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Jia-Ming Liu - Principles of Photonics-Cambridge University Press (2016)

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Principles of Photonics
With this self-contained and comprehensive text, students will gain a detailed understanding of
the fundamental concepts and major principles of photonics. Assuming only a basic background in optics, readers are guided through key topics such as the nature of optical fields, the
properties of optical materials, and the principles of major photonic functions regarding the
generation, propagation, coupling, interference, amplification, modulation, and detection of
optical waves or signals. Numerous examples and problems are provided throughout to enhance
understanding, and a solutions manual containing detailed solutions and explanations is
available online for instructors.
This is the ideal resource for electrical engineering and physics undergraduates taking introductory, single-semester or single-quarter courses in photonics, providing them with the
knowledge and skills needed to progress to more advanced courses on photonic devices,
systems, and applications.
Jia-Ming Liu is Distinguished Professor of Electrical Engineering and Associate Dean for
Academic Personnel of the Henry Samueli School of Engineering and Applied Science at the
University of California, Los Angeles. Professor Liu has published over 250 scientific papers
and holds 12 US patents, and is the author of Photonic Devices (Cambridge, 2005). He is a
fellow of the Optical Society of America, the American Physical Society, the IEEE, and the
Guggenheim Foundation.
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ENDORSEMENTS FOR LIU, PRINCIPLES OF PHOTONICS
“With much thoughtfulness and a rigorous approach, Prof. Jia-Ming Liu has put
together an excellent textbook to introduce students to the principles of photonics. This
book covers a comprehensive list of subjects that allow students to learn the
fundamental properties of light as well as key phenomena and functions in photonics.
Compared to other textbooks in classical optics, this book places the necessary
emphasis on photonics for readers who want to learn about this field. Compared to other
textbooks introducing photonics, this book is carefully and well written, with ample
examples, illustrations, and well-designed homework problems. Instructors will find
this book very helpful in teaching the subjects, and students will find themselves
gaining solid understanding of the materials by reading and working through the book.”
Lih Lin, University of Washington
“For a long while the photonics community has been waiting for a new textbook which
is informative, comprehensive, and also contains practical examples for students; in
other words, one which describes fundamental concepts and provides working
principles in optics. Professor Jia-Ming Liu’s book, Principles of Photonics, serves very
well for these purposes – it covers optical phenomena and optical properties of
materials, as well as the basic principles behind light emitting, modulation,
amplification and detection devices that are commonly used nowadays in
communications, displays, and sensing. A distinguishing feature of this book is its
seamless use of “additional space” to ensure that each concept is sufficiently explained
in words, coupled with mathematics, simple yet illustrative figures, and/or examples.
Each chapter ends with questions/problems followed by key references, making it very
self-contained and very easy to follow.”
Paul Yu, University of California, San Diego
“A pedagogical tour-de-force. Professor Liu covers the principles of photonics with
extreme attention to notation, completeness of derivations, and clear examples matched
to the concepts being taught. This is a book one can really learn from.”
Jeffrey Tsao, Sandia National Lab
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Cambridge Books Online © Cambridge University Press, 2016
Principles of
Photonics
JIA-MING LIU
University of California
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Cambridge Books Online © Cambridge University Press, 2016
University Printing House, Cambridge CB2 8BS, United Kingdom
Cambridge University Press is part of the University of Cambridge.
It furthers the University’s mission by disseminating knowledge in the pursuit of
education, learning and research at the highest international levels of excellence.
www.cambridge.org
Information on this title: www.cambridge.org/9781107164284
© Jia-Ming Liu 2016
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2016
Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall
A catalog record for this publication is available from the British Library
Library of Congress Cataloging-in-Publication data
Names: Liu, Jia-Ming, 1953- author.
Title: Principles of photonics / Jia-Ming Liu.
Description: Cambridge, United Kingdom : Cambridge University Press, [2016] | Includes bibliographical
references and index.
Identifiers: LCCN 2016011758 | ISBN 9781107164284 (Hard back : alk. paper)
Subjects: LCSH: Photonics.
Classification: LCC TA1520 .L58 2016 | DDC 621.36/5–dc23 LC record available at
https://lccn.loc.gov/2016011758
ISBN 978-1-107-16428-4 Hardback
Additional resources for this publication at www.cambridge.org/9781107164284
Cambridge University Press has no responsibility for the persistence or accuracy
of URLs for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
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Cambridge Books Online © Cambridge University Press, 2016
To Vida and Janelle
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Cambridge Books Online © Cambridge University Press, 2016
CONTENTS
Preface
Partial List of Symbols
1
Basic Concepts of Optical Fields
1.1 Nature of Light
1.2 Optical Fields and Maxwell’s Equations
1.3 Optical Power and Energy
1.4 Wave Equation
1.5 Harmonic Fields
1.6 Polarization of Optical Fields
1.7 Optical Field Parameters
Problems
Bibliography
2
Optical Properties of Materials
2.1 Optical Susceptibility and Permittivity
2.2 Optical Anisotropy
2.3 Resonant Optical Susceptibility
2.4 Optical Conductivity and Conduction Susceptibility
2.5 Kramers–Kronig Relations
2.6 External Factors
2.7 Nonlinear Optical Susceptibilities
Problems
Bibliography
3
Optical Wave Propagation
3.1 Normal Modes of Propagation
3.2 Plane-Wave Modes
3.3 Gaussian Modes
3.4 Interface Modes
3.5 Waveguide Modes
3.6 Phase Velocity, Group Velocity, and Dispersion
3.7 Attenuation and Amplification
Problems
Bibliography
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Cambridge Books Online © Cambridge University Press, 2016
page xi
xiii
1
1
4
8
10
11
13
18
20
21
22
22
24
32
38
44
44
55
60
65
66
66
73
86
92
108
122
129
132
139
viii
Contents
4 Optical Coupling
4.1 Coupled-Mode Theory
4.2 Two-Mode Coupling
4.3 Codirectional Coupling
4.4 Contradirectional Coupling
4.5 Conservation of Power
4.6 Phase Matching
Problems
Bibliography
5 Optical Interference
5.1 Optical Interference
5.2 Optical Gratings
5.3 FabryPérot Interferometer
Problems
Bibliography
6 Optical Resonance
6.1 Optical Resonator
6.2 Longitudinal Modes
6.3 Transverse Modes
6.4 Cavity Lifetime and Quality Factor
6.5 FabryPérot Cavity
Problems
Bibliography
7 Optical Absorption and Emission
7.1 Optical Transitions
7.2 Transition Rates
7.3 Attenuation and Amplification of Optical Fields
Problems
Bibliography
8 Optical Amplification
8.1 Population Rate Equations
8.2 Population Inversion
8.3 Optical Gain
8.4 Optical Amplification
8.5 Spontaneous Emission
Problems
Bibliography
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141
141
147
154
156
159
160
165
168
169
169
183
191
200
203
204
204
207
211
214
216
221
223
224
224
234
241
245
248
249
249
251
259
265
267
270
273
Contents
9
Laser Oscillation
9.1 Conditions for Laser Oscillation
9.2 Mode-Pulling Effect
9.3 Oscillating Laser Modes
9.4 Laser Power
Problems
Bibliography
10
Optical Modulation
10.1 Types of Optical Modulation
10.2 Modulation Schemes
10.3 Direct Modulation
10.4 Refractive External Modulation
10.5 Absorptive External Modulation
Problems
Bibliography
11
Photodetection
ix
274
274
277
279
285
293
296
297
297
298
308
319
344
353
361
11.1 Physical Principles of Photodetection
11.2 Photodetection Noise
11.3 Photodetection Measures
Problems
Bibliography
362
362
375
382
391
395
Appendix A
Appendix B
Appendix C
Appendix D
Index
396
403
405
406
409
Symbols and Notations
SI Metric System
Fundamental Physical Constants
Fourier-Transform Relations
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Cambridge Books Online
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Principles of Photonics
Jia-Ming Liu
Book DOI: http://dx.doi.org/10.1017/CBO9781316687109
Online ISBN: 9781316687109
Hardback ISBN: 9781107164284
Chapter
Preface pp. xi-xii
Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.001
Cambridge University Press
PREFACE
The field of photonics has matured into an important discipline of modern engineering and
technology. Its core principles have become essential knowledge for all undergraduate students
in many engineering and scientific fields. This fact is fully recognized in the new curriculum of
the Electrical Engineering Department at UCLA, which makes the principles of photonics a
required course for all electrical engineering undergraduate students. Graduate students studying in areas related to photonics also need this foundation.
The most fundamental concepts in photonics are the nature of optical fields and the properties
of optical materials because the entire field of photonics is based on the interplay between
optical fields and optical materials. Any photonic device or system, no matter how simple or
sophisticated it might be, consists of some or all of these functions: the generation, propagation,
coupling, interference, amplification, modulation, and detection of optical waves or signals.
The properties of optical fields and optical materials are addressed in the first two chapters of
this book. The remaining nine chapters cover the principles of the major photonic functions.
This book is written for a one-quarter or one-semester undergraduate course for electrical
engineering or physics students. Only some of these students might continue to study advanced
courses in photonics, but at UCLA we believe that all electrical engineering students need to
have a basic understanding of the core knowledge in photonics because it has become an
established key area of modern technology. Many universities already have departments that
are entirely devoted to the field of photonics. For the students in such photonics-specific
departments or institutions, the subject matter in this book is simply the essential foundation
that they must master before advancing to other photonics courses. Based on this consideration,
this book emphasizes the principles, not the devices or the systems, nor the applications.
Nevertheless, it serves as a foundation for follow-up courses on photonic devices, optical
communication systems, biophotonics, and various subjects related to photonics technology.
Because this book is meant for a one-quarter or one-semester course, it is kept to a length that
can be completed in a quarter or a semester. Because it likely serves the only required
undergraduate photonics course in the typical electrical engineering curriculum, it has to cover
most of the essential principles. The chapters of this book are organized based on the major
principles of photonics rather than based on device or system considerations. These attributes
are the key differences between this book and other books in this field.
Through my teaching experience on this subject over many years, I find a need for a textbook
that has the following features.
1. It is self-contained, and its prerequisites are among the required core courses in the typical
electrical engineering curriculum.
2. It covers the major principles in a single book that can be completely taught in a one-quarter
or one-semester course. And it treats these subjects not superficially but to a sufficient depth
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xii
Preface
for a student to gain a solid foundation to move up to advanced photonics courses, if the
student stays in the photonics field, or for a student to gain a useful understanding of
photonics, if the student moves on to a different field.
3. It has ample examples that illustrate the concepts discussed in the text, and it has plenty of
problems that are closely tied to these concepts and examples.
This book is written with the above features to serve the need for a book covering a core
photonics course in a modern electrical engineering curriculum.
There are two prerequisites for a course that uses this book: (1) basic electromagnetics up to
electromagnetic waves and (2) basic solid-state physics or solid-state electronics. No advanced
background in optics beyond what a student normally learns in general physics is required. At
UCLA, this course is taught as a required course in the Electrical Engineering Department to
undergraduate juniors and seniors. The materials of this book have been test taught for a few
years in this one-quarter course, which has 38 hours of lectures, excluding the time for the
midterm and final exams. This course is followed by elective courses on photonic devices and
circuits, photonic sensors and solar cells, and biophotonics.
Carefully designed examples are given at proper locations to illustrate the concepts discussed
in the text and to help students apply what they learn to solving problems. Each example is tied
closely to one or more concepts discussed in the text and is placed right after that text; its
solution does not simply give the answer but presents a detailed explanation as part of the
teaching process. An ample number of problems are given at the end of each chapter. The
problems are labeled with the corresponding section numbers and are arranged in the sequence
of the material presented in the text. The entire book has 100 examples and 247 problems.
The materials in this book are selected and structured to suit the purpose of a course on the
principles of photonics. Besides the newly written materials, text and figures are adopted from
my book Photonic Devices wherever suitable. All examples and problems, except for the very
few that illustrate key concepts, are newly designed specifically to meet the pedagogical
purpose of this book.
This book was developed through test teaching a course in the new curriculum at UCLA. In
this process, I received much feedback from my colleagues and my students. I would like to
thank my editor, Julie Lancashire, for her help at every stage during the development of this
book, and my content manager, Jonathan Ratcliffe, for taking care of the production matters of
this book. I would like to express my loving appreciation to my daughter, Janelle, who took a
special interest in this project and shared my excitement in it. Special thanks are due to my wife,
Vida, who gave me constant support and created an original oil painting for the cover art of
this book.
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PARTIAL LIST OF SYMBOLS
Symbol
Unit
Meaning; derivatives
References1
a
none
round-trip intracavity field amplification factor
(6.4)
aE , aM
none
asymmetry factors for TE and TM modes
(3.130)
~
A, A
W1=2
mode amplitude
(4.23), (4.26)
Av
W1=2
amplitude of mode ν
(4.3)
A21
s1
Einstein A coefficient
(7.21)
A
m2
area
(11.59)
b
m
confocal parameter of Gaussian beam
(3.69)f
b
none
normalized guide index
(3.129)
b
none
linewidth enhancement factor
(9.39)
~
B, B
W1=2
mode amplitude
(4.24), (4.27)
B
Hz
bandwidth
(11.1)
B12 , B21
m3 J1 s1
Einstein B coefficients
(7.19), (7.20)
B
T
real magnetic induction in the time domain
(1.3)
B, B
T
complex magnetic induction
(1.41)
c
m s1
speed of light in free space
(1.1)b, (1.39)
cvμ
none
overlap coefficient between modes v and μ
(4.19)
cijkl
m2 A2
quadratic magneto-optic coefficient
(2.77)
d
m
thickness or distance; d g , dQW
(3.127)
d, d0
m
beam spot size diameter, d ¼ 2w, d 0 ¼ 2w0
(3.69)b
dE , dM
m
effective waveguide thicknesses for TE and TM modes
(3.138), (3.143)
D
none
group-velocity dispersion; D1 , D2 , Dβ
(3.167)
D
W1
detectivity
(11.58)
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xiv
Partial List of Symbols
(cont.)
Symbol
Unit
Meaning; derivatives
References1
Dλ
s m2
group-velocity dispersion; Dλ ¼ D=cλ
(3.168)
D∗
m Hz1=2 W1
specific detectivity
(11.59)
D
C m2
real electric displacement in the time domain
(1.2)
D, D
C m2
complex electric displacement; De , Do , Dþ , D
(1.42), (1.51)
D, D
C m2
slowly varying amplitudes of D and D; De , D0
(3.57)
DR
dB
dynamic range
(11.62)
e
C
electronic charge
(2.30)f
^e
none
unit vector of electric field polarization; ^e e , ^e o , ^e þ , ^e (1.61)
E1 , E2
eV
energies of levels j1i and j2i
(7.1)
Ec , Ev
eV
conduction-band and valence-band edges
(10.106)
EF
eV
Fermi energy
(11.5)b
Eg
eV
bandgap
(10.105), (11.7)
Eth
eV
threshold photon energy
(11.5)
E
V m1
real electric field in the time domain
(1.2)
E0 , E 0
V m1
static or low-frequency electric field
(2.54)
Ee , Eh
V m1
electric fields seen by electrons and holes
(10.106)
E, E
V m1
complex electric field
(1.40)
Ev , E v
V m1
complex electric field of mode v
(3.1)
E, E
V m1
slowly varying amplitudes of E and E; E e , E o , E þ , E (1.52)
Ev, E v
V m1
complex electric field profile of mode v
(3.1)
^v
E
V m1 W1=2
normalized electric mode field distribution, E v ¼ Av E^ v
(3.18)
ER
dB
extinction ratio
(10.18)
f
Hz
acoustic or modulation frequency, f ¼ Ω=2π
(2.79)b, (10.27)b
f 3dB
Hz
3-dB modulation bandwidth or cutoff frequency
(10.31), (11.64)
fK
m
Kerr focal length
(10.115)
f ijk
m A1
linear magneto-optic coefficient, Faraday coefficient
(2.76), (10.77)
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Partial List of Symbols
xv
(cont.)
Symbol
Unit
Meaning; derivatives
References1
fr
Hz
relaxation resonance frequency, f r ¼ Ωr =2π
(10.41)
F
none
excess noise factor
(11.38)
F
none
finesse of interferometer or optical cavity
(5.49), (6.12)
Fðz; z0 Þ
none
forward-coupling matrix for codirectional coupling
(4.48)
g, g ðvÞ
m1
gain coefficient, amplification coefficient
(3.183)f, (7.46)
g0
m1
unsaturated gain coefficient
(8.22)
g th
m1
threshold gain coefficient
(9.9), (9.19)
g^ ðvÞ
s
lineshape function
(7.2)
g
none
degeneracy factor; g1 , g2
(7.1)f, (7.28)
g
s1
gain parameter
(9.18)
g0
s1
unsaturated gain parameter
(9.22)
gn
m3 s1
differential gain parameter
(10.36)
gp
m3 s1
nonlinear gain parameter
(10.36)
gth
s1
threshold gain parameter
(9.20), (10.34)
G
none
cavity round-trip field gain; Gc , Gmn , Gcmn
(6.4)
G
none
photodetector current gain
(11.4)f, (11.36)
G, G0
none
optical amplifier power gain, G0 for unsaturated gain
(8.39)
h, ℏ
Js
Planck’s constant, ℏ ¼ h=2π
(1.1)
h1 , h2 , h3
m1
transverse oscillation parameters of mode field
(3.104), (3.133)
H
m
height of acousto-optic transducer
(10.89)
H ðÞ
none
Heaviside step function
(2.24)
H m ðÞ
none
Hermite function
(3.73)f
H
A m1
real magnetic field in the time domain
(1.3)
H0 , H 0
A m1
static or low-frequency magnetic field
(2.68)
H, H
A m1
complex magnetic field
(1.42)
Hv , H v
A m1
complex magnetic field of mode v
(3.2)
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xvi
Partial List of Symbols
(cont.)
Symbol
Unit
Meaning; derivatives
References1
H, H
A m1
slowly varying amplitudes of H and H
(3.5)
Hv , Hv
A m1
complex magnetic field profile of mode v
(3.2)
^ν
H
A m1 W1=2
normalized electric mode field distribution,
^
H ¼AH
(3.18)
v
v
v
i
none
pffiffiffiffiffiffiffi
1
i
A
current; ib , id , in , iph , is
(11.4)f
I
A
injection current; I 0 , I m , I th
(10.22)
I
W m2
optical intensity; I 0 , I i , I in , I out , I r , I t
(1.56)
I0
A
reverse current
(11.15)
I ðvÞ
W m2 Hz1
optical spectral intensity distribution
(7.17)
Ip, Is
W m2
optical pump and signal intensities
(8.36)
I sat
W m2
saturation intensity
(8.22)
J, J
A m2
real current density
(1.5)
J
A m2
complex current density
(2.35)
k
m1
propagation constant, wavenumber; k 0 , k i , k r , k t
(1.84)
kB
J K1
Boltzmann constant
(7.14), (7.25)
ke , ko
m1
propagation constants of extraordinary and ordinary
waves
(3.57)
k 0 , k 00
m1
real and imaginary parts of k, k ¼ k 0 þ ik 00
(3.180)
kx , ky , kz
m1
propagation constants of x, y, and z polarized fields
(2.15)
kX , kY , kZ
m1
propagation constants of X, Y, and Z polarized fields (2.67)
kþ , k
m1
propagation constants of circularly polarized fields
(2.21)
k^
none
unit vector in the k direction
(1.84)
k
m1
wavevector; ki , kr , kt , kq
(1.1)b, (1.52)
ke , ko
m1
wavevectors of extraordinary and ordinary waves
(3.56)f, (3.57)
kx , ky , kz
m1
wavevectors of x, y, and z polarized fields
(3.48)f
kþ , k
m1
wavevectors of circularly polarized fields
(10.74)
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Partial List of Symbols
xvii
(cont.)
Symbol
Unit
Meaning; derivatives
References1
K
m1
wavenumber of acoustic wave or grating, K ¼ 2π=Λ
(2.79)f, (4.35)
K
s
K factor of a semiconductor laser
(10.43)
K
m1
wavevector of acoustic wave
(2.79)
l
m
length or distance
(3.185)
lc
m
coupling length; lPM
c
(4.56)
lRT
m
round-trip optical path length
(6.1)
lλ=4 , lλ=2
m
quarter-wave and half-wave lengths
(3.49), (3.50)
L
m
length of acousto-optic transducer
(10.89)
m
none
transverse mode index associated with x
(3.1)f
m
none
modulation index
(10.27)
m0
kg
free electron rest mass
Fig. 11.1
m∗
kg
effective mass of carriers
(2.31)
∗
m∗
e , mh
kg
effective masses of electrons and holes
(10.107)
M
kg
atomic or molecular mass
(7.14)
M TE , M TM
none
numbers of guided TE and TM modes
(3.152), (3.153)
Ms
A m1
saturation magnetization
(10.78)
M
A m1
real magnetic polarization in the time domain
(1.3)
M0 , M 0
A m1
static or low-frequency magnetization
(2.70)
n
none
transverse mode index associated with y
(3.1)f
n
none
index of refraction; nβ , n
(1.84)
n
m3
electron concentration
(11.9)
n0
m3
equilibrium concentration of electrons
(11.9)f
n1 , n2 , n3
none
refractive indices of waveguide layers, n1 > n2 > n3
(3.125)
n2
m2 W1
coefficient of intensity-dependent index change
(10.101)
ne , no
none
extraordinary and ordinary indices of refraction
(2.15)f, (3.56)
nx , ny , nz
none
principal indices of refraction
(2.14)
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xviii
Partial List of Symbols
(cont.)
Symbol
Unit
Meaning; derivatives
References1
nX , nY , nZ
none
new principal indices of refraction
(2.66)
nþ , n
none
principal indices of refraction for circular polarized
modes
(2.20)
n? , njj
none
indices of second-order magneto-optic effect
(2.16)
n0 , n00
none
real and imaginary parts of refractive index,
n ¼ n0 þ in00
(3.181)
n^
none
unit normal vector
(1.23)
N
none
some number
(5.21)
N
none
group index; N 1 , N 2 , N β
(3.171)
N
m3
carrier density
(2.31)
N
m3
effective population inversion
(8.4)
N1, N2,
Nt
m3
population densities in levels j1i, j2i, and all levels
(7.26), (8.12)
N sp
none
spontaneous emission factor
(9.14)
N
none
number of charge carriers
(11.3)
NEP
W
noise equivalent power
(11.55)
p
none
probability
(11.18)
p
none
cross-section ratio for pumping
(8.13)
p
m3
hole concentration
(11.9)
p0
m3
equilibrium concentration of holes
(11.9)f
pijkl p0ijkl
none
elasto-optic and rotation-optic coefficients
(2.83)
pðvk Þ
Hz1
probability density function
(7.10)
P
W
power; Pa , Pin , Pout , Ppk , Pth , Pv
(3.17)
Pp , Ps
W
tr
in
out
pump and signal powers; Pth
p , Pp , Ps , Ps
(9.27), (8.37)
Psat
W
saturation power
(8.37)
Psp
W
spontaneous emission power
(8.44)
Ptrsp
W
critical fluorescence power
(8.46)
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Partial List of Symbols
xix
(cont.)
Symbol
Unit
Meaning; derivatives
References1
^ sp
P
W m3
spontaneous emission power density
(8.43)
p^trsp
W m3
critical fluorescence power density
(8.45)
P
C m2
real electric polarization in the time domain
(1.2)
P, P
C m2
complex electric polarization
(1.50)
PðnÞ
C m2
nth-order nonlinear real electric polarization
(2.91)
PðnÞ , PðnÞ
C m2
nth-order nonlinear complex electric polarization
(2.91)
Pres
C m2
complex electric polarization from resonant transition (7.47)b
q
none
longitudinal mode index
(5.47), (6.9)
q
none
order of coupling or diffraction
(4.36), (5.24)
q
C
charge
(2.30)
qðzÞ
m
complex radius of curvature of a Gaussian beam
(3.75)
Q
none
quality factor of resonator; Qmnq
(6.26), (6.30)
Q
none
acousto-optic diffraction parameter
(10.83)
r
m
radial coordinate, radial distance
r
none
reflection coefficient; r1 , r 2 , rp , rs
(3.91), (4.67)
r
none
pumping ratio of a laser
(9.26)
rijk , rαk
m V1
linear electro-optic coefficients, Pockels coefficients
(2.58), (2.60)
rðf Þ, rðΩÞ
none
complex modulation response function
(10.29), (10.40)
r
m
spatial vector
(1.2)
R
none
reflectance, reflectivity; R1 , R2 , Rp , Rs
(3.93)
R
Ω
resistance; Ri , RL
(11.16)
R
m3 s1
effective pumping rate for population inversion
(8.6)
R1 , R2
m3 s1
pumping rates for levels j1i and j2i
(8.1), (8.2)
Rðf Þ
none
electrical power spectrum of modulation response
(10.30), (10.44)
Rðz; 0; l Þ
none
R, Rij
none
reverse-coupling matrix for contradirectional coupling (4.59)
(2.82)
rotation tensor and elements, R ¼ Rij
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xx
Partial List of Symbols
(cont.)
Symbol
Unit
Meaning; derivatives
References1
R
m
radius of curvature; R1 , R2
(3.71), (6.31)
R
none
amplitude of rotation; Rij
(2.87)
R
A W1
responsivity of photodetector with current output; R0
(11.50)
R
V W1
responsivity of photodetector with voltage output
(11.51)
s
m
separation, spacing; se
Fig. 4.2, (10.69)
s
none
signal; sn
(11.18)
sijkl , sαkl
m2 V2
quadratic electro-optic coefficients, Kerr coefficients
(2.58), (2.60)
S
m3
photon density
(9.21)
Ssat
m3
saturation photon density
(9.24)
S
W m2
real Poynting vector
(1.32)
S
W m2
(1.54)
S, Sij
none
complex Poynting vector; Se , So
strain tensor and elements, S ¼ Sij
(2.81)
S
none
amplitude of strain; S ij
(2.87)
S
none
number of photons
(11.2)
SNR
none, dB
signal-to-noise ratio
(11.26)
t
s
time
t
none
transmission coefficient; tp , ts
(3.92)
tr , tf
s
risetime and falltime
(11.63)b
T
K
temperature
(7.14)
T
s
time interval
(1.53)
T
s
round-trip time of optical cavity
(6.1)
T
none
transmittance, transmissivity; T p , T s
(3.94), (10.108)
u, u0
J m3
electromagnetic energy density
(7.16), (1.33)
uðvÞ
J m3 Hz1
spectral energy density
(7.16)
u, ui
m
elastic deformation wave and its components
(2.79), (2.81)
U
J
optical energy; U mode
(9.28)
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Partial List of Symbols
xxi
(cont.)
Symbol
Unit
Meaning; derivatives
References1
U
m
amplitude of elastic wave
(2.79)
v
V
voltage; v n , v out , v s
(11.16)
v
m s1
velocity
Fig. 11.1
va
m s1
acoustic wave velocity
(2.80)b
vg
m s1
group velocity; v gβ
(3.165)
vp
m s1
phase velocity; v pβ
(3.162)
V
none
normalized frequency and waveguide thickness, V
number
(3.128)
V
rad A1
Verdet constant
(10.77)
V
V
voltage; V m , V π , V π=2
(10.51), (11.15)
Vc
none
cutoff V number; V cm
(3.147)
V
m3
volume; V gain , V mode
(1.31)b, (6.2)
w, w0
m
Gaussian beam radius, spot size
(3.69), (3.70)
W
m
width of acousto-optic cell
(10.91)
W
s1
transition probability rate; W 12 , W 21 , W p , W sp
(7.22)(7.24)
W p, W m
W m3
power densities expended by EM field on P and M
(1.34), (1.35)
W ðvÞ
none
transition rate per unit frequency;
W 12 ðvÞ, W 21 ðvÞ,W sp ðvÞ
(7.19)(7.21)
x
m
spatial coordinate
^x
none
unit coordinate vector or principal dielectric axis
X
m
^
spatial coordinate along X
^
X
none
new principal dielectric axis
y
m
spatial coordinate
^y
none
unit coordinate vector or principal dielectric axis
Y
m
spatial coordinate along Y^
Y^
none
new principal dielectric axis
z
m
spatial coordinate
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(1.62), (2.13)b
(2.65)b
(1.62), (2.13)b
(2.65)b
xxii
Partial List of Symbols
(cont.)
Symbol
Unit
Meaning; derivatives
References1
^z
none
unit coordinate vector or principal dielectric axis
(3.16), (2.13)b
zR
m
Rayleigh range of a Gaussian beam
(3.69)
Z
m
spatial coordinate along Z^
Z^
none
new principal dielectric axis
(2.65)b
α
rad
field polarization angle
(1.64)
α
rad
walk-off angle of extraordinary wave
(3.60)
α, αðvÞ
m1
attenuation coefficient, absorption coefficient
(3.180), (7.45)
α0
m1
unsaturated absorption coefficient
(10.110)
αc
m1
propagation parameter for contradirectional coupling
(4.61)
β
none
bottleneck factor
(8.7)
β
m1
propagation constant of a mode; βmn , βTE , βTM
(3.1)
β0 , β00
m1
real and imaginary parts of β, β ¼ β0 þ iβ00
(3.184)
βc
m1
propagation parameter for codirectional coupling
(4.50)
γ
s1
relaxation rate, decay rate; γ21 , γi , γout
(2.23)
γ1 , γ2 , γ3
m1
transverse decay parameters of mode field
(3.118), (3.131)
γa
s1
acoustic decay rate
(10.93)
γc
s1
cavity decay rate, photon decay rate; γcmnq
(6.25)
γn
s1
differential carrier relaxation rate
(10.37)
γp
s1
nonlinear carrier relaxation rate
(10.37)
γr
s1
total carrier relaxation rate
(10.42)
γs
s1
spontaneous carrier relaxation rate
(10.42)
Γ
none
overlap factor
(6.2)
δ
m1
phase mismatch parameter for phase mismatch of 2δ
(4.31)
δωmnq
rad s1
frequency shift of mode pulling
(9.12)
Δn, Δp
m3
excess electron and hole concentrations
(11.10)
ΔP
C m2
change in electric polarization
(4.8)
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Partial List of Symbols
xxiii
(cont.)
Symbol
Unit
Meaning; derivatives
References1
Δt
s
pulsewidth or time duration; Δt ps
(10.117), (11.1)
Δϵ, Δϵ
F m1
change or variation in electric permittivity
(2.55), (4.12)
~
Δϵ , Δ~ϵ
F m1
amplitudes of Δϵ and Δϵ
(2.88)
Δη, Δηij
none
change or variation in relative impermeability
(2.58)
Δθ
rad
divergence angle of a Gaussian beam
(3.72)
Δλ
m
spectral width; Δλg
Table 7.1
Δv
Hz
optical linewidth, bandwidth; ΔvD , Δvg , Δvinh , Δvh
(7.4)
Δvc
Hz
longitudinal mode linewidth
(6.18)
ΔvL
Hz
longitudinal mode frequency spacing
(6.17)
Δvmnq
Hz
oscillating laser mode linewidth
(9.13)
ΔvST
Hz
SchawlowTownes linewidth of laser mode; ΔvST
mnq
(9.14)
Δφ
rad
phase shift or phase retardation
(10.13)
Δφc
rad
phase width of a cavity resonance peak
(6.11)
ΔφL
rad
phase spacing between cavity resonance peaks
(6.10)
Δχ, Δχ
none
change or variation in electric susceptibility
(2.54)
Δω
rad s1
optical linewidth, bandwidth, Δω ¼ 2πΔv; Δωinh , Δωh (7.3)f, (7.13)
ϵ
F m1
electric permittivity
(2.11), (3.4)
ϵ0
F m1
electric permittivity of free space
(1.2)
ϵ 0 , ϵ 00
F m1
real and imaginary parts of ϵ, ϵ ¼ ϵ 0 þ iϵ 00
(3.179)
ϵx, ϵy, ϵz
F m1
principal dielectric permittivities
(2.13)
ϵX , ϵY , ϵZ
F m1
new principal dielectric permittivities
(2.65)
ϵþ, ϵ
F m1
principal dielectric permittivities of circular
polarizations
(2.17)
ϵðr, tÞ
F m4 s1
real permittivity tensor in the real space and time
domain
(1.21)
ϵ ðωÞ, ϵ ij
F m1
complex permittivity tensor in the frequency domain
(1.60)
ϵ res ðωÞ
F m1
permittivity of resonant transition
(6.36)
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xxiv
Partial List of Symbols
(cont.)
Symbol
Unit
Meaning; derivatives
References1
ε
rad
ellipticity of polarization ellipse
(1.68)
ζ mn ðzÞ
rad
phase variation of Gaussian mode field; ζ RT
mn
(3.76)
η
none
coupling efficiency; ηPM
(4.55)
ηc
none
power conversion efficiency
(9.37)
ηcoll
none
collection efficiency
(11.48)
ηe
none
external quantum efficiency
(10.24), (11.48)
ηi
none
internal quantum efficiency
(11.48)
ηinj
none
injection efficiency
(10.22)
ηs
none
slope efficiency, differential power conversion
efficiency
(9.38)
ηt
none
transmission efficiency
(11.48)
η, ηij , ηα
none
relative impermeability tensor and its elements,
η ¼ ½ηij (2.57)
θ
rad
angle, spherical angular coordinate
(3.51)
θ
rad
orientation of the polarization ellipse
(1.69)
θB
rad
Brewster angle or Bragg angle
(3.100), (10.88)
θc
rad
critical angle
(3.102)
θd
rad
angle of diffraction
(10.87)
θdef
rad
deflection angle
Example 10.9
θF
rad
Faraday rotation angle
(10.75)
θi , θr , θt
rad
angles of incidence, reflection, and refraction
(transmitted)
(3.88)
κ
m1
coupling coefficient; κvμ
(4.13)
~κ
m1
coupling coefficient; κ~vμ
(4.20)
λ
m
optical wavelength in free space
(1.1)
λc
m
cutoff wavelength; λcm
(3.151)
λth
m
threshold wavelength
(11.5)
Λ
m
acoustic wavelength or grating period
(2.79)b, (4.35)
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Partial List of Symbols
(cont.)
Symbol
Unit
Meaning; derivatives
References1
μ0
H m1
magnetic permeability of free space
(1.3)
μe , μh
m2 V1 s1
electron and hole mobilities
(11.9)
v
Hz
optical frequency
(1.1)
v0
Hz
center optical frequency
(2.27)f, (7.12)
v21
Hz
resonance frequency between levels j1i and j2i
(7.1)
ξ
none
duty factor
Fig. 4.3
ξ, ξ ðM 0z Þ
none
permittivity tensor elements for circular birefringence
(2.16), (2.78)
ρ
C m3
charge density
(1.6)
ρF
rad m1
specific Faraday rotation
(10.79)
σ
S m1
conductivity; σ 0
(2.33), (11.9)
σ 12 , σ 21
m2
transition cross sections
(7.36), (7.37)
σa , σe
m2
absorption and emission cross sections
(7.38), (7.39)
σ 2s
none
variance of s
(11.19)
τ
s
lifetime, decay time, or time constant
(2.30), (7.6)
τ1, τ2
s
fluorescence lifetimes of levels j1i and j2i
(7.6), (7.8)
τc
s
photon lifetime; τ cmnq
(6.23)
τs
s
saturation lifetime or spontaneous carrier lifetime
(8.23), (10.23)
τ sp
s
spontaneous radiative lifetime
(7.32)
ϕ
rad
azimuthal angle, azimuthal angular coordinate
(3.52)
ϕ
V
work function potential; eϕ ¼ work function
(11.6)
φ
rad
phase or phase shift
(1.63), (1.83)
χ
none
electric susceptibility
(2.11)
χ
V
electron affinity potential; eχ ¼ electron affinity
(11.7)
χ res
none
resonant electric susceptibility
(2.25), (2.26)
χx, χy, χz
none
principal dielectric susceptibilities
(2.15)f
χ 0 , χ 00
none
real and imaginary parts of χ, χ ¼ χ 0 þ 1χ 00
(2.7)b
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xxv
xxvi
Partial List of Symbols
(cont.)
Symbol
Unit
Meaning; derivatives
References1
χðr; tÞ
m3 s1
real susceptibility tensor in the real space and time
domain
(1.20)
χðωÞ, χ ij
none
complex susceptibility tensor in the frequency domain (1.59)
ð2 Þ
m V1
second-order nonlinear susceptibility in the frequency (2.98), (2.100)
domain
χð3Þ , χ ijkl
ð3 Þ
m2 V2
third-order nonlinear susceptibility in the frequency
domain
(2.99), (2.101)
ψ
rad
spatial phase of mode field distribution
(3.107)
ψe
rad
angle between Se and optical axis of crystal
(3.60)
ω
rad s1
optical angular frequency; ω ¼ 2πv
(1.1)b
ω0
rad s1
center optical angular frequency; ω0 ¼ 2πv0
(2.22), (7.13)
ω21
rad s1
resonance angular frequency between levels j1i and
j2i
(2.22)
ωc
rad s1
cutoff frequency; ωcm
(3.151)
Ω
rad s1
acoustic or modulation angular frequency; Ω ¼ 2πf
(2.79), (10.27)
Ωr
rad s1
relaxation resonance frequency; Ωr ¼ 2πf r
(10.41)
χð2Þ , χ ijk
1
Suffixes, f “forward” and b “backward,” on the equation number indicate symbols explained for the first
time in the text immediately after or before the equation cited.
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Cambridge Books Online
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Principles of Photonics
Jia-Ming Liu
Book DOI: http://dx.doi.org/10.1017/CBO9781316687109
Online ISBN: 9781316687109
Hardback ISBN: 9781107164284
Chapter
1 - Basic Concepts of Optical Fields pp. 1-21
Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.002
Cambridge University Press
1
1.1
Basic Concepts of Optical Fields
NATURE OF LIGHT
..............................................................................................................
Photonics addresses the control and use of light for various applications. Light is electromagnetic radiation of frequencies in the range from 1 THz to 10 PHz, corresponding to wavelengths
between 300 μm and 30 nm in free space, which is generally divided into the infrared,
visible, and ultraviolet regions. In this spectral region, the electromagnetic radiation exhibits the
dual nature of photon and wave. The photon nature has to be considered in the generation,
amplification, frequency conversion, or detection of light, whereas the wave nature is important
in all processes but especially in the propagation, transmission, interference, modulation, or
switching of light.
1.1.1 Photon Nature of Light
The energy of a photon is determined by its frequency ν or, equivalently, its angular frequency
ω ¼ 2πν. Associated with its particle nature, a photon has a momentum determined by its
wavelength λ or, equivalently, its wavevector k. These characteristics are summarized below for
a photon in free space:
speed
energy
momentum
c ¼ λν;
hν ¼ ℏω ¼ pc;
p ¼ hν=c ¼ h=λ,
p ¼ ℏk.
The energy of a photon that has a wavelength of λ in free space can be calculated using the
formula:
hν ¼
1:2398
1239:8
μm eV ¼
nm eV:
λ
λ
(1.1)
The photon energy at the optical wavelength of 1 μm is 1.2398 eV, and its frequency is
300 THz.
EXAMPLE 1.1
The visible spectrum ranges from 700 nm wavelength at the red end to 400 nm wavelength at
the violet end. What is the frequency range of the visible spectrum? What are the energies of
visible photons?
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2
Basic Concepts of Optical Fields
Solution:
The 700 nm optical wavelength at the red end has a frequency of
νred ¼
c
λred
¼
3 108 m s1
¼ 429 THz
700 nm
and a photon energy of
hνred ¼
1239:8
1239:8
nm eV ¼
eV ¼ 1:77 eV:
λred
700
The 400 nm optical wavelength at the violet end has a frequency of
νviolet ¼
c
λviolet
¼
3 108 m s1
¼ 750 THz
400 nm
and a photon energy of
hνviolet ¼
1239:8
1239:8
nm eV ¼
eV ¼ 3:10 eV:
λviolet
400
Therefore, the frequency range of the visible spectrum is from 429 THz to 750 THz. Visible
photons have energies in the range from 1.77 eV to 3.10 eV.
The energy of a photon is determined only by its frequency or, equivalently, by its free-space
wavelength, but not by the light intensity. The intensity, I, of monochromatic light is related to
the photon flux density, or the number of photons per unit time per unit area, by
photon flux density ¼
I
I
¼
:
hν ℏω
The photon flux, or the number of photons per unit time, of a monochromatic optical beam is
related to the beam power P by
photon flux ¼
P
P
¼
:
hν ℏω
EXAMPLE 1.2
Find the photon flux of a monochromatic optical beam that has a power of P ¼ 1 W by taking
its wavelength at either end of the visible spectrum. What are the momentum carried by a red
photon and the momentum carried by a violet photon? What is the total momentum carried by
the beam in a time duration of Δt ¼ 1 s?
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1.1 Nature of Light
3
Solution:
From Example 1.1, the photon energy of the 700 nm wavelength at the red end is
hνred ¼ 1:77 eV, and that of the 400 nm wavelength at the violet end is hνviolet ¼ 3:10 eV.
Therefore, the photon flux of a beam that has a power of P ¼ 1 W at the 700 nm red
wavelength is
red photon flux ¼
P
1
¼
s1 ¼ 3:53 1018 s1 ,
hνred 1:77 1:6 1019
and the photon flux of a beam that has a power of P ¼ 1 W at the 400 nm violet wavelength is
violet photon flux ¼
P
hνviolet
¼
1
s1 ¼ 2:02 1018 s1 :
3:10 1:6 1019
The momentum carried by a red photon is
pred ¼
hνred 1:77 1:6 1019
N s ¼ 9:44 1028 N s,
¼
c
3 108
and that carried by a violet photon is
pviolet ¼
hνviolet 3:10 1:6 1019
¼
N s ¼ 1:65 1027 N s:
c
3 108
The total momentum carried by an optical beam that has a power of P during a time duration of
Δt is independent of the optical wavelength:
total momentum ¼ ðphoton fluxÞpΔt ¼
P hν
PΔt
Δt ¼
:
hν c
c
Therefore, irrespective of whether the wavelength of the beam is at the red or the violet end, the
total momentum carried by the beam in a time duration of Δt ¼ 1 s is
total momentum ¼
PΔt
11
¼ 3:33 109 N:
¼
c
3 108
1.1.2 Wave Nature of Light
An optical wave is characterized by the space and time dependence of the optical field, which is
composed of coupled electric and magnetic fields governed by Maxwell’s equations. It varies
with time at an optical carrier frequency, and it propagates in a spatial direction determined by a
wavevector. The behavior of an optical wave is strongly dependent on the optical properties of
the medium. An optical field is a vectorial field characterized by five parameters: polarization,
magnitude, phase, wavevector, and frequency. Polarization and wavevector are vectorial
quantities; magnitude, frequency, and phase are scalar quantities. The general properties of
optical fields are described in the following sections.
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4
Basic Concepts of Optical Fields
1.2
OPTICAL FIELDS AND MAXWELL’S EQUATIONS
..............................................................................................................
An electromagnetic field in a medium is characterized by four vectorial fields:
electric field
electric displacement
magnetic field
magnetic induction
Eðr; t Þ
Dðr; t Þ
H ðr; t Þ
Bðr; t Þ
V m1 ,
C m2 ,
A m1 ,
T or Wb m2 :
The response of a medium to an electromagnetic field generates the polarization and the
magnetization:
polarization (electric polarization)
magnetization (magnetic polarization)
Pðr; tÞ C m2 ,
M ðr; t Þ A m1 :
The electric field Eðr; tÞ and the magnetic induction Bðr; t Þ are the macroscopic forms of the
microscopic fields seen by the charge and current densities in the medium. The polarization
Pðr; t Þ and the magnetization M ðr; t Þ are the macroscopically averaged densities of microscopic
electric dipoles and magnetic dipoles that are induced by the presence of the electromagnetic
field in the medium. These macroscopic forms are obtained by averaging over a volume that is
small compared to the dimension of the optical wavelength but is large compared to the atomic
dimension. The electric displacement Dðr; tÞ and the magnetic field H ðr; t Þ are macroscopic
fields defined as
Dðr; t Þ ¼ ϵ 0 Eðr; t Þ þ Pðr; tÞ,
(1.2)
and
H ðr; tÞ ¼
1
Bðr; t Þ M ðr; t Þ,
μ0
(1.3)
where ϵ 0 1=36π 109 F m1 ¼ 8:854 1012 F m1 is the electric permittivity of free
space and μ0 ¼ 4π 107 H m1 is the magnetic permeability of free space. In addition to the
induced charge density and current density that respectively generate electric dipoles and
magnetic dipoles for Pðr; t Þ and M ðr; t Þ, an independent charge or current density, or both,
from external sources may exist:
charge density
current density
ρðr; tÞ C m3 ,
J ðr; t Þ A m2 :
The behavior of a space- and time-varying electromagnetic field in a medium is governed by
space- and time-dependent macroscopic Maxwell’s equations:
∇E¼
∇H ¼
∂B
,
∂t
∂D
þ J,
∂t
Faraday’s law;
Ampère’s law;
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(1.4)
(1.5)
1.2 Optical Fields and Maxwell’s Equations
∇ D ¼ ρ,
∇ B ¼ 0,
5
Gauss’s law; Coulomb’s law;
(1.6)
absence of magnetic monopoles:
(1.7)
Note that Gauss’s law in the form of (1.6) is equivalent to Coulomb’s law because one can be
derived from the other. The current and charge densities are constrained by the continuity
equation:
∇J þ
∂ρ
¼ 0,
∂t
conservation of charge:
(1.8)
The total current density in an optical medium has two contributions: the polarization current
from the bound charges of the medium and the current from free charge carriers, thus
Jtotal ¼ J bound þ J free . The free-carrier current has two possible origins, one from the response
of the conduction electrons and holes of the medium to the optical field and the other from an
external current source: J free ¼ J cond þ J ext . Both J bound and J cond are induced by the optical
field; thus
J total ¼ J bound þ J free ¼ J bound þ Jcond þ J ext ¼ Jind þ J ext ,
(1.9)
where J ind ¼ J bound þ J cond : Similarly, the total charge density can be decomposed as
ρtotal ¼ ρbound þ ρfree ¼ ρbound þ ρcond þ ρext ¼ ρind þ ρext :
(1.10)
In an optical medium, charge conservation requires that an increase of charge density induced
by an optical field at a location is always accompanied by a reduction at another location,
resulting in no net macroscopic induced charge density. Therefore, ρind ¼ 0 and ρtotal ¼ ρext for
a macroscopic optical field. By contrast, an induced macroscopic current density of J ind 6¼ 0
can exist in an optical medium.
In an optical medium that is free of external sources, J ext ¼ 0 and ρtotal ¼ ρext ¼ 0, but
Jtotal ¼ J bound þ J cond ¼ J ind 6¼ 0: Both J bound and Jcond are induced currents in response to an
optical field. The bound-electron polarization current J bound is a displacement current that is
always included in the ∂D=∂t term but not in the J term in (1.5). The conduction current J cond is
also an induced current, but it is carried by free charge carriers in the medium. In the case when
both external current and external charge are absent, the form of Maxwell’s equations depends
on how the conduction current is treated. There are generally two alternatives.
1. Being an induced current, J cond can be considered as a displacement current to be included
in the ∂D=∂t term so that J ¼ 0 in (1.5). Then, Maxwell’s equations are
∇E¼
∇H ¼
∂B
,
∂t
∂D
,
∂t
(1.11)
(1.12)
∇ D ¼ 0,
(1.13)
∇ B ¼ 0,
(1.14)
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6
Basic Concepts of Optical Fields
where D is the electric displacement that includes optical-field-induced responses from all
bound and conduction charges in the medium.
2. Being a current carried by free charge carriers, J cond can be separated from the ∂D=∂t term so
that J ¼ J cond in (1.5). Then, Maxwell’s equations have the form:
∇E¼
∇H ¼
∂B
,
∂t
∂Dbound
þ J cond ,
∂t
(1.15)
(1.16)
∇ Dbound ¼ 0,
(1.17)
∇ B ¼ 0,
(1.18)
with ∇ J cond ¼ 0, where Dbound is the electric displacement that includes only the contribution from bound charges and excludes that from the conduction current.
These two alternative forms of Maxwell’s equations are equivalent. The form using (1.16) is
taken only when a specific effect of the conduction current is considered, as in Section 2.4.
Otherwise, the form using (1.12) is generally taken. Therefore, we use the general form given in
(1.11)–(1.14) unless the situation calls for specific attention to a conduction current.
1.2.1 Transformation Properties
Maxwell’s equations and the continuity equation are the basic physical laws that govern the
behavior of electromagnetic fields. They are invariant under the transformation of space
inversion, in which the spatial vector r is changed to r0 ¼ r, i.e., r ! r, or ðx; y; zÞ !
ðx; y; zÞ, and under the transformation of time reversal, in which the time variable t is
changed to t 0 ¼ t, i.e., t ! t: This means that the form of these equations is not changed
when we perform the space-inversion transformation or the time-reversal transformation, or
both together.
The field quantities that appear in Maxwell’s equations, however, do not have to be invariant
under space inversion or time reversal. Their transformation properties are summarized as
follows.
1. Electrical fields: The electric field vectors E, D, and P are polar vectors associated with the
charge-density distribution. They change sign under space inversion but not under time
reversal.
2. Magnetic fields: The magnetic field vectors B, H, and M are axial vectors associated with
the current-density distribution. They change sign under time reversal but not under space
inversion.
3. Charge density: The charge density ρ is a scalar. It does not change sign under either space
inversion or time reversal.
4. Current density: The current density J is a polar vector that is the product of charge density
and velocity: J ¼ ρv. It changes sign under either space inversion or time reversal following
the sign change of the velocity vector under either transformation.
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1.2 Optical Fields and Maxwell’s Equations
7
1.2.2 Optical Response of a Medium
Polarization and magnetization are generated in a medium by the response of the medium to the
electric and magnetic fields, respectively: Pðr; tÞ depends on Eðr; t Þ, and M ðr; t Þ depends on
Bðr; tÞ: At an optical frequency, the magnetization vanishes: M ¼ 0: Therefore, it is always true
for an optical field that
Bðr; t Þ ¼ μ0 Hðr; tÞ:
(1.19)
Because μ0 is a constant that is independent of the medium, the magnetic induction Bðr; t Þ can
be replaced by μ0 H ðr; t Þ for any equations that describe optical fields, including Maxwell’s
equations, thus effectively eliminating one field variable. Note that this is not true at DC or low
frequencies, however, because a nonzero DC or low-frequency magnetization, M 6¼ 0, can exist
in any material. Indeed, it is possible to change the optical properties of a medium through a
magnetization induced by a DC or low-frequency magnetic field, leading to the functioning of
magneto-optics. It should be noted that even for magneto-optics, the magnetization is induced
by a DC or low-frequency magnetic field that is separate from the optical field. No magnetization is induced by the magnetic component of the optical field.
The optical properties of a material are completely determined by the relation between Pðr; tÞ
and Eðr; tÞ: This relation is generally characterized by an electric susceptibility tensor, χ,
through the following definition for electric polarization,
ðt ððð
Pðr; t Þ ¼ ϵ 0
χðr r0 ; t t 0 Þ Eðr0 , t 0 Þdr0 dt 0:
(1.20)
∞ all r0
The relation between Dðr; t Þ and Eðr; t Þ is characterized by the electric permittivity tensor, ϵ, of
the medium:
ðt ððð
Dðr; t Þ ¼ ϵ 0 Eðr; tÞ þ Pðr; t Þ ¼
ϵ ðr r0 ; t t 0 Þ Eðr0 , t 0 Þdr0 dt 0:
(1.21)
∞ all r0
From (1.20) and (1.21), the relationship between χ and ϵ in the real space and time domain is
ϵ ðr; t Þ ¼ ϵ 0 ½δðrÞδðt ÞI þ χðr; tÞ,
(1.22)
where I is the identity tensor that has the form of a 3 3 unit matrix and the delta functions are
ÐÐÐ
Ð∞
Dirac delta functions: all r δðrÞdr and ∞ δðtÞdt ¼ 1. The relation in (1.22) indicates that χ and
ϵ contain exactly the same information about the medium: one is known when the other is known.
Because χ and, equivalently, ϵ represent the response of a medium to an optical field and thus
completely characterize the macroscopic electromagnetic properties of the medium, (1.20) and
(1.21) can be regarded as the definitions of Pðr; t Þ and Dðr; t Þ, respectively.
1.2.3 Boundary Conditions
At the interface of two media of different optical properties, as shown in Fig. 1.1, the optical
field components must satisfy certain boundary conditions. These boundary conditions can be
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8
Basic Concepts of Optical Fields
Figure 1.1 Boundary between two media of different optical properties.
derived from Maxwell’s equations given in (1.11)–(1.14). From (1.11) and (1.12), the tangential components of the fields at the boundary satisfy
n^ E1 ¼ n^ E2 ,
(1.23)
n^ H 1 ¼ n^ H2 ,
(1.24)
where n^ is the unit vector normal to the interface as shown in Fig. 1.1. From (1.13) and (1.14),
the normal components of the fields at the boundary satisfy
n^ D1 ¼ n^ D2 ,
(1.25)
n^ B1 ¼ n^ B2 :
(1.26)
The tangential components of E and H are continuous across an interface, while the normal
components of D and B are continuous. Because B ¼ μ0 H at an optical frequency, as discussed
above, (1.24) and (1.26) also imply that the tangential component of B and the normal
component of H are also continuous. Consequently, all of the magnetic field components in
an optical field are continuous across a boundary. Possible discontinuities in an optical field
exist only in the normal component of E or in the tangential component of D.
1.3
OPTICAL POWER AND ENERGY
..............................................................................................................
Taking the dot product of H and (1.4) and that of E and (1.5) yields
H ð∇ EÞ ¼ H E ð∇ H Þ ¼ E ∂B
,
∂t
∂D
þ E J:
∂t
(1.27)
(1.28)
Using the vector identity B ð∇ AÞ A ð∇ BÞ ¼ ∇ ðA BÞ, (1.27) and (1.28) can be
combined to give
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1.3 Optical Power and Energy
9
Figure 1.2 Boundary surface enclosing a volume element.
∇ ðE H Þ ¼ E J þ E ∂D
∂B
þH
:
∂t
∂t
Using (1.2) and (1.3) and rearranging (1.29), we obtain
∂P
∂ ϵ 0 2 μ0
∂M
2
E J ¼ ∇ ðE HÞ þ μ0 H :
jEj þ jH j E 2
∂t 2
∂t
∂t
(1.29)
(1.30)
Recall that power in an electric circuit is given by voltage times current and has the unit of
W ¼ V A (watts = volts amperes). Similarly, in an electromagnetic field E J is the power
density and has the unit of V A m3 , or W m3 . From (1.30), the total power dissipated by an
electromagnetic field in a volume of V is simply the integral of E J over the volume:
ð
þ
ð
ð
∂ ϵ 0 2 μ0
∂P
∂M
2
E JdV ¼ E H n^da E þ μ0 H dV , (1.31)
jEj þ jH j dV 2
2
∂t
∂t
∂t
V
A
V
V
where the first term on the right-hand side is a surface integral over the closed surface A of the
volume V and n^ is the outward-pointing unit normal vector of the surface, as shown in Fig. 1.2.
Each term in (1.31) has the unit of power, and each has an important physical meaning.
1. The vectorial quantity
S¼EH
(1.32)
is called the Poynting vector of the electromagnetic field. It represents the instantaneous
magnitude and direction of the power flow of the field.
2. The scalar quantity
u0 ¼
ϵ 0 2 μ0
jEj þ jH j2
2
2
(1.33)
has the unit of energy per unit volume and is the energy density stored in the propagating
field. It consists of two components, thus accounting for energies stored in both electric and
magnetic fields at any instant of time.
3. The last term in (1.31) also has two components associated with electric and magnetic fields,
respectively. The quantity
Wp ¼ E ∂P
∂t
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(1.34)
10
Basic Concepts of Optical Fields
is the power density expended by the electromagnetic field on the polarization. It is the rate
of energy transfer from the electromagnetic field to the medium on inducing the electric
polarization in the medium. Similarly, the quantity
W m ¼ μ0 H ∂M
∂t
(1.35)
is the power density expended by the electromagnetic field on the magnetization.
With these physical meanings attached to the terms in (1.31), it can be seen that (1.31) simply
states the law of conservation of energy in any arbitrary volume element V in the medium. The
total electromagnetic energy in the medium equals that contained in the propagating field plus
that stored in the electric and magnetic polarizations.
For an optical field, E J ¼ 0 and W m ¼ 0 because J ¼ 0 and M ¼ 0, as discussed above.
Then, (1.31) becomes
þ
ð
ð
∂
S n^da ¼
u0 dV þ W p dV ,
(1.36)
∂t
V
A
V
which states that the total optical power flowing into volume V through its boundary surface A
is equal to the rate of increase with time of the energy stored in the propagating fields in V plus
the power transferred to the polarization of the medium in this volume.
1.4
WAVE EQUATION
..............................................................................................................
By applying ∇ to (1.11) and using (1.19) and (1.12), we obtain the wave equation:
∇ ∇ E þ μ0
∂2 D
¼ 0:
∂t 2
(1.37)
By using (1.2), the wave equation can be expressed as
∇∇Eþ
1 ∂2 E
∂2 P
¼
μ
,
0
c2 ∂t 2
∂t 2
(1.38)
where
1
c ¼ pffiffiffiffiffiffiffiffiffi 3 108 m s1
μ0 ϵ 0
(1.39)
is the speed of light in free space.
The wave equation in (1.38) describes the space-and-time evolution of the electric field of the
optical wave. Its right-hand side can be regarded as the driving source for the optical wave; that
is, the polarization in a medium drives the evolution of an optical field. This wave equation can
take on various forms depending on the characteristics of the medium, as will be seen on
various occasions later. Here we leave it in this general form.
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1.5 Harmonic Fields
1.5
11
HARMONIC FIELDS
..............................................................................................................
Optical fields are harmonic fields that vary sinusoidally with time. The field vectors defined in
the preceding section are all real quantities. For harmonic fields, it is always convenient to use
complex fields. We define the space- and time-dependent complex electric field, Eðr; tÞ, through
its relation to the real electric field, Eðr; t Þ:1
Eðr; t Þ ¼ Eðr; tÞ þ E∗ ðr; tÞ ¼ Eðr; t Þ þ c:c:,
(1.40)
where c.c. means the complex conjugate. In our convention, Eðr; tÞ contains the complex field
components that vary with time as exp ðiωtÞ with ω having a positive value, while E∗ ðr; tÞ
contains those components that vary with time as exp ðiωt Þ with positive ω. The complex fields
of other field quantities are similarly defined.
With this definition for the complex fields, all of the linear field equations retain their forms.
In terms of complex optical fields, Maxwell’s equations in the form of (1.11)–(1.14) are
∇E¼
∇H¼
∂B
,
∂t
∂D
,
∂t
(1.41)
(1.42)
∇ D ¼ 0,
(1.43)
∇ B ¼ 0;
(1.44)
and those in the form of (1.15)–(1.18) are
∇E¼
∇H¼
∂B
,
∂t
∂Dbound
þ Jcond ,
∂t
(1.45)
(1.46)
∇ Dbound ¼ 0,
(1.47)
∇ B ¼ 0:
(1.48)
The wave equation in terms of the complex electric field is
∇∇Eþ
1
1 ∂2 E
∂2 P
¼
μ
,
0
c2 ∂t2
∂t 2
(1.49)
In some literature, the complex field is defined through a relation with the real field as Eðr; t Þ ¼ ½Eðr; tÞ þ E∗ ðr; tÞ=2,
which differs from our definition in (1.40) by the factor 1=2. The magnitude of the complex field defined through this
alternative relation is twice that of the complex field defined through (1.40). As a result, expressions for many quantities
may be different under the two different definitions. An example is the time-averaged Poynting vector given in (1.53),
which would be changed to S ¼ ReðE H∗ Þ=2 in this alternative definition of the complex field.
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12
Basic Concepts of Optical Fields
while
ðt ððð
Pðr; t Þ ¼ ϵ 0
χðr r0 ; t t 0 Þ Eðr0 , t 0 Þdr0 dt0
(1.50)
∞ all r0
and
ðt ððð
Dðr; t Þ ¼ ϵ 0 Eðr; t Þ þ Pðr; t Þ ¼
∞ all
ϵ ðr r0 ; t t0 Þ Eðr0 , t0 Þdr0 dt0:
(1.51)
r0
It is important to note that while E, D, and P are complex, χðr r0 ; t t 0 Þ and ϵ ðr r0 ; t t0 Þ
in (1.50) and (1.51) are always real functions of space and time and are the same as those in
(1.20) and (1.21).
The complex electric field of a harmonic optical field that has a carrier wavevector of k and a
carrier angular frequency of ω can be further expressed as
Eðr; tÞ ¼ E ðr; t Þ exp ðik r iωtÞ ¼ ^e E ðr; t Þ exp ðik r iωt Þ,
(1.52)
where E ðr; t Þ is the space- and time-dependent amplitude of the field, and ^e is the unit
polarization vector of the field. The vectorial field amplitude E ðr; t Þ is generally a complex
vectorial quantity that has a magnitude, a phase, and a polarization. Other complex field
quantities, such as Dðr; t Þ, Bðr; t Þ, and Hðr; t Þ, can be similarly expressed. The space- and
time-dependent phase factor in (1.52) indicates the direction of wave propagation:
ik r iωt
for a wave propagating in the k direction;
ik r iωt for a wave propagating in the k direction.
1.5.1 Light Intensity
The light intensity, or irradiance, is the power density of the harmonic optical field. It can be
calculated by time averaging the Poynting vector over one wave cycle:
ðT
1
S¼
Sdt ¼ 2Re E H∗ ,
T
(1.53)
0
where Reð Þ means taking the real part. We can define a complex Poynting vector:
S ¼ E H∗
(1.54)
so that
∗
S ¼ S þ S∗ ¼ S þ S ,
(1.55)
which has the same form as the relation between the real and complex fields defined in (1.40)
except that the Poynting vector in this relation is time averaged. In the case of a coherent
monochromatic wave, E H∗ ¼ E H∗ ; then, (1.55) can be written as S ¼ S þ S∗ . The
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1.6 Polarization of Optical Fields
13
light intensity, I, on a surface is simply the magnitude of the real time-averaged Poynting vector
projected on the surface:
∗
I ¼ S n^ ¼ ðS þ S Þ n^,
(1.56)
where n^ is the unit normal vector of the projected surface and I is in watts per square meter.
1.5.2 Fields in Momentum Space and Frequency Domain
For harmonic optical fields, it is often useful to consider the complex fields in the momentum
space and frequency domain defined by the following Fourier-transform relations:
ð∞ ððð
Eðr; t Þ exp ðik r þ iωtÞdrdt,
Eðk; ωÞ ¼
for ω > 0,
(1.57)
∞ all r
Eðr; t Þ ¼
1
ð2π Þ4
ð∞ ððð
Eðk; ωÞ exp ðik r iωt Þdkdω:
(1.58)
0 all k
Note that Eðk; ωÞ in (1.57) is only defined for ω > 0; therefore, the integral for the time
dependence of Eðr; t Þ in (1.58) only extends over positive values of ω. This is in accordance
with the convention we used to define the complex field Eðr; t Þ in (1.40). All other space- and
time-dependent quantities, including other field vectors and the permittivity and susceptibility
tensors, are transformed in a similar manner.
Through the Fourier transform, the convolution integrals in real space and time become
simple products in the momentum space and frequency domain. Consequently, we have
Pðk; ωÞ ¼ ϵ 0 χðk; ωÞ Eðk; ωÞ
(1.59)
Dðk; ωÞ ¼ ϵ 0 ½1 þ χðk; ωÞ Eðk; ωÞ ¼ ϵ ðk; ωÞ Eðk; ωÞ:
(1.60)
and
Note that in the real space and time domain Pðr; t Þ and Dðr; tÞ are connected to Eðr; t Þ through
convolution integrals in space and time, whereas in the momentum space and frequency domain
Pðk; ωÞ and Dðk; ωÞ are connected to Eðk; ωÞ through direct products.
1.6
POLARIZATION OF OPTICAL FIELDS
..............................................................................................................
The polarization state of an optical field is determined by the vectorial nature of the optical field.
It is characterized by the unit polarization vector ^e of the complex electric field expressed in
(1.52). Consider a monochromatic plane optical wave that has a complex electric field of
Eðr; t Þ ¼ E exp ðik r iωt Þ ¼ ^e E exp ðik r iωtÞ,
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(1.61)
14
Basic Concepts of Optical Fields
where E is a constant independent of r and t, and ^e is its unit vector. The polarization state of
the optical field is characterized by the unit vector ^e . The optical field is linearly polarized, also
called plane polarized, if ^e can be expressed as a constant, real vector. Otherwise, the optical
field is elliptically polarized in general, and is circularly polarized in some special cases.
For the convenience of discussion, we take the direction of wave propagation to be the z
direction so that k ¼ k^z and assume that both E and H lie in the xy plane. Then, we have
E ¼ ^x E x þ ^y E y ¼ ^x jE x jeiφx þ ^y E y eiφy ,
(1.62)
where E x and E y are space- and time-independent complex amplitudes, with phases φx and φy ,
respectively. The polarization state of the wave is completely characterized by the phase
difference and the magnitude ratio between the two field components E x and E y :
φ ¼ φ y φx ,
π < φ π,
(1.63)
π
0α :
2
(1.64)
and
1
α ¼ tan
E y ,
jE x j
Because only the relative phase φ matters, we can set φx ¼ 0 and take E ¼ jE j to be real in the
following discussion. Then E from (1.62) can be written as
E ¼ ^e E,
with ^e ¼ ^x cos α þ ^y eiφ sin α:
(1.65)
Using (1.40), the space- and time-dependent real field is
Eðz; t Þ ¼ 2E ½^x cos α cos ðkz ωtÞ þ ^y sin α cos ðkz ωt þ φÞ:
(1.66)
At a fixed z location, say z ¼ 0, we see that the electric field varies with time as
Eðt Þ ¼ 2E ½^x cos α cos ωt þ ^y sin α cos ðωt φÞ:
(1.67)
1.6.1 Elliptic Polarization
In general, E x and E y have different phases and different magnitudes. Therefore, the values of φ
and α can be any combination. At a fixed point in space, both the direction and the magnitude of
the field vector E in (1.67) can vary with time. Except when the values of φ and α fall into one
of the special cases discussed below, the tip of this vector generally describes an ellipse, and the
wave is said to be elliptically polarized. Note that we have assumed that the wave propagates in
the positive z direction. When we view the ellipse by facing against this direction of wave
propagation, we see that the tip of the field vector rotates counterclockwise, or left handedly, if
φ > 0; and it rotates clockwise, or right handedly, if φ < 0: Figure 1.3 shows the ellipse traced
by the tip of the rotating field vector at a fixed point in space. Also shown in the figure are the
relevant parameters that characterize elliptic polarization.
In the description of the polarization characteristics of an optical field, it is sometimes
convenient to use, in place of φ and α, a set of two other parameters, θ and ε, which specify
the orientation and ellipticity of the ellipse, respectively. The orientational parameter θ is the
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1.6 Polarization of Optical Fields
15
Figure 1.3 Ellipse described by the tip of
the field of an elliptically polarized
optical wave at a fixed point in space.
Also shown are relevant parameters
characterizing the state of polarization.
The propagation direction is assumed to
be the positive z direction, and the ellipse
is viewed by facing against this
direction.
directional angle measured from the x axis to the major axis of the ellipse. Its range is taken to
be 0 θ < π for convenience. The ellipticity ε is defined as
ε¼
b
tan1 ,
a
π
π
ε ,
4
4
(1.68)
where a and b are the major and minor semiaxes, respectively, of the ellipse. The plus sign for
ε > 0 is taken to correspond to φ > 0 for left-handed polarization, whereas the minus sign
for ε < 0 is taken to correspond to φ < 0 for right-handed polarization. The two sets of
parameters ðα; φÞ and ðθ; εÞ have the following relations:
tan 2θ ¼ tan 2α cos φ,
(1.69)
sin 2ε ¼ sin 2α sin φ:
(1.70)
Either set is sufficient to completely characterize the polarization state of an optical field.
Elliptic polarization can be considered as the general polarization state for any combination of α
and φ values, whereas linear polarization and circular polarization are special cases of elliptic
polarization for specific combinations of α and φ values.
1.6.2 Linear Polarization
An optical field is linearly polarized when φ ¼ 0 or π for any value of α. It is also characterized
by ε ¼ 0 and θ ¼ α, if φ ¼ 0; or by ε ¼ 0 and θ ¼ π α, if φ ¼ π. Clearly, the ratio E x =E y is
real in this case; therefore, linear polarization is described by a constant, real unit vector as
^e ¼ ^x cos θ þ ^y sin θ:
(1.71)
It follows that Eðt Þ described by (1.67) reduces to
Eðt Þ ¼ 2E^e cos ωt:
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(1.72)
16
Basic Concepts of Optical Fields
Figure 1.4 Field of a linearly polarized
optical wave.
The tip of this vector traces a line in space at an angle of θ with respect to the x axis, as shown in
Fig. 1.4.
1.6.3 Circular Polarization
An optical field is circularly polarized when φ ¼ π=2 or π=2, and α ¼ π=4. It is
also
characterized by ε ¼ π=4 or π=4, and θ ¼ 0. Because α ¼ π=4, we have jE x j ¼ E y ¼
pffiffiffi
E= 2. There are two different circular polarization states.
1. Left-circular polarization: For φ ¼ π=2, also ε ¼ π=4, the wave is left circularly polarized
if it propagates in the positive z direction. The complex field amplitude in (1.65) becomes
^x þ i^y
E ¼ E pffiffiffi ¼ E^e þ ,
2
and Eðt Þ described by (1.67) reduces to
pffiffiffi
EðtÞ ¼ 2E ð^x cos ωt þ ^y sin ωt Þ:
(1.73)
(1.74)
As we view against the direction of propagation ^z , we see that the field vector EðtÞ rotates
counterclockwise at an angular frequency of ω. The tip of this vector describes a circle. This
is shown in Fig. 1.5(a). This left-circular polarization is also called positive helicity. Its unit
vector is
^e þ
^x þ i^y
pffiffiffi :
2
(1.75)
2. Right-circular polarization: For φ ¼ π=2, also ε ¼ π=4, the wave is right circularly
polarized if it propagates in the positive z direction. We then have
^x i^y
E ¼ E pffiffiffi ¼ E^e 2
(1.76)
and
EðtÞ ¼
pffiffiffi
2E ð^x cos ωt ^y sin ωt Þ:
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(1.77)
1.6 Polarization of Optical Fields
17
Figure 1.5 (a) Field of a left circularly polarized wave. (b) Field of a right circularly polarized wave.
The tip of this field vector rotates clockwise in a circle, as shown in Fig. 1.5(b). This rightcircular polarization is also called negative helicity. Its unit vector is
^e ^x i^y
pffiffiffi :
2
(1.78)
As can be seen, neither ^e þ nor ^e is a real vector. Note that the identification of ^e þ , defined
in (1.75), with left-circular polarization and that of ^e , defined in (1.78), with right-circular
polarization are based on the assumption that the wave propagates in the positive z direction.
For a wave that propagates in the negative z direction, the handedness of these unit vectors
changes: ^e þ becomes right-circular polarization, while ^e becomes left-circular polarization.
1.6.4 Orthogonal Polarizations
Two polarizations are orthogonal if they are normal to each other. The unit polarization vector ^e
can be either a real vector, for a linearly polarized wave, or a complex vector, for a circularly or
elliptically polarized wave. Each unit polarization vector is normalized to be a unit vector
according to the relation:
^e ^e ∗ ¼ 1:
(1.79)
Two polarizations, ^e 1 and ^e 2 , are orthogonal if
^e 1 ^e ∗
2 ¼ 0:
(1.80)
Note that normalization is not performed by ^e ^e ¼ 1, and orthogonality is not defined by
^e 1 ^e 2 ¼ 0.
EXAMPLE 1.3
Consider the two circularly polarized unit vectors ^e þ and ^e that are given in (1.75) and (1.78),
respectively. Show that they are normalized unit vectors that are orthogonal to each other.
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18
Basic Concepts of Optical Fields
Solution:
Using (1.79) for normalization, we find that
^x þ i^y
^x þ i^y ∗
^x þ i^y
^x i^y
∗
pffiffiffi pffiffiffi
pffiffiffi pffiffiffi
^e þ ^e þ ¼
¼
¼1
2
2
2
2
and
^e ^e ∗
^x i^y
^x i^y ∗
^x i^y
^x þ i^y
pffiffiffi pffiffiffi
pffiffiffi pffiffiffi
¼
¼
¼ 1:
2
2
2
2
Therefore, both ^e þ and ^e are normalized unit vectors. Using (1.80) for orthogonality, we find
that
^x þ i^y
^x i^y ∗
^x þ i^y
^x þ i^y
∗
pffiffiffi pffiffiffi
pffiffiffi pffiffiffi
^e þ ^e ¼
¼0
¼
2
2
2
2
and
^e ^e ∗
þ
^x i^y
^x þ i^y ∗
^x i^y
^x i^y
pffiffiffi pffiffiffi
pffiffiffi pffiffiffi
¼ 0:
¼
¼
2
2
2
2
Therefore, ^e þ and ^e are normalized unit vectors that are orthogonal to each other. The two
circular polarizations are orthogonal to each other. Note that ^e þ ^e þ ¼ ^e ^e ¼ 0 6¼ 1 and
^e þ ^e ¼ ^e ^e þ ¼ 1 6¼ 0, which can be easily verified.
1.7
OPTICAL FIELD PARAMETERS
..............................................................................................................
As stated in Section 1.1, an optical field is characterized by the five parameters of polarization
^e , magnitude jE j, phase φE , wavevector k, and frequency ω:
Eðr; t Þ ¼ E ðr; t Þ exp ðik r iωt Þ
¼ ^e E ðr; tÞ exp ðik r iωt Þ
(1.81)
¼ ^e jE ðr; t ÞjeiφE ðr;tÞ exp ðik r iωtÞ,
where E ¼ ^e E is the vectorial complex field amplitude that contains the field polarization ^e and
the scalar complex field amplitude E. The scalar complex field amplitude E ¼ jEjeiφE has a
magnitude of jE j and a phase of φE . Note that in general, jE j and φE can vary with space and
time, as indicated above in (1.81). Among the five parameters, ^e and k are vectors, while jE j,
φE , and ω are scalars.
The unit polarization vector ^e fully characterizes the polarization state of an optical field. It
can be real, for linearly polarized light, or complex, for elliptically or circularly polarized light.
The details are discussed in the preceding section.
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1.7 Optical Field Parameters
19
The magnitude jE j of the complex field amplitude defines the strength of the optical field. For
simplicity of discussion, consider a linearly polarized wave so that the unit polarization vector ^e
is a real vector. Then the complex field given in (1.81) yields the following real field,
(1.82)
Eðr; t Þ ¼ Eðr; t Þ þ E∗ ðr; t Þ ¼ 2jE ðr; t Þj^e cos k r ωt þ φE ðr; t Þ :
Therefore, under our definition of the complex field through (1.40), the amplitude of the real
field is 2jE ðr; t Þj. Note that this field amplitude can be a function of space and time to describe
the modulation on the field strength in space and time. It describes an envelope of the field on
the optical carrier.
The phase φE of the complex field amplitude is the phase shift with respect to the space- and
time-varying phase factor, k r ωt. As seen in (1.82), the total phase of the field is
φðr; tÞ ¼ k r ωt þ φE ðr; t Þ:
(1.83)
In the case when φE is a constant that is independent of both space and time, it has physical
meaning only when it is compared to a reference, such as the phase of another field. An
unreferenced constant phase can be eliminated by redefining the origin of the space or time
coordinate. Nevertheless, as expressed in (1.81) and (1.82), this phase can be a function of
space or time, or both: φE ðr; tÞ: The spatial dependence of φE ðr; t Þ leads to a shift of the
wavevector from the carrier wavevector k; the temporal dependence of φE ðr; t Þ leads to a shift
of the frequency from the carrier frequency ω:
The wavevector k defines the spatial variation and the propagation direction of the optical
carrier field. Its value, k, known as the propagation constant or the wavenumber, is determined
by the wavelength, or equivalently the frequency, of the optical wave and the refractive index of
the medium:
2πn ^ nω ^
k ¼ kk^ ¼
k¼
k,
λ
c
(1.84)
where n is the refractive index of the medium. From (1.82), it can be seen that k defines the
spatial variation of the optical carrier field. The propagation direction of a wave is defined as
the direction normal to the wavefront of the wave, and a wavefront is the surface of a constant
phase: φðr; t Þ ¼ constant: With φðr; tÞ ¼ k r ωt þ φE ðr; t Þ from (1.83), the space-dependent
wavevector is
kðrÞ ¼ ∇φ ¼ k þ ∇φE :
(1.85)
Thus, the space-dependent wave propagation direction can be found as
kðrÞ
k^ðrÞ ¼
:
k ðrÞ
(1.86)
In the case when φE is independent of space so that ∇φE ¼ 0, such as the case of a plane wave,
the wave propagates with a space-independent propagation constant k in a space-independent
propagation direction defined by the constant unit vector k^ ¼ k=k. In the case when φE varies
across space so that ∇φE 6¼ 0, such as the case of a spatially diverging or converging wave,
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20
Basic Concepts of Optical Fields
either one or both of the propagation constant kðrÞ and the propagation direction defined by
k^ðrÞ ¼ kðrÞ=k ðrÞ vary from one spatial location to another.
The frequency ω defines the temporal variation of the optical carrier field. It is the optical
angular frequency that is related to the field oscillation frequency ν as ω ¼ 2πν; ν has the unit
of hertz ðHzÞ while ω has the unit of radians per second ðrad s1 Þ. As an optical wave
propagates through different media of different refractive indices, its wavelength, thus the
value of k, changes with the changing refractive indices, but its frequency remains unchanged.
The angular frequency of a wave is defined by the temporal variation of its phase. With
φðr; t Þ ¼ k r ωt þ φE ðr; tÞ from (1.83), the angular frequency can be found as
ωðt Þ ¼ ∂φ
∂φ
¼ω E:
∂t
∂t
(1.87)
The frequency of the wave is the constant ω in the case when φE is independent of time so that
∂φE =∂t ¼ 0, such as the case of a monochromatic wave. In the case when φE varies with time,
such as the case of a phase-modulated wave, the frequency ωðtÞ is a function of time with a shift
of ∂φE =∂t from the constant frequency ω.
Problems
1.1.1 At room temperature, diamond transmits optical waves of wavelengths longer than
227 nm but absorbs shorter wavelengths. What is the bandgap energy of diamond at
room temperature?
1.1.2 At room temperature, the bandgap energy of Ge is 0.66 eV. It absorbs photons of energies
above its bandgap and transmits those of energies below its bandgap. What is the cutoff
wavelength for light to be transmitted through a thick piece of pure Ge?
1.1.3 Find the wavelength and photon energy of a terahertz wave at a frequency of 5 THz.
1.1.4 The optical window for long-distance optical communications is at the 1.55 μm wavelength. What are the optical frequency and the photon energy?
1.1.5 A red laser pointer emits a red beam of P ¼ 1 mW power at the λ ¼ 635 nm wavelength.
What are the photon energy, the photon momentum, and the photon flux of this beam? If
it illuminates a totally absorbing surface, what is the force exerted by the beam on the
absorbing surface? If it illuminates a totally reflecting surface, what is the force exerted
by the beam on the reflecting surface?
1.2.1 Verify that Maxwell’s equations and the continuity equation, given in (1.4)–(1.8), are
invariant under (a) the transformation of space inversion, (b) the transformation of time
reversal, and (c) the simultaneous transformation of space inversion and time reversal.
1.4.1 Derive the optical wave equation given in (1.37) in the case when J ¼ 0 so that
Maxwell’s equations take the form of (1.11)–(1.14). Show that in this case the optical
wave equation can be expressed in the form of (1.38).
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Bibliography
21
1.4.2 In the case when a conduction current Jcond is explicitly separated from the ∂D=∂t term so
that Maxwell’s equations take the form of (1.15)–(1.18), rewrite the optical wave
equation given in (1.37) and that given in (1.38) to explicitly account for Jcond .
1.5.1 By taking the Fourier transform on the relation given in (1.50) between Pðr; t Þ and Eðr; tÞ
in the real space and time domain, verify the relation given in (1.59) between Pðk; ωÞ and
Eðk; ωÞ in the momentum space and frequency domain.
1.6.1 As discussed in the text, any polarization state in the xy plane can be generally considered
as elliptic polarization represented by the unit polarization vector ^e ¼ ^x cos α þ ^y eiφ sin α
given in (1.65) with proper choices of α and φ for a particular polarization state. Because
the xy plane is a two-dimensional space, a basis set of unit polarization vectors consists of
two orthonormal vectors. Find the other unit polarization vector ^e ⊥ that forms a basis
together with ^e .
1.6.2 The circularly polarized unit vectors ^
e þ and ^e given in (1.75) and (1.78) are each
expressed in terms of the linearly polarized unit vectors ^x and ^y . Each pair form a basis
for representing any polarization state in the xy plane. Show that each of the linearly
polarized unit vectors ^x and ^y can be represented in terms of a linear superposition of two
circularly polarized components on the basis of ^e þ and ^e .
1.6.3 Express the general linearly polarized unit vector ^
e ¼ ^x cos θ þ ^y sin θ given in (1.71) as
a linear superposition of two circularly polarized components on the basis of the
circularly polarized unit vectors ^e þ and ^e given in (1.75) and (1.78), respectively.
Bibliography
Born, M. and Wolf, E., Principles of Optics: Electromagnetic Theory of Propagation, Interference and
Diffraction of Light, 7th edn. Cambridge: Cambridge University Press, 1999.
Fowler, G. R., Introduction to Modern Optics, 2nd edn. New York: Dover, 1975.
Iizuka, K., Elements of Photonics in Free Space and Special Media, Vol. I. New York: Wiley, 2002.
Jackson, J. D., Classical Electrodynamics, 3rd edn. New York: Wiley, 1999.
Liu, J. M., Photonic Devices. Cambridge: Cambridge University Press, 2005.
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Cambridge Books Online
http://ebooks.cambridge.org/
Principles of Photonics
Jia-Ming Liu
Book DOI: http://dx.doi.org/10.1017/CBO9781316687109
Online ISBN: 9781316687109
Hardback ISBN: 9781107164284
Chapter
2 - Optical Properties of Materials pp. 22-65
Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.003
Cambridge University Press
2
2.1
Optical Properties of Materials
OPTICAL SUSCEPTIBILITY AND PERMITTIVITY
..............................................................................................................
The electric susceptibility, χ, and the electric permittivity, ϵ, of an optical medium characterize
the intrinsic response of the medium to an optical field. They are respectively defined in (1.20) for
the relation between Pðr; t Þ and Eðr; t Þ and in (1.21) for the relation between Dðr; t Þ and Eðr; tÞ:
ðt ððð
Pðr; tÞ ¼ ϵ 0
χðr r0 ; t t 0 Þ Eðr0 ; t 0 Þdr0 dt 0,
(2.1)
∞ all r0
ðt ððð
Dðr; t Þ ¼ ϵ 0 Eðr; t Þ þ Pðr; t Þ ¼
ϵ ðr r0 ; t t0 Þ Eðr0 ; t 0 Þdr0 dt0:
(2.2)
∞ all r0
These relations can be expressed in terms of the complex field:
ðt ððð
Pðr; t Þ ¼ ϵ 0
χðr r0 ; t t0 Þ Eðr0 ; t 0 Þdr0 dt 0
(2.3)
∞ all r0
ðt ððð
Dðr; t Þ ¼ ϵ 0 Eðr; t Þ þ Pðr; t Þ ¼
ϵ ðr r0 ; t t0 Þ Eðr0 ; t 0 Þdr0 dt0:
(2.4)
∞ all r0
The relations in the momentum space and frequency domain, obtained by taking the Fourier
transform on (2.3) and (2.4), are direct products, given in (1.59) and (1.60):
Pðk; ωÞ ¼ ϵ 0 χðk; ωÞ Eðk; ωÞ
(2.5)
Dðk; ωÞ ¼ ϵ 0 ½1 þ χðk; ωÞ Eðk; ωÞ ¼ ϵ ðk; ωÞ Eðk; ωÞ:
(2.6)
The real-space and time-domain relations given in (2.1)(2.4) are convolution integrals over
real space and time. The convolution in time accounts for the fact that the response of a medium
to the stimulation by an electric field is generally not instantaneous, or local, in time and does
not vanish for some time after the stimulation is over. Because time is unidirectional, causality
exists in physical processes. An earlier stimulation can influence the property of a medium at a
later time, whereas a later stimulation does not have any effect on the medium at an earlier time.
Therefore, the upper limit in the time integral is t, not infinity. By contrast, the convolution in
space accounts for the spatial nonlocality of the material response. Stimulating a medium at a
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2.1 Optical Susceptibility and Permittivity
23
Figure 2.1 Nonlocal responses in (a) time and (b) space.
location r0 can result in a change in the property of the medium at another location r: For
example, the property of a semiconductor at one location can be changed by electric or optical
excitation at another location through carrier diffusion. There is generally no spatial causality
because space is not unidirectional; therefore, spatial convolution is integrated over the entire
space. Figure 2.1 shows the temporal and spatial nonlocality of responses to electromagnetic
excitations. The temporal nonlocality of the optical response of a medium makes the optical
property of the medium dependent on the optical frequency, a phenomenon known as frequency
dispersion, whereas the spatial nonlocality makes the optical property of the medium dependent
on the optical wavevector, a phenomenon known as momentum dispersion. The frequency
dispersion and the momentum dispersion of a medium are respectively characterized by the
frequency dependence and the momentum dependence of χðk; ωÞ and ϵ ðk; ωÞ. Because χðk; ωÞ
and ϵ ðk; ωÞ are respectively the Fourier transforms of χðr; tÞ and ϵ ðr; t Þ, it is clear that the
frequency dispersion and the momentum dispersion of a medium respectively originate from
the temporal nonlocality and the spatial nonlocality of its response to an optical stimulation.
The susceptibility tensor χðr; t Þ and the permittivity tensor ϵ ðr; tÞ of real space and time are
always real quantities though the optical fields in the real space and time domain can be
expressed either as real fields, as in (2.1) and (2.2), or as complex fields, as in (2.3) and (2.4).
This statement is true even when the medium exhibits an optical loss or gain. However, the
susceptibility tensor χðk; ωÞ and the permittivity tensor ϵ ðk; ωÞ in the momentum space and
frequency domain are generally complex. If an eigenvalue χ i of χðk; ωÞ is complex, the
corresponding eigenvalue ϵ i of ϵ ðk; ωÞ is also complex, and their imaginary parts have the
same sign because ϵ ðk; ωÞ ¼ ϵ 0 ½1 þ χðk; ωÞ. The signs of the imaginary parts of such eigenvalues tell whether the medium provides an optical gain or loss. In our convention, we write, for
example, χ i ¼ χ 0i þ iχ 00i in the frequency domain. Then, χ 00i ðωÞ > 0 indicates an optical loss or
absorption, while χ 00i ðωÞ < 0 represents an optical gain or amplification.
The fact that χðr; t Þ and ϵ ðr; t Þ are real quantities leads to the following symmetry relations
for the tensor elements of χðk; ωÞ and ϵ ðk; ωÞ:
∗
χ∗
ij ðk; ωÞ ¼ χ ij ðk; ωÞ and ϵ ij ðk; ωÞ ¼ ϵ ij ðk; ωÞ,
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(2.7)
24
Optical Properties of Materials
which are called the reality condition. The reality condition implies that χ 0ij ðk; ωÞ ¼ χ 0ij ðk; ωÞ
and χ 00ij ðk; ωÞ ¼ χ 00ij ðk; ωÞ: The real and imaginary parts of ϵ ij ðk; ωÞ have similar properties.
Therefore, the real parts of χ ij ðk; ωÞ and ϵ ij ðk; ωÞ are even functions of k and ω, whereas the
imaginary parts are odd functions of k and ω. If a tensor element χ ij ðk; ωÞ or ϵ ij ðk; ωÞ has any
constant term that is independent of k and ω, the constant term can only appear in its real part
because a constant value is an even function of k and ω. As a result, the imaginary part is always a
function of k or ω, or both. The optical loss, or gain, in a medium is associated with the imaginary
part of an eigenvalue of χðk; ωÞ or ϵ ðk; ωÞ; consequently, a medium that absorbs or amplifies light
is inherently dispersive. Any other effect that can be described by the imaginary part of an eigenvalue
of χðk; ωÞ or ϵ ðk; ωÞ is also inherently dispersive in either momentum or frequency, or both.
In addition to the nonlocality of medium response, it is also important to consider the
inhomogeneity of a medium, in both space and time. Spatial inhomogeneity exists in every
optical structure, such as an optical waveguide, where the optical property is a function of
space. Temporal inhomogeneity exists when the optical property of a medium varies with time,
for example, because of modulation by a low-frequency electric field or by an acoustic wave.
The space and time variables characterizing nonlocality are relative space and time of the
medium response with respect to an optical stimulation, whereas those characterizing inhomogeneity are absolute space and time measured with respect to a reference point in space and a
reference point in time. When both response nonlocality and medium inhomogeneity are
considered, the response nonlocality is commonly characterized in the momentum space and
frequency domain as a function of k and ω by taking the Fourier transform on the relative space
and time, whereas the medium inhomogeneity is characterized in the real space and time
domain as a function of the absolute space and time variables r and t; therefore, χðk; ω; r; t Þ
and, correspondingly, ϵ ðk; ω; r; t Þ.
In a linear medium, changes in the wavevector of an optical wave, or coupling between
waves of different wavevectors, can occur only if the optical property of the medium in which
the wave propagates is spatially inhomogeneous such that χðk; ω; r; t Þ is a function of space.
Likewise, changes in the frequency of an optical wave, or coupling between waves of different
frequencies, are possible in a linear medium only if the optical property of the medium is time
varying such that χðk;ω;r;tÞ varies with time. A change in the wavevector of an optical wave
^ as in the case of reflection or
can take the form of a change in the wave propagation direction k,
diffraction of an optical wave, or in the propagation constant k through a change in the optical
wavelength, as in the case when a wave propagates from one part of the medium to another part
of a different refractive index. A change in the frequency of an optical wave results in the
generation of other frequencies or the conversion to a completely different frequency.
2.2
OPTICAL ANISOTROPY
..............................................................................................................
In general, both χ and ϵ are tensors because the P and D vectors are not necessarily parallel to
the E vector due to material anisotropy. In the case of an isotropic medium, both χ and ϵ reduce
to the scalars χ and ϵ, respectively. In the case of a linear anisotropic medium, both χ and ϵ are
second-order tensors. They can be expressed in the matrix form:
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2.2 Optical Anisotropy
0
χ 11
@
χ ¼ χ 21
χ 31
χ 12
χ 22
χ 32
1
0
χ 13
ϵ 11
A
@
and ϵ ¼ ϵ 21
χ 23
χ 33
ϵ 31
1
ϵ 13
ϵ 23 A:
ϵ 33
ϵ 12
ϵ 22
ϵ 32
25
(2.8)
Each of the relationships P ¼ ϵ 0 χ E and D ¼ ϵ E is carried out as the product between a
tensor and a column vector:
0
1
0
P1
χ 11
@ P2 A ¼ ϵ 0 @ χ 21
P3
χ 31
χ 12
χ 22
χ 32
10 1
0 1 0
E1
D1
ϵ 11
χ 13
χ 23 A@ E 2 A and @ D2 A ¼ @ ϵ 21
χ 33
E3
D3
ϵ 31
ϵ 12
ϵ 22
ϵ 32
10 1
E1
ϵ 13
ϵ 23 A@ E 2 A: (2.9)
ϵ 33
E3
In general, the matrices in (2.8) representing the χ and ϵ tensors are not diagonal when they are
expressed using an arbitrarily chosen coordinate system. When optical field vectors are
projected on the axes of this coordinate system, a component of P or D does not necessarily
contain only the corresponding component of E but can also contain one or both of the other
two E components. For example, P1 and D1 are functions of E 2 or E3 , or both, unless
χ 12 ¼ χ 13 ¼ 0, in which case ϵ 12 ¼ ϵ 13 ¼ 0 as well, because P1 ¼ ϵ 0 ðχ 11 E 1 þ χ 12 E 2 þ χ 13 E 3 Þ
and D1 ¼ ϵ 11 E 1 þ ϵ 12 E 2 þ ϵ 13 E 3 .
Because χ and ϵ are physical quantities, they are diagonalizable matrices that can always be
diagonalized by a proper set of eigenvectors, yielding
0
χ1
χ¼@0
0
0
χ2
0
1
0
ϵ1
0
0 A and ϵ ¼ @ 0
χ3
0
0
ϵ2
0
1
0
0 A:
ϵ3
(2.10)
Here χ i and ϵ i are, respectively, the eigenvalues of χ and ϵ with corresponding eigenvectors ^e i
such that
χ ^e i ¼ χ i ^e i and ϵ ^e i ¼ ϵ i ^e i , for i ¼ 1, 2, 3:
(2.11)
The characteristics of the eigenvalues χ i and ϵ i , as well as their eigenvectors ^e i , depend on the
symmetry properties of χ and ϵ. The two matrices representing χ and ϵ have the same symmetry
properties because ϵ ¼ ϵ 0 ð1 þ χÞ, where 1 has the form of a 3 3 identity matrix when it
is added to the χ tensor. Therefore, χ and ϵ are diagonalized by the same set of eigenvectors.
When an optical field is projected on these eigenvectors, each component of P or D depends
only on the corresponding component of E but not on the other two E components; that is,
Pi ¼ ϵ 0 χ i E i and Di ¼ ϵ i E i .
The three eigenvectors ^e i define the principal polarization states for proper decomposition of
optical field vectors so that each component has a well-defined susceptibility χ i and permittivity
ϵ i . They are the principal normal modes of polarization satisfying the orthonormality condition:
1, for i ¼ j;
∗
^e i ^e j ¼ δij ¼
(2.12)
0, for i 6¼ j:
As discussed in Section 1.6, a real eigenvector represents linear polarization, while a complex
eigenvector represents elliptic or circular polarization. The characteristics of these eigenvectors
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26
Optical Properties of Materials
are determined by the symmetry properties of χ and ϵ, which are determined by the properties
of the medium. Because χ and ϵ have the same properties and the same eigenvectors, only ϵ is
mentioned in the following discussion while all conclusions apply equally to χ.
2.2.1 Reciprocal Media
Nonmagnetic materials that are not subject to an external magnetic field are reciprocal media.
In a reciprocal medium, the Lorentz reciprocity theorem of electromagnetics holds; consequently, the source and the detector of an optical signal can be interchanged for the same
function of an optical system. If such a material is not optically active, its optical properties are
described by a symmetric ϵ tensor: ϵ ij ¼ ϵ ji . The eigenvectors ^e i of a symmetric tensor are
always real vectors. They can be chosen to be ^x , ^y , and ^z of a rectilinear coordinate system in
real space. This is true even when ϵ is complex.
1. If a nonmagnetic medium does not have any optical loss or gain, its ϵ tensor is Hermitian,
∗
∗
i.e., ϵ ij ¼ ϵ ∗
ji . A symmetric Hermitian tensor is real and symmetric: ϵ ij ¼ ϵ ij ¼ ϵ ji ¼ ϵ ji : The
eigenvectors ^e i are real vectors representing linear polarization states, and all three eigenvalues ϵ i have real values.
2. If a nonmagnetic medium has an optical loss or gain, its ϵ tensor is still symmetric but is
complex and non-Hermitian: ϵ ij ¼ ϵ ji but ϵ ij 6¼ ϵ ∗
e i are real
ji : Then, the eigenvectors ^
vectors representing linear polarization states, but at least one of the eigenvalues ϵ i is
complex. The sign of the imaginary part, ϵ 00i , indicates whether the medium has a loss or
gain for an ^e i -polarized optical wave: ϵ 00i > 0 for a loss and ϵ 00i < 0 for a gain, as discussed
in Section 2.1 in terms of χ 00i .
3. If a nonmagnetic medium is optically active, it is still reciprocal although its ϵ tensor is not
symmetric. The eigenvectors ^e i are complex vectors representing elliptic or circular polarization states, but the eigenvalues can be real, if the medium has no loss or gain, or complex,
if the medium has an optical loss or gain.
2.2.2 Nonreciprocal Media
Magnetic materials, and nonmagnetic materials that are subject to an external magnetic
field, are nonreciprocal media. In such a medium, no symmetry exists when the source and
the detector of an optical signal are interchanged. The ϵ tensor describing the optical
properties of such a material is not symmetric: ϵ ij 6¼ ϵ ji . The eigenvectors ^e i of a nonsymmetric matrix are generally complex vectors. Therefore, they are not ordinary coordinate axes
in real space.
1. For a magnetic medium that has no optical loss or gain, ϵ is Hermitian: ϵ ij ¼ ϵ ∗
ji : The
eigenvalues ϵ i are real even though the eigenvectors ^e i are complex vectors representing
elliptic or circular polarization states.
2. For a magnetic medium that has an optical loss or gain, ϵ is nonsymmetric and nonHermitian. The eigenvectors ^e i and the eigenvalues ϵ i are generally complex.
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27
2.2 Optical Anisotropy
EXAMPLE 2.1
At a given optical wavelength, the permittivity tensors of several optical materials are obtained
with respect to an arbitrary set of rectilinear coordinates in real space. From each of the
permittivity tensors shown below, identify each material as being (i) reciprocal or nonreciprocal
and (ii) lossless or lossy. Here “lossless” means having no loss or gain, and “lossy” means
having a loss or gain.
0
0
1
2:3 þ i0:3
0
C
A;
0
3:2 þ i0:1
3:4 þ i0:2 0:7 i0:1
B
A : ϵ ¼ ϵ 0 @ 0:7 þ i0:1
0
0
2:25
B
C : ϵ ¼ ϵ 0 @ i0:35
0
i0:35
2:20
0
1
0
C
0 A;
0
4:79
0:17
B
B : ϵ ¼ ϵ 0 @ 0:17
4:49
0
0
B
D : ϵ ¼ ϵ 0 @ 0:02 4:88
2:30
0
4:91 0:02
B
E : ϵ ¼ ϵ 0 @ 0:20 þ i0:18
0:20 i0:18
0
2:72
i0:22
1
C
0:05 A;
0:05
5:01
0
1
0:01
C
A;
0:01 4:58 þ i0:02
0
2:74
0
0
1
C
i0:22 A:
2:38
Solution:
The permittivity tensor of a reciprocal material is symmetric with ϵ ij ¼ ϵ ji , and that of a lossless
medium is Hermitian with ϵ ij ¼ ϵ ∗
ji . The properties of each material can be determined by
examining its permittivity tensor using these two characteristics. A, nonreciprocal and lossy; B,
reciprocal and lossless; C, nonreciprocal and lossless; D, reciprocal and lossy; E, nonreciprocal
and lossless.
2.2.3 Linear Birefringence and Linear Dichroism
For a reciprocal material that is not optically active, the eigenvectors ^e i of ϵ for proper
decomposition of optical field vectors are real unit vectors representing three linearly polarized
principal normal modes. These three orthogonal real unit vectors can be labeled as ^x , ^y , and ^z ,
which can be used to define the axes of a rectilinear coordinate system in real space.
Noncrystalline materials are generally isotropic, for which the choice of the orthogonal coordinate axes ^x , ^y , and ^z is arbitrary. For a crystal, these unique ^x , ^y , and ^z coordinate axes are called
the principal dielectric axes, or simply the principal axes, of the crystal. In the coordinate
system defined by these principal axes, ϵ is diagonalized with eigenvalues ϵ x , ϵ y , and ϵ z , known
as the principal permittivities. The properly decomposed components of D and E along these
axes have the following simple relations,
Dx ¼ ϵ x E x , Dy ¼ ϵ y E y , Dz ¼ ϵ z E z :
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(2.13)
28
Optical Properties of Materials
The values ϵ x =ϵ 0 , ϵ y =ϵ 0 , and ϵ z =ϵ 0 are the eigenvalues of the dielectric constant tensor, ϵ=ϵ 0 ,
and are called the principal dielectric constants. They define three principal indices of
refraction:
rffiffiffiffiffi
rffiffiffiffiffi
rffiffiffiffiffi
ϵx
ϵy
ϵz
nx ¼
, ny ¼
, nz ¼
:
(2.14)
ϵ0
ϵ0
ϵ0
The propagation constants for the ^x , ^y , and ^z principal normal modes of polarization are,
respectively,
kx ¼
nx ω
ny ω
nz ω
, ky ¼
, kz ¼
:
c
c
c
(2.15)
When ϵ is diagonalized, χ is also diagonalized along the same principal axes with corresponding principal dielectric susceptibilities, χ x , χ y , and χ z . The principal dielectric susceptibilities of
any dielectric material of no loss or gain always have real, positive values; therefore, the
principal dielectric constants of a lossless dielectric material are always greater than unity.
In an anisotropic crystal, the properly decomposed optical field components in two different
principal normal modes of polarization defined by two different eigenvectors ^e i and ^e j have
different indices of refraction, i.e., ni 6¼ nj , and thus different propagation constants, i.e.,
k i 6¼ kj , when the eigenvalues ϵ i and ϵ j are different for the two polarization states. This
phenomenon is known as birefringence. A crystal that shows birefringence is a birefringent
crystal. Two principal normal modes of polarization experience different degrees of optical loss
or gain when their principal dielectric constants have different imaginary parts. This phenomenon is known as dichroism.
The birefringence of an anisotropic nonmagnetic crystal causes two different linearly polarized principal normal modes to propagate with different propagation constants; this is known as
linear birefringence. The dichroism of an anisotropic nonmagnetic crystal appears between two
linearly polarized principal normal modes; this is known as linear dichroism.
The state of polarization of an optical wave generally varies along its path of propagation
through an anisotropic crystal unless it is linearly polarized in the direction of a principal axis.
However, in an anisotropic crystal with nx ¼ ny 6¼ nz , a wave propagating in the z direction
does not see the anisotropy of the crystal because in this situation the x and y components of the
field have the same propagation constant. This wave maintains its original polarization as it
propagates through the crystal. Evidently, this is true only for propagation along the z axis in
such a crystal. Such a unique axis in a crystal along which an optical wave can propagate with
an index of refraction that is independent of its polarization state is called the optical axis of
the crystal.
An anisotropic crystal that has only one distinctive principal index among its three principal
indices is called a uniaxial crystal because it has only one optical axis, which coincides with the
axis of the distinctive principal index of refraction. It is customary to assign ^z to this unique
principal axis such that nz is the distinctive index with nx ¼ ny 6¼ nz . The two identical principal
indices of refraction are called the ordinary index, no , and the distinctive principal index of
refraction is called the extraordinary index, ne . Thus, nx ¼ ny ¼ no and nz ¼ ne . The crystal is
called positive uniaxial if ne > no ; it is negative uniaxial if ne < no . A birefringent crystal of
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2.2 Optical Anisotropy
29
three distinct principal indices of refraction is called a biaxial crystal because it has two optical
axes, neither of which coincides with any of the principal axes.
EXAMPLE 2.2
At the 1 μm optical wavelength, the permittivity tensor of the KDP crystal represented in an
arbitrarily chosen Cartesian coordinate system defined by ^x 1 , ^x 2 , and ^x 3 unit vectors, with
^x 1 ^x 2 ¼ ^x 3 to satisfy the right-hand rule, is found to be
0
1
2:19
0
0:05196
A:
0
2:28
0
ϵ ¼ ϵ0@
0:05196
0
2:25
Find the principal indices of refraction and the corresponding principal axes ^x , ^y , and ^z in terms
of the coordinate axes ^x 1 , ^x 2 , and ^x 3 . Is KDP uniaxial or biaxial? If it is uniaxial, is it positive or
negative uniaxial?
Solution:
The given ϵ tensor is symmetric and Hermitian because KDP is a nonmagnetic dielectric crystal
that has a negligible optical loss at the 1 μm optical wavelength. Diagonalization of the matrix
yields the eigenvalues 2.28, 2.28, and 2.16 for the principal dielectric constants. Thus, the
crystal is uniaxial. By convention we assign the distinctive dielectric constant of 2.16 to be
associated with the z principal axis. The principal indices of refraction and the corresponding
principal axes are
pffiffiffiffiffiffiffiffiffi
nx ¼ 2:28 ¼ 1:51, ^x ¼ 0:500^x 1 0:866^x 3 ;
pffiffiffiffiffiffiffiffiffi
ny ¼ 2:28 ¼ 1:51, ^y ¼ ^x 2 ;
pffiffiffiffiffiffiffiffiffi
nz ¼ 2:16 ¼ 1:47, ^z ¼ 0:866^x 1 þ 0:500^x 3 :
Note that ^x ^y ¼ ^z to satisfy the right-hand rule. The KDP crystal is negative uniaxial because
nx ¼ ny > nz so that no > ne .
The optical anisotropy of a crystal depends on its structural symmetry. Crystals are classified
into seven systems according to their symmetry. The linear optical properties of these seven
systems are summarized in Table 2.1. Some important remarks regarding the relation between
the optical properties and the structural symmetry of a crystal are as follows.
1. A cubic crystal does not have an isotropic structure although its linear optical properties are
isotropic. For example, most III–V semiconductors, such as GaAs, InP, InAs, AlAs, etc., are
cubic crystals with isotropic linear optical properties. Nevertheless, they have well-defined
^ and ^c . They are also polar semiconductors, which have anisotropic
crystal axes, a^, b,
nonlinear optical properties.
