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Sub-wavelength grating components for silicon-on-insulator platform Wang, Yun 2016

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Sub-wavelength Grating Componentsfor Silicon-on-insulator PlatformbyYun WangB.Sc., Shenzhen University, 2011M.ASc., The University of British Columbia, 2013A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2016c© Yun Wang 2016AbstractThis dissertation is a theoretical and experimental study of sub-wavelengthgrating (SWG) based photonic devices for the silicon-on-insulator (SOI)platform, including high-efficiency sub-wavelength grating couplers (SWGCs),broadband SWGCs, broadband SWG directional couplers, and an SWG po-larization splitter-rotator.High-efficiency SWGCs with improved operating bandwidths and sup-pressed back reflections have been demonstrated to couple light into and outof SOI based photonic integrated circuits (PICs). One-dimensional SWGshave been proposed and experimentally demonstrated for the first time tomake fully-etched grating couplers, which have performances comparable tothe state-of-the-art fully-etched grating couplers, but with better fabricationtolerance, reduced fabrication complexity, and less cost.A theoretical study of the operating bandwidths for grating couplershas been presented and a design methodology has been demonstrated fordesigning SWGCs with design-intent operating bandwidths. SWGCs with1-dB bandwidths up to 90 nm have been demonstrated, which have improvedthe operating bandwidth of fully-etched grating couplers by a factor of 3.Such broadband SWGCs are essential components for applications such aswavelength-division multiplexing (WDM) PICs and bio-sensing.Compact directional couplers, with dimensions about 10 times smalleriiAbstractthan its alternatives, i.e., adiabatic couplers and multimode interferencecouplers have been demonstrated for various power splitting ratios. Theoperating bandwidths of our directional couplers have been improved by afactor of 2 as compared to conventional directional couplers. The dispersionproperties of SWGs have been explored and applied to engineer the wave-length dependancy of conventional directional couplers for broad operatingbandwidths, which is the first experimental demonstration of such devices.Polarization splitter and polarization rotators are essential componentsto address the polarization diversity of PICs. An ultra-compact mode-coupling based polarization splitter-rotator (PSR), which combines func-tionalities of a polarization splitter and a polarization rotator, with dimen-sions 15-20 times smaller than its alternative, i.e., mode-evolution basedPSRs, has been experimentally demonstrated for the first time, where anasymmetric waveguide system consisting of a strip waveguide and an SWGwaveguide were used to improve the fabrication tolerance of such devices. Ameasured peak polarization conversion efficiency of −0.3 dB with crosstalksbelow −10 dB over the C-band has been achieved.iiiPrefaceParts of the dissertation are based on the author’s manuscripts, which havebeen or will be published, resulting from collaborations with multiple re-searchers. A complete list of publications is given in Appendix A.A version of Section 2.1 has been published:1. Yun Wang, Xu Wang, Jonas Flueckiger, Han Yun, Wei Shi, RichardBojko, Nicolas A. F. Jaeger, and Lukas Chrostowski, “Focusing sub-wavelength grating coupler with low back reflections for rapid proto-typing of silicon photonic circuits”, Optics Express, Vol. 22, Issue 17,pp.20652-20662, 2014The author contributed the idea, conducted the devices’ design, con-ducted the measurement, conducted the data processing and analysis,and wrote the manuscript. Xu Wang helped with many discussionson the design methodology. Jonas Flueckiger helped on building themeasurement setup. Han Yun helped with the measurement. WeiShi helped with discussion on the design and assisted in editing themanuscript. Richard Bojko fabricated the design and took SEM pic-tures of the fabricated devices. Prof. Chrostowski and Prof. Jaegerhelped with many suggestions in the course of the project and assistedin editing the manuscript.A version of Section 2.2 has been published:ivPreface2. Yun Wang, Han Yun, Zeqin Lu, Richard Bojko, Wei Shi, Xu Wang,Jonas Flueckiger, Fan Zhang, Michael Caverley, Nicolas A. F. Jaeger,Lukas Chrostowski, “Apodized focusing fully etched sub-wavelengthgrating couplers”, Photonics Journal, IEEE, Vol. 7, No. 3, 2400110,2015The author contributed the idea, conducted the devices’ design, con-ducted the measurement, conducted the data processing and analysis,and wrote the manuscript. Han Yun and Zeqin Lu helped with dis-cussions on the design. Richard Bojko fabricated the design and tookSEM pictures of the fabricated devices. Wei Shi and Xu Wang as-sisted in editing the manuscript. Jonas Flueckiger helped with themeasurement setup. Fan Zhang and Michael Caverley helped with themeasurement. Prof. Chrostowski and Prof. Jaeger helped with manysuggestions in the course of the project and assisted in editing themanuscript.A version of Chapter 3 has been published:3. Yun Wang, Wei Shi, Xu Wang, Zeqin Lu, Michael Caverley, RichardBojko, Lukas Chrostowski, and Nicolas A. F. Jaeger, “Design of broad-band sub-wavelength grating couplers with low back reflection”, OpticsLetters, Vol. 40, Issue 20, pp.4647-4650, 2015The author contributed the idea, conducted the devices’ design, con-ducted the measurement, conducted the data processing and analysis,and wrote the manuscript. Wei Shi helped with the motivation withthe design. Xu Wang helped with the simulation of the design. ZeqinLu helped with discussion about the design. Michael Caverley helpedwith the measurement. Richard Bojko fabricated the design and tookvPrefaceSEM pictures of the fabricated devices. Prof. Chrostowski and Prof.Jaeger helped with many suggestions in the course of the project andassisted in editing the manuscript.A version of Chapter 4 has been published:4. Yun Wang, Zeqin Lu, Minglei Ma, Han Yun, Fan Zhang, Nicolas A.F. Jaeger, and Lukas Chrostowski, “Compact Broadband DirectionalCouplers Using Sub-wavelength Gratings,” Photonics Journal, IEEE,Vol. 8, Issue 3, 2016The author contributed the idea, conducted the devices’ design, con-ducted the measurement, conducted the data processing and analysis,and wrote the manuscript. The author Zeqin Lu and Minelei Mahelped with the mask layout of the design. Han Yun and Fan Zhanghelped with the measurement. Prof. Chrostowski and Prof. Jaegerhelped with many suggestions in the course of the project and assistedin editing the manuscript.A version of Chapter 5 has been published:5. Yun Wang, Minglei Ma, Han Yun, Zeqin Lu, Xu Wang, Nicolas A.F. Jaeger, and Lukas Chrostowski, “Ultra-Compact Sub-WavelengthGrating Polarization Splitter-Rotator for Silicon-on-Insulator Platform,”Photonics Journal, IEEE, Vol. 8, Issue 6, 2016The author contributed the idea, conducted the devices’ design, con-ducted the measurement, conducted the data processing and analysis,and wrote the manuscript. The author Minelei Ma helped with themask layout and the measurement. Han Yun helped with the mea-surement, Zeqin Lu helped with the layout. Xu Wang helped withviPrefacethe simulation methodology. Prof. Chrostowski contributed on thedesign methodology. Prof. Chrostowski and Prof. Jaeger helped withmany suggestions in the course of the project and assisted in editingthe manuscript.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . xixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Silicon Photonics . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Silicon-on-Insulator . . . . . . . . . . . . . . . . . . . 21.1.2 Challenges and State of the Art . . . . . . . . . . . . 21.2 Sub-wavelength Structures . . . . . . . . . . . . . . . . . . . 81.2.1 The Sub-wavelength Regime . . . . . . . . . . . . . . 81.2.2 Effective Medium Theory . . . . . . . . . . . . . . . . 101.2.3 Applications of Sub-wavelength Structures on SOI . . 11viiiTable of Contents1.3 About This Dissertation . . . . . . . . . . . . . . . . . . . . 131.3.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.2 Dissertation Organization . . . . . . . . . . . . . . . . 132 High-Efficiency Fully-etched Sub-wavelength Grating Cou-plers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1 Uniform Sub-wavelength Grating Couplers . . . . . . . . . . 162.1.1 Challenges of Fully-etched Grating Couplers . . . . . 162.1.2 2D SWGs vs. 1D SWGs . . . . . . . . . . . . . . . . 182.1.3 Design and Simulation . . . . . . . . . . . . . . . . . 212.1.4 Fabrication and Measurement . . . . . . . . . . . . . 312.2 Apodized Sub-wavelength Grating Couplers . . . . . . . . . 392.2.1 Design and Simulation . . . . . . . . . . . . . . . . . 392.2.2 Design of Apodized SWGCs . . . . . . . . . . . . . . 412.2.3 Measurement Results . . . . . . . . . . . . . . . . . . 443 Broadband Sub-wavelength Grating Couplers . . . . . . . . 493.1 Bandwidth Analysis . . . . . . . . . . . . . . . . . . . . . . . 493.2 Design Methodology . . . . . . . . . . . . . . . . . . . . . . . 513.3 Measurement Results . . . . . . . . . . . . . . . . . . . . . . 564 Broadband Sub-wavelength Directional Couplers . . . . . . 614.1 Wavelength Dependency of Directional Couplers . . . . . . 624.2 Dispersion Engineering . . . . . . . . . . . . . . . . . . . . . 644.3 Measurement Results . . . . . . . . . . . . . . . . . . . . . . 705 Sub-wavelength Polarization Splitter Rotator . . . . . . . . 765.1 Design of SWG PSR . . . . . . . . . . . . . . . . . . . . . . . 77ixTable of Contents5.2 Fabrication and Measurement . . . . . . . . . . . . . . . . . 856 Conclusion and Future Work . . . . . . . . . . . . . . . . . . 886.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.1.1 Technical Contributions . . . . . . . . . . . . . . . . . 886.1.2 Design Methodology Contributions . . . . . . . . . . 906.1.3 Theoretical Contributions . . . . . . . . . . . . . . . . 936.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97AppendicesA Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113A.1 Book Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . 113A.2 Journal Publications . . . . . . . . . . . . . . . . . . . . . . . 113A.3 Conference Proceedings . . . . . . . . . . . . . . . . . . . . . 116xList of Tables2.1 Simulated and measured sensitivities of the central wave-length as a function of Λ, ΛH, and lsub. . . . . . . . . . . . . . 332.2 Λ and lsub values for the apodized SWGCs. . . . . . . . . . . 412.3 Parameters used for corner analysis for SWGCs. . . . . . . . 463.1 Optimization steps and design parameters . . . . . . . . . . . 543.2 Design variations . . . . . . . . . . . . . . . . . . . . . . . . . 575.1 Parameter Variations. . . . . . . . . . . . . . . . . . . . . . . 856.1 Comparison of SWGCs for the SOI platform. CE, couplingefficiency; BW, bandwidth. . . . . . . . . . . . . . . . . . . . 90xiList of Figures1.1 Schematic of the cross-section of a silicon-on-insulator wafer. 21.2 A schematic of the cross-section of an SOI waveguide withdimensions of 500 nm x 220 nm and an optical fiber with adiameter of 10 µm drawn to scale. . . . . . . . . . . . . . . . . 31.3 Schematics of (a) a shallow-etched grating coupler and (b) aregular fully-etched grating coupler. . . . . . . . . . . . . . . 51.4 Schematics of (a) an adiabatic coupler, (b) an multimode in-terference coupler, and (c) a directional coupler. (drawn toscale) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Schematic of a polarization diverse PICs. . . . . . . . . . . . 71.6 Schematic of the cross-section of a periodic waveguide structure. 91.7 SEM images of (a) an SWG waveguide, and (b) an SWGwaveguide with an SWG waveguide taper. . . . . . . . . . . . 121.8 SEM images of (a) a directional coupler consisting of stripwaveguides, and (b) a directional coupler consisting of a stripwaveguide and an SWG waveguide. . . . . . . . . . . . . . . . 131.9 SEM images of (a) a straight SWGC with two-dimensionalSWGs and (b) a focusing SWGC with one-dimensional SWGs. 14xiiList of Figures2.1 Measured transmission spectra for a pair of shallow-etchedgrating couplers and fully-etched grating coupler. . . . . . . . 172.2 A Schematics of a fully-etched grating couplers with effectiveindex regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Schematics of a fully-etched SWGCs with one-dimensionalSWGs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 (a) 2D energy distribution of an SWGC formed by two-dimensionalSWGs; (b) energy distribution along the 1-D cutline acrossthe 2-D simulation of an SWGC formed by two-dimensionalSWGs; (c) 2D energy distribution of an SWGC formed byone-dimensional SWGs; (d) energy distribution along 1-Dcutlines across the 2-D simulations of SWGCs formed by one-dimensional SWGs. . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Schematic of the cross-section of an SWGC with (a) a positiveincident angle and (b) a negative incident angle. . . . . . . . 222.6 Simulations of SWGCs in FDTD Solutions. . . . . . . . . . . 242.7 Simulated coupling efficiency and back reflection for the TESWGC and a shallow-etched grating coupler. SWGC-T andSWGC-R denote the coupling efficiency and the back reflec-tion of the optimized TE SWGC, Shallow-T and Shallow-Rdenote the coupling efficiency and the back reflection of theoptimized shallow-etched grating coupler. . . . . . . . . . . . 25xiiiList of Figures2.8 The simulated coupling efficiency and the back reflection forthe TM SWGC and a shallow-etched grating coupler. SWGC-T and SWGC-R denote the coupling efficiency and the backreflection of the optimized TM SWGC, Shallow-T and Shallow-R denote the coupling efficiency and the back reflection of theoptimized shallow-etched grating coupler. . . . . . . . . . . . 262.9 (a) Schematic of the input-waveguide-output circuit in In-terconnect, and (b) simulated transmission spectra of input-waveguide-ouput circuits for TE and TM SWGCs. . . . . . . 282.10 Mask layouts of (a) a grating coupler with straight gratinglines and a linear taper, and (b) an SWGC with focusinggrating lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.11 Schematics for SWGCs with focusing grating lines and (a)positive incident angles, and (b) negative incident angles. . . 302.12 SEM images of (a) the top view of a fabricated SWGC and(b) the side view of a fabricated SWGC. . . . . . . . . . . . . 322.13 (a) Simulated sensitivities of Λ, ΛH, and lsub for the TESWGCs; (b) measured sensitivities for the TE SWGCs; (c)simulated sensitivities for the TM SWGCs; (d) measured sen-sitivities for the TM SWGCs. . . . . . . . . . . . . . . . . . 342.14 Simulated and measured transmission spectra of a fabricatedTE SWGC assuming lsub = 89nm; inset is the spectrum nearthe central wavelength. . . . . . . . . . . . . . . . . . . . . . . 36xivList of Figures2.15 (a) Measured spectra of 11 TM SWGCs; (b) comparisonof measured and simulated coupling efficiencies of the TMSWGCs; (c) comparison of the measured and simulated 1-dBbandwidths of the TM SWGCs; (d) measured spectra of TMSWGCs with various ΛH values; (e) measured spectra of TMSWGCs with various Λ values; (f) measured spectra of TMSWGCs with various lsub values. . . . . . . . . . . . . . . . . 382.16 α as functions of lsub and ΛH for the TE SWGCs, and (b) αas functions of lsub and ΛH for the TM SWGCs. . . . . . . . . 402.17 Simulated coupling efficiencies and back reflections of theapodized and the un-apodized (a) TE SWGCs using 2D FDTDsimulation, and (b)TM SWGCs using 2D FDTD simulation. . 422.18 Simulated transmission spectra of (a) the apodized TE SWGCs,and (b) the apodized TM SWGCs. . . . . . . . . . . . . . . . 432.19 (a) Measured spectra of an apodized TE SWGC and an un-apodized TE SWGC; (b) coupling efficiencies of apodized andun-apodized TE SWGCs; (c) 1-dB bandwidths of apodizedand un-apodized TM SWGCs; (d) central wavelengths of theapodized and un-apodized TE SWGCs. . . . . . . . . . . . . 452.20 (a) Measured spectra of an apodized TM SWGC and an un-apodized TM SWGC; (b) coupling efficiencies of apodized andun-apodized TM SWGCs; (c) 1-dB bandwidths of apodizedand un-apodized TM SWGCs; (d) central wavelengths of theapodized and un-apodized TM SWGCs. . . . . . . . . . . . . 473.1 Schematic of the cross-section of a broadband SWGC withone-dimensional SWGs. . . . . . . . . . . . . . . . . . . . . . 50xvList of Figures3.2 Optimal incident angle, θopt, as a function of the group indexof a grating coupler, ng. . . . . . . . . . . . . . . . . . . . . . 523.3 Schematics of (a) an SWGC with refractive index regions ofnH and nL, (b) an SWGC with one-dimensional SWGs. . . . 533.4 Simulated transmission and reflection spectra of the SWGCwith the design parameters shown in Table 3.1. . . . . . . . . 553.5 Simulated 1-dB bandwidth and peak coupling efficeincy as afunction of ffL. . . . . . . . . . . . . . . . . . . . . . . . . . . 563.6 SEM images of the as-fabricated focusing and straight SWGCs.(For the straight SWGC, the taper is not shown.) . . . . . . . 573.7 Transmission spectra of the simulated SWGC (red), the mea-sured focusing SWGC (blue), and the measured straight SWGC(green). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.8 Measured (a) 1-dB bandwidths and (b) coupling efficienciesof the fabricated focusing SWGCs with ffL ranging from 0.1to 0.19; measured (c) 1-dB bandwidths and (d) coupling effi-ciencies of the fabricated straight SWGCs with ffL rangingfrom 0.1 to 0.18. . . . . . . . . . . . . . . . . . . . . . . . . . 604.1 Schematic of an SWG DC with design parameters labelled. . 624.2 Field profiles of (a) the TE0 mode, and (b) the TE1 mode fora conventional DC. . . . . . . . . . . . . . . . . . . . . . . . 634.3 (a) neff1 and neff2 as functions of λ for a conventional DC, (b)Lpi as a function of λ for a conventional DC. The simulatedDC has w = 450 nm, g = 220 nm, based on an SOI wafer witha silicon layer of 220 nm. . . . . . . . . . . . . . . . . . . . . 644.4 neff as a function of λ for an SWG with Λ = 285 nm, ff = 0.5. 65xviList of Figures4.5 (a) Schematic of the simulated structure for an SWG DC inFDTD Solutions, (b) simulated band diagram for an SWGDC with Λ = 285 nm and ff = 0.28. . . . . . . . . . . . . . 664.6 (a) Simulated transmission spectra from the cross ports ofSWG DCs with ff = 0.25, NG = 47, and various Λ values,(b) simulated transmission spectra from the cross ports ofSWG DCs with Λ = 285 nm, NG = 47, and various ff values. 674.7 (a) neff1 and neff2 as functions of λ for an SWG DC withΛ = 285 nm and ff = 0.28, (b) calculated Lpi as a functionof λ based on the nneff1 and neff2 shown in (a). . . . . . . . . 684.8 (a) Simulated spectra for an SWG DC with a designed powersplitting ratio of 50/50 (η1 and η2 are the normalized outputpower for the cross and through ports, respectively), (b) sim-ulated IL as a function of λ for the SWG DC with a designedpower splitting ratio of 50/50. . . . . . . . . . . . . . . . . . . 694.9 Simulated ILs as a function of ff for SWG DCs with Λ =285 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.10 Mask layout of the test structure for SWG DCs. . . . . . . . 714.11 SEM images of (a) a fabricated SWG DC, (b) a zoom-in ofthe central portion of the fabricated SWG DC with the designparameters labelled. . . . . . . . . . . . . . . . . . . . . . . . 734.12 (a) Measured spectra for a test structure with SWG DCswhich had power splitting ratios close to 50/50, and (b) nor-malized optical powers for a fabricated SWG DC using themeasurement data shown in (a). . . . . . . . . . . . . . . . . 744.13 Normalized optical powers for fabricated SWG DCs with powersplitting ratios of (a) 40/60, (b) 30/70, and (c) 20/80. . . . . 75xviiList of Figures5.1 Schematic of the top view of an SWG PSR. . . . . . . . . . . 785.2 neff-TM as a function of WA for a strip waveguide and neff-TEas a function of WB for an SWG waveguide. . . . . . . . . . 