Springer optical properties of photonic crystals sakoda; springer; 2001

Eigenmodes of Photonic Crystals As the first step of the analysis of the radiation field in a photonic crystal, we will formulate the eigenvalue problem of the wave equation and give a g

professor Kazuaki Sakoda

Optical properties of photonic crystals 1 Kazuaki Sakoda

p m - (Springer series in optical science$, ISSN 034%-4ln ; 80)

Includes bibliagraphical references and index

ISBN 3.540411992 (acid~free paper)

L Photons 2 Crystal optics I Title 11 Springer series in optical sciences; v 80

Q C ~ ~ ) I P ~ W St4 2001 ~ R ' g - d ~ l l (10-0661~8

parts thereofis permitted only under the provisions ofthe German Copyright Law ofSeptember 9.1965, in its

current verrion, and permission for use must always be obtained from Springer-Verlag Violations are liable

for prosecutioll under the Gerrr~arr Copyright Ldw

Springer-Verlag Berlin Heidelberg New York

a member of Bertelsmannspringer Science+R~~rin~sr Mpdir GmhH

Springer Series in

OPTICAL SCIENCES

ÿ he Springer Series m Optical Sciences, under the leadership ot Editor-in-Chief williarn T ~hoder,

Georgia Institute of Technology, USA, and Georgia Tech Lorraine, France, provides an expanding

selection ofresearch monographs in all majorareas ofoptics: lasers and quantum optics, ultrafast p h e ~

nomena, optical spectroscopy techniques, optoelectronics, quantum information, information optics,

applied laser technology, industrial applications, and other topics of contemporary interest

With this broad coverageoftopics, the series is ofuse to all researchscientistsandengineerr who need

up-to-date reference books

The editors encourage prospective authors to correspond with them in advance ofsubmitting a manu-

script Submission of manuscripts should be made to the Editor-in-Chiefor one of the ~ d i t o r s see also

1-1, MinamiG26, Nishi n, Chuo-ku

Sapporo, Hokkaido 064-0926, Tapan

Phone: + I 404 894 %929

F u : + L 404 894 4641

E-mail: bill.rhodes@ece.gatech.edu URL: http:llwww.ece.gatech.edu/prohIesi

F u : +49 0 0 ) 314 27850

E-mail: weber@physik.tu-berlin.de URL: http:l/wulv.ph~ik.t~~berlin,deIinstituteI OllWeberlWebhome.htm

of all nleteorological phenomena

The fundamentals of the radiation field and its interaction with matter were clarified by classical electromagnetism and quantum electrodynamics These theories, we believc, explain all electromagnetic phenomena They not only provide a firm basis for contemporarv physics but also generate a vast range of technological applications Thrsc include television, radar, optical and microwave telecorr~rnu~~ications lasers, light-emitting diodes, solar cells, etc

Now the interaction between the radiation field and matter is so funda- memt,al that it may seem universal and invariant But in fact it is controllable

This discovery has been the great motivating force of intensive investigations

in optical physics during the last three decades In this book, I will show how

it is controlled using photonic crystals, a remarkable inm:ntion realized by the combination of optical physics and contemporary microfabrication tech- niques I will also show how the controlled radiation field alters the optical properties of atoms and molecules embedded in t,he photonic crystals and what kinds of new phenomena and new physics are expected t o manifest

t hemselves

This book was written to serve as a comprehensive textbook covering the optical properties of photonic cryst,als I t deals not ullly will, (Ire proper- ties of the radiation field inside the photonic crystals but also t,heir peculiar optical response t o external fields Only an elementary knowledge of electro- magnetism, quantum mechanics solid-stntc physics, and complex analysis is required of the readcr Therefore, undergraduate students in physics, applird physics optics electronics, and electrical engi~~eering in the final year shoilld

be ahle to read this book withont difficl~lt,~ Since tlir main recent dcvrl-

VlII Preface

oprnents such as t he enhancement of stimulated e~nission second harmonic

generation and quadrature-phase squeezing are also treated in a dctailed and

~n~derstandable manner this book also provides important ideas for graduate

students and researchers in this field

1 would like t o thank Professor Eiichi Hanamura and Professor Kuon

Inoile who gave me a wonderful introdnct ion t o this excit ing field I would

also like t o thank Professor Kmno Ohtaka and Professor Josepf~ W Haus

n h o gave me many suggestions on important problems I am grateful to

Professor Sajeev John for giving me the opportunity to visit his laboratory

in Toronto University in the summer of 1998 During t h a t period I learned

much ahout the qrlant urn optics of photonic crystals I am also grateful t o

Dr Kurt Busch who helped me t o confirm that the group theory worked well

for the fcc s t r u c t ~ ~ r e I acknowledge Professor T n h Koda for his continuous

encouragement since I was an undergraduate student of Tokyo University I

deeply acknowledge Professor Toshimitsu Asakura and Dr Claus Ascheron

for giving me thc opport unity t o write this hook I am grateful to many of my

graduate students for their efforts in numerical calculations In particular Ms

Hitomi Shiroma-Hirata Ms Noriko Kawai and Mr Takunori Ito did many

of the calcnlatioms that are shown in this hook I am also grateful t o Dr

Tetsuyuki Ochiai who made the main contribution t o the study of photonic

crystal slabs prcscntcd in Chapter 8

Sapporo

February 2001

Kazuakz Sakoda

1 I n t r o d u c t i o n 1

2 E i g e n m o d e s of P h o t o n i c C r y s t a l s 13

2.1 Wave Equations and Eigenvalue Problems 13

2.2 Eigenvalue Problems in Twc+Dimensional Crystals 19

2.3 Scaling Law and Time Reversal Symmetry 21 2.4 Photonic Band Calculation 23

2.4.1 Fourier Expansion of Dielectric Rinct ions 23 2.4.2 Some Examples 26

2.5 Phase Velocity, Group Velocity., and Encrgy Velocity 30 2.6 Calculation of Group Velocity 32

2.7 Complete Set of Eigenfunctions 34

2.8 Retarded Green's Function 39

3 S y m m e t r y of E i g e n m o d e s 43

3.1 Group Theory for TwwDimensional Crystals 43 3.2 Classification of Eigenmodes in the Square Lattice 55

3.3 Classification of Eigenmodes in the Hexagonal Lattice 57

3.4 Group Theory for Three-Dimensional Crystals 62 3.5 Classification of Eigenmodes in the Simple Cubic Lattice 65

3.6 Classification of Eigcnmodcs in thc fcc Latticc 75 4 T r a n s m i s s i o n S p e c t r a 81

4.1 Light Transmission and Bragg Reflection 81

4.2 Field Equations 83

4.2.1 E Polarization 8:l ! 4.2.2 H Polarization 85

4 3 Foriricr Transform of the Dielectric Function 87

4.3.1 Square Lattice 87 4.3.2 Hexagonal Lattice 89

4.4 Some Examples g l 4.4.1 SquareLattice 91

4.4.2 Hexagonal Latticp 91

4.5 Refraction Laxv for Photonic Crystals 95

5 O p t i c a l R e s p o n s e of P h o t o n i c C r y s t a l s 99

5.1 Solutions of Inhomogeneous Equations 99

5.2 Dipole Radiation 102

5.3 Stimulated Emission 105

5.4 Sum-Frequency Gcr~cratio~i 109

5.4.1 Three-Dimensional Case 109

5.4.2 Two-Dimensional Case 112

5.5 SHG in the Square Lattice 116

5.6 Free Indiiction Decay 121

6 D e f e c t M o d e s i n P h o t o n i c C r y s t a l s 125

6.1 General Properties 125

6.2 Principle of Calculation 128

6.3 Point Defects in a Square Lattice 131

6.4 Point Defects in a Hexagonal Lattice 134

6.5 Line Defects in a Square Lattice 142

6.6 Dielectric Loss and Quality Factor 146

7 B a n d C a l c u l a t i o n w i t h F r e q u e n c y - D e p e n d e n t D i e l e c t r i c C o n s t a n t s 151

7.1 Principlc of Calculation 151

7.2 hlodificd Plane iVaves in hIetallic Crystals 154

7 3 Surface Plasmon Polaritons 161

7.3.1 Plasmun Polnritons on Flat Surface 162

7.3.2 Plasnlon Resonance on a Metallic Cylinder 165

7.3.3 Symn~ctrv of Plasmon Polaritons 169

7.3.4 Plasrrion Bands in a Square Lattice 171

8 P h o t o n i c C r y s t a l S l a b s 177

8.1 Eigcnmodcs of Uniform Slabs 177

8 2 Symmetry of Eigenmodrs 181

8.3 Photonic Band Structure and Trallslnission Spectra 183

8.4 Q11;rlity Factor 185

9 L o w - T h r e s h o l d L a s i n g D u e to Group- Velocity A n o m a l y 189

9 1 Enhanced Stimulated Elnission 189

9.2 Lasing Threshold 193

9.2.1 -4llalytical Expression 191

9.2.2 Numerical Estimation 195

10 Q u a n t u m O p t i c s i n P h o t o n i c C r y s t a l s 201

10.1 Quantization of the Electromagnetic Field 201 10.2 Qua<lraton Phase Squeezing 203

10.3 Interaction Hamiltonian 207 10.4 Lamb Shift 208

11 E p i l o g u e 213

As is well known, there is the following relation between the frequency u, the velocity c; and the wavelength Xo,

When we define the wave number k by

I we obtain the relation between the angular frequency w and k:

This equation is called the dispersion relation of the radiation field If one thinks of the radiation field in a uniform material with refractive index q ,

one can obtain its dispersion relation by replacing c by v = c / q and Xo by

X = X o l q in (1.2) and (1.3) The density of states of the radiation field in the volume V of free space, D ( w ) , is proportional t o w 2 (Fig I 1):

The density of states in the uniform material is obtained by replacing c by

u in this equation.' The optical properties of atoms and molecules strongly depend on D ( w ) As an example, let us consider the spontaneous emission of

a photon from an electronic excited state of an atom or a molecule Quantum mechanics tells us that the rate of the spontaneons emission is proportional t o

w D ( w ) Since the spontaneous emission is an origin of the energy dissipation and the fluctuation of the radiation field, it suppresses the occr~rrence of lwer oscillations This suppression is marked in the high frequency region since D ( w ) is proportional to wZ This is one of the reasons why t,hr lascr oscillation is difficult t o realize a t high frequencies

Now: if we can design and modify D ( w ) , we can substantially change the optical properties of atorris and molecules [I] This is a key idea of con- temporary optical physics, and it is possible One method is to use optical

microcavities while another is to use photonic crystals In this hook, it will be

shown how the charncterist,ics of the radiation field are modified in photanic

' The derivation of (1.4) will be giver, in Sect 5.2

2 1 Introduction 1 Introduction 3

Defect mode

(a) Free space (b) Photonic Crystal

Fig 1.1 Schematic illustration of the density of states of the radiation field (a) in

frer apace and (b) in R photonic crystal In the illustration for the photonic crystal

a photonic bandgap and a localized defect mode with a delta-function like dcnsity

is included (see text)

