Electromagnetic theory and applications for photonic crystals

451 Pages·2006·30.509 MB·English

by  Kiyotoshi Yasumoto

Electromagnetic Theory
and Applications
for Photonic Crystals
edited by
Kiyotoshi Yasumoto
Kyushu University
Fukuoka, Japan
Boca Raton London New York
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© 2006 by Taylor & Francis Group, LLC
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Library of Congress Cataloging-in-Publication Data
Electromagnetic theory and applications for photonic crystals / edited by Kiyotoshi Yasumoto.
p. cm. -- (Optical engineering)
Includes bibliographical references and index.
ISBN 0-8493-3677-5 (alk. paper)
1. Photonics--Materials. 2. Crystal optics--Materials. 3. Electrooptics. I. Yasumoto, Kiyotoshi. II.
Optical engineering (Marcel Dekker, Inc.)
TA1522.E24 2005
621.36--dc22 2005041895
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Preface
Photonic crystals are periodic dielectric or metallic structures that are artificially
designed to control and manipulate the propagation of light. A photonic crystal can
be made either by arranging a lattice of air holes on a transparent background dielec-
tricor by forming a lattice of high refractive index material embedded in a trans-
parent medium with a lower refractive index. The lattice size may be roughly
estimated to be the wavelength of light in the background medium.
The behavior of light propagating in a photonic crystal can be intuitively under-
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 the
formation of allowed and forbidden energy states of electrons. The light propagating
in a photonic crystal interacts with the periodic modulation of refractive index. This
results 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 wave-
lengths to propagate freely. The band structures depend on the specific geometry
and composition of the photonic crystal such as the lattice size, the diameter of
the lattice elements, and the contrast in refractive index. It is possible to create
allowed bands within the photonic bandgaps by introducing point defects or line
defects in the lattice of photonic crystals. Light will be strongly confined within the
defects for wavelengths in the bandgap of the surrounding photonic crystals. The
point defects and line defects can be used to make optical resonators and photonic
crystal waveguides, respectively.
The photonic crystals with bandgaps are expected to be new materials for future
optical circuits and devices, which can control the behavior of light in a micron-
sized scale. Although there is continuing interest in new findings of photonic
bandgap structures associated with particular lattice configurations, recent attention
has been focused on the engineering applications of the photonic crystals. To be
able to create photonic-crystal-based optical circuits and devices, their electromag-
netic modeling has become a much more important area of research. From the
viewpoint of electromagnetic field theory, the photonic crystals are optical materi-
als with periodic perturbation of macroscopic material constants. Fortunately we
have a great deal of knowledge about the electromagnetic theory forperiodic 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 provide
the electromagnetic theoretical methods that can be effectively applied to the
modeling of photonic crystals and related optical devices.
This book consists of eight chapters that are ordered in a reasonably logical
manner from analytical methods to computational methods. Each chapter starts
with a brief introduction and a description of the method, followed by detailed
formulations for practical applications. Chapter 1 describes the scattering matrix
method based on multipole expansions, its extension combined with the method
© 2006 by Taylor & Francis Group, LLC
of fictitious sources, and the Bloch modes approach. The applications of the methods
and numerical examples are presented with a particular emphasis on phenomenaof
anomalous refraction and control of light emission in photonic crystals.
In Chapter 2, the multipole theory of scattering by a finite cluster of cylinders
is discussed to investigate the propagation of light in photonic crystal fibers and
the radiation dynamics of photonic crystals. The multipole method combined
with 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
photonic crystals are formulated in Chapter 3, using a model of multilayered periodic
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 and
transmission matrices in a layered system.
Chapter 4 is devoted to the method of multiple multipole program applied to
the simulation of photonic crystal devices. The method comprises a modeling of
periodic structures using the concepts of fictitious boundaries and periodic boundary
conditions, novel eigenvalue solvers, a so-called connections scheme that is a unique
macro feature of the method, and eigenvalue and parameter estimation techniques.