2. Although the principal axes may coincide with the crystal axes in certain crystals, they are
^ and
not the same concept and are not necessarily the same. The crystal axes, denoted by a^, b,
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30
Optical Properties of Materials
Table 2.1 Linear optical properties of crystals
Crystal symmetry
Optical property
Cubic
Isotropic: nx ¼ ny ¼ nz
Trigonal, tetragonal, hexagonal
Uniaxial: nx ¼ ny 6¼ nz
Orthorhombic, monoclinic, triclinic
Biaxial: nx 6¼ ny 6¼ nz
^c , are defined by the structural symmetry of a crystal, whereas the principal axes, denoted by
^x , ^y , and ^z , are determined by the symmetry of ϵ. The principal axes of a crystal are
orthogonal to one another, but the crystal axes are not necessarily so.
2.2.4 Circular Birefringence and Circular Dichroism
For a nonreciprocal material or an optically active reciprocal material, the eigenvectors ^e i of ϵ
for proper decomposition of optical field vectors are generally complex unit vectors representing orthogonal elliptic polarization states, which may reduce to linear or circular polarization
states in particular cases. Optical activity is the phenomenon that a linearly polarized optical
wave remains linearly polarized but with its plane of polarization rotating about the direction of
propagation as it travels through a material. Natural optical activity that appears in a nonmagnetic reciprocal material not subject to a magnetic field was first discovered in quartz. It
occurs in many organic materials such as solutions of sugar or amino acids. Nonreciprocal
materials of interest in photonics can be magnetic with an intrinsic magnetization, M 0 , or
nonmagnetic but subject to a static or low-frequency external magnetic field, H 0 ; these
materials exhibit magnetically induced optical activity for magneto-optics applications, such
as optical isolation and optical circulation.
Consider, for simplicity, a nonsymmetric ϵ that has only two off-diagonal elements:
0
n2⊥
ϵ ¼ ϵ 0 @ iξ
0
iξ
n2⊥
0
1
0
0 A,
n2k
where n⊥ , nk , and ξ can be real or complex. The eigenvalues are
ϵ þ ¼ ϵ 0 n2⊥ ξ , ϵ ¼ ϵ 0 n2⊥ þ ξ , ϵ z ¼ ϵ 0 n2k ;
(2.16)
(2.17)
and the corresponding eigenvectors are
1
1
^e þ ¼ pffiffiffi ð^x þ i^y Þ, ^e ¼ pffiffiffi ð^x i^y Þ, ^z :
2
2
(2.18)
The complex eigenvectors, ^e þ and ^e are respectively the left and right circularly polarized unit
vectors defined in (1.75) and (1.78). These two eigenvectors are complex unit vectors because
the ϵ tensor in (2.16) is not symmetric. If n⊥ , nk , and ξ are all real, the eigenvalues are all real
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2.2 Optical Anisotropy
31
because then ϵ is Hermitian. If n⊥ , nk , or ξ is complex, the eigenvalues are also complex
because then ϵ is non-Hermitian.
It is clearly not possible to attach the meaning of the principal axes in real space to the
complex eigenvectors given in (2.18). Nonetheless, these eigenvectors still define the principal
normal modes of polarization for proper decomposition of optical field components:
Dþ ¼ ϵ þ E þ , D ¼ ϵ E , Dz ¼ ϵ z E z :
(2.19)
Therefore, ϵ þ =ϵ 0 , ϵ =ϵ 0 , and ϵ z =ϵ 0 are the principal dielectric constants for the three normal
modes. They define the following three principal indices of refraction:
nþ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ξ
ξ
n2⊥ ξ n⊥ , n ¼ n2⊥ þ ξ n⊥ þ
, nz ¼ nk ,
2n⊥
2n⊥
(2.20)
where the approximate expansion of the square root is valid for ξ=2n⊥ n⊥ : The propagation
constants for the principal normal modes of polarization are
kþ ¼
nþ ω
n ω
nz ω
, k ¼
, kz ¼
:
c
c
c
(2.21)
When an optical wave propagates along the z axis, in either the positive z or the negative z
direction, the principal normal modes of polarization are the circularly polarized modes ^e þ and
^e , which have different propagation constants k þ and k , respectively. This phenomenon that
the two circularly polarized modes have different propagation constants is called circular
birefringence. In the presence of an optical loss or gain, both nþ and n become complex no
matter whether the optical loss or gain is characterized by the nonzero imaginary part of a
complex n⊥ or ξ, or both. When the imaginary parts of nþ and n have different values, the two
circularly polarized normal modes experience different degrees of optical loss or gain. This
phenomenon is called circular dichroism, as distinct from the linear dichroism between two
linearly polarized modes.
Circular birefringence caused by the magneto-optic effect in a magnetic material or in a
nonmagnetic material subject to a magnetic field is known as magnetic circular birefringence.
Circular birefringence in a nonmagnetic reciprocal material that has natural optical activity is
known as natural circular birefringence. Circular dichroism caused by a loss or gain associated
with the magneto-optic effect in a magnetic material or in a nonmagnetictic material subject to a
magnetic field is known as magnetic circular dichroism. Circular dichroism due to a loss or
gain in a nonmagnetic reciprocal material that has natural optical activity is known as natural
circular dichroism.
The similarities between the two phenomena of natural optical activity and magnetically
induced optical activity are that both have circularly polarized normal modes and both can
cause circular birefringence and circular dichroism. In both cases, the plane of polarization of a
linearly polarized wave can be rotated as the wave travels through the material. The fundamental difference between the two phenomena is that natural optical activity is reciprocal, so that a
round trip through the medium cancels the polarization rotation, whereas magnetically induced
optical activity is nonreciprocal, so that a round trip through the medium does not cancel but
doubles the polarization rotation. In the simplest case of the nonsymmetric ϵ tensor of the form
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32
Optical Properties of Materials
given in (2.16), natural optical activity can be described by ξ ¼ γk^ ^z , which depends on the
propagation direction k^ and on a characteristic constant γ of the medium, whereas magnetically
induced optical activity is described by ξ ðM 0z Þ or ξ ðH 0z Þ, which is a linear function of M 0z or
^ Whereas all materials exhibit magneticH 0z but is independent of the propagation direction k.
ally induced optical activity in the presence of a magnetization or a magnetic field, natural
optical activity cannot exist in centrosymmetric materials. In an otherwise centrosymmetric
medium, such as a liquid, the addition of molecules, such as sugar molecules, that cause optical
activity breaks the centrosymmetry of the system.
2.3
RESONANT OPTICAL SUSCEPTIBILITY
..............................................................................................................
Frequency dispersion of a medium is caused by the fact that the response of the medium to an
optical field does not end instantaneously but relaxes over time after the optical stimulation. The
root of the optical response is the interaction between the electrons in the material and the
optical field. The electrons in a material can be either valence electrons, which are localized
bound electrons, or conduction electrons, which are nonlocalized free electrons. The electrons
in atoms and molecules are bound electrons that have discrete energy levels. In a condensed
matter, such as a solid material, the electronic states form energy bands. Separate impurity
atoms or molecules that are embedded in an insulator or a semiconductor as dopants can have
discrete energy levels inside an energy band or in the gap between two energy bands of the
host solid. The electrons in a valence band of a solid material, which can be an insulator,
a semiconductor, or a metal, are bound electrons. An electron in a conduction band of a
semiconductor or a metal behaves like a free electron, but it has an effective mass that is
determined by the structure of the conduction band and is different from the electron mass in
free space. A hole in a valence band of a semiconductor behaves like a free positive charge
carrier with an effective mass that is determined by the structure of the valence band.
Resonant interaction involves the transition of an electron, stimulated by an optical field,
between two discrete energy levels or between two energy bands. Nonresonant interaction can
take place between an electron in a conduction band, or a hole in a valence band, and an optical
field while the electron or hole stays in the same band without making a transition to another
band. Both resonant and nonresonant interactions contribute to material dispersion, but their
characteristics are different. In this section, the salient characteristics of resonant interactions
involving valence electrons are considered. The dispersion characteristics of nonresonant
interactions involving free charge carriers are considered in the next section.
A given material generally has many transition resonances across the electromagnetic
spectrum; each resonance is characterized by a resonance frequency, ω0 , and a relaxation rate, γ.
A resonant interaction involves two separate energy states: a lower energy state j1i of energy
E1 and population density N 1 , and an upper energy state j2i of energy E2 and population
density N 2 . The energy states j1i and j2i are discrete energy levels in an atom or molecule, or
specific states in different energy bands of a condensed matter. The population densities N 1 and
N 2 are the number of electrons per unit volume in states j1i and j2i, respectively. When a
material is in thermal equilibrium with its background environment, i.e., in its normal state, the
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33
2.3 Resonant Optical Susceptibility
Figure 2.2 Discrete energy levels for
resonant interaction.
laws of population distribution require that its lower energy level be more populated than its
upper energy level such that N 1 > N 2 : Population inversion with N 2 > N 1 is possible only
when a material is sufficiently pumped to bring it far away from thermal equilibrium.
Because the focus of this section is on the salient features of resonant susceptibility, we
consider the simple case of the resonant interaction involving two discrete energy levels as
shown in Fig. 2.2. The transition resonance frequency is determined by the energy separation of
the two levels,
ω0 ¼ ω21 ¼
E2 E1
,
ℏ
(2.22)
and the relaxation rate is the total susceptibility relaxation rate contributed by various relaxation
mechanisms involving the two energy levels,
γ ¼ γ21 :
(2.23)
Note that the susceptibility relaxation rate γ ¼ γ21 discussed here is the rate of relaxation of the
optical polarization induced by the optical field, which is generally different from the population decay rates of the two energy levels. The details of such differences are discussed in
Section 7.1.
The resonant susceptibility associated with two discrete energy levels can be obtained by
quantum mechanical calculation through the density matrix formalism. Quantum mechanical
calculation allows the accurate treatment of the susceptibility as a tensor; it can be extended to a
complex system that has multiple energy levels or energy bands. A classical Lorentz model that
describes the single-resonance system as a one-dimensional damped oscillator is often used to
obtain the key features of the resonant susceptibility. (See Problem 2.3.1.)
The quantum mechanical result of the resonant susceptibility tensor as a function of the
response time t with respect to an optical excitation at time zero is
2ðN 1 N 2 Þp12 p12 γ21 t
e
sin ω21 t H ðt Þ
ϵ0ℏ
8
< 2ðN 1 N 2 Þp12 p12 γ21 t
sin ω21 t, t 0;
e
¼
ϵ0ℏ
: 0,
t < 0;
χres ðt; ω21 Þ ¼
(2.24)
where the Heaviside step function H ðt Þ has the values of H ðtÞ ¼ 1 for t 0 and H ðt Þ ¼ 0 for
t < 0; and p12 ¼ h1jp^j2i is the matrix element of the electric-dipole operator p^ ¼ e^
x for the
transition between states j1i and j2i, where e is the electronic charge and x^ is the displacement
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34
Optical Properties of Materials
operator. We consider the eigenvalue of the susceptibility tensor for a normal mode of
polarization ^e . For simplicity, we express it in terms of ω0 and γ by applying (2.22) and (2.23):
8
< 2ΔNp2 γt
2
2ΔNp γt
(2.25)
χ res ðt; ω0 Þ ¼ e sin ω0 t H ðt Þ ¼ ϵ 0 ℏ e sin ω0 t, t 0;
:
ϵ0ℏ
0,
t < 0;
where ΔN ¼ N 2 N 1 is the population difference between the upper and the lower energy
levels, and p ¼ p12 ^e is the electric-dipole strength of the resonant transition. Note that
χ res ðt Þ ¼ 0 for t < 0 because a medium can respond only after, but not before, an excitation.
This is the causality condition, which applies to all physical systems.
The Fourier transform of (2.25) to the frequency domain yields
ð∞
χ res ðω; ω0 Þ ¼ χ res ðt; ω0 Þeiωt dt
∞
ΔNp2
1
1
¼
ϵ 0 ℏ ω ω0 þ iγ ω þ ω0 þ iγ
ΔNp2
1
:
ϵ 0 ℏ ω ω0 þ iγ
(2.26)
In (2.26), we have taken the so-called rotating-wave approximation by keeping only the
resonant term that contains ω ω0 in the denominator and dropping the nonresonant term that
contains ω þ ω0 in the denominator because for a frequency ω in the optical spectral region it is
always valid that ω þ ω0
jω ω0 j near resonance. The real and imaginary parts of this
resonant susceptibility are
χ 0res ðωÞ ¼
ΔNp2
ω ω0
ΔNp2
γ
00
,
χ
ð
ω
Þ
¼
,
res
2
2
ϵ 0 ℏ ðω ω0 Þ þ γ
ϵ 0 ℏ ðω ω0 Þ2 þ γ2
(2.27)
which are plotted in Fig. 2.3.
Figure 2.3 Real and imaginary parts, χ 0res ðωÞ and χ 00res ðωÞ, respectively, of susceptibility for a medium that
shows (a) a loss and (b) a gain near a resonance frequency at ω0 .
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2.3 Resonant Optical Susceptibility
35
The imaginary part χ 00res ðωÞ of the resonant susceptibility has a Lorentzian lineshape, which has
a full width at half-maximum (FWHM) of Δω ¼ 2γ. In terms of the frequency ν ¼ ω=2π, the
lineshape has a center frequency at ν0 ¼ ω0 =2π and a FWHM of Δν ¼ Δω=2π ¼ γ=π. The sign
of χ 00res ðωÞ depends on that of ΔN. When the material is in its normal state in thermal equilibrium
with the surrounding, the lower energy level is more populated than the upper level so that
ΔN < 0; thus, the material shows an optical loss near resonance with χ 00res ðωÞ > 0. This characteristic results in the absorption of light at the resonance frequency ω ¼ ω0 when the material is
in thermal equilibrium with its background environment. When population inversion is accomplished so that ΔN > 0, the material shows an optical gain with χ 00res ðωÞ < 0, resulting in the
amplification of light at ω ¼ ω0 due to stimulated emission, such as in the case of an optical
amplifier or a laser. Note that both χ 0res ðωÞ and χ 00res ðωÞ are proportional to ΔN. Therefore, when
χ 00res ðωÞ changes sign with ΔN, χ 0res ðωÞ also changes sign. When χ 00res ðωÞ > 0, for ΔN < 0, χ 0res ðωÞ
is positive for ω < ω0 and negative for ω > ω0 , as is shown in Fig. 2.3(a); when χ 00res ðωÞ < 0, for
ΔN > 0, χ 0res ðωÞ is negative for ω < ω0 and positive for ω > ω0 , as is shown in Fig. 2.3(b).
A medium generally has many resonance frequencies, each corresponding to an absorption
frequency for the medium in its normal state. The permittivity of the medium due to all bound
electrons is the sum of all resonance susceptibilities:
"
#
X
X ΔN i p2 1
1
i
: (2.28)
χ res ðω; ω0i Þ ¼ ϵ 0 þ
ϵ bound ðωÞ ¼ ϵ 0 1 þ
ℏ
ω ω0i þ iγi ω þ ω0i þ iγi
i
i
Note that the rotating-wave approximation is not taken in the above expression because a frequency
ω near one resonance frequency can be very far away from another resonance frequency. For this
reason, the rotating-wave approximation is not generally valid across a broad spectrum. The
characteristics of the real and imaginary parts of ϵ bound ðωÞ for a medium in its normal state as a
function of ω over a spectral range covering a few resonances are illustrated in Fig. 2.4. Some
important dispersion characteristics of χ res ðωÞ and ϵ bound ðωÞ are summarized below.
1. It can be seen from Fig. 2.3(a) that for a material in its normal state, χ 0res ðω < ω0 Þ is always
larger than χ 0res ðω > ω0 Þ. Therefore, around any single resonance frequency, ϵ 0bound ðωÞ at
any frequency on the low-frequency side has a value greater than that at any frequency on
the high-frequency side.
2. From (2.28), it is found that
ϵ bound ð0Þ ¼ ϵ 0 X ΔN i p2 2ω0i
i
> ϵ 0 and ϵ bound ð∞Þ ¼ ϵ 0 :
ℏ
ω20i þ γ2i
i
(2.29)
We see that because ΔN i < 0 for a material in thermal equilibrium, the DC susceptibility
contributed by all bound electrons in a material is real and positive so that the DC
permittivity ϵ bound ð0Þ due to all bound electrons is always real and larger than ϵ 0 . At a very
high frequency that is well above all resonance frequencies, such as one in the hard X-ray
region, all bound electrons stop responding to the high-frequency field so that the medium
behaves much like free space to the high-frequency field; thus ϵ bound ð∞Þ ¼ ϵ 0 . At a finite
frequency of ω that is far away from any resonance frequency, ϵ 00bound ðωÞ 0 so that
ϵ bound ðωÞ ϵ 0bound ðωÞ and ϵ bound ð0Þ > ϵ bound ðωÞ. Therefore, the permittivity of an insulator,
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36
Optical Properties of Materials
Figure 2.4 Real and imaginary parts of ϵ bound as a function of ω for a medium in its normal state over a spectral
range covering a few resonance frequencies.
which does not have free charge carriers, at a frequency that is far away from all resonances
is always smaller than its DC permittivity.
3. A medium is said to have normal dispersion in a spectral region where ϵ 0 ðωÞ increases with
frequency so that dϵ 0 =dω > 0. It is said to have anomalous dispersion in a spectral region where
ϵ 0 ðωÞ decreases with increasing frequency so that dϵ 0 =dω < 0. Because dn=dω and dϵ 0 =dω
have the same sign, the index of refraction also increases with frequency in a spectral region of
normal dispersion and decreases with frequency in a spectral region of anomalous dispersion.
4. It can be seen from Fig. 2.4 that when a material is in its normal state in thermal equilibrium,
normal dispersion appears everywhere except in the immediate neighborhood within the
FWHM of a resonance frequency, where anomalous dispersion occurs. This characteristic
can be reversed near a resonance frequency where resonant amplification, rather than
absorption, takes place due to population inversion.
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2.3 Resonant Optical Susceptibility
37
5. In most materials that are transparent in the visible spectral region, such as glass and water,
normal dispersion appears in the visible region and may extend to the near-infrared and nearultraviolet regions.
Only transitions between discrete energy levels are considered above. In a solid material
where electronic states form energy bands, transitions between separate energy bands, called
band-to-band transitions or interband transitions, contribute to the resonant susceptibility of the
material. The susceptibility is found by integrating over the electronic states in the two bands
involved in the transitions; the integration takes into account the population distribution probability of electrons in each band. The general concepts described above are still valid, except that
the susceptibility contributed by band-to-band transitions does not show the characteristic sharp
resonance peaks of transitions between discrete energy levels seen in Figs. 2.3 and 2.4.
EXAMPLE 2.3
An atomic absorption spectral line associated with an optical transition from the ground state to
an excited state is found to appear at a center wavelength of λ ¼ 800 nm with a FWHM spectral
width of Δλ ¼ 0:48 nm. Find the energy of the excited state above the ground state. Find the
resonance frequency and the polarization relaxation rate associated with this transition. Where
can we find anomalous dispersion caused by this atomic transition when the atoms are in their
normal state in thermal equilibrium with the surrounding?
Solution:
The energy of the excited state above the ground state is the photon energy of the absorption
wavelength at λ ¼ 800 nm:
E2 E1 ¼ hν ¼
1239:8
1239:8
nm eV ¼
eV ¼ 1:55 eV:
λ
800
The resonance frequency is
c 3 108 m s1
ν0 ¼ ¼
¼ 375 THz ;
λ 800 109 m
Because λ
ω0 ¼ 2πν0 ¼ 2:36 1015 rad s1 :
Δλ, we can use the approximation Δν=ν0 Δλ=λ to find that
Δν ¼
Δλ
0:48
ν0 ¼
375 THz ¼ 225 GHz:
λ
800
Thus, the relaxation rate is
γ ¼ πΔν ¼ 7:07 1011 s1 :
When the atoms are in their normal state in thermal equilibrium with the surrounding, the
ground state is more populated than the excited state. In this situation, anomalous dispersion
caused by this transition is found within the FWHM of the spectral line, in the wavelength
range of λ Δλ=2 ¼ 800 0:24 nm, corresponding to the frequency range of ν0 Δν=2 ¼
375 THz 112:5 GHz.
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38
Optical Properties of Materials
2.4
OPTICAL CONDUCTIVITY AND CONDUCTION SUSCEPTIBILITY
..............................................................................................................
An electron in a conduction band of a semiconductor or a metal behaves like a free electron
with an effective mass, while a hole in a valence band of a semiconductor behaves like a free
positive charge carrier with an effective mass. The response of these free charge carriers to an
optical field can be treated using quantum mechanics by considering induced transitions within
a band, known as intraband transitions, or using a classical Drude model by considering an
induced conduction current J cond as discussed in Section 1.1. Because the quantum mechanical
approach involves the consideration of the band structure, we use the classical Drude model for
simplicity. In this classical approach, the effective mass m∗ of the charge carrier accounts for
the effect of the energy band; clearly, the value of m∗ depends on the structure of the energy
band on which the charge carrier lies.
In the Drude model, conduction electrons, and holes in a semiconductor, are treated as
independent free charge carriers. The momentum, p, of a charge carrier is driven by the force
of an electric field, F ¼ qE, and is damped by random collisions with the ions of the medium,
characterized by an average momentum relaxation time τ. Therefore,
dp
p
¼ qE ,
dt
τ
(2.30)
where q ¼ e for an electron and q ¼ e for a hole. The conduction current density is
J cond ¼ Nqv ¼
Nqp
,
m∗
(2.31)
where N is the density of the free charge carriers. By combining (2.30) and (2.31), we have the
equation for the conduction current that is induced by an electric field:
dJ cond J cond Ne2
þ
¼ ∗ E,
dt
τ
m
(2.32)
where q2 ¼ e2 is used for the charge carriers to be either electrons or holes.
The general solution of (2.32) can be expressed as a convolution integral:
ðt
J cond ðtÞ ¼
σ ðt t 0 ÞEðt 0 Þdt 0,
(2.33)
∞
where
8
2
< Ne t=τ
Ne t=τ
e ,
σ ðtÞ ¼ ∗ e H ðt Þ ¼
m∗
:
m
0,
2
for t 0;
(2.34)
for t < 0:
Note that Jcond ðt Þ and Eðt Þ are real fields in the real space and time domain. The relation in
(2.33) defines the optical conductivity σ ðt Þ in the real space and time domain, as seen in (2.34).
For simplicity, their spatial dependence is ignored. In terms of the complex field,
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2.4 Optical Conductivity and Conduction Susceptibility
ðt
Jcond ðt Þ ¼
σ ðt t 0 ÞEðt0 Þdt 0,
39
(2.35)
∞
where σ ðt Þ is the same as that in (2.34). The frequency domain relation is obtained by taking the
Fourier transform on (2.35):
Jcond ðωÞ ¼ σ ðωÞEðωÞ,
(2.36)
where
ð∞
σ ðtÞeiωt dt ¼
σ ðω Þ ¼
∞
Ne2 τ 1
:
m∗ 1 iωτ
(2.37)
This frequency-dependent optical conductivity, also known as the AC conductivity, can be
expressed in terms of the DC conductivity:
σ ðωÞ ¼
σ ð0Þ
,
1 iωτ
(2.38)
Ne2 τ
:
m∗
(2.39)
where σ ð0Þ is the DC conductivity,
σ ð0Þ ¼
As discussed in Section 1.1, there are two alternative, but equivalent, ways to described the
optical response of free charge carriers: (1) by treating it as part of the total susceptibility and
total permittivity in the total displacement D, as in (1.12); or (2) by treating it as an optical
conductivity through an explicit conduction current Jcond , as in (1.16). The discussion above
follows the second alternative, which allows us to find the optical conductivity in (2.38). By
equating the two alternative approaches, the conduction susceptibility, χ cond , due to the free
charge carriers can be found.
Equating (1.12) and (1.16) but expressing them in complex fields, we have
∂D ∂Dbound
þ Jcond :
¼
∂t
∂t
(2.40)
Converting this relation to the frequency domain, we find
iωDðωÞ ¼ iωDbound ðωÞ þ Jcond ðωÞ:
(2.41)
By using the relations DðωÞ ¼ ϵ ðωÞEðωÞ, DðωÞbound ¼ ϵ bound ðωÞEðωÞ, and Jcond ðωÞ ¼
σ ðωÞEðωÞ from (2.36), we find the total permittivity that includes all contributions from bound
and free charges in a material:
ϵ ðωÞ ¼ ϵ bound ðωÞ þ
iσ ðωÞ
σ ð0Þ
¼ ϵ bound ðωÞ ,
ω
ωðωτ þ iÞ
(2.42)
where ϵ bound ðωÞ is the permittivity from bound charges discussed in Section 2.3. Therefore, we
can identify the conduction susceptibility due to the free charge carriers:
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Optical Properties of Materials
Figure 2.5 Real and
imaginary parts, χ 0cond ðωÞ and
χ 00cond ðωÞ, respectively, of the
conduction susceptibility,
normalized to σ ð0Þτ=ϵ 0 , as a
function of ωτ.
χ cond ðωÞ ¼
iσ ðωÞ
σ ð0Þτ
1
¼
:
ϵ0ω
ϵ 0 ωτ ðωτ þ iÞ
(2.43)
The real and imaginary parts of this conduction susceptibility are
χ 0cond ðωÞ ¼ σ ð0Þτ
1
σ ð0Þτ
1
, χ 00cond ðωÞ ¼
,
2
2
2
ϵ0 ω τ þ 1
ϵ 0 ωτ ðω τ 2 þ 1Þ
(2.44)
which are plotted in Fig. 2.5.
At an optical frequency that is far away from any resonance transition frequency,
00
ϵ bound ðωÞ 0 so that ϵ bound ðωÞ ϵ 0bound ðωÞ. In this case, the real and imaginary parts of the
total permittivity given in (2.42) are
ϵ 0 ðωÞ ¼ ϵ bound ðωÞ σ ð0Þτ
,
þ1
ω2 τ 2
ϵ 00 ðωÞ ¼
σ ð0Þτ
:
ωτ ðω2 τ 2 þ 1Þ
(2.45)
We find that due to the effect of the conduction electrons, the real part of the total susceptibility
vanishes, i.e., ϵ 0 ðωÞ ¼ 0, at the frequency ωp , known as the plasma frequency:
ω2p ¼
σ ð0Þ
ϵ bound τ
1
Ne2
1
σ ð0Þ
Ne2
¼
¼
:
τ 2 ϵ bound m∗ τ 2 ϵ bound τ ϵ bound m∗
(2.46)
Because it is almost always true that ωp τ
1 for most conducting materials, the plasma
frequency is generally defined by neglecting the 1=τ 2 term in (2.46). The permittivity ϵ bound
in (2.46) is taken to be a constant that has the value in the frequency range of interest. In terms
of ω2p , the total permittivity can be expressed as
"
ϵ ðωÞ ¼ ϵ bound 1 ω2p τ 2
ωτ ðωτ þ iÞ
#
"
¼ ϵ bound 1 ω2p τ 2
ω2 τ 2 þ 1
þi
ω2p τ 2
ωτ ðω2 τ 2 þ 1Þ
The real and imaginary parts of this total permittivity are plotted in Fig. 2.6.
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#
:
(2.47)
2.4 Optical Conductivity and Conduction Susceptibility
41
Figure 2.6 Real and imaginary parts, ϵ 0 ðωÞ and ϵ 00 ðωÞ, respectively, of the total permittivity, normalized to
ϵ bound , as a function of frequency ω showing (a) low-frequency characteristics and (b) high-frequency
characteristics. The value of ωp τ ¼ 10 is used for this plot.
Some important characteristics are summarized below.
1. For all frequencies, the real part χ 0cond ðωÞ of the conduction susceptibility is negative, and the
imaginary part χ 00cond ðωÞ is positive. Thus the conduction susceptibility only contributes to
optical loss and never contributes to optical gain, and it makes possible a negative real part
for the permittivity, as discussed below.
2. At low frequencies for which ωτ 1, ϵ 0 ðωÞ=ϵ bound 1 ω2p τ 2 approaches a constant
but ϵ 00 ðωÞ=ϵ bound ω2p τ=ω becomes inversely proportional to frequency so that jϵ 00 ðωÞj
jϵ 0 ðωÞj. Then,
!
2
ω
τ
p
:
(2.48)
ϵ ðωÞ ϵ bound 1 ω2p τ 2 þ i
ω
These low-frequency characteristics are seen in Fig. 2.6(a).
3. At high frequencies for which ωτ
1, ϵ 0 ðωÞ=ϵ bound 1 ω2p =ω2 and ϵ 00 ðωÞ 0 so that
jϵ 0 ðωÞj
jϵ 00 ðωÞj. Then,
!
ω2p
ϵ ðωÞ ϵ bound 1 2 :
(2.49)
ω
These high-frequency characteristics are seen in Fig. 2.6(b).
4. At all frequencies, the imaginary part of the permittivity is positive because χ 00cond ðωÞ is
positive: ϵ 00 ðωÞ > 0 for all ω.
5. At frequencies below the plasma frequency, the real part of the permittivity is negative:
ϵ 0 ðωÞ < 0 for ω < ωp . This leads to high reflectivity on the surface and low penetration of
the optical field in the medium, which are the common properties of metallic surfaces.
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42
Optical Properties of Materials
6. At frequencies above the plasma frequency, the real part of the permittivity is positive while
the positive imaginary part decreases quickly with increasing frequency. Consequently, the
contribution of the conduction susceptibility quickly diminishes. Then the medium behaves
optically like an insulator, allowing a high-frequency optical field to penetrate through with
little attenuation except when the optical frequency comes close to a transition resonance.
7. For a perfect conductor, only free conduction electrons contribute to the optical response
so that the permittivity has no contribution from bound electrons; thus, ϵ bound ¼ ϵ 0 . For this
reason, it is a good approximation to take ϵ bound ¼ ϵ 0 for a metal that has a high conductivity, such as Ag, Au, Cu, and Al. For such a metal, it is also a good approximation to take the
effective electron mass as the free electron mass, m∗ ¼ m0 , when applying (2.46).
8. For a semiconductor where electrons and holes both contribute to the conduction susceptibility, the total permittivity is
ϵ ðωÞ ¼ ϵ bound ðωÞ σ e ð0Þ
σ h ð0Þ
,
ωðωτ e þ iÞ ωðωτ h þ iÞ
(2.50)
where
σ e ð0Þ ¼
N e e2 τ e
N h e2 τ h
and
σ
ð0Þ
¼
:
h
m∗
m∗
e
h
(2.51)
The plasma frequency is found at ϵ 0 ðωÞ ¼ 0 to be
ω2p ¼
σ e ð0Þ
1
σ h ð0Þ
1
N e e2
N h e2
2þ
2
þ
:
ϵ bound m∗
ϵ bound τ e τ e ϵ bound τ h τ h ϵ bound m∗
e
h
(2.52)
EXAMPLE 2.4
Silver is one of the best conductors such that the free-electron Drude model describes its optical
properties reasonably well. In this model, the free electron density of Ag is found to be N ¼
5:86 1028 m3 . The DC conductivity of Ag at T ¼ 273 K is σ ð0Þ ¼ 6:62 107 S m1 . Find
the plasma frequency ωp and the relaxation time τ for Ag at T ¼ 273 K. Also find the cutoff
optical frequency νp and the cutoff wavelength λp . For what optical wavelengths is Ag expected
to be highly reflective? For what wavelengths is it expected to become transmissive?
Solution:
For Ag, it is a good approximation to take ϵ bound ¼ ϵ 0 and m∗ ¼ m0 . Then, using (2.46), we
find that
ω2p
2
5:86 1028 1:6 1019
Ne2
¼
¼
rad2 s2 ¼ 1:86 1032 rad2 s2
12
31
ϵ 0 m∗
8:854 10
9:1 10
) ωp ¼ 1:36 1016 rad s1 ,
τ¼
σ ð0Þ
6:62 107
¼
s1 ¼ 4:02 1014 s ¼ 40:2 fs:
ϵ 0 ω2p 8:854 1012 1:86 1032
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2.4 Optical Conductivity and Conduction Susceptibility
43
The cutoff frequency and cutoff wavelength are those at the plasma frequency:
νp ¼
ωp
¼ 2:17 PHz,
2π
λp ¼
c
¼ 138 nm:
νp
Ag is highly reflective for λ > λp , corresponding to ν < νp ; it becomes transmissive for λ < λp ,
corresponding to ν > νp .
EXAMPLE 2.5
GaAs is a direct-gap semiconductor that has an electron effective mass of m∗
e ¼ 0:067m0 and a
hole effective mass of m∗
¼
0:52m
,
where
m
is
the
mass
of
a
free
electron.
Its low-frequency
0
0
h
dielectric constant is 10.9. Find the plasma frequency, the cutoff frequency, and the cutoff
wavelength for (a) an n-type GaAs sample that has an electron density of N e ¼ 1 1024 m3 ,
(b) a p-type GaAs sample that has a hole density of N h ¼ 1 1024 m3 , and (c) a GaAs sample
that is injected with an equal electron and hole density of N e ¼ N h ¼ 1 1024 m3 .
Solution:
As we will see below, the plasma frequency is much lower than the bandgap frequency of GaAs,
which corresponds to a wavelength of λg ¼ 871 nm. Therefore, the low-frequency dielectric
constant is used for ϵ bound ¼ 10:9ϵ 0 . Then, the plasma frequency is found using (2.52).
(a) For the n-type GaAs with N e ¼ 1 1024 m3 , the hole density is negligibly small so that
ω2p N e e2
ϵ bound m∗
e
2
1 1024 1:6 1019
¼
rad2 s1 ¼ 4:35 1027 rad2 s2 :
10:9 8:854 1012 0:067 9:1 1031
Therefore, ωp ¼ 6:60 1013 rad s1 , νp ¼ 10:5 THz, and λp ¼ 28:6 μm.
(b) For the p-type GaAs with N h ¼ 1 1024 m3 , the electron density is negligibly small so that
ω2p N h e2
ϵ bound m∗
h
2
1 1024 1:6 1019
¼
rad2 s1 ¼ 5:60 1026 rad2 s2 :
10:9 8:854 1012 0:52 9:1 1031
Therefore, ωp ¼ 2:37 1013 rad s1 , νp ¼ 3:77 THz, and λp ¼ 79:6 μm.
(c) For the injected GaAs with N e ¼ N h ¼ 1 1024 m3 ,
ω2p N e e2
N h e2
þ
ϵ bound m∗
ϵ bound m∗
e
h
¼ 4:35 1027 rad2 s2 þ 5:60 1026 rad2 s2 ¼ 4:91 1027 rad2 s2 :
Therefore, ωp ¼ 7:01 1013 rad s1 , νp ¼ 11:2 THz, and λp ¼ 26:8 μm.
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44
Optical Properties of Materials
2.5
KRAMERS–KRONIG RELATIONS
..............................................................................................................
It can be seen from the above discussion that the real and imaginary parts of χ ðωÞ, or those of
ϵ ðωÞ, are not independent of each other. The susceptibility of any physical system has to satisfy
the causality requirement in the time domain. This requirement leads to a general relationship
between χ 0 ðωÞ and χ 00 ðωÞ in the frequency domain:
ð∞
ð∞
2 ω0 χ 00 ðω0 Þ 0
2
ωχ 0 ðω0 Þ
00
dω
,
χ
ð
ω
Þ
¼
dω0 ,
P
χ ðωÞ ¼ P 0 2
π ω ω2
π ω0 2 ω2
0
0
(2.53)
0
where the principal values are taken for the integrals. These relations are known as the Kramers–
Kronig relations. They are valid for any χ ðωÞ that represents a physical process, such as the
resonant susceptibility χ res ðωÞ in Section 2.3 and the conduction susceptibility χ cond ðωÞ in
Section 2.4. Therefore, once the real part of χ ðωÞ for any physical process is known over the
entire spectrum, its imaginary part can be found, and vice versa. Note that the relations in (2.53)
are consistent with the fact that χ 0 ðωÞ is an even function of ω and χ 00 ðωÞ is an odd function of ω,
as discussed in Section 2.1. The contradiction to this statement seen in (2.27) for χ 0res ðωÞ and
χ 00res ðωÞ is only apparent but not real; it is caused by the rotating-wave approximation taken in
(2.26). There is no contradiction when the rotating-wave approximation is removed and exact
expressions are used for χ 0res ðωÞ and χ 00res ðωÞ. For χ 0cond ðωÞ and χ 00cond ðωÞ given in (2.44), it is clear
that χ 0cond ðωÞ is an even function of ω and χ 00cond ðωÞ is an odd function of ω.
2.6
EXTERNAL FACTORS
..............................................................................................................
The optical property of a material is influenced by external factors, such as an electric field, a
magnetic field, an acoustic wave, an injection current, a pressure, or a temperature change. The
dependence of the optical property on an externally controllable factor allows the active control
and modulation of an optical wave; this is the basis for active photonic devices. On the other
hand, this characteristic is passively used in a photonic sensor which optically senses the
parameter that causes a change in the optical property of the sensor material. Some of the
major effects that cause changes in the permittivity of an optical material are discussed below.
2.6.1 Electro-optic Effect
The optical property of a dielectric material can be changed by a static or low-frequency electric
field E0 through an electro-optic effect. The result is a field-dependent susceptibility and thus a
field-dependent permittivity:
Pðω; E0 Þ ¼ ϵ 0 χðω; E0 Þ EðωÞ ¼ ϵ 0 χðωÞ EðωÞ þ ϵ 0 Δχðω; E0 Þ EðωÞ
(2.54)
Dðω; E0 Þ ¼ ϵ ðω; E0 Þ EðωÞ ¼ ϵ ðωÞ EðωÞ þ Δϵ ðω; E0 Þ EðωÞ,
(2.55)
and
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2.6 External Factors
45
where the field-independent susceptibility χðωÞ ¼ χðω; E0 ¼ 0Þ and permittivity ϵ ðωÞ ¼
ϵ ðω; E0 ¼ 0Þ represent the intrinsic linear response of the material at the optical frequency ω,
whereas Δχðω; E0 Þ and Δϵ ðω; E0 Þ represent the changes induced by the electric field E0 . We
can define the electric-field-induced polarization change as ΔPðω; E0 Þ ¼ ϵ 0 Δχðω; E0 Þ EðωÞ to
express the total field-dependent displacement as Dðω; E0 Þ ¼ DðωÞ þ ΔPðω; E0 Þ. The total
permittivity of the material in the presence of the electric field is then
ϵ ðω; E0 Þ ¼ ϵ ðωÞ þ Δϵ ðω; E0 Þ ¼ ϵ ðωÞ þ ϵ 0 Δχðω; E0 Þ:
(2.56)
In the discussion of electro-optic effects, it is necessary to introduce the relative impermeability tensor, which is the inverse of the dielectric constant tensor:
1
ϵ
:
(2.57)
η¼
ϵ0
The reason for considering the relative impermeability tensor is historical because electro-optic
effects are traditionally not expressed as Δϵ ðω; E0 Þ or Δχðω; E0 Þ but are defined in terms of the
changes in the elements of the relative impermeability tensor as ηðE0 Þ ¼ η þ ΔηðE0 Þ, which is
expanded as
X
X
ηij ðE0 Þ ¼ ηij þ Δηij ðE0 Þ ¼ ηij þ
r ijk E 0k þ
sijkl E 0k E 0l þ ,
(2.58)
k
k, l
where the first term ηij is the field-independent component, the elements of the third-order rijk
tensor are the linear electro-optic coefficients known as the Pockels coefficients, and those of
the fourth-order sijkl tensor are the quadratic electro-optic coefficients known as the electrooptic Kerr coefficients. The first-order electro-optic effect characterized by the linear dependence of ηij ðE0 Þ on E0 through the coefficients r ijk is called the linear electro-optic effect, also
known as the Pockels effect. The second-order electro-optic effect characterized by the quadratic field dependence through the coefficients sijkl is called the quadratic electro-optic effect,
also known as the electro-optic Kerr effect. In (2.58), indices i and j are associated with optical
fields, whereas indices k and l are associated with the low-frequency applied field. Because the
ϵ tensor of a nonmagnetic electro-optic material is symmetric, the η tensor as defined in (2.57)
is also symmetric; thus ηij ¼ ηji and Δηij ¼ Δηji . The symmetric indices i and j can be
contracted to reduce the double index ij to a single index α using the index contraction rule:
ij :
or ij :
α:
11 22 33 23, 32 31, 13 12, 21
xx yy zz yz, zy zx, xz xy, yx
1 2 3
4
5
6
Using index contraction, (2.58) is expressed as
X
X
ηα ðE0 Þ ¼ ηα þ Δηα ðE0 Þ ¼ ηα þ
r αk E 0k þ
sαkl E 0k E 0l þ ,
k
k, l
(2.59)
(2.60)
where α ¼ 1, 2, . . . , 6 and k, l ¼ 1, 2, 3 or x, y, z:
The Pockels effect does not exist in a centrosymmetric material, which is a material that
possesses inversion symmetry. The structure and properties of such a material remain
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46
Optical Properties of Materials
unchanged under the transformation of space inversion, which changes the signs of all
rectilinear spatial coordinates from ðx; y; zÞ to ðx; y; zÞ, and the signs of all polar vectors.
As discussed in Section 1.1, an electric field vector is a polar vector that changes sign under the
transformation of space inversion. By simply considering the effect of space inversion, it is
clear that the electro-optically induced changes in the optical property of a centrosymmetric
material are not affected by the sign change in the applied field from E0 to E0 , meaning that
ηij ðE0 Þ ¼ ηij ðE0 Þ. As can be seen from (2.58), this condition requires that the Pockels
coefficients r ijk vanish, but it does not require the electro-optic Kerr coefficients sijkl to vanish.
Consequently, the Pockels effect exists only in noncentrosymmetric materials, whereas the
electro-optic Kerr effect exists in all materials, including centrosymmetric ones. Structurally
isotropic materials, including all gases, liquids, and amorphous solids such as glass, show no
Pockels effect because they are centrosymmetric.
The majority of electro-optic devices are based on the Pockels effect because the electro-optic
Kerr coefficients are generally very small. For this reason, practical electro-optic applications
usually require noncentrosymmetric crystals in order to make use of the Pockels effect. Among
the 32 point groups in the 7 crystal systems, 11 are centrosymmetric, and the remaining 21 are
noncentrosymmetric. It is important to note that the linear optical property of a crystal is
determined only by its crystal system, as mentioned in Section 2.2 and summarized in Table 2.1,
but the electro-optic property further depends on its point group.
Because the electro-optic coefficients are traditionally defined through the changes in the
relative impermeability tensor, as expressed in (2.58), the field-induced changes in the permittivity tensor have to be found through the relation between Δϵ ij ðE0 Þ and Δηij ðE0 Þ. Using the
relation η ϵ=ϵ 0 ¼ 1, the relation between Δϵ and Δη can be found:
Δϵ ¼ 1
1
ϵ Δη ϵ and Δη ¼ η Δϵ η:
ϵ0
ϵ0
(2.61)
As discussed in Section 2.2, the intrinsic permittivity tensor ϵ ðωÞ of a crystal in the absence of
the electric field is diagonal with eigenvalues ϵ x , ϵ y , and ϵ z in the coordinate system defined by
the intrinsic principal dielectric axes ^x , ^y , and ^z , which are determined by the structural
symmetry of the crystal lattice. In this coordinate system, the relations in (2.61) can be written
explicitly as
Δηij
Δϵ ij
Δϵ ij
¼ ϵ 0 n2i n2j Δηij and Δηij ¼ ϵ 0
¼
,
(2.62)
ηi ηj
ϵiϵj
ϵ 0 n2i n2j
pffiffiffiffiffiffiffiffiffiffi
where ηi ¼ ϵ 0 =ϵ i are the eigenvalues of the η tensor and ni ¼ ϵ i =ϵ 0 are the principal indices
of refraction.
Δϵ ij ¼ ϵ 0
EXAMPLE 2.6
LiNbO3 is a negative uniaxial crystal with nx ¼ ny ¼ no > nz ¼ ne . Being a crystal of the 3m
symmetry group, it has eight nonvanishing Pockels coefficients of four distinct values:
r 12 ¼ r22 , r 13 , r 22 , r 23 ¼ r 13 , r 33 , r42 , r 51 ¼ r42 , r 61 ¼ r22 . Find the field-induced permittivity change Δϵ ðE0 Þ for an applied DC electric field of E0 ¼ E 0x ^x þ E 0y ^y þ E 0z^z .
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2.6 External Factors
47
Solution:
According to (2.58), the field-induced impermeability change due to the Pockels effect is
X
Δηα ðE0 Þ ¼
rαk E 0k ,
k
which can be expressed in the matrix form as
0
Δη1
1
0
r 12
r 11
B
C B
B Δη2 C B r 21
B
C B
B Δη C B r 31
B 3C B
B
C¼B
B Δη4 C B r 41
B
C B
B
C B
@ Δη5 A @ r 51
Δη6
1
r 52
C
r 33 C0
1
C E0x
C
r 33 CB
C
C@ E 0y A:
r 43 C
C E
0z
C
r 53 A
r 62
r 63
r 22
r 32
r 42
r 61
r 13
Using the given nonvanishing Pockels coefficients for LiNbO3 , we have
0
Δη1
1
0
B
C B
B Δη2 C B
B
C B
B Δη C B
B 3C B
B
C¼B
B Δη4 C B
B
C B
B
C B
@ Δη5 A @
Δη6
0
r22
0
r22
0
0
0
r42
r 42
0
r 22
0
r 13
1
0
r 22 E 0y þ r13 E 0z
B
C
r 13 C0
1 B r 22 E 0y þ r13 E 0z
B
C E 0x
B
r 33 C
r 33 E 0z
C B
CB
C@ E 0y A ¼ B
B
0 C
r 42 E 0y
B
C E
0z
B
C
0 A
r 42 E 0x
@
1
C
C
C
C
C
C:
C
C
C
A
r 22 E 0x
0
By the index contraction rule, Δη1 ¼ Δηxx , Δη2 ¼ Δηyy , Δη3 ¼ Δηzz , Δη4 ¼ Δηyz ¼ Δηzy ,
Δη5 ¼ Δηzx ¼ Δηxz , Δη6 ¼ Δηxy ¼ Δηyx . Using (2.62), we find
Δϵ xx ¼ ϵ 0 n4x Δηxx ¼ ϵ 0 n4o r 22 E 0y ϵ 0 n4o r 13 E 0z ,
Δϵ yy ¼ ϵ 0 n4y Δηyy ¼ ϵ 0 n4o r 22 E 0y ϵ 0 n4o r 13 E 0z ,
Δϵ zz ¼ ϵ 0 n4z Δηzz ¼ ϵ 0 n4e r 33 E 0z ,
Δϵ yz ¼ Δϵ zy ¼ ϵ 0 n2y n2z Δηyz ¼ ϵ 0 n2o n2e r 42 E 0y ,
Δϵ zx ¼ Δϵ xz ¼ ϵ 0 n2x n2z Δηyz ¼ ϵ 0 n2o n2e r 42 E 0x ,
Δϵ xy ¼ Δϵ yx ¼ ϵ 0 n2x n2y Δηxy ¼ ϵ 0 n4o r22 E 0x :
Expressed in the matrix form, the field-induced permittivity change is
0 4
1
no r 22 E 0y n4o r 13 E 0z
n4o r22 E 0x
n2o n2e r 42 E 0x
B
C
Δϵ ðE0 Þ ¼ ϵ 0 @
n4o r 22 E 0x
n4o r22 E 0y n4o r13 E 0z n2o n2e r 42 E 0y A:
n2o n2e r42 E 0x
n2o n2e r 42 E 0y
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n4e r 33 E 0z
48
Optical Properties of Materials
The electric-field-induced changes represented by Δϵ ðω; E0 Þ usually generate off-diagonal
elements besides changing the diagonal elements:
0
1
0
1
Δϵ xy
Δϵ xz
ϵ x þ Δϵ xx
ϵx 0 0
ϵ y þ Δϵ yy
Δϵ yz A (2.63)
ϵ ðωÞ ¼ @ 0 ϵ y 0 A while ϵ ðω; E0 Þ ¼ @ Δϵ yx
Δϵ zx
Δϵ zy
ϵ z þ Δϵ zz
0 0 ϵz
in the coordinate system of the principal ^x , ^y , and ^z axes. As discussed in Section 2.2, ϵ of a
nonmagnetic material is a symmetric tensor. This remains true for a nonmagnetic material
subject to an applied electric field; thus, for ϵ ðω; E0 Þ in (2.63),
ϵ ij ðω; E0 Þ ¼ ϵ ji ðω; E0 Þ and Δϵ ij ðω; E0 Þ ¼ Δϵ ji ðω; E0 Þ:
(2.64)
Because the field-dependent nondiagonal permittivity tensor is symmetric, it can be
diagonalized to find a new set of eigenvalues ϵ X , ϵ Y , and ϵ Z with corresponding real
^ , Y^ , and Z^ , which represent a new set of linearly polarized principal normal
eigenvectors X
modes for defining the new principal dielectric axes of the material in the presence of the
electric field E0 . In general, the new principal axes depend on the direction and, in certain
cases, the magnitude of E0 . Thus,
0
1
ϵX 0 0
(2.65)
ϵ ðω; E0 Þ ¼ @ 0 ϵ Y 0 A:
0 0 ϵZ
The propagation characteristics of an optical wave in the presence of an electro-optic effect are
then determined by ϵ X , ϵ Y , and ϵ Z , which define the principal indices of refraction,
rffiffiffiffiffi
rffiffiffiffiffi
rffiffiffiffiffi
ϵX
ϵY
ϵZ
, nY ¼
, nZ ¼
,
(2.66)
nX ¼
ϵ0
ϵ0
ϵ0
and the propagation constants,
kX ¼
nX ω
nY ω
nZ ω
, kY ¼
, kZ ¼
,
c
c
c
(2.67)
^ Y^ , and Z^ principal normal modes of polarization. Note that these three new principal
for the X,
normal modes of polarization are linearly polarized. Therefore, electrically induced
birefringence and dichroism due to an electro-optical effect are linear birefringence and linear
dichroism.
EXAMPLE 2.7
At the 1 μm optical wavelength, LiNbO3 has the refractive indices of no ¼ 2:238 and
ne ¼ 2:159. The four distinct values of its Pockels coefficients are r 13 ¼ 8:6 pm V1 ,
r 22 ¼ 3:4 pm V1 , r 33 ¼ 30:8 pm V1 , and r 42 ¼ 28 pm V1 . Use the results from Example
2.6 to answer the following questions. Is it possible to apply a DC electric field to change the
principal indices of refraction through the Pockels effect without rotating the principal axes? If
this is possible, find the changes in the principal indices of refraction caused by an applied
electric field of E 0 ¼ 5 MV m1 .
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2.6 External Factors
49
Solution:
For the Pockels effect to cause only changes in the principal indices of refraction without rotating
the principal axes, an applied electric field has to generate changes only in the diagonal elements,
but not in the off-diagonal elements, of Δϵ ðE0 Þ. By examining Δϵ ðE0 Þ obtained in Example 2.6
for LiNbO3 , we find that this is possible if the DC electric field is applied only along the direction
of the z principal axis such that E0 ¼ E 0^z for E 0z ¼ E 0 and E0x ¼ E 0y ¼ 0. Then,
0 2
1
no n4o r13 E 0
0
0
A:
ϵ ðE0 Þ ¼ ϵ þ Δϵ ðE0 Þ ¼ ϵ 0 @
0
n2o n4o r 13 E 0
0
2
4
0
0
ne ne r33 E 0
Because ϵ ðE0 Þ is diagonal in the coordinate system of the original principal axes, all principal
axes remain unchanged:
^ ¼ ^x , Y^ ¼ ^y , Z^ ¼ ^z :
X
Using (2.65) and (2.66), we find the new principal indices of refraction:
nX ¼ nY ¼ ðn2o n4o r 13 E 0 Þ1=2 no n3o r 13
n3 r 33
E 0 , nZ ¼ ðn2e n4e r 33 E 0 Þ1=2 ne e E 0 :
2
2
Clearly, the crystal remains negative uniaxial. The changes in the principal indices of refraction
caused by an applied electric field of E 0 ¼ 5 MV m1 are
ΔnX ¼ ΔnY ¼ Δno ¼ n3o r13
2:2283 8:6 1012
E0 ¼ 5 106 ¼ 2:41 104
2
2
for the ordinary index and
ΔnZ ¼ Δne ¼ n3e r 33
2:1593 30:8 1012
E0 ¼ 5 106 ¼ 7:75 104
2
2
for the extraordinary index.
2.6.2 Magneto-optic Effect
A material can be either diamagnetic or paramagnetic. A diamagnetic material does not contain
intrinsic magnetic dipole moments; a paramagnetic material consists of atoms or ions that have
intrinsic magnetic dipole moments. A paramagnetic material can be either magnetically disordered, when its intrinsic magnetic dipole moments are randomly oriented, or magnetically
ordered. A magnetically ordered material is ferromagnetic if all of its intrinsic dipole moments
line up in the same direction; it is ferrimagnetic if it contains different types of intrinsic dipole
moments that line up in alternating antiparallel directions but do not cancel each other; it is
antiferromagnetic, also called antiferrimagnetic, if different types of intrinsic dipole moments line
up in alternating antiparallel directions and cancel each other. Below a critical temperature, known
as the Curie temperature for a ferromagnetic material and the Néel temperature for a ferrimagnetic
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50
Optical Properties of Materials
material, the magnetic ordering in a ferromagnetic or ferrimagnetic material generates a spontaneous magnetization M 0 . No spontaneous magnetization exists in a diamagnetic material, in a
magnetically disordered paramagnetic material, or in an antiferromagnetic material.
As mentioned in Section 1.1, at an optical frequency μ ¼ μ0 and thus BðωÞ ¼ μ0 HðωÞ; the
response of a material, irrespective of whether it is magnetic or nonmagnetic, to an optical field
at an optical frequency of ω is fully described by its electric susceptibility χðωÞ and, equivalently, by its electric permittivity ϵ ðωÞ. Nevertheless, a nonmagnetic material that does not have
a spontaneous magnetization still responds to a static or low-frequency magnetic field, H 0 . Its
optical property can be changed by H 0 , resulting in a magnetic-field-dependent susceptibility
and a magnetic-field-dependent permittivity:
Pðω; H 0 Þ ¼ ϵ 0 χðω; H 0 Þ EðωÞ ¼ ϵ 0 χðωÞ EðωÞ þ ϵ 0 Δχðω; H 0 Þ EðωÞ
(2.68)
Dðω; H 0 Þ ¼ ϵ ðω; H 0 Þ EðωÞ ¼ ϵ ðωÞ EðωÞ þ Δϵ ðω; H 0 Þ EðωÞ,
(2.69)
and
where χðωÞ ¼ χðω; H 0 ¼ 0Þ and ϵ ðωÞ ¼ ϵ ðω; H 0 ¼ 0Þ represent the intrinsic properties of the
material in the absence of the static or low-frequency magnetic field. In the case of a ferromagnetic or ferrimagnetic material, in which a static magnetization M 0 exists, the properties of the
material at an optical frequency are dependent on M 0 . Then, instead of (2.68) and (2.69), we
have magnetization-dependent susceptibility and magnetization-dependent permittivity:
Pðω; M 0 Þ ¼ ϵ 0 χðω; M 0 Þ EðωÞ ¼ ϵ 0 χðωÞ EðωÞ þ ϵ 0 Δχðω; M 0 Þ EðωÞ
(2.70)
Dðω; M 0 Þ ¼ ϵ ðω; M 0 Þ EðωÞ ¼ ϵ ðωÞ EðωÞ þ Δϵ ðω; M 0 Þ EðωÞ:
(2.71)
and
While χ and ϵ are changed by H 0 or M 0 , the magnetic permeability of the material at an
optical frequency remains the constant μ0 , and the relation between BðωÞ and HðωÞ remains
independent of H 0 or M 0 : BðωÞ ¼ μ0 HðωÞ. Therefore, magneto-optic effects are completely
characterized by ϵ ðω; H 0 Þ, if no internal magnetization is present, or by ϵ ðω; M 0 Þ, if an internal
magnetization is present. In general, these effects are weak perturbations to the optical properties of the material. The first-order magneto-optic effect, or linear magneto-optic effect, is
characterized by a linear dependence of ϵ on H 0 or M 0 , and the second-order magneto-optic
effect, or quadratic magneto-optic effect, causes a quadratic dependence of ϵ on H 0 or M 0 .
We first consider the magneto-optic effects in a material that has no spontaneous magnetization, i.e., a diamagnetic material, a magnetically disordered paramagnetic material, or an antiferromagnetic material. In such a material, operation of the time-reversal transformation yields
ϵ ij ðω; H 0 Þ ¼ ϵ ji ðω; H0 Þ
(2.72)
when the material is subject to an external magnetic field H 0 . This relation is generally true
regardless of the symmetry property of the material. If the material is lossless, then its dielectric
permittivity tensor is Hermitian:
ϵ ij ðω; H 0 Þ ¼ ϵ ∗
ji ðω; H 0 Þ:
(2.73)
If we express the real and imaginary parts of ϵ explicitly by writing ϵ ij ¼ ϵ 0ij þ iϵ 00ij , we find by
combining these two relations that
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2.6 External Factors
51
ϵ 0ij ðω; H 0 Þ ¼ ϵ 0ij ðω; H 0 Þ ¼ ϵ 0ji ðω; H 0 Þ ¼ ϵ 0ji ðω; H 0 Þ,
(2.74)
ϵ 00ij ðω; H 0 Þ ¼ ϵ 00ij ðω; H0 Þ ¼ ϵ 00ji ðω; H 0 Þ ¼ ϵ 00ji ðω; H 0 Þ:
(2.75)
As a result, the magneto-optic effects in a lossless material that has no spontaneous magnetization can be generally described as
X
X
f ijk H 0k þ ϵ 0
cijkl H 0k H 0l þ ,
(2.76)
ϵ ij ðH 0 Þ ¼ ϵ ij þ Δϵ ij ðH 0 Þ ¼ ϵ ij þ iϵ 0
k
k, l
where f ijk and cijkl are real quantities that satisfy the following relations:
f ijk ¼ f jik ,
cijkl ¼ cjikl ¼ cijlk ¼ cjilk :
(2.77)
Because magnetic fields have transformation symmetry properties that are very different
from those of electric fields, magneto-optic effects also have properties very different from
those of electro-optic effects.
1. Because a magnetic field does not change sign under space inversion, the linear magnetooptic effect does not vanish, thus f ijk 6¼ 0, in a centrosymmetric material. By comparison, the
linear electro-optic effect vanishes, thus r ijk ¼ 0, in a centrosymmetric material because an
electric field changes sign under space inversion.
2. Because a magnetic field changes sign under time reversal, the linear magneto-optic effect is
nonreciprocal, thus f ijk ¼ f jik . By comparison, the linear electro-optic effect is reciprocal,
thus rijk ¼ r jik , because an electric field does not change sign under time reversal.
3. Because the product of two electric field components, E 0k E 0l , and the product of two
magnetic field components, H 0k H 0l , both do not change sign under space inversion or time
reversal, the quadratic electro-optic effect and the quadratic magneto-optic effect both exist
in centrosymmetric materials and are both reciprocal, thus sijkl ¼ sjikl ¼ sijlk ¼ sjilk and
cijkl ¼ cjikl ¼ cijlk ¼ cjilk .
4. Both linear and quadratic magneto-optic effects exist in all materials, i.e., f ijk 6¼ 0 and
cijkl 6¼ 0 in all materials, including all solids, liquids, and gases.
5. When a magnetically induced optical loss exists in the linear magneto-optic effect, f ijk
becomes complex with an imaginary part that characterizes the loss. When it exists in the
quadratic magneto-optic effect, cijkl becomes complex with an imaginary part that characterizes the loss.
The magneto-optic effects in magnetically ordered crystals have the same general properties as
discussed above, but their details can be rather complicated due to the magnetic symmetry
properties of such crystals.
In reality, the magneto-optic effects are relatively weak in comparison to, and tend to be
obscured by, any natural or structural birefringence that might exist in a material. Fortunately,
both first- and second-order magneto-optic effects exist in nonbirefringent materials, which
have isotropic linear optical properties, including noncrystals and cubic crystals. For these
reasons, materials of particular interest and practical importance for magneto-optic effects and
their applications are those in which birefringence originating from other effects, such as
material anisotropy or inhomogeneity, does not exist or, if it exists, does not dominate the
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52
Optical Properties of Materials
particular magneto-optic effect of interest. Such materials include isotropic materials and, in
some cases, uniaxial crystals subject to a magnetic field or a magnetization that is parallel to the
optical axis. For magneto-optic effects in these materials, we can take the direction of H 0 or M 0
to be the z direction without loss of generality, i.e., H0 ¼ H 0z^z or M 0 ¼ M 0z^z . Then, ϵ ðH 0 Þ or
ϵ ðM 0 Þ can be generally expressed in the form of (2.16):
0
n2⊥
ϵ ðH 0 Þ or ϵ ðM 0 Þ ¼ ϵ 0 @ iξ
0
iξ
n2⊥
0
1
0
0 A,
n2k
(2.78)
where ξ represents the first-order effect, and n2⊥ and n2k account for the second-order effect. In
the case of ϵ ðH 0 Þ, ξ ¼ f 123 H 0z , n2⊥ ¼ n2o þ c1133 H 20z ¼ n2o þ c2233 H 20z , and n2k ¼ n2o þ c3333 H 20z .
In the case of ϵ ðM 0 Þ, ξ is linearly proportional to M 0z with the symmetry of ξ ðM 0z Þ ¼
ξ ðM 0z Þ, and n2⊥ and n2k are functions of M 20z .
The linear dependence of ϵ ij ðH 0 Þ on the magnetic field, or that of ϵ ij ðM 0 Þ on the magnetization, appears only as antisymmetric components in the off-diagonal elements of the permittivity
tensor. In the absence of a magnetically induced optical loss, these off-diagonal elements are
purely imaginary; then this first-order magneto-optic effect results in a magnetically induced
circular birefringence, discussed in Section 2.2. When this first-order magneto-optic effect
induces an optical loss, these off-diagonal elements become complex, resulting in a magnetically induced circular dichroism, also discussed in Section 2.2. The linear magneto-optic effect
has two notable phenomena: the Faraday effect and the magneto-optic Kerr effect. The Faraday
effect is manifested in the propagation and transmission of an optical wave through a material
subject to a magnetic field or a magnetization; the magneto-optic Kerr effect is manifested in
the reflection of an optical wave from the surface of such a material. The first-order magnetooptic effect and these phenomena resulting from it are nonreciprocal.
By contrast, the quadratic dependence on the magnetic field or the magnetization appears as
symmetric components in the permittivity tensor elements. This second-order magneto-optic
effect is reciprocal and is called the Cotton–Mouton effect. In the absence of a magnetically
induced optical loss, it causes a magnetically induced linear birefringence in the material and is
analogous to, but much weaker than, the electro-optic Kerr effect. When this second-order
magneto-optic effect causes an optical loss, the symmetric permittivity tensor elements are
complex, resulting in a magnetically induced linear dichroism.
2.6.3
Acousto-optic Effect
An acoustic wave in a medium is an elastic wave of space- and time-dependent periodic
deformation in the medium. A traveling plane acoustic wave of a wavelength Λ ¼ 2π=K and
a frequency f ¼ Ω=2π can be expressed as
uðr; t Þ ¼ U cos ðK r Ωt Þ,
(2.79)
where U is the amplitude vector of the deformation that defines the polarization of the acoustic
^ is the acoustic wavevector
wave, Ω is the angular frequency of the acoustic wave, and K ¼ K K
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2.6 External Factors
53
^ describes the propagation direction and K ¼ 2π=Λ ¼ Ω=v a is the propagation constant
where K
with v a being the acoustic velocity. A standing plane acoustic wave is a combination of two
contrapropagating traveling waves of equal amplitude, wavelength, and frequency:
uðr; tÞ ¼ U cos ðK rÞ cos Ωt:
(2.80)
An acoustic wave polarized in the direction of K is known as a longitudinal wave, while one
with a polarization perpendicular to K is called a transverse wave. For any given direction of
propagation in a medium, there are three orthogonal acoustic normal modes of polarization: one
longitudinal or quasi-longitudinal mode, and two transverse or quasi-transverse modes.
The mechanical strains associated with deformation are described by a symmetric strain
tensor, S ¼ Sij , defined by
1 ∂ui ∂uj
,
(2.81)
þ
Sij ¼
2 ∂xj ∂xi
where the indices i, j ¼ x, y, z. The three tensor elements Sxx , Syy , and Szz are tensile strains,
while the other elements Syz ¼ Szy , Szx ¼ Sxz , and Sxy ¼ Syx are shear strains. In addition, there
is an antisymmetric rotation tensor, R ¼ Rij , defined by
1 ∂ui ∂uj
Rij ¼
:
(2.82)
2 ∂xj ∂xi
Clearly, Rxx ¼ Ryy ¼ Rzz ¼ 0, while Ryz ¼ Rzy , Rzx ¼ Rxz , and Rxy ¼ Ryx . For elastic
deformation caused by an acoustic wave, all of the strain and rotation tensor elements are
space- and time-dependent quantities.
Mechanical strain in a medium causes changes in the optical property of the medium due to
the photoelastic effect. The basis of acousto-optic interaction is the dynamic photoelastic effect
in which the periodic time-dependent mechanical strain and rotation caused by an acoustic
wave induce periodic time-dependent variations in the optical properties of the medium. The
photoelastic effect is traditionally defined in terms of changes in the elements of the relative
impermeability tensor:
X
ηij ðS; RÞ ¼ ηij þ Δηij ðS; RÞ ¼ ηij þ
(2.83)
pijkl Skl þ p0ijkl Rkl ,
k, l
where pijkl are dimensionless elasto-optic coefficients, also called strain-optic coefficients or
photoelastic coefficients, and p0ijkl are dimensionless rotation-optic coefficients. Both are fourth
order tensors. Because ηij ¼ ηji and Skl ¼ Slk , the pijkl tensor is symmetric in i and j and in k
and l. Because ηij ¼ ηji and Rkl ¼ Rlk , the ½p0ijkl tensor is symmetric in i and j but is
antisymmetric in k and l.
The photoelastic effect exists in all matter, including centrosymmetric crystals and isotropic
materials, because the pijkl tensor never vanishes in any material though the ½ p0ijkl tensor
vanishes in isotropic materials and cubic crystals. Acousto-optic interactions are not precluded
by any symmetry property of a material. The tensor form of pijkl for a crystal is determined by the
point group of the crystal. The ½p0ijkl tensor elements of a crystal are determined by the birefringence of the crystal. If the indices i, j, k, are l referenced to the principal axes of a crystal, we have
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54
Optical Properties of Materials
p0ijkl
!
1 1
1 ¼
δ
δ
δ
,
δ
ik
jl
il
jk
2 n2i n2j
(2.84)
where ni and nj represent the principal indices of refraction of the crystal. It is clear that p0ijkl
vanishes in an isotropic material or a cubic crystal.
It is desirable to formally express the photoelastic effect caused by strain and rotation in a
medium in terms of a change in the permittivity of the medium as
ϵ ðω; S; RÞ ¼ ϵ ðωÞ þ Δϵ ðω; S; RÞ ¼ ϵ ðωÞ þ ϵ 0 Δχðω; S; RÞ,
(2.85)
where ϵ ðωÞ is the dielectric permittivity tensor of the medium in the absence of strain and
rotation fields. Using (2.62), the elements of Δϵ can be found from those of Δη in (2.83):
X
Δϵ ij ¼ ϵ 0 n2i n2j Δηij ¼ ϵ 0 n2i n2j
pijkl Skl þ p0ijkl Rkl ,
(2.86)
k, l
where for an acoustic wave, Skl and Rkl are functions of space and time. For a traveling wave
characterized by a wavevector of K and a frequency of Ω as described by (2.79), Skl and Rkl can
be found by using (2.81) and (2.82), respectively. They have the form:
Skl ¼ S kl sin ðK r ΩtÞ, Rkl ¼ Rkl sin ðK r ΩtÞ,
(2.87)
where S kl is the amplitude of the strain and Rkl is the amplitude of the rotation. Therefore, the
photoelastic permittivity tensor is a function of space and time:
Δϵ ¼ Δe
ϵ sin ðK r Ωt Þ,
where Δe
ϵ is the amplitude of Δϵ, and its elements are
X
Δe
ϵ ij ¼ ϵ 0 n2i n2j
pijkl S kl þ p0ijkl Rkl :
k, l
(2.88)
(2.89)
EXAMPLE 2.8
Silica glass is an isotropic material. An acoustic wave propagating in any direction in silica
glass can have two transverse modes and one longitudinal mode. The two transverse modes
have the same acoustic wave velocity of v Ta ¼ 5:97 km s1 , whereas the longitudinal mode has
an acoustic wave velocity of v La ¼ 3:76 km s1 . Take the acoustic wave propagation direction
to be the z direction. How does each mode of an acoustic wave at a frequency of 500 MHz
modulate the optical permittivity in space and time?
Solution:
All three modes modulate the optical permittivity at the same frequency of f ¼ 500 MHz,
thus Ω ¼ 1 109 π rad s1 , in time, but they modulate the optical permittivity differently
in space. Because the wave propagates in the z direction, the longitudinal mode is polarized
in the z direction while the two transverse modes are polarized in the x and y directions,
respectively.
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2.7 Nonlinear Optical Susceptibilities
55
For the longitudinal mode, v La ¼ 3:76 km s1 . Thus,
ΛL ¼
v La 3:76 103
2π
m ¼ 7:52 μm and K L ¼
¼ 8:36 105 m1 :
¼
6
f
Λ
500 10
L
The wavevector of the longitudinal mode is K ¼ K L^z . The optical permittivity that is modulated by the longitudinal acoustic wave varies in space and time with K L ¼ 8:36 105 m1 and
Ω ¼ 1 109 π rad s1 as
Δϵ ðz; t Þ ¼ Δe
ϵ sin ðK L z Ωt Þ:
For both transverse modes, v Ta ¼ 5:97 km s1 . Thus,
ΛT ¼
v Ta 5:97 103
2π
m ¼ 11:94 μm and K T ¼
¼ 5:26 105 m1 :
¼
6
f
ΛT
500 10
The wavevectors of both transverse modes are K ¼ K T^z . The optical permittivity that is
modulated by either of the transverse acoustic waves varies in space and time with K T ¼
5:26 105 m1 and Ω ¼ 1 109 π rad s1 as
Δϵ ðz; t Þ ¼ Δe
ϵ sin ðK T z Ωt Þ:
The permittivity tensor Δe
ϵ is a constant that does not vary with space or time, but it has different
forms for different acoustic modes.
2.7
NONLINEAR OPTICAL SUSCEPTIBILITIES
..............................................................................................................
The origin of optical nonlinearity is the nonlinear response of electrons in a material to an
optical field as the strength of the field is increased. Macroscopically, the nonlinear optical
response of a material is described by a polarization that is a nonlinear function of the optical
field. In general, such nonlinear dependence on the optical field can take a variety of forms. In
particular, it can be very complicated when the optical field becomes extremely strong.
In most situations of interest, with the exception of saturable absorption, the perturbation
method can be used to expand the total optical polarization in terms of a series of linear and
nonlinear polarizations:
Pðr; t Þ ¼ Pð1Þ ðr; tÞ þ Pð2Þ ðr; t Þ þ Pð3Þ ðr; t Þ þ ,
(2.90)
where Pð1Þ is the linear polarization, and Pð2Þ and Pð3Þ are the second- and third-order nonlinear
polarizations, respectively. Except in some special cases, nonlinear polarizations of the fourth
and higher orders are usually not important and thus can be ignored. Note that the space- and
time-dependent polarizations in (2.90) are complex polarizations defined with respect to the
corresponding real polarizations according to the definition of the complex field in (1.40):
PðnÞ ðr; t Þ ¼ PðnÞ ðr; t Þ þ PðnÞ∗ ðr; tÞ ¼ PðnÞ ðr; t Þ þ c:c:,
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(2.91)
56
Optical Properties of Materials
where PðnÞ ðr; t Þ is the nth-order real nonlinear polarization and PðnÞ ðr; t Þ is the nth-order
complex polarization.
The optical field involved in a nonlinear interaction usually contains multiple, distinct
frequency components. Such a field can be expanded in terms of its frequency components:
X
X
Eðr; t Þ ¼
Eq ðrÞ exp iωq t ¼
E q ðrÞ exp ikq r iωq t ,
(2.92)
q
q
where E q ðrÞ is the slowly varying amplitude and kq is the wavevector of the ωq frequency
component. The nonlinear polarizations also contain multiple frequency components and can
be expanded as
X
PðnÞ ðr; t Þ ¼
PðqnÞ ðrÞ exp iωq t :
(2.93)
q
Note that we do not attempt to further express PðqnÞ ðrÞ in terms of a slowly varying polarization
amplitude multiplied by a fast varying spatial phase term, as is done for Eq ðrÞ. The reason is
that the wavevector that characterizes the fast varying spatial phase of a nonlinear polarization
PðqnÞ ðrÞ is not simply determined by the frequency ωq but is dictated by the fields that generate
the nonlinear polarization. In the discussion of nonlinear polarizations, we also use the
notations E ωq and PðnÞ ωq defined respectively as
E ωq ¼ Eq ðrÞ and PðnÞ ωq ¼ PðqnÞ ðrÞ:
(2.94)
Field and polarization components of negative frequencies are interpreted as
E ωq ¼ E∗ ωq and PðnÞ ωq ¼ PðnÞ∗ ωq :
(2.95)
The frequency-domain nth-order nonlinear susceptibility characterizing the nonlinear
response of a material to optical fields at frequencies ω1 , ω2 , . . . , ωn is a function of these
optical frequencies: χðnÞ ðω1 ; ω2 ; ; ωn Þ. In the momentum space and frequency domain, the
nonlinear susceptibility is also a function of wavevectors: χðnÞ ðk1 ; ω1 ; k2 ; ω2 ; ; kn ; ωn Þ. The
reality condition discussed in Section 2.1 and expressed explicitly in (2.7) for the linear
susceptibility can be generalized for each nonlinear susceptibility. This reality condition leads
to the following relation for a nonlinear susceptibility:
χðnÞ∗ ðk1 ; ω1 ; k2 ; ω2 ; ; kn ; ωn Þ ¼ χðnÞ ðk1 ; ω1 ; k2 ; ω2 ; ; kn ; ωn Þ:
(2.96)
When expressing the nonlinear polarization that is generated at a frequency of ωq ¼ ω1 þ ω2 þ
þ ωn by the nonlinear optical interaction of the optical fields at frequencies ω1 , ω2 , . . . , ωn ,
the following notation for the nonlinear susceptibility is used:
χðnÞ ωq ¼ ω1 þ ω2 þ þ ωn ¼ χðnÞ ðω1 ; ω2 ; ; ωn Þ:
(2.97)
Using the expansions of the complex fields and polarizations in (2.92) and (2.93), we have
the expressions for the second- and third-order nonlinear polarizations:
X
Pð2Þ ωq ¼ ϵ 0
χð2Þ ωq ¼ ωm þ ωn : Eðωm ÞEðωn Þ
m, n
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(2.98)
57
2.7 Nonlinear Optical Susceptibilities
and
X
χð3Þ ωq ¼ ωm þ ωn þ ωp : Eðωm ÞEðωn ÞE ωp :
Pð3Þ ωq ¼ ϵ 0
m, n, p
(2.99)
The summation is performed for a given ωq over all positive and negative values of frequencies
that satisfy the constraint of ωm þ ωn ¼ ωq in the case of (2.98) and the constraint of ωm þ
ωn þ ωp ¼ ωq in the case of (2.99). More explicitly, by performing the summation over
positive frequencies only and by expanding the tensor products, we have
X X h ð2Þ ð2Þ χ ijk ωq ¼ ωm þ ωn E j ðωm ÞE k ðωn Þ
Pi ωq ¼ ϵ 0
j, k ωm , ωn >0
ð2 Þ þ χ ijk ωq ¼ ωm ωn E j ðωm ÞE ∗
k ðωn Þ
i
ð2Þ þχ ijk ωq ¼ ωm þ ωn E ∗
ð
ω
ÞE
ð
ω
Þ
(2.100)
m
k
n
j
and
ð3Þ Pi
X X h ð3Þ ωq ¼ ϵ 0
χ ijkl ωq ¼ ωm þ ωn þ ωp Ej ðωm ÞE k ðωn ÞE l ωp
j, k , l ωm , ωn , ωp >0
ð3Þ þ χ ijkl ωq ¼ ωm þ ωn ωp Ej ðωm ÞE k ðωn ÞE ∗
l ωp
ð3Þ þ χ ijkl ωq ¼ ωm ωn þ ωp Ej ðωm ÞE ∗
k ðωn ÞE l ωp
ð3Þ þ χ ijkl ωq ¼ ωm þ ωn þ ωp E ∗
j ðωm ÞE k ðωn ÞE l ωp
ð3Þ ∗
þ χ ijkl ωq ¼ ωm ωn ωp Ej ðωm ÞE ∗
k ðωn ÞE l ωp
ð3Þ ∗
þ χ ijkl ωq ¼ ωm þ ωn ωp E ∗
j ðωm ÞE k ðωn ÞE l ωp
i
ð3Þ ∗
ð
ω
ÞE
ð
ω
ÞE
ωp :
þχ ijkl ωq ¼ ωm ωn þ ωp E ∗
m
n
l
j
k
(2.101)
Usually only a limited number of frequencies participate in a given nonlinear optical interaction. Consequently, only one or a few terms among those listed in (2.100) or (2.101)
contribute to a particular nonlinear polarization.
EXAMPLE 2.9
Three optical fields at the wavelengths of λ1 ¼ 300 nm, λ2 ¼ 750 nm, and λ3 ¼ 1500 nm,
corresponding to the frequencies of ω1 ¼ 2πc=λ1 , ω2 ¼ 2πc=λ2 , and ω3 ¼ 2πc=λ3 , respectively, are involved in second-order nonlinear
optical interactions. The optical fields at the three
pffiffiffi
frequencies are E ðω1 Þ ¼ E 1 ð^x þ ^y Þ= 2, E ðω2 Þ ¼ E 2^z , and Eðω3 Þ ¼ E 3^z , where ^x , ^y , and ^z are
the x, y, and z principal axes of the nonlinear crystal. Find the nonlinear polarization Pð2Þ ðω4 Þ at
the frequency of ω4 ¼ 2πc=λ4 where λ4 ¼ 375 nm. Express the components of Pð2Þ ðω4 Þ
explicitly in terms of the elements of χð2Þ and the given magnitudes, E 1 , E2 , and E3 , of the
three optical fields.
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58
Optical Properties of Materials
Solution:
1
1
1
1
Because λ1
1 λ3 ¼ λ2 þ λ2 ¼ λ4 , we find that ω4 ¼ ω1 ω3 ¼ ω2 þ ω2 . Therefore, the
second-order nonlinear polarization at the frequency ω4 is
Pð2Þ ðω4 Þ ¼ ϵ 0 χð2Þ ðω4 ¼ ω1 ω3 Þ : Eðω1 ÞE∗ ðω3 Þ þ χð2Þ ðω4 ¼ ω3 þ ω1 Þ : E∗ ðω3 ÞEðω1 Þ
i
þχð2Þ ðω4 ¼ ω2 þ ω2 Þ : Eðω2 ÞEðω2 Þ :
Note that there are two terms from the mixing of ω1 and ω3 because of permutation, but there is
only one term from ω2 mixing with itself. Using the given fields at the three frequencies, we can
express the components of Pð2Þ ðω4 Þ as
E1 E∗
E1 E∗
2Þ
2Þ
ðω4 ¼ ω1 ω3 Þ pffiffiffi3 þ χ ðxyz
ðω4 ¼ ω1 ω3 Þ pffiffiffi3
Pðx2Þ ðω4 Þ ¼ ϵ 0 χ ðxxz
2
2
E∗
E∗
ð2Þ
ð2Þ
3 E1
3 E1
ffiffiffi
þ χ xzx ðω4 ¼ ω3 þ ω1 Þ pffiffiffi þ χ xzy ðω4 ¼ ω3 þ ω1 Þ p
2
i 2
ð2Þ
2
þχ xzz ðω4 ¼ ω2 þ ω2 ÞE 2 ,
E1 E∗
E1 E∗
2Þ
2Þ
ðω4 ¼ ω1 ω3 Þ pffiffiffi3 þ χ ðyyz
ðω4 ¼ ω1 ω3 Þ pffiffiffi3
Pðy2Þ ðω4 Þ ¼ ϵ 0 χ ðyxz
2
2
∗
E
E
E∗
2Þ
2Þ
3 1
3 E1
ffiffiffi þ χ ðyzy
ffiffiffi
ðω4 ¼ ω3 þ ω1 Þ p
ðω4 ¼ ω3 þ ω1 Þ p
þ χ ðyzx
2
2
i
ð2Þ
2
þχ yzz ðω4 ¼ ω2 þ ω2 ÞE 2 ,
E1 E∗
E1 E∗
2Þ
2Þ
Pðz2Þ ðω4 Þ ¼ ϵ 0 χ ðzxz
ðω4 ¼ ω1 ω3 Þ pffiffiffi3 þ χ ðzyz
ðω4 ¼ ω1 ω3 Þ pffiffiffi3
2
2
∗
E3 E1
E∗
2Þ
2Þ
3 E1
ffiffiffi þ χ ðzzy
ffiffiffi
þ χ ðzzx
ðω4 ¼ ω3 þ ω1 Þ p
ðω4 ¼ ω3 þ ω1 Þ p
2
2
i
2Þ
þχ ðzzz
ðω4 ¼ ω2 þ ω2 ÞE 22 :
As discussed in Section 2.2, the form of the linear susceptibility tensor is determined by the
symmetry property of the material. The forms of the nonlinear susceptibility tensors of a
material also reflect the spatial symmetry property of the material structure. As a result, some
elements in a nonlinear susceptibility tensor may be zero and others may be related in one
way or another, greatly reducing the total number of independent tensor elements. The linear
susceptibility tensor has its form determined only by the crystal system of a material, whereas
the form of a nonlinear susceptibility tensor further depends on the point group of the
material.
Within the 7 crystal systems, there are 32 point groups. Among the 32 point groups, 21 are
noncentrosymmetric and 11 are centrosymmetric. All gases, liquids, and amorphous solids are
centrosymmetric. Centrosymmetric materials possess space-inversion symmetry. In the electricdipole approximation, nonlinear optical effects of all even orders, but not those of the odd
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2.7 Nonlinear Optical Susceptibilities
59
orders, vanish identically in a centrosymmetric material. Therefore, χð2Þ contributed by electricdipole interaction is identically zero in a centrosymmetric material, whereas a nonzero χð3Þ
exists in any material. This fact can be easily verified by considering the effect of space
inversion on the nonlinear polarizations Pð2Þ and Pð3Þ given in (2.98) and (2.99), respectively.
The space-inversion transformation can be performed on a centrosymmetric material without
changing the properties of the material. Being polar vectors, Pð2Þ , Pð3Þ , and E all change sign
under such a transformation. From (2.98), we then find that Pð2Þ ¼ Pð2Þ . Therefore, Pð2Þ
cannot exist and χð2Þ has to vanish identically in a centrosymmetric material. No such conclusion is drawn for Pð3Þ or χð3Þ as we examine (2.99) following the same procedure. Comparing
the above discussion with that in Section 2.6 for the Pockels coefficients r ijk , which vanish in
centrosymmetric materials, and the electro-optic Kerr coefficients sijkl , which exist in any
material, we find the similarity between the χð2Þ and r ijk , and that between χð3Þ and sijkl . Indeed,
they are directly related:
r ijk ¼ 2 ð2Þ
χ ðω
2
ni n2j ijk
¼ ω þ 0Þ ¼ 2 ð2 Þ
χ ð0
2
ni n2j kij
¼ ω ωÞ
(2.102)
and
sijkl ¼ 3 ð3Þ
χ ðω
2
ni n2j ijkl
¼ ω þ 0 þ 0Þ:
(2.103)
EXAMPLE 2.10
The BBO crystal structure belongs to the 3m point group, for which the only nonvanishing
2Þ
2Þ
2Þ
2Þ
2Þ
2Þ
2Þ
2Þ
2Þ
2Þ
2Þ
χð2Þ elements are χ ðxzx
¼ χ ðyzy
, χ ðxxz
¼ χ ðyyz
, χ ðyyy
¼ χ ðyxx
¼ χ ðxxy
¼ χ ðxyx
, χ ðzxx
¼ χ ðzyy
, and χ ðzzz
.
If the nonlinear interaction considered in Example 2.9 takes place in a BBO crystal, what
are the expressions of the components of Pð2Þ ðω4 Þ in terms of the nonvanishing elements
of χð2Þ ?
Solution:
By keeping the terms that contain only the nonvanishing χð2Þ elements in each of the components of Pð2Þ ðω4 Þ obtained in Example 2.9, we find that
E1 E∗
E∗
2Þ
2Þ
3 E1
ffiffiffi ,
Pðx2Þ ðω4 Þ ¼ ϵ 0 χ ðxxz
ðω4 ¼ ω1 ω3 Þ pffiffiffi3 þ χ ðxzx
ðω4 ¼ ω3 þ ω1 Þ p
2
2
E1 E∗
E∗
2Þ
2Þ
3 E1
ffiffiffi ,
Pðy2Þ ðω4 Þ ¼ ϵ 0 χ ðyyz
ðω4 ¼ ω1 ω3 Þ pffiffiffi3 þ χ ðyzy
ðω4 ¼ ω3 þ ω1 Þ p
2
2
2Þ
ðω4 ¼ ω2 þ ω2 ÞE22 :
Pðz2Þ ðω4 Þ ¼ ϵ 0 χ ðzzz
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60
Optical Properties of Materials
Problems
2.1.1 Verify the relations given in (2.7) that are required by the reality condition.
2.2.1 At a given optical frequency, the optical susceptibility tensors of several materials are
measured with respect to an arbitrary rectilinear coordinate system in space, as listed
below. Identify each material as (1) a dielectric or magnetic material and (2) an optically
lossless or lossy material.
0
1
0
1
2:3
0:1 þ i0:2
0
2:0 þ i0:1 i0:3
0
1 þ i0:2 0 A;
A : χ ¼ @ 0:1 þ i0:2
2:7
i0:2 A; B : χ ¼ @ i0:3
0
i0:2
2:4
0
0
1:5
0
1
0
1
1:59 0:13 0:16
1:9 0:2
0:3
@
A
@
A;
C:χ¼
0:13 1:59 0:11 ; D : χ ¼ 0:2 2:8
0:1
0:16 0:11 1:41
0:3 0:1 2:5 þ i0:2
0
1
1:30 i0:35
0
E : χ ¼ @ i0:35
1:25
0:15 A:
0
0:15
1:40
2.2.2 Represented in an arbitrarily chosen right-handed Cartesian coordinate system defined by
the unit vectors ^x 1 , ^x 2 , and ^x 3 , with ^x 1 ^x 2 ¼ ^x 3 , the permittivity tensor of a crystal at
λ ¼ 0:50 μm is
0
1
5:481
0
0
ϵ ¼ ϵ0@ 0
5:267 0:214 A:
0
0:214 5:267
(a) Find the principal indices and the corresponding principal axes of the crystal at this
wavelength.
(b) Is this crystal birefringent or nonbirefringent? If it is birefringent, is it uniaxial or
biaxial? If it is uniaxial, is it positive or negative uniaxial?
2.2.3 At the λ ¼ 1:300 μm optical wavelength, the permittivity tensor of a LiNbO3 crystal
represented in an arbitrarily chosen ðx1 ; x2 ; x3 Þ rectilinear coordinate system with
^x 1 ^x 2 ¼ ^x 3 is found to be
0
1
4:938
0
0
4:770 0:168 A:
ϵ ¼ ϵ0@ 0
0
0:168 4:770
(a) Find the principal indices and the corresponding principal axes of the LiNbO3 crystal
at this wavelength.
(b) Is it possible to send an optical wave at this wavelength through a LiNbO3 crystal of
arbitrary thickness in such a manner that the polarization of the wave is maintained
throughout its path no matter how the wave is initially polarized? If the answer is yes,
how can this be arranged? If the answer is no, why is it not possible?
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Problems
61
2.2.4 Represented in an arbitrarily chosen right-handed ðx1 ; x2 ; x3 Þ Cartesian coordinate system
with ^x 1 ^x 2 ¼ ^x 3 , the permittivity tensor of a KTP crystal at λ ¼ 1:0 μm is
0
1
3:035
0
0
3:210 0:147 A:
ϵ ¼ ϵ0@ 0
0
0:147 3:210
(a) Find the principal indices and the corresponding principal axes of the crystal at this
wavelength.
(b) Is the crystal birefringent or nonbirefringent? If it is birefringent, is it uniaxial or
biaxial? If it is uniaxial, is it positive or negative uniaxial?
2.2.5 What is the difference between linear birefringence and circular birefringence?
2.2.6 What is the difference between linear birefringence and linear dichroism? What is the
difference between circular birefringence and circular dichroism?
2.2.7 In a properly chosen xyz Cartesian coordinate system, a particular medium has a
symmetric but non-Hermitian permittivity tensor of the form:
0 2
1
n þ iς iξ þ γ 0
(2.104)
ϵ ¼ ϵ 0 @ iξ þ γ n2 þ iς 0 A,
2
0
0
nz
where n, ς, ξ, and γ are all real and positive numbers with n
ς, ξ, γ. Find the principal
refractive indices and the corresponding principal normal modes of polarization. Show
that this medium is linearly birefringent and linearly dichroic.
2.2.8 In a properly chosen xyz Cartesian coordinate system, a particular medium has an
asymmetric and non-Hermitian permittivity tensor of the form:
0 2
1
n þ iς iξ þ γ 0
(2.105)
ϵ ¼ ϵ 0 @ iξ γ n2 þ iς 0 A,
0
0
n2z
where n, ς, ξ, and γ are all real and positive numbers with n
ς, ξ, γ. Find the principal
refractive indices and the corresponding principal normal modes of polarization. Show
that this medium is circularly birefringent and circularly dichroic.
2.3.1 Lorentz model: The resonant susceptibility given in (2.26) for an atomic system that has a
single resonance frequency at ω0 and a relaxation rate of γ can be obtained using a classical
Lorentz model by considering a one-dimensional damped oscillator for the bound electrons
of this system. The system consists of N oscillating electrons, each of which has a charge of
q ¼ e and an effective mass of m∗ . The displacement of the oscillating electron in
response to the force of an optical field is described by the Lorentz oscillator equation:
d2 x
dx
F
þ 2γ þ ω20 x ¼ ∗ ,
dt2
dt
m
(2.106)
where xðt Þ ¼ xðt Þ^x and FðtÞ ¼ qEðt Þ ¼ eEeiωt þ c:c: ¼ eE^x eiωt þ c:c: The electricdipole polarization due to the electron displacement induced by the optical field is defined as
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62
Optical Properties of Materials
Pðt Þ ¼ NexðtÞ:
(2.107)
The induced electron displacement and the corresponding optical polarization in response
to the optical field at the frequency ω can be expressed as xðt Þ ¼ xðt Þ^x ¼
xðωÞ^x eiωt þ c:c: and PðtÞ ¼ PðωÞ^x eiωt þ c:c:
(a) Solve the Lorentz oscillator equation by using the above expression for xðtÞ to find xðωÞ.
(b) Use the definition of the electric-dipole polarization and the frequency-domain
relation PðωÞ ¼ ϵ 0 χ ðωÞE ðωÞ, as given in (1.59), between the optical polarization
and the optical field to find χ ðωÞ, which is the resonant susceptibility χ res ðω; ω0 Þ.
(c) Compare the result obtained in (b) with the resonant susceptibility given in (2.26) by
taking ΔN ¼ N 2 N 1 N because the atomic system considered here is in the
thermal-equilibrium state so that it is almost all populated in the ground level.
Identify the electric-dipole moment p in (2.26) and express it in terms of the
parameters of the Lorentz oscillator model.
2.3.2 All susceptibilities and permittivities of physical materials have to satisfy the reality
condition given in (2.7).
(a) Show that the real and imaginary parts of the resonant susceptibility given in (2.27)
do not satisfy the reality condition. Explain this apparent issue.
(b) Show that the resonant susceptibility given in (2.26) satisfies the reality condition
before the rotating-wave approximation is applied but not after that.
2.3.3 The absorption spectral line of Yb3þ : Al2 O3 due to the optical transition from the 2 F7=2
ground level to the 2 F5=2 upper level appears at a center wavelength of λ ¼ 974:5 nm with
a FWHM spectral width of Δλ ¼ 7:4 nm. Find the energy separation between the two
levels. Find the resonance frequency and the polarization relaxation rate associated with
this transition. Where is anomalous dispersion caused by this transition found when the
Yb3þ ions are in their normal state in thermal equilibrium with the surrounding?
2.4.1 Drude model: The Drude model considers free-moving electrons or holes that, unlike
bound electrons, do not have resonant oscillation frequencies.
(a) Show that the Drude model given in (2.30) can be obtained by setting ω0 ¼ 0 and
2γ ¼ 1=τ for the Lorentz model in Problem 2.3.1.
(b) Show that χ cond ðωÞ given in (2.43) can be obtained from the expression of χ res ðωÞ
found in Problem 2.3.1 by setting ω0 ¼ 0 and 2γ ¼ 1=τ.
2.4.2 Show that the conduction susceptibility given in (2.43) and its real and imaginary parts
given in (2.44) all satisfy the reality condition.
2.4.3 Aluminum is a good conductor. The free-electron Drude model describes its optical
properties reasonably well with a free electron density of N ¼ 1:81 1029 m3 . The DC
conductivity of Al at T ¼ 273 K is σ ð0Þ ¼ 4:08 107 S m1 . Find the plasma frequency
ωp and the relaxation time τ for Al at T ¼ 273 K. Also find the cutoff optical frequency
νp and the cutoff wavelength λp . For what wavelengths is Al expected to be highly
reflective? For what wavelengths is it expected to become transmissive?
∗
2.4.4 Si has an electron effective mass of m∗
e ¼ 1:08m0 and a hole effective mass of mh ¼
0:56m0 , where m0 is the mass of a free electron. Its low-frequency dielectric constant is
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Problems
63
11.8. Find the plasma frequency, the cutoff frequency, and the cutoff wavelength for
(a) an n-type Si sample that has an electron density of N e ¼ 1 1024 m3 , (b) a p-type Si
sample that has a hole density of N h ¼ 1 1024 m3 , and (c) a Si sample that is injected
with an equal electron and hole density of N e ¼ N h ¼ 1 1024 m3 .
2.5.1 Show that the Kramers–Kronig relations given in (2.53) satisfy the reality condition.
2.5.2 Do the real part χ 0res ðωÞ and the imaginary part χ 00res ðωÞ of the exact χ res ðωÞ given in (2.26)
before making the rotating-wave approximation satisfy the Kramers–Kronig relations?
Do the real and imaginary parts, given in (2.27), of the χ res ðωÞ obtained under the
rotating-wave approximation satisfy the Kramers–Kronig relations?
2.5.3 Do the real part χ 0cond ðωÞ and the imaginary part χ 00cond ðωÞ of the conduction susceptibility
given in (2.44) satisfy the Kramers–Kronig relations?
2.6.1 LiNbO3 is a negative uniaxial crystal with nx ¼ ny ¼ no > nz ¼ ne . Being a crystal of the
3m symmetry group, it has eight nonvanishing Pockels coefficients of four distinct
values: r12 ¼ r 22 , r 13 , r 22 , r 23 ¼ r13 , r 33 , r42 , r 51 ¼ r 42 , and r61 ¼ r 22 . At the 1 μm
optical wavelength, no ¼ 2:238 and ne ¼ 2:159, and the four distinct values of its
Pockels coefficients are r13 ¼ 8:6 pm V1 , r 22 ¼ 3:4 pm V1 , r 33 ¼ 30:8 pm V1 , and
r 42 ¼ 28 pm V1 . Use the results from Example 2.6 to find the new principal axes and
the changes in the principal indices of refraction caused by an electric field of E 0 ¼
5 MV m1 that is applied along the y principal axis.
2.6.2 InP is a cubic crystal of the 43m symmetry group with nx ¼ ny ¼ nz ¼ no and three
nonvanishing Pockels coefficients of the same value: r 41 ¼ r52 ¼ r 63 . At the 1:55 μm
optical wavelength, no ¼ 3:166 and r 41 ¼ 1:6 pm V1 . Because of the symmetry among
the three principal axes, an electric field applied along any principal axis results in a
similar effect. Consider a DC electric field of E0 ¼ 10 MV m1 applied along the z
principal axis. Find the new principal axes and the changes in the principal indices of
refraction caused by the applied field due to the Pockels effect.
2.6.3 KTP is a biaxial crystal of the mm2 symmetry group with nx 6¼ ny 6¼ nz and five
nonvanishing Pockels coefficients of distinct values: r 13 , r 23 , r 33 , r42 , and r 51 . Find the
field-induced permittivity change Δϵ ðE0 Þ for an applied DC electric field of
E0 ¼ E 0x ^x þ E 0y ^y þ E 0z^z .
2.6.4 At the 1 μm optical wavelength, the principal indices of KTP are nx ¼ 1:742, ny ¼ 1:750,
and nz ¼ 1:832; the nonvanishing Pockels coefficients are r 13 ¼ 8:8 pm V1 ,
r 23 ¼ 13:8 pm V1 , r 33 ¼ 35 pm V1 , r 42 ¼ 8:8 pm V1 , and r51 ¼ 6:9 pm V1 . Is it
possible to apply a DC electric field to change the principal indices of refraction through
the Pockels effect without rotating the principal axes? If this is possible, find the changes in
the principal indices of refraction caused by an applied electric field of E 0 ¼ 12 MV m1 .
2.6.5 Magneto-optic effect can lead to circular birefringence and circular dichroism. For
simplicity, consider a material for which the only optical loss is magnetically induced
so that ϵ ij ¼ ϵ ∗
ji in the absence of a magnetic field or a magnetization but
∗
ϵ ij ðH0 Þ 6¼ ϵ ∗
ji ðH 0 Þ in the presence of a magnetic field and ϵ ij ðM 0 Þ 6¼ ϵ ji ðM 0 Þ in the
presence of a magnetization.
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64
Optical Properties of Materials
(a) Show for the case of a magnetic-field-induced loss that the relations in (2.76) and
(2.77) are still valid but f ijk or cijkl , or both, are complex. Thus, the magneto-optic
permittivity tensor given in (2.78) can be generalized to the form:
0
n2⊥ þ iς
ϵ ¼ ϵ 0 @ iξ 0 þ ξ 00
0
iξ 0 ξ 00
n2⊥ þ iς
0
1
0
0 A,
n2k
(2.108)
where ξ 0 ¼ f 0123 H 0z , ξ 00 ¼ f 00123 H 0z , n2⊥ ¼ n2o þ c01234 H 20z , and ς ¼ c001234 H 20z . The same
concept is applicable to a magnetization-induced optical loss for which ξ 0 and ξ 00 are
linearly proportional to M 0z , and n2⊥ and ς are functions of M 20z .
(b) Show that the first-order magneto-optic effect results in circular birefringence and, in
the situation when ξ 00 6¼ 0 with a magnetically induced loss, circular dichroism.
(c) Show, by setting ξ 0 ¼ ξ 00 ¼ 0 to mathematically turn off the first-order magneto-optic
effect, that the second-order magneto-optic effect does not cause circular birefringence, or circular dichroism, but only linear birefringence or linear dichroism.
2.7.1 Three optical fields at the wavelengths of λ1 ¼ 1200 nm, λ2 ¼ 600 nm, and λ3 ¼ 800 nm,
corresponding to the frequencies of ω1 ¼ 2πc=λ1 , ω2 ¼ 2πc=λ2 , and ω3 ¼ 2πc=λ3 ,
respectively, are involved in second-order nonlinear optical interactions. The
pffiffiffioptical
fields at the three frequencies are E ðω1 Þ ¼ E 1 ^x , E ðω2 Þ ¼ E 2 ð^y þ ^z Þ= 2, and
E ðω3 Þ ¼ E 3^z , where ^x , ^y , and ^z are the x, y, and z principal axes of the nonlinear crystal.
(a) Find the nonlinear polarization Pð2Þ ðω4 Þ at the frequency of ω4 ¼ 2πc=λ4 where
λ4 ¼ 400 nm. Express each of the components of Pð2Þ ðω4 Þ explicitly in terms of
the elements of χð2Þ and the given magnitudes, E1 , E2 , and E 3 , of the three optical
fields.
(b) If the nonlinear interaction takes place in a KTP crystal, what are the expressions of
the components of Pð2Þ ðω4 Þ in terms of the nonvanishing elements of χð2Þ ? Note that
KTP belongs to the mm2 point group, for which the only nonvanishing χð2Þ elements
2Þ
2Þ
2Þ
2Þ
2Þ
2Þ
2Þ
are χ ðxzx
, χ ðxxz
, χ ðyyz
, χ ðyzy
, χ ðzxx
, χ ðzyy
, and χ ðzzz
.
2.7.2 Three optical fields at the wavelengths of λ1 ¼ 1200 nm, λ2 ¼ 600 nm, and λ3 ¼ 800 nm,
corresponding to the frequencies of ω1 ¼ 2πc=λ1 , ω2 ¼ 2πc=λ2 , and ω3 ¼ 2πc=λ3 ,
respectively, are involved in second-order nonlinear optical interactions. The
pffiffiffioptical
^
^
fields at the three frequencies are E ðω1 Þ ¼ E 1 x , E ðω2 Þ ¼ E 2 ðy þ ^z Þ= 2, and
E ðω3 Þ ¼ E 3^z , where ^x , ^y , and ^z are the x, y, and z principal axes of the nonlinear crystal.
(a) Find the nonlinear polarization Pð2Þ ðω4 Þ at the frequency of ω4 ¼ 2πc=λ4 where
λ4 ¼ 2400 nm. Express each of the components of Pð2Þ ðω4 Þ explicitly in terms of
the elements of χð2Þ and the given magnitudes, E1 , E2 , and E 3 , of the three optical
fields.
(b) If the nonlinear interaction takes place in a KTP crystal, what are the expressions of
the components of Pð2Þ ðω4 Þ in terms of the nonvanishing elements of χð2Þ ? Note that
KTP belongs to the mm2 point group, for which the only nonvanishing χð2Þ elements
2Þ
2Þ
2Þ
2Þ
2Þ
2Þ
2Þ
are χ ðxzx
, χ ðxxz
, χ ðyyz
, χ ðyzy
, χ ðzxx
, χ ðzyy
, and χ ðzzz
.
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Bibliography
65
2.7.3 Two optical fields at the wavelengths of λ1 ¼ 500 nm and λ2 ¼ 1500 nm, corresponding
to the frequencies of ω1 ¼ 2πc=λ1 and ω2 ¼ 2πc=λ2 , respectively, are involved in
second-order nonlinear optical interactions. The optical fields at the two frequencies are
E ðω1 Þ ¼ E 1 ^x and E ðω2 Þ ¼ E 2 ^y , where ^x , ^y , and ^z are the x, y, and z principal axes of the
nonlinear crystal.
(a) Find the nonlinear polarization Pð2Þ ðω3 Þ at the frequency of ω3 ¼ 2πc=λ3 where
λ3 ¼ 750 nm. Express each of the components of Pð2Þ ðω3 Þ explicitly in terms of
the elements of χð2Þ and the given magnitudes, E 1 and E 2 , of the two optical fields.
(b) If the nonlinear interaction takes place in a LiNbO3 crystal, what are the expressions
of the components of Pð2Þ ðω3 Þ in terms of the nonvanishing elements of χð2Þ ? Note
that LiNbO3 belongs to the 3m point group, for which the only nonvanishing
2Þ
2Þ
2Þ
2Þ
2Þ
2Þ
2Þ
2Þ
2Þ
2Þ
¼ χ ðyzy
, χ ðxxz
¼ χ ðyyz
, χ ðyyy
¼ χ ðyxx
¼ χ ðxxy
¼ χ ðxyx
, χ ðzxx
¼ χ ðzyy
,
χð2Þ elements are χ ðxzx
2Þ
and χ ðzzz
.
Bibliography
Altman, C. and Suchy, K., Reciprocity, Spatial Mapping and Time Reversal in Electromagnetics, 2nd edn.
Dordrecht: Springer, 2001.
Bloembergen, N., Nonlinear Optics, 4th edn. Singapore: World Scientific, 1996.
Born, M. and Wolf, E., Principles of Optics: Electromagnetic Theory of Propagation, Interference and
Diffraction of Light, 7th edn. Cambridge: Cambridge University Press, 1999.
Boyd, R. W., Nonlinear Optics, 3rd edn. Boston, MA: Academic Press, 2008.
Butcher, P. N. and Cotter, D., The Elements of Nonlinear Optics. Cambridge: Cambridge University Press,
1990.
Davis, C. C., Lasers and Electro-Optics: Fundamentals and Engineering, 2nd edn. Cambridge: Cambridge
University Press, 2014.
Fowler, G. R., Introduction to Modern Optics, 2nd edn. New York: Dover, 1975.
Fox, M., Optical Properties of Solids, 2nd edn. Oxford: Oxford University Press, 2010.
Iizuka, K., Elements of Photonics in Free Space and Special Media, Vol. I. New York: Wiley, 2002.
Jackson, J. D., Classical Electrodynamics, 3rd edn. New York: Wiley, 1999.
Korpel, A., Acousto-Optics, 2nd edn. New York: Marcel Dekker, 1997.
Landau, L. D. and Lifshitz, E. M., Electrodynamics of Continuous Media. Oxford: Pergamon, 1960.
