{"http:\/\/dx.doi.org\/10.14288\/1.0392500":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Applied Science, Faculty of","type":"literal","lang":"en"},{"value":"Electrical and Computer Engineering, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Ma, Minglei","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2020-07-22T19:11:42Z","type":"literal","lang":"en"},{"value":"2020","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Doctor of Philosophy - PhD","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"Wavelength and polarization are the fundamental and key properties of lightwave transmissions. Photonic integrated circuits (PICs) on silicon-on-insulator (SOI) provide a low-cost, large-scale, and on-chip solution for the increasingly demanding optical interconnects. This thesis presents a theoretical and experimental study on wavelength and polarization manipulations for silicon photonic receiver applications. Firstly, wavelength and polarization components in SOI platforms are demonstrated; then, active subsystems based on the developed components are proposed and tested, in which mathematical methods-based, automated control algorithms are also investigated.\r\n\t\r\nFor the development of SOI components, first, narrow-band, Gaussian-apodized, spiral Bragg grating waveguides (SBGWs) are demonstrated for the use of wavelength filters. The fabricated apodized SBGWs perform smoother spectra with higher side-lobe suppression ratios, which prove that the applied apodization scheme can reduce the channel crosstalk of the multiple wavelength filters. Second, broadband, sub-wavelength grating (SWG)-assisted, adiabatic polarization splitter-rotators (PSRs) are experimentally demonstrated using an electron-beam lithography process and an optical lithography process, respectively. The SWG-PSRs are more compact than previously reported entirely adiabatic PSRs, and the PSRs fabricated using the optical lithography is the first implementation of an SWG-based structure in a standard, complementary metal-oxide semiconductor (CMOS) compatible fabrication process.\r\n\t\r\nFor the active subsystems demonstration, an automated polarization receiver (PR), formed by the adiabatic components, is proposed to overcome any arbitrary input polarization state from a standard optical fiber. Through the fabricated PR, high-speed transmission experiments are implemented to demonstrate the automated control process. Four control algorithms - greedy linear descent-based, basic gradient descent-based, two-point step size gradient descent-based, and two-stage optimization method-based control algorithm are developed. We implemented high-speed experimental to achieve automated adaptations and compared the control algorithms' performance as regards the iteration number and the output responses. Then, automated wavelength and polarization control in a wavelength-division multiplexing polarization receiver (WDM PR) are developed. We have designed and fabricated a two-channel WDM PR to implement the automated control process. The demonstrated gradient descent-based control algorithm is utilized for the automated adaptations of any arbitrary input polarization states, and, simultaneously, used for the automated stabilization of the channels.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/75248?expand=metadata","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"Wavelength and Polarization Controlfor Silicon Photonic ReceiverApplicationsbyMinglei MaB. Sc., Xidian University, 2010M. Sc., Beijing University of Aeronautics and Astronautics, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)July 2020\u00a9 Minglei Ma 2020The following individuals cer3fy that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the disserta3on en3tled: Examining Commi+ee: Wavelength and polariza3on control for silicon photonic receiver applica3onssubmiAed by Minglei Ma in par3al fulfillment of the requirements forthe degree of Doctor of Philosophyin Electrical EngineeringLukas Chrostowski, Professor, Electrical and Computer Engineering, UBCSupervisor Nicolas A. F. Jaeger, Professor, Electrical and Computer Engineering, UBCSupervisory CommiAee Member Sudip Shekhar, Associate Professor, Electrical and Computer Engineering, UBCSupervisory CommiAee MemberShuo Tang, Professor, Electrical and Computer Engineering, UBCUniversity ExaminerRyozo Nagamune, Associate Professor, Mechanical Engineering, UBCUniversity ExamineriiAbstractWavelength and polarization are the fundamental and key properties oflightwave transmissions. Photonic integrated circuits (PICs) on silicon-on-insulator (SOI) provide a low-cost, large-scale, and on-chip solution for theincreasingly demanding optical interconnects. This thesis presents a the-oretical and experimental study on wavelength and polarization manipu-lations for silicon photonic receiver applications. Firstly, wavelength andpolarization components in SOI platforms are demonstrated; then, activesubsystems based on the developed components are proposed and tested, inwhich mathematical methods-based, automated control algorithms are alsoinvestigated.For the development of SOI components, first, narrow-band, Gaussian-apodized, spiral Bragg grating waveguides (SBGWs) are demonstrated forthe use of wavelength filters. The fabricated apodized SBGWs performsmoother spectra with higher side-lobe suppression ratios, which prove thatthe applied apodization scheme can reduce the channel crosstalk of the mul-tiple wavelength filters. Second, broadband, sub-wavelength grating (SWG)-assisted, adiabatic polarization splitter-rotators (PSRs) are experimentallydemonstrated using an electron-beam lithography process and an opticallithography process, respectively. The SWG-PSRs are more compact thanpreviously reported entirely adiabatic PSRs, and the PSRs fabricated us-ing the optical lithography is the first implementation of an SWG-basedstructure in a standard, complementary metal-oxide semiconductor (CMOS)compatible fabrication process.For the active subsystems demonstration, an automated polarizationreceiver (PR), formed by the adiabatic components, is proposed to over-come any arbitrary input polarization state from a standard optical fiber.iiiThrough the fabricated PR, high-speed transmission experiments are imple-mented to demonstrate the automated control process. Four control algo-rithms - greedy linear descent-based, basic gradient descent-based, two-pointstep size gradient descent-based, and two-stage optimization method-basedcontrol algorithm are developed. We implemented high-speed experimentalto achieve automated adaptations and compared the control algorithms\u2019 per-formance as regards the iteration number and the output responses. Then,automated wavelength and polarization control in a wavelength-divisionmultiplexing polarization receiver (WDM PR) are developed. We have de-signed and fabricated a two-channel WDM PR to implement the automatedcontrol process. The demonstrated gradient descent-based control algorithmis utilized for the automated adaptations of any arbitrary input polarizationstates, and, simultaneously, used for the automated stabilization of the chan-nels.ivLay SummaryAn integrated data receiver is one of the essential building blocks for on-chipoptical interconnects. Silicon photonic integrated circuits allow for low-costand dense communications for optical data signals. Based on this promisingand mature technology, this thesis presents wavelength and polarizationcontrol solutions to high-speed optical communication in silicon photonicreceivers.As regards the design contributions, this thesis develops novel siliconphotonic components and sub-systems. As the contributions to the designmethodologies, efficient simulation approaches that cost less computationalefforts to large structures are also introduced. In terms of the control tech-nique contributions, various automated control algorithms based on mathe-matical optimization theories are studied and used to realize proposed auto-mated control for on-chip data processing systems. The research work pavesa way to develop low-cost, high-speed, intelligent, fully integrated system forfuture on-chip optical interconnects.vPrefaceThe content of this thesis is mainly based on the manuscripts, which havebeen or will be published, resulting from collaborations with other researchers.Note that only publications directly arising from the research work presentedin this thesis are listed here. A complete list of publications is given in Ap-pendix A.1. M. Ma, K. Murray, M. Ye, S. Lin, Y. Wang, Z. Lu, H. Yun, R. Hu,N. A. F. Jaeger, and L. Chrostowski, \u201cSilicon Photonic PolarizationReceiver with Automated Stabilization for Arbitrary Input Polariza-tions,\u201d In Conference on Lasers and Electro-Optics (CLEO), pp. STu4G.8, 2016.L. Chrostowski, and M. Ma contributed the idea, M. Ma conductedthe device design, performed the measurements and data analysis, anddrafted the manuscript. K. Murray assisted the simulations for the sys-tem model. M. Ye and R. Hu assisted the edge-coupling measurements.N. A. F. Jaeger helped editing the manuscript, and L. Chrostowski su-pervised the project. All authors commented on the manuscript.Location: Chapter 4.2. M. Ma, Z. Chen, H. Yun, Y. Wang, X. Wang, N. A. F. Jaeger,and L. Chrostowski, \u201cApodized Spiral Bragg Grating Waveguides inSilicon-on-Insulator,\u201d IEEE Photonics Technology Letters, vol. 30, no. 1,pp. 111-114, 2017.M. Ma contributed the idea, conducted the devices\u2019 design, imple-mented the measurement, the data processing, data analysis, andviwrote the manuscript. Z. Chen helped with many discussions on thedesign methodology. H. Yun assisted the measurements. N. A. F. Jaegerhelped structuring the manuscript, and L. Chrostowski supervised theproject. All authors commented on the manuscript.Location: Chapter 2.3. M. Ma, A. H. K. Park, Y. Wang, H. Shoman, F. Zhang, N. A. F. Jaeger,and L. Chrostowski, \u201cSub-wavelength grating-assisted polarization splitter-rotators for silicon-on-insulator platforms,\u201d Optics Express, vol. 27,no. 13, pp. 17581-17591, 2019.M. Ma contributed the idea, conducted the devices\u2019 design, performedthe measurement, the data processing and analysis, and drafted themanuscript. A. H. K. Park helped with the design of the edge-couplers,and the simulation methodology. Y. Wang helped with discussionabout the sub-wavelengh grating designs. H. Shoman assisted edit-ing the optical micro-graph. N. A. F. Jaeger helped structuring themanuscript, and L. Chrostowski supervised the project. All authorscommented on the manuscript.Location: Chapter 3.4. M. Ma, H. Shoman, K. Tang, S. Shekhar, N. A. F. Jaeger, and L. Chros-towski, \u201cAutomated control algorithms for silicon photonic polariza-tion receiver,\u201d Optics Express, vol. 28, no. 2, pp. 1885-1896, 2020.M. Ma contributed the idea, developed the automated control algo-rithms, performed the measurement, the data processing and analysis,and drafted the manuscript. H. Shoman helped discussing the con-trol algorithm, wire-bonding the chip, and setting up the high-speedexperiments. K. Tang assisted developing the two-stage control algo-rithm and debugging the Python scripts. S. Shekhar helped editingthe manuscript. N. A. F. Jaeger and L. Chrostowski supervised theproject. All authors commented on the manuscript.viiLocation: Chapter 4.5. M. Ma, H. Shoman, S. Shekhar, N. A. F. Jaeger, and L. Chros-towski, \u201cAutomated Adaptation and Stabilization of a Tunable WDMPolarization-Independent Receiver on Active Silicon Photonic Plat-form,\u201d IEEE Photonics Journal, accepted and to be published, 2020.M. Ma and H. Shoman equally contributed to this work. M. Macame up with the idea and the devices\u2019 design, performed the experi-ments, the data processing and analysis, and drafted the manuscript.H. Shoman drew and submitted the design layouts to the fabricationfoundry, wire-bonded the chips, helped setting up the high-speed ex-periments, and assisted drafting the manuscript. N. A. F. Jaeger andS. Shekhar helped editing the manuscript. L. Chrostowski supervisedthe project. All authors commented on the manuscript.Location: Chapter 5.viiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . xxiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Silicon photonics for optical interconnects . . . . . . . . . . . 11.2 SOI integrated wavelength filters . . . . . . . . . . . . . . . . 31.3 Optical interface and on-chip polarization control . . . . . . 61.4 About this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.2 Thesis organization . . . . . . . . . . . . . . . . . . . 102 Apodized spiral Bragg grating waveguides . . . . . . . . . . 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Design and Simulations . . . . . . . . . . . . . . . . . . . . . 14ix2.3 Measurement and results . . . . . . . . . . . . . . . . . . . . 192.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Sub-wavelength grating-assisted polarization splitter-rotators253.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 SWG-assisted PSR using an E-beam lithography process . . 263.2.1 Operating principle and design . . . . . . . . . . . . . 263.2.2 Simulation methodology and results . . . . . . . . . . 293.2.3 Measurement results . . . . . . . . . . . . . . . . . . 313.3 SWG-assisted PSR using a standard optical lithography pro-cess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.1 Operating principle and design . . . . . . . . . . . . . 333.3.2 Simulation and experiment results . . . . . . . . . . . 353.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 Active polarization receiver and automated control algo-rithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1 Polarization receiver in active silicon photonic platform . . . 424.1.1 Operating principle and design . . . . . . . . . . . . . 424.1.2 System modelling and analysis . . . . . . . . . . . . . 434.1.3 Experiment and results . . . . . . . . . . . . . . . . . 464.2 Automated control algorithms for the PR . . . . . . . . . . . 474.2.1 Greedy linear descent- and gradient descent-based con-trol algorithms . . . . . . . . . . . . . . . . . . . . . 474.2.2 Two stage method-based control algorithm . . . . . . 504.3 High-speed experiment and results . . . . . . . . . . . . . . . 524.3.1 Automated control for an arbitrary input polarization 544.3.2 Real-time automated control test . . . . . . . . . . . 574.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . 574.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59x5 Automated wavelength and polarization control of WDMPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.1 Two-channel design and operating principle . . . . . . . . . . 625.2 Experiments and results . . . . . . . . . . . . . . . . . . . . . 645.2.1 Experiment preparation and automated control . . . 645.2.2 Long-duration, continuous, automated adaptation andstabilization test . . . . . . . . . . . . . . . . . . . . . 705.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . 745.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756 Conclusion and future work . . . . . . . . . . . . . . . . . . . 766.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84AppendicesA Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102A.1 Journal Publications . . . . . . . . . . . . . . . . . . . . . . . 102A.2 Conference Proceedings . . . . . . . . . . . . . . . . . . . . . 103B Derivation of the theoretical model for a tunable MZI . . 106C Phase shift - current relation derivation . . . . . . . . . . . 108D Code listing of the automated control algorithms . . . . . 110D.1 Python code for greedy linear descent method . . . . . . . . 110D.2 Python code for basic gradient descent method . . . . . . . . 111D.3 Python code for two-point step size gradient descent method 113D.4 Python code for two-stage optimization method . . . . . . . 114xiList of Tables3.1 Measurement configurations of the GC ports. . . . . . . . . . 313.2 Summary of the optical I\/O ports used in our measurement. . 393.3 Figure of Merits of the Fabricated Mode-evolution-based PSRs. 406.1 A summary of recently published BGW designs on SOI plat-forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79xiiList of Figures1.1 Silicon photonic integrated circuit applications for typical op-tical transceiver communications. Major optical componentsin the transmitters and receivers are also listed: optical inter-face (I\/O), power splitter\/combiner (PS), high-speed modu-lators (MOD), wavelength filters (Mux\/DeMux), polarizationcontrol (PC), and photodetectors (PD). . . . . . . . . . . . . 11.2 A schematic of cross-section of a standard silicon-on-insulatorplatform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 (a) A schematic of a silicon BGW where \u039b is the gratingperiod, \u2206W is the corrugation width, N is the period num-ber, and W is the average waveguide width. And an SEMimage of a fabricated device. (b) An example of normalizedtransmission and reflection responses of a BGW across C-band. 41.4 (a) A schematic and an SEM image of the MRR-based add-drop filter. (b) Normalized transmission spectra at the drop-port and through-port of a MRR filter. . . . . . . . . . . . . . 51.5 (a) An schematic of edge-coupling interface and a top viewof a nano-taper and a TE mode polarization splitter-rotatorstructure. (b) A schematic of surface vertical-coupling inter-face and a top view of a polarization splitting-grating coupler. 71.6 (a) A schematic of a polarization diversity circuit and (b) aschematic of active polarization control circuit. . . . . . . . . 8xiii2.1 (a) Schematic of the Gaussian-apodized SBGW. The totallength is 3 mm long. (b), (c) Zoom-in showing the gratingperiod, the corrugation width, \u2206W = (Wmax\u2212Wmin)\/2, andthe spacing between two spiral waveguides, g. . . . . . . . . . 142.2 Calculated apodized sidewall corrugation width (in blue) anddistributed coupling coefficient (in orange) as functions of thelongitudinal position. The index of Gaussian function is cho-sen as 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Simulated (a) reflection and (b) transmission spectra withthe proposed uniform (in orange) and Gaussian-apodized (inblue) SBGW. . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 (a) Transmission ER (in blue), reflection SLSR (in orange),and (b) bandwidth versus various Gaussian indices of apodiz-ing \u2206W and \u03ba, respectively. . . . . . . . . . . . . . . . . . . . 172.5 Schematic of a period-chirped SBGW. . . . . . . . . . . . . . 182.6 Simulated reflection spectrum (in blue) and group delay (inorange) of the proposed (a) uniform and (b) Gaussian-apodizedC-SBGW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.7 SEM images of an apodized SWBG device. (a) The completeapodized SWBG with input and output GCs. (b) Zoom-inof the Archimedean SWBG with Gaussian apodization. (c)Zoom-in of the center of the S-shaped WBG. . . . . . . . . . 202.8 (a) An automated measurement setup and zoom-in of siliconchip and fiber array. (b) A block diagram of the automatedmeasurement setup. . . . . . . . . . . . . . . . . . . . . . . . 212.9 Measured (a) reflection and (b) transmission spectra for theuniform (in orange) and Gaussian-apodized (in blue) SWBGs.(c) Zoom-in of the measured reflection spectra for the uniform(in orange) and Gaussian-apodized (in blue) SWBGs. TheSLSR is 0.9 dB for the uniform SWBG and 13.2 dB for theapodized one. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22xiv2.10 (a) Gaussian noise introduced period lengths as a functionof the period number. (b) Simulated reflection with randomgrating period noise (in red, purple, and green) and our mea-sured spectrum (in blue). . . . . . . . . . . . . . . . . . . . . 232.11 Measured reflection spectrum (in blue) and group delay (inorange) for the (a) uniform and (b) Gaussian-apodized C-SBGWs.The group delay outside the passband (in grey) refersto data with a large measurement uncertainty. . . . . . . . . 233.1 (a) 3D view of our air-clad PSR with mode profiles at variouspoints along the device; (b) top view of the adiabatic nano-taper; (c) top view of the SWG-assisted adiabatic couplerwith labelled design parameters. The parameters in thesesegments are as follows: in (b), W1 = 600 nm, W2 = 700nm, W3 = 900 nm (given that Winput = 450 nm), L1 = 6 \u00b5m,L2 = 30 \u00b5m, and L3 = 15 \u00b5m; in (c), L4 = 100 \u00b5m, \u039b = 200nm, g = 100 nm, W3 = 900 nm, W4 = 200 nm, W5 = 650 nm,and W6 = 800 nm. . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Calculated effective indices for the first three eigenmodes ofan air-clad strip waveguide, as a function of the waveguidewidth. The center wavelength is 1550 nm. . . . . . . . . . . . 283.3 Simulation circuit model of the proposed PSR in INTER-CONNECT (INTC), and the extracted S-parameter compactmodels (ATaper and ASplitter) calculated by FDTD. . . . . . 303.4 Simulated transmission spectra of the proposed PSRs, with(a) TM00 mode and (b) TE00 mode inputs. . . . . . . . . . . 303.5 SEM images of a fabricated PSR. (a) The complete test struc-ture of PSR, including the GC ports and their calibrationstructures. (b) The entire structure of the PSR. (c) Zoom-inof the blunt tip on the SWG-AC. (d) Zoom-in of the middlesection of the SWG-AC. (e) Zoom-in of the S-bend and theSWG taper on the splitting section. . . . . . . . . . . . . . . 32xv3.6 Transmission spectra of the fabricated PSRs, with (a) TM00mode and (b) TE00 mode inputs. . . . . . . . . . . . . . . . . 333.7 (a) A 3D view of our oxide-clad PSR (For purpose of clar-ity, oxide cladding layer is not drawn and the layers\u2019 thick-ness\/length ratios are enlarged); Schematic of the adiabaticbi-level taper (b) top view with mode profiles at differentpoints along the bi-level taper, and (c) calculated effective in-dices of the first three modes along the first half of the bi-leveltaper. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.8 (a) TM00 to TE01 mode conversion efficiency for various lengthsof LA and LB in the bi-level taper. Inset Fig.: spectra of themode conversion efficiency for a TM00 mode and a TE00 modeinput. The lengths LA and LB are chosen to be LA = 35 \u00b5mand LB = 30 \u00b5m. Top view of the simulated electric fielddistribution for the bi-level taper when launching (b) a TM00mode and (c) a TE00 mode. . . . . . . . . . . . . . . . . . . . 363.9 Normalized transmission for various lengths of L4 in the adia-batic coupler (AC) section, with a TE01 mode and TE00 modeinputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.10 Simulated transmission spectra of the proposed PSRs, with(a) TM00 mode and (b) TE00 mode inputs. . . . . . . . . . . 