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Spiral Bragg gratings for TM mode silicon photonics Chen , Zhitian 2015

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Spiral Bragg Gratings for TM ModeSilicon PhotonicsbyZhitian ChenB.A.Sc., The University of British Columbia, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September 2015c© Zhitian Chen 2015AbstractIn this thesis, we demonstrate long transverse magnetic (TM) Bragg grat-ings wrapped compactly using spiral waveguides on the silicon-on-insulator(SOI) platform. We developed three types of TM spiral Bragg grating waveg-uides (SBGWs) including uniform spiral Bragg grating (U-SBGWs), phase-shifted spiral Bragg grating (P-SBGWs), and chirped spiral Bragg gratings(C-SBGWs). Our spiral waveguides are space-efficient, requiring an area ofonly 189×189 µm2 to accommodate 1 cm long Bragg grating waveguidesand, thus, are less susceptible to fabrication non-uniformities. Due to thesefactors, the TM U-SBGWs are able to successfully obtain narrow band-widths and high extinction ratios (ERs), as narrow as 0.09 nm and as largeas 52 dB respectively. Also, the TM P-SBGWs can obtain sharp resonancepeaks with high quality factors of 78790. Finally, we demonstrate the TMC-SBGWs, which exhibit group delays that are linear functions of the wave-lengths over their passbands.Traditionally, due to the large coupling coefficients and the flexibility forachieving desired spectral characteristics, short Bragg grating waveguides fortransverse electric (TE) modes on the SOI platform have been developed forapplications in optical communication and sensing systems. In contrast, TMmodes Bragg gratings on SOI platform have small coupling coefficients and,therefore, the grating lengths need to be much longer than TE mode devices,in order to obtain large ERs. However, such TM mode Bragg gratings canachieve very narrow bandwidths. Creating long gratings in regular straightwaveguides suffer from the fabrication non-uniformity effects caused by thewafer thickness. As is shown here, spiral-shaped waveguides can be used toincrease the grating length, while still being made using little on chip realestate, thus reducing the effects of fabrication non-uniformity.iiPrefaceThis thesis is based mostly on the conference publication entitled, “BraggGrating Spiral Waveguide Filters for TM modes”. I am the main author ofboth publications. I proposed to investigate the long Bragg gratings for TMmodes. Prof. Lukas Chrostowski suggested me to design the long gratingsusing spiral waveguides. I then modeled and simulated the device, usingnumerical methods. I also created scripts in Matlab and Pyxis that gen-erated mask layouts for various spiral Bragg grating designs. Our deviceswere fabricated using eBeam lithography at the University of Washington.Also for fabrication, ePIXfab technology was used by IMEC in Leuven, Bel-gium. The devices were measured using an automated measurement system,developed by Jonas Flueckiger and Charlie Lin. Prof. Nicolas Jaeger, mysupervisor, and Prof. Lukas Chrostowski supervised and guided the project.I have also been involved in other research projects during my Master’sstudies. I participated in designs and measurements of the projects includ-ing broadband silicon photonic directional couplers and anti-coupling SOIstrip waveguides. I am one of the co-authors in the journal and conferencepublications related to these projects.The list of publications as main author includes:1. Zhitian Chen, Jonas Flueckiger, Xu Wang, Han Yun, Yun Wang, ZeqinLu, Fan Zhang, Nicolas A. F. Jaeger, and Lukas Chrostowski, “BraggGrating Spiral Strip Waveguide Filters for TM Modes,” in CLEO 2015,San Jose, CA, May 2015, pp. SM3I-7.The list of publications as co-authors include:1. Zeqin Lu, Han Yun, Yun Wang, Zhitian Chen, Fan Zhang, Nicolas A.F. Jaeger, and Lukas Chrostowski, “Broadband silicon photonic direc-iiiPrefacetional coupler using asymmetric-waveguide based phase control,” OpticsExpress 23, no. 3 (2015): 3795-3808.2. Zeqin Lu, Han Yun, Yun Wang, Zhitian Chen, Fan Zhang, Nicolas A. F.Jaeger and Lukas Chrostowski, “2×2 Asymmetric-waveguide-assisted 3-dB Broadband Directional Coupler,” in CLEO 2015, San Jose, CA, May2015, pp. SM1I-8.3. Fan Zhang, Han Yun, Valentina Donzella, Zeqin Lu, Yun Wang, ZhitianChen, Lukas Chrostowski, and Nicolas A. F. Jaeger, “Sinusoidal Anti-coupling SOI Strip Waveguides,” in CLEO 2015, San Jose, CA, May2015, pp. SM1I-7.4. Han Yun, Jonas Flueckiger, Zhitian Chen, Yun Wang, Nicolas A. F.Jaeger, and Lukas Chrostowski, “A wavelength-selective polarization ro-tating reflector using a partially-etched asymmetric Bragg grating on anSOI strip waveguide,” in IEEE International Conference on Group IVPhotonics (GFP), FA3 (2015).ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Silicon Photonics . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation and Objectives . . . . . . . . . . . . . . . . . . . 31.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . 72 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1 Mode Analysis for Strip Waveguides . . . . . . . . . . . . . . 82.2 Silicon Bragg Grating Waveguides . . . . . . . . . . . . . . . 122.3 Transfer Matrix Analysis . . . . . . . . . . . . . . . . . . . . 153 Design and Simulations . . . . . . . . . . . . . . . . . . . . . . 213.1 Spiral Grating Waveguide Design . . . . . . . . . . . . . . . 213.1.1 Spiral Grating Schematic . . . . . . . . . . . . . . . . 213.1.2 Spiral Waveguide Design . . . . . . . . . . . . . . . . 24vTable of Contents3.2 FDTD Simulations . . . . . . . . . . . . . . . . . . . . . . . 273.3 Grating Design and Simulations . . . . . . . . . . . . . . . . 283.3.1 Uniform Spiral Bragg Gratings . . . . . . . . . . . . . 293.3.2 Phase-shifted Spiral Bragg Gratings . . . . . . . . . . 323.3.2.1 Basics . . . . . . . . . . . . . . . . . . . . . 323.3.2.2 Simulations . . . . . . . . . . . . . . . . . . 333.3.2.3 Quality Factor . . . . . . . . . . . . . . . . . 393.3.3 Chirped Spiral Bragg Gratings . . . . . . . . . . . . . 404 Experimental Results and Discussion . . . . . . . . . . . . . 464.1 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . 484.2.1 Measurements . . . . . . . . . . . . . . . . . . . . . . 484.2.2 Waveguide Propagation Loss . . . . . . . . . . . . . . 504.2.3 Uniform Spiral Bragg Gratings . . . . . . . . . . . . . 544.2.3.1 U-SBGWs fabricated by eBeam lithography 544.2.3.2 U-SBGW fabricated by 193 nm DUV lithog-raphy . . . . . . . . . . . . . . . . . . . . . . 584.2.3.3 Thermal Sensitivity . . . . . . . . . . . . . . 604.2.3.4 Discussion . . . . . . . . . . . . . . . . . . . 614.2.4 Phase-shifted Spiral Bragg Gratings . . . . . . . . . . 624.2.4.1 Design Variations . . . . . . . . . . . . . . . 624.2.5 Chirped Spiral Bragg Gratings . . . . . . . . . . . . . 674.2.5.1 Design Variations . . . . . . . . . . . . . . . 695 Summary, Conclusion, and Future work . . . . . . . . . . . 725.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3 Suggestions for Future work . . . . . . . . . . . . . . . . . . 74Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75viList of Tables3.1 Comparison of packing efficiencies of curved Bragg gratingSOI waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1 Comparison of propagation losses between TE and TM spi-ral waveguides fabricated using both eBeam lithography and193 nm DUV lithography . . . . . . . . . . . . . . . . . . . . 514.2 Comparison between TM U-SBGW and straight, uniform TESOI Bragg grating . . . . . . . . . . . . . . . . . . . . . . . . 59viiList of Figures1.1 Cross section view of silicon-on-insulator (SOI) wafer. . . . . 21.2 Electric field distributions of (a) the fundamental TE modeand (b) the fundamental TM mode in a 500 nm×220 nm stripwaveguide with SiO2 cladding. . . . . . . . . . . . . . . . . . 52.1 Cross section view of a 500 nm×220 nm SOI strip waveguide. 82.2 Electric field distributions of (a) straight waveguide and curvedwaveguides with radii of curvatures of (b) 15 µm, (c)10 µm,and (d) 5 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Waveguide effective refractive index varies with the wave-length for various radii of curvatures, as compared with thatof a straight waveguide. . . . . . . . . . . . . . . . . . . . . . 112.4 Waveguide effective refractive index varies with the wave-length for various waveguide widths. . . . . . . . . . . . . . . 112.5 Illustration of a uniform Bragg grating. neff1 and neff2 arethe effective indices of the low and high index sections, re-spectively. neff is the average effective index of the gratingwaveguide, which is equal to (neff1+neff2)/2. Λ is the grat-ing period. R and T are the grating reflection and trans-mission. The 180◦ arrows indicate the numerous reflectionsthroughout the grating. . . . . . . . . . . . . . . . . . . . . . 122.6 Typical spectral response of a uniform Bragg grating includ-ing reflection and transmission spectra. . . . . . . . . . . . . . 132.7 Illustration of dielectric media with refractive indices of neff1and neff2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15viiiList of Figures2.8 A multi-layer dielectric film showing the transmission andpropagation matrices. . . . . . . . . . . . . . . . . . . . . . . 173.1 (a) SEM image and (b) schematic of a 4 mm long TM SBGWwrapped into an area of 131×131 µm2. (c) Zoom-in showingthe grating period, Λ, the corrugation width, Wcorr=(Wmax-Wmin)/2, and the spacing between two spiral waveguides, g. . 223.2 Simulated effective refractive index variations (blue curve)and calculated packing efficiencies (green curve) for spiralwaveguides with R0s ranging from 5 µm to 60 µm. . . . . . . 253.3 Comparison of simulated spectra between straight BGW andSBGW. The simulated spectrum is not greatly affected by thesmall δneff , causing a bandwidth increase of 0.02 nm. . . . . 263.4 Magnitudes of the time domain signals of three devices withcorrugation widths of 60 nm, 100 nm, and 140 nm, obtainedusing FDTD simulations with Bloch boundary conditions.The wavelength range between the two resonant peaks indi-cates the bandwidth, ∆λ, of the device, and the Bragg wave-length, λ0, is located at the center between the two resonantpeaks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.5 (a) Schematic of a U-SBGW. (b) Enlarged plot shows theperiod, Λ, with a duty cycle of 50%. . . . . . . . . . . . . . . 293.6 (a) Simulated reflection and transmission spectra for a TMU-SBGW in (a) linear scale and (b) dB scale. The device hasparameters: Λ=444 nm, N=400, and Wcorr=20 nm. . . . . . 303.7 Simulated transmission spectra for the TM U-SBGWs withdifferent period. They have the same Wcorr=20 nm and N=400. 313.8 Simulated transmission spectra for the TM U-SBGWs withdifferent corrugation widths. They have the same Λ=444 nmand N=400. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.9 (a) Schematic of a TM P-SBGW. (b) Enlarged plot shows acavity with one period length at the center area of the grating. 32ixList of Figures3.10 Simulated (a) reflection and (b) transmission spectra of aTM P-SBGW. The device has Λ=438 nm, Wcorr=80 nm, andN=100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.11 Tmax as a function of R for different values of G. . . . . . . . 363.12 Transmission spectra for many Wcorr values of (a) 80 nm, (b)120 nm, and (c) 160 nm. The devices have the same N=100,and Λ=444 nm, and αL=3 dB/cm. . . . . . . . . . . . . . . . 373.13 Transmission spectra for different N of (a) 100, (b) 125, and(c) 150. The devices have the same Wcorr=80 nm, Λ=444 nm,and αL=3 dB/cm. . . . . . . . . . . . . . . . . . . . . . . . . 383.14 Illustration of a TM C-SBGW, where Λ changes along thelength of the grating. . . . . . . . . . . . . . . . . . . . . . . 403.15 Simulated reflection spectra (blue curves) and group delay(green curves) of the TM C-SBGWs with (a) dΛ/dL=6 nm/cmand (b) dΛ/dL =-6 nm/cm. They have the same parameters:L= 4 mm, Wcorr=5 nm, initial Λ= 444 nm, and αL=2 dB/cm. 433.16 Gaussian apodization profile for a 4 mm long grating. . . . . 443.17 Simulated group delays for unapodized C-SBGW (green curve)and apodized C-SBGW (red curve). . . . . . . . . . . . . . . 443.18 (a) Simulated reflection spectra of the TM C-SBGWs withlengths of 2 mm, 3 mm, and 4 mm. The devices have the fixedparameters: αL=2 dB/cm and dΛ/dL =-6 nm/cm. (b) Simu-lated reflection spectra of the TM C-SBGWs with chirp ratesof -4 nm/cm, -5 nm/cm, and -6 nm/cm. The devices havethe fixed parameters: αL=2 dB/cm, initial Λ= 444 nm, andL=4 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1 Mask layout of an U-SBGW with input and output GCs, anda 3-dB Y-branch. . . . . . . . . . . . . . . . . . . . . . . . . . 47xList of Figures4.2 SEM images of an U-SBGW. (a) The complete U-SBGW withinput and output GCs. (b) Zoom-in of the center of the S-shaped grating waveguide. (c) Zoom-in of the Archimedeanspiral grating waveguides. Device is fabricated using eBeamlithography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 SEM image shows the spiral grating waveguides of a SBGWfabricated using 193 nm DUV lithography. . . . . . . . . . . . 484.4 Full view of the automated measurement system . . . . . . . 494.5 (a) Measured transmission spectra for the TM mode andTE mode spiral waveguides fabricated by eBeam lithography.The waveguide lengths vary from 1 cm to 3 cm. (b) Averagetransmission losses for TM spiral waveguides are comparedto the TE spiral waveguides’ losses. . . . . . . . . . . . . . . . 