Liu, J. M., Photonic Devices. Cambridge: Cambridge University Press, 2005.
Nye, J. F., Physical Properties of Crystals. London: Oxford University Press, 1957.
Post, E. J., Formal Structure of Electromagnetics. Amsterdam: North-Holland, 1962.
Saleh, B. E. A. and Teich, M. C., Fundamentals of Photonics. New York: Wiley, 1991.
Sapriel, J., Acousto-Optics. New York: Wiley, 1979.
Shen, Y. R., The Principles of Nonlinear Optics. New York: Wiley, 1984.
Sugano, S. and Kojima, N., eds., Magneto-Optics. Berlin: Springer, 2000.
Wooten, F., Optical Properties of Solids. New York: Academic Press, 1972.
Zernike, F. and Midwinter, J. E., Applied Nonlinear Optics. New York: Wiley, 1973.
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Cambridge Books Online
http://ebooks.cambridge.org/
Principles of Photonics
Jia-Ming Liu
Book DOI: http://dx.doi.org/10.1017/CBO9781316687109
Online ISBN: 9781316687109
Hardback ISBN: 9781107164284
Chapter
3 - Optical Wave Propagation pp. 66-140
Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.004
Cambridge University Press
3
3.1
Optical Wave Propagation
NORMAL MODES OF PROPAGATION
..............................................................................................................
The propagation of an optical wave is governed by Maxwell’s equations. The propagation
characteristics depend on the optical property and the physical structure of the medium. They
also depend on the makeup of the optical wave, such as its frequency content and its temporal
characteristics. In this chapter, we discuss the basic propagation characteristics of a monochromatic optical wave in three basic categories of medium: an infinite homogeneous
medium, two semi-infinite homogeneous media separated by an interface, and an optical
waveguide defined by a transverse structure. Some basic effects of dispersion and attenuation
on the propagation of an optical wave are discussed in Sections 3.6 and 3.7.
The optical property of a medium at a frequency of ω is fully described by its permittivity
ϵ ðωÞ, which is a tensor for an anisotropic medium but reduces to a scalar for an isotropic
medium. For a homogeneous medium, ϵ ðωÞ is a constant of space; for an optical structure, it is
a function of space variables. Without loss of generality, we designate the z coordinate axis to
be the direction of optical wave propagation in an isotropic medium; thus the longitudinal axis
of an optical waveguide that is fabricated in an isotropic medium is the z axis. For this reason,
ϵ ðωÞ has only transverse spatial variations that are functions of the transverse coordinates,
which are x and y in the rectilinear coordinate system, or ϕ and r in the cylindrical coordinate
system. We use the rectilinear coordinates for our general discussion. The exception is optical
wave propagation in an anisotropic crystal, for which the natural coordinate system is that
defined by its principal axes but an optical wave does not have to propagate along its principal
z axis.
For the following discussion in this section, we consider propagation in an isotropic medium,
which is not necessarily homogeneous in space. The wave propagates in the z direction, and the
possible inhomogeneity characterizing the optical structure is described by a scalar permittivity
ϵ ðx; yÞ, as illustrated in Fig. 3.1. If the medium is homogeneous, then ϵ ðx; yÞ ¼ ϵ is a constant
of space, as shown in Fig. 3.1(a). If the medium is inhomogeneous in only one transverse
dimension, then it has a planar optical structure, such as a planar interface shown in Fig. 3.1(b)
or a planar waveguide shown in Fig. 3.1(c); in these cases, we take the structural variation to be
in the x direction for ϵ ðx; yÞ ¼ ϵ ðxÞ to be independent of the y variable. If structural variations
exist in two dimensions, then the medium has a nonplanar optical structure with ϵ ðx; yÞ being a
function of both x and y, such as the single-core nonplanar waveguide shown in Fig. 3.1(d).
In any event, there is no structural variation in the direction of propagation; therefore, ϵ ðx; yÞ
is never a function of the z variable.
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3.1 Normal Modes of Propagation
67
Figure 3.1 (a) Homogeneous medium. (b) Planar interface. (c) Planar waveguide. (d) Nonplanar waveguide.
The normal modes of propagation for an optical wave in a medium are the characteristic
solutions of Maxwell’s equations subject to the boundary conditions that are defined by the
physical structure of the medium and are fully described by ϵ ðx; yÞ. Each characteristic solution
has an eigenvalue, which gives the propagation constant, and an eigenfunction, which gives the
field pattern of the normal mode. Therefore, each normal mode is defined by a specific
propagation constant β and a pair of specific electric and magnetic mode field profiles E ðx; yÞ
and Hðx; yÞ. It is possible for two or more degenerate normal modes to have the same
propagation constant but different field profiles. By contrast, two normal modes of different
propagation constants cannot share the same field profile. Because electric and magnetic fields
are vectorial fields, a mode field is defined by a specific amplitude and polarization pattern of
E ðx; yÞ and Hðx; yÞ. A mode index ν is used to label a mode when the optical structure supports
multiple normal modes. Therefore, the space- and time-dependent electric and magnetic fields
of a normal mode at a frequency of ω are expressed as
Eν ðr; tÞ ¼ E ν ðx; yÞ exp ðiβν z iωt Þ,
(3.1)
Hν ðr; t Þ ¼ Hν ðx; yÞ exp ðiβν z iωtÞ,
(3.2)
where βν is the propagation constant of the mode. If the cylindrical coordinate system is used,
then the mode fields in (3.1) and (3.2) are expressed as functions of ϕ and r: E ν ðϕ; rÞ and
Hν ðϕ; r Þ.
The characteristic of the mode index ν depends on the transverse boundary conditions
imposed on the mode field. For an optical medium that imposes two-dimensional boundary
conditions in the transverse xy plane, the mode field profiles are functions of two transverse
spatial variables: E ν ðx; yÞ and Hν ðx; yÞ. Therefore, the mode index ν consists of two parameters
for characterizing the variations of the mode fields in these two transverse dimensions. Then ν
represents two mode numbers or symbols: ν ¼ mn. This is the case for an optical structure that
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68
Optical Wave Propagation
provides two-dimensional transverse optical confinement, such as the nonplanar waveguide in
Fig. 3.1(d). Another example is a collimated Gaussian mode in a homogeneous medium, which
has a two-dimensional transverse profile. For an optical medium that imposes boundary conditions in only one transverse direction, such as that in Fig. 3.1(b) or (c), the mode field profiles are
functions of only one transverse spatial variable: E ν ðxÞ and Hν ðxÞ. In this case, the mode index ν
consists of only one parameter for characterizing the variations of the mode fields in the transverse
dimension x. Then ν represents only one mode number or symbol: ν ¼ m. For discrete modes, i.e.,
modes of discrete propagation constants, the mode index numbers are discrete numbers, which
are normally integers. For continuous modes, i.e., modes of continuously distributed propagation
constants, the mode index numbers are continuously distributed numbers.
3.1.1 Mode Types
For an optical structure in an isotropic medium, which is characterized by a spatial permittivity
distribution of scalar ϵ ðx; yÞ, Maxwell’s equations for wave propagation take the form:
— E ¼ μ0
—H¼ϵ
∂H
,
∂t
∂E
:
∂t
(3.3)
(3.4)
For the mode fields of the form of (3.1) and (3.2), these two equations can be expressed in terms
of the components of the mode field profiles as
∂E z
iβE y ¼ iωμ0 Hx ,
∂y
∂E z
¼ iωμ0 Hy ,
∂x
∂E y ∂E x
¼ iωμ0 Hz ,
∂x
∂y
iβE x (3.5)
(3.6)
(3.7)
and
∂Hz
iβHy ¼ iωϵE x ,
∂y
(3.8)
∂Hz
¼ iωϵE y ,
∂x
(3.9)
∂Hy ∂Hx
¼ iωϵE z :
∂x
∂y
(3.10)
iβHx From these equations, the transverse components of the electric and magnetic mode fields can
be expressed in terms of the longitudinal components:
∂E z
∂Hz
k2 β2 E x ¼ iβ
þ iωμ0
,
∂x
∂y
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(3.11)
3.1 Normal Modes of Propagation
69
2
∂E z
∂Hz
iωμ0
,
k β2 E y ¼ iβ
∂y
∂x
(3.12)
2
∂Hz
∂E z
k β2 Hx ¼ iβ
iωϵ
,
∂x
∂y
(3.13)
2
∂Hz
∂E z
k β2 Hy ¼ iβ
þ iωϵ
,
∂y
∂x
(3.14)
k 2 ¼ ω2 μ0 ϵ ðx; yÞ
(3.15)
where
is a function of x and y to account for the transverse spatial inhomogeneity of the structure.
The relations in (3.11)–(3.14) are generally valid for a longitudinally homogeneous structure
of any transverse geometry and any transverse index profile, for which ϵ ðx; yÞ is not a function
of z. In a structure of cylindrical symmetry, such as an optical fiber, the x and y coordinates of
the rectilinear system can be transformed to the ϕ and r coordinates of the cylindrical system for
similar relations. It is clear from (3.11)–(3.14) that once the longitudinal mode field components, E z and Hz , are known, all mode field components can be obtained. Therefore, a normal
mode can be classified based on the characteristics of its longitudinal field components, as
follows.
1.
2.
3.
4.
A transverse electromagnetic mode, or TEM mode, has E z ¼ 0 and Hz ¼ 0.
A transverse electric mode, or TE mode, has E z ¼ 0 and Hz 6¼ 0.
A transverse magnetic mode, or TM mode, has Hz ¼ 0 and E z 6¼ 0.
A hybrid mode has both E z 6¼ 0 and Hz 6¼ 0.
Several comments can be made.
1. Any dielectric optical structure that has an inhomogeneous transverse profile does not
support TEM modes. For such an optical structure, k2 ¼ ω2 μ0 ϵ ðx; yÞ is not a constant of
space but β2 is always a constant; therefore, all field components vanish when E z ¼ 0 and
Hz ¼ 0, as can be seen from (3.11)–(3.14).
2. TEM modes exist in (a) a homogeneous dielectric medium without any conductors, (b) the
outside of a single-conductor transmission line in a homogeneous dielectric medium, and (c)
a waveguide consisting of multiple separate conductors in a homogeneous dielectric
medium. For a TEM mode to exist, (3.11)–(3.14) require that ϵ ðx; yÞ ¼ ϵ be a constant
of space so that k2 ¼ β2 . Therefore, the propagation constant of a TEM mode is simply that
pffiffiffiffiffiffiffi
of the dielectric medium: β ¼ k ¼ ω μ0 ϵ .
3. Only TE and TM modes are allowed in (a) a planar dielectric structure of ϵ ðx; yÞ ¼ ϵ ðxÞ and
(b) the inside of a hollow metallic waveguide.
4. TE and TM modes are allowed but are not the only modes in (a) a planar metallic waveguide
consisting of two parallel plates, which also supports TEM modes, and (b) a nonplanar
dielectric waveguide, which also supports hybrid modes.
5. Hybrid modes are allowed in nonplanar dielectric waveguides, but not in planar dielectric
structures. The HE and EH modes of optical fibers are hybrid modes.
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70
Optical Wave Propagation
6. From the above discussion, planar dielectric optical structures only have TE and TM modes,
whereas nonplanar dielectric optical structures only have TE, TM, and hybrid modes. None
of them have TEM modes.
EXAMPLE 3.1
Find the general relations between the transverse components of the electric field and those of
the magnetic field for (a) a TEM mode, (b) a TE mode, (c) a TM mode, and (d) a hybrid mode.
Solution:
The general relations between the transverse electric-field components, E x and E y , and the transverse
magnetic-field components, Hx and Hy , for each type of mode can be found from (3.5)–(3.10).
(a) TEM modes: For a TEM mode, E z ¼ 0 and Hz ¼ 0. Therefore,
Hx ¼ β
ωϵ
E y ¼ E y,
ωμ0
β
β
ωϵ
Ex ¼
E x:
ωμ0
β
pffiffiffiffiffiffiffi
From these relations, it is always true that β ¼ ω ϵμ0 ¼ k for a TEM mode.
(b) TE modes: For a TE mode, E z ¼ 0 but Hz 6¼ 0. Therefore,
Hy ¼
Hx ¼ β
ωϵ
E y 6¼ E y ,
ωμ0
β
β
ωϵ
E x 6¼
E x:
ωμ0
β
pffiffiffiffiffiffiffi
From these relations, it is always true that β 6¼ ω ϵμ0 for a TE mode.
(c) TM modes: For a TM mode, Hz ¼ 0 but E z 6¼ 0. Therefore,
Hy ¼
Hx ¼ ωϵ
β
E y,
E y 6¼ β
ωμ0
ωϵ
β
E x:
E x 6¼
β
ωμ0
pffiffiffiffiffiffiffi
From these relations, it is always true that β 6¼ ω ϵμ0 for a TM mode.
(d) Hybrid modes: For a hybrid mode, E z 6¼ 0 and Hz 6¼ 0. Therefore,
Hy ¼
Hx 6¼ β
ωϵ
E y 6¼ E y ,
ωμ0
β
β
ωϵ
E x 6¼
E x:
ωμ0
β
pffiffiffiffiffiffiffi
From these relations, it is always true that β 6¼ ω ϵμ0 for a hybrid mode.
Hy 6¼
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3.1 Normal Modes of Propagation
71
3.1.2 Power and Orthonormalization of Modes
The intensity distribution of a normal mode ν projected on a transverse plane, which has a
surface normal of n^ ¼ ^z , is given by
∗
z ¼ E ν H∗
z,
I ν ¼ S ν ^z ¼ Sν þ S∗
ν ^
ν þ E ν Hν ^
(3.16)
which is a function of x and y. The power, Pν , of the mode is obtained by integrating I ν ðx; yÞ
over the entire transverse cross-sectional plane. It can be seen from (3.16) that the longitudinal
components, E z and Hz , of the mode fields do not contribute to the mode intensity or the mode
power. Because different normal modes are orthogonal to each other, the mode fields of a
lossless isotropic structure satisfy the orthogonality relation:
ð∞ ð∞ ∗
^z dxdy ¼ Pν δνμ :
E ν H∗
þ
E
H
ν
μ
μ
(3.17)
∞ ∞
where δνμ is the Kronecker delta function for discrete modes, with ν and μ representing discrete
numbers; but δνμ is the Dirac delta function δðν μÞ for continuous modes, with ν and μ
representing continuous numbers. For a nonplanar structure, ν ¼ mn and μ ¼ m0 n0 ; hence
δνμ ¼ δmm0 δnn0 . For a planar structure, ν ¼ m and μ ¼ m0 ; then, δνμ ¼ δmm0 .
The normal mode fields are normalized according to the following orthonormality relation:
ð∞ ð∞ ^∗ H
^ν H
^∗þE
^ ν ^z dxdy ¼ δνμ :
E
μ
μ
(3.18)
∞ ∞
This orthonormality relation defined in terms of cross products based on the form of the
Poynting vector is valid for all types of modes. Simplified relations in terms of dot products
exist for TE, TM, and TEM modes.
For TE modes, (3.17) can be reduced to
2βν
ωμ0
ð∞ ð∞
TE
Eν E∗
μ dxdy ¼ Pν δνμ :
(3.19)
∞ ∞
Therefore, as an alternative to (3.18), the orthonormality relation among TE modes can also be
written as
2βν
ωμ0
ð∞ ð∞
^ ∗ dxdy ¼ δνμ :
^ν E
E
μ
(3.20)
1
TM
H ν H∗
μ dxdy ¼ Pν δνμ :
ϵ ðx; yÞ
(3.21)
∞ ∞
For TM modes, (3.17) can be reduced to
2βν
ω
ð∞ ð∞
∞ ∞
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72
Optical Wave Propagation
As an alternative to (3.18), the orthonormality relation among TM modes can also be written as
2βν
ω
ð∞ ð∞
∞ ∞
1 ^ ^∗
H ν H μ dxdy ¼ δνμ :
ϵ ðx; yÞ
(3.22)
The simplified relations for TE modes and those for TM modes are both valid for TEM modes
because a TEM mode is both TE and TM. As discussed above, a TEM mode exists only when
ϵ ðx; yÞ ¼ ϵ is a constant of space. Therefore, for TEM modes,
2βν
ωμ0
ð∞ ð∞
Eν E∗
μ dxdy
∞ ∞
2β
¼ ν
ωϵ
ð∞ ð∞
TEM
Hν H∗
δνμ :
μ dxdy ¼ Pν
(3.23)
∞ ∞
There are two equivalent dot-product orthonormality relations among TEM modes:
2βν
ωμ0
ð∞ ð∞
^ ∗ dxdy
^ν E
E
μ
¼ δνμ
∞ ∞
2βν
and
ωϵ
ð∞ ð∞
^ ∗ dxdy ¼ δνμ :
^ ν H
H
μ
(3.24)
∞ ∞
The orthogonality relation in (3.17) and the orthonormality relation in (3.18) indicate that
power cannot be transferred between different normal modes in a linear, lossless structure
of isotropic dielectric medium. For anisotropic or lossy structures, (3.17) and (3.18) do
not apply, neither do the other simplified relations for TE, TM, and TEM modes. The
orthogonality conditions and orthonormality relations for modes of such structures have
other forms.
3.1.3 Mode Expansion
The normal modes are orthogonal and can be normalized with the general orthonormality
relation given in (3.18). They form a basis for linear expansion of any optical field at a
frequency of ω propagating in the optical medium:
X
^ ν ðx; yÞ exp ðiβ z iωt Þ,
Eðr; t Þ ¼
Aν E
(3.25)
ν
ν
Hðr; t Þ ¼
X
ν
^ ν ðx; yÞ exp ðiβν z iωt Þ,
Aν H
(3.26)
where the summation symbol sums over all discrete indices of the discrete modes and
integrates over all continuous indices of the continuous modes. In a linear structure where
the normal modes are defined, these modes propagate independently without exchanging
power. Therefore, the expansion coefficients Aν are constants that are independent of x, y, and z.
According to (3.17) and (3.18), the normal modes are normalized such that the mode power
is simply
P ν ¼ jA ν j2 :
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(3.27)
3.2 Plane-Wave Modes
3.2
73
PLANE-WAVE MODES
..............................................................................................................
A plane wave has wavefronts of infinite parallel planes. As defined in Section 1.7, a
wavefront is the surface of a constant phase, and the wavevector is the gradient of the phase,
which is normal to the wavefront. Therefore, a monochromatic plane wave that propagates
in a homogeneous medium is defined by one constant frequency ω and one constant
wavevector k:
Eðr; tÞ ¼ E exp ðik r iωt Þ,
(3.28)
Hðr; tÞ ¼ H exp ðik r iωt Þ,
(3.29)
where both E and H are constants of space and time. The electric displacement and the magnetic
induction of the plane wave have similar forms: Dðr; t Þ ¼ ϵ Eðr; t Þ ¼ D exp ðik r iωt Þ and
Bðr; tÞ ¼ μ0 Hðr; t Þ ¼ B exp ðik r iωt Þ, where D and B are constants of space and time.
When operating on the fields of a plane wave, the space operator — always yields ik and the
time operator ∂=∂t always yields iω. Therefore, for a plane wave propagating in a homogeneous
medium, the following replacements can be made:
— ! ik,
∂
! iω:
∂t
(3.30)
A monochromatic plane wave is a normal mode of propagation in a homogeneous medium
because it has a well-defined wavevector, thus a well-defined propagation constant. In an
isotropic medium, the propagation constant of a plane wave does not depend on the polarization
of the wave; therefore, a plane wave of any polarization has the same well-defined propagation
constant and is a normal mode. In an anisotropic medium, only fields of certain polarizations
have well-defined propagation constants, as discussed in Section 2.2. Plane-wave normal
modes in a homogeneous anisotropic medium have specific polarization characteristics and
polarization-dependent propagation constants that are determined by both the property of the
medium and the direction of wave propagation.
In any event, for a monochromatic plane-wave normal mode, Maxwell’s equations as given
in (1.41)–(1.44) can be expressed in the algebraic form:
k E ¼ ωμ0 H,
(3.31)
k H ¼ ωD,
(3.32)
k D ¼ 0,
(3.33)
k H ¼ 0:
(3.34)
Note that the relation B ¼ μ0 H, as is always true for optical fields, is used for the above
equations. The wave propagation direction is defined by the wavevector k, whereas the power
flow direction is defined by the Poynting vector from (1.54):
S ¼ E H∗ :
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(3.35)
74
Optical Wave Propagation
By combining (3.31) and (3.32) to eliminate the magnetic field H, the algebraic form of the
wave equation for a plane wave is obtained:
k k E þ ω2 μ0 D ¼ 0:
(3.36)
A plane-wave normal mode is characterized by six vectors: E, D, H, B, k, and S. Their relations
found from (3.31)–(3.35) are summarized as follows.
1. From (3.31) and (3.35), the three vectors E, H, and S are always mutually orthogonal for a
plane wave in any homogeneous medium.
2. From (3.32)–(3.34), the three vectors D, H, and k are always mutually orthogonal for a plane
wave in any homogeneous medium.
3. In any optical medium BkH is always true because B ¼ μ0 H. Both are orthogonal to all of
the other four vectors E, D, k, and S.
4. In a homogeneous isotropic medium, DkE because D ¼ ϵE. Both are orthogonal to all of the
other four vectors H, B, k, and S.
5. In a homogeneous anisotropic medium, D is not necessarily parallel to E because D ¼ ϵ E.
Both D and E are orthogonal to H and B, but E is not necessarily orthogonal to k while D is
not necessarily orthogonal to S.
As expressed in (3.28) and (3.29), a true plane wave transversely extends to infinity in
space, which is unrealistic. It is a good approximation if a medium is homogeneous in all
directions over dimensions that are very large compared to the wavelength. Because the
field amplitude of every plane wave is a constant of space, the difference between two plane
waves of the same frequency that propagate in the same direction is only in their polarization characteristics. Orthogonality between two such plane-wave modes is determined
only by the orthogonality of their polarization states but not by the spatial integral
of their field overlap. Therefore, for a given wave propagation direction, there are only
two orthogonally polarized plane-wave modes. Furthermore, because a plane wave has a
constant amplitude extending throughout the transverse plane, the integrals that define
mode normalization in Section 3.1 cannot be performed. For these reasons, the actual
amplitude of each wave is used in the field expansion though a unit polarization vector is
often used to represent the polarization state of a plane wave. The plane wave basis
for linear expansion of any optical field that has a frequency of ω and propagates in the
k^ direction through a homogeneous optical medium consists of only two orthogonally
polarized elements:
Eðr; t Þ ¼ E1 ðr; t Þ þ E2 ðr; t Þ ¼ E 1 exp iβ1 k^ r iωt þ E 2 exp iβ2 k^ r iωt ,
(3.37)
Hðr; t Þ ¼ H1 ðr; t Þ þ H2 ðr; tÞ ¼ H1 exp iβ1 k^ r iωt þ H2 exp iβ2 k^ r iωt ,
(3.38)
where E 1 , H1 , E 2 , and H2 are constants of space; β1 and β2 are the propagation constants of
the two plane-wave modes; and the two modes satisfy the polarization orthogonality relations:
∗
∗
∗
E1 E∗
2 ¼ E 1 E 2 ¼ 0 and H1 H2 ¼ H1 H2 ¼ 0:
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(3.39)
3.2 Plane-Wave Modes
75
Figure 3.2 Relationships among the
directions of E, D, H, B, k, and S in free
space or in an isotropic medium.
In a homogeneous medium, the propagation constants are determined by the material properties
and the polarization states of the waves but not by any optical structure. Therefore, β1 ¼ k1 and
β2 ¼ k2 . The two propagation constants are the same if the medium is isotropic, but they are
generally different if the medium is anisotropic, as discussed below.
3.2.1 Isotropic Medium
The permittivity tensor of a homogeneous isotropic medium reduces to a scalar ϵ that is
independent of spatial location and direction. Free space is a special case of homogeneous
isotropic medium with ϵ ¼ ϵ 0 . Figure 3.2 shows the relations among the six vectors E, D, H, B,
k, and S of a plane wave that propagates in a homogeneous isotropic medium. For this plane
wave, EkD⊥k because D ¼ ϵE. A plane-wave normal mode of a homogeneous isotropic
medium is a TEM wave because its E and H fields are both orthogonal to its wavevector k.
With E⊥k, we find that k k E ¼ k2 E. By using this relation and D ¼ ϵE, the wave
equation in (3.36) is reduced to
k2 E þ ω2 μ0 ϵE ¼ 0,
(3.40)
which yields the eigenvalue equation:
k 2 ¼ ω2 μ0 ϵ:
(3.41)
Therefore, the propagation constant of the wave in the medium is
pffiffiffiffiffiffiffi nω 2πnν 2πn
k ¼ ω μ0 ϵ ¼
¼
¼
,
c
c
λ
(3.42)
where ν is the frequency of the optical wave, λ is its wavelength,
1
c ¼ pffiffiffiffiffiffiffiffiffi
μ0 ϵ 0
(3.43)
rffiffiffiffiffi
ϵ
¼ ðdielectric constantÞ1=2
n¼
ϵ0
(3.44)
is the speed of light in free space, and
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76
Optical Wave Propagation
is the index of refraction, or refractive index, of the isotropic medium. Because k is proportional
to 1=λ, it is also called the wavenumber. In a medium that has an index of refraction of n, the
optical frequency is still ν, but the optical wavelength is λ=n, and the speed of light is v ¼ c=n.
Regardless of the propagation direction or the polarization state, all plane waves of the same
frequency ω in a homogeneous isotropic medium are degenerate and have the same propagation
^ any two orthogonally polarized
constant k found in (3.42). For any given propagation direction k,
plane waves that propagate in the k^ direction can be used as the basis for linear expansion. Both
^ as is seen in Fig. 3.2. Because
are TEM waves and are orthogonal to the propagation direction k,
the medium is isotropic, the coordinates can be chosen such that the z axis is in the direction of
^ Then the field expansion of (3.37) and (3.38) takes the form:
wave propagation, i.e., ^z ¼ k.
Eðr; tÞ ¼ E 1 exp ðikz iωt Þ þ E 2 exp ðikz iωt Þ ¼ ðE 1 þ E 2 Þ exp ðikz iωtÞ,
(3.45)
Hðr; tÞ ¼ H1 exp ðikz iωtÞ þ H2 exp ðikz iωt Þ ¼ ðH1 þ H2 Þ exp ðikz iωtÞ:
(3.46)
For propagation in the z direction with k^ ¼ ^z as considered here, any two orthogonal polarization
states in the xy plane can be used as the basis set for the field expansion. For example, the basis
set can be formed by the two linearly polarized waves E x ^x and E y ^y , by the two circularly
polarized waves E þ ^e þ and E ^e , or by any two orthogonal elliptically polarized waves. It can
be seen from (3.45) and (3.46) that the linear superposition of two plane-wave normal modes of
a homogeneous isotropic medium is also a normal mode of the same propagation constant.
Hence any plane wave of a given frequency ω traveling in a homogeneous isotropic medium is
a normal mode with the same propagation constant k. This is not true for plane waves traveling
in a homogeneous anisotropic medium, which is discussed below.
EXAMPLE 3.2
GaAs is a cubic crystal. At the λ ¼ 900 nm wavelength, its principal indices of refraction
are nx ¼ ny ¼ nz ¼ 3:593. A circularly polarized wave and a linearly polarized wave at this
wavelength propagate along the z and x principal axes, respectively. What are the propagation
constants and the wavelengths of these two waves in the GaAs crystal?
Solution:
Though GaAs has well-defined principal axes, it is optically isotropic because nx ¼ ny ¼
nz ¼ n. Therefore, a plane wave of any polarization state propagating in any direction
is a normal mode that has a refractive index of n. At λ ¼ 900 nm, n ¼ 3:593. For both waves,
we find the propagation constant to be
k¼
2πn 2π 3:593
¼
¼ 2:51 107 m1
λ
900 nm
and the wavelength in GaAs to be
λGaAs ¼
λ 900 nm
¼
¼ 250:5 nm:
n
3:593
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3.2 Plane-Wave Modes
77
3.2.2 Anisotropic Medium
As discussed in Sections 2.2, 2.6, and 2.7, the anisotropy of a medium can be intrinsic, such as
that of an anisotropic crystal, or it can be induced by an external factor, such as that caused by
an electro-optic, magneto-optic, acousto-optic, or nonlinear optical effect. The principal normal
modes associated with linear or circular birefringence have already been discussed in Section
2.2. Here we consider only linear birefringence of an anisotropic crystal characterized by a
symmetric dielectric tensor ϵ whose eigenvectors define the principal axes ^x , ^y , and ^z with
eigenvalues ϵ x , ϵ y , and ϵ z , respectively.
Plane-wave normal modes still exist for wave propagation in a homogeneous anisotropic
medium. However, their characteristics depend on the direction of propagation with respect to
the principal axes of the medium. In contrast to plane-wave normal modes in an isotropic
medium, all of which are degenerate with the same propagation constant, plane-wave normal
modes in an anisotropic medium are generally nondegenerate. Their polarization states and
propagation constants are specific to each propagation direction. Three general cases are
discussed in the following.
Propagation along an Optical Axis
In the special case of propagation along an optical axis, the crystal appears to be isotropic to the
wave. For a uniaxial crystal, the optical axis is one of the principal axes, taken to be the z
principal axis by convention. For a biaxial crystal, neither of the two optical axes is a principal
axis. In any event, by the definition of optical axis, a wave does not experience any birefringence when it propagates along an optical axis. Then the plane-wave normal modes have the
same characteristics as those discussed above for an isotropic medium. All plane waves
polarized in the plane normal to an optical axis are normal modes of propagation along this
optical axis, and any two of them that are orthogonally polarized can be used as the basis for
linear expansion.
EXAMPLE 3.3
LiNbO3 is a negative uniaxial crystal that has principal refractive indices of nx ¼ ny ¼ no ¼
2:238 and nz ¼ ne ¼ 2:159 at the λ ¼ 1 μm wavelength. Find the possible arrangements for (a)
a linearly polarized wave and (b) a circularly polarized wave to propagate through LiNbO3 with
a propagation constant defined by either no or ne . In each case, find the propagation constant
and the wavelength for the wave in LiNbO3 .
Solution:
The refractive index seen by a wave is determined by the polarization of the wave. Then, the
possible direction of propagation is constrained by a given polarization. Because the z principal
axis of the uniaxial LiNbO3 crystal is an optical axis, a wave that propagates along the z
direction with its polarization in the xy plane sees the crystal as optically isotropic with no
without seeing ne .
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78
Optical Wave Propagation
(a) A linearly polarized wave at λ ¼ 1 μm sees no ¼ 2:238 if it is polarized in any direction in
the xy plane. This is always true when the wave propagates along the z principal axis. Then,
it has
ko ¼
2πno 2π 2:238
¼
¼ 1:41 107 m1
λ
1 μm
and
λo ¼
λ
1 μm
¼
¼ 446:8 nm:
no 2:238
A linearly polarized wave sees ne ¼ 2:159 if it is polarized along the z principal axis. This is
possible only when the wave propagates in a direction that lies in the xy plane. Then, it has
ke ¼
2πne 2π 2:159
¼
¼ 1:36 107 m1
λ
1 μm
and
λe ¼
λ
1 μm
¼
¼ 463:2 nm:
ne 2:159
(b) A circularly polarized wave at λ ¼ 1 μm sees no ¼ 2:238 if its circular polarization lies in
the xy plane. For this to happen, the wave has to propagate along the z principal axis. It has
ko ¼
2πno 2π 2:238
¼
¼ 1:41 107 m1
λ
1 μm
and
λo ¼
λ
1 μm
¼
¼ 446:8 nm:
no 2:238
There is no possible arrangement for a circularly polarized wave to propagate in a uniaxial
crystal with a propagation constant defined by ne .
Propagation along a Principal Axis
When an optical wave propagates in a direction other than that along an optical axis, the index
of refraction depends on the direction of its polarization. In this situation, there exist two normal
modes of linearly polarized waves, each of which has a unique index of refraction. If the
propagation direction is along a principal axis that is not an optical axis, the two normal modes
are simply the principal modes of polarization that are linearly polarized along the other two
principal axes. Each principal mode of polarization has its characteristic principal index of
refraction.
Without loss of generality, take the principal axis along which the wave propagates to be the z
^ z . In the case when the z principal axis is not an optical axis, the other
principal axis so that kk^
two principal axes ^x and ^y , which are orthogonal to the propagation direction, are birefringent
with different principal permittivities, ϵ x 6¼ ϵ y , thus different propagation constants: k x 6¼ k y ,
where kx ¼ nx ω=c and ky ¼ ny ω=c as defined in (2.15). Note that kx and ky are the propagation
constants of the x- and y-polarized principal normal modes, respectively, not to be confused
with the x and y components of a wavevector k, which are normally expressed as kx and ky :
These two plane wave principal normal modes are
E 1 ¼ ^x E 1 ¼ ^x E x ,
E 2 ¼ ^y E 2 ¼ ^y E y ,
H1 ¼ ^y H1 ¼ ^y Hy ,
H2 ¼ ^x H2 ¼ ^x Hx ,
k1 ¼ β1 k^ ¼ k x ^z ,
k2 ¼ β2 k^ ¼ k y ^z :
(3.47)
In the form of (3.37) and (3.38), these two normal modes form the basis for linear decomposition of any plane wave that propagates along the z principal axis.
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3.2 Plane-Wave Modes
79
Figure 3.3 Evolution of the polarization state of an optical wave propagating along the principal axis ^z of an
anisotropic crystal that has nx 6¼ ny . Only the evolution over one half-period is shown here. (a) The optical
wave is initially linearly polarized at an arbitrary angle θ with respect to the principal axis ^x . (b) The optical
wave is initially polarized at 45 with respect to ^x .
For a plane wave propagating along ^z , the electric field can be expressed as
Eðr; t Þ ¼ E1 ðr; t Þ þ E2 ðr; t Þ ¼ ^x E x exp ðikx z iωt Þ þ ^y E y exp ðiky z iωt Þ:
(3.48)
Because the wave propagates in the z direction, the wavevectors are kx ¼ kx ^z for the x-polarized
field and ky ¼ k y ^z for the y-polarized field. The field expressed in (3.48) has the following
propagation characteristics.
1. If Eðr; t Þ is originally linearly polarized along one of the principal axes, i.e., E y ¼ 0 for
Eðr; t Þ ¼ E1 ðr; t Þk^x or E x ¼ 0 for Eðr; t Þ ¼ E2 ðr; t Þk^y , it remains linearly polarized in the
same direction as it propagates.
2. If Eðr; t Þ is originally linearly polarized at an angle of θ ¼ tan1 E y =E x with respect to the
x axis with E1 ðr; t Þ 6¼ 0 and E2 ðr; t Þ 6¼ 0, its polarization state varies periodically along z
with a period of 2π=jk y kx j because the two normal modes propagate with different
propagation constants. In general, its polarization follows a sequence of variations from
linear to elliptic to linear in the first half-period and then reverses the sequence back to linear
in the second half-period. At the half-period position, it is linearly polarized at an angle of θ
on the other side of the x axis. Thus the polarization is rotated by 2θ from the original
direction, as shown in Fig. 3.3(a). In the special case when θ ¼ 45 , the wave is circularly
polarized at the quarter-period point and is linearly polarized at the half-period point with its
polarization rotated by 90 from the original direction, as shown in Fig. 3.3(b).
These characteristics have very useful applications. A plate of an anisotropic material that has
a quarter-period thickness of
lλ=4 ¼
1
2π
λ
y
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(3.49)
80
Optical Wave Propagation
is called a quarter-wave plate. It can be used to convert a linearly polarized wave to circular or
elliptic polarization, and vice versa. A plate that has a thickness of 3lλ=4 or 5lλ=4 , or any odd
integral multiple of lλ=4 , also has the same function. By contrast, a plate that has a half-period
thickness of
lλ=2 ¼
1
2π
λ
y
x ¼ 2 jk k j 2 ny nx (3.50)
is called a half-wave plate. It can be used to rotate the polarization direction of a linearly
polarized wave by any angular amount by properly choosing the angle θ between the direction
of the incident linear polarization and the principal axis ^x , or ^y , of the crystal. A plate of a
thickness that is any odd integral multiple of lλ=2 has the same function. Note that though the
output from a quarter-wave or half-wave plate can be linearly polarized, the wave plates are not
polarizers. Wave plates and polarizers are based on different principles and have completely
different functions. For the quarter-wave and half-wave plates discussed here, nx 6¼ ny . Between
the two principal axes ^x and ^y , the one with the smaller index is called the fast axis, while the
other, with the larger index, is the slow axis.
EXAMPLE 3.4
At λ ¼ 1 μm, the principal indices of refraction of the KTP crystal are nx ¼ 1:742,
ny ¼ 1:750, and nz ¼ 1:832. Is the crystal uniaxial or biaxial? If you want to propagate a
linearly polarized wave through it, how do you arrange it so that its linear polarization is
maintained throughout the propagation path in the crystal? If the crystal is used to make a
half-wave plate for λ ¼ 1 μm, what is the minimum thickness of the plate? In which direction
must the wave propagate to use this half-wave plate? Note that there is only one possible
minimum thickness.
Solution:
Because nx 6¼ ny 6¼ nz , the crystal is biaxial. To maintain linear polarization throughout, the
wave has to be linearly polarized along one of the principal axes while propagating along
a direction that is perpendicular to its polarization direction. Its propagation constant is
determined by its polarization direction but not by its propagation direction. For example, it
can be polarized in the x direction while propagating in any direction in the yz plane. In this
case, the wave sees nx and has a propagation constant of kx ¼ 2πnx =λ.
Because the largest difference between two principal refractive indices is nz nx ¼
1:832 1:742 ¼ 0:09, the wave must propagate along the y axis of the crystal and have
its polarization in the zx plane, but not along the x or z axis, to utilize this birefringence for
the minimum thickness of the half-wave plate:
lλ=2 ¼
λ
1:00
μm ¼ 5:56 μm:
¼
2jnz nx j 2j1:832 1:742j
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3.2 Plane-Wave Modes
81
Figure 3.4 Relationships among the direction of wave
propagation and the polarization directions of the
ordinary and extraordinary waves.
Propagation in a General Direction
In the general case when the propagation direction is neither along an optical axis nor
along a principal axis, there still exist two linearly polarized normal modes. For simplicity,
the propagation in a uniaxial crystal is considered. The z principal axis of the uniaxial
crystal is the optical axis, and the wave propagation direction k^ is at an angle of θ with
respect to the z principal axis and at an angle of ϕ with respect to the x principal axis, as
shown in Fig. 3.4.
One of the normal modes is the polarization that is perpendicular to the optical axis. This
normal mode is called the ordinary wave. We use ^e o to indicate its direction of polarization.
The other normal mode is clearly perpendicular to ^e o because the two normal-mode polarizations are orthogonal to each other. This normal mode is called the extraordinary wave,
and we use ^e e to indicate its direction of polarization. Note that these are the directions
of D rather than those of E. For the ordinary wave, ^e o kDo kEo . For the extraordinary wave,
^e e kDe =
kEe except when ^e e is parallel to a principal axis. Both ^e o and ^e e , being the unit vectors
of Do and De , are perpendicular to the propagation direction k^ because D is always perpen^ From this understanding, both ^e o and ^e e can be found if both k^ and the optical
dicular to k.
axis ^z are known:
^e o ¼
1 ^
k ^z ,
sin θ
^
^e e ¼ ^e o k:
(3.51)
These vectors are illustrated in Fig. 3.4. They can be expressed as
k^ ¼ ^x sin θ cos ϕ þ ^y sin θ sin ϕ þ ^z cos θ,
(3.52)
^ ϕ,
^e o ¼ ^x sin ϕ y cos
(3.53)
^e e ¼ ^x cos θ cos ϕ ^y cos θ sin ϕ þ ^z sin θ:
(3.54)
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82
Optical Wave Propagation
Figure 3.5 Determination of the indices of refraction for the
ordinary and extraordinary waves in a uniaxial crystal using
index ellipsoid.
The indices of refraction associated with the ordinary and extraordinary waves can be found
by using the index ellipsoid defined as
x2 y2 z2
þ þ ¼ 1:
n2x n2y n2z
(3.55)
The index ellipsoid for the uniaxial crystal under consideration is illustrated in Fig. 3.5 with
nx ¼ ny ¼ no and nz ¼ ne . The intersection of the index ellipsoid and the plane normal to k^ at
the origin of the ellipsoid defines an index ellipse. The principal axes of this index ellipse are in
the directions of ^e o and ^e e , and their half-lengths are the corresponding indices of refraction.
For a uniaxial crystal, the index of refraction for the ordinary wave is simply no . The index of
refraction for the extraordinary wave depends on the angle θ and is given by
1
cos2 θ sin2 θ
¼
þ 2 ,
n2e ðθÞ
n2o
ne
(3.56)
which can be seen from Fig. 3.5. We see that ne ð0 Þ ¼ no and ne ð90 Þ ¼ ne . For θ ¼ 0 , the
propagation direction k^ is along the optical axis. For θ ¼ 90 , the propagation direction k^ lies
in the plane perpendicular to the optical axis; in a uniaxial crystal, this situation is the same as
when k^ is along a principal axis that is not the optical axis.
Each of the two normal modes has a well-defined propagation constant; the ordinary
wave has k o ¼ no ω=c and the extraordinary wave has ke ¼ ne ðθÞω=c. Maxwell’s equations
in the form of (3.31)–(3.34) have to be separately written with different values of k for
the ordinary and the extraordinary normal modes; no such form applies to a wave that is a
^ for the extraordinary way,
mixture of the two modes. For the ordinary way, k ¼ ko ¼ ko k;
^
k ¼ ke ¼ k e k.
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3.2 Plane-Wave Modes
83
EXAMPLE 3.5
LiNbO3 is a negative uniaxial crystal that has principal refractive indices of nx ¼ ny ¼ no ¼
2:238 and nz ¼ ne ¼ 2:159 at the λ ¼ 1 μm wavelength. Find the polarization directions ^e o and
^e e , and the corresponding propagation constants k o and ke , of the ordinary and extraordinary
normal modes for a propagation direction k^ that makes an angle of ϕ ¼ 30 with respect to the
x principal axis and an angle of θ ¼ 45 with respect to the z principal axis.
Solution:
With ϕ ¼ 30 and θ ¼ 45 , we find by using (3.52)–(3.54) that
pffiffiffi
pffiffiffi
pffiffiffi
pffiffiffi
pffiffiffi
pffiffiffi
pffiffiffi
3
1
6
2
2
6
2
2
^x þ
^y þ
^e o ¼ ^x ^x ^y þ
^y , ^e e ¼ ^z ,
^z :
k^ ¼
4
4
2
4
4
2
2
2
At θ ¼ 45 , we find by using (3.56) that
2 cos 45
sin2 45
þ
ne ð45 Þ ¼
2:2382
2:1592
1=2
¼ 2:197:
Therefore, the propagation constants of the two normal modes are, respectively,
ko ¼
ke ¼
2πno 2π 2:238
¼
¼ 1:41 107 m1 ,
λ
1 μm
2πne ð45 Þ 2π 2:197
¼
¼ 1:38 107 m1 :
λ
1 μm
Because D is always perpendicular to the propagation direction, D⊥k for both ordinary and
extraordinary waves. For an ordinary wave, Eo ⊥ko because Eo kDo . Therefore, the relationships shown in Fig. 3.6(a) among the field vectors for an ordinary wave in an anisotropic
medium are the same as those shown in Fig. 3.2 for a wave in an isotropic medium. For an
extraordinary wave, in general Ee ⊥k
= e because Ee =
kDe ; thus Se is not necessarily parallel to ke .
This means that Ee is not transverse to ke but has a longitudinal component in the ke direction.
The only exception is when ^e e is parallel to a principal axis. As a result, the direction of power
flow, which is that of Se , is not the same as the direction of wave propagation, which is that
of ke and is normal to the wavefronts, i.e., the planes of constant phase. Their relationship is
shown in Fig. 3.6(b) together with the relationships among the directions of the field vectors.
Note that Ee , De , ke , and Se lie in the plane normal to He because Be kHe . Though it is still true
that Ee ⊥He because ke Ee kHe according to (3.31), ke He =
kEe because ke He kDe
according to (3.32).
These two plane-wave normal modes have the following characteristics:
E o ¼ ^e o E o ,
Do ¼ ^e o Do ,
Ho ¼ ^e e Ho ,
^
ko ¼ ko k;
^ k
E e ¼ ^e e E ⊥
e þ kE e ,
De ¼ ^e e De ,
He ¼ ^e o He ,
^
ke ¼ ke k;
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(3.57)
84
Optical Wave Propagation
Figure 3.6 Relationships among the directions of E, D, H, B, k, and S in an anisotropic medium for (a) an ordinary
wave and (b) an extraordinary wave. In both cases, the vectors E, D, k, and S lie in a plane normal to H.
where E ⊥
e e and E ke ¼ E e k^ are, respectively, the transverse and longitudinal compone ¼ Ee ^
ents of the electric field of the extraordinary wave. Note that only E e has a longitudinal
component, and this component vanishes when ^e e is parallel to a principal axis. Note also that
Ho kk^ ^e o ¼ ^e e and He kk^ ^e e ¼ ^e o because ωμ0 H ¼ k E for each mode, according to
(3.31). In the form of (3.37) and (3.38), these two normal modes form the basis for the linear
expansion of any plane wave propagating along the k^ direction:
Eðr; t Þ ¼ Eo ðr; tÞ þ Ee ðr; t Þ ¼ E o exp ik o k^ r iωt þ E e exp ik e k^ r iωt ,
(3.58)
Hðr; t Þ ¼ Ho ðr; t Þ þ He ðr; tÞ ¼ Ho exp iko k^ r iωt þ He exp ik e k^ r iωt :
(3.59)
If the electric field of an extraordinary wave is not parallel to a principal axis, its Poynting
vector is not parallel to its propagation direction because Ee is not parallel to De . As a result,
its energy flows away from its direction of propagation. This phenomenon is known as spatial
beam walk-off. If this characteristic appears in one of the two normal modes of an optical wave
propagating in an anisotropic crystal, the optical wave splits into two beams that have parallel
wavevectors but separate, nonparallel traces of energy flow.
Consider a plane wave that propagates in a uniaxial crystal along a general direction k^ at an angle
of θ with respect to the optical axis ^z ; this wave consists of both ordinary and extraordinary waves,
as described by (3.58) and (3.59). Clearly, there is no walk-off for the ordinary wave because
^ For the extraordinary wave, Se is not parallel to k^ but points in a direction at an
Eo kDo so that So kk.
angle of ψ e with respect to the optical axis. Figure 3.7(a) shows the relationships among these
^ which is defined as α ¼ ψ e θ, is called the walk-off angle
vectors. The angle α between Se and k,
of the extraordinary wave. Note that α is also the angle between Ee and De , as is seen in Fig. 3.7(a).
Because neither Ee nor De is parallel to any principal axis, their relationship is found through their
projections on the principal axes: Dez ¼ ϵ 0 n2e E ez and Dex, y ¼ ϵ 0 n2o E ex, y . Using these two relations and
the definition of α in Figs. 3.6(b) and 3.7(a), it is found that the walk-off angle is given by
2
no
α ¼ ψ e θ ¼ tan
tan θ θ:
n2e
1
(3.60)
If the crystal is negative uniaxial, α as defined in Fig. 3.6(b) is positive. This means that k^
is between Se and ^z for a negative uniaxial crystal. If the crystal is positive uniaxial, α is
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3.2 Plane-Wave Modes
85
Figure 3.7 (a) Wave propagation and walk-off in a uniaxial crystal. (b) Birefringent plate acting as a polarizing
beam splitter for a normally incident wave. The ^x , ^y , and ^z unit vectors indicate the principal axes of the
birefringent plate.
negative and Se is between k^ and ^z . No walk-off appears if an optical wave propagates along
any of the principal axes of a crystal.
A birefringent crystal can be used to construct a simple polarizing beam splitter by taking
advantage of the walk-off phenomenon. For such a purpose, a uniaxial crystal is cut into a plate
whose surfaces are at an oblique angle with respect to the optical axis, as shown in Fig. 3.7(b).
When an optical wave is normally incident on the plate, it splits into ordinary and extraordinary
waves in the crystal if its original polarization contains components of both polarizations.
The extraordinary wave is separated from the ordinary wave because of spatial walk-off, creating
two orthogonally polarized beams. Because of normal incidence, both ke and ko are parallel to k^
although they have different magnitudes. When both beams reach the other side of the plate, they
are separated by a distance of d ¼ l tan jαj, where l is the thickness of the plate. After leaving the
plate, the two spatially separated beams propagate parallel to each other in the same k^ direction
because the directions of their wavevectors have not changed, as also shown in Fig. 3.7(b).
EXAMPLE 3.6
LiNbO3 is a negative uniaxial crystal that has principal refractive indices of nx ¼ ny ¼
no ¼ 2:238 and nz ¼ ne ¼ 2:159 at the λ ¼ 1 μm wavelength. Find the walk-off angle of
α of the extraordinary wave in LiNbO3 for a propagation direction k^ that makes an angle
of ϕ ¼ 30 with respect to the x principal axis and an angle of θ ¼ 45 with respect to
the z principal axis. If a collimated optical beam that consists of both ordinary and
extraordinary components at this wavelength propagates in this direction through a
LiNbO3 plate, how thick must the plate be for the ordinary and extraordinary beams
to be separated by at least 100 μm?
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86
Optical Wave Propagation
Solution:
The walk-off angle for θ ¼ 45 is found by using (3.60) to be
2
1 2:238
tan 45 45 ¼ 2:06 :
α ¼ tan
2:1592
For the ordinary and extraordinary beams to be separated by at least 100 μm,
d ¼ l tan α > 100 μm ) l >
100 μm
¼ 2:78 mm:
tan 2:06
Thus, the thickness of the plate has to be at least 2:78 mm.
3.3
GAUSSIAN MODES
..............................................................................................................
A monochromatic optical wave propagating in a homogeneous isotropic medium is
governed by Maxwell’s equations for wave propagation given in (3.3) and (3.4). In this
situation, ϵ is a scalar constant so that D ¼ ϵE and — E ¼ — D=ϵ ¼ 0: Then,
— — E ¼ — — E r2 E ¼ r2 E. By using this relation while combining (3.3) and
(3.4), we obtain the simple wave equation that is specific for the propagation of a monochromatic wave in a homogeneous isotropic medium:
r2 E þ ω2 μ0 ϵE ¼ 0,
(3.61)
where the substitution of ∂=∂t ! iω is taken for the monochromatic wave at the frequency ω.
Because every term in (3.61) has the same constant unit vector, the vectorial wave equation can
be reduced to the scalar Helmholtz equation:
r2 E þ k2 E ¼ 0,
(3.62)
where k2 ¼ ω2 μ0 ϵ, as defined in (3.41). A similar equation can be written for the magnetic field.
Clearly, a monochromatic plane wave of the form in (3.28) and (3.29) is a solution of the
equations for wave propagation given in (3.3) and (3.4), which in this case reduce to the simple
form of (3.31) and (3.32) with D ¼ ϵE; thus, it is a solution of the wave equation in (3.61).
Therefore, plane waves are normal modes of propagation in a homogeneous isotropic medium.
They are not the only normal modes, however, as the equations that govern wave propagation
in such a medium have other normal-mode solutions.
One important set of modes is the Gaussian modes. Like plane waves, Gaussian modes are
normal modes of wave propagation in a homogeneous isotropic medium. Different from a plane
wave, a Gaussian mode has a finite cross-sectional field distribution defined by its spot size. Being
an unguided field that has a finite spot size, a Gaussian mode differs from a waveguide mode,
discussed in Section 3.5, in that its spot size varies along its longitudinal axis, taken to be the
z axis, of propagation though its pattern remains unchanged. Its transverse field distribution also
changes with z though the field pattern does not change. The beam has a finite divergence angle, Δθ.
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3.3 Gaussian Modes
87
For a collimated Gaussian beam that has a small divergence angle such that the paraxial
approximation
sin Δθ Δθ 1
(3.63)
is valid, the propagation constant of the Gaussian normal mode is β ¼ k. Therefore, rather
than those in (3.1) and (3.2), the electric and magnetic fields of a monochromatic Gaussian
mode at a frequency of ω can be expressed as
Emn ðr; t Þ ¼ E mn ðx; y; zÞ exp ðikz iωt Þ ¼ ^e E mn ðx; y; zÞ exp ðikz iωtÞ,
(3.64)
Hmn ðr; tÞ ¼ Hmn ðx; y; zÞ exp ðikz iωtÞ ¼ k^ ^e Hmn ðx; y; zÞ exp ðikz iωt Þ,
(3.65)
where m and n are mode indices associated with the two transverse dimensions x and y,
respectively. The paraxial approximation requires that
2 ∂ E ∂E k and ∂E , ∂E , ∂E jkE j
(3.66)
∂z2 ∂z ∂x ∂y ∂z for the electric field amplitude, and there are similar relations for the magnetic field amplitude.
In this approximation, the Helmholtz equation in (3.62) reduces to
∂2 E ∂2 E
∂E
þ 2 þ i2k
¼0
2
∂x
∂y
∂z
(3.67)
for the electric field amplitude in (3.64). The magnetic field amplitude in (3.65) satisfies an
equation in H of the same form.
In the paraxial approximation, a Gaussian mode field is a TEM mode that has only transverse
electric and magnetic field components; it has neither longitudinal electric nor longitudinal
magnetic field components. Then, the unit polarization vector ^e for the electric mode field in
(3.64) is polarized in the transverse xy plane; the unit vector k^ ^e for the magnetic mode field
in (3.65) is also polarized in the transverse xy plane because k^ ¼ ^z . The paraxial approximation
is not valid when a Gaussian beam is very tightly focused to the extent that its spot size is on the
order of its optical wavelength. In this situation, the longitudinal electric and magnetic field
components cannot be ignored; such a Gaussian mode field is not truly TEM.
The electric mode fields of Gaussian modes in the paraxial approximation are eigenfunctions
of (3.67); the corresponding magnetic mode fields have the same form because they are
eigenfunctions of an equation of H that has the same form as (3.67). As TEM modes, they
can be normalized by the dot-product orthonormality relations given in (3.24):
2k
ωμ0
ð∞
^ ∗0 0 ðx; y; zÞdxdy
^ mn ðx; y; zÞ E
E
mn
∞
2k
¼
ωϵ
ð∞
^ mnðx; y; zÞ H
^ ∗0 0 ðx; y; zÞdxdy ¼ δmm0 δnn0 :
H
mn
(3.68)
∞
The Gaussian beam eigenfunctions of (3.67) in the paraxial approximation have several salient
characteristics. A Gaussian beam has a finite spot size that varies with location along the
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88
Optical Wave Propagation
Figure 3.8 Gaussian beam characteristics.
propagation axis. The location where the smallest spot size of the beam occurs is known as the
waist of the Gaussian beam. This beam waist location is taken to be z ¼ 0 for a beam that
propagates in the direction along the z axis. The minimum Gaussian beam spot size, w0 , is
defined as the e1 radius of the Gaussian beam electric field magnitude profile, i.e., the e2
radius of the Gaussian beam intensity profile, at the beam waist. The diameter of the beam waist
is d 0 ¼ 2w0 : As illustrated in Fig. 3.8, a Gaussian beam has a plane wavefront at its beam waist.
The beam remains well collimated within a distance of
zR ¼
kw20 πnw20
¼
,
2
λ
(3.69)
pffiffiffiffiffiffiffi
known as the Rayleigh range, on either side of the beam waist. In (3.69), k ¼ ω μ0 ϵ ¼ 2πn=λ
is the propagation constant of the optical beam in a medium of a refractive index n. The
parameter b ¼ 2zR is called the confocal parameter of the Gaussian beam.
Because of diffraction, a Gaussian beam diverges away from its waist and acquires a
spherical wavefront at a far-field distance, where jzj zR . As a result, both its spot size,
wðzÞ, and the radius of curvature, RðzÞ, of its wavefront are functions of the distance z from its
beam waist:
"
#1=2
1=2
z2
2z 2
wðzÞ ¼ w0 1 þ 2
¼ w0 1 þ
(3.70)
zR
kw20
and
"
2 2#
z2R
kw0
RðzÞ ¼ z 1 þ 2 ¼ z 1 þ
:
(3.71)
z
2z
pffiffiffi
We see from (3.70) that w ¼ 2w0 at z ¼ zR . At jzj zR , far away from the beam waist, we
find that RðzÞ z and wðzÞ 2jzj=kw0 . Therefore, the far-field beam divergence angle is
Δθ ¼ 2
wðzÞ
4
2λ
¼
:
¼
kw0 πnw0
jzj
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(3.72)
3.3 Gaussian Modes
89
For the far field at jzj zR , we find that the beam spot size wðzÞ is inversely proportional to
the beam waist spot size w0 but is linearly proportional to the distance jzj from the beam waist.
This characteristic does not exist for the near field at jzj zR :
From (3.72), it can be seen that the paraxial approximation sin Δθ Δθ 1 expressed in
(3.63) is valid when the beam is well collimated so that the spot size is much larger than the
optical wavelength in the medium: w0 λ=n. Then the Gaussian mode fields are TEM modes.
This is normally the case for Gaussian wave propagation. The Gaussian mode fields are not
TEM when the beam is tightly focused such that the spot size is on the order of the optical
wavelength. In this situation, w0 λ=n, and the paraxial approximation is invalid.
EXAMPLE 3.7
A Gaussian beam from a Nd:YAG laser at the λ ¼ 1:064 μm wavelength propagates in free
space with a beam divergence of 1 mrad. Find the beam waist spot size, the Rayleigh range,
and the confocal parameter of the beam. What are the spot sizes and the radii of curvature of
the beam at the distances of 10 cm, 1 m, 10 m, and 1 km, respectively?
Solution:
Given λ ¼ 1:064 μm and Δθ ¼ 1 mrad, we find from (3.72) that the beam waist spot size is
w0 ¼
2λ
2 1:064 μm
¼ 677 μm:
¼
πΔθ π 1 103
From (3.69), the Rayleigh range and the confocal parameter are found:
2
πw20 π 677 106
zR ¼
m ¼ 1:35 m and b ¼ 2zR ¼ 2:7 m:
¼
λ
1:064 106
By using (3.70) and (3.71), the spot sizes and the radii of curvature at different locations are
found:
w ¼ 695 μm
w ¼ 843 μm
w ¼ 5:06 mm
w ¼ 50:1 cm
R ¼ 18:33 m at z ¼ 10 cm,
R ¼ 2:82 m
at z ¼ 1 m,
R ¼ 10:18 m at z ¼ 10 m,
R ¼ 1 km
at z ¼ 1 km:
Within the Rayleigh range, both the spot size and the radius of curvature vary nonlinearly with
distance; the spot size increases slowly, whereas the radius of curvature decreases with distance.
At a large distance, both the spot size and the radius of curvature increase approximately linearly
with distance as the Gaussian beam approaches a spherical wave.
A complete set of Gaussian modes in the paraxial approximation includes the fundamental
TEM00 mode and high-order TEMmn modes. The specific forms of the mode fields depend
on the transverse coordinates of symmetry: the mode fields are described by a set of
Hermite–Gaussian functions in the rectilinear coordinates, whereas they are described by the
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90
Optical Wave Propagation
Laguerre–Gaussian functions in the cylindrical coordinates. Both sets are equally valid in free
space or in a homogeneous isotropic medium because there is no structurally defined symmetry.
Usually the Hermite–Gaussian functions in the rectilinear coordinates are used. In a transversely isotropic and homogeneous medium, a normalized TEMmn Hermite–Gaussian mode
field propagating along the z axis can be expressed as
pffiffiffi
pffiffiffi
2x
2y
Cmn
k x2 þ y2
^
Hm
exp ½iζ mn ðzÞ
Hn
exp i
E mn ðx; y; zÞ ¼
wðzÞ
wðzÞ
wðzÞ
2 qðzÞ
(3.73)
pffiffiffi
pffiffiffi
2
2
2
2
2x
2y
C mn
x þy
kx þy
¼
Hm
exp i
exp ½iζ mn ðzÞ ,
Hn
exp 2
wðzÞ
w ðzÞ
wðzÞ
wðzÞ
2 RðzÞ
^ mn ðx; y; zÞ ¼ k E^ mn ðx; y; zÞ,
H
ωμ0
(3.74)
1=2
is the normalization constant, H m is the Hermite
where Cmn ¼ ðωμ0 =πk Þ1=2 ð2mþn m!n!Þ
polynomial of order m, qðzÞ is the complex radius of curvature of the Gaussian wave given by
qðzÞ ¼ z izR or
1
1
2
¼
þi 2 ,
qðzÞ RðzÞ
kw ðzÞ
and ζ mn ðzÞ is a mode-dependent on-axis phase variation along the z axis given by
2z
1 z
1
ζ mn ðzÞ ¼ ðm þ n þ 1Þtan
¼ ðm þ n þ 1Þ tan
:
zR
kw20
(3.75)
(3.76)
The Hermite polynomials can be obtained using the following relation:
2
dm eξ
:
H m ðξ Þ ¼ ð1Þ e
dξ m
m ξ2
(3.77)
Some low-order Hermite polynomials are
H 3 ðξ Þ ¼ 8ξ 3 12ξ:
(3.78)
We see from (3.73) and (3.78) that the transverse field distribution E^ 00 ðx; yÞ of the
fundamental TEM00 Gaussian mode at a fixed longitudinal location z is simply a Gaussian
1=2
function of the transverse radial distance r ¼ ðx2 þ y2 Þ and that the spot size wðzÞ is the e1
radius of this Gaussian field distribution at z. The transverse field distribution of a high-order
TEMmn mode is the Gaussian function spatially modulated by the Hermite polynomials H m ðxÞ
and H n ðyÞ in the x and y directions, respectively. As a result, its field distribution spreads out
radially farther than that of the fundamental TEM00 mode. In general, the higher the order of a
mode is, the farther its transverse field distribution spreads out. The intensity patterns of some
low-order Hermite–Gaussian modes are shown in Fig. 3.9. The Hermite–Gaussian modes are
defined in the rectilinear ðx; y; zÞ coordinates. Because a homogeneous isotropic medium is also
cylindrically symmetric with respect to the wave propagation direction, it is also possible to
define a complete set of the TEM Gaussian modes, known as the Laguerre–Gaussian modes, in
the cylindrical ðr; ϕ; zÞ coordinates with z being the longitudinal wave propagation direction.
The Hermite–Guassian modes have rectilinear symmetry in the transverse plane, whereas the
H 0 ðξ Þ ¼ 1,
H 1 ðξ Þ ¼ 2ξ,
H 2 ðξ Þ ¼ 4ξ 2 2,
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3.3 Gaussian Modes
91
Figure 3.9 Intensity patterns of some low-order Hermite–Gaussian modes.
Laguerre–Gaussian modes have circular and radial symmetry in the transverse plane. Each set
is a complete set of modes for field expansion, and one set can be mathematically transformed
to the other set by linear expansion.
EXAMPLE 3.8
Find the transverse intensity distribution of the fundamental Gaussian mode as a function of the
distance z from the beam waist. Given a fundamental Gaussian beam of a power P, find the
intensity I 0 ðzÞ at the beam center as a function of the distance z. Express P and I 0 ðzÞ in terms of
the beam spot sizes w0 at the beam waist and wðzÞ at the location z.
Solution:
For the fundamental Guassian mode, m ¼ n ¼ 0. Because the zeroth-order Hermite function is
a constant, H 0 ðxÞ ¼ H 0 ðyÞ ¼ 1, we find from (3.73) that the fundamental Guassian mode field
1=2
varies with x and y as x2 þ y2 so that E^ 00 ðx; y; zÞ ¼ E^ 00 ðr; zÞ, where r ¼ ðx2 þ y2 Þ is the
transverse radial coordinate variable. Because a Guassian mode is a TEM mode, its field
2
intensity is I ðr; zÞ / E^ 00 ðr; zÞ . Then, using (3.73), we can express I ðr; zÞ as
I ðr; zÞ ¼ I 0 ðzÞexp 2r2
,
w2 ðzÞ
where I 0 ðzÞ is the intensity at the beam center r ¼ 0. The power of the beam is found by
integrating the intensity distribution over the transverse plane:
ð∞
ð∞
2r 2
πw2 ðzÞ
P ¼ I ðr; zÞ2πrdr ¼ I 0 ðzÞ exp 2
2πrdr ¼
I 0 ðzÞ:
w ðzÞ
2
0
0
Note that the power of a beam is a constant that does not vary with the propagation distance z.
By contrast, the intensity at the beam center varies with z as
I 0 ðzÞ ¼
2P
:
πw2 ðzÞ
In terms of the parameters at the beam waist,
P¼
πw20
w2
I 0 ð0Þ and I 0 ðzÞ ¼ 2 0 I 0 ð0Þ:
2
w ðzÞ
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92
Optical Wave Propagation
For Gaussian beam propagation in a homogeneous isotropic medium along the longitudinal
coordinate axis ^z , any two mutually orthogonal unit polarization vectors ^e 1 and ^e 2 in the
transverse xy plane can be chosen as the polarization basis for linear decomposition of the wave
polarization. Thus, the linear expansion of a Gaussian beam field can be expressed as
X
X
Eðr; tÞ ¼ ^e 1
Amn, 1 E^ mn ðx;y;zÞ exp ðikz iωt Þ þ ^e 2
Amn, 2 E^ mn ðx;y;zÞ exp ðikz iωtÞ, (3.79)
m, n
m, n
Hðr; tÞ ¼
k ^
k
^z Eðr; tÞ,
k Eðr; tÞ ¼
ωμ0
ωμ0
(3.80)
where ^e 1 ^z ¼ ^e 2 ^z ¼ 0 and ^e i ^e ∗
j ¼ δij .
The concept discussed above can be extended to Gaussian beam propagation in a homogeneous anisotropic crystal. For simplicity, consider the case when the propagation direction k^ is
along a principal axis ^z that is not an optical axis so that nx 6¼ ny . As discussed in Section 3.2,
the two principal modes of polarization, ^x and ^y , form the unique basis for polarization
decomposition of TEM waves propagating along the z axis, when the x and y principal axes
are birefringent. In this situation, the Gaussian field is decomposed into two linearly polarized
components that propagate with different propagation constants: k x ¼ nx ω=c and ky ¼ ny ω=c
for the x and y polarizations, respectively. The linear expansion of such a Gaussian beam field
can be expressed as
Eðr; t Þ ¼ Ex ðr; t Þ þ Ey ðr; t Þ
X
X
¼ ^x
Amn, x E^ mn, x ðx; y; zÞ exp ðik x z iωtÞ þ ^y
Amn, y E^ mn, y ðx; y; zÞ exp ðiky z iωt Þ,
m, n
m, n
(3.81)
Hðr; t Þ ¼
kx
ky
^z Ex ðr; tÞ þ
^z Ey ðr; tÞ:
ωμ0
ωμ0
(3.82)
Because all of the characteristic parameters defined in (3.69)–(3.72) for a Gaussian mode
field are functions of the refractive index n, the two polarization modes in (3.81) have different
Gaussian beam parameters besides having different propagation constants. Therefore, in addition to changing its polarization state along the propagation axis as was the case for the plane
wave discussed in Section 3.2, a Gaussian beam that propagates in an anisotropic medium can
have two different spot sizes, two different divergence angles, and two different radii of
curvature between the two principal polarization modes. The beam typically has an elliptic
cross-sectional profile. When focused by a spherical lens, the two polarization modes are
focused at different focal points with different beam waist spot sizes.
3.4
INTERFACE MODES
..............................................................................................................
The simplest optical structure is a planar interface separating two semi-infinite homogeneous
media, as shown in Fig. 3.1(b). The coordinates are chosen as shown in Fig. 3.1(b), with
the interface located at x ¼ 0 such that ϵ ðxÞ ¼ ϵ 1 for x > 0 and ϵ ðxÞ ¼ ϵ 2 for x < 0. The
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3.4 Interface Modes
93
permittivities ϵ 1 and ϵ 2 of the two media are scalar constants, whereas the permeabilities
are simply μ0 at optical frequencies. As discussed in Section 3.1, only TE and TM modes
are possible for this structure. Take the z axis to be the wave propagation direction.
Then, because the index profile is independent of the y coordinate and the wavevector has
no y component, all field components have no variations in the y direction: ∂E=∂y ¼ 0 and
∂H=∂y ¼ 0.
1. TE mode: For any TE mode of a planar structure, E z ¼ 0. It can be seen from (3.11)–(3.14)
that E x ¼ 0, and Hy ¼ 0 as well because ∂Hz =∂y ¼ 0. The only nonvanishing field
components are Hx , E y , and Hz . Once the only nonvanishing electric field component E y
is found for a TE mode, the two nonvanishing magnetic field components can be obtained
by using (3.5) and (3.7):
β
E y,
ωμ0
(3.83)
1 ∂E y
:
iωμ0 ∂x
(3.84)
Hx ¼ Hz ¼
2. TM mode: For any TM mode of a planar structure, Hz ¼ 0. It can be seen from (3.11)–
(3.14) that Hx ¼ 0, and E y ¼ 0 as well because ∂E z =∂y ¼ 0. The only nonvanishing
field components are E x , Hy , and E z . Once the only nonvanishing magnetic field component
Hy is found for a TM mode, the two nonvanishing electric field components can be obtained
by using (3.8) and (3.10):
Ex ¼
Ez ¼ β
Hy ,
ωϵ
(3.85)
1 ∂Hy
:
iωϵ ∂x
(3.86)
In the case of a planar structure, it is convenient to solve for the unique transverse field
component first: E y for a TE mode and Hy for a TM mode. The other field components,
including the longitudinal component, then follow directly.
3.4.1 Reflection and Refraction
We first consider the simple case of reflection and refraction of plane waves at the planar
interface of two media as shown in Fig. 3.1(b). With the coordinates described above, the
interface is located at x ¼ 0 and the plane of incidence is the xz plane so that all wavevectors
have no y component. We assume that the optical wave is incident from the medium of ϵ 1 with
a wavevector of ki , while the reflected wave has a wavevector of kr and the transmitted wave
has a wavevector of kt .
Because an optical wave varies with exp ðik r iωt Þ, the condition
ki r ¼ kr r ¼ kt r
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(3.87)
94
Optical Wave Propagation
Figure 3.10 Reflection and
refraction of a TE-polarized wave at
the interface of two isotropic
dielectric media. The three vectors
ki , kr , and kt lie in the plane of
incidence. The relationship between
θi and θt shown here is for the case
of n1 < n2 :
Figure 3.11 Reflection and
refraction of a TM-polarized wave at
the interface of two isotropic
dielectric media. The three vectors
ki , kr , and kt lie in the plane of
incidence. The relationship between
θi and θt shown here is for the case
of n1 < n2 :
is required at the interface x ¼ 0 for the boundary conditions described by (1.23)–(1.26) to be
satisfied at all points along the interface at all times. This condition implies that the three vectors
ki , kr , and kt lie in the same plane known as the plane of incidence, as shown in Figs. 3.10 and
3.11. The projections of these three wavevectors on the interface are all equal so that
ki sin θi ¼ kr sin θr ¼ kt sin θt
(3.88)
where θi is the angle of incidence, and θr and θt are the angle of reflection and the angle of
refraction, respectively, for the reflected and transmitted waves. All three angles are measured
with respect to the normal n^ of the interface, as is shown in Figs. 3.10 and 3.11. Because ki ¼ kr
and ki =kt ¼ n1 =n2 , (3.88) yields the relation
θi ¼ θr
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(3.89)
3.4 Interface Modes
95
for reflection, and the familiar Snell’s law for refraction:
n1 sin θi ¼ n2 sin θt :
(3.90)
By expressing H in terms of k E in the form of (3.31) with appropriate values of k for the
incident, reflected, and refracted fields, respectively, the amplitudes of the reflected and transmitted fields can be obtained from the boundary conditions n^ E1 ¼ n^ E2 and n^ H1 ¼ n^ H2
given in (1.23) and (1.24). There are two different modes of field polarization.
TE Polarization (s Wave, σ Wave)
For the transverse electric (TE) polarization, or the perpendicular polarization, the electric
field is linearly polarized in a direction perpendicular to the plane of incidence while the
magnetic field is polarized parallel to the plane of incidence, as shown in Fig. 3.10. This wave is
also called s polarized, or σ polarized. For the TE-polarized wave, the reflection coefficient, r,
and the transmission coefficient, t, of the electric field are respectively given by the following
Fresnel equations:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E r n1 cos θi n2 cos θt n1 cos θi n22 n21 sin2 θi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,
rs ¼ ¼
(3.91)
¼
E i n1 cos θi þ n2 cos θt n1 cos θi þ n22 n21 sin2 θi
ts ¼
Et
2n1 cos θi
2n1 cos θi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 þ r s :
¼
¼
E i n1 cos θi þ n2 cos θt n1 cos θi þ n22 n21 sin2 θi
(3.92)
The intensity reflectance and transmittance, R and T, which are also known as reflectivity and
transmissivity, respectively, are given by
I r Sr n^ n1 cos θi n2 cos θt 2
Rs ¼ ¼
¼ jr s j2 ,
(3.93)
¼
Ii
n1 cos θi þ n2 cos θt
Si n^
I t St n^
¼ 1 Rs 6¼ jt s j2 :
(3.94)
Ts ¼ ¼ I i S n^
i
TM Polarization (p Wave, π Wave)
For the transverse magnetic (TM) polarization, or the parallel polarization, the electric field is
linearly polarized in a direction parallel to the plane of incidence while the magnetic field is
polarized perpendicular to the plane of incidence, as shown in Fig. 3.11. This wave is also
called p polarized, or π polarized. For the TM-polarized wave, the reflection and transmission
coefficients of the electric field are respectively given by the following Fresnel equations:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E r n2 cos θi n1 cos θt n22 cos θi n1 n22 n21 sin2 θi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,
rp ¼ ¼
(3.95)
¼
E i n2 cos θi þ n1 cos θt n22 cos θi þ n1 n22 n21 sin2 θi
tp ¼
Et
2n1 cos θi
2n1 n2 cos θi
n1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼
¼
¼ 2
1
þ
r
:
p
E i n2 cos θi þ n1 cos θt n2 cos θi þ n1 n22 n21 sin2 θi n2
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(3.96)
96
Optical Wave Propagation
The intensity reflectance and transmittance for the TM polarization are given, respectively, by
I r Sr n^ n2 cos θi n1 cos θt 2 2
¼
Rp ¼ ¼ ¼ rp ,
I i Si n^ n2 cos θi þ n1 cos θt (3.97)
I t St n^
¼ 1 Rp 6¼ t p 2 :
Tp ¼ ¼ I i Si n^
(3.98)
Several important characteristics of the reflection and refraction of an optical wave at an
interface between two media are summarized below.
1. For both TE and TM polarizations, R ¼ jrj2 and R þ T ¼ 1, but T 6¼ jt j2 :
2. In the case when n1 < n2 , light is incident from a rare medium upon a dense medium; then,
the reflection is called external reflection. In the case when n1 > n2 , light is incident from a
dense medium on a rare medium; then, the reflection is called internal reflection.
3. Normal incidence: In the case of normal incidence, θi ¼ θt ¼ 0: Then, there is no difference
between TE and TM polarizations, and
n1 n2 2
, T ¼ 1 R ¼ 4n1 n2 :
R ¼ n1 þ n2 ðn1 þ n2 Þ2
(3.99)
In the case when both media are lossless so that the values of n1 and n2 are both real, there is
a π phase change for the reflected electric field with respect to the incident field for external
reflection at normal incidence, but the phase of the reflected field is not changed for internal
reflection at normal incidence. A phase change of a value between 0 and π is possible when
either or both media have an optical loss or gain so that n1 or n2 or both have complex values.
In any event, the values of R and T do not depend on the side of the interface from which the
incident wave comes from.
4. Brewster angle: For a TE wave, Rs increases monotonically with the angle of incidence. For
a TM wave, Rp first decreases then increases as the angle of incidence increases. For the
interface between two lossless media, Rp ¼ 0 at an angle of incidence of θi ¼ θB , where
θB ¼ tan1
n2
n1
(3.100)
is known as the Brewster angle. When θi ¼ θB , the angle of refraction for the transmitted
wave is
θt ¼
π
θB :
2
(3.101)
It can be shown that this angle is the Brewster angle for the same wave incident from the
other side of the interface. Thus, the Brewster angles from the two sides of an interface are
complementary angles. Figure 3.12 shows, for both the external reflection and the internal
reflection, the reflectances of TE and TM waves as functions of the angle of incidence at
the interface between two media of refractive indices of 1 and 3.5. These characteristics are
very useful in practical applications. At θi ¼ θB , a TM-polarized incident wave is totally
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3.4 Interface Modes
97
Figure 3.12 Reflectances of TE and TM waves at an interface of lossless media as functions of the angle of
incidence for (a) external reflection and (b) internal reflection. The reflective indices of the two media used for
these plots are 1 and 3.5.
transmitted, resulting in a perfect transmitting window for the TM polarization. Such
windows are called Brewster windows and are useful as laser windows. For a wave of any
polarization that is incident at θi ¼ θB , the reflected wave is completely TE polarized. Linearly
polarized light can be produced by a reflection-type polarizer based on this principle.
5. Critical angle: In the case of internal reflection with n1 > n2 , total internal reflection occurs
if the angle of incidence θi is larger than the angle
θc ¼ sin1
n2
,
n1
(3.102)
which is called the critical angle. The reflectances of TE and TM waves as functions of the
angle of incidence for internal reflection at the interface between two media of refractive
indices of 1 and 3.5 are shown in Fig. 3.12(b). Note that the Brewster angle for internal
reflection is always smaller than the critical angle.
6. At the interface of two lossless dielectric media, both of which have real refractive indices,
the transmitted field has the same phase as the incident field for both TE and TM polarizations because both ts and tp have positive, real values. For external reflection of a TE wave,
the reflected field has a π phase change at any incident angle. For internal reflection of a TE
wave, the reflected field has no phase change at any incident angle smaller than the critical
angle. For external reflection of a TM wave, the reflected field has no phase change at any
incident angle smaller than the Brewster angle, θi < θB , but has a π phase change at any
incident angle larger than the Brewster angle, θi > θB . For internal reflection of a TM wave,
the reflected field has a π phase change at any incident angle smaller than the Brewster angle,
θi < θB , but it has no phase change at any incident angle larger than the Brewster angle but
smaller than the critical angle, θB < θi < θc . (See Problem 3.4.1.)
7. The relations for the reflection and transmission coefficients and those for the reflectance and
transmittance, given in (3.91)–(3.98), remain valid if one or both media have an optical loss
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98
Optical Wave Propagation
or gain so that the refractive indices have complex values. In this situation, each of the
reflection and transmission coefficients of TE and TM waves has a phase that is different
from 0 or π.
8. If one or both media have a loss or gain, the indices of refraction become complex. In this
situation, the reflectance of the TM wave has a minimum value that does not reach zero. This
minimum value is determined by the imaginary parts of the refractive indices of both media.
9. For wave propagation in a general direction in an anisotropic medium, there are two normal
modes that have different indices of refraction. The refracted fields of these two normal
modes can propagate in different directions, resulting in the phenomenon of double
refraction. Meanwhile, the Poynting vector of a normal mode in the anisotropic medium
does not have to be in the plane of incidence.
10. Optical media are generally dispersive. Therefore, reflectance and transmittance, as well as
the direction of the refracted wave, are generally frequency dependent.
EXAMPLE 3.9
The index of refraction of water is n ¼ 1:33. The index of refraction of ordinary glass depends
on its composition and the optical wavelength but is approximately n ¼ 1:5. The refractive
indices of semiconductors, such as Si, GaAs, and InP, vary significantly with the optical
wavelength and the material composition, as well as with temperature, but they usually fall
in the range between 3 and 4. Take a nominal value of n ¼ 3:5 for the typical semiconductor.
For each material at its interface with air, find the reflectivity at normal incidence, the Brewster
angle for external reflection, and the critical angle.
Solution:
Using (3.99), the reflectivities at normal incidence are found to be R ¼ 0:02 for water, R ¼ 0:04
for glass, and R ¼ 0:31 for the semiconductor. Using (3.100), the Brewster angles for external
reflection are found to be θB ¼ 53:1 for water, θB ¼ 56:3 for glass, and θB ¼ 74 for the
semiconductor. Using (3.102), the critical angles are found to be θc ¼ 48:8 for water, θc ¼
41:8 for glass, and θc ¼ 16:6 for the semiconductor.
3.4.2 Radiation Modes
In the above, we considered the reflection and refraction at a planar interface. Here we consider
the mode fields of this structure in the form of (3.1) and (3.2) with the characteristic propagation
constants βν in the z direction along the interface but with the mode field profiles E ν ðxÞ and
Hν ðxÞ being functions of only the x coordinate. The normal modes of a single interface are
radiation modes that have a continuous spectrum of eigenvalues, i.e., continuously distributed
values of propagation constants. From (3.87), we find that the propagation constant in the z
direction is that of the common longitudinal z component of ki , kr , and kt :
β ¼ k i, z ¼ k r , z ¼ k t, z :
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(3.103)
3.4 Interface Modes
99
We assume that the two media are dielectric with ϵ 1 > ϵ 2 so that k1 ¼ n1 ω=c > k 2 ¼ n2 ω=c:
There are two different cases: (1) k 1 > β > k 2 and (2) k 1 > k2 > β, discussed below.
One-Sided Radiation Modes: k1 > β > k2
This is the case when total internal reflection occurs with θi > θc ¼ sin1 n2 =n1 , as
discussed above. Because ki, z ¼ k1 sin θi ¼ β and kr, z ¼ k1 sin θr ¼ β, the condition
k2i, x þ k2i, z ¼ k2r, x þ k2r, z ¼ k 21 requires that the transverse x components of ki and kr have the
same real value: h1 ¼ ki, x ¼ kr, x ¼ k1 cos θi . However, no real solution of θt exists for kt, z ¼
k2 sin θt ¼ β and k t, x ¼ k 2 cos θt to be valid because β > k2 in this case; therefore, no real value
for the transverse x component of kt can be found. Instead, the condition k 2t, x þ k2t, z ¼ k22
requires that k t, x ¼ iγ2 be purely imaginary. Therefore, positive real parameters h1 and γ2 can
be defined for the transverse field profiles in media 1 and 2, respectively, as
h21 ¼ k21 β2 ,
γ22 ¼ β2 k22 :
(3.104)
Using the two parameters h1 and γ2 , the reflection coefficients found in (3.91) and (3.95) for
the TE and TM polarizations can be expressed respectively as
n22 h1 in21 γ2
:
(3.105)
n22 h1 þ in21 γ2
2
As expected for total internal reflection, Rs ¼ jr s j2 ¼ 1 and Rp ¼ r p ¼ 1. However, from
(3.105), it is found that total internal reflection has the following phase shifts for the TE and
TM polarizations, respectively,
r TE ¼ r s ¼
h1 iγ2
,
h1 þ iγ2
φTE ¼ φs ¼ 2 tan1
γ2
,
h1
r TM ¼ r p ¼
φTM ¼ φp ¼ 2 tan1
n21 γ2
:
n22 h1
(3.106)
As commented in the preceding subsection, for external reflection at any incident angle or
internal reflection at an incident angle smaller than the critical angle, the reflection coefficient
of a TE or TM wave at an interface between two lossless dielectric media can only have a phase
of either 0 or π. By contrast, (3.106) indicates that total internal reflection of a TE or TM wave
can have a phase shift between 0 and π.
The fact that ki, x and kr, x both have the real value of k i, x ¼ kr, x ¼ h1 means that the transverse
field profile in medium 1 has sinusoidal variations extending to infinity in the positive x
direction. By contrast, k t, x ¼ iγ2 means that the transverse field profile in medium 2 decays
exponentially in the negative x direction away from the interface. This is a one-sided radiation
mode which is a radiation wave in medium 1 but is evanescent in medium 2, as illustrated in
Fig. 3.13. The penetration depth of the evanescent tail into medium 2 is γ1
2 .
For the TE mode, it is only necessary to find E y ; then the other two nonvanishing components
Hx and Hz can be found by using (3.83) and (3.84), respectively. The boundary conditions
require that E y , Hx , and Hz be continuous at the interface, which dictates that E y and ∂E y =∂x
be both continuous at x ¼ 0. The field profile satisfying these boundary conditions is
E y ðxÞ ¼
cos ðh1 x ψ Þ, x > 0,
cos ψ exp ðγ2 xÞ, x < 0,
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(3.107)
100
Optical Wave Propagation
Figure 3.13 Total internal reflection and
transverse field profile of one-sided
radiation mode. The fact that θr ¼ θi
is shown.
where
ψ ¼ tan1
γ2
1
¼ φTE :
h1
2
(3.108)
Note that the mode field profile E y given in (3.107) is not normalized because it extends
to infinity in the positive x direction. For x > 0, E y in (3.107) is the superposition of
an incident field of an amplitude E i ¼ ^y eiψ =2 and a wavevector ki ¼ h1 ^x þ β ^z and a
totally reflected field of an amplitude E r ¼ E i eiφTE and a wavevector kr ¼ h1 ^x þ β ^z so
that the total space- and time-varying electric field is Eðr; tÞ ¼ E i exp ðiki r iωt Þþ
E r exp ðikr r iωt Þ ¼ ^y E y ðxÞ exp ðiβz iωtÞ.
For the TM mode, it is only necessary to find Hy ; then the other two nonvanishing
components E x and E z can be found by using (3.85) and (3.86), respectively. The boundary
conditions require that Hy , E x , and E z be continuous at the interface, which dictates that Hy
and ϵ 1 ∂Hy =∂x, i.e., n2 ∂Hy =∂x, be both continuous at x ¼ 0. The field profile satisfying these
boundary conditions is
Hy ðxÞ ¼
x > 0,
cos ðh1 x ψ Þ,
cos ψ exp ðγ2 xÞ, x < 0,
(3.109)
n21 γ2
1
¼ φTM :
2
2
n2 h1
(3.110)
where
ψ ¼ tan1
Again, the mode field profile Hy given in (3.109) is not normalized because it extends to infinity in the
positive x direction. For x > 0, Hy in (3.109) is the superposition of an incident field of an amplitude
Hi ¼ ^y eiψ =2 and a wavevector ki ¼ h1 ^x þ β ^z and a totally reflected field of an amplitude Hr ¼
Hi eiφTM and a wavevector kr ¼ h1 ^x þ β ^z so that the total space- and time-varying magnetic field is
Hðr; tÞ ¼ Hi exp ðiki r iωt Þ þ Hr exp ðikr r iωt Þ ¼ ^y Hy ðxÞ exp ðiβz iωt Þ.
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3.4 Interface Modes
101
EXAMPLE 3.10
A glass plate has a refractive index of 1.5 at the λ ¼ 1 μm wavelength. Find the parameters
of the radiation modes at the air–glass interface corresponding to internal reflection at the two
different incident angles of 45 and 75 , respectively. What is the penetration depth of the
evanescent tail into the air if a radiation mode is found to be a one-sided radiation mode at a
particular incident angle? What are the phase shifts on reflection at the interface for TE and TM
waves, respectively?
Solution:
In this problem, n1 ¼ 1:5 and n2 ¼ 1 so that the critical angle of the interface is θc ¼
sin1 ð1=1:5Þ ¼ 41:8 . Because θi > θc for both incident angles, the radiation modes for both
cases are one-sided radiation modes. At λ ¼ 1 μm,
k1 ¼
2πn1
¼ 9:42 106 m1
λ
and k2 ¼
2πn2
¼ 6:28 106 m1 :
λ
For θi ¼ 45 > θc , the radiation mode is a one-sided radiation mode; the parameters of this
radiation mode are
β ¼ k1 sin θi ¼ 6:66 106 m1 ,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6
1
h1 ¼ k 1 cos θi ¼ 6:66 10 m , γ2 ¼ β2 k22 ¼ 2:22 106 m1 :
The penetration depth of the evanescent tail into the air is γ1
2 ¼ 451 nm. The phase shifts on
reflection at the interface for TE and TM waves are
φTE ¼ 2 tan1
γ2
n2 γ
¼ 0:64 rad ¼ 0:20π, φTM ¼ 2 tan1 21 2 ¼ 1:29 rad ¼ 0:41π:
h1
n2 h1
For θi ¼ 75 > θc , the radiation mode is a one-sided radiation mode; the parameters of this
radiation mode are
β ¼ k1 sin θi ¼ 9:10 106 m1 ,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6
1
h1 ¼ k1 cos θi ¼ 2:44 10 m , γ2 ¼ β2 k22 ¼ 6:59 106 m1 :
The penetration depth of the evanescent tail into the air is γ1
2 ¼ 152 nm. The phase shifts on
reflection at the interface for TE and TM waves are
φTE ¼ 2 tan1
γ2
n2 γ
¼ 2:43 rad ¼ 0:77π, φTM ¼ 2 tan1 21 2 ¼ 2:82 rad ¼ 0:90π:
h1
n2 h1
Two-Sided Radiation Modes: k1 > k2 > β
This is the case when partial reflection accompanied by refracted transmission occurs for an
incident angle of θi < θc . In this case, k i, z ¼ k1 sin θi ¼ β and kr, z ¼ k1 sin θr ¼ β so that the
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102
Optical Wave Propagation
condition k 2i, x þ k2i, z ¼ k2r, x þ k2r, z ¼ k 21 requires that the transverse x components of ki and kr
have the same real value: h1 ¼ k i, x ¼ kr, x ¼ k 1 cos θi . Meanwhile, because k 2 > β, a real
solution of θt exists for k t, z ¼ k 2 sin θt ¼ β so that the transverse x component of kt also has
a real value: h2 ¼ kt, x ¼ k2 cos θt . Therefore, positive real parameters h1 and h2 can be defined
for the transverse field profiles in media 1 and 2, respectively, as
h21 ¼ k21 β2 ,
h22 ¼ k 22 β2 :
(3.111)
Note that h1 > h2 because k1 > k 2 .
Using the two parameters h1 and h2 , the reflection coefficients found in (3.91) and (3.95)
for the TE and TM polarizations can be respectively expressed as
n22 h1 n21 h2
:
(3.112)
n22 h1 þ n21 h2
2
As expected for partial reflection, Rs ¼ jr s j2 6¼ 1 and Rp ¼ r p 6¼ 1. Because h1 > h2 , there is
no phase shift in reflection for the TE polarization: φTE ¼ φs ¼ 0. The phase shift in reflection
for the TM polarization flips at the Brewster angle: φTM ¼ φp ¼ π for θi < θB , but φTM ¼
φp ¼ 0 for θi > θB . (See Problem 3.4.1.)
The real parameters h1 ¼ ki, x ¼ kr, x and h2 ¼ kt, x characterize a two-sided radiation mode
field profile that has sinusoidal variations extending to infinity in both positive and negative x
directions, as illustrated in Fig. 3.14. This field pattern is the superposition of the incident,
reflected, and transmitted fields on each side from two incident waves, one from medium 1 and
the other from medium 2, as also illustrated in Fig. 3.14 and discussed below.
For the TE mode, the E y field profile satisfying the boundary conditions that E y and ∂E y =∂x
are continuous at x ¼ 0 is
r TE ¼ r s ¼
E y ðxÞ ¼
h1 h2
,
h1 þ h2
r TM ¼ r p ¼
cos ψ 2 cos ðh1 x ψ 1 Þ,
cos ψ 1 cos ðh2 x ψ 2 Þ,
x > 0,
x < 0,
(3.113)
where the two phase factors ψ 1 and ψ 2 are related by
h1 tan ψ 1 ¼ h2 tan ψ 2 :
(3.114)
Figure 3.14 Partial reflection
and transmission, and transverse field
profile of two-sided radiation mode. The
fact that θr ¼ θi and θt > θi for incidence
from medium 1 is shown.
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3.4 Interface Modes
103
The nonvanishing magnetic field components Hx and Hz of the TE mode are found from E y by
using (3.83) and (3.84), respectively. The mode field E y in (3.113) is not normalized because it
extends to infinity in both positive and negative x directions. For all x, E y in (3.113) is the
superposition of the incident, reflected, and transmitted fields resulting from two incident
waves: one from medium 1 that has a field amplitude of E i1 ¼ ^y cos ψ 2 eiψ 1 =2 and a wavevector
of ki1 ¼ h1 ^x þ β^z , and the other from medium 2 that has E i2 ¼ ^y cos ψ 1 eiψ 2 =2 and
ki2 ¼ h2 ^x þ β^z . Note that (3.114) eliminates one free phase parameter so that the phase relation
between the two incident waves in the composition of the TE mode field is determined.
For the TM mode, the Hy field profile satisfying the boundary conditions that Hy and
2
n ∂Hy =∂x are continuous at x ¼ 0 is
Hy ðxÞ ¼
cos ψ 2 cos ðh1 x ψ 1 Þ,
cos ψ 1 cos ðh2 x ψ 2 Þ,
x > 0,
x < 0,
(3.115)
where the two phase factors ψ 1 and ψ 2 are related by
h1
h2
tan ψ 1 ¼ 2 tan ψ 2 :
2
n1
n2
(3.116)
The nonvanishing electric field components E x and E z of the TM mode are found from Hy by
using (3.85) and (3.86), respectively. The mode field Hy in (3.115) is not normalized because it
extends to infinity in both positive and negative x directions. For all x, Hy in (3.115) is
the superposition of the incident, reflected, and transmitted fields resulting from two incident
waves: one from medium 1 that has a field amplitude of Hi1 ¼ ^y cos ψ 2 eiψ1 =2 and a wavevector
of ki1 ¼ h1 ^x þ β^z , and the other from medium 2 that has Hi2 ¼ ^y cos ψ 1 eiψ 2 =2 and
ki2 ¼ h2 ^x þ β^z . The relation in (3.116) eliminates one free phase parameter so that the phase
relation between the two incident waves in the composition of the TM mode field is determined.
EXAMPLE 3.11
The glass plate with a refractive index of 1.5 at the λ ¼ 1 μm wavelength given in Example 3.10 is
now immersed in water, which has a refractive index of 1.33. Find the parameters of the radiation
modes at the water–glass interface corresponding to internal reflection at the two different incident
angles of 45 and 75 , respectively. What is the penetration depth of the evanescent tail into the
water if a radiation mode is found to be a one-sided radiation mode at a particular incident angle?
What are the phase shifts on reflection at the interface for TE and TM waves, respectively?
Solution:
In this problem, n1 ¼ 1:5 and n2 ¼ 1:33 so that the critical angle of the interface is θc ¼
sin1 ð1:33=1:5Þ ¼ 62:5 and the Brewster angle for internal reflection is θB ¼ tan1
ð1:33=1:5Þ ¼ 41:6 < θc . At λ ¼ 1 μm,
k1 ¼
2πn1
2πn2
¼ 9:42 106 m1 and k2 ¼
¼ 6:28 106 m1 :
λ
λ
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104
Optical Wave Propagation
For θi ¼ 45 < θc , the radiation mode is a two-sided radiation mode; the parameters of this
radiation mode are
β ¼ k 1 sin θi ¼ 6:66 106 m1 ,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6
1
h1 ¼ k1 cos θi ¼ 6:66 10 m , h2 ¼ k 22 β2 ¼ 5:05 106 m1 :
Because this mode is a two-sided radiation mode, it extends to infinity on both the glass and
water sides. Because θi ¼ 45 > θB , the phase shifts of the internal reflection at the interface for
TE and TM waves are
φTE ¼ 0, φTM ¼ 0:
For θi ¼ 75 > θc , the radiation mode is a one-sided radiation mode; the parameters of this
radiation mode are
β ¼ k 1 sin θi ¼ 9:10 106 m1 ,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6
1
h1 ¼ k 1 cos θi ¼ 2:44 10 m , γ2 ¼ β2 k22 ¼ 3:59 106 m1 :
The penetration depth of the evanescent tail into the water is γ1
2 ¼ 278 nm. The phase shifts on
reflection at the interface for TE and TM waves are
φTE ¼ 2 tan1
γ2
¼ 1:95 rad ¼ 0:62π,
h1
φTM ¼ 2 tan1
n21 γ2
¼ 2:16 rad ¼ 0:69π:
n22 h1
3.4.3 Surface Plasmon Mode
In the above, we have seen that an interface between two isotropic dielectric media supports
only radiation modes. At most, it supports a one-sided radiation mode that has a localized
transverse field distribution on only one side of the interface. No localized, guided surface mode
is supported by this type of interface. Guided surface modes do exist in certain types of
interface, such as that between an isotropic dielectric medium and an anisotropic dielectric
medium or that between an isotropic dielectric medium and a plasma medium.
We consider the interface between an isotropic dielectric medium of a permittivity ϵ 1 and an
isotropic plasma medium of a permittivity ϵ 2 , as shown in Fig. 3.15. For simplicity, we take the
limit that ωτ 1 so that the permittivity of the plasma medium is that given in (2.49):
!
ω2p
ϵ2 ¼ ϵb 1 2 ,
(3.117)
ω
where ϵ b ¼ ϵ bound is the background permittivity due to bound electrons and ωp is the plasma
frequency defined in (2.46). The plasma medium can be any medium that has free charge
carriers, such as a doped semiconductor or a metal. For simplicity, we neglect the absorption
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3.4 Interface Modes
105
Figure 3.15 Surface plasmon mode at the
interface between a dielectric medium of
ϵ 1 and a plasma medium of ϵ 2 .
loss in the dielectric medium and that due to bound electrons in the plasma medium so that
both ϵ 1 and ϵ b are real and positive: ϵ 1 > 0 and ϵ b > 0. However, as discussed in Section 2.4
and seen from (3.117), at any frequency below the plasma frequency, the permittivity of
the plasma medium is negative: ϵ 2 < 0 for ω < ωp . The opposite signs of ϵ 1 and ϵ 2 in this
situation create the possibility of a guided surface plasmon mode that is supported by the
interface.
The surface plasmon mode between a dielectric medium and a plasma medium is a TM mode.
To be guided by the interface, it has to be transversely localized near the interface. Thus, it has
to decay exponentially away from the interface in both positive and negative x directions with
characteristic parameters γ1 and γ2 , respectively:
γ21 ¼ β2 k21 ,
γ22 ¼ β2 k 22 :
(3.118)
Because the surface plasmon mode is a TM mode, we find Hy with the boundary conditions
that Hy and ϵ 1 ∂Hy =∂x are continuous at the interface located at x ¼ 0.
The guided TM mode can be normalized using (3.22). The normalized field profile of the
surface plasmon mode that satisfies the boundary condition for the continuity of Hy is
^ y ðxÞ ¼ C exp ðγ1 xÞ,
H
exp ðγ2 xÞ,
x > 0,
x < 0,
(3.119)
where
ω
C¼
β
1=2 γ1 γ2 ϵ 1 ϵ 2
γ1 ϵ 1 þ γ2 ϵ 2
1=2
:
(3.120)
The boundary condition for the continuity of ϵ 1 ∂Hy =∂x at x ¼ 0 yields the eigenvalue
equation:
γ1 γ2
þ ¼ 0:
ϵ1 ϵ2
(3.121)
^y
The nonvanishing mode electric field components are E^ x and E^ z , which can be found from H
by using (3.85) and (3.86), respectively.
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106
Optical Wave Propagation
Figure 3.16 Dispersion curve for surface plasmon mode showing (a) propagation constant as a function of
frequency and (b) frequency as a function of propagation constant. At a low frequency, the surface plasmon
propagation constant β approaches the propagation constant k 1 in the dielectric medium. As the frequency
increases towards ωsp , β becomes much larger than k 1 and approaches infinity. The example in this figure ispplotted
ffiffiffi
with ϵ 1 ¼ ϵ 0 and ϵ b ¼ ϵ 0 for the surface of a perfect metal in free space. In this special case, ωsp ¼ ωp = 2:
Because γ1 > 0, γ2 > 0, and ϵ 1 > 0, it is necessary that ϵ 2 < 0 for the eigenvalue equation
to have a solution. Using the relations in (3.118), with k21 ¼ ω2 μ0 ϵ 1 and k22 ¼ ω2 μ0 ϵ 2 , the
eigenvalue equation (3.121) can be solved to find
μ ϵ1ϵ2
β¼ω 0
ϵ1 þ ϵ2
1=2
,
μ0 ϵ 21
γ1 ¼ ω
ϵ1 þ ϵ2
1=2
,
γ2 ¼ ω
μ0 ϵ 22
ϵ1 þ ϵ2
1=2
:
(3.122)
The condition for γ1 , γ2 , and β in (3.122) to have real and positive solutions is that
ϵ 2 < 0 and ϵ 1 þ ϵ 2 < 0
)
ϵ 2 < ϵ 1 < 0:
This condition limits the surface plasmon mode to the frequency range:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ϵb
ω < ωsp ¼
ωp ,
ϵ1 þ ϵb
(3.123)
(3.124)
where ωsp is known as the surface plasma frequency.
Figure 3.16 shows the relation between β and ω for the surface plasmon mode. At a low
pffiffiffiffiffiffiffiffiffi
frequency such that ω ωsp , β ω μ0 ϵ 1 ¼ k1 so that the surface plasmon propagation
constant β approaches the propagation constant k1 in the dielectric medium. As the frequency
increases, β increases and gradually becomes much larger than k 1 , β k1 , approaching infinity
as the frequency approaches ωsp . Note that ωsp < ωp , as is also shown in Fig. 3.16. The cutoff
frequency and cutoff wavelength of a surface plasmon mode are νsp ¼ ωsp =2π and
λsp ¼ c=νsp ¼ 2πc=ωsp , respectively. The surface plasmon mode can be excited only by a
TM-polarized wave of ν < νsp and λ > λsp .
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3.4 Interface Modes
107
EXAMPLE 3.12
A surface plasmon mode can exist at the interface between a silver plate and free space. The plasma
frequency of Ag found in Example 2.4 is ωp ¼ 1:36 1016 rad s1 . What is the surface plasma
frequency of this interface? What are the cutoff frequency and cutoff wavelength of the surface
plasmon mode? Does the surface plasmon mode exist at the λ ¼ 500 nm wavelength? If it exists,
find its propagation constant and characteristic parameters. Find the penetration depths of the mode
into the free space and into the silver to find its confinement at the interface.
Solution:
At the interface between free space and Ag, ϵ 1 ¼ ϵ 0 for free space and ϵ 2 is that of Ag. For Ag,
ϵ b ¼ ϵ 0 so that
!
!
!
ω2p
ω2p
λ2
ϵ2 ¼ ϵb 1 2 ¼ ϵ0 1 2 ¼ ϵ0 1 2 :
ω
ω
λp
Given ωp ¼ 1:36 1016 rad s1 for Ag, the surface plasma frequency is
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ωp
ϵb
ϵ0
ωp ¼
ωp ¼ pffiffiffi ¼ 9:62 1015 rad s1 :
ωsp ¼
ϵ0 þ ϵb
ϵ0 þ ϵ0
2
Therefore, the cutoff frequency and cutoff wavelength are, respectively,
ωsp
c
¼ 196 nm:
¼ 1:53 1015 Hz ¼ 1:53 PHz, λsp ¼
νsp ¼
2π
νsp
The surface plasmon mode exists at the λ ¼ 500 nm wavelength because λ > λsp .
For ωp ¼ 1:36 1016 rad s1 , we find λp ¼ 138 nm. Therefore, for λ ¼ 500 nm,
!
λ2
5002
¼ 12:13ϵ 0 :
ϵ2 ¼ ϵ0 1 2 ¼ ϵ0 1 1382
λp
Then, by using (3.122), we find
μ ϵ1ϵ2
β¼ω 0
ϵ1 þ ϵ2
1=2
2π ðϵ 1 =ϵ 0 Þðϵ 2 =ϵ 0 Þ
¼
λ ϵ 1 =ϵ 0 þ ϵ 2 =ϵ 0
1=2
2π
12:13
¼
9 1 12:13
500 10
¼ 1:31 107 m1 ,
"
#1=2
1=2
μ0 ϵ 21
2π
ðϵ 1 =ϵ 0 Þ2
2π
1
¼
¼
γ1 ¼ ω
9
ϵ1 þ ϵ2
λ ϵ 1 =ϵ 0 þ ϵ 2 =ϵ 0
1 12:13
500 10
1=2
1=2
m1
m1
¼ 3:77 106 m1 ,
"
#1=2
2π
ðϵ 2 =ϵ 0 Þ2
2π
12:132
¼
¼
λ ϵ 1 =ϵ 0 þ ϵ 2 =ϵ 0
500 109 1 12:13
¼ 4:57 107 m1 :
μ0 ϵ 22
γ2 ¼ ω
ϵ1 þ ϵ2
1=2
1=2
m1
1
The penetration depths are γ1
1 ¼ 265 nm into the free space and γ2 ¼ 22 nm into the silver.
1
Therefore, the confinement of the surface plasmon mode at the interface is γ1
1 þ γ2 ¼ 287 nm.
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108
Optical Wave Propagation
3.5
WAVEGUIDE MODES
..............................................................................................................
The basic structure of a dielectric optical waveguide consists of a longitudinally extended
high-permittivity, thus high-index, optical medium, called the core, which is transversely
surrounded by low-permittivity, thus low-index, media, called the cladding. We consider a
straight waveguide whose longitudinal direction is taken to be the z direction, as shown in
Figs. 3.1(c) and (d).