Fig 1.2 Schematic illustration oi one-dimensional (ID), two-dimensional (2D)

and tlir~e-dimensional (30) photonic crystals a is the lattice constant

Fig 1.3 SEA1 nnagc of a :ill pllotc,llir crystal composed of an fcc array of SiOl

splierps with L diatnrtcr of 300 nm (After [ 2 ] )

Fig 1.4 Sb:hl ilr~a-P of a :ID pilotonic rt.ystnl made of silicon (After [ 3 ] )

crystals and how the optical vroverties of atoms and molecules embedded in

them are altered

Photonic crystals are regular arrays of materials with different rafractive

indices Figure 1.2 shows the simplest case in which two materials denoted

by A and B are stacked alternately The spatial period of t,he stack is called

thc lattice constant, since it corresponds t o the lattice constant of ordinary

crystals composrd of a regular array of atoms Actually, many basic ideas

are connnon t,o hot,h rrystals and thry will be utilized t o huild fundamental

theories of thc photonic crystals, n? will he shown in the following chapters

However, one big difference hetureen them is the scale of the lattice con-

stant In t h r case of ordinary cryst,als, the lattice constant is on the order of

angstroms On the other hand it is on the order of the wavelength of the

relevant electromagnetic waves for the photonic crystals For example it is

ahont 1 nlrn or less for visible light, a r ~ d is abont 1 cm for microu~aves

Photonic crystals are classified mainly into three categories, that is one- dimensional (lD), twodimensional (2D) and thrcc-dimcnsiond (3D) crystals according to the dimensionality of the stack (see Fig 1.2) The photonic crystals t,hat work in the microwave and far-infrared regions are r~latively easy t o fabricate Those that work in the visible region, especially 3D ones are difficult t o fabricate because of their small lattice constants However various technologies have been developed and applied t o their fabrication

in the last ten years, and man)- good crystals with a lattice constant less than 1 mm are now available For example, Fig 1.3 shows an SEM (scanning electron micrograph) image of an fcc (face-centered-cubic) lattice c o n ~ p i ~ s ~ d

of silica spheres [Z] The diarneter of the spheres is 0.3 mnl On the other hand Fig 1.4 is an SEhl image of a 3D st,rncture made of Si [3]

4 1 Introduction 1 Introduction 5

Fig 1.5 (a) Schematic illustration of a phatanic crystal slab and (b) the scanning

electron micrograph of a n i ~ c t ~ ~ a l specimen fabricated on a Si snhstratc (After (41)

If we design a 3D phot,onir cryst,al appropriately there appears a fre-

quency range where no electromagnetic eigenmode exists Frequency ranges

of this kind are called photonic bandgaps, since they correspond t o bandgaps

of elect,ronic eigenstates in ordinary crystals Moreover, if we introduce a dis-

order into the regular dielectric structure of the photonic crystal, we may

obt,ain midgap modes whose eigenfunctions are strongly localized around the

disorder These modes are called localized defect modes The density of states

for a 3D photonic crystal with a photonic bandgap and a localized defect

mode is schematically illustrated in Fig l l ( b ) If the emission frequency

of an atom or a molecule embedded in t h e photonic crystal lies just in the

photonic bandgap the spontaneous emission of a photon from its electronic

excited state is con~pletely forbidden, since there exists no photon in the

gap On the other hand, if the emission frequency coincides with the eigen-

freq~incy of the localized mode and the atom is locat,ed near the defect, the

spontaneous emission is accelerated

Another class of photonic crystals known as photonic c ~ s t a l slabs 14-131 is

illustrated in Fig 1.5 Photonic crystals of this type are usually fabricated on

a substrate made of a semiconductor or an insulator They have been investi-

gated energetically in recent years, h e c a ~ ~ s e many snphistirat,ed t,echnologies

such as electron beam lithography and thin-layer formation developed in the

field of electronics and opto-electronics can be applied t o their fabrication

hIonolayers made of polymer micrrrspheres 114-161 may also be regarded as

photonic crystal slabs

In order to gain an intuitive understanding of thc photonic bands and

bandgaps, we examine 1D crystals in some detail The 1D photonic crystals

are traditionally called dielectric multilayers and thcir optical properties are

wcll-known 1171 We take the x axis in the direction perpendicular t o the

surface of the diclectric layers as shown in Fig 1.6 We only dcnl with elec-

tromxgnet,ic waves propagated in the z direction and polarized linearly here

\Ve take the y axis in the directin11 of thc polarization The electric field of thc

Fig 1.6 Geometry of the calculation of the photon~c band structure, or the d i p perqlon relation, of a 1D photonic crystal

propagated wave is denoted by a complex function E ( z t ) for convenience The actual electric field is, of course, a real quantity I t is given by the real part of E ( x , t )

Sow, the wave equation for E ( x , t ) is given by

where E(Z) denotes the position-dependent relative dielectric constant of the

I D photorlic crystal, which will hc callcd thc diclcctric function h e r e a f t e ~ ~

In (1.5), we assumed that the magnetic permeability of tlie pl~otonic crystal

is equal t o that in free space, since we do not treat magnetic materials in this book Because ~ ( z ) is a pcriodic function of z,

E-'(x) is also periodic and can be expanded in a Fourier series:

E ( x t) = & ( z , t ) = uk(z) exp {i(kz - &kt)} ; (1.8)

T h r derivation of the wave equation from hlaxwell's equations will he describ~d

in Chap 2

Thr proof =.ill he given in Sect 2.1

where {E,,) are the Fourier coefficients

Now, we assume for simplicity that only components with m = 0 and f 1

are dominant in the expansion (1.7):

When we substitute (1.10) and (1.11) into the wave equation (1.5) we obtain

For m = 0,

F o r m = -1

Therefore, if k I k 2 ~ l a (i.e k = xla), and if w: F tioc2k2, EO and E-I

are dornir~ar~t in thc expansior~ (1.10) In this case wc neglect all other terms

and obtain the following coupled equations:

These linear equations have a nontrivial sohition when t h e dct,crminant of

coefficients uli~ishr,~:

Fig 1.7 Dispersion relation for a 1D photonic crystal (solid lznes) The boundary

of the first Brillouin zone is denoted by two vertical lincs The dispersion lines in thc uniform material are denoted by dadhed lznes They are folded into the first Brillouin zone taking into account the identity of the wave numbers which differ from each other by a multiple of 2nla When two dispersion lines cross, they repel each other and a photonic bandgap appears

If we introduce h = k - n/a; the solutions are given by

as far as h <<n/a So, there is no mode in the interval

This gap disappears when nl = 0 This result can be interpreted that the niodes with k F n/a and k - -n/a were mixed with each other in the presence of the periodic rnodulation of t,lle dielectric constant and this mixing led t o a frequency splitting

In general, those wave vectors which diffcr from each otilrr b.y a ar~ulliplt:

of 2n/a should be regarded as the same because of the presence of the periodic spatial lnodulation of the dielectric constant When the spatial modulation

is small, the dispersion relation in the photonic crystal is not so far fro111

d = vk but it should thus be expressed with the wave vector in the first Brillouin zone [ - ~ / a : a / a ] ~ In addition, if two dispersion lines cross each other, a frequency gap appears All these things are schematically illustrated

in Fig 1.7 There are an infinite number of frequency gaps in the spectrnm However we should note that this is true only as far as we deal with optical This is the so-ca1lc.d reduced zone schen~c

1 Introduction 9

8 1 Introduction

k ,

Fig 1.8 Reciprocal lattice space of the 2D square photonic clystsi with the lattice

constant o The first Brillouin zone is surrounded by a dashed line

waves travelling in the x direction, and t h a t there is no gap when we take

into consideration those modes travelling in other directions

1D phot,onic crystals or dielectric ml~ltilayers, have many important ap-

plications For example, they are utilized as high-reflection mirrors and anti-

reflection coatings However, most investigations related t o t h e control of the

radiation field in recent years have been concentrated on 2D and 3D photonic

crystals because the," offer a rich variety of new physics of the r d i a t i o n field,

as will be described in this book One example is the photonic bandgaps that

may be realized in a certain class of 3D crystals Another example is the

group-velocity ano~r~nl?, that is realized in all 2D and 3D crystals We con-

sider this point in some detail here

Like the 1D crystals, the dispersion relation of a 2D crystal with a small

spatial variation of the dielectric constant can be well-represmted by the

folding of t h e dispersion line of the uniform material, w = v k , into the first

Brillouin zone The fact that all wave vectors are not parallel t o each other in

two dimensions, however, n~akes a large difference As an example, Fig 1.8

shows the reciprocal lattice space of a 2D square lattice The lattice constant

is der~oted bv a as before Thc first Brillouin zone is surrounded by a dashed

line in t,his figure The elementary lattice vectors are

and the elementary reciprocal lattice vectors are

Those wave vectors that differ from each other by the sum of a multiple of

bl and that of b2 should be regarded as the same There are three highly symmetric points in thc first Brilloui~r zone, i.e., the r point, ( 0 , 0 ) , the X

point, ( n l a 0 ) , and the hf point, la, la) r' and X' shown in Fig 1.8 are equivalent t o r and X , respectively The wave vect,ors hetween the I' and the X points are called the A point On the other hand, that between the r

and the Af points is called the 2 point

Now, the anomalous group velocit,y takes place when the original wave vector in the uniform material is not parallel t o either bl or b2 We examine this point for the wave vectors between r' and X ' , which are denoted by A'

in Fig 1.8 Those wave vect,ors are

The original dispersion relation is given by

and this should also he regarded as a good approximation t o the rigourous dispersion relation of the photonic crystal for the corresponding wave vectors

k in the first Brillouin zone:

The dispersion relation for the square lattice with an infinitesimally small

m o d ~ ~ l a t i o n of the dielectric constant thus obtained is shown in Fig 1.9 The averaged refractive index is assumed t o bc 1.28 in this fignre The band that

we have just discussed is the third lowest one on the D point This band has a very small slope compared with the rest This is a consequence of the fact that the original wave vector is not parallel t o the elementary reciprocal lattice vectors as was mentioned above

The group velocity u, of the radiational eigenmode is given by the slope

of the dispersion corve:

Hence, the third lowest band ha.? a small group vclocity over its entire fre quency range We refer to the small group velocity of this type as the group-

energy velocity in the photonic crystal.%he small group velocity implies that the interaction time between the radiational mode and the matter system is long [19] This leads t o a large effective coupling hetween them and various optical processes are enhanced I will show later the enhancement of stim- ulated emission; sum-frequency genprat,ion (Chap 5), and qnadrature-phase

