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Photonic Crystals
C O P Y R I G H T 2 0 0 8 , P R I N C ETO N U N IVERS ITY PR ES S i

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C O P Y R I G H T 2 0 0 8 , P R I N C ETO N U N IVERS ITY PR ES S ii

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Photonic Crystals
Molding the Flow of Light Second Edition
John D. Joannopoulos Steven G. Johnson Joshua N. Winn Robert D. Meade
PRINCETON UNIVERSITY PRESS • PRINCETON AND OXFORD
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Copyright c 2008 by Princeton University Press

Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY

All Rights Reserved

Library of Congress Cataloging-in-Publication Data

Joannopoulos, J. D. (John D.), 1947-

Photonic crystals: molding the ﬂow of light/John D. Joannopoulos . . . [et al.].

p. cm.

Includes bibliographical references and index.

ISBN: 978-0-691-12456-8 (acid-free paper)

1. Photons. 2. Crystal optics. I. Joannopoulos, J. D. (John D.), 1947- II. Title.

QC793.5.P427 J63 2008

548 .9–dc22

2007061025

British Library Cataloging-in-Publication Data is available

This book has been composed in Palatino

Printed on acid-free paper. ∞

press.princeton.edu

Printed in Singapore

10 9 8 7 6 5 4 3 2 1

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To Kyriaki and G. G. G.
C O P Y R I G H T 2 0 0 8 , P R I N C ETO N U N IVERS ITY PR ES S v

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To see a World in a Grain of Sand, And a Heaven in a Wild Flower, Hold Infinity in the palm of your hand And Eternity in an hour.
— William Blake, Auguries of Innocence (1803)
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CONTENTS

Preface to the Second Edition

xiii

Preface to the First Edition

xv

1 Introduction

1

Controlling the Properties of Materials

1

Photonic Crystals

2

An Overview of the Text

3

2 Electromagnetism in Mixed Dielectric Media

6

The Macroscopic Maxwell Equations

6

Electromagnetism as an Eigenvalue Problem

10

General Properties of the Harmonic Modes

12

Electromagnetic Energy and the Variational Principle

14

Magnetic vs. Electric Fields

16

The Effect of Small Perturbations

17

Scaling Properties of the Maxwell Equations

20

Discrete vs. Continuous Frequency Ranges

21

Electrodynamics and Quantum Mechanics Compared

22

Further Reading

24

3 Symmetries and Solid-State Electromagnetism

25

Using Symmetries to Classify Electromagnetic Modes

25

Continuous Translational Symmetry

27

Index guiding

30

Discrete Translational Symmetry

32

Photonic Band Structures

35

Rotational Symmetry and the Irreducible Brillouin Zone

36

Mirror Symmetry and the Separation of Modes

37

Time-Reversal Invariance

39

Bloch-Wave Propagation Velocity

40

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viii
Electrodynamics vs. Quantum Mechanics Again Further Reading

CONTENTS
42 43

4 The Multilayer Film: A One-Dimensional Photonic

Crystal

44

The Multilayer Film

44

The Physical Origin of Photonic Band Gaps

46

The Size of the Band Gap

49

Evanescent Modes in Photonic Band Gaps

52

Off-Axis Propagation

54

Localized Modes at Defects

58

Surface States

60

Omnidirectional Multilayer Mirrrors

61

Further Reading

65

5 Two-Dimensional Photonic Crystals

66

Two-Dimensional Bloch States

66

A Square Lattice of Dielectric Columns

68

A Square Lattice of Dielectric Veins

72

A Complete Band Gap for All Polarizations

74

Out-of-Plane Propagation

75

Localization of Light by Point Defects

78

Point defects in a larger gap

83

Linear Defects and Waveguides

86

Surface States

89

Further Reading

92

6 Three-Dimensional Photonic Crystals

94

Three-Dimensional Lattices

94

Crystals with Complete Band Gaps

96

Spheres in a diamond lattice

97

Yablonovite

99

The woodpile crystal

100

Inverse opals

103

A stack of two-dimensional crystals

105

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CONTENTS

ix

Localization at a Point Defect

109

Experimental defect modes in Yablonovite

113

Localization at a Linear Defect

114

Localization at the Surface

116

Further Reading

121

7 Periodic Dielectric Waveguides

122

Overview

122

A Two-Dimensional Model

123

Periodic Dielectric Waveguides in Three Dimensions

127

Symmetry and Polarization

127

Point Defects in Periodic Dielectric Waveguides

130

Quality Factors of Lossy Cavities

131

Further Reading

134

8 Photonic-Crystal Slabs

135

Rod and Hole Slabs

135

Polarization and Slab Thickness

137

Linear Defects in Slabs

139

Reduced-radius rods

139

Removed holes

142

Substrates, dispersion, and loss

144

Point Defects in Slabs

147

Mechanisms for High Q with Incomplete Gaps

149

Delocalization

149

Cancellation

151

Further Reading

155

9 Photonic-Crystal Fibers

156

Mechanisms of Conﬁnement

156

Index-Guiding Photonic-Crystal Fibers

158

Endlessly single-mode ﬁbers

161

The scalar limit and LP modes

163

Enhancement of nonlinear effects

166

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x
Band-Gap Guidance in Holey Fibers Origin of the band gap in holey ﬁbres Guided modes in a hollow core
Bragg Fibers Analysis of cylindrical ﬁbers Band gaps of Bragg ﬁbers Guided modes of Bragg ﬁbers
Losses in Hollow-Core Fibers Cladding losses Inter-modal coupling
Further Reading

CONTENTS
169 169 172 175 176 178 180 182 183 187 189

10 Designing Photonic Crystals for Applications

190

Overview

190

A Mirror, a Waveguide, and a Cavity

191

Designing a mirror

191

Designing a waveguide

193

Designing a cavity

195

A Narrow-Band Filter

196

Temporal Coupled-Mode Theory

198

The temporal coupled-mode equations

199

The ﬁlter transmission

202

A Waveguide Bend

203

A Waveguide Splitter

206

A Three-Dimensional Filter with Losses

208

Resonant Absorption and Radiation

212

Nonlinear Filters and Bistability

214

Some Other Possibilities

218

Reﬂection, Refraction, and Diffraction

221

Reﬂection

222

Refraction and isofrequency diagrams

223

Unusual refraction and diffraction effects

225

Further Reading

228

Epilogue

228

A Comparisons with Quantum Mechanics

229

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CONTENTS

xi

B The Reciprocal Lattice and the Brillouin Zone

233

The Reciprocal Lattice

233

Constructing the Reciprocal Lattice Vectors

234

The Brillouin Zone

235

Two-Dimensional Lattices

236

Three-Dimensional Lattices

238

Miller Indices

239

C Atlas of Band Gaps

242

A Guided Tour of Two-Dimensional Gaps

243

Three-Dimensional Gaps

251

D Computational Photonics

252

Generalities

253

Frequency-Domain Eigenproblems

255

Frequency-Domain Responses

258

Time-Domain Simulations

259

A Planewave Eigensolver

261

Further Reading and Free Software

263

Bibliography

265

Index

283

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PREFACE TO THE SECOND EDITION
We were delighted with the positive response to the ﬁrst edition of this book. There is, naturally, always some sense of trepidation when one writes the ﬁrst text book at the birth of a new ﬁeld. One dearly hopes the ﬁeld will continue to grow and blossom, but then again, will the subject matter of the book quickly become obsolete? To attempt to alleviate the latter, we made a conscious effort in the ﬁrst edition to focus on the fundamental concepts and building blocks of this new ﬁeld and leave out any speculative areas. Given the continuing interest in the ﬁrst edition, even after a decade of exponential growth of the ﬁeld, it appears that we may have succeeded in this regard. Of course, with great growth come many new phenomena and a deeper understanding of old phenomena. We felt, therefore, that the time was now ripe for an updated and expanded second edition.
As before, we strove in this edition to include new concepts, phenomena and descriptions that are well understood—material that would stand the test of advancements over time.
Many of the original chapters are expanded with new sections, in addition to innumerable revisions to the old sections. For example, chapter 2 now contains a section introducing the useful technique of perturbation analysis and a section on understanding the subtle differences between discrete and continuous frequency ranges. Chapter 3 includes a section describing the basics of index guiding and a section on how to understand the Bloch-wave propagation velocity. Chapter 4 includes a section on how to best quantify the band gap of a photonic crystal and a section describing the novel phenomenon of omnidirectional reﬂectivity in multilayer ﬁlm systems. Chapter 5 now contains an expanded section on point defects and a section on linear defects and waveguides. Chapter 6 was revised considerably to focus on many new aspects of 3D photonic crystal structures, including the photonic structure of several well known geometries. Chapters 7 through 9 are all new, describing hybrid photonic-crystal structures consisting, respectively, of 1D-periodic dielectric waveguides, 2D-periodic photonic-crystal slabs, and photonic-crystal ﬁbers. The ﬁnal chapter, chapter 10 (chapter 7 in the ﬁrst edition), is again focused on designing photonic crystals for applications, but now contains many more examples. This chapter has also been expanded to include an introduction and practical guide to temporal coupled-mode theory. This is a very simple, convenient, yet powerful analytical technique for understanding and predicting the behavior of many types of photonic devices.
Two of the original appendices have also been considerably expanded. Appendix C now includes plots of gap size and optimal parameters vs. index contrast for both 2D and 3D photonic crystals. Appendix D now provides a completely new description of computational photonics, surveying computations in both the frequency and time domains.
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xiv

PREFACE TO THE SECOND EDITION

The second edition also includes two other major changes. The ﬁrst is a change to SI units. Admittedly, this affects only some of the equations in chapters 2 and 3; the “master equation” remains unaltered. The second change is to a new color table for plotting the electric and magnetic ﬁelds. We hope the reader will agree that the new color table is a signiﬁcant improvement over the old color table, providing a much cleaner and clearer description of the localization and signdependence of the ﬁelds.
In preparing the second edition, we should like to express our sincere gratitude to Margaret O’Meara, the administrative assistant of the Condensed Matter Theory Group at MIT, for all the time and effort she unselﬁshly provided. We should also like to give a big Thank You! to our editor Ingrid Gnerlich for her patience and understanding when deadlines were not met and for her remarkable good will with all aspects of the process.
We are also very grateful to many colleagues: Eli Yablonovitch, David Norris, Marko Loncˇar, Shawn Lin, Leslie Kolodziejski, Karl Koch, and Kiyoshi Asakawa, for providing us with illustrations of their original work, and Yoel Fink, Shanhui Fan, Peter Bienstman, Mihai Ibanescu, Michelle Povinelli, Marin Soljacic, Maksim Skorobogatiy, Lionel Kimerling, Lefteris Lidorikis, K. C. Huang, Jerry Chen, Hermann Haus, Henry Smith, Evan Reed, Erich Ippen, Edwin Thomas, David Roundy, David Chan, Chiyan Luo, Attila Mekis, Aristos Karalis, Ardavan Farjadpour, and Alejandro Rodriguez, for numerous collaborations.
Cambridge, Massachusetts, 2006

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PREFACE TO THE FIRST EDITION
It is always difﬁcult to write a book about a topic that is still a subject of active research. Part of the challenge lies in translating research papers directly into a text. Without the beneﬁt of decades of classroom instruction, there is no existing body of pedagogical arguments and exercises to draw from.
Even more challenging is the task of deciding which material to include. Who knows which approaches will withstand the test of time? It is impossible to know, so in this text we have tried to include only those subjects of the ﬁeld which we consider most likely to be timeless. That is, we present the fundamentals and the proven results, hoping that afterwards the reader will be prepared to read and understand the current literature. Certainly there is much to add to this material as the research continues, but we have tried to take care that nothing need be subtracted. Of course this has come at the expense of leaving out new and exciting results which are a bit more speculative.
If we have succeeded in these tasks, it is only because of the assistance of dozens of colleagues and friends. In particular, we have beneﬁted from collaborations with Oscar Alerhand, G. Arjavalingam, Karl Brommer, Shanhui Fan, Ilya Kurland, Andrew Rappe, Bill Robertson, and Eli Yablonovitch. We also thank Paul Gourley and Pierre Villeneuve for their contributions to this book. In addition, we gratefully thank Tomas Arias and Kyeongjae Cho for helpful insights and productive conversations. Finally, we would like to acknowledge the partial support of the Ofﬁce of Naval Research and the Army Research Ofﬁce while this manuscript was being prepared.
Cambridge, Massachusetts, 1995
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Photonic Crystals
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1
Introduction
Controlling the Properties of Materials
Many of the true breakthroughs in our technology have resulted from a deeper understanding of the properties of materials. The rise of our ancestors from the Stone Age through the Iron Age is largely a story of humanity’s increasing recognition of the utility of natural materials. Prehistoric people fashioned tools based on their knowledge of the durability of stone and the hardness of iron. In each case, humankind learned to extract a material from the Earth whose ﬁxed properties proved useful.
Eventually, early engineers learned to do more than just take what the Earth provides in raw form. By tinkering with existing materials, they produced substances with even more desirable properties, from the luster of early bronze alloys to the reliability of modern steel and concrete. Today we boast a collection of wholly artiﬁcial materials with a tremendous range of mechanical properties, thanks to advances in metallurgy, ceramics, and plastics.
In this century, our control over materials has spread to include their electrical properties. Advances in semiconductor physics have allowed us to tailor the conducting properties of certain materials, thereby initiating the transistor revolution in electronics. It is hard to overstate the impact that the advances in these ﬁelds have had on our society. With new alloys and ceramics, scientists have invented high-temperature superconductors and other exotic materials that may form the basis of future technologies.
In the last few decades, a new frontier has opened up. The goal in this case is to control the optical properties of materials. An enormous range of technological developments would become possible if we could engineer materials that respond to light waves over a desired range of frequencies by perfectly reﬂecting them, or allowing them to propagate only in certain directions, or conﬁning them within a speciﬁed volume. Already, ﬁber-optic cables, which simply guide light, have revolutionized the telecommunications industry. Laser engineering, high-speed computing, and spectroscopy are just a few of the ﬁelds next in line to reap the
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2

CHAPTER 1

beneﬁts from the advances in optical materials. It is with these goals in mind that this book is written.

Photonic Crystals
What sort of material can afford us complete control over light propagation? To answer this question, we rely on an analogy with our successful electronic materials. A crystal is a periodic arrangement of atoms or molecules. The pattern with which the atoms or molecules are repeated in space is the crystal lattice. The crystal presents a periodic potential to an electron propagating through it, and both the constituents of the crystal and the geometry of the lattice dictate the conduction properties of the crystal.
The theory of quantum mechanics in a periodic potential explains what was once a great mystery of physics: In a conducting crystal, why do electrons propagate like a diffuse gas of free particles? How do they avoid scattering from the constituents of the crystal lattice? The answer is that electrons propagate as waves, and waves that meet certain criteria can travel through a periodic potential without scattering (although they will be scattered by defects and impurities).
Importantly, however, the lattice can also prohibit the propagation of certain waves. There may be gaps in the energy band structure of the crystal, meaning that electrons are forbidden to propagate with certain energies in certain directions. If the lattice potential is strong enough, the gap can extend to cover all possible propagation directions, resulting in a complete band gap. For example, a semiconductor has a complete band gap between the valence and conduction energy bands.
The optical analogue is the photonic crystal, in which the atoms or molecules are replaced by macroscopic media with differing dielectric constants, and the periodic potential is replaced by a periodic dielectric function (or, equivalently, a periodic index of refraction). If the dielectric constants of the materials in the crystal are sufﬁciently different, and if the absorption of light by the materials is minimal, then the refractions and reﬂections of light from all of the various interfaces can produce many of the same phenomena for photons (light modes) that the atomic potential produces for electrons. One solution to the problem of optical control and manipulation is thus a photonic crystal, a low-loss periodic dielectric medium. In particular, we can design and construct photonic crystals with photonic band gaps, preventing light from propagating in certain directions with speciﬁed frequencies (i.e., a certain range of wavelengths, or “colors,” of light). We will also see that a photonic crystal can allow propagation in anomalous and useful ways.
To develop this concept further, consider how metallic waveguides and cavities relate to photonic crystals. Metallic waveguides and cavities are widely used to control microwave propagation. The walls of a metallic cavity prohibit the propagation of electromagnetic waves with frequencies below a certain threshold frequency, and a metallic waveguide allows propagation only along its axis. It would be extremely useful to have these same capabilities for electromagnetic

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INTRODUCTION

3

waves with frequencies outside the microwave regime, such as visible light. However, visible light energy is quickly dissipated within metallic components, which makes this method of optical control impossible to generalize. Photonic crystals allow the useful properties of cavities and waveguides to be generalized and scaled to encompass a wider range of frequencies. We may construct a photonic crystal of a given geometry with millimeter dimensions for microwave control, or with micron dimensions for infrared control.
Another widely used optical device is a multilayer dielectric mirror, such as a quarter-wave stack, consisting of alternating layers of material with different dielectric constants. Light of the proper wavelength, when incident on such a layered material, is completely reﬂected. The reason is that the light wave is partially reﬂected at each layer interface and, if the spacing is periodic, the multiple reﬂections of the incident wave interfere destructively to eliminate the forward-propagating wave. This well-known phenomenon, ﬁrst explained by Lord Rayleigh in 1887, is the basis of many devices, including dielectric mirrors, dielectric Fabry–Perot ﬁlters, and distributed feedback lasers. All contain low-loss dielectrics that are periodic in one dimension, and by our deﬁnition they are onedimensional photonic crystals. Even these simplest of photonic crystals can have surprising properties. We will see that layered media can be designed to reﬂect light that is incident from any angle, with any polarization—an omnidirectional reﬂector—despite the common intuition that reﬂection can be arranged only for near-normal incidence.
If, for some frequency range, a photonic crystal prohibits the propagation of electromagnetic waves of any polarization traveling in any direction from any source, we say that the crystal has a complete photonic band gap. A crystal with a complete band gap will obviously be an omnidirectional reﬂector, but the converse is not necessarily true. As we shall see, the layered dielectric medium mentioned above, which cannot have a complete gap (because material interfaces occur only along one axis), can still be designed to exhibit omnidirectional reﬂection—but only for light sources far from the crystal. Usually, in order to create a complete photonic band gap, one must arrange for the dielectric lattice to be periodic along three axes, constituting a three-dimensional photonic crystal. However, there are exceptions. A small amount of disorder in an otherwise periodic medium will not destroy a band gap (Fan et al., 1995b; Rodriguez et al., 2005), and even a highly disordered medium can prevent propagation in a useful way through the mechanism of Anderson localization (John, 1984). Another interesting nonperiodic class of materials that can have complete photonic band gaps are quasi-crystalline structures (Chan et al., 1998).

An Overview of the Text
Our goal in writing this textbook was to provide a comprehensive description of the propagation of light in photonic crystals. We discuss the properties of photonic crystals of gradually increasing complexity, beginning with the simplest

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1-D

2-D

CHAPTER 1 3-D

periodic in one direction

periodic in two directions

periodic in three directions

Figure 1: Simple examples of one-, two-, and three-dimensional photonic crystals. The different colors represent materials with different dielectric constants. The deﬁning feature of a photonic crystal is the periodicity of dielectric material along one or more axes.

case of one-dimensional crystals, and proceeding to the more intricate and useful properties of two- and three-dimensional systems (see ﬁgure 1). After equipping ourselves with the appropriate theoretical tools, we attempt to convey a useful intuition about which structures yield what properties, and why?
This textbook is designed for a broad audience. The only prerequisites are a familiarity with the macroscopic Maxwell equations and the notion of harmonic modes (which are often referred to by other names, such as eigenmodes, normal modes, and Fourier modes). From these building blocks, we develop all of the needed mathematical and physical tools. We hope that interested undergraduates will ﬁnd the text approachable, and that professional researchers will ﬁnd our heuristics and results to be useful in designing photonic crystals for their own applications.
Readers who are familiar with quantum mechanics and solid-state physics are at some advantage, because our formalism owes a great deal to the techniques and nomenclature of those ﬁelds. Appendix A explores this analogy in detail. Photonic crystals are a marriage of solid-state physics and electromagnetism. Crystal structures are citizens of solid-state physics, but in photonic crystals the electrons are replaced by electromagnetic waves. Accordingly, we present the basic concepts of both subjects before launching into an analysis of photonic crystals. In chapter 2, we discuss the macroscopic Maxwell equations as they apply to dielectric media. These equations are cast as a single Hermitian differential equation, a form in which many useful properties become easy to demonstrate: the orthogonality of modes, the electromagnetic variational theorem, and the scaling laws of dielectric systems.
Chapter 3 presents some basic concepts of solid-state physics and symmetry theory as they apply to photonic crystals. It is common to apply symmetry arguments to understand the propagation of electrons in a periodic crystal potential. Similar arguments also apply to the case of light propagating in a photonic crystal. We examine the consequences of translational, rotational,

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INTRODUCTION

5

mirror-reﬂection, inversion, and time-reversal symmetries in photonic crystals, while introducing some terminology from solid-state physics.
To develop the basic notions underlying photonic crystals, we begin by reviewing the properties of one-dimensional photonic crystals. In chapter 4, we will see that one-dimensional systems can exhibit three important phenomena: photonic band gaps, localized modes, and surface states. Because the index contrast is only along one direction, the band gaps and the bound states are limited to that direction. Nevertheless, this simple and traditional system illustrates most of the physical features of the more complex two- and three-dimensional photonic crystals, and can even exhibit omnidirectional reﬂection.
In chapter 5, we discuss the properties of two-dimensional photonic crystals, which are periodic in two directions and homogeneous in the third. These systems can have a photonic band gap in the plane of periodicity. By analyzing ﬁeld patterns of some electromagnetic modes in different crystals, we gain insight into the nature of band gaps in complex periodic media. We will see that defects in such two-dimensional crystals can localize modes in the plane, and that the faces of the crystal can support surface states.
Chapter 6 addresses three-dimensional photonic crystals, which are periodic along three axes. It is a remarkable fact that such a system can have a complete photonic band gap, so that no propagating states are allowed in any direction in the crystal. The discovery of particular dielectric structures that possess a complete photonic band gap was one of the most important achievements in this ﬁeld. These crystals are sufﬁciently complex to allow localization of light at point defects and propagation along linear defects.
Chapters 7 and 8 consider hybrid structures that combine band gaps in one or two directions with index-guiding (a generalization of total internal reﬂection) in the other directions. Such structures approximate the three-dimensional control over light that is afforded by a complete three-dimensional band gap, but at the same time are much easier to fabricate. Chapter 9 describes a different kind of incomplete-gap structure, photonic-crystal ﬁbers, which use band gaps or indexguiding from one- or two-dimensional periodicity to guide light along an optical ﬁber.
Finally, in chapter 10, we use the tools and ideas that were introduced in previous chapters to design some simple optical components. Speciﬁcally, we see how resonant cavities and waveguides can be combined to form ﬁlters, bends, splitters, nonlinear “transistors,” and other devices. In doing so, we develop a powerful analytical framework known as temporal coupled-mode theory, which allows us to easily predict the behavior of such combinations. We also examine the reﬂection and refraction phenomena that occur when light strikes an interface of a photonic crystal. These examples not only illustrate the device applications of photonic crystals, but also provide a brief review of the material contained elsewhere in the text.
We should also mention the appendices, which provide a brief overview of the reciprocal-lattice concept from solid-state physics, survey the gaps that arise in various two- and three-dimensional photonic crystals, and outline the numerical methods that are available for computer simulations of photonic structures.