In a planar waveguide, which has optical confinement in only one transverse dimension, the
core is sandwiched between cladding layers in only one dimension, designated the x dimension,
with a permittivity profile of ϵ ðxÞ, thus an index profile of nðxÞ, as shown in Fig. 3.1(c). The core
of a planar waveguide is also called the film, while the upper and lower cladding layers are called
the cover and the substrate, respectively. Optical confinement is provided only in the x dimension
by the planar waveguide. A waveguide in which the index profile has abrupt changes between the
core and the cladding is called a step-index waveguide, while one in which the index profile varies
gradually is called a graded-index waveguide. Figure 3.17 shows examples of step-index and
graded-index planar waveguides. In a nonplanar waveguide of two-dimensional transverse
optical confinement, the core is surrounded by the cladding in all transverse directions, with
ϵ ðx; yÞ and nðx; yÞ being functions of both x and y coordinates. A nonplanar waveguide can
also have a step-index or graded-index profile. As discussed in Section 3.1, a planar dielectric
waveguide supports only TE and TM modes, whereas a nonplanar dielectric waveguide supports
TE, TM, and hybrid modes. No TEM modes exist in dielectric waveguides.
To get a general idea of the modes of a dielectric waveguide, it is instructive to consider
the qualitative behavior of an optical wave in the asymmetric planar step-index waveguide
shown in Fig. 3.17(a), where n1 > n2 > n3 . For an optical wave of an angular frequency ω and
a free-space wavelength λ, the media in the three different regions of the waveguide define three
propagation constants:
k1 ¼
n1 ω 2πn1
,
¼
λ
c
k2 ¼
n2 ω 2πn2
,
¼
λ
c
k3 ¼
n3 ω 2πn3
,
¼
λ
c
(3.125)
where k1 > k 2 > k3 .
Figure 3.17 Index profiles of (a) a step-index planar waveguide and (b) a graded-index planar waveguide.
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3.5 Waveguide Modes
109
An intuitive picture of waveguide modes can be obtained from studying ray optics by
considering the path of an optical ray, or a plane optical wave, in the waveguide, as shown
in the central column of Fig. 3.18. There are two critical angles associated with the internal
reflections at the lower and upper interfaces:
θc2 ¼ sin1
n2
n1
and
θc3 ¼ sin1
n3
,
n1
(3.126)
respectively, where θc2 > θc3 because n2 > n3 . The characteristics of the reflection and
refraction of the ray at the interfaces depend on the incident angle θ and the polarization of
the wave.
Guided Modes
For a ray that has an incident angle of θ > θc2 > θc3 at the interfaces of the waveguide,
the wave inside the core is totally reflected at both interfaces and is trapped by the core,
resulting in a guided mode when the resonance condition described below is satisfied. As the
wave is reflected back and forth between the two interfaces, it interferes with itself. A guided
mode can exist only when a transverse resonance condition is satisfied so that the repeatedly
reflected wave constructively interferes with itself. In the core region, the x component of
the wavevector is h1 ¼ k1 cos θ, and the z component is β ¼ k 1 sin θ. The phase shift caused
by a round-trip transverse passage of the field in the core that has a thickness of d is
2h1 d ¼ 2k1 dcos θ. In addition, the internal reflection at the lower interface causes a localized
phase shift of φ2 as given in (3.106), and that at the upper interface causes a phase shift of φ3 ,
which can be found by replacing γ2 with γ3 in (3.106). The phase shifts φ2 and φ3 are
functions of the incident angle θ; for a given θi ¼ θ > θc2 > θc3 , each of them has different
values for TE and TM waves.
The transverse resonance condition for constructive interference is that the total phase shift in
a round-trip transverse passage is
2h1 d þ φ2 ðθÞ þ φ3 ðθÞ ¼ 2k1 d cos θ þ φ2 ðθÞ þ φ3 ðθÞ ¼ 2mπ,
(3.127)
where m is an integer. Because m takes only integral values, only certain discrete values of θ
satisfy (3.127). This condition results in discrete values of the propagation constant βm for
guided modes identified by the mode number m. From (3.106), we find that π < φ2 < 0
and π < φ3 < 0 so that 2π < φ2 þ φ3 < 0. Therefore, the smallest value of m for (3.127) to
have a solution is m ¼ 0; no negative values of m are allowed. The guided mode with m ¼ 0 is
the fundamental mode, and those with m 6¼ 0 are high-order modes.
Though the critical angles, θc2 and θc3 , do not depend on the polarization of the wave, the
phase shifts, φ2 ðθÞ and φ3 ðθÞ, caused by internal reflection at a given angle θ depend on
the polarization, as seen in (3.106). Therefore, (3.127) have different solutions for TE and TM
waves, resulting in different values of βm and different mode characteristics for TE and
TM modes of a given mode number m. Because φTM < φTE < 0 as seen from (3.106), the
TM
solution of (3.127) yields θTE > θTM for a given value of m; thus, βTE
m > βm .
For a given polarization, the solution of (3.127) yields a smaller value of θ and a correspondingly smaller value of βm for a larger value of m. Therefore, among guided modes of different
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110
Optical Wave Propagation
Figure 3.18 Modes of an asymmetric planar step-index waveguide where n1 > n2 > n3 . The range of the
propagation constants, the zig-zag ray pictures, and the field patterns are shown correspondingly for
(a) the guided fundamental mode, (b) the guided first high-order mode, (c) a substrate radiation mode for
β ¼ 1:3k3 , and (d) a substrate–cover radiation mode for β ¼ 0:3k3 . The waveguide structure is chosen so that
it supports only two guided modes. The mode field profiles are calculated mode field distributions that are
normalized to their respective peak values.
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3.5 Waveguide Modes
111
orders but of the same polarization that are supported by a waveguide, the fundamental mode
has the largest propagation constant β0 ; that is, β0 > β1 > . . . for a given polarization, as shown
in Figs. 3.18(a) and (b).
Substrate Radiation Modes
When θc2 > θ > θc3 , total reflection occurs only at the upper interface and not at the lower
interface. As a result, an optical wave incident from either the core or the substrate is refracted
and transmitted at the lower interface. This wave is not confined to the core, but is transversely
extended to infinity in the substrate. It is called a substrate radiation mode. In this case, the
angle θ is not dictated by a resonance condition like (3.127) but can take any value in the range
of θc2 > θ > θc3 . As a result, the allowed values of β form a continuum between k2 and k3
such that the modes are not discrete. The characteristics of a substrate radiation mode are
illustrated in Fig. 3.18(c).
Substrate–Cover Radiation Modes
When θc2 > θc3 > θ, no total reflection occurs at either interface. An optical wave incident
from either side is refracted and transmitted at both interfaces; thus, it transversely extends
to infinity on both sides of the waveguide, resulting in a substrate–cover radiation mode. These
modes are not discrete; their values of β form a continuum between k 3 and 0. The characteristics of a substrate–cover radiation mode are illustrated in Fig. 3.18(d).
In addition to the three types of modes discussed above, there are also evanescent radiation
modes, which have purely imaginary values of β that are not discrete. Their fields decay
exponentially along the z direction. Because the dielectric waveguide considered here is lossless
and does not absorb energy, the energy of an evanescent mode transversely radiates away from
the waveguide. A lossless waveguide cannot generate energy, either. Therefore, evanescent
modes do not exist in a perfect, longitudinally infinite waveguide. They exist at a longitudinal
junction or imperfection of a waveguide, as well as at the terminals of a realistic waveguide
that has a finite length. By comparison, a substrate radiation mode or a substrate–cover
radiation mode has a real β; therefore, its energy does not diminish as it propagates. Like a
plane wave, its power flows in the z direction, though its field transversely extends to infinity
because the power flowing away from the center of the waveguide in the transverse direction is
equal to that flowing toward the center.
The approach of ray optics used above gives an intuitive picture of the waveguide modes and
their key characteristics. Nevertheless, this approach has many limitations. In more sophisticated waveguide geometries such as that of a circular fiber, the idea of using the resonance
condition based on total internal reflection to find the allowed values of β for the guided modes
does not necessarily yield correct results. For a complete description of the waveguide fields,
rigorous electromagnetic analyses as illustrated below are required.
3.5.1 Step-Index Planar Waveguides
A step-index planar waveguide is also called a slab waveguide. The general structure and
parameters of a three-layer slab waveguide are shown in Fig. 3.17(a), which has a core
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112
Optical Wave Propagation
thickness of d and a step-index profile of n1 > n2 > n3 . In the above, the approach of ray
optics was used to illustrate an intuitive picture and some basic mode characteristics of a
slab waveguide. Further understanding requires quantitative analyses of the mode fields
discussed below.
Normalized Waveguide Parameters
The mode properties of a waveguide are commonly characterized in terms of a few dimensionless normalized waveguide parameters. The normalized frequency and waveguide thickness,
also known as the V number, of a step-index planar waveguide is defined as
V¼
2π
d
λ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ω
n21 n22 ¼ d n21 n22 ,
c
(3.128)
where d is the thickness of the waveguide core. The propagation constant β can be represented
by the following normalized guide index,
β2 k22 n2β n22
b¼ 2
¼
,
k1 k22 n21 n22
(3.129)
where nβ ¼ cβ=ω ¼ βλ=2π is the effective refractive index of the waveguide mode that has
a propagation constant of β. The measure of the asymmetry of the waveguide is represented by
an asymmetry factor a, which depends on the polarization of the mode under consideration:
aE ¼
n22 n23
for TE modes,
n21 n22
aM ¼
n41 n22 n23
for TM modes:
n43 n21 n22
(3.130)
Note that aM > aE for a given asymmetric structure. For a symmetric waveguide, aM ¼ aE ¼ 0
because n3 ¼ n2 .
Mode Parameters
For a guided mode, positive real parameters h1 , γ2 , and γ3 exist such that
h21 ¼ k21 β2 ,
γ22 ¼ β2 k22 ,
γ23 ¼ β2 k23
(3.131)
because k1 > β > k2 > k3 . From the ray-optics approach discussed above and from (3.131),
the transverse component of the wavevector in the core region of a refractive index n1 is
h1 ¼ k 1 cos θ. For a guided mode, the transverse components of the wavevectors in the
1=2
1=2
substrate and cover regions are h2 ¼ k22 β2
¼ iγ2 and h3 ¼ k 23 β2
¼ iγ3 , respectively, which are purely imaginary because β > k 2 > k3 . Thus, the field of the guided mode has
to exponentially decay in the transverse direction with decay constants γ2 and γ3 in the substrate
and cover regions, respectively.
For a substrate radiation mode, h2 can be chosen to be real and positive because
k 1 > k 2 > β > k 3 ; thus, (3.131) is replaced by
h21 ¼ k 21 β2 ,
h22 ¼ k22 β2 ,
γ23 ¼ β2 k23 :
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(3.132)
3.5 Waveguide Modes
113
For a substrate–cover radiation mode, both h2 and h3 are real and positive because
k1 > k2 > k 3 > β; thus, (3.131) is replaced by
h21 ¼ k21 β2 ,
h22 ¼ k22 β2 ,
h23 ¼ k23 β2 :
(3.133)
The transverse field pattern of a mode is characterized by the transverse parameters h1 , γ2 (or
h2 ), and γ3 (or h3 ). Because k1 , k 2 , and k3 are specified parameters of a given slab waveguide,
the only parameter that has to be determined for a particular waveguide mode is the longitudinal
propagation constant β. Once the value of β is found, all parameters that characterize the
transverse field pattern are completely determined. Therefore, a waveguide mode is completely
specified by its β. Alternatively, because of the definite relations between β and the transverse
parameters, a mode is completely specified, and the value of its β determined, if any one of the
transverse parameters is known. In most cases, rather than directly solving for β, it is more
convenient to solve an eigenvalue equation for h1 , as seen below.
EXAMPLE 3.13
A step-index planar waveguide of the structure shown in Fig. 3.17(a) is made of glass of
slightly different compositions for the core and the substrate so that n1 ¼ 1:54 for the core and
n2 ¼ 1:47 for the substrate. The cover is simply air so that n3 ¼ 1:00. The exact values of the
parameters for the guided modes depend on the core thickness, but the propagation constant of
any guided mode at a given wavelength is bounded within a range irrespective of the core
thickness. In what range can the propagation constant of a guided mode, if it exists, be found
at the λ ¼ 1 μm wavelength? For what wavelengths can a guided mode be found to have a
propagation constant of β ¼ 1:5 107 m1 ? What will happen to the answers if the structure
is immersed in water so that n3 ¼ 1:33? What will happen if it is immersed in benzene so that
n3 ¼ 1:50? What will happen if it is immersed in CS2 so that n3 ¼ 1:63?
Solution:
With n1 ¼ 1:54, n2 ¼ 1:47, and n3 ¼ 1:00, we have k 1 > k2 > k3 so that the propagation
constant β of any guided mode, if it exists, has to be in the range of k1 > β > k2 . At
λ ¼ 1 μm, we find that
2πn1
2πn2
>β>
λ
λ
)
9:68 106 m1 > β > 9:24 106 m1 :
The wavelength of a guided mode that has a propagation constant of β ¼ 1:5 107 m1 falls in
the range:
2πn1
2πn2
>λ>
β
β
)
645:1 nm > λ > 615:8 nm:
If the structure is immersed in water so that n3 ¼ 1:33, we still find that k1 > k 2 > k3
because n1 > n2 > n3 . Therefore, there are no changes in the answers obtained above.
If the structure is immersed in benzene so that n3 ¼ 1:50, then k 1 > k3 > k2 because
n1 > n3 > n2 . Then, at λ ¼ 1 μm,
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Optical Wave Propagation
2πn1
2πn3
>β>
λ
λ
)
9:68 106 m1 > β > 9:43 106 m1 :
And the wavelength of a guided mode that has a propagation constant of β ¼ 1:5 107 m1
falls in the range:
2πn1
2πn3
>λ>
β
β
)
645:1 nm > λ > 632:8 nm:
If the structure is immersed in CS2 so that n3 ¼ 1:63, then k 3 > k 1 > k2 because n3 > n1 > n2 .
In this situation, the structure does not have any guided mode because the core has a lower
refractive index than the cover. Only cover radiation modes and substrate–cover radiation
modes can be found for this structure.
Guided TE Modes
For a TE mode, it is only necessary to find E y ; then the other two nonvanishing field
components Hx and Hz can be found by using (3.83) and (3.84), respectively. The boundary
conditions require that E y , Hx , and Hz be continuous at the interfaces at x ¼ d=2 between
layers of different refractive indices. From (3.83) and (3.84), it can be seen that these boundary
conditions are equivalent to requiring E y and ∂E y =∂x be continuous at these interfaces.
For a guided mode, we know that the transverse field patterns in the core, substrate, and cover
regions are respectively characterized by the transverse field parameters h1 , γ2 , and γ3 , given in
(3.131). A guided TE mode field distribution that satisfies the boundary conditions for the
continuity of E y at x ¼ d=2 has the form:
8
< cos ðh1 d=2 ψ Þ exp ½γ3 ðd=2 xÞ , x > d=2,
(3.134)
d=2 < x < d=2,
E^ y ¼ CTE cos ðh1 x ψ Þ,
:
cos ðh1 d=2 þ ψ Þ exp ½γ3 ðd=2 þ xÞ , x < d=2:
Application of the other two boundary conditions for the continuity of ∂E y =∂x at x ¼
yields two eigenvalue equations:
d=2
h1 ðγ2 þ γ3 Þ
h21 γ2 γ3
(3.135)
h1 ðγ2 γ3 Þ
:
h21 þ γ2 γ3
(3.136)
tan h1 d ¼
and
tan 2ψ ¼
A guided TE mode can be normalized using the orthonormality relation in (3.20) for
rffiffiffiffiffiffiffiffi
ωμ0
,
(3.137)
C TE ¼
βd E
where
dE ¼ d þ
1 1
þ
γ2 γ3
is the effective waveguide thickness for a guided TE mode.
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(3.138)
3.5 Waveguide Modes
115
Guided TM Modes
For a TM mode, it is only necessary to find Hy ; then the other two nonvanishing field
components E x and E z can be found by using (3.85) and (3.86), respectively. The boundary
conditions require that Hy , ϵE x , and E z be continuous at the interfaces at x ¼ d=2 between
layers of different refractive indices. From (3.85) and (3.86), it can be seen that these boundary
conditions are equivalent to requiring Hy and ϵ 1 ∂Hy =∂x, or n2 ∂Hy =∂x, be continuous at
these interfaces.
For a guided mode, we know that the transverse field patterns in the core, substrate, and cover
regions are respectively characterized by the transverse field parameters h1 , γ2 , and γ3 , given in
(3.131). A guided TM mode field distribution that satisfies the boundary conditions for the
continuity of Hy at x ¼ d=2 has the form:
8
< cos ðh1 d=2 ψ Þ exp ½γ3 ðd=2 xÞ , x > d=2,
^ y ¼ C TM cos ðh1 x ψ Þ,
d=2 < x < d=2,
H
:
cos ðh1 d=2 þ ψ Þ exp ½γ3 ðd=2 þ xÞ , x < d=2:
(3.139)
Application of the other two boundary conditions for the continuity of n2 ∂Hy =∂x at x ¼ d=2
yields two eigenvalue equations:
h1 =n21 γ2 =n22 þ γ3 =n23
(3.140)
tan h1 d ¼ 2
h1 =n21 γ2 γ3 =n22 n23
and
h1 =n21 γ2 =n22 γ3 =n23
tan 2ψ ¼ :
2
h1 =n21 þ γ2 γ3 =n22 n23
(3.141)
A guided TM mode can be normalized using the orthonormality relation in (3.22) for
CTM
sffiffiffiffiffiffiffiffiffiffiffiffi
ωμ0 n21
,
¼
βd M
(3.142)
where the effective waveguide thickness for a guided TM mode is
dM ¼ d þ
1
1
β2 β2
þ
, where q2 ¼ 2 þ 2 1 and
γ2 q2 γ3 q3
k1 k2
q3 ¼
β2 β2
þ 1:
k 21 k23
(3.143)
Modal Dispersion
Guided modes have discrete allowed values of β. They are determined by the allowed values of
h1 because β and h1 are directly related to each other through (3.131). Because γ2 and γ3 are
uniquely determined by β through (3.131), they are also uniquely determined by h1 :
γ22 d 2 ¼ β2 d 2 k22 d 2 ¼ V 2 h21 d 2 ,
(3.144)
γ23 d 2 ¼ β2 d 2 k23 d 2 ¼ ð1 þ aE ÞV 2 h21 d 2 :
(3.145)
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Optical Wave Propagation
Figure 3.19 Allowed values of normalized guide index b as a function of the V number and the asymmetry
factor aE for the first three guided TE modes. The cutoff value V c for a mode is the value of V at the intersection
of its dispersion curve with the horizontal axis.
Figure 3.20 Propagation constants of guided modes as functions of optical frequency for a given step-index
dielectric waveguide.
Therefore, there is only one independent variable h1 in the eigenvalue equations. The solutions
of (3.135) yield the allowed parameters for guided TE modes, while those of (3.140) yield the
parameters for guided TM modes. A transcendental equation such as (3.135) or (3.140) is usually
solved numerically, or graphically by plotting its left- and right-hand sides as a function of
h1 d while using (3.144) and (3.145) to replace γ2 and γ3 by expressions in terms of h1 d. The
solutions yield the allowed values of β, or the normalized guide index b, as a function of the
parameters a and V. The results for the first three guided TE modes are shown in Fig. 3.19.
For a given waveguide, a guided TE mode has a larger propagation constant than the TM
mode of the same order:
TM
βTE
m > βm :
(3.146)
TM
However, the difference between βTE
m and βm is very small for modes of an ordinary dielectric
waveguide, where n1 n2 n1 . Then Fig. 3.19 can be used approximately for TM modes
with a ¼ aM .
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3.5 Waveguide Modes
117
For a given waveguide, the values of aE and aM , as well as those of d and n21 n22 , are
completely specified. Then, β of any guided mode is a function of the optical frequency ω
because V is a function of ω. Figure 3.20 illustrates the typical relation between β and ω for
guided modes of different orders.
Comparing β, k 1 , and k2 in Fig. 3.20, it is seen that the propagation constant of a waveguide
mode has a frequency dependence that is contributed by the structure of the waveguide besides
that due to material dispersion. This extra contribution also causes different modes to have
different dispersion properties, resulting in the phenomenon of modal dispersion. Polarization
dispersion also exists because TE and TM modes generally have different propagation constants.
Polarization dispersion is very small in a weakly guiding waveguide for which n1 n2 n1 .
Cutoff Conditions
As discussed above, γ2 and γ3 of a guided mode are real and positive so that the mode field
exponentially decays in the transverse direction outside the core region and remains bound to
the core. This characteristic of a guided mode is equivalent to the condition that θ > θc2 > θc3
in the ray optics picture illustrated in Fig. 3.18 so that the ray in the core is totally reflected by
both interfaces. Because θc2 > θc3 , the transition from a guided mode to an unguided radiation
mode occurs when θ ¼ θc2 . This transition point corresponds to the condition that β ¼ k2 and
γ2 ¼ 0. As can be seen from the mode field solutions given in (3.134) and (3.139), the
field extends to infinity on the substrate side when γ2 ¼ 0. This defines the cutoff condition
for a guided mode. The cutoff condition is determined by γ2 ¼ 0, rather than by γ3 ¼ 0, because
γ3 > γ2 so that γ2 reaches zero first as their values are reduced.
At cutoff, V ¼ V c . The cutoff value V c of a particular guided mode is the value of V at
the point where the curve of its b versus V dispersion relation, shown in Fig. 3.19, intersects
with the horizontal axis b ¼ 0. From (3.144) and (3.145), we find by setting γ2 ¼ 0 that, at
cutoff,
pffiffiffiffiffi
h1 d ¼ V c and γ3 d ¼ aE V c :
(3.147)
Substituting (3.147) and γ2 ¼ 0 into (3.135) for a guided TE mode yields
pffiffiffiffiffi
tanV c ¼ aE :
(3.148)
Therefore, the cutoff condition for the mth guided TE mode is
pffiffiffiffiffi
V cm ¼ mπ þ tan1 aE , m ¼ 0, 1, 2, . . . :
(3.149)
Substituting (3.147) and γ2 ¼ 0 into (3.140) yields the cutoff condition for the mth guided
TM mode:
pffiffiffiffiffiffi
V cm ¼ mπ þ tan1 aM , m ¼ 0, 1, 2, . . . :
(3.150)
Using the definition of the V number given in (3.128), we can write
V cm ¼
2π
d
λcm
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ωc
n21 n22 ¼ m d n21 n22
c
(3.151)
where λcm is the cutoff wavelength and ωcm is the cutoff frequency of the mth mode. The mth
mode is not guided at a wavelength longer than λcm , or a frequency lower than ωcm .
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118
Optical Wave Propagation
For given waveguide parameters, (3.149) and (3.150) can be used, respectively, to determine the
cutoff wavelengths, and the corresponding cutoff frequencies, of TE and TM modes from (3.151).
For a given optical wavelength, they can be used to determine the waveguide parameters that allow
the existence of a particular guided mode. For given waveguide parameters and optical wavelength,
they can be used to determine the number of guided modes for the waveguide. Therefore, the total
number of guided TE modes supported by a given waveguide at a given optical wavelength is
M TE ¼
pffiffiffiffiffi
V 1
tan1 aE
π π
int
pffiffiffiffiffiffi
V 1
tan1 aM
π π
int
,
(3.152)
,
(3.153)
and that of guided TM modes is
M TM ¼
where ½ int takes the nearest integer larger than the value in the bracket.
Because aM > aE 6¼ 0 for an asymmetric waveguide, the value of V cm for the mth-order TM
mode is larger than that for the mth-order TE mode. Furthermore, both TE0 and TM0 modes
pffiffiffiffiffi
pffiffiffiffiffiffi
have cutoff: V cTE0 ¼ tan1 aE for the TE0 mode and V cTM0 ¼ tan1 aM for the TM0 mode,
with V cTM0 > V cTE0 . An asymmetric waveguide of a V number such that V cTM0 > V cTE0 > V
supports no guided modes, neither TE nor TM. An asymmetric waveguide of a V number such
that V cTM0 > V > V cTE0 supports the TE0 mode but not the TM0 mode. For V > V cTM0 > V cTE0 ,
both TE0 and TM0 modes are supported. As the V number increases, additional high-order
modes are supported in the sequence: TE1 , TM1 , TE2 , TM2 , . . .. As the V number decreases,
the highest order TM mode is cut off before the TE mode of the same order.
A waveguide that supports only one mode is called a single-mode waveguide. A waveguide
that supports more than one mode is a multimode waveguide. From the above discussion, a truly
single-mode asymmetric waveguide is one that supports only the TE0 mode but not the TM0
mode. However, a waveguide that supports only the fundamental TE0 and TM0 modes is often
called a single-mode waveguide, particularly in the situation of a symmetric waveguide, for
which the two fundamental modes both have no cutoff, as discussed below.
EXAMPLE 3.14
The step-index planar glass waveguide considered in Example 3.13 has n1 ¼ 1:54 for the core, n2 ¼
1:47 for the substrate, and n3 ¼ 1:00 for the cover. Consider the λ ¼ 1 μm wavelength. What is the
range of core thickness for the waveguide to support the TE0 mode but not the TE1 mode? What is
the range of core thickness for the waveguide to support the TM0 mode but not the TM1 mode? What
is the range of core thickness for the waveguide to support the TE0 mode but not the TM0 mode?
Solution:
With n1 ¼ 1:54, n2 ¼ 1:47, and n3 ¼ 1:00, we find that
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2π
V ¼ d n21 n22 ¼ 2:884d, where d is in μm;
λ
aE ¼
n22 n23
¼ 5:51,
n21 n22
aM ¼
n41 n22 n23
¼ 31:
n43 n21 n22
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119
3.5 Waveguide Modes
For the waveguide to support the TE0 mode but not the TE1 mode,
pffiffiffiffiffi
V 1
tan1 aE 1 )
π π
1:168
μm < d
1:168 < V 4:310 )
2:884
405 nm < d 1:494 μm:
M TE ¼ 1
)
)
)
0<
pffiffiffiffiffi
tan1 aE < V
4:310
μm
2:884
pffiffiffiffiffi
π þ tan1 aE
For the waveguide to support the TM0 mode but not the TM1 mode,
pffiffiffiffiffiffi
pffiffiffiffiffiffi
V 1
tan1 aM 1 ) tan1 aM < V
π π
1:393
4:535
1:393 < V 4:535 )
μm < d
μm
2:884
2:884
483 nm < d 1:572 μm:
M TM ¼ 1
)
)
)
0<
pffiffiffiffiffiffi
π þ tan1 aM
For the waveguide to support the TE0 mode but not the TM0 mode,
pffiffiffiffiffi
pffiffiffiffiffiffi
M TE ¼ 1 and M TM ¼ 0 ) tan1 aE < V < tan1 aM ) 405 nm < d
483 nm:
3.5.2 Symmetric Slab Waveguides
For a symmetric slab waveguide, n3 ¼ n2 , aE ¼ aM ¼ 0, and γ3 ¼ γ2 . Then, it can be seen from
(3.136) and (3.141) that for both TE and TM modes,tan 2ψ ¼ 0 so that
ψ¼
mπ
,
2
m ¼ 0, 1, 2, . . . :
(3.154)
Therefore, the mode field patterns of a symmetric waveguide given by (3.134) and (3.139) are
either even functions of x, varying in space as cos h1 x in the core region d=2 < x < d=2, for
even values of m, or odd functions of x, varying in space as sin h1 x in the core region
d=2 < x < d=2, for odd values of m. This characteristic is expected because the mode field
pattern in a symmetric structure is either symmetric or antisymmetric. Figure 3.21 shows the
field patterns and the corresponding intensity distributions of the first few guided modes of a
symmetric slab waveguide.
By using the identity tan 2θ ¼ 2 tan θ=ð1 tan2 θÞ ¼ 2 cot θ=ð cot2 θ 1Þ while equating γ3
to γ2 , the eigenvalue equation in (3.135) for guided TE modes can be transformed to two
equations:
tan
h1 d γ2
¼ ,
h1
2
for even modes;
cot
h1 d γ2
¼ ,
h1
2
for odd modes:
(3.155)
These two equations can be combined in one eigenvalue equation for all guided TE modes:
h1 d mπ
tan
2
2
¼
γ2
¼
h1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
V 2 h21 d 2
h1 d
,
m ¼ 0, 1, 2, . . . ,
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(3.156)
120
Optical Wave Propagation
Figure 3.21 (a) Field patterns and (b) intensity distributions of the first few guided modes of a symmetric slab
waveguide.
Figure 3.22 Graphic solutions for the eigenvalues of guided TE and TM modes of a symmetric waveguide of
V ¼ 5π. The intersections of dashed and solid curves yield the values of h1 d for eigenmodes.
where m is the same mode number as the one in (3.154). Using (3.140), a similar procedure
yields the eigenvalue equation for all guided TM modes:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2
2
2
2
h1 d mπ
n1 γ2 n1 V h1 d
tan
, m ¼ 0, 1, 2, . . . :
(3.157)
¼ 2 ¼ 2
2
2
h1 d
n2 h1 n2
For a given value of the waveguide parameter V, the solutions of (3.156) yield the allowed values
of h1 d for both even and odd TE modes, and those of (3.157) yield the allowed values of h1 d for
both even and odd TM modes. Figure 3.22 shows an example for V ¼ 5π. Because n1 > n2 , it can
be seen from comparing (3.156) with (3.157) and from the graphic solution shown in Fig. 3.22
TE
TM
TM
that for modes of the same order, hTE
1 < h1 ; thus βm > β m . This observation is consistent
with the conclusion obtained from the above general discussion on asymmetric waveguides.
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3.5 Waveguide Modes
121
Because aE ¼ aM ¼ 0, TE and TM modes of a symmetric waveguide have the same cutoff
condition:
V cm ¼ mπ
(3.158)
for the mth TE and TM modes. This can also be seen in Fig. 3.22. Because m ¼ 0 for the
fundamental modes, neither the fundamental TE mode nor the fundamental TM mode of a
symmetric waveguide has cutoff. Any symmetric planar dielectric waveguide supports at least
one TE and one TM mode. The number of TE modes supported by a given symmetric
waveguide is the same as that of the TM modes, which is simply
M TE ¼ M TM ¼
V
π
int
:
(3.159)
For this reason, a symmetric waveguide is never truly single mode because it supports at least
both TE0 and TM0 modes no matter how small its V number is, as long as V > 0. Often, a
symmetric slab waveguide that has V < π is loosely called a single-mode waveguide because it
supports only the fundamental TE0 and TM0 modes. These conclusions are unique to symmetric waveguides. They are not true for an asymmetric waveguide. For example, an asymmetric
slab waveguide might not support any guided mode at a given optical wavelength because both
its fundamental TE and TM modes have a nonzero cutoff.
EXAMPLE 3.15
The step-index planar glass waveguide considered in Example 3.14 is made symmetric by using
the substrate material for the cover so that n2 ¼ n3 ¼ 1:47 for the substrate and the cover while
keeping n1 ¼ 1:54 for the core. Consider the λ ¼ 1 μm wavelength. What is the range of core
thickness for the waveguide to support the TE0 mode but not the TE1 mode? What is the range
of core thickness for the waveguide to support the TM0 mode but not the TM1 mode? What is
the range of core thickness for the waveguide to support the TE0 mode but not the TM0 mode?
Solution:
With n1 ¼ 1:54 and n2 ¼ n3 ¼ 1:47, we find that
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2π
V ¼ d n21 n22 ¼ 2:884d, where d is in μm; aE ¼ 0,
λ
For the waveguide to support the TE0 mode but not the TE1 mode,
aM ¼ 0:
V
0<
1 ) 0<V π
π
π
) 0<d
μm ) 0 < d 1:089 μm:
2:884
For the waveguide to support the TM0 mode but not the TM1 mode,
M TE ¼ 1
)
V
0<
1 ) 0<V π
π
π
) 0<d
μm ) 0 < d 1:089 μm:
2:884
It is not possible for a symmetric waveguide to support the TE0 mode but not the TM0 mode
because they both have no cutoff.
M TM ¼ 1
)
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122
Optical Wave Propagation
3.6
PHASE VELOCITY, GROUP VELOCITY, AND DISPERSION
..............................................................................................................
Phase velocity, group velocity, and dispersion are important parameters that characterize the
propagation of an optical wave. Phase velocity determines the rate of phase variation in wave
propagation. Group velocity determines the speed of transmission of an optical signal. Dispersion is the primary cause of limitation on the bandwidth of the transmission of optical signals.
As discussed in Chapter 2, the susceptibility χðωÞ, thus the permittivity ϵ ðωÞ, of a medium is
a function of the optical frequency. This is the origin of material dispersion. In a homogeneous
anisotropic medium, normal modes of different polarizations have different characteristic
refractive indices, and thus different propagation constants, resulting in polarization dispersion.
In an optical structure, there are waveguide dispersion and modal dispersion besides material
dispersion. Both material dispersion and waveguide dispersion are examples of chromatic
dispersion because both are frequency dependent. Waveguide dispersion is caused by the
frequency dependence of the propagation constant of a specific mode due to the waveguiding
effect. The combined effect of material dispersion and waveguide dispersion for a particular
mode alone is called intramode dispersion. Modal dispersion is also called intermode
dispersion because it is caused by the variation in propagation constant between different
modes. Modal dispersion appears only when more than one mode is excited in a multimode
waveguide; it exists even when chromatic dispersion disappears.
To illustrate the concepts of phase velocity, group velocity, and dispersion, we first consider a
plane-wave normal mode of a homogeneous medium that has a characteristic propagation
constant of k ðωÞ ¼ nðωÞω=c, where nðωÞ is the frequency-dependent characteristic refractive
index of the normal mode. Without loss of generality, the z coordinate direction is taken to be
along the propagation direction. The electric field of such a monochromatic plane optical wave
can be written as
E ¼ E exp ðikz iωt Þ,
(3.160)
where E is a constant vector independent of space and time.
The field expressed in (3.160) represents a sinusoidal wave that has a phase varying with z and t as
φ ¼ kz ωt:
(3.161)
A point of constant phase on the space- and time-varying field is defined by φ ¼ constant, thus
dφ ¼ kdz ωdt ¼ 0. If we track this point of constant phase as the wave propagates, we find
that it moves with a velocity of
vp ¼
dz ω
¼ :
dt
k
(3.162)
This is called the phase velocity of the wave. Note that the phase velocity is a function of
the optical frequency because the refractive index nðωÞ is a function of frequency. There is
phase-velocity dispersion due to the fact that dn=dω 6¼ 0. In the case of normal dispersion,
dn=dω > 0 and dn=dλ < 0; in the case of anomalous dispersion, dn=dω < 0 and dn=dλ > 0.
As discussed in Section 2.3, normal dispersion and anomalous dispersion are associated with
resonant transitions in a material.
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3.6 Phase Velocity, Group Velocity, and Dispersion
123
Figure 3.23 Wave packet composed of two frequency components showing the carrier and the envelope. The
carrier travels at the phase velocity, whereas the envelope travels at the group velocity.
In practice, a propagating optical wave rarely contains only one frequency. It usually consists of
many frequency components that are grouped around some center frequency, ω0 . For the simplicity
of illustration, we consider a wave packet traveling in the z direction that is composed of two plane
waves of equal real amplitude E. The frequencies and propagation constants of the two components are
ω1 ¼ ω0 þ dω, k1 ¼ k0 þ dk,
ω2 ¼ ω0 dω, k2 ¼ k0 dk:
(3.163)
The space- and time-dependent total real field of the wave packet is then given by
E ¼ E exp ðik 1 z iω1 t Þ þ c:c: þ E exp ðik 2 z iω2 t Þ þ c:c:
n
o
¼ 2E cos ðk0 þ dkÞz ðω0 þ dωÞt þ cos ðk 0 dk Þz ðω0 dωÞt
(3.164)
¼ 4E cos ðdkz dωtÞ cos ðk0 z ω0 tÞ:
As illustrated in Fig. 3.23, the resultant wave packet has a carrier, which has a frequency of ω0 and
a propagation constant of k0 , and an envelope, which varies in space and time as cosðdkz dωtÞ.
Therefore, a fixed point on the envelope is defined by dkz dωt ¼ constant, which travels with a
velocity of
dω
vg ¼
:
(3.165)
dk
This is the velocity of the wave packet and is called the group velocity.
Because the energy of a harmonic wave is proportional to the square of its field amplitude,
the energy carried by a wave packet that is composed of many frequency components is
concentrated in the regions where the amplitude of the envelope is large. Therefore, the energy
in a wave packet is transported at the group velocity v g . Because a wave package carries an
optical signal, thus information, optical signals and optical information are transmitted at the
group velocity. The constant-phase wavefront travels at the phase velocity, but optical energy
and information are transmitted at the group velocity.
In reality, the group velocity is usually a function of the optical frequency. Then,
d2 k
d 1
¼
v 6¼ 0,
2
dω
dω g
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(3.166)
124
Optical Wave Propagation
which represents group-velocity dispersion. A dimensionless coefficient for group-velocity
dispersion is defined as
D ¼ cω
d2 k 2πc2 d2 k
¼
:
dω2
λ dω2
(3.167)
Group-velocity dispersion is an important consideration in the propagation of optical pulses, which
can represent information bits of an optical signal. It can cause broadening of an individual pulse, as
well as changes in the time delay between pulses of different frequencies. The sign of the groupvelocity dispersion can be either positive or negative. In the case of positive group-velocity
dispersion, d2 k=dω2 > 0 and D > 0, a long-wavelength, or low-frequency, pulse travels faster
than a short-wavelength, or high-frequency, pulse. By contrast, a short-wavelength pulse travels
faster than a long-wavelength pulse in the case of negative group-velocity dispersion, d2 k=dω2 < 0
and D < 0. In a given material, the sign of D generally depends on the spectral region of concern.
Group-velocity dispersion and phase-velocity dispersion discussed above have different meanings.
When measuring the transmission delay or the broadening of optical signals or pulses due
to the dispersion in a medium that has a large transmission length, such as an optical fiber,
another group-velocity dispersion coefficient defined as
Dλ ¼ 2πc d2 k
D
¼
2 dω2
cλ
λ
(3.168)
is usually used. This coefficient is generally expressed as a function of wavelength in the unit
of picoseconds per kilometer per nanometer ps km1 nm1 . It is a direct measure of the
chromatic pulse transmission delay over a unit transmission length.
To summarize, the propagation constant of a plane-wave normal mode is
k¼
ω
nðωÞ:
c
(3.169)
vp ¼
ω c
¼ ,
k n
(3.170)
dω c
¼ ,
dk N
(3.171)
dn
dn
¼nλ
dω
dλ
(3.172)
Therefore, the phase velocity is
and the group velocity is
vg ¼
where
N ¼nþω
is called the group index. Using (3.167) and (3.168), the group-velocity dispersion coefficient
can be expressed as
DðλÞ ¼ λ2
d2 n
λ d2 n
or
D
ð
λ
Þ
¼
:
λ
c dλ2
dλ2
(3.173)
Figure 3.24 shows, as an example, the dispersion properties of pure silica glass and germania–
silica glass.
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3.6 Phase Velocity, Group Velocity, and Dispersion
125
Figure 3.24 (a) Index of refraction n and group index N and (b) group-velocity dispersion D as functions of
wavelength for pure silica (solid curves) and germania–silica containing 13.5 mol% GeO2 (dashed curves).
Zero group-velocity dispersion appears at 1:284 μm for pure silica.
EXAMPLE 3.16
The index of refraction of pure silica in the wavelength range between 1:0 and 1:6 μm varies
with wavelength approximately as
n ¼ 1:4507 þ 0:00301λ2 0:00332λ2 :
(a) Within this wavelength range, where does silica have normal dispersion? Where does it
have anomalous dispersion?
(b) Within this wavelength range, where does silica have positive group-velocity dispersion?
Where does it have negative group-velocity dispersion?
(c) Find the refractive index, the group index, and the group-velocity dispersion of silica at the
three wavelengths of λ ¼ 1:0 μm, 1:3 μm, and 1:6 μm.
(d) Express the group-velocity dispersion as Dλ in the unit of ps km1 nm1 .
Solution:
With the given wavelength dependence of the refractive index, we find
dn
¼ 0:00602λ3 0:00664λ,
dλ
N ¼nλ
dn
¼ 1:4507 þ 0:00903λ2 þ 0:00332λ2 ,
dλ
D ¼ λ2
d2 n
¼ 0:01806λ2 0:00664λ2 :
2
dλ
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Optical Wave Propagation
(a) From the above, we find that dn=dλ < 0 for all wavelengths in the wavelength range
between 1:0 and 1:6 μm. Therefore, silica has normal dispersion throughout this
wavelength range.
(b) The wavelength dependence of D obtained above indicates that it can be zero at the
wavelength:
D ¼ 0 ) λ ¼ 1:284 μm:
It is found that silica has positive group-velocity dispersion with D > 0 for λ < 1:284 μm,
and it has negative group-velocity dispersion with D < 0 for λ > 1:284 μm.
(c) Using the wavelength dependence of each parameter obtained above, we find
λ
1:0 μm
1:3 μm
1:6 μm
n
N
D
1:450 1:463
0:01142
1:447 1:462 0:00054
1:443 1:463 0:00994:
(d) Using (3.168) and the values of D obtained in (c), we find
λ
1:0 μm
1:3 μm
1:6 μm
D
0:01142
0:00054
0:00994
Dλ
38 ps km1 nm1
1:4 ps km1 nm1
21 ps km1 nm1 :
3.6.1 Waveguide Dispersion
The propagation constant β of a mode of an optical structure is determined both by the
parameters of the optical structure and by the material properties. As seen in Figs. 3.16 and
3.20, due to the waveguiding effect, the frequency dependence of β can be very different from
that of the k constants of the materials that form the optical structure. Therefore, β of a mode has
mixed contributions from both material dispersion and waveguide dispersion. It is in fact more
convenient to directly consider the combined effect. To do so, we only have to replace k of a
plane-wave normal mode in all of the formulas obtained in the above by β of the waveguide
mode under consideration, thus defining the effective refractive index nβ , the effective group
index N β , and the effective group-velocity dispersion Dβ for the mode:
nβ ¼
Nβ ¼ c
cβ
,
ω
(3.174)
dnβ
dβ
,
¼ nβ λ
dλ
dω
(3.175)
2
d2 β
2 d nβ
¼
λ
:
dω2
dλ2
(3.176)
Dβ ¼ cω
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3.6 Phase Velocity, Group Velocity, and Dispersion
127
Figure 3.25 (a) Effective index of refraction and group index and (b) group-velocity dispersion of the
fundamental mode as a function of wavelength. The solid curves show the effective parameters of the mode
with both material and waveguide contributions. The dashed curves show only the material contribution
to the core and cladding regions, labeled 1 and 2, respectively.
The phase velocity and group velocity of the mode are, respectively,
ω
c
¼ ,
β nβ
(3.177)
dω
c
:
¼
dβ N β
(3.178)
v pβ ¼
and
v gβ ¼
As an example of the contributions of the waveguiding effect to the dispersion parameters,
Fig. 3.25 shows nβ , N β , and Dβ of the fundamental mode of a circular optical fiber in
comparison to the parameters of its core and cladding materials.
3.6.2 Modal Dispersion
The frequency dependence of the propagation constant β of a mode discussed above is the total
intramode dispersion that includes material and waveguide contributions for the mode. Different normal modes of an anisotropic medium or an optical structure have different propagation
constants at a given optical frequency. Such differences lead to modal dispersion among
different modes, which is intermode dispersion.
For plane waves or Gaussian modes propagating in a homogeneous anisotropic medium,
modal dispersion exists due to different propagation constants for normal modes of different
polarizations, such as k x , ky , and k z of the linearly birefringent principal normal modes of
polarization given in (2.15), k þ and k of the circularly birefringent principal normal modes
of polarization given in (2.21), or k o and ke of the ordinary and extraordinary waves in (3.57).
Such modal dispersion causes polarization dispersion.
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128
Optical Wave Propagation
For normal modes in an optical structure, such as an interface or a waveguide, modal
dispersion exists among modes of the same polarization but of different order, such as the
different propagation constants of TE modes of different orders shown in Fig. 3.19. In general,
at a given frequency, a lower order mode has a larger propagation constant, as seen in Figs. 3.19
and 3.20. This dispersion is not caused by polarization or frequency but is purely imposed by
the optical structure. This type of modal dispersion is mode-order dispersion. Modal dispersion
in an optical structure also exists among modes of the same order but of different polarizations,
such as that between TEm and TMm modes of a planar waveguide. As discussed in Section 3.5
TM
and expressed in (3.146), βTE
m > βm for any given order m. This type of modal dispersion is
polarization-mode dispersion.
EXAMPLE 3.17
An optical pulse has a pulse duration of Δt ps ¼ 20 ps and a spectral width of Δλps ¼ 0:1 nm. It
is transmitted through a silica fiber over a distance of 10 km. Use the data of silica obtained in
Example 3.16 for the silica fiber to find the transmission time and the temporal broadening of
the pulse due to group-velocity dispersion at the transmission end in the case when the center
wavelength of the pulse is at λ ¼ 1:0 μm, 1:3 μm, or 1:6 μm. How does the group-velocity
dispersion temporally spread the pulse spectrum in each case?
Solution:
For a transmission distance of l, the transmission time ttr is
t tr ¼
l
N
¼ l
vg
c
and the temporal pulse broadening ΔtGVD due to group-velocity dispersion is
Δt GVD ¼ jDλ jΔλps l:
At λ ¼ 1:0 μm, N ¼ 1:463 and Dλ ¼ 38 ps km1 nm1 . Thus, for l ¼ 10 km,
ttr ¼
N
1:463
10 103 s ¼ 48:8 μs,
l¼
c
3 108
ΔtGVD ¼ jDλ jΔλps l ¼ 38 0:1 10 ps ¼ 38 ps:
At λ ¼ 1:3 μm, N ¼ 1:462 and Dλ ¼ 1:4 ps km1 nm1 . Thus, for l ¼ 10 km,
ttr ¼
N
1:462
10 103 s ¼ 48:7 μs,
l¼
8
c
3 10
ΔtGVD ¼ jDλ jΔλps l ¼ 1:4 0:1 10 ps ¼ 1:4 ps:
At λ ¼ 1:6 μm, N ¼ 1:463 and Dλ ¼ 21 ps km1 nm1 . Thus, for l ¼ 10 km,
ttr ¼
N
1:463
10 103 s ¼ 48:8 μs,
l¼
c
3 108
ΔtGVD ¼ jDλ jΔλps l ¼ 21 0:1 10 ps ¼ 21 ps:
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3.7 Attenuation and Amplification
129
We find that the transmission time is about the same for all three wavelengths because the
group index is about the same for all three wavelengths. However, the temporal pulse
broadening varies much among the three wavelengths because of the different values of
group-velocity dispersion. At the low group-velocity dispersion point of 1:3 μm, the pulse is
only slightly broadened. At the other two wavelengths, the broadening is larger than the original
pulse duration. Group-velocity dispersion causes frequency chirping in an optical pulse. At
λ ¼ 1:0 μm, the broadening causes the long-wavelength component of the pulse to move to the
temporal leading edge of the pulse because of positive group-velocity dispersion with D > 0
and Dλ < 0, making the pulse positively chirped with its frequency increasing with time within
the pulse. At λ ¼ 1:3 μm and 1:6 μm, the broadening causes the short-wavelength component
of the pulse to move to the temporal leading edge of the pulse because of negative groupvelocity dispersion with D < 0 and Dλ > 0, making the pulse negatively chirped with
its frequency decreasing with time within the pulse.
3.7
ATTENUATION AND AMPLIFICATION
..............................................................................................................
As discussed in Section 2.1, a complex eigenvalue of χðωÞ, thus that of ϵ ðωÞ, signifies an
optical loss or gain for the corresponding principal mode of polarization of the medium, with
χ 00 > 0 and ϵ 00 > 0 for optical loss, and χ 00 < 0 and ϵ 00 < 0 for optical gain. For a plane-wave
normal mode characterized by a complex eigenvalue ϵ,
k 2 ¼ ω2 μ0 ϵ ¼ ω2 μ0 ðϵ 0 þ iϵ 00 Þ:
(3.179)
Therefore, the propagation constant k becomes complex:
α
k ¼ k0 þ ik00 ¼ k0 þ i :
2
(3.180)
The index of refraction also becomes complex:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ϵ 0 þ iϵ 00
:
(3.181)
n ¼ n þ in ¼
ϵ0
The relation k ¼ nω=c between k and n is still valid.
If we choose k0 to be positive, the sign of α is the same as that of ϵ 00 . Then, k0 and n0 are both
positive, and k 00 and n00 also have the same sign as ϵ 00 . Taking the z coordinate direction to be
along the propagation direction, the electric field of a monochromatic plane optical wave as
expressed in (3.160) is
0
00
E ¼ E exp ðikz iωt Þ ¼ E eαz=2 exp ðik0 z iωt Þ:
(3.182)
It can be seen that the wave has a phase that varies sinusoidally with a period of 2π=k0 along z.
However, because of the nonvanishing imaginary part k00 ¼ α=2 of the propagation constant,
the magnitude jEj of the electric field is not constant but varies exponentially with z.
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130
Optical Wave Propagation
The intensity of an optical field projected on a surface is defined in (1.56):
∗
I ¼ S n^ ¼ S þ S n^, where n^ is the unit normal vector of the projected surface. Note
that the intensity of a given optical field depends on the projected surface on which the intensity
is measured. Note further that for an extraordinary wave in an anisotropic medium, the
^ These factors have
Poynting vector S is not generally parallel to the propagation direction k.
to be considered when calculating the intensity. For a monochromatic plane-wave normal mode
that has an optical field given in (3.182), we can use the relation k E ¼ ωμ0 H given in (3.31)
to find that its intensity projected on the surface that is normal to the propagation direction k^
can be expressed as
I¼
2k 0 jE⊥ j2 2k 0 jE ⊥ j2 αz
¼
e ,
ωμ0
ωμ0
(3.183)
where E⊥ ¼ E E k^ k^ is the component of the optical field that is transverse to the
^ For a plane wave in an isotropic
propagation direction defined by k^ and E ⊥ ¼ E E k^ k.
medium or an ordinary wave in an anisotropic medium, E⊥ ¼ E because E k^ ¼ 0. For an
extraordinary wave in an anisotropic medium, E⊥ 6¼ E because E k^ 6¼ 0. In any event, the
optical intensity varies exponentially with z when α 6¼ 0.
Clearly, k 0 is the wavenumber in this situation, and the sign of α determines the attenuation
or amplification of the optical wave.
1. If χ 00 > 0, then ϵ 00 > 0 and α > 0. As the optical wave propagates, its field amplitude and
intensity decay exponentially along the direction of propagation. Therefore, α is called the
absorption coefficient or attenuation coefficient.
2. If χ 00 < 0, then ϵ 00 < 0 and α < 0. The field amplitude and intensity of the optical wave grow
exponentially. Then, we define g ¼ α as the gain coefficient or amplification coefficient.
Both α and g have the unit of per meter, often also quoted per centimeter.
EXAMPLE 3.18
A Si crystal has a complex refractive index of n ¼ 4:30 þ i0:073 at the λ ¼ 500 nm wavelength. Find the absorption coefficient and the absorption depth of Si at this wavelength. What
is the complex susceptibility?
Solution:
From (3.180), the absorption coefficient is
α ¼ 2k00 ¼
4πn00 4π 0:073 1
m ¼ 1:835 106 m1 :
¼
λ
500 109
The absorption depth is α1 ¼ 545 nm. Because 1 þ χ ¼ ϵ=ϵ 0 ¼ n2 , the complex susceptibility is
χ ¼ n2 1 ¼ ð4:30 þ i0:073Þ2 1 ¼ 17:48 þ i0:628:
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3.7 Attenuation and Amplification
131
3.7.1 Attenuation and Amplification of Waveguide Modes
Several factors contribute to the attenuation of the power of an optical wave propagating in an
optical structure. Besides the loss or gain contributed by the material as discussed above, the
imperfections of an optical structure, such as the roughness of its interfaces and the irregularity
of its geometric shape, cause additional losses. Furthermore, the distribution of optical loss or
gain might not be uniform across an optical structure because different regions of an optical
structure generally have different optical properties. In any event, the normal mode of an optical
structure is characterized by a unique, well-defined propagation constant β. The attenuation or
amplification of the normal mode while it propagates through the structure is characterized by a
complex β in the same manner as the complex k for a plane wave. Thus,
α
β ¼ β0 þ iβ00 ¼ β0 þ i :
2
(3.184)
As described above, a positive α is the absorption coefficient or attenuation coefficient of the
mode, whereas g ¼ α is the gain coefficient or amplification coefficient of the mode.
For a guided mode, attenuation or amplification affects the mode across its entire profile even
though it does not have a uniform field profile across the transverse plane. Therefore, the
attenuation or amplification of a guided mode is measured with respect to the change of its mode
power rather than its intensity: PðzÞ / eαz . The attenuation of optical power over a propagation
distance of l in an optical structure for a mode that has an attenuation coefficient of α is given by
Pout ¼ Pin eαl :
(3.185)
The input and output powers of the mode, Pin and Pout , respectively, are measured in watts,
while α is given per meter. The power is often measured in milliwatts or microwatts in lowpower applications, and in kilowatts or megawatts in high-power applications. In practical
applications, α is also measured per centimeter or per kilometer when l is measured in
centimeters or kilometers.
In practical engineering applications, it is convenient to use decibels (dB) as a measure of
relative changes of quantities. The attenuation coefficient α is then measured in decibels per
meter or decibels per kilometer when l is measured in meters or kilometers:
1
Pout
1
Pout
α dB m1 ¼ , α dB km1 ¼ ,
10 log
10 log
Pin
Pin
lðmÞ
lðkmÞ
(3.186)
where Pin and Pout are measured in the same unit which can be watts, milliwatts, or microwatts.
In the case of a low-loss fiber, the propagation length l in the fiber is usually measured in
kilometers, and α is conventionally given in decibels per kilometer. Comparing (3.185) with
(3.186), we find that
α dB km1 ¼ 4:32α km1 and α km1 ¼ 0:23α dB km1 :
(3.187)
Power can also be measured in decibels and has the unit of decibel-watts (dBW), decibelmilliwatts (dBm), or decibel-microwatts (dBμ), defined as
PðdBWÞ ¼ 10 log PðWÞ, PðdBmÞ ¼ 10 log PðmWÞ, PðdBμÞ ¼ 10 log PðμWÞ:
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(3.188)
132
Optical Wave Propagation
When power is given in decibel-watts or decibel-milliwatts and the attenuation coefficient is in
decibels per kilometer, (3.185) can be expressed as
Pout ðdBWÞ ¼ Pin ðdBWÞ α dB km1 lðkmÞ
(3.189)
Pout ðdBmÞ ¼ Pin ðdBmÞ α dB km1 lðkmÞ:
(3.190)
or, equivalently,
A similar formula can be written for power measured in decibel-microwatts. These formulas are
convenient and useful in practical applications as they relate the input power, output power, and
attenuation in a simple arithmetic relation.
EXAMPLE 3.19
An optical fiber has an attenuation coefficient of α ¼ 0:4 dB km1 at λ ¼ 1:3 μm. An optical
signal at an input power level of Pin ¼ 10 mW is transmitted through this fiber over a distance
of l ¼ 100 km. What is the output power? If the attenuation coefficient is slightly reduced to
α ¼ 0:35 dB km1 , what is the output power?
Solution:
The input power is Pin ¼ 10 mW ¼ 10 dBm. With α ¼ 0:4 dB km1 , the output power is
Pout ¼ Pin αl ¼ 10 dBm 0:4 dB km1 100 km ¼ 30 dBm ¼ 103 mW ¼ 1 μW:
If the attenuation coefficient is slightly reduced to α ¼ 0:35 dB km1 , the output power is
Pout ¼ Pin αl ¼ 10 dBm 0:35 dB km1 100 km ¼ 25 dBm ¼ 102:5 mW ¼ 3:16 μW:
For a transmission distance of 100 km, the output power is increased by more than 200% when
the attenuation coefficient is reduced by only 0:05 dB km1 .
Problems
3.1.1 Explain why a TEM mode field can exist only in an optically homogeneous space where
ϵ is a constant of space, and not in an optically inhomogeneous space where ϵ varies
in space.
3.1.2 Can a dielectric waveguide support TEM modes? Explain.
3.1.3 Can a planar optical structure support hybrid modes? Explain.
3.1.4 What types of guided modes does each of the following structure support: (a) a planar
metallic structure, (b) a planar dielectric structure, (c) a hollow cylindrical metallic
structure, and (d) a cylindrical dielectric structure?
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Problems
133
3.1.5 Show that (a) the dot-product orthonormality relation of (3.20) applies to TE modes, (b)
the dot-product orthonormality relation of (3.22) applies to TM modes, and (c) both
relations apply to TEM modes.
3.2.1 The principal indices of refraction of InP, which is a cubic crystal, at the λ ¼ 1:3 μm
wavelength are nx ¼ ny ¼ nz ¼ 3:205. Find the propagation constant and the wavelength
in the crystal for an optical wave at λ ¼ 1:3 μm that propagates through an InP crystal
under each of the following conditions. In each case, does the polarization state change
as the wave propagates through the crystal?
(a) Linearly polarized along ^x , propagating along ^y .
(b) Linearly polarized along ^y , propagating along ^z .
(c) Linearly polarized along ^z , propagating along ^x .
(d) Circularly polarized in the xy plane, propagating along ^z .
(e) Circularly polarized in the yz plane, propagating along ^x .
3.2.2 The principal indices of refraction of LiNbO3 , which is a negative uniaxial crystal,
at the λ ¼ 1:3 μm wavelength are nx ¼ ny ¼ no ¼ 2:222 and nz ¼ ne ¼ 2:145. Find
the propagation constant and the wavelength in the crystal for an optical wave at λ ¼
1:3 μm that propagates through a LiNbO3 crystal under each of the following conditions. In each case, does the polarization state change as the wave propagates through
the crystal?
(a) Linearly polarized along ^x , propagating along ^y .
(b) Linearly polarized along ^y , propagating along ^z .
(c) Linearly polarized along ^z , propagating along ^x .
(d) Circularly polarized in the xy plane, propagating along ^z .
(e) Circularly polarized in the yz plane, propagating along ^x .
3.2.3 The principal indices of refraction of KTP, which is a biaxial crystal, at the λ ¼
1:3 μm wavelength are nx ¼ 1:734, ny ¼ 1:742, and nz ¼ 1:822. Find the propagation
constant and the wavelength in the crystal for an optical wave at λ ¼ 1:3 μm
that propagates through a KTP crystal under each of the following conditions.
In each case, does the polarization state change as the wave propagates through the
crystal?
(a) Linearly polarized along ^x , propagating along ^y .
(b) Linearly polarized along ^y , propagating along ^z .
(c) Linearly polarized along ^z , propagating along ^x .
(d) Circularly polarized in the xy plane, propagating along ^z .
(e) Circularly polarized in the yz plane, propagating along ^x .
3.2.4 The principal indices of refraction of LiNbO3 at λ ¼ 1:3 μm are nx ¼ ny ¼ no ¼ 2:222
and nz ¼ ne ¼ 2:145. Design a waveplate based on LiNbO3 for rotating the polarization
direction of a linearly polarized wave at λ ¼ 1:3 μm by 30o . Give the possible thicknesses
of the plate and the arrangement for this purpose.
3.2.5 The principal indices of refraction of LiNbO3 at λ ¼ 1:3 μm are nx ¼ ny ¼ no ¼ 2:222
and nz ¼ ne ¼ 2:145. Design a waveplate based on LiNbO3 for converting a linearly
polarized wave into a circularly polarized wave at λ ¼ 1:3 μm. Give the possible thicknesses of the plate and the arrangement for this purpose.
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134
Optical Wave Propagation
3.2.6 The permittivity tensor of a KDP crystal at λ ¼ 1 μm in an arbitrarily chosen Cartesian
coordinate system is found to be
0
1
2:174
0
0:039
ϵ ¼ ϵ0@ 0
2:280
0 A:
0:039
0
2:266
3.2.7
3.2.8
3.2.9
3.2.10
(a) Is the KDP crystal birefringent or nonbirefringent? If it is birefringent, is it uniaxial
or biaxial? What are its principal indices of refraction?
(b) If it is used to make a half-wave plate at λ ¼ 1 μm, what is the thickness of the plate?
(c) If it is used to make a quarter-wave plate at λ ¼ 1 μm, what is the thickness of the plate?
The principal indices of refraction of quartz at the λ ¼ 600 nm wavelength are nx ¼
ny ¼ 1:544 and nz ¼ 1:553.
(a) Quartz is clearly a birefringent crystal, is it positive or negative uniaxial?
(b) What kind of quartz plate can be used to rotate the polarization direction of a
linearly polarized wave by 90 to its orthogonal linear polarization? Describe the
arrangement for this function and find the thickness of the plate.
(c) What kind of quartz plate can be used to convert a circularly polarized wave into a
linearly polarized wave? Describe the arrangement for this function and find the
thickness of the plate. How is the direction of the output linear polarization determined?
The principal indices of refraction of BBO, which is a negative uniaxial crystal, are nx ¼
ny ¼ no ¼ 1:677 and nz ¼ ne ¼ 1:557 at the λ ¼ 500 nm wavelength. Consider a
propagation direction k^ that makes an angle of ϕ ¼ 45 with respect to the x principal
axis and an angle of θ ¼ 60 with respect to the z principal axis.
(a) Find the polarization directions ^e o and ^e e , and the corresponding propagation
constants k o and ke , of the ordinary and extraordinary normal modes.
(b) Find the walk-off angle α of the extraordinary wave. What is the separation of
the ordinary and extraordinary beams if an optical wave that consists of both
ordinary and extraordinary components at this wavelength propagates in this direction through a BBO crystal over a distance of 3 mm?
The principal indices of refraction of quartz, which is a positive uniaxial crystal, are
nx ¼ ny ¼ no ¼ 1:544 and nz ¼ ne ¼ 1:553 at the λ ¼ 600 nm wavelength. Consider a
propagation direction k^ that makes an angle of ϕ ¼ 60 with respect to the x principal
axis and an angle of θ ¼ 30 with respect to the z principal axis.
(a) Find the polarization directions ^e o and ^e e , and the corresponding propagation
constants k o and ke , of the ordinary and extraordinary normal modes.
(b) Find the walk-off angle α of the extraordinary wave. What is the separation of the
ordinary and extraordinary beams if an optical wave that consists of both ordinary
and extraordinary components at this wavelength propagates in this direction
through a quartz crystal over a distance of 5 mm?
Show that there is no walk-off for an extraordinary wave when it propagates in any
direction that lies in the xy plane of a uniaxial crystal, for which the z principal axis is the
unique optical axis.
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Problems
135
3.3.1 Give two examples of TEM modes that are not plane waves: (a) one example in purely
dielectric medium and (b) another example not in purely dielectric medium.
3.3.2 A fundamental Gaussian beam from an Er:fiber laser at the λ ¼ 1:53 μm wavelength exits
the fiber with a spot size of w0 ¼ 8 μm, which is determined by the fiber core radius. The
beam then propagates in free space without being collimated. Find the beam divergence
angle, the Rayleigh range, and the confocal parameter of the beam. What are the spot
sizes and the radii of curvature of the beam at the distances of 1 mm, 1 cm, 10 cm, and
1 m, respectively, from the end of the fiber?
3.3.3 A Gaussian beam of an unknown wavelength in free space is found to have spot sizes of
w0 ¼ 100 μm at the beam waist and wðzÞ ¼ 300 μm at a distance of z ¼ 15 cm from the
beam waist. Find the wavelength, the Rayleigh range, and the divergence angle of
the beam.
3.3.4 A fundamental Gaussian laser beam that has a power of P ¼ 10 W at a wavelength of
λ ¼ 600 nm is focused to a small spot size for an intensity at the beam center of I 0 ¼
2:5 MW cm2 at its beam waist. What is the beam-waist radius w0 of the beam? What
is the divergence angle of the beam? What are its spot size and beam-center intensity at
a distance of 5 m from the beam waist? If the spot size is increased to w0 ¼ 50 μm
at the beam waist, what are the changes in the beam-center intensities at the beam waist
and at 5 m from the waist, respectively?
3.4.1 Consider reflection and transmission of TE and TM waves at the interface of two lossless
dielectric media that have real refractive indices of n1 and n2 , respectively. Use (3.91) and
(3.95) to show the following facts.
(a) For external reflection of a TE wave, the reflected field has a π phase change at any
incident angle. For internal reflection of a TE wave, the reflected field has no phase
change at any incident angle that is smaller than the critical angle.
(b) For external reflection of a TM wave, the reflected field has no phase change at
any incident angle that is smaller than the Brewster angle, θi < θB , but has a π phase
change at any incident angle that is larger than the Brewster angle, θi > θB . For
internal reflection of a TM wave, the reflected field has a π phase change at any
incident angle that is smaller than the Brewster angle, θi < θB , but has no phase
change at any incident angle that is larger than the Brewster angle and smaller than
the critical angle, θB < θi < θc .
3.4.2 When a collimated beam of broadband white light covering the spectrum from red to
violet is incident at an oblique angle from free space on a flat surface of ordinary glass,
the transmitted beam is no longer collimated. Sketch how the spectral components of the
transmitted beam spread from red to violet. Give a brief explanation why they spread in
that manner.
3.4.3 The refractive index of a glass plate is 1.5. It can be used as a reflection-type polarizer
so that if a beam is incident on its surface at a proper angle, the reflected beam is
always linearly polarized no matter what the polarization of the incident beam is. If the
glass plate is placed in air, what is this proper incident angle from the air? What is the
polarization of the reflected beam at this incident angle?
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136
Optical Wave Propagation
3.4.4 The refractive index of diamond at λ ¼ 1:0 μm is n ¼ 2:39. What is the reflectivity
of the diamond surface at normal incidence? At a particular incident angle, a specific
linearly polarized optical wave at λ ¼ 1:0 μm is completely transmitted through a
diamond surface exposed to air. What are this incident angle and the specific polarization
of the incident wave that make this happen?
3.4.5 The refractive index of water is 1.33. For the λ ¼ 600 nm wavelength, find the parameters
of the radiation modes at the air–water interface for internal reflection at the two different
incident angles of 45 and 75 , respectively. What is the penetration depth of
the evanescent tail into the air if a radiation mode is found to be a one-sided radiation
mode at a particular incident angle? What are the phase shifts on reflection at the interface
for TE and TM waves, respectively?
3.4.6 At the λ ¼ 1:5 μm wavelength, the refractive index of intrinsic GaAs is 3.38. Find the
parameters of the radiation modes at the air–GaAs interface for internal reflection at the
two different incident angles of 30 and 60 , respectively. What is the penetration depth
of the evanescent tail into the air if a radiation mode is found to be a one-sided radiation
mode at a particular incident angle? What are the phase shifts on reflection at the interface
for TE and TM waves, respectively?
3.4.7 Consider the interface between SiO2 and silver. The refractive index of SiO2 is 1.46 in the
visible spectral region. Use the plasma frequency ωp ¼ 1:36 1016 rad s1 of Ag to find
the surface plasma frequency of this interface. What are the cutoff frequency and cutoff
wavelength for the surface plasmon mode? Does the surface plasmon mode exist at the
λ ¼ 500 nm wavelength? If it exists, find its propagation constant and characteristic
parameters. Find the penetration depths of the mode into the SiO2 and the silver to find
its confinement at the interface.
3.4.8 Consider the interface between GaAs and silver. The refractive index of GaAs varies
with optical wavelength, increasing with decreasing wavelength. For simplicity, take the
refractive index of GaAs to be 3.51 at λ ¼ 1 μm. Use the plasma frequency ωp ¼ 1:36 1016 rad s1 of Ag to find the surface plasma frequency of this interface. What are the
cutoff frequency and cutoff wavelength for the surface plasmon mode? Does the surface
plasmon mode exist at the λ ¼ 500 nm and λ ¼ 1 μm wavelengths, respectively? If it
exists, find its propagation constant and characteristic parameters. Find the penetration
depths of the mode into the GaAs and the silver to find its confinement at the interface.
3.5.1 A step-index planar GaAs=AlGaAs waveguide has a GaAs core and AlGaAs cover
and substrate. At λ ¼ 900 nm, the GaAs core has n1 ¼ 3:593, the AlGaAs substrate
has n2 ¼ 3:409, and the AlGaAs cover of a different composition has n3 ¼ 3:261. In
what range can the propagation constant of a guided mode, if it exists, be found at the
λ ¼ 900 nm wavelength? Ignoring wavelength-dependent changes in the refractive
indices, for what wavelengths can a guided mode be found to have a propagation constant
of β ¼ 2:5 107 m1 ? What happens to the answers if the AlGaAs composition for the
cover is changed so that n3 ¼ 3:453?
3.5.2 A step-index planar glass waveguide has a glass core of n1 ¼ 1:54, a glass substrate
of a different composition of n2 ¼ 1:47, and a free-space cover of n3 ¼ 1:00. The core
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Problems
3.5.3
3.5.4
3.5.5
3.5.6
3.5.7
3.5.8
3.5.9
137
thickness is d ¼ 1:5 μm. What is the range of optical wavelength for the waveguide
to support the TE0 mode but not the TE1 mode? What is the range of optical wavelength for the waveguide to support the TM0 mode but not the TM1 mode? What is the
range of optical wavelength for the waveguide to support the TE0 mode but not the
TM0 mode?
A step-index planar glass waveguide has a glass core of n1 ¼ 1:54 and a substrate and a
cover of n2 ¼ n3 ¼ 1:47. The core thickness is d ¼ 1:5 μm. What is the range of optical
wavelength for the waveguide to support the TE0 mode but not the TE1 mode? What is
the range of optical wavelength for the waveguide to support the TM0 mode but not
the TM1 mode? What is the range of optical wavelength for the waveguide to support the
TE0 mode but not the TM0 mode?
What is the most outstanding difference between symmetric and asymmetric waveguides
in terms of finding guided modes?
A planar dielectric waveguide supports exactly three modes among all types of modes.
Name these modes. Which mode has the largest propagation constant? Which one has
the smallest propagation constant?
An asymmetric InGaAsP=InP waveguide has a refractive index of n1 ¼ 3:432 for its
core, and indices of n2 ¼ 3:354 and n3 ¼ 3:166 for its two cladding layers. What is
the required core thickness for the waveguide to have one and only one guided mode at
λ ¼ 1:55 μm, including modes of all different polarizations?
A symmetric step-index planar InGaAsP=InP waveguide has the high-index InGaAsP for
its core and the low-index InP for its cladding layers. At λ ¼ 1:55 μm, the core index is
n1 ¼ 3:432 and the cladding index is n2 ¼ n3 ¼ 3:166. If a single-mode waveguide
is desired, what is the required core thickness? Is the waveguide truly single-mode if
this requirement is met? Name the mode or modes.
A symmetric step-index planar InGaAsP=InP waveguide has a core index of n1 ¼ 3:438
and a cladding index of n2 ¼ 3:205. The core thickness is d ¼ 0:60 μm.
(a) At the λ ¼ 1:30 μm wavelength, how many guided modes are supported by the
waveguide? What are they?
(b) At what wavelengths does the waveguide support only one TE mode and one
TM mode?
A symmetric step-index planar GaAs=Al0:3 Ga0:7 As waveguide has the high-index GaAs
for its core and the low-index Al0:3 Ga0:7 As for its two cladding layers. At λ ¼ 1:5 μm,
the core index is n1 ¼ 3:38 and the cladding index is n2 ¼ 3:22.
(a) If a single-mode waveguide is desired, what is the required core thickness? Is the
waveguide truly single-mode if this requirement is met? Name the mode or modes.
(b) If the core thickness is chosen to be d ¼ 2 μm, how many guided modes are
supported by the waveguide? What are they?
(c) If the waveguide thickness is kept at d ¼ 2 μm, but its structure is made asymmetric
by lowering the index of only one cladding layer, would existing modes start
disappearing or new modes start appearing if that index is sufficiently reduced? What
is the first mode to disappear or appear if this happens?
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138
Optical Wave Propagation
3.6.1 The effective index of refraction of a single-mode optical fiber as a function of optical
wavelength around λ ¼ 1:3 μm is found to be approximated as nβ ¼ 1:465 0:0114
ðλ 1:3Þ 0:004ðλ 1:3Þ3 , where λ is in micrometers.
(a) Characterize the phase-velocity dispersion of this fiber at λ ¼ 1:2 μm and λ ¼ 1:5 μm,
respectively.
(b) Find and characterize the group-velocity dispersion of this fiber at λ ¼ 1:2 μm and
λ ¼ 1:5 μm, respectively.
(c) Express the group-velocity dispersion as Dλ in the unit of ps km1 nm1 at λ ¼ 1:2 μm
and λ ¼ 1:5 μm, respectively.
3.6.2 The fiber described in Problem 3.6.1 is used to transmit two optical pulses at λ ¼ 1:2 μm
and λ ¼ 1:5 μm, respectively. Each pulse has a pulse duration of Δt ps ¼ 5 ps and a
spectral width of Δλps ¼ 1 nm. Find the temporal widths of these two pulses after
propagating over a distance of 5 km in the fiber.
3.6.3 How far can the pulse at each of the three wavelengths described in Example 3.17
propagate through that fiber before the pulse broadening caused by group-velocity
dispersion is larger than the original pulse duration?
3.6.4 The ordinary and extraordinary indices of refraction of LiNbO3 in the wavelength range
between 1:0 and 2:0 μm vary with wavelength approximately as
no ¼ 2:2158 þ 0:00286λ2 0:0062λ2 ,
ne ¼ 2:1395 þ 0:00247λ2 0:0052λ2 :
(3.191)
Answer each of the following questions for the ordinary and extraordinary waves,
respectively.
(a) Within this wavelength range, where does LiNbO3 have normal dispersion? Where
does it have anomalous dispersion?
(b) Within this wavelength range, where does LiNbO3 have positive group-velocity
dispersion? Where does it have negative group-velocity dispersion?
(c) Find the refractive index, the group index, and the group-velocity dispersion of
LiNbO3 at the three wavelengths of λ ¼ 1:0 μm, 1:5 μm, and 2:0 μm.
(d) Express the group-velocity dispersion as Dλ in the unit of fs cm1 nm1 .
3.6.5 An optical pulse has a pulse duration of Δt ps ¼ 100 fs and a spectral width of
Δλps ¼ 75 nm. Use the values of Dλ obtained in Problem 3.6.4(d) for LiNbO3 to find
the pulse broadening caused by group-velocity dispersion after the pulse propagates over
1 cm in LiNbO3 . Find also the distance that the pulse can propagate in LiNbO3 before
its pulse duration doubles. Answer both questions for the pulse polarized in the ordinary
and extraordinary axes, respectively, and for its center wavelength at λ ¼ 1:0 μm, 1:5 μm,
and 2:0 μm, respectively.
^j given in (1.56)
3.7.1 By using the definition of the optical intensity I ¼ jS n^j ¼ jðS þ S Þ n
for a coherent wave and the equation k E ¼ ωμ0 H given in (3.31), show that the
optical intensity of a plane-wave mode projected on the surface that is normal to its
propagation direction k^ is given by the expression in (3.183).
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Bibliography
139
3.7.2 Show that under the condition that ϵ 00 ϵ 0 , so that χ 00 χ 0 , n00 n0 , and α k 0 , the
absorption coefficient can be approximated as
α k0
00
00
ϵ 00
2π χ 00
0 χ
0 χ
¼
k
k
¼
:
ϵ0
1 þ χ0
n02
λ n0
(3.192)
3.7.3 At the λ ¼ 300 nm wavelength, Si has a complex refractive index of n ¼ 5:0 þ i4:16, and
GaAs has n ¼ 3:73 þ i2:0. Find the absorption coefficients and the absorption depths of
Si and GaAs at this wavelength. What is the complex susceptibility for each material at
this wavelength?
3.7.4 The complex susceptibility of GaAs is χ ¼ 17:31 þ i3:70 at λ ¼ 500 nm and χ ¼ 12:55
þi0:63 at λ ¼ 800 nm. Find the absorption coefficient and the absorption depth of GaAs
at these wavelengths.
3.7.5 At λ ¼ 800 nm, Si has an absorption depth of α1 ¼ 9:8 μm and a reflectivity of 32:9% at
normal incidence on its surface exposed to air. Find its complex refractive index and
complex susceptibility at this wavelength.
3.7.6 An optical fiber of a length l ¼ 120 km has an attenuation coefficient of 0:3 dB km1 at
λ ¼ 1:3 μm and 0:15 dB km1 at λ ¼ 1:55 μm. If 2 mW of optical power at each
wavelength is launched into the fiber, what is the output power at each wavelength?
3.7.7 An optical fiber has an attenuation coefficient of 0:5 dB km1 at λ ¼ 1:3 μm and
0:2 dB km1 at λ ¼ 1:55 μm. If 1 mW of optical power at each wavelength is launched
into the fiber and the detection limit of a detector at each wavelength is 1 μW, what is the
maximum length of the fiber for the power at each wavelength to be detectable by the
detector?
Bibliography
Born, M. and Wolf, E., Principles of Optics: Electromagnetic Theory of Propagation, Interference and
Diffraction of Light, 7th edn. Cambridge: Cambridge University Press, 1999.
Buckman, A. B., Guided-Wave Photonics. Fort Worth, TX: Saunders College Publishing, 1992.
Davis, C. C., Lasers and Electro-Optics: Fundamentals and Engineering, 2nd edn. Cambridge: Cambridge
University Press, 2014.
Fowler, G. R., Introduction to Modern Optics, 2nd edn. New York: Dover, 1975.
Ebeling, K. J., Integrated Optoelectronics: Waveguide Optics, Photonics, Semiconductors. Berlin: SpringerVerlag, 1993.
Haus, H. A., Waves and Fields in Optoelectronics. Englewood Cliffs, NJ: Prentice-Hall, 1984.
Hunsperger, R. G., Integrated Optics: Theory and Technology, 5th edn. New York: Springer-Verlag, 2002.
Iizuka, K., Elements of Photonics, Vols. I and II. New York: Wiley, 2002.
Jackson, J. D., Classical Electrodynamics, 3rd edn. New York: Wiley, 1999.
Kasap, S. O., Optoelectronics and Photonics: Principles and Practices, 2nd edn. Upper Saddle River, NJ:
Prentice-Hall, 2012.
Liu, J.M., Photonic Devices. Cambridge: Cambridge University Press, 2005.
Marcuse, D., Theory of Dielectric Optical Waveguides, 2nd edn. Boston, MA: Academic Press, 1991.
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140
Optical Wave Propagation
Nishihara, H., Haruna, M., and Suhara, T., Optical Integrated Circuits. New York: McGraw-Hill, 1989.
Pollock, C. R. and Lipson, M, Integrated Photonics. Boston, MA: Kluwer, 2003.
Saleh, B. E. A. and Teich, M. C., Fundamentals of Photonics. New York: Wiley, 1991.
Syms, R. and Cozens, J., Optical Guided Waves and Devices. London: McGraw-Hill, 1992.
Yariv, A. and Yeh, P., Photonics: Optical Electronics in Modern Communications. Oxford: Oxford University
Press, 2007.
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Cambridge Books Online
http://ebooks.cambridge.org/
Principles of Photonics
Jia-Ming Liu
Book DOI: http://dx.doi.org/10.1017/CBO9781316687109
Online ISBN: 9781316687109
Hardback ISBN: 9781107164284
Chapter
4 - Optical Coupling pp. 141-168
Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.005
Cambridge University Press
4
4.1
Optical Coupling
COUPLED-MODE THEORY
..............................................................................................................
Coupled-mode theory deals with the coupling of normal modes of propagation due to spatially
dependent perturbations. The theory has broad applicability. It applies to the coupling of spatial
modes in various optical structures, including Gaussian spatial modes in a homogeneous
medium, interface modes, and waveguide modes.
The space- and time-dependent electric and magnetic fields of a normal mode at a given
frequency ω are expressed in the form of (3.1) and (3.2). Because the coupled-mode theory
describes mode coupling caused by spatially dependent perturbations, no temporal changes are
involved. Therefore, the time dependence of all fields remains exp ðiωtÞ throughout the
interaction so that it can be ignored in the expressions of the fields while ∂=∂t is replaced
by iω in Maxwell’s equations. Then, the two Maxwell equations for wave propagation can be
written in the form:
∇ E ¼ iωμ0 H,
(4.1)
∇ H ¼ iωϵ E:
(4.2)
The normal modes of an unperturbed optical structure are governed by (4.1) and (4.2). They
are mutually orthogonal and are normalized through the orthonormality relation given in (3.18).
These normal modes form a basis for linear expansion of any optical field at the frequency ω in
the optical structure:
X
^ ν ðx; yÞ exp ðiβ zÞ,
EðrÞ ¼
Aν E
(4.3)
ν
ν
HðrÞ ¼
X
ν
^ ν ðx; yÞ exp ðiβν zÞ,
Aν H
(4.4)
^ ν and H
^ ν are normalized mode fields; the linear expansion sums over all discrete
where E
indices of the guided modes and integrates over all continuous indices of the radiation and
evanescent modes. In the original, unperturbed structure where these modes are defined, the
normal modes do not couple because they are mutually orthogonal. Then, the expansion
coefficients Aν are constants that are independent of x, y, and z, as discussed in Section 3.1.
In the presence of a spatially dependent perturbation to an optical structure, the modes
defined by the original structure are not exact normal modes of the perturbed structure. For
this reason, the perturbation can cause coupling of these modes as they propagate. As a result, if
an optical field in the perturbed structure is expanded in terms of the normal modes of the
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142
Optical Coupling
unperturbed structure, the expansion coefficients are not constants of propagation but vary with
z as the optical field propagates through the structure:
EðrÞ ¼
X
ν
HðrÞ ¼
X
ν
^ ν ðx; yÞ exp ðiβ zÞ,
Aν ðzÞE
ν
(4.5)
^ ν ðx; yÞ exp ðiβν zÞ:
Aν ðzÞH
(4.6)
Because the power in a normal mode is given by Pν ¼ jAν j2 , according to (3.27), the z
dependence of Aν ðzÞ in the above indicates that the power of a mode that is coupled to another
mode does not remain a constant of propagation. Thus, coupling of modes leads to exchange of
mode power.
4.1.1 Single-Structure Mode Coupling
We first consider the coupling between normal modes in a single optical structure, such as a
single waveguide, that is subject to some perturbation. By single structure, we mean that the
entire optical structure is considered in defining the normal modes characterized by normalized
^ ν, H
^ ν of propagation constants βν . The structure can be a simple structure, such
mode fields E
as a homogeneous medium, a single interface, or a single waveguide; or it can be a compound
structure that consists of multiple interfaces or multiple waveguides. In any event, no matter
how complicated the structure might be, it is considered as a single entity and is described with
a single ϵ ðrÞ to define the normal modes.
A spatially dependent perturbation to the structure at a frequency of ω can be represented
by a single perturbing polarization, ΔPðrÞ, so that the equations in (4.1) and (4.2) are modified
as
∇ E ¼ iωμ0 H,
(4.7)
∇ H ¼ iωϵ E iωΔP:
(4.8)
Any optical field propagating in this perturbed structure can be expanded as (4.5) and (4.6)
while its propagation is governed by these two equations with ΔP 6¼ 0. Meanwhile, the normal
mode fields defined by the unperturbed structure, which are defined by (4.1) and (4.2), also
satisfy these two equations with ΔP ¼ 0.
Applying (4.7) and (4.8) to two arbitrary sets of fields, ðE1 ; H1 Þ and ðE2 ; H2 Þ, with respective
perturbations of ΔP1 and ΔP2 , we find the Lorentz reciprocity theorem:
∗
∗
∗
∇ E1 H ∗
2 þ E2 H1 ¼ iω E1 ΔP2 E2 ΔP1 ,
(4.9)
which holds for any two sets of fields that are respectively associated with two arbitrary
perturbations. To derive the couple-mode equation, we take ðE1 ; H1 Þ to be the optical field
propagating in the perturbed structure with ΔP1 ¼ ΔP, which can be expanded as (4.5) and
^ ν, H
^ ν defined by the unperturbed structure
(4.6), and ðE2 ; H2 Þ to be the normal mode fields E
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4.1 Coupled-Mode Theory
143
with ΔP2 ¼ 0. By substituting these into (4.9) and integrating both sides of the resultant
equation over the cross section of the waveguide, we find
ð∞ ð∞ ð∞ ð∞
Xd
iðβν βμ Þz
∗
∗
iβμ z
^
^
^ ∗ ΔPdxdy:
^
^
E ν H μ þ E μ H ν ^z dxdy ¼ iωe
E
Aν ðzÞe
μ
dz
ν
∞ ∞
∞ ∞
(4.10)
By applying the orthonormality relation given in (3.18) to (4.10), we find the general form of
the coupled-mode equations:
dAν
¼ iωeiβν z
dz
ð∞ ð∞
^ ∗ ΔPdxdy,
E
ν
(4.11)
∞ ∞
where the plus sign is used when βν > 0 for mode ν to be forward propagating in the positive z
direction, and the minus sign is used when βν < 0 for mode ν to be backward propagating in the
negative z direction.
The general form of the coupled-mode equations expressed in (4.11) is applicable to mode
coupling caused by any kind of spatially dependent perturbation on any feature of the optical
structure. For example, ΔP can be a perturbing polarization at the frequency ω on the fields in a
waveguide due to any of the external effects discussed in Section 2.6 or due to any nonlinear
optical susceptibility discussed in Section 2.7.
For the simple case where the perturbation can be represented by a change in the linear
polarization as
X
^ ν eiβν z ,
ΔP ¼ Δϵ E ¼ Δϵ Aν E
(4.12)
ν
the coupled-mode equations can be expressed in the form:
dAν X
iκνμ Aμ eiðβμ βν Þz ,
¼
dz
μ
(4.13)
where
ð∞ ð∞
κνμ ¼ ω
^ μdxdy
^ ∗ Δϵ E
E
ν
(4.14)
∞ ∞
is the coupling coefficient between mode ν and mode μ. This result is applicable to isotropic and
anisotropic structures. For an optical structure made of isotropic media, Δϵ simply reduces to a
^ ∗ Δϵ E
^ μ ¼ ΔϵE
^∗ E
^ μ in (4.14). For a lossless optical structure, the
scalar Δϵ so that E
ν
ν
dielectric tensor is a Hermitian matrix so that Δϵ ij ¼ Δϵ ∗
ji , as discussed in Section 2.2. Consequently, mode coupling in a lossless dielectric single structure is symmetric with
κνμ ¼ κ∗
μν :
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(4.15)
144
Optical Coupling
EXAMPLE 4.1
Any physical mechanism that creates a change in the optical permittivity of a material can
possibly be a perturbation for the coupling of two modes in a waveguide. Is the mode coupling
caused by the electro-optic Pockels effect symmetric? Is that caused by optical absorption in a
semiconductor due to current injection symmetric?
Solution:
The Pockels effect mainly changes the permittivity tensor without causing additional optical
loss. The permittivity change is Hermitian: Δϵ ¼ Δϵ † . Thus the mode coupling caused by this
effect is symmetric:
ð∞ ð∞
κνμ ¼ ω
^
E^ ∗
ν Δϵ E μ dxdy
∞ ∞
0
1∗
ð∞ ð∞
¼ @ω
A
E^ ν Δϵ ∗ E^ ∗
μ dxdy
∞ ∞
0
¼ @ω
ð∞ ð∞
1∗
† ^
A
E^ ∗
μ Δϵ E ν dxdy
∞ ∞
0
¼ @ω
ð∞ ð∞
)
κνμ ¼ κ∗
μν :
1∗
^
A
E^ ∗
μ Δϵ E ν dxdy
∞ ∞
¼
κ∗
μν
The permittivity change associated with optical absorption is not Hermitian: Δϵ 6¼ Δϵ † . Thus
the mode coupling caused by this effect is not symmetric:
ð∞ ð∞
κνμ ¼ ω
^
E^ ∗
ν Δϵ E μ dxdy
∞ ∞
0
¼ @ω
0
∞ ∞
ð∞ ð∞
¼ @ω
0
ð∞ ð∞
∞ ∞
6¼ @ω
ð∞ ð∞
1∗
A
E^ ν Δϵ ∗ E^ ∗
μ dxdy
1∗
† ^
A
E^ ∗
μ Δϵ E ν dxdy
)
κνμ 6¼ κ∗
μν :
1∗
^
A
E^ ∗
μ Δϵ E ν dxdy
∞ ∞
¼
κ∗
μν
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145
4.1 Coupled-Mode Theory
4.1.2 Multiple-Structure Mode Coupling
In a compound optical structure, such as a structure that consists of more than one waveguide, there
are two alternative approaches to the analysis of the characteristics of optical fields that propagate in
the structure. One approach is to treat the compound structure as a single super structure by expanding
any optical field in terms of its normal modes, known as the super modes, which are found by solving
Maxwell’s equations directly with the boundary conditions defined by the entire super structure. The
alternative approach is to divide the compound structure into separate substructures, expand the fields
in terms of the normal modes of the individual substructures, and treat the problem with a coupledmode approach. The first approach can yield exact solutions and is sometimes desirable. However, it
is not generally possible to obtain the exact super-mode solutions for complicated structures. The
coupled-mode approach yields approximate solutions, but it can be applied to most structures without
difficulty. In addition, it gives an intuitive picture of how optical waves interact in a compound
structure. Here we consider the coupled-mode formulation for multiple substructures.
The concept of dividing a super structure into a combination of individual substructures is
illustrated in Fig. 4.1. In this illustration, the individual waveguides are the substructures of the
multiple-waveguide super structure. The multiple-waveguide super structure is described by
ϵ ðx; yÞ, whereas the individual waveguides are described by ϵ a ðx; yÞ, ϵ b ðx; yÞ, ϵ c ðx; yÞ, and so on.
The normal modes are solved for each individual substructure. The fields in the entire structure
can be expanded in terms of these normal modes in the same form as (4.5) and (4.6) but with the
summation over the index ν covering all the modes of every substructure. From the standpoint of
any substructure, the presence of other substructures is a perturbation to it. Thus, for substructure
i that is described by ϵ i ðx; yÞ, the entire structure looks like ϵ i ðx; yÞ plus a perturbation of
Δϵ i ðx; yÞ ¼ ϵ ðx; yÞ ϵ i ðx; yÞ:
(4.16)
The coupled-mode equations for the multiple-structure scenario can be obtained by using the
reciprocity theorem of (4.9) and then following a procedure similar to that taken above to obtain
Figure 4.1 Schematic diagram of three coupled waveguides showing the decomposition into individual
waveguides, in solid curves, plus the corresponding perturbation, in dashed curves, for each of them.
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146
Optical Coupling
the coupled-mode equations for the single structure. Because the mathematics is quite involved,
only the results are given in the following without detailed derivation.
The coupled-mode equations for multiple substructures can still be written in the same form
as that of (4.13):
dAν X
iκνμ Aμ eiðβμ βν Þz ,
¼
dz
μ
(4.17)
where the plus sign is taken if mode ν is forward propagating, and the minus sign is used if it is
backward propagating. It is noted that the summation over the index μ runs through the modes
of every substructure, not just the modes of one single substructure. In contrast to that for
single-structure coupling discussed above, the coupling coefficients κνμ for multiple-structure
coupling have a complicated form and are best expressed in terms of matrix elements:
~ νμ ,
(4.18)
κνμ ¼ cνν c1 κ
where cνν ¼ 1 if mode ν is forward propagating and cνν ¼ 1 if it is backward propagating, as
~ ¼ κ~νμ are given,
can be seen from (4.19) below. The elements of the matrices c ¼ cνμ and κ
respectively, by
cνμ ¼
ð∞ ð∞ ∗
∗
^
^
^
^
E ν H μ þ E μ H ν ^z dxdy ¼ c∗
μν
(4.19)
∞ ∞
and
ð∞ ð∞
κ~νμ ¼ ω
^ μdxdy:
^ ∗ Δϵ μ E
E
ν
(4.20)
∞ ∞
Note that Δϵ μ in (4.20) is the perturbation, defined in (4.16), to the substructure that defines the
^ μ, H
^ μ of normal mode μ. The coefficient cνμ represents the overlap coefficient of
fields E
^ ν, H
^ μ, H
^ ν and E
^ μ , which can be the mode fields of different substructures in the super
E
structure. In general, cνμ 6¼ 0 because modes of different substructures are not necessarily
orthogonal to each other. Because the mode fields used in (4.19) are normalized, we have
cνν ¼ 1 or cνν ¼ 1, depending
on whether mode ν is forward or backward propagating as
mentioned above, and cνμ 1 for any ν and μ. Note also the difference between the form of κ~νμ
expressed in (4.20) and that of the single-structure coupling coefficients κνμ given in (4.14).
As discussed above and expressed in (4.15), the coupling between modes of a single structure
is always symmetric with κνμ ¼ κ∗
μν if the structure is dielectric and lossless. By contrast, the
coupling between modes of different substructures in a super structure, such as those of
different individual waveguides in a multiple-waveguide structure, is generally asymmetric:
∗
κ~νμ 6¼ κ~∗
μν and κνμ 6¼ κμν
(4.21)
where ν and μ refer to modes of two different substructures. Indeed, it can be shown by using
the reciprocity theorem that
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4.2 Two-Mode Coupling
κ~νμ κ~∗
μν ¼
cνμ þ c∗
μν
βν βμ ¼ cνμ βν βμ :
2
147
(4.22)
This relation indicates that there is a direct relationship between the coupling coefficients and
the propagation constants. It has the following implications.
1. Unless βν ¼ βμ or cνμ ¼ c∗
μν ¼ 0, coupling between two modes is not symmetric, i.e.,
∗
κνμ 6¼ κμν , because the normal modes of different substructures are not necessarily orthogonal to each other.
2. The coupling of modes of the same order between two identical substructures is always
∗
symmetric because βν ¼ βμ , resulting in κ~νμ ¼ κ~∗
μν and κνμ ¼ κμν .
3. The relation in (4.22) applies to modes of a single structure as well. In this situation,
~νμ ¼ κνμ . Therefore, κνμ ¼ κ∗
cνμ ¼ c∗
μν ¼ 0 if ν 6¼ μ, and κ
μν in (4.15) holds true for the
normal modes of the same structure because they are mutually orthogonal.
4. It is not possible to change the coupling between two modes without simultaneously
changing their overlap coefficient or their propagation constants.
4.2
TWO-MODE COUPLING
..............................................................................................................
The coupling between two modes is the simplest and most common situation of mode coupling.
It includes coupling between two modes of the same structure, such as mode coupling in a
single waveguide that is modulated by a grating, or coupling between modes of two substructures, such as mode coupling in a directional coupler that is formed by two parallel waveguides.
For two-mode coupling, the coupled-mode equations can be written in a simple form that can
be analytically solved. In this section, we consider the general formulation of two-mode
coupling.
We have shown that both coupling among modes of a single structure and coupling among
modes of different substructures can be described by coupled-mode equations of the same form
as given in (4.13) and (4.17). The only difference is that the coupling coefficients in (4.17) for
multiple-structure mode coupling are defined differently from those in (4.13) for singlestructure mode coupling. This commonality is convenient because the general solutions of
the coupled-mode equations can be applied to both cases. For a particular problem, we only
have to calculate the coupling coefficients that are specific to the problem under consideration.
For two-mode coupling either in a single structure or between two different substructures, the
field expansion in (4.5) and (4.6) consists of only two modes, designated as mode a and mode b
of amplitudes A and B, respectively. Thus, coupled-mode equations of the form given in (4.13)
or (4.17) reduce to the following two coupled equations:
dA
¼ iκaa A þ iκab Beiðβb βa Þz ,
dz
(4.23)
dB
¼ iκbb B þ iκba Aeiðβa βb Þz :
dz
(4.24)
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148
Optical Coupling
For coupling between two modes of a single structure, the coupling coefficients in these
equations are given by (4.14), which are always symmetric with κab ¼ κ∗
ba if the structure is
dielectric and lossless. For coupling between modes of two different substructures, the coupling
coefficients are given by (4.18), which can be explicitly expressed as
κaa ¼
κba
κ~aa cab κ~ba =cbb
κ~ab cab κ~bb =cbb
, κab ¼
,
1 cab cba =caa cbb
1 cab cba =caa cbb
κ~ba cba κ~aa =caa
κ~bb cba κ~ab =caa
¼
, κbb ¼
:
1 cab cba =caa cbb
1 cab cba =caa cbb
(4.25)
As discussed earlier and expressed in (4.21), in general κab 6¼ κ∗
ba for coupling between modes
of two different substructures.
The iκaa A and iκbb B terms in the coupled equations (4.23) and (4.24) are self-coupling terms.
These terms are caused by the fact that the normal modes see in the perturbed structure an index
profile that is different from the index profile of the unperturbed original structure where the
modes are defined. They can be removed from the equations by expressing the normal-mode
expansion coefficients as
2 z
3
ð
~ ðzÞ exp 4i κaa ðzÞdz5,
(4.26)
AðzÞ ¼ A
0
2
3
ðz
~ ðzÞ exp 4i κbb ðzÞdz5,
BðzÞ ¼ B
(4.27)
0
where a plus or minus sign is chosen for a forward-propagating or backward-propagating mode,
respectively. Then (4.23) and (4.24) can be transformed into two coupled equations in terms of
~ and B
~ to remove the self-coupling terms:
A
~
dA
~ iφðzÞ ,
¼ iκab ðzÞBe
dz
(4.28)
~
dB
~ iφðzÞ ,
¼ iκba ðzÞAe
dz
(4.29)
where
2
ðz
3
2
ðz
3
φðzÞ ¼ 4βb z κbb ðzÞdz5 4βa z κaa ðzÞdz5:
0
(4.30)
0
As shown in (4.28)(4.30), we have to consider the fact that each coupling coefficient can be
a function of z because Δϵ can be a function of z but the integration in (4.14) and (4.20) is
carried out only over x and y. In the case when κab ðzÞ and κba ðzÞ are arbitrary functions of z, the
coupled-mode equations cannot be analytically solved. In this situation, there is no need to
further simplify the coupled-mode equations because they can only be numerically solved.
However, for optical structures of practical interest that are designed for two-mode coupling, Δϵ
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4.2 Two-Mode Coupling
149
is usually either independent of z or periodic in z. Then, the coupling coefficients are either
independent of z or periodic in z. In either case, (4.28) and (4.29) can be reduced to the
~ and B
~ with κab and κba being constants that are independfollowing general form in terms of A
ent of z:
~
dA
~ i2δz ,
¼ iκab Be
dz
(4.31)
~
dB
~ i2δz :
¼ iκba Ae
dz
(4.32)
The parameter 2δ is the phase mismatch between the two modes. Perfectly phase-matched
coupling of two modes with δ ¼ 0 is always symmetric with κab ¼ κ∗
ba irrespective of whether
these two modes belong to the same structure or two different substructures.
The general form of (4.31) and (4.32) applies to both cases of uniform and periodic
perturbations, but the details of the parameters vary between the two cases.
4.2.1 Uniform Perturbation
In this case, Δϵ is only a function of x and y but is not a function of z. Then all of the coupling
coefficients κaa , κbb , κab , and κba in (4.28)(4.30) are constants that are independent of z. We
then find that
~ ðzÞeiκaa z ,
AðzÞ ¼ A
~ ðzÞeiκbb z ,
BðzÞ ¼ B
(4.33)
and 2δz ¼ φðzÞ ¼ ½ðβb κbb Þ ðβa κaa Þz for (4.30) so that
2δ ¼ ðβb κbb Þ ðβa κaa Þ:
(4.34)
The choice of sign in each in (4.33) and (4.34) is consistent with that in (4.26) and (4.27)
discussed above. The physical meaning of the self-coupling coefficients, κaa and κbb , is a
change in the propagation constant of each normal mode. While the propagation constants of
the normal modes in the original unperturbed structure are βa and βb , their values are changed
by the perturbation characterized by Δϵ. These modes now propagate with the modified
propagation constants βa κaa and βb κbb , respectively, which take into account the effect
of the perturbation on the structure. In addition, they couple to each other through κab and κba .
With the simple transformation of (4.33) and the phase mismatch 2δ given in (4.34), twomode coupling due to a uniform perturbation is described by the general form of (4.31) and
(4.32) with constant values of κab and κba . A good example of two-mode coupling due to a
uniform perturbation is that in a two-channel directional coupler, which consists of two parallel
single-mode waveguides, as shown schematically in Fig. 4.2. This is the case of multiplestructure coupling. If the two waveguides are not identical, the directional coupler is not
symmetric. Then, in general κba 6¼ κ∗
ab , as discussed in Section 4.1. Furthermore, 2δ 6¼ 0 except
for a certain possible phase-matched optical frequency because κaa 6¼ κbb and βa 6¼ βb in
general. If the two waveguides are identical, the directional coupler is symmetric. Then,
κba ¼ κ∗
ab , κaa ¼ κbb , and βa ¼ βb so that 2δ ¼ 0 for all frequencies.
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150
Optical Coupling
Figure 4.2 Schematic diagram of (a) a two-channel directional coupler of a length l consisting of two parallel
waveguides and (b) its index profile assuming two step-index waveguides on the same substrate. The coupler is
symmetric if na ¼ nb ¼ n1 and d a ¼ d b ¼ d.
4.2.2 Periodic Perturbation
In this case, Δϵ is a periodic function of z, and so are the coupling coefficients κaa ðzÞ, κbb ðzÞ, κab ðzÞ,
and κba ðzÞ in (4.28)(4.30). The periodic perturbation has a period of Λ and a wavenumber of
2π
:
(4.35)
Λ
Each coupling coefficient, being periodic in z with a period of Λ, can be expanded in a Fourier series:
X
X
κνμ ðqÞ exp ðiqKzÞ ¼
κνμ ðqÞ exp ðiqKzÞ
(4.36)
κνμ ðzÞ ¼
K¼
q
q
where q represents the order of coupling, the summation over q runs through all integers, and
ðΛ
1
κνμ ðqÞ ¼
κνμ ðzÞ exp ðiqKzÞdz:
Λ
(4.37)
0
Using (4.36) for κab ðzÞ and κba ðzÞ, (4.28) and (4.29) can be expressed as
X
~
dA
~ iφðzÞþiqKz ,
κab ðqÞBe
¼i
dz
q
(4.38)
X
~
dB
~ iφðzÞiqKz :
κba ðqÞAe
¼i
dz
q
(4.39)
For κaa ðzÞ and κbb ðzÞ, we find that
ðz
X κνν ðqÞ κνν ðzÞdz ¼ κνν ð0Þz þ
eiqKz 1 :
iqK
q6¼0
0
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(4.40)
4.2 Two-Mode Coupling
151
The κνν ð0Þ term represents a possible uniform perturbation that might exist due to a uniform
bias in the periodic Δϵ. It can be removed by redefining Δϵ or by considering it separately. In
any event, for Kz 1,
X κ ðqÞ νν
iqKz
1 Kz:
(4.41)
e
q6¼0 iqK
Therefore, the contributions of the q 6¼ 0 terms of κaa ðzÞ and κbb ðzÞ to the z-dependent phases in
(4.38) and (4.39) are negligible so that
2
3 2
3
ðz
ðz
φðzÞ þ qKz ¼ 4βb z κbb ðzÞdz5 4βa z κaa ðzÞdz5 þ qKz
(4.42)
0
0
f½βb κbb ð0Þ ½βa κaa ð0Þ þ qK gz:
With this approximation, the coupled-mode equations in the case of a periodic perturbation can
be expressed as
X
~
dA
~ iφðzÞþiqKz
κab ðqÞBe
¼i
dz
q
X
~
dB
~ iφðzÞiqKz
κba ðqÞAe
¼i
dz
q
~ i2δz ,
iκab ðqÞBe
(4.43)
~ i2δz ,
iκba ðqÞAe
(4.44)
where
2δ ¼ ½βb κbb ð0Þ ½βa κaa ð0Þ þ qK:
(4.45)
Note that only one term in the Fourier series that yields a minimum value for jδj is kept in each
of the two coupled-mode equations expressed in (4.43) and (4.44) because only this term will
effectively couple the two modes. Thus, the coupled-mode equations in (4.43) and (4.44) have
the general form of (4.31) and (4.32) with κab ¼ κab ðqÞ and κba ¼ κba ðqÞ being constants that
are independent of z, where q is the integer chosen to minimize the phase mismatch given
in (4.45).
The most common periodic perturbations are gratings. The simplest gratings are onedimensional gratings. For our purpose, such one-dimensional gratings are structures that are
periodic only in the longitudinal direction, which is taken to be the z direction. Grating
waveguide couplers have many useful applications and are one of the most important kinds
of waveguide couplers. They consist of periodic fine structures that form gratings in waveguides. The grating in a waveguide can take the form of either periodic index modulation or
periodic structural corrugation. Periodic index modulation can be permanently written in a
waveguide by periodically modulating the doping concentration in the waveguide medium, for
example, or it can be created by an electro-optic, acousto-optic, or nonlinear optical effect.
Figure 4.3 shows some examples of planar grating waveguide couplers in single waveguides. In
these examples, there is no uniform perturbation apart from the periodic perturbation; therefore,
κaa ð0Þ ¼ κbb ð0Þ ¼ 0 in (4.45) for these single-waveguide grating couplers.
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152
Optical Coupling
Figure 4.3 Structures of planar grating waveguide couplers with (a) and (b) periodic index modulation, (c), (d),
(e), and (f) periodic structural corrugation.
EXAMPLE 4.2
Find the qth-order coupling coefficient κνμ ðqÞ for a sinusoidal grating that has a period of Λ, as
shown in Fig. 4.3(c), such that κνμ ðzÞ ¼ a cos Kz, where K ¼ 2π=Λ. Find it for a squarefunction grating that has a period of Λ and a duty factor of ξ, as shown in Fig. 4.3(d), such
that κνμ ðzÞ ¼ a for 0 < z < ξΛ and κνμ ðzÞ ¼ a for ξΛ < z < Λ within each period. In each
case, which orders are useful for mode coupling?
