Electromagnetic Theory and Applications for Photonic Crystals

Electromagnetic Theory and Applications for Photonic Crystals © 2006 by Taylor & Francis Group, LLC OPTICAL SCIENCE AND ENGINEERING Founding Editor Brian J Thompson University of Rochester Rochester, New York 10 11 12 13 14 15 16 17 18 19 Electron and Ion Microscopy and Microanalysis: Principles and Applications, Lawrence E Murr Acousto-Optic Signal Processing: Theory and Implementation, edited by Norman J Berg and John N Lee Electro-Optic and Acousto-Optic Scanning and Deflection, Milton Gottlieb, Clive L M Ireland, and John Martin Ley Single-Mode Fiber Optics: Principles and Applications, Luc B Jeunhomme Pulse Code Formats for Fiber Optical Data Communication: Basic Principles and Applications, David J Morris Optical Materials: An Introduction to Selection and Application, Solomon Musikant Infrared Methods for Gaseous Measurements: Theory and Practice, edited by Joda Wormhoudt Laser Beam Scanning: Opto-Mechanical Devices, Systems, and Data Storage Optics, edited by Gerald F Marshall Opto-Mechanical Systems Design, Paul R Yoder, Jr Optical Fiber Splices and Connectors: Theory and Methods, Calvin M Miller with Stephen C Mettler and Ian A White Laser Spectroscopy and Its Applications, edited by Leon J Radziemski, Richard W Solarz, and Jeffrey A Paisner Infrared Optoelectronics: Devices and Applications, William Nunley and J Scott Bechtel Integrated Optical Circuits and Components: Design and Applications, edited by Lynn D Hutcheson Handbook of Molecular Lasers, edited by Peter K Cheo Handbook of Optical Fibers and Cables, Hiroshi Murata Acousto-Optics, Adrian Korpel Procedures in Applied Optics, John Strong Handbook of Solid-State Lasers, edited by Peter K Cheo Optical Computing: Digital and Symbolic, edited by Raymond Arrathoon © 2006 by Taylor & Francis Group, LLC 20 Laser Applications in Physical Chemistry, edited by D K Evans 21 Laser-Induced Plasmas and Applications, edited by Leon J Radziemski and David A Cremers 22 Infrared Technology Fundamentals, Irving J Spiro and Monroe Schlessinger 23 Single-Mode Fiber Optics: Principles and Applications, Second Edition, Revised and Expanded, Luc B Jeunhomme 24 Image Analysis Applications, edited by Rangachar Kasturi and Mohan M Trivedi 25 Photoconductivity: Art, Science, and Technology, N V Joshi 26 Principles of Optical Circuit Engineering, Mark A Mentzer 27 Lens Design, Milton Laikin 28 Optical Components, Systems, and Measurement Techniques, Rajpal S Sirohi and M P Kothiyal 29 Electron and Ion Microscopy and Microanalysis: Principles and Applications, Second Edition, Revised and Expanded, Lawrence E Murr 30 Handbook of Infrared Optical Materials, edited by Paul Klocek 31 Optical Scanning, edited by Gerald F Marshall 32 Polymers for Lightwave and Integrated Optics: Technology and Applications, edited by Lawrence A Hornak 33 Electro-Optical Displays, edited by Mohammad A Karim 34 Mathematical Morphology in Image Processing, edited by Edward R Dougherty 35 Opto-Mechanical Systems Design: Second Edition, Revised and Expanded, Paul R Yoder, Jr 36 Polarized Light: Fundamentals and Applications, Edward Collett 37 Rare Earth Doped Fiber Lasers and Amplifiers, edited by Michel J F Digonnet 38 Speckle Metrology, edited by Rajpal S Sirohi 39 Organic Photoreceptors for Imaging Systems, Paul M Borsenberger and David S Weiss 40 Photonic Switching and Interconnects, edited by Abdellatif Marrakchi 41 Design and Fabrication of Acousto-Optic Devices, edited by Akis P Goutzoulis and Dennis R Pape 42 Digital Image Processing Methods, edited by Edward R Dougherty © 2006 by Taylor & Francis Group, LLC 43 Visual Science and Engineering: Models and Applications, edited by D H Kelly 44 Handbook of Lens Design, Daniel Malacara and Zacarias Malacara 45 Photonic Devices and Systems, edited by Robert G Hunsberger 46 Infrared Technology Fundamentals: Second Edition, Revised and Expanded, edited by Monroe Schlessinger 47 Spatial Light Modulator Technology: Materials, Devices, and Applications, edited by Uzi Efron 48 Lens Design: Second Edition, Revised and Expanded, Milton Laikin 49 Thin Films for Optical Systems, edited by Francoise R Flory 50 Tunable Laser Applications, edited by