373.11 (a) A photo of the edge-coupling test stage and zoom-in ofthe transposer fiber-array aligning to the chip. A cross-sectionview (b) and a top view (c) of the alignment between trans-poser and the chip. . . . . . . . . . . . . . . . . . . . . . . . . 383.12 Measured transmission spectra of the fabricated PSRs, with(a) TM00 mode and (b) TE00 mode inputs. . . . . . . . . . . 384.1 A block diagram showing that the arbitrary input polariza-tion states (red trace on LHS Poincare\u00b4 sphere) are convertedto a TE polarization state (red point on RHS Poincare\u00b4 sphere)through the proposed polarization receiver. . . . . . . . . . . 42xvi4.2 (a) Schematic of the polarization receiver consisting of an edgecoupler, a PSR, a balanced MZI including two thermal phaseshifters, two broadband 3dB couplers, and a photodetector(PD). (b) Optical micrograph of a fabricated polarization re-ceiver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 Normalized power at the output port as a function of thethermal phase shifts, \u2206\u03c61 and \u2206\u03c62, of the MZI for four inputpolarization ratios: (a) 100% TE mode input, (b) 100% TMmode input, (c) 50% TE and 50% TM mode input, and (d)75% TE and 25% TM mode input. . . . . . . . . . . . . . . . 444.4 Measured optical power at the output port with various tun-ing powers applied to the thermal phase shifters, H1 and H2,for four input polarization ratios: (a) 100% TE mode input,(b) 100% TM mode input, (c) 50% TE and 50% TM modeinput, and (d) 75% TE and 25% TM mode input. . . . . . . . 454.5 (a) Optical power of the output and feedback ports of the po-larization receiver versus time (b) Recorded eye-diagrams ofoutput signal at each experimental stage: (i) before (polarization-scramble state), (ii) during, and (iii) after the automated op-timization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.6 A flow diagram illustrating the GLD-based minimization methodfor the control algorithm. . . . . . . . . . . . . . . . . . . . . 484.7 A flow diagrams illustrating the basic GD- and B-B GD-basedminimization method for the control algorithms. . . . . . . . 494.8 Flow diagram illustrating the minimization process for thesingle-iteration, two-stage control algorithm: (a) an \u201dISTM\u201dmethod for Stage 1 and (b) a dynamic minimum trackingmethod for Stage 2. . . . . . . . . . . . . . . . . . . . . . . . 514.9 (a) Schematic of the experimental setup for eye diagram andBER measurements. (b) A photo of the alignment betweenthe transposer fiber array and chip, while the electrical prob-ings are positioned. . . . . . . . . . . . . . . . . . . . . . . . . 53xvii4.10 Measurement results for four control algorithms: (a) BERsversus iteration and (b) normalized optical power at the out-put and feedback ports of the PR . . . . . . . . . . . . . . . . 554.11 (a) BER versus optical input power (markers for measuredBERs and solid lines for the polynomial fittings) and (b) somemeasured eye diagrams with four different polarization statesin the optical fiber. The polarization angles of the polariza-tion state 1 - 4: 3.0 degree, 13.5 degree, 22.8 degree, and 84.6degree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.12 Measurement results of tracking and stabilization of continu-ous changed input polarization states: (a) BERs versus track-ing time, a reference BER level (dash line), and optical powerpenalty to achieve a BER of 10-10; (b) Normalized opticalpower at feedback port of the PR versus tracking time; (c)Polarization angles and extinction ratios of the transmittedlight in the optical fiber versus tracking time, and electricfield distributions of slow-axis and fast-axis polarization state. 585.1 (a) A block diagram showing that the arbitrary input po-larization states for two wavelength channels (red and purpletraces on LHS Poincare\u00b4 sphere) are both converted to TE po-larization states in separate outputs (the mixed color pointon RHS Poincare\u00b4 sphere) through the proposed WDM polar-ization receiver. Proposed WDM polarization control for Nchannel data processing circuits: (b) A polarization diversityscheme using a PSR (PSGC) and two demultiplexers (De-Muxes), and (c) a WDM, active polarization control schemeusing a PSR (PSGC), two demultiplexers (DeMuxes), and Ntunable MZIs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61xviii5.2 (a) Schematic of the two-channel WDM polarization receiverfor two wavelength-channels each having arbitrarily polarizedinputs (SiO2 cladding layer is not included in this figure).Inset figure: Zoom-in of a MRR filter with an IRPH. (b) Anoptical micro-graph of a fully fabricated two-channel WDM-polarization receiver. . . . . . . . . . . . . . . . . . . . . . . . 635.3 Normalized drop-port spectra as-fabricated and after tuningan MRR first to Ch. 1 and then to Ch. 2. Inset Fig.: Pho-tocurrent IPD measured in the MRR\u2019s IRPH while tuning toCh. 1 and to Ch. 2. Applied voltages V1 and V2 are foundwhen locating the maximum photocurrent of the IRPH toCh. 1 and to Ch. 2, respectively. . . . . . . . . . . . . . . . . 655.4 (a) A schematic of the experimental setup for eye diagram andBER measurement. (b) A photo of the actual experimentalsetup and (c) a zoom-in of the tested wire-bonded chip withthe chip carrier. . . . . . . . . . . . . . . . . . . . . . . . . . . 665.5 Recorded eye diagrams for Ch. 1, (a) - (d) OOK and (e) - (h)PAM-4 output signal at each control stage: In Stage 0, (a),(e), all of control was offline; In Stage 1, (b), (f), the desiredchannel (Ch. 1) was found by configuring MRR\u2019s IRPH; InStage 2, eye opening (c), (g) during and (d), (h) after theautomated polarization adaptation. . . . . . . . . . . . . . . . 675.6 (a) The HLP, VLP, and RCP polarization states recorded onthe Poincare\u00b4 sphere, and normalized optical powers at outputand feedback ports,by sweeping the thermal phase shifter H21(b), (d), (f), and H11(c), (e), (g) for each polarization state. . 695.7 (a) The measured coordinates of the monitored polarizationstates on the Poincare\u00b4 sphere and the corresponding normal-ized Stokes parameters for (b) Ch. 2 and (c) Ch. 1 versustracking time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71xix5.8 (a) Chip temperature versus tracking time; (b) Wavelengthaligning and stabilization: applied voltages to the MRRs\u2019IRPHs versus tracking time. Inset Figure: Initial IRPH\u2019sphotocurrents (IPD) measured while tuning the MRRs for thetwo channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.9 (a) BERs versus tracking time, a reference BER level (dashline), and normalized optical powers at feedback ports of theWDM PR versus tracking time; (b) Recorded applied electricpower to the MZIs\u2019 thermal phase shifters versus trackingtime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.1 A schematic of demonstrated polarizaiton receiver (a) andthe extended phase shifter (H1) design with the endless phaseshifting structure (b). . . . . . . . . . . . . . . . . . . . . . . 816.2 A schematic of the proposed CWDM polarization receiverbased on a contra-DC, a PRBG, and a tunable MZI. . . . . . 826.3 Schematic of (a) an expanded N-channel WDM PR and (b)a proposed improved N-channel design. The single ring canbe replaced with a double ring filter to improve the channelisolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83B.1 A schematic of the tunable MZI in the PR. . . . . . . . . . . 106xxList of AbbreviationsB-B GD Barzilai and Borwein Gradient DescentBER Bit Error RatioBGW Bragg grating WaveguideBOX Buried OxideBW BandwidthCDC Contra-directional CouplerCMOS Complementary Metal Oxide SemiconductorCWDM Coarse Wavelength-Division MultiplexingDWDM Dense Wavelength-Division MultiplexingED Error DetectorEDFA Erbium Doped Fiber AmplifierER Extinction RatioFDTD Finite-Difference Time-DomainFSR Free Spectral RangeFOM Figure of MeritsGC Grating CouplerGLD Greedy Linear DescentxxiIRPH In Resonator Photoconductive HeaterISTM Initializing, Sampling, Training, MinimizingMRR Micro-ring ResonatorMZI Mach-Zehnder InterferometerNRZ Non-return to ZeroOOK On-off KeyingPA Polarizaion AnalyzerPAM-4 Pulse-amplitude Modulation 4-levelPD Photo-detectorPDL Polarization Dependent LossPIC Photonic Integrated CircuitPC Polarization ContorllerPEM Polarization Extinction-ratio MeterPM Polarization-maintainingPR Polarization ReceiverPRBS Pseudo Random Binary SequencePRBG Polarization Rotating Bragg gratingPSR Polarization Splitter-rotatorPSGC Polarization Splitting Grating CouplerPPG Pulse Pattern GeneratorRx High-speed Photo-receiverSBGW Sprial Bragg grating WaveguidexxiiSLSR Side-lobe Suppression RatioSWG Sub-wavelength GratingSWG-AC Subwavelength grating-assisted adiabatic couplerTEC Thermo-electric CoolerTMM Transfer-matrix MethodVOA Variable Optical AttenuatorWDM Wavelength-Division MultiplexingxxiiiAcknowledgementsI would like to sincerely thank my supervisor Prof. Lukas Chrostowski forthe guidance throughout this long yet wonderful journey. Without his greatsupports and patience, I could not imagine any achievements that I hadin my Ph.D. study. Beyond that, his great mentorship, encouraging me tonever give up when facing frustrations and keep working hard when havingany success, builds up my mindset to be a qualified Ph.D. researcher. Hisattitudes and passions towards every aspect - work, family, and outdooradventures - always influence and inspire me to chase outstanding in life.I want to express my deep gratitude to Prof. Nicolas A. F. Jaeger forhis academic guidance throughout the years. Looking back at the process ofmy research projects, Dr. Jaeger spent number of hours providing dedicatedhelps to every project. Those precious times when I revised papers anddiscussed problems with him not only help me strengthen my academicskills, but also make me truly understand what critical thinking, honesty,and communication mean to my work and future career. For me, it is ahonoured privilege to be his mentee in my Ph.D. program.I am thankful to Prof. Sudip Shekhar for being my Ph.D. committeemember, and giving me financial and great academic support in the lastyear of my Ph.D. study. I also appreciate Dr. Dylan Logan and Dr. EdgarHuante-Ceron for their generous helps during my internship at Ranovus Inc.The six months\u2019 experience gave me a real-world industrial vision and anexcellent opportunity to know the city of Ottawa.I thank the NSERC Si-EPIC program for financial support and train-ing, and CSC scholarship for financial support during my Ph.D. study. Thefabrication of our active chips and the silicon photonic workshops held inCanada cannot happen without the supports of CMC Microsystems. Par-xxivticularly, I acknowledge Dr. Dan Deptuck and Dr. Jessica Zhang from CMCMicrosystems for their assistance with the tape-outs.Lastly, I would like to thank all my colleagues with whom I have beenworking in UBC photonic research group, especially Hossam Shoman, Dr. YunWang, Zhitian Chen, Anthony Park, Dr. Han Yun, Dr. Zeqin Lu, Fan Zhang,Kyle Murray, Dr. Xu Wang, Dr. Mengyuan Ye, Norhan Eid, Stephen H. Lin,and Michael Caverley for their supports and helps. I also want to acknowl-edge my friends Keyi Tang for the great collaboration of the automatedcontrol algorithms project and Fan Weng for the priceless friendship andsupports, which help me going through those hard times.xxvDedicationTo my parents.xxviChapter 1Introduction1.1 Silicon photonics for optical interconnectsOptical interconnects have been widely used in high-speed data commu-nications, over a broad range of transmission distances such as long-haul(500 km - 6000 km), metro (10 km - 500 km), rack (100 m - 10 km), back-plane (10 cm - 100 m), and chip (mm - cm) communications [1]. IntegratingFigure 1.1: Silicon photonic integrated circuit applications for typical opticaltransceiver communications. Major optical components in the transmittersand receivers are also listed: optical interface (I\/O), power splitter\/combiner(PS), high-speed modulators (MOD), wavelength filters (Mux\/DeMux), po-larization control (PC), and photodetectors (PD).1the optical components for communication systems onto photonic integratedcircuit (PIC) significantly saves cost and footprint [2]. Silicon photonics isnow a mature and key technology for making PICs used in various researchareas. In the past decade, numerous research work utilizing this technologyhas been published and numbers of industry companies have released theircommercialized silicon photonics products [3]. Owing to their compatibilitywith CMOS electronics fabrication processes, silicon PICs are also widely ac-cepted and used to realize high-speed optical interconnects with high-densityintegration and low manufacturing costs [4]. Fig. 1.1 1 shows typical high-speed optical communications using integrated optical transceivers (trans-mitters and receivers), where silicon PICs are used to develop the essentialoptical components (e.g., optical couplers (I\/O), wavelength filters (Mux-es\/DeMuxes), high-speed modulators (MODs), and photodetectors (PDs)).Figure 1.2: A schematic of cross-section of a standard silicon-on-insulatorplatform.Silicon-on-insulator (SOI) is the typical platform for such PICs devel-opments. Its high refractive index contrast between silicon waveguides andsurrounding medium allows for the creation of compact devices. SOI wafersprovide a way to implement dense and monolithic integration of various op-tical components with minimum footprints. A cross-section of a standardSOI platform with multi-layers and several commonly used optical compo-1The PIC photo is provided by [5]2nents is presented in Fig. 1.2 [6]. Silicon substrate (\u223c700 \u00b5m thick), buriedoxide (BOX) layer (2 to 3 \u00b5m thick), and silicon layer (220 nm and 90 nmthick) form the basic passive SOI wafer. The optical components are de-fined in the top silicon layer and are normally covered by a silicon-dioxidecladding layer for protection. Light propagates through the low-loss siliconlayer over a broad operating bandwidth from 1260 nm to 1620 nm [5]. Aswe mentioned above, the passive components, such as edge couplers, gratingcouplers (GCs), power or polarization beam splitters\/combiners, and wave-length filters, are generally designed by engineering the effective index of thewaveguides. Moreover, having other material layers and carrier injection inthe silicon waveguides, one can use the thermal-optic and plasma-dispersioneffect to design active components such as phase shifters (metal heatersor n doped waveguides), high-speed modulators (p-n junctions), and pho-todetectors (Ge PDs). Eventually, an on-chip transceiver subsystem can beachieved by assembling all the essential passive and active PICs, as shownin Fig. 1.1.1.2 SOI integrated wavelength filtersOne crucial metric of silicon photonics is the capability of wavelength-division multiplexing (WDM), which is one of the typical technologies toincrease the aggregate bandwidth of an optical communication system [7, 8].Thus, SOI integrated wavelength filters, i.e., wavelength Muxes\/DeMuxes,are the necessary elements for high-speed optical interconnects. SiliconBragg grating waveguide (BGW) is one promising option among the WDMfilters. As shown in Fig. 1.3(a), a standard BGW can be implemented byphysically corrugating the silicon waveguides on an SOI platform. Fig. 1.3(b)shows the typical spectra response of a uniform BGW designed for C-band.Utilizing the coupled-mode theory and transfer-matrix method (TMM) [9],one can decide the BGW\u2019s geometry parameters to achieve the aimed spec-tral properties, such as Bragg wavelength, bandwidth, and extinction ratio(ER) [10]. An SOI BGW enables us to design wide or narrow bandwidthfilters and, more importantly, it also has an unlimited free spectral range3Figure 1.3: (a) A schematic of a silicon BGW where \u039b is the grating period,\u2206W is the corrugation width, N is the period number, and W is the averagewaveguide width. And an SEM image of a fabricated device. (b) An exampleof normalized transmission and reflection responses of a BGW across C-band.(FSR). These advantages facilitate designing various wavelength filters forcoarse-WDM (CWDM) or dense-WDM (DWDM) applications. For exam-ple, silicon integrated BGWs with \u223c0.4 nm narrow bandwidths were demon-strated in [11]; The BGW assisted contra-directional couplers for CWDMnetworks were reported in [12, 13]; Long spiral BGWs with high ERs weredeveloped in [14, 15]; And apodized BGWs with high side-lobe suppressionratios (SLSRs) were demonstrated in [16]. In this thesis, we will demon-4strate a novel apodized spiral BGW filter with a high ER and a high SLSR,see Chapter 2.Another option for on-chip WDM filters is silicon micro-ring resonators(MRRs). Its compact footprint size and spectral characteristics (e.g., highextinction ratios and bandwidths) enable highly efficient integration andcontrol (modulation or wavelength tuning) [17], especially for multi-channelDWDM PICs [18, 19]. As shown as Fig. 1.4(a) 2, an MRR based add-dropFigure 1.4: (a) A schematic and an SEM image of the MRR-based add-dropfilter. (b) Normalized transmission spectra at the drop-port and through-port of a MRR filter.filter is formed by two straight, one looped waveguide, and two couplingregions. The input light is injected into the bottom bus-waveguide and2The MRR\u2019s SEM image is provided from [20]5passes to the coupling region, where light is then transmitted into the ring.When the lightwave travelling inside the ring builds up a round-trip phaseshift of integer times 2pi, it constructively interferes with the input lightso that the resonance at the center wavelength occurs in the cavity. Withthe second coupling region, the top bus-waveguide partially drops the res-onating light to the output-port (drop-port). Fig. 1.4(b) shows the typicaltransmission spectra at the drop-port and through-port in C-band, wherewe can see the three adjacent channels with an identical FSR. Each chan-nel\u2019s bandwidth usually needs to be optimized for certain high-speed datatransmission as a multi-channel Mux\/DeMux [21]. However, it should bementioned that a common drawback to MRR\u2019s implementation is its res-onance (filtering) susceptibility to fabrication variations and backgroundtemperature fluctuations, which can hold back its large commercialization[22]. To improve such drawbacks, additional components (e.g., photodetec-tors and heaters) need to be included in order to sense and tune its resonancewavelengths actively [23, 24]. Alternatively, an in-resonator photoconduc-tive heater (IRPH), which has been demonstrated in MRRs [25], allowsone to simultaneously sense (by measuring the photocurrent) and control(via the applied heater voltage) a filter\u2019s resonance condition. Therefore,using the IRPH in an MRR provides a promising approach, with minimalfootprint, to automatically align and stabilize the transmitted light at thedesired wavelength channel. In Chapter 5, we will use such approach in theMR-based filters to demonstrate a novel automated control for an on-chipWDM receiver.1.3 Optical interface and on-chip polarizationcontrolIn order to transfer the data on and off silicon chips, PICs need to be in-terfaced with optical fibers. As shown in Fig. 1.5 3, surface (vertical)- andedge-coupling are the two main approaches to the interface between optical3The schematic cartoons are revised from [5]6fibers and PICs. Various edge or surface interfacing designs and techniqueshave been recently developed, such as polarization insensitive, low-loss edgecouplers [26, 27], wide-band, sub-wavelength grating (SWG) based nano-tapers [28, 29] and GCs [30\u201333], low-cost, fiber-to-chip packaging using mi-croelectronic or flip-chip technologies [34\u201336], and multi-channel, low-lossinterface using photonic wire bonding [37, 38]. However, another challengethat needs to tackle is the polarization mismatch between an optical fiberand an on-chip coupler\/waveguide. When the input light in an optical fiberpasses through the on-chip interface, it couples into two orthogonal polar-ization modes, i.e., transverse electric (TE) and transverse magnetic (TM)modes of a silicon waveguide. Due to the difference of their mode properties,such as effective index and mode field confinement, a silicon PIC typicallyoperates in either of these two polarizations, in other words, it may not workidentically for both TE and TM modes. Thus, if the polarization state in(a)Optical fiber(b)Optical fiberPSGCTETETETETE+TMTE+TMTE+TMTE+TMFigure 1.5: (a) An schematic of edge-coupling interface and a top view ofa nano-taper and a TE mode polarization splitter-rotator structure. (b) Aschematic of surface vertical-coupling interface and a top view of a polariza-tion splitting-grating coupler.an optical fiber is randomly changed by induced stress or temperature fluc-tuations, indicating that the injected polarization state may be no longer7aligned to the desired mode of the silicon waveguide, the performance of thePICs can be detrimentally affected. To solve this problem, we need to eitherdesign a polarization control PIC, or maintain the polarization states in thefiber. Using polarization-maintaining (PM) fiber can reduce the polarizationchanges in the fiber, but, due to its large dispersion, high cost, and highlydemanded alignment to the PICs [5], PM fiber can not be a sustainablesolution in real-world optical interconnects. Therefore, considering a cost-effective solution, a polarization transparent PIC is worthwhile investigating,where a polarization splitter-rotator (PSR) or a polarization splitting grat-ing coupler (PSGC) is one of the essential passive components. As shownin Fig. 1.5, a PSR and a PSGC are normally used in the edge-coupling andvertical-coupling interface, respectively. Using various techniques, recentlynumbers of PSRs [39\u201349] and PSGCs [50\u201356] have been demonstrated forbroadband, high-coupling efficiency, low-loss, compact devices. In Chap-ter 3, we will also demonstrate novel broadband, adiabatic PSRs in two SOIplatforms.Figure 1.6: (a) A schematic of a polarization diversity circuit and (b) aschematic of active polarization control circuit.Polarization diversity is a typical approach to realize the polarizationtransparency scheme in a silicon PIC. It simply splits the input light into8two branches using a PSR or a PSGC, as shown in Fig. 1.6(a), and thenimplements two identical PICs separately. This method has been recentlyused in various demonstrated WDM receivers [57\u201365], coherent receivers[66\u201369], and optical switches [70\u201373]. However, this polarization diversitycircuit has several disadvantages. For example, the requirement for anothercopy of the circuit doubles footprint; The copies of some components needto match each other (e.g., MRR-based or BGW-based filters), which maybe achieved by active tuning, but this increases the power consumption[5]. Although some photonic integrated devices, such as wavelength add-drop filters [64, 65], can be used in a waveguide loop design that is formedbetween two output ports of a PSR (PBS or PSGC), so that one can re-usethe devices, many PICs still need to have two identical copies, which is notefficient as regards the required on-chip real estate.An alternative approach is to develop an active polarization controller[5, 39, 49, 74\u201376]. As shown in Fig. 1.6(b), a PSGC or a PSR is still neededto manage the input polarization states and transmit them into two separatesilicon waveguides. Then, instead of duplicating the PIC, a tunable Mach-Zehnder interferometer (MZI), mainly formed by two phase shifters (PS1 andPS2 in Fig. 1.6 (b)) and two 3 dB couplers, is connected to the two outputsof the first device (i.e., the PSGC or PSR). The phase shifters in the MZIare used to adjust the phase of the incoming light, such that a desired singlepolarization can be obtained at one output waveguide of the MZI, and finallypass into the main PIC. A photo-detector (PD in Fig. 1.6(b)) is used at thebottom port to monitor optical power as a feedback loop control. Comparedto the polarization diversity scheme, this method does not need two copies ofthe PIC, and, thus, it reduces the chip area, saving significant footprint andcost, especially for complex PIC designs that require many electrical signals[77\u201379]. Also, it should be mentioned that a robust control algorithm is animportant part of active polarization management. In Chapter 4, we willinvestigate several automated control algorithms for an on-chip polarizationreceiver.91.4 About this thesis1.4.1 ObjectiveThe objective of this thesis is to investigate wavelength and polarizationdevices, circuits, and control methodologies, and eventually provide a siliconphotonic, systematic solution for on-chip receiver applications. In specific,this thesis will focus on:\u2022 Compact wavelength filters and broadband polarization managementdevices on SOI platforms;\u2022 Active polarization PICs and automated control methodologies;\u2022 WDM polarization PICs, and a fully automated adaptation and sta-bilization for wavelength and polarization.The further objective of this research is to develop a fully CMOS driven, au-tomated WDM polarization on-chip control PIC for large-scale, high-speed,optical interconnects, such as complex coherent systems and multi-channeloptical switch matrices.1.4.2 Thesis organizationIn Chapter 2, we demonstrate an apodization technique by tapering thecorrugations of spiral Bragg grating waveguides on the silicon-on-insulatorplatform, for the fundamental transverse magnetic mode. The transmissionand reflection responses of uniform and apodized spiral Bragg grating waveg-uides are experimentally compared to show higher sidelobe suppression ra-tios by the proposed apodization scheme. We also present an apodized,period-chirped spiral Bragg grating waveguide, and the group delay of theproposed device has been measured; the results show an efficient suppressionin the ripples of the group delay response.In Chapter 3, we propose and demonstrate broadband, entirely mode-evolution-based, PSR using SWG-assisted adiabatic waveguides for two SOIplatforms. Our PSRs are more compact than previously demonstrated en-tirely mode-evolution-based designs. The devices were fabricated using two10fabrication processes and, in both cases, the measured spectra show closematches to the simulation results. One of the processes uses standard opticallithography and, hence, this is the first time that an SWG-based PSR hasbeen experimentally implemented using such a process. More importantly,it can also be used as the passive polarization component in an active, siliconphotonic polarization receiver.In Chapter 4, we experimentally demonstrate an automated polariza-tion receiver that couples light from arbitrary polarization states from asingle-mode fiber into the fundamental TE modes of a single-mode waveg-uide in silicon PICs. First, theoretical model of the proposed circuit isbuilt and experimentally verified. Then, we performed a high-speed ex-periment for arbitrary input polarization adaptation in our polarization re-ceiver. Furthermore, to investigate the automated control algorithms for thepolarization receiver, we present greedy linear descent-based, basic gradientdescent-based, two-point step size gradient descent-based, and two-stage op-timization method-based automated control algorithms and examine theirperformance for use with a silicon photonic polarization receiver. Withan active feedback loop control process, time-varying arbitrary polarizationstates from an optical fiber can be automatically adapted and stabilizedto the transverse-electric mode of a single-mode silicon waveguide. Usingthe proposed control algorithms, we successfully realize automated adapta-tions for a 10 Gb\/s on-off keying signal in the polarization receiver. Basedon the large-signal measurement results, the control algorithms are exam-ined and compared with regard to the iteration number and the outputresponse. In addition, we implemented a long-duration experiment to track,adapt, and stabilize arbitrary input polarization states using the two-pointstep size gradient descent-based and two-stage optimization method-basedcontrol algorithms. The experimental results show that these control algo-rithms enable the polarization receiver to achieve real-time and continuouspolarization management.In Chapter 5, we demonstrate automated adaptation and stabilizationfor a silicon photonic wavelength-division multiplexing (WDM) polarizationreceiver. A two-channel, tunable WDM polarization receiver is designed11and used to demonstrate the automated WDM polarization control. Us-ing a control algorithm based on Barzilai and Borwein\u2019s two-point step sizegradient descent method, we realize automated polarization adaptation andwavelength stabilization for two arbitrarily polarized input data streams. 10Gb\/s on-off keying (OOK) and 20 Gb\/s pulse-amplitude modulation 4-level(PAM-4) formats are generated as the high-speed input data streams. In ad-dition, we implement a long-duration bit-error-ratio (BER) experiment forcontinuously varying polarization states and changing chip temperatures.The experimental results show that, with the automated control, the WDMpolarization receiver can adapt, stabilize, and track the arbitrary input po-larization states from a standard optical fiber into the transverse-electricmode of a silicon waveguide, and simultaneously stabilize the transmittedwavelength-channels at various chip temperatures. We also show how thepresented WDM polarization receiver scales with N channels, and proposean improved design for large-scale WDM applications.In Chapter 6, we summarize the contributions and address the valuablepoints of this research. Some discussions and proposed designs are also madefor future research directions.12Chapter 2Apodized spiral Bragggrating waveguidesIn this chapter, we demonstrate apodized, TM mode, SOI strip spiral BGWs(SBGWs). Our technique is based on modulating the sidewall corrugationwidth of the SBGW with a Gaussian apodization window function. Anapodized, period-chirped SBGW (C-SBGW) is also presented to show theeffect of apodization on the group delay response. Our apodized SBGWswere fabricated and their spectral responses were measured. In addition,we present measurement results of group delay response for both apodizedand uniform C-SBGWs, demonstrating the elimination of the ripples bythe apodization used. The study on apodized SBGWs helps building morecompact wavelength filters, particularly for BGW-assisted add-drop filters,which can be used in a CWDM receivier design proposed in Chapter 6.2.1 IntroductionIntegrated BGWs in SOI platforms have been used in various optical devicesfor communication [80], biosensing [81], and microwave photonic signal pro-cessing [82]. Over the last few years, in order to achieve the small couplingcoefficients and large ER necessary for applications such as channel selectionfor WDM [11, 83] and optical dispersion compensation [84\u201387], Bragg grat-ing based structures with long lengths have been developed. When usinglong BGWs, SBGWs offer several advantages over straight BGWs as re-gards fabrication uniformity and efficient use of on-chip real estate [14, 15].In addition, as compared to TE mode BGWs, TM mode BGWs have asmoother spectral response due to lower backscattering, lower propagation13losses, and smaller coupling coefficients. Hence, TM mode SBGWs can beused as bandpass filters in WDM applications and as compensators for thegroup velocity dispersion in optical fibers [14]. However, the strong side-lobes of uniform BGWs can result in high crosstalk between channels inWDM applications. This crosstalk can be mitigated by using appropriateapodization of the BGWs; apodization has been shown to be a practical wayto suppress the sidelobes of straight TE mode Bragg grating devices on SOIplatforms [16, 83].2.2 Design and SimulationsIn our design, we wrap long apodized BGW into compact areas using spiralwaveguides. Fig. 2.1(a) shows the layout of an apodized SBGW. A 3 mmlong apodized SBGW can be wrapped into a circular spiral with a maximumradius of 59 \u00b5m. There is an S-shaped waveguide formed by two semi-circular-60 -40 -20 0 20 40 60X (\u00b5m)-60-40-200204060Y (\u00b5m)(a)\u039b\u2206WWmin Wmax(b)(c)\u2206W\u2193Port 1Port 2gR0R0Figure 2.1: (a) Schematic of the Gaussian-apodized SBGW. The totallength is 3 mm long. (b), (c) Zoom-in showing the grating period, thecorrugation width, \u2206W = (Wmax \u2212Wmin)\/2, and the spacing between twospiral waveguides, g.14waveguides at the center of the device. Two interleaved Archimedean spiralsare connected to the S-shaped waveguide. Both semi-circular waveguideshave the same radius of curvature, R0, we choose R0 = 15 \u00b5m [14]. Thischoice of R0 provides a good balance between efficient packing and curvaturerelated changes in the effective refraction index and in the insertion loss. Thegap between the waveguides, g, is chosen to be 2 \u00b5m. Each apodized SBGWhas two ports, Port 1 and Port 2. A Gaussian window function is appliedto modulate the sidewall corrugation widths of the SBGW:\u2206W = \u2206W0e\u221212[2\u03b1(z\u2212L2)L]2(2.1)where the apodized corrugation width, \u2206W , is varied according to its cor-responding longitudinal position, z, and L is the total length of the SBGW.The index, \u03b1, determines the curvature of the Gaussian window function andthus the apodization strength. \u2206W0 corresponds to the maximum value ofthe corrugation width, which is located at the center of the apodized SBGW,shown in Fig. 2.1(c).0 1 2 3z (mm)01020304050W (nm)010002000300040005000 (m-1)Figure 2.2: Calculated apodized sidewall corrugation width (in blue) anddistributed coupling coefficient (in orange) as functions of the longitudinalposition. The index of Gaussian function is chosen as 3.To fully understand the proposed apodization method\u2019s effect on the15BGWs, we investigated an accurate simulation approach using finite-differencetime-domain (FDTD) calculations and the TMM. To begin with, we calcu-lated the coupling coefficient \u03ba versus \u2206W , using an FDTD bandstructurecalculation with Bloch boundary conditions [88]. A fine mesh size of 2 nmwas chosen to fit the calculation range of \u2206W from 20 nm to 100 nm. Next,we used a second-order polynomial fit to the simulated \u03ba values,\u03ba = a\u2206W 2 + b\u2206W , (2.2)where a = 2.0043 \u00d7 1018 m-3 and b = 1.01415 \u00d7 1010 m-2. Then, by sub-stituting \u2206W in Eq. 2.2 and using Eq. 2.1, we can obtain the apodized \u03baas a function of z, as shown in Fig. 2.2. Here we used \u2206W0 = 45 nm anda grating period, \u039b, of 440 nm and assumed an average waveguide width,W0 = (Wmax + Wmin)\/2, of 500 nm. We can see that \u03ba approaches zero atthe two ends of the BGW and that it has a peak of 4600 m\u22121 at the center ofthe BGW. Thus, we conclude that an apodized corrugation width results inan apodized coupling strength. In addition, in a high-index contrast Bragggrating, a change in \u2206W results in both a change of \u03ba, and a small parasiticchange in the Bragg wavelength, \u03bbB [89]. Therefore, since \u2206W determines\u03ba (Eq. 2.2), changes in \u2206W result in small shifts in \u03bbB, i.e., changing \u2206Wchirps \u03bbB, which needs to be accounted for. Hence, we also simulated thischirping effect on the local Bragg wavelength of our apodized BGW, where\u2206\u03bbB = \u03bbB \u2212 \u03bb0 = c\u2206W 2 + d\u2206W + e, (2.3)c = \u22124.4524 \u00d7 105 m-1, d = \u22120.0115, e = 1.5 \u00d7 10\u22129 m, and the designedBragg wavelength \u03bb0 = 1.554\u00d710\u22126 m, for \u2206W0 = 45 nm. Finally, based onthe simulated coupling coefficients and the shifts of the center wavelength,we calculated the spectral responses shown in Fig. 2.3, using the TMM. Wecan see that the uniform BGW has strong sidelobes, while the apodized onepresents a nearly ideal filter shape: the sidelobe suppression ratio (SLSR) ofthe reflection response is greater than 62 dB for \u03b1 = 3, versus 1.8 dB for theuniform BGW. The transmission response also shows essentially eliminated16-5 -2.5 0 2.5 5Wavelength detuning (nm)-60-50-40-30-20-100Transmission (dB)ApodizedUniform-5 -2.5 0 2.5 5Wavelength detuning (nm)-60-50-40-30-20-100Reflection (dB)ApodizedUniform(a) (b)Figure 2.3: Simulated (a) reflection and (b) transmission spectra with theproposed uniform (in orange) and Gaussian-apodized (in blue) SBGW.sidelobes but a comparatively lower ER (i.e. \u223c 32 dB for the apodized BGWcompared with \u223c 60 dB for the uniform one) due to the weakened couplingstrength. The propagation loss is assumed to be 2 dB\/cm. In addition, By1 2 3 4 5 Apodization index050100150Transmission ER (dB)050100150Reflection SLSR (dB)Apodize WApodize Apodize WApodize 1 2 3 4 5Apodization index012345Bandwidth (nm)Apodize WApodize (a) (b)Figure 2.4: (a) Transmission ER (in blue), reflection SLSR (in orange), and(b) bandwidth versus various Gaussian indices of apodizing \u2206W and \u03ba,respectively.sweeping the Gaussian apodization indices for \u2206W and \u03ba, we investigated17the changes of the spectral properties, as shown in Fig. 2.4. Various SLSRs(Fig. 2.4(a)), transmission ERs (Fig. 2.4(a)), and bandwidths (Fig. 2.4(c))of the reflection responses can be observed. We choose \u03b1 = 3 as a goodtrade-off between the SLSR and ER.-40 40-5050InputL\uf04c1 L\uf04c2. . . .l\uf06c1l\uf06c2. . . .L\uf04cm-1L\uf04cm-1l\uf06cm-1l\uf06cm-1........L\uf04c1 > L\uf04c2 > ... > L\uf04cm-1 > L\uf04cm-1l\uf06c1 > l\uf06c2 > ... > l\uf06cm-1 > l\uf06cm-1ReflectionFigure 2.5: Schematic of a period-chirped SBGW.As presented in Fig. 2.5, our apodized C-SBGWs are period-chirped,such that in them, \u039b varies linearly along the length of the device. Todemonstrate a negative dispersion, we designed a 4 mm long C-SBGW thathad a negative chirp rate, d\u039b\/dL = -14 nm\/cm, starting from a gratingperiod of 440 nm. Our Gaussian-apodization scheme was applied here witha \u2206W0 = 60 nm. Using the FDTD and TMM method described previously,we calculated the reflection spectrum of the C-BGWs. Also, the group delay\u03c4 of the device can be estimated by taking the derivative of the phase of thereflection coefficient, \u03d5, with respect to the frequency, \u03c9, or respect to thewavelength, \u03bb, which is given by [90]\u03c4 =d\u03d5d\u03c9= \u2212 \u03bb22picd\u03d5d\u03bb. (2.4)Hence, Figs. 2.6(a) and 2.6(b) present the simulated reflection response andgroup delay of the uniform and apodized BGWs, respectively. As can beseen, the average slope of the group delay is negative, with a slope of -1118ps\/nm over the passband. This also indicates a negative chromatic disper-sion of -2.75 ps\/(nm\u00b7mm), which can be used as an on-chip dispersion com-pensator for the optical fiber transmissions that have positive dispersions[91]. The loss increases as the wavelength decreases due to the increasedpropagation distance [87]. As expected, we can see that the group delay ofthe uniform C-BGW exhibits ripples whereas that of the apodized C-BGWpresents a smoothed curve.-10 -5 0 5 10Wavelength detuning (nm)-30-20-100Reflection (dB)0306090Group Delay (ps)-10 -5 0 5 10Wavelength detuning(nm)-30-20-100Reflection (dB)0306090Group Delay (ps)(a) (b)Figure 2.6: Simulated reflection spectrum (in blue) and group delay (inorange) of the proposed (a) uniform and (b) Gaussian-apodized C-SBGW.2.3 Measurement and resultsOur SBGWs were fabricated by Applied Nanotools Inc., using electron-beamlithography with a 5 nm grid spacing. The fabrication used a single etch pro-cess on an SOI wafer with 220 nm thick silicon. A 2 \u00b5m thick silicon dioxidecladding layer was deposited on the etched sample. Figures 2.7(a) - 2.7(c)show SEM images of a fabricated SBGW. Three full-etched GCs [32], wereused to couple light into and out of our test structures from a fiber arraywith a 127 \u00b5m fiber-to-fiber pitch. A low loss Y-junction power splitter [92]was used to connect the input and output GCs to each SBGW in order totransfer the injected light to the device and to collect the reflected light. As19shown by the zoom-in SEM images, Figs. 2.7(b) and 2.7(c), the designedrectangular corrugations on the apodized SBGWs have been accurately fab-ricated. To measure the fabricated devices, a custom-built test setup is used(a) (b)(c)Y \u00a0JunctionInput \u00a0GCTransmission \u00a0GCReflection \u00a0GC1 \u00a0\ud835\udf07\ud835\udc5aFigure 2.7: SEM images of an apodized SWBG device. (a) The com-plete apodized SWBG with input and output GCs. (b) Zoom-in of theArchimedean SWBG with Gaussian apodization. (c) Zoom-in of the centerof the S-shaped WBG.[5], as shown in Fig. 2.8. The setup mainly consists of an Agilent 81600Btunable laser source, Agilent 81635A optical power sensors, a multi-channelfiber array, and an automated stage where places the silicon photonic chip.An SRS LDC501 thermoelectric cooler (TEC) is used to control the chiptemperature.The measured reflection and transmission spectra of the fabricated uni-form and apodized SBGWs are shown in Fig. 2.9, where the insertion lossesfrom the Y-junctions, input and output GCs have been calibrated out. \u2206Wof the uniform SBGW and \u2206W0 of the apodized SBGW are 45 nm, andboth of the devices have a length of 3 mm. Fig. 2.9(a) shows that theuniform SBGWs have very strong sidelobes and rippled reflection spectra,and Fig. 2.9(c) is a zoom in of Fig. 2.9 that shows the minimum measuredSLSR for the uniform SBGW is 0.9 dB. As also observed in Fig. 2.9(a), theapodized SBGW shows a sidelobe suppression and less rippled reflection, andFig. 2.9(d) shows a minimum SLSR of 13.2 dB. Additionally, the smoother2015342 1. Tunable laserInput GCDeviceOutput GC2. Fiber array (input channel)5. optical power sensorSiP Chip2. Fiber array (output channel)3. Automated stage4. TEC(a) (b)Figure 2.8: (a) An automated measurement setup and zoom-in of siliconchip and fiber array. (b) A block diagram of the automated measurementsetup.spectrum of the apodized SBGW can be seen in the measured transmissionin Fig. 2.9(b). However, the experimental results still do not match thesimulation, where the SLSR of the reflection spectrum are below 60 dB.We attribute this performance deterioration to phase noise induced by thesidewall roughness and wafer non-uniformity [93]. Therefore, to investigatethis effect, we applied a Gaussian noise function to the grating periods tosimulate the effective deviations of this parameter due to all the variationsin the waveguide parameters that can affect the optical length of the gratingperiod, as shown in Fig. 2.10(a). The spectra were then simulated based onthe Gaussian noise introduced period lengths. In Fig. 2.10(b), it is clearto see that the SLSR is reduced due to the introduced noise with a devia-tion \u03c3 = 1.3 nm, which corresponds to the SLSR that we measured. Also,because the simulations have not involved the noise factor into some otherparameters of the BGW (e.g., corrugation width, grating ratio, etc.), some21-1.2 -0.6 0 0.6 1.2Wavelength detuning (nm)-25-20-15-10-50Reflection (dB)-0.8 -0.4 0 0.4 0.8Wavelength detuning (nm)-10-8-6-4-20Reflection (dB)(a) (b)-5 -2.5 0 2.5 5Wavelength detuning (nm)-30-25-20-15-10-50Reflection (dB)ApodizedUniform-5 -2.5 0 2.5 5Wavelength detuning (nm)-50-40-30-20-100Transmisson (dB)ApodizedUniform(c) (d)0.9 dB1.8 dB 18.1 dB 13.2 dBFigure 2.9: Measured (a) reflection and (b) transmission spectra for theuniform (in orange) and Gaussian-apodized (in blue) SWBGs. (c) Zoom-inof the measured reflection spectra for the uniform (in orange) and Gaussian-apodized (in blue) SWBGs. The SLSR is 0.9 dB for the uniform SWBG and13.2 dB for the apodized one.high-frequency oscillations in measurement spectrum have not been seen inthe simulated spectra.We have investigated TM apodized C-SBGWs experimentally. Fig. 2.11shows the measured reflection spectrum and the group delay of the uniformC-SBGW (a) and apodized C-SBGW (b). The uniform C-SBGW has \u2206W =60 nm and the apodized C-SBGW has \u2206W0 = 60 nm, and both have a220 2000 4000 6000Period number435437439441443445Period length (nm)JitteredAverage-5 -2.5 0 2.5 5Wavelength detuning (nm)-30-25-20-15-10-50Reflection (dB)Sim1  = 1.3 nmSim2  = 1.3 nmSim3  = 1.3 nmMeasurement(a) (b)Figure 2.10: (a) Gaussian noise introduced period lengths as a function ofthe period number. (b) Simulated reflection with random grating periodnoise (in red, purple, and green) and our measured spectrum (in blue).-10 -5 0 5 10Wavelength detuning(nm)-30-20-100Reflection (dB)0306090Group Delay (ps)-10 -5 0 5 10Wavelength detuning(nm)-30-20-100Reflection (dB)0306090Group Delay (ps)(a) (b)Figure 2.11: Measured reflection spectrum (in blue) and group delay (inorange) for the (a) uniform and (b) Gaussian-apodized C-SBGWs.The groupdelay outside the passband (in grey) refers to data with a large measurementuncertainty.length of 4 mm. The group delays were measured using an Optical VectorAnalyzerTM STe from Luna Innovations Inc. As shown in Fig. 2.11(a), thepassband of the uniform C-SBGWs is 8.8 nm wide and, within it, the group23delay has an average slope of -11 ps\/nm. Strong ripples are observed inthe group delay, which are reduced in the apodized C-SBGW, as seen inFig. 