Principles of Photonics [1 ed.] 1107164281, 978-1-107-16428-4, 9781316687109, 1316687104, 141-141-147-1

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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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Principles of Photonics JIA-MING LIU University of California

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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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To Vida and Janelle

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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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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 Fabry
Pérot 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 Fabry
Pérot 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 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 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

s
1

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 J
1 s
1

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 s
1

speed of light in free space

(1.1)b, (1.39)

cvμ

none

overlap coefficient between modes v and μ

(4.19)

cijkl

m2 A
2

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

W
1

detectivity

(11.58)

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xiv

Partial List of Symbols

(cont.) Symbol

Unit

Meaning; derivatives

References1

Dλ

s m
2

group-velocity dispersion; Dλ ¼
D=cλ

(3.168)

D∗

m Hz1=2 W
1

specific detectivity

(11.59)

D

C m
2

real electric displacement in the time domain

(1.2)

D, D

C m
2

complex electric displacement; De , Do , Dþ , D


(1.42), (1.51)

D, D

C m
2

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 m
1

real electric field in the time domain

(1.2)

E0 , E 0

V m
1

static or low-frequency electric field

(2.54)

Ee , Eh

V m
1

electric fields seen by electrons and holes

(10.106)

E, E

V m
1

complex electric field

(1.40)

Ev , E v

V m
1

complex electric field of mode v

(3.1)

E, E

V m
1

slowly varying amplitudes of E and E; E e , E o , E þ , E


(1.52)

Ev, E v

V m
1

complex electric field profile of mode v

(3.1)

^v E

V m
1 W
1=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 A
1

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

m
1

gain coefficient, amplification coefficient

(3.183)f, (7.46)

g0

m
1

unsaturated gain coefficient

(8.22)

g th

m
1

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

s
1

gain parameter

(9.18)

g0

s
1

unsaturated gain parameter

(9.22)

gn

m3 s
1

differential gain parameter

(10.36)

gp

m3 s
1

nonlinear gain parameter

(10.36)

gth

s
1

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

m
1

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 m
1

real magnetic field in the time domain

(1.3)

H0 , H 0

A m
1

static or low-frequency magnetic field

(2.68)

H, H

A m
1

complex magnetic field

(1.42)

Hv , H v

A m
1

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 m
1

slowly varying amplitudes of H and H

(3.5)

Hv , Hv

A m
1

complex magnetic field profile of mode v

(3.2)

^ν H

A m
1 W
1=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 m
2

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 m
2 Hz
1

optical spectral intensity distribution

(7.17)

Ip, Is

W m
2

optical pump and signal intensities

(8.36)

I sat

W m
2

saturation intensity

(8.22)

J, J

A m
2

real current density

(1.5)

J

A m
2

complex current density

(2.35)

k

m
1

propagation constant, wavenumber; k 0 , k i , k r , k t

(1.84)

kB

J K
1

Boltzmann constant

(7.14), (7.25)

ke , ko

m
1

propagation constants of extraordinary and ordinary waves

(3.57)

k 0 , k 00

m
1

real and imaginary parts of k, k ¼ k 0 þ ik 00

(3.180)

kx , ky , kz

m
1

propagation constants of x, y, and z polarized fields

(2.15)

kX , kY , kZ

m
1

propagation constants of X, Y, and Z polarized fields (2.67)

kþ , k


m
1

propagation constants of circularly polarized fields

(2.21)

k^

none

unit vector in the k direction

(1.84)

k

m
1

wavevector; ki , kr , kt , kq

(1.1)b, (1.52)

ke , ko

m
1

wavevectors of extraordinary and ordinary waves

(3.56)f, (3.57)

kx , ky , kz

m
1

wavevectors of x, y, and z polarized fields

(3.48)f

kþ , k


m
1

wavevectors of circularly polarized fields

(10.74)

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Partial List of Symbols

xvii

(cont.) Symbol

Unit

Meaning; derivatives

References1

K

m
1

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

m
1

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 m
1

saturation magnetization

(10.78)

M

A m
1

real magnetic polarization in the time domain

(1.3)

M0 , M 0

A m
1

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

m
3

electron concentration

(11.9)

n0

m
3

equilibrium concentration of electrons

(11.9)f

n1 , n2 , n3

none

refractive indices of waveguide layers, n1 > n2 > n3

(3.125)

n2

m2 W
1

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

m
3

carrier density

(2.31)

N

m
3

effective population inversion

(8.4)

N1, N2, Nt

m
3

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

m
3

hole concentration

(11.9)

p0

m
3

equilibrium concentration of holes

(11.9)f

pijkl p0ijkl

none

elasto-optic and rotation-optic coefficients

(2.83)

pðvk Þ

Hz
1

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 m
3

spontaneous emission power density

(8.43)

p^trsp

W m
3

critical fluorescence power density

(8.45)

P

C m
2

real electric polarization in the time domain

(1.2)

P, P

C m
2

complex electric polarization

(1.50)

PðnÞ

C m
2

nth-order nonlinear real electric polarization

(2.91)

PðnÞ , PðnÞ

C m
2

nth-order nonlinear complex electric polarization

(2.91)

Pres

C m
2

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 V
1

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

m
3 s
1

effective pumping rate for population inversion

(8.6)

R1 , R2

m
3 s
1

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 W
1

responsivity of photodetector with current output; R0

(11.50)

R

V W
1

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 V
2

quadratic electro-optic coefficients, Kerr coefficients

(2.58), (2.60)

S

m
3

photon density

(9.21)

Ssat

m
3

saturation photon density

(9.24)

S

W m
2

real Poynting vector

(1.32)

S

W m
2

(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 m
3

electromagnetic energy density

(7.16), (1.33)

uðvÞ

J m
3 Hz
1

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 s
1

velocity

Fig. 11.1

va

m s
1

acoustic wave velocity

(2.80)b

vg

m s
1

group velocity; v gβ

(3.165)

vp

m s
1

phase velocity; v pβ

(3.162)

V

none

normalized frequency and waveguide thickness, V number

(3.128)

V

rad A
1

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

s
1

transition probability rate; W 12 , W 21 , W p , W sp

(7.22)
(7.24)

W p, W m

W m
3

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

m
1

attenuation coefficient, absorption coefficient

(3.180), (7.45)

α0

m
1

unsaturated absorption coefficient

(10.110)

αc

m
1

propagation parameter for contradirectional coupling

(4.61)

β

none

bottleneck factor

(8.7)

β

m
1

propagation constant of a mode; βmn , βTE , βTM

(3.1)

β0 , β00

m
1

real and imaginary parts of β, β ¼ β0 þ iβ00

(3.184)

βc

m
1

propagation parameter for codirectional coupling

(4.50)

γ

s
1

relaxation rate, decay rate; γ21 , γi , γout

(2.23)

γ1 , γ2 , γ3

m
1

transverse decay parameters of mode field

(3.118), (3.131)

γa

s
1

acoustic decay rate

(10.93)

γc

s
1

cavity decay rate, photon decay rate; γcmnq

(6.25)

γn

s
1

differential carrier relaxation rate

(10.37)

γp

s
1

nonlinear carrier relaxation rate

(10.37)

γr

s
1

total carrier relaxation rate

(10.42)

γs

s
1

spontaneous carrier relaxation rate

(10.42)

Γ

none

overlap factor

(6.2)

δ

m
1

phase mismatch parameter for phase mismatch of 2δ

(4.31)

δωmnq

rad s
1

frequency shift of mode pulling

(9.12)

Δn, Δp

m
3

excess electron and hole concentrations

(11.10)

ΔP

C m
2

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 m
1

change or variation in electric permittivity

(2.55), (4.12)

~ Δϵ , Δ~ϵ

F m
1

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

Schawlow
Townes 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 s
1

optical linewidth, bandwidth, Δω ¼ 2πΔv; Δωinh , Δωh (7.3)f, (7.13)

ϵ

F m
1

electric permittivity

(2.11), (3.4)

ϵ0

F m
1

electric permittivity of free space

(1.2)

ϵ 0 , ϵ 00

F m
1

real and imaginary parts of ϵ, ϵ ¼ ϵ 0 þ iϵ 00

(3.179)

ϵx, ϵy, ϵz

F m
1

principal dielectric permittivities

(2.13)

ϵX , ϵY , ϵZ

F m
1

new principal dielectric permittivities

(2.65)

ϵþ, ϵ


F m
1

principal dielectric permittivities of circular polarizations

(2.17)

ϵðr, tÞ

F m
4 s
1

real permittivity tensor in the real space and time domain

(1.21)

ϵ ðωÞ, ϵ ij

F m
1

complex permittivity tensor in the frequency domain

(1.60)

ϵ res ðωÞ

F m
1

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

m
1

coupling coefficient; κvμ

(4.13)

~κ

m
1

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 m
1

magnetic permeability of free space

(1.3)

μe , μh

m2 V
1 s
1

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 m
3

charge density

(1.6)

ρF

rad m
1

specific Faraday rotation

(10.79)

σ

S m
1

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

m
3 s
1

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 V
1

second-order nonlinear susceptibility in the frequency (2.98), (2.100) domain

χð3Þ , χ ijkl

ð3 Þ

m2 V
2

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 s
1

optical angular frequency; ω ¼ 2πv

(1.1)b

ω0

rad s
1

center optical angular frequency; ω0 ¼ 2πv0

(2.22), (7.13)

ω21

rad s
1

resonance angular frequency between levels j1i and j2i

(2.22)

ωc

rad s
1

cutoff frequency; ωcm

(3.151)

Ω

rad s
1

acoustic or modulation angular frequency; Ω ¼ 2πf

(2.79), (10.27)

Ωr

rad s
1

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 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 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; Amp`ere’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 b ε ¼ 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 ^x þ i^y ^e þ 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 ^x  i^y ^e  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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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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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

ω0 ¼ 2πν0 ¼ 2:36  1015 rad s1 :

Because λ 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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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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40

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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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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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)

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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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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 ð2Þ 2Þ ðω4 ¼ ω1  ω3 Þ pffiffiffi3 þ χ ðxyz ðω4 ¼ ω1  ω3 Þ pffiffiffi3 Px 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 ,

Pðy2Þ ðω4 Þ

E1 E∗ E1 E∗ 2Þ 2Þ ¼ ϵ 0 χ ðyxz ðω4 ¼ ω1  ω3 Þ pffiffiffi3 þ χ ðyyz ðω4 ¼ ω1  ω3 Þ pffiffiffi3 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 ,

Pðz2Þ ðω4 Þ

E1 E∗ E1 E∗ 2Þ 2Þ 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∗ ð2 Þ ð2Þ 3 3 E1 ¼ ϵ 0 χ xxz ðω4 ¼ ω1  ω3 Þ pffiffiffi þ χ xzx ðω4 ¼ ω3 þ ω1 Þ pffiffiffi , 2 2

E1 E∗ E∗ ð2Þ ð2 Þ ð2Þ 3 3 E1 Py ðω4 Þ ¼ ϵ 0 χ yyz ðω4 ¼ ω1  ω3 Þ pffiffiffi þ χ yzy ðω4 ¼ ω3 þ ω1 Þ pffiffiffi , 2 2 Pðx2Þ ðω4 Þ

2Þ ðω4 ¼ ω2 þ ω2 ÞE22 : Pðz2Þ ðω4 Þ ¼ ϵ 0 χ ðzzz

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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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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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(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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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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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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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)

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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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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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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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3.2.2 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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(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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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 x ¼  4 jk  k j 4 ny  nx 

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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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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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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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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  1=2 cos 45 sin2 45 þ ¼ 2:197: ne ð45 Þ ¼ 2:2382 2:1592 

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)

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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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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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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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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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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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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

 2r2 , I ðr; zÞ ¼ I 0 ðzÞexp  2 w 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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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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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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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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3.4 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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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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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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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)

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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 cos ðh1 x  ψ Þ, x > 0, E y ðxÞ ¼ (3.107) cos ψ exp ðγ2 xÞ, x < 0,

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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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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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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 x > 0, cos ψ 2 cos ðh1 x  ψ 1 Þ, E y ðxÞ ¼ (3.113) x < 0, cos ψ 1 cos ðh2 x  ψ 2 Þ, r TE ¼ r s ¼

h1  h2 , h1 þ h2

r TM ¼ r p ¼

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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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 x > 0, cos ψ 2 cos ðh1 x  ψ 1 Þ, (3.115) Hy ðxÞ ¼ x < 0, cos ψ 1 cos ðh2 x  ψ 2 Þ, 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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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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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 exp ðγ1 xÞ, x > 0, ^ H y ðxÞ ¼ C (3.119) x < 0, exp ðγ2 xÞ, where  1=2  ω γ1 γ2 ϵ 1 ϵ 2 1=2 : C¼ γ1 ϵ 1 þ γ2 ϵ 2 β

(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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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



,

μ0 ϵ 22 γ2 ¼ ω 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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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 1=2 2π 12:13 1=2 1 ¼ ¼ m λ ϵ 1 =ϵ 0 þ ϵ 2 =ϵ 0 500  109 1  12:13

¼ 1:31  107 m1 , " #1=2  1=2  1=2 μ0 ϵ 21 2π ðϵ 1 =ϵ 0 Þ2 2π 1 ¼ ¼ m1 γ1 ¼ ω ϵ1 þ ϵ2 λ ϵ 1 =ϵ 0 þ ϵ 2 =ϵ 0 500  109 1  12:13 ¼ 3:77  106 m1 ,

" #1=2  1=2 2π ðϵ 2 =ϵ 0 Þ2 2π 12:132 ¼ ¼ m1 9 1  12:13 λ ϵ 1 =ϵ 0 þ ϵ 2 =ϵ 0 500  10 7 1 ¼ 4:57  10 m : 

μ0 ϵ 22 γ2 ¼ ω ϵ1 þ ϵ2

1=2

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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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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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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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. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016

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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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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)

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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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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 d=2 yields two eigenvalue equations: 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)

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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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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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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 . Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016

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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

 V 1 1 pffiffiffiffiffi M TE ¼ aE , (3.152)  tan π π int and that of guided TM modes is M TM



 V 1 1 pffiffiffiffiffiffi ¼  tan aM , π π int

(3.153)

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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For the waveguide to support the TE0 mode but not the TE1 mode, pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi V 1  tan1 aE 1 ) tan1 aE < V π þ tan1 aE π π 1:168 4:310 μm < d

μm 1:168 < V 4:310 ) 2:884 2:884 405 nm < d 1:494 μm:

M TE ¼ 1 ) )

)

0
β m . This observation is consistent with the conclusion obtained from the above general discussion on asymmetric waveguides.

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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

 V M TE ¼ M TM ¼ : (3.159) π int 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 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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(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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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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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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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)

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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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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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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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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 ð
iωtÞ 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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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 iωe
iβν 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)

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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 c
1  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 ð
qÞ exp ð
iqKzÞ (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 ð
iqKzÞdz: 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 ~
iφðzÞ
iqKz : κba ð
qÞAe 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 iκab ðqÞBe ~ i2δz , κab ðqÞBe ¼i dz q

(4.43)

X ~ dB ~
iφðzÞ
iqKz iκba ð
qÞAe ~
i2δz , κba ð
qÞAe ¼i dz q

(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 ð
qÞ 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 ð
iqKzÞdz Λ 0

ðΛ

¼

1 a cos Kz exp ð
iqKzÞdz Λ 0

¼

a Λ

ðΛ 0

exp ðiKz
iqKzÞ þ exp ð
iKz
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 ð
1Þ 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 ð
iqKzÞdz Λ 0 ξΛ ð

ðΛ 1 1 ¼ a exp ð
iqKzÞdz
a exp ð
iqKzÞdz: 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 ðz
z0 Þ
iδ sin βc ðz
z0 Þ iδðz
z0 Þ iκab iδðzþz0 Þ e sin β ð z
z Þe 0 c 6 7 βc βc 7 Fðz;z0 Þ ¼ 6 4 iκba βc cos βc ðz
z0 Þþiδ sin βc ðz
z0 Þ
iδðz
z0 Þ 5
iδðzþz0 Þ sin βc ðz
z0 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.



iδ ~ ~ A ðzÞ ¼ A ð0Þ cos βc z
sin βc z eiδz , βc

iκba ~ ~ sin βc z e
iδz : B ðzÞ ¼ B ð0Þ βc

(4.51) (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 iδðl
zÞ 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 e
iδz : 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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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



π
1 δ
1 δ φ ¼ þ φκba
tan tanh αc l ¼ φPM
tan tanh αc l , 2 αc 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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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  ieiφ sin jκjðz
z0 Þ cos jκjðz
z0 Þ FPM ðz; z0 Þ ¼ , ie
iφ sin jκjðz
z0 Þ cos jκjðz
z0 Þ 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)

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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:



1 1 1 1 π 2
1 η3dB ¼ sin jκjl3dB ¼  pffiffiffi ¼ m þ ) l3dB ¼ sin for m ¼ 0, 1, 2, . . . 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 tanh
1 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 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2 η¼ (4.78) sin jκjl 1 þ jδ=κj2 : 1 þ jδ=κj2 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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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 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

η¼ : 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 m
1 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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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 m
1 . It is also found that the maximum absolute value of the coupling coefficient of this grating is jκjmax ¼ 1:0  104 m
1 , 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 m
1 , 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 m
1 and βb ¼ 5:849  106 m
1 , respectively, and the coupling coefficient between the two fiber modes is κ ¼ κab κba ¼ 2  103 m
1 . 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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(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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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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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)

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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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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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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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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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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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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, , Δym 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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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, 2, . . .

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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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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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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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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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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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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)

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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, 1, 2,    , sin2 ðx þ qπ Þ sin2 x

(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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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)

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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

qλ 20 651 109 ¼ m ¼ 75 μm: sin 10 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 : 6 75 10

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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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

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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 850 109 ¼ sin1 n2 Λ 1:5 2 106

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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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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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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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    

2 1=2 1=2 iφRT 1=2 1=2 iφRT iωt ¼ E0e 1 þ R1 R2 e þ R1 R2 e þ  ¼ E 0 eiωt

1 1=2 1=2

1  R1 R2 eiφRT

,

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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 ð1  R1 Þð1  R2 Þ T max

2 : FP ¼ 1=2 1=2 1  R1 R2

(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 φ þ φ2

c φRT ¼ 2qπ, νq ¼ q  1 , (5.47) 2π 2nl cos θ 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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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)

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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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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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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 φ þ φ2 c c c 2noil l ) λdark ¼ ¼ νq ¼ q  1 ¼q , 2π 2nl 2noil l 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 1:4 100 nm ¼ 560 nm: 2q  1

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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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

199

       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 3  ¼      R1 ¼  1 þ 1:38 ¼ 0:0255, R2 ¼ nf þ ng  ¼ 1:38 þ 1:5 ¼ 1:736 10 : nair þ nf  The reflectivity of the coated surface is ð1  R1 Þð1  R2 Þ RFP ¼ 1  T max

2 ¼ 1  0:986 ¼ 0:014: FP ¼ 1  1=2 1=2 1  R1 R2 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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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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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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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

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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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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 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016

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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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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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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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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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 nl 3:5  300  106 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ 2:9 ps: pffiffiffiffiffiffiffiffiffiffi ¼ τc 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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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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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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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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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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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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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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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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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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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)

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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)

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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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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)

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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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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)

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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 quickly relaxes to the ground level because level j1i has a short lifetime of τ 1 ¼ 200 ps τ 2 . 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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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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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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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 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 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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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 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016

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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

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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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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

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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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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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8.2.4 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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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

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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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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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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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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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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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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 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. 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 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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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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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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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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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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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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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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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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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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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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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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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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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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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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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

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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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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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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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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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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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10.2.4 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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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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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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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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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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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)

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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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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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(c) From (10.29), we find m 0:1 2 jr j ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi ¼ 8:47  10 , 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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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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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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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 : 2 16π 4 f 2  f 2r þ 4π 2 f 2 γ2r 

As shown in Fig. 10.8, this spectrum has a resonance peak at

1=2 γ2r 2 f pk ¼ f r  2 8π

(10.44)

(10.45)

and a 3-dB modulation bandwidth of f 3dB

1=2 pffiffiffi 1=2 2 γ2r ¼ 1þ 2 f r  pffiffiffi  1:554 f pk : 8 2π 2

(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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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



 Ωγr π 1 1 Ωγr ¼ rad: ¼ π  tan φ ¼ π  tan 2 2 0 2 Ω  Ωr

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),

1=2 pffiffiffi 1=2 2 γ2r f 3dB ¼ 1 þ 2 f r  pffiffiffi 8 2π 2

1=2 pffiffiffi 1=2  45:52 2 ¼ 1þ 2 10  pffiffiffi GHz, 8 2π 2 ¼ 14 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

 2 m2 γ2c γ2n ¼ jr ðf Þj2 ¼ 1:37  101 ¼ 1:88  102 : 2 2 2 4π f γr

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

 2 Rðf Þ 4π 2 f 2r 4π 2 f 2r 4π 2  10  109 ¼ 2 2 ¼ 2 ¼  2 ¼ 1:91, γr Rð0Þ f γr 4:55  1010

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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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 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016

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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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



 ω ω n3e ω n3e l Z ne l  r 33 E 0z l ¼ ne l  r 33 V , φZ ¼ k l ¼ nZ l ¼ 2 2 c c 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

n3e r33

(10.54)

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



 ω ω n3o ω n3o l Y φY ¼ k l ¼ nY l ¼ no l  r13 E 0z l ¼ no l  r 13 V : (10.55) 2 2 c c c d 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



 ω n3o ω n3o φX ¼ φ Y ¼ no l  r 13 E 0z l ¼ no l  r 13 V 2 2 c 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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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 Y Z 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 l π 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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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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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 voltage of this Mach–Zehnder waveguide interferometer for the TE-like mode is that given in (10.69) because the optical field is primarily polarized in the z direction. Using the LiNbO3 parameters given in Example 10.5, we find that Vπ ¼

λ se 850  109 10  106 V ¼ 2:7 V: ¼  2n3e r 33 ΓTE l 2  2:1703  30:8  1012  0:5 1  102

For an applied voltage of V ¼ 1 V, the phase difference between the two arms is Δφ ¼ π

V π ¼ : V π 2:7

Therefore, the transmittance is found from (10.71) to be 1 1 π T ¼ ð1 þ cos ΔφÞ ¼ 1 þ cos ¼ 69:8%: 2 2 2:7

10.4.2 Magneto-optic Modulation As discussed in Section 2.6, the optical property of a material can be changed by a magnetization, M 0 , in a ferromagnetic or ferrimagnetic material or by an externally applied static or lowfrequency magnetic field, H 0 , in a nonmagnetic material. Practical magneto-optic modulators use the first-order magneto-optic effect, i.e., the linear magneto-optic effect, because the second-order magneto-optic effect is relatively weak. The linear magneto-optic effect, in which the electric permittivity tensor ϵðωÞ at an optical frequency of ω is a linear function of M 0 or H 0 , results in magnetically induced circular birefringence. Then, if the propagation direction is taken to be along the z direction, the normal modes of propagation are circularly polarized with

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the unit polarization vectors ^e þ and ^e  given in (2.18) and the corresponding propagation constants kþ and k given in (2.21): kþ ¼

nþ ω ξ , with nþ ¼ n⊥  c 2n⊥

k ¼

n ω ξ ; with n ¼ n⊥ þ c 2n⊥

(10.72)

where nþ and n are, respectively, the principal refractive indices seen by the ^e þ and ^e  normal modes. The parameter ξ quantifying the linear magneto-optic effect on the refractive indices is defined in (2.78). For an optical wave propagating along the z direction, ξ is a linear function of the z component M 0z with ξ ðM 0z Þ ¼ ξ ðM 0z Þ in the case of a magnetization, and ξ ðH 0z Þ ¼ f 123 H 0z is linearly proportional to the z component H 0z in the case of an applied magnetic field. Magneto-optic refractive modulation on an optical wave can be performed through the dependence of nþ and n on the magnetization or on the applied magnetic field by varying the magnetization or the applied magnetic field. For magneto-optic phase modulation, the optical wave has to be a circularly polarized normal mode rather than linearly polarized as in the case of electro-optic phase modulation. A circularly polarized normal mode, either ^e þ or ^e  , remains in the same polarization state as its phase is modulated when the wave is reflected from or is transmitted through a linear magneto-optic material. If an optical wave is initially linearly or elliptically polarized, its field is a superposition of the two circularly polarized normal modes. This field then decomposes into two circularly polarized orthogonal components that propagate with different propagation constants of kþ and k , respectively. Magneto-optic modulation on this wave causes differential phase modulation on the two orthogonal circularly polarized modes, resulting in magneto-optic polarization modulation on a wave that is initially linearly or elliptically polarized. The polarization change caused by the linear magneto-optic effect on an optical wave that propagates through a magneto-optic material is known as the Faraday effect. The polarization change caused by the linear magneto-optic effect on an optical wave that is reflected from the surface of a magneto-optic material is known as the magneto-optic Kerr effect. Both effects lead to magneto-optic polarization modulation. The Faraday effect is the mechanism used for optical isolators and optical circulators. Magneto-optic polarization modulation can be converted into magneto-optic amplitude modulation using polarizers by following the same principles as discussed above for electro-optic modulation. The Faraday effect can then be used for magneto-optic spatial light modulation. The magneto-optic Kerr effect is used for magnetooptic recording. Of special interest is the Faraday rotation of a linearly polarized optical wave propagating in a magneto-optic medium. Assume, without loss of generality, that the wave is initially linearly polarized in the x direction at an arbitrary initial position taken to be z ¼ 0: E Eð0; tÞ ¼ ^x Eeiωt ¼ pffiffiffi ð^e þ þ ^e  Þeiωt , 2

(10.73)

pffiffiffi with E þ ¼ E  ¼ E= 2. Both circularly polarized components propagate as normal modes with their respective propagation constants. When the wave propagates a distance of l in the positive z direction, we have

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Eðl; t Þ ¼ ^e þ E þ exp ½ikþ  ^z ðl  0Þ  iωt  þ ^e  E  exp ½ik  ^z ðl  0Þ  iωt  ¼ ^e þ E þ exp ðikþ l  iωt Þ þ ^e  E  exp ðik l  iωtÞ

þ

i   E h ikþ l ^x e þ eik l þ i^y eik l  eik l eiωt ¼ 2



þ  k  kþ k  kþ k þ k l þ ^y sin l exp i l  iωt : ¼ E ^x cos 2 2 2

(10.74)

The optical field clearly remains linearly polarized because its x and y components are in phase, but its plane of polarization is rotated by an angle of θF ¼ tan1

E y k  kþ π πξ l ¼ ðn  nþ Þl ¼ ¼ l: Ex 2 λ λn⊥

(10.75)

This magnetically induced rotation of the plane of linear polarization is called Faraday rotation, and this phenomenon is the Faraday effect. It can be shown that the plane of polarization rotates by the same amount in the same sense if the wave propagates in the negative z direction for the same distance of l. Therefore, the sense of Faraday rotation is independent of the direction of wave propagation. A device that provides the function of Faraday rotation is called a Faraday rotator. In a paramagnetic or diamagnetic material, which has no internal magnetization, the Faraday rotation for a linearly polarized wave propagating over a distance of l is linearly proportional to the externally applied magnetic field. The Faraday rotation angle in this case is generally expressed as θF ¼ VH 0z l,

(10.76)

where V¼

ωf 123 πf 123 ¼ 2cn⊥ λn⊥

(10.77)

is the Verdet constant, measured in radians per ampere (rad A1 ). In practice, the Faraday rotation angle is often expressed as θF ¼ VB0z l in terms of the magnetic flux; in this case, the Verdet constant is measured in radians per tesla per meter (rad T1 m1 ). In a ferromagnetic or ferrimagnetic material, which has an internal magnetization, the total Faraday rotation angle for an optical wave traveling over a distance of l through such a material is simply θF ¼ ρF

M 0z l, Ms

(10.78)

where M 0z M s is the existing magnetization in the material and M s is the saturation magnetization of the material. The Faraday rotation can be small if the material is not sufficiently magnetized; it is maximized only when the material is fully magnetized to reach its saturation magnetization. The Faraday rotation is thus characterized by the following specific Faraday rotation, or rotatory power, ρF ¼

ωξ ðM s Þ πξ ðM s Þ ¼ , 2cn⊥ λn⊥

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(10.79)

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which is the amount of rotation per unit length traversed by the optical wave in the material at the saturation magnetization. The Faraday effect is nonreciprocal. It has the characteristic that the sense of the Faraday rotation in a specific material is independent of the direction of wave propagation and is determined only by the direction of the external magnetic field, or that of the magnetization in a ferromagnet or ferrimagnet. The expression of θF in (10.76) holds true for propagation in both the parallel and the antiparallel directions with respect to H 0 , and that of ρF in (10.79) is also valid for propagation in both directions with respect to M 0 . In the case when H0 , or M 0 , is not ^ only the longitudinal component of the magnetic aligned with the wave propagation direction k, field, or that of the magnetization, in the k^ or k^ direction counts because the transverse components that are perpendicular to the wave propagation direction do not contribute to Faraday rotation. The amount of Faraday rotation is doubled, rather than canceled, when an optical wave passing through a magneto-optic material is reflected to retrace its original path in the opposite direction back to the starting point. This phenomenon is a consequence of the fact that the propagation constant of each circularly polarized normal mode is independent of the wave propagation direction and, therefore, is not changed by reflection. The Faraday rotation is positive when the value of θF , or that of ρF , is positive, meaning that the rotation is counterclockwise when viewed in the direction against that of H 0 , or that of M 0 when an internal magnetization exists. Therefore, the sense of positive Faraday rotation is the same as that of the electric current that generates H 0 or the current that can be conceptually associated with M 0 in the case of a ferromagnet or ferrimagnet. Using the right-hand rule, the axial vector corresponding to a positive Faraday rotation points in the same direction as that of the H 0 or M 0 causing the Faraday effect. For negative Faraday rotation, the sense of rotation is opposite to that of positive Faraday rotation. Figure 10.13 summarizes these concepts. The nonreciprocity of Faraday rotation is important for optical isolation. Indeed, the Faraday effect remains the unique physical mechanism for optical isolators and optical circulators which are necessary components in sophisticated optical systems and networks. The basic structure of an optical isolator consists of a Faraday rotator that has a total Faraday rotation angle of θF ¼ 45 and two linear polarizers with axes oriented at 45 with respect to each other, as shown in Fig. 10.14(a). An optical wave entering the device in the forward direction through the input polarizer is linearly polarized by this polarizer. The linearly polarized wave emerging from the Faraday rotator is transmitted by the output polarizer. For reverse isolation, an optical wave of

Figure 10.13 Positive Faraday rotation for an optical wave propagating in (a) a parallel direction and (b) an antiparallel direction with respect to H 0 or M 0 . The sense of positive rotation is the same as the electric current that can be associated with H 0 or with M 0 . For negative Faraday rotation, the sense of rotation is just the opposite.