524.6 (a) Measured transmission spectra for the TM mode and TEmode spiral waveguides fabricated by 193 nm DUV lithog-raphy. The waveguide lengths vary from 0.5 mm to 10 mm.(b) Average transmission losses for TM spiral waveguides arecompared to the TE spiral waveguides’ losses. . . . . . . . . . 534.7 (a) Measured transmission spectra for the TM U-SBGWs,with grating periods of 432 nm, 438 nm, and 444 nm. (b)The measured and simulated transmission spectra, where theaverage neff of the waveguide was adjusted by 0.84×10−3 inthe simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 554.8 Measured transmission spectra of TM U-SBGWs with variouscorrugation widths. The grating period of each TM U-SBGWis 432 nm. The grating lengths of the TM U-SBGWs withWcorr > 40 nm were 1 mm. The TM U-SBGW with Wcorr =40 nm had a grating length of 3 mm. . . . . . . . . . . . . . . 56xiList of Figures4.9 Simulation and experimental results for the Bragg gratings.(a) The Bragg gratings centre wavelength shifts to shorterwavelengths as the corrugation width increases. The verticaloffset between simulation and experimental results is due toa difference between the fabrication and the target waveg-uide design. (b) The grating strength (coupling coefficient)increases as a function of the corrugation width. . . . . . . . 574.10 Measured reflection spectrum of a TM U-SBGW with a cor-rugation width of 20 nm, a length of 4 mm, and a period of444 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.11 Measured transmission spectrum of the TM U-SBGW fabri-cated using 193 nm DUV lithography. The device has param-eters: Wcorr= 20 nm, L= 1 cm, and Λ=440 nm. . . . . . . . . 594.12 Measured transmission spectra of a TM U-SBGW at varioustemperatures. The device has a length of 1 mm, a gratingperiod of 438 nm, and a corrugation width of 80 nm. . . . . . 604.13 SEM image of the center of a TM P-SBGW. . . . . . . . . . . 634.14 Transmission and reflection spectra of a TM P-SBGW, whichhas parameters: Wcorr= 100 nm, L= 250 µm, and Λ=444 nm. 634.15 Measured transmission spectra of TM P-SBGW with variouscorrugations. The devices have the same parameters: Λ=444nm, Wcorr=100 nm. . . . . . . . . . . . . . . . . . . . . . . . 644.16 Measured transmission spectra of TM P-SBGW with variouslengths. The devices have the same parameters: Λ=444 nm,L=400 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.17 Measured Q factor versus (a) grating length and (b) corruga-tion width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.18 Measured maximum Q-factor. The device has parameters:Λ=432 nm, L=900 µm, and Wcorr=80 nm. . . . . . . . . . . 674.19 Reflection spectrum of the TM C-SBGW. The group delaydecreases linearly as the wavelength increases within the pass-band. The device has parameters: Wcorr=60 nm, L=4 mm,and dΛ/dL =-6 nm/cm. . . . . . . . . . . . . . . . . . . . . . 68xiiList of Figures4.20 Measured reflection spectrum of the TM C-SBGW from Fig. 4.19is compared with the simulated reflection spectrum. The av-erage neff of the waveguide was adjusted by 0.98×10−3 inthe simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 684.21 Measured reflection spectra of the TM C-SBGWs with lengthsof 2 mm, 3 mm, and 4 mm. Devices have fixed parameters:dΛ/dL =-6 nm/cm, Wcorr = 60 nm. . . . . . . . . . . . . . . 704.22 Measured reflection spectra of the TM C-SBGWs with chirprates of -4 nm/cm, -5 nm/cm, and -6 nm/cm. Devices havefixed parameters: L=4 mm, Wcorr = 60 nm. . . . . . . . . . . 70xiiiAcknowledgementsI would like to thank my parents for their continuous support, encourage-ment, and their love throughout the years of my education.I would like to express my sincere gratitude and thanks to my supervisor,Dr. Nicolas A. F. Jaeger, for all the support and guidance he gave meduring my research. I would also like to thank Dr. Lukas Chrostowski, whoprovided insight and expertise that greatly assisted the research.My thanks goes out to those students (including former students) inour research group who helped me during my research, especially DoctorsWei Shi, Xu Wang, and Sahba T. Fard, as well as Jonas Flueckiger, YunHan, Fan Zhang, Yun Wang, Zeqin Lu, Minglei Ma, Charlie Lin, Miguel A.Guillen-Torres, Michael Caverley, Robert Boeck.I acknowledge the Natural Sciences and Engineering Research Council ofCanada and CMC Microsystems for their financial support. I would like tothank Richard Bojkos assistance with the fabrication (from the WashingtonNanofabrication Facility (WNF) which is part of the University of Washing-ton). I also would like to acknowledge Lumerical Solutions, Inc. and MentorGraphics, Corp. for the design software.xivChapter 1Introduction1.1 Silicon PhotonicsSilicon photonics is the name used to describe the research, design, andmanufacture of optical devices on a silicon platform. Such optical devices,with sub-micrometer size, can be integrated with advanced microelectronicson a single chip. These electronic-photonic circuits have the potential ofbeing used for high-speed communications for mobile devices, optical com-munications within computers and within data centres, sensor systems, andmedical applications [1].Over the last decade, the field of silicon photonics has been developingquickly. One critical reason is that low cost, high quality silicon wafersallow complex photonic circuits to be made inexpensively. Also, exist-ing CMOS (Complementary Metal-Oxide-Semiconductor) fabrication tech-niques are compatible with silicon photonics. As a result, optical devicesand integrated optical circuits can be made at low-cost and using reliablemanufacturing.In recent years, many applications of silicon photonics have been demon-strated. For example, Intel, the world’s largest semiconductor chip manufac-turer, has demonstrated the world’s first silicon-based optical data connec-tion at 50 Gbit/s using high-speed optical modulators and photo-detectors [2].Also, Luxtera, one of the world leaders in silicon photonics, has releasednew monolithic opto-electronic devices manufactured at a low cost, using aCMOS process [3]. In addition, various silicon waveguide components, in-cluding optical filers [4, 5], optical switches [6, 7], and optical sensors [8, 9],have been successfully demonstrated. All of these achievements confirmthat silicon photonics will play a significant role in optical communication11.1. Silicon Photonicssystems and optical sensor systems.The basic component of the integrated circuit, building block of siliconphotonic circuits is the waveguide. Since the silicon has a band-gap of1.12 eV, the silicon becomes transparent over the wavelengths longer than1100 nm. This makes them suitable for operation in telecommunicationwindows around 1310 nm to 1550 nm. To make these waveguides, specialsilicon-on-insulator (SOI) wafers are used. A typical SOI wafer consists of220 nm crystalline silicon on top of 2 µm or 3 µm silicon dioxide supportedby a 725 µm thick silicon substrate, see Fig. 1.1. These dimensions used inthese work.The high refractive index contrast between silicon (n=3.48) and sili-con dioxide (n=1.44) provides strong optical confinement, and allows sub-micrometer scale waveguide geometries. For instance, a 500 nm×220 nmstrip waveguide in SOI is often used to guide the fundamental TE and TMmodes.Waveguides fabricated in the SOI platform can be thermally tuned dueto the temperature-dependents of the band-gap [1, 10]. A change of the tem-perature will shift the spectral responses of the silicon photonic devices, in-cluding Bragg gratings, ring resonators, MachZehnder interferometers (MZI)and so on. Crystalline Silicon, 220 nm   Silicon Dioxide, 2 μm or 3 μm   Silicon Substrate, 725 μm  Figure 1.1: Cross section view of silicon-on-insulator (SOI) wafer.21.2. Motivation and Objectives1.2 Motivation and Objectives1.2.1 MotivationThere is an increasing need to develop components for processing transversemagnetic (TM) modes in waveguides fabricated on a SOI platform. Numer-ous components can be made more compactly or to give better performancewhen designed for TM mode operation. For example, directional couplerscan be made smaller due to the larger coupling coefficients achieved for TMmodes [11]. Optical sensors utilizing the evanescent fields [12–15] of TMmodes can have higher sensitivities, as compared to sensors using transverseelectric (TE) modes. Given the increased packing efficiency and robustnesspossible using edge coupling to SOI chips, there is current interest in sepa-rating, and separately processing, the TE and TM modes that are coupledfrom standard fibers onto SOI chips using edge couplers [16]. To this end,existing TE and TM polarizers [17–19], polarization rotators (PRs) [20–23], polarization splitters (PSs) [24, 25], and polarization splitter-rotators(PSRs) [26–28] allow us to efficiently integrate TM devices with TE devices.For example, the low loss PR in Ref. [20] is able to convert the fundamentalTE mode to the fundamental TM mode with an insertion loss below 0.5 dB,and converts the fundamental TM mode to the fundamental TE mode withan insertion loss below 1 dB, over a 200 nm wavelength range that includesthe C and L bands. The relatively low insertion loss and large wavelengthrange allows TM devices to be efficiently integrated with TE devices.In this work, we have found several advantages to using TM modes, ascompared to TE modes. The TM modes show smoother spectral responsesand lower propagation losses than TE modes in the same waveguides. Also,the coupling coefficients in Bragg gratings can be smaller for TM modesthan for TE modes. As a result, uniform Bragg gratings can benefit byusing TM modes to obtain high quality spectral responses, such as narrowbandwidths and large extinction ratios (ERs). Waveguides using TM modesare also suited for chirped Bragg gratings, since these gratings require smallcoupling coefficients and low propagation losses. Such TM chirped Bragggratings can be designed to achieve linear group delays with negative slopes31.2. Motivation and Objectivesover broad passbands. Additionally, integrated microwave photonics signalprocessing requires high quality, low propagation loss, and low couplingcoefficient gratings [29] and, thus, can benefit by using TM modes.Bragg grating waveguides (BGWs) on the SOI platform have been at-tracting interest for optical communication and sensing applications. Untilnow, most of the work has focused on BGWs designed for TE mode op-eration [30–32]. Typically, such devices have relatively large grating cou-pling coefficients, which means that they can be relatively short and haverelatively large bandwidths. However, small coupling coefficients and longgrating lengths are required for such devices to achieve narrower bandwidths(< 1 nm) and large ERs. There are several approaches for TE BGWs toachieve small coupling coefficients. For instance, one approach is to use smallsidewall corrugations on the strip waveguide, i.e., below 10 nm [31]. Suchsmall corrugations are challenging to manufacture as they are limited bythe fabrication lithography. Another approach is to use rib waveguides withlarger corrugations on the rib and/or the slab sidewalls. However, using ribwaveguides requires two etch steps [33, 34], which increases the fabricationcost. In contrast, transverse magnetic (TM) mode Bragg gratings allow forsmall coupling coefficients to be achieved for relatively large corrugations onstrip waveguides due to the fact that the TM mode is less optically confinedto the waveguide. Figures. 1.2(a) and 1.2(b) illustrate the confinement ofthe fundamental TE and TM modes of a 500 nm×220 nm strip waveguidewith SiO2 cladding, respectively. As mentioned above, such TM BGWswith small coupling coefficients require long grating lengths, on the order ofmillimeters, in order to achieve large ERs in their spectral responses. How-ever, the main challenge in making long TM gratings on a straight siliconwaveguide is the non-uniformity in the waveguide thickness [35, 36], whichadversely affects the effective refractive indices over the length of the device.To reduce the effects of the waveguide thickness variations, we have madeour long TM gratings into spirals.Spiral configurations have been shown to be very efficient for packinglong waveguides in compact areas [37–41]. By wrapping a long grating intoa spiral, the straight-line distance between any two points on the grating is41.2. Motivation and Objectivesminimized which, in turn, minimizes the waveguide thickness variation inthe grating. Hence, the adverse effects caused by waveguide thickness varia-tions can be significantly reduced using spirals. TE spiral BGWs (SBGWs)have been demonstrated on rib waveguides [39] and relatively wide (largerthan 1000 nm) strip waveguides [40] that reduce the sensitivity to side-wall roughness as compared to a TE mode in a regular 500 nm×220 nmstrip waveguide. It should be mentioned that the sidewall roughness notonly increases scattering losses, but also results in Fabry-Perot effects whichcause undesirable variations in the spectral responses [42, 43]. Instead ofhaving to use rib waveguides or relatively wide strip waveguides to achievethese improvements, a TM mode propagating in a regular 500 nm×220 nmstrip waveguide will have a similar reduction in the scattering losses and,therefore, reduced variations in the spectral responses as compared to a TEmode propagating in the same waveguide. These improvements are realizedbecause the TM mode interacts mostly with the smooth top and bottomsurfaces of the waveguide and not as much with the rough sidewalls [44].Hence, by implementing TM Bragg gratings on spiral strip waveguides, theadverse effects of non-uniformity caused by both sidewall roughness andwaveguide thickness variations can be reduced. Therefore, TM SBGWs areable to obtain high quality spectral responses and low propagation losses.