795.3 The X and Y field distributions for the first three super-modes. (a) Y component of the Exy0 mode, (b) X componentof the Exy0 mode; (c) X component of the Exy1 mode, (d) Ycomponent of the Exy1 mode; (e) X component of the Exy2mode, (f) Y component of the Exy2 mode. . . . . . . . . . . . 815.4 neff as a function of λ for the Exy1 and the Exy2 modes of thetwo waveguide system comprising the coupling section of theSWG PSR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.5 Schematic of the top view of an SWG taper. . . . . . . . . . . 835.6 (a) Simulated ILTE-TE and PCETM-TE as functions of λ, and(b) simulated XTTE-TE and XTTM-TM as functions of λ. . . 845.7 PCETM-TE as functions of δWX and δWY-Z. . . . . . . . . . 855.8 SEM images of a fabricated SWG PSR with zoomed images ofthe coupling section, the SWG taper, and the S bend waveg-uide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.9 (a) Measured ILTE-TE and PCETM-TE as functions of λ, and(b) measured XTTE-TE and XTTM-TM as functions of λ. . . . 876.1 Schematic of the cross-section of an SWGC with sub-wavelengthstructures comprising both the coupling region and the taperregion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95xviiiList of AbbreviationsBOX Buried OxideCMOS Complementary Metal-Oxide-SemiconductorDC Directional CouplerEME Eigen Mode ExpansionEMT Effective Medium TheoryER Extinction RatioFDTD Finite-Difference Time-DomainFOM Figure-of-MeritFSR Free Spectrum RangeIL Insertion LossMMI Multimode InterferometerMPW Multi-Project WaferMZI Mach Zehnder InterferometerNA Numerical ApertureNG Number of GratingPCE Polarization Conversion EfficiencyPIC Photonic Integrated CircuitPM Polarization MaintainingPSA Particle Swarm AlgorithmPSF Point Spread FunctionxixList of AbbreviationsPSR Polarization Splitter-RotatorSEM Scanning Electron MicroscopeSOI Silicon-on-InsulatorSWG Sub-Wavelength GratingSWGC Sub-Wavelength Grating CouplerTE Transverse ElectricTM Transverse MagneticXT CrosstalkxxAcknowledgementsI would like to thank my supervisor Dr. Lukas Chrostowski for guiding methrough a fantastic journey that I have enjoyed so much. I want to thankhim for supporting me from every aspects he can and providing me with thefreedom to do the research that I am interested. To me, he is a passionateresearcher, a great supervisor, a talented educator and a good friend.I would like to thank Dr. Nicolas A. F. Jaeger for the dedicated helphe provided through my PhD journey. Part of the most enjoyable timeduring my PhD was sitting with Dr. Jaeger in his front yard, editing mymanuscripts and discussing various research ideas. Beyond the help fromthe research side, he has been a mentor to me since I joined the photonicsresearch group at UBC. Through out the years, he has shown me what iswisdom, kindness, and generosity.I would also like to thank all my colleagues that I have been workingwith for their help and support, especially Dr. Wei Shi, Dr. Xu Wang, andDr. Jonas Flueckiger.xxiDedicationTo my familiy.xxiiChapter 1Introduction1.1 Silicon PhotonicsSilicon photonics is an emerging technology for building complex photonicintegrated circuits (PICs). The great promise of silicon photonics lies inintegrating multiple functions into a single package, and manufacturingmost or all of them using the same fabrication facilities that are used tobuild advanced microelectronics, namely the complementary metal-oxide-seminconductor (CMOS) technology, as part of a single chip or chip stack[19]. Historically, photonic devices with different functionalities are designedand fabricated based on different materials, i.e., lithium niobate for highspeed modulators, indium phosphide for lasers, germanium for photodec-tors, glass for optical multiplexers. Though discrete photonic devices can beconnected using standard optical fibers, a large fraction of the final devicecost emerges from the the photonic packaging processes [19]. Silicon pho-tonics provides the possibility to integrate all of these functionalities intothe same platform, which is the silicon-on-insulator (SOI) platform. Duringthe past two decades, various photonic designs at the component level, withcompetitive performances, have been demonstrated for the SOI platform,including low loss waveguides [63], multiplexers and demultiplexers [67, 76],high-speed electrooptic modulators [61], photodetectors [52], etc.11.1. Silicon Photonics1.1.1 Silicon-on-InsulatorA schematic of the cross-section of an SOI wafer is shown in Fig. 1.1, whichconsists of a silicon substrate for mechanical support, a buried oxide (BOX)layer acting as the insulator layer, and another thin silicon layer on top ofthe BOX where the waveguides and devices are defined. A cladding oxideare normally applied on top of the top silicon layer for protection. Thethickness of silicon substrate is about 700 µm, the thickness of the BOXlayer is normally between 1 µm to 3 µm, as long as it is thick enough toavoid penetration loss of optical waves from the top silicon layer into thesilicon substrate. The thickness of the top silicon layer is about few hundrednanometers and the optimal thickness is application dependent [92]. Thedesigns shown in this dissertation are based on SOI wafers with a 220 nmsilicon layer and a 3 µm BOX layer. The thickness of the top silicon layer weused are the same with those provided by the multiple-project wafer (MPW)foundries, such as imec, LETI, IME and IHP [1, 36, 39, 42].Figure 1.1: Schematic of the cross-section of a silicon-on-insulator wafer.1.1.2 Challenges and State of the ArtAlthough significant progress has been achieved, many challenges remain forthe silicon photonics community to be solved. Due to the fact that silicon21.1. Silicon Photonicsis an indirect-band semiconductor which is inefficient at light generation,external lasers are often used as the power supplies for the SOI PICs, whereoptical fibers are often used to couple light into and out of the SOI PICs. Dueto the large refractive index contrast between the top silicon layer and itscladding, propagation modes can be highly confined within the waveguides,with cross-sectional dimensions on the order of a few hundred nanometers.However, the small feature sizes of the waveguide raise the problem of largemode mismatch when coupling light from an optical fiber to the sub-micronsilicon waveguide core. A schematic of the cross-section of an SOI waveguideand an optical fiber drawn to scale is shown in Fig. 1.2.Figure 1.2: A schematic of the cross-section of an SOI waveguide with di-mensions of 500 nm x 220 nm and an optical fiber with a diameter of 10 µmdrawn to scale.Edge coupling and surface coupling are the two approaches that havebeen used to address the mode mismatch issue, and both approaches havedemonstrated below 1 dB loss per device [75, 99]. Using edge coupling tech-nique has the advantages of broad operating bandwidth with ultra-low cou-31.1. Silicon Photonicspling loss. It also couples both the transverse electric (TE) mode and thetransverse magnetic (TM) mode at the same time. In addition, using edgecoupling technique can take advantages of the mature packaging solution.Compared with the edge coupling technique, surface coupling using grat-ing couplers has the following advantages: firstly, the cost of using grat-ing couplers is lower because post processing of the chip, i.e., deep etchingon the edges of chips and edge polishing, and expensive components, i.e.,lensed fibers or high numerical aperture (NA) fibers, are not required for themeasurement; secondly, grating couplers are easy to align to during mea-surement; thirdly, grating couplers have more compact design shape and alsoprovide the flexibility to be located anywhere on a chip, which enables betterarchitectural design; last but not least, grating couplers enable wafer-scaleautomated measurement, without the need to dice the wafer.Depending on the required etch-steps for fabrication, grating couplerscan be divided into shallow-etched grating couplers and fully-etched gratingcouplers. Schematics of a shallow-etched grating coupler and a fully-etchedgrating coupler are shown in Fig. 1.3. Coupling efficiency and operatingbandwidth are the two most important figures-of-merits (FOMs) for a grat-ing coupler. Both shallow-etched grating couplers and fully-etched gratingcouplers with coupling efficiencies higher than −1 dB have been demon-strated [5, 24, 99]. However, fabricating shallow-etched grating couplers,with various etch depths, increases both the fabrication complexity and thecost. When combining fundamental building blocks that can be fabricatedin a single, fully-etched step, having fully-etched grating couplers providesan efficient and economical solution for rapid prototyping. High-efficiencyfully-etched grating couplers will be demonstrated in Chapter 2 of this dis-sertation and fully-etched grating couplers with ultra-broad operating band-41.1. Silicon Photonicswidths will be demonstrated in Chapter 3 of this dissertation.(a) (b)Figure 1.3: Schematics of (a) a shallow-etched grating coupler and (b) aregular fully-etched grating coupler.Optical couplers are essential for coupling light between waveguides inPICs, which are often used in applications such as wavelength-division-multiplexing [12] and optical switching [2]. Dealing with the dispersioneffects in various optical couplers is a great challenge since we are interestedin designing devices for a variety of wavelengths, and the wavelength depen-dence of the devices should be taken into account. There are two sources forthe dispersion effects: the material dispersion, i.e., dispersion comes from thematerial property, and the waveguide dispersion, i.e., dispersion comes fromthe geometry of the waveguide. Such dispersion effects limit the operatingbandwidth of a device. Adiabatic couplers [71, 98], multimode interference(MMI) couplers [48, 70], and directional couplers (DCs) [47, 55] are threetypes of optical coupler that are often used in PICs, and the schematics ofthem are shown in Fig. 1.4. Adiabatic couplers have the advantage of broadoperating bandwidth, and they have been used to form 3-dB couplers [98]and thermo-optic switches [86]. However, in order to avoid the excitationof higher order modes, the lengths of adiabatic couplers are much longerthan MMI couplers and DCs. MMI couplers are shorter than adiabatic cou-plers, but are much larger than DCs. In addition, MMI based devices are51.1. Silicon Photonicsmuch less predictable when used for low coupling ratios, i.e., power splittingratios of 2/98 or 5/95 [21]. Conventional DCs, consisting of two parallelwaveguides, are very compact in size, but they are also very sensitive to thewavelength and polarization state of the light. Several methods have beenproposed to improve the bandwidths of DCs, such as connecting couplers ina Mach-Zehnder interferometer (MZI)[44], using a plasmonic waveguide tocompensate the wavelength dependancy [55], or using cascaded DCs withadditional phase shifter [47], but at a cost of increased device footprint,increased fabrication complexity, and increased cost. Compact directionalcouplers, with a single etch step for fabrication, which have broad operatingbandwidths will be demonstrated in Chapter 4 of this dissertation.(a)(b)(c)Figure 1.4: Schematics of (a) an adiabatic coupler, (b) an multimode inter-ference coupler, and (c) a directional coupler. (drawn to scale)Dealing with the polarization is another challenge. The high index con-trast and aspect ratio of SOI waveguides result in a large modal birefrin-61.1. Silicon Photonicsgence, which means SOI based photonic devices often behave differentlywith different polarizations of light. Such birefringence makes SOI PICs in-compatible with optical fiber systems that use non-polarization-maintaining,single mode fibers, in which the polarization state of the optical modes atthe outputs of the fibers can change randomly. A polarization diversity [3]approach has been demonstrated to address the modal birefringence issueof the SOI PICs, which takes the advantages of polarization splitter [88]and polarization rotator [87]. Recently, polarization splitter-rotators havebeen demonstrated for the SOI platform [20, 23, 45, 65, 80, 81, 97], whichcombines the two functionalities, i.e., polarization splitting and polarizationrotating, into one device. An ultra-compact polarization splitter-rotatorwith improved fabrication tolerance will be demonstrated in Chapter 5 ofthis dissertation.Figure 1.5: Schematic of a polarization diverse PICs.71.2. Sub-wavelength Structures1.2 Sub-wavelength StructuresSub-wavelength periodic structures are arrangements of different materialswith a pitch small enough to suppress the diffraction effects arising fromtheir periodicity [32]. With the development of high-resolution lithography,sub-wavelength structures have seen widespread application in photonic in-tegrated circuits for the SOI platform. Currently, sub-wavelength structureshave been used to optimize grating couplers [4–6, 14, 16, 17, 22, 24, 29, 33,46, 59, 83–85, 93–95, 101], wavelength multiplexers [13], and to design low-loss waveguide crossings [9], high Q resonators [26, 82], biosensors [25, 26],broadband power splitters and combiners [30, 48], and polarization splitter-rotators (PSRs) [91], etc.1.2.1 The Sub-wavelength RegimeThe electromagnetic properties of the sub-wavelength structures consideredin this dissertation are fully described by the theory developed for photoniccrystals [37] and the principle and applications of sub-wavelength structuresfor the SOI platform are detailed in a review paper by Halir [32]. Here,we only provide a simplified theory about the sub-wavelength structure tofacilitate the understanding of this dissertation. Consider the periodic grat-ing structures for the SOI platform as shown in Fig. 1.6, which consists ofrectangles with alternating indices of n1 and n2, and a substrate which hasan index of n3. In our case, n1 = 3.47, n3 = 1.45, and n2 = 1.45 or 1 whenoxide/air cladding is used. The fill factor, or duty cycle of the structure,ff , is defined as ff = l/Λ, where l denotes the length of one grating and Λdenotes the period of the grating. The thickness of the top silicon layer isdenoted by t, and t = 220 nm in our case. Depending on the ratio of Λ and81.2. Sub-wavelength Structuresthe operating wavelength, λ, the periodic structure shown in Fig. 1.6 cangenerally operate in the following regimes [32]:• the diffraction regime, where the incoming beam is scattered in differ-ent orders;• the reflection regime, where the incoming beam is reflected backwards;• the sub-wavelength regime, where diffraction effects due to the peri-odicity of the structure are suppressed.Figure 1.6: Schematic of the cross-section of a periodic waveguide structure.When light incident on the grating structure shown in Fig 1.6, the angleof diffraction orders can be expressed as:nd · sin(θk) = k · λΛ(1.1)where θk denote the diffraction angle, k denotes the diffraction order, andnd = {n2, n3} for diffractions into the cladding and substrate, respectively[60]. For the SOI platform, all diffraction orders will be eliminated whenλΛ> nd = 1.45 (1.2)91.2. Sub-wavelength StructuresIn this dissertation, a periodic structure where all diffraction orders are sup-pressed is referred to as a sub-wavelength structure. In the sub-wavelengthregime, both the reflection and the diffraction effects are suppressed andlight will propagate through the sub-wavelength structure without affectedby the discontinuities along the propagation direction.1.2.2 Effective Medium TheorySub-wavelength structures can be approximated as a homogeneous mate-rial with an equivalent refractive index when λ  Λ, and the equivalentrefractive index of the sub-wavelength structure can be calculated with azeroth-order approximation given by Rytov [64]:n(0)‖ =[n21 · ff + n22 · (1− ff)]1/2(1.3)1n(0)⊥=[ffn21+(1− ff)n22]1/2(1.4)where n(0)‖ denotes the zeroth order approximation of the equivalent refrac-tive index for the polarization along the x direction, and n(0)⊥ denotes thezeroth order approximation of the equivalent refractive index for the po-larization along the z direction. In the case when the ratio of λ and Λ aresmaller, higher order approximations can be used to calculate the equivalentrefractive indices more accurately [40, 41]. It should be noted that in thecalculation given by Rytov, the assumption has been made that the struc-ture is infinite in the x and y directions. When designing a sub-wavelengthstructure, Rytov’s formulas can be used as a starting point for further opti-mizations of the design using more accurate numerical calculations.101.2. Sub-wavelength Structures1.2.3 Applications of Sub-wavelength Structures on SOIOne of the fundamental building blocks, using sub-wavelength structures,is the sub-wavelength grating (SWG) waveguide. Scanning electron micro-scope (SEM) images of an SWG waveguide and an SWG waveguide taperare shown in Fig. 1.7. A theoretical study has confirmed that the SWGwaveguide is lossless in the absence of fabrication imperfections [69]. Ascompared to the conventional silicon wire waveguides, SWG waveguides pro-vide some features that are beneficial in specific applications. Due to thefact that SWG waveguides consist of rectangles with alternating indices, SU-8 polymer has been used as the cladding material to make athermal SWGwaveguides [35, 66]. Due to the fact that SU-8 has a negative thermo-opticcoefficient while silicon has a positive thermo-optic coefficient, a athermalSWG waveguide can be created by interleaving specific amounts of thesetwo materials. Silicon oxide is normally used as the cladding material forSOI devices, and the material dispersion of silicon oxide is 6X lower thanin silicon. Therefore, SWG waveguides are less wavelength dependent thansilicon waveguides. Experimental demonstration of SWG waveguide, whichhas propagation loss as low as 2.1 dB/cm with negligible polarization andwavelength dependent loss has been demonstrated in [8]. In addition, SWGwaveguides also have a much smaller effective indices as compared to theconventional silicon wire waveguide. The reduced effective indices leads toloosely confined optical modes in the waveguides, which are ideal for applica-tions such as bio-sensing [25]. Fig. 1.8 shows the SEM images of a directionalcoupler consisting of strip waveguides and a directional coupler consistingof a strip waveguide and an SWG waveguide, where SWG waveguide is usedto improve the sensitivity of bio-sensors [27].111.2. Sub-wavelength Structures(a) (b)Figure 1.7: SEM images of (a) an SWG waveguide, and (b) an SWG waveg-uide with an SWG waveguide taper.Sub-wavelength structures have also been used to optimize grating cou-plers. When combining fundamental building blocks that can be fabricatedin a single, fully-etched step, having fully-etched sub-wavelength gratingcouplers (SWGCs) provides an efficient and economical solution for rapidprototyping. Various types of sub-wavelengths structures have been used ingrating couplers [4–6, 14, 16, 17, 22, 24, 29, 33, 46, 59, 83–85, 93–95, 101].Depending on the dimensions of the sub-wavelength structures, SWGs canbe divided as two-dimension SWGs and one-dimensional SWGs. Dependingon the shape of an SWGC, they can also be divided as straight SWGCs andfocusing SWGCs. SEMs images of a straight SWGC with two-dimensionSWGs and a focusing SWGC with one-dimensional SWGs are shown inFig. 