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Figure 10.14 (a) Basic structure and principle of a polarization-dependent optical isolator, which changes the polarization direction at the output. (b) Two-stage cascaded optical isolator that does not change the polarization direction at the output.

any polarization entering the Faraday rotator from the output end is linearly polarized by the output polarizer. Because Faraday rotation is independent of the wave propagation direction, the backward-propagating wave emerging from the Faraday rotator has a linear polarization that is orthogonal to the axis of the input polarizer and is thus blocked. Figure 10.14(b) shows a two-stage cascaded optical isolator that has input and output waves linearly polarized in the same direction. EXAMPLE 10.8 An optical isolator of the configuration shown in Fig. 10.14(a) consists of a Faraday rotator made of a TGG crystal, which has a length of l ¼ 5 cm. The TGG crystal has a Verdet constant of V ¼ 40 rad T1 m1 at the λ ¼ 1:064 μm wavelength of the Nd:YAG laser and V ¼ 190 rad T1 m1 at the λ ¼ 532 nm wavelength. What is the required strength of the magnetic induction B0z along the wave propagation direction for the isolator to function at each of the two wavelengths, respectively? Solution: For the optical isolator of the configuration shown in Fig. 10.14(a), which has the polarizers arranged such that the linear polarization rotates in the sense of a positive θF , the required Faraday rotation angle for a single pass is θF ¼ π=4. Therefore, the required magnetic induction along the wave propagation direction for λ ¼ 1:064 μm is B0z ¼

θF π=4 ¼ T ¼ 393 mT, Vl 40  5  102

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and that required for λ ¼ 532 nm is B0z ¼

θF π=4 T ¼ 82:7 mT: ¼ Vl 190  5  102

Note that the magnetic induction has to point in the direction opposite to the forwardpropagating direction of the optical wave because the Verdet constant of TGG is negative.

10.4.3 Acousto-optic Modulation An acoustic wave causes a space- and time-dependent periodic permittivity change in a medium, as discussed in Section 2.6. For a traveling acoustic wave of the form expressed in (2.79), which has a wavevector of K and a frequency of Ω, the acousto-optically induced permittivity change can be generally expressed in the form of (2.88) as Δϵ ¼ Δ~ϵ sin ðK  r  Ωt Þ,

(10.80)

where the acoustic wavevector K depends on both the polarization and the propagation direction of the acoustic wave. The wavenumber of the acoustic wave is K ¼ 2π=Λ ¼ 2πf =v a , where v a is the acoustic wave velocity, f ¼ Ω=2π is the acoustic wave frequency, and Λ ¼ v a =f is the acoustic wavelength. The space- and time-dependent periodic permittivity change Δϵ is effectively a timedependent grating, which diffracts an optical wave. In general, as expressed in (2.89), Δ~ϵ is a function of the strain and the rotation generated by the acoustic wave in the medium, the elastooptic coefficients of the medium, the mode and direction of the acoustic wave, and the frequency and polarization of the optical wave, but it is independent of the values of K and Ω. When an optical wave at a frequency of ω is incident on this medium, the interaction between the optical wave and the periodic modulation described by (10.80) can generate diffracted optical waves at the frequencies of ω  Ω. The diffracted waves at ω þ Ω can successively be diffracted to generate waves at the frequencies of ω  2Ω. If this process is allowed to cascade, it is possible to generate a series of diffracted optical waves at the frequencies of ω þ qΩ, where q admits both positive and negative integers and is the order of acousto-optic diffraction: ωq ¼ ω þ qΩ:

(10.81)

The phase-matching condition for the qth-order diffraction is kq ¼ ki þ qk,

(10.82)

where ki is the wavevector of the incident optical wave. Acousto-optic modulation is fundamentally diffraction modulation. Because the frequency of each diffraction order is shifted by an integral multiple of the acoustic frequency, digital frequency modulation can be accomplished by switching between discrete acoustic frequencies while analog frequency modulation can be performed using a continuously time-varying acoustic frequency of Ωðt Þ. Because Δ~ϵ is generally a tensor, even when the medium is an isotropic material, the polarization of a diffracted wave can be different from that of the incident wave. Therefore, polarization modulation of a desired polarization change from the incident

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wave to a diffracted wave can be accomplished by properly arranging the parameters of the acoustic wave. By generating diffracted waves, acousto-optic modulation reduces the power of the undiffracted optical wave at the original frequency ω and wavevector ki , thus imposing amplitude modulation on this optical wave. To encode information through acousto-optic amplitude modulation, the power of the acoustic wave has to be modulated with a timevarying signal by modulating the amplitude of the acoustic wave. For the qth-order diffraction to occur, the frequency-shift condition given in (10.81) has to be strictly obeyed, but the phase-matching condition given in (10.82) does not have to be exactly satisfied. As we have learned from Section 4.6, perfect phase matching is necessary for the maximum efficiency, but a small phase mismatch does not completely prohibit the process though it reduces the efficiency. The degree of phase mismatch that can be tolerated in acoustooptic diffraction depends on the length of interaction between the optical wave and the acoustic wave. The criterion is quantified by the factor: Q¼

K 2l , k

(10.83)

where K is the wavenumber of the acoustic wave, k is the propagation constant of the optical wave, and l is the interaction length between the two waves. Based on this criterion, there are two separate regimes of acousto-optic diffraction. In the regime of Raman–Nath diffraction, where Q  1, multiple diffraction orders can take place simultaneously, as shown in Fig. 10.15. Raman–Nath diffraction occurs only when the optical wave propagates in a direction that is normal, or nearly normal, to the propagation direction K of the acoustic wave. Phase matching in the direction parallel to K is exactly satisfied for each diffraction order that occurs, but a phase mismatch in the direction perpendicular to K can be tolerated because of the short interaction length. In the regime of Bragg diffraction, where Q  1, the phase-matching condition has to be satisfied for a diffraction order to occur in response to an acoustic wave. In practice, it is often

Figure 10.15 (a) Configuration and (b) wavevector diagram for Raman–Nath diffraction in an isotropic medium. Phase matching in the x direction determines the propagation angles of the diffracted waves. Phase mismatch exists only in the z direction.

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necessary to have Q 4π for clean Bragg diffraction. In its interaction with a traveling acoustic wave, an incident optical wave, of the zeroth order with a wavevector of ki and a frequency of ω, is directly coupled only to the two diffraction orders of q ¼ 1 and q ¼ 1. It can be seen from (10.82) that the phase-matching condition for the generation of the diffraction order q ¼ 1 at the up-shifted frequency of ω1 ¼ ω þ Ω is kd ¼ k1 ¼ ki þ K,

(10.84)

whereas that for the generation of the diffraction order q ¼ 1 at the down-shifted frequency of ω1 ¼ ω  Ω is kd ¼ k1 ¼ ki  K:

(10.85)

For Bragg diffraction through the interaction with a given acoustic wave, the angle of incidence of the incoming optical wave is not arbitrary but is determined by the phasematching condition. The required angle of incidence, θi , for the incoming optical wave and the angle of diffraction, θd , at which the diffracted wave appears can be found by solving (10.84) for the up-shifted diffraction, or (10.85) for the down-shifted diffraction. In the case when the acoustic medium is an anisotropic crystal, the refractive indices ni and nd that are respectively seen by the incident and diffracted optical waves can be different because the two waves might have different polarizations. The solutions of the required incident angle and the resulting diffraction angle are 1

  K 2 þ k2i  k2d v 2a  2 1 λf 2 1 þ 2 2 ni  nd , ¼ sin 2ki K 2ni v a λf

(10.86)

1

  K 2 þ k2d  k2i v 2a  2 1 λf 2 1 þ 2 2 nd  ni , ¼  sin 2kd K 2nd v a λf

(10.87)

θi ¼ sin θd ¼  sin

where the upper signs are for up-shifted diffraction and the lower signs are for down-shifted diffraction. For up-shifted diffraction, θi < 0 and θd > 0. For down-shifted diffraction, θi > 0 and θd < 0. Note that θi and θd are both measured with respect to the z direction, which is normal to the K vector of the acoustic wave, and each of them can be either positive or negative, as shown in Fig. 10.15 for the case of a positive θi . In the case when the acousto-optic diffraction takes place in an isotropic medium, ni ¼ nd ¼ n. Then, jθi j ¼ jθd j ¼ θB ¼ sin1

K λ λf , ¼ sin1 ¼ sin1 2k 2nΛ 2nv a

(10.88)

where θB is the Bragg angle. In this case, θi ¼ θB and θd ¼ θB for up-shifted diffraction, and θi ¼ θB and θd ¼ θB for down-shifted diffraction. For any diffraction order q to be generated, the phase-matching condition given in (10.82) has to be satisfied for ωq ¼ ω þ qΩ. In addition, because each diffraction order is directly coupled only to its neighboring orders, Bragg diffraction at a high order requires the successive generation of low diffraction orders, thus requiring simultaneous satisfaction of the corresponding phase-matching conditions. Except for some very special cases, these requirements cannot be fulfilled. Consequently, only one diffraction order, either q ¼ 1 or q ¼ 1, is usually

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generated in Bragg diffraction from a traveling acoustic wave. Bragg diffraction occurs in both isotropic and anisotropic media when the phase-matching condition in (10.84) or (10.85) is satisfied. Unlike Raman–Nath diffraction, the incident optical wave does not have to propagate in a direction that is normal, or nearly normal, to the direction of propagation of the acoustic wave. For Bragg diffraction, the optical wave can propagate in any direction, including the K direction or the K direction, if the phase-matching condition for q ¼ 1 or q ¼ 1 can be satisfied. The characteristics of acousto-optic diffraction from a standing acoustic wave are quite different. A standing acoustic wave can be considered as a linear superposition of two contrapropagating traveling waves with both K and K simultaneously present for phase matching. The implication of this situation is two-fold. (1) Both up-shifted and down-shifted frequencies are simultaneously generated in each phase-matched diffraction direction, and (2) each shifted optical frequency generated by diffraction can be diffracted back to the direction of the incident wave with a further shift in frequency. This process cascades. For Raman–Nath diffraction from a standing acoustic wave, each of the even spatial orders, including the undiffracted zeroth order, consists of all of the frequencies up-shifted or down-shifted by the even multiples of Ω, whereas each of the odd spatial orders consists of all of the frequencies up-shifted or downshifted by the odd multiples of Ω. For Bragg diffraction from a standing acoustic wave, the undiffracted beam in the ki direction contains a series of even side bands at ω  2mΩ, and the diffracted beam in the kd direction contains the odd side bands at ω  ð2m þ 1ÞΩ. Traveling-Wave Acousto-optic Modulator The majority of practical acousto-optic modulators take the configuration of small-angle Bragg interaction, with the incident optical wavevector ki normal or nearly normal to the acoustic wavevector K so that the diffraction angle is small. Figure 10.16 shows a traveling-wave

Figure 10.16 Representative solid-state traveling-wave acousto-optic modulator operating in the Bragg regime. Up-shifted diffraction is illustrated here. For an anisotropic acousto-optic modulator, jθd j 6¼ jθi j. For an isotropic acousto-optic modulator, jθd j ¼ jθi j ¼ θB ¼ sin1 ðK=2kÞ.

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acousto-optic modulator operating under small-angle Bragg diffraction. Up-shifted diffraction takes place in this configuration, where ki  K < 0. Down-shifted diffraction occurs if the optical wave is incident from a direction such that ki  K > 0. The diffraction efficiency of a Bragg-type traveling-wave modulator with perfect phase matching can be expressed as "

1=2 # M2 2 π l , (10.89) ηPM ¼ sin Pa λ 2HL where λ is the optical wavelength; M 2 is the acousto-optic figure of merit determined by the properties of the material, the mode of the acoustic wave, and the polarizations of the incident and diffracted optical waves; H and L are respectively the height and the length of the transducer that generates the acoustic wave; Pa is the power of the acoustic wave; and l is the length of interaction between the optical wave and the acoustic wave. In the configuration of small-angle Bragg interaction shown in Fig. 10.16, l  L. In the low-efficiency limit, the diffraction efficiency is linearly proportional to the acoustic power: ηPM 

π 2 M 2 l2 Pa , 2λ2 HL

if

ηPM  1:

(10.90)

EXAMPLE 10.9 A traveling-wave acousto-optic modulator made of silica glass is used to modulate an optical wave at λ ¼ 1:3 μm using a longitudinal acoustic wave at an acoustic frequency of f ¼ 100 MHz. The silica glass has a refractive index of n ¼ 1:447 at λ ¼ 1:3 μm; it has an acoustic wave velocity of v a ¼ 5:97 km s1 and a figure of merit of M 2 ¼ 1:50  1015 m2 W1 for a longitudinal acoustic wave and an optical wave polarized in a direction that is perpendicular to the propagation direction K of the acoustic wave. The transducer that generates the acoustic wave has the dimensions of L ¼ 1:5 cm and H ¼ 2 mm; it delivers an acoustic power of Pa ¼ 500 mW. (a) Does this modulator operate in the Raman–Nath regime or the Bragg regime? (b) What is the deflection angle between the diffracted beam and the incident beam? (c) What is the diffraction efficiency? Solution: The wavelength of the acoustic wave is Λ¼

v a 5:97  103 m ¼ 59:7 μm: ¼ f 100  106

For this acousto-optic modulator, the incident angle has to be small so that the interaction length is approximately l  L. (a) The Q factor is Q¼

K 2 l 2πλl 2π  1:3  106  1:5  102 ¼ ¼ 23:8 > 4π: ¼  2 k nΛ2 1:447  59:7  106

Therefore, the modulator works in the Bragg regime.

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(b) The silica glass is isotropic. In the Bragg regime, the phase-matching condition requires that the angles of incidence and diffraction have the same magnitude but opposite signs: jθi j ¼ jθd j ¼ θB ¼ sin1

K λ 1:3  106 ¼ 0:43 : ¼ sin1 ¼ sin1 2k 2nΛ 2  1:447  59:7  106

Because θi and θd have opposite signs in both up-shifted and down-shifted diffraction, the deflection angle between the diffracted and incident beams for both cases is jθdef j ¼ jθd  θi j ¼ 2θB ¼ 0:86 : (c) The diffraction efficiency is small and can be found using (10.90): ηPM

 2 π 2  1:50  1015  1:5  102 π 2 M 2 l2  2  500  103 ¼ 1:6%: Pa ¼  6 2 3 2 2λ HL 2  1:3  10  2  10  1:5  10

Standing-Wave Acousto-optic Modulator A standing-wave acousto-optic modulator provides sinusoidal amplitude modulation at a very high frequency. It differs from a traveling-wave acousto-optic modulator in many important aspects, ranging from the device structure to the performance characteristics. Figure 10.17 shows a standing-wave acousto-optic modulator operating under small-angle Bragg diffraction. In order to create a standing acoustic wave, the acousto-optic cell is made to be a resonant acoustic cavity. Instead of the angled surface of the acousto-optic cell of a traveling-wave

Figure 10.17 Representative solid-state standing-wave acousto-optic modulator operating in the Bragg regime. For an anisotropic acousto-optic modulator, jθd j 6¼ jθi j. For an isotropic acousto-optic modulator, jθd j ¼ jθi j ¼ θB ¼ sin1 ðK=2kÞ.

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modulator, the surface at the far end across the cell width is made parallel to the near end, which is attached to the piezoelectric transducer, as seen in Fig. 10.17. With a given cell width W measured in the direction of the acoustic wave, a standing acoustic wave is formed only when the acoustic wavelength satisfies the condition: Λ W¼m , 2

m ¼ integer:

(10.91)

Therefore, the device functions only at the discrete acoustic resonance frequencies of f ¼m

va , 2W

m ¼ integer,

(10.92)

which are determined by the cell width and the acoustic velocity v a . The diffraction efficiency of a standing-wave acousto-optic modulator with perfect phase matching is "

# 1=2 M2va 2 π Pa l cos Ωt , (10.93) ηPM ¼ sin λ HLWγa where γa is the decay rate of the acoustic energy in the acoustic cavity. In the low-efficiency limit, we have ηPM 

π 2 M 2 l2 v a π 2 M 2 l2 v a 2 P cos Ωt ¼ Pa ð1 þ cos 2Ωt Þ, a λ2 HLWγa 2λ2 HLWγa

if

ηPM  1:

(10.94)

Again, l  L in the configuration of small-angle Bragg diffraction. As can be seen from (10.94), the diffracted beam varies with time at a modulation frequency of f m ¼ 2f , which is twice the frequency f of the acoustic wave. Therefore, the undiffracted beam is loss modulated at f m . EXAMPLE 10.10 A standing-wave acousto-optic modulator made of silica glass is used to modulate an optical wave at λ ¼ 1:3 μm using a longitudinal acoustic wave at an acoustic frequency of f ¼ 100 MHz. The silica glass has a refractive index of n ¼ 1:447 at λ ¼ 1:3 μm; it has an acoustic wave velocity of v a ¼ 5:97 km s1 and a figure of merit of M 2 ¼ 1:50  1015 m2 W1 for a longitudinal acoustic wave and an optical wave polarized in a direction perpendicular to the propagation direction K of the acoustic wave. The transducer that generates the acoustic wave has the dimensions of L ¼ 1:5 cm and H ¼ 2 mm; it delivers an acoustic power of Pa ¼ 500 mW. The acoustic cavity has a cell width of W ¼ 2 cm and a decay rate of γa ¼ 6  104 s1 . (a) Does this modulator operate in the Raman–Nath regime or the Bragg regime? (b) What is the deflection angle between the diffracted beam and the undiffracted beam? (c) What is the modulation frequency at which the diffracted and undiffracted beams are modulated? (d) What is the peak value of the diffraction efficiency? Solution: The material and the parameters of the optical and acoustic waves are the same as those described in Example 10.9 for the traveling acousto-optic modulator. Therefore, the answers to (a) and (b) are the same as those in Example 10.9.

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(a) The modulator works in the Bragg regime because Q ¼ 23:8 > 4π: (b) The deflection angle between the diffracted and incident beams is jθdef j ¼ jθd  θi j ¼ 2θB ¼ 0:86 : (c) The modulation frequency is twice that of the acoustic frequency: f m ¼ 2f ¼ 200 MHz: (d) The diffraction efficiency is found using (10.93). The peak efficiency is ηpk PM

"

1=2 # π M 2va  sin Pa l λ HLWγa " #

1=2 π 1:50  105  5:97  103  500  103 2 2 ¼ sin   1:5  10 2  103  1:5  102  2  102  6  104 1:3  106 2

¼ 15:5%:

10.4.4 All-Optical Refractive Modulation All-optical modulation is accomplished through a nonlinear optical process based on a thirdorder susceptibility of the form χð3Þ ðω ¼ ω þ ω0  ω0 Þ that causes optical-field-induced permittivity changes in a nonlinear optical material. In the case when ω0 ¼ ω, the process can involve only one beam, as illustrated in Fig. 10.18(a), or two physically distinguishable beams of the same frequency, as illustrated in Fig. 10.18(b). In the case when ω0 6¼ ω, there are always two optical beams in the interaction, as illustrated in Fig. 10.18(c). All-optical modulation can be based on either self modulation or cross modulation. In the case of self modulation, only one optical beam is present, and the modulation on the beam is a function of the characteristics of the beam itself. In the case of cross modulation, two or more optical beams are present, and the beam of interest is modulated by one or more other beams that carry the modulation signals. In either case, no electric, magnetic, or acoustic field is needed. Therefore, optical modulation and switching based on nonlinear optical processes are known as all-optical modulation and alloptical switching, respectively. There are two fundamentally different types of nonlinear optical modulation and switching. One is the dispersive, or refractive, type, which is based on the optical-field-induced changes in the real part of the permittivity of a material. The other is the absorptive type, which relies on an intensity-dependent absorption coefficient caused by the nonlinear characteristics of the imaginary part of the permittivity. All-optical refractive modulation is discussed here; all-optical absorptive modulation is discussed in the following section. In the case of one-beam interaction, we find by using (2.101) and the intrinsic permutation ð3Þ symmetry of χ ijkl ðω ¼ ω þ ω  ωÞ that

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10.4 Refractive External Modulation

Figure 10.18 Third-order processes for field-induced susceptibility changes: (a) one-beam interaction, (b) interaction of two beams of the same frequency, and (c) interaction of two beams of different frequencies. ð3Þ

Pi ðωÞ ¼ 3ϵ 0

X j, k , l

ð3 Þ

χ ijkl ðω ¼ ω þ ω  ωÞE j ðωÞE k ðωÞE ∗ l ðωÞ:

(10.95)

In the case of two-beam interaction, for either ω0 ¼ ω or ω0 6¼ ω, we have X ð3Þ ð3Þ χ ijkl ðω ¼ ω þ ω  ωÞE j ðωÞE k ðωÞE ∗ Pi ðωÞ ¼ 3ϵ 0 l ðωÞ j, k , l X ð3 Þ 0 χ ijkl ðω ¼ ω þ ω0  ω0 ÞE j ðωÞE k ðω0 ÞE∗ þ 6ϵ 0 l ðω Þ j, k , l ð3Þ

ð3Þ

(10.96)

ð3 Þ

and a similar expression for Pi ðω0 Þ in terms of χ ijkl ðω0 ¼ ω0 þ ω0  ω0 Þ and χ ijkl ðω0 ¼ ω0 þ ð1Þ

ð3Þ

ω  ωÞ. By identifying the total polarization at the frequency ω as Pi ðωÞ ¼ Pi ðωÞ þ Pi ðωÞ, we find that the total optical-field-dependent permittivity tensor can be expressed as (10.97) ϵ ij ðω; EÞ ¼ ϵ ij ðωÞ þ Δϵ ij ðω; EÞ, h i ð1Þ where ϵ ij ðωÞ ¼ ϵ 0 1 þ χ ij ðωÞ represents the field-independent linear permittivity tensor of the medium and Δϵ ij ðω; EÞ accounts for the optical-field-dependent change induced by nonlinear optical interaction. For one-beam interaction, X ð3Þ χ ijkl ðω ¼ ω þ ω  ωÞE k ðωÞE ∗ (10.98) Δϵ ij ðω; EÞ ¼ 3ϵ 0 l ðωÞ: k, l For two-beam interaction, X ð3Þ Δϵ ij ðω; EÞ ¼ 3ϵ 0 χ ijkl ðω ¼ ω þ ω  ωÞE k ðωÞE ∗ l ðωÞ k, l X ð3Þ (10.99) 0 χ ijkl ðω ¼ ω þ ω0  ω0 ÞE k ðω0 ÞE∗ þ 6ϵ 0 l ðω Þ: k, l The field-dependent permittivity described here is the basis for various forms of all-optical modulation. Because Δϵ ij is a tensor, the nonlinear process discussed here generally leads to an

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optical-field-induced birefringence, known as the optical Kerr effect. The phase of an optical field can be modulated by itself through self-phase modulation or by another optical field through cross-phase modulation. All-optical polarization modulation can be accomplished through optical-field-induced birefringence. Such polarization modulation can be either selfinduced in a one-beam interaction or cross-induced in a two-beam interaction. The simplest case involves a single linearly polarized optical wave in an isotropic medium with the optical field polarized in any fixed direction, or in a cubic crystal with the optical field polarized along one of the principal axes. Then Pð3Þ is parallel to E of the optical field, and the ð3Þ only susceptibility element that contributes to this interaction is χ 1111 ðω ¼ ω þ ω  ωÞ. Thus, the permittivity seen by the optical field is ð3Þ

ϵ ðω; EÞ ¼ ϵ ðωÞ þ 3ϵ 0 χ 1111 jEðωÞj2 ¼ ϵ ðωÞ þ

ð3Þ

3χ 1111 I ðωÞ, 2cn0

(10.100)

where n0 is the linear refractive index of the medium and I ðωÞ is the intensity of the optical ð3 Þ beam. We find from this relation that the real part of χ 1111 ðω ¼ ω þ ω  ωÞ leads to the intensity-dependent index of refraction: n ¼ n0 þ n2 I ðωÞ,

(10.101)

where ð3Þ0

n2 ¼

3χ 1111 4cϵ 0 n20

(10.102)

is the coefficient of intensity-dependent index change. The intensity-dependent index of refraction expressed in (10.101) represents the simplest case of the optical Kerr effect. After the optical wave propagates through such a nonlinear medium over a distance of l in the z direction, the total phase shift is Δφðx; y; t Þ ¼

2π ½n0 þ n2 I ðx; y; t Þl ¼ φ0 þ φK ðx; y; t Þ: λ

(10.103)

The intensity-dependent Kerr phase change, φK ðx; y; t Þ ¼

2π n2 I ðx; y; t Þl, λ

(10.104)

is the space- and time-dependent self-phase modulation because it depends on the intensity of the optical wave itself. Depending on the material properties, the spatial and temporal profiles of the optical intensity, and the experimental conditions, this all-optical self-phase modulation leads to the phenomena of self focusing, self defocusing, spectral broadening of optical pulses, Kerr-lens mode locking, and optical solitons. EXAMPLE 10.11 At λ ¼ 1:3 μm, silica glass has a linear refractive index of n0 ¼ 1:45 and a nonlinear susceptið3Þ0 bility of χ 1111 ¼ 1:8  1022 m2 V1 . A laser pulse that has a wavelength of λ ¼ 1:3 μm, a Gaussian pulse shape with a FWHM pulsewidth of Δt ps ¼ 100 fs, and a peak power of

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10.4 Refractive External Modulation

343

Ppk ¼ 1 kW propagates through a silica fiber that has an effective core radius of a ¼ 5 μm and a length of l ¼ 1 m. In answering the following questions, ignore the effect of temporal pulse broadening caused by the dispersion in the fiber while the pulse propagates through the 1-m fiber because silica has zero group-velocity dispersion near λ ¼ 1:3 μm. (a) Find the value of n2 for silica. (b) Find the optical-field-induced index change at the peak of the pulse. (c) Find the self-phase modulation due to the intensity-dependent Kerr phase change. (d) The timedependent phase modulation leads to frequency modulation. Find the percent of frequency shifts, measured with respect to the original optical frequency, at the two half-width points of the pulse. Does the frequency shift up or down on the leading and trailing edges of the pulse, respectively? Solution: The Gaussian laser pulse of a peak power Ppk propagating in a fiber of an effective core radius a has a temporal intensity profile of ! ! t2 P0 t2 I ðtÞ ¼ I 0 exp 4 ln 2 2 ¼ 2 exp 4 ln 2 2 : Δt ps πa Δt ps (a) The value of n2 for silica is found using (10.102): ð3Þ 0

n2 ¼

3χ 1111 3  1:8  1022 ¼ m2 W1 ¼ 2:4  1020 m2 W1 : 4cϵ 0 n20 4  3  108  8:85  1012  1:452

(b) The optical-field-induced index change at the peak of the pulse is Δn ¼ n2 I pk ¼ n2

Ppk 1  103 20 7 ¼ 2:4  10   ¼ 3:1  10 : 6 2 πa2 π  5  10

(c) The self-phase modulation due to the intensity-dependent Kerr phase change is 2π n2 I ðt Þl λ ! 2n2 lP0 t2 ¼ exp 4 ln 2 2 λa2 Δt ps

φK ðt Þ ¼

2  2:4  1020  1  1  103 t2 exp 4 ln 2 ¼  2 Δt 2ps 1:3  106  5  106 ! t2 ¼ 1:48 exp 4 ln 2 2 rad, Δt ps

! rad

where Δtps ¼ 100 fs. (d) Though φK has a peak value of only 1:48 rad, the frequency modulation is in fact pretty large because this self-phase modulation takes place within the pulse duration of Δt ps ¼ 100 fs. Using (10.7), we find that

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Optical Modulation

! ∂φK n2 lP0 t n2 lP0 t ωðtÞ ¼ ω  ¼ ω 1 þ 8 ln 2 ¼ ω þ 16 ln 2 : ∂t λa2 Δt2ps cπa2 Δt2ps On the leading edge of the pulse, ωðt Þ < ω because t < 0; thus, the frequency shifts down. On the trailing edge of the pulse, ωðt Þ > ω because t > 0; thus, the frequency shifts up. This results in positive chirping, characterized by a frequency that increases with time. At the two half-width points, t  ¼ Δt ps =2, we find that

 n2 lP0 ωðt  Þ ¼ ω 1  4 ln 2 cπa2 Δt ps ! 2:4  1020  1  1  103 : ¼ ω 1  4 ln 2   2 3  108  π  5  106  100  1015 ¼ ωð1  2:8%Þ The frequency shifts down by 2:8% at the half-width point of t ¼ Δtps =2 on the leading edge of the pulse; it shifts up by 2:8% at the half-width point tþ ¼ Δt ps =2 on the trailing edge of the pulse.

