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Sinusoidal anti-coupling symmetric strip waveguides on a silicon-on-insulator platform Zhang, Fan 2017

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Sinusoidal Anti-coupling SymmetricStrip Waveguides on aSilicon-on-insulator PlatformbyFan ZhangB.A.Sc., The University of British Columbia, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2017c© Fan Zhang 2017AbstractSinusoidal anti-coupling (AC) symmetric waveguides provide a means to de-sign dense waveguide arrays that have minimal inter-waveguide crosstalk forhigh-density integration of photonic circuits. Also, the polarization sensitiv-ity of sinusoidal AC symmetric waveguides and the reduction of wavelengthdependence that is achieved by the sinusoidal waveguides can be used todesign broadband polarization beam splitters (PBSs) for polarization di-versity systems. In this thesis, I demonstrate the use of sinusoidal bendsto suppress the optical power exchange between pairs of symmetric stripwaveguides for both transverse-electric (TE) and transverse-magnetic (TM)modes as well as to separate the TE and TM modes into two output sym-metric strip waveguides on a silicon-on-insulator platform. I design, model,simulate, and analyze sinusoidal AC symmetric waveguide pairs for both theTE and TM modes. Then, based on the TE sinusoidal AC waveguide struc-ture, I design, simulate, and analyze a PBS using a symmetric directionalcoupler (DC) with sinusoidal bends. I also compare the modal dispersionsof the sinusoidally-bent symmetric DC, which is used in the PBS, with themodal dispersions of an equivalent straight symmetric DC. I measure thefabricated test devices and evaluate their performances.The TE sinusoidal AC device, which has a gap width of 200 nm, has anaverage crosstalk suppression ratio (SR) of 38.2 dB, and the TM sinusoidaliiAbstractAC device, which has a gap width of 600 nm, has an average crosstalk SRof 34.9 dB over an operational bandwidth of 35 nm. The PBS has a smallcoupler length of 8.55 µm, has average extinction ratios of 12.0 dB for theTE mode and of 20.1 dB for the TM mode, and has average polarizationisolations of 20.6 dB for the through port (the TE mode over the TM mode)and of 11.5 dB for the cross port (the TM mode over the TE mode) overa broad operational bandwidth of 100 nm. All of my devices are easy tofabricate and compatible with complementary metal-oxide-semiconductortechnologies.iiiLay SummaryIn modern communications systems, light is often used to carry informa-tion via optical waveguides in photonic circuits. When straight symmetricwaveguides are placed near each other, light will transfer, or couple, betweenthe waveguides. In this thesis, I present sinusoidal anti-coupling symmetricsilicon waveguide pairs that eliminate this coupling. Without the coupling,the footprints of photonic circuits can be reduced.There are two light propagation modes in the waveguides with rectan-gular cross-sections that are often used in photonic circuits: transverse-electric and transverse-magnetic modes. Sinusoidal anti-coupling symmet-ric waveguides can be used to either separate or combine these two typesof modes. Hence, in this thesis, I also present a silicon polarization beam-splitter/combiner using sinusoidal anti-coupling symmetric waveguides. Thisdevice can be utilized to enhance the information-carrying capacity of opti-cal communications systems.ivPrefaceI am the main author of a journal paper, “Compact broadband polarizationbeam splitter using a symmetric directional coupler with sinusoidal bends[1], and a conference paper, “Sinusoidal anti-coupling SOI strip waveguide”[2]. I presented a sinusoidal anti-coupling (AC) symmetric strip waveguidepair for transverse-electric mode and a polarization beam splitter (PBS)using a symmetric strip waveguide directional coupler (DC) with sinusoidalbends on an SOI platform.I calculated the design parameters for my devices using an analyti-cal model and a finite-difference eigenmode solver (MODE Solutions fromLumerical Solutions, Inc.). For the PBS, I also derived the modal disper-sion to analyze the wavelength dependence of a sinusoidally-bent symmetricDC, which was used in the PBS. I optimized the design parameters for fab-rication using a finite-difference time-domain solver (FDTD Solutions fromLumerical Solutions, Inc.). Then, I drew the test devices on mask layouts. Inorder to evaluate the performance of the TE sinusoidal AC device, I added astraight symmetric DC and a straight AC asymmetric waveguide pair, whichhad the same gap width and coupler length as the TE sinusoidal AC device,to the mask layouts. The test devices were fabricated using electron-beamlithography at the University of Washington. I measured the fabricated testdevices using an automated probe station, which was developed by Han Yun,vPrefaceJonas Flu¨ckiger, Charlie Lin, Stephen Lin, and Michael Caverley, in our labat the University of British Columbia. Finally, I evaluated the performancesof those devices using the simulation and measurement data.In this thesis, Section 1.2.2 of Chapter 1, Sections 2.1, 2.3, 2.4 and 2.5 ofChapter 2, Section 3.5 of Chapter 3, Sections 4.1, 4.2, and 4.3.3 of Chapter 4,and Section 5.3 of Chapter 5 are based on the work in the paper [1]. Alsoin this thesis, Section 1.2.1 of Chapter 1, Sections 2.1 and 2.4 of Chapter 2,Section 3.3 of Chapter 3, and Sections 4.1, 4.2, and 4.3.1 of Chapter 4 arebased on the work in the paper [2].My supervisor, Dr. Nicolas A. F. Jaeger, provided me with essentialguidance, advice, and support for my research and helped me write andedit my papers. Dr. Lukas Chrostowski provided me with valuable insightsregarding design and fabrication issues and useful suggestions to help meimprove my papers. My colleague, Han Yun, also helped me edit my papers.Here is a copyright notice for the contents that are reproduced in thisthesis with permissions from my papers [1, 2]:c©2015 and 2017 Optical Society of America. One print or electroniccopy may be made for personal use only. Systematic reproduction and dis-tribution, duplication of any material in this paper for a fee or for commercialpurposes, or modifications of the content of this paper are prohibited.Here is a list of my publications as the main author:1. Fan Zhang, Han Yun, Yun Wang, Zeqin Lu, Lukas Chrostowski,and Nicolas A. F. Jaeger. Compact broadband polarization beam splitterusing a symmetric directional coupler with sinusoidal bends. Optics Letters,42(2):235-238, Optical Society of America, 2017.2. Fan Zhang, Han Yun, Valentina Donzella, Zeqin Lu, Yun Wang,Zhitian Chen, Lukas Chrostowski, and Nicolas A. F. Jaeger. SinusoidalviPrefaceanti-coupling SOI strip waveguides. In CLEO: Science and Innovations,pages SM1I-7. Optical Society of America, 2015.During my Master’s studies, I participated in various research projects.In these projects, I worked with my colleagues to design, analyze, and mea-sure various optical devices, including broadband couplers, photonic crys-tals, optical pulse shaping devices, polarization beam splitters, Bragg gratingfilters, and sub-wavelength grating couplers. A list of publications for theprojects is given in Appendix B.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . xxvAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . .xxviiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Silicon Photonics in Optical Communications . . . . . . . . . 11.2 Thesis Motivation and Objective . . . . . . . . . . . . . . . . 31.2.1 Sinusoidal Anti-coupling Symmetric Waveguides . . . 31.2.2 Polarization Beam Splitter Using a Symmetric Direc-tional Coupler with Sinusoidal Bends . . . . . . . . . 4viiiTable of Contents1.3 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . 52 Design and Theory . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Design Overview . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Analytical Model of a Straight Symmetric DC . . . . . . . . 92.3 Optical Transmissions of a Symmetric DC . . . . . . . . . . 172.4 Sinusoidal AC Bends for a Symmetric Waveguide Pair . . . . 172.5 Modal Dispersions between the Supermodes of SymmetricDCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Simulation and Analysis . . . . . . . . . . . . . . . . . . . . . 223.1 Simulation and Analysis Overview . . . . . . . . . . . . . . . 233.2 Single Straight Waveguide . . . . . . . . . . . . . . . . . . . 263.3 Coupling and AC Devices for TE Operation . . . . . . . . . 293.3.1 TE Straight Symmetric DC . . . . . . . . . . . . . . 323.3.2 TE Sinusoidal AC Symmetric Waveguide Pair . . . . 343.3.3 Study for TM Operation . . . . . . . . . . . . . . . . 373.4 Coupling and AC Devices for TM Operation . . . . . . . . . 433.4.1 Selection of Gap Width . . . . . . . . . . . . . . . . . 433.4.2 TM Straight Symmetric DC . . . . . . . . . . . . . . 503.4.3 TM Sinusoidal AC Symmetric Waveguide Pair . . . . 523.5 PBS using a Symmetric DC with Sinusoidal Bends . . . . . . 543.5.1 Optimization . . . . . . . . . . . . . . . . . . . . . . . 543.5.2 Modal Dispersion . . . . . . . . . . . . . . . . . . . . 603.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62ixTable of Contents4 Fabrication, Measurement, and Demonstration . . . . . . . 634.1 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3 Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3.1 Sinusoidal AC Symmetric Waveguide Pair for TE Op-eration . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3.2 Sinusoidal AC Symmetric Waveguide Pair for TM Op-eration . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3.3 PBS Using a Symmetric DC with Sinusoidal Bends . 784.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825 Summary, Conclusions, and Suggestions for Future Work 835.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.3 Suggestions for Future Works . . . . . . . . . . . . . . . . . . 85Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89AppendicesA Derivation of the Propagation Constant Difference of a Sinusoidally-bent Symmetric DC . . . . . . . . . . . . . . . . . . . . . . . . 99B Additional Publications . . . . . . . . . . . . . . . . . . . . . . 111xList of Tables3.1 Simulation parameters. . . . . . . . . . . . . . . . . . . . . . . 233.2 Analysis parameters. . . . . . . . . . . . . . . . . . . . . . . . 243.3 Design parameters. . . . . . . . . . . . . . . . . . . . . . . . . 253.4 Simulated and calculated values of the neff,wgs, ng,wgs, andβwgs for both mode types at λ0 = 1550 nm of the singlestraight waveguide. . . . . . . . . . . . . . . . . . . . . . . . . 283.5 Simulated and calculated values of the neff s and βs for boththe TE and TM supermodes at λ0 = 1550 nm of the straightsymmetric waveguide pair, which has G = 200 nm. . . . . . . 313.6 Calculated and optimized values of the Lc,mins for both TEoperation and TM operation at λ0 = 1550 nm of the straightsymmetric DCs, which have the same G = 200 nm (see Ref. [2]). 383.7 Calculated values of the ATMmins and optimized ATMcross,mins forthe operation at λ0 = 1550 nm of the sinusoidal symmetricwaveguide pairs, which have G = 200 nm and different Λs. . . 403.8 Simulated and calculated values of the neff s and βs for boththe even and odd TM supermodes at λ0 = 1550 nm of thestraight symmetric waveguide pairs, which have different Gs. 43xiList of Tables3.9 Calculated and optimized values of the LTMc,mins for the oper-ation at λ0 = 1550 nm of the straight symmetric DCs, whichhave different Gs. . . . . . . . . . . . . . . . . . . . . . . . . . 463.10 Calculated and optimized values of the ATMmins for the opera-tion at λ0 = 1550 nm of the sinusoidal AC symmetric waveg-uide pairs, which have different Gs. . . . . . . . . . . . . . . . 503.11 Calculated values of the ∆βbents and ∆βstraights and absolutevalues of their ratios for both mode types at λ0 = 1550 nmof the bent and straight DCs. . . . . . . . . . . . . . . . . . . 593.12 Calculated values of the Dbents and Dstraights and absolutevalues of their ratios for both mode types at λ0 = 1550 nmof the bent and straight DCs. . . . . . . . . . . . . . . . . . . 604.1 Design parameters of the TE and TM sinusoidal AC symmet-ric waveguide pairs and a PBS using a symmetric DC withsinuosidal bends (see Refs. [1, 2]). . . . . . . . . . . . . . . . . 704.2 Minimum, average, and maximum values of the SRs of boththe TE sinusoidal and straight AC devices for TE operationover the C-band (see Ref. [2]). . . . . . . . . . . . . . . . . . . 714.3 Minimum, average, and maximum values of the SRs of boththe TM sinusoidal and straight AC devices for TM operationover the C-band. . . . . . . . . . . . . . . . . . . . . . . . . . 754.4 Minimum, average, and maximum values of the ERs and PIsof the PBS for both TE operation and TM operation over awavelength range from 1470 nm to 1570 nm. Adapted withpermission from Ref. [1], c©2017 Optical Society of America. 80xiiList of Figures2.1 Cross-sectional view of a pair of parallel symmetric strip waveg-uides on an SOI platform. Adapted with permission fromRef. [1], c©2017 Optical Society of America. . . . . . . . . . . 82.2 Top view of a full period of a symmetric DC with sinusoidalbends. Adapted with permission from Ref. [1], c©2017 OpticalSociety of America. . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Top view of a straight symmetric DC. . . . . . . . . . . . . . 102.4 1-D normalized transverse field distributions for the even TEsupermode (left) and odd TE supermode (right), ψe(x) andψo(x), over their respective maximum values, ψe,max and ψo,max. 112.5 1-D normalized transverse field distributions for the even TMsupermode (left) and odd TM supermode (right), ψe(x) andψo(x), over their respective maximum values, ψe,max and ψo,max. 122.6 1-D normalized transverse field distributions for the TE localnormal modes in waveguide cores a (left) and waveguide coreb (right), ψa(x) and ψb(x), over their respective maximumvalues, ψa,max and ψb,max. . . . . . . . . . . . . . . . . . . . . 13xiiiList of Figures2.7 1-D normalized transverse field distributions for the TM localnormal modes in waveguide core a (left) and waveguide coreb (right), ψa(x) and ψb(x), over their respective maximumvalues, ψa,max and ψb,max. . . . . . . . . . . . . . . . . . . . . 142.8 Bessel functions of the first kind of orders 0 and 1 with respectto a variable, xvar, ranging from 0 to 10. . . . . . . . . . . . . 183.1 Port configuration in my simulations. . . . . . . . . . . . . . . 263.2 Perspective and cross-sectional views of a single straight waveg-uide, which has W = 500 nm and H = 220 nm. . . . . . . . . 273.3 Simulated wavelength-dependent effective refractive indicesfor both the fundamental TE and TM modes of the singlestraight waveguide. . . . . . . . . . . . . . . . . . . . . . . . . 273.4 Simulated 2-D mode profiles for the fundamental TE mode(left) and the fundamental TM mode (right) at λ0 = 1550 nmof the single straight waveguide, which has W = 500 nm andH = 220 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.5 Perspective and cross-sectional views of a straight symmetricwaveguide pair, which has W = 500 nm, H = 220 nm, andG = 200 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.6 Simulated wavelength-dependent effective refractive indicesfor (a) the even TE and TM supermodes and (b) the oddTE and TM supermodes of the straight symmetric waveguidepair, which has G = 200 nm. . . . . . . . . . . . . . . . . . . 30xivList of Figures3.7 Simulated 2-D mode profiles for the even and odd TE su-permodes (upper left and right, respectively) and the evenand odd TM supermodes (lower left and right, respectively)at λ0 = 1550 nm of the straight symmetric waveguide pair,which has G = 200 nm. . . . . . . . . . . . . . . . . . . . . . 313.8 Simulated optical transmissions of the cross and through portsas functions with respect to L for TE operation at λ0 = 1550 nmof straight symmetric DCs, which have G = 200 nm. . . . . . 333.9 Simulated 2-D power distribution profile for the operationat λ0 = 1550 nm of the straight symmetric DC, which hasG = 200 nm and L = 32.88 µm, when a fundamental TEmode is launched into Port 1 of the device. . . . . . . . . . . 343.10 Simulated optical transmissions of the cross and through portsas functions with respect toA for TE operation at λ0 = 1550 nmof sinusoidal symmetric waveguide pairs, which have theG = 200 nmand Λ = 16.44 µm. . . . . . . . . . . . . . . . . . . . . . . . . 353.11 Simulated 2-D power distribution profiles for the operation atλ0 = 1550 nm of the sinusoidal AC symmetric waveguide pair,which has G = 200 nm, L = 32.88 µm, and A = 932 nm, whena fundamental TE mode is launched into each of (a) Port 1and (b) Port 3 of the same device. Adapted with permissionfrom Ref. [2], c©2015 Optical Society of America. . . . . . . . 353.12 Perspective view of the TE sinusoidal AC device. Adaptedwith permission from Ref. [2], c©2015 Optical Society of Amer-ica. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36xvList of Figures3.13 Simulated optical transmissions of the cross and through portsas functions with respect to L for TM operation at λ0 = 1550 nmof straight symmetric DCs, which have G = 200 nm. . . . . . 