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Modeling the Optical Processes in Semiconductor Lasers

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Modeling the Optical Processes in Semiconductor Lasers Research Collection Doctoral Thesis Modeling the optical processes in semiconductor lasers Author(s): Witzig, Andreas Publication Date: 2002 Permanent Link: https://doi.org/10.3929/ethz-a-004407405 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Diss. ETH No. 14694 Modeling the Optical Processes in Semiconductor Lasers A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH for the degree of Doctor of Technical Science presented by ANDREAS WITZIG Dipl. El.-Ing. ETH born 26 10 1973 citizen of Laufen-Uhwiesen, Switzerland accepted on the recommendation of Prof. Dr. Wolfgang Fichtner, examiner Prof. Dr. Hans Melchior, co-examiner 2002 Contents Acknowledgements vii Abstract ix Zusammenfassung xi 1 Introduction 1 1.1 Motivation 1 1.2 Active Optoelectronic Devices 2 1.3 Optoelectronic Device Simulation 6 1.4 Contents 7 2 Physics of Optoelectronic Devices 9 2.1 Introduction 9 2.2 Basic Semiconductor Laser Physics 10 2.3 Empirical Laser Model: The Lumped Laser Equations 11 2.4 Eigenmode Laser Equations 18 2.4.1 Electro-Thermal Model 18 2.4.2 Optical Model 19 2.4.3 Local Electron-Photon Interaction 23 2.4.4 Device-Specific Models 26 2.4.5 Extension of the Eigenmode Laser Equations . 28 2.4.6 Beyond the Adiabatic Approximation 30 2.5 More Fundamental Laser Models 31 2.5.1 Semiconductor Bloch Equations 32 2.5.2 Slowly Varying Amplitude Approximation ... 32 i ii CONTENTS 2.6 Characteristic Time Scales 33 2.7 Diffraction Theory 36 2.8 Discussion 39 3 Optical Cavities and Waveguides 41 3.1 Introduction 41 3.2 Optical Cavities 42 3.2.1 Empirical Cavity Model 42 3.2.2 Cavity Eigenvalue Equation 44 3.2.3 Mode Designation for VCSELs 46 3.2.4 ID Cavity 49 3.2.5 Spectral Properties 52 3.3 Optical Waveguides 54 3.3.1 Waveguide Eigenvalue Equation 54 3.3.2 Slab Waveguides 55 3.3.3 Waveguide Spectrum 56 3.4 Mathematical Background 59 3.4.1 Eigenvalues and Eigenvectors 59 3.4.2 Quasi-Normal Modes 60 3.5 Discussion 62 4 Implementation 65 4.1 Introduction 65 4.2 Analysis of the Governing Equations 66 4.3 Traveling-Wave Optics 67 4.3.1 Finite-Difference Time-Domain Method .... 67 4.3.2 Transfer-Matrix Method 68 4.4 Hierarchy of Approximations for Waveguide Solutions 70 4.5 Numerical Eigenmode Solution 73 4.6 Eigenmode Laser Equations 74 4.7 Discussion 78 5 Results 81 5.1 Introduction 81 5.2 Edge-Emitting Laser Diodes 82 5.2.1 Radiation Leakage into Substrate 82 5.2.2 Spontaneous Emission Coupling 83 CONTENTS m 5.3 Surface-Emitting Lasers 89 5.3.1 Finite-Element Optical Mode Calculation ... 89 5.3.2 Optical Modes Derived by FDTD 90 5.4 Discussion 95 6 Conclusion and Outlook 99 6.1 Major Results 99 6.2 Further Development 100 A Maxwell's Equations 105 B Semiconductor Dispersion Relations 107 C Coupled Semiconductor Equations 111 D Notation 115 Bibliography 120 Curriculum Vitae 131 List of Figures 1.1 Edge-Emitting Laser 4 1.2 Vertical-Cavity Surface-Emitting Laser 5 1.3 Semiconductor Laser Design Methodology 6 2.1 Laser Characteristics 13 2.2 Modulation Response 14 2.3 Time-Domain Response 15 2.4 3D Optical Field in a DFB Laser 29 2.5 Schematic Device Physics for Diode Lasers 35 2.6 Geometry Definition for the Diffraction Theory .... 36 3.1 Body-of-Revolution Expansion of VCSEL Modes ... 48 3.2 Optical Eigenmode of a ID Cavity 50 3.3 Spectral Properties of Bragg Mirrors 52 3.4 Slab Waveguide 55 3.5 Waveguide Spectrum of Slab Structure 56 4.1 Simple Simulation Flow 76 4.2 Simulation Flow for Optical Re-Solving 76 5.1 Radiation Leakage into Substrate 82 5.2 Results of the Waveguide Eigenvalue Problem 84 5.3 Setup and Results of the FDTD Simulation 85 5.4 EEL-Simulation: Time-Domain Results 86 5.5 EEL-Simulation: Relative Power vs. Distance 87 5.6 Finite-Element Result of VCSEL Eigenmodes 90 5.7 FDTD Result of Steady-State VCSEL Field 91 v VI LIST OF FIGURES 5.8 Mode Pattern in the Active Region 92 5.9 Comparing Devices with Different Aperture Diameters 93 5.10 3D Simulation of a Tilted-Pillar VCSEL 94 Acknowledgements First of all, I would like to thank Prof. Dr. Wolfgang Fichtner for his support and for his confidence in my work. He provided an excellent working environment and put a special effort into our group. Only through him, we got to establish the connections to leading research groups and industrial partners which proved to be essential for our work. I also wish to thank Prof. Dr. Hans Melchior for accepting to be the co-examiner of this thesis and for carefully reviewing the manuscript. My interest in numerical simulations was initially sparked in an undergraduate student project. I am grateful to Dr. Pascal Leucht¬ mann who supported this project actively and taught me the secrets of electromagnetics. Furthermore, one of the reasons I finally ended up in optoelectronics was an ERASMUS exchange project with Bristol University, UK. I would like to thank Prof. Dr. John Gowar for giving me this opportunity and for spending a lot of time in discussions with me. My start at the Integrated Systems Laboratory (Institut für Inte¬ grierte Systeme, IIS) has been in the group of Dr. Peter Regli from whom I learned a lot about programming and software integration. The close collaboration with Dr. Christian Schuster, Dr. Thomas Körner, and Andreas Christ was fruitful and both a personal and professional enrichment. At this time, Dr. Bernd Witzigmann started with the laser extension of the device simulator DESSIS-ISE. I ad¬ mire Bernd for many decisions that he made in the early stage of this project. I wish to thank him for his patience and for explaining over and over again the details of semiconductor physics. Michael Pfeiffer, Matthias Streiff, Lutz Schneider, and Serguei Vll vm Acknowledgements Chevtchenko joined the group during the last three years. The in¬ teraction within our group has become considerable, and the Opto¬ electronics Modeling Group gained momentum. A great part of the motivation in my work I had from the good atmosphere in our office. I thank them all very much! As the Optoelectronics Modeling Group expanded, administrative issues and funding got some importance. I wish to express my grat¬ itude to Dr. Dölf Aemmer, the coordinator of the research activities in technology computer aided design (TCAD) at IIS. He was always there for us and I learned a lot from him about the formulation and planning of new projects. From the beginning of the optoelectronics activities at IIS, the col¬ laboration with industry was very important. First of all comes the ISE Integrated Systems Engineering AG who provided the basis for a professional software development. I am grateful to the ISE soft¬ ware developers who showed a lot of confidence in our work. On the hardware side, we had the chance to collaborate with a laser manufac¬ turer that was an off-spring of the IBM Research Lab in Riischlikon, later owned by JDS Uniphase and then by Nortel Networks. The collaboration with the strong R&D group was very profitable. Later, collaborations with Avalon Photonics Inc. and Opto Speed AG have been established. I want to thank all these partners for their active support. During my work at IIS I had the opportunity to advise or sev¬ eral students performing undergraduate research in our group. There has been considerable output from their work and we had many in¬ teresting discussions. I wish to thank Volker Mornhinweg, Thomas Eberle, Pamela Lässer, Kersten Schmidt and Adrian Bregy. In regard of the student projects, I must also thank Dr. Norbert Felber, who coordinated them. In an advanced stage of my work, Prof. Dr. Joachim Piprek invited me to the University of California in Santa Barbara (UCSB). The work at UCSB was funded by a joint ETH-UCSB scholarship that supports research collaborations between the two institutions. It was an important experience for myself and an invaluable enrichment for the development of our simulator. Abstract In modern telecommunication systems, semiconductor lasers are key components. This doctoral thesis presents a physics-based approach to simulate active optoelectronic devices. It covers the analysis of a broad range of devices, including Edge-Emitting Lasers (EELs) and Vertical-Cavity Surface-Emitting Lasers (VCSELs). A numeri¬ cal solver for the optical field has been implemented, and an interface to an electro-thermal device simulator has been built. Contributions have been made in the formulation of the physical models, and in the coupling scheme between electronics and optics. For EELs, the optical field can be separated into transverse waveg¬ uide modes and longitudinal cavity modes. The transverse eigen¬ modes have been solved in 2D applying a general but efficient finite- element method. The longitudinal modes have been calculated using the transfer-matrix method. For VCSELs, a separation into trans¬ verse and longitudinal direction is not possible. Instead, the fields can be expanded into a Fourier series and the eigenmodes are solved in 2D. In addition to the finite-element solution of the optical eigenvalue problem, the finite-difference time-domain method has been applied for the calculation of the electromagnetic wave propagation. In a post¬ processing step, the optical eigenmodes of general VCSELs have been obtained. Furthermore, the spontaneous emission coupling factor is calculated rigorously by a full-3D treatment of an EEL structure.
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