10.5 ABSORPTIVE EXTERNAL MODULATION

.............................................................................................................. In contrast to refractive modulation, the basic principle of absorptive modulation is to modulate the imaginary part of a principal dielectric constant, thus modulating the absorption coefficient, of an optical medium. By modulating the absorption coefficient, the desired effect of absorptive modulation is clearly amplitude modulation on an optical wave that propagates through the medium. Absorptive modulation is usually accompanied by a significant change in the refractive index, creating undesirable phase modulation that leads to frequency chirping. As discussed in the preceding section, modulating the real part of a dielectric constant for the purpose of refractive modulation also changes the imaginary part because they are intrinsically related through the Kramers–Kronig relations. For the same reason, modulating the imaginary part for absorptive modulation also changes the real part. For refractive modulation, undesirable accompanying absorptive modulation can be avoided, or at least minimized, by operating a modulator in a spectral region far away from any of the transition resonance frequencies of the material, as discussed in Section 2.4. By contrast, undesirable accompanying refractive modulation is difficult to eliminate in the case of absorptive modulation because the absorption of a photon takes place through a resonant optical transition at or near the resonance frequency where the real part of the optical susceptibility varies together with the imaginary part, as seen in Fig. 2.3. Unless the optical frequency is tuned exactly at an isolated transition resonance frequency, where the real part of the resonant susceptibility is zero, the change in the real part of the resonant susceptibility is directly proportional to the change in the imaginary part. However, practically it is not realistic to use this possibility in order to avoid the undesirable

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345

accompanying refractive modulation for two reasons. (1) The material, such as a semiconductor, that is used for absorptive modulation often has a continuous absorption band rather than isolated, discrete absorption frequencies. (2) In the case when absorptive modulation is performed based on a transition between two discrete energy levels, such as the absorption line of an exciton, the transition resonance frequency shifts under modulation. Because the refractive index near a resonance frequency varies nonlinearly with the optical frequency, as can be seen in Fig. 2.3, the frequency chirping is often nonlinear and difficult to compensate. As discussed in Section 10.3, this is also the case for direct modulation. Indeed, direct modulation is a form of absorptive modulation where the carrier density of a semiconductor gain medium is modulated. Modulating the gain coefficient is the same as modulating the absorption coefficient because a gain coefficient is simply a negative absorption coefficient: g ¼ α. Absorptive modulation can cause different changes in the imaginary parts of the three principal dielectric constants of a material, resulting in induced linear or circular dichroism, which makes the absorption coefficients different for different normal modes of polarization, as discussed in Section 2.2. Whether induced dichroism occurs or not depends on the properties of the material used and the physical mechanism responsible for the absorptive modulation. Induced dichroism has to be avoided to prevent undesirable polarization changes when an optical wave is amplitude modulated through absorptive modulation. On the other hand, induced dichroism can be used to accomplish desired polarization modulation at the same time when an optical wave is amplitude modulated. In principle, any physical mechanism that can cause a change in the absorption coefficient of an optical medium can be used for absorptive modulation. Absorptive modulation is most often implemented with semiconductor materials either using a modulation current, known as current modulation, to change the carrier density, or using a modulation electric field, known as electro-absorption modulation, to change the energy bandgap of a bulk semiconductor, through the Franz–Keldysh effect, or the quantized energy subbands of a quantum-well structure, through the quantum-confined Stark effect. All-optical absorptive modulation is possible through the nonlinear optical effect of absorption saturation, or gain saturation in the case of a gain medium. The principle of current modulation has already been discussed under direct modulation in Section 10.3. Though current modulation can also be used for external absorptive modulation, the principle is the same and thus is not further discussed here. The principle of electro-absorption modulation and that of all-optical absorptive modulation through saturable absorption are discussed below.

10.5.1 Electro-absorption Modulation Practical electro-absorption modulators are semiconductor devices based on the Franz–Keldysh effect, for a bulk semiconductor, or based on the quantum-confined Stark effect, for a semiconductor quantum-well structure. Both effects are prominent near the bandgap. In addition, the electro-absorption change caused by the quantum-confined Stark effect is significantly enhanced by excitons. The absorption coefficient of a semiconductor is primarily determined by band-to-band absorption, which excites an electron from a valence band to a conduction band, though

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absorption involving impurity states is also possible. Band-to-band absorption creates free electron–hole pairs. It takes place only when the photon energy is larger than the bandgap of the semiconductor, as shown in Fig. 10.19(a). For a direct-gap bulk semiconductor, such as GaAs or InP, the absorption coefficient has the following characteristics: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αðωÞ / ℏω  E g for ℏω > E g , (10.105) where Eg is the bandgap of the bulk semiconductor, ω is the optical frequency, and ℏω is the photon energy. For an indirect-gap semiconductor, such as Si or Ge, band-to-band absorption near the bandgap is assisted by phonon emission or phonon absorption, thus αðωÞ / ðℏω  E g þ Ephonon Þ2 for ℏω > E g  Ephonon near the bandgap, where E phonon is the phonon energy. The electric fields seen by electrons and holes can be respectively expressed in terms of the spatial gradients of the conduction- and valence-band edges as: Ee ¼

∇E c e

Eh ¼

∇E v , e

(10.106)

where Ec and E v are the conduction- and valence-band edges, respectively, and e is the electronic charge. In the presence of an applied electric field, E, the conduction- and valence-band edges

Figure 10.19 (a) Band-to-band absorption of a semiconductor in the absence of an applied electric field. (b) Band-to-band absorption of a semiconductor in the presence of an applied electric field. (c) Change in the absorption coefficient due to the Franz–Keldysh effect for a direct-gap semiconductor.

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remain parallel to each other and are tilted in the direction of the applied field such that Ee ¼ Eh ¼ E, as shown in Fig. 10.19(b). As a result, the wavefunctions of the electrons in the conduction band and those of the holes in the valence band penetrate into the bandgap, creating the probability for an electron and a hole to recombine through quantum mechanical tunneling at an energy that is lower than the bandgap energy, as illustrated in Fig. 10.19(b). This effect is known as the Franz–Keldysh effect; it can be understood as electric-field-assisted absorption or photon-assisted tunneling for band-to-band transition at a photon energy below Eg . Figure 10.19(c) shows the change in the absorption coefficient due to the Franz–Keldysh effect for a direct-gap semiconductor. In a quantum-well structure, the electrons and holes are spatially confined within the finite width, d QW , of the quantum well. This localization leads to the quantization of momentum in the direction perpendicular to the quantum-well boundaries, resulting in discrete quantized energy levels associated with the motion of the electrons and holes in this direction, as shown in Fig. 10.20(a). In the horizontal dimensions, electrons and holes remain free and form energy bands. As a result, both conduction and valence bands are split into a number of subbands corresponding to the quantized levels. The minimum photon energy required for band-to-band absorption in a quantum-well structure is the effective bandgap, EQW g , of a quantum well, which is no longer the bandgap Eg of the semiconductor material in the quantum well but is the separation between the lowest subband of the conduction band and the highest subband of the valence band: hν > E QW ¼ Eg þ g

h2 h2 þ , 2 2 8m∗ 8m∗ e d QW h d QW

(10.107)

∗ where h is the Planck constant, and m∗ e and mh are, respectively, the effective mass of electrons in the conduction band and that of holes in the valence band.

EXAMPLE 10.12 The bandgap of GaAs at room temperature is E g ¼ 1:424 eV, corresponding to the energy of a photon that has a wavelength of λg ¼ 870:6 nm. A GaAs/AlGaAs quantum well has GaAs in the quantum-confined well region; the width of the quantum well is d QW ¼ 20 nm. The electron ∗ and hole effective masses for GaAs are m∗ e ¼ 0:067 m0 and mh ¼ 0:52 m0 , respectively, where m0 is the free electron mass. Find the effective bandgap increase caused by the quantum confinement of the quantum well. What are the energy and the corresponding optical wavelength of a photon that can be absorbed by the quantum well? Solution: The effective bandgap increase of the quantum well is h2 h2 þ 2 2 8m∗ 8m∗ e d QW h d QW  2

 6:626  1034 1 1 1 ¼ eV þ   2  31 9 0:067 0:52 1:6  1019 8  9:11  10  20  20 ¼ 15:9 meV:

E QW  Eg ¼ g

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For a photon to be absorbed by the quantum well, the photon energy has to be ¼ E g þ 15:9 meV ¼ 1:440 eV, hν > E QW g for a corresponding wavelength of λ
0, the medium has a linear loss; in this case, χ ð3Þ00 < 0 so that the nonlinear susceptibility causes an intensity-dependent reduction of the loss, resulting in absorption saturation. When χ ð1Þ00 < 0, the medium has a gain; then, χ ð3Þ00 > 0, and it causes intensity-dependent gain saturation. A saturable absorber has an absorption coefficient that decreases with increasing light intensity, such as that characterized by (10.109) with χ ð1Þ00 > 0 and χ ð3Þ00 < 0. Note, however, that the relation in (10.109) originates from taking the leading terms of the power series expansion of linear and nonlinear polarizations expressed in (2.90). Because absorption saturation necessarily occurs at a resonant transition between two energy levels, the perturbation approach taken for power series expansion is not valid at a sufficiently high intensity. Instead, a full analysis of the resonant absorption has to be carried out. Such an analysis results in an intensity-dependent absorption coefficient characterized by the relation: α¼

α0 , 1 þ I=I sat

(10.110)

where α0 is the unsaturated absorption coefficient and I sat is the saturation intensity. The saturation intensity is a characteristic of the resonant transition that is responsible for the absorption. For I < I sat , the relation in (10.110) can be expanded: " #

2 3 I I I α ¼ α0 1  þ  þ  : (10.111) I sat I sat I sat Only when I  I sat can α be accurately approximated by the first two terms of this expansion, resulting in a linear dependence on I like the relation in (10.109). In general, the relation in (10.110) has to be used because the light intensity encountered in a practical device that uses a saturable absorber can easily be comparable to or higher than I sat . The propagation of an optical wave through a saturable absorber that has an absorption coefficient given in (10.110) is described by dI α0 I: ¼ 1 þ I=I sat dz

(10.112)

This equation can be integrated to obtain the relation: I ðzÞeI ðzÞ=I sat ¼ I ð0ÞeI ð0Þ=I sat eα0 z ,

(10.113)

where I ð0Þ is the input light intensity at z ¼ 0. The transmittance of an optical wave through a saturable absorber of a thickness l is

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Figure 10.21 Transmittance of an optical wave through a saturable absorber that has a thickness of l and an unsaturated absorption coefficient of α0 as a function of the input light intensity normalized to the saturation intensity. The curves are plotted for different values of α0 l in terms of T 0 ¼ eα0 l .

I out I ðlÞ ¼ , (10.114) I in I ð0Þ which is a nonlinear function of the input intensity and can be found by numerically solving (10.113). The transmittance is plotted in Fig. 10.21 as a function of the input light intensity, normalized to the saturation intensity, for a few different values of α0 l represented in terms of T 0 ¼ eα0 l . As Fig. 10.21 shows, the optical transmittance through a saturable absorber increases nonlinearly as the input intensity is increased, and it approaches unity at high input intensities. In a specific application of a saturable absorber, the value of α0 l has to be properly chosen for a desired difference between the maximum transmittance at high intensities and the minimum transmittance at low intensities. Saturable absorbers have many useful applications for self-intensity modulation of optical beams or optical pulses. A saturable absorber can be used as a spatial light filter, which blocks low-intensity stray light or background optical noise but transmits a high-intensity signal beam. It can be used as an optical discriminator, which transmits optical pulses of intensities above a certain threshold and suppresses those below. A saturable absorber is also commonly used as a passive Q switch in a Q-switched laser or as a passive mode locker in a mode-locked laser for the generation of very short laser pulses. The saturable absorber in this kind of application functions as a passive optical switch in the time domain. It is switched open by the rising intensity of a laser pulse but closes through its own relaxation after the pulse passes. T¼

EXAMPLE 10.14 A saturable absorber is used for all-optical binary modulation by switching between two levels of the optical intensity for a maximum transmittance of T max at a high input intensity of I in ¼ I high and a minimum transmittance of T min at a low input intensity of I in ¼ I low . Find the input intensity required for a transmittance of T when the saturable absorber has an unsaturated

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Problems

353

absorption coefficient of α0 and a thickness of l. What is the minimum value of α0 l that is required for this modulator to function? Solution: Using (10.113) for the output intensity and (10.114) for the transmittance, we have     I ðl Þ I ð0Þ  I ðlÞ I in ð1  T Þ  α0 l ,  α0 l ¼ exp T¼ ¼ exp I sat I ð0Þ I sat where I in ¼ I ð0Þ . Therefore, the required input intensity is α0 l þ ln T I sat : 1T A sufficiently large value of α0 l is required for I in 0 for both high and low input intensities. Because I low < I high , the value required for α0 l is found by making sure that I low 0. Thus, I in ¼

α0 l ln T min ) ðα0 lÞmin ¼ ln T min : Once the desired values of T max and T min are determined, a proper value of α0 l can be chosen for the saturable absorber. Then the required input intensities I high and I low can be determined with a known saturation intensity I sat of the absorber.

Problems 10.1.1 What is the difference between analog optical modulation and digital optical modulation? How does the nonlinearity in the modulation response limit each type of modulation? 10.1.2 What is the difference between direct modulation and external modulation? What are the advantages and disadvantages between the two? 10.1.3 What is the basic difference between refractive modulation and absorptive modulation? Generally speaking, without considering specific physical mechanisms or device structures, which one is expected to have a faster modulation response? 10.2.1 Which modulation scheme is the most fundamental among all of the optical modulation schemes? Why is it fundamental? 10.2.2 Briefly describe how frequency modulation, polarization modulation, amplitude modulation, spatial modulation, and diffraction modulation can each be accomplished through phase modulation. 10.2.3 Briefly describe how information is coded on an optical carrier wave through each of the following modulation schemes: (a) BPSK and QPSK, (b) BFSK and QFSK, (c) BPolSK, and (d) OOK. 10.2.4 An optical field is initially linearly polarized in the direction of the unit polarization vector pffiffiffi ^e ¼ ð^x þ ^y Þ= 2. The two linearly polarized components of the mutually orthogonal x and

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y polarizations are differentially phase modulated for polarization modulation of the optical field. Find its orthogonal unit polarization vector ^e ⊥ on the f^x ; ^y g basis. 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? 10.2.5 An optical field is initially linearly polarized in the direction of the unit polarization pffiffiffi  vector ^e ¼ ^x þ ^y Þ= 2. The polarized pffiffiffi two circularly pffiffiffi components of the mutually   orthogonal ^e þ ¼ ^x þ i^y Þ= 2 and ^e  ¼ ^x  i^y Þ= 2 polarizations are differentially phase modulated for polarization modulation of the optical field. Find its orthogonal unit polarization vector ^e ⊥ on the f^e þ ; ^e  g basis. 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? 10.2.6 Describe two approaches to modulating the output intensity of a waveguide structure by modulating the phase of a waveguide mode. Give an example for each approach. 10.2.7 An optical wave is normally incident at θi ¼ 0 on a diffraction grating. A diffraction order q appears at the diffraction angle of θq ¼ 30 . If the grating period Λ can be varied within a range of 10% for diffraction modulation, what is the angular range of variations for this diffraction order? 10.3.1 For the direct current modulation of an LED with a sinusoidal modulation current as given in (10.27), show that the output power of the LED has the modulation response given in (10.28) with the complex response of (10.29) as a function of the modulation index m and the modulation frequency Ω. 10.3.2 For the LED described in Example 10.3, it is desired that the amplitude of the modulated output power be Pm ¼ 500 μW when it is modulated with a modulation index of m ¼ 10% at a modulation frequency of f ¼ f 3dB ¼ 15:9 MHz. This goal can be accomplished by adjusting the bias current I 0 and correspondingly the amplitude I m of the modulation current. Find the values of I 0 and I m for this purpose. 10.3.3 To increase the 3-dB modulation bandwidth of an LED, the spontaneous carrier lifetime τ s of the LED can be prescribed by properly controlling the impurity concentration in the LED. If a 3-dB bandwidth of 50 MHz is desired for an LED, what is the required value for τ s ? What is its normalized modulation response measured in dB at a modulation frequency of 20 MHz? 10.3.4 An LED emitting at a center wavelength of λ ¼ 1:3 μm has an external quantum efficiency of ηe ¼ 26%. Its spontaneous carrier lifetime is τ s ¼ 3 ns. The LED is biased at a DC injection current of I 0 ¼ 10 mA and is modulated at a modulation frequency of f ¼ 40 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. (e) At this modulation frequency, what is the modulation response in the electrical power spectrum of the photodetector that is used to measure the LED output? What is the normalized modulation response measured in dB?

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10.3.5 For the direct current modulation of a semiconductor laser with a sinusoidal modulation current as given in (10.38), show that the output power of the laser has the modulation response given in (10.39) with the complex response of (10.40) as a function of the modulation index m and the modulation frequency Ω. 10.3.6 The semiconductor laser described in Example 10.4 is biased at a DC injection current level of I 0 so that its output power is P0 ¼ 5 mW at this DC bias point. It is then modulated with a modulation current at a modulation frequency of f ¼ 10 GHz for a modulation index of m ¼ 10%. (a) Find the required DC bias injection current level I 0 . (b) What is the amplitude I m of the modulation current required for a modulation index of m ¼ 10%? (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 at the modulation frequency of f ¼ 10 GHz? (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. (f) At the modulation frequency of f ¼ 10 GHz, what is the modulation response in the electrical power spectrum of the photodetector that is used to measure the laser output? What is the normalized modulation response measured in dB? 10.3.7 The 3-dB bandwidth of a semiconductor laser can be increased by increasing the output power of the laser at the bias point through increasing the bias injection current. A GaAs/AlGaAs quantum-well semiconductor laser emitting at λ ¼ 827:6 nm has the following parameters: cavity decay rate γc ¼ 2:4  1011 s1 , spontaneous carrier relaxation rate γs ¼ 1:458  109 s1 , differential carrier relaxation rate γn ¼ 1:55P0  108 s1 , and nonlinear carrier relaxation rate γp ¼ 2:8P0  108 s1 , where P0 is the laser output power at the bias point measured in mW. (a) Find the relaxation resonance frequency f r and the total carrier relaxation rate γr of this laser as functions of the laser output power P0 . (b) Find the value of the K factor for this laser. (c) What is the 3-dB modulation bandwidth of the laser when it is biased at an output power of P0 ¼ 10 mW? (d) What is the laser output power at the bias point required for the laser to have a 3-dB modulation bandwidth of f 3dB ¼ 5 GHz? 10.3.8 A semiconductor laser emitting at λ ¼ 1:3 μm has an external quantum efficiency of

ηe ¼ 21:5%. It has a cavity decay rate of γc ¼ 5:36  1011 s1 , a spontaneous carrier relaxation rate of γs ¼ 5:96  109 s1 , a differential carrier relaxation rate of γn ¼ 1:67P0  109 s1 , and a nonlinear carrier relaxation rate of γp ¼ 4:24P0  109 s1 , where P0 is the laser output power measured in mW. The laser has a threshold current of I th ¼ 18 mA. It is biased at a DC injection current of I 0 ¼ 50 mA and is current modulated with a modulation index of m ¼ 10% at a modulation frequency of f ¼ 10 GHz.

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(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. (f) At this modulation frequency, what is the modulation response in the electrical power spectrum of the photodetector that is used to measure the laser output? What is the normalized modulation response measured in dB? 10.4.1 LiNbO3 is a negative uniaxial crystal, which has nx ¼ ny ¼ no ¼ 2:222 and nz ¼ ne ¼ 2:145 at the λ ¼ 1:3 μm wavelength. It has eight nonvanishing Pockels coefficients, which are r13 ¼ r 23 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 r 51 ¼ 28 pm V1 . For electro-optic phase modulation using a LiNbO3 electro-optic modulator, the half-wave voltage V π can be significantly reduced by transverse modulation in a waveguide structure while using the largest Pockels coefficient. How can the lowest possible V π be accomplished by properly arranging the optical wave and the applied voltage with respect to the crystal axes and the waveguide structure? What is this lowest value of V π for a waveguide that has the dimensions of l ¼ 3 mm and d ¼ 2 μm? 10.4.2 KTP is a biaxial crystal of the mm2 symmetry group, which has nx ¼ 1:742, ny ¼ 1:750, and nz ¼ 1:832 at the λ ¼ 1:0 μm optical wavelength. Its only 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 r 51 ¼ 6:9 pm V1 . A KTP electro-optic transverse phase modulator has the configuration shown in Fig. 10.9(a) with the modulation voltage applied along the z principal axis. Answer each of the following questions for an optical wave at λ ¼ 1:0 μm that is linearly polarized in the z direction and propagates in the x direction. (a) Find the phase modulation depth φm as a function of the parameters of KTP, the dimensions of the modulator, and the peak modulation voltage V m . What is the halfwave voltage required for a modulated phase shift of π? (b) Find the half-wave voltage V π required for a modulated phase shift of π for a bulk modulator that has the dimensions of d ¼ 3 mm and l ¼ 6 mm. (c) Find the half-wave voltage V π required for a modulated phase shift of π for a waveguide modulator that has the dimensions of d ¼ 3 μm and l ¼ 6 mm. Figure 10.22 GaAs electro-optic longitudinal modulator.

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10.4.3 Consider a GaAs longitudinal electro-optic modulator as shown in Fig. 10.22. GaAs is a nonbirefringent cubic crystal of the 43m symmetry group. At λ ¼ 1:0 μm, it has nx ¼ ny ¼ nz ¼ no ¼ 3:5, and its only nonvanishing Pockels coefficients are

r 41 ¼ r 52 ¼ r 61 ¼ 1:2 pm V1 . In the illustration, ^x , ^y , and ^z are the intrinsic principal ^ , Y^ , and Z^ are the new principal axes of GaAs without an applied voltage, whereas X axes when a voltage is applied in the z direction as shown. ^ and Y^ , (a) For two linearly polarized input optical fields that are polarized along X respectively, find the phase changes ΔφX and ΔφY in the two fields at the output as functions of the parameters of GaAs, the dimensions of the modulator, and the modulation voltage V. (b) This device can be used as an electro-optic polarization modulator. Describe how the input field polarization has to be arranged for the device to function as a voltagecontrolled half-wave plate that rotates the optical field polarization direction of a linearly polarized input field by 90 at the output. (c) Find the half-wave voltage V π required for the modulator to function as a polarization modulator as described in (b) if its dimensions are d ¼ 3 mm and l ¼ 1 cm. 10.4.4 Answer the questions in Example 10.7 for the TM-like mode instead of the TE-like mode considered in Example 10.7. What is the lowest voltage required for a transmittance of 50%? 10.4.5 Consider an x-cut, y-propagating KTP Mach–Zehnder waveguide interferometer in the push–pull configuration as shown in Fig. 10.12 for the LiNbO3 interferometer. KTP is a biaxial crystal of the mm2 symmetry group, which has nx ¼ 1:742, ny ¼ 1:750, and nz ¼ 1:832 at the λ ¼ 1:0 μm optical wavelength. Its only 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 r 51 ¼ 6:9 pm V1 . The KTP Mach–Zehnder waveguide interferometer has identical single-mode waveguides for both arms, which have confinement factors of ΓTE ¼ ΓTM ¼ 0:7 for λ ¼ 1:0 μm. The electrodes have an equal length of l ¼ 3 mm and an equal separation of se ¼ 8 μm. (a) Find the half-wave voltage of this amplitude modulator for the TE-like mode at λ ¼ 1:0 μm. What is the lowest voltage required for a transmittance of 30%? (b) Answer the questions in (a) for the TM-like mode. 10.4.6 A Faraday rotator consists of a TGG crystal in a magnetic field that has a flux density of B0z ¼ 0:35 T along the longitudinal axis of the crystal. The Verdet constant of TGG is V ¼ 80 rad T1 m1 at the 750 nm wavelength and V ¼ 65 rad T1 m1 at the 800 nm wavelength. If a linearly polarized optical wave at the 750 nm wavelength is sent through the TGG Faraday rotator, what is the required length of the crystal for a Faraday rotation angle of 45 in a single pass? In which sense does the polarization rotate? With this magnetic field and this crystal length, what is the Faraday rotation angle in a single pass for a linearly polarized wave at the 800 nm wavelength? 10.4.7 Ce3+–P glass has a Verdet constant of V ¼ 94:7 rad T1 m1 at the 500 nm optical wavelength. A Ce3+–P glass rod of a length l ¼ 5 cm is placed between two cross polarizers, which have orthogonally oriented transmission polarization directions. An

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optical beam at the 500 nm wavelength is polarized in the transmission polarization direction of the input polarizer and is sent through this system. (a) What is the required magnetic flux density B0 that has to be applied along the propagation direction of the beam for the beam to be completely transmitted at the output of the system, i.e., at the output end of the second cross polarizer? (b) If the magnetic flux density is half of that found in (a), what is the percentage transmission of the beam through the system? (c) If the magnetic flux density is twice that found in (a), what is the percentage transmission of the beam through the system? (d) If the magnetic flux density remains that found in (a) but the direction of the magnetic field is oriented at 45 with respect to the propagation direction of the beam, what is the percentage transmission of the beam through the system? 10.4.8

Show that as required by the phase-matching condition given in (10.84) for up-shifted diffraction and that in (10.85) for down-shifted diffraction in the Bragg regime, the angle of incidence for the incoming optical wave is that given in (10.86) and the angle of diffraction for the diffracted wave is that given in (10.87) with the upper signs for up-shifted diffraction and the lower signs for down-shifted diffraction.

10.4.9

LiNbO3 is a negative uniaxial crystal, which has nx ¼ ny ¼ no ¼ 2:222 and nz ¼ ne ¼ 2:145 at the λ ¼ 1:3 μm wavelength and nx ¼ ny ¼ no ¼ 2:291 and nz ¼ ne ¼ 2:201 at the λ ¼ 632:8 nm wavelength. It is a very good acousto-optic crystal that can be used at high acoustic frequencies up to 5 GHz. The acoustic wave velocity in LiNbO3 depends on the polarization and propagation directions of the acoustic wave. The acoustic wave velocity is v La ¼ 6:57 km s1 for a longitudinal acoustic wave that propagates in the ½100 crystal direction such that K ¼ K L ^x , and it is v Ta ¼ 3:59 km s1 for a transverse acoustic wave that propagates in the ½001 crystal direction such that K ¼ K T^z . Consider the diffraction of an optical wave by a traveling acoustic wave through an interaction length of l ¼ 5 mm. (a) The optical wave is linearly polarized along the z principal axis such that E ¼ E^z and the acoustic wave is the longitudinal wave with K ¼ K L ^x . For each optical wavelength, find the minimum acoustic frequency required for the diffraction to be in the Bragg regime. (b) The optical wave is linearly polarized along the x principal axis such that E ¼ E^x and the acoustic wave is the transverse wave with K ¼ K T^z . For each optical wavelength, find the minimum acoustic frequency required for the diffraction to be in the Bragg regime.

10.4.10 Silica glass has a refractive index of n ¼ 1:452 at λ ¼ 850 nm. It has an acoustic wave velocity of v a ¼ 5:97 km s1 and an acousto-optic figure of merit of M 2 ¼ 1:50 

1015 m2 W1 for a longitudinal acoustic wave and an optical wave polarized in a direction perpendicular to the propagation direction K of the acoustic wave. A traveling-wave acousto-optic modulator made of silica glass in the configuration shown in Fig. 10.16 is used to modulate an optical wave at λ ¼ 850 nm. The transducer that generates a longitudinal acoustic wave has the dimensions of L ¼ 1 cm and H ¼ 3 mm; it delivers an acoustic power of Pa ¼ 300 mW.