F J Duarte 51 Acousto-Optic Signal Processing: Theory and Implementation, Second Edition, edited by Norman J Berg and John M Pellegrino 52 Handbook of Nonlinear Optics, Richard L Sutherland 53 Handbook of Optical Fibers and Cables: Second Edition, Hiroshi Murata 54 Optical Storage and Retrieval: Memory, Neural Networks, and Fractals, edited by Francis T S Yu and Suganda Jutamulia 55 Devices for Optoelectronics, Wallace B Leigh 56 Practical Design and Production of Optical Thin Films, Ronald R Willey 57 Acousto-Optics: Second Edition, Adrian Korpel 58 Diffraction Gratings and Applications, Erwin G Loewen and Evgeny Popov 59 Organic Photoreceptors for Xerography, Paul M Borsenberger and David S Weiss 60 Characterization Techniques and Tabulations for Organic Nonlinear Optical Materials, edited by Mark G Kuzyk and Carl W Dirk 61 Interferogram Analysis for Optical Testing, Daniel Malacara, Manuel Servin, and Zacarias Malacara 62 Computational Modeling of Vision: The Role of Combination, William R Uttal, Ramakrishna Kakarala, Spiram Dayanand, Thomas Shepherd, Jagadeesh Kalki, Charles F Lunskis, Jr., and Ning Liu 63 Microoptics Technology: Fabrication and Applications of Lens Arrays and Devices, Nicholas Borrelli © 2006 by Taylor & Francis Group, LLC 64 Visual Information Representation, Communication, and Image Processing, edited by Chang Wen Chen and Ya-Qin Zhang 65 Optical Methods of Measurement, Rajpal S Sirohi and F S Chau 66 Integrated Optical Circuits and Components: Design and Applications, edited by Edmond J Murphy 67 Adaptive Optics Engineering Handbook, edited by Robert K Tyson 68 Entropy and Information Optics, Francis T S Yu 69 Computational Methods for Electromagnetic and Optical Systems, John M Jarem and Partha P Banerjee 70 Laser Beam Shaping, Fred M Dickey and Scott C Holswade 71 Rare-Earth-Doped Fiber Lasers and Amplifiers: Second Edition, Revised and Expanded, edited by Michel J F Digonnet 72 Lens Design: Third Edition, Revised and Expanded, Milton Laikin 73 Handbook of Optical Engineering, edited by Daniel Malacara and Brian J Thompson 74 Handbook of Imaging Materials: Second Edition, Revised and Expanded, edited by Arthur S Diamond and David S Weiss 75 Handbook of Image Quality: Characterization and Prediction, Brian W Keelan 76 Fiber Optic Sensors, edited by Francis T S Yu and Shizhuo Yin 77 Optical Switching/Networking and Computing for Multimedia Systems, edited by Mohsen Guizani and Abdella Battou 78 Image Recognition and Classification: Algorithms, Systems, and Applications, edited by Bahram Javidi 79 Practical Design and Production of Optical Thin Films: Second Edition, Revised and Expanded, Ronald R Willey 80 Ultrafast Lasers: Technology and Applications, edited by Martin E Fermann, Almantas Galvanauskas, and Gregg Sucha 81 Light Propagation in Periodic Media: Differential Theory and Design, Michel Nevière and Evgeny Popov 82 Handbook of Nonlinear Optics, Second Edition, Revised and Expanded, Richard L Sutherland 83 Polarized Light: Second Edition, Revised and Expanded, Dennis Goldstein © 2006 by Taylor & Francis Group, LLC 84 Optical Remote Sensing: Science and Technology, Walter Egan 85 Handbook of Optical Design: Second Edition, Daniel Malacara and Zacarias Malacara 86 Nonlinear Optics: Theory, Numerical Modeling, and Applications, Partha P Banerjee 87 Semiconductor and Metal Nanocrystals: Synthesis and Electronic and Optical Properties, edited by Victor I Klimov 88 High-Performance Backbone Network Technology, edited by Naoaki Yamanaka 89 Semiconductor Laser Fundamentals, Toshiaki Suhara 90 Handbook of Optical and Laser Scanning, edited by Gerald F Marshall 91 Organic Light-Emitting Diodes: Principles, Characteristics, and Processes, Jan Kalinowski 92 Micro-Optomechatronics, Hiroshi Hosaka, Yoshitada Katagiri, Terunao Hirota, and Kiyoshi Itao 93 Microoptics Technology: Second Edition, Nicholas F Borrelli 94 Organic Electroluminescence, edited by Zakya Kafafi 95 Engineering Thin Films and Nanostructures with Ion Beams, Emile Knystautas 96 Interferogram Analysis for Optical Testing, Second Edition, Daniel Malacara, Manuel Sercin, and Zacarias Malacara 97 Laser Remote Sensing, edited by Takashi Fujii and Tetsuo Fukuchi 98 Passive