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Feasibility of optical gyroscopic sensors in silicon-on-insulator technology Guillén-Torres, Miguel Ángel 2015

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Feasibility of Optical GyroscopicSensors in Silicon-On-InsulatorTechnologybyMiguel A´ngel Guille´n-TorresB.Sc., National Autonomous University of Mexico, 2004M.Phil., The University of Cambridge, 2007A 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)August 2015c© Miguel A´ngel Guille´n-Torres 2015AbstractIn the last decade, silicon photonics has become a strategic technology forthe development of telecommunications and sensors. Due to its compatibil-ity with well-developed complementary metal oxide semiconductor (CMOS)fabrication processes, silicon on insulator (SOI) wafers can be processedto create thousands of devices per die in a fast and inexpensive way. Be-ing solid state devices with no movable parts, optical gyroscopes have longerlife expectancies and shock resistance compared to micro-electro-mechanicalgyroscopes. Thus, the implementation of SOI-based gyroscopes is desirablefor large-scale, low-cost production.This thesis presents a study of the feasibility of implementing opticalgyroscopes in SOI technology. A comprehensive theoretical study has beencarried out to develop a device-level optimization and robustness analysis,showing that the most crucial resonator parameter is the propagation loss,followed by length and coupling. For a given propagation loss, there isan optimal resonator size, beyond which the angular speed resolution isseverely degraded. On the system level, the impact of signal-to-noise ratioand insertion loss on the resolution are described.Given that the propagation loss is the most important parameter, strate-gies were proposed to reduce it as much as possible while still using CMOS-compatible processes. The quality factor, Q, was chosen as the figure ofmerit to be maximized during the design iterations. As a result, the largestQ factors reported to date on SOI, using standard CMOS-compatible pro-cesses, were achieved. These Q factors are comparable to, or exceed, thoseof optical resonators intended for gyroscopic applications that are fabricatedin materials such as indium phosphide (InP). Innovative approaches to com-pensate for fabrication variations are proposed, such as thermally-tuneableiiAbstractcoupling and reference rings for differential measurements.Complex mechano-opto-electrical measurement setups were designed andimplemented to characterize SOI gyroscopes, both at rest and under rota-tion. As a result, the Microsystem Integration Platform for Silicon-Photonics(Si-P MIP) was created. This characterization platform is now being com-mercialized by CMC Microsystems for academic and industrial applications.The main practical and theoretical challenges regarding the implemen-tation of optical ring gyroscopes on SOI have been identified. Schemes toaddress them and suggestions for future work are proposed.iiiPrefaceThis thesis work is partially based on the publications listed below, some ofwhich resulted from collaborations with other researchers. Note that onlythe publications directly linked to this thesis work are shown below. Itshould also be noted that some numerical simulations and experiments havebeen re-made since their first publication, to yield more insightful resultsand comparisons, and may not necessarily be identical to those previouslypublished. An additional list of publications, not directly related to thiswork, is given in Appendix A.Journal Publications1. M. A. Guillen-Torres, E. Cretu, N. A. F. Jaeger, and L. Chrostowski,“Ring Resonator Optical Gyroscopes - Parameter Optimization andRobustness Analysis,” J. Lightw. Technol., vol. 30, no. 12, pp. 1802-1817, 2012I conceived the idea of a theoretical study about the robustness of ringresonator gyroscopes to variations in their design parameters. I carriedout the analytical and numerical modelling, and wrote the manuscript.L. Chrostowski provided deep insight into how ring resonators workand how to model them. N. A. F. Jaeger provided invaluable insightsthroughout the course of numerous discussions to further understandthe principles of the Sagnac Effect, as well as the effect of waveguidecoupling on resonance depth and quality factor. E. Cretu suggestedvaluable analytical and numerical approaches, such as parameter nor-malization, in order to find generalized optimal parameters. L. Chros-towski, N. A. F. Jaeger, and E. Cretu supervised the work, providedivPrefacefeedback and advice regarding visualization and interpretation of nu-merical simulation results, and contributed to editing the manuscript.Chapter 2 is mostly based on this publication.Conference Proceedings1. M. A. Guillen-Torres, M. Almarghalani, E. H. Sarraf, M. Caverley,N. A. F. Jaeger, E. Cretu, and L. Chrostowski, “Silicon photonicscharacterization platform for gyroscopic devices,” in SPIE PhotonicsNorth, Proc. SPIE, vol. 9288, Montre´al, Canada, pp. 1-8, May 2014I conceived the idea of a rotational setup for gyroscope device charac-terization, built up the optomechanical setup, coordinated the softwaredevelopment, performed the measurements, post-processed the data,and wrote the manuscript. N. A. F. Jaeger suggested using specialcoupler designs for splitting and combining of optical signals, and pro-vided the optical instruments and opto-mechanics required for creatingthe apparatus. L Chrostowski suggested the use of straight multimodewaveguides for resonator roundtrip loss reduction and quality factorimprovement. I implemented these ideas, and created scripts for semi-automatic device layout generation with optimum coupling values, asper the findings of our previous journal publication. L. Chrostowskiand N. A. F. Jaeger provided advice regarding various optical devicetest procedures. E. Cretu provided special equipment and guidance forthe creation of the platform computerized control, and contributed tothe conception of tests for apparatus characterization. M. Almargha-lani created hardware and software interfaces for motor control, aswell as codes for rotational patterns. E. H. Sarraf created interfacesfor data acquisition and instrument communication. M. Caverley con-tributed to post-processing and data analysis. L. Chrostowski, N. A.F. Jaeger, and E. Cretu supervised the work. All authors commentedand assisted in editing the manuscript. Part of Chapter 3 is based onthis publication.2. M. A. Guillen-Torres, M. Caverley, E. Cretu, N. A. F. Jaeger, andvPrefaceL. Chrostowski, “Large-area, high-Q SOI ring resonators”, in IEEEPhotonics Conference, San Diego, California, USA, pp. 1-2, Oct. 2014L. Chrostowski, N. A. F. Jaeger, and I conceived the idea. L Chros-towski suggested the use of Be´zier bends for the resonator corners forfurther improvement of the resonator quality factor values, as well asthe creation of robust test structures for multimode waveguide propa-gation loss characterization. N. A. F. Jaeger provided valuable insightinto the interplay of key parameters that influence the value of theQ factor. I implemented the idea and conducted the device layoutdesign and performed the measurements. M. Caverley contributed todevice designs, assisted with measurements. M. Caverley and I wrotethe manuscript. L. Chrostowski, N. A. F. Jaeger, and E. Cretu su-pervised the work and assisted in editing the manuscript. All authorscommented on the manuscript. Part of Chapter 3 is also based on thispublication.3. M. A. Guillen-Torres, L. Chrostowski, E. Cretu, and N. A. F. Jaeger,“Ring Resonator Gyroscope: System Level Analysis and ParameterOptimization,” Canadian Semiconductor Science and Tech. Conf., MPA,p. 1, Vancouver, Canada, 2011I conceived the idea of a theoretical study about the obtention of op-timal ring resonator gyroscope design parameters. I carried out theanalytical and numerical modelling, and wrote the manuscript. L.Chrostowski provided insight into the main figures of merit of ringresonators. N. A. F. Jaeger provided insights into the Sagnac effectand concepts related to waveguide coupling. E. Cretu helped developgeneralized analytical equations for modelling double-bus-waveguidering resonators, and suggested numerical approaches and tools for pa-rameter optimization. L. Chrostowski, N. A. F. Jaeger, and E. Cretusupervised the work, provided feedback and advice regarding the inter-pretation of results, and contributed to editing the manuscript. Chap-ter 2 is partially based on this publication.viPrefaceThe following work, currently unpublished, was the result of collaborationswith various members of the UBC Micro- and Nano-technology group:1. L. Chrostowski conceived the idea of resonators with thermally-tuneablecouplers. M. Caverley and I created the device layout designs. K.Murray and I created the theoretical device models. M. Caverley andI performed the device measurements. Part of Chapter 3 is based onthis work.2. Theoretical models were created in collaboration with K. Murray todescribe the behaviour of various device designs, as well as to analyzethe effect of backscattering in them. Part of Chapter 3 is also basedon this work.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.1 Motivation and Potential Applications . . . . . . . . 71.2 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . 72 Resonator Simulation and Parameter Optimization . . . . 92.1 Waveguide Coupling . . . . . . . . . . . . . . . . . . . . . . . 102.2 Sagnac Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.1 Sagnac Effect in Vacuum . . . . . . . . . . . . . . . . 142.2.2 Sagnac Effect in a Dielectric Medium . . . . . . . . . 152.3 Analytical and Numerical Modelling . . . . . . . . . . . . . . 182.3.1 Through- and Drop-Port Power Transmission . . . . 21viiiTable of Contents2.3.2 Figures of Merit of the Spectral Response . . . . . . . 222.3.3 Loss Impact on Performance . . . . . . . . . . . . . . 232.3.4 Spectral Response for the Through Port . . . . . . . 262.3.5 Spectral Response for the Drop Port . . . . . . . . . 292.3.6 Noise Analysis . . . . . . . . . . . . . . . . . . . . . . 312.3.7 Resonator Gyroscope Resolution Estimations . . . . . 372.4 Resonator Parameter Optimization . . . . . . . . . . . . . . 382.4.1 Local and Global Optimization . . . . . . . . . . . . . 382.4.2 Chip-Sized vs. Globally-Optimized Gyroscopes . . . . 452.4.3 Target Applications . . . . . . . . . . . . . . . . . . . 452.4.4 Design Robustness . . . . . . . . . . . . . . . . . . . . 472.4.5 Predictions with Experimental SNRs . . . . . . . . . 522.5 Phase Modulation Requirements . . . . . . . . . . . . . . . . 543 Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.1 First Design Cycle . . . . . . . . . . . . . . . . . . . . . . . . 603.1.1 Layout Design . . . . . . . . . . . . . . . . . . . . . . 613.1.2 Setup Design . . . . . . . . . . . . . . . . . . . . . . 653.1.3 Measurements . . . . . . . . . . . . . . . . . . . . . . 673.1.4 Fibre Attachment . . . . . . . . . . . . . . . . . . . . 693.1.5 Iteration Challenges and Conclusions . . . . . . . . . 703.2 Second Design Cycle . . . . . . . . . . . . . . . . . . . . . . . 713.2.1 Layout Design . . . . . . . . . . . . . . . . . . . . . . 723.2.2 Setup Improvements . . . . . . . . . . . . . . . . . . . 743.2.3 Measurements . . . . . . . . . . . . . . . . . . . . . . 753.2.4 Fibre Attachment . . . . . . . . . . . . . . . . . . . . 793.2.5 Iteration Challenges and Conclusions . . . . . . . . . 803.3 Third Design Cycle . . . . . . . . . . . . . . . . . . . . . . . 823.3.1 Layout Design . . . . . . . . . . . . . . . . . . . . . . 833.3.2 Setup Design . . . . . . . . . . . . . . . . . . . . . . . 923.3.3 Measurements . . . . . . . . . . . . . . . . . . . . . . 963.3.4 Fibre Attachment . . . . . . . . . . . . . . . . . . . . 1013.3.5 Iteration Challenges and Conclusions . . . . . . . . . 107ixTable of Contents3.4 Fourth Design Cycle . . . . . . . . . . . . . . . . . . . . . . . 1093.4.1 Layout Design . . . . . . . . . . . . . . . . . . . . . . 1103.4.2 Setup Design . . . . . . . . . . . . . . . . . . . . . . . 1223.4.3 Measurements . . . . . . . . . . . . . . . . . . . . . . 1273.4.4 Iteration Challenges and Conclusions . . . . . . . . . 1473.5 Fifth Design Cycle . . . . . . . . . . . . . . . . . . . . . . . . 1483.5.1 Layout Designs . . . . . . . . . . . . . . . . . . . . . 1493.5.2 Waveguide Parameters . . . . . . . . . . . . . . . . . 1503.5.3 Measurements . . . . . . . . . . . . . . . . . . . . . . 1513.5.4 Effects of Backscattering . . . . . . . . . . . . . . . . 1613.5.5 Iteration Challenges and Conclusions . . . . . . . . . 1754 Summary, Conclusions, and Suggestions for Future Work 1784.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1784.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1824.3 Suggestions for Future Work . . . . . . . . . . . . . . . . . . 1854.3.1 Sinusoidal Phase Modulation . . . . . . . . . . . . . . 185Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189AppendixA Other Publications . . . . . . . . . . . . . . . . . . . . . . . . . 206A.1 Journal Publications . . . . . . . . . . . . . . . . . . . . . . . 206A.2 Conference Proceedings . . . . . . . . . . . . . . . . . . . . . 206A.3 Conference Presentations . . . . . . . . . . . . . . . . . . . . 206B Frequency-Stepped Sinusoidal Patterns . . . . . . . . . . . . 208C Time-Domain Measurements in Selected Components . . 211D Transfer Functions of Resonators with Thermally-TuneableCouplers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213xList of Tables1.1 Performance requirements of different gyroscope grades [143]. 52.1 Most frequent variables . . . . . . . . . . . . . . . . . . . . . 112.2 Parameters for different lengths and coupling conditions, throughport, all-pass configuration, for αdB = 0.06 dB/cm . . . . . . 292.3 List of Simulation Parameters . . . . . . . . . . . . . . . . . . 342.4 Global optimum parameters and resolutions for different portconfigurations and losses . . . . . . . . . . . . . . . . . . . . . 432.5 Resolutions for LOC resonator gyroscopes with L = Lmax chip 462.6 Optimum resolutions and lengths for GOC resonator gyroscopes 462.7 Resolution requirements for different classes of gyroscopes . . 472.8 Comparison with commercially available gyroscopes . . . . . 472.9 3-deciBel cut-off normalized lengths and length bandwidthfor LOC resonators . . . . . . . . . . . . . . . . . . . . . . . . 493.1 Parameters for WDCs made with 500-nm wide strip SMWGs,with Lc = 10 µm, at λ0 = 1550 nm. . . . . . . . . . . . . . . . 643.2 Numerical results for mode power coupling between 500 nm-wide straight and bent strip waveguides (R=20 µm), at λ0 =1550 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.3 Length and gap ranges for various resonator groups shown inFigure 3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.4 Cross-over length and selected parameters at λ0 = 1550 nm,for air-clad strip WDCs with various gaps. . . . . . . . . . . . 733.5 Cross-over length and selected parameters at λ0 = 1550 nm,for glass-clad strip WDCs with various gaps. . . . . . . . . . 74xiList of Tables3.6 As-fabricated widths for 500 nm WGs, standard (air-clad)and custom (glass-clad) wafers. . . . . . . . . . . . . . . . . . 753.7 Through-port spectral curve-fit parameters for a 5.91 mm-long resonator (Ring 2, Std. Chip R0C−3), with a correlationvalue between measured and fit data r2 = 0.856 . . . . . . . . 773.8 Effective and group indices at λ0 = 1550 nm for strip waveg-uides of various geometries. In all cases the height is H = 220nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.9 Coupling between straight and bent strip SMWG modes atλ0 = 1550 nm. W=500 nm, R3 = 6 µm, and R5 = 200µm. . . 903.10 A-priori propagation loss estimates for air-clad strip waveguides 913.11 Theoretical field coupling for air-clad strip SMWG directionalcouplers, g = 500 nm (L⊗ = 1058.7 µm) . . . . . . . . . . . . 913.12 Parameter estimations for different glass-clad waveguides . . 1203.13 Theoretical values for effective and group indices at λ0 = 1550nm, for rib waveguides of various geometries. In all casesthe strip height is Hstrip = 220 nm and the slab height isHslab = 90 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . 1213.14 Theoretical values of effective and group indices at λ0 =1550 nm, for strip waveguides of various geometries. In allcases the strip height is Hstrip = 220 nm. . . . . . . . . . . . . 1503.15 Parameter estimations for different glass-clad waveguides . . 151xiiList of Figures1.1 Gyro technology requirement by application, based on [11,35, 110]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Resolution vs. lengthscale for selected optical and MEMS-based gyroscopes, either reported in the literature, [20, 108],or commercially available, [1, 2, 37], and theoretical resolu-tion estimations (hollow markers) for ring resonator gyro-scopes fabricated in different materials and port configura-tions. For silicon nitride (SiN), αdB1 = 0.06 dB/cm [91],and αdB2 = 0.12 dB/cm [53]. For silicon on insulator (SOI),αdB3 = 3 dB/cm [13]. For SOI resonators with differentwaveguide widths for different segments, the projected prop-agation loss estimations are between αdB4 = 1 dB/cm andαdB5 = 1.46 dB/cm. LoptT1 = 2.77 m; LoptT2 = 1.39 m;LoptT3 = 55.5 mm; and the perimeter of a chip is Lchip = 114 mm.The insertion losses (ILs) for the All-Pass (AP) and the Drop-Port rings are assumed to be ILT = 7 dB and ILD = 3 dB,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Schematic diagram of the All-Pass (top) and Drop-Port (bot-tom) configurations for a racetrack resonator gyroscope sys-tem. Light is split and injected in opposite directions intothe racetrack resonators. The output light in each directionis then directed to a photodetector. Circulators are requiredfor proper interrogation in the all-pass configuration, but canbe replaced by elements such as Y-branches for easier inte-gration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12xiiiList of Figures2.2 Sagnac effect in a clockwise (CW)-rotating resonator. . . . . 152.3 Normalized spectra of the counter-propagating light beamsin an all-pass (single-bus) racetrack resonator. Solid curve:resonator at rest. Dashed curves: spectra for the co-rotating(red curve, red-shifted) and the counter-rotating (cyan curve,blue-shifted) beams. For this example, αdB = 0.01 dB/cm,L = 1 m, κ = 1/√2. A Sagnac phase shift δφ = 0.1pi rad hasbeen used to produce a noticeable phase shift. . . . . . . . . . 202.4 Magnitude of field coupling coefficient of waveguide a, (κa),as a function of coupling coefficient κb, for critically coupled,50-mm long ring resonators, for propagation loss levels of dif-ferent technologies: αdB = 0.06 dB/cm (SiN , [91]), αdB =0.12 dB/cm(SiN , [53]), αdB = 3.0 dB/cm(SOI, [13]), andαdB = 1.17 dB/cm (estimate for SOI material with waveguidewidth variation, proposed in the present work as a low-lossalternative for SOI technology). . . . . . . . . . . . . . . . . . 242.5 Finesse as a function of coupling coefficient κb, for criticallycoupled, 50-mm long ring resonators, for propagation loss lev-els of different technologies: αdB = 0.06 dB/cm (Si3N4, [91]),αdB = 0.12 dB/cm(Si3N4, [53]), αdB = 3.0 dB/cm(SOI, [13]),and αdB = 1.17 dB/cm (proposed SOI design with waveguidewidth variation). . . . . . . . . . . . . . . . . . . . . . . . . . 252.6 Through port response, |S21|2 (solid), and first derivative,∂|S21|2∂φ (dashed), as a function of φ/pi, for αdB = 0.06 dB/cmand L = Lopt21CC = 1.63 m. In all cases, κb → 0 (all-passresonators). In spite of lacking a zero output at resonance, theOC case shows a larger maximum slope, at a smaller detuningin comparison to the CC case (see Table 2.2). . . . . . . . . . 27xivList of Figures2.7 Slope of the optimized all-pass frequency response, ∂|S21|2∂φ , asa function of φ/pi, for αdB = 0.06 dB/cm (i.e., α = 0.69 m−1),L = 1 mm (solid) and L = Lopt21 = 1.63 m (dashed). Blackcurves: OC; Light curves: CC. The small resonators yieldlarger slopes at smaller detunings. However, as shown in Ta-ble 2.2, they do not yield the best resolutions. . . . . . . . . . 282.8 Drop-port frequency response, |S41|2 (solid), and first deriva-tive, ∂|S41|2∂φ , (dashed), as a function of φ/pi, for αdB = 0.06 dB/cmand L = Lopt41CC = 0.93 m, for OC and CC cases. The pa-rameters are: κbCC = 0.697⇒ κaCC = 0.926, κaOC = κbOC =0.779. From the dotted curves, it is evident that the OCcase has a larger maximum spectral slope, which occurs at asmaller normalized detuning. . . . . . . . . . . . . . . . . . . 312.9 Signal-to-noise ratio (SNR) of various noise components (ther-mal noise, laser noise, and shot noise) and total SNR as func-tions of input power, for a photodetector with the parametersshown in Table 2.3 (see legend for proper identification). . . . 332.10 Normalized spectrum and its first derivative as a functionof normalized detuning, all-pass configuration, for αdB =0.06 dB/cm, Lopt21OC = 2.78 m, and κa = 0.805, for threedifferent linewidth values. Notice how the curves for ∆ν = 0Hz and 100 kHz are practically identical. . . . . . . . . . . . . 362.11 Normalized resolution as a function of normalized length forall-pass (solid) and drop (dashed) configurations, using theparameters shown in Table 2.3, with the laser linewidth, ∆ν,as a parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . 372.12 Optimum angular rate resolution as a function of the insertionloss, ILdB, for optimized all-pass (solid) and drop (dashed)configurations, for two different values of propagation losses. . 39xvList of Figures2.13 Angular rate resolution as a function of resonator length forthe all-pass (solid) and drop port (dashed) configurations ofracetrack resonator gyroscopes, for three different values ofpropagation losses. In all cases, the parameters φ and κ arefixed at their global optimum values. . . . . . . . . . . . . . 402.14 (a, b) Resolution |δΩ|, (c, d) optimized couplings κa and κb,and (e, f) optimized normalized detuning φn = φ/pi as func-tions of resonator length L, for various port and optimizationconditions. LOC: Locally-optimized coupling (computed ateach value of L). GOC: Globally-optimized coupling. In allcases, αdB = 0.06 dB/cm. . . . . . . . . . . . . . . . . . . . . 412.15 Resolution for LOC resonators as a function of normalizedlength, Ln, for all-pass (solid) and drop port (dashed) rings,for αdB1 = 0.06 dB/cm, αdB2 = 0.12 dB/cm and αdB3 =3 dB/cm. Lopt is different for each value of α and port con-figuration, as shown in Fig. 2.16. . . . . . . . . . . . . . . . . 442.16 Optimum resonator length Lopt as a function of average waveg-uide propagation loss αdB for the through (solid) and the drop(dashed) ports of LOC ring resonators. The value of Lopt isIL-independent. . . . . . . . . . . . . . . . . . . . . . . . . . . 442.17 Optimum resolution as a function of average waveguide prop-agation loss αdB [dB/cm] for all-pass (solid) and drop (dashed)LOC ring resonators. . . . . . . . . . . . . . . . . . . . . . . . 452.18 Normalized resolution |δΩ|norm versus normalized length Ln = L/Loptfor all-pass (solid) and drop-port (dashed) LOC resonators.Due to normalization, all plots coincide for all values of α andare IL-independent. . . . . . . . . . . . . . . . . . . . . . . . . 482.19 Global optimum values (stars), locally-optimized (LOC, solid),3-dB (dashed), and 6-dB (dash-dotted) contour plots for thecoupling coefficients of all-pass (κa) and drop-port (κa =κb) resonators, as a function of the normalized length Ln =L/Lopt. Due to normalization, all plots coincide for all valuesof α and are IL-independent. . . . . . . . . . . . . . . . . . . 49xviList of Figures2.20 Global optimum values (stars), locally-optimized (LOC, solid),3-dB (dashed), and 6-dB (dash-dotted) contour plots for φ/pi,as a function of the normalized length Ln = L/Lopt, for all-pass and drop-port resonators. Due to normalization, all plotscoincide for all α values, and are IL-independent. . . . . . . . 502.21 Resolution vs. αdB for an all-pass (solid) and a drop-port(dashed) resonator gyroscope optimally designed for αdB = 1dB/cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.22 Optimized detuning as a function of αdB for an all-pass (solid)and a drop (dashed) resonator gyroscope optimally designedfor αdB = 1 dB/cm. . . . . . . . . . . . . . . . . . . . . . . . 512.23 Angular speed resolution as a function of SNR for two all-pass, large-area resonators, both with 114 mm in length, andMMWG propagation losses of 0.085 dB/cm (solid curve) and0.026 dB/cm (dashed curve). . . . . . . . . . . . . . . . . . . 532.24 Frequency tracking using serrodyne phase modulation. . . . . 582.25 Resonance frequency tracking using modulation techniques. . 593.1 Wafer dicing schematic. Chips are identified according totheir position in the row and column pattern. . . . . . . . . . 613.2 (a) First device layout panoramic schematic and (b) Zoom-in to top left nested rings. Text tags are only shown forillustration purposes. . . . . . . . . . . . . . . . . . . . . . . 623.3 (a) TE mode profile for a 220-nm high, 500-nm wide, air-cladstrip SMWG. (b) Effective index (green) and group index(blue) curve-fits. . . . . . . . . . . . . . . . . . . . . . . . . . 633.4 (a) Effective indices for the even (solid) and odd (dashed)modes for WDCs made with 500-nm wide, strip SMWGs.(b) Corresponding WDC cross-over lengths as a function ofwavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64xviiList of Figures3.5 First characterization setup. (a) Block diagram. (b) Op-tomechanics assembly. (1) PM input fibre. (2) MM outputfibre. (3) Pedestal on XYZ stage. (4a) Fiber XYZ Stages.(4b) Fiber chuck. (5) TEC. (6) Microscope. Image: G.Sterling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.6 (a) Drop-port spectrum for a 3.3 mm-long resonator, showingmaxima (red stars) and minima (green stars). (b) Q factorfor each resonance (blue) and average Q (red). (c) Resonancecurve-fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.7 Comparison of spectra for a 16−mm long SMWG serpentineduring a fibre attachment experiment. . . . . . . . . . . . . . 703.8 Second iteration mask designs (not to scale). a Standard pro-cess (air-clad) designs. b Custom process (glass-clad) designs. 733.9 (a) Effective index (green) and group index (blue) curve-fits,and (b) Cross-over length as a function of wavelength, forglass-clad strip WDCs with various gaps. . . . . . . . . . . . 743.10 Through-port spectrum and curve-fits for a symmetrically-coupled, air-clad, 5.91 mm-long resonator (Ring 2, Std. ChipR0C−3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.11 Spectra and Q for an air-clad, 5.91 mm-long resonator (ChipR0C−3, Dev. 2) with nominal field coupling values κa = 0.656(through port), and κb = 0.292 (drop port). (a) Through-port transmission. (b) Drop-port transmission. (c)Q vs. wave-length, and average. . . . . . . . . . . . . . . . . . . . . . . . 783.12 Fibre holders and support designs for 3D printing. Dimen-sions in mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.13 (a) Fibre holder glueing platform. (b) Fibre attachment toholder. (c) Polishing station and jigs. (d) Finished holders. 81xviiiList of Figures3.14 (a) Splitting coupler schematic. I/O GC- Main Input/OutputGrating Coupler. 1- Central Y-branch. 2- Straight SMWGs.3- SMWG 180◦ bends (R3 = 6 µm). 4- Tap Y-branches. 5and 7- SMWG S-bends (R5 = R7 = 200 µm). 6- SMWGdirectional coupler. Tap GCs- GCs for CW- and CCW-resonance monitoring. (b) Mask layout. Span: ∼ 1.5×0.12 mm. 843.15 Gyro resonator schematic and interrogation block diagram.1 through 7- See nomenclature in Fig. 3.14. 8- Linear waveg-uide tapers. 9-Straight MMWGs. 10-SMWG 90◦ bends(R10 = 20 µm), with 15 µm-long straight stubs on both ends. 853.16 Theoretical power levels for the tap and merged outputs ver-sus ring normalized detuning, φring/pi, at rest (dashed curves)and under CW rotation (solid curves) with a Sagnac phaseshift ΦS = 0.1pi rad, and ILdB = 0 dB. Resonator param-eters: L = 7.5 mm, κa = 0.255, and average propagationloss αavg dB = 1 dB/cm. Dotted orange curve: CW andCCW taps at rest. Solid red and blue curves: CW andCCW taps, under rotation. Dotted brown curve: Mergedoutput at rest. Solid green curve: merged output, underrotation. Cyan dot-dashed curve: Tap power ratio underrotation, PCW(ΦS = 0.02pi)/PCCW(ΦS = 0.02pi). Magentadot-dashed curve: merged output power ratio, Pmerged(ΦS =0.02pi)/Pmerged(ΦS = 0). . . . . . . . . . . . . . . . . . . . . . 883.17 Effective and group index curve fits for air-clad strip waveg-uides of different strip widths. Also shown, original datapoints and fitted values for λ0 = 1550 nm. . . . . . . . . . . . 893.18 First E-beam layout design. . . . . . . . . . . . . . . . . . . 923.19 Mini-breadboard characterization setup. (a) Initial benchtopconfiguration. (b) On rotary platform, showing on-board ref-erence gyroscope (bottom left), and off-platform microscope(top left). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.20 SOI gyroscope characterization platform block diagram. . . . 95xixList of Figures3.21 Average angular speed (dots) and noise level (error bars), asa function of normalized speed, Sn. . . . . . . . . . . . . . . 973.22 Input and output signals for a sinusoid of frequency fin =0.885 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.23 Turntable frequency response and first-order model fitting.(a): Magnitude response. (b) Phase response. . . . . . . . . 993.24 Fibre alignments upon splitting coupler in (a) dry conditions,and (b) UV curable adhesive. (c) Spectra and (d) Q factorfor the CW and CCW resonances of a 7.4-mm-long resonator. 1003.25 (a) Reading on VI front panel, and (b) input and outputsignals during a sinusoidal rotation test with frequency fin =0.885 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.26 (a) Second fibre holder design schematic. (b) Holder withattached fibre on polishing jig. . . . . . . . . . . . . . . . . . 1033.27 Sample pedestal versions. . . . . . . . . . . . . . . . . . . . . 1033.28 Selected spectra during adhesive deposition and curing on anair-clad SMWG structure. Chip IMEC2009-R6C5. . . . . . . 1043.29 Selected spectra and variations during curing on a glass-cladSMWG structure. Sample: Imec Glass cladding, R−3C−6. . 1063.30 Splitting/merging coupler based on adiabatic splitter. (a)Schematic. 1- Straight strip SMWGs. 2- Strip SMWG bends.3- Adiabatic 50/50 splitter/merger, input and output portslabelled in blue [145]. 4- Straight rib SMWGs. 5- Rib SMWGbends. (b) Layout schematic. . . . . . . . . . . . . . . . . . 1113.31 Large-area resonator, formed by straight SMWGs (1), stripSMWG adiabatic bends (2) [21], rib SMWG directional cou-pler (3), MMWGs (4), linear tapers (5) for SMWG to MMWGconversion, and adiabatic 50/50 splitter (6) [145]. GC: grat-ing couplers [137]. Inset: Test structure for MMWG propa-gation loss characterization. . . . . . . . . . . . . . . . . . . . 112xxList of Figures3.32 Comparison of theoretical output power levels P1 and P2as functions of the ring normalized detuning, at rest andunder CW rotation, for an all-pass resonator with lengthL = 37 mm, coupling κa = 0.29, average propagation lossαdB = 0.085 dB/cm, negligible IL and splitting loop losses(ILdB = 0 dB, αscr = 0 m−1), and ΦS = 0.1pi rad. Dashedbrown curve: P1 at rest. Solid green curve: P1 under rota-tion. Orange dashed curve: P2 at rest. Purple solid curve:P2 under rotation. . . . . . . . . . . . . . . . . . . . . . . . . 1143.33 Schematic of a thermally-tuneable splitting/merging couplerfor an IME resonator. . . . . . . . . . . . . . . . . . . . . . . 1153.34 Thermally-tuneable coupler test structure. LMZI = 200 µm. . 1163.35 Resonator with thermally-tuneable coupler. . . . . . . . . . . 1173.36 Spectral simulation for a resonator with a thermally-tuneablecoupler at various MZI phase detuning conditions, fed andinterrogated through GC1. Parameters: ηθ = 24 mW/pi. L =37 mm. ∆θ = 0.1pi, α = 0.3 m−1. Thermal phase shifterpower: Pθ = 0 mW (brown dashed). Pθ = 2 mW (greensolid). Pθ = 4 mW (purple dashed). Pθ = 6 mW (red solid).Pθ = 8 mW (orange dashed). Pθ = 10 mW (blue solid).Pθ = 12 mW (magenta dashed). . . . . . . . . . . . . . . . . . 1183.37 Simulations of the spectral response for a resonator with athermally-tuneable coupler, at rest (dashed curves), and un-der rotation (solid curves, ΦS = 0.1pi). The input signalis injected into GC1, and the device is interrogated at bothports. Parameters: L = 37 mm, ∆θ = 0.1pi (Pθ = 2.4 mW),α = 0.3 m−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.38 Landmark device set, for marking chip corners and correlatinglayout coordinates to motor coordinates. Input GC name tags(illegible due to layout snapshot settings) are shown only forillustration purposes. . . . . . . . . . . . . . . . . . . . . . . . 120xxiList of Figures3.39 Curve-fitted effective index (green curves) and group index(blue curves) for glass-clad rib waveguides. (a) SMWG. (b)MMWG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213.40 (a) Cross-over length vs. wavelength for rib SMWG direc-tional couplers with various gap values. (b) Variation of thefield cross-coupling versus wavelength for rib SMWG direc-tional couplers of various gaps, all designed for κ = 1/√2 atλ0 = 1550 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . 1223.41 Block diagram of second rotary characterization setup. . . . . 1233.42 (a) Automated stage, bench-top configuration. (b) Samplepedestal and fibre array. (c) Microscope image of fibre arraynear chip alignment features. (d) Spectra of an 84 µm-longring resonator alignment feature. . . . . . . . . . . . . . . . . 1243.43 (a) Compact configuration of the characterization setup, withinturntable chamber. (b) Sample pedestal and improved fibrearray holder. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263.44 Spectra of test structures with various MMWG lengths, foran IME run. Also shown, spectrum of a reference loop-back waveguide coupled to an 84-micrometer long racetrackresonator. Inset: Insertion loss versus length, showing anMMWG propagation loss of 0.085 dB/cm. . . . . . . . . . . . 1283.45 Experimental block diagram for characterization of resonatorswith splitting/merging couplers. . . . . . . . . . . . . . . . . . 1293.46 Spectra for a 37.6 mm-long ring resonator for the mixedthrough and the return signal ports, detected at PD1 andPD2 according to Fig. 3.45. . . . . . . . . . . . . . . . . . . 1293.47 Spectra for a 37 mm-long resonator for various wavelengthstep values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130xxiiList of Figures3.48 Figures of merit for a 37.6 mm-long resonator with unbiasedtuneable coupler, in various wavelength ranges. (a) Q fac-tor. (b) ER. (c) and (d) FSR and group index, respectively,extracted from full spectrum curve fit data. (e) Curve fitsfor each resonance trough. (f) Coupling and roundtrip losscurve fit parameters, for each resonance. . . . . . . . . . . . 1323.49 T-MZI test structure experimental results. (a) V-I curve todetermine metal heater resistance, R=1106.5 Ω. (b) Normal-ized optical output power vs. heater power, at λ = 1530 nm.Minimum IL: 16.3 dB. . . . . . . . . . . . . . . . . . . . . . . 1333.50 Resonator spectra during coupler thermal tuning, with ther-mal phase shifter power as a parameter. Heater resistance:1100 Ω. Phase shifter current range: 0 to 4 mA, in 0.5 mAsteps. Dotted curves: Return signal (on circulator’s port 3).Dashed curves: Mixed-through port signal. . . . . . . . . . . 1343.51 Comparison of FWHM values for rings with various couplingconditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1363.52 Selected spectra and figures of merit for a 37 mm-long res-onator with thermally tuneable coupler. (a) Return signalspectra. (b) Extinction ratio (ER) vs. wavelength. (c) Av-erage ER vs. thermal phase shifter power, Pθ. (d) Aver-age Q-factor vs. Pθ. (e) Average FSR vs. Pθ. (e) Averagestraight-through field transmission, t(θ), and roundtrip loss,τ , vs. Pθ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373.53 Resonances for a static, 38 mm-long ring at various inputpower levels, on a temperature-controlled pedestal, at 25◦C. . 1393.54 Resonance dip stability test for an acetylene (C2H2) cell. (a)300 superimposed spectra, showing markers tracking trans-mitted power at resonance (red asterisks) and an arbitraryoff-resonance wavelength (blue stars). (b) Comparison oftransmitted power levels at selected wavelengths, and theirratio, over time. (c) Resonance wavelength over time. (d)Resonance wavelength histogram. . . . . . . . . . . . . . . . . 141xxiiiList of Figures3.55 Resonance dip stability test for a 37 mm-long resonator. (a)300 superimposed spectra, showing markers tracking trans-mitted power at resonance (red asterisks) and off-resonance(blue stars). (b) Comparison of transmitted power levels atselected wavelengths, and their ratio, over time. (c) Reso-nance wavelength over time. (d) Resonance wavelength his-togram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1423.56 Front panel of the LabVIEWTM VI for time-domain measure-ments with the N7744a photodetector. . . . . . . . . . . . . . 1433.57 (a) N7744a photodetector noise floor at various photodetec-tor sensitivity values, for 22000 samples at a 50 µs averagingtime (11 s measurements). (b) Insertion Loss and SNR as afunction of input power for a 3 m long PM patch cord, forvarious integration times. . . . . . . . . . . . . . . . . . . . . 1443.58 Comparison of noise PSD plots normalized to unit power,based on autocorrelations various time-domain tests for a 3m-long patch cord. The legend shows the integration time foreach run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1453.59 Comparison of noise PSD plots normalized to unit power,based on autocorrelations of time-domain tests for a loopbackdevice, with the pedestal vacuum pump turned on and off.Integration time: 50 µs. . . . . . . . . . . . . . . . . . . . . . 1463.60 Comparison of time-domain signals for a loopback device withvacuum pump turned on and off. For both tests the integra-tion time is 50 µs. IL: 10.9 dB with vacuum pump on, 10.8dB with vacuum off. . . . . . . . . . . . . . . . . . . . . . . . 1463.61 Dual resonator set. The smaller ring was created as a refer-ence for tracking environment-related common-mode signals. 1493.62 Spectra of similar MMWG propagaqtion loss test structures,for e-beam samples, with a SMWG loopback as a zero-lengthreference. Insertion loss as a function of length, showing anMMWG propagation loss of 0.55 dB/cm. . . . . . . . . . . . 151xxivList of Figures3.63 (a) Experimental block diagram, and (b) spectra at rest atvarious TLS sweep speeds, for a dual resonator system fab-ricated in e-beam technology. Gyro resonator length: L1 =32.8 mm. Reference resonator length: L2 = 11.4 mm. . . . . . 1533.64 Figures of merit for a 32.1 mm-long resonator fabricated usinge-beam lithography. (a) Q factor, (b) ER, (c) FSR, and (d)group index. (e) Curve fits for various resonance troughs.(f) Curve fit parameters. . . . . . . . . . . . . . . . . . . . . 1543.65 Experimental block diagram for a dual resonator system. . . 1553.66 Spectra for the forward- and back-propagating signals of adual resonator device. L1 ≈ 32 mm, L2 ≈ 12 mm. For bothresonators, the back-propagating signals are ∼ 15 dB weakerthan the forward-propagating signals. . . . . . . . . . . . . . 