373.14 Simulated 2-D power distribution profile for the operation atλ0 = 1550 nm of the TE straight symmetric DC, which hasG = 200 nm and L = 32.88 µm, when a fundamental TMmode is launched into Port 1 of the device. . . . . . . . . . . 383.15 Simulated optical transmissions of the cross and through portsas functions with respect toA for TM operation at λ0 = 1550 nmof sinusoidal symmetric waveguide pairs, which haveG = 200 nmand have (a) Λ = 2 µm, (b) Λ = 4 µm, (c) Λ = 8 µm, and(d) Λ = 16.44 µm. . . . . . . . . . . . . . . . . . . . . . . . . 393.16 Simulated 2-D power distribution profiles for the operation atλ0 = 1550 nm of the sinusoidal symmetric waveguide pairs,which haveG = 200 nm and have (a) Λ = 4 µm, (b) Λ = 8 µm,and (c) Λ = 16.44 µm, when a fundamental TM mode islaunched into Port 1 of each device. . . . . . . . . . . . . . . 413.17 Simulated 2-D power distribution profile for the operation atλ0 = 1550 nm of the TE sinusoidal AC symmetric waveguidepair, which has G = 200 nm, Λ = 16.44 µm, A = 932 nm,and L = 32.88 µm, when a fundamental TM mode is launchedinto Port 1 of the device. . . . . . . . . . . . . . . . . . . . . . 423.18 Simulated wavelength-dependent effective refractive indicesfor both the even TE and TM supermodes of the straightsymmetric waveguide pairs, which have (a) G = 300 nm, (b)G = 400 nm, (c) G = 500 nm, and (d) G = 600 nm. . . . . . 44xviList of Figures3.19 Simulated wavelength-dependent effective refractive indicesfor both the odd TE and TM supermodes of the straightsymmetric waveguide pairs, which have (a) G = 300 nm, (b)G = 400 nm, (c) G = 500 nm, and (d) G = 600 nm. . . . . . 453.20 Simulated optical transmissions of the cross and through portsas functions with respect to L for TM operation at λ0 = 1550 nmof straight symmetric DCs, which have (a) G = 300 nm, (b)G = 400 nm, (c) G = 500 nm, and (d) G = 600 nm. . . . . . 473.21 Simulated optical transmissions of the cross and through portsas functions with respect toA for TM operation at λ0 = 1550 nmof sinusoidal symmetric waveguide pairs, which have (a)G = 300 nm,(b) G = 400 nm, (c) G = 500 nm, and (d) G = 600 nm. . . . 483.22 Simulated 2-D power distribution profiles for the operationat λ0 = 1550 nm of the sinusoidal AC symmetric waveguidepairs, which have (a) G = 300 nm, (b) G = 400 nm, (c)G = 500 nm, and (d) G = 600 nm, when a fundamental TMmode is launched into Port 1 of each device. . . . . . . . . . . 493.23 Simulated 2-D mode profiles for the even and odd TE su-permodes (upper left and right, respectively) and the evenand odd TM supermodes (lower left and right, respectively)at λ0 = 1550 nm of the straight symmetric waveguide pair,which has G = 600 nm. . . . . . . . . . . . . . . . . . . . . . 513.24 Perspective and cross-sectional views of a straight symmetricwaveguide pair, which has W = 500 nm, H = 220 nm, andG = 600 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51xviiList of Figures3.25 Simulated 2-D power distribution profile for the operationat λ0 = 1550 nm of the straight symmetric DC, which hasG = 600 nm and L = 39.72 µm, when a fundamental TMmode is launched into Port 1 of the device. . . . . . . . . . . 523.26 Simulated 2-D power distribution profiles for the operationat λ0 = 1550 nm of the sinusoidal AC symmetric waveg-uide pair, which has , Λ = 19.86 µm, ATMmin = 1025 nm, andL = 39.72 µm, when a fundamental TM mode is launchedinto each of (a) Port 1 and (b) Port 3 of the same device. . . 533.27 Perspective view of the TM sinusoidal AC device. . . . . . . . 533.28 Perspective view of the PBS. Adapted with permission fromRef. [1], c©2017 Optical Society of America. . . . . . . . . . . 553.29 Simulated optical transmissions (on a linear scale) of the crossand through ports as functions with respect to L for (a) TEoperation and (b) TM operation at λ0 = 1550 nm of sym-metric DCs with sinusoidal bends, which have G = 200 nm,Λ = 16.44 µm, and A = 932 nm. Adapted with permissionfrom Ref. [1], c©2017 Optical Society of America. . . . . . . . 553.30 Simulated optical transmissions (on a logarithmic scale) of thecross and through ports as functions with respect to L for (a)TE operation and (b) TM operation at λ0 = 1550 nm of sym-metric DCs with sinusoidal bends, which have G = 200 nm,Λ = 16.44 µm, and A = 932 nm. Adapted with permissionfrom Ref. [1], c©2017 Optical Society of America. . . . . . . . 56xviiiList of Figures3.31 Simulated 2-D power distribution profiles for the operation atλ0 = 1550 nm of the PBS, when (a) a fundamental TE modeand (b) a fundamental TM mode are launched into Port 1of the same device. Adapted with permission from Ref. [1],c©2017 Optical Society of America. . . . . . . . . . . . . . . . 563.32 Simulated 2-D power distribution profiles for the operation atλ0 = 1550 nm of the PBS, when (a) a fundamental TE modeand (b) a fundamental TM mode are launched into Port 2 ofthe same device. . . . . . . . . . . . . . . . . . . . . . . . . . 573.33 Simulated 2-D power distribution profiles for the operation atλ0 = 1550 nm of the PBS, when (a) a fundamental TE modeand (b) a fundamental TM mode are launched into Port 3 ofthe same device. . . . . . . . . . . . . . . . . . . . . . . . . . 583.34 Simulated 2-D power distribution profiles for TM operation atλ0 = 1550 nm of the PBS, when (a) a fundamental TE modeand (b) a fundamental TM mode are launched into Port 4 ofthe same device. . . . . . . . . . . . . . . . . . . . . . . . . . 583.35 Calculated wavelength-dependent propagation constant dif-ferences for both the TE and TM modes of the sinusoidally-bent and straight symmetric DCs, which have G = 200 nm. . 593.36 Calculated wavelength-dependent modal dispersions for boththe TE and TM modes of the sinusoidally-bent and straightsymmetric DCs, which have G = 200 nm. . . . . . . . . . . . 614.1 Design layout (left) of the TE sinusoidal AC symmetric waveg-uide pair with an SEM image (right) of the sinusoidal sym-metric waveguide structure in the device. . . . . . . . . . . . 64xixList of Figures4.2 Design layout (left) of the TM sinusoidal AC symmetric waveg-uide pair with an SEM image (right) of the sinusoidal sym-metric waveguide structure in the device. . . . . . . . . . . . 654.3 Design layouts of the PBSs for TE operation (upper left)and TM operation (lower left) with an SEM image (right)of the sinusoidal symmetric waveguide structure in one ofthe devices. Adapted with permission from Ref. [1], c©2017Optical Society of America. . . . . . . . . . . . . . . . . . . . 664.4 Schematic of an SOI Platform. . . . . . . . . . . . . . . . . . 664.5 Design layout (left) of the TE reference device with a close-upview (right) of one of the TE SWGCs that were used in theTE reference device and the TE sinusoidal AC device. . . . . 674.6 Design layout (left) of the TE reference device with a close-upview (right) of one of the TE SWGCs that were used in theTE reference device and the PBS for TE operation. . . . . . . 674.7 Design layout (left) of the TM reference device with a close-up view (right) of one of the TM SWGCs that were used inthe TM reference device, the TM sinusoidal AC device, andthe PBS for TM operation. . . . . . . . . . . . . . . . . . . . 68xxList of Figures4.8 Automated optical fibre probe station: 1. Agilent 8164Ameasurement system; 2. Agilent 81635A dual optical powersensors; 3. Agilent 81682A tunable laser source; 4. PLCConnections and OZ Optics polarization-maintaining fibres;5. PLC Connections optical fibre array; 6. Thorlabs BBD203motor controller; 7. Standford Research Systems LDC501temperature controller; 8. metallic stage; 9. Tucsen micro-scope camera; 10. AmScope microscope lamp; 11. desktopcomputer; 12. Newport RS4000 optical table. . . . . . . . . . 694.9 Simulated and normalized measured optical transmission spec-tra for the through and cross ports of the TE sinusoidal ACsymmetric waveguide pair as well as normalized measuredoptical transmission spectrum for the cross port of the equiv-alent TE straight symmetric DC. Adapted with permissionfrom Ref. [2], c©2015 Optical Society of America. . . . . . . . 714.10 Normalized measured optical transmission spectra for the throughand cross ports of the TE straight AC asymmetric waveguidepair as well as for the cross port of the equivalent TE straightsymmetric DC. Adapted with permission from Ref. [2], c©2015Optical Society of America. . . . . . . . . . . . . . . . . . . . 724.11 SR spectra for both the TE sinusoidal and straight AC de-vices. Adapted with permission from Ref. [2], c©2015 OpticalSociety of America. . . . . . . . . . . . . . . . . . . . . . . . . 724.12 Relative SR spectrum of the TE sinusoidal AC device as com-pared to the equivalent TE straight AC device. Adapted withpermission from Ref. [2], c©2015 Optical Society of America. 73xxiList of Figures4.13 Simulated and normalized measured optical transmission spec-tra for the through and cross ports of the TM sinusoidal ACsymmetric waveguide pair as well as normalized measuredoptical transmission spectrum for the cross port of the equiv-alent TM straight symmetric DC. . . . . . . . . . . . . . . . . 754.14 Normalized measured optical transmission spectra for the throughand cross ports of the TM straight AC asymmetric waveguidepair as well as for the cross port of the equivalent TM straightsymmetric DC. . . . . . . . . . . . . . . . . . . . . . . . . . . 764.15 SR spectra of both the TM sinusoidal and straight AC devices. 764.16 Relative SR spectrum of the TM sinusoidal AC device ascompared to the equivalent TM straight AC device. . . . . . 774.17 Normalized measured optical transmission spectra of the throughand cross ports of the PBS for TE operation. Adapted withpermission from Ref. [1], c©2017 Optical Society of America. 784.18 Normalized measured optical transmission spectra of the throughand cross ports of the PBS for TM operation. Adapted withpermission from Ref. [1], c©2017 Optical Society of America.. 794.19 ER spectra for both the TE and TM modes. Adapted withpermission from Ref. [1], c©2017 Optical Society of America. 794.20 PI spectra for both the through and cross ports. Adaptedwith permission from Ref. [1], c©2017 Optical Society of Amer-ica. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.1 Top view of a TE sinusoidal AC symmetric SOI waveguidearray, which has G = 200 nm. . . . . . . . . . . . . . . . . . . 87xxiiList of Figures5.2 Simulated 2-D power distribution profile for the operation atλ0 = 1550 nm of the TE sinusoidal AC waveguide array, whichhas G = 200 nm, when a fundamental TE mode is launchedinto the waveguide in the middle of the array. . . . . . . . . . 88A.1 Cross-sectional view of a symmetric DC on an SOI platform(also Fig. 2.1 in Chapter 2). Adapted with permission fromRef. [1], c©2017 Optical Society of America. . . . . . . . . . . 100A.2 Top view of a straight symmetric DC (also Fig. 2.3 in Chap-ter 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100A.3 1-D normalized transverse field distributions for the TE localnormal modes of waveguide cores a (left) and waveguide coreb (right), ψa(x) and ψb(x), over their respective maximumvalues, ψa,max and ψb,max (also Fig. 2.6 in Chapter 2). . . . . 101A.4 1-D normalized transverse field distributions for the TM localnormal modes of waveguide core a (left) and waveguide coreb (right), ψa(x) and ψb(x), over their respective maximumvalues, ψa,max and ψb,max (also Fig. 2.7 in Chapter 2). . . . . 102A.5 Top view of a full period of a sinusoidally-bent symmetric DC(also Fig. 2.2 in Chapter 2). Adapted with permission fromRef. [1], c©2017 Optical Society of America. . . . . . . . . . . 103A.6 1-D normalized transverse field distributions for the even TEsupermode (right) and odd TE supermode (right), ψe(x) andψo(x), over their respective maximum values, ψe,max and ψo,max(also Fig. 2.4 in Chapter 2). . . . . . . . . . . . . . . . . . . . 103xxiiiList of FiguresA.7 1-D normalized transverse field distributions for the even TMsupermode (left) and odd TM supermode (right), ψe(x) andψo(x), over their respective maximum values, ψe,max and ψo,max(also Fig. 2.5 in Chapter 2). . . . . . . . . . . . . . . . . . . . 104A.8 Probability density function of the 1-D normal distribution,N(x|0, σ), which is centered at x = 0, over its maximumvalue, Nmax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105A.9 1-D normalized transverse field distributions for the approxi-mated ψa(x) and ψb(x), which are centered at x = µa = −W+G2and at x = µb = +W+G2 , over their ψa,max and ψb,max, re-spectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106A.10 1-D normalized transverse field distributions for the approx-imated ψe(x) and ψo(x), over their ψe,max and ψo,max, re-spetively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106A.11 Bessel function of the first kind of order 0 with respect to avariable, xvar, ranging from 0 to 10. . . . . . . . . . . . . . . 109xxivList of AbbreviationsAC anti-couplingCMOS complementary metal-oxide-semiconductorDC directional couplerER extinction ratioE-Beam electron-beamFDE finite-difference eigenmodeFDTD finite-difference time-domainIC integrated circuitPI polarization isolationPBS polarization beam splitterSEM scanning-electron-microscopeSOI silicon-on-insulatorSR suppression ratioSWGC sub-wavelength grating couplerxxvList of AbbreviationsTE transverse-electricTM transverse-magnetic1-D one-dimensional2-D two-dimensional3-D three-dimensionalxxviAcknowledgementsI would like to thank my parents for encouraging me throughout my under-graduate and graduate studies. They are my role models who taught me todream big and work hard. Thank you so much for always believing in meand supporting me.I would like to express my sincere gratitude to my supervisor, Dr. NicolasA. F. Jaeger, for guiding my research, suggesting my projects, and helpingme write and edit my papers. I am very thankful to Dr. Lukas Chrostowskifor providing help and advice regarding my projects and papers.I greatly appreciate the help for my research from my current and formercolleagues in the Electrical Engineering and Computer Engineering Depart-ment of University of British Columbia, in particular Drs Yun Wang, XuWang, Jonas Flu¨ckiger, Valentina Donzella, Miguel A´ngel Guille´n Torres,Wei Shi, Robert Boeck, Sahba Talebi Fard, and Mikael C. Rechtsman aswell as Han Yun, Zeqin Lu, Zhitian Chen, Minglei Ma, Charlie Lin, StephenLin, and Michael Caverley.I would like to acknowledge Natural Sciences and Engineering ResearchCouncil of Canada and CMC Microsystems for their financial support. Iwould like to acknowledge Han Yun, Jonas Flu¨ckiger, Charlie Lin, StephenLin, and Michael Caverley for the development of automated probe stationsin our lab. I would like to acknowledge Richard J. Bojko and N. ShanexxviiAcknowledgementsPatrick at the Washington Nanofabrication Facility at the University ofWashington, a part of the National Nanotechnology Infrastructure Networksupported by National Science Foundation, for the fabrication of my devices.I also would like to acknowledge Lumerical, Inc., MathWorks, Inc., andMentor Graphics, Inc. for making their design software available to me.xxviiiDedicationTo my mother, Guozhen Liu, and to my father, Jianmin Zhang, I am verygrateful for your unconditional love, continuous support, and hearty encour-agement that help me fulfill my dreams, complete my studies, and overcomechallenges in my life.xxixChapter 1Introduction1.1 Silicon Photonics in Optical CommunicationsSilicon photonics continues to evolve and to have a huge impact on the ad-vancement of optical communications technologies. Silicon has many uniqueattributes that make it a versatile material for both photonics and electron-ics. For example, silicon optical waveguides have a wide, low-loss trans-mission window within the infrared spectrum, specifically, from 1100 nm to8600 nm according to Ref. [3]. Furthermore, silicon, the second most abun-dant element in the crust of the Earth, is a semiconductor that has beenused by electronics industry for decades. Existing complementary metal-oxide-semiconductor (CMOS) design tools and fabrication processes can beused to build photonic integrated circuits (ICs) on silicon-on-insulator (SOI)platforms, and high quality SOI wafers are available at low prices. More-over, high refractive index contrast between silicon waveguide cores andsilicon dioxide or air claddings on an SOI platform allows compact waveg-uides and waveguide bends with small radii to have good optical confinementfor single-mode operation and, therefore, SOI optical components can havesmall feature sizes and low propagation losses. As a result, the footprintsof SOI ICs can be greatly reduced as compared to other photonic ICs, e.g.,indium-phosphide-based ICs.11.1. Silicon Photonics in Optical CommunicationsThe integration of silicon photonic and microelectronic circuits is crucialfor numerous applications of silicon photonics in optical communications.Using mature hybrid integration techniques, many leading semiconductorand telecommunications companies, such as Luxtera, Finisar, Cisco, andIntel, have introduced optical transceiver modules that are based on sili-con photonics, with a wide range of data rates up to 200 Gbps, for opticalinterconnects in data centers according to Refs. [4–7]. In last few years,there have also been important technological breakthroughs for the mono-lithic integration of photonic and