for producing
micron-sized bends in low index contrast (e.g. silica) waveguide platforms, with low loss
and low reflection. The main improvement of this design on previous uses of the air trench
is the elimination of junction loss using a new "cladding taper". Dimensions of the total air
trench bend (which contains the cladding tapers) can be chosen to meet any desired total
loss. In that sense, this air trench bend geometry is general.
Examples of air trench bends were shown for three index contrasts with silica cladding,
A = 0.25%, 0.7% (silica PLC index contrasts) and A = 7%. In the silica index contrasts
(0.25%-0.7%), a size reduction of the bend by a factor of 30-60 with respect to a conventional
silica bend in the same index contrast was predicted; i.e. by a factor of 900-3600 by area.
The order of size reduction, governed by the index contrast with the trench material (air in
this case), can quickly be arrived at in a few steps without the need for detailed optimization.
Cladding tapers considered in this thesis used a two-section linear taper geometry. The
tapers are short, and provide an adiabatic evolution of the fundamental mode from the
silica single mode waveguide to air trench single mode waveguide. It was also noted that
the cladding tapers aid in curving the phase front via the index profile in the taper. In this
123
4.2 Concluding Remarks
(a)
Transmission
Backscattering
20
-100
0O
C
0
25
S95
0
0
C
0
95%
Fn90
1-
S85
|-
30
U)
a
35
CU
Reflection (-41 dB)
an
0-
S40
1530
1540
1550
Wavelength (nm)
(b)
1560
1570
M5 0
1540
1550
1560
1570
Wavelength (nm)
(c)
Figure 4-1: Waveguide T-splitter based on the A = 0.7% Air Trench Bend in Fig. 3-18b. (a)
Field plot of TE 2D FDTD simulation, with inset showing the splitting section; (b)
transmitted power spectrum through output plane, and insertion loss to the destination
mode at 1550nm; (c) backscattered power spectrum and reflection to input mode.
Bends
Beyond Air TFrench
Beyond Air Trench Bends
124
124
(a)
(b)
Figure 4-2: SEM of the air trench masks for waveguide T-splitters based on the bends in Fig. 3-24.
way, the necessary length of the taper is reduced as the mode is focussed, and converges to
the small air trench waveguide partially on its own.
Finally, the truncated air trench waveguide's substrate loss is considered. It is shown
that in straight air trench waveguides the substrate loss can be reduced to arbitrarily small
values by etching a deeper trench, well below the core. For a substrate loss of 0.01dB, trench
depths on the order of 15-30pm are necessary. Due to the challenge in fabricating such high
aspect ratio air trenches, it may prove necessary for a chosen total loss budget to design
the air trench bend to have a much lower taper and bend loss, and allow the majority of
the loss to be in the substrate loss, for purposes of a minimum depth trench. Further, the
trench depth estimated above was a simple estimate assuming vertical symmetry about the
core axis. The trench extent above the core can be significantly reduced, because no leakage
occurs to the top. However, the mode must not be pushed out of the core (which has low
vertical index contrast), and other modes of the air trench must be kept suppressed.
4.2.1
Future Directions
In future work, it is of interest to study the cladding taper to isolate the fundamental
mechanisms of its operation so that (close to) optimal tapers of this kind can be designed,
without resorting to brute force numerical optimization.
The cladding taper, or a variant of it, could also find application in fiber-to-chip coupling
(a major research problem in high integration density integrated optics). It has a short
length and a single mode-to-single mode transition with low loss (with gradual effective
4.2 Concluding Remarks
125
index change from the input to the output waveguide). Directly, it can serve as a fiber-tochip coupler for a high index contrast circuit using the high-aspect-ratio air trench waveguide
as its guiding structure. Such optical circuits may not be practical. It may be possible to
use the laterally compressed output mode of the cladding taper in a type of coupler to excite
a small conventional high index contrast waveguide. This is another avenue of future work.
The enhanced vertical extent of the guided mode in the air trench region can also
produce a vertical (inter-chip-layer) coupler for two layers of a multilayer optical circuit.
This coupler was suggested by K. Wada, and would involve a straight waveguide section (in
both layers) sandwiched between input and output cladding tapers spanning both layers.
In this way regular silica waveguides would not couple strongly between layers, but the air
trench waveguides in the coupler would.