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2
Electromagnetism in Mixed Dielectric Media

IN ORDER TO STUDY the propagation of light in a photonic crystal, we begin with
the Maxwell equations. After specializing to the case of a mixed dielectric medium, we cast the Maxwell equations as a linear Hermitian eigenvalue problem. This brings the electromagnetic problem into a close analogy with the Schrödinger equation, and allows us to take advantage of some well-established results from quantum mechanics, such as the orthogonality of modes, the variational theorem, and perturbation theory. One way in which the electromagnetic case differs from the quantum-mechanical case is that photonic crystals do not generally have a fundamental scale, in either the spatial coordinate or in the potential strength (the dielectric constant). This makes photonic crystals scalable in a way that traditional crystals are not, as we will see later in this chapter.

The Macroscopic Maxwell Equations

All of macroscopic electromagnetism, including the propagation of light in a photonic crystal, is governed by the four macroscopic Maxwell equations. In SI units,1 they are

∇ ·B = 0

∇

×

E

+

∂B ∂t

=

0

(1)

∇ ·D = ρ

∇

×H

−

∂D ∂t

=

J

1 The ﬁrst edition used cgs units, in which constants such as ε0 and µ0 are replaced by factors of 4π and c here and there, but the choice of units is mostly irrelevant in the end. They do not effect the form of our “master” equation (7). Moreover, we will express all quantities of interest—frequencies, geometries, gap sizes, and so on—as dimensionless ratios; see also the section The Size of the Band Gap of chapter 4.

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ELECTROMAGNETISM IN MIXED DIELECTRIC MEDIA

7

ε 6 ε 3 ε 1

ε 5

ε 2

ε 3

ε 6 ε 4

Figure 1: A composite of macroscopic regions of homogeneous dielectric media. There are
no charges or currents. In general, ε(r) in equation (1) can have any prescribed spatial
dependence, but our attention will focus on materials with patches of homogeneous dielectric, such as the one illustrated here.

where (respectively) E and H are the macroscopic electric and magnetic ﬁelds, D and B are the displacement and magnetic induction ﬁelds, and ρ and J are the free charge and current densities. An excellent derivation of these equations from their microscopic counterparts is given in Jackson (1998).
We will restrict ourselves to propagation within a mixed dielectric medium, a composite of regions of homogeneous dielectric material as a function of the (cartesian) position vector r, in which the structure does not vary with time, and there are no free charges or currents. This composite need not be periodic, as illustrated in ﬁgure 1. With this type of medium in mind, in which light propagates but there are no sources of light, we can set ρ = 0 and J = 0.
Next we relate D to E and B to H with the constitutive relations appropriate for our problem. Quite generally, the components Di of the displacement ﬁeld D are related to the components Ei of the electric ﬁeld E via a power series, as in Bloembergen (1965):

Di/ε0 = εijEj + χijk EjEk + O(E3),

(2)

j

j,k

where ε0 ≈ 8.854 × 10−12 Farad/m is the vacuum permittivity. However, for many dielectric materials, it is reasonable to use the following approximations. First,
we assume the ﬁeld strengths are small enough so that we are in the linear regime, so that χijk (and all higher-order terms) can be neglected. Second, we assume the material is macroscopic and isotropic,2 so that E(r, ω) and D(r, ω) are related by ε0 multipled by a scalar dielectric function ε(r, ω), also called

2 It is straightforward to generalize this formalism to anisotropic media in which D and E are related by a Hermitian dielectric tensor ε0εij.

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CHAPTER 2

the relative permittivity.3 Third, we ignore any explicit frequency dependence
(material dispersion) of the dielectric constant. Instead, we simply choose the
value of the dielectric constant appropriate to the frequency range of the physical
system we are considering. Fourth, we focus primarily on transparent materials, which means we can treat ε(r) as purely real4 and positive.5
Assuming these four approximations to be valid, we have D(r) = ε0ε(r)E(r). A similar equation relates B(r) = µ0µ(r)H(r) (where µ0 = 4π × 10−7 Henry/m is the vacuum permeability), but for most dielectric materials of interest the relative magnetic permeability µ(r) is very close to unity and we may set B = µ0H for simplicity.6 In that case, ε is the square of the refractive index n that may
bne=fa√mεiµli.a) r from Snell’s law and other formulas of classical optics. (In general,
With all of these assumptions in place, the Maxwell equations (1) become

∇ · H(r, t) = 0

∇

×

E(r,

t)

+

µ0

∂H(r, ∂t

t)

=

0

(3)

∇ · [ε(r)E(r, t)] = 0

∇

×

H(r,

t)

−

ε

0ε(r)

∂E(r, ∂t

t)

=

0.

The reader might reasonably wonder whether we are missing out on interesting physical phenomena by restricting ourselves to linear and lossless materials. We certainly are, and we will return to this question in the section The Effect of Small Perturbations and in chapter 10. Nevertheless, it is a remarkable fact that many interesting and useful properties arise from the elementary case of linear, lossless materials. In addition, the theory of these materials is much simpler to understand and is practically exact, making it an excellent foundation on which to build the theory of more complex media. For these reasons, we will be concerned with linear and lossless materials for most of this text.
In general, both E and H are complicated functions of both time and space. Because the Maxwell equations are linear, however, we can separate the time dependence from the spatial dependence by expanding the ﬁelds into a set of harmonic modes. In this and the following sections we will examine the restrictions that the Maxwell equations impose on a ﬁeld pattern that varies

3 Some authors use εr (or K, or k, or κ) for the relative permittivity and ε for the permittivity ε0εr. Here, we adopt the common convention of dropping the r subscript, since we work only with the dimensionless εr.
4 Complex dielectric constants are used to account for absorption, as in Jackson (1998). Later, in the
section The Effect of Small Perturbations, we will show how to include small absorption losses. 5 A negative dielectric constant is indeed a useful description of some materials, such as metals. The
limit ε → −∞ corresponds to a perfect metal into which light cannot penetrate. Combinations of
metals and transparent dielectrics can also be used to create photonic crystals (for some early work
in this area, see e.g., McGurn and Maradudin, 1993; Kuzmiak et al., 1994; Sigalas et al., 1995; Brown
and McMahon, 1995; Fan et al., 1995c; Sievenpiper et al., 1996), a topic we return to in the subsection
The scalar limit and LP modes of chapter 9. 6 It is straightforward to include µ = 1; see footnote 17 on page 17.

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sinusoidally (harmonically) with time. This is no great limitation, since we know by Fourier analysis that we can build any solution with an appropriate combination of these harmonic modes. Often we will refer to them simply as modes or states of the system.
For mathematical convenience, we employ the standard trick of using a complex-valued ﬁeld and remembering to take the real part to obtain the physical ﬁelds. This allows us to write a harmonic mode as a spatial pattern (or “mode proﬁle”) times a complex exponential:
H(r, t) = H(r)e−iωt (4)
E(r, t) = E(r)e−iωt.

To ﬁnd the equations governing the mode proﬁles for a given frequency, we insert the above equations into (3). The two divergence equations give the conditions

∇ · H(r) = 0, ∇ · [ε(r)E(r)] = 0,

(5)

which have a simple physical interpretation: there are no point sources or sinks of displacement and magnetic ﬁelds in the medium. Equivalently, the ﬁeld conﬁgurations are built up of electromagnetic waves that are transverse. That is, if we have a plane wave H(r) = a exp(ik · r), for some wave vector k, equation (5) requires that a · k=0. We can now focus our attention only on the other two of the Maxwell equations as long as we are always careful to enforce this transversality requirement.
The two curl equations relate E(r) to H(r):

∇ × E(r) − iωµ0H(r) = 0 (6)
∇ × H(r) + iωε0ε(r)E(r) = 0.

We can decouple these equations in the following way. Divide the bottom equation

of (6) by ε(r), and then take the curl. Then use the ﬁrst equation to eliminate E(r).

Morever, light, c =

t1h/e√cεo0nµs0t.aTnhtse

ε0 and µ0 can be combined to yield the result is an equation entirely in H(r):

vacuum

speed

of

∇×

1 ε(r)

∇

×

H(r)

=

ω

2
H(r).

c

(7)

This is the master equation. Together with the divergence equation (5), it tells us everything we need to know about H(r). Our strategy will be as follows: for a given structure ε(r), solve the master equation to ﬁnd the modes H(r) and the
corresponding frequencies, subject to the transversality requirement. Then use

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CHAPTER 2

the second equation of (6) to recover E(r):

E(r)

=

i ωε0ε(r)

∇

×

H(r).

(8)

Using this procedure guarantees that E satisﬁes the transversality requirement ∇ · εE = 0, because the divergence of a curl is always zero. Thus, we need only impose one transversality constraint, rather than two. The reason why we chose to formulate the problem in terms of H(r) and not E(r) is merely one of mathematical convenience, as will be discussed in the section Magnetic vs. Electric Fields. For now, we note that we can also ﬁnd H from E via the ﬁrst equation of (6):

H(r)

=

−

i ωµ0

∇

×

E(r).

(9)

Electromagnetism as an Eigenvalue Problem

As discussed in the previous section, the heart of the Maxwell equations for a harmonic mode in a mixed dielectric medium is a differential equation for H(r),
given by equation (7). The content of the equation is this: perform a series of operations on a function H(r), and if H(r) is really an allowable electromagnetic mode, the result will be a constant times the original function H(r). This situation
arises often in mathematical physics, and is called an eigenvalue problem. If
the result of an operation on a function is just the function itself, multiplied by some constant, then the function is called an eigenfunction or eigenvector7 of that
operator, and the multiplicative constant is called the eigenvalue. In this case, we identify the left side of the master equation as an operator Θˆ
acting on H(r) to make it look more like a traditional eigenvalue problem:

Θˆ H(r) =

ω

2
H(r).

(10)

c

We have identiﬁed Θˆ as the differential operator that takes the curl, then divides by ε(r), and then takes the curl again:

Θˆ H(r)

∇×

1 ε(r)

∇

×

H(r)

.

(11)

The eigenvectors H(r) are the spatial patterns of the harmonic modes, and the eigenvalues (ω/c)2 are proportional to the squared frequencies of those modes. An important thing to notice is that the operator Θˆ is a linear operator. That is,

7 Instead of eigenvector, physicists tend to stick eigen in front of any natural name for the solution. Hence, we also use terms like eigenﬁeld, eigenmode, eigenstate, and so on.

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ELECTROMAGNETISM IN MIXED DIELECTRIC MEDIA

11

any linear combination of solutions is itself a solution; if H1(r) and H2(r) are both solutions of (10) with the same frequency ω, then so is αH1(r) + βH2(r), where α and β are constants. For example, given a certain mode proﬁle, we can construct another legitimate mode proﬁle with the same frequency by simply doubling the ﬁeld strength everywhere (α = 2, β = 0). For this reason we consider two ﬁeld patterns that differ only by an overall multiplier to be the same mode.
Our operator notation is reminiscent of quantum mechanics, in which we obtain an eigenvalue equation by operating on the wave function with the Hamiltonian. A reader familiar with quantum mechanics might recall some key properties of the eigenfunctions of the Hamiltonian: they have real eigenvalues, they are orthogonal, they can be obtained by a variational principle, and they may be catalogued by their symmetry properties (see, for example Shankar, 1982).
All of these same useful properties hold for our formulation of electromagnetism. In both cases, the properties rely on the fact that the main operator is a special type of linear operator known as a Hermitian operator. In the coming sections we will develop these properties one by one. We conclude this section by showing what it means for an operator to be Hermitian. First, in analogy with the inner product of two wave functions, we deﬁne the inner product of two vector ﬁelds F(r) and G(r) as

(F, G)

d3r F∗(r) · G(r),

(12)

where “∗” denotes complex conjugation. Note that a simple consequence of this deﬁnition is that (F, G) = (G, F)∗ for any F and G. Also note that (F, F) is always real and nonnegative, even if F itself is complex. In fact, if F(r) is a harmonic mode of our electromagnetic system, we can always set (F, F) = 1 by using our freedom to scale any mode by an overall multiplier.8 Given F (r) with (F , F ) = 1,
create

F(r) =

F (r) .

(13)

(F , F )

From our previous discussion, F(r) is really the same mode as F (r), since it
differs only by an overall multiplier, but now (as the reader can easily verify) we have (F, F) = 1. We say that F(r) has been normalized. Normalized modes are
very useful in formal arguments. If, however, one is interested in the physical
energy of the ﬁeld and not just its spatial proﬁle, the overall multiplier is important.9
Next, we say that an operator Ξˆ is Hermitian if (F, Ξˆ G) = (Ξˆ F, G) for any vector ﬁelds F(r) and G(r). That is, it does not matter which function is operated upon
before taking the inner product. Clearly, not all operators are Hermitian. To show

8 The trivial solution F = 0 is not considered to be a proper eigenfunction. 9 This distinction is discussed again after equation (24).

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CHAPTER 2

that Θˆ is Hermitian,10 we perform an integration by parts11 twice:

(F, Θˆ G) =

d3r F∗ · ∇ ×

1 ε

∇

×

G

=

d3r

(∇

×

F)∗

·

1 ε

∇

×

G

(14)

=

d3r

∇×

1 ε

∇

×

F

∗
· G = (Θˆ F, G).

In performing the integrations by parts, we neglected the surface terms that involve the values of the ﬁelds at the boundaries of integration. This is because in all cases of interest, one of two things will be true: either the ﬁelds decay to zero at large distances, or the ﬁelds are periodic in the region of integration. In either case, the surface terms vanish.

General Properties of the Harmonic Modes

Having established that Θˆ is Hermitian, we can now show that the eigenvalues of Θˆ must be real numbers. Suppose H(r) is an eigenvector of Θˆ with eigenvalue (ω/c)2. Take the inner product of the master equation (7) with H(r):

Θˆ H(r) = (ω2/c2)H(r)

=⇒ (H, Θˆ H) = (ω2/c2)(H, H)

(15)

=⇒ (H, Θˆ H)∗ = (ω2/c2)∗(H, H).

Because Θˆ is Hermitian, we know that (H, Θˆ H) = (Θˆ H, H). Additionally, from the deﬁnition of the inner product we know that (H, Ξˆ H) = (Ξˆ H, H)∗ for any operator Ξˆ . Using these two pieces of information, we continue:

(H, Θˆ H)∗ = (ω2/c2)∗(H, H) = (Θˆ H, H) = (ω2/c2)(H, H) (16)
=⇒ (ω2/c2)∗ = (ω2/c2).
It follows that ω2 = (ω2)∗, or that ω2 is real. By a different argument, we can also show that ω2 is always nonnegative for ε >0. Set F=G=H in the middle equation

10 The property that Θˆ is Hermitian is closely related to the Lorentz reciprocity theorem, as described
in the section Frequency-Domain Responses of appendix D. 11 In particular, we use the vector identity that ∇ · (F × G) = (∇ × F) · G − F · (∇ × G). Integrating
both sides and applying the divergence theorem, we ﬁnd that F · (∇ × G) = (∇ × F) · G plus a surface term, from the integral of ∇ · (F × G), that vanishes as described above.

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ELECTROMAGNETISM IN MIXED DIELECTRIC MEDIA

13

of (14), to obtain

(H, H) ω 2 = (H, Θˆ H) = c

d3r

1 ε

|∇

×

H|2.

(17)

Since ε(r) > 0 everywhere, the integrand on the right-hand side is everywhere nonnegative. The operator Θˆ is said to be positive semi-deﬁnite. Therefore all of the eigenvalues ω2 are nonnegative, and ω is real.
In addition, the Hermiticity of Θˆ forces any two harmonic modes H1(r) and H2(r) with different frequencies ω1 and ω2 to have an inner product of zero. Consider two normalized modes, H1(r) and H2(r), with frequencies ω1 and ω2:

ω12(H2, H1) = c2(H2, Θˆ H1) = c2(Θˆ H2, H1) = ω22(H2, H1) (18)
=⇒ (ω12 − ω22)(H2, H1) = 0.
If ω1=ω2, then we must have (H1, H2)=0 and we say H1 and H2 are orthogonal modes. If two harmonic modes have equal frequencies ω1=ω2, then we say they are degenerate and they are not necessarily orthogonal. For two modes to be degenerate requires what seems, on the face of it, to be an astonishing coincidence: two different ﬁeld patterns happen to have precisely the same frequency. Usually there is a symmetry that is responsible for the “coincidence”. For example, if the dielectric conﬁguration is invariant under a 120◦ rotation, modes that differ only by a 120◦ rotation are expected to have the same frequency. Such modes are degenerate and are not necessarily orthogonal.
However, since Θˆ is linear, any linear combination of these degenerate modes is itself a mode with that same frequency. As in quantum mechanics, we can always choose to work with linear combinations that are orthogonal (see, e.g., Merzbacher, 1961). This allows us to say quite generally that different modes are orthogonal, or can be arranged to be orthogonal.
The concept of orthogonality is most easily grasped by considering onedimensional functions. What follows is a brief explanation (not mathematically rigorous, but perhaps useful to the intuition) that may help in understanding the signiﬁcance of orthogonality. For two real one-dimensional functions f (x) and g(x) to be orthogonal means that

( f , g) = f (x)g(x)dx = 0.

(19)

In a sense, the product f g must be negative at least as much as it is positive over
the interval of interest, so that the net integral vanishes. For example, the familiar set of functions fn(x) = sin(nπx/L) are all orthogonal in the interval from x = 0 to x = L. Note that each of these functions has a different number of nodes (locations where fn(x) = 0, not including the end points). In particular, fn has n − 1 nodes. The product of any two different fn is positive as often as it is negative, and the inner product vanishes.

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CHAPTER 2

The extension to a higher number of dimensions is a bit unclear, because the integration is more complicated. But the notion that orthogonal modes of different frequency have different numbers of spatial nodes holds rather generally. In fact, a given harmonic mode will generally contain more nodes than lower-frequency modes. This is analogous to the statement that each vibrational mode of a string with ﬁxed ends contains one more node than the one below it. This will be important for our discussion in chapter 5.

Electromagnetic Energy and the Variational Principle

Although the harmonic modes in a dielectric medium can be quite complicated,
there is a simple way to understand some of their qualitative features. Roughly,
a mode tends to concentrate its electric-ﬁeld energy in regions of high dielectric
constant, while remaining orthogonal to the modes below it in frequency. This
useful but somewhat vague notion can be expressed precisely through the elec-
tromagnetic variational theorem, which is analogous to the variational principle of quantum mechanics. In particular, the smallest eigenvalue ω02/c2, and thus the lowest-frequency mode, corresponds to the ﬁeld pattern that minimizes the
functional:

Uf (H)

(H, Θˆ H) (H, H)

.

(20)

That is, ω02/c2 is the minimum of Uf (H) over all conceivable ﬁeld patterns H (subject to the transversality constraint ∇ · H = 0). The functional Uf is sometimes called the Rayleigh quotient, and appears in a similar variational theorem for any
Hermitian operator. We will refer to Uf as the electromagnetic “energy” functional, in order to stress the analogy with analogous variational theorems in quantum and
classical mechanics that minimize a physical energy.
To verify the claim that Uf is minimized for the lowest-frequency mode, we consider how small variations in H(r) affect the energy functional. Suppose that we perturb the ﬁeld H(r) by adding a small-amplitude function δH(r). What is the resulting small change δUf in the energy functional? It should be zero if the energy functional is really at a minimum, just as the ordinary derivative of a function vanishes at an extremum. To ﬁnd out, we evaluate the energy functional at H + δH and at H, and then compute the difference δUf :

Uf (H + δH)

=

(H + δH, Θˆ H + Θˆ δH) (H + δH, H + δH)

Uf (H)

=

(H, Θˆ H) (H, H)

(21)

δUf (H) Uf (H + δH) − Uf (H)

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Ignoring terms higher than ﬁrst order in δH, we can write δUf in the form δUf ≈ [(δH, G) + (G, δH)]/2, where G is given by

G(H)

=

2 (H, H)

Θˆ H −

(H, Θˆ H) (H, H)

H

.