electronic circuits on SOI platforms. In2015, a large-scale integration of silicon photonic and electronic circuits ona single chip was demonstrated using 45-nm CMOS fabrication processesin Ref. [8]. In 2016, a quantum dot laser, which was directly grown on asilicon substrate, was demonstrated using standard CMOS fabrication pro-cesses in Ref. [9]. In 2016, IBM demonstrated a multichannel 56 Gbps siliconphotonic transmitter by integrating a silicon optical modulator and CMOSdriver circuits on a single chip using 90-nm CMOS fabrication processes inRef. [10]. The continuously advancing silicon photonics and microelectron-ics fabrication technologies lay the foundation for next generation opticalcommunications systems with wide operational bandwidths and low powerconsumptions.Fundamental building blocks of silicon photonics include passive compo-nents (such as ring resonators, multi-mode interference couplers, directionalcouplers, Bragg grating filters, and grating couplers) and active compo-nents (such as modulators, photodetectors, and lasers). There are numerousdemonstrations of these photonic devices in Refs. [11–19] on SOI platformswithin standard telecommunications bands, from 1260 nm to 1675 nm.21.2. Thesis Motivation and Objective1.2 Thesis Motivation and Objective1.2.1 Sinusoidal Anti-coupling Symmetric WaveguidesSOI platforms allow us to integrate large-scale photonic circuits and realizevarious functionalities on a single chip. In SOI ICs, we use routing waveg-uides to connect photonic components. Nevertheless, we need to keep rout-ing waveguides far apart to minimize undesired inter-waveguide crosstalk.However, the use of large gaps between routing waveguides can increase thefootprints of SOI photonic circuits and limit the number of devices that canbe integrated on a chip.In order to use on-chip real estate more efficiently, we can design spe-cial waveguide structures to suppress inter-waveguide crosstalk (see Ref. [2]).Stefano proposed to suppress the crosstalk using sinusoidal symmetric waveg-uides on a titanium lithium-niobate platform in Refs. [20, 21] and demon-strated sinusoidal anti-coupling (AC) symmetric waveguides on an erbium-ytterbium phosphate platform in Ref. [22]. Nicolas demonstrated a Mach-Zehnder modulator using sinusoidal AC symmetric waveguides on a titaniumlithium-niobate platform in Ref. [23]. As an alternative AC waveguide struc-ture, straight asymmetric waveguides were demonstrated on SOI platformsin Refs. [24, 25].In this thesis, I present sinusoidal AC symmetric waveguide pairs onan SOI platform for transverse-electric (TE) or transverse-magnetic (TM)modes, each with design, model, simulation, analysis, and demonstration.The demonstrated devices provide large crosstalk suppressions for both theTE and TM modes over the entire C-band. They are also compatible withCMOS fabrication processes and easy to fabricate. Therefore, they can beused to design high-density routing waveguides in photonic circuits.31.2. Thesis Motivation and Objective1.2.2 Polarization Beam Splitter Using a SymmetricDirectional Coupler with Sinusoidal BendsIn optical communications systems based on silicon photonics, light is usu-ally transmitted between photonic chips via optical fibres (see Ref. [1]).However, the polarization states of optical signals at the outputs of opticalfibres can change randomly according to Ref. [26]. Due to geometric birefrin-gence, optical signals in SOI waveguides are prone to undesired polarization-dependent modal dispersion, central wavelength shift, and phase shift ac-cording to Refs. [27–29]. Polarization beam splitters (PBSs) are essen-tial components in polarization management systems to solve the problemscaused by the polarization dependence of SOI waveguides (see Ref. [1]).There are many different configurations of PBSs that have been demon-strated on SOI platforms, including arrayed waveguide gratings, directionalcouplers (DCs), Mach-Zehnder interferometers, photonic crystals, two-modeinterference couplers, and multi-mode interference couplers in Refs. [28, 30–40]. Zeqin demonstrated a broadband PBS using point-symmetric cascadedcouplers in Ref. [40], but it has a relatively long coupler length of 87.4 µm.Jian demonstrated a compact PBS using an asymmetric DC in Ref. [36],but it has strong wavelength dependencies for both the TE and TM modes.In this thesis, I present the design, simulation, analysis, and demon-stration of a PBS using a symmetric DC with sinusoidal bends on an SOIplatform. The device has a small coupler length of 8.55 µm and a broadoperational bandwidth for both TE operation and TM operation. It is alsocompatible with CMOS fabrication processes and easy to fabricate. There-fore, it can be used for polarization management in optical communicationssystems.41.3. Thesis Organization1.3 Thesis OrganizationThis thesis consists of the following five chapters and one appendix:In Chapter 1, I review the development of silicon photonics in opticalcommunications. I also describe the motivation, objective, and organizationof this thesis.In Chapter 2, I present the design and model of a straight symmetric DCand a sinusoidally-bent symmetric DC. I also derive the modal dispersionbetween the supermodes of the sinusoidally-bent symmetric DC in order tocompare its wavelength dependence with an equivalent straight symmetricDC (which has the same gap width and coupler length as the bent symmetricDC).In Chapter 3, I present the simulation and analysis of a single straightwaveguide, straight symmetric DCs, sinusoidal AC symmetric waveguidepairs, and a PBS using a symmetric DC with sinusoidal bends for TE and/orTM operation on an SOI platform. I compare the wavelength dependenciesof the sinusoidally-bent symmetric DC, which is used in the PBS, with anequivalent straight symmetric DC for the TE and TM modes.In Chapter 4, I present the fabrication, measurement, and demonstrationof sinusoidal AC symmetric waveguide pairs and a PBS using a symmetricDC with sinusoidal bends for TE and/or TM operation on an SOI platform.I compare the sinusoidal AC symmetric waveguide pairs with the straightsymmetric DCs and straight AC asymmetric waveguide pairs (which havethe same gap widths and couplers lengths as the sinusoidal AC devices),each for the TE and TM modes. I also evaluate the performance of thePBS.In Chapter 5, I summarize and conclude my thesis. I also provide sug-51.3. Thesis Organizationgestions for future research on sinusoidal symmetric SOI waveguides.In Appendix A, I present a derivation of the propagation constant dif-ference of a sinusoidally-bent symmetric DC in terms of the propagationconstant difference of an equivalent straight symmetric DC.6Chapter 2Design and TheoryIn this chapter, I design and model a symmetric DC with sinusoidal bendsfor TE and/or TM operation on an SOI platform. I describe the waveg-uide structures and the principles of operation for my devices. I derive ananalytical model of a straight symmetric DC and use the analytical modelto obtain the optical transmissions of symmetric DCs as well as obtainingthe formulas of essential design parameters for my devices. I also derive themodal dispersions between the supermodes of a symmetric DC in order tostudy the wavelength dependence of the device.2.1 Design OverviewAs shown in Fig. 2.1, each of my devices consists of a pair of parallel sym-metric strip waveguides that are separated by a gap of width, G, and havethe same width, W , and height, H. Since a sinusoidally-bent symmetricwaveguide pair has the same cross-sectional structure as a straight sym-metric DC, I can also refer to it as a symmetric DC with sinusoidal bends.Each sinusoidal bend is defined as a function of z, fx(z) (see Fig. 2.2), as inRef. [1]:fx(z) = A cos(2piΛz), (2.1)72.1. Design OverviewTop Oxide Cladding Layer    Buried Oxide Layer   Silicon Silicon W W G H y x z Figure 2.1: Cross-sectional view of a pair of parallel symmetric strip waveg-uides on an SOI platform. Adapted with permission from Ref. [1], c©2017Optical Society of America.W G Λ 𝒙 = 𝒇𝒙 𝒛  y z 2A x 𝒂 𝒃 (𝒙 = 𝟎, 𝒛 = 𝟎) Figure 2.2: Top view of a full period of a symmetric DC with sinusoidalbends. Adapted with permission from Ref. [1], c©2017 Optical Society ofAmerica.82.2. Analytical Model of a Straight Symmetric DCwhere A and Λ are the amplitude and period of a sinusoid. The choices ofGs depend on design requirements and constraints.The use of sinusoidal bends in parallel symmetric waveguides can sup-press the optical power exchange between the waveguides for either TE orTM operation. Therefore, I can design sinusoidal AC symmetric waveguidepairs. Based on this idea, I have demonstrated a sinusoidal AC symmetricSOI strip waveguide pair for TE operation in Ref. [2], and I demonstrate asinusoidal AC symmetric SOI strip waveguide pair for TM operation in thisthesis. Compact routing waveguides and dense waveguide buses are two ofthe potential applications of sinusoidal AC symmetric waveguides.Sinusoidal AC symmetric waveguides can also be used to design a PBS.In a sinusoidal AC symmetric waveguide pair that is designed for TE oper-ation, optical power exchange between the waveguides still occurs for TMoperation. Such a device can be regarded as an “anti-coupler” for TE oper-ation and as a coupler for TM operation. The device can have a small gapwidth and a small coupler length. The sinusoidal bends can also allow asymmetric DC to have broadband performances for both TE operation andTM operation. Based on these ideas, I have designed a compact broadbandPBS using a symmetric DC with sinusoidal bends for both TE operationand TM operation on an SOI platform in Ref. [1].2.2 Analytical Model of a Straight SymmetricDCA straight symmetric DC consists of a pair of parallel identical waveguideswhich are close to each other. According to Refs. [41, 42], the device canbe analyzed in terms of even (symmetric) and odd (anti-symmetric) super-92.2. Analytical Model of a Straight Symmetric DCmodes. When light is injected into one of the waveguide cores, it excites oneeven supermode and one odd supermode in the two-waveguide structure.The supermodes propagating in the DC each travel at their own speeds.The difference in their speeds causes the optical power to oscillate from onewaveguide core to the other along the length of the device, L.W G L 𝒂 𝒃 Input Isolation Through Cross (𝒙 = 𝟎, 𝒛 = 𝟎) y z x Figure 2.3: Top view of a straight symmetric DC.As shown in Fig. 2.3, I have labelled the waveguide cores a and b and theinput, isolation, through, and cross ports, which I will use to model a straightsymmetric DC. I assume that the DC is lossless, that the electromagneticfields propagating in the device are time-harmonic, and that the transversefield distributions for the even and odd supermodes (in the x − y plane)are invariant along the propagation direction (along the z axis). Basedon these assumptions, the electric field distributions for the even and oddsupermodes in the device, Ψe(x, y, z) and Ψo(x, y, z), can be represented interms of their respective transverse field amplitudes at z = 0, Ae and Ao,their respective normalized transverse field distributions at z = 0, ψe(x, y)and ψo(x, y), and their respective propagation constants, βe and βo. I can use102.2. Analytical Model of a Straight Symmetric DC-1 -0.5 0 0.5 1-W-G/2-(W+G)/2-G/20+G/2+(W+G)/2+W+G/2x, nme(x)/e,max-1 -0.5 0 0.5 1-W-G/2-(W+G)/2-G/20+G/2+(W+G)/2+W+G/2x, nmo(x)/o,maxTEEvenTEOddFigure 2.4: 1-D normalized transverse field distributions for the even TEsupermode (left) and odd TE supermode (right), ψe(x) and ψo(x), overtheir respective maximum values, ψe,max and ψo,max.the effective index method to collapse the two-dimensional (2-D) transversefield distribution into an one-dimensional (1-D) one, so that ψe(x, y) andψo(x, y) become ψe(x) and ψo(x) (see Figs. 2.4 and 2.5), respectively, and,as a result, Ψe(x, y, z) and Ψo(x, y, z) become Ψe(x, z) and Ψo(x, z), alsorespectively. Therefore, when light is injected into only waveguide core a ,Ψe(x, z) can be expressed as:Ψe(x, z) = Aeψe(x)e−jβez, (2.2)and Ψo(x, z) can also be expressed as:Ψo(x, z) = Aoψo(x)e−jβoz, (2.3)where ψe(x) and ψo(x) are normalized such that∫ +∞−∞ |ψe(x)|2dx+∫ +∞−∞ |ψo(x)|2dx = 1. As the superposition of the even and odd supermodes, the 2-D112.2. Analytical Model of a Straight Symmetric DC-1 -0.5 0 0.5 1-W-G/2-(W+G)/2-G/20+G/2+(W+G)/2+W+G/2x, nme(x)/e,max-1 -0.5 0 0.5 1-W-G/2-(W+G)/2-G/20+G/2+(W+G)/2+W+G/2x, nmo(x)/o,maxTMEvenTMOddFigure 2.5: 1-D normalized transverse field distributions for the even TMsupermode (left) and odd TM supermode (right), ψe(x) and ψo(x), overtheir respective maximum values, ψe,max and ψo,max.electric field distribution of a straight symmetric DC, ΨDC(x, z), can beobtained as:ΨDC(x, z) = Ψe(x, z) + Ψo(x, z) = Aeψe(x)e−jβez +Aoψo(x)e−jβoz, (2.4)which shows the overall interference between the even and odd supermodesalong the device.In order to study the local interference for the even and odd supermodesin waveguide cores a and b, I will also analyze the local normal modes.According to Refs. [21, 41, 42], the model of the DC that is based on thesupermodes and the one that is based on the coupled modes are correlated,so that I can express the 2-D electric field distributions for local normalmodes in waveguide cores a and b, Ψa(x, z) and Ψb(x, z), in terms of theirrespective z-dependent transverse field amplitudes, Aa(z) and Ab(z), their122.2. Analytical Model of a Straight Symmetric DC-1 -0.5 0 0.5 1-W-G/2-(W+G)/2-G/20x, nma(x)/a,max-1 -0.5 0 0.5 10+G/2+(W+G)/2+W+G/2x, nmb(x)/b,maxbaTE TEFigure 2.6: 1-D normalized transverse field distributions for the TE localnormal modes in waveguide cores a (left) and waveguide core b (right),ψa(x) and ψb(x), over their respective maximum values, ψa,max and ψb,max.respective 1-D normalized transverse field distributions, ψa(x) and ψb(x)(see Figs. 2.6 and 2.7), and their respective propagation constants, βa andβb. Therefore, Ψa(x, z) can be expressed as:Ψa(x, z) = Aa(z)ψa(x)e−jβaz, (2.5)and Ψb(x, z), can be also be expressed as:Ψb(x, z) = Ab(z)ψb(x)e−jβbz. (2.6)where ψa(x) is normalized such that∫ +∞−∞ |ψa(x)|2dx = 1, and ψb(x) is nor-malized such that∫ +∞−∞ |ψb(x)|2dx = 1. Here, referring to Eq. 2.4, ΨDC(x, z)can be rewritten as:ΨDC(x, z) =[Aeψe(x)e−j∆β2z +Aoψo(x)ej∆β2z]e−jΣβ2z, (2.7)132.2. Analytical Model of a Straight Symmetric DC-1 -0.5 0 0.5 1-W-G/2-(W+G)/2-G/20+G/2+(W+G)/2+W+G/2x, nma(x)/a,max-1 -0.5 0 0.5 1-W-G/2-(W+G)/2-G/20+G/2+(W+G)/2+W+G/2x, nmb(x)/b,maxTM TMbaFigure 2.7: 1-D normalized transverse field distributions for the TM localnormal modes in waveguide core a (left) and waveguide core b (right), ψa(x)and ψb(x), over their respective maximum values, ψa,max and ψb,max.where ∆β = βe − βo and Σβ = βe + βo are the difference and the sum of βeand βo, respectively.For an arbitrary integer, n, when ∆βz = 2npi, light is localized in waveg-uide core a , |Aa(z)| reaches its maximum magnitude, Aa,M , at z = 2npi∆β , andΨDC(x, 2npi∆β)= ±[Aeψe(x) +Aoψo(x)]e−jnpi∆βΣβ. Thus, referring to Eqs. 2.5and 2.7, the correlation among ψa(x), ψe(x), and ψo(x) at z =2npi∆β can begiven as:Aa,Mψa(x) = Aeψe(x) +Aoψo(x), (2.8)and, when ∆βz = (2n+ 1)pi, light is localized in waveguide core b, |Ab(z)|reaches its maximum magnitude, Ab,M , at z =(2n+1)pi∆β , andΨDC[x, (2n+1)pi∆β]= ±(−j)[Aeψe(x) − Aoψo(x)]e−j(n+ 12 )pi∆βΣβ. Thus, refer-ring to Eqs. 2.6 and 2.7, the correlation among ψb(x), ψe(x), and ψo(x)142.2. Analytical Model of a Straight Symmetric DCat z = (2n+1)pi∆β can be given as:Ab,Mψb(x) = Aeψe(x)−Aoψo(x). (2.9)Then, adding Eqs. 2.8 and 2.9, ψe(x) can be expressed, in terms of ψa(x)and ψb(x), as:ψe(x) =Aa,Mψa(x) +Ab,Mψb(x)2Ae, (2.10)and, subtracting Eq. 2.9 from Eq. 2.8, ψo(x) can also be expressed, in termsof ψa(x) and ψb(x), as:ψo(x) =Aa,Mψa(x)−Ab,Mψb(x)2Ao. (2.11)Subsequently, substituting Eqs. 2.10 and 2.11 into Eq. 2.7, ΨDC(x, z) canbe rewritten, in terms of ψa(x) and ψb(x), as:ΨDC(x, z) =[Aa,max(e−j∆β2z + ej∆β2z2)ψa(x)+Ab,max(e−j∆β2z − ej∆β2 z2)ψb(x)]e−jΣβ2z.(2.12)Since, in a lossless symmetric DC, Aa,M = Ab,M = AM , βwg is the prop-agation constant of a single straight strip waveguide with the cross-sectionaldimensions (W and H), and Σβ2 ≈ βa = βb = βwg, referring to Eq. 2.12,Ψa(x, z) can be given as:Ψa(x, z) = AM(e−j∆β2z + ej∆β2z2)ψa(x)e−j Σβ2z = AM cos(∆β2z)ψa(x)e−jβwgz,(2.13)and, also referring to Eq. 2.12, Ψb(x, z) can be given as:Ψb(x, z) = AM(e−j∆β2z − ej∆β2 z2)ψb(x)e−j Σβ2z = −jAM sin(∆β2z)ψb(x)e−jβwgz.