Other avenues of future pursuit in the context of the air trench bends proposed here are
the use of a low-index material in place of air for the trench, and single boundary air trench
bends. Filled trenches, using a very low index material could be used to reduce environment effects on performance, such as that of humidity. The dependence of the achievable
(minimum) size of air trench bends on the core-trench index contrast is of interest, where
the total size involves the dependence of the bend and that of the cladding taper. Single
boundary bends eliminate the need for an inner trench, thus simplifying the fabrication process. They will, however, have a more complex geometry and thus will be more difficult to
design. A circular single boundary bend would present excessive junction loss. Single-sided
cladding tapers will need to be merged with the bend and the three be considered as a unit
in the context of the silica input/output waveguides.
126
127
Appendix A
Reflection of Complex Waves at a
Planar Boundary
We define a complex wave as a wave in a lossless medium which has a real and imaginary
part of the propagation constant.
The propagation (real) and attenuation (imaginary)
directions of the complex wave are orthogonal to each other as illustrated in Fig. A-1. In
the special case where the propagation direction of the complex wave is the same as the
propagation direction of the entire mode of interest, such as in the cladding region of the
slab waveguide, the complex wave is commonly referred to as an evanescent wave [23].
We don't discuss general reflection of complex waves at a planar boundary, but rather
the limiting case of interest for the problem of the layered slab (1D) leaky waveguide. In this
case, we are interested in incident and reflected waves which are approximately evanescent
waves, and a general complex wave in the region of transmission (as in Fig. A-1b).
Planar boundary reflection for plane waves with real propagation constants is covered
in numerous texts including [17, 23]. For conciseness, we refer the reader to Haus [17] for
this problem, and use the same notation (but keep subscripts consistent with the rest of
the thesis) for the complex wave problem. For the TE polarization in non-magnetic media
([11,2 = [O),
the electric field reflection coefficient at the boundary is given by [17],
kiz - k2z
kiz + k 2 z
By analytic extension, the reflection coefficient holds for complex k 1 ,2 . We define angles
01, 02 as the angles between the boundary normal and the real components of the complex
k-vectors, consistent with the plane wave problem. The k-vector z-components are kmz
=
MM
__ __
- -
-
__
-
---
11 1 .
-
Reflection of Complex Waves at a Planar Boundary
Reflection of Complex Waves at a Planar Boundary
128
128
(b)
(a)
Figure A-1: Normal and complex waves for (a) lowest-order slab mode, and (b) lowest-order leaky
mode of a five-layer waveguide. Variables of incident, reflected and transmitted waves
are indexes according to the regions (1,2) under consideration.
kmR cos Om + ikmi sin 0m, m = 1, 2. In the low loss case for leaky waveguides, 01 ;
k 21
90 ,
< kii, and k2R ~ k2 so the reflection coefficient becomes
1
iki - k2 cos 02
ikii+ k 2cos 02
The reflection coefficient has unity magnitude in this approximation and can be written as,
F
-
exp 2i tan-
1
k2 COS 02
.2
-
-
- .-
.-
__
IR
Bibliography
[1] C.M. Bender and S.A. Orszag.
Advanced Mathematical Methods for Scientists and
Engineers: Asymptotic Methods and PerturbationTheory. Springer-Verlag, New York,
NY, 1999.
[2] T.M. Benson. Etched-wall bent-guide structure structure for integrated optics in the
iii-v semiconductors. Journal of Lightwave Technology, LT-2(1):31-34, Feb 1984.
[3] J.-P. Berenger. A perfectly matched layer for the absorption of electromagnetic waves.
Journal of Computational Physics, 114(2):185-200, Oct 1994.
[4] W. Berglund and A. Gopinath. Wkb analysis of bend losses in optical waveguides.
Journal of Lightwave Technology, 18(8):1161-1166, Aug 2000.
[5] P. Bienstman.
Rigorous and efficient modeling of wavelength scale photonic compo-
nents. PhD thesis, Ghent University, Ghent, Belgium, 2001.
[6] G.L. Bona, R. Germann, F. Horst, B.J. Offrein, and H.W.M. Salemink.
Versatile
silicon-oxynitride planar lightwave circuits for interconnect applications. In Proc. 6th
Intl. Conf. on ParallelInterconnects, pages 145-148, 1999.
[7] I.N. Bronshtein and K.A. Semendyayev.
Handbook of Mathematics. Verlag Harri
Deutsch, New York, NY, USA, 3rd edition, 1985.
[8] J.C. Chen and C. Dragone. A study of fiber-to-fiber losses in waveguide grating routers.
Journal of Lightwave Technology, 15(10):1895-1899, Oct 1997.
[9] W.C. Chew.
