Electromagnetic Theory and Applications for Photonic Crystals

Furthermore, we will confine the theory to the particular case of 2D photonic crystals with identical circular cylinders illuminated with s-polarized light electric field parallel to the

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Electromagnetic Theory

and Applications for Photonic Crystals

edited by

Kiyotoshi Yasumoto

Kyushu UniversityFukuoka, Japan

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dielec-The behavior of light propagating in a photonic crystal can be intuitively stood by comparing it to that of electrons in solid-state materials The electrons pass-ing through a lattice of the atoms interact with a periodic potential This results in theformation of allowed and forbidden energy states of electrons The light propagating

under-in a photonic crystal under-interacts with the periodic modulation of refractive under-index Thisresults in the formation of allowed bands and forbidden bands in optical wave-lengths The photonic crystal prohibits any propagation of light with wavelengths

in the forbidden bands, i.e., the photonic bandgaps, while allowing other lengths to propagate freely The band structures depend on the specific geometryand composition of the photonic crystal such as the lattice size, the diameter ofthe lattice elements, and the contrast in refractive index It is possible to createallowed bands within the photonic bandgaps by introducing point defects or linedefects in the lattice of photonic crystals Light will be strongly confined within thedefects for wavelengths in the bandgap of the surrounding photonic crystals Thepoint defects and line defects can be used to make optical resonators and photoniccrystal waveguides, respectively

wave-The photonic crystals with bandgaps are expected to be new materials for futureoptical circuits and devices, which can control the behavior of light in a micron-sized scale Although there is continuing interest in new findings of photonicbandgap structures associated with particular lattice configurations, recent attentionhas been focused on the engineering applications of the photonic crystals To beable to create photonic-crystal-based optical circuits and devices, their electromag-netic modeling has become a much more important area of research From theviewpoint of electromagnetic field theory, the photonic crystals are optical materi-als with periodic perturbation of macroscopic material constants Fortunately wehave a great deal of knowledge about the electromagnetic theory for periodic struc-tures During the past few decades, various analytical or computational tech-niques have been developed to formulate the electromagnetic scattering, guiding,and coupling problems in periodic structures The aim of this book is to providethe electromagnetic theoretical methods that can be effectively applied to themodeling of photonic crystals and related optical devices

This book consists of eight chapters that are ordered in a reasonably logicalmanner from analytical methods to computational methods Each chapter startswith a brief introduction and a description of the method, followed by detailed

method based on multipole expansions, its extension combined with the methodformulations for practical applications Chapter 1 describes the scattering matrix

and numerical examples are presented with a particular emphasis on phenomena ofanomalous refraction and control of light emission in photonic crystals.

is discussed to investigate the propagation of light in photonic crystal fibers andthe radiation dynamics of photonic crystals The multipole method combinedwith the notions of lattice sums and Bloch modes is presented to model the scatter-ing and guidance in various photonic crystal devices The scattering and guidance by

parallel or crossed arrays of circular cylinders standing in free-space or embedded

in a dielectric slab The method uses the aggregate transition matrix for a cluster

of cylinders within a unit cell, the lattice sums, and the generalized reflection andtransmission matrices in a layered system

the simulation of photonic crystal devices The method comprises a modeling ofperiodic structures using the concepts of fictitious boundaries and periodic boundaryconditions, novel eigenvalue solvers, a so-called connections scheme that is a uniquemacro feature of the method, and eigenvalue and parameter estimation techniques

ined A novel technique for mode matching combined with the generalized ing matrix method is presented to deal with the scattering and guidance bymetallic photonic crystals with lattice elements of arbitrary cross sections

scatter-algorithms for solving electromagnetic guiding problems The mathematical formulation and analysis procedure based on the generalized transmission line equations are discussed The results of applications are demonstrated for photoniccrystal devices consisting of various bends, junctions, and their concatenations

treated The absorbing boundary conditions, the periodic boundary conditions, and

an interface condition for dielectric interfaces with curvature are implemented in thefinite-difference scheme The method is applied to the analysis of photonic crystal

describes the finite-difference time-domain method based on the principles of multidimensional wave digital filters The method employs the finite differenceschemes using the trapezoidal rule for discretizing Maxwell’s equations that hasadvantages with regard to numerical stability and robustness Numerical examplesare presented for various photonic crystal waveguide devices

It is hoped that the material is sufficiently detailed both for readers involvedwith the physics of photonic bandgap structures and for those working on theapplications of photonic crystals to optical circuits and devices

Finally, I would like to thank the authors for their excellent contributions It

is also a pleasure for me to acknowledge Jill J Jurgensen and Taisuke Soda ofCRC Press, Taylor & Francis Group, for their help throughout the preparation ofthis book

Kiyotoshi Yasumoto

In Chapter 2, the multipole theory of scattering by a finite cluster of cylinders

photonic crystals are formulated in Chapter 3, using a model of multilayered periodic

Chapter 4 is devoted to the method of multiple multipole program applied to

In Chapter 5, the mode-matching method for periodic metallic structures is

reexam-Chapter 6 describes the method of lines, which is one of the efficient numerical