1.9.121.3. About This Dissertation(a) (b)Figure 1.8: SEM images of (a) a directional coupler consisting of strip waveg-uides, and (b) a directional coupler consisting of a strip waveguide and anSWG waveguide.1.3 About This Dissertation1.3.1 ObjectiveThe objective of this dissertation is to investigate the properties of sub-wavelength structures on the SOI platform and use the design flexibility theyprovide to improve the performance of existing photonic devices and designnovel photonic devices on the SOI platform. The long-term objective ofthis work is to apply the benefits from sub-wavelength structures to developlarge-scale PICs with enhanced performance and enriched functionalities forvarious applications, such as high-speed optical interconnects and enhancedbio-sensing systems.1.3.2 Dissertation OrganizationIn Chapter 2 we demonstrate compact, high-efficiency, fully-etched sub-wavelength grating couplers (SWGCs) for both the fundamental transverseelectric, TE0, mode and the fundamental transverse magnetic, TM0, mode.131.3. About This Dissertation(a) (b)Figure 1.9: SEM images of (a) a straight SWGC with two-dimensional SWGsand (b) a focusing SWGC with one-dimensional SWGs.There are different forms of mode notations used in literatures. In thisdissertation, we used the same form as Yariv [96], where the notation de-notes the number of nodes for a mode. The design methodology for SWGCswith one-dimensional sub-wavelength gratings (SWGs) are demonstrated.The index profile of an SWGC is engineered using SWGs for improved cou-pling efficiency and suppressed back reflections. The design, simulation,and characterization of SWGCs with uniform gratings are demonstrated inthe Section 2.1 of Chapter 2 and apodized SWGCs with improved couplingefficiency are demonstrated in Section 2.2 of Chapter 2.In Chapter 3, we demonstrate compact SWGCs with ultra-broad operat-ing bandwidths for the SOI platform. We first derive the analytical expres-sion for the bandwidth of a grating coupler, in which we show the relationbetween the group index and the operating bandwidth of a grating cou-pler. Based on the theoretical study, we propose a methodology to designbroadband SWGCs with design-intent operating bandwidth. The design,simulation, and characterization of broadband SWGCs are demonstrated.In addition, broadband SWGCs with different design shapes, i.e., SWGCs141.3. About This Dissertationwith straight gratings lines and focusing grating lines, are compared.In Chapter 4, we demonstrate compact, broadband directional couplersusing SWGs for the SOI platform. The dispersion properties of the opticalmodes in a conventional directional coupler are engineered using SWGs,which allows broadband operation. Finite-difference time-domain (FDTD)based band structure calculations, with reduced simulation time, are usedto analyze the design, which includes both the material and the waveguidedispersions. Compact broadband direction couplers, with device lengthsshorter than 14 µm, which cover bandwidths of 100 nm, for power splittingratios of 50/50, 40/60, 30/70, and 20/80, are designed and fabricated forthe TE0 mode with λ of 1550 nm.In Chapter 5, we demonstrate an ultra-compact SWG polarization splitter-rotator for the SOI platform, where an SWG waveguide is used to managethe polarization of light. An asymmetric directional coupler, consists of aregular strip waveguide and an SWG waveguide, is used to couple the TM0mode of the strip waveguide into the TE0 mode of the SWG waveguide.The SWGs provides the freedom to engineer the dispersion properties ofthe supermodes, which allow the design to be fabrication tolerant to thewaveguide width variations. A high-efficiency, space-efficient SWG taper isalso demonstrated to couple the light from the SWG waveguide into a stripwaveguide. The design, simulation, and characterization of the SWG PSRare shown as well.The dissertation is concluded in Chapter 6 with a brief summary anddiscussions about future research directions.15Chapter 2High-Efficiency Fully-etchedSub-wavelength GratingCouplersIn this chapter, we demonstrate high-efficiency, fully-etched sub-wavelengthgrating couplers (SWGCs) for the TE0 mode and the TM0 mode. From hereon, we call the SWGCs designed for the TE0 mode as TE SWGCs and theSWGCs designed for the TM0 mode as TM SWGCs for simplicity. SWGCswith uniform gratings are shown in Section 2.1 and apodized SWGCs withimproved coupling efficiencies are shown in Section 2.2.2.1 Uniform Sub-wavelength Grating Couplers2.1.1 Challenges of Fully-etched Grating CouplersThe critical issues faced by fully-etched grating couplers include high inser-tion loss and strong back reflections from the grating to the waveguide. Thehigh insertion loss is mainly caused by the penetration loss into the substrateand the mode mismatch between the optical fiber and the grating. The largeback reflection results mainly from Fresnel reflections caused by the refrac-tive index contrast between the high and low index regions in the grating.162.1. Uniform Sub-wavelength Grating CouplersFig. 2.1 shows the typical measurement spectra for a pair of shallow-etchedgrating couplers and fully-etched grating couplers, as shown in Fig. 1.3. Thecalculated Fresnel reflection coefficients for the shallow-etched grating, withan etch depth of 70 nm into a 220 nm-thick silicon layer, and a fully-etchedgrating, with a 220 nm silicon layer and oxide cladding, at 1550 nm, are 0.6%and 17%, respectively, which correspond to oscillation ripples with extinc-tion ratios (ERs) of 0.1 dB and 3 dB, respectively. The calculated ERs ofthe ripples agree with those of the the measured results shown in Fig. 2.1.1500 1520 1540 1560 1580 1600−40−35−30−25−20−15−10Wavelength (nm)Power (dB)  fully−etchedshallow−etchedFigure 2.1: Measured transmission spectra for a pair of shallow-etched grat-ing couplers and fully-etched grating coupler.The refractive index contrast in a regular fully-etched grating coupler ismuch larger than that of a shallowed-etched grating coupler, which moti-vates the need for an effective index material using sub-wavelength gratings(SWGs) to approximate the shallow-etched regions in a fully-etched gratingcoupler. A schematic of a fully-etched grating coupler with effective indexregions is shown in Fig. 2.2, where nH and nL denote the effective refractiveindices of the high and low index regions, respectively. Two dimensionalSWGs have been demonstrated to obtain the low index regions in a gratingcoupler [4–6, 14, 16, 17, 22, 24, 29, 33, 46, 59, 93–95, 101]. Here, we demon-172.1. Uniform Sub-wavelength Grating Couplersstrated grating couplers using one-dimensional SWGs to obtain the low indexregions for the first time [84, 85]. A schematic of a fully-etched grating cou-pler with low index regions obtained using one-dimensional SWGs is shownin Fig. 2.3. The high index regions, denoted by nH in Fig. 2.3 is silicon inthis case. From here on, we will simply refer to the fully-etched SWGCs asSWGCs in the rest of this dissertation.Figure 2.2: A Schematics of a fully-etched grating couplers with effectiveindex regions.Figure 2.3: Schematics of a fully-etched SWGCs with one-dimensionalSWGs.2.1.2 2D SWGs vs. 1D SWGsIt is advantageous to use one-dimensional SWGs (sub-wavelength gratinglines), as compared to two-dimensional SWGs (sub-wavelength quasi-squares),to achieve the low effective index regions in SWGCs for several reasons.182.1. Uniform Sub-wavelength Grating CouplersFrom the fabrication perspective, the one-dimensional SWGs are less chal-lenging to produce because they benefit from higher exposure contrasts [84]and shorter fabrication times, resulting in higher fabrication accuracy andlower fabrication cost. Figure 2.4 shows the simulated energy distributionfrom BEAMER, a electron-beam lithography software from GenISys GmbH[34], for fabricating two-dimensional SWGs and one-dimensional SWGs us-ing electron beam lithography. The simulator used a defined Point SpreadFunction (PSF) for 100 kV electrons on silicon, then produced a 2-D plot ofenergy distribution after exposure for the pattern data. The PSF shows en-ergy as a function of radial distance from the electron beam, which is causedby electron backscattering and secondary electron generation in the silicon.The lithography simulation applies the PSF to the pattern data, to pro-duce the 2-D energy distribution for different geometries. Figure 2.4(a) and(c) show the respective energy distributions of the two-dimensional SWGs,sub-wavelength holes with uniform diameters, and one-dimensional SWGs,where the blue denotes the least energy and the red denotes the most en-ergy. The 1-D cutlines across the 2-D simulations show the differences as-sociated with fabricating different type of structures. We get 30% exposurein the two-dimensional SWGs as shown in Fig. 2.4(b) where no exposure isrequired, whereas for the one-dimensional SWGs, we only get about 10% ex-posure as shown in Fig. 2.4(d) in the region where no exposure is required.Even though a high contrast resist process was used, the 30% exposureshown in the two-dimensional SWGs is still enough energy to at least par-tially expose the resist, which results in a high variability in feature size. Itcan be noted that the exposure contrast in the low effective index regionshas been improved by implementing one-dimensional SWGs. From the de-sign perspective, due to the use of one-dimensional SWGs that stretch the192.1. Uniform Sub-wavelength Grating Couplersentire width of the coupler, our SWGCs are simple and straightforward todraw and subsequently, to simulate in both 2D and 3D. By contrast, ef-fective refractive indices were used in 2D simulations to approximate theeffective index regions in SWGCs using two-dimensional SWGs, which oversimplifies the situation and results in mismatches between simulations andmeasurements. In addition, SWGCs with one-dimensional SWGs also en-able focusing gratings with smaller footprint using the method demonstratedin [77], which are difficult to be achieved with two-dimensional SWGs.(a) (b)(c) (d)Figure 2.4: (a) 2D energy distribution of an SWGC formed by two-dimensional SWGs; (b) energy distribution along the 1-D cutline acrossthe 2-D simulation of an SWGC formed by two-dimensional SWGs; (c) 2Denergy distribution of an SWGC formed by one-dimensional SWGs; (d) en-ergy distribution along 1-D cutlines across the 2-D simulations of SWGCsformed by one-dimensional SWGs.202.1. Uniform Sub-wavelength Grating Couplers2.1.3 Design and SimulationAs shown in Fig. 2.3, our SWGC has a grating period of Λ, and each gratingperiod consist of a high index region with a width of ΛH and a low indexregion with a width of ΛL. The fill factor, ff , of the SWGC shown inFig. 2.3 is defined as the ratio of ΛH to Λ. The length of an SWG in the lowindex regions is denoted by lsub.Incident AngleThe incident angle of an SWGC is determined by the phase-match conditionbetween the grating mode and the fiber mode. As shown in Fig. 2.5, if wedenote θ as the incident angle, which is defined as the angle between thedirection of the out-coupled wave and the normal to the grating, then thephase-match condition can be expressed as:neff · Λ = nf · Λ · sinθ +N · λ (2.1)where neff is the effective index of the mode propagating in the grating, nfis the effective index of the fiber mode, λ is the operating wavelength, andN is the diffraction order. It should be noted that θ can be either positiveor negative. θ is positive when the out-coupled wave has a component in thesame direction as the input wave, as shown in shown in Fig. 2.5(a), whileθ is negative when the out-coupled wave has a component in the oppositedirection to the input wave, as shown in Fig. 2.5(b).212.1. Uniform Sub-wavelength Grating Couplers(a) (b)Figure 2.5: Schematic of the cross-section of an SWGC with (a) a positiveincident angle and (b) a negative incident angle.Parameter OptimizationOur SWGCs are optimized using the finite-difference time-domain (FDTD)method combined with the particle swarm algorithm (PSA) [62]. PSA is apopulation based stochastic optimization technique, inspired by the socialbehavior of flocks of birds or schools of fish [57, 62], and has widely been usedfor various kinds of design optimization problems, including nanophotonicdesigns [49, 53, 68, 100]. In PSA, the potential solutions, called particles,are initialized at random positions, and then move within the parametersearch space. The particles are subject to three forces as they move: a.spring force towards the personal best position, p, ever achieved by thatindividual particle; b. spring force towards the global best position, g, everachieved by any particle; c. a frictional force, proportional to the velocity. Acommercial FDTD-method Maxwell equation solver, FDTD Solutions [72],from Lumerical Solutions, Inc. with the built-in PSA was used to conductthe design optimizations.The schematic of the simulated structures are shown in Fig. 2.6. First,the light was launched from the waveguide, as shown in Fig. 2.6(a), anddiffracted by the grating. A power monitor was positioned on top of thegrating to measure the power diffracted upward by the grating. Three de-sign parameters, i.e., Λ, ΛH, and lsub as defined in Fig. 2.3, have been222.1. Uniform Sub-wavelength Grating Couplersoptimized to achieve the highest directionality, which is defined as the ratioof the optical power diffracted upward by the grating to the input power.After we got the highest directionality, a far field calculation was done by thepower monitor to calculate the diffraction angle of the light from the grat-ing, which is the same as the incident angle of the SWGC. The optimizedincident angles for the TE and TM SWGCs were -25◦ and 10◦, respectively.Then, the light was launched from the fiber with the optimized angle, shownin Fig. 2.6(b). A power monitor was positioned in the waveguide to monitorthe total power and another mode expansion monitor was used to calculatethe power coupled into a specific mode. Again, Λ, ΛH, and lsub were op-timized using PSA in FDTD Solutions. Eight simulation generations witheight variations within each generation were used to obtain SWGCs withthe highest coupling efficiencies for the TE0 mode and the TM0 mode, re-spectively. The optimized TE SWGC has Λ = 593 nm, ΛH = 237 nm, andlsub = 74 nm. The optimized TM SWGC has Λ = 960 nm, ΛH = 575 nm,and lsub = 140 nm. The incident angles for the TE and TM SWGCs are-25◦ and 10◦, respectively.The simulated coupling efficiency and the back reflection of the optimizedTE SWGC are shown in Fig. 2.7. For comparison purposes, the simulatedcoupling efficiency and the back reflection of a shallow-etched grating cou-pler, with a partial etching layer of 70 nm, are also shown in Fig. 2.7. Thecoupling efficiencies are simulated using the structure shown in Fig. 2.6(b),where a mode expansion monitor is used to calculate the power coupled tothe fundamental TE mode. The back reflections are simulated using thestructure shown in Fig. 2.6(a), where the mode expansion monitor was po-sitioned in the opposite direction of the input mode to calculate the powercoupled back forward into the fundamental TE mode. The shallow-etched232.1. Uniform Sub-wavelength Grating Couplers(a)(b)Figure 2.6: Simulations of SWGCs in FDTD Solutions.grating coupler was optimized based on the same wafer type using the samealgorithm as the TE SWGC. The optimized TE SWGC shows a simulatedpeak coupling efficiency of −2.2 dB with a 1-dB bandwidth of 32.5 nm (3-dB bandwidth of 58 nm), which are comparable to the performance of theoptimized shallow-etched grating coupler.The simulated coupling efficiency and the back reflection of the optimizedTM SWGC are shown in Fig. 2.8. Again, the coupling efficiency and theback reflection for an optimized shallow-etched grating coupled for the sameλ and polarization of light are shown in the same graph for comparisonpurposes. The TM SWGC not only has a higher coupling efficiency, butalso shows a broader bandwidth than the shallow-etched grating coupler.The TM SWGC shows a simulated peak coupling efficiency of −2.7 dB witha 1-dB bandwidth of 57 nm (3-dB bandwidth of 92 nm). The optimization242.1. Uniform Sub-wavelength Grating CouplersWavelength (nm)1450 1500 1550 1600 1650Power (dB)-30-25-20-15-10-50SWGC-TSWGC-RShallow-TShallow-RFigure 2.7: Simulated coupling efficiency and back reflection for the TESWGC and a shallow-etched grating coupler. SWGC-T and SWGC-R de-note the coupling efficiency and the back reflection of the optimized TESWGC, Shallow-T and Shallow-R denote the coupling efficiency and theback reflection of the optimized shallow-etched grating coupler.was based on uniform gratings, where further improvement can be obtainedby apodizing the grating periods to achieve better mode overlap betweenthe grating and the optical fiber.It should be noted that the peak coupling efficiencies and operatingbandwidths of the TE and TM SWGCs are different. The difference inthe coupling efficiencies is caused by the different coupling strength of thetwo grating couplers. The power in the waveguide undergoes an exponentialdecay due to the presence of the grating:Pwg(z) = Pwg(z = 0) · exp(−2αz) (2.2)where α is the coupling strength or leakage factor of the grating, Pwg(z)denotes the power of the mode at z, assuming the wave is propagatingalong the z-axis. The directionality, which is the power diffracted upward252.1. Uniform Sub-wavelength Grating CouplersWavelength (nm)1450 1500 1550 1600 1650Power (dB)-25-20-15-10-50SWGC-TSWGC-RShallow-TShallow-RFigure 2.8: The simulated coupling efficiency and the back reflection for theTM SWGC and a shallow-etched grating coupler. SWGC-T and SWGC-Rdenote the coupling efficiency and the back reflection of the optimized TMSWGC, Shallow-T and Shallow-R denote the coupling efficiency and theback reflection of the optimized shallow-etched grating coupler.by a grating coupler, of the TE and TM SWGCs are similar, which areabout 69%. However, the α values of the TE and TM SWGCs are different,therefore, the coupling losses caused by the mode mismatches between thefiber and grating are different, which lead to the difference in the peakcoupling efficiencies. According to [16], the 1-dB bandwidth of a gratingcoupler can be expressed as:∆λ = η1dB|dλdθ| = η1dB|−Λ · nc · cos(θ)1− Λ · dneff(λ)dλ| (2.3)where η1dB is the 1-dB bandwidth coefficient, nc is the refractive index ofthe cladding material, neff is the effective index of the grating. The period ofthe TM SWGC is much larger than that of the TE SWGC, and the θ of theTM SWGC is smaller than the TE SWGC, which contribute to a broaderbandwidth for the TM SWGC.262.1. Uniform Sub-wavelength Grating CouplersSystem simulationIn actual photonic circuits, a pair of grating couplers are typically used: oneinput grating coupler and one output grating coupler. A Fabry-Perot cavityforms between the input and output grating couplers, where the reflectedwave propagates between the two grating couplers and introduces ripplesin the spectral responses of the photonic circuits. Scattering parameters(S-parameters) can be used to describe the behaviour of the grating coupler[18]. Simulation of a full input-waveguide-output circuit has been performedusing Interconnect [58], and Fig. 2.9(a) shows a schematic of the input-waveguide-output circuit in Interconnect. A virtual optical network analyzeris used to generate and measure the optical signal. The grating couplersare represented by two 2-port S-parameter matrices, which were exportedfrom FDTD Solutions. The waveguide is a component that can be eitherloaded from Interconnect or imported from Mode Solutions, an eigenmodesolver from Lumerical Solutions, Inc. The simulated transmission spectrafor the input-waveguide-ouput circuits are shown in Fig. 2.9(b). The blueline shows the simulated transmission spectrum with two TE SWGCs, with127 µm pitch, connected by a silicon wire waveguide, and the red line showsthe same simulation results for the TM SWGCs. As shown in Fig. 2.9(b),the ERs of the ripples shown for our TE and TM SWGCs are only 0.3 dBand 0.15 dB, respectively.Focusing Grating LinesSo far we have addressed the SWGC as a two-dimension structure and Equa-tion 2.1 has assumed that the grating is linearly extended in the lateral di-rection, where a linear taper is required to couple the light from the grating272.1. Uniform Sub-wavelength Grating Couplers(a)1500 1520 1540 1560 1580 1600−30−25−20−15−10−50Wavelength (nm)Insertion loss (dB)  TE−transmissionTM−transmission1540 1550 1560−7−6−5−4(b)Figure 2.9: (a) Schematic of the input-waveguide-output circuit in Inter-connect, and (b) simulated transmission spectra of input-waveguide-ouputcircuits for TE and TM SWGCs.to a regular silicon wire waveguide. The layout of a grating coupler withstraight grating lines and an a linear taper is shown in Fig. 2.10(a). Toachieve adiabatic coupling, the required length for the taper is on the orderof a few hundred microns. However, the long adiabatic tapers are not spaceefficient, consuming valuable on-chip real estate. Focusing grating lines areused as an alternative to achieve efficient mode size conversion [77]. The lay-out of a grating coupler with focusing grating lines is shown in Fig. 2.10(b).The phase-match condition for a focusing SWGC can be expressed as:neff · r = nf · r · sinθ · cosφ+N · λ (2.4)where neff denotes the effective index of the grating, nf is the effectiveindex of the fiber mode, θ denotes the incident angle, r denote the distancefrom the end of the silicon-wire waveguide (O as shown in Fig. 2.11) toan arbitrary point (P shown in Fig. 2.11) in the X-Z plan, φ denotes theangle between the OP and the z-axis. From Equation 2.4, we can get the282.1. Uniform Sub-wavelength Grating Couplersexpression for r as:r =N · λneff − nf · sinθ · cosφ (2.5)(a)(b)Figure 2.10: Mask layouts of (a) a grating coupler with straight grating linesand a linear taper, and (b) an SWGC with focusing grating lines.Equation 2.4 defines a family of confocal ellipses with one of the focalpoints at the end of the waveguide. Schematics of the focusing grating linesfor SWGCs are shown in Fig. 2.11. It should be noted that the gratingpattern of the SWGCs with positive and negative incident angles are dif-ferent. For SWGCs with positive incident angles, as shown in Fig. 2.11(a),the waveguide end overlaps with the left focal point of the confocal ellipses.For SWGCs with negative incident angles, as shown in Fig. 2.11(b), thewaveguide end overlaps with the right focal point of the confocal ellipses. Aparameterized device cell has been created in Mentor Graphics’s Pyxis [18]292.1. Uniform Sub-wavelength Grating Couplersto generate the layout files for various SWGCs. The layout of an focusingSWGC is shown in Fig. 2.10(b).