In Chapter 5, the mode-matching method for periodic metallic structures is reexam-
ined. A novel technique for mode matching combined withthe generalized scatter-
ing matrix method is presented to deal with the scattering and guidance by
metallic photonic crystals with lattice elements of arbitrary cross sections.
Chapter 6 describes the method of lines, which is one of the efficient numerical
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 photonic
crystal devices consisting of various bends, junctions, and their concatenations.
In Chapter 7, the full-vectorial finite-difference frequency-domain method is
treated. The absorbing boundary conditions, the periodic boundary conditions, and
an interface condition for dielectric interfaces with curvature are implemented in the
finite-difference scheme. The method is applied to the analysis of photonic crystal
fibers, photonic crystal planar waveguides, and bandgap structures. Chapter 8
describes the finite-difference time-domain method based on the principles of
multidimensional wave digital filters. The method employs the finite difference
schemes using the trapezoidal rule for discretizing Maxwell’s equations that has
advantages with regard to numerical stability and robustness. Numerical examples
are presented for various photonic crystal waveguide devices.
It is hoped that the material is sufficiently detailed both for readers involved
with the physics of photonic bandgap structures and for those working on the
applications 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 of
CRC Press, Taylor & Francis Group, for their help throughout the preparation of
this book.
Kiyotoshi Yasumoto
© 2006 by Taylor & Francis Group, LLC
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 Science
and Communication Engineering. He was a visiting professor at the Department of
Electrical and Computer Engineering, University of Wisconsin in Madison in 1989
and a visiting fellow at the Institute of Solid State Physics, Bulgarian Academy of
Science and Institute of Radiophysics and Electronics, Czechoslovakian Academy
of Science, in 1990.
He is a fellow of the Institute of Electronics, Information and Communication
Engineers (IEICE) of Japan, a fellow of the Optical Society of America (OSA),
a fellow of the Chinese Institute of Electronics (CIE), a senior member of IEEE AP,
MTT, and LEOS Societies, and a member of IEE Japan and the Electromagnetic
Academy. His research interests are in electromagnetic wave theory, analytical and
numerical techniques in microwave and photonics, and radiation and scattering in
electron beam–plasma systems.
He has served as a member of organizing, steering, technical program, and
international 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.
© 2006 by Taylor & Francis Group, LLC
Contributors
Ara A. Asatryan Christian Hafner
Centre for Ultrahigh-Bandwidth Computational Optics Group
Devices for Optical Systems and Laboratory for Electromagnetic
Department of Mathematical Fields and Microwave
Sciences Electronics
University of Technology ETH Zentrum
Sydney, Australia Zurich, Switzerland
Lindsay C. Botten
Stefan F. Helfert
Centre for Ultrahigh-Bandwidth
Allgemeine und Theoretische
Devices for Optical Systems and
Elektrotechnik
Department of Mathematical
University of Hagen
Sciences
Hagen, Germany
University of Technology
Sydney, Australia
Hiroyoshi Ikuno
Hung-Chun Chang Department of Electrical and
Department of Electrical Engineering Computer Engineering
Graduate Institute of Kumamoto University
Communication Engineering Kurokami, Japan
National Taiwan University
Taipei, Taiwan, Republic of China Hongting Jia
Department of Computer Science
Stefan Enoch and Communication
Faculté des Sciences de Saint Jérôme Engineering
Institut Fresnel Kyushu University
Université Paul Cézanne-Aix- Fukuoka, Japan
Marseille III
Marseille, France Boris T. Kuhlmey
Centre for Ultrahigh-Bandwidth
Daniel Erni
Devices for Optical Systems and
Communication Photonics Group
School of Physics
Laboratory for Electromagnetic
University of Sydney
Fields and Microwave Electronics
Sydney, Australia
ETH Zentrum
Zurich, Switzerland
Timothy N. Langtry
David P. Fussell Centre for Ultrahigh-Bandwidth
Centre for Ultrahigh-Bandwidth Devices for Optical Systems
Devices for Optical Systems and and Department of Mathematical
School of Physics Sciences
University of Sydney University of Technology
Sydney, Australia Sydney, Australia
© 2006 by Taylor & Francis Group, LLC
Daniel Maystre Geoffrey H. Smith
