Optical Microresonators Theory Fabrication and Applications

Springer Series in

OPTICAL SCIENCES Founded by H.K.V. Lotsch Editor-in-Chief: W.T. Rhodes, Atlanta Editorial Board: A. Adibi, Atlanta T. Asakura, Sapporo T.W. Hänsch, Garching T. Kamiya, Tokyo F. Krausz, Garching B. Monemar, Linköping H. Venghaus, Berlin H. Weber, Berlin H. Weinfurter, Munich

138

Springer Series in

OPTICAL SCIENCES The Springer Series in Optical Sciences, under the leadership of Editor-in-Chief William T. Rhodes, Georgia Institute of Technology, USA, provides an expanding selection of research monographs in all major areas of optics: lasers and quantum optics, ultrafast phenomena, optical spectroscopy techniques, optoelectronics, quantum information, information optics, applied laser technology, industrial applications, and other topics of contemporary interest. With this broad coverage of topics, the series is of use to all research scientists and engineers who need up-to-date reference books. The editors encourage prospective authors to correspond with them in advance of submitting a manuscript. Submission of manuscripts should be made to the Editor-in-Chief or one of the Editors. See also www.springer.com/series/624

Editor-in-Chief William T. Rhodes Georgia Institute of Technology School of Electrical and Computer Engineering Atlanta, GA 30332-0250, USA E-mail: [email protected]

Editorial Board Ali Adibi School of Electrical and Computer Engineering Van Leer Electrical Engineering Building Georgia Institute of Technology 777 Atlantic Drive NW Atlanta, GA 30332-0250, USA E-mail: [email protected]

Toshimitsu Asakura Hokkai-Gakuen University Faculty of Engineering 1-1, Minami-26, Nishi 11, Chuo-ku Sapporo, Hokkaido 064-0926, Japan E-mail: [email protected]

Theodor W. Hänsch Max-Planck-Institut für Quantenoptik Hans-Kopfermann-Strasse 1 85748 Garching, Germany E-mail: [email protected]

Takeshi Kamiya Ministry of Education, Culture, Sports Science and Technology National Institution for Academic Degrees 3-29-1 Otsuka, Bunkyo-ku Tokyo 112-0012, Japan E-mail: [email protected]

Ferenc Krausz Ludwig-Maximilians-Universität München Lehrstuhl für Experimentelle Physik Am Coulombwall 1 85748 Garching, Germany and Max-Planck-Institut für Quantenoptik Hans-Kopfermann-Strass 1 85748 Garching, Germany E-mail: [email protected]

Bo Monemar Department of Physics and Measurement Technology Materials Science Division Linköping University 58183 Linköping, Sweden E-mail: [email protected]

Herbert Venghaus Heinrich-Hertz-Institut für Nachrichtentechnik Berlin GmbH Einsteinufer 37 10587 Berlin, Germany E-mail: [email protected]

Horst Weber Technische Universität Berlin Optisches Institut Strasse des 17. Juni 135 10623 Berlin, Germany E-mail: [email protected]

Harald Weinfurter Ludwig-Maximilians-Universität München Sektion Physik Schellingstrasse 4/III 80799 München, Germany E-mail: harald.weinfurter@physik. uni-muenchen.de

John Heebner • Rohit Grover • Tarek Ibrahim

Optical Microresonators Theory, Fabrication, and Applications

ABC

John Heebner Lawrence Livermore National Laboratory 7000 East Ave Livermore, CA 94550 [email protected]

Tarek Ibrahim RA3-353 Intel Corporation 5200 NE Elam Young Parkway Hillsboro, OR 97124 [email protected]

Rohit Grover MS F15-58 Intel Corporation 3585 SW 198th Ave Aloha, OR 97007 [email protected]

ISBN: 978-0-387-73067-7

e-ISBN: 978-0-387-73068-4

DOI: 10.1007/978-0-387-73068-4 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2007940365 c 2008 Springer-Verlag London Limited
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 Springer Science+Business Media springer.com

Preface

In writing this book we sought to describe some of the important aspects and applications found in the wonderful world of optical microresonators. Of course we tell it from our respective points of view. These vantage points have been clearly biased by the specific roads we took during our investigations. We only hope that it does not detract from the ideas and information collected in this research monologue. We would never admit to perfection and cannot claim mathematical rigor in the theoretical chapters nor detailed process recipes in the chapter on fabrication. These circulating fields and their interactions have kept us busy and entertained both conceptually and in the lab for the better part of a decade. When we started out in this field, there was no textbook to consult. It is our hope that students and researchers entering this field now have such a guide. We dedicate this effort to Erika, Priya, Rhea, Uma, Dalia, and Mariam.

Acknowledgments

John E. Heebner graduated first in his class with a B.E. in engineering physics from Stevens Institute of Technology, Hoboken, NJ, in 1996. He then conducted graduate studies at the Institute of Optics, University of Rochester, where he received a Ph.D. in optics in 2003. Dr. Heebner’s doctoral research on the topic of nonlinear optical effects in microring resonators was carried out under the supervision of Prof. Robert W. Boyd. Much of the material for this book derives from those investigations supported by the National Science Foundation (NSF) and the Defense Advanced Research Projects Agency (DARPA). Dr. Heebner is currently employed as a Senior Optical Scientist by Lawrence Livermore National Laboratory in Livermore, CA. John thanks his graduate advisor and skiing instructor Prof. Robert W. Boyd. Additionally, much appreciation goes out to all his mentors, colleagues, and friends both at the Institute and in the greater optics, photonics, and fabrication community, including Govind Agrawal, Gary Wicks, Turan Erdogan, Richart Slusher, John Sipe, Mark Bowers, Deborah Jackson, Young Kwon Yoon, Nick Lepeshkin, Mark Sanson, Brian Soller, Dawn Gifford, Vincent Wong, Sean Bentley, Ryan Bennink, Matt Bigelow, Giovanni Piredda, Aaron Schweinsberg, Collin O’Sullivan-Hale, Petros Zerom, Laura Allaire, Alan Heaney, Mike Koch, Alan Bleier, John Treichler, Philip Chak, Rob Ilic, and Ellen Chang, newfound colleagues and friends in the LLNL ETD photonics group and NIF. Finally, he thanks his parents Harry and Lily for their endless support and encouragement. Rohit Grover completed his B.Tech. in engineering physics from the Indian Institute of Technology Bombay, Mumbai, in 1997. He then received both an M.S. and Ph.D. in electrical and computer engineering from the University of Maryland, College Park, in 1999 and 2004, respectively. Dr. Grover’s research on semiconductor optical microresonators was carried out primarily at the Laboratory for Physical Sciences, College Park, MD. This book draws, in part, on Rohit’s work while at the University of Maryland, where he received support from a Graduate Research Assistantship (1998-2000), Distinguished Graduate Research Assistantship (2000-2003), and an IEEE-LEOS Graduate Student Fellowship (2001). Dr. Grover is currently a Senior Process Integration Engineer at

viii

Acknowledgments

Intel Corporation, Hillsboro, OR, where he is studying die-package interactions for the 45-nm node. Rohit thanks the staff and his many coworkers while at the Laboratory for Physical Sciences, and also the staff of the Cornell NanoScale Facility. In particular, he thanks his advisor, Prof. Ping-Tong Ho, colleagues Philippe Absil, Kuldeep Amarnath, John Hryniewicz, Tarek Ibrahim (coauthor for this book), Yongzhang Leng, and Vien Van. He also thanks his family for their support through his Ph.D. and during his professional life thus far (and hopefully beyond). Tarek A. Ibrahim received B.S. and M.S. degrees in electrical engineering from Cairo University, Giza, Egypt, in 1996 and 1999, respectively. He then joined the Laboratory for Physical Sciences, University of Maryland, College Park, as a graduate student and completed his Ph.D. in 2004. His research while at the University of Maryland was on the topic of all-optical signal processing using nonlinear semiconductor microring resonators. Dr. Ibrahim received the University of Maryland Graduate Fellowship in 1999 and the Distinguished Electrical and Computer Engineering Graduate Research Assistantship in 2001. He is currently a Senior Process Integration Engineer at Intel Corporation. Tarek would like to thank the staff of the Laboratory for Physical Sciences. In particular, he thanks his advisors, Prof. J. Goldhar and Prof. Ping-Tong Ho, and his colleagues, Philippe Absil, Kuldeep Amarnath, John Hryniewicz, Rohit Grover, Vien Van, Ken Ritter, and Marshall Saylors. He also thanks his family for their support throughout his Ph.D. and during his professional life. He also thanks his advisor at Cairo University, Prof. Adel El-Nadi, for his great support and continuous encouragement.

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Optical Microresonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Putting the “Micro” in “Microring” . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Nonlinear Optics in Microresonators . . . . . . . . . . . . . . . . . . . . . 1.5 Book Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 4 5 7

2.

Optical Dielectric Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Total Internal Reflection and Waveguide Confinement . . . . 2.2 The Paraxial Waveguiding Equation . . . . . . . . . . . . . . . . . . . . . . 2.3 The Planar Slab Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 TE Mode Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 TM Mode Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Planar Waveguide Dispersion Relations . . . . . . . . . . . . 2.3.4 Normalized Planar Waveguide Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Analysis Methods for Rectangular Dielectric Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Marcatili’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Effective Index Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Goell’s Circular Harmonic Method . . . . . . . . . . . . . . . . . 2.4.4 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Beam Propagation Method . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Finite-Difference Time-Domain Method . . . . . . . . . . . . 2.5 Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Perturbation Method for Deriving Coupling Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Coupling Between Symmetric TE Planar Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Coupled Wave Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 The Scattering Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Optimized Coupling for Waveguides and Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Bending Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Whispering Gallery Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 TM Whispering Gallery Modes . . . . . . . . . . . . . . . . . . . . . 2.7.2 TE Whispering Gallery Modes . . . . . . . . . . . . . . . . . . . . . . 2.7.3 Radiation Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 12 13 13 15 18

ix

19 20 22 23 24 25 25 26 26 27 29 30 31 31 36 38 39 41 42

x

Contents

2.7.4 WGM Dispersion Relations (Resonance Maps) . . . . . . 2.7.5 Normalized WGM Dispersion Relations (Resonance Maps) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.6 Spheres, Rings, and Disks . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Scattering Losses Resulting From Edge Roughness . . . . . . . 2.8.1 Volume Current Method Formulation for Scattering Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Current Density Contributions . . . . . . . . . . . . . . . . . . . . 2.8.3 Spectral Density Formulation for Edge Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.4 Far-Field Scattered Power . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.5 TM Scattering Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.6 TE Scattering Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.7 Normalized Formulation for Edge Scattering Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 46 48 50 50 50 52 53 55 56 61 70

3.

Optical Microresonator Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.1 Resonator Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.1.1 Fabry–Perot Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.1.2 Gires–Tournois Resonators . . . . . . . . . . . . . . . . . . . . . . . . 73 3.1.3 Ring Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.2 All-Pass Ring Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.2.1 Intensity Buildup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.2.2 Finesse F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.2.3 Effective Phase Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.2.4 Group Delay and Group Delay Dispersion . . . . . . . . . . 79 3.2.5 Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.3 Add–Drop Ring Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.3.1 Intensity Buildup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.3.2 Add–Drop Resonance Width ∆ω or ∆λ . . . . . . . . . . . . 87 3.3.3 Free Spectral Range (FSR) . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.3.4 Finesse F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.4 More on Concepts Associated with Resonators . . . . . . . . . . . 89 3.4.1 Quality Factor Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.4.2 Physical Significance of F and Q . . . . . . . . . . . . . . . . . . 91 3.4.3 Phasor Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.4.4 Kramers–Kronig Relations . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.5 Higher Order Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.

Microring Filters: Experimental Results . . . . . . . . . . . . . . . . . . . . . 4.1 Passive Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 GaAs-AlGaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 GaInAsP-InP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 107 107

Contents

xi

4.2 Active Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Electro-Optic Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Material Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Tuning Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 110 112 116 121

Nonlinear Optics with Microresonators . . . . . . . . . . . . . . . . . . . . . 5.1 Nonlinear Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Resonator Enhanced χ (3) Nonlinear Effects . . . . . . . . . . . . . . . 5.2.1 Enhanced Nonlinear Phase Shift . . . . . . . . . . . . . . . . . . . 5.2.2 Nonlinear Pulsed Excitation . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Kerr Effect in Solid State Materials below Mid-Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Experimental Enhancement of the Kerr Effect . . . . . . 5.2.5 Nonlinear Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6 Multistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.7 Fabry–Perot, Add–Drop, and REMZ Switching . . . . . . 5.2.8 Reduced Nonlinear Enhancement via Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.9 Nonlinear Figures of Merit (FOMs) . . . . . . . . . . . . . . . . . 5.2.10 Inverted Effective Nonlinearity . . . . . . . . . . . . . . . . . . . . 5.3 Resonator-Enhanced Free Carrier Refraction . . . . . . . . . . . . . 5.4 Enhanced Four-Wave Mixing Efficiency in Microring Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 124 124 125

147 148

6.

All-Optical Switching and Logic using Microresonators . . . . . . 6.1 All-Optical Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Linear Device Characteristics . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Nonlinear Device Characteristics . . . . . . . . . . . . . . . . . . 6.2 Thresholding and Pulse Reshaping . . . . . . . . . . . . . . . . . . . . . . . 6.3 Time-Division Demultiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Linear Device Characteristics . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Nonlinear Device Characteristics . . . . . . . . . . . . . . . . . . 6.4 All-Optical Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 AND/NAND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 NOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149 149 149 151 152 154 156 158 159 160 163 163 169 173

7.

Distributed Microresonator Systems . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Linear Propagation in Distributed Microresonators . . . . . . . 7.2.1 Group Velocity Reduction . . . . . . . . . . . . . . . . . . . . . . . . .

175 175 175 177

5.

127 128 132 136 137 138 142 144 144

xii

Contents

7.3

7.4 7.5 7.6

7.7

7.8 8.

7.2.2 Group Velocity Dispersion . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Higher Order Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear Propagation in Distributed Microresonators . . . . 7.3.1 SCISSOR Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Induced Self-Steepening . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Pulse Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Nonlinear Detuning and Multistability . . . . . . . . . . . . . 7.3.5 Nonlinear Frequency Mixing . . . . . . . . . . . . . . . . . . . . . . . Limited Depth of Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attenuation in Distributed Microresonators . . . . . . . . . . . . . . Slow and Fast Light in SCISSORs . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Slow Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Tunable Optical Delay Lines . . . . . . . . . . . . . . . . . . . . . . . 7.6.3 Fast Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized Periodic Resonator Systems . . . . . . . . . . . . . . . . . 7.7.1 Bloch-Matrix Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.2 Directly Coupled Resonators . . . . . . . . . . . . . . . . . . . . . . 7.7.3 Single-Channel SCISSORs . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.4 Double-Channel SCISSORs . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.5 Twisted Double-Channel SCISSORs . . . . . . . . . . . . . . . . 7.7.6 Bandgap Engineering in Distributed Feedback Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.7 Slow Light, Group Velocity Dispersion, and Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fabrication Techniques for Microresonators . . . . . . . . . . . . . . . . 8.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 III–V Semiconductors for Active and Passive Microrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Growing a Waveguide Stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 III–V Semiconductors (InP and GaAs) . . . . . . . . . . . . . . 8.3.4 Silicon Oxynitride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Feature Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Polymers, Glass, SiON, SiN . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 III–V Semiconductors (InP and GaAs) . . . . . . . . . . . . . . 8.5 Multilayer Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Laterally Coupled III–V Passive Microresonators . . . . . . . . . . 8.7 Polymer-Bonded, III–V Vertically Coupled Passive Microresonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Active III–V Laterally Coupled Microresonators . . . . . . . . . . . 8.9 Other Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

178 179 179 181 184 185 186 187 189 190 191 192 195 197 197 200 202 204 206 208 210 213 215 217 217 218 219 219 220 220 220 220 221 221 222 225 227 234 238

Contents

xiii

8.10 Polymer Microrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 8.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

1. Introduction

1.1 Optical Microresonators Optical microresonators have demonstrated great promise as fundamental building blocks for a variety of applications in photonics. They can be implemented for such diverse applications such as lasers, amplifiers, sensors, optical channel dropping filters (OCDFs), optical add/drop (de)multiplexers (OADMs), switches, routers, logic gates, and artificial media. For brevity and in keeping with their current usage in the literature of this field, we specialize the term “microresonators” and generalize the term “microring resonators.” We use these terms interchangeably in this book to refer to any of a number of compact geometries that support cyclically propagating modes that close on themselves in a ring-like geometry. One particular embodiment of a microring resonator consists of an ordinary waveguide that channels light in a closed loop. But in general, the loop can take the form of other closed shapes, such as a disk, racetrack, or ellipse. In the case of a ring, the microresonator is simply a curved waveguide closed onto itself forming a resonant cavity that supports both transverse and longitudinal (here azimuthal) modes. The confinement and channeling of light in this closed geometry, however, does not require an inner dielectric boundary. This is evidenced by the existence of optical “whispering gallery” modes in a microdisk or microsphere resonator. Placement of a microresonator near one or two waveguides (Fig. 1.1) enables access to modes of the resonant cavity. In this particular arrangement, the resonant modes are accessed through evanescent coupling — a phenomena analogous to tunneling in solid-state physics. Component wavelengths of an optical signal channeled in a waveguide are resonant with the cavity if its (effective) circumference supports an integer number of wavelengths. For these spectral components of the signal, an increased circulation of intensity can build up in the resonator. The presence of a second waveguide coupled to the ring enables extraction of the resonant, circulating signal. Component wavelengths that do not resonate with the ring bypass it altogether. Thus, at their most fundamental level, microring resonators act as a spectral filter and a temporary compressor of energy density. These properties are not unique to microring resonators.

2

1. Introduction

Fig. 1.1. Schematic of an optical microring resonator add/drop filter.

Rather, they are common to all resonant cavities such as the well known Fabry–Perot resonator. Although functionally similar to Fabry–Perots, microring resonators offer several advantages. First, their planar nature is naturally compatible with monolithic microfabrication technologies. Second, high finesse operation does not require multilayer or distributed Bragg reflectors but is rather achieved by increasing the gap widths of evanescent couplers. Third, because the equivalent injected, transmitted, and reflected waves occupy spatially distinct channels, the need for costly Faraday circulators is eliminated. Fourth, for the same reason, although there is only one natural way to sequence arrays of Fabry–Perots (into multilayer stacks) three altogether new possible arrangements for arrays of resonators are enabled that differ qualitatively in many ways. The small scale-size of microresonators currently achievable by stateof-the-art fabrication methods is important for many reasons of which we highlight two. First, because the propagation velocity of light is of the order of a few hundred µm per ps in most optical materials of interest, high bandwidths (GHz to THz) are naturally attainable. Second, their small dimensions allow the integration of many devices on the same chip, enabling high–level functionalities such as ultrafast all-optical signal processing at a heretofore unrealized compact scale. Because of these inherent advantages, the very large–scale integration (VLSI) of high– bandwidth photonics may rely on optical microresonators. In the next section, we offer our perspective on how microresonators came to be important components in the photonic toolbox.

1.2 Historical Perspective

3

1.2 Historical Perspective The “whispering gallery” effect was analyzed (with wave approaches) as early as 1910 by Lord Rayleigh [1]. His analysis of the channeling of acoustic waves by the dome of St. Paul’s cathedral in London is a precursor to similar methods applied to electromagnetic waves. Ring and disk resonators for electromagnetic waves have since been implemented in microwave applications starting in the early 1960s. In the optical domain, integrated ring resonators were proposed in 1969 by Marcatili at Bell Labs [2]. The first guided optical ring resonator was demonstrated by Weber and Ulrich in 1971 [3–5]. Weber and Ulrich’s device consisted of a 5-mm-diameter glass rod (n = 1.47) coated with Rhodamine-6G-doped polyurethane (n = 1.55), for a resonator circumference of 31.4 mm. Light was coupled in and out of the resonator with a prism. By pumping the polymer with light from a N2 laser (λ = 337.1 nm), they obtained laser operation. The next relevant demonstration was by Haavisto and Pajer in 1980 [6]. Their device was the first to incorporate integrated bus waveguides made with a doped polymethyl methacrylate (PMMA) film on quartz substrate. A significant feature of their work was that the device was fabricated without lithography by using direct-writing with a 325-nm He–Cd laser. Although they demonstrated low-loss waveguides and rings, the ring was quite large (circumference 28.3 cm). Nevertheless coupling to the ring was via evanescent coupling to integrated bus waveguides, and the basic idea had been established. In 1982, Stokes, Chodorow, and Shaw [7] demonstrated the first optical glass fiber ring resonator, operating at λ = 632.8 nm. Fibers unfortunately, do not lend themselves to compact integrated optics; their resonator had a circumference of 3 m. Between 1982 and 1990, numerous groups demonstrated integrated ring resonators based on glass. Early efforts used ion-exchange from AgNO3 , KNO3 , and similar compounds that modify the index of the glass, to make the waveguide core. Walker and Wilkinson demonstrated a ring resonator with silver ion-exchanged glass in 1983 [8] (circumference 3.1 mm, operating at λ = 632.8 nm), as did Mahapatra and Connors in 1986 [9,10] (circumference 4.1 mm, operating at λ = 632.8 nm). Honda, Garmire, and Wilson demonstrated a ring resonator with potassium ion-exchanged glass in 1984 [11] (circumference 25.1 cm). A related effort by Naumaan and Boyd in 1986 [12] used CVD phosphosilicate glass films. Other efforts used Ti-exchanged LiNbO3 (Tietgen, 1984 [13]), and proton-exchanged LiNbO3 (Mahapatra and Robinson, 1985 [14]). Tietgen’s work is especially significant as it represents the first demonstration of a tunable ring resonator. Instead of a circular ring, he used a waveguide loop with two 3-dB couplers. His device used electro-optic tuning, had a circumference of a little over 24 mm, and operated at λ = 790 nm.

4

1. Introduction

Since the early efforts outlined above, there have been numerous works in various doped and undoped silica-based glasses [15–25], Si(Si3 N4 , SiON, SiO2 ) [26–37], and polymers [38–41] in the past decade. Many of these studies have reported multiring filters, temperature-insensitive operation and so on. Oda’s work with TiO2 -doped silica-glass rings represents the first demonstration of serially cascaded rings, with increased free spectral range over single-ring devices. Rabiei’s work with polymer rings represents the first passive and active polymer ring resonator. Microresonators constructed in III–V semiconductors began “seeing light” in the early 1990s. Several groups demonstrated optically pumped microdisk lasers in both GaInAsP-InP and III–Nitrides using the whispering-gallery; the smallest reported disks had circumferences of ∼15 µm [42–50]. Most of these early efforts did not incorporate bus waveguides and relied on fibers to directly collect light from the disk. The first GaAsAlGaAs microring resonator laterally coupled to bus waveguides was demonstrated by Rafizadeh et al. in 1997 at Northwestern University, Evanston, IL [51,52]. Their smallest ring had a circumference of 32.8 µm. Since then, members of Ping-Tong Ho’s group at the Laboratory for Physical Sciences (LPS), College Park, MD, have demonstrated both laterally and vertically coupled rings in GaAs-AlGaAs acting as multi ring devices, switches, routers, and mux/demux operation [53–61]. The GaInAsP-InP material system has proven problematic for passive microrings because of processing difficulties resulting in high device losses. Nevertheless, the first vertically coupled passive InP-based rings were demonstrated by Ho’s group [59, 62, 63]. Other groups have concentrated on disk resonators; the group at the University of Southern California, for example, has demonstrated active and passive vertically coupled microdisk resonators [64, 65].

1.3 Putting the “Micro” in “Microring” Initial efforts toward the fabrication of integrated ring resonators produced very large devices because the index contrast ∆n between the core and cladding was small, and operation with large radii of curvature minimizes bending losses [2, 66–69]. For example, the device fabricated by Honda et al. had 0.052 < ∆n < 0.067 and a radius of 4.5 cm [11]. Also, the device was multi mode, and there was considerable mode mixing, leading to a large background of non resonant light and the convolution of resonances from multiple modes. The construction of microrings 10-µm in diameter or smaller is a challenging effort often requiring high index contrast, anisotropically etched pedestal waveguide designs with ultrasmooth sidewalls. In addition, precise coupling gap widths with extremely tight tolerances are demanded

1.4 Nonlinear Optics in Microresonators

5

to engineer coupling strengths. This is accomplished laterally using highresolution lithographically or vertically by material growth. In the past decade, these technologies have reached maturity. The confluence of the many mature and maturing technologies has led to the demonstration of single-mode microring resonators both laterally and vertically coupled to a bus waveguide in GaAs-AlGaAs [51] and Si-SiO2 [21]. Advanced functions, such as high-order filtering for dense wavelength division multiplexing applications, notch filters, and wavelength-selective mirrors, have since been demonstrated [55, 57, 60]. The lateral-coupling approach in III–V semiconductors demands strict patterning tolerances typically requiring e-beam lithography followed by advanced etching techniques. Nevertheless, in 2000, Griffel and coworkers at Sarnoff Corp., Princeton, NJ, and Riverside Research Institute, New York, NY, demonstrated a laterally coupled ring laser with integrated bus waveguides with stepper lithography by employing a bi-level etching technique [70]. They were able to demonstrate devices with circumferences of 1000 µm or more. In 2001, Rabus and coworkers at the Heinrich-Hertz Institute (HHI), Berlin, Germany, demonstrated multimode interference (MMI) coupled rings in GaInAsP-InP, and in 2002 they demonstrated active rings by integrating semiconductor optical amplifiers and on-chip platinum heaters [71–73]. The HHI work features very large ring circumference; their smallest rings had circumferences as large as 928 µm, which resulted in extremely narrow free spectral range (around 0.8 nm). Laterally coupled ring resonators in GaInAsP-InP were also demonstrated by Rommel and coworkers from the University of Illinois, Urbana-Champaign, IL, and Sarnoff in 2002 [74]. They demonstrated a ring resonator with circumference 274.2 µm and free spectral range a little over 2 nm. Ho’s group recently demonstrated microring resonator notch filters, both passive and electro-optically tuned, with the smallest turning radius to date [75–77]. Their smallest device had a circumference of just 20.1 µm, with a free spectral range much greater than 30 nm.

1.4 Nonlinear Optics in Microresonators All optical materials display a change in their optical properties under high-intensity illumination. Fundamentally, this change results from is due to a nonlinear response of the material polarization to the applied electromagnetic field. As a concise and practical definition, nonlinear optics is study of high intensity phenomena in which the optical response of a medium differs from its low-intensity limits. The narrowing of a microring’s resonance is accompanied by a proportional increase in the coherent buildup of intensity circulating within the cavity. Although this buildup is of no direct consequence to the linear transmission properties,

6

1. Introduction

it has a dramatic effect on the nonlinear (intensity-dependent) transmission properties. In 1969, Szöke et al. appreciated this effect and proposed inserting a nonlinear material (saturable absorber) between the mirrors of a Fabry– Perot; they also described optical multistability with possible applications for optical logic [78]. The first optical bistability experiments were later performed by McCall and Gibbs et al. in a sodium-filled Fabry–Perot [79, 80]. Marburger and Felber later developed the theory of a Fabry–Perot resonator containing a medium possessing a nonlinear (intensity-dependent) refractive index and demonstrated that under certain operating conditions, the device could be implemented as an all-optical switch [81]. Miller etal later performed the first two-beam a.c. amplification experiment [82] and derived the optimum parameters for all-optical switching implementing refractive bistability in a lossy Fabry–Perot [83]. In the late 1980s and 1990s, ring resonators were used in various experiments in nonlinear and quantum optics. Shelby et al. observed a finesse-squared enhancement dependence of squeezed light generation in a fiber ring resonator [84, 85]. In 1989, Braginsky et al. studied the nonlinear properties of optical whispering-gallery modes [86]. Braginsky proposed the application of such modes to the lofty goal of switching with a single quantum [87]. In 1996, Vernooy and Kimble performed cavity quantum electrodynamic (QED) experiments that exploited the small mode volumes and high field strengths associated with the modes of microcavities [88]. In 1997 Chang et al. demonstrated Q-switching using enhanced saturable absorption associated with the WGM resonances of microdroplets [89]. In 1997, Blom et al. proposed the development of an integrated all-optical switch based upon the high-Q whispering gallery modes of a nonlinear polymer disk [90, 91]. In 1999, Rosenberger [92] furthered the study of nonlinear optical effects in microspheres. More recently, enhanced Raman interactions in glass microspheres were demonstrated [93, 94]. Cavity QED phenomena [95, 96] still continue to be explored in microresonators because of their potential for high quality factors and ultrasmall mode volumes. Despite all this research, microresonators implemented as nonlinear optical elements continue to be a tremendous technological challenge in real applications, because nonlinear optical effects are extremely weak, particularly with respect to their electronic counterparts. In contrast, electrons interact with each other very strongly because of Coulombic interaction. At the most fundamental level, photons do not manipulate photons without first interacting with electrons in practical nonlinear optical materials. The result is that photons are extremely adept in the high-fidelity transmission of information, but they possess an underlying handicap associated with the manipulation of information — as in active routing or performing logic.

1.5 Book Overview

7

Much like the field of electronics exploded when nonlinear elements (the transistor) were integrated at a compact scale, the field of photonics could be waiting to blossom away from traditional optics when nonlinear photonic devices at the chip level become commonplace. This promise has been held back in large part by the lack of an optimal material system with fast nonlinear response and direct compatibility with a mature fabrication technology. Some III–V semiconductors such as AlGaAs come closest to the mark, but they are nevertheless several orders of magnitude weaker than comparable nonlinear responses in electronics. Microresonators enable the augmentation of the intrinsic nonlinear optical properties. Much like quantum dots and photonic crystals, microresonators offer another method for creating engineerable materials with nonlinear responses appreciable enough to construct photonic devices with advanced functionalities operating at ultrafast speeds.

1.5 Book Overview In the rest of this book, we present an overview of the theory, fabrication, and application of optical microresonators. Along the way, where relevant, we present the results of our research efforts along with those from other groups. In chapter 2 we introduce the equations describing confinement and propagation in optical dielectric waveguides. We then go on to present methods for analyzing coupling, losses, and whispering galley effects relevant to microresonators. In chapter 3 we delve into the theoretical framework describing the properties associated with resonators in a generalized manner. Chapter 4 presents the results of passive, linear optical experiments with fabricated microring devices operating as spectral filters. In chapter 5, we revisit the theoretical framework with the inclusion of nonlinear optical phenomena associated with microresonators. Chapter 6 follows up with demonstrations of all-optical switching and logic. In chapter 7, we consider both linear and nonlinear optical propagation phenomena in sequences of multiple microresonators and draw comparisons to related photonic crystal systems. Finally, in chapter 8, we discuss fabrication processes and techniques involved in the construction of microresonators.

2. Optical Dielectric Waveguides

In this chapter, we introduce several analytical and numerical methods used to model the confinement of light in optical dielectric waveguides. We then go on to develop methods for calculating the coupling strength between waveguides and for predicting losses resulting from bending loss. These losses are then considered in the regime of the optical whispering gallery, and we conclude with a treatment of scattering losses resulting from edge roughness in this regime. The ideas presented in this chapter are of fundamental importance to the practical implementation of optical microresonators.

2.1 Total Internal Reflection and Waveguide Confinement Conventional dielectric waveguides channel light through transverse confinement in a dielectric “core,” of refractive index n1 surrounded by a “cladding” often of lower refractive-index n2 . The cladding may consist of a different material, a differently doped region, or in some cases the surrounding air. Guidance in dielectric waveguides results from the phenomenon of total internal reflection (TIR). Light inside the core experiences total internal reflection when striking the core-cladding interface at angles greater (with respect interface normal) than the critical angle. For most conventional waveguides, this condition is satisfied when light enters the core at angles smaller than the guide’s acceptance angle (with respect to the guide’s propagation axis) or numerical aperture NA. Provided that the waveguide cross section does not vary significantly or rapidly along the direction of propagation, the light will be localized over long lengths with low loss. Figure 2.1 depicts some of the more common waveguide cross sections. The critical angle arises from the Fresnel equations that derive from applying boundary conditions to the electric and magnetic fields in conjunction with Maxwell’s equations. The Fresnel reflectivity expressions applicable to angles of incidence within the critical angle for TIR are given as
n1 cos θ1 − i n21 sin2 θ1 − n22
rTE = ≡ e−iφTE (2.1) 2 2 2 n1 cos θ1 + i n1 sin θ1 − n2

10

2. Optical Dielectric Waveguides

Fig. 2.1. Common waveguide geometries. Darker regions represent higher refractive index.

rTM


n i n12 n21 sin2 θ1 − n22 − n2 cos θ1
= ≡ e−iφTM . n1 2 2 2 i n2 n1 sin θ1 − n2 + n2 cos θ1

(2.2)

Here, transverse electric (TE) refers to linearly polarized light where the electric field is perpendicular to the plane of incidence and transverse magnetic (TM) refers to the case where the magnetic field is perpendicular to the plane of incidence. It follows from these equations that for angles of incidence satisfying (sin θ1 > n2 /n1 ), the modulus of the reflectivity is unity. Thus, the critical angle for total internal reflection is defined solely by the refractive index ratio as θc = arcsin(n2 /n1 ). For angles above θc , the complex reflectivities can be expressed in phasor representation with unit amplitudes and phase shifts acquired upon reflection1 : ⎞ ⎛
  n21 sin2 θ1 − n22 ⎠ = 2 arctan γx (2.3) φTE = 2 arctan ⎝ n1 cos θ1 kx
⎞ ⎛  2 n21 sin2 θ1 − n22 n 1 ⎠ = 2 arctan n1 γx . (2.4) φTM = 2 arctan ⎝ n2 n2 cos θ1 n22 kx 1

In the final forms of the expressions, kx = n1 k0 sin θ1 refers to the perpendicular component of the propagation vector k1 = n1 k0 incident on the interface and γx refers to the decay constant associated with the evanescent tail of the field that extends beyond the interface.

2.1 Total Internal Reflection and Waveguide Confinement

11

Fig. 2.2. Light confinement and guiding by total internal reflection. The zigzag path can be decomposed into a transverse wavevector, kx describing purely oscillatory behavior and a longitudinal propagation vector, β along the guide axis.

As we will see later, the angular dependence of the acquired phase shift in reflection plays an important role in dictating the dispersion relation for guided modes. In Fig. 2.2, light guided by total internal reflection is depicted as resulting from the interference of plane waves alternately reflecting from both core-cladding interfaces of a planar slab waveguide of thickness d. The dotted line represents the direction of the total wavevector k1 . The wave-vector can be decomposed into a component along the guide propagation direction termed the effective propagation constant β and another in the transverse direction kx . For an eigenmode solution of a slab waveguide, the transverse field profile is oscillatory and does not vary with propagation. Hence, the wave must experience a phase shift of 2π m, where m is an integer, in one round-trip between the two core-cladding interfaces. The phase shift consists of a contribution from the transverse component of the propagation constant along with the Fresnel phase shift acquired at each interface. The round-trip phase requirement implies that for a given frequency ω, only certain discrete incidence angles θm are allowable for the wavevector; each angle solution is associated with a transverse mode of the guide. Thus we arrive at a very simple geometric derivation for the dispersion relation of modes in a planar or slab waveguide for the TE case: n1 k0 2d cos θm − 2φTE [θm ] = m2π ⎞ ⎛
2 2 2
β − n k m 2 0 ⎠ = m2π 2d n21 k20 − β2m − 4 arctan ⎝
n21 k20 − β2m

(2.5) (2.6)

and for the TM case: n1 k0 2d cos θm − 2φTM [θm ] = m2π
⎞ ⎛
2 β 2 − n2 k 2 m n 2 0 1 2 2 2 ⎠ = m2π . 2d n1 k0 − βm − 4 arctan ⎝ 2
n2 n2 k2 − β2m 1 0

(2.7) (2.8)

12

2. Optical Dielectric Waveguides

This derivation has intuitive appeal in describing the allowable propagation constants for a given frequency (ω = ck0 ) in terms of reflection angles for geometric ray paths and Fresnel reflection phase shifts. From these discrete propagation constants, the mode profile in the core and cladding can be obtained. Unfortunately, this technique is of limited use and is generally not directly applicable to the derivation of dispersion relations associated with waveguides of a two-dimensional cross section. The source of the limitation is that the Fresnel equations are directly applicable only for plane waves incident on planar interfaces in which the angle of incidence is discrete. In the case of a planar waveguide, fields confined inside the core can be described by two interfering plane waves and the Fresnel equations are applicable. The intuition acquired through this description is sometimes useful for quickly analyzing other guiding structures, although with some degree of caution. In order to analyze more generalized waveguide cross sections, we turn to a physical optics description of wave propagation.

2.2 The Paraxial Waveguiding Equation A physical optics model for propagation in a waveguide begins with Maxwell’s two first-order vector curl equations for the electric and magnetic fields in dielectric media: ∂H ∂t ∂E ∇×H=ε ∂t

∇ × E = −µ0

(2.9) (2.10)

These equations are combined into a single second-order wave equation, ∇2 U −

n2 ∂ 2 U =0 c 2 ∂t 2

(2.11)

where U represents a component of the electric E or magnetic H field. We assume that the fields are harmonic in time t and that propagation takes place along the z axis. A temporal Fourier transform of the wave equation results in the time-independent Helmholtz equation ∂2U + ∇2T U + k2 U = 0, ∂z2

(2.12)

where the Laplacian ∇2 operator has been decomposed into longitudinal ∂ 2 /∂z2 and transverse ∇2T = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 components. Next, we assume a field solution such that for all points transverse to the direction of propagation, the field accumulates phase uniformly with propagation

2.3 The Planar Slab Waveguide

13

constant β according to U (x, y, z) = A(x, y)eiβz , where A(x, y) represents the stationary transverse mode profile. Substituting this expression into Eq. 2.12 results in [97] ∂2A ∂A + ∇2T A + (n2 k20 − β2 )A = 0. + i2β ∂z2 ∂z

(2.13)

Since we wish to find the stationary transverse modes of the structure, we set both longitudinal derivatives to zero and are left with the equation to be solved: ∇2T A + (n2 k20 − β2 )A = 0. (2.14) This equation is equivalent to the time-independent Schrödinger equation and is solved in both the core and cladding with the appropriate boundary conditions to obtain β. For guided modes, the value of β lies between the propagation constants of the core and cladding. This ensures that light is confined to the core where the solutions to Eq. 2.14 (for which β < nk0 ) are oscillatory. The evanescent tail of the mode in the cladding (for which β > nk0 ) is described by solutions exhibiting exponential decay away from the core. For a guided wave, the quantity neff = β/k0 is termed the “effective index” of the waveguide, since it represents the ratio of the speed of light in vacuum to that of the propagating mode. In certain special cases, analytic solutions for Eq. 2.14 are possible. In general, however, approximate analytic expressions deliver quick and intuitive understanding, whereas numerical methods are indispensable in achieving sufficient accuracy for most modern waveguiding structures of interest. In the following section we revisit the planar slab waveguide within the physical optics formalism.

2.3 The Planar Slab Waveguide For a planar slab waveguide, confinement exists only in one direction (here along the x axis). Consequently, there is in infinite quantity of optical power in the mode, although a power per unit length along the translation invariant direction (y axis) can be ascribed. Localized solutions require that the field decays to 0 at x → ±∞. We examine separately the derivation [98] of the field profiles for TE and TM guided modes (for which k2 < β < k1 ). 2.3.1 TE Mode Profiles For TE polarized modes, Maxwell’s vector curl equations reduce from six to three equations involving three field components (Ey , Hx , Hz ): −iβEy = iωµ0 Hx

(2.15)

14

2. Optical Dielectric Waveguides

∂Ey ∂x ∂Hx ∂Hz − ∂z ∂x

=

iωµ0 Hz

(2.16)

=

−iωεEy

(2.17)

that reduces to a single wave equation for the transverse electric field component:   ∂ 2 Ey ∂Hx ∂Hx + iωεE − ω2 µ0 εEy = iωµ 0 y = iωµ0 ∂x 2 ∂z ∂z

 = β2 − k2 Ey

(2.18)

that can be separated into transverse Helmholtz equations for each of the core and the two cladding regions. For simplicity, we assume a symmetric cladding configuration.  ∂2 2 + kx Ey = 0 (x < −d/2, x > d/2) (2.19) ∂x 2  ∂2 − γx2 Ey = 0 (−d/2 < x < d/2) (2.20) ∂x 2 resulting in field profiles for the three regions here for even-order modes.2

2

n1 − n2 eff −γx (x−d/2) Ey = Ey0  2 e (x > d/2) (2.21) n1 − n22 (x > −d/2, x < d/2) = Ey0 cos (kx x)

2

n1 − n2 eff γx (x+d/2) e (x < −d/2). = Ey0  2 n1 − n22

(2.22) (2.23)

The magnetic field components are then easily derived from this electric field solution through the following relationships: neff Ey Z0 i ∂Ey . Hz = − k0 Z0 ∂x

Hx = −

(2.24) (2.25)

The power per unit length carried by the mode can be obtained by integrating the time-averaged Poynting vector (S = 21 {E × H∗ }) across the waveguide dimension. 2

For odd-order modes, the cosine is replaced with a sine and the two cladding amplitudes are oppositely signed.

2.3 The Planar Slab Waveguide

P/L

1 =− 2 =

=

+∞ 

dxEy Hx∗

−∞ +∞  neff

2Z0

15

(2.26)

 2   dx Ey 

(2.27)

−∞

 2    neff Ey0  4Z0

2 d+ γx

 .

(2.28)

The power per unit length thus reduces to an intuitive expression equal 2   ing half the time-averaged peak intensity (I0 = neff Ey0  /2Z0 ) times the effective width of the mode (deff = d + 2/γx ). In terms of the power per unit length, the field amplitudes are given by  4Z0 P/L (2.29) Ey0 = neff deff  4neff P/L Hx0 = (2.30) Z0 deff with the relationship: Ey0 Hx0 = 4P/L /deff . As an example, the mode profile for an even mode is shown in Fig. 2.3a, and that for an odd mode is shown in Fig. 2.3b. Shown are the lowestorder modes, i.e., the modes with the minimum number of transverse oscillations, that also have the highest effective indices. The thickness d of the waveguide in this example is 1000 nm, and n2 = 3.17, n1 = 3.35. 2.3.2 TM Mode Profiles The expressions for TM polarized modes have a similar derivation. The Maxwell vector curl equations again reduce from six to three equations involving three field components (Hy , Ex , Ez ): −iβHy = −iωεEx ∂Hy = −iωεEz ∂x ∂Ex ∂Ez − = iωµ0 Hy ∂z ∂x

(2.31) (2.32) (2.33)

that reduces to a single wave equation for the transverse magnetic field component:   ∂ 2 Hy ∂Ex ∂Ex − iωµ − ω2 µ0 εHy = −iωε H 0 y = −iωε 2 ∂x ∂z ∂z 

= β2 − k 2 Hy (2.34)

16

2. Optical Dielectric Waveguides

(a)

(b)

Fig. 2.3. (a) An even and (b) an odd mode of a slab waveguide. The dotted lines represent the waveguide boundaries.

that can be separated into transverse Helmholtz equations for each of the core and the two (here symmetric) cladding regions:  ∂2 2 + kx Hy = 0 (x < −d/2, x > d/2) (2.35) ∂x 2  ∂2 − γx2 Hy = 0 (−d/2 < x < d/2), (2.36) ∂x 2 resulting in the following transverse electric field profiles for the three regions for even-order modes.3

3

For odd-order modes, the cosine is replaced with a sine and the two cladding amplitudes are oppositely signed.

2.3 The Planar Slab Waveguide



Hy = Hy0 

1 1+

n41 n2eff −n22 n42 n21 −n2eff

e−γx (x−d/2)

(x > d/2)

(2.37)

(x > −d/2, x < d/2) = Hy0 cos (kx x)

1

= Hy0 eγx (x+d/2) (x < −d/2),  n41 n2eff −n22 1 + n4 n2 −n2 2

1

17

(2.38) (2.39)

eff

where field components are related through the following equations: neff Z0 Hy n2 i ∂Ex iZ0 ∂Hy = . Ez = 2 k0 n ∂x k0 neff ∂x

Ex =

(2.40) (2.41)

Note that unlike the TE case, the transverse component of the electric field Ex in the TM case displays a discontinuous increase of

n21 n22

from the

core to the cladding. This asymmetry results because we are only considering dielectric waveguides where only the permittivity varies while the permeability remains constant. There is also a discontinuity in the magnetic field gradient for the TM mode that forces it to have a lower effective index and a longer evanescent tail. The power per unit length carried by the mode is again obtained by integrating the time-averaged Poynting vector across the waveguide dimension.

P/L

1 = 2 =

+∞ 

dxEx Hy∗

−∞ +∞  neff

2Z0

dx

−∞

n2 |Ex |2 n2eff ⎡

2

=

neff |Ex0 | 4Z0

(2.42)

⎢ n21 ⎢ ⎣ n2 d eff

+

(2.43)

n22 n2eff

2 + γx 1+

n41 n42 n41 n42

⎤ −1 n2eff −n22 n21 −n2eff

n22 n2eff

2 ⎥ ⎥. γx ⎦

(2.44)

The power per unit length again reduces to an expression equaling half the peak intensity times the effective width (contained in square brackets). The effective width has a more complicated expression than in the TE case but reduces to the same value in the limit of low index contrast (n1 ≈ n2 ≈ neff ). In terms of the power per unit length, the field amplitudes are given by  4Z0 P/L Ex0 = (2.45) neff deff

18

2. Optical Dielectric Waveguides

Hy0 =

n21 n2eff



4neff P/L . Z0 deff

(2.46)

2.3.3 Planar Waveguide Dispersion Relations The previous sections used a physical optics description to determine the mode profiles for a planar waveguide assuming a known propagation constant. We return to solve for the value and frequency dependence of the propagation constant βm (ω) that will differ for each nondegenerate mode m. For TE polarization, four equations and four unknowns result from ensuring the continuity of the transverse electric field Ey and the (boundary) transverse magnetic field Hz (related to ∂Ey /∂x) at each interface. For clarity, we depart from the previous notation and label the four field amplitudes as A1 , A2 , A3 , and A4 corresponding respectively to the +x, −x propagating (oscillatory) fields in the core and the +x, −x evanescently decaying fields in the claddings. The resulting four equations are, thus, A1 e−ikx d/2 + A2 e+ikx d/2 2 −ikx d/2 A1 e − A2 e+ikx d/2 γx A4 = ikx 2 A1 e+ikx d/2 + A2 e−ikx d/2 A3 = 2 +ikx d/2 A1 e − A2 e−ikx d/2 . −γx A3 = ikx 2 A4 =

(2.47) (2.48) (2.49) (2.50)

These equations can be solved simultaneously resulting in the equation:  2 (kx − iγx ) +ikx d e = 1 = eim2π . (2.51) (kx + iγx ) that can be rewritten in the form of a transcendental expression relating the propagation constant βm to the radian frequency ω for TE modes:

2


βm − n2 ω2 /c 2 2 d n21 ω2 /c 2 − β2m − 2 arctan  2 = mπ . (2.52) n1 ω2 /c 2 − β2m For completeness, we repeat the derivation of the slab waveguide dispersion relation this time for the TM modes. A1 e−ikx d/2 + A2 e+ikx d/2 2 −ikx d/2 1 A1 e − A2 e+ikx d/2 1 γ A = 2 ikx 2 x 4 2 n2 n1 A4 =

(2.53) (2.54)

2.3 The Planar Slab Waveguide

A1 e+ikx d/2 + A2 e−ikx d/2 2 A1 e+ikx d/2 − A2 e−ikx d/2 1 1 . − 2 γx A3 = 2 ikx 2 n2 n1 A3 =

19

(2.55) (2.56)

These equations can be solved simultaneously resulting in the equation: ⎡



⎤2 k − x ⎢ ⎥ im2π ⎢  e+ikx d ⎥ . ⎣ ⎦ =1=e n21 kx + i n2 γx n2 i n12 γx 2

(2.57)

2

that is again rewritten in the form of a transcendental expression relating the propagation constant βm to the radian frequency ω for TM modes:
2 n β2m − n22 ω2 /c 2 d n21 ω2 /c 2 − β2m − 2 arctan 12  2 = mπ (2.58) n2 n1 ω2 /c 2 − β2m Note that these dispersion relations Eqs. 2.52 and 2.58 are equivalent to Eqs. 2.6 and 2.8 derived earlier using the Fresnel equations in conjunction with a geometrical optics description. 2.3.4 Normalized Planar Waveguide Dispersion Relations Kogelnik and Ramaswamy presented a universal set of normalized curves from which the dispersion relation of an arbitrary one-dimensional (1D) slab waveguide can be mapped [99]. Under this formalism, a normalized frequency V incorporates the wave frequency ω, core dimension d, and


numerical aperture n21 − n22 , whereas a normalized propagation constant b is a shifted and scaled form of the propagation constant:
ω V = d n21 − n22 (2.59) c β2 − k22 b= 2 . (2.60) k1 − k22

Useful relations between the real and normalized parameters are given by  

neff = n22 + b n21 − n22 (2.61) √ V 1−b kx = (2.62) d

20

2. Optical Dielectric Waveguides

√ V b γx = d   2 deff = d 1 + √ V b √ V b + 2b √ Γ = , V b+2

(2.63) (2.64) (2.65)

where Γ represents the confinement factor or proportion of optical power contained in the waveguide core. Implementing these relations with Eq. 2.52 results in the normalized dispersion relation for a symmetric TE planar waveguide,   b = mπ . (2.66) V 1 − b − 2 arctan 1−b This transcendental relation between V and b is plotted in Fig. 2.4. Note that single-mode operation is guaranteed for V < π .

2.4 Analysis Methods for Rectangular Dielectric Waveguides The previous sections were intended to refresh the reader with the properties of one-dimensional planar waveguides. However, the guiding structures associated with ring resonators are generally two-dimensional (2D). The most common 2D dielectric waveguide for which analytic solutions exist is the circular waveguide or optical fiber. This results from the symmetry of the waveguide cross section. The geometry can be defined independently along radial and azimuthal dimensions, and this fact is in turn reflected in the separability of field solutions along each dimension. It would seem to follow that other cross sections that possess high degrees of symmetry might also support analytic solutions. Waveguides of rectangular cross section are typically employed in planar geometries and in particular for the construction of microring resonators. A rectangular cross section, of course, possesses Cartesian symmetry. A rectangular metallic waveguide possesses separable sinusoidal field variation along two orthogonal dimensions. The same, however, is not true of a rectangular dielectric waveguide. The reason is that the fields extend beyond the core into a region where the geometry cannot be defined independently along orthogonal directions. Thus, the problem of the rectangular dielectric waveguide has been addressed via a large number of approximate analytic techniques and numerical methods. Although somewhat dated, Saad presents an excellent review [100]. In the following discussion, we describe two approximate analytical techniques: Marcatili’s method and the effective index method, which through approximation, reduce 2D

2.4 Analysis Methods for Rectangular Dielectric Waveguides

21

Fig. 2.4. The dispersion relation for a planar slab waveguide with n1 = 3.35, n1 = 1.0, and d = 240 nm. For these parameters, light at λ = 1.55 µm, ω/c = 4.05 µm−1 , is just barely single-moded. The TE solutions are in bold linetype, and the TM solutions are dashed. The gray region in the top plot represents the guided wave solution space bounded by core and cladding light lines. The lower plot is a normalized version of the dispersion relation.

structures with rectangular symmetry to equivalent 1D structures separable along each dimension. Guidance along each dimension can then be analyzed using the planar slab waveguide formalism derived earlier. These techniques have their limitations, and so we follow up with a discussion of numerical techniques that have been implemented to solve the waveguide equation.

22

2. Optical Dielectric Waveguides

2.4.1 Marcatili’s Method Marcatili introduced a method whereby rectangular waveguides can be analyzed through a slight alteration of the refractive index geometry [101]. This modification forces the field solutions along Cartesian axes to be separable. The technique works well for 2D rectangular waveguides where the index contrast along each dimension is of similar magnitude. Such is the case for a buried-channel waveguide as depicted in Fig. 2.5. The corresponding Marcatili approximation and equations are shown in the lower figure. Note that the relative permittivities (square of refractive indices) of the guiding channel and four cladding regions at its edge boundaries remain unchanged, but the four corner regions are artificially reduced by the permittivity difference n21 − n22 . For low index contrast guidance of moderate to strongly confined modes, the field is expected to be small in these corner regions and the approximation works rather well. Notice that the same permittivity difference exists across each of the three regions of each of the horizontal and vertical slab waveguide components. Thus, this trick successfully separates the geometry into two orthogonal slab waveguides, each of which can be solved independently. Because the index in the corner regions is artificially lowered, the

Fig. 2.5. Marcatili approximation for a rectangular buried channel waveguide.

2.4 Analysis Methods for Rectangular Dielectric Waveguides

23

result is an underestimated propagation constant. The severity of the error is reduced proportional to the index contrast of the guide and the confinement of the mode. Kumar et al. [102] extended Marcatili’s method by taking the resultant field of the Marcatili solution and reintroducing the corner permittivity difference n21 − n22 as a perturbation. Applying this perturbative method results in greater accuracy albeit with increased complexity. 2.4.2 Effective Index Method The effective index method (EIM) was introduced by Knox and Toulios [103] as an alternative analytic method to that of Marcatili’s for the modeling of rectangular dielectric waveguides. The method is extremely simple to implement and to this day retains popular appeal as a method for quick estimation. Unlike the Marcatili method that solves each dimension in parallel, the EIM proceeds in series solving one dimension first and analyzing the second as a perturbation of the first. Kumar et al. [102] pointed out that ambiguous results are obtained when trying to implement this method for waveguides of square cross section. The method is better suited for modeling rib or ridge waveguides. In these structures, the guidance is nearly a slab waveguide in one dimension with a small perturbation to index or core thickness that achieves guidance in the other dimension. The EIM begins by analyzing the structure for each segmented slab waveguide section and assigning effective indices for each section. Next, the effective indices for each slab segment are used to solve for the resulting slab waveguide geometry in the slightly perturbed dimension. As an example, we apply the method to solve for the dispersion relation and mode profiles of a rib waveguide. Figure 2.6 shows the structure along with the first and second approximation steps. The first step results in three normalized frequencies (V2 , V1 , V2 ) where here, due to the symmetry, the first and third are equal. Using Kogelnik’s generalized dispersion relation allows us to solve for (b2 , b1 , b2 ) and then the effective refractive indices (neff,2 , neff,1 , neff,2 ) of each segment. Next, these auxilliary effective indices define a net normalized propagation constant (V ) that leads to a unique effective index for the mode. In contrast to Marcatili’s method, the EIM overestimates the coupling constant [104] but is more accurate for analyzing the fundamental mode solution near cutoff. Lee et al. [105] pointed out that this is because the effective index method makes the one-mode approximation. Numerous extensions of the EIM have been developed [106–108]. Chiang [108], for example, introduced a correction factor that greatly improves the accuracy of the method. Because modern computers and commercial software can readily solve for modes of waveguides of arbitrary cross section, overrefinement of this method is usually unnecessary. The usefulness of the original method lies in its simplicity.

24

2. Optical Dielectric Waveguides

n3 Rib waveguide n2

Effective index approximation (step 1)

Effective index approximation (step 2)

a2 a 1

n1 b

n3

n3

n3

n1

n1

n1

n2

n2

n2

a2 a1

neff,2 neff,1 neff,2

b Fig. 2.6. Effective index approximation for a rib waveguide.

2.4.3 Goell’s Circular Harmonic Method Goell [109] introduced the first accurate method for computing the propagation constants and mode profiles of rectangular dielectric waveguides. The method is based on the expansion of the mode fields in circular harmonics (Bessel functions in radius and sinusoids in azimuth). For a time, it was the gold standard method against which all other approximate methods were compared. Today if one requires a high degree of numerical accuracy, finite element methods can generally provide high accuracy with better flexibility.

2.4 Analysis Methods for Rectangular Dielectric Waveguides

25

2.4.4 Finite Element Method The finite element method (FEM) applied to waveguide analysis is a variational method for solving the Helmholtz equation along faceted elements of a 2D mesh filling a waveguide cross section and surrounding cladding. The accuracy depends on the number of elements used to approximate the structural geometry. It has the advantage of being able to discretize the region of an arbitrarily shaped waveguide cross section in an adaptive manner. This allows increased numerical accuracy by adding more elements in regions where the fields are expected to be rapidly varying (such as near dielectric discontinuities). Although somewhat difficult to implement, it is both flexible and highly accurate particularly for computing the full vectorial nature of guided mode fields. Many commercial mode-solving packages implement this technique. 2.4.5 Beam Propagation Method The beam propagation method is the name often given to fast Fourier transform (FFT) methods of solving the paraxial diffraction equation [97, 110]. As the name implies, this is a method that can be used to predict the evolution of an arbitrary field distribution injected into one end of a waveguiding structure. The scalar Fresnel paraxial diffraction equation applied to waveguides results from allowing the transverse mode profile A to evolve slowly in z by dropping the second longitudinal derivative in Eq. 2.13 ∂A(x, y, z) i 2 i = ∇ A+ (n(x, y)2 k20 − β2 )A. ∂z 2β T 2β ∂A(kx , ky , z) i i =− (k2 + k2y )A + (n(kx , ky )2 k20 − β2 ) A. ∂z 2β x 2β (2.67) Here, we present the form of the equation in both the spatial (x, y) and the wavevector (kx , ky ) or Fourier domains. The wavevector spectrum associated with a field is directly related to the angular spectrum through the transformation n sin(θx,y )/λ = kx,y . The first term on the right side of each equation corresponds to Fresnel diffraction. In the spatial domain, the second-order derivatives associated with the transverse Laplacian operator are cumbersome to compute. In the wavevector domain, however, the plane wave or angular components of the field advance with a propagation constant that exhibits a simple quadratic dependence with spatial frequency. Hence it is both natural and simple to numerically compute Fresnel diffracted fields in the Fourier domain. The second term on the right side of each equation corresponds to the offset in propagation constant from the assumed β value in each region of the waveguide structure. The index variation associated with a waveguide is represented in

26

2. Optical Dielectric Waveguides

this term and is of course defined locally in space. It follows that the spatial domain is the natural domain in which to apply this perturbation (the numerically taxing convolution operation denoted by is avoided). The BPM takes advantage of the natural solution domains and solves the equation alternately in the Fourier domain for Fresnel diffraction and in the spatial domain for refractive index waveguiding. The main advantage of this method is its speed. Limitations of the method are restriction to paraxiality and low index contrasts as well as an inability in its basic form to handle retro-reflections (which of course are highly non-paraxial). The modes of a waveguide may be determined through use of the BPM via impulse response methods. A spatially and temporally localized field (such as a short-pulsed point source) is introduced into the waveguide and is allowed to propagate. The spatial spectral content that does not lie within one of the modes of the structure radiates into continuum modes and what is left behind after a suitable propagation length is a superposition of all the exited modes of the structure. By Fourier transforming the resulting field distribution both along the propagation axis, and in time, the modes are clearly defined as sharp peaks in the propagation constant spectrum for each spectral frequency. 2.4.6 Finite-Difference Time-Domain Method The finite-difference time domain (FDTD) method typically refers to the Yee method of solving Maxwell’s equations [111, 112]. As a result, it can be very accurate for the solution of compact photonic structures, particularly those involving fully vectorial, highly nonparaxial propagation (associated with high index contrasts) and rapidly time varying (ultrafast) pulse envelopes. The primary drawback associated with this method is its computational intensiveness. For example, 1/20th wave resolution at λ = 1 µm, over a 10-µm2 window results in a 40,000 count pixel matrix for 2D (and 8,000,000 for 3D) in which the discretized form of Maxwell’s equations must be solved every 1/20th of an optical cycle to satisfy numerical dispersion conditions (Courant). Another drawback is a somewhat inflexible grid. Despite these limitations, FDTD methods have become very common for simulating the transient evolution of fields in ring resonators and photonic crystals. The modes of a photonic structure can be obtained much in the same way as with the BPM by introducing temporal impulses and by spectrally resolving the fields after some suitable propagation distance or time to determine the dispersion relations.