(a) (b)Figure 2.11: Schematics for SWGCs with focusing grating lines and (a)positive incident angles, and (b) negative incident angles.The shape of the taper in a focusing SWGC needs to be optimized inorder to avoid excess loss in the taper. In Eq. 2.4, N decides the taper lengthand for a given N, φ decides the taper width, as shown in Fig. 2.10(b). 3DFDTD simulations have been done to optimize the tapers of our SWGCs inthe following steps. First, we fixed the value of N for a taper length of 22 µm,then we varied the value of φ. The highest coupling efficiency were achievedfor both the TE and TM SWGCs when the taper widths are about 10.6 µm,which is similar to the diameter of the optical mode from the PM fiber weused. Next, we simulated SWGCs with different combinations of N and φwith the same taper width of 10.6 µm. The highest coupling efficiencies wereachieved for the TE and TM SWGCs when the length to width ratios of thetapers were 2.76 and 2.24, respectively. The simulated coupling efficienciesof the optimized TE and TM SWGCs in 3D FDTD simulations are −2.5 dBand −2.8 dB, which are 0.3 dB and 0.1 dB lower than the simulated results in2D FDTD simulations, respectively. Such differences are due to propagation302.1. Uniform Sub-wavelength Grating Couplerslosses in the tapers, which can be improved by further design optimizationsof the tapers.2.1.4 Fabrication and MeasurementTest structures, consisting of an input SWGC and an output SWGC, with127 µm pitch, connected by a strip waveguide have been fabricated usingelectron beam lithography at the University of Washington [11]. The fab-rication used SOI wafers with 220 nm thick silicon on 3 µm thick silicondioxide. The substrates were 25 mm squares diced from 150 mm wafers.Oxide cladding was used for the protection of the chip. After a solvent rinseand hotplate dehydration bake, hydrogen silsesquioxane resist (HSQ, Dow-Corning XP-1541-006) was spin-coated at 4000 rpm, then hotplate bakedat 80 ◦C for 4 minutes. Electron beam lithography was performed using aJEOL JBX-6300FS system operated at 100 kV energy, 8 nA beam current,and 500 µm exposure field size. The machine grid used for shape placementwas 1 nm, while the beam stepping grid, the spacing between dwell pointsduring the shape writing, was 6 nm. An exposure dose of 2800 µC/cm2 wasused. The resist was developed by immersion in 25% tetramethylammoniumhydroxide for 4 minutes, followed by a flowing deionized water rinse for 60 s,an isopropanol rinse for 10 s, and then blown dry with nitrogen. The sil-icon was removed from unexposed areas using inductively coupled plasmaetching in an Oxford Plasmalab System 100, with a chlorine gas flow of20 sccm, pressure of 12 mT, ICP power of 800 W, bias power of 40 W, anda platen temperature of 20 ◦C, resulting in a bias voltage of 185 V. Duringetching, chips were mounted on a 100 mm silicon carrier wafer using perflu-oropolyether vacuum oil. The scanning electron microscope (SEM) images312.1. Uniform Sub-wavelength Grating Couplersof the fabricated SWGCs are shown in Fig. 2.12.(a) (b)Figure 2.12: SEM images of (a) the top view of a fabricated SWGC and (b)the side view of a fabricated SWGC.To characterize the devices, a custom-built automated test setup [19]was used. An Agilent 81600B tunable laser, with a polarization maintain-ing (PM) fiber, was used as the input source, and Agilent 81635A opticalpower sensors, also with PM fibers, were used as the output detectors. Awavelength sweep from 1500 nm to 1600 nm in 10 pm steps was performed.Fabrication ToleranceVariations were applied to Λ, ΛH, and lsub to understand the sensitivities ofthese design parameters to the fabrication imperfections. Figure 2.13 showsthe central wavelength shifts as a function of various design parameter vari-ations for the TE and TM SWGCs, respectively. Figure 2.13(a) shows thesimulated central wavelength offsets as a function of Λ, ΛH, and lsub for theTE SWGC. The blue circles indicate the central wavelength shift as a func-tion of δΛ, the red crosses indicate the central wavelength shift as a functionof the δΛH, and the purple diamonds indicate the central wavelength shiftas a function of δlsub. Figure 2.13(b) shows the measurement results for322.1. Uniform Sub-wavelength Grating Couplersthe as-fabricated TE SWGCs with the same design parameters. The sameanalysis has been applied to the TM SWGCs and the results are shown inFig. 2.13(c) and Fig. 2.13(d). Linear-fits have been applied to compare thesensitivities of Λ, ΛH, and lsub and the corresponding slopes are shown inTable 2.1.Table 2.1: Simulated and measured sensitivities of the central wavelengthas a function of Λ, ΛH, and lsub.TE TMsimulation experiment simulation experiment(nm/nm) (nm/nm) (nm/nm) (nm/nm)Λ 1.3 1.0 0.59 0.43ΛH 0.73 0.67 0.39 0.35lsub 2.1 2.06 0.31 0.24It can be noted that the slopes extracted from the experimental resultswere slightly smaller than the simulated results for both the TE and TMSWGCs, but the comparative relation between those values were the samein both cases. From Table 2.1 we can see that Λ and lsub were the keyparameters affecting the central wavelength of the SWGCs. In addition, wecan notice that the slopes of various design parameters for the TM SWGCswere smaller than that of the TE SWGCs, which indicate a lower sensitivityto fabrication imperfections.TE SWGCThe measurement transmission spectrum of a TE SWGC is denoted by thered line shown in Fig. 2.14. The simulated transmission spectrum for thesame design is denoted by the blue line shown in Fig. 2.14 for comparisonpurposes. The best device has a measured coupling efficiency of −4.1 dB332.1. Uniform Sub-wavelength Grating Couplers−15 −10 −5 0 5 10 15−40−30−20−10010203040Dimension Offset (nm)Simulated wavelength offset (nm)  GratingPeriodGratingWidthSubWidth(a)−15 −10 −5 0 5 10 15−40−30−20−10010203040Dimension Offset (nm)Experimental wavelength offset(nm)  GratingPeriodGratingWidthSubWidth(b)−60 −40 −20 0 20 40 60−40−30−20−10010203040Dimension Offset (nm)Simulated wavelength offset (nm)  GratingPeriodGratingWidthSubWidth(c)−60 −40 −20 0 20 40 60−40−30−20−10010203040Dimension Offset (nm)Experimental wavelength offset(nm)  GratingPeriodGratingWidthSubWidth(d)Figure 2.13: (a) Simulated sensitivities of Λ, ΛH, and lsub for the TESWGCs; (b) measured sensitivities for the TE SWGCs; (c) simulated sensi-tivities for the TM SWGCs; (d) measured sensitivities for the TM SWGCs.with a 1-dB bandwidth of 30.6 nm (3-dB bandwidth of 52.3 nm). The mea-sured peak coupling efficiency is lower than the simulated value, which maycome from two major sources. First, the insertion losses from the fiber arrayand all the optical connections, i.e., connection between the laser and thefiber array and connection between the fiber array and the detector, werenot calibrated out from the measurement result. We did not calibrate outthe insertion losses from the measurement system because we found that theuncertainty of the calibration process is around 1 dB, i.e., the tightness of342.1. Uniform Sub-wavelength Grating Couplersthe connectors, the position of the fibers, the vibrations of the setup, etc. allhave impacts on the accuracy of the calibration process. Given that the mis-match of between the simulated coupling efficiency and measured couplingefficiency is below 2 dB, we found it is not reasonable to calibrate the losswith an uncertainty at the same level as the loss we are trying to calibrate.In addition, the fiber array used for our measurement were intentionallypolished to a larger angle than the required incident angle of the designedgrating coupler. Therefore, the tip of the fiber array was not parallel to thesurface of the grating coupler during measurement and an air gap existedbetween the fiber array and the chip surface. Such an air gap not onlyintroduced excess loss but also narrowed the operating bandwidth of themeasured grating coupler. The back reflection from the SWGC has beenhighly eliminated by using the SWGs, hence the ripples in our transmis-sion spectrum caused by the Fabry-Perot cavity are significantly suppressed(< 0.25 dB), which is comparable to that of a shallow-etched grating cou-pler (about 0.1 dB). Due to fabrication inaccuracies, the measured centralwavelength shifted to 1580 nm.As shown in Table 2.1, the slope of lsub is much larger than that of Λ andΛH. In addition, the value of lsub (74 nm) is much smaller than the valuesof Λ (593 nm) and ΛH (237 nm). The red shift of the spectrum is mainlycaused by the fabrication inaccuracy in lsub, though it may also be causedby the fabrication inaccuracies in Λ and ΛH. The simulation results shownin Fig. 2.14 assumed that the central wavelength shift was caused by thefabrication imperfection in the SWGs, and a δlsub = 15 nm was assumedwhen the simulated central wavelength matches the measurement results.There were also mismatches in the coupling efficiency and the bandwidthbetween the simulated and measured results, which are mainly caused by352.1. Uniform Sub-wavelength Grating Couplers1520 1540 1560 1580 1600 1620 1640−25−20−15−10−50Wavelength (nm)Insertion loss (dB)  measurementsimulation1560 1580 1600−7−6−5−4−31dB bandwidth =30.6 nmFigure 2.14: Simulated and measured transmission spectra of a fabricatedTE SWGC assuming lsub = 89nm; inset is the spectrum near the centralwavelength.the fiber array we used. We intentionally polished our fiber array with alarge angle so that it can accommodate for various incident angles. Therequired θ for the TE SWGC and the TM SWGC were smaller than thepolished angle, so the chip surface and the fiber tip were not parallel duringthe measurement. Therefore, an inevitable gap was introduced between thefiber tip and measured chip, which leads to a lower coupling efficiency anda smaller bandwidth. In addition, extra losses were also introduced fromthe transmission lines (optical fibbers) and the interfaces of the connectorsused in the measurement system, which have not been calibrated out fromthe measurement results.TM SWGCThe measurement results for the TM SWGCs are shown in Fig. 2.15. Themeasured transmission spectra of 11 TM SWGCs with the same design pa-rameters are shown in Fig. 2.15(a). The inset of the graph shows the zoomed362.1. Uniform Sub-wavelength Grating Couplersspectra near the central wavelength. Fig. 2.15(b) and (c) show the compar-ison of the measured coupling efficiencies and 1-dB bandwidths of the 11TM SWGCs with the simulated results. The green lines denote the simu-lation results and the blue diamonds and circles denote the measurementresults extracted from the spectra shown in Fig. 2.15(a). The coupling effi-ciencies of the 11 TM SWGCs ranged from −3.7 dB to −3.8 dB with 1-dBbandwidth ranging from 45.8 nm to 47.5 nm, which shows good performancestability and device repeatability. The best device has an coupling efficiencyof −3.7 dB with a 1-dB bandwidth of 47.5 nm (3-dB bandwidth of 81.5 nm).Figure 2.15(d)-(f) show the measured transmission spectra of TM SWGCswith various values of Λ, ΛH, and lsub. The offsets versus central wave-length shifts of those design parameters have been shown in Fig. 2.13 andthe linear-fits of those design parameters have been given in Table. 2.1. Thecentral wavelengths of the TM SWGCs were proportional to Λ, ΛH, andlsub. The TM SWGCs did not have large central wavelength shifts as theTE SWGCs did for two reasons. The first reason is that the feature sizes ofthe SWGs in the TM SWGCs were much larger than the TE SWGC, whichmakes it less challenging for the fabrication process. The second reason isthat, according to the linear-fits shown in Table. 2.1, the central wavelengthof the TM SWGC is less sensitive to variations in Λ, ΛH, and lsub, which iscaused by the weak confinement of the TM mode.To summarize, in this section, we have demonstrated compact SWGCsfor both the TE0 mode and the TM0 mode. The back reflections fromour SWGCs have been significantly suppressed compared to the regularfully-etched grating couplers. It has also been shown by the simulationand experimental results that one-dimensional SWGs is an alternative tothe two-dimensional SWGs to achieve effective index medium with better372.1. Uniform Sub-wavelength Grating Couplers1460 1480 1500 1520 1540 1560 1580 1600 1620−25−20−15−10−50Wavelength (nm)Insertion Loss (dB)1530 1540 1550 1560 1570−5−4.5−4−3.5(a)0 2 4 6 8 10 12−5−4.5−4−3.5−3−2.5−2Device−IDInsertion Loss (dB)  measurementsimulation(b)0 2 4 6 8 10 1235404550556065Device−ID1−dB bandwidth (nm)  measurementsimulation(c)1460 1480 1500 1520 1540 1560 1580 1600 1620−35−30−25−20−15−10−50Wavelength (nm)Insertion Loss (dB)  Period=900nmPeriod=924nmPeriod=948nmPeriod=972nmPeriod=996nmPeriod=1020nm(d)1460 1480 1500 1520 1540 1560 1580 1600 1620−35−30−25−20−15−10−50Wavelength (nm)Insertion Loss (dB)  Grating−width=515nmGrating−width=539nmGrating−width=563nmGrating−width=587nmGrating−width=611nmGrating−width=635nm(e)1460 1480 1500 1520 1540 1560 1580 1600 1620−35−30−25−20−15−10−50Wavelength (nm)Insertion Loss (dB)  Sub−width=80nmSub−width=104nmSub−width=128nmSub−width=152nmSub−width=176nmSub−width=200nm(f)Figure 2.15: (a) Measured spectra of 11 TM SWGCs; (b) comparison ofmeasured and simulated coupling efficiencies of the TM SWGCs; (c) com-parison of the measured and simulated 1-dB bandwidths of the TM SWGCs;(d) measured spectra of TM SWGCs with various ΛH values; (e) measuredspectra of TM SWGCs with various Λ values; (f) measured spectra of TMSWGCs with various lsub values.382.2. Apodized Sub-wavelength Grating Couplersfabrication accuracy and less complexity. Our TE SWGC had a measuredpeak coupling efficiency of −4.1 dB with a 1-dB bandwidth of 30.6 nm (3-dBbandwidth of 52.3 nm) and the TM SWGC had a measured peak couplingefficiency of −3.7 dB with a 1-dB bandwidth of 47.5 nm (3-dB bandwidthof 81.5 nm). Further improvement can be made by apodizing the gratingperiods, which will be shown in the next section.2.2 Apodized Sub-wavelength Grating CouplersUniform gratings have been used in Section 2.1, where higher coupling ef-ficiencies can be obtained for both the TE SWGC and the TM SWGC byapodizing the gratings to achieve better mode match between the gratingand the optical fiber. The maximum theoretical coupling efficiency for agrating coupler with uniform gratings is nearly 80% [56], which means thatthe coupling efficiency of a grating couplers with uniform gratings can beimproved by about 1dB using non-uniform gratings. In this section, wecompare two different approaches to apodize SWGCs to reduce the modemismatch loss and we experimentally demonstrate apodized SWGCs withimproved coupling efficiencies for both the TE0 and the TM0 modes.2.2.1 Design and SimulationTo achieve a Gaussian output beam, the leakage factor, α, is given by [74]:α(z) =G2(z)2 · (1− ∫ z0 G2(t)dt) (2.6)where G(z) is a normalized Gaussian profile determined by the fiber modeand z denotes the distance in the propagation direction. α of an SWGC392.2. Apodized Sub-wavelength Grating Couplerscan be tuned by changing ΛH and lsub of the SWGC. Figure. 2.16(a) showsα as functions of ΛH and lsub for the TE SWGC that we obtained fromSection 2.1. Figure. 2.16(b) shows α as functions of ΛH and lsub for the TMSWGC that we obtained from Section 2.1. To obtain the simulation resultsshown in Fig. 2.16, we kept all the other design parameters as constantswhen we swept ΛH and lsub, respectively. From Fig. 2.16 we can see that bychanging lsub, a larger tuning range for α can be obtained for our SWGCs,so we used the lsub as the tuning factor to modify the α of our SWGCs.Feature sizes of the gratings (nm)50 100 150 200 250 300 350 400Leakage factor (/um)00.050.10.150.20.25lsub$H(a)Feature sizes of the gratings (nm)100 200 300 400 500 600 700Leakage factor (/um)00.050.10.150.2lsub$H(b)Figure 2.16: α as functions of lsub and ΛH for the TE SWGCs, and (b) α asfunctions of lsub and ΛH for the TM SWGCs.With the relations shown in Fig. 2.16 and Eq. 2.2, we can get the requiredlsub as a function of z. It should be noted that Eq. 2.2 is only accurate forlong gratings with small α, which is not the case for our SWGCs. So weused the calculated lsub as the starting point for further optimization. Inorder to achieve the phase-match condition between the adjacent gratingperiods, Λ also need to be modified. We used PSA in FDTD Solutions tooptimize the values of lsub and Λ for each grating period to achieve themaximum coupling efficiency. Each grating period has been treated as a402.2. Apodized Sub-wavelength Grating Couplersseparate grating section for optimization starting from the interface of thewaveguide and the grating. The optimized Λ and lsub for each apodizedgrating section are shown in Table 2.2. Eight grating periods are requiredto be apodized for the TE SWGC and four grating periods are required tobe apodized for the TM SWGC, which is due to the fact that the Λ of theTM SWGC is larger than that of the TE SWGC.Table 2.2: Λ and lsub values for the apodized SWGCs.TE SWGC TM SWGCΛ1 (nm) 480 lsub1 (nm) 120 Λ1 (nm) 895 lsub1 (nm) 220Λ2 (nm) 516 lsub2 (nm) 120 Λ2 (nm) 895 lsub2 (nm) 189Λ3 (nm) 500 lsub3 (nm) 120 Λ3 (nm) 917 lsub3 (nm) 173Λ4 (nm) 521 lsub4 (nm) 120 Λ4 (nm) 914 lsub4 (nm) 200Λ5 (nm) 500 lsub5 (nm) 120Λ6 (nm) 560 lsub6 (nm) 108Λ7 (nm) 500 lsub7 (nm) 100Λ8 (nm) 560 lsub8 (nm) 712.2.2 Design of Apodized SWGCsA parameterized device cell has been created in Mentor Graphics’s Pyxis[18] to generate the layout files for various apodized SWGCs. The masklayouts were first exported from Mentor Graphics’s Pyxis and then importedinto FDTD Solutions for 3D simulation. 