2.11(b). A delay of 31.2 ps at the center wavelength of the apodized C-SBGW is measured, which is close to our simulated value of 32 ps. However,unexpected ripple and power reductions are seen at the edges of the passbandof the measured reflection spectra for both the uniform and apodized C-SBGWs. This also causes a deterioration of the apodized group delay. Webelieve that the phase noise is responsible for the group delay deteriorationand for the distortion of the reflection spectrum. It should be pointed outthat the responses at shorter and longer wavelengths, as compared to thecenter wavelength, are more affected due to the reduced coupling coefficientswhich result from small \u2206W s that occur in apodized C-SBGWs.2.4 SummaryIn this chapter, we have experimentally demonstrated apodized, TM SBGWs.We have verified that apodized, TM SBGWs have higher SLSRs and smootherspectra, compared to uniform, TM SBGWs. We conclude that the apodiza-tion can improve the channel crosstalk for the cascaded SBGWs, whichcan be used for wavelength DeMux. In addition, we have demonstrated anapodized TM C-SBGW. We presented measurements taken on the TM C-SBGWs with a negative group delay slope, which showed that the rippleson the group delay can be efficiently reduced by our apodization scheme.Such a smoothed, negative group delay should be useful for compensatingdispersion in optical fibers.24Chapter 3Sub-wavelengthgrating-assisted polarizationsplitter-rotatorsIn this chapter, we propose and demonstrate Sub-wavelength grating (SWG)-assisted, mode-evolution-based polarization splitter-rotators (PSRs) usingtwo SOI processes: an air cladding, single etch-step, electron-beam (E-beam) lithography process and a SiO2 cladding, multiple etch-step, CMOS-compatible, optical lithography process. Our SiO2-clad PSR is the firstimplementation of an SWG-based PSR in a standard optical lithographyprocess. The test devices were fabricated in the two SOI processes andtheir spectra responses were measured. As a preparation of the next stepresearch, the designs will be used in implementing on-chip polarization con-trol (Chapter 4).3.1 IntroductionDue to the aspect ratio of SOI waveguides (typically \u223c2:1) and the high-index contrast between the silicon and the surrounding media, integratedphotonic devices suffer from sizeable modal birefringences and, therefore,typically require control of the input polarization if one desires that only TEor TM modes are launched. This can be achieved using properly orientedPM fiber, however, this is a costly solution for real-world optical commu-nication systems. Therefore, in order to allow standard (non-PM) fiber tobe used at the input, a polarization-transparent integrated photonic device,25i.e., a photonic polarization receiver, is needed [57, 75, 76]. A PSR is an es-sential element in such a polarization receiver. We can categorize the PSRsthat have been recently experimentally demonstrated into three types: thosethat are entirely mode-coupling based devices [40\u201344], those that are hybridmode-evolution-based\/mode-coupling-based devices [45\u201347], and those thatare entirely mode-evolution-based devices [39, 48]. The hybrid devices usemode evolution for the rotation function and mode coupling for the splittingfunction, whereas the other two types typically use either mode coupling ormode evolution for both functions. All of the devices that use mode couplinghave lower tolerances to fabrication variations, have limited bandwidths,but are relatively compact in their sizes, compared to the entirely mode-evolution-based devices which have greater tolerances to fabrication varia-tions, wider bandwidths, but are less compact. Nevertheless, it has beentheoretically shown that more compact entirely mode-evolution-based PSRsare possible [94, 95]. SWG waveguides have been used in various photonicdevices due to the ability to engineer the index and dispersion properties ofmetamaterial-based waveguides [96, 97]. SWG waveguides also lead to morecompact and broadband devices, compared to regular SOI waveguide-baseddevices [42, 98\u2013102]. Therefore, they provide a pathway to improve thedesigns of mode-evolution-based PSRs, which is also the important compo-nent of a polarization receiver, as we discussed in Chapter 1, for use in thefront-ends of PICs intended for high-speed optical data processing systems.3.2 SWG-assisted PSR using an E-beamlithography process3.2.1 Operating principle and designThe schematic of the proposed PSR is shown in Fig. 3.1. An adiabatic nano-taper, an SWG-assisted adiabatic coupler, and an adiabatic mode splittingsection are the three main parts of the proposed PSR. Air cladding is usedto break the vertical symmetry so that efficient polarization conversion canbe achieved. As shown in Fig. 3.1(a), light injected into the first-order TE26mode (the TE00 mode) at the left-hand side (LHS) of the adiabatic nano-taper is expanded into the TE00 mode at the right-hand side (RHS) of thenano-taper, whereas light injected into the first-order TM mode (the TM00mode) at the LHS of the nano-taper evolves into the second-order TE mode(the TE01 mode) at the RHS of the nano-taper. Fig. 3.1(b) shows the threesegments of the nano-taper. The widths and lengths are optimized to max-imize the mode-conversion efficiency and minimize losses in the nano-taper,see for example in [45]. To understand the mode expansion\/evolution in theTM00TE00TE00TE00TE00TE01TE01TE00Adiabatic Taper SWG-Assisted Adiabatic CouplerL1 L2 L3Winput W1 W2 W3(a)(b). . .. . . W5\u039bgW4W3W6L4W4(c)Through-portCross-portInput-portFigure 3.1: (a) 3D view of our air-clad PSR with mode profiles at variouspoints along the device; (b) top view of the adiabatic nano-taper; (c) topview of the SWG-assisted adiabatic coupler with labelled design parameters.The parameters in these segments are as follows: in (b), W1 = 600 nm, W2 =700 nm, W3 = 900 nm (given that Winput = 450 nm), L1 = 6 \u00b5m, L2 = 30 \u00b5m,and L3 = 15 \u00b5m; in (c), L4 = 100 \u00b5m, \u039b = 200 nm, g = 100 nm, W3 = 900nm, W4 = 200 nm, W5 = 650 nm, and W6 = 800 nm.adiabatic nano-taper, we calculated the effective indices of the eigenmodesin an air-clad, SOI strip waveguide with a thickness of 220 nm. Fig. 3.2shows the effective indices of the first three eigenmodes in the waveguide asa function of the waveguide\u2019s width. It can be seen that there is a hybridmode region between the TM00 mode and the TE01 mode when the waveg-uide width increases from 0.62 \u00b5m to 0.72 \u00b5m. Thus, this leads to mode27evolution from the TM00 mode to the TE01 mode when a TM00 mode propa-gates along the adiabatic nano-taper. In contrast, the effective index of theTE00 mode in the strip waveguide increases but shows no mode conversionas the strip waveguide widens.0.4 0.5 0.6 0.7 0.8 0.9 1Waveguide width ( m)1.451.651.852.052.252.452.652.85Effective indexHybrid mode regionTE01TM00TE01TM00TE00Figure 3.2: Calculated effective indices for the first three eigenmodes of anair-clad strip waveguide, as a function of the waveguide width. The centerwavelength is 1550 nm.An SWG-assisted, adiabatic mode evolution section, which we will referto as an SWG-assisted adiabatic coupler (SWG-AC), is designed such thatthe TE00 and TE01 modes on the RHS of the nano-taper are well matchedto the TE00 and TE01 modes on the LHS of the two-waveguide system,respectively. This two-waveguide system consists of the upper, regular stripwaveguide and the lower, SWG-assisted strip waveguide, and, eventually, atthe outputs of the SWG-AC, the TE00 and TE01 modes of the two-waveguidesystem are predominately located in the upper waveguide and in the lowerwaveguide, respectively. The two waveguides are tapered to minimize thelosses in the two system modes, as their mode distributions evolve from theLHS of the SWG-AC to the RHS of the SWG-AC. Specifically, to achievethe optimal field distributions of the two TE system modes in the SWG-AC,we use the finite-difference time-domain (FDTD) bandstructure calculations[42] to determine an appropriate period, \u039b = 200 nm, and fill factor, ff =280.6. In addition, as regards the mode splitting section, the TE00 mode at theRHS of the SWG-AC is directed into the TE00 mode of an isolated, 450 nmwide, strip waveguide using a low loss S-bend, and, a high-efficiency SWGtaper is used to evolve the TE01 mode at RHS of the SWG-AC into theTE00 mode of the second isolated, 450 nm wide, strip waveguide. Overall,the injected TE00 mode is received at the through-port of the device, whereasthe injected TM00 mode is converted to the TE00 mode at the cross-port ofthe device. Compared with the reported adiabatic PSR in [39], our SWG-based PSR is also adiabatic but requires a shorter coupler length (L4 = 100\u00b5m here versus L = 300 \u00b5m in [39]). Further details will be discussed inSection 3.3.1.3.2.2 Simulation methodology and resultsTo fully understand our proposed PSR, we investigated an efficient simula-tion approach based on FDTD S-parameter matrix calculations and a circuitmodelling method, as shown in Fig. 3.3. Initially, using Lumerical FDTD So-lutions, we obtained the S-parameter matrices for each section of the device,i.e., the nano-taper (\u201cAtaper.dat\u201d in Fig. 3.3) and the SWG-AC combinedwith the splitting section (\u201cAsplitter.dat\u201d in Fig. 3.3), and created compactmodel representations in Lumerical INTERCONNECT (INTC). Then, webuilt the circuit model and calculated the transmission response of the entirePSR in INTC. This approach is more computationally efficient than runninga full 3D FDTD simulation for the entire structure. The simulated trans-mission spectra are shown in Fig. 3.4. In Fig. 3.4(a), we can see that for aninjected TM00 mode, the output light was mainly received at the cross-portwith less than 0.5 dB insertion loss. At the through-port, the TE00 modecrosstalk was less than \u221218 dB over a 120 nm bandwidth, from 1500 nm to1620 nm, and the TM00 mode crosstalk stayed less than \u221219 dB in the wave-length range from 1546 nm to 1620 nm. However, the TM00 mode crosstalkrose to nearly \u221215 dB below 1530 nm, which can be reduced by replacingthe S-bend with a compact broadband polarization beam splitter [103, 104].For an injected TE00 mode, on the other hand, the input light propagated to29\u201cAtaper.dat\u201d \u201cAsplitter.dat\u201dFDTD S-parameterInterconnect circuit modelFDTDINTCFigure 3.3: Simulation circuit model of the proposed PSR in INTERCON-NECT (INTC), and the extracted S-parameter compact models (ATaperand ASplitter) calculated by FDTD.the through-port with less than 0.4 dB insertion loss, and less than \u221225 dBcrosstalk was received at the cross-port over a broad wavelength range from1500 nm to 1620 nm, as shown in Fig. 3.4(b).(a) (b)1500 1520 1540 1560 1580 1600 1620Wavelength (nm)-40-35-30-25-20-15-10-50Transmission (dB)Cross-port (TE00)Through-port (TE00)Through-port (TM00)1500 1520 1540 1560 1580 1600 1620Wavelength (nm)-40-35-30-25-20-15-10-50Transmission (dB)Through-port (TE00)Cross-port (TE00)Figure 3.4: Simulated transmission spectra of the proposed PSRs, with (a)TM00 mode and (b) TE00 mode inputs.303.2.3 Measurement resultsOur air-clad, single etch-step PSRs were fabricated by Applied NanotoolsInc. using electron-beam lithography (EBL) with a 5 nm grid spacing.Figs. 3.5(a)-3.5(e) show scanning electron microscope (SEM) images of thefabricated structures for the proposed PSR. Broadband grating couplers(GCs) were used to couple light into and out of the devices. Since the GCswere optimized for TE0 or TM0 mode coupling, three configurations withdifferent GCs are required to measure each of the devices, as shown in Ta-ble 3.1. For calibration purposes, we used two GC pairs fabricated nextto the test structures. Also, it can be seen that the designed rectangularcorrugations on the SWG-AC and the splitting section have been accuratelyfabricated, as shown in Figs. 3.5(c)-3.5(e).Configuration Input-port Through-port Cross-port1 TM00 TE00 TE002 TM00 TM0 TE003 TE00 TE00 TE00Table 3.1: Measurement configurations of the GC ports.The fabricated devices were experimentally characterized using a custom-built test setup[5], as we introduced in Chapter 2. An Agilent 81600B tun-able laser was used as the light source and Agilent 81635A optical powersensors were used as the output detectors. Broadband grating couplers(GCs)[105] were used to couple light into and out of the devices. Figs. 3.6(a)and 3.6(b) show the measurement results for a fabricated PSR when theTM00 and the TE00 modes were launched, respectively (here, the transmis-sion spectra have been calibrated using the responses of the GC calibrationpairs). As shown in Fig. 3.6(a), when the TM00 mode was launched, the out-put light was mainly collected at the cross-port and the measured crosstalkat the through-port, for both the TM00 mode and the TE00 mode, had wave-length dependent responses, that were similar to the simulated spectra. Asshown in Fig. 3.6(b), the launched TE00 mode mainly propagated to thethrough-port, and the crosstalk was less than -20 dB over a 120 nm wave-311\u00b5m 1\u00b5m200nm(b)(c) (d) (e)10\u00b5m20 \u00b5mGC Calibration(Through port)GC Calibration(Cross port)Input-port GCThrough-port GCCross-port GC(a)Figure 3.5: SEM images of a fabricated PSR. (a) The complete test structureof PSR, including the GC ports and their calibration structures. (b) Theentire structure of the PSR. (c) Zoom-in of the blunt tip on the SWG-AC.(d) Zoom-in of the middle section of the SWG-AC. (e) Zoom-in of the S-bendand the SWG taper on the splitting section.length range (1500 nm to 1620 nm), which is close to the simulation results.It should be noted that the measured optical powers for the transmission ofboth the TE00 mode at the cross-port (in Fig. 3.6(a)) and the TE00 mode atthe through-port (in Fig. 3.6(b)) are reasonably well matched to the simu-lation results. The insertion loss was less than 1.4 dB from 1530 nm to 1620nm. The errors in the measured transmissions were approximately \u00b10.5 dBfor the TE00 mode measurements and \u00b11.0 dB for TM00 mode measure-ments, respectively. We attribute this due to the measurement alignmentinaccuracies and calibration errors in the GC port measurements.32(a) (b)1500 1520 1540 1560 1580 1600 1620Wavelength (nm)-40-35-30-25-20-15-10-50Transmission (dB)Cross-port (TE00)Through-port (TE00)Through-port (TM00)1500 1520 1540 1560 1580 1600 1620Wavelength (nm)-40-35-30-25-20-15-10-50Transmission (dB)Through-port (TE00)Cross-port (TE00)Figure 3.6: Transmission spectra of the fabricated PSRs, with (a) TM00mode and (b) TE00 mode inputs.3.3 SWG-assisted PSR using a standard opticallithography process3.3.1 Operating principle and designFor the first time, we demonstrate a broadband PSR using SWG-assistedadiabatic waveguides on a standard optical lithography platform. Insteadof an air-clad nano-taper, an adiabatic bi-level taper (see Figs. 3.7(a) and3.7(b)) was designed as the first section of our proposed PSR. A shallow-etched silicon slab waveguide, with a 90 nm thickness, was used to breakthe vertical symmetry of the waveguide for efficient polarization conversion.This enabled us to use a symmetric SiO2 cladding so that our device wouldbe fully compatible with standard foundry processes and, hence, would allowus to integrate our device in active PICs. Following the bi-level taper werean SiO2-clad SWG-AC and a splitting section, similar to the air-clad PSRshown in Figs.3.1(a) and 3.1(c). However, due to both our use of the SiO2cladding layer and the fabrication process rules, the design parameters ofthe SiO2-clad SWG-AC were optimized and found to be W3 = 850 nm, W4= 150 nm, W5 = 450 nm, W6 = 550 nm, L4 = 100 \u00b5m, g = 180 nm, \u039b =300 nm, and ff = 0.5.330.45 0.475 0.5 0.525 0.55Waveguide width ( m)1.41.61.822.22.42.62.8Effective index0.45 0.725 1 1.275 1.55Partially-etched slab width ( m)TE01TM00TE01TM00TE00(c)TE01TM00\u27f6TE01TE00TE00LBLA450 nm 550 nm 850 nm1.55\u00b5mTE00TM00(b)Full-etched Si (220 nm)Shallow-etched Si (90 nm)(a)Adiabatic Bi-taper SWG-Assisted Adiabatic Coupler Through-portCross-portInput-portFigure 3.7: (a) A 3D view of our oxide-clad PSR (For purpose of clarity,oxide cladding layer is not drawn and the layers\u2019 thickness\/length ratiosare enlarged); Schematic of the adiabatic bi-level taper (b) top view withmode profiles at different points along the bi-level taper, and (c) calculatedeffective indices of the first three modes along the first half of the bi-leveltaper.Similar to the analysis of the nano-taper, the effective indices of the firstthree eigenmodes along the first half of the bi-level taper were calculated andare presented in Fig. 3.7(c). Again, there is a hybrid mode region, in whichthe mode conversion between the TM00 mode and the TE01 mode occurs.Accordingly, in the first half of the bi-level taper, the rib and slab waveguideswiden from 0.45 \u00b5m and 0.45 \u00b5m to 0.55 \u00b5m and 1.55 \u00b5m, respectively. Onthe other hand, we can see that the TE00 mode simply propagates alongthe bi-level taper without any mode hybridization. In the second half ofthe bi-level taper, which is designed to ensure efficient TM00 to TE01 mode34conversion and to provide a transition to the following SWG-AC in stripwaveguides, the width of the rib waveguide increases linearly to 0.85 \u00b5mand the width of the slab waveguide decreases linearly to the same value.Also, to optimize the TM00 to TE01 mode conversion in the bi-level taper,in our simulations we swept the lengths of the two sections of the taper, LAand LB, as shown in Fig. 3.8(a). Here, LA = 35 \u00b5m and LB = 30 \u00b5m wasfound to result in a 99.8% mode conversion efficiency at a center wavelengthof 1550 nm. Finally, using the 3D FDTD solver, we simulated the modeevolutions in the bi-level taper for both TM00 mode and TE00 mode inputs,as shown in Fig. 3.8(b) and 3.8(c), respectively.Additionally, we swept the length of the adiabatic coupler section(L4 inour SWG-AC) for our SWG-based design and the reported design in [39].Based on the simulation results shown in Fig. 3.9, we can see that, forthe TE00 mode, both of the designs have similar transmission responses atvarious coupler lengths. This is because the mode evolution is predominantlyconfined to the upper strip waveguide for the TE00 mode. However, for theTE01 mode, the mode evolution happens in both waveguides of the adiabaticcoupler section. Due to the smaller effective index of TE01 mode in our SWG-AC, the field of the TE01 mode, in the two-waveguide system, is less confinedto the silicon region, i.e., stronger mode overlap occurs. Therefore, comparedwith the adiabatic coupler section with two regular strip waveguides in [39],the mode evolution along our SWG-AC will occur more rapidly for the TE01mode, i.e., our SWG-AC requires a shorter coupler length (\u223c90 \u00b5m) toachieve a full mode evolution.3.3.2 Simulation and experiment resultsUsing the simulation method previously introduced in Section 2.2, we alsosimulated the transmission spectra of the SiO2-clad PSR. As shown inFig. 3.10(a), where a TM00 mode is injected at the input-port, the out-put light, in the TE00 mode at the cross-port, has less than 0.5 dB insertionloss. The crosstalk at the through-port is obtained separately for the TE00mode and the TM00 mode. As shown in Fig. 3.10(b), where a TE00 mode35(b)(a) (c)10 20 30 40 50 60LA ( m)0.950.960.970.980.991The mode conversion efficiencyLB = 10 mLB = 20 mLB = 30 mLB = 40 mLB = 50 mLB = 60 m1.5 1.55 1.6Wavelength ( m)0.980.991TE00       TE00TM00      TE01Figure 3.8: (a) TM00 to TE01 mode conversion efficiency for various lengthsof LA and LB in the bi-level taper. Inset Fig.: spectra of the mode conversionefficiency for a TM00 mode and a TE00 mode input. The lengths LA and LBare chosen to be LA = 35 \u00b5m and LB = 30 \u00b5m. Top view of the simulatedelectric field distribution for the bi-level taper when launching (b) a TM00mode and (c) a TE00 mode.Figure 3.9: Normalized transmission for various lengths of L4 in the adiabaticcoupler (AC) section, with a TE01 mode and TE00 mode inputs.36is injected at the input-port, the output, in the TE00 mode at the through-port, has a less than 0.3 dB insertion loss, and the crosstalk at the cross-portwas also obtained for the TE00 mode across a wavelength range from 1500nm to 1620 nm.(a) (b)1500 1520 1540 1560 1580 1600 1620Wavelength (nm)-40-35-30-25-20-15-10-50Transmisson (dB)Cross-port (TE00)Through-port (TE00)Through-port (TM00)1500 1520 1540 1560 1580 1600 1620Wavelength (nm)-40-35-30-25-20-15-10-50Transmission (dB)Through-port (TE00)Cross-port (TE00)Figure 3.10: Simulated transmission spectra of the proposed PSRs, with (a)TM00 mode and (b) TE00 mode inputs.Our PSRs were fabricated on an SOI platform using 193 nm opticallithography at the Institute of Microelectronics (IME), Singapore. To ex-perimentally characterize the fabricated devices, we used an edge couplersetup with a fiber array having a mode-reducing transposer (made by PLCConnections Inc.) to couple light into and out of our devices, as shown inFig. 3.11. Fig. 3.12 shows the measured transmission spectra of the test de-vice for both TM00 mode and TE00 mode inputs. The transmission spectrahave been calibrated using the responses of edge coupler calibration pairs.We can see that the measurement results, i.e., the optical power measuredat the target output-ports and the crosstalk-ports for both TM00 mode andTE00 mode inputs, are close to the simulation results shown in Fig. 3.10.The insertion loss was less than 1.3 dB over a wavelength range from 1500nm to 1620 nm. However, we should also note the slight optical powerdiscrepancies between the simulations and measurements, which are mainlyattributed to measurement alignment errors in edge coupling and to fabri-cation imperfections. In addition, instead of the flat responses seen in the37Fiber arrayOn-chipwaveguideTransposer Chip(b)(a) (c)Figure 3.11: (a) A photo of the edge-coupling test stage and zoom-in ofthe transposer fiber-array aligning to the chip. A cross-section view (b) anda top view (c) of the alignment between transposer and the chip.