(2.14)From now on, the subscript, wg, is used to indicate that a particular variableis related to a single straight waveguide. Hence, referring to Eqs. 2.5 and152.2. Analytical Model of a Straight Symmetric DC2.13, Aa(z) can be represented as:Aa(z) = AM cos(∆β2z), (2.15)and, referring to Eqs. 2.6 and 2.14, Ab(z) can also be represented as:Ab(z) = −jAM cos(∆β2z). (2.16)Subsequently, referring to Eq. 2.13, the 1-D local electric field distributionsat the input port, Ψinput(x), in waveguide core a at z = 0 can be solved as:Ψinput(x) = Ψa(x, 0) = AMψa(x), (2.17)referring to Eq. 2.14, the 1-D local electric field distribution at the isolationport, Ψisolation(x), in waveguide core b at z = 0 can be solved as:Ψisolation(x) = Ψb(x, 0) = 0, (2.18)referring to Eq. 2.13, the 1-D local electric field distribution at the throughport, Ψthrough(x), in waveguide core a at z = L can be solved as:Ψthrough(x) = Ψa(x, L) = AM cos(∆β2L)ψa(x)e−jβwgL, (2.19)and, referring to Eq. 2.14, the 1-D local electric field distribution at the crossport, Ψcross(x), in waveguide core b at z = L can be solved as:Ψcross(x) = Ψb(x, L) = −jAM sin(∆β2L)ψb(x)e−jβwgL. (2.20)As the value of L varies, Ψthrough(x, z) and Ψcross(x, z) change sinusoidallyin the waveguide cores.162.3. Optical Transmissions of a Symmetric DC2.3 Optical Transmissions of a Symmetric DCNow, referring to Eqs. 2.17 and 2.19, the optical transmission to the throughport of a symmetric DC, Tthrough, can be obtained as in Ref. [1]:Tthrough =PthroughPinput∝∫ +∞−∞ Ψthrough(x)Ψ∗through(x)dx∫ +∞−∞ Ψinput(x)Ψ∗input(x)dx= cos2(∆β2L),(2.21)and, referring to Eqs. 2.17 and 2.20, the optical transmission to the crossport of the DC, Tcross, can also be obtained as in Ref. [1]:Tcross =PcrossPinput∝∫ +∞−∞ Ψcross(x)Ψ∗cross(x)dx∫ +∞−∞ Ψinput(x)Ψ∗input(x)dx= sin2(∆β2L), (2.22)where Pinput, Pthrough, and Pcross are the optical powers at the input, through,and cross ports of the device, respectively. When light is injected into onlyone waveguide of the device, referring to Eq. 2.22, the smallest value of cross-over length that allows the maximum optical power transfer to an adjacentwaveguide, Lc,min, at a given wavelength in vacuum, λ0, can be given as:Lc,min =pi∆β. (2.23)I will use Eq. 2.23 to calculate the Lc,min of a straight symmetric DC foreither TE or TM operation.2.4 Sinusoidal AC Bends for a SymmetricWaveguide PairAccording to Refs. [20, 21, 43], the ∆β of a sinusoidally-bent symmetric DC,∆βbent, can be given (see Appendix A) as in Ref. [1]:∆βbent ∼= ∆βstraightJ0[2piA(W +G)Λβwg], (2.24)172.4. Sinusoidal AC Bends for a Symmetric Waveguide Pair0 2 4 6 8 10-0.6-0.4-0.200.20.40.60.811.2xvarJn  J0J1xvar = 2.405Figure 2.8: Bessel functions of the first kind of orders 0 and 1 with respectto a variable, xvar, ranging from 0 to 10.where J0 is the Bessel function of the first kind of order 0 (see Fig. 2.8).When W , H, G, Λ, and βwg have been determined, I can find a value ofA to obtain the first positive root of J0 and to set the ∆βbent and Tcross ofthe sinusoidally-bent symmetric DC to be zero (see Refs. [1, 2]). As shownin Fig. 2.8, the value of this root is approximately 2.405 and, therefore, thesmallest value of A, Amin, that can suppress the optical power exchangebetween the sinusoidal waveguides can be given as in Refs. [1, 2]:Amin ∼= 2.405Λ2pi(W +G)βwg. (2.25)Since the derivation of ∆βbent does not depend on mode type (either theTE or TM mode), Eqs. 2.24 and 2.25 can be used with either mode type.Thus, I will use Eq. 2.25 to calculate the Amin of a sinusoidal AC symmetricwaveguide pair for either TE or TM operation. I will also use Eq. 2.24 toanalyze bent and straight symmetric DCs.182.5. Modal Dispersions between the Supermodes of Symmetric DCs2.5 Modal Dispersions between the Supermodesof Symmetric DCsIn order to study the wavelength dependence of a symmetric DC with sinu-soidal bends, I will derive the modal dispersions between the even and oddsupermodes of a symmetric DC. The phase velocity for the even supermode,vp,e, can be defined as:vp,e =2picλ0βe, (2.26)the phase velocity for the odd supermode, vp,o, can also be defined as:vp,o =2picλ0βo, (2.27)where c = 299.8 µm/ps is the speed of light in vacuum. The arrival time forthe even supermode, τe, can be defined as:τe =Lvp,e, (2.28)and the arrival time for the odd supermode, τo, can also be defined as:τo =Lvp,o. (2.29)Then, substituting Eqs. 2.26 and 2.27 into Eqs. 2.28 and 2.29, respectively,I can write the difference between τo and τe, ∆τ , as in Ref. [1]:∆τ = τo − τe = −Lλ02pic∆β, (2.30)and, therefore, I can obtain the modal dispersion of a symmetric DC, D, asin Ref. [1]:D =1Ld∆τdλ0=−12pic(∆β + λ0d∆βdλ0). (2.31)192.6. SummarySubsequently, the D of a straight symmetric DC, Dstraight, can be givenas:Dstraight =−12pic(∆βstraight + λ0d∆βstraightdλ0), (2.32)and the D of a sinusoidally-bent symmetric DC, Dbent, can also be given as:Dbent =−12pic(∆βbent + λ0d∆βbentdλ0). (2.33)Hence, substituting Eq. 2.24 into Eq. 2.33, I can rewrite Dbent, in terms of∆βstraight, as:Dbent =−12pic[(∆βstraight+λ0d∆βstraightdλ0)J0(Kβwg)+(2piλ0Kng,wg)∆βstraightJ1(Kβwg)],(2.34)where K = 2piA(W+G)Λ , J1 is the Bessel function of the first kind of order 1(see Fig. 2.8), and ng,wg is the group index of a straight strip waveguide withthe cross-sectional dimensions (W and H). Then, referring to Eqs. 2.32 and2.34, I can express the absolute value of the ratio between Dbent and Dstraightas in Ref. [1]:∣∣∣∣ DbentDstraight∣∣∣∣ = J0(Kβwg)+(2piλ0Kng,wg)∆βstraightJ1(Kβwg)∆βstraight + λ0d∆βstraightdλ0. (2.35)I will use Eq. 2.35 to calculate the∣∣∣∣ DbentDstraight∣∣∣∣ at λ0 and use the result toevaluate the wavelength dependency of a sinusoidally-bent symmetric DCas compared to an equivalent straight symmetric DC, which has the sameW , H, G and L, for either the TE or TM mode.2.6 SummaryIn this chapter, the design and modelling methods for a straight symmet-ric DC and for a sinusoidally-bent symmetric DC were presented. The202.6. Summarymodal dispersion was also derived to study the wavelength dependence ofthe sinusoidally-bent symmetric DC and the equivalent straight symmetricDC.21Chapter 3Simulation and AnalysisIn this chapter, I simulate and analyze a single straight waveguide, straightsymmetric DCs, and sinusoidal AC symmetric waveguide pairs. All of mysimulations and analyses are based on the assumption that an SOI platformis being used for those devices. Using a finite-difference eigemode (FDE)solver, I simulate a single straight waveguide for both the fundamental TEand TM modes and simulate straight symmetric waveguide pairs, which areseparated by gaps of different widths, for both the TE and TM supermodes.Referring to the formulas which have been derived from the previous chap-ter, I calculate the design parameters of the straight symmetric DCs andsinusoidal AC waveguide pairs for both TE operation and TM operation.Using a finite-difference time-domain (FDTD) solver, I optimize the designsof straight symmetric DCs and sinusoidal AC symmetric waveguide pairs.Then, I optimize the design of a PBS using a symmetric DC with sinusoidalbends using the FDTD solver, in which the suppression of optical powerexchange between the waveguides is optimal only for TE operation but notfor TM operation. Subsequently, using the simulation data that have beencollected, I analyze the wavelength dependencies of the sinusoidally-bentsymmetric DC, which is used in the PBS, for both the TE and TM modes.223.1. Simulation and Analysis Overview3.1 Simulation and Analysis OverviewTable 3.1: Simulation parameters.Parameters Namesλ0, nm Wavelength in vacuumneff Effective refractive index at λ0neff,wg The neff of a single straight waveguidenTEeff,wg The neff,wg for the fundamental TE modenTMeff,wg The neff,wg for the fundamental TM modeneff,e The neff of a pair of straight symmetric waveguides forthe even supermodenTEeff,e The neff,e for the TE supermodenTMeff,e The neff,e for the TM supermodeneff,o The neff of a pair of straight symmetric waveguides forthe odd supermodenTEeff,o The neff,o for the TE supermodenTMeff,o The neff,o for the TM supermodeBoth the FDE and FDTD solvers (MODE Solutions and FDTD Solu-tions, respectively, from Lumerical Solutions, Inc.) use the finite-differencemethod for simulation, the FDE solver simulates optical modes on a cross-section of a waveguide structure by solving Maxwell’s equations on a 2-Dplanar mesh according to Ref. [44], and the FDTD solver simulates thepropagation of optical modes in a waveguide structure by solving Maxwell’sequations in a 3-D cuboidal mesh with respect to both space and time ac-cording to Ref. [45]. The ambient temperature in my simulations is assumed233.1. Simulation and Analysis OverviewTable 3.2: Analysis parameters.Parameters Namesng Group index at λ0ng,wg The ng of a single straight waveguidenTEg,wg The ng,wg for the fundamental TE modenTMg,wg The ng,wg for the fundamental TM modeβ, nm−1 Propagation constant at λ0βwg, nm−1 The β of a single straight waveguideβTEwg , nm−1 The βwg for the fundamental TE modeβTMwg , nm−1 The βwg for the fundamental TM modeβe, nm−1 The β of a pair of straight symmetric waveg-uides for the even supermodeβTEe , nm−1 The βe for the TE supermodeβTMe , nm−1 The βe for the TM supermodeβo, nm−1 The β of a pair of straight symmetric waveg-uides for the odd supermodeβTEo , nm−1 The βo for the TE supermodeβTMo , nm−1 The βo for the TM supermodeD, ps·µm−1·nm−1 Modal dispersion at λ0Dstraight, ps·µm−1·nm−1 The D of a straight symmetric DCDTEstraight, ps·µm−1·nm−1 The Dstraight for the TE modeDTMstraight, ps·µm−1·nm−1 The Dstraight for the TM modeDbent, ps·µm−1·nm−1 The D of a sinusoidal symmetric DCDTEbent, ps·µm−1·nm−1 The Dbent for the TE modeDTMbent, ps·µm−1·nm−1 The Dbent for the TM mode243.1. Simulation and Analysis OverviewTable 3.3: Design parameters.Parameters NamesG, nm Gap widthW , nm Waveguide widthH, nm Waveguide heightL, µm Length of a deviceLc,min, µm The smallest cross-over L of a straight symmetric DCthat is optimized for the operation at λ0LTEc,min, µm The Lc,min for TE operationLTMc,min, µm The Lc,min for TM operationΛ, µm Period of a sinusoidA, nm Amplitude of a sinusoidAmin, nm The smallest A of a sinusoidal AC symmetric waveguidepair that is optimized for the operation at λ0ATEmin, nm The Amin for TE operationATMmin, nm The Amin for TM operation253.2. Single Straight WaveguideDevice 3 1 2 4 Figure 3.1: Port configuration in my simulations.to be at 25 ◦C, and the central wavelength of operation for each device ischosen to be at 1550 nm. In Tables 3.1, 3.2, and 3.3, I list the parametersthat will be used for my simulations, analyses, and designs. In my simula-tions, Ports 1, 2, 3, and 4 are defined as in Fig. 3.1 to indicate the inputsand outputs of each device. In my simulations, Port 1 is used as the defaultinput port, unless another port is specified to be used as an input port.3.2 Single Straight WaveguideAs shown in Fig. 3.2, the SOI strip waveguides, which are used in my devices,are surrounded by silicon dioxide, and all of these waveguides have the samecross-sectional dimensions, W = 500 nm and H = 220 nm, and supportboth single-mode TE operation and single-mode TM operation.Using the FDE solver to simulate the cross-section of the single straightwaveguide, I can obtain the neff,wg for the waveguide structure. Hereafter,263.2. Single Straight WaveguideCrystalline Silicon W = 500 nm H = 220 nm Figure 3.2: Perspective and cross-sectional views of a single straight waveg-uide, which has W = 500 nm and H = 220 nm.1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 15701.41.61.822.22.42.62.80, nmEffective Refractive Index  TETMnTMeff,wg = 1.773nTEeff,wg = 2.446Figure 3.3: Simulated wavelength-dependent effective refractive indices forboth the fundamental TE and TM modes of the single straight waveguide.273.2. Single Straight WaveguideTE TM W = 500 nm H = 220 nm W = 500 nm H = 220 nm Figure 3.4: Simulated 2-D mode profiles for the fundamental TE mode (left)and the fundamental TM mode (right) at λ0 = 1550 nm of the single straightwaveguide, which has W = 500 nm and H = 220 nm.Table 3.4: Simulated and calculated values of the neff,wgs, ng,wgs, and βwgsfor both mode types at λ0 = 1550 nm of the single straight waveguide.Modes TE TMneff,wg 2.446 1.773ng,wg 4.209 3.763βwg, nm−1 0.009916 0.007188283.3. Coupling and AC Devices for TE Operationthe subscript, wg, is used when I intend to indicate that a particular vari-able is related to the single straight waveguide. Subsequently, ng,wg can becalculated, with respect to neff,wg, as:ng,wg = neff,wg − λ0dneff,wgdλ0, (3.1)and βwg can also be calculated, with respect to neff,wg, as:βwg =2piλ0neff,wg. (3.2)Figure 3.3 shows the wavelength-dependent effective refractive indices forboth the fundamental TE and TM modes of the single straight waveguide,and, referring to Eqs. 3.1 and 3.2 and Fig. 3.3, the values of the neff,wgs,ng,wgs, and βwgs for both mode types at λ0 = 1550 nm of the waveguidestructure are obtained and listed in Table 3.4. As shown in Fig. 3.4, thefundamental TE mode is better confined to the waveguide core than the fun-damental TM mode at λ0 = 1550 nm. Due to stronger confinement for thefundamental TE mode than for the fundamental TM mode to the waveguidecore, the nTEeff,wg is greater than the nTMeff,wg at λ0 = 1550 nm (see Fig. 3.3and Table 3.4).3.3 Coupling and AC Devices for TE OperationBased on a symmetric two-waveguide structure, a straight symmetric DCand a sinusoidal AC symmetric waveguide pair can be designed for TEoperation. The TE straight coupling device can be used as a reference toevaluate the performance of the TE sinusoidal AC device.293.3. Coupling and AC Devices for TE OperationCrystalline Silicon G = 200 nm W = 500 nm H = 220 nm W = 500 nm Figure 3.5: Perspective and cross-sectional views of a straight symmetricwaveguide pair, which has W = 500 nm, H = 220 nm, and G = 200 nm.1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 15701.41.61.822.22.42.62.80, nmEffective Refractive Index  TETMnTEeff,e = 2.458nTMeff,e = 1.826(a)1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 15701.41.61.822.22.42.62.80, nmEffective Refractive Index  TETMnTEeff,o = 2.437nTMeff,o = 1.714(b)Figure 3.6: Simulated wavelength-dependent effective refractive indices for(a) the even TE and TM supermodes and (b) the odd TE and TM super-modes of the straight symmetric waveguide pair, which has G = 200 nm.303.3. Coupling and AC Devices for TE OperationTE Even G = 200 nm TM Even G = 200 nm TE Odd G = 200 nm TM Odd G = 200 nm Figure 3.7: Simulated 2-D mode profiles for the even and odd TE super-modes (upper left and right, respectively) and the even and odd TM super-modes (lower left and right, respectively) at λ0 = 1550 nm of the straightsymmetric waveguide pair, which has G = 200 nm.Table 3.5: Simulated and calculated values of the neff s and βs for boththe TE and TM supermodes at λ0 = 1550 nm of the straight symmetricwaveguide pair, which has G = 200 nm.Modes TE TMneff,e 2.458 1.826neff,o 2.437 1.714βe, nm−1 0.009963 0.007404βo, nm−1 0.009879 0.006948313.3. Coupling and AC Devices for TE Operation3.3.1 TE Straight Symmetric DCThe value of G is chosen to be 200 nm for the TE straight symmetric DC.Using the FDE solver to simulate the cross-section of a straight symmetricwaveguide pair, which has W = 500 nm, H = 220 nm, and G = 200 nm(see Fig. 3.5), I can obtain the neff,e and neff,o for the waveguide structure.Subsequently, βe can be calculated, with respect to neff,e, as:βe =2piλ0neff,e, (3.3)and βo can be calculated, with respect to neff,o, as:βo =2piλ0neff,o. (3.4)Figures 3.6a and 3.6b show the wavelength-dependent effective refractiveindices for both the TE and TM supermodes of the straight symmetricwaveguide pair. Referring to Eqs. 3.3 and 3.4 and Figs. 3.6a and 3.6b, thevalues of the neff,es, neff,os, βes, and βos for both the TE and TM super-modes at λ0 = 1550 nm of the waveguide structure are obtained and listed inTable 3.5. As shown in Fig. 3.7, the even and odd TE supermodes are betterconfined to the waveguide cores than the TM even and odd supermodes atλ0 = 1550 nm. Due to stronger confinement for the TE supermodes than forthe TM supermodes to the waveguide cores, the nTEeff,e and nTEeff,o are greaterthan the nTMeff,e and nTMeff,o, respectively, at λ0 = 1550 nm (see Figs. 3.6a and3.6b and Table 3.5).Referring to Eq. 2.23 (Lc,min =pi∆β ), I can calculate the Lc,min to allowthe maximum optical power transfer to the cross port of a straight symmetricDC for either TE or TM operation. Thus, referring to Eq. 2.23 and Table 3.5,I calculate the LTEc,min = 37.40 µm for the operation at λ0 = 1550 nm of theDC, which has G = 200 nm. Since the circularly-bent branches are used323.3. Coupling and AC Devices for TE Operation30 31 32 33 34 35 36 37-40-30-20-10010Opitcal Transmission, dBL, m  ThroughCrossTEFigure 3.8: Simulated optical transmissions of the cross and through portsas functions with respect to L for TE operation at λ0 = 1550 nm of straightsymmetric DCs, which have G = 200 nm.in my devices either to bring the waveguides closer or to separate them,from now on, my simulations include the coupling and insertion losses thatare caused by the bifurcating branches. Then, using the FDTD solver, Isimulate straight symmetric DCs, which have G = 200 nm, for TE operationand obtain an optimized value of the LTEc,min to be 32.88 µm (see Fig. 3.8and Ref. [2]). Figure 3.9 shows the 2-D power distribution profile for theoperation at λ0 = 1550 nm of the TE straight symmetric DC, which hasG = 200 nm and L = LTEc,min = 32.88 µm, and TE optical power crosses overto the adjacent waveguide in the TE coupling device, when a fundamentalTE mode is launched into Port 1 of the device.333.3. Coupling and AC Devices for TE OperationPower Amplitude Max. Min. 3 1 2 Input Through 4 Cross TE Figure 3.9: Simulated 2-D power distribution profile for the operation atλ0 = 1550 nm of the straight symmetric DC, which has G = 200 nm andL = 32.88 µm, when a fundamental TE mode is launched into Port 1 of thedevice.3.3.2 TE Sinusoidal AC Symmetric Waveguide PairReferring to Eq. 2.25 (Amin ∼= 2.405Λ2pi(W+G)βwg ), I can calculate the Amin tosuppress the optical power exchange between the waveguides of a sinusoidalAC device for either TE or TM operation. I choose a value of the Λ tobe 16.44 µm (by default, I choose Λ =Lc,min2 for my sinusoidal AC de-vices) and, referring to Eq. 2.25 and Table 3.4, calculate the ATEmin = 906 nmfor the operation at λ0 = 1550 nm of the sinusoidal AC device, which hasG = 200 nm and Λ = 16.44 µm. Then, using the FDTD solver, I sim-ulate sinusoidal symmetric waveguide pairs, which have G = 200 nm andΛ = 16.44 µm, for TE operation and obtain an optimized value of the ATEminto be 932 nm (see Fig. 3.10 and Ref. [2]). Figures 3.11a and 3.11b showthe 2-D power distribution profiles for the operation at λ0 = 1550 nm of343.3. Coupling and AC Devices for TE Operation900 910 920 930 940 950 960-50-40-30-20-10010Optical Transmission, dBA, m  ThroughCrossTEFigure 3.10: Simulated optical transmissions of the cross and throughports as functions with respect to A for TE operation at λ0 = 1550 nmof sinusoidal symmetric waveguide pairs, which have the G = 200 nm andΛ = 16.44 µm.Power Amplitude Max. Min. 3 4 1 2 Input Through TE Cross (a)Power Amplitude Max. Min. TE Input Cross 3 4 1 2 Through (b)Figure 3.11: Simulated 2-D power distribution profiles for the operation atλ0 = 1550 nm of the sinusoidal AC symmetric waveguide pair, which hasG = 200 nm, L = 32.88 µm, and A = 932 nm, when a fundamental TEmode is launched into each of (a) Port 1 and (b) Port 3 of the same device.Adapted with permission from Ref. [2], c©2015 Optical Society of America.353.3. Coupling and AC Devices for TE Operation3 1 2 4  TE TE z x y Figure 3.12: Perspective view of the TE sinusoidal AC device. Adaptedwith permission from Ref. [2], c©2015 Optical Society of America.the TE sinusoidal AC symmetric waveguide pair, which has G = 200 nm,L = 2Λ = LTEc,min = 32.88 µm, and A = ATEmin = 932 nm, when a fundamen-tal TE mode are launched into each of Ports 1 and 3 of the same device. Incomparison with the equivalent TE straight symmetric DC (see Figs. 3.9),Fig. 3.11a illustrates that TE optical power exchange between the waveg-uides is suppressed and that TE optical power transfers from an input portto an output port along the same waveguide in the TE AC device, when afundamental TE mode is launched into Port 1 of each device. As shown inFig. 