2.5 Coupling When two waveguiding cores are situated in close proximity to each other, optical power can be exchanged between their supported modes. The analysis of this problem can proceed by considering the collective normal

2.5 Coupling

27

Fig. 2.7. Coupling of two slab waveguides. The horizontal lines represent the interfaces between the core and the cladding. The direction of propagation is left-to-right (or vice versa).

modes of the two waveguides to be superpositions of the modes of the individual guides. In the case of single-moded waveguides, both symmetric and antisymmetric solutions exist for the coupled structure. In general, the symmetric and antisymmetric modes possess different propagation constants. If all the power is initially in one waveguide, the field distribution can be represented as one particular superposition of the symmetric and antisymmetric modes adding coherently in that waveguide’s core. Since the two modes have different phase velocities, over some length (the “beat” length), optical power is transferred to the other waveguide and continues to slosh back and forth while the two waveguide cores maintain their close proximity (Fig. 2.7). The beat length depends on the separation of the two waveguide cores. If the cores are far apart, the beat length is infinitely long, and for all practical purposes, the modes will not couple. Coupling coefficients can be calculated from the effective indices of the symmetric and antisymmetric modes or by using the modes of the individual waveguides and applying a perturbation-based approach described in what follows. 2.5.1 Perturbation Method for Deriving Coupling Coefficients There are a variety of methods for computing distributed coupling coefficients κj,k . Finite-difference time-domain methods can give this result directly but are computationally intensive. Finite element methods are often better suited to waveguides with non-Cartesian or noncircular cross sections. By solving for the normal modes of the composite structure, one can obtain the propagation constants for symmetric and antisymmetric modes that are related to the coupling coefficient via κ = |∆β|/2. We next

28

2. Optical Dielectric Waveguides

present an analytic method that can be employed if the fields associated with the coupled modes can be defined exactly or approximately with analytic expressions. The derivation of the coupling coefficients between two modes proceeds assuming that the field associated with mode j induces a polarization within the core of mode k. We begin by writing Maxwell’s vector curl equations for the two modes. ∇ × Ek = +iωµ0 Hk

(2.68)

∇ × Ej = +iωµ0 Hj

(2.69)

∇ × Hk = −iωεEk

(2.70)

∇ × Hj = −iωεEj − iωPper,j ,

(2.71)

where the asymmetry results from the fact that we are allowing mode k to be perturbed by mode j inducing polarization Pper,j . Combining these equations results in:

 



 ∗ ∗ ∇ · Ej × H∗ (2.72) k = +iω µ0 Hj · Hk − ε Ej · Ek

 



  ∗ ∗ ∗ ∇ · E∗ (2.73) k × Hj = −iω µ0 Hj · Hk − ε Ej · Ek − Pper,j · Ek Adding these equations and integrating both sides across the transverse dimension results in:   

∗ dxdy = iω Pper,j · E∗ + E × H (2.74) ∇ · E j × H∗ j k k k dxdy. Gauss’s theorem eliminates the transverse components of the divergence operator yielding: 



  ∂ ∗ ∗ EjT × HkT + EkT × HjT dxdy = iω Pper,j · E∗ k dxdy. (2.75) ∂z where the “T” subscripts refer to the transverse components of the fields. The orthonormality of modes can be written in the following manner: 

   1

z · ejT x, y × hkT x, y (2.76) dxdy = δj,k , 2 where the variables e and h represent the field strengths for modes normalized to possess unit power per length. The Kronecker delta δj,k is zero for j ≠ k and unity for j = k. Implementing these orthonormal fields and extracting the longitudinal field dependence for the perturbed mode as Ak (z), the expression for the longitudinal variation in field strength of the perturbed field begins to take form: 

 ∂ iω Ak = Pper,j · Ek ∗ dxdy. (2.77) ∂z 4

2.5 Coupling

29

Finally, implementing orthonormal fields for the induced polarization and extracting the longitudinal field dependence for the perturbing mode Aj (z): ∂ iω Ak = ∂z 4

 ! ∆εejT Aj eiβj z +

 " ε∆ε −iβk z dxdy ejz Aj eiβj z · e∗ e k ε + ∆ε

(2.78) ⎧ ⎫ ε ∆ε  ∗ ∗ r r ⎨ ik ⎬ ∆εr ejT · ekT + εr +∆εr ejz · ekz ∂ 0 Ak = dxdy Aj ei(βj −βk )z . ⎩ 2 ⎭ ∂z 2Z0 (2.79) From this expression we see that a coupling coefficient can be extracted as: ⎧ ⎫ εr ∆εr  ∗ ∗ ⎬ k0 ⎨ ∆εr ejT · ekT + εr +∆εr ejz · ekz κjk = dxdy . (2.80) ⎭ 2 ⎩ 2Z0 This overlap integral must be solved to determine the coupling per unit length. In the following section, we apply this integral to the coupling between the lowest order modes of two symmetric TE planar waveguides. 2.5.2 Coupling Between Symmetric TE Planar Waveguides The expression for the coupling coefficient between symmetric TE planar waveguides results from the following overlap integral: d

κ21 =

k0 dx 2 0



 ∗ n21 − n22 Ey1 Ey2 2Z0 P/L

.

(2.81)

Substituting the field profile expressions derived earlier for a core separation of s yields:
d  |E |2  n21 − n2eff −γ (s+x−d/2) 0 dx cos (kx x)
e x κ21 = k0 n21 − n22 2 2 4Z0 P/L n − n 1 2 0 (2.82) that can be shown to reduce to the following expression:



 2 n2eff − n22 n21 − n2eff e−γx s 2b (1 − b) n21 − n22

 κ21 = = k0

e−γx s . √  1 neff n21 − n22 deff 2+V b b + n2 /n2 −1

1

2

(2.83)

30

2. Optical Dielectric Waveguides

Making the following substitutions: n ≡ n1 /n2

(2.84)

V1 ≡ n1 k0 d = √ G≡

s , d

V 1 − n−2

(2.85) (2.86)

the coupling coefficient can be written in a slightly different normalized form: √ √ √ √ √ −2 κ21 2 b (1 − b) e−V bG 2b (1 − b) 1 − n−2 e−V1 1−n bG


, =

→ √ √ √  1 n→∞ n1 k0 2+V b 2 + V1 1 − n−2 b b + 2 n −1

(2.87) where the last form of the expression refers to the high index contrast limit applicable to pedestal waveguides. 2.5.3 Coupled Wave Formalism We next introduce a set of first-order coupled-wave equations that model the exchange of power between two waves of arbitrary propagation constants β1 , β2 and coupling coefficients κi,j . d A1 = iκ11 A1 + iκ21 A2 ei(β2 −β1 )z dz d A2 = iκ12 A1 ei(β1 −β2 )z + iκ22 A2 . dz

(2.88) (2.89)

These equations can be combined into a single second-order differential )2 = A2 e−i(κ22 +δ)z )1 = A1 e−i(κ11 −δ)z or A equation for either A  

 d2 2 )j = 0, A + δ + κ κ (2.90) 12 21 dz2 where the detuning parameter is defined as 2δ = (β1 + κ11 ) − (β2 + κ22 ). Integration of this equation along the mutual propagation direction z yields lumped self- and cross-coupling coefficients:         δ2 2 2 + κ2 z δ2 + κ 2 z + 2 sin δ (2.91) r 2 = cos2 δ + κ2     κ2 t2 = 2 sin2 δ2 + κ 2 z . (2.92) 2 δ +κ Here, it is assumed that κ12 = κ21 ≡ κ. Note that the lumped crosscoupling efficiency can be limited to a value below 100% for fields that are phase mismatched (δ ≠ 0). The maximum achievable coupling effi2 ciency (tmax = κ√2 /(δ2 + κ 2 )) is achieved at a minimum interaction length of zmax = π /(2 δ2 + κ 2 ). Extensive treatment of this coupled mode formalism can be found in the literature [113–116].

2.5 Coupling

a1

b1

a2

b2

31

Fig. 2.8. Coupling of fields in a 2 × 2 directional coupler. 2.5.4 The Scattering Matrix The lumped coupling coefficients are typically implemented in a scattering matrix formalism. Consider a set of coupled waveguides (Fig. 2.8). The output fields are b = [b1 , b2 ]t , and the input fields are a = [a1 , a2 ]t . The scattering matrix S = sij , that relates the output fields to the input fields, is b = Sa. (2.93) Then, using power conservation and time-reversal, we can show that the scattering matrix is symmetric and has the form [117]   rc tc , (2.94) S= t tc −rc∗ tc∗c where

tc∗ rc + rc∗ tc = 1.

(2.95)

We can choose our reference planes such that rc = −rc∗ tc /tc∗ = r ∈ R (where R is the set of real numbers). Then, we can define t ∈ R such that tc = ±it. Then r 2 + t 2 = 1 for lossless coupling, and eq. 2.94 becomes   r it S= . (2.96) it r 2.5.5 Optimized Coupling for Waveguides and Resonators In practice, the coupling strength may need to be optimized so as to to achieve the greatest possible coupling coefficient to keep device dimensions small. An accurate modeling tool (such as FDTD) should be performed as a final check, but there are fundamental limitations on the maximum achievable coupling per unit length. At best, a distributed coupling of no more than κ ≈ 2n1 /λ is achievable. The required distributed coupling is obtained from the lumped cross-coupling coefficient t from t = sin(κzint ), where zint is the effective interaction length. An

32

2. Optical Dielectric Waveguides

analytic expression for the TM4 distributed coupling in terms of normalized waveguide parameters (derived earlier) is given by √ √ 2 b (1 − b) e−V b(s/d) √ κ → n1 k0 . (2.97) n1 >>n2 2+V b Two crucial implications result from this high index contrast limit [118]. The first is that in the high contrast approximation, the coupling coefficient is independent of index contrast (although it implicitly appears in the normalized frequency V -parameter). The second is that for high contrast laterally coupled TM modes,5 the normalized frequency V and normalized gap width G completely dictate the coupling. Thus, practically, there are only two parameters that are critical and that can be chosen to determine the coupling. Provided that the material parameters and wavelength are fixed, both the guide width and the gap width must be chosen appropriately. Even if the guides are touching, their coupling may still be weaker than needed over a short distance because the guide dimension may be too wide. The TE distributed coupling is more complicated and cannot be written in such a compact form. Figure 2.9 shows the variation of coupling strength versus the normalized frequency and gap width in the high index contrast planar waveguide approximation for TM modes. Note that it predicts a maximum value of the coupling strength lying between V = 0 and V = π in the region of single-moded behavior. Recall that for evanescent coupling, the evanescent tail of a mode extending beyond one guide excites the material polarization in a second guide that in turn reradiates into its mode. For large enough guides, the portion of confined optical power present in the evanescent tail is negligibly small leading to low overlap. In the other limit, for small enough guides, the evanescent tail becomes the dominant portion of the confined optical power and is spread widely throughout space such that only a small fraction of it overlaps the other waveguide. In a racetrack geometry, the coupling region is extended with a straight waveguide section to greatly increase the coupling, at a modest reduction of the FSR and introduction of some junction loss [119]. This workaround can achieve a low loading finesse in small resonators where the effective interaction length is limited. Figure 2.10 displays the required interaction length to achieve a given loading finesse associated with a single coupler. Other considerations for tweaking the coupling strength lie in the choice of air-cladding, filled cladding, or rib waveguiding. In order to confine light in a microresonator, it is necessary that the lateral index contrast 4 5

TM with respect to the substrate (vertical electric field) is really TE with respect to the coupling interfaces for laterally coupled waveguides. Again, to avoid confusion, the convention for distinguishing between TE and TM modes is with reference to the substrate. A mode with a dominant electric field perpendicular to the substrate surface is denoted TM. Such a mode field is, however, parallel with respect to the lateral interface, and thus, the TE Fresnel reflection and planar waveguide laws apply.

Normalized coupling, κ / k1

2.5 Coupling

33

d

0.3 s/d=0

s

0.2 s/d=1 0.1 increasing gap/width ratio 0 0

π/2 π 3π/2 Normalized frequency, V=k0d NA

Fig. 2.9. Normalized coupling strength (κ/k1 ) versus normalized frequency (V = k0 dNA), and normalized gap (s/d) in the limit of high index contrast for the planar waveguide approximation to TM guide-to-disk coupling.

Fig. 2.10. The required interaction lengths for TM waveguides in the high index contrast limit. Shown are the lengths required to achieve a resonator loading finesse of 10, 100, and 1000. Note that in many cases, the required interaction length varies greatly with differing gap and guide widths indicating a need for extremely fine tolerances on design geometry.

34

2. Optical Dielectric Waveguides

be high. This implies that, in order to make the guides single-moded laterally and to maintain high lateral coupling by exposing a higher fraction of the evanescent tail of the modes, the guides would have to be very thin in the lateral dimension. This still makes it difficult to couple a significant fraction of light from free space or low NA single-mode fiber. However, although it is difficult to taper the guides out in the vertical dimension, it is generally a straightforward task in the lateral dimension. Thus, a lateral-tapering guide with high lateral index contrast that is initially highly multi-moded can be used. In the vertical dimension, however, there is no requirement for high index contrast. As such, the input coupling efficiency can be improved by making the guides possess a reasonably large vertical core dimension that is maintained single-moded by employing low vertical index contrast. A finite-element solver can be used to solve for the 2D modes and coupling coefficients associated with the cross section of a pair of waveguides [Fig. 2.11]. This can be most useful for determining the coupling coefficient (per unit length) that can then be integrated (along z) to find the required coupling length for a desired coupling coefficient. This method is appropriate for modeling the couplers in a Mach–Zehnder or the coupling to and from racetrack resonators in which the gap per unit length does not vary longitudinally over most of the coupling region. For more complicated geometries, such as that of bent resonator-to-straight waveguide coupling, numerical techniques are generally required. The three-dimensional nature of the problem can sometimes be reduced to two dimensions by employing the effective index method. This method is implemented by first solving for the propagation constants of equivalent planar waveguides of regions that possess similar vertical guidance. Once the propagation constants for differing regions is known, an effective refractive index is assigned to them and the problem eliminates any further need to calculate along the vertical dimension. These effective indices are then used to model the photonic structure in only two dimensions by using an FDTD solver. By measuring the power associated with fields coupled into a ring or disk resonator after a single round-trip, the FDTD method readily gives estimates of the coupling coefficient. Phase matching the waveguide mode to the mode of the ring or disk may be an important consideration when requiring a high lumped coupling strength. Specifically, ensuring that the modes are phase matched is important when the difference in propagation constants ∆β is greater than a critical value determined by the coupling per unit length, κ and the desired lumped coupling coefficient t 2 such  that ∆β < κ (1 − t 2 )/t 2 . Fortunately, for small enough resonators, phase matching is not a significant issue. The tolerance on maximum allowable beta-mismatch increases in inverse proportion to the interaction length of guide and resonator. The tolerance on the gap dimension and fidelity, however, remain quite strict.

2.5 Coupling

35

Fig. 2.11. Finite element simulation (in the commercial software package Comsol) of the normal modes of a coupled pedestal waveguiding structure. The n = 3.4 cores are 1-µm tall and 0.5-µm wide supported and capped by n = 3.2 cladding layers. For a gap width of 100 nm, the symmetric and antisymmetric modes at λ = 1.55 µm propagate with different propagation constants (βsym = 12.85 µm−1 , βantisym = 12.62 µm−1 ). The spatial beat length is directly related to the coupling strength and this configuration exchanges 100% power over 13.7 µm. In these plots, the contours represent the electric field magnitude, arrows represent the electric field direction (TM), and the shading represents the intensity.

36

2. Optical Dielectric Waveguides

2.6 Bending Loss A critical factor in the fabrication of small ring resonators is controlling the loss in the ring. Loss in microrings and pedestal waveguides is typically dominated by leakage to the substrate, scattering or contradirectional coupling due to edge roughness [120–122] and bending [2, 66–69]. Loss due to substrate leakage can be solved by waveguide design, and that from edge surface roughness can be solved with process improvements, such as high-resolution lithography, vertical photoresist sidewalls, and hard masks. However, bending losses are dictated by the refractive index contrast associated with the choice of materials. The index contrast prescribes a minimum size for ring resonators in a material system. The index contrast that determines the minimum bending radius is that in the lateral direction (the plane of the ring). The cladding in the plane of the ring, for a pedestal waveguide, is typically air, a lowindex polymer, like amorphous polytetrafluoroethane (better known by the DuPont brand Teflon, n = 1.33) or benzocyclobutene (n = 1.51), SiO2 (n ∼ 1.5), spin-on glasses (n ∼ 1.5), SiON (n ∼ 1.5–2.1), or polyimides (n ∼ 1.7). Since the core in the case of III–V waveguides typically has a refractive index >3, the minimum bend radius possible is <1 µm, and in practice bending losses are rarely a concern with proper design. Before delving into the origin of bending loss [2, 66–69], let us examine the effect of index contrast and waveguide type (strip-loaded versus pedestal) on the mode profile (intensity distribution across the waveguide); the next section establishes the connection between the index contrast and the bending loss. For optical fibers that have a (doped) silica core, the index contrast is very small — around a few tenths of a percent. For rib or strip-loaded III–V semiconductor waveguides, the index contrast is generally higher around 5 − 10% in the vertical dimension. Along the lateral dimension, the index contrast is provided by the presence of three slab-like regions with small effective-index difference, with the central slab having slightly higher effective index than the outer slabs. The confinement in the lateral dimension is therefore weaker than in the vertical [Fig. 2.12(a)]. The waveguide mode is very asymmetric and elliptical in shape. By contrast in the case of pedestal waveguides, the index contrast is high in both dimensions but particularly in the lateral dimension. The mode is well-confined in the waveguide core [Fig. 2.12(b)]. Bending losses result from the inability of light to remain confined inside a bending open waveguide. Radiation takes the form of an outgoing cylindrical accompanied by an azimuthal component resulting in spiral phase contours. In the low Q limit, the bending loss associated with a cylindrical geometry can be computed exactly. In the high Q limit, it makes sense to employ other methods. In a finite 2D geometry, there is some out-of-plane radiation loss.

Strong confinement

2.6 Bending Loss

37

Air n3

GaInAsP-InP:

Core, n2

n1

n1 ∼ n3 ≈ 3.17 (InP) n2 ∼ 3.3 − 3.5 Weak confinement (a)

Strong confinement

Air Air

n3

n1

Air

Core, n2

Strong confinement GaInAsP-InP: n1 ∼ n3 ≈ 3.17 (InP) n2 ∼ 3.3 − 3.5 (b)

Fig. 2.12. Schematic of (a) a strip loaded waveguide and corresponding mode profile and (b) a pedestal waveguide and corresponding mode profile. Shown in each case are nine contours, with the innermost representing the locus of points where the intensity falls to 0.9 times the maximum, and the outermost representing the locus of points where the intensity falls to 0.1 times the maximum. Analytical solutions to the problem of bending loss can be obtained by using a conformal transformation to convert the problem to an equivalent slab waveguide with a graded index [67]. Marcuse used a different approach to provide an analytical form; he calculated the loss from the

Minimum radius for 10 % loss in one wavelength, µm

38

2. Optical Dielectric Waveguides

100 ncore = 1.5 10 ncore = 2.0 1

ncore = 3.5

1

1.5

2

2.5

3

3.5

Cladding refractive index Fig. 2.13. Minimum radius for 10% loss in one wavelength versus cladding index for various core indices. radial component of the field at large distances from the waveguide [69]. Marcuse’s analytical form is used to plot Fig. 2.13. Regardless of the approach used, we can estimate trends in waveguide behavior from index contrast (the difference in index between core and cladding). Since pedestal waveguides provide the strong index contrast necessary for minimizing bending loss, they are are preferred for photonic circuits with very small bending radii. Pedestal waveguides are referred to as “tight-confining” or “strong-confining” because of the corresponding mode shape. Although it may seem like a good idea to use ever-higher index contrasts, in practice, high index contrast waveguides pose serious problems. Fabrication tolerances are reduced because the guide dimensions must be much smaller than low-index-contrast waveguides to maintain single-mode behavior. Finally, since the modes are much smaller than in conventional fibers, coupling inefficiencies can be quite high. In the next section, we reexamine the phenomena of bending loss from the formalism of whispering gallery modes.

2.7 Whispering Gallery Modes In 1912, Lord Raleigh analyzed the curiosity of the “whispering gallery” that involved the propagation of acoustic waves skimming along the interior of the dome of St. Paul’s cathedral [1]. The term has since come to be applied to a family of modes of a cyclical curved interface such as a

2.7 Whispering Gallery Modes

39

Fig. 2.14. Geometry for a TM whispering gallery mode. cylindrical or spherical surface, where wave propagation is confined primarily to the inside surface of an interface and guided by it by repeated reflection. Light propagating in a conventional optical waveguide is confined via total internal reflection by two dielectric interfaces (along each dimension) to a region of high refractive index. The guiding region of high refractive index is formally equivalent to a potential well wherein the electric fields may be decomposed into eigenmodes that are solutions of the Schrödinger equation. Light propagating in a curved waveguide is still guided via TIR at the outer interface, but it no longer demands an inner interface to complete the confinement. Elimination of the inner boundary leaves a dielectric disk that supports whispering gallery modes. These modes consist of azimuthally propagating fields guided by TIR at the dielectric interface and “optical inertia” that prevents the field from penetrating inward beyond a fixed radius termed the inner caustic. Mathematically, a whispering gallery mode (WGM) is a solution of the Helmholtz equation in a curved coordinate geometry as in Fig. 2.14. Attention is restricted to a cylindrical geometry appropriate for the analysis of planar disk and ring resonators. 2.7.1 TM Whispering Gallery Modes  The Helmholtz equation ∇2 + k2 Ez = 0, written in cylindrical coordinates for the axial field of a TM6 whispering gallery mode is  ∂2 1 ∂2 1 ∂ 2 (2.98) + 2 + + k Ez (r , ϕ) = 0. ∂r 2 r ∂r r ∂ϕ2 The method of separation of variables can be employed to simplify this equation by splitting it into two separate equations for radial and azimuthal dependance. An integer7 parameter m is introduced to connect the two equations. This parameter physically corresponds to the 6

7

Once again, the accepted nomenclature for assigning TE or TM designations for cylindrical whispering gallery modes, is to follow the conventions for that of planar waveguides. For example, in the limit of an infinitely long cylinder, only one electric field component exists in the TM mode solution - the one directed axially. As a result, the TE and TM designations with respect to the sidewall seem to be reversed. The restriction that m be an integer assumes a resonance.

40

2. Optical Dielectric Waveguides

number of optical cycles (effective wavelengths) the field undergoes in space when completing one revolution around the disk. The azimuthal equation takes the form,  ∂2 2 (2.99) + m Ez (ϕ) = 0, ∂ϕ2 and has solutions that are simply complex exponentials: Ez (ϕ) = e±imϕ .

(2.100)

The resulting (Bessel) equation for the radial field dependence becomes:  ∂2 1 ∂ m2 2 (2.101) + k − 2 Ez (r ) = 0. + ∂r 2 r ∂r r The solutions of Bessel’s equation are the Bessel functions of the first Jm and second Ym kind. Because the second kind function is singular at the origin, only the first kind function is retained inside the disk. Outside the disk, both functions are well behaved8 and must be retained. The Hankel functions are linear superpositions of the two Bessel function (1) (2) solutions corresponding to outward Hm = Jm + iYm and inward Hm = Jm − iYm propagating cylindrical waves. The analysis of waves arriving back at the resonator from the radial horizon is not considered here, (1) and thus, only the Hankel function of the first kind, Hm is retained. The appropriate solutions for the radial field dependence both interior (r < R) and exterior (r > R) to the dielectric disk boundary [123] are )1 r ) Ez (r < R) = Am Jm (k Ez (r > R) =

(1) ) B m Hm (k2 r ),

(2.102) (2.103)

) j = nj ω */c are where a complex propagation constant and frequency k introduced for reasons that will become apparent later. The complete axial electric field interior and exterior to the disk is constructed from the azimuthal and radial solutions, including the boundary condition at the interface (r = R) that forces the tangential electric field to be continuous:

 ωt ) )1 r ei(±mϕ−* (2.104) Ez (r , ϕ) = Am Jm k 

)1 R

 Jm k ωt ) (1) )  Hm k2 r ei(±mϕ−* . (2.105) Ez (r , ϕ) = Am (1)

)2 R Hm k 8

The fields outside the disk are not modified Bessel functions of the first kind, Km as in the case of bound modes of a circular dielectric waveguide or optical fiber because the absence of an axial propagation constant eliminates the possibility of modified Bessel function solutions.

2.7 Whispering Gallery Modes

41

Finally, the radial and azimuthal magnetic field components are easily derived from the axial electric field by use of Maxwell’s equations −i 1 ∂ m Ez = Ez )0 r ∂ϕ )0 r Z0 k Z0 k ∂ i = Ez . )0 ∂r Z0 k

Hr =

(2.106)

Hϕ

(2.107)

2.7.2 TE Whispering Gallery Modes The Helmholtz equation for the axial field of a TE whispering gallery mode is  ∂2 1 ∂2 1 ∂ 2 (2.108) + 2 + + k Hz (r , ϕ) = 0. ∂r 2 r ∂r r ∂ϕ2 The equation is again simplified via the method of separation of variables, and the azimuthal equation takes the form,  ∂2 2 (2.109) + m Hz (ϕ) = 0, ∂ϕ2 possessing complex exponential solutions, Hz (ϕ) = e+imϕ , e−imϕ . The radial equation:  ∂2 1 ∂ m2 2 (2.110) + k − 2 Hz (r ) = 0, + ∂r 2 r ∂r r is Bessel’s equation with equivalent solutions as in the TM case. The appropriate solutions for the radial dependence of the magnetic field are )1 r ) Hz (r < R) = Am Jm (k Hz (r > R) =

(1) ) Bm H m (k2 r ).

(2.111) (2.112)

The complete axial magnetic field interior and exterior to the disk is constructed from the azimuthal and radial solutions, including the boundary condition at the interface (r = R) that forces the tangential magnetic field to be continuous:

 ωt ) )1 r ei(±mϕ−* Hz (r , ϕ) = Am Jm k (2.113) 

)1 R 

Jm k ωt ) (1) )  Hm k2 r ei(±mϕ−* . (2.114) Hz (r , ϕ) = Am (1)

) Hm k 2 R Finally, the radial and azimuthal electric fields are easily derived from the axial magnetic field by use of Maxwell’s equations,

42

2. Optical Dielectric Waveguides

iZ0 1 ∂ −mZ0 Hz = Hz 2 ) )0 r r ∂ϕ n k0 n2 k −iZ0 ∂ Hz . = )0 ∂r n2 k

Er =

(2.115)

Eϕ

(2.116)

2.7.3 Radiation Loss For an open-boundary structure such as a dielectric disk,9 whispering gallery modes are inherently “leaky” [124]. The mechanism for loss is a tunneling or coupling of the azimuthally guided mode into radially outward-going radiation modes. This phenomenon is directly related to bending loss and is sometimes referred to as whispering gallery or radiation loss. It is a property of any curved open boundary waveguide configuration. A convenient measure parameterizing this loss is the intrinsic quality factor that quantifies the number of optical cycles a mode will last confined within a resonator. Theoretical predictions show that the intrinsic quality factor (Q) can be as high as 1011 , and experiments have confirmed this fact [125]. Intuitively, the WGM is confined within a radial potential well (m/nr )2 that can also be expressed as a effective radial index nr :

m2 nr = n2 − 2 . (2.117) k0 r 2 The inner caustic boundary, below which the “optical inertia” is too great, is defined at R1 = nm . There is an outer radiation boundary, beyond 1 k0 which the azimuthal phase velocity exceeds the speed of light in vacuum is defined at R2 = nm . Figure 2.15 depicts an optical whispering gallery 2 k0 mode with the associated radial potential well and illustrates the regions where the fields in the mode are bounded, evanescent, and radiative (outwardly propagating). The radiation loss associated with whispering gallery resonators may be viewed physically as a tunneling of the confined field through a potential barrier defined by the disk edge and radiation boundary into a region of lower potential. Beyond the radiation boundary, the radially evanescent tail of the field becomes propagating again. For a typical high Q disk resonator, the field has decayed to such a low value that its leakage into cylindrical radiating waves is very small. The predominant propagation direction for a WGM is, of course, primarily in the azimuthal direction such that the phase contours behave like revolving spokes of a wheel. The pattern revolves about the disk center with a angular frequency of ω/m. Because the azimuthal phase contours increase in separation with radius, the azimuthal phase velocity likewise increases 9

As opposed to a closed-boundary structure, i.e., dielectric guiding region with perfectly conducting walls.

2.7 Whispering Gallery Modes

43

Fig. 2.15. Propagation constant kφ as a function of radial distance from disk axis for a whispering gallery mode. The circulating power is confined between an inner caustic and the disk edge. Beyond the outer radiation boundary, the radially evanescent field becomes propagating and acts as a loss mechanism. without bound. At the radiation boundary, the azimuthal phase velocity is equal to the phase velocity in the surrounding medium. Beyond this radius, the fields cannot keep up and thus spiral away. Figure 2.16 graphically illustrates this fact for a very low Q WGM. 2.7.4 WGM Dispersion Relations (Resonance Maps) In order to calculate the mode propagation constants and quality factors for particular WGMs, one must solve the complex WGM dispersion relation. The dispersion relation for whispering gallery modes is similar to the relation for fiber modes with the exception that the axial propagation constant kz is much smaller and perhaps negligible. The dominant propagation constant is of course directed in the azimuthal direction. For an infinite cylinder and a longitudinal propagation constant of zero, the dispersion relation has eigensolutions with complex propagation constants, which implies that either the refractive index and/or the frequency

44

2. Optical Dielectric Waveguides

Fig. 2.16. Plot of the electric field associated with the sixth order TM (axial E-field) whispering gallery mode. Here n1 = 2, n2 = 1, and the resonant radius (solid line) is 1.04 µm at λ = 1.55 µm. This configuration was chosen because it is poorly confined with a low Q of 64 and thus allows easy visualization of the super-evanescent component of the field below the caustic radius (inner dotted line) and the radiating component of the field past the radiation boundary (outer dotted line).

must be complex — a consequence of the radiation losses present in the system. Should the disk possess a negative imaginary component of the refractive index (representing gain), solutions exist that maintain a constant field energy in the resonator while power is steadily radiated. In the case of a lossless/gainless dielectric resonator, however, solutions of the equation necessarily involve a complex frequency, implying a decay rate of energy confined within the resonator. The complex frequency is γ * = ω − i 2 , and a qualrelated to the temporal decay rate γ according to ω ω ity factor may be ascribed Q = γ defined as the characteristic number

2.7 Whispering Gallery Modes

45

of optical cycles before confined energy is lost to the radiation continuum. The Pythagorean theorem does not apply for the variables ω, γ as it does for the traditional dispersion relation variables for a waveguide β, kx . This fact along with the presence of discrete resonances give the dispersion relation a different character — thus, perhaps “complex resonance map” is a more appropriate description. Obtaining solutions involves solving for the complex roots of a complex equation. Many approximations have been employed to simplify this equation, for example, conformal transformation [67, 68, 126], WKB [127], and volume current methods [120, 128]. Because the validity of these methods are in doubt when the resonator circumference approaches a small number of optical cycles, we choose to solve the equation numerically from the dispersion relation. For an infinite cylinder of dielectric material with negligible absorption and no axial component of the propagation vector, we proceed to derive the TM WGM dispersion relation. The field matching equations at the disk boundary (r = R) are expressed as:



  )1 R = A2m H (1) k )2 R A1m Jm k (2.118) m  



 (1) )1 A1m J ) ) )2 R . k (2.119) k m k1 R = k2 A2m Hm The TM dispersion relation can be written as 



(1) ) ) 2 Hm )1 R )1 J  k k k2 R k m



 =  . (1) ) )1 R Jm k Hm k 2R

(2.120)

Next, we proceed to derive the TE WGM dispersion relation. The field matching equations at the disk boundary (r = R) are expressed as:



  )1 R = A2m H (1) k )2 R A1m Jm k (2.121) m  



1  )1 R = 1 A2m H (1) k )2 R . A1m Jm (2.122) k m ) )2 k1 k The TE dispersion relation can be written as  



(1) )  )1 R Jm Hm k k 2R



 = . (1) ) )1 R )1 Jm k ) 2 Hm k2 R k k

(2.123)

The complete vector WGM dispersion relation is obtained by combining the TE and TM equations  ⎤ ⎡   ⎤





⎡ (1) ) (1) )  )1 J  k ) 2 Hm )1 R )1 R k k Hm k k2 R k Jm 2R m ⎣





 −  ⎦⎣ −  ⎦ = 0. (1) ) (1) ) )1 R )1 R ) 1 Jm k ) 2 Hm k2 R Jm k Hm k k k 2R (2.124)

46

2. Optical Dielectric Waveguides

Note the equivalence with the complete vector dispersion relation for circular step index fibers (for β = 0):  ⎤⎡  ⎤



 

⎡   )2 K  −ik )1 J  k )1 R )2 R )2 R )1 R k Km k −ik k Jm m m ⎣





 +  ⎦⎣ +  ⎦ = 0, )1 R )2 R )1 R )2 R )2 Km −ik ) 1 Jm k Jm k −iKm −ik −ik k (2.125) where the following relations hold: Km (−iz) =

 −iKm (−iz) =

iπ i mπ (1) e 2 Hm (z) 2 iπ i mπ (1) e 2 Hm (z) 2

(2.126)

(2.127)

(1)

 Km (−iz) Hm (z) . = (1) Km (−iz) −iHm (z)

(2.128)

2.7.5 Normalized WGM Dispersion Relations (Resonance Maps) In terms of the quality factor, the TM and TE dispersion relations can be written, respectively, as 

 

 (1) 2π R 1 1  n2 Hm n2 2πλ R 1 − i 2Q n1 Jm n1 λ 1 − i 2Q 

 = 

 (2.129) (1) 2π R 2π R 1 1 Jm n1 λ 1 − i 2Q Hm n2 λ 1 − i 2Q 



  (1) 1 1  Jm Hm n2 2πλ R 1 − i 2Q n1 2πλ R 1 − i 2Q 

 = 

 . (1) 2π R 2π R 1 1 n1 Jm n1 λ 1 − i 2Q n2 Hm n2 λ 1 − i 2Q

(2.130)

For a given index ratio, n = n1 /n2 and azimuthal mode number m, a normalized radius X = n1 2π R/λ, and intrinsic quality factor Q may be obtained for a particular WGM solution. Equations 2.130 can be rewritten in terms of the normalized radius and the intrinsic, radiation-limited Q or finesse (F = Q/m). 

 

 (1) 1 1  Jm Hm X/n 1 − i 2mF X 1 − i 2mF 

 = 

 n (2.131) (1) 1 1 Jm X 1 − i 2mF Hm X/n 1 − i 2mF 

 

 (1) 1 1  Jm Hm X/n 1 − i 2mF X 1 − i 2mF 

 = n 

 . (1) 1 1 Jm X 1 − i 2mF Hm X/n 1 − i 2mF

(2.132)

To solve for the roots of these equations, a global optimization scheme can be used to minimize the absolute value of each equation over two

2.7 Whispering Gallery Modes

47

variables: the real and imaginary parts of the propagation constant or X and Q in the normalized formulation. Using this method, generalized plots of limiting finesse against normalized radius may be obtained for the whispering gallery modes of a dielectric cylinder. Figure 2.17 displays the Radiation-loss-limited finesse vs. normalized radius for a variety of azimuthal mode numbers and index ratios.

9 8

Log10 F

7

TE WGMs Hz

n1/n2 = 3.5 Eφ Er

6

3.0 2.5 2.0

5 4

1.7

3 2

1.5

1

1.35 1.2530 20

0

m = 2 5 10 0 20

9 8

Log10 F

7 6

80 40

50

60

90

100

70

40 60 80 Normalized radius, (n1ω/c)R

100

TM WGMs Ez

n1/n2 = 3.5 3.0 Hφ 2.5 Hr

5 4

2.0 1.7

3 2

1.5

1

1.35 1.25 30 20

0 m = 2 5 10 0

20

80 40

50

60

90

100

70

40 60 80 Normalized radius, (n1ω/c)R

100

Fig. 2.17. Radiation-loss-limited finesse of the lowest order radial TE and TM whispering gallery modes of a dielectric cylinder of index n1 in a medium of index n2 plotted against normalized radius (n1 ω/c) R. The family of diagonal lines represents varying refractive index contrast (n1 /n2 ). The family of nearly vertical lines corresponds to whispering gallery mode resonances, each characterized by an azimuthal mode number m. The plots were obtained by numerically solving the dispersion relation for whispering-gallery modes. (After [129], ©2002, Optical Society of America.)

48

2. Optical Dielectric Waveguides

2.7.6 Spheres, Rings, and Disks The quality factors associated with silica microspheres [86–88, 94, 125, 130–138] have attracted much attention for their ultra-high Q factors. Their traditional method of production relies on melting silica (usually a fiber tip) and allowing surface tension forces to reshape the liquid glass into a sphere that is then allowed to cool. Unfortunately, this method does not allow for the fabrication of spheres of reproducible diameters, nor does it allow for spheres much smaller than 20 µm. Recent advances, however, have shown that selective reflow of patterned silica disk edges can result in resonators possessing ultra-high Q factors typically found in microspheres [139]. Propagating whispering gallery modes have been imaged by scanning a near-field probe across the surface of fused-silica microspheres [140, 141], and silicon nitride cylindrical disks [142] whereby light is collected (as it is injected [143]) via frustrated total internal reflection. In the past decade WGMs have attracted increasing attention for their ability to sustain very high Q-factors and low mode volumes. Microdisks and microrings constructed using the techniques of microfabrication and nanofabrication, however, can be readily constructed in a reproducible manner to designed dimensions down to dimensions less than 1 µm. The main drawback of planar fabricated microresonators is the surface roughness left on the edges due to etching processes that result in resonators of much lower quality factors. A ring has two bounding edges that can be used to properly design a single-moded guide. A disk only has one bounding edge; the other boundary is an effective one arising from an inner caustic. As a result, a disk can possess less scattering loss than that of a corresponding ring geometry but is multi moded. If the FSR is high, and/or the side coupling guides preferentially couple to one of the radial modes of the disk, then disks are preferable. If not, then a ring geometry is better and the extra loss must be taken into account [144]. A mode may be considered a “whispering galley mode” if the confinement along some dimension is provided by only a single reflective interface. For a given core-to-cladding refractive-index difference, the loss at a given bend radius decreases with increasing waveguide width until a limit is reached where only the outer core interface is important for guiding. A mode of a curved waveguide (forming a ring) defined by two interfaces would be considered a whispering gallery mode if the inner caustic radius (defined by the azimuthal index m and the wavelength) lies between the inner and outer interfaces. In this regime, light cannot penetrate (towards the origin) beyond the inner caustic, and thus, the interior interface plays a negligible role in the guidance. Thus, a curved waveguide will have similar bending loss per unit radian as a disk with the same exterior radius. It is worthwhile pointing out the advantages and disadvantages of disks and rings. A microdisk may possess higher order radial modes primarily depending on the location of the inner caustic. These radial modes possess differing resonant wavelengths and thus

2.7 Whispering Gallery Modes

120

Disk Singlemoded ring

Radial potential, µm-2

100

Azimuthal # m=8

Radial potential well

80

Index contrast n = 2.5 : 1.0

60 3rd mode

λ=874nm, Q=72

40 2nd mode

λ=1079nm, Q=224

20

49

1st mode

0 0

λ=1438nm, Q=3247

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Radius, µm

Fig. 2.18. A plot of the three supported TM radial modes of an m = 8, 1-µm radius microdisk resonator with index contrast n = 2.5 : 1. The resonance wavelengths for the first, second, and third modes are 1438, 1079, and 874 nm, respectively. Mode quality factors are 3247, 224, and 72, respectively. The suppression of the higher order modes can be achieved by eliminating the interior of the disk to form a ring, here illustrated by a dashed line. might be discriminated against by properly choosing the excitation wavelength. Depending on the quality factors associated with the modes, the resonant wavelength of a particular mode of a microdisk (defined by a radial and azimuthal number) might still overlap another. This may be a problem if only a single-mode is desired.10 A properly designed ring can be used instead to force single-radial-mode operation. However, as will be investigated in more detail later, the presence of an extra sidewall contributes to additional scattering losses. Figure 2.18 shows the radial field distribution associated with the TM modes of a 1-µm-radius disk resonator supporting three radial modes. Each mode possesses an “energy,” 10

Although the modes may overlap, the higher order modes will typically posses a lower coupling coefficient and higher loss.

50

2. Optical Dielectric Waveguides

k20 that lies within the potential energy well (m/nr )2 , which is dictated by the azimuthal number and structure.11 A ring resonator formed by removing the material within a 0.65-µm radius would cut off the secondand third-order radial modes. Its potential energy distribution, shown as a dashed line, only supports the lowest order mode.

2.8 Scattering Losses Resulting From Edge Roughness For microresonators constructed using planar fabrication technology, the dominant loss mechanism is scattering due to edge roughness. In the following discussion we develop further the volume current-based analysis of Kuznetsov, Haus, and Little [120, 128], Borselli et al. [145], and Rabiei [146]. For the design of other waveguide geometries, generalized formulations such as that of Kogelnik and Ramaswamy [99] have been instrumental both in fostering intuition and in making early, high-level design choices. The following derivation proceeds in the same spirit. 2.8.1 Volume Current Method Formulation for Scattering Losses The volume current method is used to determine edge scattering losses associated with microdisk resonators in the whispering gallery mode regime. Only the fundamental whispering gallery mode characterized by a single radial lobe is considered. The time-independent modal field amplitude for a WGM can be decomposed into a transverse mode profile and an azimuthal phase dependence: E(r  , z )eimϕ



(2.133)

where E is the cross section of the modal field amplitude and m is the azimuthal quantization number. A dielectric perturbation on the disk edge may be written as a spatially dependent permittivity distribution ∆ε(r  , z , ϕ ). The dielectric perturbation introduces dipole currents that contribute to outwardly radiated (scattered) fields. The expressions for the perturbed current densities manifest themselves in the form of surface-parallel and surface-perpendicular contributions. 2.8.2 Current Density Contributions The surface-parallel current contribution is readily derived starting from the curl of the Maxwell equation: 11

Strictly speaking, these “energies” do not have the correct units of energy. However, they are formally mathematically equivalent to the energies involved in the solution of the radial Helmholtz equation.

2.8 Scattering Losses Resulting From Edge Roughness

∇×J

=

∂ ∇×D+∇×∇×H ∂t +iω∇ × (εE) + ∇ × ∇ × H

=

+iω∇ε × E + iωε∇ × E + ∇ × ∇ × H

=

−

2

51

(2.134) (2.135) (2.136)

=

+iω∇ε × E − ω µ0 εH + ∇ × ∇ × H

(2.137)

=

+iω∇ε × E.

(2.138)

By definition, the permittivity gradient is oriented normal to the interface; taking the line integral from just below to just above the interface eliminates the curl:  +  + ˆ × (∇ × J) = lim J = lim du u du iωˆ u × (∇ε × E ) = −iω∆ε E . →0

−

→0

−

(2.139) The surface-perpendicular current contribution is readily derived starting from the continuity relation in the absence of free charges. In combination with the expanded Maxwell equation for the divergence of the displacement vector: ∇·J

=

∇·D

=

∇·J

=

∂ρ = iωε0 ∇ · E ∂t ε∇ · E + ∇ε · E = 0 ∇ε −iωε0 2 · D. ε −

(2.140) (2.141) (2.142)

Again, the permittivity gradient is oriented normal to the interface; taking the integral eliminates the divergence:   

1 ˆ du∇ · J = iωε0 du∇ D⊥ = iωε0 ∆ ε−1 D⊥ . (2.143) J⊥ = u ε The current density associated with boundary-continuous parallel electric and perpendicular displacement fields in the presence of the dielectric perturbation is thus given as 

   J(r  , z , ϕ ) = −iω ∆εE (r  , z ) − ε0 ∆ ε−1 D⊥ (r  , z ) eimϕ . (2.144) The dielectric perturbation at the disk boundary can be written as a radial step variation at the interface between the two dissimilar refractive indices (n1 for the core, and n2 for the cladding), + , ∆εin = ε0 (n22 − n21 ) step −∆R(z , ϕ ) (2.145) + , 2 2   ∆εout = ε0 (n1 − n2 ) step ∆R(z , ϕ ) (2.146)  , + 1 1 1 −1 ) = ∆(εin − 2 step −∆R(z , ϕ ) (2.147) 2 ε0 n2 n1  , + 1 1 1 −1 ∆(εout ) = − 2 step ∆R(z , ϕ ) . (2.148) 2 ε0 n1 n2

52

2. Optical Dielectric Waveguides

The interpretation is that the field just inside the interface is perturbed by a lower permittivity when the radial variation is negative and that the field just outside the interface is perturbed by a higher permittivity when the radial variation is positive. 2.8.3 Spectral Density Formulation for Edge Roughness The mode is primarily affected by perturbations along the direction in which the propagation vector is dominant — here azimuthal. Moreover, etch processes tend to deliver uniform corrugations along height, z. This justifies a decomposition of the radial variation into a Fourier series expansion of corrugation harmonics of azimuthal quantization number M ∞ -

∆R(z , ϕ ) =



∆RM (z )e−iMϕ .

(2.149)

M=−∞

The amplitude of each corrugation harmonic may be determined from an experimental measurement of the corrugation. Typically, Gaussian statistics apply as in Borselli [145] where the roughness can be characterized by two parameters: the rms value of the roughness σ and its correlation length Sc . Here, the correlation function for edge roughness, C(s) is related to the measured variation in radius ∆R(s) with respect to arc length s at the disk edge over a suitable measured total arc length Smeas . C(s) =

1 Smeas

Smeas 

ds  ∆R(s  )∆R(s  − s).

(2.150)

0

The value of the correlation function at zero is equal to the mean squared roughness C(0) = σ 2 . A Gaussian correlation function can be defined as: C(s) = σ 2 e

−π



s Sc

2

.

(2.151)

When written in this form, the correlation length is within a factor of  π /4 ln 2 = 1.064 of a full width at half maximum (FWHM) definition. The result of further manipulations is also cleaner, hence, the motivation. The spectral density is equal to the Fourier transform of the correlation function, which is a Gaussian function of spatial frequency variable fs , 2

C(fs ) = σ 2 Sc e−π (Sc fs ) .

(2.152)

If the entire circumference of a disk were to be mapped, a Fourier series representation with harmonics of integer azimuthal quantization numbers would emerge naturally. In practice, it is not often feasible to measure the entire circumference; thus, the amplitude coefficients of the

2.8 Scattering Losses Resulting From Edge Roughness

53

Fourier series expansion of the corrugation must be extrapolated from the limited data. To obtain those amplitudes, the spatial frequency variable is expressed as fs = M/2π R and the spectral density is integrated around each integer M. 2 ∆RM

=

=

≈

1 2π R

σ2

σ2

M+ 12



 dM C

1 M− 2

Sc 2π R

M 2π R

M+ 12



dM e M− 12

Sc −π e 2π R



−π

Sc 2π R M





Sc 2π R M

(2.153)

2

(2.154)

2

.

(2.155)

For most cases of interest, in comparison with the disk circumference, the correlation length is very small Sc /2π R  1 allowing the final trapezoidal approximation to the integral to hold. 2.8.4 Far-Field Scattered Power Returning now to the electrodynamics of scattering, the vector potential in the far field consists of the volume-integrated current density vector with a retardation phasor term to account for coherent interaction among the current density elements,  µ0  dV  J(r  , z , ϕ )eikr cos ψ . (2.156) A= 4π r Here, the volume integral is represented in cylindrical coordinates appropriate to the geometry of the circulating mode while the far-field scattering direction is represented in spherical coordinates (see Fig. 2.19). The

Fig. 2.19. The geometry used in the volume current method formulation for edge scattering losses in microresonators, here shown for a microdisk. The roughness perturbations on the disk edge are parameterized in cylindrical coordinates (r  , z , ϕ ) whereas the scattered radiation is parameterized in spherical coordinates (r , θ, ϕ).

54

2. Optical Dielectric Waveguides

angle cosine between the current density element and the observation point is expanded in spherical coordinates as cos ψ ≈ cos(θ) cos(θ  ) + sin(θ) sin(θ  ) cos(ϕ − ϕ ).

(2.157)

For most geometries in which the disk height is smaller than the radius d  R, the small polar angle approximations about θ = 90◦ can be made, sin(θ  ) ≈ 1 z cos(θ  ) ≈  . r

(2.158) (2.159)

Incorporating these approximations results in a volume integral written completely in cylindrical coordinates: µ0 A= 4π r

+d/2 

dz −d/2



∞ 

dr 0



2π 

r  dϕ J(r  , z , ϕ )eikz



cos θ ikr  sin(θ) cos(ϕ−ϕ )

e

.

0

(2.160) Note that the integrated vector potential will be azimuthally independent due to the inherent symmetry of the geometry yet retain a polar dependence. In the far field, the electric and magnetic fields and Poynting vector are expressed in terms of the vector potential as EFF

=

HFF

=

SFF

=

iωˆ r × (A × ˆ r)
r) −iω ε2 /µ0 (A × ˆ EFF × HFF =

(2.161) (2.162)

2

ω 2 |ˆ r × A| ˆ r. 2µ0 c

(2.163)

The power radiates as transverse electromagnetic waves into all angles of the far field. The scattered power per unit solid angle thus consists of only polar and azimuthal contributions,    dPs Z0  2 Nϕ 2 , = r 2 SFF · ˆ r= | + |N (2.164) θ dΩ 8λ2 where the radiation vector N = 4π r A/µ0 is introduced for convenience [146]. The total scattered power results from the solid angle integral:  Ps =

 dϕ

dPs = 2π sin θ dθ dΩ

π 

sin θ dθ 0

dPs . dΩ

(2.165)

Finally, the loss per unit length attributed to scattering is directly related to the scattered power according to αs =

1 Ps , 2π R Pg

where Pg is the power in the guided mode.

(2.166)

2.8 Scattering Losses Resulting From Edge Roughness

55

2.8.5 TM Scattering Losses For TM WGM modes, the modal electric field is perpendicular to the plane of the disk (directed along the z-axis) and thus does not give rise to radiated fields polarized in the azimuthal direction. Thus, the radiation vector only possesses a polar component (the projection of the z-component), Nθ = − sin θ Nz ,

(2.167)

resulting in +d/2 

Nθ = iω

dz



2π 

−d/2

dϕ 0



∞ 

r  dr 

0 

∆ε sin θ Ez (r  , z )eimϕ eikz



cos θ ikr  sin(θ) cos(ϕ−ϕ )

e

.

(2.168)

The resulting scattered field retains the orthogonality of the corrugation harmonics; thus, the integral for the radiation vector can be treated separately for each harmonic and later summed incoherently. The modal field is continuous across the interface simplifying greatly the result. Incorporating the unit step perturbation as a limit in the integral results in +d/2 

NθM = iωε0 sin θ −d/2

(n21

−

dz

2π 

dϕ

R+∆RM (z   )e−iMϕ



r  dr 

R

0

    n22 )Ez (r  , z )eikz cos θ eimϕ eikr sin(θ) cos(ϕ−ϕ ) .

(2.169)

Because the perturbation is small ∆RM  R and localized to the disk edge surface all radial variables are replaced with the nominal disk radius and the integral is collapsed: +d/2 

NθM

=

iωε0 R(n21

−

n22 ) sin θ

dz ∆RM (z )Ez (R, z )eikz



cos θ

−d/2 2π 





dϕ ei(m−M)ϕ eikR sin(θ) cos(ϕ−ϕ ) .

(2.170)

0

Implementing the identity: 2π 





dϕ ei(m−M)ϕ eikR sin(θ) cos(ϕ−ϕ ) = 2π im−M Jm−M (kR sin θ) ei(m−M)ϕ

0

(2.171)

56

2. Optical Dielectric Waveguides

and retaining only the square modulus of the polar radiation vector component yields  

2  M 2 Nθ  = 2π ωε0 R(n21 − n22 ) sin2 θ |Jm−M (kR sin θ)|2  2 +d/2          ikz cos θ   .  dz ∆RM (z )Ez (R, z )e    −d/2

(2.172)

Next the scattered power is calculated assuming that the field and corrugation are z-independent: Ps =

∞ -

2π R 2

M=−∞

k40 (n21 − n22 )2 2 ∆RM |Ez (R)|2 8Z0

 2 +d/2      2 3  ikz cos θ   . dz e dθ sin θ |Jm−M (kR sin θ)|      0 −d/2

π 

(2.173)

The calculation of the z integral is straightforward and results in a polar sinc pattern. Incorporating both this and the Gaussian correlation function, the expression for the scattering loss becomes αs = R

2 ∞ Sc k40 (n21 − n22 )2 σ 2 Sc 1 λd |Ez (R)|2 −π 2π RM e 4 2π R Pg 2Z0 M=−∞ π 

dθ sin3 θ |Jm−M (kR sin θ)|2

0

  d cos θ d sinc2 . λ λ

(2.174)

2.8.6 TE Scattering Losses The edge scattering loss derivation for TE WGM modes is considerably more complicated. First, the modal electric field lies in the plane of the disk with both radial Er and azimuthal Eϕ components. Although the azimuthal component may be negligible for very low index contrast microresonators, in general it can be quite strong and cannot be neglected. Second, each of these components couples to both polar and azimuthal components of the radiated fields. Third, discontinuities in the planar electric field components exist unless the roughness is locally flat [147]. Fortunately, this approximation is valid for typical (shallow) roughness distributions where the corrugation depth is much smaller than the correlation length (σ  Sc ). For a treatment of deep perturbations, see Johnson et al. [148].

2.8 Scattering Losses Resulting From Edge Roughness

57

The modal field amplitudes projected in Cartesian coordinates are Ex

=

cos ϕ Er − sin ϕ Eϕ

(2.175)

Ey

=

sin ϕ Er + cos ϕ Eϕ .

(2.176)

The radiation vector will consist of both polar θ and azimuthal ϕ components each arising from these modal field components.

 (2.177) Nθ = cos θ cos ϕ Nx + sin ϕ Ny =

Nϕ

− sin ϕ Nx + cos ϕ Ny .

(2.178)

For convenience, field projection variables are defined: Kθ (θ, ϕ, r  , z , ϕ ) = . /   cos θ cos ϕ − ϕ ε0 ∆ε−1 Dr + sin ϕ − ϕ ∆εEϕ

(2.179)

Kϕ (θ, ϕ, r  , z , ϕ ) = . /   − sin ϕ − ϕ ε0 ∆ε−1 Dr + cos ϕ − ϕ ∆εEϕ ,

(2.180)

resulting in a compact expression for the radiation vector components +d/2 

Nθ = −iω

dz



2π 

−d/2

dϕ



∞ 

0

r  dr 

0 



Kθ (θ, ϕ, r , z , ϕ )eimϕ eikz +d/2 

Nϕ = −iω

dz −d/2





2π 

dϕ



0

∞ 



cos θ ikr  sin(θ) cos(ϕ−ϕ )

e

(2.181)

r  dr 

0 





Kϕ (θ, ϕ, r , z , ϕ )eimϕ eikz



cos θ ikr  sin(θ) cos(ϕ−ϕ )

e

.

(2.182)

The integral for the radiation vector is treated separately for each harmonic. Furthermore, for convenience, the radial and azimuthal field contributions can be treated separately and summed later. Incorporating the unit step perturbation as a limit in the integral results in +d/2  M Nθ,r

= −iω cos θ +

dz −d/2



2π 

dϕ 0





R+∆RM(z )e−iMϕ 

r dr R





1 1 − 2 n21 n2



 ,     cos ϕ − ϕ Dr (r  , z ) eikz cos θ eimϕ eikr sin(θ) cos(ϕ−ϕ )

(2.183)

58

2. Optical Dielectric Waveguides +d/2 

M Nθ,ϕ = −iωε0 cos θ

dz

2π 

−d/2

R+∆RM (z )e−iMϕ





dϕ r  dr  (n21 − n22 ) R

0

, +      sin ϕ − ϕ Eϕ (r  , z ) eikz cos θ eimϕ eikr sin(θ) cos(ϕ−ϕ ) +d/2 

dz

M Nϕ,r = −iω

+

−d/2



− sin ϕ − ϕ

cos ϕ − ϕ



1 1 − 2 n21 n2

dϕ r  dr  R



,     Dr (r , z ) eikz cos θ eimϕ eikr sin(θ) cos(ϕ−ϕ )

dz

−d/2 

R+∆RM(z )e−iMϕ



M Nϕ,ϕ = −iωε0





0



+d/2 

+

2π 

2π 

R+∆RM (z )e−iMϕ



(2.184)

(2.185)



dϕ r  dr  (n21 − n22 ) R

0

,     Eϕ (r , z ) eikz cos θ eimϕ eikr sin(θ) cos(ϕ−ϕ ) . 



(2.186)

Because the perturbation is small ∆RM  R and localized to the disk edge surface, all radial variables can be replaced with the disk radius to collapse the integral.  M Nθ,r

= −iωR 2π 

1 1 − 2 2 n1 n2



+d/2 

cos θ

dz ∆RM (z )Dr (R, z )eikz



cos θ

−d/2

+  ,   dϕ cos ϕ − ϕ ei(m−M)ϕ eikR sin(θ) cos(ϕ−ϕ )

(2.187)

0 +d/2  M Nθ,ϕ

=

−iωε0 R(n21

−

n22 ) cos θ

dz ∆RM (z )Eϕ (R, z )eikz



cos θ

−d/2 2π 

+  ,   dϕ sin ϕ − ϕ ei(m−M)ϕ eikR sin(θ) cos(ϕ−ϕ )

(2.188)

0

 M Nϕ,r

= −iωR 2π 

0

1 1 − 2 n21 n2

+d/2 

dz ∆RM (z )Dr (R, z )eikz



cos θ

−d/2

, +    dϕ − sin ϕ − ϕ ei(m−M)ϕ eikR sin(θ) cos(ϕ−ϕ )

(2.189)

2.8 Scattering Losses Resulting From Edge Roughness +d/2 

dz ∆RM (z )Eϕ (R, z )eikz

M = −iωε0 R(n21 − n22 ) Nϕ,ϕ



59

cos θ

−d/2 2π 

+  ,   dϕ cos ϕ − ϕ ei(m−M)ϕ eikR sin(θ) cos(ϕ−ϕ ) .