3D FDTD simulations were donefor the focusing SWGCs for both the TE0 and the TM0 modes. Thesesimulations have been done for both apodized and un-apodized SWGCs forcomparison purposes. Fig. 2.17(a) shows the simulated transmission andreflection spectra of the apodized and un-apodized SWGCs for the TE0mode. The apodized TE SWGC has a peak coupling efficiency of −2.1 dB,0.6 dB lower than the un-apodized design, and a 1-dB bandwidth of 42 nm.412.2. Apodized Sub-wavelength Grating CouplersThe reflections from the apodized and un-apodized SWGCs are both below−20 dB. Fig. 2.17(b) shows the simulated transmission and reflection spectraof the apodized and un-apodized SWGCs for the TM0 mode. The apdizedTM SWGC has a peak coupling efficiency of −2.2 dB, 0.7 dB lower than theun-apodized design, and a 1-dB bandwidth of 45 nm.Wavelength (nm)1500 1520 1540 1560 1580 1600Power (dB)-10-7.5-5-2.50Reflection (dB)-40-30-20-100T-ApodizedR-ApodizedT-UniformR-UniformReflection (dB)(a)Wavelength (nm)1500 1520 1540 1560 1580 1600Power (dB)-10-7.5-5-2.50Reflection (dB)-40-30-20-100T-ApodizedR-ApodizedT-UniformR-UniformReflection (dB)(b)Figure 2.17: Simulated coupling efficiencies and back reflections of theapodized and the un-apodized (a) TE SWGCs using 2D FDTD simulation,and (b)TM SWGCs using 2D FDTD simulation.The principal loss in our SWGCs is the penetration loss to the substrate,which can be reduced by optimizing the thickness of the BOX or by addinga bottom mirror at the interface of the BOX and the silicon substrate. Theoptimization of our design parameters are based on the silicon wafer used forfabrication, which has a 220 nm silicon layer and a 3 µm BOX layer. Boththe silicon thickness and the BOX thickness are not optimized to achievehigh coupling efficiencies for our SWGCs. If we include the thickness of theBOX layer in our optimization process, then the coupling efficiencies of theSWGCs can be improved to be −1.5 dB and −1.7 dB, respectively (shown inFig. 2.18). Furthermore, if we include a bottom mirror (i.e., either a metalmirror or a Bragg reflector) in our SWGCs, then the coupling efficiencies of422.2. Apodized Sub-wavelength Grating Couplersour TE SWGC and TM SWGC can be improved to −0.34 dB and −0.41 dB,respectively (show in Fig. 2.18). The back reflections can be further reducedusing the technique demonstrated in [78], which is focusing the reflected lightaway from the entrance waveguide. It should be noted that the operatingbandwidths of SWGCs are affected by apodizations. According to [16], thebandwidth of the grating coupler is proportional to Λ of the grating coupler.The Λ values in the apodized region of the TM SWGC decreases from 960 nmto 872 nm, which causes the decrease in bandwidth. Compared to the TMSWGC, the Λ values in the apodized regions of the TE SWGC only decreasesfrom 556 nm to 505 nm. In addition, the back reflection from the apodizedTM SWGC is larger than that of the un-apodized design, which is causedby the increased index contrast in the apodized region.Wavelength (nm)1500 1520 1540 1560 1580 1600Transmission (dB)-12-10-8-6-4-20220 Si with 3 um BOX220 Si with optimized BOX220 Si with optimized BOX + bottom mirror(a)Wavelength (nm)1500 1520 1540 1560 1580 1600Transmission (dB)-12-10-8-6-4-20220 Si with 3 um BOX220 Si with optimized BOX220 Si with optimized BOX + bottom mirror(b)Figure 2.18: Simulated transmission spectra of (a) the apodized TE SWGCs,and (b) the apodized TM SWGCs.432.2. Apodized Sub-wavelength Grating Couplers2.2.3 Measurement ResultsTE SWGCsFigure. 2.19(a) shows the calibrated transmission spectra of an apodized anda un-apodized TE SWGCs. An HP 81525A high power optical head wasused to calibrate the loss from the measurement system, including the loss inthe fiber array, the connectors, and the additional fibres used for extension.The apodized design has a peak coupling efficiency of −3.2 dB, 0.6 dB higherthan the un-apodized one, with a 1-dB bandwidth of 36 nm, which is similarto that of the un-apodized design. The ripple at the central wavelengthof the apodized TE SWGC is about 0.07 dB, which corresponds to a backreflection of −24 dB. The highly suppressed back reflection from our SWGCis comparable to the state of the art shallow-etched grating couplers [51].The same designs are fabricated multiple times on the same chip to test theperformance stability. Figures 2.19(b)-(d) show the coupling efficiencies, 1-dB bandwidths, and the central wavelengths of apodized and un-apodizedTE SWGCs having the same designs and measured at different locations on aparticular chip. It should be noted from this comparison, that, the stabilitiesof the coupling efficiency, the bandwidth, and the central wavelength of theun-apodzied SWGCs are better than their stabilities for the apodized ones.This is the case because the apodized grating will only work if all of theapodized grating periods are in phase. Even a few nanometers offset in lsubcan degrade the performance of the SWGCs. Never the less, and despite thereduced stability, the average coupling efficiency of the apodized SWGCswas about 0.6 dB higher than the average coupling efficiency of un-apodizedSWGCs with similar bandwidths and central wavelengths.Corner analysis has been applied to predict the range of the key char-442.2. Apodized Sub-wavelength Grating CouplersWavelength (nm)1540 1560 1580 1600 1620-12.5-10-7.5-5-2.5Transmission (dB)ApodizationUn-apodized1560 1570 1580 1590 1600-6-5-4-31dB bandwidth =36 nm(a)Device-ID0 2 4 6 8 10Insertion Loss (dB)-6-5-4-3-2-1ApodizedUn-apodizedCornerAnalaysis(b)Device-ID0 2 4 6 8 101-dB bandwidth (nm)2030405060ApodizedUn-apodizedCornerAnalysis(c)Device-ID0 2 4 6 8 10Central Wavelength (nm)14801500152015401560158016001620ApodizedUn-apodizedCornerAnalysis(d)Figure 2.19: (a) Measured spectra of an apodized TE SWGC and an un-apodized TE SWGC; (b) coupling efficiencies of apodized and un-apodizedTE SWGCs; (c) 1-dB bandwidths of apodized and un-apodized TM SWGCs;(d) central wavelengths of the apodized and un-apodized TE SWGCs.acteristics of the as-designed SWGCs. Six parameters, θ, ΛH, lsub, thethickness of the silicon layer (Si), thickness of the BOX, and thickness ofthe oxide cladding (SiO2) have been used in the corner analysis as shownin Table 2.3. The first three parameters shown in Table 2.3 mainly affectthe central wavelength and the bandwidth of an SWGC, and the last threeparameters shown in Table 2.3 mainly affect the coupling efficiencies of anSWGC, since they change the interference condition of the light diffractedby the grating. The gap between the fiber tip and the chip surface is another452.2. Apodized Sub-wavelength Grating Couplersimportant parameter that affect the coupling efficiency and bandwidth ofan SWGC. Given the fact that we use angle polished fibers and leaving suf-ficient space to avoid scratching the chip surface, the gap between the fibercore and the chip surface is kept at about 15 µm. A 15 µm gap is also used inthe corner analysis (Table 2.3). According to our simulation, the extra losscaused by this gap is about 0.5 dB. The measured coupling efficiencies canbe further improved by polishing the fiber array to a particular angle so thatthe fiber tip and the chip surface can be parallel during the measurement.The dashed green lines in Figs. 2.19(b)-(d) denote the simulated boundariesin the corner analysis.Table 2.3: Parameters used for corner analysis for SWGCs.δθ δΛH δlsub δSi δBOX δSiO2± 2◦ ± 10 nm ± 10 nm ± 10 nm ± 20 nm ± 100 nmTM SWGCFigure 2.20 shows the calibrated measurement results for the TM SWGCs.Figure. 2.20(a) shows the measured transmission spectra of an apodizedTM SWGC and a un-apodized TM SWGC. The apodized TM SWGC has acoupling effciency of −3.3 dB, 0.6 dB higher than the un-apodized one, witha 1-dB bandwidth of 37 nm. The ripple at the central wavelength is about0.15 dB, which corresponds to a back reflection of−21 dB. Figures 2.20(b)-(d)show the coupling efficiencies, 1-dB bandwidths, and the central wavelengthsof the apodized and un-apodized TM SWGCs having the same designs andmeasured at different locations on a particular chip. Both the apodized andun-apodzied TM SWGCs have good stability and reproducibility. This is462.2. Apodized Sub-wavelength Grating Couplersthe case because the TM0 mode is less sensitive than the TE0 mode to thedesign parameter variations. The average coupling efficiency of the apodizedTM SWGCs is about 0.6 dB higher than the average coupling efficiency ofthe un-apodized TM SWGCs. Same corner analysis have been done for theTM SWGCs, and a 30 µm gap was assumed in the simulation. The enlargedgap is caused by the increased angle difference between the polished angleand the required incident angle by the TM SWGCs.Wavelength (nm)1500 1520 1540 1560 1580-12.5-10-7.5-5-2.5Transmission (dB)ApodizationUn-apodized1530 1550 1570-7-6-5-4-31dB bandwidth = 37 nm(a)Device-ID0 2 4 6 8 10Insertion Loss (dB)-6-5-4-3-2-1ApodizedUn-apodizedCornerAnalysis(b)Device-ID0 2 4 6 8 101-dB bandwidth (nm)25303540455055ApodizedUn-apodizedCornerAnalysis(c)Device-ID0 2 4 6 8 10Central Wavelength (nm)15201540156015801600ApodizedUn-apodizedCornerAnalysis(d)Figure 2.20: (a) Measured spectra of an apodized TM SWGC and anun-apodized TM SWGC; (b) coupling efficiencies of apodized and un-apodized TM SWGCs; (c) 1-dB bandwidths of apodized and un-apodizedTM SWGCs; (d) central wavelengths of the apodized and un-apodized TMSWGCs.472.2. Apodized Sub-wavelength Grating CouplersAs compared to the measured spectral response of the TE SWGCs, anasymmetry can be observed in the spectral response of the TM SWGCs,which is due to the fact that the surface of the fiber tip and the gratingsurface were not parallel during the measurement. The fiber ribbon used inour measurement has a polish angle of 23.2◦, which means that we need totilt the fiber ribbon by 24◦ to get the required incident angle for the TMSWGCs. Therefore, an acute angle exists between the fiber tip and thesurface of the grating. Incident waves at different wavelengths are diffractedat different angles by the grating. Longer wavelengths are not coupled intothe fiber as efficiently as shorter wavelengths, which results in the asymmetryobserved in the spectra.To summarize, in this section, we have experimentally demonstrated theapodized focusing SWGCs for both the TE0 and TM0 modes, which showa consistent improvement over the un-apodized designs. Corner analysishave been applied to the SWGCs, which shows that our devices are robustconsidering the manufacturing variations assumed. As the resolution ofthe CMOS fabrication becomes smaller, those SWGCs can even becomealternatives to the shallow-etched grating couplers; therefore the fabricationcost and complexity can be reduced.48Chapter 3Broadband Sub-wavelengthGrating CouplersSignificant effort has been devoted to improve the coupling efficiency ofgrating couplers [24, 29, 50, 54, 79, 84, 85, 99], while only a few attempts havebeen made to improve the operating bandwidths [16, 89, 90, 95, 101]. In thischapter, the study on the operating bandwidth for a grating coupler is shownfirst. Then we present a methodology to obtain SWGCs with design-intentbandwidths. Finally, we experimentally demonstrate broadband SWGCswith both straight and focusing SWGs.3.1 Bandwidth AnalysisThorough bandwidth analyses of grating couplers have been presented in[89, 90], where the bandwidth of a grating coupler was attributed to themismatch of the effective indices between the diffracted beam and the actualgrating structure. Alternatively [16], the bandwidth of a grating coupler canbe attributed to the wavelength dependent diffraction angle of the gratingcoupler. SWGs have been used to improve the bandwidths of grating cou-plers [16, 95, 101]. A schematic of the cross-section of a broadband SWGCis shown in Fig. 3.1.493.1. Bandwidth AnalysisFigure 3.1: Schematic of the cross-section of a broadband SWGC with one-dimensional SWGs.The phase-match condition for grating couplers, shown in Eq. 2.1, canalso be expressed as [96]:k0 · neff = k0 · nc · sin(θ) +m · 2piΛ(3.1)where k0 =2piλ , neff is the effective index of the grating, nc is the refractiveindex of the cladding, θ is the incident angle in free space, Λ is the gratingperiod, and m is an integer denoting the diffraction order (here m=1). Weestimate the 1-dB bandwidth of a grating coupler using the wavelengthdependent diffraction angle:∆λ1dB = ∆θ1dB · 2|dλdθ| (3.2)where ∆θ1dB is a constant that depends solely on the fiber parameters andthe calculated results match well with the measurement when ∆θ1dB equals0.047. |dλdθ | can be obtained from Equation 3.1 and the factor of 2 accountsfor the 1-dB bandwidth including the wavelength range both above andbelow the central operating wavelength. From Eq. 3.1, Λ can be expressed503.2. Design Methodologyas:Λ =λneff − nc · sin(θ) (3.3)We also know that ng is defined as [96] :ng = neff − λdneffdλ(3.4)Using Eqs.2.2 - 2.4, ∆λ1dB becomes:∆λ1dB = ∆θ1dB · 2|− nc · cos(θ) · λng − nc · sin(θ) | (3.5)Given that ∆θ1dB and nc are constants, we can see that the bandwidth isonly dependent on ng and θ for a given λ. Equation 3.5 shows us that the1-dB bandwidth of a grating coupler can be increased by reducing the ng ofthe grating. For a given ng, an optimal θ, that gives the largest bandwidth,can also be calculated using Eq. 3.5.3.2 Design MethodologyOur design methodology follows a four-step process. In the first step, wehave derived an expression for the 1-dB bandwidth of a grating coupler,that depends on the group index, ng, and θ of a grating coupler. Based onthe derived expression, a specific θ is chosen for further design optimization.Then three more steps are followed to finalize various design parametersusing the particle swarm algorithm [62] and effective medium theory (EMT)[28].The optimal incident angle, θopt, as a function of ng is shown in Fig. 3.2,for values of ng between 2.2 and 4.2; the simulated ng for a un-etched silicon513.2. Design Methodologywaveguide is about 4.2, and the simulated ng for a one-dimensional SWGwith an overall fill factor of 0.2 is about 2.2. (where the material dispersionwas not included in the calculation due to the fact that we used a constantrefractive index for the estimation based on EMT). As mentioned above, ifwe wish to increase the bandwidth, a larger θ is required. Here, we choseθ = 25◦, which is the optimal θ for an ng near the middle of the ng rangefrom 2.2 to 4.2.ng2.2 3.2 4.23 opt (°)152025303540Figure 3.2: Optimal incident angle, θopt, as a function of the group index ofa grating coupler, ng.As will become apparent from the discussion, the subsequent optimiza-tions of our SWGC design parameters begins with the structure shown inFig. 3.3(a). Our SWGCs are designed based on SOI wafers with 220 nm sil-icon layer and 3 µm BOX layer. Here, the grating of our SWGC is modelledas alternating regions of high and low refractive indices, where each gratingperiod consists of a high index region with a refractive index of nH and alow index region with a refractive index of nL. The lengths of nH and nLare denoted as ΛH and ΛL. The length of the grating period is Λ, whereΛ = ΛH + ΛL. The lengths of the SWGs in the high and the low index523.2. Design Methodologyregions are denoted as lH and lL, as shown in Fig. 3.3(b). The fill factor ofthe grating coupler, ff , is defined as the ratio of ΛH to Λ. The numbersof gratings in each high and low index region are NH and NL, respectively.The fill factors of the high and low index regions, denoted as ffH and ffL,are defined as NH ∗ lH/ΛH and NL ∗ lL/ΛL, respectively.(a)(b)Figure 3.3: Schematics of (a) an SWGC with refractive index regions of nHand nL, (b) an SWGC with one-dimensional SWGs.The two-dimensional FDTD method is used to optimize the design pa-rameters of our SWGCs. A figure of merit (FOM), defined as the product ofthe 1-dB bandwidth and the coupling efficiency, is used to predict the per-formance of an SWGC. Having chosen θ = 25◦, we have completed the firststep in our design methodology. We now optimize our SWGC design in threemore steps. In the second step, we optimized four design parameters, Λ, ff ,nH , and nL, to achieve the maximum FOM using the PSA in FDTD Solu-533.2. Design Methodologytions. The design parameters, with the largest FOM for θ =25◦ are shownin row 2 of Table 3.1. In the third step, using our optimized nH and nLfrom the second step, ffH and ffL are calculated using zeroth-order EMT[14, 28]. The dimensions of the SWGs have both a lower limit, which comesfrom the fabrication limitations, and an upper limit, which is determinedusing EMT. Since the calculated ffL and ffH are only approximations,further optimizations with boundary conditions are done using the PSA, asbefore, with the optimized values shown in row 3 of Table 3.1. Finally, wedetermined the number of SWGs in the high and low index regions, i.e.,NH and NL. The dimensions of the SWGs should be small enough to allowEMT to work and large enough to be fabricated. Based on the optimizedvalues of ffH and ffL, we simulated the allowed combinations of NH andNL, and have given the optimized values in the last row of Table 3.1.Table 3.1: Optimization steps and design parametersStep 1: θ = 25◦Step 2: Λ =1130 nm ff =0.5 nH = 2.78 nL = 1.8Step 3: ffH=0.49 ffL= 0.13Step 4: NH= 3 NL=2The simulated transmission and reflection spectra of the optimized SWGC,having the parameters given in Table 3.1, are shown in Fig. 3.4. Since allof the design optimizations are done using 2D FDTD simulations, we thenverify the designs using 3D simulations. The simulated transmission spectrafor both the 2D and the 3D simulations are shown in Fig. 3.5(a). The simu-lated reflection spectra of the straight and focusing SWGCs are also shown inFig. 3.5(a). As compared to the straight SWGC, the back reflection from thefocusing SWGC is smaller, which means that the focusing design not onlyreduces the footprint, but also suppresses the back reflection. The SWGC543.2. Design Methodologyhas an coupling efficiency of −3.6 dB and a 1-dB bandwidth of 84 nm. TheFOM is defined as:FOM =1-dB bandwidth (nm)100 (nm)· CE(%) (3.6)which takes both the bandwidth and coupling efficiency into consideration,the result being that the SWGC with the largest FOM may not necessarilyhave the largest bandwidth.6 (nm)1500 1550 1600 1650 1700CE (dB)-20-15-10-50T-2DT-3DR-straightR-focusingFigure 3.4: Simulated transmission and reflection spectra of the SWGC withthe design parameters shown in Table 3.1.Based on the design parameters in Table 3.1, we simulated the 1-dBbandwidths and the peak coupling efficiencies as we varied ffL. As we variedffL, ffH was also varied such that the operating wavelength of the gratingcoupler remained constant. Fig. 3.5 shows the simulated 1-dB bandwidthand peak coupling efficiency as a function of ffL. We can see that thebandwidth of the grating coupler is inversely proportional to ffL, and thatbroader bandwidths can be achieved with smaller ffL values. However, theminimum ffL is determined by the minimum feature size provided by thefabrication facility.553.3. Measurement Resultsff L0.1 0.12 0.14 0.16 0.181-dB Bandwidth (nm)60657075808590Peak CE (dB)-4.6-4.4-4.2-4-3.8-3.6-3.41-dB BandwidthCEFigure 3.5: Simulated 1-dB bandwidth and peak coupling efficeincy as afunction of ffL.3.3 Measurement ResultsTest structures, consisting of an input SWGC and an output SWGC, con-nected by a strip waveguide, were fabricated using electron beam lithographyat the University of Washington [11]. Since it is known that focusing SWGCsare more space-efficient than straight SWGCs, both types were fabricatedfor comparison purposes. The purpose of comparing them is to determinewhich has the higher FOM and, also, to confirm whether the focusing SWGChas a lower back reflection than the straight SWGC. The straight SWGCsare 12 µm wide and in order to keep the total device length to a reasonablenumber, we used 150 µm long tapers to couple the light from the grating tothe 500 nm wide waveguide. The focusing SWGCs were generated using themethod demonstrated in [77, 85]. Scanning electron microscope (SEM) im-ages of the as-fabricated straight and focusing SWGCs are shown in Fig. 3.6.Based on the simulation results shown in Fig. 3.5(b), design variationswere fabricated as shown in Table 3.2. The fabricated devices were mea-sured using our fiber-array-based automated measurement setup, the fiber563.3. Measurement Resultsarray was polished at 24.7◦ to accommodate the required incident angles.Figure 3.7 shows the measured transmission spectra of the focusing andstraight SWGCs with ffL=0.1 and ffH=0.52. The simulated transmissionspectrum, using the same grating design as the measured device, is shownin the same figure. The measured focusing and straight SWGCs, have max-imum coupling efficiencies of −5.5 dB and −5.8 dB, respectively. Both have1-dB bandwidths of 90 nm, with central wavelengths of about 1578 nm.Figure 3.6: SEM images of the as-fabricated focusing and straight SWGCs.