Solution:
For the sinusoidal grating, we find by using (4.37) that
ðΛ
1
κνμ ðqÞ ¼
κνμ ðzÞ exp ðiqKzÞdz
Λ
0
ðΛ
¼
1
a cos Kz exp ðiqKzÞdz
Λ
0
¼
a
Λ
ðΛ
0
exp ðiKz iqKzÞ þ exp ðiKz iqKzÞ
dz:
2
a
¼ δq, 1 þ δq, 1 ,
2
where δq, 1 and δq, 1 are the Kronecker delta functions. Therefore, only the order q ¼ 1 and
q ¼ 1 the order are useful for mode coupling because only these two orders have a nonzero
coupling coefficient of κνμ ð1Þ ¼ κνμ ð1Þ ¼ a=2.
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4.2 Two-Mode Coupling
153
For the square-function grating, we find by using (4.37) that
ðΛ
1
κνμ ðqÞ ¼
κνμ ðzÞ exp ðiqKzÞdz
Λ
0
ξΛ
ð
ðΛ
1
1
¼
a exp ðiqKzÞdz a exp ðiqKzÞdz:
Λ
Λ
0
¼ 2a
sin ξqπ iξqπ
:
e
qπ
ξΛ
We find that κμν ðqÞ for a given value of q can be made nonzero by an appropriate choice of the
duty factor ξ. Therefore,
any order can be used if the value of ξ is properly chosen to maximize
the value of κνμ ðqÞ for a given q. However, it is possible to have κνμ ðqÞ ¼ 0 for certain
combinations of the values
of q and ξ, such as q ¼ 2 and ξ ¼ 1=2, or q ¼ 3 and ξ ¼ 1=3, etc.
The largest value of κνμ ðqÞ appears when q ¼ 1 or q ¼ 1 while ξ ¼ 1=2 so that
κνμ ðqÞ ¼ 2a=π.
A grating can also be used in a multiple-structure coupler. Figure 4.4 shows an example of a
grating placed in a dual-channel coupler that consists of two waveguides. The two waveguides
can be either identical, as in a symmetric structure, or nonidentical, as in an asymmetric
structure. In both cases, the phase mismatch of this dual-channel coupler with a grating is that
given in (4.45) with κaa ð0Þ 6¼ 0 and κbb ð0Þ 6¼ 0 due to the uniform perturbation on one
waveguide by the other waveguide, as in the directional coupler shown in Fig. 4.2.
EXAMPLE 4.3
Find the grating period for perfect phase matching of two modes a and b.
Solution:
For perfect phase matching, the phase mismatch given in (4.45) between two modes a and b of
propagation constants βa and βb has to be made zero by the perturbation of a grating:
2δ ¼ ½ βb κbb ð0Þ ½ βa κaa ð0Þ þ qK ¼ 0
)
)
)
qK ¼ ½ βa κaa ð0Þ ½ βb κbb ð0Þ
2π
¼ ½ βa κaa ð0Þ ½ βb κbb ð0Þ
Λ
2qπ
Λq ¼
,
½ βa κaa ð0Þ ½ βb κbb ð0Þ
q
where ½ βa κaa ð0Þ ½ βb κbb ð0Þ is the total phase mismatch including all uniform perturbations on the structure, and the sign of q is chosen to be the sign of ½ βa κaa ð0Þ ½ βb κbb ð0Þ
so that the grating period Λ has a positive value.
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154
Optical Coupling
Figure 4.4 Dual-channel directional coupler with a
grating of period Λ.
With the above general considerations, (4.31) and (4.32) represent the most general coupled
equations for two-mode coupling in structures of practical interest. They can be analytically
solved; their solutions apply to various two-mode coupling problems.
4.3
CODIRECTIONAL COUPLING
..............................................................................................................
First, we consider the coupling of two modes that propagate in the same direction, taken to be
the positive z direction, over a length of l, as is shown in Fig. 4.5. In this case, βa > 0 and
βb > 0. The coupled equations are
~
dA
~ i2δz ,
¼ iκab Be
dz
(4.46)
~
dB
~ i2δz :
¼ iκba Ae
dz
(4.47)
The equations for codirectional coupling are generally solved as an initial-value problem with
~ ðzÞ and B
~ ðz0 Þ and B
~ ðz0 Þ at z ¼ z0 to find the values of A
~ ðzÞ at any other
given initial values of A
location z. The general solution can be expressed in the matrix form:
"
#
"
#
~ ðzÞ
~ ðz0 Þ
A
A
¼ Fðz; z0 Þ
,
(4.48)
~ ðzÞ
~ ðz0 Þ
B
B
where the forward-coupling matrix Fðz; z0 Þ relates the field amplitudes at the location z0 to
those at the location z. It has the form:
2
3
βc cos βc ðzz0 Þiδ sin βc ðzz0 Þ iδðzz0 Þ
iκab
iδðzþz0 Þ
e
sin
β
ð
zz
Þe
0
c
6
7
βc
βc
7
Fðz;z0 Þ ¼ 6
4
iκba
βc cos βc ðzz0 Þþiδ sin βc ðzz0 Þ iδðzz0 Þ 5
iδðzþz0 Þ
sin βc ðzz0 Þe
e
βc
βc
(4.49)
where
1=2
:
βc ¼ κab κba þ δ2
(4.50)
We consider a simple case when power is launched only into mode a at z ¼ 0. Then the initial
~ ð0Þ 6¼ 0 and B
~ ð0Þ ¼ 0. By applying these conditions to (4.48) and taking z0 ¼ 0 in
values are A
(4.49), we find that
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4.3 Codirectional Coupling
155
Figure 4.5 Codirectional coupling between two modes of propagation constants βa and βb (a) in the same
waveguide and (b) in two parallel waveguides. A perturbation is required for codirectional coupling in the same
waveguide but is not required for codirectional coupling between two waveguides.
Figure 4.6 Periodic power exchange between two codirectionally coupled modes for (a) the phase-mismatched
condition δ 6¼ 0 and (b) the phase-matched condition δ ¼ 0. The solid curves represent Pa ðzÞ=Pa ð0Þ, and the
dashed curves represent Pb ðzÞ=Pa ð0Þ.
~ ðzÞ ¼ A
~ ð0Þ cos βc z iδ sin βc z eiδz ,
A
βc
(4.51)
~ ðzÞ ¼ B
~ ð0Þ iκba sin βc z eiδz :
B
βc
(4.52)
The power in the two modes varies with z as
~ ðzÞ 2 κab κba
Pa ðzÞ A
δ2
2
¼
cos
β
z
þ
,
¼
c
~ ð0Þ
Pa ð0Þ
β2c
β2c
A
(4.53)
~ ðzÞ 2 jκba j2
Pb ðzÞ B
¼
¼
sin2 βc z:
2
~ ð0Þ
Pa ð0Þ A
βc
(4.54)
The coupling efficiency for codirectional coupling over a length of l is
η¼
Pb ðlÞ jκba j2 2
¼ 2 sin βc l:
Pa ð0Þ
βc
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(4.55)
156
Optical Coupling
Thus, power is exchanged periodically between two modes with a coupling length of
lc ¼
π
,
2βc
(4.56)
where maximum power transfer occurs. Figure 4.6 shows the periodic power exchange between
the two coupled modes as a function of z. As can be seen from Fig. 4.6, complete power transfer
can occur only in the phase-matched condition when δ ¼ 0.
EXAMPLE 4.4
Find the maximum coupling efficiency for codirectional coupling and the length of a codirectional coupler that reaches this efficiency. What happens if the phase mismatch is large such
that δ2 > κab κba ?
Solution:
From (4.55), the maximum efficiency for codirectional coupling is
ηmax ¼
jκba j2
jκba j2
¼
,
β2c
κab κba þ δ2
which is reached when sin2 βc l ¼ 1. Because sin2 βc l is periodic, sin2 βc l ¼ 1 has many
solutions. The length to reach the maximum efficiency is any of
lmax ¼ ð2m þ 1Þ
π
¼ ð2m þ 1Þlc for m ¼ 0, 1, 2, . . .
2βc
The formulas obtained above remain valid for δ2 > κab κba . There are no qualitative changes,
but only quantitative changes, when the phase mismatch is large such that δ2 > κab κba . The
maximum coupling efficiency decreases with increasing phase mismatch because βc increases
with δ2 . The length lmax to reach the maximum efficiency also decreases with increasing phase
mismatch because the coupling length lc decreases with increasing βc .
4.4
CONTRADIRECTIONAL COUPLING
..............................................................................................................
We now consider the coupling of two modes that propagate in opposite directions over a length
of l, as is shown in Fig. 4.7 where mode a is forward propagating in the positive z direction and
mode b is backward propagating in the negative z direction. In this case, βa > 0 and βb < 0.
Thus, the coupled equations are
~
dA
~ i2δz ,
¼ iκab Be
dz
(4.57)
~
dB
~ i2δz :
¼ iκba Ae
dz
(4.58)
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4.4 Contradirectional Coupling
157
Figure 4.7 Contradirectional coupling between two modes of propagation constants βa and βb (a) in the same
waveguide and (b) in two parallel waveguides. A significant perturbation is required for contradirectional
coupling in both cases.
The equations for contradirectional coupling are generally solved as a boundary value problem with
~ ð0Þ at one end and B
~ ðzÞ and B
~ ðlÞ at the other end to find the values of A
~ ðzÞ
given boundary values of A
at any location z between the two ends. The general solution can be expressed in the matrix form:
"
#
"
#
~ ðzÞ
~ ð0Þ
A
A
¼ Rðz; 0; lÞ
(4.59)
~ ðzÞ
~ ðlÞ
B
B
~ ð0Þ at z ¼ 0 and B
~ ðlÞ
where the reverse-coupling matrix Rðz; 0; lÞ relates the field amplitudes A
at z ¼ l to those at any location z. It has the form:
2
3
αc cosh αc ðl zÞ þ iδ sinh αc ðl zÞ iδz
iκab sinh αc z
iδðlþzÞ
e
e
6
7
αc cosh αc l þ iδ sinh αc l
αc cosh αc l þ iδ sinh αc l
6
7
Rðz; 0; lÞ ¼ 6
7
4
iκba sinh αc ðl zÞ
αc cosh αc z þ iδ sinh αc z iδðlzÞ 5
iδz
e
e
αc cosh αc l þ iδ sinh αc l
αc cosh αc l þ iδ sinh αc l
(4.60)
where
1=2
:
αc ¼ κab κba δ2
(4.61)
We consider a simple case when power is launched only into mode a at z ¼ 0 but not into
~ ð0Þ 6¼ 0 and B
~ ðlÞ ¼ 0. By applying these
mode b at z ¼ l. Then the boundary values are A
conditions to (4.59), we find that
~ ðzÞ ¼ A
~ ð0Þ αc cosh αc ðl zÞ þ iδ sinh αc ðl zÞ eiδz ,
A
αc cosh αc l þ iδ sinh αc l
~ ð0Þ
~ ðzÞ ¼ A
B
iκba sinh αc ðl zÞ
eiδz :
αc cosh αc l þ iδ sinh αc l
The power in the two contradirectionally coupled modes varies with z as
~ ðzÞ 2 cosh2 αc ðl zÞ δ2 =κab κba
Pa ðzÞ A
¼
,
¼
~ ð0Þ
Pa ð0Þ A
cosh2 αc l δ2 =κab κba
~ ðzÞ 2 κ∗
Pb ðzÞ B
sinh2 αc ðl zÞ
¼ ba
¼
:
~ ð0Þ
κab cosh2 αc l δ2 =κab κba
Pa ð0Þ
A
(4.62)
(4.63)
(4.64)
(4.65)
Because mode b is propagating backward with no input at z ¼ l but with an output at z ¼ 0, the
coupling efficiency for contradirectional coupling over a length of l is
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158
Optical Coupling
Figure 4.8 Power exchange between two contradirectionally coupled modes for (a) the phase-mismatched
condition δ 6¼ 0 and (b) the phase-matched condition δ ¼ 0. The solid curves represent Pa ðzÞ=Pa ð0Þ, and the
dashed curves represent Pb ðzÞ=Pa ð0Þ.
~ ð0Þ2 κ∗
Pb ð0Þ B
sinh2 αc l
¼ ba
η¼
:
¼
~ ð0Þ
Pa ð0Þ
κab cosh2 αc l δ2 =κab κba
A
(4.66)
Figure 4.8 shows the power exchange between the two contradirectionally coupled modes as a
2
function of z. Power transfer approaches 100% as l ! ∞ if κab ¼ κ∗
ba and δ < κab κba .
~ ð0Þ 6¼ 0 and B
~ ðlÞ ¼ 0, as considered above, contradirectional coupling can
In the case when A
~ ð0Þ at z ¼ 0 with a reflection coefficient of
be viewed as reflection of the field amplitude A
~ ð0Þ
iκba sinh αc l
B
r ¼ jr jeiφ ¼
:
(4.67)
¼
~ ð0Þ αc cosh αc l þ iδ sinh αc l
A
The reflectivity is R ¼ jr j2 ¼ η as is given in (4.66). The phase shift is
φ¼
π
δ
tanh αc l
þ φκba tan1
2
αc
¼ φPM tan1
δ
tanh αc l ,
αc
(4.68)
where φκba is the phase angle of κba , and φPM ¼ π=2 þ φκba is the phase shift at the phasematched point where δ ¼ 0.
EXAMPLE 4.5
Find the maximum coupling efficiency for contradirectional coupling and the length of a
contradirectional coupler that reaches this efficiency. What happens if the phase mismatch is
large such that δ2 > κab κba ?
Solution:
In the case when δ2 < κab κba , the parameter αc given in (4.61) has a real, positive value. Then,
sinh αc l and cosh αc l are both monotonic functions with sinh αc l ! 1 and cosh αc l ! 1 as
l ! ∞. From (4.66), the maximum efficiency for contradirectional coupling in the case when
δ2 < κab κba is
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4.5 Conservation of Power
ηmax ¼
159
κ∗
ba
,
κab
which can only be asymptotically reached when l ! ∞. Therefore, lmax ¼ ∞ when δ2 < κab κba .
In the case when δ2 > κab κba , we find that the parameter αc given in (4.61) becomes purely
imaginary:
1=2
1=2
¼ iγc with γc ¼ δ2 κab κba
:
αc ¼ κab κba δ2
Then the coupling efficiency given in (4.66) becomes
η¼
κ∗
sin2 γc l
ba
:
κab δ2 =κab κba cos2 γc l
We find that η varies with l periodically. By taking dη=dðγc lÞ ¼ 0, the maximum value of η is
found when 2γc l ¼ ð2m þ 1Þπ. Thus, it takes place when sin2 γc l ¼ 1 and cos2 γc l ¼ 0 with
ηmax ¼
jκba j2
:
δ2
The length to reach this maximum efficiency is any of
lmax ¼ ð2m þ 1Þ
π
ð2m þ 1Þπ
¼ 2γc 2 δ2 κab κba 1=2
for m ¼ 0, 1, 2, . . .
For contradirectional coupling, there is a qualitative change in the coupling efficiency when the
phase mismatch becomes large so that δ2 > κab κba .
4.5
CONSERVATION OF POWER
..............................................................................................................
Conservation of power requires that in a lossless structure the net power flowing across any
cross section of the structure be a constant that does not vary along the longitudinal direction of
the structure. For codirectional coupling between two modes with the power initially launched
into only one mode such that Pa ð0Þ 6¼ 0 but Pb ð0Þ ¼ 0, this requirement suggests that the sum
of power in the two waveguides, Pa ðzÞ þ Pb ðzÞ, be a constant independent of z because the
power in the two modes flows in the same direction. For contradirectional coupling with the
power launched into only one mode such that Pa ð0Þ 6¼ 0 and Pb ðlÞ ¼ 0, this requirement
suggests that Pa ðzÞ Pb ðzÞ be a constant independent of z because the power in mode b flows
in the backward direction while that in mode a flows in the forward direction. These conclusions are correct for mode coupling in a single structure, such as a single waveguide, but they
do not generally hold for coupling between modes of two different substructures, such as two
separate waveguides.
It can be seen from (4.53) and (4.54) that Pa ðzÞ þ Pb ðzÞ is not a constant of z for codirectional
coupling unless κab ¼ κ∗
ba . Similarly, from (4.64) and (4.65), it is also found that Pa ðzÞ Pb ðzÞ
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Optical Coupling
is not a constant of z for contradirectional coupling when κab 6¼ κ∗
ba . It seems that the total
power is not conserved in a lossless structure in the case of asymmetric coupling with
κab 6¼ κ∗
ba . A close examination reveals that because cab 6¼ 0 in the case of asymmetric
coupling, the two interacting modes are not orthogonal to each other. For this reason, the total
power flow cannot be fully accounted for by gathering the power in each individual mode as if
the modes were mutually orthogonal. Indeed, by expanding the total electric and magnetic
fields in the structure as a linear superposition of the two modes in the form of (4.5) and (4.6) to
calculate the power of the entire structure, we find that the total power as a function of space is
PðzÞ ¼ caa jAðzÞj2 þ cbb jBðzÞj2 þ 2Re cab A∗ ðzÞBðzÞeiΔβz
(4.69)
¼ caa Pa ðzÞ þ cbb Pb ðzÞ þ Pab ðzÞ,
where Pab ðzÞ ¼ 2Re cab A∗ ðzÞBðzÞeiΔβz can be considered as the power residing between the
two nonorthogonal modes of the two different substructures. As defined in Section 4.1, cνν ¼ 1
if mode ν is forward propagating and cνν ¼ 1 if mode ν is backward propagating. It can be
shown, using (4.53) and (4.54) for the case of codirectional coupling and using (4.64) and
(4.65) for the case of contradirectional coupling, that PðzÞ given in (4.69) is a constant
∗
independent of z no matter whether κab ¼ κ∗
ba or κ ab 6¼ κba . Therefore, conservation of power
holds as expected.
It can be shown simply by applying conservation of power that the coupling is symmetric
with κab ¼ κ∗
ba when Pab ðzÞ ¼ 0. Conversely, if the coupling is symmetric, Pab ðzÞ always
vanishes even when mode a and mode b are not orthogonal to each other. Two conclusions
can thus be made.
1. If mode a and mode b are orthogonal to each other with cab ¼ 0, then Pab ðzÞ ¼ 0 and
κab ¼ κ∗
ba even when the two modes are not phase matched so that δ 6¼ 0.
2. If mode a and mode b are phase matched with δ ¼ 0, then Pab ðzÞ ¼ 0 and κab ¼ κ∗
ba even
when the two modes are not orthogonal to each other with cab 6¼ 0.
Consequently, coupling between two modes a and b is symmetric with κab ¼ κ∗
ba if these two
modes are orthogonal to each other or if they are phase matched.
4.6
PHASE MATCHING
..............................................................................................................
As can be seen from Figs. 4.6 and 4.8, power transfer is most efficient when δ ¼ 0. The
parameter δ is a measure of phase mismatch between the two modes being coupled. For the
simple case when 2δ ¼ Δβ ¼ βb βa , the phase-matching condition δ ¼ 0 is achieved when
βa ¼ βb . Then, the two modes are synchronized to have the same phase velocity. In the case
when δ includes a contribution from additional structural perturbation, such as a periodic
grating, phase matching of the two modes being coupled can be accomplished by compensating
for the difference Δβ ¼ βb βa with a perturbation phase factor to make δ ¼ 0, as can be seen
in (4.34) for a uniform perturbation and in (4.45) for a periodic perturbation. When considering
phase matching between two modes, it is important to always include all sources of contribution to the phase-mismatch parameter δ. When all contributions to the phase mismatch are
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4.6 Phase Matching
161
considered and their effects on the coupling coefficients are accounted for, the coupling
coefficients and the phase mismatch have a relation similar to (4.22):
∗
κab κ∗
(4.70)
ba ¼ cab þ cba δ ¼ cab 2δ:
Phase-matched coupling is always symmetric because κab ¼ κ∗
ba whenever δ ¼ 0, as seen in
(4.70). This statement is true even when cab 6¼ 0 and βa 6¼ βb . However, symmetric coupling
does not necessarily imply a phase-matched condition because symmetric coupling can be
accomplished by having cab ¼ 0 when δ 6¼ 0, as also seen in (4.70). Therefore, though δ ¼ 0
∗
always implies κab ¼ κ∗
ba , the converse is not true; it is possible to have κab ¼ κ ba when δ 6¼ 0.
The clearest example of this situation is the coupling between two phase-mismatched modes in
the same waveguide.
4.6.1 Phase-Matched Coupling
When perfect phase matching is accomplished so that δ ¼ 0, we can take
iφ
κab ¼ κ∗
ba ¼ κ ¼ jκje :
(4.71)
βc ¼ αc ¼ jκj:
(4.72)
Because δ ¼ 0, we find that
With these relations, the matrix Fðz; z0 Þ for codirectional coupling is reduced to
FPM ðz; z0 Þ ¼
ieiφ sin jκjðz z0 Þ
cos jκjðz z0 Þ
,
ie sin jκjðz z0 Þ
cos jκjðz z0 Þ
iφ
and the matrix Rðz; 0; lÞ for contradirectional coupling is reduced to
2
3
cosh jκjðl zÞ
iφ sinh jκjz
ie
6
cosh jκjl
cosh jκjl 7
6
7
RPM ðz; 0; lÞ ¼ 6
7:
4 iφ sinh jκjðl zÞ
cosh jκjz 5
ie
cosh jκjl
cosh jκjl
(4.73)
(4.74)
For perfectly phase-matched codirectional coupling, the coupling efficiency is
ηPM ¼ sin2 jκjl,
(4.75)
as shown in Fig. 4.9(a), and the coupling length is
lPM
c ¼
π
:
2jκj
(4.76)
PM
By choosing the interaction length to be l ¼ lPM
c , or any odd multiple of lc , 100% power
transfer from one mode to the other with ηPM ¼ 1 can be accomplished.
For perfectly phase-matched contradirectional coupling, the coupling efficiency is
ηPM ¼ tanh2 jκjl,
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(4.77)
162
Optical Coupling
Figure 4.9 Coupling efficiency ηPM as a function of the normalized coupling length jκjl for (a) perfectly phasematched codirectional coupling and (b) perfectly phase-matched contradirectional coupling.
as shown in Fig. 4.9(b). For an interaction length of l ¼ lPM
defined in (4.76), phase-matched
c
contradirectional coupling has a coupling efficiency of ηPM ¼ 84%. Although complete power
transfer with 100% efficiency cannot be accomplished for contradirectional coupling, most
power is transferred in a length comparable to the coupling length of codirectional coupling if
perfect phase matching is accomplished.
EXAMPLE 4.6
The coupling efficiency of a contradirectional coupler never reaches 100% but only approaches
100% as the length of the coupler approaches infinity: η ! 1 as l ! ∞. For a practical
application, η ¼ 99% might be as good. Find the length of a perfectly phase-matched contradirectional coupler that has η ¼ 99%.
Solution:
The length for a perfectly phase-matched contradirectional coupler that has η ¼ 99% is
found as
2
η99% ¼ tanh jκjl99% ¼ 0:99 )
l99%
pffiffiffiffiffiffiffiffiffi 3:0
1
1
¼ tanh
0:99 ¼
:
jκ j
jκ j
EXAMPLE 4.7
A 3-dB coupler is one that has a coupling efficiency of η ¼ 50%. Consider a 3-dB codirectional
coupler and a 3-dB contradirectional coupler. Both have perfect phase matching and have the
same coupling coefficient of κ. Find the length l3dB of each phase-matched 3-dB coupler in
terms of jκj?
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4.6 Phase Matching
163
Solution:
Using (4.75), the length of a phase-matched 3-dB codirectional coupler is found to be one of the
many values:
η3dB ¼ sin2 jκjl3dB ¼
1
2
) l3dB ¼
1
1
1 π
sin1 pffiffiffi ¼ m þ
for m ¼ 0, 1, 2, . . .
2 2jκj
jκ j
2
Using (4.77), the length of a phase-matched 3-dB contradirectional coupler is found to have
only one value:
η3dB ¼ tanh2 jκjl3dB ¼
1
2
)
l3dB ¼
1
1
0:88
tanh1 pffiffiffi ¼
:
jκ j
jκ j
2
The values of l3dB found above for codirectional and contradirectional coupling can be seen in
Figs. 4.9(a) and (b), respectively.
4.6.2 Phase-Mismatched Coupling
In the presence of phase mismatch with δ 6¼ 0, symmetric coupling with κab ¼ κ∗
ba is still true
for coupling between two modes in the same structure but is not necessarily true for coupling
between two different substructures. Nevertheless, to illustrate the effect of phase mismatch on
the coupling efficiency between two modes, we consider the simple case that κ ¼ κab ¼ κ∗
ba , as
expressed in (4.71).
For codirectional coupling with a phase mismatch of δ, the coupling efficiency obtained in
(4.55) can be written in terms of jκjl and jδ=κj as
η¼
1
1 þ jδ=κj2
2
sin
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jκjl 1 þ jδ=κj2 :
(4.78)
The maximum efficiency is
ηmax ¼
1
1 þ jδ=κj2
(4.79)
at a coupling length of
lPM
c
:
lc ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
1 þ jδ=κj
(4.80)
The maximum coupling efficiency is clearly less than unity when δ 6¼ 0. As shown in
Fig. 4.10(a), both lc and ηmax decrease as jδ=κj increases. If the interaction length is fixed at
l ¼ lPM
c , the efficiency drops quickly as jδ=κj increases, as shown in Fig. 4.10(b).
For contradirectional coupling with a phase mismatch of δ, the coupling efficiency can be
expressed in terms of jκjl and jδ=κj as
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164
Optical Coupling
Figure 4.10 Effect of phase mismatch on codirectional coupling showing, as a function of jδ=κj, (a) the
coupling length lc , normalized as lc =lPM
c , and the maximum coupling efficiency ηmax and (b) the coupling
PM
efficiency for fixed interaction lengths of l ¼ lPM
, 3lPM
c
c ,5lc .
Figure 4.11 Effect of phase mismatch on contradirectional
coupling showing the coupling efficiency for a few different
values of jκjl as a function of jδ=κj.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sinh jκjl 1 jδ=κj2
η¼
:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
cosh jκjl 1 jδ=κj jδ=κj
2
(4.81)
The coupling efficiency decreases as phase mismatch increases, as seen in Fig. 4.11. It
decreases monotonically with increasing jδ=κj for jδ=κj < 1; it decreases nonmonotonically
but oscillatorily for jδ=κj > 1.
In summary, to accomplish efficient coupling between two waveguide modes, the following
three parameters have to be considered.
1. Coupling coefficient: The coupling coefficient κ has to exist and be sufficiently large.
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Problems
165
2. Phase matching: The phase mismatch has to be minimized so that jδ=κj is made as small as
possible. Ideally, perfect phase matching with δ ¼ 0 is desired.
3. Interaction length: For codirectional coupling, because the efficiency oscillates with
interaction length, the length has to be properly chosen. An overly large length is neither
required nor beneficial. For contradirectional coupling, because the efficiency monotonically
increases with the interaction length, the length has to be sufficiently large but does not have
to be critically chosen. A very large length is not necessary, either.
Problems
4.1.1 Is the mode coupling caused by introducing an optical gain to a single waveguide
symmetric? Is the mode coupling caused by a slight structural change in the waveguide
symmetric?
4.1.2 Show that the general formulation for multiple-structure mode coupling is applicable to
the coupling of modes in a single waveguide.
4.2.1 Show that symmetric mode coupling in a single waveguide remains symmetric when a
lossless grating is introduced for phase matching.
4.2.2 Find the qth-order coupling coefficient κνμ ðqÞ for a saw-tooth grating, as shown in
Fig. 4.3(f), that has a period of Λ and a duty factor of ξ such that
8
>
> 2z ξΛ a,
for 0 < z < ξΛ;
<
ξΛ
κνμ ðzÞ ¼ ð1 þ ξ ÞΛ 2z
(4.82)
>
>
a, for ξΛ < z < Λ;
:
ð1 ξ ÞΛ
with K ¼ 2π=Λ. Which orders are useful for mode coupling?
4.2.3 A single-mode GaAs/AlGaAs waveguide supports a mode that has a propagation constant of β ¼ 2:5 107 m1 at λ ¼ 900 nm. To make a waveguide reflector, the forwardpropagating wave in this mode has to be coupled to the backward-propagating wave of
the same mode. A grating is incorporated into the waveguide for phase matching. Ignore
any zeroth-order effect of the grating. Find the first-order grating period and the secondorder grating period for this purpose.
4.2.4 A dual-channel directional coupler consists of two parallel InGaAsP/InP waveguides for
the two channels. A grating is fabricated in the space between the two channels to phase
match the waveguide modes of the two channels, as shown in Fig. 4.4. At λ ¼ 1:55 μm,
the modes have effective indices of nβa ¼ 3:40 and nβb ¼ 3:35, respectively. Ignore any
zeroth-order effect of the grating. Find the first-order grating period and the second-order
grating period for phase matching the modes of the two channels in the same direction.
Find those values for phase matching the modes in the two channels for them to
propagate in opposite directions.
4.3.1 Find the length of a codirectional coupler that has a coupling efficiency of half of the
maximum possible efficiency for given coupling coefficients of κab and κba and phase
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166
Optical Coupling
mismatch of δ between two modes in the case when the phase mismatch is small such that
δ2 < κab κba . What happens if the phase mismatch is large such that δ2 > κab κba ?
4.3.2 Find the length of a codirectional coupler that has a coupling efficiency of 25% of the
maximum possible efficiency for given coupling coefficients of κab and κba and phase
mismatch of δ between two modes in the case when the phase mismatch is small such that
δ2 < κab κba . What happens if the phase mismatch is large such that δ2 > κab κba ?
4.4.1 Find the length of a contradirectional coupler that has a coupling efficiency of half of the
maximum possible efficiency for given coupling coefficients of κab and κba and phase
mismatch of δ between two modes in the case when the phase mismatch is small such that
δ2 < κab κba .
4.4.2 Find the length of a contradirectional coupler that has a coupling efficiency of half of the
maximum possible efficiency for given coupling coefficients of κab and κba and phase
mismatch of δ between two modes in the case when the phase mismatch is large such that
δ2 > κab κba .
4.5.1 Show that in the case of symmetric coupling with κab ¼ κ∗
ba , the powers of the two
codirectionally coupled modes given in (4.53) and (4.54) for the condition of Pa ð0Þ 6¼ 0
and Pb ð0Þ ¼ 0 satisfy the power conservation relation PðzÞ ¼ Pa ðzÞ þ Pb ðzÞ ¼ Pa ð0Þ
with Pab ðzÞ ¼ 0.
4.5.2 Show that in the case of symmetric coupling with κab ¼ κ∗
ba , the powers of the two
contradirectionally coupled modes given in (4.64) and (4.65) for the condition of Pa ð0Þ
6¼ 0 and Pb ðlÞ ¼ 0 satisfy the power conservation relation PðzÞ ¼ Pa ðzÞ Pb ðzÞ ¼
Pa ð0Þ Pb ð0Þ with Pab ðzÞ ¼ 0. Show also that Pa ðlÞ þ Pb ð0Þ ¼ Pa ð0Þ for the total
power to be conserved.
4.6.1 Two optical waves of exactly the same wavelength and the same power are respectively
launched into the two input ports of a perfectly phase-matched 3-dB directional coupler at
the same time, as shown in Fig. 4.12. What are the possible power ratios between the two
output ports? What factor determines this ratio?
4.6.2 If the length of the coupler shown in Fig. 4.12 is doubled so that it becomes a coupler of
100% efficiency, what are the possible power ratios between the two output ports? What
factor determines this ratio?
Figure 4.12 3-dB directional coupler.
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Problems
167
4.6.3 A waveguide distributed Bragg reflector (DBR) has a grating of square corrugation as
shown in Fig. 4.3(d). The period of the grating is Λ, and its duty factor is ξ. It is found that
the propagation constant of the fundamental TE0 mode of the waveguide at the λ ¼
1:0 μm optical wavelength is β ¼ 1:0 107 m1 . It is also found that the maximum
absolute value of the coupling coefficient of this grating is jκjmax ¼ 1:0 104 m1 ,
which is obtained when the parameters of the grating are properly chosen. Assume that
the waveguide structural parameters and the grating depth are fixed. Only the period Λ
and the duty factor ξ of the grating are varied.
(a) What are the optimal choices of the period Λ and the duty factor ξ for the grating to
have the maximum coupling coefficient jκjmax ? What is the length of the DBR if 50%
reflectivity is desired?
(b) If a second-order grating has to be used, what are the best choices of its period Λ and
its duty factor ξ for the highest efficiency? What is the length of the DBR if 50%
reflectivity is desired in this case?
4.6.4 A waveguide Bragg reflector is fabricated with a grating of a period Λ in a symmetric
planar semiconductor waveguide, which has a core index of 3.25 and a cladding index of
3.20 for the wavelength of λ ¼ 1:55 μm.
(a) Estimate the required grating period for a first-order grating and that for a secondorder grating.
(b) Between the sinusoidal and the square gratings, choose a combination of shape and
duty factor for a first-order grating that has a maximized coupling efficiency for a
given modulation depth.
(c) If the grating chosen in (b) has a coupling coefficient of jκj ¼ 1:0 104 m1 , what is
the required length of the grating for the Bragg reflector to have a 90% reflectivity?
4.6.5 A fiber-optic frequency filter is made of two single-mode fibers of different mode propagation constants. They are placed in close contact over a length of l, as shown in
Fig. 4.13. At the λ ¼ 1:55 μm optical wavelength, the effective indices for the two fiber
modes are βa ¼ 5:959 106 m1 and βb ¼ 5:849 106 m1 , respectively, and the
coupling coefficient between the two fiber modes is κ ¼ κab κba ¼ 2 103 m1 .
A grating that has a period of Λ is built into the fibers in the coupling section. The input
port of the device is port 1. The device is to function as an optical filter for separating the
1:55 μm wavelength from other wavelengths.
(a) If the device is to direct all of the optical power at the 1:55 μm wavelength to port
4 and to dump all other wavelengths to port 3, what is the maximum possible
coupling efficiency for the 1:55 μm wavelength without the grating?
(b) With a first-order grating, what are the values of Λ and l that have to be selected to
obtain the best efficiency for directing the power at the 1:55 μm wavelength to port 4?
What is the maximum efficiency if the parameters of the grating are properly chosen?
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168
Optical Coupling
(c) If the device is to direct the power at the 1:55 μm wavelength to port 2, what is the
maximum possible coupling efficiency without the grating?
(d) With a first-order grating, what should the choice of the grating period Λ be in order
to get the highest efficiency for directing the power at the 1:55 μm wavelength to
port 2? In this case, if the length l of the coupler remains the same as that found in (b),
what is the efficiency of directing the 1:55 μm light from port 1 to port 2?
Figure 4.13 Fiber-optic frequency filter
consisting of two single-mode fibers and a
grating.
4.6.6 In designing an efficient waveguide coupler of any geometry, what are the three major
parameters that have to be considered in order to have a good efficiency? In what order of
priority do they have to be considered?
Bibliography
Buckman, A. B., Guided-Wave Photonics. Fort Worth, TX: Saunders College Publishing, 1992.
Chuang, S. L., Physics of Photonic Devices, 2nd edn. New York: Wiley, 2009.
Hunsperger, R. G., Integrated Optics: Theory and Technology, 5th edn. New York: Springer-Verlag, 2002.
Liu, J.M., Photonic Devices. Cambridge: Cambridge University Press, 2005.
Marcuse, D., Theory of Dielectric Optical Waveguides, 2nd edn. Boston, MA: Academic Press, 1991.
Nishihara, H., Haruna, M., and Suhara, T., Optical Integrated Circuits. New York: McGraw-Hill, 1989.
Pollock, C. R. and Lipson, M., Integrated Photonics. Boston, MA: Kluwer, 2003.
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Cambridge Books Online
http://ebooks.cambridge.org/
Principles of Photonics
Jia-Ming Liu
Book DOI: http://dx.doi.org/10.1017/CBO9781316687109
Online ISBN: 9781316687109
Hardback ISBN: 9781107164284
Chapter
5 - Optical Interference pp. 169-203
Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.006
Cambridge University Press
5
5.1
Optical Interference
OPTICAL INTERFERENCE
..............................................................................................................
An optical field is a sinusoidal wave that has a space- and time-varying phase. The complex
electric field of an optical wave that propagates in a homogeneous medium can be generally
expressed in the form of (1.81):
Eðr; t Þ ¼ E ðr; t Þ exp ðik r iωt Þ ¼ ^e jE ðr; t ÞjeiφE ðr;tÞ exp ðik r iωt Þ,
(5.1)
which has a total space- and time-dependent phase as given in (1.83):
φðr; tÞ ¼ k r ωt þ φE ðr; t Þ:
(5.2)
For a waveguide mode that propagates along the longitudinal waveguide axis, taken to be the z
axis, the complex electric field takes the form of (3.1):
Eν ðr; tÞ ¼ E ν ðr; t Þ exp ðiβν z iωtÞ ¼ ^e jE ν ðr; t ÞjeiφE ν ðz;tÞ exp ðiβν z iωt Þ,
(5.3)
which has a total space- and time-dependent phase of
φν ðz; t Þ ¼ βν z ωt þ φE ν ðz; tÞ:
(5.4)
The wave nature of an optical field is fully characterized by its total space- and time-dependent
phase factor. Because φν ðz; tÞ in (5.4) for a waveguide mode is mathematically a special form of
φðr; tÞ in (5.2), by taking k to be βν^z and φE ðr; t Þ to be φE ν ðz; t Þ, in the following discussion we
consider only optical waves in a homogeneous medium. The general concept applies equally to
waveguide modes. Unless otherwise specified, we also consider a lossless medium for simplicity so that the propagation constant k has a real value.
One phenomenon that clearly demonstrates the wave nature of optical fields is optical
interference of two or more fields of different phases. In this section, we consider the interference of two fields that are superimposed only once. In Section 5.2, the concept of an optical
grating based on the interference of multiple waves that emerge from a periodic optical
structure is discussed. Multiple interference leading to optical resonance and optical filtering
is discussed in Section 5.3.
Consider the superposition of two optical fields, E1 and E2 . The total field is the linear vector
sum of the two:
E ¼ E1 þ E2 ¼ ^e 1 jE 1 jeiφ1 þ ^e 2 jE 2 jeiφ2 ,
(5.5)
where φ1 ¼ k1 r ω1 t þ φE 1 and φ2 ¼ k2 r ω2 t þ φE 2 are the total phases of the two fields
E1 and E2 , respectively. According to (3.183), the intensity of an optical field is proportional to
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Optical Interference
jE⊥ j2 . Though (3.183) is strictly only applicable to a plane-wave normal mode that has a unique
wavevector of k and a unique frequency of ω while the composite field E in (5.5) might not be a
normal mode because k1 and k2 might not be the same and ω1 and ω2 might not be the same, it
is clear that the intensity of the composite field E is not simply the sum of the intensities of the
component fields E1 and E2 because
∗
jEj2 ¼ jE1 j2 þ jE2 j2 þ E1 E∗
2 þ E1 E2
iðφ1 φ2 Þ
¼ jE 1 j2 þ jE 2 j2 þ 2jE 1 jjE 2 jRe ^e 1 ^e ∗
:
2e
(5.6)
The interference between the two fields E1 and E2 arises from the term
iðφ1 φ2 Þ
2jE 1 jjE 2 jRe ^e 1 ^e ∗
in (5.6). Clearly, interference does not exist between two orthog2e
onally polarized fields for which ^e 1 ^e ∗
2 ¼ 0. Note that the orthogonality between two optical
fields is defined by ^e 1 ^e ∗
¼
0,
as
given
in (1.80), but not by ^e 1 ^e 2 ¼ 0. This is important for
2
circularly polarized or elliptically polarized fields, which have complex unit polarization
vectors. Interference occurs only when two fields are not orthogonally polarized so that
^e 1 ^e ∗
2 6¼ 0.
Using the time-averaged
Poynting vector S defined in (1.53) and the definition of the light
intensity I ¼ S n^j while assuming that the angle between k1 and k2 is small, the intensity of
the total field can be expressed as
I ¼ I 1 þ I 2 þ I 12 cos ðφ1 φ2 þ φ^e 1 ^e ∗2 Þ
¼ I 1 þ I 2 þ I 12 cos ðk1 k2 Þ r ðω1 ω2 Þt þ φE 1 φE 2 þ φ^e 1 ^e ∗2 ,
(5.7)
where I 1 ¼ 2k 1 jE 1⊥ j2 =ω1 μ0 and I 2 ¼ 2k 2 jE 2⊥ j2 =ω2 μ0 are respectively the intensities of E1 and
E2 alone,
k1
k2
I 12 ¼ 2
(5.8)
þ
jE 1⊥ E 2⊥ j^e 1 ^e ∗
2 0
ω1 μ0 ω2 μ0
is the intensity magnitude of the interference between the two fields, φ^e 1 ^e ∗2 is the phase of
^e 1 ^e ∗
Þ is the time average of cos ðφ1 φ2 þ φ^e 1 e^∗2 Þ over one
e∗
e1 ^
2 , and cos ðφ1 φ2 þ φ^
2
wave cycle, as defined in (1.53) for the time-averaged Poynting vector S. The phase factor
φ^e 1 ^e ∗2 matters only when the two polarizations ^e 1 and ^e 2 are not mutually orthogonal and at
least one of them is not linearly polarized because φ^e 1 ^e ∗2 ¼ 0 when ^e 1 ^e ∗
e 1 and
2 ¼ 0 or both ^
^e 2 are real vectors. With this understanding, in the following we consider for simplicity only the
case when the two component fields have the same polarization, i.e., ^e 1 ¼ ^e 2 , so that
^e 1 ^e ∗
¼ 0. Then,
2 ¼ 1 and φ^e 1 ^e ∗
2
I ¼ I 1 þ I 2 þ I 12 cos ðφ1 φ2 Þ
¼ I 1 þ I 2 þ I 12 cos ðk1 k2 Þ r ðω1 ω2 Þt þ φE 1 φE 2 ,
(5.9)
and
I 12
k1
k2
¼2
þ
jE 1⊥ E 2⊥ j > 0:
ω1 μ0 ω2 μ0
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(5.10)
5.1 Optical Interference
171
As seen from (5.9), I 1 þ I 2 I 12 I I 1 þ I 2 þ I 12 . Depending on the total phase difference φ1 φ2 , the total intensity I of the composite field can be higher or lower than, or equal to,
the sum of the intensities I 1 and I 2 of the individual component fields. Because I 12 I 1 þ I 2 ,
maximum interference takes place when the two component fields have the same polarization
and the same amplitude so that I 12 ¼ I 1 þ I 2 .
1. Constructive interference occurs when the phase difference φ1 φ2 is such that the total
intensity I is higher than the sum of the intensities I 1 and I 2 of the individual component
fields: I 1 þ I 2 < I I 1 þ I 2 þ I 12 . Complete constructive interference happens when the
two component fields are in phase, i.e., φ1 φ2 ¼ 2qπ, where q is an integer, so that
I ¼ I 1 þ I 2 þ I 12 . Partial constructive interference happens when the phase difference is
such that 2qπ π=2 < φ1 φ2 < 2qπ þ π=2 but φ1 φ2 6¼ 2qπ so that I 1 þ I 2 < I < I 1 þ
I 2 þ I 12 . These concepts of constructive interference are illustrated in Fig. 5.1 for the case
when the two component fields have the same frequency.
2. Destructive interference occurs when the phase difference φ1 φ2 is such that the total
intensity I is lower than the sum of the intensities I 1 and I 2 of the individual component
fields: 0 I 1 þ I 2 I 12 I < I 1 þ I 2 . Complete destructive interference happens when
Figure 5.1 Constructive interference between two fields of the same frequency but of different amplitudes
showing the individual fields (dashed curves) and the composite field (solid curve). The two component fields
have amplitudes of jE 1 j ¼ E 0 and jE 2 j ¼ 0:8E 0 in this example. (a) Complete constructive interference for
φ1 φ2 ¼ 0. In this case, I 1 ¼ I 0 , I 2 ¼ 0:64I 0 , and I ¼ 3:24I 0 > I 1 þ I 2 because the amplitude of the
composite field is jE j ¼ 1:8E 0 . (b) Partial constructive interference for φ1 φ2 ¼ π=4 as an example. In this
case, I 1 ¼ I 0 , I 2 ¼ 0:64I 0 , and I 2:77I 0 > I 1 þ I 2 because the amplitude of the composite field is
jE j 1:665E 0 .
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172
Optical Interference
the two component fields are completely out of phase, i.e., φ1 φ2 ¼ ð2q þ 1Þπ, and they
have the same amplitude to completely cancel each other so that I ¼ I 1 þ I 2 I 12 ¼ 0.
Partial destructive interference happens when the two fields cancel each other only partially
but not completely so that I 6¼ 0 but 0 < I < I 1 þ I 2 . Partial destructive interference occurs
under one of the two following different situations. The two fields are completely out of
phase, φ1 φ2 ¼ ð2q þ 1Þπ, but they do not have the same amplitude, jE 1⊥ j 6¼ jE 2⊥ j, so
that I 12 < I 1 þ I 2 ; or the phase difference is such that ð2q þ 1Þπ π=2 < φ1 φ2 <
ð2q þ 1Þπ þ π=2 but φ1 φ2 6¼ ð2q þ 1Þπ. These concepts of destructive interference are
illustrated in Fig. 5.2 for the case when the two component fields have the same frequency.
Figure 5.2 Destructive interference between two fields of the same frequency showing the fields and
intensities of the individual fields (dashed curves) and the composite field (solid curve). (a) Complete
destructive interference for φ1 φ2 ¼ π and jE 1⊥ j ¼ jE 2⊥ j so that I ¼ 0. (b) Partial destructive interference
for φ1 φ2 ¼ π but jE 1⊥ j 6¼ jE 2⊥ j so that I 6¼ 0 but 0 < I < I 1 þ I 2 . (c) Partial destructive interference
for φ1 φ2 ¼ 3π=4 and jE 1⊥ j ¼ jE 2⊥ j so that I 6¼ 0 but 0 < I < I 1 þ I 2 .
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5.1 Optical Interference
173
Interference between two optical fields can create intensity patterns that vary in space or time,
or both, because the phase difference φ1 φ2 can be a function of space or time, or both. As
seen in (5.9), the phase difference φ1 φ2 ¼ ðk1 k2 Þ r ðω1 ω2 Þt þ φE 1 φE 2 has three
components.
1. When k1 ¼
6 k2 , the spatially varying phase factor ðk1 k2 Þ r creates periodic spatial interference fringes that have a period of Λ ¼ 2π=jk1 k2 j along the k1 k2 direction. These
interference fringes disappear when k1 ¼ k2 : When ω1 ¼ ω2 and φE 1 φE 2 is time independent, these interference fringes in space are stationary patterns that do not vary with time.
Figure 5.3 shows the stationary periodic fringes produced by the interference between two
fields of the same polarization, same amplitude, and same frequency, but different wavevectors.
2. When ω1 6¼ ω2 , the temporally varying phase factor ðω1 ω2 Þt causes periodic temporal
beats that have a frequency of f ¼ jω1 ω2 j=2π. In the case when jω1 ω2 j ω1 and
jω1 ω2 j ω2 , these beats create a detectable temporal intensity variation at the frequency
f . This periodic temporal intensity variation disappears when ω1 ¼ ω2 . When k1 ¼ k2 and
φE 1 φE 2 is space independent, these periodic beats in time are spatially uniform patterns
that do not vary in space. Figure 5.4 shows the periodic temporal beats produced by the
interference between two fields of the same polarization, same amplitude, and same wavevector, but different frequencies.
3. The phase factor φE 1 φE 2 depends on the phases of the two optical fields E 1 and E 2 . It defines
the coherence between the two fields. The two fields are temporally coherent with each other if
φE 1 φE 2 is a constant of time; they are spatially coherent if φE 1 φE 2 is a constant of space.
The two fields are temporally incoherent if φE 1 φE 2 varies randomly with time on the scale of
the optical cycle; they are spatially incoherent if φE 1 φE 2 varies randomly with space on the
scale of the optical wavelength. Between the extremes of complete coherence and complete
incoherence, the two fields can be partially coherent to different degrees in time, space, or both.
Figure 5.3 Stationary periodic fringes produced by the interference between two optical fields of the same
polarization, same amplitude, and same frequency, but different wavevectors.
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174
Optical Interference
Figure 5.4 Periodic temporal beats (sold curve) produced by the interference between two fields (dashed
curves) of the same polarization, same amplitude, and same wavevector, but different frequencies. The
envelope of the beat notes is shown in dashed gray curves.
The time average cos ðk1 k2 Þ r ðω1 ω2 Þt þ φE 1 φE 2 depends strongly on the
degree of coherence. When the two fields are coherent, φE 1 φE 2 does not vary on the
time scale of the optical cycle or on the space scale of the optical wavelength, but it can still
vary in time or space slowly so that cos ðk1 k2 Þ r ðω1 ω2 Þt þ φE 1 φE 2 ¼
cos ðk1 k2 Þ r ðω1 ω2 Þt þ φE 1 φE 2 : The phase factor φE 1 φE 2 is a constant of both
space and time when the phases of the field amplitudes E 1 and E 2 are constants or vary in the
same manner with space and time. It varies with space or time when the phases of the two field
amplitudes vary differently with space or time; it varies with both space and time when the phases
of the field amplitudes have different spatial variations and different temporal variations. Thus, a
modulation on the total intensity I in space or time, or both, can be accomplished by properly
modulating this phase factor. The principles of most interferometers are based on this concept.
EXAMPLE 5.1
A glass wedge of a refractive index n has a small wedge angle of α as shown in Fig. 5.5. It has a
length of l in the x direction and a height of h in the y direction. A monochromatic plane optical
wave at the wavelength λ vertically illuminates the wedge from above. If the optical wave is
coherent, find the locations of the bright and dark fringe lines when viewed from above. What is
the period of the fringes? How many periods of interference fringes appear on the top surface of
the wedge? What happens to the fringes if the light is not completely coherent?
Solution:
The incident wave propagates in the negative y direction with a wavevector of ki ¼ k^y . When
viewed from above, there are two reflected waves, from the two surfaces of the glass wedge,
respectively. The first is reflected from the top wedge surface; it has a wavevector of
k1 ¼ k sin 2α^x þ k cos 2α^y at an angle of 2α from the y direction. The second is reflected
from the bottom wedge surface; it has a wavevector of k2 ¼ k^y in the y direction. Thus,
k1 k2 ¼ k sin 2α^x þ kð cos 2α 1Þ^y :
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5.1 Optical Interference
175
Figure 5.5 Interference
fringes formed by reflected
waves from the two
surfaces of a glass wedge.
Because the two reflected waves are from the same source, they have the same frequency:
ω1 ¼ ω2 . However, the two reflected waves have different phases because the top reflection is
external reflection at nearly normal incidence with a phase change of π for the electric field,
whereas the bottom reflection is internal reflection at normal incidence with no phase change.
If the incident optical wave is coherent, the phase of the two reflected waves does not vary
with time so that φE 1 φE 2 ¼ π. Then,
cos ðk1 k2 Þ r ðω1 ω2 Þt þ φE 1 φE 2 ¼ cos ð2kx sin α þ π Þ ¼ cos ð2kx sin αÞ:
Therefore,
I ¼ I 1 þ I 2 I 12 cos ð2kx sin αÞ:
Bright fringe lines appear at the locations where cos ð2kx sin αÞ ¼ 1 so that I ¼ I 1 þ I 2 þ I 12 ;
dark fringe lines appear where cos ð2kx sin αÞ ¼ 1 so that I ¼ I 1 þ I 2 I 12 . We find that a dark
fringe line appears at the tip of the wedge at x ¼ 0. Therefore, the dark and bright fringe lines
appear, respectively, at the locations:
π
λ
λl
¼m
m
,
m ¼ 0, 1, 2 . . .
k sin α
2n sin α
2nh
1
π
1
λ
1 λl
b
xm ¼ m þ
¼ mþ
mþ
,
m ¼ 0, 1, 2 . . .
2 k sin α
2 2n sin α
2 2nh
xdm ¼ m
where we take sin α h=l for a small angle of α. The period Λ of the fringes is found for
2kΛ sin α ¼ 2π:
Λ¼
π
λ
λl
¼
:
k sin α 2n sin α 2nh
The number of periods over the length is
M¼
l 2nl sin α 2nh
¼
:
Λ
λ
λ
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176
Optical Interference
If the incident optical wave is not coherent, then φE 1 φE 2 is not a constant of time. Because
the two reflected waves are from the same source, whether they will create interference fringes
or not depends on the coherence time of the incident wave, i.e., the degree of coherence or
incoherence of the wave. The difference in the optical path lengths between the two reflected
waves depends on the location of the fringe. It is Δy ¼ 2nh for the last fringe located at the end
of the wedge at x ¼ l, and it is
8
mλ,
for the mth dark fringe,
<
xm
Δym ¼ Δy ¼
1
: m þ λ, for the mth bright fringe:
l
2
The corresponding time difference of the two waves for the last fringe located at the end of the
wedge is
Δt ¼
Δy 2nh M
¼
¼ ,
c
c
ν
and it is
8m
> ,
Δym <ν
Δt m ¼
¼
1 1
>
c
: mþ
,
2 ν
for the mth dark fringe,
for the mth bright fringe:
For the mth fringe to appear, the coherence time τ coh of the incident optical wave has to be such
that τ coh > Δtm , which means that τ coh is longer than m optical cycles for the mth dark fringe and
longer than m þ 1=2 cycles for the mth bright fringe. If the coherence time is sufficiently long
such that τ coh > Δt, then all fringes on the surface of the wedge appear.
5.1.1 Double-Slit Interference
Young’s double-slit experiment established the wave nature of light. Figure 5.6 illustrates the
double-slit interference. We consider a monochromatic plane wave of a frequency ω and a
wavevector ki ¼ k^x , which is normally incident on two identical slits separated at a spacing of
Λ in the z direction. The observation point is in the direction that makes an angle of θ with
respect to the x axis and is on a plane at a distance of l from the plane of the slits. The optical
path lengths from the two slits to the observation point are r 1 and r 2 , respectively. In the limit
that l Λ, the path difference is
r 2 r 1 Λ sin θ:
(5.11)
Because the incoming wave is normally incident on the plane of the slits, the fields that emerge
from the two slits have the same phase at the exit plane of the slits. Because the two slits have
the same geometrical dimensions, these fields have the same polarization and the same
amplitude such that E 1 ¼ E 2 ¼ ^e E 0 . The total field at the observation point is the linear
superposition of the fields from the two slits:
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5.1 Optical Interference
177
Figure 5.6 Double-slit interference.
E ¼ E 1 eikr1 iωt þ E 2 eikr2 iωt ¼ ^e E 0 eikr1 iωt 1 þ eiδ ,
(5.12)
δ ¼ k ðr 2 r 1 Þ kΛ sin θ
(5.13)
where
is the phase difference at the observation point between the two fields that come from the two
slits. The intensity at the observation point is
I ¼ 4I 0 cos2
δ
,
2
(5.14)
where I 0 / jE 0 j2 is the intensity contributed by a single slit alone. This result can be obtained
from (5.9) because I 1 ¼ I 2 ¼ I 0 , I 12 ¼ I 1 þ I 2 ¼ 2I 0 , and φ1 φ2 ¼ δ.
EXAMPLE 5.2
Find the angles at which the double-slit interference from normal incidence of a plane wave
shows bright interference fringes. Find the locations of the bright fringes on a screen that is at a
distance of l from the slits.
Solution:
The intensity pattern of the double-slit interference from normal incidence of a plane wave is
that given in (5.14). A bright interference fringe appears when
cos 2
δ
¼1
2
)
δ ¼ 2qπ for q ¼ 0,
1,
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2, . . .
178
Optical Interference
Using (5.13), the qth-order bright interference fringe appears at the angles θq :
kΛ sin θq ¼ 2qπ
)
sin θq ¼
2qπ qλ
¼
kΛ
nΛ
)
θq ¼ sin1
qλ
,
nΛ
where n is the refractive index of the medium. On a screen that is located at a distance of l from
the slits, the qth-order bright fringe is found at a distance of
zq ¼ l sin θq ¼ q
λl
nΛ
from the zeroth-order bright fringe, which is located at z ¼ 0.
5.1.2 Optical Interferometers
Optical interference has been developed into many advanced concepts and applications. One
important application is interferometry, which uses optical interference to interrogate the
characteristics, including the polarization state, the wavevector, the frequency, and the phase,
of an optical wave with respect to a reference wave. Many types of interferometers have been
developed. The most important ones for photonics applications include the Michelson interferometer, MachZehnder interferometer, and FabryPérot interferometer. The Michelson
interferometer and the MachZehnder interferometer are illustrated below. The FabryPérot
interferometer is discussed in Section 5.3.
Michelson Interferometer
The Michelson interferometer was used in the historical MichelsonMorley experiment.
Figure 5.7 shows its basic structure. The single beam splitter in this structure defines four
optical paths. The two paths that are respectively on the left of and below the beam splitter
Figure 5.7 Michelson
interferometer. BS, beam splitter.
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5.1 Optical Interference
179
define two ports, each of which serves as a port for both input and output. Input light can be sent
into either port or into both ports, but usually only one input is supplied, as shown in Fig. 5.7
where only port 1 receives an input of an intensity I in while port 2 receives no input. By
contrast, both ports always function as output ports with output intensities of I out, 1 and I out, 2 ,
respectively, though the output intensity at a port can be zero when totally destructive interference occurs at the port.
The input wave is split by the beam splitter into two waves, each of which enters one of the
two internal paths that are respectively above and on the right of the beam splitter. Each internal
path ends with a totally reflective mirror, which reflects the light back to the beam splitter. The
beam splitter again divides each returning wave into one reflected wave and one transmitted
wave for the two output ports. Each output field is the combination of one reflected field from
one internal path and one transmitted field from the other internal path: The output field at port
1 is the linear superposition of the reflected field from the vertical internal path and the
transmitted field from the horizontal internal path, whereas the output field at port 2 is the
linear superposition of the transmitted field from the vertical internal path and the reflected field
from the horizontal internal path.
Though the two component fields of each output field come from different internal paths, they
have the same polarization, the same frequency, and the same wavevector because they both
originate from the same input field and they propagate in the same direction. Their phase
difference depends only on the optical length difference of the two internal paths and the phase
change caused by reflection or transmission at the beam splitter. Because the phase change at
the beam splitter has a fixed value, the output intensity at a port can be varied by varying the
optical length difference of the two internal paths. Note that what matters is not the physical
length difference of the paths but the optical length difference. The optical length difference can
be varied by varying the physical length difference, through moving one or both mirrors, or by
varying the refractive index along one or both paths, through modulating the medium using any
of the effects discussed in Sections 2.6 and 2.7.
The beam splitter is partially reflective and partially transmissive. In practice, it has negligible
absorption so that R þ T 1. The beam splitter can have any reflectance/transmittance ratio,
but complete destructive interference is possible only when it is a 50/50 beam splitter so that the
reflected field and the transmitted field have the same magnitude though possibly different
phases. Conservation of energy requires that I out, 1 þ I out, 2 ¼ I in when there is no loss in the
system. Clearly, I out, 1 ¼ 0 and I out, 2 ¼ I in when complete destructive interference occurs
at port 1, whereas I out, 2 ¼ 0 and I out, 1 ¼ I in when complete destructive interference occurs at
port 2. Thus, complete constructive interference occurs at one output port when complete
destructive interference occurs at the other output port. This condition is clearly required by
conservation of energy, but it is not trivial if we take a closer look. It implies that the two
component fields for the total output field at port 1 are completely in phase when those at
port 2 are completely out of phase. This seems puzzling: each output field is the combination of
one reflected field and one transmitted field through the beam splitter, but one combination is
constructive while the other is destructive at the same time.
To resolve this puzzle, we have to pay attention to two key properties of the functioning of an
optical beam splitter. (1) An optical beam splitter always has a layer of properly designed and
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180
Optical Interference
accurately implemented coating on one of its two surfaces to accomplish the desired reflectance/transmittance ratio. The other surface is often antireflection coated to eliminate unwanted
reflection. In any event, reflection takes place on only one surface of the beam splitter. Because
the two waves returning from the two different internal paths reach the beam splitter from
different sides, one undergoes external reflection while the other undergoes internal reflection.
(2) For any polarization, a transmitted field through a lossless dielectric interface has no phase
change with respect to the incident field. A reflected field may have either no phase change or a
phase change of π, depending on its polarization, its incident angle, and whether it undergoes
external reflection or internal reflection; in any case, the phase difference between external
reflection and internal reflection for a given polarization at a given incident angle is always π.
(See Problem 3.4.1.) Considering the above two characteristics, it is clear that the phase
difference between the two field components at one output port is always different by a phase
factor of π from that at the other output port because the reflected field component for one
output port comes from external reflection and that for the other output port is from internal
reflection. For this reason, constructive interference happens at one output port when destructive interference takes place at the other output port, ensuring conservation of energy.
Assume that the beam splitter has the reflective surface on the left side. Then, reflection on
the left side of the beam splitter is external reflection with a phase change of π and reflection on
the right side of the beam splitter is internal reflection with no phase change. If the beam splitter
is a 50/50 splitter, the output intensities of the two output ports are
I out, 1 ¼ I in cos2
Δφ
,
2
I out, 2 ¼ I in sin2
Δφ
,
2
(5.15)
where Δφ is the phase difference of the two optical paths. In the case when the two paths are
filled with the same uniform medium, Δφ ¼ 2kðla lb Þ, where la and lb are respectively the
lengths of the two arms, and the factor 2 accounts for the fact that the wave in each arm travels
through the arm twice before returning to the beam splitter.
Mach–Zehnder Interferometer
Figure 5.8 shows the basic structure of the MachZehnder interferometer. With two beam
splitters, this structure is different from that of the Michelson interferometer in two basic
features: The output ports are separate from the input ports, and light propagates through each
of the two separate internal paths only once. Despite these differences, the fundamental concepts
discussed above for the Michelson interferometer are applicable to the MachZehnder interferometer. The output intensity at a given port can be varied by varying the difference of the optical
path lengths between the two paths, which can be accomplished by varying the physical length
difference between the two paths or by varying the refractive index in the medium along one or
both paths. When constructive interference occurs at one output port, destructive interference
happens at the other output port. Thus, I out, 1 þ I out, 2 ¼ I in for a lossless system.
Assume that each beam splitter has the reflective surface on the left side. Then, reflection on
the left side of each beam splitter is external reflection with a phase change of π and reflection
on the right side of each beam splitter is internal reflection with no phase change. If both beam
splitters are 50/50 splitters, the output intensities of the two output ports are
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5.1 Optical Interference
181
Figure 5.8 MachZehnder interferometer. BS, beam splitter.
Figure 5.9 MachZehnder interferometers in the waveguide form using (a) two Y-junction waveguides
and (b) two directional couplers. Only one input is supplied in this illustration. In general, the lengths of the
two arms are not identical.
I out, 1 ¼ I in sin2
Δφ
,
2
I out, 2 ¼ I in cos2
Δφ
,
2
(5.16)
where Δφ is the phase difference of the two optical paths. In the case when the two paths are
filled with the same uniform medium, Δφ ¼ k ðla lb Þ, where la and lb are respectively the
lengths of the two arms, and the wave in each arm travels through the arm only once before
reaching the output beam splitter.
The MachZehnder interferometer can be implemented in various waveguide forms.
Figure 5.9 shows two common forms using (a) Y-junctions and (b) directional couplers for
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182
Optical Interference
the beam-splitting function. In the case when the Y-junctions and the directional couplers are all
3-dB couplers such that ξ ¼ 1=2, we find that
T ¼ cos2
Δφ
2
(5.17)
for the interferometer using 3-dB Y-junctions shown in Fig. 5.9(a), and
T ¼ sin2
Δφ
2
(5.18)
for the interferometer using 3-dB directional couplers shown in Fig. 5.9(b), where Δφ ¼ φa φb
is the phase difference of the two optical paths.
5.1.3 Standing Wave
In the analysis and discussion presented above following (5.7), we have assumed that the angle
between the two wavevectors k1 and k2 of the interfering waves is small, or zero as in the case
of the interferometers. In the case when the angle between k1 and k2 is large, the principle of
linear superposition expressed as (5.5) is still valid and interference between two fields still
occurs, but the intensity of the combined field expressed as (5.7) is not valid. Here we consider
the special case when two waves have the same polarization, ^e 2 ¼ ^e 1 ¼ ^e , the same amplitude,
E 2 ¼ E 1 ¼ E, and the same frequency, ω2 ¼ ω1 ¼ ω, but they propagate in opposite directions, k2 ¼ k1 ¼ k, so that E1 ¼ ^e 1 E 1 eik1 riω1 t ¼ ^e Eeik riωt and E2 ¼ ^e 2 E 2 eik2 riω2 t ¼
^e Eeik riωt . The linear superposition of these two fields yields
Eðr; t Þ ¼ E1 ðr; t Þ þ E2 ðr; t Þ ¼ ^e Eeik riωt þ ^e Eeik riωt ¼ 2^e Eeiωt cos ðk rÞ:
(5.19)
For simplicity of discussion without loss of generality, we assume linear polarization and a
field amplitude of E ¼ jEj by taking its phase to be zero. Then the real field of the combined
field can be expressed as
Eðr; t Þ ¼ Eðr; t Þ þ E ðr; tÞ ¼ 4^e jEj cos ωt cos ðk rÞ:
(5.20)
The spatial variation of this field is decoupled from the temporal variation. We find that
Eðr; t Þ vanishes for all times at the fixed locations, known as nodes, where k r ¼ ð2q þ 1=2Þπ
for integers q so that cos ðk rÞ ¼ 0, as shown in Fig. 5.10. The nodes are periodically
distributed along the line defined by k^ at a spacing of π=k ¼ λ=2n, where λ=n is the
wavelength of the optical field in the medium of a refractive index n. At the locations where
k r ¼ 2qπ so that cos ðk rÞ ¼ 1, we find that Eðr; tÞ ¼ 4^e jE j cos ðωt Þ; such locations are
known as antinodes. The antinodes are also periodically distributed along the line defined
by k^ at a spacing of π=k ¼ λ=2n. An antinode is found at the midpoint between two
neighboring nodes.
Because the nodes and antinodes are fixed in space, the field given in (5.20) appears to stand
still in space. It does not travel but only oscillates in time. Therefore, the interference of the two
contrapropagating waves of the same polarization, the same frequency, and the same amplitude
results in a standing wave.
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5.2 Optical Gratings
183
Figure 5.10 Standing wave. Nodes, labeled with N, are periodically distributed along the line defined by k^ at a
spacing of π=k ¼ λ=2n. Each antinode, labeled with A, is located at the midpoint between two neighboring
nodes. A standing wave oscillates in time but appears to stand still in space.
5.2
OPTICAL GRATINGS
..............................................................................................................
An optical grating is a periodic optical structure. Some waveguide grating structures are
illustrated in Fig. 4.3. The functioning of an optical grating can be understood from the
viewpoint of phase matching, as discussed in Chapter 4, or from the viewpoint of optical
interference. In this section, we make the connection between these two viewpoints.
The concept of double-slit interference discussed in the preceding section can be extended to
equally spaced multiple slits of identical geometrical parameters, which form a periodic
structure of a period Λ in the z direction, as shown in Fig. 5.11. The slits are on the yz plane,
which is normal to the x axis. Being a periodic optical structure, this multiple-slit structure can
be considered a grating. Indeed, it functions as a transmissive diffraction grating, also called a
transmission grating.
5.2.1 Normal Incidence
We first consider normal incidence of a monochromatic plane wave of a frequency ω and a
wavevector ki ¼ k^x on the periodic multiple-slit structure, as shown in Fig. 5.11. Because the
incoming plane wave is normally incident on the plane of the slits, the fields that emerge
from all of the slits have the same phase at the exit plane of the slits, which is perpendicular
to ki . They also have the same polarization and the same amplitude because the slits have the
same geometrical dimensions. Therefore, on the exit plane of the slits, E 1 ¼ E 2 ¼ ¼
E N ¼ ^e E 0 :
As seen in Fig. 5.11, at a distant point in the direction at an angle of θ with respect to the x
axis, the phases of the rays coming from different slits increase between successive slits by the
amount of δ ¼ kΛ sin θ given in (5.13). Following the same reasoning for the double slits, the
total field at the distant point in this direction is the linear superposition of the fields coming
from all slits to the point:
E ¼ E 1 eikr1 iωt þ E 2 eikr2 iωt þ þ E N eikrN iωt
¼ ^e E 0 eikr1 iωt 1 þ eiδ þ þ eiðN1Þδ
¼ ^e E 0 eikr1 iωt
iNδ
1e
:
1 eiδ
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(5.21)
184
Optical Interference
Figure 5.11 Normal incidence of a monochromatic plane wave on a periodic multiple-slit structure.
The intensity at the distant observation point is
I ¼ I0
sin2 ðNδ=2Þ
,
sin2 ðδ=2Þ
(5.22)
where I 0 / jE 0 j2 is the intensity contributed by a single slit alone. Using the mathematical
relations
lim
x!0
sin2 Nx
¼ N 2 and
sin2 x
sin2 ½N ðx þ qπ Þ
sin2 Nx
¼
for q ¼ 0,
sin2 ðx þ qπ Þ
sin2 x
1,
2, ,
(5.23)
we find that the intensity I has maxima of the value N 2 I 0 when
δ ¼ kΛ sin θq ¼ 2qπ,
(5.24)
where q is an integer that represents the order of diffraction.
Figure 5.12 shows the intensity distribution given in (5.22) for the multiple-slit structure as
a function of the phase factor δ ¼ kΛ sin θ, which varies with the angle θ for fixed values of k
and Λ. The primary maxima that have the peak intensity of N 2 I 0 appear at the angles that satisfy
the condition given in (5.24). Secondary maxima of lower peak intensities exist between
primary maxima. As the number N of the periods in the structure increases, the peak intensity
of each primary maximum increases quadratically as N 2 while the width decreases linearly as
N 1 ; meanwhile, the peak intensities of all secondary maxima decrease.
The periodic multiple-slit structure functions as a transmissive diffraction grating that has a
wavenumber of
K¼
2π
Λ
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(5.25)
5.2 Optical Gratings
185
Figure 5.12 Intensity distribution as a function of the phase factor δ ¼ kΛ sin θ for a multiple-slit structure
functioning as a transmission grating. As the number N of periods increases, the primary maxima representing
the diffraction orders have peak intensities increasing as N 2 and widths decreasing as N 1 while the peak
intensities of all secondary maxima decrease.
in the z direction. Each primary maximum in the spatial intensity distribution of the transmitted
light represents a diffraction order. The qth-order diffracted beam has a wavevector of
kq ¼ k cos θq ^x þ k sin θq ^z :
(5.26)
Using the relations in (5.24) and (5.25), we find that
k sin θq ¼ qK:
(5.27)
Because the wavevector of the incident wave is ki ¼ k^x , there is a phase mismatch of
(5.28)
Δkq ¼ kq ki ¼ k cos θq 1 ^x þ k sin θq ^z ¼ k cos θq 1 ^x þ qK^z
between the qth-order diffracted beam and the incident wave.
A phase mismatch between two waves is a momentum difference between two photons of the
two waves. Clearly from (5.28), except for the zeroth order, an incident photon acquires
momentum changes in both x and z directions in the process to exit as a diffracted photon.
Because of conservation of momentum, any momentum change of a photon has to be compensated by an opposite momentum change of another physical object. The momentum change
Δkq, x ¼ k cos θq 1 in the x direction is easily compensated by an opposite momentum
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186
Optical Interference
change of the entire multiple-slit structure in the x direction because it is in the direction normal
to the plane of the structure and it is negligibly small for the mass of the structure. In the z
direction, however, no such momentum compensation is possible if the slits are absent from the
structure because no force in the direction parallel to the plane of the structure can be exerted on
the structure. In the presence of the periodic slits, the periodicity along the z direction provides
the necessary compensation for the momentum change of Δkq, z ¼ k sin θq in the z direction
when the phase-matching condition k sin θq ¼ qK of (5.27) is satisfied. Therefore, constructive
interference for a diffracted beam is equivalent to phase matching for the beam.
5.2.2 Oblique Incidence
In the above, we considered a monochromatic plane wave that is normally incident on the
multiple-slit structure at an incident angle of θi ¼ 0 so that ki ¼ k^x . The equivalent concepts of
constructive interference and phase matching for the diffraction orders can also be applied to
oblique incidence at a nonzero incident angle of θi 6¼ 0 so that ki ¼ k i, x ^x þ k i, z^z ¼ k cos θi ^x þ
k sin θi^z 6¼ k^x , as shown in Fig. 5.13. With an incident wave of this wavevector, the field
emerging from the slits has a phase shift of k i, z Λ ¼ kΛ sin θi from one slit to the next in
the z direction so that E 1 ¼ ^e E 0 , E 2 ¼ E 1 eiki, z Λ ¼ ^e E 0 eikΛ sin θi , . . . , E N ¼ E 1 eiðN1Þki, z Λ ¼
^e E 0 eiðN1ÞkΛ sin θi . Applying these relations to (5.21), we find that the phase factor δ ¼ kΛ sin θ
for normal incidence at θi ¼ 0 is generalized to
δ ¼ kΛ sin θ kΛ sin θi
(5.29)
for oblique incidence at θi 6¼ 0. Therefore, the condition given in (5.24) for finding the maxima
of the diffracted intensity distribution is generalized to the condition:
δ ¼ kΛ sin θq kΛ sin θi ¼ 2qπ,
where q is an integer.
Figure 5.13 Oblique incidence of a monochromatic plane wave on a periodic multiple-slit structure.
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(5.30)
187
5.2 Optical Gratings
From the phase-matching point of view, the condition in (5.30) can be easily obtained from
the condition for phase matching assisted by the grating of a wavenumber K in the z direction:
Δkz ¼ k q, z k i, z ¼ qK,
(5.31)
which is identical to (5.30) in the form of
k sin θq ¼ k sin θi þ qK:
(5.32)
As discussed above for the case of normal incidence, it is also true for oblique incidence that
phase matching in the x direction normal to the plane of the grating structure does not set a
required condition because it is automatically satisfied by a compensating momentum change of
the massive structure. Note that the zeroth order takes place at θ0 ¼ θi .
EXAMPLE 5.3
A monochromatic plane wave at the λ ¼ 651 nm wavelength is normally incident on the plane
of an array of equally spaced slits. The 20th-order diffraction peak is found at the angle of
θ20 ¼ 10 . Find the spacing Λ between neighboring slits. If a plane wave at λ ¼ 488 nm is
normally incident on the slits, what is the diffraction angle of the 20th-order diffraction peak? If
it is obliquely incident for the 20th-order diffraction peak to appear at θ20 ¼ 10 , what is the
required incident angle?
Solution:
For normal incidence with λ ¼ 651 nm and θ20 ¼ 10 , (5.27) requires that
k sin θq ¼ qK
)
sin θq q
¼
λ
Λ
)
Λ¼
651 109
m ¼ 75 μm:
sin 10
qλ
20
¼
sin θq
For λ ¼ 488 nm at normal incidence, the 20th-order diffraction peak appears at
k sin θq ¼ qK
)
θq ¼ sin1
qλ
Λ
)
θ20 ¼ sin1
20
488 109
¼ 7:48 :
75 106
For oblique incidence, the incident angle is found using (5.32) as
qλ
1
sin θq k sin θq ¼ k sin θi þ qK ) θi ¼ sin
:
Λ
For the 20th-order diffraction peak of λ ¼ 488 nm to appear at θ20 ¼ 10 , we find
20 488 109
1
¼ 2:49 :
sin 10 θi ¼ sin
75 106
5.2.3 Grating at an Interface
When an optical wave is incident on a grating at an interface between two different optical
media, as shown in Fig. 5.14, diffraction orders in reflection and in transmission can both
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188
Optical Interference
Figure 5.14 (a) Optical grating at an interface. (b) Phase-matching conditions for the reflective and
transmissive diffraction orders.
appear. Assuming that the incident wave comes from medium 1, which has a refractive index of
n1 , at an incident angle of θi with respect to the normal of the interface, the diffraction orders on
both sides of the interface are determined by the phase-matching conditions:
k1 sin θ1q ¼ k1 sin θi þ qK
(5.33)
for the reflective diffraction orders in medium 1, and
k2 sin θ2q ¼ k2 sin θi þ qK
(5.34)
for the transmissive diffraction orders in medium 2. Note that for the zeroth order, θ10 ¼ θi in
reflection and n2 sin θ20 ¼ n1 sin θi in transmission, which are just those required by Snell’s law
for a flat surface when the grating does not exist. Here we only consider the phase-matching
conditions that determine the direction of each diffraction order; whether a diffraction order
appears or not also depends on the shape and the geometrical parameters of the grating, as
discussed in Example 4.2.
EXAMPLE 5.4
A grating that has a period of Λ ¼ 2 μm is fabricated on the surface of a glass plate, which has a
refractive index of 1:5. It is exposed to air. A laser beam at the wavelength of λ ¼ 850 nm is
normally incident on the grating from the air side. How many diffraction orders are possible on
each side? What is the diffraction angle of each order?
Solution:
For normal incidence, θi ¼ 0 . Thus, the phase-matching conditions in (5.33) and (5.34) reduce
to k1 sin θ1q ¼ qK and k2 sin θ2q ¼ qK, which can be expressed as
sin θ1q ¼
qλ
qλ
and sin θ2q ¼
:
n1 Λ
n2 Λ
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5.2 Optical Gratings
189
Every diffraction angle is required to be within the range between 90 and 90 , i.e., 1 sin θ1q 1 and 1 sin θ2q 1.
On the air side, n1 ¼ 1; thus
1 sin θ1q ¼
qλ
1
n1 Λ
)
0 jqj n1 Λ 1 2 106
¼ 2:35:
¼
λ
850 109
There are five diffraction orders on the air side for q ¼ 2,1, 0, 1, 2. The diffraction angles
with respect to the surface normal are
θ1q ¼ sin1
)
qλ
q 850 109
¼ sin1
n1 Λ
1 2 106
θ1q ¼ 58:21 , 25:15 , 0 , 25:15 , 58:21 :
On the glass side, n2 ¼ 1:5; thus
1 sin θ2q ¼
qλ
1
n2 Λ
)
0 jqj n2 Λ 1:5 2 106
¼ 3:52:
¼
λ
850 109
There are seven diffraction orders on the glass side for q ¼ 3, 2, 1, 0, 1, 2, 3. The
diffraction angles with respect to the surface normal are
θ2q ¼ sin1
)
qλ
q 850
¼ sin1
n2 Λ
1:5 2
109
106
θ1q ¼ 58:21 , 34:52 , 16:46 , 0 , 16:46 , 34:52 , 58:21 :
5.2.4 Surface Grating–Waveguide Coupling
A grating fabricated on the surface of a waveguide can couple a radiation field that propagates
in the homogeneous space on one side of the waveguide into a waveguide mode. In reverse
operation, it can also couple a waveguide mode into a radiation field from the surface of the
waveguide. These concepts are illustrated in Fig. 5.15.
For this purpose, it is necessary to phase match the radiation field with the waveguide mode
in the longitudinal direction of the waveguide, which is taken to be the z direction. For coupling
Figure 5.15 Surface
grating for (a) input
coupling and (b) output
coupling of a
waveguide mode.
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190
Optical Interference
with a waveguide mode that has a propagation constant of β, the incident optical wave has to
satisfy the phase-matching condition:
k2 sin θ2q þ qK ¼ β
(5.35)
if the wave is incident from the substrate side of a refractive index n2 at an incident angle of
θ2q , or
k3 sin θ3q þ qK ¼ β
(5.36)
if the wave is incident from the cover side of a refractive index n3 at an incident angle of θ3q .
The same phase-matching conditions are used to determine the directions of output coupling.
Note that the phase-matching conditions given in (5.35) and (5.36) only determine the
directions of the radiation fields that can be coupled into or out from a waveguide mode, but
they do not tell us the efficiency of the coupling. The coupling efficiency is determined by the
coupling coefficient, which depends on the shape, the depth, and other geometrical parameters
of the grating, as discussed in Example 4.2.
EXAMPLE 5.5
A sinusoidal grating that can only serve as a first-order grating is fabricated on the surface of a
GaAs slab waveguide as shown in Fig. 5.15. The cover of the waveguide is simply air so that
n3 ¼ 1. At the wavelength of λ ¼ 1:3 μm, the propagation constant of the TE0 mode of this
waveguide is β ¼ 1:62 107 nm, corresponding to an effective index of nβ ¼ 3:35. If it is
desired that a laser beam at this wavelength be coupled into this guided mode through the
surface grating at an incident angle of θi ¼ 45 , what is the required period of the grating?
Solution:
Because a sinusoidal grating can be used only as a first-order grating, it is necessary that the
phase-matching condition is satisfied for q ¼ 1 or q ¼ 1. Because the wave is incident from
the cover side, the condition is that from (5.36) with q ¼ 1:
k 3 sin θ31 þ K ¼ β
)
n3
1 nβ
sin θ31 þ ¼
λ
λ
Λ
)
Λ¼
λ
nβ n3 sin θ31
:
With λ ¼ 1:3 μm, nβ ¼ 3:35, n3 ¼ 1, and θ31 ¼ θi ¼ 45 , the required grating period is
Λ¼
λ
nβ n3 sin θ31
¼
1:3 106
m ¼ 492 nm:
3:35 1 sin 45
5.2.5 Flat Interface
The phase-matching concept can be applied to reflection and refraction at a flat, smooth
interface between two media of different indices n1 and n2 to obtain Snell’s law discussed in
Section 3.4. For a smooth surface that is not modified by any periodic structure, we can take the
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5.3 Fabry–Pérot Interferometer
191
limit of an infinitely large period, Λ ! ∞, thus a zero wavenumber of K ¼ 0. Then, by applying
(5.31) with K ¼ 0 to the reflected and transmitted waves, we obtain the following phasematching condition:
k r, z ki, z ¼ k t, z ki, z ¼ 0,
(5.37)
which yields the condition of ki sin θi ¼ k r sin θr ¼ k t sin θt given in (3.88) and Snell’s law
expressed in (3.89) and (3.90).
5.3
FABRY–PÉROT INTERFEROMETER
..............................................................................................................
The basic principle of the Fabry–Pérot interferometer is the interference of multiple reflections from
two partially reflective parallel surfaces. The desired reflectivity for each of these two surfaces can
be obtained by proper coating. The basic structure of the Fabry–Pérot interferometer takes two
different forms. The first form shown in Fig. 5.16(a) consists of two partially reflective mirrors on
the parallel inner surfaces of two dielectric plates; the outer surfaces of the plates are antireflection
coated and often wedged to prevent unwanted reflection from these surfaces. In the second form
shown in Fig. 5.16(b), the two partially reflective surfaces are the parallel surfaces of a transparent
dielectric plate; a Fabry–Pérot interferometer of this form is usually called a Fabry–Pérot etalon.
The two structures shown in Figs. 5.16(a) and (b) have the same interferometric characteristics despite the differences in their detailed structures. For both structures, we consider a
physical spacing of l that is filled with a medium of a refractive index n between the two
partially reflective surfaces, as shown in Fig. 5.16. The direction normal to the reflective
surfaces is taken to be the z direction. We consider for generality oblique incidence of a
Figure 5.16 (a) Fabry–Pérot interferometer. The outer surfaces of the wedged plates are antireflection coated.
(b) Fabry–Pérot etalon.
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192
Optical Interference
monochromatic plane wave of a frequency ω and a wavelength λ. The wavevector of the wave
that is transmitted through the first partially reflective surface makes an angle of θ with respect
to the normal of the reflective surface; this angle is not necessarily the same as the incident
angle of the wave coming from outside because the refractive index of the outside medium is
not necessary the same as that inside the interferometer. The field-amplitude reflection coefficients r 1 and r2 of the left and right mirrors, respectively, can be expressed as
1=2
1=2
r1 ¼ R1 eiφ1 ,
r 2 ¼ R2 eiφ2 ,
(5.38)
where R1 and R2 are the intensity reflectivities of the left and right reflective surfaces,
respectively, and φ1 and φ2 are the phase changes of the optical fields upon reflection on these
surfaces. As discussed in Section 3.4, the reflection coefficients r 1 and r 2 are functions of the
incident angle θ and the polarization of the optical field.
Multiple partial reflections inside the interferometer take place at the two partially reflective
surfaces, as seen in Fig. 5.16. Between the two reflective surfaces, all forward-propagating
waves have the same wavevector at an angle of θ with respect to the z direction so that
k z ¼ k cos θ, and all backward-propagating waves have the same wavevector at an angle of π θ with respect to the z direction so that k z ¼ k cos ðπ θÞ ¼ k cos θ. Each forward or backward pass through the spacing of a length l causes a phase shift of kl cos θ. Each time a wave
reaches a reflective surface, part of it is transmitted and the rest of it is reflected; multiple
reflections by the reflective surfaces produce multiple transmitted waves. At a given location
on the outside of the interferometer, each successive transmitted field is related to the
preceding transmitted field by a factor of
1=2 1=2
1=2 1=2
r 1 r 2 ei2kl cos θ ¼ R1 R2 eið2kl cos θþφ1 þφ2 Þ ¼ R1 R2 eiφRT ,
(5.39)
where
φRT ¼ 2kl cos θ þ φ1 þ φ2 ¼ 4π
νnl
nl
cos θ þ φ1 þ φ2 ¼ 4π cos θ þ φ1 þ φ2
c
λ
(5.40)
is the total phase shift caused by a round-trip passage between the two reflective surfaces. This
phase shift includes the phase shift of 2kl cos θ from the double passes through the medium in
the spacing and the localized phase shifts of φ1 and φ2 from reflections at the two reflective
surfaces.
The interferometer has two output ports: one in the forward direction for the total transmitted
field and the other in the backward direction for the total reflected field. The total transmitted
field through the interferometer at the forward output port is the linear sum of all transmitted
fields through the second reflective surface:
Etout ¼ E 0 eiωt þ E 1 eiωt þ E 2 eiωt þ 1=2 1=2
1=2 1=2
¼ E 0 eiωt 1 þ R1 R2 eiφRT þ R1 R2 eiφRT
¼ E 0 eiωt
1
1=2 1=2
1 R1 R2 eiφRT
2
þ ,
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(5.41)
5.3 Fabry–Pérot Interferometer
193
where E 0 is the transmitted field that directly passes through the two reflective surfaces, E 1 is
the transmitted field after one reflection by each reflective surface, and E 2 is the transmitted
field after two reflections by each reflective surface, and
2
. so forth.
1=2 1=2 iφRT t
From (5.41), the total transmitted intensity is I out ¼ I 0 1 R1 R2 e , where the intensity
I 0 of the directly transmitted field E 0 is related to the input intensity as I 0 ¼ ð1 R1 Þð1 R2 ÞI in .
Therefore, the transmittance of a lossless Fabry–Pérot interferometer for the forward output
port is
T FP ¼
I tout
ð1 R1 Þð1 R2 Þ
ð1 R1 Þð1 R2 Þ
:
¼
2 ¼
2
I in
1=2 1=2
1=2 1=2
1=2 1=2 iφ 2
RT
R
þ
4R
R
sin
ð
φ
=2
Þ
1
R
R
e
1
R
RT
1
2
1
2
1
2
(5.42)
The reflectance of the Fabry–Pérot interferometer for the backward output port is
RFP ¼
I rout
¼ 1 T FP :
I in
(5.43)
The maximum transmittance of the Fabry–Pérot interferometer is
T max
FP ¼
ð1 R1 Þð1 R2 Þ
1=2 1=2
1 R1 R2
2
:
(5.44)
The maximum transmittance is T max
FP ¼ 1 for a lossless symmetric Fabry–Pérot interferometer
max
that has R1 ¼ R2 , but T FP < 1 for an asymmetric Fabry–Pérot interferometer that has R1 6¼ R2 .
We can define a normalized transmittance as
T FP
T^ FP ¼ max ¼
T FP
where
1
2
4F
1 þ 2 sin2 ðφRT =2Þ
π
,
(5.45)
1=4 1=4
F¼
πR1 R2
1=2 1=2
1 R1 R2
(5.46)
is the finesse of the lossless Fabry–Pérot interferometer. As expressed in (5.46) and plotted in
Fig. 5.17, the finesse of a lossless Fabry–Pérot interferometer is a nonlinear function of the
product, R1 R2 , of the reflectivities of the two reflective surfaces that form the interferometer.
The normalized transmittance T^ FP of a lossless Fabry–Pérot interferometer expressed in
(5.45) is plotted in Fig. 5.18 as a function of the round-trip phase shift φRT for a few values
of the finesse of the interferometer. The strong dependence of T^ FP on φRT is the consequence of
the interference of the multiple reflections between the two reflective surfaces. The transmittance peaks appear at
φRT ¼ 2qπ,
νq ¼ q φ1 þ φ2
c
,
2π
2nl cos θ
(5.47)
where q is an integer so that all transmitted fields resulting from multiple reflections in the
interferometer are in phase for constructive interference. The separation between two
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Optical Interference
Figure 5.17 Finesse, F, of a lossless Fabry–Pérot interferometer as a function of the product, R1 R2 , of the
reflectivities of the two reflective surfaces of the interferometer.
Figure. 5.18 Normalized transmittance T^ FP of a lossless Fabry–Pérot interferometer as a function of the roundtrip phase shift φRT for a few values of the finesse of the interferometer.
neighboring peaks in the spectrum is called the free spectral range, which has a round-trip
phase difference of ΔφFSR and a frequency difference of ΔνFSR :
c
ΔφFSR ¼ 2π, ΔνFSR ¼
:
(5.48)
2nl cos θ
Away from the peaks, the transmittance is low because the transmitted fields are out of phase,
resulting in destructive interference. Each transmittance peak has a finite FWHM linewidth,
Δφline , measured in terms of the shift in the round-trip phase, or Δνline , measured in terms of the
optical frequency. Actually, the finesse is defined as the ratio of the free spectral range to the
linewidth:
F¼
ΔφFSR ΔνFSR
¼
:
Δφline
Δνline
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(5.49)
5.3 Fabry–Pérot Interferometer
195
The relation given in (5.46) for a lossless Fabry–Pérot interferometer is a valid approximation
for F 1: Therefore, the linewidth decreases with increasing finesse, which in turn increases
nonlinearly with the value of R1 R2 .
As seen in (5.40), the round-trip phase φRT is a function of the wavelength λ of the optical
wave, the physical spacing l of the interferometer, the refractive index n of the medium between
the two reflective surfaces, and the angle θ at which the wave propagates inside the interferometer and is incident on the reflective surfaces. The transmittance of a Fabry–Pérot interferometer can be varied by varying any of these physical parameters. The strong dependence of
the transmittance on the optical wavelength, thus on the optical frequency, allows a high-finesse
Fabry–Pérot interferometer to be used as an optical spectrum analyzer. A high finesse leads to a
narrow linewidth for the transmittance peaks, thus a high resolution for the optical spectrum
analyzer. Further detailed characteristics of the Fabry–Pérot interferometer used as an optical
resonator are discussed in Chapter 6.
EXAMPLE 5.6
What happens to the maximum transmittance T max
FP , the finesse F, the frequencies νq at which
the peak transmittance occurs, the free-spectral range ΔνFSR , and the spectral linewidth Δνline of
a Fabry–Pérot interferometer in each of the following situations? (a) The reflectivity R1 or R2 is
increased, or both are increased. (b) The spacing l is increased. (c) The index n of the medium
between the reflective surfaces is increased. (d) The angle θ at which the wave propagates
between the reflective surfaces is increased.
Solution:
The transmittance of a Fabry–Pérot interferometer is a direct function of only three parameters,
R1 , R2 , and φRT , as seen in (5.42); however, φRT is a function of the parameters l, n, θ, and the
optical frequency ν. Each of the other characteristics of the Fabry–Pérot interferometer depends
on some of these parameters but is independent of the other parameters.
(a) The reflectivity R1 or R2 is increased, or both are increased. From (5.44), we find that T max
FP
does not monotonically vary with R1 or R2 . Indeed, we find
max max dT FP
dT FP
¼ signðR2 R1 Þ and sign
¼ signðR1 R2 Þ:
sign
dR1
dR2
Therefore, T max
FP increases with increasing R1 if R1 < R2 , but it decreases with R1 if
max
R1 R2 , including when R1 ¼ R2 because T max
FP reaches its largest value of T FP ¼ 1 when
R1 ¼ R2 . Similarly, T max
FP increases with increasing R2 if R1 > R2 , but it decreases with R2 if
R1 R2 , including when R1 ¼ R2 . From (5.46), we find that the finesse F monotonically
increases with the product R1 R2 ; therefore, it increases when R1 R2 is increased through
increasing either R1 or R2 , or both. From (5.47) and (5.48), we find that both νq and ΔνFSR
do not vary with R1 or R2 . From (5.49), we find that Δνline decreases when the product R1 R2
is increased because Δνline ¼ ΔνFSR =F.
(b) The spacing l is increased. From (5.44) and (5.46), we find that both T max
FP and F are
independent of the spacing l; they do not change as l is increased. From (5.47) and (5.48),
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Optical Interference
we find that both νq and ΔνFSR decrease when the spacing l is increased. From (5.49), we
find that Δνline decreases with increasing l because Δνline ¼ ΔνFSR =F.
(c) The index n of the medium between the reflective surfaces is increased. From (5.40), (5.47),
and (5.48), we find that the index n and the spacing l always appear together in the form of
their product nl. Indeed what counts is the optical path length nl, rather than the physical
length. Therefore, increasing the index n has exactly the same consequences as increasing
the spacing l discussed in (b).
(d) The angle θ at which the wave propagates between the reflective surfaces is increased.
From (5.40), (5.47), and (5.48), we find that actually the angle θ always appears together
with the index n and the spacing l in the form of nl cos θ. Increasing θ reduces the effective
optical path length nl cos θ. Therefore, increasing θ is equivalent to reducing the spacing l
or the refractive index n: Both T max
FP and F do not change with θ; both νq and ΔνFSR increase
with increasing θ; Δνline increases with increasing θ.
5.3.1 Optical Thin Films
Optical thin films are thin layers of optical materials that have thicknesses on the order of the
optical wavelength. An optical thin film can be either a free-standing layer in a homogeneous
medium, such as the film of a soap bubble in air, or a layer deposited on a substrate of a
different optical property, such as a thin SiO2 layer on a silicon substrate. A sophisticated thinfilm structure can be composed of multiple thin layers of different optical properties. Figure 5.19
shows some examples of optical thin films.
Figure 5.19 Examples of optical thin films.
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5.3 Fabry–Pérot Interferometer
197
A single optical thin film has the structure, thus the basic optical property, of a Fabry–Pérot
interferometer in the etalon form. The two surfaces of the thin film act as the two partially
reflective surfaces of the interferometer. Multiple reflections take place in the thin film between
these two surfaces. Therefore, the reflectance and transmittance of an optical thin film are
functions of the optical wavelength, the incident angle, the thickness and refractive index of the
thin film, and the refractive indices of the media on the two sides of the thin film. An optical
thin film often exhibits a color because of the strong wavelength dependence of its reflectance
and transmittance. A thin film that has a spatially varying thickness can produce a spectrum of
spatially distributed colors, as often seen in soap bubbles or oil slicks.
EXAMPLE 5.7
An oil film of a uniform thickness l ¼ 100 nm floats on water. The refractive index of the oil
film is noil ¼ 1:40 and that of water is nw ¼ 1:33. When it is illuminated by white light at
normal incidence, which wavelength in the visible spectral range shows the highest reflection?
What color does it appear to be? If the same film is coated on a glass surface of a refractive
index ng ¼ 1:50, does it show the same high reflection?
Solution:
For the oil film on water, we find that noil > nw > nair . Therefore, for the wave inside the oil
film as an interferometer, the reflection at the air–film interface and that at the film–water
interface are both internal reflection with no phase changes so that φ1 ¼ φ2 ¼ 0. Then,
according to (5.47), for normal incidence the peak transmittance for dark reflection occurs at
νq ¼ q φ 1 þ φ2 c
c
¼q
2π
2nl
2noil l
)
λdark ¼
c
2noil l
¼
,
νq
q
and the minimum transmittance for bright reflection occurs at
1 φ 1 þ φ2 c
1
c
c
4noil l
νq1=2 ¼ q ¼
) λbright ¼
¼ q
:
2π
2
2nl
2 2noil l
νq1=2 2q 1
With noil ¼ 1:40 and l ¼ 100 nm, we find that the only λbright that falls within the 400 to 700 nm
visible spectral range is found for q ¼ 1 at
λbright ¼
4noil l
¼ 4noil l ¼ 4
2q 1
1:4
100 nm ¼ 560 nm:
The next bright reflection takes place for q ¼ 2 at 186:7 nm, which is in the deep UV.
Therefore, the film appears to be green.
If the same film is coated on a glass surface of a refractive index ng ¼ 1:50, then
ng > noil > nair . In this situation, the reflection at the air–film interface is still internal reflection
with φ1 ¼ 0, but that at the film–glass interface is external reflection with φ1 ¼ π. Then,
according to (5.47), for normal incidence the peak transmittance for dark reflection occurs at
φ1 þ φ2 c
1
c
c
4noil l
νq ¼ q ) λdark ¼ ¼
¼ q
,
2π
2nl
2 2noil l
νq 2q 1
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Optical Interference
and the minimum transmittance for bright reflection occurs at
1 φ1 þ φ2 c
c
c
2noil l
νq1=2 ¼ q ¼
) λbright ¼
¼ ðq 1Þ
:
2π
2
2nl
2noil l
νq1=2 q 1
With noil ¼ 1:40 and l ¼ 100 nm, we find that no λbright falls within the 400 to 700 nm visible
spectral range because the largest value for λbright is found for q ¼ 2 at 280 nm, which is in the
UV. Therefore, this film appears to be colorless on glass.
A thin film on an optical surface can dramatically change the reflection and transmission
properties of the surface. Thin-film coating is an important technology for designing and
achieving desired reflection and transmission properties of an optical surface, and thin-film
optics has been developed into an important field in optics. Sophisticated thin films consisting
of multiple layers of different thicknesses and different refractive indices are used for advanced
optical coatings. A desired reflection property, such as broadband antireflection, broadband
total reflection, narrowband antireflection, or narrowband high reflection, can be obtained by
coating an optical surface with a properly designed thin-film structure. Applications of thin-film
optical coatings range from high-precision coatings for optical filters and laser mirrors to lowemission glass panes for house windows.
EXAMPLE 5.8
A uniform thin film of MgF2 , which has a refractive index of nf ¼ 1:38 is deposited on the
surface of a glass lens, which has a refractive index of ng ¼ 1:50, to serve as an antireflective
coating at the wavelength of λ ¼ 552 nm. What is the minimum thickness of the thin film?
What other thicknesses can be chosen? How effective is this thin film as an antireflective
coating? How can the thin-film material be chosen to further increase the effectiveness of the
antireflective coating?
Solution:
There are two interfaces: the air–MgF2 interface and the MgF2–glass interface. Because the
refractive index increases from one medium to the next with nair ¼ 1, nf ¼ 1:38, and ng ¼ 1:50,
for the wave inside the thin film as an interferometer, the reflection at the air–film interface is
internal reflection with no phase change and that at the film–glass interface is external reflection
with a phase change of π; thus φ1 ¼ 0 and φ2 ¼ π. For the film to serve as an antireflective
coating, it is desired that T FP ¼ T max
FP , which takes place at the optical frequencies νq given
in (5.47):
φ 1 þ φ2
c
1 c
νq ¼ q ¼ q
2π
2nl cos θ
2 2nf l
for normal incidence. With the given wavelength at λ ¼ c=ν ¼ 552 nm, the acceptable thicknesses are
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5.3 Fabry–Pérot Interferometer
lq ¼
1
c
1 λ
1
552
1
q
¼ q
¼ q
nm ¼ 200 q nm:
2 2nf ν
2 2nf
2 2 1:38
2
Therefore, the minimum thickness is lmin ¼ 100 nm for q ¼ 1, and any thickness that is larger
than the minimum thickness by an integral multiple of 200 nm, such that l ¼ 100ð2m þ 1Þ nm,
also works.
Without the coating, the reflectivity at the air–glass interface is
nair ng 2 1 1:52
¼ 0:04:
R¼
¼
nair þ ng 1 þ 1:5
With the thin-film coating, the reflectivities at the two interfaces are
nair nf 2 1 1:382
nf ng 2 1:38 1:52
¼
R1 ¼ 1 þ 1:38 ¼ 0:0255, R2 ¼ nf þ ng ¼ 1:38 þ 1:5 ¼ 1:736
nair þ nf 103 :
The reflectivity of the coated surface is
RFP ¼ 1 T max
FP ¼ 1 ð1 R1 Þð1 R2 Þ
1=2 1=2
1 R1 R2
2
¼ 1 0:986 ¼ 0:014:
Therefore, the thin-film coating cuts the reflectivity by 65% from 0:04 to 0:014.
To increase the effectiveness of the antireflective coating, the material of the thin film has to
be chosen so that R1 and R2 have closer values. The coating results in total antireflection with
RFP ¼ 0 when R1 ¼ R2 so that T max
FP ¼ 1. This can be accomplished by choosing the refractive
pffiffiffiffiffiffiffiffiffiffiffi
index of the thin film
to be ffinf ¼ nair ng . For this thin film to be totally antireflective, a material
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
of an index nf ¼ 1 1:5 ¼ 1:225 has to be chosen for the film.
5.3.2 Interference Filters
A high-finesse Fabry–Pérot interferometer can be used as an interference filter to selectively
transmit a desired wavelength. The wavelength selectivity of the filter is determined by its free
spectral range; a larger free spectral range allows fewer transmission wavelengths within a given
spectrum. For a desired transmission wavelength λ, the largest spectral range for an interference
filter is ΔνFSR ¼ ν ¼ c=λ, which is achieved when the optical path length of the interferometer is
half the optical wavelength: nl ¼ λ=2. For such a filter, the next transmission peak occurs at the
second harmonic frequency, 2ν, of the desired transmission frequency ν, i.e., at the wavelength λ=2
that is half the desired transmission wavelength λ, if the dispersion of the refractive index n is
negligible between ν and 2ν. The pass band around the transmission frequency is determined by
the linewidth of the interferometer. As discussed above, for a given free spectral range the
linewidth can be reduced by increasing the finesse through increasing the product R1 R2 of the
reflectivities of the reflective surfaces. By properly coating the two reflective surfaces for high
reflectivities, an interference filter of a narrow linewidth on the order of a nanometer or an
angstrom can be obtained. Such a highly selective, narrow-linewidth filter is also called a line filter.
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Optical Interference
Problems
5.1.1 Show that in the case when the angles between the wavevectors k1 and k2 of two optical
fields is small, the intensity of the combined optical field projected on a plane that is
normal to k1 þ k2 is approximately that given in (5.7).
5.1.2 A glass wedge of a refractive index n ¼ 1:5 as shown in Fig. 5.5 has a length of l ¼ 5 cm
and a height of h ¼ 1 mm. It is vertically illuminated with coherent light at the λ ¼
600 nm wavelength. What is the period of the interference fringes? How many dark and
bright interference fringes appear on the surface of the wedge?
5.1.3 If the incident light in Problem 5.1.2 is not completely coherent, what is the minimum
coherence time of the wave for all of the interference fringes to appear on the wedge? If
1000 periods of interference fringes appear, what is the coherence time of the incident light?
5.1.4 An air wedge is formed between two flat glass plates by making them in contact at one
end but separated by the thickness of a piece of paper at the other end. When it is
vertically illuminated with monochromatic coherent light at the λ ¼ 500 nm wavelength,
exactly 400 periods of interference fringes are seen. What is the thickness of the paper?
5.1.5 A laser beam at the λ ¼ 532 nm wavelength is normally incident on two slits that are
spaced at Λ ¼ 200 μm. What is the angle between the two bright interference fringes of
the diffraction orders q ¼ 10? On a screen that is at a distance of l ¼ 2 m from the slits,
what is the separation of these two fringes?
5.1.6 Two slits separated by Λ ¼ 100 μm are illuminated with a laser beam at normal incidence. On a screen that is at a distance of l ¼ 2:5 m from the slits, it is found that the
separation between two neighboring dark fringes is 12:2 mm, what is the wavelength of
the laser light?
5.1.7 A laser beam is sent into a Michelson interferometer that is constructed in free space, as
shown in Fig. 5.7.
(a) When the mirror of one arm is moved to increase the length of the arm by 0:5 mm
while the other arm is fixed, the intensity pattern at each output port repeats itself
1880 times. Find the wavelength of the laser beam.
(b) The two arms are adjusted such that I out, 1 ¼ I in and I out, 2 ¼ 0. Then, a thin glass
plate that has a refractive index of n ¼ 1:46 and a thickness of d ¼ 1 mm is inserted
perpendicularly to the beam path into one of the two arms without changing the
optical alignment. What are the output intensities I out, 1 and I out, 2 now?
5.1.8 A laser beam is sent into a Mach–Zehnder interferometer that is constructed in free space,
as shown in Fig. 5.8.
(a) When the mirror of one arm is moved to increase the length of the arm by 0:5 mm
while the other arm is fixed, the intensity pattern at each output port repeats itself 940
times. Find the wavelength of the laser beam.
(b) The two arms are adjusted such that I out, 1 ¼ I in and I out, 2 ¼ 0. Then, a thin glass
plate that has a refractive index of n ¼ 1:46 and a thickness of d ¼ 1 mm is inserted
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Problems
201
perpendicularly to the beam path into one of the two arms without changing the
optical alignment. What are the output intensities I out, 1 and I out, 2 now?