In Chapter 7, the full-vectorial finite-difference frequency-domain method is

fibers, photonic crystal planar waveguides, and bandgap structures Chapter 8

The Editor

Kiyotoshi Yasumoto earned the B.E., M.E., and D.E degrees in communication

engineering from Kyushu University, Fukuoka, Japan, in 1967, 1969, and 1977,respectively In 1969, he joined the faculty of engineering of Kyushu University,where since 1988 he has been a professor of the Department of Computer Scienceand Communication Engineering He was a visiting professor at the Department ofElectrical and Computer Engineering, University of Wisconsin in Madison in 1989and a visiting fellow at the Institute of Solid State Physics, Bulgarian Academy ofScience and Institute of Radiophysics and Electronics, Czechoslovakian Academy

He has served as a member of organizing, steering, technical program, andinternational advisory committees and as a session organizer for various interna-tional conferences He has published more than 250 papers in various interna-tional journals and conference proceedings

Ara A Asatryan

Centre for Ultrahigh-Bandwidth

Devices for Optical Systems and

Centre for Ultrahigh-Bandwidth

Devices for Optical Systems and

National Taiwan University

Taipei, Taiwan, Republic of China

Communication Photonics Group

Laboratory for Electromagnetic

Fields and Microwave Electronics

ETH Zentrum

Zurich, Switzerland

David P Fussell

Centre for Ultrahigh-Bandwidth

Devices for Optical Systems and

ElectronicsETH ZentrumZurich, Switzerland

Stefan F Helfert

Allgemeine und TheoretischeElektrotechnik

University of HagenHagen, Germany

Hiroyoshi Ikuno

Department of Electrical andComputer EngineeringKumamoto UniversityKurokami, Japan

Hongting Jia

Department of Computer Scienceand Communication

EngineeringKyushu UniversityFukuoka, Japan

Boris T Kuhlmey

Centre for Ultrahigh-BandwidthDevices for Optical Systems andSchool of Physics

University of SydneySydney, Australia

Timothy N Langtry

Centre for Ultrahigh-BandwidthDevices for Optical Systems and Department of MathematicalSciences

University of TechnologySydney, Australia

Contributors

Faculté des Sciences de Saint

Centre for Ultrahigh-Bandwidth

Devices for Optical Systems and

Centre for Ultrahigh-Bandwidth

Devices for Optical Systems and

University of TechnologySydney, Australia

C Martijn de Sterke

Centre for Ultrahigh-BandwidthDevices for Optical Systems andSchool of Physics

University of SydneySydney, Australia

University of SydneySydney, Australia

Kiyotoshi Yasumoto

Department of Computer Scienceand Communication EngineeringKyushu University

Chapter 1 Scattering Matrix Method Applied to

Photonic Crystals 1

Daniel Maystre, Stefan Enoch, and Gérard Tayeb

Chapter 2 From Multipole Methods to Photonic Crystal

Device Modeling 47

Lindsay C Botten, Ross C McPhedran, C Martijn de Sterke, Nicolae A Nicorovici, Ara A Asatryan, Geoffrey H Smith, Timothy N Langtry, Thomas P White, David P Fussell, and Boris T Kuhlmey

Chapter 3 Modeling of Photonic Crystals by Multilayered

Periodic Arrays of Circular Cylinders 123

Kiyotoshi Yasumoto and Hongting Jia

Chapter 4 Simulation and Optimization of Photonic

Crystals Using the Multiple Multipole Program 191

Christian Hafner, Jasmin Smajic, and Daniel Erni

Chapter 5 Mode-Matching Technique Applied to

Metallic Photonic Crystals 225

Hongting Jia and Kiyotoshi Yasumoto

Chapter 6 The Method of Lines for the Analysis of

Photonic Bandgap Structures 295

Reinhold Pregla and Stefan F Helfert

Chapter 7 Applications of the Finite-Difference

Frequency-Domain Mode Solution Method to

Photonic Crystal Structures 351

Chin-Ping Yu and Hung-Chun Chang

Chapter 8 Finite-Difference Time-Domain Method Applied

to Photonic Crystals 401

Hiroyoshi Ikuno and Yoshihiro Naka

Table of Contents

1 Scattering Matrix Method

Applied to Photonic

Crystals

Daniel Maystre, Stefan Enoch, and

Gérard Tayeb

CONTENTS

1.1 Introduction 2

1.2 Scattering Matrix Method 3

1.2.1 Presentation of the Problem and Notation 3

1.2.2 Fourier–Bessel Expansions of the Field inside the Cylinders 5

1.2.3 Fourier–Bessel Expansions of the Field outside the Cylinders 7

1.2.4 First Set of Equations: Causality Property for Each Cylinder 11

1.2.5 Second Set of Equations: Introducing the Coupling between Cylinders 12

1.2.6 Final Equation 15

1.3 Combination of Scattering Matrix and Fictitious Sources Methods 16

1.3.1 Introduction 16

1.3.2 Setting of the Problem 17

1.3.3 The Method of Fictitious Sources (MFS) 18

1.3.4 Implementation of the Scattering Matrix Method (SMM) 22

1.3.5 Hybrid Method Using MFS and SMM 23

1.3.6 Numerical Example 24

1.4 Dispersion Relations of Bloch Modes 25

1.4.1 Infinite Structure 26

1.4.2 Finite-Size Photonic Crystals 30

1.5 Theoretical and Numerical Studies of Photonic Crystal Properties 35

1.5.1 Ultrarefraction with Dielectric Photonic Crystals 35

1.5.2 Ultrarefraction with Metallic Photonic Crystals 36

1.5.3 Negative Refraction by a Dielectric Slab Riddled with Galleries 39

1.6 Conclusion 42

References 43

1

1.1 INTRODUCTION

The scattering matrix method (SMM) is one of the most efficient methods for ing a problem of scattering by a large but finite number of objects It basically takesinto account separately the specific scattering properties of each object and thenevaluates the coupling phenomena between them Even though it can be presented