(a) (b)1500 1520 1540 1560 1580 1600 1620Wavelength (nm)-40-35-30-25-20-15-10-50Transmission (dB)Cross-port (TE00)Through-port (TE00)Through-port (TM00)1500 1520 1540 1560 1580 1600 1620Wavelength (nm)-40-35-30-25-20-15-10-50Transmission (dB)Through-port (TE00)Cross-port (TE00)Figure 3.12: Measured transmission spectra of the fabricated PSRs, with(a) TM00 mode and (b) TE00 mode inputs.simulations, oscillations can be observed in the measured spectra, which aremainly due to the Fabry-Perot effect created by reflections between the chip38facets at the input and outputs ports of the test device [39].It is necessary to address the optical I\/O ports that we chose for ourmeasurements. Table 3.2 as shown below gives a brief summary of theoptical I\/O ports used in our measurements. For the proposed PSRs usingE-beam lithography, we chose grating couplers [105] since they were moreconvenient to characterize the test devices using an automated measurementstage [5]. On the other hand, in order to implement the active polarizationexperiment with PSRs using optical lithography, i.e., couple any arbitrarilypolarized input light into the chip, edge couplers are needed as the interface.Type Designparameter3 dB BW(nm)Couplingloss (dBm)*Grating coupler(Air-clad, EBL)SWGC [105]45 \u00b5m\u00d720 \u00b5m50 (TE00 mode),60 (TM00 mode)\u223c -6.0 (TE00mode), \u223c -5.1(TM00 mode)Edge coupler(SiO2-clad,opticallithography)Nano-taper(180 nm tip,200 \u00b5mlength)>120 (TE00 andTM00 modemode)\u223c -4.4 (TE00mode), \u223c -5.6(TM00 mode)* The loss is at center wavelength of 1550 nm.Table 3.2: Summary of the optical I\/O ports used in our measurement.3.4 SummaryTo wrap up, we have demonstrated broadband polarization splitter-rotatorsusing sub-wavelength grating-assisted adiabatic waveguides in two silicon-on-insulator fabrication processes: an air-clad, single etch-step process anda SiO2-clad, multiple etch-step, CMOS compatible process. The SiO2-claddevice is the first demonstration of a fabricated polarization splitter-rotator39based on sub-wavelength grating waveguides using a standard optical lithog-raphy process. We measured the spectra of both fabricated sub-wavelengthgrating-assisted adiabatic polarization splitter-rotators and compared themwith the simulation results. A figure of merits of the fabricated mode-evolution-based PSRs were summarized as shown in Table. 3.3. We can seethat, compared to the previous reported design, our devices perform simi-lar extinction ratios and low insertion loss but have much shorter lengths.Therefore, our polarization splitter-rotators provide a valuable option toa more compact, entire mode-evolution-based, CMOS compatible, passivepolarization component. In the next chapter, we will use our polarizationsplitter-rotator to design an active polarization receiver intended for use inhigh-speed optical communication systems.Type Length(\u00b5m)Extinction ratio (1530nm - 1600 nm)InsertionlossSWG-based PSR(Air-clad, EBL)171 TE00 mode input: >19dB; TM00 mode input:>15 dB<1.4 dBSWG-based PSR(SiO2-clad, opticallithography)185 TE00 mode input: >14dB; TM00 mode input:>15 dB<1.3 dBPSR in [39](SiO2-clad, opticallithography)475 TE00 mode input: >13dB; TM00 mode input:>15 dB<1.5 dBTable 3.3: Figure of Merits of the Fabricated Mode-evolution-based PSRs.40Chapter 4Active polarization receiverand automated controlalgorithmsIn this chpater, we demonstrate a silicon photonic, tunable polarization re-ceiver (PR) which can automatically overcome the polarization mismatch be-tween a standard optical fiber and an SOI waveguide, as shown in Fig. 4.14.The SWG-assisted PSR demonstrated in Chapter 3 is used in our PR. As wediscussed in Chapter 1, a PR does not need to copy the main PIC. Therefore,compared to the polarization diversity scheme, it can reduce the footprint,and significantly save the cost, especially for some complex, large-scale PICs.Also, a robust control algorithm is a necessary part of active polarizationcontrol. Investigations into the performance of the control algorithms, suchas automation, iteration cost of the optimization, and optimization accuracy,are important if one wants to use this approach in real-world optical inter-connects. However, although several studies have recently demonstratedactive polarization management [74\u201376], not much effort has been spent oncomparing the performance of control algorithms for use in on-chip polariza-tion controllers. Hence, in this chapter, we also experimentally demonstrategreedy linear descent-based, basic gradient descent-based, two-point stepsize gradient descent-based, and two-stage optimization method-based con-trol algorithm to automatically adapt an arbitrary input polarization stateinto the TE mode of, and to continuously stabilize the polarization state in,a single-mode, SOI waveguide of an active PR. Large-signal measurement4The Poincare\u00b4 sphere cartoon is provided by [106]41results are presented to examine the performance of each studied control al-gorithm. In addition, by continuously changing the input polarization state,we also implement an experiment to evaluate the stabilization capabilitiesof the control algorithms used in the PR over a longer duration of time.Figure 4.1: A block diagram showing that the arbitrary input polarizationstates (red trace on LHS Poincare\u00b4 sphere) are converted to a TE polarizationstate (red point on RHS Poincare\u00b4 sphere) through the proposed polarizationreceiver.4.1 Polarization receiver in active siliconphotonic platform4.1.1 Operating principle and designA schematic of our polarization receiver is shown in Fig. 4.2(a). An adia-batic, SWG-assisted PSR (see Chapter 3), and an MZI form the proposedsystem. The MZI consists of two thermal phase shifters and two 3-dB adi-abatic couplers (ACs) [107] (in fact, all of the components of the PR areadiabatic). Nano-tapered edge couplers are used as the optical I\/O ports ofthe system. The input light, in an arbitrary polarization state, is coupledinto the TE and TM modes of the edge coupler and propagates into thePSR. The PSR passively evolves these two modes into the fundamental TEmodes of its two output waveguides and then passes them into the balancedMZI. Acting on the outputs of the PSR, the two thermal phase shifters, H142and H2, control the outputs of the MZI, so that the optical power at theoutput port is optimized by minimizing the optical power at the feedbackport. In this way, an arbitrary input polarization state can be adapted tothe fundamental TE mode of the output waveguide. An optical micrographof a fully fabricated PR is shown in Fig. 4.2(b).PSR 3dB AC 3dB ACH1 H2(a)(b)50 \ud835\udf41mFigure 4.2: (a) Schematic of the polarization receiver consisting of an edgecoupler, a PSR, a balanced MZI including two thermal phase shifters, twobroadband 3dB couplers, and a photodetector (PD). (b) Optical micrographof a fabricated polarization receiver.4.1.2 System modelling and analysisTo investigate our PR\u2019s ability to adapt any arbitrary input polarizationstate to the output TE mode, firstly, we developed and analyzed the theo-retical model of the tunable MZI using the TMM. The output power versusthe phase tunings, \u2206\u03c61 and \u2206\u03c62, for four general input cases were calcu-lated and shown in Fig. 4.3. The total power of the MZI\u2019s two inputs (i.e.,the PSR\u2019s two outputs) are normalized without loss of generality. Their ini-tial phase difference was adjusted by giving the first phase shift \u2206\u03c61, whichdetermines the power ratio of the first AC\u2019s outputs. Similarly, the secondphase shift \u2206\u03c62, is tuned so that the power ratio of the second AC\u2019s outputs,i.e., the outputs of the MZI, is accordingly changed. Therefore, as you cansee in all four cases, a local maximum power of 100% at the output can be43(a) (b)(c) (d)(dB)Figure 4.3: Normalized power at the output port as a function of the thermalphase shifts, \u2206\u03c61 and \u2206\u03c62, of the MZI for four input polarization ratios:(a) 100% TE mode input, (b) 100% TM mode input, (c) 50% TE and 50%TM mode input, and (d) 75% TE and 25% TM mode input.obtained with an appropriate choice of \u2206\u03c61 and \u2206\u03c62. The mathematicalderivation of the system model is presented in Appendix B. Furthermore,we also experimentally verified the above simulations by tuning the thermalphase shifters, H1 and H2, as shown in Fig. 4.4. Four input polarizationstates, resulting in four different TE\/TM ratios at the input of the edgecoupler, were injected into the PR. We can see that the power distributionof the measured outputs closely matched the simulation results in Fig. 4.3.The distorted regions, such as the \u201dwavy\u201d lower optical power region shownin Fig. 4.4(a), were mainly attributed to the optical crosstalk in the PSR.44(a) (b)(c) (d)(dB)Figure 4.4: Measured optical power at the output port with various tuningpowers applied to the thermal phase shifters, H1 and H2, for four inputpolarization ratios: (a) 100% TE mode input, (b) 100% TM mode input,(c) 50% TE and 50% TM mode input, and (d) 75% TE and 25% TM modeinput.The thermal crosstalk in the thermal phase shifters may also cause a shiftof the diagrams. In all four input cases, maximum power at the output portcan be obtained with appropriate control of H1 and H2, which means that aminimum power was recorded at the feedback port to maximize the outputpower of the PR. This ability also provides a prerequisite to implement anautomated control process, indicating an automated power minimization atthe feedback port of the PR.454.1.3 Experiment and resultsThe arbitrary polarization state of the input light, as shown in Fig. 4.2, wasgenerated using a HP11896A polarization controller (PC) in the input path.The input light was coupled into the polarization receiver using a taperededge coupler.We generated a non-return-to-zero (NRZ) 12.5 Gbps 231 \u2212 1pseudorandom binary sequence (PRBS) optical data stream using a Mach-Zehnder modulator, which was then passed through the PC. The automatedcontrol, which is based on the greedy linear descent method, was then im-plemented (the control algorithm will be discussed in next section). Theoptical power at both the output port and the feedback port were recordedbefore, during, and after the automated optimization process, as shown inFig. 4.5(a). The optical light collected at the output port was detected byan external high-speed photodetector (PD) and finally sent to an Agilent86100A sampling oscilloscope with a 50 GHz 83484A sampling head. Thedisplayed eye diagrams of the data stream before, during, and after the opti-mization process were also recorded and are shown in Fig. 4.5(b): before theautomated optimization, the polarization receiver\u2019s MZI was not functionaland the arbitrary polarization state led to a small eye amplitude (58 mVin Fig. 4.5(b)(i)); during the optimization process, the eye started chang-ing (Fig. 4.5(b)(ii))indicating that the optimization process was affectingthe eye; after the optimization process, a large eye amplitude (124.9 mVin Fig. 4.5(b)(iii)) was obtained and stabilized using our automated tuningsystem. The experimental results indicate an automated control that cansuccessfully optimize the output data of the PR. However, for the optimiza-tion process, the consumed time, i.e., the iteration number, is still possible toreduce by modifying the control algorithms (not including the instrumentallimitation). Therefore, it is necessary to investigate the control algorithmsbased on other mathematical methods. In the next section, we demonstratetwo types of control algorithms based on two different optimization meth-ods, which treated the system model of the PR, in one type, as an implicitfunction, i.e., a \u201dblack box,\u201d and, in the other type, as a known analyticfunction.460 5 10 15 20 25 30 35Time (s)-40-35-30-25-20-15-10-50Optical Power (dB) OutputFeedbackEye height: 58 mV 12.5 Gb\/s Eye height: 124.9 mV 12.5 Gb\/sAutomated optimization(a)(b)1.5 dB30.3 dB(i) (ii) (iii)Figure 4.5: (a) Optical power of the output and feedback ports of the polar-ization receiver versus time (b) Recorded eye-diagrams of output signal ateach experimental stage: (i) before (polarization-scramble state), (ii) during,and (iii) after the automated optimization.4.2 Automated control algorithms for the PR4.2.1 Greedy linear descent- and gradient descent-basedcontrol algorithmsThe first type of control algorithm, that treats the entire system as a blackbox, is generally based on direct minimum search methods. We studiedthree minimization methods to track the minimum power at the feedbackport of the proposed PR: the greedy linear descent (GLD) method, the basicgradient descent (GD) method, and the two-point step size gradient descent(Barzilai and Borwein gradient descent method, B-B GD) method [108].These methods are illustrated in the flow diagrams shown in Fig. 4.6. Forthe GLD method shown in Fig. 4.6, the variables x1,i, x2,i, and y correspondto currents injected into H1and H2 and the measured optical power at the47feedback port of the PR, respectively. \u2206p represents the phase tuning stepsize, that corresponds to the current step being injected into the thermalphase shifters. A closed control loop is used to keep tracking and minimiz-ing the feedback power. The GLD method provides an easy to implement\ud835\udc65!,#$! = \ud835\udc65!,# + \u2206\ud835\udc5dNo Yes\ud835\udc56 = \ud835\udc56 + 1YesNo\ud835\udc65%,#$! = \ud835\udc65%,# + \u2206\ud835\udc5d\ud835\udc65!,#$! = \ud835\udc65!,# \u2212 \u2206\ud835\udc5d\ud835\udc66 \ud835\udc65!,#$!\t, \ud835\udc65%,# < \ud835\udc66(\ud835\udc65!,# \t, \ud835\udc65%,#)\ud835\udc66 \ud835\udc65!,#$!\t, \ud835\udc65%,#$! < \ud835\udc66(\ud835\udc65!,#$!\t, \ud835\udc65%,#)\ud835\udc65%,#$! = \ud835\udc65%,# \u2212 \u2206\ud835\udc5dFigure 4.6: A flow diagram illustrating the GLD-based minimization methodfor the control algorithm.minimum search algorithm for the PR. However, the GLD method\u2019s use ofa fixed current step size may result in a long convergence time for a smallstep size, or a failure to converge for a large step size. Therefore, to reducethe iteration number and improve the robustness of the control algorithm,we used the basic GD method. As shown in Fig. 4.7, instead of a uniformnumber, the step size of the variable vector xi (xi = [x1,i , x2,i]) is relatedto a term \u00b1 \u03b1 \u00b7 gi, where the gradient gi was re-calculated in each iteration,and the scalar factor \u03b1 is set to a small real number. Also, a thresholdnumber \u03c3 is given to end the control loop when the norm of gi decreases tothe convergence level. Consequently, given a large initial step size, the basic48GD method will be able to locate the minimum feedback power within feweriterations as compared to the GLD method. However, the constant scalarfactor \u03b1 is still a limitation as regards decreasing the convergence time of thealgorithm. Therefore, we employed a known, improved GD method, the B-BGD method [108], in which a new \u03b1, \u03b1i, is calculated for each iteration usinga two-point approximation to the secant equation underlying quasi-Newtonmethods. Now, the scalar factor \u03b1i is calculated by\u03b1i = \u2206xi \u00b7\u2206gTi \/\u2016\u2206gi\u20162 (4.1)where \u2206xi = xi \u2212 xi\u22121, \u2206gi = gi \u2212 gi\u22121, and, therefore, the step size,\u00b1 \u03b1i \u00b7 gi, would depend on the adjacent points. It has been mathematicallyproven [108] that the B-B GD method requires less computational effortand is less sensitive to ill-conditioning than the basic GD method, and,thus, the B-B GD-based control algorithm will outperform the basic one.The experimental demonstration on the PR will be presented in Section 4.3.\ud835\udc56 = \ud835\udc56 + 1\ud835\udc88! = \t\ud835\udefb\ud835\udc66(\ud835\udc99!)\ud835\udc88! < \ud835\udf0e\ud835\udc99!\"# = \ud835\udc99! \u2212 \ud835\udefc$ 0 \ud835\udc88!\ud835\udc66 \ud835\udc99!\"# < \ud835\udc66(\ud835\udc99!)\ud835\udc99!\"# = \ud835\udc99! + \ud835\udefc 0 \ud835\udc88! (basic GD)or \ud835\udc99!\"# = \ud835\udc99! + \ud835\udefc$ 0 \ud835\udc88! (B-B GD)No YesYesNoContinuous trackingFigure 4.7: A flow diagrams illustrating the basic GD- and B-B GD-basedminimization method for the control algorithms.494.2.2 Two stage method-based control algorithmThe second type of control algorithm is a single-iteration, two-stage methodbased on mathematical optimization functions. As shown in Fig. 4.8(a), afour-step workflow - initializing, sampling, training, and minimizing (\u201dISTM\u201d)forms the first stage of the control process. As we mentioned previously, aknown, systematic, analytic function, f(x, h1, h2), is needed and given byf(x, h1, h2) = E1(x, h1, h2) , (4.2)Eout(x, h1, h2) =[E1(x, h1, h2)E2(x, h1, h2)]= M(x) \u00b7 Ein(h1, h2) (4.3)where the 2 \u00d7 2 matrix, M , and the 2 \u00d7 1 matrix, Ein, correspond to thetheoretical model and the inputs of the MZI, i.e., the outputs of the PSR,respectively. Using the TMM, the 2 \u00d7 1 output matrix of the MZI, Eout,is derived using Eq. 4.3. The vector x still corresponds to the currentsinjected into H1 and H2, and there are two \u201dhidden\u201d parameters, h1 and h2,which are related to the power percentage in one input of the MZI and thephase difference between the two inputs of the MZI, respectively. Exceptfor h1 and h2, all of the other parameters in f(x, h1, h2) were obtainedusing the design parameters and the simulation results for the PR (e.g., S-parameter matrices for the ACs). In addition, it should be mentioned thatan experimental curve fitting correlation between the phase shift and thecurrent injections was also made (see Appendix C), which relates the phasechange in the analytic function to the injected currents into H1 and H2.Hence, in order to automatically locate a minimum power at the feed-back port of the PR, the control algorithm was started by following the\u201dISTM\u201d steps in Stage 1 (in Fig. 4.8(a)). To begin with, two initial num-bers for h1 and h2 were given to the analytic function f(x, h1, h2). Next, agroup of sample points (xi, yi), where yi is the sampled optical power at thefeedback port, were chosen and measured as a preparation for the followingoptimization process. Then, for the training step, we defined and minimizedthe amplitude of the \u201dresidual\u201d function \u2016yi\u2212f(x, h1, h2)\u2016, such that h1 andh2 could be trained simultaneously and finally optimized. Here, a non-linear50(a)Sampling (\ud835\udc5b\u00d7\ud835\udc5b points): \ud835\udc99! , \ud835\udc66! , \ud835\udc56 = 1\u2026\ud835\udc5bTraining: optimize \u210e\", \u210e# by minimizing\ud835\udc66! \u2212 \ud835\udc53(\ud835\udc99! , \u210e\", \u210e#)Minimizing: locate \ud835\udc99 to the \ud835\udc53$!%(\ud835\udc99, \u210e\", \u210e#)Initializing:the hidden parameters \u210e\", \u210e#(b)Stage 1 Stage 2Optimizing: update \u210e\u2032\", \u210e\u2032# by solving differential equations (Eq. 4)Minimizing: locate \ud835\udc99 to the \ud835\udc53$!%(\ud835\udc99, \u210e\u2032\", \u210e\u2032#)Measuring:the \u2206\ud835\udc87 = \ud835\udc53 \ud835\udc99 + \ud835\udf16 \u2212 \ud835\udc53(\ud835\udc99)Continuous trackingRe-execute Stage 1\ud835\udc66 \u2212 \ud835\udc53$!% \ud835\udc99, \u210e&\", \u210e&# < \ud835\udefeNo (Abrupt change)YesFigure 4.8: Flow diagram illustrating the minimization process for the single-iteration, two-stage control algorithm: (a) an \u201dISTM\u201d method for Stage 1and (b) a dynamic minimum tracking method for Stage 2.least-squares solver [109] was employed to realize a rapid optimization pro-cess. Lastly, with the optimized h1 and h2, one of the minimum points forf(x, h1, h2) could be quickly located using the Newton-Conjugate-Gradient(Newton-CG) method [110], which told us the currents that should be in-jected into H1 and H2 in order to obtain the minimum optical power at thefeedback port.However, as the input polarization state may vary continuously, h1 andh2 can drift overtime. Thus, in Stage 2, a dynamic tracking method is neededto keep updating the function parameters (h1 and h2) without repeatingthe sampling step. As shown in Fig. 4.8(b), firstly, output differences, \u2206f ,were measured by adding a perturbation \u000f to the variable vector x. Then,the instantaneous h\u20321 and h\u20322 were determined by the following differentialequation:\u2206f =\u2202f(x, h\u20321, h\u20322)\u2202x\u00b7 \u000f (4.4)51where\u2202f(x, h\u20321, h\u20322)\u2202xis the partial derivative of the analytic function. Usingthe Newton-Raphson method [111], we could solve the equations numericallyto obtain h\u20321 and h\u20322 . Finally, with the updated parameters, we adjustedthe currents injected into H1 and H2 to maintain the minimum power atfeedback port. Here, it should be noted that, if some abrupt change occurs(e.g., a discontinuous change of the polarization state or a major opticalpower fluctuation) during the tracking process, i.e., the measured opticalpower at the feedback port (y in Fig. 4.8(b)) is significantly larger thanthe minimum optical power at feedback port (fmin(x, h\u20321, h\u20322) in Fig. 4.8(b)),Stage 1 needs to be re-executed to update the analytic function. A threshold\u03b3 needs to be given to determine whether re-executing Stage 1, as shown inFig. 