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(a) What is the acoustic frequency required for this modulator to operate in the Bragg regime? (b) If the acoustic frequency is chosen to be f ¼ 300 MHz, what is the Bragg angle? What is the deflection angle between the diffracted beam and the incident beam? (c) What is the diffraction efficiency? 10.4.11 Silica glass has a refractive index of n ¼ 1:452 at λ ¼ 850 nm. It has an acoustic wave velocity of v a ¼ 5:97 km s1 and an acousto-optic figure of merit of M 2 ¼ 1:50 

1015 m2 W1 for a longitudinal acoustic wave and an optical wave polarized in a direction perpendicular to the propagation direction K of the acoustic wave. A standing-wave acousto-optic modulator made of silica glass in the configuration shown in Fig. 10.17 is used to modulate an optical wave at λ ¼ 850 nm. The transducer that generates a longitudinal acoustic wave has the dimensions of L ¼ 1 cm and H ¼ 3 mm; it delivers an acoustic power of Pa ¼ 300 mW. The acoustic cavity has a cell width of W ¼ 1 cm and a decay rate of γa ¼ 9:1  104 s1 . (a) It is desired that the optical wave is modulated at a modulation frequency of f m ¼ 300 MHz. What is the acoustic frequency required for this purpose? (b) At the acoustic frequency found in (a), what is the minimum required acoustooptic interaction length l for Bragg diffraction? Does the acousto-optic modulator satisfy this requirement? (c) What is the deflection angle between the diffracted beam and the undiffracted beam? (d) What is the peak value of the diffraction efficiency? 10.4.12 A laser beam that has a transverse spatial intensity distribution can cause a spatially varying intensity-dependent Kerr phase change as expressed in (10.104) through selfphase modulation. For a circular beam that propagates along a longitudinal direction taken to be the z direction, the transverse spatial intensity profile can be expressed as 1=2

I ðr Þ as a function of the radial variable r ¼ ðx2 þ y2 Þ . (a) The radially varying intensity-dependent Kerr phase has the same effect as a thin lens. The effective focal length f K of the Kerr lens is given by the relation:  1 c d2 φK  ¼ a , (10.115) fK ω dr 2 r¼0 where a is a correction factor that depends on the profile of the circular optical beam. Find the relation between f K and the intensity profile I ðr Þ. (b) The intensity profile of a fundamental circular Gaussian beam at a fixed z location is

 r2 I ðr Þ ¼ I 0 exp 2 2 , (10.116) w where I 0 is the intensity at the beam center and w is the beam spot size at the given z location. Find the Kerr focal length f K for the Gaussian beam as a function of the beam parameters. Express it also in terms of the power P ¼ πw2 =2 of the beam. (c) For a circular Gaussian beam, a ¼ 1:723. An ultrashort laser pulse at λ ¼ 532 nm has a peak power of Ppk ¼ 10 kW. It has a fundamental circular Gaussian beam

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intensity profile and is focused on a thin silica piece of l ¼ 1 mm to a focused spot size of w ¼ 12 μm. For silica, the coefficient of intensity-dependent index change is n2 ¼ 2:4  1020 m2 W1 . Find the Kerr focal length f K for the Gaussian beam at its pulse peak. 10.4.13 For a cubic crystal of the 43m symmetry group, the only nonvanishing third-order ð3 Þ ð3Þ ð3Þ ð3Þ nonlinear susceptibilities are the elements of the types χ 1111 , χ 1122 , χ 1212 and χ 1221 , ð3Þ ð3Þ 3Þ 3Þ 3Þ 3Þ 2Þ where χ 1111 represents χ ðxxxx ¼ χ ðyyyy ¼ χ ðzzzz and χ 1122 represents χ ðxxyy ¼ χ ðyyzz ¼ 3Þ ¼   , and so forth. Consider cross-phase modulation on an optical wave at a χ ðzzxx frequency of ω by another optical wave at a frequency of ω0 , both propagating along the z principal axis of the crystal. The optical field at ω is initially linearly pffiffimodulated ffi  polarized as EðωÞ ¼ E ðωÞ ^x þ ^y Þ= 2, and the modulating optical field at ω0 is linearly polarized along the x principal axis as Eðω0 Þ ¼ Eðω0 Þ^x . The modulating field at ω0 is so much stronger than the modulated field at ω that the self-phase modulation of the field at ω can be ignored in comparison to the cross-phase modulation by the field at ω0 . (a) Find the intensity-dependent refractive indices nx and ny seen by the x and y components of the modulated field at ω as functions of the intensity I ðω0 Þ of the modulating field at ω0 . (b) What is the minimum intensity of the modulating field required pffiffifor ffi the modulated  field to be linearly polarized in a direction parallel to ^x  ^y Þ= 2 after both waves propagate through the crystal over a distance of l?

10.5.1

The bandgap of GaAs at room temperature is Eg ¼ 1:424 eV, corresponding to the energy of a photon that has a wavelength of λg ¼ 870:6 nm. A GaAs/AlGaAs quantum well has GaAs in the quantum-confined well region that has a width of d QW . The ∗ electron and hole effective masses in GaAs are m∗ e ¼ 0:067 m0 and mh ¼ 0:52 m0 , respectively, where m0 is the free electron mass. It is desired that at room temperature, the absorption edge of the quantum well be at the optical wavelength of λ ¼ 840 nm when no voltage is applied to the structure. Find the required width d QW of the quantum well.

10.5.2

A waveguide electro-absorption modulator for λ ¼ 1:55 μm is based on the quantum Stark effect. It consists of 10 InGaAs/InGaAsP quantum wells. The optical path length in the waveguide is l ¼ 250 μm. The voltage-dependent effective absorption coefficient of the TE waveguide mode has the values of αð0Þ ¼ 400 m1 , αð2 VÞ ¼ 5 103 m1 , αð4VÞ ¼ 9:6  103 m1 , αð6 VÞ ¼ 1:34  104 m1 , and αð8 VÞ ¼ 1:6  104 m1 . (a) What is the highest possible transmittance T high ? What is the voltage for this transmittance? (b) With respect to the highest transmittance found in (a), a low transmittance can be chosen by applying a specific voltage. Find the low-transmittance value for each possible voltage listed above other than the high-transmittance voltage found in (a). Find the extinction ratio, in dB, for each case. (c) What is the extinction ratio if the device is switched between the two voltages of V ¼ 2 V and V ¼ 4 V?

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Bibliography

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10.5.3 An all-optical switch uses a saturable absorber that has an unsaturated absorption coefficient of α0 , a saturation intensity of I sat , and a thickness of l in the optical path. It is desired that the transmittance be switched between the two levels of T high 90% and T low 10% for the high and low input light intensities, respectively. (a) What is the minimum value of α0 l required for this function? (b) If the value of α0 l is chosen to be 10% above the minimum value found in (a), i.e., α0 l ¼ 1:1ðα0 lÞmin , what are the required high and low input intensities for T high 90% and T low 10%, respectively? 10.5.4 The temporal intensity profile of a Gaussian optical pulse that has a FWHM pulsewidth of Δtps is described as ! t2 I ðt Þ ¼ I pk exp 4 ln 2 2 , (10.117) Δt ps

where I pk is the intensity at the temporal pulse peak, which is taken to be at t ¼ 0. Such a Gaussian pulse is passed through a saturable absorber that has an unsaturated absorption coefficient of α0 , a saturation intensity of I sat , and a thickness of l. With a peak in intensity of I in pk ¼ 10I sat and a pulsewidth of Δt ps for the pulse at the input end, it is found that the transmittance at the pulse peak is T pk ¼ 90% such that the pulse at the in output end has a peak intensity of I out pk ¼ 0:9I pk ¼ 9I sat . The nonlinear response of the saturable absorber to the temporally varying intensity of the pulse results in a reduction of the pulsewidth at the output. Find the pulsewidth Δtout ps of the output pulse in as a percentage of the input pulsewidth Δtps .

Bibliography Boyd, R. W., Nonlinear Optics, 3rd edn. Boston, MA: Academic Press, 2008. 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. Davis, C. C., Lasers and Electro-Optics: Fundamentals and Engineering, 2nd edn. Cambridge: Cambridge University Press, 2014. 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 for Fiber and Integrated Optics, Vol. II. New York: Wiley, 2002. Korpel, A., Acousto-Optics, 2nd edn. New York: Marcel Dekker, 1997. Liu, J. M., Photonic Devices. Cambridge: Cambridge University Press, 2005. 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. Sugano, S. and Kojima, N., eds., Magneto-Optics. Berlin: Springer, 2000. 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 11 - Photodetection pp. 362-395 Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.012 Cambridge University Press

11

Photodetection

11.1 PHYSICAL PRINCIPLES OF PHOTODETECTION

.............................................................................................................. Photodetection converts an optical signal into a signal of another form. Most photodetectors convert optical signals into electrical signals that can be further processed or stored. All photodetectors are square-law detectors that respond to the power or intensity, rather than the field amplitude, of an optical signal. The electrical signal generated by an optical signal is either a photocurrent or a photovoltage that is proportional to the power of the optical signal. Based on the difference in the conversion mechanisms, there are two classes of photodetectors: photon detectors and thermal detectors. Photon detectors are quantum detectors based on the photoelectric effect, which converts a photon into an emitted electron or an electron–hole pair; a photon detector responds to the number of photons absorbed by the detector. Thermal detectors are based on the photothermal effect, which converts optical energy into heat; a thermal detector responds to the optical energy, rather than the number of photons, absorbed by the detector. Because of this fundamental difference, the general characteristics of these two classes of photodetectors have a number of important differences. The response of a photon detector is a function of the optical wavelength with a longwavelength cutoff, whereas that of a thermal detector is wavelength independent. A photon detector can be much more responsive than a thermal detector in a particular spectral region, which typically falls somewhere within the range from the near ultraviolet to the near infrared. By comparison, a thermal detector normally covers a wide spectral range from the deep ultraviolet to the far infrared with a nearly constant response. Photon detectors can be made extremely sensitive. Some of them have a photon-counting capability that is not possible for a thermal detector. A photon detector can be designed to have a high response speed capable of following very fast optical signals. Most thermal detectors are relatively slow in response because the speed of a thermal detector is limited by thermalization through heat diffusion and by heat dissipation when the power of an optical signal varies. For these reasons, photon detectors are suitable for detecting optical signals in photonic systems, whereas thermal detectors are most often used for optical power measurement or infrared imaging. In this chapter, only the basic principles of photon detectors are discussed because our major concern is photodetection for photonics applications. Photon detectors can be classified into two groups: one based on the external photoelectric effect and the other based on the internal photoelectric effect. Photodetectors based on the external photoelectric effect are photoemissive devices, such as vacuum photodiodes and photomultiplier tubes, in which photoelectrons are ejected from the surface of a photocathode. Photodetectors based on the internal photoelectric effect are semiconductor devices, in which electron–hole pairs are generated through the absorption of incident photons.

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A host of such devices have been developed, such as photoconductors, junction photodiodes, many photovoltaic devices, phototransistors, and charge-coupled devices. In the discussion of photodetection, we consider an input optical signal with an optical power Ps . The detection system has an electrical response bandwidth of Δf ¼ B to effectively sample the optical signal within a rectangular time interval of Δt ¼

1 : 2B

(11.1)

The total number of photons received by the photodetector within this time interval is S¼

Ps Ps Δt ¼ : hv 2Bhv

(11.2)

If the photodetector has an external quantum efficiency of ηe , the total number of charge carriers generated in the photodetector by the photoelectric effect upon receiving the photons within the time interval Δt is N ¼ ηe S ¼ ηe

Ps , 2Bhv

(11.3)

where 0  ηe  1. Consequently, the photocurrent is iph ¼

eN ePs , ¼ 2eBN ¼ ηe hv Δt

(11.4)

where e is the electronic charge. The signal current is is ¼ iph for a photodetector that has no internal gain. The signal current is is ¼ Giph for a photodetector that has an internal gain of G.

11.1.1 External Photoelectric Effect Photoemissive detectors are based on the external photoelectric effect. Photoelectrons are emitted when the surface of a metal or a semiconductor, known as a photocathode in this situation, is illuminated with light of a sufficient photon energy. The lowest vacuum energy level, Evac , for an electron that is free from the confinement of a material is higher than the Fermi level in the material. For both a metal and a semiconductor, the energy barrier between the lowest vacuum level and the Fermi level is defined as the work function, eϕ ¼ Evac  EF , of the material. For a semiconductor, the difference between the lowest vacuum level and the conduction-band edge is known as the electron affinity, eχ ¼ Evac  Ec , of the semiconductor. The parameters ϕ and χ have the physical property of an electric potential measured in volts. The work function and the electron affinity are normally measured in electronvolts. Photoemission from a given material occurs only when the incident photon has an energy higher than a certain threshold photon energy, Eth , corresponding to an optical wavelength shorter than a threshold wavelength, λth : hv  E th , for λ  λth ¼

hc 1:2398 ¼ μm eV: E th Eth

The values of E th and λth are the characteristics of a given material.

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(11.5)

Figure 11.1 Photon energy requirement for photoemission from the surface of (a) a metal, (b) a nondegenerate semiconductor, (c) an n-type degenerate semiconductor, and (d) a p-type degenerate semiconductor.

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1. Metal: In a metal, shown in Fig. 11.1(a), electrons occupy all energy levels below the Fermi level. The threshold photon energy for the emission of a photoelectron from a metal is E th ¼ eϕ:

(11.6)

2. Nondegenerate semiconductor: In a nondegenerate semiconductor, shown in Fig. 11.1(b), not all energy levels below the Fermi level, but only those below the valence-band edge, are occupied by electrons because the Fermi level lies within the bandgap. The threshold photon energy for photoemission from a nondegenerate semiconductor that has a bandgap of E g is E th ¼ eχ þ Eg > eϕ

if χ > 0:

(11.7)

3. Degenerate semiconductor: In a degenerate semiconductor, the highest level occupied by electrons is the Fermi level. Therefore, the threshold photon energy for photoemission from a degenerate semiconductor is the work function, just like that given in (11.6) for a metal. For an n-type degenerate semiconductor, E th ¼ eϕ < eχ, as shown in Fig. 11.1(c), because the Fermi level lies in the conduction band. For a p-type degenerate semiconductor, E th ¼ eϕ > eχþ Eg , as shown in Fig. 11.1(d), because the Fermi level lies in the valence band.

EXAMPLE 11.1 Among all elemental metals, Cs has the lowest work function of 2.14 eV. What is the threshold wavelength for an optical wave to cause photoemission from a Cs surface? If a Cs surface is illuminated with a laser beam at the 400 nm wavelength, what is the highest kinetic energy of the photoemitted electrons? Solution: With a work function of eϕ ¼ 2:14 eV, the threshold photon energy for photoemission is Eth ¼ eϕ ¼ 2:14 eV because Cs is a metal. Therefore, the threshold wavelength is λth ¼

1239:8 1239:8 nm eV ¼ nm ¼ 579:3 nm: E th 2:14

The kinetic energy of a photoemitted electron is T ¼ m0 v 2 =2  hv  Eth . When a Cs surface is illuminated with photons at λ ¼ 400 nm, the highest kinetic energy of the photoemitted electrons is T max ¼ hv  E th ¼

1239:8 eV  2:14 eV ¼ 959:5 meV: 400

EXAMPLE 11.2 At room temperature, silicon has an electron affinity of eχ ¼ 4:05 eV and a bandgap of Eg ¼ 1:12 eV. (a) The Fermi level of intrinsic Si lies at EF ¼ E c  572:8 meV ¼ E v þ 547:2 meV, where E c and E v are the conduction-band and valence-band edges, respectively. Find the work function of intrinsic Si. What is the threshold photon energy and the threshold

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wavelength for an optical wave to cause photoemission from its surface? (b) Find the work function, the threshold photon energy, and the threshold wavelength for a lightly doped n-type silicon crystal that has a Fermi level at EF ¼ E c  200 meV. (c) Find the work function, the threshold photon energy, and the threshold wavelength for a heavily doped n-type silicon crystal that has a Fermi level at E F ¼ E c þ 200 meV. Solution: The work function of a semiconductor is eϕ ¼ Evac  E F , and the electron affinity is eχ ¼ E vac  Ec : (a) The work function of intrinsic Si is eϕ ¼ Evac  EF ¼ E vac  E c þ 572:8 meV ¼ eχ þ 547:2 meV ¼ 4:5972 eV: Because the Fermi level lies in the bandgap, the threshold photon energy is not the work function but is Eth ¼ eχ þ Eg ¼ 4:05 eV þ 1:12 eV ¼ 5:17 eV > eϕ: The threshold wavelength is λth ¼

1239:8 1239:8 nm eV ¼ nm ¼ 239:8 nm: E th 5:17

(b) The work function of the lightly doped n-type silicon with E F ¼ E c  200 meV is eϕ ¼ Evac  E F ¼ E vac  Ec þ 200 meV ¼ eχ þ 200 meV ¼ 4:25 eV: This lightly doped Si is nondegenerate because its Fermi level lies in the bandgap. Therefore, the threshold photon energy is not the work function but is Eth ¼ eχ þ Eg ¼ 4:05 eV þ 1:12 eV ¼ 5:17 eV > eϕ: The threshold wavelength is λth ¼

1239:8 1239:8 nm eV ¼ nm ¼ 239:8 nm: E th 5:17

(c) The work function of the heavily doped n-type silicon with E F ¼ E c þ 200 meV is eϕ ¼ Evac  E F ¼ E vac  Ec  200 meV ¼ eχ  200 meV ¼ 3:85 eV: This heavily doped Si is degenerate because its Fermi level lies above the conduction-band edge. Therefore, the threshold photon energy is the work function: E th ¼ eϕ ¼ 3:85 eV < eχ: The threshold wavelength is λth ¼

1239:8 1239:8 nm eV ¼ nm ¼ 322 nm: E th 3:85

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The work functions of elemental metals are in the range of 2–6 eV. The lowest is that of Cs at 2.14 eV. Elemental metals have poor quantum efficiencies. Ordinary group IV and III–V, semiconductors, including Si, Ge, GaAs, and InP, have work functions typically in the range of 4–5 eV. Because of their high threshold photon energies and low quantum efficiencies, elemental metals and ordinary semiconductors are not useful for photocathodes in the visible and infrared spectral regions. There are two groups of practical photocathodes that have both high quantum efficiencies and low threshold photon energies. One group consists of compounds of alkaline metals and cesiated silver oxides that are usually labeled using a standard international designation of spectral response and window type, such as S-1 (AgOCs), S-4 (Cs3Sb), S-10 (AgBiOCs), S-11 (Cs3Sb), S-20 (Na2KCsSb), and S-24 (Na2KSb). These compounds are semiconductors that have low threshold photon energies in the range of 1–2 eV because of their small bandgaps and small electron affinities. Another group consists of negative electron affinity (NEA) photocathodes. An NEA photocathode is made by depositing a very thin n-type layer on the surface of a p-type semiconductor to cause a large downward band bending at the surface. The photocathode has a negative effective affinity if the band bending is sufficiently large that the conduction-band edge of the p-type semiconductor lies above the vacuum level, as shown in Fig. 11.2. Practical NEA photocathodes have been developed for a few III–V semiconductors by depositing a thin layer of Cs or Cs2O on the surface; these include GaAs:Cs2O, InGaAs:Cs, and InAsP:Cs. As can be seen in Fig. 11.2, once an electron is excited to the conduction band of an NEA photocathode, it has sufficient energy to be emitted by tunneling through the thin surface layer because E c > E vac . Therefore, the threshold photon energy for photoemission from an NEA photocathode is simply the bandgap of the semiconductor: E th ¼ E g :

(11.8)

Figure 11.3 shows the spectral responsivity, which is defined in (11.50) in Section 11.3, of representative photocathodes. The spectral responsivity of a photoemissive device has a longwavelength cutoff determined by the threshold wavelength of the photocathode material and a

Figure 11.2 Energy levels and photoemission in an NEA photocathode.

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Figure 11.3 Spectral responsivity of representative photocathodes. The quantum efficiency is indicated by the gray curves.

short-wavelength cutoff determined by the window material. The standard international designation with the letter S, such as S-1, includes both the response of the photocathode material and the transmission of the window material. Among the practical photocathodes, including alkaline compounds and NEA semiconductors, S-1 has the lowest threshold energy of approximately 1:1 eV, corresponding to a threshold wavelength around 1:1 μm. Currently no photocathode can respond at wavelengths longer than 1:2 μm. Therefore, no photoemissive detectors exist for the infrared at wavelengths longer than 1:2 μm.

11.1.2 Photoconductivity Photoconductive detectors are based on the phenomenon of photoconductivity. The conductivity of a photoconductor, which is usually a semiconductor but can sometimes be an insulator, increases with optical illumination due to the photogenerated excess carriers. The conductivity of a semiconductor that has electron and hole concentrations of n and p, respectively, is σ ¼ eðμe n þ μh pÞ,

(11.9)

where e is the electronic charge, and μe and μh are the electron and hole mobilities, respectively. In the absence of optical illumination, a semiconductor has a dark conductivity of σ 0 ¼ eðμe n0 þ μh p0 Þ because the electron and hole concentrations in this situation are the equilibrium concentrations, n0 and p0 , respectively. When a semiconductor is illuminated with light of a sufficient photon energy, carriers in excess of the equilibrium concentrations are generated. The photoconductivity is the additional conductivity contributed by these photogenerated excess carriers: Δσ ¼ σ  σ 0 ¼ eðμe Δn þ μh ΔpÞ,

(11.10)

where Δn ¼ n  n0 and Δp ¼ p  p0 are the photogenerated excess electron and hole concentrations, respectively. Similar to photoemission, photoconductivity also has a threshold photon energy, E th , and a corresponding threshold wavelength, λth , that are the characteristics of a given photoconductor. Depending on the processes involved in the photogeneration of free carriers, there are two

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Figure 11.4 Optical transitions for (a) intrinsic photoconductivity, (b) n-type extrinsic photoconductivity, and (c) p-type extrinsic photoconductivity.

principal types of photoconductivity. The intrinsic photoconductivity is contributed by the excess electrons and holes that are generated by band-to-band absorption of the incident photons, as shown in Fig. 11.4(a). The threshold photon energy of intrinsic photoconductivity is clearly the bandgap energy of the photoconductor: Eth ¼ E g :

(11.11)

The extrinsic photoconductivity is contributed by carriers that are generated by optical transitions associated with impurity levels within the bandgap of an extrinsic semiconductor. In an ntype extrinsic photoconductor, the impurity levels have an energy of E i ¼ E d below the conduction-band edge; electrons are excited from these donor levels to the conduction band, as shown in Fig. 11.4(b). In a p-type extrinsic photoconductor, the impurity levels have an energy of E i ¼ E a above the valence-band edge; electrons are excited from the valence band to these acceptor levels, as shown in Fig. 11.4(c). Thus, the threshold photon energy of extrinsic photoconductivity for both n-type and p-type photoconductors is E th ¼ E i :

(11.12)

The spectral response of a photoconductor is determined by its threshold wavelength and the spectral dependence of its absorption coefficient. Photoconductors cover a broad spectral range from the ultraviolet to the far infrared. In particular, there are many sensitive photoconductors in the infrared region at wavelengths longer than 1.2 μm where no photoemissive detectors exist. Both direct-gap and indirect-gap semiconductors are used for photoconductors. All group IV semiconductors, III–V and II–VI compound semiconductors, and IV–VI compound semiconductors can be used as intrinsic photoconductors. Among them, intrinsic silicon photoconductors are the most important photoconductive detectors in the visible and near-infrared spectral regions at wavelengths shorter than 1.1 μm, while intrinsic germanium photoconductors are the most important photoconductive detectors in the near-infrared region at wavelengths up to 1.8 μm. In the mid-infrared region between 2 and 7 μm wavelength, one finds intrinsic photoconductors based on InAs, InSb, PbS, and PbSe. Extrinsic photoconductors are available for these mid-infrared wavelengths as well as for longer wavelengths well into the far-infrared region. The most important extrinsic photoconductive detectors are p-type germanium photoconductors, such as Ge:Au, Ge:Hg, Ge:Cd, Ge:Cu, and Ge:Zn. Figure 11.5 shows the specific detectivity, which is defined in (11.59) in Section 11.3, of representative infrared photoconductive detectors as a function of the optical wavelength.

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Figure 11.5 Specific detectivity, D∗ , of representative infrared photoconductive detectors as a function of optical wavelength. The gray curve shows for comparison the ideal D∗ for a background-limited photoconductor of unity quantum efficiency.

EXAMPLE 11.3 At T ¼ 300 K, intrinsic Si has the same electron and hole concentration of n0 ¼ p0 ¼ ni ¼ 7:0  1015 m3 in thermal equilibrium. It has an electron mobility of μe ¼ 0:135 m2 V1 s1 and a hole mobility of μh ¼ 0:048 m2 V1 s1 . An intrinsic Si crystal that is used as a photoconductor is uniformly illuminated with an optical beam to generate electron–hole pairs such that the electrons and holes have the same concentration of n ¼ p ¼ 2:0  1020 m3 . Find the dark conductivity and the photoconductivity. Solution: The dark conductivity is σ 0 ¼ eðμe n0 þ μp p0 Þ ¼ eðμe þ μp Þni ¼ 1:6  1019  ð0:135 þ 0:048Þ  7:0  1015 S m1 ¼ 1:57  103 S m1 : Because n ¼ p ¼ 2:0  1020 m3 is more than four orders of magnitude larger than n0 ¼ p0 ¼ ni ¼ 7:0  1015 m3 , we find that Δn ¼ n  n0 ¼ p  p0  n ¼ 2:0  1020 m3 . Therefore, the photoconductivity is Δσ ¼ σ  σ 0 ¼ eðμe Δn þ μp ΔpÞ  eðμe þ μp Þn ¼ 1:6  1019  ð0:135 þ 0:048Þ  2:0  1020 S m1 ¼ 5:856 S m1 :

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Figure 11.6 Spectral responsivity of representative photodiodes as a function of optical wavelength at 300 K. The quantum efficiency is indicated by the gray curves.

11.1.3 Photodiodes Every junction diode has a photoresponse that can be utilized for optical detection. Junction photodiodes are the most commonly used photodetectors in the photonics industry. They can take many different forms, including semiconductor homojunctions, semiconductor heterojunctions, and metal–semiconductor junctions. Similar to that of a photoconductor, the photoresponse of a photodiode results from the photogeneration of electron–hole pairs. In contrast to a photoconductor, which can be of either intrinsic or extrinsic type, a photodiode is normally of intrinsic type, in which electron–hole pairs are generated through band-to-band optical absorption. Therefore, the threshold photon energy of a semiconductor photodiode is the bandgap energy of its active region: Eth ¼ E g :

(11.13)

Junction photodiodes cover a wide spectral range from the ultraviolet to the infrared. All of the semiconductor materials used for intrinsic photoconductors discussed in the preceding section can be used for photodiodes with similar spectral characteristics. Figure 11.6 shows the spectral responsivity, which is defined in (11.50) in Section 11.3, of representative photodiodes as a function of the optical wavelength at 300 K. All junction photodiodes share some basic principles and characteristics. Therefore, we consider a simple p–n homojunction photodiode for a general discussion of the common principles and characteristics. In a semiconductor photodiode, the generation of electron–hole pairs by optical absorption can take place in any of the different regions: the depletion layer, the diffusion regions, and the homogeneous regions. In the depletion layer of a diode, the immobile space charges create an internal electric field that has a polarity in the direction from the n side to the p side, resulting in an electron energy-band gradient shown in Fig. 11.7. When an electron–hole pair is generated in the depletion layer by photoexcitation, the internal field sweeps the electron to the n side and the hole to the p side, as illustrated in Fig. 11.7. This process results in a drift current that flows in the reverse direction from the cathode on the n side to the anode on the p side. If a photoexcited electron–hole

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Figure 11.7 Photoexcitation and energy-band gradient of a p–n photodiode.

pair is generated within one of the diffusion regions at the edges of the depletion layer, the minority carrier, which is the electron in the p-side diffusion region or the hole in the n-side diffusion region, can reach the depletion layer by diffusion and then be swept to the other side by the internal field, as also illustrated in Fig. 11.7. This process results in a diffusion current that also flows in the reverse direction. For an electron–hole pair generated by absorption of a photon in the p or n homogeneous region, no current is generated because there is no internal field to separate the charges and a minority carrier generated in a homogeneous region cannot diffuse to the depletion layer before recombining with a majority carrier. Because photons absorbed in the homogeneous regions do not generate any photocurrent, the active region of a photodiode consists of only the depletion layer and the diffusion regions. For a high-performance photodiode, the diffusion current is undesirable and is minimized. Therefore, the active region mainly consists of the depletion layer where a drift photocurrent is generated. The external quantum efficiency, ηe , of a photodiode is the fraction of total incident photons absorbed in the active region that actually contributes to the photocurrent. There are two contributions to the photocurrent in a junction photodiode: a drift current from photogeneration in the depletion layer and a diffusion current from photogeneration in the diffusion regions. The homogeneous regions on the two ends of the diode act like blocking layers for the photogenerated carriers because carriers neither drift nor diffuse through these regions. Consequently, a junction photodiode acts like a photoconductor with two blocking contacts, with the external signal current being equal to the photocurrent: ePs : (11.14) hv This photocurrent is a reverse current that depends only on the power of the optical signal. When a bias voltage is applied to the photodiode, the total current of the photodiode is the combination of the diode current and this reverse photocurrent is ¼ iph ¼ ηe

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Figure 11.8 Current–voltage characteristics of a junction photodiode at various power levels of optical illumination. The basic circuitry and load line are shown for the photodiode (a) in the photoconductive mode and (b) in the photovoltaic mode.



 ePs , iðV; Ps Þ ¼ I 0 eeV=akB T  1  is ¼ I 0 eeV=akB T  1  ηe hv

(11.15)

which is a function of both the bias voltage V and the optical signal power Ps . The dark characteristics for Ps ¼ 0 are simply those of an unilluminated diode, with I 0 being the reverse current and a being a device-specific factor that has a value between 1 and 2 for a realistic diode. Figure 11.8 shows the current–voltage characteristics of a junction photodiode at various power levels of optical illumination. According to (11.15), the current–voltage characteristics of an illuminated photodiode shift downward from the dark characteristics by the amount of the photocurrent, which is linearly proportional to the optical power but is independent of the bias voltage. As shown in Fig. 11.8, there are two modes of operation for a junction photodiode. The device functions in the photoconductive mode in the third quadrant of its current–voltage characteristics, including the short-circuit condition on the vertical axis for V ¼ 0. It functions in the photovoltaic mode in the fourth quadrant, including the open-circuit condition on the horizontal axis for i ¼ 0. The mode of operation is determined by the external circuitry and the bias condition.