(2.190)

0

Implementing the identity 2π 







dϕ e±i(ϕ−ϕ ) ei(m−M)ϕ eikR sin(θ) cos(ϕ−ϕ ) =

0

2π im−M±1 Jm−M±1 (kR sin θ) ei(m−M)ϕ

(2.191)

and retaining only the square modulus of the radiation vector components yields    M 2 Nθ,r  =



 2π ωR

1 1 − 2 2 n1 n2

2 cos2 θ

|Jm−M+1 (kR sin θ) − Jm−M−1 (kR sin θ)|2 4 2   +d/2        ikz cos θ    dz ∆RM (z )Dr (R, z )e    −d/2

(2.192)

 

2  M 2 Nθ,ϕ  = 2π ωε0 R(n21 − n22 ) cos2 θ |Jm−M+1 (kR sin θ) + Jm−M−1 (kR sin θ)|2 4 2   +d/2        ikz cos θ    dz ∆RM (z )Eϕ (R, z )e    −d/2    M 2 Nϕ,r  =



 2π ωR

1 1 − 2 n21 n2

(2.193)

2

|Jm−M+1 (kR sin θ) + Jm−M−1 (kR sin θ)|2 4 2   +d/2        ikz cos θ    dz ∆RM (z )Dr (R, z )e    −d/2

(2.194)

60

2. Optical Dielectric Waveguides

 

2  M 2 Nϕ,ϕ  = 2π ωε0 R(n21 − n22 ) |Jm−M+1 (kR sin θ) − Jm−M−1 (kR sin θ)|2 4 2   +d/2        ikz cos θ   .  dz ∆RM (z )Eϕ (R, z )e    −d/2

(2.195)

Assuming that the field and corrugation are z-independent, the scattered power is given as Ps =

∞ -

2π R 2

M=−∞

k40 (n21 − n22 )2 2 ∆RM 8Z0

0

π 

|Jm−M+1 (kR sin θ) − Jm−M−1 (kR sin θ)|2 4 0  |Jm−M+1 (kR sin θ) + Jm−M−1 (kR sin θ)|2 |Dr (R)|2 + sin θ

2 4 n2 n2 ε sin θ cos2 θ

dθ

1



2 0

|Jm−M+1 (kR sin θ) + Jm−M−1 (kR sin θ)|2 4 1  2 |Jm−M+1 (kR sin θ) − Jm−M−1 (kR sin θ)|2    + sin θ Eϕ (R) 4 2   +d/2      ikz cos θ   . (2.196)  dz e    −d/2 + sin θ cos2 θ

The calculation of the z integral is straightforward and, again, results in a polar sinc pattern. Incorporating both this and the Gaussian correlation function, the expression for the scattering loss results, here split into azimuthal and radial field contributions: αs = R

k40 (n21 − n22 )2 σ 2 Sc 4⎧ 2π R

2 ∞ ⎨ 1 λd |D (R)|2 Sc −π 2π RM

r  e ⎩ Pg 2Z n4 n4 ε2 0 1 2 0 M=−∞ π 

dθ 0



  2 sin θ cos2 θ Jm−M (kR sin θ)

2.8 Scattering Losses Resulting From Edge Roughness

+ sin θ

(m − M)2 |Jm−M (kR sin θ)|2



61

  d d cos θ sinc2 λ λ

2

(kR sin θ)  2 ∞

2 Sc 1 λd Eϕ (R) −π 2π RM e + Pg 2Z0 M=−∞  π  (m − M)2 |Jm−M (kR sin θ)|2 dθ sin θ cos2 θ 2 (kR sin θ) 0  2   2  d d cos θ sinc2 . + sin θ Jm−M (kR sin θ) λ λ

(2.197)

2.8.7 Normalized Formulation for Edge Scattering Losses The expressions for the edge scattering loss scale with the inverse fourth power of the wavelength, typical of scattering processes. They also predict that the loss is strongly dependent on index contrast in proportion to at least the square of the permittivity difference. Finally, the expression is insensitive to disk height d when the height exceeds the wavelength as we will show later. The quality factor can be expressed in normalized units, from which useful limiting approximate forms can be derived. The quality factor is given by the radians per cycle divided by the fractional loss per cycle. Its association with the scattering loss is given by Qs−1 = αs R/m, 1 3 n1 2π R TM = 4π mλ Qs

 1−

n22

2 

n21 ∞ -

e

−π

σ λ/n1

2

Sc 1 d |Ez (R)|2 λ/n1 Pg 4kZ0

2 Sc λ λ/n1 n1 2π R M

M=−∞ π 

  d 2 d cos θ sinc dθ sin θ |Jm−M (kR sin θ)| λ λ 2

3

0

1 n1 2π R = 4π 3 mλ QsTEr

 1−

n22

2 

n21

σ λ/n1

2

Sc λ/n1

∞ Sc 1 d |Dr (R)|2 −π λ/n 

1 e 4 4 2 Pg 4kZ0 n n ε M=−∞ 1 2 0 π 

dθ

2 λ n1 2π R M

   2 sin θ cos2 θ Jm−M (kR sin θ)

0

+ sin θ

(2.198)

(m − M)2 |Jm−M (kR sin θ)|2 (kR sin θ)

2



  d d cos θ sinc2 λ λ

(2.199)

62

2. Optical Dielectric Waveguides

1 TEϕ Qs



n22

2 

σ λ/n1

2

Sc λ/n1 n21  2 ∞

2 Sc λ 1 d Eϕ (R) −π λ/n n1 2π R M 1 e Pg 4kZ0 M=−∞  π  (m − M)2 |Jm−M (kR sin θ)|2 dθ sin θ cos2 θ 2 (kR sin θ) 0     2  d d cos θ sinc2 . + sin θ Jm−M (kR sin θ) λ λ

= 4π

3 n1 2π R

mλ

1−

(2.200)

These expressions can be written in a compact form by defining normalized units:   1 1 2 2 p p p 3 X (2.201) 1 − 2 ξ c Γz Γr Gm , p = 4π m n Qs where X, ξ, c , respectively, are normalized quantities representing the radius X = n1 2π R/λ, roughness ξ = n1 σ /λ, and correlation length c = n1 Sc /λ. The superscript symbol p refers to the polarization state p (TM, TEr , or TEϕ ). The quantity Γz = Pc /Pg is the ratio of power vertically confined to total power guided or simply the vertical confinement factor. The usual confinement factor for a planar waveguide can be applied here when using the effective index method. This requires solution of the simple planar slab waveguide dispersion relation (kz vs. ω) as in the Kogelnik p formulation [99]. The quantity Γr is the edge confinement factor and is defined as the ratio of the intensity at the disk edge (|E(R)|2 /2Z0 ) to the characteristic intensity of the mode (2kPc /d). This requires solution of the simple infinite cylinder whispering gallery mode dispersion relation (X, Qi ) vs. (m, n) vs. (m, n). The radial mode profile is normalized to a power per unit disk height of Pc /d by integrating the azimuthal component of the Poynting vector along the radial dimension from the disk center out to the radiation boundary (Rr = mλ/2π n2 ). Finally, a geometric factor resulting from the sum/integral of the far field scattering pattern 2

∞ 3 p p −π Xc M into azimuthal and polar angles is defined, Gm = e Pm−M , p

M=−∞

where Pm−M refers to the associated polar integral. There are useful limits to consider: thick and thin cylinder. In the limit of a thick cylinder (d  λ, but where d  R such that the small polar angle approximation, Eq. 2.159 still holds), the scattered radiation is directed outward at θ = 90◦ . The following expressions result for the polar integrals where the derivation is assisted by making the change of variables d cos θ/λ ≡ τ  ):

2.8 Scattering Losses Resulting From Edge Roughness

63

π 

  d cos θ d dθ sin3 θ |Jm−M (kR sin θ)|2 sinc2 λ ⎞  ⎛ λ  d/λ  0  2 2 2       λ λ  2   Jm−M ⎝kR 1 − τ τ  ⎠ dτ 1− =   sinc τ d d   ∞ −d/λ  → dτ  |Jm−M (kR)|2 sinc2 τ  = |Jm−M (kR)|2 (2.202)

TM Pm−M

=

dλ

π  TEr Pm−M

=

−∞



  2 sin θ cos2 θ Jm−M (kR sin θ)

dθ 0



  d 2 d cos θ sinc + sin θ 2 λ λ (kR sin θ) ⎡   ⎛ ⎞  d/λ   2  2 2    λ λ ⎢   ⎝ ⎠ τ  dτ  ⎣ =  Jm−M kR 1 − d τ d   −d/λ ⎤      2  2

  ⎥  ⎥ (m − M)2 Jm−M kR 1 − dλ τ    ⎥ ⎥ sinc2 τ  +   2 ⎥

2 ⎦ λ kR 1 − d τ  (m − M)2 |Jm−M (kR sin θ)|2

∞ 

→

dλ

(m − M)2 |Jm−M (kR)|2

dτ 

(kR)

−∞

2

sinc2 τ  =

(m − M)2 |Jm−M (kR)|2 (kR)

2

(2.203)

TEϕ Pm−M



π 

=

dθ

sin θ cos2 θ

0

(m − M)2 |Jm−M (kR sin θ)|2 (kR sin θ)

2

    2  d 2 d cos θ   sinc + sin θ Jm−M (kR sin θ) ⎡  λ   λ    2

λ  2  2 ⎢ d/λ    2 (m − M) Jm−M kR 1 − d τ ⎢   ⎢ λ  τ dτ ⎢ =   2 ⎢ d 2

⎣ λ −d/λ kR 1 − d τ   ⎛  ⎞ ⎤  2 2     λ 2   ⎝ ⎠ ⎥ +  ⎦ sinc τ Jm−M kR 1 − d τ   ∞      2 2 → dτ  Jm−M (kR) sinc2 τ  = Jm−M (kR) .

dλ

−∞

(2.204)

64

2. Optical Dielectric Waveguides

In the limit of a thin cylinder (d  λ), the following expressions result for the polar integrals: TM Pm−M

π  d → dθ sin3 θ |Jm−M (kR sin θ)|2 dλ λ

(2.205)

0

TEr Pm−M

π     2 d → dθ sin θ cos2 θ Jm−M (kR sin θ) dλ λ 0

+ sin θ

TEϕ Pm−M

(m − M)2 |Jm−M (kR sin θ)|2



2

(kR sin θ)

 π  d (m − M)2 |Jm−M (kR sin θ)|2 → dθ sin θ cos2 θ 2 dλ λ (kR sin θ) 0   2  + sin θ Jm−M (kR sin θ) .

(2.206)

(2.207)

The polar integrals for scattering as a function of corrugation order are plotted for comparison in Fig. 2.20. The scattering as a function of corrugation order can be expressed as scattering as a function of local azimuthal coordinate ∆ϕ through a modified grating equation, " !  M . (2.208) ∆ϕ = arccos n 1 − m If the correlation length is small such that the width of the spectral density of the roughness distribution (Xλ/n1 Sc ) is much wider than that of the polar integral (2X/n), then the summation term is simplified greatly. For typical fabrication processes, the relation (Sc  λ/2n2 ) is generally the case. ∞ p p Gm → Pm−M . (2.209) Sc λ/2n2

M=−∞ p

This allows the further reduction of the Gm parameter to simple limiting values: TM,TEr ,TEϕ

Gm

TM,TEr ,TEϕ

Gm

THICK

→

dλ THIN

→

dλ

1 1 , 2 2 4 4 4 δ, δ, δ. 3 3 3

1,

(2.210) (2.211)

Here, a normalized thickness has been defined as δ = d/λ. As a final useful approximation, the edge confinement factor may be approximated in the high index contrast limit as Γr ≈ 1/X. This simple inverse radius

2.8 Scattering Losses Resulting From Edge Roughness

65

Fig. 2.20. Distribution of the polar integral terms Pm−M versus corrugation order M for (a) TM, (b) TE radial, and (c) TE azimuthal. Here, the azimuthal order for the mode, and hence the center for the distribution is m = 50. The index ratio (for this example n = 3) restricts the participating corrugation orders from the full 0 < M < 2m because of Snell’s law or phase matching conditions. For λ = 1.55 µm, the resonant radii for TM and TE at m = 50 are 4.605 and 4.686 µm respectively. The total sums Gm are shown for each component in the thick and thin limits. The thick/thin cylinder limits are denoted by thick/thin linewidths, respectively. The thin curves have been normalized by factoring out the normalized thickness δ = d/λ parameter. The scattering distributions are also plotted as an angular distribution on the right. (After [296], ©2002, Optical Society of America.)

66

2. Optical Dielectric Waveguides

Fig. 2.21. Variation in the edge confinement factor associated with the electric field amplidudes of (a) TM, (b) TE radial, (c) TE azimuthal, and (d) TE net as a function of normalized radius for varying index contrasts (n = 1.25, 1.35, 1.5, 1.7, 2.0, 2.5, 3.0, 3.5). Note that (a) and (d) approach the approximate form 1/X for high index contrasts. (After [296], ©2002, Optical Society of America.) 106

n1/n2=1.5

Finesse

n1/n2=3.0

104

102

100

σ=1nm

0 20 40 60 80 100 Normalized radius, X = n12πR/λ

Fig. 2.22. Finesse limited by bending and edge scattering losses, for TM polarization, λ = 1.55 µm, d = 300 nm, n1 = 1.5, 3.0, n2 = 1, σ = 1 nm, Sc = 75 nm. Note in the asymptotic limit, the validity of the edge scattering limited finesse approximation (dashed line) for thin microresonator disks. (After [296], ©2002, Optical Society of America.)

2.8 Scattering Losses Resulting From Edge Roughness

67

dependence is found for both TM and TE when the radial and azimuthal contributions are summed, although it is dominated by the azimuthal component in the high index contrast limit. For a breakdown of these contributions, see Fig. 2.21. Because the edge confinement is inversely related to the normalized disk radius, it directly cancels the increased circumferential path length per round-trip. Ultimately this leads to an edge scattering limited finesse value that is independent of radius. Incorporating all these approximations results in simple forms for the edge scattering limited finesse. For thick, vertically extended (d > λ) microresonators, the expressions for TM and TE differ slightly:

Fig. 2.23. Finesse limited by bending and edge scattering losses, for both TM and TE polarization, λ = 1.55 µm, d = 300 nm, n2 = 1, σ = 1, 10 nm, Sc = 75 nm. The index ratios are n = 1.25, 1.35, 1.5, 1.7, 2.0, 2.5, 3.0, 3.5. Note the clamping of finesse with increasing normalized radius in the edge scattering limited regime. (After [296], ©2002, Optical Society of America.)

68

2. Optical Dielectric Waveguides

  1 2 2 1 3 1 − ξ c = 4π n2 FsTM   1 2 2 1 3 ξ c . 1 − = 2π n2 FsTE

(2.212) (2.213)

However, for thin, thumbtack-like (d < λ) microresonators, they are equivalent:   1 2 2 16π 3 1 = ξ c Γz δ. 1 − (2.214) TM/TE 3 n2 Fs In practice it is often found that TE operation results in higher losses. This has been attributed to a variety of interpretations such as a higher field strength at the sidewall and arguments involving erroneous derivations of geometric factors. The derivation shows that neither is the case.

Fig. 2.24. Quality factor limited by bending and edge scattering losses, for both TM and TE polarization, λ = 1.55 µm, d = 300 nm, n2 = 1, σ = 1, 10 nm, Sc = 75 nm. The index ratios are n = 1.25, 1.35, 1.5, 1.7, 2.0, 2.5, 3.0, 3.5.

2.8 Scattering Losses Resulting From Edge Roughness

69

Rather, it is likely that other mechanisms are responsible for the discrepancy such as the lower vertical confinement factors associated with TM operation. The validity of this approximation for a thin disk is shown by plotting the bending and edge scattering limited finesse as a function of normalized radius in Fig. 2.22. Figures 2.23 and 2.24 show the TM and TE finesse and Q curves for two different values of roughness across a variety of refractive index ratios. Note that at small radii, the advantage of high indices to combat bending loss is apparent as higher indices lead to higher finesse. For larger radii, however, where bending loss is insignificant, the ordering of the curves interchanges where edge scattering limited operation becomes prevalent. Figure 2.25 displays the tradeoff from a different perspective, plotting the finesse versus refractive index ratio for a few radii (parameterized by m). With increasing refractive index, bending losses decrease whereas edge scattering losses increase. The tradeoff is thus clear: if an index ratio is a design variable, one desires enough to negate the effects of bending loss, but only just enough, as any more leads to increased edge scattering loss in a practical device. To summarize, this generalized formulation predicts an edge scattering limited finesse that is independent of disk radius and polarization in the thin disk regime. The derived simplified expressions should be

Finesse

10

6

m= 100

10

4

10

2

10

0

1.0

50 25

Scatterin

15

g loss li mited

5

1.5

2.0 2.5 Index ratio, n1/n2

3.0

3.5

Fig. 2.25. The tradeoff between edge scattering and bending loss as a function of index contrast. An optimum index contrast exists whose value increases as the resonator is made smaller (lower azimuthal number m). Specific choices made for this plot are: TM polarization, λ = 1.55 µm, d = 300 nm, n2 = 1, σ = 3 nm, Sc = 75 nm. (After [296], ©2002, Optical Society of America.)

70

2. Optical Dielectric Waveguides

widely applicable both in predicting the impact of fabrication imperfections and/or in selecting the lateral index contrast for optimizing the performance of microresonators.

2.9 Summary In this chapter we introduced the basic formalisms for analyzing propagation, coupling, and loss mechanisms in optical waveguides and whispering gallerys. In the next chapter, we examine a generic microresonator from a slightly higher vantage point where these distributed parameters are condensed into the single-pass phase shift, coupling constant, and loss parameter.

3. Optical Microresonator Theory

Optical resonators manifest themselves everyday in our world: from the lasers in supermarket scanners to the colorful patterns reflected from oil slicks. They all share the ability to selectively modify certain optical properties within narrow spectral ranges. In this chapter, we introduce fundamental concepts associated with optical microresonators. Although introduced in the context of optical microring resonators, the concepts may be applied to a variety of resonant physical systems independent of application or scale. Many of the ideas presented in the chapter are thus equally applicable to analogous microwave and acoustic resonators.

3.1 Resonator Fundamentals 3.1.1 Fabry–Perot Resonators Optical resonators were employed as useful devices as early as 1899, when Fabry and Perot described the use of a parallel-plate resonator as a multipass interferometer [149]. Light incident on this Fabry–Perot resonator is split into transmitted and reflected components with power fractions that depend on many variables. If temporally incoherent (“white”) light is incident on the resonator, then the transmission and reflection coefficients depend only on the mirror reflectivities. The total reflected power consists of the power reflected from the first mirror plus all the multiple reflections between the mirrors that contribute to overall reflection. The sum of these contributions takes the form: R = R1 + T12 R2

∞ -

(R1 R2 )m−1 =

m=1

R1 − 2R1 R2 + R2 1 − R1 R 2

→

R1 =R2 ≡R

2R . (3.1) 1+R

Likewise, the transmitted power fraction takes the form: T = T1 T2

∞ m=1

(R1 R2 )m−1 =

T1 T2 1 − R1 R 2

→

R1 =R2 ≡R

T2 1−R . = 1 − R2 1+R

(3.2)

If, however, the incident light consists of a temporally coherent (monochromatic) plane wave, then the reflected power will be proportional to

72

3. Optical Microresonator Theory

the square of the coherent sum of all reflected fields. Because the fields carry phase information in addition to their amplitude, the fraction of reflected and transmitted light depends not only on the mirror reflectivities, but also on the mirror spacing and excitation wavelength. The coherent sum of fields is maximized when all the fields interfere constructively (in phase) and minimized when they interfere destructively (out of phase). Phase accumulates with propagation distance as φ(z) = βz and may also be acquired upon interaction with the mirrors. The coherent versions of Eqs. 3.1 and 3.2 include an accumulated phase factor per round-trip that can be interpreted as a normalized detuning φ = TR ω, where TR is the cavity transit time, TR = neff L/c for the circumference, L and effective index neff . Now, r˜ represents the complex reflectivity:

 ∞

m−1 r 1 − e+iφ r1 − r2 eiφ 2 imφ imφ → , = r1 r2 e r˜ = r1 −t1 r2 e 1 − r1 r2 eiφ r1 =r2 ≡r 1 − r 2 e+iφ m=1 (3.3) and t˜ represents the complex transmittivity:  φ − 1 − r 2 eim 2 t˜ = −t1 t2 e → . r1 r2 e r1 =r2 ≡r 1 − r2 m=1 (3.4) The square modulus of these complex quantities gives the reflection R and transmission T coefficients. Antiresonant wavelengths are more strongly reflected than in the incoherent case, whereas resonant wavelengths are transmitted 100% for balanced reflectors (r1 = r2 ). For a fixed mirror spacing, the transmission and reflection spectra thus exhibit periodic peaks and valleys. Figure 3.1 displays the transmission and reflection spectra for a lossless, balanced Fabry–Perot resonator. The fraction of reflected and transmitted power for incoherent excitation is equivalent to the respective spectrally averaged reflection and transmission across a period of the spectrum. φ

im 2

∞

-

imφ

m−1

φ

−t1 t2 eim 2 = 1 − r1 r2

Fig. 3.1. The transmission and reflection spectra for a Fabry–Perot resonator.

3.1 Resonator Fundamentals

73

3.1.2 Gires–Tournois Resonators A lossless Fabry–Perot resonator with a 100% reflecting rear mirror constitutes a device that is 100% reflecting at all frequencies. Nevertheless, resonant frequencies spend more time circulating in the resonator and experience longer group delays than do non-resonant frequencies. This deceptively simple device, termed a Gires–Tournois resonator, provides a means for preserving the spectral power of light reflected from it while modifying its phase. Consequently it is often referred to as a “phase-only” filter. The complex reflectivity of this device may be taken as a limiting case of Eq. 3.3 with r1 ≡ r and r2 = 1: r˜ = r − t 2 eimφ

∞

-

r eimφ

m=1

m−1

=

r − eiφ . 1 − r eiφ

(3.5)

The square modulus of this expression is unity for all values of the detuning parameter, φ. The phase argument of this expression, however, varies with detuning according to Φ = π + φ + 2 arctan

r sin(φ) . 1 − r cos(φ)

(3.6)

For a fixed mirror spacing, the spectral phase exhibits a staircase-like ascension with frequency. Figure 3.2 displays the spectral phase for a lossless Gires–Tournois resonator. The increase in phase sensitivity (slope) near each resonance is related to an increased group delay (to be investigated in a later section). 3.1.3 Ring Resonators Fabry–Perot and Gires–Tournois resonators are extremely versatile devices finding application as spectroscopic tools, add–drop filters, dispersion compensators, and laser cavities. Unfortunately, their free-space

Fig. 3.2. The spectral phase versus detuning for a Gires–Tournois resonator.

74

3. Optical Microresonator Theory

Fig. 3.3. Two ring resonator devices and their free-space embodiments.

mirror embodiment is incompatible with planar integrated technology. Devices nearly equivalent to these resonators can be constructed in a planar waveguiding geometry through the use of a ring waveguide coupled to one or two ordinary waveguides. For these ring resonators, the coupling strengths play the analogous role of mirror transmission coefficients. Figure 3.3 shows schematics of these two ring resonator devices and their free-space embodiments.

3.2 All-Pass Ring Resonators A simple ring resonator is created by taking one output of a generic directional coupler and feeding it back into one input (see Fig. 3.4). Such a device exhibits a periodic cavity resonance when light traversing the ring acquires a phase shift corresponding to an integer multiple of 2π radians. The resonator is mathematically formulated from two components: a coupling strength and a feedback path. Contrary to the infinite sum derivations performed earlier for the Fabry–Perot and Gires–Tournois, in what follows we derive the basic spectral properties by assuming steadystate operation and matching fields. Although both methods are equally valid, the field-matching method has the advantage of simplicity.

3.2 All-Pass Ring Resonators

75

Fig. 3.4. Fields associated with an all-pass ring resonator. The basic relations amongst the incident E1 , transmitted E2 , and circulating E3 , E4 fields of a single resonator are derived by combining the relations for the coupler with that of the feedback path. In the spectral domain, the fields exiting the coupling region1 are related to the input fields via the following unitary matrix (section 2.5.4):    r it E3 (ω) E4 (ω) = , (3.7) E2 (ω) E1 (ω) it r where the lumped self- and cross-coupling coefficients r and t are assumed to be independent of frequency 2 and satisfy the relation r 2 +t 2 = 1. The feedback path (of length 2π R) connects the output from port 4 back into input port 3 where the field is expressed as E3 = e −

αring 2

2π R ik2π R

e

E4 ≡ aeiφ E4 .

(3.8)

Here, a represents the single-pass amplitude transmission and φ represents the single-pass phase shift. Because adding or subtracting an integer number m of 2π radians from the single-pass phase shift does not change the value of the function, the single-pass phase shift for all resonances is defined such that its value is zero for a local resonance of interest. Furthermore, because the single-pass phase shift is directly related to the radian frequency as φ = ωTR , where TR is the transit time of the resonator and φ is clearly representative of a normalized frequency detuning. 3.2.1 Intensity Buildup As we will see in the next section, all-pass resonators delay incoming signals via the temporary storage of optical energy within the resonator. Constructive interference at the coupler port entering the ring ensures that circulating optical intensity is built up to a higher value than that 1

2

In the case of a microresonator, although the coupling is distributed over a significant angular portion of the disk, the coupling can be treated as being lumped and localized to a single point without loss of generality. In most cases, this is valid for microresonators since the coupling interaction length is much smaller than the cavity circumference, yielding a much broader bandwidth for the coupling interaction than for the cavity resonance.

76

3. Optical Microresonator Theory

initially injected. A coherent source is essential to achieving a buildup of intensity. In the case of perfectly incoherent excitation of an all-pass resonator (and no attenuation) the intensity in the cavity equalizes with the incident intensity. This result is analogous to the equalization of pressure between coupled pipelines in hydrostatics. Implementation of a coherent source does not change the average intensity buildup across a free spectral range. Rather, with coherent excitation, the buildup can greatly exceed unity near resonances at the expense of being reduced below unity away from them such that the average never deviates from the incoherent case. A coherent buildup of intensity can lead to a dramatically enhanced nonlinear response. We will return to this point later in this volume. Equations 3.7 and 3.8 are solved to obtain an expression for the ratio of the circulating field to the incident field: E3 itaeiφ = . E1 1 − r aeiφ

(3.9)

The ratio of circulating intensity to incident intensity, or the buildup factor B, is given by the squared modulus of this result,  2  E3  I3  B= = E  I1  1 2 2 1−r a = 1 − 2r a cos φ + r 2 a2 1+r 4 ≈ 2, → t φ=m2π ,a=1 1 − r

(3.10) (3.11) (3.12)

where the last result refers to the situation in which the incident light is resonant with the ring (φ = m2π ) and attenuation is negligible (a = 1). A passive ring resonator under these conditions attains the maximum ratio of circulating power to incident power that can be achieved, which is illustrated in Fig. 3.5 for two successive resonances. For cross-coupling

Fig. 3.5. A plot of the buildup factor versus detuning for an all-pass ring resonator.

3.2 All-Pass Ring Resonators

77

Fig. 3.6. Finite-difference time-domain simulation demonstrating coherent buildup of intensity at 1.5749 µm. Guide and ring widths are 0.4 µm. Resonator radii are 2.5 µm (outer) and 2.1 µm (inner). All guiding structures have a refractive index of 2.5 that is cladded by air. values of 10% (t 2 = 0.1), the intensity in the ring can be 40 times higher than the intensity incident on the resonator in the input waveguide. Since the intensity in the ring can be much higher than in the bus, ring resonators can be used for nonlinear optics applications with moderate input intensities. Figure 3.6 displays the coherent buildup of intensity in an all-pass microring resonator. 3.2.2 Finesse F The spectral shape of the buildup factor displays sharply peak resonances that are characterized by a finesse parameter. The finesse is defined as the free spectral range (FSR) between resonance peaks divided

78

3. Optical Microresonator Theory

by the full width at half depth (FWHD) of a resonance. Implementing this definition in the absence of internal losses results in an expression for the finesse of an all-pass resonator: F=

2π  2 arccos

2r 1+(r )2

 →

r ≈1

π 2π ≈ 2 . 1−r t

(3.13)

It is through the coherent constructive interference of recirculating feedback that resonators are able to increase the effective path length (and interaction time) of light traversing them by a factor equivalent to the finesse. 3.2.3 Effective Phase Shift An examination of the transfer characteristics of the resonator reveals another periodically resonant feature. Equations 3.7 and 3.8 are solved to obtain the ratio of the transmitted field to the incident field: a − r e−iφ E2 = ei(π +φ) . E1 1 − r ae+iφ

(3.14)

The intensity transmission is given by the squared modulus of this quantity. For negligible attenuation, i.e., a = 1, the equation predicts a unit intensity transmission for all values of detuning φ. Such a device is directly useless as an amplitude filter, allowing 100% transmission for all frequencies and is aptly termed an “all-pass” filter. This result is satisfying from an intuitive standpoint because light is offered only two choices: leave the device at port 2 or (re)enter the resonator at port 4. No mechanism exists for light to exit via port 1, and thus, in steady state, the optical powers entering and exiting the resonator are equal. The device, however, does not impose a uniform requirement of constant phase across all frequencies. The phase of the transmitted light, as in the Gires–Tournois resonator, can be dramatically different for different frequencies, especially those near resonance. The effective phase shift is defined as the phase argument of the field transmission factor and is the phase shift acquired by light in crossing the coupler from port 1 to port 2: Φ = π + φ + arctan

r sin(φ) r a sin(φ) + arctan a − r cos(φ) 1 − r a cos(φ) r sin(φ) . → π + φ + 2 arctan a=1 1 − r cos(φ)

(3.15)

A plot of the effective phase shift versus the single-pass phase shift φ for different values of r 2 is shown in Fig. 3.7. Near resonances (φ ≈ m2π ) the slope of the curve becomes very steep indicating that the phase that the

3.2 All-Pass Ring Resonators

79

Fig. 3.7. A plot of the effective phase shift versus the single-pass phase shift or normalized detuning for an all-pass ring resonator. Note the increasing sensitivity near resonance for increasing self-coupling parameter r 2 . device imparts is sensitively dependent on the normalized detuning [150, 151]. The phase sensitivity is obtained by differentiating the effective phase shift with respect to the detuning to obtain Φ = =

dΦ dφ

1 − 2r a cos(φ)



(

1+a2 2

1 − r 2 a2

 2 ) + r 2 a2 + sin2 (φ) (1 − a2 ) r 2 − (1 − a2 ) →

φ=m2π ,a=1

1+r . 1−r

(3.16)

The last form of this result refers to the situation in which the incident light is resonant and attenuation is negligible (a = 1). A comparison of Eqs. 3.16 and 3.12 reveals that under these conditions, the level of phase sensitivity is exactly equal to the level of intensity buildup across the entire spectrum. 3.2.4 Group Delay and Group Delay Dispersion The increased phase sensitivity is directly related to the increase in effective path length. This correspondence can be observed by examining the delay imposed by the resonator on a resonant pulse. The group delay for a linear device is given by the radian frequency derivative of the phase of the transfer function, dΦ = Φ  TR . (3.17) TD = − dω Because the detuning is related to the radian frequency as φ = ωTR , the group delay can be expressed as the cavity transit time enhanced by the phase sensitivity. Although a pulse is being delayed in a resonator, it’s

80

3. Optical Microresonator Theory

energy is stored inside the cavity; hence, the group delay is also equal to the cavity lifetime. Conversely, the phase sensitivity is interpreted as the effective number of round-trips light traverses in the resonator. Because the group delay associated with an all-pass filter is a frequencydependent function, its transmission characteristics are inherently dispersive. The group delay dispersion (GDD) for a linear device is defined as the radian frequency derivative of the group delay, GDD =

d2 Φ = Φ  TR2 . dω2

(3.18)

The GDD can be strong enough to significantly disperse a pulse. On resonance, the GDD (and all even dispersive orders) is zero although higher √ 3 where it order dispersion exists. The GDD has extrema at φ = ±π /F √ attains the value ∓3 3F 2 TR2 /4π 2 . Assuming a Gaussian pulse of FWHM equal to the cavity lifetime of a resonator, the quadratic depth of phase

 1 2 ln 2 2 imparted across the FWHM of the spectrum is equal to 2 GDDmax F TR or approximately 0.1265 radians. A convenient parameter characterizing the depth of the spectral quadratic phase resulting from GDD across a pulse spectrum is the chirp parameter. The chirp parameter (C) is defined by the following expression for the complex spectral amplitude of a Gaussian pulse: E(ω) = E0 e

− 1+iC 2



ω ωp

2

.

(3.19)

At a spectral spreads such that its peak intensity √ chirp of unity, the pulse 3 value . The maximum chirp per resonator falls to 1/ 2 of its minimum √ then is of the order of 3 3 ln 2/π 2 or approximately 0.365. Thus, approximately three resonators are required to impart a chirp of unity. Because the properties of resonators are periodic in frequency, a possibility exists for imparting equivalent phase profiles across multiple spectral bands. In this manner, ring resonators may be used to perform dispersion compensation across multiple wavelength division multiplexed channels simultaneously [152]. Similarly, a delay line may be built that operates for multiple wavelength multiplexed channels simultaneously. Furthermore, carrying this concept over for time-division multiplexed signals, it also seems that it is possible to operate a resonator with extremely short pulses (much sorter than the usual cavity lifetime imposed restriction) as long as the pulses are spaced by the round-trip time. This method of operation is termed synchronous pumping. Such a pulse train possesses a wideband spectrum but only in the discrete sense. That is, the pulse-train spectrum may completely lie within multiple resonance bandwidths and can thus take advantage of the phase-enhancing properties of the resonator. However, unless changes (on–off switches) 3

This is analogous to the Raleigh range for a Gaussian beam.

3.2 All-Pass Ring Resonators

81

in the pulse train take place at a time scale that is longer than the cavity lifetime, spectral components will be present in the signal that will fall outside the resonance bandwidths and the synchronous operation will fail. Thus, it is fallacious to think that employing a synchronous operation can circumvent the bandwidth limitation imposed by a periodic resonance on an information-carrying pulse train. For excitation pulse widths less than or equal to the cavity lifetime, all orders of dispersion imparted by the resonator are important. Expanding the phase response as a Taylor series does not make much sense in this regime. The high orders of dispersion are responsible for introducing ringing and stepwise behavior that is intuitively understood in the time-domain. Figure 3.8 demonstrates the rect and Gaussian responses for comparable pulse widths slightly larger than the resonator cavity

Power, arb. units

10

Circulating pulse

8 6 4 Output pulse

2 0

Input pulse 0

10 Power, arb. units

Rect response

2

4

6 Time, ps

8

10

12

14

Gaussian response

8 Circulating pulse

6 4 2 0 0

Input pulse Output pulse 2

4

6 Time, ps

8

10

12

14

Fig. 3.8. Rect response and Gaussian response of a ring resonator with r = 0.8, TR = 0.131 ps, TC = 1.9 ps, and TP = 4 ps.

82

3. Optical Microresonator Theory Anti-resonant

.035 ps (0.01 TC) 2

4

6 Time, ps

Amplitude, arb.

(b)

8

10

4

6 Time, ps

Amplitude, arb.

(c)

8

10

12

2

4

2

4

(f)

3.50 ps (1.00 TC) 0

4

6 Time, ps

8

10

12

8

10

12

6 Time, ps

8

10

12

8

10

12

3.50 ps (1.00 TC) 0

2

4

6 Time, ps

effective phase a -20 -15 -10

b

c

-5 0 5 10 Frequency, THz

15

20

Spectrum, arb.

(h)

Spectrum, arb.

(g)

6 Time, ps

0.35 ps (0.10 TC) 0

Amplitude, arb.

2

2

(e)

0.35 ps (0.10 TC) 0

.035 ps (0.01 TC) 0

12

Amplitude, arb.

0

Resonant

(d) Amplitude, arb.

Amplitude, arb.

(a)

effective phase d -20 -15 -10

e

f

-5 0 5 10 Frequency, THz

15

20

Fig. 3.9. Simulations of interfering output field amplitudes for 6 input pulse cases. Pulse widths of 3.5, 0.35, and 0.035 ps are injected into a 10 µm diameter resonator (n = 3, r = 0.75). In (a), (b), and (c), the carrier frequency is completely detuned from resonance. In (d), (e), and (f), the carrier frequency is tuned directly on resonance. Note that for ultrashort pulse excitation, the output pulses are representative of the impulse response of an all-pass resonator. For pulse widths of the order of the cavity lifetime (TC ), the pulse remains mostly undistorted. Plots g and h show the corresponding pulse spectra superimposed on the effective phase of the transfer function. lifetime. The hard edges associated with the rect pulse lead to ringing behavior in the transmitted pulse. The adiabatically changing Gaussian pulse, however, passes delayed with minor distortion. Figure 3.9 shows the transmitted field amplitudes for resonant and antiresonant cases for pulse widths that are 0.01, 0.1, and 1.0 times the cavity lifetime. Note how the subsequent impulses interfere destructively, forming an undisturbed

3.2 All-Pass Ring Resonators

83

pulse in the antiresonant case and interfere constructively, forming a delayed (and inverted) pulse in the resonant case. 3.2.5 Attenuation In reality, internal attenuation mechanisms are always present and thus render limitations as to when a ring resonator may closely approximate a true all-pass, phase-only filter. In particular, near resonance, the internal attenuation is increased such that dips appear in the transmission spectrum: a2 − 2r a cos φ + r 2 T = . (3.20) 1 − 2r a cos φ + (r a)2 The attenuation at the dips is equal to the single-pass attenuation magnified by the phase sensitivity (for r < a). The width of the resonance also broadens, lowering the finesse: F=

2π  2 arccos

2r a 1+(r a)2

 →

r a≈1

π 2π ≈ 2 , 1 − ra t + αL

(3.21)

where for small losses, αL is the fraction of power lost per round-trip. If the attenuation is comparable to the cross-coupling, light is resonantly attenuated strongly. Under critical coupling, (r = a) or (t 2 = αL) the finesse drops by a factor of 2 and more importantly the transmission at resonances drops to zero. The circulating intensity peaks are diminished and the phase sensitivity is paradoxically increased (Eq. 3.16). At resonances, the phase sensitivity increases without bound at the expense of a decreasing transmitted signal until the transmission is zero and the phase sensitivity is infinite. Of course the phase only undergoes a finite and discrete phase jump at this point. If the resonator is dominated by bending or scattering loss, then the waveguide mode is coupled perfectly to the continuum of outward propagating waves outside the resonator. Undercoupling occurs when the loss exceeds the coupling strength (r > a) or (t 2 < αL). Many counter-intuitive effects may take place in this regime such as the inversion of the phase sensitivity. Overcoupling occurs when the round-trip loss does not exceed the coupling strength (r < a) or (t 2 > αL) and is the conventional mode of operation for an all-pass resonator. Figure 3.10 displays the effect of attenuation on the transmission and build-up for the overcoupled, critically-coupled, and undercoupled regimes. An experimental demonstration of the transmission characteristics in each of these regimes for a fiber ring resonator can be found in [153]. Finally it is worth examining the introduction of gain. Gain may be implemented, if possible, to offset loss mechanisms and to restore the all-pass nature of a normally lossy ring resonator. Of course if the round trip gain exceeds the net round-trip loss due to attenuation and coupling, the resonator can exceed the threshold for lasing.

84

3. Optical Microresonator Theory

1.0

7 a = 0.90

a = 0.99 a = 0.99

0.9 a = 0.75

6 a = 0.96

0.8

a = 0.96

a = 0.50

5 a = 0.20

Build-up factor

Transmission

0.7 0.6 0.5

a = 0.00

0.4 0.3

4 a = 0.90

3

2 a = 0.75

0.2 1 a = 0.50

0.1 0.0

a = 0.20

0

-2 0 2 Normalized detuning, rad

-2 0 2 Normalized detuning, rad

Fig. 3.10. A plot of the (a) net transmission and (b) buildup versus normalized detuning for an all-pass resonator with r = 0.75 and varying loss. The single-pass field transmission a is displayed for each curve in the figure.

3.3 Add–Drop Ring Resonators The direct waveguide analogy of a free-space Fabry–Perot is obtained by adding a second guide that side-couples to the resonator as in Fig. 3.11. Because this configuration behaves as a narrow-band amplitude filter that can add or drop a frequency band from an incoming signal, it is commonly termed an add–drop filter. Because this configuration is mathematically equivalent to the extensively studied classic Fabry–Perot interferometer, the equations for transmission coefficients are simply stated: The intensity throughput coefficient corresponding to light bypassing the lower excitation waveguide is r22 a2

r12

− 2r1 r2 a cos φ + I2 T1 = = I1 1 − 2r1 r2 a cos φ + (r1 r2 a)2

4r 2

→

a=1,r1 =r2 ≡r

2 1−r 2

(

1+

)

sin2

4r 2

(

2 1−r 2

)

 φ 2

sin2

. φ 2

(3.22)

3.3 Add–Drop Ring Resonators

85

Fig. 3.11. Fields associated with an add–drop ring resonator.

This corresponds to a transmitted signal modified such that a narrow frequency band4 has been extracted; see Fig. 3.1. The extracted band exits at the drop port with transmission coefficient:



 1 − r12 1 − r22 a 1 I5

. = → T2 = 2 2 a=1,r =r ≡r 4r 2 φ I1 1 2 1 − 2r1 r2 a cos φ + (r1 r2 a) 1+ sin 2 2 (1−r 2 ) (3.23) Although there are similar enhancements (as in the all-pass resonator) in the effective phase shifts at the two output ports, they are intermingled with the dominant amplitude effects. Hence, although the phase response should not be ignored (particularly for multiple interacting resonators) it is rarely implemented for its phase properties. It is worth stating an interpretation of the add–drop configuration. Whereas in the lower coupler, interference exists between the input and the circulating field, the upper coupler does not display any interference (provided that excitation is from the lower guide only). The upper coupler may thus be viewed simply as a “tap” waveguide that leaks power out of the cavity into the drop port. This tap is formally equivalent to a lumped loss. For an add–drop filter, it is desirable to operate at critical coupling for complete extinguishment of a band in the through guide. Here the sum of all losses incurred in the resonator including at the out-coupled drop port must be taken into account. 3.3.1 Intensity Buildup The buildup factor for an add–drop resonator is given by  2  E3  I3  = B= E  I1 1

 1 − r12 r22 a2 = 1 − 2r1 r2 a cos φ + (r1 r2 a)2 4

(3.24) (3.25)

The shape of the transmission curve, as with the buildup, is approximately Lorentzian centered on a resonance.

86

3. Optical Microresonator Theory



→

a=1,r1 =r2 ≡r

→

φ=m2π

1 − r2 r2 1 − 2r 2 cos φ + r 4

1 r2 ≈ 2, 2 1−r t

(3.26)

(3.27)

where the second-to-last result refers to the situation where attenuation is negligible and the resonator is coupled critically through balanced couplers (a = 1,r1 = r2 ). The last result refers to the situation in which additionally, the incident light is resonant with the ring (φ = m2π ). A comparison with the buildup expression for an all-pass resonator (Eq. 3.12) reveals that the intensity buildup in the add–drop configuration is only 1/4-th that in the all-pass configuration for the same coupling strength. Figure 3.12 displays the coherent buildup of intensity in an add–drop microring resonator. Figure 3.13 compares the transmission, phase, intensity buildup, and group delay for all-pass and add–drop resonators with common coupling strengths.

Fig. 3.12. Finite-difference time-domain simulation demonstrating resonator buildup and rerouting (channel dropping) at 1.5736 µm. Parameters are the same as in Fig. 3.6.

3.3 Add–Drop Ring Resonators All-pass resonator

87

Add-drop resonator T2 , Φ 2

φ, B

(e)

T1

5π

T2

4π

.75 .50

3π Φ

Φ2

Φ1

π

.25

0

0

-π

Build-up

(b) 20

(d)

20

(f) dΦ1/dφ

15 10

2π

15

B dΦ/dφ

10

5

B

0 -π 0 π 2π 3π Norm. detuning, φ =TRω

B

5 dΦ2/dφ

0 0 π 2π 3π -π 0 π 2π 3π -π Norm. detuning, φ =TRω Norm. detuning, φ =TRω

Effective phase shift

(c)

T

T1 , Φ1

Input

Norm. group delay

Transmission

(a) 1.0

φ, B

T,Φ

Input

Fig. 3.13. Amplitude transmission (solid lines) and effective phase shift (dashed lines) for (a) an all-pass resonator, (c) through port of an add– drop resonator, and (e) drop port of an add–drop resonator. Plots (b), (d), and (f) display the coherent intensity buildup (solid lines) and group delay normalized with respect to the cavity transit time (dashed lines) for the same ports as in (a), (c), and (e), respectively. The independent variable on all plots is the normalized detuning, and all coupling coefficients are t 2 = 0.1814.

3.3.2 Add–Drop Resonance Width ∆ω or ∆λ The resonance width is defined as the FWHD of the resonance lineshape. Using the expression for the drop-port output of an add–drop resonator, we can write the transmission as t12 t22 a 1 − 2r1 r2 a cos φ + (r1 r2 a)2

=

t12 t22 a 1 . (3.28) 2 1 − 2r1 r2 a + (r1 r2 a)2

Solving this equation for the FWHD 2φ = ∆ωTR results in ∆ω =

2 arccos

1−(1−r1 r2 a)2 2r1 r2 a

TR

.

(3.29)

88

3. Optical Microresonator Theory

Per the Euler formula, for small φ, cos φ = 1 − φ2 /2, so φ2 =

(1 − r1 r2 a)2 . r1 r2 a

(3.30)

In the case where the loss is negligible, i.e., a = 1, and the coupling is symmetric, i.e., r1 = r2 ≡ r , the RHS of Eq. 3.30 is (1 − r 2 )2 /r 2 . Then, 

1/2

φ

=

(1 − r 2 )2 r2

⇒ ∆ω

=

2(1 − r 2 ) , r TR

(3.32)

2t 2 c . Lneff

(3.33)

λ20 ∆ω, 2π c

(3.34)

(3.31)

and for weak coupling, ∆ω = Translating to wavelength, ∆λ ≈

where λ0 is the free-space wavelength, and we have assumed λ0  ∆λ. Then, t 2 λ20 . (3.35) ∆λ ≈ π Lneff A more elegant expression for ∆ω may be obtained by treating the coupling to the bus waveguides as a distributed loss. We define αdis as αdis = αring + αthrough + αdrop , exp(−αthrough L) = r12 = 1 − t12 , and exp(−αdrop L) = r22 = 1 − t22 . Then, eq. 3.30 gives ∆ωTR 1 − exp(−αdis L/2) = 2 exp(−αdis L/4)

(3.36)

If αdis L  1, we get ∆ω

= =

2 αdis L TR 2 αdis L cαdis = . TR neff

(3.37)

The above equation reduces to Eq. 3.33 for t1 = t2 ≡ t  1 since exp(αdis L) = 1 − αdis L ⇒ αdis L = 2t 2 .

(3.38)

3.4 More on Concepts Associated with Resonators

89

3.3.3 Free Spectral Range (FSR) The separation of successive resonances is termed the FSR. For negligible internal GDD, ring resonators possess a discrete impulse response and hence a periodic spectral response. At resonance, ωTR = m2π , where TR is the round-trip time and m is an integer. Two successive resonances, ω1 and ω2 , are then related as FSRfrequency

=

ω2 − ω1 =

=

2π c . Lneff

2π TR (3.39)

Translating to wavelength, we get FSRwavelength =

λ20 . Lneff

(3.40)

3.3.4 Finesse F The finesse F is defined as the ratio of FSR and resonance width. It is thus a convenient measure of the sharpness of resonances relative to their spacings. Using Eqs. 3.37 and 3.39, F

= =

1 cαdis neff

2π c Lneff

2π . αdis L

(3.41)

In the case where internal loss is negligible and coupling to the bus waveguides is symmetric and weak (t1 = t2 ≡ t  1), using Eq. 3.38, we get π F = 2. (3.42) t A particularly compact approximate expression for the drop-port transmission incorporating the finesse results from making the Lorentzian approximation: 1 T2 → (3.43) 2 .

φ a=1,r1 =r2 ≡r ≈1 1 + π /F

3.4 More on Concepts Associated with Resonators 3.4.1 Quality Factor Q The quality factor of a resonator is a measure of the sharpness of the resonance relative to its central frequency. The Q is formally defined

90

3. Optical Microresonator Theory

as the ratio of the stored energy circulating inside the resonator to the energy lost per optical cycle: Stored energy . Power loss

Q = ω0

(3.44)

Since power loss is a temporal phenomenon, here, we must examine the transient response. Let us consider the behavior of a ring that has been charged to an intensity |E0 |2 , after which time the input is abruptly switched off. For a circularly symmetric microring resonator, the location inside the resonator where we measure the circulating intensity is arbitrary. The intensity after the nth round-trip is: |En |2

=

exp(−αdis L)|En−1 |2

(3.45)

=

2

(3.46)

exp (−nαdis L) |E0 | .

If n is large, we can treat it as a continuous variable and get d|En |2 = −αdis L|En |2 . dn

(3.47)

We can now relate this to the power loss, as each round-trip takes time TR . Since the power loss is energy lost per unit time, d|En |2 /dt = (1/TR )d|En |2 /dn. Therefore, Q

=

|En |2 −d|En |2 /dt ω0 T R . αdis L

ω0

=

(3.48)

From Eqs. 3.37 and 3.48, Q=

ω λ ≈ . ∆ω ∆λ

(3.49)

Implementing Eq. 3.41, we arrive at the relationship between the quality factor and the finesse: ω0 T R Q= F (3.50) 2π or Q

= ≈ =

ω0 Lneff F 2π c neff L F λ0 mF .

(3.51) (3.52) (3.53)

For most optical resonators of interest, the optical path length within a cavity cycle or ring circumference is typically many wavelengths long.

3.4 More on Concepts Associated with Resonators

91

The order (azimuthal number) of a particular resonance is a measure of the number of wavelengths within the circumference, m = neff L/λ0 . The order is also indicative of the mth peak in the spectrum and directly relates the quality factor to the finesse. A more rigorous approach in deriving the quality factor uses the Laplace transform to go from the steady-state frequency response to the transient response. We consider the circulating field E4 in an add–drop resonator: it1 E4 = . (3.54) E1 1 − r1 r2 aei(δω)TR Although the choice of E4 is arbitrary, for a low-loss ring, E4 is a reasonable approximation to the field everywhere in the ring. We can identify iδω as s and use the Euler formula to write exp[i(δω)TR ] = exp(sTR ) = 1 + sTR . Then, it1 /TR E4 , (3.55) = E1 (1 − A )/TR − A s where A = r1 r2 A. If we switch off the input, the transient response is given by L−1 1/[(1 − A )/TR − A s], so that E4 (t)

1 (1 − A )/TR − A s   1 − A t . exp −  A TR L−1

∼ ∼

(3.56) (3.57)

Identifying the coupling to the bus waveguides as a distributed loss, we get A = exp(−αdis L/2) = 1 − αdis L/2. The Q is then Q

= = =

|E4 |2 −d|E4 |2 /dt A T R ω0 2 (1 − A ) ω0 TR , αdis L

ω0

(3.58)

which is the same result as obtained in Eq. 3.48. 3.4.2 Physical Significance of F and Q To find the physical meaning of the finesse and Q, we consider the number of round-trips made by the energy in the resonator before being lost to internal loss and the bus waveguides. If we define N as the number of round-trips required to reduce the energy to 1/e of its initial value, we get

92

3. Optical Microresonator Theory

exp (−αdis NL)

=

⇒N

=

⇒F

=

1 e 1 αdis L 2π N.

(3.59) (3.60) (3.61)

Equation 3.61 tells us that the finesse represents, within a factor of 2π , the number of round-trips made by light in the ring. Similarly, Q = ω0 TR N

(3.62)

tells us that Q represents the number of oscillations of the field before the circulating energy is depleted to 1/e of the initial energy. In summary, the expressions for buildup, finesse, and Q for all-pass and add–drop resonators are related in the following manner: π Q Ball-pass = F = m (3.63) 2 Q (3.64) π Badd–drop = F = m . Thus, the finesse and Q represent metrics for the intensity buildup and effective interaction time, respectively, in a microresonator. Light interacts with the coupling interface for a finesse number of times while interacting with the cavity interior for a Q number of cycles. This insight has implications for the design of applications relying on the change in the transfer characteristics of a microresonator brought about by variations to the microresonator constituents. For instance, to construct an enhanced microresonator-based sensor or switch operating on the variation of a distributed optical property such as the refractive index, it is beneficial to make the cavity both large and with a high finesse. For this application, the figure of merit characterizing the potential enhancement is the quality factor. Alternatively, to construct a sensor or switch operating on the variation of a localized optical property such as the displacement of a coupler or end mirror, it is no longer beneficial to make the cavity large. Rather, it is often detrimental due to the increased sensitivity to thermal and vibrational noise. For such applications, the finesse completely characterizes the enhancement. 3.4.3 Phasor Representation Phasor diagrams are highly intuitive representations of the interference of successively delayed complex field amplitudes in a resonator. Figure 3.14 graphically depicts the sum of phasors contributing to the net complex output field for both an all-pass resonator and a critically coupled or add–drop resonator. Note that for small detunings near resonance (φ = m2π ), the net phasor in each case sweeps through π radians very rapidly. In the critically coupled case, the output amplitude is zero on resonance and grows as the net phasor sweeps away from the origin.

3.4 More on Concepts Associated with Resonators

1.5

All-pass resonator, r=0.9, a=1 −π / F

1.0 Imaginary( E2 / E1 )

93

−π/2F

0.5

−4π/F

0.0 φ = 0 +4π/F

-0.5 -1.0

+π/2F +π/F

-1.5 -1.5 -1.0 -0.5 0.0 0.5 Real( E2 / E1 )

1.5

1.0

1.5

Critically coupled resonator, r1=r2=0.9

Imaginary( E2 / E1 )

1.0 0.5 0.0

-0.5

−π/2F

−π/F

−4π/F

φ=0 +π/2F

+4π/F +π/F

-1.0 -1.5 -1.5 -1.0 -0.5 0.0 0.5 Real( E2 / E1 )

1.0

1.5

Fig. 3.14. Graphical sum of phasors contributing to the transmitted complex field for a single all-pass (r = 0.9) and critically coupled (r1 = r2 = 0.9 or r = a = 0.9 ) resonator. Shown are the phasor sums for equal increments of π /2F . Note that the effective (net) phase sweeps through π radians over the ±π /F bandwidth.

94

3. Optical Microresonator Theory

3.4.4 Kramers–Kronig Relations A true all-pass resonator is necessarily over-coupled to achieve lossless operation. The spectral phase response of a lossless ring resonator coupled to a single waveguide varies dramatically with the coupling coefficient. In the limit of 100% cross-coupling, the ring is traversed only once and the resonator simply imparts a linear phase and group delay that corresponds to the single-pass transit time. Seemingly paradoxically, as the cross-coupling coefficient t 2 is reduced from 100%, the phase sensitivity and group delay are increased near resonance. Intuitively, this increase occurs because although smaller fractions of power are initially injected into the ring, larger fractions of power are subsequently coherently maintained within the ring after each pass. As the coupling coefficient approaches zero, the slope or group delay increases without bound for a lossless resonator. The phase difference through a resonance, however, remains clamped at 2π radians for a single resonator. In practice, the group delay cannot be made arbitrarily small due to attenuation mechanisms that can dramatically reduce throughput to zero at critical coupling. Because a resonator introduces a frequency-dependent group delay peaking at resonance, it necessarily introduces GDD. Depending on the application, this may be seen as an intended feature or an undesirable side effect. From a theoretical standpoint, since a phase-only filter modifies the spectral phase of a signal without modification to its spectral amplitude, it seems at first glance that this device has violated the well known Hibert and Kramers–Kronig relations that hold for a causal device. However, it is also clear from examination of the impulse response (a comb of weighted delta functions appearing only for positive times) that causality has not been not violated. Let us examine this apparent inconsistency in more detail. If the response of system is causal (i.e., causes precede effects), then the impulse response function is zero for all time values prior to zero. This allows one to separate any function into even and odd components that are exact although of opposite sign for all t < 0 and are identical for t > 0. Thus, for all t < 0, they add destructively to yield zero, and for all t > 0, they add constructively in equal proportions to form the impulse response function. h(t) = heven (t) + hodd (t). (3.65) The even and odd components are simply related in the time-domain by the signum function heven (t) = sgn(t)hodd (t)

(3.66)

hodd (t) = sgn(t)heven (t).

(3.67)

3.4 More on Concepts Associated with Resonators

95

If, furthermore, the system is linear, and time-translation invariant, then a frequency-dependent transfer function can be assigned to spectral components traversing the system. The Fourier transform of the impulse response equations may be employed to generate the Hilbert transform relations: 2 Hreal (ω) = + Himag (ω) = +2 ω

+∞ 

dω

−∞ +∞ 

Himag (ω) = −

2

Hreal (ω) = −2 ω

−∞

dω

Himag (ω ) ω − ω

Hreal (ω ) . ω − ω

(3.68)

(3.69)

These relations apply to the real and imaginary parts of the transfer functions of any causal signal. They are also known as one form of the Kramers–Kronig relations [154, 155]. The impulse response function of a ring resonator with no internal dispersion (i.e., impulses propagate within the ring without dispersing) is a weighted sum of equally spaced delta functions. ∞

 m−1 h(t) = r δ(t) − 1 − r 2 e+iφ0 δ(t − mTR ). r e+iφ0

(3.70)

m=1

An impulse response function consisting of a weighted sum of delta impulses spaced by the cavity transit time TR in the time domain can be interpreted as the Fourier series for some periodic function in the frequency domain. The periodic function is of course the transfer function whose fundamental period or FSR is equal to the inverse of the transit time. The functions form a Z-transform pair that is simply a time-domain version of a Fourier series equation pair for discrete time signals. The transfer function is given by the following expression that may be simplified by taking the limiting value of the infinite series (provided it is convergent, satisfied by r < 1): ∞

 m−1 r − e+iωTR H(ω) = r − 1 − r 2 e+iωTR = r e+iωTR . 1 − r e+iωTR m=1

(3.71)

Because the system is linear, time-translation invariant, and causal, the real and imaginary parts of its complex transfer function form a Hilbert transform pair − cos ωTR + 2r − r 2 cos ωTR 1 − 2r cos ωTR + r 2  − 1 − r 2 sin ωTR = . 1 − 2r cos ωTR + r 2

Hreal(ω) = Himag(ω)

(3.72) (3.73)

96

3. Optical Microresonator Theory

Fig. 3.15. (a) Plots of the real and imaginary components of the complex field transfer function for an all-pass resonator with r = 0.9. (b) Plots of the amplitude square modulus (transmission) and phase. Note that although an all-pass resonator selectively modifies the phase spectrally preserving flat, unit transmission, there is no violation of Hilbert or Kramers–Kronig relations. The real and imaginary components of the transmissivity do satisfy these relations but in a manner that results in a flat square modulus transmission response.

Thus, the real and imaginary parts of the transfer function do in fact satisfy the Kramers–Kronig relations. But although the real and imaginary parts of the transfer function vary in a complicated manner, they do so in such a way that the amplitude is always preserved to be unity. Figure 3.15 illustrates this process graphically. In certain cases it is possible to formulate Kramers–Kronig relations for the amplitude and phase of a transfer function. It is accomplished by taking the natural logarithm of the transfer function — a procedure that maps the amplitude and phase into real and imaginary components: |H(ω)| = 1

 r sin ωTR arg [H(ω)] = π + ωTR + 2 arctan . 1 − r cos ωTR 

(3.74) (3.75)

3.5 Higher Order Filters

97

Thus, in certain systems, if the natural logarithm of the transfer function is analytic in the upper half complex frequency plane, the amplitude and phase of a transfer function form a Hilbert transform pair as well. For all values of gain and some values of attenuation, with the exception of the undercoupled regime, zeros are present in the upper half complex frequency plane of the response of the all-pass resonator. The natural logarithm of the response is nonanalytic at the zeros. As a result, Kramers–Kronig relations cannot be applied to the amplitude and phase response of an all-pass resonator [156–158]. Nevertheless, In the undercoupled regime, Kramers–Kronig relations for amplitude and phase do in fact exist. These concepts apply equally to add–drop resonators treating the drop coupling as an internal (tap) loss. Useful phase-modifying devices can also be constructed from under-coupled ring resonators. Such devices possess an inverted phase response near resonance, meaning that the phase decreases with increasing frequency. However, high losses necessarily accompany operation in this regime.