3.12, the device is symmetric with respect to the x axis, the injectionof an optical signal into Port 1 is equivalent to that into Port 2, and theinjection of an optical signal into Port 3 is equivalent to that into Port 4.Thus, the TE sinusoidal AC device can work as a four-port device.363.3. Coupling and AC Devices for TE Operation3.3.3 Study for TM Operation2 2.5 3 3.5 4 4.5 5 5.5 6-40-30-20-10010Opitcal Transmission, dBL, m  ThroughCrossTMFigure 3.13: Simulated optical transmissions of the cross and through portsas functions with respect to L for TM operation at λ0 = 1550 nm of straightsymmetric DCs, which have G = 200 nm.Here, I study the TE coupling and AC devices, which were studied above,for TM operation. Referring to Eq. 2.23 and Table 3.5, I calculate the valueof LTMc,min for the operation at λ0 = 1550 nm of a straight symmetric DC,which has G = 200 nm (see Table 3.6). Then, using the FDTD solver, I sim-ulate straight symmetric DCs, which have G = 200 nm, for TM operationand obtain an optimized value of the LTMc,min (see Fig. 3.13 and Table 3.6).The calculated and optimized values of the LTEc,mins are also listed in Ta-ble 3.6. Due to weaker confinement for the TM supermodes than for the TEsupermodes to the waveguide cores, TM optical power can transfer to theadjacent waveguide in a shorter length, the Lc,min, than TE optical power.As shown in Fig. 3.14, when a fundamental TM mode is launched into Port 1373.3. Coupling and AC Devices for TE OperationTM Power Amplitude Max. Min. 3 1 2 Input Through 4 Cross Figure 3.14: Simulated 2-D power distribution profile for the operation atλ0 = 1550 nm of the TE straight symmetric DC, which has G = 200 nmand L = 32.88 µm, when a fundamental TM mode is launched into Port 1of the device.Table 3.6: Calculated and optimized values of the Lc,mins for both TEoperation and TM operation at λ0 = 1550 nm of the straight symmetricDCs, which have the same G = 200 nm (see Ref. [2]).Operation TE TMLc,min, µmCalculated 37.40 6.89Optimized 32.88 4383.3. Coupling and AC Devices for TE Operationof the TE straight symmetric DC (G = 200 nm and L = 32.88 µm), whichwas studied above, the optical power crosses over between the waveguidesin a much shorter length than when a fundamental TE mode is launchedinto Port 1 of the same device (see Fig. 3.9).0 20 40 60 80 100-40-30-20-10010Optical Transmission, dBA, m  ThroughCrossTM(a)220 240 260 280 300 320-40-30-20-10010Optical Transmission, dBA, m  ThroughCrossTM(b)620 640 660 680 700 720-40-30-20-10010Optical Transmission, dBA, m  ThroughCrossTM(c)1200 1220 1240 1260 1280-40-30-20-10010Optical Transmission, dBA, m  ThroughCrossTM(d)Figure 3.15: Simulated optical transmissions of the cross and through portsas functions with respect to A for TM operation at λ0 = 1550 nm of si-nusoidal symmetric waveguide pairs, which have G = 200 nm and have (a)Λ = 2 µm, (b) Λ = 4 µm, (c) Λ = 8 µm, and (d) Λ = 16.44 µm.Thus, I choose the values of the Λs to be 2 µm, 4 µm, 8 µm, and16.44 µm. Referring to Eq. 2.25 and Table 3.4, I calculate the values of the393.3. Coupling and AC Devices for TE OperationTable 3.7: Calculated values of the ATMmins and optimized ATMcross,mins for theoperation at λ0 = 1550 nm of the sinusoidal symmetric waveguide pairs,which have G = 200 nm and different Λs.Λ, µm 2 4 8 16.44ATMmin, nmCalculated 152 304 608 1250Optimized - 270 670 1240Amins of the sinusoidal symmetric waveguide pairs, which have G = 200 nmand different Λs (see Table 3.7). Then, using the FDTD solver, I simulatethe sinusoidal symmetric waveguide pairs, which have G = 200 nm anddifferent Λs, and obtain the optimized values of the Amins (see Figs. 3.15a,3.15b, 3.15c, and 3.15d, and Table 3.7).As shown in Figs. 3.15a, the bending amplitudes that are near the calcu-lated ATMmin for Λ = 2 µm yield larger optical transmissions to the cross portsthan the optical transmissions to the through ports. Thus, there is no practi-cal ATMmin for the sinusoidal AC device, which has G = 200 nm and Λ = 2 µm.Figures 3.16a, 3.16b, and 3.16c show the 2-D power distribution profiles forthe operation at λ0 = 1550 nm of the TM sinusoidal symmetric waveguidepairs, which have G = 200 nm, different Λs, and L = 2Λ = LTMc,min, when afundamental TM mode is launched into Port 1 of each device. As shown inFigs. 3.15b, 3.15c, 3.15d, 3.16a, 3.16b, and 3.16c, those sinusoidal devicessuffer from significant insertion losses and inter-waveguide crosstalk. Thecrosstalk is also becoming more intensive as the bending periods increase inthose devices. As a result, there is no practical ATMmin for those sinusoidaldevices, which have G = 200 nm and various Λs.Therefore, the sinusoidal bends need to have a Λ that is smaller than403.3. Coupling and AC Devices for TE OperationPower Amplitude Max. Min. 3 4 1 2 Input Through Cross TM (a)Power Amplitude Max. Min. 3 1 2 Input Through 4 Cross TM (b)Power Amplitude Max. Min. 3 1 2 Input Through 4 Cross TM (c)Figure 3.16: Simulated 2-D power distribution profiles for the operationat λ0 = 1550 nm of the sinusoidal symmetric waveguide pairs, which haveG = 200 nm and have (a) Λ = 4 µm, (b) Λ = 8 µm, and (c) Λ = 16.44 µm,when a fundamental TM mode is launched into Port 1 of each device.413.3. Coupling and AC Devices for TE Operationor equal to the Lc,min to allow for the suppression of optical power ex-change between the waveguides. Meanwhile, due to weak confinement tothe waveguide cores, the TM modes are much more susceptible to bend-ing losses than the TE modes. Bending losses is proportional to bendingcurvature according to Ref. [46], and the bending curvature of a sinusoidis inversely proportional to the Λ (when the A remains the same). Thus, Ineed to choose an appropriate G for the TM sinusoidal AC device, whichallows for sufficiently large LTMc,min and Λ, so that the sinusoidal AC bendscan be designed to have low insertion loss.Power Amplitude Max. Min. 3 1 2 Input 4 Through TM Cross Figure 3.17: Simulated 2-D power distribution profile for the operation atλ0 = 1550 nm of the TE sinusoidal AC symmetric waveguide pair, whichhas G = 200 nm, Λ = 16.44 µm, A = 932 nm, and L = 32.88 µm, when afundamental TM mode is launched into Port 1 of the device.As shown in Fig. 3.17, when a fundamental TM mode is launched intoPort 1 of the TE sinusoidal AC symmetric waveguide pair (G = 200 nm,Λ = 16.44 µm, A = 932 nm, and L = 32.88 µm), which was also studied423.4. Coupling and AC Devices for TM Operationabove, TM optical power exchange still takes place between the waveguidesof the device. Hence, the AC effect of sinusoidal bends is sensitive to thepolarization.3.4 Coupling and AC Devices for TM OperationAlso based on a symmetric two-waveguide structure, a straight symmetricDC and a sinusoidal AC symmetric waveguide pair can be designed for TMoperation. The TM straight coupling device can be used as a reference toevaluate the performance of the TM sinusoidal AC device. According to thestudy above, the selection of gap width for the TM sinusoidal AC devicerequires further investigation.3.4.1 Selection of Gap WidthTable 3.8: Simulated and calculated values of the neff s and βs for both theeven and odd TM supermodes at λ0 = 1550 nm of the straight symmetricwaveguide pairs, which have different Gs.G, nm 300 400 500 600nTMeff,e 1.806 1.794 1.786 1.781nTMeff,o 1.736 1.750 1.759 1.764βTMe , nm−1 0.007320 0.007270 0.007240 0.007221βTMo , nm−1 0.007039 0.007096 0.007131 0.007153Using the FDE solver to simulate the cross-sections of straight symmetricwaveguide pairs, which have different Gs of 300 nm, 400 nm, 500 nm, and600 nm, I can obtain the neff,e and neff,o for each of them. Figures 3.18a,433.4. Coupling and AC Devices for TM Operation1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 15701.41.61.822.22.42.62.80, nmEffective Refractive Index  TETMnTEeff,e = 2.451nTMeff,e = 1.806(a)1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 15701.41.61.822.22.42.62.80, nmEffective Refractive Index  TETMnTEeff,e = 2.448nTMeff,e = 1.794(b)1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 15701.41.61.822.22.42.62.80, nmEffective Refractive Index  TETMnTEeff,e = 2.447nTMeff,e = 1.786(c)1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 15701.41.61.822.22.42.62.80, nmEffective Refractive Index  TETMnTMeff,e = 1.781nTEeff,e = 2.447(d)Figure 3.18: Simulated wavelength-dependent effective refractive indices forboth the even TE and TM supermodes of the straight symmetric waveguidepairs, which have (a) G = 300 nm, (b) G = 400 nm, (c) G = 500 nm, and(d) G = 600 nm.443.4. Coupling and AC Devices for TM Operation1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 15701.41.61.822.22.42.62.80, nmEffective Refractive Index  TETMnTEeff,o = 2.442nTMeff,o = 1.736(a)1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 15701.41.61.822.22.42.62.80, nmEffective Refractive Index  TETMnTEeff,o = 2.444nTMeff,o = 1.75(b)1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 15701.41.61.822.22.42.62.80, nmEffective Refractive Index  TETMnTEeff,o = 2.446nTMeff,o = 1.759(c)1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 15701.41.61.822.22.42.62.80, nmEffective Refractive Index  TETMnTEeff,o = 2.446nTMeff,o = 1.764(d)Figure 3.19: Simulated wavelength-dependent effective refractive indices forboth the odd TE and TM supermodes of the straight symmetric waveguidepairs, which have (a) G = 300 nm, (b) G = 400 nm, (c) G = 500 nm, and(d) G = 600 nm.453.4. Coupling and AC Devices for TM Operation3.18b, 3.18c, 3.18d, 3.19a, 3.19b, 3.19c, and 3.19d show the wavelength-dependent effective refractive indices for both the TE and TM supermodesof those straight symmetric waveguide pairs. Referring to Eqs. 3.3 and 3.4and Figs. 3.18a, 3.18b, 3.18c, 3.18d, 3.19a, 3.19b, 3.19c, and 3.19d, thevalues of the neff,es, neff,os, βes, and βos for both the even and odd TMsupermodes at λ0 = 1550 nm of the waveguide structures are obtained andlisted in Table 3.8.Table 3.9: Calculated and optimized values of the LTMc,mins for the operationat λ0 = 1550 nm of the straight symmetric DCs, which have different Gs.G, nm 300 400 500 600LTMc,min, µmCalculated 11.18 18.06 28.82 46.20Optimized 8 14 24 39.72Referring to Eq. 2.23 and Table 3.8, I calculate the values of the LTMc,minsfor the operation at λ0 = 1550 nm of the straight symmetric DCs, whichhave different Gs (see Table 3.9). Then, using the FDTD solver, I simulatestraight symmetric DCs, which have different Gs, for TM operation andobtain the optimized values of the LTMc,mins (see Figs. 3.20a, 3.20b, 3.20c, and3.20d and Table 3.9).The chosen values of the Λs and the calculated values of the ATMmins(referring to Eq. 2.25 and Table 3.4) are listed in Table 3.10. Then, using theFDTD solver, I simulate the sinusoidal symmetric waveguide pairs, whichhave different Gs, for TM operation and obtain the optimized values of theATMmins (see Table 3.10). Figures 3.22a, 3.22b, 3.22c, and 3.22d show the 2-Dpower distribution profiles for the operation at λ0 = 1550 nm of the TMsinusoidal AC symmetric waveguide pairs, which have different Gs, Λs, and463.4. Coupling and AC Devices for TM Operation6 6.5 7 7.5 8 8.5 9 9.5 10-40-30-20-10010Optical Transmission, dBL, m  ThroughCrossTM(a)12 12.5 13 13.5 14 14.5 15 15.5 16-40-30-20-10010Optical Transmission, dBL, m  ThroughCrossTM(b)21 22 23 24 25 26-40-30-20-10010Opitcal Transmission, dBL, m  ThroughCrossTM(c)36 37 38 39 40 41-40-30-20-10010Optical Transmission, dBL, m  ThroughCrossTM(d)Figure 3.20: Simulated optical transmissions of the cross and throughports as functions with respect to L for TM operation at λ0 = 1550 nmof straight symmetric DCs, which have (a) G = 300 nm, (b) G = 400 nm,(c) G = 500 nm, and (d) G = 600 nm.473.4. Coupling and AC Devices for TM Operation180 200 220 240 260 280-40-30-20-10010Optical Transmission, dBA, m  ThroughCrossTM(a)350 400 450-40-30-20-10010Optical Transmission, dBA, m  ThroughCrossTM(b)640 660 680 700 720-40-30-20-10010Optical Transmission, dBA, m  ThroughCrossTM(c)980 1000 1020 1040 1060-40-30-20-10010Optical Transmission, dBA, m  ThroughCrossTM(d)Figure 3.21: Simulated optical transmissions of the cross and throughports as functions with respect to A for TM operation at λ0 = 1550 nmof sinusoidal symmetric waveguide pairs, which have (a) G = 300 nm, (b)G = 400 nm, (c) G = 500 nm, and (d) G = 600 nm.483.4. Coupling and AC Devices for TM OperationPower Amplitude Max. Min. 3 4 1 2 Input Through Cross TM (a)Power Amplitude Max. Min. 3 1 2 Input Through 4 Cross TM (b)Power Amplitude Max. Min. 3 1 2 Input Through 4 Cross TM (c)Power Amplitude Max. Min. 3 1 2 Input Through 4 Cross TM (d)Figure 3.22: Simulated 2-D power distribution profiles for the operation atλ0 = 1550 nm of the sinusoidal AC symmetric waveguide pairs, which have(a) G = 300 nm, (b) G = 400 nm, (c) G = 500 nm, and (d) G = 600 nm,when a fundamental TM mode is launched into Port 1 of each device.493.4. Coupling and AC Devices for TM OperationTable 3.10: Calculated and optimized values of the ATMmins for the operationat λ0 = 1550 nm of the sinusoidal AC symmetric waveguide pairs, whichhave different Gs.G, nm 300 400 500 600Λ, µm 4 7 12 19.86ATMmin, nmCalculated 266 414 639 961Optimized 230 430 680 1025L = 2Λ = LTMc,min, when a fundamental TM mode is launched into Port 1 ofeach device. As shown in Figs. 3.21a, 3.21b, 3.21c, 3.21d, 3.22a, 3.22b, 3.22c,and 3.22d, the insertion losses are getting closer to 0 dB and the crosstalksuppressions are getting larger as the values of Gs increase from 300 nm to600 nm for those devices.A larger G results in a larger LTMc,min, a larger Λ, and a larger devicefootprint. As shown in Figs. 3.21c, 3.21d, 3.22c, and 3.22d, there is littleimprovement in either the insertion loss or the crosstalk suppression for theTM sinusoidal AC devices when the Gs increase from 500 nm to 600 nm.Hence, I choose G = 600 nm, which allows the TM sinusoidal AC device tohave a compact device footprint, a large crosstalk suppression, and a lowinsertion loss.3.4.2 TM Straight Symmetric DCAs shown in Fig. 3.23, the even and odd TM supermodes of a straightsymmetric waveguide pair, which has W = 500 nm, H = 220 nm, andG = 600 nm (see Fig. 3.24), are less confined to the waveguide cores thanthe even and odd TE supermodes at λ0 = 1550 nm. Due to weaker confine-503.4. Coupling and AC Devices for TM OperationTE Even TE Odd TM Even G = 600 nm G = 600 nm G = 200 nm TM Odd G = 600 nm G = 600 nm Figure 3.23: Simulated 2-D mode profiles for the even and odd TE super-modes (upper left and right, respectively) and the even and odd TM super-modes (lower left and right, respectively) at λ0 = 1550 nm of the straightsymmetric waveguide pair, which has G = 600 nm.Crystalline Silicon G = 600 nm W = 500 nm H = 220 nm W = 500 nm Figure 3.24: Perspective and cross-sectional views of a straight symmetricwaveguide pair, which has W = 500 nm, H = 220 nm, and G = 600 