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The circuitry for the photoconductive mode, shown in Fig. 11.8(a), normally consists of a reverse bias voltage of V ¼ V r and a load resistance of RL . In this mode of operation, it is necessary to keep the output voltage, v out , smaller than the bias voltage, V r , so that a reverse voltage is maintained across the photodiode. This requirement can be fulfilled if the bias voltage is sufficiently large while the load resistance is smaller than the internal resistance of the photodiode in reverse bias, as illustrated with the load line in the third quadrant of Fig. 11.8. In the photoconductive mode under the conditions that RL < Ri and v out < V r , a photodiode has the following linear response before it saturates:   ePs v out ¼ ðI 0 þ is ÞRL ¼ I 0 þ ηe RL : (11.16) hv The circuitry for the photovoltaic mode, shown in Fig. 11.8(b), does not require a bias voltage but requires a large load resistance. In this mode of operation, the photovoltage appears as a forward bias voltage across the photodiode. As illustrated with the load line in the fourth quadrant of Fig. 11.8, the load resistance is required to be much larger than the internal resistance of the photodiode in forward bias, RL  Ri , so that the current i flowing through the diode and the load resistance is negligibly small. In the photovoltaic mode under this condition, the response of the photodiode is not linear but is logarithmic to the optical signal:     akB T is akB T ePs v out  ¼ , (11.17) ln 1 þ ln 1 þ ηe e e I0 hvI 0 where a is the realistic diode factor in the diode equation of (11.15). In the photoconductive mode, electric energy supplied by the bias voltage source is delivered to the photodiode. In the photovoltaic mode, electric energy generated by the optical signal can be extracted from the photodiode to the external circuit. Solar cells are basically semiconductor junction diodes operated in the photovoltaic mode for converting solar energy into electricity. EXAMPLE 11.4 A Si photodiode has a reverse current of I 0 ¼ 10 nA and a realistic diode factor of a ¼ 1:2 at T ¼ 300 K. For detection of optical signals at the λ ¼ 850 nm wavelength, its external quantum efficiency is ηe ¼ 0:6. It is illuminated with an optical signal at λ ¼ 850 nm that has a power of Ps ¼ 1 mW. (a) If the photodiode is operated in the photoconductive mode with a load resistance of RL ¼ 50 Ω and a reverse bias voltage of V r ¼ 5 V, what is the output voltage across the load resistor? Is the bias voltage sufficient for the photodiode to operate in the linear regime at the input signal level of Ps ¼ 1 mW? (b) If the photodiode is operated in the photovoltaic mode with a very large load resistance, what is the output voltage? Solution: The photon energy for λ ¼ 850 nm is hv ¼

1239:8 eV: 850

The signal current for Ps ¼ 1 mW is

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11.2 Photodetection Noise

is ¼ ηe

375

ePs 850 ¼ 0:6   1  103 A ¼ 411 μA: hv 1239:8

(a) The output voltage in the photoconductive mode with RL ¼ 50 Ω is found using (11.16) to be   v out ¼ ðI 0 þ is ÞRL ¼ 10  109 þ 411  106  50 V ¼ 20:6 mV: Because V r =v out  240  1, the bias voltage of V r ¼ 5 V is sufficient for the photodiode to operate in the linear regime at the input power level of Ps ¼ 1 mW. (b) At T ¼ 300 K, we have k B T=e ¼ 25:9 mV. The output voltage in the photovoltaic mode with a very large load resistance is found using (11.17) to be     akB T is 411  106 ¼ 1:2  25:9  ln 1 þ mV ¼ 330 mV: v out  ln 1 þ I0 e 10  109

11.2 PHOTODETECTION NOISE

.............................................................................................................. Noise is one of the most fundamental phenomena in nature. It is ubiquitous. The noise in a photodetector sets the fundamental limit on the detectivity of the photodetector, thus determining the usefulness of the photodetector for a particular application. In terms of their physical nature, there are a few different types of noise relevant for photodetectors. Two types of noise, quantum noise and thermal noise, originate from the basic physical laws of nature. Quantum noise, described as shot noise of electrons or photons in electronics and photonics, results from the statistical nature of a quantum event dictated by the uncertainty principle. Thermal noise, known as Johnson noise or Nyquist noise in electronics and photonics, is the consequence of thermal fluctuations and is directly associated with thermal radiation. Noise of such fundamental nature can only be minimized but can never be completely eliminated. The noise in a photodetector can come from a few physical sources: the photodetector itself, the possible amplifier used in conjunction with the photodetector, and the circuit used to extract the electrical signal from the photodetector. Noise appears in a signal as random fluctuations about the mean value of the signal. A measured signal s has a mean value of s defined as X s¼ pðsÞs, (11.18) s

where pðsÞ is the probability for the measured signal to have a value of s and the sum is carried out over all possible values obtained from measuring the signal. This mean value s is the expected value, or the ensemble average, of the variable s. The variance, or the mean square deviation, of the signal s is σ 2s ¼ ðs  s Þ2 ¼ s2  s 2 : The noise in a signal s can be expressed by a random variable sn defined as

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(11.19)

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sn ¼ s  s:

(11.20)

The noise represented by the random variable sn has a few general characteristics. As can be clearly seen from (11.20), it has a zero mean value: sn ¼ 0:

(11.21)

From (11.19) and (11.20), we find that the mean square value of sn is equal to the variance of s: s2n ¼ σ 2s ¼ s2  s 2 :

(11.22)

The mean square value of the noise in a signal is simply the mean square deviation of the signal. Because sn ¼ 0 but s2n 6¼ 0, the average amplitude of the noise vanishes but the power of the noise does not. Therefore, the magnitude of the noise is not measured by its average value but rather by its root mean square (rms) value defined as rmsðsn Þ ¼ s2n

1=2

:

(11.23)

Noise characterized by random fluctuations is incoherent. If two or more independent noise sources, sn1 , sn2 ,   , are simultaneously present in a signal s, their combined effect is not found by adding their amplitudes but is obtained by adding their mean square values, or their powers: s2n ¼ s2n1 þ s2n2 þ   

(11.24)

The total noise from different independent sources then has an rms value of
1=2 1=2 : rmsðsn Þ ¼ s2n ¼ s2n1 þ s2n2 þ   

(11.25)

One important figure of merit for a detection system is the signal-to-noise ratio (SNR or S/N). It is defined as the ratio of the power of a signal to the power of its noise or, equivalently, the ratio of the mean square of a signal to the mean square of its noise: SNR ¼

s2 s2n

¼

s2 , or σ 2s

SNR ¼ 10 log

s2 s2n

¼ 10 log

s2 ðdBÞ: σ 2s

(11.26)

The SNR defined above is also known as the signal-to-noise power ratio, to be distinguished from the signal-to-noise current ratio or the signal-to-noise voltage ratio defined as SNRcurrent ¼

s s2n

1=2

¼

s or σs

SNRvoltage ¼

s s2n

1=2

¼

s σs

(11.27)

for a photocurrent signal or a photovoltage signal, respectively. Without specification, however, the SNR of a detection system generally refers to the signal-to-noise power ratio defined in (11.26). In a photodetection system, a signal can take the form of photon number or photon flux as the input optical signal. It can also take the form of photocurrent or photovoltage as the output electrical signal. Therefore, the signal s can represent photon number, photon flux, photocurrent, or photovoltage. The general characteristics discussed above for the noise sn apply to every case.

11.2.1 Shot Noise The shot noise in a photodetector results from the quantum nature of the photons in the optical input and that of the charge carriers generated in the photodetector. Due to the quantum mechanical probabilistic nature of photons, the photons in an optical signal are not distributed

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11.2 Photodetection Noise

377

uniformly in time but arrive at the photodetector randomly in time. Therefore, both the power Ps of the optical signal and the number S of photons received in a given time interval Δt fluctuate randomly around their respective average values of Ps and S. The random fluctuations of the photon numbers are characterized by Poisson statistics. In any given time interval Δt, the probability of receiving S photons is given by the Poisson probability distribution: S

S eS pðS Þ ¼ : S! The mean square noise in photon number fluctuations can then be calculated as X  2 S 2n ¼ σ 2S ¼ pðS Þ S  S ¼ S :

(11.28)

(11.29)

S

This photon contribution to the noise of a photodetector is independent of the physical properties of the photodetector because it is external to the photodetector. It is the ultimate lower limit of the noise in an optical detection system. It sets the fundamental limit on the detectivity of a photodetector. The photons received by a photodetector are converted to photoelectrons or electron–hole pairs, depending on the type of photodetector, through the photoelectric effect. With a quantum efficiency of ηe , which has a value between 0 and 1, the number of photoelectrons generated is only a fraction of that of the photons received by the photodetector. Because a given photon can only generate either one or no electron, but not a fraction of an electron, the photoelectric process is clearly quantum mechanical and probabilistic. The shot noise associated with this process has to be considered if the quantum efficiency is less than unity. This effect is fully accounted for by considering the statistics of the number N of charge carriers that are generated in the photodetector with a quantum efficiency ηe . The random fluctuations of the photogenerated charge carriers are also characterized by the Poisson probability distribution: N

N eN pðN Þ ¼ , N!

(11.30)

where N ¼ ηe S as given in (11.3). We find, through a procedure similar to that used in (11.29), that the mean square noise in the number of photogenerated carriers is 2

N n ¼ σ 2N ¼ N :

(11.31)

Because N < S when ηe < 1, the noise is actually reduced by an imperfect quantum efficiency. This result seems odd. However, what really counts in a detection system is not the noise alone, but the SNR. While the noise is reduced by an imperfect quantum efficiency of ηe < 1, the signal is reduced even more. Consequently, a photodetector that has a poorer quantum efficiency has a lower SNR. We consider here a photodetector that has no internal gain, such that is ¼ iph . Using (11.4) and (11.31), we find the shot current noise in the photodetector: i2n, sh ¼ 4e2 B2 N 2n ¼ 4e2 B2 N ¼ 2eBis :

(11.32)

We then find the mean square current fluctuations for the shot noise in a photodetector that receives an optical power of Ps from an input optical signal:

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i2n, sh ¼ 2eBis ¼ 2ηe e2 B

Ps : hv

(11.33)

From this relation, we have 2

i2s ¼ is þ 2eBis :

(11.34)

In practice, there are other sources that also contribute to the shot noise in a photodetector. One important source is the photons from the background radiation that impinge on the photodetector. The contribution of this noise source can be minimized by reducing the aperture of the photodetector to the minimum needed to receive the optical signal. It cannot be completely eliminated, however, because at the very minimum there is still background thermal radiation, which can only be reduced by reducing the temperature of the environment surrounding the photodetector. Another important source of shot noise is the dark current of the photodetector. The dark current is the current in a photodetector when it is not illuminated with any optical input. In a semiconductor device, the dark current is normally caused by thermal generation of electron–hole pairs and by leakage currents due to surface defects of the device. When these additional noise sources are considered, the total shot noise in a photodetector is given by   i2n, sh ¼ 2eBi ¼ 2eB is þ ib þ id , (11.35) where ib is the photocurrent generated by background radiation and id is the dark current of the photodetector.

11.2.2 Excess Shot Noise In a photodetector that has an internal gain, such as a photomultiplier, a photoconductor, or an avalanche photodiode, both signal and noise are amplified. For a photodetector that has a gain of G, the signal current, the background radiation current, and the dark current are all amplified by the factor G: is ¼ Giph ¼ Gηe

ePs hv

(11.36)

and ib ¼ Gib0 ,

id ¼ Gid0 ,

(11.37)

where ib0 and id0 are unamplified background and dark currents, respectively, and ib and id are amplified currents that can be directly measured externally. The shot noise is also amplified through a process of random multiplication of the noise electrons. The statistical nature of this random multiplication process results in an excess noise factor, F, which is a function of the material, the structure, and the gain of a photodetector. As a consequence, the mean square shot noise current for a photodetector that has an internal gain can be expressed as     i2n, sh ¼ 2eBG2 F iph þ ib0 þ id0 ¼ 2eBGF is þ ib þ id , (11.38) 2

where the excess noise factor is a function of the gain, defined as F ¼ G2 =G . For a photodetector that has no internal gain, we find that G ¼ 1 and F ¼ 1; then, the shot noise given in (11.38) reduces to that in (11.35), as expected.

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11.2.3 Thermal Noise Thermal noise results from random thermal motions of the electrons in a conductor. It is associated with the blackbody radiation of a conductor at the radio or microwave frequency range of the signal. Because only materials that can absorb and dissipate energy can emit blackbody radiation, thermal noise is generated only by the resistive components of the photodetector and its circuit. Capacitive and inductive components do not generate thermal noise because they neither dissipate nor emit energy. The energy of the thermal noise generated by a resistive element is independent of the detailed physical properties of the resistor and is dictated only by the law of blackbody radiation. At a temperature of T, the thermal noise power in a small frequency interval of df centered at f is Pn, th ð f Þdf ¼

4hf

df : (11.39) ehf =kB T  1 In the normal operation of most photodetectors, f kB T=h. Thus, the frequency dependence of the thermal noise power is negligible, resulting in Pn, th ð f Þdf  4kB Tdf :

(11.40)

Then, the total thermal noise power for a detection system of a bandwidth B is simply Pn, th ¼ 4kB TB: (11.41) For a resistor that has a resistance of R, the thermal noise can be treated as either current or voltage noise through the relation Pn, th ¼ i2n, th R ¼ v 2n, th =R. Then, we have i2n, th ¼

4k B TB R

(11.42)

and v 2n, th ¼ 4kB TBR:

(11.43)

For an optical detection system, the resistance R is the total equivalent resistance, including the internal resistance of the photodetector and the load resistance from the circuit, at the output of the photodetector. For a photodetector that has a current signal, (11.42) is used. In this case, the thermal noise is determined by the lowest shunt resistance to the photodetector, which is often the load resistance of the photodetector. The thermal noise can be reduced by increasing this resistance at the expense of reducing the response speed of the system. For a photodetector that has a voltage signal, (11.43) is used. In this situation, the thermal noise is determined by the largest series resistance to the photodetector, which again is often the load resistance of the photodetector. The thermal noise can now be reduced by decreasing this resistance, but at the expense of reducing the output voltage signal.

11.2.4 Signal-to-Noise Ratio There are other noise sources, such as the 1=f noise, but they are usually not important for the normal operation of a photodetector. Therefore, the total noise in a photodetector, whether it has an internal gain or not, is basically the sum of its shot noise and thermal noise: i2n ¼ i2n, sh þ i2n, th :

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(11.44)

380

Photodetection

A photodetector is said to function in the quantum regime if i2n, sh > i2n, th . A photodetector operating in the quantum regime is shot-noise limited because shot noise is the primary source of noise in this regime. A photodetector is in the thermal regime if i2n, th > i2n, sh . A photodetector operating in the thermal regime is thermal-noise limited because its thermal noise is dominant compared with shot noise in this regime. For a photodetector that has no internal gain, the SNR is given by SNR ¼

i2s i2n

¼

i2ph P2s R2     ¼ , 2eB iph þ ib þ id þ 4k B TB=R 2eB Ps R þ ib þ id þ 4k B TB=R

(11.45)

where R ¼ ηe e=hv is the responsivity of a photodetector without an internal gain, defined in the following section. For a photodetector that has an internal gain of G, the SNR is SNR ¼

i2s i2n

¼



G2 i2ph



2eBG2 F iph þ ib0 þ id0 þ 4k B TB=R

¼

P2s R2  , 2eBGF Ps R þ ib þ id þ 4kB TB=R 

(11.46) where R ¼ Gηe e=hv is the responsivity of a photodetector with an internal gain of G, also defined in the following section. The relations in (11.45) and (11.46) apply to photodetectors that have current signals at the output. For a photodetector that has an output voltage signal, the SNR is defined as SNR ¼

v 2s P2s R2 ¼ , v 2n v 2n

(11.47)

where R is the responsivity of a photodetector that has an output voltage signal, defined in the following section.

EXAMPLE 11.5 The Si photodiode described in Example 11.4 is operated in the photoconductive mode with a load resistance of RL ¼ 50 Ω and a reverse bias voltage of V r ¼ 5 V. The total equivalent resistance is R  RL ¼ 50 Ω. The photodetector has a bandwidth of B ¼ 150 MHz. The dark current of the photodetector is its reverse current, which has the values of I 0 ¼ 10 nA at T ¼ 300 K and I 0 ¼ 4 nA at T ¼ 273 K. When the photodetector is illuminated with an optical signal of Ps ¼ 1 mW at λ ¼ 850 nm, a photocurrent of is ¼ 411 μA is generated. (a) Find the shot noise when the photodetector is operated at T ¼ 300 K and T ¼ 273 K, respectively. (b) Find the thermal noise at T ¼ 300 K and T ¼ 273 K, respectively. (c) Find the SNR at T ¼ 300 K and T ¼ 273 K, respectively. (d) The photocurrent is proportionally reduced to is ¼ 41:1 μA for an optical signal of Ps ¼ 100 μW. Find the SNR for this case at T ¼ 300 K and T ¼ 273 K, respectively? Solution: In this example, two temperatures are considered. Both shot noise and thermal noise vary with temperature. At T ¼ 300 K, id ¼ I 0 ¼ 10 nA and kB T ¼ 25:9 meV. At T ¼ 273 K, id ¼ I 0 ¼ 4 nA and kB T ¼ 23:5 meV.

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(a) With is ¼ is ¼ 411 μA, the shot noise at T ¼ 300 K is   i2n, sh ¼ 2eBi ¼ 2eB is þ id   ¼ 2  1:6  1019  150  106  411  106 þ 10  109 A2 ¼ 1:97  1017 A2 : The shot noise at T ¼ 273 K is   i2n, sh ¼ 2eBi ¼ 2eB is þ id   ¼ 2  1:6  1019  150  106  411  106 þ 4  109 A2 ¼ 1:97  1017 A2 : The shot noise is the same at T ¼ 300 K and T ¼ 273 K because id is . (b) With R  RL ¼ 50 Ω, the thermal noise at T ¼ 300 K is i2n, th ¼

4kB TB 4  25:9  103  1:6  1019  150  106 2 A ¼ 4:97  1014 A2 : ¼ 50 R

The thermal noise at T ¼ 273 K is i2n, th ¼

4kB TB 4  23:5  103  1:6  1019  150  106 2 A ¼ 4:51  1014 A2 : ¼ 50 R

The thermal noise at T ¼ 300 K is proportionally higher than that at T ¼ 273 K. (c) At T ¼ 300 K, the SNR is  2 411  106 i2s i2s ¼ ¼ 3:40  106 ¼ 65:3 dB: SNR ¼ ¼ 17 14 2 2 2 1:97  10 þ 4:97  10 in in, sh þ in, th At T ¼ 273 K, the SNR is SNR ¼

i2s i2n

¼

i2s i2n, sh þ i2n, th

 2 411  106 ¼ ¼ 3:75  106 ¼ 65:7 dB: 1:97  1017 þ 4:51  1014

(d) When the photocurrent is proportionally reduced to is ¼ 41:1 μA for an optical signal of Ps ¼ 100 μW, the shot noise is reduced but the thermal noise remains unchanged. From (c), we find that the SNR is almost entirely determined by the thermal noise because the shot noise is negligibly small compared to the thermal noise at is ¼ 411 μA. At is ¼ 41:1 μA, the shot noise is even smaller, with i2n, sh  1:97  1018 A2 . At T ¼ 300 K, the SNR is  2 41:1  106 i2s i2s SNR ¼  ¼ ¼ 3:40  104 ¼ 45:3 dB: 18 þ 4:97  1014 i2n i2n, th 1:97  10 At T ¼ 273 K, the SNR is SNR ¼

i2s i2n



i2s i2n, th



2 41:1  106 ¼ ¼ 3:75  104 ¼ 45:7 dB: 1:97  1018 þ 4:51  1014

Because the photodetector is thermal-noise limited, the SNR is reduced by two orders of magnitude, i.e., by 20 dB, when the signal current is reduced by one order.

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382

Photodetection

11.3 PHOTODETECTION MEASURES

.............................................................................................................. Several parameters are commonly used to define the performance characteristics of a photodetector. These parameters can be considered as the figures of merit of a photodetector. They are used for comparing one photodetector to another and for determining the suitability of a photodetector for a particular application. In this section, the basic concepts of these parameters are defined and discussed.

11.3.1 Spectral Response Because the response of a photon detector is wavelength dependent, a given photodetector is responsive only within a finite, specific range of the optical spectrum. The spectral range of response for a photodetector is determined by the material, the structure, and the packaging of the photodetector. The spectral response of a photodetector is usually specified in terms of the spectral responsivity or the spectral detectivity of the photodetector. In choosing a photodetector for an application, the match between the spectral content of the optical signal and the spectral response of the photodetector is the first thing to be verified.

11.3.2 Quantum Efficiency Quantum efficiency is the probability of generating a charge carrier in a photodetector for each photon that is incident on the photodetector. The external quantum efficiency, ηe , is the probability for each photon incident from outside the photodetector to generate a charge carrier that is externally measured, whereas the internal quantum efficiency, ηi , is the probability for each photon that actually reaches the active region of the photodetector to be absorbed and converted to an internal charge carrier. The external quantum efficiency of a photodetector is reduced from its internal quantum efficiency by the transmission efficiency, ηt , which accounts for the probability of an externally incident photon to reach the active region of the photodetector, and by the collection efficiency, ηcoll , which accounts for the efficiency of the photogenerated electrical carriers being collected into a photocurrent. Thus, we can express the external quantum efficiency of a photodetector as ηe ¼ ηcoll ηt ηi :

(11.48)

As expressed in (11.3), the external quantum efficiency can be defined as the ratio of the number of photogenerated charge carriers, in the form of either photoelectrons or electron–hole pairs, that actually contribute to the photocurrent to the number of incident photons: ηe ¼ N =S. According to (11.4), the external quantum efficiency of a photodetector can then be expressed in terms of the incident optical power and the photocurrent as ηe ¼

iph =e hviph : ¼ Ps =hv ePs

(11.49)

The quantum efficiency of a photodetector is a function of the wavelength of the incident photons because of the spectral response of the photodetector. Its wavelength dependence arises

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383

not only from its explicit dependence on the optical frequency v seen in (11.49) but also from the wavelength dependence of the ratio iph =Ps defined below as the responsivity of the photodetector.

EXAMPLE 11.6 A photocurrent of 800 μA is generated in a Ge photodetector when it is illuminated with an optical signal of 1 mW power at the 1:55 μm wavelength. Find the external quantum efficiency of the photodetector at this wavelength. Solution: At λ ¼ 1:55 μm, we have hv 1:2398 ¼ V ¼ 0:8 V: e 1:55 With iph ¼ 800 μA for Ps ¼ 1 mW, the external quantum efficiency is ηe ¼

hviph 800  106 ¼ 0:8  ¼ 64%: ePs 1  103

11.3.3 Responsivity Responsivity is an important parameter for a photodetector. It allows one to determine the available output signal of a photodetector for a given input optical signal. The responsivity of a photodetector is defined as the ratio of the output current or voltage signal to the power of the input optical signal. For a photodetector that has an output current signal, the responsivity is defined as R¼

is : Ps

(11.50)

For a photodetector that has an output voltage signal, the responsivity is defined as R¼

vs : Ps

(11.51)

Because most of the commonly used photodetectors have output current signals, we consider in further detail the responsivity of these types of photodetectors in the following. Similar concepts can be extended to photodetectors that have output voltage signals. For a photodetector that has no internal gain, the signal current is simply the photocurrent, is ¼ iph . Using (11.49) and (10.50), we find the expression for its responsivity: R¼

iph e ¼ ηe : Ps hv

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(11.52)

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Photodetection

For a photodetector that has an internal gain, the signal current is amplified by the gain, is ¼ Giph ; then, the responsivity is R¼

Giph e ¼ Gηe ¼ GR0 , Ps hv

(11.53)

where R0 is the intrinsic responsivity of the photodetector defined as R0 ¼

iph e ¼ ηe : Ps hv

(11.54)

The responsivity of a photodetector that has no internal gain is simply its intrinsic responsivity, R ¼ R0 , whereas a photodetector that has an internal gain has a responsivity of R ¼ GR0 . The spectral response of a photodetector is usually characterized by the responsivity of the photodetector as a function of the optical wavelength, RðλÞ, which is known as the spectral responsivity. In addition, the responsivity of a photodetector is also a function of the modulation-signal frequency f . Its frequency dependence, Rðf Þ, characterizes the frequency response of the photodetector, as discussed later.

EXAMPLE 11.7 Find the responsivity at λ ¼ 1:55 μm of the Ge photodetector that is described in Example 11.6. What is the responsivity at λ ¼ 1:3 μm if the external quantum efficiency remains the same for both wavelengths? Solution: From Example 11.6, iph ¼ 800 μA for Ps ¼ 1 mW at λ ¼ 1:55 μm. Thus the responsivity at λ ¼ 1:55 μm is R¼

iph 800  106 ¼ AW1 ¼ 0:8 A W1 : Ps 1  103

If the external quantum efficiency at λ ¼ 1:3 μm remains ηe ¼ 64% as found in Example 11.6, the responsivity at λ ¼ 1:3 μm is R¼

iph e 1:3 ¼ ηe ¼ 0:64  A W1 ¼ 0:67 A W-1 : Ps hv 1:2398

11.3.4 Noise Equivalent Power The noise equivalent power (NEP) of a photodetector is defined as the input power required of the optical signal for the signal-to-noise ratio to be unity, SNR ¼ 1, at the photodetector output. Then, using the relations in (11.45) and (11.46), the NEP of a photodetector that has an output current signal, with or without an internal gain, can be defined as

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385

1=2

i2 rmsðin Þ NEP ¼ n ¼ , R R

(11.55)

where i2n is the mean square noise current at an input optical power level for SNR ¼ 1 and R is the responsivity defined in (11.50). Using the relation in (11.47), the NEP of a photodetector that has an output voltage signal can be defined as 1=2

v2 rmsðv n Þ NEP ¼ n ¼ , R R

(11.56)

where v 2n is the mean square noise voltage at an input optical power level for SNR ¼ 1 and R is the responsivity defined in (11.51). For most detection systems at the low input signal level for SNR ¼ 1, the shot noise from the input optical signal is negligible compared to both the shot noise from other sources and the thermal noise of the photodetector. In this case, the NEP of a photodetector that has an output current signal but no internal gain can be expressed as 

2eib þ 2eid þ 4kB T=R NEP ¼ R

1=2 B1=2 :

(11.57)

The most fundamental limit is the noise contributed by the ubiquitous blackbody radiation in the background. This background radiation sets the absolute minimum of NEP for a photodetector. It is often the limitation for photodetectors in the mid- and far-infrared spectral regions, but it is normally not important for photodetectors in the visible and ultraviolet spectral regions. For most photodetectors responding to optical wavelengths shorter than 3 μm, the noise from background blackbody radiation is dominated by that from the dark current or that from resistive thermal noise, or both. For such a photodetector, the intrinsic NEP is that defined by its dark current when the load resistance is sufficiently large if the photodetector generates a photocurrent signal, or when the load resistance is sufficiently small if it generates a photovoltage signal. However, in order to reduce its RC time constant, a high-speed photodetector that has a current signal normally has a small area, thus a small dark current, but it requires a small load resistance, thus a large thermal noise. Therefore, the NEP of a high-speed photodetector is usually limited by the thermal noise from its external load resistance rather than by the shot noise from its internal dark current. Because the mean square noise is proportional to the photodetector bandwidth, i2n / B and v 2n / B, the NEP of a photodetector is proportional to the square root of the photodetector bandwidth: NEP / B1=2 . Therefore, the NEP of a photodetector is often specified in terms of the NEP for a bandwidth of 1 Hz as NEP=B1=2 , in the unit of W Hz1=2 . EXAMPLE 11.8 A Ge photodiode has a dark current of id ¼ 15 μA and a negligible background current at T ¼ 300 K. It has a total equivalent resistance of R ¼ 2 kΩ, a response bandwidth of B ¼ 5 kHz, and a responsivity of R ¼ 0:8 A W1 at λ ¼ 1:55 μm. Find its NEP=B1=2 and NEP at T ¼ 300 K for optical signals at λ ¼ 1:55 μm.