(a) (b)Figure 1.2: Electric field distributions of (a) the fundamental TE modeand (b) the fundamental TM mode in a 500 nm×220 nm strip waveg-uide with SiO2 cladding.51.2. Motivation and Objectives1.2.2 ObjectivesDue to the small coupling coefficients, low scattering losses, and reducedeffects of fabrication non-uniformity, such a TM spiral waveguide wouldallow one to implement various types of Bragg gratings. In this thesis, Ireport on three types of TM Bragg gratings that were designed using spiralSOI strip waveguides. They are uniform SBGWs (U-SBGWs), phase-shiftedSBGWs (P-SBGWs), and chirped SBGWs (C-SBGWs).TM U-SBGWs have been designed to be notch filters at the transmissionport. Particularly, the small coupling coefficients allow one to achieve narrowbandwidths over wide wavelength ranges. Such devices require long gratinglengths to achieve large ERs. Hence, they can benefit from the spirals dueto the reduced non-uniformity effects caused by the wafer thickness. As aresult, high quality spectral responses can be obtained.TM P-SBGWs have been designed to generate sharp resonance peakswith large quality factors (Q factors). Specifically, Q factors are limited bywaveguide losses. TM waveguides, with low propagation losses, are suitablefor implementing such devices. Also, TM P-SBGWs require long gratinglengths to obtain large Q factors. They can benefit from the spiral waveg-uides as mentioned above. TM P-SBGWs, with large Q factors, have manyapplications, such as microwave photonics [29, 45], ultrafast optical signalprocessing [46], and biosensing [9].In addition, we have designed TM C-SBGWs to achieve linear groupdelays with negative slopes, over a broad passband. These chirped gratingsneed very long grating lengths to obtain large group delays. Using TM spiralwaveguides, these long chirped gratings can be warped into small areas. Asstated previously, the effect caused by the wafer thickness along the gratinglength can be reduced. Such TM C-SBGWs, with negative group delayslopes, can be used for optical dispersion compensation applications [47–50].61.3. Thesis Organization1.3 Thesis OrganizationThis thesis consists of five chapters.In Chapter 1 we briefly introduce the development and advantages ofsilicon photonics. We then present the motivation and objectives of thisthesis.In Chapter 2 the mode analysis of the curved strip waveguides for variousradii of curvature and widths is first presented. Then, the principles andcharacteristics of silicon waveguide Bragg gratings are reviewed. We alsointroduce the transfer matrix method that can be used to investigate thecharacteristics of Bragg gratings.In Chapter 3 we first present the design of the spiral grating waveg-uide, including the spiral waveguide schematics and the spiral waveguidedesign. Then, we describe a numerical simulation method that can be usedto simulate TM U-SBGWs. Next, we present designs and simulations ofTM U-SBGWs, TM P-SBGWs, and TM C-SBGWs. A number of designvariations for these TM SBGWs are theoretically analyzed and simulated.To verify that TM spiral waveguides have lower propagation losses than TEspiral waveguides, we also designed TE and TM spiral waveguides withoutgratings with various lengths in this chapter.Chapter 4 begins with a discussion regarding how our devices were fab-ricated and a description of the automated measurement system used tocharacterize the devices. Then, the propagation losses, measured for bothTE and TM spiral waveguides are compared. Next, the experimental re-sults for the TM U-SBGWs, the TM P-SBGWs, and the TM C-SBGWs arepresented. These results are analyzed and compared with the simulationresults.Chapter 5 provides the summary and conclusions for our work as wellas some suggestions for future research into SBGWs.7Chapter 2Theory2.1 Mode Analysis for Strip WaveguidesThe spiral waveguide is based on a 500 nm×220 nm strip waveguide layingon top of a buried oxide layer, as shown in Fig. 2.1. This SOI waveguidegeometry supports a single TM mode [1]. The buried oxide layer is chosen tohave a thickness of 3 µm, in order to isolate the silicon waveguide from thesilicon substrate, and, thus, reduce the mode leakage loss. The top claddinglayer is typically made in SiO2 with a thickness chosen from 1 µm to 2 µm.SiO2 Si Cladding 500 nm 220 nm 3 μm > 1 μm Si substrate Figure 2.1: Cross section view of a 500 nm×220 nm SOI strip waveg-uide.To design a spiral waveguide, it is important to analyze the effects ofrefractive index variations caused by different curvatures. MODE Solutions,built from Lumerical Solutions, Inc., [51], was used to calculate the effectiverefractive index, neff , of the TM modes in the proposed strip waveguides forvarious radii of curvature. Fig. 2.2 shows the electric field distributions of82.1. Mode Analysis for Strip Waveguidesthe TM modes for a radius of curvature of Fig. 2.2(a) infinity (i.e, straightwaveguide), Figures 2.2(b), 2.2(c), and 2.2(d) are the curved waveguideswith radius of 15 µm, 10µm, and 5 µm. The color bars in Fig. 2.2 showdifferent electric field intensities transferred in the waveguides.As can be seen, such TM mode strip waveguides have large electric fieldson the top and bottom surfaces of the silicon but have relatively small electricfields inside the silicon. The electric field on the straight waveguide showsa symmetric distribution with respect to the center path of the waveguide,see Fig. 2.2(a). However, when the waveguide has a radius of curvatureof 5 µm, the electric field distribution has a relatively large shift towardsthe edge of the silicon waveguide, see Fig. 2.2(d). Such curved waveguideshave large refractive index variations (see below) and result in relativelylarge mode mismatch losses due to the imperfect mode overlap betweenthe straight and the bent waveguides [1]. Hence, they are not suited for TMspiral waveguides. As the radius of curvature increases to 15 µm, the curvedwaveguide shows an electric field distribution similar to that of the straightwaveguide, see Figures 2.2(a) and 2.2(b). This means that the waveguiderefractive index variations and mode mismatch losses are reduced by largerradii of curvatures.Since silicon waveguides have material and waveguide dispersion, theneff values of silicon waveguides are wavelength dependent. The neff valuesof the silicon waveguides also vary with the radius of curvature due to theshift of the electric field distributions, as shown in Fig. 2.2. To verify thatneff is a function of the radius of curvature, TM strip waveguides withvarious radii of curvature were modeled using eigenmode solver from MODESolutions [51]. Fig. 2.3 shows the neff values for the four TM waveguidesover the wavelength range from 1500 nm to 1600 nm. Their neff valuesall decrease linearly as the wavelength increases. As also shown in Fig. 2.3,the smaller the radius of curvature, the smaller the neff . As can be seen inFig. 2.3, when the radius of curvature is 15 µm, the simulated neff closelymatches that of the straight waveguide. Hence, for TM strip waveguide withradii of curvature greater than 15 µm, their neff values can be approximatedusing that of a straight waveguide.92.1. Mode Analysis for Strip WaveguidesIn addition, the group index can be calculated using Eq. (2.1) [52]:ng = neff − λdneffdλ(2.1)and ng of the straight TM waveguide at 1550 nm is found to be approxi-mately 3.71. Once we have ng, we can extract the coupling coefficients usingEq. (2.9).Straight Waveguide  Center path  (a)R=15 μm (b)R=10 μm (c)R=5 μm (d)Figure 2.2: Electric field distributions of (a) straight waveguide andcurved waveguides with radii of curvatures of (b) 15 µm, (c)10 µm,and (d) 5 µm.As the gratings are recessed onto the sidewalls of the waveguides, it is alsonecessary to calculate neff values of silicon waveguides with various widths,see Fig. 2.4. The neff values increase as the waveguide width increases.These neff values can be further applied to simulate the devices describedin Section 2.3.102.1. Mode Analysis for Strip Waveguides1500 1520 1540 1560 1580 16001.651.71.751.81.85Wavelength (nm)neff  StraightR=15 mR=10 mR=5 m1530 1532 15341.761.771.78Wavelength (nm)neff  Figure 2.3: Waveguide effective refractive index varies with the wave-length for various radii of curvatures, as compared with that of astraight waveguide.1500 1520 1540 1560 1580 16001.61.651.71.751.81.851.9Wavelength (nm)neff  W=420nmW=460nmW=500nmW=540nmW=580nmFigure 2.4: Waveguide effective refractive index varies with the wave-length for various waveguide widths.112.2. Silicon Bragg Grating Waveguides2.2 Silicon Bragg Grating WaveguidesIn this section, I would like to introduce a basic structure called the uniformBragg grating on the SOI platform. It is a simple structure, consisting of nu-merous unit cells with periodic modulations of the neff of the optical mode.The geometry of such a structure is sketched in Fig. 2.5. Such Bragg gratingsare fundamental components for achieving selective wavelength functions,such as reflectors and filters. The neff modulation is typically obtainedR T neff1 neff2 ȿ neff Figure 2.5: Illustration of a uniform Bragg grating. neff1 and neff2are the effective indices of the low and high index sections, respectively.neff is the average effective index of the grating waveguide, which isequal to (neff1+neff2)/2. Λ is the grating period. R and T arethe grating reflection and transmission. The 180◦ arrows indicate thenumerous reflections throughout the grating.by varying the physical dimensions of the waveguides. This work focusedon Bragg gratings with corrugations on both sidewalls of the waveguide, asshown in Fig. 2.5. Intuitively, when a light beam is injected into the Bragggrating waveguide, each grating period reflects only a very small fraction ofthe incident light wave. In principle, the phase of the reflected light wave atthe input/output port is determined by the wavelength of the light and thegrating period. The total reflected light consists of a superposition of all ofthe partially reflected light waves. For uniform grating, the reflected lightonly interferes constructively at one particular wavelength, which is calledthe Bragg wavelength (or the center wavelength). Since the neff modulationof the Bragg grating is periodic, and repeated, the light signal at the Bragg122.2. Silicon Bragg Grating Waveguideswavelength is strongly reflected. For other the wavelengths, the reflectedlight signals interfere destructively and cancel each other out. As a result,these light signals are transmitted through the grating.1545 1550 155500.20.40.60.81ResponseWavelength (nm)  ReflectionTransmissionO0'OFigure 2.6: Typical spectral response of a uniform Bragg grating in-cluding reflection and transmission spectra.Fig. 2.6 shows the typical transmission and reflection spectra of a finitelength uniform Bragg grating. In the figure, the Bragg wavelength, λ0, isdetermined by the phase-match condition:λ0 = 2Λneff (2.2)where Λ is the grating period and neff is the average effective index of thegrating waveguide, see Fig. 2.5. By following coupled-mode theory [52–54],the power reflectivity of a finite length uniform Bragg grating is given by:R =|κ|2sinh2(sL)s2cosh2(sL) + (∆β/2)2sinh2(sL)(2.3)where κ is the coupling coefficient, L is the grating length, and ∆β is thephase mismatch in the contra-directional coupling, which is related to theparameter s and κ through the expression:s2 = |κ|2 − (∆β/2)2 (2.4)At the Bragg wavelength, ∆β = 0, a perfect phase matching occurs. Based132.2. Silicon Bragg Grating Waveguideson the Eq (2.3), R can be simplified as:R = tanh2(|κ|L) (2.5)where κ can be interpreted as the amount of the reflection per unit length [1].Accordingly, based on the Fresnel equations at the interfaces between thewaveguide sections with effective indices (neff1 and neff2), κ can be found.In one grating period, the reflection at each interface can be expressedas: (neff1-neff2)/(neff1+neff2). Since each grating period includes twointerfaces, the coupling coefficient can be written as:κ = 2neff1 − neff2neff1 + neff21Λ=∆nneffΛ=2neff∆nneffλ0=2∆nλ0(2.6)Another important parameter is the bandwidth, ∆λ, of the Bragg grat-ing. It is determined by the first nulls around the resonance, as shown inFig. 