Practical Applications of Microresonators in Optics and Photonics

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PRACTICAL APPLICATIONS OF MICRORESONATORS IN OPTICS AND PHOTONICS

© 2009 by Taylor & Francis Group, LLC

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OPTICAL SCIENCE AND ENGINEERING Founding Editor Brian J. Thompson University of Rochester Rochester, New York

1. Electron and Ion Microscopy and Microanalysis: Principles and Applications, Lawrence E. Murr 2. Acousto-Optic Signal Processing: Theory and Implementation, edited by Norman J. Berg and John N. Lee 3. Electro-Optic and Acousto-Optic Scanning and Deflection, Milton Gottlieb, Clive L. M. Ireland, and John Martin Ley 4. Single-Mode Fiber Optics: Principles and Applications, Luc B. Jeunhomme 5. Pulse Code Formats for Fiber Optical Data Communication: Basic Principles and Applications, David J. Morris 6. Optical Materials: An Introduction to Selection and Application, Solomon Musikant 7. Infrared Methods for Gaseous Measurements: Theory and Practice, edited by Joda Wormhoudt 8. Laser Beam Scanning: Opto-Mechanical Devices, Systems, and Data Storage Optics, edited by Gerald F. Marshall 9. Opto-Mechanical Systems Design, Paul R. Yoder, Jr. 10. Optical Fiber Splices and Connectors: Theory and Methods, Calvin M. Miller with Stephen C. Mettler and Ian A. White 11. Laser Spectroscopy and Its Applications, edited by Leon J. Radziemski, Richard W. Solarz, and Jeffrey A. Paisner 12. Infrared Optoelectronics: Devices and Applications, William Nunley and J. Scott Bechtel 13. Integrated Optical Circuits and Components: Design and Applications, edited by Lynn D. Hutcheson 14. Handbook of Molecular Lasers, edited by Peter K. Cheo 15. Handbook of Optical Fibers and Cables, Hiroshi Murata 16. Acousto-Optics, Adrian Korpel 17. Procedures in Applied Optics, John Strong 18. Handbook of Solid-State Lasers, edited by Peter K. Cheo 19. Optical Computing: Digital and Symbolic, edited by Raymond Arrathoon 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


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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 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 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


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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 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. Laser Beam Shaping Applications, edited by Fred M. Dickey, Scott C. Holswade and David L. Shealy 103. Electromagnetic Theory and Applications for Photonic Crystals, Kiyotoshi Yasumoto


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104. Physics of Optoelectronics, Michael A. Parker 105. Opto-Mechanical Systems Design: Third Edition, Paul R. Yoder, Jr. 106. Color Desktop Printer Technology, edited by Mitchell Rosen and Noboru Ohta 107. Laser Safety Management, Ken Barat 108. Optics in Magnetic Multilayers and Nanostructures, Sˇtefan Viˇsˇnovsky’ 109. Optical Inspection of Microsystems, edited by Wolfgang Osten 110. Applied Microphotonics, edited by Wes R. Jamroz, Roman Kruzelecky, and Emile I. Haddad 111. Organic Light-Emitting Materials and Devices, edited by Zhigang Li and Hong Meng 112. Silicon Nanoelectronics, edited by Shunri Oda and David Ferry 113. Image Sensors and Signal Processor for Digital Still Cameras, Junichi Nakamura 114. Encyclopedic Handbook of Integrated Circuits, edited by Kenichi Iga and Yasuo Kokubun 115. Quantum Communications and Cryptography, edited by Alexander V. Sergienko 116. Optical Code Division Multiple Access: Fundamentals and Applications, edited by Paul R. Prucnal 117. Polymer Fiber Optics: Materials, Physics, and Applications, Mark G. Kuzyk 118. Smart Biosensor Technology, edited by George K. Knopf and Amarjeet S. Bassi 119. Solid-State Lasers and Applications, edited by Alphan Sennaroglu 120. Optical Waveguides: From Theory to Applied Technologies, edited by Maria L. Calvo and Vasudevan Lakshiminarayanan 121. Gas Lasers, edited by Masamori Endo and Robert F. Walker 122. Lens Design, Fourth Edition, Milton Laikin 123. Photonics: Principles and Practices, Abdul Al-Azzawi 124. Microwave Photonics, edited by Chi H. Lee 125. Physical Properties and Data of Optical Materials, Moriaki Wakaki, Keiei Kudo, and Takehisa Shibuya 126. Microlithography: Science and Technology, Second Edition, edited by Kazuaki Suzuki and Bruce W. Smith 127. Coarse Wavelength Division Multiplexing: Technologies and Applications, edited by Hans Joerg Thiele and Marcus Nebeling 128. Organic Field-Effect Transistors, Zhenan Bao and Jason Locklin 129. Smart CMOS Image Sensors and Applications, Jun Ohta 130. Photonic Signal Processing: Techniques and Applications, Le Nguyen Binh 131. Terahertz Spectroscopy: Principles and Applications, edited by Susan L. Dexheimer 132. Fiber Optic Sensors, Second Edition, edited by Shizhuo Yin, Paul B. Ruffin, and Francis T. S. Yu 133. Introduction to Organic Electronic and Optoelectronic Materials and Devices, edited by Sam-Shajing Sun and Larry R. Dalton 134. Introduction to Nonimaging Optics, Julio Chaves 135. The Nature of Light: What Is a Photon?, edited by Chandrasekhar Roychoudhuri, A. F. Kracklauer, and Katherine Creath 136. Optical and Photonic MEMS Devices: Design, Fabrication and Control, edited by Ai-Qun Liu 137. Tunable Laser Applications, Second Edition, edited by F. J. Duarte 138. Biochemical Applications of Nonlinear Optical Spectroscopy, edited by Vladislav Yakovlev 139. Dynamic Laser Speckle and Applications, edited by Hector J. Rabal and Roberto A. Braga Jr. 140. Slow Light: Science and Applications, edited by Jacob B. Khurgin and Rodney S. Tucker 141. Laser Safety: Tools and Training, edited by Ken Barat 142. Near-Earth Laser Communications, edited by Hamid Hemmati


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143. Polarimetric Radar Imaging: From Basics to Applications, Jong-Sen Lee and Eric Pottier 144. Photoacoustic Imaging and Spectroscopy, edited by Lihong V. Wang 145. Infrared-Visible-Ultraviolet Devices and Applications, Second Edition, edited by William Nunley and Dave Birtalan 146. Practical Applications of Microresonators in Optics and Photonics, edited by Andrey B. Matsko


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PRACTICAL APPLICATIONS OF MICRORESONATORS IN OPTICS AND PHOTONICS

Edited by

ANDREY B. MATSKO

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business


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Cover images reprinted from: Central figure: Savchenkov, A.A., Matsko, A. B., Grudinin, I., Savchenkova, E. A., Strekalov, D., and Maleki, L., Optical vortices with large orbital momentum: generation and interference. Opt. Express, 14, 2888-2897 (2006). Top and bottom figures: Savchenkov, A.A., Matsko, A. B., Ilchenko, V. S., Strekalov, D., and Maleki, L. Direct observation of stopped light in a whispering-gallery-mode microresonator. Phys. Rev., A 76, 023816 (2007).

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-6578-7 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, 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. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com


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Contents Preface...............................................................................................................................................xi Editor...............................................................................................................................................xv Contributors................................................................................................................................. xvii 1. Ultrahigh-Q Photonic Crystal Nanocavities and Their Applications..........................1 Takasumi Tanabe, Eiichi Kuramochi, Akihiko Shinya, and Masaya Notomi 2. Pillar Microcavities for Single-Photon Generation........................................................ 53 Charles Santori, David Fattal, Jelena Vucˇ kovi´c, Matthew Pelton, Glenn S. Solomon, Edo Waks, David Press, and Yoshihisa Yamamoto 3. Crystalline Whispering Gallery Mode Resonators in Optics and Photonics.......................................................................................................... 133 Lute Maleki, Vladimir S. Ilchenko, Anatoliy A. Savchenkov, and Andrey B. Matsko 4. Microresonator-Based Devices on a Silicon Chip: Novel Shaped Cavities and Resonance Coherent Interference................................. 211 Andrew W. Poon, Xianshu Luo, Linjie Zhou, Chao Li, Jonathan Y. Lee, Fang Xu, Hui Chen, and Nick K. Hon 5. Electro-Optic Polymer Ring Resonators for Millimeter-Wave Modulation and Optical Signal Processing................................................................... 265 William H. Steier, Byoung-Joon Seo, Bart Bortnik, Hidehisa Tazawa, Yu-Chueh Hung, Seongku Kim, and Harold R. Fetterman 6. Organic Micro-Lasers: A New Avenue onto Wave Chaos Physics............................ 317 Melanie Lebental, Eugene Bogomolny, and Joseph Zyss 7. Optical Microfiber Loop and Coil Resonators.............................................................. 355 Misha Sumetsky 8. Optofluidic Ring Resonator Biological and Chemical Sensors................................. 385 Xudong Fan, Ian M. White, Siyka I. Shopova, Hongying Zhu, Jonathan D. Suter, Yuze Sun, and Gilmo Yang 9. A Non-Electronic Wireless Receiver with Immunity to Damage by Electromagnetic Pulses...................................................................................................... 421 Bahram Jalali, Ali Ayazi, Rick Hsu, Andrew Yick, William H. Steier, and Gary Betts 10. Cavity Opto-Mechanics..................................................................................................... 447 Tobias Jan Kippenberg and Kerry J. Vahala ix © 2009 by Taylor & Francis Group, LLC