(22)

This G can be interpreted as the gradient (rate of change) of the functional Uf with respect to H.12 At an extremum, δUf must vanish for all possible shifts δH, which implies that G = 0. This implies that the parenthesized quantity in (22) is zero, which is equivalent to saying that H is an eigenvector of Θˆ . Therefore, Uf is at an extremum if and only if H is a harmonic mode. More careful considerations show that the lowest-ω electromagnetic eigenmode H0 minimizes Uf . The next-lowest-ω eigenmode will minimize Uf within the subspace of functions that are orthogonal to H0, and so on.
In addition to providing a useful characterization of the modes of Θˆ , the
variational theorem is also the source of the heuristic rules about modes that
were alluded to earlier in this section. This is most easily seen after rewriting the
energy functional in terms of E. Beginning with an eigenmode H that minimizes
Uf , we rewrite the numerator of (20) using (11), (8), and (9), and we rewrite the denominator using (17) and (8). The result is:

Uf (H)

=

(∇ × E, ∇ × E) (E, ε(r)E)

(23)

=

d3r |∇ × E(r)|2 d3r ε(r)|E(r)|2 .

From this expression, we can see that the way to minimize Uf is to concentrate the electric ﬁeld E in regions of high dielectric constant ε (thereby maximizing the denominator) and to minimize the amount of spatial oscillations (thereby minimizing the numerator) while remaining orthogonal to lower-frequency modes.13 Although we derived (23) by starting with an eigenmode H and rewriting the (minimized) energy functional in terms of E, it can be shown (using the E eigenproblem of the next section) that (23) is also a valid variational theorem: the lowest-frequency eigenmode is given by the E ﬁeld that minimizes (23), subject to ∇ · εE = 0.
The energy functional must be distinguished from the physical energy stored in the electromagnetic ﬁeld. The time-averaged physical energy can be separated

12 The analogous and perhaps more familiar expression for functions f (x) of a real vector x is δ f ≈ δx · ∇ f = [δx · ∇ f + ∇ f · δx]/2, in terms of the gradient ∇ f . This is the ﬁrst-order change in f when x is perturbed by a small amount δx.
13 The analogous heuristic rule in quantum mechanics is to concentrate the wave function in regions
of low potential energy, while minimizing the kinetic energy and remaining orthogonal to lower-
energy eigenstates.

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CHAPTER 2

into a contribution from the electric ﬁeld, and a contribution from the magnetic ﬁeld:

UE

ε0 4

d3r ε(r)|E(r)|2

(24)

UH

µ0 4

d3r |H(r)|2.

In a harmonic mode, the physical energy is periodically exchanged between the electric and magnetic ﬁelds, and one can show that UE = UH.14 The physical energy and the energy functional are related, but there is an important difference. The energy functional has ﬁelds in both the numerator and the denominator, and is therefore independent of the ﬁeld strength. The physical energy is proportional to the square of the ﬁeld strength. In other words, multiplying the ﬁelds by a constant affects the physical energy but does not affect the energy functional. If we are interested in the physical energy, we must pay attention to the amplitudes of our modes, but if we are interested only in mode proﬁles, we might as well normalize our modes.
Finally, we should also mention the expression for the rate of energy transport, which is given by the Poynting vector, S:

S 1 Re [E∗ × H] ,

(25)

2

where Re denotes the real part. This is the time-average ﬂux of electromagnetic energy in the direction of S, per unit time and per unit area, for a time-harmonic ﬁeld. We also sometimes refer to the component of S in a given direction as the light intensity. The ratio of the energy ﬂux to the energy density deﬁnes the velocity of energy transport, a subject we return to in the section Bloch-Wave Propagation Velocity of chapter 3.15

Magnetic vs. Electric Fields

We digress here to answer a question that commonly arises at this stage: why work with the magnetic ﬁeld instead of the electric ﬁeld? In the previous sections, we reformulated the Maxwell equations as an eigenvalue equation for the harmonic modes of the magnetic ﬁeld H(r). The idea was that for a given frequency, we could solve for H(r) and then determine the E(r) via equation (8). But we could

14 This can be shown from equations (8) and (9) combined with the fact that ∇× is a Hermitian

operator

(see

footnote

11

on

page

12).

Thus,

(µ0H,

H)

=

(µ0H,

−

i ωµ0

∇

×

E)

=

(+ε0ε

i ωε0ε

∇

×

H,

E)=

(ε0εE, E).

15 These equations for energy density and ﬂux are derived in, for example, Jackson (1998) from the

principle of conservation of energy. Note that the energy equations change in the presence of

nonnegligible material dispersion.

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have equally well tried the alternate approach: solve for the electric ﬁeld in (6) and then determine the magnetic ﬁeld with (9). Why didn’t we choose this route?
By pursuing this alternate approach, one ﬁnds the condition on the electric ﬁeld to be

∇ × ∇ × E(r) =

ω

2
ε(r)E(r).

(26)

c

Because there are operators on both sides of this equation, it is referred to as a generalized eigenproblem. It is a simple matter to convert this into an ordinary eigenproblem by dividing (26) by ε, but then the operator is no longer Hermitian. If we stick to the generalized eigenproblem, however, then simple theorems analogous to those of the previous section can be developed because the two operators of the generalized eigenproblem, ∇ × ∇ × and ε(r), are easily shown to be both Hermitian and positive semi-deﬁnite.16 In particular, it can be shown that ω is real, and that two solutions E1 and E2 with different frequencies satisfy an orthogonality relation: (E1, εE2) = 0.
For some analytical calculations, such as the derivation of the variational equation (23) or the perturbation theory discussed in the next section, the E eigenproblem is the most convenient starting point. However, it has one feature that turns out to be undesirable for numerical computation: the transversality constraint ∇ · εE = 0 depends on ε.
We can restore a simpler transversality constraint by using D instead of E, since ∇ · D = 0. Substituting D/ε0ε for E in (26) and dividing both sides by ε (to keep the operator Hermitian) yields

1 ε(r)

∇

×

∇

×

1 ε(r)

D(r)

=

ω c

2

1 ε(r)

D(r).

(27)

This is a perfectly valid formulation of the problem, but it seems unnecessarily complicated because of the three factors of 1/ε (as opposed to the single factor in
the H or E formulations). For these reasons of mathematical convenience, we tend to prefer the H form for numerical calculations.17

The Effect of Small Perturbations

A perfectly linear and lossless material is a very useful idealization, and many real materials are excellent approximations of this idealization. But of course no material is perfectly linear and transparent. We can enlarge the scope of our formalism

16 The ε(r) operator on the right-hand side is actually positive deﬁnite: (E, εE) is strictly positive for

any nonzero E. This is necessary for the generalized eigenproblem to be well behaved.

17 If a relative permeability µ = 1 is included, the E and H eigenproblems take on similar forms. In that

case,

∇

×

1 ε

∇

×

H

=

ω c

2

µH

with

∇

·

µH

=

0,

compared

to

∇

×

1 µ

∇

×

E

=

ω c

2 εE with ∇ · εE = 0.

See, e.g., Sigalas et al. (1997) and Drikis et al. (2004).

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CHAPTER 2

considerably by allowing for small nonlinearities and material absorption, using the well-developed perturbation theory for linear Hermitian eigenproblems. More generally, we may be interested in many types of small deviations from an initial problem. The idea is to begin with the harmonic modes of the idealized problem, and use analytical tools to approximately evaluate the effect of small changes in the dielectric function on the modes and their frequencies. For many realistic problems, the error in this approximation is negligible.
The derivation of perturbation theory for a Hermitian eigenproblem is straightforward and is covered in many texts on quantum mechanics, such as Sakurai (1994). Suppose a Hermitian operator Oˆ is altered by a small amount ∆Oˆ . The resulting eigenvalues and eigenvectors of the perturbed operator can be written as series expansions, in terms that depend on increasing powers of the perturbation strength ∆Oˆ . The resulting equation can be solved order-by-order using only the eigenmodes of the unperturbed operator.
Since we are interested in changes ∆ε(r), the combination of ε(r) with curls in equation (7) is inconvenient, and it turns out to be easier to work with equation (26). By applying the perturbation procedure to equation (26), we obtain a simple formula for the frequency shift ∆ω that results from a small perturbation ∆ε of the dielectric function:

∆ω = − ω 2

d3r ∆ε(r)|E(r)|2 d3r ε(r)|E(r)|2

+

O(∆ε2).

(28)

In this equation, ω and E are the frequency and the mode proﬁle for the perfectly linear and lossless (unperturbed) dielectric function ε. The error in this approximation is proportional to the square of ∆ε and can be neglected in many practical cases, for which |∆ε|/ε is 1% or smaller.
Although we refer the reader to other texts for a rigorous derivation of equation (28), we point out an intu√itive interpretation. Consider the case of a material with a refractive index n = ε, in which the index is perturbed in some regions by an amount ∆n. The volume integral in the numerator of equation (28) has nonzero contributions only from the perturbed regions. Writing ∆ε ≈ ε · 2∆n/n, and supposing that ∆n/n is the same in all the perturbed regions (and can therefore be brought outside the integral), we obtain

∆ω ω

≈

− ∆n n

·

(fraction

of

ε|E|2 in the perturbed regions ).

(29)

We see that the fractional change in frequency is equal to the fractional change in index multiplied by the fraction of the electric-ﬁeld energy inside the perturbed regions. The minus sign appears because an increase in the refractive index lowers the mode frequencies, as can be understood from the variational equation (23).
A small absorption loss can be represented by a small imaginary part of the dielectric function. This does not present any obstacle to the perturbation theory, which requires only that the unperturbed problem be Hermitian; the perturbation

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can be non-Hermitian. Thus, a small imaginary ∆ε = iδ leads to a small imaginary ∆ω = −iγ/2, where γ = ω |E|2δ/ ε|E|2. This corresponds to a ﬁeld that is exponentially decaying in time as e−γt/2. The factor γ is the decay rate for the energy of the mode. It is also possible to consider a gain medium, in which an external energy source is pumping atomic or molecular excited states, by reversing the sign of the imaginary ∆ε. The corresponding modes experience exponential growth, although in any real system the growth must eventually halt at some ﬁnite value.
If a material is only weakly nonlinear, then there will be a small shift ∆ε in the dielectric function that is proportional to either the amplitude of the ﬁeld or its intensity (depending on the material). Perturbation theory is nearly exact for many problems with optical nonlinearities because the maximum changes in the refractive index are typically much less than 1%. Despite this small perturbation strength, the consequences can be profound and fascinating if the perturbations are allowed to accumulate for a long time. A full appraisal of the riches of nonlinear systems is generally beyond the scope of this book, although we will examine restricted examples in chapters 9 and 10.
The formula (28) is applicable to a wide range of possible perturbations. Some of the most interesting of these are time-variable external perturbations, such as are imposed by an external electromagnetic ﬁeld or the variation of dielectric constant with temperature. However, we warn that there are cases in which the formula is not applicable. For example, a small displacement of the boundary between two materials certainly counts as a small perturbation of the system, but if the materials have highly dissimilar dielectric constants ε1 and ε2, then the moving discontinuity in the dielectric function renders equation (28) invalid. In this case, if a block of the ε1-material is moved towards the ε2-material by a distance ∆h (perpendicular to the boundary), the correct expression for the frequency shift involves a surface integral over the interface (Johnson et al., 2002a):

∆ω = − ω 2

d2r

(ε1 − ε2)|E (r)|2 −

1 ε1

−

1 ε2

d3r ε(r)|E(r)|2

|εE⊥(r)|2 ∆h + O(∆h2).

(30)

In this expression, E is the component of E that is parallel to the surface, and εE⊥ is the component of εE that is perpendicular to the surface. (Both of these components are guaranteed to be continuous across a dielectric interface.) This expression assumes that ∆h is small compared to the transverse extent of the shifted portion of the material. If instead the surface-parallel extent of the shifted material is comparable to ∆h or smaller (so that the perturbation is more like a “bump” than a shifted interface), then a more complicated correction is needed (Johnson et al., 2005).
The preceding example is one of several new developments in perturbation theory that can be found in the literature. New twists on the classic perturbative approaches have been required to deal correctly with the high material contrasts and strong periodicities that characterize photonic crystals.

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CHAPTER 2

Scaling Properties of the Maxwell Equations

One interesting feature of electromagnetism in dielectric media is that there is no fundamental length scale other than the assumption that the system is macroscopic. In atomic physics, the spatial scale of the potential function is generally set by the fundamental length scale of the Bohr radius. Consequently, conﬁgurations of material that differ only in their overall spatial scale nevertheless have very different physical properties. For photonic crystals, there is no fundamental constant with the dimensions of length—the master equation is scale invariant. This leads to simple relationships between electromagnetic problems that differ only by a contraction or expansion of all distances.
Suppose, for example, we have an electromagnetic eigenmode H(r) of frequency ω in a dielectric conﬁguration ε(r). We recall the master equation (7):

∇×

1 ε(r)

∇

×

H(r)

=

ω

2
H(r).

c

(31)

Now suppose we are curious about the harmonic modes in a conﬁguration of dielectric ε (r) that is just a compressed or expanded version of ε(r): ε (r) = ε(r/s) for some scale parameter s. We make a change of variables in (31), using r = sr and ∇ = ∇/s:

s∇ ×

1 ε(r /s)

s∇

× H(r

/s)

=

ω

2
H(r /s).

c

(32)

But ε(r /s) is none other than ε (r ). Dividing out the s’s shows that

∇×

1 ε (r

)∇

× H(r

/s)

=

ω

2
H(r /s).

cs

(33)

This is just the master equation again, this time with mode proﬁle H (r ) = H(r /s) and frequency ω = ω/s. What this means is that the new mode proﬁle and its corresponding frequency can be obtained by simply rescaling the old mode proﬁle and its frequency. The solution of the problem at one length scale determines the solutions at all other length scales.
This simple fact is of considerable practical importance. For example, the microfabrication of complex micron-scale photonic crystals can be quite difﬁcult. But models can be easily made and tested in the microwave regime, at the much larger length scale of centimeters, if materials can be found that have nearly the same dielectric constant. The considerations in this section guarantee that the model will have the same electromagnetic properties.
Just as there is no fundamental length scale, there is also no fundamental value of the dielectric constant. Suppose we know the harmonic modes of a system with dielectric conﬁguration ε(r), and we are curious about the modes of a system

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with a dielectric conﬁguration that differs by a constant factor everywhere, so that ε (r) = ε(r)/s2. Substituting s2ε (r) for ε(r) in (31) yields

∇×

ε

1 (r)

∇

×

H(r)

=

sω

2
H(r).

c

(34)

The harmonic modes H(r) of the new system are unchanged,18 but the frequencies are all scaled by a factor s: ω → ω = sω. If we multiply the dielectric constant everywhere by a factor of 1/4, the mode patterns are unchanged but their frequencies double.
Combining the above two relations, we see that if we scale ε by s2 and also rescale the coordinates by s, the frequency ω is unchanged. This simple scaling invariance is a special case of more general coordinate transformations. Amazingly, it turns out that any coordinate transformation can be replaced simply by a change of ε and µ while keeping ω ﬁxed (Ward and Pendry, 1996). This can be a powerful conceptual tool, because it allows one to warp and distort a structure in complicated ways while retaining a similar form for the Maxwell equations. In general, however, this change in ε and µ is not merely a multiplication by a constant, as it is here.

Discrete vs. Continuous Frequency Ranges
The spectrum of a photonic crystal is the totality of all of the eigenvalues ω. What does this spectrum look like? Is it a continuous range of values, like a rainbow, or do the frequencies form a discrete sequence ω0, ω1, . . ., like the vibration frequencies of a piano string? The next chapter will feature some speciﬁc examples of spectra, but in this section we discuss this question in general terms.
The answer depends on the spatial domain of the mode proﬁles H(r) (or E). If the ﬁelds are spatially bounded, either because they are localized around a particular point or because they are periodic in all three dimensions (and therefore represent a bounded proﬁle repeated indeﬁnitely), then the frequencies ω form a discrete set. Otherwise they can form a single continuous range, a set of continuous ranges, or a combination of continuous ranges and discrete sets (for a combination of localized and extended modes).
This property is quite general for many Hermitian eigenproblems. We will argue below that it follows from the orthogonality of the modes. A host of seemingly unrelated physical phenomena can be attributed to this abstract mathematical result: from the discrete energy levels in the spectrum of hydrogen gas (in which the electron wave functions are localized around the nucleus) to the distinct overtones of an organ pipe (in which the vibrating modes dwell within a ﬁnite length). Other cases that are familiar to physics students are the
18 Note, however, that the relationship between E and H has changed by a factor of s, from equation (8).

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CHAPTER 2

quantum-mechanical problem of a particle in a box (as in Liboff, 1992) and the electromagnetic problem of microwaves in a metallic cavity (as in Jackson, 1998). We will see in the chapters to come that this result, applied to photonic crystals, leads to the concepts of discrete frequency bands and of localized modes near crystal defects.
An intuitive explanation for the relation between the bounded spatial domain of the eigenmodes and the discreteness of the frequency spectrum is as follows.19 Suppose that we have a continuous range of eigenvalues, so that we can vary the frequency ω continuously and get some eigenmode Hω(r) for each ω. We now argue that this continuum cannot be the spectrum of spatially bounded modes. It is reasonable to suppose that, as we change ω continuously, the ﬁeld Hω can be made to change continuously as well, so that for an arbitrarily small change δω there is a correspondingly small change δH. Any drastic difference in the ﬁelds would correspond to a very different value of the electromagnetic energy functional and hence of the frequency. (An exception is made for systems with spatial symmetries that produce degeneracies, as discussed in the next chapter, but a similar argument implies that a bounded system has at most a ﬁnite number of degenerate modes with a given eigenvalue.) On the other hand, two spatially bounded modes H and H + δH that are arbitrarily similar cannot also be orthogonal: their inner product is (H, H) + (H, δH), where the ﬁrst term is positive and the second term is arbitrarily small for integration over a ﬁnite domain, i.e. a system with spatially bounded modes. Thus, the continuous spectrum is incompatible with the required orthogonality of the modes, unless the modes are of unbounded spatial extent.
We will see in the next chapter that many interesting electromagnetic systems exhibit both discrete localized modes and a continuum of extended states. This is not too different from the case of a hydrogen atom, which has both bound electron states with discrete energy levels and also a continuum of freely propagating states for electrons with a kinetic energy greater than the ionization energy.

Electrodynamics and Quantum Mechanics Compared
As a compact summary of the topics in this chapter, and for the beneﬁt of those readers familiar with quantum mechanics, we now present some similarities between our formulation of electrodynamics in dielectric media and the quantum mechanics of noninteracting electrons (see table 1). This analogy is developed further in appendix A.
In both cases, we decompose the ﬁelds into harmonic modes that oscillate with a phase factor e−iωt. In quantum mechanics, the wave function is a complex scalar ﬁeld. In electrodynamics, the magnetic ﬁeld is a real vector ﬁeld and the complex exponential is just a mathematical convenience.
19 For a more formal discussion, see e.g. Courant and Hilbert (1953, chap. 6).

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23

Table 1

Field Eigenvalue problem Hermitian operator

Quantum Mechanics Ψ(r, t) = Ψ(r)e−iEt/

Hˆ Ψ = EΨ

Hˆ

=

−

2
2m

∇

2

+

V(r)

Electrodynamics

H(r, t) = H(r)e−iωt

Θˆ H =

ω c

2H

Θˆ

=

∇

×

1 ε(r)

∇

×

Comparison of quantum mechanics and electrodynamics.

In both cases, the modes of the system are determined by a Hermitian eigenvalue equation. In quantum mechanics, the frequency ω is related to the eigenvalue via E = ω, which is meaningful only up to an overall additive constant V0.20 In electrodynamics, the eigenvalue is proportional to the square of the frequency, and there is no arbitrary additive constant.
One difference we did not discuss, but is apparent from Table 1, is that in quantum mechanics, the Hamiltonian is separable if V(r) is separable. For example, if V(r) is the sum of one-dimensional functions Vx(x) + Vy(y) + Vz(z), then we can write Ψ as a product Ψ(r) = X(x)Y(y)Z(z) and the problem separates into three more manageable problems, one for each direction. In electrodynamics, such a factorization is not generally possible: the differential operator, Θˆ , couples the different coordinates even if ε(r) is separable. This makes analytical solutions rare, and generally conﬁned to very simple systems.21 To demonstrate most of the interesting phenomena associated with photonic crystals, we will usually make use of numerical solutions.
In quantum mechanics, the lowest eigenstates typically have the amplitude of the wave function concentrated in regions of low potential, while in electrodynamics the lowest modes have their electric-ﬁeld energy concentrated in regions of high dielectric constant. Both of these statements are made quantitative by a variational theorem.
Finally, in quantum mechanics, there is usually a fundamental length scale that prevents us from relating solutions to potentials that differ by a scale factor. Electrodynamics is free from such a length scale, and the solutions we obtain are easily scaled up or down in length scale and frequency.

20 Here h/2π is given by Planck’s constant h, a fundamental constant with an approximate value h ≈ 6.626 × 10−34 J sec.
21 It is possible to achieve a similar separation of the Maxwell equations in two dimensions, or in
systems with cylindrical symmetry, but even in these cases the separation is usually achieved
only for a particular polarization (Chen, 1981; Kawakami, 2002; Watts et al., 2002). In these special
cases, the Maxwell equations can be written in a Schrödinger-like form. [The separable cases of the Schrödinger equation were enumerated by Eisenhart (1948).] On the other hand, if ε does
not depend on a particular coordinate, then that particular dimension of the problem is always
separable, as we will see in the section Continuous Translational Symmetry of chapter 3.