solv-in a quite rigorous form, it is based on a physically solv-intuitive approach to the lem of scattering from a set of objects One of its advantages is to be accessible topostgraduate students Moreover, the numerical implementation does not presentmajor difficulties In contrast to other classical methods like Finite Difference TimeDomain method (FDTD) or the finite element method, it becomes much simpler

prob-in the case of two-dimensional (2D) photonic crystals with circular cross sections

or 3D photonic crystals formed by spherical inclusions It deals with crystals offinite size regardless of whether or not they have defects of periodicity

The first achievement relating to that method should be attributed to LordRayleigh, who dealt with the electrostatic case [1] The electromagnetic version ofthis method has been developed since the 1980s in various forms by differentgroups working independently [2–9] Their studies deal with 2D or 3D, dielectric,metallic, or perfectly conducting objects placed in space in a periodic or randomway, but the essence of each of the approaches remains the same

However, the SMM is not able to deal with some interesting configurations,especially when the set of scatterers is surrounded by a jacket To extend themethod to more complicated structures, it is possible to combine the SMM withthe method of fictitious sources (MFS) The MFS is another rigorous method that

is able to solve the problem of scattering from arbitrary scatterers In MFS, thefield in each medium is represented as the field radiated by a set of fictitioussources with initially unknown intensities These intensities are obtained byimposing the boundary conditions for the fields on the surfaces of the scatterers

By combining these two methods, we concurrently procure their advantages Themethod is described in Section 1.3 in a 2D case and can, for instance, addressstructures such as a finite dielectric body pierced to form galleries, such as a pho-tonic crystal made with macroporous silicon More generally, the method could

be useful for the study of problems dealing with small clusters of buried objects.These two methods enable one to deal with a large range of two-dimensionalphotonic crystals However, the phenomena generated by photonic crystals are sosurprising and so complex that a theoretician needs a preliminary phenomeno-logical approach For this purpose, the notion of Bloch modes provides a valuabletool We will describe such modes, and we will show that their dispersion curvesenable one to predict most of the properties of photonic crystals Indeed, thesedispersion curves enlighten us on quantities such as the average energy velocityand provide an intuitive way of grasping the vital notion of effective optical index

in the case of a heterogeneous material

A special emphasis will be put on anomalous refraction phenomena and thecontrol of light emission Two kinds of anomalous refraction phenomena can bedistinguished In the phenomenon of ultrarefraction, a photonic crystal simulates

a material having an optical index between zero and unity, while in negativerefraction, it acts like a material having a negative index These phenomena will

be predicted from three-dimensional dispersion diagrams and illustrated using thetwo methods It will be shown that these phenomena can lead to new optical com-ponents: directive sources, microlenses, and so on

1.2 SCATTERING MATRIX METHOD

The method discussed here can deal with collections of objects having differentelectromagnetic parameters or different shapes However, for simplicity, we willlimit ourselves to a set of identical objects, which is in general the case for photonic crystals Furthermore, we will confine the theory to the particular case of 2D

photonic crystals with identical circular cylinders illuminated with s-polarized light (electric field parallel to the cylinder axes) The generalization to p-polarized

light (magnetic field parallel to the cylinder axes) or to the case in which the ders are different is straightforward, while the extension to 3D photonic crystalsleads to a considerable increase in the complexity of the algebraic developments

The scattering problem is represented in Figure 1.1 An incident monochromaticplane wave with wavelength 0 2/k0in vacuum propagates in a homogeneous

x

y

Z O

Incident

field

inc

FIGURE 1.1 Presentation of the problem of scattering: an s-polarized incident plane wave

illuminates N cylinders (here N 6).

material of relative permittivity  r,ext The generalization to an incident beam or to a

field generated by a 2D antenna is not problematic It illuminates under s-polarization

and incidence angle  inc

a set of N identical dielectric cylinders of radius R

and relative permittivity  r,int We work under the assumption that all the materialsare nonmagnetic, and the optical indices of the materials outside and inside

the cylinders are denoted respectively by next  r,extand nint  r,int Using

a time dependence in exp(it) and denoting by xˆ, yˆ and zˆ the unit vectors of the

three axes in the Cartesian coordinate system xyz, the incident electric field is

boundary conditions at the interfaces between different materials (continuity of E z

and of its normal derivative)