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x

Contents

11. Optical Frequency Comb Generation in Monolithic Microresonators....................483 Olivier Arcizet, Albert Schliesser, Pascal Del’Haye, Ronald Holzwarth, and Tobias Jan Kippenberg 12. Bit Rate Limitations in Single and Coupled Microresonators................................... 507 Jacob Khurgin 13. Linear and Nonlinear Localization of Light in Optical Slow-Wave Structures.......................................................................................... 529 Shayan Mookherjea


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Preface The race for compactness and scalability of optical and photonic devices calls for the development of efficient micro- and nano-optical elements. The optical resonators are important here because they can be used in the optical signal processing systems as modulators, filters, delay lines, switches, sensors, and so on. The number of the different types of resonators increases every day, and the basic research of their properties is gradually and steadily substituted with applied research. Such practical issues as efficient packaging and robust coupling, as well as integration of the resonators into complex optical systems, become especially important when one tries to bridge the gap between the fundamental research and practical implementation. There are many scientific books and reviews discussing the properties of optical microresonators, and I believe that at this stage it is important to have a collection reviewing the basic directions in the development of the practical applications of the microresonators, which is my goal with this book. Though it is practically impossible to cover the whole field with several contributions, I hope that this collection will provide readers with the flavor of the applied studies in the field and will convince them that systems containing microresonators will soon become as common and widespread as electronic devices containing quartz oscillators. I also hope that this book will attract the attention of a general audience dealing with R&D in broadly defined physics/electrical engineering areas to the fascinating world of the microresonators. The chapters are written by brilliant scientists and engineers working in the field and can be understood by any graduate student in the field. Traditional mirrored optical resonators are utilized in all branches of optics where, for example, multiple recirculation of optical power is required to maintain laser oscillation, to increase the effective path length in spectroscopic or resolution in interferometric measurements, and to enhance wave mixing interactions. Crucial properties of the resonators, such as high quality (Q) factor and finesse, can be achieved with the highest reflectivity and low-loss mirrors. Despite their versatility, these resonators have remained fairly complex devices. They are prone to vibration instabilities because of relatively low-frequency mechanical resonances. Stability and small modal volume are of great importance for practical applications; however, miniaturization of conventional Fabry–Perot resonators is either complicated and expensive, or yields rather low Q-factors. This book contains several reports on the progress in the rapidly growing field of monolithic micro- and nano-resonators. Such resonators do not have localized mirrors as such. The light is confined inside these resonators due to their morphology. The monolithic resonators are characterized by the unique combination of properties unreachable in other resonator structures. They have tiny volumes along with huge finesse and Q factors. The modal spectrum of the resonators can be efficiently engineered. These properties make the resonators extremely efficient in multiple applications. The first chapter in this book, authored by Takasumi Tanabe et al. (NTT Corporation, Japan), is devoted to photonic crystal-based resonators (nanocavities). Among various microresonators, photonic crystal nanocavities have the smallest mode volume (V) and nearly the highest Q/V value. High Q/V devices are attracting considerable attention because they enable multiple quantum and nonlinear optics applications. Recent progress on these ultrahigh-Q photonic crystal nanocavities is also discussed in the first chapter. xi © 2009 by Taylor & Francis Group, LLC

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xii

Preface

Various designs of photonic crystal nanocavities, fabrication and the characterization technologies are reviewed. In addition, various applications like light buffering, slow light propagation, all-optical switching, and bistable memory operation are discussed by the authors. The second chapter, authored by Charles Santori et al. (Hewlett-Packard Laboratories, USA), is devoted to the discussion of applications of a particular type of distributed feedback microresonators called “pillar microcavities”. These microcavities are well suited for efficient coupling of dipole emitters to a single mode in free space and thus are suitable for generation of photons on demand. The design, fabrication and characterization of singlephoton devices based on single InAs quantum dots coupled to pillar microcavities formed from AlAs/GaAs distributed-Bragg-reflector mirrors are described in this chapter. Several applications including quantum cryptography and entanglement formation through twophoton interference are presented. Future applications that could be developed as the devices improve are also discussed. Chapters 3 through 11 deal with the resonators in which the closed trajectories of light are supported by a variety of total internal reflection in curved and polygonal transparent dielectric structures. The circular optical modes in such resonators, frequently called whispering gallery modes (WGMs), can be understood as closed circular beams supported by total internal reflections from the boundaries of the resonators. High values of Q-factor can be achieved in WGMs of very small volume, in certain cases as small as cubic wavelength, with appropriately designed dielectric interface and with use of transparent materials. Applications of the microresonators made of various materials, including silicon, fused silica, fluorite, lithium niobate, and polymers are discussed in these chapters. These resonators have cylindrical, spherical, spheroidal, toroidal, ring, and other shapes and topologies. When the reflecting boundary has high index contrast, and radius of curvature exceeds several wavelengths, the radiative losses, similar to bending losses of a waveguide with high refractive index contrast, become very small, and the Q factor of the resonators becomes limited only by and material attenuation scattering caused by geometrical imperfections (e.g. surface roughness). Fabrication of the open dielectric resonators can be simple and inexpensive, and they lend themselves to integration. The unique combination of very high Q (as high as 1011) and very small volume has attracted interest in the applications of the resonators in fundamental science and engineering. Small size also results in excellent mechanical stability and easy control of the resonator parameters. The authors describe applications of the resonators for filtering and modulating light, for detecting chemical and biological substances. Various lasers and oscillators based on the resonators are also discussed. Namely, Lute Maleki et al. (OEwaves Inc., USA) discuss application of crystalline WGM resonators in filtering and laser stabilization in Chapter 3. Applications of polygonalshaped microdisk resonators are studied in Chapter 4, authored by Andrew W. Poon et al. (The Hong Kong University of Science and Technology, People’s Republic of China). Applications of electro-optic polymer ring resonators for millimeter-wave modulation and optical signal processing are reviewed by William H. Steier et al. (University of Southern California, USA) in Chapter 5. Chapter 6, authored by Melanie Lebental et al. (Ecole Normale Superieure de Cachan, France), is devoted to the discussion of properties of organic micro-lasers. Practical applications of optical microfiber loop and coil resonators are described in Chapter 7 by Misha Sumetsky (OFS Laboratories, USA). Chapter 8, authored by Xudong Fan (University of Missouri, Columbia, USA), deals with optofluidic ring resonator biological and chemical sensors. An application of crystalline microresonators for fabrication of a non-electronic wireless receiver with immunity to damage by


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Preface

xiii

electromagnetic pulses is introduced in Chapter 9 by Bahram Jalali et al. (the University of California, Los Angeles, USA). Properties and applications of cavity enhanced optomechanics are studied in Chapter 10, authored by Tobias Kippenberg (Max Planck Institut für Quantenoptik, Garching, Germany) and Kerry Vahala (California Institute of Technology, USA). Generation of optical frequency combs in optical microresonators and applications of the combs are described in Chapter 11, authored by Oliver Arcizet et al. (Max Planck Institut für Quantenoptik, Garching, Germany). The last two chapters are devoted to the theoretical discussion of the properties of long chains of coupled microresonators. Though the fabrication of such chains is still problematic because of technological immaturity, the theoretical studies shed light on the problems and phenomena one needs to expect once the chain fabrication becomes feasible. Chapter 12, written by Jacob Khurgin (Johns Hopkins University, USA), is focused on the two most important factors limiting the performance of linear and nonlinear optical devices based on coupled resonator structures. These factors are, respectively, dispersion of loss and dispersion of group velocity. Chapter 13, authored by Shayan Mookherjea (University of California, San Diego, USA), deals with linear and nonlinear localization of light in chains of nearly identical resonators. I hope that this book will help accelerate the already rapid pace of the research and developments in the exciting field of the applications of optical microresonators. I would like to thank the authors for their contributions making this book a success. This book would not have been possible without the assistance of my colleagues Lute Maleki, Anatoliy Savchenkov, and Vladimir Ilchenko. I am also thankful to Allyson Beatrice for her assistance. Andrey B. Matsko


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Editor Andrey B. Matsko (MS, 1994 and PhD, 1996, Moscow State University, Russia) has been a principal engineer with OEwaves Inc. since 2007. He joined the company after six year employment as a senior/principal member of technical staff at Jet Propulsion Laboratory (JPL) and four year post-doctoral training at the Department of Physics, Texas A&M University. He has numerous publications in the field and holds several patents. His current research interests include, but are not restricted to, applications of whispering gallery mode resonators in quantum and nonlinear optics, and photonics; coherence effects in resonant media; and quantum theory of measurements. He is a member of the Optical Society of America and a member of the Program Committee of Photonics West: Laser Resonators and Beam Control Conference. He received JPL’s Lew Allen Award for excellence in 2005.

xv © 2009 by Taylor & Francis Group, LLC

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Contributors Olivier Arcizet Max Planck Institut für Quantenoptik Garching, Germany

David Fattal Hewlett-Packard Laboratories Palo Alto, California

Ali Ayazi Electrical Engineering University of California Los Angeles, California

Harold R. Fetterman Department of Electrical Engineering University of California Los Angeles Los Angeles, California