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CHAPTER 2

Further Reading
A particularly lucid undergraduate text on electromagnetism is Grifﬁths (1989). A more advanced and complete treatment of the macroscopic Maxwell equations, including a derivation from their microscopic counterparts, is contained in Jackson (1998). To explore the analogy between our formalism and the Schrödinger equation of quantum mechanics, consult the ﬁrst few chapters of any introductory quantum mechanics text. In particular, Shankar (1982), Liboff (1992), and Sakurai (1994) develop the properties of the eigenstates of a Hermitian operator with proofs very similar to our own. The ﬁrst two are undergraduate texts; the third is at the graduate level. A more formal mathematical approach to the subject of Hermitian operators leads to the ﬁeld of functional analysis, as introduced in, e.g., Gohberg et al. (2000).

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3
Symmetries and Solid-State Electromagnetism
IF A DIELECTRIC STRUCTURE has a certain symmetry, then the symmetry offers
a convenient way to categorize the electromagnetic modes of that system. In this chapter, we will investigate what various symmetries of a system can tell us about its electromagnetic modes. Translational symmetries (both discrete and continuous) are important because photonic crystals are periodic dielectrics, and because they provide a natural setting for the discussion of band gaps. Some of the terminology of solid-state physics is appropriate, and will be introduced. We will also investigate rotational, mirror, inversion, and time-reversal symmetries.
Using Symmetries to Classify Electromagnetic Modes
In both classical mechanics and quantum mechanics, we learn the lesson that the symmetries of a system allow one to make general statements about that system’s behavior. Because of the mathematical analogy we pursued in the last chapter, it is not too surprising that careful attention to symmetry also helps to understand the properties of electromagnetic systems. We will begin with a concrete example of a symmetry and the conclusion we may draw from it, and will then pass on to a more formal discussion of symmetries in electromagnetism.
Suppose we want to ﬁnd the modes that are allowed in the two-dimensional metal cavity shown in ﬁgure 1. Its shape is somewhat arbitrary, which would make it difﬁcult to write down the exact boundary condition and solve the problem analytically. But the cavity has an important symmetry: if you invert the cavity about its center, you end up with exactly the same cavity shape. So if, somehow, we ﬁnd that the particular pattern H(r) is a mode with frequency ω, then the pattern H(−r) must also be a mode with frequency ω. The cavity cannot distinguish between these two modes, since it cannot tell r from −r.
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+

+

+

CHAPTER 3

Figure 1: A two-dimensional metallic cavity with inversion symmetry. Red and blue suggest
positive and negative ﬁelds. On the left, an even mode occupies the cavity, for which H(r) = H(−r). On the right, an odd mode occupies the cavity, for which H(r) = −H(−r).

Recall from chapter 2 that different modes with the same frequency are said to be degenerate. Unless H(r) is a member of a degenerate family of modes, then if H(−r) has the same frequency it must be the same mode. It must be nothing more than a multiple of H(r): H(−r) = αH(r). But what is α? If we invert the system twice, picking up another factor of α, then we return to the original function H(r). Therefore α2H(r) = H(r), and we see that α = 1 or −1. A
given nondegenerate mode must be one of two types: either it is invariant under inversion, H(−r) = H(r), and we call it even; or, it becomes its own opposite, −H(−r) = H(r), and we call it odd.1 These possibilities are depicted in ﬁgure 1.
We have classiﬁed the modes of the system based on how they respond to one of
its symmetry operations.
With this example in mind, we can capture the essential idea in more abstract language. Suppose I is an operator (a 3 × 3 matrix) that inverts vectors (3 × 1 matrices), so that Ia = −a. To invert a vector ﬁeld, we use an operator Oˆ I that inverts both the vector f and its argument r: Oˆ If(r) = If(Ir).2 What is the mathematical expression of the statement that our system has inversion
symmetry? Since inversion is a symmetry of our system, it does not matter whether we operate with Θˆ or we ﬁrst invert the coordinates, then operate with Θˆ , and then change them back:

Θˆ = Oˆ −I 1Θˆ Oˆ I.

(1)

1 This dichotomy is not automatically true of degenerate modes. But we can always form new modes
that are even or odd, by taking appropriate linear combinations of the degenerate modes. 2 This is a special case of the operator deﬁned later in equation (14). There is a minor complication
here because H is a pseudovector and E is a vector, as proved in Jackson (1998). This means that
H transforms with a plus sign (IH = +H), while E transforms with a minus sign (IE = −E). That is, Oˆ I H(r) = +H(−r), and Oˆ I E(r) = −E(−r). An even mode is deﬁned as one that is invariant under the inversion Oˆ I , which means that an even mode has H(r) = H(−r) and E(r) = −E(−r). Similarly, an odd mode is deﬁned as one that acquires a minus sign under the inversion Oˆ I , so that H(r) = −H(−r) and E(r) = E(−r).

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27

This equation can be rearranged as Oˆ I Θˆ − Θˆ Oˆ I = 0. Following this cue, we deﬁne the commutator [Aˆ, Bˆ ] of two operators Aˆ and Bˆ just like the commutator in
quantum mechanics:

[Aˆ, Bˆ ] Aˆ Bˆ − Bˆ Aˆ.

(2)

Note that the commutator is itself an operator. We have shown that our system is symmetric under inversion only if the inversion operator commutes with Θˆ ; that is, we must have [Oˆ I, Θˆ ] = 0. If we now operate with this commutator on any mode of the system H(r), we obtain

[Oˆ I, Θˆ ]H = Oˆ I (Θˆ H) − Θˆ (Oˆ IH) = 0

=⇒

Θˆ (Oˆ IH) = Oˆ I (Θˆ H) =

ω2 c2

(Oˆ I

H).

(3)

This equation tells us that if H is a harmonic mode with frequency ω, then Oˆ IH is also a mode with frequency ω. If there is no degeneracy, then there can only be one mode per frequency, so H and Oˆ IH can be different only by a multiplicative factor: Oˆ IH = αH. But this is just the eigenvalue equation for Oˆ I, and we already know that the eigenvalues α must be either 1 or −1. Thus, we can classify the eigenvectors H(r) according to whether they are even (H → +H) or odd (H → − H) under the inversion symmetry operation Oˆ I.
What if there is degeneracy in the system? Then two modes may have the same
frequency, but might not be related by a simple multiplier. Although we will not
demonstrate it, we can always form linear combinations of such degenerate modes
to make modes that are themselves even or odd.
Generally speaking, whenever two operators commute, one can construct
simultaneous eigenfunctions of both operators. One reason why this is convenient is that eigenfunctions and eigenvalues of simple symmetry operators like Oˆ I are easily determined, whereas those for Θˆ are not. But if Θˆ commutes with a symmetry operator Sˆ, we can construct and catalogue the eigenfunctions of Θˆ using their Sˆ properties. In the case of inversion symmetry, we can classify the Θˆ eigenfunctions as either odd or even. We will ﬁnd this approach useful in later
sections when we introduce translational, rotational, and mirror symmetries.

Continuous Translational Symmetry
Another symmetry that a system might have is continuous translation symmetry. Such a system is unchanged if we translate everything through the same distance in a certain direction. Given this information, we can determine the functional form of the system’s modes.
A system with translational symmetry is unchanged by a translation through a displacement d. For each d, we can deﬁne a translation operator Tˆd which, when

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CHAPTER 3

operating on a function f(r), shifts the argument by d. Suppose our system is translationally invariant; then we have Tˆdε(r) = ε(r − d) = ε(r), or equivalently, [Tˆd, Θˆ ] = 0. The modes of Θˆ can now be classiﬁed according to how they behave under Tˆd.
A system with continuous translation symmetry in the z direction is invariant under all of the Tˆd’s for that direction. What sort of function is an eigenfunction of all the Tˆd’s? We can prove that a mode with the functional form eikz is an
eigenfunction of any translation operator in the z direction:

Tˆdzˆ eikz = eik(z−d) = (e−ikd)eikz.

(4)

The corresponding eigenvalue is e−ikd. With a little more work, one can show the converse, too: any eigenfunction of Tˆd for all d = dzˆ must be proportional to eikz for some k.3 The modes of our system can be chosen to be eigenfunctions of all the Tˆd’s, so we therefore know they should have a z dependence of the functional form eikz (the z dependence is separable). We can classify them by the particular values
for k, the wave vector. (k must be a real number in an inﬁnite system where we
require the modes to have bounded amplitudes at inﬁnity.)
A system that has continuous translational symmetry in all three directions is a homogeneous medium: ε(r) is a constant ε (= 1 for free space). Following a line of
argument similar to the one above, we can deduce that the modes must have the
form

Hk(r) = H0eik·r,

(5)

where H0 is any constant vector. These are plane waves, polarized in the direction of H0. Imposing the transversality requirement—equation (5) of chapter 2—gives the further restriction k · H0 = 0. The reader can also verify that these plane waves are in fact solutions of the master equation√with eigenvalues (ω/c)2 = |k|2/ε, yielding the dispersion relation ω = c|k|/ ε. We classify a plane wave by its
wave vector k, which speciﬁes how the mode is transformed by a continuous
translation operation.
Another simple system with continuous translational symmetry is an inﬁnite
plane of glass, as shown in ﬁgure 2. In this case, the dielectric function varies in the z direction, but not in the x or y directions: ε(r) = ε(z). The system is invariant
under all of the translation operators of the xy plane. We can classify the modes according to their in-plane wave vectors, k = kxxˆ + kyyˆ . The x and y dependence must once again be a complex exponential (a plane wave):

Hk(r) = eik·ρh(z).

(6)

3 If f (x) = 0 is such an eigenfunction, then f (x − d) = λ(d) f (x) for all d and some eigenvalues λ(d). Scale f (x) so that f (0) = 1 and thus f (x) = f (0 − [−x]) = λ(−x). Therefore, f (x + y) = f (x) f (y), and the only anywhere-continuous functions with this property are f (x) = ecx for some constant c
(see, e.g., Rudin, 1964, ch. 8 exercise 6).

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29

z ρ

Figure 2: A plane of glass. If the glass extends much farther in the x and y directions than in the z direction, we may consider this system to be one-dimensional: the dielectric function ε(r) varies in the z direction, but has no dependence on the in-plane coordinate ρ.
This is the ﬁrst of many occasions in which we will use the symbol ρ to denote a vector that is conﬁned to the xy plane. The function h(z) (which depends on k) cannot be determined by this line of reasoning, because the system does not have translational symmetry in that direction. (The transversality condition does imply one restriction on h: substitution of (6) into ∇ · Hk = 0 gives k · h = i∂hz/∂z.)
The reason why the modes are described by equation (6) can also be understood with an intuitive argument. Consider three non-collinear neighboring points at r, r + dx, and r + dy, all of which have the same z value. Due to symmetry, these three points should be treated equally, and should have the same magnetic ﬁeld amplitude. The only conceivable difference could be the variation in the phase between the points. But once we choose the phase differences between these three points, we set the phase relationships between all the points. We have effectively speciﬁed kx and ky at one point, but they must be universal to the plane. Otherwise we could distinguish different locations in the plane by their phase relationships. Along the z direction, however, this restriction does not hold. Each plane is at a different distance from the bottom of the glass structure and can conceivably have a different amplitude and phase.
We have seen that we can classify the modes by their values of k. Although we cannot yet say anything about h(z), we can nevertheless line up the modes (whatever they may be) in order of increasing frequency for a given value of k. Let n stand for a particular mode’s place in line of increasing frequency, so that we can identify any mode by its unique name (k, n). If there is degeneracy, then we might have to include an additional index to name the degenerate modes that have the same n and k.
We call n the band number. If the spectrum is discrete for a given k, we can use integers for n, but sometimes the band number is actually a continuous variable. As the n value grows, so too does the frequency of the mode. If we make a plot of wave vector versus mode frequency for the plane of glass, the different bands correspond to different lines that rise uniformly in frequency. This band structure (also called a band diagram or dispersion relation) is shown in ﬁgure 3 and is considered in more detail below. We computed it by solving the master equation (7) of chapter 2 numerically.

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1.5
Extended states
1

CHAPTER 3

Frequency ωa/2πc

0.5

n = 7

n = 3

n = 1

0 0 0.5 1 1.5 2 2.5 3 3.5 4
Parallel wave vector ka/2π
Figure 3: Harmonic mode frequencies for a plane of glass of thickness a and ε = 11.4.
Blue lines correspond to modes that are localized in the glass. The shaded blue region is a continuum of states that extend into both the glass and the air around it. The red line is the
light line ω = ck. This plot shows modes of only one polarization, for which H is perpendicular to both the z and k directions.

Another way to state the signiﬁcance of continuous translational symmetry
is that the components of the wave vector k along the symmetry directions are conserved quantities. If a ﬁeld pattern starts out with a particular eigenvalue e−ikd of Tˆd (which commutes with Θˆ ), then it will have that eigenvalue at all future
times. Such conservation laws have far-reaching consequences, as we will see in subsequent sections.4

Index guiding
Returning to the inﬁnite plane of glass, we now discuss one of the most wellknown phenomena in classical optics, total internal reﬂection. The familiar description of this phenomenon is that light rays within the glass that strike the interface with the air (or any lower-index medium) at too shallow an angle are totally reﬂected, and remain conﬁned to the glass (forming a planar waveguide).
4 For experts, we can state more generally that the irreducible representation of the symmetry group is conserved in a linear system. This is easily proved from the fact that the projection operator for the group representation (see, e.g. Inui et al., 1996) commutes with the time-evolution operator.

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k||

k

θ1 θ2

ε1 ε2
Figure 4: For a ﬂat interface between two dielectrics ε1 and ε2, light can be described by a ray with an incident angle θ1 and a refracted angle θ2 given by Snell’s law. When ε2 < ε1, we can have no solution θ2 for certain θ1, and the light undergoes total internal reﬂection. A generalization of this result follows from translational symmetry, which tells us that k is
conserved.

In this section, we will describe this familiar phenomenon with the symmetry

language developed earlier in this chapter. We will see that, in a sense, the

conﬁnement of light is a consequence of translational symmetry. Moreover,

symmetry considerations lead to a more general concept of index guiding than the

ray-optics picture might suggest, a result that forms the foundation of chapters 7

and 8.

The refraction of a light ray at an interface between two dielectrics ε1 and ε2,

illustrated in ﬁgure n2 sin θ2, where ni is

4, is the

ruesfruaacltliyvedeinscdreibxe√d εiinatnedrmθis

of Snell’s law: is the angle the

n1 sin θ1 = ray makes

with the normal to the interface. If θ1 > sin−1(n2/n1), then the law would demand

sin θ2 > 1, for which there is no real solution; the interpretation is that the ray is

totally reﬂected. The critical angle θc = sin−1(n2/n1) exists only for n2 < n1, so

total internal reﬂection occurs only within the higher-index medium. Snell’s law,

however, is simply the combination of two conservation laws that follow from

symmetry: conservation of frequency ω (from the linearity and time-invariance of

the Maxwell equations), and conservation of the component k of k that is parallel

to the interface (from the continuous translational symmetry along the interface, as

we noted above). In particular, k = |k| sin θ, and |k| = nω/c from the dispersion

relation. We obtain Snell’s law by setting k equal on both sides of the interface.

The advantage of this way of thinking about the problem is that we are now in a

position to generalize beyond the ray-optics regime (which is valid only on length

scales much larger than the wavelength of light).

Let us now be a little more concrete. Consider a plane of glass of width a

centered about the origin. We now wish to understand the band structure of

the electromagnetic modes, by which we mean the frequency ω versus the wave

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CHAPTER 3

vector k (both of which are conserved quantities). This plot, shown in ﬁgure 3, is described below.
First, consider the modes that are not conﬁned to the glass, and extend into the air and out to inﬁnity. Far away from the glass, these modes must closely resemble free-space plane waves. These are superpositions of plane waves with
ω = c|k| = c k2 + k2⊥ for some perpendicular real wave vector component k⊥. For
a given value of k , there will be modes with every possible frequency greater than ck , because k⊥ can take any value. Thus the spectrum of states is continuous for all frequencies above the light line ω = ck , which is marked with a red line in ﬁgure 3. The region of the band structure with ω > ck is called the light cone. The modes in the light cone are solutions of Snell’s law (less than the critical angle).
In addition to the light cone, the glass plate introduces new electromagnetic solutions that lie below the light line. Because ε is larger in the glass than in air, these modes have lower frequencies relative to the values the corresponding modes would have in free space (as demanded by the variational theorem, equation (23) of chapter 2). These new solutions must be localized in the vicinity of the glass. Below the light line, the only solutions in air are those with imaginary k⊥ =
±i k2 − ω2/c2, corresponding to ﬁelds that decay exponentially (are evanescent)
away from the glass. We call these the index-guided modes, and from the section Discrete vs. Continuous Frequency Ranges of chapter 2 we expect that for a given k they form a set of discrete frequencies, because they are localized in one direction. Thus, we obtain the discrete bands ωn(k ) below the light line in ﬁgure 3. In the limit of larger and larger |k |, one obtains more and more guided bands, and eventually one approaches the ray-optics limit of totally internally reﬂected rays with a continuum of angles θ > θc.

Discrete Translational Symmetry
Photonic crystals, like traditional crystals of atoms or molecules, do not have continuous translational symmetry. Instead, they have discrete translational symmetry. That is, they are not invariant under translations of any distance, but rather, only distances that are a multiple of some ﬁxed step length. The simplest example of such a system is a structure that is repetitive in one direction, like the conﬁguration in ﬁgure 5.
For this system we still have continuous translational symmetry in the x direction, but now we have discrete translational symmetry in the y direction. The basic step length is the lattice constant a, and the basic step vector is called the primitive lattice vector, which in this case is a = ayˆ . Because of this discrete symmetry, ε(r) = ε(r ± a). By repeating this translation, we see that ε(r) = ε(r + R) for any R that is an integral multiple of a; that is, R = a, where
is an integer.

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z
y x
a
Figure 5: A dielectric conﬁguration with discrete translational symmetry. If we imagine
that the system continues forever in the y direction, then shifting the system by an integral multiple of a in the y direction leaves it unchanged. The repeated unit of this periodic system
is framed with a box. This particular conﬁguration is employed in distributed-feedback lasers, as in Yariv (1997).

The dielectric unit that we consider to be repeated over and over, highlighted
in ﬁgure 5 with a box, is known as the unit cell. In this example, the unit cell is an xz slab of dielectric material with width a in the y direction.
Because of the translational symmetries, Θˆ must commute with all of the
translation operators in the x direction, as well as the translation operators for lattice vectors R = ayˆ in the y direction. With this knowledge, we can identify the modes of Θˆ as simultaneous eigenfunctions of both translation operators. As before, these eigenfunctions are plane waves:

Tˆdxˆ eikx x = eikx(x−d) = (e−ikxd)eikx x TˆReikyy = eiky(y− a) = (e−iky a)eikyy.

(7)

We can begin to classify the modes by specifying kx and ky. However, not all values of ky yield different eigenvalues. Consider two modes, one with wave vector ky and the other with wave vector ky + 2π/a. A quick insertion into (7) shows that they have the same TˆR eigenvalues. In fact, all of the modes with wave vectors of the form ky + m(2π/a), where m is an integer, form a degenerate set; they all have the same TˆR eigenvalue of e−i(ky a). Augmenting ky by an integral multiple of b = 2π/a leaves the state unchanged. We call b = byˆ the primitive reciprocal
lattice vector.
Since any linear combination of these degenerate eigenfunctions is itself an
eigenfunction with the same eigenvalue, we can take linear combinations of our

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CHAPTER 3

original modes to put them in the form

Hkx,ky (r) = eikxx

cky ,m (z)ei(ky +mb)y

m

= eikxx · eikyy ·

cky ,m (z)eimby

(8)

m

= eikxx · eikyy · uky (y, z),

where the c’s are expansion coefﬁcients to be determined by explicit solution, and u(y, z) is (by construction) a periodic function in y: by inspection of equation (8), we can verify that u(y + a, z) = u(y, z).
The discrete periodicity in the y direction leads to a y dependence for H that is simply the product of a plane wave with a y-periodic function. We can think of it as a plane wave, as it would be in free space, but modulated by a periodic function because of the periodic lattice:

H(. . . , y, . . .) ∝ eikyy · uky (y, . . .).

(9)

This result is commonly known as Bloch’s theorem. In solid-state physics, the
form of (9) is known as a Bloch state (as in Kittel, 1996), and in mechanics as a Floquet mode (as in Mathews and Walker, 1964). We will use the former name.5
One key fact about Bloch states is that the Bloch state with wave vector ky and the Bloch state with wave vector ky + mb are identical. The ky’s that differ by integral multiples of b = 2π/a are not different from a physical point of view. Thus, the mode frequencies must also be periodic in ky: ω(ky) = ω(ky + mb). In fact, we need only consider ky to exist in the range −π/a < ky π/a. This region of important, nonredundant values of ky is called the Brillouin zone.6 Readers unfamiliar with the notion of a reciprocal lattice or a Brillouin zone might ﬁnd
appendix B a useful introduction to that material.
We digress brieﬂy to make the analogous statements that apply when the
dielectric is periodic in three dimensions; here we skip the details and summarize
the results. In this case the dielectric is invariant under translations through
a multitude of lattice vectors R in three dimensions. Any one of these lattice
vectors can be written as a particular combination of three primitive lattice vectors (a1, a2, a3) that are said to “span” the space of lattice vectors. In other words, every R = a1 + ma2 + na3 for some integers , m, and n. As explained in appendix B, the vectors (a1, a2, a3) give rise to three primitive reciprocal lattice vectors (b1, b2, b3) deﬁned so that ai · bj = 2πδij. These reciprocal vectors span a reciprocal lattice of their own which is inhabited by wave vectors.

5 In fact, the essentials of this theorem were discovered independently at least four different times
(in four languages), by Hill (1877), Floquet (1883), Lyapunov (1892), and Bloch (1928). With this in
mind, it is not surprising that the nomenclature is often confusing. 6 Strictly speaking, this is known as the ﬁrst Brillouin zone.