Micro-Optical Alignment Methods, edited by Robert A Boudreau and Sharon M Boudreau 99 Organic Photovoltaics: Mechanism, Materials, and Devices, edited by Sam-Shajing Sun and Niyazi Serdar Saracftci 100.Handbook of Optical Interconnects, edited by Shigeru Kawai 101.GMPLS Technologies: Broadband Backbone Networks and Systems, Naoaki Yamanaka, Kohei Shiomoto, and Eiji Oki 102.Electromagnetic Theory and Applications for Photonic Crystals, Kiyotoshi Yasumoto © 2006 by Taylor & Francis Group, LLC Electromagnetic Theory and Applications for Photonic Crystals edited by Kiyotoshi Yasumoto Kyushu University Fukuoka, Japan Boca Raton London New York A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc © 2006 by Taylor & Francis Group, LLC Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S Government works Printed in the United States of America on acid-free paper 10 International Standard Book Number-10: 0-8493-3677-5 (Hardcover) International Standard Book Number-13: 978-0-8493-3677-5 (Hardcover) Library of Congress Card Number 2005041895 This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe 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) Photonics Materials Crystal optics Materials Electrooptics I Yasumoto, Kiyotoshi II Optical engineering (Marcel Dekker, Inc.) TA1522.E24 2005 621.36 dc22 2005041895 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group is the Academic Division of Informa plc © 2006 by Taylor & Francis Group, LLC and the CRC Press Web site at http://www.crcpress.com 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 dielectric or by forming a lattice of high refractive index material embedded in a transparent 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 understood by comparing it to that of electrons in solid-state materials The electrons passing 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 wavelengths The photonic crystal prohibits any propagation of light with wavelengths in the forbidden bands, i.e., the photonic bandgaps, while allowing other wavelengths 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 micronsized 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 electromagnetic modeling has become a much more important area of research From the viewpoint of electromagnetic field theory, the photonic crystals are optical materials with periodic perturbation of macroscopic material constants Fortunately we have a great deal of knowledge about the electromagnetic theory for periodic structures During the past few decades, various analytical or computational techniques 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 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 phenomena of 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 scattering 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 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 reexamined A novel technique for mode matching combined with the generalized scattering matrix method is presented to deal with the scattering and guidance by metallic photonic crystals with lattice elements of arbitrary cross sections Chapter 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 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 430 Electromagnetic Theory and Applications for Photonic Crystals (a) (b) (c) FIGURE 8.18 Electric field intensity and Poynting vector of (a) 90°, (b) 120°, and (c) 60° bent waveguide The incident frequencies va/2pc are (a) 0.389 in square lattice and 0.407 for (b) and (c) in triangular lattice sharply bent waveguide More precisely, in the transmitted region the mode profile shows the same transverse profile as that of input region In other words, there is no mode conversion, and the eigenmode profile remains even if the light propagates through the sharply bent waveguide Next we try to control the transmission frequency range of bent waveguides using the property of an unloaded microcavity The resonant frequency of an unloaded microcavity can be adjusted by adding another dielectric pillar in that microcavity [24] We place an additional pillar in the unloaded microcavity as shown at the top of Figure 8.19 The radius and relative permittivity of additional © 2006 by Taylor & Francis Group, LLC Finite-Difference