2.6. Theoretically, ∆λ is determined by the coupling coefficient and thegrating length via the expression as [54]:∆λ =λ20ping√κ2 + (pi/L)2 (2.7)By considering an ideal case, the grating length L is very long, and, thus,κ >> pi/L. Equation (2.7) can be simplified to be:∆λ =λ20κping(2.8)which means that the bandwidth can be controlled by the coupling coeffi-cient. According to Eq. (2.8), κ can also be written as:κ = ping∆λλ20(2.9)which can be used to extract the real coupling coefficients of the Bragggratings from the measured ∆λ and λ0 in the spectral responses [55].142.3. Transfer Matrix Analysis2.3 Transfer Matrix AnalysisSince a Bragg grating can be considered to be a multi-layer structure, a2×2 transfer matrix method (TMM) [53, 56] can be applied to simulateits reflection and transmission properties. The following has been takenfrom Ref. [53] and included here for readers’ convenience. This structurebegins with a dielectric medium, as shown in Fig.2.7. By considering normalA 1 B 1 n ef f 1 ι Medium 1  ι n ef f 2 A 2 B 2 r 12 t 12 t 21 r 21 Medium2  Figure 2.7: Illustration of dielectric media with refractive indices ofneff1 and neff2.incidence, A1 and A2 are the electric fields of the waves traveling to the right.B1 and B2 are the electric fields of the waves traveling to the left. r12 isthe reflection coefficient of the right traveling wave at the interface of thedielectric medium and r21 is the reflection coefficient of the left travelingwave at the same interface. Similarly, t12 is the transmission coefficient ofthe right traveling wave at the interface and t21 is the transmission coefficientof the left traveling wave at the interface.At the both sides of the dielectric interface, the electric fields are related152.3. Transfer Matrix Analysisto each other through the following expressions:A2 = t12A1 + r21B2 (2.10a)B1 = r12A1 + t21B2 (2.10b)where t12, t21, r12, r21 are given by [53]:r12 =neff1 − neff2neff1 + neff2(2.11a)r21 =neff2 − neff1neff1 + neff2(2.11b)t12 =2neff1neff1 + neff2(2.11c)t21 =2neff2neff1 + neff2(2.11d)According to Eq. (2.10), the A1 and B1 can be rewritten in terms of A2and B2 and become:A1 =1t12A2 −r21t12B2 (2.12a)B1 =r12t12A2 + (t21 −r12r21t12B2) (2.12b)Using Eq. (2.11), we finally obtain:A1 =1t12A2 +r12t12B2 (2.13a)B1 =r12t12A2 +1t12B2 (2.13b)which can be equivalently expressed using a 2×2 matrix:A1B1 =1t12r12t12r12t121t12A2B2 = T12A2B2 (2.14)162.3. Transfer Matrix Analysiswhere T12 is the interface transmission matrix from medium1 to medium2.Additionally, the propagation matrix in a homogeneous medium with lengthl can be written as:Phw =ejβl 00 e−jβl (2.15)where β is the complex propagation constant in the medium and is givenby [53]:β =2pineffλ− jαL2(2.16)where αL is the waveguide propagation loss.Using Eqs. (2.14) and (2.15), we can construct the 2×2 transfer matrixfor a multi-layer structure. Here, we consider a uniform Bragg grating withN unit cells (i.e., N pairs of layers), see Fig. 2.8. Each unit cell can ben ef f 1 ι n ef f 2 T0 1  ι T1 2  P hw1  n ef f 0  P hw2  T2 1  B 0  n ef f 0  B 0  ͛ ͙͘͘ A 0  A 0  ͛ n ef f 1 ι P hw1  T1 2  n ef f 1 ι n ef f 2 T2 1  ι T1 2  P hw1  P hw2  T2 0  n ef f 2 ι P hw2  T2 1  1  2  N  Figure 2.8: A multi-layer dielectric film showing the transmission andpropagation matrices.considered to be consist of two a dielectric layers having neff1 and neff2,respectively. The input and output media of the uniform Bragg gratinghave the same effective refractive index, neff0. A transfer matrix, M , isthen created to connect the input and output fields through the expression:A0B0 = MA′0B′0 (2.17)172.3. Transfer Matrix AnalysiswhereM = T01Phw1T12Phw2T21Phw1 • • • T12Phw2T20 =M11 M12M21 M22 (2.18)Based on Eqs. (2.14) and (2.11), T01, T12, T21 and T20 are given by:T01 =neff0+neff12neff0neff0−neff12neff0neff0−neff12neff0neff0+neff12neff0 (2.19)T12 =neff1+neff22neff1neff1−neff22neff1neff1−neff22neff1neff1+neff22neff1 (2.20)T21 =neff2+neff12neff2neff2−neff12neff2neff2−neff12neff2neff2+neff12neff2 (2.21)T20 =neff2+neff02neff2neff2−neff02neff2neff2−neff02neff2neff2+neff02neff2 (2.22)Using Eqs. (2.15) and (2.16), Phw1 and Phw1 can be obtained by:Phw1 =ej(2pineff1λ −jαL2 )l 00 e−j(2pineff1λ −jαL2 )l (2.23)Phw2 =ej(2pineff2λ −jαL2 )l 00 e−j(2pineff2λ −jαL2 )l (2.24)Then, the reflection and transmission coefficients, r and t, of the complete182.3. Transfer Matrix Analysisstructure, are:r =B0A0∣∣∣∣B′0=0=M21A′0M11A′0=M21M11(2.25)t =A′0A0∣∣∣∣B′0=0=A′0M11A′0=1M11(2.26)In this case, the electric field of the light traveling from the right, B′0, isconsidered to be 0. Finally, the reflectance, R, and transmittance, T , canbe calculated using the following equations:R = (M21M11)2 (2.27)T = (1M11)2 (2.28)where R and T satisfy the conservation of energy:R+ T = 1 (2.29)Using the TMM, the spectral responses of Bragg gratings can be calcu-lated using the following steps:Step 1: Obtain the effective refractive indices for each grating period us-ing Lumerical MODE Solutions, see Fig. 2.4, and calculate theirpropagation constants based on Eq. (2.16).Step 2: Construct the propagation matrix for each grating period usingEq. (2.15).Step 3: Calculate the reflection and transmission coefficients of each gratingperiod using Eq. (2.11).Step 4: Construct the interface transmission matrix for each grating pe-riod using the reflection and transmission coefficients obtained fromStep 3.Step 5: Construct the transfer matrix for the complete grating by properlycascading the interface transmission matrices and the propagation192.3. Transfer Matrix Analysismatrices.Step 6: Use Eqs. (2.27) and (2.28) to extract the reflectance and transmit-tance values for different wavelengths and plot their spectra.The TMM is a flexible method that allows us to simulate long Bragg grat-ings. More specifically, the transfer matrices from Step 2 to Step 5 can beadjusted to fit various types of Bragg grating depending on their intendedfunctions. As will be seen, the TMM is used to calculate the spectral re-sponses for our TM U-SBGWs, TM P-SBGWs, and TM C-SBGWs.20Chapter 3Design and SimulationsIn this chapter, I describe the design of the TM spiral grating waveguideand the simulations used for various TM U-SBGWs, TM P-SBGWs, andTM C-SBGWs.3.1 Spiral Grating Waveguide DesignIn this section, I present a method of compactly wrapping long BGWs intospirals. The electric field distributions of TM mode strip waveguides are an-alyzed for various radii of curvature and compared to a straight waveguide.Then, the refractive index variation is analyzed as a function of the radiusof curvature to justify modeling a spiral waveguide using a straight waveg-uide approximation. Such a straight waveguide approximation can then beapplied to numerically simulate the TM SBGWs and to design subsequentdevices.3.1.1 Spiral Grating SchematicOur goal is to wrap long Bragg gratings into compact areas using spiralwaveguides. Figures 3.1(a) and 3.1(b) are, respectively, a scanning elec-tron microscope (SEM) image and a schematic representation of an SBGW.There are two semi-circular waveguides forming an S-shaped waveguide inthe center of the device. Another two interleaved Archimedean spirals areconnected to the S-shaped waveguide. Both semi-circular waveguides havethe same radius of curvature, R0. Each SBGW has two ports, Port A andPort B, see Fig. 3.1(b). Depending on the intended function of the grating,either reflected light or transmitted light can be emitted from either port.213.1. Spiral Grating Waveguide Design(a)-80 -60 -40 -20 0 20 40 60 80-80-60-40-20020406080x (m)y (m)R0Port APort B(b)gWminWmax=2 mWcorr/(c)Figure 3.1: (a) SEM image and (b) schematic of a 4 mm long TMSBGW wrapped into an area of 131×131 µm2. (c) Zoom-in showingthe grating period, Λ, the corrugation width, Wcorr=(Wmax-Wmin)/2,and the spacing between two spiral waveguides, g.223.1. Spiral Grating Waveguide DesignThe complete spiral waveguide consists of the S-shaped waveguide andthe two Archimedean spirals. We start our design by laying out the S-shapedwaveguide. In Cartesian coordinates, the center of the waveguide for oneof the semi-circular sections, see the red semi-circular section in Fig. 3.1(a),can be expressed as:{xS(θ) = R0 · sin(θ)yS(θ) = R0 · (1− cos(θ))(3.1)where R0 is designed to be 15 µm (see below). Since the two semi-circularsections are centrosymmetric with respect to the center of the spiral, thecoordinates of the green semi-circular section in Fig. 3.1(b) can be obtainedfrom those of the red semi-circular section by simply inverting the coordi-nates in Eq. (3.1).The two interleaved Archimedean spirals are also centrosymmetric withrespect to the center of the spiral, as can be seen by the red and greenArchimedean spirals in Fig. 3.1(b). The coordinates of the center of thewaveguides for the red Archimedean spiral are given by:{xA(θ) = (Bθ +Rc) · cos(θ)yA(θ) = (Bθ +Rc) · sin(θ)(3.2)where the parameter B controls the distance between the two Archimedeanspirals through the expression B = (g + Wmax)/pi, where g is the spacingbetween the spirals and Wmax is the maximum width of the grating, asshown in Fig. 3.1(c). The parameter Rc controls the starting point of thered Archimedean spiral, and is equal to 2R0. Due to the symmetry betweenthe red and the green Archimedean spirals, the coordinates of the greenspiral can be calculated by inverting the coordinates in Eq. (3.2).Once the layout of the centers of the waveguides for the complete SBGWwas determined, we added normal vectors to the inner and outer sides ofthe paths of the centers of the waveguides, which allowed us to calculate theexact positions of the sidewalls and/or corrugations for each waveguides.For SBGWs, the amplitude of each normal vector is determined by half of233.1. Spiral Grating Waveguide Designthe grating width, i.e., Wmax/2 and Wmin/2, as shown in Fig. 3.1(c). Usingthis approach, we were able to place arbitrary Bragg gratings onto the spiralwaveguides, by appropriately adjusting the grating periods and widths.3.1.2 Spiral Waveguide DesignThe main challenge encountered when designing a grating on a spiral waveg-uide is that the effective refractive index, neff , of the waveguide varies withthe radius of curvature of the spiral. The radius of curvature of the spiralneeds to be designed to reduce the neff variations. Based on the modeanalysis of the curved waveguides from the last section, we calculate thespiral waveguide index variation, δneff , for various radii of curvature, ascompared to a straight waveguide. As can be seen from the blue curve inFig. 3.2, δneff dramatically increases as the radius of curvature decreases.However, when the radius of curvature is greater than or equal to 15 µm,δneff is small and exhibits a low rate of change as the radius of curvatureincreases. Also, the reflection caused by the mode mismatch at the centerof the S-shaped waveguide is calculated to be smaller than -28.4 dB usingMODE Solutions. Hence, as long as R0 ≥ 15 µm, δneff and the mode mis-match reflection are small. neff can be approximated to be that of a straightwaveguide. However, increasing R0 also reduces the packing efficiency of aspiral waveguide (see below) and, thus, increases the non-uniformity effectscaused by the waveguide thickness variation. The packing efficiency, α, canbe defined as L/√A, where L is the grating length and A is the area ofthe spiral waveguide [57]. As can be seen in Figures 3.1(b) and 3.1(c), thespiral waveguide area is manly affected by g and R0. In order to increase α,while preventing evanescent coupling of the optical modes between the twoArchimedean spirals, g was chosen to be 2 µm. By considering a 1 cm longSBGW, the packing efficiencies for various R0s are shown in Fig. 3.2. Ourgoal is to keep δneff small while achieving a large packing efficiency. As canbe seen from Fig. 3.2, this criterion is best satisfied for R0=15 µm.For R0=15 µm, we obtain a small δneff of 0.78×10−3. Theoretically,such a small δneff can be compensated for by an appropriate change of the243.1. Spiral Grating Waveguide Design0 5 10 15 20 25 30 35 40 45 50 55 6002468x 10-3Radius ( m)G neff3238445056DFigure 3.2: Simulated effective refractive index variations (blue curve)and calculated packing efficiencies (green curve) for spiral waveguideswith R0s ranging from 5 µm to 60 µm.grating period, Λ, based on [58]:neffcΛc = neffΛ (3.3)where the Λc is the local period with compensation, and the neffc andneff are the effective refractive indices of the curved waveguide and thestraight waveguide, respectively. However, for a TM SBGW with a radiusof curvature greater than 15 µm, the change of the designed grating period,Λ-Λc, is less than 0.27 nm. Such a small change in the grating period isbelow the fabrication resolution for a single period and, therefore, has beenignored in the design. Also, according to our calculations, using the transfermatrix method [56], the simulated spectrum is not greatly affected by thesmall value of δneff , in fact, the δneff calculated here causes a bandwidthincrease of only 0.02 nm, see Fig. 3.3. In other words, since R0=15 µm,we have not needed to compensate for the small waveguide index variations253.1. Spiral Grating Waveguide Designcaused by the increasing radius of curvature.1549.6 1549.8 1550 1550.2 1550.4 1550.6-40-35-30-25-20-15-10-5Transmission (dB)Wavelength (nm)  Straight BGWSBGW, R0=15 mFigure 3.3: Comparison of simulated spectra between straight BGWand SBGW. The simulated spectrum is not greatly affected by thesmall δneff , causing a bandwidth increase of 0.02 nm.We also compare the packing efficiency of the proposed device with re-sults found in the literature as shown in Table 3.1. Due to the relativelysmall waveguide width and the spacing between two adjacent spiral waveg-uides, the packing efficiency of our device reaches 52.9, which is much largerthan the curved or spiral Bragg gratings reported in [39, 40, 57]. As a re-sult, the non-uniformity effects caused by silicon waveguide thickness canbe reduced.263.2. FDTD SimulationsTable 3.1: Comparison of packing efficiencies of curved Bragg gratingSOI waveguidesReference Grating length Area Packing efficiencyThis work 10 mm 189×189 µm2 52.9[57] 0.92 mm 190×114 µm2 6.2[40] 2 mm 200×190 µm2 10.2[39] 3.12 mm 141×146 µm2 21.83.2 FDTD SimulationsTo design Bragg gratings, it is important to predict their characteristics andperformance using effective simulation methods. In this section, a