(For the straight SWGC, the taper is not shown.)Table 3.2: Design variationsFocusing Grating ffL: 0.1 - 0.19 ffH : 0.52 - 0.43Straight Grating ffL: 0.1 - 0.18 ffH : 0.52 - 0.44The coupling efficiency of the straight SWGC is lower than that of thefocusing one because we used a short linear taper in it. The length of thelinear tapers used in our straight SWGC are 150 µm, which is not long573.3. Measurement Resultsenough to allow adiabatic coupling between the grating and the waveguide.It should be noted that the oscillation ripples in the spectrum of the focusingSWGC are smaller than those in the spectrum of the straight SWGC, whichconfirms that the focusing design not only reduces the footprint of the device,but also suppresses the back reflection from the grating. This is the casebecause the focal point of the reflected wave is away from the entranceof the waveguide, and such technique is detailed in [78]. The ERs of theripples near the central wavelength of the focusing SWGC are about 0.08 dB,which correspond to back reflections of less than −23 dB, whereas the ERsof the ripples near the central wavelength of the straight SWGC are about0.2 dB, which correspond to back reflections of less than−19 dB. In addition,the back reflection of the focusing SWGCs are more consistent over the1-dB bandwidth wavelength range, while the back reflection of the straightSWGCs increases gradually as the wavelength decreases, which matches thesimulation results shown in Fig. 3.4.Wavelength (nm)1500 1550 1600 1640CE (dB)-15-12-9-6-3Measured 1dB bandwidth =90 nmSimulated 1dB bandwidth =89 nmSimulationFocusingStraightFigure 3.7: Transmission spectra of the simulated SWGC (red), the mea-sured focusing SWGC (blue), and the measured straight SWGC (green).Figure 3.8 shows the measured 1-dB bandwidths and coupling efficiencies583.3. Measurement Resultsof the fabricated focusing and straight SWGCs with the design variationsshown in Table 3.2. For the focusing SWGCs, as ffL changed from 0.1 to0.19, the 1-dB bandwidths changed from 90 nm to 48 nm and the couplingefficiencies varied between −5.5 dB and −3.8 dB. For the straight SWGCs,as ffL changed from 0.1 to 0.18, the 1-dB bandwidths changed from 90 nmto 48 nm and the coupling efficiencies varied between −5.8 dB and −4.2 dB.The measured 1-dB bandwidths of both focusing and straight SWGCs followthe same trend as shown in Fig. 3.5(b) except that the slope of the measured1-dB bandwidth variations were slightly larger than those of the simulations.The increased slope may result from the fact that the ∆θ1dB of the fibermodel that we used in the simulations is smaller than the ∆θ1dB of the fibersthat we used in the measurements. The measured coupling efficiencies of thefocusing and straight SWGCs are lower than the simulated results, whichmay come from several sources, such as the measurement system, fabricationerrors, etc. In addition, the measured peak coupling efficiency correspondsto a larger ffL than was used in the simulation, which may come from thefact that the ffLs of the fabricated devices were smaller than the designvalues.In conclusion, in this chapter we presented a methodology to designbroadband SWGCs using one dimensional SWGs. Both straight SWGCsand focusing SWGCs with 1-dB bandwidths of 90 nm are designed and fab-ricated. Back reflections from the SWGCs are suppressed by using focusingSWG designs, and a measured back reflection below −23 dB is achieved,which is comparable to state of the art shallow-etched grating couplers [78].593.3. Measurement Resultsff L0.1 0.12 0.14 0.16 0.181-dB bandwidth (nm)405060708090100(a)ff L0.1 0.12 0.14 0.16 0.18CE (dB)-5.5-5-4.5-4-3.5(b)ff L0.1 0.12 0.14 0.16 0.181-dB bandwidth (nm)405060708090100(c)ff L0.1 0.12 0.14 0.16 0.18CE (dB)-6-5.5-5-4.5-4(d)Figure 3.8: Measured (a) 1-dB bandwidths and (b) coupling efficiencies ofthe fabricated focusing SWGCs with ffL ranging from 0.1 to 0.19; measured(c) 1-dB bandwidths and (d) coupling efficiencies of the fabricated straightSWGCs with ffL ranging from 0.1 to 0.18.60Chapter 4Broadband Sub-wavelengthDirectional CouplersSub-wavelength gratings (SWGs) provide the flexibility to engineer boththeir index profiles and their dispersion properties, which have been pro-posed to engineer the wavelength dependency of a conventional directionalcoupler (DC) for broad operating bandwidth [30]. However, the theoreticalstudy shown in [30] was based on SOI wafers with 260 nm silicon layer, whichis different than the 220 nm SOI wafters that are more commonly used byMPW foundry services. In addition, the minimum feature size in the pro-posed structure is only about 60 nm, which is challenging to fabricate evenwith the most advanced electron beam lithography [11].In this chapter, we elaborate on the proposed work in [30] by experi-mentally demonstrating more compact broadband DCs using SWGs, withvarious power splitting ratios, for SOI wafers with silicon layers of 220 nm.The minimum feature size of our design is increased to 80 nm, which canbe easily fabricated with electron beam lithography. The design can bealso modified to have minimum feature size around 100 nm, which can befabricated with CMOS technology using optical lithography [43]. In addi-tion, a design methodology using FDTD-based band structure calculationsis presented to design our SWG DCs.614.1. Wavelength Dependency of Directional CouplersA schematic of our SWG DC is shown in Fig. 4.1, which consists of aconventional DC with waveguide widths, w, separated by a spacing, g, andSWGs with a period, Λ, and a fill factor, ff , where ff is defined as theratio of lsub to Λ, and lsub is the length of one grating tooth. The SWGs areextended outside of the waveguides by a distance, t. S bend waveguides areused at the input and output ports to decouple the two parallel waveguides.Figure 4.1: Schematic of an SWG DC with design parameters labelled.4.1 Wavelength Dependency of DirectionalCouplersA conventional DC consists of two parallel waveguides, where the couplingcoefficient is controlled by both the length of the coupling region and thespacing between the two waveguides. The behaviour of a DC can be ex-plained based on the phase matching condition between the TE0 mode andthe TE1 mode of the two waveguide system. The TE0 mode is also knowas the even “supermode” or the symmetric mode and the TE1 mode isalso known as the odd “supermode” or the antisymmetric mode of the twowaveguide system. The field profile for the TE0 mode and the TE1 mode624.1. Wavelength Dependency of Directional Couplersfor a conventional DC are shown in Fig. 4.2.-1 0 1X Position (um)-0.500.51Y Position (um)-0.500.5(a)-1 0 1X Position (um)-0.500.51Y Position (um)-0.500.5(b)Figure 4.2: Field profiles of (a) the TE0 mode, and (b) the TE1 mode for aconventional DC.As the TE0 and the TE1 modes travel, the optical power appears to beatback and forth between the two waveguides. The cross-over length, Lpi, isa function of the difference in propagation constants, β1 and β2 of the TE0and the TE1 modes, respectively, and is the minimum length required forthe maximum optical power transfer from one waveguide to the other:Lpi =piβ1 − β2 =λ/2neff1 − neff2 (4.1)where λ is the operating wavelength, neff1 and neff2 are the effective indices ofthe TE0 and TE1 modes, respectively. The simulated neff1 and neff2 values,as functions of λ for a DC with w = 450nm and g = 220nm, are shown inFig 4.3(a). It can be seen that both the TE0 mode and the TE1 mode have634.2. Dispersion Engineeringnormal dispersions (dn/dλ < 0), and the slope of neff2 is larger than thatof neff1. The difference between neff1 and neff2, δn, increases as λ increases.Such a wavelength dependent δn leads to a wavelength dependent Lpi, asshown in Fig. 4.3(b), which limits the operating bandwidth of conventionalDCs.1500 1525 1550 1575 16006 (nm)2.252.32.352.42.45neffneff1neff2(a)1500 1525 1550 1575 16006 (nm)2025303540L: (um)(b)Figure 4.3: (a) neff1 and neff2 as functions of λ for a conventional DC, (b) Lpias a function of λ for a conventional DC. The simulated DC has w = 450 nm,g = 220 nm, based on an SOI wafer with a silicon layer of 220 nm.4.2 Dispersion EngineeringSWGs can be used to engineer the slope of neff1 so that it is matched to thatof neff2; therefore, a δn that is less wavelength dependent in the wavelengthrange that we are interested in can be obtained. As shown in Fig. 4.4,when λ is approaching the Bragg wavelength, λB = 2 · neff · Λ, of an SWG,from the long wavelength side, the neff of the fundamental Floquet mode[7] increases dramatically. When the SWGs are applied to a conventionalDC, the index perturbation of neff2 is cancelled in the central region becausethe TE1 mode has an antisymmetric field profile. On the other hand, neff1644.2. Dispersion Engineering1500 1550 1600 16506 (nm)2.12.22.32.42.5neff(a)Figure 4.4: neff as a function of λ for an SWG with Λ = 285 nm, ff = 0.5.increases dramatically as λ approaches λB. Therefore, the slope of neff1 canbe increased to match with that of neff2. There are five parameters that needto be determined for SWG DCs: Λ, ff , g, t, and the number of grating, NG.In order to avoid the coupling in the S bend waveguides regions, we usedg = 500 nm in our design. It has been found [30] that t should be greater than400 nm to sufficiently suppress the optical power coupled to the higher ordermodes and we used t = 500 nm in our design. We used the “Bandstructure”model from the knowledge base provided by Lumerical Solutions, Inc. toanalyze our designs. Compared to the commonly used eigenmode expansion(EME) method, in which the SWGs are estimated by the effective mediumtheory [64], our approach has the following advantages. Firstly, FDTD-based band structure calculations simulate the actual structure and takethe material dispersion, the structure dispersion, and the Bragg effect intoconsideration. Our SWG DC can be simulated with using the EMT since theBragg effect from the SWGs was omitted by the EMT method where SWGsare treated as uniform material with equivalent refractive indices. Secondly,654.2. Dispersion Engineeringband structure calculations only require simulations of one unit cell of thestructure, which significantly reduces simulation times as compared to full-structure FDTD simulations. A full-structure 3D FDTD simulation can takea few hours, while it only take a few minutes to simulate the structure using3D FDTD band structure calculations, which makes device optimizationpossible. Our SWG DCs are designed for the TE0 mode with λ of 1550 nm.A schematic of the simulation structure for an SWG DC is shown inFig. 4.5(a). Only one unit cell, one period of the SWG DC, was simulatedand Bloch boundary conditions were used in the z direction, which is thepropagation direction of the optical modes. The simulations were launchedin the time domain and then transferred into the frequency domain usingthe Fourier transform. The band diagram of an SWG DC can be obtainedby running a parameter sweep for various wave vectors, kz = 2piΛ . The banddiagram of an SWG DC is shown in Fig. 4.5(b).(a)0.45 0.455 0.46kz (2 :/a)190192194196f (THz)-20246810(b)Figure 4.5: (a) Schematic of the simulated structure for an SWG DC inFDTD Solutions, (b) simulated band diagram for an SWG DC with Λ =285 nm and ff = 0.28.The simulated transmission spectra from the cross ports as functions ofλ for SWG DCs with ff = 0.25 and various Λ values, are shown Fig. 4.6(a),664.2. Dispersion Engineeringwhere the troughs correspond to λB and the peak regions denote the wave-length regions where δn is over compensated. In order to achieve a broadoperating bandwidth, in the wavelength range from 1500 nm to 1600 nm,we used Λ = 285 nm in our designs. The ff of the SWG DC has both alower limit, which comes from the fabrication limitation, and an upper limit,which determines the optical power coupled into the higher order mode of theSWG [30]. Simulations shows that as the ff of the SWG DC increases, morepower will coupled into the third order mode. In our case, 0.2 < ff < 0.3.The simulated transmission spectra from the cross ports for SWG DCs withΛ = 285 nm, and various ff values are shown in Fig. 4.6(b). It can be seenthat SWG DCs with larger ff values have stronger coupling, which resultsin smaller Lpi. For a given Λ, ff determines neff, which in turn determinesλB. By comparing the wavelength dependent δn, over the wavelength rangefrom 1500 nm to 1600 nm, we can find the SWG DC with an optimized ff ,with the least wavelength dependency. In our case, ff = 0.28.1400 1500 1600 1700Wavelength (nm)-25-20-15-10-50Cross Coupling (dB)$=275nm$=280nm$=285nm$=290nm$=295nm(a)1450 1500 1550 1600 1650Wavelength (nm)-20-15-10-50Cross Coupling (dB)ff=0.2ff=0.22ff=0.24ff=0.26ff=0.28ff=0.3(b)Figure 4.6: (a) Simulated transmission spectra from the cross ports ofSWG DCs with ff = 0.25, NG = 47, and various Λ values, (b) simulatedtransmission spectra from the cross ports of SWG DCs with Λ = 285 nm,NG = 47, and various ff values.674.2. Dispersion EngineeringFrom the band structure diagram, we can extract neff1 and neff2 values asfunctions of λ. Therefore, Lpi can be calculated. The neff1 and neff2 values, asfunctions of λ for the SWG DC with Λ = 285 nm and ff = 0.28, are shownin Fig. 4.7(a) and the corresponding Lpi, as a function of λ, is shown inFig. 4.7(b). It can be seen that by using SWGs, the wavelength dependencyof Lpi has been reduced from ±7 µm to ±1.5 µm. The λB, determined by Λand neff, is about 1425 nm. It should be noted that the optimized ff is forthe given Λ used in our design, other combinations of Λ and ff with thesame λB can provide SWG DCs with similar bandwidths. NG determinesthe coupling length, which in turn determines the power splitting ratio ofthe design.1500 1525 1550 1575 16006 (nm)2.42.452.52.552.6neffneff1neff2(a)1500 1525 1550 1575 16006 (nm)242526272829L : (um)(b)Figure 4.7: (a) neff1 and neff2 as functions of λ for an SWG DC with Λ =285 nm and ff = 0.28, (b) calculated Lpi as a function of λ based on thenneff1 and neff2 shown in (a).After we designed the SWG DC using band structure calculations, full-strucuture simulations were used to verify the design. The light was launchedin the upper waveguide from the left and measured at the cross and throughports on the right. We define the normalized output power for the cross andthrough ports as η1 = 10∗ log10(P1/(P1 +P2)) and η2 = 10∗ log10(P2/(P1 +P2)), where P1 and P2 are the output power measured from the cross and684.2. Dispersion Engineeringthe through ports, respectively. Figure 4.8(a) shows the full-structure FDTDsimulation results for the SWG DC with a device length of 13.4 µm, whichwas designed to have a power splitting ratio of 50/50. The designed SWG DCcovers a bandwidth of 100 nm, from 1500 nm to 1600 nm, with an imbalanceof ±0.55 dB. The simulated insertion loss, IL, as a function of λ for the SWGDC with the designed power splitting ratio of 50/50 is shown in Fig. 4.8(b).The IL as a function of ff for SWG DCs with Λ = 285 nm is shown inFig. 4.9. It should be noted that there is a trade-off between the bandwidthand IL and here we chose bandwidth over IL. In addition, by using a largeff for our design, we also benefit from a small Lpi, which means that thedesign is space efficient.1500 1525 1550 1575 16006 (nm)-6-5-4-3-2-10Power (dB)2122(a)1500 1525 1550 1575 16006 (nm)00.20.40.60.81Insertion Loss (dB)(b)Figure 4.8: (a) Simulated spectra for an SWG DC with a designed powersplitting ratio of 50/50 (η1 and η2 are the normalized output power for thecross and through ports, respectively), (b) simulated IL as a function of λfor the SWG DC with a designed power splitting ratio of 50/50.694.3. Measurement Results0.2 0.22 0.24 0.26 0.28 0.3ff0.30.40.50.60.70.8Insertion Loss (dB)Figure 4.9: Simulated ILs as a function of ff for SWG DCs with Λ = 285 nm.4.3 Measurement ResultsA waveguide-based Mach-Zehnder interferometer (MZI) was used to measurethe wavelength dependent power coupling coefficient, K, of the designedSWG DCs. The mask layout of the test structure is shown in Fig. 4.10.Broadband SWG grating couplers [83] were used to couple the light intoand out of the test structures. Identical SWG DCs were used as the powersplitter and the combiner in the MZI and were connected by two waveguideswith a δL of 250 µm. When both the splitter and the combiner have a powersplitting ratio of 50/50, complete destructive interference occurs when theoptical waves from the two arms of the MZI are out of phase. In the case inwhich the splitting or combining ratios are not 50/50, K can be extractedfrom adjacent maxima and minima of the spectrum [10], assuming that theSWG DCs are lossless.For a real field coupling coefficient, k, for both the splitter and thecombiner, and lossless waveguide, the transfer matrix of the test structure704.3. Measurement ResultsFigure 4.10: Mask layout of the test structure for SWG DCs.shown in Fig. 4.10 is [96]: Eout1Eout2 = √1− k2 jkjk√1− k2 e−jφ1 00 e−jφ2 √1− k2 jkjk√1− k2 Ein0where φ1 = βL1, φ2 = βL2, L1 and L2 are the lengths of the waveguidescomprising the two arms of the MZI. Eout1 and Eout2 are the output electricfields from the two output ports of the MZI, which can be calculated fromthe above matrix to be:Eout1 = [(1− k2) · e−jφ1 + (jk)2 · e−jφ2 ] · Ein (4.2)andEout2 = [jk√1− k2 · e−jφ1 + jk√1− k2 · e−jφ2 ] · Ein (4.3)From the electric fields, we can obtain the ERs of the normalized power714.3. Measurement Resultsintensity for the two output ports as:ER1 = 10log10(11− 4k2(1− k2)) (4.4)ER2 = 10log10(4k2(1− k2)0) =∞ (4.5)From Eq. 5 we can see that ER2 is independent of the splitting ratio, andK can only be extracted from ER1 and is given by:K = k2 =12± 12√110ER110(4.6)SWG DC test structures, with various power splitting ratios, were fabri-cated on an SOI wafer with a 220 nm silicon layer on a 3 µm silicon dioxide.A JEOL JBX-6300FS system, operating at 100 keV energy, and with an8nA beam current and a 500 µm exposure field was used to perform elec-tron beam lithography [11]. The samples were on 25 mm squares diced from150 mm wafers. The machine grid used for shape placement was 1 nm, whilethe beam stepping grid, the spacing between dwell points during the shapewriting, was 6 nm. To characterize the devices, a custom-built automatedtest setup [19] was used. An Agilent 81600B tunable laser, with a PM fiber,was used as the input source, and Agilent 81635A optical power sensors,also with PM fibers, were used as the output detectors. A wavelength sweepfrom 1500 nm to 1600 nm in 10 pm steps was performed. SEM images of afabricated SWG DC are shown in Fig. 4.11.Due to the fact that the actual thickness of the silicon layer for theSOI wafer used in our fabrication was 206 nm, the optical modes were lessconfined within the waveguides. Thus, the coupling between the two waveg-724.3. Measurement Results(a)(b)Figure 4.11: SEM images of (a) a fabricated SWG DC, (b) a zoom-in ofthe central portion of the fabricated SWG DC with the design parameterslabelled.uides of the fabricated SWG DCs was stronger than the design values. Fig-ure 4.12(a) shows the measured spectra for a fabricated test structure withSWG DCs which had power splitting ratios close to 50/50. The measuredSWG DCs had g = 540 nm, Λ = 285 nm, ff = 0.28, and NG = 47. Theg of the fabricated SWG DCs with power splitting ratio close to 50/50 was40 nm larger than the designed value, which compensated for the increasedcoupling resulting from the reduced thickness of the silicon layer. Output 1and Output 2 denote the two output ports of the MZI as shown in Fig. 4.10.Figure 4.12(b) shows the normalized powers from the two ports of a fabri-cated SWG DC, which were calculated using Eq.