nm.513.4. Coupling and AC Devices for TM OperationPower Amplitude Max. Min. TM 3 1 2 Input Through 4 Cross Figure 3.25: Simulated 2-D power distribution profile for the operation atλ0 = 1550 nm of the straight symmetric DC, which has G = 600 nm andL = 39.72 µm, when a fundamental TM mode is launched into Port 1 of thedevice.ment for the TM supermodes than for the TE supermodes to the waveg-uide cores, the nTMeff,e and nTMeff,o are smaller than the nTEeff,e and nTEeff,o,respectively, at λ0 = 1550 nm (see Figs. 3.18d and 3.19d and Table 3.8).Figure 3.25 shows the 2-D power distribution profile for the operation atλ0 = 1550 nm of the TM straight symmetric DC, which has G = 600 nmand L = LTMc,min = 39.72 µm (see Table 3.9), and TM optical power crossesover to the adjacent waveguide in the TM coupling device, when a funda-mental TM mode is launched into Port 1 of the device.3.4.3 TM Sinusoidal AC Symmetric Waveguide PairFigures 3.26a and 3.26b show the 2-D power distribution profiles for the op-eration at λ0 = 1550 nm of the TM sinusoidal AC symmetric waveguide pair,523.4. Coupling and AC Devices for TM OperationPower Amplitude Max. Min. TM 3 1 2 Input Cross 4 Through (a)Power Amplitude Max. Min. TM 3 1 2 Input Cross 4 Through (b)Figure 3.26: Simulated 2-D power distribution profiles for the operation atλ0 = 1550 nm of the sinusoidal AC symmetric waveguide pair, which has ,Λ = 19.86 µm, ATMmin = 1025 nm, and L = 39.72 µm, when a fundamentalTM mode is launched into each of (a) Port 1 and (b) Port 3 of the samedevice.3 1 2 4 TM TM z x y Figure 3.27: Perspective view of the TM sinusoidal AC device.533.5. PBS using a Symmetric DC with Sinusoidal Bendswhich has G = 600 nm, L = LTMc,min = 39.72 µm, and A = ATMmin = 1025 nm(see Tables 3.9 and 3.7), when a fundamental TM modes is launched intoeach of Ports 1 and 3 of the same device. In comparison with the equivalentTM straight symmetric DC (see Fig. 3.25), Fig. 3.26a illustrates that TMoptical power exchange between the waveguides is suppressed and that TMoptical power transfers from an input port to an output port along the samewaveguide in the TM AC device, when a fundamental TM mode is launchedinto Port 1 of the device. As shown in Fig. 3.27, the device is symmetric withrespect to the x axis. Thus, the TM sinusoidal AC symmetric waveguidepair can work as a four-port device.3.5 PBS using a Symmetric DC with SinusoidalBendsThe polarization sensitivity of the sinusoidal AC bends can be used to designpolarization selective devices (eg. a PBS). In the TE sinusoidal AC device,TE optical power propagates along the same waveguide (Fig. 3.11a), and TMoptical power can transfer to the adjacent waveguide in a relatively shortlength (see Fig. 3.17). Therefore, I can design a compact PBS using the TEsinusoidal AC symmetric waveguide structure (G = 200 nm, Λ = 16.44 µm,and A = 932 nm), which was studied above.3.5.1 OptimizationUsing the FDTD solver, I simulate symmetric DCs with sinusoidal bendsand the optimize the L of the devices, so that the maximum TM opticalpower transfer to the adjacent waveguide is allowed and TE optical power543.5. PBS using a Symmetric DC with Sinusoidal Bends3 1 2 4 TE TM TM TE z x y Figure 3.28: Perspective view of the PBS. Adapted with permission fromRef. [1], c©2017 Optical Society of America.6 6.5 7 7.5 8 8.5 9 9.5 1000.20.40.60.81L, mOptical Transmission  ThroughCrossTE(a)6 6.5 7 7.5 8 8.5 9 9.5 1000.20.40.60.81L, mOptical Transmission  ThroughCrossTM(b)Figure 3.29: Simulated optical transmissions (on a linear scale) of the crossand through ports as functions with respect to L for (a) TE operation and(b) TM operation at λ0 = 1550 nm of symmetric DCs with sinusoidal bends,which have G = 200 nm, Λ = 16.44 µm, and A = 932 nm. Adapted withpermission from Ref. [1], c©2017 Optical Society of America.553.5. PBS using a Symmetric DC with Sinusoidal Bends6 6.5 7 7.5 8 8.5 9 9.5 10-30-25-20-15-10-50510L, mOptical Transmission, dB  ThroughCrossTE(a)6 6.5 7 7.5 8 8.5 9 9.5 10-30-25-20-15-10-50510L, mOptical Transmission, dB  ThroughCrossTM(b)Figure 3.30: Simulated optical transmissions (on a logarithmic scale) of thecross and through ports as functions with respect to L for (a) TE operationand (b) TM operation at λ0 = 1550 nm of symmetric DCs with sinusoidalbends, which have G = 200 nm, Λ = 16.44 µm, and A = 932 nm. Adaptedwith permission from Ref. [1], c©2017 Optical Society of America.Power Amplitude Max. Min. TE 3 1 2 Input Cross 4 Through (a)Power Amplitude Max. Min. TM 3 1 2 Input Cross 4 Through (b)Figure 3.31: Simulated 2-D power distribution profiles for the operationat λ0 = 1550 nm of the PBS, when (a) a fundamental TE mode and (b) afundamental TM mode are launched into Port 1 of the same device. Adaptedwith permission from Ref. [1], c©2017 Optical Society of America.563.5. PBS using a Symmetric DC with Sinusoidal Bendsexchange between the waveguides is suppressed in the device (see Fig. 3.28).Referring to Figs. 3.29a, 3.29b, 3.30a, and 3.30b, I choose L = 8.55 µmfor my PBS (see Ref. [1]). Figures 3.31a and 3.31b show the 2-D powerdistribution profiles for the operation at λ0 = 1550 nm of the PBS, and theTE and TM modes diverge into two different waveguides in the device, whenboth the fundamental TE and TM modes are launched into Port 1 of thesame device.Power Amplitude Max. Min. TE 3 1 2 4 Input Cross Through (a)Power Amplitude Max. Min. TM 3 1 2 4 Input Cross Through (b)Figure 3.32: Simulated 2-D power distribution profiles for the operation atλ0 = 1550 nm of the PBS, when (a) a fundamental TE mode and (b) afundamental TM mode are launched into Port 2 of the same device.In addition, Figs. 3.32a, 3.32b, 3.33a, 3.33b, 3.34a, and 3.34b also il-lustrate that the fundamental TE and TM are separated into two differentports of the device, when both of the fundamental modes are launched intoeach of Ports 2, 3, and 4 of the same device. Hence, the PBS using a sym-metric DC with sinusoidal bends can work as a four-port device and canalso work as a polarization beam combiner.573.5. PBS using a Symmetric DC with Sinusoidal BendsPower Amplitude Max. Min. TE 3 1 2 Cross 4 Through Input (a)Power Amplitude Max. Min. TM 3 1 2 Input Cross 4 Through (b)Figure 3.33: Simulated 2-D power distribution profiles for the operation atλ0 = 1550 nm of the PBS, when (a) a fundamental TE mode and (b) afundamental TM mode are launched into Port 3 of the same device.Power Amplitude Max. Min. TE 3 1 2 Input Cross 4 Through (a)Power Amplitude Max. Min. TM 3 1 2 4 Input Cross Through (b)Figure 3.34: Simulated 2-D power distribution profiles for TM operation atλ0 = 1550 nm of the PBS, when (a) a fundamental TE mode and (b) afundamental TM mode are launched into Port 4 of the same device.583.5. PBS using a Symmetric DC with Sinusoidal Bends1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 1570-2-10123456x 10-40, nmPropagation Constant Difference, nm-1  TEbentTEstraightTMbentTMstraightTEbent = -2.879X10-6 nm-1TMstraight = 4.559X10-4 nm-1TEstraight = 8.321X10-5 nm-1TMbent = 1.571X10-4 nm-1Figure 3.35: Calculated wavelength-dependent propagation constant differ-ences for both the TE and TM modes of the sinusoidally-bent and straightsymmetric DCs, which have G = 200 nm.Table 3.11: Calculated values of the ∆βbents and ∆βstraights and absolutevalues of their ratios for both mode types at λ0 = 1550 nm of the bent andstraight DCs.Modes TE TM∆βbent, nm−1 −2.879× 10−6 1.571× 10−4∆βstraight, nm−1 8.321× 10−5 4.559× 10−4| ∆βbent∆βstraight | 0.03460 0.3446593.5. PBS using a Symmetric DC with Sinusoidal Bends3.5.2 Modal DispersionReferring to Eqs. 3.3, and 3.4 and Figs. 3.3, 3.6a, and 3.6b, the wavelength-dependent propagation constant differences for both the TE and TM modesof a straight symmetric DC, which has G = 200 nm, are calculated andshown in Fig. 3.35. Then, referring to Eq. 2.24 (∆βbent ∼= ∆βstraightJ0[2piA(W+G)Λβwg]) and 3.2 and Figs. 2.8, 3.3, and 3.35, the wavelength-dependent propa-gation constant differences for both the TE and TM modes of the sinusoidally-bent symmetric DC, which is used in my PBS, are calculated and shown inFig. 3.35. Therefore, referring to Fig. 3.35, the values of the ∆βbents and∆βstraightss and the absolute values of their ratios for both mode types atλ0 = 1550 nm of the bent and straight DCs are obtained and listed in Ta-ble 3.11. Referring to Fig. 3.35 and Table 3.11, the sinusoidal bends reduceTE coupling between the waveguides by approximately 96.5 % and reduceTM coupling between the waveguides by approximately 65.5 % for the bentDC as compared to the equivalent straight DC, thus, the sinusoidally-bentsymmetric DC in my PBS only allows TM optical power to cross over tothe adjacent waveguide and has a cross-over length that is greater than theequivalent straight DC for TM operation (see Ref. [1]).Table 3.12: Calculated values of the Dbents and Dstraights and absolute valuesof their ratios for both mode types at λ0 = 1550 nm of the bent and straightDCs.Modes TE TMDbent, ps·µm−1·nm−1 −8.257× 10−8 −9.321× 10−7Dstraight, ps·µm−1·nm−1 −3.491× 10−7 −1.153× 10−6| DbentDstraight | 0.2365 0.8084603.5. PBS using a Symmetric DC with Sinusoidal Bends1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 1570-1.2-1-0.8-0.6-0.4-0.20x 10-60, nmModal Dispersion, psm-1nm-1  DTEbentDTEstraightDTMbentDTEstraightDTEbent = -8.257X10-8 psm-1nm-1DTEstraight = -3.491X10-7 psm-1nm-1DTMbent = -9.321X10-7 psm-1nm-1DTEstraight = -1.153X10-6 psm-1nm-1Figure 3.36: Calculated wavelength-dependent modal dispersions for boththe TE and TM modes of the sinusoidally-bent and straight symmetric DCs,which have G = 200 nm.Referring to Eq. 2.31 (D = −12pic(∆β+λ0d∆βdλ0)) and Fig. 3.35, the wavelength-dependent modal dispersions for both the TE and TM modes of the bentand straight DCs are calculated and shown in Fig. 3.36. The values of theDbents and Dstraights and the absolute values of their ratios for both modetypes at λ0 = 1550 nm of the bent and straight devices are obtained andlisted in Table 3.12. Referring to Fig. 3.36 and Table 3.12, the sinusoidally-bent symmetric DC in my PBS has lower wavelength dependencies andbetter broadband performances than the equivalent straight symmetric DCfor both the TE and TM modes (see Ref. [1]).613.6. Summary3.6 SummaryIn this chapter, a single straight waveguide, straight symmetric DCs, sinu-soidal AC symmetric waveguide pairs, and a PBS using a symmetric DCwith sinusoidal bends were simulated for TE and/or TM operation on anSOI platform. The design parameters for the sinusoidal AC devices and thePBS were optimized using the simulation data. The wavelength dependen-cies of the sinusoidally-bent symmetric DC, which was used in the PBS,were also compared with the equivalent straight symmetric DC for the TEand TM modes.62Chapter 4Fabrication, Measurement,and DemonstrationIn this Chapter, I present the fabrication, measurement, and demonstrationof the TE and TM sinusoidal AC symmetric waveguide pairs and a PBSusing a symmetric DC with sinusoidal bends. Using the optimized designparameters that I obtained from the previous chapter, I created the masklayouts for the fabrication of those devices. I also added the straight sym-metric DCs and straight AC asymmetric waveguides to the mask layoutsin order to evaluate the performances of the sinusoidal AC devices. All ofthe devices were fabricated using electron-beam (E-Beam) lithography onan SOI platform. Using an automated optical fibre probe station in our lab,I measured the fabricated test devices. Then, using the measurement data,I demonstrate a sinusoidal AC waveguide pair for TE operation, a sinusoidalAC waveguide pair for TM operation, and a PBS using a symmetric DC withsinusoidal bends for both TE operation and TM operation. I also evaluatetheir performances accordingly.11Sections 4.1, 4.2, and 4.3.3 are based on the work in the published paper [1], andSections 4.1, 4.2, and 4.3.1 are based on the work in the published paper [2].634.1. Fabrication4.1 FabricationTE I: Input Port,  T: Through Port, C: Cross Port. I T C SWGCs Figure 4.1: Design layout (left) of the TE sinusoidal AC symmetric waveg-uide pair with an SEM image (right) of the sinusoidal symmetric waveguidestructure in the device.The mask layouts of my test devices (see Figs. 4.1, 4.2, and 4.3) weredrawn using Pyxis layout editor (from Mentor Graphics, Inc.). The testdevices were fabricated on a 220-nm-thick crystalline silicon layer of an SOIwafer (see Fig. 4.4) using E-Beam lithography, and a layer of 2-µm-thicksilicon dioxide were deposited over the devices at the University of Wash-ington. As shown in Fig. 4.4, the wafer also consists of a 3-µm-thick silicondioxide layer and a 675-µm-thick silicon layer. The fabrication processes arecompatible with CMOS technologies. Scanning-electron-microscope (SEM)images in Figs. 4.1, 4.2, and 4.3 show the sinusoidal symmetric waveguidestructures of the fabricated test devices (the TE sinusoidal AC device inRef. [2] and the PBS in Ref. [1]).644.2. MeasurementTM I: Input Port,  T: Through Port, C: Cross Port. I T C SWGCs Figure 4.2: Design layout (left) of the TM sinusoidal AC symmetric waveg-uide pair with an SEM image (right) of the sinusoidal symmetric waveguidestructure in the device.4.2 MeasurementAs shown in Figs. 4.1, 4.2, and 4.3, sub-wavelength grating couplers (SWGCs)in Refs. [47, 48] were used to couple light in and out of each of the test de-vices. A reference device is a pair of SWGCs, which are directly connectedto each other by a waveguide, for either TE or TM operation. There is onereference device, near each of the test devices, and its spectral response canbe used to normalize the measurement data of the test device for the cor-responding mode type (see Refs. [1, 2]). Figure 4.5 shows the TE referencedevice for the TE sinusoidal AC device, as well as one of the TE SWGCsthat were used in these devices. Figure 4.6 shows the reference device forthe PBS for TE operation, as well as one of the TE SWGCs that were usedin these devices. Figure 4.7 shows the reference device for the TM sinusoidal654.2. Measurement                                                         TE                                                           TM             I T C I T C I: Input Port, T: Through Port, C: Cross Port. SWGCs SWGCs Figure 4.3: Design layouts of the PBSs for TE operation (upper left) and TMoperation (lower left) with an SEM image (right) of the sinusoidal symmetricwaveguide structure in one of the devices. Adapted with permission fromRef. [1], c©2017 Optical Society of America. Silicon Dioxide: 2 μm     Silicon Dioxide: 3 μm    Crystalline Silicon: 220 nm   Silicon: 675 μm Wafer Figure 4.4: Schematic of an SOI Platform.664.2. Measurement                                           TE                  SWGC Figure 4.5: Design layout (left) of the TE reference device with a close-upview (right) of one of the TE SWGCs that were used in the TE referencedevice and the TE sinusoidal AC device.                                         