Time-Domain Method Applied to Photonic Crystals 431 a a a2 a2 0.46 Normalized frequency a/2 c 0.44 0.42 0.4 0.38 Triangular lattice 0.36 Square lattice 0.34 0.32 1.5 2.5 3.5 Relative permittivity of additional pillar 4.5 a2 FIGURE 8.19 Resonant frequency of cavity as a function of relative permittivity of additional pillar the pillar are denoted by ra2 and ea2, respectively First we change the relative permittivity of the additional pillar Figure 8.19 shows the resonant frequency of the unloaded microcavities when relative permittivity of the additional pillar is changed The radius of the additional pillar is ra2/a ϭ 0.175 The resonant frequencies are decreased by increasing the relative permittivity of the additional pillar for both a square and a triangular lattice microcavity By using this property, we can control the transmission frequency range of the bent waveguide Figure 8.20 shows the optical power transmission characteristics of 90° and 120° bent waveguides when an additional pillar is placed in the center of the cavity We can see that the resonant frequency decreases as the relative permittivity of additional pillar ea2 becomes larger Next we vary another parameter, such as the radius of additional pillar [24] Figures 8.21 and 8.22 show the resonant frequency of an unloaded microcavity as a function of the radius of an additional pillar placed in the center of the © 2006 by Taylor & Francis Group, LLC 432 Electromagnetic Theory and Applications for Photonic Crystals 2.25 1.56 3.06 4.00 a2 = 1.00 0.9 a 0.8 a2 Transmissivity 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.34 0.36 0.38 0.4 0.42 0.44 Normalized frequency a/2 c (a) 4.00 3.06 2.25 1.56 a2 = 1.00 0.9 a 0.8 Transmissivity 0.7 a2 0.6 0.5 0.4 0.3 0.2 0.1 0.36 0.38 0.4 0.42 0.44 Normalized frequency a/2 c (b) FIGURE 8.20 Optical power transmission spectra when relative permittivity of additional pillar is changed: (a) square lattice (b) triangular lattice © 2006 by Taylor & Francis Group, LLC Finite-Difference Time-Domain Method Applied to Photonic Crystals Dipoles Monopole ra2 433 Hexapoles Quadrupoles 0.46 Normalized frequency a/2 c 0.44 Octupoles 2nd order monopole Dipoles 0.42 0.4 Hexapoles Monopole 0.38 Quadrupoles 2nd Order dipoles 0.36 0.34 0.32 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Radius of additional pillar ra2 /a FIGURE 8.21 Resonant frequency of square lattice photonic crystal unloaded microcavity as a function of radius of additional pillar cavity for a square lattice and a triangular lattice photonic crystal microcavity, respectively The relative permittivity of the additional pillar is ea2 ϭ 11.56 We can find from these figures that resonant frequency is decreased by increasing the radius of the additional pillar and the higher order resonance modes appear when the radius of the additional pillar exceeds that of the pillar of the photonic crystal The electric field distributions of the resonance modes are shown at the tops of these figures The resonance modes have symmetry in accordance with the symmetry of the structure The resonance modes are labeled by the number of nodes of the electric field [24], e.g., the dipole resonance mode has two nodes © 2006 by Taylor & Francis Group, LLC 434 Electromagnetic Theory and Applications for Photonic Crystals ra2 Monopole Dipoles Quadrupoles Hexapoles 0.5 Normalized frequency a/2 c 2nd Order monopole Hexapoles 0.45 Monopole 0.4 Dipoles Quadrupoles 0.35 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Radius of additional pillar ra2/a FIGURE 8.22 Resonant frequency of triangular lattice photonic crystal unloaded microcavity as a function of radius of additional pillar Figure 8.23 shows the optical power transmission characteristics of a 90° bent waveguide when an additional dielectric pillar whose radius is ra2/a ϭ 0.475 is placed in the center of the cavity We can see that optical power is transmitted completely at the frequency that is the same as that of the higher order resonance mode of the unloaded cavity as shown in Figure 8.21 Figure 8.24 shows the electric field intensity profiles at the resonant frequency The incident frequencies are va/2pc ϭ 0.391 and 0.372 We can see that the second order monopole and