specificFinite-Difference Time-Domain (FDTD) method [1, 59, 60] based on Blochboundary conditions [51, 61] is described, which can be used to simulateU-SBGWs.FDTD is a numerical method, which is often used to analyze the behav-ior of light in complicated structures by solving 3D Maxwell’s equations inthe time domain [1, 59, 60]. However, the FDTD simulations can be com-putationally intensive as they require small mesh sizes. As a result, largecomputer memories and long simulation times are often required to obtainaccurate results.To reduce the simulation time, Bloch boundary conditions can be ap-plied to the FDTD simulations and are used to calculate the band structureof periodic waveguides. The Bloch boundary condition method allows forthe FDTD simulation of an infinitely long periodic structure using only oneunit cell [51, 61]. As a result, the simulation time, and the usage of com-puter memory are significantly reduced. Since our TM U-SBGWs are verylong periodic structures, Bloch boundary conditions were used in the FDTDsimulations [61]. In the simulations, the Bloch boundaries are used to locatethe band gap of the TM U-SBGWs, from which λ0 and the bandwidth, ∆λ,273.3. Grating Design and Simulationscan be obtained, as shown in Fig. 3.4. Then, the coupling coefficient can becalculated using Eq. (2.9). The group index of a waveguide has an averagewidth, i.e., having Wave = (Wmax+Wmin)/2. As will be seen in Fig. 4.9(b),calculated κ values for the TM U-SBGWs with various Wcorr values arecompared to measured values.1515 1520 1525 1530 1535 1540 1545 1550 155500.511.52x 1010Wavelength (nm)Magnitude  Wcorr=60nmWcorr=100nmWcorr=140nm'OO0Figure 3.4: Magnitudes of the time domain signals of three deviceswith corrugation widths of 60 nm, 100 nm, and 140 nm, obtained usingFDTD simulations with Bloch boundary conditions. The wavelengthrange between the two resonant peaks indicates the bandwidth, ∆λ,of the device, and the Bragg wavelength, λ0, is located at the centerbetween the two resonant peaks.3.3 Grating Design and SimulationsIn this section, I investigate the basic characteristics of the TM U-SBGWs,the TM P-SBGWs, and the TM C-SBGWs, using the TMM simulations.As our TM spiral waveguides have small effective refractive index variations,as compared to straight waveguides, see Section 3.1.2, a straight waveguideapproximation is used to simulate and design our devices.283.3. Grating Design and Simulations3.3.1 Uniform Spiral Bragg GratingsThe TM U-SBGWs are realized by introducing periodic corrugations on bothsidewalls of the spiral waveguide, as shown in Fig. 3.6. The grating periodis kept constant, which determines the center wavelength of the spectralresponse according to the phase-match condition λ0=2Λneff . The dutycycle of the grating is 50%. Using the TMM, the reflection and transmissionx ( μm)  (a)^  x ( μm)  (b)Figure 3.5: (a) Schematic of a U-SBGW. (b) Enlarged plot shows theperiod, Λ, with a duty cycle of 50%.spectra of a U-SBGW were obtained and one shown in Fig. 3.6(a) on a linearscale and in Fig. 3.6(b) on a dB scale. As can be seen, the center wavelengthof the spectral response is 1550 nm for the grating period of 438 nm used inthe simulations. To account for variations in the fabrication process, we alsovaried the grating period by ±6 nm, i.e., included two extra gratings withperiods of 432 nm and 444 nm. As shown in Fig. 3.7, the center wavelengthshifts linearly to shorter wavelength as the grating period decreases, whilethe bandwidths remain, essentially, the same.As stated previously, the bandwidths were determined by the couplingcoefficients. According to the Eqs. (2.9) and (2.6), the coupling coefficientcan be controlled by the sidewall corrugations on the waveguide. Thus, the293.3. Grating Design and Simulations1510 1520 1530 1540 1550 1560 1570 1580 159000.10.20.30.40.50.60.70.80.91ResponseWavelength (nm)  ReflectionTransmission(a)1510 1520 1530 1540 1550 1560 1570 1580 1590-70-60-50-40-30-20-100Response (dB)Wavelength (nm)  ReflectionTransmission(b)Figure 3.6: (a) Simulated reflection and transmission spectra for aTM U-SBGW in (a) linear scale and (b) dB scale. The device hasparameters: Λ=444 nm, N=400, and Wcorr=20 nm.303.3. Grating Design and Simulationsbandwidth can be controlled by properly adjusting the sidewall corrugations.The TMM was used to calculate the transmission spectra of the U-SBGWswhose corrugation widths range from 20 nm to 60 nm. As shown in Fig. 3.8,the bandwidth increases as the corrugation width increases. Since largercorrugations also create larger reflections, the extinction ratios (ERs) of thetransmission spectra are increased, as shown in Fig. 3.8.1500 1510 1520 1530 1540 1550 1560 1570 1580-60-50-40-30-20-100Transmission (dB)Wavelength (nm)  /=444nm/=438nm/=432nmFigure 3.7: Simulated transmission spectra for the TM U-SBGWs withdifferent period. They have the same Wcorr=20 nm and N=400.1510 1520 1530 1540 1550 1560 1570 1580 1590-70-60-50-40-3-20-100Transmission (dB)Wavelength (nm)  Wcorr=20nmWcorr=40nmWcorr=60nmFigure 3.8: Simulated transmission spectra for the TM U-SBGWswith different corrugation widths. They have the same Λ=444 nmand N=400.313.3. Grating Design and Simulations3.3.2 Phase-shifted Spiral Bragg GratingsA phase-shifted Bragg grating can be realized by introducing a pi-phase shiftat the center of the uniform Bragg grating. Fig.3.9 shows the schematic of aTM P-SBGW. The length of the phase shift is equal to the grating period.Such a structure can be considered to be a Fabry-Perot (FP) etalon witha one period-long cavity. The front and rear sides of the cavity within thegrating, work as the mirrors of the etalon with large reflections. As a result,the whole structure works as a FP Bragg grating that has a narrow bandpassresonance appearing at the center of the spectral response window [52].x ( μm)  (a)^  x ( μm)  (b)Figure 3.9: (a) Schematic of a TM P-SBGW. (b) Enlarged plot showsa cavity with one period length at the center area of the grating.3.3.2.1 BasicsHere, we would like to briefly introduce the basic characteristics of phase-shifted Bragg gratings by following the analysis of a FP etalon as describedin [52, 53]. We can first consider that input light with an intensity of Iiis launched into the etalon. The reflected intensity and transmitted lightintensity are Ir and It respectively. The fraction of the reflected power is323.3. Grating Design and Simulationsgiven by:IrIi=4Rsin2(δ/2)(1−R)2 + 4Rsin2(δ/2)(3.4)while the fraction of the transmitted power is given by:ItIi=(1−R)2(1−R)2 + 4Rsin2(δ/2)(3.5)where R is the mirror reflectivity, and δ is the round-trip phase shift throughthe etalon and for guided waves can be calculated by:δ =4pineff letalonλ(3.6)where letalon is the length of the etalon.According to Eqs. (3.4) and (3.5), sin2(δ/2)=0 can result in the mini-mum reflection and the maximum transmission, which means that the res-onance peak occurs at sin(δ/2)=0. Since sin(δ/2) is periodic, we can solvethe resonance wavelength through:δ =4pineff letalonλm=4pineffΛλm= 2(m+ 1)pi, m = 0, 1, 2, 3...... (3.7)where letalon is equal to Λ. For lowest order phase-shifted Bragg grating,m is chosen to be 0. We can then obtain: λ0 = 2neffΛ, which is the sameas the Bragg condition, see Eq. (2.2). Therefore, the resonance peak of thephase-shifted Bragg grating occurs at the Bragg wavelength.3.3.2.2 SimulationsTo perform the TMM simulation, the TM P-SBGW is considered to be aFP Bragg grating with N front periods and N rear periods. According toEqs. (2.15), the transfer matrix of such a P-SBGW can be expressed as:TP−SBGW = T01MfrontPP−SBGWhw1 T12MrearT10 (3.8)333.3. Grating Design and Simulationswhere PP−SBGWhw1 is the propagation matrix in the FP cavity. Since thecavity has letalon = Λ, PP−SBGWhw1 is expressed as:PP−SBGWhw1 =ejβΛ 00 e−jβΛ (3.9)Mfront and Mrear are the transfer matrices of the gratings for the front andrear sides of the cavity. They are given by:Mfront = (Phw1T12Phw2T21)N (3.10)Mrear = (Phw2T21Phw1T12)N−1Phw2T21Phw1 (3.11)Based on the transfer matrix from Eq. (3.8), the reflection and transmis-sion spectra of a TM P-SBGW are simulated and shown in Fig 3.10. Thisdevice has Λ=444 nm, Wcorr=80 nm, and N=100. The narrow resonantpeaks are observed at the center wavelength of 1550 nm.The model shown above does not contain loss mechanisms. Hence, themaximum transmission shown in Fig. 3.10(b) is 1. However, in a practi-cal case, silicon waveguides have losses, which reduce the maximum trans-missions. By taking into account the loss per pass, G, defined as G =Ioutput/Iinput, the fraction of intensity transmission becomes [52]:ItIi=(1−R)2G(1−GR)2 + 4GRsin2(δ/2)(3.12)As the maximum transmission occurs when sin(δ/2)=0, the maximum trans-mission can be written as [52]:Tmax = (ItIi)max =(1−R)2G(1−GR)2(3.13)G is determined by the absorption coefficient in the medium, α, and thelength of the medium, l, then can be written as G=exp(-αl) [52]. For passivewaveguides, G is less than 1, which results in the maximum transmission343.3. Grating Design and Simulations1500 1520 1540 1560 1580 160000.51ReflectionWavelength (nm)(a)1500 1520 1540 1560 1580 160000.51TransmissionWavelength (nm)(b)Figure 3.10: Simulated (a) reflection and (b) transmission spectra of aTM P-SBGW. The device has Λ=438 nm, Wcorr=80 nm, and N=100.353.3. Grating Design and Simulationsalso being less than 1. Fig. 3.11 plots Tmax versus R for several values of G.As can be seen, Tmax decreases dramatically as R approaches 1.0.8 0.85 0.9 0.95 1-100-80-60-40-200RTmax (dB)  G=0.98G=0.94G=0.9G=0.86G=0.82Figure 3.11: Tmax as a function of R for different values of G.To verify that the maximum transmission is limited by the waveguideloss, a propagation loss of 3 dB/cm, is introduced to the TMM simulationsfor various TM P-SBGWs. Also, according to Eq. (2.5), R can be increasedby increasing either κ or L of the Bragg grating. As the larger Wcorr leadsto lager κ, the TM P-SBGWs are simulated for various Wcorr values rangingfrom 80 nm to 120 nm. These devices have the same N=100 and Λ=444 nm.Their transmission spectra are shown in Fig. 3.12. As can be seen, the peakof the transmission resonance decreases as Wcorr increases. The bandwidthof the stop band becomes broader when applying larger Wcorr, which isanalogous to the U-SBGWs, see Fig. 3.8.The TM P-SBGWs are also simulated for different lengths by increasingthe number of grating periods from 100 to 150. The grating periods andcorrugation widths remain 444 nm and 80 nm, respectively. Their transmis-sion spectra are as shown in Fig. 3.13. Again, the peak of the transmissionresonance is decreased when increasing the grating length. Fig. 3.13 alsoshows that the devices have the same bandwidths of the stop bands due tothe similar κ perturbed by the fixed Wcorr.363.3. Grating Design and Simulations1500 1520 1540 1560 1580 160000.20.40.60.81TransmissionWavelength (nm)  Wcorr=80nm(a)1500 1520 1540 1560 1580 160000.20.40.60.81TransmissionWavelength (nm)  Wcorr=120nm(b)1500 1520 1540 1560 1580 160000.20.40.60.81TransmissionWavelength (nm)  Wcorr=160nm(c)Figure 3.12: Transmission spectra for many Wcorr values of (a) 80 nm,(b) 120 nm, and (c) 160 nm. The devices have the same N=100, andΛ=444 nm, and αL=3 dB/cm.373.3. Grating Design and Simulations1500 1520 1540 1560 1580 160000.20.40.60.81TransmissionWavelength (nm)  N=100(a)1500 1520 1540 1560 1580 160000.20.40.60.81TransmissionWavelength (nm)  N=125(b)1500 1520 1540 1560 1580 160000.20.40.60.81TransmissionWavelength (nm)  N=150(c)Figure 3.13: Transmission spectra for different N of (a) 100, (b) 125,and (c) 150. The devices have the same Wcorr=80 nm, Λ=444 nm,and αL=3 dB/cm.383.3. Grating Design and Simulations3.3.2.3 Quality FactorThe sharpness of the resonance peak is evaluated using the parameter calledquality factor (Q factor), which is given by:Q =λ0∆λ3dB(3.14)where ∆λ3dB is the 3-dB bandwidth of the resonant peak and λ0 is thecenter wavelength of the resonance peak.Theoretically, the Q factor is determined by the angular frequency, ω0,and the photon life time, τp, through the expression of Q = ω0τp [53]. Sincethe TM P-SBGW can be considered to be an FP etalon, τp is given by:τp =ngc(α+ αm)(3.15)where c is the speed of the light, α is the waveguide propagation loss, andαm is the average mirror loss of the FP etalon, which is given by:αm =ln√R1R2letalon(3.16)where R1 and R2 are the mirror reflectivities. When R1 and R2 are bothapproximately 1, αm approaches zero. As a result, the Q factor reaches itsmaximum value through the expression [62, 63]:Qmax = ω0τp =2pic · ngλ · c · α[m−1]=2pi · ng · 4.34λ · 100 · α[dB/cm](3.17)Particularly, α[dB/cm] is the waveguide propagation loss in dB/cm. Qmax isalso know as the unloaded Q factor. Assuming that the average propagationloss is 2.6 dB/cm and that ng is 3.71, Qmax is calculated to be approximate250000 at 1550 nm. To obtain high Q factors, a number of design variationsin grating lengths and corrugation widths have been applied to the TMP-SBGWs. The grating length increases from 200 µm to 1000 µm. Thecorrugation width increases from 80 nm to 120 nm. Their experimentalresults for these cases are presented and analyzed in Section 4.2.4.393.3. Grating Design and Simulations3.3.3 Chirped Spiral Bragg GratingsThe C-SBGWs are chirped, such that in them Λ varies linearly along thelength of the device. In principle, as Λ changes along the length, the wave-length of the reflected light also changes according to the phase-match con-dition, λm=2Λmneff , where Λm is the local grating period and λm is thecorresponding center wavelength of the reflection, as shown in Fig. 3.14.Therefore, a C-SBGW can be considered to be the sum of a large numberof uniform gratings with different periods. Such a C-SBGW can providebroader bandwidth spectrum as compared to a uniform grating. The band-width of the reflection spectrum can be controlled by two parameters: thechirp rate of the grating period, dΛ/dL, and the total grating length, L.Light in  Figure 3.14: Illustration of a TM C-SBGW, where Λ changes alongthe length of the grating.403.3. Grating Design and SimulationsSince light at different frequencies propagates different distances beforebeing reflected, a C-SBGW can be used to vary the group delay as a functionof wavelength over the passband. Also, the slope of the group delay, as afunction of the wavelength, can be controlled by the chirp rate. For instance,when the chirp rate is negative, the grating period decreases linearly alongthe grating length. Due to the increased propagation distance, the opticalwaves at shorter wavelengths experience larger group delay as compared tothose at longer wavelengths. A negative group delay slope can be obtainedwithin the passband. In contrast, a positive chirp rate can result in a positivegroup delay slope over the passband.The TMM can be used to obtain the group delay for a C-SBGW. First,the TMM calculates the reflection coefficient, rC−SBGW , while the phase ofthe reflection coefficient, ϕ(rC−SBGW ), can also be obtained. Afterwards,the group delay, τC−SBGW , can be calculated by taking the derivative ofϕ(rC−SBGW ) with respect to ω, or respect to λ, as follows:τC−SBGW =dϕ(rC−SBGW )dω= −λ22picdϕ(rC−SBGW )dλ(3.18)Figures. 