In outline, the method can be divided into three steps The first step consists ofshowing that the total field can be expressed in the form of Fourier–Bessel series.Outside a cylinder, the series can be separated into two parts: the first part describesthe total incident field on the cylinder The total incident field includes not only theincident plane wave given by Equation (1.2), but also the field scattered by the othercylinders in the direction of the cylinder that is considered The second part repre-sents the field scattered by the cylinder The second step is achieved by requiringthat a relation of causality exists between the field scattered by a cylinder and thetotal incident field that illuminates the same cylinder This relationship can beexpressed in terms of the Fourier–Bessel coefficients through the notion of the scat-tering matrix The third step addresses the fact that the total incident field on acylinder is the sum of a known component (the incident plane wave) and of thefields scattered by the other cylinders A second relation between these two parts ofthe field surrounding each cylinder can thereby be obtained In contrast to the sec-ond step, the last step expresses the coupling phenomena between all the cylinders

e

ee

r

r r

E

 ˆ

1.2.2 F OURIER –B ESSEL E XPANSIONS OF THE F IELD INSIDE THE C YLINDERS

The expansion of the field in Fourier–Bessel series inside and outside the cylindersconstitutes the basis of the formalism, allowing the initial problem, namely thedetermination of the field at any point of space, to be reduced to the evaluation of

a set of complex coefficients First, let us establish that the field inside each cylinder

is described by a Fourier–Bessel series This can be demonstrated using a local

system of polar coordinates (r j, j ) with an origin located at the center O jof eachcylinder (Figure 1.2) It can be shown that this property extends to noncircularcylinders, but in that case the domain of validity of the Fourier–Bessel series doesnot include the entire cross section of the cylinder

Obviously, the electric field is, for a given value of r j, a periodic function of

 jwith period 2, and thus it can be represented by a Fourier series:

FIGURE 1.2 Domains of validity of the Fourier–Bessel series inside and outside the jth

cylinder The deep gray region represents the domains of validity of the Fourier–Bessel series inside the cylinder (with Bessel functions of the first kind) The light gray domain represents the domain of validity of the Fourier–Bessel series around the cylinder (with Hankel func-

tions and Bessel functions of the first kind) The arrows l → j and j → l show the fields

scat-tered by the lth and jth cylinders, respectively, which propagate toward the interior (and exterior) of the light gray ring.

Using the expression of the scalar Laplacian in polar coordinates:

(1.7)

Then substituting the expression of the field given by Equation (1.5) into Equation(1.6), a straightforward calculation shows that the Fourier coefficients satisfy thefollowing equation:

m r

z m

j j

z m j

r j

2

2 ,int 2

the cylinder, all the coefficients d j,mvanish This is a straightforward consequence

of the fact that the Hankel function H m(1)(s) has a singularity for s 0 and that the

field should not be singular inside the cylinder Thus,

(1.15)

Finally, the determination of the field inside the cylinders reduces to the

calcula-tion of the coefficients c j,m of the Fourier–Bessel expansion on the right-hand side

of Equation (1.15)

The calculations we have done for the expression of the field inside a cylinder can

be reproduced almost identically for the regions surrounding the cylinders Thefield outside the cylinders satisfies the following Helmholtz equation:

(1.16)

We consider the ring surrounding the jthcylinder and extending to the nearest point

nate system as in the preceding section, following the same mathematical lines, and

remarking that for a given value of r j, the value of the optical index remains equal

to next, it can be shown that inside this region the field takes the form of anotherFourier–Bessel series having the same form as that given in Equation (1.14):

(1.17)

By contrast with the expansions inside the cylinders, it is not possible to show that

the coefficients b j,mvanish, because no physical argument prevents the expression

of the field in the light gray ring from being singular at the center of the cylinder,due to the fact that this center does not belong to the ring

Now, it turns out that the properties of Bessel functions allow us to give aphysical meaning to each of the terms on the right-hand side of Equation (1.17).Indeed, from an intuitive physical viewpoint, the field in the ring can be separatedinto three different parts:

• The incident plane wave (source) given by Equation (1.2)

• The fields scattered by all the other cylinders toward the jthcylinder,which behave for this cylinder as incident fields These fields add to theincident plane wave (source) to constitute the total incident field on the

( ,u ) ( 0 ext ) exp( u )

,

Z

∑

0 2

coordi-In order to identify in Equation (1.17) each physical component of the field,

let us note first that, obviously, the total incident field on the jthcylinder has beengenerated by sources located outside this cylinder (at infinity for the incident planewave and inside the other cylinders for the complementary part) Thus, it must sat-isfy inside this cylinder a Helmholtz equation and cannot have any singularity.This property allows us to deduce that the total incident field is contained insidethe first sum of Equation (1.17) Consequently the second sum represents a scat-

tered field of the jthcylinder The question that now arises is whether this second

sum represents the totality or only a part of the field scattered by the jthcylinder

In the first case, the first sum represents the total incident field on this cylinder(and only the total incident field on this cylinder) In the second case, the first sumrepresents not only the total incident field on the cylinder but also a part of the scat-tered field Causality properties provide the answer to this question The field scat-

tered by the jth cylinder must propagate away from it It emerges that the onlyFourier–Bessel functions that satisfy this condition are those containing Hankelfunctions of the first kind Indeed, the asymptotic expression at infinity of Hankelfunctions of the first kind is given by [10]:

(1.18)

and they satisfy, at any order q, the radiation condition at infinity So the second sum of Equation (1.17) does indeed represent a field scattered by the jthcylinder.Conversely, an arbitrary term of the first sum cannot represent a scattered field

because, from Equation (1.12), J m (s) is given by:

(1.19)

with H–m(1)complex conjugate of H–m(1)(s) Obviously, Equation (1.18) shows that

H–(1)m (s) does not satisfy the radiation condition at infinity As a consequence, the first sum in Equation (1.17) actually represents the total incident field illuminating the jth

cylinder, and the second sum represents the field scattered by the same cylinder.Now let us calculate the contribution in the total incident field of the incidentplane wave (source), as given by Equation (1.2) With this aim, let us express the

(1.20)

where the incident wavevector k has components (k0nextsin inc, k0nextcos inc)