Faculté des Sciences de Saint Centre for Ultrahigh-Bandwidth
Jérôme Devices for Optical Systems and
Institut Fresnel Department of Mathematical
Université Paul Cézanne-Aix- Sciences
Marseille III University of Technology
Marseille, France Sydney, Australia
Ross C. McPhedran C. Martijn de Sterke
Centre for Ultrahigh-Bandwidth Centre for Ultrahigh-Bandwidth
Devices for Optical Systems and Devices for Optical Systems and
School of Physics School of Physics
University of Sydney University of Sydney
Sydney, Australia Sydney, Australia
Gérard Tayeb
Yoshihiro Naka
Faculté des Sciences de Saint Jérôme
Department of Electrical and
Institut Fresnel
Computer Engineering
Université Paul Cézanne-Aix-
Kumamoto University
Marseille III
Kurokami, Japan
Marseille, France
Nicolae A. Nicorovici
Thomas P. White
Centre for Ultrahigh-Bandwidth
Centre for Ultrahigh-Bandwidth
Devices for Optical Systems and
Devices for Optical Systems and
Department of Mathematical
School of Physics
Sciences
University of Sydney
University of Technology
Sydney, Australia
Sydney, Australia
Kiyotoshi Yasumoto
Reinhold Pregla Department of Computer Science
Allgemeine und Theoretische and Communication Engineering
Elektrotechnik Kyushu University
University of Hagen Fukuoka, Japan
Hagen, Germany
Chin-Ping Yu
Jasmin Smajic Graduate Institute of Electro-Optical
ABB Switzerland, Ltd Engineering
Corporate Research National Taiwan University
Baden-Dättwil, Switzerland Taipei, Taiwan, Republic of China
© 2006 by Taylor & Francis Group, LLC
Table of Contents
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
© 2006 by Taylor & Francis Group, LLC
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
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2 Electromagnetic Theory and Applications for Photonic Crystals
1.1 INTRODUCTION
The scattering matrix method (SMM) is one of the most efficient methods for solv-
ing a problem of scattering by a large but finite number of objects. It basically takes
into account separately the specific scattering properties of each object and then
evaluates the coupling phenomena between them. Even though it can be presented
in a quite rigorous form, it is based on a physically intuitive approach to the prob-
lem of scattering from a set of objects. One of its advantages is to be accessible to
postgraduate students. Moreover, the numerical implementation does not present
major difficulties. In contrast to other classical methods like Finite Difference Time
Domain method (FDTD) or the finite element method, it becomes much simpler
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 of
finite size regardless of whether or not they have defects of periodicity.
The first achievement relating to that method should be attributed to Lord
Rayleigh, who dealt with the electrostatic case [1]. The electromagnetic version of
this method has been developed since the 1980s in various forms by different
groups working independently [2–9]. Their studies deal with 2D or 3D, dielectric,
metallic, or perfectly conducting objects placed in space in a periodic or random
way, 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 the
method to more complicated structures, it is possible to combine the SMM with
the 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, the
field in each medium is represented as the field radiated by a set of fictitious
sources with initially unknown intensities. These intensities are obtained by
imposing the boundary conditions for the fields on the surfaces of the scatterers.
By combining these two methods, we concurrently procure their advantages. The
method is described in Section 1.3 in a 2D case and can, for instance, address
structures 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-dimensional
photonic crystals. However, the phenomena generated by photonic crystals are so
surprising and so complex that a theoretician needs a preliminary phenomeno-
logical approach. For this purpose, the notion of Bloch modes provides a valuable
tool. We will describe such modes, and we will show that their dispersion curves
enable one to predict most of the properties of photonic crystals. Indeed, these
dispersion curves enlighten us on quantities such as the average energy velocity
and 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 the
control of light emission. Two kinds of anomalous refraction phenomena can be
distinguished. In the phenomenon of ultrarefraction, a photonic crystal simulates
© 2006 by Taylor & Francis Group, LLC