3.5 Higher Order Filters The design of an optical amplitude filter base on ring resonators typically involves fixed trade-offs among extinction, ripple, and complexity. The filter-order describes how the filter response behaves away from the resonant wavelength or frequency. The drop-port amplitude for a single-ring OCDF behaves as 1/{1 − r1 r2 A[1 + i(ω − ω0 )TR ]}, so the drop-port intensity behaves as 1/[(r1 r2 ATR )2 (ω − ω0 )2 + (1 − r1 r2 A)2 ] (near ω0 ), a Lorentzian. By higher order response, we refer to the roll-off compared to a single resonator device. A single-ring device has a rolloff of 20 dB/decade and is referred to as a first-order filter, because the drop-port amplitude behaves as 1/ω. In some applications, such as bandpass filters, we require improved rejection out-of-band (away from resonance). We can draw an analogy between the ring as a reactive optical element and an electronic filter with LC elements; cascading reactive elements makes the filter response steeper away from resonance. Rings can be cascaded serially (Fig. 3.16) or in parallel (Fig. 3.17), allowing the fabrication of higher order filters with increased FSR and faster roll-off compared with a single ring resonator device. In both cases, the Vernier effect can enhance certain resonances and suppress others, resulting in a wide free spectral range. In a serial cascade, the resonators are coupled directly to each other. In a parallel cascade the resonators are coupled via the bus, and the filter response depends on the length of the bus waveguide between the rings. For example, Fig. 3.18 compares the rolloff of a parallel-cascade of three rings and a single-ring filter. The x-axis is normalized to the bandwidths of the two filters.

98

3. Optical Microresonator Theory

Fig. 3.16. Serially cascaded rings.

Fig. 3.17. Rings cascaded in parallel.

Fig. 3.18. A comparison of the roll-off for a first- and third-order optical ring resonator filter.

3.5 Higher Order Filters

99

Mathematically, the reponse of some filters may be written as [152, 159] P1 (s) H(s) = , (3.76) P2 (s) where P1 and P2 are polynomials, and the filter response is evaluated near a resonance; sTR  1, s = −i(ω−ω0 ), where ω0 is a resonant frequency. For a single-ring OCDF, the drop-port response is −t1 t2 A1/2 (1 + s) Ed . = Ei (1 − r1 r2 A) − sr1 r2 A

(3.77)

The zeros of P1 are called the zeros of the filter, and the zeros of P2 are called the poles of the filter. If we assume that P1 and P2 have no zeros in common, the filter-order is determined from the order of P2 ; in a first-order filter, the polynomial P2 is of the form (s − p), i.e., a firstorder polynomial; the single-ring OCDF response shown above, therefore, qualifies as a first-order response. For an nth order filter, the polynomial 4 P2 is of the form n (s −pn )n , where the poles pn may be degenerate. The asymptotic behavior of the filter is s −n , so it has a sharper roll-off than a first-order filter. At the same time, by using interference effects between multiple filters, a flatter top compared with lower order filters can be obtained. Sharper roll-off is desirable in applications such as band-pass filters for better isolation between channels. A flat-top minimizes signal distortion in similar applications. We go over parallel-cascaded ring filters here. Serially coupled ring filters have been analyzed elsewhere [15,152,160,161] and demonstrated by several groups [15,18,55,162]. Parallel-cascaded ring filters have been discussed by Little et al. [163], Griffel [164], Melloni [165], and Grover et al. [63], and have been demonstrated by a few groups [22, 63]. The analysis follows [63]. The single microring device with the transfer matrix formalism is represented as:      Ei T11 T12 Et = , where (3.78) Ed T21 T22 Ea T11

=

T12

=

T21

=

T22

=

1 − r1 r2 AΦ , r1 − r2 AΦ t1 t2 A1/2 Φ1/2 , r1 − r2 AΦ −T12 , and r1 r2 − AΦ . r1 − r2 AΦ

(3.79) (3.80) (3.81) (3.82)

Now, consider a parallel cascade of microresonators separated by Λ (Fig. 3.17). The transfer matrix of each bus section of length Λ, Tφ , is

100

3. Optical Microresonator Theory

⎡ Tbus = ⎣

 Λ ∗ Φbus exp −αbus 2 0

⎤ 0

 ⎦, Φbus exp −αbus Λ 2

(3.83)

where Φbus = exp(iβbus Λ), βbus is the propagation constant in the bus waveguide, and αbus is the loss per unit length in the bus. Using the transfer matrix defined in Eq. 3.78,     Ei1 EtN = T1 · Tbus · T2 · Tbus . . . TN , (3.84) EaN Ed1 where Eim is the input-port field for the m-th resonator, Edm is the dropport field, Etm is the through-port field, and Eam is the add-port field. To show that the response of a multiring cascade has higher order behavior, we only need to show that a double-ring cascade has secondorder behavior; the behavior of cascades with more rings will follow by induction. The response for a double ring cascade is:     Ei1 Et2 = T1 · Tbus · T2 . (3.85) Ed1 Ea2 If we assume that the two rings are similar and assume that the bus waveguide has no loss, the net transfer function T2RP is given as

Fig. 3.19. A triple ring parallel cascade (3R) has faster roll-off than a single ring (1R). The simulation parameters are αring = 5 × 10−8 nm−1 , L = √ ring 60004 nm, t1 = t2 = 0.1, neff = 3.1, αbus = 0, Λ = 39000 nm, nbus eff = 3.15. The x-axis has been normalized to the bandwidths of the two filters.

3.5 Higher Order Filters

101

FSRring

FSRbus

FSRmulri−ring

Wavelength, a. u. Fig. 3.20. Vernier effect with a multi-ring cascade. The resonances of the rings that coincide with those of the bus are enhanced while the others are suppressed.  T2RP

=  =  =

T11 T21

T12 T22

T11 T21

T12 T22

 

∗ Φbus 0

0 Φbus

∗ T11 Φbus ∗ T21 Φbus

2 2 ∗ + T21 )Φbus (T11 ∗ (T11 + T22 )T21 Φbus



T11 T12 T21 T22 

T12 Φbus T22 Φbus

(T11 + T22 )T12 Φbus 2 2 (T22 + T21 )Φbus



 .

(3.86)

102

3. Optical Microresonator Theory

Normalized power

1

0.1

0.01 1

0.1

0.01 1550

1560

1570

1580

1590

Wavelength, nm Fig. 3.21. Simulated drop port response of triple ring parallel cascade OCDF (bottom) and single ring OCDF (top). The simulation parameters ring are αring = 5 × 10−6 nm−1 , L = 60004 nm, t1 = t2 = 0.1, neff = 3.1, αbus = 10−6 nm−1 , Λ = 39368 nm, nbus eff = 3.15. With appropriate choice of bus length, it is possible to suppress some resonances, giving a wide FSR.

Then Ed2 Ei2

= =

2RP T12 2RP T11 (T11 + T22 )T12 Φbus . 2 2 ∗ (T11 + T21 )Φbus

(3.87)

We can now substitute the expressions for the various terms on the RHS from Eq. 3.78 and see that the numerator and denominator on the RHS have no common factors. So the roll-off near a resonance is determined by the denominator. Since both T11 and T21 are polynomials of first-order in Φ = exp(−iωTR ) ≈ 1 − i(ω − ω0 )TR , or s = −i(ω − ω0 ), the denominator

3.6 Summary

103

is a polynomial of second order in s. Thus the double-ring parallel cascade behaves as a second-order filter. Similarly, a single ring cascaded in parallel with an n-ring cascade (with n-th order response) has a response of order (n + 1), etc. In other words, a parallel-cascaded N-ring OCDF has a roll-off N-times faster than a single-ring OCDF. The faster roll-off possible with a multi ring cascade is shown in Figs. 3.18 and 3.19 for the case where there are three rings. If the spacing, Λ, between the resonators is chosen such that Nr · FSRr = Nb · FSRb = FSRa , (3.88) where FSRr is the FSR of the resonator, FSRb is the FSR of the bus section between the resonators, and Nr and Nb are integers, then the Vernier effect causes the suppression of transmission peaks within the spectral range FSRa , resulting in an Nr -fold increase of the effective FSR. Since FSRr = 2π c/nr L, and FSRb = π c/nb Λ, Eq. 3.88 gives Λ=

Nb nr L Nr n b 2

(3.89)

for an effective Nr -fold increase in the FSR of the filter. The Vernier effect is illustrated in Fig. 3.20, and the simulated drop-port response of a triplering parallel-cascade OCDF is shown in Fig. 3.21. There are significant practical limitations to the number of resonators that can be cascaded in parallel or series. Both resonator and bus loss limit the number of resonators, as does the fabrication accuracy, in turn affecting the ability to exactly tailor the resonator and bus response for using the Vernier effect. In a parallel cascade, if the loss in the bus is high, the amount of light reaching the rings down the cascade will not be enough to enable them to affect the filter response; in that case, the response will be of a lower order than designed. In a serial cascade, if the loss in the ring is high, no power will reach the output bus.

3.6 Summary In this chapter we introduced fundamental concepts associated with optical microresonators. We began with a treatment of the traditional Fabry– Perot resonator, using it as a familiar starting point to present the key concepts associated with all-pass and add–drop microresonators. We concluded with a treatment of higher order filters - a topic we will return to albeit with a different slant in the chapter 7.

4. Microring Filters: Experimental Results

This chapter presents the results of experiments performed on both passive and actively tunable microring resonators functioning as optical filters.

4.1 Passive Resonators The results presented in this section represent a small subset of work in the linear optics regime on microresonators from our group and other groups. We refer the reader to [51, 55, 57, 58, 60, 62–64, 71, 76] for more. Examples of linear performance are summarized in Table 4.1. Optical microring resonators require a mechanism for coupling light into and out of a mode of the ring. This coupling is best accomplished through evanescent coupling of the resonator mode to that of the bus waveguide(s) situated in close proximity. Microring resonators can be made with laterally (in-plane) or vertically (out-of-plane) coupled bus waveguides (Fig. 4.1). III–V semiconductor devices based on lateral coupling require the use of advanced fabrication technologies (high-resolution e-beam lithography, facet-quality-dry-etching with etch rate independent of trench width) to achieve reproducible filter bandwidths and high dropping efficiencies. In the lateral coupling case, the separation between the waveguides and the ring, typically less than 0.3 µm, is achieved by ebeam lithography. E-beam lithography is affected by processing conditions (humidity, temperature), the age of the cathode, and time taken by the machine to be stable. These problems are exacerbated by the requirements of small gaps that can be difficult to etch, making reproducible bandwidth and high dropping efficiency challenging to achieve. Vertical coupling is a more robust and reproducible architecture as it can be accomplished with standard optical lithography. In the vertical coupling approach, the coupling gap is defined by material growth or deposition. Furthermore, the inherently symmetric structure with respect to resonator input and output coupling can result in better transfer efficiency, as demonstrated in glass with a process that requires redeposition and planarization [21]. Vertical coupling typically requires wafer bonding and growth-substrate removal, which can be problematic and

106

4. Microring Filters: Experimental Results

(a) Lateral coupling

(b) Vertical coupling

Fig. 4.1. Schematic of a microring with (a) laterally and (b) vertically coupled bus waveguides. The waveguide core is shown in red, and the cladding is shown in blue.

4.2 Active Resonators

107

Table 4.1. Examples of device performance in the linear optical regime. L: circumference, L-GaAs: laterally coupled GaAs-AlGaAs, V-GaAs: vertically coupled GaAs-AlGaAs, L-InP: laterally coupled GaInAsP-InP, V-InP: vertically coupled GaInAsP-InP. Type L-GaAs V-GaAs L-InP V-InP

L, µm 51 63 82 29

∆λ, nm 1.3 0.22 0.25 0.24

FSR, nm 18 10.4 8 24

Q 1200 7040 6250 6200

F 14 47 32 100

Reference [173] [173] [174] [174]

may limit yield in commercial applications by dictating a maximum size for void-free bonding. Wafer bonding can be achieved with polymers like benzocyclobutene or polyimides [166]; planarization followed by solders like Pb-Sn [167] Pd [168], Au-Sn [169], or Pd-In [170, 171]; and wafer fusion [172]. The use of solders and wafer fusion provides a conducting interface, which is advantageous for active devices, as contacts can be made to the (doped) substrate. Laterally coupled microring resonators remain important for the eventual realization of large-scale integrated photonic circuits, where we must integrate active and passive devices on the same chip. By combining the vertical- and lateral-coupling approaches, all the passive elements can be made with lateral coupling on the same layer as the (passive) bus, whereas the active elements can be coupled to the bus vertically. 4.1.1 GaAs-AlGaAs Fabricated GaAs-AlGaAs devices possess a GaAs core and an AlGaAs cladding. The response of single- and double-ring GaAs-AlGaAs laterally coupled resonators is shown in Fig. 4.2. Note the Lorentzian first-order response, and the steeper roll-off for the double ring device. The response of a vertically coupled device is shown in Fig. 4.3. 4.1.2 GaInAsP-InP Fabricated GaInAsP-InP devices possess a GaInAsP core and InP cladding. The response of a single-ring GaInAsP-InP laterally coupled resonator is shown in Fig. 4.4. The response of a vertically coupled device is shown in Fig. 4.5.

4.2 Active Resonators Most devices demonstrated so far have been static passive. Static passive devices can function only at certain wavelengths that cannot be changed

108

4. Microring Filters: Experimental Results

Fig. 4.2. Single and multiple ring GaAs-AlGaAs laterally coupled microring add/drop filters (left), and single and double-ring filter drop-port responses (right). Copyright Philippe P. Absil, 2000, used with permission.

Fig. 4.3. Optical micrograph of vertically coupled GaAs-AlGaAs microring add/drop filter (right), and 10 µm-radius ring filter response (left). Copyright Philippe P. Absil, 2000, used with permission.

once the device has been fabricated. This limitation is serious, because it is not always possible to fine-tune the fabrication process for the exact filter wavelength. Even small variations (tens of nanpmeters) in waveguide width can affect the resonance wavelength. However, if it were possible to modify the wavelength of operation after fabrication — say, when the device is part of an optical network — fabrication errors might be trimmed. Also through the use of gain and loss modulation, resonators can be implemented as switches, amplifiers,

4.2 Active Resonators

109

Fig. 4.4. Scanning electron micrograph of a laterally coupled microring add/drop filter (upper), and drop port response of an InP-based laterally coupled add/drop filters (lower).

Fig. 4.5. Optical micrograph of vertically coupled GaInAsP-InP microring add/drop filter (left), and 10 µm-radius ring filter response (right).

and routers. Thus, active microring resonator-based devices continue to be an important topic of research. The first active ring resonator was demonstrated by Tietgen in 1984 [13]. Since then, Rabiei et al. have

110

4. Microring Filters: Experimental Results

demonstrated active polymer microring resonators [40], Rabus et al. have demonstrated active GaInAsP-InP microring resonators [73], and Grover et al. have demonstrated active GaInAsP-InP microring resonators [77]. Microresonators can be tuned with temperature [52, 71]. Some of the first devices on the market (from Little Optics, Inc.) also use thermal tuning. Thermal tuning has the advantage that it has no polarization dependence. However, it can be very slow (tuning speeds of milliseconds), and can create polarization dependence in waveguide properties by causing stress and consequent bireferingence. The refractive index can also be altered via by carrier injection. Such resonators have been demonstrated by the group at the University of Southern California [65, 175]. The electro-optic effect has the highest potential for low-loss and highspeed operation. The electro-optic effect is related to the nonlinear susceptibility of the material, so we can use the values of χ (2) or r (the linear EO coefficient) and χ (3) or s (the quadratic EO coefficient) to choose the material system (GaAs or InP in the case of III–V semiconductors) for operation at the signal wavelength. For GaInAsP, Adachi and Oe provide analytical expressions for the linear electro-optic effect in [176], and the quadratic electro-optic in [177]. However, published data are of limited usefulness because of the wide range of parameters in the material and waveguide design and the detuning of signal wavelength from the bandgap. As detuning from the bandedge is reduced, the electro-optic effect becomes stronger, albeit at the expense of increased insertion loss. Nevertheless, the published data can prove useful as a sanity check on experimental results. 4.2.1 Electro-Optic Tuning The general form of the nonlinear polarization is [178] 

(2) (3) PiNL (t) = 4π χijk Ej Ek + χijkl Ej Ek El ,

(4.1)

where repeated indices imply summation and c.g.s. units are used. InP and its quaternaries have a tetragonal structure, and belong to the crystal class 422. The nonvanishing elements of the tensor χ (2) for this class are xyz = −yxz, xzy = −yzx, zxy = −zyx. Of the 81 elements of the tensor χ (3) , 21 are nonvanishing. They are xxxx = yyyy, zzzz yyzz = xxzz, yzzy = xzzx, xxyy = yyxx, zzyy = zzxx, yzyz = xzxz, xyxy = yxyx, zyyz = zxxz, zyzy = zxzx, xyyx = yxxy. Consider the case where the direction of propagation is z, in the plane of the wafer, and the x-axis is the direction of crystal growth, (100), i.e.,

4.2 Active Resonators

111

out-of-wafer-plane. In the presence of a constant electric field along e1 , and harmonic field Ei cos(ωt)e1 , the nonlinear polarization, P1NL (t), has the form . / (2) (3) P1NL (t) = 4π χ111 E1 (t)2 + χ1111 E1 (t)3 . / (2) (3) = 4π χ111 [Ei cos(ωt) + Edc ]2 + χ1111 [Ei cos(ωt) + Edc ]3 . /  (2) (3) 2 P1NL (ω, t) = 4π χ111 [2Edc Ei cos(ωt)] + χ1111 3Edc Ei cos(ωt) . (4.2) In Eq. 4.2, the P1NL (ω, t) notation indicates that we are considering only that part of the polarization that oscillates at the signal frequency; the other components oscillate at 2ω or just add a constant background. The nonlinear polarization has two terms: the first is linear in Edc and gives rise to the linear electro-optic effect. The second term is quadratic in Edc and causes the quadratic electro-optic effect. For III–V semiconductors, (2) where χ111 = 0, the relevant effect for the polarization and direction of electric field chosen is the quadratic effect. Then, since n2 = 1 + 4π χeff , (3)

n(ω) = n0 (ω) +

2 6π χ1111 Edc , n0 (ω)

(4.3)

for the direction of the constant electric field and the polarization chosen. This result is referred to as the quadratic electro-optic effect since the refractive index can be varied as the square of the electric field. Near the band-edge, the quadratic effect is caused primarily by the Franz–Keldysh effect in bulk and the quantum-confined Stark effect (QCSE) in multiple-quantum-well (MQW) waveguides. The two effects are related; Miller, Chemla, and Schmitt-Ring have shown that the FranzKeldysh effect is a limiting case of the QCSE [179]. Typical values of χ (3) for InP are ∼10−13 cm2 /W [180]. The QCSE develops from bandgap shrinkage shifting the excitonic resonance closer to the signal wavelength [181, 182]. The resultant change in the absorption coefficient causes a change in refractive index via the Kramers–Kronig relations. The Franz–Keldysh effect has a similar mechanism, except that excitons are not involved; when a bias is applied, the carriers move apart in bulk materials destroying the exciton resonance. The exciton resonance is maintained in quantum wells due to carrier confinement [183]. The quantum-confined effect is stronger than the bulk effect. Although both the linear and the quadratic electro-optic effect can be described by the nonlinear susceptibility, they have traditionally been described in terms of the linear and quadratic electro-optic coefficients [178, 184] given as 1 1 ∆n1 = − Γ n30,1 r11 E1 − Γ n30,1 s1111 E12 , 2 2

(4.4)

112

4. Microring Filters: Experimental Results

where E1 e1 is the electric field, n0,1 is the refractive index in the absence of field for the out-of-plane polarization, Γ is the confinement factor, r11 is the linear electro-optic coefficient (contracted notation), and s1111 is an element of the quadratic electro-optic coefficient tensor. 4.2.2 Material Characteristics To demonstrate electro-optic tuning, Grover et al. use a p-i-n diode structure, with the intrinsic region forming the core of the waveguide. Their intrinsic region consists of a 25-quantum-well superlattice, to maximize refractive index change via the QCSE. To confirm that the wafer behaved as a diode, the electrical characteristics of the wafer were tested by putting contacts on either side of the wafer, and by using a semiconductor parameter analyzer (Agilent 4155B). The characteristics are shown in Fig. 4.6 and validate the electrical design, but they introduce a wrinkle in the form of a high reverse leakage current leading to significant thermal effects playing a role in devices employing reverse bias. To find the electro-optic coefficient of the material, a free-space Mach–Zehnder interferometer was constructed (Fig. 4.7) to test a slab waveguide made from the material. Composite images of the interference pattern versus voltage for the S (out-of-wafer-plane) and P (in-wafer-plane) polarizations are shown in Fig. 4.8. As can be observed, the interference pattern shifts with voltage for both polarizations. For the S-polarization, at 5 V, the phase change is 2π because the maxima and minima coincide with those at 0 V. If we assume that the effect is purely quadratic in nature (such is the case for the quadratic EO effect in InP for the S-polarization), we obtain

Fig. 4.6. Electrical characteristics of diode made with wafer used for electro-optically tuned ring resonators. The diode area is 100×2000 µm2 .

4.2 Active Resonators

113

Fig. 4.7. Mach-Zehnder interferometer for EO coefficient measurement.

∆n

=

2π

=

1 3 2 n sE , and 2 2π L . ∆n λ

(4.5) (4.6)

Using λ = 1.55 µm, L = 0.2 cm, E = 10 V/µm (5 V), and n = 3.22 (effective index of slab) gives s = 2.3 × 10−15 cm2 /V2 .

(4.7)

At this stage, it is tempting to identify s with the quadratic electro-optic coefficient, but the high reverse leakage current means that the effect may well be thermal — the voltage curve for the material is approximately linear below 0 V to around 6 V, so both the thermal and the EO effect are quadratic with voltage for the S-polarization. To distinguish between the thermal and the electro-optic effects, we require high-speed (pulsed) testing. If the absorption changes at high speeds, it implies that the QCSE is changing absorption; hence, it is at least partly responsible for the index change. Since the EO coefficient was measured under dc conditions, the modulation depth should match low-frequency measurements. First, the impedance of the device was measured up to 2 MHz by placing it in series with a 10 Ω resistor as shown in Fig. 4.9(a). The resistor had an inductance of 4 µH (as measured by an inductance meter), so it is depicted as two elements, L2 and R2 . The device is represented as a resistor

114

4. Microring Filters: Experimental Results

(a) S-polarization (out-of-wafer-plane)

(b) P-polarization (in-wafer-plane)

Fig. 4.8. Interference pattern versus voltage. The empty rectangular box represents the orientation of the slab, and the arrow indicates the E-field polarization.

4.2 Active Resonators

115

(a)

(b)

Fig. 4.9. (a) Setup for impedance measurement of slab waveguide, and (b) voltage across R2 and L2 versus frequency. R1 in parallel with a capacitance C. The capacitance is calculated from the dimensions of the slab (100 µm × 2 mm plate, separated by 0.5 µm, n2 = 3.3). R1 is calculated from the voltage drop across the resistor at low frequencies. The measured data and the theoretical prediction are shown in Fig. 4.9(b). Since the predicted curve provides a reasonable fit, it can be concluded that the device impedance does not change appreciably up to 2 MHz. Next, the waveguide was modulated with sinusoidal signals from 1 Hz to 2 MHz. The response was measured below 2 MHz with a 2 V peak-topeak signal offset below 0 by 1 V, a large-area photodetector (ThorLabs PDA400) and a lock-in amplifier (Perkin-Elmer 7081 DSP Lock-in amplifier); the frequency response is shown in Fig. 4.10. The thermal roll-off frequency for the slab structure can be estimated to be 14 kHz or less by solving the heat diffusion equation. Since the frequency response is flat to 2 MHz, which is considerably higher than the maximum predicted thermal roll-off frequency of 14 kHz,

116

4. Microring Filters: Experimental Results

Fig. 4.10. Frequency response of slab waveguide.

it is a reasonable conclusion that the absorption modulation is not thermal in nature. It can be concluded that the absorption modulation is from the quantum-confined Stark effect and that the index change observed in the waveguide is due to the electro-optic effect. Since there is a large leakage current, the electro-optic coefficient measured (s) represents a lower bound for the quadratic electro-optic coefficient of the waveguide. 4.2.3 Tuning Measurements For the tuning measurements, the device was tested without antireflection coatings. Amplified spontaneous emission (ASE) from two cascaded erbium-doped fiber amplifiers (EDFAs) was used as a source in obtaining the spectrum of the device at various voltages. The spectrum was normalized to account for the input EDFA spectrum, and increased absorption with bias in the bus waveguide due to QCSE or increased temperature. The latter effect can be mitigated by patterning the contact so that it is only on the ring, and by using a waveguide core with larger bandgap. The data prior to normalization for two voltages is shown in Fig. 4.11. To normalize the data to the EDFA spectrum, the measured output power from the device was divded into a polynomial fit to the EDFA power reading at each measurement wavelength (Fig. 4.12). Finally, to account for the changing loss in the bus, the power was normalized to the maximum power over the wavelength range at each voltage. The device spectrum at two voltages is shown in Fig. 4.13. Although the input has a mixed polarization, the device has very high loss for the inplane polarization, effectively filtering the in-plane polarization. So the response shown in Fig. 4.13 is for the out-of-plane polarization. The free spectral range of the device is >30 nm, so other peaks lie outside the EDFA band. Due to the fused bus and ring, the resonator possesses high loss, increasing the bandwidth to a few nanometers; as a

4.2 Active Resonators

117

Fig. 4.11. Raw data for active resonator at two voltages.

Fig. 4.12. EDFA spectrum and polynomial fit used to normalize the raw spectral data for the active resonator.

Normalized power

1

0.1

0.01 1548

+ + + ♦ ♦ + + ♦ + + + ♦ ♦ + + + ♦ + ♦ ♦ + ♦ ♦ ♦ ♦ + + ♦ ♦ ♦ + + ♦ + ♦ ♦ + ♦ ♦ ♦ ♦ + ♦ ♦ ♦ + ♦ + + + ♦ + ♦ + + + ♦ + + ♦ ♦ + + + + + ♦ ♦ ♦ + + ♦ ♦ ♦ + ♦ ♦ + ♦ + + ♦ + + + + + ♦ ♦ ♦ ♦ ♦ ♦ + + + ♦ + + + ♦ ♦ + ♦ ♦ ♦ ♦ ♦ + ♦ ♦ + + + ♦ + ♦ + + ♦ +♦ ♦ + ♦ + ♦ + ♦ ♦ + ♦ + + + ♦ + ♦ + ♦ ♦ + + ♦ ♦ + ♦ ♦ ♦ ♦ ♦ + + ♦ + ♦ ♦ ♦ + + ♦ ♦ + + + + + ♦ + + + ♦ + + ♦ ♦+ + ♦ + ♦ + ♦ + ♦ ♦ + ♦ + ♦ ♦ ♦ + ♦ ♦ ♦ ♦ ♦ + + ♦ ♦ + ♦ + + ♦+ ♦ + ♦ + + + + ♦ + ♦ + + + ♦ ♦ + ♦ + ♦ ♦ + ♦ + ♦ + ♦+ + ♦ ♦ ♦ + ♦ + ♦ + ♦ + ♦ + 0V ♦ ♦ ♦ + + ♦ ♦+ + ♦ ♦+ + ♦ 6V + + + ♦ + ♦ ♦ +♦ ♦+ + ♦ + ♦ +♦ + + ♦ ♦+ ♦ +♦ + ♦ ♦+ + ♦ ♦ ♦ ♦ ♦ + + + + + + + 1550

1552

1554

1556

1558

Wavelength, nm Fig. 4.13. Change in resonance wavelength with voltage.

1560

1562

4. Microring Filters: Experimental Results

♦

2.8882

Refractive index

2.888

♦

♦

Experiment Quadratic fit

1553.4

♦

2.8878

1553.3

♦

2.8876

♦

2.8874

1553.2

♦ ♦

1553.1

♦ ♦

2.8872 2.887

1553.5

♦

1553

♦ ♦ ♦ ♦ ♦ ♦ 0 1 2

3

4

5

6

7

Resonance wavelength, nm

118

8

Reverse bias, V Fig. 4.14. Effective refractive index and resonance wavelength versus voltage. The step-like behavior is from the resolution of the optical spectrum analyzer used to acquire the device spectrum.

result, the Q of the resonator is low, about a few hundred. The resonance wavelength of the resonator is red-shifted with reverse bias. For clarity, the spectra for only two voltages are shown. Figure 4.14 shows the change in effective refractive index with voltage. The resonance locations were determined by a minimum search routine employing polynomial fits on each resonance shape to overcome noise, and then they were searched for the local minimum of the polynomial. The resonance locations are stepped because of the resolution of the optical spectrum analyzer (0.08 nm) used to collect the spectra. The corresponding value of the resonance wavelength is shown on the right side. The refractive index in the absence of a reverse bias is obtained from simulations using a commercial waveguide mode solver (Optical Waveguide Mode Solver from Apollo Photonics). Using Eq. 4.4 and a quadratic fit to the data in Fig. 4.14, the quadratic electro-optic coefficient for the waveguide can be obtained. The value of the coefficient for the waveguide is a lower bound for the quadratic electro-optic coefficient of the core because of the high leakage current in the material when under reverse bias. The tuning range of 0.8 nm, or 100 GHz, means that the device is suitable for dense wavelength division multiplexing applications. The curve-fit in Fig. 4.14 allows us to calculate the quadratic electrooptic coefficient for the waveguide as 4.9 × 10−15 cm2 /V2 at ∼ 65 meV from the band edge for the waveguide. Since the quantum wells occupy roughly half the core of the waveguide, the quadratic electro-optic

4.2 Active Resonators

119

coefficient for the wells is double that value; i.e., sqw = 9.8×10−15 cm2 /V2 . The obtained quadratic electro-optic coefficient value compares favorably with the value of 5 × 10−15 cm2 /V2 reported by Fetterman et al. for a similar MQW structure, at 113 meV from the band edge [184]. Higher values of the quadratic coefficient may be obtained by operating closer to the band edge; however, that has the deleterious effect of decreasing the signal transmission substantially. Zucker et al. report a much higher value for their structure by operating 32 meV from the band edge [185]. The change in throughput with voltage at off-resonance wavelengths allows us to estimate the increase in loss with electric field (Fig. 4.15). The increased loss is due to bandgap shrinkage in the quantum wells [186]. The loss in the absence of any field is due to scattering by rough sidewalls and mode-mismatch between the modes of the straight and curved sections of the resonator [122]. In earlier work, Grover et al. demonstrated resonance bandwidths as narrow as 0.25 nm in the same material system, so it is possible to make substantial improvements in the filter characteristics. Imperfect mask definition in the coupling region during fabrication of the current series of devices resulted in fused bus and resonators. The effect was a high coupling between the bus and the resonator, giving us the broad (a few nanometers) resonance of the device in this report. As an example, let us consider the behavior of a high-finesse microring resonator add–drop filter made from the same layer structure described in this chapter. We can use the index change from Fig. 4.14 and the loss from Fig. 4.15, τ1 = τ2 = 0.989, and A = 0.99 to predict the behavior of a ring resonator with achievable behavior — the corresponding 30

Loss, cm−1

25

20

15

10 0

2

4

6

8

10

12

14

16

Electric field, V/µm

Fig. 4.15. Measured loss versus electric field in straight waveguides. The loss in the absence of electric field is estimated as 15 cm−1 based on results from similar passive waveguides. The change in loss will be the same in the ring as in the bus, as the electric field affects only the internal loss.

120

4. Microring Filters: Experimental Results

bandwidth is a little over 0.5 nm. The simulated drop-port response for an add/drop filter at two voltages is shown in Fig. 4.16. The increased loss affects the bandwidth and decreases the drop-port amplitude, and it changes the extinction on the through port (not shown in the graph). The simulated bandwidth versus electric field is shown in Fig. 4.17. Since the bandwidth increases drastically over the tuning range, the material bandgap must be modified to decrease the change in loss. Ultra-compact tunable microring notch filters have been demonstrated in a p-i-n diode geometry, with the intrinsic region forming the waveguide core. The intrinsic region is composed of a superlattice of quantum wells. By applying a reverse bias, the resonance wavelength was made tunable over 0.8 nm (100 GHz) by the application of over 8 V. Direct material property measurements indicate that the effect is electro-optic, and that the quadratic electro-optic coefficient is at least 2.3×10−15 cm2 /V2 , which is consistent with values reported in literature. The quadratic electro-optic coefficient for the waveguide is estimated as 4.9 × 10−15 cm2 /V2 ; the corresponding value for the wells that comprise the waveguide compares favorably with previous reports. The tuning range is suitable for wavelength-division multiplexing applications.

Normalized power

1

7.5 V

0V

0.1

0.01 -3

-2

-1

0

1

2

3

Detuning, nm Fig. 4.16. Simulated change in spectral response with reverse bias for the layer structure used in this chapter.

4.3 Summary

0.8

♦

Bandwidth, nm

0.75

♦

0.7

♦ ♦

0.65 ♦

0.6 0.55

121

♦ ♦ ♦ ♦ ♦ ♦

♦ ♦

♦

♦

♦

0.5 0

1

2

3

4

5

6

7

8

Voltage, V Fig. 4.17. Simulated change in bandwidth with electric field across the core for the layer structure used in this chapter.

4.3 Summary In this chapter we presented experimental results of microrings acting as both passive and active tunable filters. The performance associated with passive microring resonators is limited by fabrication imperfections. Actively tunable (trimmable) microrings not only forgive fabrication errors but also allow the possibility of dynamic switching and routing.

5. Nonlinear Optics with Microresonators

The previous chapters were concerned with the fundamental linear optical properties of microresonators. In this section, we examine mechanisms whereby the coherent buildup of intensity and increased interaction length can enhance the nonlinear or intensity-dependent optical properties of microresonators. As we shall see, such enhancements can lead to all-optical functionalities in an ultra-compact geometry.

5.1 Nonlinear Susceptibility We proceed to describe the formalism of nonlinear optics through the perturbation approach. The material polarization P may be expanded in a Taylor expansion of electric field strength E as follows [178]: P = χ (0) + χ (1) E + χ (2) E 2 + χ (3) E 3 . . .

(5.1)

where χ (N) refers to the N th -order susceptibility. The first-order susceptibility gives rise to the linear refractive index " !
(5.2) n =  1 + χ (1) and linear attenuation coefficient, " !
α =  1 + χ (1) 4π /λ.

(5.3)

The second-order polarization term cannot have a frequency component that is composed of field components at that same frequency. This term is responsible for describing second-harmonic generation, the more generalized sum-frequency mixing, degenerate one-half subharmonic generation, the more general difference frequency mixing, optical rectification, and the electro-optic (Pockels) effect. In the case of a centrosymmetric material, the second-order polarization and all subsequent even orders vanish. The third-order polarization term describes third harmonic generation, four-wave mixing, intensitydependent refractive index, saturable absorption, and two-photon absorption

124

5. Nonlinear Optics with Microresonators

P (3) (ω = ω1 + ω2 + ω3 ) = χ (3) (ω1 , ω2 , ω3 )E(ω1 )E(ω2 )E(ω3 ). (5.4) The last of these phenomena can occur for all three fields at the same frequency and has a degeneracy factor of 3 P (3) (ω) = 3χ (3) (ω, −ω, ω)E(ω)E ∗ (−ω)E(ω).

(5.5)

This equation gives rise to an intensity-dependent refractive index also known as the optical Kerr effect, n(I) = n + n2 I,

(5.6)

where the nonlinear coefficient is related to the third-order susceptibility as n2 = 3[χ (3) ]/n2 ε0 c. (5.7) Two-photon absorption is a nonlinear effect by which two-photons arrive within a coherence time of each other and can be simultaneously absorbed exciting an electron in a material at twice the photon energy. This process gives rise to either induced or saturable absorption depending on the sign of the imaginary part of the third-order susceptibility: α(I) = α + α2 I,

(5.8)

where the two-photon attenuation coefficient is related to the third-order susceptibility as α2 = 12π [χ (3) ]/n2 ε0 cλ. (5.9)

5.2 Resonator Enhanced χ (3) Nonlinear Effects 5.2.1 Enhanced Nonlinear Phase Shift If an all-pass microresonator is constructed with a material that possesses a third-order nonlinearity manifested as an intensity-dependent refractive index, then the single-pass phase shift acquires a powerdependent term, φ = φ0 + 2π Ln2 P2 /λAeff where φ0 , is a linear phase offset. The derivative of the effective phase shift with respect to input power gives a measure of the power-dependent accumulated phase. This derivative can be expressed as dΦ dφ dP3 dΦ = dP1 dφ dP3 dP1



 1+r 2 1−r φ=m2π ,a=1   2 2 2 2π Ln2 2 π F F , = = λAeff π Pπ π →

2π Ln2 λAeff

(5.10)

5.2 Resonator Enhanced χ (3) Nonlinear Effects

125

where Pπ = λAeff /2n2 L is the threshold power required to achieve a nonlinear phase shift of π radians. The effect of the resonator is to introduce two separate enhancements for which the combined action on resonance yields an overall nonlinear response enhanced quadratically by the finesse [151]. The dual effect can be understood intuitively noting that an increased interaction length develop from the light recirculation, and an increased field intensity develops from the coherent buildup of the optical field. The increased interaction length in a microresonator of course comes with the penalty of a reduction in bandwidth. 5.2.2 Nonlinear Pulsed Excitation In the previous sections, resonator enhancement of nonlinearity was derived in a steady-state basis. The steady-state analysis presented earlier breaks down when the bandwidth of the optical field incident on a microring resonator is of the order of or greater than the cavity bandwidth. To simulate the time-dependent nature of the resonator response, a recirculating sum of successively delayed and interfered versions of the incident pulse must be performed numerically. The circulating field after M passes is built-up from successive and increasingly delayed coupler splittings: A3,M (t) = it

M -

(r a)m eiφm (t) A1 (t − mTR ) ,

(5.11)

m=0

where the phase cannot be treated as a constant due to the timedependence of the nonlinear phase shift. The phase shift at each pass is computed as φm=0 (t) = 0 (5.12) φm (t) − φm−1 (t) = φ0 + γ

 2π R

 2 dz A3,M (t) e−αz 0   2 a2 − 1 = φ0 + γ2π R A3,M (t) , 2 ln a

(5.13)

where γ is the self-phase modulation coefficient. Finally, the transmitted intensity is computed as A2,M (t) = r A1 (t) − t 2 a

M -

(r a)m−1 eiφm (t) A1 (t − mTR ) .

(5.14)

m=1

To preserve pulse fidelity and achieve finesse-squared enhancement, it is necessary to operate the device with pulses of widths that are greater than or equal to the cavity lifetime.

126

5. Nonlinear Optics with Microresonators

We next examine the trade-off between bandwidth and nonlinear response for enhanced phase accumulation in a single microresonator. The bandwidth of a resonator is primarily governed by its radius and finesse, ∆ν = c/ (2π RnF ) .

(5.15)

For a single add–drop resonator, this bandwidth corresponds to the width of the narrow-band add or drop transmission windows. For a single allpass resonator, this bandwidth corresponds to the frequency interval over which the phase varies sensitively and in a nearly linear manner over π radians. Outside this interval, the sensitivity falls and the phase significantly departs from linear behavior such that a pulse with a larger bandwidth can become severely distorted by higher order dispersive terms. There is an exact trade-off for linear properties. A resonator’s lifetime, group delay, and interaction length may be increased at the expense of bandwidth in direct proportion to the finesse. Such trade-offs can be circumvented fortuitously for certain nonlinear properties. The strength of the enhanced self-phase modulation may be characterized by how much power is required to achieve a nonlinear phase shift of π radians in a structure composed of a few resonators. To good approximation, the threshold power required to achieve a π nonlinear phase shift in just a single all-pass resonator is given by Pπ ≈

λAeff . 4F 2 n2 R

(5.16)

The ratio of the reduced threshold power to the reduced bandwidth for a resonator of a given finesse is a form of figure of merit and is related to the threshold pulse energy. The minimum pulse energy required to achieve the π nonlinear phase shift is obtained when the pulse width is of the order of the inverse of the resonator bandwidth. This process is easily understood because a longer pulse width with the same peak power will carry more energy but not be any more effective at accumulating nonlinear phase. A shorter pulse width will not allow the resonator sufficient time to buildup in intensity and thus will experience a weakened nonlinear response in addition to being severely distorted. For a high contrast dielectric waveguide, the effective area in which the power is confined may be as small as λ2 /8n2 , where n is the refractive index of the guiding layer. The threshold energy required to achieve a π nonlinear phase shift is accordingly reduced in linear proportion to finesse:  λ3 π ln(2) . (5.17) Eπ = 16F nn2 c To reduce the parameter space, we make some practical choices (n = 3, n2 = 1.5 × 10−17 m2 /W) corresponding to AlGaAs [187, 188] or chalcogenide [189, 190] glass waveguides operating near 1.55 µm. Figure 5.1

5.2 Resonator Enhanced χ (3) Nonlinear Effects

127

Fig. 5.1. The inherent trade-off between bandwidth and energy required to achieve a π nonlinear phase shift in a single add–drop microresonator. The diagonal lines correspond to constant resonator diameter for AlGaAs or chalcogenide-based systems near 1.55µm. Increasing finesse is directly proportional to decreasing energy.

displays the trade-off between the energy requirement for a π radian nonlinear phase shift per resonator and the bandwidth for resonators of varying diameter. It is of technological interest to note that a π nonlinear phase shift is obtainable with a 1ps, 1pJ pulse by use of a single, ultracompact microresonator of moderate finesse. Being able to accomplish useful tasks with low loaded finesse relaxes tolerances on the design and fabrication of microresonators that require λ/nF precision. Fortunately, although low-finesse devices do not make good high resolution add–drop filters or sensors, they still can possess strong nonlinear effects due to these favorable scaling laws. 5.2.3 Kerr Effect in Solid State Materials below Mid-Gap The Kramers–Kronig relation connects the refractive and absorptive spectra of a linear optical material with a causal response  c ∞ α (ω ) n (ω) − 1 = dω 2 . (5.18) π 0 ω − ω2 Changes in each of these quantities due to an external perturbation are causal and similarly related. However, in a strict mathematical sense, the Kramers–Kronig relations are invalid for degenerate third-order processes such as self-induced changes in refractive index because of the existence of a pole in the upper half complex frequency plane [178]. Nevertheless, it has been demonstrated that application of the Kramers–Kronig relation to the two-photon absorption spectrum of a material correctly predicts the magnitude and dispersion of the Kerr effect in solids [191,192].

128

5. Nonlinear Optics with Microresonators

The spectra of nonlinear refraction and two-photon absorption are thus related as  c ∞ α2 (ω ) n2 (ω) = dω 2 . (5.19) π 0 ω − ω2 A two-band model is generally sufficient for accurately predicting the third-order, nonlinear absorptive and refractive properties of semiconductors. Under this model, the two-photon absorption coefficient is given as  3/2 2

2|pvc | ω 9 4 − 1 2 2 πe Eg m0 α2 (ω) = √ . (5.20)

 ω 5 5 m0 c 2 n20 Eg3 2 Eg

For most direct gap semiconductors, many parameters are constant such that to good approximation, the two-photon absorption spectrum can be simplified to be only dependent on the bandgap energy and the linear refractive index as 3/2

ω 1.42 × 10−7 2 Eg − 1 α2 (ω) ≈ m/W, (5.21)

5 n20 Eg3 2 ω Eg where Eg is assumed to be given in eV. Equations 5.19 and 5.21 can then be used to calculate the nonlinear refractive index spectrum. This procedure results in a bandgap scaling law of Eg−4 for the magnitude of n2 . This scaling explains why chalcogenide and AlGaAs materials possessing bandgaps much smaller than silica possess 100–1000 times higher Kerr nonlinearities. Moreover, operation just below the half-gap ensures that one- and two-photon absorption processes are negligible; yet, a reasonably high and ultra-fast nonlinear Kerr coefficient is retained. Figure 5.2 displays the predicted two-photon absorption and nonlinear refractive index for a two-band model of AlGaAs. Although ignoring Urbach tail absorption and competing nonlinearities, this model provides excellent intuition for identifying the ideal spectral regimes in which to operate. 5.2.4 Experimental Enhancement of the Kerr Effect Heebner et al. constructed and characterized a 10-µm-diameter microring resonator for the purpose of verifying the enhancement of the Kerr nonlinearity in AlGaAs [193]. Many proposed applications of the Kerr effect require nonlinear phase shifts near π radians to become practical, and such was the goal of this experimental work. The microring resonator was constructed in AlGaAs and probed at a photon energy (hc/λ = 0.8 eV) below the half-gap (Eg /2 = 0.97 eV) of the material. The motivation for this choice was to maximize the ultra-fast

5.2 Resonator Enhanced χ (3) Nonlinear Effects

129

Fig. 5.2. Plots of the two-photon absorption coefficient and associated Kerr nonlinear refractive index of SiO2 , GaAs, and Al0.36 Ga0.64 As versus wavelength. The plot for two-photon absorption is derived from a two– band model [192]. The Kramers–Kronig relation applied to the third-order susceptibility is subsequently used to generate the plot for the nonlinear refractive index in each case. The dip on the log plot of n2 corresponds to a reversal of sign for the nonlinear refractive index, and gray line-strokes correspond to negative nonlinear refractive indices.

bound (Kerr) nonlinearities resulting from virtual transitions while minimizing the two-photon contribution to carrier generation [194]. Although this effect is a relatively weak nonlinearity requiring relatively high circulating intensities, the deposition of heat resulting from two-photon absorption can be substantially minimized. In addition, the lower limit of the response time is not dictated by the carrier recombination lifetime but only by the cavity lifetime and hence the structural design. Due to an increased phase sensitivity (or increased effective path length) near resonance, the resonator need only be detuned by 2π /F radians to achieve a net effective phase shift of π . Furthermore, the power threshold requirement to achieve this detuning via an intensitydependent refractive index change is reduced due to a circulating intensity buildup of B = 2F /π so that Pthreshold ≈

π λAeff , 2n2 LF 2

(5.22)

where Aeff is the effective mode area and L is the circumference. For a typical AlGaAs channel waveguide with nonlinear coefficient of n2 =

130

5. Nonlinear Optics with Microresonators

10−17 m2 /W, an effective mode area of 0.5-µm2 ring diameter of 10µm and finesse of 10, the threshold peak power is about 40 W at 1.55 micr on. Note that in comparison with standard single-mode silica fiber with a threshold power-length product of 500 Wm, this situation corresponds to 1200 Wm. This nearly six-order-of-magnitude reduction is attributed to a stronger nonlinear coefficient, a tighter mode area, and resonant enhancement. To demonstrate this enhancement, a device was fabricated in a manner that would display the nonlinear phase shift interferometrically by coupling an all-pass microring resonator to one arm of a Mach–Zehnder interferometer [195]. First, the vertical structure was grown via molecular beam epitaxy on a GaAs substrate providing a 1-µm-thick guiding layer. The arrangement of the layers and their composition are shown in Fig. 5.3. The narrow gap region was extended along the propagation direction by 4 µm into a racetrack geometry to increase the guide-to-ring coupling. Due to the 80-nm-gap feature, the device was patterned using e-beam lithography. An intermediate oxide-chromium mask was used because the resist itself was not sufficiently robust to serve as a high-quality mask for deep anisotropic etching. Lift-off of the bilayer PMMA resist yielded a 40 nm chromium mask for reactive ion etching of the underlying 800 nm oxide layer. Finally, a highly anisotropic chlorine-based ICP etch transferred the pattern directly into the AlGaAs with highly vertical sidewalls. Figure 5.4 displays a scanning electron microscrope (SEM) image of a resulting device. Figure 5.5 displays the spectral transmission data for an output port of a nearly balanced Mach–Zehnder interferometer containing an

Fig. 5.3. Design of a resonator structure fabricated to demonstrate the enhancement of the Kerr effect in AlGaAs. The vertical structure is formed by molecular beam epitaxy and the horizontal structure by nanolithography. All dimensions are in µm. (After Ref. [193], ©2002, Optical Society of America.)

5.2 Resonator Enhanced χ (3) Nonlinear Effects

131

Fig. 5.4. SEM image of an all-pass microring resonator coupled to one arm of a Mach–Zehnder interferometer. (After Ref. [193], ©2002, Optical Society of America.)

Fig. 5.5. Measured transmission spectrum (left axis) of the device shown in Fig. 5.4 with a theoretical fit. The resonator-induced phase shift as inferred from the interferogram is also shown in the plot (right axis). (After Ref. [193], ©2002, Optical Society of America.)

all-pass resonator in one arm as shown in Fig. 5.4. In this configuration, the overcoupled phase response of the 5-µm-radius racetrack resonator could be inferred from the amplitude response of the composite resonator enhanced Mach–Zehnder (REMZ). A measured bandwidth of 240 GHz and free spectral range of 2.3 THz results in a finesse of about 10 (Q = 810), dictated primarily by the coupling. Scattering-limited losses are estimated at 11% per round-trip. Also shown in the figure is the phase response obtained from a fit to the data displaying increased sensitivity near each of two resonances. Next, the intensity-dependent transmission of the REMZ was measured using a 10 Hz, 30 ps Nd:YAG pumped optical parametric generator (Ekspla) source at 1.545 µm. The low average power of the source ensured that thermal effects could be ignored. Optical self-switching was clearly observed in some samples. Figure 5.6 displays transverse slices of the imaged outputs of the REMZ in Fig. 5.4. Three traces are shown each with increasing pulse energies. Incomplete extinction resulted from higher losses in the arm containing the ring.

132

5. Nonlinear Optics with Microresonators

Fig. 5.6. Demonstration of all-optical switching between ports of a Mach– Zehnder interferometer resulting from the accumulation of a π -radian phase shift in an AlGaAs microring resonator (device shown in Fig. 5.4). (After Ref. [193], ©2002, Optical Society of America.) The actual pulse energies injected into the resonators were of the order of 1 nanojoule. The peak power associated with the pulse is thus of the order of the 40 W threshold requirement as predicted for a finesse of 10. Because the relative distribution of output power is clearly seen to shift from one output guide to the other as the pulse energy is increased a phase shift advance of approximately π radians is inferred. The experiment shows that previous demonstrations of large Kerr nonlinear phase shifts in 5 mm or longer waveguide lengths [187,196] can be compressed into devices 100 or more times shorter through the use of microring resonators. At photon energies below the half-gap, the boundelectron Kerr nonlinearity is essentially instantaneous whereas carrier generation due to one- and two-photon absorption is negligible. Ultracompact devices constructed from these building-blocks thus have the potential to be engineered into ultra-fast nonlinear photonic devices that generate negligible heat. For example, a device similar to the one presented but with a finesse of 100 could still support 12.5-ps pulses and switch with energies as low as 4 pJ. Additional optimization of the guiding confinement area and nonlinear response can produce devices with a 1-THz bandwidth and 1-pJ energy threshold. 5.2.5 Nonlinear Saturation An implicit assumption in the derivations thus far is the fact that the resonator does not get power-detuned or pulled away from its initial detuning. One might call this the non-pulling pump approximation (NPPA). The power-dependent pulling away from resonance decreases the nonlinear enhancement such that the process may be described mathematically as a saturation of the effective nonlinearity. Figure 5.7 displays the results of a simulation involving the interaction of a resonant pulse with a

5.2 Resonator Enhanced χ (3) Nonlinear Effects

Circulating power, Watts

50 0 0

200

5 10 15 Incident power, Watts

20

pre

dic

tio n

200

π 2

100 Time, ps

R

100

d) π

0

na

o es

150

π

0

nt

200

c cir

i

at

ul

250

po

are d

100 Time, ps

300

ng

π 2

ess esq u

0

r

we

350

Fin

Effective phase shift, rad

c)

b) 18 16 14 12 10 8 6 4 2 0

Effective phase shift, rad

Incident & transmitted power, Watts

a)

133

0

0

5 10 15 Incident power, Watts

20

Fig. 5.7. Simulations demonstrating the dynamic accumulation of nonlinear phase shift across a resonant pulse interacting with a nonlinear resonator. A 50-ps (10TC ), 16-W peak power pulse (corresponding to twice the reduced threshold power) interacts with a 5-µm diameter nonlinear resonator with r = 0.905 (B = 20). (a) incident and transmitted pulses, (b) circulating power versus incident power in steady state showing the nonlinear pulling (dashed line corresponds to resonant slope), (c) accumulated effective phase shift, and (d) effective phase shift versus incident power (the dashed line corresponds to resonant slope). Notice that the effective phase shift accumulation at the center of the pulse falls short of reaching π radians. nonlinear resonator. The phase accumulation (c) across the pulse tracks the pulse intensity but falls short of reaching π radians even though its peak power corresponds to twice the value predicted by the NPPA. Figures 5.7(b) and (d) illustrate why this is the case. As the incident intensity rises, the resonator is indeed pulled off resonance and the circulating intensity and effective phase shift are pulled away from their respective NPPA predicted slopes (dashed lines). A greater nonlinear phase shift may be extracted from the resonator by employing a small amount of initial detuning [197]. This can be predicted first by examining the phase transfer function vs. detuning in the linear regime as in Fig. 5.8. This is faithfully represented in the nonlinear regime [Fig. 5.9], where an initially detuned pulse pulls itself through resonance and in the process acquires a nonlinear phase shift of π radians.

Effective phase shift, Φ

134

5. Nonlinear Optics with Microresonators

2π

π

π 2π F

0 -4π F

-2π F

2π 0 F Normalized detuning, φ

4π F

Fig. 5.8. Linear effective phase shift showing the optimum amount of detuning such that the dynamic range of the effective phase shift is efficiently implemented near resonance.

Circulating power, Watts

50 0 0

20

pre

dic

tio

n

5 10 15 Incident power, Watts

red

200

200

c

a on

100

π 2

100 Time, ps

u irc

s

Re

150

d) π

0

nt

200

π

0

tin

la

250

π 2

ua

100 Time, ps

g

300

po

sse

-sq

0

r

we

350

Fin e

Effective phase shift, rad

c)

b) 18 16 14 12 10 8 6 4 2 0

Effective phase shift, rad

Incident & transmitted power, Watts

a)

0

0

5 10 15 Incident power, Watts

20

Fig. 5.9. Simulations demonstrating the dynamic accumulation of nonlinear phase shift across an optimally detuned pulse interacting with a nonlinear resonator. All parameters with the exception of the detuning (φ0 = −π /F ) are the same as in Fig. 5.7. Notice that the effective phase shift accumulation at the center of the pulse reaches π radians.

5.2 Resonator Enhanced χ (3) Nonlinear Effects

135

The value of the detuning chosen (φ0 = −π /F ) corresponds to the point at which a symmetric pull-through resonance results in a π radian phase shift. If the detuning is increased some more, the circulating versus incident intensity relation exhibits a very sharp upward sloping curve. The curve in fact can become √ steeper than that predicted by the NPPA. At a detuning of (φ0 = − 3π /F ), the curve becomes infinitely steep over a narrow range, which corresponds to an operating point just below the threshold of optical bistability. Operation in this regime will be examined more closely in the next section. For certain applications, this might be more or less useful than the previous case. Figure 5.10(c) demonstrates the phase accumulation across such a pulse. The phase shift actually can exceed π radians but is confined to a narrower range near the peak of the pulse. Devices operating in this regime of high differential phase gain are referred to as transphasors in analogy to the very high differential gains exhibited by transistors in electronics [82].

Circulating power, Watts

100 50 0 0

200

20

pre

dic t

ion

5 10 15 Incident power, Watts

red

200

π 2

100 Time, ps

so

Re

150

d) π

0

ir

tc

n na

200

π

0

l cu

g

in

at

250

π 2

ua

100 Time, ps

300

po

ess

esq

0

r

we

350

Fin

Effective phase shift, rad

c)

b) 18 16 14 12 10 8 6 4 2 0

Effective phase shift, rad

Incident & transmitted power, Watts

a)

0

0

5 10 15 Incident power, Watts

20

Fig. 5.10. Simulations demonstrating the dynamic accumulation of nonlinear phase shift across a pulse detuned just below the limit of bistability interacting with a nonlinear √ resonator. All parameters with the exception of the detuning (φ0 = − 3π /F ) are the same as in Fig. 5.7. Notice the very sharp jump in phase.

136

5. Nonlinear Optics with Microresonators

5.2.6 Multistability The inclusion of nonlinearity changes the input–output relations in a qualitatively different manner. As shown in Fig. 5.11, multistable branches in the input–output relationships of a resonator become possible [80, 198]. The circulating intensity is a function of detuning, which in turn is a function of the circulating intensity and so on ad infinitum. Mathematically, the circulating intensity can be expressed either as an infinitely nested expression or more compactly as an implicit relation between the incident and the circulating intensities.  1 − r2 

I3 = I1 . (5.23) 2π (2π R) 1 − 2r cos φ0 + n2 I3 + r 2 λ

a) Input/output pulse

80

400

60

300

40

200

20

100 0

0 0

50

100 Time, ps

150

c) Accumulated phase shift

6

0

50

150

100 Time, ps

d) Circulating vs incident power

600

Circulating power, Watts

Accumulated phase, rad

b) Circulating pulse

500 Circulating power, Watts

Output power, Watts

100

500

4

400 300

2

200 100 0

0 0

50

100 Time, ps

150

0

40 20 Incident power, Watts

60

Fig. 5.11. Simulation demonstrating bistability in a 5-µm-radius nonlinear resonator with r = 0.818, φ0 = −3π /F , Pπ = 1644.6 W, and a 50-ps, and 66-W peak power pulse. (a) input/output pulse, (b) circulating pulse, (c) accumulated phase, (d) incident versus circulating bistable power relation. Notice the fast switch between stable states, when the input pulse power reaches the hysteresis “jump up” and “jump down” points at 53 and 30 W.

5.2 Resonator Enhanced χ (3) Nonlinear Effects

137

This implicit relation is easily converted into a single-valued function if the circulating intensity is considered the independent variable and the incident intensity the dependent variable. Multistability is always present in such a lossless nonlinear resonator above some threshold intensity. One way to reduce this threshold intensity is to red-detune the resonator slightly so that the resonator is initially off resonance but forms two stable circulating intensity levels √ for a low intensity. The onset of multistability takes place at φ0 = − 3π /F . Optical bistability in a resonator allows for the construction of all-optical flip-flops and other devices exhibiting dynamic optical memory. 5.2.7 Fabry–Perot, Add–Drop, and REMZ Switching The placement of a nonlinear all-pass resonator within one arm of an interferometer, for example, a Mach–Zehnder as in Fig. 5.12, allows for the conversion of phase modulation into amplitude modulation serving as a basis for a nonlinear all-optical switch [150,151]. One might ask why one would want to use this indirect method of implementing a microresonator as an amplitude switching device when a nonlinear add–drop resonator accomplishes the same task in a more direct manner. The properties of an add–drop filter are analogous to that of a traditional Fabry– Perot interferometer which offers light the choice of two output ports and can display nonlinear amplitude switching between them. A careful examination of a nonlinear add–drop resonator as a switching device, however, reveals a fundamental limitation. The switching curve cannot

a)

c) E3

E4

E1

E2

EB2 EA1

EB1

E5 b)

d) E3

E4

E1

E2

EB2 EA1

EB1

Fig. 5.12. (a) An all-pass microresonator, (b) an add–drop microresonator (or Fabry–Perot), (c) a Mach–Zehnder interferometer (MZI), and (d) an REMZ interferometer.