1563.67 Normalized power spectral density comparison with micro-scope light on and off, for (a) forward-propagating and (b)back-propagating signals in a 32 mm-long gyro resonator. . . 1563.68 Experimental block diagram for rotational tests a dual res-onator system. . . . . . . . . . . . . . . . . . . . . . . . . . . 1573.69 Gyro and reference resonator spectra of forward- and back-propagating signals. (a) Before adhesive deposition. (b) Af-ter adhesive deposition, before curing. (c) After adhesivecuring. (d) Narrow-range sweep after adhesive curing. . . . 1583.70 (a) Spectra for the forward- and backward-propagating sig-nals of a 32 mm-long gyro resonator at various input powerlevels, after fibre attachment. (b) Narrow spectral sweep ofthe forward-propagating signals for the same gyro resonator,and its reference ring. . . . . . . . . . . . . . . . . . . . . . . 1593.71 Comparison of normalized time-domain signals and theirPSD plots. Time-domain plots (a) and PSDs (b) for theangular speed and the unfiltered optical power signal. Time-domain plots (c) and PSDs (d) for the angular speed and thefiltered and shifted optical power signal. . . . . . . . . . . . 160xxvList of Figures3.72 Measured wavelength spectra for the return signal (top, pink)and mixed-through signal (bottom, blue) of a device with a37 mm-long resonator. . . . . . . . . . . . . . . . . . . . . . . 1613.73 Q factor and ER as functions of wavelength for the returnsignal spectrum of Fig. 3.72. . . . . . . . . . . . . . . . . . . 1623.74 Backscattering model schematic. . . . . . . . . . . . . . . . . 1633.75 Schematic of resonator formed by rib SMWGs (brown), ribMMWGs (cyan), and linear SM to MM converters (purple).The total resonator length is 37 mm. The variable z de-notes the position along the length of the ring, starting atthe point coupler. Left inset: Adiabatic splitting/mergingcoupler, formed by strip (orange) and rib (blue) waveguides.Right inset: Point coupler model. . . . . . . . . . . . . . . . . 1643.76 Point coupler with straight through transmission t, cross-coupling κ, back-reflection , and contra-directional couplingγ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1683.77 Normalized measured spectra for the DUT. . . . . . . . . . . 1693.78 Simulated spectra with ∆λ = 0.1 pm for a DUT with abackscatter-free ring and a perfect adiabatic coupler. . . . . 1703.79 Theoretical spectra for a DUT with Tac = 0.5, t = 0.905,σb˜SM = 5.8 mm−1, σb˜MM = 0.084 mm−1, γ = 0, and ∆λ =0.2 pm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1723.80 Theoretical spectra for a DUT with Tac = 0.57, t = 0.905,σb˜SM = 18 mm−1, σb˜MM = 0.522 mm−1, γ = 0, and ∆λ =0.1 pm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1723.81 Theoretical spectra for a DUT with Tac = 0.57, t = 0.932,σb˜SM = 18 mm−1, σb˜MM = 0.522 mm−1, γ = 0.1, and ∆λ =0.1 pm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1733.82 Simulated spectra for a DUT with Tac = 0.57, t = 0.938,σb˜ SM(z) = 18 mm−1, σb˜ MM(z) = 0.522 mm−1, γ = 0.1, and∆λ = 0.1 pm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 173xxviList of Figures3.83 Theoretical spectra for a 37 mm-long ring resonator witht = 0.938, σb˜ SM(z) = 18 mm−1, σb˜ MM(z) = 0.522 mm−1,γ = 0.1, and ∆λ = 0.1 pm. (a) Normalized transmittedpower in forward- and backward-propagating directions forCW- and CCW-direction input beams. (b) Total outputspectra in CW and CCW directions for simultaneous counter-propagation excitation. The power is referred to the totalinput power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1743.84 Comparison of measured values of (a) Q factor, and (b) ERwith those obtained in various simulations. In all cases, Tac =0.57, σb˜ SM(z) = 18 mm−1, σb˜ MM(z) = 0.522 mm−1, and∆λ = 0.1 pm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1754.1 (a) Average ring propagation loss, and (b) average ring Qfactor vs. design cycle. . . . . . . . . . . . . . . . . . . . . . 1804.2 Lock-in amplifier output vs. resonance frequency detuning,with modulation signal frequency as a parameter. . . . . . . . 1884.3 Lock-in amplifier output slope vs. modulation signal frequency.188B.1 (a) Normalized angular speed, Sn, as a function of samplenumber, ns (bottom axis), and PWM cycle number, npwm(top axis), for pmax = 1, M = 5, and K = 40. (b) Compar-ison of input and output signals for a sinusoid of frequencyfin = 0.083 Hz. The red curve is the ideal PWM duty cycle,the black curve is the rotation direction signal, and the bluecurve is the experimental PWM duty cycle. . . . . . . . . . . 209C.1 Linear power vs. time and FFT spectra for two PM circu-lators. (a, b) Red (first) circulator. (c, d) Blue (second)circulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212xxviiGlossaryAcronym MeaningAOM(s) Acusto-Optic Modulator(s)AP All-PassAR Aspect RatioAWG Arbitrary Waveform GeneratorBOX Burried OXide layerCC Critical CouplingCMC Canadian Microsystems CorporationCMOS Complementary Metal Oxide SemiconductorCPU Central Processing UnitCSS Continuous Spectral SweepCCW Counter-clock-wiseCW Clock-wiseDMA FIFO Direct Memory Access - First Input/First OutputDRC Design Rule CheckingDUT Device Under TestDUV Deep Ultra-VioletER Extinction RatioFFT Fast Fourier TransformFOG(s) Fiber-Optic Gyroscope(s)FPGA Field-Programmable Gate ArrayFSIL COM Fast Spectral Insertion Loss COM port driversFSR Free Spectral RangeFWHM Full Width at Half MaximumContinued in next page...xxviiiGlossaryAcronym MeaningGC Grating CouplerGCC Globally-optimized Critical CouplingGOC Global Optimal CouplingGPIB General Purpose Interface BusGUI Graphical User InterfaceI/O GC Input/Output Grating CouplerIFOG(s) Interferometric Fiber-Optic Gyroscope(s)IL(s) Insertion Loss(es)IME Institute of Micro ElectronicsIMEC Interuniversity Microelectronics CentreIMOG(s) Interferometric Micro-Optic Gyroscope(s)InP Indium PhosphideIOG Integrated Optic GyroscopeLIA(s) Lock-In Amplifier(s)LOC Locally-Optimized, under-Coupled ringLPF Low-Pass FilterMEMS Micro Electro-Mechanical SystemMIG MEMS Interferometric GyroscopeLCC Locally-optimized, Critical CouplingLOC Locally-optimized, Optimal under-CouplingMM Multi-ModeMMWG Multi-Mode WaveguideMZI Mach-Zehnder InterferometerOC Optimal under-CouplingPCI Peripheral Component InterconnectPD Photo-DetectorPM Polarization-MaintainingPSD power spectral densityPWM Pulse-Width ModulationPXI PCI eXtensions for InstrumentationRF Radio FrequencyContinued in next page...xxixGlossaryAcronym MeaningSiN Silicon NitrideSi-P MIP Microsystem Integration Platform for Si-PhotonicsSM Single-ModeSMWG Single-Mode WaveguideSNR Signal-to-Noise RatioSOI Silicon On InsulatorSSE Spontaneous Source EmissionTE Transverse ElectricTM Transverse MagneticTS Test StructureT-MZI Tuneable Mach-Zehnder InterferometerTEC Thermo-Electric Peltier CoolerTLS Tuneable Laser SourceUW University of WashingtonVI Virtual InstrumentWDC Waveguide Directional CouplerZRO Zero-Rate OutputxxxAcknowledgementsTo my three co-supervisors, Dr. Lukas Chrostowski, Dr. Nicolas A. F. Jaeger,and Dr. Edmond Cretu, for all their guidance, knowledgeable advice, early-morning and late-night brainstorming sessions, invaluable support, patience,kind mentoring, encouragement, and friendship during my graduate years atUBC. A very special mention to Dr. Nicolas Jaeger for the countless hoursinvested on the detail-oriented read-through sessions during the editing ofthis thesis document.Thanks to CMC Microsystems through Dr. Robert Mallard, for provid-ing me with the equipment to create the Microsystem Integration Platformfor Si-Photonics (Si-P MIP), which constitutes our current device character-ization platform. Thanks to the Natural Science and Engineering ResearchCouncil of Canada, as well as the National Science and Technology Councilof Mexico.To each and every one of the members and collaborators of the threeoutstanding research labs where I have done my theoretical and experimentalwork, for all their help, fruitful discussions, and support, as well as manyfaculty and staff members from Electrical and Computer Engineering andAMPEL Laboratories, who contributed to this work with either discussions,tools, or suggestions:I thank Dr. B. Faraji, Dr. W. Shi, Dr. R. Vafaei, and Dr. S. Talebi-Fard,for our fruitful discussions regarding the initial modelling of ring resonators.I thank R. Boeck for insightful discussions regarding the state of the art.Thanks to A. Sharkia, M. Finnis, Dr. M. Beaudoin, Dr. M. Greenberg,and D. Dawson, for their help and advice during the design and fabrica-tion of custom parts for various characterization setups. Many thanks toE. Herna´ndez, M. Almarghalani, Dr. E. Sarraf, and C. Gerardo for theirxxxiAcknowledgementscollaboration, brainstorming, and feedback during the implementation ofthe time-domain interfaces in LabVIEWTM. Thanks to Dr. R. Rosales forvarious discussions regarding signal processing.A very special mention to K. Murray for his collaboration, feedback,and invaluable help during the creation and verification of theoretical devicemodels, as well as during experimental data processing and interpretation. Ithank H. Yun for discussions regarding the behaviour of his adiabatic split-ting coupler, and for facilitating its layout design, used as a subcomponentin some of my devices. I appreciate the contributions of H. Jayatilleka andM. Caverley with ideas and collaboration during measurements, as well asinsightful discussions and suggestions to increase efficiency during experi-ments and data processing. Last, but not least, I thank K. Khondoker forhis help and feedback during the creation and de-bugging of MATLABTMscripts for data post-processing.xxxiiDedicationTo God, my wife, my family, and my friends, for their love and supportthroughout this journey.xxxiiiChapter 1IntroductionFor decades, gyroscopes have been mounted on both military and civilianvehicles such as airplanes, submarines, satellites, and missiles, to name afew. Depending on its specific purpose, each vehicle has different require-ments for angular speed resolution, bias drift (i.e., the variation of its outputover time), and its scale factor (defined as the variation of the output signalper unit change in rotation speed [143]), form factor (i.e., size and weight),and power consumption. For instance, inertial grade applications such assatellite orbit control and submarine navigation require very stringent per-formance, whereas automotive and consumer electronics applications allowfor more relaxed specifications [35, 143].Figure 1.1 shows a comparison between the bias stability and resolu-tion requirements in the early 2000s and the present, based on informationavailable in [11, 35, 110]. From this figure, one can see the growth in gyroapplications, as well as the advent of MEMS gyroscopic devices. In the early2000s the aerospace and defence applications were dominant, requiring pri-marily inertial- and tactical-grade gyroscopes [110]. Now, the availability ofsmaller, inexpensive MEMS devices has made them the sensor of choice forrate-grade applications such as robotics, automotive safety systems, medicalinstrumentation, and even general consumer products [35].The high-end gyroscope applications belong to aerospace and defencesectors, where considerable efforts have been made to create optical gyro-scopes since the mid 1970s. See, for instance, the pioneering fibre opticinterferometer experiments of Vali and Shorthill in 1976 [130] and the free-space experiments to create a passive ring resonator gyroscope by Ezekieland Balsamo in 1977 [45]. Thanks to the low propagation loss levels achievedin optical fibres and glass waveguides, glass became the material of choice1Chapter 1. Introductionfor passive resonator optical gyroscopes.Efforts to integrate micro electro-mechanical system (MEMS) with op-tical technology, in order to create micro opto-electro-mechanical system(MOEMS) gyroscopes, started in the late 1970s. According to Liu et al.[81], Northrop started its silicon waveguide investigations in 1978, leadingto the development in 1991 of a MOEMS gyroscope with a resolution of 10deg/h.The use of silicon (Si) and silicon nitride (SiN) waveguide resonatorsfor sensing and telecommunication applications has been extensively inves-tigated in the past two decades [4, 18, 80]. Special attention has beendevoted to silicon-on-insulator (SOI) and silicon-nitride-on-silica (SiN) tech-nologies, in which a waveguide (made of Si or SiN, respectively) is ontop of a silica (SiO2) layer. These particular materials exhibit relativelylow losses in the C-band, with single-mode waveguide losses ranging from0.27 dB/cm [15] to 3 dB/cm [38] for SOI, and ranging from 0.7 dB/m[12] to 6 dB/m [91] for SiN waveguides. In particular, SOI technologyshows good optical confinement due to its high refractive index contrast(cf. nSi = 3.48 vs. nSiO2 = 1.45; nSi3N4 = 2), as well as CMOS fabricationtechnology compatibility [4, 85, 106, 111].As shown in Fig. 1.2, Interferometric Fiber-Optic Gyroscopes (IFOGs)and Ring Laser Gyroscopes (RLGs) are by far the most sensitive rotationsensors to date, with sensitivities in the range of 1 to 100 µdeg/s√Hzfor high-endsystems [20, 37, 108]. However, the optical path length required to achievesuch high sensitivities (e.g. ∼ 102 to ∼ 103 m) require complicated fibrespool winding and thermal control schemes [32, 64, 74, 88], which preventtheir further miniaturization and cost reduction. On the other hand, MEMS-based vibratory gyroscopes allow for small size designs, but at the expense ofresolution, usually 2 to 4 orders of magnitude worse than those of IFOGs andRLGs, and with proof-masses driven very close to their resonance frequency,which impacts the expected lifespan of the device.2Chapter 1. IntroductionBias drift [°/h]Robotics ApplicationsAerospace and defenceResolution [°/h]Mechanical Technologies  RLGsBias drift [°/h]ApplicationsAerospace and defenceResolution [°/h]Technologiesc. 2003AutomotiveAutomotive10001x10-71x10-7RoboticsConsumer electronicsc. 20141x104 RLGsIFOGs and QuartzQuartz10001 10 1001x10-5 1x10-4 0.001 0.01 0.1 1 101x10-6 1x10-5 1x10-4 0.001 0.01 0.11x10-41x10-51x10-61x104IFOGs MEMS0.001 0.01 0.1 1 10 100100 100010 100 10001x10-5 1x10-4 0.001 0.01 0.1 1Figure 1.1: Gyro technology requirement by application, based on [11, 35,110].The most important performance figures for a gyroscope are its resolu-tion, scale factor, zero-rate output (ZRO), and bias drift. Resolution is theminimum detectable angular rate. Scale factor is the amount of change inthe output signal per unit change of angular speed. Since optical gyroscopesusually have voltage outputs, this is usually expressed in V/(deg/s) [143].The ZRO is the random output of the sensor in the absence of rotation,and is the sum of white noise and a slowly varying random function. Thenoise defines the resolution of the sensor, expressed in units of angular speedper square root of bandwidth, e.g., deg/s√Hz. The slow varying function definesthe drift of the gyroscope, usually expressed in units of deg/s or deg/h, de-pending on the gyroscope grade [81, 143]. Table 1.1 shows the performancerequirements of different gyroscope grades [143].3Chapter 1. Introduction10−2 100 102 10410−610−410−2100δΩ/B1/2 [deg/s/Hz1/2]Lengthscale [m]  Honeywell IFOG [9]IFOG−EDFA [10]Honeywell GG1320AN01 Laser [12]Honeywell GG5300 MEMS [12]Melexis MLX90609 MEMS [13]A.D. ADXRS450 MEMS [11]SiN AP, αdB1, LoptT1, ILTSiN AP, αdB2, LoptT2, ILTSOI, AP, αdB3, LoptT3, ILTSiN AP, αdB1, Lchip, ILTSiN Drop, αdB1,Lchip, ILDSOI AP, αdB4, Lchip, ILTSOI AP, αdB5, Lchip, ILTFigure 1.2: Resolution vs. lengthscale for selected optical and MEMS-basedgyroscopes, either reported in the literature, [20, 108], or commercially avail-able, [1, 2, 37], and theoretical resolution estimations (hollow markers) forring resonator gyroscopes fabricated in different materials and port configu-rations. For silicon nitride (SiN), αdB1 = 0.06 dB/cm [91], and αdB2 = 0.12dB/cm [53]. For silicon on insulator (SOI), αdB3 = 3 dB/cm [13]. ForSOI resonators with different waveguide widths for different segments, theprojected propagation loss estimations are between αdB4 = 1 dB/cm andαdB5 = 1.46 dB/cm. LoptT1 = 2.77 m; LoptT2 = 1.39 m; LoptT3 = 55.5 mm;and the perimeter of a chip is Lchip = 114 mm. The insertion losses (ILs) forthe All-Pass (AP) and the Drop-Port rings are assumed to be ILT = 7 dBand ILD = 3 dB, respectively.41.1. State of the ArtTable 1.1: Performance requirements of different gyroscope grades [143].Parameter Rate grade Tactical grade Inertial gradeAngle Random Walk [◦/√h] > 0.5 0.5 - 0.05 < 0.001Bias Drift [◦/h] 10 - 1000 0.1 - 10 < 0.01Scale Factor Accuracy [%] 0.1 - 1 0.01 - 0.1 < 0.001Full Scale Range [◦/s] 50 - 1000 > 500 > 400Max. Shock in 1 ms [g] 103 103 - 104 103Bandwidth [Hz] > 70 100 1001.1 State of the ArtCompared with ring laser gyroscopes (RLGs) and fiber-optic gyroscopes(FOGs), MOEMS gyroscopes replace the long fibre coils with optical devicesor cavities, offering the advantages of small size and lighter weight. MOEMSgyroscopes can be categorized into interferometric micro-optic gyroscopes(IMOGs) and resonant micro-optic gyroscopes (RMOGs).IMOGs can be fabricated with waveguides, mirror arrays, proof-masses,or a combination thereof, on silicon substrates [81]. Design and simulationefforts to implement optical gyroscopic devices using micro-mirrors startedin the early 2000s. For instance, in 2000, the Air Force Institute of Tech-nology of the United States of America proposed a MEMS interferometricgyroscope (MIG), in which mirrors were placed on a silicon die to create twospiral paths with an increased path length [119]. Also in 2000, the Univer-sity of Alabama proposed a 3-axis, monolithic, all-reflective gyroscope basedon simulation of parabolic reflectors to create long multi-turn helical pathsin free space, with the objective of enhancing the Sagnac effect [25]. Thesimulation results predicted a resolution of 0.001 deg/h for a structure ofapproximately 7 cm in diameter, with a 1-W input power at a wavelengthof 0.5 µm. However, there is no evidence in the literature of any fabricationefforts to create such a device.Obstacles such as high mirror losses, fragility, stringent misalignmenttolerances, and fabrication complexity [94], preclude any significant ad-51.1. State of the Artvancement of micro-mirror-based devices for gyroscopic applications. OtherIMOG designs rely on interferometric techniques to determine the displace-ment of a movable proof-mass, usually fabricated in SOI technology. Thishas allowed for interferometric techniques to determine the spectral responseof the proofmasses, [5, 6], and experimental angular rate sensitivities as lowas 27 deg/h/√Hz have been reported [95].RMOGs principle of operation is an optical micro-resonator, which in-herently has the advantage of not requiring movable masses. RMOGs canbe fabricated with optical waveguide resonators on various materials, silicabeing the most commonly used [48, 57, 82, 84, 120, 121]. Recent effortsfocused on the design and fabrication of optical resonators using silica [48]and alternative materials such as InP [22] have lead to a decrease in propa-gation loss values and improvement in resonator quality factor (Q). Severaldesign proposals compatible with fabrication technologies in materials suchas silicon nitride, silicon oxynitride, and SOI exist in the literature, for in-stance, see [58, 109, 113], but to the best of my knowledge, there has notbeen a study of the feasibility, nor a fully demonstrated device of such kind,in SOI.As mentioned earlier, after an early start, Northrop developed a MOEMSgyroscope in 1991. Honeywell and the University of Minnesota also devel-oped special components for RMOGs in 2000, such as alumina (Al2O3)and zirconia (ZrO2) low-loss (0.1 dB/cm) trench waveguides, ion-beam de-posited to form a 2-cm diameter cavity on an ultra-low-expansion substrate.Rare-earth doping was used to allow for optical cavity gain. A theoreticalresolution range between 0.1 and 1 deg/h was predicted [50].In 2003, an integrated optic gyroscope was presented by the Universityof Arizona [19]. It was fabricated in Schott IOG-10 glass [129], using aAg+−Na+ ion exchange process [135]. It had a 28-mm diameter resonator,designed with single-mode waveguides for an operating wavelength of 1550nm. Based on spectral characterization at rest and geometrical parameters,a shot-noise limited resolution of 170 deg/h was predicted. However, to thebest of my knowledge, there are no reports in the literature regarding anydynamic tests of this particular device.61.2. Structure of the ThesisIn 2003, Litton systems proposed and patented an integrated optic gy-roscope (IOG) using a multi-layer waveguide coil instead of a fiber coil [52],which can in principle reduce the cost of waveguide coils, although it wouldincrease their fabrication complexity. Another possible approach consists ofutilizing waveguide crossings (such as those investigated by our team [112]),to create multi-turn planar coils. However, the integration of such crossingsis, to the best of my knowledge, still at the theoretical design stage, e.g., see[116], and could require stringent fabrication restrictions.1.1.1 Motivation and Potential ApplicationsThe main objective of this thesis is to explore the feasibility of using SOI forfabricating rate-grade gyroscopes. If tactical- and rate-grade optical gyro-scopes can be implemented in SOI technology, these sensors could be used inhigh-volume applications such as automotive or consumer electronics. Thisthesis provides insight into theoretical limitations as well as specific prac-tical challenges regarding the design, fabrication, and interrogation of SOIgyroscopic devices.1.2 Structure of the ThesisThe remainder of this thesis is organized in three chapters. Chapter 2 fo-cuses on the theoretical analysis of optical gyroscopic devices, their workingprinciples, the key optical resonator parameters and their interdependenceto achieve optimal angular resolution, the impact of parameter variationsin the device performance, and the requirements to implement frequencytracking using phase modulation.Chapter 3 guides the reader through the iterative process carried outfor the design, fabrication, and characterization of devices, as well as theconstruction and improvement of the characterization setups, which hasbeen divided in five design cycles. Thus, Chapter 3 is divided in five sections,stating for each cycle the initial objectives, describing device designs, thedevice characterization. A critical evaluation of the results is conducted at71.2. Structure of the Thesisthe end of each iteration, in order to identify problems, plan approaches tosolve them, and derive conclusions that help improve both the devices andthe characterization setup during the next iteration.Chapter 4 summarizes the main conclusions and describes the futurework required for further improvement of the device readout and the char-acterization setup.8Chapter 2Resonator Simulation andParameter OptimizationIn this chapter, a theoretical analysis of the angular speed resolution and therobustness of waveguide resonator devices for gyroscopic applications is car-ried out, taking into account various propagation losses and refractive indexvalues, corresponding to various currently available materials and waveguidefabrication technologies.Although several theoretical studies regarding the use of micro-ring res-onators for gyroscopic applications have been carried out [109, 146], to thebest of my knowledge there had not yet been a thorough study of the in-terdependence of the values of propagation loss, resonator length, couplingcoefficients, and off-resonance detuning necessary to achieve a truly opti-mized angular speed resolution (as mentioned in Chapter 1, resolution isthe minimum detectable angular rate). As will be shown in Subsection2.3.4, the optimal resolution is inversely proportional to the product of thespectral slope and the square of the resonator length. Optimized parametervalues for some particular cases have been depicted in references such as[127], but subject to restrictions of a small length-propagation loss product,L · α. This precludes a proper analysis for the case of a gyroscope, wherelong lengths are desirable to enhance the Sagnac effect [101].In this chapter we propose and analyze optical resonator gyroscopes withmillimetre-range optical paths, in either an all-pass (Gires-Tournois, single-bus), or a drop-port (double-bus) configuration, which could be fabricatedon a Silicon-on-Insulator (SOI) platform, using currently existing CMOS-compatible techniques, with theoretical resolutions comparable to those of92.1. Waveguide CouplingMEMS-based systems. For computational and modelling simplicity, theresolutions are calculated assuming a circular area for the ring resonators.Since the Sagnac effect is proportional to the area enclosed by the resonator,rather than the resonator length, for a fixed length value, the use of a rect-angular shape with a particular aspect ratio will affect the resolution, butdoes not modify the relationships nor the optimum values for any other pa-rameters. For an easier reading, the most frequent acronyms and variablesare summarized in the Glossary and Table 2.1, respectively.As shown in Fig. 2.1, light from a laser source is injected into, and ex-tracted from, low-loss ring resonators via one or two bus waveguides, whichare interrogated with photodetectors, either at the through port in an all-pass configuration, or at the drop port in a double-bus configuration, re-spectively. A 50%-50% splitter is used to allow for counter-propagatinglight injection. The output light is directed to two photodetectors.In the all-pass configuration, there is only one bus waveguide, so tb = 1.Hence, two circulators are required for proper signal detection, which intro-duces slight insertion losses. Circulators can also be replaced by Y-branchcouplers, which also produce insertion losses. In contrast, for the drop portconfiguration no circulators are required. In both alternatives, the counter-propagating beams undergo different phase shifts whenever the sensor ro-tates, which allow for differential measurements.2.1 Waveguide CouplingFor a system formed by two straight parallel waveguides with identical cross-sections, the modal powers in the injection (Pin) and the coupled (Pc) waveg-uides are functions of wavelength, waveguide geometry, core and claddingrefractive indices, separation (gap) between the waveguides, and positionalong the axial coordinate of the waveguides. Power is exchanged betweenthe waveguides obeying a sine-squared law along the propagation direction[141]. Assuming lossless coupling, at a distance L⊗, known as the cross-overlength, all the energy is transferred to the coupled waveguide. According tothe so-called supermode theory [42, 79], the cross-over length is:102.1. Waveguide CouplingTable 2.1: Most frequent variablesSymbol MeaningPin Input power, injection power, [W] or [dBm]Pn Noise equivalent power, [W] or [dBm]λ0 Laser central wavelength, [m]ν0 Laser central frequency, [Hz]∆ν Laser spectral linewidth, [Hz]L= ΣLi Total resonator length, [m]TR Resonator roundtrip time, [s]L⊗ Crossover length, [m]Lca,cb Waveguide coupling length, region a, b, [m]Lopt Optimum resonator length, [m]Ln = LLopt Normalized resonator lengthα Field propagation loss [m−1]αdB Power propagation loss, [dB]τ = e−ΣLiαi Round-trip field amplitude attenuation coefficientta,b Transmission amplitude coefficient, region a, bκa,b Cross-coupling amplitude coefficient,, region a, bΥ = tatbτ ∆ν-independent transmission-attenuation productΨ = Υe−2pi∆ν ∆ν-dependent transmission-attenuation productφ Detuning [rad]|φHMT | Half-Maximum detuning, Through port [rad]|φHMD | Half-Maximum detuning, Drop port [rad]φn = φ/pi Normalized detuningCIL Insertion loss coefficientILdB Insertion loss, [dB]Ω Angular rate, [rad/s]Ωdps Angular rate, [deg/s]|δΩ| Angular rate resolution, [rad/s]|δΩ|opt Optimum angular rate resolution, [rad/s]|δΩ|norm =|δΩ||δΩ|optNormalized angular rate resolution112.1. Waveguide Coupling!"#$%&'(&')*+,--$%!"#"$. $%$&'/$-01 /$-02"$3,%45+"-6%76%789:%";4< 3,%45+"-6%76%789:%";4<!7=72 >?27!3(a) All-pass configuration!"#$%&'(&')*+,--$%!"#"$. $%#%$&'#/$-/0/$-/1. #"$"#!2321 4512!6&'$(b) Drop-port configurationFigure 2.1: Schematic diagram of the All-Pass (top) and Drop-Port (bottom)configurations for a racetrack resonator gyroscope system. Light is split andinjected in opposite directions into the racetrack resonators. The outputlight in each direction is then directed to a photodetector. Circulators arerequired for proper interrogation in the all-pass configuration, but can bereplaced by elements such as Y-branches for easier integration.122.2. Sagnac EffectL⊗ =piβ1sym − β1asym=λ02(neff 1sym − neff 1asym), (2.1)where β1sym and β1asym are the propagation constants, and neff 1sym andneff 1asym are the effective indices of the first (also known as first even) andsecond (also known as first odd) supermodes of the coupled system, respec-tively.For identical coupled waveguides, the modulus of the (dimensionless)field cross-coupling amplitude coefficient, κ, is a function of both L⊗ andthe coupling region length, Lc, given by [141]:κ =∣∣∣∣∣√PcPin∣∣∣∣∣=∣∣∣∣sin(pi2·LcL⊗)∣∣∣∣ . (2.2)Assuming reciprocity and negligible backscattering in the coupling re-gions of the rings shown in Fig. 2.1, after obtaining the effective indices ofthe first two supermodes of the straight couplers a and b, with respectivewaveguide gaps, ga and gb, the cross-over lengths L⊗ a,b are obtained for eachcoupler using Eq. (2.1), and the values of κa and κb can be determined usingEq. (2.2). Notice that for the all-pass configuration, gb → ∞;⇒ κb = 0.Although these moduli are real numbers, a phase shift factor must accom-pany them to account for the relative phase-shift between the injected andthe coupled waves [141], [60]. Assuming Lc  L, the phase shift betweenthe injected and the coupled waves can be considered constant, and equalto 90 degrees, hence the ”−j” phase shift factors in Fig. 2.1.For lossless coupling conditions, the straight-through transmission am-plitude coefficient is t =√1− κ2. However, if the couplers are lossy, thent2 = 1−κ2−γ2, where γ2 is a coefficient representing the existence of lossesin the coupler.2.2 Sagnac EffectIn any resonator that undergoes a rotation, there is a phase difference pro-portional to the dot product of the angular velocity vector ~Ω and the area132.2. Sagnac Effectvector ~A of the enclosed optical path [101, 104]. Firstly the equations of theSagnac effect in vacuum will be derived. Secondly, the case of a resonatormade using a waveguide with effective index neff will be analyzed.2.2.1 Sagnac Effect in VacuumConsider the resonator (assumed circular for simplicity) with radius R,shown in Fig. 2.2, and assume it is built in vacuum, which could be achieved,for instance, by a set of mirrors and a 50/50 beam splitter located at point X.When static, the transit time t = 2piR/c for the light to make one round-tripin the ring at a speed c = 3× 108m/s is identical for both beams. However,if the resonator is rotated at an angular speed Ω in a clockwise (CW) direc-tion, the counter-rotating wave, travelling in the counter-clockwise (CCW)direction, represented by the dotted line, will be enhanced in phase whenreaching the injection point at position X’, as it reaches the displaced beamsplitter before geometrically closing the circular path. Conversely, the co-rotating wave, represented by the dashed line, will be retarded in phase whenreaching the injection point, as it travels a longer path to reach the coupler,at point X”. This can be regarded as an effective travel length change foreach beam [8, 44], which will produce a blue- and a red-shift in the reso-nances of the counter- and co-rotating beams, respectively. To a first orderapproximation in terms of ΩR/c the effective travel length changes are:Lccw = 2piR−RΩtccw = 2piR− δLccw = ctccw (2.3)Lcw = 2piR+RΩtcw = 2piR+ δLcw = ctcw, (2.4)where tccw and tcw are the travel times of the beams in the CCW and CWdirection, respectively.The travel times for the counter-propagating waves can be expressed asfollows:tccw =2piRc+RΩ=2piRc1 + RΩc≈2piRc(1−RΩc)(2.5)tcw =2piRc−RΩ=2piRc1− RΩc≈2piRc(1 +RΩc), (2.6)142.2. Sagnac EffectFigure 2.2: Sagnac effect in a clockwise (CW)-rotating resonator.where first-order Taylor expansions have been used for the last terms of theequalities. The travel time difference is therefore:∆t ≈4piR2Ωc2(2.7)Also to a first order approximation in terms of ΩR/c, valid for ΩR c,both path length differences are equal in magnitude, i.e., points X’ and X”are at the same position:∆L = Lcw − Lccw ≈4piR2Ωc. (2.8)δL = δLccw = δLcw ≈2piR2Ωc=∆L2(2.9)2.2.2 Sagnac Effect in a Dielectric MediumWhen the light travels in a waveguide fabricated with a material of refractiveindex n = n(λ), the waveguide will exhibit a wavelength-dependent effectiverefractive index neff = neff(λ) [39]. In this case, Eqs. (2.3) and (2.4) become:Lccw n = 2piRneff − δLccw n ≈ (2piR− δL)neff (2.10)152.2. Sagnac EffectLcw n = 2piRneff + δLcw n ≈ (2piR+ δL)neff. (2.11)Since the medium is not vacuum, and considering the postulates of spe-cial relativity, it is no longer possible to consider that the speed of light isthe same for both counter-propagating waves as we did before. The speed oflight in the medium, v = c/neff, will be the same for both directions only ifthe observer is moving along with the rotating medium. Therefore, in orderto estimate the speed of light as observed in the laboratory, a relativisticaddition of speeds [103, 131] is necessary. The velocities of light ucw anduccw for the CW and the CCW waves as measured by a stationary observerare given by:u =c/neff ± ΩR1± c/neffΩRc2(2.12)To a first order approximation with respect to ΩR:ucw =cneff+ ΩRζ (2.13)uccw =cneff− ΩRζ, (2.14)where the term ζ = 1− 1n2effis known as the Fresnel-Fizeau Drag coefficient[101, 131]. The travelling times for the CW and CCW waves then are:tcw n =Lcw nucw=neff(2piR+ δL)cneff+ ΩRζ(2.15)tccw n =Lccw nuccw=neff(2piR− δL)cneff− ΩRζ. (2.16)Therefore, from Eqs. (2.15) and (2.16), the travel time difference is, to afirst order approximation (valid for ΩR c):∆tn ≈4piR2Ωc2n2eff(1− ζ) =4piR2Ωc2= ∆t, (2.17)which is the same travel time difference as for vacuum, shown in Eq. (2.7).Thus, to a first order approximation in terms of ΩR/c, the phase shift ex-162.2. Sagnac Effectperienced by each beam, either in vacuum or in a dielectric medium, isindependent of the refractive index. If the medium is a fiber wound in a coilof N turns, then Eq. (2.7 will be multiplied by N [44], but for the case ofa resonant optical waveguide gyroscope, N = 1, and the resonator lengthis L = 2piR. Since ω = ∆φ/∆t and ω = 2pic/λ, where λ is the free-spacewavelength of light, the total phase shift between both beams due to theSagnac effect for a resonant optical waveguide gyroscope is:|∆φ| =8picλ0−→A ·−→Ω =8pi2R2Ωcλ=2L2Ωcλ. (2.18)In the particular case of a resonant optical waveguide gyroscope, the variableφ, known as the detuning [60, 127], is different for each propagating beam.The detuning for the co-rotating (counter-rotating) beam, φ, is the algebraicsum of the phase undergone by each beam in one roundtrip through thering, φring, identical for both beams, plus (minus) the Sagnac phase shift,δφ = L2Ωcλ , proportional to the path length elongation (reduction) by δL:φ = φring ± δφ =2pineff(λ)Lλ±L2Ωcλ. (2.19)The electric field of the laser light can be described in terms of time andspace variables, t and z, respectively, as E = E0ej(ωt−2pineffλ z), where ω isthe central angular frequency of the laser light. In the case of a resonator(see Fig. 2.2), a roundtrip is completed whenever z = ML = M2piR.Equivalently, a roundtrip is completed whenever t increases its value bythe roundtrip time TR, i.e., t = MTR, as the total phase of the complexexponential remains the same. Therefore, the electric field in the resonatorcan be writtten as:E = E0ej(ωt−φ) (2.20)The first term of Eq. (2.19) can also be written as its time-domain equiv-alent, i.e., 2piLneff(λ)/λ = ωTR, where ω is the central angular frequencyof the laser light, and TR = Lneff(λ)/c is the time required to complete aroundtrip in the resonator. This substitution will prove useful for the phasenoise analysis (Section 2.3.6).172.3. Analytical and Numerical ModellingEqs. 