Electromagnetic theory on a lattice.
75(10):4843-4850, May 1994.
Journal of Applied Physics,
130
BIBLIOGRAPHY
[10] M.K. Chin and S.T. Ho. Design and modeling of waveguide-coupled single-mode microring resonators. Journal of Lightwave Technology, 16(8):1433-1446, Aug 1998.
[11] L.A. Coldren and S.W. Corzine. Diode lasers and photonic integrated circuits. Wiley,
New York, NY, 1995.
[12] J.H. den Besten, M.P. Dessens, C.G.P. Herben, X.J.M. Leijtens, F.H. Groen, M.R.
Leys, and M.K. Smit. Low-loss, compact, and polarization independent phasar demultiplexer fabricated by using a double-etch process. IEEE Photonics Technology Letters,
14(1):62-64, Jan 2002.
[13] C. Dragone. An n x n optical multiplexer using a planar arrangement of two star
couplers. IEEE Photonics Technology Letters, 3(9):812-815, Sep 1991.
[14] R. Germann, H. W. M. Salemink, R. Beyeler, G. L. Bona, F. Horst, I. Massarek,
and B. J. Offrein.
Silicon oxynitride layers for optical waveguide applications.
J.
Electrochem. Soc., 147(6):2237-41, June 2000.
[15] J.E. Goell. A circular-harmonic computer analysis of rectangular dielectric waveguides.
The Bell System Technical Journal, 48:2133-2160, Sep 1969.
[16] I.C. Goyal, R.L. Gallawa, and A.K. Ghatak. Bent planar waveguides and whispering
gallery modes: A new method of analysis. Journal of Lightwave Technology, 8(5):768-
774, May 1990.
[17] H.A. Haus.
Waves and Fields in Optoelectronics. Solid State Physical Electronics.
Prentice-Hall, 1984.
[18] M. Heiblum and J.H. Harris.
Analysis of curved optical waveguides by conformal
transformation. IEEE Journal of Quantum Electronics, QE-11:75-83, Feb 1975.
[19] W.P. Huang, C.L. Xu, W. Lui, and K. Yokoyama. The perfectly matched layer boundary condition for modal analysis of optical waveguides: Leaky mode calculations. IEEE
Photonics Technology Letters, 8(5):652-654, May 1996.
[20] W.B. Jones Jr. Introduction to Optical Fiber Communication Systems. Oxford Uni-
versity Press, New York, NY, USA, 1988.
BIBLIOGRAPHY
131
[21] P.C. Kendall, M.S. Stern, and P.N. Robson. A new curvature loss formula of huygenstype for rib waveguides. IEE ProceedingsPart J, 135(1):11-16, 1988.
[22] R.M. Knox and P.P. Toulios. Integrated circuits for the millimeter through optical
frequency range.
In J. Fox, editor, Proceedings of the symposium on submillimeter
waves, volume XX of Microwave research institute symposia series, pages 497-516,
New York, NY, 1970. Polytechnic Press.
[23] J.A. Kong. Electromagnetic Wave Theory. EMW Publishing, Cambridge, MA, USA,
1999.
[24] S.K. Korotky, E.A.J. Marcatili, J.J. Veselka, and R.H. Bosworth. Greatly reduced losses
for small-radius bends in ti:linbo3 waveguides. Applied Physics Letters, 48(2):92-94,
Jan 1986.
[25] F. Ladouceur and P. Labeye. A new general approach to optical waveguide path design.
Journal of Lightwave Technology, 13(3):481-492, Mar 1995.
[26] K.K. Lee, D.R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L.C. Kimerling. Effect
of size and roughness on light transmission in a si/sio2 waveguide: experiments and
model. Applied Physics Letters, 77(11):1617-1619, Sep 2000.
[27] K.K. Lee, D.R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L.C. Kimerling. Erratum:
"effect of size and roughness on light transmission in a si/sio2 waveguide: experiments
and model". Applied Physics Letters, 77(14):2258, Oct 2000.
[28] Y.P. Li and C.H. Henry.
Silica-based optical integrated circuits.
IEE Proc.-
Optoelectron., 143(5):263-280, Oct 1996.
[29] B.E. Little, S.T. Chu, W. Pan, and Y. Kokobun. Microring resonator arrays for vlsi
photonics. IEEE Photonics Technology Letters, 12(3):323-325, Mar 2000.
[30] C. Manolatou and H.A. Haus.
Passive Components for Dense Optical Integration.
Kluwer Academic Publishers, Boston, MA, 2002.