138

5. Nonlinear Optics with Microresonators

be made to perform a complete switch within the phase-sensitive region near resonance because the center of the phase-sensitive region (at a resonance frequency) directly coincides with a minimum or maximum of the transmission spectrum. Only half of the sensitive region is usable as a switch, the other half being wasted. It would be advantageous to shift the peak in phase sensitivity away from the transmission minimum such that it coincides with the linear portion of the transmission curve. However, there is simply no way to accomplish this shift in an add–drop resonator or Fabry–Perot. The missing degree of freedom that can provide this capability can be found in a device formed from a nonlinear all-pass resonator coupled to one arm of an interferometer. An REMZ, for example, can be used to set independently the peaks of nonlinear phase sensitivity and transmission. Figure 5.13 displays the linear transmission characteristics against detuning for a REMZ with offset bias phase between arms of φB = 0, π /2, π , and 3π /2. Optimized switching characteristics are obtained when the peak of the phase sensitivity coincides with the linear portion of the switching curve φB = π /2,3π /2. The characteristic switching curve of a Mach–Zehnder is cosine-squared. To build a effective switch it is important to switch in a region of this curve where the maximum change in transmission is brought about by the minimum change in accumulated phase. This is at the point of maximum switching sensitivity, related to the phase sensitivity as dT dΦ = − 12 sin(Φ − φB ) . dφ dφ

(5.24)

Figure 5.14 compares the switching characteristics for an add–drop resonator, an unbiased REMZ, and a properly biased REMZ. For the add– drop resonator, the buildup and finesse are each lower by a factor of four in comparison with an all-pass resonator with the same coupling strength. In a REMZ, however, only half of the power enters the arm containing the resonator and its phase contributes in a manner that another factor of two is lost. Still, it results in a net improvement in switching threshold by a factor of four that can be observed by comparing the curves in the figure. It might be argued that this analysis is not valid since the finesse is not maintained equal in the comparison. When the finesse is maintained equal, the curves are in fact equivalent. More significant, however, is the improvement in the shape of the switching curve when the REMZ is properly biased. Proper biasing is achieved by tuning the peak of the nonlinear phase sensitivity to the 50% transmission operating point of the unloaded Mach–Zehnder. 5.2.8 Reduced Nonlinear Enhancement via Attenuation As shown, the presence of attenuation mechanisms in the resonator lead to a reduction in peak buildup, a paradoxical increase in phase sensitivity,

5.2 Resonator Enhanced χ (3) Nonlinear Effects

139

Fig. 5.13. A comparison of the linear transmission spectra for an REMZ interferometer with varying phase shift bias differences between arms.

and a reduction in transmission. The dependence of the nonlinear enhancement on the round-trip attenuation is thus non-trivial. The peak buildup (and phase sensitivity) with no loss is termed B0 for simplicity in the following comparisons all taken on resonance. The buildup varies with the round-trip amplitude transmission, a as, B(a)

=

1 − r2 → 1 B0 , (1 − r a)2 r =a 4

B(α2π R)

≈

B0 − 12 B20 (α2π R) +

(5.25) 3 3 2 16 B0 (α2π R)

+ O(3), (5.26)

140

5. Nonlinear Optics with Microresonators

Fig. 5.14. A comparison of the nonlinear switching characteristics of an add–drop resonator (Fabry–Perot), an unbiased, and a properly biased REMZ interferometer. Here, the coupling parameter is held constant for all three cases of resonators. If the finesse is held constant, the nonlinear response of the add–drop resonator matches that of the unbiased REMZ.

where the second expression is a Taylor expansion for loss near zero. The buildup drops with decreasing a until reaching critical coupling (r = a) where it drops to 1/4 its lossless value. In contrast, the phase sensitivity increases from the lossless value without bound at critical coupling, S(a)

=

a(1 − r 2 ) → ∞, (a − r )(1 − r a) r =a

S(α2π R)

≈

B0 +

1 3 2 16 B0 (α2π R)

+ O(3).

(5.27) (5.28)

The nonlinear enhancement is equal to the product of the buildup and phase sensitivity. Additionally, since the intensity drops continuously when traversing the resonator, a correction factor is introduced, C(a) = 1−a2 ln a−2 , such that the effective nonlinear phase shift induced by a much weaker single-pass nonlinear phase shift is ∆ΦNL (a) = B(a)S(a)C(a)∆φNL .

(5.29)

Dividing this quantity by the the shift obtained for the lossless case results in a normalized nonlinear enhancement N (a)

=

∆ΦNL (a) . ∆ΦNL (a = 1)

(5.30)

N (α2π R)

≈

1 − 12 B0 (α2π R) + 14 B20 (α2π R)2 + O(3).

(5.31)

The lowest order variation in phase sensitivity with loss is quadratic. Thus, for small losses, the nonlinear enhancement is reduced primarily because of the reduction in peak buildup. Near critical coupling, the phase sensitivity increase without bound results in a growing enhancement as

5.2 Resonator Enhanced χ (3) Nonlinear Effects

141

2.0

1.5 Normalized nonlinear enhancement

1.0

0.5 Net transmission 0 0

0.02

0.04

0.06

0.08

0.10

Single-pass loss (1-a2) Fig. 5.15. Variation in net transmission and normalized enhancement with loss for a resonator with r 2 = 0.9. For zero loss, both are equal to unity. The transmission steadily drops to zero at critical coupling (r = a), here at 10% loss. By contrast, the normalized enhancement is somewhat impervious to loss and never dips below a certain value due to an increasing phase sensitivity. Although the enhancement diverges at critical coupling, it is of little use to operate there since the transmission drops to zero. well but at the expense of rapidly decreasing attenuation to zero at critical coupling. For comparison, the transmission varies as T (a)

=

(a − r )2 → 0, (1 − r a)2 r =a

(5.32)

T (α2π R)

≈

1 1 − B0 (α2π R) + 2 B20 (α2π R)2 + O(3).

(5.33)

Figure 5.15 shows the variation of normalized nonlinear enhancement and transmission with respect to loss. Notice that although the transmission steadily drops with increasing attenuation, the normalized enhancement is somewhat more stable and never dips below a certain value.1 For a nonlinear device, both enhancement and transmission are generally important and how their importance is weighted will affect how much loss can be tolerated. 1

0.592 in the high-finesse limit.

142

5. Nonlinear Optics with Microresonators

5.2.9 Nonlinear Figures of Merit (FOMs) As will be examined in the next chapter, nonlinear photonic devices hold great promise for the implementation of densely integrated all-optical signal processing. Fundamental restrictions imposed by material properties have not allowed the promise to be fulfilled. Most refractive optical processing devices require a π radian nonlinear phase shift for successful operation that may in turn demand impractically high optical powers. In many material systems, strong nonlinear refractive indices are often accompanied by strong linear and nonlinear absorption that limit the achievable phase shift in many cases far below the target of π radians. The limitations to the achievable nonlinear phase shift imposed by linear and nonlinear absorption thus merit examination. Linear absorption limits the interaction length to an effective length given by  1 − e−αL Lα = . (5.34) α Because the effective length is independent of intensity, in theory, any desired nonlinear phase shift may still be obtained by making the intensity high enough. In practice of course, this may not be practical because all materials possess some threshold for optical damage [178]. Several nonlinear figures of merit are useful for comparing the relative strength of a nonlinear coefficient to absorption. A common definition is the nonlinear coefficient divided by the linear attenuation (M1 = 4π n2 /λα). In general, this parameter is material property with some fixed value at a given wavelength. Most highly nonlinear materials are inherently absorptive because they operate via enhancement near some atomic or molecular resonance. For this reason, M1 is useful for comparing the nonlinearities of different material systems provided that interaction length is not a limitation. Silica single-mode fiber, for example, has a very high M1 despite a low intrinsic nonlinearity because its attenuation is extremely low (5.3 × 10−8 m2 /W). Clever arrangements exist whereby M1 can be modified from its traditionally fixed value granted by nature. The techniques of electromagnetically induced transparency (EIT), for instance, hold the promise of maintaining a strong nonlinearity and canceling the linear absorption via quantum interference. In the case of a nonlinear resonator, although the attenuation is increased in proportion to the effective number of round-trips or finesse, the third order nonlinearity is increased in quadratic proportion. Two-photon absorption imposes a stricter limitation on the achievable nonlinear phase shift. In the presence of linear and nonlinear absorption, the reduced wave equation for nonlinear phase evolution (self-phase modulation) takes the form:

 ∂ 1 A = iγ 1 + iM−1 (5.35) |A|2 A − αA. 2 ∂z 2

5.2 Resonator Enhanced χ (3) Nonlinear Effects

143

Where a second FOM is introduced proportional to the ratio of the nonlinear refractive index to the two-photon absorption coefficient, M2 = 4π n2 /λα2 . Equation 5.35 has exact solutions for the transmitted intensity and nonlinear phase shift after propagating for a distance L. The fractional intensity remaining after propagating is T =

e−αL . 1 + α2 ILα

(5.36)

Because two-photon absorption increases in proportion to the nonlinear refractive index, the achievable nonlinear phase shift saturates logarithmically with increasing intensity:    M1

M2 M2 φNL = ln (1 + 2γILα /M2 ) = ln 1 + 1 − e−2γL/M1 I , (5.37) 2 2 M2  M1

→ γILα = (5.38) 1 − e−2γL/M1 I, M2 →∞ 2 → γIL.

M1 →∞

(5.39)

Within a resonator, two-photon absorption2 is likewise enhanced in proportion to the the nonlinear refractive index,3 i.e., quadratically with finesse. As a result, M2 would not be modified by the use of a resonator. In general, however, both linear and nonlinear absorptive processes are detrimental to switching devices based on the intensity-dependent refractive index. When considering both linear and nonlinear absorption, constructing a device that delivers a π radian nonlinear phase shift using a resonator can exceed in performance; an equivalent device is formed from a simple waveguide. A third FOM, proportional to the nonlinear 

 ∆φNL |E2 |2 phase shift multiplied by the transmission [199], M3 = e |E |2 π 1

is deemed more useful in such a comparison. This FOM is defined in such a way that a π phase change over a 1/e intensity falloff results in M3 = 1. This FOM can be favorably modified by use of a resonator. Figure 5.16 compares M3 for a waveguide of length L, 10L, F L, with that of a resonator. There are of course many other ways of characterizing the trade-offs between attenuation and nonlinearity. Perhaps the most useful definition is the interferometric contrast between a pulse that has acquired a given phase with some some unavoidable attenuation and a reference copy of the same pulse. 2 3

It is expected that higher photon number absorptive processes of order N would be enhanced by a value of F N . Two-photon absorption is not always deleterious. A nonlinear refractive response in a semiconducting material can be obtained through the generation of carriers usually leading to a stronger, albeit slower, response limited by the carrier recombination time. We shall investigate this phenomena further in the next chapter.

Nolinear device figure of merit, M3

144

5. Nonlinear Optics with Microresonators

L

100 10 L Resonator

10-1

10

FL

-2

10-3 10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Nonlinear refractive index, n2I Fig. 5.16. Comparison of the nonlinear device figure of merit (M3 ) for a waveguide of length L, 10L, F L, with that of a resonator. (After Ref. [199], ©2002, Optical Society of America.)

5.2.10 Inverted Effective Nonlinearity Most refractive nonlinear materials behave in such a way that increasing intensity brings about a positive change in index. Negative nonlinear refractive indices are rare but nevertheless can be found in thermal (effectively) nonlinear phenomena and in semiconductors well above the half-gap. Neither case is useful for high-speed all-optical switching. Many proposed applications, such as nonlinear management, would greatly benefit from a negative nonlinear material. A negative effective nonlinearity may be achieved by use of a nonlinear microring resonator. In the undercoupled regime, the sign of the input–output phase relationship is inverted and thus allows for the possibility of inverting the sign of the intrinsic nonlinearity of the resonator medium. Figure 5.17 demonstrates a pulse propagating through a single resonator and acquiring a phase shift of negative π /2 radians.

5.3 Resonator-Enhanced Free Carrier Refraction The increased circulating intensity in a semiconductor microresonator can also be used to enhance charge carrier-based nonlinear optical phenomena. Carriers may be excited through a variety of processes,

5.3 Resonator-Enhanced Free Carrier Refraction a) 0.10

b)

30 25 Power, Watts

Transmission

0.08 0.06 0.04 0.02 0.00 0 c)

5 10 15 20 Incident power, Watts

15 10

0

25

Transmitted pulse 0

20

40

60 80 100 120 140 Time, ps

π 20

20

40

60 80 100 120 140 Time, ps

d) Effective phase shift, rad

Effective phase shift, rad

Incident pulse

20

5

0

π 4

π 20

145

5

10 15 20 Incident power, Watts

25

π 4

0

π 4

Fig. 5.17. A resonator can display negative effective nonlinearity even if the intrinsic material nonlinearity is positive, which is possible only in the undercoupled regime and thus requires a lossy resonator or imbalanced add–drop resonator. (a) Transmission versus incident power. (b) Power versus time. (c) Effective phase shift versus incident power. (d) Effective phase shift versus time. The parameters used in these simulations are r = 0.9, a = 0.875, φ0 = −0.0115π , P = 20W, Pπ = 3183 W, and TP = 20TC = 35 ps. including two-photon absorption. The presence of free carriers modifies the refractive index of a probe signal via the plasma effect.4 The recovery time associated with refractive index changes brought about by carrier generation depends on the lifetime of carriers in the device. Since the width of microring waveguides is usually <1 µm, diffusing carriers may quickly find a boundary where surface recombination becomes an important consideration. The carrier lifetime in this regime can be approximated by the time taken to reach the surface. Holes have much lower mobility than electrons, and in intrinsic materials, the carrier lifetime is dominated by the hole lifetime. This phenomenon is known as “ambipolar diffusion” [201,202]. Several methods can be used to shorten 4

This effect has been implemented in silicon microrings through linear absorptive effects as well [200].

146

5. Nonlinear Optics with Microresonators

carrier lifetime: a bias to sweep the carriers out of the waveguide core and a p-doped core, among others [203]. If the pump pulse is longer than the cavity lifetime, a quasi-steadystate analysis can be used to solve for the interaction between the pump and the probe pulses. To find the change in refractive index due to twophoton absorption, we must first find the change in index from carriers, i.e., the contribution of free carriers to refractive index change. The refractive index n is given as n2 (ω) = (ω) = b + 4π χf ,

(5.40)

where  is the relative permittivity of the medium, and b = 1 + 4π χb ; χb is the contribution of bound charges to the susceptibility, and χf is the contribution of free charges. The equation of motion for carriers in a harmonic electric field, E exp(iωt), is [181] ¨ = −eE exp(iωt), m∗ x

(5.41)

where m∗ is the effective mass of the carrier and e is the electron charge. We assume that the damping term is negligible. The solution to this equation is eE x= exp(iωt). (5.42) m∗ ω2 The corresponding contribution to polarization may be written as P

= =

4π Nfree e2 E m∗ ω2 4π χf E, −

(5.43) (5.44)

where Nfree is the free carrier density. We define the plasma frequency ωp as 4π Nfree e2 , (5.45) ω2p = m∗ which gives ω2p (5.46) 4π χf = − 2 , ω so ω2p n2 = (ω) = b − 2 . (5.47) ω Since there is no imaginary term above, the refractive index is real. 5 Then,

⇒ 2n 5

n2

=

dn dωp

=

b −

ω2p

ω2 ωp −2 2 . ω

(5.48) (5.49)

Realistic, noninfinite carrier mobilities lead to free-carrier-induced changes in the absorption as well.

5.4 Enhanced Four-Wave Mixing Efficiency in Microring Resonators

Since ωp  ω for optical frequencies, 1/2    2ωp 1 dn = − dωp ω2 b − ω2p /ω2  ω3p 2ωp 1 + = −√ . b ω2 b ω 4

147

(5.50) (5.51)

Now that we know the dependence of refractive index on carriers (via the plasma frequency), we have to consider the dependence of carrier concentration on intensity for two-photon absorption — the effect of interest. The absorption cross section for two-photon absorption increases linearly with laser intensity, so the atomic transition rate increases as the square of intensity [178]. If we define a two-photon absorption coefficient α2 we can write the rate equation for the free carrier density Nfree as Nfree (t) dNfree (t) α2 2 (t) − , = I dt 2ω pump τfree

(5.52)

where ω is the pump beam frequency, Ipump is the pump beam intensity, and τfree is the free carrier lifetime. From Eqs. 5.45, 5.51, and 5.52, dn ∝ B3 . dPbus

(5.53)

Since the change in refractive index scales as B3 , the switching power can be reduced dramatically for moderate intensity buildup. We will return to this in experimental demonstrations of optical logic later in the following chapter.

5.4 Enhanced Four-Wave Mixing Efficiency in Microring Resonators We next consider the case of four-wave mixing in a microresonator adapted from Absil et al. [56]. Intuitively, we can expect that the intensity buildup in microrings boosts four-wave mixing as B4 , since all four waves involved experience an intensity buildup. Here, we show that this is indeed the case. Four-wave mixing is due to χ (3) . In the presence of a pump beam E1 (ω1 ) and a signal E2 (ω2 ), a converted signal E3 (ω3 = 2ω1 − ω2 ) is generated: ∂E1 α1 =− E1 ∂z 2 ∂E2 α2 =− E2 ∂z 2 ∂E3 α3 =− E3 + γE12 E2∗ exp(i∆βz), ∂z 2

(5.54) (5.55) (5.56)

148

5. Nonlinear Optics with Microresonators

where ∆β = β3 − (2β1 − β2 ), ∆β = 0 due to dispersion in the material. The equation in E3 is a nonhomogeneous differential equation, and has the solution [204]   z  α3 α3 dz exp z z γE12 E2∗ exp(i∆βz). (5.57) E3 = E3 (t = 0) exp − 2 2 0 However, this just represents the response in the absence of feedback (via resonance). If the effect of dispersion is small enough to ensure that ω3 is on resonance, then E3 is enhanced by 1/(1 − ar ), and the field coupled out is E3,out

 L   α3 t α3 exp − L z γE12 E2∗ exp(i∆βz). (5.58) = dz exp 1 − ar 2 2 0

Since we are concerned with the intensity, we consider the squared magnitude of the right-hand-side; comparing with the expressions for add– drop buildup we see that |t/(1 − ar )|2 ∝ B, noting that the pump and signal intensities are enhanced as B, we get the result that conversion efficiency associated with four-wave mixing is enhanced as B4 .

5.5 Summary In this chapter we introduced the nonlinear optical properties of microresonators. The finesse-scaling laws associated with resonant enhancement of third-order nonlinearities were derived. Experimental demonstrations of the enhancement of the Kerr effect were presented. In the following chapter we describe the results of all-optical switching and logic experiments based on two-photon absorption-induced changes in the refractive index of microrings.

6. All-Optical Switching and Logic using Microresonators

In this chapter we present the results of all-optical switching and logic experiments carried out in III–V microring resonators. All-optical data processing devices are key components for future integrated photonic circuits. One category of such devices is switching and logic. Architectures have been proposed based on intensity-dependent all-optical logic switches and gates constructed from nonlinear planar waveguide devices such as Mach-Zehnder interferometers and Distributed Bragg gratings [205, 206]. Microring resonators can be used to build elementary switches and logic gates with the added advantages of compact size [207] and reduced switching threshold [56, 151]. The transmittance of a microring coupled to a single optical bus, shown in Fig. 6.1, can be altered by changing either the absorption or the refractive index of the microring, or both. For GaAs/AlGaAs microrings operated at the 1.55-µm wavelength, the absorption and index changes can be accomplished via two-photon absorption (TPA) of a high-intensity pump beam tuned to one of the microring’s resonance modes. The subsequently induced change in the transmittance is then used to switch a low-intensity probe beam tuned to the vicinity of a different resonance mode.

6.1 All-Optical Switching Through the implementation of refractive index changes near critical coupling, one can go from low to high transmission if initially on-resonance, or from high to low transmission if initially off-resonance. 6.1.1 Theory The interaction between the pump and the probe beams as they circulate in the microring can be described by a coupled set of nonlinear propagation equations in the slowly varying envelope approximation. Let Ar (z, t) and Br (z, t) represent the field envelope in the ring of the pump and the probe signals, respectively, and designate z as the linear coordinate along

150

6. All-Optical Switching and Logic using Microresonators

Fig. 6.1. Optical micrograph of a vertically coupled GaAs/AlGaAs microring resonator. the ring path. Assuming |Br |  |Ar |, the propagation of the pump and the probe beams in the presence of TPA-induced nonlinearity is given by dAr n0 dAr + dz c dt

=

−α0 Ar − (α2 + in2 k0 )Ip Ar −(∆αfc + i∆nf c k0 )Ar ,

n0 dBr dBr + dz c dt

=

(6.1)

−α0 Br − (α2 + in2 k0 )Ip Br −(∆αfc + i∆nf c k0 )Ar ,

(6.2)

In the above, Ip ∝ |Ar |2 is the pump beam intensity; n0 is the waveguide effective index; α0 is the linear absorption coefficient; α2 and n2 are the nonlinear TPA loss and refraction coefficients, respectively; and ∆αfc and ∆nfc are the changes in the absorption and refractive index induced by free carriers generated from the TPA process. The contributions ∆αfc and ∆nfc are proportional to the free-carrier density, Nfc , via ∆αfc = σa Nfc and ∆nfc = σr Nfc , with σa and σr being the absorption cross section and refraction volume, respectively. The free-carrier density generated by TPA evolves according to dNfc α2 2 Nfc = I − , dt 2ω p τfc

(6.3)

where τfc is the carrier relaxation time. Coupling among the input pump Ai (t), the output pump Ao (t), and the circulating pump beam Ar (z, t) in the ring is described by the set of equations,  Ao (t) = r Ai (t) − i 1 − r 2 Ar (t − TR )eiφ , (6.4)  iφ 2 Ar (t) = r Ar (t − TR )e − i 1 − r Ai (t), (6.5) where r is the lumped self-coupling coefficient, TR is the cavity transit time, and φ is the round-trip phase. A similar relationship exists for the probe signals Bi (t), Bo (t), and Br (t). Equations. 6.2, 6.3, and 6.5 are timedifference equations that are numerically integrated to simulate the passage of the pump and the probe beams through the microring resonator.

6.1 All-Optical Switching

151

6.1.2 Linear Device Characteristics The microresonators used in the experiment consist of 10-µm-radius GaAs/AlGaAs microrings vertically coupled to a straight bus waveguide. The waveguiding structure has high lateral index contrast (3.37:1.5) to minimize the bending loss. A high-index, epitaxially-grown AlGaAs mid-layer allows for efficient coupling between the ring and the bus waveguides. Figure 6.2 shows the measured spectral response of the microresonator at the 1562-nm resonance and the theoretical fit. At resonance, an extinction notch of 8.5 dB in the signal is evidence of near critical coupling. For the device shown, assuming negligible propagation loss in the bus waveguide, a coupling coefficient of 3% and a round-trip power loss of 6% were obtained for the microring, giving a 3-dB bandwidth of 0.16 nm and a quality factor of 9800. The finesse of the device as determined from the measured 3-dB bandwidth and free spectral range was 68. The device has a computed field enhancement factor of 3.7 and a cavity lifetime of 42 ps.

1

Transmittance

0.8

0.6

0.4

0.2

0

measaured model fit 0.4

0.2

0

0.2

0.4

Wavelength detuning, nm Fig. 6.2. Measured spectral response and its theoretical fit of a 10-µmradius GaAs/AlGaAs microring resonator at the 1562-nm resonance. The discrepancy between measurement and model on both sides of the resonance peak is due to the Fabry–Perot modulations in the bus waveguide.

152

6. All-Optical Switching and Logic using Microresonators

6.1.3 Simulations A discretized form of Eqs. 6.2, 6.3, and 6.5 enables numerical simulation of the nonlinear propagation of a 300-ps pump pulse through the microring. The wavelength of the pulse was set at 1561.97 nm, that is, 0.1-nm blue detuned with respect to the cavity resonance mode at 1562.07 nm. The parameters used in the simulation are α0 = 10.2 cm−1 , α2 = 24 cm/GW, n2 = 1.5 × 10−13 cm2 /W, σa = 1.5 × 10−16 cm2 , σr = 10−20 cm3 , and τfc = 50 ps. The free carrier lifetime τfc was an adjusted parameter in the simulations to fit measured transient responses. Figure 6.3 shows the evolution of the normalized pump intensity, Ip , inside the microring and the output pump intensity. The circulating pump intensity in the ring is normalized to its peak value and is actually 12 times higher than the peak input intensity, due to coherent buildup 1

Input pump Circulating pump Normalized Intensity

0.8

0.6

Output pump

0.4

0.2

0 0

200

400

600

800

Time, ps Fig. 6.3. Simulated nonlinear propagation of a 300-ps pump pulse through a 10-µm-radius GaAs-AlGaAs microring resonator. The output intensity is normalized with respect to the peak input intensity. The pump intensity circulating inside the resonator is normalized with respect to its peak value, that is 12 times higher than the peak intensity of the input pump. The input pump is a real experimental pulse captured by the oscilloscope to be used in the simulation.

6.1 All-Optical Switching

153

Normalized Intensity

in the resonator. As can be observed in Fig. 6.3, the circulating pump intensity is lagging the input pump intensity by approximately the cavity charging time. It was observed in the simulation that as the intensity in the ring builds up, carriers generated from TPA cause a net decrease in the refractive index of about 2 × 10−4 , which has the effect of shifting the resonance mode toward shorter wavelengths by nearly 0.1 nm. Consequently, the output pump is observed to rise initially with the field in the ring, but then dip to a minimum as the microring resonance is pulled closer to the pulse wavelength. We note that this dip coincides with the peak of the pump intensity in the microring. At the falling edge of the input pulse, the field in the microring begins to discharge, and the output pump intensity is observed to rise again as the cavity resonance mode returns to its initial position. Next, a low-intensity continuous wave (CW) probe signal tuned to the resonance mode at 1550.92 nm is launched into the microring. By tuning the probe beam with respect to the resonance and looking at the transmitted probe, one can quantify the refractive index change and its sign. This is demonstrated in the three-dimensional plot shown in Fig. 6.4. When the probe beam is tuned to 1550.6 nm, far from resonance, it is completely transmitted to the throughput port or in the “ON” state. As the probe beam is tuned closer to resonance, the

0.2

0

0.2

800 600 400

Time, ps

1551.4 1551.2 200

1551 0

1550.8 1550.6

Probe wavelength, nm

Fig. 6.4. Three-dimensional plot of the simulation results for the throughput probe intensity versus wavelength and time. The dc level has been removed for clarity.

154

6. All-Optical Switching and Logic using Microresonators

probe experiences an increased attenuation as the pump pulse causes the resonance modes of the microring to shift to shorter wavelengths, effectively pulling the probe beam into resonance. When the probe wavelength is tuned to resonance at 1550.92 nm and prior to the arrival of the pump pulse, the transmitted probe signal is attenuated strongly. When the pump pulse arrives, it shifts the resonance away from the probe beam, and as a result, we see a rapid increase in the transmission of the probe signal as it switches on. In Fig. 6.8, we plot the input pump intensity and the switched probe for the two cases when the probe beam is initially blue-detuned off resonance at 1550.75 nm, and when it is initially tuned to resonance at 1550.92 nm. These represents the ideal scenarios for the “ON-OFF” and “OFF-ON” switching, respectively. 6.1.4 Nonlinear Device Characteristics The experimental setup for the pump and the probe experiment is shown in Fig. 6.6. An externally modulated laser diode was used to produce 1

(a)

Normalized Intensity

0.8

0.6

0.4

(b) 0.2

0 0

200

400

600

800

Time, ps Fig. 6.5. Simulated pump-probe interactions. (a) Probe beam initially off resonance. (b) Probe beam initially on resonance. The probe beam intensities are normalized with respect to the maximum transmittance at resonance.

6.1 All-Optical Switching

155

Fig. 6.6. Experimental setup for the pump-probe technique to demonstrate all-optical switching using a GaAs-AlGaAs 10-µm-radius microring resonator. PG: pulse generator. FPC: fiber polarization controller. MZI: Mach-Zehnder interferometer. EDFA: Erbium doped fiber amplifier. D: detector. BPF: band-pass filter. a 300-ps pump pulse signal with a 20-MHz repetition rate. The pump beam was then amplified using an erbium doped fiber amplifier (EDFA) to compensate for the external modulator and fiber to waveguide coupling losses. The average input power was 8 mW at the device input waveguide, giving an estimated peak intensity of 4.8 GW/cm2 in the microring at resonance. The pump wavelength was set at 1562.0 nm, slightly blue-detuned from the resonance. A 3-mW CW probe signal tuned to the next higher resonance at 1550.9 nm was coupled to the pump beam via a 50/50 fiber coupler and fed to the device input using a conically tipped fiber to minimize the mode-mismatch between the fiber and the waveguide. At the output, the pump and the probe signals were collected using another conically tipped fiber, optically amplified, then separated by a band-pass filter, and detected using a 40-GHz detector and a 50-Ghz oscilloscope. In Fig. 6.7, we plot a three-dimensional graph for the measured throughput probe intensity as a function of time and probe wavelength. It is in very good agreement with the simulated 3-dimensional graph shown earlier

Normalized Intensity

156

6. All-Optical Switching and Logic using Microresonators

0.2

0

-0.2

800 600

Time, ps

1551.4 400

1551.2 1551

200 1550.8 0

1550.6

Probe wavelength, nm

Fig. 6.7. Three-dimensional plot of the measured throughput probe intensity versus wavelength and time. The dc level has been removed for clarity.

in Fig. 6.4. Fig. 6.8 show the time traces of the input pump, the output pump, and the transmitted probe power for the switch-on and switchoff cases. The measured responses of both the pump and the probe signals are in excellent agreement with the simulated responses shown in Fig. 6.5. The pump pulse is strongly distorted and absorbed due to critical coupling and TPA in the microring. The on and off switching times of the probe beam are slightly less than 100 ps. Also, it is observed that the probe beam quickly returns to the original state after the pump beam has passed. It is postulated that this fast recovery time of the order of 50 ps is due to surface-state recombination at the microring sidewalls.

6.2 Thresholding and Pulse Reshaping The nonlinear transmission transfer function of a notch filter microring resonator can be used for thresholding. If an input signal is tuned exactly to a critically coupled microring resonance, it is initially absorbed by the resonator and no output is transmitted. If the signal intensity is increased, the self-induced phase shift in the resonator will result in a detuning in the microring resonance wavelength. If such detuning is

6.2 Thresholding and Pulse Reshaping 1

Input pump Output probe

0.8

Normalized Intensity

157

0.6

(a)

Output pump

0.4

0.2

0 0 1

200

400

600

800

Input pump 0.8

Normalized Intensity

Output probe

0.6 (b)

0.4

Output pump

0.2

0 0

200

400

Time, ps

600

800

Fig. 6.8. Measured time traces of the pump and the probe beam intensities with (a) probe beam initially off resonance (high transmission) and (b) probe beam initially on resonance (low transmission).

158

6. All-Optical Switching and Logic using Microresonators

1

Normalized Intensity

Output Input

0.75

0.5

0.25

0

-100

-50

0

50

100

Time, ps Fig. 6.9. Time traces of the input and output pulses of a GaAs-AlGaAs microring resonator. Input and output intensities are normalized by their respective peak values. The delayed output pulse is shifted in time by 25 ps to overlap with the input pulse.

enough, the signal will be out of resonance and no longer absorbed by the microresonator and totally transmitted to the output port. This effect can be used for thresholding applications such as digital level restoration. Figure 6.9 shows the input and output time traces of a GaAs-AlGaAs microring notch filter excited by a 35-ps laser pulse carrying approximately 50 pJ of energy. The input signal has a rise time of 40 ps, whereas the rise time of the output pulse was measured to be 20 ps. The output pulse was delayed with respect to the input pulse by 20 ps corresponding to the charging time of the resonator. We plot both pulses to show the pulse reshaping.

6.3 Time-Division Demultiplexing A routing switch connects the input port to one of several output ports and physically moves photons from one port to another. Routing can be based on either the intensity of the input signals or an external

6.3 Time-Division Demultiplexing

159

control beam. Typically, the control beam is in a different physical format than the data. For example, the pump beam might have a different wavelength or polarization than the data beam. Nonlinear directional Couplers [208], distributed-feedback waveguides [209], are examples of passive all-optical routing and demultiplexing switches. When a microring resonator is used in the add–drop configuration, the pump and probe method can be used to spatially route an incoming signal to the through port or drop port of the device depending on the presence or absence of the control beam [210]. If the incoming signal consists of a train of different data pulses or channels multiplexed in time, the pump pulse can be applied at the appropriate time slots to pick out an individual data pulse or channel and drop it at the drop port, while allowing the rest of the pulses to pass on via the through port. The device in this case functions as an optical time-division demultiplexer. In a similar manner, data pulses can also be multiplexed onto the incoming signal through the add port of the device. 6.3.1 Linear Device Characteristics The microresonator used in the experiment consists of a single, 10-µm radius GaAs/AlGaAs microring vertically coupled to two straight waveguide buses as shown in Fig. 6.10. The wavelength spectrum of the both the

Drop

20 micron

Add

Input

Through

Fig. 6.10. Optical micrograph of a GaAs vertically coupled OCDF microring resonator.

160

6. All-Optical Switching and Logic using Microresonators

Fig. 6.11. Normalized measured spectral response and for the throughput and drop ports for the OCDF microring resonator.

through and the drop ports of the device are shown in Fig. 6.11. The resonator has a bandwidth of 0.22 nm, a finesse of 45, and a quality factor of 7000. 6.3.2 Nonlinear Device Characteristics The response time of the microring was measured as follows. A semiconductor CW source, tunable in the 1.5 − 1.6-µm range, provided a low-power (signal) beam with typical coupled power of 1 mW. A second semiconductor laser diode is gain-switched at 8.5 GHz to produce 25-ps pulses and is fed to an external modulator to reduce the repetition rate to 125 MHz. It was subsequently amplified by an EDFA to 10-mW average power. The CW signal (probe) beam was blue-tuned to the microring resonance at 1555 nm, whereas the control beam was tuned to the next higher microring resonance at 1544 nm. Wavelength tuning was achieved by both tuning the input laser sources wavelengths and by thermally tuning the microring resonator using a thermoelectric cooler. The temperature sensitivity of the microring was 0.11 nm/◦ C. Both input beams were coupled together using a 10:90 fiber coupler and were fed to the device input waveguide using a conically tipped single-mode fiber. The dropped and through beams were collected using conically tipped fibers, spectrally filtered from the pump beam using a band-pass filter centered at the probe beam wavelength, and analyzed as functions of time with a 45-GHz

6.3 Time-Division Demultiplexing

161

Fig. 6.12. Time traces for (a) the input pump pulse at 1545 nm and (b) the output dropped signal initially blue-tuned to the 1555-nm resonance.

detector and a 40-GHz digitizing oscilloscope. Thus, measurements were instrument limited. Fig. 6.12 shows both the input pump pulse at 1545 nm and the output dropped signal whose wavelength was slightly tuned off resonance at 1555 nm, thus initially undropped. When the 25-ps pump beam arrives, it is partially absorbed by the microring material via TPA, and free carriers are generated causing a change in both the absorption and the refractive index coefficients. This leads to a shift in the microring resonance toward shorter wavelengths temporarily overlapping the CW signal pulse and resulting in high transmission at the drop port. It is observed that the dropped signal exhibits a switching window of 30-ps pulse width and approximately 8-dB switching contrast, demonstrating that the device is capable of switching data up to a rate of 30 Gb/s. In the second part of the experiment, the use of the microring resonator as a spatial pulse router is demonstrated. Specifically, the microring routes incoming data to either the drop port or the through port, depending on the presence or absence of a control pulse. The signal (probe) beam was a 5-GHz, 40-ps RZ data stream tuned to the microring resonance at 1555 nm. For the control beam, an external modulator was used to chop a CW beam that was tuned to the next microring resonance at 1544 nm. This resulted in a train of 150-ps control pulses at 80-MHz repetition rate. The signals detected at the drop and through ports are shown in Fig. 6.13.

162

6. All-Optical Switching and Logic using Microresonators

Fig. 6.13. Time traces for (a) the input 5-GHZ RZ data stream slightly detuned to the resonance at 1555 nm, (b) the control signal at the 1545 nm resonance, (c) unswitched data at the throughput port, and (d) demultiplexed data at the drop port.

It can be observed from the figure that the drop port signal is the demultiplexed data from the input data stream, whereas the remaining unswitched data are passed on to the through port where further demultiplexing can be performed. The cross talk at the drop port was measured to be less than 8 dB. This cross talk is primarily due to the asymmetry between the input and the output coupling coefficients resulting from fabrication imperfections. This asymmetry in the coupling coefficients is the reason for incomplete extinction at resonance for the

6.4 All-Optical Logic

163

throughput spectrum shown in Fig. 6.11. Higher contrast and therefore less cross talk can be obtained by better matching of the input and output coupling coefficients and reducing the round-trip loss to result in a critically coupled OCDF resonator. The pulse energy required for switching was approximately 50 pJ/pulse at the input of the waveguide. The actual energy coupled to the resonator is much less due to scattering and waveguide tapering losses and is difficult to measure. Engineering the material bandgap and increasing the resonator field enhancement will lower the switching power. It should be noted that the device can be also used in a similar manner to multiplex data onto the incoming data stream via the add port of the device.

6.4 All-Optical Logic We next consider the implementation of nonlinear switching mechanisms in microresonators for the construction of all-optical logic gates. By injecting two signals in a microresonator copropagating at two different wavelengths or counter-propagating at the same wavelength, we can switch on or switch off an auxiliary signal. Careful preparation can enable AND/NAND or OR logical operations when the injected signal pulses coincide spatially and temporally. 6.4.1 AND/NAND In this section, all-optical AND and NAND logic gate operation via nonlinear absorption is demonstrated in single semiconductor micro-racetrack resonators [211]. Operation near critical coupling allows a high switching contrast [57]. The device is pumped with two counter-propagating data streams tuned to one of its resonance wavelengths and is probed at the next higher resonance wavelength. The performance of both InP and GaAs devices are compared in terms of switching energy and speed. The nonlinear optcal mechanism implemented in the gates is again the change in refractive index from free carriers generated by TPA [212]. The number of free carriers generated inside the resonator is quadratically proportional to the pump intensity inside the ring, which is in turn proportional to the resonator intensity buildup factor. In addition, the phase change needed for switching is reduced by the resonator finesse. The input optical energy needed for switching is, thus, reduced by the third power of the intensity buildup. On the other hand, thermal nonlinearity is only proportional to the intensity inside the resonator. The guiding layer of the InP and GaAs devices are designed to have bandgap energies at 1380 and 800 nm, respectively. The 1550-nm pump pulses are absorbed through TPA inside the resonator, and free carriers

164

6. All-Optical Switching and Logic using Microresonators

are generated. This results in a temporal decrease in the refractive index of the resonator waveguide and, thus, a blue shift in its resonance wavelengths. In a critically coupled ring, when a probe beam is initially tuned to a resonance, it is highly attenuated at the output representing a logical “0.” If “A” and “B” are the input data streams, then when either “A” or “B” is “1,” the phase shift acquired by the resonator is not enough to switch the probe out of resonance. However, when both signals are “1,” the number of generated carriers is four times higher because of the quadratic dependence of the TPA. Moreover, the nonlinear transmission of the resonator enhances the switching and brings the probe beam out of resonance to “1” or high transmission. This is functionally representative of AND logic operation. If the probe beam is initially out of resonance, then the device can be operated as a NAND logic gate. The InP devices [76] were tested first. These consisted of a laterally coupled micro-racetrack resonator with a 10-µm radius and a 3-µm straight coupling section [Fig. 6.14(a)]. The waveguide core consists of GaInAsP and possesses a cross section of 0.5 × 0.5 µm2 . Figure 6.14(b) shows the measured spectral response at the throughput port of the

Fig. 6.14. (a) Scanning electron microscope picture of the InP device and (b) its measured (solid line) and fitted (dashed line) spectral response at the throughput port.

6.4 All-Optical Logic

165

Fig. 6.15. Experimental setup for optical logic AND gate using the microracetrack resonator. PG: Pattern generator. PC: Polarization controller. BPF: Band-pass filter.

racetrack resonator. For the resonance at 1550 nm, the extinction is 20 dB, the round-trip power loss is 40% and the coupling coefficient is 46%. The ring resonator has a 3-dB bandwidth of 1.8 nm, a free spectral range of 10 nm, a finesse of 6, and an estimated of 1.5. Fig. 6.15 displays a schematic diagram for the experimental setup used to demonstrate the logic gate operation. A pump beam source, consisting of a CW external cavity laser diode tuned at one of the resonator resonant wavelengths at 1550 nm, was externally modulated by a pattern generator. The pump pulses were return-to-zero (RZ) data generated at 0.5 GHz with 500-ps pulse duration. They were split by a 50/50 coupler into two counter-propagating data streams channels. Each channel was separately amplified by two EDFAs and filtered with a band-pass filter then injected into one port of the device using conically tipped fibers. The pump pulse energy was 18 pJ at the input of the device. A variable optical delay line was used to adjust the arrival of one data channel with respect to the other at the resonator. The probe beam, a CW signal tuned to the resonator resonance at 1560 nm, was amplified separately, and coupled with one data channel using a 50/50 coupler. The probe power was 10 mW at the device input. The output probe was collected using a fiber circulator, band-pass filtered at the probe beam wavelength, optically amplified and fed into a 40-GHz detector and a 50-GHz oscilloscope.

166

6. All-Optical Switching and Logic using Microresonators

Fig. 6.16. Time traces showing an AND logic gate using the InP microresonator: (a) “A” and (b) “B” are the two input pumps tuned to the resonance at 1550 nm and (c) “F = A AND B” is the output probe signal tuned to the next higher resonance at 1560 nm.

Fig. 6.16 displays the time traces of the input and output data patterns illustrating AND gate operation. The probe beam is initially tuned to resonance and, hence, suffers low transmission representing a logical “0.” The estimated pump intensity required to switch the probe beam out of resonance is 0.4 cm, when the pump beam is tuned to the max slope of the resonator where the switching sensitivity is maximum [212]. We have adjusted the intensity and wavelength of the pump beams, “A” and “B,” such that either one alone cannot switch the probe beam out of resonance. When both “A” and “B” are logical “1”, the intensity is four times higher and is enough to switch the probe out of resonance to the ON or “1” logic state. The speed of the device was investigated using a mode-locked laser at 5.6GHz and externally modulated at 140MHz. The pump pulse possessed a pulse width of 35 ps and energy of 20 pJ. A switching window of 100 ps

6.4 All-Optical Logic

167

(limited by the carrier lifetime) and a 13-dB switching contrast (limited by the noise floor of the receiver) were measured. From measured data and a simulation model, we estimate a 0.9-nm temporal tuning of the microring resonator. Next, similar GaAs devices [57] were tested. The device had the same configuration as the one shown in Fig. 6.14(a). It had a resonance at 1548 nm with an extinction of 19.5 dB, a round-trip power loss of 20%, a coupling coefficient of 25%, a 3-dB bandwidth of 0.78 nm, a free spectral range of 11 nm, and a finesse of 14. We used a similar setup to the one shown in Fig. 6.15. To obtain high-speed data, a gain-switched laser diode at 8.4 GHz and externally modulated at 140 MHz was used for the data source. The pump pulses had energy of 80 pJ per pulse at the input waveguide of the device. Again, the actual pulse energy coupled to the device is hard to measure. The pump data were tuned to the device resonance at 1548 nm, whereas the probe was tuned to its resonance at 1559 nm. The probe beam had a power of 10 mW at the input waveguide. Figure 6.17(a) and (b) display the time traces for the inputs “A” and “B.” Figure 6.17(c), displays the output probe “F” when the probe beam was initially tuned to resonance. It shows that the output is only “1” when both “A” and “B” are “1,” demonstrating AND gate operation. Figure 6.17(d) displays the output “F” when the probe beam was initially 0.4 nm bluetuned to resonance. Initially, the probe beam is highly transmitting or “1.” When both “A” and “B” are “1”, the generated carriers shift the resonance wavelength enough to bring the probe beam in resonance and, thus, impart low transmission resulting in a logical “0.” In both cases, the output “F” can be observed to follow the inputs instantaneously. We measured the switching window of the device to be 35 ps limited by the carrier lifetime and the detection system. To characterize the performance of the devices, two additional measurements were performed: the TPA coefficient and the carrier lifetime. In straight waveguides, the single-beam transmittance technique [213] was used to measure the TPA coefficient at approximately 18 and 43 cm/GW for GaAs and InP, respectively. Based on these values, the assumption of constant round-trip loss inside the resonator is valid for the pump intensity that we used for the logic gate experiment. In other words, the nonlinear loss contribution is neglected. The carrier lifetime of the materials was measured by pumping a straight waveguide section and probing the induced absorption at a different wavelength. Since the materials are intrinsic and bulk, ambipolar diffusion is expected. The mobility of GaAs was estimated as 500 cm2 /Vs and of InP as 210 cm2 /Vs, consistent with 780 and 310 cm2 /Vs, as calculated from published data [201]. A measured quadratic dependence of the carrier lifetime on the straight waveguide width is a signature of the diffusion process. Faster response might be achieved by shortening the

168

6. All-Optical Switching and Logic using Microresonators

Fig. 6.17. Time traces showing logic operation using the GaAs microresonator: (a) and (b) are the inputs “A” and “B” tuned to the resonance at 1548 nm; (c) output “F” when the probe was initially in resonance at 1559 nm; and (d) output “F” when the probe was initially blue-tuned out of resonance at 1558.6 nm. carrier lifetime, e.g., by sweep-out [202], or doping the guiding layer with p-type carriers 1 Engineering the bandgap through tensile strain can also enhance the hole mobility and, thus, shorten the carrier lifetime. The generated free carrier density was calculated from the carrier lifetime measurements and compared with the estimated value from the index change. The refractive index change was estimated by measuring the shift in the microresonator resonance wavelength and was estimated to be approximately −10−3 . The free carrier density calculated by absorption is two orders of magnitude higher than that estimated. This 1

This results in a reduced n-type minority carrier lifetime, which is much shorter than the ambipolar lifetime.

6.4 All-Optical Logic

169

discrepancy is likely due to the difficulty in estimating the input power at the coupling section of the microring because of coupling losses and tapering waveguide losses. 6.4.2 NOR In this section, all-optical NOR logic gate operation is demonstrated. Again, the employed mechanism is nonlinear absorption albeit this time in symmetric GaAs-AlGaAs microring resonators whose resonances are closely matched [214]. Two input pump data streams are tuned close to one resonance of the symmetric microrings to switch a probe beam tuned to another resonance by two-photon absorption. The use of two rings provides for better cascading in photonic logic circuits because of the higher number of available ports. All-optical AND and NAND logic gates based on single-ring devices were discussed previously [211]. However, multiple ring-based logic gates are better suited for cascading in photonic logic circuits because they provide more ports, which provide for better separation of single channels and prevent backfeeding or crossfeeding of optical power, and minimize the need for additional external components such as circulators, isolators, and band-pass filters. An optical micrograph of the NOR gate is shown in Fig. 6.18. It consists of two symmetric 9.6-µm-radius microring resonators vertically coupled to a throughput bus waveguide. Each ring has a drop port separated by 250 µm from the throughput port. These extra input and output ports eliminate the need for any external fiber splitter and circulator, as were required for the single microring logic gate [211]. The 250-µm-separation was chosen to match the pitch distance of a conically tipped fiber array used to couple light in and out of the device. The throughput and drop waveguides are tapered from 2.5 µm at the cleaved facets to 0.8 µm at the coupling section to the resonators over a length of 200 µm. The 0.8-µm width at the coupling sections of the microrings maintains single-mode behavior of the bus waveguide. The microring level was first fabricated and then flip-bonded to a new substrate via polymer wafer bonding. The growth substrate was then etched away, and the bus waveguides were fabricated. Fig. 6.19 displays the spectra of the throughput and both drop ports of the device. The resonance wavelengths of the two microrings were designed to coincide with each other. The resonators have 3 dB bandwidths of 0.22 and 0.30 nm; finesse of 47 and 35; quality factors of 7100 and 5200, respectively; and a free spectral range of 10.42 nm. Assuming symmetric coupling for each resonator, a coupling coefficient of 5.3% and a round-trip loss of 2.3% are estimated for the first microring resonator, and a coupling coefficient of 7.2% and a round-trip loss of 2.9% are estimated for the second microring resonator. The small differences in the

170

6. All-Optical Switching and Logic using Microresonators

Fig. 6.18. Optical micrograph of the NOR logic gate. The two microring resonators are symmetric and of 9.6-µm radii. The etched features appear lighter, whereas the waveguides appear as dark lines and curves. Inset, magnified picture of the microring resonators underneath the bus waveguides. A, B, pump inputs; C, probe input; F, probe output.

two microrings arise from bus-waveguide alignment asymmetry. Because of lithographic nonuniformity, there is a 0.2-nm mismatch between the resonators for the 1557-nm peak, which results in a higher than expected switching energy. For a pump pulse that is much longer than the cavity lifetime, the quasi-steady-state analysis can again be used to solve for the interaction between the pump and the probe electric fields circulating inside a microring resonator. As the guiding layer of the GaAs microrings is designed to have bandgap energy at 800 nm, the 1550-nm pump pulses are partially absorbed through TPA inside the resonator, and free carriers are generated according to Eq. 6.3. The TPA coefficient was found to be 18-cmGW, based on transmittance measurements of straight GaAs waveguides, and the effective carrier lifetime was found by pump-probe measurements of similar straight and tapered waveguide sections to be 40 ps. The effective carrier lifetime is determined by carrier diffusion and surface recombination at the sidewalls. The change in refractive index

6.4 All-Optical Logic

171

Fig. 6.19. Normalized spectrum for (a) the throughput port, (b) the drop port of the first microring resonator, and (c) the drop port of the second microring resonator.

resulting from the generated free carriers causes the microring resonance peaks to shift and thus can be used to switch a probe beam on or off. Fig. 6.20 displays a spectral schematic for one of the microrings in the NOR logic gate operation. The pump beam (A or B) is slightly blue-tuned to the first peak, whereas the probe beam (C) is initially blue-detuned to the second (or any other) peak in the spectrum (shown as a solid curve) and hence passes unaffected to the through port. When the pump beam (A or B) is switched on, it induces free carriers in the ring resonator by TPA, and the resultant refractive-index change causes the spectrum to blueshift (shown as a dashed curve). This spectral shift causes the second peak to coincide with the probe wavelength, resulting in its being dropped from the output. The first peak moves past the pump wavelength, causing it to be slightly red-detuned. Note that the pump wavelength is initially only slightly blue-detuned [0.5 full width at half maximum (FWHM)], whereas the probe wavelength has a larger gap (1 FWHM). This exact wavelength placement is critical to device operation, as the pump wavelength has to overlap the first peak at all times for any reasonable intensity buildup within the ring.

172

6. All-Optical Switching and Logic using Microresonators

Fig. 6.20. Spectral schematic of NOR logic gate operation. Solid curves, spectral characteristics of intensity inside the ring when the pump beam is off. Dashed curves, the blueshifted spectral characteristics when the pump is on. The positions of the pump and the probe wavelengths are indicated by dotted vertical lines.

We estimate that the intensity in the ring needed to produce a spectral shift of 0.30 nm (1 FWHM) is 2.0×108 W/cm2 . The pulse energy in the bus waveguide is estimated to be 6 pJ for a pulse width of 35 ps. Compared with that of conventional two-armed interferometers such as Michelson and Mach-Zehnder types, the switching energy is reduced by a factor proportional to the third power of the microring cavity finesse. The input and output ports and signals are as shown in Fig. 6.18. The pump pulses are provided by a laser diode tuned close to the device resonance at 1546 nm and gain switched at 8.5 GHz. The repetition rate is reduced by use of an external modulator at 140 MHz. The pulses are further split by a 50/50 fiber splitter, and one arm is delayed by an optical delay to provide the two independent input data pulses, A and B. Each data stream is amplified separately by an EDFA and then filtered to reduce the amplified spontaneous emission noise. The CW probe beam (C) is provided by a tunable external cavity laser diode that is tuned to the device resonance at 1557 nm. Input pump data and the probe beam are then coupled to the device by use of two conically tipped fiber arrays, and the output light is collected in the same manner. The output probe signal is then spectrally filtered and optically amplified by another EDFA. When either beam A or beam B is switched on, it induces free carriers in the corresponding ring by TPA and causes the resonance peaks to blueshift and overlap the probe wavelength, resulting in its being dropped from the output (F; Fig. 6.18). Thus the logic function can be written as F = A NOR B. In Fig. 6.21 we plot the time traces for input data streams A and B and output probe F, showing all possible binary combinations. The switching

6.5 Summary

173

Fig. 6.21. Time traces for the input pump data streams (a) A and (b) B at 1546 nm and (c) the output probe signal at 1557 nm, showing the NOR logic operation. contrast is 6 dB and is limited by the scattering loss in the rings, which results in noncritical coupling. The measured switching energy is estimated to be 20 pJ per pulse at the input of the waveguide, which is higher than expected for an ideal case calculated by use of device and material parameters. We believe that two factors contribute to the higher than estimated switching energy: the slight mismatch between the two resonators, and the fiber-to-waveguide coupling efficiency. The 0.2-nm resonance mismatch between the two microring resonators results in nonoptimal placement of the pump wavelength with respect to the peaks. Also, the fiber coupling efficiency was measured based on experiments in which a single fiber was coupled to a single test waveguide. Aligning a fiber array to multiple waveguides is inherently more difficult and resulted in lower coupling efficiency. Decreasing the pump pulse widths to lower values will result in distortion of the output pulses. Specifically, the output pulse width will be determined by carrier diffusion times and not by the input pulse width anymore. Assuming ideal input pulses, the rate of free-carrier generation by TPA, and hence the index fall time, is limited by the photon cavity lifetime (time for intensity to build up in the ring). However, the carrier removal rate, and hence the index rise or recover time, is limited by diffusion and surface recombination.

6.5 Summary In this chapter we gave an overview of the theoretical framework along with experimental demonstrations of all-optical switches and logic devices in III–V microring resonators by TPA using the pump-probe method.

174

6. All-Optical Switching and Logic using Microresonators

The dominant nonlinear effect was observed experimentally and confirmed by simulation to be carrier-induced change in the refractive index. Also demonstrated was the use of a single microring resonator notch filter in pulse reshaping. Moreover, using a microring resonator in the OCDF configuration, we demonstrated time-division demultiplexing and spatial pulse routing. As the switching speed of these devices is currently limited by the carrier lifetime and not the microring cavity lifetime, the performance may be improved further by a variety of known methods for reducing carrier lifetimes.2 The pump energy required for switching can also be reduced by more efficient absorption mechanisms such as operating near the band edge at the expense of increased losses, which ultimately limit device cascadability. Consequently, although much weaker, the best promise for cascadable all-optical logic continues to be the Kerr effect.

2

Methods for reducing carrier lifetimes in III–V semiconductors include proton/ion damage and dc biasing to sweep out free carriers.

7. Distributed Microresonator Systems

7.1 Introduction In the previous chapters, the linear and nonlinear transfer characteristics of a single (or a few) microresonator were examined in detail. In this chapter, the linear and nonlinear propagation characteristics of a distribution of microresonators are examined. Due to their inherent simplicity, we begin with a treatment of the linear pulse propagation characteristics of distributed all-pass resonators, move on to an examination of their nonlinear propagation characteristics, including the prediction of solitons, and make some comments on the limitations of these structures. Finally, we conclude with a generalized treatment of periodically distributed microresonators. These structures may be considered to be a new family of artificial media [215, 216] with analogies to photonic crystals [217–219]. Note: Both in this chapter and in the literature cited here, the terms SCISSOR (side-coupled integrated spaced sequence of resonators) and CROW (coupled-resonator optical waveguide) are used, which refer, respectively, to parallel and serial cascaded rings.

7.2 Linear Propagation in Distributed Microresonators A pulse propagating though a single all-pass resonator acquires a frequency-dependent phase shift that serves to delay and/or distort the pulse shape. By arranging a sequence of resonators coupled to an ordinary waveguide as in Fig. 7.1 (SCISSOR or a parallel-cascade), the effective propagation constant of the guide can be modified. The modified effective propagation constant can be defined as the accumulated phase per unit length and is composed of the propagation constant of the waveguide itself plus a contribution from the effective phase shift of the resonators. For a resonator spacing of L, the effective propagation constant becomes keff (ω) = n0 ω/c + Φ (ω) /L.

(7.1)

A plot of the dispersion relation (effective propagation constant against radian frequency) for various values of the self-coupling parameter r is

176

7. Distributed Microresonator Systems

E3

E4

R

E1

E2

L

Wavevector, keff −k0

Fig. 7.1. A microresonator-structured, fully transmissive waveguide consisting of a SCISSOR. E1 is the incident field, E4 is the field injected into the disk, E3 is the field after one pass around the resonator, and E2 is the transmitted field.

kR + 2π L kR

r2 r2 r2 r2

kR − 2π L ωR −ncR 0

ωR Frequency, ω

= = = =

0.00 0.25 0.75 0.95 ωR + ncR 0

Fig. 7.2. The dispersion relation for light propagation in a SCISSOR with differing values of the self-coupling coefficient r . For generality, the waveguide contribution of constant slope k0 has been subtracted from the effective propagation constant keff . shown in Fig. 7.2. The deviation in the curve from the light line of the ordinary waveguide takes the form of periodic changes in the group velocity and group velocity dispersion with a periodicity of c/n0 R. Here, the material and waveguide dispersion are assumed to be negligible. In fact, it will be shown later that the dispersive nature of the resonators in general dominates the intrinsic dispersion by many orders of magnitude. A pulsed waveform can be decomposed into the product of a slowly varying envelope A (t) and a carrier wave with frequency ω0 as E (t) ≡ 12 A (t) exp (−iω0 t) + c.c.. The relationship of the carrier frequency to some resonance frequency controls the central operating point on the dispersion relation curve and thus sets the normalized detuning φ0 = (ω0 − ωR ) TR , where TR = FSR−1 is the resonator transit time and ωR is the closest resonance frequency. The transfer function of a single resonator can be expanded into two embedded Taylor’s series: one for

7.2 Linear Propagation in Distributed Microresonators

177

the effective phase shift (expanded about the normalized detuning φ0 ), and one for the exponential (expanded about the effective phase shift of the carrier Φ0 ): ⎧ ⎡ ⎤n ⎫ ∞ ∞ n ⎨ ⎬ m 1 Φ d i m ⎣ ⎦ − φ H (ω) = eiΦ = eiΦ0 1 + . (7.2) ) (φ 0 ⎩ ⎭ n! m! dφm φ n=1

m=1

0

Using this formal expansion, the transmitted field is related to the incident field with the assumption that the effective phase shift induced by each resonator is distributed over the spacing L so that the effective propagation constant is independent of propagation distance at the macroscopic scale. The field at some point zj+1 separated a small distance δz from the field at another zj is given by a similar equation that distributes the resonator contribution and includes that of the waveguide:

nω

i

0

+

Φ0



δz

L Ej+1 (ω) = e c × ⎧ ⎡ ⎤n ⎫ ∞ n ∞ ⎬ ⎨ i ⎣ n0 1 δz dm Φ m⎦ ∆ωδz + E (ω) . 1+ (φ − φ0 ) m ⎭ j ⎩ n! c m! L dφ φ0 n=1 m=1

(7.3) Taking the Fourier transform of this equation results in a difference equation relating the pulse envelopes at the two points: Aj+1 (t) = Aj (t) ⎡ ⎤n   ∞ n ∞ ∂ m⎦ i ⎣ n0 1 δz dm Φ ∂ i + i δz + Aj (t) . (7.4) n! c ∂t m=1 m! L dωm φ0 ∂t n=1   Finally, the differential approximation Aj+1 (t) − Aj (t) /δz → dA/dz is made and δz is allowed to go to zero.1 This procedure yields a linear propagation equation for the pulse envelope: ⎡ ⎤   ∞ ∂ m⎦ 1 1 dm Φ dA ⎣ n0 ∂ A. (7.5) i = − +i dz c ∂t m! L dωm φ0 ∂t m=1 The different terms in this equation are isolated and examined in what follows. 7.2.1 Group Velocity Reduction The increased phase sensitivity on resonance is related to an increased group delay per resonator. This extra delay distributed among the resonators is responsible for a slower group velocity. The inverse of the 1

Implicit in this assumption is that each resonator is not strongly driven; i.e., the effective phase shift Φ per resonator is small with respect to unity.