TE                  SWGC Figure 4.6: Design layout (left) of the TE reference device with a close-upview (right) of one of the TE SWGCs that were used in the TE referencedevice and the PBS for TE operation.674.2. Measurement                                       TM                  SWGC Figure 4.7: Design layout (left) of the TM reference device with a close-upview (right) of one of the TM SWGCs that were used in the TM referencedevice, the TM sinusoidal AC device, and the PBS for TM operation.AC device and the PBS for TM operation, as well as one of the TM SWGCsthat were used in these devices.The fabricated test devices were measured using an automated opticalfibre probe station in our lab (see Ref. [49]). As shown in Fig. 4.8, theprobe station consists of a laser and detector mainframe (Agilent 8164A), atunable laser source module (Agilent 81682A), a dual optical power sensormodule (Agilent 81635A), polarization-maintaining fibres (from PLC Con-nections, LLC. and OZ Optics, Ltd.), an optical fibre array (from PLCConnections, LLC.), a motor controller (Thorlabs BBD203), a temperaturecontroller (Standford Research Systems LDC501), a metallic stage, a micro-scope camera (from Tucsen Photonics Co., Ltd.), a microscope lamp (fromAmScope/United Scope, LLC.), a desktop computer, and an optical table(Newport RS4000). Since thermal stability is essential for consistent opti-684.2. Measurement1 6 5 9 7 4 8 10 11 12 2 3 Figure 4.8: Automated optical fibre probe station: 1. Agilent 8164A mea-surement system; 2. Agilent 81635A dual optical power sensors; 3. Ag-ilent 81682A tunable laser source; 4. PLC Connections and OZ Opticspolarization-maintaining fibres; 5. PLC Connections optical fibre array; 6.Thorlabs BBD203 motor controller; 7. Standford Research Systems LDC501temperature controller; 8. metallic stage; 9. Tucsen microscope camera; 10.AmScope microscope lamp; 11. desktop computer; 12. Newport RS4000optical table.694.3. Demonstrationcal measurement, a room temperature of 25 ◦C was maintained during mymeasurement using the temperature controller.4.3 DemonstrationTable 4.1: Design parameters of the TE and TM sinusoidal AC symmetricwaveguide pairs and a PBS using a symmetric DC with sinuosidal bends(see Refs. [1, 2]).DevicesParametersW , nm H, nm G, nm Λ, µm L, µm A, nmTE sinusoidal AC [2] 500 220 200 16.44 32.88 932TM sinusoidal AC 500 220 600 19.86 39.72 1025PBS [1] 500 220 200 16.44 8.55 932The TE and TM sinusoidal AC symmetric waveguide pairs and a PBSusing a symmetric DC with sinusoidal bends are demonstrated using themeasurement data. The design parameters of those devices are listed inTable 4.1.4.3.1 Sinusoidal AC Symmetric Waveguide Pair for TEOperationI measured my TE sinusoidal AC test device (see Fig. 4.1) for TE operationover the C-band (a wavelength range from 1530 nm to 1565 nm) and normal-ized the measurement data using the TE reference device (see Fig. 4.5). Astraight AC asymmetric waveguide pair, which consisted of a 450-nm-widewaveguide and a 550-nm-wide waveguide, was also fabricated and measured.704.3. Demonstration1530 1535 1540 1545 1550 1555 1560 1565-70-60-50-40-30-20-10010  Normalized Optical Transmission, dB0, nmThrough, Sinusoidal AC Symmetric-SimulatedCross, Sinusoidal AC Symmetric-SimulatedThrough, Sinusoidal AC Symmetric-MeasuredCross, Sinusoidal AC Symmetric-MeasuredCross, Straight Coupling Symmetric-MeasuredTEFigure 4.9: Simulated and normalized measured optical transmission spectrafor the through and cross ports of the TE sinusoidal AC symmetric waveg-uide pair as well as normalized measured optical transmission spectrum forthe cross port of the equivalent TE straight symmetric DC. Adapted withpermission from Ref. [2], c©2015 Optical Society of America.Table 4.2: Minimum, average, and maximum values of the SRs of both theTE sinusoidal and straight AC devices for TE operation over the C-band(see Ref. [2]).Devices Sinusoidal Symmetric AC Straight Asymmetric ACSR, dBMinimum Average Maximum Minimum Average Maximum26.8 38.2 56.5 16.9 17.8 18.6714.3. Demonstration1530 1535 1540 1545 1550 1555 1560 1565-70-60-50-40-30-20-10010  Normalized Optical Transmission, dB0, nmThrough, Straight AC AsymmetricCross, Straight AC AsymmetricCross, Straight Coupling SymmetricTEFigure 4.10: Normalized measured optical transmission spectra for thethrough and cross ports of the TE straight AC asymmetric waveguide pairas well as for the cross port of the equivalent TE straight symmetric DC.Adapted with permission from Ref. [2], c©2015 Optical Society of America.1530 1535 1540 1545 1550 1555 1560 15650102030405060700, nmSuppression Ratio, dB  Sinusoidal SymmetricStraight AsymmetricTEFigure 4.11: SR spectra for both the TE sinusoidal and straight AC devices.Adapted with permission from Ref. [2], c©2015 Optical Society of America.724.3. Demonstration1530 1535 1540 1545 1550 1555 1560 15650102030405060700, nmRelative Suppression Ratio, dB  TEFigure 4.12: Relative SR spectrum of the TE sinusoidal AC device as com-pared to the equivalent TE straight AC device. Adapted with permissionfrom Ref. [2], c©2015 Optical Society of America.Each of the two TE AC devices has the same G = 200 nm and has the sameL = 32.88 µm that was chosen to allow for the maximum TE optical powertransfer to the cross port of the equivalent TE straight symmetric DC. Fig-ure 4.9 shows the normalized measured optical transmission spectra for boththe through and cross ports of the TE sinusoidal AC symmetric waveguidepair as compared to the results from FDTD simulations, and Fig. 4.10 showsthe normalized measured optical transmission spectra for both the throughand cross ports of the TE straight AC asymmetric waveguide pair. Both ofthe figures also show the normalized measured optical transmission spectrumfor the cross port of the equivalent TE straight symmetric DC.Each of the AC devices was designed to suppress the optical transmissionto the cross port, TACcross, while the equivalent DC was designed to maximize734.3. Demonstrationthe optical transmission to the cross port, TDCcross. Therefore, the TDCcross can beused as a reference to evaluate the suppression of inter-waveguide crosstalkthat is achieved by each of the AC devices, and I define the suppression ratio(SR) for each of the AC devices as:SR = 10 log10(TDCcrossTACcross). (4.1)Figure 4.11 shows the SR spectra of both the TE sinusoidal and straightAC devices for TE operation over the C-band. The minimum, average, andmaximum values of the SRs of the two TE AC devices within the C-bandare listed in Table 4.2. In order to compare the crosstalk suppression of thesinusoidal AC device with the straight AC device, I also define the relativeSR as the ratio (difference in dB) of the SR of the TE sinusoidal AC deviceto the SR of the TE straight AC device at each λ0. As shown in Fig. 4.12,the TE sinusoidal AC device has higher relative SR than the TE straightAC device over the C-band. A minimum value of the relative SR is 8.9 dB,an average value of the relative SR is 20.4 dB, and a maximum value of therelative SR is 38.6 dB (see Ref. [2]). Hence, the TE sinusoidal AC symmetricwaveguides provide excellent crosstalk suppression over the entire C-band.4.3.2 Sinusoidal AC Symmetric Waveguide Pair for TMOperationI measured my TM sinusoidal AC test device (see Fig. 4.2) for TM opera-tion over the C-band and normalized the measurement data using the TMreference device (see Fig. 4.7). A straight AC asymmetric waveguide pair,which consisted of a 450-nm-wide waveguide and a 550-nm-wide waveguide,was also fabricated and measured. Each of the two AC devices has the sameG = 600 nm and has the same L = 39.72 µm that was chosen to allow for744.3. Demonstration1530 1535 1540 1545 1550 1555 1560 1565-70-60-50-40-30-20-10010Normalized Optical Transmission, dB0, nm  Through, Sinusoidal AC Symmetric-SimulatedCross, Sinusoidal AC Symmetric-SimulatedThrough, Sinusoidal AC Symmetric-MeasuredCross, Sinusoidal AC Symmetric-MeasuredCross, Straight Coupling Symmetric-MeasuredTMFigure 4.13: Simulated and normalized measured optical transmission spec-tra for the through and cross ports of the TM sinusoidal AC symmetricwaveguide pair as well as normalized measured optical transmission spec-trum for the cross port of the equivalent TM straight symmetric DC.Table 4.3: Minimum, average, and maximum values of the SRs of both theTM sinusoidal and straight AC devices for TM operation over the C-band.Devices Sinusoidal Symmetric Straight AsymmetricSR, dBMinimum Average Maximum Minimum Average Maximum23.1 34.9 44.4 28.3 41.1 52.8754.3. Demonstration1530 1535 1540 1545 1550 1555 1560 1565-70-60-50-40-30-20-10010Normalized Optical Transmission, dB0, nm  Through, Straight AC AsymmetricCross, Straight AC AsymmetricCross, Straight Coupling SymmetricTMFigure 4.14: Normalized measured optical transmission spectra for thethrough and cross ports of the TM straight AC asymmetric waveguide pairas well as for the cross port of the equivalent TM straight symmetric DC.1530 1535 1540 1545 1550 1555 1560 1565010203040506070Suppression Ratio, dB0, nm  Sinusoidal SymmetricStraight AsymmetricTMFigure 4.15: SR spectra of both the TM sinusoidal and straight AC devices.764.3. Demonstration1530 1535 1540 1545 1550 1555 1560 1565-30-20-1001020300, nmRelative Suppression Ratio, dB  TMFigure 4.16: Relative SR spectrum of the TM sinusoidal AC device as com-pared to the equivalent TM straight AC device.the maximum TM optical power transfer to the cross port of the equivalentTM straight symmetric DC. Figure 4.13 shows the normalized measuredoptical transmission spectra for both the through and cross ports of the TMsinusoidal AC symmetric waveguide pair as compared to the results fromFDTD simulations, and Fig. 4.14 shows the normalized measured opticaltransmission spectra for the through and cross ports of the TM straight ACasymmetric waveguide pair. Both of the figures also show the normalizedmeasured optical transmission spectrum for the cross port of the equivalentTM straight symmetric DC.Referring to Eq. 4.1, the SR for each of the TM AC devices can becalculated. Figure 4.15 shows the SR spectra of both the TM sinusoidaland straight AC devices for TM operation over the C-band. The minimum,average, and maximum values of the SRs of the two TM AC devices within774.3. Demonstrationthe C-band are listed in Table 4.3. Even though the TM sinusoidal ACdevice has lower relative SR than the TM straight AC device over the C-band, the TM sinusoidal AC symmetric waveguides still provide a largecrosstalk suppression over the C-band (see Fig. 4.16).4.3.3 PBS Using a Symmetric DC with Sinusoidal Bends1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 1570-30-25-20-15-10-50510Normalized Optical Transmission, dB0, nm  ThroughCrossTEFigure 4.17: Normalized measured optical transmission spectra of thethrough and cross ports of the PBS for TE operation. Adapted with per-mission from Ref. [1], c©2017 Optical Society of America.I measured the test devices of my PBSs (see Fig. 4.3) for both TE oper-ation and TM operation over a wavelength range from 1470 nm to 1570 nmand normalized the measurement data using the TE and TM reference de-vices (see Figs. 4.6 and 4.7), respectively. Figures 4.17 and 4.18 show thenormalized measured optical transmission spectra for the through and crossports for both TE operation and TM operation. The average excess losses784.3. Demonstration1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 1570-30-25-20-15-10-50510Normalized Optical Transmission, dB0, nm  ThroughCrossTMFigure 4.18: Normalized measured optical transmission spectra of thethrough and cross ports of the PBS for TM operation. Adapted with per-mission from Ref. [1], c©2017 Optical Society of America..1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 1570051015202530Extinction Ratio, dB0, nm  TETMFigure 4.19: ER spectra for both the TE and TM modes. Adapted withpermission from Ref. [1], c©2017 Optical Society of America.794.3. Demonstration1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 1570051015202530Polarization Isolation, dB0, nm  ThroughCrossFigure 4.20: PI spectra for both the through and cross ports. Adapted withpermission from Ref. [1], c©2017 Optical Society of America.Table 4.4: Minimum, average, and maximum values of the ERs and PIs ofthe PBS for both TE operation and TM operation over a wavelength rangefrom 1470 nm to 1570 nm. Adapted with permission from Ref. [1], c©2017Optical Society of America.Modes TE TMER, dBMinimum Average Maximum Minimum Average Maximum10.5 12.0 17.0 16.4 20.1 26.1Ports Through CrossPI, dBMinimum Average Maximum Minimum Average Maximum16.6 20.6 26.5 9.9 11.5 16.5804.3. Demonstrationof the PBS are 0.84 dB for TE operation and 1.33 dB for TM operation (seeRef. [1]).In my PBS, the T TEthrough and TTEcross are TE optical transmissions to thethrough and cross ports, respectively, and the T TMthrough and TTMcross are TMoptical transmissions to the through and cross ports, also respectively. Asshown in Figs. 4.17 and 4.18, the T TEthrough is high as compared to the TTEcrossand T TMthrough, while the TTMcross is high as compared to the TTMthrough and TTEcross.Hence, the extinction ratios (ERs) between the through and cross ports forthe TE and TM modes, ERTE and ERTM , respectively, are defined as inRef. [40]:ERTE = 10 log10(T TEthroughT TEcross), (4.2a)andERTM = 10 log10(T TMcrossT TMthrough), (4.2b)which indicate how much TE optical power is transmitted to the throughport as compared to the cross port and how much TM optical power istransmitted to the cross port as compared to the through port, respectively.The polarization isolations (PIs) between the TE and TM modes for thethrough and cross ports, PIthrough and PIcross, respectively, are defined asin Ref. [40]:PIthrough = 10 log10(T TEthroughT TMthrough), (4.3a)andPIcross = 10 log10(T TMcrossT TEcross), (4.3b)which indicate how well the TE mode is isolated from the TM mode at thethrough port and how well the TM mode is isolated from the TE modeat the cross port, respectively. Figures. 4.19 and 4.20 show the ER and814.4. SummaryPI spectra of the PBS over a wavelength range from 1470 nm to 1570 nm.The minimum, average, and maximum values of the ERs and PIs withinthe demonstrated wavelength range are listed in Table 4.4. As shown inFigs. 4.19 and 4.20 and Table 4.4, the PBS has broadband performance forboth TE operation and TM operation.4.4 SummaryIn this chapter, the sinusoidal AC symmetric waveguide pairs and a PBSusing a symmetric DC with sinusoidal bends were demonstrated for TEand/or TM operation on an SOI platform. The fabrication and measurementof the test devices were also described. Each of the sinusoidal AC deviceswas compared with the straight coupling and AC devices, which had thesame gap width and coupler length. The performance of the PBS was alsoevaluated for both the TE and TM modes.82Chapter 5Summary, Conclusions, andSuggestions for Future Work5.1 SummaryI have designed, simulated, analyzed, and demonstrated the TE and TMsinusoidal AC symmetric waveguide pairs and a PBS using a symmetric DCwith sinusoidal bends in this thesis. Using the supermodes of a symmetrictwo-waveguide structure, I derived the optical transmission of a symmetricDC. Using the coupled modes of individual waveguides in the DC, I derivedthe optical transmission of each waveguide in the device. Based on theanalyses for the supermodes and local normal modes of the symmetric DC,I derived the design parameters of sinusoidal bends that suppressed theoptical power exchange between a pair of symmetric waveguides.Using the FDE solver, I simulated the optical modes of single-waveguideand two-waveguide structures and obtained their characterization data forboth the TE and TM modes. Then, using the characterization data, I de-signed the coupling and AC devices, each for TE or TM operation. The TMmodes were more sensitive to the bending losses than the TE modes. There-fore, using the FDTD solver, I simulated straight and sinusoidal symmetricdevices that had different gap widths and chose an appropriate gap width835.1. Summaryfor the TM sinusoidal AC device. Using the FDTD solver, I optimized thedesign parameters of the coupling and AC devices and simulated the devices,each for TE or TM operation.I also designed a PBS using a symmetric DC with sinusoidal bends. Thesinusoidal bends suppressed the optical power exchange between the waveg-uides for TE operation and can still allowed for the maximum optical powertransfer to an adjacent waveguide for TM operation. While a sinusoidal ACdevice was designed to suppress either TE or TM optical power exchange be-tween the waveguides, the PBS was designed to split the TE and TM modesinto two output waveguides. I also derived and calculated the modal disper-sions of the sinusoidally-bent symmetric DC, which was used in the PBS,and the modal dispersions of the equivalent straight symmetric DC. Then, Icompared the wavelength dependencies of the sinusoidally-bent symmetricDC and the wavelength dependencies of the equivalent straight symmetricDC for both the TE and TM modes according to their modal dispersions.Using the FDTD solver, I optimized the design parameters of the PBS andsimulated the device for both TE operation and TM operation.I created the mask layouts of the test devices for fabrication. I also addedstraight symmetric DCs and straight AC asymmetric waveguide pairs thathad the same gap widths and coupler lengths as the sinusoidal AC devices,each for TE or TM operation, to the mask layouts. The test devices werefabricated using E-Beam lithography at the University of Washington andmeasured using an automatic probe station in our lab. The TE sinusoidalAC symmetric waveguide pair, which had a gap width of 200 nm, had anaverage crosstalk SR of 38.2 dB and an average improvement in crosstalkSR of 20.4 dB, as compared to the equivalent TE straight AC asymmetricwaveguide pair, over the entire C-band. The TM sinusoidal AC symmetric845.2. Conclusionswaveguide pair, which had a gap width of 600 nm, had an average crosstalkSR of 34.9 dB over the entire C-band. The PBS, which had a small couplerlength of 8.55 µm, had an average ER of 12.0 dB for the TE mode and anaverage ER of 20.1 dB for the TM mode and had an average PI of 20.6 dBfor the through port and an average PI of 11.5 dB for the cross port overa wavelength range from 1470 nm to 1570 nm, which covered the entireC-band.5.2 ConclusionsIn conclusion, I have demonstrated the TE and TM sinusoidal AC waveg-uide pairs and a PBS using a symmetric DC with sinusoidal bends an SOIplatform. My sinusoidal AC symmetric waveguides have large suppressionof optical power exchange between the waveguides over the entire C-band.Hence, the sinusoidal waveguides can be used to design compact AC routingwaveguides and dense waveguide buses. My PBS has a small coupler lengthand shows a broad operational bandwidth for both TE operation and TMoperation. Hence, the device can be used to separate/combine the TE andTM modes in polarization diversity systems. All of my devices were easy tofabricate and compatible with CMOS technologies.5.3 Suggestions for Future WorksAs shown in Figs. 3.11a, 3.11b, 3.22d, and 3.26b in Chapter 3, small amountsof crosstalk still occurs between the waveguides over half of the bending pe-riod. Since the cross-over length is proportional to the gap width of a straightsymmetric DC, larger gap widths can be used to design sinusoidal AC sym-855.3. Suggestions for Future Worksmetric waveguides to achieve larger crosstalk suppression. The bifurcatingbranches that were