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Solution: At the noise-equivalent power level, the shot noise for this photodetector is contributed only by the dark current because the background current is negligible. Thus, the shot noise is i2n, sh ¼ 2eBid ¼ 2  1:6  1019  15  106 B A2 Hz1 ¼ 4:8  1024 B A2 Hz1 : With R ¼ 2 kΩ, the thermal noise at T ¼ 300 K, for which kB T ¼ 25:9 meV, is i2n, th ¼

4kB TB 4  25:9  103  1:6  1019 B A2 Hz1 ¼ 8:3  1024 B A2 Hz1 : ¼ R 2  103

The total noise is i2n ¼ i2n, sh þ i2n, th ¼ 4:8  1024 B A2 Hz1 þ 8:3  1024 B A2 Hz1 ¼ 1:31  1023 B A2 Hz1 : Therefore, 1=2

NEP i2n ¼ ¼ B1=2 RB1=2

 1=2 1:31  1023 W Hz1=2 ¼ 4:52 pW Hz1=2 : 0:8

With B ¼ 5 kHz, the total NEP over the entire bandwidth is NEP ¼

 1=2 NEP  B1=2 ¼ 4:52  1012  5  103 W ¼ 320 pW: 1=2 B

11.3.5 Detectivity The detectivity characterizes the ability of a photodetector to detect a small optical signal. It is defined as the inverse of the NEP of the photodetector: D¼

1 , NEP

(11.58)

which has the unit of W1 . As discussed above, NEP / B1=2 . The shot noise from the input optical signal at the NEP level is negligible compared to the shot noise from the background radiation current, ib , and that from the dark current, id , both of which are often proportional to the surface area, A, of a photodetector. Therefore, when ib and id are the dominant sources of noise for a photodetector, the intrinsic noise characteristics of the photodetector can be better quantified by normalizing NEP to ðABÞ1=2 . A useful intrinsic parameter of a photodetector is the specific detectivity, D∗ , defined as D∗ ¼

ðABÞ1=2 , NEP

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(11.59)

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387

which has the unit of m Hz1=2 W1 , often quoted in cm Hz1=2 W1 . Then, for a dark-currentlimited photodetector that has no internal gain, we have A1=2 R D∗   1=2 : 2eid

(11.60)

The specific detectivity D∗ is independent of the area of the photodetector. It is a measure of the intrinsic detection capability of the material and the structure of the photodetector. The detectivity of a photodetector is a function of the optical wavelength. The spectral characteristics of the detectivity, given as DðλÞ or D∗ ðλÞ, reflect the spectral response of a photodetector. The detectivity is also a function of the modulation frequency f of a signal that is modulated on the optical carrier. EXAMPLE 11.9 The Ge photodetector described in Example 11.8 has a circular surface area that has a diameter of 2r ¼ 5 mm. Find its detectivity and specific detectivity. Solution: From Example 11.8, we have NEP=B1=2 ¼ 4:52 pW Hz1=2 and NEP ¼ 320 pW for this photodetector. The detection surface area is 

5  103 A ¼ πr ¼ π  2 2

2

m2 ¼ 1:96  105 m2 :

Therefore, the detectivity is D¼

1 1 W1 ¼ 3:13  109 W1 , ¼ NEP 320  1012

and the specific detectivity is  1=2 1:96  105 ðABÞ1=2 D ¼ m Hz1=2 W1 ¼ 9:8  108 m Hz1=2 W1 : ¼ 12 NEP 4:52  10 ∗

11.3.6 Linearity and Dynamic Range Linearity of a photodetector is defined by the linear response of the photodetector, meaning that its output current or voltage signal is linearly proportional to its input optical signal. Linear response is required for a photodetector to faithfully convert the modulation waveform of an input optical signal to an output electrical signal without distortion. When a photodetector has a linear response, its quantum efficiency ηe and responsivity R defined above are constants that are independent of the power Ps of the input optical signal. However, every practical photodetector only has a finite range of linear response, as shown in Fig. 11.9. As the power of

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Figure 11.9 Typical response characteristics as a function of the power of the input optical signal for (a) a photodetector with an output current signal and (b) a photodetector with an output voltage signal.

the input optical signal reaches a certain level, the response of a photodetector starts to saturate, thereby deviating from linearity. The maximum acceptable power of the input signal is determined by the maximum deviation from the linear response of a photodetector that can be tolerated in a particular application. Given the maximum tolerable deviation from linearity to be δ (for 100δ%), the saturation signal power, Psat s , for the photodetector in the application is the corresponding maximum acceptable input power. As illustrated in Fig. 11.9, the value of Psat s can be found from   dis  dv s  ¼ ð1  δÞR or ¼ ð1  δÞR, (11.61)  dPs Ps ¼Psat dPs Ps ¼Psat s s where R is the responsivity of the photodetector in the linear range. The usefulness of a photodetector for detecting an optical signal is clearly limited by its saturation, which is quantified by Psat s , at the large-signal end and by its detectivity, which is determined by the NEP of the photodetector, at the small-signal end. The range of the input signal power above the NEP but below Psat s in the linear-response region is the useful range of operation for a photodetector. This range is known as the dynamic range (DR) of the photodetector, as indicated in Fig. 11.9. The dynamic range is usually quantified, in dB, as DR ¼ 10 log

Psat s : NEP

(11.62)

Alternatively, the dynamic range of a photodetector is also frequently stated in terms of the number of orders of magnitude in the input optical power from the NEP to Psat s . EXAMPLE 11.10 The Ge photodetector described in Example 11.8 has NEP ¼ 320 pW and a responsivity of R ¼ 0:8 A W1 at λ ¼ 1:55 μm. It saturates at a signal current level of 80 mA. Find its saturation optical power at λ ¼ 1:55 μm and its linear dynamic range.

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389

Solution: 1 With a saturation signal current of isat s ¼ 80 mA and a responsivity of R ¼ 0:8 A W , the saturation optical power is Psat s ¼

isat 80 s ¼ mW ¼ 100 mW: R 0:8

Therefore, the linear dynamic range is DR ¼ 10 log

Psat 100  103 s dB ¼ 85 dB: ¼ 10  log NEP 320  1012

11.3.7 Speed and Frequency Response The response speed of a photodetector is directly related to its frequency response. It determines the ability of a photodetector to follow a fast-varying optical signal. To faithfully record an optical signal, a photodetector must have a speed higher than the fastest temporal variations in the signal or, equivalently, a frequency response that has a bandwidth covering the entire bandwidth of the signal. In the time domain, the speed of a photodetector is characterized by the risetime, tr , and the falltime, t f , of its response to an impulse signal or a square-pulse signal, as shown in Fig. 11.10. The risetime is defined as the time interval for the response to rise from 10% to 90% of its peak value, whereas the falltime is defined as the time interval for the response to decay from 90% to 10% of its peak value. Generally, the overall speed of a photodetector is determined by both its intrinsic bandwidth and its circuit-limited RC bandwidth. The risetime of the impulse response is determined by the intrinsic bandwidth of a photodetector, and that of the square-pulse response is determined by the circuit-limited RC bandwidth of the photodetector. The risetime and its corresponding bandwidth have the relation:

Figure 11.10 Typical responses of a photodetector to (a) an impulse signal and (b) a square-pulse signal.

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tr ¼

0:35 , f 3dB

(11.63)

where f 3dB is the 3-dB cutoff frequency defined below. The frequency response, which is characterized by the frequency dependence of the responsivity Rðf Þ at a given optical wavelength, can be obtained by simply taking the Fourier transform of the impulse response or by registering the response of the photodetector at one modulation-signal frequency at a time while sweeping this frequency. Note that Rðf Þ is the current or voltage response spectrum of the photodetector because the responsivity of a photodetector is defined in terms of the output current or voltage signal of the photodetector. The output electrical power spectrum of the photodetector is R2 ðf Þ, which defines a 3-dB cutoff frequency, or 3-dB bandwidth, for a photodetector as 1 R2 ðf 3dB Þ ¼ R2 ð0Þ: 2

(11.64)

Considering the rectangular time interval of Δt that is used to define the bandwidth B, we have the relation between f 3dB and B of a photodetector: f 3dB ¼ 0:886B ¼

0:443 : Δt

(11.65)

The 3-dB bandwidth of a photodetector is a function of the combined effect of a few different physical factors that determine the speed and the frequency response of the photodetector. These factors and their relative importance depend on the type of the photodetector. EXAMPLE 11.11 The Ge photodetector described in Example 11.8 has a response bandwidth of B ¼ 5 kHz. Find its 3-dB cutoff frequency. What is the risetime of the photodetector response to an impulse signal? Solution: The 3-dB cutoff frequency is f 3dB ¼ 0:886B ¼ 0:886  5 kHz ¼ 4:43 kHz: The risetime of the photodetector response to an impulse signal is tr ¼

0:35 0:35 ¼ s ¼ 79 μs: f 3dB 4:43  103

EXAMPLE 11.12 A photodetector is used to detect an optical pulse that has a pulse duration of 500 ps and a pulse risetime of 200 ps. What is the minimum bandwidth of the photodetector required to detect the pulse? What is the minimum bandwidth required for resolving the pulse risetime?

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Solution: The minimum bandwidth required for detecting the pulse duration of Δt ¼ 500 ps is found using (11.1) or (11.65) as Bmin ¼

1 1 Hz ¼ 1 GHz: ¼ 2Δt 2  500  1012

The minimum bandwidth required for resolving the pulse risetime of tr ¼ 200 ps is found using (11.63) and (11.65) as Bmin ¼

f min 0:35 0:35 3dB ¼ ¼ Hz ¼ 1:98 GHz: 0:886 0:886t r 0:886  200  1012

It is clear that a larger bandwidth is needed to resolve the pulse risetime.

Problems 11.1.1 Alkaline metals have low work functions. Besides Cs, which has the lowest work function of 2.14 eV, as described in Example 11.1, the work functions are 2.29 eV for K, 2.36 eV for Na, and 2.90 eV for Li. What is the threshold wavelength for an optical wave to cause photoemission from the surface of each alkaline metal? If the surface of each metal is illuminated with a laser beam at the 500 nm wavelength, what is the highest kinetic energy of the photoemitted electrons? 11.1.2 The work function of Ag varies from 4.26 to 4.74 eV, depending on the crystallographic orientation of the Ag surface. When a specific Ag surface is illuminated with a laser beam at the 260 nm wavelength, the highest kinetic energy of the photoemitted electrons is found to be T max ¼ 168 meV. What is the work function of this Ag surface? 11.1.3 The work function of Au depends on the crystallographic orientation of the Au surface. Experimental data on various Au surfaces show threshold wavelengths varying between 226:7 nm and 243:1 nm. Find the work function range of Au. 11.1.4 At room temperature, silicon has an electron affinity of eχ ¼ 4:05 eV and a bandgap of E g ¼ 1:12 eV. (a) Find the work function, the threshold photon energy, and the threshold wavelength for a lightly doped p-type silicon crystal that has a Fermi level at EF ¼ E v þ 200 meV. (b) Find the work function, the threshold photon energy, and the threshold wavelength for a heavily doped p-type silicon crystal that has a Fermi level at E F ¼ E v  200 meV. 11.1.5 At room temperature, GaAs has an electron affinity of eχ ¼ 4:07 eV and a bandgap of Eg ¼ 1:424 eV. The Fermi level of GaAs at room temperature lies within the bandgap at EF ¼ E c  672:2 meV ¼ E v þ 751:8 meV, where E c and E v are the conduction-band and valence-band edges, respectively. Find the work function of intrinsic GaAs. What is the threshold photon energy and the threshold wavelength for an optical wave to cause photoemission from its surface?

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11.1.6

At room temperature, GaAs has an electron affinity of eχ ¼ 4:07 eV and a bandgap of E g ¼ 1:424 eV. (a) Find the work function, the threshold photon energy, and the threshold wavelength for a lightly doped n-type GaAs crystal that has a Fermi level at EF ¼ E c  300 meV. (b) Find the work function, the threshold photon energy, and the threshold wavelength for a lightly doped p-type GaAs crystal that has a Fermi level at EF ¼ E v þ 300 meV.

11.1.7

At room temperature, GaAs has an electron affinity of eχ ¼ 4:07 eV and a bandgap of E g ¼ 1:424 eV. (a) Find the work function, the threshold photon energy, and the threshold wavelength for a heavily doped n-type GaAs crystal that has a Fermi level at EF ¼ E c þ 300 meV. (b) Find the work function, the threshold photon energy, and the threshold wavelength for a lightly doped p-type GaAs crystal that has a Fermi level at E F ¼ E v  300 meV.

11.1.8

The intrinsic electron and hole concentrations of GaAs in thermal equilibrium at room temperature are n0 ¼ p0 ¼ ni ¼ 2:33  1012 m1 . It has an electron mobility of μe ¼ 0:85 m2 V1 s1 and a hole mobility of μh ¼ 0:04 m2 V1 s1 . An intrinsic GaAs crystal used as a photoconductor is uniformly illuminated with an optical beam to generate electron–hole pairs for total electron and hole concentrations of n  p  1:0  1020 m3 . Find the dark conductivity and the photoconductivity.

11.1.9

The intrinsic electron and hole concentrations of Ge in thermal equilibrium at room temperature are n0 ¼ p0 ¼ ni ¼ 1:95  1019 m1 . It has an electron mobility of μe ¼ 0:39 m2 V1 s1 and a hole mobility of μh ¼ 0:19 m2 V1 s1 . An intrinsic Ge crystal used as a photoconductor is uniformly illuminated with an optical beam to generate electron–hole pairs. Find the dark conductivity. What are the required concentrations of the photogenerated electrons and holes for the photoconductivity to be 20 times the dark conductivity?

11.1.10 A Si photodiode at T ¼ 300 K has a reverse current of I 0 ¼ 10 nA and a realistic diode factor of a ¼ 1:2. For the detection of optical signals at the λ ¼ 532 nm wavelength, its external quantum efficiency is ηe ¼ 0:7. It is illuminated with an optical signal that has a power of Ps ¼ 200 μW at λ ¼ 532 nm. (a) If the photodiode is operated in the photoconductive mode with a reverse bias voltage of V r ¼ 5 V, what is the required load resistance for the output voltage to be at least 100 mV? (b) If the photodiode is operated in the photovoltaic mode with a very large load resistance, what is the output voltage? (c) What are the output voltages for Ps ¼ 5 mW in the photoconductive mode with the load resistance found in (a) and in the photovoltaic mode, respectively? 11.1.11 A Ge photodiode has a reverse current of I 0 ¼ 2 μA and a realistic diode factor of a ¼ 1:1 at T ¼ 300 K. Its external quantum efficiency is ηe ¼ 0:54 for an optical signal at λ ¼ 1:55 μm. The power of the optical signal varies between 0:5 mW and 5 mW. (a) The photodiode is operated in the photoconductive mode with a reverse bias voltage of V r ¼ 10 V and a load resistance of RL ¼ 50 Ω. What is the range of

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Problems

393

the output voltage? What is the range of the signal voltage? Is the bias voltage sufficient for the photodiode to function in the linear regime for the whole range of signal powers? (b) If the photodiode is operated in the photovoltaic mode with a very large load resistance, what is the range of the signal voltage? 11.2.1 The Si photodiode described in Example 11.5 has a dark current of id ¼ 10 nA at T ¼ 300 K and a signal current of is ¼ 411 μA when it is illuminated with an optical signal of Ps ¼ 1 mW at λ ¼ 850 nm. With a total resistance of R  RL ¼ 50 Ω, it has a bandwidth of B ¼ 150 MHz, and its SNR at T ¼ 300 K is limited by thermal noise with i2n, sh ¼ 1:97  1017 A2 and i2n, th ¼ 4:97  1014 A2 . Clearly, the SNR can be increased by reducing the thermal noise, at least until it reaches the level of the shot noise. How can this be accomplished? Find the parameter changes needed to reduce the thermal noise to the level of the shot noise. What price has to be paid in doing so? 11.2.2 A large-area Ge photodetector has a dark current of id ¼ 10 μA at T ¼ 300 K and a signal current of is ¼ 400 μA when it is illuminated with an optical signal of Ps ¼ 500 μW at λ ¼ 1:55 μm. The total equivalent resistance is R  RL ¼ 1 kΩ, and the bandwidth is B ¼ 10 kHz. Find the shot noise, the thermal noise, and the SNR of the photodetector in this operating condition. Which noise source sets the primary limit on the SNR? 11.2.3 The signal current generated in the Ge photodetector described in Problem 11.2.2 is proportional to the power of the optical signal. Answer the questions raised in Problem 11.2.2 for (a) an optical power of Ps ¼ 5 μW generating a photocurrent of is ¼ 4 μA and (b) an optical power of Ps ¼ 50 μW generating a photocurrent of is ¼ 40 μA. 11.3.1 A photodetector has an InGaAs active layer, which absorbs optical signals to be detected. The active layer has a bandgap of 0.75 eV. The incoming optical beam has to pass through an InGaAsP top layer of a higher bandgap of 0.95 eV before reaching the active layer. What is the optical spectral bandwidth of this photodetector, i.e., the wavelength range that can be detected by this photodetector? 11.3.2 An uncoated surface of Si has a reflectivity of R ¼ 32:6% at λ ¼ 850 nm. A Si photodiode has an active region that absorbs 90% of light at λ ¼ 850 nm that reaches this region. Almost all photogenerated carriers contribute to the photocurrent. (a) If the surface of the Si photodiode is not coated, what is the largest possible external quantum efficiency it can have? What is the largest possible photocurrent for an optical signal at λ ¼ 850 nm that has an input power of 1 mW? (b) If it is desired that a photocurrent of at least 600 μA be generated with an input power of 1 mW for an optical signal at λ ¼ 850 nm, what is the required external efficiency? How can this efficiency be accomplished by properly coating the surface of the Si photodetector? 11.3.3 The maximum possible external quantum efficiency is clearly ηe ¼ 1 for any photodetector. For this reason, the intrinsic responsivity for any photodetector has a maximum possible value of Rmax 0 , which is a function of only the wavelength of the optical signal.

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How does the value of Rmax vary with the wavelength of the optical signal? What is 0 the range of its values for optical signals in the visible spectral region? What are its values at the three common near-infrared wavelengths of 850 nm, 1:3 μm, and 1:55 μm that are used in optical communication systems? 11.3.4

An InGaAs/InP avalanche photodiode (APD) has an internal gain of G, which can be varied up to about G ¼ 20 by varying the bias voltage that is applied to the APD. The circuitry of the APD has a small load resistance of RL ¼ 50 Ω for fast response to highfrequency optical signals. For an optical signal at λ ¼ 1:55 μm, the external quantum efficiency of the APD is ηe ¼ 64%. (a) Find the intrinsic responsivity of the APD at λ ¼ 1:55 μm. (b) By applying a certain bias voltage for an internal gain, a signal voltage of v s ¼ 15 mV on the load resistance is observed with an optical signal of Ps ¼ 25 μW at λ ¼ 1:55 μm. Find the responsivity and the gain of the APD at this operating point.

11.3.5

A Ge photodiode has a negligible background current. Its dark current is id ¼ 10 μA at T ¼ 300 K and id ¼ 20 nA at T ¼ 250 K. It has a total equivalent resistance of R ¼ 20 kΩ, a response bandwidth of B ¼ 1 kHz, and a responsivity of R ¼ 0:9 A W1 at λ ¼ 1:55 μm. (a) Find its NEP=B1=2 and NEP at T ¼ 300 K for optical signals at λ ¼ 1:55 μm. Which noise source limits the NEP at this temperature? (b) Find its NEP=B1=2 and NEP at T ¼ 250 K for optical signals at λ ¼ 1:55 μm. Which noise source limits the NEP at this temperature?

11.3.6

A Si photodiode has a total equivalent resistance of R ¼ 50 Ω and a bandwidth of B ¼ 100 MHz. At T ¼ 300 K, it has a negligible background current and a dark current of id ¼ 10 nA. It has a circular surface area that has a diameter of 2r ¼ 1 mm. Its responsivity at λ ¼ 850 nm is R ¼ 0:52 A W1 . Find its NEP, detectivity, and specific detectivity at λ ¼ 850 nm.

11.3.7

A Si photodiode saturates at a signal photocurrent of isat s ¼ 16 mA. Find the saturation optical power for an optical signal at λ ¼ 850 nm, where the photodiode has a responsivity of R ¼ 0:45 A W1 . If it has an NEP of 150 nW, what is its linear dynamic range?

11.3.8

A photodetector has an NEP of 1:6 nW and a linear dynamic range of 67 dB for optical signals at the λ ¼ 1:3 μm wavelength. What is the maximum optical signal power allowed for the photodetector to respond linearly?

11.3.9

When an optical pulse that has a temporal duration of 1 ps is detected by a photodetector, the electrical response output of the photodetector shows a pulse that has a risetime of 180 ps. What is the 3-dB cutoff frequency and the electrical response bandwidth of this photodetector?

11.3.10 A photodetector has a bandwidth of B ¼ 8 GHz. What is the duration of the shortest optical pulse that can be clearly detected using this photodetector? What is the fastest pulse risetime of an optical pulse that can be resolved by this photodetector?

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Bibliography

395

Bibliography Bhattacharya, P., Semiconductor Optoelectronic Devices, 2nd edn. Englewood Cliffs, NJ: Prentice-Hall, 1997. Bube, R. H., Photoconductivity of Solids. New York: Wiley, 1960. Chuang, S. L., Physics of Photonic Devices, 2nd edn. New York: Wiley, 2009. Davis, C. C., Lasers and Electro-Optics: Fundamentals and Engineering, 2nd edn. Cambridge: Cambridge University Press, 2014. Donati, S., Photodetectors: Devices, Circuits, and Applications. Upper Saddle River, NJ: Prentice-Hall, 2000. Haus, H. A., Waves and Fields in Optoelectronics. Englewood Cliffs, NJ: Prentice-Hall, 1984. Iizuka, K., Elements of Photonics for Fiber and Integrated Optics, Vol. II. New York: Wiley, 2002. 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. Nalwa, H. S., ed., Photodetectors and Fiber Optics. San Diego, CA: Academic Press, 2001. 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. 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 Appendix A - Symbols and Notations pp. 396-402 Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.013 Cambridge University Press

APPENDIX A

Symbols and Notations

A.1

FIELDS

.............................................................................................................. Field vectors and their scalar magnitudes are represented using a consistent system of symbols and fonts. All vectors except for unit vectors are represented in bold-face fonts, whereas all scalar quantities are represents in nonbold fonts. This system is illustrated in the following using the electric field as an example.

A.1.1 Real Fields All real fields are defined in the real space and time domain only. All real field vectors are represented in the italic bold capital Roman font, such as Eðr; tÞ,

(A.1)

for the real electric field vector. Other real field vectors are Hðr; tÞ, Dðr; t Þ, Bðr; t Þ, Pðr; t Þ, M ðr; t Þ, J ðr; t Þ, and Sðr; t Þ. Except for the current density, all real fields are always represented in the vector form without separate symbols defined for their scalar magnitudes. The scalar magnitude of J is represented as J.

A.1.2 Complex Fields All complex field vectors are represented in the nonitalic bold capital Roman font. All complex field vectors in the real space and time domain are defined in relation to their respective real field vectors, such as Eðr; t Þ defined in (1.40) for the complex electric field vector: Eðr; t Þ ¼ Eðr; tÞ þ E∗ ðr; tÞ ¼ Eðr; t Þ þ complex conjugate:

(A.2)

Other complex field vectors defined in a similar manner are Hðr; t Þ, Dðr; t Þ, Bðr; t Þ, Pðr; t Þ, Mðr; t Þ, and Jðr; tÞ. The complex Poynting vector is defined differently, as given in (1.54) and (1.55): ∗

S ¼ E  H∗ so that S ¼ S þ S :

(A.3)

The scalar magnitude of a complex field vector is represented in the nonbold mathematical capital Roman font, for example E for the magnitude of E: E ¼ E^e ,

(A.4)

where ^e is the unit vector of E. Other scalar field magnitudes represented in a similar manner are H, D, B, P, and M. No scalar complex current density at an optical frequency is used; the

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scalar J represents the magnitude of a real current density vector J at DC or a low frequency. No scalar magnitude of the complex Poynting vector S is used.

A.1.3 Complex Field Amplitudes The slowly varying amplitude vector of a complex field vector is represented in the bold capital script font, such as E for the slowly varying amplitude of E. It is defined as the slow variation of the field envelope on its carrier frequency through the relation Eðr; t Þ ¼ E ðr; tÞ exp ðikr  iωtÞ,

(A.5)

as expressed in (1.52) for the electric field. Other slowly varying field amplitude vectors defined in a similar manner are H, D, B, P, M, and J , but not all of them are used in the text. No slowly varying field amplitude is defined for the Poynting vector. The scalar magnitude of a slowly varying field amplitude vector is represented in the nonbold capital script font, for example E for the magnitude of E: E ¼ E^e :

(A.6)

Other scalar magnitudes of slowly varying field amplitudes represented in a similar manner are H, D, B, P, M, and J , but not all of them are used in the text.

A.1.4 Mode Fields Complex mode field vectors are represented as Ev ðr; t Þ and Hv ðr; tÞ with their scalar magnitudes represented as E v ðr; tÞ and H v ðr; t Þ, respectively, where the subscript index v represents a compound mode index such as m or mn for waveguide modes, or mn for Gaussian modes. The vectorial field profiles of a waveguide mode characterize the transverse spatial distributions of the mode fields. A waveguide mode field profile is a function of the transverse spatial coordinates only. The vectorial waveguide mode field profiles are represented as E v ðx; yÞ and Hv ðx; yÞ, or E v ðϕ; r Þ and Hv ðϕ; r Þ, with their scalar magnitudes represented as E v ðx; yÞ and Hv ðx; yÞ, or E v ðϕ; r Þ and Hv ðϕ; r Þ. Normalized vectorial mode field patterns, defined in (3.18), are repre^ v ðx; yÞ and H ^ v ðϕ; r Þ and H ^ v ðx; yÞ, or E ^ v ðϕ; r Þ. Gaussian modes are represented sented as E using similar symbols, but they are functions of both transverse and longitudinal spatial coordinates, x, y, and z, as seen in (3.73).

A.2

VECTORS AND TENSORS

.............................................................................................................. All vectors are represented in bold face, with the exceptions of unit vectors, and their magnitudes are represented with corresponding symbols in nonbold fonts. A vector is also represented in the form of a 3  1 column matrix. Besides the field vectors and their magnitudes described in the preceding section, we have k, k; K, K; r, r; u, u; U, U; Δk, Δk:

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398

Appendix A

All tensors and transformation matrices are represented in bold face or in terms of their elements with subscript indices. Second-order tensors and transformation matrices are also represented in the form of 3  3 square matrices. The tensors used include h i



 f ijk , pijkl , p0ijkl , r ijk , cijkl , h i



 R ¼ Rij , sijkl , S ¼ Sij , ϵ ¼ ϵij , η ¼ ηij , h i h i h i h i
 ð2Þ ð3 Þ ð2Þ ð3 Þ χ ¼ χ ij , χ ¼ χ ijk , χ ¼ χ ijkl , Δϵ ¼ Δϵij , Δη ¼ Δηij : The transformation matrices used in the text include Fðz; z0 Þ, Rðz; 0; lÞ:

A.3

FOURIER-TRANSFORM PAIRS

.............................................................................................................. The same symbol is used for a quantity in real space and its counterpart in the momentum space, or one in the time domain and its counterpart in the frequency domain. The difference is indicated by expressing a quantity as a function of r or k, or as a function of t or ω. Note that the unit of a quantity is multiplied by a length unit of a meter each time one of the three spatial dimensions is transformed from the real space to the momentum space, and is multiplied by a time unit of a second when the quantity is transformed from the time domain to the frequency domain. For example, the electric field E ðr; t Þ in the real space and time domain has the unit of volts per meter (V m1 ), but E ðk; t Þ has the unit of volt square meters (V m2 ), E ðr; ωÞ has the unit of volt seconds per meter (V s m1 ), and E ðk; ωÞ has the unit of volt second square meters (V s m2 ).

A.4

SPECIAL NOTATIONS

.............................................................................................................. A few special notations are used to label symbols for special meanings.

A.4.1 Unit Vectors and Normalized Quantities Unit vectors are denoted with a hat on top of a symbol. The unit vectors that appear in the text are ^ n^, ^r , u^, ^x , ^y , ^z , X ^ , Y^ , Z^ : ^e , k, Normalized quantities are also denoted with a hat on top of a symbol. The normalized mode field profiles appear in both vector and scalar forms: ^ v , E^ v , H ^ v, H ^ v: E Other normalized quantities that appear in the text include ^ sp , T^ FP , T^ c : g^ ðvÞ, P

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Symbols and Notations

399

A.4.2 Modified Quantities A quantity that is modified from the original quantity in some manner is denoted with a tilde on top of a symbol. Modified quantities that appear in the text include ~ B, ~ Δ~ϵ , Δ~ϵ , ~ A, κ:

A.4.3 Average Values The spatial average, temporal average, weighted average, or mean value of a quantity is denoted with a bar on top of a symbol, such as i, i2 , k, n, N , N 2 , s, s2 , S, S, S 2 , v 2 , W p , α:

A.5

SUBSCRIPTS AND SUPERSCRIPTS

.............................................................................................................. Various fonts and notations are used for subscripts and superscripts. They include numerals, the mathematical font, the Greek font, coordinate symbols, and the Roman font. Bare numerals, mathematical font letters, and Greek letters that represent indices or variables are used only for subscripts. Roman letters and some special notations that have literal meanings can appear either as subscripts or as superscripts.