Gary Betts Photonic Systems Inc. Burlington, Massachusetts Eugene Bogomolny Laboratoire de Physique Théorique et Modéles Statistiques Université Paris-Sud Orsay, France Bart Bortnik Department of Electrical Engineering University of California Los Angeles Los Angeles, California Hui Chen Photonic Device Laboratory Department of Electronic and Computer Engineering The Hong Kong University of Science and Technology Clear Water Bay, Hong Kong, People’s Republic of China

Ronald Holzwarth Max Planck Institut für Quantenoptik Garching, Germany Nick K. Hon Photonic Device Laboratory Department of Electronic and Computer Engineering The Hong Kong University of Science and Technology Clear Water Bay, Hong Kong, People’s Republic of China Rick C. J. Hsu Broadcom Corporation Irvine, California Yu-Chueh Hung Department of Electrical Engineering University of California Los Angeles Los Angeles, California

Pascal Del’Haye Max Planck Institut für Quantenoptik Garching, Germany

Vladimir S. Ilchenko OEwaves Inc. Pasadena, California

Xudong Fan Biological Engineering Department University of Missouri Columbia, Missouri

Bahram Jalali Electrical Engineering University of California Los Angeles, California

xvii © 2009 by Taylor & Francis Group, LLC

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Jacob Khurgin Department of Electrical and Computer Engineering Johns Hopkins University Baltimore, Maryland Seongku Kim Department of Electrical Engineering University of California Los Angeles Los Angeles, California Tobias Jan Kippenberg Max Planck Institut für Quantenoptik Garching, Germany Eiichi Kuramochi Optical Science Laboratory NTT Basic Research Laboratories NTT Corporation Atsugi, Japan Melanie Lebental Laboratoire de Photonique Quantique et Moléculaire Ecole Normale Supérieure of Cachan Cachan, France and Laboratoire de Physique Théorique et modéles statistiques Université Paris-Sud Orsay, France Jonathan Y. Lee Photonic Device Laboratory Department of Electronic and Computer Engineering The Hong Kong University of Science and Technology Clear Water Bay, Hong Kong, People’s Republic of China Chao Li Photonic Device Laboratory Department of Electronic and Computer Engineering The Hong Kong University of Science and Technology Clear Water Bay, Hong Kong, People’s Republic of China

Contributors

Xianshu Luo Photonic Device Laboratory Department of Electronic and Computer Engineering The Hong Kong University of Science and Technology Clear Water Bay, Hong Kong, People’s Republic of China Lute Maleki OEwaves Inc. Pasadena, California Andrey B. Matsko OEwaves Inc. Pasadena, California Shayan Mookherjea Department of Electrical Engineering University of California San Diego, California Masaya Notomi Optical Science Laboratory NTT Basic Research Laboratories NTT Corporation Atsugi, Japan Matthew Pelton Center for Nanoscale Materials Argonne National Laboratory Argonne, Illinois Andrew W. Poon Photonic Device Laboratory Department of Electronic and Computer Engineering The Hong Kong University of Science and Technology Clear Water Bay, Hong Kong, People’s Republic of China David Press Edward L. Ginzton Laboratory Stanford University Stanford, California


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Contributors

Charles Santori Hewlett-Packard Laboratories Palo Alto, California Anatoliy A. Savchenkov OEwaves Inc. Pasadena, California

Jonathan D. Suter Biological Engineering Department University of Missouri Columbia, Missouri

Albert Schliesser Max Planck Institut für Quantenoptik Garching, Germany

Takasumi Tanabe Optical Science Laboratory NTT Basic Research Laboratories NTT Corporation Atsugi, Japan

Byoung-Joon Seo Department of Electrical Engineering University of California Los Angeles Los Angeles, California

Hidehisa Tazawa Department of Electrical Engineering University of Southern California Los Angeles, California

Akihiko Shinya Optical Science Laboratory NTT Basic Research Laboratories NTT Corporation Atsugi, Japan

Kerry J. Vahala California Institute of Technology Pasadena, California

Siyka I. Shopova Biological Engineering Department University of Missouri Columbia, Missouri Glenn S. Solomon Joint Quantum Institute National Institute of Standards and Technology and University of Maryland Gaithersburg, Maryland William H. Steier Department of Electrical Engineering University of Southern California Los Angeles, California Misha Sumetsky OFS Laboratories Somerset, New Jersey Yuze Sun Biological Engineering Department University of Missouri Columbia, Missouri

Jelena Vucˇ kovic’ Edward L. Ginzton Laboratory Stanford University Stanford, California Edo Waks Institute for Research in Electronics and Applied Physics University of Maryland College Park, Maryland Ian M. White Biological Engineering Department University of Missouri Columbia, Missouri Fang Xu Photonic Device Laboratory Department of Electronic and Computer Engineering The Hong Kong University of Science and Technology Clear Water Bay, Hong Kong, People’s Republic of China


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xx

Yoshihisa Yamamoto Edward L. Ginzton Laboratory Stanford University Stanford, California

Gilmo Yang Biological Engineering Department University of Missouri Columbia, Missouri

Andrew Yick Department of Electrical Engineering University of Southern California Los Angeles, California

Contributors

Linjie Zhou Photonic Device Laboratory Department of Electronic and Computer Engineering The Hong Kong University of Science and Technology Clear Water Bay, Hong Kong, People’s Republic of China Hongying Zhu Biological Engineering Department University of Missouri Columbia, Missouri Joseph Zyss Laboratoire de Photonique Quantique et Moléculaire Ecole Normale Supérieure of Cachan Cachan, France


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1 Ultrahigh-Q Photonic Crystal Nanocavities and Their Applications Takasumi Tanabe, Eiichi Kuramochi, Akihiko Shinya, and Masaya Notomi NTT Basic Research Laboratories NTT Corporation

Contents 1.1 Introduction.............................................................................................................................2 1.2 Small Optical Cavities Fabricated on 2D Photonic Crystal Slabs....................................3 1.2.1 2D and 3D Photonic Crystals....................................................................................3 1.2.2 Ultrasmall Cavity: Photonic Crystal Nanocavity..................................................6 1.3 Designing High-Q Photonic Crystal Nanocavities............................................................7 1.3.1 Design of High-Q Photonic Crystal Nanocavity....................................................7 1.3.2 Waveguide-Coupled High-Q Photonic Crystal Nanocavity................................8 1.3.3 Various Types of High-Q Photonic Crystal Cavities.............................................9 1.3.3.1 Line Defect Cavities with Modulated End-Holes....................................9 1.3.3.2 Point Defect Hexapole Cavity with Rotational Symmetry Confinement............................................................................................... 12 1.3.3.3 Width-Modulated Line Defect Cavity with Mode-Gap Confinement............................................................................................... 16 1.3.3.4 Other Photonic Crystal Nanocavities..................................................... 17 1.3.4 Discussion of Structural Error and Q.................................................................... 19 1.3.5 Fabrication of Photonic Crystal Slabs.................................................................... 20 1.4 Characterization of Ultrahigh-Q Photonic Crystal Nanocavities.................................. 20 1.4.1 Spectral Domain Measurement.............................................................................. 20 1.4.1.1 Spectrum Measurement with Frequency Tunable Laser.............................................................................................. 21 1.4.1.2 Spectrum Measurement using Electro-Optic Frequency Shifter....................................................................................... 21 1.4.2 Time Domain Measurement................................................................................... 23 1.4.3 Technical Issues Related to Obtaining Accurate Q.............................................. 25 1.5 Applications of High-Q Photonic Crystal Nanocavities................................................. 26 1.5.1 Caging Light and Slow Light.................................................................................. 26 1.5.1.1 Caging Light Using Ultrahigh-Q Photonic Crystal Nanocavity.................................................................................... 26 1.5.1.2 Slow Light with Photonic Crystal Nanocavity...................................... 26 1.5.2 Compact Optical Add-Drop Filter..........................................................................30 1.5.3 All-Optical Switching.............................................................................................. 31

1 © 2009 by Taylor & Francis Group, LLC

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Practical Applications of Microresonators in Optics and Photonics

1.5.3.1 Switching by Thermo-Optic Effect.......................................................... 32 1.5.3.2 Switching by Carrier Plasma Dispersion Effect....................................34 1.5.3.3 Numerical Study of Carrier Dynamics in Silicon Photonic Crystal......................................................................................... 35 1.5.3.4 5-GHz Return-to-Zero Pulse Train Modulation.................................... 38 1.5.3.5 Accelerating the Speed of All-Optical Switches using Ion-Implantation Technology................................................................... 39 1.5.4 Ultra-Low Power Bistable Memory........................................................................ 41 1.5.5 Optical Logic-On Chip............................................................................................. 45 1.5.5.1 Optical Flip-Flop........................................................................................ 45 1.5.5.2 Pulse Retiming Circuit.............................................................................. 47 1.6 Summary................................................................................................................................ 48 References........................................................................................................................................ 48

1.1 Introduction Light is fast and thus can carry large amounts of data in a very short time, which makes photonic technology a promising communication tool. In fact, photonic technologies, such as optical fiber, that support a high transmittance speed1 are becoming more important in our lives. However, these technologies have had limited practical applications in data transmission, and signal processing has yet to find a commercial use. Although all-optical signal processing has been widely studied for several decades,2 it has been difficult to employ in practical systems, often because the required operating energy was too large.3,4 This is due to the fundamental nature of light. In other words, light is fast but difficult to store or confine in a small space. This makes photonic approaches difficult to handle. In contrast, there is a growing demand for a practical all-optical signal processor because today the network system bandwidth is often limited by the speed of the electronics used at network nodes. Optical nonlinearities, which can change such material characteristics as refractive index or absorption, are key phenomena in terms of achieving optical signal processing.5–7 But their coefficients are usually small. As a result, a high input power is required, which makes the device impractical. However, the input power can be significantly reduced if we can achieve strong light confinement. Photonic crystals8,9 are attracting considerable attention because of their strong light confinement and small structure. It has already been shown that photonic crystals can confine light in a very small space if we construct ultrahigh-Q (quality factor) photonic crystal nanocavities.10,11 Since the photon density in these small cavities is extremely high, various optical nonlinearities occur at a small input energy, which enables the fabrication of practical all-optical switches12 and logic gates.13 Indeed all-optical switching at an input energy of less than 100 fJ has been demonstrated using a two-dimensional (2D) silicon photonic crystal nanocavity.12 In this chapter, we describe recent progress on 2D photonic crystal nanocavities, and introduce some applications. These applications include add-drop filters for wavelength division multiplexing, optical buffers, slow light, all-optical switching, bistable memory operation, and logic gates such as flip-flop and pulse retiming circuits.