the quadrupole resonance mode are excited in the microcavity Figure 8.25 shows the optical power transmission characteristics of 120/60° bent waveguides when an additional dielectric pillar whose radius is ra2 /a ϭ 0.575 is placed in the center of the cavity We can find that a hexapole resonance mode is excited in the microcavity because another resonance mode cannot be excited by the fundamental © 2006 by Taylor & Francis Group, LLC Finite-Difference Time-Domain Method Applied to Photonic Crystals 435 0.9 0.8 Transmissivity 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.34 0.36 0.38 0.4 Normalized frequency a/2 c 0.42 0.44 FIGURE 8.23 Optical power transmission spectrum of square lattice bent waveguide when additional dielectric pillar whose radius is ra2/a ϭ 0.475 is placed in the cavity (a) (b) FIGURE 8.24 Electric field intensity of square lattice bent waveguide when additional dielectric pillar whose radius is ra2/a ϭ 0.475 is placed in the cavity The incident frequencies are (a) va/2pc ϭ 0.391 and (b) va/2pc ϭ 0.372 mode of the input waveguide Figure 8.26 shows the electric field intensity profiles of 120/60° bent waveguides at the resonant frequency The incident frequency is va/2pc ϭ 0.415 for both bent waveguides We can see that the hexapole resonance mode is excited in the microcavity © 2006 by Taylor & Francis Group, LLC 436 Electromagnetic Theory and Applications for Photonic Crystals Transmissivity 0.8 0.6 0.4 0.2 0.36 0.38 0.4 0.42 0.44 Normalized frequency a/2 c FIGURE 8.25 Optical power transmission spectrum of triangular lattice bent waveguide when additional dielectric pillar whose radius is ra2/a ϭ 0.575 is placed in the cavity (a) (b) FIGURE 8.26 Electric field intensity of square lattice bent waveguide when additional dielectric pillar whose radius is ra2 /a ϭ 0.575 is placed in the cavity The incident frequency is va/2pc ϭ 0.415 (a) 60° bend, (b) 120° bend 8.4.2.2 Air-Hole Type Photonic Crystal Bent Waveguide The air-hole type photonic crystal sharply bent waveguide is designed as shown in Figure 8.27 In the corner region are placed two additional air-holes whose radius is ra2 Figure 8.28 shows the electric field intensity and Poynting vector of an L-shaped bent waveguide at the resonant frequency va/2pc ϭ 0.248 The radius of the air-hole is ra2/a ϭ 0.35 The waveguide width and shift are W/a ϭ 0.85 and D/a ϭ 0.8, respectively We can see that there are no reflected © 2006 by Taylor & Francis Group, LLC Finite-Difference Time-Domain Method Applied to Photonic Crystals 437 W ra2 a x y z FIGURE 8.27 Photonic crystal L-shaped bent waveguide with additional air-holes in the corner region (a) (b) FIGURE 8.28 Electric field intensity and Poynting vector of L-shaped bent waveguide at the resonant frequency va/2pc ϭ 0.248 The radius and relative permittivity of additional air-hole are ra2/a ϭ 0.35 and ea2 ϭ 1.0, respectively (a) Electric field intensity, (b) Poynting vector © 2006 by Taylor & Francis Group, LLC 438 Electromagnetic Theory and Applications for Photonic Crystals 1.00 ra2 /a = 0.250 ra2 /a = 0.350 Transmissivity 0.75 0.50 ra2 /a = 0.150 0.25 Without air-holes 0.00 0.246 0.248 0.250 0.252 0.254 0.256 0.258 Normalized frequency a/2 c FIGURE 8.29 Optical power transmission characteristics of L-shaped bent waveguide when the radius of additional air-holes ra2 is changed waves in the input side and electric fields concentrate at the corner region due to resonant tunneling The optical power flows efficiently through the right-angle bend, and it makes some vortices at the corner region as shown in Figure 8.28(b) We can see that the mode field profile in the transmitted region shows the same transverse profile as that of input region Figure 8.29 shows the optical power transmission characteristics when the radius of an additional air-hole is changed By using additional air-holes we can completely eliminate reflected waves from the right-angle corner at the resonant frequency We can see from the figure that for an increase of radius ra2 the resonant frequency shifts to a lower side and its quality factor increases In other words, transmission frequency ranges can be controlled by changing the radii of additional air-holes 8.5 WAVELENGTH