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

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

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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

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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

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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 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