4.8(b).Benefiting from the use of the analytic function, the two-stage controlalgorithm has several advantages as compared with the first type of controlalgorithms i.e., the GLD, basic GD, and B-B GD method. In Stage 1,the more sampling points we have, the higher measurement tolerance thesystem model will have. This provides us with more flexibility to efficientlyadjust the accuracy based on the measurement tolerances such as opticalalignment errors, fluctuations of the output power, and noise level of theoptical feedback power. In Stage 2, the output response of the PR can bestabilized by introducing only a small perturbation. However, any of thefirst type control algorithms can be used in Stage 2. In addition, it shouldbe noted that some experimental factors, such as thermal drift and powerfluctuations, may effect the tracking results.4.3 High-speed experiment and resultsThe test structures were fabricated using 193 nm optical lithography at theIME, Singapore. As shown in Fig. 4.9, a high-speed experimental setupwas built to test the performance of the fabricated PRs using the controlalgorithms. A NRZ 231 \u2212 1 PRBS data stream, provided by a pulse pat-tern generator (PPG), was applied to a LiNbO3 Mach-Zehnder modulator52(MZM) to generate 10 Gbps modulated optical signals. An off-chip PC wasthen used to generate the arbitrary polarization states injected into the PR,and, in order to monitor these polarization states in the optical fiber, we useda 10\/90 polarization-maintaining fiber-optic tap to couple 10 percent of thetransmitted light into a polarization extinction ratio meter (PEM). Here, itMZM PCPEMDUTPPGOscilloscopeEDFAEDRxLaser SourceVOARFCurrent Source MeterControl algorithms PD90%10%(b)(a)Figure 4.9: (a) Schematic of the experimental setup for eye diagram andBER measurements. (b) A photo of the alignment between the transposerfiber array and chip, while the electrical probings are positioned.should be noted that, although we used PM fiber throughout our measure-ment setup, other than the fiber in the PC, the polarization states that weobtained from the PEM are, in general, different from the polarization statesinjected into the chip. However, the two orthogonal s and p polarizationswill be maintained by the PM fiber and only the phase difference betweenthem will change. Therefore, when only one of these two polarization states(s and p) is measured by the PEM, we know that it is the polarization stateinjected into our chip; the orthogonal states shown in Fig. 4.12(c) representthe s and p polarization states with \u223c15 dB polarization extinction ratios.The other polarization states recorded in Fig. 4.12(c) provide evidence that53a range of arbitrary polarization states were injected into our PR. For theon-chip control process, a current source meter and an external photode-tector (PD) were employed to tune the thermal phase shifters and read theoptical power at the feedback port of the PR. The output light was detectedby a high-speed photoreceiver (Rx), and an erbium-doped fiber amplifier(EDFA) and a variable optical attenuator (VOA) were placed ahead of theRx to set a desirable optical power level. The detected RF output was fi-nally sent to an oscilloscope for the eye diagram measurement and an errordetector (ED) for the bit-error ratio (BER) measurement.4.3.1 Automated control for an arbitrary input polarizationFigure 4.10 shows the measured BERs and optical powers for each of thecontrol algorithms applied to the DUT. An arbitrary polarization state thatwould result in a high optical power at the feedback port, i.e., a low opticalpower and high BER measured at the output port, was launched into thePR. This initial input case can be considered as an abrupt change of thepolarization state in an optical fiber. Then, we implemented the control al-gorithms. As shown in Fig. 4.10, by implementing the B-B GD-based, basicGD-based, GLD-based, or ISTM-based control algorithm, the optical powerat the feedback port was minimized to obtain at least a \u223c30 dB power ex-tinction ratio relative to the power at the output port, and correspondinglylow BERs (10-9\u223c10-10) at the output port. Here, we should also note thatfluctuations of the optimized BER data were mainly due to a short averag-ing time (1s) of the ED and minor changes in the optical coupling. Moreimportantly, based on the BERs in Fig. 4.10 (a), we can also see a signifi-cant difference in the number of iterations that were needed to accomplishthe optimization process: 19 iterations for the GLD method, 6 iterationsfor the basic GD method, 3 iterations for the B-B GD method, and 1 itera-tion for the ISTM method. These measurement results indicate that, for anarbitrary input polarization state, all of the automated control algorithmsaccomplish a similar level of optimized accuracy, i.e., they can adapt theinput into a desired, optimized TE output mode, while the B-B GD-based540 5 10 15 20 25 30Iteration-30-25-20-15-10-50Normalized optical power (dB) Feedback (B-B GD)Output (B-B GD)Feedback (Classical GD)Output (Classical GD)Feedback (GLD)Output (GLD) Feedback (ISTM)Output (ISTM)0 2 4 6 8 10-2-1.5-1-0.500 5 10 15 20 25 30-10-9-8-7-6-5-4-3Log 10(BER)B-B GDClassical GDGLDISTM(b)(a)Figure 4.10: Measurement results for four control algorithms: (a) BERsversus iteration and (b) normalized optical power at the output and feedbackports of the PRversus iteration. Inset Fig.: Zoom-in of the measured optical power at theoutput port.and the ISTM-based control algorithms take fewer iterations to obtain theoptimized output. Related Python codes for all the control algorithms arelisted in Appendix D.In addition, due to the polarization-dependent loss (PDL), mainly com-ing from the edge couplers and PSR, the optimized optical output powerwould vary with different input polarization state, even for constant inputpowers. Thus, we also measured BERs and eye diagrams versus the input55(b)Pol. state 1Pol. state 3Pol. state 4Pol. state 2-0.5 dBm -1.5 dBm -2.0 dBm16.3 ps\/div(a)-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5Optical input power (dBm)-10-9-8-7-6-5-4-3-2Log 10(BER)Pol. state 1 (measured points)Pol. state 1 (curve-fit)Pol. state 2 (measured points)Pol. state 2 (curve-fit)Pol. state 3 (measured points)Pol. state 3 (curve-fit)Pol. state 4 (measured points)Pol. state 4 (curve-fit)Figure 4.11: (a) BER versus optical input power (markers for measuredBERs and solid lines for the polynomial fittings) and (b) some measuredeye diagrams with four different polarization states in the optical fiber. Thepolarization angles of the polarization state 1 - 4: 3.0 degree, 13.5 degree,22.8 degree, and 84.6 degree.power for four polarization states (Pol. state 1 - 4 in Fig. 4.11). As shownin Fig. 4.11(a), by changing the polarization angle of the polarization statein the fiber, i.e., rotating a slow-axis polarization state (\u201dTE-like\u201d polarizedmode) to a fast-axis polarization state (\u201dTM-like\u201d polarized mode) coupled56into our input waveguide, larger BERs were obtained. This increase in BERwas due to higher losses for the TM-like mode. On the other hand, basedon the curve-fittings (Pol. state 1 and 4), we could also estimate \u223c1.5 dBPDL in our test PR. The PDL impact is reflected by the amplitude decreaseof the eyes when changing from TE- to TM-like polarized modes. This wasalso observed for various input power levels, see Fig. 4.11(b).4.3.2 Real-time automated control testFurthermore, we implemented a tracking and stabilization test by contin-uously changing the input polarization states using the external PC. Asshown in Fig. 4.12, the test PR optimized and stabilized the arbitrary in-put polarization states using both the B-B GD-based and two-stage controlalgorithm. In Fig. 4.12(c), the polarization angles and extinction ratios,monitored by the PEM, provided the angle of the major axes and elliptic-ities of the polarizations measured. As we mentioned previously, for the sand p polarizations on their own, it can also indicates the power launchedinto either the TE-like mode or the TM-like mode of the on-chip waveg-uide. The BER and the optical power at the feedback port are presented inFigs. 4.12(a) and 4.12(b). As a reference, the output responses without anycontrol algorithm were also measured. We can see that the feedback powerwas constantly minimized and stabilized by using both the B-B GD-basedand two-stage control algorithm, and, thus, the BER stayed at a lower levelas compared to the response without any control. However, it should benoted that, due to the PDL impact mentioned previously, the optimizedBER for both methods exhibited a higher level (10-6\u223c10-8) when the lightinjected into our PR was primarily in the TM-like modes. We can reducethis PDL impact by compensating the optical input power.4.3.3 DiscussionWhile this chapter is focused on our automated control algorithms, it isworth discussing the active polarization receiver that we used. First, theelectric powers that need to be applied to the phase shifters of the MZI570 100 200 300 400 500 600 700 800 900 1000-40-30-20-100Normalized optical power (dB)B-B GDTwo-stageNone0 100 200 300 400 500 600 700 800 900 1000-200-150-100-50050Polarization angle (degree)051015202530Polarization extinction ratio (dB)Measurement time (second)0\u00b090\u00b00\u00b090\u00b0(b)(c)(a)Slow-axis polarization stateFast-axis polarization state 100 200 300 400 500 600 700 800 900 1000-10-8-6-4-20Log 10(BER)B-B GDTwo-stageNonePower penalty (dB)00.380.931.72.63Figure 4.12: Measurement results of tracking and stabilization of contin-uous changed input polarization states: (a) BERs versus tracking time, areference BER level (dash line), and optical power penalty to achieve aBER of 10-10; (b) Normalized optical power at feedback port of the PRversus tracking time; (c) Polarization angles and extinction ratios of thetransmitted light in the optical fiber versus tracking time, and electric fielddistributions of slow-axis and fast-axis polarization state.58depend on the input polarization state. Given that the input polarizationstate is also wavelength dependent, this leads to the electric power thatneeds to be applied to the phase shifters also being wavelength-dependent.Also, adapting a particular polarization state to the TE-like output moderequires cancelling the group delay difference between the outputs of thePSR. It has been suggested that implementing a variable optical delay lineafter one of the outputs of PSR can be used to compensate for such groupdelay difference [39, 78]. Second, the polarization receiver used here is not anendless polarization tracking design [77, 78]. Additional MZIs are requiredto implement such endless tracking. Nevertheless, the automated controlalgorithms demonstrated in the chapter can easily be adapted to be usedwith any polarization receiver design.4.4 SummaryTo summarize, we have experimentally demonstrated greedy linear descent-based, basic gradient descent-based, two-point step size gradient descent-based, and two-stage optimization method-based automated control algo-rithms for use with an active polarization receiver. High-speed, large-signalmeasurements, for all of the control algorithms, show similar output re-sponses after the automated control, while different iterations are neededfor the various control process to converge. The PDL impact on the large-signal outputs has also been verified and discussed. Furthermore, using thetwo-point step size gradient descent-based and the two-stage optimizationmethod-based control algorithm, we have presented long-duration, large-signal measurements. The results indicate that these two control algorithmscan enable the polarization receiver to continuously adapt and stabilize thearbitrary polarization states, which change from a standard optical fiber,into the TE-like mode of our output waveguide. The demonstrated auto-mated control algorithms can also be used for WDM system applications. Anexpanded polarization receiver, which combines wavelength de-multiplexersto separate the input signal into individual transmitted wavelength-channels,each with its own polarization receiver, would be required.59Chapter 5Automated wavelength andpolarization control oftunable WDM polarizationreceiverAs we mentioned in Chapter 1, for efficient data transmission, WDM tech-nique is used to increase the data handling capacity in a single optical fiberlink. This, however, requires a receiver that can resolve the wavelength atwhich each data stream is transmitted. Moreover, since low-cost, commercialoptical fibers are not polarization-maintaining, on-chip receivers should alsobe able to adapt the polarization state of each input wavelength-channel.Polarization transparent schemes are typically realized using polarizationdependent components (mostly designed for TE mode operation) [5, 8, 57].For example, over the last decade, polarization diversity schemes, as shownin Fig. 5.1(b), have been intensively used and reported to develop variouson-chip WDM receivers [58\u201365]. Alternatively, in Chapter 4, we have intro-duced an active, on-chip polarization controller [39, 49, 76] uses a tunableMZI to actively manage the input polarization of each channel and passesthe optimized output light to the data-processing circuits. This approachcan enable the automated on-chip polarization control and also save the costof duplicating the circuits, especially for the complex or large scale circuitsthat may need many electrical controls[2, 67, 68, 78, 79, 112\u2013116], as shownin Fig. 5.1(c). Therefore, implementing on-chip WDM polarization controlbased on the above schemes is a promising and essential approach to realize60real-estate-efficient, fully-monolithic systems designed to process data formultiple wavelength channels simultaneously.Figure 5.1: (a) A block diagram showing that the arbitrary input polar-ization states for two wavelength channels (red and purple traces on LHSPoincare\u00b4 sphere) are both converted to TE polarization states in separateoutputs (the mixed color point on RHS Poincare\u00b4 sphere) through the pro-posed WDM polarization receiver. Proposed WDM polarization control forN channel data processing circuits: (b) A polarization diversity scheme us-ing a PSR (PSGC) and two demultiplexers (DeMuxes), and (c) a WDM,active polarization control scheme using a PSR (PSGC), two demultiplexers(DeMuxes), and N tunable MZIs.In this chapter, we experimentally demonstrate the automated adap-tation and stabilization of a silicon photonic WDM polarization receiver(WDM PR). A two-channel, tunable WDM PR was designed and used to61demonstrate the automated wavelength and polarization control, as illus-trated in Fig. 5.1(a). Due to our ability to simultaneously sense (by measur-ing the photocurrent) and control (via the applied voltage) their resonancecondition, compact micro-ring resonators (MRRs) with in-resonator photo-conductive heaters (IRPHs) [25] are used for the wavelength filters in ourWDM PR. The outputs of the MRR filters with IRPHs are then combinedby the tunable MZI, which removes the need to duplicate the data process-ing circuit. Therefore, using MRRs with IRPHs and tunable MZIs enablesus to to automatically adapt, and simultaneously stabilize any arbitrary in-put polarization state into the TE mode of the output waveguide for eachwavelength-channel. Also, a control algorithm based on Barzilai and Bor-wein\u2019s two-point step size gradient descent method (B-B GD) [108] is used torealize the automated wavelength and polarization control. By generatingarbitrarily polarized, modulated data streams (on-off keying (OOK) andpulse-amplitude modulation 4-level (PAM-4)) in a standard optical fiber,we implement the automated polarization and wavelength adaptation inour WDM PR. To assess the practicality of our approach, the stabilizationcapabilities of the automated control for both wavelength-channels are alsoevaluated over a long period of time in which the input polarization statesand the chip temperature of our WDM PR are continuously changed.5.1 Two-channel design and operating principleFig. 5.2(a) shows a schematic of our two wavelength-channel WDM PR.The proposed system consists of a broadband PSR [39], four tunable MRRswith IRPHs, an optical crossing [117], and two tunable MZIs that includebroadband 3 dB adiabatic couplers (ACs) [107]. Each of the componentsused has a low insertion loss of less than 0.5 dB; less than 0.2 dB for thePSR, less than 0.4 dB for an MRR with an IRPH, less than 0.5 dB foran optical crossing, and less than 0.2 dB for an AC. We choose broadbandnano-tapers [5] as the optical interfaces between the optical fibers and ouron-chip system. The input light, in two wavelength-channels, each havingan arbitrary polarization state, is predominately coupled into the TE00 and62(a)(b)Ch.1data2data1Ch.1 Ch.2data2data1PSRMRRdMRRaMRRcMRRbH11H21H12H223dB ACOutput1Feedback1Feedback2Output2Ch.2Crossing 3dB ACSiO2 Si (220 nm) Si (90 nm)metal n++ via to metaln50 \ud835\udf07mFigure 5.2: (a) Schematic of the two-channel WDM polarization receiverfor two wavelength-channels each having arbitrarily polarized inputs (SiO2cladding layer is not included in this figure). Inset figure: Zoom-in of aMRR filter with an IRPH. (b) An optical micro-graph of a fully fabricatedtwo-channel WDM-polarization receiver.TM00 modes in the nano-taper and propagates into the PSR. The adia-batic PSR evolves these TE00 and TM00 modes into the TE00 modes of itstwo, single-mode, output waveguides, and then transmits these TE00 outputmodes to the MRR filters (MRRa to MRRd in Fig. 5.2(a)). By configuringthe voltages applied to the IRPHs (zoom-in in Fig. 5.2(a)), the MRRs can betuned to separate the two channels, for each of the PSR outputs, and guidethem to the MRRs\u2019 drop-ports. Following the MRR filters, the outputs forone channel (Ch. 1) are guided to the upper MZI, and the outputs for theother channel (Ch. 2) are guided to the lower MZI; to accomplish this, abroadband optical crossing is used in one of the optical paths for each chan-nel. Then, by tuning the thermal phase shifters (H11, H21, H12, and H22 inFig. 5.2(a)), the optical power at the feedback port of each MZI is minimizedso that the optical power at its output port is maximized. In this way, anarbitrary input polarization state from the optical fiber can be adapted tothe TE00 mode of the output waveguide for each wavelength-channel. TheB-B GD-based control algorithm, demonstrated in Chapter 4, is employed63to realize automated adaptation and stabilization for both wavelength andpolarization. Therefore, by automatically controlling the voltages appliedto the IRPHs and the currents passed through the thermal phase shifters,we can track and stabilize the maximum photo-current in the IRPHs of ourMRR filters and the minimum power at the feedback port of our WDMPR, respectively. An optical micrograph of a fully fabricated two-channelWDM PR is shown in Fig. 5.2(b). Low resistance aluminum wire-bonds areused to connect the on-chip bond-pads to the electrical power sources fortuning the MRRs\u2019 IRPHs and the MZIs\u2019 thermal phase shifters, as shown inFig. 5.2(b) and Fig. 5.4(c). Here, it should be noted that the automated con-trol algorithm that stabilizes the MRR filters will reach a \u201chalt\u201d when thephotocurrent read by an IRPH drops below a specified \u201clow\u201d level thresh-old, that can occur due to the changing input polarization state, and thealgorithm will continue when the IRPH\u2019s photocurrent again rises above thelow level threshold (e.g., 10% of the maximum photocurrent). Therefore,halting can result in small input power penalties that depend on the chosenthreshold.5.2 Experiments and results5.2.1 Experiment preparation and automated controlThe test structures were fabricated using 193 nm optical lithography atthe IME, Singapore. We chose two neighbouring ITU wavelength-channels,1554.13 nm (Ch. 1) and 1555.75 nm (Ch. 2) on a 200 GHz grid, as theWDM input signals. Therefore, by locating the voltages that maximizethe photocurrents measured by the MRRs\u2019 IRPHs, the desired wavelength-channels can be obtained at the drop-ports of the corresponding MRR filters.As an example, Fig. 5.3 shows the as-fabricated and tuned spectra of anMRR\u2019s calibration structure, and the locating voltages (V1 to V2) appliedto the IRPHs for both chosen wavelength-channels.A high-speed experimental setup was built to characterize the automatedadaptation and stabilization for the tested WDM PR, as shown in Fig. 5.4.641550 1552 1554 1556 1558 1560Wavelength (nm)-30-25-20-15-10-50Normalized transmission (dB)As-fabricated Ch. 1 Ch. 22 2.5 3 3.5 4V (V)1020304050I PD (A)Ch. 1Ch. 2V1 V2Figure 5.3: Normalized drop-port spectra as-fabricated and after tuning anMRR first to Ch. 1 and then to Ch. 2. Inset Fig.: Photocurrent IPD measuredin the MRR\u2019s IRPH while tuning to Ch. 1 and to Ch. 2. Applied voltagesV1 and V2 are found when locating the maximum photocurrent of the IRPHto Ch. 1 and to Ch. 2, respectively.Two optical light sources (Laser 1 and Laser 2) are provided by a KeysightN7714A multi-channel laser source. Then, two NRZ 231 \u2212 1 PRBS datastreams, generated by two pulse pattern generators, an Anritsu MU183020A(PPG1) and an Anritsu MP1763B (PPG2), were applied to two LiNbO3Mach-Zehnder modulators (MZM1 and MZM2) to generate the data streamfor each wavelength-channel. In one channel path, a HP 11896A polariza-tion controller (PC) was used to generate the arbitrary polarization states,and, a 10\/90 PM fiber-optic tap was used to couple 10 percent of the trans-mission into a HP 8509B polarization analyzer (PA), which can give thelocations of the input polarization state on the Poincare\u00b4 sphere. We cal-ibrated the PA using s and p polarized input light. Other than the fiberin the PC, all of the fibers used throughout our test setup were PM fibers.Here, it should be mentioned that, for arbitrary polarization states recordedby the PA, the ratios of the s and p polarizations and their phase differ-ences determine the points on the Poincare\u00b4 sphere. Hence, by using a PMfiber tap together with the PA, we can know relative amounts of s and ppolarizations, but we cannot know their phase differences at the input to65Figure 5.4: (a) A schematic of the experimental setup for eye diagram andBER measurement. (b) A photo of the actual experimental setup and (c) azoom-in of the tested wire-bonded chip with the chip carrier.the chip. Nevertheless, the arbitrary polarization states recorded on the66Poincare\u00b4 sphere provide evidence that a range of polarization states wereinjected into our on-chip system. Then, the two wavelength-channel inputs,each having identical optical power, were combined using a PM 3 dB fiber-optic coupler. An SRS LDC501 thermoelectric cooler (TEC) was used tomanage the temperature of the tested chip. For the on-chip active controlprocess, Keithley 2604B SMUs and an HP 81635A PD were used to tunethe thermal phase shifters and read the optical power at the feedback portof our WDM PR, respectively. The output optical signal was then detectedusing an HP 11982 high-speed Rx, and an EDFA and a VOA were placedahead of the Rx to set a desired optical power level. Finally, the detectedRF output was sent to an Agilent 86100A sampling oscilloscope with a HP83484A 50 GHz sampling head for the eye diagram measurements and anAnritsu MU183040B ED for the BER measurements.54.9 mV\/div 10.0Gb\/s49.3 mV\/div 20.0Gb\/s 49.3 mV\/div 20.0Gb\/s 49.3 mV\/div 20.0Gb\/s54.9 mV\/div 10.0Gb\/s 54.9 mV\/div 10.0Gb\/s54.9 mV\/div 10.0Gb\/s49.3 mV\/div 20.0Gb\/s(a) (b) (c) (d)(e) (f) (g) (h)Stage 0 Stage 1 Stage 2\u201c00\u201d\u201c01\u201d\u201c10\u201d\u201c11\u201d\u201c0\u201d\u201c1\u201dFigure 5.5: Recorded eye diagrams for Ch. 1, (a) - (d) OOK and (e) - (h)PAM-4 output signal at each control stage: In Stage 0, (a), (e), all of controlwas offline; In Stage 1, (b), (f), the desired channel (Ch. 1) was found byconfiguring MRR\u2019s IRPH; In Stage 2, eye opening (c), (g) during and (d),(h) after the automated polarization adaptation.Figs. 5.5(a)-(d) show a 10 Gb\/s OOK data stream at Ch. 1 measuredat various stages of the automated control process when a second 10 Gb\/sOOK data stream was present at Ch. 2. Similarly, Figs. 5.5(e)-(h) show a20 Gb\/s PAM-4 data stream (generated using two synchronized, PPGs and67combined using a Wilkinson\u2019s combiner [118]) at Ch. 1 at various stagesof the automated control process when a 10 Gb\/s OOK data stream wastransmitted at Ch. 2. In Stage 0, as shown in Figs. 5.5(a) and 5.5(e), sincethe active controls were offline and the input channels did not match the as-fabricated resonances of the MRR filters, closed eyes for Ch. 1 were recorded.Then, in Stage 1, by configuring the applied voltages for the correspondingMRRs\u2019 IRPHs (MRRa and MRRd in Fig. 5.2(a)), i.e., maximizing the mea-sured photocurrents, the tuned resonances of the MRR filters were aligned toCh. 1; this led to the output signal having a relatively open eye, i.e., a \u223c71.5mV eye height (from level \u201c0\u201d to level \u201c1\u201d) for OOK transmission and eyeheights of \u223c0 mV, \u223c17.9 mV, and \u223c12.2 mV (from level \u201c00\u201d to level \u201c11\u201d)for PAM-4 transmission, see Figs. 5.5(b) and 5.5(f). After the wavelengthchannel tuning, i.e., in Stage 2 as shown in Figs. 5.5(c) and 5.5(d), and inFigs. 