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Analysis and Modeling of Lossy Planar Optical Waveguides and Application to Silicon-Based Structures Redacted for Privacy Abstract Approved AN ABSTRACT OF THE THESIS OF Catherine A. Rem ley for the degree of Master of Science in Electrical and Computer Engineering presented on June 20, 1995. Title: Analysis and Modeling of Lossy Planar Optical Waveguides and Application to Silicon-Based Structures Redacted for Privacy Abstract approved. Andreas Weisshaar This work is concerned with the modeling and analysis of lossy planar dielectric optical waveguides. Loss mechanisms which affect propagation characteristics are reviewed, and various representations of the propagation constant in the lossy case are defined. Waveguide structures which are susceptable to absorption and/or to leakage loss, in particular silicon-based structures, are discussed. The modeling and analysis of these waveguides by various computational techniques is considered. Two computational methods, the commonly used transfer matrix method and the recently developed impedance boundary method of moments (IBMOM), are re- viewed and extended to the complex domain. A third computational method, which offers improved convergence of the IBMOM for structures with large stepwise changes in refractive index, is formulated. In this approach, the regions containing refractive index discontinuities are replaced by equivalent extended impedance boundary con- ditions, and expansion of the transverse field in the remaining region of continuous refractive index profile is carried out. A significant increase in the rate of conver- gence is demonstrated for various waveguide structures, including an anti-resonant reflecting optical waveguide (ARROW) structure. Two applications of the IBMOM with extended impedance boundary conditions are presented. In the first, the method is applied to the design of a chemical sensor. The sensor, a silicon-based ARROW structure, is designed to measure the refractive index of certain chemical substances with a high degree of accuracy. In a second application, graded index SiON waveguides fabricated at Oregon State University are characterized and compared to the theoretical model. Excellent agreement between the theoretical and measured coupling angles is shown. © Copyright by Catherine A. Rem ley June 20, 1995 All Rights Reserved Analysis and Modeling of Lossy Planar Optical Waveguides and Application to Silicon-Based Structures by Catherine A. Rem ley A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Master of Science Completed June 20, 1995 Commencement June 1996 Master of Science thesis of Catherine A. Rem ley presented on June 20, 1995 APPROVED: Redacted for Privacy Major Professor, representing Electrical and Computer Engineering Redacted for Privacy Head of Department of Electrical and Computer Engineering Redacted for Privacy Dean of Gradu chool I understand that my thesis will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my thesis to any reader upon request. Redacted for Privacy( Catherine A. RemleWAuthor ACKNOWLEDGEMENTS I would like to express my sincere appreciation to a number of people for their con­ tributions to the progress of this work. First and foremost, thanks are due my major advisor Dr. Andreas Weisshaar for guidance and support as well as many helpful discussions through the various stages of this project. With Professor Weisshaar's encouragement, I applied for and was honored to receive, a National Science Founda­ tion Graduate Research Fellowship, which provides financial support for three years. I am grateful to the members of my graduate committee, Drs. Vijai K. Tripathi, John F. Wager, and Mary E. Flahive, for their time and suggestions. Thanks also to Dr. Satish Reddy for helpful discussions on numerical methods. The experimental work was made possible through the efforts of J. John Marlia and Kevin Lite, who fabricated the SiON waveguides. Drs. Jong Wu and Martin Wybourne at the University of Oregon provided the electron beam lithography. The support of friends has been invaluable throughout my studies, with special thanks to Liz Wise, Jon Dorbolo, Peggy Hinsman, Mark Dow, Dave and Trisch Burke, Nanci Snyder, and Anu Krishnaswamy. Finally, a heartfelt thank you to my parents Fred and Anne Rem ley, whose unconditional support and encouragement have been constant sources of inspiration. TABLE OF CONTENTS Page 1 Introduction 1 2 Analysis of Lossy Waveguides 5 2.1 General Modeling Considerations 5 2.2 Loss Mechanisms in Dielectric Waveguides 6 2.2.1 Absorption 7 2.2.2 Scattering 8 2.2.3 Leakage 9 3 Analysis Tools 12 3.1 The Matrix Method 12 3.1.1 Absorbing Structures 16 3.1.2 Leaky Structures 17 3.1.3 Complex Root Search 17 3.2 Impedance Boundary Method of Moments 20 3.2.1 Absorbing Structures 23 3.2.2 Leaky Structures 24 3.3 Extended Boundary Conditions in the