2.19 and 2.20 allow one to see the phase retardation and enhance-ment caused by the Sagnac effect for each beam. For the co-rotating beam,the effective cavity length increases (positive sign in Eq. 2.19), produc-ing a red shift in the resonance. In order to maintain a constant phase inthe argument of Eq. 2.20, more time has to elapse, i.e., the wave is phaseretarded. Conversely, for the counter-rotating beam, the effective cavitylength decreases (negative sign in Eq. 2.19) producing a blue-shift, and inorder to maintain a constant phase in the argument of Eq. 2.20, less timeshould elapse, i.e., the wave is phase-enhanced. The red- and blue-shifts areillustrated in Fig. 2.3.2.3 Analytical and Numerical ModellingBy computing an infinite sum of successive round-trip field couplings forthe racetrack resonators shown in Figs. 2.1(a) and 2.1(b), in a similar wayas done by Vorckel et al. [133], but maintaining the phase sign conventionof Eq. 2.20, the S parameters for the through- and drop-port transmissionfields, S21 and S41, are given by:S21 =ta − tbτe−jφ(1− γ2a)√CIL(1−Υe−jφ)(2.21)S41 = −κaκb√τe−jφ2√CIL(1−Υe−jφ), (2.22)where τ = e−ΣLiαi is the round-trip amplitude (field) propagation attenua-tion, Li is the length of the waveguide segment i, i = 1, 2, ...N , L =∑Ni=1 Liis the length of the resonator, αi denotes the field propagation loss of thewaveguide segment i, in units of m−1, Υ = tatbτ , ta and tb are the (dimen-sionless) field through-transmission amplitude coefficients of regions a andb, κa and κb are the field coupling coefficients of regions a and b; γa and γbare the coefficients representing the existence of coupling losses, such thatt2a = 1 − κ2a − γ2a and t2b = 1 − κ2b − γ2b ; CIL = 10ILdB10 eαLbus is a coefficientrepresenting the amplitude decrease due to insertion loss, ILdB, (in dB),182.3. Analytical and Numerical Modellingas well as the propagation losses in the bus waveguide(s). Assuming thatthe waveguide directional coupler (WDC) is small in comparison to the to-tal resonator length, its phase shift contribution has been neglected. In allcases, α ≥ 0 m−1; 0 ≤ ta,b ≤ 1; 0 ≤ κa,b ≤ 1; 0 ≤ γa,b ≤ 1; ILdB ≥ 0 dB;and CIL ≥ 1, are real numbers. In the case of lossless coupling, γa,b = 0 andt2a,b = 1− κ2a,b.Fig. 2.3 illustrates the resonant Sagnac gyroscope concept. At rest, thespectra for light beams propagating in the resonator in opposite directionsare identical, and depicted by the solid line. As per Fig. 2.2, and Eqs. (2.3),(2.4), (2.9), (2.19), and (2.20), for a CW-rotating resonator, the co-rotating(i.e., CW) beam will be red-shifted due to its longer equivalent resonatorlength, LCW = Lring + δL. Conversely, the counter-rotating (i.e., CCW)beam will be blue-shifted due to its shorter equivalent resonator length,LCW = Lring − δL. The co-rotating beam is considered to be retarded inphase, and its resonance shifts towards more negative values of φring (reddashed curve). The counter-rotating beam is considered to be enhanced inphase, and its resonance shifts towards more positive values of φring (bluedashed curve). This produces a difference between the light intensities ofboth counter-propagating beams, which can be used for a differential mea-surement of the angular velocity.192.3. Analytical and Numerical Modelling-1.0 -0.5 0.0 0.5 1.00.30.40.50.60.70.80.91.0{ring/o|S 212(a) Amplitude response-1.0 -0.5 0.0 0.5 1.0-1.0-0.50.00.51.0{ring/oÝ(S21)/o(b) Phase responseFigure 2.3: Normalized spectra of the counter-propagating light beams inan all-pass (single-bus) racetrack resonator. Solid curve: resonator at rest.Dashed curves: spectra for the co-rotating (red curve, red-shifted) andthe counter-rotating (cyan curve, blue-shifted) beams. For this example,αdB = 0.01 dB/cm, L = 1 m, κ = 1/√2. A Sagnac phase shift δφ = 0.1pi radhas been used to produce a noticeable phase shift.202.3. Analytical and Numerical ModellingAlthough the proper units of α are m−1, in the literature propagationlosses are usually reported in units of dB/cm. In the present work, α repre-sents field attenuation, not power attenuation. In order to avoid confusions,a “dB” subscript will be added in the case of power attenuations, reported indB/cm, and the equation below will be used to transform one to the other:α[m−1] = 5 · ln(10) · (αdB[dB/cm]) (2.23)2.3.1 Through- and Drop-Port Power TransmissionThe expressions for the through- and drop-port power transmission ampli-tudes are, based on Eqs. (2.21) and (2.22):|S21|2 =t2a − 2ΥΓa cos (φ) + t2bΓ2aτ2CIL [1− 2Υ cos(φ) + Υ2](2.24)|S41|2 =κ2aκ2bτCIL [1− 2Υ cos(φ) + Υ2], (2.25)where Γa = 1 − γ2a. Since the amplitude responses |S21|2 and |S41|2 repre-sent normalized power transmissions, they can also represent the normalizedcurrent in a photodetector. From the numerator of Eq. (2.21), at resonantsteady state (i.e., φ = 2mpi, m is an integer), and assuming lossless coupling(i.e., γa,b = 0), it is possible to see that whenever:ta = tbτ, (2.26)the through-port signal is cancelled, i.e., S21 = 0, and for an add-drop ring,the output at the drop port is maximal. This particular case, known ascritical coupling (CC), reduces to ta = τ for the single-bus waveguide case(tb = 1), a particular case depicted in [141]. For a symmetrically-coupledring (i.e., ta = tb), critical coupling is strictly possible only if the waveguidesare lossless (α = 0 m−1).In contrast to the single-bus waveguide case, where the critical couplingcondition requires a unique value of ta (and hence a unique value of κa), for212.3. Analytical and Numerical Modellingthe double-bus resonator case, Eq. (2.26) is satisfied for an infinite numberof values of κb. In fact, for any given α, the slope of the spectral notch(through port) or the spectral spike (drop port) of any double-bus resonatordepends on L, κa, κb and φ. Our initial, intuitive assumption was thatthe CC condition would show the largest extinction ratio due to its nullthrough-port output at resonance. However, as shown in the modelling ofSections 2.3.4 and 2.3.5, this is not the case.2.3.2 Figures of Merit of the Spectral ResponseExtinction RatioFrom Eqs. (2.24) and (2.25), the extinction ratios ERdB21 and ERdB41 , indB, for the through and the drop ports are, respectively:ERdB21 = 10 log10[(ta + tbτ)2(1−Υ)2(ta − tbτ)2(1 + Υ)2](2.27)ERdB41 = 10 log10[(1 + Υ)2(1−Υ)2], (2.28)Full Width at Half MaximumThe full width at half-maximum (FWHM) is defined as the difference of thevalues of the independent variable (either frequency, wavelength, or phase),for which the spectral response is at the midpoint between the normalizedbaseline and the resonance trough or dip, for the through and the drop port,respectively.Quality FactorIn this work, we followed the definition of quality factor as the resonancewavelength divided by the (wavelength) FWHM: Q = λ0∆λ . In order toderive it, we firstly obtained the values of φ that yield, for each port, half ofthe maximum normalized amplitude, |φHM|. After algebraic manipulationof Eq. (2.25), based on the definition of FWHM described above, we have222.3. Analytical and Numerical Modellingthe same result for both ports:|φHMT | = |φHMD | = |φHM| = cos−1[1− 2(1−Υ)2 + Υ22Υ]. (2.29)Thus, the bandwidth in terms of wavelength is:∆λ =λ2φHMpiΣ (ngiLi)(2.30)where ngi and Li are the group index and length of the ith waveguide seg-ment of the racetrack resonator. Finally, the quality factor is:Q=piΣ (ngiLi)λ0 cos−1[1−2(1−Υ)2+Υ22Υ] (2.31)Free Spectral RangeThe free spectral range (FSR) [60] of the resonator can be defined in termsof the (group) optical path and the resonant wavelength for a resonator withN segments of different lengths Li and group indices ngi , i = 1, 2, ..., N , as:FSR =λ20∑Ni=1 ngiLi(2.32)FinesseThe finesse of the ring resonators is the ratio of the FSR and the FWHM,which can be expressed as:F =FSRFWHM=Qλ0ngL=picos−1[1−2(1−Υ)2+Υ22Υ] (2.33)2.3.3 Loss Impact on PerformanceFor the CC double-bus case, κb was chosen as our independent couplingvariable. Fig. 2.4 shows κa as a function of κb at CC, assuming losslesscoupling (i.e., γ2a = γ2b = 0), for a resonator with a length L = 50 mm.232.3. Analytical and Numerical ModellingFour different propagation loss values are used in this figure. The firstthree, αdB = 0.06 dB/cm, αdB = 0.12 dB/cm, and αdB = 3.0 dB/cm,correspond respectively to propagation loss levels reported for waveguidesfabricated in SiN [53, 91] and SOI [13] technologies. The fourth value is atheoretical average propagation loss value of αdB = 1.17 dB/cm, achievableby using SOI waveguide segments of different widths (and thus, propagationlosses) to create the resonator. For small propagation losses, κa tends toa non-zero value for low values of κb, and behaves almost as an identityfunction (i.e., κa ≈ κb) for medium to large values of κb. In the limit whenα→ 0⇒ κa → κb.0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0NbN aCC ĮdB=0.06  dB/cmĮdB=0.12  dB/cmĮdB=1.17  dB/cmĮdB=3.00  dB/cmFigure 2.4: Magnitude of field coupling coefficient of waveguide a, (κa),as a function of coupling coefficient κb, for critically coupled, 50-mmlong ring resonators, for propagation loss levels of different technolo-gies: αdB = 0.06 dB/cm (SiN , [91]), αdB = 0.12 dB/cm(SiN , [53]),αdB = 3.0 dB/cm(SOI, [13]), and αdB = 1.17 dB/cm (estimate for SOImaterial with waveguide width variation, proposed in the present work as alow-loss alternative for SOI technology).If the product αL is too large, most of the energy will be lost after afew roundtrips in the resonator, as can be concluded based on the finessevalues of Fig. 2.5, which hinders the resolution. In particular for the CC242.3. Analytical and Numerical Modelling0.0 0.2 0.4 0.6 0.8 1.01251020Finesse CCĮdB=0.06  dB/cmĮdB=0.12  dB/cmĮdB=1.17  dB/cmĮdB=3.00  dB/cmNbFigure 2.5: Finesse as a function of coupling coefficient κb, for criticallycoupled, 50-mm long ring resonators, for propagation loss levels of differenttechnologies: αdB = 0.06 dB/cm (Si3N4, [91]), αdB = 0.12 dB/cm(Si3N4,[53]), αdB = 3.0 dB/cm(SOI, [13]), and αdB = 1.17 dB/cm (proposed SOIdesign with waveguide width variation).case, as the product αL increases, the value of κa becomes larger and lessdependent on κb in order to fulfill Eq. (2.26), describing an almost-flat linethat quickly tends to unity (see Fig. 2.4). For αL ≈ 2.5, we can considerthat the value of κa is already unity regardless of the value of κb. As willbe shown in the following section, large coupling values imposed by the CCcondition imply that energy is drained out of the resonator faster than ifthe ring is optimally coupled (OC) below the CC value.For both the CC and the OC case, ever decreasing losses are requiredso that the resonator length and enclosed area can be increased, and thecoupling values can be decreased, thus enhancing the Sagnac effect andachieving better resolution values. For any given propagation loss α, (dic-tated by the material and fabrication technology), there is an optimal set ofvalues for L, κa, κb and φ that globally optimize the resolution, even thoughthey do not yield the absolute maximum in the spectral response slope.Such parameters are different for the through and the drop configurations252.3. Analytical and Numerical Modellingbecause the optimal configuration for the through-port cases is always aGires-Tournois (all-pass) [60] configuration, which implies that the resonantcavity is less loaded due to the absence of a drop port.2.3.4 Spectral Response for the Through PortIn this section, the spectral characteristics of the through port will be de-scribed, and expressions for finding the value of φ that maximizes the slopeof the spectral response as a function of the resonator length and couplingswill be derived. For simplicity, lossless coupling and negligible ILs are as-sumed from now on, unless otherwise stated, i.e., γa = γb = 0; CIL = 1.The ILs can be incorporated later on without loss of generality. The firstand second derivatives of the through-port spectral response are:∂|S21|2∂φ=2κ2aΥ(1− t2bτ2)sin (φ)[1− 2Υ cos(φ) + Υ2]2(2.34)∂2|S21|2∂φ2=2Aκ2aΥ [B cos (φ) + Υ[cos (2φ)− 3]][1− 2Υ cos(φ) + Υ2]3, (2.35)where A = 1− τ2t2b and B = 1 + Υ2. For a given length L and propagationloss α, there is a certain combination of couplings κa, κb and detuning φthat produces a maximum slope in the spectral response. At such a pointthe numerator of Eq. (2.35) is zero. By solving for φ (modulo 2pi), anddiscarding any complex roots:φMST = ± cos−1[−1−Υ2 +√1 + 34Υ2 + Υ44Υ]. (2.36)Eq. (2.36) yields the same results as Eq. (6) in [127], but it is a moregeneral expression, due to the inclusion of the drop port (κb) in the model.Fig. 2.6 shows the through-port spectral response and its first derivativeas a function of the normalized detuning, φ/pi, for two all-pass resonators1with identical propagation loss αdB = 0.06 dB/cm and identical length, in1As will be shown in Subsection 2.3.7, all optimized through-port resonators are all-passresonators, i.e., κb = 0.262.3. Analytical and Numerical ModellingCC and OC conditions, respectively. The length L = 1.63 m = Lopt21CCcorresponds to the optimum length for the CC resonator. It is noticeablethat, as long as the length is the same, the maximum OC slope is almosttwice as large as the maximum CC slope, even though the OC resonatorlacks zero response at resonance. This trend is the same for all values ofpropagation loss, demonstrating that OC resonators perform better thantheir CC counterparts.-1.0 -0.5 0.0 0.5 1.0-0.20.00.20.40.6fpÈS 212 ,¶S 212¶f          OC, Lopt21CC          CC, Lopt21CC                OC, Lopt21CC                CC, Lopt21CCS21 2S21 2¶ S21 2/¶f¶ S21 2/¶fFigure 2.6: Through port response, |S21|2 (solid), and first derivative,∂|S21|2∂φ(dashed), as a function of φ/pi, for αdB = 0.06 dB/cm and L = Lopt21CC =1.63 m. In all cases, κb → 0 (all-pass resonators). In spite of lacking azero output at resonance, the OC case shows a larger maximum slope, at asmaller detuning in comparison to the CC case (see Table 2.2).A resonator with a steeper slope for a fixed resonator length value willindeed yield a better resolution. This fact initially led us to the intuitivemisconception that ever growing slopes would always yield better perfor-mance. In fact, as shown in Fig. 2.7 and in Table 2.2, for low-loss waveg-uides (αdB = 0.06 dB/cm), the largest slopes occur at very small resonatorlengths, but from a gyroscopic viewpoint, the optimum length is larger. Even272.3. Analytical and Numerical Modelling¶ ¶ffp–510–310–310–410 0.01 0.1 1–5100.1101000S 212 L= 1 mm, OCL=1 mm, CCL=1.63 m, OCL=1.63 m, CCFigure 2.7: Slope of the optimized all-pass frequency response, ∂|S21|2∂φ , as afunction of φ/pi, for αdB = 0.06 dB/cm (i.e., α = 0.69 m−1), L = 1 mm(solid) and L = Lopt21 = 1.63 m (dashed). Black curves: OC; Light curves:CC. The small resonators yield larger slopes at smaller detunings. However,as shown in Table 2.2, they do not yield the best resolutions.if the spectral slope (and thus the extinction ratio) is considerably larger forlengths L < Lopt, the resolution at Lopt will always be the best, since theresolution is inversely proportional not only to the spectral slope, but alsoto L2 (see subsection 2.3.7). Therefore, maximizing the product L2 ∂|S21|2∂φ ,rather than just the slope, is necessary to ensure the best resolution.In order to compare the slopes of the four resonators considered inFig. 2.7, logarithmic scales are needed, hence, only positive detuning valueswere plotted. However, the ordinate-axis-symmetry of the spectrum andthe origin-symmetry of its derivative, are conserved in a linear scale. Theslopes of the small resonators are almost four orders of magnitude largerthan those of the large resonators. However, as shown in Table 2.2, theresolutions of the small resonators are ∼ 3 orders of magnitude worse than282.3. Analytical and Numerical ModellingTable 2.2: Parameters for different lengths and coupling conditions, throughport, all-pass configuration, for αdB = 0.06 dB/cmPPPPPPPPParam.L1 mm (CC) 1 mm (OC) 1.63 m (CC) 1.63 m (OC)κa opt 0.037 0.0263 0.945 0.759φMTpi 250× 10−6 190× 10−6 0.373 0.277Max. slope[rad−1]470.7 558 0.181 0.276|δΩ| [deg/s] 3.98 3.37 0.00392 0.00253those of the large resonators. Once again it is possible to observe that foridentical values of α and L, the maximum spectral slope for the OC case(solid) is always larger than that of the CC case (dashed), and it occurs forsmaller values of φ, despite not having a zero output at resonance, due tothe under-coupling of any OC case.2.3.5 Spectral Response for the Drop PortFor the drop port case, the OC condition also shows larger slopes than theCC condition for resonators with identical lengths and propagation losses.The first and second derivatives of the drop-port response are:∂|S41|2∂φ=−2κ2aκ2bτΥ sin (φ)[1− 2Υ cos(φ) + Υ2]2(2.37)∂2|S41|2∂φ2=2κ2aκ2bτΥ [B cos (φ) + Υ[cos (2φ)− 3]][1− 2Υ cos(φ) + Υ2]3(2.38)Due to its numerator structure, the roots (real, modulo 2pi) of Eq. (2.38)are the same as those for Eq. (2.36):φMSD = ± cos−1[−1−Υ2 +√1 + 34Υ2 + Υ44Υ]. (2.39)However, the spectral response values (Eqs. (2.24) and (2.25)) and their292.3. Analytical and Numerical Modellingfirst derivatives (Eqs. (2.34) and (2.37)) are quite different for the throughand drop ports, and thus a new optimization is necessary to find the com-bination of L, κa,b and φMD that yields the best resolution.Fig. 2.8 shows the drop-port normalized spectral response (|S41|2, solid),and its first derivative with respect to φ (dashed) as a function of the nor-malized detuning, φn = φ/pi, for an OC (black) and a CC (light) resonator,both with identical lengths and propagation losses.From Figs. 2.6 and 2.8, and as will be shown in detail in Section 2.4, itis possible to conclude that the all-pass configuration offers a higher perfor-mance thanks to its longer optimum length, larger maximum slope and thusbetter resolution. Its lower cavity loading contributes to smaller round triplosses, which in turn allow for larger slopes and larger optimum lengths. Inboth the all-pass and the drop cases, OC resonators perform better than CCresonators, even at the optimum CC lengths, Lopt21CC and Lopt41CC , respec-tively. For the sake of clarity and briefness, only OC cases will be plottedand discussed from now on, unless stated otherwise.We will see in the next subsection that the best resolution, |δΩ|opt fora particular α, will depend upon the optimization of L, φ, κb and κa for theparticular port under study. The examples depicted in Figs. 2.6 and 2.8were obtained with globally optimized all-pass- and drop-port parameters,respectively; i.e., the only restrictions for the values of the parameters werethose imposed by the propagation losses and the physical model of the sys-tem. This is useful for illustration purposes, but rather impractical forfabrication, as the lengths are not feasible in a standard 200-mm SOI wafer.In case of lengths smaller than the optimum length Lopt, the parametersφ, κb and κa can, and should still be, tuned for achieving the best possibleresolution. The best achievable resolution in such a case will obviouslybe worse than the global, unconstrained optimum, due to the decrease oflength and thus smaller Sagnac phase shift. The parameters φ, κb and κaare identical for different values of α if expressed in terms of a normalizedlength Ln = L/Lopt.302.3. Analytical and Numerical ModellingÈS 412 ,¶S 412¶ffp          OC, Lopt41CC          CC, Lopt41CC                OC, Lopt41CC                CC, Lopt41CCS41 2S41 2¶ S41 2/¶f¶ S41 2/¶f-1.0 -0.5 0.0 0.5 1.0-0.10.00.10.20.3Figure 2.8: Drop-port frequency response, |S41|2 (solid), and first deriva-tive, ∂|S41|2∂φ , (dashed), as a function of φ/pi, for αdB = 0.06 dB/cm andL = Lopt41CC = 0.93 m, for OC and CC cases. The parameters are:κbCC = 0.697 ⇒ κaCC = 0.926, κaOC = κbOC = 0.779. From the dot-ted curves, it is evident that the OC case has a larger maximum spectralslope, which occurs at a smaller normalized detuning.2.3.6 Noise AnalysisIn order to compute the minimum resolvable angular rate δΩn, it is necessaryto first determine the amplitude noise level of the system, as well as the effectof the laser linewidth (phase noise) on the resonator spectral response.Amplitude NoiseThe amplitude noises that were taken into account in the present section arethe shot noise, thermal noise and laser noise [141]. Amplitude noise effectsrelated to Rayleigh backscattering [27, 67, 90, 92, 124] were neglected in thissection, as spectral modulation techniques [30, 62, 63, 82] allow for reducingbackscattering noise and achieve shot-noise limited resolution. Based on312.3. Analytical and Numerical Modellingthe formulae for the aforementioned noise components (e.g., see [141]) andconsidering them statistically independent, the standard deviation of thephotocurrent (i.e., the noise rms current) is:δi =√(2qiD +4kBTRL+ i2DRIN)B, (2.40)where q = 1.6 × 10−19 C is the fundamental electric charge, iD = RPinis the maximum photodiode current, Pin is the maximum power incidentonto the photodetector, R = qηhν , in units of A/W , is the responsivity of thephotodiode, h = 6.626 × 10−34 J · s is Planck’s constant, ν is the opticalwave frequency in Hz, η is the quantum efficiency, B is its bandwidth in Hz,kB = 1.38×10−23 J/K is Boltzman’s constant, RL denotes the photodetectorload resistance, and RIN is the relative intensity noise of the laser, usuallyexpressed in dB/Hz [141].Fig. 2.9 shows the different noise components, the total noise and theSNR as a function of the power incident upon the photodetector, for the casedepicted in Table 2.3. For Pin = 0 dBm and RIN=−145 dB/Hz (consistentwith our Agilent 81682A tuneable laser), the system is theoretically laser-noise limited, with SNR≈ 67 dB for a 10-Hz bandwidth. According to oursimulations, a RIN≈ −160 dB/Hz is required to achieve shot-noise limit.This can be achieved using low-RIN laser sources [16, 47]. The remainderof this Section and Section 2.4 show resolution estimations for a shot-noiselimited case, as originally published in [54]. Subsection 2.4.5 shows theimpact of SNR degradation on the achievable resolution.Phase NoiseDue to phase fluctuations associated with spontaneous emission and carrierdensity fluctuations [24], laser sources exhibit a (normalized) Lorentzianoptical power spectral density of the form:P(ν) =2∆ω(∆ω)2 + (ω − ω0)2, (2.41)322.3. Analytical and Numerical Modellingwhere ω = 2piν is the instantaneous angular frequency, ω0 = 2piν0 is thecentral angular frequency, ∆ω = 2pi∆ν, and ∆ν is the full linewidth at halfmaximum [24, 97], expressed in Hz. The electric field in the time domain is:Total SNRPin /Pn shotPin /Pn thermalPin /Pn laserFigure 2.9: Signal-to-noise ratio (SNR) of various noise components (thermalnoise, laser noise, and shot noise) and total SNR as functions of input power,for a photodetector with the parameters shown in Table 2.3 (see legend forproper identification).Ein(t) = E0ejω0t+ϑ(t), (2.42)where ϑ(t) is the stochastic phase variation, and E0 is the electric fieldmodulus. It is possible to characterize the behaviour of this field using itsnormalized autocorrelation function [99]:CEE(TR) = E(t)E (t+ TR)∗/|E(t)|2, (2.43)where the upper bar denotes time averaging, the asterisk (∗) represents thecomplex conjugate, and TR is the correlation time shift. As thoroughly332.3. Analytical and Numerical ModellingTable 2.3: List of Simulation ParametersParameter Symbol Value UnitsGeneral parametersLight speed in vacuum c 3× 108 m/sTemperature T 298.15 KPhotodetectorInput power Pin −100 to 20 dBmResponsivity R 0.85 A/WQuantum efficiency η = Rhcqλ0 0.6813Integration time τPD 50 msBandwidth BHz = 12τPD 10 HzThermal noise power PnTh 2.182 fWNoise Figure F 1 (ideal)Dark current Idark 0 (ideal) mALaserFree space wavelength λ0 1.55 µmRelative Intensity Noise RIN −145 dBOutput power Pin −100 to 20 dBmdescribed in [96, 97], taking into account that the time average of the au-tocorrelation function and the normalized power density spectrum form aFourier transform pair, and considering the phase noise ϑ(t) to be an er-godic random process, it is proven that the variance of the phase fluctuationis proportional to the spectral width, and it is possible to obtain the twofollowing equations:CEE(TR) = ejω0TRe−2pi∆ν|TR|, (2.44)ejϑ(t−TR)e−jϑ(t) = e−2pi∆νTR . (2.45)Recomputing the fields at the through- and drop-port of the resonatorsshown in Fig. 2.1, now in the form of the time average of infinite summa-tions of time-dependent round-trip field components, after proper algebraicmanipulation and using Eq. (2.45) under the assumption of phase noise342.3. Analytical and Numerical Modellingergodicity [96, 97, 99], we have, neglecting ILs:|S21|2 =t2a − 2Ψcos(φ) + Ψ2(k2a + 1) +(κ2atbτ)2GD[1− 2Ψcos(φ) + Ψ2], (2.46)|S41|2 =κ2aκ2bτD[G1− 2Ψcos(φ) + Ψ2](2.47)where the detuning φ is now expressed in the time domain, as φ = ω0TR ±L2Ωcλ0, TR is the resonator roundtrip time, Ψ = Υe−2pi∆νTR , D = 1−Υ2 andG = 1 − Ψ2. Notice the intentional use of TR for both the roundtrip timeand the phase noise autocorrelation timeshift in Eqs. (2.43) through (2.45).The spectral slopes are then:∂|S21|2∂φ=−2Ψ[(tbτ)2κ4aG+D(κ2aΨ2 + t2a − 1)]sin(φ)D [1 + Ψ2 − 2Ψ cos(φ)]2(2.48)∂|S41|2∂φ=−2κ2aκ2bτΨG sin(φ)D [1 + Ψ2 − 2Ψ cos(φ)]2. (2.49)In the ideal case (∆ν → 0 Hz), Ψ→ Υ, G→ D, and Eqs. (2.46) through(2.49) become their phase-noiseless counterparts, obtained in Sections 2.3.4and 2.3.5. Ψ decays exponentially as the product ∆ν · TR grows, and thusthe effect of the linewidth can be regarded as a flattening of the spectrum,as exemplified in Fig. 2.10. Notice that the difference between the spectraand slopes for ∆ν = 0 Hz (ideal) and ∆ν = 100 kHz (nominal linewidthfor an Agilent 81682A tuneable laser) is negligible for the selected length of∼ 2.8 m, while the maximum slope for ∆ν = 2.5 MHz is only ∼ 50% of theideal value. Since the argument of the exponential includes both ∆ν andTR, the longer the resonator, the smaller the linewidth must be. This is whyinterferometric fiber gyroscopes with several meters in length require laserswith sharp linewidths. The narrow linewidth requirement is less stringent asthe resonator length decreases, but so does the Sagnac effect in a quadraticlaw with length, and thus the resolution can be severely hindered for too352.3. Analytical and Numerical Modellingsmall a length.-1.0 -0.5 0.0 0.5 1.0-0.10.00.10.20.30.4fpÈS 212 ,¶S 212¶f, Dn=2.5 MHz¶ S21 2/¶f, Dn=100 kHzS21 2, Dn=2.5 MHzS21 2, Dn=0 HzS21 2, Dn¶ S21 2/¶f =0 Hz¶ S21 2/¶f, Dn=100 kHzFigure 2.10: Normalized spectrum and its first derivative as a functionof normalized detuning, all-pass configuration, for αdB = 0.06 dB/cm,Lopt21OC = 2.78 m, and κa = 0.805, for three different linewidth values.Notice how the curves for ∆ν = 0 Hz and 100 kHz are practically identical.Fig. 2.11 shows the normalized resolution (considering also amplitudenoise, section 2.3.6) as a function of normalized length for the aforemen-tioned linewidth examples, for Locally-Optimized, under-Coupled rings (LOC,see Section 2.4.1). As predicted by the spectra, the resolutions for ∆ν = 0 Hz(ideal) and ∆ν = 100 kHz are practically the same. Only for Ln > 2.5 thedifference starts to be noticeable. In contrast, for ∆ν = 2.5 MHz the dete-rioration is evident at all lengths, and worsens remarkably for Ln > 1. Thedependence of the phase noise on the round-trip time makes the phase noisemore deleterious for smaller propagation losses, which, as will be explainedin Section 2.4, allow for larger values of Lopt.The present work focuses on chips with Lmax chip ≈ 114 mm, twentytimes smaller than the length considered for Figs. 2.10 and 2.11. This362.3. Analytical and Numerical Modelling0.2 0.5 1.0 2.0 5.0 101100104106108LLoptÈdW normDn=0Hz, All−pass, LOCDn=0Hz, Drop, LOCDn=100 kHz, All−pass, LOCDn=100 kHz, Drop, LOCDn=2.5 MHz, All−pass, LOCDn=2.5 MHz, Drop, LOCFigure 2.11: Normalized resolution as a function of normalized length for all-pass (solid) and drop (dashed) configurations, using the parameters shownin Table 2.3, with the laser linewidth, ∆ν, as a parameter.means that the linewidth ∆ν = 100 kHz of our chosen tuneable laser source(Agilent 81682A) will produce negligible deterioration. Hence, from now onphase noise is neglected unless stated otherwise.2.3.7 Resonator Gyroscope Resolution EstimationsFrom Eqs. (2.24) and (2.25), assuming lossless coupling, the photocurrentsfor the through- and the drop-port are, respectively:i21 = iD|S21|2 = iDt2a − 2Υcos(φ) + t2bτ2CIL21 [1− 2Υ cos(φ) + Υ2](2.50)i41 = iD|S41|2 = iDκ2aκ2bτCIL41 [1− 2Υ cos(φ) + Υ2], (2.51)where CIL21 and CIL41 represent the ILs of the through and drop port con-figurations, which are considered different due to the use of circulators orY-branches for the former. When the intensity of the light source fluctuatesdue to noise, this cannot be distinguished from a variation in intensity due372.4. Resonator Parameter Optimizationto rotation. In order to estimate the minimum detectable angular rate, theuncertainty in light intensity (or in photocurrent) has to be translated to anuncertainty in the phase shift δφ = δim where δi is the noise rms current andm is the slope of the current-phase curves, given by m = iDCIL21∂|S21|2∂φ andm = iDCIL41∂|S41|2∂φ for the through and the drop ports, respectively.Assuming a constant resonator length and for simplicity, an invariantwavelength, for each of the counter-propagating beams we have:δφ =L2cλ∆Ω⇒ δΩ =cλL2δφ =cλL2δim(2.52)Using both counter-propagating beams, the resolution for the through portis:|δΩ21| =cλ2L2CIL21δiiD∂|S21|2∂φ, (2.53)and for the drop port is:|δΩ41| =cλ2L2CIL41δiiD∂|S41|2∂φ. (2.54)2.4 Resonator Parameter Optimization2.4.1 Local and Global OptimizationFor any given propagation loss value α, the values of L, φ, κb and κa canbe optimized for maximizing the products ∂|S21|2∂φ L2 and ∂|S41|2∂φ L2, thus ob-taining the global minima of Eqs. (2.53) and (2.54), respectively. On theother hand, CILδiiD depends on the input power and the ILs. Fig. 2.12 showsthe effect of ILs on the resolution. The optimum parameters remain thesame, but the increase of CIL produces an exponential deterioration of theresolution.The ILs depend on the light injection/probing methods. In the presentwork, an intrinsic IL of 3 dB will be considered, consistent with low-ILgrating couplers. This is a relatively low, and yet conservative theoretical382.4. Resonator Parameter Optimization0 2 4 6 8 10 12 1410-40.0010.010.1110100ILdB @dBDÈdWÈ@degsD α =0.06 dB/cm, OCdBα =3 dB/cm, OCdBFigure 2.12: Optimum angular rate resolution as a function of the insertionloss, ILdB, for optimized all-pass (solid) and drop (dashed) configurations,for two different values of propagation losses.level, considering that the lowest IL reported are in the order of 1 dB, forapodized grating couplers [87]. The use of Y-branches as an in-chip al-ternative to circulators increases the insertion loss by ILY ≈ 4 dB [105].Optical PM fibre 1550-nm circulators offer smaller ILs than Y-branches,(e.g., ILcirc = 1.5 dB, AC Photonics PMOC315P), but have the disadvan-tage of not being integrated on the chip. Therefore, the total ILs for theall-pass and the drop-port cases will be considered to be ILT = 7 dB andILT = 3 dB, respectively, unless explicitly stated otherwise.Fig. 2.13 shows the resolution |δΩ|, in deg/s, as a function of L for threepropagation loss values, assuming the noise parameters of Table 2.3 andPin = 0 dBm, for all-pass and drop-port configurations. For each αdB andeach port configuration, the optimum values Lopt, φopt, κaopt and κbopt werenumerically found to obtain the global minimum of |δΩ|, and then L wasvaried, while the other parameters remained constant. Despite their greaterIL levels, all-pass resonators perform better than drop-port resonators.It is possible to observe that Lopt is larger for smaller propagation losses.For every value of α, |δΩ21|min < |δΩ41|min, by approximately one order of392.4. Resonator Parameter Optimization0.001 0.01 0.1 1 100.0010.1101000L @mDÈdWÈ@degsDAll-pass, α =0.06 dB/cmdBDrop, α =0.06 dB/cmdBAll-pass, α =0.12 dB/cmdBDrop, α =0.12 dB/cmdBAll-pass, α =3 dB/cmdBDrop, α =3 dB/cmdBFigure 2.13: Angular rate resolution as a function of resonator length for theall-pass (solid) and drop port (dashed) configurations of racetrack resonatorgyroscopes, for three different values of propagation losses. In all cases, theparameters φ and κ are fixed at their global optimum values.magnitude. In all cases, for L > Lopt the resolution deteriorates veryrapidly as L increases (notice the graphic has a logarithmic scale and yetthe plot describes an exponential curve). This is explained by the fact thatthe Sagnac effect is proportional to the area of the resonator, (i. e. ∝ L2),whereas the propagation losses increase exponentially with the length, (i.e.,∝ e2 ΣαiLi). As previously mentioned, the deterioration of the resolutionfor L < LoptCC is due to the decrease of the Sagnac effect phase shift forsmaller resonator lengths.Fig. 2.13 properly depicts the abrupt deterioration of the resolution forlengths beyond each optimum length Lopt. However, the Figure can bemisleading if one observes the sensitivities for small lengths: For each prop-agation loss value, the global optimum values of φ, κa, and κb are foundand remain constant as L is varied. Therefore, at short enough lengths for402.4. Resonator Parameter Optimization10−4 10−3 10−2 10−1 10010−2100102104L [m]|δΩ|21 [°/s]  |δΩ|21, LOC|δΩ|21 GOC(a) All-pass resolution10−4 10−3 10−2 10−1 10010−2100102104L [m]|δΩ|41 [°/s]  |δΩ|41, LOC|δΩ|41 GOC(b) Drop-port resolution10−4 10−3 10−2 10−1 10000.10.20.30.40.50.60.70.80.91L [m]κa21  κa21, LOCκa21, GOC(c) All-pass couplings10−4 10−3 10−2 10−1 10000.10.20.30.40.50.60.70.80.91L [m]κa41; κb41  κa41, LOCκb41, LOCκa41, GOCκb41, GOC(d) Drop-port couplings10−4 10−3 10−2 10−1 10000.050.10.150.20.250.30.350.40.450.5L [m]φ / π  φ21/π, LOCφ21/π, GOC(e) All-pass normalized detuning10−4 10−3 10−2 10−1 10000.050.10.150.20.250.30.350.40.450.5L [m]φ / π  φ41/π, LOCφ41/π, GOC(f) Drop-port normalized detuningFigure 2.14: (a, b) Resolution |δΩ|, (c, d) optimized couplings κa andκb, and (e, f) optimized normalized detuning φn = φ/pi as functions ofresonator length L, for various port and optimization conditions. LOC:Locally-optimized coupling (computed at each value of L). GOC: Globally-optimized coupling. In all cases, αdB = 0.06 dB/cm.412.4. Resonator Parameter Optimizationeach value of α, the unvaried values of coupling and detuning (not optimalat such a length) allow for better sensitivities in the drop-port configura-tion instead of the all-pass configuration, incorrectly suggesting that highervalues of α allow for better resolutions at short lengths. This is due to thefact that for each propagation loss value, the globally optimum values of φ,κb, and κa were kept constant, and thus they yield resolution minima onlyat each Lopt. In order to achieve the best possible resolution for particu-lar values of propagation loss α and arbitrary length L, it is necessary tomaximize the products L2 ∂|S21|2∂φ and L2 ∂|S41|2∂φ , for the all-pass and the dropconfigurations, respectively, in terms of the variables φ, κa, and κb.Figs. 2.14(a) and 2.14(b) show the resolution for the all-pass (solid) andthe drop-port (dashed) configurations, with αdB = 0.06 dB/cm, for twodifferent cases: 1) OC, local optimization (LOC) of φ, κa and κb at eachvalue of L (bottom line); and 2) OC, with φ, κb and thus κa fixed at theirglobal optimum values (GOC, top, black line). The latter are the same blackcurves shown in Fig. 2.13. For the sake of image clarity, only the plots forαdB = 0.06 dB/cm are shown, but the trend is similar for different prop-agation loss values. For either configuration, LOC rings offer considerablybetter resolutions than their globally optimized counterparts.Figs. 2.14(c) and 2.14(d) show respectively the all-pass and the drop-port coupling coefficients κa and κb for each ring configuration. For theall-pass cases, (Fig. 2.14(c)), κb = 0 so it has been omitted in the plots.From Fig. 2.14(d) it is possible to conclude that the LOC for the drop portconfiguration is symmetric. Figs. 2.14(e) and 2.14(f) show the normalizeddetuning for each case, for the all-pass and the drop-port configurations,respectively. The optimum values of φ for the drop port are always largerthan their through-port counterparts for a given length, and in all cases,φ→ pi2 as L→∞.Table 2.4 summarizes the global optimum (GOC) values of resolution,resonator length, coupling coefficients and normalized detuning for propa-gation loss values of αdB = 0.06 dB/cm, 0.12 dB/cm, and 3 dB/cm, forall-pass and drop-port configurations. The GOC values of κa,b and φ arethe same for all values of α, and the product αLopt is a constant for the422.4. Resonator Parameter Optimizationsame port and coupling conditions. As can be observed in such a table, theall-pass configuration always offers the best resolution.Table 2.4: Global optimum parameters and resolutions for different portconfigurations and lossesXXXXXXXXXXXParameterαdB 0.06 dB/cm 0.12 dB/cm 3.0 dB/cmα[m−1] 0.69 1.382 34.5All-pass, GOC, ILT = 7 dB|δΩ| [deg/s] 2.1× 10−3 8.42× 10−3 5.26Lopt [m] 2.77 1.386 55.5× 10−3κa opt 0.805 0.805 0.805φopt/pi 0.393 0.393 0.393αLopt 1.915 1.915 1.915Drop port, GOC, ILD = 3 dB|δΩ| [deg/s] 13× 10−3 52.3× 10−3 32.6Lopt [m] 1.263 0.63 25.3× 10−3κa opt 0.779 0.779 0.779κb opt 0.779 0.779 0.779φopt/pi 0.336 0.336 0.336αLopt 0.871 0.871 0.871Fig. 2.15 compares the resolutions for LOC all-pass and drop-port res-onators as a function of the normalized length Ln = L/Lopt, for the threevalues of α in Table 2.4. LOC all-pass rings offer the best resolution for anyparticular propagation loss value, by approximately one order of magnitudeat all lengths, in comparison to the drop-port LOC configuration.The deterioration of the resolution due to length variations is less tolerantfor larger propagation losses, as the values of Lopt, shown in Fig. 2.16, andthe values of |δΩ|opt, shown in Fig. 2.17, are considerably smaller and larger,respectively.Materials with smaller propagation losses can achieve considerably bettersensitivities: as shown in Fig. 2.16, if the average propagation loss decreasesbelow 1 dB/cm, there is an abrupt enhancement of both the resolution and432.4. Resonator Parameter Optimization0.1 0.2 0.5 1.0 2.0 5.0 10.00.1101000105LnÈdWÈ@degsD α =3 dB/cmdBα =0.12 dB/cmdBα =0.06 dB/cmdBFigure 2.15: Resolution for LOC resonators as a function of normal-ized length, Ln, for all-pass (solid) and drop port (dashed) rings, forαdB1 = 0.06 dB/cm, αdB2 = 0.12 dB/cm and αdB3 = 3 dB/cm. Loptis different for each value of α and port configuration, as shown in Fig. 2.16.0.5 1.0 1.5 2.0 2.5 3.00.010.050.100.501.005.0010.00L opt[m]All-pass, LOCDrop, LOCαdB [dB/cm]Figure 2.16: Optimum resonator length Lopt as a function of average waveg-uide propagation loss αdB for the through (solid) and the drop (dashed) portsof LOC ring resonators. The value of Lopt is IL-independent.442.4. Resonator Parameter Optimization0.5 1.0 1.5 2.0 2.5 3.010-40.0010.010.1110100adB @dBcmDdÈWÈ@degsDAll-pass, LOCDrop, LOCFigure 2.17: Optimum resolution as a function of average waveguide prop-agation loss αdB [dB/cm] for all-pass (solid) and drop (dashed) LOC ringresonators.the optimum length value, as can be concluded from the exponential shapeof the plots in both figures for αdB < 1 dB/cm, in spite of the fact thatboth plots are already in a log-log scale.2.4.2 Chip-Sized vs. Globally-Optimized GyroscopesBased on the constant α ·L products for GOC (see Table 2.4) and the max-imum on-chip resonator length, Lmax chip = 114 mm, the largest permissibleaverage propagation losses for the all-pass and drop-port configurations areαdB maxAP = 1.46 dB/cm and αdB max41 = 0.66 dB/cm, respectively. Ta-ble 2.5 compares the resolutions of chip-sized ring gyroscopes for differentvalues of αdB, and Table 2.6 shows the optimum resolutions and lengths forthe same values of αdB.2.4.3 Target ApplicationsTable 2.7, adapted from [143] following the formulae derived in [139], sum-marizes the resolution requirements for three existing standard gyroscope452.4. Resonator Parameter OptimizationTable 2.5: Resolutions for LOC resonator gyroscopes with L = Lmax chip``````````````αdB [dB/cm]|δΩ| [deg/s] All-pass Drop0.06 0.0234 0.06190.12 0.047 0.1280.66 (αdB max41) 0.314 1.601 0.595 5.371.2 0.844 10.561.46 (αdB maxAP) 1.25 25.13 16.91 4133Table 2.6: Optimum resolutions and lengths for GOC resonator gyroscopesPPPPPPPPαdBParam. All-pass Drop|δΩ| [◦/s] Lopt [m] |δΩ| [◦/s] Lopt [m]0.06 dB/cm 0.0021 2.78 0.013 1.2620.12 dB/cm 0.0084 1.386 0.052 0.630.66 dB/cm 0.258 0.251 1.60 0.1141 dB/cm 0.585 0.166 3.63 0.0751.2 dB/cm 0.843 0.139 5.22 0.0631.46 dB/cm 1.25 0.114 7.72 0.05183 dB/cm 5.26 0.0555 32.6 0.025grades. It is possible to conclude that if αdB is in the range of ∼ 0.7 to2 dB/cm, resonators with lengths in the range of Lmax chip would achieveresolutions appropriate for tactical- and rate-grade applications.Table 2.8 compares the performance of commercially available MEMSgyroscopes from Analog Devices (A.D., [37]), Melexis [2], and ST Micro-electronics (STM, [117]) against the theoretical performance of LOC opticalresonator gyroscopes.One can see that the propagation losses are the critical parameter that,if properly dealt with, allow optical gyroscopes to offer similar resolutionsto those of commercially available devices for rate- and tactical-grade appli-462.4. Resonator Parameter OptimizationTable 2.7: Resolution requirements for different classes of gyroscopesRate grade Tactical grade Inertial grade|δΩ| [◦/h] > 300 30 to 300 < 1Table 2.8: Comparison with commercially available gyroscopesDevice |δΩ| [deg/s] GradeA.D. ADXRS450 0.14 RateSTM L3G3250A 0.18 RateMelexis MLX90609 0.30 RateAP ring, 1 dB/cm,Lmax chip0.595 RateAP ring, 0.06 dB/cm,Lmax chip0.0234 Tacticalcations. If properly integrated with the light source and readout circuitry,these optical gyroscopes would then have the advantage of a longer lifetimethanks to the lack of moving parts. As