3.15(a) and 3.15(b) show the simulated reflection spectra andgroup delays for two C-SBGWs that have the same parameters, except thatthe sign of the chirp rate is reversed in relation to each other. Both ofthe devices have 4 mm lengths and corrugation widths of 60 nm. Theirgrating periods are chirped with dΛ/dL = ±6 nm/cm starting from a gratingperiod of 444 nm. As can be seen, both of the C-SBGWs have the samebandwidths. However, the excess losses over the passbands are different.For the C-SBGW with the positive chirp rate, the excess loss near the bandedge at longer wavelengths is lower than that at shorter wavelengths dueto the increased propagation distance, see Fig. 3.15(a). On the other hand,for the C-SBGW with the negative chirp rate, the excess loss near the bandedge at longer wavelengths is higher than that at shorter wavelengths, seeFig. 3.15(b).Also, the C-SBGW with the positive chirp rate experiences a positivegroup delay, with a slope of 11 ps/nm over the passband, see Fig. 3.15(a),413.3. Grating Design and Simulationswhile the C-SBGW with the negative chirp rate experiences a negative groupdelay, with a slope of -11 ps/nm over the passband, see Fig. 3.15(b). It canbe seen that the group delay exhibits ripples. These ripples are due to ouruse of constant corrugation widths. These ripples can be eliminated by usingapodization [49, 52, 64]. To verify this, the grating was apodized using aGaussian function of position, which is given by [65]:δneff (z) = δneff0 · exp[−4ln(2)(z − L/2L/3)2] (3.19)where δneff0 is the maximum effective index change for a Wcorr of 60 nm.Fig. 3.16 shows a Gaussian apodization profile for a 4 mm long grating.As can be seen, the effective index change is smoothly reduced to zero atboth ends of the grating. Using this Gaussian profile, we have designedan apodized C-SBGW with the chirp rate of -6 nm/cm starting from agrating period of 444 nm. Fig. 3.17 shows the simulated group delays forthis apodized C-SBGW and of and an unapodized C-SBGW. As can beenseen, the ripples have been significantly reduced by the apodization.The C-SBGWs are then simulated for various lengths and their reflectionspectra are shown in Fig. 3.18(a). With a fixed dΛ/dL =-6 nm/cm, thebandwidth increases as the grating length increases. Also, the C-SBGWsare simulated by increasing the magnitudes of the chirp rates, while the signof the chip rates remains negative. Their reflection spectra are shown inFig. 3.18(b). The bandwidth also increases when the chirp rate increases.423.3. Grating Design and Simulations1540 1545 1550 1555 1560-200Wavelength (nm)Reflection (dB)0100Group delay (ps)Slope=11ps/nm (a)1540 1545 1550 1555 1560-200Wavelength (nm)Reflection (dB)0100Group delay (ps)Slope=- 11ps/nm (b)Figure 3.15: Simulated reflection spectra (blue curves) and group delay(green curves) of the TM C-SBGWs with (a) dΛ/dL=6 nm/cm and(b) dΛ/dL =-6 nm/cm. They have the same parameters: L= 4 mm,Wcorr=5 nm, initial Λ= 444 nm, and αL=2 dB/cm.433.3. Grating Design and Simulations0 0.5 1 1.5 2 2.5 3 3.5 400.20.40.60.81Normalized Gneffz (mm)Figure 3.16: Gaussian apodization profile for a 4 mm long grating.1544 1546 1548 1550 1552 1554-50050100Wavelength (nm)Group delay (ps)  Unapo dizedA po dizedFigure 3.17: Simulated group delays for unapodized C-SBGW (greencurve) and apodized C-SBGW (red curve).443.3. Grating Design and Simulations1540 1545 1550 1555 1560-25-20-15-10-50Wavelength (nm)Reflection (dB)  L=2mmL=3mmL=4mm(a)1540 1545 1550 1555 1560-25- 0-15- 0-50Wavelength (nm)Reflection (dB)  d //dL=-4nm/cmd //dL=-5nm/cmd //dL=-6nm/cm(b)Figure 3.18: (a) Simulated reflection spectra of the TM C-SBGWswith lengths of 2 mm, 3 mm, and 4 mm. The devices have the fixedparameters: αL=2 dB/cm and dΛ/dL =-6 nm/cm. (b) Simulatedreflection spectra of the TM C-SBGWs with chirp rates of -4 nm/cm,-5 nm/cm, and -6 nm/cm. The devices have the fixed parameters:αL=2 dB/cm, initial Λ= 444 nm, and L=4 mm.45Chapter 4Experimental Results andDiscussionIn this chapter, first, the layouts of the devices are presented and the fab-rication processes are described. Next, the measured spectral responses ofthe fabricated devices are presented and compared with the simulations.4.1 FabricationTo fabricate the devices, we need to first draw the mask layouts for ourdevices. Mentor Graphics software [66] was used to draw the mask layouts ofour SBGWs, and to generate the GDSII files for fabrication. Fig. 4.1 showsa mask layout of an U-SBGW device. Three grating couplers (GCs) [67],aligned on a 127 µm pitch, were used to couple light into and out of ourdevices through a fiber array with a 127 µm fiber-to-fiber pitch. A low lossY-junction power splitter [68] was used to connect the input and outputGCs to the SBGWs in order to transfer the injected light to the device andto collect the reflected light.The designed devices were then fabricated using two fabrication pro-cesses: (1) electron beam lithography (eBeam) and (2) IMEC ePIXfabtechnology using 193 nm deep ultra-violet (DUV) lithography. Most ofthe SBGWs were fabricated using eBeam lithography at the University ofWashington [69]. Fig. 4.2 shows SEM images of a fabricated U-SBGW de-vice. As can be seen, the designed rectangular corrugations on the SBGWshave been efficiently fabricated, see the zoom-in SEM images, Figures 4.2(b)and 4.2(c), on the right side of Fig. 4.2. Alternatively, some U-SBGW de-vices were fabricated using 193 nm DUV lithography at IMEC in Leuven,464.1. Fabrication^ Input GC  Outpu t GC  Outpu t GC  127 μm  127 μm  Y - branch  Figure 4.1: Mask layout of an U-SBGW with input and output GCs,and a 3-dB Y-branch.Belgium [70]. Fig. 4.3 shows an SEM image of an SBGW fabricated using193 nm DUV lithography. As can be seen, the waveguides have relativelyrounded corrugations, as compared with the rectangular corrugations fabri-cated using eBeam lithography. Such rounded corrugations can reduce thecoupling coefficients and result in smaller bandwidths.Reflection  Input GC  Transmission  Reflection GC  Input GC  Transmission GC  Y - junction  (a)  (b)  (c)  Figure 4.2: SEM images of an U-SBGW. (a) The complete U-SBGWwith input and output GCs. (b) Zoom-in of the center of the S-shapedgrating waveguide. (c) Zoom-in of the Archimedean spiral gratingwaveguides. Device is fabricated using eBeam lithography.474.2. Experimental ResultsFigure 4.3: SEM image shows the spiral grating waveguides of aSBGW fabricated using 193 nm DUV lithography.4.2 Experimental ResultsThis section first introduces the measurement setup for our devices. Then,we compare the measured transmission spectra of both TE and TM spiralwaveguides and shows that the TM spiral waveguides have lower propagationlosses. Then, the measured spectral responses of various TM SBGWs aredemonstrated and analyzed.4.2.1 MeasurementsTo measure our devices, I used the automated measurement system in ourlab. In it, after the chip is properly aligned, light from the swept laser (Agi-lent 81600B) is injected into the device and a wavelength sweep is performed.At the same time, dual channel optical power sensors (Agilent 81635A) de-tect and record the measured data for each device. The full view of theautomated measurement system is shown in Fig. 4.4.As stated previously, silicon devices are sensitive to temperature, hence,it is necessary to maintain a constant operating temperature during themeasurements. Under the stage, a thermoelectric copper plate is attached484.2. Experimental Results4  2 5  6  7  8  9  1. Stage       2. Chip        3. Fiber array         4. Micros cope        5. Stage m otion controller  6. Optical fibers         7. Tem perature controller (SRS LDC - 501)  8. Tunab le lase r source (Agilent 81600B)           9. Optical power senso rs (Agilent 81635A )    1  3  Figure 4.4: Full view of the automated measurement system494.2. Experimental Resultsto maintain the chip at a stabilized operating temperature through a tem-perature controller (SRS LDC-501). Typically, the operating temperatureof the chip was maintained at 25◦C during the measurements. The oper-ating temperature can also be varied when implementing experiments forthermally tuning devices.4.2.2 Waveguide Propagation LossAs mentioned previously, it was important to verify that TM spiral waveg-uides had less sensitivity to sidewall roughness and had lower propagationlosses than TE spiral waveguides. Since the devices were fabricated usingtwo different processes, the sensitivities to the sidewall roughness and thepropagation losses would depend on the process. In this section, we first fo-cused on the devices fabricated using eBeam lithography. The transmissionspectra of the spiral waveguides were compared for both the TE and TMmodes. Then, we compared the transmission spectra between the TE andTM spiral waveguides fabricated using 193 nm DUV lithography.For eBeam lithography, spiral waveguides without gratings were de-signed, for both the TE and TM modes, varying in length from 0.5 cmto 3 cm. Fig. 4.5(a) shows the comparison of the measured transmissionspectra among the three TE spiral waveguides and three TM spiral waveg-uides over the C-band. They had equivalent lengths of 1 cm, 2 cm and3 cm. It was observed that the TM spiral waveguides had much smoothertransmission spectra, which means that they had reduced sensitivities to FPeffects caused by the sidewall roughness. Using the measured transmissionspectra, the average transmission losses were extracted from the C-band.Fig. 4.5(b) shows that the average transmission losses, for both the TE andTM spiral waveguides, increased as the length increased. The rate of changein the average transmission losses, as a function of the length, gave us theaverage propagation loss of each spiral waveguide. Using linear fits to thedata in Fig. 4.5(b), the average propagation loss of the TM spiral waveguidesand of TE spiral waveguides were found to be 2.6 dB/cm and 6 dB/cm, re-spectively. This confirms that TM spiral waveguides have lower propagation504.2. Experimental Resultslosses than TE spiral waveguides.For 193 nm DUV lithography, both the TE and TM spiral waveguideswithout gratings were designed with lengths ranging from 0.05 cm to 1 cm.Their measured transmission spectra are compared and shown in Fig. 4.6(a).The TM spiral waveguides have smoother transmission spectra as comparedto the TE spiral waveguides. The average transmission losses were alsoextracted for both the TE and TM spiral waveguides, as shown in Fig. 4.6(b).The average propagation loss of a TM spiral waveguide was found to be1 dB/cm, which was lower than the average propagation loss of 2.5 dB/cmfor a TE spiral waveguide.In summary, for both eBeam lithography and 193 nm DUV lithography,the TM spiral waveguides were found to have smoother transmission spec-tra and lower propagation losses than the TE spiral waveguides. As willbe seen in the next sections, this increased smoothness in the transmissionspectra can be observed in the results presented for all TM SBGWs. Also,for both the TE and TM mode, spiral waveguides fabricated using 193 nmDUV lithography have lower propagation losses than those fabricated usingeBeam lithography, see Table 4.1. This indicates that the spiral waveg-uides fabricated using 193 nm DUV lithography have smoother sidewalls ascompared with these fabricated using eBeam lithography.Table 4.1: Comparison of propagation losses between TE and TM spi-ral waveguides fabricated using both eBeam lithography and 193 nmDUV lithographyDevice type Fabrication Propagation lossTE spiral waveguides eBeam lithography 6 dB/cmTM spiral waveguides eBeam lithography 2.6 dB/cmTE spiral waveguides 193 nm DUV lithography 2.5 dB/cmTM spiral waveguides 193 nm DUV lithography 1 dB/cm514.2. Experimental Results1530 1535 1540 1545 1550 1555 1560 1565-80-60-40-20T