A.5.1 Numerals Bare numerals are used only for subscripts. The following four numbers have special meanings in a proper context: 0 base value (α0 , m0 ), constant value (P0 , S0 ), free-space value (ϵ 0 , μ0 ), center value (v0 , ω0 ), unsaturated value (g 0 , g0 ), equilibrium value (n0 , p0 ), beam waist (w0 ), or static field (E0 , H 0 ); 1 parameters for waveguide core (n1 , N 1 , D1 , k1 , h1 ) or parameters for the lower laser level j1i (E 1 , N 1 , R1 ); 2 parameters for waveguide substrate (n2 , N 2 , D2 , k2 , γ2 ) or parameters for the upper laser level j2i (E 2 , N 2 , R2 ); 3 parameters for waveguide cover (n3 , N 3 , D3 , k 3 , γ3 ) or parameters for the energy level j3i. Note that the same symbol can have different meanings in different contexts. For example, n2 in nonlinear optics also represents the coefficient of intensity-dependent index change defined in (10.101). The numbers 1, 2, and 3 are also used as subscripts to represent the orthogonal coordinates of a general three-dimensional spatial coordinate system. The numbers 1 through 6 are also used as subscripts representing double indices to label tensor elements under the index contraction rule defined in (2.59): xx yy 1 2

zz 3

yz, zy zx, xz xy, yx 4 5 6

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400

Appendix A

A numeral in the superscript is always placed in parentheses so that it is never confused with an exponent. It represents a perturbation order or the order of an interaction process. For example, χ ð1Þ is a linear susceptibility, χ ð2Þ is a second-order nonlinear susceptibility, χ ð3Þ is a third-order nonlinear susceptibility, and so forth.

A.5.2 Mathematical and Greek Subscripts Mathematical and Greek fonts are used only for subscripts. They represent variable indices with the following well-defined meanings: a, b, c i, j, k, l m, n, p, q m, n q α, β μ, v

general indices or general mode indices; integers or coordinate indices; integers or frequency component indices; transverse mode indices, each labeling a spatial dimension; longitudinal mode index or diffraction order; contracted indices representing double coordinate indices; compound transverse mode indices, each representing a mode.

Some Greek subscripts do not represent indices or variables but express literal meanings. They include: β, λ, λ=2, λ=4, π, π=2:

A.5.3 Coordinate Labels General orthogonal spatial coordinates are labeled as 1, 2, and 3. Specific coordinates include the rectilinear coordinates ðx; y; zÞ, the cylindrical coordinates ðr; ϕ; zÞ, and the spherical coordinates ðr; θ; ϕÞ. One set of special rectilinear coordinates, ðX; Y; Z Þ, is used for the new ^ , Y^ , and Z^ of a crystal transformed under the Pockels effect. Two orthogonal principal axes X unit vectors, ^e þ and ^e  defined in (1.75) and (1.78), are used for left-circular and right-circular polarizations, respectively. Two special symbols are also used to represent directions: ? for perpendicular and jj for parallel. Coordinate labels generally appear as subscripts with commonly accepted meanings, with one exception. This exception takes place when labeling a propagation constant k and the corresponding wavevector k of an optical field that has a particular normal mode polarization. Because kx conventionally represents the x component of the k vector, meaning kx ¼ k^x , the propagation constant of an x-polarized optical field that can propagate in any direction perpendicular to ^x is represented as kx in order to avoid confusion. To be consistent, the corresponding wavevector is labeled as kx . Thus, kx ¼ nx ω=c 6¼ k x , and kx ¼ k x k^ where k^ ? ^x . Such superscript coordinate labeling for k and k applies only to the following: kx , kx ,

ky , ky ,

kz , kz ,

kX , kX ,

kY , kY ,

kZ , kZ ,

kþ , kþ ,

k , k :

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Symbols and Notations

401

A.5.4 Roman Labels All superscript and subscript labels in the Roman font have literal meaning. A given Roman label can appear either as a subscript or as a superscript, depending on the convenience of the situation, with exactly the same meaning. Among all subscript and superscript labels, only the Roman labels have such flexibility. With only a few exceptions to avoid confusion, the conventional rules for abbreviations are largely followed: (1) abbreviations for common words are in the lower case, with the exceptions of E for TE, M for TM, L for longitudinal or load, and T for transverse; (2) abbreviations for proper nouns are in the upper case; and (3) acronyms are all in the upper case. The Roman labels used in this book are listed below.

Label

Meaning

Label

Meaning

a

absorption, acceptor, acoustic

h

hole, homogeneous

b

background, bias, bound

i

B

Boltzmann, Bragg, Brewster

incidence, initial, internal, intrinsic, impurity

c

carrier, cavity, center, characteristic, conduction band, conversion, coupling, critical, cutoff

in

input

ind

induced

inh

inhomogeneous

inj

injection

K

Kerr

L

load, longitudinal

m

magnetization, modulation

coll

collection

cond

conduction

d

dark, dielectric, diffraction, donor

D

Doppler

def

deflection

M

TM mode

e

electrical, electrode, electron, emission, external, extraordinary

max

maximum

E

TE mode

min

minimum

ext

external

n

n-type, noise

f

fall

nonrad

nonradiative

F

Faraday, Fermi

o

ordinary

FP

Fabry–Pérot

out

output

FSR

free spectral range

p

g

bandgap, gain, gap, glass, grating, group

p-polarized (TM), p-type, parallel, phase, plasma, polarization, pump

ph

photo, photon

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402

Appendix A

Label

Meaning

Label

Meaning

pk

peak

sat

saturation

PM

phase matched

sh

shot

ps

pulse

sp

spontaneous, surface plasma

QW

quantum well

ST

Shawlow–Townes

r

radiation, reduced, recombination, reflection, relaxation, reversed, rise

t

total, transmitted

T

transverse

R

Rayleigh

tr

transit, transparency

rad

radiative

th

thermal, threshold

res

resonance

v

valence band

RT

round trip

vac

vacuum

s

s-polarized (TE), saturation, series, signal, slope, spontaneous, source, switching

w

water

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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 Appendix B - SI Metric System pp. 403-404 Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.014 Cambridge University Press

APPENDIX B

SI Metric System

Table B.1 SI base units Quantity

Name

Symbol

Length

meter

m

Mass

kilogram

kg

Time

second

s

Electric current

ampere

A

Temperature

kelvin

K

Amount of substance

mole

mol

Luminous intensity

candela

cd

Table B.2 SI derived units Quantity

Name

Symbol

Equivalent

Plane angle

radian

rad

mm

Solid angle

steradian

sr

m2 m

Frequency

hertz

Hz

s

Force

newton

N

kg m s

Pressure

pascal

Pa

Nm

Energy

joule

J

kg m2 s

Power

watt

W

Js

Electric charge

coulomb

C

As

Electric potential

volt

V

J C 1, W A

Magnetic flux

weber

Wb

Vs

Magnetic flux intensity

tesla

T

Wb m

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1

=1 2

=1

1 2

2 2

1

2

1

404

Appendix B

Table B.2 (cont.) Quantity

Name

Symbol

Equivalent

Resistance

ohm

Ω

VA

Conductance

siemens

S

A V 1, Ω

Capacitance

farad

F

CV

Inductance

henry

H

Wb A

Luminous flux

lumen

lm

cd sr

Illuminance

lux

lx

lm m

Source: Nelson, R. A., Guide for metric practice, Physics Today BG15–BG16, August, 2002.

Table B.3 Metric prefixes Name

Symbol

Factor

Exa

E

1018

Peta

P

1015

Tera

T

1012

Giga

G

109

Mega

M

106

Kilo

k

103

Hecto

h

102

Deca

da

10

Unit

1

Deci

d

10

1

Centi

c

10

2

Milli

m

10

3

Micro

μ

10

6

Nano

n

10

9

Pico

p

10

12

Femto

f

10

15

Atto

a

10

18

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1

1 1

2

1

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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 Appendix C - Fundamental Physical Constants pp. 405-405 Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.015 Cambridge University Press

APPENDIX C

Fundamental Physical Constants

Table C.1 Physical constants Quantity

Symbol

Value

Unit

Speed of light in free space

c

2.997 924 58
108

m s1

Magnetic permeability of free space

μ0

4π
107 1.256 637 061 4
106

H m1 H m1

Electric permittivity of free space

ϵ0

8.854 187 817
1012

F m1

Impedance of free space ðμ0 =ϵ 0 Þ1=2

Z0

376.730 313 461

Ω

Planck constant

h

6.626 068 765 2
1034 4.135 667 271 6
1015

Js eV s

Reduced Planck constant h/2π

ħ

1.054 571 596 8
1034 6.582 118 892 6
1016

Js eV s

Elementary charge

e

1.602 176 462 6
1019

C

Electron rest mass

m0

9.109 381 887 2
1031

kg

Proton rest mass

mp

1.672 621 581 3
1027

kg

Atomic mass unit

mu

1.660 538 731 3
1027

kg

Boltzmann constant

kB

1.380 650 324
1023 8.617 342 15
105

J K1 eV K1

Thermal energy at T = 300 K

kBT

2.585 202 645
102

eV

Photon constant hc ¼ λhv

hc

1.239 841 86
106

eV m

Source: Mohr, P. J. and Taylor, B. N., The fundamental physical constants, Physics Today BG6–BG13, August, 2002.

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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 Appendix D - Fourier-Transform Relations pp. 406-408 Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.016 Cambridge University Press

APPENDIX D

Fourier-Transform Relations

According to the discussion in Chapter 1, we define the Fourier transform between the time domain and the frequency domain in terms of the angular frequency as follows ð∞ E ðωÞ ¼ F fE ðt Þg ¼

E ðt Þeiωt dt

(D.1)

E ðωÞeiωt dω:

(D.2)

∞

and E ðtÞ ¼ F

1

1 fE ðωÞg ¼ 2π

ð∞ ∞

In terms of the real frequency v ¼ ω=2π, we have ð∞

E ðt Þei2πvt dt

(D.3)

E ðvÞei2πvt dv:

(D.4)

E ðvÞ ¼ ∞

and ð∞ E ðtÞ ¼ ∞

The Fourier-transform relations for common functions encountered in the description of various waveforms are listed in Table D.1. In this table, the Heaviside step function H ðxÞ is defined as  1, if x  0, (D.5) H ðxÞ ¼ 0 if x 1=2;

(D.6)

and the triangular function ΛðxÞ is defined as  ΛðxÞ ¼

1  jxj,

if jxj  1,

0

if jxj > 1:

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(D.7)

407

Fourier-Transform Relations

Table D.1 Fourier-transform relations Function form

E(t)

Gaussian

et

sech

sechðt=τÞ

Infinite impulse sequence

 X t δ m τ m

πτ sechðπωτ=2Þ  X ωτ τ δ n 2π n

Complex exponential

eiω0 t

2πδðω  ω0 Þ

delta

Double-sided exponential

ejt=τj

2τ 1 þ ω2 τ 2

Lorentzian

Single-sided exponential

et=τ H ðt Þ

τ 1  iωτ

complex Lorentzian

Rectangular

Πðt=τÞ

τ

Triangular

Λðt=τÞ

τ

Convolution

f ðtÞ∗gðtÞ

f ðωÞgðωÞ

product

Product

f ðtÞgðtÞ

1 f ðωÞ∗gðωÞ 2π

convolution

Complex conjugate

f ∗ ðtÞ

½ f ðωÞ∗

2

=τ 2

E(ω)

Function form

pffiffiffi ω2 τ2 =4 π τe

Gaussian sech infinite impulse sequence

sin ðωτ=2Þ ωτ=2

sinc

sin2 ðωτ=2Þ

sinc2

ðωτ=2Þ2

The convolution integral is defined as ð∞

0

0

0

ð∞

f ðx  x Þgðx Þdx ¼

f ðxÞ∗gðxÞ ¼ ∞

f ðx0 Þgðx  x0 Þ dx0 :

(D.8)

∞

Using the Fourier-transform relation between f ðt Þ∗gðt Þ and f ðωÞgðωÞ and that between f ðt Þ and ½f ðωÞ∗ shown in Table D.1, some useful relations can be obtained. ∗

ð∞ Correlation theorem : ∞

1 f ðtÞgðt þ τ Þ dt ¼ 2π

ð∞ Autocorrelation theorem : ∞

ð∞

∗

(D.9)

∞

1 f ðt Þf ðt þ τ Þ dt ¼ 2π ∗

f ∗ðωÞgðωÞeiωτ dω,

ð∞

  f ðωÞj2 eiωτ dω,

∞

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(D.10)

408

Appendix D

ð∞

1 f ðtÞgðt Þ dt ¼ 2π ∗

Power theorem : ∞

ð∞ Parseval’s theorem : ∞

ð∞

f ∗ðωÞgðωÞ dω,

(D.11)

∞

1 j f ðt Þj dt ¼ 2π

ð∞

2

  f ðωÞj2 dω:

(D.12)

∞

Using (D.3) and (D.4), Parseval’s theorem can be written as ð∞

2

ð∞

jE ðt Þj dt ¼ ∞

∞

1 jE ðvÞj dv ¼ 2π 2

ð∞

jE ðωÞj2 dω:

∞

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(D.13)

INDEX

absorption, 23, 35, 224 band-to-band, 345, 369 absorption coefficient, 130, 242–243, 250 intensity-dependent, 351 of direct-gap semiconductor, 346 of indirect-gap semiconductor, 346 of waveguide mode, 131 unsaturated, 351 absorption cross section, 238, 250 pump, 254 absorption saturation, 351 absorption transition, 236 absorptive external modulation, 344 absorptive modulation, 297, 305 all-optical, 340, 345, 350 external, 344 AC conductivity, 39 acoustic frequency, 52 normal mode, 53 longitudinal, 53 quasi-longitudinal, 53 quasi-transverse, 53 transverse, 53 wave, 52 longitudinal, 53 standing, 53 transverse, 53 traveling, 52 wavelength, 52 acousto-optic amplitude modulation, 334 acousto-optic diffraction Bragg, 334 order, 333 Raman–Nath, 334 acousto-optic effect, 52 acousto-optic modulation, 297, 320, 333 acousto-optic modulator standing-wave, 338 traveling-wave, 336 acousto-optic polarization modulation, 333 active region of photodiode, 372 all-optical absorptive modulation, 340, 345, 350 all-optical dispersive modulation, 340 all-optical modulation, 297, 320, 340, 350

all-optical phase modulation, 342 all-optical polarization modulation, 342 all-optical refractive modulation, 340 all-optical switching, 340 AM, 298, See amplitude modulation Ampere’s law, 4 amplification, 23 of optical field, 241 amplification coefficient, 130, 242 of waveguide mode, 131 amplification factor round-trip, 207, 274 amplified spontaneous emission, 269 amplitude modulation, 297–298, 305, 309, 320, 326, 344 acousto-optic, 334 analog, 305–306 digital, 305–306 electro-optic, 326 magneto-optic, 329 amplitude modulator, 326 amplitude-shift keying, 298, 305, See ASK binary, 305 analog amplitude modulation, 305–306 analog frequency modulation, 300, 333 analog modulation, 297–298, 306, 310, 314 analog polarization modulation, 302 analyzer, 326 angle of diffraction, 335 of incidence, 94, 335 of reflection, 94 of refraction, 94 angular frequency, 1 anisotropic crystal, 28 anisotropic medium, 24, 77 anisotropy, 24 anomalous dispersion, 36, 122 antiferrimagnetic material, 49 antiferromagnetic material, 49 antiguidance factor, 294 ASK, 298, 305, See amplitude-shift keying asymmetric coupling, 144, 146, 160 asymmetric waveguide, 118 attenuation of optical field, 241

attenuation coefficient, 130, 242 of waveguide mode, 131 attenuation factor round-trip, 207 autocorrelation theorem, 407 axial vector, 6 bandgap, 365 of photoconductor, 369 of quantum well, 347 of semiconductor, 346, 365, 371 band-to-band absorption, 345, 369 band-to-band transition, 37 bandwidth, 122 3-dB, 311, 317, 390 gain, 240, 281, 283 modulation. See modulation bandwidth of detection system, 363, 379 of LED, 311 of photodetector. See photodetector bandwidth of semiconductor laser, 317 beam waist, 88 BFSK. See binary frequency-shift keying biaxial crystal, 29, 77 binary amplitude-shift keying, 305 binary frequency-shift keying, 300 binary phase-shift keying, 300 binary polarization-shift keying, 302 birefringence, 28 circular, 31, 52 electrically induced, 48 linear, 28, 52 magnetically induced, 52 optical-field-induced, 342 birefringent crystal, 28 blackbody radiation, 235, 379 bleached condition, 257 bottleneck factor, 251, 256–257, 260 boundary conditions, 7, 67 BPolSK. See binary polarization-shift keying BPSK. See binary phase-shift keying Bragg angle, 335 Bragg diffraction, 334 down-shifted, 335 up-shifted, 335

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410

Index

Brewster angle, 96 Brewster window, 97 bulk modulator, 297 carrier frequency, 3, 12, 19, 123, 300–301, 305 carrier injection efficiency, 309 carrier relaxation rate differential, 315 nonlinear, 315 spontaneous, 315 total, 316 carrier wavevector, 19 causality, 22, 44 causality condition, 34 cavity cold, 214 Fabry–Pérot, 204 finesse, 209 folded, 204 laser, 216 length, 206 linear, 204, 206, 212 non-Fabry–Pérot, 204 optical, 204 passive, 207 resonance condition, 209, 211 ring, 204, 206, 212 cavity decay rate, 214, 315 of Fabry–Pérot cavity, 219 cavity lifetime, 214 centrosymmetric, 46, 58 material, 32, 45, 51, 59 charge carrier excess, 308, 368 free, 5, 38, 104 number, 363 photogenerated, 363, 368, 376–377, 382 positive, 32, 38 charge density, 4 charge-coupled device, 363 chromatic dispersion, 122 circular birefringence, 31 magnetic, 31 magnetically induced, 52, 328 natural, 31 circular dichroism, 31 magnetic, 31 magnetically induced, 52 natural, 31 circular polarization, 16, 25 circularly polarized, 14 left, 16 right, 16 codirectional coupling, 154 coherence, 173

spatial, 173 temporal, 173 cold cavity, 214 collection efficiency, 382 complex field, 11 complex field amplitude, 18 conduction current, 5, 38 conduction electron, 32 conduction susceptibility, 39 conductivity AC, 39 dark, 368 DC, 39 of photoconductor, 368 of semiconductor, 368 optical, 38 confocal parameter, 88 conservation of charge, 5 conservation of power, 159 constructive interference, 171, 204 complete, 171 partial, 171 continuity equation, 5 contradirectional coupling, 156 convolution integral, 407 correlation theorem, 407 Cotton–Mouton effect, 52 Coulomb’s law, 5 coupled-mode equations, 143 for multiple substructures, 146 for single structure, 143 for two-mode coupling, 147 codirectional, 154 contradirectional, 156 coupled-mode theory, 141 coupler 3-dB, 182, 327 asymmetric, 149 directional, 147, 149, 182 grating, 151 symmetric, 149 waveguide, 151 Y-junction, 182, 327 coupling coefficient, 143, 146, 161, 164 for multiple-structure coupling, 146 two modes, 148 for periodic structure, 150 for single-structure coupling, 143 qth order, 150 self, 149 coupling efficiency, 155, 157, 190, 306 for codirectional coupling, 155 perfectly phase-matched, 161 phase-mismatched, 163 for contradirectional coupling, 157 perfectly phase-matched, 161 phase-mismatched, 163

coupling length, 156 perfectly phase-matched, 161 phase-mismatched, 163 critical angle, 97 critical fluorescence power, 268 critical fluorescence power density, 268 cross modulation, 340 cross section absorption, 238, 250 emission, 238, 250 transition, 238 cross-phase modulation, 342 crystal anisotropic, 28 biaxial, 29 birefringent, 28 cubic, 29 hexagonal, 30 monoclinic, 30 negative uniaxial, 28 orthorhombic, 30 positive uniaxial, 28 structural symmetry, 29 tetragonal, 30 triclinic, 30 trigonal, 30 uniaxal, 28 crystal axis, 29 crystal system, 46, 58 Curie temperature, 49 current conduction, 5, 38 dark, 378 displacement, 5 induced, 5, 38 current density, 4 current modulation, 345 direct, 308 cutoff frequency 3-dB, 390 of guided mode, 117 of surface plasmon mode, 106 cutoff wavelength of guided mode, 117 of surface plasmon mode, 106 dark conductivity, 368 dark current, 378 DC conductivity, 39 degeneracy in energy level, 224, 251 degeneracy factor, 224, 236 degenerate semiconductor, 365 n-type, 365 p-type, 365 depletion layer, 371

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Index destructive interference, 171 complete, 171 partial, 172 detectivity, 386 specific, 369, 386 detector photoconductive, 368 photoemissive, 363 photon, 362 quantum, 362 square-law, 362 thermal, 362 diamagnetic material, 49 dichroism, 28 circular, 31, 52 electrically induced, 48 linear, 28, 52 magnetically induced, 52 dielectric constant principal, 28 tensor, 28 differential carrier relaxation rate, 315 differential gain parameter, 315 differential phase modulation, 302, 324, 326, 329 differential power conversion efficiency, 291 diffraction modulation, 297, 299, 307, 333 diffraction order, 184–185 reflective, 188 transmissive, 185, 188 diffusion current, 372 diffusion region, 371 digital amplitude modulation, 305–306 digital frequency modulation, 300, 333 digital modulation, 297–298, 350 digital polarization modulation, 302 direct current-modulation, 308 direct modulation, 297, 299, 305, 308, 345 direct-gap semiconductor, 249, 346, 369 directional coupler, 147, 149, 182 asymmetric, 149 symmetric, 149 two-channel, 149 discrete energy level, 32 dispersion, 122 anomalous, 36, 122 chromatic, 122 frequency, 23 group-velocity, 124 coefficient, 124 effective, 126 negative, 124 positive, 124 intermode, 122, 127 intramode, 122, 127 material, 122, 126

modal, 115, 117, 122, 127 mode-order, 128 momentum, 23 normal, 36, 122 of surface plasmon mode, 106 phase-velocity, 122 polarization, 117, 122, 127 polarization-mode, 128 waveguide, 122, 126 displacement current, 5 distributed feedback, 204 distributed feedback laser, 275 distributed loss, 218, 276 divergence angle, 86, 88 Doppler broadening, 231 Doppler effect, 230 double refraction, 98 double-slit interference, 176 drift current, 371–372 Drude model, 38, 62 dynamic range of photodetector, 387–388 effective group index, 126 effective group-velocity dispersion, 126 effective mass, 32, 38, 347 effective population inversion, 250 effective refractive index, 126 of waveguide mode, 112 effective waveguide thickness for guided TE mode, 114 for guided TM mode, 115 EH mode, 69 Einstein A coefficient, 226, 234 Einstein B coefficient, 234 elastic wave, 52 elasto-optic coefficient, 53 electric displacement, 4 electric field, 4 complex, 12, 18, 169 electric permittivity, 4, 7 of free space, 4 tensor, 7 electric polarization, 4, 7 electric susceptibility, 7 tensor, 7 electric-dipole approximation, 58 electric-dipole interaction, 59 electric-dipole operator, 33 electro-absorption modulation, 345 electro-absorption modulator, 345, 349 electromagnetic field, 4 electron bound, 32 conduction, 32 free, 32 valence, 32 electron affinity, 363

411

electron mobility, 368 electro-optic amplitude modulation, 326 electro-optic coefficient linear, 45 quadratic, 45 electro-optic effect, 44 first order, 45 linear, 45 quadratic, 45 second order, 45 electro-optic Kerr coefficient, 45, 59 electro-optic Kerr effect, 45 electro-optic modulation, 297, 320 electro-optic modulator, 320 electro-optic phase modulation, 321, 328 electro-optic polarization modulation, 324 elliptic polarization, 14, 25 elliptically polarized, 14 ellipticity, 14 emission spontaneous, 224 stimulated, 35, 224 emission cross section, 238, 250 homogeneously broadened medium, 239 inhomogeneously broadened medium, 239 pump, 254 energy band, 32 energy density of optical radiation, 234 energy level, 249 ground, 255 lower, 224 upper, 224 vacuum, 363 envelope, 19, 123 Er:fiber, 239 etalon, 191 evanescent radiation mode, 111 excess noise factor, 378 excess shot noise, 378 exciton, 345 free, 348 external modulation, 297 absorptive, 344 refractive, 319 external modulator, 306 external photoelectric effect, 362–363 external quantum efficiency of LED, 309 of photodetector, 363, 382 of photodiode, 372 of semiconductor laser, 314 external reflection, 96 extinction ratio, 307 extraordinary index, 28

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412

Index

extraordinary wave, 81 extrinsic photoconductivity, 369 extrinsic photoconductor n-type, 369 p-type, 369 extrinsic semiconductor, 369

frequency-shift keying, 298, See FSK binary. See BFSK quadrature. See QFSK Fresnel equations, 95 FSK, 298, See frequency-shift keying fundamental mode, 109, 118, 121

Fabry–Pérot cavity, 204, 216 stability criterion, 216 Fabry–Pérot etalon, 191 Fabry–Pérot interferometer, 178, 191, 204, 209 Fabry–Pérot laser, 275 falltime, 389 Faraday effect, 52, 329–330 Faraday rotation, 329–330 sense, 331 specific, 330 Faraday rotator, 330 Faraday’s law, 4 fast axis, 80 Fermi level, 363, 365 ferrimagnetic material, 49 ferromagnetic material, 49 field amplitude complex, 18 scalar, 18 vectorial, 12, 18 filling factor of gain medium, 206 finesse, 194, 209 of lossless Fabry–Pérot interferometer, 193, 219 of lossy Fabry–Pérot cavity, 219 of optical cavity, 209 flat interface, 190 fluorescence, 226 fluorescence lifetime, 227, 239, 250 FM, 298, See frequency modulation folded cavity, 204 forward-coupling matrix, 154, 161 Fourier-transform relations, 13, 406 four-level system, 256, 259–260 ideal, 256 Franz–Keldysh effect, 345–346 free spectral range, 194, 210 frequency chirping, 129, 309 negative, 129 positive, 129 frequency dispersion, 23 frequency modulation, 297–298, 300, 309 analog, 300, 333 digital, 300, 333 frequency response of LED, 311 of photodetector, 389 of semiconductor laser, 316

gain bandwidth, 240, 281, 283 gain coefficient, 130, 242, 244, 250, 259 of waveguide mode, 131 small-signal, 263 unsaturated, 259, 263 gain compression, 315 gain factor round-trip, 207, 274 gain filling factor, 206 gain parameter, 286 differential, 315 nonlinear, 315 of gain medium, 286 of laser mode, 286 of semiconductor laser, 315 saturated, 287 unsaturated, 287 gain saturation, 264, 281–282, 351 Gauss’s law, 5 Gaussian beam, 87 complex radius of curvature, 90 confocal parameter, 88 divergence angle, 86, 88 radius of curvature, 88 Rayleigh range, 88 spot size, 86, 88 waist, 88 Gaussian lineshape, 231 Gaussian mode, 86, 211 basis for linear expansion, 89, 92 orthonormality relation, 87 propagation constant, 87 graded-index waveguide, 108 grating, 151, 183 at interface, 187 one-dimensional, 151 period, 153 sinusoidal, 152 square-function, 153 surface, 189 transmission, 183 transmissive diffraction, 183–184 grating waveguide coupler, 151 ground level, 255 group index, 124 effective for waveguide mode, 126 group IV semiconductor, 367 group velocity, 122–123 of waveguide mode, 127 group-velocity dispersion, 124

coefficient, 124 effective for waveguide mode, 126 negative, 124 positive, 124 guided mode, 109, 112 cutoff condition, 117, 121 cutoff frequency, 117 cutoff wavelength, 117 fundamental, 109, 118, 121 high-order, 109 number, 109, 118, 121 half-wave plate, 80 half-wave voltage, 322 harmonic field, 11 HE mode, 69 Heaviside step function, 406 helicity negative, 17 positive, 16 Helmholtz equation, 86 Hermite polynomial, 90 Hermite–Gaussian function, 89 Hermite–Gaussian mode, 90 high-order mode, 109 hole mobility, 368 homogeneous broadening, 225 homogeneous region, 371 homogeneous system, 225 homogeneously broadened laser, 280 hybrid mode, 69, 108 IM, 305, See intensity modulation impulse response, 389 index contraction, 45 index ellipsoid, 82 index of refraction, 76 complex, 129 effective for waveguide mode, 112 extraordinary, 28 for extraordinary wave, 82 for ordinary wave, 82 intensity-dependent, 342 ordinary, 28 principal, 28 indirect-gap semiconductor, 346, 369 induced transition, 224 induced transition rate, 235 inhomogeneous broadening, 230 inhomogeneous system, 230 inhomogeneously broadened laser, 282 intensity, 2 light, 12–13 of normal mode, 71 of optical field, 130 intensity modulation, 305