(6) and the maxima andminima shown in Fig. 4.12(a). The ILs from the input and output grat-734.3. Measurement Resultsing couplers have been calibrated out. The measured SWG DC covered abandwidth of 100 nm with an imbalance of ±0.7 dB, which agrees with thesimulation results shown in Fig. 4.8. The power splitting ratios of an SWGDC can be controlled by either changing NG or ff . However, as we men-tioned earlier, increasing ff may results in power coupling into the higherorder mode of the SWG DC, which is unwanted. Therefore, we used NGto control K. Test structures for SWG DCs with power spitting ratios of40/60, 30/70, and 20/80 have also been fabricated and measured, with theirmeasurement results shown in Figs. 4.13(a)-(c), respectively.1500 1525 1550 1575 16006 (nm)-50-40-30-20-1001020Power (dB)Output 1Output 2(a)1500 1525 1550 1575 16006 (nm)-6-5-4-3-2-10Normalized PowerPort 1Port 2(b)Figure 4.12: (a) Measured spectra for a test structure with SWG DCs whichhad power splitting ratios close to 50/50, and (b) normalized optical powersfor a fabricated SWG DC using the measurement data shown in (a).744.3. Measurement Results1500 1525 1550 1575 16006 (nm)020406080100Normalized Power (%) Port 1Port 2(a)1500 1525 1550 1575 16006 (nm)020406080100Normalized Power (%) Port 1Port 2(b)1500 1525 1550 1575 16006 (nm)020406080100Normalized Power (%) Port 1Port 2(c)Figure 4.13: Normalized optical powers for fabricated SWG DCs with powersplitting ratios of (a) 40/60, (b) 30/70, and (c) 20/80.75Chapter 5Sub-wavelength PolarizationSplitter RotatorThe polarization splitter-rotator (PSR) demonstrated in this chapter is acomponent that changes the polarization of the TM input mode to the TEmode, i.e., polarization rotated by 90 degrees, while maintaining the polar-ization of the TE input mode as the same. The TE and TM input modesinject into the PSR from the same input port and split into two output portsas two TE modes. PSRs have been used to construct polarization-insensitvePICs [3] in the strong confinement limit. Mode-evolution-based PSRs havebeen demonstrated [20, 23, 65, 81, 97], in which the TM0 mode was firstconverted to the first order quasi-transverse electric, TE1, mode, and thenthe TE1 mode and the TE0 modes were coupled into the TE0 modes of twoseparate waveguides, using an asymmetric directional coupler [20], an asym-metric Y-branch [81, 97], a phase-shifted Y-branch and an MMI [23], or anadiabatic coupler [65]. The mode-evolution-based devices are less sensitiveto fabrication imperfections and have broad operating bandwidths, but atthe cost of large device footprints, typically on the order of a few hundredmicrometers. Mode-coupling-based devices are more compact in size [45, 80],but the requirement of phase-matched modes makes such devices sensitiveto both fabrication imperfections and wavelength. A mode-coupling-based765.1. Design of SWG PSRSWG PSR has been theoretically proposed [91], where SWGs were used toreduce the sensitivity to manufacturing variations of the waveguides widths.Although improved stability was quoted as the advantage of such device,the effects that induces the greatest variability, i.e., change in the fill factorof the SWG waveguide and the gap between the waveguides were ignored.In addition, the SWG taper and S bend waveguide used in [91] were muchlarger than the PSR itself, which significantly increases the overall size ofthe proposed device.In this chapter, we extend the idea proposed in [91]. First, we explorethe fabrication tolerance of our SWG PSR to various parameter variations,including the gap variations between the waveguides and the fill factor vari-ations of the SWG waveguide; second, we design a high-efficiency and com-pact SWG taper that couples the optical modes from the SWG waveguidein our PSR into a strip waveguide without significant increase of the devicefootprint; third, we engineer the SWG PSR for broader operating band-width; fourth, the experimentally demonstrate the SWG PSR. In addition,an efficient modelling methodology based ons FDTD band structure calcula-tions is presented in this chapter to design the coupling section of our SWGPSR, which dramatically reduced the computation time and design effortas compared to brute-force simulations of the full device using 3D FDTDsimulations.5.1 Design of SWG PSRA schematic of our SWG PSR is shown in Fig. 5.1, which consists of threeparts: a coupling section consisting of two parallel waveguides, one stripwaveguide with a width, WA and an SWG waveguide with a width, WB,775.1. Design of SWG PSRwhich separated by a gap, g; an SWG taper which couples the light fromthe SWG waveguide into the strip waveguide; and an S bend waveguidewhich decouples the two output waveguides. The SWG waveguide consistsof SWGs with period, Λ, and fill factor, ff , which is defined as the ratio ofl to Λ, as shown in Fig. 5.1. Again, our SWG PSR was designed for an SOIplatform having silicon layers of 220 nm on a 3 µm BOX layer. The design ofour SWG PSR was carried out by following a four-step process as describedbelow.Figure 5.1: Schematic of the top view of an SWG PSR.In the first step, we calculate the effective index of the TM0 mode,neff-TM, for the strip waveguide as a function of WA, which is denoted by theblue curve shown in Fig. 5.2. The dimensions of the strip waveguide we usedis 450 nm x 220 nm, which has neff-TM = 1.545 at 1550 nm. In the secondstep, we treat the SWG waveguide as an equivalent wire waveguide with arefractive index of nB and a width of WB. The effective index of the TE0mode for the equivalent waveguide, neff-TE, is calculated as a function of WBand nB using MODE Solutions (an eigenmode solver from Lumerical Solu-tions, Inc.). An SWG waveguide is then found that has the same neff (i.e.,neff-TE = neff-TM) and slope of neff (i.e., δnneff-TE/δWB = δnneff-TM/δWA) at1550 nm as the TM0 mode of the strip waveguide that we calculated in thefirst step. This SWG waveguide has an nB = 2.43 and a WB = 670 nm. The785.1. Design of SWG PSRsimulated neff-TE as a function of WB for this SWG waveguide is denotedby the red curve in Fig. 5.2. Due to the fact that the neff for the stripand the SWG waveguides have the same slopes, the phase-match conditionwill be preserved if the waveguide widths are changed by the same amountsduring fabrication. Here, in the first two steps, we used the phase-matchcondition of the two local normal modes in the strip and the SWG waveg-uide to obtain preliminary design parameters for the SWG waveguide giventhe dimensions of the strip waveguide. The design will be further optimizedusing supermodes theory, which is presented below.WG width (nm)300 400 500 600 700 800neff1.41.451.51.551.61.651.7neff-TMneff-TEFigure 5.2: neff-TM as a function of WA for a strip waveguide and neff-TE asa function of WB for an SWG waveguide.In the third step, we determine the Λ and the ff of the SWG waveguidebased on the nB we obtained from the previous two steps. FDTD bandstructure calculations are used to obtain the band diagram of the SWGwaveguide, from which we extract neff-TE as a function of λ for the SWGwaveguide. For these calculations, we use FDTD Solutions. We set Λ =300 nm in our simulation and by varying ff , we obtained SWG waveguidesequivalent to wire waveguides with various values of nB. In our case, ff =0.5 for nB = 2.43, calculated using three dimensional FDTD simulations.795.1. Design of SWG PSRThe field distributions of the first three supermodes for the two waveguidesystems comprising our SWG PSR are shown in Fig. 5.3, where the red dashlines denote the edges of the waveguides and the top of the BOX layer. Dueto the fact that the supermodes are hybrid modes, and have electric fieldsin both the X and Y directions, we refer to the first three supermdoes asExy0 , Exy1 , and Exy2 .In the fourth step, we calculate the length of the coupling section forthe PSR. When the TE0 mode is launched into the two waveguide system,it couples to the highly asymmetric Exy0 mode (shown in Figs. 5.3(a)-(b)),in which the power is primarily confined to the strip waveguide and whichpropagates to the output of the two waveguide system with minimum powerin the SWG waveguide. When the TM0 mode is launched into the twowaveguide system, it couples into both the Exy1 (shown in Figs. 5.3(c)-(d))and the Exy2 (shown in Figs. 5.3(e)-(f)) modes in nearly equal portions sothat after propagating one beat length, or Lpi, in the two waveguide system,the Y -polarized electric fields in the two modes destructively interfere onthe strip waveguide side of the two waveguide system while the X-polarizedelectric fields constructively interfere on the SWG waveguide side of the twowaveguide system. The effective indices of the Exy1 and the Exy2 modes, neff,1and neff,2, as functions of λ are shown in Fig. 5.4. The length of the couplingsection for our SWG PSR is equal to Lpi of the Exy1 and the Exy2 modes, whichis calculated using:Lpi =λ/2neff,1 − neff,2 (5.1)A g = 100 nm was used in our simulation to achieve a comparatively shortLpi and a larger Lpi will be required for a coupling section with a larger g. Inour case, neff,1 = 1.5843 and neff,2 = 1.5528, which lead to a Lpi = 24.6 µm.805.1. Design of SWG PSR-1 0 1Y Position [um]-0.500.51X Position [um]-0.500.51(a)-1 0 1Y Position [um]-0.500.51X Position [um]-0.500.51(b)-1 0 1Y Position [um]-0.500.51X Position [um]-0.500.51(c)-1 0 1Y Position [um]-0.500.51X Position [um]-0.500.51(d)-1 0 1Y Position [um]-0.500.51X Position [um]-0.500.51(e)-1 0 1Y Position [um]-0.500.51X Position [um]-0.500.51(f)Figure 5.3: The X and Y field distributions for the first three supermodes.(a) Y component of the Exy0 mode, (b) X component of the Exy0 mode; (c)X component of the Exy1 mode, (d) Y component of the Exy1 mode; (e) Xcomponent of the Exy2 mode, (f) Y component of the Exy2 mode.815.1. Design of SWG PSR1500 1525 1550 1575 16006 (nm)1.51.551.61.65neffExy1Exy2(a)Figure 5.4: neff as a function of λ for the Exy1 and the Exy2 modes of the twowaveguide system comprising the coupling section of the SWG PSR.A high-efficiency, space-efficient SWG taper is required to couple theoptical mode from the SWG waveguide into the strip waveguide. As shownin Fig. 5.5, our SWG taper consists of two parts: SWGs with period, Λt, andfill factor, fft = lt/Λt and tapered bridge sections connecting the SWGs.In order to suppress the reflection at the interface of the SWG waveguideand the SWG taper, we use Λ = Λt and ff = fft in our design. Thewidths of the SWGs comprising the taper linearly decrease from WB =670 nm to WA = 450 nm. The bridge sections are used between the SWGsto reduce the effective index differences between adjacent sections, hence, tosuppress the reflections between adjacent SWGs. The widths of the bridgesections are linearly increased from t1 to WA, where t1 is the minimumfeature size provided by the fabrication process. In our case, t1 = 60 nm.We use an SWG taper with 30 periods, which has a simulated IL of lessthan 0.02 dB over the wavelength range from 1500 nm to 1600 nm using3D FDTD simulation. The test structure of our design were fabricated825.1. Design of SWG PSRusing electron beam lithography [11], which has an achievable minimumfeature size of about 40 nm. In order to make the structure compatible withCMOS technology using optical lithography via MPW foundry services [43],which have minimum feature sizes of about 100 nm, t1 can be increased ata cost of larger ILs. For t1 = 100 nm and other parameters stay the same,the simulated IL of the taper increased from 0.02 dB to 0.04 dB over thewavelength range from 1500 nm to 1600 nm.Figure 5.5: Schematic of the top view of an SWG taper.Finally, the full structure of the designed SWG PSR is verified using 3DFDTD simulation, maintaining the SWG structure in the simulation andusing SWG taper described above. The S bend waveguide used in the simu-lation has a height of 3 µm and a length of 5 µm. For the TE0 mode input, wedefine the IL at P1 (shown in Fig. 5.1) as ILTE-TE = P1out,TE/Pin,TE and thecrosstalk (XT) at P2 (shown in Fig. 5.1) as XTTE-TE = P2out,TE/Pin,TE.For the TM0 mode input, we define the polarization conversion efficiency(PCE) at P2 as PCETM-TE = P2out,TE/Pin, TM and the XT at P1 asXTTM-TM = P1out,TM/Pin,TM. The most important FOMs for a PSR are theILTE-TE and the PCETM-TE, which are shown in Fig. 5.6(a). The simulatedPSR has a negligible ILTE-TE, a peak PCETM-TE of −0.46 dB at 1550 nmwith a 1dB bandwidth in excess of 80 nm. The PCE is limited by the modedistributions of the Exy1 and the Exy2 modes as shown in Fig.5.3(c)-(f). Dueto the fact that the X and Y components of the Exy1 and the Exy2 modes arenot fully overlapped, when the X components of Exy1 and the Exy2 modes are835.1. Design of SWG PSRout of phase, there is still a small portion of optical power resides on thestrip waveguide side of the two waveguide system, which cause the loss ofPCE. The PCE can be improved by optimizing the mode overlaps betweenthe Exy1 and the Exy2 modes. The simulated XTTM-TM and XTTE-TE of thedesigned PSR are shown in Fig. 5.6(b).1500 1525 1550 1575 1600 16256 (nm)-4-3-2-10Transmission (dB)1dB bandwidth =83 nmILTE-TEPCETM-TE(a)1500 1525 1550 1575 1600 16256 (nm)-25-20-15-10-50Transmission (dB)XTTM-TMXTTE-TE(b)Figure 5.6: (a) Simulated ILTE-TE and PCETM-TE as functions of λ, and (b)simulated XTTE-TE and XTTM-TM as functions of λ.The fabrication tolerance of the designed SWG PSR, with the parametervariations shown in Table 5.1, has also been explored. The silicon layerthickness variation is denoted by δWX and the variations in the other twodimensions are denoted by δWY-Z. δWX is decided by the wafer uniformityused for fabrication. We used SOI wafers with silicon layers of 220 nm on a3 µm BOX, provide by SOITEC, which has a guaranteed 3σ = ±10 nm [92].However, the central thickness of the wafer is not necessary to be 220 nm.Based on our experiences, the thickness of the silicon layer is 220±15 nm, andwe used a variation range of ±20 nm to predict the worst possible situationin our simulations. When we explored the design tolerance to δWY-Z, weused ±δWY-Z = ±δWA = ±δWB = ∓12δg = ±δl = ±δlt to mimic the845.2. Fabrication and Measurementsituation during fabrication. The simulated PCETM-TE as functions of δWXand δWX are shown in Fig. 5.7 as the green and the red curves. It has beendiscussed earlier that the designed SWG PSR has a high tolerance to δWAand δWB, which is achieved by designing the SWG waveguide to maintainthe phase-match condition as the waveguide width varies. Therefore, thereduced PCETM-TE caused by δWY-Z is mainly the result of the variationsin g and l (or ff), which have a strong impact on the coupling strength ofthe designed SWG PSR.Table 5.1: Parameter Variations.Parameters δWX δWY-ZVariation Range ±20 nm ±20 nm-20 -10 0 10 20Dimension offsets (nm)-4-3-2-10PCE TM-TE (dB)/ WX/ WY-ZFigure 5.7: PCETM-TE as functions of δWX and δWY-Z.5.2 Fabrication and MeasurementTest structures of the designed PSR, using broadband SWGCs as the in-put/output [83], were fabricated by electron beam lithography [11]. SEMimages of a fabricated SWG PSR are shown in Fig. 5.8. Due to the fact thatgrating couplers are polarization sensitive, we designed grating couplers with855.2. Fabrication and Measurementthe same θ to couple the TE0 and the TM0 modes into the test structures,respectively. To characterize the devices, a custom-built test setup [19] wasused. An Agilent 81600B tunable laser was used as the light source andAgilent 81635A optical power sensors were used as the output detectors. Awavelength sweep from 1500 nm to 1600 nm in 10 pm steps was performed.(a)(b) (c)Figure 5.8: SEM images of a fabricated SWG PSR with zoomed images ofthe coupling section, the SWG taper, and the S bend waveguide.The measured transmission spectra of a fabricated SWG PSR is shownin Fig. 5.9, where the ILs from the input and output grating couplers havebeen calibrated out. The measured ILTE-TE and PCETM-TE are denoted bythe blue and red curves shown in Fig. 5.9(a), respectively. The measuredSWG PSR had an ILTE-TE close to 0dB over most of the wavelength rangefrom 1500 nm to 1600 nm, and the ripples shown in the curve was causedby the non-uniformity of the grating couplers. The fabricated SWG PSRhad a peak PCETM-TE of −0.3 dB at 1538 nm and a 1-dB bandwidth over50 nm. The measured XTs of the SWG PSR are shown in Fig. 5.9(b) bythe red and blue curves, respectively. The XTTE-TE is below −15 dB over865.2. Fabrication and Measurementthe wavelength range from 1500 nm to 1600 nm, and the XTTM-TM is below−10 dB over the wavelength range from 1503 nm to 1588 nm.1500 1525 1550 1575 1600Wavelength (nm)-10-8-6-4-202Power (dB)TE-TE ThrouTM-TE Cross(a)1500 1525 1550 1575 1600Wavelength (nm)-50-40-30-20-100Power (dB)TM-TM ThrouTE-TE Cross(b)Figure 5.9: (a) Measured ILTE-TE and PCETM-TE as functions of λ, and (b)measured XTTE-TE and XTTM-TM as functions of λ.To summarize, in this chapter, we have experimentally demonstrated acompact SWG PSR for an SOI platform with a peak PCETM-TE of −0.3 dB,a 1-dB bandwidth over 50 nm. The designed SWG PSR had a compactsize of about 35 µm x 5 µm, which was achieved by using a space-efficientSWG taper with a length less than 10 µm. The fabrication tolerance of thedesigned SWG PSR to the waveguide width variations have been improvedand the fabrication tolerance to the dimension variations and wafer thicknessvariations have been studied.87Chapter 6Conclusion and Future Work6.1 ConclusionTo conclude, we will summarize the contributions of the works shown in thisdissertation from three aspects: the technical contributions, design method-ology contributions, and the theoretical contributions.6.1.1 Technical Contributions• First demonstration of compact, high-efficiency SWGCs using one-dimensional SWGs for both the TE0 mode and the TM0 modes withsuppressed back reflections. Our SWGCs for the TE and TM SWGCsshow respective measured coupling efficiencies of −4.1 dB and −3.7 dBwith 1-dB bandwidths of 30.6 nm (3-dB bandwidth of 52.3 nm) and47.5 nm (3-dB bandwidth of 81.5 nm), respectively. The back reflec-tions for the TE0 and TM0 modes have been significantly suppressedto −16.2 dB and −20.8 dB, respectively.• Demonstration of apodized focusing SWGCs with improved couplingefficiencies for both the TE and TM modes. A measured measuredcoupling efficiency of −3.2 dB with a 1-dB bandwidth of 36 nm hasbeen obtained for the TE SWGC and a measured coupling efficiencyof −3.3 dB with a 1-dB bandwidth of 37 nm has been obtained for the886.1. ConclusionTM SWGC. Back reflections for the TE SWGC and TM SWGC havebeen suppressed to −24 dB and -21 dB, respectively.