previously stated, the use of a rect-angular reticle will worsen by ∼ 22% the value of the resolution, but thisdoes not affect the target application grades for these devices.2.4.4 Design RobustnessBased on the model developed so far, it is possible to vary some of the pa-rameters and observe their impact on the resolution. In this way, parameterspread ranges can be defined, to assess the effect of possible variations intuning and fabrication conditions. Thus, the 3-dB bandwidths ∆L, ∆φ and∆κ will define ranges for such parameters around their optimum values,within which the resolution deteriorates by a factor of two.Fig. 2.18 shows a plot of the normalized resolution, |δΩnorm| = |δΩ/δΩopt|,as a function of the normalized length Ln = L/Lopt for the all-pass (solid)and drop-port (dashed) configurations of LOC resonators. Due to normal-472.4. Resonator Parameter Optimization0.01 0.05 0.10 0.50 1.0 5.0 10.01101001000104105106LLopt|dW|   normAll-pass, LOCDrop, LOCFigure 2.18: Normalized resolution |δΩ|norm versus normalized lengthLn = L/Lopt for all-pass (solid) and drop-port (dashed) LOC resonators.Due to normalization, all plots coincide for all values of α and are IL-independent.ization, these plots are valid for all values of α, and they are independent ofthe value of IL. The 3-dB-cutoff normalized lengths for through- and drop-port LOC resonators are shown in Table 2.9. These normalized cutoff valuesare all the same for all values of α. However, smaller propagation losses willallow for more flexible design constraints, as Lopt increases considerably fordecreasing values of α, as shown in Fig. 2.16.Fig. 2.19 shows the optimum values of the coupling coefficient, κ, as afunction of the normalized length Ln for all-pass (solid) and drop (dashed)LOC resonators. It also shows contour plots of the values of κ at whichthe resolution is two times (3-dB) and four times (6-dB) worse than theoptimum resolution for each port.482.4. Resonator Parameter OptimizationTable 2.9: 3-deciBel cut-off normalized lengths and length bandwidth forLOC resonatorsXXXXXXXXXXXParameterPortAll-pass DropLn low 0.30505 0.2869Ln high 2.1502 2.2719∆Ln 1.84515 1.98510−1 1 1000.10.20.30.40.50.60.70.80.91Ln=L/Lopt k  ka, All−pass, LOCka kb, Drop, LOC-6 dBOptimaGlobal-3 dBFigure 2.19: Global optimum values (stars), locally-optimized (LOC, solid),3-dB (dashed), and 6-dB (dash-dotted) contour plots for the coupling coef-ficients of all-pass (κa) and drop-port (κa = κb) resonators, as a function ofthe normalized length Ln = L/Lopt. Due to normalization, all plots coincidefor all values of α and are IL-independent.In all cases, φ is locally optimized, to emulate the the fact that in anyexperiment, the user would still tune the wavelength of the light source inthe best possible way, despite the imperfections of the fabricated device.492.4. Resonator Parameter Optimization10−1 1 1000.10.20.30.40.50.60.70.80.91Ln=L/L  p  pf/p-6 dBOptimaGlobal-3 dBFigure 2.20: Global optimum values (stars), locally-optimized (LOC, solid),3-dB (dashed), and 6-dB (dash-dotted) contour plots for φ/pi, as a func-tion of the normalized length Ln = L/Lopt, for all-pass and drop-port res-onators. Due to normalization, all plots coincide for all α values, and areIL-independent.Both κa = κb are detuned simultaneously for the drop port, as in reality,fabrication errors and tolerances affect both coupling regions.Fig. 2.20 shows the optimum values of the normalized detuning, φn = φ/pi,as well as 3- and 6-dB contour plots for φn versus the normalized length Ln.Once again, the largest tolerance occurs for values in the vicinity of Ln = 1.As expected, these contour plots are independent of α and IL. It is possibleto notice that in all cases, the detuning bandwidth favours the region towardssmaller lengths, consistent with the smaller deterioration rate for Ln < 1shown in Fig. 2.18. The farther from the optimum point, the smaller therange within which the coupling coefficient and detuning can vary.Fig. 2.21 shows the resolution as a function of propagation loss for anall-pass (solid) and a drop-port (dashed) ring gyroscope, each optimized for502.4. Resonator Parameter Optimization0.5 1.0 1.5 2.0 2.5 3.00.10.51.05.010.050.0100.0ΑdB￿dB￿￿∆￿￿￿deg￿s￿Figure 2.21: Resolution vs. αdB for an all-pass (solid) and a drop-port(dashed) resonator gyroscope optimally designed for αdB = 1 dB/cm.0.5 1.0 1.5 2.0 2.5 3.00.150.200.250.300.350.400.450.50ΑdB￿dB￿Φ￿ΠFigure 2.22: Optimized detuning as a function of αdB for an all-pass(solid) and a drop (dashed) resonator gyroscope optimally designed forαdB = 1 dB/cm.512.4. Resonator Parameter Optimizationa propagation loss αdB = 1 dB/cm. In each case, L = Lopt and κa,b = κa,bopt ,since after fabrication, only the detuning can be optimized, as shown in Fig.2.22. As a result, it is noticeable the greater deterioration for increasingpropagation losses in comparison to the trends shown in Fig. 2.17, and eventhough the resolution is better for decreasing propagation losses, there is avalue (αdB ≈ 0.1) below which the sensitivity actually starts to deterioratefor the all-pass configuration, due to the fixed length and couplings.2.4.5 Predictions with Experimental SNRsIn the previous sections we have assumed a close-to-ideal SNR. Here, welook at the impact of lower SNR values on the gyro resolution. For the all-pass resonator case, by substituting the expression CIL21iD/δi in equation(2.53) with the signal to noise ratio, SNR = 10SNRdB/10, we have:|δΩ21| =cλ2L2 ∂|S21|2∂φ1SNR, (2.55)Figure 2.23 shows the best theoretically achievable resolution as a func-tion of SNR, for two resonators of identical length, L = 114 mm, but differ-ent propagation losses. The resonator geometry used for this modelling isconsistent with the design described in Section 3.4, with 200 µm-long lineartapers for SMWG to MMWG conversion. For both resonators, the lengthis Lmax chip = 114 mm, with an SMWG length fraction of 0.1%. The prop-agation losses for rib and strip SMWG are considered to be 1.4 dB/cm [15]and 2 dB/cm [78], respectively. The rib MMWG propagation losses usedfor the first resonator (dashed curve) are 0.026 dB/cm [78], whereas, for thesecond resonator (solid curve), a value of 0.085 dB/cm is used, based onour experimental results [56]. In practice, the SNR can be estimated basedon time-domain experimental data, by calculating the ratio of the meanphotodetector power divided by its standard deviation, as shown in Section3.4.3.Phase modulation techniques, primarily intended for frequency tracking,should be used in order to reduce undesirable effects produced by backscat-522.4. Resonator Parameter Optimization10 15 20 25 30 35101001000104105SNRdB@dBDÈ∆WÈ@degsDFigure 2.23: Angular speed resolution as a function of SNR for two all-pass,large-area resonators, both with 114 mm in length, and MMWG propagationlosses of 0.085 dB/cm (solid curve) and 0.026 dB/cm (dashed curve).tering. Otherwise, the contribution of backscattering to the overall res-olution error can be significant, if not dominant [27]. Frequency trackingcan be achieved by acusto-optic modulation [30, 45, 46] or phase modulation[62, 118]. Since silicon photonic waveguides have a smaller cross-section thanlow-contrast waveguides and optical fibres, their backscattering level is con-siderably greater. This is exacerbated for small cross-section SMWG wires,as surface imperfections are the source of propagation loss and backscatter-ing, contributing to backscattering levels in the order of −30 dB, accordingto the literature [92].In the specific case of SOI devices interrogated via grating couplers, de-pending on the grating coupler designs one can expect backreflections at thefibre-coupler interface ranging from −30 dB to −16 dB [136, 138]. Thesebackreflections, if not phase-modulated, will produce undesirable interfer-ence with the light propagating in the same direction. Also, due to thesmaller size of the SOI chips, the roundtrip time is much shorter than thatof optical fibre gyros, and despite their high Q factors, the frequency shiftsrequired for appropriate frequency tracking (in the order of hundreds ofMHz) pose challenges for large ramp signal generation.532.5. Phase Modulation Requirements2.5 Phase Modulation RequirementsThe reason for requiring frequency tracking techniques for enhancing sensi-tivity is the minute order of magnitude of the wavelength shift due to theSagnac effect, which is, in general, too small a change to track based solelyon amplitude measurements. From equation (2.19), at resonance we have:φres = 2Mpi =2pineff(λ0)Lλ0±L2Ωcλ0, (2.56)where the integer M denotes the optical resonator mode. At rest, this num-ber can be obtained as:M =neff(λ0)Lλ0. (2.57)For an arbitrary angular speed, the resonance wavelength for mode numberM will undergo a shift, depending on the propagation direction of the light,given by:λ0 new = λ0 ±L2Ωneff2picMng. (2.58)The second term in equation (2.58) is a very small quantity, especially forsmall footprint resonators. As an example, let us consider a resonator witha length L = 37.7 mm, λ0 = 1550.2 nm, neff(λ0) ≈ 2.83, and ng(λ0) ≈ 4,consistent with a 12-mm diameter resonator design in an SOI chip. Then,M=68833 (rounded to the nearest integer). For an angular speed of 1 dps,the resonance wavelength shift for each beam is δλ0 ≈ 1.353 × 10−19 m.The total resonance wavelength difference is thus ∆λ0 ≈ 2.7 × 10−7 pm,equivalent to a frequency difference ∆f = | − c∆λλ20| ≈ 34 Hz. This resultis consistent with the expression in the literature for the total frequencydifference due to the Sagnac effect [46]:∆f =4AΩλP, (2.59)where A denotes the area within the resonator, and P its perimeter. Thefrequency shift can be considered to vary linearly with angular speed. Fora shot-noise limited system, the resolution for an optical gyroscope is given542.5. Phase Modulation Requirementsby [22, 45, 107]:δΩ =picQL√2cBhCILληPin(2.60)where Pin is the optical input power and CIL represents the insertion losses.As previously mentioned, if shaped as a rectangle rather than a circle,the resonator aspect ratio will reduce the enclosed area, thus affecting thegyroscope resolution. Considering these small frequency shifts, relying ex-clusively on amplitude techniques can render the device insensitive to smallangular speeds, especially considering waveguides with high backscatteringlevels. Backscattering in optical fibres and waveguides has been studied forseveral years [10, 76, 89, 90, 92, 98], and is an important if not dominanterror source in passive resonator-based optical gyroscopes [67, 71, 82, 124].In order to reduce its effects and achieve shot-noise limited performance,fibre optic gyroscopes typically use frequency-domain techniques for track-ing the resonance of each counterpropagating light beam [63, 82, 120, 121],and read out the angular speed of the device based on the frequency differ-ence between these two signals. The first techniques used for this purpose,based on frequency shifting using acusto-optic modulators (AOMs), wereimplemented in optical fiber gyroscopes with spool lengths of the order ofhundreds of meters [31], and in fibre resonators with lengths of the order of1 meter [46]. However, due to inherent AOMs disadvantages such as bulk-iness and unintended intensity modulation, integrated-optics phase modu-lators have been preferred for frequency tracking and feedback techniques[62, 118]. The phase modulation implementation known as the serrodynemodulation technique [62] consists of creating an alternating frequency shiftin the optical signal by driving the phase modulators with ramp functionsof constant amplitude and two alternating frequencies.Figures 2.24 and 2.25 illustrate the principle of operation of serrodynephase modulation for a passive resonator optical gyroscope, described as wellin [62, 118]. As shown, we assume that the optical gyroscope shown in Figure2.24 rotates in CCW direction. The optical signal of a tuneable laser sourceis equally split at the 50/50 polarization maintaining beam splitter. Each552.5. Phase Modulation Requirementsbeam is then phase modulated by separate phase modulators, each fed withfrequency-modulated ramp signals of constant amplitude Vp = 2Vpi, whereVpi is the half-wave voltage of the phase modulators. As previously explained,the corrotating (CCW) light, detected in Photodetector 2, experiences aneffective elongation of the resonator due to the Sagnac effect, so it is retardedin phase. Its resonance wavelength is increased, or equivalently, its resonancefrequency is decreased by an amount given by Eq. (2.59).The laser wavelength, λlaser should be initially tuned at a value slightlygreater than the resonance at rest, so that its frequency, flaser, is tunedbelow resonance. The lock-in amplifier LIA 1b is used to tune the laserfrequency to track the CCW resonance. Phase modulator 1 is driven byserrodyne ramps with alternating frequencies f1 and 2f1 for equal periods.The value of f1 depends on the finesse of the resonator, and is usuallyon the order of hundreds of MHz. The driving signal is switched betweenfrequencies f1 and 2f1 at the lower frequency fs1, which is on the order ofkHz. Therefore, the optical frequency switches symmetrically at a (slow)frequency fs1 around the optical frequency flaser + 32f1. If such a frequencydoes not exactly coincide with the CCW resonance, the photodetector PD2creates rectangular pulses of frequency fs1. This frequency is used as thereference frequency in the lock-in amplifier LIA 1b, which creates a feedbacksignal Vfb1 whenever the amplitudes at points B and C in Fig. 2.25 aredifferent. This feedback signal is used to adjust the laser frequency. Whenthe laser is perfectly tuned, the amplitude at both points B and C in figure2.25 is exactly the same, which implies that the photodetector PD2 producesa pure DC output, and thus V1b is zero. The lock-in amplifier LIA 1acorrects for imperfections in the serrodyne amplitude (i.e., deviations from2Vpi), which generate exponentially decaying pulses at the frequency of theserrodyne signal being generated [118], in this case, either f1 or 2f1.On the other branch, the counter-rotating (CW) beam, detected in pho-todetector PD1, experiences an effective shrinkage of the resonator lengthdue to the Sagnac effect, so it is enhanced in phase. As its resonance wave-length is shortened, its resonance frequency is increased. Thus, the phasemodulator PM2 must apply a frequency f2, slightly greater than f1, in order562.5. Phase Modulation Requirementsto track the CW resonances rising edge, depicted by point D in Fig. 2.25.The ramp frequency is switched between f2 and f2 + f1 at a slow frequencyfs2, for tracking the resonance in a similar way to the CCW case. If f2 isimproperly tuned, PD1 produces a rectangular wave output of frequencyfs2, which is also the reference frequency of the lock-in amplifier LIA 2b,thus producing a strong error signal V2b, used to adjust the value of f2 untilthe amplitudes of points D and E are the same. The lock-in amplifier LIA2a corrects for amplitude imperfections to achieve a voltage excursion ofexactly 2Vpi.572.5.PhaseModulationRequirements  Arbitrary Waveform Generator 2  Arbitrary Waveform Generator 1TuneableLaser Source50/50 Pol. Maint. Fibre SpliterPhase Modulator 2Phase Modulator 1t [s]Vawg1V2t [s]Vawg2V2P1c1P3c1P2c1P3c2Photo-detector 1Photo-detector 2LIA 2bLIA 1aLIA 1bV1bLIA 2aTurntableV1aV2bV2aFeedback for f2valueFeedback for V2amplitudeOn-boardopto-mechanicsMicroscopeP1c2P2c2Rotary platformOn-board opto-mechanicsFigure 2.24: Frequency tracking using serrodyne phase modulation.582.5. Phase Modulation RequirementsResonance at restCo-rotating (red shift)Counter-rotating(blue shift)AD EPoutCBfPointABCDEFrequencyflaserflaser+f1flaser+2f1flaser+f2flaser+f1+f2Figure 2.25: Resonance frequency tracking using modulation techniques.59Chapter 3Design ProcessDuring this work, cycles of device design, fabrication, characterization, andcritical evaluation were conducted iteratively, in order to identify problems,envision approaches to solve them, and derive conclusions that lead to im-provements for the next device generation. This iterative process also in-volved the creation and evolution of the experimental setups used to char-acterize the devices.Taking into consideration that UBC does not currently have a CMOS-compatible nanofabrication facility capable of creating SOI devices, all ourdesigns were submitted for fabrication as part of multi-project wafer (MPW)shuttles to various foundries around the world, such as the InteruniversityMicroelectronics Centre (IMEC), in Belgium, the Institute of Micro Elec-tronics (IME), in Singapore, and the Micro and Nanofabrication Facility atthe University of Washington (UW), in the USA. Each of these foundrieshas its own design specifications, rules, and fabrication technologies. Thus,the time from design to measurement and evaluation could range from a fewmonths to a year.This chapter will guide the reader through the iterative process, statingfor each cycle the initial objectives, the device designs, the device character-ization, the evaluation of the results, the difficulties encountered along theway, and the improvements planned for the next cycle.3.1 First Design CycleThe main objectives during this cycle were the creation of large resonatorsand test structures for determining insertion and propagation losses, as wellas the design and assembly of a characterization setup. The first devices603.1. First Design Cyclewere fabricated using an IMEC epixFab process, based on 193-nm deep ultra-violet (DUV) lithography, to fabricate air-clad strip waveguides with heightsof 220 nm, on SOI wafers with a 2 µm-thick buried oxide layer (BOX). TheSOI wafers were diced into chips arranged in a number of columns and rows,as depicted in Fig. 3.1. -3-4 -1-2 10 32 4Rows-4-3-2-101234ColumnsFigure 3.1: Wafer dicing schematic. Chips are identified according to theirposition in the row and column pattern.The UV illumination dose for each column was intentionally varied, toproduce waveguide width and gap variations for devices located in differentcolumns across the wafer. Chips from various columns were then shipped toeach participant in the MPW shuttle.3.1.1 Layout DesignIMEC provided the grating coupler (GC) design to interrogate the devices.Each GC required a 1 mm-long linear waveguide taper. Since in practiceoptimal coupling conditions would be difficult to achieve due to fabricationimperfections, a methodology of parameter variation was adopted for ourmask design. In order to vary the coupling to the various resonators studied,the gaps in the WDCs were varied between 150 nm and 450 nm, while theWDC lengths were fixed at Lc = 10 µm.613.1. First Design CycleThe multi-user nature of the MPW required the designs to be space-efficient and rectangular, rather than circular, in order to efficiently framedevices designed by other users. Figure 3.2 shows a panoramic view of thelarge resonator designs, and a zoom-in to a subset of them. Nested sets ofrings were created using single-mode (SM) strip waveguides. Their lengthsranged from 3 to 33 mm, with their aspect ratios dictated by the availablelayout space. To vary the amount of coupling to the resonators, the WDCgaps were varied as described above.For the sake of space efficiency, and due to the considerable length ofthe GC tapers, both the input and the output GC for each resonator wereoriented in the same direction, as shown in Fig. 3.2(b). However, thisproved to be a difficulty with regard to interrogation, as the dimensions ofthe fibre holders and positioning opto-mechanics prevented the fibres frombeing located on the same side of the chip in close proximity (see Fig. 3.5).(a)(b)Figure 3.2: (a) First device layout panoramic schematic and (b) Zoom-into top left nested rings. Text tags are only shown for illustration purposes.623.1. First Design CycleWaveguide ParametersTheoretical estimations of the effective and group indices for these waveg-uides were also necessary in order to estimate resonator parameters such asQ and FSR, as well as cross-over length values for the WDCs. The effectiveand group indices for different air-clad waveguide geometries were obtainedusing MODE SolutionsTM eigenmode solver. Figure 3.3(a) shows the TEmode profile for a 500-nm wide, 220-nm tall strip SMWG. The effective andgroup indices for this waveguide were obtained using five equally spacedwavelength values, and curve-fitted using a third-order polynomial model.Fig. 3.3(b) shows the resulting polynomial curve-fits for the effective andgroup indices. The values of the effective and group indices at λ0 = 1550 nmare neff = 2.3826 and ng = 4.3547, respectively.(a)1.5 1.52 1.54 1.56 1.58 1.6 1.6222.533.544.5neff(λ0)=2.3826ng(λ0)=4.3547Wavelength [µm]n  neff fitneff(λ0)ng fitng(λ0)(b)Figure 3.3: (a) TE mode profile for a 220-nm high, 500-nm wide, air-cladstrip SMWG. (b) Effective index (green) and group index (blue) curve-fits.The effective indices for two-waveguide systems with varying gaps wereobtained for discrete wavelength values and curve-fitted, as shown in 3.4(a)for 500 nm-wide strip WDC. Using these curve fits and equation (2.1, thecross-over lengths, L⊗, for WDCs with various gap values were obtained as afunction of wavelength, as shown in Fig. 3.4(b). Table 3.1 shows theoreticalvalues of L⊗ and κ for WDCs with constant length Lc = 10 µm and variousgap values, used in various resonator designs.633.1. First Design Cycle1.5 1.52 1.54 1.56 1.58 1.6 1.622.362.382.42.422.442.462.482.52.52Wavelength [µm]n eff  neff e1, gap=170 nmneff o1, gap=170 nmneff e1, gap=210 nmneff o1, gap=210 nmneff e1, gap=250 nmneff o1, gap=250 nmneff e1, gap=290 nmneff o1, gap=290 nm(a)1.5 1.52 1.54 1.56 1.58 1.6 1.622030405060708090100110Wavelength [µm]L ⊗ [µm]  gap=170 nmgap=210 nmgap=250 nmgap=290 nm(b)Figure 3.4: (a) Effective indices for the even (solid) and odd (dashed) modesfor WDCs made with 500-nm wide, strip SMWGs. (b) Corresponding WDCcross-over lengths as a function of wavelength.Table 3.1: Parameters for WDCs made with 500-nm wide strip SMWGs,with Lc = 10 µm, at λ0 = 1550 nm.Gap [nm] L⊗ κ(Lc, λ0)150 32.5 0.465200 51.52 0.3250 81.4 0.192300 128.52 0.122350 202.88 0.077In order to create the resonator corners, 90◦ waveguide bends with con-stant radii were used. Since bend losses are greater for smaller radii, all 90◦bends were designed with a constant radius of 20 µm, in order to ensurenegligible bending losses (expected to be below 0.01 dB [13]). Table 3.2summarizes the results of eigenmode solver simulations to assess the theo-retical coupling between the modes of straight and bent strip waveguides,with horizontal and vertical meshing steps dx=15 nm and dy=20 nm, re-spectively. Other structures, such as long waveguide serpentines and small,circular ring resonators were also created in the mask layout in collaboration643.1. First Design Cyclewith other group members, in order to assess SMWG propagation losses.Table 3.2: Numerical results for mode power coupling between 500 nm-widestraight and bent strip waveguides (R=20 µm), at λ0 = 1550 nmStraight waveguide mode Bent waveguide mode Coupling [dB]Fundamental (TE-like) Fundamental (TE-like) −0.0002Fundamental (TE-like) Mode 2 (TM-like) −56.9Mode 2 (TM-like) Fundamental (TE-like) −53.5Mode 2 (TM-like) Mode 2 (TE-like) −0.0053.1.2 Setup DesignFigure 3.5(a) shows a block diagram of the first measurement setup, whichconsisted of a sweep-tuneable laser source (TLS), optical power sensors, bareoptical fibers, and manual XYZ stages for sample and fibre positioning. Inthis figure, the fibres are mounted on brass fibre chucks, and these are at-tached to XYZ stages. A sample pedestal is placed on a shorter XYZ stage,on top of a thermo-electric Peltier cooler (TEC) set to a constant temper-ature (usually 25 ◦C) using a Stanford Research Systems SRS LDC500TM36-W TEC controller [122].653.1. First Design CycleXYZ stageXYZ stageOpto -mechanicsOptomechanicsSampleTECI np u t f ib rePCM ic ros c o peUSBGPI BLaserTECController XYZ stageMonitorPhoto -detectorI np u t f ib r eh o l d e rXYZ stageOpto -mechanicsI npu tf ibreO utp ut f ib r eh ol d er(a)(b)Figure 3.5: First characterization setup. (a) Block diagram. (b) Optome-chanics assembly. (1) PM input fibre. (2) MM output fibre. (3) Pedestalon XYZ stage. (4a) Fiber XYZ Stages. (4b) Fiber chuck. (5) TEC. (6)Microscope. Image: G. Sterling.663.1. First Design CycleIn the actual setup, shown in Fig. 3.5(b), the optical signal from the TLSwas injected into the chip using a polarization maintaining (PM) fibre. Theoutput optical signal was collected by a multimode (MM) output fibre anddirected to a photodetector. The reason for using a MM output fibre was tomaximize the collected power. Both fibres were mounted at a 10◦-angle withrespect to the vertical, in agreement with the specifications of IMEC’s 1Dgrating coupler design [13]. The TLS and the photodetectors were controlledusing MATLABTM via General Purpose Interface Bus (GPIB), to sweepthe wavelength and to record the received power. The microscope cameraimages were sent to the system computer via Universal Serial Bus (USB).All alignments were performed manually, using the microscope images as avisual aid. The alignments were considered optimal when the power readingat the photodetector mainframe was maximized, within a 0.1-dB accuracy.3.1.3 MeasurementsFor this run, typical IL values ranged from −25 to −35 dB, with an averageILavg ≈ −30 dB). The fibre positions drifted over a period of several min-utes. After realignment, the IL could vary significantly. This was attributedto the fact that the alignment was performed manually.Ring ResonatorsAfter several modifications to the opto-mechanical setup, it was possibleto access the GCs of only some resonators on these chips. Figure 3.6(a)shows the spectrum for a 3.3-mm long resonator (see layout in Fig. 3.2(b)).The quality factor, Q, was calculated by dividing the resonance wavelengthand the FWHM at each resonance. Fig 3.6(b) shows Q as a function ofwavelength, and its average value, Qavg ≈ 28000. Based on the resonatorlength and the FSR value, the group index for this particular device wasng ≈ 4.086.673.1. First Design CycleWavelength [nm]1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555IL [dB]-42-40-38-36-34-32-30(a)6 [nm]1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555Q#10422.22.42.62.833.23.4(b)Wavelength [nm]1549.75 1549.8 1549.85 1549.9 1549.95 1550 1550.05Transmission [mW]#10-412345678ExperimentalFit(c)Figure 3.6: (a) Drop-port spectrum for a 3.3 mm-long resonator, show-ing maxima (red stars) and minima (green stars). (b) Q factor for eachresonance (blue) and average Q (red). (c) Resonance curve-fit. 683.1. First Design CycleThe spectrum was curve-fitted using equation (2.25), considering a sym-metrical coupled case (κa = κb = κ), neglecting coupler losses (γ = 0),and using the nominal coupling as the initial guess (κdesign = 0.465 for thisparticular device). The curve fit shown in Figure 3.6(c) suggests a couplingκ ≈ 0.38, and a propagation loss αdB =20.2 dB/cm. However, different ini-tial guess values yielded significantly different values for coupling, propaga-tion loss, and IL. Further tests to verify these parameters were not possible,due to the lack of a through-port GC in these particular devices.3.1.4 Fibre AttachmentIn order to reduce the alignment variations over time, and foreseeing even-tual rotary tests, I decided to perform fibre attachment tests with some ofthe chips, using optically-transparent, ultra-violet-curable adhesives. TheDUTs for these experiments were 16-mm-long SMWG serpentines.Firstly, in dry conditions, the input and output fibres were adjusted to apitch angle θp air = 10◦ with respect to the vertical. The fibres were alignedto the GCs of the device under test (DUT), and the spectrum was recorded.Secondly, the fibres were raised to allow for the deposition of dropletsof NorlandTM NOA 61 UV-curable adhesive on top of the input and out-put GCs of the DUT. The nominal refractive index of the adhesive wascalculated based on the data provided by the manufacturer, as per theequation nglue = A + B/λ2nm − C/λ4nm, where A = 1.5375, B = 8290.45,C = 2.11046× 108, and λnm is the wavelength expressed in nm. In order tocompensate for the transmission peak shift produced by the adhesive, thepitch angle was adjusted as follows, based on Snell’s law:θp glue = arcsin(sin(θp air)nglue)= arcsin(sin 10◦1.5409)≈ 6.5◦ (3.1)The fibres were submerged in the glue droplets, and realigned. The inser-tion losses in wet (uncured) adhesive showed an improvement between 5and 6 dB. However, the simultaneous 1-minute-long curing of both adhesivedroplets produced misalignment, increasing the IL. Figure 3.7 shows a com-693.1. First Design Cycleparison of the spectra for a particular DUT prior to applying any adhesive(blue), after applying the adhesive but prior to curing it (green), and aftercuring the adhesive (red).Figure 3.7: Comparison of spectra for a 16 −mm long SMWG serpentineduring a fibre attachment experiment.In order to detach the fibres, it was necessary to cut the fibre tips,dismount the fibres from the chucks, strip them, cleave them, mount themin the chucks, and in the case of PM fibres, perform axial alignment toensure proper mode injection into the GCs.3.1.5 Iteration Challenges and ConclusionsThe main difficulty faced during this design iteration was the lack of through-port output GCs for the large resonator designs. This prevented the disam-703.2. Second Design Cyclebiguation of experimental values of IL and coupling. No extracted parametervalues other than the FSR, ng, and Q factor could be trusted based solely ondrop-port spectra. The GC orientation in the mask layout (Fig. 3.2(b), andthe geometrical constraints imposed by the opto-mechanics (Fig. 3.5(b))required modifications to reduce the distance between fibres. Therefore,the designs for the next iteration would have the GCs facing in oppositedirections.Due to adhesive viscosity and fibre flexibility, proper alignment of barefibres was considerably more difficult in the uncured adhesive than withoutit. In a wet (uncured) adhesive, the fibres would not return to the sameposition after equal translations in opposite directions, i.e., they showedpositional hysteresis. I also realized that samples with uncured adhesive hadto be protected from light sources other than the UV curing gun light, asthese light sources could initiate curing of the adhesive. Long curing cyclesshould be avoided early in the procedure, to allow for necessary positionaladjustments. The minimum tip separation of fibres pointing in oppositedirections was affected when performing wet alignments, as the pitch angleswere closer to the vertical in that case. Finally, the fibre detachment wasa destructive procedure, and preparing the fibres for the next alignmentinvolved several delicate, time-consuming tasks.3.2 Second Design CycleIn the second iteration, one year later, IMEC offered two different fabricationbatches, one known as the standard batch, with air cladding, and anotherone called the custom batch, which had a glass cladding and allowed forgreater design parameter flexibility. The glass cladding offered the advantageof protecting the waveguides from scratches and contaminants such as dust.Once again, the UV illumination dose was intentionally varied during thelithography process.713.2. Second Design Cycle3.2.1 Layout DesignFigures 3.8(a) and 3.8(b) show panoramic views of the designs included onchips with air and glass cladding, respectively. Table 3.3 summarizes theranges of length, theoretical τ (assuming propagation losses of 3 dB/cm),and gap values for each resonator group. With a few exceptions, all gapswere multiples of 100 nm, and all coupler lengths were Lc = 200 µm.Table 3.3: Length and gap ranges for various resonator groups shown inFigure 3.8Resonator group L range [mm] τ range g range [nm]1 (custom) 14.27 to 16.37 0.607 to 0.564 200 to 7002 (custom) 14.15 to 14.96 0.609 to 0.592 400 to 6003 (custom) 16.49 to 17.02 0.561 to 0.551 150 to 7004 (custom) 15.88 to 16.50 0.574 to 0.561 500 to 7005 (custom) 6.47 to 7.90 0.797 to 0.758 200 to 500A (std) 5.91 (all) 0.813 200 to 600Waveguide ParametersIn addition to air-clad SMWG parameters, glass-clad SMWG parameterswere required. These were obtained in a similar fashion as was done for theair-clad waveguides during the first iteration. Curve fits for the effective andgroup indices for glass-clad strip SMWGs are shown in 3.9(a), and cross-overlength values for glass-clad strip SMWG directional couplers with variousgap values are shown in Fig. 3.9(b). Tables 3.4 and 3.5 show theoreticalvalues of L⊗ for various gap values, for air- and glass-clad directional cou-plers, respectively. These tables also show the Lc value required to achievea 50% power coupling at λ0 = 1550 nm, as well as the theoretical couplingfor a fixed coupler length Lc = 200 µm. By comparing these two tables, aswell as Fig. 3.9 to its first iteration counterpart (Section 3.1), one can seethat the glass cladding produces an increase in neff, a slight decrease in ng,and a decrease in the cross-over length values for the same gap.723.2. Second Design Cycle(a) Standard (air-clad) chip designs12345(b) Custom (glass-clad) chip designsFigure 3.8: Second iteration mask designs (not to scale). a Standard process(air-clad) designs. b Custom process (glass-clad) designs.Table 3.4: Cross-over length and selected parameters at λ0 = 1550 nm, forair-clad strip WDCs with various gaps.Gap [nm] L⊗ [µm] Lc(κ = 1√2) [µm] κ(Lc = 200 µm)150 46.2 23.1 0.492200 73.6 36.8 0.902300 180.7 90.4 0.986400 438.7 219.3 0.656500 1058.7 529.4 0.292600 2543.1 1271.5 0.1232733.2. Second Design Cycle1.5 1.52 1.54 1.56 1.58 1.6 1.6222.533.544.5neff(λ0)=2.4434ng(λ0)=4.1772Wavelength [µm]n  neff fitneff(λ0)ng fitng(λ0)(a)Wavelength [6m]1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.6L# [6m]020040060080010001200140016001800L# forStrip Dir Coupler, WG width=500 nmgap=200 nmgap=300 nmgap=400 nmgap=500 nmgap=600 nm(b)Figure 3.9: (a) Effective index (green) and group index (blue) curve-fits,and (b) Cross-over length as a function of wavelength, for glass-clad stripWDCs with various gaps.Table 3.5: Cross-over length and selected parameters at λ0 = 1550 nm, forglass-clad strip WDCs with various gaps.Gap [nm] L⊗ [µm] Lc(κ = 1√2) [µm] κ(Lc = 200 µm)170 28.7 14.4 0.998200 37.6 18.8 0.877210 41.2 20.6 0.974250 58.6 29.3 0.797290 83.0 41.5 0.60300 90.5 45.3 0.324400 214.3 107.2 0.995500 503.7 251.8 0.584600 1177.0 588.5 0.2643.2.2 Setup ImprovementsIn an effort to decrease the ILs, lensed fibres were used as the input fibresfor measurements being made by several group members. Initially, I usedthem as well for dry alignments (i.e., with no adhesive), however, due tothe special geometry, fragility, and high cost of lensed fibres, attaching them743.2. Second Design Cyclewith adhesive would not have been viable, as this would have unavoidablyaffected their lensing properties. Also, as the realignment and detachmentprocedures required me to cut the fibre tips, one would not be able to recoverfrom a misalignment at a reasonable cost. Therefore, in parallel to standardalignment and measurements (Subsection 3.2.3), I developed a set of 3Dprinted supports to glue my fibres prior to alignment (Subsection 3.2.4 ),as an alternative to the destruction and excessive cutting of fibres, with theidea that the supports would reduce position hysteresis during alignment,and would facilitate attachment, detachment, and fibre tip polishing.3.2.3 MeasurementsTable 3.6 shows the as-fabricated widths of waveguides designed to be 500 nmwide, based on IMEC metrology information relayed by Dr. D. Deptuck,from CMC Microsystems. The on-target doses were in column −2 for thestandard (air-clad) wafer, and in column −1 for the custom (glass-clad)wafer.Table 3.6: As-fabricated widths for 500 nm WGs, standard (air-clad) andcustom (glass-clad) wafers.Air-clad Glass-cladColumn Width [nm] Column Width [nm]−6 554.69 −6 532.37−5 542.81 −5 526.97−4 525.08 −4 520.04−3 515.47 −3 514.95−2 502.49 −2 504.31−1 496.04 −1 503.240 486.63 0 494.541 479.62 1 489.962 465.11 2 482.843 463.28 3 477.334 441.49 4 467.335 452.69 5 465.846 433.07 6 458.63753.2. Second Design CycleFigure 3.10 shows the measured spectrum (dotted curve) for the through-port of an asymmetrically-coupled 5.91 mm-long resonator (Fig. 3.8(a)),from the air-clad chip at row “0”, column “−3” (R0C−3). Both couplershad identical lengths Lc = 200 µm, and their gaps were ga = 400 nm andgb = 500 nm. As shown in Table 3.4, the as-fabricated width for 500 nm-wide waveguides for that particular column was 502.49 nm. Hence, thenominal field cross-coupling values were κa = 0.656, κb = 0.292. The re-spective through-coupling values, ta = 0.754 and tb = 0.956, were used asinitial curve-fit guesses. Based on Eq. (2.24), assuming lossless directionalcoupling, the following equation was used for curve-fitting purposes:|S21|2 =t2a − 2tatbτ cos (φ) + t2bτ2CIL[1− 2tatbτ cos(φ) + t2at2bτ2] (3.2)The curve-fit algorithms are based on non-linear least-square-fits, com-puted using MATLABTM. The initial algorithms proved very sensitive tothe initial guess values. Considerable refinement efforts of the curve-fit al-gorithms were conducted by myself and various members of the researchgroup. The red curve in Fig. 3.10 shows the curve-fit obtained with an al-gorithm developed by Dr. L. Chrostowski that estimates the refractive indexand its dispersion based on the FSR of the DUT prior to performing thecurve-fit based on Eq. (3.2). These refractive index parameters are usedto create wavelength-dependent guess values of φ that allow for repeatablecurve-fits encompassing several resonances. Although the ER is not per-fectly matched, there is a high correlation between the measured and the fitdata, regardless of the initial guess values. Table 3.7 shows the extractedparameter values and compares it to the nominal expected values. Figure3.11 shows the measurement results for the through- and the drop-ports ofthe aforementioned ring.763.2. Second Design CycleWavelength [nm]1550 1550.1 1550.2 1550.3 1550.4 1550.5 1550.6 1550.7 1550.8Transmission [dB]-50-45-40-35MeasuredGuessFitFigure 3.10: Through-port spectrum and curve-fits for a symmetrically-coupled, air-clad, 5.91 mm-long resonator (Ring 2, Std. Chip R0C−3).Table 3.7: Through-port spectral curve-fit parameters for a 5.91 mm-longresonator (Ring 2, Std. Chip R0C−3), with a correlation value betweenmeasured and fit data r2 = 0.856.Parameter Nominal FitILdB ∼ 10 32αdB 3 4.65ng 4.3547 ∼ 4.46κa 0.656 0.57κb 0.292 0.07ta 0.754 0.821tb 0.956 0.997773.2. Second Design CycleWavelength [nm]1550 1550.2 1550.4 1550.6 1550.8 1551Transmission [dBm]-60-55-50-45-40-35-30(a)Wavelength [nm]1550 1550.1 1550.2 1550.3 1550.4 1550.5 1550.6 1550.7 1550.8 1550.9 1551Transmission [dBm]-50-45-40-35(b)Wavelength [nm]1550 1550.1 1550.2 1550.3 1550.4 1550.5 1550.6 1550.7 1550.8 1550.9 1551Q#1045.566.577.588.599.510QQavg(c)Figure 3.11: Spectra and Q for an air-clad, 5.91 mm-long resonator (ChipR0C−3, Dev. 2) with nominal field coupling values κa = 0.656 (throughport), and κb = 0.292 (drop port). (a) Through-port transmission. (b)Drop-port transmission. (c) Q vs. wavelength, and average. 