Karlsruhe Series in Photonics & Communications, Vol. 4 Karlsruhe Series in Photonics & Communications, Vol. 4 Edited by Prof. J. Leuthold and Prof. W. Freude Universität Karlsruhe (TH) Institute of High-Frequency and Quantum Electronics (IHQ) Germany Jan-Michael Brosi Photonic crystals are nano-optical devices that have the potential to slow down the propagation of light. “Slow light” is not only useful for generating true time delays, but is also good for enhancing nonlinear effects. With photonic crystals it is basically possible to tailor dispersion. Devices can be integrated in silicon- on-insulator photonic chips and could also be combined with electronics. For Slow-Light Photonic Crystal fabrication, the well-established CMOS process is available. Devices for High-Speed This book discusses design, modeling, and the characterization of fabricated slow-light photonic crystal waveguides, where applications aim for high-speed optical signal processing. Slow-light photonic crystal waveguides are fabrica- Optical Signal Processing ted both at optical and microwave frequencies. Microwave model measure- ments serve as a highly accurate reference, based on which various numerical methods for calculating photonic crystals are evaluated and compared. Guide- lines are developed to obtain slow-light waveguides with broadband characte- ristics. Losses by fabrication imperfections are numerically determined and minimized. Nonlinearities are enhanced by a proper waveguide design and by additionally employing nonlinear organic materials. Three functional devices are proposed and studied: A tunable dispersion compensator, a tunable opti- cal delay line, and a high-speed electro-optic modulator. Optical measure- ments confirm the designs. About the Author Jan-Michael Brosi was born in 1978 in Filderstadt, Germany. In 2002 he recei- ved the M.Sc. degree in Electrical Engineering from Georgia Institute of Tech- Slow-Light PC Devices for High-Speed Optical Signal Processing nology, Atlanta, USA, and in 2008 the Dr.-Ing. (Ph.D.) degree from University of Karlsruhe, Germany. His research is focused on modeling, processing and characterization of photonic crystals and integrated optical devices. ISSN: 1865-1100 Jan-Michael Brosi ISBN: 978-3-86644-313-6 www.uvka.de 4 Jan-Michael Brosi Slow-Light Photonic Crystal Devices for High-Speed Optical Signal Processing Karlsruhe Series in Photonics & Communications, Vol. 4 Edited by Prof. J. Leuthold and Prof. W. Freude Universität Karlsruhe (TH), Institute of High-Frequency and Quantum Electronics (IHQ), Germany Slow-Light Photonic Crystal Devices for High-Speed Optical Signal Processing by Jan-Michael Brosi Dissertation, Universität Karlsruhe (TH) Fakultät für Elektrotechnik und Informationstechnik, 2008 Impressum Universitätsverlag Karlsruhe c/o Universitätsbibliothek Straße am Forum 2 D-76131 Karlsruhe www.uvka.de Dieses Werk ist unter folgender Creative Commons-Lizenz lizenziert: http://creativecommons.org/licenses/by-nc-nd/2.0/de/ Universitätsverlag Karlsruhe 2009 Print on Demand ISSN: 1865-1100 ISBN: 978-3-86644-313-6 Slow-Light Photonic Crystal Devices for High-Speed Optical Signal Processing Zur Erlangung des akademischen Grades eines DOKTOR-INGENIEURS von der Fakultat¨ fur¨ Elektrotechnik und Informationstechnik der Universitat¨ Karlsruhe (TH) genehmigte DISSERTATION von Jan-Michael Brosi, M. Sc. geboren in Filderstadt Tag der mundlichen¨ Prufung:¨ 18. Juli 2008 Hauptreferent: Prof. Dr.-Ing. Dr. h. c. Wolfgang Freude Korreferenten: Prof. Dr. sc. nat. Jurg¨ Leuthold Prof. Dr. rer. nat. Ulrich Lemmer Contents Zusammenfassung (Deutsch) 1 Achievements and Limitations 5 Summary 9 1 Fundamentals 13 1.1 Wave Propagation in Dielectric Media . 14 1.1.1 Maxwell’s Equations and Scaling Laws . 14 1.1.2 Modes of Translational Invariant Waveguides . 15 1.2 2D Photonic Crystals . 