MULTI/DEMULTIPLEXER As a practical signal processing device, we propose and design a wavelength multi/demultiplexer A compact size wavelength multi/demultiplexer is desired for a wavelength division multiplexed (WDM) communication system To handle many wavelengths, the size of a wavelength multi/demultiplexer made with conventional dielectric waveguides becomes several centimeters square because bent waveguides in the device have large curvatures to suppress radiation loss Using the photonic crystal bent waveguide, we can reduce the size of the device © 2006 by Taylor & Francis Group, LLC Finite-Difference Time-Domain Method Applied to Photonic Crystals 439 L1 Port 1 Port Port IN Port L2 FIGURE 8.30 The structure of three-wavelength multi/demultiplexer Black circles denote additional pillars whose radii and relative permittivity are ra2 /a ϭ 0.175 and ea2 ϭ 1.69, respectively Lengths of coupling region are L1/a ϭ 48.0 and L2 /a ϭ 24.0, respectively The parameters of photonic crystal /a ϭ 0.175 and ea ϭ 11.56 are used The wavelength multi/demultiplexer designed here is composed of directional couplers and reflectionless right-angle bent waveguides as shown in Figure 8.30 The number of channels of the wavelength multi/demultiplexer determines the number of directional couplers, i.e., M-wavelengths are multi/demultiplexed by using (M Ϫ 1) directional couplers, which are connected sequentially As an example, we design a three-wavelength multi/demultiplexer using two directional couplers The directional coupler is composed of two waveguides which are single mode waveguides whose width is W/a ϭ 1.65 The interval between the two waveguides is d/a ϭ 0.35, which corresponds to one row of crystals 8.5.1 DESIGN PARAMETERS For analyzing this device, we need to estimate several parameters of the directional coupler Figure 8.31 shows the normalized coupling length of the directional coupler L/a as a function of wavelength In the figure, integral multiples of coupling length nL (n ϭ 1, 2, 3, …) are plotted for determining the length of the directional coupler If the length of the directional coupler is chosen as an evennumbered multiple of the coupling length 2nL (n ϭ 1, 2, …), the optical power for the appropriate wavelength is finally transmitted to the straight port Considering the desired wavelengths that are filtered by the directional couplers, we choose the lengths of the first and second directional couplers as L1/a ϭ 48 and L2/a ϭ 24, respectively From Figure 8.31 we can see that the light of wavelength l1 (ϭ2.68a) is completely transmitted across the port of the first directional coupler and two different lights of wavelength l2 (ϭ2.60a) and l3 (ϭ2.74a) that are first transmitted to the straight port of the first directional coupler are finally divided into two output ports through the second directional coupler © 2006 by Taylor & Francis Group, LLC 440 Electromagnetic Theory and Applications for Photonic Crystals 50 Coupling length of directional coupler L/a 7L 45 40 6L 35 5L 30 5L 25 20 3L 15 2L 10 2.50 Coupling length L 2.55 2.60 2.65 2.70 2.75 2.80 Wavelength /a FIGURE 8.31 Coupling length of the directional coupler as a function of wavelength The directional coupler is composed of two single mode waveguides whose width is W/a ϭ 1.65 The interval between two waveguides is d/a ϭ 0.35 8.5.2 NUMERICAL RESULTS Using these parameters we assemble directional couplers and L-shaped bent waveguides into the three-wavelength multi/demultiplexer as shown in Figure 8.30 In the corner region we place additional pillars denoted by black circles in the figure In order to obtain wide transmission frequency characteristics, we choose the radii and relative permittivity of additional pillars as ra2 /a ϭ 0.175 and ea2 ϭ 1.69, respectively The transmission spectra of the three-wavelength multi/demultiplexer are shown in Figure 8.32 Here we evaluate the characteristics of the threewavelength multi/demultiplexer quantitatively by using the extinction ratio and insertion loss defined by  P ′  P  Extinction ratio ϭϪ10 log  i  , Insertion loss ϭϪ10 log  i  (8.30)  Pin   Pi  where Pin is input power flow, Pi and P′i indicate output power at port i for a selected port and an unselected