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The modes of a three-dimensional periodic system are Bloch states that can be labelled by a Bloch wave vector k = k1b1 + k2b2 + k3b3 where k lies in the Brillouin zone. For example, for a crystal in which the unit cell is a rectangular box, the Brillouin zone is given by |ki| 1/2. Each value of the wave vector k inside the Brillouin zone identiﬁes an eigenstate of Θˆ with frequency ω(k) and an
eigenvector Hk of the form

Hk(r) = eik·ruk(r),

(10)

where uk(r) is a periodic function on the lattice: uk(r) = uk(r + R) for all lattice vectors R.
Just as continuous translational symmetry leads to the conservation of the wave vector, a corollary of Bloch’s theorem is that k is a conserved quantity in a periodic system, modulo the addition of reciprocal lattice vectors. Addition of a reciprocal lattice vector does not change an eigenstate or its propagation direction; it is essentially a mere change of label, as discussed further in the section Bloch-Wave Propagation Velocity. This is quite different from the free-space case, in which all wave vectors represent physically distinct states. In the section The Physical Origin of Photonic Band Gaps of chapter 4, we return to this question by considering a plane wave in free space and imagining that a periodicity is slowly turned on by gradually increasing the strength of a periodic dielectric perturbation.

Photonic Band Structures

From very general symmetry principles, we have just suggested that the electro-
magnetic modes of a photonic crystal with discrete periodicity in three dimensions
can be written as Bloch states, as in equation (10). All of the information about such a mode is given by the wave vector k and the periodic function uk(r). To solve for uk(r), we insert the Bloch state into the master equation (7) of chapter 2:

Θˆ Hk = (ω(k)/c)2Hk

∇

×

1 ε(r)

∇

×

eik·r uk (r)

=

(ω(k)/c)2eik·ruk(r)

(11)

(ik

+

∇)

×

1 ε(r)

(ik

+

∇)

×

uk(r)

=

(ω(k)/c)2uk(r)

Θˆ kuk(r) = (ω(k)/c)2uk(r).

Here we have deﬁned Θˆ k as a new Hermitian operator that appears in this substitution and depends on k:

Θˆ k

(ik

+

∇)

×

1 ε(r)

(ik

+

∇)

×

.

(12)

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CHAPTER 3

The function u, and therefore the mode proﬁles, are determined by the eigenvalue problem in the fourth equation of (11), subject to transversality (ik + ∇) · uk = 0 and the periodicity condition

uk(r) = uk(r + R).

(13)

Because of this periodic boundary condition, we can regard the eigenvalue problem as being restricted to a single unit cell of the photonic crystal. As was discussed in the section Discrete vs. Continuous Frequency Ranges of chapter 2, restricting a Hermitian eigenvalue problem to a ﬁnite volume leads to a discrete spectrum of eigenvalues. We can expect to ﬁnd, for each value of k, an inﬁnite set of modes with discretely spaced frequencies, which we can label by a band index n.
Since k enters as a continuous parameter in Θˆ , we expect the frequency of each band, for a given n, to vary continuously as k varies. In this way we arrive at the description of the modes of a photonic crystal: They are a family of continuous functions, ωn(k), indexed in order of increasing frequency by the band number. The information contained in these functions is called the band structure of the photonic crystal. Studying the band structure of a crystal supplies us with most of the information we need to predict its optical properties, as we will see.
For a given photonic crystal ε(r), how can we calculate the band structure functions ωn(k)? Powerful computational techniques are available for the task, but we will not discuss them extensively. The focus of this text is on concepts and results, not on the numerical studies of the equations. A brief outline of the technique that was used to generate the band structures in this text is in appendix D. In essence, the technique relies on the fact that the last equation of (11) is a standard eigenvalue equation that is readily solvable by an iterative minimization technique for each value of k.

Rotational Symmetry and the Irreducible Brillouin Zone

Photonic crystals might have symmetries other than discrete translations. A given
crystal might also be left invariant after a rotation, a mirror reﬂection, or an
inversion is performed. To begin, we examine the conclusions we can draw about
the modes of a system with rotational symmetry. Suppose the operator (3 × 3 matrix) R(nˆ , α) rotates vectors by an angle α about
the nˆ axis. Abbreviate R(nˆ , α) by R. To rotate a vector ﬁeld f(r), we take the vector f and rotate it with R to give f = Rf. We also rotate the argument r of the vector ﬁeld: r = R−1r. Therefore f (r ) = Rf(r ) = Rf(R−1r). Accordingly, we deﬁne the vector ﬁeld rotator Oˆ R as

Oˆ R · f(r) = Rf(R−1r).

(14)

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If rotation by R leaves the system invariant, then we conclude (as before) that [Θˆ , Oˆ R] = 0. Therefore, we may carry out the following manipulation:

Θˆ (Oˆ RHkn) = Oˆ R(Θˆ Hkn) =

ωn(k) c

2
(Oˆ RHkn).

(15)

We see that Oˆ RHkn also satisﬁes the master equation, with the same eigenvalue
as Hkn. This means that the rotated mode is itself an allowed mode, with the same frequency. We can further prove that the state Oˆ RHkn is none other than the Bloch state with wave vector Rk. To do this, we must show that Oˆ RHkn is an eigenfunction of the translation operator TˆR with eigenvalue e−iRk·R where R is a lattice vector. We can do just that, using the fact that Θˆ and Oˆ R commute and thus R−1R must also be a lattice vector:

TˆR(Oˆ RHkn) = Oˆ R(TˆR−1RHkn) = Oˆ R(e−i(k·R−1R)Hkn) = e−i(k·R−1R)(Oˆ RHkn) = e−i(Rk·R)Oˆ RHkn.

(16)

Since Oˆ RHkn is the Bloch state with wave vector Rk and has the same eigenvalue as Hkn, it follows that

ωn(Rk) = ωn(k).

(17)

We conclude that when there is rotational symmetry in the lattice, the frequency bands ωn(k) have additional redundancies within the Brillouin zone. In a similar manner, we can show that whenever a photonic crystal has a rotation, mirrorreﬂection, or inversion symmetry, the ωn(k) functions have that symmetry as well. This particular collection of symmetry operations (rotations, reﬂections, and
inversions) is called the point group of the crystal. Since the functions ωn(k) possess the full symmetry of the point group, we need
not consider them at every k point in the Brillouin zone. The smallest region within the Brillouin zone for which the ωn(k) are not related by symmetry is called the irreducible Brillouin zone. For example, a photonic crystal with the symmetry of a simple square lattice has a square Brillouin zone centered at k = 0, as depicted
in ﬁgure 6. (See appendix B for a fuller discussion of the reciprocal lattice and the
Brillouin zone.) The irreducible zone is a triangular wedge with 1/8 the area of the
full Brillouin zone; the rest of the Brillouin zone consists of redundant copies of
the irreducible zone.

Mirror Symmetry and the Separation of Modes
Mirror reﬂection symmetry in a photonic crystal deserves special attention. Under certain conditions it allows us to separate the eigenvalue equation for Θˆ k into two

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Real lattice

CHAPTER 3
Brillouin zone of
reciprocal lattice M

r

Γ

X

k

Figure 6: Left: A photonic crystal made using a square lattice. An arbitrary vector r is shown. Right: The Brillouin zone of the square lattice, centered at the origin (Γ). An arbitrary wave vector k is shown. The irreducible zone is the light blue triangular wedge. The special points at the center, corner, and face are conventionally known as Γ, M, and X.

separate equations, one for each ﬁeld polarization. As we will see, in one case Hk is perpendicular to the mirror plane and Ek is parallel; while in the other case, Hk is in the plane and Ek is perpendicular. This simpliﬁcation is convenient, because it provides immediate information about the mode symmetries and also
facilitates the numerical calculation of their frequencies.
To show how this separation of modes comes about, let us turn again to the
dielectric system illustrated in ﬁgure 5, the notched dielectric. This system is
invariant under mirror reﬂections in the yz and xz planes. We focus on reﬂections Mx in the yz plane (Mx changes xˆ to −xˆ and leaves yˆ and zˆ alone).7 In analogy with our rotation operator, we deﬁne a mirror reﬂection operator Oˆ Mx , which reﬂects a vector ﬁeld by using Mx on both its input and its output:

Oˆ Mx f(r) = Mxf(Mxr).

(18)

Two applications of the mirror reﬂection operator restore any system to its original state, so the possible eigenvalues of Oˆ Mx are +1 and −1. Because the dielectric is symmetric under a mirror reﬂection in the yz plane, Oˆ Mx commutes with Θˆ : [Θˆ , Oˆ Mx ] = 0. As before, if we operate on Hk with this commutator we can show that Oˆ Mx Hk is just the Bloch state with the reﬂected wave vector Mxk:

Oˆ Mx Hk = eiφHMxk.

(19)

Here, φ is an arbitrary phase. This relation does not restrict the reﬂection properties of Hk very much, unless k happens to be pointed in such a way that Mxk = k.

7 Note that any slice perpendicular to the x axis is a valid mirror plane for our system. Thus for any r in the crystal we can always ﬁnd a plane such that Mxr = r. This is not true for My.

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39

When this is true, (19) becomes an eigenvalue problem and, using (18), we see that Hk must obey

Oˆ Mx Hk(r) = ±Hk(r) = MxHk(Mxr).

(20)

Although we will not show it explicitly, the electric ﬁeld Ek obeys a similar equation, so that both the electric and magnetic ﬁelds must be either even or odd under the Oˆ Mx operation. But Mxr = r for any r in our dielectric (taken in the two-dimensional yz plane). Therefore, since E transforms like a vector and
H transforms like a pseudovector (see footnote 2 on page 26), the only nonzero ﬁeld components of an Oˆ Mx -even mode must be Hx, Ey, and Ez. The odd modes are described by the components Ex, Hy, and Hz.
In general, given a reﬂection M such that [Θˆ , Oˆ M] = 0, this separation of modes is only possible at Mr = r for Mk = k. Note from (11) that Θˆ k and Oˆ M will not commute unless Mk = k. It appears that the separation of polarizations holds
only under fairly restricted conditions and is not that useful for three-dimensional
photonic-crystal analyses. (However, a generalization is discussed in the section
Symmetry and Polarization of chapter 7.)
On the other hand, these conditions can always be met for two-dimensional
photonic crystals. Two-dimensional crystals are periodic in a certain plane, but
are uniform along an axis perpendicular to that plane. Calling that axis the z axis, we know that the operation zˆ → −zˆ is a symmetry of the crystal for any choice of origin. It also follows that Mzk = k for all wave vectors k in the two-dimensional Brillouin zone. Thus the modes of every two-dimensional photonic crystal can be classiﬁed into two distinct polarizations: either (Ex, Ey, Hz) or (Hx, Hy, Ez). The former, in which the electric ﬁeld is conﬁned to the xy plane, are called transverse-electric (TE) modes. The latter, in which the magnetic ﬁeld is conﬁned to the xy plane, are called transverse-magnetic (TM) modes.8

Time-Reversal Invariance

We will discuss one more symmetry in detail, and it is of global signiﬁcance: the time-reversal symmetry. If we take the complex conjugate of the master equation for Θˆ [equation (7) of chapter 2], and use the fact that the eigenvalues are real for lossless materials, we obtain

(Θˆ Hkn)∗

=

ωn2 (k)
c2

H∗kn

(21)

Θˆ H∗kn

=

ωn2 (k)
c2

H∗kn

.

8 Classical waveguide theory sometimes uses different conventions for the meaning of “TE” and “TM.” Our notation is common in the photonic-crystal literature, but it is wise to be aware that other authors use different nomenclatures.

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CHAPTER 3

Bveyrtyhsisammeaneiipguenlavtaiolune, .wBeusteferotmhat(1H0)∗knwseatsieseﬁeths atht eHs∗kanmisejeuqsut aatBiolnocahs

Hkn, state

with the at −k. It

follows that

ωn(k) = ωn(−k).

(22)

The above relation holds for almost all photonic crystals.9 The frequency bands have inversion symmetry even if the crystal does not. Taking the complex conjugate of Hkn is equivalent to reversing the sign of time t in the Maxwell equations, as can be veriﬁed from equation (5) of chapter 2. For this reason, we say that (22) is a consequence of the time-reversal symmetry of the Maxwell equations.

Bloch-Wave Propagation Velocity

At this point, some remarks are warranted on the physical interpretation of the Bloch state, to ward off common points of confusion. The Bloch state Hk(r)e−iωt is a plane wave ei(k·r−ωt) multiplied by a periodic “envelope” function uk(r). It propagates through the crystal without scattering, because k is conserved (apart from addition of reciprocal lattice vectors, which is merely a relabeling). Or, equivalently, all of the scattering events are coherent and result in the periodic shape of uk.
In the familiar case of a homogeneous, isotropic medium, k is the direction in which the wave propagates, but this is not necessarily true in a periodic medium. Rather, the direction and the speed with which electromagnetic energy passes through the crystal are given by the group velocity v, which is a function of both the band index n and the wave vector k:

vn(k)

∇k ωn

∂ωn ∂kx

xˆ

+

∂ωn ∂ky

yˆ

+

∂ωn ∂kz

zˆ ,

(23)

where ∇k is the gradient with respect to k. The group velocity is the energytransport velocity whenever the medium is lossless, the material dispersion is small, and the wave vector is real. This fact is often derived, in a homogeneous medium, by considering the propagation of a broad pulse of energy through the medium (as in Jackson, 1998).10 In our case, it is perhaps easier to employ the

9 As an exception, magneto-optic materials can break time-reversal symmetry. Such materials are described by a dielectric tensor ε that is a 3 × 3 complex Hermitian matrix (Landau et al., 1984). In this case, ε∗ = ε even for a lossless medium.
10 Several classic papers on wave velocity are collected in Brillouin (1960). Early discussions of group
velocity in periodic structures can be found in Brillouin (1946) and Yeh (1979). Beware that the
issue of the energy-transport velocity is greatly complicated if one includes nonnegligible losses, a frequency-dependent ε (material dispersion), or complex values of k (evanescent modes, such as
those of the section Evanescent Modes in Photonic Band Gaps of chapter 4).

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SYMMETRIES AND SOLID-STATE ELECTROMAGNETISM

41

formalism we have already developed in previous chapters. We differentiate the eigenequation Θˆ kuk = (ω/c)2uk with respect to k, and then take the inner product with uk on both sides:11

uk, ∇k Θˆ kuk

ω2 = uk, ∇k c2 uk

,

(24)

giving

uk, [∇kΘˆ k]uk + Θˆ k∇kuk

=

ω

ω2

uk, 2 c2 vuk + c2 ∇kuk

.

(25)

The ∇kuk terms on both sides cancel one another, because the Hermitian operator Θˆ k that appears in the second term on the left-hand side can operate leftwards and produce (ω/c)2, matching the second term on the right-hand side. The remaining
terms can be solved for v = ∇kω, yielding

v

=

c2 2ω

uk, [∇kΘˆ k]uk (uk, uk)

.

(26)

The right-hand side can now be rearranged to a more suggestive form by writing uk = e−ik·rHk from equation (10). The denominator then becomes 4UH/µ0 = 2(UE + UH)/µ0, given the deﬁnitions [equation (24) of chapter 2] of the timeaveraged electric and magnetic energy. The numerator (with the c2/2ω factor) is in
fact 2/µ0 times the average electromagnetic energy ﬂux: that is, it is (2/µ0) S = (2/µ0) Re d3r E∗ × H/2 (the integral of the Poynting vector S from equation (25) of chapter 2). This can be seen by differentiating Θˆ k from (12) and applying
equation (8) of chapter 2. The ﬁnal result is that v is the ratio of the energy ﬂux
to the energy density (averaged in time and over the unit cell):

1 Re

d3r E∗ × H

d3r S

∇kω = v = 1 4

2 d3r (µ0|H|2

+

ε0ε|E|2)

=

UE

+

, UH

(27)

which by deﬁnition is the velocity of energy propagation. For a real k and a real dielectric function ε 1 that is independent of frequency, this speed |v| is always
c.12 (In more general cases, the deﬁnition of velocity itself is more subtle.)

11 As in solid-state physics, the process of differentiating the eigenequation with k is the entry point

into “k · p theory” (Sipe, 2000), and is also equivalent to the Hellman–Feynman theorem of quantum

mechanics. The result is analogous to the ﬁrst-order perturbation theory described in the section The

Effect of Small Perturbations of chapter 2, but with ∆k replacing ∆ε as the perturbation.

12 |v| c for ε 1 can be proved by applying a sequence of elementary inequalities to the numerator

of |v| = |ﬂux|/energy. In particular, Re E∗ × H

|E∗ × H|

√ | εE| · |H|

ε|E|2

|H|2 = 4cUH (where the ﬁnal was the Cauchy–Schwarz inequality), which cancels the denominator 2(UE + UH) = 4UH, leaving c.

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42

Table 1

Discrete translational symmetry Commutation relationships Bloch’s theorem

Quantum Mechanics
V(r) = V(r + R) [Hˆ , TˆR] = 0
Ψkn(r) = ukn(r)eik·r

Quantum mechanics vs. electrodynamics in periodic systems.

CHAPTER 3
Electrodynamics ε(r) = ε(r + R)
[Θˆ , TˆR] = 0 Hkn(r) = ukn(r)eik·r

Another concept with which the reader may be familiar is the phase velocity, usually given by ωk/|k|2. For a photonic crystal, however, the phase velocity is difﬁcult to deﬁne because k is not unique: it is equivalent to k + G for any
reciprocal lattice vector G. Equivalently, because of the periodic envelope function
uk modulating the plane wave, it is hard to identify unique phase fronts whose velocity is to be measured.13 In contrast, adding a reciprocal lattice vector to k has
no effect on the group velocity.

Electrodynamics vs. Quantum Mechanics Again
As in the previous chapter, we summarize by way of analogy with quantum mechanics. Table 1 compares the system containing an electron propagating in a periodic potential with the system of electromagnetic modes in a photonic crystal. appendix A develops this analogy further.
In both cases, the systems have translational symmetry: in quantum mechanics the potential V(r) is periodic, and in the electromagnetic case the dielectric function ε(r) is periodic. This periodicity implies that the discrete translation operator commutes with the major differential operator of the problem, whether with the Hamiltonian or with Θˆ .
We can index the eigenstates (Ψkn or Hkn) using the translation operator eigenvalues. These can be labelled in terms of the wave vectors and bands in the Brillouin zone. All of the eigenstates can be cast in Bloch form: a periodic function modulated by a plane wave. The ﬁeld can propagate through the crystal in a coherent manner, as a Bloch wave. The understanding of Bloch waves for electrons explained one of the great mysteries of nineteenth-century physics: Why do electrons behave like free particles in many examples of conducting crystals? In the same way, a photonic crystal provides a synthetic medium in which light can propagate, but in ways quite different from propagation in a homogeneous medium.
13 For purposes of qualitative description, one sometimes deﬁnes a “phase velocity” in a photonic crystal by arbitrarily restricting k to the ﬁrst Brillouin zone, or alternatively by picking the k + G corresponding to the largest Fourier component. Care must be taken in interpreting such a quantity, however.

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Further Reading
The study of symmetry in the most general context falls under the mathematical subject of group theory or, more speciﬁcally, group representation theory. Perhaps most useful references on this subject are texts that apply the formalism of group theory to speciﬁc physical disciplines. For example, Tinkham (2003) and Inui et al. (1996) cover both molecular and solid-state applications, and include useful tables of the possible crystallographic symmetry groups.
Readers completely unfamiliar with concepts like the reciprocal lattice, the Brillouin zone, or Bloch’s theorem might ﬁnd it useful to consult the ﬁrst few chapters of Kittel (1996). There, the concepts are introduced where they ﬁnd common use, in conventional solid-state physics. Additionally, appendix B of this text contains a brief introduction to the reciprocal lattice and the Brillouin zone. The earliest works on waves in periodic media are reviewed in, for example, Brillouin (1946).

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4
The Multilayer Film: A One-Dimensional Photonic Crystal
WE BEGIN OUR STUDY of photonic crystals by considering the simplest possible
case, a one-dimensional system, and applying the principles of electromagnetism and symmetry that we developed in the previous chapters. Even in this simple system, we can discern some of the most important features of photonic crystals in general, such as photonic band gaps and modes that are localized around defects. The optical properties of a one-dimensional layered system may be familiar, but by expressing the results in the language of band structures and band gaps, we can discover new phenomena such as omnidirectional reﬂectivity, as well as prepare for the more complicated two- and three-dimensional systems that lie ahead.
The Multilayer Film
The simplest possible photonic crystal, shown in ﬁgure 1, consists of alternating layers of material with different dielectric constants: a multilayer ﬁlm. This arrangement is not a new idea. Lord Rayleigh (1887) published one of the ﬁrst analyses of the optical properties of multilayer ﬁlms. As we will see, this type of photonic crystal can act as a mirror (a Bragg mirror) for light with a frequency within a speciﬁed range, and it can localize light modes if there are any defects in its structure. These concepts are commonly used in dielectric mirrors and optical ﬁlters (as in, e.g., Hecht and Zajac, 1997).
The traditional way to analyze this system, pioneered by Lord Rayleigh (1917), is to imagine that a plane wave propagates through the material and to consider the sum of the multiple reﬂections and refractions that occur at each interface. In this chapter, we will use a different approach—the analysis of
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THE MULTILAYER FILM

45

z

a

y x

Figure 1: The multilayer ﬁlm, a one-dimensional photonic crystal. The term “one-dimensional”
is used because the dielectric function ε(z) varies along one direction (z) only. The system
consists of alternating layers of materials (blue and green) with different dielectric constants,
with a spatial period a. We imagine that each layer is uniform and extends to inﬁnity along the x and y directions, and we imagine that the periodicity in the z direction also extends
to inﬁnity.

band structures—that is easily generalized to the more complex two- and threedimensional photonic crystals.1
We begin in the spirit of the previous chapter. By applying symmetry arguments, we can describe the electromagnetic modes sustainable by the crystal. The material is periodic in the z direction, and homogeneous in the xy plane. As we saw in the previous chapter, this allows us to classify the modes using k , kz, and n: the wave vector in the plane, the wave vector in the z direction, and the band number. The wave vectors specify how the mode transforms under translation operators, and the band number increases with frequency. We can write the modes in the Bloch form:

Hn,kz,k (r) = eik ·ρeikzzun,kz,k (z).