178

7. Distributed Microresonator Systems

group velocity is proportional to the frequency derivative of the propagation constant, keff

n0 1 dΦ n0 dkeff = + = = dω c L dω c



 1 − r2 1 − 2r cos φ0 + r 2   n0 4R → 1+ F . (7.6) L φ0 =0,r ≈1 c

2π R 1+ L



The group velocity 1/keff is observed to be composed of contributions from propagation in the waveguide and discrete delays introduced by the resonators. The component of the group velocity reduction introduced by the resonators is proportional to the finesse and can dominate the waveguide component for moderate values of the finesse. 7.2.2 Group Velocity Dispersion Although propagation in the waveguide is assumed to be dispersionless, strong dispersive effects are induced by the resonator contribution. It has been argued that of all photonic devices, microresonators currently provide the highest dispersion per unit volume [207]. The group velocity dispersion (GVD) is proportional to the second frequency derivative of the effective propagation constant, k eff

dk2eff TR2 1 d2 Φ = = = dω2 L dω2 L



  −2r 1 − r 2 sin φ0 (1 − 2r cos φ0 + r 2 )

2

√ 3 3F 2 TR2 →√ . ∓ 4π 2 L φ0 =±π /F 3,r ≈1

(7.7)

Although the GVD coefficient is zero on resonance, appreciably strong normal (positive) or anomalous (negative) values of the dispersion can be obtained on the red (lower) or blue (higher) side of resonance, √ respectively. The dispersion maxima occur at detunings φ0 = ±π /F 3 where the magnitude of the group velocity dispersion is proportional to the square of the finesse. This induced structural dispersion can be many orders of magnitude greater than the material dispersion of typical optical materials [220]. For example, a 10 ps optical pulse propagating in a sequence of resonators with a finesse of 10π , free spectral range of 10 THz (∼5 µm diameter), and a spacing of 10 µm experiences a group 2 velocity dispersion coefficient k eff of roughly 100 ps per mm. In general, this structural dispersion can be as much as seven orders of magnitude greater than material dispersion in conventional materials such as silica fiber (20 ps2 per km). Figure 7.3 shows the pulse evolution for a 10 ps pulse propagating through 100 resonators, each tuned to the anomalous dispersion maxima.

Power, mW

7.3 Nonlinear Propagation in Distributed Microresonators

179

0.3 0.2 0.1 0

0

0 100

20

40

60

80

z (resonator #)

200

100

Time, ps

Fig. 7.3. A weak pulse tuned to the dispersion maxima disperses while propagating in a SCISSOR. A 10-ps FWHM hyperbolic secant pulse tuned for maximum anomalous GVD (B = 0.13) enters the system consisting of 100 resonators each with a 5-µm diameter and finesse of 10π , spaced by 10 µm. Note that the peak power is reduced by a factor of about 4 after propagating only 1 mm as a consequence of the strong induced dispersion. (After Ref. [129], ©2002, Optical Society of America.)

7.2.3 Higher Order Dispersion Higher orders of dispersion may be derived from Eq. 7.5, with each subsequently possessing a maximum that is proportional to the cavity lifetime F TR to the nth power. Specifically, the third-order dispersion coefficient is k eff

TR3 1 d3 Φ ≡ = L dω3 L



 + , −2r 1 − r 2 1 + r 2 cos φ0 − 3r + r cos 2φ0 (1 − 2r cos φ0 + r 2 ) →

φ0 =0,r ≈1

−

3

4 F 3 TR3 . π3 L

(7.8)

It is important to note that all orders of dispersion in the Taylor expansion become significant when the pulse bandwidth is nearly as wide as the resonance bandwidth or equivalently when the pulse is nearly as short as the cavity lifetime.

7.3 Nonlinear Propagation in Distributed Microresonators In addition to inducing a strong group delay and dispersion, a resonator may enhance a weak nonlinearity. If the resonator possesses a nonlinear refractive index, i.e., Kerr nonlinearity, then the internal phase shift will be intensity dependent. For simplicity the nonlinearity of the waveguide

180

7. Distributed Microresonator Systems

is neglected in what follows since it is expected to be small. The intensitydependent contribution of the resonators to the internal phase shift is given by γ2π R|E3 |2 , where γ represents the strength of the intrinsic nonlinear propagation constant. This parameter is of course traditionally fixed for a given material system. The more useful parameter γ/Aeff can be as low as 0.002 m−1 W−1 for standard single-mode silica fiber or as high as 102 m−1 W−1 in an air-clad GaAs or chalcogenide glass based waveguide.2 Near resonance, the effective phase shift is sensitively dependent upon the internal phase shift that is in turn dependent on an enhanced circulating intensity. The combined action of these effects gives rise to a dually-enhanced effective nonlinear propagation constant γeff , calculated from the derivative of the effective phase shift with respect to the input intensity:

γeff

1 dΦ dφ d|E3 |2 γ2π R 1 dΦ ≡ = = 2 2 2 L d|E1 | L dφ d|E3 | d|E1 | L



1 − r2 1 − 2r cos φ0 + r 2 →

φ0 =0,r ≈1

γ

8R 2 F . πL

2

(7.9)

As can be observed from this equation, the increased phase sensitivity (or group velocity reduction) and the buildup of intensity contribute equally to enhance quadratically the nonlinear propagation constant with respect to the finesse [151]. To properly account for the all the third-order Kerr nonlinear contributions of the spectral components of three fields, a double convolution of the three interacting fields is performed in the spectral domain. In the time-domain, the double convolution operation is reduced to simple multiplication. This allows for the straightforward addition of a nonlinear contribution [97] to the internal phase shift term in the linear propagation equation (Eq. 7.5): ⎧ ⎫ ! " ∞ dA ⎨ n0 ∂ 1 1 dm Φ ∂ m⎬ 2 γ2π RB|A| + iTR = − +i A. ⎩ c ∂t ⎭ dz m! L dφm φ ∂t m=1

0

(7.10) For two nonlinearly interacting resonant pulses, the results here derived for the self-phase modulation (SPM) effect similarly apply to the effect of cross-phase modulation (XPM) although with an extra degeneracy factor of two [178].

2

For the purpose of quoting the material nonlinearities, we default to the more useful parameter γ whereby the nonlinear propagation constant is scaled by the effective mode area such that γP L is the nonlinear phase shift acquired for a power level of P over a distance L.

7.3 Nonlinear Propagation in Distributed Microresonators

181

7.3.1 SCISSOR Solitons Next, the nonlinear propagation equation that retains only the lowest order dispersive and nonlinear terms in Eq. 7.10 is examined. The time

 coordinate is shifted to the reference frame of the pulse t  = t − keff z . It is found that, in this limit, the pulse evolution is governed by a nonlinear Schrödinger equation (NLSE) with effective GVD and SPM parameters: ∂2 1 ∂ A = −i k A + iγeff |A|2 A. eff ∂z 2 ∂t 2

(7.11)

Soliton solutions for this equation exist provided that the enhanced nonlinearity and induced dispersion are of opposite sign [221–223]. Although the sign of the enhanced nonlinearity is predetermined by the sign of the intrinsic nonlinearity in the overcoupled regime, the sign of the induced dispersion is as shown previously in Eq. 7.7, assumes the opposite sign of the normalized detuning from resonance. Fig. 7.4 shows the frequency dependence of the lowest order GVD k eff and enhanced nonlinearity γeff and also indicates the optimum detuning for soliton propagation. The fundamental soliton solution for this equation is [224]   1 2 A z, t  = A0 sech t  /TP ei 2 γeff |A0 | z ,

(7.12)

2 where   the amplitude and pulse width are related according to |A0 | =    keff  /γeff TP2 , below which the pulse is severely distorted by all orders of dispersion. The finite response time of the resonator places a lower to be the rabound on the pulse width TP . A scaling factor B is defined  √ 2 tio of the pulse bandwidth 2 arcsech 1/ 2 /π TP to the resonator

1.0 0.5

Soliton condition (GVD)

(SPM)

'' keff (2FT/π)2/L

γeff γ (2F/π)2 (2πR/L)

0.0 -0.5 -4π F

-2π 2π 0 π F F 3F Normalized detuning, φ0

4π F

Fig. 7.4. A SCISSOR soliton is created from a balance between resonator enhanced nonlinearity and resonator-induced anomalous dispersion.

182

7. Distributed Microresonator Systems



√   bandwidth (1/F TR ), such that B = 2 arcsech 1/ 2 /π 2 F TR /TP . A  nonlinear strength parameter Γ = 4/π 2 F 2 γ |A0 |2 R is also defined. With these definitions, a simple relation holds between Γ and B for the fundamental soliton operating at the anomalous dispersion peak3 : Γ =

π

√  B2 ≈ B2. 2 3 arcsech2 1/ 2 √

(7.13)

Power, W

Higher order dispersive and nonlinear terms become increasingly significant when either B and/or Γ approach unity. Time-domain simulations confirm the validity of this approximation.4 Figure 7.3 shows the pulse evolution of a low-power 10 ps FWHM hyperbolic secant pulse tuned for maximum anomalous GVD (B = 0.13) in an AlGaAs or chalcogenide glass-based system. The system consists of 100 resonators spaced by 10 µm each with a 5-µm diameter and finesse of 10π . As can be observed, the temporal pulse profile is greatly dispersed. Figure 7.5 shows the pulse evolution for the same system, but with a peak power of 125 mW, corresponding to the fundamental

0.3 0.2 0.1 0

0

0 20

100 40

60

80

z (resonator #)

100

200

Time, ps

Fig. 7.5. A pulse with amplitude corresponding to the fundamental soliton propagates in a SCISSOR without dispersing. The same parameters were used as in Fig. 7.3, but with a peak power of 125 mW (Γ = 0.0196) in a chalcogenide glass-based system. (After Ref. [129], ©2002, Optical Society of America.) 3

4

The values of keff and γeff are, respectively, lowered by factors of 3/4 and 9/16 from their given maximum values when operating at dispersion extremum points. The simulations used to study pulse evolution in a sequence of waveguidecoupled resonators are carried out using an iterative method in which each iteration consisted of linear and nonlinear phase accumulation during one round-trip within the resonator followed by interference at the coupler. Traditional beam or pulse propagation split-step Fourier methods are unnecessary as both nonlinear phase accumulation and structural dispersion resulting from a discrete impulse response are more readily treated in the time-domain.

7.3 Nonlinear Propagation in Distributed Microresonators

183

SCISSOR soliton (Γ = 0.0196). As can be observed, the pulse shape is well preserved upon propagation. Many of the familiar characteristics of fundamental solitons such as robustness, reshaping, pulse compression, and pulse expansion have been observed in simulations to carry over from the continuous-medium case. The strengths of dispersion and optical nonlinearity are two key parameters that control the length scale over which solitons evolve with a F 2 reduction. Higher order solitons, satisfying Γ ≈ NS2 B 2 , where NS is an integer are readily observed in simulations, but they are unstable because of higher order dispersive nonlinear effects present in this system.5 Dark solitons consist of an intensity dip in an otherwise uniform continuous-wave field. They can also be supported if the enhanced nonlinearity and induced dispersion are of the same sign (on the other side of resonance). The fundamental dark soliton possesses a hyperbolic tangent field distribution [97],   2 (7.14) A z, t  = A0 tanh t  /TP eiγeff |A0 | z .

Power, mW

Fig. 7.6 shows the propagation of the fundamental dark SCISSOR soliton tuned to the normal dispersion peak.

30 20 10 0 0

20 40 60 80 z (resonator #) 100 400

300

200

100

0

Time, ps

Fig. 7.6. A negative pulse in a uniform intensity background with parameters corresponding to the fundamental dark soliton propagates in a SCISSOR without dispersing. The incident field distribution was a hyperbolic tangent with twice the pulse width of the bright soliton and a background power that was one fourth that of its peak power in Fig. 7.5. (After Ref. [129], ©2002, Optical Society of America.)

5

Additionally, higher order dispersive and/or nonlinear effects render the scattering of solitons to be inelastic. Under these conditions, the term solitary wave is more appropriate.

184

7. Distributed Microresonator Systems

7.3.2 Induced Self-Steepening In the previous section, frequency dependence of γeff was neglected. One of the effects resulting from the frequency-dependent nature of γeff is an intensity-dependent group velocity. This effect leads to the phenomenon of self-steepening (SS) of a pulse where the peak of a pulse travels slower than (+SS) or faster than (−SS) its wings. The self-steepening coefficient s may be derived6 from Eq. 7.10, but it is more readily obtained from the frequency derivative of the nonlinear coefficient: √  γeff 3F TR 2 dB s= →√ . (7.15) = ∓ γeff B dω φ0 =±π /F 3,r ≈1 π

Power, W

Although this effect is not easily isolated from the induced GVD in a sequence of resonators to form pulses with steep edges, it plays an important role in the breakup of higher order solitons. The known phenomenon of soliton decay [97, 225] involves the breakup of an NS order breathing soliton into NS fundamental solitons of differing pulse amplitudes and widths. Fig. 7.7 shows a situation in which a second-order SCISSOR soliton

1.2 0.8 0.4 0

0

0 20

100 40

60

80

z (resonator #)

100

200

Time, ps

Fig. 7.7. A higher order breathing soliton is unstable under the influence of the resonator-induced intensity-dependent group velocity (self-steepening). Here a second-order soliton splits into two stable fundamental solitons upon propagation in a SCISSOR. The incident field distribution was the same as in Fig. 7.5 but with four times the peak power. (After Ref. [129], ©2002, Optical Society of America.) 6

So as to correctly expand Eq. 7.10, the B term also must be expanded generating more time derivative terms within the square brackets. Thus, the selfsteepening contribution will not only consist of two m = 2 terms but also of one m = 1 term. For terms such that m > 1, the time derivatives implicitly appear to the far left of each term when the square brackets are expanded.

7.3 Nonlinear Propagation in Distributed Microresonators

185

with a launched peak power of 500 mW undergoes decay and splits into two stable fundamental SCISSOR solitons. The solitons are well isolated in time and uncorrupted by a background or pedestal. One of them possesses a higher peak power and narrower width than the original demonstrating the potential for pedestal-free optical pulse compression. The effects of induced self-steepening in a sequence of resonators can take place for picosecond and even nanosecond pulses because unlike in the case of intrinsic self-steepening, the relative strength of SS to SPM is not governed by how close the pulse width is to becoming a single optical cycle, 2π /ω0 but rather how close the pulse width is to becoming a single cavity lifetime F TR . For the 10 ps pulse propagating in a SCISSOR with the above parameters, the nondimensional self-steepening parameter (s/TP ) is 0.173. To observe the same effect with traditional intrinsic self-steepening, a six-cycle or 30-femtosecond pulse would be required. 7.3.3 Pulse Compression More complicated interaction can exist between structural dispersion and enhanced self-phase modulation in a SCISSOR structure. The soliton order NS , proportional to the injected amplitude, is a convenient measure of the relative strengths of the two processes. If the relative strengths of the processes are balanced (NS = 1), it was shown previously that solitonlike pulses may be propagated. If, however, the nonlinearity dominates the dispersion (NS > 1), pulse compression can result. Because the enhanced nonlinearity and induced dispersion are each proportional to the finesse squared [129], the soliton order for a SCISSOR can be set by simply choosing the ratio of the pulse width to the microring transit time (TR ),  n2 I0 R TP 8π NS = 3/4 . (7.16) 3 λ TR Figure 7.8 demonstrates the result of a simulation in which a 5-ps Fourier transform-limited pulse is injected into a ten-resonator SCISSOR tuned above resonance at the peak of anomalous dispersion. The injected NS = 5 pulse fractures into multiple solitons of differing amplitudes along with some dispersing waves. The soliton with the largest amplitude emerges compressed by a factor of approximately five. It is worth noting that although this method can be used to compress a pulse on a very short distance scale, the amount of pulse compression is ultimately limited by the bandwidth of SCISSOR. The process of temporal imaging is related closely to pulse compression. With temporal imaging [226], intra-pulse structure can be preserved and magnified (or demagnified). An imaging condition must be satisfied in a setup that consists of an initial dispersive segment, a directly imposed quadratic phase, and a final dispersive segment of opposite sign

186

7. Distributed Microresonator Systems

90 80

Output

Input

70 Power, W

60 50

Output (1ps)

40 30 20 10 0

0

Input (5ps) 10

20 30 Time, ps

40

50

Fig. 7.8. A highly compact microresonator-based 5× pulse compressor. A 25 W transform-limited input pulse (NS = 5) of 5-ps width is compressed to 1 ps. In the process, some energy is shed in the form of other nondispersing pulses that walk away from each other linearly in time owing to an intensity-dependent group velocity. These extra pulses might be eliminated through the use of a saturable absorbing material that may even be microresonator-based. Here, 10 AlGaAs or chalcogenide-based microresonators of 10-µm-diameter form a SCISSOR. The resonators possess a finesse of 5π , coherent intensity buildup of 10, and nonlinear enhancement of 100. to that of the initial. Owing to the large dispersive and nonlinear properties of microresonators, temporal imagers might be fabricated on an integrated photonic chip. Each of the three sections might be composed of differently tuned SCISSOR elements. Sign selection of the dispersive segments is easily accomplished in SCISSORs tuned below (normally dispersive) or above (anomalously dispersive) resonances. The quadratic phase chirp providing the temporal lensing may be accomplished by enhanced cross-phase modulation in a resonant SCISSOR. 7.3.4 Nonlinear Detuning and Multistability Equation 7.9 was implicitly derived with the assumption of low intensity. Next, the intensity dependence of γeff is examined. It is found that the

7.3 Nonlinear Propagation in Distributed Microresonators

187

circulating intensity and γeff are in fact interdependent. The circulating intensity depends on the buildup factor, which in turn depends on the circulating intensity from the nonlinear detuning contribution. As a result nonlinear resonators can possess memory and multistable branches in the input–output relationships within certain operating regimes [80,198]. When operated near resonance, the onset of multistability occurs when the circulating power is high enough to generate a single-pass nonlinear phase shift of 2π radians. Saturation resulting from intensity-dependent detuning pulling the resonator off resonance generally takes place well before this effect. Working on the lower branch of the multistable relation for positive detunings, the saturation of the effective nonlinear propagation constant is well fitted by a (1 + I/Is )−1 type of saturation model given explicitly as γ 2πL R B2φ0 , (7.17) γeff → γ2π RB2φ r ≈1 1 + 2π −Φ0 0 |A|2 where the saturating intensity near resonance is |AS |2 = π /γeff L. The saturating intensity is lower for higher detunings from resonance. A generalized, nonlinear Schrödinger equation incorporating every effect discussed so far takes the following form: ∂ ∂ A + keff A = ∂z ∂t   ∂2 1  ∂ 3 ∂ γeff |A|2 1 k A + A + i 1 + is A. − i k 2 eff ∂t 2 6 eff ∂t 3 ∂t 1 + |A|2

(7.18)

2

|AS |

7.3.5 Nonlinear Frequency Mixing The characteristics of frequency mixing processes such as harmonic generation and four-wave mixing can also be enhanced via waveguidecoupled resonators. As a general rule, the scaling of the enhancement of these processes can be inferred by including contributions from each intensity involved (lying within a resonance) and the interaction length. Each contributes a factor proportional to the finesse. Four-wave mixing is a third-order nonlinear process that annihilates two photons at one frequency and generates two photons at a higher and lower frequency. Four-wave mixing can give rise to modulation instability whereby a low contrast amplitude ripple grows via the amplification of sidebands at the expense of the central frequency. In a dispersive medium, four-wave mixing is stifled due to phase mismatch. However, in an anomalous dispersive medium, positive self-phase modulation generates new frequency components that compensate for the mismatch [178]. If the process is allowed to continue, the modulation depth increases until

188

7. Distributed Microresonator Systems

Power, W

2.0 1.5 1.0 0.5 0 0

20

40

60

z (resonator #)

30 20 10 0 70 60 50 40

Time, ps

Fig. 7.9. Demonstration of modulation instability in a SCISSOR. The input field consists of 800 mW of CW power with a 1% power ripple. The SCISSOR parameters have been chosen such that the peak of the instability gain is at the input modulation frequency of 100 GHz. Note that the modulation frequency given by Eq. 7.19 need not be a resonance frequency of the structure. (After Ref. [129], ©2002, Optical Society of America.)

a train of solitons stabilizes. Figure 7.9 shows the increase in modulation depth for a seeded 1% amplitude ripple of 100 GHz with a propagation distance for a sequence of 60 resonators [129]. The peak of the effective instability gain gm = 2γeff |A0 |2 occurs at some modulation frequency      Ωm = ± 2γeff |A0 |2 / k (7.19) eff  provided that this value does not exceed the resonance bandwidth. The gain is enhanced by the square of the finesse. Up until this point, attention has been restricted to pulses whose bandwidth is of the order or less than that of a single resonance peak. By copropagating pulses with carrier frequencies lying within differing resonance peaks, four-wave mixing processes can be enhanced with frequency separations of pump and signal equal to an integer number of free spectral ranges (FSRs). Because the efficiency of idler generation depends on the pump intensity, signal intensity, and grows quadratically with length, the efficiency scales as the fourth power of the finesse [56]. We expect such effects to be important in systems that have low intrinsic dispersion such that the FSR is independent of frequency so that the three enhancement linewidths coincide with signal, pump, and idler frequencies. Finally, the efficiency of harmonic generation processes may be increased. The efficiency of second harmonic generation (SHG), for example, would be enhanced cubically with the finesse.

7.4 Limited Depth of Phase

189

7.4 Limited Depth of Phase Within a free spectral range, a single resonator can impart only a maximum phase depth of 2π radians. This limitation has important implications for the maximum delay, chirp, and a nonlinear phase that can be imposed on a pulse per resonator. As the imparted phase nears only π /2 radians, higher order effects become increasingly significant such that the system can no longer be treated perturbatively. The extent of group velocity reduction that can be achieved in a SCISSOR is limited by how high the finesse can be made. A SCISSOR with an ultra-high value of finesse can be used to slow a pulse appreciably but that pulse must be long enough such that it is at least of the order of the finesse times the transit time of a single resonator. Thus, the maximum delay per resonator is fixed and equal to one pulse width at best. The same is true for the induced group velocity dispersion. A high GVD coefficient (proportional to F 2 ) can be obtained by making the finesse very large. However, the increasing finesse places an increasing restriction on the pulse bandwidth ∆ω (proportional to 1/F ). As a result, the imposed spectral chirp per 2 resonator 1/2k eff ∆ω L is independent of finesse and only dependent on the scaling factor B. If the requirement is to broaden a pulse by N pulse widths, then the minimum number of resonators needed (occurring at B ∼ 1) is roughly N. This is an important point: An ultra-high finesse is not required for designing dispersive devices based on resonators. However, while reducing the resonator size and increasing the finesse in inverse proportion maintains the same resonator bandwidth and thus the same linear properties, the nonlinear properties are enhanced. This is of fundamental importance since a low threshold power and small number of resonators is desirable practically. As a result of saturation, it is very difficult to achieve an effective nonlinear phase shift of π radians from a single resonator when operating on resonance. It is achieved only in the limit as the resonator’s internal phase shift is power-detuned completely by π radians resulting in an external phase shift of π radians as well. As a result of this saturation, one completely loses the advantage of resonant enhancement. Achieving a phase shift of π /2 is, however, h much easier to attain before the saturation takes place and requires a power-detuning of only ∆φ = π /F . A effective nonlinear phase shift of π may be obtained readily from a single resonator taking advantage of enhancement by ensuring that the resonator is red-detuned initially by π /F and allowing the resonator to be power-detuned though resonance for a total value of π radians [227].

190

7. Distributed Microresonator Systems

7.5 Attenuation in Distributed Microresonators Attenuation in microresonators is in general detrimental. Internal attenuation reduces the net transmission, buildup, and (in general) nonlinear response. It also broadens the resonance limiting the achievable finesse. Attenuation in microresonators typically arises from three mechanisms: intrinsic absorption, radiation loss [133, 135], and scattering. Intrinsic absorption can typically be rendered insignificant over millimeter-sized propagation distances by choosing an appropriate material system at a given wavelength. Additionally, since the circulating intensity can greatly exceed the incident intensity, intensity-dependent absorption processes such as two-photon absorption may be significant in resonators. Twophoton absorption may be minimized (if desired) by proper selection of a material with a bandgap that is greater than twice the incident photon energy [189, 228]. Whispering-gallery modes of microdisks and microrings suffer from bending or radiation loss that is increasingly important for small resonators with low refractive index contrast. Scattering can take place in the bulk or on the edges. Edge scattering is typically the dominant loss mechanism that results from roughness on the microresonator edges that in practice cannot be made perfectly smooth [229]. The surface perturbations phase-match the guided mode to radiating modes outside the disk structure [120] or into the contra-directional mode [121]. In the planar waveguide approximation, the attenuation constant associated with scattering loss is derived as [98]  k

 x 2 2 αscatt = σ 2 n21,eff − n22,eff k20 + E(x=+d/2) E(x=−d/2) . kz

(7.20)

Where σ is the RMS edge roughness. Applying this result to a microdisk or microring results in the expression [122]

 k

 x,eff 2 2 αscatt = σ 2 n21,eff − n22,eff k20 + E E(r =R1 ) (r =R2 ) . kz,eff

(7.21)

Because the strength of the scattering loss coefficient increases with the square of the refractive index difference, it can be expected that high-contrast microring and microdisk microresonators will possess high scattering losses. Figure 7.10(a) shows a finite-difference time-domain (FDTD) [112] simulation of a 5-resonator SCISSOR with 50-nm sidewall roughness displaying strong scattering losses and weak circulating intensity. In Fig. 7.10(b), a lower sidewall roughness of 30 nm results in negligible scattering loss and strong intensity buildup. Gain may be implemented where possible to combat attenuation. More interestingly, a dispersion decreasing system may be used to propagate a SCISSOR soliton in an attenuating structure. In this case, the pulse width is kept constant as the amplitude decreases via an exponential decrease

7.6 Slow and Fast Light in SCISSORs

191

Fig. 7.10. FDTD method of solving Maxwell’s equations for a SCISSOR structure composed of five microresonators. A TE field of wavelength 1.55 µm is launched into the 0.4-µm-wide waveguide evanescently sidecoupled to a disk with diameter of 5.1 µm. The refractive index of the air-clad disk and guide is 2. Exclusive coupling to the m = 16 azimuthal whispering-gallery mode (WGM) is achieved by careful selection of parameters. In (a), strong scattering losses result due to roughness associated with a 50-nm grid. In (b), scattering losses are made negligible by using a 30-nm grid. Consequently, a buildup factor of 16 and finesse of 25 are achieved in this structure. (After Ref. [129], ©2002, Optical Society of America.)

in dispersion down the length of the structure. In general, however, lowloss propagation through a SCISSOR constructed from N resonators can be ensured if the single-pass attenuation satisfies α2π R  1/NF .

7.6 Slow and Fast Light in SCISSORs In recent years there has been a flurry of activity aimed at the development of techniques that can lead to a significant modification of the group velocity of propagation of a light pulse through a material medium [230]. Proposed applications of these procedures include the development of optical delay lines [220] and the “storage” of light pulses [231, 232] with possible applications in optical communications and quantum information. Most of this research has made use of the response of resonant media [233], and much of it has made use of the concept of electromagnetically induced transparency [231, 234] and other quantum coherence effects [235, 236].

192

7. Distributed Microresonator Systems

7.6.1 Slow Light Many properties of the SCISSOR system are in fact analogous to those found in atomic systems. In both cases, light is coupled into and out of discrete resonators without loss or dispersion, but with delay. It is known from studies of slow light propagation in atomic systems displaying electromagnetically induced transparency (EIT) that the width of the induced transparency window in atomic systems with N interacting atoms is re√ duced by a factor that scales as 1/ N [237]. This fundamental limitation on bandwidth results from the fact that near the frequency of maximum transmission, the transmission decreases quadratically with detuning. Such a limitation is absent in a fully transmissive SCISSOR geometry used to propagate solitons, but a fundamental limitation is imposed ultimately by fourth-order dispersion. This point is next examined in detail. The effective group index associated with a SCISSOR takes on its maximum value when the optical wave is tuned to a cavity resonance (φ = 0), and it can be expressed in any of the forms       4R 2π R 2π R 1 + r =n 1+ B0 = n 1 + F (7.22) ng = n 1 + L 1−r L L Although the steep slope of the dispersion-relation curve near resonance is responsible for the reduced group velocity, the transition from the flat sections of the dispersion curve to the steep section is necessarily curved and introduces GVD. On resonance, the lowest-order GVD parameter k eff (and all other even orders) was shown previously to be zero. However, the distance over which pulses can propagate is limited ultimately by broadening induced by third-order dispersion. It is better to propagate off resonance and sacrifice some enhancement in order to gain much in terms of the maximum possible propagation √ distance. Specifically, by tuning slightly above resonance (φ0 = π /F 3), third-order dispersion can be eliminated. The lowest order dispersion that is introduced by operating off resonance can be compensated for by the enhanced nonlinear response of the structure. The negative, lowest order GVD occurring at this operating point can be balanced precisely by the nonlinearity to form a SCISSOR soliton. In the following paragraphs, this conjecture will be proven. An all-pass resonator is inherently a phase-only filter and possesses a field amplitude transmission function that is a pure phasor that can be expanded at resonance about φ0 = 0: e

iΦ

=e

! dΦ i Φ0 + dφ φ+ 12

d2 Φ dφ2

φ2 + 16

d3 Φ dφ3

" φ3 +...

.

(7.23)

On resonance, the delay associated with the resonator is given by the 2 value of the first normalized frequency derivative term (TD = π F TR ). Because no amplitude filtering function is associated with the operation

7.6 Slow and Fast Light in SCISSORs

193

of this device, the normalized bandwidth is determined by the phase error induced by higher order dispersive terms. On resonance, the even-order terms such as the group-delay dispersion vanish and the next nonzero higher order limiting term is third-order dispersion. The dominant phase error on resonance is thus   3   1 d3 Φ  11 2  3 3 |Φerror | =  F φ . φ = (7.24) 3  6 dφ  62 π Assuming that the phase error of 1 radian is the maximum tolerable error,

1/3 1 3 1 the usable bandwidth is restricted to ∆ν = 2 F TR ≈ F TR . A SCISSOR, composed of a sequence of N all-pass resonators, possesses an effective phase shift equal to the single device phase shift multiplied by N. It follows that the net accumulated delay for such a system is simply equal to N times the single-device delay. Fortunately, the net accumulated phase errors do not scale in the same fashion. That is, the phase-error limitation on usable bandwidth (governed on resonance by third-order √ dispersion) 3 is not simply inversely proportional to N, but rather as ∆ν/ N. As discussed previously, soliton propagation at the maximally dispersive frequency allows for the cancellation of group velocity dispersion and the elimination of third-order dispersion. This leaves fourth-order dispersion as the limiting phase error. The dominant phase error in this regime is thus      √    4 4    1  d Φ 3 27 2 1   4 4    φ = F φ  |Φerror | =  (7.25) 4   24 dφ 24 128 π   φD 

4 1/8

2 1 1 ≈ F TR for a with a similar bandwidth restriction of ∆ν = 35 F TR single resonator. It is easily shown that a less restricting scaling law of √ 4 ∆ν/ N applies to propagation in the SCISSOR soliton regime. Next, the analysis derived in the previous paragraph is tested via rigorous simulations. Fig. 7.11 compares three approaches to attempting to propagate slow light in a SCISSOR with a group velocity of approximately c/(100n). In Fig. 7.11(a), a weak 100-ps pulse tuned to resonance is delayed greatly, but broadens and acquires ripples associated with negative third-order dispersion. In Fig. 7.11(b), the pulse frequency is tuned above resonance to the extremum of the lowest order GVD. At this frequency, the third-order GVD necessarily vanishes, and pulse distortion of the sort shown in part (a) is decreased noticeably. However, the pulse broadens considerably as a result of non-vanishing second-order (lowest order) dispersion. In Fig. 7.11(c), the same pulse but with a peak power corresponding to that of the fundamental SCISSOR soliton is observed to propagate with a preserved pulse shape. The group velocity reduction in this case is 75× as opposed to 100× in part (a), but the high fidelity of pulse propagation makes this strategy seem to be superior.

194

7. Distributed Microresonator Systems

Fig. 7.11. Numerical results showing the advantage of using optical nonlinearity on the propagation of slow light through a SCISSOR structure. The SCISSOR consists of 100, 10-µm diameter resonators spaced by 10π µm with r = 0.98 corresponding to a group velocity reduction of 100×. (a) A weak, 100-ps resonant pulse propagates at a group velocity of (c/n)/100 through the SCISSOR and is corrupted by resonator induced third-order dispersion. (b) The same pulse, but with carrier frequency tuned to the anomalous GVD maximum propagates with a group velocity of (c/n)/75 but is greatly broadened. (c) A 6.4-mW peak power, 100-ps pulse tuned to the anomalous GVD maximum propagates as the fundamental SCISSOR soliton with a group velocity of (c/n)/75 and is well preserved. Here, parameters typical of a GaAs or chalcogenide-glass based waveguiding structure, (γ/Aeff = 60m− 1W− 1) were employed. A sequence of resonators might someday be useful in studying the properties of slow light in the regime where acoustic and optical group velocities are of the same order of magnitude [238, 239]. However, in order to slow the group propagation to this level in silica, an ultra-high finesse of about 105 is required.

7.6 Slow and Fast Light in SCISSORs

195

7.6.2 Tunable Optical Delay Lines Because the group delay can in practice be controlled by detuning resonators thermally [152], electro-optically, by carrier injection [65], electroabsorption [240], or other means, tunable optical delay lines may be constructed from microresonators. The net group delay TD in a SCISSOR increases linearly with the number of resonators, TD = N

dΦ . dω

(7.26)

The net group delay is inversely proportional to the average group velocity that can be minimized by coupling the resonators in a manner that yields the maximum possible finesse. Under this condition, however, the bandwidth may be restricted severely. For example, reducing the group velocity in a SCISSOR by six orders of magnitude results in a bandwidth of less than 10 MHz for 10-µm-diameter resonators. As a result, designing a SCISSOR for the minimum attainable group velocity (in practice limited by attenuation) is not desirable. The fractional group delay, defined as the group delay normalized to one pulse width TP , is a convenient figure of merit characterizing a resonator-based delay line. A fractional group delay near unity corresponds to a delay by one pulse width. This delay is the greatest that can be achieved in a single resonator without the introduction of higher order dispersive effects. These higher order dispersive effects result from frequency content in the spectral wings experiencing a lower delay. Fractional group delays greater than unity can be obtained by use of more resonators. However, dispersive effects accumulate and eventually severely distort pulses. For a desired net fractional delay, these phase errors might be reduced by relaxing the fractional delay per resonator and by increasing the number of resonators. Alternatively, the phase errors might be corrected by some means of dispersion compensation, thereby increasing the system complexity. In principle, there is an advantage to using nonlinear soliton propagation in a SCISSOR. Figure 7.12 displays the results of a simulation of a tunable delay line based on a SCISSOR. Eight bits of a 160-Gb/s pulse train are delayed by one and four bit-slots. For simplicity, the delay associated with propagation in the side-coupling waveguide is removed. In (a), six resonators are sufficient to delay the pulse-train by a single bit-slot while suffering minor third-order phase distortion. In (b), a four bit-slot delay is achieved via the use of 26 resonators, but the pulses emerge greatly dispersed. By tuning the resonator and pulse energy to parameters near the SCISSOR soliton for the structure, pulse fidelity can be improved as demonstrated in (c). Note that a delay of four bit-slots corresponds to a fractional delay of 12 for the duty cycle of 1/3 employed. For comparison, the same pulse train is sent through the SCISSORs with the resonators tuned far away from resonance. The delay accumulated for both weak and strong pulses in

196

7. Distributed Microresonator Systems

Fig. 7.12. A microresonator-based tunable delay line. Shown are eight bits of a 160-Gb/s (duty cycle = 1/3) pulse train. In (a), six resonators achieve a delay of a single bit-slot with some minor distortion resulting from third-order dispersion. In (b) 26 resonators achieve a delay of four bit-slots, although with increased distortion. In (c), propagation near parameters associated with a SCISSOR soliton results in noticeably less distortion. Also shown in (b) and (c) are the same pulse trains when the resonator is detuned. In all cases, a finesse of 10π and 4 µm diameters are used. The bit sequence is 10100111.

the off-resonance state, where pulses effectively bypass the resonators, is negligible. This demonstrates the possibility for tunable delay. Most importantly, this simulation demonstrates that a unit fractional delay does not require an ultra-high finesse, but rather it is achieved when the pulse bandwidth is of the order of the resonator bandwidth. In order that sequences of resonators forming the SCISSOR overlap in their delay bandwidths, their effective optical circumferences must be made reproducibly

7.7 Generalized Periodic Resonator Systems

197

to within λ/F . If the reproducibility does not conform to this standard, a Doppler-like broadening of the Lorentzian resonance linewidths will result in lowered group delay and broader bandwidth. 7.6.3 Fast Light Lossy resonators forming a SCISSOR structure can be implemented to propagate light superluminally [241]. If the coupling strength is chosen to be weaker than the round-trip loss, the resonator is said to be undercoupled. The dispersion relation for the SCISSOR in this regime is qualitatively different from that found in the overcoupled regime. Figure 7.13(a) displays the transmission for a single resonator in the three regimes. In addition, Fig. 7.13(b) displays the resonator contribution to the dispersion relation for all three regimes and displays the reversal of sign near resonance. This negative slope implies that a pulse exiting from a resonator will emerge with its center advanced in time with respect to the incident pulse. Causality is maintained because the discrete impulse response of the resonator does not display any advanced impulses. A pulse displays superluminal propagation but quickly attenuates in a multi-resonator undercoupled SCISSOR. Gain (assumed flat across the pulse bandwidth) may be incorporated into the straight waveguide to offset the losses associated with the undercoupled configuration. The group index near resonance in the undercoupled regime is given by [242]  a(1 − r 2 ) 2π R ng = n 1 − , (7.27) L (r − a)(1 − r a) where a = e−απ R . Figure 7.13(c) demonstrates superluminal propagation of a linear, resonant pulse through undercoupled resonators situated near an amplifying waveguide. The exiting pulse experiences a negative time delay or, equivalently, a time advance. Moreover, the theory also predicts superluminal propagation for a SCISSOR in which each resonator is constructed of an amplifying medium in which the round-trip gain is greater than the coupling strength7 ; however, simulations suggest that propagation is highly unstable in this regime.

7.7 Generalized Periodic Resonator Systems We now compare and contrast the geometry and propagation characteristics of periodic SCISSORs with that of other well-known systems in optical physics that possess photonic bandgaps. In certain respects, the SCISSOR soliton is analogous to gap [243], Bragg [244], and discrete solitons [245], 7

This occurs in the equivalent undercoupled regime accompanying gain.

198

7. Distributed Microresonator Systems

Fig. 7.13. (a) The transmission for a single resonator with r = 0.9 in the overcoupled (a = 0.96), critically coupled (a = 0.9), and undercoupled (a = 0.84) regimes. (b) The dispersion relation for the same three cases. Note the change in the sign of the slope of the curve near resonance in the undercoupled case. In the critically coupled case, the curve undergoes a π /L phase jump (where the transmission is zero) whereas in the overcoupled case, the second half of curve has been cut and displaced down by 2π /L for generality. The frequency units of δω correspond to the nonattenuating resonance bandwidth. (c) Numerical simulation demonstrating superluminal propagation of 36-ps pulses through a SCISSOR structure composed of twenty lossy undercoupled resonators with 10-µm diameter, spaced by 10π µm. Gain has been added to the straight waveguide section to maintain pulse power.

which result from nonlinear pulse propagation within or near the photonic bandgap [217–219] of a distributed feedback structure. Many photonic crystal systems possess similar phase [246] and intensity enhancing properties [247–250] but in general possess bandgaps. Because each constituent resonator of a SCISSOR is an all-pass filter, feedback is present

7.7 Generalized Periodic Resonator Systems

199

within each resonator but not among resonators. Alternatively, there is no intended mechanism for light to couple into the counterpropagating modes of the microresonator or guide.8 As a result, there is no frequency at which light is restricted from propagating and thus the structure cannot possess a photonic bandgap (PBG). Nevertheless, the SCISSOR structure displays enhanced nonlinear optical response for much the same reason that a PBG structure can produce enhanced nonlinearity. We will thus take a step back and examine the relationships between periodically distributed microresonator systems and photonic crystals. We will accomplish this by comparing and contrasting the light propagation characteristics (dispersion relations) associated with different connection geometries for sequences of microring resonators. We restrict our attention to four archetypal building blocks from which sequences can be constructed [252]. For each of these four microring-based units, a direct analog in the form of a traditional Fabry–Perot cavity is depicted in Fig. 7.14. With the insight provided by these Fabry–Perot analogues, it is clear that these geometries identify the four ways of connecting four-port resonators9 to form sequences. A sequence based on Fig. 7.14(a), (cavity version) represents the traditional multilayered structure or Bragg grating. The microring-based version of such a multilayer is a sequence of directly coupled microrings [253] often termed a coupled resonator optical waveguide (CROW) [254, 255]. Another way to couple the rings is indirectly through a common waveguide as depicted in Fig. 7.14(b). Such is the case in a SCISSOR. The other two ways (c and d) are characterized by indirect coupling of rings via two common waveguides. Note that these four structures are identical in terms of their fundamental building block and differ only in the manner in which they are interconnected. Of the cavity versions, only geometry (a) forms a building block that is naturally sequenced as in a multilayered structure. The other three geometries possess interchanged ports that do not sequence naturally in their cavity embodiments and can be constructed only with greater complexity. Microring resonators are, however, more naturally suited to sequencing in any of these geometries because the two input and two output ports occupy spatially distinct channels. Furthermore, their planar nature is compatible with current fabrication technologies.

8

9

Counterpropagating waves in a microring resonator can be coupled via either surface roughness or a small perturbation, such as a notch. When side coupled to a guide, such a notched resonator can approximate a general second-order Chebyschev reflection response, because a single notched microring with one guide is equivalent to two coupled rings between two guides [150, 251]. Strictly speaking, the first configuration is constructed from two-port resonators and is a special case.

200

7. Distributed Microresonator Systems

Ring version

Cavity version

Bj

Bj+1

Bj

Bj+1

Aj

Aj+1

Aj

Aj+1

Aj

Aj+1

Aj

Aj+1

Bj

Bj+1

Bj

Bj+1

Aj

Aj+1

Aj

Aj+1

Bj

Bj+1

Bj

Bj+1

Aj

Aj+1

Aj

Aj+1

a)

b)

c)

d)

Fig. 7.14. Illustrations of four archetypal microring resonator unit cells for (a) a CROW, (b) a single-channel SCISSOR, (c) a double-channel SCISSOR, and (d) a twisted double-channel SCISSOR — along with their cavity versions. These unit cells can serve as building blocks for constructing sequences of resonators that serve as novel photonic guidance architectures. 7.7.1 Bloch-Matrix Formalism In previous sections, the frequency analysis of periodically sequenced allpass resonators was straightforward because of the serial feed-forward nature of the transmission. That is, after the fields leave a resonator unit of a SCISSOR, they never return to it and phase accumulation is simply cumulative. Two other archetypes do not share this property (CROW and double-channel SCISSOR). Thus, a more generalized description of periodically sequenced microresonator systems is required. We adopt a matrix formalism used to analyze photonic bandgap structures based on Bloch’s theorem with the assumption that the resonator sequences are infinite and periodic. Using this formalism, the dispersion relations for the four archetypal sequenced-resonator geometries are readily derived.

7.7 Generalized Periodic Resonator Systems

201

Table 7.1. Port-to-port relations for the building blocks in Fig. 7.14. M11 (a) (b) (c) (d)

−ei

M12



(φ2 +φ1 )

(φ2 −φ1 ) 2 +r 2 e−i 2 t (1−r e−iφ )eiθ r −e−iφ (1−r1 r2 e−iφ )eiθ r1 −r2 e−iφ

φ2 ei 2

r

2

φ2 −e−i 2



t2

0

∗ M12

∗ M11 φ

−t1 t2 ei 2 r2 −r1 eiφ

−t1 t2 e−i 2 r1 −r2 e−iφ

−t1 t2 ei 2 eiθ 1−r1 r2 eiφ

−t1 t2 ei 2 eiθ 1−r1 r2 eiφ

φ

1−r1 r2 eiφ

M22

0 φ

(r1 −r2 eiφ )eiθ

M21

φ

0 (1−r1 r2 eiφ )e−iθ r2 −r1 eiφ

(r2 −r1 eiφ )eiθ 1−r1 r2 eiφ

We begin with the scattering matrix equations for a single coupler. E3

=

r E1 + itE2 ,

E4

=

itE1 + r E2 .

Neglecting loss allows the simple relation r 2 + t 2 = 1 to hold for each coupler. Using these relations for each coupling point as required, we can calculate the matrix M that relates the fields at the two right ports of a coupled resonator, Aj+1 and Bj+1 , to the fields at the two left ports Aj and Bj . For a generic four-port optical device, this takes the form 

Aj+1 Bj+1



 =

M11 M21

M12 M22



Aj Bj

.

(7.28)

Table 7.1 shows the components Mij that represent the port-to-port relations associated with each of the building blocks in Fig. 7.14. We are now ready to consider an infinite sequence of resonators, each of the same building block and connected to its neighbors, with spatial periodicity L. Bloch’s theorem states that we can find solutions for the fields that at periodic intervals in the infinite lattice are simply related by a phase factor:   Aj+1 Aj = eikeff L . (7.29) Bj Bj+1 That is, eikeff L must be an eigenvalue of the matrix formed from Mij ; for this to hold, we require  M11 − eikeff L M12 det = 0. (7.30) M21 M22 − eikeff L The quadratic formula is applied to the expanded form of this equation to obtain the dispersion relation (keff vs. ω) based on the matrix coefficients that in general are frequency dependent:

202

7. Distributed Microresonator Systems

ei2keff L − (M11 + M22 ) eikeff L + (M11 M22 − M12 M21 ) = 0, ⎡  1/2 ⎤ 2 1 + M − M (M (M ) ) 11 22 11 22 ⎦. keff = arg ⎣ ± + M12 M21 L 2 4

(7.31) (7.32)

In the following sections, we apply this technique to derive dispersion relations for the four archetypal infinite periodic lattices that result from connecting the unit cells in Fig. 7.14. 7.7.2 Directly Coupled Resonators A sequence of directly coupled resonators, or serial-cascaded rings [253– 255], with no auxiliary waveguides is depicted in Fig. 7.15. One interpretation of this geometry results from considering every other resonator to act as two auxiliary waveguides that connect the unit cells of Fig. 7.14(a) together in sequence. Consequently, the optical properties are exactly analogous to a multilayered structure. To allow for a richness of greater complexity, we consider a structure with two length scales where the resonator circumference alternates between 2π R1 and 2π R2 . In what follows we will restrict ourselves to the case where light propagates clockwise in the larger resonators and counterclockwise in the smaller resonators. Implementing Eq. 7.32 and Table 7.1 entry (a), the dispersion relation for an infinite sequence is given by keff =

!!  "   −1 r φ1 + φ2 φ2 − φ1 −i arg + ± cos cos 2(R1 + R2 ) t2 2 t2 2 ⎤ 1/2      −1 r φ1 + φ2 φ2 − φ1 2 ⎦ , (7.33) + 2 cos cos −1 t2 2 t 2

where the two phase degrees of freedom φj = 2π Rj nω/c are proportional to the two characteristic length scales. The two solutions represent the forward- and backward-going waves throughout the structure. Due to distributed feedback, photonic bandgaps emerge where the Bragg condition is satisfied (2π Rj = (m + 1/2)λ/n). The dispersion relation, normalized group index ((c/n)dkeff /dω), and group velocity dispersion (d2 keff /dω2 ) are plotted in Fig. 7.15 for a sequence of resonators at wavelengths near 1.55 µm. The shaded areas represent bandgaps while the bands are labelled according to the nearest resonance order (m). The structure may be used to support Bragg soliton propagation [244], particularly near the band edges where the group velocity dispersion is strong [256] and the nonlinearity is enhanced [257].

7.7 Generalized Periodic Resonator Systems

203

L R1

R2

Coupled-resonator optical waveguide

Wavelength (µm)

a) Low finesse 1.498 1.522

R1 32, R2 48 common band

1.546

R2 band 47

1.571 1.597 10 8 6 4 2 0 - Lπ Norm. group index (ng/n)

R1 band 31 R2 band 46 π -2L

π 0 2L Bloch vector (keff)

π L -10 -5

0 5 10 GVD (ps2/mm)

Wavelength (µm)

b) High finesse 1.498 1.522

R1 32, R2 48 common band

1.546

R2 band 47 R1 band 31

1.571

R2 band 46

1.597

50 40 30 20 10 0 - Lπ Norm. group index (ng/n)

π -2L

π 0 2L Bloch vector (keff)

π L -400

0 400 GVD (ps2/mm)

Fig. 7.15. The dispersion relation, normalized group index, and GVD for (a) a low-finesse and (b) a high-finesse coupled resonator optical waveguide. The dispersion relation is analogous to that of a multilayered structure with alternating layer indices and/or thicknesses. Bandgaps are always of the direct type and result from distributed Bragg reflection. Parameters include a refractive index of n = 3.1, alternating radii of R1 = 2.5 µm and R2 = 1.5R1 . Resonances mR1 = 31 and 32 and mR2 = 46, 47 and 48 are shown. In (a), a high coupling strength, t 2 = 0.75, results in narrow bandgaps, whereas in (b), a low coupling strength, t 2 = 0.1814, results in wide bandgaps. These qualitative features are directly opposite those found in the double-channel SCISSOR. (After Ref. [252], ©2004, Optical Society of America.)

204

7. Distributed Microresonator Systems

7.7.3 Single-Channel SCISSORs Next we revisit the single-channel SCISSOR [129, 224, 252] from the perspective of the Bloch formalism. Because no mechanism for contradirectional coupling is present, light of all frequencies is simply transmitted in a feed-forward sequential manner from resonator to resonator “pausing” for localized feedback at each. As a result, no photonic bandgaps can exist in this geometry. The optical properties are in fact independent of whether all the spacings between neighboring resonators are the same; only the average density of resonators in a given length (or simply the total number) dictates the optical properties of the structure. For an infinite periodic SCISSOR, the only surviving matrix coefficient from Table 7.1 is M11 . The dispersion relation (Eq. 7.32) takes the form   r − eiφ 1 Φ (ω) n n , (7.34) keff = ω + arg = ω+ iφ c L 1 − re c L where φ = 2π Rnω/c, and Φ(ω) is the effective phase shift of a single resonator. The dispersion relation, normalized group index, and GVD are plotted in Fig. 7.16 for a series of 2.5-µm radius resonators at wavelengths near 1.55 µm. To avoid confusion, the dispersion relation is shown only for forward-going propagation defined by the arrows in the schematic. For resonance frequencies, light experiences group delays of the order of the cavity lifetime (2F /π )2π nR/c at each resonator. From a macroscopic point of view, the discrete delays may be considered to be distributed along the channel resulting in an effective group velocity that can be slower than that in the common waveguide. On resonance, the group index scales directly with finesse as ng = (1 + (4R/L)F ) n for r ≈ 1. Due to the frequency-dependent nature of the effective group velocity peaking at resonance, the lowest order dispersion vanishes on resonance but can be very strongly normal (positive) or anomalous (negative) below or above each resonance, respectively. Additionally, as we have shown earlier, nonlinear processes such as self-phase modulation are enhanced dramatically. Other coupled-cavity structures have been shown to enhance nonlinearities [258] and support soliton propagation. [245] The SCISSOR structure is interesting in its ability to slow, disperse, and/or intensify light pulses while maintaining a transmission spectrum free from bandgaps — a feature that, to the best of our knowledge, does not bear direct analogy with any traditional artificial medium. Fundamentally, this results from disabling distributed feedback throughout the structure and localizing it to the all-pass resonators. Recently, it has been shown that photonic crystals can also be used to synthesize distributed all-pass resonant structures [259].

7.7 Generalized Periodic Resonator Systems

205

L R Single-channel SCISSOR

Wavelength (µm)

a) Low finesse 1.498 1.522 1.546 1.571 1.597 10 8 6 4 2 0 - Lπ Norm. group index (ng/n)

π -2L

π 0 2L Bloch vector (keff)

π L -10 -5

π -2L

π L -400

0 5 10 GVD (ps2/mm)

Wavelength (µm)

b) High finesse 1.498 1.522 1.546 1.571 1.597

50 40 30 20 10 0 - Lπ Norm. group index (ng/n)

π 0 2L Bloch vector (keff)

0 400 GVD (ps2/mm)

Fig. 7.16. The dispersion relation, normalized group index, and GVD for (a) a low-finesse and (b) a high-finesse single-channel SCISSOR. Note that the dispersion relation does not display photonic bandgaps. Nevertheless, at the resonances (λm = 2π nR/m), the group index (and the intensity buildup) is maximized. Parameters include a refractive index of n = 3.1, and a radius of R = 2.5 µm. Resonances mR = 31 and 32 at 1.571 µm and 1.522 µm are shown. In (a), a high coupling strength, t 2 = 0.75, results in a wide bandwidth, whereas in (b), a low coupling strength, t 2 = 0.1814, results in a narrow bandwidth. To avoid redundancy, and because the forward and backward traveling waves do not couple, only the dispersion relation for the forward-traveling wave is shown. (After Ref. [252], ©2004, Optical Society of America.)

206

7. Distributed Microresonator Systems

7.7.4 Double-Channel SCISSORs The addition of a second common coupling channel to a SCISSOR as depicted in Fig. 7.17 enables distributed feedback. The unit cell for this structure is an add–drop resonator oriented as in Fig. 7.14(c). In contrast to the single-channel case, the spacing between resonators is now important because resonances can develop not only within the resonators but also among them. Photonic bandgaps emerge around each type of resonance in the dispersion relation of an infinite periodic structure [260]. One type of bandgap emerges when the periodicity of the structure satisfies the Bragg condition (2L = mB λ/n, with mB an integer), and another type emerges when the circumference of the microresonator is an integer multiple of the wavelength (2π R = mR λ/n, with mR an integer). Compared with a multilayered structure or CROW, the output port connections (drop and through) at each unit cell are reversed. This reversal couples what were distributed Bragg reflections to the feed-forward direction and diverts resonance frequencies to the retro-reflected direction. In the case of a multilayered structure or CROW, high reflectivity can be achieved by distributed feedback from many partially reflecting unit cells and 100% transmission is always obtained at a resonant frequency. In contrast, in a double-channel SCISSOR, 100% reflection is achieved readily at any of the resonator’s resonant frequencies irrespective of the coupling parameter r , provided that the coupling coefficients are matched. This result is attributed to the well-known phenomena of critical coupling, whereby incident light is attenuated completely at the resonance frequency due to complete destructive interference of the bypassed incoming field with the outcoupled circulating field. For this condition to be met, the sum of the tap coupling due to the second channel (and losses if present in the microring) must equal the cross-coupling from the excitation channel. Implementing Eq. 7.32, the dispersion relation for the double-channel SCISSOR structure takes the form  1 1 − r1 r2 e−iφ iθ 1 1 − r1 r2 eiφ −iθ 1 keff = arg e + e L 2 r1 − r2 e−iφ 2 r2 − r1 eiφ ⎛ 2 1 1 − r1 r2 e−iφ iθ 1 1 − r1 r2 eiφ −iθ ⎝ ± e − e 2 r1 − r2 e−iφ 2 r2 − r1 eiφ



 ⎞1/2 ⎤ 1 − r12 1 − r22 ⎠ ⎥  + ⎦ , (7.35) r2 − r1 eiφ r1 − r2 e−iφ where the two phase degrees of freedom, φ = 2π Rnω/c and θ = Lnω/c, are proportional to the two characteristic length scales. The two solutions represent the bottom channel forward- and top channel backwardgoing waves in the photonic structure. The dispersion relation, group index, and GVD are plotted in Fig. 7.17 for 2.5-µm-radius resonators

7.7 Generalized Periodic Resonator Systems

207

L R Double-channel SCISSOR

Wavelength (µm)

a) Low finesse 1.498 1.522

Bragg gap 48 + resonator gap 32

1.546 1.571 1.597 10 8 6 4 2 0 - Lπ Norm. group index (ng/n)

Bragg gap 47 resonator gap 31 Bragg gap 46 π -2L

π 0 2L Bloch vector (keff)

π L -10 -5

0 5 10 GVD (ps2/mm)

Wavelength (µm)

b) High finesse 1.498 1.522

Bragg gap 48 + resonator gap 32

1.546

Bragg gap 47

1.571 1.597

50 40 30 20 10 0 - Lπ Norm. group index (ng/n)

res. gap 31 Bragg gap 46 π -2L

π 0 2L Bloch vector (keff)

π L -400

0 400 GVD (ps2/mm)

Fig. 7.17. The dispersion relation, normalized group index, and GVD for (a) a low-finesse and (b) high-finesse double-channel SCISSOR. Note that unlike the dispersion relation for the single-channel SCISSOR, the double-channel variety displays photonic bandgaps. Two qualitatively different bandgaps manifest themselves. For spectral components satisfying the Bragg condition (λmB = 2nL/mB ), the bandgap is direct and results from distributed Bragg reflection. At the resonances of the rings (λmR = 2π nR/mR ), the bandgap is indirect and results from strong resonator-mediated back-coupling. Parameters were chosen such that one Bragg gap was coincident with one resonator gap within the figure: refractive index n = 3.1, radii R = 2.5 µm, and spacing L = 1.5π R. Resonator resonances mR = 31 and 32 and Bragg resonances mB = 46, 47 and 48 are shown. The coincident resonator (mR = 32) and Bragg (mB = 48) resonance results in a wide direct gap. In (a) a high coupling strength, t 2 = 0.75, results in wide bandgaps, whereas in (b) a low coupling strength, t 2 = 0.1814, results in narrow bandgaps. (After Ref. [252], ©2004, Optical Society of America.)