port, respectively In fact, the extinction ratio and insertion loss for ports 1, 2, and are almost greater than 19.6 dB and less than ϫ 10Ϫ2 dB, respectively, as shown in Table 8.1 We can see from the figure and table that additional pillars in right-angle bends realize a high extinction ratio and low insertion loss Figure 8.33 shows electric field intensity of the directional coupler for wavelength l/a ϭ 2.68, 2.60, and 2.74 We can see that a low insertion © 2006 by Taylor & Francis Group, LLC Finite-Difference Time-Domain Method Applied to Photonic Crystals 441 Transmittance (dB) −5 −10 −15 Port1 Port2 Port3 −20 2.50 2.55 2.60 2.65 2.70 2.75 2.80 Wavelength /a FIGURE 8.32 Transmission spectra of the three-wavelength multi/demultiplexer TABLE 8.1 Extinction Ratio and Insertion Loss of the Three-Wavelength Multi/ Demultiplexer Wavelength L/a Extinction ratio [dB] 2.68 2.60 2.74 Ͼ27.5 Ͼ18.5 Ͼ19.6 Insertion loss [dB] Ϫ2 3.0 ϫ 10 8.5 ϫ 10Ϫ2 6.5 ϫ 10Ϫ2 Output Port Port Port loss and high-extinction-ratio three-wavelength multi/demultiplexer whose overall length is approximately 35lm, where lm is a maximum wavelength of incident wave, can be realized in all incident wavelengths 8.6 CONCLUSION We have presented the finite-difference time-domain method based on the multidimensional wave digital filters and shown several advantages of the MD-WDFs over the Yee algorithm We have newly proposed a modified form of the Maxwell’s equations in the perfectly matched layers and its MD-WDFs’ representation by using the current controlled voltage sources We have evaluated the © 2006 by Taylor & Francis Group, LLC 442 Electromagnetic Theory and Applications for Photonic Crystals (a) (b) (c) FIGURE 8.33 Electric field intensity of the three-wavelength multi/demultiplexer The wavelengths of incident wave are (a) l/a ϭ 2.66, (b) l/a ϭ 2.58, and (c) l/a ϭ 2.72 numerical errors concerning phase velocity in the plane wave propagation problem by examining the numerical dispersion relation and pointed out the capability of the MD-WDFs compared with the Yee algorithm We have clarified the propagation characteristics of two-dimensional photonic crystal optical waveguide devices By examining the eigenmode propagation in the photonic crystal waveguides constructed by photonic crystals, we can find that electric and magnetic field intensities in the waveguides oscillate according to the period of the lattice constant of the photonic crystal We have checked the dispersion relation of the waveguide and confirmed differences in the performance of a single mode propagation in air-hole type photonic crystal waveguide and a pillar type waveguide Next, we have analyzed typical waveguide devices such as a directional coupler and a sharply bent waveguide First, we have simulated a directional coupler constructed by photonic crystal We have also checked the dispersion relation of the directional coupler and confirmed that the even and odd modes in the air-hole type photonic crystal directional coupler cause cutoff at each frequency band because each mode couples onto each higher-order mode As a result, the frequency spectrum of coupling length shows different characteristics from that of a pillar type photonic crystal directional coupler, which has the same characteristics as © 2006 by Taylor & Francis Group, LLC Finite-Difference Time-Domain Method Applied to Photonic Crystals 443 the conventional dielectric waveguide Second, we have proposed a reflectionless sharply bent waveguide with a microcavity in the corner region Numerical results show that reflected waves from the sharp bend can be completely eliminated by adding additional pillars/air-holes due to resonant tunneling, and its transmission bandwidth can be controlled by changing the structure of the microcavity Finally, we have designed a wavelength multi/demultiplexer that consists of a directional coupler and reflectionless right-angle bent waveguides and demonstrated that the device can work as a low-insertion-loss and high-extinction-ratio device REFERENCES [1] E Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys Rev Lett., 58, 2059–2062, 1987 [2] J.D Joannopoulos, P.R Villeneuve, and S Fan, Photonic crystals: putting a new twist on light, Nature, 