(1)

The function u(z) is periodic, with the property u(z) = u(z + R) whenever R is an integral multiple of the spatial period a. Because the crystal has continuous translational symmetry in the xy plane, the wave vector k can assume any value. However, the wave vector kz can be restricted to a ﬁnite interval, the one-dimensional Brillouin zone, because the crystal has discrete translational symmetry in the z direction. Using the prescriptions of the previous chapter, if the primitive lattice vector is azˆ then the primitive reciprocal lattice vector is (2π/a)zˆ and the Brillouin zone is −π/a < kz ≤ π/a.

1 Interestingly, in Lord Rayleigh’s ﬁrst attack on the problem in 1887, he used a cumbersome predecessor of Bloch’s theorem that had been worked out by Hill (1877). In modern terminology, Rayleigh was able to show that any one-dimensional photonic crystal has a band gap. When he returned to the problem in 1917, however, he switched to the sum-of-reﬂections technique.

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CHAPTER 4

Frequency ωa/2πc

GaAs Bulk
0.30 0.25 0.20 0.15

GaAs/GaAlAs Multilayer
0.30
0.25

GaAs/Air Multilayer
0.30 n=2
0.25

0.20

n=2

0.20

Photonic Band Gap

0.15

Photonic Band Gap

0.15

0.10

0.10

0.10

n=1

n=1

0.05

0.05

0.05

0.00

0.00

0.00

-0.5 -0.25 0 0.25 0.5

-0.5 -0.25 0 0.25 0.5

-0.5 -0.25 0 0.25 0.5

Wave vector ka/2π

Wave vector ka/2π

Wave vector ka/2π

Figure 2: The photonic band structures for on-axis propagation, as computed for three
different multilayer ﬁlms. In all three cases, each layer has a width 0.5a. Left: every layer has the same dielectric constant ε = 13. Center: layers alternate between ε of 13 and 12. Right: layers alternate between ε of 13 and 1.

The Physical Origin of Photonic Band Gaps

For now, consider waves that propagate entirely in the z direction, crossing the sheets of dielectric at normal incidence. In this case, k = 0 and only the wave vector component kz is important. Without possibility of confusion, we can abbreviate kz by k.
In ﬁgure 2, we plot ωn(k) for three different multilayer ﬁlms. The left-hand plot is for a system in which all of the layers have the same dielectric constant; the medium is actually uniform in all three directions. The center plot is for a structure with alternating dielectric constants of 13 and 12, and the right-hand plot is for a structure with a much higher dielectric contrast of 13 to 1.2
The left-hand plot is for a homogeneous dielectric medium for which we have arbitrarily assigned a periodicity of a. But we already know that in a homogeneous medium, the speed of light is reduced by the index of refraction. The modes lie along the light line (as in the subsection Index guiding of chapter 3), given by

ω(k)

=

√ck ε

.

(2)

2 We use these particular values because the static dielectric constant of gallium arsenide (GaAs) is about 13, and for gallium aluminum arsenide (GaAlAs) it is about 12, as reported in Sze (1981). These and similar materials are commonly used in devices. Air has a dielectric constant very nearly equal to 1.

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Because we have insisted that k repeat itself outside the Brillouin zone, the light line folds back into the zone when it reaches an edge. One can regard this as simply an eccentric way of relabeling of the solutions, in which k + 2π/a is replaced by k.3 The center plot, which is for a nearly-homogeneous medium, looks like the homogeneous case with one important difference: there is a gap in frequency between the upper and lower branches of the lines. There is no allowed mode in the crystal that has a frequency within this gap, regardless of k. We call such a gap a photonic band gap. The right-hand plot shows that the gap widens considerably as the dielectric contrast is increased.
We will devote a considerable amount of attention to photonic band gaps, and with good reason. Many of the promising applications of two- and threedimensional photonic crystals to date hinge on the location and width of photonic band gaps. For example, a crystal with a band gap might make a very good, narrow-band ﬁlter, by rejecting all (and only) frequencies in the gap. A resonant cavity, carved out of a photonic crystal, would have perfectly reﬂecting walls for frequencies in the gap.
Why does the photonic band gap appear? We can understand the gap’s physical origin by considering the electric ﬁeld mode proﬁles for the states immediately above and below the gap. The gap between bands n = 1 and n = 2 occurs at the edge of the Brillouin zone, at k = π/a. For now, we focus on the band structure in the center panel of ﬁgure 2, corresponding to the conﬁguration that is a small perturbation of the homogeneous system. For k = π/a, the modes have a wavelength of 2a, twice the crystal’s spatial period (or lattice constant). There are two ways to center a mode of this type. We can position the nodes in each low-ε layer, as in ﬁgure 3(a), or in each high-ε layer, as in ﬁgure 3(b). Any other position would violate the symmetry of the unit cell about its center.
In our study of the electromagnetic variational theorem, in the section Electromagnetic Energy and the Variational Principle of chapter 2, we found that the lowfrequency modes concentrate their energy in the high-ε regions, and the highfrequency modes have a larger fraction of their energy (although not necessarily a majority) in the low-ε regions. With this in mind, it is understandable why there is a frequency difference between the two cases. The mode just under the gap has more of its energy concentrated in the ε = 13 regions as shown in ﬁgure 3(c), giving it a lower frequency than the next band, most of whose energy is in the ε = 12 regions as shown in ﬁgure 3(d).
The bands above and below the gap can be distinguished by where the energy of their modes is concentrated: in the high-ε regions, or in the low-ε regions. Often, especially in the two- and three-dimensional crystals of the later chapters, the low-ε regions are air regions. For this reason, it is convenient to refer to the band above a photonic band gap as the air band, and the band below a gap as the dielectric band. The situation is analogous to the electronic band structure

3 The reader may be familiar with the relabeling of k + 2π/a as k from the phenomenon of quasi– phase-matching, in which states at the same frequency can couple to one another if their k values differ by multiples of 2π/a, when a weak periodicity a is introduced into a medium.

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(a) E-field for mode at top of band 1 ε=13 ε=12
(b) E-field for mode at bottom of band 2
(c) Local energy density in E-field, top of band 1
(d) Local energy density in E-field, bottom of band 2
Figure 3: The modes associated with the lowest band gap of the band structure plotted
in the center panel of ﬁgure 2, at k = π/a. (a) Electric ﬁeld of band 1; (b) electric ﬁeld of band 2; (c) electric-ﬁeld energy density ε |E|2/8π of band 1; (d) electric-ﬁeld energy density
of band 2. In the depiction of the multilayer ﬁlm, blue indicates the region of higher dielectric
constant (ε = 13).
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THE MULTILAYER FILM

49

of semiconductors, in which the conduction band and the valence band bracket the fundamental gap.
This heuristic, based on the variational theorem, can be extended to describe the conﬁguration with a large dielectric contrast. In this case, we ﬁnd that the ﬁeld energy for both bands is primarily concentrated in the high-ε layers, but in different ways—the ﬁrst band being more concentrated than the second. These ﬁelds are shown in ﬁgure 4, corresponding to the right panel of ﬁgure 2. The gap arises from this difference in ﬁeld energy location. Consequently, we will still refer to the upper band as the air band and the lower as the dielectric band.
We conclude this section with the observation that in one dimension, a gap usually occurs between every set of bands, at either the Brillouin zone’s edge or its center.4 This is illustrated for the band structure of a multilayer ﬁlm in ﬁgure 5. Finally, we emphasize that band gaps always appear in a one-dimensional photonic crystal for any dielectric contrast. The smaller the contrast, the smaller the gaps, but the gaps open up as soon as ε1/ε2 = 1. This statement is quantiﬁed in the following section.

The Size of the Band Gap
The extent of a photonic band gap can be characterized by its frequency width ∆ω, but this is not a really useful measure. Remember from the section Scaling Properties of the Maxwell Equations of chapter 2 that all of our results are scalable. If the crystal were expanded by a factor s, the corresponding band gap would have a width ∆ω/s. A more useful characterization, which is independent of the scale of the crystal, is the gap–midgap ratio. Letting ωm be the frequency at the middle of the gap, we deﬁne the gap–midgap ratio as ∆ω/ωm, generally expressed as a percentage (e.g., a “10% gap” refers to a gap–midgap ratio of 0.1). If the system is scaled up or down, all of the frequencies scale accordingly, but the gap–midgap ratio remains the same. Thus, when we refer to the “size” of a gap, we are generally referring to the gap–midgap ratio. For the same reason, in the band diagrams in ﬁgure 2, as well as all of the other band diagrams in this book, the frequency and wave vector are plotted in dimensionless units ωa/2πc and ka/2π. The dimensionless frequency is equivalent to a/λ, where λ is the vacuum wavelength (given by λ = 2πc/ω).
We are emphasizing general principles of periodic systems that will apply equally well to the more complicated two- and three-dimensional structures of the later chapters. It is worthwhile, however, to point out a few exceedingly useful analytical results that are only possible for the special case of one-dimensional problems.
4 There is a special exception for the quarter-wave stack described in the next section. In that case, while there is always a gap at the edge of the Brillouin zone, there is no gap at the center, because every successive pair of bands is degenerate at k = 0.

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(a) E-field for mode at top of band 1 ε=13 ε=1

CHAPTER 4

(b) E-field for mode at bottom of band 2

(c) Local energy density in E-field, top of band 1

(d) Local energy density in E-field, bottom of band 2
Figure 4: The modes associated with the lowest band gap that is shown in the band
structure of the right-hand panel of ﬁgure 2, at k = π/a. The situation is similar to that of ﬁgure 3, but the dielectric contrast is larger. The blue and green regions correspond to ε of
13 and 1, respectively.
In a multilayer ﬁlm with weak periodicity, we can derive a simple formula for the size of the band gap from the perturbation theory of the section The Effect of Small Perturbations of chapter 2. Suppose that the two materials in a multilayer ﬁlm have dielectric constants ε and ε + ∆ε, and thicknesses a − d and d. If either the dielectric contrast is weak (∆ε/ε 1) or the thickness d/a is small, then the

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1

Frequency ωa/2πc

0.8

0.6 air band

0.4 Photonic Band Gap

0.2
0 -0.5

dielectric band

-0.25

0

0.25

0.5

Wave vector ka/2π

Figure 5: The photonic band structure of a multilayer ﬁlm with lattice constant a and alternating layers of different widths. The width of the ε = 13 layer is 0.2a, and the width of the ε = 1 layer is 0.8a.

gap–midgap ratio between the ﬁrst two bands is approximately

∆ω ωm

≈

∆ε ε

·

sin(πd/a) π.

(3)

This quantiﬁes our previous statement that even an arbitrarily weak periodicity
gives rise to a band gap in a one-dimensional crystal. For one of the structures considered in the previous section, with ∆ε/ε = 1/12 and d = 0.5a (see the center
panel of ﬁgure 2), the perturbative formula (3) predicts a 2.65% gap, which is in
good agreement with the results of a more accurate numerical calculation (2.55%).
Equation (3) would predict that the gap–midgap ratio is maximized for d = 0.5a, but this is valid only for small ∆ε/ε. More generally, one can obtain a number of analytical results for arbitrary ∆ε/ε, which we summarize here and√are derived in, for example, Yeh (1988). For two materials with refractive indices ( ε) n1 and n2 and thicknesses d1 and d2 = a − d1, respectively, the normal-incidence gap is maximized when d1n1 = d2n2, or, equivalently, d1 = an2/(n1 + n2). In this speciﬁc case, it can be shown that the midgap frequency ωm is

ωm

=

n1 + n2 4n1n2

·

2πc a

.

(4)

The corresponding vacuum wavelength λm = 2πc/ωm satisﬁes the relations λm/n1 = 4d1 and λm/n2 = 4d2, which means that the individual layers are exactly

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CHAPTER 4

a quarter-wavelength in thickness. For this reason, this type of multilayer ﬁlm is called a quarter-wave stack. The reason why the gap is maximized for a quarterwave stack is related to the property that the reﬂected waves from each layer are all exactly in phase at the midgap frequency. For the gap between the ﬁrst two bands of a quarter-wave stack, the gap–midgap ratio is

∆ω ωm

=

4 π

sin−1

|n1 − n2| n1 + n2

.

(5)

Returning to ﬁgure 2, the case that is shown in the right-hand panel is a multilayer ﬁlm with a dielectric contrast of 13:1 and d1 = d2 = 0.5a, which is not a quarterwave stack. Numerically, we ﬁnd that this structure produces a 51.9% gap. If instead we had chosen d1 ≈ 0.217, the structure would be a quarter-wave stack with a 76.6% gap, as computed from equation (5). Figure 5 shows the results for d1 = 0.2a, which is nearly a quarter-wave stack, and has a computed band gap of 76.3%. Note also the small gap at k = 0, which would go to zero for a quarter-wave
stack.

Evanescent Modes in Photonic Band Gaps

The key observation of the section The Physical Origin of Photonic Band Gaps was that the periodicity of the crystal induced a gap in its band structure. No electromagnetic modes are allowed to have frequencies in the gap. But if this is indeed the case, what happens when we send a light wave (with a frequency in the photonic band gap) onto the face of the crystal from outside?
No purely real wave vector exists for any mode at that frequency. Instead, the wave vector is complex. The wave amplitude decays exponentially into the crystal. When we say that there are no states in the photonic band gap, we mean that there are no extended states like the mode given by equation (1). Instead, the modes are evanescent, decaying exponentially:

H(r) = eikzu(z)e−κz.

(6)

They are just like the Bloch modes we constructed in equation (1), but with a complex wave vector k + iκ. The imaginary component of the wave vector causes the decay on a length scale of 1/κ.
We would like to understand how these evanescent modes originate, and what determines κ. This can be accomplished by examining the bands in the immediate
vicinity of the gap. Return to the right-hand plot of ﬁgure 2. Suppose we try to approximate the second band near the gap by expanding ω2(k) in powers of k about the zone edge k = π/a. Because of time-reversal symmetry, the expansion
cannot contain odd powers of k, so to lowest order:

∆ω = ω2(k) − ω2

π a

≈α

k− π

2
= α(∆k)2,

a

(7)

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ω

Real ∆k

Imaginary ∆k

Figure 6: Schematic illustration of the complex band structure of the multilayer ﬁlm. The
upper and lower blue lines correspond to the bottom of band 2 and the top of band 1,
respectively. The evanescent states occur on the red line, which extends along the
imaginary-k axis running out of the page. The maximum decay occurs roughly at the center
of the gap.
where α is a constant depending on the curvature of the band (i.e., the second derivative).
Now we can see where the complex wave vector originates. For frequencies slightly higher than the top of the gap, ∆ω > 0. In this case, ∆k is purely real, and we are within band 2. However, for ∆ω < 0, when we are within the gap, ∆k is purely imaginary.5 The states decay exponentially since ∆k = iκ. As we traverse the gap, the decay constant κ grows as the frequency reaches the gap’s center, then disappears again at the lower gap edge. This behavior is depicted in ﬁgure 6. By the same token, larger gaps usually result in a larger κ at midgap, and thus less penetration of light into the crystal; for a multilayer ﬁlm, minimal penetration is therefore achieved in the quarter-wave stack described by the preceding section.
Although evanescent modes are genuine solutions of the eigenvalue problem, they diverge as z goes to ±∞ (depending on the sign of κ). Consequently, there is no physical way to excite them within an idealized crystal of inﬁnite extent. However, a defect or an edge in an otherwise perfect crystal can terminate this exponential growth and thereby sustain an evanescent mode. If one or more evanescent modes is compatible with the structure and symmetry of a given crystal defect, we can then excite a localized mode within the photonic band gap. And, as a general rule of thumb, we can localize states near the middle of the gap much more tightly than states near the gap’s edge.6
5 Technically, we are exploiting concepts from complex analysis. The eigenvalues ω are generally analytic functions of any smooth parameter of the operator Θˆ k, so we can use the analytic continuation of ω(k) into the complex k domain via its Taylor expansion.
6 There are subtle exceptions to this rule. For example, with certain band structures in two and three dimensions, saddle points in the bands can lead to strong localization away from midgap (Ibanescu et al., 2006).

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CHAPTER 4

surface

a

defect

z
y x
Figure 7: Schematic illustration of possible sites of localized states for a one-dimensional photonic crystal. The states are planar and would be localized differently near the differently
colored regions, which break the symmetry in the z direction. We will call a mode at the
edge of the crystal (green) a surface state, and a mode within the bulk of the crystal (blue) a defect state.
Of course, one-dimensional photonic crystals can localize states only in one dimension. The state is conﬁned to a given plane, as shown in ﬁgure 7. In the section Localized Modes at Defects, we will discuss the nature of such states when they lie deep within the bulk of a photonic crystal. In certain circumstances, however, an evanescent mode can exist at the face of the crystal. We will also discuss these surface states in the section Surface States.
Off-Axis Propagation
So far, we have considered the modes of a one-dimensional photonic crystal which happen to have k = 0; that is, modes that propagate only in the z direction. In this section we will discuss off-axis modes. Figure 8 shows the band structure for modes with k = kyyˆ for the one-dimensional photonic crystal described in the caption of ﬁgure 5.
The most important difference between on-axis and off-axis propagation is that there are no band gaps for off-axis propagation when all possible ky are included. This is always the case for a multilayer ﬁlm, because the off-axis direction contains no periodic dielectric regions to coherently scatter the light and split open a gap. (Despite this, we will see in the section Omnidirectional Multilayer Mirrors that it is

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1

Frequency ωa/2πc

0.8

0.6

n=2

0.4

n=1

0.2

TM: x - polarized

TE: yz - polarized

0

-0.5

kz

0

0.5

1

Wave vector ka/2π

ky 1.5

Figure 8: The band structure of a multilayer ﬁlm. The on-axis bands (0, 0, kz) are shown on the left side, and an off-axis band structure (0, ky, 0) is displayed on the right. On-axis, the bands overlap—they are degenerate. Along ky, the bands split into two distinct polarizations. Blue indicates TM modes polarized so that the electric ﬁeld points in the x direction, and red indicates TE modes polarized in the yz plane. The layered structure is the same as the one
described in the caption of ﬁgure 5.

still possible to design a multilayer ﬁlm that reﬂects external plane waves that are incident from any angle.)
Another difference between the on-axis and off-axis cases involves the degeneracy of the bands. For the case of on-axis propagation, the electric ﬁeld is oriented in the xy plane. We might choose the two basic polarizations as the x and y directions. Since those two modes differ only by a rotational symmetry which the crystal possesses, they must be degenerate. (How could the crystal distinguish between the two polarizations?)
However, for a mode propagating with an arbitrary direction of k, this symmetry is broken. The degeneracy is lifted. There are other symmetries; for example, notice that the system is invariant under reﬂection through the yz plane. For the special case of propagation down the dielectric sheets, in the y direction, we know from the symmetry discussion of chapter 3 that the possible polarizations are in the x direction (TM) or in the yz plane (TE). But there is no rotational symmetry relationship between these two bands, so they will generally have different frequencies. All of these phenomena are displayed in ﬁgure 8.
Although ω(k) for the two different polarizations have different slopes, both are approximately linear at long wavelengths (ω → 0). This long-wavelength

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x z

Figure 9: A sketch of the displacement ﬁeld lines for a long-wavelength mode traveling in
the y direction (out of the page). In the left ﬁgure, the ﬁelds are oriented along x. In the right ﬁgure, the ﬁelds are oriented primarily along z. The blue regions correspond to high ε.

behavior is characteristic of all photonic crystals, regardless of geometry or

dimensionality:

ων(k) = cν(kˆ )k.

(8)

Here ν is a label that refers to one of the two possible polarizations or, equivalently, one of the ﬁrst two bands. In general, cν will depend on both the direction of k and on ν.
Why is the dispersion relation always linear at long wavelengths? At long wavelengths, the electromagnetic wave doesn’t probe the ﬁne structure of the crystal lattice. Instead, the light effectively sees a homogeneous dielectric medium, with an effective dielectric constant that is a weighted average over all of the “microscopic” variations in ε.
In many cases, the averaged dielectric constant will be a function of the polarization (the direction of E). In those cases, the effective medium is anisotropic and the dielectric function is a 3-by-3 matrix, a tensor. The “principal axes” are the basic symmetry axes for which the dielectric tensor is diagonalized. Procedurally, we can imagine measuring the effective dielectric constant along each of three directions by applying a static ﬁeld in a capacitance measurement. A general analytical expression for the effective dielectric constants of a photonic crystal is not available, but the constants can be calculated numerically.7
Returning to the multilayer ﬁlm, we would like to understand why modes polarized in the x direction (band 1 in ﬁgure 8) have a lower frequency than modes polarized in the yz plane (band 2). Once again we use our heuristic: the lower modes concentrate their electrical energy in the high-ε regions. In this case, we focus on the long-wavelength limit of each mode. The ﬁelds for both bands are shown schematically in ﬁgure 9. For the x-polarized wave, the displacement ﬁelds lie in the high-ε regions. But at long wavelengths, the polarization of

7 One useful analytical constraint for the effective dielectric constant of a general photonic crystal is provided by the Weiner bounds, as in Aspnes (1982). Speciﬁcally, for a two-material composite, each effective dielectric constant εα is bounded by
( f1ε−1 1 + f2ε−2 1)−1 ≤ εα ≤ f1ε1 + f2ε2
where f1 and f2 are the volume fractions of the materials with dielectric constants ε1 and ε2.