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Ultrahigh-Q Photonic Crystal Nanocavities and Their Applications

3

1.2 Small Optical Cavities Fabricated on 2D Photonic Crystal Slabs 1.2.1 2D and 3D Photonic Crystals Artificial dielectric structures with periodically modulated refractive indexes at a structural dimension close to an optical wavelength can alter the density of photonic states, thus allowing the creation of unique photonic band diagrams in terms of frequency and wavevector.14 For those with 2D or 3D structures whose refractive index modulation contrast is relatively large (i.e. between air (n = 1) and semiconductors such as silicon (n = 3.4), GaAs, or InP) are called photonic crystals,15 by analogy with the electrical property of natural crystals. Photonic crystals enable us to control the spontaneous emission or propagation of light.8 For instance, light rapidly decays exponentially, as evanescent wave, when the photon energy (light frequency) is within the photonic forbidden band. In other words, light cannot penetrate a photonic crystal, which enables the fabrication of perfect mirrors. If we surround transparent material with photonic crystals, light cannot escape towards the outside, and this allows us to confine light securely in a tiny space. From this point of view, 3D photonic crystals are ideal because light propagation can be altered or prohibited in all dimensions. However, such crystals are difficult to fabricate, because complex woodpile structure16 or highly sophisticated novel 3D material processing17 is required. Fortunately, 2D semiconductor photonic crystals are relatively straightforward to fabricate, because we can utilize mature planar semiconductor processing technologies. Although 3D photonic crystals have been considered essential in terms of taking full advantage of the various unique properties of photonic crystals, recent studies have revealed that a 2D structure is sufficient for various applications such as optical switching12 or even for spontaneous emission control18 if they are designed carefully. Low-loss waveguides and ultrahigh-Q nanocavity fabrication play important roles with respect to these applications. Figure 1.1 shows an example of 2D photonic crystals fabricated on a silicon slab, which was designed for telecom wavelengths. It should be noted that silicon is transparent at telecom wavelengths. The photonic crystals consist of hexagonal arrays with air rods. The calculated energy band diagram for a hexagonal photonic crystal is shown in Figure 1.2. The graph shows a clear forbidden band where no light propagation is allowed. Indeed, the experiment in Figure 1.3 shows low optical transmittance in the Γ–M and Γ–K directions for wavelengths of several hundred nanometers, which is clear evidence of the presence of a photonic band gap. A low-loss waveguide is essential if we are to use photonic crystals as a platform for on-chip photonic integration. For photonic crystals with an air-rod array with a hexagonal lattice, one row with no holes can act as an optical waveguide, because light can propagate along this line defect. Figure 1.4a shows an example illustration of a photonic crystal line-defect waveguide fabricated on a silicon-on-insulator wafer, and Figure 1.4b shows an electron scanning microscope image of a fabricated air-bridge type photonic crystal waveguide. Figure 1.4c shows the band diagram for these structures. It should be noted that the vertical confinement of a 2D photonic crystal is achieved by total reflection at the slab surface. The critical angle of the total reflection is given by Snell’s law. Starting from Snell’s law, the condition for a 1D case is described as:

ω=

c k n

(1.1)


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Г–K Г–M Si SiO2

Figure 1.1 Scanning electron microscope image of a 2D photonic crystal fabricated on a silicon slab. Hexagonal arrays of air holes were fabricated by using electron beam lithography and dry etching. The diameter of the air hole is 200 nm and the lattice constant is 400 nm. The silicon slab is about 200 nm thick.

Frequency (λ/a)

0.4

0.3

PBG (0.255–0.305)

0.2

0.1

0

LL(air)

M

Г

K

G M

Г–K

Г–M

Г

K

1st BZ Figure 1.2 Energy band diagram of 2D photonic crystals with hexagonal air holes. LL: light line. PBG: photonic band gap. BZ: Brillouin zone.

where c is the light velocity and n is the refractive index of the cladding. Equation 1.1 is shown as a line, known as a light line, in a band diagram. Photons that are beyond this line satisfy the total reflection condition; hence they are vertically confined. The photonic crystal waveguide is theoretically lossless for a frequency component that is below the light line. According to Equation 1.1, the slope of the light line becomes steeper as the refractive


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Transmittance (dB)


–10

Г–K

Г–M PBG

–20 –30

1200

1300

1400 1500 Wavelength (nm)

1600

1700

Figure 1.3 Transmittance spectrum of 2D photonic crystal in Γ–M and Γ–K directions.

(a)

Photonic crystal slab

Line defect core (row w/o holes)

(c)

Light line(air) Light line (SiO2)

0.35

Even

(b)

Normalized frequency

0.3

Odd PBG

0.25 Even

0.2 0.15 0.1

Even

0.05 0

0

0.1

0.2

kx

0.3

0.4

0.5

Figure 1.4 (a) Schematic image of a line defect silicon photonic crystal waveguide fabricated on a silicon-on-insulator wafer. (b) Scanning electron microscope image of a fabricated air-bridge type silicon photonic crystal waveguide. (c) Band diagram of a line defect photonic crystal waveguide.

index contrast between the slab and cladding increases. Indeed, Figure 1.4c shows light lines for air cladding (n ~ 1) and SiO2 cladding (n ~ 2.4), where it can be seen that a larger light component will be confined in the air cladding case. Obviously, a steeper light line is preferred because it yields greater tolerance as regards the structural design. For this reason, an air-bridge structure, where the photonic crystal slab is sandwiched by air, is often preferred to a SiO2 cladding structure. The lowest reported propagation loss for an air-bridge photonic crystal waveguide is 2 dB/cm,19,20 which is sufficiently small if we take the total size of the device into account. Since on-chip photonic devices will be much smaller than a centimeter in size, the total loss is less than 1 dB.


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1.2.2 Ultrasmall Cavity: Photonic Crystal Nanocavity Let us provide a simplified discussion to consider the smallest size optical cavity. Figure 1.5 is a schematic illustration of 1D Fabry–Pérot optical cavities of different lengths. The length of a cavity is multiples of a half wavelength, because it must satisfy the standing wave condition. Since, the wavelength of the light in a material with a refractive index of n is written as λ/n, the cavity length is given as N(λ/2n), where N is a positive integer. Thus the shortest length possible for a 1D optical cavity is λ/2n, i.e. N = 1, as shown in Figure 1.5c. Hence, the smallest volume of an optical cavity should be close to (λ/2n)3. It should be noted that this discussion is just a simplified one, and the value is not accurately the theoretical limit of the small modal volume. However, this simplified picture helps us to understand intuitively the smallest size of an optical cavity. There are various approaches that can be used to achieve small optical cavities. These include microrings,21 microdiscs,22 micro-pillars,23 spherical,24 and toroidal25 cavities. However, in terms of size, photonic crystal nanocavities can achieve the smallest optical mode volumes. In fact, a cavity with a size of just 1.18(λ/2n)3 has recently been reported26 using a point shifted cavity27 on 2D photonic crystals. This value is close to the above discussed volume. Other types fabricated on 2D photonic crystals also have small volumes. For instance the mode volume is 1.6 (λ/n)3 for a hexapole cavity and 1.1(λ/n)3 for a waveguide width modulated cavity, both of which are very close to an optical wavelength in size. In addition to the small size, photonic crystal nanocavities are suitable for the integration on a chip. Microcavities such as microdiscs, micro-pillars, spherical, and toroidal cavities often couples light directly into space or into an optical fiber, which makes the integration

(a)

x

0

N·

(c)

(b)

x

0

λ n

λ n

x

0

λ 2n

Figure 1.5 (a) Schematic illustration of an optical cavity with a length of Nλ/n. (b) Optical cavity with a length of λ/n. (c) Smallest optical cavity with a length of λ/(2n).


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7

relatively difficult. On the other hand, photonic crystal nanocavities can couple light through photonic crystal waveguides, which are fabricated on the same chip. As a result, a large number of cavities can be connected in tandem or in parallel though photonic crystal waveguides. The surface of the 2D photonic crystal is flat, which makes the mechanical strength high. As a result, the optical coupling strength between the cavities and the waveguides is stable, once they are determined at the designing stage. Small size, ultrahigh-Q and the ability of making integrated photonic circuit makes photonic crystal nanocavities attractive for various studies.