"ee Sect 2.5

Fig 1.9 Dispersiun relation for a 2D photonic crystal with a n infinitesimally small

spatial variation of the dielectric constant The abscissa rcprcscnts the wave vector

in thc first Brillouin zone of the 2D square lattice The ordinate is the normalized

irrqoency where a and c stand far t,hr latticc constant and the light velocity in free

spscp

squeezing (Chap 10) Light absorption is also enhanced As for nonlinear

optical processes s ~ r c l ~ as sum-freqnency- generation, the phase-matching cori-

dition which is an important character t o obtaiii a large outprtt, is given by

thc coriservation of the crystalline momentnm (see Sect 5.4):

where k, (kr) dcnotes the initial (final) wave vector of t h e radiation field

whereas G denotes a reciprocal lattice vcctor As for the uniform material,

This equation may he regarded as a special case of (1.26) for which G = 0

Bcca~ise G is an arbitrary reciprocal lattice vector for the photonic crystals

the phase-matching condition is relaxed compared with the iiniform materi-

als The group-velocity anomaly and the relaxed phase-niatcliiiig condition

are very iniportant aspects of nonlinear optical properties of the photoriic

crystals

This book is organized as follows The eigenvalue problems for the radia-

tion field in the photonic crystals will be fori~rulatcd in Chap 2 An Hermitian

operator related t o the electric field will he introduced; and the properties of

ils eigenfilnctions will bc studicd in detail The retarded Green's function of

this Hrlrmitian operator will be derived for later use In Chap 3; group theory

will be forrnlrlated to analyze tlie symmetry of the eigenmodes, and will be

il~pliod to some examples Tlie uncoupled modes, that is, those rigenmodes

which callnot be excited by external plane waves doe t o the mismatching

of thc symmetry property, will he identified by group tlmury In Chap 4, a

rluirierical met,hod t o calc~llatt! tlrf: trans~rrittarice anrl t h r Bragg reflect,ancr

will h t ~ forrn~llated and applied t o some e x m p l c s T h r prcrise coil~cidelice between the calculatrd spectra alid the band structi~rc will be shown In Chap 5 , a general formula for the description of the optical rpsponsc of tlie photonic crystals will he derived based on the mcthod of Green's function Four examples of its application i.e., dipole radiation, stimnlated emission, srlrrl-hequency generation: and free indrlct,ion drcay will be presented The phase-matching condition and the arlditional selection rulc for nonlinear opti- cal processes peculiar to photonic crystals will be given there An examplp of the nunierical analysis of tlic sllrrl-frequency generation will also be presented

In Chap 6 an efficient numerical method t o calculate the localized d e f ~ c t , modes will be derived based on the general formula t o describe> the dipole radiation in the pliotonic crystals given in Chap 5 An excellent agreement

between the calculated eigenfrcqr~encies and the experimental ohscrvation will be shown In Chap 7 I will present a general and accurate numerical method t o calculate the band structure with freqnencv-dependent dielectric constants This mcthod will be applied to a photonic cryst,al with metallic components Tlie presence of ext,remely flat hands related to surface plasmon polaritons will be shourn In Chap 8, t h e optical properties of photoiric crys- tal slabs will he examined The low-threshould lasing due t o the enhanced stimulated etriission caused by the group-vclocity airomaly will bc discussed in Cliap 9 In addition t o the analyt,ical formula describing the onset of lasing the lasirig thresliould will he analyzed numerically The radiation field will

be quantized in Chap 10 Two qilantunl optical propcrtics: i.e quadrature- phase squeezing due to optical parametric amplrfication and Lanib shift of atoms in the photonic crystal urill bc examined The enhancement of the for- mer due to thc anomalous group velocity and thwt of tlie latter due t,o t,hr anomalous densitv of st,at,es of the radiation field will be shown

2 Eigenmodes of Photonic Crystals

As the first step of the analysis of the radiation field in a photonic crystal, we will formulate the eigenvalue problem of the wave equation and give a general numerical method to solve it Based on the complete set of the eigenfunctions,

we will also derive the expression for the retarded Green's function related

to the electric field In addition to the three-dimensional (3D) case, we will

treat the two-dimensional (2D) crystals, for which the vectorial wave equation reduces to two independent scalar equations and the relevant expressions are simplified

2.1 Wave Equations and Eigenvalue Problems

We hegin with hlaxwell's equations Because we are interested in the eigen- modes of the radiation field, and the interaction between the field and matter will be discussed in later chapters, we assume here that free charges and the electric current are absent In this caye, Maxwell's equations in the most general form are given in MKS units as follows

a

V x H ( r , t ) = - D ( r ; dt t ) (2.4)

The standard notations for the electric field ( E ) , the magnetic field (H) the electric displacement ( D ) : and the magnetic induction ( B ) are used in t,hese eqnations

In order to solve the wave equations derived from Maxwell's equations, we need spcalled constitutive equations that relate D t o E and B to H Since

we do not deal with magnetic materials in this book, we assume that the magnetic pernleahility of the photonic crystal is equal to that in free space,

Pa:

B ( r , t ) = p ~ H ( r , t ) (2.5)

Eigerirrlodes of Photonic Crystals

As for the dielectric constant, we assume in this chapter that it is real,

isotropir, perfectly prriodic u.it,h respect t o the spatial coordinate r , and

does not depend on frequency Those cases in which the dielectric constant is

complex, has a disorder, or dcpcnds on frequency, which are very interesting

and important, will be dealt with in Chaps 5-9 Readers who are interested

in the treatment of anisotropic dielectric constants may consult [ 2 0 ] \Ve de-

note the dielectric constant of free space by i o arid the relative dielrct,ric

constant of the photonic crystal by ~ ( r ) The electric displacen~ent is thus

given by

D ( r , t ) = ~ a ~ ( r ) E ( r , t ) ( 2 6 )

The periodicity of ~ ( r ) implies

~ ( r + a i ) = ~ ( r ) (i = 1 , 2 ; 3 )

where { a , ) are the elernentary lattice vectors of the photonic crystal Because

of this spatial periodicity, we can expand ~ - ' ( r ) in a Fourier series For this,

we introduce the clcmentary reciprocal latt,ice vectors { b i ; i = 1 , 2 3 } arid the

reciprocal lattice vectors { G } :

wliere { l i } are arbitrary integers and hL, is Kronecker's delta E - ' ( r ) is ex-

pressed a s

1

- = 1 i;(G) exp ( i G r ) ( 2 1 0 )

4 ~ )

Because we assumed that the dielectric fui~ctior~ is real K ( - G ) = K * ( G )

Whcn rue substitute ( 2 5 ) and ( 2 6 ) into ( 2 1 ) - ( 2 d ) : we ohtain

\!;hen we rli~rli~iate E ( r t ) or H ( r t ) in ( 2 1 3 ) and ( 2 1 4 ) ; we obtain thc

following wave equations:

-V x {V x E ( r , t ) ] = - E ( r , I )

r 2 at2 ( 2 1 5 )

2.1 Wave Equations atid Eigenvalue Problems 15

where c stands for the light velocity in free spacc

where the two differential operators C E and C H are defined by the first equality in each of t h e above equations

Becanse E is a periodic function of the spatial coordinate r , we can apply Bloch's theorem t o ( 2 2 0 ) and ( 2 2 1 ) as in the case of the electronic wave

equation in ordinary crystals with a periodic potential due t o t,llr regular

array of atoms.' E ( r ) and H ( r ) arc thus characterized by a wave vector k

in the first Brillonin zone and a band index n and expressed as

where u k n ( r ) and u k n ( r ) are periodic vectorial functions that satisfy the

following relations:

Because of the spatial periodicity of these functions, thry can be expanded

in Fourier series likc c r 1 ( r ) in ( 2 1 0 ) This Fourier expansion leads to the

follou~ing form of the eigenfunctions

16 2 Eigenmodes of Photonic Crystals

The expansion coefficients in reciprocal lattice space, i.e Ek,(G) and

Hk,,(G) are denoted by the same symbols as the origillal ones in rcul space

Substituting (2.10), (2.26), and (2.27) into (2.20) and (2.21), we obtain the

following eigenvalue equations for the expansion coefficients { E h n ( G ) } and

solving one of these two sets of equations numerically we can obtain the

dispersion relation of the eigenmodes, or the photonic band structure [21, 221

This nuinerical mcthod, which is based on the Fourier expansion of the

electromagnetic field and the dielectric function, is callcd the planewave ex-

pansion method In the actual numerical calculation of t h e photonic hands,

the summation in (2.28) or (2.29) is calculated up t o a sufficiently large

nun~ber N of G', and an eigenvalue problem for each k is solved, which is

equivalent t o the diagonalization of the matrix defined by the left-hand side

of (2.28) or (2.29) The di~nerrsiun of the rnnt,rix that should be diagonalized

is 3 N for {Ek,(G) : G ) On the other hand, it is 2 N for {Hk,(G) : G ) ,

sincf H k , ( G ) shonld be perpendicular t o k + G according t o (2.12) and

(2.27), and its degree of freedom is two The CPU time that is necessary for

the diagnalizatiou is usually proportional t o the cube of its dimension Hence

the CPU time for the photonic band calculation by means of the plane-wave

expansion method is proportional t o N 3 This fact sometimes leads t,o a seri-

ous constraint of the accuracy of the calculation In fact, the convergence of

the plane-wave expansion method is not good when the amplitude of the spa-

tial variation of the dielectric constant is large, and numerical error cxceeds

5% in certain cases even though we take N greater than 3000 The numeri-

cal error also depends on the dimensionality of the crystal The convcrgence

of tlrc plane-wavc cxpansion method has been discussed by several authors

Readers who are interested in this problem should consult [23-261

The poor convergence of the plane-wave expansion method ha.? been im-

proved by several means Wc may use spherical waves instead of plane waves

as a basis set if the photonic crystal is composed of dielectric spheres or cir-

cular cylinders This method is called the spherical-wave expansion method

or the vector KKR (Koringa-Kohn-Rostker) method Although it can treat

only spherical arid cylindrical dielectric structures, t h e convergence of the

nurnerical calculation is usually much bet,ter than the plane-wavc expansion

method The reader may consult [27 301

2.1 Wave Eqnations and Eigenvalnr Prublerrlv 17

Wc should not,e that E k n ( r ) and H r , ( r ) have, of course, the same eige~i- frequency since they are related t o each other 1?y (2.13) arid (2.11) There is

a difference in the number of the eigenvalues between E ~ , ( T ) aud Hk.,(r) since the dimension of the matrix that should be diagonalized is 3A' for the former and 2 N for the latter Actually N of 3 N eigenvalues of E k o ( r ) are equal t o zero This point will be discussed again in Sect 2.7 in connection with thc quasi-longitudinal modcs of the radiation field

Because Hk,(G) is perpendicular t o k + G , it can he expressed by a linear combination of two orthogonal normal vectors e c , and e c 2 :

H r , ( G ) = h F d e c ~ + h F : e ~ * (2.30)

\Ve assume without the loss of generality that

constitute a right-hand system It is easy t o derive the following equation from (2.29)

where hlk(G G') is given by I\.Ik(G,G1) = lk + Gllk + G ' ~ K ( G - G ' )

which is an Hermitian matrix:

( G , G ' ) = hl:"*(G1, G ) The eigenvectors { h k n } are thus orthogonal t o each other:

From this equation, we obtain

18 2 Eigenmades of Photonic Crystals

Recause both k and k' are wave vertors in the first Brilloilin zone, k' -

k cannot coincide with a reciprocal latt,ice rector The periodicity of the

fnnction u ; l , ( r ) u k , , , ( r ) thus leads t o thc above integral vanishing Hence,

xv finally obtain

This orthogonality is a direct consequence of the fact that the matrix XIk

defined by ( 2 3 3 ) is Hermitian The latter is on the other hand, a consequence

of thc fact that LH defined by ( 2 2 5 ) is an Hermitian operator As for L E , it,

is not Hermitian, and so its eigelifuuctio~ls are not necessarily orthogonal to

each other This point will be described in more detail in Sect 2 7

P r o o f of Bloch's T h e o r e m

Bloch's theorem holds for the eigenmodes of the regular photonic crystals;

here we will prove ( 2 2 2 ) Eqnat,ion ( 2 2 3 ) can be proved in a similar manner

First, we express the eigenfunction of t h e electric field by a Fourier integral:

We write ( 2 2 0 ) in a slightly different way:

Here we used the same notation for the expansion coefficients a? the origi-

nal dielectric f u n d o n When we substitute ( 2 3 9 ) and ( 2 4 1 ) into ( 2 4 0 ) we

obtain

( 2 4 2 )

Since this equation holds for all T , the integrand should vanish:

This equation implies that only thusc Fourier components that are related

by the reciprocal lattice vectors constitnte the eigenvalue problen~, that is a

2.2 F;igcnvalue Problcnls in Two-Dimcnsiu~lal Crystals 19

set of lincar eigenmhc equations Hcnce, only thosc Fourier conlponellt,s are necessary to express the eigenfunction in ( 2 3 9 ) :

tions, we distinguish them by a subscript n Hence, we obtain ( 2 2 2 )

2.2 Eigenvalue Problems in Two-Dimensional Crystals

For two-dimensional ( 2 D ) crystals, the eigerlvalue eqnnt,ions are much simpli-

fied if the k vector is parallel t o t h e 2 D plane We examine this case here In

the 2 D crystal, the dielectric strnctnre is uniform in the z direct,ion (see Fig

1 2 ) The electromagnetic waves travel in the z - y plane and are also uniform

in the z direction Hcnce, ~ ( r ) , E ( r ) , and H ( T ) are independent of the z

coordinate in (2.13) and (2.14) In this case, these vectorial equations are decoupled t o two independent sets of eqnations The first is

and the second 1s

20 2 Eigennlodes uf Photonic Crystals

From t h r scromd set we obtain the wave equation for H , ( r l t ) : I

\Ve seek, as before, t h e solutions of these equations of t h e form

E , ( r i l , t) = ~ , ( r ~ ) e C ' " ~ , (2.55)

H , ( r l i : t ) = H , ( r g ) e - ' " ' (2.56)

The eigenvalue equations are thus given by

I where the two differential operators C g ' and ~ E ' f o r t h e 2D case are defined

by the first equality in each of the above two equations These two kinds of

eigenfunctio~is represcnt two independent polarizations; one is called the E

polarization for which the electric field is parallel to t h e z axis, and the other

is called the H polarization for which the magnetic field is parallel t o the z

axis

When we apply Bloch's theorem as before, we can express E , ( r l ) and

H z ( r l ) as

where k I and G I art: t h e wave vector and the reciprocal lattice vector in two

dimensions Substituting (2.59) and (2.60) into ( 2 5 7 ) and ( 2 5 8 ) , we obtain

the following eigelivalue equations for t h e expansion coefficients:

2.3 Scaling Law and Time Reversal Symmetry 21

The eigenvalue equation: (2.62), is thus expressed as

As we derived the orthogonality of H k n ( r ) in ( 2 3 8 ) , we can prove that

where V ( ' ) denot,es the 2D volume of the photonic crystal This orthogonality relation is a consequence of the fact that c is an Hermitian operator On the other hand, C g ) is not Hermitian, and so its eigenfunctions are not necessarily orthogonal to each other This point will also be described in more detail in Sect 2.7

For the detailed numerical method, see [31] and 1321 Also, see [33] for the off-plane dispersion of the 2D crystals

2.3 Scaling Law and Time Reversal Symmetry

There are two useful properties of the photonic bands One property is the scaling law and the othcr is thc time reversal symmetry of the wave equation The scaling law tells us that two photonic crystals which are similar t o each other essentially have the same photonic band structure, that is, the difference between the two band structures is simply the scales of frequency and the wave vector On the other hand, t h e time reversal symmetry tells us that any photonic band structure hay inversion symmetry even though the crystal structure does not have inversion symmetry The proof of these properties is given in the following

First, the following scale transformation t o (2.15) is performed:

22 2 Eigenrnodes of Photonic Crystals

- = r and - t = t'

The new variables r' and t' are dimensionless When we define anew dielect,ric

functio~l E,, and a new vector field E,, by

where V' stands for the differentiation with respect t o r ' Hence, if the struc-

tures of two photonic crystals are sinlilar t o each other and their difference is

simply the scale of Llle length, i.e.? the lattice constant, then thcir wave equa-

tions are attributed t o the same dimensionless wave equation by the scale

transformation

Now, we denote the dimensionless wave vector and the dimensionless

eigen-angular frequency in the (r', t') space by k' and wk.,,, respectively

Because lc and w have the inverse dimensions of r and t, respectively, the

following transformations are necesary t o return to real space:

k = - k and w = - w , a

where we included the factor 1 / 2 ~ according lu Lht: cunvention in this field

Therefore, if we measure the wave vector in units of 2 ~ l a and the angular

frequency in units of 277cla: all dispersion curves are the same for those

crystals which have similar dielectric functions Thanks t o this scaling law,

we can conduct siru~~lation experiments for crystals with a lattice constant of

about 1 mm for example, using specimens with a lattice constant of 1 cml

Since the fabrication of specimens with smal lattice constants is usually a

difficult task, the scaling law is very usuful t o accelerate experimental studics

and t o confirm theoretical predictions

As for t,hc tirne reversal symmetry, we should note that the wave equation,

(2.15); is invariant when we change the sign of the time vnril~ble \?'Ire11 wp

define a new variable t' and a new vector field Etr by

Equation (2.78) implies that the dispersion relation has inversion symmetry This property is irrespective of whether the structure of the photonic crystal has inversion symmetry

2.4 Photonic Band Calculation

2.4.1 F o u r i e r E x p a n s i o n of D i e l e c t r i c F u n c t i o n s

I t is necessary t o calculate the expansion coefficients { K ( G ) } in (2.10) for the band calculation by the plane-wave cxpansion method The inverse Fourier transform gives

where Vo denotes the volume of the unit cell of the photonic crystal In gen- eral, this integral should he evaluated numerically However, if the shapes of the dielectric components in the unit cell are simple enough, we can calculate

it analytically In what follows, we treat two such cases One is the three-

dimensional (3D) crystal whose unit cell contains one dielectric sphere, and

the other is the 2D crystal whose unit cell contains one circular dielectric rod

D i e l e c t r i c S p h e r e

We denote the radius and the dielectric constant of the sphere by r, and E,,

respectively, and the dielectric constant of the backgro~~nd mnterial by cb

24 2 Eigenmodes of Photonic Crystals

Substituting (2.81) and (2.82) into (2.80),

In order t o calculate the integral in (2.83), we use spherical coordinates

( r 0 , ~ ) We take the direction with 0 = 0 as the direction of vector G

For G # 0, the integral is thus modified t o

Because the structure is uniform in the z direction for this case, the integral

in (2.80) is equal t o zero if G, # 0 So, we restrict our discussion t o 2D

vectors {GI} If we denote the 2D unit cell by v?' as before,

If we demote the radius and the dielectric constant of the circular rod by r,

and E,, respectively, and the dielectric constant of the background material

In order t o calculate the integral in (2.92), we use polar coordinates ( T , p )

We take the direction with y = 0 as the direction of vector G g For G g # 0 the integral is modified t o

vd"

If we denote the volume fraction of the circular rod by f ,

we finally obtain for G I # 0

For G l = 0;

26 2 Eigennlodes of Photonic Crystals

as before

2.4.2 S o m e E x a m p l e s

We consider two examples of tlie band diagranls calculated by the plane-wave

expansion method here First, we examine the band structure of a simple

cubic lattice composed of dielectric spheres We assume that one dielectric

sphere is located a t each lattice point The following values were used for the

numerical calculation: t h e dielectric constant of the sphrres and the back-

ground are 13.0 and 1.0, and the ratio of the lattice constant t o the radius of

the sphere is 1:0.3 Figure 2.1 shows t h e first Brillouin zone of the simple cu-

bic lattice Highly symmetric points are denoted by thc standard notations in

this figure Figure 2.2 sliows the photonic band structure obtained by solving

(2.32) with 1174 plane waves (N = 587) The ordinatc is the normalized fre-

quency whcre u and c denote the lattice const,ant and t h e light velocity in free

space The CPU time necessary for t,hc whole calculation was about 23 min-

utes when we used a supercomputer with a vector processor The numerical

error was esti~nated to be better than 5% for lower frequencies

For this structure, there is no complete bandgap t h a t extends throughout

the Brillouin zone However, ure can find several in~portaut properties peculiar

t o the photonic crystal in Fig 2.2 First, there are partial bandgaps for optical

waves travelling along the (1, 0, 0) direction, i e , those waves which have wave

vectors on t h e A point, The frequency ranges of the partial gaps are

Fig 2.1 First Brillauin zone of the simple cuhic latticc

2.4 I'hotouic Band Calculation 27

Fig 2.2 Photonic band structure of a simple cubic lattice with a dielectric sphere

at each lattice point The ordinate is the normalized frequency The following values

were assumed for the numerical calculation: the dielectric constant of the spheres and the background are 13.0 and 1.0: and the ratio of the lattice constant t o the radius of the sphere is 1:0.3

symmetric points such a s the r and the R points In addition, several bands '

on t h e A and the T points are doubly degenerate as a whole This property is closely related to the symmetry of t,he crystal structrire and will be examined

in detail by using g r o l ~ p theory in Chap 3

Next, we consider t h e band structure of a 2D cryst,al composed of a regu- lar square array of circular dielectric cylinders Figure 2.3 shows tlie structure

of t h r photonic crystal and the 2D first Brilloui~~ cone of tlie sqnare latticc

Fig 2.3 (a) Intersection of the 2D square lattice composed of circular cylinders

(b) First Brillouin zone of the square lattice

We restrict our discnssion to the case for which the wave vectors of the eigen-

modes lie in the 2D z-y plane As was explained in Sect 2.2, the eigenmodes

are classified into two categories according t o thcir polarization for this case,

that is, the E polarization for which the electric field is perpendicular to the

z-y plane and the H polarization for which the magnetic field is perpendicu-

lar t o the x-y plane We treat the E polarization here The following values

were assumed for the numerical calculation: the dielectric constants of the

cylinders and the background are 9.0 and 1.0, and the ratio of the lattice

constant to the radius of the cylinder is 1:0.38 Figure 2.4 shows the photonic

band structure obtained by solving (2.61) with 441 plane waves The nu-

merical error was estimated t o he about 1% The ordinate is the normalized

frequency as in Fig 2.2 The left-hand side of this figure shows the photonic

band structure of the E polarization, whereas the right-hand side shows the i

density of states of t h e radiational modes that was obtained by examining

t h ~ dist,rihiit,ion of the eigenfrequencies for 16 000 different wave vectors in

the Brillouin zone

I t is clearly seen that there are three bandgaps that extend throughout

the 2D Brillouin zone whose frequency ranges are

polarization and those with off-plane wave vectors have eigcnfrequencies in

2.4 Photonir Rand Calculation 29

the above gaps Therefore, the complete inhibition of spontaneous emission of photons from atoms embbeded in t,he photonic crystal, for example, cannot he expected However, it can be partially inhibited The partial inhibition may be effective when the transition dipole moment of the embedded atoms is aligned perpendicular to the x-y plane Situations of this kind may be realized with artificial layered structures such as quantum wells made of semiconductors