[31] C. Manolatou, S.G. Johnson, S. Fan, P.R. Villeneuve, H.A. Haus, and J.D. Joannopoulos. High-density integrated optics. Journal of Lightwave Technology, 17(9):1682-1692,
Sep 1999.
BIBLIOGRAPHY
132
[32] E.A.J. Marcatili. Bends in optical dielectric waveguides.
The Bell System Technical
Journal, 48:2103-2132, Sep 1969.
[33] E.A.J. Marcatili.
Dielectric rectangular waveguide and directional coupler for inte-
grated optics. The Bell System Technical Journal,48:2071-2102, Sep 1969.
[34] E.A.J. Marcatili and S.E. Miller. Improved relations describing directional control in
electromagnetic wave guidance. The Bell System Technical Journal,48:2161-2188, Sep
1969.
[35] D. Marcuse.
Mode conversion caused by surface imperfections of a dielectric slab
waveguide. The Bell System Technical Journal,48(10):3187-3215, Dec 1969.
[36] D. Marcuse. Radiation losses of dielectric waveguides in terms of the power spectrum
of the wall distortion function. The Bell System Technical Journal,48(10):3233-3242,
Dec 1969.
[37] D. Marcuse. Radiation losses of tapered dielectric slab waveguides. The Bell System
Technical Journal,49:273-290, Feb 1970.
[38] D. Marcuse. Light Transmission Optics. Von Nostrand Reinhold, New York, NY, 1982.
[39] D. Marcuse.
Theory of Dielectric Optical Waveguides. Academic Press, New York,
NY, 2nd edition, 1991.
[40] A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Joannopoulos. High
transmission through sharp bends in photonic crystal waveguides.
Physical Review
Letters, 77(18):3787-3790, Oct 1996.
[41] S.E. Miller. Integrated optics: An introduction. The Bell System Technical Journal,
48(1):2059-2069, Sep 1969.
[42] T. Miya. Silica-based planar lightwave circuits: passive and thermally active devices.
IEEE J. of Sel. Topics in Quan. Electron., 6:38-45, Jan 2000.
[43] M. Mrozowski.
Guided Electromagnetic Waves - Properties and Analysis. Research
Studies Press, Taunton, 1997.
BIBLIOGRAPHY
133
[44] T.E. Murphy. Integrated optical grating-based matched filters for fiber-optic communications. Master's thesis, Massachusetts Institute of Technology, Feb 1996.
[45] E.G. Neumann. Curved dielectric optical waveguides with reduced transition loss. IEE
Proceedings Part H, 129(5):278-280, 1982.
[46] F.P. Payne and J.P.R. Lacey. A theoretical analysis of scattering loss from planar
optical waveguides. Optical and Quantum Electronics, 26(10):977-986, Oct 1994.
[47] E.C.M. Pennings.
Bends in Optical Ridge Waveguides: Modeling and Experiments.
PhD thesis, T.U. Delft, The Hague, Netherlands, 1990.
[48] S.P. Pogossian, L. Vescan, and A. Vonsovici.
The single-mode condition for semi-
conductor rib waveguides with large cross section. Journal of Lightwave Technology,
16(10):1851-1853, Oct 1998.
[49] M. Popovi6, K. Wada, S. Akiyama, H.A. Haus, and J. Michel. Micron-size bending
radii in silica-based waveguides.
In Proceedings of the SPIE Photonics West 2002
Optoelectronics Symposium (to be published), San Jose, CA, Jan 2002.
[50] D.M. Pozar. Microwave Engineering. Wiley, New York, NY, 2nd edition, 1998.
[51] K. Radhakrishnan and W.C. Chew. An efficient krylov-subspace-based algorithm to
solve the dielectric-waveguide problem. IEEE Transactions on Microwave Theory and
Techniques, 49(7):1345-1348, July 2001.
[52] R. Ramaswami and K.N. Sivarajan. Optical Networks: A PracticalPerspective.Morgan
Kaufmann Publishers, San Francisco, CA, 2nd edition, 2002.
[53] D. Rowland. Nonperturbative calculation of bend loss for a pulse in a bent planar
waveguide. IEE Proc.-Optoelectron., 144(2):91-96, Apr 1997.
[54] E.B. Saff and A.D. Snider. Fundamentals of Complex Analysis for Mathematics, Science and Engineering. Prentice Hall, Upper Saddle River, NJ, 2nd edition, 1993.