1.3 Designing High-Q Photonic Crystal Nanocavities 1.3.1 Design of High-Q Photonic Crystal Nanocavity This section offers a quick review of the method used for designing a high-Q photonic crystal nanocavity system on a 2D photonic crystal slab. Q is determined by Q = E/Pout, where E is the energy stored in the cavity and Pout is the energy flux toward the outside of the cavity. Therefore, it is essential to reduce the optical loss (i.e. reduce the Pout ) of the cavity to obtain a high Q. Here, let us consider 2D photonic crystals with no structural errors. The theoretical Q of the nanocavity can be obtained by 3D finite difference time domain (FDTD) calculations. Since the nanocavity is surrounded by ideal photonic crystals on the horizontal plane, which has a perfect photonic bandgap, the light cannot leak towards the outside as regards the in-plane direction. Therefore, the sole loss is the out-of-slab radiation. Because the vertical confinement is achieved by total reflection, it is essential to look for a condition where as far as possible the light in the nanocavity satisfies the total reflection condition. The idea of the light line provides a guideline for the design. To obtain stronger vertical confinement, it is essential to reduce the optical component that is above the light line. As discussed above, an air-bridge structure is preferred because the refractive index contrast between the semiconductor and the air is relatively large. To reduce the number of optical components above the light line, Srinivasan and Painter proposed a momentum-space design,28 where they considered the k-vector (the Fourier space of the real coordinate) distribution of the optical mode. To describe this briefly; if 2 we divide the wavevector k into two components k// and k⊥ , it can be given as k 2 = k// + k⊥2 , where k// and k⊥ are the wavevectors of the in-plane and out-of-plane directions. Since 2 the light line is given by Equation 1.1, k⊥2 = ( ω c ) describes the cone in three dimensions for the air-cladding slab. Therefore, the strategy for achieving low vertical loss, hence for achieving an ultrahigh-Q in a 2D photonic crystal nanocavity system, is to find a structure where the Fourier transform of the optical mode yields very few components inside the light cone. Examples of the Fourier space distribution of the optical mode of the point defect photonic crystal are shown in Figure 1.6. In terms of k-space design, the ideal optical mode profile is a sinc function, because it has a square shaped function in the k-space.29 However it is not possible to find a structure that can exhibit a perfect sinc shaped optical mode, because sinc function requires infinite endpoints to define. Akahane et al. used a more convenient strategy to obtain a high-Q mode, namely they used a Gaussian function as a figure of merit.30 Since the


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Light cone of air 1.0 0.1 0.01 E-3 E-4 Dipole mode Q = 3.0×102

Monopole mode Q = 1.6×104

Hexapole mode Q = 2.3×106

Figure 1.6 The k-space distributions of three different modes of a point defect photonic crystal nanocavity. The Q values calculated by 3D FDTD are shown in the panel.

Fourier function of a Gaussian curve is also Gaussian and its overlap with the light cone is relatively small, a Gaussian shape is a good practical shape for obtaining a high-Q. So far this strategy appears to work very well; in fact, a theoretical Q of about 2 × 107 has been obtained based on this strategy.29 1.3.2 Waveguide-Coupled High-Q Photonic Crystal Nanocavity If we are to develop on-chip all-optical signal processing by using photonic crystal nanocavity systems, it is essential that we connect various nanocavities in tandem or in parallel by using in-plane photonic crystal waveguides. It should be noted that the Q values of waveguide-coupled cavities are different from that of an isolated cavity that has no input/ output. The Q of an isolated cavity is called the unloaded Q. It is usually determined by the out-of-slab radiation loss, because the horizontal loss is negligible when the wavelength of the light is within the photonic bandgap. Hence the unloaded Q is written as,

Qunloaded = Qv

(1.2)

where Qv is the Q determined by the out-of-slab radiation loss. On the other hand, the Q of the cavity coupled to the waveguides is called the loaded Q, and its value depends on the out-of-slab radiation loss and the coupling between the waveguides. The loaded Q can be written as

- 1 - 1 Q - 1 loaded = Qv + Q h

(1.3)

where Qh is the Q value determined by the coupling strength with the waveguides. Figure 1.7 shows this schematically. Note that the unloaded Q is identical to Qv for an isolated cavity because Qh- 1 is null. By employing this equation and simple coupled mode theory,31,32 we can derive the relationship between Q and the transmittance. The transmittance T of a cavity/waveguide system is given as,33 2

2

Q   Q -1  T =  loaded  =  -1 h -1   Qh   Qv + Qh 

(1.4)


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9

Qv : vertical radiation Qh: coupling with WG

PhC WG

PhC WG PhC cavity

Figure 1.7 Schematic illustration of photonic crystal nanocavity coupled to photonic crystal waveguides.

For a given Qv, Equations 1.3 and 1.4 tell us that the loaded Q and the transmittance are in a tradeoff relationship. However, if we can design a high Qv, it should be possible to achieve a high transmittance and a high loaded Q simultaneously. Therefore it is extremely important to find designs that can exhibit a high Qv. 1.3.3 Various Types of High-Q Photonic Crystal Cavities 1.3.3.1 Line Defect Cavities with Modulated End-Holes Of the various designs of photonic crystal nanocavities, line defect cavities with modulated end holes have been widely studied.30,34,35 A schematic illustration of a shifted end-hole cavity is shown in Figure 1.8. To achieve a high Q, the position and the diameter of the end-holes are varied from those of the other holes. The position is slightly shifted towards the outside, and the hole diameter is smaller than that of the other holes. When the line defect consists of two missing holes, we call it an L2 cavity. When the line defect consists of x missing holes, it is called an Lx cavity. Figure 1.9a shows the k-space (in-plane wavevectors) distribution of an L2 cavity with shifted end holes. The component in the light cone is smaller for an end-hole shift of 0.199a than for a 0.064a shift. As a result, it exhibits a higher Q. The Q reaches its maximum value of 2.9 × 104 at a shift of ~ 0.2a as shown in Figure 1.9b. Further optimization is possible by fine-tuning the end-holes, for example, by changing their diameter. Q exhibited a value of 3.8 × 105 when the innermost hole diameter was changed to 0.4a.36 Various sophisticated fine-tuning techniques have been reported in order to achieve higher Q values.37 Figure 1.10 shows an example of the fabricated L3 and L4 silicon photonic crystal nanocavities coupled to input and output photonic crystal waveguides. In addition to the modulation of the cavity end holes (r-holes), the end holes at both the input and output waveguides (c-holes) are modulated to control the coupling between the cavity and the waveguides. When the cavities and waveguides are placed in a straight line, this is referred to as in-line coupling. On the other hand, the L4 cavity shown in Figure 1.11 is coupled with the waveguides at an angle of 60 degrees, which we call a shoulder coupled configuration. Although the fabricated in-line coupled and shoulder coupled Lx cavities exhibit fairly similar optical properties, shoulder coupling has certain advantages in the design stage. As regards the design, shoulder coupling is more straightforward and easier to understand. This becomes clear when we consider the field distribution of the resonance for a Lx nanocavity. The electrical field of the cavity mode decays smoothly at an


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Figure 1.8 Schematic illustration of a line defect photonic crystal nanocavity with shifted end-holes.

0.064a shift

0.199a shift

100 10–1 10–2

Light cone Qv ~ 2.7×103

(b) 105 Isolated Q

(a)

104

10–3 Qv ~ 2.9×104

10–4

103

0.1 0.2 0.3 End hole shift (× a)

Figure 1.9 (a) Fourier space distribution (k-space) of the spatial mode of the L2 cavity with different end-hole shifts. The light cone is indicated by the dotted circle. a is the lattice constant. (b) Isolated Q (Qv) of the L2 cavity with different end-hole shifts. Q reaches a maximum value of 2.9×104 at an end-hole shift of ~0.2a. The slab thickness is 0.5a, and the hole diameter is 0.55a.

angle of 60 degrees as shown in Figure 1.11, which makes it easier for the cavity field to overlap the waveguide ends when the waveguides are placed in a shoulder coupled direction. In addition, the optical field of the waveguide termination exhibits gradual decay in the 60-degree direction.38 As a result the coupling between the cavity and the waveguides always becomes smaller as the cavity/waveguide distance is increased for shoulder coupling. This is more complicated with in-line coupling. Figure 1.12 shows the calculated and measured spectra of the L3 cavity shown in Figure 1.10a and b. The resonance at 1550.36 nm yields a theoretical Qv of ~ 3 × 104 compared with 1547.68 nm and Q = 1.84 × 104 for the experiment. Since the transmittance for the fabricated sample was very small (T = ~ 2.1%) the measured Q should be almost identical to the Qv of the cavity. The mode volume calculated using 3D FDTD is 7.2 × 10 -2 µm3. Figure 1.13 is the measured spectrum of the L4 cavity shown in Figure 1.10b. It has two resonant modes, which we call mode C and mode S. The Q values are Qc = 1.15 × 104 for mode C and Q S = 2.3 × 104 for mode S. The corresponding photon lifetimes are τc = 9.3 ps and τS = 19.1 ps.


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11

(b)

(a) Г–M Г–K

c-hole

r-hole t-hole

Figure 1.10 Scanning electron microscope image of fabricated silicon photonic crystal nanocavities. The lattice constant for both samples is a = 420 nm with a hole diameter of d = 0.55a. (a) An L3 nanocavity is fabricated with c-, t-, and r-hole diameters of 0.25a, 0.57a, and 0.3a, respectively. The r-holes are shifted 40 nm in the t-hole direction. (b) Four point defect nanocavity with c-, t-, and r-hole diameters of 0.45a, 0.57a, and 0.3a, respectively. The r-hole shift is 60 nm in the t-hole direction.