used in the AC devices can be further optimized to elim-inate the crosstalk that occurs within the branches. Asymmetric waveguidescan also be used with sinusoidal bends to enhance crosstalk suppression.As shown in Fig. 4.20, my PBS has a relatively low PI for the cross port.However, the device is sufficiently compact to allow for several devices to beconnected in series in order to obtain better broadband performance thanother published broadband PBSs that were based on DCs (see Ref. [1]).Since the PBS is based on the TE sinusoidal AC symmetric waveguide pair,the suggestions for improvements of the sinusoidal AC waveguides above alsoapply to the improvements of the PBS. Moreover, the operational bandwidthof my PBS is limited by the operational bandwidth of the SWGCs that wereused in the test device, and, thus, edge couplers can be used for the PBS toachieve a wider measurable wavelength range than the SWGCs.There are many potential applications of my sinusoidal AC symmetricwaveguides. The sinusoidally-bent AC symmetric waveguide pair can beused to design polarization rotators and splitters, which have been demon-strated using straight and circularly-bent asymmetric waveguides on anSOI platform in Refs. [50, 51]. It has been shown in Chapter 3 that thesinusoidal bends reduced the wavelength dependence of a symmetric DCwith sinusoidal bends as compared to an equivalent straight symmetric DC.Therefore, the sinusoidally-bent waveguide pair can also be used to de-sign broadband DCs, which have been proposed and demonstrated usingsinusoidally-bent symmetric waveguides on a titanium lithium-niobate plat-form in Ref. [52] and using circularly-bent symmetric waveguides on an SOIplatform in Ref. [53]. Moreover, the optical propagation in a symmetric DCwith sinusoidal bends can also be analyzed using a conformal transformation865.3. Suggestions for Future Workstechnique as in Refs. [54, 55].Input Through Figure 5.1: Top view of a TE sinusoidal AC symmetric SOI waveguide array,which has G = 200 nm.In addition, multiple sinusoidally-bent symmetric waveguides (see Figs. 5.1and 5.2) can be used for various applications, such as dense AC waveguidebuses and mode-division and polarization-division demultiplexer/multiplexer,which have been demonstrated using straight asymmetric and circularly-bent symmetric waveguides on an SOI platform in Refs. [25, 56, 57]. Thesinusoidally-bent symmetric waveguide array can also be used to control theoptical propagation in photonic lattices on an SOI platform as suggested inRefs. [58–60]. The sinusoidal AC symmetric waveguides can also be used todesign active and thermal optical switches, which have been demonstratedusing sinusoidal symmetric waveguides on a titanium lithium-niobate plat-form as in Ref. [23] and using straight asymmetric waveguides an SOI plat-form in Ref. [61].875.3. Suggestions for Future WorksPower Amplitude Max. 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In a straight symmetric DC(see Fig. A.2), the 2-D electric field distribution for local normal mode ofwaveguide core a , Ψa,straight(x, z), can be given as:Ψa,straight(x, z) = Aa,straight(z)ψa(x)e−jβwgz, (A.1)and the 2-D electric field distribution for local normal mode of waveguidecore b, Ψb,straight(x, z), can also be given as:Ψb,straight(x, z) = Ab,straight(z)ψb(x)e−jβwgz, (A.2)99Appendix A. Derivation of the Propagation Constant Difference of a Sinusoidally-bent Symmetric DCTop Oxide Cladding Layer    Buried Oxide Layer   Silicon Silicon W W G H y x z Figure A.1: Cross-sectional view of a symmetric DC on an SOI platform(also Fig. 2.1 in Chapter 2). Adapted with permission from Ref. [1], c©2017Optical Society of America.W G L 𝒂 𝒃 Input Isolation Through Cross (𝒙 = 𝟎, 𝒛 = 𝟎) y z x Figure A.2: Top view of a straight symmetric DC (also Fig. 2.3 in Chapter 2).100Appendix A. Derivation of the Propagation Constant Difference of a Sinusoidally-bent Symmetric DC-1 -0.5 0 0.5 1-W-G/2-(W+G)/2-G/20x, nma(x)/a,max-1 -0.5 0 0.5 10+G/2+(W+G)/2+W+G/2x, nmb(x)/b,maxbaTE TEFigure A.3: 1-D normalized transverse field distributions for the TE localnormal modes of waveguide cores a (left) and waveguide core b (right),ψa(x) and ψb(x), over their respective maximum values, ψa,max and ψb,max(also Fig. 2.6 in Chapter 2).where Aa,straight(z) and Ab,straight(z) are their respective z-dependent trans-verse field amplitudes, ψa(x) and ψb(x) are their respective 1-D normalizedtransverse field distributions (see Figs. A.3 and A.4), and βwg is the prop-agation constant of a straight strip waveguide core with cross-sectional di-mensions (W and H). Since Aa,straight(z) can be given as:Aa,straight(z) = AM cos(∆βstraight2z), (A.3)and Ab,straight(z) can also be given as:Ab,straight(z) = −jAM sin(∆βstraight2z), (A.4)where AM is the maximum magnitude of both Aa,straight(z) and Ab,straight(z)along the z-axis, and ∆βstraight is the difference in the propagation con-stants for the even (symmetric) and odd (anti-symmetric) supermodes of101Appendix A. Derivation of the Propagation Constant Difference of a Sinusoidally-bent Symmetric DC-1 -0.5 0 0.5 1-W-G/2-(W+G)/2-G/20+G/2+(W+G)/2+W+G/2x, nma(x)/a,max-1 -0.5 0 0.5 1-W-G/2-(W+G)/2-G/20+G/2+(W+G)/2+W+G/2x, nmb(x)/b,maxTM TMbaFigure A.4: 1-D normalized transverse field distributions for the TM localnormal modes of waveguide core a (left) and waveguide core b (right), ψa(x)and ψb(x), over their respective maximum values, ψa,max and ψb,max (alsoFig. 2.7 in Chapter 2).the straight DC. Then, referring to Eqs. A.3 and A.4, the first derivative ofAa,straight(z) with respect to z can be solved and represented, in terms ofAb,straight(z), as:∂Aa,straight(z)∂z= −j∆βstraight2[−jAM sin(∆βstraight2z)]= −j∆βstraight2Ab,straight(z),(A.5)and, also referring to Eqs. A.3 and A.4, the first derivative of Ab,straight(z)with respect to z can be solved and represented, in terms of Aa,straight(z),as:∂Ab,straight(z)∂z= −j∆βstraight2[AM cos(∆βstraight2z)]= −j∆βstraight2Aa,straight(z),(A.6)where both of them show the relations between individual local normalmodes of the DC.102Appendix A. Derivation of the Propagation Constant Difference of a Sinusoidally-bent Symmetric DCW G Λ 𝒙 = 𝒇𝒙 𝒛  y z 2A x 𝒂 𝒃 (𝒙 = 𝟎, 𝒛 = 𝟎) Figure A.5: Top view of a full period of a sinusoidally-bent symmetric DC(also Fig. 2.2 in Chapter 2). Adapted with permission from Ref. [1], c©2017Optical Society of America.-1 -0.5 0 0.5 1-W-G/2-(W+G)/2-G/20+G/2+(W+G)/2+W+G/2x, nme(x)/e,max-1 -0.5 0 0.5 1-W-G/2-(W+G)/2-G/20+G/2+(W+G)/2+W+G/2x, nmo(x)/o,maxTEEvenTEOddFigure A.6: 1-D normalized transverse field distributions for the even TE su-permode (right) and odd TE supermode (right), ψe(x) and ψo(x), over theirrespective maximum values, ψe,max and ψo,max (also Fig. 2.4 in Chapter 2).103Appendix A. Derivation of the Propagation Constant Difference of a Sinusoidally-bent Symmetric DC-1 -0.5 0 0.5 1-W-G/2-(W+G)/2-G/20+G/2+(W+G)/2+W+G/2x, nme(x)/e,max-1 -0.5 0 0.5 1-W-G/2-(W+G)/2-G/20+G/2+(W+G)/2+W+G/2x, nmo(x)/o,maxTMEvenTMOddFigure A.7: 1-D normalized transverse field distributions for the even TMsupermode (left) and odd TM supermode (right), ψe(x) and ψo(x), over theirrespective maximum values, ψe,max and ψo,max (also Fig. 2.5 in Chapter 2).Again, as discussed in Chapter 2, a sinusoidally-bent symmetric DC hasthe same cross-sectional dimensions, W , H, and G, (see Fig. A.1) and hasthe same device length as an equivalent straight symmetric DC. Each of thebends is defined as a function of z, fx(z) (see Fig. A.5), as in Ref. [1]:fx(z) = A cos(2piΛz), (A.7)where A and Λ are the amplitude and period of a sinusoid. Accordingto Refs. [54, 55, 62], the refractive index profiles and electromagnetic fielddistributions of a bent waveguide are skewed as compared to the ones of astraight waveguide, and, according to Refs. [20, 21, 43], in a symmetric DCwith sinusoidal bends, the effects of the sinusoidal bends on the local normalmode of waveguide core a can be approximated using an envelope function,104Appendix A. Derivation of the Propagation Constant Difference of a Sinusoidally-bent Symmetric DCFa(z):Fa(z) = e+jβwgu∂fx(z)∂z , (A.8)and the effects of the sinusoidal bends on the local normal mode of waveguidecore b can also be approximated using an envelope function, Fb(z):Fb(z) = e−jβwgu ∂fx(z)∂z , (A.9)where u =∫+∞−∞ xψe(x)ψo(x)dx√∫+∞−∞ ψ2e(x)dx∫+∞−∞ ψ2o(x)dx, ∂fx(z)∂z = −2piΛ A sin(2piΛ z), ψe(x) andψo(x) are the 1-D normalized transverse field distributions for the even andodd supermodes (see Figs. A.6 and A.7).0 0.5 1-W/20W/2x, nmN(x|0,)/NmaxFigure A.8: Probability density function of the 1-D normal distribution,N(x|0, σ), which is centered at x = 0, over its maximum value, Nmax.Here, I will use the 1-D normal distributions to approximate ψa(x) andψb(x), and each of the distributions is defined by a probability density func-tion, N(x|µlnm, σ):N(x|µlnm, σ) = e−(x−µlnm)22σ2√2piσ2, (A.10)105Appendix A. Derivation of the Propagation Constant Difference of a Sinusoidally-bent Symmetric DC-1 -0.5 0 0.5 1-W-G/2-(W+G)/2-G/20x, nma(x)/a,max-1 -0.5 0 0.5 10+G/2+(W+G)/2+W+G/2x, nmb(x)/b,maxabFigure A.9: 1-D normalized transverse field distributions for the approx-imated ψa(x) and ψb(x), which are centered at x = µa = −W+G2 and atx = µb = +W+G2 , over their ψa,max and ψb,max, respectively.-1 -0.5 0 0.5 1-W-G/2-(W+G)/2-G/20+G/2+(W+G)/2+W+G/2x, nme(x)/e,max-1 -0.5 0 0.5 1-W-G/2-(W+G)/2-G/20+G/2+(W+G)/2+W+G/2x, nmo(x)/o,maxEven OddFigure A.10: 1-D normalized transverse field distributions for the approxi-mated ψe(x) and ψo(x), over their ψe,max and ψo,max, respetively.106Appendix A. Derivation of the Propagation Constant Difference of a Sinusoidally-bent Symmetric DCwhere its mean, µlnm, determines the location of its center in the x-axis,and its standard deviation, σ, determines its shape. I can set σ = W2√2 ln(2)such that N(x|µlnm, σ) has a maximum value, Nmax, and N(−W2 |0, σ) =N(+W2 |0, σ) = Nmax2 (see Fig. A.8). Thus, ψa(x) can be approximated as:ψa(x) ≈√N(x|µa, σ) = e(x−µa)2σ24√2piσ2, (A.11)and ψb(x) can also be approximated as:ψb(x) ≈√N(x|µb, σ) = e(x−µb)2σ24√2piσ2, (A.12)where ψa(x) is centered at x = µa = −W+G2 , and ψb(x) is centered atx = µb = +W+G2 (see Fig. A.9). Since µa = −µb = −W+G2 = −µ, ψe(x) canbe approximated, in terms of ψa(x) and ψb(x), as:ψe(x) ≈ AM2Ae[ψa(x) + ψb(x)] =AM2Ae[e(x−µ)2σ2 + e(x+µ)2σ24√2piσ2], (A.13)ψo(x) can also be approximated, in terms of ψa(x) and ψb(x), as:ψo(x) ≈ AM2Ao[ψa(x)− ψb(x)] = AM2Ao[e(x−µ)2σ2 − e (x+µ)2σ24√2piσ2], (A.14)where both of them are shown in Fig. A.10. Subsequently, I can estimateu ≈ −W+G2 = −µ.Therefore, in the sinusoidally-bent symmetric DC, referring to Eqs. A.1,A.3, and A.8, the 2-D electric field distribution for local normal mode ofwaveguide core a , Ψa,bent(x, z), can be obtained as:Ψa,bent(x, z) = Ψa,straight(x, z)Fa(z) ∼=[AM cos(∆βstraight2z)e−jβwgµ∂fx(z)∂z]ψa(x)e−jβwgz,(A.15)107Appendix A. Derivation of the Propagation Constant Difference of a Sinusoidally-bent Symmetric DCand, referring to Eqs. A.2, A.4, and A.9, the 2-D electric field distribution forlocal normal mode of waveguide core b, Ψb,bent(x, z), can also be obtainedas:Ψb,bent(x, z) = Ψb,straight(x, z)Fb(z) ∼=[−jAM sin(∆βstraight2z)e+jβwgµ∂fx(z)∂z]ψb(x)e−jβwgz.(A.16)Then, referring to Eq. A.15, the z-dependent transverse field amplitude forlocal normal mode of waveguide core a of the bent DC, Aa,bent(z), can begiven, in terms of Aa,straight(z), as:Aa,bent(z) =[AM cos(∆βstraight2z)]e−jβwgµ∂fx(z)∂z = Aa,straight(z)e−jβwgµ ∂fx(z)∂z ,(A.17)and, referring to Eq. A.16, the z-dependent transverse field amplitude forlocal normal mode of waveguide core b of the bent DC, Ab,bent(z), can begiven, in terms of Ab,straight(z), as:Ab,bent(z) =[−jAM sin(∆βstraight2z)]e+jβwgµ∂fx(z)∂z = Ab,straight(z)e+jβwgµ∂fx(z)∂z .(A.18)Alternatively, referring to Eq. A.17, Aa,straight(z), can be expressed, in termsof Aa,bent(z), as:Aa,straight(z) = Aa,bent(z)e+jβwgµ∂fx(z)∂z , (A.19)and, referring to Eq. A.18, Ab,straight(z) can be expressed, in terms ofAb,bent(z), as:Ab,straight(z) = Ab,bent(z)e−jβwgµ ∂fx(z)∂z . (A.20)Thus, referring to Eq. A.17, the first derivative of Aa,bent(z) with respect toz can be solved and represented, in terms of Aa,straight(z), as:∂Aa,bent(z)∂z=[∂Aa,straight(z)∂z− jµβwgC(z)Aa,straight(z)]e−jµβwg∂fx(z)∂z ,(A.21)108Appendix A. Derivation of the Propagation Constant Difference of a Sinusoidally-bent Symmetric DCand, referring to Eq. A.18, the first derivative of Ab,bent(z) with respect toz can be solved and represented, in terms of Ab,straight(z), as:∂Ab,bent(z)∂z=[∂Ab,straight(z)∂z+ jµβwgC(z)Ab,straight(z)]e+jµβwg∂fx(z)∂z ,(A.22)where C(z) = ∂2fx(z)∂z2= 4pi2Λ2A cos(2piΛ z)is the curvature of fx(z).0 2 4 6 8 10-0.6-0.4-0.200.20.40.60.811.2xvarJ0  Figure A.11: Bessel function of the first kind of order 0 with respect to avariable, xvar, ranging from 0 to 10.Since C(z) ≈ 0 over a Λ that is much larger than the A of the sinusoidalbends, referring to Eq. A.21,∂Aa,bent(z)∂z can be approximated as:∂Aa,bent(z)∂z∼= ∂Aa,straight(z)∂ze−jµβwg∂fx(z)∂z , (A.23)and, referring to Eq. A.22,∂Ab,bent(z)∂z can also be approximated as:∂Ab,bent(z)∂z∼= ∂Ab,straight(z)∂ze+jµβwg∂fx(z)∂z . (A.24)109Appendix A. Derivation of the Propagation Constant Difference of a Sinusoidally-bent Symmetric DCThen, substituting Eqs. A.5 and A.20 into Eq. A.23,∂Aa,bent(z)∂z becomes:∂Aa,bent(z)∂z∼= −j∆βstraight2Ab,straight(z)e−jµβwg ∂fx(z)∂z = −j[∆βstraighte−2jµβwg ∂fx(z)∂z]2Ab,bent(z),(A.25)and, substituting Eqs. A.6 and A.19 into Eq. A.24,∂Ab,bent(z)∂z becomes:∂Ab,bent(z)∂z∼= −j∆βstraight2Aa,straight(z)e+jµβwg∂fx(z)∂z = −j[∆βstraighte+2jµβwg∂fx(z)∂z]2Aa,bent(z).(A.26)Hence, referring to Eqs. A.25 and A.26, I can find the complex z-dependent∆β of the bent DC, ∆β˜bent(z), as:∆β˜bent(z) = ∆βstraighte−j2µβwg ∂fx(z)∂z = ∆βstraightej2piA(W+G)Λβwg sin(2piΛz).(A.27)Subsequently, calculating a running average of ∆β˜bent(z) over one Λ alongthe z-axis according to Ref. [63], the effective ∆β of the bent DC, ∆βbent,can be estimated as:∆βbent ∼= 1Λ∫ Λ0∆β˜bent(z)dz = ∆βstraightJ0[2piA(W +G)Λβwg], (A.28)where J0 is the Bessel function of the first kind of order 0 (see Fig. A.11).I will use Eq. A.28 to design and analyze sinusoidal AC symmetric stripwaveguide pairs and a PBS using a symmetric DC with sinusoidal bends.110Appendix BAdditional PublicationsI am one of the co-authors of the following publications:1. Han Yun, Yun Wang, Fan Zhang, Zeqin Lu, Stephen Lin, LukasChrostowski, and Nicolas A. F. Jaeger. Broadband 2×2 adiabatic 3 dB cou-pler using silicon-on-insulator sub-wavelength grating waveguides. OpticsLetters, 41(13):3041-3044, 2016.2. Matthew J. Collins, Fan Zhang, Richard J. Bojko, Lukas Chrostowski,and Mikael C. Rechtsman. Integrated optical Dirac physics via inversionsymmetry breaking. Physical Review A, 94(6):063827, 2016.3. Hamed Pishvai Bazargani, Maurizio Burla, Zhitian Chen, Fan Zhang,Lukas Chrostowski, and Jose´ Azan˜a. Long-duration optical pulse shapingand complex coding on SOI. IEEE Photonics Journal, 8(4):1-7, 2016.4. Yun Wang, Zeqin Lu, Minglei Ma, Han Yun, Fan Zhang, Nicolas A.F. Jaeger, and Lukas Chrostowski. Compact broadband directional couplersusing subwavelength gratings. IEEE Photonics Journal, 8(3):1-8, 2016.5. Matthew Collins, Jack Zhang, Richard J. Bojko, Lukas Chrostowski,and Mikael C. Rechtsman. Dirac physics in silicon via photonic boron ni-tride. In CLEO: QELS-Fundamental Science, pages FM3A.4. Optical Soci-ety of America, 2016.6. Zeqin Lu, Yun Wang, Fan Zhang, Nicolas A. F. Jaeger, and LukasChrostowski. Wideband silicon photonic polarization beamsplitter based on111Appendix B. Additional Publicationspoint-symmetric cascaded broadband couplers. Optics Express, 23(23):29413-29422, 2015.7. Zhitian Chen, Jonas Flueckiger, Xu Wang, Fan Zhang, Han Yun,Zeqin Lu, Michael Caverley, Yun Wang, Nicolas A. F. Jaeger, and LukasChrostowski. Spiral Bragg grating waveguides for TM mode silicon photon-ics. Optics Express, 23(19):25295-25307, 2015.8. Zeqin Lu, Han Yun, Yun Wang, Zhitian Chen, Fan Zhang, NicolasA. F. Jaeger, and Lukas Chrostowski. Asymmetric-waveguide-assisted 3-dBbroadband directional coupler. In CLEO: Science and Innovations, pagesSM1I.8. Optical Society of America, 2015.9. Yun Wang, Han Yun, Zeqin Lu, Richard J. Bojko, Fan Zhang, MichaelCaverley, Nicolas A. F. Jaeger, and Lukas Chrostowski. Apodized focusingfully etched sub-wavelength grating couplers with ultra-low reflections. InCLEO: Science and Innovations, pages SM1I.6. Optical Society of America,2015.10. Zhitian Chen, Jonas Flueckiger, Xu Wang, Han Yun, Yun Wang,Zeqin Lu, Fan Zhang, Nicolas A. F. Jaeger, and Lukas Chrostowski. Bragggrating spiral strip waveguide filters for TM modes. In CLEO: Science andInnovations, pages SM3I.7. Optical Society of America, 2015.11. Yun Wang, Han Yun, Zeqin Lu, Richard J. Bojko, Wei Shi, Xu Wang,Jonas Flueckiger, Fan Zhang, Michael Caverley, Nicolas A. F. Jaeger, andLukas Chrostowski. Apodized focusing fully etched subwavelength gratingcouplers. IEEE Photonics Journal, 7(3):1-10, 2015.12. Zeqin Lu, Han Yun, Yun Wang, Zhitian Chen, Fan Zhang, Nicolas A.F. Jaeger, and Lukas Chrostowski. Broadband silicon photonic directionalcoupler using asymmetric-waveguide based phase control. Optics Express,23(3):3795-3808, 2015.112

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