1341 ~,

As in Fig 2.2, there are several extremely flat bands in Fig 2.4 These bands can be nsed t o enhance various optical processes such as stimulated emission, second harmonic generation, and quadrature-phase squeezing For all of thesc optical processes, the incident electromagnetic field is usually pre- pared with a laser Hence, t,he incident beam has a well-defined propagation direction, and we can propagate it along the x-y plane very easily There- fore, the 2D band structures: such a s Fig 2.4, have not only mathematical interest but also practical importance Especially where the laser oscillation

is concerned, it takes place at such a freqnency and in such a direction that the optical gain due t o the stimulated emiasion is a maximun~ and the optical loss by transmission a t the surface of the crystal is a minimum It is well rec- ognized that the transmission coefficients for thoscs modes with small group velocities are usually small The lasing thus readily takes place for the flat bands This point will be examined in detail in Chap 9

As for ihe localized defcct modes, the presence of the partial bandgaps shown in Fig 2.4 is enough for their existence In fact, hlcCall et al [35]

observed a defect modc in this 2D crystal by removing a single cylinder Thc defect modes usually have quite large quality factors and they can be nsed as guod resonat,ors Since the interaction between t h r radiational field

30 2 Eigenmodes of Photonic Crystals

and iriatt,er depends st,rongly on the field strength and the interaction time,

the largc quality factor also enhannces the optical interact,ions An efficient

numerical method t o deal with the defect modes will be given in Chap 6

2.5 Phase Velocity, Group Velocity, and Energy Velocity

In the previous three sections, we learned how t o calculate the dispersion re-

lations of the radiation modes in the photonic crystals \Ve can also calculate

their wave functions In addition t o the eigenfrequencies and eigenfunctions,

therc are several parameters that characterize the radiational waves One of

then1 is t h e wave velocity In contrast t o the case of particles for which the

velocity has a single meaning, waves have three different kinds of velocities,

i.e t h e phase velocity, the group velocity, and the energy velocity These ve-

lucities are equal t o each othcr in uniform materials with dielectric cnnst,ant,s

which are real and independent of frequency

The phase velocity is defined as t h e velocity of the propagation of an equi-

phase surface This velocity has a definite meaning for example, for plane

waves and spherical waves for which the equi-phase surface can be defined

without ambiguity In the photonic crystal, howeverl the equi-pha~e surface

cannot bc defined rigorously, since its eigenfunction is a superposition of

plane waves, as can be seen in (2.26) and (2.27) This means that the phase

velocity cannot be defined appropriately in the photonic crystal

O n the other hand, the group velocity, which is t h e velocity of the prop-

agation of a wave packet, can be defined as usual:

aw

- a k '

The energy velocity is defined as the velocity of the propagation of the electrrr

magnetic energy The propagation of the electromagnetic energy is described

by Poynting's vector The time-averaged Poynting's vect,or S k n ( ~ ) is given

Thus, t h e energy velocity v , is defined as

where ( - .) means the spatial average

2 3 Phase Vrlocity, Group Velocity, and Energy Velority 31 Now, the group velocit,y is eqllal to the erlcrgy velocity even though the dielectric const,ant is modulated periodically The proof, which we follow

was given by Yeh 1361 If we substitute (2.1R), (2.19), (2.22) and (2.23) into hlaxwell's equations (2.13) and ( 2 1 4 ) , we have

V x ~ k n ( ~ ) + ik X u k n ( T ) = i ~ ~ ~ k n ~ k n ( ~ ) , (2.106)

Supposc that k is changed by an infinitesimal amount 6k This results in

changes i11 wk,, ur,,, and v k n , which are denoted by 6wk,, sub,, and 6vk,& From (2.106) and (2.1071, we have

+i6ukn (PO 1vk,l2 + E O E jukn2) (2.110)

If we calcl~late 6u;, (2.107) and b u r , (2.106)*,

~ E O E W O ~ ~ U ; , u k , , = - 6 4 , ( V x W E , ) - ik ( v k , x 6u;,), (2.111)

Using (2 111) and (2.112), (2.110) is written as

v;,, (V x 6 u k , ) + u k , (V x 6v;,,) + 2i6k ( u k , , x v;,) +6u;, (V x v k n ) + 6 v k n (V x u;,)

+ik ( 6 X u;, ~ - ~ 6 v ; , ~X U k , + V k n X 6 ~ ; " - uin X 6 v k n )

= i6&kn ( P O I ~+ E O E I U ~ ~ I ' ) ~ ~ ~ I ~ (2.113) The right-hand sidc of (2.113) is purely imaginary Hence, we take the imag- inary part of (2.113) by calculating (2.113) - (2,113)':

v;, (V x hub,) - b u r , - (V x v;,)

+ U k n (v X 6 ~ ; ~ ) - 6 ~ ; ~ (V X U k , , )

+Su;, (V x u k n ) - vkn (V x bu;,,) +6vr,, (V x u;,) - u;, (V x 6 v k n )

+4i6k R e [ u k , x v;,]

= 2ifiukn (po I V ~ , ~ I ~ + E O E l u k n l 2 ) (2.114)

32 2 Eigenrnodes of Photonic Crystals

If we define a periodic function F by

F = SUE., X w i n + h u t , X u k , + Vk, X J u t , + u;, X 6vk,, (2.115)

(2.114) is modified t o

2

V F + 4 i 6 k R e [ u r n x u;,] = 2i6wk, (fro lvrnl + E O E ( U ~ ~ ~ ~ ) ,(2.116)

where we have used t h e following vector identity:

V ( A x B ) = B ( V x A ) - A ( V x B ) (2.117)

If we perform an integration over a unit cell, we have

( V F ) + 4 i J k (Re [ubn x v;,])

= 2i6Ukn (Po luknl + EoE b k n l ) (2.118)

Because of the periodic nature of F ,

In this equation, Vo is the volume of t h e unit cell and So is its surface F,

denotes the outward normal component of F on So We used Gauss's theorem

to derive the second equality in (2.119) Using the definition of v,, (2.105),

we finally have

From the definition of the group velocity, (2.102), we have

2.6 Calculation of Group Velocity

As we learned in the last section, the group velocity of the radiation modes

has very important role in light propagation and optical response in the

photonic crystals IIence, t h e calculation of thc group velocity is an essencial

task for the understanding of their optical properties Since the group velocity

is defined as thc dcrivative of the angular frequency with respect t o t h e wave

vector (see (2.102)) we may calculate it by numerical differentiation That

is: we may actually evaluate the following limit:

Wk+Ak,n - Wk.n

v - lim

- a k i o A k

This nrlrnerical differentiation needs a limiting procedure for which we have

t o know a series of eigenfrequencies as a function of the wave vector

There is quite a convenient method t o avoid this procedure and give an

accurate evaluation of t,he group velocity \Ire use t h e Hellmann-Feynman

2.6 Calculation of Group Velocity 33

theorem for this purpose, with which the readers may be familiar as it re- lates t,o quantum mechanical calculations In the case of quantum mechanics, the Hellmann-Feyn%an theorem is stated as follows First, we assume an

Hermitian operator H that depends on an external variable a , and denote it

by H, We also & s p i t ~ t l that we kuow the orthonormal set composed of the eigenfunctions of H,, which we denote by {Ian) ; n = 1 2 , .):

h~ the above two equations, we used_the standard notation of "bra" and "ket" vectors A,, is the eigenvalue of H, for state J a n ) and ( I ) denotes

t h e inner product Now, our problem is t o calculate the derivative of A,,

with respect to a Because the state vector (or the eigenfunction) Ian) is normalized t o unity (2.124), the following holds

a

=A,,- aa ( a n l a n ) +

Once we know the analytical expression of a E , / a a , aA,,/aa can readily be obtained using this equation without the limiting procedure in (2.122) Now, we will see how t o use the Hellmann-Feynman theorem for the calcu- lation of the group velocity Here, we show the method for the E polarization The dispersion relation is obtained by solving the eigenvalue equation (2.61) When we define a column vector Akln by

Ak,,(Gl) = Iky + G l l E z k l ~ ( G l ) (2.61) is transformed t o

where Mkl is a kl-dependent matrix whose ( G b , G)) component is given by the following equation:

The ky-dfpendent vector Arl, gives th? eigenfunctlon E,.k,,(rll) as

34 2 Eigenrnodes of Phutanic Crystals

and it is normalized t o unity, i.e Aklnl = 1 In ( 2 1 2 9 ) h is the normaliza-

tion constallt

Here, we assume that E ( T / ) is real Then, K ( - G I ) = n * ( G g ) and M r ,

is an Hermitian matrix Therefore, we can apply the Hellmann-Feynman

theorem t o the present problem, and we obtain

where t denotes the transposed matrix and

a h f k / f G l , G ) )

a k !