783.2. Second Design Cycle3.2.4 Fibre AttachmentIn order to minimize breakage and shortening of fibres during detachment,Dr. H. Kato (CEO of Versawave Inc., and a collaborator with our researchgroup) suggested using optical adhesives with relatively low glass transitiontemperatures, Tg ≤ 100 ◦C, and heating the chips to a temperature above Tgto detach the fibres. The value of Tg depends on the specific adhesive, beingusually in the range between 80 ◦C and 100 ◦C [40, 61]. I designed fibre hold-ers with various end-face angles in an effort to reduce insertion losses andto provide support for the fibre tips, thus decreasing the tip bending duringalignment and attachment. Mr. A. Sharkia and I laid out and 3D-printed thefibre holder designs shown in Fig. 3.12. We also printed supports for mount-ing these holders to more compact opto-mechanics, e.g., mini-goniometres,that allowed for more precise angular adjustments. Metallic and plastic pol-ishing jigs, with suitable angles, were also machined and 3D-printed. Eachfibre holder had 3 trenches to allow multiple fibres to be attached. Afterfabrication of the initial holders, I noticed only 2 trenches, instead of 3, wereprinted on each holder. These trenches were shallower than expected, andtheir separation was not very repeatable. It was still possible to align andattach single fibres. Nevertheless, we considered that this approach shouldwork for precision-machined holders, which we eventually acquired in theform of fibre arrays, as described in Section 3.4. Figure 3.13(a) shows theFigure 3.12: Fibre holders and support designs for 3D printing. Dimensionsin mm.793.2. Second Design Cyclestation I created to align and attach the fibres to the holders. UV curableadhesive was deposited along the holder trench. A long-working-distancemicroscope, a fibre rotator, and two XYZ stages were used to align strippedfibres along the holders. This was followed by UV curing, as shown in Fig.3.13(b). Initially I envisioned attaching PM fibres to these holders as well,however, aligning the polarization axes with those of the holders proveddifficult. Thus, these holders were used exclusively with SM fibres. Afterfibre attachment, the end faces of the holders were polished in a portablefibre polishing station, shown in Fig. 3.13(c) in 30-minute coarse- and fine-grit cycles, with intermediate iso-propyl alcohol (IPA) and de-ionized water(DIW) washing cycles, to avoid grit cross-contamination. They were thenviewed under a stereoscopic microscope, to inspect the polishing quality.Figure 3.13(d) shows fibre holders mounted on mini-goniometres, which are,in turn, mounted on manual linear stages using aluminium spacers.3.2.5 Iteration Challenges and ConclusionsThe experimental quality factors of devices fabricated in this iteration weremore than double than those from the previous iteration (cf. Qavg 1 ≈ 2.8×104 to Qavg 2 ≈ 8× 104). The finesse of these resonators is F =Qλ0ngL≈ 5.A 2D GC design provided by IMEC, and intended for on-chip opti-cal power splitting, was used in most of my glass-clad resonator designs.However, the as-fabricated GCs showed uneven power splitting ratios andconsiderable IL, which rendered these devices inoperable.Until that point in time, the spectral measurements in the characteriza-tion setup were slow, as the TLS wavelength tuning and the detector poweracquisition was made through MATLABTM by sending GPIB commandsfor point-by-point tuning and recording. This caused, for instance, singlespectral sweeps with over 1000 data points to take longer than 15 minutes tobe acquired. Based on written and oral communications with applicationsengineers from Agilent Technologies Inc. (now Keysight Technologies Inc.),I learned of alternative, faster spectral sweeps, based on a proprietary Plugand Play driver known as “hp816x PnP”, that could be used on various803.2. Second Design Cycle(a) (b)(c) (d)Figure 3.13: (a) Fibre holder glueing platform. (b) Fibre attachment toholder. (c) Polishing station and jigs. (d) Finished holders.platforms, such as MATLABTM, LabVIEWTM , and Python. Thus, duringthe next iteration, I implemented faster measurements using these drivers.The 3D printed holders helped prevent the fibre tips from vibratingor bending during alignment. However, the far end of the holders, nearthe fibre jacket, had a brittle zone where bare fibre was exposed, whicheasily broke during either polishing, mounting, or alignment. Adjustmentsto the trench dimensions were also required, since the as-printed parts hadshallower trench depths than the designs. Therefore, I improved the designsby adding 20-mm long, 1.2-mm wide channels at the far end of the holders,for robust fibre jacket attachment, as shown in Subsection 3.3.4.813.3. Third Design Cycle3.3 Third Design CycleIn our previous devices, we had resonators with Q factors in the range of6×104 to 1×105, with an average value Qavg = 8×104. Assuming the ring isnear critical coupling condition (based on the large ER values) the Q factorand the average field attenuation, αavg, are related as follows [68, 141]:Q =ping2αavgλ0⇒ αavg =ping2λ0Q(3.3)For Qavg = 8 × 104, λ0 = 1.55 µm, and ng ≈ 4, the average field atten-uation value is αavg ≈ 50 m−1 = 0.05 mm−1. This translates to a powerattenuation αp ≈ 0.1 mm−1, which means that for a 6 mm-long resonator,the optical signal makes less than two roundtrips (1.67 roundtrips) beforeits original intensity decreases by a factor of e. Thus, in order to enhancethe sensitivity of the devices, it was imperative to increase the Q factor ofour resonators.Since absorption in silicon is negligible for infrared wavelengths in therange from ∼ 1 µm to ∼ 4 µm [43], the main source of losses in the SMWGsis sidewall scattering [17, 78, 114]. As discussed by Payne and Lacey [98],Yap et al. [140], and Li et al. [78], for a fixed sidewall roughness value(determined by the fabrication process), the scattering losses decrease ex-ponentially for increasing waveguide widths, as the field intensity of thefundamental mode at the sidewall edges is considerably smaller for widemultimode waveguides (MMWGs) as compared to SMWGs.Therefore, we decided to use MMWGs as the straight segments of ourresonators, which account for most of their length. SMWGs were still usedfor the directional couplers and corner bends, and were connected to theMMWG segments using linear tapers, as to avoid excitation of higher ordermodes [115]. This contributed to a significant reduction of the roundtrip lossand the achievement of greater Q values. Specifically, as shown in Section3.3.3, the Q values for our strip waveguide resonators in this iteration rangedfrom 2×105 to 6×105, with an average Qavg ≈ 3.4×105. As shown in Section3.4.3, the eventual use of rib waveguides led to even lower roundtrip losses.823.3. Third Design CycleIn this way, we achieved Q values as high as ∼ 4.5 × 106, with an averageQavg ≈ 1.7× 106, the largest Q values obtained to date using standard SOItechniques. Our results were published in [56].Foreseeing eventual rotary tests, considerable improvements were madeto the setup hardware and software. Specifically, I assembled a new opti-cal characterization setup on a portable optical breadboard, and eventuallyplaced it on top of a turntable, for which I designed the rotary control andsignal acquisition system, in collaboration with other colleagues, as describedin Subsection 3.3.2 and published in [55].3.3.1 Layout DesignDuring this iteration, we submitted designs to the Nanofabrication Facilityat the University of Washington, which had a single-etch process and useda 100 keV electron beam (e-beam) lithography system. Although still indevelopment at the time, this particular process had turnaround times ofa few weeks, rather than a year or longer. The GC designs for this fabri-cation run were provided by Dr. M. Hochberg’s group, at the Universityof Washington. These were air-clad focusing GCs with a nominal incidenceangle of 40◦, and did not require long waveguide tapers. Depending on therefractive index of the adhesives used, an incidence angle between 23◦ and25◦ would be required to shift the transmission peak back to a wavelengthnear 1550 nm, according to Eq. (3.1). The e-beam runs offered more layoutarea, thus allowing me to create larger resonators, with lengths ranging from7 to 70 mm, and aspect ratios (ARs) closer to 1 as compared to resonatorsfrom previous runs, i.e., with shapes closer to a square, to maximize thearea for a given length.The idea behind the splitting/merging coupler design for these resonators,shown in Figure 3.14, is to split on chip the optical input signal, to injecttwo counter-propagating signals into the resonator, and to merge the twocorresponding output signals upon exit. We expect intensity variation ofthe optical signals during rotation, as described below.The optical input signal is coupled into the device using the main in-833.3. Third Design CycleGCGCGC13 24CCW tapGC CW tapGC65 7I/ O GC(a) Schematic(b) Mask layoutFigure 3.14: (a) Splitting coupler schematic. I/O GC- Main Input/OutputGrating Coupler. 1- Central Y-branch. 2- Straight SMWGs. 3- SMWG 180◦bends (R3 = 6 µm). 4- Tap Y-branches. 5 and 7- SMWG S-bends (R5 =R7 = 200 µm). 6- SMWG directional coupler. Tap GCs- GCs for CW- andCCW-resonance monitoring. (b) Mask layout. Span: ∼ 1.5× 0.12 mm.put/output grating coupler (I/O GC). The central Y-branch (1) splits theinput signal upon entrance, and merges the output signals upon exit. Oneach branch, the split signals travel through straight SMWGs (2) and 6-µm radius, 180◦ SMGW bends (3) towards 6◦-angle Y-branches (4) thatserve as output signal “taps”, to independently monitor the spectra of thetwo optical signals travelling in opposite directions. SMWG S-bends with200 µm radii and 20 µm vertical offsets (5) connect the stems of these tapY-branches to the SMWG directional coupler (6), which injects light into,and collects light from, the large area resonator (grayed out) in oppositedirections. The directional coupler lengths are designed to achieve optimalcoupling depending on the resonator length, SMWG-to-MMWG length ra-tio, and a-priori propagation loss estimations, as per the optimization study843.3. Third Design Cycledeveloped in Section 2.4 and published in [54].Each counter-propagating output signal exiting the directional coupleris split by its corresponding tap Y-branch. Each top branch is connectedvia an S-bend (7) to a tap GC. This allows for monitoring the CCW andCW ring resonances using the left and right GC, respectively. The rationalebehind the output S-bends (7) is to create sufficient separation between thetap GCs and the large resonator waveguides, thus avoiding damage to theresonator during fibre alignments. The bottom branch of each tap Y directsa fraction of each counter-propagating output signal towards the centralY-branch (1), where both are interfered.Figure 3.15 shows the schematic of a resonator with its splitting/mergingcoupler and its interrogation block diagram. The I/O GC is interrogatedusing a polarization maintaining optical circulator. The tap GCs are inter-rogated using single mode fibres. Each signal is injected in a photodetectorfor intensity readout.PD 1LaserGCI/ OGCGCGC 13 2 4CCW“tap” CW“tap”106 95 78PD 2PMCircPD 3Figure 3.15: Gyro resonator schematic and interrogation block diagram.1 through 7- See nomenclature in Fig. 3.14. 8- Linear waveguide tapers.9-Straight MMWGs. 10-SMWG 90◦ bends (R10 = 20 µm), with 15 µm-longstraight stubs on both ends.853.3. Third Design CycleAssuming a CW rotation, based on Eqs. (2.21) and (2.22), the electricfields at the CW and CCW taps are described by the following equations:ECCW =e−jΦsct2√2√CIL· T (−ΦS) , (3.4)ECW =e−jΦsct2√2√CIL· T (ΦS) , (3.5)where Φsct = −jαscLsct + φsct, αsc is the average propagation loss of thesplitting coupler waveguides, Lsct and φsct are, respectively, the waveguidepath length and the phase shift undergone from the central Y-branch to eachtap GC, CIL = 10ILdB10 is a coefficient corresponding to an insertion loss ILdBin dB, and the function T (ΦS) represents the Sagnac-phase-shifted transferfunction of the resonator, given by:T (ΦS) =t− e−j(Φring+ΦS)1− te−j(Φring+ΦS), (3.6)where t =√1− κ2, κ is the field cross-coupling of the WDC depicted by(6) in Fig. 3.15, Φring = −jαringL + φring, αring is the average propagationloss of the ring resonator, L is the resonator length, φring = 2pineffL/λ is thephase undergone by the beams due to the resonator optical path, ΦS is theSagnac phase shift.The merged output signal exits the device via the I/O GC, and showsan amplitude modulation that depends on the phase difference between theCW and CCW resonances due to δφ. Based on Eqs. (3.4) and (3.5), theelectric field of the merged output signal is:Emerged =e−jΦscl4√CIL· (T (ΦS) + T (−ΦS))=e−jΦscl2√CIL·t(ejΦring + e−jΦring)−(t2 + 1)cos(ΦS)ejΦring − 2t cos(ΦS) + t2e−jΦring(3.7)where Φscl = −jαscLscl + φscl, Lscl is the length of the waveguide loop be-863.3. Third Design Cycletween the arms of the central Y-branch, and φscl is the phase shift undergoneby traversing this loop. The power for each port, PCCW, PCW, and Pmerged,can be obtained as the modulus squared of the expressions given by Eqs.(3.4) through (3.7).Figure 3.16 compares the theoretical output power levels at rest and un-der rotation with a Sagnac phase shift ΦS = 0.02pi rad, as functions of thering normalized detuning, φring/pi, for a resonator with length L = 7.5 mm,coupling κa = 0.255, average propagation loss αdB = 1 dB/cm, and negligi-ble attenuation and IL at the splitting coupler (αsc = 0 m−1, ILdB = 0 dB).The brown dotted curve shows the power at the merged output at rest,Pmerged(ΦS = 0). The orange dotted curve shows power of both tap outputsat rest, PCCW(ΦS = 0) = PCW(ΦS = 0). The green solid curve shows thepower of the merged output under rotation Pmerged(ΦS = 0.02pi), and thered and blue solid curves show the phase-shifted power spectra for the CWtap, PCW(ΦS = 0.02pi), and the CCW tap, PCCW(ΦS = 0.02pi), respectively.The magenta dash-dotted curve shows the ratio between the merged outputpower at rest and under rotation, Pmerged(ΦS = 0.02pi)/Pmerged(ΦS = 0).The cyan dash-dotted curve shows the ratio between the CW and the CCWtap under rotation, PCW(ΦS = 0.02pi)/PCCW(ΦS = 0.02pi). The CW andCCW spectra shift under rotation, and this produces a power variation be-tween them at each φring. The maximum and minimum ratio occur at a valueof φring that is phase-shift dependent, as shown by the cyan dash-dottedcurve, and shift further away from the static resonance null for greater δφ.Frequency tracking is required in order to follow the resonance shifts. How-ever, the resonant frequency difference created by the Sagnac effect is muchsmaller than the temperature-induced drifts [102], which require frequencyand phase spectroscopy techniques [82, 118, 121, 144]. As shown by the ma-genta dash-dotted curve, the phase-shift-dependent power variation of themerged output has its greatest variation at the original resonance null, andas it does not require frequency tracking techniques, we decided to use it asour sensing signal at this stage.873.3. Third Design Cycle-0.4 -0.2 0.0 0.2 0.4-15-10-505{ring/oTransmission[dB]Figure 3.16: Theoretical power levels for the tap and merged outputs ver-sus ring normalized detuning, φring/pi, at rest (dashed curves) and underCW rotation (solid curves) with a Sagnac phase shift ΦS = 0.1pi rad, andILdB = 0 dB. Resonator parameters: L = 7.5 mm, κa = 0.255, and averagepropagation loss αavg dB = 1 dB/cm. Dotted orange curve: CW and CCWtaps at rest. Solid red and blue curves: CW and CCW taps, under rotation.Dotted brown curve: Merged output at rest. Solid green curve: mergedoutput, under rotation. Cyan dot-dashed curve: Tap power ratio under ro-tation, PCW(ΦS = 0.02pi)/PCCW(ΦS = 0.02pi). Magenta dot-dashed curve:merged output power ratio, Pmerged(ΦS = 0.02pi)/Pmerged(ΦS = 0).Waveguide ParametersWe chose a MMWG width of 3 µm, i.e., one micrometre wider than thewaveguides studied by Yap et al [140]. The widths of the SMWGs remained500 nm. All waveguides had heights of 220 nm, and all were air-clad stripwaveguides, due to the single-etch option. 200 µm-long linear tapers wereused for conversion between SMWGs and MMWGs. Figure 3.17 shows thepolynomial curve-fits for the effective and group indices as functions of wave-length for straight strip SM and MMWGs. Table 3.8 summarizes the effec-tive and group index values at λ = 1550 nm, for the fundamental modesof strip waveguides with various widths and radii, obtained using MODE883.3. Third Design CycleSolutionsTM eigenmode solver with a 15-nm mesh.1.5 1.52 1.54 1.56 1.58 1.6 1.6222.533.544.5neff(λ0)=2.3826ng(λ0)=4.3547Wavelength [µm]n  neff dataneff fitneff(λ0)ng datang fitng(λ0)(a) W=500 nm1.5 1.52 1.54 1.56 1.58 1.6 1.622.62.833.23.43.63.84neff(λ0)=2.8171ng(λ0)=3.7259Wavelength [µm]n  neff dataneff fitneff(λ0)ng datang fitng(λ0)(b) W=3 µmFigure 3.17: Effective and group index curve fits for air-clad strip waveg-uides of different strip widths. Also shown, original data points and fittedvalues for λ0 = 1550 nm.Table 3.9 shows the results of numerical simulations carried out to assessthe cross-talk between the various modes of straight SMWGs and those ofbent waveguides with radii R3 = 6 µm and R5 = 200 µm. Each waveguidesupports two guided modes with different polarizations, a TE-like mode anda TM-like mode. The cross-talk between straight SMWGs and bent SMWGswith 20 µm radius are shown in Table 3.2. An initial estimate for the prop-agation loss values of the SMWGs and MMWGs, shown in Table 3.10, wasrequired to determine the coupling for each resonator, as the optimum cou-pling depends mainly on the resonator length and roundtrip loss, as per ouroptimization study in [54]. I created algorithms to determine the optimumcoupling values based on resonator port configuration, loss parameters, AR,total length, and SM-to-MM length ratio. For simplicity, and consideringtheir short lengths in comparison to the resonator dimensions, the indicesand propagation loss of linear tapers were approximated to be the averagesof the respective SM and MM values.893.3. Third Design CycleTable 3.8: Effective and group indices at λ0 = 1550 nm for strip waveguidesof various geometries. In all cases the height is H = 220 nmWidth [nm] Radius [µm] Mode neff ng500 ∞ TE-like 2.3826 4.3547500 ∞ TM-like 1.5821 3.3895500 200 TE-like 2.3823 4.3546500 200 TM-like 1.5821 3.4395500 20 TE-like 2.3824 4.1977500 20 TM-like 1.5827 3.4276500 6 TE-like 2.3976 4.1967500 6 TM-like 1.728 2.1191750 ∞ TE-like 2.7942 3.75431750 ∞ TM-like 2.6876 3.90293000 ∞ TE-like 2.8171 3.72513000 ∞ TM-like 2.7813 3.7726Table 3.9: Coupling between straight and bent strip SMWG modes at λ0 =1550 nm. W=500 nm, R3 = 6 µm, and R5 = 200µm.Straight WG mode Bend radius, mode Coupling [dB]TE-like R3, TE-like −0.0027TE-like R3, TM-like −56.9TM-like R3, TE-like −42.7TM-like R3, TM-like −0.012TE-like R5, TE-like −4.34× 10−6TE-like R5, TM-like −75.0TM-like R5, TE-like −71.3TM-like R5, TM-like −9.9× 10−5The WDCs in this run had 500 nm gaps, with a theoretical L⊗ = 1058.7 µm(see Table 3.4). Table 3.11 shows the theoretical field coupling for variouscoupler lengths used in the devices.903.3. Third Design CycleTable 3.10: A-priori propagation loss estimates for air-clad strip waveguidesXXXXXXXXXXXParameterKindSM strip MM stripαdB [dB/cm] 4 1α [1/m] 46.05 11.513Table 3.11: Theoretical field coupling for air-clad strip SMWG directionalcouplers, g = 500 nm (L⊗ = 1058.7 µm)Lc [µm] κ90 0.1331174 0.255315 0.451400 0.0.559435 0.602527 0.705Layout ScriptingConsidering the significant amount of time invested on manually drawing thelayouts in the first two iterations, we considered more efficient to create codesfor semi-automatic layout generation. During this design iteration, I usedMATLABTM codes to generate and place basic waveguide shapes in WieWebCleWinTM, to create resonator shapes. Various elaborate structures such asY branches, S bends, and GCs were created separately (either manually orby script), saved as layout cells, and instanced in the main layout code.Figure 3.18(a) shows a panoramic view of an e-beam layout design, andFig. 3.18(b) shows a zoomed-in view of the area enclosed by the blue rect-angle. The resonator lengths in this design range from 7 mm to 70 mm.Several straight SM and MM waveguides with various lenghts were includedacross the layout, in an effort to experimentally estimate the propagationlosses in both SMWGs and MMWGs.913.3. Third Design Cycle(a) Panoramic view (b) Zoom inFigure 3.18: First E-beam layout design.3.3.2 Setup DesignDuring my literature review, I noticed that many research groups workingon this subject inject artificial rotation signals into their devices, rather thanfacing the challenges posed by mechanical rotation, vibration, and packag-ing, see for example [34, 36, 83, 121]. In contrast, I decided to create amechano-opto-electrical setup for characterizing SOI gyroscopes under ac-tual rotation conditions. The first priority was, thus, to create a new opticalcharacterization setup assembled on an optical breadboard. The opticalbreadboard was portable so that it could, eventually, be placed on top ofa turntable. The second priority was to improve the fibre attachment sup-ports.First Rotary SetupI built the opto-mechanical setup shown in Fig. 3.19(a) on a portable mini-breadboard, using several mounting posts and five manual-adjustment trans-lation stages. Three linear stages are used for positioning optical fibre hold-ers and rotators. Another stage is used for mounting a heatsink and a TECfor temperature control of the sample pedestal. A fifth translation stage is923.3. Third Design Cycleused as a detachable jig for adhesive dispensing and curing. As a result, theopto-mechanical setup weighs over 20 kg and spans 46×30×60 cm in length,width and height, respectively. This setup was eventually transferred on topof a rotary platform, shown in Fig. 3.19(b). Additional payload capabilityand off-breadboard area were required for mounting a long-working-distancemicroscope, used during fibre alignment and attachment (Fig. 3.24(b)). Thefibre attachment in this setup was carried out with bare fibres, as the secondfibre holder generation was being developed in parallel.In order to support the optical breadboard, a 52-cm diameter acrylicplatform was mounted on a 38-cm diameter ball bearing ring, which in turnwas attached to a wooden frame. A timing belt was glued to the outer edgeof the ball bearing ring and was actuated by a pulley, mounted on a DCmotor with a nominal working voltage of 12 V, and a torque of 16.7 kg-cm. Motor drivers and a microcontroller were also attached to the woodenframe for angular speed and direction control. A MEMS gyroscope with asensing range of ±400 dps, a sensitivity of 0.02 dps · Hz−12 , a bandwidth of140 Hz, and a nominal noise level of 0.236 dps [117], was mounted on therotary platform as a calibration reference. The turntable was modelled as afirst-order system with a transfer function of the form:H(ω) =A · ejφ0jω/ω0 + 1, (3.8)where ω = 2pif is the angular frequency in rad/s, f is the frequency in Hz,ω0 = 2pif0 is the cut-off angular frequency, f0 is the system bandwidth,in Hz, A is the DC gain, and φ0 is a phase offset to represent a constantdelay between the input and output signals. This transfer function has amagnitude and a phase response given respectively by:|H(ω)|2 =A2(ω/ω0)2 + 1, (3.9)]H(ω) = φ0 − arctan(ω/ω0). (3.10)933.3. Third Design Cycle(a) (b)Figure 3.19: Mini-breadboard characterization setup. (a) Initial benchtopconfiguration. (b) On rotary platform, showing on-board reference gyro-scope (bottom left), and off-platform microscope (top left).System ControlThe control system uses an NITM PXIe 1062Q mainframe with a PXIe 8133central processing unit (CPU), a DC power supply, and an PXI 7852R Field-Programmable Gate Array (FPGA) module with multiple digital and ana-logue I/O channels [65, 66]. As depicted in Fig. 3.20, the PXIe 1062Qcontrols an optical mainframe (laser source and photodetectors) as well asthe TEC driver using a General-Purpose Interface Bus (GPIB) interface,and communicates with the microcontroller via RS-232.In order to simultaneously monitor signals and send commands to therotation microcontroller, a Graphical User Interface (GUI) was implementedusing LabVIEWTM-FPGA. The FPGA target-host communication is madethrough a 128-bit Direct Memory Access First Input/First Output block(DMA FIFO) for recording the motor control digital signals, as well as theanalogue signals from the DUT and reference gyroscope.The optical input signal is generated by a C-band Tuneable Laser Source(TLS). The light is coupled to the input GC using polarization-maintaining943.3. Third Design Cycle(PM) patch cords. Depending on the specific interrogation configuration, theoptical output signal(s) are collected using either a PM optical circulator,or independently-positioned multi-mode output fibre patch cords. Up tothree photodetectors are used simultaneously. To avoid fibre entanglement,the turntable was operated using bow-tie rotation patterns, consisting ofperiodic, angle-limited movements.The micro-controller generates two digital signals: a slowly varying sig-nal which defines the motor rotation direction, and a 500-Hz Pulse-Width-Modulated (PWM) signal, whose duty cycle is directly proportional to thenormalized angular speed, Sn. During rotation electrical signals were sentto, and collected from, the reference gyroscope and the TEC, via a slip ring.Off-board optomechanicsData acqui-sitionFPGAGyro SampleTECDC power supplyInput fibreOutput fibre(s)NI PXI RS-232MicroscopeUSB GPIBLaserTECControllerMicro-ControllerDC Motor driversRotary platformDC MotorMonitor Photo-detectorRef .  Gyr oFigure 3.20: SOI gyroscope characterization platform block diagram.953.3. Third Design CycleRotation PatternsFour different rotation patterns, which are described below, were programmedinto the micro-controller.1. Continuous rotation: The rotation direction and the value of Sn aredefined in the GUI, in order to rotate the platform at the specifiedspeed and direction. This pattern was mainly used to characterize theturntable speed response.2. Rectangular bow-tie rotation: The user defines values of Sn and travelangle, θ. The turntable will rotate in the clock-wise (CW) direction atthe given speed until the travel angle is reached, then stop and rotatethe same amount in the counter-clock-wise (CCW) direction. Thismotion creates a rectangular bow-tie angular speed pattern, which isrepeated until stopped by the user via the GUI.3. Sinusoidal bow-tie rotation: A sinusoidal bow-tie speed pattern, ratherthan a rectangular one, was necessary for obtaining the frequency re-sponse of the turntable, and characterizing the DUT. Appendix Bshows a detailed explanation of the generation of these patterns.4. User-defined rotation: created using an array of angular speed andangular travel values. In this case, a sign is added to the normalizedspeed for defining the rotation direction. The angular travel is alwaysa positive number, in degrees. The motion stops after executing asmany steps as entered in the array.3.3.3 MeasurementsAs the rotational pattern would be sinusoidal, we wished to have a quanti-tative idea of the frequency and speed ranges at which our turntable setupwould be able to work. We carried out tests to determine these parameters,followed by static and rotational tests on our photonic devices.963.3. Third Design CycleSpeed Range and Noise LevelWe measured the turntable step-response, i.e., going from rest to continuousrotation, at different speeds. As shown in Fig. 3.21, the relationship be-tween the turntable angular speed and the value of Sn was non-linear. Eventhough angular speed values below 20 dps could be achieved, these could notbe sustained, because vibrations and the payload occasionally hindered orstopped the movement. These problems did not arise at high angular speedsnor during unloaded motor tests, suggesting that this may be caused by acombination of insufficient motor torque, unbalanced payload coupling, anda weak motor base condition [49]. The minimum repeatable angular speedwas Ωmin ≈ 27 dps (for Sn = 0.175), whereas the maximum angular speedwas Ωmax ≈ 74.3 dps (for Sn = 1). The angular acceleration values rangefrom 9.4 to 171.2 dps2. As shown by the error bars in Fig. 3.21, the noiselevel decreased as the normalized angular speed increased. For example, thenoise level was δΩ = 2 dps for Sn = 0.175, and δΩ = 0.73 dps for Sn = 1.0 0.2 0.4 0.6 0.8 1020406080Normalized angular speedAngular speed [dps]Figure 3.21: Average angular speed (dots) and noise level (error bars), as afunction of normalized speed, Sn.973.3. Third Design CycleFrequency ResponseWe operated the turntable in a sinusoidal bow-tie fashion, and used thereference gyroscope output signal to characterize the frequency response ofthe turntable. The input consisted of over forty sinusoidal pulses, steppedin frequency between 0.05 and 5 Hz. To avoid damaging the optical setup,a maximum normalized angular speed value Snmax = 0.4 was used for allfrequencies. Figure 3.22 shows the input and output signals for a frequencyfin = 0.89 Hz. The PWM duty cycle (red curve), is signed according to therotation direction (black curve) for fitting purposes and easier visualization.For each sinusoid frequency, we curve-fitted both the normalized angularspeed of the motor (red curve), and the reference gyroscope output signal(orange curve), in order to obtain the phase shift between both signals. Wealso extracted the magnitude of the reference gyroscope signal, as explainedin Appendix B. The magnitude and phase responses of the turntable areshown in Fig. 3.23. We curve-fitted both the magnitude and the phaseresponses using Eqs. (3.9) and (3.10). According to the fit parameters,f0 = 0.54 Hz, A = 0.383 (AdB = −8.33 dB), and φ0 = −11.9 degrees.Signed PWMduty cycleGyro VoutVout curve fittime [s]DirectionSigned duty cycleOutput voltage [V]Figure 3.22: Input and output signals for a sinusoid of frequency fin =0.885 Hz.983.3. Third Design Cycle10−1 100−25−20−15−10Frequency [Hz]Normalized magnitude [dB](a)10−1 100−120−100−80−60−40−20Frequency [Hz]Phase [deg](b)Figure 3.23: Turntable frequency response and first-order model fitting.(a): Magnitude response. (b) Phase response.Static Optical TestsDue to space constraints, stripped fibres were used to interrogate these sam-ples. Figures 3.24(a) and 3.24(b) show microscope images of optical fibresaligned to the GCs of the splitting coupler of a gyro resonator before andafter permanent attachment. The central fibre, aligned upon the main I/OGC, is PM, whereas the two lateral fibres, aligned upon the tap GCs, areSM. For initial dry alignments, a pitch angle θp = 40◦ was used for all fibres.Norland NOA 61 adhesive was deposited afterwards, and the angle was setto θp = 25◦. Fine angle and position adjustments were made to minimizethe IL.After fibre attachment, spectra were obtained with the turntable at rest.Figure 3.24(c) shows the spectra for the CW and CCW resonances of an all-pass, 7.4-mm-long resonator, which exhibits an insertion loss (IL) of approx-imately 25 dB, a free spectral range (FSR) of 81-pm, extinction ratio valuesranging from 8 to 15 dB. The FSR value suggests an average group indexof ngavg ≈ 4, consistent with our theoretical group indices of ngSM = 4.177for SMWGs, and ngMM = 3.706 for MMWGs, shown in Table 3.8. Figure3.24(d) shows the Q as a function of wavelength for the CW and CCW res-onances, as well as their respective averages. The average quality factor is993.3. Third Design Cycle(a) (b)Wavelength [nm]1550 1550.5 1551 1551.5 1552 1552.5 1553Transmission [dBm]-45-40-35-30-25-20ccwcw(c)Wavelength [nm]1550 1550.5 1551 1551.5 1552 1552.5 1553Q#10522.533.544.555.56ccwccwavgcwcwavg(d)Figure 3.24: Fibre alignments upon splitting coupler in (a) dry conditions,and (b) UV curable adhesive. (c) Spectra and (d) Q factor for the CWand CCW resonances of a 7.4-mm-long resonator.Qavg ≈ 3.4× 105, more than 4 times greater than the Q of resonators fromprevious iterations.Dynamic TestsBased on spectra such as that shown in Fig. 3.24(c), the TLS wavelength wastuned to single wavelength values, to perform time-domain measurements.Wavelength values with large spectral slope near resonances were tested insearch for greater sensitivity to manual turntable rotations. The turntable1003.3. Third Design Cyclewas then rotated using sinusoidal bow-tie patterns. Figure 3.25(b) shows thesigned PWM duty cycle, as well as the corresponding output voltages of thereference gyroscope and the photodetector connected to the PM circulator.The experimental optical gyroscope resolution, limited by the setup per-formance, was δΩopt ≈ 27 dps. However, the amplitude variations showedby the optical output signal were considerably larger than those expectedsolely to the Sagnac effect. We associated this variations with the vibrationsof the characterization setup. The optical output signal also showed drift,likely due to thermal variations across the chip, despite having temperaturecontrol in the sample pedestal.3.3.4 Fibre AttachmentIn order to improve the robustness of the 3D-printed fibre holders to be usedwith future samples, I designed a second holder generation with 20-mm long,1.2-mm wide channels on their back end, for a firm and safe grasp of thefibre jacket, as shown in Fig. 3.26. The fibre trenches were deeper (0.3mm radius) and more widely spaced (2.5 mm separation) in comparison tothe first designs. Foreseeing the need for various incidence angles, I createdholders and polishing jigs with various front end angles, namely, 6, 10, 23,25, and 39 degrees. The fibres were attached to the holders and polishedusing the same setup and procedure as was described in Subsection 3.2.4.As the detachment would be heat assisted from now on, two adhesiveswith different glass transition temperatures (Tg) were used. An adhesivewith a relatively high Tg (Loctite 3492TM, Tg = 64◦C [61]) was dispensedover the trenches to attach the fibres to the holders, whereas an adhesivewith a lower Tg (Dymax 429TM, Tg = 59◦C [40]), was used for attachmentbetween fibres and samples.1013.3. Third Design Cycle(a)Signed duty cycleMEMS gyro voltage Optical gyro voltagetime [s]Output voltage [V](b)Figure 3.25: (a) Reading on VI front panel, and (b) input and outputsignals during a sinusoidal rotation test with frequency fin = 0.885 Hz.1023.3. Third Design CycleSMWG test structures from previous fabrication runs were chosen toperform the attachment tests. Samples were mounted on top of the temper-ature controlled sample pedestal shown in Fig. 3.27(a) using double-sidedadhesive tape. This was eventually replaced by a metallic vacuum chuck,as shown in Fig. 3.27(b), since the tape elasticity allowed for undesirabledisplacements during fibre positioning and adhesive curing.(a) (b)Figure 3.26: (a) Second fibre holder design schematic. (b) Holder withattached fibre on polishing jig.(a) Adhesive tape chuck (b) Vacuum chuckFigure 3.27: Sample pedestal versions.Figure 3.28 illustrates the evolution of the fibre attachment process foran air-clad device with a 10◦ incidence angle in air. SM fibres were attachedto both the input and the output fibre holders, and these were polished at1033.3. Third Design Cycle6.5◦ angles. The pitch angles and positions were adjusted as to maximizethe transmitted power without adhesive, with uncured adhesive, and duringthe curing cycles. The optimum pitch angle for each case is depicted in thefigure legends. As shown in Fig. 3.28(a), the IL improved approximately 12dB after depositing adhesive on top of the input GC, thanks to a smallerindex mismatch with the fibre. The spectral peak shifted due to the angleand position adjustments.