5.1.9 A waveguide Mach–Zehnder interferometer uses Y-junction couplers for its input and
output ports, as shown in Fig. 5.9(a). It has a symmetric structure with an equal length of
la ¼ lb ¼ l for the two arms. The two Y-junctions are both 3-dB couplers. Thus, Δφ ¼ 0,
and the transmittance is T ¼ 1. By changing the refractive index of the medium in one
arm with respect to the other through the Pockels effect, for example, the phase shifts
through the two arms can be made different for Δφ 6¼ 0 so that T 6¼ 1. Find the minimum
necessary index difference Δn between the two arms for T ¼ 0 at an optical wavelength
of λ. At λ ¼ 1 μm, what is the minimum value of Δn for an equal arm length of
l ¼ 1 mm? If the Mach–Zehnder interferometer has a symmetric structure with la ¼ lb ¼
l using two 3-dB directional couplers, as shown in Fig. 5.9(b), the transmittance is T ¼ 0
with Δφ ¼ 0. Then, what is the minimum necessary index difference Δn between the two
arms for T ¼ 1 at an optical wavelength of λ? At λ ¼ 1 μm, what is the minimum value of
Δn for an equal arm length of l ¼ 1 mm?
5.2.1 Identical slits in an array are equally spaced at Λ ¼ 20 μm. A plane wave at the λ ¼
532 nm wavelength is normally incident on the slits. How many diffraction peaks can be
found in transmission within the range of angles between 30 and 30 ? If the wave is
obliquely incident at an angle of θi ¼ 15 , how many diffraction peaks can be found in
transmission within the range of angles between 30 and 30 ?
5.2.2 Three perfectly aligned plane optical waves at λ1 ¼ 450 nm, λ2 ¼ 550 nm, and λ3 ¼ 650 nm
are normally incident at the same time on an array of identical slits that are equally spaced at
Λ. The diffraction peaks in transmission are examined. It is clear that the zeroth-order peaks
for all three wavelengths completely overlap at θq ¼ 0 for q1 ¼ q2 ¼ q3 ¼ 0.
(a) What are the lowest nonzero diffraction orders q1 and q2 for λ1 and λ2 , respectively,
that have exactly overlapped peaks? What is the minimum slit spacing Λ for this to be
possible?
(b) Answer the questions in (a) for λ2 and λ3 .
(c) Answer the questions in (a) for λ1 and λ3 .
(d) What are the nonzero diffraction orders q1 , q2 , q3 for λ1 , λ2 , λ3 , respectively, that
have exactly overlapped peaks? What is the smallest slit spacing Λ for this to be
possible?
5.2.3 A grating on the surface of a glass plate has a period of Λ ¼ 800 nm. The glass plate has a
refractive index of 1:5. A laser beam is normally incident on the grating from the air.
Only two nonzero diffraction orders, for q ¼ 1 and q ¼ 1, are allowed on the glass side,
but no nonzero diffraction orders are allowed on the air side. What is the possible
wavelength of the incident laser light?
5.2.4 A collimated laser beam at λ ¼ 800 nm is incident on a grating at an air–glass interface
from the air side. The refractive index of this glass is 1.5. At normal incidence, three
diffraction peaks for q ¼ 1, 0, and 1 are found on the glass side. By carefully varying
the incident angle of the laser beam, it is found that the q ¼ 1 diffraction peak just
disappears when the incident angle is θi ¼ 12:1 . Find the grating period. How many
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Optical Interference
diffraction peaks can be found at an incident angle of θi ¼ 10 from the air and glass
sides, respectively? At what angles are these diffraction peaks found?
5.2.5 Consider the waveguide and the grating of a period Λ ¼ 492 nm found in Example 5.5.
The waveguide supports the TE0 mode at the λ ¼ 1:55 μm wavelength. The effective
index of this mode at this wavelength is nβ ¼ 3:33. Find the incident angle for a laser
beam at λ ¼ 1:55 μm to be coupled into this guided mode.
5.2.6 A surface grating that has a period of Λ ¼ 300 nm is fabricated on the surface of a GaAs/
AlGaAs slab waveguide as shown in Fig. 5.15. The cover of the waveguide is simply air
with n3 ¼ 1. At the wavelength of λ ¼ 900 nm, the GaAs core has n1 ¼ 3:59 and the
AlGaAs substrate has n2 ¼ 3:39. The waveguide supports only the TE0 mode of an
unknown propagation constant. If it is found that a laser beam at λ ¼ 900 nm can be
coupled into this guided mode through the surface grating at an incident angle of
θi ¼ 30 , what is the propagation constant of the mode? What grating period will allow
coupling of this laser beam into this waveguide mode at normal incidence with θi ¼ 0 ?
5.3.1 A laser beam is sent at normal incidence into a Fabry–Pérot interferometer that is constructed in free space with R1 ¼ R2 ¼ 0:5.
(a) When one reflective surface is fixed in location but the other is moved to increase the
spacing between them by 0:5 mm, the transmitted intensity pattern repeats itself 1880
times. Find the wavelength of the laser beam.
(b) The interferometer is adjusted such that T FP ¼ 1. Then, a thin glass plate that has a
refractive index of n ¼ 1:46 and a thickness of d ¼ 1 mm is inserted perpendicularly
to the beam path into the spacing without changing the optical alignment. What is the
transmittance of the interferometer now?
5.3.2 A lossless Fabry–Pérot interferometer consists of two highly reflective surfaces with
R1 ¼ 95% and R2 ¼ 90%, which are separated by a spacing of l in free space. What are
the maximum transmittance and the finesse of this interferometer? It is used as an optical
spectrum analyzer. If a spectral resolution with a linewidth of Δλline ¼ 0:1 nm at the λ ¼
500 nm wavelength is desired, what is the required spacing l of the interferometer? What
is the wavelength separation ΔλFSR between neighboring transmission peaks? If a higher
resolution is needed, how should the spacing be changed in order to reduce the spectral
linewidth by half to Δλline ¼ 0:05 nm?
5.3.3 A Fabry–Pérot etalon consists of a thin glass plate that has a refractive index of n ¼ 1:50
and a thickness of l ¼ 100 μm. Its surfaces are coated such that its peak transmittance is
100% and it has a spectral linewidth of Δνline 5 GHz for high spectral resolution. Find
the values of R1 and R2 that allow the etalon to have these properties.
5.3.4 An oil film that has a refractive index of noil ¼ 1:40 floats on a smooth water surface,
which has nw ¼ 1:33. It reflects most strongly at the 672 nm red wavelength and appears
to have no reflection at the 504 nm blue wavelength. What is the thickness of the oil film?
5.3.5 A material that has a refractive index of nf ¼ 1:25 is used for the thin film discussed in
Example 5.8, which is deposited on the surface of a glass lens that has a refractive index
of ng ¼ 1:50. To serve as an antireflective coating at the wavelength of λ ¼ 552 nm, what
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Bibliography
203
is the minimum thickness required for the thin film? What other thicknesses can be
chosen? How effective is this thin film as an antireflective coating?
5.3.6 The refractive index of Si at the λ ¼ 1:0 μm wavelength is nSi ¼ 3:61. If an antireflective
thin film is to be coated on a smoothly polished Si surface, how should the refractive
index of the thin-film material be chosen so that the coated surface is totally antireflective
when exposed to air? What should the refractive index of the thin film be chosen if the
surface is to become totally antireflective in water, which has a refractive index of
nw ¼ 1:33?
Bibliography
Born, M. and Wolf, E., Principles of Optics: Electromagnetic Theory of Propagation, Interference and
Diffraction of Light, 7th edn. Cambridge: Cambridge University Press, 1999.
Fowler, G. R., Introduction to Modern Optics, 2nd edn. New York: Dover, 1975.
Haus, H. A., Waves and Fields in Optoelectronics. Englewood Cliffs, NJ: Prentice-Hall, 1984.
Liu, J. M., Photonic Devices. Cambridge: Cambridge University Press, 2005.
Serway, R. A. and Jewett, J. W., Physics for Scientists and Engineers, 9th edn. Boston, MA: Brooks Cole,
2013.
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Cambridge Books Online
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Principles of Photonics
Jia-Ming Liu
Book DOI: http://dx.doi.org/10.1017/CBO9781316687109
Online ISBN: 9781316687109
Hardback ISBN: 9781107164284
Chapter
6 - Optical Resonance pp. 204-223
Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.007
Cambridge University Press
6
6.1
Optical Resonance
OPTICAL RESONATOR
..............................................................................................................
As discussed in Section 5.3, multiple reflections take place between the two reflective surfaces
of a Fabry–Pérot interferometer, resulting in multiple transmitted fields. A transmittance peak
occurs when the round-trip phase shift φRT between the two reflective surfaces is an integral
multiple of 2π so that all of the transmitted fields are in phase. From the viewpoint of the field
inside the interferometer, this condition results in optical resonance between the two reflective
surfaces. Thus a Fabry–Pérot interferometer behaves as an optical resonator, also called a
resonant optical cavity. At resonance, the field amplitude inside an optical resonator reaches a
peak value due to constructive interference of multiple reflections. The optical energy stored in
an optical cavity peaks at its resonance frequencies.
An optical cavity can take a variety of forms. Figure 6.1 shows the schematic structures of a
few different forms of optical cavities. Though an optical cavity has a clearly defined longitudinal axis, the axis can lie on a straight line, as in Fig. 6.1(a), or it can be defined by a folded
path, as in Figs. 6.1(b), (c), and (d). A linear cavity defined by two end mirrors, as in Fig. 6.1(a),
is known as a Fabry–Pérot cavity because it takes the form of the Fabry–Pérot interferometer.
A folded cavity can simply be a folded Fabry–Pérot cavity that supports a standing intracavity
field, as in Fig. 6.1(b). A folded cavity can also be a non-Fabry–Pérot ring cavity that supports
two independent, contrapropagating intracavity fields, as in Figs. 6.1(c) and (d).
An optical cavity provides optical feedback to the optical field in the cavity. Optical
resonance occurs when the optical feedback is in phase with the intracavity optical field. The
optical feedback in a Fabry–Pérot cavity is provided simply by the two end mirrors that have
the reflective surfaces perpendicular to the longitudinal axis, as in Figs. 6.1(a) and (b). In a ring
cavity, it is provided by the circulation of the laser field along a ring path defined by mirrors, as
in Fig. 6.1(c), or a ring path defined by an optical fiber, as in Fig. 6.1(d). The cavity can also be
constructed with an optical waveguide, as in the case of a semiconductor laser or a fiber laser. In
the following discussion, we take the coordinate defined by the longitudinal axis to be the z
coordinate, and the transverse coordinates that are perpendicular to the longitudinal axis to be
the x and y coordinates. In a folded cavity, the z axis is thus also folded along with the
longitudinal optical path. Sophisticated optical cavities can use gratings to provide distributed
feedback; such advanced cavities are not shown in Fig. 6.1 and are not discussed in this chapter.
In a ring cavity, an intracavity field completes one round trip by circulating inside the
cavity in only one direction. The two contrapropagating fields that circulate in a ring cavity
in opposite directions are independent of each other even when they have the same frequency.
In a Fabry–Pérot cavity, an intracavity field has to travel the length of the cavity twice in
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6.1 Optical Resonator
205
Figure 6.1 Schematics of a few different forms of optical cavities: (a) linear Fabry–Pérot cavity with end mirrors;
(b) folded Fabry–Pérot cavity with end mirrors; (c) three-mirror ring cavity with two independent, contrapropagating
fields; and (d) ring cavity with two independent, contrapropagating fields guided by an optical-fiber waveguide.
opposite directions to complete a round trip. The time it takes for an intracavity field to
complete one round trip in the cavity is called the round-trip time,
T¼
round-trip optical path length lRT
,
¼
c
c
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(6.1)
206
Optical Resonance
Figure 6.2 Passive laser cavities with a gain filling factor Γ under optical injection: (a) a Fabry–Perot cavity and (b)
a ring cavity. The refractive index of the gain medium is n, while that of the background medium in the cavity is n0 .
A laser cavity is simply a passive optical cavity when its gain medium is absent or is present but not pumped.
where the round-trip optical path length lRT takes into account the refractive index of the
medium inside the cavity.
The space inside an optical cavity can be filled with a variety of optical media of different
properties. For example, a laser cavity contains at least a gain medium. The gain medium may fill
up the entire length of the cavity, or it may occupy a fraction of the cavity length. For a laser cavity
of a length l that contains a gain medium of a length lg , as shown in Fig. 6.2, we can define an
overlap factor between the gain medium and the intensity distribution of the laser mode as the ratio
ððð
jEj2 dxdydz
V gain
lg
gain
:
Γ ¼ ððð
(6.2)
V mode
l
2
jEj dxdydz
cavity
This ratio is commonly known as the gain filling factor for a gain medium that takes up only a
fraction of the length of the laser cavity, whereas it is related to the mode confinement factor in
a waveguide laser, such as a fiber laser or a semiconductor laser. When the gain medium fills up
an optical cavity and covers the entire intracavity field distribution, Γ ¼ 1; otherwise, Γ < 1.
Take the refractive index of the gain medium to be n and that of the intracavity medium
excluding the gain medium to be n0 ; then, the round-trip optical path length can be expressed as
2½Γnl þ ð1 ΓÞn0 l ¼ 2nl, for a linear cavity;
(6.3)
lRT ¼
for a ring cavity;
Γnl þ ð1 ΓÞn0 l ¼ nl,
where n ¼ Γn þ ð1 ΓÞn0 is the weighted average index of refraction throughout the laser
cavity. When an optical cavity contains optical elements other than a gain medium, n is still the
weighted average index throughout the cavity with n0 being the weighted average index of the
background medium and these optical elements.
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6.2 Longitudinal Modes
207
Consider an intracavity field, Ec ðzÞ, at any location z along the longitudinal axis inside an
optical cavity. When this field completes a round trip in the cavity and returns back to the
location z, it is amplified or attenuated by a factor a to become aEc ðzÞ. The complex
amplification or attenuation factor a can be generally expressed as
a ¼ GeiφRT ,
(6.4)
where G is the round-trip gain factor for the field amplitude, equivalent to the power gain in a
single pass through a linear Fabry–Pérot cavity, and φRT is the round-trip phase shift for the
intracavity field. Both G and φRT have real values, and G 0. For a cavity that has a net optical
gain, G > 1, and the intracavity field is amplified. For a cavity that has a net optical loss, G < 1,
and the intracavity field is attenuated.
EXAMPLE 6.1
Consider a linear cavity, as shown in Fig. 6.1(a), and a ring cavity, as shown in Fig. 6.1(c). The
linear cavity has two mirrors with R1 ¼ R2 ¼ 0:9, which are separated at l ¼ 1:5 m. The ring
cavity has three mirrors with R1 ¼ R2 ¼ 0:9 and R3 ¼ 1, which are separated at l12 ¼ 0:7 m
and l23 ¼ l31 ¼ 0:4 m. Find the physical length, the round-trip length lRT , the round-trip time T,
and the round-trip gain factor G of each cavity.
Solution:
For the linear cavity, the physical length is simply l ¼ 1:5 m defined by the separation of the
two mirrors. The round-trip length and the round-trip time are, respectively,
llinear
¼ 2l ¼ 3 m, T linear ¼ RT ¼ 10 ns:
c
In a round trip through the linear cavity, the intracavity intensity changes by a factor of R1 R2
because the intracavity light is reflected once by each of the two mirrors in each round trip.
Therefore, the round-trip gain factor for the field amplitude is
pffiffiffiffiffiffiffiffiffiffi
Glinear ¼ R1 R2 ¼ 0:9:
llinear
RT
For the ring cavity, the physical length is simply l ¼ l12 þ l23 þ l31 ¼ 1:5 m defined by the ring
length. The round-trip length and the round-trip time are, respectively,
lring
¼ l ¼ l12 þ l23 þ l31 ¼ 1:5 m, T ring ¼ RT ¼ 5 ns:
c
In a round trip through the ring cavity, the intracavity intensity changes by a factor of R1 R2 R3
because the intracavity light is reflected once by each of the three mirrors in each round trip.
Therefore, the round-trip gain factor for the field amplitude is
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Gring ¼ R1 R2 R3 ¼ 0:9:
lring
RT
6.2
LONGITUDINAL MODES
..............................................................................................................
We first consider the resonant characteristics of a passive optical cavity. A passive cavity
cannot generate or amplify an optical field; thus G < 1. In order to maintain an intracavity field
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208
Optical Resonance
in such a cavity, it is necessary to constantly inject an input optical field, Ein , into the cavity. As
shown in Fig. 6.2, the forward-traveling component of the intracavity field at the location z1 just
inside the cavity next to the injection point is the sum of the transmitted input field and the
fraction of the intracavity field that returns after one round trip through the cavity:
Ec ðz1 Þ ¼ t in Ein þ aEc ðz1 Þ,
(6.5)
where t in is the complex transmission coefficient for the input field. We find that
t in
(6.6)
Ec ðz1 Þ ¼
Ein :
1a
The transmitted output field, Eout , is proportional to the intracavity field: Eout / Ec ðz1 Þ. Therefore, the output intensity is proportional to the input intensity through the following relationship,
I out /
I in
j1 aj
2
¼
I in
2
ð1 GÞ þ 4G sin2 ðφRT =2Þ
:
(6.7)
The proportionality constant of this relationship depends on the transmittance of the output mirror
and the intracavity attenuation over the distance from the point at z1 to the output point. The
transmittance of the cavity is T c ¼ I out =I in , which is scaled by the value of this proportionality
constant. For our discussion in the following, this proportionality constant is irrelevant. Therefore,
we only have to consider the normalized transmittance of the passive cavity:
T^ c ¼
1
1
h
i
h
i
¼
,
2
2
2
1 þ 4G=ð1 GÞ sin ðφRT =2Þ 1 þ ð4=GÞ=ð1 1=GÞ sin2 ðφRT =2Þ
(6.8)
which is obtained by normalizing T c to its peak value.
Clearly, T^ c has a peak value of unity, as expected for a normalized quantity. In Fig. 6.3, T^ c is
plotted as a function of the round-trip phase shift φRT for a few different values of G. We find that
Figure 6.3 Normalized transmittance of an optical cavity as a function of the round-trip phase shift in the
cavity. In a resonator that has a fixed, frequency-independent optical path length, the round-trip phase shift
is directly proportional to the optical frequency. The longitudinal mode frequencies are defined by the
frequencies corresponding to the resonance peaks. The spectral shape for a gain factor of G is the same as that
for a gain factor of 1=G. Thus, the curve for G ¼ 0:1 is the same as that for G ¼ 10, that for G ¼ 0:5 is the
same as that for G ¼ 2, and so on.
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6.2 Longitudinal Modes
209
the spectral shape for a gain factor of G is the same as that for a gain factor of 1=G. Therefore, a
passive cavity that has a gain factor of Gp ¼ G < 1 has the same spectral characteristics as an
active cavity that has a gain factor of Ga ¼ 1=G > 1. Note that the characteristics of T^ c shown in
Fig. 6.3 are the same as those of T^ FP shown in Fig. 5.18 because a Fabry–Pérot interferometer
can be considered as an optical cavity. Clearly, T^ FP given in (5.45) for a Fabry–Pérot interferometer can be identified with T^ c in (6.8) for a general optical cavity by properly relating the
finesse F of a cavity to the gain factor G, as is given below in (6.12).
At a given input field intensity, the intracavity field intensity of a passive cavity is proportional to T^ c because the transmitted output field intensity is directly proportional to the
intracavity field intensity while it is also proportional to T^ c . Therefore, resonances of the cavity
occur at the peaks of T^ c , where the intracavity intensity reaches its maximum level with respect
to a constant input field intensity. As can be seen from Fig. 6.3, the resonance condition of the
cavity is that the round-trip phase shift is an integral multiple of 2π:
φRT ¼ 2qπ,
q ¼ 1, 2, . . . :
(6.9)
From (6.9) and Fig. 6.3, we find that the separation between two neighboring resonance peaks
of T^ c is
ΔφL ¼ 2π
(6.10)
and that the FWHM of each resonance peak is
1G
:
G1=2
The finesse, F, of the cavity is the ratio of the separation to the FWHM of the peaks:
Δφc ¼ 2
(6.11)
ΔφL πG1=2
:
(6.12)
¼
Δφc 1 G
In the simplest situation that the optical field is a plane wave at a frequency of ω, the roundtrip phase shift can be generally expressed as
F¼
φRT ¼
ω
lRT þ φlocal ,
c
(6.13)
where the first term on the right-hand side is the phase shift contributed by the propagation of
the optical field over an optical path length of lRT , and the second term, φlocal , is the sum of all
the localized, and usually fixed, phase shifts such as those caused by reflection from the mirrors
of a cavity. In the case when the frequency of the input field is fixed, the resonance condition
given in (6.9) can be satisfied by varying the optical path length lRT of the cavity, either by
varying the physical length of the cavity or by varying the refractive index of the intracavity
medium, or both. The optical cavity then functions as an optical interferometer, which is used to
accurately measure the frequency and the spectral width of an optical wave.
When both the optical path length and the localized phase shifts are fixed, as is typically the
case for a laser resonator, the resonance condition of φRT ¼ 2qπ is satisfied only if the optical
frequency satisfies the condition:
ωq ¼
c
lRT
ð2qπ φlocal Þ,
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(6.14)
210
Optical Resonance
or
νq ¼
c lRT
q
φlocal :
2π
(6.15)
These discrete resonance frequencies are the longitudinal mode frequencies of the optical
resonator because they are defined by the resonance condition of the round-trip phase shift
along the longitudinal axis of the cavity. The frequency spacing, ΔνL , between two neighboring
longitudinal modes is known as the free spectral range, also called the longitudinal mode
frequency spacing, of the optical resonator. The FWHM of a longitudinal mode spectral peak is
Δνc , which is known as the longitudinal mode width of the cavity. If the values of lRT and φlocal
are independent of frequency, then ΔνL / ΔφL and Δνc / Δφc . Therefore, the finesse of an
optical resonator is the ratio of its free spectral range to its longitudinal mode width:
F¼
ΔφL ΔνL
¼
:
Δφc Δνc
(6.16)
From (6.15), we find that the longitudinal mode frequency spacing is related to the round-trip
time as
ΔνL ¼ νqþ1 νq ¼
c
1
¼ :
lRT T
(6.17)
The longitudinal mode width of the cavity can be expressed as
Δνc ¼
ΔνL 1 G
¼
ΔνL :
F
πG1=2
(6.18)
EXAMPLE 6.2
Find the finesse F, the longitudinal mode frequency spacing ΔνL , and the longitudinal mode
width Δνc of the linear and ring cavities that are considered in Example 6.1.
Solution:
For the linear cavity, the finesse is
1=2
F linear
πGlinear
π 0:91=2
¼
¼
¼ 29:8:
1 Glinear
1 0:9
The longitudinal mode frequency spacing is
¼
Δνlinear
L
1
T linear
¼
1
Hz ¼ 100 MHz:
10 109
The longitudinal mode width is
Δνlinear
¼
c
Δνlinear
100
L
¼
MHz ¼ 3:36 MHz:
F linear
29:8
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6.3 Transverse Modes
211
For the ring cavity, the finesse is
1=2
F ring ¼
πGring
1 Gring
¼
π 0:91=2
¼ 29:8:
1 0:9
The longitudinal mode frequency spacing is
Δνring
L ¼
1
T ring
¼
1
Hz ¼ 200 MHz:
5 109
The longitudinal mode width is
Δνring
¼
c
6.3
Δνring
200
L
¼
MHz ¼ 6:71 MHz:
F ring
29:8
TRANSVERSE MODES
..............................................................................................................
Any realistic optical cavity has a finite transverse cross-sectional area. Therefore, the resonant
optical field inside a realistic optical cavity cannot be a plane wave. Indeed, there exist certain
normal modes for the transverse field distribution in a given optical cavity. Such transverse field
patterns are known as the transverse modes of a cavity. A transverse mode of an optical cavity
is a stable transverse field pattern that reproduces itself after each round-trip pass in the cavity,
except that it might be amplified or attenuated in magnitude and shifted in phase.
The transverse modes of an optical cavity are defined by the transverse boundary conditions
that are imposed by the transverse cross-sectional index profile of the cavity. For a cavity that
utilizes an optical waveguide for lateral confinement of the optical field, the transverse modes
are the waveguide modes, such as the TE and TM modes of a slab waveguide or the TE, TM,
HE, and EH modes of a cylindrical fiber waveguide. For a nonwaveguiding cavity, the
transverse modes are TEM fields determined by the shapes and sizes of the end mirrors of
the cavity, as well as by the properties of the medium and any other optical components inside
the cavity. The Gaussian modes discussed in Section 3.3 are an important set of such unguided
TEM modes.
In an optical cavity that supports multiple transverse modes, the round-trip phase shift is
generally a function of the transverse mode indices m and n. Therefore, the resonance condition
can be explicitly written as
φRT
mn ¼ 2qπ:
(6.19)
As a result, the resonance frequencies of the cavity, ωmnq or νmnq , are dependent on both
longitudinal and transverse mode indices. When the frequency spacing between neighboring
transverse modes is smaller than that between neighboring longitudinal modes, multiple
resonance frequencies of different transverse modes can exist for each longitudinal mode, as
illustrated schematically in Fig. 6.4.
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212
Optical Resonance
Figure 6.4 Cavity resonance frequencies associated with different longitudinal and transverse modes. For
clarity, the heights of the transverse modes are made arbitrarily decreasing.
In a cavity that consists of an optical waveguide, the propagation constant βmn ðωÞ is a
function of the waveguide mode. If the physical length of the waveguide cavity is l, the
effective round-trip optical path length of a waveguide mode is
8 c
>
< 2 βmn ðωÞl, for a linear cavity;
ω
RT
(6.20)
lmn ¼ c
>
: βmn ðωÞl,
for a ring cavity:
ω
The round-trip optical path length lRT
mn generally varies from one mode to another due to the
modal dispersion of the waveguide. In addition, the localized phase shift can also be mode
dependent. Therefore, instead of the resonance frequencies ωq given by (6.14) for a plane wave,
the resonance frequencies ωmnq of a waveguide cavity are found by solving, for integral values
of q, the following resonance condition,
φRT
mn ¼
ω RT
l þ φlocal
mn ¼ 2qπ:
c mn
(6.21)
In a nonwaveguiding cavity, the propagation constant, k, is a property of only the medium and
is not mode dependent. Nevertheless, a mode-dependent on-axis phase variation ζ mn ðzÞ does
exist, which is given in (3.76) for a Hermite–Gaussian mode as discussed in Section 3.3. The
total on-axis phase variation of the TEMmn Gaussian mode is φmn ðzÞ ¼ kz þ ζ mn ðzÞ, which
includes the mode-independent phase shift kz and the mode-dependent phase shift ζ mn ðzÞ.
Consequently, the cavity resonance condition for a Gaussian mode is a modification of that for
a plane wave made by adding the round-trip contribution of the mode-dependent phase shift:
φRT
mn ¼
ω
local
lRT þ ζ RT
mn þ φmn ¼ 2qπ,
c
(6.22)
where the localized phase shift can, in general, also be mode dependent.
It is clear from the above discussion that the qth longitudinal mode frequency of a given
longitudinal mode index q varies among different transverse modes, as illustrated in Fig. 6.4.
For transverse modes defined by a waveguide structure, the longitudinal mode frequency
spacing ΔνLmn ¼ νmnðqþ1Þ νmnq between two neighboring longitudinal modes, q and q þ 1,
of the same transverse mode mn varies slightly among different transverse modes, as illustrated
in Example 6.3. Because a higher-order transverse waveguide mode has a smaller propagation
constant, thus a smaller effective index of refraction, ΔνLmn is generally larger for a higher-order
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6.3 Transverse Modes
213
transverse mode. By comparison, the longitudinal mode frequency spacing ΔνLmn stays constant
for different transverse Gaussian modes defined in free space because all Gaussian modes are
TEM modes of the same propagation constant. The mode-dependent phase shift ζ mn ðzÞ only
changes the mode frequency νmnq but not the difference ΔνLmn between two neighboring
longitudinal modes mnq and mnðq þ 1Þ.
EXAMPLE 6.3
A GaAs/AlGaAs semiconductor optical cavity has the longitudinal structure of a linear Fabry–
Pérot cavity and the transverse structure of a slab waveguide. The cavity has a physical length
of l ¼ 500 μm. The GaAs/AlGaAs slab waveguide supports three TE modes at the λ ¼ 870 nm
wavelength, with propagation constants of βTE0 ¼ 2:61 107 m1 , βTE1 ¼ 2:58 107 m1 ,
and βTE2 ¼ 2:53 107 m1 for the TE0 , TE1 , and TE2 modes, respectively. The end surfaces of
the cavity are not coated. Find the effective round-trip optical path length lRT
m , the round-trip
L
time T m , the longitudinal mode frequency spacing Δνm , and the longitudinal mode width Δνcm
for each transverse mode.
Solution:
For the linear cavity, the effective round-trip optical path length of each transverse waveguide
mode is found using (6.20):
c
λβ l
RT
RT
βm l ¼ m
) lRT
TE0 ¼ 3614 μm, lTE1 ¼ 3572 μm, lTE2 ¼ 3503 μm:
ω
π
The round-trip time of the cavity for each transverse waveguide mode is
lRT
m ¼2
lRT
m
) T TE0 ¼ 12:05 ps, T TE1 ¼ 11:91 ps, T TE2 ¼ 11:68 ps:
c
The longitudinal mode frequency spacing for each transverse waveguide mode is
Tm ¼
ΔνLm ¼
1
Tm
)
ΔνLTE0 ¼ 83:0 GHz, ΔνLTE1 ¼ 84:0 GHz, ΔνLTE2 ¼ 85:6 GHz:
To find Δνcm , it is necessary to find the finesse. The effective refractive index for each mode is
found, which is used to find the reflectivities of the cavity and the finesse:
nβm ¼
R1, m ¼ R2, m ¼ RTEm
1=4
Fm ¼
λβm
) nTE0 ¼ 3:61, nTE1 ¼ 3:57, nTE2 ¼ 3:50;
2π
1 nβm 2
) RTE ¼ 32:1%, RTE ¼ 31:6%, RTE ¼ 30:9%;
¼ 0
1
2
1 þ nβ m 1=4
πR1, m R2, m
1=2
1=2
1 R1, m R2, m
1=2
πRTEm
¼
1 RTEm
)
F TE0 ¼ 2:62, F TE1 ¼ 2:58, F TE2 ¼ 2:53:
The longitudinal mode width Δνcm for each transverse waveguide mode is
Δνcm ¼
ΔνLm
Fm
)
ΔνcTE0 ¼ 31:7 GHz, ΔνcTE1 ¼ 32:6 GHz, ΔνcTE2 ¼ 33:8 GHz:
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214
Optical Resonance
6.4
CAVITY LIFETIME AND QUALITY FACTOR
..............................................................................................................
Here we consider some important parameters of a passive optical cavity of zero optical gain so
that χ res ¼ 0, thus g ¼ 0. Such a passive optical cavity is known as a cold cavity. To be specific,
we identify the round-trip gain factor for the field amplitude in a cold cavity as Gc , or as Gcmn for
the transverse mode mn.
Because there is no optical gain in a cold cavity, the cavity has a net loss from finite
transmission through the end mirrors and various passive loss mechanisms so that Gc < 1.
Any optical field that initially exists in the cavity gradually decays as it circulates inside the
cavity. Because the field amplitude is attenuated by a factor of Gc per round trip, the intensity
and thus the number of intracavity photons are attenuated by a factor of G2c per round trip. We
can define a photon lifetime, also called cavity lifetime, τ c , and a cavity decay rate, γc , for a cold
cavity through the relation:
G2c ¼ eT=τc ¼ eγc T :
(6.23)
Therefore, the cavity lifetime is found as
τc ¼ T
:
2 ln Gc
(6.24)
The cavity decay rate is the decay rate of the optical energy stored in a cavity and is given by
γc ¼
1
2
¼ ln Gc :
τc
T
(6.25)
In general, the value of Gc for a given cavity is mode dependent. Usually, the fundamental
transverse mode has the lowest loss because its field distribution is transversely most concentrated toward the center along the longitudinal axis of the cavity. As the order of a mode
increases, its loss in the cavity increases due to the increased diffraction loss caused by the
transverse spreading of its field distribution. Consequently, both τ c and γc are also mode
dependent: τ cmnq and γcmnq . Unless a specific mode-discriminating mechanism is introduced in
a cavity, either intentionally or unintentionally, the fundamental mode generally has the largest
τ c and, correspondingly, the lowest γc .
The quality factor, Q, of a resonator is generally defined as the ratio of the resonance
frequency, ωres , to the energy decay rate, γ, of the resonator:
energy stored in the resonator
ωres
:
(6.26)
Q ¼ ωres
¼
γ
average power dissipation
Therefore, the quality factor of a cold cavity is
Q¼
ωq
¼ ωq τ c ,
γc
(6.27)
where ωq is the longitudinal mode frequency. For a low-loss, high-Q cavity, Gc is not much less
than unity; then, it can be shown by using (6.17), (6.18), and (6.23) that
Δνc 1
γ
¼ c
2πτ c 2π
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(6.28)
6.4 Cavity Lifetime and Quality Factor
215
and
Q
νq
:
Δνc
(6.29)
Note that though it is not explicitly spelled out in (6.27) and (6.29), the quality factor is a
function of not only the longitudinal-mode index q but also the transverse-mode indices m and
n: Q ¼ Qmnq . To be precise, (6.27) should be written as
Qmnq ¼
ωmnq
¼ ωmnq τ c :
γc
(6.30)
For an optical cavity, the dependence of Qmnq on the longitudinal-mode index q is generally
insignificant because q is a very large number except in the case of a very short microcavity. By
comparison, the dependence of Qmnq on the transverse-mode indices m and n cannot be ignored.
Indeed, Q00q for the fundamental transverse mode is generally larger than Qmnq for any highorder transverse mode because the fundamental transverse mode generally has the lowest loss.
EXAMPLE 6.4
Find the photon lifetime τ c , the cavity decay rate γc , and the quality factor Q at the λ ¼ 500 nm
wavelength of the linear and ring cavities that are considered in Example 6.1.
Solution:
For the linear cavity, the photon lifetime is
¼
τ linear
c
T linear
10
¼
ns ¼ 47:5 ns:
linear
2 ln 0:9
2 ln Gc
The cavity decay rate is
¼
γlinear
c
1
τ linear
c
¼
1
s1 ¼ 2:1 107 s1 :
47:5 109
The quality factor Q at λ ¼ 500 nm is
2πc linear 2π 3 108
¼
47:5 109 ¼ 1:79 108 :
τ
λ c
500 109
For the ring cavity, the photon lifetime is
Qlinear ¼ ωτ linear
¼
c
τ ring
¼
c
T ring
5
¼
ns ¼ 23:7 ns:
ring
2 ln 0:9
2 ln Gc
The cavity decay rate is
γring
¼
c
1
τ ring
c
¼
1
s1 ¼ 4:2 107 s1 :
9
23:7 10
The quality factor Q at λ ¼ 500 nm is
Qring ¼ ωτ ring
¼
c
2πc ring 2π 3 108
23:7 109 ¼ 8:93 107 :
τ ¼
λ c
500 109
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216
Optical Resonance
6.5
FABRY–PÉROT CAVITY
..............................................................................................................
The most common type of optical cavity is the Fabry–Pérot cavity, which consists of two end
mirrors in the form of the Fabry–Pérot interferometer and, in the case when it is used as a laser
cavity, an optical gain medium, as shown in Fig. 6.5. The radii of curvature of the left and right
mirrors are R1 and R2 , respectively. The sign of the radius of curvature is taken to be positive
for a concave mirror and negative for a convex mirror. For example, the cavity shown in
Fig. 6.5 has R1 > 0 and R2 > 0 because it is formed with two concave mirrors.
6.5.1 Stability Criterion
Most of the important features of a nonwaveguiding Fabry–Pérot cavity can be obtained by
applying the following simple concept. For the cavity to be a stable cavity in which a Gaussian
mode can be established, the radii of curvature of both end mirrors have to match the wavefront
curvatures of the Gaussian mode at the surfaces of the mirrors: Rðz1 Þ ¼ R1 and Rðz2 Þ ¼ R2 ,
where z1 and z2 are, respectively, the coordinates of the left and right mirrors measured from the
location of the Gaussian beam waist. Based on this concept, we have from (3.71) two relations:
z1 þ
z2R
z2
¼ R1 and z2 þ R ¼ R2 :
z1
z2
(6.31)
From these relations, we find that
z2R ¼
lðR1 lÞðR2 lÞðR1 þ R2 lÞ
ðR1 þ R2 2lÞ2
,
(6.32)
where l ¼ z2 z1 is the length of the cavity defined by the separation between the two end
mirrors.
Given the values of R1 , R2 , and l, stable Gaussian modes exist for the cavity if both relations
in (6.31) can be satisfied with a real and positive parameter of zR > 0 from (6.32) for a finite,
positive beam -waist spot size w0 according to (3.69). Then the cavity is stable. If the relations
in (6.31) cannot be simultaneously satisfied with a real and positive value for zR , then the cavity
is unstable because no stable Gaussian mode can be established in the cavity. Application of
this concept yields the stability criterion for a Fabry–Pérot cavity:
Figure 6.5 Fabry–Pérot cavity
containing an optical gain
medium with a filling factor Γ.
Changes of Gaussian beam
divergence at the boundaries of
the gain medium are ignored in
this plot.
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6.5 Fabry–Pérot Cavity
0
l
1
R1
l
1
1:
R2
217
(6.33)
In a stable Fabry–Pérot cavity, the mode-dependent on-axis phase shift in a single pass through
the cavity from the left mirror to the right mirror is simply ζ mn ðz2 Þ ζ mn ðz1 Þ for the TEMmn
Hermite–Gaussian mode. Therefore, the round-trip mode-dependent on-axis phase shift is
ζ RT
mn ¼ 2½ζ mn ðz2 Þ ζ mn ðz1 Þ:
(6.34)
With proper modifications, the above concept can be used to find the characteristics and
stability criterion of a cavity that has multiple mirrors, such as a folded Fabry–Pérot cavity or a
ring cavity.
EXAMPLE 6.5
A two-mirror Fabry–Pérot cavity as shown in Fig. 6.5 has a cavity length of l ¼ 1 m. One
mirror has a radius of curvature of R1 ¼ 2 m. Find the condition that the radius of curvature R2
of the other mirror has to satisfy in order for the cavity to be stable. Choose a proper value for
R2 so that the cavity is stable and is most symmetric. Find the beam spot size w0 at the beam
waist for a Gaussian beam at λ ¼ 600 nm that is stably established in the cavity. Where is the
beam waist located?
Solution:
With l ¼ 1 m and R1 ¼ 2 m, the stability condition in (6.33) requires that
l
l
1
l
0 1
1
1 ) 0
1 ) jR2 j l ¼ 1 m:
1
R1
R2
2
R2
Under this condition, R2 can be either positive or negative but its magnitude has to be larger
than 1 m. For the cavity to be stable and most symmetric, we can choose R2 ¼ R1 ¼ 2 m.
Then, using (6.32), we find the Rayleigh range:
pffiffiffi
3
lðR1 lÞðR2 lÞðR1 þ R2 lÞ 3 2
2
m:
¼ m ) zR ¼
zR ¼
2
2
4
ðR1 þ R2 2lÞ
The spot size at the beam waist is
w0 ¼
λzR
π
1=2
pffiffiffi1=2
600 109 3
¼
m ¼ 407 μm:
2π
Because R2 ¼ R1 , by symmetry the beam waist must be located right at the center of the
cavity.
6.5.2 Characteristic Parameters
We consider a cavity that contains an isotropic gain medium with a filling factor of Γ. The
surfaces of the gain medium are antireflection coated so that there is no reflection inside the
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218
Optical Resonance
cavity other than the reflection at the two end mirrors. If the gain medium fills up the entire
cavity, we simply make Γ ¼ 1 in the results obtained below. The Fabry–Pérot cavity has a
physical length of l between the two end mirrors. The field reflection coefficients are r 1 and r 2
for the left and right mirrors, respectively. They are generally complex to account for the phase
changes on reflection, φ1 and φ2 , respectively, and can be expressed as
1=2
r 1 ¼ R1 eiφ1 ,
1=2
r 2 ¼ R2 eiφ2 ,
(6.35)
where R1 and R2 are the reflectivities of the left and right mirrors, respectively.
The dielectric property of the intracavity gain medium includes the permittivity of the
background material and a resonant susceptibility χ res ðωÞ that characterizes the laser transition. To clearly identify the effect of each contribution, it is instructive to explicitly express
the permittivity of the gain medium, including the contribution of the resonant laser transition, as
ϵ res ðωÞ ¼ ϵ ðωÞ þ ϵ 0 χ res ðωÞ,
(6.36)
where ϵ ¼ ϵ 0 n2 is the background permittivity of the gain medium excluding the resonant
susceptibility. Because χ res ¼ 0 for a cold cavity, the weighted average of the propagation
constant for the intracavity field in a cold cavity is
k¼
nω
¼ Γk þ ð1 ΓÞk0 ,
c
(6.37)
where k ¼ nω=c is the propagation constant in the gain medium and k0 ¼ n0 ω=c is that in the
surrounding medium. The round-trip optical path length in this cavity is lRT ¼ 2nl.
Usually there is an intracavity background loss contributed by a variety of mechanisms that
are irrelevant to the laser transition, such as scattering or absorption. In addition, modedependent diffraction losses exist for the intracavity optical field due to the finite sizes of the
end mirrors. The combined effect of these losses can be accounted for by taking a spatially
averaged, mode-dependent loss coefficient, α mn , so that the effective propagation constant is
complex with a mode-dependent imaginary part: k þ iα mn =2. This loss is known as the
distributed loss of the cavity mode. In general, α mn k for a practical optical cavity.
By following a mode field through one round trip in the cavity, we find that
a ¼ r 1 r 2 exp i2kl α mn l þ iζ RT
mn
(6.38)
for the TEMmn Hermite–Gaussian mode. Therefore, by using (6.4) and (6.35), we find that both
the round-trip gain factor and the round-trip phase shift are mode dependent:
1=2 1=2
Gcmn ¼ R1 R2 eα mn l
(6.39)
RT
φRT
mn ¼ 2kl þ ζ mn þ φ1 þ φ2 :
(6.40)
and
Using (6.40) for the resonance condition given in (6.19), we find the resonance frequencies of
the cold Fabry–Pérot cavity:
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6.5 Fabry–Pérot Cavity
ωcmnq
c
¼
2qπ ζ RT
mn φ1 φ2 ,
2nl
νcmnq
c
ζ RT
mn þ φ1 þ φ2
¼
¼
q
,
2π
2π
2nl
ωcmnq
219
(6.41)
where the superscript c indicates the fact that the frequencies are those for a cold cavity with
χ res ¼ 0. These frequencies are clearly functions of the transverse-mode indices because of the
RT
mode-dependent phase shift ζ RT
mn . However, because ζ mn is not a function of the longitudinalmode index q, the frequency separation between two neighboring longitudinal modes of the
same transverse mode group is a mode-independent constant:
ΔνL ¼ νcmn, qþ1 νcmnq ¼
c
1
¼ :
2nl T
(6.42)
Here we assume that the background optical property of the medium is not very dispersive so
that the background refractive index n can be considered a constant that is independent of
optical frequency in the narrow range between neighboring modes of interest.
Using (6.12) and (6.39), the finesse of the lossy Fabry–Pérot cavity is
1=4 1=4
F¼
πR1 R2 eα mn l=2
1=2 1=2
1 R1 R2 eα mn l
,
(6.43)
which is mode dependent due to the mode-dependent loss α mn . The longitudinal mode width,
Δνc ¼ ΔνL =F, is also mode dependent for the same reason. For a cavity that has a negligible
loss, we can take α mn ¼ 0; then, (6.43) reduces to the familiar formula for the finesse of a
lossless Fabry–Pérot interferometer as given in (5.46):
1=4 1=4
F¼
πR1 R2
1=2 1=2
1 R1 R2
:
(6.44)
Therefore, for a nondispersive, lossless Fabry–Pérot cavity, ΔνL , F, and Δνc are all independent
of the longitudinal and transverse mode indices though the mode frequency νmnq is a function of
all three mode indices.
Using (6.24) and (6.39), the mode-dependent photon lifetime of the Fabry–Pérot cavity can
be expressed as
τ cmnq ¼
nl
pffiffiffiffiffiffiffiffiffiffi ,
cðαmn l ln R1 R2 Þ
and the mode-dependent cavity decay rate can be expressed as
c
1 pffiffiffiffiffiffiffiffiffiffi
c
α mn ln R1 R2 :
γmnq ¼
n
l
(6.45)
(6.46)
Clearly, both τ cmnq and γcmnq are also mode dependent due to the mode-dependent distributed loss
α mn . However, they are independent of the longitudinal mode index q under the assumption that
the background refractive index n, the loss α mn , and the mirror reflectivities R1 and R2 are not
sensitive to the frequency differences among different longitudinal modes. If any of these
parameters vary significantly within the range of the longitudinal modes of interest, then the
dependence of τ cmnq and γcmnq on the index q cannot be ignored.
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220
Optical Resonance
A Fabry–Pérot cavity that is used as a laser cavity has a Q value ranging from the order of 103
for a cavity of a high-gain laser that has low mirror reflectivities to the order of 108 for a cavity
of a low-gain laser that has high mirror reflectivities. A Fabry–Pérot cavity that is used as a
high-resolution optical spectrum analyzer can have an even higher Q value.
EXAMPLE 6.6
The Fabry–Pérot cavity of a high-gain InGaAsP/InP semiconductor laser emitting at the 1.3 μm
wavelength has an effective average refractive index of n ¼ nβ ¼ 3:5 defined by the InGaAsP/
InP waveguide mode, a physical length of l ¼ 300 μm, and mirror reflectivities of R1 ¼
R2 ¼ 0:3. The structure supports only one transverse mode. Assume a negligibly small α for
simplicity. Find the round-trip time, the longitudinal mode frequency spacing, the finesse, the
longitudinal mode width, the photon lifetime, the cavity decay rate, and the quality factor of this
cavity as a cold cavity.
Solution:
The round-trip time of the cavity is
T¼
2nl 2 3:5 300 106
s ¼ 7 ps:
¼
c
3 108
The longitudinal mode frequency spacing is
ΔνL ¼
1
1
Hz ¼ 142:9 GHz:
¼
T 7 1012
Assuming no distributed loss, the finesse of the cavity is
1=4 1=4
F¼
πR1 R2
1=2 1=2
1 R1 R2
¼
π 0:31=4 0:31=4
¼ 2:46:
1 0:31=2 0:31=2
The longitudinal mode width is
Δνc ¼
ΔνL 142:9
GHz ¼ 58:1 GHz:
¼
2:46
F
The photon lifetime is
τc ¼
nl
3:5 300 106
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ 2:9 ps:
pffiffiffiffiffiffiffiffiffiffi ¼
c ln R1 R2
3 108 ln 0:3 0:3
The cavity decay rate is
γc ¼
1
1
¼
s1 ¼ 3:4 1011 s1 :
τ c 2:9 1012
To find the quality factor, we note that the frequency is found using ω ¼ 2πc=λ for the given
optical wavelength of λ ¼ 1:3 μm. Thus, using (6.27), we find the quality factor of this cavity to
be
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Problems
Q ¼ ωτ c ¼
221
2πc
2π 3 108
2:9 1012 ¼ 4:2 103 :
τc ¼
λ
1:3 106
The approximate relation (6.29) yields a slightly smaller value of Q ¼ 4:0 103 . A Q value on
the order of 103 is relatively low for a laser cavity. Even so, the difference between (6.27) and
(6.29) is only about 5%.
Problems
6.1.1 A folded Fabry–Pérot cavity as shown in Fig. 6.1(b) has two end mirrors with R1 ¼ R2 ¼
0:8 and a middle mirror with Rm ¼ 0:9 for folding the cavity, which is separated from the
two end mirrors at l1m ¼ 0:8 m and l2m ¼ 0:3 m, respectively. A glass rod that has a
length of lg ¼ 0:2 m and a refractive index of ng ¼ 1:5 is placed along the beam path
between the two mirrors of R1 and Rm . Find the physical length, the round-trip length lRT ,
the round-trip time T, and the round-trip gain factor G of the cavity.
6.1.2 A ring cavity as shown in Fig. 6.1(c) has three mirrors with R1 ¼ R2 ¼ 0:8 and R3 ¼ 0:9,
which are separated at l12 ¼ 0:5 m and l23 ¼ l31 ¼ 0:3 m. A glass rod that has a length of
lg ¼ 0:2 m and a refractive index of ng ¼ 1:5 is placed along the beam path between the
two mirrors of R1 and R2 . Find the physical length, the round-trip length lRT , the roundtrip time T, and the round-trip gain factor G of the cavity.
6.1.3 An optical-fiber ring cavity as shown in Fig. 6.1(d) has one input–output coupler that has
a coupling efficiency of η ¼ 20%. The fiber loop has a length of l ¼ 2 m, and the
effective index of the fiber mode is n ¼ 1:47. Find the physical length, the round-trip
length lRT , the round-trip time T, and the round-trip gain factor G of the cavity.
6.2.1 Find the finesse F, the longitudinal mode frequency spacing ΔνL , and the longitudinal
mode width Δνc of the folded Fabry–Pérot cavity considered in Problem 6.1.1.
6.2.2 Find the finesse F, the longitudinal mode frequency spacing ΔνL , and the longitudinal
mode width Δνc of the ring cavity considered in Problem 6.1.2.
6.2.3 Find the finesse F, the longitudinal mode frequency spacing ΔνL , and the longitudinal
mode width Δνc of the fiber ring cavity considered in Problem 6.1.3.
6.3.1 An InP/InGaAsP semiconductor optical cavity has the longitudinal structure of a linear
Fabry–Pérot cavity and the transverse structure of a slab waveguide. The cavity has a
physical length of l ¼ 400 μm. The slab waveguide supports two TE and two TM modes
at the λ ¼ 1:3 μm wavelength, with propagation constants of βTE0 ¼ 1:67 107 m1 ,
βTM0 ¼ 1:65 107 m1 , βTE1 ¼ 1:57 107 m1 , and βTM1 ¼ 1:56 107 m1 for the
TE0 , TM0 , TE1 , and TM1 modes, respectively. The end surfaces of the cavity are not
coated. Find the effective round-trip optical path length lRT , the round-trip time T, the
longitudinal mode frequency spacing ΔνL , and the longitudinal mode width Δνc for each
transverse mode.
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222
Optical Resonance
6.4.1 Find the photon lifetime τ c , the cavity decay rate γc , and the quality factor Q at the λ ¼
850 nm wavelength of the folded Fabry–Pérot cavity considered in Problems 6.1.1 and 6.2.1.
6.4.2 Find the photon lifetime τ c , the cavity decay rate γc , and the quality factor Q at the
λ ¼ 850 nm wavelength of the ring cavity considered in Problems 6.1.2 and 6.2.2.
6.4.3 Find the photon lifetime τ c , the cavity decay rate γc , and the quality factor Q at the
λ ¼ 850 nm wavelength of the fiber ring cavity considered in Problems 6.1.3 and 6.2.3.
6.4.4 An optical cavity has two characteristic time constants: the round-trip time T and the
photon lifetime τ c . Once they are known, most of the other characteristic parameters of
the cavity can be found. Find the cold-cavity field-amplitude gain factor Gc , the finesse F,
the longitudinal mode frequency spacing ΔνL , the longitudinal mode width Δνc , the cavity
decay rate γc , and the quality factor Q at the λ ¼ 1:3 μm wavelength for an optical cavity
that has T ¼ 1 ns and τ c ¼ 20 ns.
6.4.5 An optical cavity has two characteristic spectral parameters: the longitudinal mode
frequency spacing ΔνL and the longitudinal mode width Δνc . Once they are known, most
of the other characteristic parameters of the cavity can be found. Find the finesse F, the
cold-cavity field-amplitude gain factor Gc , the round-trip time T, the photon lifetime τ c ,
the cavity decay rate γc , and the quality factor Q at the λ ¼ 1:064 μm wavelength for an
optical cavity that has ΔνL ¼ 150 MHz and Δνc ¼ 5 MHz.
6.4.6 An optical cavity has two characteristic quality factors: the finesse F and the quality factor
Q at a specific resonance frequency. Once they are known, most of the other characteristic
parameters of the cavity can be found. Find the cold-cavity field-amplitude gain factor Gc ,
the photon lifetime τ c , the cavity decay rate γc , the round-trip time T, the longitudinal mode
frequency spacing ΔνL , and the longitudinal mode width Δνc for an optical cavity that has a
finesse of F ¼ 100 and a quality factor of Q ¼ 2 108 at the λ ¼ 532 nm wavelength.
6.5.1 Show for a linear Fabry–Pérot cavity of a length l as shown in Fig. 6.5 that the locations
of the left and right end mirrors measured from the beam waist are, respectively,
lðR2 lÞ
lðR1 lÞ
, z2 ¼
,
(6.47)
R1 þ R2 2l
R1 þ R2 2l
where R1 and R2 are the radii of curvature of the left and right mirrors, respectively.
Show also that the Rayleigh range of a stable Gaussian beam defined by the cavity is that
given by (6.32).
z1 ¼ 6.5.2 A linear Fabry–Pérot cavity in free space has a concave left mirror that has a radius of
curvature of R1 ¼ 2 m and a convex right mirror that has a radius of curvature of
R2 ¼ 1 m. The cavity length is l ¼ 1:5 m. Is the cavity stable? If it is stable, where
is the Gaussian beam waist located? What is the beam waist spot size?
6.5.3 A symmetric linear Fabry–Pérot cavity in free space has a cavity length of l and two
mirrors of the same radius of curvature of R1 ¼ R2 ¼ R ¼ 1 m.
(a) In what range can the cavity length be chosen to make the cavity stable?
(b) For different choices of the cavity length, where is the location of the beam waist of
the Gaussian beam that is defined by the cavity?
(c) Find the cavity length that maximizes the waist spot size of the Gaussian beam? What
is this spot size for an optical wavelength of λ ¼ 1:064 μm?
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Bibliography
223
(d) For a beam waist spot size of w0 ¼ 350 μm, what is the cavity length that has to be
chosen?
(e) If the cavity length is chosen to be l ¼ 1:5 m, is the cavity stable? If it is stable, what
is the beam waist spot size?
6.5.4 The length of the InGaAsP/InP Fabry–Pérot cavity described in Example 6.6 is doubled
to l ¼ 600 μm. At the λ ¼ 1:3 μm wavelength, the effective index of n ¼ nβ ¼ 3:5 and
the mirror reflectivities of R1 ¼ R2 ¼ 0:3 remain unchanged, while the distributed loss is
still negligible. Find the round-trip time, the longitudinal mode frequency spacing, the
finesse, the longitudinal mode width, the photon lifetime, the cavity decay rate, and the
quality factor of this cavity. How are these parameters changed as compared to those
found in Example 6.6?
6.5.5 The length of the InGaAsP/InP Fabry–Pérot cavity described in Example 6.6 remains
l ¼ 300 μm. At the λ ¼ 1:3 μm wavelength, the effective index of n ¼ nβ ¼ 3:5 and the
mirror reflectivities of R1 ¼ R2 ¼ 0:3 remain unchanged, but the cavity now has a small
distributed loss of α ¼ 10 cm1 . Find the round-trip time, the longitudinal mode frequency spacing, the finesse, the longitudinal mode width, the photon lifetime, the cavity
decay rate, and the quality factor of this cavity. How are these parameters changed as
compared to those found in Example 6.6?
6.5.6 An optical-fiber Fabry–Perot cavity has a physical length of l ¼ 20 m, an averaged
intracavity refractive index of n ¼ 1:45, a distributed loss of α ¼ 0:005 m1 , and mirror
reflectivities of R1 ¼ R2 ¼ 80%.
(a) What are the round-trip optical path length, the round-trip time, and the longitudinal
mode frequency spacing of this cavity?
(b) Find the free spectral range, the finesse, and the longitudinal mode width of this cavity.
(c) What are the cavity decay rate, the photon lifetime, and the quality factor for
λ ¼ 1:3 μm?
Bibliography
Davis, C. C., Lasers and Electro-Optics: Fundamentals and Engineering, 2nd edn. Cambridge: Cambridge
University Press, 2014.
Fowler, G. R., Introduction to Modern Optics, 2nd edn. New York: Dover, 1975.
Haus, H. A., Waves and Fields in Optoelectronics. Englewood Cliffs, NJ: Prentice-Hall, 1984.
Iizuka, K., Elements of Photonics in Free Space and Special Media, Vol. I. New York: Wiley, 2002.
Liu, J. M., Photonic Devices. Cambridge: Cambridge University Press, 2005.
Milonni, P. W. and Eberly, J. H., Laser Physics. New York: Wiley, 2010.
Saleh, B. E. A. and Teich, M. C., Fundamentals of Photonics. New York: Wiley, 1991.
Siegman, A. E., Lasers. Mill Valley, CA: University Science Books, 1986.
Silfvest, W. T., Laser Fundamentals. Cambridge: Cambridge University Press, 1996.
Svelto, O., Principles of Lasers, 5th edn. New York: Springer, 2010.
Verdeyen, J. T., Laser Electronics, 3rd edn. Englewood Cliffs, NJ: Prentice-Hall, 1995.
Yariv, A. and Yeh, P., Photonics: Optical Electronics in Modern Communications. Oxford: Oxford University
Press, 2007.
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Cambridge Books Online
http://ebooks.cambridge.org/
Principles of Photonics
Jia-Ming Liu
Book DOI: http://dx.doi.org/10.1017/CBO9781316687109
Online ISBN: 9781316687109
Hardback ISBN: 9781107164284
Chapter
7 - Optical Absorption and Emission pp. 224-248
Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.008
Cambridge University Press
7
7.1
Optical Absorption and Emission
OPTICAL TRANSITIONS
..............................................................................................................
Optical absorption and emission occur through the interaction of optical radiation with electrons in a material system that defines the energy levels of the electrons. Depending on the
properties of a given material, electrons that interact with optical radiation can be either those
bound to individual atoms or those residing in the energy-band structures of a material such as a
semiconductor. In any event, the absorption or emission of a photon by an electron is associated
with a resonant transition of the electron between a lower energy level j1i of energy E1 and an
upper energy level j2i of energy E 2 , as illustrated in Fig. 7.1. The resonance frequency, ν21 , of
the transition is determined by the separation between the energy levels:
v21 ¼
E2 E1
:
h
(7.1)
In an atomic or molecular system, a given energy level usually consists of a number of
degenerate quantum mechanical states that have the same energy. The degeneracy factors g1
and g2 account for the degeneracies in the energy levels j1i and j2i, respectively.
There are three basic types of processes associated with resonant optical transitions of electrons
between two energy levels: absorption, stimulated emission, and spontaneous emission, which are
illustrated in Figs. 7.1(a), (b), and (c), respectively. Absorption and stimulated emission of a photon
are both associated with induced transitions between two energy levels caused by the interaction of
an electron with existing optical radiation. An electron that is initially in the lower level j1i can
absorb a photon to make a transition to the upper level j2i. An electron that is initially in the upper
level j2i can be stimulated by the optical radiation to emit a photon while making a downward
transition to the lower level j1i. By contrast, spontaneous emission is not induced. Irrespective of
the presence or absence of existing optical radiation, an electron initially in the upper level j2i can
spontaneously relax to the lower level j1i by emitting a spontaneous photon.
Figure 7.1 (a) Absorption, (b) stimulated emission, and (c) spontaneous emission of photons resulting from
resonant transitions of electrons in a material.
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7.1 Optical Transitions
225
A photon that is emitted through stimulated emission has the same frequency, phase,
polarization, and propagation direction as the optical radiation that induces the process. By
contrast, spontaneously emitted photons are random in phase and polarization, and they are
emitted in all directions, though their frequencies are still dictated by the separation between the
two energy levels, subject to a degree of uncertainty determined by the linewidth of the
transition. Therefore, stimulated emission results in the amplification of an optical signal,
whereas spontaneous emission merely adds noise to an optical signal. Absorption simply leads
to the attenuation of an optical signal.
7.1.1 Spectral Lineshape
A resonant transition is selective of the frequency of the interacting optical field because the
process is associated with the absorption or emission of a photon that has a frequency
determined by the energy change of the electron making the transition, as indicated in (7.1).
The spectral characteristic of a resonant transition is never infinitely sharp, however. The finite
spectral width of a resonant transition is dictated by the uncertainty principle of quantum
mechanics, but it can be intuitively understood using the reasoning in Section 2.3. One
important conclusion learned from the discussion in Section 2.3 is that any response that has
a finite relaxation time in the time domain must have a finite spectral width in the frequency
domain. As we shall see later, the induced transition rates of both absorption and stimulated
emission between two energy levels in a given system are directly proportional to the spontaneous emission rate from the upper to the lower of the two levels. Therefore, it is a basic law of
physics that any allowed resonant transition between two energy levels has a finite relaxation
time because at least the upper level has a finite lifetime due to spontaneous emission.
Consequently, every optical process associated with a resonant transition between two specific
energy levels is characterized by a lineshape function, g^ðvÞ or g^ ðωÞ, of a finite linewidth. The
lineshape function is generally normalized as
ð∞
ð∞
g^ðvÞdv ¼ g^ðωÞdω ¼ 1, where g^ðvÞ ¼ 2π^
g ðωÞ:
0
(7.2)
0
7.1.2 Homogeneous Broadening
If all of the atoms in a material that participate in a resonant interaction associated with the
energy levels j1i and j2i are indistinguishable, their responses to an electromagnetic field are
characterized by the same transition resonance frequency ν21 and the same relaxation rate γ21 .
Note that γ21 is the phase relaxation rate of the resonant interaction between the electromagnetic field and the two energy levels. In such a homogeneous system, the physical mechanisms
that broaden the linewidth of the transition affect all atoms equally. Spectral broadening caused
by such mechanisms is called homogeneous broadening.
From the discussion in Section 2.3, the spectral characteristics of a damped response that is
characterized by a single resonance frequency and a single relaxation rate, such as that of a
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226
Optical Absorption and Emission
resonant interaction in a homogeneously broadened system, are described by the resonant
susceptibility given in (2.26), with its real and imaginary parts given in (2.27). As discussed
later in Section 7.2, in the interaction of an optical field with a material, the absorption and
emission of optical energy are characterized by the imaginary part χ 00res of the resonant susceptibility of the material. Therefore, the spectral characteristics of resonant optical absorption and
emission in a homogeneously broadened medium are described by the Lorentzian lineshape
function of χ 00res ðωÞ given in (2.27). The normalized Lorentzian lineshape function, which is
normalized using (7.2), for the resonant transitions between j1i and j2i has the form:
1
γ21
g^ðωÞ ¼ ,
π ðω ω21 Þ2 þ γ221
(7.3)
which has a FWHM of Δωh ¼ 2γ21 , or
g^ ðvÞ ¼
Δvh
2π½ðv v21 Þ2 þ ðΔvh =2Þ2 ,
(7.4)
where
Δvh ¼
γ21
π
(7.5)
is the FWHM of g^ðvÞ. We see that the spectrum has a finite width that is determined by the
relaxation rate γ21 .
The fundamental mechanism for homogeneous broadening is lifetime broadening due to the
finite lifetimes, τ 1 and τ 2 , respectively, of the energy levels, j1i and j2i, that are involved in the
resonant transitions. The population in an energy level can relax through both radiative and
nonradiative transitions to lower levels. Radiative relaxation is associated with population
relaxation through spontaneous emission of radiation. The radiative relaxation rate of the
transition from level j2i to level j1i is characterized by a rate constant A21 , known as the
Einstein A coefficient, which defines a time constant τ sp ¼ 1=A21 , known as the spontaneous
radiative lifetime, between j2i and j1i. Both A21 and τ sp are discussed in further detail later.
The total radiative relaxation rate, γrad
all radiative
2 , of level j2i is the sum of the rates ofX
rad
spontaneous transitions from j2i to all levels of lower energies: γ2 ¼
A . The
i 2i
nonrad
nonradiative relaxation rate, γ2
, accounts for all other population relaxation mechanisms
that do not result in the emission of photons. The total relaxation rate, γ2 , of level j2i is the sum
of its radiative and nonradiative relaxation rates, and the lifetime of the energy level has both
radiative and nonradiative contributions:
nonrad
γ2 ¼ γrad
,
2 þ γ2
1
1
1
¼ rad þ nonrad ,
τ2 τ2
τ2
(7.6)
rad
nonrad
¼ 1=γnonrad
. This concept can be applied to level j1i
where τ 2 ¼ 1=γ2 , τ rad
2 ¼ 1=γ2 , and τ 2
2
to obtain similar relations for γ1 and τ 1 .
Though τ 2 has contributions of both radiative and nonradiative relaxations, the fluorescence
due to spontaneous emission from level j2i decays in time at the total relaxation rate γ2 because
its strength is proportional to the population in level j2i, which relaxes at the total relaxation
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7.1 Optical Transitions
227
rate. Therefore, the decay time constant of the fluorescent emission from level j2i is τ 2 , not τ rad
2 .
For this reason, the total lifetimes τ 1 and τ 2 are known as the fluorescence lifetimes of energy
levels j1i and j2i, respectively. The contributions of various relaxation rates to the radiative and
nonradiative lifetimes, and to the fluorescence lifetimes, of the upper and lower energy levels
are summarized in Fig. 7.2.
The nonradiative relaxation rate of an energy level is a function of extrinsic factors, such as
collisions and thermal vibrations. It can therefore be changed by varying the conditions of the
surrounding environment. The minimum broadening is called natural broadening, which is
caused only by radiative relaxation when all nonradiative processes are eliminated. The linewidth due to natural broadening is determined by the radiative phase relaxation rate caused by
radiative decays of the two energy levels:
1 rad
1 1
1
natural
rad
rad
γ21
(7.7)
¼ γ21 ¼ ðγ1 þ γ2 Þ ¼
þ rad :
2
2 τ rad
τ2
1
The total phase relaxation rate that characterizes lifetime broadening of the linewidth accounts for
the lifetimes of the two energy levels due to both radiative and nonradiative relaxation processes:
1
1 1
1
life
þ
:
(7.8)
γ21 ¼ ðγ1 þ γ2 Þ ¼
γnatural
21
2
2 τ1 τ2
and γlife
The contributions to γnatural
21
21 are also summarized in Fig. 7.2. Note that the linewidth is
determined by the lifetimes of both upper and lower levels. In the case when the lower level j1i
is the ground level of an atomic system, we have γ1 ¼ 0 and τ 1 ¼ ∞. Then, the linewidth due to
lifetime broadening is solely determined by the lifetime of the upper level, τ 2 .
Other mechanisms that affect all atoms equally can further increase the homogeneous linewidth without changing the fluorescence lifetime of either the upper or the lower level. One
Figure 7.2 Contributions of various relaxation rates to the radiative and nonradiative lifetimes, and to the
fluorescence lifetimes, of the upper and lower energy levels. The homogeneous natural linewidth is determined
by the radiative lifetimes, whereas the lifetime-broadened linewidth is determined by the fluorescence lifetimes.
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228
Optical Absorption and Emission
important mechanism is collision-induced phase randomization of the emitted radiation. Collisions among atoms in a gas or liquid and collisions between atoms and phonons in a solid
normally have two possible effects. One effect is to reduce the fluorescence lifetimes of the
upper and lower levels by increasing the nonradiative relaxation rates. Such a process increases
life
nonrad
lifetime broadening; its effect is included in γlife
and
21 through the dependence of γ21 on γ1
nonrad
contained in γ1 and γ2 , respectively. Collisions can also increase a homogeneous lineγ2
width without reducing the fluorescence lifetimes by simply interrupting the phase of the
radiation emitted through radiative relaxation. This dephasing process, quantified by a
linewidth-broadening factor γdephase
, is often more important than the lifetime-reduction pro21
cess, resulting in a homogeneous linewidth that is significantly broader than the linewidth due
to lifetime broadening. Therefore, the homogeneous linewidth can increase with both pressure
and temperature in a gas medium, and with active-ion concentration and temperature in a liquid
or solid medium. In general, the homogeneous linewidth including the contributions of
such extrinsic mechanisms is a function of pressure, P, active-ion concentration, N, and
temperature, T:
dephase
natural
γ21 ðP, N, TÞ ¼ γlife
γlife
:
21 þ γ21
21 γ21
(7.9)
EXAMPLE 7.1
The energy levels of Nd:YAG are shown in Fig. 7.3. The highest level 4 F3=2 of the active Nd3þ
ion relaxes to four lower levels at different radiative relaxation rates characterized by the
Einstein A coefficients shown for different emission wavelengths. The lowest level 4 I9=2 is
the ground level, which does not relax to any other level. The dominant transition of this system
is that associated with the well-known Nd:YAG emission wavelength of λ ¼ 1:064 μm, which
takes place between the upper level 4 F3=2 , labeled j2i, and the lower level 4 I11=2 , labeled j1i.
The upper level 4 F3=2 has a lifetime of τ 2 ¼ 240 μs predominantly due to radiative relaxation;
Figure 7.3 Energy levels of Nd:YAG.
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7.1 Optical Transitions
229
the lower level 4 I11=2 has a lifetime of τ 1 ¼ 200 ps purely from nonradiative relaxation. (a) Find
the radiative, nonradiative, and total relaxation rates for the upper and lower levels, j2i and j1i,
respectively. (b) Find the natural linewidth and the lifetime-broadened linewidth for the λ ¼
1:064 μm emission line. If no other mechanisms further broaden this line, what is its lineshape
and linewidth? (c) At room temperature, dephasing due to phonon collisions contributes a
dephasing rate of γdephase
¼ 3:75 1011 s1 to the linewidth. What is the homogeneous line21
width of this emission line at room temperature?
Solution:
All of the processes considered here cause homogeneous broadening because they are
common to all Nd3þ ions. Inhomogeneous broadening mechanisms are not considered in
this example.
(a) The upper level j2i relaxes both radiatively and nonradiatively to four lower levels, but the
lower level j1i relaxes only nonradiatively to the ground level. The total relaxation rates of
the two levels are, respectively,
γ2 ¼
1
1
¼
s1 ¼ 4167 s1 ,
τ 2 240 106
γ1 ¼
1
1
¼
s1 ¼ 5 109 s1 :
τ 1 200 1012
The radiative relaxation rates of the two levels are, respectively,
X
A2i ¼ 3868 s1 , γrad
γrad
2 ¼
1 ¼ 0:
i
The nonradiative relaxation rates of the two levels are, respectively,
1
γnonrad
¼ γ2 γrad
2
2 ¼ 299 s ,
9 1
γnonrad
¼ γ1 γrad
1
1 ¼ 5 10 s :
(b) Using the results from (a), we find that
1
1
rad
1
¼ ðγrad
¼ 1934 s1 ,
γnatural
21
1 þ γ2 Þ ¼ ð0 þ 3868Þ s
2
2
1
1
9
1
¼ 2:5 109 s1 :
γlife
21 ¼ ðγ1 þ γ2 Þ ¼ ð5 10 þ 4167Þ s
2
2
The natural linewidth and the lifetime-broadened linewidth are, respectively,
Δνnatural ¼
γnatural
21
¼ 616 Hz,
π
Δνlife ¼
γlife
21
¼ 796 MHz:
π
If no other mechanisms further broaden this line, this emission line has a Lorentzian
lineshape that has a homogeneously broadened linewidth of Δνh ¼ Δνlife ¼ 796 MHz:
(c) With a dephasing rate of γdephase
¼ 3:75 1011 s1 , the total phase relaxation rate is
21
dephase
¼ 3:775 1011 s1 :
γ21 ¼ γlife
21 þ γ21
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230
Optical Absorption and Emission
Thus, the homogeneous linewidth is broadened to
Δνh ¼
γ21
¼ 120 GHz:
π
The linewidth is further broadened by inhomogeneous mechanisms discussed below. For the
λ ¼ 1:064 μm line of Nd:YAG, the total linewidth varies with temperature and with the
quality of the YAG crystal. Increasing temperature increases the homogeneous linewidth,
whereas a poorer crystal quality leads to a larger inhomogeneous linewidth. In any event, this
emission line of Nd:YAG is predominantly homogeneously broadened at room temperature.
7.1.3 Inhomogeneous Broadening
A resonant transition can be further broadened by inhomogeneous broadening if certain
physical mechanisms exist that do not equally affect all atoms, causing energy levels j1i or
j2i, or both, to shift differently among different groups of atoms. The resulting inhomogeneous
shifts of the transition resonance frequency cause inhomogeneous broadening of the transition
spectrum on top of the original homogeneous broadening.
If we express the homogeneous lineshape function given in (7.4) as g^h ðν, ν21 Þ to explicitly
indicate that its transition resonance frequency is ν21 , the homogeneously broadened spectrum
of a group of atoms whose resonance frequency is shifted from ν21 to νk is g^h ðν, νk Þ. The
distribution of atoms in an inhomogeneous system can be described by a probability density
function pðνk Þ with
ð∞
pðνk Þdνk ¼ 1:
(7.10)
0
The probability that the resonance frequency of a given atom falls in the range between νk and
νk þ dνk is pðνk Þdνk . Then, the overall spectral lineshape of the inhomogeneously broadened
transition is
ð∞
g^ðνÞ ¼ pðνk Þ^
g h ðν, νk Þdνk :
(7.11)
0
The overall lineshape function obtained from (7.11) depends on the degree of inhomogeneous
broadening in comparison to homogeneous broadening. Mathematically, it depends on the
spread of the distribution function pðνk Þ in comparison to the homogeneous linewidth.
One possibility for inhomogeneous broadening is the existence of different isotopes, which
have slightly different resonance frequencies for a given resonant transition. In this situation,
pðνk Þdνk represents the percentage of each isotope group among all atoms and (7.11) becomes
simply the weighted sum of the isotope groups.
Other mechanisms for inhomogeneous broadening include the Doppler effect in a gaseous
medium at a low pressure and the random distribution of active impurity atoms doped in a solid
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7.1 Optical Transitions
231
host. The inhomogeneous frequency shifts caused by these mechanisms are usually randomly
distributed, resulting in a Gaussian functional distribution for pðνk Þ. In an extremely inhomogeneously broadened system, the spread of this distribution dominates the homogeneous linewidth. Then, the transition is characterized by a normalized Gaussian lineshape:
"
#
2ðln 2Þ1=2
ðν ν0 Þ2
g^ðνÞ ¼ 1=2
exp 4 ln 2
,
(7.12)
π Δνinh
Δν2inh
where ν0 is the center frequency and Δνinh is the FWHM of the inhomogeneously broadened
spectral distribution. In terms of the angular frequency, the normalized Gaussian lineshape is
"
#
2ðln 2Þ1=2
ðω ω0 Þ2
g^ðωÞ ¼ 1=2
exp 4 ln 2
,
(7.13)
π Δωinh
Δω2inh
where ω0 ¼ 2πν0 and Δωinh ¼ 2πΔνinh .
Whether a medium is homogeneously or inhomogeneously broadened is often a function of
pressure and temperature. In a gas at a low pressure, the velocity distribution of the gas
molecules in thermal equilibrium is characterized by the Maxwellian velocity distribution,
which is a Gaussian function. This velocity distribution leads to a Gaussian distribution of
Doppler frequency shifts with a linewidth of ΔνD given by
3=2
ΔνD ¼ 2
ðln 2Þ
1=2
k B T 1=2 23=2 ðln 2Þ1=2 k B T 1=2
ν
¼
,
λ
Mc2
M
(7.14)
where λ is the emission wavelength, kB is the Boltzmann constant, T is the temperature in kelvin, and
M is the mass of the atom or molecule that emits the radiation. When this Doppler-broadening effect
dominates, the Gaussian lineshape has an inhomogeneous linewidth of Δνinh ¼ ΔνD .
Figure 7.4 Normalized Lorentzian (solid curves) and Gaussian (dashed curves) lineshape functions of the same
FWHM with (a) a normalized area as g^ ðνÞ is defined and (b) a normalized peak value. For the Lorentzian
lineshape, ν0 ¼ ν21 and Δν ¼ Δνh . For the Gaussian lineshape, Δν ¼ Δνinh .
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232
Optical Absorption and Emission
The normalized Lorentzian lineshape function and the normalized Gaussian lineshape function of the same FWHM are compared in Fig. 7.4. In Fig. 7.4(a), we show g^ðνÞ as expressed in
(7.4) for the Lorentzian lineshape and in (7.12) for the Gaussian lineshape, both with a
normalized area as defined in (7.2). In Fig. 7.4(b), the lineshapes are normalized to have the
same peak value.
EXAMPLE 7.2
The transition for the well-known He–Ne emission wavelength of λ ¼ 632:8 nm takes place
between the 3s2 level, which is the upper level j2i, and the 2p4 level, which is the lower
level j1i, of the Ne atom. The upper and lower levels for this emission both relax
20
rad
radiatively, with τ 2 ¼ τ rad
and
2 ¼ 30 ns and τ 1 ¼ τ 1 ¼ 10 ns. Two Ne isotopes, Ne
22
20
Ne , contribute to this emission, with more than 90% due to Ne . For simplicity, we
take the atomic mass number of Ne to be 20. The typical He–Ne laser medium operates at a
temperature of T ¼ 400 K and a low gas pressure of P ¼ 2:5 torr. (a) Find the radiative,
nonradiative, and total relaxation rates for the upper and lower levels, j2i and j1i,
respectively. (b) Find the natural linewidth and the lifetime-broadened linewidth of the
emission line. (c) Find the linewidth caused by Doppler broadening. (d) What is the
lineshape and linewidth of this emission line?
Solution:
Natural broadening and lifetime broadening are homogeneous broadening mechanisms,
whereas Doppler broadening is an inhomogeneous broadening mechanism. Pressure-induced
broadening is a homogeneous mechanism, but it can be ignored in this problem because of the
low gas pressure of P ¼ 2:5 torr.
(a) Both the upper level j2i and the lower level j1i relax radiatively. For each level, the total
relaxation rate is the same as the radiative relaxation rate:
γ2 ¼ γrad
2 ¼
1
1
¼
s1 ¼ 3:3 107 s1 ,
τ 2 30 109
γ1 ¼ γrad
1 ¼
1
1
¼
s1 ¼ 1 108 s1 :
τ 1 10 109
The nonradiative relaxation rates of the two levels are both zero:
nonrad
γnonrad
¼ γ2 γrad
¼ γ1 γrad
2
2 ¼ 0, γ1
1 ¼ 0:
(b) Using the results from (a), we find that
1
1
rad
8
7
1
¼ ðγrad
¼ 6:7 107 s1 ,
γnatural
21
1 þ γ1 Þ ¼ ð1 10 þ 3:3 10 Þ s
2
2
1
1
8
7
1
γlife
¼ 6:7 107 s1 :
21 ¼ ðγ1 þ γ2 Þ ¼ ð1 10 þ 3:3 10 Þ s
2
2
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7.1 Optical Transitions
233
The natural linewidth and the lifetime-broadened linewidth are the same:
Δνlife ¼
γlife
γnatural
21
¼ Δνnatural ¼ 21
¼ 21:2 MHz:
π
π
If no other mechanisms further broaden this line, this emission line has a Lorentzian
lineshape that has a homogeneously broadened linewidth of Δνh ¼ Δνlife ¼ 21:2 MHz.
(c) The mass of a Ne atom is M ¼ 20 1:66 1027 kg ¼ 3:32 1026 kg for a mass
number of 20. Therefore, the Doppler-broadened linewidth at T ¼ 400 K is
1=2
23=2 ðln 2Þ1=2 k B T 1=2
23=2 ðln 2Þ1=2 1:38 1023 400
¼
Hz ¼ 1:5 GHz:
ΔνD ¼
λ
M
632:8 109
3:32 1026
(d) Because ΔνD Δνlife , the homogeneous lifetime broadening is completely dominated by
the inhomogeneous Doppler broadening. Therefore, the lineshape of this emission line is
Gaussian with a linewidth of Δνinh ΔνD ¼ 1:5 GHz:
7.1.4 Mixed Broadening
When the pressure of a gaseous medium is increased, frequent collisions among the gas
molecules shorten the lifetimes of the excited states of the molecules. This effect reduces τ 2 ,
and it can also reduce τ 1 if the lower level is not the ground level. The resulting pressureinduced lifetime broadening causes the homogeneous linewidth to increase. At a certain
pressure, the homogeneous linewidth Δνh finally dominates the Doppler linewidth ΔνD . Then
the medium becomes predominantly homogeneously broadened.
Another good example is the linewidth associated with the impurity ions doped in a solid
host, such as Nd:YAG or Nd:glass. At a low temperature, the homogeneous linewidth of
the Nd3þ ions is narrow. The lineshape is dominated by inhomogeneous shifts of the
resonance frequency due to variations in the local environment of individual Nd3þ ions.
As a result, the lineshape function is inhomogeneously broadened. As the temperature
increases, the homogeneous linewidth increases because of increased collisions of phonons
with the ions. At room temperature, the spectral line of Nd:YAG at 1.064 μm has a total
linewidth of Δν 120 to 180 GHz with an inhomogeneous component of only about
6 to 30 GHz. Therefore, as illustrated in Example 7.1, Nd:YAG is pretty much homogeneously broadened at room temperature. In comparison, Nd:glass has a much larger
inhomogeneous linewidth than Nd:YAG because the glass host provides a larger range of
local variations than the YAG crystal. At room temperature, the same spectral line of Nd:
glass appears at 1.054 μm with a total linewidth of Δν 5 to 7 THz, which is almost all
inhomogeneously broadened.
Clearly a lineshape can be neither Lorentzian nor Gaussian when the homogeneously
broadened linewidth Δνh and the inhomogeneously broadened linewidth Δνinh of an emission
line are on the same order of magnitude. In this situation, the line profile is a convolution of the
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234
Optical Absorption and Emission
Lorentzian profile of a width Δνh and the Gaussian profile of a width Δνinh . The result is a Voigt
lineshape that has a linewidth of
Δν 0:5346Δνh þ ð0:2166Δν2h þ Δν2inh Þ1=2 :
7.2
(7.15)
TRANSITION RATES
..............................................................................................................
The probability per unit time for a resonant optical process to occur is measured by the
transition rate of the process. Because of the resonant nature of the interaction, the transition
rate of an induced process is a function of both the spectral distribution of the optical radiation
and the spectral characteristics of the resonant transition.
The spectral distribution of an optical field is characterized by its spectral energy density,
uðνÞ, which is the energy density of the optical radiation per unit frequency interval at the
optical frequency ν. The total energy density of the radiation is
ð∞
u ¼ uðνÞdν:
(7.16)
0
The spectral intensity distribution, IðνÞ, of the radiation is related to uðνÞ by the relation
c
IðνÞ ¼ uðνÞ,
n
(7.17)
where n is the refractive index of the medium, and the total intensity is simply
ð∞
I ¼ IðνÞdν:
(7.18)
0
Because an induced transition is stimulated by optical radiation, its transition rate is proportional to the energy density of the optical radiation within the spectral response range of the
transition. The transition rate for the upward transition from j1i to j2i, associated with
absorption, in the frequency range between ν and ν þ dν is
W 12 ðνÞdν ¼ B12 uðνÞ^
g ðνÞdν,
(7.19)
whereas that for the induced downward transition from j2i to j1i, associated with stimulated
emission, in the frequency range between ν and ν þ dν is
W 21 ðνÞdν ¼ B21 uðνÞ^
g ðνÞdν:
(7.20)
Because the spontaneous emission rate is independent of the energy density of the radiation, the
spontaneous emission spectrum is determined solely by the lineshape function of the transition:
W sp ðνÞdν ¼ A21 g^ðνÞdν:
(7.21)
The A and B constants defined above are known as the Einstein A and B coefficients,
respectively. The rates associated with the transitions between two atomic levels j1i and j2i
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7.2 Transition Rates
235
Figure 7.5 Resonant transitions in the interaction of a radiation field with two atomic levels j1i and j2i of
population densities N 1 and N 2 , respectively.
in the interaction with a radiation field of an energy density uðνÞ are summarized in Fig. 7.5.
The total induced transition rates are
ð∞
ð∞
W 12 ¼ W 12 ðνÞdν ¼ B12 uðνÞ^
g ðνÞdν
and
0
0
ð∞
ð∞
W 21 ¼ W 21 ðνÞdν ¼ B21 uðνÞ^
g ðνÞdν:
0
(7.22)
(7.23)
0
The total spontaneous emission rate is
ð∞
W sp ¼ W sp ðνÞdν ¼ A21 :
(7.24)
0
The induced and spontaneous transition rates of a given system are not independent of each
other but are directly proportional to each other. Their relationship was first obtained by
Einstein by considering the interaction of blackbody radiation with an ensemble of identical
atomic systems in thermal equilibrium. The spectral energy density of blackbody radiation at a
temperature T is given by Planck’s formula:
uðνÞ ¼
8πn3 hν3
1
,
3
hν=k
T 1
B
c
e
(7.25)
where k B is the Boltzmann constant.
As shown in Fig. 7.5, the population densities per unit volume of the atoms in levels j2i and
j1i are N 2 and N 1 , respectively. The number of atoms per unit volume making the downward
transition per unit time accompanied by the emission of radiation in a frequency range from ν to
ν þ dν is N 2 ½W 21 ðνÞ þ W sp ðνÞdν, and the number of atoms per unit volume making the upward
transition per unit time through the absorption of radiation in the same frequency range is
N 1 W 12 ðνÞdν. In thermal equilibrium, both the spectral density of blackbody radiation and the
atomic population density in each energy level reach a steady state, meaning that
N 2 ½W 21 ðνÞ þ W sp ðνÞ ¼ N 1 W 12 ðνÞ:
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(7.26)
236
Optical Absorption and Emission
This relation spells out the principle of detailed balance in thermal equilibrium. Therefore, the
steady-state population distribution in thermal equilibrium satisfies
N2
W 12 ðνÞ
B12 uðνÞ
¼
¼
:
N 1 W 21 ðνÞ þ W sp ðνÞ B21 uðνÞ þ A21
(7.27)
In thermal equilibrium at a temperature T, however, the population ratio of the atoms in the
upper and the lower levels follows the Boltzmann distribution. Taking into account the
degeneracy factors, g2 and g1 , of these energy levels, we have
N 2 g2 hv=kB T
¼ e
N 1 g1
(7.28)
for the population densities associated with a transition energy of hν. Combining (7.27) and
(7.28), we have
uðνÞ ¼
A21 =B21
:
ðg1 B12 =g2 B21 Þehv=kB T 1
(7.29)
Identifying (7.29) with (7.25), we find that
A21 8πn3 hν3
¼
B21
c3
(7.30)
g1 B12 ¼ g2 B21 :
(7.31)
and
The spontaneous radiative lifetime of the atoms in level j2i associated with the radiative
spontaneous transition from j2i to j1i is
τ sp ¼
1
1
¼
:
W sp A21
(7.32)
The spectral dependence of the spontaneous emission rate can be expressed as
W sp ðνÞ ¼
1
g^ ðνÞ:
τ sp
(7.33)
According to the relations in (7.30) and (7.31), the transition rates of both of the induced
processes of absorption and stimulated emission are directly proportional to the spontaneous
emission rate. In terms of τ sp , the spectral dependence of the stimulated-emission transition
from j2i to j1i can be generally expressed as
W 21 ðνÞ ¼
c3
c2
uðνÞ^
g
ðνÞ
¼
IðνÞ^
g ðνÞ,
8πn3 hv3 τ sp
8πn2 hv3 τ sp
(7.34)
and that for the absorption transition from j1i to j2i can be found as
W 12 ðνÞ ¼
g2
W 21 ðνÞ:
g1
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(7.35)
7.2 Transition Rates
237
Because WðνÞ is the transition rate per unit frequency according to the definition in (7.19)–
(7.21), we have WðνÞdν ¼ WðωÞdω. Therefore, W sp ðνÞ ¼ 2πW sp ðωÞ, W 21 ðνÞ ¼ 2πW 21 ðωÞ,
and W 12 ðνÞ ¼ 2πW 12 ðωÞ.
EXAMPLE 7.3
A cylindrical Nd:YAG rod has a length of l ¼ 5 cm and a diameter of d ¼ 6 mm. The Nd3þ
ions are doped in the YAG host at 1.2% atomic concentration for a total concentration of
N t ¼ 1:66 1020 cm3 . The rod is uniformly pumped such that 1% of the Nd3þ ions are
excited to the 4 F3=2 level and then left to relax spontaneously. Use the parameters given in
Fig. 7.3 for the energy levels of Nd:YAG to answer the following questions regarding the
emission at the two lines of λ ¼ 1:064 μm and λ ¼ 1:34 μm. (a) Find the spontaneous
radiative lifetimes for the transitions of the two emission lines, respectively. (b) What are
the decay times of the spontaneous emission at the two emission lines, respectively? (c)
What are the optical energies of the spontaneous emission at the two wavelengths,
respectively? (d) What are the powers of the spontaneous emission at the two wavelengths,
respectively?
Solution:
The Nd:YAG rod has a volume of
V ¼ πðd=2Þ2 l ¼ πð6 103 =2Þ2 5 102 m3 ¼ 1:41 106 m3 :
It is pumped to have a concentration in the upper level j2i of
N 2 ¼ 1%N t ¼ 1:66 1018 cm3 ¼ 1:66 1024 m3 :
(a) The spontaneous radiative lifetime of each transition is determined by the A coefficient of
the transition. From Fig. 7.3, we find A1:064 ¼ 1940 s1 and A1:34 ¼ 493 s1 . Therefore, the
spontaneous radiative lifetimes are, respectively,
τ sp
1:064 ¼
1
1
¼
s ¼ 515 μs,
A1:064 1940
τ sp
1:34 ¼
1
1
¼
s ¼ 2:03 ms:
A1:34 493
(b) Because the spontaneous emission at both emission lines results from the population in level
j2i, the number density S1:064 of the spontaneous photons that are emitted at λ ¼ 1:064 μm and
the number density S1:34 of the spontaneous photons emitted at λ ¼ 1:34 μm are both proportional to N 2 . Therefore, the fluorescence at both wavelengths decays at the same rate as that of
N 2 . The fluorescence time is the same for both wavelengths and is the lifetime τ 2 ¼ 240 μs of
level j2i, given in Fig. 7.3.
(c) Though the number densities S1:064 and S1:34 of the spontaneous photons emitted at λ ¼
1:064 μm and λ ¼ 1:34 μm, respectively, are both proportional to N 2 and both decay at the
same decay time, their magnitudes are respectively proportional to the spontaneous radiative relaxation rates, A1:064 and A1:34 , of their transitions:
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238
Optical Absorption and Emission
S1:064 ¼
A1:064
N 2 ¼ A1:064 τ 2 N 2 ¼ 1940 240 106 1:66 1024 m3 ¼ 7:73 1023 m3 ,
γ2
S1:34 ¼
A1:34
N 2 ¼ A1:34 τ 2 N 2 ¼ 493 240 106 1:66 1024 m3 ¼ 1:96 1023 m3 :
γ2
The photon energies at the two wavelengths are, respectively,
hv1:064 ¼
1:2398
1:2398
eV, hv1:34 ¼
eV:
1:064
1:34
The spontaneous optical energies emitted at the two wavelengths are, respectively,
U 1:064 ¼ hv1:064 S1:064 V ¼
U 1:34 ¼ hv1:34 S1:34 V ¼
1:2398
1:6 1019 7:73 1023 1:41 106 J ¼ 203 mJ;
1:064
1:2398
1:6 1019 1:96 1023 1:41 106 J ¼ 41 mJ:
1:34
Because these optical energies both decay at the fluorescence time of τ 2 ¼ 240 μs,
P1:064 ¼
P1:34 ¼
U 1:064 203 103
¼
W ¼ 846 W,
τ2
240 106
U 1:34
41 103
¼
W ¼ 17 W:
τ2
240 106
7.2.1 Transition Cross Section
It is often useful to express the transition probability of an atom in its interaction with optical
radiation at a frequency of ν in terms of the transition cross section, σðνÞ. For transitions
between energy levels j1i and j2i, the transition cross sections σ 21 ðνÞ and σ 12 ðνÞ are defined
through the following relations to the transition rates,
W 21 ðνÞ ¼
IðνÞ
σ 21 ðνÞ
hν
(7.36)
W 12 ðνÞ ¼
IðνÞ
σ 12 ðνÞ:
hν
(7.37)
and
The transition cross section σ 21 ðνÞ, which is associated with stimulated emission, is also called
the emission cross section, σ e ðνÞ, whereas σ 12 ðνÞ, which is associated with absorption, is also
called the absorption cross section, σ a ðνÞ. From (7.34), we find that
σ e ðνÞ ¼ σ 21 ðνÞ ¼
c2
g^ ðνÞ:
8πn2 ν2 τ sp
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(7.38)
7.2 Transition Rates
239
According to (7.35), we find that g1 σ 12 ¼ g2 σ 21 . Therefore,
σ a ðνÞ ¼ σ 12 ðνÞ ¼
g2
g
σ 21 ðνÞ ¼ 2 σ e ðνÞ:
g1
g1
(7.39)
The transition cross sections have the unit of area in square meters but are often quoted in
square centimeters. Note that σðνÞ ¼ σðωÞ because σðνÞ is simply defined as the value of the
transition cross section at the frequency ν rather than as that per unit frequency, but WðνÞ ¼
2πWðωÞ and g^ ðνÞ ¼ 2π^
g ðωÞ. Therefore, in terms of ω,
σ e ðωÞ ¼ σ 21 ðωÞ ¼
π 2 c2
g^ ðωÞ
n2 ω2 τ sp
and σ a ðωÞ ¼
g2
σ e ðωÞ:
g1
(7.40)
For the ideal Lorentzian and Gaussian lineshapes expressed in (7.4) and (7.12), respectively,
the peak value of g^ ðνÞ occurs at the center of the spectrum and is a function of the linewidth Δν
only. By applying this fact to (7.38), the peak value of the emission cross section at the center
wavelength λ of the spectrum can be expressed as
σ he ¼
λ2
4π 2 n2 Δνh τ sp
(7.41)
for a homogeneously broadened medium that has an ideal Lorentzian lineshape, and as
σ inh
e ¼
ðln 2Þ1=2 λ2
4π 3=2 n2 Δνinh τ sp
(7.42)
for an inhomogeneously broadened medium that has an ideal Gaussian lineshape. In
practice, the experimentally measured peak emission cross section usually differs from
that calculated using these formulas because the spectral lineshape of a realistic gain
medium is generally determined by a combination of many different mechanisms and,
consequently, is rarely ideal Lorentzian or ideal Gaussian. Nevertheless, these formulas
provide a good estimate for the peak value of the emission cross section. They also clearly
indicate that the emission cross section varies quadratically with the emission wavelength
but is inversely proportional to both the emission linewidth and the spontaneous radiative
lifetime of the transition.
The characteristics of some representative laser materials are listed in Table 7.1. As seen in
Table 7.1, the parameters vary over a wide range among different types of optical gain
media. For example, the peak value of the emission cross section varies from 6 1025 m2
for Er:fiber to 2:5 1016 m2 for the Ar-ion laser, whereas the spontaneous emission linewidth varies from 60 MHz for CO2 to 100 THz for Ti:sapphire. The fluorescence lifetime
varies from the order of 1 ns for a semiconductor gain medium to the order of 10 ms for
Er:fiber.
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240
Optical Absorption and Emission
Table 7.1 Characteristics of some laser materials
Gain medium
Wavelength System
λ (μm)
a
Cross section
σe (m2)
Spontaneous
linewidthc
b
Lifetimesd
Δν
Δλ (nm)
τ sp
τ2
Index
n
He–Ne
0.6328
I,4
3.0 1017
1.5 GHz
0.002
300 ns 30 ns
1
Ar ion
0.488
I,4
2.5 1016
2.7 GHz
0.004
13 ns
10 ns
1
CO2
10.6
I,4
3.0 1022
60 MHz
0.02
4s
1 μs
1
Copper vapor
0.5105
I,3
8.6 1018
2.3 GHz
0.002
500 ns 500 ns
1
KrF excimer
0.248
H,3
2.6 1020
10 THz
2
10 ns
8 ns
1
R6G dye
0.57–0.65
H/I,Q2
2.3 1020
30 THz
33
6 ns
4 ns
1.4
Rubye
0.6943
H,3
1.25–2.5 1024 330 GHz
0.53
3 ms
3 ms
1.76
Nd:YAG
1.064
H,4
2–10 1023
150 GHz
0.56
515 μs 240 μs
1.82
Nd:glass
1.054
I,4
4.0 1024
6 THz
22
330 μs 330 μs
1.53
Er:fiber
1.53
H/I,3
6.0 1025
5 THz
40
10 ms
10 ms
1.46
0.66–1.1
H,Q2
3.4 1023
100 THz
180
3.9 μs 3.2 μs
1.76
0.78–1.01
H,Q2
4.8 1024
83 THz
200
67 μs
67 μs
1.4
H/I,Q2
1–5 1020
10–20 THz 20–100
1 ns
1 ns
3–4
Ti:sapphire
Cr:LiSAF
f
Semiconductor 0.37–1.65
a
H, homogeneously broadened; I, inhomogeneously broadened; Q2, quasi-two-level system; 3, three-level system;
4, four-level system.
b
Both the absorption and emission cross sections depend on the optical frequency. The absorption and emission
cross sections generally have different peak values and different spectral dependences. Listed is the peak value of
the emission cross section.
c
The spontaneous linewidth determines the gain bandwidth of a medium when population inversion is achieved.
d
The spontaneous lifetime τ sp is related to the transition rate, whereas the fluorescence lifetime τ 2 is related to the
upper-level population relaxation. The fluorescence lifetime of a gaseous medium varies with temperature and
pressure; that of a liquid or solid medium varies with temperature, the host material, and the concentration of the
active ions or molecules. For example, τ 2 of CO2 varies from 100 ns to 1 ms depending on temperature and pressure.
e
Ruby is sapphire (Al2O3) doped with Cr3+ ions. The sapphire crystal is uniaxial. For ruby, the value of σe for
emission with E⊥c, which is listed, is larger than that for Ekc.
f
For Ti:sapphire, the value of σ e for Ekc, which is listed, is larger than that for E⊥c.
EXAMPLE 7.4
The λ ¼ 1:064 μm emission line of Nd:YAG considered in Example 7.1 has a predominantly
homogeneously broadened total linewidth of 150 GHz and a spontaneous radiative relaxation
rate of A ¼ 1940 s1 . The refractive index of the YAG crystal is n ¼ 1:82. The λ ¼ 632:8 nm
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241
7.3 Attenuation and Amplification of Optical Fields
emission line of He–Ne considered in Example 7.2 has a predominantly inhomogeneously
broadened total linewidth of 1.5 GHz and a spontaneous radiative lifetime of τ sp ¼ 300 ns. The
refractive index of the low-pressure He–Ne gas is n ¼ 1. Find the peak emission cross sections
for these two lines.
Solution:
For the λ ¼ 1:064 μm emission line of Nd:YAG, we take Δνh ¼ 150 GHz to be the homogeneous linewidth as an approximation because this line is predominantly homogeneously
broadened. The spontaneous radiative lifetime is τ sp ¼ 1=A ¼ 515 μs. Then, using (7.41), the
emission cross section is found to be
σ he ¼
λ2
ð1:064 106 Þ2
¼
m2 ¼ 1:12 1022 m2 ,
4π 2 n2 Δνh τ sp 4π 2 1:822 150 109 515 106
which is slightly larger than, but consistent with, the value listed in Table 7.1.
For the λ ¼ 632:8 nm emission line of He–Ne, we take Δνinh ¼ 1:5 GHz to be the inhomogeneous linewidth as an approximation because this line is predominantly inhomogeneously
broadened. With a spontaneous radiative lifetime of τ sp ¼ 300 ns, the emission cross section is
found using (7.42) to be
σ inh
e ¼
ðln 2Þ1=2 λ2
ðln 2Þ1=2 ð632:8 109 Þ2
¼
m2 ¼ 3:33 1017 m2 ,
4π 3=2 n2 Δνinh τ sp 4π 3=2 12 1:5 109 300 109
which is slightly larger than, but consistent with, the value listed in Table 7.1.
7.3
ATTENUATION AND AMPLIFICATION OF OPTICAL FIELDS
..............................................................................................................
Optical absorption results in the attenuation of an optical field, whereas stimulated emission
leads to the amplification of an optical field. To quantify the net effect of a resonant transition
on the attenuation or amplification of an optical field, we consider the interaction of a
monochromatic plane optical field at a frequency of ν with a material that consists of electronic
or atomic systems with population densities N 1 and N 2 in energy levels j1i and j2i, respectively. Because the spectral intensity distribution of the monochromatic plane optical field that
has an intensity of I is simply IðνÞ ¼ Iδðν0 νÞ, the total induced transition rates between
energy levels j1i and j2i in this interaction are
I
I
(7.43)
σ e ðνÞ and W 12 ¼ σ a ðνÞ:
hν
hν
The net power that is transferred from the optical field to the material is the difference
between that absorbed by the material and that emitted due to stimulated emission:
W 21 ¼
W p ¼ hνW 12 N 1 hνW 21 N 2 ¼ ½N 1 σ a ðνÞ N 2 σ e ðνÞI:
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(7.44)
242
Optical Absorption and Emission
In the case when W p > 0, there is net power absorption by the medium from the optical field
due to resonant transitions between energy levels j1i and j2i. The absorption coefficient, also
called attenuation coefficient, is
g1
αðνÞ ¼ N 1 σ a ðνÞ N 2 σ e ðνÞ ¼ N 1 N 2 σ a ðνÞ:
(7.45)
g2
In the case when W p < 0, net power is transferred from the medium to the optical field,
resulting in the amplification of the optical field. The gain coefficient, also called the
amplification coefficient, is
g2
gðνÞ ¼ N 2 σ e ðνÞ N 1 σ a ðνÞ ¼ N 2 N 1 σ e ðνÞ:
(7.46)
g1
The coefficients α and g have the unit of per meter, also often quoted per centimeter. Note that
αðνÞ ¼ αðωÞ and gðνÞ ¼ gðωÞ because σðνÞ ¼ σðωÞ. Note also that αðνÞ ¼ gðνÞ because a
negative gain is a positive loss, and vice versa.
According to (7.43), both σ e ðνÞ and σ a ðνÞ have positive values because W 21 0 and W 12 0
by definition. Therefore, αðνÞ > 0 and gðνÞ < 0 if N 1 > ðg1 =g2 ÞN 2 , whereas gðνÞ > 0 and
αðνÞ < 0 if N 2 > ðg2 =g1 ÞN 1 . A material in its normal state in thermal equilibrium absorbs
optical energy because the lower energy level is more populated than the upper energy level. In
order to provide a net optical gain to the optical field, a material has to be in a nonequilibrium
state of population inversion for the upper level to be more populated than the lower level.
EXAMPLE 7.5
The λ ¼ 1:064 μm emission line of Nd:YAG has τ 2 ¼ 240 μs for the upper level j2i and
τ 1 ¼ 200 ps for the lower level j1i, as shown in Fig. 7.3. We consider here the Nd:YAG rod
in Example 7.3, which is doped with Nd3þ ions at 1.2% atomic concentration for a total
concentration of N t ¼ 1:66 1020 cm3 . If it is not pumped, what is its absorption coefficient
at λ ¼ 1:064 μm at T ¼ 300 K? If the rod is uniformly pumped such that 1% of the total Nd3þ
ions are excited to level j2i, what is the absorption or gain coefficient at λ ¼ 1:064 μm?
Solution:
The lower level j1i is not the ground level. From Fig. 7.3, we find that its energy above the
ground level is
ΔE 10 ¼
1:2398
1:2398
eV eV ¼ 0:21 eV:
0:9
1:064
At T ¼ 300 K, k B T ¼ 25:9 meV. Thus, the population density of Nd3þ ions in this level is
approximately
0:21
ΔE 10 =kB T
N 1 N te
¼ 3 104 N t
¼ N t exp 25:9 103
which is negligibly small because level j1i lies sufficiently high above the ground level.
Therefore, the absorption coefficient at λ ¼ 1:064 μm is negligibly small: α 0.
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7.3 Attenuation and Amplification of Optical Fields
243
When 1% of the total Nd3þ ions are excited to level j2i, we have
N 2 ¼ 1%N t ¼ 1:66 1018 cm3 ¼ 1:66 1024 m3 :
In this situation, the excited ions can relax from level j2i to level j1i, but any ion reaching level j1i
τ2.
quickly relaxes to the ground level because level j1i has a short lifetime of τ 1 ¼ 200 ps
Therefore, level j1i remains almost empty, N 1 0, as compared to level j2i. The emission cross
section of the λ ¼ 1:064 μm line found in Example 7.4 is σ e ¼ 1:12 1022 m2 . Consequently,
at λ ¼ 1:064 μm the Nd:YAG rod has a gain coefficient of
g ¼ N 2 σ e N 1 σ a N 2 σ e ¼ 1:66 1024 1:12 1022 m1 ¼ 186 m1 :
This is a very large gain coefficient even though only 1% of the total Nd3þ ions are excited. In
practice, depending on the design of the laser cavity, only a smaller percentage of ions has to be
excited for laser action.
7.3.1 Resonant Optical Susceptibility
The macroscopic optical properties of a medium are characterized by its electric susceptibility.
As seen in Section 2.3, resonances in an optical medium contribute to the dispersion in the
susceptibility of the medium. Clearly, the optical properties of a medium are functions of the
resonant optical transitions between the energy levels of the electrons in the medium.
