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Coupled Wave Analysis of Two-Dimensional Second Order COUPLED WAVE ANALYSIS OF TWO-DIMENSIONAL SECOND ORDER SURFACE-EMITTING DISTRIBUTED FEEDBACK LASERS Thesis Submitted to The School of Engineering of the UNIVERSITY OF DAYTON In Partial Fulfillment of the Requirements for The Degree of Master of Science in Electro-Optics By Yangfei Shen UNIVERSITY OF DAYTON Dayton, Ohio May, 2016 COUPLED WAVE ANALYSIS OF TWO-DIMENSIONAL SECOND ORDER SURFACE-EMITTING DISTRIBUTED FEEDBACK LASERS Name: Shen, Yangfei APPROVED BY: ____________________________ ____________________________ Andrew Sarangan, Ph.D. ,P.E. Partha Banerjee, Ph.D. Advisory Committee Chairman Committee Member Professor Director and Professor Electro-Optics Program Electro-Optics Program ____________________________ Qiwen Zhan, Ph.D. Committee Member Professor Electro-Optics Program ____________________________ ____________________________ John G. Weber, Ph.D. Eddy M. Rojas, Ph.D., M.A., P.E. Associate Dean Dean, School of Engineering School of Engineering ii © Copyright by Yangfei Shen All rights reserved 2016 iii ABSTRACT COUPLED WAVE ANALYSIS OF TWO-DIMENSIONAL SECOND ORDER SURFACE-EMITTING DISTRIBUTED FEEDBACK LASERS Name: Shen, Yangfei University of Dayton Advisor: Dr. Andrew Sarangan A distributed feedback laser (DFB) is a type of semiconductor laser where the cavity is periodically structured as a diffraction grating. This allows the same grating region to act as a resonator, out-coupler and the gain region simultaneously. While conventional DFB lasers emit light along its edges, by modifying the grating structure, it is possible to make the output appear from the top surface. This is known as the surface- emitting DFB laser. This thesis will discuss the case in which a two-dimensional grating (which consists of a 2D grid of holes or pillars) etched into the top cladding surface of the waveguide. The thesis consists of four main parts. For the first part, we give the background principles and the coupled wave equations with radiation modes. In the iv second part, we derive the equations for a waveguide that utilizes a grating strip for guidance. The effective index of this guide is then used in the 2D DFB crossed grating structure. The third part begins with calculations of a particular GaAs:GaAlAs waveguide geometry with a grating. We present the fundamental resonant modes by utilizing the numerical shooting method. Then, the matrix solution method is applied in the calculation of the higher order guided and radiation modes. In the last part, we solve the transcendental equations for the eigenvalues to get the threshold gain and near field radiation pattern from the DFB surface. v Dedicated to my parents vi ACKNOWLEDGEMENTS I am greatly indebted to the people who have helped me over the years. Above all, I would like to acknowledge my advisor, Dr. Andrew Sarangan, for his tremendous amount of help and support through my master period. I couldn’t thank him enough. I would like to thank him for his paper 《the novel waveguide utilizing a grating strip for guidance 》 which contributes the key point in my thesis. He also provides very pragmatic methods to calculate random shape of gratings and higher order grating field. Additionally, I am grateful to Jack O'Gorman who enthusiastically helps me in the library searching system, which enables me to search related resources much effectively. Last but not least, I would like to thank my family. My parents are always there for me. None of this could have happened without their support. vii TABLE OF CONTENTS ABSTRACT ....................................................................................................................... iv DEDICATION .................................................................................................................. vii ACKNOWLEDGEMENTS .............................................................................................. vii LIST OF FIGURES ............................................................................................................ x CHAPTER 1 INTRODUCTION OF COUPLE MODE THEORY FOR DISTRIBUTED FEEDBACK ( DFB) STRUCTURES ................................................................................ 1 CHAPTER 2 LATERAL WAVEGUIDING IN A PERIODIC STRUCTURE ............... 11 2.1 Introduction of lateral grating waveguide .............................................................. 11 2.2 Analysis.................................................................................................................. 12 2.3 Applicaion for 2D-DFB ......................................................................................... 21 CHAPTER 3 COUPLED MODE DESCRIPTION OF THE TWO-DIMENSIONAL DFB CAVITY ................................................................................................................. 