Therefore, the group velocity u , ( u , n) can be readily evaluated once the eigen-

vector A k , " and the e i g e n v a ~ u e u ( ~ ) ~ / c ~ are obtained by the band calculation

k i n

b a ~ e d on the plane-wave expansioil method

2.7 Complete Set of Eigenfunctions

When we derive the Green's function related t o the electric field and dis-

cuss t h e optical response of the photonic crystals, u e need a complete set

of eigenfunctions concerned with the electric field However, LE defined by

( 2 2 0 ) is not an Hermitian operator, hence its eigenfunctions { E k n ( ~ ) } are

not complete or orthogonal Here, we define a differential operator that is

Hermitian and examine the properties of its eigenfunctions 137-391 We in-

troduce a cornplex vectorial function Q ( T , t ) and a differential operator n:

2.7 Complete Set of Eigcnfunctions 35

We assume the periodic boundary condition for Q ( r t ) as usual We will verify that X is an Hermitian operator Then, its eigenfunctions form an orthogonal complete set

Now, the inner product of two complex vectorial functions Q l ( r ) and

Q z ( T ) is defined by

where V is the volume on which the periodic boundary condition is imposed

As a vector identity, the next equation holds

V ( A x B ) = ( V x A ) B - A ( V x B ) ( 2 1 3 6 )

Using this identity, we obtain

where S denotes the surface of V and the first integral on the right-hand side

is thc surfacc integral of the normal component of the integrand This surface integral is equal t o zero because of thp periodic boundary condition Then,

applying the identity (2.136) again, we obtain

where we have again used the fact that the surface integral is equal to zero This equation implies that 'H is an Hermitian operator

Next, we can err& show that ~ k ' ( r ) : given by

Q ~ ; ! ( T ) = c\/ ' S e x p { i ( k + G , ) r ) ,

h ( T ) I k + G,l (2.13Y)

where G , is a reciprocal lattice vector and C is a ~lormalization constant satisfies

36 2 Eigenmodcs of Photonic Crystals

Then, from the definit,ion of 'U,

( L )

This equat,ion implies that Qkn ( 7 ) is an eigenfunction of 'U and its eigen-

angular frequency is zero There exists one such solution for each k and n

In general,

V - { , - ( T ) E E ( ~ ) } = V { m Q E ( r ) )

I L )

We call Q ~ : ( T ) a quasi-longitudinal solution Strictly speaking, the Q k , > ( r )

defined by ( 2 1 3 9 ) are not orthogonal t o each other but their orthogonal-

ization can be readily accomplished by Schmidt's method That is, we can

take linear combinations of the &E(T) t o form an orthonormal set We de-

note the orthogonaliaed quasi-longitudinal solutions by the same symbols

Because the Q ~ ( T ) do not satisfy (2.111, they are uuphysical solutions for

the present prohlem However, they are important mathematically, since we

cannot obtain a complete set without them

On the other hand, there exist qila9i-transverse solutions that correspond

to the transverse plane waves in a uniform crystal They satisfy the following

equation:

Thc rigen-angislar frequency WE' is generally non-zero From ( 2 1 4 3 ) and

( 2 1 3 3 ) ,

ZTote that QE' or QL) are not purely transverse nor longitudinal because

of the spatial variation of € ( T I The terms "quasi-transverse'' and "quasi-

longitndinal" are used here t o emphasize that ( 2 1 4 4 ) and ( 2 1 4 0 ) reduce

when C ( T ) is a constant, t o the usual relations concerned with transverse and

longitudinal waves; respectively

Now we normalize these eigenfunctions as follows:

where a: 0 = T or L Note that QL'(r) and E ~ ) ( T ) (= Q L i ( r ) / m )

art: dimensionless by this definition The complete~iess of the eigenfunctions

elements of two vectors i.e., ( A 8 B ) i j = A , B j , and I is the unit tensor

6 ( r ) is Dirac's delta filnction

We make one remark on ( 2 1 4 6 ) here Readers who are familiar with

non-relativistic quantum mechanics may wonder why the tensors introduced

in the above equation are necessary to express the completeness In non- relativistic quantum mechanics, the wave function that describes an electron

is a compl_ex scalar function UTe denote the eigenfunctions of a Hamiltonian operator H by ( 4 , ; n = 1 , 2 , .) We normalize them as

The completeness is thus expressed as

C ~ , ( T ) ~ ~ ( T ' ) = V ~ ( T - r')

n

From this equation, we obt,aiu for any complex function f i r )

This equation means that any complex function can be expanded with

{ d , ( r ) } This is what t,he complet,eness inlplies for a set of scalar functions

As for a set of vectorial functions, they should also be able t o expand any vectors in 3D Euclidean space As an example, consider the unit vectors e l :

e Z , and es which are parallel t o the r , y , and z axes, respectively Thr:? are

a complete set in 3D Euclidean space, since any vector v can be expanded with them:

v = c e n ( e n v ) ( 2 1 5 0 )

"I

h e can write this e uation in a sllghtly * , R different manner We ddrnr three tensor?, T I T z , and T ? as

38 2 Eigcnmodcs of Photonic Crystals

is the condition for the completeness in the 3D Euclidean space As a conc111-

sion, (2.146) is the condition for the completeness both in functional space

and in Euclidean spare

As for the 2D crystal, we examine the case of t h e E polarization here

The H polarization can be treated in a similar manner First, we define a

function Q , ( r l l : t ) and a diff~rential operator ?Lf1(') such that

Then (2.53) leads t o

Itre can verify a s before that ~ f 1 ( ~ ) defined hy (2.159) is an Hermitian operator

Its eigenfunctions { Q , , ~ , , ( T ~ ) } thus form an orthogonal complete set We

normalize t,hrse eigpnfirnctions as f01lou.s:

where \I(') is the 2D volume on which the periodic boundary condition is

imposed The completeness of the eigenfunctions leads to

x Q : k , 7 , ( ~ ~ ~ ) Q ; , k l n ( ~ ) j = v ( ~ ) ~ ( T ! - T ) ) (2.161)

kin

in this case sirlcr { Q i C , n ( ~ l l ) ] arc scalar functions

2.8 Rclarded Green's hlnrt,ion :%I

2.8 Retarded Green's Function

In later chapters, wc will deal with various optical processes in photonic crystals For that purpose it is essential t o calculate the electromagnetic f i ~ l d radiated frum oscillating polarization fields This task can enerally be per- W

formed by means of a retarded Green's (tensor) function G ( T , T ' , t ) 137-391

that satisfies thc following t,wo equations

Then from (2.141), (2.143), and (2.146), we can obtain its explicit expression:

Here; 6 is a positive infinitesimal that assures the causality, i.e., (2.163) For

t 2 0, the inverse transform of (2.166) gives

40 2 Eigcnmodes of Photurlic Crystals

Fig 2.5 Contour of the integration in (2.167) for t > 0 For t < 0 , the contour

should enclose the uppcr half plane

where the contour C is shown in Fig 2.5 For t > 0 , we close the path of

integration in the lower half of the complex w plane For t < 0: we close it in

the upper half plane, and we obtain ( 2 1 6 3 )

As for t h e E polarization in the 2 D crystal, we define the retarded Green's

function of ( 2 1 5 9 ) by

In this case, the Fourier transform of the Green's fnnction, G ( 2 ! ( ~ g , T ) , w ) : is

givcn by

This satisfies the following equation:

The inverse transforn~ of (2.170) leads to

for t > 0 These Green's functions will be used in later chapters t o analyze the optical response and the defect modes in photonic crystals

3 Symmetry of Eigenmodes

The symmetry of the eigcnmodes of p h o t o ~ ~ i c crystals plays an important role in their optical response In this chapter, we will formulate the group theory of the radiation field in the photonic crystals in order to classify the spatial symmetry of their cigenfunctions An cfficient method Iur the sym- metry assignment that excludes unphysical quasi-longitudinal modes, which was discussed in the last chapter, will be presented The existence of such eigenmodes that cannot be excited by external plane waves because of the mismatching of the symmetry will be shown This method will be applied to some examples

3.1 Group Theory for Two-Dimensional Crystals

The symmetry of t h e eigenfilnctions of the photonic cryst,als plays an impor- tant role in their optical response In fact, Robertson et al 140, 411 showed

clear evidence of the presence of such eigenn~odes in two-dirnensional (2D)

photonic crystals that cannot be excited by an external plane wave due to the mismatching of their spatial symmetry We refer t o these modes nn- coupled modes On the othcr hand, the symmetry of the eigenmodes leads

to an additional selection rule for nonlinear optical processes and absence

of diffraction loss in the photonic crystal (see Chaps 5 and 8) The clwsi- fication of the eigenmodes according t o their spatial symmetry by means of xronp theory is thns necessary and often very powerful t o undersla~ld the o p tical properties of the photonic crystals In particular, when we compare the transmission spectra observed rxperimentally with the calculated photollic band structure, the knowledge of the uncoupled modes is indispensable Sillce it is easier to deal with scalar waves, wc begin with the grollp theory [or 2D photonic crystals 1421 As w a shown in Sect 2.2, the vectorial wave

equations derived from hlaxwell's equations are reduced t o two independent scalar equations as long as the wave vector lies in the 2D 2 - y plane These two waves are called the E polarization for which the electric field is parallel

t o the z axis and the H polaricat~iorl for which the magnetic field is parallel

to t,he z axisl respectively The ~igenf~lnctions satisfy the following eigenvahle eqnations:

44 3 Symmetry of Eigenmodes

Y

Fig 3.1 Symmetry operations for the square array of dielectric cylinders

nrhcre operators c(EZ' and L$) are defined by (2.57) and (2.58)

Now, all 2D photonic crystals have translational syninletry with respect

to t h e elementary lattice vectors, a l l and a z l :

whcrc 1 and m are integers In addition t o this translational symmetry the 2D

photonic crystal often has other spatial symmetries As an example, consider

the square array of dielectric cylinders discussed in Sect 2.2 We take the x

arid y axes as in Fig 3.1 It is apparent that the structure is invariant when

we change x t o x :

If we denote the symmetry operation that changes x to -x by a,, we say

t,hat the st,rllct,llrc is invariant under the mirror reflection a, Equation (3.4)

may be rewritten as

[nZ.](rl) = E ( U ; ~ T ~ ~ ) = E ( T / ) (3.5)

The convention of group theory is t o use riot n,, hot 0 in t,lie middle term

3.1 Group Theory for TwoDimensior~al Crystals 45

Table 3.1 Products of the operations in the C d u point group This table shows the product R I R z where R 1 denotes an operation in the first columrl and R2 denotes

an operation in thc first row

is also invariant under the mirror reflection a, that changes y to -y There

is another set of mirror reflections under whiqh the structure is invariant They are 02, which changes (x, y) t o ( y , x ) , and a&', which changes (x, y) to

(-g, -x) The structure is also invariant when it is rotated by 90, 1801 or

270 degrees counterclockwise about the origin These symmetry operations are denoted by CJ, Cz(- C j ) , and c;'(= C:) C, generally means rotation

by 2 n l n radian

Together with the identity operation E that keeps the structure as it is, all these symmetry operations constitute the CqU point group:

Namely, (1) any two subsequent symmetry operations in C4,: are equivalent

t o a single operation in it, (2) any three symmetry operations {R1,Rz, R3} have the property that Rl(RzR3) = (R1Rz)Rs and (3) the inverse of any symmetry operation in C4, is also an operation in it These conditions may easily be confirmed As for item ( I ) , Table 3.1 shows the products of two operations in C4v If the following relation holds, we say that R1 is conjugate with Rz:

46 3 Symmctry of Eigcnmodcs

then a, becon~es a, and a, becomes a, a, and rr, are soniet,imes denoted

by a,, and a& and n i are denoted by a,,

Now: any rotation or mirror reflection in two dirr~erisions can be repre-

sented by a 2 x 2 matrix As for the Cd, point group, we have

Tl'e denote the matrix representation of a 2D symmetry operation R of an

where Rt is the transposed matrix of R, and I is the unit matrix For an

arhitrary function f of r l , the action of R on f is defined by

Hence, t_he o r d e ~ o f R1 and R2 when they operate on f is consistent with

that of R1 and Rz when they operate on r , / This is a consequence of the

*

definition (3.13), where R-' is operated on T!

In general, we assume that the 2D photonic crystal is i n ~ a r i a n t under any

symmetry operation R that belongs t o t,he point gronp M:

[ R E ] ( T , ~ ) i ( R ' r / / ) = E ( T ! ) (VR E M ) (3.15)

1-king this equation we ran prove that R commutes with c:) and c:':

This implies t,hat the conventional classifiration of the eigenmodes according

to the irreducible representations of the k-group M k can he applicahle t o

rhe present problem.'