(a) During input GC adhesive deposition (b) Input GC adhesive curing(c) Output GC adhesive curing (d) Spectrum after each subcycleFigure 3.28: Selected spectra during adhesive deposition and curing on anair-clad SMWG structure. Chip IMEC2009-R6C5.The IL increased during the curing cycles due to adhesive shrinkage.Depending on the curing time, the position of the fibres could shift and1043.3. Third Design Cycleshow hysteresis, requiring position adjustments to minimize the IL. After arest period of 5 minutes, the power increased between 1 and 3 dB for short-(10 s) and medium- (20 s) curing cycles without requiring re-positioning.Therefore, the attachment procedure was divided into several short- andmedium-length-curing cycles (10 s and 20 s, respectively), followed by long-curing cycles (60 s), carried out sequentially for the input and output fibres.Spectra obtained before, during, and after the adhesive curing cycles areshown in Figures 3.28(a), 3.28(b), and 3.28(c). Figure 3.28(d) shows spectraat the end of each cycle, as well as the spectrum after fibre detachment anda 15-minute ultrasonic bath in warm acetone.The greater IL increase and misalignment observed for longer curingcycles motivated a study of the insertion loss and peak wavelength valuesover time, for which glass-clad SMWG test structures were used. Additionalspectra were recorded during the sequential curing cycles. Figure 3.29(a)compares the spectra of alignments performed without adhesive to thosehaving uncured adhesive at the input GC. An improvement of ∼ 2 dB wasobserved after angular and positional adjustments. As expected due to therelatively small index contrast between the adhesive and the cladding, theoptimal incidence angle was only slightly changed. Figure 3.29(b) comparesspectra before and after curing the adhesive at the input GC. Five 20-s curingcycles were performed, each with a 5-minute rest. One 60-s curing cyclefollowed, and after several hours of rest, a small realignment was required.A final 60-s curing cycle followed, requiring no further realignment.Figures 3.29(c) and 3.29(d) show, respectively, the evolution over timeof spectral peak power and wavelength during the first 20-s curing cycle.Immediately after this curing cycle there was an excess loss of ∼ 1 dB.After a 5 minute rest period, the excess loss decreased to ∼ 0.5 dB with noalignment required. As per the procedure described earlier, four 20-s andtwo 60-s curing cycles followed. Figures 3.29(e) and 3.29(f) show the spectralpeak power and wavelength evolution over time for the last 60-s curing cycle.One can see that by this last cycle, the peak power and wavelength valueexcursions are smaller, and follow smoother trends.1053.3. Third Design Cycle(a) Spectra before input GC curing (b) Spectra after input GC curing(c) Peak IL, first 20-s curing (d) Peak wavelength, first 20-s curing•– –•(e) Peak IL, second 60-s curing (f) Peak wavelength, second 60-s curingFigure 3.29: Selected spectra and variations during curing on a glass-cladSMWG structure. Sample: Imec Glass cladding, R−3C−6.1063.3. Third Design Cycle3.3.5 Iteration Challenges and ConclusionsIn this iteration, we designed resonators with a combination of SMWGs andMMWGs allowed for a reduction of the overall roundtrip loss. Our fabricatedresonators showed Q factors ranging from 2×105 to 6×105, with an averageQavg ≈ 3.4 × 105. Using equation 3.3 with λ0 = 1.55 µm, and ng ≈ 4, theaverage field and power attenuation values are αavg ≈ 0.024 mm−1 andαp ≈ 0.048 mm−1, respectively. Comparing to the resonators of the pre-vious iteration, for a 6-mm length, the optical signal makes approximately3.5 roundtrips before its intensity decreases by a factor of e. For the res-onator described in Fig. 3.24, with L = 7.4 mm, the finesse is F ≈ 18.This improvement by a factor of ∼ 2 with respect to our previous iterationencouraged us to find ways to further reduce the roundtrip loss. This wasachieved using rib MMWGs, and adiabatic strip SMWG bends, as will beshown in the next subsection.Regarding layout creation, the use of scripts considerably accelerated theprocess in comparison to the manual drawing techniques used in previousiterations. The compatibility of CleWinTM with MATLABTM allowed forflexibility and a relatively flat learning curve. However, issues such as brokenwaveguides or feature spacing violations had to be identified by visual in-spection, as there is no embedded design rule checking (DRC) in CleWinTM.To avoid these issues, the layout scripting was migrated to Mentor Graph-ics PyxisTM, which despite its complexity and steep learning curve, provedto be a much more robust layout tool with capabilities such as DRC andparametric cells, that allow for creation and modification of parametrizeddevices.A turntable for testing SOI gyroscopes was built. Custom operationmodes were implemented for characterizing the apparatus and the SOI de-vices. The turntable exhibited a bandwidth of 0.54 Hz, angular speeds rang-ing from 27 to 74.3 dps, angular acceleration values from 9.4 to 171.2 dps2,and noise levels of 2 and 0.73 dps at its minimum and maximum speed,respectively. Static and dynamic tests were carried out on SOI gyroscoperesonators fabricated using an air-clad e-beam process, showing an average1073.3. Third Design Cycleresonator Q of 3.4× 105, and a resolution of 27 dps, as published in [55].The variations in the measured signals of our gyroscopes were muchlarger than those expected due to the Sagnac effect, and were attributed tovibration and mechanical stress at the fibre-DUT bonds. Although somegroups report performing rotational tests (e.g., [70]), they do not specify in-terrogation interface details, nor do they specify any measures to deal withvibrational noise. The construction of our turntable was a valuable expe-rience. It helped to identify challenges and key parameters that needed tobe addressed in subsequent iterations. The results of our tests suggestedthe need to reduce mechanical vibrations and to increase the angular speedrange. An enclosure was also required in order to reduce the deleteriousimpact of ambient temperature variations. Thus, we acquired a turntablesystem within a temperature-controlled chamber and the necessary equip-ment to create a characterization setup within it.During attachment experiments with 3D printed holders I observed thatthe optimum incidence angle changed after each facet re-polishing (θ∆p ≈± 1◦), suggesting variations due to polishing and fastening inaccuracies. Ialso noticed that occasionally the edge of the holder end face made contactwith the sample before the fibre reached the surface, thus affecting the opti-mum holder angle and position. These issues occurred in spite of performingshallow-angle polishing on acute angle holders (e.g., 6.5◦ polishing on a 25◦facet) to reduce the holder footprint.Due to 3D printing dimension tolerances (e.g., ∼ 10% variations in2.5-mm feature spacings) the distance variations between fibres precludedmounting more than one fibre per holder to form fibre arrays. Since the in-put and output GCs in these samples were separated by several millimetres,this required separate adhesive dispensing, manual alignment, and UV cur-ing of the input and output fibres. Even though the fibre holders eliminatedthe fibre bending, observed when attaching unsupported bare fibres, thealignment of separate, bulky holders occasionally produced sample shiftingand unintended fibre-sample separation. The manual fibre positioning stilllimited the position repeatability, which negatively impacted the IL leveland overall device characterization. In order to solve these issues during the1083.4. Fourth Design Cyclefollowing iteration, we used compact fibre arrays [126] instead of individualfibres, and replaced the manual positioning stages with computer-controlledmicro-positioners.3.4 Fourth Design CycleThe Q factor increase achieved in our previous iteration as a result of usingstraight MMWGs motivated us to find ways to further reduce the overallrountrip loss of our resonators. As shown in Yap et al., rib waveguides havelower propagation losses than strip waveguides due to factors that dependon the etch depth [140]. Bogaerts et al. reported losses as low as 0.27 dB/cmusing a combination straight rib MMWGs and strip SMWG adiabatic bends[15]. We decided to adopt a similar approach and submitted our designsfor fabrication to the Institute of Microelectronics (IME), Singapore, wheredevices were fabricated using a CMOS-compatible SOI process with twoetch depths: a 220-nm full etch for strip waveguides, and a 130-nm partial(shallow) etch for rib waveguides and GCs. This eventually allowed us toachieve Q factor values as high as ∼ 4.5 × 106, with an average value of∼ 1.7× 106, as shown in Subsection 3.4.3, and published in [56].Compact arrays of four PM fibres with 127-µm-pitch and custom polishangles were acquired from PLC Connections, LLC (now PLCC2, LLC [3]),for simultaneous alignment of multiple input and output fibres. Thus, fromthis iteration onwards, all of our device GCs had the same orientation and a127-µm-pitch. Unless explicitly stated otherwise, from this iteration onwardsall samples had glass cladding, which contributed to waveguide protection,IL reduction, and smaller angular adjustments during fibre attachment.A computer-controlled micro-positioning system was acquired for faster,more accurate positioning across the chips. In this new setup, the sam-ple pedestal is translated in-plane, and the fibre array is translated only inthe vertical direction. A graphical user interface (GUI) for instrument con-trol and automatic alignment was developed by members of UBC and UWresearch groups using MATLABTM.The setup is capable of automatically positioning the chip according1093.4. Fourth Design Cycleto pre-defined coordinates contained in the layout files, of performing au-tomated fine alignments, and of measuring and recording spectra for var-ious devices sequentially. In order to extend the angular speed range incomparison to that of our first rotary table, and also to reduce the expo-sure of our chips to ambient illumination and temperature variations, anIdeal Aerosmith 1291BLTM single-axis automatic turntable system with atemperature-controlled chamber was acquired.3.4.1 Layout DesignThe GCs for each device were oriented in the same direction and spacedwith a 127 µm pitch, so as to ensure compatibility with our fibre arrays. Theresonator designs combine SMWGs and MMWGs, as done in the previousiteration. However, as previously mentioned, strip waveguides would bereplaced with rib waveguides to the extent possible and used only for small-radius bends wherever required for space efficiency. To minimize the modemismatch losses, the bends consisted of adiabatic Be´zier bends with 5-µmequivalent radii [21], and 5-µm long strip SMWG stubs at both ends. 100-µm long linear tapers were used for conversion from strip SMWGs to ribMMWGs, and 50-µm long linear tapers were used to convert from stripSMWGs to rib SMWGs.Gyroscope Devices and Propagation Loss Test StructuresOn-chip splitting/merging was implemented in these designs as well. How-ever, as shown in Figure 3.30, the splitting/combining coupler was modifiedfor space efficiency, and the signal taps previously used were eliminated todecrease the excess loss. In the figure, the main I/O GC shows an arrowdepicting the optical signal entering the device. Straight strip SMWGs (1)and strip SMWG Be´zier bends (2) are used for short, space-efficient waveg-uide routing. The central Y-branch of our previous design was replaced byan adiabatic 50/50 splitter/merger, (labelled as 3 in Fig. 3.30) which usesa combination of strip waveguides (orange) and rib waveguides (drawn asblue strips with pink slabs) to evenly split the signal(s) injected to the input1103.4. Fourth Design Cyclestrip SMWG(s) [145]. Straight rib waveguides and 20µm-radius rib SMWGbends (4 and 5 respectively, drawn as brown strips with pink slabs) wereused to create a waveguide loop. A waveguide directional coupler (6) wasincorporated in the loop and is regarded as a point coupler for modellingpurposes. The directional coupler injects light into, and collects light from,the resonator (grayed out) in both propagation directions. Figure 3.31 showsthe schematic of a gyroscope resonator with its splitting/merging coupler.Only GC2 is used as an input and both GC1 and GC2 are used to detectthe optical signals from the two outputs of the device. The output of GC2is interrogated via a PM circulator.3GC 1GC 2412 56L r1L r2L r4Directional coupler-jttL r3(a)(b)Figure 3.30: Splitting/merging coupler based on adiabatic splitter. (a)Schematic. 1- Straight strip SMWGs. 2- Strip SMWG bends. 3- Adiabatic50/50 splitter/merger, input and output ports labelled in blue [145]. 4-Straight rib SMWGs. 5- Rib SMWG bends. (b) Layout schematic.Figure 3.31 also shows, as an inset, the schematic of a test structure forMMWG propagation loss characterization. Each test structure consists of afixed number of straight rib MMWG segments, Be´zier bends, and waveguidetapers. Specifically, each test structure has forty 100-µm waveguide tapers,1113.4. Fourth Design Cycleforty 90◦ Be´zier bends with 5µm equivalent radii [21], and twenty straightrib MMWG segments. Four different test structures were included in thelayout design, with total MMWG length values of 2, 56, 86, and 116 mm,respectively. In principle, these length values allow one to observe an ILdifference between the shortest and the longest test structure of ∼ 0.35 dBin the best expected scenario (αdB ∼ 0.03 dB/cm [78]), and of ∼ 11 dBin the worst expected scenario (αdB ∼ 1 dB/cm [78]). The experimentalresults obtained with these structures are described in Subsection 3.4.3.3GC 1GC 2412 561GCGC82 71287Figure 3.31: Large-area resonator, formed by straight SMWGs (1), stripSMWG adiabatic bends (2) [21], rib SMWG directional coupler (3),MMWGs (4), linear tapers (5) for SMWG to MMWG conversion, and adi-abatic 50/50 splitter (6) [145]. GC: grating couplers [137]. Inset: Teststructure for MMWG propagation loss characterization.The matrix equations governing the electric fields on the way in (left toright) and on the way out of the adiabatic splitting/merging coupler shown1123.4. Fourth Design Cyclein Fig. 3.30 are, respectively:E+5E+6== ME+3E+4 (3.11)E−3E−4== M−1E−5E−6 (3.12)where:M =1√2[1 1−1 1]. (3.13)Although during experiments only one input was excited at a time, thefields exiting the device will be described in terms of both possible inputfields. For a CW rotation of the device depicted in Fig. 3.31, using Eqs.(3.35) through (3.13), we have:E3− =ejΦascl2√CIL[−E3+ [T (ΦS) + T (−ΦS)] + E4+ [T (ΦS)− T (−ΦS)]](3.14)E−4 =ejΦascl2√CIL[−E+3 [T (ΦS)− T (−ΦS)] + E+4 [T (ΦS) + T (−ΦS)]](3.15)where Φascl = jαscrLascl − φascl, Lascl is the path length of the rib SMWGloop (drawn as brown strips on pink slabs in Fig. 3.30(a)), αscr is the ribSMWG propagation loss, φascl is the phase shift produced by the rib SMWGloop, and T (ΦS) is the phase-shifted ring transfer function defined in Eq.(3.6). The sum and difference of T (ΦS) and T (−ΦS) can be expanded asfollows:T (ΦS) + T (−ΦS) = 2 ·t(ejΦring + e−jΦring)−(t2 + 1)cos(ΦS)e−jΦring − 2t cos(ΦS) + t2ejΦring(3.16)T (ΦS)− T (−ΦS) = 2 ·j(t2 − 1)sin (ΦS)e−jΦring − 2t cos(ΦS) + t2ejΦring. (3.17)1133.4. Fourth Design CycleBased on Eqs. (3.14) through (3.17), Fig. 3.32 shows the theoretical powerspectra of the signals exiting the device, P1 =∣∣E−1∣∣2 and P2 =∣∣E−2∣∣2, whenE+1 = 0 and E+2 = 1. Since theoretically∣∣E−1∣∣2 = 0 at rest (ΦS = 0), anarbitrary noise floor of −70 dB, consistent with the noise floor of our currentmeasurement system, has been added for plotting purposes.-1.0 -0.5 0.0 0.5 1.0-70-60-50-40-30-20-100{ring/oTransmission[dB]Figure 3.32: Comparison of theoretical output power levels P1 and P2 asfunctions of the ring normalized detuning, at rest and under CW rotation,for an all-pass resonator with length L = 37 mm, coupling κa = 0.29, averagepropagation loss αdB = 0.085 dB/cm, negligible IL and splitting loop losses(ILdB = 0 dB, αscr = 0 m−1), and ΦS = 0.1pi rad. Dashed brown curve: P1at rest. Solid green curve: P1 under rotation. Orange dashed curve: P2 atrest. Purple solid curve: P2 under rotation.Resonators with Thermally-Tuneable CouplingSince fabrication imperfections can create discrepancies between the as-designed and the as-fabricated power coupling coefficients, we created amodified version of the splitting/merging coupler design (Fig. 3.30) to allowfor thermally-tuneable coupling. The thermooptic effect is the phenomenonby which the refractive index of a medium changes as a result of a change inits temperature [75, 100]. By locally heating a waveguide, its optical lengthcan be modified. This can be exploited in a Mach-Zehnder interferometerto produce a tuneable coupler as described below.1143.4. Fourth Design CycleThe splitting/merging coupler schematic is shown in Fig. 3.33. It con-sists of two sections. The first one is an adiabatic coupler, used for inputsignal splitting. The second section is the tuneable coupler. It consists ofa thermally-tuneable Mach-Zehnder interferometer (T-MZI), formed by twoadiabatic couplers and two arms consisting of straight rib SMWGs. A metalstrip is deposited on top of one of the arms to act as a resistive heater, whichmodifies the optical phase difference between the T-MZI arms due to thethermooptic effect. Since this subcomponent is based on an MZI structure,rather than a resonator, and it occupies negligible area in comparison tothat of the ring resonator (cf. ∼ 0.034 mm2 vs. ∼ 90 mm2), the effects ofrotation on its behaviour are negligible.Figure 3.33: Schematic of a thermally-tuneable splitting/merging couplerfor an IME resonator.As derived in Appendix D, the matrix equation describing the relation-ship between the fields at the ports of a T-MZI with perfectly balancedadiabatic couplers is:F1+F2+ = e−j(θ1+θ2)2[t(∆θ) jκ(∆θ)jκ(∆θ) t(∆θ)]·C1+C2+ , (3.18)where θ1 and θ2 are, respectively, the optical phases of the top and bottomarms of the T-MZI, ∆θ = θ2 − θ1 is the arm phase imbalance, produced bythe optical path difference between both arms, and the tuneable through-1153.4. Fourth Design Cycleand cross-coupling coefficients are defined, respectively, as:t(∆θ) = cos (∆θ/2) (3.19)κ(∆θ) = sin (∆θ/2) . (3.20)Figure 3.34 shows the schematic of a T-MZI test structure, created todetermine the phase shifter efficiency, ηθ, measured in units of power for a piphase shift. In this particular design only one input port is used, hence thetapered waveguide terminator at the unused input port. Our as-fabricatedT-MZI couplers have an efficiency ηθ = 24 mW/pi, according to our experi-mental data (see Subsection 3.4.3).Figure 3.34: Thermally-tuneable coupler test structure. LMZI = 200 µm.As fully derived in Appendix D, when the T-MZI coupler is connected tothe resonator, as shown in Fig. 3.35, it is possible to express F1+ in termsof C1+ as:F1+ = Ttc(ΦS)C1+, (3.21)whereTtc(ΦS) = e−jΦ(θ) ·t(∆θ)− e−j(Φring+Φ(θ)+ΦS)1− t(∆θ)e−j(Φring+Φ(θ)+ΦS)(3.22)and Φ(θ) = (θ1 + θ2)/2. By comparing Eqs. (3.22) and (3.6), one cansee that the transfer functions for the resonators with tuneable and with1163.4. Fourth Design CycleFigure 3.35: Resonator with thermally-tuneable coupler.fixed couplers have similar structures. However, for the tuneable case, thethrough-coupling, cross-coupling, resonance wavelength, and extinction ra-tio vary as functions of the phase imbalance.Figure 3.36 compares theoretical spectra for a 37 mm-long ring at variousphase detuning conditions as a function of the power delivered to the thermaltuner. The optical signal is fed and interrogated through the same GC1. Aphase shifter efficiency of ηθ = 24 mW/pi was considered in the model, basedon the experimental measurements shown in Section 3.4.3.Figure 3.37 compares theoretical spectra at both ports for a gyroscoperesonator with a tuneable splitting/merging coupler at rest and under rota-tion. The simulation parameters are: length L = 37 mm, phase imbalance∆θ = 0.1pi produced by a thermal phase shifter power Pθ = 2.4 mW, prop-agation loss α = 0.3 m−1, noise floor Nf = −70 dB, and a Sagnac phaseshift ΦS = 0.1pi in the rotational simulation. As expected from Eq. (3.22),this plot is similar to Fig. 3.32, but with values of resonance shift and ERdependent on the phase imbalance.1173.4. Fourth Design Cycle-0.4 -0.2 0.0 0.2 0.4-15-10-50ϕring/πTransmission[dB]Figure 3.36: Spectral simulation for a resonator with a thermally-tuneablecoupler at various MZI phase detuning conditions, fed and interrogatedthrough GC1. Parameters: ηθ = 24 mW/pi. L = 37 mm. ∆θ = 0.1pi,α = 0.3 m−1. Thermal phase shifter power: Pθ = 0 mW (brown dashed).Pθ = 2 mW (green solid). Pθ = 4 mW (purple dashed). Pθ = 6 mW (redsolid). Pθ = 8 mW (orange dashed). Pθ = 10 mW (blue solid). Pθ = 12 mW(magenta dashed).-1.0 -0.5 0.0 0.5 1.0-70-60-50-40-30-20-100ϕring/πTransmission[dB]Figure 3.37: Simulations of the spectral response for a resonator with athermally-tuneable coupler, at rest (dashed curves), and under rotation(solid curves, ΦS = 0.1pi). The input signal is injected into GC1, and thedevice is interrogated at both ports. Parameters: L = 37 mm, ∆θ = 0.1pi(Pθ = 2.4 mW), α = 0.3 m−1.1183.4. Fourth Design CycleAuto-Alignment Landmark DevicesFor coordinate mapping purposes, from this iteration onwards, all layoutdesign files were created with a layer dedicated for device name tags, lo-cated at the apex of the input GCs. The input GC tags must adhere toa specific syntax convention, in order to be recognized by a special scriptdeveloped in KlayoutTM by group colleagues to create text files with devicecoordinates and names. These text files were used by the software interfacefor coordinate mapping and device data identification during automatedmeasurements.The coordinate mapping procedure requires a manual alignment upon,and recording of the location of, three unambiguous landmark devices oneach chip. Recording the location of three different landmark devices acrossthe chip (preferably near three chip corners) allows one to correlate thelayout coordinates of these devices to the actual X and Y stage positions.The system is then able to carry out interpolations to the rest of the listeddevices, perform automated translations to their locations, and record mea-surements made on them automatically. Figure 3.38 shows a typical land-mark device set. Besides its coordinate mapping purpose, these devicesproved quite useful as zero-length devices during propagation loss charac-terization (Subsection 3.4.3). The rationale for having more than one device,each with a different shape, is to unambiguously identify copies of the setacross the chip. Specifically, for the set shown in Fig. 3.38, the devicespectra observed moving from top to bottom should be a through-port ringresponse, a 50%-50% power-split loopback response, and a simple loopbackresponse.Waveguide ParametersAs previously mentioned, strip SMWGs were used for straight stubs andcompact waveguide bends, whereas rib SMWGs were used in the WDCregion, and rib MMWGs were used for the straight resonator segments (la-belled 6 and 8 in Fig. 3.31, respectively). All rib waveguides had slab heightsof 90 nm, and all waveguides had strip heights of 220 nm. Table 3.12 shows1193.4. Fourth Design CycleFigure 3.38: Landmark device set, for marking chip corners and correlatinglayout coordinates to motor coordinates. Input GC name tags (illegible dueto layout snapshot settings) are shown only for illustration purposes.the strip and slab widths of various waveguides used in these designs, as wellas a-priori propagation loss ranges, based on experimental results reportedin the literature [9, 15, 78].Table 3.12: Parameter estimations for different glass-clad waveguidesXXXXXXXXXXXParameterWGSM strip SM rib MM ribStrip width (nm) 500 500 3000Slab width (nm) - 2600 5100αdB(dBcm)[9, 15, 78] 2.4 - 3 2.2 - 3.4 0.026 - 1Table 3.13 shows the effective and group index values for various ribwaveguide geometries used in the designs, obtained using MODE SolutionsTM eingenmode solver. The radius of the rib SMWG bends was chosen tobe 20 µm to minimize bending and mode mismatch losses [29]. The modemismatch loss was estimated to be −0.0065 dB, according to our simulations.1203.4. Fourth Design CycleTable 3.13: Theoretical values for effective and group indices at λ0 = 1550nm, for rib waveguides of various geometries. In all cases the strip height isHstrip = 220 nm and the slab height is Hslab = 90 nm.Strip width [nm] Radius [µm] Mode neff ng500 ∞ TE-like 2.5660 3.8797500 ∞ TM-like 2.1144 3.1898500 20 TE-like 2.5663 3.8790500 20 TM-like 2.3072 3.31901750 ∞ TE-like 2.8150 3.72731750 ∞ TM-like 2.7215 3.83673000 ∞ TE-like 2.8348 3.70493000 ∞ TM-like 2.8017 3.7456Figures 3.39(a) 3.39(b) show curve fits for the effective and group in-dices versus wavelength for rib waveguides consistent with the geometriesdescribed in Table 3.12. However, for simulation purposes, the rib waveg-uides were considered to have infinite slab widths. Each plot highlights thevalue of the effective and group indices at λ0 = 1550 nm.1.5 1.52 1.54 1.56 1.58 1.6 1.622.533.54neff(λ0)=2.566ng(λ0)=3.8797Wavelength [µm]n  neff dataneff fitneff(λ0)ng datang fitng(λ0)(a) Rib SMWG, W= 500 nm1.5 1.52 1.54 1.56 1.58 1.6 1.622.833.23.43.63.8neff(λ0)=2.8348ng(λ0)=3.7049Wavelength [µm]n  neff dataneff fitneff(λ0)ng datang fitng(λ0)(b) Rib MMWG, W= 3 µmFigure 3.39: Curve-fitted effective index (green curves) and group index(blue curves) for glass-clad rib waveguides. (a) SMWG. (b) MMWG.Figure 3.40(a) shows the cross-over length versus wavelength for rib1213.4. Fourth Design CycleSMWG directional couplers with various gaps, and Fig. 3.40(b) shows thevariation of the field-cross coupling, κ, as a function of wavelength, for cou-plers with various gaps but all of them with κ = 1/√2 at λ0 = 1550 nm.Since smaller gaps produce less coupling variation versus wavelength, a 210-nm gap was used for the layout designs.1.5 1.52 1.54 1.56 1.58 1.6 1.621015202530Wavelength [µm]L ⊗ [µm]  gap=170 nmgap=210 nmgap=250 nmgap=290 nm(a)1.5 1.52 1.54 1.56 1.58 1.6 1.620.620.640.660.680.70.720.740.760.780.8 Field κ vs. λ,Ridge DrCplr, WG width=500 nmWavelength [µm]Field cross−coupling, κ  κdesign(λ0)=0.707gap=170 nm, Lc=6.39 µmgap=210 nm, Lc=8.15 µmgap=250 nm, Lc=10.38 µmgap=290 nm, Lc=13.21 µm(b)Figure 3.40: (a) Cross-over length vs. wavelength for rib SMWG directionalcouplers with various gap values. (b) Variation of the field cross-couplingversus wavelength for rib SMWG directional couplers of various gaps, alldesigned for κ = 1/√2 at λ0 = 1550 nm.3.4.2 Setup DesignAs previously mentioned, a new characterization stage was built to improveon the opto-mechanics and data acquisition. Initially, a benchtop versionof this new stage was created, and, eventually, I built a compact versioninside a turntable chamber. Figure 3.41 shows the block diagram of thissetup. Major improvements included the use of a fibre array instead ofindividual fibres, computer-controlled positioners and a newer PXI controllerfor automated measurements, fast data recording using special drivers forfaster spectral sweeps, a more powerful TEC controller, a larger TEC for agreater temperature range, and an improved vacuum chuck for holding thesamples.1223.4. Fourth Design CycleZ-axis postMicroscope postFPGA Data acquisitionSampleTECPXI MicroscopeMulti-port detectorTECControllerXYZ positionerControllerRotary platformTurntable controllerMonitor 1 Tuneable laserFibre array XY positioners Vacuum chuckLeveling stagesZ positionerTranslation stagesAngle stagesMonitor 2 Translation stagesTurntable PCTurntable chamber Monitor 3 Single detector(s)GPIBUSBFigure 3.41: Block diagram of second rotary characterization setup.Optomechanics ImprovementsFigure 3.42(a) shows the benchtop configuration of the stage. The manuallinear stages formerly used for fibre and sample positioning were substitutedwith a computer-controlled Micronix SMCorvusTM positioning system. TheX- and Y-axis stages were assembled together to control the in-plane po-sition of a temperature-controlled sample vacuum chuck. The Z-axis stagewas mounted in an independent post to control the height of the fibre array.A two-axis tilt stage was used to perform horizon-level adjustments after1233.4. Fourth Design Cycleloading each sample. Figure 3.42(b) shows our custom 127-µm pitch, 22◦polish-angle, lidless fibre array on top of the sample vacuum chuck. This par-ticular array has four equally-oriented polarization maintaining (PM) fibres,for proper excitation of the DUT’s polarization-sensitive GCs. The same fig-ure shows the aluminium sample pedestal, with a push-to-connect fitting forvacuum connection, and wires protruding from a perforation where a ther-mistor has been permanently attached. The perforations allow the thermis-tor to be approximately 2 mm away from the sample without affecting thevacuum.(a) (b)(c) (d)Figure 3.42: (a) Automated stage, bench-top configuration. (b) Samplepedestal and fibre array. (c) Microscope image of fibre array near chipalignment features. (d) Spectra of an 84 µm-long ring resonator alignmentfeature.1243.4. Fourth Design CycleFigure 3.42(c) shows a microscope image of the fibre array tip on top ofthe GCs of the landmark device set shown in Fig. 3.38, and Fig. 3.42(d)shows the spectrum of the ring resonator landmark device, obtained usingthe automated measurement GUI. After successful automated alignmenttests using the benchtop configuration, I re-assembled the characterizationsetup on a rotary platform within a temperature-controlled chamber. Giventhe dimension restrictions, elements such as the microscope tube, posts,and post holders were substituted with shorter elements, as shown in Fig.3.43(a). I designed custom 3D-printed spacers shown in Fig. 3.43(b), tomount the fibre array on a set of two compact goniometres. These goniome-tres, in combination with a precision rotary platform, allowed for more re-peatable and precise manual adjustment of the pitch, roll, and yaw anglesthan those obtained on the benchtop configuration. Since continuous rota-tion is not possible, due to the risk of fibre entanglement, fibres and cableswithin the chamber were laid down and clamped so as to allow for rotationin a sinusoidal pattern.Instrument ImprovementsA PXIe-1062Q chassis and a PXIe-8135 2.3GHz quad-core controller controlthe setup. The Agilent 81682A TLS used in the previous setup was now usedwith a four-channel Agilent N7744a photodetector, allowing for simultaneousmeasurements of up to four device ports. Eventually, an Agilent 81960aTLS and a 8163B mainframe were acquired, allowing for faster continuoussweep rates in a wider spectral range. However, the 81960a TLS does nothave a low output power option (Pin min=6 dBm), and due to the lack ofpolarization-maintaining attenuators, the 81682A TLS was kept for testswith input power levels ranging from −13 to 6 dBm. A Stanford ResearchSystems TEC controller PTC10 [123] was acquired as to allow for a widerpedestal temperature range.1253.4. Fourth Design CycleSoftware ImprovementsInterfaces were created on various software platforms for open- and closed-loop sample positioning. Specifically, the CorvusTM positioning systemcould be controlled using either MATLABTM or LabVIEWTM for open-looppositioning, shifting each axis with user-defined displacements.(a) (b)Figure 3.43: (a) Compact configuration of the characterization setup, withinturntable chamber. (b) Sample pedestal and improved fibre array holder.The TLSs and the quad-port N7744a photodetector could be interfacedwith either MATLABTM or LabVIEW. However, being explicitly intendedfor time-domain signal acquisition and control, of the two software environ-ments, LabVIEWTM proved more reliable and efficient at acquiring time-domain data. In contrast, the automatic alignment and spectral data record-ing algorithms had already been developed in the UBC-UW MATLABTMGUI. Attempts to embed the MATLABTM GUI within LabVIEWTM as analignment sub-routine proved futile. Due to timing and driver access issues,these two interfaces could not be open simultaneously.In order to investigate resonance stability over time, I created continu-ous spectral sweep (CSS) algorithms using MATLABTM and Agilent’s FastSpectral Insertion Loss (FSIL) COM drivers. This allowed for spectrum1263.4. Fourth Design Cyclecapturing and recording at rates between 0.5 to 5 seconds per sweep. TheFSIL COM drivers were incompatible with the drivers used in the UBC-UWMATLABTM GUI, so these two elements could not be open simultaneouslyeither. Therefore, the MATLABTM GUI was used for all fine alignments, andwas closed prior to any CSS or LabVIEWTM time-domain characterization.Once the time-domain or CSS characterization started, the LabVIEWTMopen-loop positioning VI could be used for any necessary manual adjust-ment without causing any conflict.3.4.3 MeasurementsAfter the preliminary automated alignment and measurement tests were car-ried out in the benchtop configuration (e.g., see Fig. 3.42(d)), I re-assembledthe setup inside the turntable chamber and characterized the test structuresto determine the propagation loss of MMWGs. This was followed by char-acterizing the large resonators.MMWG Propagation Loss CharacterizationAs mentioned in Section 3.4.1, four MMWG test structures (MMWG TSs)with rib MMWG length values of 2, 56, 86, and 116 mm were created onthe layout design. Neighbouring loopback and ring landmark structureswere used as “zero-length” references. The MATLABTM GUI was usedto perform sequential alignments upon the MMWG TSs and neighbouringlandmark devices in various chips. At least twenty measurement cycles wererecorded for each chip. I developed automated Matlab scripts to sort thedata for each chip, obtain the average IL for each device, plot the averageIL vs. MMWG TS length, and perform propagation loss curve fits.Figure 3.44 shows the average IL as a function of wavelength for theMMWG TSs of a single chip, and the figure inset shows the correspond-ing linear fit. The average propagation losses ranged between 0.077 and0.085 dB/cm across various chips. The excess loss caused by 40 adiabaticbends and tapers was estimated to be ∼ 0.07 dB, by comparing the extrap-olated zero-length loss of the MMWG test structures to that of a reference1273.4. Fourth Design Cycleloop-back device, also shown in the Figure.1500 1510 1520 1530 1540 1550 1560 1570−60−55−50−45−40−35−30−25−20−15−10Wavelength [nm]Transmission [dBm]  L = 2 mmL = 56 mmL = 86 mmL = 116 mmReference ring0 2 4 6 8 10 12−21.4−21.2−21−20.8−20.6−20.4−20.2Length [cm]Average Insertion loss [dB]Slope = 0.085 dB/cm  Experimental dataLinear fitFigure 3.44: Spectra of test structures with various MMWG lengths, for anIME run. Also shown, spectrum of a reference loopback waveguide coupledto an 84-micrometer long racetrack resonator. Inset: Insertion loss versuslength, showing an MMWG propagation loss of 0.085 dB/cm.Resonator CharacterizationFigure 3.45 shows the block diagram of the resonator characterization tests.Resonators with both fixed and thermally-tuneable couplers were character-ized using this experimental configuration. Light was injected into the DUTvia a PM circulator and two detectors were used to record the output powerof the mixed-through port, P1 =∣∣A−1∣∣2 and the signal exiting through theinput port, P2 =∣∣A−2∣∣2. These spectra were saved for each device, and au-tomated codes were developed to obtain figures of merit (e.g., Q factor, ER,FSR, and group index) and to perform spectrum normalization and curvefitting for each resonance of the signal P2. Figure 3.46 shows the spectrafor a 37.6 mm-long tuneable-coupler resonator, with null bias current. Thetotal wavelength span (not shown in the figure for clarity) ranged from 1548to 1552 nm and encompassed over 200 resonances.1283.4. Fourth Design Cycle1 3 481 2 562 7GC 1GC 2PD 2FibrearrayLaserPD 1GPIBUSBFigure 3.45: Experimental block diagram for characterization of resonatorswith splitting/merging couplers.Wavelength [nm]1549.7 1549.8 1549.9 1550 1550.1 1550.2Transmission [dB]-45-40-35-30-25-20-15-10Figure 3.46: Spectra for a 37.6 mm-long ring resonator for the mixed throughand the return signal ports, detected at PD1 and PD2 according to Fig. 3.45.1293.4. Fourth Design CycleAs one can see in Fig. 3.46, although the resonator was at rest, a sig-nal was detected in the mixed-through port. One also can see in Fig. 3.48that the values of Q, ER, and the curve fit parameters show considerablefluctuations. We initially associated this to non-idealities of the adiabaticsplitters. However, as will be shown in Subsection 3.5.3, these fluctuationsalso appeared for rings with directional couplers without adiabatic splitters.A later study (see Subsection 3.5.4) showed that the peaks in the mixed-through port are consistent with the existence of backscattering in the ring[10]. The variations of the extinction ratio can be partially explained bythe limited resolution of the wavelength sweep step of the laser. Figure 3.47illustrates this effect by comparing simulated spectra with various wave-length step values. One can see that the variation of the ER increases as thewavelength step, ∆λ, increases. This artifact, along with the backscatteringeffects, can explain the observed ER variations in our experimental spectra.Figure 3.47: Spectra for a 37 mm-long resonator for various wavelength stepvalues.Figures 3.48(a), 3.48(b), 3.48(c), and 3.48(d) show, respectively, the Q,ER, FSR, and group index as functions of wavelength for this device. Thewavelength sweep step was 0.2 pm. The Q factor in this experiments reached1303.4. Fourth Design Cyclevalues as high as ∼ 2 × 106, with an average Qavg ≈ 1.7 × 106. Eachspectral trough was curve-fitted to Eq. (2.24) in search for the round-triploss, τ , the through transmission, ta, and the insertion loss coefficient, CIL.The detuning, φ was obtained as a function of wavelength based on theseparation of adjacent resonances. Since these were all-pass resonators, tb =1, and neglecting losses in the directional couplers, γa = 0. Thus, Eq. (2.24)became:|S21|2 =t2a − 2taτ cos (φ) + τ2CIL [1− 2taτ cos(φ) + t2aτ2](3.23)Figure 3.48(e) shows the normalized spectrum for the aforementionedresonator and superimposed curve fits for each resonance obtained usingEq. (3.23). Figure 3.48(f) shows the fit values as functions of wavelengthfor ta and τ . Due to the structure of Eq. (3.23), the round-trip loss τand the coupling ta are interchangeable parameters. In order to distinguishthem, the idea was to observe the trends of the parameters as functions ofwavelength, as τ should be fairly constant. However, it was difficult to finda trend versus wavelength due to the fluctuations in the fitted values.However, since the T-MZI current was null, it was possible to associatethe fitted values closer to 1 with the value of ta. Thermal tuning experiments(see Subection 3.4.3) allowed us to confirm that for current values of 1 mAand 3.7 mA (equivalent to thermal phase shifter power values of 1.1 mW and14.4 mW, respectively), the resonator was near critical coupling condition,and the ER was maximized.1313.4. Fourth Design CycleWavelength [nm]1548 1548.5 1549 1549.5 1550 1550.5 1551 1551.5 1552Q factor104105106107Q, Gyro9 P2avg(Q)(a) Q factorWavelength [nm]1548 1548.5 1549 1549.5 1550 1550.5 1551 1551.5 1552ER [dB]5101520253035ER1 Gyro 9, D2avg(ER1)(b) ERWavelength [nm]1550.05 1550.1 1550.15 1550.2 1550.25FSR [pm]17.0517.117.1517.217.25(c) FSRWavelength [nm]1550.05 1550.1 1550.15 1550.2 1550.25Group index, ng3.773.783.793.83.81(d) Group indexWavelength [nm]1549.8 1549.9 1550 1550.1 1550.2Normalized transmission [dB]-20-18-16-14-12-10-8-6-4-20(e) Curve fitsWavelength [nm]1548 1548.5 1549 1549.5 1550 1550.5 1551 1551.5 1552Fit parameters0.840.860.880.90.920.940.960.981fit param 1avg(fit param 1)fit param 2avg(fit param 2)(f) Fit parametersFigure 3.48: Figures of merit for a 37.6 mm-long resonator with unbiasedtuneable coupler, in various wavelength ranges. (a) Q factor. (b) ER. (c)and (d) FSR and group index, respectively, extracted from full spectrumcurve fit data. (e) Curve fits for each resonance trough. (f) Coupling androundtrip loss curve fit parameters, for each resonance. 