17 1.2.1 Bulk Photonic Crystal . 17 1.2.2 Photonic Crystal Waveguide . 19 1.3 Group Velocity and Chromatic Dispersion . 20 1.4 Mode Gaps and Mini-Stop Bands in PC Waveguides . 22 1.5 Excitation of Modes . 24 1.6 Losses in Photonic Crystals . 25 1.7 Tuning of Photonic Crystals . 27 1.7.1 Thermal Tuning . 27 1.7.2 Free-Carrier Injection . 27 1.7.3 Electro-Optic Effect . 28 2 Slow-Light Waveguides 31 2.1 Design . 32 2.1.1 Waveguide Geometries . 32 Line-Defect Waveguide . 33 Slot Waveguide . 34 Center-Hole Waveguide . 35 2.1.2 Design Principles . 36 2.1.3 Group Velocity and Dispersion Engineering . 39 Broadband Slow Light . 39 Negative Chromatic Dispersion . 40 2.1.4 Efficient Mode Excitation . 41 Adiabatic Tapers . 41 Counter-Propagating Mode Coupling . 43 i ii CONTENTS 2.1.5 Conclusion . 45 2.2 Imperfections . 47 2.2.1 Loss Parameters and Loss Model . 48 2.2.2 Numerical Methods . 50 2.2.3 Broadband Slow-Light Structures . 50 2.2.4 Loss Simulation Results . 52 FIT Method . 54 GME Method . 57 Comparison . 58 2.2.5 Dependance of Losses on Group Velocity . 59 2.2.6 Conclusion . 61 3 Devices 63 3.1 Tunable Dispersion Compensator . 63 3.1.1 Principle of Operation . 64 3.1.2 PC-WG Design . 66 3.1.3 Simulation of Device Characteristics . 67 3.1.4 Conclusion . 68 3.2 Tunable Optical Delay Line . 69 3.2.1 Principle of Operation . 69 3.2.2 PC-WG Design . 71 Constant Negative Chromatic Dispersion . 71 Constant Positive Chromatic Dispersion . 72 3.2.3 Optimization . 73 3.2.4 Conclusion . 75 3.3 Compact High-Speed Electro-Optic Modulator . 77 3.3.1 The Modulator . 77 3.3.2 Mach-Zehnder Optimization Strategy . 79 3.3.3 Slow-Wave Phase Modulator . 79 PC Slot Waveguide . 80 PC Slot Waveguide with Dispersion Engineering . 81 3.3.4 Modulator Performance Parameters . 81 Modulation Bandwidth of Mach-Zehnder Modulator . 81 p -Voltage Up of Phase Modulator . 83 3.3.5 Optimized Mach-Zehnder Modulator . 84 3.3.6 Slow-Light Coupling Structure . 85 3.3.7 Conclusion . 86 4 Experiments and Modeling 89 4.1 Optical Experiments . 89 4.1.1 Measurement Setups . 90 4.1.2 Strip Waveguide . 91 4.1.3 Broadband Slow Light Waveguide . 93 4.1.4 Waveguide with Linearly Varying Chromatic Dispersion . 95 4.1.5 Slot Waveguide . 97 CONTENTS iii 4.1.6 Conclusion . 99 4.2 Microwave Experiments . 101 4.2.1 Microwave Model . 102 Dielectric Microwave Materials . 102 Excitation of the Waveguide Mode . 102 Material Characterization . 104 4.2.2 Broadband Slow Light Waveguide . 105 Structure and Design . 105 Device Characteristics . 106 Pulse Transmission and Delay-Bandwidth Product . 108 Field Distribution . 109 Disorder Influence . 109 4.2.3 Conclusion . 112 4.3 Verification of Numerics . 113 4.3.1 Benchmark of Different Design Tools . 113 4.3.2 Simulation of Disorder-Induced Losses . 114 Numerical Calibration Procedure . 115 Comparison of Ensemble Averaged Results . 116 Comparison of Single Realization . 117 4.3.3 Conclusion . 117 Appendix 119 A.1 Mathematical Transformations and Signal Representation . 119 A.1.1 Fourier Transform . 119 A.1.2 Signal Representation . 119 A.1.3 Laplace Transform . 120 A.2 Numerical Modeling Tools . 121 A.3 Mode Orthogonality and Group Velocity . 123 A.4 Through-Reflect-Line (TRL) Calibration . 127 A.5 Coupling of Counter-Propagating Modes . 131 A.6 Loss Modeling . 135 A.7 Determination of Modulator Characteristics . 137 A.7.1 Propagation Equation and Field Interaction Factor G . 137 A.7.2 Modulator Walk-Off Bandwidth . 137 A.7.3 Modulator Bandwidth Limitations by RC-Effects . 138 A.8 Microwave Measurement Techniques . 141 A.8.1 Continuous Wave Measurements . 141 A.8.2 Pulse Measurements . 142 A.8.3 Near-Field Measurements . 143 Glossary 145 Acronyms . 145 Symbols . 147 Bibliography 161 iv CONTENTS Acknowledgments 163 List of Own Publications 165 Curriculum Vitae 169 List of Figures 1.1 Schematic of a strip waveguide. ..