From the viewpoint of the macroscopic optical properties of a medium, the interaction
between an optical field and a medium is characterized by the polarization induced by the
optical field in the medium. The power exchange between the optical field and the medium is
given by (1.34). For the resonant interaction of an isotropic medium with a monochromatic
plane optical field at a frequency of ω ¼ 2πν, we have EðtÞ ¼ Eeiωt þ E∗ eiωt and
∗ iωt
Pres ðtÞ ¼ ϵ 0 ½ χ res ðωÞEeiωt þ χ ∗
res ðωÞE e , where Pres is the polarization contributed by the
resonant transitions and χ res is the resonant susceptibility. Using (1.34), we find that the timeaveraged power density absorbed by the medium is
ω
W p ¼ 2ωϵ 0 χ 00res ðωÞjEj2 ¼ χ 00res ðωÞI:
(7.47)
nc
By identifying (7.47) with (7.44), we find that the imaginary part of the susceptibility contributed by the resonant transitions between energy levels j1i and j2i is
χ 00res ðωÞ ¼
nc
½N 1 σ a ðωÞ N 2 σ e ðωÞ:
ω
(7.48)
The real part χ 0res ðωÞ of the resonant susceptibility can be found through the Kramers–Kronig
relations given in (2.53).
As discussed in Sections 2.1 and 2.3, a medium causes an optical loss if χ 00 > 0, and it
provides an optical gain if χ 00 < 0. It is also clear from (7.47) that there is a net power loss from
the optical field due to absorption by the medium if χ 00res > 0, but there is a net power gain for
the optical field if χ 00res < 0. By comparing (7.48) with (7.45) and (7.46), we find that the
medium has an absorption coefficient given by
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244
Optical Absorption and Emission
αðωÞ ¼
ω 00
χ ðωÞ
nc res
(7.49)
in the case of normal population distribution when χ 00res > 0, whereas it has a gain coefficient
given by
gðωÞ ¼ ω 00
χ ðωÞ
nc res
(7.50)
in the case of population inversion so that χ 00res < 0.
Note that the material susceptibility characterizes the response of a material to the excitation
of an electromagnetic field. Therefore, the magnitude of the resonant susceptibility χ 00res only
accounts for the contributions from the induced processes of absorption and stimulated
emission, and not that from the process of spontaneous emission. Spontaneous emission causes
natural broadening of the spectral width of χ 00res ðωÞ, as discussed in Section 7.1. The resonant
susceptibility contributed by the induced transitions between two energy levels is proportional
to the population difference between the two levels, but the power density of the optical
radiation due to spontaneous emission is a function of the population density in the upper
energy level alone.
The coefficients α and g respectively characterize the attenuation and growth of the optical
intensity per unit length traveled by the optical wave in a medium. The intensity of a
monochromatic plane wave at the resonance frequency varies with distance along its propagation direction, taken to be the z direction, as
dI
¼ αI
dz
(7.51)
in the case of optical attenuation when χ 00res > 0, and
dI
¼ gI
dz
(7.52)
in the case of optical amplification when χ 00res < 0.
EXAMPLE 7.6
What is the imaginary part χ 00res of the resonant susceptibility, at λ ¼ 1:064 μm, of the pumped
Nd:YAG rod considered in Example 7.5? The refractive index of Nd:YAG is n ¼ 1:82. The rod
has a length of l ¼ 5 cm. If a beam at λ ¼ 1:064 μm that has a power of Pin ¼ 1 mW is sent into
one end of the Nd:YAG rod uniformly over the cross-sectional area of the rod, what is the
optical power coming out at the other end?
Solution:
From Example 7.5, the gain coefficient at λ ¼ 1:064 μm for the pumped Nd:YAG rod is
g ¼ 186 m1 . Using (7.50), we find the imaginary part of the resonant susceptibility at
λ ¼ 1:064 μm:
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Problems
χ 00res ¼ 245
nc
nλ
1:82 1:064 106
186 ¼ 5:73 105 :
g¼ g¼
2π
ω
2π
For uniform illumination, (7.52) can be written in terms of the optical power to find the output
power as
2
dP
¼ gP ) Pout ¼ Pin egl ¼ 1 103 e186510 W ¼ 10:9 W:
dz
Problems
7.1.1 A ruby laser rod is a sapphire crystal doped with active Cr3þ ions. The upper level j2i of
the transition for the ruby emission wavelength of λ ¼ 694:3 nm is the E level of the Cr3þ
ion, and the lower level j1i is the 4 A2 ground level. The population in the E level relaxes
only to the 4 A2 ground level, and the relaxation is purely radiative. The upper level
lifetime is τ 2 ¼ 3 ms. At room temperature, this emission line has a predominantly
homogeneous linewidth of Δν ¼ 330 GHz.
(a) Find the radiative, nonradiative, and total relaxation rates for the upper and lower
levels, j2i and j1i, respectively.
(b) Find the natural linewidth and the lifetime-broadened linewidth for the λ ¼ 694:3 nm
emission line. If no other mechanisms further broaden this line, what are its lineshape
and linewidth?
(c) The homogeneous broadening at room temperature is contributed by dephasing due
to phonon collisions. What is the dephasing rate γdephase
?
21
7.1.2 Ti:sapphire and Cr:LiSAF are solid-state laser media. Ti:sapphire contains active Ti3þ
ions doped in a sapphire crystal, and Cr:LiSAF contains active Cr3þ ions doped in a
LiSAF crystal. The fluorescence lifetime of Ti:sapphire is τ 2 ¼ 3:2 μs, and that of Cr:
LiSAF is τ 2 ¼ 67 μs. For both systems, the lower level j1i is the ground level. Both
media have very broad spontaneous linewidths that are predominantly homogeneously
broadened, with Δν 100 THz for Ti:sapphire and Δν 83 THz for Cr:LiSAF. What
are the expected lifetime-broadened homogeneous linewidths of these two media?
Explain why these two media have such broad homogeneous linewidths.
7.1.3 The CO2 laser gain medium contains the gas mixture of CO2 , N2 , and He with about the
same fractional ratio of CO2 and N2 , and somewhat more He. The λ ¼ 10:6 μm emission
takes place between two vibrational levels of the CO2 molecule. The upper level j2i has a
radiative lifetime of τ rad
2 ¼ 4 s, and the lower level j1i has a radiative lifetime of
rad
τ 1 ¼ 200 ms. The N2 molecules help to pump the CO2 molecules to the upper level
j2i, while the He atoms help to de-excite the N2 and CO2 molecules back to their
respective ground levels. The collisions of the CO2 molecules with the N2 molecules
and the He atoms change the lifetimes τ 2 of the upper level and τ 1 of the lower level by
inducing nonradiative relaxations from these levels. As a result, τ 2 and τ 1 depend on the
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246
Optical Absorption and Emission
pressure and temperature of the gas mixture. The working temperature of a CO2 laser
ranges from 400 K to 700 K. The working gas pressure varies from below 50 torr to
760 torr for different CO2 lasers.
(a) Find the radiative relaxation rates for the upper and lower levels, j2i and j1i,
respectively. What is the natural linewidth of the emission line?
(b) The molecular mass number of CO2 is 44. Find the range of the Doppler-broadened
linewidth for the CO2 lasers.
(c) Consider a CO2 laser medium of a relatively low pressure working at T ¼ 400 K,
which has τ 2 ¼ 10 μs and τ 1 ¼ 1 μs. Find the nonradiative and total relaxation rates
for the upper and lower levels, j2i and j1i, respectively. What are the homogeneously and inhomogeneously broadened linewidths of the emission line? What are
the lineshape and the total linewidth? Is it homogeneously or inhomogeneously
broadened?
(d) Consider a CO2 laser medium of a high pressure working at T ¼ 700 K, which
has τ 2 ¼ 100 ns and τ 1 ¼ 1 ns. Find the nonradiative and total relaxation rates for
the upper and lower levels, j2i and j1i, respectively. What are the homogeneously and inhomogeneously broadened linewidths of the emission line? What
are the lineshape and the total linewidth? Is it homogeneously or inhomogeneously broadened?
7.1.4 The argon-ion laser has two emission lines at 488 nm and 514:5 nm. Both lines are
almost entirely broadened by Doppler broadening at the typical operating temperature of
T ¼ 1200 C. The Ar atom has an atomic mass number of 40. Find the linewidths and the
lineshapes of the two emission lines, respectively.
7.2.1 A cylindrical ruby rod, which is a sapphire crystal doped with active Cr3þ ions, has a
length of l ¼ 6 cm and a diameter of d ¼ 5 mm. The Cr3þ ions has a total concentration
of N t ¼ 1:58 1019 cm3 . The upper level j2i of the transition for the ruby emission
wavelength of λ ¼ 694:3 nm relaxes only radiatively through this emission line with a
3þ
lifetime of τ 2 ¼ τ rad
ions
2 ¼ 3 ms. The rod is uniformly pumped such that 50% of the Cr
are excited to the upper level and then left to relax spontaneously.
(a) Find the spontaneous radiative lifetime for the transition of this emission line. What is
the decay time of the spontaneous emission?
(b) What are the optical energy and the power of the spontaneous emission?
7.2.2 Two emission lines have exactly the same wavelength and the same linewidth, but one
has a Lorentzian lineshape while the other has a Gaussian lineshape. If the optical
transitions for both emission lines have the same spontaneous lifetime and the two media
have the same refractive index, do they have the same peak emission cross section? If
they do not have the same peak emission cross section, which one has a larger cross
section? What is the difference?
7.2.3 Two emission lines have exactly the same center wavelength, the same linewidth, the
same peak emission cross section, and they take place in two media that have the same
refractive index, but one has a Lorentzian lineshape and the other has a Gaussian lineshape. What is the possible parameter that has different values for these two transitions?
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Problems
247
7.2.4 Are the emission cross section and the absorption cross section of the same spectral line
associated with the transitions between the same pair of energy levels necessarily the
same? Explain.
7.2.5 The upper level j2i of the transition for the ruby emission wavelength of λ ¼ 694:3 nm
is the E level of the active Cr 3þ ions doped in the ruby crystal, which has a degeneracy
of g2 ¼ 2, and the lower level j1i is the 4 A2 ground level, which has a degeneracy of
g1 ¼ 4. The population in the E level relaxes radiatively only through this emission line
to the 4 A2 ground level with τ 2 ¼ τ rad
2 ¼ 3 ms. At room temperature, this emission line
has a homogeneous linewidth of Δν ¼ 330 GHz. The refractive index of the ruby
crystal is n ¼ 1:76. Find the peak emission and absorption cross sections for this
spectral line.
7.2.6 The λ ¼ 510:5 nm emission line of the copper vapor laser has a linewidth of 2:3 GHz,
which is almost entirely caused by Doppler broadening, and a spontaneous radiative
lifetime of τ sp ¼ 500 ns. The refractive index of the low-pressure gaseous medium is
n 1. Find the peak emission cross section of this line.
7.3.1 A large absorption cross section of Ti:sapphire appears at the wavelength of λa ¼ 490 nm
with σ a ðλa Þ ¼ 6:4 1024 m2 , while σ e ðλa Þ 3 1028 m2 . The peak emission cross
section appears at the wavelength of λe ¼ 795 nm with σ e ðλe Þ ¼ 3:4 1023 m2 , while
σ a ðλe Þ 8 1026 m2 . The lower level is the ground level. A Ti:sapphire rod that is not
pumped is found to have an absorption coefficient of αðλa Þ ¼ 200 m1 at λa ¼ 490 nm.
(a) Find the total doping concentration N t of the active Ti3þ ions in this rod.
(b) If a gain coefficient of gðλe Þ ¼ 20 m1 is desired at λe ¼ 795 nm, what percent of the
Ti3þ ions have to be excited to the upper level?
7.3.2 Ti:sapphire has a refractive index of n ¼ 1:76. A Ti:sapphire rod has a length of
l ¼ 10 cm.
(a) When it is not pumped, it has an absorption coefficient of αðλa Þ ¼ 200 m1 at
λa ¼ 490 nm. Find the imaginary part χ 00res of the resonant susceptibility at this
wavelength. If a beam that has a power of Pin ðλa Þ ¼ 1 W at λa ¼ 490 nm is sent into
the rod from one end, what is the output power at the other end? How much of the
power is absorbed?
(b) It is pumped so that it has a gain coefficient of gðλe Þ ¼ 20 m1 at λe ¼ 795 nm. Find
the imaginary part χ 00res of the resonant susceptibility at this wavelength. If a beam that
has a power of Pin ðλe Þ ¼ 1 mW at λe ¼ 795 nm is sent into the rod from one end,
what is the output power at the other end? How much of the power is emitted through
stimulated emission?
7.3.3 Because the lower level of the He–Ne emission line at λ ¼ 632:8 nm is not the ground
level, an unexcited Ne atom does not absorb light at this wavelength. The emission cross
section of this emission line is σ e ¼ 3 1017 m2 . An optical beam at λ ¼ 632:8 nm is
sent through a uniformly pumped He–Ne tube that has a length of l ¼ 1 m. If the output
power is 120% of the input power, what is the population density of the excited Ne atoms
in the upper level of the emission line?
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248
Optical Absorption and Emission
7.3.4 An Er:fiber is doped with an Er3þ ion concentration of N t ¼ 2:2 1024 m3 . It is found
to have an absorption cross section of σ a ¼ 5:7 1025 m2 and an emission cross section
of σ e ¼ 7:9 1025 m2 at the λ ¼ 1:53 μm wavelength. The lower level is the ground
level. Assume uniform pumping throughout the fiber. Assume also that all Er3þ ions are
distributed only between the two levels of the λ ¼ 1:53 μm transition.
(a) What is its intrinsic absorption coefficient α0 at this wavelength when the Er:fiber is
not pumped?
(b) What percent of the Er3þ ions have to be pumped to the upper level for the fiber to be
transparent with α ¼ g ¼ 0?
(c) What percent of the Er3þ ions have to be pumped to the upper level for a gain
coefficient of g ¼ 0:2 m1 ?
(d) What percent of the Er3þ ions have to be pumped to the upper level for a gain
coefficient of g ¼ α0 ?
(e) What is the maximum gain coefficient g max when all Er3þ ions are pumped to the
upper level? Compare it to the intrinsic absorption coefficient α0 , which is the
maximum value of the absorption coefficient.
Bibliography
Davis, C. C., Lasers and Electro-Optics: Fundamentals and Engineering, 2nd edn. Cambridge: Cambridge
University Press, 2014.
Liu, J. M., Photonic Devices. Cambridge: Cambridge University Press, 2005.
Milonni, P. W. and Eberly, J. H., Laser Physics. New York: Wiley, 2010.
Rosencher, E. and Vinter, B., Optoelectronics. Cambridge: Cambridge University Press, 2002.
Saleh, B. E. A. and Teich, M. C., Fundamentals of Photonics. New York: Wiley, 1991.
Siegman, A. E., Lasers. Mill Valley, CA: University Science Books, 1986.
Silfvest, W. T., Laser Fundamentals. Cambridge: Cambridge University Press, 1996.
Svelto, O., Principles of Lasers, 5th edn. New York: Springer, 2010.
Verdeyen, J. T., Laser Electronics, 3rd edn. Englewood Cliffs, NJ: Prentice-Hall, 1995.
Yariv, A. and Yeh, P., Photonics: Optical Electronics in Modern Communications. Oxford: Oxford University
Press, 2007.
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Cambridge Books Online
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Principles of Photonics
Jia-Ming Liu
Book DOI: http://dx.doi.org/10.1017/CBO9781316687109
Online ISBN: 9781316687109
Hardback ISBN: 9781107164284
Chapter
8 - Optical Amplification pp. 249-273
Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.009
Cambridge University Press
8
8.1
Optical Amplification
POPULATION RATE EQUATIONS
..............................................................................................................
From the discussion in the preceding chapter, it is clear that population inversion is the basic
condition for an optical gain. For any system in its normal state in thermal equilibrium, a lowenergy level is always more populated than a high-energy level, hence there is no population
inversion. Population inversion in a system can only be accomplished through a process called
pumping by actively exciting the atoms in a low-energy level to a high-energy level. If left
alone, the atoms in a system relax to thermal equilibrium. Therefore, population inversion is a
nonequilibrium state that cannot be sustained without active pumping. To keep a constant
optical gain, continuous pumping is required to maintain population inversion. This condition is
clearly consistent with the law of conservation of energy: amplification of an optical wave leads
to an increase in optical energy, which is possible only if the required energy is supplied by a
source.
Pumping is the process that supplies energy to the gain medium for the amplification of an
optical wave. There are many different pumping techniques, including optical excitation,
electric current injection, electric discharge, chemical reaction, and excitation with particle
beams. The use of a specific pumping technique depends on the properties of the gain medium
being pumped. The lasers and optical amplifiers of particular interest in photonic systems are
made of either dielectric solid-state media doped with active ions, such as Nd:YAG and Er:
glass fiber, or direct-gap semiconductors, such as GaAs and InP. For a dielectric gain medium,
the most commonly used pumping technique is optical pumping using either an incoherent light
source, such as a flashlamp or a light-emitting diode, or a coherent light source from another
laser. A semiconductor gain medium can also be optically pumped, but it is usually pumped by
electric current injection. In this section, we consider the general conditions for pumping to
achieve population inversion. Detailed pumping mechanisms and physical setups are not
addressed here because they depend on the specific gain medium used in a particular
application.
The net rate of increase of the population density in a given energy level is described by a rate
equation. As we shall see below, pumping for population inversion in any practical gain
medium always requires the participation of more than two energy levels. In general, a rate
equation has to be written for each energy level that is involved in the process. For simplicity
but without loss of validity, however, we shall explicitly write down only the rate equations for
the two energy levels, j2i and j1i, that are directly associated with the resonant transition of
interest. We are not interested in the population densities of other energy levels but only in how
they affect N 2 and N 1 .
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250
Optical Amplification
In the presence of a monochromatic optical wave that has an intensity of I at a frequency of v,
the rate equations that govern the temporal evolution of N 2 and N 1 are
dN 2
N2
I
ðN 2 σ e N 1 σ a Þ,
¼ R2 dt
τ 2 hv
(8.1)
dN 1
N1 N2
I
¼ R1 þ
þ ðN 2 σ e N 1 σ a Þ,
dt
τ 1 τ 21 hv
(8.2)
where R2 and R1 are the total rates of pumping into energy levels j2i and j1i, respectively, and
τ 2 and τ 1 are the fluorescence lifetimes of levels j2i and j1i, respectively. The total rate of
population relaxation, including radiative and nonradiative spontaneous relaxations, from level
j2i to level j1i is τ 1
21 . Because it is possible for the population in level j2i to also relax to other
1
energy levels, the total population relaxation rate of level j2i is τ 1
2 τ 21 . Therefore, in
general, we have
τ 2 τ 21 τ sp :
(8.3)
1
Note that τ 1
21 is not the same as γ21 defined in (7.9): τ 21 is purely the rate of population
relaxation from level j2i to level j1i, whereas γ21 is the rate of phase relaxation of the
polarization associated with the transition between these two levels. For an optical gain
medium, level j2i is known as the upper laser level, and level j1i is known as the lower laser
level. The fluorescence lifetime τ 2 of the upper laser level is an important parameter that
determines the effectiveness of a gain medium. Generally speaking, for a gain medium to be
useful, the upper laser level has to be a metastable state that has a relatively large τ 2 .
To account for the difference between the emission cross section and the absorption cross
section, the effective population inversion can be more accurately defined as
N ¼ N2 σa
N 1:
σe
(8.4)
With this definition for the effective population inversion, the gain coefficient is simply
g ¼ σ e N ¼ α:
(8.5)
This relation is also valid for finding the absorption coefficient. A positive gain coefficient
g > 0 is found when the system reaches effective population inversion so that N > 0; it has a
negative gain coefficient, i.e., a positive absorption coefficient, α ¼ g > 0 when effective
population inversion is not accomplished so that N < 0.
For the different systems discussed in the following section, the two rate equations given in
(8.1) and (8.2) for N 2 and N 1 can be combined into one equation for the effective population
inversion N:
dN
N
I
¼ R β σ e N,
dt
τ2
hv
(8.6)
where R is the effective pumping rate for population inversion and
β ¼1þ
σa
1
σe
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(8.7)
8.2 Population Inversion
251
is the bottleneck factor that characterizes the effectiveness of pumping a system for population
inversion. It is more difficult to reach population inversion in a system that has a larger value of
β. Note that the detailed form of the effective pumping rate R depends on the pumping
mechanism and the pumping scheme. It can be a function of the effective population inversion
N, as in the situation when the gain medium contains a fixed density of active atoms or
molecules. In this case, the pumping rate R cannot be generally taken as an independent
external parameter. However, it is possible in a different situation that the pumping rate can
be taken as an independent external parameter, such as in the case of a semiconductor gain
medium that is pumped by current injection where the pumping rate is determined by the
injection current. In the following section, we consider the case when a gain medium contains a
fixed, finite concentration of active atoms or molecules so that the pumping rate R is a function
of the effective population inversion N.
EXAMPLE 8.1
A Nd:YAG crystal is doped with 1 at.% of Nd3þ ions for a concentration of N t ¼
1:38 1026 m3 . For its λ ¼ 1:064 μm laser line, the emission cross section is found to be σ e ¼
4:5 1023 m2 and the absorption cross section is σ a ¼ 0 because the lower laser level of this
laser line is effectively empty all the time. A ruby crystal is doped with 0.05 wt.% of Cr3þ ions for
a concentration of N t ¼ 1:58 1025 m3 . For its λ ¼ 694:3 nm laser line, the emission cross
section is found to be σ e ¼ 1:34 1024 m2 and the absorption cross section is
σ a ¼ 1:25 1024 m2 . The variations in the measured emission and absorption cross sections
of these gain media are caused by the population ratios in the degenerate states of each laser level,
which vary with doping and temperature. Find the bottleneck factors for these two laser media.
Solution:
The bottleneck factor of this Nd:YAG crystal at λ ¼ 1:064 μm is
β ¼1þ
σa
0
¼1þ
¼ 1:
σe
4:5 1023
The bottleneck factor of this ruby crystal at λ ¼ 694:3 nm is
β ¼1þ
σa
1:25 1024
¼1þ
¼ 1:93:
σe
1:34 1024
The λ ¼ 1:064 μm laser line of Nd:YAG has the smallest possible bottleneck factor of β ¼ 1
because σ a ¼ 0. The λ ¼ 694:3 nm laser line of ruby has a bottleneck factor of β ¼ 1:93, which
is close to 2, because σ a is comparable to σ e .
8.2
POPULATION INVERSION
..............................................................................................................
Population inversion between the upper laser level j2i of a degeneracy g2 and the lower laser
level j1i of a degeneracy g1 in a medium is generally defined as
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252
Optical Amplification
N2 N1
g
g
>
so that N 1 < 1 N 2 and N 2 > 2 N 1 :
g2
g1
g2
g1
(8.8)
According to (7.45) and (7.46), this condition makes αðvÞ < 0 and g ðvÞ > 0 so that the medium
shows a positive optical gain. However, in many systems, the degenerate states in level j1i or
j2i, or both, are split into closely spaced sublevels to form small energy bands. When the
energy spread of the sublevels in a laser level is sufficiently large, the population in the level
can be distributed unevenly so that (7.39) is not valid, i.e., σ a ðvÞ 6¼ ðg2 =g1 Þσ e ðvÞ. In this
situation, the second equal sign in (7.45) and (7.46) is not valid though the first equal sign is
still valid:
g1
(8.9)
αðvÞ ¼ N 1 σ a ðvÞ N 2 σ e ðvÞ 6¼ N 1 N 2 σ a ðvÞ
g2
and
g ðvÞ ¼ N 2 σ e ðvÞ N 1 σ a ðvÞ 6¼
g2
N 2 N 1 σ e ðvÞ:
g1
(8.10)
For this reason, when the condition for population inversion given in (8.8) is achieved in a
medium, we might find σ a ðvÞ ðg2 =g1 Þσ e ðvÞ for an optical gain at an optical frequency v while
at the same time we might find σ a ðv0 Þ > ðg2 =g1 Þσ e ðv0 Þ for an optical loss at another frequency
v0 . Therefore, the population inversion condition in (8.8) does not guarantee an optical gain at a
particular optical frequency v in the case when the population in level j1i or j2i is distributed
unevenly among its sublevels so that σ a ðvÞ 6¼ ðg2 =g1 Þσ e ðvÞ.
What really matters to an optical wave at a given frequency is the optical gain at that specific
frequency. For this reason, in the following discussion, we shall consider, instead of the
condition in (8.8), the condition that guarantees an optical gain at the frequency v,
g ðvÞ ¼ N 2 σ e ðvÞ N 1 σ a ðvÞ ¼ Nσ e ðvÞ > 0,
(8.11)
as the effective condition for population inversion as far as an optical signal at the frequency v is
concerned. Clearly, by defining the effective population inversion N as in (8.4), the effective
condition for population inversion is simply N > 0. This population inversion condition can be
reached even when N 2 < N 1 in the case when σ a < σ e . On the other hand, if σ a > σ e , it is
possible that N 2 > N 1 but N 2 is not sufficiently large so that N < 0 and effective population
inversion for an optical gain is not reached.
The pumping requirement for the condition in (8.11) to be satisfied depends on the properties of
a medium. For atomic and molecular media, there are three different basic systems. Each has a
different pumping requirement to reach effective population inversion for an optical gain. The
pumping requirement can be found by solving the coupled rate equations given in (8.1) and (8.2).
EXAMPLE 8.2
Use the parameters given in Example 8.1 to find the effective population inversion required to
have a gain coefficient of g ¼ 10 m1 for the λ ¼ 1:064 μm laser line of Nd:YAG and that
required for the λ ¼ 694:3 nm laser line of ruby.
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8.2 Population Inversion
253
Solution:
For the λ ¼ 1:064 μm laser line of Nd:YAG, σ e ¼ 4:5 1023 m2 . Therefore, the required
effective population inversion is
g
10
¼
m3 ¼ 2:22 1023 m3 :
23
σ e 4:5 10
For the λ ¼ 694:3 nm laser line of ruby, σ e ¼ 1:34 1024 m2 . Therefore, the required effective population inversion is
N¼
g
10
¼
m3 ¼ 7:46 1024 m3 :
σ e 1:34 1024
For the same gain coefficient, the population inversion required for the ruby laser line is about
34 times that required for the Nd:YAG laser line because the emission cross section of the
Nd:YAG laser line is about 34 times that of the ruby laser line.
N¼
8.2.1 Two-Level System
When the only energy levels involved in the pumping and the relaxation processes are the upper
and lower laser levels, j2i and j1i, the system can be considered as a two-level system, as
shown in Fig. 8.1. In such a system, level j1i is the ground level, which has τ 1 ¼ ∞, and level
j2i relaxes only to level j1i, so that τ 21 ¼ τ 2 . The total population density is N t ¼ N 1 þ N 2 .
While a pumping mechanism excites atoms from the lower laser level to the upper laser level
of a two-level system, the same pump also stimulates atoms in the upper laser level to relax to
the lower laser level. Therefore, irrespective of the specific pumping technique used, it is
always true that R2 ¼ R1 ¼ W p12 N 1 W p21 N 2 , where W p12 and W p21 are the pumping transition
probability rates, or simply the pumping rates, from j1i to j2i and from j2i to j1i, respectively.
Under these conditions, (8.1) and (8.2) are equivalent to each other. The upward and downward
pumping rates are not independent of each other but are directly proportional to each other
because both are associated with the interaction between the same pump source and a given pair
of energy levels. We take the upward pumping rate to be W p12 ¼ W p and the downward
Figure 8.1 (a) Pumping scheme of a true two-level system. (b) Pumping scheme of a quasi-two-level system.
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254
Optical Amplification
pumping rate to be W p21 ¼ pW p , where p is a constant that depends on the detailed characteristics of the two-level atomic system and the pump source. In the steady state when
dN 2 =dt ¼ dN 1 =dt ¼ 0 , we then find that
g ¼ N 2σe N 1σa ¼
W p τ 2 ðσ e pσ a Þ σ a
Nt:
1 þ ð1 þ pÞW p τ 2 þ ðIτ 2 =hvÞðσ e þ σ a Þ
Using the relation in (7.43), we find that, in the case of optical pumping,
W p21 σ pe σ e λp
p ¼ p ¼ p ¼ ,
W 12 σ a σ a λp
(8.12)
(8.13)
where σ pa and σ pe are the absorption and emission cross sections, respectively, at the pump
wavelength.
In a true two-level system, shown in Fig. 8.1(a), the energy levels j2i and j1i can respectively
be degenerate with degeneracies g2 and g1 , but the population density in each level is evenly
distributed among the degenerate states in the level. In this situation, p ¼ σ pe =σ pa
¼ g1 =g2 ¼ σ e =σ a . Then, we find from (8.12) that
g ¼ N 2σe N 1σa ¼
σ a
N t < 0:
1 þ Iτ 2 =hv þ W p τ 2 =σ a ðσ e þ σ a Þ
(8.14)
No matter how a true two-level system is pumped, it is clearly not possible to achieve population
inversion for an optical gain in the steady state. This situation can be understood by considering
the fact that the pump for a two-level system has to be in resonance with the transition between the
two levels, thus simultaneously inducing downward and upward transitions. In the steady state,
the two-level system reaches thermal equilibrium with the pump at a finite temperature, resulting
in a Boltzmann population distribution of the form given in (7.28) without population inversion.
As discussed above and illustrated in Fig. 8.1(b), however, in many systems an energy level
is actually split into a band of closely spaced, but not exactly degenerate, sublevels with its
population density unevenly distributed among these sublevels. This type of system is not a true
two-level system, but is known as a quasi-two-level system, if either or both of the two levels
are split in such a manner. By properly pumping a quasi-two-level system, it is possible to reach
the needed population inversion in the steady state for an optical gain at a particular laser
frequency v because the ratio p ¼ σ pe =σ pa at the pump frequency vp can now be made different
from the ratio σ e =σ a at the laser frequency v due to the uneven population distribution among
the sublevels within an energy level. From (8.12), we find that the pumping requirements for a
quasi-two-level system to have a steady-state optical gain are
p¼
σ pe σ e
σa
and W p >
:
p <
τ 2 ðσ e pσ a Þ
σa σa
(8.15)
Because the absorption spectrum is generally shifted to the short-wavelength side of the
emission spectrum, these conditions can be satisfied by pumping sufficiently strongly at a
higher transition energy than the photon energy at the peak of the emission spectrum. In the
case of optical pumping, this condition means that the pump wavelength has to be shorter than
the emission wavelength. Figure 8.1(b) illustrates such a pumping scheme for a quasi-two-level
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8.2 Population Inversion
255
system. Indeed, many laser gain media, including laser dyes, semiconductor gain media, and
vibronic solid-state gain media, are often pumped as a quasi-two-level system.
8.2.2 Three-Level System
Population inversion in the steady state is possible for a system that has three energy levels
involved in the process. Figure 8.2 shows the energy-level diagram of an idealized three-level
system. The lower laser level j1i is the ground level, E 1 ¼ E 0 , or is very close to the ground
level, within an energy separation of ΔE10 ¼ E 1 E0 k B T from the ground level, so that it is
initially populated. The atoms are pumped to an energy level j3i above the upper laser level j2i.
An effective three-level system satisfies the following conditions.
1. Population relaxation from level j3i to level j2i is very fast and efficient, ideally
τ 2 τ 32 τ 3 , so that the atoms excited by the pump quickly end up in level j2i.
2. Level j3i lies sufficiently high above level j2i with ΔE32 ¼ E 3 E2 k B T so that the
population in level j2i cannot be thermally excited back to level j3i.
3. The lower laser level j1i is the ground level, or its population relaxes very slowly if it is not the
ground level, so that τ 1 ∞. Furthermore, level j2i relaxes mostly to level j1i so that τ 21 τ 2 .
Under these conditions, R2 W p N 1 , R1 W p N 1 , and N 1 þ N 2 N t . The parameter W p is
the effective pumping rate of exciting atoms in the ground level to eventually reach the upper
laser level. It is proportional to the pump power.
In the steady state under constant pumping, W p is a constant and dN 2 =dt ¼ dN 1 =dt ¼ 0.
With these conditions, we find that
g ¼ N 2 σe N 1 σa ¼
W pτ2σe σa
Nt:
1 þ W p τ 2 þ ðIτ 2 =hvÞðσ e þ σ a Þ
(8.16)
Therefore, the pumping requirement for a positive optical gain under steady-state population
inversion is
Wp >
σa
:
τ2σe
(8.17)
This condition sets the minimum pumping requirement for a three-level system to have a
positive optical gain. This requirement can be understood by considering the fact that almost
Figure 8.2 Energy levels of a three-level
system.
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256
Optical Amplification
all of the population initially resides in the lower laser level j1i. To achieve effective population
inversion, the pump has to be strong enough to sufficiently depopulate level j1i while the
system has to be able to keep the excited atoms in level j2i. In the case when σ a ¼ σ e , for a
bottleneck factor of β ¼ 2, no population inversion occurs before at least half of the total
population is transferred from level j1i to level j2i. This is the bottleneck effect that limits the
energy conversion efficiency of a three-level laser system as compared to a quasi-two-level or
four-level system.
8.2.3 Four-Level System
A four-level system, shown schematically in Fig. 8.3, is more efficient than a three-level
system. A four-level system differs from a three-level system in that the lower laser level j1i
lies sufficiently high above the ground level j0i with ΔE10 ¼ E 1 E0 k B T so that in thermal
equilibrium the population in level j1i is negligibly small compared to that in level j0i.
Pumping takes place from level j0i to level j3i.
An effective four-level system also has to satisfy the conditions concerning levels j3i and j2i
discussed above for an effective three-level system. In addition, it has to satisfy the condition
that the population in level j1i relaxes very quickly to the ground level, ideally τ 1 τ 10 τ 2 ,
so that level j1i remains relatively unpopulated in comparison to level j2i when the system is
pumped. Under these conditions, N 1 0 and R2 W p ðN t N 2 Þ, where the effective pumping
rate W p is again proportional to the pump power. Because N 1 0, (8.2) can be ignored. For a
four-level system, we can also take σ a ¼ 0, for a bottleneck factor of β ¼ 1, because its
absorption coefficient at the laser wavelength is zero even when it is not pumped.
In the steady state with a constant W p , we find by taking dN 2 =dt ¼ 0 for (8.1) and taking
σ a ¼ 0 that
g ¼ N 2σe ¼
W pτ2σe
Nt:
1 þ W p τ 2 þ σ e Iτ=hv
(8.18)
This result indicates that there is no minimum pumping requirement for an ideal four-level
system that satisfies the conditions discussed above. No bottleneck effect limits an ideal fourlevel system because level j1i is initially empty in such a system. Real systems are rarely ideal,
but a practical four-level system is still much more efficient than a three-level system.
Figure 8.3 Energy levels of a four-level
system.
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8.2 Population Inversion
257
8.2.4 Transparency
When the gain coefficient is zero, g ¼ 0, the medium becomes transparent, or bleached, to the
optical signal, neither absorbing nor amplifying it. An ideal four-level system is transparent at
no pumping. A quasi-two-level or three-level system reaches transparency, or the bleached
condition, at the transparency pumping rate:
W trp ¼
σa
β1
¼
,
τ 2 ðσ e pσ a Þ τ 2 ½1 pðβ 1Þ
(8.19)
where β is the bottleneck factor defined in (8.7). This relation is valid for all systems though it is
obtained for a two-level or three-level system. For a four-level system, we simply find from
(8.19) that W trp ¼ 0 because σ a ¼ 0 and β ¼ 1 for the system. For a system to have an optical
gain, the pumping rate has to be higher than the transparency pumping rate: W p > W trp . For a
four-level system, any pumping leads to a gain because it is always true that W p > W trp ¼ 0 as
long as the system is pumped. For a two-level or three-level system, which has σ a 6¼ 0 so that
β > 1, it is possible for the system to have no optical gain but optical attenuation when it is not
sufficiently pumped such that W trp > W p > 0.
The relation in (8.19) gives the necessary pumping effort for a system to reach transparency
and then an optical gain above it. Another useful measure is the population density N 2 that has
to be pumped to the upper laser level in order for a system to have an optical gain. For a twolevel or three-level system, N 1 þ N 2 N t . By simultaneously solving N 1 þ N 2 N t and
N 2 σ e N 1 σ a ¼ g, the population of the upper laser level is found:
σaN t þ g
1
N
N2 ¼
¼ 1 Nt þ :
(8.20)
σe þ σa
β
β
Though this relation is obtained by using N 1 þ N 2 N t , which is not valid for a four-level
system, the relation is still valid for a four-level system because it reduces to N 2 ¼ g=σ e in the
case of a four-level system, for which σ a ¼ 0. Therefore, this relation is valid for all systems.
The relation given in (8.20) is valid for any valid value of g, which can be positive, zero, or
negative. In the case of a four-level system, it is always true that g 0. In the case of a quasi-twolevel or three-level system, g ¼ α < 0 when the medium is not sufficiently pumped to reach
transparency. Because the maximum value of the absorption coefficient for a two-level or threelevel system is α0 ¼ σ a N t while α0 g α0 σ e =σ a , we find from (8.20) that N 2 0 for
any values of g, including g < 0 when the system has a positive absorption coefficient of α ¼
g > 0 for optical attenuation, g ¼ 0 when the system neither attenuates nor amplifies the optical
signal, and g > 0 when the system has a positive gain coefficient for optical amplification.
Because g ¼ 0 and N ¼ 0 at transparency, the transparency population density for the upper
laser level is obtained from (8.20) as
σa
1
tr
N2 ¼
N t ¼ 1 Nt:
(8.21)
σe þ σa
β
Population inversion with N > 0 for a positive optical gain of g > 0 is reached when N 2 > N tr2
so that the system is above transparency. Clearly, the bottleneck factor gives a measure of the
ease or difficulty in reaching the transparency point. For a four-level system, such as the
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258
Optical Amplification
Nd:YAG laser, β ¼ 1 because σ a ¼ 0; thus N tr2 ¼ 0. In this situation, any population density N 2
pumped to the upper laser level contributes to an optical gain even when most of the active
atoms remain in the ground level, which is not the lower laser level of the system. For a twolevel or three-level system, β > 1; thus N tr2 > 0. In this situation, a population density of
N 2 > N tr2 > 0 in the upper laser level is required for the system to have an optical gain, and
it increases with the value of β. In many three-level systems, such as the ruby laser, the value of
β is close to 2; in this situation, about half of all active atoms have to be pumped to the upper
laser level before the system can have any optical gain. In some quasi-two-level systems,
however, the value of β is close to 1 though larger than 1; then it is relatively easy, though not
as easy as for a four-level system, for the system to reach population inversion for a positive
optical gain.
EXAMPLE 8.3
Consider the Nd:YAG and ruby crystals that have the parameters given in Example 8.1. Find the
population density of the upper laser level required for the Nd:YAG crystal to reach transparency at its λ ¼ 1:064 μm laser line and that required for the ruby crystal to reach transparency at
its λ ¼ 694:3 nm laser line. What percent of all active ions are excited in each case?
Solution:
For the Nd:YAG crystal, we have β ¼ 1 and N t ¼ 1:38 1026 m3 from Example 8.1. The
population density of the upper laser level required for the Nd:YAG crystal to reach transparency at its λ ¼ 1:064 μm laser line is found using (8.21) to be
1
1
tr
N2 ¼ 1 Nt ¼ 1 1:38 1026 m3 ¼ 0:
β
1
The percentage of all active ions that are excited to the upper laser level is 0%.
For the ruby crystal, we have β ¼ 1:93 and N t ¼ 1:58 1025 m3 from Example 8.1. The
population density of the upper laser level required for the ruby crystal to reach transparency at
its λ ¼ 694:3 nm laser line is found using (8.21) to be
1
1
tr
N2 ¼ 1 Nt ¼ 1 1:58 1025 m3 ¼ 7:61 1024 m3 :
β
1:93
The percentage of all active ions that are excited to the upper laser level is
N tr2 7:61 1024
¼
¼ 48%:
N t 1:58 1025
We find that no active ions have to be excited for the Nd:YAG crystal to reach the
transparency point because it is a four-level system that has a bottleneck factor of β ¼ 1. By
comparison, as many as 48% of all active ions have to be excited to the upper laser level for the
ruby crystal to reach transparency because it is a three-level system that has a large bottleneck
factor of β ¼ 1:93.
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8.3 Optical Gain
8.3
259
OPTICAL GAIN
..............................................................................................................
When the condition in (8.11) is satisfied for a system, an optical gain coefficient at a specific
optical frequency v can be found as g ðvÞ ¼ N 2 σ e ðvÞ N 1 σ a ðvÞ. The optical gain coefficient is a
function of the optical signal intensity, I, as a result of the dependence of N 2 and N 1 on I due to
stimulated emission, which changes the population densities by causing downward transitions
from level j2i to level j1i. This effect causes saturation of the optical gain coefficient by the
optical signal. For all three basic systems discussed above, the optical gain coefficient can be
expressed as a function of the optical signal intensity I:
g¼
g0
,
1 þ I=I sat
(8.22)
where g 0 is the unsaturated gain coefficient, which is independent of the optical signal
intensity, and I sat is the saturation intensity of a medium, which can be generally
expressed as
I sat ¼
hv
:
τsσe
(8.23)
The time constant τ s is an effective saturation lifetime of the population inversion. It can be
considered as an effective decay time constant for the optical gain coefficient through the
relaxation of the effective population inversion. Both g 0 and τ s are functions of the intrinsic
properties of a gain medium, as well as of the pumping rate. They can be found from (8.12),
(8.16), and (8.18) for the quasi-two-level, three-level, and four-level systems, respectively. The
results are summarized below.
Quasi-two-level system:
g0 ¼ W pτsσe σa N t,
τs ¼ τ2
Three-level system:
1 þ σ a =σ e
:
1 þ ð1 þ pÞW p τ 2
g0 ¼ W pτsσe σa N t,
τs ¼ τ2
1 þ σ a =σ e
:
1 þ W pτ2
(8.24)
(8.25)
(8.26)
(8.27)
Four-level system:
g0 ¼ W pτsσeN t,
(8.28)
τ2
:
1 þ W pτ2
(8.29)
τs ¼
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260
Optical Amplification
The minimum pumping requirement for a medium to have an optical gain is clearly g 0 > 0.
This is the condition for reaching transparency discussed in Section 8.2. For a desired unsaturated gain coefficient of g 0 , the required pumping rate can be found by solving (8.24) and (8.25)
for a quasi-two-level system, (8.26) and (8.27) for a three-level system, and (8.28) and (8.29)
for a four-level system. The results are summarized below.
Quasi-two-level system:
Wp ¼
1
σaN t þ g0
:
τ 2 ðσ e pσ a ÞN t ð1 þ pÞg 0
(8.30)
Wp ¼
1 σaN t þ g0
:
τ2 σeN t g0
(8.31)
Wp ¼
1
g0
:
τ2 σeN t g0
(8.32)
Three-level system:
Four-level system:
The different forms of unsaturated gain coefficient g 0 and saturation lifetime τ s found above
for different systems can be expressed in a general form for all systems by using the parameter p
and the bottleneck factor β to account for the differences among the systems. Meanwhile, the
required pumping rate for an unsaturated gain coefficient of g 0 can be found expressed in a
general form for all systems. They are given below.
General forms for all systems:
g0 ¼ W pτs þ 1 β σeN t,
(8.33)
β
,
1 þ ð1 þ pÞW p τ 2
(8.34)
1
ðβ 1Þσ e N t þ g 0
:
τ 2 ½1 pðβ 1Þ σ e N t ð1 þ pÞg 0
(8.35)
τs ¼ τ2
Wp ¼
For a quasi-two-level system, p 0 and β 1. When using a specific quasi-two-level system, it
is desirable to make p as small as possible by properly choosing the pumping parameters and it
is desirable to make β as close to unity as possible by properly choosing the laser emission
wavelength. For a three-level system, p ¼ 0 and β > 1; the value of β is usually close to 2 for
the typical three-level system, but it can be less than 2 or sometimes greater than 2. The large
bottleneck factor makes a three-level system inefficient, as discussed in Section 8.2. For a
four-level system, p ¼ 0 and β ¼ 1, making the system most efficient in pumping for an
optical gain.
In the limit when p ! 0, a quasi-two-level system is identical to a three-level system. In the
limit when p ! 0 and σ a ! 0 ðβ ! 1Þ, a quasi-two-level system behaves like a four-level
system. In the limit when σ a ! 0 ðβ ! 1Þ, a three-level system behaves like a four-level
system.
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8.3 Optical Gain
261
EXAMPLE 8.4
The Nd:YAG laser crystal described in Example 8.1 has τ 2 ¼ 240 μs for its λ ¼ 1:064 μm laser line.
The ruby laser crystal described in Example 8.1 has τ 2 ¼ 3 ms for its λ ¼ 694:3 nm laser line. (a)
Find the pumping rates for the λ ¼ 1:064 μm Nd:YAG laser line to reach transparency and to have an
unsaturated gain coefficient of g 0 ¼ 10 m1 , respectively. What are the saturation lifetime and the
saturation intensity in each case? (b) Answer the same questions for the λ ¼ 694:3 nm ruby laser line.
Solution:
The two laser media belong to different systems and have different parameters.
(a) The Nd:YAG at λ ¼ 1:064 μm is a four-level system with σ e ¼ 4:5 1023 m2 and σ a ¼ 0.
The doping density is N t ¼ 1:38 1026 m3 . The photon energy is
hv ¼
1:2398
eV ¼ 1:165 eV:
1:064
Using (8.32), (8.29), and (8.23) for a four-level system, we find the pumping rate, the
saturation lifetime, and the saturation intensity for g 0 ¼ 0 at transparency to be
W trp ¼ 0,
τ trs ¼ τ 2 ¼ 240 μs,
I trsat ¼
hv
1:165 1:6 1019
¼
W m2 ¼ 17:3 MW m2 :
6
23
tr
τ s σ e 240 10 4:5 10
The parameters for an unsaturated gain coefficient of g 0 ¼ 10 m1 are
Wp ¼
1
g0
1
10
s1 ¼ 6:72 s1 ,
¼
τ 2 σ e N t g 0 240 106 4:5 1023 1:38 1026 10
τs ¼
I sat ¼
τ2
240 106
¼
μs ¼ 239:6 μs,
1 þ W p τ 2 1 þ 6:72 240 106
hv
1:165 1:6 1019
¼
W m2 ¼ 17:3 MW m2 :
τ s σ e 239:6 106 4:5 1023
(b) The ruby at λ ¼ 694:3 nm is a three-level system with σ e ¼ 1:34 1024 m2 and σ a ¼
1:25 1024 m2 . The doping density is N t ¼ 1:58 1025 m3 . The photon energy is
hv ¼
1239:8
eV ¼ 1:786 eV:
694:3
Using (8.31), (8.27), and (8.23) for a three-level system, we find the pumping rate, the
saturation lifetime, and the saturation intensity for g 0 ¼ 0 at transparency to be
W trp ¼
1 σa
1
1:25 1024 1
¼
s ¼ 311 s1 ,
τ 2 σ e 3 103 1:34 1024
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262
Optical Amplification
τ trs ¼ τ 2
I trsat ¼
1 þ σ a =σ e
¼ τ 2 ¼ 3 ms,
1 þ W trp τ 2
hv
1:786 1:6 1019
¼
W m2 ¼ 71:1 MW m2 :
τ trs σ e 3 103 1:34 1024
The parameters for an unsaturated gain coefficient of g 0 ¼ 10 m1 are
Wp ¼
1 σaN t þ g0
1
1:25 1024 1:58 1025 þ 10 1
¼
s ¼ 888 s1 ,
τ 2 σ e N t g 0 3 103 1:34 1024 1:58 1025 10
τs ¼ τ2
I sat ¼
1 þ σ a =σ e
1 þ 1:25=1:34
¼ 3 103 s ¼ 1:58 ms,
1 þ W pτ2
1 þ 888 3 103
hv
1:786 1:6 1019
¼
W m2 ¼ 139:4 MW m2 :
τ s σ e 1:58 103 1:34 1024
EXAMPLE 8.5
The Nd:YAG crystal considered in Example 8.4 can be optically pumped with an absorption
cross section of σ pa ¼ 3:0 1024 m2 at the λp ¼ 808 nm pump wavelength, whereas the ruby
crystal considered in Example 8.4 can be optically pumped with an absorption cross section of
σ pa ¼ 2:0 1023 m2 at the λp ¼ 554 nm pump wavelength. Assume a 100% pump quantum
efficiency for the following questions. (a) Find the required pump intensities at λp ¼ 808 nm to
pump the λ ¼ 1:064 μm Nd:YAG laser line to transparency and to have an unsaturated gain
coefficient of g 0 ¼ 10 m1 , respectively. (b) Find the required pump intensities at λp ¼ 554 nm
to pump the λ ¼ 694:3 nm ruby laser line to transparency and to have an unsaturated gain
coefficient of g 0 ¼ 10 m1 , respectively.
Solution:
The pumping transition probability rate W p determines the number per second of active atoms
excited by the pump to the upper laser level. If the pump has a pump quantum efficiency of ηp
when N p pump photons are absorbed, only ηp N p atoms are excited. Thus, the required pump
intensity for a pumping transition probability rate of W p is
Ip ¼
1 hvp
W p:
ηp σ pa
With ηp ¼ 1 assumed in this example, we have
Ip ¼
hvp W p
:
σ pa
(a) For the Nd:YAG crystal, λp ¼ 808 nm and σ pa ¼ 3:0 1024 m2 . The pump photon energy is
hvp ¼
1239:8
eV:
808
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8.3 Optical Gain
263
From Example 8.4, the transparency pumping rate is W trp ¼ 0 and the pumping rate for
g 0 ¼ 10 m1 is W p ¼ 6:72 s1 . Therefore, the required pump intensity for transparency is
I trp
¼
hvp W trp
σ pa
¼ 0,
and that for g 0 ¼ 10 m1 is
Ip ¼
hvp W p 1239:8
6:72
¼
W m2 ¼ 550 kW m2 :
1:6 1019 p
24
808
σa
3:0 10
(b) For the ruby crystal, λp ¼ 554 nm and σ pa ¼ 2:0 1023 m2 . The pump photon energy is
hvp ¼
1239:8
eV:
554
From Example 8.4, the transparency pumping rate is W trp ¼ 311 s1 and the pumping rate for
g 0 ¼ 10 m1 is W p ¼ 888 s1 . Therefore, the required pump intensity for transparency is
I trp ¼
hvp W trp
σ pa
¼
1239:8
311
W m2 ¼ 5:57 MW m2 ,
1:6 1019 554
2:0 1023
and that for g 0 ¼ 10 m1 is
Ip ¼
hvp W p 1239:8
888
¼
W m2 ¼ 15:9 MW m2 :
1:6 1019 p
554
σa
2:0 1023
8.3.1 Unsaturated Gain
The unsaturated gain coefficient g 0 is also known as the small-signal gain coefficient because it
is the gain coefficient of a weak optical signal that does not saturate the gain medium. At
transparency, g 0 ¼ 0 because g ¼ 0. For a four-level system, g 0 > 0 as long as the medium is
pumped because there is no minimum pumping requirement for transparency. For a quasi-twolevel or three-level system, g 0 > 0 only when the pumping level exceeds its minimum pumping
requirement for transparency; below that, the medium has absorption because g 0 < 0.
It can be seen from (8.24)–(8.29) that for any system, g 0 increases with pump power less than
linearly because τ s decreases with the pump power though W p is linearly proportional to the
pump power. This dependence of τ s on the pump power is caused by the fact that as the pump
excites atoms from the ground state to any excited state to eventually reach the upper laser
level, it depletes the population in the ground level. Consequently, as the pump power
increases, fewer atoms remain available for excitation in the ground level, thus reducing the
differential increase of the effective population inversion with respect to the increase of the
pump power.
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264
Optical Amplification
8.3.2 Gain Saturation
The optical gain coefficient is a function of the intensity of the optical wave that propagates in
the gain medium; it decreases as the optical signal intensity increases. According to (8.22), the
optical gain coefficient g is reduced to half of the unsaturated gain coefficient g 0 when the
optical signal intensity reaches the saturation intensity such that I ¼ I sat . The smaller the value
of I sat , the easier it is for the gain to be saturated. For a quasi-two-level system,
τ s ¼ τ 2 ð1 pσ a =σ e Þ at transparency. For a three-level or four-level system, τ s ¼ τ 2 at transparency. For all three systems, τ s < τ 2 when the gain medium is pumped above transparency for a
positive gain coefficient. Therefore, I sat increases as the gain medium is pumped harder for a
larger unsaturated gain coefficient.
EXAMPLE 8.6
The Nd:YAG laser crystal considered in Example 8.4 has a saturation intensity of
I sat ¼ 17:3 MW m2 when it is pumped to have an unsaturated gain coefficient of
g 0 ¼ 10 m1 at λ ¼ 1:064 μm. The ruby laser crystal also considered in Example 8.4 has a
saturation intensity of I sat ¼ 139:4 MW m2 when it is pumped to have an unsaturated gain
coefficient of g 0 ¼ 10 m1 at λ ¼ 694:3 nm. Two Gaussian laser beams of the same power
of P ¼ 1:5 W at these two wavelengths are both collimated to have the same spot size of
w0 ¼ 300 μm in each crystal. Find the saturated gain coefficient for each crystal when the
beam at the respective wavelength is sent through each crystal.
Solution:
Each Gaussian beam has a cross-sectional area of
2
πw20 π 300 106
A¼
¼
m2 ¼ 1:4 107 m2 :
2
2
The peak intensity of each beam is
I¼
P
1:5
W m2 ¼ 10:7 MW m2 :
¼
A 1:4 107
For the Nd:YAG laser crystal, the saturated gain coefficient is
g¼
g0
¼
1 þ I=I sat
10
m1 ¼ 6:18 m1 :
10:7
1þ
17:3
For the ruby laser crystal, the saturated gain coefficient is
g¼
g0
¼
1 þ I=I sat
10
m1 ¼ 9:29 m1 :
10:7
1þ
139:4
The gain coefficient of the Nd:YAG laser line is more saturated than that of the ruby laser line
because the saturation intensity of the Nd:YAG laser line is lower than that of the ruby laser line.
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8.4 Optical Amplification
8.4
265
OPTICAL AMPLIFICATION
..............................................................................................................
Any medium that has an optical gain can be used to amplify an optical signal. Depending on the
physical mechanism that is responsible for the optical gain, there are two different categories of
optical amplifiers: nonlinear optical amplifiers and laser amplifiers. The optical gain of a
nonlinear optical amplifier originates from a nonlinear optical process in a nonlinear medium,
whereas the gain of a laser amplifier results from the population inversion in a gain medium as
discussed in the preceding section.
Ignoring the effect of noise, the amplification of the intensity, I s , of an optical signal
propagating in the z direction through a laser amplifier can be described by
dI s
g 0 ðzÞ
I s,
¼ gI s ¼
dz
1 þ I s =I sat
(8.36)
where g 0 ðzÞ is the unsaturated gain coefficient and I sat is the saturation intensity of the gain
medium, both defined in the preceding section. Here we assume transverse uniformity but consider
the possibility of longitudinal nonuniformity by taking the unsaturated gain coefficient g 0 ðzÞ to be a
function of z. Such a longitudinally nonuniform gain distribution is a common scenario for an
amplifier under longitudinal optical pumping because of pump absorption by the gain medium.
In the following discussion, we assume for simplicity that the signal beam is collimated
throughout the length of the amplifier such that its divergence is negligible. This assumption
allows us to express (8.36) in terms of the power, Ps , of the optical signal as
dPs
g 0 ðzÞ
¼ gPs ¼
Ps ,
dz
1 þ Ps =Psat
(8.37)
where Psat is the saturation power obtained by integrating I sat over the cross-sectional area of
the signal beam. By integrating (8.37), the following relation is obtained:
ðz
Ps ðzÞ
Ps ðzÞ Ps ð0Þ
exp
(8.38)
¼ exp g 0 ðzÞdz,
Ps ð0Þ
Psat
0
where Ps ð0Þ is the power of the signal beam at z ¼ 0. When Ps Psat , the power of the optical
signal grows exponentially with distance. The growth slows down as Ps approaches the value of
Psat . Eventually, the signal grows only linearly with distance when Ps Psat .
The power gain of a signal is defined as
Pout
s
,
(8.39)
Pin
s
out
where Pin
s and Ps are the input and output powers of the signal, respectively. By using the
relation in (8.38) while identifying Pout
and Pin
s
s with Ps ðlÞ and Ps ð0Þ, respectively, for an
amplifier that has a length of l, an implicit relation is found for the power gain of the signal:
Pin
s
G ¼ G0 exp ð1 GÞ
,
(8.40)
Psat
G¼
where G0 is the unsaturated power gain, or the small-signal power gain. For a single pass
through the amplifier, G0 is given by
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266
Optical Amplification
Figure 8.4 Gain, normalized to the unsaturated gain as G=G0 , of a laser amplifier as a function of the input signal
power, normalized to the saturation power as Pin
s =Psat , for different values of the unsaturated power gain G0 .
ðl
G0 ¼ exp g 0 ðzÞdz:
(8.41)
0
Note that, according to (8.40), G0 G > 1 because g 0 > 0 for an amplifier. For a small optical
out
signal such that Pin
s < Ps Psat , the power gain is simply the small-signal power gain so that
G ¼ G0 . If the signal power approaches or even exceeds the saturation power of the amplifier,
the relation in (8.40) clearly indicates that G < G0 because of gain saturation. In this situation,
the overall gain G can be found by solving (8.40) when the values of Pin
s and Psat , as well as that
of G0 , are given. Figure 8.4 shows the amplifier gain as a function of the input signal power for
a few different values of the unsaturated power gain G0 .
EXAMPLE 8.7
A Nd:YAG laser rod and a ruby laser rod with the properties described in the preceding
examples both have a length of l ¼ 10 cm and a cross-sectional diameter of d ¼ 6 mm. The
refractive index of Nd:YAG is 1.82, and that of ruby is 1.76. Each is uniformly pumped to
have an unsaturated gain coefficient of g 0 ¼ 10 m1 at its laser wavelength, λ ¼ 1:064 μm
for Nd:YAG and λ ¼ 694:3 nm for ruby. The saturation intensities at g 0 ¼ 10 m1 are
found in Example 8.4 to be I YAG
¼ 1:73 MW m2 for the Nd:YAG laser line and
sat
2
I ruby
for the ruby laser line. Two collimated Gaussian signal beams at the
sat ¼ 139:4 MW m
two laser wavelengths that have the same spot size of w0 ¼ 400 μm in the rod and the same
power of Pin
s ¼ 5 W are respectively sent through the Nd:YAG and ruby rods for amplification.
What are the output signal powers from the Nd:YAG and ruby amplifiers, respectively?
Solution:
The primary difference between the Nd:YAG amplifier and the ruby amplifier is their different
saturation intensities. Because their signal wavelengths are different, the two Gaussian beams
have different Rayleigh ranges when their spot sizes are the same. With w0 ¼ 400 μm, the
Rayleigh ranges of the two beams are
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8.5 Spontaneous Emission
267
2
πnw20 π 1:82 400 106
zR ¼
m ¼ 86 cm for λ ¼ 1:064 μm,
¼
λ
1:064 106
2
πnw20 π 1:76 400 106
¼
m ¼ 1:27 m for λ ¼ 694:3 nm:
zR ¼
λ
694:3 109
Both Rayleigh ranges are much larger than the l ¼ 10 cm length of each rod, and the spot size
of each beam is much smaller than the cross-sectional diameter of each rod. Therefore, each
Gaussian beam can be considered to be collimated throughout each rod with an approximate
beam cross-sectional area of
2
πw20 π 400 106
A¼
¼
m2 ¼ 2:51 107 m2 :
2
2
Then, the saturation powers are
6
7
¼ I YAG
W ¼ 4:34 W for the Nd:YAG amplifier,
PYAG
sat
sat A ¼ 17:3 10 2:51 10
ruby
6
7
W ¼ 35 W for the ruby amplifier:
Pruby
sat ¼ I sat A ¼ 139:4 10 2:51 10
With l ¼ 10 cm and a uniform unsaturated gain coefficient of g 0 ¼ 10 m1 for both rods, both
amplifiers have the same unsaturated power gain of
G0 ¼ exp ðg 0 lÞ ¼ e1:0 :
Using (8.40), the power gain for an input signal power of Pin
s ¼ 5 W can be found for each
amplifier:
Pin
5
1:0
s
GYAG ¼ G0 exp ð1 GYAG Þ YAG ¼ e exp ð1 GYAG Þ
) GYAG ¼ 1:51,
4:34
Psat
"
#
Pin
5
1:0
s
Gruby ¼ G0 exp 1 Gruby ruby ¼ e exp 1 Gruby
) Gruby ¼ 2:27:
35
Psat
Thus, the output signal powers are
in
Pout
s, YAG ¼ GYAG Ps ¼ 1:51 5 W ¼ 7:55 W for the Nd:YAG amplifier,
in
Pout
s, ruby ¼ Gruby Ps ¼ 2:27 5 W ¼ 11:35 W for the ruby amplifier:
8.5
SPONTANEOUS EMISSION
..............................................................................................................
Spontaneous emission occurs whenever the upper laser level of a system is populated, irrespective of the lower-level population. The population of the upper laser level for any system is
given in (8.20):
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268
Optical Amplification
N2 ¼
σaN t þ g
,
σe þ σa
(8.42)
where g > 0 when the system is above transparency with an optical gain, g ¼ 0 when the
system is at transparency, and g < 0 when the system is below transparency with an optical
attenuation coefficient of α ¼ g.
According to the discussion in Section 7.1, the spontaneous emission power is proportional to
N 2 but is independent of N 1 . Therefore, regardless of whether the medium has a gain or a loss,
the spontaneous emission power density, which is defined as the spontaneous emission power
per unit volume of the medium in watts per cubic meter, is
^ sp ¼ hv N 2 ¼ hv σ a N t þ g ,
P
τ sp
τ sp σ e þ σ a
(8.43)
where g can be positive for a medium pumped above transparency, zero for a system at
transparency, or negative for a medium below transparency. For a gain volume of V, the
spontaneous emission power is
^ sp V:
Psp ¼ P
(8.44)
The spontaneous emission power density at transparency, which is known as the critical
fluorescence power density, is
^ trsp ¼ hv N 2 ¼ hv σ a N t :
P
τ sp
τ sp σ e þ σa
(8.45)
The critical fluorescence power for a gain volume of V is
^ trsp V:
Ptrsp ¼ P
(8.46)
^ trsp ¼ 0 and Ptrsp ¼ 0 because σ a ¼ 0 so that it is transparent
For an ideal four-level system, P
^ trsp 6¼ 0 and Ptrsp 6¼ 0
without pumping. For a quasi-two-level system or a three-level system, P
^ trsp and Ptrsp are
because σ a 6¼ 0. A practical quasi-two-level system usually has σ a σ e so that P
^ sp and Psp when the medium is pumped for a positive gain of
respectively much smaller than P
^ sp and Psp
^ trsp and Ptrsp are often respectively comparable to P
g > 0. For a three-level system, P
when the medium is pumped for a positive gain of g > 0 because σ a and σ e are of the same
order of magnitude.
When an optical medium is pumped below transparency, it can still emit light through
spontaneous emission as long as N 2 > 0 though N 2 < N tr2 in this situation. Even when an
optical medium is pumped above transparency, spontaneous emission still occurs, and the
power of spontaneous emission can still dominate that of stimulated emission before laser
action takes place. Such spontaneous emission power is the basis of incoherent luminescent
light sources. For example, light-emitting diodes are solid-state light sources that emit spontaneous emission generated by electroluminescence through radiative relaxation of electron–hole
pairs that are injected by an electric current.
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8.5 Spontaneous Emission
269
In a laser amplifier that amplifies an optical signal through stimulated emission, the spontaneous emission is also amplified, resulting in amplified spontaneous emission. Amplified
spontaneous emission is the major source of optical noise for a laser amplifier. It is also the
major source of optical noise for a laser oscillator.
EXAMPLE 8.8
Consider the Nd:YAG and ruby crystals that have the characteristics described in the preceding
examples. As found in Example 8.3, the population density of the upper laser level required for
the λ ¼ 1:064 μm Nd:YAG laser line to reach transparency is N tr2 ¼ 0, whereas that required for
the λ ¼ 694:3 nm ruby laser line to reach transparency is N tr2 ¼ 7:61 1024 m3 . The spontaneous lifetimes are τ sp ¼ 515 μs for the Nd:YAG laser line and τ sp ¼ 3 ms for the ruby laser line.
A Nd:YAG laser rod and a ruby laser rod both have a length of l ¼ 10 cm and a cross-sectional
diameter of d ¼ 6 mm. Find the critical fluorescence power density and the critical fluorescence
power for each rod.
Solution:
The volume of each rod is
2
2
d
6 103
V¼π
l¼π
10 102 m3 ¼ 2:83 106 m3 :
2
2
For the Nd:YAG rod, because N tr2 ¼ 0, both the critical fluorescence power density and the
critical fluorescence power are zero:
^ trsp ¼ 0 and Ptrsp ¼ 0:
P
For the ruby rod, N tr2 ¼ 7:61 1024 m3 , τ sp ¼ 3 ms, and the photon energy is
hv ¼
1239:8
eV ¼ 1:786 eV:
694:3
Therefore, the critical fluorescence power density and the critical fluorescence power for the
ruby rod are, respectively,
hv tr 1:786 1:6 1019
tr
^
¼
N ¼
7:61 1024 W m3 ¼ 725 MW m3
P sp
3
τ sp 2
3 10
and
^ trsp V ¼ 725 106 2:83 106 W ¼ 2:05 kW:
Ptrsp ¼ P
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270
Optical Amplification
Problems
8.1.1 Show that the rate equation given in (8.6) for the effective population inversion is valid
for all systems if the differences among the systems are accounted for by using the
bottleneck factor defined in (8.7). Show also that the effective pumping rate is
R ¼ βR2 ðβ 1Þ
Nt
:
τ2
(8.47)
Hint: Use (8.20) directly for the relation between the population density of the upper laser
level and the gain coefficient defined in (8.5).
8.1.2 A Ti:sapphire crystal is doped with 0.024 wt.% of Ti2 O3 for a Ti3þ ion concentration of
N t ¼ 7:9 1024 m3 . At the λ ¼ 800 nm wavelength, it has an emission cross section of
σ e ¼ 3:4 1023 m2 and an absorption cross section of σ a ¼ 8 1026 m2 . Find its
bottleneck factor at this laser wavelength.
8.1.3 An Er:fiber is doped with an Er3þ ion concentration of N t ¼ 2:2 1024 m3 . It has an
absorption cross section of σ a ¼ 5:7 1025 m2 and an emission cross section of σ e ¼
7:9 1025 m2 at the λ ¼ 1:53 μm wavelength. Find its bottleneck factor at this laser
wavelength. What is the effective population inversion for a gain coefficient of g ¼
0:3 m1 at λ ¼ 1:53 μm?
8.2.1 Verify the relation given in (8.20) for the population density of the upper laser level for a
gain coefficient of g at an effective population inversion of N.
8.2.2 A Nd:YAG crystal is doped with 1 at.% of Nd3þ ions for a concentration of
N t ¼ 1:38 1026 m3 . For its λ ¼ 1:064 μm laser line, the emission cross section is found
to be σ e ¼ 4:5 1023 m2 and the absorption cross section is σ a ¼ 0 because the lower
laser level of this laser line is effectively empty all the time. A ruby crystal is doped with
0.05 wt.% of Cr3þ ions for a concentration of N t ¼ 1:58 1025 m3 . For its λ ¼ 694:3 nm
laser line, the emission cross section is found to be σ e ¼ 1:34 1024 m2 and the absorption cross section is σ a ¼ 1:25 1024 m2 . Find the effective population inversion and the
population density of the upper laser level required for the λ ¼ 1:064 μm Nd:YAG laser
line to have a gain coefficient of g ¼ 6 m1 . Find those values required for the
λ ¼ 694:3 nm ruby laser line to have a gain coefficient of g ¼ 6 m1 . What percent of
all active ions are excited in each case? Explain the difference between the two media.
8.2.3 A Ti:sapphire crystal is doped with 0.03 wt.% of Ti2 O3 for a Ti3þ ion concentration of
N t ¼ 1:0 1025 m3 . At the λ ¼ 800 nm wavelength, it has an emission cross section of
σ e ¼ 3:4 1023 m2 and an absorption cross section of σ a ¼ 8 1026 m2 .
(a) Find the population density of the upper laser level required for this Ti:sapphire crystal
to reach transparency at λ ¼ 800 nm. What percent of all active ions are excited?
(b) What is the effective population inversion for a gain coefficient of g ¼ 15 m1 at
λ ¼ 800 nm? What is the population density of the upper laser level for this effective
population inversion? What percent of all active ions are excited? What percent of the
excited ions effectively contribute to the population inversion?
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Problems
271
8.2.4 An Er:fiber is doped with an Er3þ ion concentration of N t ¼ 2:2 1024 m3 . It has an
absorption cross section of σ a ¼ 5:7 1025 m2 and an emission cross section of σ e ¼
7:9 1015 m2 at the λ ¼ 1:53 μm wavelength.
(a) Find the population density of the upper laser level required for this Er:fiber to reach
transparency at λ ¼ 1:53 μm. What percent of all active ions are excited?
(b) What is the effective population inversion required for a gain coefficient of g ¼
0:3 m1 at λ ¼ 1:53 μm? What is the population density of the upper laser level for
this effective population inversion? What percent of all active ions are excited? What
percent of the excited ions effectively contribute to the population inversion?
8.3.1 With a constant upward pumping transition probability rate of W p into the upper laser
level j2i by depleting the population in the lower laser level j1i, and a constant downward
pumping transition probability rate of pW p that depletes the population in the upper level,
the total pumping rate to the upper laser level is R2 ¼ W p ðN 1 pN 2 Þ. Show by using
N 1 þ N 2 N t and (8.20) that the effective pumping rate found in Problem 8.1.1 can be
expressed in terms of the total population N t and the effective population inversion N as
β1
N t ð1 þ pÞW p N:
(8.48)
R ¼ ½1 ðβ 1Þp W p τ2
Use this pumping rate and the rate equation given in (8.6) for the effective population
inversion to show that in the steady state the gain coefficient can be expressed in the form
of (8.22) with the saturation intensity I sat taking the form of (8.23), the unsaturated gain
coefficient g 0 having the form of (8.33), and the saturation lifetime τ s having the form of
(8.34).
8.3.2 By using (8.33) and (8.34), show that the required pumping probability rate for an
unsaturated gain coefficient of g 0 is that given in (8.35).
8.3.3 By using the general expression in (8.34), find the saturation lifetime at the transparency
point for all systems.
8.3.4 A Ti:sapphire crystal is doped with 0.03 wt.% of Ti2 O3 for a Ti3þ ion concentration of
N t ¼ 1:0 1025 m3 . At the λ ¼ 800 nm wavelength, it has an emission cross section of
σ e ¼ 3:4 1023 m2 and an absorption cross section of σ a 8 1026 m2 . It has an
upper laser level lifetime of τ 2 ¼ 3:2 μs. It can be optically pumped at the pump wavelength of λp ¼ 532 nm, where the absorption cross section is σ pa ¼ 7:4 1024 m2 and the
emission cross section is σ pe 3 1026 m2 . The pump quantum efficiency is ηp ¼ 0:9.
(a) Find the pumping rates for this Ti:sapphire to reach transparency and to have an
unsaturated gain coefficient of g 0 ¼ 15 m1 at λ ¼ 800 nm, respectively. What are
the saturation lifetime and the saturation intensity in each case?
(b) Find the required pump intensities at λp ¼ 532 nm to pump this Ti:sapphire to
transparency and to have an unsaturated gain coefficient of g 0 ¼ 15 m1 , respectively.
(c) When this Ti:sapphire is pumped to have an unsaturated gain coefficient of g 0 ¼
15 m1 at λ ¼ 800 nm, a collimated Gaussian laser beam at this wavelength that has a
power of P ¼ 1 W and a spot size of w0 ¼ 200 μm is sent through this crystal. Find
the saturated gain coefficient.
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272
Optical Amplification
8.3.5 An Er:fiber is doped with an Er3þ ion concentration of N t ¼ 2:2 1024 m3 in its core.
This fiber is a cylindrical waveguide that has a core radius of a ¼ 4:5 μm. At the λ ¼
1:53 μm wavelength, the Er:fiber has an absorption cross section of σ a ¼ 5:7 1025 m2 ,
an emission cross section of σ e ¼ 7:9 1025 m2 , and an upper laser level lifetime of
τ 2 ¼ 10 ms. It can be optically pumped as a three-level system at the pump wavelength of
λp ¼ 980 nm, where the absorption cross section is σ pa ¼ 2:58 1025 m2 . At the signal
wavelength of λ ¼ 1:53 μm and the pump wavelength of λp ¼ 980 nm, the guided signal
and pump waves respectively have effective mode radii of ρ ¼ 4:1 μm and ρp ¼ 3:3 μm
for their intensity profiles. The fractions of the signal and pump intensities that overlap
with the core doped with active ions are determined by the confinement factors, which are
Γ ¼ 0:70 and Γp ¼ 0:72, respectively. The pump quantum efficiency is ηp ¼ 0:8.
(a) Find the pumping rates for this Er:fiber to reach transparency and to have an
unsaturated gain coefficient of g 0 ¼ 0:3 m1 , respectively, at λ ¼ 1:53 μm. What
are the saturation lifetime and the saturation intensity in each case?
(b) Find the required pump intensities at λp ¼ 980 nm to pump this Er:fiber to transparency and to have an unsaturated gain coefficient of g 0 ¼ 0:3 m1 , respectively.
(c) Find the required pump powers for transparency and for g 0 ¼ 0:3 m1 by accounting
for the overlap between the guided pump beam and the active core.
(d) When this Er:fiber is pumped to have an unsaturated gain coefficient of g 0 ¼ 0:3 m1
at λ ¼ 1:53 μm, a guided laser beam at this wavelength that has a power of P ¼
1 mW is sent through this fiber. Find the saturated gain coefficient by accounting for
the overlap between the guided signal beam and the active core.
8.4.1 If the spot sizes of both beams in Example 8.6 are increased to w0 ¼ 800 μm, what is the
output power from each amplifier?
8.4.2 A Ti:sapphire laser rod of the characteristics described in Problem 8.3.4 has a length of
l ¼ 4 cm and a cross-sectional diameter of d ¼ 3 mm. The refractive index of sapphire is
1.76. The laser rod is uniformly pumped to have an unsaturated gain coefficient of g 0 ¼
15 m1 at the wavelength of λ ¼ 800 nm. The saturation intensity at g 0 ¼ 15 m1 is
I sat > 2 GW m2 . A collimated Gaussian signal beam at λ ¼ 800 nm that has a spot size
of w0 ¼ 300 μm in the rod and a power of Pin
s ¼ 1 W is sent through the Ti:sapphire
amplifier. What is the output signal power from this Ti:sapphire amplifier?
8.4.3 An Er:fiber amplifier of the characteristics described in Problem 8.3.5 has a length of
l ¼ 10 m. It is uniformly pumped to have an unsaturated gain coefficient of g 0 ¼ 0:3 m1 at
its laser wavelength of λ ¼ 1:53 μm. After accounting for the overlap between the guided
signal beam and the active core, the saturation power at g 0 ¼ 0:3 m1 is Psat ¼ 1:49 mW. If
a guided signal beam at λ ¼ 1:53 μm that has a power of Pin
s ¼ 10 μW is sent through the
Er:fiber amplifier, what is the amplified output signal power? What is the output signal
power if the input signal power is increased to Pin
s ¼ 1 mW?
8.5.1 A Nd:YAG crystal is doped with a Nd3þ concentration of N t ¼ 1:38 1026 m3 . For its
λ ¼ 1:064 μm laser line, the emission cross section is σ e ¼ 4:5 1023 m2 , the absorption cross section is σ a ¼ 0, and the spontaneous lifetime is τ sp ¼ 515 μs. A ruby crystal
is doped with a Cr3þ concentration of N t ¼ 1:58 1025 m3 . For its λ ¼ 694:3 nm laser
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Bibliography
273
line, the emission cross section is σ e ¼ 1:34 1024 m2 , the absorption cross section is
σ a ¼ 1:25 1024 m2 , and the spontaneous lifetime is τ sp ¼ 3 ms. The refractive index
of Nd:YAG is 1.82, and that of ruby is 1.76. A Nd:YAG laser rod and a ruby laser rod
both have a length of l ¼ 10 cm and a cross-sectional diameter of d ¼ 6 mm. Find the
spontaneous emission power density and the spontaneous emission power of each rod
when each is uniformly pumped to have an unsaturated gain coefficient of g 0 ¼ 10 m1 .
8.5.2 A Ti:sapphire laser rod has a length of l ¼ 4 cm and a cross-sectional diameter of
d ¼ 3 mm. It is doped with a Ti3þ ion concentration of N t ¼ 1:0 1025 m3 . At the
λ ¼ 800 nm wavelength, it has an emission cross section of σ e ¼ 3:4 1023 m2 and an
absorption cross section of σ a 8 1026 m2 . Its upper laser level for the λ ¼ 800 nm
emission has a total lifetime of τ 2 ¼ 3:2 μs and a spontaneous lifetime of τ sp ¼ 3:9 μs.
(a) Find the critical fluorescence power density and the critical fluorescence power of
the rod.
(b) Find the spontaneous emission power density and the spontaneous emission power of
the rod when it is uniformly pumped to have an unsaturated gain coefficient of g 0 ¼
15 m1 at λ ¼ 800 nm.
8.5.3 An Er:fiber that has a length of l ¼ 10 m is doped with an Er3þ ion concentration of N t ¼
2:2 1024 m3 in its core, which has a radius of a ¼ 4:5 μm. It has an absorption cross
section of σ a ¼ 5:7 1025 m2 and an emission cross section of σ e ¼ 7:9 1025 m2 at
the λ ¼ 1:53 μm wavelength. Its upper laser level for the λ ¼ 1:53 μm emission has the
same total lifetime and spontaneous lifetime of τ 2 ¼ τ sp ¼ 10 ms.
(a) Find the critical fluorescence power density and the critical fluorescence power of
the fiber.
(b) Find the spontaneous emission power density and the spontaneous emission power
of the fiber when it is uniformly pumped to have an unsaturated gain coefficient of
g 0 ¼ 0:3 m1 at λ ¼ 1:53 μm.
Bibliography
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University Press, 2014.
Iizuka, K., Elements of Photonics for Fiber and Integrated Optics, Vol. II. New York: Wiley, 2002.
Liu, J. M., Photonic Devices. Cambridge: Cambridge University Press, 2005.
Milonni, P. W. and Eberly, J. H., Laser Physics. New York: Wiley, 2010.
Saleh, B. E. A. and Teich, M. C., Fundamentals of Photonics. New York: Wiley, 1991.
Siegman, A. E., Lasers. Mill Valley, CA: University Science Books, 1986.
Silfvest, W. T., Laser Fundamentals. Cambridge: Cambridge University Press, 1996.
Svelto, O., Principles of Lasers, 5th edn. New York: Springer, 2010.
Verdeyen, J. T., Laser Electronics, 3rd edn. Englewood Cliffs, NJ: Prentice-Hall, 1995.
Yariv, A. and Yeh, P., Photonics: Optical Electronics in Modern Communications. Oxford: Oxford University
Press, 2007.
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Cambridge Books Online
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Principles of Photonics
Jia-Ming Liu
Book DOI: http://dx.doi.org/10.1017/CBO9781316687109
Online ISBN: 9781316687109
Hardback ISBN: 9781107164284
Chapter
9 - Laser Oscillation pp. 274-296
Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.010
Cambridge University Press
9
9.1
Laser Oscillation
CONDITIONS FOR LASER OSCILLATION
..............................................................................................................
The word laser is the acronym of light amplification by stimulated emission of radiation.
A medium that is pumped to population inversion has an optical gain to amplify an optical
field through stimulated emission. Besides optical amplification, however, positive optical
feedback is normally required for laser oscillation. This requirement is fulfilled by placing
the gain medium in an optical resonator. One major characteristic of laser light is that it is
highly collimated and is spatially and temporally coherent. The directionality of laser light is a
direct consequence of the fact that laser oscillation takes place only along a longitudinal axis
defined by the optical resonator. The spatial and temporal coherence results from the fact that a
photon emitted by stimulated emission is coherent with the photon that induces the emission.
The gain medium emits spontaneous photons in all directions, but only the radiation that
propagates along the longitudinal axis within a small divergence angle defined by the resonator
obtains sufficient regenerative amplification through stimulated emission to reach the threshold
for oscillation. In order for the oscillating laser field to be most efficiently amplified in the
longitudinal direction, any spontaneous photons emitted in a direction outside of that small
angular range must not be allowed to compete for the gain. For this reason, a functional laser
oscillator is necessarily an open cavity that provides optical feedback only along the longitudinal axis. Most of the randomly directed spontaneous photons quickly escape from the cavity
through the open sides. Only a very small fraction of them that happen to be emitted within the
divergence angle of the laser field mix with the coherent oscillating laser field to become the
major incoherent noise source of the laser.
A laser is basically a coherent optical oscillator, and the basic function of an oscillator is to
generate a coherent signal through resonant oscillation without an input signal. No external
optical field is injected into the optical cavity for laser oscillation. The intracavity optical field
has to grow from the field that is generated by spontaneous emission from the intracavity gain
medium. When steady-state oscillation is reached, the coherent laser field at any given location
inside the cavity has to be a constant of time in both phase and magnitude. In the model shown
in Fig. 9.1, the situation of steady-state laser oscillation requires that Ein ¼ 0 while Ec ðzÞ 6¼ 0 at
any intracavity location z does not change with time. By applying this concept to (6.5) while
using (6.4), we find the condition for steady-state laser oscillation:
a ¼ G exp ðiφRT Þ ¼ 1,
(9.1)
where a is the round-trip complex amplification factor for the intracavity field, G is the roundtrip gain factor for the intracavity field amplitude, and φRT is the round-trip phase shift for the
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9.1 Conditions for Laser Oscillation
275
Figure 9.1 Fabry–Pérot laser.
intracavity field, as defined in (6.4). This general condition for laser oscillation applies to lasers
of various cavity structures that use different feedback mechanisms, including Fabry–Pérot
lasers, ring lasers, and distributed-feedback lasers. To illustrate the implications of this condition, we consider in the following the simple Fabry–Pérot laser shown in Fig. 9.1 that contains
an isotropic gain medium with a filling factor of Γ.
The total permittivity of the gain medium, including the contribution of the resonant laser
transition, is ϵ res ¼ ϵ þ ϵ 0 χ res , as given in (6.36). Therefore, the total complex propagation
constant of the gain medium, including the contribution from the resonant transition, is
g
1=2
(9.2)
kg ¼ ωμ0 ðϵ þ ϵ 0 χ res Þ1=2 ¼ k þ Δkres i ,
2
where
χ 0res
ω 0
¼
(9.3)
χ ,
2
2n
2nc res
χ 00
ω
¼ χ 00res :
(9.4)
g k res
2
n
nc
Here g is the gain coefficient of the laser medium, which is identified in (7.50), and Δkres is the
corresponding change in the propagation constant caused by the change in the refractive index
of the gain medium due to the changes in the population densities of the laser levels. When
population inversion is achieved, χ 00res < 0 so that the gain coefficient g has a positive value.
By replacing k for a cold medium with k g for a pumped gain medium, we find that k given in (6.38)
for a cold cavity has to be replaced with k þ ΓΔk res iΓg=2 when an actively pumped laser cavity is
considered. We then find for an active laser cavity the mode-dependent round-trip gain factor,
Δkres k
1=2 1=2
Gmn ¼ R1 R2
exp ½ðΓmn g αmn Þl,
(9.5)
and the mode-dependent round-trip phase shift,
RT
φRT
mn ¼ 2ðk þ ΓΔk res Þl þ ζ mn þ φ1 þ φ2 :
(9.6)
Because both Gmn and φRT
mn are real parameters, the oscillation condition given in (9.1) can be
satisfied for a given laser mode to oscillate only if the gain condition
Gmn ¼ 1
(9.7)
and the phase condition
φRT
mn ¼ 2qπ,
q ¼ 1, 2, . . .
are simultaneously fulfilled. Note that both Gmn and φRT
mn are frequency dependent.
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(9.8)
276
Laser Oscillation
9.1.1 Laser Threshold
The condition in (9.7) implies that there exist a threshold gain and a corresponding threshold
pumping level for laser oscillation. For the Fabry–Pérot laser shown in Fig. 9.1, which has a
length of l and contains a gain medium of a length lg for a filling factor of Γ ¼ lg =l, the
threshold gain coefficient, g th
mn , of the TEMmn mode is given by
1 pffiffiffiffiffiffiffiffiffiffi
ln R1 R2 ,
l
(9.9)
pffiffiffiffiffiffiffiffiffiffi
g th
mn lg ¼ αmn l ln R1 R2 :
(9.10)
Γg th
mn ¼ αmn or
Because the distributed loss αmn is mode dependent, the threshold gain coefficient g th
mn varies
from one transverse mode to another. In addition, the effective gain coefficient can be different
for different transverse modes because different transverse modes have different field distribution patterns and thus overlap with the gain volume differently. The transverse mode that has
the lowest loss and the largest effective gain at any given pumping level reaches threshold first
and starts oscillating at the lowest pumping level. In the typical laser, the transverse mode that
reaches threshold first is normally the fundamental TEM00 mode.
Unless a frequency-selecting mechanism is placed in a laser to create a frequencydependent loss that varies from one longitudinal mode to another, the threshold gain coefficient g th
mn varies little among the mnq longitudinal modes of different q values that share the
common mn transverse mode pattern. It is possible, however, to introduce a frequencyselecting device to a laser cavity to make αmn and, consequently, g th
mn of a given mn transverse
mode highly frequency dependent for the purpose of selecting or tuning the oscillating laser
frequency.
The power required to pump a laser to reach its threshold is called the threshold pump
power, Pth
p . Because the threshold gain coefficient is mode dependent and frequency
dependent, the threshold pump power is also mode dependent and frequency dependent.
The threshold pump power of a laser mode can be found by calculating the power required
for the gain medium to have an unsaturated gain coefficient equal to the threshold gain
coefficient of the mode: g 0 ¼ g th
mn ðωmnq Þ, assuming uniform pumping throughout the gain
medium. For a quasi-two-level or three-level laser, there is also a transparency pump power,
Ptrp , for g 0 ¼ 0, assuming uniform pumping. In the situation of nonuniform pumping, these
conditions for reaching threshold and transparency have to be modified. Clearly, Ptrp < Pth
p by
definition.
EXAMPLE 9.1
A Nd:YAG laser for the λ ¼ 1:064 μm laser wavelength consists of a Nd:YAG laser rod of a
length lg ¼ 3 cm as a gain medium in a Fabry–Pérot cavity, which is formed by two mirrors of
reflectivities R1 ¼ 90% and R2 ¼ 100% at a physical spacing of l ¼ 10 cm. The surfaces of the
laser rod are antireflection coated to eliminate losses and undesirable effects. The crosssectional area of the laser rod is larger than that of the TEM00 Gaussian laser mode. This laser
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9.2 Mode-Pulling Effect
277
mode has a distributed optical loss of α ¼ 0:1 m1 . Find the threshold gain coefficient of this
laser mode.
Solution:
Using (9.10), we find with the given parameters that the threshold gain coefficient of the TEM00
Gaussian laser mode is
g th ¼
9.2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffi
1
1
ðαl ln R1 R2 Þ ¼
ð0:1 0:1 ln 0:9 1Þ m1 ¼ 2:09 m1 :
lg
0:03
MODE-PULLING EFFECT
..............................................................................................................
Comparing (9.6) for an active Fabry–Pérot laser with (6.40) for its cold cavity, we find that,
through its dependence on Δkres , the round-trip phase shift of a field in a laser cavity is a
function of χ 0res . Consequently, the longitudinal mode frequencies ωmnq at which a laser
oscillates are not exactly the same as the longitudinal mode frequencies ωcmnq given in (6.41)
for the cold Fabry–Pérot cavity.
Using (9.6) and (9.8), we find that the longitudinal mode frequencies of a Fabry–Pérot laser
are related to those of its cold cavity by the relation:
ωmnq ¼
ωcmnq
χ 0res 1
χ 0res
c
:
1þ
ωmnq 1 2nn
2nn
(9.11)
Clearly, the laser mode frequencies ωmnq differ from the cold-cavity mode frequencies because they
vary with the resonant susceptibility, which depends on the level of population inversion in the gain
medium. This dependence of the laser mode frequencies on the population inversion in the gain
medium is caused by the fact that the refractive index and the gain of the medium are directly connected
to each other, as is dictated by the Kramers–Kronig relation. This effect causes a frequency shift of
δωmnq ¼ ωmnq ωcmnq χ 0res c
ω
2nn mnq
(9.12)
for the oscillation frequency of mode mnq. Because of the frequency dependence of χ 0res , the
dependence of this frequency shift on χ 0res results in the mode-pulling effect demonstrated in
Fig. 9.2. Near the transition resonance frequency, ω21 , of the gain medium, χ 0res is highly dispersive.
When a medium is pumped to have population inversion for a transition that has a resonance
frequency of ω21 , χ 00res ðωÞ < 0 for either ω < ω21 or ω > ω21 , but χ 0res ðωÞ < 0 for ω < ω21 and
χ 0res ðωÞ > 0 for ω > ω21 . As a result, ωmnq > ωcmnq for ωcmnq < ω21 , whereas ωmnq < ωcmnq for
ωcmnq > ω21 . Therefore, in comparison to the resonance frequencies of the cold cavity, the mode
frequencies of a laser are pulled toward the transition resonance frequency of the gain medium. In
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278
Laser Oscillation
Figure 9.2 Frequency-pulling effect for laser modes. Compared to the resonance frequencies of the cold cavity
shown as dotted lines, the mode frequencies of an active laser shown as solid lines are pulled toward the
transition resonance frequency of the gain medium in the situation of population inversion. The real and
imaginary parts of the gain susceptibility as a function of optical frequency are shown.
addition, the longitudinal modes belonging to a common transverse mode are no longer equally
spaced in frequency. In a laser of a relatively high gain and a large dispersion, such as a
semiconductor laser, this effect can result in a large variation in the frequency spacing between
neighboring laser modes.
Because of the frequency dependence of the gain coefficient g due to the frequency
dependence of χ 00res , different longitudinal modes not only experience different values of
refractive index but also see different values of gain coefficient, as also illustrated in Fig. 9.2.
A longitudinal mode that has a frequency close to the gain peak at the transition resonance
frequency has a higher gain than one that has a frequency far away from the gain peak.
EXAMPLE 9.2
A Nd:YAG laser contains a Nd:YAG rod described in Example 8.1 in a cavity described in
Example 9.1. The refractive index of the Nd:YAG crystal is n ¼ 1:82. Find the largest
frequency shift of the longitudinal mode frequencies of the Nd:YAG laser due to the modepulling effect. How large is this frequency shift compared to the longitudinal mode frequency
spacing?
Solution:
From Example 9.1, we find that the gain coefficient is g ¼ g th ¼ 2:09 m1 when the TEM00
laser mode is pumped to its threshold. The overlap factor is Γ ¼ lg =l ¼ 0:3; thus, the weighted
average refractive index seen by the laser mode is
n ¼ 0:3 1:82 þ ð1 0:3Þ 1 ¼ 1:246:
With λ ¼ 1:064 μm at the transition frequency ω21 , we find that the maximum value of the
imaginary part of the resonant susceptibility associated with this laser transition is
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9.3 Oscillating Laser Modes
χ 00res ðω21 Þ ¼ 279
nc
nλ
1:82 1:064 106
g¼ g¼
2:09 ¼ 6:44 107 ,
2π
ω21
2π
which appears at the line center. Because this laser transition is a discrete atomic transition, the
real part χ 0res has the largest absolute value at two frequencies. With χ 00res ðω21 Þ < 0, χ 0res has the
largest negative value of χ 0res ðω Þ ¼ χ 00res ðω21 Þ=2 at the frequency ω ¼ ω21 γ and the largest
positive value of χ 0res ðωþ Þ ¼ χ 00res ðω21 Þ=2 at ωþ ¼ ω21 þ γ, as seen in Figs. 2.3 and 9.2. Thus,
jχ 0res jmax ¼ jχ 00res ðω21 Þ=2j ¼ 3:22 107 :
For a Nd:YAG laser at λ ¼ 1:064 μm, γ=ω21 2 104 because the gain linewidth is about
Δvg ¼ γ=π 120 GHz, whereas the laser frequency is v21 ¼ ω21 =2π ¼ c=λ 283 THz. Therefore, we can take the approximation that ωc ¼ ω ¼ ω21 γ ω21 for (9.12) to find the
absolute value of the largest frequency shift caused by mode pulling:
jδvjmax ¼
jδωjmax jχ 0res jmax
3:22 107
ν21 ¼
283 1012 Hz ¼ 20:1 MHz:
2π
2nn
2 1:82 1:246
This is the largest amount of frequency shift, which occurs for a longitudinal mode that has a coldcavity mode frequency at either the positive or negative half-width points vc, ¼ v21 Δvg =2. As
shown in Fig. 9.2, the mode that is closest to the lower frequency, vc, ¼ v21 Δvg =2, is pulled
up by an amount of approximately jδvjmax , whereas the mode that is closest to the higher
frequency, νc, þ ¼ v21 þ Δνg =2, is pulled down by an amount of approximately jδvjmax .
The longitudinal mode frequency spacing is
ΔνL ¼
c
3 108
Hz ¼ 1:204 GHz:
¼
2nl 2 1:246 10 102
Thus, the percentage of the maximum mode-pulling frequency shift is
jδνjmax
20:1 106
¼
1:67%:
ΔνL
1:204 109
This frequency shift is appreciable though small. It is small because the dispersive effect of the
optical gain is small in the Nd:YAG medium. It can be much larger in a highly dispersive gain
medium, such as a semiconductor laser gain medium.
9.3
OSCILLATING LASER MODES
..............................................................................................................
Because the gain coefficient is a function of frequency, the net gain coefficient, g g th
mn , of a
laser mode is always frequency dependent and varies among different transverse modes and
among different longitudinal modes no matter whether the threshold gain coefficient g th
mn of a
transverse mode is frequency dependent or not. At a low pumping level before the laser starts
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280
Laser Oscillation
oscillating, the net gain is negative for all laser modes. As the pumping level increases, the
mode that first reaches its threshold starts to oscillate.
Once a laser starts oscillating in one mode, whether any other longitudinal or transverse modes
have the opportunity to oscillate through further increase of the pumping level is a complicated
issue of mode interaction and competition that depends on a variety of factors, including the
properties of the gain medium, the structure of the laser, the pumping geometry, the nonlinearity
in the system, and the operating condition of the laser. Here we only discuss some basic concepts
in the situation of steady-state oscillation of a CW laser. Interaction and competition among laser
modes are more complicated when a laser is pulsed than when it is in CW operation. Therefore,
some of the conclusions obtained below may not be valid for a pulsed laser.
The gain condition in (9.7) implies that once a given laser mode is oscillating in the steady state,
the gain that is available to this mode does not increase with increased pumping above the threshold
pumping level because Gmn has to be kept at unity for the steady-state oscillation of a laser mode.
Thus the effective gain coefficient of an oscillating mode is “clamped” at the threshold level of the
mode as long as the pumping level is kept at or above threshold. The mechanism for holding down
the gain coefficient at the threshold level is the effect of gain saturation discussed in Section 8.3. An
increase in the pumping level above threshold only increases the field intensity of the oscillating
mode in the cavity, but the gain coefficient is saturated at the threshold value by the high intensity of
the intracavity laser field. The fact that the gain of a laser mode oscillating in the steady state is
saturated at the threshold value has a significant effect on the mode characteristics of a CW laser.
9.3.1 Homogeneously Broadened Lasers
When the gain medium of a laser is homogeneously broadened, all modes that occupy the same
spatial gain region compete for the gain from the population inversion in the same group of active
atoms. As the mode that first reaches threshold starts oscillating, the entire gain curve supported by
this group of atoms saturates. Because this oscillating mode is normally the one that has a
longitudinal mode frequency closest to the gain peak and a transverse mode pattern of the lowest
loss, the gain curve is saturated in such a manner that its value at this longitudinal mode frequency is
clamped at the threshold value of the transverse mode that has the lowest threshold gain coefficient
among all transverse modes. If the gain peak does not happen to coincide with this mode frequency,
it still lies above the threshold when the gain curve is saturated, as shown in Fig. 9.3. Nevertheless,
all other longitudinal modes belonging to this transverse mode have frequencies away from the gain
peak. Therefore, even with increased pumping, they do not have sufficient gain to reach threshold
because the entire gain curve shared by these modes is saturated, as illustrated in Fig. 9.3. Other
transverse modes that are supported solely by this group of saturated, homogeneously broadened
atoms do not have the opportunity to oscillate either, because the gain curve is saturated below their
respective threshold levels. Nevertheless, because different transverse modes have different spatial
field distributions, a high-order transverse mode may draw its gain from a gain region outside of the
region that is saturated by a low-order transverse mode. Therefore, when the pumping level is
increased, a high-order transverse mode may still reach its relatively high threshold for oscillation if
a low-order transverse mode of a low threshold is already oscillating.
Consequently, for a homogeneously broadened CW laser in steady-state oscillation, only one
among all of the longitudinal modes belonging to a particular transverse mode will oscillate, but
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9.3 Oscillating Laser Modes
281
Figure 9.3 Gain saturation in a homogeneously broadened laser. Only one longitudinal mode whose frequency
is closest to the gain peak oscillates. The entire gain curve is saturated such that the gain at the single oscillating
frequency remains at the loss level.
it is possible for more than one transverse mode to oscillate simultaneously at a high pumping
level. Note that this conclusion does not hold true for a pulsed laser. It is possible for multiple
longitudinal modes belonging to the same transverse mode to oscillate simultaneously in a
pulsed laser even when its gain medium is homogeneously broadened.
EXAMPLE 9.3
The Nd:YAG laser described in Examples 9.1 and 9.2 has a Lorentzian gain lineshape that has a
bandwidth of Δλg ¼ 0:45 nm for the laser line at λ ¼ 0:064 μm. It is pumped at a level such that
the peak unsaturated gain coefficient is twice the threshold gain coefficient: g max
¼ 2g th . How
0
many longitudinal modes have their unsaturated gain coefficients pumped above the threshold?
How many longitudinal modes oscillate?
Solution:
The gain bandwidth in terms of frequency is
Δν Δλ g g
¼
:
ν
λ
With Δλg ¼ 0:45 nm and λ ¼ 1:064 μm,
ν
c
3 108
0:45 109 Hz ¼ 119:25 GHz:
Δνg ¼ Δλg ¼ 2 Δλg ¼
λ
λ
ð1:064 106 Þ2
¼ 2g th , the two frequencies at the two ends of the
When the laser is pumped such that g max
0
FWHM of the gain bandwidth have an unsaturated gain coefficient of g 0 ¼ g th . Therefore,
every mode that has a frequency within the FWHM, Δνg ¼ 119:25 GHz, of the gain bandwidth
has an unsaturated gain coefficient above the threshold value. From Example 9.2, the longitudinal mode frequency spacing is
ΔνL ¼
c
3 108
Hz ¼ 1:204 GHz:
¼
2nl 2 1:246 10 102
Then,
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282
Laser Oscillation
Δνg 119:25
¼
¼ 99:04:
ΔνL
1:204
Therefore, depending on where the longitudinal mode frequencies are located with respect to
the gain peak, 99 or 100 longitudinal modes have unsaturated gain coefficients that are above
the threshold value.
Because the gain spectrum has a Lorentzian lineshape, the laser is homogeneously broadened.
Therefore, ideally only one longitudinal mode oscillates. Though 99 or 100 longitudinal modes are
each pumped to have an unsaturated gain coefficient above the threshold value, all of them except
the oscillating mode are saturated below the threshold by the oscillating mode, which reaches the
threshold first. In practice, however, we often find that a Nd:YAG laser oscillates steadily in more
than one mode because it is not completely homogeneously broadened though it is predominantly
so. The degree of inhomogeneous broadening determines the number of oscillating modes.
9.3.2 Inhomogeneously Broadened Lasers
In a laser that has an inhomogeneously broadened gain medium, there are different groups of active
atoms in the same spatial gain region. Each group saturates independently. Two modes occupying
the same spatial gain region do not compete for the same group of atoms if the separation of their
frequencies is larger than the homogeneous linewidth of each group of atoms. When one longitudinal mode reaches threshold and oscillates, the gain coefficient is saturated only within the spectral
range of a homogeneous linewidth around its frequency, while the gain coefficient at frequencies
outside this small range continues to increase with increased pumping. As the pumping level
increases, other longitudinal modes can successively reach threshold and oscillate. As a result, at a
sufficiently high pumping level, multiple longitudinal modes belonging to the same transverse
mode can oscillate simultaneously. The saturation of the gain coefficient in a small spectral range
within a homogeneous linewidth around each of the frequencies of these oscillating modes, but not
across the entire gain curve, creates the effect of spectral hole burning in the gain curve of an
inhomogeneously broadened laser medium, as illustrated in Fig. 9.4. Different transverse modes
Figure 9.4 Spectral hole burning effect in the gain saturation of an inhomogeneously broadened laser. Multiple
longitudinal modes oscillate simultaneously at a sufficiently high pumping level. The gain at each oscillating
frequency is saturated at the loss level. The mode-pulling effect is ignored in this illustration.
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9.3 Oscillating Laser Modes
283
also saturate independently in an inhomogeneously broadened medium if their frequencies are
sufficiently separated. Therefore, an inhomogeneously broadened laser can also oscillate in
multiple transverse modes.
EXAMPLE 9.4
A He–Ne laser has a Doppler-broadened gain bandwidth of Δνg ¼ 1:5 GHz at its laser
wavelength of λ ¼ 632:8 nm. The laser has a cavity length of l ¼ 32 cm. It is pumped at a
level such that the peak unsaturated gain coefficient is twice the threshold gain coefficient:
g max
¼ 2g th . How many longitudinal modes have their unsaturated gain coefficients pumped
0
above the threshold? How many longitudinal modes oscillate?
Solution:
When the laser is pumped such that g max
¼ 2g th , the two frequencies at the two end of the
0
FWHM Δvg of the gain bandwidth have an unsaturated gain coefficient of g 0 ¼ g th . Therefore,
the laser has a bandwidth of Δv ¼ Δvg ¼ 1:5 GHz. Every mode that has a frequency within this
bandwidth has an unsaturated gain coefficient above the threshold value. With l ¼ 32 cm and
n 1 for the gaseous He–Ne laser gain medium, the longitudinal mode frequency spacing is
ΔνL ¼
c
3 108
¼
Hz ¼ 468:75 MHz:
2nl 2 1 32 102
Then,
Δν
1:5 109
¼
¼ 3:2:
ΔνL 468:75 106
Therefore, three or four longitudinal modes have unsaturated gain coefficients that are above
the threshold value, depending on where the longitudinal mode frequencies are located with
respect to the gain peak. Because the gain spectrum is Doppler broadened, the laser is
inhomogeneously broadened. All longitudinal modes above threshold oscillate.
9.3.3 Laser Linewidth
The linewidth of an oscillating laser mode is still described by (6.18):
Δνmnq ¼
1 Gmnq L
Δνmn ,
πGmnq
(9.13)
where the longitudinal mode frequency spacing ΔνLmn might vary for different transverse modes.
From this relation, we see that in practice the round-trip field gain factor Gmnq of a laser mode in
steady-state oscillation cannot be exactly equal to unity because the laser linewidth cannot be
zero, due to the existence of spontaneous emission. In reality, in steady-state oscillation the
value of Gmnq is slightly less than unity, with the small difference made up by spontaneous
emission. Clearly, the linewidth of an oscillating laser mode is determined by the amount of
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284
Laser Oscillation
spontaneous emission that is channeled into the laser mode. Therefore, (9.13) is not very useful
for calculating the linewidth of a laser mode in steady-state oscillation without knowing the
exact value of Gmnq in the presence of spontaneous emission.
A detailed analysis taking into account spontaneous emission yields the Schawlow–Townes
relation for the linewidth of a laser mode in terms of the laser parameters:
ΔνST
mnq ¼
2πhvðΔνcmnq Þ2
Pout
mnq
N sp ¼
hv
2πðτ cmnq Þ2 Pout
mnq
N sp ,
(9.14)
where Δνcmnq and τ cmnq are respectively the cold-cavity linewidth and the photon lifetime of the
oscillating mnq mode, Pout
mnq is the output power of the oscillating laser mode, and
N sp ¼
σeN 2
σeN 2 N 2
¼
¼
σeN 2 σaN 1
g
N
(9.15)
is the spontaneous emission factor that measures the degree of the effective population inversion
in the gain medium. The effective population inversion defined as N ¼ g=σ e in (8.5) is the
population density that is able to contribute to the coherent stimulate emission, which does not
broaden the laser linewidth, whereas all of the upper level population N 2 contributes to the
incoherent spontaneous emission, which broadens the laser linewidth. The effect of spontaneous
emission on the linewidth of an oscillating laser mode enters the relation in (9.14) through the
population densities of the laser levels in the form of the spontaneous emission factor.
Because N sp 1, the ultimate lower limit of the laser linewidth, which is known as the
Schawlow–Townes limit, is that given in (9.14) for N sp ¼ 1. It can also be seen that the
linewidth of a laser mode decreases as the laser power increases. This phenomenon is easily
understood. Because the gain of an oscillating laser mode is clamped at its threshold level,
increased pumping above threshold does not increase the population inversion, and thus does
not increase the spontaneous emission, which is proportional to the population of the upper
laser level. When the power of an oscillating laser mode increases with increased pumping, the
coherent stimulated emission increases proportionally but the incoherent spontaneous emission
is clamped at its threshold level. As a result, the linewidth of the laser mode decreases with
increasing laser power.