Expressing OP as the sum of OOjet OjP yields:

 exp(ik0nextr jsin( inc uj )) exp(ik0nextr j (P) sin(  inc uj (P))) (1.21)

z (P)  exp(ik0nextr jsin( inc uj))

m ∈Zexp(im  inc ) J m (k0nextr j (P)) exp( im j (P)) (1.23)

Bearing in mind that J m (s) (1)m J m (s), m can be replaced by m in Equation

(1.23), enabling us to obtain an expression that can be compared directly with thefirst sum of Equation (1.17):

E z i (P)  exp(ik0nextr jsin( inc uj

))

m ∈Z()m

exp(im  inc

) J m (k0nextr j (P)) exp(im  j (P)) (1.24)

Equations (1.23) and (1.24) provide the development of the source incident

field in Fourier–Bessel series in the coordinate system linked to the jthcylinder.Coming back to Equation (1.17), it becomes possible to distinguish the coeffi-cients of the Fourier–Bessel series associated with the source field from thoseassociated with the field scattered by the other cylinders They will be denoted by

a j,msourceand a j,mrods, respectively, their sum being equal to a j,m Equation (1.24) allows

us to identify a j,msource:

sin( inc uj)im inc) (1.25)

Finally, the three components of the field in the ring surrounding the jthcylindercan be expressed respectively by the following Fourier–Bessel series:

• The source term (incident plane wave):

0 ext) exp

fol-• The field scattered by the jthcylinder:

(1.28)

It has been established that the total field in the ring can be expressed from

three series of coefficients: a j,msource(known), a j,mrods(unknown) and b j,m(unknown).Now let us demonstrate a vital property of the field scattered by a cylinder

The expression of the field scattered by the jthcylinder in the ring around thiscylinder given by Equation (1.28) extends in fact to the entire space surroundingthis cylinder This fundamental property is very intuitive: the field scattered by

the jthcylinder is produced by sources located inside this cylinder and thus can bedefined in the entire space around the cylinder In order to give a mathematicaldemonstration of this property, let us rewrite Equation (1.3) in the form:

z 0 and subtracting this equation from Equation (1.29), it can

be derived that the field E d

zscattered by the entire set of cylinders, defined at anypoint of space as the difference between the total field and the source incidentfield (the incident plane wave), satisfies the equation:

(1.30)

Hence, the scattered field at any point P of space outside the cylinders can be

expressed using Green’s theorem:

(1.31)

with x and y coordinates of M in the general coordinate system Notice that the

inte-gral on the right-hand side of Equation (1.31) can be restricted to the set of cylinderssince r,ext r (M ) vanishes outside these cylinders Consequently, the scattered

field can be represented as a sum of integrals on the cylinders:

0 2

0 2

0 2

r z d

e ,ext ( e ,ext e ( , ))

0 2

0 2

0 2

E z k er,extE z (k er,ext k er( , ))x y E z

j j j m m

the jthcylinder the Helmholtz equation:

(1.34)

As a consequence, the expression of the field scattered by the jthcylinder given

by Equation (1.28) extends to the entire space surrounding this cylinder, and fromEquation (1.32), the total scattered field is obtained by summing the fields scat-tered by the entire set of cylinders:

(1.36)

In this equation, we have introduced the infinite column matrices ajand bjwith

components a j,m and b j,m, and Sjis a square matrix of infinite dimension

When the cylinders have circular cross section s, the scattering matrix can be

calculated in closed form by writing the boundary conditions on the periphery of acylinder Equation (1.3) being valid in the sense of distributions, the electric fieldand its normal derivative are continuous across this boundary (this property can also

be proved from the continuity of the tangential components of the electric and netic fields) Using the expansions of the electric field inside and outside the cylin-der given by Equations (1.15) and (1.17), then identifying the Fourier coefficients

mag-on both sides for r j  R, it can be deduced that:

, e,ext ,

Eliminating coefficients c j,mbetween Equations (1.37) and (1.38), it turns out that:

(1.39)

Note that the variable r j in this equation can be replaced by a variable r independent

of the cylinder Indeed, since the cylinders are identical, this relation of causality is

independent of the particular cylinder considered Thus, all the matrices Sjdefined

by Equation (1.36) are diagonal and equal to the matrix S defined by:

(1.40)and

(1.41)

Obtaining the scattering matrix for noncircular cylinders is much more cult because this calculation is no longer analytic In this case, numerical tech-niques can be employed instead, for example the FDTD method, a finite elementmethod, or the method of fictitious sources The primary advantage of the scat-tering matrix method is that the use of a classical way of considering scattering isrestricted to a single cylinder rather than to the entire set of cylinders When thecylinders do not have the same radii, or the same permittivities, the scatteringmatrices cease to be identical, but they nevertheless remain diagonal if the cylin-

diffi-ders are circular The extension of the method to the p-polarization case is

straightforward: the only change is in the calculation of the Sjmatrices

First let us give the intuitive physical background of the introduction of the couplingphenomena between cylinders The total incident field contains two components:

the source term (known) and the field scattered by the other cylinders toward the jth

cylinder (unknown) The latter, represented by coefficients a j,mrods, is nothing but the

field scattered by all the other cylinders toward the jthcylinder Now, the field

0

1 int

k n r dr

scattered by an arbitrary cylinder is given by the second series of Equation (1.17);