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1

0.8

Frequency ωa/2πc

0.6

0.4

0.2

0

0

0.5

1.0

1.5

2.0

2.5

Wave vector kya/2π

Figure 10: Two superimposed x-polarized band structures of a multilayer ﬁlm, showing how the bandwidths vary with ky. The blue lines refer to bands along (0, ky, 0), while the green lines beside them refer to the same bands along (0, ky, π/a). The regions in between are shaded gray to indicate where the continuum of bands for intermediate kz would lie. Only modes with electric ﬁeld oriented along the x direction are shown. The straight red line is the light line ω = cky. The layered material is the same as the one described in the caption
of ﬁgure 5.

band 2 is almost entirely along the z direction, crossing both the low-ε and the high-ε regions. Continuity forces the ﬁeld to penetrate the low-ε region, leading to a higher frequency.
We can also understand the short-wavelength, large-k limit of the band structure. In ﬁgure 8, notice that the width of each band is determined by the difference between frequencies at the zone center (kz = 0) and the zone edge (kz = π/a). As ky is increased, the bandwidths decrease to zero. This is illustrated in ﬁgure 10, which shows the superposition of two band structures. The blue lines represent states along k = (0, ky, 0) and the green lines represent states along k = (0, ky, kz = π/a). As we saw for the case of a plane of glass, once the frequency is below the light line ω = cky, the modes are index guided and decay exponentially into the air region. As ky is increased, the overlap between modes within neighboring layers of high-ε material goes exponentially to zero. The coupling between neighboring planes becomes small, and each plane essentially guides its own mode independent of its neighbors.8 In this case, the mode
8 In solid-state physics, the analogous system is the tight-binding model in the limit of small hopping. See, for example, Harrison (1980).

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Figure 11: A defect in a multilayer ﬁlm, formed by doubling the thickness of a single low-ε
layer in the structure of ﬁgure 5. Note that this can be considered to be an interface between two perfect multilayer ﬁlms. The red curve is the electric-ﬁeld strength of the defect state associated with this structure (for on-axis propagation).
frequency becomes independent of the on-axis wave vector, and every mode in the band becomes the frequency of a guided mode that is trapped by the high-ε layers.
Localized Modes at Defects
Now that we understand the band structure of a perfectly periodic system, we can examine systems in which the translational symmetry has been broken by a defect. Suppose that the defect consists of a single layer of the one-dimensional photonic crystal that has a different width than the rest. Such a system is shown in ﬁgure 11. We no longer have a perfectly periodic lattice. However, we expect intuitively that if we move many wavelengths away from the defect, the modes should look similar to the corresponding modes of a perfect crystal.
For now, we restrict our attention to on-axis propagation and consider a mode with frequency ω in the photonic band gap. There are no extended modes with frequency ω inside the periodic lattice. Introducing the defect does not change that fact. The destruction of periodicity prevents us from describing the modes of the system with wave vector k, but we can still employ our knowledge of the band structure to determine whether a certain frequency will support extended states inside the rest of the crystal. In this way, we can divide up the frequency range into regions in which the states are extended and regions in which they are evanescent, as in ﬁgure 12.
Defects may permit localized modes to exist, with frequencies inside photonic band gaps. If a mode has a frequency in the gap, then it must exponentially decay once it enters the crystal. The multilayer ﬁlms on both sides of the defect behave like frequency-speciﬁc mirrors. If two such ﬁlms are oriented parallel to one another, any z-propagating light trapped between them will just bounce back and forth between these two mirrors. And because the mirrors localize light within a ﬁnite region, the modes are quantized into discrete frequencies, as described in the section Discrete vs. Continuous Frequency Ranges of chapter 2.
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Extended States

Density of States
Photonic Band Gap Photonic Band Gap

ω
Defect State Evanescent States
Figure 12: The division of frequency space into extended and evanescent states. In this sketch, the density of states (the number of allowed modes per unit frequency) is zero in the band gaps of the crystal (yellow). Modes are allowed to exist in these regions only if they are evanescent, and only if the translational symmetry is broken by a defect. Such a mode is shown in red.
Consider the family of localized states generated by continuously increasing the thickness of the defect layer (shifting the crystal to either side of the defect). The bound mode(s) associated with each member of this family will have a different frequency. As the thickness is increased, the frequency will decrease, because the mode has more space to oscillate; this decreases the numerator of the variational theorem and thus the frequency (all other things equal). In fact, as the thickness increases, a sequence of discrete modes are pulled down into the gap from the upper bands. The ﬁrst such mode is shown in ﬁgure 11, pulled down by doubling the thickness of a low-ε layer. On the other hand, if we were to keep the thickness ﬁxed and either increase or decrease the dielectric constant ε of a single layer, we would decrease or increase the frequency, respectively, due to the change in the denominator of the variational theorem. So, in general, a defect may either pull modes down into the gap from the upper bands, or push modes up into the gap from the lower band. Moreover, the degree of localization of the defect mode will be greatest when the frequency is near the center of the gap, as shown in ﬁgure 6. States with frequencies in the center of the gap will be most strongly attached to the defect.
The density of states of a system is the number of allowed states per unit interval of ω. If a single state is introduced into the photonic band gap, then the density of states of the system in ﬁgure 12 is zero in the photonic band gap,
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Figure 13: The electric ﬁeld strength associated with a localized mode at the surface of a
multilayer ﬁlm. In particular, the mode at ky = 2π/a from ﬁgure 14 is shown. (This mode
actually oscillates in sign with each period of the crystal, but with an amplitude too small to see clearly here.)
except for a single peak associated with the defect.9 This property is exploited in the bandpass ﬁlter known as the dielectric Fabry–Perot ﬁlter, a case that will be discussed in chapter 10. It is particularly useful at visible-light frequencies because of the relatively low losses of dielectric materials.
Similarly, if we include off-axis wave vectors, we obtain states that are localized in the z direction, but that propagate (are guided) along the interface (kz = iκ, k = 0). These guided modes, forming a planar waveguide, can differ markedly from the conventional solution of total internal reﬂection, discussed in the subsection Index guiding of chapter 3; for example, as in ﬁgure 11, they can be guided in a lower-ε region.10 This idea can be generalized to include the case of an interface between two different multilayer ﬁlms with different spatial periods. Localized states can exist as long as the band gaps of the two photonic crystals overlap.
Surface States
We have seen under what conditions we can localize electromagnetic modes at defects in a multilayer ﬁlm. In a similar fashion, we can also localize modes at its surface, called surface states. In the previous section, the mode was bound because its frequency was within the photonic band gap of the ﬁlms on both sides. But at the surface, there is a band gap on only one side of the interface. The exterior medium (air, in our typical example) does not have a band gap.
In this case, light is bound to the surface if its frequency is below the light line. Such a wave is index-guided, a phenomenon that is a generalization of total internal reﬂection (see the subsection Index guiding of chapter 3). An example is depicted in ﬁgure 13. At a surface, we must consider whether the modes are extended or localized in both the air region and the layered material, and we
9 This peak is described by a Dirac delta function in the density of states. 10 This was pointed out by Yeh and Yariv (1976). We return to an analogous three-dimensional
structure, the Bragg ﬁber, in chapter 9.

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1

0.8

Frequency ωa/2πc

0.6

0.4

ED states

DE states

EE states 0.2
DD surface states

0

0

0.5

1

1.5

2

2.5

Wave vector kya/2π

Figure 14: The x-polarized band structure at the surface of a multilayer ﬁlm. The shaded
regions describe states that are extended in the air region (blue, ED), in the layered material (pink, DE), or in both (purple, EE). The green (DD) line represents a band of surface states conﬁned at the interface. The layered material is the same as the one described in ﬁgure 5.
The surface is terminated by a layer of high dielectric with a width of 0.1a (half of its usual
thickness), as shown in ﬁgure 13.

must consider all possibilities for k . The appropriate band structure is shown in ﬁgure 14 (restricting ourselves to xˆ polarizations as in ﬁgure 10). We divide the band structure into four regions, depending on whether they are localized or extended in the air and crystal regions. For example, the label “DE” means that modes in that region Decay in the air region, and are Extended in the crystal region.
The EE modes are spatially extended on both sides of the surface, the DE modes decay into the air region and are extended into the crystal, and the ED modes are extended in the air region, but decay inside the crystal. Only when a mode is evanescent on both sides of the surface—it is both in a band gap and below the light line—can we have a surface wave. This is possible in any of the white regions of ﬁgure 14, and one such localized mode is labelled DD. In fact, every periodic material has surface modes for some choice of termination, a phenomenon we shall discuss again in the next two chapters.

Omnidirectional Multilayer Mirrors
As described in the section Off-Axis Propagation, a multilayer ﬁlm does not have a complete band gap, once one allows for a component of the wave vector that is

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parallel to the layers. Another way to state this is that for every choice of ω, there exist extended modes in the ﬁlm for some wave vectors (k , kz). Given this fact, it may seem paradoxical that a properly designed multilayer structure can still reﬂect light waves that are incident from any angle, with any polarization, if it has a frequency that is within a speciﬁed range.
Such a device, an omnidirectional mirror, relies on two physical properties. First, k is conserved at any interface parallel to the layers, if the light source is far enough away that it does not interrupt the translational symmetry of the structure in the direction. Second, light that is incident from air must have ω > c|k |, corresponding to the freely propagating modes above the light line, just as we saw for index guiding and surface states (see the subsection Index guiding of chapter 3 and the section Surface States). Modes that are below the light line are evanescent modes that cannot reach the mirror from a faraway source. Because of these two properties, the modes that the crystal harbors below the light line are irrelevant for the purpose of reﬂection.
To investigate omnidirectional reﬂection, we plot ω vs. ky, just as we did in ﬁgure 14. The yz plane is the plane of incidence, with the y direction parallel to the layers, and the z direction perpendicular to the layers. Now, however, we must consider both of the possible polarization states: TM, in which the electric ﬁeld is perpendicular to the plane of incidence (as in ﬁgure 14); and TE, in which the electric ﬁeld is within the plane of incidence. In this context, it is common to refer to TM modes as s-polarized and TE modes as p-polarized. Our example structure, here, is a quarter-wave stack (see the section The Size of the Band Gap) consisting of layers with ε of 13 and 2 , rather than 13 and 1 as before. The resulting band diagram is shown in ﬁgure 15. Indeed, there is a range of frequencies (yellow) within which all of the modes of the multilayer ﬁlm are below the light line. Within this range, any incident plane waves cannot couple to the extended states in the layers. Instead, their ﬁelds must decay exponentially within the quarterwave stack. The transmission through such a mirror will drop exponentially with the number of layers: the light will be perfectly reﬂected, insofar as material absorption can be neglected.
Omnidirectional reﬂection is not a general property of one-dimensional photonic crystals. There are two necessary conditions. First, the dielectric contrast between the two mirror materials must be sufﬁciently large so that the point labelled U (upper) is above the point L (lower) in ﬁgure 15. If the band gap is too narrow, we will hit the top U of the ky = 0 gap before the bottom of the gap has exited the light cone at L. Second, the smaller of the two dielectric constants (ε1) must be larger than the dielectric constant of the ambient medium (εa) by a critical amount. This critical contrast with the ambient medium is reached when the p (TE) bands are pulled down in frequency far enough that point B in ﬁgure 15 (where the ﬁrst and second p bands intersect) does not fall above the light line. This second criterion is why we chose ε1 = 2 rather than 1 in our example.

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1.4

1.2

p-polarized

s-polarized

Frequency ωa/2πc

1

0.8

0.6

0.4
B 0.2

L

U

0

1

0.8 0.6 0.4 0.2

0

0.2 0.4 0.6 0.8

1

Wave vector kya/2π

Figure 15: Extended modes (shaded regions) for off-axis propagation vectors (0, ky, kz) in a quarter-wave stack with ε of 13 and 2. The right side (blue) indicates modes with E ﬁelds polarized in the x direction (TM or s-polarized), similar to ﬁgure 10. The left side (green) indicates modes with ﬁelds polarized in the yz incidence plane (TE or p-polarized). The straight red line is the light line ω = cky, above which extended modes exist in air. In yellow
is shaded the ﬁrst frequency range of omnidirectional reﬂection (with lower and upper
edges at L and U, respectively). The dashed white line corresponds to Brewster’s angle, which gives rise to the crossing at B.

The B point falls on a line that corresponds to Brewster’s angle, at which p-polarized light has no reﬂection at the ε1/ε2 interface.11 The lack of reﬂection is what permits the bands to intersect. Combining these two criteria, ﬁg√ure 16 show√s the size of the “omnidirectional gap” as a function of the ratios ε2/ε1 and ε1/εa, for the case of quarter-wave stacks. Strictly speaking, a quarter-wave stack does not maximize the size of the omnidirectional gap, but in practice it very nearly does so. In ﬁgure 16, had we used the optimal layer spacings instead of the quarter-wave spacings, the contours would be displaced by less than about 2% along either axis.
A suitably designed multilayer ﬁlm can therefore function as an omnidirectional mirror, but there are some things it cannot do. Its reﬂective property depends on the translational symmetry of the interface, and consequently it cannot
11 The existence of Brewster’s angle (see, e.g., Jackson, 1998) is related to the observation that light reﬂected obliquely from water, glass, or even asphalt is preferentially s-polarized. This, in turn, is why polarizing sunglasses are effective at reducing glare from the road.

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3.5
3 Si/SiO2 in air
2.5
2 εa
1.5 ε2 ε1
1

CHAPTER 4
70%
60%
50% 40% 30% 20% 10% 0%

0.5

0

0

0.5

1

1.5

2

2.5

3

3.5

ε1/ εa

Figure 16: The size of the omnidirectional gap as a function of the dielectric constants of the layers, for a quarter-wave stack. The gap–midgap ratios are labelled on the right side of each curve. The system is illustrated in the inset, in which light is incident from an ambient medium
with dielectric constant εa. The two materials of the ﬁlm have dielectric constants ε1 and ε2, with ε1 < ε2. It does not matter which material forms the edge of the mirror. The pink shaded
area is the region in which there is a nonzero omnidirectional gap. Some common materials, such as the silicon/silica/air combination indicated by the arrow, fall within this region.

conﬁne a mode in three dimensions. In addition, if the interface is not ﬂat, or if there is an object (or a light source) close to the surface, then k is not conserved. In that case, light will generally couple to extended modes propagating in the mirror and will be transmitted. There is an interesting exception, however: if the mirror is curved around a hollow sphere or cylinder, then the continuous rotational symmetry can substitute for translational symmetry, and light can be localized within the core. As with the planar mirror, the leakage rate from the core to the exterior decreases exponentially with the number of layers. The cylindrical case (discussed further in chapter 9) was called a Bragg ﬁber by Yeh et al. (1978), and the spherical case has been dubbed a Bragg onion by Xu et al. (2003). Here, one did not require omnidirectional mirrors to obtain localized modes, because a mode’s rotational symmetry imposes restrictions on the angles that it can escape into at large radii.

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Further Reading
Many of the theorems we have developed and the properties we have observed for photonic crystals have analogues in quantum mechanics and solid-state physics. For readers familiar with those ﬁelds, appendix A provides a comprehensive listing of these analogies.
The conventional treatment of the multilayer ﬁlm, including the calculation of absorption and reﬂection coefﬁcients, can be found in Hecht and Zajac (1997). The use of the multilayer ﬁlms in optoelectronic devices is widespread in current literature. For example, Fowles (1975) outlines their use in Fabry–Perot ﬁlters, and Yeh (1988, p. 337) explains how they are incorporated into distributed feedback lasers. A historical review of the study of electromagnetic waves in onedimensionally periodic structures can be found in Elachi (1976). Despite the long history of multilayer ﬁlms, the possibility of achieving omnidirectional reﬂection was not apparent until 1998, when the idea was proposed (Winn et al., 1998) and realized experimentally (Fink et al., 1998).
The details of the computational scheme we used to compute band structures and eigenmodes can be found in Johnson and Joannopoulos (2001), and are also summarized in appendix D along with various alternative methods.

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5
Two-Dimensional Photonic Crystals
NOW THAT WE HAVE discussed some interesting properties of one-dimensional
photonic crystals, in this chapter we will see how the situation changes when the crystal is periodic in two directions and homogeneous in the third. Photonic band gaps appear in the plane of periodicity. For light propagating in this plane, the harmonic modes can be divided into two independent polarizations, each with its own band structure. As before, we can introduce defects in order to localize light modes, but in this case we can localize a mode in two dimensions, rather than just one dimension.
Two-Dimensional Bloch States
A two-dimensional photonic crystal is periodic along two of its axes and homogeneous along the third axis. A typical specimen, consisting of a square lattice of dielectric columns, is shown in ﬁgure 1. We imagine the columns to be inﬁnitely tall; the case of a ﬁnite extent in the third direction is treated in chapter 8. For certain values of the column spacing, this crystal can have a photonic band gap in the xy plane. Inside this gap, no extended states are permitted, and incident light is reﬂected. Unlike the multilayer ﬁlm, this two-dimensional photonic crystal can prevent light from propagating in any direction within the plane.
As always, we can use the symmetries of the crystal to characterize its electromagnetic modes. Because the system is homogeneous in the z direction, we know that the modes must be oscillatory in that direction, with no restrictions on the wave vector kz. In addition, the system has discrete translational symmetry in the xy plane. Speciﬁcally, ε(r) = ε(r + R), as long as R is any linear combination of the primitive lattice vectors axˆ and ayˆ . By applying Bloch’s theorem, we can focus our attention on the values of k that are in the Brillouin zone. As before, we use the label n (band number) to label the modes in order of increasing frequency.
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a r

z
y x

Figure 1: A two-dimensional photonic crystal. This material is a square lattice of dielectric
columns, with radius r and dielectric constant ε. The material is homogeneous along the z direction (we imagine the cylinders are very tall), and periodic along x and y with lattice constant a. The left inset shows the square lattice from above, with the unit cell framed in red.

Indexing the modes of the crystal by kz, k , and n, they take the now-familiar form of Bloch states

H(n,kz,k )(r) = eik ·ρeikzzu(n,kz,k )(ρ).

(1)

In this equation, ρ is the projection of r in the xy plane and u(ρ) is a periodic function, u(ρ) = u(ρ + R), for all lattice vectors R. The modes of this system look similar to those of the multilayer ﬁlm that we saw in equation (1) of chapter 4. The key difference is that in the present case, k is restricted to the Brillouin zone and kz is unrestricted. In the multilayer ﬁlm, the roles of these two wave vectors were reversed. Also, u is now periodic in the plane, and not in the z direction as before.
Any modes with kz = 0 (i.e. that propagate strictly parallel to the xy plane) are invariant under reﬂections through the xy plane. As discussed in chapter 3, this mirror symmetry allows us to classify the modes by separating them into two distinct polarizations. Transverse-electric (TE) modes have H normal to the plane, H = H(ρ)zˆ, and E in the plane, E(ρ) · zˆ = 0. Transverse-magnetic (TM) modes have just the reverse: E = E(ρ)zˆ and H(ρ) · zˆ = 0.
The band structures for TE and TM modes can be completely different. It is possible, for example, that there are photonic band gaps for one polarization but not for the other polarization. In the coming sections, we will investigate the TE and TM band structures for two different two-dimensional photonic crystals, mainly restricting ourselves to in-plane (kz = 0) propagation. The results will provide some useful insights into the appearance of band gaps.

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0.8

0.7

Frequency ωa/2πc

0.6

0.5

0.4
TE modes 0.3

M 0.2

0.1

TM modes

Γ

X

0

Γ

X

M

Γ

Figure 2: The photonic band structure for a square array of dielectric columns with
r = 0.2a. The blue bands represent TM modes and the red bands represent TE modes. The
left inset shows the Brillouin zone, with the irreducible zone shaded light blue. The right inset
shows a cross-sectional view of the dielectric function. The columns (ε = 8.9, as for alumina) are embedded in air (ε = 1).

A Square Lattice of Dielectric Columns
Consider light that propagates in the xy plane of a square array of dielectric columns, such as the structure depicted in ﬁgure 1, with lattice constant a. The band structure for a crystal consisting of alumina (ε = 8.9) rods in air, with radius r/a = 0.2, is plotted in ﬁgure 2. Both the TE and the TM band structures are shown. (As described in the section The Size of the Band Gap of chapter 4, the frequency is expressed as a dimensionless ratio ωa/2πc.) The horizontal axis shows the value of the in-plane wave vector k . As we move from left to right, k moves along the triangular edge of the irreducible Brillouin zone, from Γ to X to M, as shown in the inset to ﬁgure 2.
Since this is the ﬁrst photonic crystal we have encountered that exhibits a complicated band structure, we will discuss it in detail. Speciﬁcally, we will describe the nature of the modes when k is right at the special symmetry points of the Brillouin zone, and we will investigate the appearance of the band gaps. The reason why we have plotted k only along the edge of the Brillouin zone is that the minima and the maxima of a given band (which determine the band gap) almost always occur at the zone edges, and often at a corner. While this is not guaranteed,

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it is true in most cases of interest, such as all of the structures discussed in this chapter.
The square lattice array has a square Brillouin zone, which is illustrated in the inset panel of ﬁgure 2. The irreducible Brillouin zone is the triangular wedge in the upper-right corner; the rest of the Brillouin zone can be related to this wedge by rotational symmetry. The three special points Γ, X, and M correspond (respectively) to k = 0, k = π/axˆ, and k = π/axˆ + π/ayˆ . What do the ﬁeld proﬁles of the electromagnetic modes at these points look like?
The ﬁeld patterns of the TM modes of the ﬁrst band (dielectric band) and second band (air band) are shown in ﬁgure 3. For modes at the Γ point, the ﬁeld pattern is exactly the same in each unit cell. For modes at the X point (the zone edge), the ﬁelds alternate in sign in each unit cell along the direction of the wave vector kx, forming wave fronts parallel to the y direction. For modes at the M point, the signs of the ﬁelds alternate in neighboring cells, forming a checkerboard pattern. Although the X and M patterns may look like wave fronts of a propagating wave, in fact the modes at these particular k points do not propagate at all—they are standing waves with zero group velocity. The ﬁeld patterns of the TE modes at the X point for the ﬁrst and second bands are shown in ﬁgure 4.
For the TM modes, this photonic crystal has a complete band gap between the ﬁrst and second bands, with a 31.4% gap–midgap ratio (deﬁned as in the section The Size of the Band Gap of chapter 4). In contrast, for the TE modes there is no complete band gap. We should be able to explain such a signiﬁcant fact, and we can, by examining the ﬁeld patterns in ﬁgures 3 and 4. The ﬁeld associated with the lowest TM mode (the dielectric band) is strongly concentrated in the dielectric regions. This is in sharp contrast to the ﬁeld pattern of the air band. There, a nodal plane cuts through the dielectric columns, expelling some of the displacement ﬁeld amplitude from the high-ε region.
As we found in chapter 2, a mode concentrates most of its electric-ﬁeld energy in the high-ε regions in order to lower its frequency, but upper bands must be orthogonal to lower bands. This statement of the variational theorem explains the large splitting between these two bands. The ﬁrst band has most of its energy in the dielectric regions, and has a low frequency; the second must have a nodal plane in order to be orthogonal to the ﬁrst, and thus has most of its energy in the air region with a correspondingly higher frequency. We can quantify this statement. An appropriate measure of the degree of concentration of the electric ﬁelds in the high-ε regions is the concentration factor, which we deﬁne as

concentration factor

ε=8.9 d3r ε(r)|E(r)|2 d3r ε(r)|E(r)|2

.