1323.4. Fourth Design CycleTuneable Coupling ExperimentsThe samples with thermally-tuneable couplers were tested in an alternativecharacterization setup, as the electrical probes used during these experi-ments could not be accommodated within the turntable chamber withoutmajor modifications. Curves of voltage as a function of current (V-I curves)were obtained to determine the resistance of the 200 µm-long metallic ther-mal heaters. Their resistance ranged from 1070 Ω to 1200 Ω. Figure 3.49shows experimental results obtained with a T-MZI test structure similarto that shown in Fig. 3.34. The electrical current of the metal heater wasstepped, and the voltage and optical spectrum were recorded at each cur-rent value. During post-processing, the optical transmitted power at a singlewavelength (λ = 1530 nm) was plotted as a function of the thermal phaseshifter power, Pθ. Figure 3.49(a) shows the metal heater V-I curve, andFigure 3.49(b) shows the normalized optical power at the through and dropports of the test structure as a function of Pθ. The metal heater resistance,insertion loss, and phase shifter efficiency for this device were, respectively,R = 1106.5 Ω, IL=16.3 dB, and ηθ = 24 mW/pi.Current [mA]0 2 4 6 8 10Voltage [V]0246810R=1106.5 +(a)Heater Power [mW]0 20 40 60 80 100 120Normalized Output Power00.20.40.60.81Through portCross port(b)Figure 3.49: T-MZI test structure experimental results. (a) V-I curveto determine metal heater resistance, R=1106.5 Ω. (b) Normalized opticaloutput power vs. heater power, at λ = 1530 nm. Minimum IL: 16.3 dB.Spectral sweeps over a 300-pm span were obtained for a 37 mm-long1333.4. Fourth Design Cycleresonator with tuneable coupler. The phase shifter currents ranged from 0to 7.9 mA, in 0.1-mA steps. Figure 3.50 shows the spectra of the resonator’sreturn signal via the PM circulator (dotted curves) and the mixed-throughport signal (dashed curves) with the thermal phase shifter power, Pθ, as aparameter. For clarity, the plot range and has been restricted to 80 pm, andthe values Pθ shown correspond to shifter currents ranging from 0 to 4 mA,in 0.5-mA steps. One can see changes in the return signal extinction ratioand the resonance wavelength as the current is increased. One can also seesimilar changes for the mixed-through signal, which is expected to be zeroin the absence of rotation. Its appearance can be explained based on thebackscattering model shown in Subsection 3.5.4.Wavelength [nm]1550 1550.02 1550.04 1550.06 1550.08Transmission [dB]-60-50-40-30-200.00 mW0.28 mW1.11 mW2.49 mW4.43 mW6.92 mW9.96 mW13.55 mW17.70 mWFigure 3.50: Resonator spectra during coupler thermal tuning, with thermalphase shifter power as a parameter. Heater resistance: 1100 Ω. Phase shiftercurrent range: 0 to 4 mA, in 0.5 mA steps. Dotted curves: Return signal(on circulator’s port 3). Dashed curves: Mixed-through port signal.1343.4. Fourth Design CycleFigure 3.52 shows the evolution of the spectra and figures of merit ofthe return signal as the phase shifter power Pθ is varied. Figures 3.52(a)and 3.52(b) show, respectively, the spectra and ER versus wavelength forPθ from 0 to 6.37 mW, corresponding to thermal phase shifter current (iθ)values from 0 to 2.4 mA, with steps ∆iθ = 0.2 mA). One can see that themeasurement for Pθ = 1.11 mW is the closest to critical coupling condition,as the ER is maximum. The remaining subfigures of Fig. 3.52 show resultsfor a wider Pθ range, from 0 to 27.7 mW, using a finer current step ∆iθ =0.1 mA. For clarity, these subfigures show only average values of each figureof merit as functions of Pθ. Figs. 3.52(c), 3.52(d), 3.52(e), and 3.52(f) show,respectively, the average ER, average Q factor, average FSR, and the averagecurve fit parameters corresponding to the straight-through field transmissiont(θ), and roundtrip loss τ , for Pθ ranging from 0 to 27.7 mW.Based on Fig. 3.52(f) one can unambiguously discern between t(θ) andτ , as only t(θ) varies sinusoidally as Pθ is increased, whereas τ is fairlyconstant. The resonator is initially overcoupled, reaches its maximum ERat critical coupling, near Pθ = 1.1 mW and Pθ = 13.6 mW. The ring isundercoupled for 1.1 mW < Pθ < 13.6 mW, and overcoupled for 13.6 mW <Pθ < 27.7 mW. The imperfect fits for t(θ) and τ near critical coupling aredue to the fact that an infinite extinction ratio would be required to makeboth parameters identical. This is also true for non-fit-based parameterestimations, such as those described in [86].To properly estimate the Q factor in heavily over- and under-coupledconditions, it is important to point out that for an all-pass resonator, theFWHM is located at half the depth of the spectral trough in a linear scale [14,33], as shown in Fig. 3.51, rather than being fixed 3 dB below the maximumtransmission value. Figure 3.51 depicts two cases: one slightly overcoupled(t=0.8, τ = 0.88), and the other one, heavily overcoupled (t=0.42, τ = 0.88).As one can see in the figure, the denomination of −3 dB linewidth canlead to erroneous estimations, as strictly, the FWHM can only be locatedat the −3 dB line for a critically coupled resonator. For high extinctionratio resonators (ER∼ 1), the −3 dB approximation is valid. However, theseparation between the two points intersected by a horizontal line at −3 dB1353.4. Fourth Design Cyclebecomes narrower as the extinction ratio decreases. Thus, the FWHM andQ factor can be mistakenly under- and over-estimated, respectively. Takinginto account the finite extinction ratio ER < 1, the normalized transmissionat FWHM for an all-pass resonator is, in linear scale:T (φfwhm1,2) = 1− 0.5 · ER, (3.24)where φfwhm1,2 are the two detuning values at FWHM, ER = 1 − T (φres)is the extinction ratio (expressed in linear scale), and φres is the normalizedpower transmission at resonance. When processing experimental data, theFWHM is obtained as the difference of the wavelength values correspondingto φfwhm1,2, and Q is obtained as Q = λ0/FWHM. All Q factor estimationsshown in Fig. 3.52(d) have been obtained using this procedure. The high-est quality factors were Q ≈ 4.5 × 106, achieved in heavily undercoupledconditions.Figure 3.51: Comparison of FWHM values for rings with various couplingconditions.1363.4. Fourth Design CycleWavelength [nm]1550 1550.0187 1550.0375 1550.0562 1550.0749Transmission [dB]-40-35-30-25-20-150.00 mW0.04 mW0.18 mW0.40 mW0.71 mW1.11 mW1.59 mW2.17 mW2.83 mW3.59 mW4.43 mW5.36 mW6.37 mW(a)Wavelength [nm]1550 1550.05 1550.1 1550.15 1550.2 1550.25 1550.3ER [dB]051015200.00 mW0.04 mW0.18 mW0.40 mW0.71 mW1.11 mW1.59 mW2.17 mW2.83 mW3.59 mW4.43 mW5.36 mW6.37 mW(b)Thermal power [mW]0 5 10 15 20 25 30Average ER [dB]05101520(c)Thermal power [mW]0 5 10 15 20Average Q factor [dB]#10611.522.533.54(d)Thermal power [mW]0 5 10 15 20 25 30Average FSR [pm]17.117.1517.217.2517.3(e)Thermal power [mW]0 5 10 15 20 25 30Average fit value0.30.40.50.60.70.80.91ta=(f)Figure 3.52: Selected spectra and figures of merit for a 37 mm-long resonatorwith thermally tuneable coupler. (a) Return signal spectra. (b) Extinctionratio (ER) vs. wavelength. (c) Average ER vs. thermal phase shifter power,Pθ. (d) Average Q-factor vs. Pθ. (e) Average FSR vs. Pθ. (e) Averagestraight-through field transmission, t(θ), and roundtrip loss, τ , vs. Pθ. 1373.4. Fourth Design CycleThe average values of the the roundtrip loss and the group index were,respectively, τavg = 0.925 and ng avg = 3.8. Thus, αavg = − ln(τ)/L =2.118 m−1, and:Qi =pingαavgλ0≈ 3.63× 106 (3.25)The coupling quality factor, Qc, and t(θ) are related by [21, 68, 141]:Qc =−pingLλ0 ln(t(θ)), (3.26)and the total Q factor is:Q =QiQcQi +Qc. (3.27)Fast Spectral SweepsAs mentioned earlier, automated measurements were carried out sequentiallyfor several devices on each chip. Any particular device was re-measured af-ter a period that depended on the number of devices in the run. On aspecific chip, we carried out repeated tests at various input power levelson multiple devices. Figure 3.53 shows the measurements carried out ona 38 mm long resonator at various input power levels. The measurementsshown in this figure were taken over a time span of 1 hour. These measure-ments showed resonance shifts in spite of the fact that the sample pedestalwas temperature-controlled. Similar resonance shifts were observed betweenmeasurements made on a single device at different times, even using a con-stant input power (0 dBm). If no translation was necessary, the MATLABTMGUI measured a spectrum in approximately 60 to 90 seconds. In order toobserve the resonant behaviour over time, measurements at faster sweeprates than those achievable with the MATLABTM GUI were necessary.1383.4. Fourth Design Cycle1549.9 1549.92 1549.94 1549.96 1549.98 1550 1550.02 1550.04 1550.06 1550.08 1550.1-60-55-50-45-40-35-30GyroRing7, A2 Chip19, Through PortPin= -13 dBm Pin= -10 dBmPin= -8 dBmPin= -6 dBmPin= -3 dBmPin= 0 dBm1549.9 1549.92 1549.94 1549.96 1549.98 1550 1550.02 1550.04 1550.06 1550.08 1550.1-30-25-20-15GyroRing7, A2 Chip19, Circulator Port 3Pin= -13 dBm Pin= -10 dBmPin= -8 dBmPin= -6 dBmPin= -3 dBmPin= 0 dBmFigure 3.53: Resonances for a static, 38 mm-long ring at various input powerlevels, on a temperature-controlled pedestal, at 25◦C.Based on information provided by applications engineers from KeysightTechnologies, Inc., I created scripts for fast sweeps and data recordingat high repetition rates using proprietary COM drivers compatible withMATLABTM. The sweep rate for each run depended on its wavelengthspan, wavelength sweep step, and sweep speed. The recording rates rangedfrom 0.5 to 5 seconds per sweep.To investigate whether the laser could be a source of noise, stability testswere carried out with a large ring resonator and an acetylene cell (C2H2), agas with stable absorption lines in the C band [93]. These tests were carriedout using a 0.1 pm wavelength step, the smallest possible with our TLSs.Figure 3.54(a) shows the superimposed spectra for a 300-sweep test usingthe C2H2 cell. The input power was 6 dBm, and the sweep rate was 40 nm/s.The average sweep time was 0.7 s. During post-processing, two points weretracked in each spectrum to obtain statistical data. The resonance wave-length and its transmitted power were recorded for each spectrum. A fixedoff-resonance wavelength value was chosen in the first spectrum, and thetransmitted power at that wavelength was tracked in the remaining spec-1393.4. Fourth Design Cycletra to assess the off-resonance power variation over time. Figure 3.54(b)compares the transmitted power vs. time for the selected wavelength values.Figure 3.54(c) shows the resonance wavelength value over time. The averageresonance wavelength was λ0 = 1530.3651 nm, and the standard deviationof the resonance wavelength was σ(λ0) = 0.239 pm. The relative powerfluctuation off-resonance, σ(P0)/P0, was 3.44 × 10−4. Fig. 3.54(d) shows ahistogram for the resonance wavelength values.Figure 3.55 shows plots for a similar test using a 37 mm-long ring res-onator. The resonance wavelength average value was 1550.230 nm, thestandard deviation of the resonance wavelength was 0.255 pm. The rela-tive power fluctuation off-resonance for the resonator was 3.53× 10−2. Fig.3.54(d) shows a histogram for the resonance wavelength values.The resonance wavelength variation for the acetylene cell and the res-onator were similar, suggesting that the variation was mainly caused bythe laser wavelength accuracy rather than by an effect internal to the res-onator. Furthermore, additional tests with the acetylene cell showed thatthe resonance wavelength fluctuation depended on the sweep speed of thelaser. However, the off-resonance power fluctuations cannot be explainedby the laser power stability, because the variations were much larger for theresonator than for the acetylene cell.1403.4. Fourth Design Cycle(a) (b)(c) (d)Figure 3.54: Resonance dip stability test for an acetylene (C2H2) cell. (a)300 superimposed spectra, showing markers tracking transmitted power atresonance (red asterisks) and an arbitrary off-resonance wavelength (bluestars). (b) Comparison of transmitted power levels at selected wavelengths,and their ratio, over time. (c) Resonance wavelength over time. (d) Reso-nance wavelength histogram.1413.4. Fourth Design Cycle(a) (b)(c) (d)Figure 3.55: Resonance dip stability test for a 37 mm-long resonator. (a)300 superimposed spectra, showing markers tracking transmitted power atresonance (red asterisks) and off-resonance (blue stars). (b) Comparison oftransmitted power levels at selected wavelengths, and their ratio, over time.(c) Resonance wavelength over time. (d) Resonance wavelength histogram.Time-Domain MeasurementsFigure 3.56 shows the front panel of a LabVIEWTM interface developed formonitoring and recording the power over time at a single wavelength forour Agilent N7744a quadruple photodetector. Statistical analysis featureswere also created to automatically calculate the mean, standard deviation,and SNR of linear scale power measurements. The experimental SNR values1423.4. Fourth Design Cyclewere obtained as the ratio of the mean of the power divided by its standarddeviation. For consistency, all measurements taken had an approximatelyequal number of data points.Figure 3.56: Front panel of the LabVIEWTM VI for time-domain measure-ments with the N7744a photodetector.The noise floor was estimated at each value of the photodetector powersensitivity, by connecting light dumps to its ports and carrying out 22000-point measurements using an integration time of 50 µs. Figure 3.57(a)shows the noise floor as a function of the photodetector sensitivity. Sincethe most commonly used sensitivity settings in our experiments were either−20 dBm or −10 dBm, our experimental noise floor value was consideredto be −70 dBm.The greatest achievable SNR at various input power levels was esti-1433.4. Fourth Design Cycle−30 −25 −20 −15 −10 −5 0 5 10−85−80−75−70−65−60−55−50−45Detector range [dBm]Noise floor [dBm](a) Noise floor−40 −30 −20 −10 0 100510152025303540Input Power [dBm]SNR, IL [dB]  SNRdB, tavg=50 µsILdB, tav=50 µsSNRdB, tavg=100 µsILdB, tav=100 µsSNRdB, tavg=500 µsILdB, tav=500 µsSNRdB, tavg=1000 µsILdB, tav=1000 µs(b) patch cord IL and SNRFigure 3.57: (a) N7744a photodetector noise floor at various photodetectorsensitivity values, for 22000 samples at a 50 µs averaging time (11 s mea-surements). (b) Insertion Loss and SNR as a function of input power for a3 m long PM patch cord, for various integration times.mated by connecting the laser to the photodetector using a connectorizedpolarization-maintaining (PM) patch cord. The patch cord length was 3meters, similar to that of the PM fibre array. SNR and IL tests, with inte-gration times ranging from 5 µs to 2 ms, were carried out for input powerlevels ranging from −30 to 10 dBm at a single wavelength (λ = 1556 nm),as shown in Fig. 3.57(b).As expected, the IL was fairly constant, and the SNR was slightly im-proved for longer integration times. Considering the trade-off between SNRimprovement and acquisition time, integration times in the range of 50 to200 µs were deemed appropriate for subsequent experiments, and the re-maining tests for tavg ≥ 500 µs were discontinued. Similar SNR tests werecarried out with two PM circulators, showing that they did not significantlyimpact the achievable SNR. Details can be found in Appendix C.Further time-domain tests were carried out with PM patch cords in or-der to determine the frequency content of the signals at the photodetectors.Measurements were recorded over a period of 2 minutes, using a constantinput power and wavelength, with averaging times of 50 and 100 µs. Duringpost-processing, the power spectral density (PSD) plots, normalized to unit1443.4. Fourth Design Cyclepower, were obtained. Figure 3.58 compares PSD plots obtained in variousexperiments using the two aforementioned averaging times. The experi-ments were repeated several times to ensure consistency. Spurious signalsare consistently observed at 120 Hz, due to the cavity vibration caused bythe TLS wavelength tuning control. Similar tests were carried out on aloopback device to compare the PSDs of signals obtained with the pedestalvacuum pump on and off. The SNR values for the pump on and off are,respectively, 24.2 and 28.8 dB. Figure 3.59 shows the PSDs for each case.When the vacuum pump is on, the noise level is greater in the frequencyrange between ∼ 10 and ∼ 100 Hz. The time-domain plots in Fig. 3.60shows a slight IL decrease when the vacuum is turned off. Thus, the vac-uum pump should be turned off once the fibre array was properly attachedto the samples.Figure 3.58: Comparison of noise PSD plots normalized to unit power,based on autocorrelations various time-domain tests for a 3 m-long patchcord. The legend shows the integration time for each run.1453.4. Fourth Design CycleFigure 3.59: Comparison of noise PSD plots normalized to unit power, basedon autocorrelations of time-domain tests for a loopback device, with thepedestal vacuum pump turned on and off. Integration time: 50 µs.Figure 3.60: Comparison of time-domain signals for a loopback device withvacuum pump turned on and off. For both tests the integration time is50 µs. IL: 10.9 dB with vacuum pump on, 10.8 dB with vacuum off.1463.4. Fourth Design Cycle3.4.4 Iteration Challenges and ConclusionsIn this iteration, we designed resonators with a combination of strip SMWGs,rib SMWGs, and rib MMWGs. This allowed us to further reduce the prop-agation losses, as compared to our previous designs. The propagation lossesof our rib MMWGs ranged from 0.077 to 0.085 dB/cm. Without thermaltuning of the coupling, our as-fabricated resonators show Q factors as highas ∼ 2× 106, with an average Qavg ≈ 1.7× 106, as published in [56]. Thisis, to the best of our knowledge, the highest Q values reported to date forresonators fabricated using a standard CMOS-compatible process. Q factorsas high as ∼ 4.5× 106 were achieved by thermal tuning of the coupling.Using equation (3.3) with λ0 = 1.55 µm, and ng ≈ 3.7, the averagefield and power attenuation values are αavg ≈ 0.004 mm−1 and αp ≈0.008 mm−1, respectively. As compared to the previous iteration’s res-onators, for a 6-mm length, the optical signal makes approximately 19roundtrips before its intensity decreases by a factor of e. This was a sig-nificant improvement, by a factor of ∼ 5, with respect to those previousresonators. For the resonator described in Fig. 3.46, with L ≈ 37.6 mm,the optical signal makes approximately 3 roundtrips before its intensity de-creases by a factor of e. The average finesse of these rings is 18.5 withoutthermal tuning.Mathematical models were developed to describe the theoretical be-haviour of devices designed for this run with fixed and with thermally tune-able splitting/merging couplers. Thermal tuning experiments allowed us toconfirm the expected behaviour, as well as to compare figures of merit forresonators in over-coupled, under-coupled, and critically-coupled conditions.The Q factor estimations at critical coupling were in average Qcc ≈ 1.9×106,with a finesse F ≈ 22.5. For heavy under-coupling, the average Q value ap-proached the estimated intrinsic Q value of the resonator, Qi ≈ 3.7× 106.The use of computer controlled positioning allowed for automated mea-surements of several devices across each chip, and for obtaining the spectrumof each device in a period of 60 to 90 seconds. Continuous spectral sweep(CSS) algorithms were implemented in order to perform faster sweeps at1473.5. Fifth Design Cyclerates of 0.5 to 5 seconds per sweep, depending on the sweep parameters.This reduced the time required to measure devices across a chip from a fewdays to several hours.The standard deviation of the resonance wavelength in the long-termcontinuous spectral sweep (CSS) tests (σ(λ0) ≈ 0.44 pm) was greater thanwhat was expected according to the 0.2-mK standard deviation of the pedestaltemperature (i.e., σT (λ0) ≈ 0.014 pm). The tests carried out with theacetylene absorption cell suggested that this variation was produced by thewavelength accuracy of the tuneable laser source. The variation of the trans-mitted power off-resonance for the large resonators suggested non-thermalsources of power variation, such as vibration and position drift.A time-domain interface was developed using LabVIEWTM. This inter-face was used to obtain the experimental SNR values for different systemcomponents and devices at constant values of power and wavelength. Thebest SNR value achieved with the experimental setup was ∼ 40 dB, ob-tained with connectorized PM patch cords. Power spectral density (PSD)plots were obtained for signals through the PM patch cords and SMWGloopback devices. The PSDs showed that over most of the frequency range,the laser noise is considerably smaller than that of the characterization stage.Thus, we concluded that the laser noise is not the main limitation for thesystem performance.3.5 Fifth Design CycleDuring our experiments in the previous iteration, the resonators showedvariations in the transmitted power that could not be attributed to thelaser power stability. During this iteration, we designed devices consistingof two adjacent resonators with GCs laid out so as to allow for simultaneousmeasurements. Performing simultaneous measurements in both resonatorscould help us determine whether or not the transmitted power variationswere correlated across different zones of the chip as well as help us eliminateundesirable common-mode signals.During this iteration, we also developed a lumped-reflector backscatter-1483.5. Fifth Design Cycleing model for the ring resonators, motivated by the non-zero signals detectedat the mixed-through port of the devices from the previous iteration and bythe backscattered signals detected in experiments carried out during thisfifth cycle.3.5.1 Layout DesignsFigure 3.61 shows the schematic of the resonator devices created duringthis iteration, combining SMWGs and MMWGs as previously done. Sincethe lengths and aspect ratios of the resonators could not be equal due tolayout area constraints, we decided to maximize the enclosed area of thelarger resonator to make them more sensitive to rotation, while the smallerresonators could be used as a reference to eliminate deleterious common-mode signals.5 55 55 55 522 21GCGC111 1134GCGC5554Reference ringGyroscope ring35Figure 3.61: Dual resonator set. The smaller ring was created as a referencefor tracking environment-related common-mode signals.These designs had simple SMWG directional couplers to allow for estima-tions of the backscattering in our rings and also to investigate the feasibilityof external phase modulation for backscattering noise reduction [120], whichwas not possible to assess with the splitting/merging couplers used in theprevious design cycle. MMWG propagation loss test structures similar tothose shown in the inset of Fig. 3.31 were created to assess the propagation1493.5. Fifth Design Cycleloss in samples fabricated using e-beam lithography.3.5.2 Waveguide ParametersDesigns were submitted for fabrication to IME Singapore (IME) and theNanofabrication Facility at University of Washington (UW). The designssubmitted to IME used rib waveguides for the SMWG directional couplersand the MMWG straight segments. The designs submitted to UW wereentirely strip-waveguide based. All chips were glass clad for waveguide pro-tection and IL reduction.Table 3.14 shows the values for effective and group indices for variousstrip waveguide geometries. Table 3.15 summarizes the propagation lossparameters for various types of waveguides used during this iteration. Inthe case of the rib MMWGs, the propagation loss estimation was based onthe experimental results obtained in the previous iteration.Table 3.14: Theoretical values of effective and group indices at λ0 =1550 nm, for strip waveguides of various geometries. In all cases the stripheight is Hstrip = 220 nm.Strip width [nm] Radius [µm] Mode neff ng500 ∞ TE-like 2.4435 4.1772500 ∞ TM-like 1.7703 3.7043500 20 TE-like 2.4432 4.1769500 20 TM-like 1.7708 3.70051750 ∞ TE-like 2.8121 3.73431750 ∞ TM-like 2.7087 3.87093000 ∞ TE-like 2.8343 3.70643000 ∞ TM-like 2.7993 3.75171503.5. Fifth Design CycleTable 3.15: Parameter estimations for different glass-clad waveguides``````````````ParameterWGSM strip SM rib MM rib MM stripStrip width (nm) 500 500 3000 3000Slab width (nm) - 2600 5100 -αdB(dBcm)[9, 15, 56, 78] 2.4 - 3 2.2 - 3.4 ∼ 0.085 0.1 - 13.5.3 MeasurementsE-beam MMWG Propagation LossesFollowing the same procedure described in Subsection 3.4.3, spectral sweepswere carried out for MMWG TSs on an e-beam sample. An average propa-gation loss of 0.55 dB/cm was estimated for the e-beam MMWGs. Figure3.62 shows the spectra and the propagation loss for these devices.1500 1510 1520 1530 1540 1550 1560 1570 1580−60−55−50−45−40−35−30−25−20−15−10Wavelength [nm]Total insertion loss [dB]  L = 0 mmL = 2.34 mmL = 2.34 mmL = 2.34 mmL = 56 mmL = 116 mm(a)0 2 4 6 8 10 12−30−28−26−24−22−20Length [cm]Average insertion loss [dB]  Slope = 0.551 dB/cmExperimental dataLinear fit(b)Figure 3.62: Spectra of similar MMWG propagaqtion loss test structures,for e-beam samples, with a SMWG loopback as a zero-length reference.Insertion loss as a function of length, showing an MMWG propagation lossof 0.55 dB/cm.1513.5. Fifth Design CycleDual Resonator Device CharacterizationFigure 3.63(a) schematically depicts an experiment for characterizing thedual ring devices. The input signal is split off-chip using a 50/50 PM splitter.Spectral sweeps were carried out for both rings simultaneously. After thesespectral sweeps, the laser wavelength was tuned to fixed wavelength val-ues to perform time-domain tests under various conditions. Figure 3.63(b)shows the spectra obtained at various sweep speeds for a dual resonatorset. The gyro resonator has a length L1 = 32.8 mm, and the referenceresonator has a length L2 = 11.4 mm. One can see that the sweep speedimpacts the measurement noise in a counter-intuitive manner. Specifically,the slowest sweep speed, with the greatest integration time, produced thenoisiest spectra. Figure 3.64 shows the Q, ER, FSR, and group index asfunctions of wavelength for this device, as well as resonance curve fits andfit parameters. The wavelength sweep step was 0.1 pm. Outlying Q factorvalues above 12×106 were discarded as they were considered to be unreliableestimations, due to the minimum wavelength resolution of the laser.When compared to the resonators characterized in the previous itera-tion, one can notice that the Q factor values for these newer samples aresmaller, and that the values of the FSR and the group index show greatervariations, which impeded obtaining reliable fits of these two figures of meritas functions of wavelength. This could be due to greater levels of backscat-tering [10]. We conducted further experiments using one circulator per res-onator to assess the backscattering in our devices, as per the block diagramin Fig. 3.65. Figure 3.66 shows the experimental spectra of the forward-and backward-propagated signals for a dual resonator set. For both res-onators, the back-propagating signals are ∼ 15 dB weaker than the for-ward signals. This assessment was qualitative, as the GCs also contributeto backreflections on the order of ∼ −16 dB [138]. Nevertheless, the evi-dence of backscattering and its impact on the figures of merit motivated usto develop the backscattering model described in Subsection 3.5.4. Basedon the spectral sweep results, an appropriate wavelength value was chosen(λ = 1550.3432 nm) to record signals for both rings off-resonance in the time-1523.5. Fifth Design Cycledomain. A considerable improvement of the SNR in the forward-propagatingsignals was noticed when the microscope light was turned off, namely, from−0.4 dB to 19.25 dB. In contrast, the SNR for the back-propagating signalsdecreased slightly, from 2.21 to 2.05 dB.5 21GCGC 34GCGCReference ringGyroscope ring350/50 PMsplitterLaserDet 1Det 2 Fibrearray(a)(b)Figure 3.63: (a) Experimental block diagram, and (b) spectra at rest atvarious TLS sweep speeds, for a dual resonator system fabricated in e-beamtechnology. Gyro resonator length: L1 = 32.8 mm. Reference resonatorlength: L2 = 11.4 mm.1533.5. Fifth Design CycleWavelength [nm]1550 1551 1552 1553 1554 1555Q factor105106107Q, GyroRing 4avg(Q)(a) Q factorWavelength [nm]1550 1551 1552 1553 1554 1555ER [dB]1516171819202122232425ER,GyroRing 4avg(ER)(b) ERWavelength [nm]1550.5 1551 1551.5 1552 1552.5 1553FSR [pm]171819202122232425(c) FSRWavelength [nm]1550.5 1551 1551.5 1552 1552.5 1553Group index33.23.43.63.844.24.4(d) Group indexWavelength [nm]1551 1551.125 1551.25 1551.375 1551.5Normalized transmission [dB]-25-20-15-10-505(e) Curve fitsWavelength [nm]1550.5 1551 1551.5 1552 1552.5 1553Fit parameters0.70.750.80.850.90.951fit param 1avg(fit param 1)fit param 2avg(fit param 2)(f) Fit parametersFigure 3.64: Figures of merit for a 32.1 mm-long resonator fabricated usinge-beam lithography. (a) Q factor, (b) ER, (c) FSR, and (d) group index.(e) Curve fits for various resonance troughs. (f) Curve fit parameters.1543.5. Fifth Design CycleGCGCGCGCReference ringGyroscope ringPD 1PD 3Fibrearray50 / 50 PMsplitterLaserPD 4 PD 2DUTGPIBUSBFigure 3.65: Experimental block diagram for a dual resonator system.Figure 3.67 compares the PSD plots for time-domain signals recordedwith the microscope light on and off, for the forward- and backward-propagatingsignals for the 32 mm-long gyro resonator. The trends shown by these plotsare consistent with the SNR readings. Measurements for both rings off-resonance followed, to determine whether there was correlation between thesignals of both rings. The cross-correlation of the forward-propagating sig-nals for both resonators was 0.92 and 0.82, with the microscope light onand off, respectively. This indicated that the sources of noise for these ringswere strongly correlated, and that the microscope light was a common-mode noise source, and should be turned off during experiments. The highcross-correlation also indicated that the use of a reference ring could helpameliorate the quality of the signal reading.1553.5. Fifth Design Cycle1550.28 1550.3 1550.32 1550.34 1550.36 1550.38 1550.4−75−70−65−60−55−50−45−40−35−30−25−20Wavelength [nm]Power [dBm]  Gyro4, throughGyro4, CircP3Ref4, throughRef4, CircP3Figure 3.66: Spectra for the forward- and back-propagating signals of adual resonator device. L1 ≈ 32 mm, L2 ≈ 12 mm. For both resonators, theback-propagating signals are ∼ 15 dB weaker than the forward-propagatingsignals.Frequency [Hz]10-210-1100101102103104Normalized PSD [1/Hz]10-1510-1010-5100105Gyro4 Fwd, Light ONGyro4 Fwd, Light OFF(a) Forward-propagatingFrequency [Hz]10-210-1100101102103104Normalized PSD [1/Hz]10-2010-1510-1010-5100105Gyro4 BS, Light ONGyro4 BS, Light OFF(b) Back-propagatingFigure 3.67: Normalized power spectral density comparison with micro-scope light on and off, for (a) forward-propagating and (b) back-propagatingsignals in a 32 mm-long gyro resonator.1563.5. Fifth Design CycleFibre Array Attachment and Dynamic TestsFigure 3.68 shows the block diagram for the rotational experiments. Avoltage signal proportional to the turntable speed was monitored via theFPGA FIFO. An external photodetector was also monitored via the FPGAFIFO to measure any backscattering from the main 50/50 splitter.GCGCGCGCReference ringGyroscope ringPD 1PD 3Fibrearray50 / 50 PMsplitterLaserPD 4 PD 2Ext PD 1DUTTurntableAngularSpeedGPIBUSBFPGAFigure 3.68: Experimental block diagram for rotational tests a dual res-onator system.The chip was held with vacuum, and optimal alignment coordinates wererecorded using the LabVIEWTM positioning VI. The fibre array was raisedso as to allow for deposition of low shrinkage UV-curable adhesive (Dymax4-20418 [41]), then returned to its original optimal position. The adhesivewas cured in various cycles of progressive duration, as previously describedin Subsection 3.3.4. The vacuum pump was turned off after curing. Figure3.69 shows spectra for forward- and backward-propagating signals beforeand after fibre array attachment. An increase in the power of the backward-1573.5. Fifth Design Cyclepropagating signals was observed (cf. IL ≈ −35 dB without adhesive toIL ≈ −25 dB with cured adhesive).15101520153015401550156015701580−70−65−60−55−50−45−40−35−30−25−20Wavelength [nm]IL [dB]Sample EB 486A UR, Chip3Acetoned01, GyroandRef4 acetone1, WideRange, Tavg=12 us  FA5, 80 nm/sRedCirP3, 80 nm/sFA2, 80 nm/sBluCirP3, 80 nm/s(a) Before adhesive deposition1500151015201530154015501560−60−55−50−45−40−35−30−25−20Wavelength [nm]IL [dB]Sample EB 486A UR, Chip3Acetoned01, GyroandRef4 acetone1, WideRange, Tavg=20 us  FA5, 50 nm/sRedCirP3, 50 nm/sFA2, 50 nm/sBluCirP3, 50 nm/s(b) After adhesive deposition1500151015201530154015501560−60−55−50−45−40−35−30−25−20Wavelength [nm]IL [dB]Sample EB 486A UR, Chip3Acetoned01, GyroandRef4 acetone1, WideRange, Tavg=20 us  FA5, 50 nm/sRedCirP3, 50 nm/sFA2, 50 nm/sBluCirP3, 50 nm/s(c) After curing1512.515131513.51514−50−45−40−35−30−25−20Wavelength [nm]IL [dB]Sample EB 486A UR, Chip3Acetoned01, GyroandRef4 acetone1, NarrowRange, T avg=20 us  FA5, 5 nm/sRedCirP3, 5 nm/sFA2, 5 nm/sBluCirP3, 5 nm/s(d) After curing, zoom-inFigure 3.69: Gyro and reference resonator spectra of forward- and back-propagating signals. (a) Before adhesive deposition. (b) After adhesivedeposition, before curing. (c) After adhesive curing. (d) Narrow-rangesweep after adhesive curing.In this iteration I decided not to re-adjust the pitch angle, to ensureoptimal alignment when lowering the array back to its original position.This produced a blue shift in the working wavelenths, (cf. Figs. 3.69(a) and3.69(b)). One can see that the power of the back-propagating signals becamecomparable to those of the forward-propagating signals after adhesive de-position. Figures 3.69(c) and 3.69(d) show, respectively, wide- and narrow-1583.5. Fifth Design Cyclerange spectra after adhesive curing. The IL of the forward-propagatingsignals is approximately the same before and after fibre attachment.Subsequent spectral sweeps after fibre attachment showed no variationof the spectra at input power levels between −13 dBm and 3 dBm, as shownin Fig. 3.70(a). Figure 3.70(b) shows a narrow spectral sweep at rest, withPin = 0 dBm, for wavelength selection prior to a rotational test. For clarity,only the forward-propagating signal of each resonator is shown.1512.9 1513 1513.1 1513.2 1513.3 1513.4 1513.5−55−50−45−40−35−30−25Wavelength [nm]IL [dB]  FA5, 0 dBmCIRBS, 0 dBmFA5, 3 dBmFA5, −13 dBmCIRP3,−13 dBCIRP3, 3 dBm(a)Wavelength [nm]1513.3 1513.35 1513.4 1513.45 1513.5Transmission [dB]-50-45-40-35-30-25-20Gyro Ring 4Ref Ring 4(b)Figure 3.70: (a) Spectra for the forward- and backward-propagating signalsof a 32 mm-long gyro resonator at various input power levels, after fibreattachment. (b) Narrow spectral sweep of the forward-propagating signalsfor the same gyro resonator, and its reference ring.The TLS was tuned to a wavelength λ = 1513.3805 nm with Pin =0 dBm. The SNR at rest for the forward- and backward-propagating signalsof the gyro resonator were, respectively, 8.68 dB and 6.54 dB. The turntablewas rotated sinusoidally with a maximum angular speed Ω = 20 dps at afrequency fin = 0.5 Hz. Figure 3.71(a) compares the normalized turntableangular speed to the unfiltered normalized optical signal of the gyro res-onator. A delay of ∼ 1 s between the signals is due to the asynchronousacquisition times of the FPGA FIFO and the photodetector LabVIEWTMVIs. Fig. 3.71(b) shows the PSDs of these two normalized signals, where onecan see the optical signal has considerable high-frequency noise. Based onthese PSDs, we chose to use a low-pass filtering scheme to eliminate noise in1593.5. Fifth Design Cyclethe optical signal at frequencies above the dominant frequency componentsof the turntable signal. A 6th order Butterworth low-pass filter (LPF) witha 5 Hz cutoff frequency was implemented for this purpose.Time [s]0 5 10 15 20 25Normalized signals [a.u.]-1-0.8-0.6-0.4-0.200.20.40.60.81 Normalized Angular SpeedNormalized Optical Power(a)Frequency [Hz]10-210-1100101102103104Normalized PSD [1/Hz]10-2010-1510-1010-5100105Normalized angular speedNormalized optical power(b)Time [s]0 5 10 15 20 25Normalized signals [a.u.]-1-0.8-0.6-0.4-0.200.20.40.60.81 Normalized Angular SpeedNormalized Optical Power(c)Frequency [Hz]10-210-1100101102103104Normalized PSD [1/Hz]10-2010-1510-1010-5100105Normalized angular speedNormalized optical power(d)Figure 3.71: Comparison of normalized time-domain signals and their PSDplots. Time-domain plots (a) and PSDs (b) for the angular speed and theunfiltered optical power signal. Time-domain plots (c) and PSDs (d) forthe angular speed and the filtered and shifted optical power signal.The cross-correlation of the time-domain signals was obtained to de-termine the time delay between them (∆t = 0.9 s), and shift the filteredsignals accordingly in Fig. 3.71(c). Figure 3.71(d) compares the PSDs ofthe turntable and the filtered optical signal. According to our theoretical1603.5. Fifth Design Cyclecalculations (2.4.5), the optical signal variation is too large to be due solelyto a Sagnac phase shift. Therefore, vibration and stress have a significantinfluence on the optical amplitude variation, despite the fibre array attach-ment.3.5.4 Effects of BackscatteringAs exemplified in Figs. 3.48 and 3.64, in our measurements we observedthat the Q factor and the ER of our device spectra changed considerablyas the wavelength was varied. As shown in Fig. 3.72, for devices withsplitting/merging couplers, we observed that the mixed-through port hada non-zero transmission at rest, which exhibited resonances coinciding withthose of the return signal at the input port. As shown in Fig. 3.73, thereturn signal in these devices also showed considerable variation in its Qfactor and ER. This motivated us to develop a distributed backscatteringmodel for our resonators.Wavelength [nm]1550 1550.1 1550.2 1550.3 1550.4 1550.5 1550.6 1550.7 1550.8 1550.9 1551Normalized Transmission [dB]-60-55-50-45-40-35-30-25-20Mixed-through SignalReturn SignalFigure 3.72: Measured wavelength spectra for the return signal (top, pink)and mixed-through signal (bottom, blue) of a device with a 37 mm-longresonator.1613.5. Fifth Design CycleWavelength (nm)1550 1550.1 1550.2 1550.3 1550.4 1550.5 1550.6 1550.7 1550.8 1550.9 1551Experimental ER (dB)510152025Experimental Q×10611.52ERQFigure 3.73: Q factor and ER as functions of wavelength for the returnsignal spectrum of Fig. 3.72.Figure 3.74 shows a schematic of the reflection that occurs in a waveguidesegment of length ∆z with a backscattering per unit length b and forwardtransmission f =√1− (b∆z)2. The backscattering coefficient at each pointis considered to be a random complex number, b = b˜(z)ejθ˜(z), with amplitudeb˜(z) normally distributed with mean 0 and root mean square magnitudeσb˜(z), and with a phase θ˜(z) equal to 0 or pi with equal probability. In ourmodel, we estimate that the backscattering coefficient ratio between SMWGsand MMWGs is the same as their propagation loss ratio, since both arisefrom scattering. Based on this model, the electric fields shown in Fig. 3.74are related by the matrix equation:[E+(z+∆z)E−(z)]= B[E+(z)E−(z+∆z)], (3.28)where the matrix B is defined by:B =fe−jk+∆z b∆ze−j(k++k−2 )∆z−b∗∆ze−j(k++k−2 )∆z f∗e−jk−∆z , (3.29)1623.5. Fifth Design Cyclewhere k± = 2pineff(λ, z)/λ − jα(z) ± ΦS/L, neff(λ, z) is the effective index,and α(z) is the field loss coefficient. At rest, k+ = k− = k = 2pineff(λ, z)/λ−jα(z). By re-arranging Eq. (3.28) and taking the limit as ∆z goes to zero,we have:∂∂z[E+(z)E−(z)]= P (z)[E+(z)E−(z)], (3.30)where the propagation matrix, P (z), is given by:P (z)=−j[k+ jbjb∗ −k−][E+(z)E−(z)]. (3.31)For a z range starting at z0 over which P (z) is constant, the solution to Eq.(3.30) is given by:[E+(z)E−(z)]=e(z−z0)·P (z)[E+(z0)E−(z0)]. (3.32)fz-b * z b zf *Figure 3.74: Backscattering model schematic.By dividing the resonator of length L shown in Fig. 3.75 into N segmentsand letting N → ∞ so that each segment is considered to have a constantpropagation matrix, using Eq. (3.32) we have:1633.5. Fifth Design CyclePoint coupler -jt tGC 2GC 1Adiabatic 50/50splitting/merging couplerPD 1 FibrearrayLaserPD 20Figure 3.75: Schematic of resonator formed by rib SMWGs (brown), ribMMWGs (cyan), and linear SM to MM converters (purple). The total res-onator length is 37 mm. The variable z denotes the position along thelength of the ring, starting at the point coupler. Left inset: Adiabatic split-ting/merging coupler, formed by strip (orange) and rib (blue) waveguides.Right inset: Point coupler model.