therefore, we can deduce easily that it is possible to express the coefficients a j,mrodsin

terms of the set of coefficients b l,m with l  j However, we must overcome a

math-ematical problem: the incident field on the jthcylinder is expressed in a local dinate system linked to this cylinder, while the fields scattered by the other cylindersare expressed in the local coordinate system associated with those other cylinders

coor-To solve this problem, the fields will be expressed in a unique coordinate

system, that associated with the jthcylinder With this aim, an adequate mathematicaltool is provided by the Graf formula [10] This formula gives in a mathematical

form a very intuitive result: the field scattered by the lthcylinder toward the jthder satisfies around this last cylinder a Helmholtz equation and does not possesssingularities As a consequence, it can be expressed in the form of a Fourier–Besselseries involving exclusively Bessel functions of the first kind This formula can be

cylin-written using the notation of Figure 1.3 as: if r j (P) r l

Equations (1.17) and (1.42) enable us to express the field scattered by the lthcylinder

toward the jthcylinder in the coordinate system of origin O j Thus, if r j (P) r l

(1.45)

By adding the coefficient a j,msourceof the source field (Equation (1.25)) to the

coeffi-cients a j,mrods, it is possible to express a j,m in terms of the coefficients b l,qof the fieldsscattered by the other cylinders:

j l m

j q

sin( inc uj

)im inc

)

1.2.6 F INAL E QUATION

Equations (1.36) and (1.47) constitute a set of 2N matrix equations with 2N

unknown column matrices ajand bj A single matrix equation can be obtained by

multiplying both sides of Equation (1.47) by Sj S, Equation (1.36) being used

for simplifying the left-hand side We thus obtain

(1.50)

or, in a more explicit form:

In this way, we get an infinite linear system of equations, I being the infinite unit matrix In order to limit the size of the system, matrices S, Tj,l, Qjand blare

truncated by restricting the values of m and q between M and M, the final size

of the system being equal to N(2M + 1) The field at infinity can be deduced easily

from the column matrices bl With this aim, the expression of the field scattered side each cylinder (right-hand side of Equation (1.28)) is transformed to the unique

out-coordinate system xy The change of out-coordinate system from the local system linked with a cylinder to xy can be done by applying Graf’s formula [10]: if r  r j,

(1.52)

with r and  being the polar coordinates of a point P in space in the xy coordinate

system Thus it turns out that at a large distance from the cylinders:

0 ext

q q q

1

2

b

SQ SQ SQ

The field at infinity is obtained by using the asymptotic form of the Hankel tion at infinity, given by Equation (1.18) Equation (1.53) then becomes:

func-(1.55)

with:

(1.56)

The bistatic differential cross section, which represents the intensity of the field

at infinity, is given by:

(1.57)

When all the materials (cylinders and exterior) are lossless, the energy balancecriterion (also called the optical theorem) can be expressed in the form:

g(  inc p) 0 (1.58)This theorem can be valuable for testing the validity of a numerical code, althoughlike casting out nines, it does not provide a rigorous verification of validity

1.3 COMBINATION OF SCATTERING MATRIX AND

FICTITIOUS SOURCES METHODS

In this section, we present a method that combines the scattering matrix method(SMM) described in Section 1.2 with the method of fictitious sources (MFS) inorder to deal with more complicated structures Although the basic ideas can begeneralized to 3D cases, we will consider only 2D cases for simplicity Note thatthe SMM presented in Section 1.2 cannot deal with structures in which the set ofcylinders is embedded inside a jacket consisting of a dielectric medium that is

made to disappear by using the results of the present section

The MFS method [11] can solve the problem of scattering from arbitrarilyshaped scatterers The space is divided into different regions in which the field isrepresented as the field radiated by a set of fictitious sources with unknown inten-sities These intensities are obtained by matching the fields at the boundaries ofthe regions using a least squares technique

Returning to Figure 1.4, the basic idea of the method proposed in this section

is to use the SMM in order to build a set of functions that correctly represent the

p

pext

field inside the jacket These functions are then used to solve a fictitious sourcesproblem on the boundary 0of the jacket Combining the SMM and MFS methodsoffers the advantages of both rigorous methods and enables us efficiently to addressnew classes of problems, such as a finite dielectric body riddled by galleries.

The cylindrical scatterer is delimited by its external boundary 0 The exterior of

0is the domain e , filled with a medium of optical index n e (n emay be complex,

and we define k e  k0n e) The interior of 0is the domain i It is filled with a

medium of optical index n i (shaded region in Figure 1.4; n imay be complex, and

we put k i  k0n i) The domain i also contains cylinders with boundaries j ( j 1, 2, 3, ), filled with arbitrary media

The structure is illuminated by an incident field coming from the exterior(this assumption is made for clarity, but the method also can deal with an excita-tion from inside 0with very little change) This incident field is also assumed to

be z-independent For instance, it can be a plane wave, or the field emitted by one

(or several) line source(s) placed outside 0 It is well known that in this case the

problem can be reduced to two independent problems: the s-polarization case where the electric field is parallel to the z axis, and the p-polarization case where the magnetic field is parallel to the z axis Each of these cases leads to a scalar

problem in which the unknown u is the z component of either E or H: u  E z(for

s-polarization) or u  H z (for p-polarization) We denote by u ithe incident field,

and by u dthe scattered field, in such a way that the total field is:

((

1.3.3 T HE M ETHOD OF F ICTITIOUS S OURCES (MFS)

The MFS is a versatile and reliable method to deal with many scattering problems

It relies upon a simple idea: the electromagnetic field in the various domains of thediffracting structure can be expressed as a combination of fields radiated by suitableelectromagnetic sources These sources have no physical existence, and this is whythey are denoted as fictitious sources They are located in homogeneous regions andnot on the interfaces In other words, one can consider that they generate electro-magnetic fields that taken together faithfully reproduce the actual field and as suchform a convenient basis for this field From a numerical point of view, proper basesare those capable of representing the solution with the fewest possible functions.Obviously, the quality of the bases is closely linked with the nature of the sourcesand their location The freedom in the choice of the sources provides a great adapt-ability to various complex problems

The MFS has been developed in our laboratory in the last decade from both theoretical and numerical points of view [11–15] Almost at the same time and inde-pendently, other groups have worked on the same basic ideas [16–21], but theirapproaches are slightly different from ours In fact, one of the first attempts at usingthis method is probably due to Kupradze [22] The method has been developed andapplied to a large collection of problems, and a good review can be found in [23]

It is not our intention to depict here all the details of the MFS, and it will be cient for our present purposes to give an outline of the general principles

suffi-the interior region i is thus filled with a homogeneous material of optical index n i (Figure 1.5) Let us introduce a set of N s fictitious sources S e,n (n  1, 2, …, N s)

FIGURE 1.5 The sources S e,n (represented by dots) radiate the fields F e,n(r) used to

represent the scattered field u din e , whereas the sources S i,n(represented by stars) radiate

the fields F i,n (r) used to represent the total field uint in i.

Let us consider the same situation as in Figure 1.4 but without the inclusions:

located at N spoints re,n in i, which are supposed to radiate in a homogeneous

space filled with a medium of index n e Let us denote by F e,n(r) the field radiated

by the source S e,n By construction, the fields F e,n(r) satisfy Maxwell’s equations in

e , and a radiation condition at infinity If the coefficients c e,nare conveniently chosen,

a linear combination ∑n c e,n F e,n (r) can thus be regarded as an approximation u d(r)

of the diffracted field u d(r) in e , where c e,ncan be understood as the complex

amplitude of the source S e,n So we obtain an approximation for the field in e:

filled with a medium of index n i Let us denote by F i,n(r) the field radiated by the

source S i,n The functions F i,n(r) satisfy Maxwell’s equations in i and can be

used to get an approximate expansion uint(r) of the total field uint(r) in i:

(1.62)

where c i,n can be understood as the complex amplitude of the source S i,n.Note that the nature of the sources can be chosen arbitrarily In our case, we

choose infinitely thin line sources parallel to the z axis: S e,n(r) 4i (r  r e,n) and

S i,n(r) 4i (r  r i,n) In that case, they radiate the fields:

, in( )r  , , ( )r 

1

∑

e n n

where p is a polarization-dependent constant equal to

(1.66)

The coefficients c e,n and c i,nthat give the best approximation for the fieldsgiven by Equations (1.60) and (1.62) are those that match the boundary conditions

most satisfactorily They are obtained by minimizing the two expressions V1and

V2derived from Equation (1.65) and defined on 0:

given computation time, we obtain a better approximation for u d and uint

Of course, the efficiency of the method depends on the locations and the

num-ber N sof the fictitious sources It can be shown that the precision of the method

is related to the least-squares remainder obtained in the last step This remaindercan thus be used to quantify the quality of the solution, and this quantity is quitehelpful in the numerical implementation The interested reader will find moredetails in [11]

We have also developed some tricks in order to place the sources cally The general idea used in these tricks is to increase the density of sources inregions where the radius of curvature of 0 is lower An example is given in

automati-0mimics a rounded F letter (firstletter of Fresnel Institute), and it is given by the parametric equation

(1.68)

The values of the coefficients c nare given in the caption to Figure 1.6 We use

N s 200 sources in each region eand i , and 2N ssample points on 0 This

cylinder of index n i  1.5 lies in a vacuum (n e 1) and is illuminated with an

inci-dence angle inc 45° by a plane wave of wavelength 0 2 and unit amplitude

Note that, contrary to Section 1.2, the angles are measured with respect to the x axis.

recall the definition of D( ) already given in Section 1.2 Due to the asymptotic

N

e n i n n

Figure 1.6 The cross section of the cylinder 

Figure 1.7 gives the intensity D() scattered at infinity in the direction  Let us

FIGURE 1.6 Cross section of the cylinder and the two sets of sources The profile is given

by Equation (1.68) and the values c5 0.1134  i0.1310, c4 0.0297  i0.3238,

behavior of the Hankel function, the scattered field at infinity can be written fromEquations (1.60 and 1.63) as:

(1.69)

and the intensity scattered at infinity is

(1.70)

Finally, Figure 1.8 shows the resultant field map in the vicinity of the scatterer

j

the groundwork for section 1.3.5, we assume that the medium outside the

cylin-ders has index n idefined in section 1.3.2 Contrary to our assumption in Section1.2, we do not suppose here that the cylinders have a circular cross section All thedevelopments done in Section 1.2 are still valid, and the only difference is as follows For any cylinder j, we consider a circle j with center O j, in such a way

FIGURE 1.8 Modulus of the total field in p-polarization.