27 3.1 Utilizing numerical shooting method to calculate 퐸0 ............................................ 27 3.2 Calcultion of Higher order of 퐸푚using matrix solution method ........................... 30 3.3 Calculation of coupling coeficient and other constants ......................................... 33 3.4 Summary & Discussions ....................................................................................... 34 viii CHAPTER 4 NEAR FIELD PROFILE AND THRESHOLD GAIN OF 2-D DFB STRUCTURES ........................................................................................................................ 35 4.1 Near field and threshold gain ................................................................................. 35 4.2 Comparison of 1-D and 2-D .................................................................................. 41 4.3 Summary & Discussions ....................................................................................... 44 CHAPTER 5 THE CURRENT AND FUTURE RESEARCH .......................................... 46 REFERENCES ................................................................................................................. 48 ix LIST OF FIGURES Figure 1-1 3-D schematic view of a 2-D surface-emitting laser and second-order DFB gratings with GaAs-Au gratings ........................................................................................ 3 Figure 1-2 A guided-wave DFB structure with a grating .................................................. 5 Figure 1-3 Relationship between amplitude threshold gain and detuning factor coefficient of a mirrorless first order indux-coupled DFB laser diode ............................. 10 Figure 2-1 Lateral Grating waveguide structure .............................................................. 12 Figure 2-2 Modes of the grating waveguide at 1.0 휇푚, with 푊 = 100 휇푚, a grating −3 period of 0.5 휇푚, 푛0 = 3.2, ∆푛 = 3.2 × 10 , and for various of 휅푊 ........................... 18 Figure 2-3 First two modes of the grating waveguide for different values of 휅푊 ............ 19 Figure 2-4 Holes etched on the top of cladding surface in lab ............................................. 21 Figure 2-5 Cosine function simulation of the holes on cladding surface ........................... 22 Figure 2-6 A guided-wave laser with “cosine” shape grating, where g is the grating height while t is the outer guide thickness. ....................................................................... 22 + − Figure 2-7 Real part of 푛 (푥) and 푛 (푥) vs. depth x with 푛1, 푛3 = 3.2, 푛2 = 3.3, 휆 = 1.0 휇푚, Λ = 0.5휇푚, = 1 휇푚, 푡 = 1.5 휇푚 ..................................................................... 24 x Figure 2-8 Differ from conventional DFB, 푛+ and 푛− change with z not only x............ 24 Figure 2-9 Real part of 푛+. .............................................................................................. 25 Figure 2-10 Real part of 푛− ............................................................................................. 25 Figure 2-11 Imaginary part of 푛+ and 푛− ........................................................................ 26 Figure 3-1 퐸0 in 2-D DFB waveguide (we seperate x from -2.5-3.5 into 9999 steps) .... 29 (0) Figure 3-2 The partial waves 퐸푚 for m=-8-5 (m ≠ 0, −2) ............................................ 32 Figure 4-1 Intensity of multiple modes as a function of L for the case of 푛+ ................. 37 Figure 4-2 Intensity of multiple modes as a function of L for the case of 푛− ................. 38 Figure 4-3 Threshold gain for different L for both 푛+ (circle and diamond points) and 푛− (rectangular and star points), red lines represent symmetric modes while blue lines represent antisymmetric modes......................................................................................... 39 Figure 4-4 Graphic representation of the relationship between the wave vectors and the guided mode, the radiation modes, and the grating of a simple thin-film wave guide ..... 40 Figure 4-5 Intensity (1D-DFB) of different modes in different cavity length ................ 42 Figure 4-6 Threshold gain (1D-DFB) for different modes with the same L as 2-D case ........................................................................................................................................... 43 xi CHAPTER 1 INTRODUCTION OF COUPLED MODE THEORY FOR DISTRIBUTED FEEDBACK (DFB) STRUCTURES Distributed feedback lasers today are the workhorse of fiber communication applications because they are able to produce single frequency, single spatial mode emission. They are also dynamically stable, meaning they maintain their single frequency behavior under varying operating conditions, which is important in a fiber communication system
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