E Spiral WaveguideWavelength (nm)Transmission (dB)  L=1cmL=2cmL=3cm1530 1535 1540 1545 1550 1555 1560 1565-30-20-10T M Spiral WaveguideTransmission (dB)  L=1cmL=2cmL=3cm(a)0 0.5 1 1.5 2 2.5 3 3.5-50-45-40-35-30-25-20-15-1-5Length (cm)Transmission loss (dB)   slo pe=- 2.6 dB/cmslo pe=- 6 dB /cm TMT E(b)Figure 4.5: (a) Measured transmission spectra for the TM mode andTE mode spiral waveguides fabricated by eBeam lithography. Thewaveguide lengths vary from 1 cm to 3 cm. (b) Average transmis-sion losses for TM spiral waveguides are compared to the TE spiralwaveguides’ losses.524.2. Experimental Results1530 1535 1540 1545 1550 1555 1560 1565-26-24-22-20-18T E Spiral WaveguideTransmission (dB)  L=0.5mmL=3mmL=6mmL=10mm1530 1535 1540 1545 1550 1555 1560 1565-28-26-24-22-20-18T M Spiral WaveguideWavelength (nm)Transmission (dB)  L=0.5mmL=3mmL=6mmL=10mm(a)0 0.2 0.4 0.6 0.8 1-23.5-23-22.5-22-21.5-21-20.5-20Length (cm)Transmission (dB)  T MT ESlo pe=-2.51 dB/cmSlo pe=-1.02 dB/cm(b)Figure 4.6: (a) Measured transmission spectra for the TM mode andTE mode spiral waveguides fabricated by 193 nm DUV lithography.The waveguide lengths vary from 0.5 mm to 10 mm. (b) Averagetransmission losses for TM spiral waveguides are compared to the TEspiral waveguides’ losses.534.2. Experimental Results4.2.3 Uniform Spiral Bragg GratingsThis section first shows the results of experiments done on TM U-SBGWsfabricated by eBeam lithography. These devices have various corrugationwidths, grating periods, and grating lengths. Their spectral responses wereanalyzed and compared to FDTD simulations. The thermal sensitivity of aTM U-SBGW was also investigated. Then, we presented the experimentalresults done on a TM U-SBGW fabricated by 193 nm DUV lithography.The U-SBGWs exhibited narrow bandwidths of 0.09 nm (eBeam lithogra-phy) and 0.5 nm ( 193 DUV lithography) and large ERs of 52 dB (eBeamlithography) and 40 dB (DUV lithography).4.2.3.1 U-SBGWs fabricated by eBeam lithographyAccording to the phase-match condition, the center wavelength of the spec-tral response of a TM USBGW is determined by the grating period. Thiswas observed experimentally. The center wavelengths of the stop bands shiftlinearly to longer wavelengths as the grating period increases, as shown inFig. 4.7(a). The shift of the center wavelength as a function of the gratingperiod was calculated to be 1.7. When the period is increased to 444 nm, thecenter wavelength reaches 1549.3 nm, which closely matches the simulatedvalue, as shown in Fig. 4.7(b). Also, since the gratings have long lengths,the transmission spectra exhibit large ERs, as large as 52 dB, as shown inFig. 4.7(a).On the other hand, the bandwidths of the TM U-SBGWs are controlledby the corrugation widths of the gratings. Figure 4.8 shows the measuredtransmission spectra for TM U-SBGWs with various corrugation widths. Asexpected, the larger corrugation widths resulted in larger bandwidths dueto the increased coupling coefficients.Fig. 4.8 also shows that the center wavelengths of the gratings shift toshorter wavelengths as the corrugation widths increase. Such wavelengthshifts are also observed in the simulation results, as shown in Fig. 4.9(a).This indicates that larger corrugation widths result in smaller average neffvalues due to the non-linear relationship between neff and waveguide width544.2. Experimental Results1520 1530 1540 1550 1560-60-50-40-30-20-100Wavelength (nm)Transmission (dB)  /=432nm/=438nm/=444nm52 dB(a)1546 1547 1548 1549 1550 1551 1552 1553-50-40-30-20-10Transmission (dB)Wavelength (nm)  T M M SimulationMeasurement(b)Figure 4.7: (a) Measured transmission spectra for the TM U-SBGWs,with grating periods of 432 nm, 438 nm, and 444 nm. (b) The mea-sured and simulated transmission spectra, where the average neff ofthe waveguide was adjusted by 0.84×10−3 in the simulation.554.2. Experimental Results1490 1500 1510 1520 1530 1540 1550 1560-60-50-40-30-20-100Wavelength (nm)  Transmission (dB)Wcorr=40 nmWcorr=60 nmWcorr=80 nmWcorr=100 nmWcorr=160 nmFigure 4.8: Measured transmission spectra of TM U-SBGWs withvarious corrugation widths. The grating period of each TM U-SBGWis 432 nm. The grating lengths of the TM U-SBGWs with Wcorr >40 nm were 1 mm. The TM U-SBGW with Wcorr = 40 nm had agrating length of 3 mm.[71]. This effect can be compensated for by adjusting the grating widths,Wmax and Wmin, to maintain a fixed, average neff , and, thus, a fixed centerwavelength. The wavelength shift can also be compensated for by adjustingthe grating period to keep the center wavelength constant.Using the measured center wavelengths and bandwidths, the couplingcoefficients were extracted using Eq. (2.9), and then compared to FDTDsimulation results. Fig. 4.9(b) shows the simulated and measured couplingcoefficients as functions of the corrugation width. Also shown in Fig. 4.9(b)is a second-order polynomial fit to the simulated coupling coefficients.Since TM U-SBGWs can have small coupling coefficients for relativelylarge corrugations, we were able to design our U-SBGWs with narrow band-widths. Figure 4.10 shows the reflection spectrum from a TM U-SBGWwith a 3-dB bandwidth of 0.09 nm, which is smaller than the bandwidth of0.14 nm obtained by a state-of-the-art TE spiral grating using a 1200 nmwide strip waveguide in Ref. [72].564.2. Experimental Results40 60 80 100 120 140 160150015101520153015401550O0 (nm)Wcorr (nm)  MeasurementFDT D Simulation(a)40 60 80 100 120 140 160012345x 104N (m-1)Wcorr (nm)   N = 1.2*Wcorr2 + 1.3e+002*Wcorr - 4.9e+003MeasurementFDT D Simulation(b)Figure 4.9: Simulation and experimental results for the Bragg grat-ings. (a) The Bragg gratings centre wavelength shifts to shorter wave-lengths as the corrugation width increases. The vertical offset betweensimulation and experimental results is due to a difference between thefabrication and the target waveguide design. (b) The grating strength(coupling coefficient) increases as a function of the corrugation width.574.2. Experimental Results1555 1560 1565-16-14-12-10-8-6-4-20Wavelength (nm)Reflection (dB)1559.9 1560 1560.1-8-6-4-20Wavelength (nm)Reflection (dB)0.09 nmFigure 4.10: Measured reflection spectrum of a TM U-SBGW with acorrugation width of 20 nm, a length of 4 mm, and a period of 444 nm.4.2.3.2 U-SBGW fabricated by 193 nm DUV lithographyFig. 4.11 shows a normalized transmission spectrum of a TM U-SBGWfabricated using 193 nm DUV lithography. This TM U-SBGW has a cor-rugation width of 20 nm and grating length of 1 cm. As seen in 4.11,the TM U-SBGW exhibits a 3-dB bandwidth of 0.5 nm and large ER of40 dB. The experimental coupling coefficient of this device was found to be2.99×103 m−1.The TM U-SBGW was also compared with a uniform TE SOI BGW witha corrugation width of 10 nm [31], see Table 4.2. Both the TM U-SBGWand the TE straight BGW were fabricated by 193 nm DUV lithography.They both had the same waveguide width and thickness. The TM U-SBGWexhibited a smaller bandwidth for a larger corrugation width and a largergrating period. Hence, manufacturing tolerance and design flexibility canbe improved by using such TM U-SBGWs.584.2. Experimental Results1500 1510 1520 1530 1540 1550 1560-45-40-35-30-25-20-15-10-505W a ve le ng th ( nm)Transmission (dB)1529 1530 1531-40-200W a ve le ng th ( nm)Transmission (dB)0.5 nm40 dBFigure 4.11: Measured transmission spectrum of the TM U-SBGWfabricated using 193 nm DUV lithography. The device has parameters:Wcorr= 20 nm, L= 1 cm, and Λ=440 nm.Table 4.2: Comparison between TM U-SBGW and straight, uniformTE SOI Bragg gratingDevice TM U-SBGW TE straight BGW [31]Fabrication 193 nm DUV lithography 193 nm DUV lithographyWaveguide geometry 500 nm×220 nm 500 nm×220 nmPeriod 440 nm 310 nmCorrugation width 20 nm 10 nm3-dB Bandwidth 0.5 nm 0.8 nm594.2. Experimental Results4.2.3.3 Thermal SensitivityBased on the experimental results shown above, the U-SBGWs can be de-signed to be transmission filters in the C-band by choosing proper combi-nations of the grating periods and the corrugation widths. Once a device isfabricated, it may be necessary to dynamically tune the center wavelength.This can be achieved by adjusting the operating temperature of the de-vice [73, 74]. The thermal sensitivity of the Bragg grating, dλ/dT , can becalculated usingdλdT=λngdneffdndndT(4.1)where the dn/dT=1.8×10−4 K−1 [73] and dneff/dn=0.57 obtained for TMmode of a 500 nm×220 nm SOI strip waveguide using Lumerical MODE So-lutions [51]. Then, the theoretical thermally-induced variation of the Braggwavelength was calculated to be 49 pm/◦C.1534 1536 1538 1540 1542 1544-80-70-60-50-40-30-20-10Wavelength (nm)Transmission (dB)  T=30 qCT=35 qCT=40 qCT=45 qCFigure 4.12: Measured transmission spectra of a TM U-SBGW atvarious temperatures. The device has a length of 1 mm, a gratingperiod of 438 nm, and a corrugation width of 80 nm.To verify this, the TM U-SBGWs were thermally tuned using a tem-perature controller. Fig. 4.12 demonstrates that the measured transmissionspectrum of a TM U-SBGW shifts to longer wavelengths as the temperatureincreases from 30◦C to 45◦C. The measured thermally-induced variation of604.2. Experimental Resultsthe Bragg wavelength was 48 pm/◦C, which agrees with the theoretical pre-diction of 49 pm/◦C.4.2.3.4 DiscussionWe can compare these TM U-SBGWs to the previously reported designsbased on 1200 nm wide TE spiral waveguides [40]. In Ref. [40], a 2 mm longTE spiral using a 1200 nm µm×220 nm wide strip waveguide is wrappedinto an area of 200 µm×190 µm. In this work, a 2 mm long TM spiralusing regular 500 nm×220 nm strip waveguide can be wrapped into an areaof 103 µm×103 µm. Using Lumerical Mode solutions, the effective indexsensitivities to waveguide thickness variations for the two types of waveg-uides were calculated to be: a) 0.0071 nm−1 for TM 500 nm×220 nm stripwaveguide, and b) 0.0036 nm−1 for TE 1200 nm×220 nm strip waveguide.Although the TM waveguides sensitivity to thickness variation is approx-imately 2 times that of the wide TE waveguide, the spiral waveguide inthis work has an area that is four times smaller, hence the thickness varia-tion across the device is reduced. Overall, these effects cancel, resulting indevices with a narrow linewidth in a compact size.We can also compare the results obtained using eBeam lithography and193 nm DUV lithography. As mentioned previously, 193 nm DUV lithogra-phy led to rounded corrugations which could result in smaller bandwidths,as compared to the rectangular corrugations obtained using eBeam lithog-raphy. However, the results for our U-SBGWs are opposite to what wewould expected. The U-SBGW fabricated using eBeam lithography had abandwidth of 0.09 nm, which is smaller than the 0.5 nm obtained for the U-SBGW fabricated using 193 nm DUV lithography. Both of the devices hadthe same corrugation width of 20 nm. The device fabricated using 193 nmDUV lithography has a longer grating length (1 cm), as compared to the4 mm length of the device fabricated using eBeam lithography. Since thelonger U-SBGW requires a larger area, the waveguide thickness variationsare greater than those for the shorter U-SBGW, which could account for thelarger bandwidth. Additionally, other variations that can occur between614.2. Experimental Resultswafers and processes, such as differences in the average waveguide thick-ness and waveguide width, could also contribute to the different bandwidthsmeasured for the two devices.4.2.4 Phase-shifted Spiral Bragg GratingsIn this section, experimental results for many TM P-SBGWs are presented.These TM P-SBGWs were fabricated using eBeam lithography. Severalvariations in the grating length and the corrugation width were applied tothese TM P-SBGWs. Their spectral responses and Q factors were thenanalyzed and discussed.As stated previously, a TM P-SBGW has one grating period-long cavityin the center of the grating. Fig. 4.13 shows an SEM image of a fabri-cated TM P-SBGW. Based on the simulation results (see Fig. 3.10 in Sec-tion 3.3.2), the narrow resonant peak occurrs at the centre wavelength of thetransmission and the reflection spectra. This was experimentally observed.Fig. 4.14 shows the measured transmission and reflection spectra of a TMP-SBGW with a grating period of 444 nm. The resonant peak occurrs at thecenter wavelength of 1551.3 nm, which is slightly greater than the designedvalue of 1550 nm.4.2.4.1 Design VariationsAs stated above, we varied the grating length and the corrugation width ofseveral TM P-SBGWs.Fig. 4.15 shows the transmission spectra of the TM P-SBGWs with grat-ing lengths of 300 µm, 350 µm, 400 µm, and 450 µm. These devices have thesame grating period of 444 nm and the same corrugation width of 100 nm.As shown in Fig. 4.15, the transmission stop band became deeper as thegrating length increased from 300 µm to 450 µm. Due to the increasedreflection, the longer grating also resulted in sharper resonant peak. TheQ factors of these resonance peaks were extracted using Eq. 3.14. Theyincreased from 17488 to 38124 as the grating length increased, as shown inFig. 