We apply the SMM to the N cylinders  , as shown in Figure 1.9 In order to lay

that the cylinder is completely inside  (Figure 1.10) Via Equation (1.35), the SMM

provides the set of coefficients b j,m that enables us to write the total field u

every-where outside the circles j:

(1.71)

The consequence is that everywhere outside the circles j , the total field u and

also its derivatives are known closed form from Equation (1.71)

Let us come back to the original problem described in section 1.3.2 This problemcan be solved by a slight modification of the method of fictitious sources described

in section 1.3.3

Indeed, the scattered field can still be expressed as in Equation (1.60) using the

same fictitious sources S e,n (line sources that radiate F e,n(r) fields expressed as

Hankel functions exactly as in Equation (1.63)) However, in that case, the F i,n(r)

j m m i j m

functions used to expand the field in i(inside 0) must be changed Let us considerthe problem depicted in Figure 1.11 The inclusions jare immersed in a medium

with index n i(the boundary 0of the external scatterer is suppressed) We keep the

same line sources S i,n as in Section 1.3.3 The new function F i,n(r) is the total field

when the structure of Figure 1.11 is excited by the source S i,n By solving this

prob-lem using the SMM as proposed in Section 1.3.4, F i,n(r) can be expressed by the

series provided by Equation (1.71) In other words, F i,n (r) (n  1,2, … ,N s) is a set ofsolutions for the total field inside 0that are available in closed form and that can beused to expand the field in ifollowing Equation (1.62) Using Equation (1.71), we

can compute the value of F i,n(r) and its normal derivative on 0and get the hand sides of Equation (1.67) to be minimized This minimization gives the coeffi-

right-cients c e,n and c i,n, and we finally get the expressions of the total field in closed formeverywhere using Equation (1.61) in eand Equation (1.62) in i

Let us now illustrate the capabilities of the method by applying it to a concrete ple As in section 1.3.3, the cross section of the cylinder 0is a letter F, but now withsharp edges The reason for this choice is only to prove that the method also workspretty well in that case, which is more difficult to solve than a cylinder with roundedboundaries All the coordinates of 0corners in the (x, y) plane have integer values

exam-0is described by a series

identi-cal to Equation (1.68), but in this case we use a large number of c n coefficients

(n 100, 100) in order to get a quasi-polygonal shape This scatterer of index

n i  1.5 lies in vacuum (n e 1) There are also four inclusions inside 0 The tical inclusion has principal axes with half dimensions equal to 1 and 0.5; its center

ellip-is placed at coordinates (1, 4); its principal axes are rotated 45° away from the (x, y)

axes; and it is made of infinitely conducting material The rectangular inclusion is

FIGURE 1.11 Posing of the problem in order to get the functions F i,n(r).

that can be deduced from Figure 1.12 The profile 

placed in the region1.5 x 0.5 and 2 y 0 and is filled with vacuum.

One of the circular inclusions has its center in (2,1) and a radius of 0.5, and it is

filled with vacuum The second circular inclusion has its center in (3,4), a radius of0.5, and it is filled with a lossy material of optical index 0.5 2i (typical value for

a metal in the optical range) This structure is illuminated with an incidence angle

inc 45° by a plane wave of wavelength 0 2 and p-polarization In that

case, the number of sources in each region eand i is chosen to be N s 500 The

total H zfield map is shown in Figure 1.12 Note that the present version of ournumerical code does not allow the computation of the field inside a circle thatincludes elliptical or rectangular bodies This is why dark areas appear aroundthese two inclusions

1.4 DISPERSION RELATIONS OF BLOCH MODES

In this section, we develop useful theoretical tools to understand and design theproperties of devices based on photonic crystals We summarize the properties ofthe propagating Bloch modes in infinite photonic crystals and pay special atten-tion to the propagation of energy Our aim is to predict the behavior of the finitestructure from the knowledge of the properties of the infinite crystal We thenapply these tools to understand how a photonic crystal can behave as a homoge-neous material with an optical index smaller than one (to illustrate ultrarefraction)

or even with a negative optical index (to illustrate negative refraction) We consider

two-dimensional photonic crystals made of a finite number of dielectric cylinders

placed in a vacuum The cylinders are assumed to be infinite along the z-axis (the

coordinate system is defined in Figure 1.13)

We first review the properties of the allowed electromagnetic propagating modes inthe infinite photonic crystal Of course, if there are no such propagative modes for afrequency range we have detected a bandgap We assume that the crystal fills thewhole space and that there is no incident field To be consistent with the previousparts, we consider two-dimensional problems Consequently, two fundamental cases

of polarization exist and the examples are given for the s-polarization case; that is,

when the electric field is parallel to the cylinder axes If the periodicity of the

struc-ture is defined by the vectors d1 and d2, then for any integer values of l and m:

(1.72)

where  r(r) is the relative permittivity.

It is now well known that the allowed propagative modes in photonic crystalsare Bloch modes (as for electrons in crystals) As in the previous parts we assume

an exp(it) time dependence and use the usual complex amplitudes for harmonic

fields Thus, for any Bloch mode, the z-component of the electric field can be written

FIGURE 1.13 Finite-size two-dimensional photonic crystals made of a finite number of

cylinders The structures could be illuminated by an external beam The coordinate system used is also represented.

an exp(it) time dependence and use the usual complex amplitudes for harmonic... ,N s) is a set ofsolutions for the total field inside 0that are available in closed form and that can beused to expand the field in ifollowing