' See [41] and 1441 for drtails of the applicatior~ of group thpory tn solirl-state

Here, we nsed (3.19) Equation (3.17) can be verified similarly rn

Now, the k-group M E is defined as the subgroup of M that keeps the wave vector k invariant2 Consider this point with t h e square lattice as an

' Rigorously speaking we should deal with the spatial group instead of the point group to treat our problem in a rriathernatirally corrrrt mannrr Hoacvrr, as long

we treat symnlorphic spatial groups, it makes no difference to the results

48 3 Symmetry of Eigenmodes

Fig 3.2 First Brillouin zone of the 2D square lattice

example Figure 3.2 shows its first Brillouin zone The elementary lattice

vectors (all,a2/} and the elementary reciprocal lattice vectors {bll, b2!}

are

where a is the lattice constant of the square photonic crystal The r, X , and

M points in Fig 3.2 denote (O.O), ( i ~ l a , 0), and ( i n l a , *n/a), respectively

These two X points and four M points are equivalent to each other, since

the difference between them is just a linear combination of the elementary

reciprocal lattice vectors First, consider the r point We can readily see that

all symmetry operations in the C4", point group keep the r point invariant

Hence, by definition,

In the case of thc X point

Here, we should note that operations C2 and o, keep the X point invariant,

since

As for the others,

3.1 Group Thwry for Tw*Dirnensional Crystals 49

M.w = CqV,

MA = {E,oy} c ~ h ,

M E = {E, od} Clh,

M z = (E, a,) = Clh

Group theory tells us that any eigenfnnction, E z , ~ , ( r l ) or HZ,m(r6),

is an irreducible representation of M h For example, the Cq, point group has four one-dimensional irreducible representations A1, Az, B1, Bz and one two-dimensional irreducible representation E Here,"onedimensional" im- plies that the eigenmode is not degenerate and "two-dimensional" implies that the eigenmode is doubly degenerate Each irreducible representation has its own spatial symmetry which is expressed by its chamcter, X Con- sider this point with a B1 mode as an example We denote the character of the Bl representation by XB, and assume t h a t f ~ , ( r y ) is at,t,rihnt,ed t,o the

B1 representation We refer t o f5,(rl) as a basis of the BI representation Group theory tells us that for any symmetry operation R of the CqV point group,

R ~ B , ( ~ 1 ) = XBI (R)fs1 ( ~ 1 ) ~ (3.33) and

1 f o r R = E , C z , o , , o , ,

- 1 for R = Cq, C:', a;, a:

For other onedimensional irreducible representations, the same kind of equa- tion holds For t h e two-dimensional representation E, there are two eigen- functions that have the same eigenfrequency We denote them by f t ) ( r l ) and f g l ( r I ) We refer t o them as a basis set of the E representation In gen- eral, the operation of R on these functions transforms them into the following linear combinations

nf:)(rl) = Allf;)(ril) + A l z f t ) ( r l ) , (3.35)

(2)

R ~ ( ~ ) ( T ~ ) E = A Z I ~ , $ ) ( T ~ ) + A22fE ( ~ 8 ) (3.36) Group theory tells us that

The characters of these irreducible representations are listed in Table 3.2 Generally speaking, the conjugate operations such as o, and o,, or, C4 and

6;' have the same character Hence, in Table 3.2, o, and o, are denoted

by o,, o& and o&' are denoted by od, and Cq and C:' are represented by

Cq The symbol "2" before Cq, ov, and a d in the first row of this table implies that there are two such elements that are conjugate with each other and have the same character As for the CZv and Clh point groups, the irreducble representations and their characters are listed in Tables 3.3 and

Table 3.3 The character table for the Csu point group

Table 3.4 Character table for the Clh point group

1 1

3.4, respectively All irreducible representations are one-dimensional for these

two groups

For the assignment of the irreducible representation t o each mode, it is

very convenient t o use so-called compatibility relations that tell us the mutual

relations between the irreducible representations for adjacent k vectors We

examine this for the r and A points as an example As mentioned previously,

the r point has the Cq, symmetry and the A point has the Clh symmetry

Since the r point can he regarded as a special case of the A point, the irre-

ducible re~resentations for the r noint can also be regarded as those for the A .,

point The uV(rnI.) mirror reflection for t h e former is identical t o the ir mirror

reflection fur t,he latter Wlle~l we coupare the c l ~ a r a c l e ~ tables oC these two

point groups, we readily find that the Al and B1 irreducible representations

of Cd, have the same characters a? the A irreducible representation of C l h

for symmetry operations E and o,(o) Hence, we can conclude that the Al

and BI modes on the r point should connect to the A mode on the A point

Similarly, the Az and Bz modes on the former should connect to the B mode

on the latter On the other hand, a so-callcd reductzon procedure is necessary

to relate the doubly degenerate E mode on thc r point t o one-dimensional

irreducible representations on the A point, since the E n ~ o d e is a reducible

representation for t h r C l h point group The rule for t h r rednction is simple

3.1 Group T11wr.v fur Two-Dimensional Cr,ystnls 51 Table 3.5 Compatibility relations for the square lattice

Hence, it is calculated a below and we obtain one A representation

Similarly, we see t h a t the number of t,he B representation is also one As a result, we can conclnde that the doubly degenerate E mode on the r point connects t o one A nlodc alal one B mode on the A point Similar relations

can be obtained for the rest of the highly symmetric points The resrilts are summarized in Table 3.5

The assignment of the spatial symmetry t o each mode can of course:

be accomplished by examining its eigenfunctions by numerical calri~lation However, there is another simple but powerful method t o do that 1421 This

method also provides t h e information in a vr:ry intuitive manner on how many eigenmodes with a particular symmetry should appear in the band diagram The method is similar t o that used for the assignment of the symmetry of molecular orbitals composed of atomic orbitals In order t o show the detail

of this method, consider the extended zone scheme for t h r reciprocal lattice space which is shown in Fig 3.3 for the sqllare lattice

5" 3 Synlmetry of Eigenmodes

Fig 3.3 Reciprocal lattice space of the 2D square lattice in the extended zone

scheme Highly symmetric points T X, and M are denoted by dots The number

in each pair of parentheses is given in ascending order of the angular frequency of

the plane wave in free space For example, the angular frequencies that correspond

to r12), a11d rln) 0, 211c/u, and 2 \ h i r c / a , respectively Note that, for

example, there are four equivalent rI2) and r(31 points that are connected to each

other with reciprocal lattice vectors

Highly syrrimetric points r, X , and M are denoted by dots, and they

are numbered in ascending order of the angular frequency of the correspond-

ing plane wave in free space First, consider the r points, which have the

symmetry of C4", We begin with r(') that has the second lowest angular

frequency among the r points There are four equivalent r(2) points that are

connected to each other with the reciprocal lattice vectors They have the

same eigenfrequency when the spatial variation of the dielectric constant is

absent When an infinitesimally small periodic perturbation is introduced t o

the dielectric constant, these four plane waves are mixed and their particular

combinations will be the eigenmoda of the radiation field This implies, in

t h e terminology of group theory, that t,hose- fnnr plane waves are the reduciblc

four-dimensional representation of the radiation field In order to determine

what kind of symmetries the cigenmodes can have, we need t,o perform the re-

duction procedure For this purpose, it is enough t o see how many r(') points

will be invariant when the symmetry operations of the C4?: point group are a p

3.1 Group Theory for Two-Dimensional Crystals 53

Table 3.6 Characters of the lowest three representations at the r point of the square lattice

"or this case the word "invariant" should he considered exactly N R does not include the number of the wave vectors that are transtornled to their equivalent points, but is the number of those wave vrrtors that R keeps at their original

~ositions

2

rcli N R

x(')(R) x(R) rIZ) N R

x l l i ( R )

x ( R )

r(3J N R

x ( l l ( ~ ) x(R)

the transformed atomic orbitals, or, for the present problem, t h e overlapping

of the initial and the transformed polarization vectors This quantity plays an important role when we make the symmetry assignment for the 3D crystals However, for the 2D crystals, it is always equal to unity as long a s we treat

t h e electric field E, for the E polarization and the magnetic field Hz for the

H polarization, since no 2D symmetry operation changes E, or H z

The reduction procedure is similar t o that used to obtain the compatibility relations For example, in the case of the B1 representation,

the number of the B1 representation on the r(') point

- RECa,

the number of elements in Cq, '

(3.39) Hence, it is calculated as below, and we obtain one B1 representation

54 3 Syrnrnet,ry of Eigenmodes

Table 3.7 Irreducible representations given by the superposition of electrornag-

netic plane w a v ~ s in free space whose w w e vectors are regarded as those in the

reciprocal lattice space of tile square lattice The number in thr square brackets

denotes the number of the equivalent k vectors

Syrnrnetry Point Representative wa/Zrcin Irreducible

k vector free space representations

rcZi nln(2.0) [4] 1 AI + BI + E

r("1 r l a ( 4 0 ) 141 2 AI + B I + E

We also obtain one A1 mode and one E mode The irreducible representations

can be obtained for other r points in a similar mnnner T l ~ e results are

summarized in Table 3.7 The reduction procedure is the same for the h.1

points becanse they also have the Cq, symmetry On the other hand, t,he X

points have C2, symmetry Hence, their irreducible representatio~~s can be

obtained by consulting the character table of the CzI point group shown in

Table 3.3 The results for the M and X points are also listed in Table 3.7

We have so far treated the case of the dielectric function with an infinites-

imally small periodic modulation When the spatial variation of the dielectric

fi~nction becomes large, in such a manner that its symmetry is maintained,

then the angular frequency of the eigenmode will change, but the symmetry of

the eigenfunctions will reniain unchanged This is because the rigorous eige~i-

function of the photonic crystal with a particular type of symnletry should be

formed a? a linear combinatiot~ of unperturbed wave f u ~ ~ c t i o n s with t h e same

symmetry Otherwise the rigorous eigenfunction would have no symmetry

and this contrndict,~ t,he general rule that any eigenfunction of a photonic

crystal should he attributed t o one of its irreducible representations There-

fore, t h e mode assignment of the actual photonic bands can be accomplished

by means of the comparison with those in free space This task is easy espe-

cially for the low frequency region where wn/Z?ic < 1: and examples will be

given in the following sections

3.2 Classification of Eignnmo~lrs in t h r Square Lattice 55

3.2 Classification of Eigenmodes in the Square Lattice

Figure 3.4 shows the photonic band structure of the 2D crystal composed of

a square array of circular air cylinders marle in a uniform dirlect,ric material with a dielectric constant of 2.1 The filling factor, or the volnme fraction

of air cylinders, is 0.25 The dielectric constant of the air rod was taken

to be unity 2D photonic crystals of this type were fabricated and studied experimentally [45] The dispersion curves were obtained by the plane-wave

expansion method described in Sect 2.2 In order t o obtain the symmetry

nssignrncnt shown in Fig 3.4, it was sufficient t o know the symmetry of a few modes by examining their eigenfunctions by numerical calculation The rest of the assignment was conducted by consulting Tables 3.5 and 3.7 If we

compare the symmetries in the band diagram and those in Table 3.7, their

correspondence is apparent, For example, the lowest mode on the r point, with an eigenfrequency of zero comes from the l'(" point in Table 3.7 The

second t o the fifth modes on the r point come from the r(2) point Group theory tells us that two of them are non-degenerate Al and B1 modes and one of them is doubly degenerate E mode, which is actually observed in Fig 3.4 Similar correspondence is found for t,he sixth to the ninth modes on the