Output waveguide

Mode profile

Input waveguide

Figure 1.11 A silicon photonic crystal L4 cavity coupled to an input/output waveguide in the Γ–M direction. The inset is the mode profile shown on a log scale.

Historically, a line-defect cavity with shifted end holes has significantly increased the Q of 2D photonic crystal nanocavity systems and given them various possible applications.30 Therefore, although there are now a number of other types of cavities that can exhibit a much higher Q, this type of cavity has been widely employed for various applications


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1×10–3 1×10–4 1×10–5 1×10–6 1×10–7 1×10–8 1×10–9 1×10–10

(b)

101

Transmittance (%)

Transmittance (a.u.)

(a)


100

10–1

1350

1400


1550

10–2 1546.5 1547.0 1547.5 1548.0 1548.5 1549.0 Wavelength (nm)

1600

Figure 1.12 (a) Calculated transmission spectrum of an L3 cavity, which has the design parameters of the cavity shown in Figure 1.10. (b) The measured transmittance spectrum of the fabricated silicon photonic crystal L3 cavity shown in Figure 1.10. (Reprinted with permission from Tanabe, T. et al., Appl. Phys. Lett., 87(15), 151112, 2005. © American Institute of Physics.)

Transmittance (%)

101

100

Mode C

Mode S

10–1

10–2

1530

1531


1568

1569

Figure 1.13 Transmittance spectrum of the silicon photonic crystal L4 cavity shown in Figure 1.10b. The Q values of mode C and mode S are Q C = 1.15 × 104 and Q S = 2.3 × 104, respectively. (Reprinted with permission from Tanabe, T. et al., Appl. Phys. Lett., 87(15), 151112, 2005. © American Institute of Physics.)

such as quantum electro-dynamics studies,39,40 ultrasmall wavelength add-drop filters,41 all-optical switching12 and optical bistable devices.13 1.3.3.2 Point Defect Hexapole Cavity with Rotational Symmetry Confinement A schematic image of a point defect cavity is shown in Figure 1.14. The cavity consists of one missing hole, and it exhibits various optical modes, namely dipole, quadrupole, and hexapole modes. The optical field patterns for these cavity modes are shown in Figure 1.15. In particular, the hexapole mode exhibits an ultrahigh Q.42 To obtain the highest Q, the parameters of the innermost holes are slightly different from those of the other holes. By setting the radius of the innermost holes at 0.23a (the radius of the other holes is 0.25a) and shifting the holes slightly towards the outside (ck = 1.18a), a theoretical Q of 3.3 × 106 has been obtained.43 Recently, it has been found that a high Q can be obtained without changing the hole diameter, which makes fabrication much easier.44 The theoretical Q with respect to the hole shift is shown in Figure 1.16. A maximum Q of 1.6 × 106 was obtained at ck = 1.26a. The mode volume is 1.18(λ/n)3 ≈ 0.11 µm3. © 2009 by Taylor & Francis Group, LLC

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13

Cavity

ck Figure 1.14 Schematic image of a point defect photonic crystal nanocavity. The innermost holes are shifted slightly towards the outside to optimize the Q of the hexapole mode. Hz

Energy of EM field

Dipole

Hexapole

Quadrupole

Figure 1.15 Magnetic and electro-magnetic fields of the dipole, hexapole and quadrupole modes seen in a point defect photonic crystal nanocavity.

The explanation for the ultrahigh Q obtained in the hexapole mode is different from that for line-defect type photonic crystal nanocavities. Figure 1.17 shows the mode distribution and phase property of the magnetic field. The beauty of its spatial rotational symmetry originates in the matching between the symmetry of the optical mode and the pattern of the holes in the hexagonal photonic crystal lattice. In addition, because the phase of the © 2009 by Taylor & Francis Group, LLC

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2.0×106

Unloaded Q

1.5×106

1.0×106

5.0×105

1.22

1.23

1.24 1.25 Shift ck (× a)

1.26

1.27

Figure 1.16 Calculated theoretical Q of the hexapole mode for different hole shifts ck, where ck is as shown in Figure 1.14. (Reprinted with permission from Tanabe, T. et al., Appl. Phys. Lett., 91(2), 021110, 2007. © American Institute of Physics.)

–

+ –

+ –

+

Figure 1.17 Profile of hexapole mode. The + and - represent the phase of the Hz magnetic field (z: perpendicular to the slab).

nearby Hz component is reversed (shifted by π), as indicated in Figure 1.17, the optical far field is cancelled out. As a result, the out-of-slab radiation is reduced by the destructive interference effect of the far-field pattern, which was initially discussed by Johnson et al.45 Hence, the hexapole mode can exhibit an ultrahigh Q. We call this light confinement mechanism “rotational symmetric confinement”. The rotational symmetries yield different interesting characteristics of the hexapole mode when it is coupled with waveguides. Figure 1.18a and b, respectively, show the in-line coupling and side coupling of the hexapole mode with the waveguides. Owing


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(a)

(b)

Min/20

0

Max/20

Min/20

0

Max/20

Figure 1.18 (a) Butt-coupled structure of separation=7 with Hz distribution. (b) Side-coupled structure of separation=7 with Hz distribution. (From Kim, G.-H. et al., Opt. Express, 12, 6624–6631, 2004. © Optical Society of America. With permission.)

(b) V (λ/n)3

(a)

1.24 1.22 1.20 1.18 QH

Q

107

QV

106

QT

105 104

Min/20

0

Max/20

5

7

9 11 Separation

13

15

Figure 1.19 (a) Shoulder-coupled structure of separation=7 with Hz distribution. (b) Modal volume (V) and quality factors (Qs) of hexapole mode. (From Kim, G.-H. et al., Opt. Express, 12, 6624–6631, 2004. © Optical Society of America. With permission.)

to the symmetries of the optical mode, the light hardly couples with the waveguides. This characteristic enables the cavity to be extremely well isolated from closely positioned waveguides or cavities, which offers the possibility of the dense packing of cavitywaveguide systems. To couple the hexapole mode with the waveguides, the waveguides are placed in a shoulder-coupled configuration as shown in Figure 1.19. This configuration enables efficient coupling of the optical mode with the waveguides. When the separation is 7, Qloaded is almost identical to Qh. According to Equation 1.4, the transmittance is nearly 100%. For a separation of 9, QT is 1.9 × 105 and the transmittance is 88%. In fact, the transmittance of the fabricated sample was almost 100% when the separation was 9.46


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(a)

(b)


Output WG Hex cavity

Input WG

4.8 pm

Г–K Г–M 1547.48

1547.52 Wavelength (nm)

1547.56

Figure 1.20 (a) Scanning electron microscope image of a hexapole cavity with photonic crystal waveguide ends fabricated on a silicon photonic crystal. WG: waveguide, Hex: hexapole. (b) Transmittance spectrum of the hexapole mode. The solid line is the fitted Lorenz curve. (Reprinted with permission from Tanabe, T. et al., Appl. Phys. Lett., 91(2), 021110, 2007. © American Institute of Physics.)

Figure 1.20 shows a scanning electron microscope image and the transmittance spectrum of the fabricated sample. The obtained spectrum exhibits an extremely high Q of 3.2 × 105, which is the highest experimental Q yet reported for photonic crystal nanocavities, except for those with mode-gap confinement.44 This cavity is a good candidate for achieving an ultrahigh Q in photonic crystals. 1.3.3.3 Width-Modulated Line Defect Cavity with Mode-Gap Confinement Figure 1.21 shows a schematic diagram of the width-modulated line defect cavities.10,47 The air holes are shifted slightly towards the outside to modulate the width of the line defect locally. It should be recalled that a line defect exhibits propagation modes as shown in Figure 1.4c. Since the propagation mode becomes flat as kx increases, some frequency are not allowed to propagate the line defect, which is known as a mode gap. The frequency of the propagation mode can be shifted by changing the width of the line defect. Namely, a line defect with a larger width has a smaller mode frequency. This yields a shifting of mode gap when the width of a line defect is changed. As shown in Figure 1.21, the widthmodulated cavities are composed of two line defects with different widths. The optical cavity consists of a line defect with a larger width and the termination is achieved by using line defects with smaller widths. Since the optical frequency mode of the line defect used for the cavity is lower than the line defects used for the termination, the light is localized at the cavity. In addition to the in-plane confinement, it is essential to reduce the out-ofslab radiation to obtain an ultrahigh Q. For this purpose, the original lossless propagation mode profile of the line defect must be kept in the k space. Therefore, the positional shifts of the holes at the cavity are kept small and they are gradually tapered along the line defect as shown in Figure 1.21. The typical value of the shift is a few nanometers. Other types of resonators that utilize the mode gap of photonic band gap waveguides have also been discussed by Inoshita et al.48 and Song et al.49 © 2009 by Taylor & Francis Group, LLC

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A2

C C C

C B B B C

C

B

B

A A B C

Line defect

C

C

A A B C

C B B B C

C C

C C

B B B

B A A B

Line defect

A1

B A A B

B B B

C C

C C W = W0.9

X

W = W0.98 Hole shift dA dB dC

A A

Line defect

A3

Z

A A

W = W0.9

Figure 1.21 Different designs of width-modulated line defect cavity (A1, A2, A3). The width of the baseline defect is 0.98 × a 3 (W0.98) and 0.90 × a 3 (W0.90). Holes marked with A, B, and C are shifted slightly towards the outside of dA, dB, and dC, respectively. Typical values for the A1 cavity are dA = 9 nm, dB = 6 nm, dC = 3 nm, a = 420 nm, r = 108 nm, t = 204 nm, where r is the air hole radius and t is the slab thickness. (Reprinted with permission from Kuramochi, E. et al., Appl. Phys. Lett., 88(4), 041112, 2006. © American Institute of Physics.)