(2)

Comparing this expression with equation (24) of chapter 2, we see that the concentration factor measures the fraction of electric-ﬁeld energy located inside the high-ε regions. Table 1 shows the concentration factors for the ﬁelds we are considering. The dielectric-band TM mode has a concentration factor of 83%, while the air-band TM mode has a concentration factor of only 32%. This difference in the

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Dz field at Γ (TM)

band 1

band 2

Dz field at X (TM)

band 1

band 2

Dz field at M (TM)

band 1

band 2

negative

positive

Figure 3: Displacement ﬁelds of TM states inside a square array of dielectric (ε = 8.9)
columns in air. The color indicates the amplitude of the displacement ﬁeld, which points in
the z direction. Modes are shown at the Γ point (top), the X point (middle), and the M point
(bottom). In each set, the dielectric band is on the left and the air band is on the right. The
ﬁelds of the air band at the M point are from one of a pair of degenerate states.

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Hz field at X (TE)

band 1

71
band 2

negative

positive

Figure 4: Magnetic ﬁelds of X-point TE states inside a square array of dielectric (ε = 8.9)
columns in air. The column positions are indicated by dashed green outlines, and the color indicates the amplitude of the magnetic ﬁeld. The dielectric band is on the left and the air
band is on the right. Since D is largest along nodal planes of H, the white regions are where the displacement energy is concentrated. TE modes have D lying in the xy plane.
Table 1

TM

TE

Dielectric band

83%

23%

Air band

32%

9%

Concentration factors for the lowest two bands of the square lattice of dielectric rods at the X point of the Brillouin zone.

energy distribution of consecutive modes is responsible for the large TM photonic band gap.
The concentration factors for the TE modes do not contrast as strongly. This is reﬂected in the ﬁeld conﬁgurations for the lowest two bands, shown in ﬁgure 4. We have actually plotted the magnetic ﬁeld H, since it is a scalar for TE modes and easy to visualize. From equation (8) of chapter 2, we know that the displacement ﬁeld D tends to be largest along the nodal planes of the magnetic ﬁeld H. The displacement ﬁeld of both modes has a signiﬁcant amplitude in the air regions, raising the mode frequencies. But in this case there is no choice: there is no continuous pathway between the rods that can contain the ﬁeld lines of D. The ﬁeld lines must be continuous, and they are consequently forced to penetrate the air regions. This is the origin of the low concentration factors, and the explanation for the absence of a band gap for TE modes.
The vector nature of the electromagnetic ﬁeld, and in particular its discontinuous boundary conditions at material interfaces, is central to this phenomenon. When we move across a dielectric boundary from a high dielectric ε1 to some lower ε2 < ε1,

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0.8

0.7

Frequency ωa/2πc

0.6

0.5

0.4

0.3

TE modes

M

0.2

0.1

Γ

X

TM modes

0

Γ

X

M

Γ

Figure 5: The photonic band structure for the lowest-frequency modes of a square array of
dielectric (ε = 8.9) veins (thickness 0.165a) in air. The blue lines are TM bands and the red lines
are TE bands. The left inset shows the high-symmetry points at the corners of the irreducible
Brillouin zone (shaded light blue). The right inset shows a cross-sectional view of the dielectric
function.
the energy density ε|E|2 will decrease discontinuously by ε2/ε1 if E is parallel to the interface (since E is continuous) and will increase discontinuously by ε1/ε2 if E is perpendicular to the interface (since εE⊥ is continuous). In the TM case, E is parallel to all dielectric interfaces and so a large concentration factor is possible. In the TE case, however, the electric ﬁeld lines must cross a boundary at some point, forcing electric-ﬁeld energy out of the rods and preventing a large concentration factor. As a result, consecutive TE modes cannot exhibit markedly different concentration factors, and band gaps do not appear.

A Square Lattice of Dielectric Veins
Another two-dimensional photonic crystal that we will investigate is a square grid of dielectric veins (thickness 0.165a, ε = 8.9), shown as an inset in ﬁgure 5. In a sense, this structure is complementary to the square lattice of dielectric columns, because it is a connected structure. The high-ε regions form a continuous path in the xy plane, instead of discrete spots. The complementary nature is reﬂected in the band structure of ﬁgure 5. Here, there is an 18.9% gap in the TE band structure, but there is no gap for the TM modes. This is the opposite of the situation for the square lattice of dielectric columns.

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Dz field at X (TM)

band 1

73
band 2

negative

positive

Figure 6: Displacement ﬁelds of X-point TM modes for a square array of dielectric (ε = 8.9)
veins in air. The color indicates the amplitude of the displacement ﬁeld, which is oriented in
the z direction (out of the page). The dielectric band is on the left, and the air band is on the
right.

Hz field at X (TE)

band 1

band 2

negative

positive

Figure 7: Magnetic ﬁelds of X-point TE modes for a square array of dielectric (ε = 8.9) veins
in air. The green dashed lines indicate the veins, and the color indicates the amplitude of the
magnetic ﬁeld (which is oriented in the z direction). The dielectric band is on the left, and the
air band is on the right.
Again, we turn to the ﬁeld patterns of the modes in the two lowest bands to understand the appearance of the band gap. The ﬁelds are displayed for the TM and TE modes in ﬁgures 6 and 7, respectively.
Looking at the TM-ﬁeld patterns in the ﬁrst two bands, we see that both modes are mainly concentrated within the high-ε regions. The ﬁelds of the dielectric band

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Table 2

TM

TE

Dielectric band

89%

83%

Air band

77%

14%

Concentration factors for the lowest two bands of the square lattice of veins at the X point.

are conﬁned to the dielectric crosses and vertical veins, whereas the ﬁelds of the air band are concentrated in the horizontal dielectric veins connecting the square lattice sites. The consecutive modes both manage to concentrate in high-ε regions, thanks to the arrangement of the dielectric veins, so there is no large jump in frequency. This claim is veriﬁed by calculating the concentration factors for the ﬁeld conﬁgurations. The results are in table 2.
On the other hand, the TE band structure has a photonic band gap between the ﬁrst two bands. What is the difference between this lattice and the square lattice of dielectric columns, in which no TE gap appeared? In this case, the continuous ﬁeld lines of the transverse electric ﬁeld lines can extend to neighboring lattice sites without ever leaving the high-ε regions. The veins provide high-ε roads for the ﬁelds to travel on, and for n = 1 the ﬁelds stay almost entirely on them. As before, since the D ﬁeld will be largest along the nodal (white) regions of the H ﬁeld, we can infer from ﬁgure 7 that the D ﬁeld of the lowest band is strongly localized in the vertical dielectric veins.
The D ﬁeld of the next TE band (n = 2) is forced to have a node passing through the vertical high-ε region, to make it orthogonal to the previous band. Some of its energy is thereby forced into the low-ε regions. (The maxima in the displacement energy coincide with the white regions of H in ﬁgure 7.) This causes a sizable jump in frequency between the two bands. This hypothesis is supported quantitatively by concentration factor calculations in table 2. We ﬁnd a large concentration factor for the dielectric band and a small one for the air band. This jump in concentration factor between consecutive bands causes the formation of a band gap. In this case, it is the connectivity of the lattice that is crucial to the production of TE band gaps.

A Complete Band Gap for All Polarizations
In the previous two sections, we used the ﬁeld patterns as our guide to understand which aspects of two-dimensional photonic crystals lead to TM and TE band gaps. By combining our observations, we can design a photonic crystal that has band gaps for both polarizations. By adjusting the dimensions of the lattice, we can even arrange for the band gaps to overlap, resulting in a complete band gap for all polarizations.
Earlier, we found that the isolated high-ε spots of the square lattice of dielectric columns forced consecutive TM modes to have different concentration factors, due

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r a

z
y x

Figure 8: A two-dimensional photonic crystal of air columns in a dielectric substrate (which
we imagine to extend indeﬁnitely in the z direction). The columns have radius r and dielectric constant ε =1. The left inset shows a view of the triangular lattice from above, with the unit
cell framed in red. It has lattice constant a.
to the appearance of a node in the higher-frequency mode. This, in turn, led to the large TM photonic band gap. The lattice of dielectric veins presented a more spread-out distribution of high-ε material, leading to greater similarity between the concentration factors of successive modes.
On the other hand, the connectivity of the veins was the key to achieving gaps in the TE band structure. In the square lattice of dielectric rods, the TE modes were forced to penetrate the low-ε regions, since the ﬁeld lines had to cross dielectric boundaries. As a result, the concentration factors for consecutive modes were both low and not very far apart. This problem disappeared for the lattice of dielectric veins, since the ﬁelds could follow the high-ε paths from site to site, and the additional node in the higher mode corresponded to a large frequency jump.
To summarize our rule of thumb, TM band gaps are favored in a lattice of isolated high-ε regions, and TE band gaps are favored in a connected lattice.
It seems impossible to arrange a photonic crystal with both isolated spots and connected regions of dielectric material. The answer is a sort of compromise: we can imagine crystals with high-ε regions that are both practically isolated and linked by narrow veins. An example of such a system is the triangular lattice of air columns, shown in ﬁgure 8.
The idea is to put a triangular lattice of low-ε columns inside a medium with high ε. If the radius of the columns is large enough, the spots between columns look like localized regions of high-ε material, which are connected (through a narrow squeeze between columns) to adjacent spots. This is shown in ﬁgure 9. The band structure for this lattice, shown in ﬁgure 10, has photonic band gaps for both the TE and TM polarizations. In fact, for the particular radius r/a = 0.48 and dielectric constant ε = 13, these gaps overlap, and we obtain an 18.6% complete photonic band gap.

Out-of-Plane Propagation
Until now we focused exclusively on modes that propagate in the plane of periodicity, with kz = 0. However, for some applications we must understand

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spot

vein

Figure 9: The spots and veins of a triangular lattice. Between the columns are narrow veins, connecting the spots surrounded by three columns.

0.8

0.7

Frequency ωa/2πc

0.6

0.5

Photonic Band Gap

0.4

0.3

TE modes

0.2

0.1

TM modes

MK Γ

0

Γ

M

K

Γ

Figure 10: The photonic band structure for the modes of a triangular array of air columns
drilled in a dielectric substrate (ε = 13). The blue lines represent TM bands and the red lines
represent TE bands. The inset shows the high-symmetry points at the corners of the irreducible Brillouin zone (shaded light blue). Note the complete photonic band gap.
the propagation of light in an arbitrary direction. We will investigate the out-ofplane band structure by considering the kz > 0 modes of the triangular lattice of air columns, the lattice which we discussed in the previous section. The outof-plane band structure for this photonic crystal is shown in ﬁgure 11. Many

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1.5
ω = ckz
1

77
lowest photonic band

Frequency ωa/2πc
Frequency

0.5
0 0

Γ bands
K bands

Γ

MK

Γ

kza/2π

1

2

3

4

5

Perpendicular wave vector kza/2π

Figure 11: The out-of-plane band structure of the triangular lattice of air columns for the ﬁrst
few bands. The bands that start at Γ, ω(Γ, kz), are plotted with blue lines, whereas the bands that start at K, ω(K, kz), are plotted with green lines. The light line ω = ckz (red) separates the modes that are oscillatory (ω ≥ ckz) in the air regions from those that are evanescent (ω < ckz) in the air regions. The inset shows the frequency dependence of the lowest band as kz varies. Note that as kz increases, the lowest band ﬂattens.

of the qualitative features of the out-of-plane band structure for this crystal are common to all two-dimensional crystals. In fact, these features are just the natural extensions of the corresponding notions in multilayer ﬁlms, which we developed in the previous chapter.
The ﬁrst thing to notice about the out-of-plane band structure is that there are no band gaps for propagation in the z direction, just as we found for one-dimensional crystals in the section Off-Axis Propagation of chapter 4. This is a consequence of the homogeneity of the crystal in that direction. Intuitively, this is because no scattering occurs along that direction; band gaps require multiple scatterings from regions of different dielectric constant. Another important point is that there is no longer a useful distinction between the TE and TM polarizations, because the mirror symmetry is broken for kz = 0.
Also note that the bands become ﬂat with increasing kz. The inset to ﬁgure 11 shows the frequency dependence of the lowest band as kz is varied. When kz = 0, this lowest band spans a broad range of frequencies, as we saw in ﬁgure 10. As kz increases, the lowest band ﬂattens and the bandwidth (the range of allowed frequencies for a given kz) decreases to zero. Since the bandwidth is typically determined by the difference between the frequencies at Γ and K, ﬁgure 11 also

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shows both ω(Γ, kz) and ω(K, kz) for the ﬁrst few bands. As kz increases, the bandwidth of each of the bands vanishes. Why is this the case?
There is a simple explanation. For large kz, the light is trapped by index guiding inside the dielectric regions, just as it would be in an optical waveguide. The light modes that are trapped in neighboring high-ε regions have very little overlap, so the modes decouple and the bandwidth shrinks to zero. This is especially true for modes that have ω ckz.1 In this regime, the ﬁelds decay exponentially outside the high-ε regions, and the overlap between modes in neighboring high-ε regions vanishes. This behavior is displayed by the bands in ﬁgure 11, which have a large dispersion for ω > ckz, but a small dispersion for ω ckz.
In later chapters we will expand upon these notions. In chapter 8, we will consider the case of cylinders of ﬁnite height—structures that are periodic in two dimensions, but are not strictly homogeneous in the third direction. In that case, kz is not conserved, and we cannot ignore out-of-plane propagation vectors. In chapter 9, we will consider the new kinds of band gaps that can arise for large kz, distinct from the TE/TM gaps in this chapter, and which have important applications in ﬁbers with 2D-crystal cross sections.

Localization of Light by Point Defects
Previously, we found two-dimensional photonic crystals with band gaps for inplane propagation. Within the band gap, no modes are allowed; the density of states (the number of possible modes per unit frequency) is zero. By perturbing a single lattice site, we can create a single localized mode or a set of closely spaced modes that have frequencies within the gap. For the case of the multilayer ﬁlm, we found that we could localize light near a particular plane by perturbing the dielectric constant of that plane.
In two dimensions, we have many options. As depicted in ﬁgure 12, we can remove a single column from the crystal, or replace it with another whose size, shape, or dielectric constant is different than the original. Perturbing just one site ruins the translational symmetry of the lattice. Strictly speaking, we can no longer classify the modes by an in-plane wave vector. However, the mirror-reﬂection symmetry is still intact for kz = 0. Therefore we can still restrict our attention to in-plane propagation, and the TE and TM modes still decouple. That is, we can discuss the band structures for the two polarizations independently, as before. Perturbing a single lattice site causes a defect along a line in the z direction. But because we are considering propagation only in the plane of periodicity, and the perturbation is localized to a particular point in that plane, we refer to this perturbation as a point defect.
Removing one column may introduce a peak into the crystal’s density of states within the photonic band gap. If this happens, then the defect-induced state must
1 For ω > ckz, there are also modes that are mostly trapped in the low-ε regions, since Fresnel reﬂection goes to 100% in the glancing-angle ray-optics limit.

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surface

line-defect
point-defect
z
y x
Figure 12: Schematic illustration of possible sites of point, line, and surface defects. Perturbing one column in the bulk of the crystal (yellow) might allow a defect state to be
localized in both x and y. Perturbing one row in the bulk of the crystal (red) or truncating the crystal at a surface (green) might allow a state to be localized in one direction (x). The rods are assumed to extend indeﬁnitely in the z direction.
be evanescent. The defect mode cannot penetrate into the rest of the crystal, since it has a frequency in the band gap. The analysis of chapter 4 is easily generalized to the case of two dimensions, allowing us to conclude that any defect modes decay exponentially away from the defect. They are localized in the xy plane, but extend in the z direction.
We reiterate the simple explanation for the localizing power of defects: the photonic crystal, because of its band gap, reﬂects light of certain frequencies. By removing a rod from the lattice, we create a cavity that is effectively surrounded by reﬂecting walls. If the cavity has the proper size to support a mode in the band gap, then light cannot escape, and we can pin the mode to the defect.2
2 A reader familiar with semiconductor physics can understand this result by analogy with impurities in semiconductors. In that case, atomic impurities create localized electronic states in the band gap of a semiconductor (see, e.g., Pantelides, 1978). Attractive potentials create a state at the conduction band edge, and repulsive potentials create a state at the valence band edge. In the photonic case, we can put the defect mode within the band gap with a suitable choice of εdefect. In the electronic case, we use the effective-mass approximation to predict the frequencies and wave functions of defect modes. Within each unit cell the wave functions are oscillatory, but the oscillatory functions are modulated by an evanescent envelope. An analogous treatment is applicable to the photonic case (Istrate and Sargent, 2006).

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0.5

Frequency ωa/2πc

0.4

0.3

0.2 M

0.1

TM modes

Γ

X

0

Γ

X

M

Γ

Figure 13: The TM modes of a square array of dielectric (ε = 8.9) columns in air, with r = 0.38a.
The center inset shows the irreducible Brillouin zone (shaded light blue). The other two insets suggest the ﬁeld patterns of the modes, within each column (red/blue for positive/negative
ﬁelds). the left inset shows the π-like pattern for band 3; the right shows the δ-like pattern for
the bottom of band 4.

We illustrate this discussion of localized modes in two-dimensional photonic crystals with a system that has been studied both experimentally3 and theoretically4: a square lattice of alumina columns in air. Unlike the inﬁnitely long columns of ﬁgure 1, the real columns were sandwiched between metal plates, which introduced a cutoff TE frequency that was larger than the frequencies being investigated. The plates also insured kz = 0 propagation. In this way, the experimentalists created a system that displays only TM modes with kz = 0.
The in-plane band structure of this system is reproduced in ﬁgure 13. The system has a 10.1% photonic band gap between the third and fourth bands. By examining the ﬁeld patterns, as we did in the ﬁrst two sections of this chapter, we would ﬁnd that the ﬁrst band is composed primarily of states that have no nodal planes passing through the high-ε columns. In analogy with the nomenclature of molecular orbitals, we describe such a nodeless ﬁeld pattern as a “σ-like” band. The second and third bands are composed of “π-like” bands, with one nodal plane passing through each column. The bottom of the fourth band has a “δ-like” pattern with two nodal planes per column (see insets to ﬁgure 13). Remember
3 See McCall et al. (1991). 4 See Meade et al. (1993).

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negative

positive
Dz

Figure 14: Displacement ﬁelds (Dz) of states localized about a defect in a square lattice of alumina rods (ε = 8.9) in air. The color indicates the magnitude of the ﬁeld. The defect on the
left was created by reducing the dielectric constant of a single rod. This mode has a monopole pattern with a single lobe in the defect and rotational symmetry. The defect on the right was created by increasing the dielectric constant of a single rod. This mode has a quadrupole pattern with two nodal planes in the defect, and transforms like the function
f (ρ) = xy under rotations.

that additional nodal planes in the high-ε regions correspond to larger amplitudes in the low-ε regions, which increases the frequency.
A defect in this array introduces a localized mode, as shown in ﬁgure 14. Experimentally, this defect was created by replacing one of the columns with a column of a different radius. Computationally, the defect was introduced by varying the d√ielectric constant of a single column. In terms of the index of refraction n = ε, the defect varied from ∆n = nalumina − ndefect = 0 to ∆n = 2 (one column completely gone). The results of the computation are shown in ﬁgure 15.
The largest photonic band gap in this structure is between the third (π-like) band of states with one nodal line and the fourth (δ-like) band of states with two nodal lines in the high-ε regions. When the index of refraction is less than 3, a state leaves the π band and enters the photonic band gap. As ∆n is increased between 0 and 0.8, this doubly degenerate mode sweeps across the gap. At ∆n = 1.4, a nondegenerate state enters the gap, sweeps across, and penetrates the δ band at ∆n = 1.8. This mode is displayed at ∆n = 1.58, before entering the δ band, in the left panel of ﬁgure 14.
Note that this state has one nodal line passing through each dielectric column other than the center, showing that it retains the essential character of the π band; however, because it has no nodal lines in the central (defect) column, we describe this state as a monopole mode. Similarly, defects with ∆n < 0 pull states out of the δ band. The resulting localized states retain their δ-like character, with two nodal

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