5.5(g) and 5.5(h), by using the B-B GD method-based control algo-rithm, the MZI\u2019s thermal phase shifters (H11 and H21 in Fig. 5.2(a)) wereautomatically tuned so that the optical power at the feedback port was min-imized and increased eye-openings were observed at the output port, wherea \u223c166.9 mV eye height (from level \u201c0\u201d to level \u201c1\u201d) for OOK transmissionand eye heights of \u223c35.9 mV, \u223c39.7 mV, and \u223c46.1 mV (from level \u201c00\u201dto level \u201c11\u201d) for PAM-4 transmission were obtained. Hence, our WDM PRachieved automated wavelength and polarization control for Ch. 1. Here, itshould be noted that the thermal fluctuations of the on-chip system, suchas environmental temperature variations, and thermal crosstalk introducedby the thermal phase shifters, will lead to a resonant wavelength-drift inthe MRRs which, in turn, may cause the eyes close at the output. Hence,automated stabilization of the configured MRRs\u2019 IRPHs was implementedfor our WDM PR.In addition, when a horizontal linear polarization state (HLP) recordedin the PA, i.e., a TE mode injected into the chip, optical power responsesshown in Figs. 5.6(b) and 5.6(c) can be obtained by tuning H21 and H11,receptively. In Figs. 5.6(b), we can see the interfering spectra with differentERs, and this is because the unbalanced coupling ratio of the AC and theunequal losses of the cross and through path; In Figs. 5.6(c), a comparatively68HLP (TE)VLP (TM)RCP(a)(b) (c)(d) (e)(f) (g)0 10 20 30 40 50Electric power (mW)-40-30-20-100Normalized optical power (dB)0 10 20 30 40 50Electric power (mW)-20-15-10-50Normalized optical power (dB)0 10 20 30 40 50Electric power (mW)-40-30-20-100Normalized optical power (dB)0 10 20 30 40 50Electric power (mW)-20-15-10-50Normalized optical power (dB)0 10 20 30 40 50Electric power (mW)-10-8-6-4-20Normalized optical power (dB)0 10 20 30 40 50Electric power (mW)-10-8-6-4-20Normalized optical power (dB)FeedbackOutputFigure 5.6: (a) The HLP, VLP, and RCP polarization states recorded onthe Poincare\u00b4 sphere, and normalized optical powers at output and feedbackports,by sweeping the thermal phase shifter H21 (b), (d), (f), and H11(c),(e), (g) for each polarization state.69flat responses were received at both ports, and, however, the optical powersat both ports were decreased with higher tuning electric powers, which ismainly due to the resonance drift of the locked MRRs caused by the thermalcrosstalk from H11. In Figs. 5.6(d) and 5.6(e), similar optical responses atswapped ports can be observed when a vertical linear polarization state(VLP) in the PA, i.e., a TM mode of a silicon waveguide, was launched.We also generated a polarized light acted as a right circular polarization(RCP) in the PA, i.e., an injected polarization with equal ratio of s and ppolarization portion, but unknown phase difference between the two vectors,and implemented the thermal tuning on H11 and H21, respectively. Theoptical responses presented in Figs. 5.6(f) and 5.6(g) show that minimumpoints at both ports can not be found by simple 1D sweeps, indicating thatoptimization algorithms are needed here to control the phase shifters.5.2.2 Long-duration, continuous, automated adaptationand stabilization testWe also implemented a long-duration adaptation and stabilization experi-ment in which we continuously varied the input polarization states and thechip temperature. The two wavelength-channel input signals, each havingequal optical power and separate data streams (10Gb\/s OOK), were simul-taneously transmitted into the chip. Here, we passed one channel throughthe PC at a time and recorded the affects of polarization and tempera-ture change for that channel. The other channel\u2019s polarization was free todrift, but was not forced to change (it was, however, subjected to the sametemperature change). The experimental results for these two channels, i.e.,Ch. 1 and Ch. 2, are shown in Fig. 5.7. As we mentioned previously, s andp polarized light was used to calibrate the PA. Then, using the externalPC, we generated arbitrary polarization states to inject into our WDM PR.In Fig. 5.7, the points, which were measured by the PA and correspond tothe normalized Stokes parameters (S1, S2, and S3 plotted on the right-handside of Fig. 5.7), provide various polarization states for Ch. 1 and Ch. 2.Simultaneously, we varied the chip temperature by controlling the TEC. In70Ch. 2Ch. 1 (a)0 100 200 300 400 500 600-1-0.500.51Stokes parameters (S) S1 S2 S30 100 200 300 400 500 600-1-0.500.51Stokes parameters (S)S1 S2 S3Time (second)(b)(c)Figure 5.7: (a) The measured coordinates of the monitored polarizationstates on the Poincare\u00b4 sphere and the corresponding normalized Stokes pa-rameters for (b) Ch. 2 and (c) Ch. 1 versus tracking time.710 100 200 300 400 500 600Time (second)2345Voltage (V)MRRa (Ch. 1)MRRd (Ch. 1)MRRc (Ch. 2)MRRb (Ch. 2)2 2.5 3 3.5 4 4.5V (V)104070100I PD (A)0 100 200 300 400 500 600242628303234Temperature ( o C)(a)(b)Va Vd VbVcFigure 5.8: (a) Chip temperature versus tracking time; (b) Wavelengthaligning and stabilization: applied voltages to the MRRs\u2019 IRPHs versustracking time. Inset Figure: Initial IRPH\u2019s photocurrents (IPD) measuredwhile tuning the MRRs for the two channels.our test, the chip temperature was set to increase linearly from \u223c26 \u25e6C to\u223c33 \u25e6C as shown in Fig. 5.8.Thus, as discussed earlier, to tackle the wavelength-drift in the tunedMRRs, i.e., stabilize the IRPHs\u2019 maximized photocurrents, the voltages ap-plied to the IRPHs were automatically adjusted using the B-B GD-basedcontrol algorithm. In this way, as shown in Fig. 5.8, when the chip temper-ature was increased, i.e., the MRRs\u2019 resonances were red-shifted, the tuning720 100 200 300 400 500 600-10-8-6-4-2Log 10(BER)-50-40-30-20-10Normalized optical power (dB)BER (Ch. 1)BER (Ch. 2)Feedback (Ch. 1)Feedback (Ch. 2)(a)(b)0 100 200 300 400 500 600Time (second)05101520253035Power (mW)H11 (Ch. 1)H21 (Ch. 1)H12 (Ch. 2)H22 (Ch. 2)Figure 5.9: (a) BERs versus tracking time, a reference BER level (dash line),and normalized optical powers at feedback ports of the WDM PR versustracking time; (b) Recorded applied electric power to the MZIs\u2019 thermalphase shifters versus tracking time.voltages applied to the MRRs\u2019 IRPHs (MRRa - MRRd) were decreased,(i.e., to compensate for the red-shift) during the automated control process.Hence, with constantly stabilized MRR filters, we could implement the au-tomated adaptation and stabilization for two arbitrarily polarized inputs.As shown in Fig. 5.9, we can see that, for both wavelength-channels, theoptical powers at the feedback port were constantly minimized and stabi-lized at a >30 dB power extinction ratio (relative to the power at the output73port) and this led to the measured BERs staying at lower levels (10-9\u223c10-10).Here, it should be noted that the fluctuations of the optimized BER datamainly resulted from minor changes in the optical coupling. Additionally,we also recorded the various electric powers applied to the MZIs\u2019 thermalphase shifters (H11, H21 for Ch. 1, and H12, H22 for Ch. 2 in Fig. 5.9).These tracks reflect the automated adaptation and stabilization process,which corresponds to the output responses as shown in Fig. 5.9. Here,since the input polarization state for each wavelength-channel was differ-ent, the control powers applied to the thermal phase shifters depended onthe wavelength-channel, i.e., different tracks for the electric powers for eachchannel were obtained.5.2.3 DiscussionThis chapter demonstrates the automated adaptation and stabilization of anon-chip WDM PR, but it is necessary to mention the limitations of this two-channel design and propose solutions to overcome these limitations. First,this design is not an endless polarization control design that can conducta phase reset without a transient response [77, 78]. Two additional activeMZIs are needed to form an endless phase shift design, which is necessaryto implement endless polarization adaptation and stabilization. Second, itshould be noted that a time delay (the \u201cTE-TM\u201d delay) can result from thePSR and the differences between the \u201cTE\u201d and \u201cTM\u201d paths leading to theinputs to the tunable MZIs. However, in previous work [65], it has beenshown that small \u201cTE-TM\u201d delays (less than 45 ps) do not detrimentallyaffect the data transmission. In other words, the receiver scheme can toleratea significant amount of delay between the signals for the \u201cTE\u201d and \u201cTM\u201dpolarizations before the BER starts to deteriorate. In Chapter 6, we willintroduce an improved, N-channel, WDM PR which will have less delay thanthe design presented in [65] even in the worst case, due to its symmetricloop design. Nevertheless, it also has been proposed that implementing avariable optical delay line after one of the outputs of the PSR[39, 78], orthat adjusting the waveguide length difference between the two paths, can74be used to compensate for such signal delays. In addition, the improved N-channel design can simplify the control of the IRPHs by ensuring indefinitelocking, i.e., it can circumvent the \u201chalt\u201d action as the same MRR is usedto combine the data carried by the same wavelength on both polarizationstates, thus ensuring a maximum photocurrent at all times.It is worth mentioning that when conducting our experiments, the re-sponse time was limited by the BER measurements, which consumed onesecond per measurement point to give the error detector a sufficient amountof time to measure the bit errors. However, in practice, the demonstratedactive approach should be able to track polarization changes on the orderof 10s to 100s of kHz. Since the IRPHs photo-detection response time isaround 280 ns, and the IRPHs heating response time is around 900 ns[113],this speed is mainly limited by the TiN heater\u2019s response time (on the orderof 10s of \u00b5s).5.3 SummaryIn this chapter, we have experimentally demonstrated automated adapta-tion and stabilization for a two-channel, tunable WDM-polarization receiver.Using a control algorithm based on Barzilai and Borwein\u2019s two-point stepsize gradient descent method, we realized automated wavelength tuningand stabilization, and automated polarization adaptation simultaneouslyfor two wavelength-channels, each having an arbitrary input polarizationstate. High-speed data was generated to modulate the optical input signals.Furthermore, we have implemented a long-duration, polarization and tem-perature varying experiment. The results present continuous polarizationadaptation and stabilization for arbitrary input polarizations from stan-dard optical fibers, and continuous stabilization of the desired wavelength-channels for on-chip temperature changes. In addition, we have proposedan improved N-channel, tunable WDM polarization receiver for practicalWDM-polarization applications. Our work paves the way toward developingfully automated, monolithic, data processing systems for multiple channelssimultaneously, such as large-scale optical switches and coherent receivers.75Chapter 6Conclusion and future work6.1 ConclusionIn this thesis, we have investigated silicon photonic wavelength and polar-ization components, subsystems, and related control techniques for on-chipreceiver applications. Apodized spiral Bragg grating waveguides, broad-band sub-wavelength grating-assisted polarization splitter-rotators, a tun-able polarization receiver with the automated control algorithms, and au-tomated wavelength and polarization control technique realized in a noveltwo-channel WDM polarization receiver, have been studied in each orga-nized chapter. For silicon photonic integrated components and circuits, wepresented the design and simulation methodologies as well as the simulatedand experimental results; For automated control techniques, we also explainthe control algorithm theories and the experimental verification in real-timeoptical interconnects. The major contributions of the research work arelisted below.\u2022 Demonstration of Gaussian-apodized, spiral Bragg grating waveguides(SBGWs) for fundamental TM modes. The measured reflection re-sponse shows a higher sidelobe suppression ratio compared to the re-sponse of a uniform SBGW. This demonstrated device can be used todesign narrow-band WDM filters. An apodized, period-chirped SBGWis also introduced and its measured group delay where the ripples areefficiently suppressed shows significant use in on-chip optical delay linedesigns.\u2022 Demonstration of SWG-assisted, fully mode-evolution-based PSRs intwo SOI platforms. Particularly, this is the first demonstration of an76SWG-assisted, adiabatic structure on a standard optical lithographyplatform. Less than \u221215 dB polarization crosstalk for fundamentalTE and TM modes was measured over a broad wavelength range from1545 to 1615 nm. The demonstrated SWG-based PSRs are more com-pact than previously reported adiabatic PSRs, and it can be used asan important polarization component in an on-chip polarization sub-system.\u2022 Realization of active silicon photonic polarization receiver with auto-mated control algorithms:\u2013 Experimental demonstration of an automated polarization re-ceiver (PR) on a standard active SOI platform. Arbitrary po-larization states from a standard optical fiber are automaticallyadapted into the fundamental TE mode of an SOI output waveg-uide of the PR.\u2013 Investigation of greedy linear descent-based, basic gradient descent-based, two-point step size gradient descent-based, and two-stageoptimization method-based automated control algorithms in thedemonstrated PR. Using all of the control algorithms, we suc-cessfully realized automated adaptations for a high-speed datastream and compared their performance with regard to the iter-ation number and the output response. The two-point step sizegradient descent-based and two-stage method-based control al-gorithms were further studied by implementing a long-durationexperiment, which shows their capabilities to achieve real-timeand continuous polarization management in the PR.\u2022 First demonstration of automated adaptation and stabilization of atunable WDM polarization receiver (WDM PR) in a standard activeSOI platform. A two-channel, tunable WDM PR is proposed andfabricated to experimentally demonstrate the automated wavelengthand polarization control. 10 Gb\/s OOK and 20 Gb\/s PAM-4 formatsare generated as the high-speed input data streams. A long-duration77BER experiment is also conducted to show that the control algorithmenables the WDM PR to adapt, stabilize, and track the arbitraryinput polarization states and simultaneously stabilize the transmittedwavelength-channels at various chip temperatures.It is also worthwhile concluding the benefits and limitations of the apodizedSBGW work in this thesis. The spiral apodization scheme used in theBGWs provides a practical and efficient approach to designing long apodizedBGWs, in order to achieve high ERs, small coupling coefficients, and smoothedgroup delays. The design and simulation method, which is based on FDTDbandstructure and TMM, provides a straightforward and accurate way todesign apodized BGWs. However, in order to receive the reflected light, weneed to generally place a Y-junction ahead of the BGW structures, whichcauses extra optical loss (> 3 dB) in the received reflected power. In fiberoptics communications applications, one can use an optical circulator toavoid such loss. Nevertheless, as we mentioned previously, designing a sili-con photonic add-drop filter can be a better on-chip option. Our apodizedSBGW research work facilities designing high performance BGW-based add-drop filters. In the recent years, numbers of add-drop filter designs based onBGWs have been reported for various WDM applications. Table 6.1 sum-marizes the research work on silicon photonic BGWs in the last decade. Wecan see that high-performance add-drop filters have been intensively demon-strated for WDM circuits. For instance, in [13, 83, 119], wide-band, highSLSR, BGW-based contra-directional coupler (CDC) add-drop filters aredemonstrated for CWDM optical links. Such designs can also be utilized inour automated WDM polarization receiver applications. In the next section,we will propose and discuss one BGW based WDM polarization receiver forCWDM networks.Overall, the above contributions of this thesis present comprehensivestudies on silicon photonic circuits from a device level to a subsystem leveland the control automation dedicated to each circuit. The wavelength andpolarization, which are the most crucial research topics on current existingSOI platforms, are investigated in this thesis and facilitate future research78Ref. Year BW\/SLSR Type & Applications[11], [15],[14]2012,2013,2015BW \u223c0.4 nm[11], \u223c0.2 nm[15], \u223c0.09 nm[14]Uniform straight [11] and spiral[15],[14] BGWs, DUVfabrication[11], [15]; Narrowbandwidth WDM filters.[81], [120] 2013,2019BW \u223c0.1 nm[81], \u223c0.2 nm[120]Phase-shifted slot-based[81] andSWG-based[120] BGWs;Label-free biosensors.[83],[87],[121],[122], [13]2013,2014,2015,2018,2019SLSR \u223c30 dB[83], BW \u223c10 nm[87], BW >10 nm[121] BW \u223c0.8nm [122], SLSR>50 dB [13]Apodized[83],[121],[122],[13],linearly chirped[87] BGW-basedcouplers, DUV fabrication;WDM add-drop filters[83],[121],[122],[13], tunableoptical delay line[87].[123] 2016 BW \u223c0.8 nm p-n junction, phase-shiftedBGW; high-speed modulators.Our work 2017 SLSR \u223c13 dB,BW \u223c1.3 nmTM mode, apodized, spiralBGWs; Narrow-band WDMfilters, linear optical delay lines.[124] 2018 BW \u223c49 pm p-n junction, phase-shiftedBGWs; Programmable opticalsignal processing[125],[119]2018,2019BW \u223c33.4 nm[125], \u223c32.6 nm[119],SWG-based contradirectionalcouplers; Broadband WDMadd-drop filters.[126] 2019 SLSR \u223c20 dB Apodized BGWs via phasemodulation; Narrow bandwidthWDM filter.[127],[128]2017,2019BW \u223c0.8 nm[127], \u223c0.26 nm[128]Polarization-rotating BGWs;Polarization-independenttransmission filtersTable 6.1: A summary of recently published BGW designs on SOI platforms.79work towards fully CMOS driven, monolithic, data processing integratedcircuits. Therefore, although some factors limit practical applications, suchas our experimental conditions (slow response speed of the instruments (or-der of kHz) and communication speeds between the instruments (order ofkHz)), the covered valuable points in this thesis will gain interests to thesilicon PIC research community.6.2 Future workBased on the achieved research work in this thesis, we propose the followingdesigns as the future work:1. An automated polarization receiver with an endless phase shifter struc-ture. Since its first demonstration in 1990 [129], endless phase shiftingarchitecture has been proposed [130] and implemented [66, 78]in sili-con photonic coherent systems. As shown in Fig. 6.1, to implement theendless phase shifting in the PR, the phase shifter H1 is replaced by theMZI switch structure in Fig. 6.1(b). H1 still controls the overall phasetuning of the endless phase shifter, and the switching action is done bytuning Ha and Hb. For instance, when H1 reaches its tuning limit, Haand Hb will simultaneously perform a phase shift of pi and \u2212pi, respec-tively. In this way, the entire MZI preserves the overall phase in H1and switches the light from H1 to the other arm. This switching actionallows H1 to re-set back to its operating range rapidly. Such endlesstracking configurations can significantly save tuning power consump-tion and, therefore, are necessary for polarization control, especiallythe long-duration, continuous polarization tracking implementations(covering the entire Poincare\u00b4 sphere). Although the polarization re-ceiver used in this thesis is not endless, the demonstrated control algo-rithms do not depend on whether the polarization control is endless.In other words, we believe that the function of the control algorithmsis easily compatible with such endless polarization control design.2. Developing a WDM polarization receiver using BGW based filters. In-80Figure 6.1: A schematic of demonstrated polarizaiton receiver (a) and theextended phase shifter (H1) design with the endless phase shifting structure(b).stead of the MRR used in this dissertation, a BGW based contra-DC,demonstrated in [13, 83], is used as the WDM filters. A polariza-tion rotating Bragg grating filter (PRBG), reported in [131], is thenconnected to rotate the input polarization in the designed wavelengthrange. As the schematic in Fig. 6.2 shows, when two-channel inputlight, each having an arbitrary polarization state, i.e., an unknownratio of TE and TM mode, is injected, the contra-DC firstly reflectthe TE mode at Ch. 1 to the its drop-port; Then, the polarized lightat other channel (Ch. 2) passes to the PRBG, where the TM portionof the input light at Ch. 1 is rotated to a TE mode and reflects backto the contra-DC, and this \u201creturned\u201d TE polarized light reflects tothe add-port of the contra-DC; Finally, together with the original TEportion light at other port, the input light at Ch. 1 is sent to thetunable MZI, where the active polarization control is implemented.Since the contra-DC and the PRBG are both required to design forthe same channel (Ch. 1) with a free FSR range, the polarized light atany other channel (Ch. 2) propagates through these two BGW basedstructures. Therefore, other channels\u2019 BGW based structures can beeasily cascaded to have an N-channel PIC. Such designs provide apath to develop multi-channel, CWDM on-chip data processing cir-cuits. The endless phase shifting structure discussed previously canalso be adapted into proposed circuits.81Figure 6.2: A schematic of the proposed CWDM polarization receiver basedon a contra-DC, a PRBG, and a tunable MZI.3. A multi-channel WDM PR with automated feedback control. Thetwo-channel WDM PR demonstrated in Chapter 5 can be extended forN-channel operation. For example, based on the two-channel scheme,an N-channel, tunable WDM PR can be realized by re-routing thewaveguides with N tunable MZIs, 2\u00d7N MRRs, and (N-1)\u00d7N\/2 op-tical crossings, see in Fig. 6.3(a). However, such a complex designmay not be an efficient option, especially for a large number of wave-length channels. Therefore, we propose an improved N-channel, tun-able WDM PR shown in Fig. 6.3(b). A waveguide loop is formedbetween two output ports of a PSR so that we can re-use MRR fil-ters [65]. In this way, for each wavelength channel, the two outputs ofthe PSR are directed to the add-port and drop-port of a single MRRfilter, respectively. This scheme simplifies the on-chip circuit designand significantly reduces the footprint and, therefore, the cost. Moreimportantly, the improved N-channel design can simplify the controlof the IRPHs by ensuring indefinite locking, i.e., it can circumvent the\u201chalt\u201d action as the same MRR is used to combine the data carriedby the same wavelength on both polarization states, thus ensuring amaximum photocurrent at all times. Also, the automated adaptation82and stabilization demonstrated in Chapter 5 can easily be adapted tosuch an N-channel WDM PR. The mode-evolution based PSR used inour WDM PR can be replaced by a more compact one demonstratedin Chapter 3.Figure 6.3: Schematic of (a) an expanded N-channel WDM PR and (b) aproposed improved N-channel design. 