I point and for the rest of this figure Hence, we may co~lclllde that group theory is quite powerful for the symmetry assignment for the photonic band structure

For the prescnt problem, t h e contrast of t h e dielectric constant, 2.1:1.0,

is not very high, and thus the dispersion cnrves are not so far from those in

a uniform material with an averaged dielectric constant, which was shown in Fig 1.9 This is why the extremely good correspondence shown above was obtained However, even if the contrast of the dielectric constant is high, such a.s 10:1, we usually also obtain a good correspondence for wa/2nc < 1

There is a very interesting phenomenon related to the symmetry of the eigenmodes that was first reported by Robertson et al [40, 411 This is con-

cerned with the B modcs on the A and the C points These modes are

antisymmetric about the mirror plane spanncd by the z and z axes (the symmetry operation is my) or by the vector (1, 1) and the 2 axis (the sym- metry operation is ud) We examine the case of the E polarization first We assume that an external plane wave polarized in the z direction is incident

on t h e photonic crystal and propagated in the x direction Hence, its electric field has only the z component of the following form:

Because t,his field does not depend on the y coordinate, it is symmetric rlnder the mirror reflection 0, On the other hand, the B modes on the A point are antisymmetric Therefore, their effective coupling on the surface of the photonir crystal is equal to zero, and the incident plane wave cannot excite the B rnodes Hence, we refcr to those B modes as lrncoupled modes or

56 3 Symmetry of Eigenmodes

"I" -

(b)

Fig 3.4 Dispersion relation of the 2D square lattice composed of circular air-rods

formed in the dielectric material with a dielectric constant of 2.1 for (a) the E

polarization and (b) the H polarization The filling factor is 0.25 (After [46])

3.3 Classification of Eigcnrnodes in the Hexagonal Lattice 57

inactive modes The same phenomenon takes place when the incident plane wave is pointed t o the (1: 1) direction or the r t o M direction It cannot excite the B modes on the C point Hence, the B modes on the C point are also uncoupled modes As for the incident plane waves polarized in the x-y plane, the same arguments can be made by considering the H polarization

For example, the incident plane wave that is propagated in the x direction now has the z component of the magnetic field of the following form:

H z = H,o exp{i(kx - w t ) } , (3.41)

which is symmetric under the mirror reflection ay By a similar discussion t o that given above for the E polarization, we can conclude that all B modes

of the H polarization on the A and the C points are uncoupled modes

Robertson e t al [40, 411 found by experimental observation t h a t these B

modes d o not cont,rihi~t,c t o t h e light propagation This was also confirmed by

numerical calculation of the transmission spectra [42, 4'71 The same feature was predicted by the vector KKR method for an fcc lattice 1301 The presence

of the uncoupled modes is peculiar t o 2D and 3D photonic crystals For 1D crystals, all eigenmodes are symmetric when it is propagated in the direction

perpendicular t o the dielectric layers In Chap 4, some examples of opaque

frequency regions in the transmission spectra due t o the uncoupled modes will be presented

3.3 Classification of Eigenmodes

in the Hexagonal Lattice

Figure 3.5 is another example that shows the symmetry operations for the hexagonal array of circular dielectric cylinders The structure is invariant when it is rotated by multiples of 60 degrees Those rotations are denoted

by Ce, C3(= C i ) , C2(= Cz), C;'(= C32), and Ccl(= C i ) The structure is also invariant when the u, or ay mirror reflection is operated I t is readily seen that there are two other mirror reflections denoted by a: and u that are equivalent to m, Thpsp t , h r ~ ~ mirror planes intersect each other a t angles

of 60 degrees ah and a; are, on the other hand, equivalent t o a, Together with the idcntity operation E, these symmetry operations constitute the C6"

58 3 Svmrnetry of Eigenrnodes

M

O x

Fig 3.5 Symmetry operations for the 2D hexagonal array of dielectric cylinders

Here, we represented Cs and Ccl by 2C6, and so on

Figure 3.6 shows the first Brillouin zone of the hexagonal lattice The

elemcntary lattice vectors {alil, as/) and the elementary reciprocal lattice

vectors {blil, b2!} are

I

Fig 3.6 First Brillouin zone of the 2D hexagonal let,tice

3.3 Classificatiorl of Eigenrnodes in the Hexagonal Lat,tice 59

whnre n is the lattice constant Therc arc three highly symmetric points: The

r point, (0, 01, the K point (411/3a,0), and the bf point, ( ~ / a , - ? i / & a ) Those points between the r and the K points are denoted by T, and those between the r and t,he M points are denoted by Z Examining the symmetry

of these points, it is apparent that

Table 3.11 Irreducible representations for the electromagnetic waves in free space,

whose wave vectors are reduced in the first Rrillouin zone of the hexagonal lattice

Symmetry Point Representative wal2rrc in Irreducible

k vector free space representations

3.3 Claqsification of Eigenmodes in t,hc Hexagonal Lattice 61

Fig 3.7 Photonic band structure of the 2D hexagonal lattice for (a) the E polar-

ization and (b) the H polarization

The character tables for the Cev and the C3, point groups are shown in Tables

3.8 and 3.9 The compatibility relations can he obtained as before The result

is summarized in Table 3.10

As we did for the square lattice, we can obtain the irreducible represen- tations that are expected t o appear in t,he band diagram of the hexagonal lattice by examirling the set of plane waves in free space with the sarnc unper- turbed eigenfrequencies The reduction procedure to obtain the irreducible representations is completely similar t o that used for the sqnare lattice The results arc shown in Table 3.11 The assignnlent of the symmetry t o each

62 3 Symmetry of Eigenmodes

mode is easily performed by consulting Tables 3.10 and 3.11 if we know t,he

symmetry of a fcw m o d ~ s by solving the eigenvalue problem by r~umerical

calculation Figure 3.7 shows an example of the band diagram and the sym-

metry assig~lmetlt that was oht,ained for the hexagonal lattice conlposed of

circular air-cylinders formed in a uniform dielectric material with a dielectric

constant of 2.72 The filling factor is 0.G.5 Photonic crystals of this typc were

fabricated and studied in 1481

3.4 Group Theory for Three-Dimensional Crystals

Now we proceed t o t h e case of 3D photonic crystals [49, 501 The eigen-

functions of thc electric field E ( T ) and the magnetic field H ( T ) satisfy the

following eig~nvalutl equations

We denote a symnletry operat~on t h a t belongs t o the point group M of the

crystal by R as before R acts on any scalar function f ( r ) such that

ti

where R is now the 3 x 3 matrix representation n of the operator R Became

R is a rotation, inversion, or their product, R is an orthogonal matrix:

Since this equation holds for arbitrary vector fields, we obtain

H

Finally we obtain

R v ~ ) R - ' J ' RLER-' = R-R- { (

Here, we used (3.61) Equation (3.65) can be proved in a similar manner

U'e denote the characters of the irreducible representations of the electric

and t h e magnetic fields by X ( E ) ( ~ , k n ) and X ( H ) ( ~ , k n ) , respectively:

3.5 Classification of Eigenmod~s in the Simple Cubic Lattice 65

This difference between X ( E ) and X(H) originates from the fact that the elec- tric field is a truc vector whereas the magnetic field is an axial vector Equa- tion (3.79) implies that the irreducible representation for the magnetic field

is generally different from that for the electric field On the other hand: the character for the electric displacement D k n ( r ) is the same as that for the electric field because the following relation holds:

Therefore, D k n ( r ) and Ek,(r) are attributed t o the same irreducible rep- resentation Because D k n ( r ) and H k n ( r ) are purely transverse waves even with the spatial variation of t h e dielectric function, we can constitute their eigenfunctions with transverse plane waves alone We can ose this fact t o obtain the irreducible representations of the eigenfunctions

3.5 Classification of Eigenmodes

in the Simple Cubic Lattice

We consider the details of the reduction procedure for the transverse vector fields with a simple cubic lattice composed of dielectric spheres a? an example Figure 3.8 shows its symmetry operations First, the structure is invariant under the rotations about thc z, y, and z axes by 90, 180, and 270 degrees These rotations are denoted by Ca, Cz, and C;' as before C4 and C c l are conjugate with each other We should also note that t h e Cq rotation about the x axis is conjugate with that about the y and the z axes Hence: these three Cq and three C;' rotations have the same characters and may

be denoted by 6C4 Similarly, three Cz rotations are conjugate with each other and denoted by 3C2 There are two other sets of rotation axes that are denoted by C; and C3 Six C; operations are conjugate with each other

As for the C3 rotation, eight operations are conjugate The structure is also invariant under the spatial inversion I The rest of the symmetry operations are given by the product of I and the operations described above: 61C4, 31Cz, 6IC;, and 8IC3 Among these, ICz and IC; are usually denoted by o h and

od, respectively They are mirror reflections about the horizontal plane and the diagonal plane Together with the identity operation E these symmetry

operations constitute the Oh point group:

O h = { E ; 6C4,3C2,3C;,8C~, I:6IC4,3uh:6nd,8IC3} (3.82) The character table of the Oh point group is shown in Table 3.12

The first Brillouin zone of t,he simple cubic lattice is depicted in Fig 2.1

We examine the symmetry of its highly symmetric points I t is apparent that the r paint has the Oh symmetry:

66 3 Symmetry of Eigenmodes

Fig 3.8 Symmetry operations for the Oh point group

Table 3.12 Character table for the Oh point group

The symmetries of other points are

Thr sy~nlnetry uperations for these points are illustrated in Fig 3.9 The

character tahlr for the Dnh point Eroup is giver, in Table 3.13 The cllaracter

3.5 Classification of Eigenmodes in the Simple Cubic Lattice 67

Fig 3.9 Symmetry operations for the highly symmetric pornts in the first Brillouin

zonp of the simple cubic lattice

tables for the C4,, Caur and Czu point groups are given in Tables 3.2, 3.9,

and 3.3, respectively Our rrexl l ~ s k is t o obtain the compatibility relations

The procedure is exactly the same as that for the 2D crystals described in the previous scctions The final results are summarized ill Tat)les 3.14 and

3.15

In Sects 3.1-3.3, we obtained the possible combinations of irreducible representations that apppar in the band diagrams of the 2D crystals by per- forming the reduction procedure for the reducible representations composed

of plane waves We wol~ld like t o do the same thing for the electromagnetic eigenmodes in the 3D crystals However, there is an apparent difference be- tween the present and the p r e v i o ~ ~ s problems That is, we now have t,o deal with vect,or fields i r ~ s t ~ a d of scalar fields Hencc we should first examine how