208

7. Distributed Microresonator Systems

spaced by 1.5π R at wavelengths near 1.55 µm. Of the two types of gaps mentioned above, those associated with the periodicity of the structure (inter-resonator spacing) are called “Bragg gaps” and are always direct. The indirect gaps are termed “resonator gaps” and are those associated with the internal resonances of the resonators. A comparison of the qualitative features of the dispersion relation reveals that the bandgaps are wider for low-finesse resonators. The interpretation is simple: In the highfinesse case, the band over which the individual resonators are reflecting is narrow whereas in the low-finesse case, it is wide. This directly carries over to the widths of the bandgaps in the infinitely periodic structure and is in stark contrast to the situation of a multilayered structure where high reflectivity results in a wider bandgap. Due to the efficient excitation of the resonators near stop-gaps, this structure is ideal for exploring nonlinear effects [261] particularly within the bandgap, such as gap solitons [243, 260]. 7.7.5 Twisted Double-Channel SCISSORs A “twist” on the double-channel SCISSOR that possesses qualitatively different optical properties is constructed by interchanging the top ports (add and drop) of each unit cell, as in Fig. 7.14(d). For optical fiber ring resonators, this configuration may be implemented by twisting each of the resonators to form figure-8 loops. In an integrated geometry, it is easier to build two resonators that are nearly 100% coupled. This “twisted” double-channel SCISSOR is depicted in Fig. 7.18. As a result of the port interchange, there is no longer any mechanism for contra-directional coupling, but codirectional coupling across the channels is now mediated by the resonators with localized feedback enhancement. Thus, light injected into one of the ports only couples to either of the channels in the forward-going direction. The structure, therefore, behaves as a resonatorenhanced directional coupler. In Table 7.1, the matrix coefficients are given taking the coefficients characterizing the couplings between the waveguides and resonators to be r1 and r2 , and assuming that the twisted structure is implemented by use of two resonators that are 100% coupled to each other. For an infinite periodic structure, the resulting dispersion relation (Eq. 7.32) is ⎡

 r1 −r2 eiφ r2 −r1 eiφ + iφ iφ 1 ⎢ 1−r1 r2 e 1−r1 r2 e keff = arg ⎣eiθ + L 2 ⎞1/2 ⎤ ⎛ 



 

 2

⎜ ±⎝

r1 − r2 eiφ − r2 − r1 eiφ + 4 1 − r12  2 4 1 − r1 r2 eiφ

1 − r22 eiφ ⎟ ⎠

⎥ ⎥, ⎦

(7.36)

7.7 Generalized Periodic Resonator Systems

209

L

Twisted double-channel SCISSOR

Wavelength (µm)

a) Low finesse 1.498 1.522 1.546 1.571 1.597 10 8 6 4 2 0 - Lπ Norm. group index (ng/n)

π -2L

π 0 2L Bloch vector (keff)

π L -10 -5

π -2L

π L -400

0 5 10 GVD (ps2/mm)

Wavelength (µm)

b) High finesse 1.498 1.522 1.546 1.571 1.597

50 40 30 20 10 0 - Lπ Norm. group index (ng/n)

π 0 2L Bloch vector (keff)

0 400 GVD (ps2/mm)

Fig. 7.18. The dispersion relation, normalized group index, and GVD for (a) a low-finesse and (b) a high-finesse twisted double-channel SCISSOR. Bandgaps are absent in the dispersion relation, which resembles that of the single-channel SCISSOR, but with the presence of a second branch. The two branches correspond to the two decoupled forward-traveling normal modes. Near the microring resonances, (λm = 2π nR/m) the two branches are strongly coupled, as in the case of a directional coupler. Parameters used are the same as in Fig. 7.17 except that there are two resonators each half in circumference and 100% coupled. Resonances mR = 31 and 32 at 1.571 µm and 1.522 µm are shown. In (a), a high coupling strength, t 2 = 0.75, results in wide-bandwidth channel-to-channel coupling whereas in (b), a low coupling strength, t 2 = 0.1814, results in narrow-bandwidth channel-to-channel coupling. To avoid redundancy, and because the two forward- and two backward-traveling waves do not couple, only the two forward-traveling dispersion relation branches are shown. (After Ref. [252], ©2004, Optical Society of America.)

210

7. Distributed Microresonator Systems

where φ = 2π Rnω/c and θ = Lnω/c. The solutions represent the two coupled forward-going waves in the photonic structure. The dispersion relation, normalized group index, and GVD are plotted in Fig. 7.18 for the symmetric case of r1 = r2 . Two branches are present, each one corresponding to one of two normal modes of the structure. 7.7.6 Bandgap Engineering in Distributed Feedback Structures Fig. 7.19 qualitatively compares the transmission properties of unit structures and infinite periodic structures for a double-channel SCISSOR and a multilayered structure. Although a multilayered structure or CROW allows propagation on resonance and takes advantage of maximum coherent buildup of intensity within the resonators, propagation is not allowed at resonances of the double-channel SCISSOR around which bandgaps form. We next compare the qualitative features of the transmission for a finite double-channel SCISSOR and CROW with the corresponding infi-

a) FabryPerot

b) Multi-layer Stack

c) Add-drop resonator

T

T

T

R

e) Add-drop f) Doubleresonator channel (re-oriented) SCISSOR

d) Coupledresonator optical waveguide

T

T

R

T

R

Fig. 7.19. A qualitative comparison of the transmission properties of three structures possessing bandgaps: (a) a Fabry–Perot, (b) a multilayered stack, (c) an add–drop resonator, (d) a CROW, (e) an add–drop resonator (reoriented), and (f) a double-channel SCISSOR. Note that the qualitative features of the transmission peaks and valleys are equivalent for multilayered stacks and CROWs but reversed for double-channel SCISSORs. With increasing finesse, the bandgap widths increase in both multilayered stacks and CROWs while they decrease in double-channel SCISSORs. (After Ref. [252], ©2004, Optical Society of America.)

7.7 Generalized Periodic Resonator Systems

211

nite periodic structures. The finite structures are termed parallel and serial resonator coupled structures respectively [22, 55, 63, 165, 262]. Figure 7.20 displays the transmission spectra for low-finesse one, five, and infinite unit-celled structures. Figure 7.21 displays the transmission spectra for equivalent high-finesse structures. The displayed plots show that the dispersion relation can provide a heuristic guide to the location and width of the transmission dips in a finite structure. It is evident that the infinite double-channel SCISSOR and CROW possess complementary transmission characteristics; that is, band and bandgap locations are interchanged. Typically, waveguide or fiber Bragg gratings typically possess low reflectivity (and thus low finesse) per unit cell and thus display small

Fig. 7.20. Transmission spectra for a low-finesse (t 2 = 0.75) finite double-channel SCISSOR and finite CROW. Parameters are the same as in Figs. 7.15 and 7.17. Here, 5-unit cells are used to approximate the structures. For comparison, the transmission for a single resonator is shown in each case and shaded regions correspond to one-dimensional photonic bandgaps in the corresponding infinite structure. The labels B and R correspond to the double-channel SCISSOR’s Bragg and resonator gaps, whereas the labels R1 and R2 correspond to the alternating CROW resonances. (After Ref. [252], ©2004, Optical Society of America.)

212

7. Distributed Microresonator Systems

Fig. 7.21. Transmission spectra for a high-finesse (t 2 = 0.1814) finite double-channel SCISSOR and finite 1D-CROW. Parameters are the same as in Figs. 7.15 and 7.17. Here, 5-unit cells are used to approximate the structures. For comparison, the transmission for a single resonator is shown in each case and shaded regions correspond to one-dimensional photonic bandgaps in the corresponding infinite structure. The labels B and R correspond to the double-channel SCISSOR’s Bragg and resonator gaps, whereas the labels R1 and R2 correspond to the alternating CROW resonances. (After Ref. [252], ©2004, Optical Society of America.)

ripple in the passbands. Here, the deep ripples in the transmission bands result from the abruptly terminated ends of the structure as encountered in unapodized high-reflectivity multilayered stacks. First, note that the ripple depth across the passbands is controlled by the transmission of a single resonator and results from the “splitting” of individual interacting resonances. Second, note that in going from low to high finesse, the width of the single resonator transmission dips (and peaks) shrink faster than the corresponding gaps (and bands) in the infinite structure. Analysis of the bandwidths show that although the single resonator dip (and peak) bandwidth scales as F −1 ≈ t 2 /π ,
the widths of infinite resonator gaps

−1 (and bands) scale as F∞ ≈ t/2 ∝ F −1 . This explains why the passband

7.7 Generalized Periodic Resonator Systems

213

ripples are of similar depth in the low-finesse cases whereas are strong only in the high-finesse case for the CROW. 7.7.7 Slow Light, Group Velocity Dispersion, and Nonlinearities The dispersion relations associated with the microresonator-based photonic structures considered here reveal greatly reduced group velocities (vg = dω/dkeff ) near structural resonances [224, 263]. The reduction of group velocity may be understood intuitively by noting that near a structural resonance light spends extra time circulating localized within the resonators or in propagating back and forth between distributed components of the periodic structure. There are several criteria by which one can compare different optical mechanisms that induce slow light effects. Furthermore, different applications demand different criteria. For some applications, the slowest attainable group velocity may be desirable. Many optical systems have demonstrated such slow light capability [230] but typically only over bandwidths that are too narrow for practical use in optical communications or logic. For such applications, a large fractional group delay (L/τ)dkeff /dω is a better measure. This dimensionless quantity is interpreted as the number of pulse widths τ by which a pulse or train of pulses can be delayed, while maintaining the integrity of the pulses. Many instances of slow light phenomena, although impressive in their ability to slow the speed of pulse propagation to a small fraction of c, would fail to delay an incoming pulse by more than a fraction of a pulse width without seriously distorting it, simply because of a large accompanying group velocity dispersion or narrow spectral window. Microresonator-based structures have been proposed [220] for use as optical delay lines because, when designed properly, they can exhibit a large fractional delay. Additionally, because the group velocity can be made tunable by shifting resonances thermally [264] or electrically [65, 77], controllable optical delay lines can be constructed. The shortest pulse for which the group delay is meaningful, and thus that for which the fractional delay is largest, is one with a pulse width that matches the cavity lifetime. Shorter pulses become distorted due to highorder dispersive effects, whereas longer pulses that experience the same physical delay exhibit a lower fractional delay. These considerations remind us that a vanishingly small group velocity is often not the sole important feature. An optical delay line or buffer based on microresonator structures will in general be useful only if slow group velocity is accompanied by high-transmission and low pulse-distorting dispersion. Near resonances of a resonances of a single-channel SCISSOR, the fractional group delay can approach unity. In distributed feedback structures, the edges of two bandgaps can be engineered so that they are close in frequency but not overlapping resulting in a flat, defect-like band generated between them. Khurgin [265] has

214

7. Distributed Microresonator Systems

proposed a combination of distributed microresonator-based structures for improved performance. Alternatively, the strong group velocity dispersion encountered at certain frequencies can be put to use for the construction of photonic devices that perform dispersion compensation in an ultra-compact geometry. Each of the archetypal structures in this can display group velocity dispersion values that greatly exceed 20 ps2 per millimeter. These are at least six orders of magnitude higher than what is conventionally measured in typical single-mode silica fiber (20 ps2 per kilometer). The situation is similar to that encountered in photonic crystal defect guides that exhibit ultra-strong dispersive effects [246]. Just as an intrinsically high group velocity reduction is not as important in many applications as a large fractional group delay, many photonics applications require strong fractional group delay dispersion irrespective of the actual group velocity dispersion. Microresonator-based structures have been implemented as dispersion compensators [266] because, when designed properly, can exhibit a fractional group delay dispersion near unity (of either sign) per resonator. Coupled microresonator structures have been shown to possess propagation characteristics associated with extended nonlinear Schrödinger equations (NLSEs). The phenomena of slow light, light trapping [267], soliton propagation [224], soliton compression [129], soliton switching [268], gap solitons [260], self-steepening, modulation instability, and four-wave mixing are all directly transferrable to the much more compact scale afforded by a microresonator-based photonic structures. The enhancement of nonlinear effects in microresonators is attributed to two different, although related, effects. First, because the group velocity is reduced in such a way that light circulates through a longer path length within a photonic structure, the interaction length (or time) for a nonlinear process is increased. This is directly related to the well-known enhancement of phase sensitivity in ring resonators [151] and photonic crystal-based [247, 269] structures. Second, because the light circulating within the resonators is built-up coherently to a higher intensity than that initially injected into the structure, stronger nonlinear effects are possible. As a result of these two effects, nonlinear processes like self-phase modulation are enhanced in proportion to the square of the resonator finesse [151]. This scaling law holds for single resonators and localized feedback structures. For example, in the case of the single-channel SCISSOR, the phase sensitivity and buildup are coincident and equal, and the enhancement of accumulated nonlinear phase shifts is directly proportional to the square of the group index, (c/vg )2 . This general result is found in many nonlinear photonic systems displaying slow light effects resulting from structural resonances.

7.8 Summary

215

7.8 Summary Passive, nonlinear resonators are still a relatively widely untapped area of research. Currently, many single microresonator systems with excellent optical properties have been constructed. In many of these cases, extending the fabrication techniques to construct long sequences of such devices to yield large-scale integration of photonic devices [207] is achievable. The aim of this chapter was to describe some of the linear and nonlinear propagation characteristics of sequences of microresonators. Figure 7.22 summarizes the detuning dependence of the linear and nonlinear propagation parameters for a single-channel SCISSOR. The application of thermal or electrical fields to the resonators makes it possible to control the detuning and/or coupling coefficients. We envision that such structures could be used as artificial media to study and apply NLSE pulse propagation effects on an integrated chip where the propagation parameters may be chosen and/or modified in realtime. Other applications might include a testbed for studies of slow-light phenomena, variable optical delay lines [220], clean pulse compression on a chip without pedestal formation via the soliton decay mechanism, and soliton-based optical switching and routing with low energy pulses. Although all of these concepts have been implemented in various geometries and material systems, the SCISSOR system has the potential for providing a highly compact, integrated optical platform for such phenomena. Furthermore, it has the potential of creating familiar phenomena in

1.0 0.5

(a)

' keff (2FT/π)/L

(b)

'' keff (2FT/π)2/L

(2F/π)2 γ 2πR/L

(c)

''' keff (2FT/π)3/L

' γeff (e) (2F/π) γ 2πRT/L

γeff

(d)

0.0 -0.5 -4π F

3

2π 0 F Normalized detuning, φ0

-2π

4π

F

F

Fig. 7.22. Functional dependence of (a) the group velocity reduction, (b) group velocity dispersion, (c) third-order dispersion, (d) self-phase modulation coefficient, and (e) self-steepening coefficient on the normalized detuning φ for a SCISSOR. The parameters have been scaled such that the curves are universal and fit within the same plot limits. (After Ref. [129], ©2002, Optical Society of America.)

216

7. Distributed Microresonator Systems

regions of parameter space that typically do not manifest it such as the case of self-steepening for pulses greater than picoseconds. As manufacturing techniques continue to improve, guiding structures based on distributed microresonators will become important photonic architectures.

8. Fabrication Techniques for Microresonators

The theories presented in this book have been around for a very long time. Only recently, due to advances in fabrication technology, have the realization of microresonators become a possibility. Indeed their great potential is tempered only by the challenges associated with their construction. In this chapter, we present the key fabrication issues involved with microresonators. Due to the authors’ expertise with III–V semiconductor fabrication techniques, detailed process flows are presented for AlGaAs- and InP-based material systems.

8.1 Materials Material systems being researched for microresonator fabrication include various glasses and polymers (on a silica-on-silicon substrate), siliconon-silica (using readily available silicon-on-insulator wafers), gallium arsenide, and indium phosphide. The ideal material, is of course, one based on silicon, since that can leverage the considerable knowledge accumulated by the electronics industry in the mass-production of silicon microprocessors. The drawback of silicon is that it is difficult to make optically active, and thus has not at the time of this writing been used for monolithic optical integration (hybrid integration, as demonstrated by the researchers at UCSB, is always a possibility). III–V semiconductors are optically active, and therefore suitable for monolithic integration, but present their own problems. The processing of III–V semiconductors is an expensive proposition, because the raw material is expensive, fragile, and requires much more careful processing than glasses and polymers. Also, III–V semiconductor wafers are still not readily available in 6” diameters (the first wafers were announced by Showa Denko in April 2002), making mass production difficult. The electronics industry, in contrast, has moved to 12” silicon wafers. This means that equipment has to be custom designed for small wafer diameters; standard commercial equipment cannot be used readily. Finally, single-mode waveguides made from both silicon and III–V semiconductors tend to have extremely small cross sections (usually less than 1 µm2 ). Together with the high effective indices of such waveguides,

218

8. Fabrication Techniques for Microresonators

this makes the problem of coupling to off-chip waveguides (fibers) extremely difficult. Although the problem of fabrication costs (for III–Vs) may be beyond a solution currently, several groups are achieving various degrees of success at tackling the problem of coupling between optical fibers and high-effective-index waveguides, notably the work from Michal’s group at Cornell. Currently, the most successful effort toward the commercial realization of microring resonator-based filters (by the now-Infinera-owned Little Optics) relies on a glass-based system. These materials offer an excellent solution to the problem of costs (the raw material — usually silica-coated-silicon substrates — is inexpensive), the equipement needed is essentially off-the-shelf equipment used by the semiconductor microelectronics industry, the waveguides themselves are made with glass, and there is a vast body of knowledge from the optical fiber industry on how to get glasses with low optical losses. The waveguide stack can be deposited easily in a chemical-vapor-deposition system, and processing can be done with dielectric etching chemistries similar to those used by the semiconductor industry. All this said, III–V semiconductors remain of great importance because some of the most innovative new devices have been demonstrated in this material system. These devices span the breadth of devices needed to realize monolithically integrated all-optical circuits, such as filters, routers, amplifiers, lasers, and logic gates. Glass-based systems may yet prove as versatile if not more so; doped glasses may provide high-opticalnonlinearities needed for photonic logic, and erbium doping may help in the realization of amplifiers. One aspect independent of material system is the geometry of the device. Since we require the resonator to be coupled to one or more bus waveguides, these waveguides must either be fused with, or placed in close proximity to, the resonator. The bus guides may be placed in-plane with the resonator (lateral coupling) or out-of-plane (vertical coupling). Future photonic circuits will almost certainly utilize a combination of the two approaches.

8.2 III–V Semiconductors for Active and Passive Microrings III–V semiconductors have high refractive indices (>3) compared with competing material systems such as glass/silica (∼1.5), silicon oxynitride (1.45 – 2.1), and LiNbO3 (∼2.2). This allows the fabrication of very high (lateral) index contrast waveguides that allows the bend radius to be as small as 1 µm before bending losses cause unacceptable deterioration in performance. Lattice matched III–V ternaries and quaternaries can be grown on both indium phosphide and gallium arsenide with a wide variety of bandgaps

8.3 Growing a Waveguide Stack

219

and refractive indices [270–275]. Using an appropriate choice of materials, we can make slab waveguides in both materials; patterning shallowetched ridges produces loosely confined waveguides, and deeply etched pedestals give us strongly confined waveguides. Since III–V semiconductors have been the subject of studies since the 1960s, a vast body of knowledge exists on fabrication technologies such as chemo-mechanical polishing [276] and selective wet-chemical etching [277–279]. Most III–V compound semiconductors have a direct bandgap. Therefore, the waveguide core and/or cladding can be active, allowing us to make optically active semiconductor devices. Modern optical communication systems operate around 1550 nm because silica optic fibers have the lowest loss around 1550 nm. InP-based materials can be used to make amplifiers, lasers, and related devices, making it the system of choice for active integrated semiconductor optical devices for the C band. So III–V semiconductors — InP and its quaternaries, and GaAs and its ternaries — are ideal for the fabrication of microring resonators if other fabrication requirements (smooth, vertical, deep, anisotropic etching, and lithography) are met. Recent demonstrations of devices in this material system at LPS have been driven by the development of technologies for in-house wafer growth by solid-source molecular beam epitaxy [280,281]; smooth, vertical dry-etching of both GaAs and InP [282–286]; and highresolution electron-beam lithography (via a collaboration with the Cornell NanoScale Facility, using the Leica VB6 HR). Commercial acceptance of III–V semiconductors for passive filters is hampered currently by the high materials cost, high equipment costs, and high optical losses in the material when compared with glass. However, III–V semiconductors are important for the development of ultra-compact monolithically integrated all-optical systems because they are optically active, so this material system is an important target for research and development.

8.3 Growing a Waveguide Stack 8.3.1 Polymers Typically, this refers to waveguides with polymer cores or refractive index ∼1.5 or more. This requirement is placed on the material because the refractive index of silica (which usually forms the lower cladding), is typically 1.45—1.51 depending on the method used (thermal/wet growth, or CVD). Given a silica-coated silicon wafer, the waveguide stack can be grown quite simply by spinning-on the appropriate polymer layers, followed by a cure step. The cure step is usually a high-temperature bake to drive off solvents and stress that may affect the polarization dependence of the waveguides, but this can usually be compensated for in the design.

220

8. Fabrication Techniques for Microresonators

8.3.2 Glass Glass waveguide stacks usually are grown with chemical vapor deposition, with silica forming the lower cladding. As with polymers, residual stresses from high temperature steps can result in stresses that affect the polarization dependence. Low-temperature deposition is possible to eliminate such stresses. Another problem arises from losses due to the vibration modes of OH ions. This can be eliminated by the use of deuterated silane [287]. 8.3.3 III–V Semiconductors (InP and GaAs) These are grown with metal-organic chemical vapor deposition (MOCVD), or molecular-beam epitaxy (MBE). These methods are significantly more costly than those used for polymers and glasses. Some of the raw materials needed (arsine, phosphine, phosphorous, arsenic, etc.) are very dangerous (inhalation hazard, flammability, etc.) and require careful handling and sophisticated safety equipment. The waveguide core/upper and lower cladding/substrate in these systems are GaAs/AlGaAs/GaAs, AlGaAs/AlGaAs/GaAs, InP/GaInAsP/InP, and GaInAsP/GaInAsP/InP, where the compositions of the ternaries and quaternaries are suitably adjusted for low/high index depending on their use in the core/cladding. The in-plane cladding is usually air, silica, spinon-glass, polyimide, or other low-index polymer to allow for tight confinement. 8.3.4 Silicon Oxynitride Silica, silicon oxynitride (SiON), and silicon nitride (SiN) are typically deposited with CVD (Low-Pressure CVD, Plasma-Enhanced CVD, etc.). By varying the oxygen content of SiON, it is possible to get a continuous variation in refractive index from around 1.45 (silica) to 2 (silicon nitride). So it makes for an attractive material system from the point of view of design. (Note: we have deliberately omitted Si:O:N ratios, since those depend on the deposition parameters.)

8.4 Feature Definition By feature definition we mean the process of forming the transverse waveguiding structure from the planar waveguide stack. This process will typically consist of a lithography step followed by an etch step.

8.4 Feature Definition

221

8.4.1 Polymers, Glass, SiON, SiN 1. Mask definition (Lithography): For these material systems, the mask definition is achieved by optical contact and/or projection lithography. Projection lithography has the advantage of greater throughput — since the mask is not in contact with the substrate, it does not need cleaning after each lithography step. Projection lithography also provides smoother waveguide sidewalls along curves for systems where the sidewall roughness is determined by the mask resolution, since the discretization on the mask is reduced by the magnification of the projection system. The mask itself could be the photoresist used in the lithography, or a hard mask of silica, aluminum, etc., defined by lift-off or dry-etching. 2. Etching: These material systems are dry-etched using one or all of sulfur hexafluoride, Cx Hy Fz (fluorinated organic gases like freon, tetrafluoromethane, etc.), hydrogen, oxygen, nitrogen, and argon. Anisotropic dry-etching is used to obtain vertical sidewalls required to prevent polarization mixing. Such etching may be performed in a parallel-plate reactive-ion-etching (RIE) chamber, inductively coupled plama RIE (ICP-RIE), electron-cyclotron resonance RIE (ECR), etc. Anisotropy is assured by the use of low-pressure (usually <10−6 T) and high substrate bias. When organic chemistries are used, verticality is assisted by the formation of polymers from the plasma, passivating the sidewalls and preventing additional etching. Nitrogen and argon are usually unreactive and are used for sputtering material off the surface (similar to ion-milling), and they stabilize the plasma and/or as a dilutant to regulate the reaction rate. 8.4.2 III–V Semiconductors (InP and GaAs) 1. Mask definition: In cases where small gaps (usually around 0.1 µm for lateral coupling) are needed, the mask is defined by electron-beam lithography. Although state-of-the-art lithography systems can provide such resolutions, most research institutions use e-beam lithography instead; even 20-year-old e-beam lithography machines can provide such resolution. Elsewhere (vertical coupling), i-line or g-line optical lithography is used. The mask itself could be the e-beam resist or photoresist used in the lithography, or a hard mask of silica, SiN, titanium, chromium, nickel, etc., or a suitable combination of them, defined by lift-off or dry-etching. 2. Etching: Both GaAs and InP-based materials can be etched with resist masks if halogenated chemistries are used (poly(methyl methacrylate) is usually used for e-beam lithography; photoresists are too numerous to list here). InP-based materials that do not contain aluminum also can be etched with a combination of methane and

222

8. Fabrication Techniques for Microresonators

hydrogen (with or without argon). In this case, anisotropy is assured by the passivation of the sidewalls with a polymer formed from the plasma. Verticality is obtained by a cleaning step used to remove the bulk of the polymer buildup from the sidewalls periodically, so that passivation is maintained without compromising the sidewall angle. The cleaning step involves an oxygen plasma, which affects resist masks, so methane-based chemistries require the use of hard masks not affected by oxygen plasmas. For this reason, masks of chromium, nickel, titanium, silica, SiN, or combinations of these materials, are used. Such masks may also be used in dry-etching with halogenated chemistries where the etch depth is needed is large enough that erosion of the resist mask in the plasma becomes a concern. For details, see [286] and the references therein. Detailed examinations of various patterning methods are beyond the scope of this volume. For that, we refer the reader to any of the excellent device processing books on the market, such as the ones by Wolf and Tauber, Campbell, and Pearton.

8.5 Multilayer Processing Multilayer processing is specific to the case of vertically coupled microresonators. Stacking multiple waveguides needs one or of: 1. Epilayer transfer using polymer-assisted, or direct wafer bonding followed by growth-substrate removal [172], or 2. Planarization and regrowth/redeposition [21]. Each method has its advantages (and disadvantages). Polymer wafer bonding is messy and introduces a material of different coefficient of thermal expansion and different chemical stability from the substrates. The different chemical stability can limit the number of high-temperature processing steps that can follow a polymer-assisted epilayer transfer. Furthermore, if volatile by-products are produced during the bonding/curing process, they can move the two substrates apart and cause angular misalignment, making later cleaving difficult. In the case where active devices are being made, the use of polymer wafer bonding will typically limit contacts to the same side (top), since polymer dielectric films used for bonding are poor conductors of electricity as they serve the added purpose of forming the side and top/bottom cladding. For a comprehensive review, see Niklaus et al. [166]. In the case of direct wafer bonding, both the transfer substrate and the epilayer have to be very clean and free of any interfacial impurities that can cause the formation of voids. Furthermore, the presence of particles or a surface with undulations can destroy waveguides (or other desirable features) on the epilayer being transferred (Fig. 8.1). When successful,

8.5 Multilayer Processing

223

Fig. 8.1. (a) Raburn’s fixture for wafer bonding, made from graphite. The solid black portions are the fasteners used to apply pressure between the two substrates. (b) A successfully fabricated double-bonded device. (c) Bonded interface. (Continued on next page.)

224

8. Fabrication Techniques for Microresonators

Fig. 8.1. (Continued from previous page.) (d) A crushed waveguide. (Adapted from Ref. [289], copyright Maura Raburn, 2003, used with permission.)

this method affords an excellent way of making both active and passive devices, and do not limit the number of waveguiding layers to two, as is the case with polymer wafer bonding. In fact, Raburn et al. have used this method to bond three waveguiding layers [172, 288, 289]. Raburn also discusses the various factors that influence successful direct wafer bonding, and achieves bond-area fractions as high as 95% in [289]. Planarization and regrowth/redeposition is another technique where an arbitrary number of waveguiding layers can be used. Thus far, most reports have used two layers, but there is no impediment to three or more (other than, perhaps, a lack of need with present designs). This method usually limits the side-cladding to materials in the same family, such as III–Vs or glasses. It is possible to achieve planarization around highindex glasses/SiON/polymers with low-index glasses/silica/polymers, so that even microring and microdisk resonators can be buried, as demonstrated by Little, Laine, and Haus [143]. However doing the same with III–V semiconductors is more difficult. Not only does regrowth of semiconductors place stringent demands on surface purity, the index contrast available between the core and side-cladding (planarization material), is typically quite small (usually <0.3). This means that it is difficult to make buried microring and microdisk resonators, as the bending losses will

8.6 Laterally Coupled III–V Passive Microresonators

225

be debilitatingly high. To get around this, Choi et al. [64] use a buried bus waveguide, and an air-clad microring/disk resonator. However, this results in a new problem — the microring and bus waveguide are no longer well matched, so it is difficult to get good coupling between the two.

8.6 Laterally Coupled III–V Passive Microresonators InP and GaAs-based laterally coupled microresonators have been demonstrated by numerous groups, notably Rafizadeh et al. [51], Absil et al. [55] (both GaAs-based), Rommel et al. [74], and Grover et al. [76]. As an example, let’s examine the process flow for InP-based microring resonators used by Grover et al. [76]. For their InP-based devices, they used wafers on which the waveguiding stack was grown with solid-source molecular beam epitaxy at the Laboratory for Physical Sciences (for example, [281]). The same process has been used successfully by the group at the Institute of Optics, Rochester, NY, to make GaAs-AlGaAs devices with BCl3 -Cl2 -Ar etching [195]. First, the blank epitaxially-grown wafer (Fig. 8.2a) is coated with 900 nm of silicon dioxide by plasma-enhanced chemical vapor deposition. A Leica VB6 e-beam lithography machine is subsequently used to pattern a bilayer film of poly(methyl methacrylate) (PMMA) as follows: 1. Spin on 100K MW 5% PMMA in Anisole at 4000 RPM for 60 s, which gives a 113-nm-thick film. 2. Bake for 15 min. at 170 ◦ C. 3. Spin on 495K MW PMMA A4 in Anisole at 2000 RPM for 60 s, which gives a 204-nm-thick film. 4. Bake for 15 min. at 170 ◦ C. 5. Write pattern in Leica VB6 HR using a dosage of 736 µC/cm2 for small features (waveguides and microrings) and 610 µC/cm2 for large features (device labels). 6. Develop with 2-Methyl-2-Pentanone (methyl isobutyl ketone, or MIBK) diluted with 2-Propanol as 1 : 3 for 2 min. 30 s. 7. Rinse with 2-Propanol and blow-dry with nitrogen. The two layers of PMMA are deliberately chosen such that the lower resolution PMMA resides on the bottom so that when developed, there is an overhang of the higher resolution PMMA above it where the pattern was written. This helps substantially in the later liftoff step, as the solvent can make its way into the gaps between the metal and the lower portion of the lip in order to dissolve both layers of PMMA. After the e-beam exposure and development, a 50-nm-thick layer of chromium is deposited and warm 1-Methyl-2-Pyrrolidone is used to lift off the PMMA resist bilayer.

226

8. Fabrication Techniques for Microresonators

Growth substrate Upper/lower cladding Core (a) Epitaxially grown wafer.

Resist and/or hard mask

(b) Pattern etch mask for resonator and bus waveguides.

Coupling gap Bus

Resonator

Bus

(c) Etch resonator and bus waveguides. This may be followed by encapsulation in a polymer, spin-on glass, polyimide, etc., to protect the waveguides.

Fig. 8.2. Process flow for the fabrication of laterally coupled microresonators.

8.7 Polymer-Bonded, III–V Vertically Coupled Passive Microresonators

227

The pattern is then transferred to the underlying silicon dioxide by dryetching in a trifluoromethane-oxygen plasma to obtain the hard mask for patterning the bus and resonator [Fig. 8.2b]. The InP and quaternary layers are etched by a methane-hydrogen plasma to a depth of 3.2 µm with the Cr-SiO2 mask. The mask is subsequently stripped with buffered hydrofluoric acid [Fig. 8.2c]. Finally, the device is prepared for testing by thinning the chip in bromine-methanol to enable good cleavage, followed by cleaving into bars, and mounting on brass mounts. A scanning electron micrograph of a waveguide cross-section is shown in Fig. 8.3, and a fabricated device is shown in Fig. 8.4. The waveguides are fused over a short region (<1 µm) because of imperfect liftoff during the mask definition. The radius of the curved sections is 2.25 µm, the straight sections are 10 µm long, the waveguide width is 0.5 µm, and the width of the coupling gap is 0.2 µm. The bus waveguide is tapered to a width of 3 µm away from the coupling region to allow increased coupling of light to and from optical fibers. The waveguides have been etched to a depth of 3.2 µm chosen to minimize losses resulting from leakage to the substrate. The pedestal design provides a high lateral index contrast allowing small bends without substantial bending loss [122]. Earlier SEM images of GaAs-AlGaAs microresonators made by the UMD group are shown in Fig. 8.5.

8.7 Polymer-Bonded, III–V Vertically Coupled Passive Microresonators Vertically coupled III–V microresonators can be fabricated with polymer wafer-bonding, direct-wafer bonding, and regrowth. We will examine the latter two methods in the following sections; here, we describe polymer-bonded vertically coupled microresonators, as demonstrated by Absil et al. [58] and Grover et al. [62]. The specific case we examine here is for InP following Grover [174]. The process begins with a SiO2 -coated waveguide stack grown in reverse order (top layers first) over an etch-stop. The etch-stop will later enable substrate removal by selective chemical etching. The general patterning method proceeds as follows: We use a 10× i-line stepper for photolithography and use the resist mask to transfer the features into a 400-nm-thick SiO2 layer with a trifluoromethane-oxygen plasma in a reactive ion etching system. Next, we use the patterned SiO2 as a mask to define the waveguides in GaInAsP-InP with a methane–hydrogen–argon plasma-based reactive ion etching process for smooth and vertical sidewalls [286]. First alignment keys are etched to a depth of 2 µm to reach the GaInAs layer; these keys are used later for processing both sides without requiring infrared back-side alignment (especially useful if the equipment

228

8. Fabrication Techniques for Microresonators

(a)

(b)

Fig. 8.3. Scanning electron micrographs of waveguide cross-section. (a) 0.5-µm-wide waveguide in the filter section, (b) 3-µm-side waveguide at the input and output(s). The dark bands are the quaternary layers that comprise the core of the waveguide.

8.7 Polymer-Bonded, III–V Vertically Coupled Passive Microresonators

229

Fig. 8.4. Scanning electron micrograph of a laterally coupled microring notch filter. available does not have IR backside illumination). Afterward, the waveguide layer is etched to a depth of 0.9 µm, and the sample is bonded to a GaAs (or InP) transfer substrate using ∼1–2-µm-thick benzocyclobutene (BCB), following the process described by Sakamoto et al. [290]. GaAs is preferred as the transfer substrate due to better mechanical properties

230

8. Fabrication Techniques for Microresonators

(a) Single-microring resonator.

(b) Serially coupled double-microring resonator.

(c) Serially coupled triple-microring resonator.

Fig. 8.5. GaAs-AlGaAs rings made by the group at the University of Maryland, College Park. Copyright Philippe P. Absil, 2000, used with permission. than InP. All but 100 µm of the growth substrate is removed by chemomechanical polishing [276]. For the remaining growth substrate, we use H3 PO4 :HCl (1:1), which selectively etches the remaining InP, stopping at the GaInAs layer [277]. Finally, the etch-stop layer is removed with

8.7 Polymer-Bonded, III–V Vertically Coupled Passive Microresonators

231

Growth substrate Etch stop Upper/lower cladding Core Midlayer (coupling layer, cladding) (a) Epitaxially grown wafer.

Resist and/or hard mask

(b) Pattern etch mask for bus waveguides.

Fig. 8.6. Process flow for the fabrication of polymer-bonded, vertically coupled microresonators. (Continued on next page.)

H2 O2 :H2 SO4 :H2 O (1:1:10), which selectively etches GaInAs over InP. This step clears the other side of the epilayer for processing. The etchedthrough alignment keys are used to align the top layer to the lower layer. The top layer is then etched to a depth of 0.9 µm. A 200-nm mid-layer is left between the waveguides to decrease losses caused by leakage to the substrate. Finally, the sample is encapsulated in BCB to ensure

232

8. Fabrication Techniques for Microresonators

Bus

Bus

(c) Etch bus waveguides.

Bus

Bus Polymer glue

Transfer substrate (d) Flip patterned chip and bond to transfer substrate.

Fig. 8.6. (Continued from previous page.) Process flow for the fabrication of polymer-bonded, vertically coupled microresonators. (Continued on next page.)

refractive-index profile symmetry, thinned, and cleaved producing highquality facets. A microring made on the chip is shown in Fig. 8.7. The rough feature near the top is etch residue, left behind after removal of SiO2 using a CHF3 -O2 plasma. (Note: RG subsequently found that there is no etch residue if the SiO2 is removed with buffered hydrofluoric acid, and adopted it as the method for such removal.) The use of BCB provides a low-refractive-index (1.53–1.55) layer between the GaAs transfer substrate and the epilayer, making tightly

8.7 Polymer-Bonded, III–V Vertically Coupled Passive Microresonators

233

Polymer glue

Transfer substrate (e) Remove growth substrate and etch stop to expose “other”side of epilayer for processing.

Polymer encapsulation Resonator

Bus

Bus Polymer glue

Transfer substrate (f) Pattern etch mask for resonator, etch resonator, and spin-on encapsulation to protect the device and to get a symmetric refractive index structure.

Fig. 8.6. (Continued from previous page.) Process flow for the fabrication of polymer-bonded, vertically coupled microresonators.

confined waveguides possible on both device layers. Also, the BCB encapsulation provides the symmetric refractive-index structure that is required for identical microrings on both layers, allowing multiring resonators with microrings on both layers. Leaving a thin layer of InP coupling layer (0.2 µm) reduces the effect of layer-to-layer misalignment [58]. This design also provides added mechanical strength to the structure for the BCB encapsulation. Finally, the alignment scheme, using a stepper and etched-through alignment keys, provides a high-quality layer-to-layer alignment with errors smaller than 0.25 µm (measured from on-chip verniers; Fig. 8.8). The fabricated single-microring optical channel-dropping filter is shown in Fig. 8.9. Residuals from removal of

234

8. Fabrication Techniques for Microresonators

Fig. 8.7. Dry etched microring.

the growth substrate caused the upper level to appear dirty. A cross section for a similar device made by Absil et al. [58], showing waveguides on either side of the epilayer, is shown in Fig. 8.10.

8.8 Active III–V Laterally Coupled Microresonators Here is an example of the process flow for active laterally coupled microresonators, adapted from [174]. The waveguide stack is MBE-grown p-InP/GaInAsP-MQW/n-InP, with the central MQW region composed of a

8.8 Active III–V Laterally Coupled Microresonators

235

Fig. 8.8. Alignment accuracy measured from on-chip verniers.

Fig. 8.9. Optical micrograph of a fabricated single-microring add–drop filter with 5-µm-radius ring on the lower level.

superlattice of quantum wells designed for maximizing the electro-optic coefficient of the waveguide. The fabrication is similar to that for passive laterally coupled microresonators, except that the waveguides are etched to a depth of 4.5 µm. Since the device needed planarization for deposition of contacts, it was coated with a thin layer of SiO2 (∼70 nm) to help with the adhesion of the BCB planarization layer. Prior to spin-coating the BCB, an adhesion promoter (AP3000 from Dow Chemical) was spun on at 2000 RPM for 30 s. The BCB used was Cyclotene 3022-46 from Dow

236

8. Fabrication Techniques for Microresonators

Fig. 8.10. Optical micrograph of the cross section of a polymer-bonded vertically coupled waveguide device. Copyright Philippe P. Absil, 2000, used with permission. Chemical, spun-on at 4000 RPM for 60 s. This was followed by etchback of benzocyclobutene and SiO2 to expose the tops of the waveguides. For the top (p-side) contact, Ti-Pt-Au was deposited. The chip was thinned via an abrasive (alumina) to enable good cleavage and the bottom (n-side) contact of Pd-Sn-Au-Pd-Au was deposited. Although the n-contact usually requires annealing [291], the BCB cannot tolerate the high temperature (∼400◦ C) required; this can be resolved by using other planarization materials like spin-on glasses or polyimides. Finally, the chip was cleaved into bars and mounted on oxygen-free copper mounts with a silver epoxy. In the device described here, Grover et al. deposited the top-side electrode globally (on both the bus and the resonator). This led to unwanted electro-absorption in the bus waveguide due to the QCSE when a reverse bias was applied. Thus, for improved performance, the bus and ring should be isolated electrically. The cross-section schematic for a device where the bus and resonator are isolated electrically is illustrated in Fig. 8.11. A scanning electron micrograph of a fabricated device prior to planarization is shown in Fig. 8.12. The microring is in the shape of a racetrack. The waveguides are fused over the short coupling region (1 µm) because of imperfect liftoff during the mask definition. The radius of the curved sections is 2.25 µm, whereas the straight sections are 1 µm long, and the waveguide width is 0.5 µm. The designed width of the coupling gap was 0.1 µm; here, since the bus and ring are fused, the coupling region has a width of 1.1 µm. The bus waveguide is tapered to a width of 3 µm away from the coupling region to allow easier coupling of light to and from optical fibers. The waveguides have been etched to a depth of

8.8 Active III–V Laterally Coupled Microresonators

237

Coupling gap

p-InP

Bus

n-InP

Resonator

Contact Planarization i-GaInAsP/InP MQW SL core InP cladding Growth substrate Fig. 8.11. Cross section schematic of an active laterally coupled resonator.

Fig. 8.12. Scanning electron micrograph of fabricated device prior to planarization and metallization.

238

8. Fabrication Techniques for Microresonators

Fig. 8.13. Cross section of input–output waveguide in fabricated device. The ridge in the center is the waveguide, the material on either side is the BCB planarization layer, and the contact is on the top. ∼4.5 µm. The tall, grass-like structures on either side of the waveguide are probably due to contaminants in the reaction chamber. The “grass” is not very close to the sidewalls, possibly because both the Cr-SiO2 -mask and the particles that caused micro-masking may have had the same sign of charge. Fig. 8.13 shows the cross section of the input–output waveguide of a fabricated device with the BCB planarization and Ti-Pt-Au top contact. As can be observed in the picture, the planarization layer began to detach from the waveguide sidewall in this particular device. This problem is serious and can severely limit the yield and reliability in a manufacturing environment, as it can create a short circuit around the diode. However, it can be addressed by using adhesion promoters, planarization materials that have better adhesion (say, polyimides), or by increasing the thickness of the SiO2 deposited to improve the adhesion of the benzocyclobutene. If the SiO2 layer were thick enough, it would prevent the diode from being short-circuited.

8.9 Other Configurations Grover et al. chose to use the lateral coupling approach for the device described in this chapter because of the simplicity of fabrication. Real-world applications typically will require the use of vertical coupling so that the bus and the ring can be of different bandgaps allowing for active-passive

8.9 Other Configurations

239

device integration. Integration of these components could be achieved by making electro-optic modulators on the same side as the bus and changing the bandgap of ring with quantum-well intermixing. Alternatively,

(a)

(b)

Fig. 8.14. Two possible configurations for active, vertically coupled microring resonator filters. The active core could enable devices like lasers, amplifiers, and electroabsorption and electro-optic modulators.

240

8. Fabrication Techniques for Microresonators

(a)

Fig. 8.15. (a) Rabiei’s process for the fabrication of large polymer microrings. (Continued on next page.)

with the direct-wafer-bonding approach, two wafer bonding steps could be used to make the electro-optically tuned components on the third layer. Two possible configurations for active vertically coupled rings are shown in Fig. 8.14. In each case, the p- and n- regions can be interchanged. The p-contacts in each case require a planarization step. The planarization and passivation layers are not shown for clarity. In Fig. 8.14(a), the wafer bonding is achieved with a suitable glue, such as BCB (not shown).

8.9 Other Configurations

241

Teflon

Si

SU-8

SU-8

Spin coating, Baking, RIE

UV 15 Protection Layer Teflon

Si

Teflon

Si

Patterning RIE Etching Spin coating, UV Cure, Baking, RIE

Teflon SU-8

SU-8

UV 15 Protection Layer

UV 15 Protection Layer

Teflon

Teflon

Si

Si Spin coating

Spin coating, UV cure, Baking

SU-8

UFC 170

UV 15 Protection Layer

Teflon

Teflon

SU-8

SU-8

Si

UV 15 Protection Layer Teflon

Pattering, UV Exposure

Si

SU-8

SU-8

RIE Etching

UV 15 Protection Layer Teflon

Teflon

Si

UFC170 SU-8

SU-8

UV 15 Protection Layer Teflon

Spin coating, Baking, RIE

Si

(b)

Fig. 8.15. (Continued from previous page.) (b) Rabiei’s process for the fabrication of small polymer microrings. Adapted from ref. [292], copyright 2003 Payam Rabiei, used with permission.

In Fig. 8.14(b), Grover et al. show the case for direct-wafer-bonding. An alternative method of achieving a conducting interface would be to planarize the bus level, etchback, and bond with a conducting material, such as Pb-Sn [167] Pd [168], Au-Sn [169], or Pd-In [170, 171].

242

8. Fabrication Techniques for Microresonators

8.10 Polymer Microrings Rabiei et al. use a multitude of polymers to fabricate active and passive polymer microrings [292]. In addition, the group at the University of Maryland, College Park, has demonstrated BCB microrings [293]. In Fig. 8.15, we show the process flow for two of Rabiei’s devices; his active rings are fabricated in a similar manner, except with contacts on either side (above and below) the resonators.

8.11 Summary This chapter examined the key elements associated with the fabrication of optical microresonators and associated devices. There are, of course, many demonstrations of these devices from various groups, each employing slightly different approaches. For example, 1. The group at University of Illinois, Urbana-Champaign [74], in collaboration with researchers from Sarnoff, uses MMI-coupling for laterally coupled microresonators. 2. The group at Twente University [294] uses SiON waveguides with a combination of low-pressure and plasma-enhanced chemical vapor deposition, as does the group at Politecnico di Milano [295]. 3. Little Optics (now Infinera) uses glass-based materials to make buried bus (and ring) resonators [37, 287]. The fabrication of microrings, microdisks, and other optical devices employing high index contrasts such as photonic crystals continues to be extremely challenging. At the time of writing, it has been over 15 years since the first microdisk lasers were demonstrated, yet they are only now finding their way into commercial products.

Bibliography

1. Lord Rayleigh, “The problem of the whispering gallery,” Philosophical Magazine, vol. xx, pp. 1001–1004, 1910. 2. E. A. J. Marcatili, “Bends in optical dielectric guides,” The Bell System Technical Journal, vol. 48, pp. 2103–2132, 1969. 3. H. P. Weber and R. Ulrich, “A thin-film ring laser,” Applied Physics Letters, vol. 19, pp. 38–40, 1971. 4. R. Ulrich and H. P. Weber, “Unidirectional thin-film ring laser,” Applied Physics Letters, vol. 20, pp. 38–40, 1972. 5. R. Ulrich and H. P. Weber, “Solution-deposited thin films as passive and active light-guides,” Applied Optics, vol. 11, pp. 428–434, 1972. 6. J. Haavisto and G. A. Pajer, “Resonance effects in low-loss ring waveguides,” Optics Letters, vol. 5, pp. 510–512, 1980. 7. L. F. Stokes, M. Chodorow, and H. J. Shaw, “All-single-mode fiber resonator,” Optics Letters, vol. 7, pp. 288–290, 1982. 8. R. G. Walker and C. D. W. Wilkinson, “Integrated optical ring resonators made by silver ion-exchange in glass,” Applied Optics, vol. 22, pp. 1029– 1035, 1983. 9. A. Mahapatra and J. M. Connors, “High finesse ring resonators fabrication and analysis,” in Integrated Optical Circuit Engineering, Innsbruck, Austria, 1986. 10. J. M. Connors and A. Mahapatra, “High finesse ring resonators made by silver ion exchange in glass,” Journal of Lightwave Technology, vol. LT-5, pp. 1686–1689, 1987. 11. K. Honda, E. M. Garmire, and K. E. Wilson, “Characteristics of an integrated optics ring resonator fabricated in glass,” Journal of Lightwave Technology, vol. LT-2, pp. 714–719, 1984. 12. A. Naumaan and J. T. Boyd, “Ring resonator fabricated in phosphosilicate glass films deposited by chemical vapor deposition,” Journal of Lightwave Technology, vol. LT-4, pp. 1294–1303, 1986. 13. K.-H. Tietgen, “Tunable integrated optical ring resonator,” in Topical Meeting on Integrated and Guided Wave Optics, Kissimee, FL, 1984. 14. A. Mahapatra and W. C. Robinson, “Integrated-optic ring resonators made by proton exchange in lithium niobate,” Applied Optics, vol. 24, pp. 2285– 2286, 1985. 15. K. Oda, N. Takato, and H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” Journal of Lightwave Technology, vol. 9, pp. 728–736, 1991. 16. T. Kominato, Y. Ohmori, N. Takato, H. Okazaki, and M. Yasu, “Ring resonators composed of GeO2 -doped silica waveguides,” Journal of Lightwave Technology, vol. 10, pp. 1781–1788, 1992.

244

Bibliography

17. W. J. Wang, S. Honkanen, S. I. Najafi, and A. Tervonen, “New integrated optical ring resonator in glass,” Electronics Letters, vol. 28, pp. 1967–1968, 1992. 18. S. T. Chu, B. E. Little, P. Wugen, T. Kaneko, and Y. Kokubun, “Cascaded microring resonators for crosstalk reduction and spectrum cleanup in adddrop filters,” IEEE Photonics Technology Letters, vol. 11, pp. 1423–1425, 1999. 19. S. T. Chu, B. E. Little, W. Pan, and Y. Kokubun, “A cross-grid array of microresonators for very large scale integrated photonic circuits,” in Conference on Lasers and Electro-Optics, San Francisco, CA, 1999. 20. S. T. Chu, P. Wugan, S. Suzuki, B. E. Little, S. Sato, and Y. Kokubun, “Temperature insensitive vertically coupled microring resonator add/drop filters by means of a polymer overlay,” IEEE Photonics Technology Letters, vol. 11, pp. 1138–1140, 1999. 21. B. E. Little, S. T. Chu, W. Pan, D. Ripin, T. Kaneko, Y. Kokubun, and E. P. Ippen, “Vertically coupled glass microring resonator channel dropping filters,” IEEE Photonics Technology Letters, vol. 11, pp. 215–217, 1999. 22. S. T. Chu, B. E. Little, W. Pan, T. Kaneko, and Y. Kokubun, “Second order filter response from parallel coupled glass microring resonators,” IEEE Photonics Technology Letters, vol. 11, pp. 1426–1428, 1999. 23. S. T. Chu, W. Pan, S. Sato, T. Kaneko, B. E. Little, and Y. Kokubun, “Wavelength trimming of a microring resonator filter by means of a UV sensitive polymer overlay,” IEEE Photonics Technology Letters, vol. 11, pp. 688–690, 1999. 24. Y. Kokubun, S. Kubota, and S. T. Chu, “Polarisation-independent vertically coupled microring resonator filter,” Electronics Letters, vol. 37, pp. 90–92, 2001. 25. Y. Yanagase, S. Suzuki, Y. Kokubun, and S. T. Chu, “Box-like filter response by vertically series coupled microring resonator filter,” in European Conference on Optical Communications, Amsterdam, The Netherlands, 2001. 26. J. Bismuth, P. Gidon, F. Revol, and S. Valette, “Low-loss ring resonators fabricated from silicon based integrated optics technologies,” Electronics Letters, vol. 27, pp. 722–724, 1991. 27. S. Suzuki, K. Shuto, and Y. Hibino, “Integrated-optic ring resonators with two stacked layers of silica waveguide on Si,” IEEE Photonics Technology Letters, vol. 4, pp. 1256–1258, 1992. 28. T. Kominato, Y. Hibino, and K. Onose, “Silica-based finesse-variable ring resonator,” IEEE Photonics Technology Letters, vol. 5, pp. 560–562, 1993. 29. S. Suzuki, K. Oda, and Y. Hibino, “Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz,” Journal of Lightwave Technology, vol. 13, pp. 1766–1771, 1995. 30. P. Katila, P. Heimala, and J. Aarnio, “Thermo-optically controlled ring resonator on silicon,” Electronics Letters, vol. 32, pp. 1005–1006, 1996. 31. J. S. Foresi, B. E. Little, G. Steinmeyer, E. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Si/SiO2 micro-ring resonator optical add/drop filters,” in Conference on Lasers and Electro-Optics, Baltimore, MD, 1997. 32. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technology Letters, vol. 10, pp. 549–551, 1998.

Bibliography

245

33. D. R. Lim, B. E. Little, K. K. Lee, M. Morse, H. H. Fujimoto, H. A. Haus, and L. C. Kimerling, “Micron-sized channel dropping filters using silicon waveguide devices,” in Optical Devices for Fiber Communication, Boston, MA, 1999. 34. D. J. W. Klunder, F. S. Tan, T. van der Veen, H. F. Bulthuis, H. J. W. M. Hoekstra, and A. Driessen, “Design and characterization of waveguide-coupled cylindrical microring resonators in Si3 N4 ,” in IEEE Lasers and Electro-Optics Society, 13th Annual Meeting, Rio Grande, Puerto Rico, 2000. 35. D. J. W. Klunder, E. Krioukov, F. S. Tan, T. van der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, “Vertically and laterally waveguide-coupled cylindrical microresonators in Si3 N4 on SiO2 technology,” Applied Physics B (Lasers and Optics), vol. B73, pp. 603–608, 2001. 36. S. Suzuki, Y. Hatakeyama, Y. Kokubun, and S. T. Chu, “Precise control of wavelength channel spacing of microring resonator add-drop filter array,” Journal of Lightwave Technology, vol. 20, pp. 745–750, 2002. 37. B. E. Little, “A VLSI photonics platform,” in Optical Fiber Communication, Atlanta, GA, 2003. 38. P. Rabiei and W. H. Steier, “Micro-ring resonators using polymer materials,” in Conference on Lasers and Electro Optics, Baltimore, MD, 2001. 39. P. Rabiei, W. H. Steier, C. Zhang, C.-G. Wang, and H. J. Lee, “Polymer microring modulator with 1 THz FSR,” in Conference on Lasers and Electro-Optics, Long Beach, CA, 2002. 40. P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer micro-ring filters and modulators,” Journal of Lightwave Technology, vol. 20, pp. 1968–1975, 2002. 41. W. Y. Chen, R. Grover, T. A. Ibrahim, V. Van, and P.-T. Ho, “Compact singlemode benzocyclobutene microracetrack resonators,” in Integrated Photonics Research, Washington, DC, 2003. 42. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Applied Physics Letters, vol. 60, pp. 289–291, 1992. 43. D. Y. Chu, M. K. Chin, N. J. Sauer, Z. Xu, T. Y. Chang, and S. T. Ho, “1.5 µm InGaAs/InAlGaAs quantum-well microdisk lasers,” IEEE Photonics Technology Letters, vol. 5, pp. 1353–1355, 1993. 44. C. R. Abernathy, S. J. Pearton, J. D. MacKenzie, J. R. Mileham, S. R. Bharatan, V. Krishnamoorthy, K. S. Jones, M. Hagerott-Crawford, R. J. Shul, S. P. Kilcoyne, J. M. Zavada, D. Zhang, and R. M. Kolbas, “Growth and fabrication of GaN-InGaN microdisk laser structures,” Solid-State Electronics, vol. 39, pp. 311–313, 1996. 45. T. Baba, P. Fujita, A. Sakai, M. Kihara, and R. Watanabe, “Lasing characteristics of GaInAsP-InP strained quantum-well microdisk injection lasers with diameter of 2 − 10 µm,” IEEE Photonics Technology Letters, vol. 9, pp. 878–880, 1997. 46. L. Dai, B. Zhang, R. A. Mair, K. Zeng, J. Lin, H. Jigang, A. Botchkarev, W. Kim, H. Morkoc, and M. A. Khan, “Optical properties and resonant modes in GaN/AlGaN and InGaN/GaN multiple quantum well microdisk cavities,” in Semiconductor Lasers III, Beijing, China, 1998. 47. R. A. Mair, K. C. Zeng, J. Y. Lin, H. X. Jiang, B. Zhang, L. Dai, A. Botchkarev, W. Kim, H. Morkoc, and M. A. Khan, “Optical modes within III–nitride multiple quantum well microdisk cavities,” Applied Physics Letters, vol. 72, pp. 1530–1532, 1998.

246

Bibliography

48. D. V. Tishinin, P. D. Dapkus, A. E. Bond, I. Kim, C. K. Lin, and J. O’Brien, “Vertical resonant couplers with precise coupling efficiency control fabricated by wafer bonding,” IEEE Photonics Technology Letters, vol. 11, pp. 1003–1005, 1999. 49. L. Djaloshinski and M. Orenstein, “Disk and ring microcavity lasers and their concentric coupling,” IEEE Journal of Quantum Electronics, vol. 35, pp. 737–744, 1999. 50. N. B. Rex, R. K. Chang, and L. J. Guido, “Threshold minimization and directional laser emission from GaN microdisks,” in Laser Resonators, San Jose, CA, 2000. 51. D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, K. A. Stair, S. T. Ho, and R. C. Tiberio, “Waveguide-coupled AlGaAs/GaAs microcavity ring and disk resonators with high finesse and 21.6 nm free spectral range,” Optics Letters, vol. 22, pp. 1244–1246, 1997. 52. D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, K. A. Stair, S. T. Ho, and R. C. Tiberio, “Temperature tuning of microcavity ring and disk resonators at 1.5 µm,” in IEEE Lasers and Electro-Optics Society, 10th Annual Meeting, San Francisco, CA, 1997. 53. J. V. Hryniewicz, P. P. Absil, B. E. Little, and P.-T. Ho, “Semiconductor microring resonators for integrated photonics,” in Integrated Photonics Research, Québec City, Canada, 2000. 54. J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, L. G. Joneckis, and P.-T. Ho, “Microring resonator notch filters,” in Conference on Lasers and Electro-Optics, San Francisco, CA, 2000. 55. J. V. Hryniewicz, P. P. Absil, B. E. Little, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photonics Technology Letters, vol. 12, pp. 320–322, 2000. 56. P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P.-T. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Optics Letters, vol. 25, pp. 554–556, 2000. 57. P. P. Absil, J. V. Hryniewicz, B. E. Little, R. A. Wilson, L. G. Joneckis, and P.-T. Ho, “Compact microring notch filters,” IEEE Photonics Technology Letters, vol. 12, pp. 398–400, 2000. 58. P. P. Absil, J. V. Hryniewicz, B. E. Little, F. G. Johnson, and P.-T. Ho, “Vertically coupled microring resonators using polymer wafer bonding,” IEEE Photonics Technology Letters, vol. 13, pp. 49–51, 2001. 59. R. Grover, P. P. Absil, V. Van, J. V. Hryniewicz, B. E. Little, O. S. King, F. G. Johnson, L. C. Calhoun, and P.-T. Ho, “Vertically coupled GaAs-AlGaAs and GaInAsP-InP microring resonators,” in Optical Fiber Communication, Anaheim, CA, 2001. 60. P. P. Absil, J. V. Hryniewicz, B. E. Little, F. G. Johnson, and P.-T. Ho, “Wavelength selective mirror using notched microring resonators,” in IEEE Lasers and Electro-Optics Society, 14th Annual Meeting, San Diego, CA, 2001. 61. V. Van, T. A. Ibrahim, P. P. Absil, R. Grover, J. V. Hryniewicz, B. E. Little, F. G. Johnson, and P.-T. Ho, “Periodically-coupled GaAs/AlGaAs microring array filter with wide free spectral range,” in Integrated Photonics Research, Monteray, CA, 2001. 62. R. Grover, P. P. Absil, V. Van, J. V. Hryniewicz, B. E. Little, O. S. King, L. C. Calhoun, F. G. Johnson, and P.-T. Ho, “Vertically coupled GaInAsP-InP microring resonators,” Optics Letters, vol. 26, pp. 506–508, 2001. 63. R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, “Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters,” Journal of Lightwave Technology, vol. 20, pp. 900–905, 2002.