[55] P.D. Sewell and T.M. Benson. Efficient curvature analysis of buried waveguides. Journal of Lightwave Technology, 18(9):1321-1329, Sep 2000.
134
BIBLIOGRAPHY
[56] M.K. Smit. A new focussing and dispersive planar component based on an optical
phased array. Electronics Letters, 24(7):385-386, Mar 1988.
[57] M.K. Smit, E.C.M. Pennings, and H. Blok. A normalized approach to the design of
low-loss optical waveguide bends. Journal of Lightwave Technology, 11(11):1737-1742,
Nov 1993.
[58] A.W. Snyder and J.D. Love. Optical Waveguide Theory. Chapman and Hall, London,
UK, 1983.
[59] L.H. Spiekman, Y.S. Oei, E.G. Metaal, F.H. Groen, P. Demeester, and M.K. Smit.
Ultrasmall waveguide bends: the corner mirrors of the future? IEE Proc.-Optoelectron.,
142:61-65, Feb 1995.
[60] L.H. Spiekman, Y.S. Oei, E.G. Metaal, F.H. Groen, I. Moerman, and M.K. Smit.
Extremely small multimode interference couplers and ultrashort bends on inp by deep
etching. IEEE Photonics Technology Letters, 6:1008-1010, Aug 1994.
[61] M.S. Stern. Semivectorial polarised finite difference method for optical waveguides with
arbitrary index profiles. IEE Proceedings Part J, 135(1):56-63, Feb 1988.
[62] G. Strang. Introduction to Linear Algebra. Wellesley-Cambridge Press, 2nd edition,
1998.
[63] A. Taflove. Computational electrodynamics: the finite-difference time-domain method.
Artech House, Boston, MA, USA, 1995.
[64] H. Takahashi, S. Suzuki, K. Kato, and I. Nishi. Arrayed-waveguide grating for wavelength division multi/demultiplexer with nanometre resolution.
Electronics Letters,
26(2):87-88, Jan 1990.
[65] C. Vassallo.
Optical Waveguide Concepts, volume 1 of Optical Wave Sciences and
Technology. Elsevier Science Publishers, Amsterdam, 1991.
[66] J. Ctyrokf. Photonic bandgap structures in planar waveguides. Journal of the Optical
Society of America A, 18(2):435-445, Feb 2001.
BIBLIOGRAPHY
[67] A.K. Vellekoop and M.K. Smit.
135
Four-channel integrated-optic wavelength demulti-
plexer with weak polarization dependence. Journal of Lightwave Technology, 9(3):310314, Mar 1991.
[68] A. Vukovi6. Fourier Transformation Analysis of Optoelectronic Components and Circuits. PhD thesis, University of Nottingham, Nottingham, UK, Apr 2000.
[69] K. Wada, M. Popovi6, S. Akiyama, H.A. Haus, and J. Michel. Micron-size bending
radii in silica-based waveguides. In Proceedings of the IEEE/LEOS Summer Topical
Meeting on WDM Components, pages 13-14, Copper Mountain, CO, Aug 2001.
[70] M.W. Watts, S.G. Johnson, H.A. Haus, and J.D. Joannopoulos. Suppression of radiation from quarter-wave shifted bragg resonators. submitted to CLEO 2002, 2002.
[71] S. Wolfram. Mathematica: A System for Doing Mathematics by Computer. AddisonWesley Pub. Co., 2nd edition, 1991.
[72] C.L. Xu, W.P. Huang, M.S. Stern, and S.K. Chaudhuri. Full-vectorial mode calculations by finite difference method. IEE Proc. Optoelectron., 141(5):281-286, Oct 1994.
[73] J. Yamauchi, M. Ikegaya, and H. Nakano. Bend loss of step-index slab waveguides with
a trench section. Microwave and Optical Technology Letters, 5(6):251-254, Jun 1992.
[74] K.S. Yee. Numerical solution of the initial boundary value problems involving maxwell's
equations in isotropic media. IEEE Transactions on Antennas and Propagation,AP-
14:302-307, May 1966.
[75] D. Yevick and B. Hermansson. Efficient beam propagation techniques. IEEE Journal
of Quantum Electronics, 26(1):109-112, Jan 1990.
[76] A.M. Zaghloul and A.A. Abou El-Fadl. A simple analytical approach to optical rib
waveguides. In Proceedings of the Sixteenth National Radio Science Conference, pages
B9/1-8, Ain Shams Univ., Cairo, Egypt, 1999. NRSC '99 (IEEE Cat. No. 99EX249).