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Index intensity-dependent absorption coefficient, 351 intensity-dependent index change, 342 intensity-dependent index of refraction, 342 interaction length, 161, 164–165, 334 interband transition, 37 interface mode, 92 interference constructive, 171, 204 complete, 171 partial, 171 destructive, 171 complete, 171 partial, 172 double-slit, 176 of two fields, 169 interference filter, 199 interference fringe, 173, 175 bright, 175 dark, 175 interferometer Fabry–Pérot, 178, 191 Mach–Zehnder, 178, 180 Michelson, 178 intermode dispersion, 122, 127 internal gain of photodetector, 363 internal photoelectric effect, 362 internal quantum efficiency of LED, 311 of photodetector, 382 internal reflection, 96 total, 97, 99 intraband transition, 38 intracavity energy growth rate, 286 threshold, 286 intracavity photon density, 286 intracavity photon growth rate, 286 intramode dispersion, 122, 127 intrinsic photoconductivity, 369 intrinsic responsivity, 384 inversion symmetry, 45 irradiance, 12 isotropic material, 46 isotropic medium, 24, 75 Johnson noise, 375 junction photodiode, 363, 371 photoconductive mode, 373–374 photovoltaic mode, 373 K factor, 316 Kerr effect electro-optic, 45 magneto-optic, 52, 329 optical, 342

Kerr-lens mode locking, 342 Kramers–Kronig relations, 44 Laguerre–Gaussian function, 90 Laguerre–Gaussian mode, 90 laser, 274 homogeneously broadened, 280 inhomogeneously broadened, 282 longitudinal mode, 280, 282 transverse mode, 280, 282 laser amplifier, 265 laser cavity, 216 laser level lower, 250, 253, 255 upper, 250, 253, 255 laser linewidth, 283 laser oscillation, 274 condition, 274 gain condition, 275 phase condition, 275 laser oscillator, 274 laser power, 285 laser system four-level, 256, 259–260 ideal four-level, 256 quasi-two-level, 254, 259 three-level, 255, 259–260 two-level, 253 laser threshold, 276 laser transition, 218, 275 LED, 297, 308–309, See light-emitting diode left-circular polarization, 16 L–I characteristics of LED, 309 of semiconductor laser, 314 lifetime cavity, 214 fluorescence, 227 of energy level, 226 photon, 214 saturation, 259 spontaneous radiative, 226, 236 lifetime broadening, 226 light intensity, 12–13 light–current characteristics of LED, 309 of semiconductor laser, 314 light-emitting diode, 297, 309, See LED line filter, 199 linear birefringence, 28, 48 electrically induced, 48 magnetically induced, 52 linear cavity, 204, 206, 212 linear dichroism, 28, 48 electrically induced, 48 magnetically induced, 52

413

linear medium, 24 linear optical property, 29, 46 linear polarization, 15, 25, 55 linear susceptibility, 22, 56 linearity of photodetector, 387 linearly polarized, 14 lineshape Gaussian, 231 homogeneously broadened, 226 inhomogeneously broadened, 230 Lorentzian, 35, 226 Voigt, 234 lineshape function, 225, 230 homogeneously broadened, 226 inhomogeneously broadened, 230 linewidth, 194 Doppler broadening, 231 homogeneous broadening, 226 inhomogeneous broadening, 231 laser mode, 283 lifetime broadening, 227 longitudinal mode, 210 mixed broadening, 234 natural broadening, 227 linewidth enhancement factor, 294 longitudinal mode, 207, 210, 280, 282 frequency spacing, 210, 219 of optical resonator, 207, 210 longitudinal mode frequency, 210 longitudinal mode width, 210 longitudinal modulation, 297, 321, 323 longitudinal modulator, 323 longitudinal phase modulation, 323 longitudinal phase modulator, 323 Lorentz model, 33, 61 Lorentz reciprocity theorem, 26, 142 Lorentzian lineshape, 35, 226 loss parameter, 286 output-coupling, 288 total, 288 lower energy state, 32 lower laser level, 250, 253, 255 Mach–Zehnder interferometer, 178, 180 waveguide, 327 magnetic field, 4 magnetic induction, 4 magnetic material, 26 magnetic permeability, 4 of free space, 4 magnetic polarization, 4 magnetically disordered, 50 magnetically ordered, 49 magnetization, 4, 7 spontaneous, 50 magneto-optic amplitude modulation, 329

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414

Index

magneto-optic effect, 49 first-order, 50, 328 linear, 50, 328 quadratic, 50 second-order, 50, 328 magneto-optic Kerr effect, 52, 329 magneto-optic modulation, 297, 320, 328 magneto-optic modulator, 328 magneto-optic phase modulation, 329 material dispersion, 122, 126 Maxwell’s equations, 4–5, 11, 142 for plane wave normal mode, 73 for wave propagation, 68 in terms of mode field components, 68 Maxwellian velocity distribution, 231 mean square noise, 377 mean square noise current, 385 mean square noise voltage, 385 metallic waveguide, 69 Michelson interferometer, 178 Michelson–Morley experiment, 178 mixed broadening, 233 modal dispersion, 115, 117, 122, 127 mode coupling, 141 asymmetric, 144, 146, 160 between two modes, 147 codirectional, 154 contradirectional, 156 multiple-structure, 145 order, 150 phase-matched, 149, 161 phase-mismatched, 163 single-structure, 142 symmetric, 143, 160 mode field normalized, 71, 141 profile, 67 mode index, 67 mode locker, 352 mode number, 109 mode power, 131, 142 mode volume, 287 mode-order dispersion, 128 mode-pulling effect, 277 modulation absorptive, 297, 305 all-optical, 340, 345, 350 external, 344 acousto-optic, 297, 320, 333 all-optical, 297, 320, 340, 342 absorptive, 340 dispersive, 340 refractive, 340 amplitude, 297–298, 305, 309, 320, 326, 344 acousto-optic, 334 analog, 305

digital, 305–306 electro-optic, 326 magneto-optic, 329 analog, 297–298, 306, 310, 314 current, 345 diffraction, 297, 299, 307, 333 digital, 297–298, 350 direct, 297, 299, 305, 308, 345 electro-absorption, 345 electro-optic, 297, 320 external, 297 absorptive, 344 refractive, 319 frequency, 297–298, 300, 309 analog, 300, 333 digital, 300, 333 longitudinal, 297, 321, 323 magneto-optic, 297, 320, 328 optical, 297 phase, 297–299, 306, 320, 344 analog, 299 differential, 302, 324, 326, 329 digital, 299 electro-optic, 321, 328 longitudinal, 323 magneto-optic, 329 transverse, 322 polarization, 297, 302, 306, 324 acousto-optic, 333 analog, 302 digital, 302 electro-optic, 324 magneto-optic, 329 refractive, 297, 300, 305, 307 all-optical, 340 electro-optic, 320 external, 319 magneto-optic, 329 small-signal, 316 spatial, 297, 299, 307 transverse, 297, 321, 323 modulation bandwidth 3-dB, 311, 317 of LED, 311 of semiconductor laser, 317 modulation current, 308, 311, 316 modulation depth, 306, 322 modulation frequency, 311, 316, 339 modulation index, 310, 316 modulation power spectrum of LED, 311 of semiconductor laser, 317 modulation response, 308 of LED, 311 of semiconductor laser, 316 modulation scheme, 297–298 modulation signal, 308

modulator acousto-optic standing-wave, 338 traveling-wave, 336 amplitude, 326 bulk, 297 electro-absorption, 345, 349 electro-optic, 320 external, 306 longitudinal, 323 magneto-optic, 328 phase, 322–323 polarization, 326 transverse, 322 waveguide, 297 momentum dispersion, 23 momentum relaxation time, 38 monochromatic optical wave, 13, 20, 73 multimode waveguide, 118 natural broadening, 227 Nd:glass, 233 Nd:YAG, 233 Nd:YAG laser, 258 NEA, 367, See negative electron affinity Néel temperature, 49 negative electron affinity, 367 NEP, 384, See noise equivalent power noise, 225 Johnson, 375 mean square, 377, 385 Nyquist, 375 of photodetector, 375 quantum, 375 shot, 375–376 thermal, 375, 379 noise equivalent power, 384 noncentrosymmetric, 46, 58 nondegenerate semiconductor, 365 nonlinear carrier relaxation rate, 315 nonlinear gain parameter, 315 nonlinear optical amplifier, 265 nonlinear optical property, 29 nonlinear optical susceptibility, 55 nonlinear polarization, 55 second-order, 55 third-order, 55 nonlinear susceptibility, 56 nth-order, 56 second-order, 59 third-order, 59 nonmagnetic material, 26 nonplanar optical structure, 66 dielectric, 70 nonplanar waveguide, 66, 108 dielectric, 69, 108 nonradiative relaxation rate, 226

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Index nonreciprocal, 51–52, 331 nonreciprocal medium, 26 normal dispersion, 36, 122 normal incidence, 96 normal mode acoustic, 53 basis for linear expansion, 72, 141 coupling, 141 degenerate, 67 extraordinary wave, 81 field profile, 67 Gaussian, 86 guided, 109, 112 fundamental, 109, 121 high-order, 109 number, 109 hybrid, 69 index, 67 intensity distribution, 71 interface, 92 of planar interface, 98 of propagation, 66, 73 ordinary wave, 81 orthogonality relation, 71 orthonormality relation, 71 plane wave, 73, 78, 83 power, 71 principal polarization, 25 propagation constant, 67 radiation, 98 substrate radiation, 111–112 substrate–cover radiation, 111, 113 super, 145 surface plasmon, 104 transverse electric, TE, 69 transverse electromagnetic, TEM, 69 transverse magnetic, TM, 69 waveguide, 107 normal mode field pattern, 67 normal state, 32 normalized frequency and waveguide thickness, 112 normalized guide index, 112 normalized mode field, 71, 141 normalized transmittance, 193, 208 of Fabry–Pérot interferometer, 193 of optical cavity, 208 Nyquist noise, 375 on-off keying, 305 OOK, 305, See on-off keying optical activity, 30 magnetically induced, 30 natural, 30 optical amplification, 129, 225, 244, 265 optical amplifier, 265 optical anisotropy, 24, 29

optical attenuation, 129, 225, 244 optical axis, 28, 77, 81 optical carrier, 3, 12, 19, 123 optical cavity, 204, 209 optical conductivity, 38 optical confinement, 108 optical discriminator, 352 optical energy, 8 density, 9 optical feedback, 204, 274 optical field angular frequency, 20 complex, 12, 18, 169 frequency, 18 harmonic, 12 magnitude, 18 phase, 18, 169 polarization, 18 real amplitude, 19 scalar complex amplitude, 18 vectorial complex amplitude, 18 wavevector, 18 optical-field-induced birefringence, 342 optical frequency, 1 optical gain, 23, 129 optical gain coefficient, 259 optical grating, 183, 307 optical interference, 169 optical interferometer, 178 optical Kerr effect, 342 optical loss, 23, 129 distributed, 218 optical medium anisotropic, 24 isotropic, 24 linear, 24 lossless, 26 lossy, 26 nonmagnetic, 26 nonreciprocal, 26 optically active, 26 reciprocal, 26 optical modulation, 297 optical noise, 225 optical nonlinearity, 55 optical path length, 176 round-trip, 206 optical power, 8, 131, 363 optical property linear, 29, 46 nonlinear, 29 optical pumping, 254 optical resonance, 204 optical resonator, 204 optical soliton, 342 optical spectrum analyzer, 195 optical switching, 298

415

optical thin film, 196 optical transition, 224 optical wave monochromatic, 13, 20 plane, 13, 19 optical wavelength in free space, 1 in homogeneous medium, 76 order of coupling, 150 order of diffraction, 184 ordinary index, 28 ordinary wave, 81 orientation, 14 orthogonal polarizations, 17 orthogonality relation of normal mode, 71 orthonormality relation of Gaussian mode, 87 of normal mode, 71 output-coupling loss parameter, 288 output-coupling rate, 288 overlap coefficient, 146 overlap factor, 206, 327 p polarized, π polarized, 95 p wave, π wave, 95 parallel polarization, 95 paramagnetic material, 49 paraxial approximation, 87 Parseval’s theorem, 408 passive cavity, 207 perfect phase matching, 153, 161–162, 165, 334 periodic index modulation, 151 periodic perturbation, 150 periodic structural corrugation, 151 permittivity, 7, 39, 122 acousto-optically induced change, 333 electric field-dependent, 44 frequency domain, 13, 23 magnetic field-dependent, 44 magnetization-dependent, 50 momentum space, 13, 23 of gain medium, 218 optical, 22 optical field-dependent, 341 photoelastic, 54 principal, 27 real space, 7, 23 rotation field-dependent, 54 scalar, 66 strain field-dependent, 54 tensor, 7 time domain, 7, 23 total, 39 perpendicular polarization, 95 perturbing polarization, 142

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416

Index

phase matched, 156, 158, 160 phase matching, 160, 165, 183, 187, 334, 336 perfect, 153, 161–162, 165, 334 phase mismatch, 149, 151, 160, 163, 165, 185, 306 phase modulation, 297–299, 306, 320, 344 all-optical, 342 analog, 299 cross, 342 depth, 322 differential, 302, 324, 326, 329 digital, 299 electro-optic, 321, 328 longitudinal, 323 magneto-optic, 329 self, 342 transverse, 322 phase relaxation, 250 phase relaxation rate, 225 phase retardation, 324 phase velocity, 122 of waveguide mode, 127 phase-matched coupling, 149, 161 phase-matching condition, 160, 186, 188, 190–191, 307, 333 for down-shifted Bragg diffraction, 335 for up-shifted Bragg diffraction, 335 phase-mismatched coupling, 163 phase-shift keying, 298, See PSK binary. See BPSK quadrature. See QPSK phase-velocity dispersion, 122 photocathode, 362–363 photoconductive detector, 368 photoconductive mode, 373–374 photoconductivity, 368 extrinsic, 369 intrinsic, 369 photoconductor, 363 extrinsic, 369 intrinsic, 369 photocurrent, 363, 372 reverse, 372 photodetector, 362 photodetector bandwidth, 385 intrinsic, 389 RC, 389 photodiode, 371 junction, 363, 371 vacuum, 362 photoelastic coefficient, 53 photoelastic effect, 53 dynamic, 53 photoelastic permittivity tensor, 54

photoelectric effect, 362 external, 362–363 internal, 362 photoelectron, 363 photoemission, 363 from degenerate semiconductor, 365 from metal, 365 from NEA photocathode, 367 from nondegenerate semiconductor, 365 photoemissive detector, 363 photoemissive device, 362 photomultiplier tube, 362 photon, 1 energy, 1 flux, 2 flux density, 2 momentum, 1 number, 363 speed, 1 photon detector, 362 photon lifetime, 214 of Fabry–Pérot cavity, 219 photothermal effect, 362 phototransistor, 363 photovoltaic device, 363 photovoltaic mode, 373 P–I characteristics of LED, 310 planar dielectric waveguide, 108 planar interface, 66, 92 planar optical structure, 66 dielectric, 70 planar waveguide, 66, 108 dielectric, 69 metallic, 69 Planck’s formula, 235 plane of incidence, 94 plane polarized, 14 plane wave, 13, 19, 73 normal mode, 73, 78, 83 basis for linear expansion, 74, 76, 78, 84 plasma frequency, 40, 104 surface, 106 PM, 298, See phase modulation p–n homojunction, 371 Pockels coefficient, 45, 59 Pockels effect, 45 point group, 46, 58 Poisson probability distribution, 377 polar semiconductor, 29 polar vector, 6 polarization, 4, 7 circular, 16, 25 elliptic, 14, 25 ellipticity, 14 left-circular, 16

linear, 15, 25, 55 nonlinear, 55 nth-order, 56 second-order, 55 third-order, 55 of optical field, 13 orientation, 14 orthogonality relation, 17, 74 principal state, 25 right-circular, 16 state, 16 unit vector, 17 polarization dispersion, 117, 122, 127 polarization modulation, 297, 302, 306, 324 acousto-optic, 333 all-optical, 342 analog, 302 digital, 302 electro-optic, 324 magneto-optic, 329 polarization modulator, 326 polarization-mode dispersion, 128 polarization-shift keying, 302, See PolSK binary. See BPolSK polarizer, 326 polarizing beam splitter, 85 PolSK, 302, See polarization-shift keying population density, 32, 249, 253 distribution, 33 population decay rate, 33 population difference, 34, 244 population inversion, 33, 242, 249, 251, 255 condition, 252 effective, 250 effective condition, 252 population relaxation, 250 power, 2 of normal mode, 71, 142 power conversion efficiency, 291 differential, 291 power density, 10 power gain, 207, 265 small-signal, 265 unsaturated, 265 power theorem, 408 power–current characteristics of LED, 309 of semiconductor laser, 314 Poynting vector, 9, 71, 73 complex, 12 time-averaged, 12 principal axis, 27, 29, 78 principal dielectric axis, 27, 46, 48 principal dielectric constant, 28

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Index principal dielectric susceptibility, 28 principal index of refraction, 28, 46, 48 principal normal mode, 25, 48, 78 principal permittivity, 27, 78 principal polarization state, 25 principle of detailed balance, 236 probability density function, 230 propagation constant, 19, 67, 75 complex, 129, 131 of extraordinary wave, 82 of Gaussian normal mode, 87 of guided mode, 109, 113 of normal mode, 67 of ordinary wave, 82 of principal normal mode, 28, 78 of substrate radiation mode, 111, 113 of substrate–cover radiation mode, 111, 113 of surface plasmon mode, 106 of waveguide mode, 126, 131 propagation direction, 19, 76 PSK, 298–299, See phase-shift keying pump power threshold, 287 transparency, 287 pumping, 249, 254 pumping rate, 250, 253 transparency, 257 pumping ratio, 287 pumping requirement, 254–256 minimum, 255–256, 260 pumping technique, 249 push–pull operation, 327 Q switch, 352 QFSK. See quadrature frequency-shift keying QPSK. See quadrature phase-shift keying quadrature frequency-shift keying, 300 quadrature phase-shift keying, 300 quality factor, 214 of cold cavity, 214 quantum detector, 362 quantum efficiency, 368, 371, 377 external, 309, 363, 382 internal, 382 of photodetector, 382 pump, 262 quantum noise, 375 quantum regime, 380 quantum-confined Stark effect, 345 quantum-well structure, 347 quarter-wave plate, 80 quasi-two-level system, 254, 259 radiation mode, 98 of planar interface, 98

one-sided, 99 substrate, 111 substrate–cover, 111 two-sided, 101 radiative relaxation rate, 226 Raman–Nath diffraction, 334 rate equations, 250 for semiconductor laser, 313 Rayleigh range, 88 reality condition, 24, 56 reciprocal, 51–52 reciprocal medium, 26 rectangular function, 406 reflectance, 95–96 of Fabry–Pérot interferometer, 193 reflection, 93 external, 96 internal, 96 partial, 101 total, 97, 99 reflection coefficient, 95 of contradirectional coupling, 158 reflection-type polarizer, 97 reflectivity, 95 of contradirectional coupling, 158 refraction, 93 double, 98 refractive external modulation, 319 refractive index, 19, 76, See index of refraction effective for waveguide mode, 112, 126 refractive modulation, 297, 300, 305, 307 all-optical, 340 electro-optic, 320 external, 319 magneto-optic, 329 relative impermeability tensor, 45, 53 relaxation oscillation, 316 relaxation rate, 32 carrier. See carrier relaxation rate nonradiative, 226 optical polarization, 33 phase, 225, 250 population, 33, 250 radiative, 226 susceptibility, 33 total, 226 relaxation resonance frequency, 316 resonance condition of optical cavity, 209, 211 for Gaussian mode, 212 for waveguide mode, 212 transverse for waveguide mode, 109 resonance frequency, 32, 204

417

of cold Fabry–Pérot cavity, 218 of optical cavity, 204, 211 transition, 33, 224–225 resonant interaction, 32 resonant laser transition, 218, 275 resonant optical cavity, 204 resonant optical susceptibility, 32, 243 resonant susceptibility tensor, 33 resonant transition, 224 response speed of a photodetector, 389 responsivity, 380, 383 intrinsic, 384 spectral, 367, 371, 384 reverse current, 372 reverse-coupling matrix, 157, 161 right-circular polarization, 16 ring cavity, 204, 206, 212 ring laser, 275 risetime, 389 rotating-wave approximation, 34 rotation tensor, 53 rotation-optic coefficient, 53 rotatory power, 330 round-trip amplification factor, 207, 274 round-trip gain factor, 207, 274 mode-dependent, 275 of cold cavity, 214 of cold Fabry–Pérot cavity, 218 round-trip optical path length, 206, 212 round-trip phase shift, 192–193, 207, 209, 274 mode-dependent, 211, 217, 275 of Gaussian mode, 218 round-trip time, 205, 210 ruby laser, 258 s polarized, σ polarized, 95 s wave, σ wave, 95 saturable absorber, 351 saturated gain parameter, 287 saturation intensity, 259, 351 saturation lifetime, 259–260 saturation output power, 288 saturation photon density, 287 saturation power, 265 scalar field amplitude, 18 Schawlow–Townes limit, 284 Schawlow–Townes relation, 284 self defocusing, 342 self focusing, 342 self modulation, 340 self-phase modulation, 342 semiconductor degenerate, 365 direct-gap, 249, 346, 369 extrinsic, 369 group IV, 367

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418

Index

semiconductor (cont.) III–V, 29, 367 indirect-gap, 346, 369 nondegenerate, 365 polar, 29 semiconductor laser, 308, 313 semiconductor photodiode, 371 shot noise, 375–376 excess, 378 of photodetector, 378 of photodetector with internal gain, 378 shot-noise limited, 380 signal current, 372 of photodetector, 363 signal-to-noise ratio, 376, 379 current, 376 power, 376 voltage, 376 single-mode waveguide, 118, 121 slab waveguide, 111 symmetric, 119 slope efficiency, 291 slow axis, 80 slowly varying amplitude, 56 small-signal gain coefficient, 263 small-signal modulation, 316 small-signal power gain, 265 Snell’s law, 95 SNR, 376, See signal-to-noise ratio space inversion, 6, 51, 59 transformation, 6, 46 spatial beam walk-off, 84 spatial light filter, 352 spatial modulation, 297, 299, 307 spatially coherent, 173 spatially incoherent, 173 specific detectivity, 369, 386 of photoconductive detector, 369 specific Faraday rotation, 330 spectral broadening, 225, 342 dephasing, 228 Doppler, 231 homogeneous, 225 inhomogeneous, 230 lifetime, 226 mixed, 233 natural, 227 spectral detectivity, 382 of photodetector, 382 spectral hole burning, 282 spectral intensity distribution, 234 spectral lineshape, 225 spectral response, 367, 382 spectral responsivity, 371, 382, 384 of photocathode, 367 of photodetector, 382 of photodiode, 371

speed of light in free space, 1, 75 in homogeneous medium, 76 spontaneous carrier recombination lifetime, 309 spontaneous carrier recombination rate, 309 spontaneous carrier relaxation rate, 315 spontaneous emission, 224, 267 amplified, 269 spontaneous emission factor, 284 spontaneous emission power, 268 spontaneous emission power density, 268 spontaneous emission rate, 234 spontaneous emission spectrum, 234 spontaneous magnetization, 50 spontaneous radiative lifetime, 226, 236 spontaneous transition, 236 spontaneous transition rate, 235 spot size, 86 square-law detector, 362 square-pulse response, 389 stability criterion for Fabry–Pérot cavity, 216 standing wave, 182 step-index waveguide, 108 planar, 111 stimulated emission, 35, 224 stimulated-emission transition, 236 strain shear, 53 tensile, 53 strain tensor, 53 strain-optic coefficient, 53 substrate radiation mode, 111–112 substrate–cover radiation mode, 111, 113 super mode, 145 super structure, 145 surface grating, 189 surface plasma frequency, 106 surface plasmon mode, 104 cutoff frequency, 106 cutoff wavelength, 106 dispersion curve, 106 propagation constant, 106 susceptibility, 7, 122 conduction, 39 electric field-dependent, 44 frequency domain, 7, 13, 23 linear, 22, 56 magnetic field-dependent, 44 magnetization-dependent, 50 momentum space, 7, 13, 23 nonlinear, 56 nth-order, 56 second-order, 59 third-order, 59

nonlinear optical, 55 optical, 22 principal, 28 real space, 23 resonant, 32, 243 rotation field-dependent, 54 strain field-dependent, 54 tensor, 7, 33 time domain, 23 switching all-optical, 340 optical, 298 symmetric coupling, 143, 160 symmetric waveguide, 112, 118–119, 121 slab, 119 TE mode, 69, 76, 114, 119, 121 TE polarization, 95 TEM mode, 69, 87, 108 TEM wave, 75 temporal beat, 173 temporally coherent, 173 temporally incoherent, 173 thermal detector, 362 thermal equilibrium, 32, 235, 249 thermal noise, 375, 379 of photodetector, 379 thermal regime, 380 thermal-noise limited, 380 three-level system, 255, 259–260 threshold carrier density, 314 threshold condition, 286 for semiconductor laser, 314 threshold current density, 314 threshold gain coefficient, 276, 280 threshold gain parameter, 286, 314 threshold injection current, 314 threshold intracavity energy growth rate, 286 threshold photon energy, 363, 368 of extrinsic photoconductivity, 369 of intrinsic photoconductivity, 369 of photoemission from degenerate semiconductor, 365 from metal, 365 from NEA photocathode, 367 from nondegenerate semiconductor, 365 of semiconductor photodiode, 371 threshold pump power, 276, 287 threshold pumping level, 276 threshold wavelength, 363, 368 Ti:sapphire, 239 time reversal, 6, 51 transformation, 6, 50 TM mode, 69, 76, 108, 115, 119, 121 TM polarization, 95

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Index total carrier relaxation rate, 316 total internal reflection, 97, 99 total relaxation rate, 226 transition absorption, 236 band-to-band, 37 energy, 236 induced, 224 interband, 37 intraband, 38 laser, 218, 275 optical, 224 resonance frequency, 33, 224 resonant, 224 spontaneous, 236 stimulated-emission, 236 transition cross section, 238 transition rate, 234 absorption, 234 induced, 235 induced downward, 234 spontaneous, 235 spontaneous emission, 234 stimulated emission, 234 upward transition, 234 transition resonance frequency, 33, 224– 225 transmission coefficient, 95 transmission efficiency, 382 transmission grating, 183 transmission line, 69 transmissive diffraction grating, 183–184 transmissivity, 95 transmittance, 95–96 normalized, 193, 208 of Fabry–Pérot interferometer, 193 transparency, 257, 260 transparency population density, 257 transparency pump power, 276, 287 transparency pumping rate, 257 transverse electric mode. See TE mode transverse electromagnetic mode. See TEM mode

transverse magnetic mode. See TM mode transverse mode, 211, 280, 282 of optical resonator, 211 transverse modulation, 297, 321, 323 transverse modulator, 322–323 transverse phase modulation, 322 transverse phase modulator, 322 triangular function, 406 two-level system, 253 quasi, 254, 259 uniaxial crystal, 28, 77, 81 negative, 28 positive, 28 uniform perturbation, 149 unit polarization vector, 17 unsaturated absorption coefficient, 351 unsaturated gain coefficient, 259–260, 263 unsaturated gain parameter, 286 unsaturated power gain, 265 upper energy state, 32 upper laser level, 250, 253, 255 population, 257, 267 transparency population density, 257 V number, 112 vacuum energy level, 363 vacuum photodiode, 362 valence electron, 32 vectorial field amplitude, 18 Verdet constant, 330 Voigt lineshape, 234 walk-off angle, 84 wave equation, 10–11 for plane wave normal mode, 74 wavefront, 19, 73 waveguide asymmetric, 118 asymmetry factor, 112 cladding, 108 core, 108 dielectric, 108

419

evanescent radiation mode, 111 graded-index, 108 guided mode, 109, 112 cutoff condition, 117, 121 cutoff frequency, 117 cutoff wavelength, 117 fundamental, 109, 118, 121 high-order, 109 number, 109, 118, 121 TE, 114, 119, 121 TM, 115, 119, 121 metallic, 69 mode, 107 effective refractive index, 112 multimode, 118 nonplanar, 66, 108 dielectric, 69 planar, 66, 108 cover, 108 dielectric, 69 film, 108 metallic, 69 step-index, 111 substrate, 108 single-mode, 118, 121 slab, 111 symmetric, 119 step-index, 108 substrate radiation mode, 111–112 substrate–cover radiation mode, 111, 113 symmetric, 112, 118–119, 121 V number, 112 weakly guiding, 117 waveguide dispersion, 122, 126 waveguide modulator, 297 wavelength acoustic, 52 optical, 1, 76 wavenumber, 19, 76, 130 wavevector, 1, 73 weakly guiding waveguide, 117 work function, 363, 367

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