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Chen, Y. Wang, J. Fluekiger, M. Caverley, L. Chros-towski, and N. A. Jaeger, \u201cPolarization-rotating, bragg-grating filterson silicon-on-insulator strip waveguides using asymmetric periodic cor-ner corrugations,\u201d Optics Letters, vol. 40, no. 23, pp. 5578\u20135581, 2015.101Appendix APublicationsA.1 Journal Publications1. Minglei Ma, Hossam Shoman, Sudip Shekhar, Nicolas A. F. Jaeger,and Lukas Chrostowski, \u201cAutomated Adaptation and Stabilization ofa Tunable WDM Polarization-Independent Receiver on Active Sili-con Photonic Platform,\u201d IEEE Photonics Journal, accepted and to bepublished, 2020.52. Minglei Ma, Hossam Shoman, Sudip Shekhar, Nicolas A. F. Jaeger,and Lukas Chrostowski, \u201cAutomated control algorithms for siliconphotonic polarization receiver,\u201d Optics Express, vol. 28, no. 2, pp. 1885-1896, 2020.3. Enxiao Luan, Han Yun, Minglei Ma, Daniel M. Ratner, Karen C. Che-ung, Lukas Chrostowski, \u201cLabel-free biosensing with a multi-box sub-wavelength phase-shifted Bragg grating waveguide,\u201d Biomedical OpticsExpress, vol. 10, no. 9, pp. 4825-4838, 2019.4. Minglei Ma, Anthony H. K. Park, Yun Wang, Hossam Shoman,Fan Zhang, Nicolas A. F. Jaeger, and Lukas Chrostowski, \u201cSub-wavelengthgrating-assisted polarization splitter-rotators for silicon-on-insulatorplatforms,\u201d Optics Express, vol. 27, no. 13, pp. 17581-17591, 2019.5. Anthony H. K. Park, Hossam Shoman, Minglei Ma, Sudip Shekhar,and Lukas Chrostowski, \u201cRing resonator based polarization diversityWDM receiver,\u201d Optics Express, vol. 27, no. 5, pp. 6147-6157, 2019.5The first two authors contributed equally to this work.1026. Mustafa Hammood, Ajay Mistry, Minglei Ma, Han Yun, Lukas Chros-towski, Nicolas AF Jaeger, \u201cCompact, silicon-on-insulator, series-cascaded,contradirectional-coupling-based filters with > 50 dB adjacent channelisolation,\u201d Optics Letters, vol. 44, no. 2, pp. 439-442, 2019.7. Saket Kaushal, Rui Cheng, Minglei Ma, Ajay Mistry, Maurizio Burla,Lukas Chrostowski, Jose\u00b4 Azan\u02dca, \u201cOptical signal processing based onsilicon photonics waveguide Bragg gratings,\u201d Frontiers of Optoelec-tronics, vol. 11, no. 2, pp. 163-188, 2019.8. Yun Wang, Luhua Xu, Han Yun, Minglei Ma, Amar Kumar, Es-lam El-Fiky, Rui Li, Nicola\u00b4s Abad\u00b4\u0131acalvo, Lukas Chrostowski, NicolasA. F. Jaeger, and David V. Plant. \u201dPolarization-independent mode-evolution-based coupler for the silicon-on-insulator platform,\u201d IEEEPhotonics Journal, vol. 10, no. 3, pp. 1-10, 2018.9. Minglei Ma, Zhitian Chen, Han Yun, Yun Wang, Xu Wang, Nico-las A. F. Jaeger, and Lukas Chrostowski, \u201cApodized Spiral BraggGrating Waveguides in Silicon-on-Insulator,\u201d IEEE Photonics Tech-nology Letters, vol. 30, no. 1, pp. 111-114, 2017.10. Yun Wang, Minglei Ma, Han Yun, Zeqin Lu, Xu Wang, Nicolas A. F. Jaeger,and Lukas Chrostowski, \u201cUltra-Compact Sub-Wavelength Grating Po-larization Splitter-Rotator for Silicon-on-Insulator Platform,\u201d IEEEPhotonics Journal, vol. 8, no. 6, pp. 1-9, 2017.11. Yun Wang, Zeqin Lu, Minglei Ma, Han Yun, Fan Zhang, Nico-las A. F. Jaeger, and Lukas Chrostowski, \u201cCompact Broadband Direc-tional Couplers Using by Sub-wavelength Gratings,\u201d IEEE PhotonicsJournal, vol. 8, no. 3, pp. 1-8, 2016.A.2 Conference Proceedings1. Minglei Ma, Hossam Shoman, Sudip Shekhar, Nicolas A. F. Jaeger,and Lukas Chrostowski, \u201cSilicon Photonic WDM polarization receiver103with Automated Feedback Control,\u201d Conference on Lasers and Electro-Optics (CLEO), accepted (Oral presentation), 2020.2. Mustafa Hammood, Ajay Mistry, Han Yun, Minglei Ma, Lukas Chros-towski, Nicolas A. F. Jaeger, \u201cFour-channel, Silicon Photonic,WavelengthMultiplexer-DemultiplexerWith High Channel Isolations,\u201d Optical FiberCommunications Conference and Exhibition (OFC), accepted, 2020.3. Mustafa Hammood, Ajay Mistry, Han Yun, Minglei Ma, Lukas Chros-towski, Nicolas A. F. Jaeger, \u201cCompact, folded, high-performance, sil-icon photonic, optical add-drop multiplexer,\u201d Photonic North, pp. 1-1,2019.4. Mustafa Hammood, Stephen Lin, Ajay Mistry, Minglei Ma, LukasChrostowski, Nicolas A. F. Jaeger, \u201cSOI Optical Add-Drop Multiplex-ers Using Apodized Spiral Contra-Directional Couplers,\u201d Conferenceon Lasers and Electro-Optics (CLEO), pp. SM3J. 7, 2019.5. Minglei Ma, Yun Wang, Anthony, H. K. Park, Han Yun, Nico-las A. F. Jaeger, and Lukas Chrostowski, \u201cBroadband polarizationsplitter-rotator using sub-wavelength grating assisted adiabatic waveg-uides,\u201d IEEE 15th International Conference on Group IV Photonics(GFP), pp. 1-2, 2018.6. Mustafa Hammood, Ajay Mistry, Minglei Ma, Lukas Chrostowski,Nicolas A. F. Jaeger, \u201cCompact Contra-Directional-Coupler-Based Fil-ters for CWDM Applications,\u201d IEEE 15th International Conference onGroup IV Photonics (GFP), pp. 1-2, 2018.7. Yun Wang, Jonas Flueckiger, Zeqin Lu, Han Yun, Minglei Ma,Fan Zhang, Valentina Donzella, Nicolas A. F. Jaeger, and Lukas Chros-towski. \u201cSub-wavelength Grating Components for the Silicon-on-insulatorPlatform (Invited Paper),\u201d The 7th International Conference on Meta-materials, Photonic Crystals and Plasmonics, 2016.8. Zeqin Lu, Minglei Ma, Han Yun, Yun Wang, Nicolas A. F. Jaeger,and Lukas Chrostowski. \u201cSilicon Photonic Polarization Beamsplitter104and Rotator for On-chip Polarization Control (Invited paper),\u201d IEEE13th International Conference on Group IV Photonics (GFP), pp. 70-71, 2016.9. Minglei Ma, Kyle Murray, Mengyuan Ye, Stephen Lin, Yun Wang,Zeqin Lu, Han Yun, Ricky Hu, Nicolas A. F. Jaeger, and Lukas Chros-towski, \u201cSilicon Photonic Polarization Receiver with Automated Sta-bilization for Arbitrary Input Polarizations,\u201d In Conference on Lasersand Electro-Optics (CLEO), pp. STu4G. 8, 2016.105Appendix BDerivation of the theoreticalmodel for a tunable MZIThis is the derivation of the MZI\u2019s model in our polarization receiver, basedon the TMM. The schematic of the MZI is shown in Fig. B.1:Figure B.1: A schematic of the tunable MZI in the PR.Here, E1 and E2 are the normalized input vectors, and \u2206\u03c60 is the phasedifference between E1 and E2. Therefore, the inputs can be explained byE1 = A1e\u2212j\u2206\u03c60 (B.1)E2 = A2 (B.2)A2 =\u221a1\u2212A21 (B.3)where A1 and A2 refer to the amplitudes for the input vector E1 and E2,respectively. Thus, based on Fig. B.1, the output vectors, E3 and E4, willbe described as [E3E4]= M \u00b7[E1E2](B.4)where M refers to the transfer matrix of the MZI. It can be further expressed106by the matrices of MZI\u2019s components, i.e., the 3dB couplers and waveguidearms, which are given by the following matrice:M1 =[e\u2212j(\u03b2L+\u2206\u03c61) 00 e\u2212j\u03b2L](B.5)M2 =[t \u2212kk t](B.6)M3 =[e\u2212j(\u03b2L+\u2206\u03c62) 00 e\u2212j\u03b2L](B.7)M4 = M2 (B.8)where \u03b2 = 2pi\/\u03bb is the propagation constant, L is the length of the MZI\u2019s arm(assumed a uniform length for all arms), \u2206\u03c61 and \u2206\u03c62 are the phase shiftsapplied to the MZI, k and t are the coupling and transmission coefficient ofthe adiabatic 3dB coupler (assumed to be\u221a1\/2). Hence, we can obtain thetransfer matrix M by multiplying all the matrices:M = M4 \u00b7M3 \u00b7M2 \u00b7M1 (B.9)Then, by replacing the M in Eq. B.4 with Eq. B.9, the output vectors isdetermined by [E3E4]= M4 \u00b7M3 \u00b7M2 \u00b7M1 \u00b7[E1E2](B.10)And using the power equation P = |E|2, we can derive the expression of theMZI\u2019s output powers, P3 and P4, which are given byP3 = 1\/2 + cos(\u2206\u03c62)\/2 +A1A2cos(\u2206\u03c60)sin(\u2206\u03c61)sin(\u2206\u03c62)+A1A2cos(\u2206\u03c61)sin(\u2206\u03c60)sin(\u2206\u03c62)\u2212A21cos(\u2206\u03c62) (B.11)P4 = 1\u2212 P3 (B.12)Finally, with Eq. B.11 and Eq. B.12, the optical power at output port andfeedback port of the polarization receiver, which are correlated to the phaseshifts (\u2206\u03c60,\u2206\u03c61,\u2206\u03c62), can be achieved.107Appendix CPhase shift - current relationderivationFor the phase shifters, a phase shift \u2206\u03d5 is normally realized by introducingan effective index change \u2206neff in the waveguide and is given by\u2206\u03d5 =2piL\u03bb0\u2206neff (C.1)where L is the length of the waveguide, \u03bb0 is the operating wavelength. Aswe use thermal phase shifters, \u2206neff can be correlated to the temperaturevariation \u2206T in the waveguide by\u2206neff =dneffdT\u2206T (C.2)wheredneffdTdepends on the waveguide material. Also, to determine theenergy \u2206Q consumed by the thermal phase shifters that corresponds tem-perature change \u2206T in the waveguide, we use the heat transfer formula,\u2206Q = mch\u2206T (C.3)where m is the mass and ch is the specific heat of the material. Overall,based on Eqs. C.2 and C.3, we can derive the injected energy per unit time,i.e., the electric power P , as follows:P = \u2206Q = mch\u2206T = mch\u2206neffdneffdT=mch\u03bbdneffdT2piL\u2206\u03d5 (C.4)108Therefore, a linear relation between the phase shift and the applied electricpower is obtained. Then, to determine \u2206\u03d5 as a function of injected current,we need to measure one of the output spectra of a 2\u00d72 MZI with a TE inputmode, and implement a curve-fit to the measured spectrum using followingequation:Pout = 0.5 \u00b7 (1 + cos(\u2206\u03d5)) (C.5)where Pout is the optical output power of the MZI. Since we know the relationin Eq. C.4, the phase shift \u2206\u03d5 can be replaced by\u2206\u03d5 = \u03b1P + \u03b2 = \u03b1(I2R) + \u03b2 (C.6)where R is the resistance of thermal phase shifter, I is the injected current,and \u03b1 and \u03b2 are determined by the curve-fit. Finally, the relation betweenthe phase shift \u2206\u03d5 and the injected current I is achieved.109Appendix DCode listing of theautomated controlalgorithmsD.1 Python code for greedy linear descentmethodimport numpy as npimport timedef greedy_linear_descent(guess, f, delta, minimum, I_max,iters):v0 = np.asarray(guess)v = np.zeros((iters+1, len(delta)))delta = np.asarray(delta)fun_values = np.zeros((iters+1, 3))v[0] = v0fun_values[0] = f(v[0])time_table = np.zeros(iters)ii = 1while((ii < iters) and(np.amax(np.absolute(v[ii]))<I_max)):f0 = f(v[ii-1])[0]if(np.abs(f0)<minimum):fun_values[ii] = f(v[ii-1])print(\u2019optimization complete\u2019)time_table[ii-1] = time.time()breakv[ii][0] = v[ii-1][0] + delta[0]v[ii][1] = v[ii-1][1]if (f(v[ii])[0]<f0):110f0 = f(v[ii])[0]v[ii][1] = v[ii-1][1] + delta[1]v[ii][0] = v[ii-1][0] + delta[0]if (f(v[ii])[0]<f0):fun_values[ii] = f(v[ii])elif (f(v[ii])[0]>=f0):v[ii][1] = v[ii-1][1] - 2*delta[1]v[ii][0] = v[ii-1][0] + delta[0]fun_values[ii] = f(v[ii])elif (f(v[ii])[0] >= f0):v[ii][0] = v[ii-1][0] - 2*delta[0]v[ii][1] = v[ii-1][1]f0 = f(v[ii])[0]v[ii][1] = v[ii-1][1] + delta[1]if (f(v[ii])[0] < f0):fun_values[ii] = f(v[ii])elif (f(v[ii])[0] >= f0):v[ii][1] = v[ii-1][1] - 2*delta[1]v[ii][0] = v[ii-1][0] - 2*delta[0]fun_values[ii] = f(v[ii])time_table[ii-1] = time.time()ii = ii + 1f(v[ii])v0 = v[ii]time_table[:] = [x-time_table[0] for x in time_table]return(v, ii, fun_values, time_table)D.2 Python code for basic gradient descentmethodimport numpy as npimport timedef gradient_descent(guess, f, delta, gamma, sigma, minimum,I_max, iters):v0 = np.asarray(guess)ii = 0grad = np.zeros((iters+1, len(delta)))v = np.zeros((iters+1, len(delta)))delta = np.asarray(delta)111fun_values = np.zeros((iters+1,3))time_table = np.zeros(iters)v[0] = v0while(ii<iters):f0 = f(v[ii])[0]for jj in xrange(0,len(delta)):temp_array = np.zeros(len(delta))temp_array[jj] = 1.0Output1 = f(v[ii]+temp_array*delta)[0]Output2 = f0grad[ii][jj] = Output1-Output2if(np.linalg.norm(grad[ii])<sigma):print(grad[ii], \u2019gradient is too small!\u2019)fun_values[ii] = f(v[ii])time_table[ii] = time.time()breakelif (np.abs(f0)<minimum):fun_values[ii] = f(v[ii])print(\u2019optimization complete\u2019)time_table[ii] = time.time()break##decrease delta if the gradient is small enoughelif (np.linalg.norm(grad[ii]) < sigma*10):delta = 0.8*deltav[ii+1] = v[ii]-gamma*grad[ii]##pertect your phase shiftersif (np.amax(np.absolute(v[ii+1])) > I_max):print(\u2019current is too high!\u2019)time_table[ii] = time.time()break##Ensure the feedback did not increaseif(f(v[ii+1])[0]<f(v[ii])[0]):print(gamma*grad[ii])print(np.linalg.norm(grad[ii]))else:v[ii+1] = v[ii] + gamma*grad[ii]print(gamma*grad[ii])print(\u2019-\u2019, np.linalg.norm(grad[ii]))time_table[ii] = time.time()fun_values[ii] = f(v[ii])ii = ii + 1f(v[ii])112v0 = v[ii]time_table[:] = [x-time_table[0] for x in time_table]return(v, ii, fun_values, time_table)D.3 Python code for two-point step size gradientdescent methodimport numpy as npimport timedef gradient_descent_BB(guess,f,delta,gamma_init, sigma,minimum, I_max, iters):v0 = np.asarray(guess)grad = np.zeros((iters+1, len(delta)))v = np.zeros((iters+1, len(delta)))gamma = gamma_initdelta = np.asarray(delta)fun_values = np.zeros((iters+1,3))time_table = np.zeros(iters)##intitialize v(0), v(1), and grad(0)v[0] = v0for jj in xrange(0,len(delta)):temp_array = np.zeros(len(delta))temp_array[jj] = 1.0Output1 = f(v[0]+temp_array*delta)[0]Output2 = f(v[0])[0]grad[0][jj] = Output1-Output2fun_values[0] = f(v[0])v[1] = v[0] - gamma*grad[0]time_table[0] = time.time()##Main optimization algorithmii = 1;while(ii<iters):f0 = f(v[ii])[0]for jj in xrange(0,len(delta)):temp_array = np.zeros(len(delta))temp_array[jj] = 1.0Output1 = f(v[ii]+temp_array*delta)[0]Output2 = f0grad[ii][jj] = Output1-Output2113if(np.linalg.norm(grad[ii]) < sigma):fun_values[ii] = f(v[ii])print(grad[ii], \u2019gradient is too small!\u2019)time_table[ii] = time.time()breakelif (np.abs(Output2)<minimum):fun_values[ii] = f(v[ii])print(\u2019optimization complete\u2019)time_table[ii] = time.time()breakelif (np.linalg.norm(grad[ii]) < sigma*10):delta = 0.8*delta##Calculate the gamma in two-step size for eachiterationgamma = np.dot((v[ii] - v[ii-1]), (grad[ii] -grad[ii-1]))\/(np.linalg.norm((grad[ii] -grad[ii-1])))**2v[ii+1] = v[ii] - gamma*grad[ii]##pertect your phase shiftersif (np.amax(np.absolute(v[ii+1])) > I_max):print(\u2019current is too high!\u2019)v[ii+1] = v[0]time_table[ii] = time.time()##Ensure the feedback did not increaseif (f(v[ii+1])[0] < f(v[ii])[0]):print(np.linalg.norm(grad[ii]))else :v[ii+1] = v[ii] + gamma*grad[ii]print(\u2019-\u2019, gamma*grad[ii])fun_values[ii] = f(v[ii])time_table[ii] = time.time()ii = ii+1f(v[ii])v0 = v[ii]return(v, ii, fun_values, time_table)D.4 Python code for two-stage optimizationmethod114import logging as logimport mathimport randomimport collectionsimport numpy as npfrom scipy.optimize import least_squares, newton_krylov,broyden1, approx_fprime, check_grad, newton, fsolveclass TwoStageFcns(object):def __init__(self, model, num_samples, num_resamples):self.model = modelself.num_samples = num_samplesself.num_resamples = num_resamplesself.model_params = Noneself.sample_queue = None## Stage 1: ISTM Functiondef ISTM(self, seed=3):log.info(\"Initializing...\")sample_inputs = list()random.seed(seed)for _ in range(self.num_samples):sample_inputs.append([random.uniform(0, math.pi),random.uniform(0, math.pi)])log.debug(\"sweeping model inputs: \" +str(sample_inputs))observes_ = [self.model.observe(input_) for input_ insample_inputs]log.debug(\"sweeping model outputs: \" + str(observes_))self.sample_queue = collections.deque([SampleRecord(sample_inputs[i], observes_[i]) for iin range(self.num_samples)])def loss_func_(model_params):loss = []for i in range(self.num_samples):pred = self.model.guess(sample_inputs[i],model_params)loss.append(self.model.residual(pred,observes_[i]))return lossinit_guess = [115[0.5, np.pi],[0.5, np.pi * 1.8],[0.1, np.pi * 0.1],[0.1, np.pi * 1.8],[0.9, np.pi * 1.8],[0.9, np.pi * 0.1]] ## Sampling points can be differentlog.info(\"searching initial model parameters...\")for ini in init_guess:results = least_squares(loss_func_, ini, verbose=1, method=\u2019trf\u2019,bounds=self.model.params_bounds,ftol=3e-16, xtol=3e-16, gtol=3e-16)if results.success:self.model_params = results.xlog.info(\"initial model parameter: \" +str(self.model_params))min_inputs =self.model.argmin(params=self.model_params)min_ = self.model.observe(min_inputs)self.sample_queue.popleft()self.sample_queue.append(SampleRecord(min_inputs,min_))returnlog.info(\"model parameters not found!\")## Stage 2: Tracking functiondef Tracking(self, epsilon=None, threshold=1E-6,num_observes=1, verbose=0):if self.sample_queue is None or self.model_params isNone:raise ValueError(\"calibrator not initialized!\")if not isinstance(num_observes, int) or num_observes <1:raise ValueError(\"number of observes must be ainteger greater than or equal to 1!\")## Estimate outputnew_output = 0for _ in range(num_observes):new_output +=self.model.observe(self.sample_queue[-1].inputs)new_output \/= num_observesif np.abs(new_output -self.model.guess(self.sample_queue[-1].inputs,116self.model_params)) < threshold:returnepsilon = np.sqrt(np.finfo(float).eps) if epsilon isNone else epsilonlog.debug(\"epsilon: \" + str(epsilon))## Estimate f_prime times epsilonnumerical_output_changes = Nonefor _ in range(num_observes):if numerical_output_changes is None:numerical_output_changes =self._approx_fprime_epsilon(self.sample_queue[-1].inputs,self.model.observe, [epsilon, epsilon])else:numerical_output_changes +=self._approx_fprime_epsilon(self.sample_queue[-1].inputs,self.model.observe, [epsilon, epsilon])numerical_output_changes \/= num_observesdef residual(new_params):log.debug(\"tried parameters: \" + str(new_params))jac_inputs =np.squeeze(self.model.guess_jac_inputs(self.sample_queue[-1].inputs, new_params))jac_residual = numerical_output_changes -jac_inputs * epsilonreturn jac_residualdef residual_jac(new_params):hes_params = self.model.guess_hes_inputs_params(self.sample_queue[-1].inputs, new_params)return - hes_params * epsilonif verbose > 0:print(\u2019tracking...\u2019)results = fsolve(residual,np.asarray(self.model_params),fprime=residual_jac, xtol=3e-16)next_model_params = results.tolist()log.debug(results)print(results)if verbose > 0:117print(\u2019searching minimum...\u2019)min_inputs =self.model.argmin(params=next_model_params,initial_inputs=self.sample_queue[-1].inputs,verbose=verbose)min_ = self.model.observe(min_inputs)log.info(\"minimum of proposed model: \" + str(min_))if min_ < new_output:self.model_params = next_model_paramsself.sample_queue.popleft()self.sample_queue.append(SampleRecord(min_inputs,min_))@staticmethoddef _approx_fprime_epsilon(xk, f, epsilon, args=(),f0=None):if f0 is None:f0 = f(*((xk,) + args))grad = np.zeros((len(xk),), float)ei = np.zeros((len(xk),), float)for k in range(len(xk)):ei[k] = 1.0d = epsilon * eigrad[k] = f(*((xk + d,) + args)) - f0ei[k] = 0.0return gradclass SampleRecord(object):def __init__(self, inputs, output):self.inputs = inputsself.output = outputself.decay_ = 1.0def decay(self, decay):self.decay_ *= decay118","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/hasType":[{"value":"Thesis\/Dissertation","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#dateIssued":[{"value":"2020-11","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt":[{"value":"10.14288\/1.0392500","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/language":[{"value":"eng","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline":[{"value":"Electrical Engineering","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/provider":[{"value":"Vancouver : University of British Columbia Library","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/publisher":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/rights":[{"value":"Attribution-NonCommercial-NoDerivatives 4.0 International","type":"literal","lang":"*"}],"https:\/\/open.library.ubc.ca\/terms#rightsURI":[{"value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/","type":"literal","lang":"*"}],"https:\/\/open.library.ubc.ca\/terms#scholarLevel":[{"value":"Graduate","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/title":[{"value":"Wavelength and polarization control for silicon photonic receiver applications","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/type":[{"value":"Text","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#identifierURI":[{"value":"http:\/\/hdl.handle.net\/2429\/75248","type":"literal","lang":"en"}]}}