Bibliography

247

64. K. Djordjev, S.-J. Choi, S.-J. Choi, and P. D. Dapkus, “High-Q vertically coupled InP microdisk resonators,” IEEE Photonics Technology Letters, vol. 14, pp. 331–332, 2002. 65. K. Djordjev, S.-J. Choi, S.-J. Choi, and P. D. Dapkus, “Microdisk tunable resonant filters and switches,” IEEE Photonics Technology Letters, vol. 14, pp. 828–830, 2002. 66. E. A. J. Marcatili, “Improved relations describing directional control in electromagnetic wave guidance,” The Bell System Technical Journal, vol. 48, pp. 2161–2188, 1969. 67. M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE Journal of Quantum Electronics, vol. QE11, pp. 75–83, 1975. 68. M. Heiblum and J. H. Harris, “Correction to ‘Analysis of curved optical waveguides by conformal transformation’,” IEEE Journal of Quantum Electronics, vol. 12, pp. 313–313, 1976. 69. D. Marcuse, “Bend loss of slab and fiber modes computed with diffraction theory,” IEEE Journal of Quantum Electronics, vol. 29, pp. 2957–2961, 1993. 70. G. Griffel, J. H. Abeles, R. J. Menna, A. M. Braun, J. C. Connolly, and M. King, “Low-threshold InGaAsP ring lasers fabricated using bi-level dry etching,” IEEE Photonics Technology Letters, vol. 12, pp. 146–148, 2000. 71. D. G. Rabus and M. Hamacher, “MMI-coupled ring resonators in GaInAsPInP,” IEEE Photonics Technology Letters, vol. 13, pp. 812–814, 2001. 72. D. G. Rabus, M. Hamacher, and H. Heidrich, “Active and passive microring resonator filter applications in GaInAsP/InP,” in Indium Phosphide and Related Materials, Nara, Japan, 2001. 73. D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High-Q channel dropping filters using ring resonators with integrated SOAs,” IEEE Photonics Technology Letters, vol. 14, pp. 1442–1444, 2002. 74. S. L. Rommel, J.-H. Jang, W. Lu, G. Cueva, L. Zhou, I. Adesida, G. A. Pajer, R. W. Whaley, A. Lepore, Z. Schellanbarger, and J. H. Abeles, “Effect of H2 on the etch profile of InP/InGaAsP alloys in Cl2 /Ar/H2 inductively coupled plasma reactive ion etching chemistries for photonic device fabrication,” Journal of Vacuum Science & Technology B, vol. 20, pp. 1327–1330, 2002. 75. R. Grover, T. A. Ibrahim, T. N. Ding, L.-C. Kuo, S. Kanakaraju, L. C. Calhoun, and P.-T. Ho, “Ultracompact single-mode GaInAsP-InP microracetrack resonators,” in Integrated Photonics Research, Washington, DC, 2003. 76. R. Grover, T. A. Ibrahim, T. N. Ding, Y. Leng, L.-C. Kuo, S. Kanakaraju, K. Amarnath, L. C. Calhoun, and P.-T. Ho, “Laterally coupled InP-based single-mode microracetrack notch filter,” IEEE Photonics Technology Letters, 2003. 77. R. Grover, T. A. Ibrahim, S. Kanakaraju, M. L. Lucas, L. C. Calhoun, and P.-T. Ho, “Tunable GaInAsP-InP optical microresonator notch filter,” submitted to IEEE Photonics Technology Letters, 2003. 78. A. Szoke and V. Daneu, “Bistable optical element and its applications,” Applied Physics Letters, vol. 15, pp. 376–379, 1969. 79. S. L. McCall, H. M. Gibbs, G. G. Churchill, and T. N. C. Venkatesan, “Optical transistor and bistability,” Bulletin of the Americal Physical Society, vol. 20, pp. 636, 1975. 80. H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometer,” Physical Review Letters, vol. 36, pp. 1135–1138, 1976. 81. J. H. Marburger and F. S. Felber, “Theory of a lossless nonlinear Fabry–Perot interferometer,” Physical Review A, vol. 17, pp. 335, 1978.

248

Bibliography

82. D. A. B. Miller and S. D. Smith, “Two-beam optical signal amplification and bistability in InSb,” Optics Communications, vol. 31, pp. 101–104, 1979. 83. D. A. B. Miller, “Refractive Fabry–Perot bistability with linear absorption: Theory of operation and cavity optimization,” IEEE Journal of Quantum Electronics, vol. 17, pp. 306–311, 1981. 84. R. M. Shelby, M. D. Levenson, D. F. Walls, A. Aspect, and G. J. Milburn, “Generation of squeezed states of light with a fiber-optic ring interferometer,” Physical Review A, vol. 33, pp. 4008–4025, 1986. 85. R. M. Shelby, M. D. Levenson, and S. H. Permutter, “Bistability and other effects in a nonlinear fiber-optic ring resonator,” Journal of the Optical Society of America B, vol. 5, pp. 347–357, 1988. 86. V. B. Braginsky and V. S. Ilchenko, “Properties of optical dielectric microresonators,” Soviet Physics Doklady, vol. 32, pp. 306–307, 1987. 87. V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Physics Letters A, vol. 137, pp. 393–397, 1989. 88. D. W. Vernooy, A. Furusawa, N. P. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high-Q whispering gallery modes,” Physical Review A, vol. 57, pp. R2293–R2296, 1998. 89. J. Popp, M. H. Fields, and R. K. Chang, “Q-switching by saturable absorption in microdroplets: elastic scattering and laser emission,” Optics Letters, vol. 22, pp. 1296–1298, 1997. 90. F. C. Blom, D. R. van Dijk, H. J. Hoekstra, A. Driessen, and T. J. A. Popma, “Experimental study of integrated-optics microcavity resonators: Toward an all-optical switching device,” Applied Physics Letters, vol. 71, pp. 747– 749, 1997. 91. F. C. Blom, H. Kelderman, H. J. W. M. Hoekstra, A. Driessen, T. J. A. Popma, S. T. Chu, and B. E. Little, “A single channel dropping filter based on a cylindrical microresonator,” Optics Communications, vol. 167, pp. 77–82, 1999. 92. A. T. Rosenberger, “Nonlinear optical effects in the whispering-gallery modes of microspheres,” in Operational Characteristics and Crystal Growth of Nonlinear Optical Materials, R. B. Lal and D. O. Frazier, Eds., vol. 3793, pp. 179–186. SPIE Proceedings, 1999. 93. D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, “Lowering of threshold conditions for nonlinear effects in a microsphere,” Physical Review A, vol. 53, pp. 3565–3572, 1996. 94. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature, vol. 415, pp. 621– 623, 2002. 95. P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A Quantum Dot Single-Photon Turnstile Device,” Science, vol. 290, pp. 2282–2285, 2000. 96. A. Kiraz, P. Michler, C. Becher, B. Gayral, A. Imamoglu, L. Zhang, E. Hu, W. V. Schoenfeld, and P. M. Petroff, “Cavity Quantum Electrodynamics Using a Single InAs Quantum Dot in a Microdisk Structure,” Applied Physics Letters, vol. 78, pp. 3932–3934, 2001. 97. G. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego, CA, 3rd edition, 2001. 98. P. K. Tien, “Light waves in thin films and integrated optics,” Applied Optics, vol. 10, pp. 2395–2413, 1971. 99. H. Kogelnik and V. Ramaswamy, “Scaling rules for thin-film optical waveguides,” Applied Optics, vol. 13, pp. 1857–1862, 1974.

Bibliography

249

100. S. M. Saad, “Review of numerical methods for the analysis of arbitrarilyshaped microwave and optical dielectric waveguides,” IEEE Transactions on Microwave Theory and Techniques, vol. 33, pp. 894–899, 1985. 101. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” The Bell System Technical Journal, vol. 48, pp. 2071– 2102, 1969. 102. A. Kumar, K. Thyagarajan, and A. K. Ghatak, “Analysis of rectangular-core dielectric waveguides: an accurate perturbation approach,” Optics Letters, vol. 8, pp. 63–65, 1983. 103. J. Fox, Ed., Integrated circuits for the millimeter through optical frequency range. Polytechnic Press, 1970. 104. G. B. Hocker and W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Applied Optics, vol. 16, pp. 113–118, 1977. 105. J.-S. Lee and S.-Y. Shin, “On the validity of the effective-index method for rectangular dielectric waveguides,” Journal of Lightwave Technology, vol. 11, pp. 1320–1324, 1993. 106. C. M. Kim, B. G. Jung, and C. W. Lee, “Analysis of dielectric rectangular waveguide by modified effective-index method,” Electronics Letters, vol. 22, pp. 296–298, 1986. 107. Y. Cai, T. Mizumoto, and Y. Naito, “Perturbation feedback method for analysing rectangular dielectric waveguides,” Electronics Letters, vol. 26, pp. 521–523, 1990. 108. K. S. Chiang, “Analysis of rectangular dielectric waveguides: Effective-index method with built-in perturbation correction,” Electronics Letters, vol. 28, pp. 388–390, 1992. 109. J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” The Bell System Technical Journal, vol. 48, pp. 2133–2160, 1969. 110. J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Applied Physics, vol. 10, pp. 129–160, 1976. 111. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Transactions on Antennas and Propagation, vol. AP-14, pp. 302–307, 1966. 112. A. Taflove and S. C. Hagness, Computational Electrodynamics, The FiniteDifference Time-Domain Method, Artech House, Boston, MA, 2000. 113. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE Journal of Quantum Electronics, vol. 9, pp. 919–933, 1973. 114. J. E. Sipe and G. I. Stegeman, “Comparison of normal mode and total field analysis techniques in planar integrated optics,” Journal of the Optical Society of America, vol. 69, pp. 1676–1683, 1979. 115. H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” Journal of Lightwave Technology, vol. 5, pp. 16–23, 1987. 116. D. G. Hall, “A comment on the coupled-mode equations used in guidedwave optics,” Optics Communications, vol. 82, pp. 82–83, 1991. 117. H. A. Haus, Waves and Fields in Optoelectronics, Prentice-Hall, Englewood Cliffs, NJ, 1984. 118. F. de Fornel, Evanescent Waves: From Newtonian Optics to Atomic Optics, Springer-Verlag, New York, NY, 1st edition, 2001.

250

Bibliography

119. M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, Z. Ren, S. L. Wu, L. Wang, Y. G. Zhao, and S. T. Ho, “GaAs microcavity channel-dropping filter based on a race-track resonator,” IEEE Photonics Technology Letters, vol. 11, pp. 1620– 1622, 1999. 120. B. E. Little and S. T. Chu, “Estimating surface-roughness loss and output coupling in microdisk resonators,” Optics Letters, vol. 21, pp. 1390–1392, 1996. 121. B. E. Little, J. P. Laine, and S. T. Chu, “Surface-roughness-induced contradirectional coupling in ring and disk resonators,” Optics Letters, vol. 22, pp. 4–6, 1997. 122. V. Van, P. P. Absil, J. V. Hryniewicz, and P.-T. Ho, “Propagation loss in single-mode GaAs-AlGaAs microring resonators: Measurement and model,” Journal of Lightwave Technology, vol. 19, pp. 1734–1739, 2001. 123. J. Sarma and K. A. Shore, “Electromagnetic theory for optical disc resonators,” IEE Proceedings Journal (Optoelectronics), vol. 132, pp. 325–330, 1985. 124. A. W. Snyder and J. D. Love, “Tunnelling leaky modes on optical waveguides,” Optics Communications, vol. 12, pp. 326–328, 1974. 125. L. Collot, V. Lefevre-Seguin, M. Brune, J.-M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhysics Letters, vol. 23, pp. 327–334, 1993. 126. M. K. Chin and S. T. Ho, “Design and modelling of waveguide-coupled singlemode microring resonators,” Journal of Lightwave Technology, vol. 16, pp. 1433–1446, 1998. 127. W. Berglund and A. Gopinath, “WKB analysis of bend losses in optical waveguides,” Journal of Lightwave Technology, vol. 18, pp. 1161–1166, 2000. 128. M. Kuznetsov and H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE Journal of Quantum Electronics, vol. 19, pp. 1505–1514, 1983. 129. J. E. Heebner, R. W. Boyd, and Q. Park, “SCISSOR solitons and other propagation effects in microresonator modified waveguides,” Journal of the Optical Society of America B, vol. 19, pp. 722–731, 2002. 130. S. Arnold, C. T. Liu, W. B. Whitten, and J. M. Ramsey, “Room-temperature microparticle-based persistent spectral hole burning memory,” Optics Letters, vol. 16, pp. 420–422, 1991. 131. A. Serpengüzel, S. Arnold, and G. Griffel, “Excitation of resonances of microspheres on an optical fiber,” Optics Letters, vol. 20, pp. 654–656, 1995. 132. N. Dubreuil, J. C. Knight, D. K. Leventhal, V. Sandoghdar, J. Hare, and V. Lefevre, “Eroded monomode optical fiber for whispering-gallery mode excitation in fused-silica microspheres,” Optics Letters, vol. 20, pp. 813– 815, 1995. 133. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Optics Letters, vol. 21, pp. 453–455, 1996. 134. V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, “Narrow-line-width diode laser with a high-Q microsphere resonator,” Optics Communications, vol. 158, pp. 305–312, 1998. 135. M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes,” Journal of the Optical Society of America B, vol. 16, pp. 147–154, 1999. 136. V. Lefevre-Seguin, “Whispering-gallery mode lasers with doped silica microspheres,” Optical Materials, vol. 11, pp. 153–165, 1999.

Bibliography

251

137. M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-galley mode system,” Physical Review Letters, vol. 85, pp. 74–77, 2000. 138. J. P. Rezac and A. T. Rosenberger, “Locking a microsphere whisperinggallery mode to a laser,” Optics Express, vol. 8, pp. 605–610, 2001. 139. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature, vol. 421, pp. 925–928, 2003. 140. J. C. Knight, N. Dubreuil, V. Sandoghdar, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Mapping whispering-gallery modes in microspheres with a near-field probe,” Optics Letters, vol. 20, pp. 1515–1517, 1995. 141. J. C. Knight, N. Dubreuil, V. Sandoghdar, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Characterizing whispering-gallery modes in microspheres by direct observation of the optical standing-wave pattern in the near field,” Optics Letters, vol. 21, pp. 698–700, 1996. 142. M. L. M. Balistreri, D. J. W. Klunder, F. C. Blom, A. Driessen, H. W. J. M. Hoekstra, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Visualizing the whispering gallery modes in a cylindrical optical microcavity,” Optics Letters, vol. 24, pp. 1829–1831, 1999. 143. B. E. Little, J. P. Laine, and H. A. Haus, “Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators,” Journal of Lightwave Technology, vol. 17, pp. 704–715, 1999. 144. S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” Journal of Lightwave Technology, vol. 15, pp. 2154–2165, 1997. 145. M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Applied Physics Letters, vol. 85, pp. 3693–3696, 2004. 146. P. Rabiei, “Calculation of losses in micro-ring resonators with arbitrary refractive index or shape profile and its applications,” Journal of Lightwave Technology, vol. 23, pp. 1295–1301, 2005. 147. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Pertubation theory for Maxwell’s equations with shifting material boundaries,” Physical Review E, vol. 65, pp. 066611, 2002. 148. S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume current methods in photonic-crystal waveguides,” Applied Physics B, vol. 81, pp. 283–293, 2005. 149. C. Fabry and A. Perot, “A Multipass Interferometer,” Ann. Chim. Phys, vol. 16, pp. 115, 1899. 150. B. E. Little, S. T. Chu, and H. A. Haus, “Track changing by use of the phase response of microspheres and resonators,” Optics Letters, vol. 23, pp. 894– 896, 1998. 151. J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Optics Letters, vol. 24, pp. 847–849, 1999. 152. C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis, John Wiley & Sons, New York, NY, 1999. 153. J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE Journal of Quantum Electronics, vol. 40, pp. 726–730, 2004. 154. La difusion del la lumiere pa les atoines, vol. 2, 1927.

252

Bibliography

155. R. Kronig, “On the theory of the dispersion of X-rays,” Journal of the Optical Society of America, vol. 12, pp. 547–557, 1926. 156. M. Beck, I. A. Walmsley, and J. D. Kafka, “Group delay measurements of optical components near 800nm,” IEEE Journal of Quantum Electronics, vol. 27, pp. 2074–2081, 1991. 157. R. H. J. Kop, P. deVries, R. Sprik, and A. Lagendijk, “Kramers-Kronig relations for an interferometer,” Optics Communications, vol. 138, pp. 118–126, 1997. 158. L. J. Wang, “Causal ‘all-pass’ filters and Kramers-Kronig relations,” Optics Communications, vol. 213, pp. 27–32, 2002. 159. A. S. Sedra and K. C. Smith, Microelectronic Circuits, Oxford University Press, New York, NY, fourth edition, 1998. 160. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” Journal of Lightwave Technology, vol. 15, pp. 998–1005, 1997. 161. G. Barbarossa and A. M. Matteo, “Novel double-ring optical-guided-wave Vernier resonator,” IEE Proceedings (Optoelectronics), vol. 144, pp. 203–208, 1997. 162. M. A. Popovic, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Optics Letters, vol. 31, 2006. 163. B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, “Filter synthesis for periodically coupled microring resonators,” Optics Letters, vol. 25, pp. 344–346, 2000. 164. G. Griffel, “Vernier effect in asymmetrical ring resonator arrays,” IEEE Photonics Technology Letters, vol. 12, pp. 1642–1644, 2000. 165. A. Melloni, “Synthesis of a parallel-coupled ring-resonator filter,” Optics Letters, vol. 26, pp. 917–919, 2001. 166. F. Niklaus, P. Enoksson, E. Kälvesten, and G. Stemme, “Low-temperature full wafer adhesive bonding,” Journal of Micromechanics and Microengineering, vol. 11, pp. 100–107, 2001. 167. P. A. Moskowitz, H. L. Yeh, and S. K. Ray, “Thermal dry process soldering,” Journal of Vacuum Science & Technology A, vol. 4, pp. 838–840, 1985. 168. I. H. Tan, C. Reaves, A. L. Holmes Jr., E. L. Hu, J. E. Bowers, and S. DenBaars, “Low-temperature Pd bonding of III–V semiconductors,” Electronics Letters, vol. 31, pp. 588–589, 1995. 169. G. R. Dohle, T. J. Drabik, J. J. Callahan, and K. P. Martin, “Low temperature bonding of epitaxial lift off devices with AuSn,” IEEE Transactions on Components, Packaging, and Manufacturing Technology, Part B, vol. 19, pp. 575–580, 1996. 170. W. S. Wong, A. B. Wengrow, Y. Cho, A. Salleo, N. J. Quitoriano, N. W. Cheung, and T. Sands, “Integration of GaN thin films with dissimilar substrate materials by Pd-In metal bonding and laser lift-off,” Journal of Electronic Materials, vol. 28, pp. 1409–1413, 1999. 171. W. S. Wong, T. Sands, N. W. Cheung, M. Kneissl, D. P. Bour, P. Mei, L. T. Romano, and N. M. Johnson, “Inx Ga1−x N light emitting diodes on Si substrates fabricated by Pd-In metal bonding and laser lift-off,” Applied Physics Letters, vol. 77, pp. 2822–2824, 2000. 172. M. A. Raburn, B. Liu, K. Rauscher, Y. Okuno, N. Dagli, and J. E. Bowers, “3-D photonic circuit technology,” IEEE Journal of Selected Topics in Quantum Electronics, vol. 8, pp. 935–942, 2002.

Bibliography

253

173. P. P. Absil, Microring resonators for wavelength division multiplexing and integrated photonics applications, Ph. D. dissertation, University of Maryland, College Park, 2000. 174. R. Grover, Indium phosphide based optical micro-ring resonators, Ph. D. dissertation, Department of Electrical and Computer Engineering, University of Maryland, College Park, 2003. 175. K. Djordjev, S.-J. Choi, S.-J. Choi, and P. D. Dapkus, “Active semiconductor microdisk devices,” Journal of Lightwave Technology, vol. 20, pp. 105–113, 2002. 176. S. Adachi and K. Oe, “Linear electro-optic effects in zincblende-type semiconductors: Key properties of InGaAsP relevant to device design,” Journal of Applied Physics, vol. 56, pp. 74–80, 1984. 177. S. Adachi and K. Oe, “Quadratic electro-optic (Kerr) effects in zincblendetype semiconductors: Key properties of InGaAsP relevant to device design,” Journal of Applied Physics, vol. 56, pp. 1499–1504, 1984. 178. R. W. Boyd, Nonlinear Optics, Academic Press, San Diego, CA, 1992. 179. D. A. B. Miller, D. S. Chemla, and S. Schmitt-Ring, “Relation between electroabsorption in bulk semiconductors and in quantum wells: The quantumconfined Franz-Keldysh effect,” Physical Review B, vol. 33, pp. 6976–6982, 1986. 180. J. P. Donnelly, H. Q. Le, E. A. Swanson, S. H. Groves, A. Darwish, and E. P. Ippen, “Nondegenerate four-wave mixing wavelength conversion in low-loss passive InGaAsP-InP quantum well waveguides,” IEEE Photonics Technology Letters, vol. 8, pp. 623–625, 1996. 181. J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York, NY, third edition, 1999. 182. S. L. Chuang, Physics of Optoelectronic Devices, John Wiley & Sons, New York, NY, 1995. 183. D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, and C. A. Burrus, “Band-edge electroabsorption in quantum well structures: the quantum-confined Stark effect,” Physical Review Letters, vol. 53, pp. 2173–2176, 1984. 184. M. Fetterman, C.-P. Chao, and S. R. Forrest, “Fabrication and analysis of high-contrast InGaAsP-InP Mach-Zehnder modulators for use at 1.55 µm wavelength,” IEEE Photonics Technology Letters, vol. 8, pp. 69–71, 1996. 185. J. E. Zucker, I. Bar-Joseph, G. Sucha, U. Koren, B. I. Miller, and D. S. Chemla, “Electrorefraction in GaInAsP/InP multiple quantum well heterostructures,” Electronics Letters, vol. 24, pp. 458–460, 1988. 186. M. J. Bloemer and K. Myneni, “Electro-optic properties near the absorption edge of GaAs/AlGaAs multiple-quantum-well waveguides,” Journal of Applied Physics, vol. 74, pp. 4849–4859, 1993. 187. S. T. Ho, C. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slusher, “Large nonlinear phase shifts in low-loss AlGaAs waveguides near half-gap,” Applied Physics Letters, vol. 59, pp. 2558–2560, 1991. 188. G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-Hemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below the half-gap: The silicon of nonlinear optical materials,” International Journal of Nonlinear Optical Physics, vol. 3, pp. 347–371, 1994. 189. S. Spalter, H. Y. Wang, J. Zimmermann, G. Lenz, T. Katsufuji, S. W. Cheong, and R. E. Slusher, “Strong self-phase modulation in planar chalcogenide glass waveguides,” Optics Letters, vol. 27, pp. 363–365, 2002.

254

Bibliography

190. J. M. Harbold, F. O. Ilday, F. W. Wise, J. S. Sanghera, V. Q. Nguyen, L. B. Shaw, and I. D. Aggarwal, “Highly nonlinear As-S-Se glasses for all-optical switching,” Optics Letters, vol. 27, pp. 119–121, 2002. 191. M. Sheik-Bahae, D. J. Hagan, and E. W. van Stryland, “Dispersion and bandgap scaling of the electronic Kerr effect in solids associated with twophoton absorption,” Physical Review Letters, vol. 65, pp. 96–99, 1990. 192. M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. van Stryland, “Dispersion of bound electronic nonlinear refraction in solids,” IEEE Journal of Quantum Electronics, vol. 27, pp. 1296–1306, 1991. 193. J. E. Heebner, N. N. Lepeshkin, A. Schweinsberg, G. W. Wicks, R. W. Boyd, R. Grover, and P.-T. Ho, “Enhanced linear and nonlinear optical phase response of AlGaAs microring resonators,” Optics Letters, vol. 29, pp. 769– 771, 2004. 194. V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two photon absorption as a limitation to all-optical switching,” Optics Letters, vol. 14, pp. 1140–1142, 1989. 195. J. E. Heebner, Nonlinear Optical Whispering Gallery Microresonators for Photonics, Ph. D. dissertation, University of Rochester, 2003. 196. A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, “Ultrafast all-optical switching in semiconductor nonlinear directional couplers at half the band gap,” Applied Physics Letters, vol. 61, pp. 147–149, 1992. 197. Y. Chen, G. Pasrija, B. Farhang-Boroujeny, and S. Blair, “Engineering the nonlinear phase shift,” Optics Letters, vol. 28, pp. 1945–1947, 2003. 198. H. M. Gibbs, Optical bistability: controlling light with light, Academic Press Inc., New York, 1985. 199. S. Blair, J. E. Heebner, and R. W. Boyd, “Beyond the absorption-limited nonlinear phase shift with microring resonators,” Optics Letters, vol. 27, pp. 357–359, 2002. 200. V. R. Almeida, C. A. Barrios, R. R. Panepucci, M. Lipson, M. A. Foster, D. G. Ouzounov, and A. L. Gaeta, “All-optical switching on a silicon chip,” Optics Letters, vol. 29, pp. 2867–2869, 2004. 201. S. M. Sze, Physics of Semiconductor Devices, John Wiley & Sons, second edition, 1981. 202. M. B. Yairi and D. A. B. Miller, “Equivalence of diffusive conduction and giant ambipolar diffusion,” Journal of Applied Physics, vol. 91, pp. 4374–4381, 2002. 203. A. Miller, “Transient grating studies of carrier diffusion and mobility in semiconductors,” in Nonlinear Optics in Semiconductors II, E. Garmire and A. Kost, Eds., vol. 59 of Semiconductors and Semimetals, pp. 287–312. Academic Press, San Diego, CA, 1999. 204. E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, New York, NY, eighth edition, 1999. 205. G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” Journal of Applied Physics, vol. 58, pp. R57–R78, 1985. 206. S. D. Smith, “Optical bistability, photonic logic, and optical computation,” Applied Optics, vol. 25, pp. 1550–1564, 196. 207. B. E. Little and S. T. Chu, “Toward very large-scale integrated photonics,” Optics & Photonics News, vol. 11, pp. 24–29, 2000. 208. C. C. Yang, A. Villeneuve, G. I. Stegeman, C. H. Lin, and H. Lin, “Measurements of two-photon absorption coefficient and induced nonlinear refractive index in GaAs/AlGaAs multiquantum well waveguides,” Electronics Letters, vol. 29, pp. 37, 1993.

Bibliography

255

209. C. Coriasso, D. Campi, C. Carcciatore, L. Faustini, L. Rigo, and A. Stano, “All optical switching and pulse routing in a distributed-feedback waveguide device,” Optics Letters, vol. 23, pp. 183–185, 1998. 210. T. A. Ibrahim, V. Van, and P.-T. Ho, “All-optical time division demultiplexing and spatial pulse routing using a GaAsAlGaAs microring resonator,” Optics Letters, vol. 27, pp. 803–805, 2002. 211. T. A. Ibrahim, R. Grover, L.-C. Kuo, S. Kanakaraju, L. C. Calhoun, and P.-T. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photonics Technology Letters, vol. 15, pp. 1422–1424, 2003. 212. V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE Journal of Selected Topics in Quantum Electronics, vol. 8, pp. 705–713, 2002. 213. J. H. Bechtel and W. L. Smith, “Two-photon absorption in semiconductors with picosecond laser pulses,” Physical Review B, vol. 13, pp. 3515–3522, 1976. 214. T. A. Ibrahim, K. Amarnath, L. C. Kuo, R. Grover, V. Van, and P.-T. Ho, “Photonic logic NOR gate based on two symmetric microring resonators,” Optics Letters, vol. 29, pp. 2779–2781, 2004. 215. G. L. Fisher, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Physical Review Letters, vol. 74, pp. 1871–1875, 1995. 216. N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Physical Review Letters, vol. 93, pp. 123902, 2004. 217. E. Yablanovitch, “Inhibited spontaneus emission in solid-state physics and electronics,” Physical Review Letters, vol. 58, pp. 2059–2062, 1987. 218. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Physical Review Letters, vol. 58, pp. 2486–2489, 1987. 219. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals, Princeton University Press, Singapore, 1995. 220. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE Journal of Quantum Electronics, vol. 37, pp. 525–532, 2001. 221. V. E. Zakharov and A. B. Shabat, “Interaction between solutions in a stable medium,” Soviet Physics JETP, vol. 37, pp. 823–828, 1973. 222. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers,” Applied Physics Letters, vol. 23, pp. 142–144, 1973. 223. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Physical Review Letters, vol. 45, pp. 1095–1099, 1980. 224. J. E. Heebner, R. W. Boyd, and Q. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Physical Review E, vol. 65, pp. 036619, 2002. 225. E. Golovchenko, E. Dianov, A. Prokhorov, and V. Serkin, “Decay of optical solitons,” JETP Letters, vol. 42, pp. 87–91, 1985. 226. C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103x magnification of femtosecond waveforms,” Optics Letters, vol. 24, pp. 783–785, 1999. 227. Y. Chen and S. Blair, “Nonlinear phase shift of cascaded microring resonators,” Journal of the Optical Society of America B, vol. 20, pp. 2125– 2132, 2003.

256

Bibliography

228. G. Lenz, J. Zimmermann, T. Katsufuji, M. E. Lines, H. Y. Wang, S. Spalter, R. E. Slusher, S. W. Cheong, J. S. Sanghera, and I. D. Aggarwal, “Large Kerr effect in bulk Se-based chalcogenide glasses,” Optics Letters, vol. 25, pp. 254–256, 2000. 229. D. Rafizadeh, J. P. Zhang, R. C. Tiberio, and S. T. Ho, “Propagation loss measurements in semiconductor microcavity ring and disk resonators,” Journal of Lightwave Technology, vol. 16, pp. 1308–1314, 1998. 230. R. W. Boyd and D. J. Gauthier, Slow and Fast Light, Progress in Optics, Elsevier Science, Amsterdam, 2002. 231. C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature, vol. 409, pp. 490–493, 2001. 232. D. F. Phillips, M. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Physical Review Letters, vol. 86, pp. 783–786, 2001. 233. S. Chu and S. Wong, “Linear pulse-propagation in an absorbing medium,” Physical Review Letters, vol. 48, pp. 738–741, 1982. 234. S. E. Harris, “Electromagnetically induced transparency,” Physics Today, vol. 50, pp. 36–42, 1997. 235. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Physical Review Letters, vol. 90, pp. 113903, 2003. 236. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science, vol. 301, pp. 200–202, 2003. 237. M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: Line narrowing and interference effects,” Physical Review Letters, vol. 79, pp. 2959–2962, 1997. 238. A. B. Matsko, Y. V. Rostovtsev, H. Z. Cummins, and M. O. Scully, “Using slow light to enhance acousto-optical effects: Application to squeezed light,” Physical Review Letters, vol. 84, pp. 5752–5755, 2000. 239. A. B. Matsko, Y. V. Rostovtsev, M. Fleischhauer, and M. O. Scully, “Anomalous stimulated brillouin scattering via ultraslow light,” Physical Review Letters, vol. 86, pp. 2006–2009, 2001. 240. K. Djordjev, S. Choi, S. Choi, and P. D. Dapkus, “Vertically coupled InP microdisk switching devices with electroabsorptive active regions,” IEEE Photonics Technology Letters, vol. 14, pp. 1115–1117, 2002. 241. R. Y. Chiao and A. M. Steinberg, Progress in Optics XXXVII, Elsevier Science, Amsterdam, 1997. 242. J. E. Heebner, R. W. Boyd, and Q. Park, “‘Slow’ and ‘fast’ light in resonatorcoupled waveguides,” Journal of Modern Optics, vol. 49, pp. 2629–2636, 2002. 243. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Physical Review Letters, vol. 58, pp. 160–163, 1987. 244. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Physical Review Letters, vol. 76, pp. 1627–1630, 1996. 245. D. N. Christodoulides and N. K. Efremidis, “Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals,” Optics Letters, vol. 27, pp. 568–570, 2002. 246. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Physical Review Letters, vol. 87, pp. 253902, 2001.

Bibliography

257

247. M. Soljacic, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” Journal of the Optical Society of America B, vol. 19, pp. 2052–2059, 2002. 248. M. D. Rahn, A. M. Fox, M. S. Skolnick, and T. F. Krauss, “Propagation of ultrashort nonlinear pulses through two-dimensional AlGaAs high-contrast photonic crystal waveguides,” Journal of the Optical Society of America B, vol. 19, pp. 716–721, 2002. 249. S. Mingaleev and Y. Kivshar, “Nonlinear transmission and light localization in photonic-crystal waveguides,” Journal of the Optical Society of America B, vol. 19, pp. 2241–2249, 2002. 250. J. E. Sipe, N. A. R. Bhat, P. Chak, and S. Pereira, “Effective field theory for the nonlinear optical properties of photonic crystals,” Physical Review E., vol. 69, pp. 016604, 2004. 251. B. E. Little, S. T. Chu, and H. A. Haus, “Second-order filtering and sensing with partially coupled traveling waves in a single resonator,” Optics Letters, vol. 23, pp. 1570–1572, 1998. 252. J. E. Heebner, P. Chak, S. Perera, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator sequences for linear and nonlinear optics,” Journal of the Optical Society of America B, vol. 21, pp. 1818–1832, 2004. 253. R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ringresonator filters for optical systems,” IEEE Photonics Technology Letters, vol. 7, pp. 1447–1449, 1995. 254. A. Yariv, X. Yong, R. K. Lee, and A. Soberer, “Coupled-resonator optical waveguide: a proposal and analysis,” Optics Letters, vol. 24, pp. 711–713, 1999. 255. Y. Xu, R. K. Lee, and A. Yariv, “Scattering theory analysis of waveguideresonator coupling,” Physical Review E, vol. 62, pp. 7389–7404, 2000. 256. S. Mookherjea, D. S. Cohen, and A. Yariv, “Nonlinear dispersion in a coupled-resonator optical waveguide,” Optics Letters, vol. 27, pp. 933–935, 2002. 257. W.-Y. Chen, R. Grover, T. A. Ibrahim, V. Van, W. N. Herman, and P.-T. Ho, “High-finesse laterally-coupled single-mode benzocyclobutene micro-ring resonators,” IEEE Photonics Technology Letters, vol. 16, pp. 470–472, 2004. 258. Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” Journal of the Optical Society of America B, vol. 17, pp. 387–400, 2000. 259. Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity,” Physical Review E., vol. 68, pp. 066616, 2003. 260. S. Pereira, J. E. Sipe, J. E. Heebner, and R. W. Boyd, “Gap solitons in a twochannel SCISSOR structure,” Optics Letters, vol. 27, pp. 536–538, 2002. 261. P. Chak, S. Pereira, and J. Sipe, “Lorentzian model for nonlinear switching in a microresonator structure,” Optics Communications, vol. 213, pp. 163– 171, 2002. 262. G. Griffel, “Synthesis of optical filters using ring resonator arrays,” IEEE Photonics Technology Letters, vol. 12, pp. 810–812, 2000. 263. A. Melloni, F. Morichetti, and M. Martinelli, “Optical slow wave structures,” Optics and Photonics News, vol. 14, pp. 44–48, 2003. 264. C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photonics Technology Letters, vol. 11, pp. 1623– 1625, 1999.

258

Bibliography

265. J. Khurgin, “Expanding the bandwidth of slow-light photonic devices based on coupled resonators,” Optics Letters, vol. 30, pp. 513–515, 2005. 266. G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE Journal of Quantum Electronics, vol. 34, pp. 1390–1402, 1998. 267. P. Chak, J. Sipe, and S. Pereira, “Depositing light in a photonic stop gap by use of Kerr nonlinear microresonators,” Optics Letters, vol. 28, pp. 1966– 1968, 2003. 268. S. Pereira, P. Chak, and J. E. Sipe, “Gap-soliton switching in short microresonator structures,” Journal of the Optical Society of America B, vol. 19, pp. 2191–2202, 2002. 269. S. Blair, “Nonlinear sensitivity enhancement with one-dimensional photonic bandgap microcavity arrays,” Optics Letters, vol. 27, pp. 613–615, 2002. 270. S. Adachi, “Material parameters of In1−x Gax Asy P1−y and related binaries,” Journal of Applied Physics, vol. 53, pp. 8775–8792, 1982. 271. S. Adachi, “Refractive indices of III–V compounds Key properties of InGaAsP relevant to device design,” Journal of Applied Physics, vol. 53, pp. 5863–5869, 1982. 272. S. Adachi, “Optical properties of InGaAsP: discussion,” in Properties of Indium Phosphide, London, UK, 1989, EMIS Datareviews Series, IEE, Inspec. 273. S. Adachi, Physical Properties of III–V Semiconductor Compounds, John Wiley & Sons, New York, NY, 1992. 274. S. Adachi, Properties of Aluminum Gallium Arsenide, EMIS Datareviews Series. IEE, Inspec, London, UK, 1993. 275. R. J. Deri and M. A. Emanuel, “Consistent formula for the refractive index of Alx Ga1−x As below the band edge,” Journal of Applied Physics, vol. 77, pp. 4667–4672, 1995. 276. M. V. Sullivan and G. A. Kolb, “The chemical polishing of gallium arsenide in bromine-methanol,” Journal of the Electrochemical Society, vol. 110, pp. 585–587, 1963. 277. S. B. Phatak and G. Kelner, “Material selective etching in the system InGaAsP/InP,” Journal of the Electrochemical Society, vol. 126, pp. 287–292, 1979. 278. G. C. DeSalvo, W. F. Tseng, and J. Comas, “Etch rates and selectivities of citric acid/hydrogen peroxide on GaAs, Al0.3 Ga0.7 As, In0.3 Ga0.8 As, In0.53 Ga0.47 As, In0.52 As0.48 As and InP,” Journal of the Electrochemical Society, vol. 139, pp. 831–835, 1992. 279. H. Fourre, F. Diette, and A. Cappy, “Selective wet etching of lattice-matched InGaAs/InAlAs on InP and metamorphic InGaAs/InAlAs on GaAs using succinic acid/hydrogen peroxide solution,” Journal of Vacuum Science & Technology B, vol. 14, pp. 3400–3402, 1996. 280. F. G. Johnson, O. S. King, F. Seiferth, D. R. Stone, R. D. Whaley, M. Dagenais, and Y. J. Chen, “Solid source molecular beam epitaxy of low threshold 1.55 µm wavelength GaInAs/GaInAsP/InP semiconductor lasers,” Journal of Vacuum Science & Technology B, vol. 14, pp. 2753–2756, 1996. 281. F. G. Johnson, O. S. King, F. Seiferth, S. Horst, D. R. Stone, R. D. Whaley, M. Dagenais, and Y. J. Chen, “Solid source MBE growth and regrowth of 1.55 µm wavelength GaInAsP/InP ridge lasers,” Journal of Crystal Growth, vol. 175, pp. 46–51, 1997. 282. J. V. Hryniewicz, Y. J. Chen, S. H. Hsu, C. H. D. Lee, and G. A. Porkolab, “Ultrahigh vacuum chemically assisted ion beam etching system with a three grid ion source,” Journal of Vacuum Science & Technology A, vol. 15, pp. 616–621, 1997.

Bibliography

259

283. S. Agarwala, S. C. Horst, O. S. King, R. A. Wilson, D. R. Stone, M. Dagenais, and Y. J. Chen, “High-density inductively coupled plasma etching of GaAs/AlGaAs in BCl3 /Cl2 /Ar: A study using a mixture design experiment,” Journal of Vacuum Science & Technology B, vol. 16, pp. 511–514, 1998. 284. S. Agarwala, O. S. King, S. C. Horst, R. A. Wilson, D. R. Stone, M. Dagenais, and Y. J. Chen, “Response surface study of inductively coupled plasma etching of GaAs/AlGaAs in BCl3 /Cl2 ,” Journal of Vacuum Science & Technology A, vol. 17, pp. 52–55, 1999. 285. R. D. Whaley, B. Gopalan, M. Dagenais, R. D. Gomez, F. G. Johnson, S. Agarwala, O. King, and D. R. Stone, “Use of atomic force microscopy for analysis of high performance InGaAsP/InP semiconductor lasers with dry-etched facets,” Journal of Vacuum Science and Technology B, vol. 16, pp. 1007–1011, 1998. 286. R. Grover, J. V. Hryniewicz, O. S. King, and V. Van, “Process development of methane-hydrogen-argon-based deep dry-etching of InP for high aspectratio structures with vertical facet-quality sidewalls,” Journal of Vacuum Science & Technology B, vol. 19, pp. 1694–1698, 2001. 287. F. G. Johnson, O. S. King, J. V. Hryniewicz, L. G. Joneckis, S. T. Chu, and D. M. Gill, “Use of deuterated gases for the vapor deposition of thin films for lowloss optical devices and waveguides,” United States Patent, vol. 6,614,977, 2003. 288. M. A. Raburn, B. Liu, P. Abraham, and J. E. Bowers, “Double-bonded InPInGaAsP vertical coupler 1 : 8 beam splitter,” IEEE Photonics Technology Letters, vol. 12, pp. 1639–1641, 2000. 289. M. Raburn, Three-dimensional wafer bonded indium phosphide photonic waveguide devices, Ph.D. thesis, Department of Electrical and Computer Engineering, University of California, Santa Barbara, 2003. 290. S. R. Sakamoto, C. Ozturk, Y. T. Byun, J. Ko, and N. Dagli, “Low-loss substrate-removed (SURE) optical waveguides in GaAs-AlGaAs epitaxial layers embedded in organic polymers,” IEEE Photonics Technology Letters, vol. 10, pp. 985–987, 1998. 291. R. Williams, Modern GaAs Processing Methods, Artech House, Boston, MA, 1990. 292. P. Rabiei, Electro-optic and thermo-optic polymer micro-ring resonators and their applications, Ph.D. thesis, University of Southern California, 2003. 293. Y. Chen and S. Blair, “Nonlinearity enhancement in finite coupled-resonator slow-light waveguides,” Optics Express, vol. 12, pp. 3353–3366, 2004. 294. A. Driessen, D. H. Geuzebroek, H. J. W. M. Hoekstra, H. Kelderman, E. J. Klein, D. J. W. Klunder, C. G. H. Roeloffzen, F. S. Tan, E. Krioukov, and C. Otto, “Microresonators as building blocks for VLSI photonics,” in Summer School on Microresonators as building blocks for VLSI Photonics, Ettore Majorana Centre for Scientific Culture, International School of Quantum Electronics, Erice, Sicily, Italy, 2003. 295. F. Morichetti, R. Siano, A. Boletti, and A. Melloni, “Optical integrated receiver for DQPSK systems,” in Transparent Optical Networks, 2005. 296. J. E. Heebner, T. C. Bond, and J. S. Kallman, “Generalized formulation for performance degradations due to bending and edge scattering loss in microdisk resonators,” Optics Express, vol. 15, pp. 4452–4473, 2007.

Index

Q – definition, 89 – disks, 48 – rings, 48 – spheres, 48 Q switching, 6 active LiNbO3 , 3 active resonators – carrier injection, 110 – electro-optic, 110 – need for, 107 – thermal tuning, 110 add–drop rings, 84 – all-optical switching, 137 – finesse, 89 – free spectral range, 89 – intensity buildup, 85 – resonance width, 87 all-optical switching, 6, 149 – add–drop, see add–drop rings – all-pass, see all-pass rings – Fabry–Perot, 137 – pump reshaping, 156 – pump-probe experiments, 154, 159 – pump-probe simulations, 152 – response time measurement, 160 – ring-enhanced MZ, 137 – routing, 161 – theory, 149 – thresholding, 156 – time-division mux/demux, 158 all-pass rings, 74, 124 – all-optical switching, 137 – attenuation in, 83 – finesse, 77 – group delay, 79 – group delay dispersion, 79 – intensity buildup, 75 – phase shift in, 78

beam propagation, 25 cavity QED, 6 coupling – coupled wave formalism, 30 – optimization of, 31 – perturbation solutions, 27 – scattering matrix, 31 – TE planar waveguides, 29 coupling of waveguides, see coupling CROW, 202 – bandgap engineering, 210 delay lines, tunable, 195 dispersion relations, 18 – WGM, 43 distributed microresonator systems, 175 – linear propagation, 175 effective index method, 23 electro-optic effect, 110 fabrication – feature definition, 221 – lateral active III–V, 234 – lateral passive III–V, 225 – multilayer, 222 – polymer rings, 242 – vertical passive III–V, 227 – wafer bonding, 222 fast light, 191, 197 figures of merit, 142 filters, 97 – parallel-cascade of rings, 99 – practical limitations to cascading, 103 – serial-cascade of rings, 99 – vernier effect, 103 finesse – enhanced nonlinearities, 124

261

262

Index

finite element method, 25 finite-difference time-domain, 26 four-wave mixing, 123 – resonator-enhanced, 147 – SCISSOR, 187 Franz–Keldysh effect, 111 free spectral range – add–drop rings, 89 GaAs-AlGaAs, see III–V semiconductors GaInAsP-InP, see III–V semiconductors Goell’s method, 24 III–V semiconductors – active, see active resonators – advantages, 218 – direct bandgap, 219 – feature definition, 221 – GaAs-AlGaAs, 107, 127, 128, 151, 159, 163, 167, 169 – GaInAsP-InP, 107, 116, 164 – history, 4 – lateral active, 116, 234 – lateral coupling, 5, 105, 107 – lateral passive, 225 – multilayer fabrication, 222 – vertical coupling, 4, 105, 107 – vertical passive, 227 – waveguide fabrication, 220 – waveguides, 218 intensity enhancement, 6 inverted effective nonlinearity, 144 Kerr effect, 123, 127 – experimental enhancement, 128 lateral coupling, 105 – relevance, 107 loss – bending, 35 – by the volume current method, 50 – effect on nonlinear enhancement, 138 – far field scattered power, 53 – mechanisms, 35 – normalized formulation for scattering loss, 61 – radiation, 42 – scattering loss, 50 – spectral density formulation, 52 – substrate leakage, 35 – TE scattering loss, 56

Mach–Zehnder interferometer, 130 Marcatili’s method, 22 material systems – fibers, 3 – glass, 3, 220 – III–V semiconductors, see III–V semiconductors – ion-exchanged glass, 3 – PMMA on quartz, 3 – polymers, 3, 242 – polyurethane on glass, 3 – silica, 3 multi-ring devices, 175 multistability, 136 nonlinear figures of merit, 142 nonlinear optics – all-pass rings, see all-pass rings – bandwidth vs. nonlinear response, 125 – challenges of, 6 – four-wave mixing, see four-wave mixing – in microrings, 5 – inverted effective nonlinearity, 144 – Kerr effect, see Kerr effect – multistability, 136 – phase shift, 124 – potential of, 6 – pulse fidelity, 125 – pulsed excitation, 125 – resonant enhancement, 124 – saturable absorption, see saturable absorption – two-photon absorption, see two-photon absorption nonlinear saturation, 132 nonlinear susceptibility, 123 – four-wave mixing, see four-wave mixing – Kerr effect, see Kerr effect – multistability, 136 – resonator enhanced χ (3) , 124 – saturable absorption, see saturable absorption – two photon absorption, see two-photon absorption normalized dispersion relations, 19 – WGM, 46 optical delay lines, 213 optical delay lines, tunable, 195 optical logic, 6, 149, 163

Index

– (N)AND, 163 – NOR, 169 – speed improvement, 173 optical microresonators – applications, 1 – compared to Fabry–Perots, 2 – definition, 1, 2 – fabrication, 217 – history, 3 – integrated, 3 – materials, 217 – size matters, 2, 4 optical multistability, 6 parallel-cascaded rings, 99, 197 – bandgap engineering, 210 – distributed resonator system, 175 – group-velocity relations, 177 – higher order dispersion, 179 – induced self-steepening, 184 – nonlinear propagation, 179 – solitons, 181 paraxial waveguiding equation, 12 passive ring resonators, 105 photonic integrated circuits, 107 planar slab waveguide, 13 polymers, 219, 242 quantum optics, 6 quantum-confined Stark effect, 111 Raman interactions, 6 rectangular dielectric waveguides, 20 resonance, 1 resonator-enhanced MZ, 130 resonators, 71 – Q, 89 – – disks, 48 – – rings, 48 – – spheres, 48 – add–drop rings, see add–drop rings – all-pass rings, see all-pass rings – characterization, 89 – Fabry–Perot, 71 – finesse, 77 – Gires–Tournois, 73 – high-order filters, 97 – physical significance, finesse and Q, 91 – ring, 73

263

saturable absorption, 123 scattering loss, see loss SCISSOR, 197 – bandgap engineering, 210 – Bloch matrix, 200 – depth of phase, 189 – double-channel, 206 – fast light, 191, 197 – four-wave mixing, 187 – group-velocity relations, 177 – higher order dispersion, 179 – induced self-steepening, 184 – linear propagation, 175 – multistability, 186 – nonlinear propagation, 179 – pulse compression, 185 – single-channel, 204 – slow light, 191 – solitons, 181 serial-cascaded rings, 99, 197, 202 – bandgap engineering, 210 slow light, 191, 213 solitons, 181 – dark, 183 squeezed light, 6 susceptibility, see nonlinear susceptibility total internal reflection, 9 two-beam a.c. amplification, 6 two-photon absorption, 124, 128, 142, 144 vertical coupling, 105 wafer bonding, 222 waveguide confinement, 9 waveguides, 9 – dispersion relations, 18 – losses in, see loss – normalized dispersion relations, 19 – planar slab, 13 – rectangular dielectric waveguides, 20 whispering gallery modes, 1, 38 – dispersion relations, 43 – normalized dispersion relations, 46 – radiation loss, 42 – TE, 41 – TM, 39

Springer Series in

OPTICAL SCIENCES Volume 1 1 Solid-State Laser Engineering By W. Koechner, 5th revised and updated ed. 1999, 472 figs., 55 tabs., XII, 746 pages

Published titles since Volume 85 85 Sensing with Terahertz Radiation By D. Mittleman (Ed.), 2003, 207 figs., 14 tabs., XVI, 337 pages 86 Progress in Nano-Electro-Optics I Basics and Theory of Near-Field Optics By M. Ohtsu (Ed.), 2003, 118 figs., XIV, 161 pages 87 Optical Imaging and Microscopy Techniques and Advanced Systems By P. Török, F.-J. Kao (Eds.), 2003, 260 figs., XVII, 395 pages 88 Optical Interference Coatings By N. Kaiser, H.K. Pulker (Eds.), 2003, 203 figs., 50 tabs., XVI, 504 pages 89 Progress in Nano-Electro-Optics II Novel Devices and Atom Manipulation By M. Ohtsu (Ed.), 2003, 115 figs., XIII, 188 pages 90/1 Raman Amplifiers for Telecommunications 1 Physical Principles By M.N. Islam (Ed.), 2004, 488 figs., XXVIII, 328 pages 90/2 Raman Amplifiers for Telecommunications 2 Sub-Systems and Systems By M.N. Islam (Ed.), 2004, 278 figs., XXVIII, 420 pages 91 Optical Super Resolution By Z. Zalevsky, D. Mendlovic, 2004, 164 figs., XVIII, 232 pages 92 UV-Visible Reflection Spectroscopy of Liquids By J.A. Räty, K.-E. Peiponen, T. Asakura, 2004, 131 figs., XII, 219 pages 93 Fundamentals of Semiconductor Lasers By T. Numai, 2004, 166 figs., XII, 264 pages 94 Photonic Crystals Physics, Fabrication and Applications By K. Inoue, K. Ohtaka (Eds.), 2004, 209 figs., XV, 320 pages 95 Ultrafast Optics IV Selected Contributions to the 4th International Conference on Ultrafast Optics, Vienna, Austria By F. Krausz, G. Korn, P. Corkum, I.A. Walmsley (Eds.), 2004, 281 figs., XIV, 506 pages 96 Progress in Nano-Electro Optics III Industrial Applications and Dynamics of the Nano-Optical System By M. Ohtsu (Ed.), 2004, 186 figs., 8 tabs., XIV, 224 pages 97 Microoptics From Technology to Applications By J. Jahns, K.-H. Brenner, 2004, 303 figs., XI, 335 pages 98 X-Ray Optics High-Energy-Resolution Applications By Y. Shvyd’ko, 2004, 181 figs., XIV, 404 pages

99 Mono-Cycle Photonics and Optical Scanning Tunneling Microscopy Route to Femtosecond Ångstrom Technology By M. Yamashita, H. Shigekawa, R. Morita (Eds.) 2005, 241 figs., XX, 393 pages 100 Quantum Interference and Coherence Theory and Experiments By Z. Ficek and S. Swain, 2005, 178 figs., XV, 418 pages 101 Polarization Optics in Telecommunications By J. Damask, 2005, 110 figs., XVI, 528 pages 102 Lidar Range-Resolved Optical Remote Sensing of the Atmosphere By C. Weitkamp (Ed.), 161 figs., XX, 416 pages 103 Optical Fiber Fusion Splicing By A.D. Yablon, 2005, 137 figs., XIII, 306 pages 104 Optoelectronics of Molecules and Polymers By A. Moliton, 2005, 229 figs., 592 pages 105 Solid-State Random Lasers By M. Noginov, 2005, 131 figs., XII, 238 pages 106 Coherent Sources of XUV Radiation Soft X-Ray Lasers and High-Order Harmonic Generation By P. Jaeglé, 2005, 150 figs., approx. 264 pages 107 Optical Frequency-Modulated Continuous-Wave (FMCW) Interferometry By J. Zheng, 2005, 137 figs., XVIII, 254 pages 108 Laser Resonators and Beam Propagation Fundamentals, Advanced Concepts and Applications By N. Hodgson and H. Weber, 2005, 497 figs., approx. 790 pages 109 Progress in Nano-Electro Optics IV Characterization of Nano-Optical Materials and Optical Near-Field Interactions By M. Ohtsu (Ed.), 2005, 123 figs., XIV, 206 pages 110 Kramers–Kronig Relations in Optical Materials Research By V. Lucarini, J.J. Saarinen, K.-E. Peiponen, E.M. Vartiainen, 2005, 37 figs., X, 162 pages 111 Semiconductor Lasers Stability, Instability and Chaos By J. Ohtsubo, 2005, 169 figs., XII, 438 pages 112 Photovoltaic Solar Energy Generation By A. Goetzberger and V.U. Hoffmann, 2005, 139 figs., XII, 234 pages 113 Photorefractive Materials and Their Applications 1 Basic Effects By P. Günter and J.P. Huignard, 2005, 169 figs., XIV, 426 pages 114 Photorefractive Materials and Their Applications 2 Materials By P. Günter and J.P. Huignard, 2005, 370 figs., XVIII, 646 pages 115 Photorefractive Materials and Their Applications 3 Applications By P. Günter and J.P. Huignard, 2005, 316 figs., X, 366 pages 116 Spatial Filtering Velocimetry Fundamentals and Applications By Y. Aizu and T. Asakura, 2006, 112 figs., XII, 212 pages 117 Progress in Nano-Electro-Optics V Nanophotonic Fabrications, Devices, Systems, and Their Theoretical Bases By Motoichi Ohtsu, 2006, 122 figs., 3 tables, 188 pages

118 Mid-infrared Semiconductor Optoelectronics By A. Krier, 2006, 443 figs., 751 pages 119 Optical Interconnects The Silicon Approach By L. Pavesi and G. Guillot, 2006, 260 figs., 384 pages 120 Relativistic Nonlinear Electrodynamics Interaction of Charged Particles with Strong and Super Strong Laser Fields By H.K. Avetissian, 2006, 23 figs., XIII, 333 pages 121 Thermal Process Using Attosecond Laser Pulses When Time Matters By M. Kozlowski and J. Marciak-Kozlowska, 2006, 46 figs., XII, 232 pages 122 Modeling and Analysis of Transient Processes in Open Resonant Structures New Methods and Techniques By Y.K. Sirenko, S. Ström, N.P. Yashina, 2006, 110 figs., XIV, 362 pages 123 Wavelength Filters in Fibre Optics By H. Venghaus, 2006, 210 figs., XXIV, 454 pages 124 Light Scattering by Systems of Particles Null-Field Method with Discrete Sources - Theory and Programs By A. Doicu, T. Wriedt and Y.A. Eremin, 2006, 119 figs., XIII, 324 pages 125 Electromagnetic and Optical Pulse Propagation 1 Spectral Representations in Temporally Dispersive Media By K.E. Oughstun, 2006, 74 figs., XVI, 460 pages 126 Quantum Well Infrared Photodetectors Physics and Applications By H. Schneider, 2006, 153 figs., XVI, 250 pages 127 Integrated Ring Resonators The Compendium By D.G. Rabus, 2007, 243 figs., 258 pages 128 High-Power Diode-Lasers Technology and Applications By F. Bachmann, P. Loosen, and R. Poprave, 2006, 543 figs., VI, 554 pages 129 Laser Ablation and its Applications By C. Phipps, 2007, 300 figs., XX, 588 pages 130 Concentrator Photovoltaics By A. Luque and V. Andreev, 2007, 250 figs., XIV, 345 pages 131 Surface Plasmon Nanophotonics By M.L. Brongersma and P.G. Kik, 2007, VII, 271 pages 132 Ultrafast Optics V By S. Watanabe and K. Midorikawa, 2007, 339 figs., 562 pages 133 Frontiers in Surface Nanophotonics Principles and Applications By D.L. Andrews and Z. Gaburro, 2007, 89 figs., 176 pages 134 Strong Field Laser Physics By T. Brabec, 2008, 150 figs., approx. 500 pages 135 Optical Nonlinearities in Chalcogenide Glasses and their Applications By A. Zakery and S.R. Elliot, 2007, 78 figs., 202 pages 136 Optical Measurement Techniques Innovations for Industry and the Life Sciences By K.E. Peiponen, R. Myllyl Ɨ and A.V. Priezzhev, 2008, 60 figs., approx. 300 pages 137 Modern Developments in X-Ray and Neutron Optics By A. Erko, M. Idir, T. Krist and A.G. Michette, 2008, 150 figs., approx. 584 pages 138 Optical Microresonators Theory, Fabrication, and Applications By J. Heebner, R. Grover, and T. Ibrahim, 2008, 100 figs., 280 pages