4.17(a).624.2. Experimental Resultsȿ 1 μm Figure 4.13: SEM image of the center of a TM P-SBGW.1545 1550 1555 1560-20-100Wavelength (nm)Reflection (dB)1545 1550 1555 1560-10-50Wavelength (nm)Transmission (dB)Figure 4.14: Transmission and reflection spectra of a TM P-SBGW,which has parameters: Wcorr= 100 nm, L= 250 µm, and Λ=444 nm.634.2. Experimental Results1540 1545 1550 1555 1560-500  L=300 m1540 1545 1550 1555-500Transmission (dB)  L=350 m1540 1545 1550 1555 1560-500  L=400 m1540 1545 1550 1555 1560-500Wavelength (nm)Transmission (dB)  L=450 mFigure 4.15: Measured transmission spectra of TM P-SBGW withvarious corrugations. The devices have the same parameters: Λ=444nm, Wcorr=100 nm.644.2. Experimental Results1540 1545 1550 1555 1560-150  Wc orr=80nm1540 1545 1550 1555 1560-200  Wc orr=100nm1540 1545 1550 1555 1560-500  Wc orr=120nm1540 1545 1550 1555 1560-500Wavelength (nm)Transmission (dB)  Wc orr=140nmFigure 4.16: Measured transmission spectra of TM P-SBGW withvarious lengths. The devices have the same parameters: Λ=444 nm,L=400 µm.654.2. Experimental ResultsFig. 4.16 shows the transmission spectra of the TM P-SBGWs with vari-ous corrugation widths of 80 nm, 100 nm, 120 nm, and 140 nm. The deviceshave the same grating period of 444 nm. The grating length remains con-stant at 400 µm. As can be seen in Fig. 4.16, the transmission stop bandbecame wider and deeper as the corrugation width increased from 80 nm to120 nm. The resonant peak also became sharper as the corrugation widthincreased due to the increased reflections. The measured Q factors increasefrom 5428 to 28640, as shown in Fig. 4.17(b).300 350 400 4501.522.533.54x 104Length ( m)Q-factor  Wcorr=100 nm(a)80 90 100 110 120 130 1400.511.522.53x 104Wcorr (nm)Q-factor  L=400 m(b)Figure 4.17: Measured Q factor versus (a) grating length and (b)corrugation widthFigures 4.16 and 4.15 also show that the maximum transmission value ofthe resonant peak was reduced when the grating length or the corrugationwidth increased. This agrees with theoretical predictions and simulationresults (see Figures 3.12 and 3.13 in Section 3.3.2). As stated previously,the reduction of the maximum transmission value was due to the waveguideloss and the increased reflections, see Eq. (3.11).After measuring many devices, a maximum Q factor of approximately78790 was obtained for a 900 µm long TM P-SBGW with a corrugationwidth of 80 nm, as shown in Fig. 4.18. This value is smaller than thetheoretical maximum Q factor of 250000. Based on the experimental results664.2. Experimental Resultsshown above, the Q factor can be increased by increasing the grating lengthand corrugation width. Hence, more design variations are required to achievethe theoretical maximum Q factor.1530 1532 1534 1536 1538-40-30-20-10010Wavelength (nm)Transmission (dB)(a)1533.95 1534 1534.05 1534.1-30-20Wavele gth (nm)Transmission (dB)Q=7879019.5 pm(b)Figure 4.18: Measured maximum Q-factor. The device has parame-ters: Λ=432 nm, L=900 µm, and Wcorr=80 nm.4.2.5 Chirped Spiral Bragg GratingsAccording to simulations, the TM C-SBGW can have a negative, average,group delay slope over a broad passband. This was observed experimentally.Fig. 4.19 shows the measured reflection spectrum and the group delay of theTM C-SBGW, which had a corrugation width of 60 nm and a grating lengthof 4 mm. It can be seen that the reflection spectrum had a 3-dB bandwidthof 11.7 nm and a center wavelength near 1548 nm. The group delay wasmeasured using an Optical Vector AnalyzerTM STe by Luna Innovations,Inc. As shown in Fig. 4.19, the group delay decreased as the wavelengthincreased over the passband. As per the design, the C-SBGW had a groupdelay with an average slope of -11 ps/nm. As discussed in [50], devices withsuch negative group delay slopes can be used to compensate for positive dis-persion. Undesired ripples were also noticed in the group delay in Fig. 4.19.As mentioned previously, the ripples can be reduced or eliminated by usingapodization, which provides a less abrupt ending to the grating [49].674.2. Experimental Results1540 1545 1550 1555 1560-200Wavelength (nm)Reflection (dB)  0100Group delay (ps)slo pe=-11 ps/nm11.7 nmFigure 4.19: Reflection spectrum of the TM C-SBGW. The group de-lay decreases linearly as the wavelength increases within the passband.The device has parameters: Wcorr=60 nm, L=4 mm, and dΛ/dL =-6 nm/cm.1540 1545 1550 1555 1560-200Wavelength (nm)Reflection (dB)  SimulationMeasurmentFigure 4.20: Measured reflection spectrum of the TM C-SBGW fromFig. 4.19 is compared with the simulated reflection spectrum. Theaverage neff of the waveguide was adjusted by 0.98×10−3 in the sim-ulation.684.2. Experimental ResultsThe measured reflection spectrum was also compared with the simulatedreflection spectrum, as shown in Fig. 4.20. The measured reflection spectrumof the TM C-SBGW closely matched the simulated reflection spectrum. Thevariation of the bandwidth as compared to the ideal design is small, and,thus, we conclude that linear chirped Bragg gratings are compatible withour TM spiral waveguides.4.2.5.1 Design VariationsBased on the theoretical analysis and simulation results, the bandwidth ofTM C-SBGWs can be increased by either increasing the grating length orincreasing the chirp rate. This was experimentally observed.Fig. 4.21 illustrates the measured reflection spectra of the TM C-SBGWswith lengths ranging from 2 mm to 4 mm. The devices had the same corru-gation width of 60 nm and the chirp rate remained constant at -6 nm/cm. Asexpected, the bandwidth increased as the grating length increased. Fig. 4.22illustrates the measured reflection spectra of the TM C-SBGWs with chirprates varying from -6 nm/cm to -4 nm/cm. These devices had the samecorrugation width of 60 nm. The grating length remained at 4 mm. Thebandwidths also increased as the chirp rates of the grating periods increased.All these measured reflection spectra confirm that the TM C-SBGWs arepromising devices that may be used as broad band filters. Their bandwidthscan be controlled by properly adjusting the grating length or the chirp rate.As can be seen in Figures. 4.19, 4.21, and 4.22, the reflection spectrasuffered unexpected ripples. It is important to note that these ripples didnot stem from the fundamental designs. Prior to our experimentation, theripples were induced by the FP effects due to reflections from the sidewallroughness, the Y-branch, and the grating couplers [29]. Particularly, for theTE mode strip waveguide, the sidewall roughness had a relatively great effecton the spectral response [42, 63]. Since the TM spiral waveguides showedsmall sensitivities to sidewall roughness, see Section 4.2.2, we attribute thoseripples to the FP effects induced by reflections from the fully-etched gratingcouplers and the Y-junction. These reflection ripples depend on particular694.2. Experimental Results1530 1535 1540 1545 1550 1555 1560 1565 1570-30-25-20-15-10-505Wavelength (nm)Reflection (dB)  L=2mmL=3mmL=4mmFigure 4.21: Measured reflection spectra of the TM C-SBGWs withlengths of 2 mm, 3 mm, and 4 mm. Devices have fixed parameters:dΛ/dL =-6 nm/cm, Wcorr = 60 nm.1530 1535 1540 1545 1550 1555 1560 1565 1570-30-25-20-15-10-505Wavelength (nm)Reflection (dB)  d //d L=4nm/cmd //d L=5nm/cmd //d L=6nm/cmFigure 4.22: Measured reflection spectra of the TM C-SBGWs withchirp rates of -4 nm/cm, -5 nm/cm, and -6 nm/cm. Devices have fixedparameters: L=4 mm, Wcorr = 60 nm.704.2. Experimental Resultsdevices’ grating couplers, and, thus, they cannot be calibrated out. However,it is possible to reduce the FP effects by using grating couplers with low backreflection, such as shallow-etched grating couplers [75], and replacing the Y-junction with alternate coupler, such as a directional coupler.71Chapter 5Summary, Conclusion, andFuture work5.1 SummaryIn this thesis, we presented the design and characterization of spiral Bragggrating waveguides on the SOI platform for the fundamental TM mode.There are three types of TM SBGWs presented, including TM U-SBGW,TM P-SBGW, and TM C-SBGW.The TM spiral waveguide was based on 500 nm×220 nm strip waveguidesand the electric field distributions for curved SOI strip waveguides were an-alyzed using Lumerical Mode solutions. The refractive indices for differentcurvatures were analyzed and compared to a 500 nm×220 nm straight waveg-uide. The effective refractive indices for different widths were also obtained.The principles and characteristics of silicon waveguide Bragg gratings werereviewed. By following the coupling mode theory, the equation for the grat-ing coupling coefficient was derived. Then, the transfer functions of thesilicon Bragg gratings were analyzed using the transfer matrix method.Then, we presented a method of wrapping long BGWs into small, com-pacted areas using an S-shaped waveguide and two interleaved Archimedeanspirals. Based on the mode analysis for the curved SOI strip waveguidesmentioned above, the effective refractive index variation, as a function of theradius of curvature, was obtained. A minimum radius of the spiral waveg-uide was then chosen to be 15 µm, which ensured that the spiral waveg-uide was approximately equivalent to a straight waveguide. Such a straightwaveguide approximation was applied to numerically simulate our TM U-725.2. ConclusionSBGW, TM P-SBGW and TM C-SBGW. These devices were designed forvariations in grating period, grating length, and corrugation width. Theirspectral responses were calculated and analyzed using the TMM. Particu-larly, an FDTD method based on Bloch boundary conditions was used toaccurately calculate the bandwidths and coupling coefficients for the TMU-SBGWs.The designed SBGWs were fabricated using eBeam lithography and/orusing IMEC ePIXfab technology, using 193 nm DUV lithography. As com-pared to 193 nm DUV lithography, eBeam lithography results in corruga-tions fabricated on a device that are closer to the corrugations used in thedesigns and in the simulations. However, due to the smoother sidewalls,193 nm DUV lithography produces devices with lower propagation losses,for both of the TE and TM spiral waveguides, as compared to eBeam lithog-raphy. The spectral responses of the TM SBGWs were measured using anautomated measurement setup in our lab. The devices’ performance wereanalyzed and compared to the simulations. The thermal sensitivity of oneof the TM U-SBGW was also measured. In addition, the group delay of aC-SBGW was measured and compared with the simulation.5.2 ConclusionIn conclusion, we have demonstrated uniform Bragg gratings, phase-shiftedBragg gratings, and chirped Bragg gratings for the fundamental TM mode incompact spiral SOI strip waveguides. Our TM spiral waveguides are space-efficient, requiring an area of only 189×189 µm2 to accommodate 1 cm longBGWs and, thus, are less susceptible to fabrication non-uniformities. Basedon the experimental results, we verified that TM spiral waveguides havesmoother transmission spectra and lower propagation losses as compared toTE spiral waveguides. Particularly, the TM spiral waveguides fabricatedusing IMEC ePIXfab technology, via 193 nm DUV lithography, had a lowpropagation loss of 1 dB/cm. The TM U-SBGWs can be used as notch filterswith narrow bandwidths and large ERs, as narrow as 0.09 nm and as largeas 52 dB, respectively. The TM U-SBGWs can also be thermally tuned735.3. Suggestions for Future workwith a temperature sensitivity of 48 pm/◦C. The TM P-SBGWs exhibitsharp resonance peaks with large Q factors, as large as 78790. In addition,we demonstrated that TM C-SBGWs are compatible with our TM spiralwaveguides. We presented the measurements taken on a TM C-SBGW hav-ing a negative, average, group delay slope of −11 ps/nm.5.3 Suggestions for Future workThe reflection spectra of the U-SBGWs often have large sidelobes, see Fig. 3.6.Apodization can be implemented to reduce the levels of the side-lobes [65,76], and, thus, increase the ERs for the reflection spectra. However, thereflection spectra of the TM SBGWs also suffered from unexpected ripplesdue to the FP effects, caused by the reflections from the fully-etched gratingcouplers and the Y-junction. In future work, such ripples could be reducedusing grating couplers with low back reflection, such as shallow-etched grat-ing couplers [75], and also by replacing the Y-junction with an alternatecoupler, such as a directional coupler.We also suggest applying apodization on C-SBGWs. As we have seen,the group delays of the C-SBGWs suffered from relatively large ripples.Based on our simulations, apodization could be used to significantly reducethese unwanted ripples. As can be seen in Fig. 3.17, a smooth group delaywas obtained by using apodization.Additionally, it is worth investigating Bragg gratings that have mis-aligned sidewall corrugations on TM spiral waveguides. TE Bragg gratingsthat have misaligned sidewall corrugations on straight SOI waveguides havebeen demonstrated to precisely control coupling coefficients. Small couplingcoefficients can be achieved using relatively large misalignments [61]. As isknown, Bragg gratings, with small coupling coefficients, require long grat-ing lengths, hence, they can benefit from spiral configurations. Since TMspiral waveguides have smoother transmission spectra and have lower prop-agation losses than TE spiral waveguides, implementing TM SBGWs withmisaligned sidewall corrugations has the potential to obtain high qualitytransmission spectra with narrow bandwidths.74Bibliography[1] L. Chrostowski and M. Hochberg, Silicon Photonics Design, CambridgeUniversity Press, 2015.[2] http://www.intel.com/content/www/us/en/research/intel-labs-silicon-photonics research.html.[3] http://www.luxtera.com/.[4] F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on asilicon chip,” Nature Photonics, vol. 1, no. 1, pp. 65–71, 2006.[5] C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets,B. Esembeson, I. Biaggio, T. Michinobu, F. 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