IBMOM 24 3.3.1 Extended Boundary Conditions Using Cascaded Trans­ mission Lines Model 26 3.3.2 Extended Boundary Conditions Using the Matrix Method 27 3.3.3 Numerical Results 28 4 Modeling of ARROW Structures 32 4.1 The ARROW Structure 32 4.2 Modeling the ARROW Structure 33 4.3 A Numerical Example 37 4.4 Conclusion 39 TABLE OF CONTENTS (CONTINUED) Page 5 Application: Design of an Optical Waveguide Sensor 41 5.1 Chemical Sensors 41 5.1.1 Methods of Sensing 42 5.1.2 Sensitivity Enhancement 45 5.2 A Silicon-based ARROW Chemical Sensor 46 5.3 Conclusion 49 6 Experimental Results: Measurements of Graded-Index Silicon-based Planar Dielectric Waveguides 51 6.1 Problem Statement 51 6.2 Waveguide Design and Fabrication 53 6.3 Determination of Effective Index 56 6.3.1 Grating Couplers 57 6.3.2 Coupling Angle Measurement 58 6.3.3 Theoretical Analysis 59 6.4 Determination of Loss 60 6.4.1 Loss Measurement Configuration 63 6.4.2 Loss Measurement Results 65 6.4.3 Probable Loss Mechanisms 67 6.5 Suggestions for Further Work 69 7 Summary and Suggestions for Future Work 70 7.1 Summary 70 7.2 Suggestions for Future Work 71 Bibliography 72 TABLE OF CONTENTS (CONTINUED) Page Appendices 83 Appendix A The Propagation Constant 84 Appendix B Absorption and Scattering Mechanisms in a Dielectric 86 LIST OF FIGURES Figure Page 2.1. A planar multilayer waveguide structure in which iii is the re­ fractive index of each homogeneous layer and may be complex, d2 is the thickness of each layer. Propagation is in the z direction. 6 2.2. Two types of leaky structures. In (a) region n3 causes leakage from region nf, and in (b) the waveguide on the right side of the interface is excited below cut-off into a leaky mode 10 3.1. A waveguide with inhomogeneous guiding region. a) Transverse field profile of the two TE modes; b) refractive index profile. 16 3.2. Bracketing scheme used in Muller's method. 18 3.3. A waveguide coupler with equivalent impedance boundary con­ ditions. 20 3.4. Two examples of extended equivalent impedance boundary con­ ditions. a) A step index waveguide coupler. b) A structure with both graded and step index regions 25 3.5. Equivalent representation of the waveguide structure for deter­ mination of extended impedance boundary conditions. 27 3.6. Extraordinary refractive index profile for a waveguide fabricated using both diffusion and proton exchange (TIPE). 30 3.7. Convergence rate of the IBMOM for the mixed structure of TIPE structure with (solid) and without (dashed) extended impedance boundary conditions. 31 3.8. Transverse field profile for the first three TE modes of the TIPE structure. 31 4.1. Refractive index profile of an ARROW structure. 33 4.2. Dispersion and attenuation curves for a typical ARROW struc­ ture. 34 4.3. Equivalent representation of an ARROW structure for analysis using cascaded transmission lines. 35 4.4. The anti-resonant reflecting principle illustrated using the Smith Chart. 36 LIST OF FIGURES (CONTINUED) Page 4.5. The ARROW structure with equivalent impedance boundary conditions. 37 4.6. The ARROW structure with an extended impedance boundary condition. 38 4.7. Dispersion curves for the ARROW structure: a) effective index and b) attenuation for varying thickness of the core layer. 40 5.1. A Mach-Zehnder Interferometer. 42 5.2. Refractive index profile of a refractometer with n, = 1.50, n2 = 1.52, n1 = 1.45pm, d2 = 1.72pm, d1 = 0.15pm 43 5.3. Sensivity of standard dielectric waveguide refractometer. 44 5.4. Change in propagation angle 9 with change in effective index. 46 5.5. Refractive index profile of the ARROW sensor. 47 5.6. Sensivity of an ARROW refractometer for various overlay thick­ nesses. 48 5.7. Comparison of effective indices and attenuation for the ARROW structure with two different cover layers. 49 5.8. Transverse field profile for an ARROW sensor with 0.018gm overlay. Cover refractive indices are a) ncover = 1.333, b) ncover = 1.45. 50 6.1. Various approximations of linear refractive index profiles. a) 5 layers, b) 9 layers, c) 57 layers, d) 111 layers. 52 6.2. The PECVD system. 55 6.3. Coupling angle measurement geometry. 56 6.4. Steps used in electron beam lithography. 58 6.5. Coupling angle measurement stage. 59 6.6. Effect of buffer layer thickness on a) effective index, b) leakage loss. 60 6.7. Loss measurement stage. 64 LIST OF FIGURES (CONTINUED) Page 6.8. Testing the loss measurement stage for repeatability of measure­ ments. 65 6.9. Loss measurement results for a) the TE0 mode of the 5 layer and the 57 layer waveguides, b) the TE0 and TM° modes of the 57 layer waveguide. 66 B.1. Energy diagram for: a) direct interband transitions, b) indirect transitions. hcope refers to a phonon emitted and hwpa refers to a phonon absorbed. After Bhattacharya [32]. 88 LIST OF TABLES Table Page 2.1. Absorption for Various Materials at Ao = 0.6328pm 8 3.1.
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