Our first generation mode-gap confined silicon photonic crystal nanocavity is shown in Figure 1.22a. The obtained Q was not very high (Figure 1.22b and c), because the nanocavity was fabricated on SiO2 cladding and the parameters were not optimized.35 Figure 1.21 shows the latest design, which exhibits a theoretical Q of 1.2 × 108 and a mode volume of 1.51(λ/n)3 for the cavity A1.47 The calculated spatial mode profile is shown in Figure 1.23a. A scanning electron microscope image is shown in Figure 1.23b, where the circled region is the cavity. Although it is hard to distinguish, the holes at the cavity are slightly shifted towards the outside. The cavity exhibits an extremely high Q of 1.2 × 106, which is one of the highest Q values achieved by photonic crystal nanocavities.10 This Q value corresponds to a photon lifetime of about 1 ns, which opens the possibility for various applications such as optical buffering and cavity quantum-electro dynamics. Note that there is an approximately two orders discrepancy between the theoretical and experimental values. This is caused by the material absorption and various types of scattering, such as absorption at the surface or scattering due to fabrication error. Of these we speculate that the variation in the hole diameter plays an important role in determining the experimental Q. 1.3.3.4 Other Photonic Crystal Nanocavities It is difficult to provide a complete review of the various types of photonic crystal nanocavities, because there are a large number of different designs. Here, we introduce two cavities that are particularly important. One is the line defect type, which is called a heterostructure nanocavity. This was originally proposed by Song et al. in 2005.49 It constituted © 2009 by Taylor & Francis Group, LLC

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(a)

(b)

0.8 Transmittance

0.6 W

0.4 W

1 Nb = 2

0.6 0.4 0.2

0.6 W

0

1480

1490

(c) 0.15

Transmittance

Nb = 3 0.1

0.05

0

1480

Wavelength (nm)

1490

Figure 1.22 Resonant-tunneling filter using the mode-gap in the width-varied waveguides. The samples are fabricated on silicon-on-insulator photonic crystal slabs, where the undercladding is SiO2. (a) Structural design. (b) Measured transmittance spectrum around the resonant wavelength for a barrier width of N b = 2. Q = 408 and T = 86%. (c) Measured transmittance spectrum for N b = 3, Q = 1350 and T = 12%. (From Notomi, M. et al., Opt. Express, 12, 1551–1561, 2004. © Optical Society of America. With permission.)

(b)

Input WG

(c)


(a)

Output WG

1.3 pm

1555.470 1555.475 1555.480 1555.485 Wavelength (nm)

Figure 1.23 (a) Mode profile of width-modulated line defect cavity with mode-gap confinement shown in Figure 1.21 A1. (b) Scanning electron microscope image of the fabricated sample on silicon photonic crystal. The circle indicates the cavity region. (c) Transmittance spectrum of the cavity shown in (b).


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a breakthrough in terms of achieving an ultrahigh Q with a value of ~ 6 × 105. The Q for this cavity is now ~ 2 × 106, which is the highest experimental Q value yet reported for a photonic crystal nanocavity.11 It is a mode-gap confinement type cavity, where the lattice of the photonic crystal is changed to achieve the mode gap of the line defect. Another important piece of work is the point-shift cavity proposed by Zhang and Qiu,27 and recently fabricated by Nozaki et al.26 It has a modal volume of only ~ 1.18(λ/2n)3, which is the smallest value reported for any micro- or nanocavity, while maintaining a reasonably high Q of 2 × 104. Owing to the high Q, they achieved continuous-wave laser operation at room temperature at a very low power. It is also the smallest reported laser. As shown by both examples above, photonic crystal nanocavities can exhibit an extremely high Q and also an ultrasmall mode volume. The optimal design has not yet been determined theoretically, and therefore there is still the possibility of finding an improved design that exhibits a higher Q and a smaller modal volume. In addition, it is also important to find a design that is as tolerant as possible in terms of such structural variations as hole diameter variation, position variation or sidewall roughness, because nanofabrication cannot produce an ideal structure. 1.3.4 Discussion of Structural Error and Q The highest experimental Q reported in a photonic crystal nanocavity is about one order lower than the theoretical value. The cavity Q decreases if material absorption or optical scattering losses are present. To understand the mechanism behind the discrepancy between the theoretical and experimental Q values, 3D FDTD calculations have been performed that consider structure randomness.20,44 Randomness was added with a Gaussian distribution to the radius of the air holes. Figure 1.24 shows the Q values for three different types of cavities with respect to the standard deviation of the Gaussian randomness. The Q value is 1.2 × 106 for a width-modulated line defect cavity at a randomness of 2.3 nm and 3.1 × 105 for a hexapole cavity at a randomness of 3 nm. These Q values are not far from 107

Qv

106

105 Waveguide-width modulated cavity Hexapole mode cavity L4 line defect cavity 104

0

1

2

σ (nm)

3

4

5

Figure 1.24 Calculated Q values for three different types of cavities (waveguide-width modulated cavity, hexapole mode cavity, and L4 line-defect cavity), with respect to radius randomness.


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(a)

EB resist

(b)

(c)

Air (n = 1)

SOI (Si, n = 3.46) SiO2 (n = 1.48)

Air (n = 1)

Si substrate

Figure 1.25 Schematic diagram of the air-bridged silicon photonic crystal slab. (a) Photonic crystals are patterned on resist by electron beam lithography. (b) Photonic crystal is formed on silicon by dry etching. (c) The underlying SiO2 layer was removed by wet-etching.

those obtained experimentally. Hence, a standard deviation of 2–3 nm provides a good fit with the experimental results. Although such a small radius variation is very difficult to estimate with a scanning electron microscopic measurement, we believe that the value is not far from the actual fabrication accuracy. Therefore we speculate that the limitation of the experimental Q results from the randomness in the air-holes radius. Further detailed discussions can be found elsewhere.20 1.3.5 Fabrication of Photonic Crystal Slabs We will briefly review the fabrication of the silicon photonic crystal slab. A schematic diagram of the fabrication process is shown in Figure 1.25. First we start with an SOI wafer spin coated with ZEP resist. Then, photonic crystals are patterned by 100-keV electron beam lithography. After the resist has been developed, the silicon is selectively etched by using inductively coupled plasma and fluorinated gas. To fabricate the air-bridge structure, the underlying SiO2 layer was removed by selective wet etching using HF solution. Note that photonic crystals can also be fabricated using other materials such as GaAs or InP. Photonic crystal fabrication using other materials such as AlGaAs is detailed elsewhere.50

1.4 Characterization of Ultrahigh-Q Photonic Crystal Nanocavities 1.4.1 Spectral Domain Measurement The Q of a cavity is given by

Q=ω×

Ecav Pout

(1.5)

where ω is the resonant angular frequency of the cavity, Ecav is the energy stored in the c avity, and Pout is the output power (i.e. optical energy loss per unit time) from the cavity. We can derive a different expression from Equation 1.1, where Q is described in the spectral domain.

Q=

ω λ ≈ ∆ω ∆λ

(1.6)


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21

Here, ∆ω and ∆λ are the linewidth of the resonant spectrum in terms of frequency and wavelength, respectively. λ is the resonant wavelength of the cavity. Obviously, Equation 1.6 suggests that the cavity Q can be determined by measuring the transmittance linewidth of the cavity when the cavity is coupled with the waveguides. Indeed the Q values of photonic crystal nanocavities have been characterized in the spectral domain. However, it should be noted that spectral domain measurement is now not an easy task because the Q of photonic crystal nanocavities has increased rapidly, and therefore, ultrahigh wavelength resolution is needed to measure the ultra-narrow transmittance linewidth of a nanocavity accurately. For a telecom wavelength, the linewidth for a cavity with a Q of 1.2 × 106 is about 1.3 pm as shown in Figure 1.22. Therefore a sub-pm wavelength resolution is required. 1.4.1.1 Spectrum Measurement with Frequency Tunable Laser For a high-Q characterization measurement, a frequency tunable narrow-linewidth laser is often used because it generally provides better wavelength resolution than a spectrum analyzer composed of gratings. The transmittance at each wavelength is monitored with an optical power meter. Indeed the spectrum shown in Figure 1.22 was obtained by this method. Since the photon density in the cavity is extremely high owing to the large Q/V, the input power must be kept sufficiently low to prevent the cavity from exhibiting optical nonlinearity. Optical nonlinearity is observed at an input power of a few µW.10 1.4.1.2 Spectrum Measurement using Electro-Optic Frequency Shifter Commercially available wavelength swept tunable laser diodes have a typical frequency resolution of a few 100 MHz. In contrast, the Q of a photonic crystal nanocavity is rapidly increasing. Therefore, it is important to develop a different measurement technique that allows higher frequency resolution. Frequency sweeping via fiber-coupled acousto-optic frequency modulators supports high frequency resolution, but the maximum achievable tuning range is limited to a few 100 MHz, and this range is often insufficient to obtain the complete transmission spectrum. A promising approach is to use an electro-optic single sideband modulator fabricated from lithium niobate as a frequency sweeper.51 By applying radio-frequency waves to the device, we can shift the frequency of the laser light according to the frequency of the radio wave. As a result, the frequency resolution with this measurement is only limited by the linewidth of the input laser light, which is typically 100 kHz or less. Figure 1.26 shows the output spectrum from a single-sideband modulator when a 10-GHz ratio-wave frequency is applied. We can obtain a suppression ratio of more than 20 dB between the 1st and –1st order single sidebands. By adjusting the DC biases applied to the single-sideband frequency modulator, we can selectively maximize the 1st or –1st order single sideband. The measured transmittance spectrum of the waveguide-modulated cavity in Figure 1.22b is shown in Figure 1.27. The data was fitted using a Lorenz curve, and we obtained a Q of 1.23 × 106 from the transmittance spectrum width. The laser frequency was set at 1564.85 nm, which is slightly detuned to a longer wavelength from the cavity resonance. The frequency of the laser light was shifted to a shorter wavelength with a singlesideband modulator. The plotting interval is 1 MHz, but it is possible to use a smaller value. The corresponding photon lifetime from the measured transmittance spectrum is 1.02 ns, which is in good agreement with the time domain measurement (1.09 ns) described in the


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100

Power (a.u.)