• Demonstration of broadband sub-wavelength grating couplers with 1-dB bandwidths ranging from 50 nm to 90 nm. Our designed SWGCshave competitive coupling efficiency, as high as −3.8 dB for the TEmode, and state-of-the-art back reflections, as low as −23 dB. Thecomparisons of the SWGCs shown in this dissertation with the state-of-the-art SWGCs demonstrated by other research groups are listed inTable. 6.1.• First experimental demonstration of broadband directional couplersusing sub-wavelength gratings. The dispersion properties of the op-tical modes are engineered using sub-wavelength gratings, which al-lows broadband operation. Compact broadband direction couplers,with device lengths shorter than 14 µm, which cover a bandwidth of100 nm, for power splitting ratios of 50/50, 40/60, 30/70, and 20/80,are designed and fabricated for the TE0 mode with a central operatingwavelength of 1550 nm.• First experimental demonstration of polarization splitter-rotator (PSR)on a silicon-on-insulator platform based on an asymmetric directionalcoupler using sub-wavelength gratings. A measured peak TM-TE po-larization conversion efficiency of −0.3 dB was achieved with TE-TEcrosstalks below −15 dB and TM-TM crosstalks below −10 dB overC-band. The designed SWG PSR has a compact size of 35 µm x 5 µm.896.1. ConclusionTable 6.1: Comparison of SWGCs for the SOI platform. CE, coupling effi-ciency; BW, bandwidth.Ref. Year CE (dB) 3-dB BW (nm) Remarks[15] 2009 -4.7 40 DUV fabrication[46] 2010 -3.8 60[29] 2010 -4.2 60 TM pol.[29] 2010 -3.7 55 TM pol.; apodized[31] 2012 -5 55 TM pol.; DUV fabrication[73] 2012 -2.4 60 optimized BOX[94] 2012 -5.6 100 reduced index[16] 2012 -3.5 80 reduced index[17] 2012 -3 50 TM pol.; apodied; focusing[95] 2013 -5.1 115 reduced index and SWG pitch[22] 2013 -1.8 60 apodized; optimized BOX[84] 2014 -3.7 52 focusing; TE pol.[84] 2014 -4.1 82 focusing; TM pol.[85] 2014 -3.2 59 focusing; apodized; TE pol.[85] 2014 -3.3 72 focusing; apodized; TM pol.[6] 2014 -2.16 64 apodized[24] 2014 -0.58 71 apodized; mirror; 250nm Si[83] 2014 -5.5 >120 focusing; TE pol.[5] 2015 -0.69 60 apodized; mirror[24] 2014 -0.43 76 mirror; 250nm Si, TE pol. (simu.)[5] 2015 -0.67 NA mirror; TE pol. (simu.)[85] 2014 -0.34 85 focusing; mirror; TE pol. (simu.)[85] 2014 -0.41 85 focusing; TM pol.; mirror(simu.)6.1.2 Design Methodology ContributionsIn the processes of my research on various projects, including the ones shownin this dissertation, the goal was not only getting one or several hero devicesfor publications, but also summarizing the design steps that I have beenfollowed and refining them to solid design methodologies that can be sharedand used by other researchers in this field. The major contributions of thisdissertation from the methodology perspective include:906.1. Conclusion• Design methodology for high-efficiency sub-wavelength grating cou-plers (SWGCs) using one-dimensional sub-wavelength gratings as shownin Chapter 2. Compared to the commonly used simulation method inwhich the high and low index regions in an SWGC were treated ashomogeneous materials using EMT, our method have several advan-tages. Firstly, our method simulates the cross-section of the actualsub-wavelength structures, which include both the material and struc-ture dispersion from an SWGC for more accurate simulation results.Secondly, our design methodology was based on two-dimensional sim-ulations, which is very efficient for design parameter optimization. Incontrast, when using EMT method, SWGCs are first optimized withvirtual refractive indices using 2D simulations. Then, EMT is used tocalculate the dimensions of the SWG structures, i.e., either 2D SWGsor 1D SWGs, where time-consuming and computational intensive 3Dsimulations are required for 2D SWGs. In case where 1D SWGs areused, using EMT is redundant in the design process since the structurecan be directly optimized using 2D simulations, as shown in this dis-sertation. The validity of the design methodology has been verified byexperimental demonstration of high-efficiency uniform and apodizedSWGCs for both the TE0 and the TM0 modes. Finally, it is easierand more straightforward to set the thresholds, minimum and max-imum dimension values, for the SWGs using our method than usingEMT. Because the optimized virtual refractive indices may requirefeature sizes that are smaller than the minimum feature sizes of thefabrication.• Design methodology for broadband SWGCs. A four-step design pro-916.1. Conclusioncess has been demonstrated in Chapter 3 of this dissertation on how todesign broadband SWGCs with design intent 1-dB bandwidths rangingfrom 50 nm to 90 nm. Again, the design methodology is based on theactual sub-wavelength structure, which provides more accurate simu-lation results and also less computationally intensive. The validity ofthe design methodology has been verified by successful experimentaldemonstration of both straight and focusing SWGCs with measured1-dB bandwidths ranging from 50 nm to 90 nm.• Design methodology for broadband SWG DCs using FDTD band struc-ture calculations. Compared to the commonly used method, in whichthe SWGs are estimated by the effective medium theory (EMT), ourapproach has the following advantages. Firstly, FDTD-based bandstructure calculations simulate the actual structure and take the ma-terial dispersion, the structure dispersion, and the Bragg effect intoconsideration. The Bragg effect from the SWGs was used to engi-neer the dispersion properties of the optical modes within our SWGDCs, which was omitted by the EMT where SWGs are treated asuniform material with equivalent refractive indices. Secondly, bandstructure calculations only require simulations of one unit cell of thestructure, which significantly reduces simulation times as comparedto full-structure FDTD simulations, i.e., hours versus minutes, andmake efficient device optimization possible. In addition, this FDTDband structure based simulation method has also been extended todesign the polarization splitter-rotator (PSR) shown in Chapter 5 ofthis dissertation.926.2. Future Work6.1.3 Theoretical ContributionsThe major theoretical contribution of this dissertation is the study on theoperating bandwidth of a grating coupler. When I first started working onthe topic of broadband grating couplers, I found that the existing theoreticalstudy was not straightforward to be used as practical guidelines from thedesign perspective. In various research papers, authors normally show theanalytical equations for the operating bandwidth of a grating coupler andthen optimizing their designs using brute-force numerical simulations thatwere not related to the analytical equations shown in their papers. We de-rived the operating bandwidth of a grating coupler based on the wavelength-dependent diffraction angle and related the bandwidth of a grating couplerto only three design parameters: the refractive index of the cladding mate-rial, the incident angle, and the group index of the grating coupler. Due tothe fact that index of the cladding is a known constant for a specific fab-rication process, the bandwidth of a grating coupler is only related to theincident angle and the group index of the grating coupler. And for a givenindent angle, the bandwidth is only related to the group index of a gratingcoupler. The above conclusions made from our analytical model providetechnical guidelines on designing broadband grating couplers in general andalso provide the possibility to design grating couplers with design intentoperating bandwidths.6.2 Future WorkWith the demonstrated works in this dissertation, suggested future workincludes:936.2. Future Work• SWGCs with coupling efficiencies below -0.5dB may be achieved byadding bottom mirrors. The major loss of the existing grating cou-plers, including the high-efficiency sub-wavelength grating couplersshown in this dissertation, is the penetration loss into the substrate.The penetration loss can be highly reduced, i.e., more than 90%, byadding a bottom mirror. The bottom mirror can obtained either byusing SOI wafers with distributed Bragg reflector at the bottom of thewafer or by deposition of a metal layer at the interface of the buriedoxide and silicon substrate.• SWGCs with ultra-low back reflections can be obtained by replacingthe taper regions of the SWGCs shown in Chapter 2 of this disser-tation with sub-wavelength structures. Schematic of the cross-sectionof a sub-wavelength grating coupler with sub-wavelength structurescomprising both the coupling region and the taper region is shown inFig. 6.1. The comparatively large index contrast at the interface ofthe coupling region, i.e., consisting of sub-wavelength gratings, and thetaper region, i.e., consisting of silicon slab, leads to a Fresnel reflectionat the interface, which results in oscillation ripples in the transmis-sion spectrum. By designing a taper region with SWGs which havea similar effective index as the coupling region, the Fresnel reflectioncan be highly suppressed. With the SWG taper, the optical mode canbe coupled directly into an SWG waveguide. For bio-sensing appli-cation, where SWG waveguide and SWG ring resonators are used forenhanced sensitivity, using the proposed SWGCs can avoid extra cou-plers that are required to coupled light between the SWG waveguidesand the strip waveguides. Therefore, the system can be more space946.2. Future Workefficient and power efficient.Figure 6.1: Schematic of the cross-section of an SWGC with sub-wavelengthstructures comprising both the coupling region and the taper region.• A CMOS-compatible broadband sub-wavelength grating directionalcoupler can be made by modifying the period and fill factor of thedesign demonstrated in Chapter 4. The minimum feature size of thebroadband sub-wavelength grating directional coupler is about 80 nm,which is not compatible with the 193 nm optical lithography providedby the MPW foundries using CMOS fabrication process. The mini-mum feature size of the device can be increased by using a smallergrating period with a larger fill factor for the sub-wavelength gratingsused in the design, thus, the modified design can be fabricated throughMPW foundries using CMOS-compatible fabrication process.• A more compact polarization splitter-rotator (PSR) can be made byreplacing the strip waveguide in the PSR demonstrated in Chapter 5with a sub-wavelength grating waveguides. By using sub-wavelengthgrating waveguides, the effective index of the waveguide can be re-duced, therefore, the optical modes are less confined within the waveg-uide, which will lead to stronger coupling between the two waveguides.Such a design also have better fabrication tolerance to the fill factorchanges as compared to the design demonstrated in Chapter 5.956.2. 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Focusing-curved subwavelengthgrating couplers for ultra-broadband silicon photonics optical inter-faces. Optics Express, 22(15):18224–18231, 2014.112Appendix APublicationsA.1 Book Chapters1. Yun Wang and Lukas Chrostowki, Chapter 5.2, Grating Couplers, InSilicon Photonics Design, Lukas Chrostowski and Michael Hochberg,Cambridge Press, 2015. pp. 178-199.A.2 Journal PublicationsFirst Author1. Yun Wang, Minglei Ma, Han Yun, Zeqin Lu, Xu Wang, Nicolas A. F.Jaeger, and Lukas Chrostowski. Ultra-Compact Sub-wavelength Grat-ing Polarization Splitter-Rotator for Silicon-on-Insulator Platform Pho-tonics Journal, IEEE 8(6), 2016.2. Yun Wang, Zeqin Lu, Minglei Ma, Han Yun, Fan Zhang, Nicolas A.F. Jaeger, and Lukas Chrostowski. Compact Broadband DirectionalCouplers Using Sub-wavelength Gratings Photonics Journal, IEEE8(3), 2016.3. Yun Wang, Wei Shi, Xu Wang, Zeqin Lu, Michael Caverley, RichardBojko, Lukas Chrostowski, Nicolas A. F. Jaeger. Design of Broadband113A.2. Journal PublicationsSub-wavelength Grating Couples with Low Back Reflection. OpticsLetters 40(20): 4647-4650, 20154. Yun Wang, Han Yun, Zeqin Lu, Richard Bojko, Wei Shi, Xu Wang,Jonas Flueckiger, Fan Zhang, Michael Caverley, Nicolas A. F. Jaeger,and Lukas Chrostowski. Apodized focusing fully etched sub-wavelengthgrating couplers. Photonics Journal, IEEE, 7(3), 20155. Yun Wang, Xu Wang, Jonas Flueckiger, Han Yun, Wei Shi, RichardBojko, Nicolas A. F. Jaeger, and Lukas Chrostowski. Focusing sub-wavelength grating couplers with low back reflections for rapid pro-totyping of silicon photonic circuits. Optics Express, 22(17): 20652–20662, 2014.Co-author6. Han Yun, Zhitian Chen, Yun Wang, Jonas Flueckiger, Michael Caver-ley, Lukas Chrostowski, and Nicolas A. F. Jaeger. Polarization-rotating, Bragg-grating filters on silicon-on-insulator strip waveguidesusing asymmetric periodic corner corrugations Optics Letters, 40(23):5578–5581, 2015.7. Han Yun, Yun Wang, Fan Zhang, Zeqin Lu, Stephen Lin, LukasChrostowski, and Nicolas A. F. Jaeger. Broadband 2x2 Adiabatic 3-dB Coupler using Silicon-on-Insulator Sub-wavelength Grating Waveg-uides Optics Letters. 41(13): 3041–3044, 2016.8. Zeqin Lu, Han Yun, Yun Wang, Zhitian Chen, Fan Zhang, Nico-las A. F. Jaeger, and Lukas Chrostowski. Broadband silicon photonic114A.2. Journal Publicationsdirectional coupler using asymmetric-waveguide based phase control.Optics Express, 23(3):3795–3808, 2015.9. Zeqin Lu, Yun Wang, Fan Zhang, Nicolas A. F. Jaeger, and LukasChrostowski. Wideband silicon photonic polarization beamsplitterbased on point-symmetric cascaded broadband couplers. Optics Ex-press, 23(23):29413–29422, 2015.10. Ahmadreza Farsaei, Yun Wang, Reza Molavi, Mohammad Beikah-madi, Amir Hossein Masnadi Shirazi, Michael Caverley, Hasitha Jay-atilleka, Lukas Chrostowski, Shahriar Mirabbasi, A Review of WirelessPhotonic Systems: Design Methodologies and Topologies, Constraints,Challenges, and Innovations in Electronics and Photonics. Journal ofOptics Communication, 373: 16-34, 2015 (Invited).11. Zhitian Chen, Jonas Flueckiger, Xu Wang, Fan Zhang, Han Yun, ZeqinLu, Michael Caverley, Yun Wang, Nicolas A. F. Jaeger, and LukasChrostowski. Spiral Bragg grating waveguides for TM mode siliconphotonics. Optics Express, 23: 25295–25307, 201512. Raphael Dube-Demers, Jonathan St-Yves, Antoine Bois, Qiuhang Zhong,Michael Caverley, Yun Wang, Lukas Chrostowski, Sophie LaRochelle,David Plant, Wei Shi. Analytical Modeling of Silicon Microring andMicrodisk Modulators with Electrical and Optical Dynamics. Journalof Lightwave Technology, IEEE, 33: 4240-4252,201513. Qiuhang Zhong, Venkat Veerasubramanian,Yun Wang, Wei Shi, DavidPatel, Samir Ghosh, Alireza Samani, Lukas Chrostowski, Richard Bo-jko, and David V Plant. Focusing-curved subwavelength grating cou-115A.3. Conference Proceedingsplers for ultra-broadband silicon photonics optical interfaces. OpticsExpress, 22(15):18224–18231, 2014.14. Xu Wang, Yun Wang, Jonas Flueckiger, Richard Bojko, Amy Liu,Adam Reid, James Pond, Nicolas A. F. Jaeger, and Lukas Chros-towski. Precise control of the coupling coefficient through destruc-tive interference in silicon waveguide bragg gratings. Optics Letters,39(19):5519–5522, 2014.A.3 Conference Proceedings1. Yun Wang, Han Yun, Zeqin Lu, Nicolas A.F. Jaeger, and LukasChrostowski, State-of-the-art Sub-wavelength Grating Couplers forSilicon-on-insulator Platform In IEEE, Canadian Conference on Elec-trical and Computer Engineering, (Invited) 2016.2. Yun Wang, Han Yun, Zeqin Lu, Nicolas A.F. Jaeger, and LukasChrostowski, Sub-wavelength Grating Components for the Silicon-on-insulator Platform In International Conference on Metamaterials,Photonic Crystals and Plasmonics, (Invited) 2016.3. Yun Wang, Han Yun, Nicolas A.F. Jaeger, and Lukas Chrostowski,Broadband Bidirectional Vertical Grating Coupler In Proc. OpticalFiber Communications (OFC) Conference, 2016.4. Zeqin Lu, Minglei Ma, Han Yun, Yun Wang, Nicolas A. F. Jaeger,and Lukas Chrostowski. Silicon Photonic Polarization Beamsplitterand Splitter-rotator for On-chip Polarization Control. In IEEE Inter-national Conference on Group IV Photonics (GFP), 2016 (Invited).116A.3. Conference Proceedings5. Minglei Ma, Kyle Murray, Mengyuan Ye, Stephen Lin, Zeqin Lu,Han Yun, Yun Wang, Richy Hu, Nicolas A.F. Jaeger, and LukasChrostowski. Sub-wavelength Grating Components for the Silicon-on-insulator Platform In CLEO: Science and Innovations, pages SM1I–6.Optical Society of America, 2016.6. Yun Wang, Han Yun, Zeqin Lu, Richard Bojko, Fan Zhang, MichaelCaverley, Nicolas A. F. Jaeger, and Lukas Chrostowski. Apodizedfocusing fully etched sub-wavelength grating couplers with ultra-lowreflections. In CLEO: Science and Innovations, pages SM1I–6. Opti-cal Society of America, 2015.7. Yun Wang, Stevan S Djordjecvic, Jin Yao, John E Cunningham,Xuezhe Zheng, Ashok V Krishnamoorthy, Michael Muller, Markus-Christian Amann, Richard Bojko, Nicolas A. F. Jaeger and LukasChrostowski. Vertical-cavity surface-emitting laser flip-chip bondingto silicon photonics chip. In Optical Interconnects Conference (OI),2015 IEEE, pages 122–123. IEEE, 2015.8. Han Yun, Jonas Flueckiger, Zhitian Chen, Yun Wang, Lukas Chros-towski, and Nicolas A. Jaeger. A Wavelength-Selective PolarizationRotating Reflector using a Partially-Etched Asymmetric Bragg Grat-ing on an SOI Strip Waveguide. In IEEE International Conference onGroup IV Photonics (GFP), 20159. Han Yun, Zeqin Lu, Yun Wang, Wei Shi, Lukas Chrostowski, andNicolas A. F. Jaeger. 2x2 Broadband Adiabatic 3-dB Couplers on SOIStrip Waveguides for TE and TM modes. In International conferenceson Laser and Electro-Optics (CLEO), 2015.117A.3. Conference Proceedings10. Zeqin Lu, Han Yun, Yun Wang, Zhitian Chen, Fan Zhang, NicolasA. F. Jaeger, and Lukas Chrostowski. Asymmetric-waveguide-assisted3-dB Broadband Directional Coupler. In International conferenceson Laser and Electro-Optics (CLEO), 2015.11. Zhitian Chen, Jonas Flueckiger, Xu Wang, Han Yun, Yun Wang,Zeqin Lu, Fan Zhang, Nicolas A. F. Jaeger, and Lukas Chrostowski.Bragg Grating Spiral Strip Waveguide Filters for TM Modes. InInternational conferences on Laser and Electro-Optics (CLEO), 201512. Fan Zhang, Han Yun, Valentina Donzella, Zeqin Lu, Yun Wang, Zhi-tian Chen, Lukas Chrostowski, and Nicolas A. F. Jaeger. SinusoidalAnti-coupling SOI Strip Waveguides. In International conferences onLaser and Electro-Optics (CLEO), 201513. Yun Wang, Jonas Flueckiger, Han Yun, Richard Bojko, Nicolas A.F. Jaeger, Lukas Chrostowski, et al. Focusing sub-wavelength gratingcoupler. In Photonics Conference (IPC), 2014 IEEE, pages 552–553.IEEE, 2014.14. J Niklas Caspers, Yun Wang, Lukas Chrostowski, and MohammadMojahedi. Active polarization independent coupling to silicon pho-tonics circuit. In SPIE Photonics Europe, pages 91330G–91330G.International Society for Optics and Photonics, 2014.15. Qiuhang Zhong, Wei Shi, Yun Wang, Lukas Chrostowski, and David VPlant. An ultra-broadband fiber grating coupler with focusing curvedsubwavelength structures. In Proc. Optical Fiber Communications(OFC) Conference, 2014.118A.3. Conference Proceedings16. Raphal Dub-Demers, Jonathan St-Yves, Antoine Bois, Qiuhang Zhong,Michael Caverley, Yun Wang, Lukas Chrostowski, Sophie LaRochelle,David Plant, and Wei Shi, Analytical Modeling for Ultra-High-SpeedMicroring Modulators with Electrical and Optical Dynamics in Euro-pean Conference on Optical Communication (ECOC), 201417. Lukas Chrostowski, Xu Wang, Jonas Flueckiger, Yichen Wu, YunWang, Sahba Talebi Fard, Impact of fabrication non-uniformity onchip-scale silicon photonic integrated circuits. In Proc. Optical FiberCommunications (OFC) Conference, 2014.18. Michael Caverley, Hasitha Jayatilleka, Yun Wang, Nicolas. A. F.Jaeger, and Lukas Chrostowski, Microring Modulator Using Drop-Port Phase Interference. In Photonics Conference (IPC), 2014 IEEE,pages 613–614. IEEE, 2014.19. Xu Wang, Michael Caverley, Jonas Flueckiger, Yun Wang, NicolasA. F. Jaeger, and Lukas Chrostowski, Silicon Photonic Bragg GratingModulators. In Photonics Conference (IPC), 2014 IEEE, pages 35.IEEE, 2014.119

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