386, 143–149, 1997 [3] S.Y Lin, E Chow, V Hietala, P.R Villeneuve, and J.D Joannopoulos, Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal, Science, 282, 274–276, 1998 [4] S Fan, P.R Villeneuve, J.D Joannopoulos, and H.A Haus, Channel drop tunneling through localized states, Phys Rev Lett., 80, 960–963, 1998 [5] H Ikuno, Y Naka, and A Yata, Analysis of optical waveguide devices using the FD-TD method based on the principles of multidimensional wave digital filters, Radio Sci., 35, 595–605, 2000 [6] Y Naka and H Ikuno Two-dimensional photonic crystal l-shaped bend waveguide and its application to wavelength multi/demultiplexer, Turkish J of Electrical Eng & Comput Sci., 10, 245–256, 2002 [7] A Fettweis, Wave digital filters: theory and practice, Proc IEEE, 74, 270–327, 1986 [8] A Fettweis, Multidimensional wave digital filters for discrete-time modelling of Maxwell’s equations, Int J Numerical Modelling, 5, 183–201, 1992 [9] J.P Berenger, A perfectly matched layer for the absorption of electromangetic waves, J Computational Phys., 114, 185–200, 1994 [10] Y Naka, H Ikuno, M Nishimoto, and A Yata, FD-TD method with PMLS ABC based on the principles of multidimensional wave digital filters for discrete-time modelling of Maxwell’s equations, IEICE Trans Electron., E81-C, 305–314, 1998 [11] D.E Johnson, J.L Hilburn, and J.R Johnson, Basic Electric Circuit Analysis, Prentice-Hall, New Jersey, 1984 [12] A Fettweis and G Nitsche, Transformation approach to numerically integrating PDES by means of WDF principles, Multidimensional Syst and Signal Process., 2, 127–159, 1991 [13] J.P Berenger, Three-dimensional perfectly matched layer for the absorption of electromagnetic waves, J Computational Phys., 127, 363–379, 1996 [14] D.R Lynch and K.D Paulsen, Origin of vector parasitics in numerical Maxwell solution, IEEE Trans Microwave Theory Tech., 39, 383–394, 1991 [15] A Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Boston, London, 1995 © 2006 by Taylor & Francis Group, LLC 444 Electromagnetic Theory and Applications for Photonic Crystals [16] M Tokushima, H Kosaka, A Tomita, and H Yamada, Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide, Appl Phys Lett., 76, 952–954, 2000 [17] M Loncar, D Nedejikovic, T Doll, J Vuckovic, A Scherer, and T.P Pearsall, Waveguiding in planar photonic crystals, Appl Phys Lett., 77, 1937–1939, 2000 [18] T Baba, A Motegi, T Iwai, N Fukaya, Y Watanabe, and A Sakai, Light propagation characteristics of straight single-line-defect waveguides in photonic crystal slabs fabricated into a silicon-on-insulator substrate, IEEE J Quantum Electron., 38, 743–752, 2002 [19] W.L Ko and R Mittra, A combination of FD-TD and Prony methods for analyzing microwave integrated circuits, IEEE Trans Microwave Theory Tech., MTT-39, 2176–2181, 1991 [20] J Wang and H Guo, Resonant tunneling through a bend in a quantum wire, Appl Phys Lett., 60, 654–656, 1992 [21] Y Naka and H Ikuno, Two-dimensional photonic crystal optical waveguides and their application to optical devices, Proc Int Union of Radio Science General Assemblies, Maastricht, Netherlands, 2002, D3.O.2 [22] Y Naka and H Ikuno, Analysis of two-dimensional photonic crystal sharply bent waveguides, Proc URSI 2004 Int Symp on Electromagnetic Theory, Pisa, Italy, 2004, 748–750 [23] S Fan, S.G Johnson, J.D Joannopoulos, C Manolatou, and H.A Haus, Waveguide branches in photonic crystals, J Opt Soc Am B, 18, 162–165, 2001 [24] P.R Villeneuve, S Fan, and J.D Joannopoulos, Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency, Phys Rev B, 54, 7837–7842, 1996 © 2006 by Taylor & Francis Group, LLC ... 102.Electromagnetic Theory and Applications for Photonic Crystals, Kiyotoshi Yasumoto © 2006 by Taylor & Francis Group, LLC Electromagnetic Theory and Applications for Photonic Crystals edited by... trademarks, and are used only for identification and explanation without intent to infringe Library of Congress Cataloging-in-Publication Data Electromagnetic theory and applications for photonic crystals. .. formation of allowed bands and forbidden bands in optical wavelengths The photonic crystal prohibits any propagation of light with wavelengths in the forbidden bands, i.e., the photonic bandgaps, while