10–1

Frequency (GHz) 9 10 11 70 pm

> 20 dB

10–2 2nd order SSB

10–3

12

Absorption (linear)

8

–2nd order SSB f0

–4

10

1st order SSB

10–5

–1st order SSB

10–6 1564.6

1564.7

1564.8 1564.9 Wavelength (nm)

1565.0

1565.1

Figure 1.26 Optical output spectrum from single-sideband modulator at a radio frequency of 10 GHz. Inset is the spect roscope image of the absorption of acetylene (13C2H2) gas at room temperature obtained with the single-sideband modulator setup to demonstrate the accurate spectrum measurement. (Reproduced by kind permission of the IET from Tanabe, T. et al., Electron. Lett., 43, 187–188, 2007.)

2 10 0

10

1

Frequency (GHz) 0

–2

Q = 1.23 × 106 τ = 1.02 ns

10 –1

Transmittance (dB)

–1

0

10 –2

10 –1

1564.75 1564.76 1564.77 1564.78 1564.79 Wavelength (nm)

10 –2 1564.75

1564.76

1564.77 Wavelength (nm)

1564.78

1564.79

Figure 1.27 Measured spectrum obtained using first-order single-sideband light scan. Inset is the measured spectrum when a –1st-order single sideband is used for the scan. The dots are the measured plot and the solid line represents the fitted Lorenz function. (Reproduced by kind permission of the IET from Tanabe, T. et al., Electron. Lett., 43, 187–188, 2007.)


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23

next section. Measurement using a single-sideband modulator provides a high resolution ( ~ 100 kHz depending on the laser linewidth) and a wide measurement bandwidth (10 GHz or more), which will enable us to measure the Q value in the spectral domain even if the value continues to increase in the future. 1.4.2 Time Domain Measurement Although the current spectral domain measurement technique is capable of measuring the highest Q yet achieved in photonic crystals, time domain measurement is an alternative and sophisticated way of characterizing the Q of a photonic crystal nanocavity. By using Equation 1.5, the Q of a cavity can be derived as, Q = ωτ

(1.7)

where τ is the photon lifetime of the cavity. It is not very easy to measure τ when Q is small; but the τ measurement becomes more accurate as Q increases. Indeed a Q of 1.2 × 106 corresponds to a photon lifetime of ~ 1 ns at telecom wavelengths. The ring-down method is a direct way to characterize the τ and Q values of a cavity. First, the cavity is charged with an input CW light, and then the input is suddenly turned off. τ can be obtained by observing the decaying optical signal at the output waveguide. A schematic diagram of the measurement is shown in Figure 1.28. Note that the signal light must be kept sufficiently low to prevent the cavity exhibiting nonlinearity such as two-photon absorption or freecarrier absorption, which may modify the cavity Q. Therefore we employ time-correlated single photon counting for the ultrahigh-Q measurement, as this allows us to measure an extremely weak signal light with a time resolution of ~ 70 ps.10 The measured signal is shown in Figure 1.29. The decay is a smooth exponential curve, and the fitted decay is 1.01 ns, which is the photon lifetime of the cavity. This value corresponds to a Q of 1.2 × 106, which is in good agreement with the value obtained from a spectral-domain measurement. Time domain measurement provides a direct view of the photon trapping by the photonic crystal nanocavity. Another aspect of the time domain measurement can be clarified by performing a ring-down measurement on a side-coupled cavity.52 Figure 1.30a and b show a schematic diagram and the transmittance spectrum of a side-coupled cavity. The transmittance

Power

Input waveform

Monitored region

∞ exp(−t/τ)

Time Output waveform Figure 1.28 Schematic diagram of the ring-down measurement. The dotted line represents a rectangular input pulse and the solid line represents the output from a cavity. (From Tanabe, T. et al., Opt. Express, 15, 7826–7839, 2007. © Optical Society of America. With permission.)


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1

Power (a.u.)

τ = 1.01 ns

0.1

Q = 1.2×106 0.01

0

1000 2000 Time (ps)

3000

Figure 1.29 Output of a ring-down waveform from the waveguide-width modulated silicon photonic crystal nanocavity shown in Figure 1.22b. (a)

(b) Output WG

Path B Path A

10–1

Input

10–2

Output

100

τ = 220 ps

Power (a.u.)


?

Cavity

Input WG

(c)

100

τ = 210 ps

10–3 10–4

1577.6 1577.8 Wavelength (nm)

10–1

0

1 6 Time (ns)

7

Figure 1.30 (a) Schematic diagram of a side-coupled cavity with a photonic crystal waveguide. (b) Transmission spectrum of the side-coupled cavity. Sometimes it is difficult to obtain the Q from the spectral bandwidth because of the background Fabry–Pérot oscillation. (c) The output waveform of a side-coupled cavity when a rectangular pulse is applied.

spectrum exhibits a dip at the resonance as shown in Figure 1.30b. Side-coupled cavity configurations are useful for constructing multi-channel add-drop filters or photonic DRAMs.20,53 The transmitted waveform of a 6-ns square input pulse is shown in Figure 1.30c, along with the output pulse. The wavelength of the input light is adjusted to the cavity resonance; therefore, the light cannot propagate though the device in a steady state. However, the pulse transmits at the rising and falling edges of the square pulse. This can be explained as follows. The light propagates until the cavity is charged and forms interference. Interference occurs between the light that travels straight from the input waveguide toward the output waveguide (path A) and the output light from the sidecoupled nanocavity (path B). When the light is turned off, path A is immediately cut off. As a result the light cannot interfere and the path B light is observed as a ring-down waveform. The discharging signal is observed after the input has been turned off, which constitutes an intuitive demonstration of light trapping in a photonic crystal nanocavity.


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1.4.3 Technical Issues Related to Obtaining Accurate Q With spectral domain measurement, it is not always easy to obtain an accurate quality factor when the transmittance spectrum is very narrow. The laser frequency must be precisely calibrated in order to obtain an accurate wavelength. In addition, we have to check the wavelength resolution of the measurement system in advance if we are to discuss the value. This wavelength resolution can be checked by measuring a reference sample, such as a high-finesse Fabry–Pérot etalon.52 Moreover, the presence of the temperature fluctuation makes measurement difficult. The thermo-optic coefficient of silicon is 3.9 × 10-5 K-1. This results in a cavity wavelength shift of about 1 pm for a small temperature change of only 17 mK. Therefore it is essential to place the photonic crystal nanocavity sample in a temperature-stabilized environment, and complete the measurement as quickly as possible. In practice, the stability and reproducibility (hence the accuracy) of the measurement can be confirmed by repeating the acquisition multiple times and obtaining the average value while also obtaining the standard deviation. Most importantly, the Q values obtained with the spectral and temporal measurements should be identical. The measurement is reliable when the two results are the same. Figure 1.31 summarizes and compares the Q values for different cavities obtained with spectral and time domain measurements. Note that with the ring-down measurement for a lower Q cavity, it is possible to use a digital sampling oscilloscope that has a bandwidth of 28 GHz instead of using a time correlated single photon counter system, because we can input more light into the cavity without inducing any nonlinearity. Qtime and Qspec show good (a)

Qtime

106

105

Qtime meas. w/ TCSPC Qtime meas. w/ DSO

(b)

1.5

Qtime / Qspec

2.0

1.0 105

Qspec

106

Figure 1.31 Qspec is the Q value obtained from the transmittance spectrum bandwidth and Qtime is the Q value obtained from the decay of the ring-down waveform. (a) Square dots show the Qtime measured using time correlated single photon counting and round dots show the Qtime measured using a digital sampling oscilloscope. The dotted line indicates the ideal case. (b) Qtime/Qspec with respect to Qspec. (From Tanabe, T. et al., Opt. Express., 15, 7826–7839, 2007. © Optical Society of America. With permission.)


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agreement, which confirms the reliability of the measurements. Figure 1.31 also suggests that a detector speed of ~ 70 ps is sufficient to measure a cavity that has a Q of >105, but a faster response is required to obtain an accurate Q for those cavities with a value of

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