EXAMPLE 9.5
Find the minimum possible linewidth that is set by the Schawlow–Townes limit for the
oscillating laser mode of the Nd:YAG laser described in Examples 9.1 and 9.2 when the laser
is pumped sufficiently above the threshold so that the output power of the mode at
λ ¼ 1:064 μm is 100 mW.
Solution:
The Nd:YAG laser described in Examples 9.1 and 9.2 has a Fabry–Pérot cavity that has a
length of l ¼ 10 cm, a weighted average index of n ¼ 1:246, a distributed loss of α ¼ 0:1 m1 ,
and mirror reflectivities of R1 ¼ 90% and R2 ¼ 100%. Therefore, from (6.45), the cold-cavity
photon lifetime of the laser mode is
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285
9.4 Laser Power
τc ¼
nl
1:246 10 102
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ 6:63 ns:
pffiffiffiffiffiffiffiffiffiffi ¼
cðαl ln R1 R2 Þ 3 108 ð0:1 10 102 ln 0:9 1Þ
Because Nd:YAG is a four-level system which has σ a ¼ 0, it has N sp ¼ 1 as can be seen from
(9.15). The photon energy at the λ ¼ 1:064 μm laser wavelength is
hv ¼
1:2398
eV ¼ 1:165 eV:
1:064
For an oscillating laser mode that has an output power of Pout ¼ 100 mW, the minimum
possible linewidth set by the Schawlow–Townes limit is found using (9.14):
ΔvST ¼
hv
1:165 1:6 1019
N
¼
1 Hz ¼ 6:7 mHz:
sp
2πτ 2c Pout
2π ð6:63 109 Þ2 100 103
This minimum possible oscillating laser mode linewidth is nine orders of magnitude smaller
than the cold-cavity longitudinal linewidth of Δvc ¼ ð2πτ c Þ1 27:9 MHz. The significant line
narrowing is caused by the coherent stimulated emission. However, the Schawlow–Townes
linewidth found above is only the fundamental lower bound limited by the spontaneous
emission noise, which can be approached if all other noise sources are eliminated in the ideal
condition. In practice, the linewidth of an oscillating laser mode is much larger than the
Schawlow–Townes linewidth, though generally much smaller than the cold-cavity linewidth,
because it is broadened by many mechanisms such as the noise from pump power fluctuations,
mechanical vibrations, and temperature fluctuations of the laser.
9.4
LASER POWER
..............................................................................................................
In this section, we consider the output power of a laser. Because the situation of a multimode
laser can be quite complicated due to mode competition, we consider for simplicity only a CW
laser that oscillates in a single longitudinal and transverse mode. The parameters mentioned in
this section are not labeled with mode indices because all of them are clearly associated with the
only oscillating mode. The simple case of a Fabry–Pérot cavity that contains an isotropic gain
medium with a filling factor of Γ as shown in Fig. 9.1 is considered. To illustrate the general
concepts, we consider the situation when the gain medium is uniformly pumped so that the
entire gain medium has a spatially independent gain coefficient.
For the single oscillating mode of the Fabry–Pérot laser considered here, the round-trip gain
factor G is that given by (9.5), and the cavity decay rate γc defined by (6.23) is that given by
(6.46). Therefore,
G2 ¼ exp ð2Γgl γc TÞ:
(9.16)
Because G2 is the net amplification factor of the intracavity field energy, which is proportional
to the intracavity photon number, in a round-trip time T of the laser cavity, we can define an
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286
Laser Oscillation
intracavity energy growth rate, or intracavity photon growth rate, Γg, for the oscillating laser
mode through the relation
G2 ¼ exp ½ðΓg γc ÞT:
(9.17)
We find, by comparing (9.17) with (9.16), the gain parameter of the gain medium:
g¼
2gl cg
¼ :
n
T
(9.18)
By comparing (6.46) with (9.9), we find that
γc ¼ Γ
2g th l
cg
¼ Γ th :
T
n
(9.19)
Note that while the unit of g and g th is per meter, the unit of g and γc is per second.
The relation in (9.18) translates the gain coefficient that characterizes spatially dependent
amplification through the gain medium of a propagating intracavity laser field into an intracavity energy growth rate that characterizes the temporal growth of the energy in a laser mode.
The relation in (9.19) clearly indicates that the threshold intracavity energy growth rate for laser
oscillation is the cavity decay rate:
Γgth ¼ γc :
(9.20)
This relation can also be obtained by applying the threshold condition of G ¼ 1 to the relation in
(9.17). It is easy to understand because for a laser mode to oscillate, the growth of intracavity
photons in that mode through amplification by the gain medium has to completely compensate for
the decay of photons caused by all the loss mechanisms. Therefore, we shall call the energy growth
rate Γg and the cavity decay rate γc , both of which are specific to a laser mode, the gain parameter
and the loss parameter, respectively, of the laser mode. Note that the gain parameter Γg of the laser
mode is reduced by the filling factor Γ from the gain parameter g of the gain medium.
By using temporal growth and decay rates instead of spatial gain and loss coefficients to
describe the characteristics of a laser, we are in effect moving from a spatially distributed
description of the laser to a lumped-device description. In the lumped-device description, a laser
mode is considered an integral entity with its spatial characteristics effectively integrated into
the parameters Γg and γc . The detailed spatial characteristics of the mode are irrelevant and are
lost in this description. Therefore, instead of the intensity of the oscillating laser field, we have
to consider the intracavity photon density, S, of the oscillating laser mode. For a Fabry–Pérot
laser that contains a gain medium of a filling factor Γ so that the average refractive index inside
the cavity is n ¼ Γn þ ð1 ΓÞn0 as defined in (6.3), the average intracavity photon density of
the laser mode is
S¼
nI
,
chv
(9.21)
where I is the spatially averaged intracavity intensity and hv is the photon energy of the
oscillating laser mode.
Because the gain parameter g is directly proportional to the gain coefficient g of the gain
medium, the relation between the unsaturated gain parameter Γg0 and the saturated gain
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9.4 Laser Power
287
parameter Γg of a laser mode in the lumped-device description can be obtained by converting
the relation between g 0 and g discussed in Section 9.3 through the relation in (9.18). Therefore,
for the gain parameter of a laser mode, we have
g¼
g0
Γg0
and Γg ¼
1 þ S=Ssat
1 þ S=Ssat
(9.22)
cg 0
n
(9.23)
where
g0 ¼
is the unsaturated gain parameter of the gain medium and
Ssat ¼
nI sat
n
¼
chv cτ s σ e
(9.24)
is the saturation photon density of the laser mode.
When a CW laser oscillates in the steady state, the value of Γg for the oscillating mode is
clamped at its threshold value of γc , just as the value of g is clamped at g th . Therefore, by setting
Γg to equal γc and using (9.22), we find that the intracavity photon density of a CW laser mode
in steady-state oscillation is
Γg0
1 Ssat ¼ ðr 1ÞSsat , for r 1:
(9.25)
S¼
γc
The dimensionless pumping ratio r represents that a laser is pumped at r times its threshold. It is
defined as
r¼
Γg0 g 0
¼
:
γc
g th
(9.26)
Assuming that the pumping efficiency is the same at transparency, at threshold, and at the
operating point, the pumping ratio can be expressed in terms of the pump power as
r¼
Pp Ptrp
tr
Pth
p Pp
,
(9.27)
where Ptrp is the pump power for the gain medium to reach transparency, Pth
p is that for the laser to
reach its threshold, and Pp is the pump power at the operating point. Note that (9.25) is valid only
for r 1 when the laser oscillates because only then is the laser gain saturated. For r < 1, the laser
does not reach threshold. The laser cavity is then filled with spontaneous photons at a density that
is small in comparison to the high density of coherent photons when the laser oscillates at r 1.
From the intracavity photon density of the oscillating laser mode, we can easily find the total
intracavity energy contained in this mode:
U mode ¼ hvV mode S,
(9.28)
where V mode is the volume of the oscillating mode. The mode volume can be found by
integrating the normalized intensity distribution of the mode over the three-dimensional
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288
Laser Oscillation
space defined by the laser cavity; it is usually a fraction of the volume of the cavity. The
output power of the laser is simply the coherent optical energy emitted from the laser per
second. Therefore, it is simply the product of the mode energy and the output-coupling rate,
γout , of the cavity:
Pout ¼ γout U mode ¼ γout hvV mode S ¼ ðr 1Þγout hvV mode Ssat :
(9.29)
The output-coupling rate is also called the output-coupling loss parameter because it contributes to the total loss of a laser cavity; it is a fraction of the total loss parameter γc . One can
indeed write γc ¼ γi þ γout , where γi is the internal loss of the laser that does not contribute to
the output coupling of the laser power.
As an example, for the Fabry–Pérot laser that has γc given by
c
1 pffiffiffiffiffiffiffiffiffiffi
(9.30)
α ln R1 R2
γc ¼
n
l
as expressed in (6.46), we have the internal loss given by γi ¼ cα=n and the output-coupling
loss given by
γout ¼ c pffiffiffiffiffiffiffiffiffiffi
c pffiffiffiffiffi c pffiffiffiffiffi
ln R1 R2 ¼ ln R1 ln R2 ¼ γout, 1 þ γout, 2 ,
nl
nl
nl
where
γout;1 ¼ c pffiffiffiffiffi
ln R1
nl
and
γout, 2 ¼ c pffiffiffiffiffi
ln R2
nl
(9.31)
(9.32)
are the output-coupling losses of mirror 1 and mirror 2, respectively. In this case, γout is the total
output-coupling loss through both mirrors. Therefore, Pout given in (9.29) is the total output
power emitted through both mirrors. For the output power emitted through each mirror, we find
that
Pout;1 ¼ U mode γout, 1 ¼
γout, 1
γ
Pout and Pout, 2 ¼ U mode γout, 2 ¼ out, 2 Pout :
γout
γout
(9.33)
It is convenient to define the saturation output power as
Psat
out ¼ γout hvV mode Ssat :
(9.34)
Using the definition of Ssat in (9.24), it can be shown that
pffiffiffiffiffiffiffiffiffiffi
Psat
out ¼ Psat ln R1 R2 ,
(9.35)
where Psat is the saturation power of the gain medium found by integrating I sat over the crosssectional area of the gain medium. Combining (9.29) with (9.34), we can express the output
laser power in terms of Psat
out as
Pout ¼ ðr 1ÞPsat
out :
(9.36)
Note that Psat
out is not the level at which the output power of a laser saturates. Its physical
meaning can be easily seen from (9.35) and (9.36). From (9.35), we find that the output power
of a laser is Psat
out when the intracavity laser power is at the level Psat of the gain medium. From
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9.4 Laser Power
289
sat
(9.36), we find that Pout ¼ Psat
out when r ¼ 2; in other words, a laser has an output power of Pout
when it is pumped at twice its threshold level.
EXAMPLE 9.6
The Nd:YAG gain medium of the laser described in Examples 9.1 and 9.2 has a saturation
intensity of I sat ¼ 17:3 MW m2 , which stays almost constant for an unsaturated gain coefficient g 0 over the range from 0 to 10 m1. With a cavity length of l ¼ 10 cm, the two cavity
mirrors are chosen such that at the λ ¼ 1:064 μm laser wavelength, the TEM00 Gaussian mode
has a beam waist spot size of w0 ¼ 500 μm located at the center of the Nd:YAG rod, which has
a length of lg ¼ 3 cm. (a) Find the pumping ratio r and the corresponding unsaturated gain
coefficient g 0 required for the laser mode to have an output power of 100 mW. (b) If the laser is
pumped at a level for an unsaturated gain coefficient of g 0 ¼ 10 m1 , what is the pumping ratio
and the output power of the laser mode?
Solution:
For the TEM00 Gaussian mode that has a beam waist spot size of w0 ¼ 500 μm in the Nd:YAG
rod, the Rayleigh range, from (3.69), is
zR ¼
πnw20 π 1:82 ð500 106 Þ2
¼
m ¼ 1:34 m:
λ
1:064 106
Because zR l > lg , the beam spot stays constant throughout the cavity. Therefore, the mode
volume of the oscillating laser mode is
V mode
πw20
π ð500 106 Þ2
¼ Al ¼
l¼
10 102 m3 ¼ 3:93 108 m3 :
2
2
The weighted average refractive index of the laser mode is n ¼ 1:246, from Example 9.2. The
photon energy for λ ¼ 1:064 μm is hv ¼ 1:165 eV, from Example 9.5. With a saturation
intensity of I sat ¼ 17:3 MW m2 , the saturation photon density of the oscillating laser mode is
Ssat ¼
nI sat
1:246 17:3 106
¼
m3 ¼ 3:85 1017 m3 :
chv 3 108 1:165 1:6 1019
The output coupling rate is
γout ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
c pffiffiffiffiffiffiffiffiffiffi
3 108
ln
0:9 1 s ¼ 1:27 108 s1 :
ln R1 R2 ¼ nl
1:246 10 102
The saturation output power is found using (9.34):
Psat
out ¼ γout hvV mode Ssat ¼ 358 mW:
(a) For an output power of Pout ¼ 100 mW, we find by using (9.36) that the required pumping
ratio is
r ¼1þ
Pout
100
¼ 1:28:
sat ¼ 1 þ
Pout
358
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290
Laser Oscillation
From Example 9.1, the threshold gain coefficient is g th ¼ 2:09 m1 . Therefore, by (9.26),
the unsaturated gain coefficient at this pumping ratio is
g 0 ¼ rg th ¼ 1:28 2:09 m1 ¼ 2:68 m1 :
(b) When the laser is pumped to have an unsaturated gain coefficient of g 0 ¼ 10 m1 , by (9.26)
the pumping ratio is
r¼
g0
10
¼
¼ 4:78:
g th 2:09
Therefore, from (9.36), the output laser power is
3
W ¼ 1:35 W:
Pout ¼ ðr 1ÞPsat
out ¼ ð4:78 1Þ 358 10
To explicitly express the output laser power as a function of the pump power, it is necessary
to specify the pumping mechanism and the pumping geometry. Irrespective of the pumping
details, it is generally true that a laser has zero coherent output power but only fluorescence
before it reaches threshold, whereas its coherent output power grows linearly with the pump
power above threshold before nonlinearity occurs at a high pump power. Upon reaching the
threshold, the output laser field also shows dramatic spectral narrowing that accompanies the
start of laser oscillation. According to (9.14) and the discussion following it, the linewidth of an
oscillating laser mode continues to narrow with increasing laser power as the laser is pumped
higher above threshold. The reason is that above threshold the coherent stimulated emission
increases with the pumping ratio, whereas the spontaneous emission, which is proportional to
the population of the upper laser level, is clamped at its threshold value. These are the unique
characteristics that distinguish a laser from other types of light sources, such as fluorescent light
emitters and luminescent light sources. However, a real laser does not have such exact ideal
characteristics, mainly because of the presence of spontaneous emission and nonlinearities in
the gain medium.
Figure 9.5 shows the typical characteristics of the output power Pout of a single-mode laser as
a function of the pump power Pp . The linear relation between Pout and Pp is a consequence of
applying the linear relation between g 0 and Pp to (9.26) for (9.27). As discussed in Section 8.3,
the linear relation between g 0 and Pp is itself an approximation near the transparency point of a
gain medium. As the pump power increases to a sufficiently high level, the unsaturated gain
coefficient of a medium cannot continue to increase linearly with the pump power because of
the depletion of the ground-level population. Therefore, we should expect that the output power
of a laser will not continue its linear increase with the pump power but will increase less than
linearly with the pump power at high pumping levels. On the other hand, once the gain medium
of a laser is pumped so that its upper laser level begins to be populated, it emits spontaneous
photons regardless of whether the laser is oscillating or not. Clearly, the output power of a laser
that is pumped below threshold is not exactly zero because fluorescence from spontaneous
emission is already emitted from the laser before the laser reaches threshold. Though this
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9.4 Laser Power
291
Figure 9.5 Typical characteristics of the output
power of a single-mode laser as a function of the
pump power.
fluorescence is incoherent and its power is generally small for a practical laser, it is significant
for a laser below and right at threshold. Above threshold, it is the major source of incoherent
noise for the coherent field of the laser output.
The overall efficiency of a laser, known as the power conversion efficiency, is
ηc ¼
Pout
:
Pp
(9.37)
The approximately linear dependence of the laser output power on the pump power above
threshold leads to the concept of the differential power conversion efficiency, also known as the
slope efficiency, of a laser, defined as
ηs ¼
dPout
:
dPp
(9.38)
Referring to the laser power characteristics shown in Fig. 9.5, the threshold of a laser can usually
be lowered by increasing the finesse of the laser cavity, thus lowering the values of γc and γout , but
only at the expense of reducing the differential power conversion efficiency of the laser. In the
linear region of the laser power characteristics, ηs is clearly a constant that is independent of the
operating point of the laser. By contrast, ηc increases with the pump power, but ηc is always
smaller than ηs in the linear region. At high pumping levels where the laser output power does not
increase linearly with the pump power because of nonlinearity, ηs is no longer independent of the
operating point. It can even become smaller than ηc in certain unfavorable situations.
EXAMPLE 9.7
The Nd:YAG laser considered in Example 9.5 is optically pumped from two sides of the laser
rod with two diode laser arrays at the 808 nm pump wavelength. Because the Nd:YAG laser is a
four-level system, its transparency pump power is zero, Ptrp ¼ 0. Furthermore, the pumping
ratio is approximately proportional to the pump power: r / Pp . It is found that the pump power
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292
Laser Oscillation
required to reach the pumping ratio for an unsaturated gain coefficient of g 0 ¼ 10 m1 is
Pp ¼ 16:5 W. Use the data obtained in Example 9.6 to answer the following questions. (a) Find
the threshold pump power. (b) Find the conversion efficiency and the slope efficiency when the
laser has an output power of Pout ¼ 100 mW as in Example 9.6(a). (c) Find the conversion
efficiency and the slope efficiency when the laser has an unsaturated gain coefficient of
g 0 ¼ 10 m1 as in Example 9.6(b).
Solution:
From Example 9.6(b), r = 4.78 for g 0 ¼ 10 m1 . Therefore, r ¼ 4:78 for Pp ¼ 16:5 W.
Because Nd:YAG is a four-level system, it is transparent without pumping. Therefore,
Ptrp ¼ 0. From (9.27), we have
r¼
Pp Ptrp
Pth
p
Ptrp
¼
Pp
,
Pth
p
and
dr
r
4:78 1
¼
¼
W ¼ 0:29 W1 :
dPp Pp 16:5
(a) The laser reaches its threshold when the pumping ratio is r th ¼ 1. Therefore, the threshold
pump power is
Pth
p ¼
rth
1
W¼
W ¼ 3:45 W:
0:29
0:29
(b) From Example 9.6(a), we find that r ¼ 1:28 for Pout ¼ 100 mW. At this pumping ratio,
Pp ¼ rPth
p ¼ 1:28 3:45 W ¼ 4:42 W:
Therefore, from (9.37), the power conversion efficiency is
ηc ¼
Pout 100 103
¼
¼ 2:26%:
Pp
4:42
From Example 9.6, we have Psat
out ¼ 358 mW. Using (9.38) and (9.36), we find that the
slope efficiency is
ηs ¼
dPout
dr sat
¼
P ¼ 0:29 358 103 ¼ 10:4%:
dPp
dPp out
(c) When the laser is pumped with a pump power of Pp ¼ 16:5 W to give an unsaturated gain
coefficient of g 0 ¼ 10 m1 , we find r = 4.78 and Pout ¼ 1:35 W from Example 9.6(b).
Therefore, from (9.37), the power conversion efficiency is
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Problems
ηc ¼
293
Pout 1:35
¼
¼ 8:18%:
Pp
16:5
The slope efficiency is the same as that found in (b):
ηs ¼
dPout
dr sat
¼
P ¼ 0:29 358 103 ¼ 10:4%:
dPp
dPp out
Problems
9.1.1 A He–Ne laser has a Fabry–Pérot cavity formed by two mirrors of reflectivities R1 ¼
95% and R2 ¼ 100% at its laser wavelength of λ ¼ 632:8 nm. The cavity length is
l ¼ 32 cm. The effective refractive index of the He–Ne gas is n 1. The TEM00
Gaussian laser mode has a distributed optical loss of α ¼ 0:05 m1 . Find the threshold
gain coefficient of this laser mode.
9.1.2 An optical-fiber laser emitting at λ ¼ 1:53 μm has a ring cavity as shown in Fig. 6.1(d). It
has one input–output coupler that has a coupling efficiency of η ¼ 10%. The fiber loop
has a total length of l ¼ 10 m, which contains a gain section of a length lg ¼ 1 m. The
effective index of the fiber laser mode is n ¼ 1:47 and the distributed loss is
α ¼ 10 dB km1 . What is the threshold gain coefficient of this laser mode?
9.1.3 A GaAs/AlGaAs semiconductor laser emitting at λ ¼ 860 nm has a Fabry–Pérot cavity
formed by two flat, cleaved surfaces of reflectivities R1 ¼ R2 ¼ 32% for the TE0 mode of
the GaAs/AlGaAs waveguide. The gain region is the GaAs waveguide core, which is
pumped uniformly throughout the cavity length such that the cavity and the gain medium
have the same length of l ¼ lg ¼ 350 μm. The laser oscillates in the single transverse TE0
waveguide mode, which has a confinement factor of Γ ¼ 0:3 defined by the overlap
factor of the TE0 mode intensity profile with the waveguide core gain region. The
distributed loss is α ¼ 25 cm1 . Find the threshold gain coefficient of this laser mode.
If one of the cleaved cavity surfaces is optically coated for 100% reflectivity, what is the
threshold gain coefficient?
9.2.1 The optical gain of a homogeneously broadened laser is contributed by a discrete optical
transition between two atomic energy levels at a transition resonance frequency of ω21 .
A longitudinal mode q of the laser has its cold-cavity frequency tuned to the transition
resonance frequency such that ωcq ¼ ω21 . When the laser is pumped above the threshold
for this mode to oscillate, what is the oscillating frequency of the laser? How much is the
frequency shift due to mode pulling?
9.2.2 The optical gain in a semiconductor laser medium is contributed by excess electrons and
holes in the conduction and valence bands, respectively, of the semiconductor. The gain
is determined by the excess carrier concentration N, which is the density of the
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294
Laser Oscillation
electron–hole pairs in excess of the thermal-equilibrium concentrations of electrons and
holes. As a result, the relationship between the real and imaginary parts of the resonant
susceptibility is not simply the Lorentzian function that characterizes a discrete atomic
transition. Nevertheless, an optical gain still causes a change in the refractive index of the
medium. This effect is usually described by an experimentally measured antiguidance
factor, also known as the linewidth enhancement factor, defined as
b¼
∂n0 =∂N
2ω ∂n0 =∂N
4π ∂n0 =∂N
¼
¼
,
∂n00 =∂N
c ∂g=∂N
λ ∂g=∂N
(9.39)
where n0 and n00 are, respectively, the real and imaginary parts of the refractive index
of the medium, and g is the gain coefficient. A GaAs/AlGaAs semiconductor laser
emitting at λ ¼ 850 nm has a Fabry–Pérot cavity, which is pumped uniformly so
that the cavity and the gain medium have the same length of l ¼ lg ¼ 300 μm. The
gain medium has an antiguidance factor of b ¼ 3:5. The effective refractive index is
n ¼ 3:65 when the laser medium is pumped to transparency at λ ¼ 850 nm. The laser
is pumped to give a gain coefficient of g ¼ 5 104 m1 . Besides shifting the frequency of each longitudinal mode, the mode-pulling effect caused by the antiguidance
factor changes the longitudinal mode frequency spacing. Find the frequency shift of a
longitudinal mode at the λ ¼ 850 nm laser wavelength. Find the change in the longitudinal mode frequency spacing.
9.3.1 A GaAs/AlGaAs vertical-cavity surface-emitting semiconductor laser emitting at λ ¼
850 nm has a very short cavity. Its gain region is composed of a few thin quantum wells,
and its reflective mirrors are distributed Bragg reflectors of periodic structures. For the
longitudinal mode frequencies, the effective physical length of the cavity is leff ¼ 1:2 μm
and the effective refractive index is neff ¼ 3:52. The laser is pumped to give a gain
bandwidth of Δλg ¼ 48 nm above the laser threshold. How many longitudinal modes
oscillate?
9.3.2 A He–Ne laser has a Doppler-broadened gain bandwidth of Δνg ¼ 1:5 GHz at its laser
wavelength of λ ¼ 632:8 nm. The laser has a cavity length of l ¼ 32 cm.
(a) It is pumped at a level such that the peak unsaturated gain coefficient is four times the
threshold gain coefficient: g max
¼ 4g th . How many longitudinal modes have their
0
unsaturated gain coefficients pumped above the threshold? How many longitudinal
modes oscillate?
(b) If a longitudinal mode frequency is tuned to the frequency of the gain peak, what is
the value of g max
for the laser to oscillate only in this mode?
0
9.3.3 An Er:fiber laser emitting at λ ¼ 1:53 μm has a cold-cavity linewidth of Δνc ¼ 520 kHz.
It is doped with an Er3þ ion concentration of N t ¼ 2:2 1024 m3 . At λ ¼ 1:53 μm, the
absorption cross section is σ a ¼ 5:7 1025 m2 , and the emission cross section is
σ e ¼ 7:9 1025 m2 . The gain coefficients of its oscillating modes are saturated at
g ¼ 0:25 m1 . The population density of the upper laser level for this gain coefficient
can be found using (8.42). What is the minimum possible linewidth set by the Schawlow–
Townes limit for an oscillating laser mode that has an output power of Pout ¼ 1 mW?
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Problems
295
9.4.1 A Ti:sapphire laser consists of a Ti:sapphire crystal of a length lg ¼ 2 cm in a Fabry–
Pérot cavity, which has a physical length of l ¼ 16 cm defined by two mirrors of
reflectivities R1 ¼ 100% and R2 ¼ 95% at the laser emission wavelength of
λ ¼ 800 nm. The TEM00 Gaussian mode defined by the cavity has a beam waist spot
size of w0 ¼ 150 μm located at the center of the Ti:sapphire crystal, which has a
refractive index of n ¼ 1:76. The end surfaces of the crystal are antireflection coated to
eliminate undesirable losses. At the λ ¼ 800 nm laser wavelength, the Ti:sapphire crystal
has an emission cross section of σ e ¼ 3:4 1023 m2 and an absorption cross section of
σ a 8 1026 m2 . Over the range of laser operation considered here, the saturation
lifetime can be taken as τ s τ 2 ¼ 3:2 μs. The distributed loss of the laser cavity,
including the absorption of the Ti:sapphire crystal at λ ¼ 800 nm, is α ¼ 0:1 m1 . The
laser is optically pumped at the pump wavelength of λp ¼ 532 nm.
(a) Find the threshold gain coefficient of this laser.
(b) Find the saturation output power of this laser.
(c) What are the pumping ratio and the unsaturated gain coefficient required for the laser
to have an output power of Pout ¼ 1 W?
(d) The transparency pump power of the laser is Ptrp ¼ 1:4 W, and the threshold pump
power is Pth
p ¼ 5:0 W. What is the pump power that is required for Pout ¼ 1 W?
(e) What are the power conversion efficiency and the slope efficiency when the laser has
an output power of Pout = 1 W?
9.4.2 The Ti:sapphire laser described in Problem 9.4.1 is pumped to have an unsaturated gain
coefficient of g 0 ¼ 5 m1 .
(a) What are the pumping ratio and the pump power?
(b) Find the output laser power at this pumping level.
(c) What are the power conversion efficiency and the slope efficiency at this
operating point?
9.4.3 A semiconductor laser is pumped by current injection. The injected current generates
excess electron–hole pairs in the active region of the laser. The excess electron–hole pairs
act as the source of the optical gain. When the details of the laser structure and the
parameters of the gain medium are known, the power and efficiency of a semiconductor
laser can be analyzed as discussed in Section 9.4. Alternatively and equivalently, the
output power of a semiconductor laser can be found by considering that one photon is
emitted when an electron–hole pair recombines radiatively. Thus, for a semiconductor
laser,
Pout ¼ ηinj
γout hv
ðI I th Þ,
γc e
(9.40)
where ηinj is the current injection efficiency, γout is the output coupling rate, γc is the
cavity decay rate, hv is the laser photon energy, e is the electronic charge, I is the
injection current, and I th is the threshold injection current for the laser to start oscillating.
The injection efficiency ηinj is the fraction of the total injection current that actually
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296
Laser Oscillation
contributes to the generation of useful electron–hole pairs in the active region of the laser.
If the bias voltage of the laser is V, the power conversion efficiency is
Pout Pout
γout hv
I th
ηc ¼
¼ ηinj
,
(9.41)
¼
1
Pp
VI
γc eV
I
and the slope efficiency is
ηs ¼
dPout dPout
γ hv
¼ ηinj out
¼
:
dPp
VdI
γc eV
(9.42)
Now, consider the GaAs/AlGaAs laser described in Problem 9.1.3 but with R1 ¼ 1 and
R2 ¼ 0:32. The effective refractive index of the laser mode is n ¼ 3:63. The injection
efficiency is ηinj ¼ 0:7, the threshold current is I th ¼ 20 mA, and the bias voltage is
V ¼ 2 V.
(a) Find the output laser power for an injection current of I ¼ 40 mA.
(b) What are the power conversion efficiency and the slope efficiency at this
operating point?
Bibliography
Davis, C. C., Lasers and Electro-Optics: Fundamentals and Engineering, 2nd edn. Cambridge: Cambridge
University Press, 2014.
Iizuka, K., Elements of Photonics for Fiber and Integrated Optics, Vol. II. New York: Wiley, 2002.
Liu, J. M., Photonic Devices. Cambridge: Cambridge University Press, 2005.
Milonni, P. W. and Eberly, J. H., Laser Physics. New York: Wiley, 2010.
Rosencher, E. and Vinter, B., Optoelectronics. Cambridge: Cambridge University Press, 2002.
Saleh, B. E. A. and Teich, M. C., Fundamentals of Photonics. New York: Wiley, 1991.
Siegman, A. E., Lasers. Mill Valley, CA: University Science Books, 1986.
Silfvest, W. T., Laser Fundamentals. Cambridge: Cambridge University Press, 1996.
Svelto, O., Principles of Lasers, 5th edn. New York: Springer, 2010.
Verdeyen, J. T., Laser Electronics, 3rd edn. Englewood Cliffs, NJ: Prentice-Hall, 1995.
Yariv, A. and Yeh, P., Photonics: Optical Electronics in Modern Communications. Oxford: Oxford University
Press, 2007.
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Cambridge Books Online
http://ebooks.cambridge.org/
Principles of Photonics
Jia-Ming Liu
Book DOI: http://dx.doi.org/10.1017/CBO9781316687109
Online ISBN: 9781316687109
Hardback ISBN: 9781107164284
Chapter
10 - Optical Modulation pp. 297-361
Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.011
Cambridge University Press
10
Optical Modulation
10.1 TYPES OF OPTICAL MODULATION
..............................................................................................................
Optical modulation allows one to control an optical wave or to encode information on a carrier
optical wave. The inverse process that recovers the encoded information is demodulation. There
are many types of optical modulation, which can be categorized in several different ways.
1. According to the particular optical-field parameter being modulated, optical modulation can be
categorized into different modulation schemes: phase modulation, frequency modulation,
polarization modulation, amplitude modulation, spatial modulation, and diffraction modulation.
2. Depending on whether the information is encoded in the analog or digital form, optical
modulation can be either analog modulation or digital modulation.
3. Optical modulation can be categorized as direct modulation or external modulation. Direct
modulation is directly performed on an optical source, which is usually a light-emitting
diode (LED) or a laser, without using a separate optical modulator. External modulation is
performed on an optical wave using a separate optical modulator to change one or more
characteristics of the wave.
4. Optical modulation is accomplished by varying the optical susceptibility of the modulator
material. Depending on whether the real or imaginary part of the susceptibility is responsible
for the functioning of the modulator, optical modulation can be categorized as refractive
modulation or absorptive modulation. Refractive modulation is performed by varying the
real part of the susceptibility, thus varying the refractive index of the material; absorptive
modulation is performed by varying the imaginary part of the susceptibility, thus varying the
absorption coefficient of the material.
5. Optical modulation can be categorized according to the physical mechanism behind the
change of the optical susceptibility, such as electro-optic modulation, acousto-optic modulation, magneto-optic modulation, all-optical modulation, and so forth.
6. Depending on the geometric relation between the modulating signal and the modulated
optical wave, optical modulation can be transverse modulation or longitudinal modulation.
In transverse modulation, the signal is applied in a direction perpendicular to the propagation
direction of the optical wave. In longitudinal modulation, the signal is applied along the
propagation direction of the optical wave.
7. Optical modulation can be performed on unguided or guided optical waves. Correspondingly, the structure of an optical modulator can take the form of a bulk or waveguide device.
A bulk modulator is used to modulate an unguided optical wave. A waveguide modulator is
used to modulate a guided optical wave.
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298
Optical Modulation
Optical switching is a special form of optical modulation. Generally speaking, optical
switching is large-signal digital optical modulation that results in the switching between two
or more discrete values of an optical parameter or between two or more optical modes. It can be
performed on any type of optical modulation. The characteristic of the optical wave being
switched can be its phase, frequency, amplitude, polarization, propagation direction, or spatial
pattern. Optical switching can also be performed between two or more normal modes in a
waveguide structure.
10.2 MODULATION SCHEMES
..............................................................................................................
As discussed in Section 1.7, an unguided optical field is characterized by its polarization ^e ,
magnitude jE j, phase φE , wavevector k, and frequency ω:
Eðr, tÞ ¼ Eðr, tÞ exp ðik r iωtÞ
¼ ^e jEðr, tÞjeiφE ðr, tÞ exp ðik r iωtÞ:
(10.1)
The total phase of this field is that given in (1.83):
φðr; t Þ ¼ k r ωt þ φE ðr; tÞ:
(10.2)
As described in (3.25), a guided optical field propagating along the z direction can be expressed
as a linear superposition of normal modes:
X
Eðr, tÞ ¼
Aν ðz, tÞE^ν ðx, yÞ exp ðiβν z iωtÞ
ν
X
(10.3)
¼
E^ν ðx, yÞjAν ðz, tÞjeiφAν ðz, tÞ exp ðiβν z iωtÞ:
ν
The field in a mode is also characterized by five field parameters: the vectorial mode field
pattern E^v ðx; yÞ, the magnitude jAv ðz; tÞj of the complex mode amplitude Av ðz; tÞ, the phase
φAv ðz; tÞ of the complex mode amplitude Av ðz; t Þ, the mode propagation constant βv , and the
frequency ω. The total phase of the field in mode v is
φv ðz; t Þ ¼ βv z ωt þ φAv ðz; t Þ:
(10.4)
Optical modulation can be performed on any of the field parameters. Therefore, there exist
many modulation techniques based on different schemes. Each modulation scheme has been
further developed into many advanced modulation formats.
In general, the concept of a modulation scheme or format that is developed for an
electromagnetic carrier wave at a low frequency, such as a radio frequency, can be adapted
and applied to optical modulation. Also common to low-frequency carriers and optical
carriers is that the modulation signal can be either analog or digital. The three basic modulation schemes for all carrier frequencies are phase modulation (PM), frequency modulation
(FM), and amplitude modulation (AM) for analog modulation, which take the forms of phaseshift keying (PSK), frequency-shift keying (FSK), and amplitude-shift keying (ASK) for
digital modulation.
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10.2 Modulation Schemes
299
Due to the differences between optical waves and low-frequency electromagnetic waves
regarding the field characteristics and the material properties in their respective spectral regions,
some schemes and certain considerations are specific to optical modulation. In addition to the
three basic modulation schemes of phase modulation, frequency modulation, and amplitude
modulation, optical modulation can also be performed on the polarization ^e of the field for
polarization modulation, on the spatial distribution jE ðr; t Þj of the field for spatial modulation,
and on the direction k^ of wave propagation for diffraction modulation.
Because of the dispersive nature and the intrinsic coupling between the real and imaginary
parts of the optical susceptibility, as well as its tensorial nature in the case of an anisotropic
crystal, a modulation signal often affects more than one parameter of the modulated optical
field. For example, amplitude modulation that is carried out by varying the absorption or
amplification coefficient, through varying χ 00 , of the material in a modulator is usually accompanied by a variation in χ 0 , thus varying the refractive index and resulting in a modulation on
the phase of the optical wave. This is the case for direct modulation discussed in Section 10.3.
As another example, phase modulation using a modulator made of an anisotropic crystal can
sometimes be accompanied by a polarization change of the optical field. In any event, a
modulation scheme is chosen based on the field parameter on which we intend to code the
information. The accompanying modulation on other field parameters is a side effect that has to
be avoided or suppressed as much as possible, if it is unavoidable.
Phase modulation is the most fundamental of all modulation schemes. By controlling the
optical phase while properly manipulating the optical wave, a desired modulation on any other
field parameter can be accomplished. On the other hand, certain field parameters can be directly
modulated without changing the optical phase. The concepts of basic optical modulation
schemes are described in the following. The techniques and the physical mechanisms that
can be used for these modulation schemes are discussed in later sections.
10.2.1 Phase Modulation
A phase-modulated optical field at a fixed location, taken to be r ¼ 0 for simplicity of
expression, is a function of time of the form:
Eð0; tÞ ¼ ^e jE j exp ½iφE ðt Þ iωt,
(10.5)
where the time-varying phase φE ðt Þ carries the encoded information, whereas ^e , jE j, and ω do
not vary with time. In analog phase modulation, φE ðt Þ is a continuous function of time; in
digital phase modulation, i.e., PSK, φE ðt Þ changes stepwise with time. The temporal characteristics of the optical field under analog and digital phase modulation are shown in Figs. 10.1(a)
and (b), respectively. The magnitude and frequency of the carrier field stay constant under
phase modulation because only the phase varies with time.
In phase modulation, the largest meaningful phase change is 2π because phase is periodic
with a period of 2π; therefore, the range of phase modulation is usually chosen to be from 0 to
2π or from π to π. In PSK, the 2π phase range is equally divided into discrete levels
representing different digital values. The phase shifts from one discrete level to another discrete
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Optical Modulation
Figure 10.1 (a) Analog phase modulation with an analog signal. (b) Digital phase modulation using two
discrete phases separated by π for BPSK. The field magnitude and the carrier frequency stay constant while the
phase varies with time.
level. In binary PSK (BPSK), two discrete phases separated by π, such as f0; π g or
fπ=2; 3π=2g, are used to respectively represent the two binary bits of 0 and 1, as shown in
Fig. 10.1(b). In quadrature PSK (QPSK), four discrete phases that are equally spaced at an
interval of π=2, such as f0; π=2; π; 3π=2g or fπ=4; 3π=4; 5π=4; 7π=4g, are used to represent
the four possible two-bit combinations of f00; 01; 10; 11g by encoding two bits with each phase.
Optical phase modulation is normally accomplished through refractive modulation. By
modulating the refractive index of a material through which an optical wave propagates, the
phase of the wave can be modulated. The physical mechanisms that can be used for this purpose
are discussed in Section 10.4.
10.2.2 Frequency Modulation
A frequency-modulated optical field has a time-varying frequency of ωðt Þ that carries the
encoded information:
Eð0; t Þ ¼ ^e jE j exp ½iφE iωðtÞt ,
(10.6)
where ^e , jE j, and φE do not vary with time. In analog frequency modulation, ωðt Þ varies
continuously with time; in digital frequency modulation, i.e., FSK, ωðt Þ shifts abruptly from
one frequency to another. In binary FSK (BFSK), two different frequencies are used to
represent the two binary bits of 0 and 1 for a digital signal. More than two frequencies can
be used to digitize a signal in multiple symbols; for example, in quadrature FSK (QFSK), four
frequencies are used to represent the four possible two-bit combinations of f00; 01; 10; 11g by
encoding two bits with each frequency.
Figures 10.2(a) and (b) show the temporal characteristics of the optical field under analog
frequency modulation and BFSK, respectively. The magnitude of the carrier field stays constant
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301
10.2 Modulation Schemes
Figure 10.2 (a) Analog frequency modulation. (b) Digital frequency modulation using two different
frequencies for BFSK. The field magnitude stays constant while the carrier frequency varies with time.
while the frequency varies with time. Note the fine differences in the characteristics of the
modulated waveforms between frequency modulation and phase modulation by comparing
Fig. 10.2 to Fig. 10.1.
Frequency modulation can be achieved by phase modulation over a large phase range
because, from (1.87),
ωðt Þ ¼ ∂φ
∂φ
¼ω E:
∂t
∂t
(10.7)
In contrast to the case for phase modulation discussed above, however, the modulated phase
change for frequency modulation is not limited to a range of 2π. Instead, the range of phase
change is a function of the magnitude and the duration of the frequency shift from the original,
unshifted carrier frequency. For example, for BFSK that shifts the frequency between ω and ω0 ,
a time-varying phase of φE ðtÞ ¼ ðω0 ωÞðt t 0 Þ has to be maintained from the time t 0 when
the frequency is shifted from ω to ω0 until the time when the frequency is shifted back to ω.
EXAMPLE 10.1
The phase of a polarized plane optical field is temporally modulated by a sinusoidal variation of
a modulation amplitude φ0 and a modulation frequency Ω as φE ðt Þ ¼ φ0 sin Ωt. What happens
to the polarization of this modulated optical field? What happens to the magnitude and intensity
of this optical field? Does this phase modulation result in frequency modulation? What happens
to the frequency of this optical field in the time domain and in the frequency domain?
Solution:
The modulation is imposed only on the phase of the field such that
E ðt Þ ¼ ^e E exp½iφE ðt Þ ¼ ^e E exp ðiφ0 sin Ωt Þ:
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Optical Modulation
Clearly, the polarization vector ^e is not affected by the phase modulation; thus, it remains a
constant of time. The field magnitude jE j is not affected by the phase modulation, either;
therefore, both the field magnitude and the intensity, which is I / jE j2 , remain constants
of time.
By contrast, this time-varying phase modulation does result in frequency modulation:
ωðt Þ ¼ ∂φ
∂φ
¼ ω E ¼ ω φ0 Ω cos Ωt:
∂t
∂t
In the time domain, we find that the frequency of this optical field varies sinusoidally with time
around the center optical carrier frequency ω as ωðt Þ ¼ ω φ0 Ω cos Ωt. To find the frequency
components in the frequency domain, we use the identity:
exp ðiφ0 sin ΩtÞ ¼
∞
X
J q ðφ0 Þ exp ðiqΩt Þ,
q¼∞
where J q is the qth-order Bessel function of the first kind, which has the property that
J q ¼ ð1Þq J q . Therefore, we can express the phase-modulated optical field as
Eðt Þ ¼ ^e jE j(
exp ðiφ0 sin Ωt iωt Þ
)
∞
h
i
X
q iðωþqΩÞt
iðωqΩÞt
iωt
þ
J q ðφ0 Þ e
þ ð1Þ e
¼ ^e jE j J 0 ðφ0 Þe
:
q¼1
It can be seen that in the frequency domain, the sinusoidal phase modulation generates a series
of side bands at the harmonics of the modulation frequency Ω on both the low-frequency and
high-frequency sides of the center optical carrier frequency ω.
10.2.3 Polarization Modulation
Information can also be encoded on the polarization of an optical field through polarization
modulation so that the polarization vector is a time-varying function:
Eð0; t Þ ¼ ^e ðt ÞjE j exp ðiφE iωt Þ,
(10.8)
where jE j, φE , and ω do not vary with time. In analog polarization modulation, ^e ðt Þ varies
continuously with time; in digital polarization modulation, known as polarization-shift keying
(PolSK), ^e ðt Þ changes abruptly from one polarization to another. In binary polarization-shift
keying (BPolSK), two orthogonal polarization states are used to represent the two binary bits of
0 and 1 for a digital signal. Multiple polarization states can be used to represent multiple
possible bit combinations; in this situation, the polarization states are not all mutually orthogonal because each polarization state has only one corresponding orthogonal polarization state.
Polarization modulation can be achieved through differential phase modulation on two
orthogonally polarized components of an optical field by using, for example, the electro-optic
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10.2 Modulation Schemes
303
Pockels effect or the magneto-optic Faraday effect. Any orthonormal set of unit polarization
vectors f^e 1 ; ^e 2 g on the plane that is normal to the wave propagation direction k^ can be used to
expand the unit polarization vector ^e on this plane as a linear superposition of two orthogonal
polarizations:
^e ¼ c1 ^e 1 þ c2 ^e 2 ,
(10.9)
where c1 and c2 are two complex constants subject to the normalization condition of
∗
c1 c∗
e 1 ; ^e 2 g basis, the unit polarization vector ^e ⊥ that is orthogonal
1 þ c2 c2 ¼ 1: On the f^
to the unit polarization vector ^e can be expressed as
^e ⊥ ¼ c∗
e 1 c∗
e2:
2^
1^
(10.10)
It is clear that f^e ; ^e ⊥ g is also an orthonormal basis because ^e ^e ∗ ¼ ^e ⊥ ^e ∗
⊥ ¼ 1 and
∗
∗
^e ^e ⊥ ¼ ^e ⊥ ^e ¼ 0. Therefore, the two unit polarization vectors ^e 1 and ^e 2 can be expressed
in terms of the f^e ; ^e ⊥ g basis as
^e 1 ¼ c∗
e þ c2 ^e ⊥ ,
1^
^e 2 ¼ c∗
e c1 ^e ⊥ :
2^
(10.11)
As an example, any polarization state on the xy plane can be represented by the unit vector
^e ¼ ^x cos α þ ^y eiφ sin α given in (1.65), which is the linear superposition of the two orthonormal linear polarization unit vectors ^x and ^y with c1 ¼ cos α and c2 ¼ eiφ sin α. In this case,
^e 1 ¼ ^x , ^e 2 ¼ ^y , and ^e ⊥ ¼ ^x eiφ sin α ^y cos α. As another
example, the linear polarization
pffiffiffi
unit vector ^x can be expressed as ^e ¼ ^x ¼ ð^e þ þ ^e Þ= 2 in terms of p
the
ffiffiffi linear superposition of
the orthonormal circular polarization unit vectors with c1 ¼ c2 ¼ 1= 2. In this case, ^e 1 ¼ ^e þ ,
pffiffiffi
^e 2 ¼ ^e , and ^e ⊥ ¼ i^y ¼ ð^e þ ^e Þ= 2.
When the phases of the two orthogonally polarized field components are differentially
modulated, the polarization vector of the modulated optical wave becomes a function of time:
h
i
^e m ðtÞ ¼ c1 eiφ1 ðtÞ ^e 1 þ c2 eiφ2 ðtÞ ^e 2 ¼ c1 ^e 1 þ c2 eiΔφðtÞ ^e 2 eiφ1 ðtÞ ,
(10.12)
where
ΔφðtÞ ¼ φ2 ðt Þ φ1 ðt Þ
(10.13)
is the time-varying phase difference due to differential phase modulation between the ^e 1 and ^e 2
components of the optical field. By substituting ^e 1 and ^e 2 of (10.11) into (10.12), we can
express the modulated time-varying unit polarization vector ^e m ðt Þ in terms of ^e and ^e ⊥ as
iφ1 ðt Þ
iφ2 ðt Þ
^e m ðtÞ ¼ c1 c∗
^e þ c1 c2 eiφ1 ðtÞ c1 c2 eiφ2 ðtÞ ^e⊥
þ c2 c∗
1e
2e
(10.14)
∗ iΔφ1 ðt Þ iφ1 ðt Þ
^e þ c1 c2 1 eiΔφðtÞ eiφ1 ðtÞ ^e⊥ :
¼ c1 c∗
e
1 þ c2 c2 e
It is clear from (10.14) that ^e m ðtÞ ^e ⊥ 6¼ 0 and ^e m ðt Þ 6¼ ^e when c1 c2 6¼ 0 and ΔφðtÞ 6¼ 2mπ,
resulting in a polarization change caused by differential phase modulation.
As discussed in Section 1.6, the polarization state of a wave depends only on the phase
difference and the magnitude ratio of the two orthogonally polarized field components.
Therefore, the polarization state defined by ^e m ðt Þ is determined by the phase difference ΔφðtÞ
and the magnitude ratio jc1 =c2 j of the ^e 1 and ^e 2 components, and is independent of the common
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304
Optical Modulation
phase factor φ1 ðtÞ. Because the magnitude ratio jc1 =c2 j is not affected by phase modulation, thus
remaining constant, the polarization state can be varied by varying only the phase difference
Δφðt Þ. Consequently, polarization modulation of an optical field can be accomplished through
differential phase modulation on two orthogonally polarized components of the field.
EXAMPLE 10.2
An optical field is initially linearly polarized in the x direction. Find two linearly polarized
components of this polarization in the xy plane that are orthogonal to each other. How does the
polarization of this field change if the two orthogonally polarized components are differentially
phase modulated by a phase difference of π=4, π=2, π, and 2π, respectively?
Solution:
In the xy plane, the two linearly polarized orthogonal components of the unit polarization vector
^e ¼ ^x can be chosen as
^x þ ^y
^x ^y
^e 1 ¼ pffiffiffi and ^e 2 ¼ pffiffiffi ,
2
2
pffiffiffi
,which are arbitrarily chosen to be real vectors such that c1 ¼ c2 ¼ 1= 2 and arbitrarily assigned
in the sequence of ^e 1 and ^e 2 . In the xy plane, the polarization that is orthogonal to ^e ¼ ^x is
^e ⊥ ¼ ^y . From (10.14), if the two orthogonally polarized components are differentially phase
modulated such that φ2 ðt Þ φ1 ðtÞ ¼ Δφðt Þ, the polarization of the field becomes
1 þ eiΔφðtÞ
1 eiΔφðtÞ
^e þ
^e ⊥ eiφ1 ðtÞ
^e m ðt Þ ¼
2
2
1 þ eiΔφðtÞ
1 eiΔφðtÞ
^x þ
^y eiφ1 ðtÞ
¼
2
2
Δφðt Þ
Δφðt Þ
^x i sin
^y eiφ1 ðtÞþiΔφðtÞ=2 :
¼ cos
2
2
The common phase factor φ1 ðtÞ þ ΔφðtÞ=2 only changes the phase of the unit polarization
vector ^e m ðtÞ and does not have an effect on the polarization state of the field. Therefore, we can
ignore this phase factor and consider only the polarization state vector of the differentially
phase-modulated field:
Δφðt Þ
ΔφðtÞ
^x i sin
^
^e 0m ðtÞ ¼ cos
y:
2
2
We find different polarization states for different phase differences:
π
π
π
For Δφ ¼ , ^e 0m ¼ cos ^x i sin ^y ¼ 0:924^x i0:383^y , elliptically polarized;
4
8
8
^x i^y
π 0
π
π
For Δφ ¼ , ^e m ¼ cos ^x i sin ^y ¼ pffiffiffi , circularly polarized;
2
4
4
2
π
π
0
For Δφ ¼ π, ^e m ¼ cos ^x i sin ^y ¼ i^y , linearly polarized parallel to ^e ⊥ ¼ ^y ;
2
2
0
For Δφ ¼ 2π, ^e m ¼ cos π^x i sin π^y ¼ ^x , linearly polarized parallel to ^e ¼ ^x .
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10.2 Modulation Schemes
305
10.2.4 Amplitude Modulation
One of the most common modulation schemes is amplitude modulation, which encodes information on the magnitude of an optical field:
Eð0; tÞ ¼ ^e jE ðtÞj exp ðiφE iωtÞ,
(10.15)
where ^e , φE , and ω do not vary with time. In analog amplitude modulation, jE ðt Þj varies
continuously with time; in digital amplitude modulation, known as amplitude-shift keying
(ASK), jE ðt Þj changes abruptly from one discrete value to another. In binary ASK, two field
magnitudes are used, with the binary bit 1 normally represented by a larger field magnitude and
the bit 0 represented by a smaller magnitude. A special case of binary ASK is on-off keying
(OOK) where the optical field is turned on at a fixed magnitude level for bit 1 and turned off for
bit 0. Multilevel ASK uses multiple discrete field magnitudes to represent multiple possible bit
combinations for each field magnitude to encode one possible combination of an equal number
of bits.
Figures 10.3(a) and (b) show the temporal characteristics of the optical field under analog
modulation and binary ASK, respectively. The magnitude of the carrier field varies with time
while the frequency and phase stay constant. Amplitude modulation leads to intensity modulation (IM), in which the intensity and the power of an optical wave are modulated because the
intensity and power of the wave are proportional to jE ðt Þj2 .
Optical amplitude modulation can be accomplished in many different ways: by direct
modulation on the optical source, as discussed in Section 10.3; by refractive modulation using
any physical mechanism discussed in Section 10.4, followed by proper manipulation of the
optical field; or by absorptive modulation of a material through which the optical wave
propagates, as discussed in Section 10.5.
Figure 10.3 (a) Analog amplitude modulation. (b) Digital amplitude modulation using two different
discrete field magnitudes. Both the carrier frequency and phase of the field stay constant while the magnitude
varies with time.
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306
Optical Modulation
Amplitude modulation on an optical field can be achieved through polarization modulation
by properly selecting a polarization component while filtering out its orthogonal component
after the field is polarization modulated. As an example, consider the polarization-modulated
optical field characterized by the time-varying unit polarization vector ^e m ðt Þ expressed in
(10.14). It is clear that by using a polarizer to select either only the ^e -polarized component or
only the ^e ⊥ -polarized component, the resulting field magnitude is modulated. For instance, by
selecting only the ^e ⊥ -polarized component, the output field has the time-varying magnitude:
h
i Δφðt Þ
iΔφðt Þ
E ¼ 2c1 c2 E sin
,
jE ⊥ ðtÞj ¼ c1 c2 1 e
2 (10.16)
where E is the time-independent field amplitude of the polarization-modulated optical field. The
intensity of this output field is modulated as
I ⊥ ðtÞ ¼ 4jc1 c2 j2 I sin2
ΔφðtÞ
,
2
(10.17)
where I / jE j2 is the time-independent intensity of the polarization-modulated optical wave.
Though polarization modulation of the optical field used in the above example is accomplished
by differential phase modulation, the concept of obtaining amplitude modulation by selecting a
polarization component while rejecting its orthogonal component is generally applicable to any
polarization-modulated optical wave.
Optical amplitude modulation can also be achieved through phase modulation to vary the
coupling or interference between different components of an optical wave.
1. By varying the phase mismatch δ through differential phase modulation on two coupled
modes in a coupler, the coupling efficiency η can be modulated, as discussed in Section 4.6.
Thus, the field amplitude of a mode is modulated. This general concept is applicable to any
mode coupler.
2. By varying the interference of two or multiple waves through differential phase modulation,
the superposition of the interfering waves can be amplitude modulated, as discussed in
Section 5.1. This general concept is applicable to any interferometer discussed in Chapter 5.
In analog amplitude modulation, the optical intensity varies continuously with time. To
faithfully encode the analog information on the carrier optical wave, linearity of the modulation
response is desired. However, as the example in (10.17) shows, the response of an amplitude
modulator generally cannot be linear over the whole range of operation. For this reason, the
linearity requirement for analog modulation often limits the modulation depth to a small linear
range of the modulation response.
In digital amplitude modulation, the optical intensity is switched between two or among
multiple discrete levels. In this case, linearity is not required, but clear separation of the discrete
levels is desired. In binary operation, where the switching takes place between a high-intensity
level of I high and a low-intensity level of I low , it is desired that the ratio I low =I high is as small as
possible while I high is sufficiently large. In digital amplitude modulation using an external
modulator, the binary states are represented by a high transmittance T high and a low
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10.2 Modulation Schemes
307
transmittance T low . The ratio of these two levels is defined as the extinction ratio, which is
usually measured in dB:
ER ¼ 10 log
I low
T low
¼ 10 log
:
I high
T high
(10.18)
A high extinction ratio allows clear separation of the two levels, thus clear identification of the
binary bits. Besides a high extinction ratio, the level of the high transmittance T high has to be
sufficiently high for good performance.
10.2.5 Spatial Modulation
Taking the propagation direction to be the z direction without loss of generality, a spatially
modulated optical field has a time-varying field pattern of E ðx; y; 0; t Þ at the fixed z ¼ 0 location
on the plane that is perpendicular to the propagation direction:
Eðx; y; 0; tÞ ¼ E ðx; y; 0; t Þexp ðiωt Þ ¼ ^e ðx; y; 0; tÞjE ðx; y; 0; t ÞjeiφE ðx;y;0;tÞ eiωt :
(10.19)
Spatial modulation can be on the field polarization, with a space- and time-varying polarization
vector ^e ðx; y; 0; tÞ; on the field magnitude, with a space- and time-varying field magnitude
jE ðx; y; 0; t Þj; or on the phase, with a space- and time-varying field phase φE ðx; y; 0; t Þ. The
spatial variation can be either a continuous function of x and y, or a digitized function of x and y.
If the spatial variation is expressed in terms of a linear superposition of transverse spatial
normal modes, then
X
Eðx; y; 0; t Þ ¼
Av ðt ÞE^v ðx; yÞ exp ðiωt Þ
(10.20)
v
according to (3.25). Thus, spatial modulation can be described as, and be accomplished
through, the temporal variations of the mode expansion coefficients Av ðt Þ.
10.2.6 Diffraction Modulation
As discussed in Section 5.2, an optical grating diffracts an incident optical wave into multiple
diffracted beams; the diffraction angle θq of the qth-order diffracted beam is determined by the
phase-matching condition given in (5.32):
k sin θq ¼ k sin θi þ qK
(10.21)
where k ¼ nω=c is the propagation constant of the optical wave, with n being the refractive
index of the medium; θi is the incident angle of the incoming wave; and K ¼ 2π=Λ is the
wavenumber of the grating, with Λ being the period of the grating. Clearly, the diffraction angle
θq , and thus the diffraction pattern, can be varied by varying the refractive index n, the incident
angle θi , the grating period Λ, or a combination of these parameters. Many refractive
modulation mechanisms, as discussed in Section 10.4, can be used to modulate the refractive
index of the grating material, thus accomplishing diffraction modulation. The grating period Λ
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Optical Modulation
can also be modulated if the grating is not a fixed structure but is generated by an acoustic wave
through an acousto-optic effect, by a low-frequency electric field through an electro-optic
effect, or by a periodic optical intensity pattern through optical interference.
10.3 DIRECT MODULATION
..............................................................................................................
The most straightforward way to encode information on an optical wave is to directly modulate
the optical source. This technique is often applied to an LED or a semiconductor laser, both of
which are current-injection devices driven by current sources. Therefore, an LED or a semiconductor laser can be directly modulated by applying the modulation signal to the injection
current, an approach known as direct current modulation. In this approach, the modulation
signal takes the form of a modulating current, which is added to the DC bias current that
supplies electrical power to the device.
Figure 10.4 shows the schematic circuitry of direct current modulation. The LED or
semiconductor laser is biased at a DC injection current level of I 0 and is modulated with a
time-varying modulation current of I m ðt Þ that carries the modulation signal. Thus the total
current injected into the device is I ðtÞ ¼ I 0 þ I m ðtÞ. The output optical power is
Pout ðt Þ ¼ P0 þ Pm ðt Þ, where P0 is the constant output optical power at the bias current level of
I 0 and Pm ðt Þ is the time-varying component of the modulated output optical power responding to
the modulation current I m ðt Þ. Though the circuitry for direct modulation is the same for an LED
and a semiconductor laser, the characteristics of their modulation responses are very different.
LEDs and semiconductor lasers are both junction diodes that usually have sophisticated
structures for improved performance. In operation as a light source, an LED or semiconductor
laser is injected with a current of I to inject excess electrons and holes, i.e., excess charge
carriers, into an active region of an area A and a thickness d. Taking into consideration the
injection efficiency of the charge carriers, the current density J that actually contributes to
carrier injection is related to the total current that is supplied to the device as
J ¼ ηinj
I
,
A
(10.22)
where ηinj is the carrier injection efficiency, which is determined by the device structure. The
injected current creates an excess carrier density of N ¼ n n0 ¼ p p0 in the active region,
where n0 and p0 are, respectively, the equilibrium electron and hole concentrations in the
absence of current injection, and n and p are the electron and hole concentrations under current
injection.
Figure 10.4 Schematic circuitry of
direct current modulation on an
LED or semiconductor laser.
A resistance in series with the
device is normally used to protect
the device.
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10.3 Direct Modulation
309
The excess carriers recombine through radiative and nonradiative mechanisms with a total
spontaneous carrier recombination rate of γs and a corresponding spontaneous carrier recombination lifetime of τ s :
γs ¼
1
:
τs
(10.23)
The output optical power of an LED is contributed by the spontaneous emission from
spontaneous radiative recombination of the excess carriers. By contrast, the output optical
power of a semiconductor laser comes from the resonant optical field undergoing stimulated
emission in the laser cavity. A semiconductor laser has a threshold for laser oscillation, but an
LED does not have a turn-on threshold. These fundamental differences lead to very different
modulation characteristics between an LED and a semiconductor laser, as discussed below.
Direct current modulation on an LED or a semiconductor laser is a technique of amplitude
modulation because its objective is the modulation of the output optical power. However, the
time-varying current also causes the refractive index of the LED or laser material to vary with
time; consequently, the phase and frequency of the output optical wave are also varied by the
modulation current. The consequence is an accompanying phase and frequency modulation that
is generally undesirable and difficult to avoid because of the nonlinearity and dispersion in the
variation of the refractive index in response to the modulation current. The temporal variation in
the optical frequency results in frequency chirping in the modulated output optical wave. This
effect is more significant for direct current modulation on a semiconductor laser than on
an LED.
10.3.1 Light-Emitting Diode
An LED converts electrical energy to optical energy through the spontaneous emission
resulting from spontaneous recombination of the excess carriers. Because spontaneous emission occurs whenever carriers are excited, an LED starts to emit light once current is injected,
i.e., there is no threshold to turn an LED on. Therefore, the output optical power Pout is directly
proportional to AdN=τ s , which is the total number of excess carriers recombining per second,
and can be expressed as
Pout ¼
ηe hvAd
N,
ηinj τ s
(10.24)
where ηe is the external quantum efficiency, ηinj is the carrier injection efficiency, both
dependent on the structure of the LED, and hv is the photon energy. The temporal variation
of the carrier density in response to the variation in the injection current I is described as
ηinj
dN
J
N
N
I ,
¼ ¼
dt
ed τ s eAd
τs
(10.25)
where e is the electronic charge and J is the injection current density given in (10.22).
The output optical power of an LED as a function of the injection current is known as the
light–current characteristics, or simply the L–I characteristics, also called the power–current
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Optical Modulation
characteristics, or simply the P–I characteristics. The steady-state solution of (10.25) for N
obtained by setting dN=dt ¼ 0 results in the ideal power–current relation for an LED in steadystate operation under DC current injection:
Pout ¼ ηe
hv
I,
e
(10.26)
which indicates that the output power of an LED increases linearly with the injection current.
The L–I characteristics of a representative LED, shown in Fig. 10.5, are not exactly linear
throughout the entire range of operation, however. These characteristics have several important
features that distinguish an LED from a laser. First, there is no threshold in the L–I characteristics of an LED, indicating that an LED is turned on and starts emitting light once it is forward
biased and injected with any amount of current. At moderate current levels, the L–I curve of an
LED is indeed quite linear, as indicated by (10.26). This linearity is useful for analog
modulation of an LED. Nonlinearities in the L–I relationship are usually found at very low
and very high current levels.
For high-speed applications, a large modulation bandwidth is desired. The intrinsic speed of
an LED is primarily determined by the lifetime of the injected carriers in the active region. For
an LED that is biased at a DC injection current level of I 0 and is modulated at a frequency of
Ω ¼ 2πf with a modulation index of m, we can express the total time-dependent current that is
injected to the LED as
I ðtÞ ¼ I 0 þ I m ðtÞ ¼ I 0 ð1 þ m cos Ωt Þ ¼ I 0 þ mI 0 cos Ωt,
Figure 10.5 Light–current characteristics and direct current modulation of a representative LED.
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(10.27)
10.3 Direct Modulation
311
where I m ðt Þ ¼ mI 0 cos Ωt is the modulation current signal, which has an amplitude of
I m ¼ mI 0 . The time-varying output optical power Pout ðt Þ of an LED in response to this
modulation can be found by using (10.24) after solving for N ðt Þ from (10.25). Note that
the time-varying Pout ðt Þ cannot be found directly from (10.26) because (10.26) is valid only
for the steady-state operation of an LED when it is only injected with a DC current. In the
linear response regime under the condition that m 1, the output optical power can be
expressed as
Pout ðt Þ ¼ P0 þ Pm ðt Þ ¼ P0 ½1 þ jr j cos ðΩt φÞ,
(10.28)
where P0 is the constant output optical power found from (10.26) at the bias current level of I 0 ,
Pm ðt Þ ¼ jr jP0 cos ðΩt φÞ is the time-varying component of the modulated output power, jr j
is the magnitude of the response to the modulation, and φ is the phase delay of the response to
the modulation signal. The characteristics of direct current modulation on an LED are illustrated
in Fig. 10.5.
For an LED that is modulated in the linear response regime, the complex response as a
function of the modulation frequency Ω is
r ðΩÞ ¼ jr ðΩÞjeiφðΩÞ ¼
m
:
1 iΩτ s
(10.29)
The frequency response and the modulation bandwidth of an LED are usually measured in
terms of the electrical power spectrum using a broadband, high-speed photodetector that
converts the output optical power of the LED into an output electrical current of the photodetector. In the linear operating regime of the detector, the detector current is linearly proportional to the optical power of the LED. Therefore, the electrical power spectrum of the detector
output is proportional to jr j2 :
Rðf Þ ¼ jr ðf Þj2 ¼
m2
m2
¼
,
1 þ 4π 2 f 2 τ 2s 1 þ f 2 =f 23dB
(10.30)
which has a 3-dB modulation bandwidth of
f 3dB ¼
1
,
2πτ s
(10.31)
as shown in Fig. 10.6.
The spontaneous carrier lifetime τ s is normally on the order of a few hundred nanoseconds
to 1 ns for an LED. Therefore, the modulation bandwidth of an LED is typically in the range
of a few megahertz to a few hundred megahertz. A modulation bandwidth up to 1 GHz can
be obtained with a reduction in the internal quantum efficiency of an LED by reducing
the carrier lifetime to the subnanosecond range. Aside from this intrinsic response speed
determined by the carrier lifetime, the modulation bandwidth of an LED can be further
limited by the parasitic effects from its electrical contacts and packaging, as well as from its
driving circuitry.
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Optical Modulation
Figure 10.6 Normalized current-modulation frequency response of an LED measured in terms of the electrical
power spectrum using a photodetector. The spontaneous carrier lifetime is taken to be τ s ¼ 10 ns for this plot.
EXAMPLE 10.3
An LED emitting at a center wavelength of λ ¼ 850 nm has an external quantum efficiency of
ηe ¼ 21%. Its spontaneous carrier lifetime is τ s ¼ 10 ns. The LED is biased at a DC injection
current of I 0 ¼ 20 mA and is modulated at a modulation frequency of f ¼ 10 MHz with a
modulation current for a modulation index of m ¼ 10%. (a) Find the output power of the LED
at the DC bias point. (b) What is the amplitude of the modulation current? (c) What are the
amplitude of the modulated output power and the phase delay of the response to the current
modulation? (d) Find the 3-dB modulation bandwidth of this LED in terms of its modulation
response in the electrical power spectrum of the photodetector output. (e) At this modulation
frequency, what is the modulation response in the electrical power spectrum of the photodetector used to measure the LED output? What is the normalized modulation response in dB?
Solution:
An LED has no threshold. Therefore, the DC output power is directly proportional to its DC
bias current I 0 , and the modulation index is defined as the ratio of the amplitude I m of the
modulation current to I 0 .
(a) The photon energy at λ ¼ 850 nm is
1239:8
eV ¼ 1:46 eV:
850
The DC output power of the LED is found using (10.26):
hv ¼
hv
I 0 ¼ 0:21 1:46 20 mW ¼ 6:13 mW:
e
(b) The amplitude of the modulation current for m ¼ 10% is
P0 ¼ ηe
I m ¼ mI 0 ¼ 10% 20 mA ¼ 2 mA:
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10.3 Direct Modulation
313
(c) From (10.29), we find
m
0:1
ffi ¼ 8:47 102 ,
jr j ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
6
9 2
1 þ ð2πf τ s Þ
1 þ 2π 10 10 10 10
φ ¼ tan1 ð2πf τ s Þ ¼ tan1 2π 10 106 10 109 ¼ 0:56 rad:
Note that it is always true that jrj < m for an LED at a nonzero modulation frequency. The
amplitude of the modulated output power is
Pm ¼ jrjP0 ¼ 8:47 102 6:13 mW ¼ 519 μW:
and the phase delay of the modulation response is φ ¼ 0:56 rad.
(d) The 3-dB modulation bandwidth of this LED is, from (10.31),
f 3dB ¼
1
1
¼
Hz ¼ 15:9 MHz,
2πτ s 2π 10 109
as seen in Fig. 10.6.
(e) At the modulation frequency of f ¼ 10 MHz, the modulation response in the electrical
power spectrum of the photodetector output is, from (10.30),
Rðf Þ ¼
m2
0:12
¼
¼ 7:2 103 :
1 þ f 2 =f 23dB 1 þ ð10=15:9Þ2
Because Rð0Þ ¼ m2 ¼ 1 102 , the normalized response is
10 log
Rðf Þ
7:2 103
¼ 1:43 dB:
¼ 10 log
Rð0Þ
1 102
10.3.2 Semiconductor Laser
For most applications, it is desired that a semiconductor laser oscillate in a single transverse
mode and a single longitudinal mode. Many practical lasers indeed have such a desirable
characteristic. For a single-mode semiconductor laser that is injected with a current of I, the
temporal characteristics of its carrier density N and its intracavity photon density S can be
described by the coupled rate equations:
ηinj
dN
J
N
N
I gS,
¼ gS ¼
eAd
dt
ed τ s
τs
(10.32)
dS
¼ γc S þ ΓgS,
dt
(10.33)
where e is the electronic charge, τ s is the spontaneous carrier lifetime, γc is the cavity decay rate,
J is the injection current density defined in (10.22), and g is the gain parameter of the gain
region defined in (9.18). The overlap factor Γ appears in the gain term of (10.33) because only
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Optical Modulation
that fraction of the laser mode volume overlaps with the gain region to receive stimulated
amplification.
The threshold condition for a semiconductor laser is that in (9.20) for any laser:
Γgth ¼ γc :
(10.34)
The gain parameter g is a function of the excess carrier density N, which in turn is determined by
the injection current I. The threshold gain parameter gth determines a threshold carrier density
N th at a threshold current density of J th that is supplied by a threshold injection current of I th .
The characteristics of a semiconductor laser in steady-state oscillation above threshold can be
obtained from the steady-state solutions of (10.32) and (10.33) by setting dN=dt ¼ dS=dt ¼ 0.
It is found that in steady-state oscillation above threshold at an injection current of I > I th , the
carrier density and the gain are clamped at their respective threshold values, N ¼ N th and
g ¼ gth , while the intracavity photon density builds up for S 6¼ 0. Most of the concepts
developed in Section 9.4 for laser power characteristics are directly applicable to a semiconductor laser. By directly applying the steady-state conditions of g ¼ gth ¼ γc =Γ and
N ¼ N th ¼ J th τ s =ed ¼ ðηinj τ s =edAÞI th to (10.32) to obtain the steady-state solution of S for
dS=dt ¼ 0, followed by using the relation J ¼ ðηinj =AÞI from (10.22) and the relation
dA ¼ V gain ¼ ΓV mode , the CW output power of a semiconductor laser in steady-state oscillation
under DC current injection can be found using (9.29) and can be expressed as a function of the
injection current:
Pout ¼ ηinj
γout hv
hv
ðI I th Þ ¼ ηe ðI I th Þ,
γc e
e
(10.35)
where ηe ¼ ηinj γout =γc is the external quantum efficiency of the semiconductor laser.
Figure 10.7 shows the power–current characteristics, i.e., the light–current characteristics, of
a representative semiconductor laser. It can be seen from (10.35) that in an ideal situation, the
output power of a semiconductor laser above threshold increases linearly with the injection
current. This characteristic is indeed observed in most semiconductor lasers over a large range
of operating conditions. This linearity is useful for analog modulation of a semiconductor laser
over a large dynamic range. Nonlinearities in the L–I characteristics appear at high injection
current levels.
Like an LED, a semiconductor laser can be directly current modulated. Unlike an LED,
however, the modulation speed of a semiconductor laser is not limited by the spontaneous
carrier lifetime τ s in the active region of the laser. This difference is due to the fact that there is
strong coupling between the carriers and the intracavity laser field. The effective lifetime of the
carriers in an oscillating laser is much shorter than the spontaneous lifetime because of the
stimulated carrier recombination that takes place in a laser. The modulation speed of a
semiconductor laser is primarily determined by the intracavity photon lifetime and the effective
carrier lifetime. Because both the photon lifetime and the effective carrier lifetime of a
semiconductor laser are generally much shorter than the spontaneous carrier lifetime, a semiconductor laser has a higher modulation speed than an LED. Because the stimulated recombination rate increases with the intracavity photon density, the modulation speed of a
semiconductor laser increases with the laser power.
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10.3 Direct Modulation
315
Figure 10.7 Light–current characteristics and direct current modulation of a representative
semiconductor laser.
When a laser is in steady-state oscillation at a DC bias injection current of I 0 > I th in the
absence of modulation, the laser gain and the carrier density are both clamped at their respective
threshold values of gth and N th , but the photon density has a value of S0 corresponding to the
laser output power P0 , which depends on the injection current at the bias point. Under the
dynamical perturbation of a modulation signal, the gain can deviate from gth due to the variations
in the carrier and photon densities caused by the external perturbation. To the first order, the
dependence of the gain parameter on the carrier and photon densities can be expressed as
g ¼ gth þ gn ðN N th Þ þ gp ðS S0 Þ,
(10.36)
where gn is the differential gain parameter characterizing the dependence of the gain parameter
on the carrier density and gp is the nonlinear gain parameter characterizing the effect of gain
compression due to the saturation of the gain by intracavity photons. It has been found
empirically that for a given laser, both gn and gp stay quite constant over large ranges of carrier
density and photon density. For most practical purposes, they can be treated as constants over
the operating range of a laser. These parameters are normally measured experimentally though
they can also be calculated theoretically. Note that gn > 0 but gp < 0.
It is convenient to define a differential carrier relaxation rate, γn , and a nonlinear carrier
relaxation rate, γp , as
γn ¼ gn S0 ,
γp ¼ Γgp S0 :
(10.37)
In addition, we have the cavity decay rate, γc ¼ 1=τ c , and the spontaneous carrier relaxation
rate, γs ¼ 1=τ s . These four relaxation rates can be directly measured for a given semiconductor
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Optical Modulation
laser. They determine the current modulation characteristics of a laser. Note that, for a given
laser, γc and γs are constants that are independent of the laser power, but γn and γp are linearly
proportional to the laser power because they are linearly proportional to the photon density, as
seen in (10.37).
Because a semiconductor laser has a threshold, the modulation index m for a laser that is
biased at a DC injection current of I 0 > I th and is modulated at a frequency of Ω ¼ 2πf is
defined as
I ðt Þ ¼ I 0 þ I m ðt Þ ¼ I th þ ðI 0 I th Þð1 þ m cos Ωt Þ ¼ I 0 þ mðI 0 I th Þ cos Ωt,
(10.38)
where I m ðt Þ ¼ mðI 0 I th Þ cos Ωt is the modulation current signal, which has an amplitude of
I m ðtÞ ¼ mðI 0 I th Þ. Note that the modulation index defined in (10.38) for a semiconductor
laser is different from that defined in (10.27) for an LED because a laser has a threshold but an
LED does not have a threshold. In the regime of linear response, the output power of the laser
can be expressed in the same form as that in (10.28) of a directly modulated LED:
Pout ðt Þ ¼ P0 þ Pm ðt Þ ¼ P0 ½1 þ jr j cos ðΩt φÞ:
(10.39)
The constant output power P0 corresponding to the DC bias current I 0 can be found from
(10.35). However, the time-varying output power Pout ðt Þ cannot be found directly from (10.35)
because the relation in (10.35) is valid only for the steady-state CW oscillation of a laser that is
injected with a DC current. When the injection current is temporally modulated, the timevarying output optical power of the laser in response to the modulation can be found by using
the relation Pout ðt Þ ¼ γout hvV mode Sðt Þ given in (9.29) after solving for the time-varying photon
density SðtÞ from the coupled equations given in (10.32) and (10.33).
For small-signal modulation of m 1, the complex response function of a laser is
r ðΩÞ ¼ jrðΩÞjeiφðΩÞ ¼ mγc γn
,
Ω Ω2r þ iΩγr
2
(10.40)
where Ωr is the relaxation resonance frequency and γr is the total carrier relaxation rate for the
relaxation oscillation of the coupling between the carriers and the intracavity laser field of the
semiconductor laser. They are related to the intrinsic dynamical parameters of the laser as
Ω2r ¼ 4π 2 f 2r ¼ γc γn þ γs γp
(10.41)
γr ¼ γs þ γn þ γp :
(10.42)
and
Because γc and γs are constants while γn and γp are linearly proportional to the laser power, Ωr
and f r are proportional to the square root of the laser power, whereas γr is a linear function of,
but not proportional to, the laser power. The relation between the relaxation resonance
frequency and the carrier relaxation rate is often characterized by a K factor that is independent
of the laser power:
K¼
γr γs
:
f 2r
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(10.43)
10.3 Direct Modulation
317
Figure 10.8 Normalized current-modulation frequency response of a semiconductor laser measured in terms of
the electrical power spectrum using a photodetector. The frequency response of a semiconductor laser depends
on the output laser power, with its 3-dB bandwidth increasing approximately with the square root of the output
pffiffiffiffiffiffiffiffi
power. These curves are generated with the relations: f r ðG H zÞ ¼ 5 Pout and γs ðns1 Þ ¼ 1:5 þ 11 Pout , where
Pout is measured in mW.
The modulation power spectrum of a semiconductor laser is
Rðf Þ ¼ jr ðf Þj2 ¼
m2 γ2c γ2n
16π 4 f 2 f 2r
2
þ 4π 2 f 2 γ2r
:
(10.44)
As shown in Fig. 10.8, this spectrum has a resonance peak at
f pk ¼
f 2r
γ2
r2
8π
1=2
(10.45)
and a 3-dB modulation bandwidth of
f 3dB ¼ 1 þ
pffiffiffi
2
1=2
γ2
f 2r pffiffirffi
8 2π 2
1=2
1:554 f pk :
(10.46)
1=2
Because f r γr =2π for most lasers and because f r / P0 , the modulation bandwidth of a
1=2
semiconductor laser increases with the output laser power and scales roughly as f 3dB / P0 .
An intrinsic modulation bandwidth on the order of a few gigahertz is common for a
semiconductor laser. A high-speed semiconductor laser can have a bandwidth larger than
20 GHz. Because the intrinsic modulation bandwidth of a semiconductor laser is significantly
larger than that of an LED, it is very important to reduce the parasitic effects from electrical
contacts and packaging for high-frequency modulation of a semiconductor laser.
EXAMPLE 10.4
A semiconductor laser emitting at λ ¼ 850 nm has a current injection efficiency of ηinj ¼ 60%
and an output coupling rate of γout ¼ 5:7 1010 s1 . Its spontaneous carrier lifetime is
τ s ¼ 6:67 ns. It has a cavity decay rate of γc ¼ 2 1011 s1 , a differential carrier relaxation rate
of γn ¼ 4:9P0 109 s1 , and a nonlinear carrier relaxation rate of γp ¼ 6:1P0 109 s1 , where
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318
Optical Modulation
P0 is the laser output power measured in mW. The laser has a threshold current of I th ¼ 12 mA. It
is biased at a DC injection current of I 0 ¼ 28 mA and is modulated with a modulation current at a
modulation frequency of f ¼ 10 GHz and a modulation index of m ¼ 10%. (a) Find the output
power of the laser at the DC bias point. (b) What is the amplitude of the modulation current?
(c) Find the relaxation resonance frequency f r and the total carrier relaxation rate γr of this laser at
this operating point. What is the value of the K factor? (d) What are the amplitude of the
modulated output power and the phase delay of the response to the current modulation? (e) Find
the 3-dB modulation bandwidth of this laser at this operating point in terms of its modulation
response in the electrical power spectrum of the photodetector output. (f) At this modulation
frequency, what is the modulation response in the electrical power spectrum of the photodetector
used to measure the laser output? What is the normalized modulation response in dB?
Solution:
A laser has a threshold. Therefore, the DC output power is not proportional to its DC bias
current but is proportional to I 0 I th , and the modulation index is defined as the ratio of the
amplitude I m of the modulation current to I 0 I th .
(a) The photon energy at λ ¼ 850 nm is
hv ¼
1239:8
eV ¼ 1:46 eV:
850
The DC output power of the laser is found using (10.35):
P0 ¼ ηinj
γout hv
5:7 1010
ðI 0 I th Þ ¼ 0:6 1:46 ð28 12Þ mW ¼ 4:0 mW:
γc e
2 1011
(b) The amplitude of the modulation current for m ¼ 10% is
I m ¼ mðI 0 I th Þ ¼ 10% ð28 12Þ mA ¼ 1:6 mA:
(c) With τ s ¼ 6:67 ns, γc ¼ 2 1011 s1 , γn ¼ 4:9P0 109 s1 , and γp ¼ 6:1P0 109 s1
given, and P0 ¼ 4:0 mW found in (a), we have
9 1
11 1
10 1
10 1
γs ¼ τ 1
s ¼ 1:5 10 s , γc ¼ 2 10 s , γn ¼ 1:96 10 s , γp ¼ 2:44 10 s :
Therefore, using (10.41) and (10.42), we find
fr ¼
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
γc γn þ γs γp ¼ 10 GHz,
2π
γr ¼ γs þ γn þ γp ¼ 4:55 1010 s1 :
The K factor is found using (10.43):
K¼
γr γs 4:55 1010 1:5 109
¼
s ¼ 440 ps:
2
f 2r
10 109
(d) For a modulation frequency of f ¼ 10 GHz, we find that f ¼ f r , thus Ω ¼ Ωr , because f r ¼
10 GHz as found in (c). Therefore, from (10.40), we find
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10.4 Refractive External Modulation
jr j ¼
319
mγc γn mγc γn 0:1 2 1011 1:96 1010
¼
¼
¼ 1:37 101 ,
Ωγr
2πf γr 2π 10 109 4:55 1010
φ ¼ π tan1
Ωγr
Ω Ω2r
2
Ωγr
0
¼ π tan1
¼
π
rad:
2
Note that jr j > m at this modulation frequency because of the response enhancement from
relaxation resonance in a semiconductor laser. The amplitude of the modulated output
power is
Pm ¼ jr jP0 ¼ 1:37 101 4 mW ¼ 548 μW,
and the phase delay of the modulation response is φ ¼ π=2 rad:
(e) The 3-dB modulation bandwidth of this laser is, from (10.46),
f 3dB
pffiffiffi
¼ 1þ 2
1=2
pffiffiffi
¼ 1þ 2
1=2
¼ 14 GHz
f 2r
γ2
pffiffirffi
8 2π 2
1=2
45:52
102 pffiffiffi
8 2π 2
1=2
GHz,
as seen in Fig. 10.8 from the 4 mW curve.
(f) At the modulation frequency of f ¼ 10 GHz ¼ f r , the modulation response in the electrical
power spectrum of the photodetector output is, from (10.44),
Rðf Þ ¼
m2 γ2c γ2n
¼ jr ðf Þj2 ¼ 1:37 101
2 2
2
4π f γr
2
¼ 1:88 102 :
From (10.46), we have
Rð0Þ ¼
m2 γ2c γ2n
,
16π 4 f 4r
Therefore, for f ¼ 10 GHz ¼ f r , we find that
Rðf Þ 4π 2 f 2r 4π 2 f 2r 4π 2 10 109
¼ 2 2 ¼ 2 ¼ 2
γr
Rð0Þ
f γr
4:55 1010
2
¼ 1:91,
and the normalized response is
10 log
Rðf Þ
¼ 10 log1:91 ¼ 2:8 dB:
Rð0Þ
10.4 REFRACTIVE EXTERNAL MODULATION
..............................................................................................................
The basic principle of refractive modulation is to modulate the real part of a principal dielectric
constant, thus modulating the corresponding principal refractive index, of an optical medium.
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320
Optical Modulation
The direct effect is phase modulation on an optical wave that propagates through the medium.
Modulating the real part of a dielectric constant also changes the imaginary part because the real
and imaginary parts are intrinsically related through the Kramers–Kronig relations. This effect
leads to undesirable amplitude modulation that appears as a side effect, which can be minimized
by operating the modulator at an optical carrier frequency that is far away from the transition
resonance frequencies of the material. For this reason, refractive modulation is generally
performed using a material that has little absorption in the spectral region of the modulated
optical wave. As discussed in Section 10.2, any other form of optical modulation can be
accomplished through phase modulation followed by properly manipulating the phasemodulated optical wave.
Refractive modulation through varying the principal refractive indices usually causes differential changes in the principal normal modes of polarization, resulting in induced linear or
circular birefringence, which can be applied to polarization modulation. The induced birefringence that is desired for a specific polarization modulation can usually be achieved by properly
choosing the parameters of the optical wave and the material. Therefore, polarization modulation can often be directly accomplished through proper refractive modulation without indirectly
manipulating a phase-modulated wave.
In principle, any physical mechanism that can cause a change in the refractive index of an
optical medium can be used for refractive modulation. Refractive modulation is most often
implemented through electro-optic modulation using the Pockels effect. It is also implemented
through magneto-optic modulation using the Faraday effect, through acousto-optic modulation
using Bragg diffraction, or through all-optical modulation using the optical-field-induced
birefringence caused by the third-order nonlinear optical susceptibility. The concepts of these
physical mechanisms are discussed in Sections 2.6 and 2.7. The principles of refractive
modulation based on these physical mechanisms are discussed in the following.
10.4.1 Electro-optic Modulation
Practical electro-optic modulators are based on the Pockels effect, which is the first-order electrooptic effect, though it exists only in noncentrosymmetric crystals, as discussed in Section 2.6. The
electro-optic Kerr effect, being a second-order effect, is relatively weak, and thus not practically
useful, though it exists in all materials. As seen in Section 2.6, depending on the direction and the
magnitude of the applied electric field with respect to the principal axes of the crystal, the linear
birefringence induced by the Pockels effect results in changes in the principal indices that might
or might not be accompanied by a rotation of the principal axes. An electro-optically induced
rotation of the principal axes is not required for the functioning of an electro-optic modulator
though it often accompanies the index changes. However, the directions of the principal axes in
the presence of an applied electric field, whether rotated or not, have to be taken into consideration in the design and operation of an electro-optic modulator. For simplicity without loss of the
general concept, we consider in the following a special case where the electro-optically induced
linear birefringence causes only index changes without rotating the principal axes.
We consider the LiNbO3 crystal, which is the most well-known electro-optic crystal. LiNbO3
is a negative uniaxial crystal of principal indices nx ¼ ny ¼ no > nz ¼ ne . Because of its 3m
symmetry, the r αk matrix defined in (2.60) for its Pockels coefficients has only eight
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10.4 Refractive External Modulation
321
nonvanishing elements with four independent values: r13 ¼ r 23 , r 12 ¼ r 61 ¼ r 22 , r 33 , and
r42 ¼ r 51 . In order that the electro-optically induced linear birefringence changes only the
values of the principal refractive indices without rotating the principal axes, the electric field
is applied along the optical axis such that E0x ¼ E 0y ¼ 0 but E 0z 6¼ 0. In this case, the changes
caused by the Pockels effect are found from (2.60) to be Δη1 ¼ Δη2 ¼ r13 E 0z , Δη3 ¼ r33 E 0z ,
and Δη4 ¼ Δη5 ¼ Δη6 ¼ 0, which can be expressed as Δηxx ¼ Δηyy ¼ r 13 E 0z , Δηzz ¼ r33 E 0z ,
and Δηij ¼ 0 for i 6¼ j by applying the index contraction rule given in (2.59). By using (2.62)
and (2.63), the field-dependent dielectric permittivity tensor can be found:
0 2
1
no n4o r 13 E 0z
0
0
A:
(10.47)
ϵ ðE 0 Þ ¼ ϵ 0 @
0
0
n2o n4o r13 E 0z
2
4
0
0
ne ne r 33 E 0z
^ ¼ ^x , Y^ ¼ ^y , and Z^ ¼ ^z . The crystal remains
Clearly, the principal axes are not rotated: X
uniaxial with the same optical axis, but the indices of refraction are changed. Since the induced
2
changes are generally so small that jr 13 E 0z j n2
o and jr 33 E 0z j ne , the new principal
indices of refraction can be expressed as
nX ¼ nY no n3o
r 13 E 0z ,
2
nZ ne n3e
r33 E 0z :
2
(10.48)
The phase of an optical wave can be electro-optically modulated. For this type of application,
the optical wave is linearly polarized in a direction that is parallel to one of the principal axes,
^ Y^ , or Z^ , of the crystal that is subjected to a modulation field. The preferred choice is a
X,
principal axis that has a large electro-optically induced index change but remains in a fixed
direction as the magnitude of the modulation electric field varies. In LiNbO3, this can be
accomplished by applying the electric field along the z axis, as discussed above and shown in
Figure 10.9. There are two possible arrangements: transverse modulation, which has the
modulation field perpendicular to the direction of optical wave propagation, as shown in
Fig. 10.9(a), and longitudinal modulation, which has the modulation field parallel to the
direction of optical wave propagation, as shown in Fig. 10.9(b).
Figure 10.9 (a) LiNbO3 transverse electro-optic phase modulator for an optical wave propagating in the X
direction. (b) LiNbO3 longitudinal electro-optic phase modulator for an optical wave propagating in the Z
direction. In both cases, the modulation field is applied in the Z direction. The ^x , ^y , and ^z unit vectors represent
the original principal axes of the crystal, and X^ , Y^ , and Z^ represent its new principal axes in the presence of
the modulation voltage.
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322
Optical Modulation
Transverse Phase Modulation
We first consider the situation of the transverse phase modulator shown in Fig. 10.9(a), where
the optical wave propagates in the X direction. In this case, the optical wave can be polarized in
either the Y or Z direction. If it is linearly polarized in the Z direction, its space and time
dependence can be written as
EðX; tÞ ¼ Z^ E exp ik Z X iωt ¼ Z^ E exp ðiφz iωtÞ:
(10.49)
For propagation through a crystal that has a length of l, the total phase shift is
φZ ¼ kZ l ¼
ω
ω
n3
nZ l ¼
ne l e r 33 E 0z l
2
c
c
¼
ω
n3
l
ne l e r 33 V
,
2
c
d
(10.50)
where V ¼ E 0z d is the voltage applied to the modulator shown in Fig. 10.9(a).
For sinusoidal modulation of a modulation frequency f ¼ Ω=2π, the modulation voltage can
be written as
V ðt Þ ¼ V m sin Ωt,
(10.51)
which has a modulation amplitude of V m . The Z-polarized optical field at the output plane,
X ¼ l, of the crystal is phase modulated:
Eðl; tÞ ¼ Z^ Eeiωne l=c exp½iðωt þ φm sin ΩtÞ,
(10.52)
ω n3e
l πn3
l Vm
π
r 33 V m ¼ e r 33 V m ¼
λ
c 2
d
d Vπ
(10.53)
where
φm ¼
is the peak modulated phase shift, known as the phase modulation depth, and
Vπ ¼
λ d
l
(10.54)
n3e r33
is the modulation voltage that is required for a phase shift of π, known as the half-wave voltage,
also denoted as V λ=2 .
If the optical field is instead linearly polarized in the Y direction, the phase shift after
propagation through the crystal is
φY ¼ kY l ¼
ω
ω
n3
nY l ¼
no l o r13 E 0z l
2
c
c
¼
ω
n3
l
no l o r 13 V
:
2
c
d
(10.55)
The phase modulation depth for the modulation voltage given in (10.51) is then
φm ¼
ω n3o
l πn3
l Vm
r 13 V m ¼ o r13 V m ¼
π,
λ
c 2
d
d Vπ
(10.56)
where the half-wave voltage for this arrangement is
Vπ ¼
λ d
:
l
n3o r 13
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(10.57)
10.4 Refractive External Modulation
323
Because no ne but r 33 3:6r13 , it can be seen by comparing (10.57) with (10.54) that for a
desired modulation depth, the modulation voltage required for a Y-polarized optical wave is
about 3.6 times that for a Z-polarized wave.
Longitudinal Phase Modulation
For the longitudinal phase modulator shown in Fig. 10.9(b), an optical wave of any polarization in the XY plane experiences the same amount of phase shift because nX ¼ nY ¼ no . For a
crystal of a length l as shown in Fig. 10.9(b), we have
φX ¼ φ Y ¼
ω
n3
no l o r 13 E 0z l
2
c
¼
ω
n3
no l o r 13 V
2
c
(10.58)
where V ¼ E 0z l for the longitudinal modulator. For a sinusoidal modulation voltage as given in
(10.51), the modulation depth of the longitudinal phase modulator is
φm ¼
ω n3o
πn3
Vm
π,
r 13 V m ¼ o r 13 V m ¼
λ
Vπ
c 2
(10.59)
where
Vπ ¼
λ
n3o r 13
:
(10.60)
Both φm and V π for longitudinal modulation are independent of the crystal length l.
It is seen that the voltage required for a given modulation depth is independent of the physical
dimensions of the modulator in the case of longitudinal modulation, whereas it is proportional
to d=l in the case of transverse modulation. One advantage of transverse modulation is that the
required modulation voltage can be substantially lowered by reducing the d=l dimensional ratio
of a transverse modulator. Another advantage is that the electrodes of a transverse modulator
can be made using standard techniques and can be patterned if desired, while those of a
longitudinal modulator have to be made of transparent conductors that can be very difficult,
if not impossible, to fabricate in the dimensions of the typical optical waveguide. However, if a
large input and output aperture is desired such that d=l > 1, it becomes advantageous to use
longitudinal modulation rather than transverse modulation.
EXAMPLE 10.5
LiNbO3 is a negative uniaxial crystal, which has nx ¼ ny ¼ no ¼ 2:251 and nz ¼ ne ¼ 2:170 at
the λ ¼ 850 nm wavelength. It has eight nonvanishing Pockels coefficients, which are r13 ¼
r23 ¼ 8:6 pm V1 , r 12 ¼ r61 ¼ r 22 ¼ 3:4 pm V1 , r 33 ¼ 30:8 pm V1 , and r 42 ¼ r51 ¼
28 pm V1 . Consider transverse and longitudinal modulation of an optical wave at
λ ¼ 850 nm using a LiNbo3 electro-optic modulator in the configurations shown in Figs. 10.9
(a) and (b), respectively. The LiNbo3 modulator has the dimensions of l ¼ 3 cm and d ¼ 3 mm.
(a) Find the values of the half-wave voltage V π for transverse and longitudinal modulation,
respectively, in the case when the optical wave is polarized along the y principal axis. (b) The
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324
Optical Modulation
largest Pockels coefficient is r 33 . If this coefficient can be used, what are the values of V π for
transverse and longitudinal modulation, respectively?
Solution:
In both configurations shown in Figs. 10.9(a) and (b), the voltage is applied in the direction
along the z principal axis. Therefore, the Pockels coefficients that are useful for the modulation
are r 13 for x-polarized wave, r23 for y-polarized wave, and r 33 for z-polarized wave. Note that
r 13 ¼ r 23 ¼ 8:6 pm V1 and r33 ¼ 30:8 pm V1 .
(a) For a y-polarized wave, we use r 23 , which is the same as r 13 . For transverse modulation in this
case, the half-wave voltage is that given in (10.57). With l ¼ 3 cm and d ¼ 3 mm, we find
Vπ ¼
λ d
850 109
3 103
V ¼ 867 V:
¼
n3o r 13 l 2:2513 8:6 1012 3 102
For longitudinal modulation, the half-wave voltage is that given in (10.60):
Vπ ¼
λ
850 109
¼
V ¼ 8:67 kV:
n3o r 13 2:2513 8:6 1012
(b) To use r 33 , the optical wave has to be polarized along the z principal axis while the applied
voltage has to be in this direction as well. This is possible for transverse modulation but is
not possible for longitudinal modulation, as can be seen by examining Figs. 10.9(a) and (b).
For transverse modulation on a z-polarized optical wave in this case, the half-wave voltage
is that given in (10.54). With l ¼ 3 cm and d ¼ 3 mm, we find
Vπ ¼
λ d
850 109
3 103
V ¼ 270 V:
¼
n3e r 33 l 2:1703 30:8 1012 3 102
This half-wave voltage is less than one third of that found in (a) for transverse modulation
on a y-polarized optical wave because r 33 is more than three times larger than r 23 .
Polarization Modulation
As discussed in Section 10.2, polarization modulation can be accomplished by differential phase
modulation between two orthogonally polarized field components. For electro-optic polarization
modulation, the optical wave is not linearly polarized in a direction that is parallel to any of the
principal axes in the presence of the modulation field. The optical field can be decomposed into
two linearly polarized normal modes. If the two normal modes see different field-induced
refractive indices, an electric-field-dependent phase retardation between the two modes occurs,
resulting in a change of the polarization of the optical wave at the output of the crystal.
The LiNbO3 transverse modulator discussed above becomes a polarization modulator if the
polarization of the input optical field is not parallel to Y^ or Z^ so that
(10.61)
Eð0; t Þ ¼ Y^ E Y þ Z^ E Z eiωt ,
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10.4 Refractive External Modulation
Figure 10.10 LiNbO3 transverse electro-optic polarization modulator. The ^x , ^y , and ^z unit vectors represent the
original principal axes of the crystal, and X^ , Y^ , and Z^ represent its new principal axes in the presence of the
modulation voltage.
where E Y 6¼ 0 and E Z 6¼ 0, as shown in Fig. 10.10. At the output, we find
Y
Z
Y
Eðl; tÞ ¼ Y^ E Y eik l þ Z^ E Z eik l eiωt ¼ Y^ E Y þ Z^ E Z eiΔφ eik l eiωt ,
where
3
π
l
3
2ðne no Þl þ no r 13 ne r 33 V
Δφ ¼ k k l ¼
λ
d
Z
Y
(10.62)
(10.63)
is the phase retardation between the Y and Z components. The polarization of the output optical
field can be electro-optically modulated by a modulation electric field of E 0z ðt Þ ¼ V ðt Þ=d that
causes a time-varying phase retardation of Δφðt Þ following the time-varying voltage V ðtÞ.
EXAMPLE 10.6
The phase retardation given in (10.63) between the Y and Z components of the optical field for
the transverse polarization modulator shown in Fig. 10.10 has a background value that is
independent of the applied voltage V because ne 6¼ no . This voltage-independent background
phase retardation can be compensated by using a DC bias voltage of V b such that Δφ ¼ Δφb ¼
2mπ when V ¼ V b . Then (10.63) can be expressed as
Δφ ¼ Δφb þ
V Vb
V Vb
π ¼ 2mπ þ
π:
Vπ
Vπ
In practice, V b can be adjusted to make sure that Δφb ¼ 2mπ. Find the expression for V π in the
above relation. Use the parameters of LiNbO3 given in Example 10.5 to find the value of V π at
λ ¼ 850 nm for a LiNbO3 polarization modulator of the dimensions of l ¼ 3 cm and d ¼ 3 mm.
Solution:
The expression for V π can be found by taking Δφ ¼ π while ignoring the voltage-independent
background term in (10.63). Thus, we find that
Vπ ¼
n3o r13
λ
d
:
3
ne r 33 l
Using the parameters given in Example 10.5, we find that
Vπ ¼
850 109
3 103
V ¼ 392 V:
2:2513 8:6 1012 2:1703 30:8 1012 3 102
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326
Optical Modulation
Amplitude Modulation
As discussed in Section 10.2, amplitude modulation can be achieved through polarization
modulation by properly selecting a polarization component of the polarization-modulated field
while filtering out its orthogonal component. This can be done by simply placing a polarization
modulator between a polarizer at the input end and another polarizer, often referred to as an
analyzer, at the output end. The axes of the polarizer and the analyzer are often orthogonally
crossed, though other arrangements are possible. Figure 10.11 shows such an arrangement
using the LiNbO3 polarization modulator discussed above and shown in Fig. 10.10.
Following the discussion in Section 10.2 on polarization modulation and amplitude modulation, here we take
^e þ ^e ⊥
^e ^e ⊥
Y^ þ Z^
Y^ Z^
^e ¼ pffiffiffi , ^e⊥ ¼ pffiffiffi , ^e 1 ¼ Y^ ¼ pffiffiffi , ^e 2 ¼ Z^ ¼ pffiffiffi ,
(10.64)
2
2
2
2
pffiffiffi
with c1 ¼ c2 ¼ 1= 2. The axis of the input polarizer is along ^e , and that of the output analyzer
is along ^e ⊥ , as shown in Fig. 10.11. The polarizer ensures that the input optical wave is linearly
polarized in the ^e direction, whereas the analyzer passes only the ^e ⊥ component of the optical
wave at the output end. Thus, the input field is
E Eð0; t Þ ¼ ^e Eeiωt ¼ pffiffiffi Y^ þ Z^ eiωt :
2
(10.65)
Then, from (10.62), the field at the end of the crystal is
Y
Y
E E 1 þ eiΔφ ^e þ 1 eiΔφ ^e⊥ eik liωt ,
Eðl; tÞ ¼ pffiffiffi Y^ þ Z^ eiΔφ eik liωt ¼
2
2
(10.66)
where Δφ is that given in (10.63). Because the analyzer passes only the ^e ⊥ component of the
optical field, the transmittance of the amplitude modulator is
I out I ⊥
Δφ 1
¼ sin2
¼
¼ ð1 cos ΔφÞ,
(10.67)
I in
I
2
2
pffiffiffi
which agrees with (10.17) for c1 ¼ c2 ¼ 1= 2.
Electro-optic amplitude modulation can also be accomplished by varying the coupling or
interference between two fields that have differential phase modulation, as discussed in Section
10.2. This concept can be implemented with many different structures, both in free space and in
waveguides. Here we illustrate the concept using a guided-wave electro-optic modulator in the
T¼
Figure 10.11 Electro-optic amplitude modulator using two cross polarizers at the input and the output of the
LiNbO3 transverse electro-optic polarization modulator shown in Fig. 10.10.
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10.4 Refractive External Modulation
327
Figure 10.12 Mach–Zehnder waveguide
interferometric modulator using
Y junctions fabricated on an x-cut, ypropagating LiNbO3 substrate.
form of the Mach–Zehnder waveguide interferometer, shown in Fig. 10.12. This structure uses
Y-junction couplers as input and output couplers. It is fabricated in an x-cut, y-propagating
LiNbO3 crystal.
In the electrode configuration shown in Fig. 10.12, the modulation voltage is applied to the
central electrode while the outer electrodes are grounded so that the upper arm sees a modulation field of E0z ¼ V=se but the lower arms sees E0z ¼ V=se , where se is the separation
between two neighboring electrodes. The modulation electric fields appearing in the two arms
point in opposite directions, resulting in a push–pull operation with equal but opposite phase
shifts in the optical waves propagating through the two arms. For an interferometer that has
identical arms, any other background phase shifts are exactly canceled. Thus the total phase
difference is twice the electro-optically induced phase shift in each arm. If the two arms are
identical single-mode waveguides, the phase difference induced by a modulation voltage V is
Δφ ¼ π
V
,
Vπ
(10.68)
where V π is the half-wave voltage for a phase difference of π between the two arms. For a TElike mode, the transverse optical field component is primarily the E z component so that
Vπ ¼
λ
se
,
2n3e r 33 ΓTE l
(10.69)
where ΓTE is the overlap factor that accounts for the overlap between the spatial distributions of
the modulation field and the TE-like mode field. For a TM-like mode, the transverse optical
field component is primarily the E x component so that
Vπ ¼
λ
2n3o r 13 ΓTM
se
,
l
(10.70)
where ΓTM is the overlap factor that accounts for the overlap between the spatial distributions of
the modulation field and the TM-like mode field.
If both input and output Y junctions of the Mach–Zehnder waveguide interferometer are ideal
3-dB couplers, i.e., the input power is split equally between the two arms and the fields from the
two arms are combined equally for the output, the power transmittance due to interference at the
output between the fields coming from the two arms is
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328
Optical Modulation
T¼
Pout
Δφ 1
¼ cos2
¼ ð1 þ cos ΔφÞ:
Pin
2
2
(10.71)
Thus, electro-optic amplitude modulation can be accomplished through electro-optic phase
modulation to create a differential phase shift of Δφ between the two arms.
EXAMPLE 10.7
The x-cut, y-propagating LiNbO3 Mach–Zehnder waveguide interferometer in the push–pull
configuration shown in Fig. 10.12 has identical single-mode waveguides for both arms, which
have confinement factors of ΓTE ¼ ΓTM ¼ 0:5 for λ ¼ 850 nm. The electrodes have an equal
length of l ¼ 1 cm and an equal separation of se ¼ 10 μm. Use the parameters of LiNbO3 given
in Example 10.5 to find the half-wave voltage of this amplitude modulator for the TE-like mode
at λ ¼ 850 nm. What is the transmittance for an applied voltage of V ¼ 1 V?
Solution:
The half-wave