[E+8E−8]=X[E+10E−10], (3.33)1643.5. Fifth Design Cyclewhere the matrix X is given by:X= limN→∞[N−1∏m=0exp{LN· P(mLN)}](3.34)In Eq. 3.34 P (mLN ) is the propagation matrix of the mth segment and theproduct of matrix exponentials is evaluated from right to left with increasingm. For computational simplicity, the horizontal and vertical MMWGs weredivided into 3500 and 4900 segments, respectively. The SMWGs and theirtapers were divided into 700 segments. The number of segments was chosenusing convergence testing.The matrix equations governing the electric fields on the way in and onthe way out of an ideal adiabatic splitting/merging coupler are, as referredto the insets of Fig. 3.75:E+5E+6=1√2[1 1−1 1]E+3E+4= ME+3E+4 (3.35)E−3E−4=1√2[1 −11 1]E−5E−6= M−1E−5E−6 (3.36)The fields at the point coupler are related by:E−7E−8=[t −jκ−jκ t]E−9E−10 =KE−9E−10 (3.37)E+9E+10= KE+7E+8 , (3.38)where t and κ are real numbers and represent, respectively, the magnitudesof the field through-coupling and the field cross-coupling. Assuming thatthe coupler is lossless, κ2 + t2 = 1.1653.5. Fifth Design CycleBased on Eqs. (3.28) through (3.38), E7 and E9 are related by:[E−7E+9]= A[E+7E−9](3.39)where:A =[−κ2(W−121 X11 +W−122 X21) t+ κ2W−122t− κ2(W−111 X11 +W−112 X21)κ2W−112](3.40)The fields entering and leaving the device are thus related by:[E−1E−2]= ΦstrM−1ΦribAΦribMΦstr[E+1E+2]. (3.41)where Φstr and Φrib are matrices that represent, respectively, the phase shiftsundergone by travelling through the strip and rib routing waveguides of thesplitting/merging coupler (see Fig. 3.30(a)), and are given by:Φstr,rib =[e−jkstr,ribLr1,r4 00 e−jkstr,ribLr2,r3], (3.42)where kstr = 2pineff str/λ − jαstr and krib = 2pineff rib/λ − jαrib are, respec-tively, the strip and rib SMWG propagation constants, Lr1 and Lr2 are,respectively, the lengths of the strip SMWGs routing the grating couplersGC1 and GC2 to the adiabatic coupler, and Lr3 and Lr4 are the lengths ofthe bottom and top segments of the rib SMWG loop.As described in Subsection 3.4.1, theoretical estimations of the effectiveindices of the SMWGs and MMWGs of these devices were carried out usingMODE SolutionsTM eigenmode solver. The field propagation loss valuesused in our model for the SMWGs and the MMWGs were, respectively,αSM = 34.5 m−1 and αMM = 1 m−1, in agreement with previously reportedexperimental results [9, 15, 56, 78].For our initial simulations, we used a SMWG backscattering value ofσb˜SM = 5.8 mm−1, estimated based on [89, 92]. However, as shown in Fig.3.80, our simulation results showed a greater agreement with the measured1663.5. Fifth Design Cyclespectra for σb˜SM = 18 mm−1 and σb˜MM = 0.522 mm−1. The values ofneff(λ, z), α(z), and σb˜(z) for the waveguide tapers were obtained by linearlyinterpolating between the SMWG and the MMWG values.Unbalanced Adiabatic CouplersAs shown in Fig. 3.79, the simulated spectra obtained assuming an ideal50%/50% power splitting ratio at the adiabatic coupler showed non-zero sig-nals in the mixed-through port due to backscattering, but did not resemblethe measured spectra. Thus, we investigated the effects of non-ideal powersplitting ratio values in the adiabatic coupler, by generalizing its matrix as:M(Tac)=[ √Tac√1− Tac−√1− Tac√Tac], (3.43)where Tac/(1− Tac) is the power splitting ratio. At this stage in the model,the phase relationships between the ports of the adiabatic coupler are stillconsidered ideal.Directional Coupler ImperfectionsFigure 3.76 shows the schematic of a waveguide directional coupler withreflections. For clarity, Fig. 3.76 only depicts the propagation directions andcorresponding S parameter coefficients for the case E+7 = 1. After algebraicmanipulation, the equation governing this subcomponent is, according tothe transfer matrix (T-matrix) formalism [51]:E−7E+9E−8E+10= −jγ t −jκt −jκ  −jγ−jγ  −jκ t−jκ t −jγ E+7E+8E−9E−10= DE+7E+8E−9E−10(3.44)In order to ensure this point coupler is passive and power-conserving, D mustbe unitary. Thus, assuming real values for t and κ, the following restrictions1673.5. Fifth Design CyclePoint-coupler with reflections -jt-jFigure 3.76: Point coupler with straight through transmission t, cross-coupling κ, back-reflection , and contra-directional coupling γ.apply:t, κ, γ,  ∈ R (3.45)t2 + κ2 + γ2 + 2 = 1 (3.46)κ =√1− γ2 − t21 + (γ/t)2(3.47) = κγ/t (3.48)In order to determine the theoretical values of the return loss (RL)and the contra-directional coupling S parameters in various WDCs, finite-difference, time-domain simulations were carried out using FDTD SolutionsTM.According to our simulations, as-designed directional couplers with 7 µm-long parallel waveguides show theoretical RL values on the order of −48 dB.In contrast, the measured RL levels are on the order of −20 dB. However,one must note that the experimental results also include the grating couplerRL, which cannot be separated from the directional coupler RL. After Sparameter obtention, the WDC phase shift contribution was neglected inthe rest of the model (i.e., the WDCs were treated as point couplers).1683.5. Fifth Design CycleMeasurements vs. SimulationsFor all the simulation results shown in this section, E+1 = 1 and E+2 = 0,and all of the results shown will be referred to a device consistent with thatshown in Fig. 3.75, with a 37 mm-long resonator, unless explicitly statedotherwise. Figure 3.77 shows normalized measured spectra for the mixed-through and the return port for the DUT. The data used to generate thisplot is the same used to create Fig. 3.72. However, the wavelength rangehas been restricted to clearly show some features of the mixed-through portspectrum, such as lobe skew and near-resonance spikes.Figure 3.77: Normalized measured spectra for the DUT.Figure 3.78 shows the theoretical spectra for a DUT with t = 0.938,∆λ = 0.1 pm, perfect adiabatic and point couplers, and a backscatter-freering, i.e., Tac = 0.5, σb˜SM = σb˜MM = 0 mm−1, γ = 0. The slight ER variationsare due to the minimum wavelength step of our laser, ∆λ = 0.1 pm.1693.5. Fifth Design CycleFigure 3.78: Simulated spectra with ∆λ = 0.1 pm for a DUT with abackscatter-free ring and a perfect adiabatic coupler.Figure 3.79 shows simulation results for a DUT with Tac = 0.5, t = 0.905,σb˜SM = 5.8 mm−1, γ = 0, and ∆λ = 0.2 pm. The noticeable ER variationsin the return signal are in this case due to both backscattering and thenon-zero value of ∆λ. A non-zero mixed-through signal is observed due tobackscattering. However, there is no resemblance of the simulated signal toits experimental counterpart. Therefore, simulations for adiabatic couplerswith various non-ideal splitting ratios were investigated until a close matchbetween the experimental and theoretical spectral baselines was found. Sev-eral simulations with various levels of backscattering were performed untilthe spectral features of our simulations resembled the phenomena observedin our measurements. The variations of Q and ER in our model are greaterfor higher backscattering levels. Since each backscattering profile is ran-dom, it is practically impossible to perfectly recreate all the spectral featuresof our measurements in one simulation. Rather, phenomenological resem-blances can be observed in each of many simulations created using the same1703.5. Fifth Design Cycleparameters. Several simulations were performed and showed resemblancesto the measured spectra, such as skewed lobes and resonance spikes insteadof troughs. However, for the sake of briefness, only selected plots have beenincluded in the document. Simulations to investigate the impact of greaterreflections at the point coupler were performed. According to our simula-tions, γ should not exceed γmax = 0.3, to ensure that the Q and ER valueranges of the simulations resemble those of our experiments.Figure 3.80 shows simulation results for a 57%/43% splitting ratio, σb˜SM =18 mm−1, t = 0.905, and γ = 0. Figure 3.81 shows simulation results for thesame parameters, except for t = 0.932, and γ = 0.1, whereas Fig. 3.82 showssimulation results for the same parameters, except for t = 0.938, and γ = 0.1.One can see resemblances between the baseline ratios and near-resonancespikes when comparing the measured spectra to each simulation. Figure3.83(a) compares the forward- and backward-propagating output signals forthe same ring backscattering profile used to create Fig. 3.82, i.e., it comparesthe signals exiting the ring for E+7 = 1, E−9 = 0 and for E+7 = 0, E−9 = 1(see Eq. (3.39)). Figure 3.83(b) shows the output spectra in the CW andthe CCW directions for the same ring, for a unit-magnitude input evenlysplit and simultaneously injected in both directions of the ring (hence the−3 dB baselines). Although the effects of γ 6= 0 are very noticeable whencomparing the output spectra for CW- and CCW-propagating signals in theresonator alone (Fig. 3.83(a)), according to our simulations the impact ofthis parameter is not as significant on the spectrum of the full device, exceptfor the need to adjust the value of t to match port baseline levels and toensure sensible values of Q and ER. Figures 3.84(a) and 3.84(b) comparethe measured values of Q and ER with those obtained in various simulationswith different parameter values and random backscattering patterns.1713.5. Fifth Design CycleFigure 3.79: Theoretical spectra for a DUT with Tac = 0.5, t = 0.905,σb˜SM = 5.8 mm−1, σb˜MM = 0.084 mm−1, γ = 0, and ∆λ = 0.2 pm.Figure 3.80: Theoretical spectra for a DUT with Tac = 0.57, t = 0.905,σb˜SM = 18 mm−1, σb˜MM = 0.522 mm−1, γ = 0, and ∆λ = 0.1 pm.1723.5. Fifth Design CycleFigure 3.81: Theoretical spectra for a DUT with Tac = 0.57, t = 0.932,σb˜SM = 18 mm−1, σb˜MM = 0.522 mm−1, γ = 0.1, and ∆λ = 0.1 pm.Figure 3.82: Simulated spectra for a DUT with Tac = 0.57, t = 0.938,σb˜ SM(z) = 18 mm−1, σb˜ MM(z) = 0.522 mm−1, γ = 0.1, and ∆λ = 0.1 pm.1733.5. Fifth Design Cycle(a)(b)Figure 3.83: Theoretical spectra for a 37 mm-long ring resonator witht = 0.938, σb˜ SM(z) = 18 mm−1, σb˜ MM(z) = 0.522 mm−1, γ = 0.1, and∆λ = 0.1 pm. (a) Normalized transmitted power in forward- and backward-propagating directions for CW- and CCW-direction input beams. (b) To-tal output spectra in CW and CCW directions for simultaneous counter-propagation excitation. The power is referred to the total input power.1743.5. Fifth Design CycleWavelength [nm]1550.4 1550.42 1550.44 1550.46 1550.48 1550.5 1550.52 1550.54 1550.56Q×10600.511.522.53 Experimental datat=0.93, γ=0.1t=0.932, γ=0.1t=0.932, γ=0.1t=0.93, γ=0.28t=0.9, γ=0(a)Wavelength [nm]1550.4 1550.42 1550.44 1550.46 1550.48 1550.5 1550.52 1550.54 1550.56Extinction Ratio [dB]051015202530 Experimental datat=0.93, γ=0.1t=0.932, γ=0.1t=0.932, γ=0.1t=0.93, γ=0.28t=0.9, γ=0(b)Figure 3.84: Comparison of measured values of (a) Q factor, and (b)ER with those obtained in various simulations. In all cases, Tac = 0.57,σb˜ SM(z) = 18 mm−1, σb˜ MM(z) = 0.522 mm−1, and ∆λ = 0.1 pm.3.5.5 Iteration Challenges and ConclusionsThe devices studied in this iteration consisted of two adjacent resonators.Both resonators combined SMWGs and MMWGs. Devices were fabricatedat two facilities: IME and UW, using two-etch and single-etch fabrication1753.5. Fifth Design Cyclemethods, respectively. Device spectra were obtained for devices from bothfabrication batches. The devices fabricated at IME showed higher propa-gation losses due to fabrication-related issues. The devices with the bestfigures of merit belonged to the e-beam lithography batch. These deviceswere used for the remainder of the static and dynamic tests. The aver-age ER and Q of these resonators were, respectively, ER ∼ 19 dB andQ ∼ 1× 106. The estimation of the propagation loss for the strip MMWGswas αdB ∼ 0.55 dB/cm. Using Eq. (3.3) with ng ≈ 3.8 at λ0 = 1.55 µm,the average field and power attenuation values are αavg ≈ 0.0077 mm−1and αp ≈ 0.0154 mm−1, respectively. As expected due to the use of e-beamlithography, the propagation losses are higher than in the previous iteration.Nevertheless, it was possible to achieve high quality factor values.For the gyro devices fabricated during this iteration, the power of thebackward-propagating signals increased approximately 10 dB after adhesivedeposition and curing. Rotational tests were performed, and the normalizedangular speed and optical signals were compared in the time- and frequency-domains. A filtering scheme (6th order Butterworth low-pass filter with a5 Hz cutoff frequency) was implemented to eliminate high frequency noise.Despite the significant improvements as regards mechanical actuationand fibre attachment (in comparison to our first rotary setup), the amplitudevariations of the optical signals are still dominated by vibration and stress.These variations are still considerably greater than those due to the Sagnaceffect.In order to improve the readout and reduce backscattering effects [67,71, 82], significant efforts were made to implement frequency tracking usingthe phase modulation techniques described in Section 2.5. Unfortunately,the available signal generators were unable to simultaneously produce rampsignals with the necessary amplitude and frequency values. Specifically, forour latest resonators, Q ≈ 1 × 106, thus FWHM≈ 200 MHz. In order toproduce the necessary frequency excursion ∆f ≈ 100 MHz between pointsB and C in Fig. 2.25, ramp signals must be generated with frequenciesf1 ≈ 100 MHz and 2f1 ≈ 200 MHz. The Vpi voltage of the ThorlabsTMLN65s-FC phase modulators intended for this purpose is Vpi = 4 V [128].1763.5. Fifth Design CycleTherefore, the required peak-to-peak ramp amplitude is Vpp = 8 V , i.e.,the sawtooth root-mean-square (RMS) voltage is VRMS = 4/√3 V ≈ 2.3 V.Since the input impedance of these modulators is RL = 50 Ω, the drivingpower required is:PdBm = 10 log(V 2RMS/RL1× 10−3[W ])= 20.3 dBm (3.49)The fastest arbitrary waveform generator (AWG) available to me was aFlukeTM 294. Although this instrument can generate signals with sufficientamplitudes, its maximum linear ramp frequency is 500 kHz [26], i.e., 400times lower than the required maximum frequency. To overcome the powerand bandwidth limitations faced when trying to implement serrodyne phasemodulation techniques used in previous work [62, 118], sinusoidal modula-tion schemes are envisioned as part of future work.A backscattering model has been developed to explain the non-zero sig-nals detected at the output ports of our devices. Our simulations indicatethat the appearance of signals at the mixed-through port is not only dueto waveguide backscattering, but it is also due to an imperfect splitting ra-tio at the as-fabricated adiabatic couplers. Suitable value ranges for keyparameters of various DUT subcomponents have been found, in order tophenomenologically match the simulations and the experimental data. Dueto the stochastic nature of the backscattering profile distribution, each simu-lation run provides a unique case in which a combination of various spectralbehaviours can be found. Our model shows that the variations of Q and ERare coupling-, backscattering-level-, and backscattering-profile-dependent,and that these variations are greater for increasing values of backscattering.177Chapter 4Summary, Conclusions, andSuggestions for Future Work4.1 SummaryIn this thesis, we have reviewed the state of the art of passive optical gyro-scopes, developed analytical models for design parameter optimization, anditeratively designed and improved both SOI gyroscopic devices and charac-terization setups.We introduced the concept of optical gyroscopes and described the stateof the art in Chapter 1. In Chapter 2 we presented a theoretical study witha thorough device-level optimization. Based on this study, we concludedthat the most crucial design parameter for the gyroscope resonators is thepropagation loss, α. This parameter dictates the optimal values for theresonator length, Lopt, the field coupling, κopt, and the detuning, φopt. Ourstudy showed that for each resonator configuration, the product αLopt is aconstant. For lengths above Lopt, the resolution deteriorates at a noticeablyfaster rate than for decreasing length values, below Lopt.Taking into account the resonator size constraints imposed by the fabri-cation technology, we obtained optimized coupling and detuning for L < Lopt,with α and L as parameters. In all cases, the best resonator design is anunder-coupled, all-pass configuration. In order to assess the design robust-ness, normalized parameter bandwidths for the length, the coupling, andthe detuning of all-pass and drop-port resonators were obtained. For eachport configuration, these bandwidths were identical for all values of α. How-ever, when these parameters are denormalized, lower values of α allow for1784.1. Summarywider tuning ranges. On the system level, the impact of insertion loss andsignal-to-noise ratio on resolution are described. Based on our theoreticalstudy, large-area resonators were designed and fabricated on SOI wafers, aspart of MPW runs.Complex mechano-opto-electrical characterization setups were createdto test the fabricated devices, and both the resonator designs and the char-acterization setups were improved during the five design iterations describedin Chapter 3. Our initial experiments motivated us to combine SMWG di-rectional couplers and SMWG resonator corners with straight MMWG seg-ments, in order to reduce the resonator round-trip loss. Figure 4.1 shows theaverage propagation loss and average Q value for each design cycle. Basedon our prior theoretical study and on the experience gained through variousfabrication iterations, we were able to design large-area resonators with Qfactors as high as 4.5× 106 and 2× 106, with and without thermally-tunedcoupling, respectively. The latter have been published in [56]. To the bestof our knowledge, these are the highest Q factor values reported to date forSOI resonators fabricated using standard CMOS-compatible processes. Ourresonators met or exceeded the specifications for Q factor and propagationloss of resonant structures reported in the literature, fabricated in othermaterials (e.g., InP [23, 28]) and also intended for gyroscopic applications.Regarding the characterization apparatus, LabVIEWTM FPGA inter-faces were created to control the rotation speed and patterns of a firstrotational setup. Preliminary gyroscopic measurements were carried out,allowing for identification of limitations regarding alignment repeatability,speed range, environmental noise, mechanical vibration, and stress. In orderto address these limitations, I built a second rotational setup with supportof CMC Microsystems. Its main features consisted of an enclosure thatreduced air motion and ambient light, auto-alignment capability, lower vi-bration, and improved rotational speed control. Continuous spectral sweepswere implemented in order to observe the behaviour of resonance over time.The initial LabVIEWTM FPGA interfaces were improved so as to al-low for observation and recording of time-domain signals from up to six1794.1. SummaryDesign Cycle1 2 3 4 5Average Ring Loss [dB/cm]05101520(a)Design Cycle1 2 3 4 5Average Q Factor104105106107(b)Figure 4.1: (a) Average ring propagation loss, and (b) average ring Qfactor vs. design cycle.different photodetector channels and from the turntable’s angular speed en-coder. Fast Fourier Transform plots and power spectral density plots werealso obtained automatically upon measurement completion. Based on thespectral characterization results, filtering schemes were implemented. Thiswork resulted in the creation of the Microsystem Integration Platform forSilicon-Photonics (Si-P MIP). This characterization platform is, to the bestof our knowledge, the only characterization platform of its kind availablein Canada, and is now being commercialized by CMC Microsystems foracademic and industrial applications.We have performed repeatable, amplitude-based tests for angular speedsabove 20 dps. However, despite the mechano-opto-electrical improvements,the detected signals were still dominated by effects of system vibration andstress at the bond between the chip and the array during rotation. Efforts toachieve better resolution by implementing frequency tracking schemes wereexplored, however, their implementations were hindered by signal distortionsin the amplification stage. RF amplifiers with gain values between 27 and30 dB were indispensable to achieve the required 2pi phase shift per rampcycle in the phase modulators. High frequency ramp signals require at least100 harmonics to ensure acceptable sharp features. Thus, a flat 30-dB gain1804.1. Summaryacross a full bandwidth from DC to 20 GHz would be required to properlyamplify the 200 MHz ramp signals required to track a resonance with a100-MHz linewidth (consistent Q ≈ 2 × 106). As none of the the availableRF amplifiers had such high, flat gain over such a wide frequency range,all high-frequency ramp signals were unavoidably distorted. An alternativescheme, based on sinusoidal signals, is proposed and analyzed in detail inSection 4.3, as part of future work suggestions.A backscattering model has been developed to explain the non-zero sig-nals detected at the output ports of our devices. Our simulations indicatethat the appearance of signals at the mixed-through port is not only dueto waveguide backscattering, but it is also due to an imperfect splitting ra-tio at the as-fabricated adiabatic couplers. Suitable value ranges for keyparameters of various DUT subcomponents have been found, in order tophenomenologically match the simulations and the experimental data. Dueto the stochastic nature of the backscattering profile distribution, each simu-lation run provides a unique case in which a combination of various spectralbehaviours can be found. Our model shows that the variations of Q and ERare coupling-, backscattering-level-, and backscattering-profile-dependent,and that these variations are greater for increasing values of backscattering.A distributed backscattering model has been developed for our largearea resonators. The phase asymmetries between the CW and CCW signalsin the resonators create differences in their spectra. The non-zero outputsignals at rest at the mixed-through port of our devices with adiabatic cou-plers are due to waveguide backscattering and also due to imperfect split-ting ratios at the as-fabricated adiabatic couplers. Due to the stochasticbackscattering profile distribution, each simulation run provides a uniquecase, in which a combination of various experimental spectral behaviourscan be found. Due to the resonator lengths and coupling values used in ourdevices, the impact of directional coupler reflections is negligible in compar-ison to those of backscattering and imperfect splitting ratios. At its presentform, our backscattering model uses a combination of S parameters for thepoint coupler and T-matrix formalisms, assuming ideal phase relationshipsbetween the adiabatic coupler ports. Further steps to add complexity to the1814.2. Conclusionsmodel include the incorporation of S parameters for the adiabatic couplers,followed by modelling of DUTs with thermally tuneable couplers.4.2 ConclusionsWe have developed models for the optimization and robustness analysis ofthe design parameters of ring resonator gyroscopes. We have achieved thenecessary Q factor values for making SOI gyroscopes feasible, and imple-mented thermal tuning methods to compensate for fabrication-related cou-pling variations. We have also created mechano-opto-electrical platformsfor frequency- and time-domain characterization of SOI gyroscopes, both atrest and under rotation.During time-domain tests, with the TLS directly connected to the detec-tors using PM patch cords, the SNR improved marginally for longer averag-ing times, and it was approximately constant as the input power was varied.This suggested that the measurements at rest were limited by laser noise.This qualitatively agrees with our initial theoretical study (see Fig. 2.9),considering that for our TLSs, the RIN=−145 dB/Hz [72]. However, the-oretically the SNR = 67 dB, whereas experimentally the SNR ≈ 30 dB.As mentioned in Subsection 2.4.5, the SNR has a significant impact on theresolution.Considering the vibration-related amplitude fluctuations, the experimen-tal SNR values (SNR ≈ 30 dB), resonator length, aspect ratio, propagationlosses, laser RIN, and lack of frequency tracking readout, the potential per-formance of the system in its present state is approximately |δΩ| > 920 ◦/s,according to estimations based on the equations developed in Chapter 2.Rotating the system at these angular rates is impractical, as both the setupand the DUTs would unavoidably be damaged or destroyed. Moreover,higher angular speeds cannot guarantee that the Sagnac-related signals willovercome vibration- or stress-related ones, as the latter will also increase inthe present setup configuration.In order to avoid the difficulties associated with amplitude variations,frequency tracking readout schemes are indispensable. Since the resonance1824.2. Conclusionslinewidth values of our resonators are on the order of 100 MHz, the imple-mentation of a sinusoidal frequency-tracking readout scheme is necessary, inorder to avoid the distortion of the RF modulation signals observed duringour experiments.Based on my findings, I conclude that with the implementation of si-nusoidal frequency tracking techniques (further discussed in Section 4.3 be-low), and provided that the mechanical stress and vibration are reduced(e.g., by releasing the fibre array from its metallic holder after curing theUV adhesive), tactical-grade resolutions (for gyroscope resolution grades seeTable 1.1) on the order of δΩ ≈ 3× 10−3 ◦/s should be feasible for square-shaped SOI resonators with lengths on the order of 70 mm and Q ≈ 2×106.This is consistent with resolution predictions based on high-quality resonantstructures fabricated in other technologies [22].Although other materials such as glass or SiN offer lower propagationlosses, the benefits of SOI technology lie on the lower fabrication costs aswell as its its compatibility with CMOS fabrication processes, which allowfor both larger production volumes and for easier integration with readoutelectronics and other on-chip functionalities. For instance, in order to reducethe resonance wavelength shifts due to ambient temperature variations, wehave developed wavelength tuning and stabilization schemes of microringsto track the input signal wavelength using photoconductive heaters [69].Similar schemes can be implemented in the large-area gyro resonators tocompensate for variations associated with temperature fluctuations. On-chip phase modulation schemes can also be implemented, as long as thelengths available for the phase modulators ensure that Vpi < 4 V and thattheir modulation bandwidth is of at least 100 MHz.Besides the implementation of frequency tracking readout techniques,the most important obstacles to overcome are the amplitude variations as-sociated with mechanical vibration and stress in the bond between the fibrearray and the DUT during rotation, as well as the low SNR values of ourcurrently laser-noise limited system.Despite the use of sturdy optomechanics, compact fibre arrays, and UV-curable adhesives, these amplitude variations remain considerably larger1834.2. Conclusionsthan any Sagnac-related variation. This results in any Sagnac-related ampli-tude variation being buried in the aforementioned vibration-related signals.As previously mentioned, due to the mass and shape of the optomechan-ics (e.g., the 40 cm-tall, 3.8 cm-wide solid aluminium post that currentlyholds the fibre array in place), as well as the use of optical fibres, continu-ous rotation and high angular speeds are impractical in our characterizationsetup.The SNR value could be improved with the use of low SpontaneousSource Emission (low SSE) laser sources, e.g., the KeysightTM 81600B [73].According to our theoretical study, a RIN ≈ −160 dB/Hz is required inorder to achieve shot-noise limit. Thus, the use of low-RIN laser sources(e.g., see [16, 47]) is advised.In order to achieve better resolution, the optical interface for deviceinterrogation must be improved further. Device packaging will very likelyhelp, as it will in principle reduce the vibration- and stress-related amplitudefluctuations. Tests should be performed before and during packaging inorder to ensure that the adhesives used for permanent fibre attachment donot produce significant reflections due to refractive index mismatch. Caremust also be taken to ensure that the packaging itself does not producesignificant time-varying nor temperature-dependent stress on the chip, asthis would negatively impact the device performance. This, in combinationwith the aforementioned on-chip functionalities of resonance stabilization,phase modulation, and gyro signal readout, would significantly increase theSNR and reduce the sensitivity to environmental fluctuations.The IL and backreflection levels must also be reduced. Due to the smallsize of the SOI waveguides, edge coupling tolerances are more stringent thanthose of GC alignments. Therefore, GC schemes with reduced backreflec-tion are advisable. GC designs with backreflections as low as −40 dB havebeen proposed [132], and GC designs with −30 dB backreflection have beendesigned and experimentally demonstrated for TE [7] and TM [59] polariza-tion. Novel space-efficient approaches, such as lateral fibre coupling [3, 77],could also be explored.Other areas of potential improvement pertain to reduction of bend loss1844.3. Suggestions for Future Workand propagation loss. Euler bends with effective radii of 100 µm and lossesof 0.005 dB/bend have been reported by the VTT Technical Research Cen-tre of Finland, where very-low loss SMWGs and custom electrical/opticalpackaging options are available [134].4.3 Suggestions for Future Work4.3.1 Sinusoidal Phase ModulationMy principal suggestion for future work is the implementation of suppressed-carrier sinusoidal phase modulation for both frequency tracking and for re-duction of backscattering-induced noise [36, 63, 82, 121]. Although the useof serrodyne and triangular wave phase modulation is considered to be moreadvantageous than sinusoidal phase modulation [144], the driving frequencyand signal bandwidth requirements of sinusoidal phase modulation are con-siderably less stringent than those imposed by serrodyne and triangularphase modulation schemes [62, 118, 125]. In this scheme, the sinusoidalsignals driving the phase modulators for the CW and CCW beams are,respectively:scw = V1 sin(2pif1t) (4.1)sccw = V2 sin(2pif2t), (4.2)where V1 and V2 are the peak voltages and f1 and f2 are the modulationfrequencies for the CW and CCW beam, respectively. The electric fields ofthe phase modulated beams entering the resonator in the CW and CCWdirections are:Ecw =Ein√2CILexp[−j(2piν1t+piVpi· scw)](4.3)Eccw =Ein√2CILexp[−j(2piν2t+piVpi· sccw)], (4.4)where Ein is the amplitude of the TLS optical input field, the factor CIL rep-resents the insertion loss, ν1 and ν2 are the central frequencies of the phase1854.3. Suggestions for Future Workmodulated beams travelling in the CW and CCW direction, respectively,and Vpi is the half-wave voltage of the phase modulators. Assuming thatV1 = V2 = Vs, the input fields can be expressed in terms of Bessel functionsas follows [142]:Ein cw =Ein√2CILe−j2piν1t∞∑u=−∞Ju(m) · e−j2piuf1t (4.5)Ein ccw =Ein√2CILe−j2piν2t∞∑u=−∞Ju(m) · e−j2piuf2t, (4.6)where Ju(m) is the uth-order Bessel function of the first kind, and m is themodulation index of the phase modulators, given by:m =piVsVpi. (4.7)In order to suppress the carrier, J0(m) must be zero, i.e., a modulationindex m = 2.405 is required [142]. In our particular case, Vpi = 4 V, so Vs ≈3.062 V, and VRMS = 2.165 V. Thus, based on Eq. (3.49), the required signalpower is PdBm = 19.72 dBm. Although this power level is similar to whatwas required for the serrodyne implementation, in this case the driving signalhas (ideally) only one spectral component, which eases the implementation.We have been able to generate sinusoidal signals of PdBm = 20 dBm witha quad-channel FlukeTM 294 signal generator [26], up to frequencies of 300MHz without distortion and without requiring additional amplification. Form = 2.405, the amplitudes of Jk(m) for |k| ≥ 4 are considered negligible.Therefore, the fields of the beams exiting the all-pass resonator are:Eout cw =Ein√2CILe−j2piν1t3∑u=−3Ju(m) · e−j2piuf1t · Scw(ν1 + kf1) (4.8)Eout ccw =Ein√2CILe−j2piν2t3∑u=−3Ju(m) · e−j2piuf2t · Sccw(ν2 + kf2), (4.9)1864.3. Suggestions for Future Workwhere Scw and Sccw are the Lorentzian spectral responses of the resonatorin the CW and CCW directions, centred at νcw and νccw, respectively, anddefined based on Eqs. (2.19),(2.20), and (2.21). Considering the 6 rele-vant spectral components for each beam, at the photodetectors we have thefollowing voltage signals [36]:Vpd1 ∝ |Eout cw|2 =6∑u=0A1u · cos(2pi · uf1 · t) (4.10)Vpd2 ∝ |Eout ccw|2 =6∑u=0A2u · cos(2pi · uf2 · t), (4.11)where A1u and A2u are the amplitudes of the spectral components of thesignals at the photodetectors Det1 and Det2. Two LIAs, with referencesignals of frequencies f1 and f2, respectively, must be used to demodulatethe signals Vpd1 and Vpd2. The output signal of LIA1 will be proportionalto the component of Vpd1 at f1. Similarly, the output signal of LIA2 will beproportional to the component of Vpd2 at f2, with amplitudes A11 and A21,respectively.Figure 4.2 shows the LIA output as a function of the frequency deviationfrom resonance due to the Sagnac effect, for several modulation frequencies.As shown in the figure, the slope and extension of the linear regime nearzero deviation varies with the modulation frequency. Figure 4.3 shows theLIA output signal slope as a function of the modulation frequency, for a 40mm-long resonator with a Q ≈ 1 × 106, consistent with our latest designs.As shown in the figure, the optimum sinusoidal modulation frequency isf1 ≈ 35 MHz, which is feasible with our currently available equipment.The output signal of LIA1 is used as an error signal in a feedback loop toeither tune the laser frequency or to impart a thermal phase shift to thering resonator. The output signal of LIA2 is used as an error signal thatis minimized by adjusting the value of f2. The rotation-induced frequencyshift ∆fSagnac is then proportional to f2.1874.3. Suggestions for Future WorkDeviation from resonance [MHz]-400 -200 0 200 400LIA output [V]-0.3-0.2-0.100.10.20.3f1=5 MHzf1=10 MHzf1=15 MHzf1=20 MHzf1=25 MHzf1=30 MHzf1=35 MHzf1=40 MHzf1=45 MHzf1=50 MHzf1=55 MHzf1=60 MHzFigure 4.2: Lock-in amplifier output vs. resonance frequency detuning, withmodulation signal frequency as a parameter.Modulation frequency [MHz]0 10 20 30 40 50 60+ VLIA/+ 6 [V/Hz]#10-911.522.533.544.5Figure 4.3: Lock-in amplifier output slope vs. modulation signal frequency.188Bibliography[1] Honeywell GG5300 MEMS gyro. 2011.[2] Melexis MEMS angular rate sensor, 2011.[3] PLC Connections, LLC, 2015.[4] Don C. Abeysinghe and Joseph T. Boyd. Micromachining Tech-niques and MEMS Structures in Optical Interferometric Sensors. InMEMS/NEMS, pages 1587–1630. Springer, 2006.[5] V. Annovazzi-Lodi, M. Benedetti, S. Merlo, and M. Norgia. Opticaldetection of multiple modes on resonant micromachined structures.IEEE Photonics Technology Letters, 16(7):1703–1705, July 2004.[6] V. 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Guille´n-Torres, M. Caverley, N.A. F. Jaeger, L. Chrostowski, and S. Shekhar, “Automated wave-length stabilization of microring filters using intra-ring photoconduc-tive heaters”, Submitted to Optics Express, July 20152. W. Shi, R. Vafaei, M. A. Guillen-Torres, N. A. F. Jaeger, and L. Chros-towski, “Design and Characterization of Microring Reflectors with aWaveguide Crossing,” Opt. Letters, vol. 35, no. 17, pp. 2901-2903,2010A.2 Conference Proceedings1. W. Shi, R. Vafaei, M. A. Guillen-Torres, N. A. F. Jaeger, and L.Chrostowski, “Ring-resonator reflector with a waveguide crossing,” inIEEE International Conference on Optical MEMS and Nanophotonics2010, Sapporo, Japan, pp. 1-2, Aug. 2010A.3 Conference Presentations1. M. A. Guillen-Torres, H. Yun, S. Grist, X. Wang, W. Shi, R. Boeck,C. Li, J. Yu, N. A. F. Jaeger, L. Chrostowski, “SOI NanophotonicsFabrication Course”, Pacific Centre for Advanced Materials and Mi-crostructures (PCAMM) Annual Meeting, 2010206A.3. Conference Presentations2. M. A. Guillen-Torres, N. Rouger, R. Vafaei, S. M. Amin, R. Boeck,B. Faraji, B. Francis, A. Kulpa, J. M. Michan, L. Chrostowski, N. A.F. Jaeger, and D. Deptuck, “SOI Nanophotonics Devices Analysis andFabrication”, Pacific Centre for Advanced Materials and Microstruc-tures (PCAMM) Annual Meeting, 20083. M. A. Guillen-Torres, M. Sharma, K. Parsa, Y. Zeng, L. Chrostowski,and E. Cretu, “Optical Readout for In-Plane Micro Displacements”,Pacific Northwest Microsystems and Nanotechnology Meeting, FridayHarbor, USA, 2008207Appendix BFrequency-SteppedSinusoidal PatternsThe PWM duty cycle of the first turntable motor control is kept constantduring an integer number M of PWM signal cycles, in order to achieve agradual speed change. Thus, the sinusoid sample number, ns, and the PWMcycle number, npwm, are related by npwm = M · ns. The input duty cycleresembles a sinusoid quantized at a sampling interval Ts, which is a multipleof the PWM signal period, Tpwm, i.e., Ts = M · Tpwm. The quantizedsinusoid, with K samples per period, is depicted in Fig. B.1(a), and has theform:Sn(ns) = pmax · sin(2piK· ns), (B.1)where Sn is the normalized angular speed and pmax is the maximum dutycycle value. The sinusoid has a period T = KTs = KMTpwm. The conditionmod (K, 4) = 0 is required to ensure proper sampling of the sinusoid extremaand zeros. At any given time, the sign of Eq. (B.1) defines the rotationdirection. The frequency of this sinusoid is:f =1T=1KMTpwm=fpwmKM, (B.2)where fpwm = 1/Tpwm is the PWM signal frequency. Figure B.1(b) showsthe ideal PWM duty cycle (red curve), the rotation direction signal (blackcurve), and the experimental PWM duty cycle (blue curve). For easierinterpretation, the duty cycle values are signed according to the rotationdirection.208Appendix B. Frequency-Stepped Sinusoidal Patterns0 5 10 15 20 25 30 35 40-1.0-0.50.00.51.00 50 100 150 200-1.0-0.50.00.51.0nsSnHnsLnpwmSnHnpwmL(a)Ideal PWM duty cycleDirectionExperimental Signed duty cycleDirectiontime [s]PWM duty cycleCWCCW(b)Figure B.1: (a) Normalized angular speed, Sn, as a function of samplenumber, ns (bottom axis), and PWM cycle number, npwm (top axis), forpmax = 1, M = 5, and K = 40. (b) Comparison of input and output signalsfor a sinusoid of frequency fin = 0.083 Hz. The red curve is the ideal PWMduty cycle, the black curve is the rotation direction signal, and the bluecurve is the experimental PWM duty cycle.209Appendix B. Frequency-Stepped Sinusoidal PatternsIn order to characterize the turntable frequency response, the sinusoid fre-quency was stepped and the inputs and output signals for each frequencyvalue were recorded as a function of time. The input sinusoid was modelledas:x(t) = Snmax sin[2pifin(t− t0in)], (B.3)where Snmax is the maximum normalized angular speed, fin is the inputfrequency, and t0in is the input time delay. The reference gyroscope outputwas curve-fitted to:y(t) = Afit sin[2piffit(t− t0fit)], (B.4)where Afit, ffit, and t0fit are the fitted magnitude, frequency, and timedelay, respectively. The fit parameters are then compared to their inputsignal counterparts. In all cases, ffit and fin were identical. The turntablefrequency response was then obtained using the following expressions:|H(fin)|2 = 20 log10[|Afit(fin)|Snmax](B.5)]H(fin) = 2pifin(t0fit − t0in), (B.6)where |H(fin)|2, in dB, is the magnitude response, and ]H(fin) is the phaseshift.210Appendix CTime-Domain Measurementsin Selected ComponentsTwo polarization-maintaining circulators were used throughout our experi-ments. They are referred to as red and blue, according to their fibre jacketcolours. Both circulators were tested injecting light in their port 1 and mea-suring the output power at their ports 2 and 3. The optical output power atport 2 for these devices showed insertion losses of ∼ 0.01 dB and ∼ 0.04 dB,respectively. The SNR values measured at port 2 were ∼ 40 dB and 34 dB,respectively. During this test, the output power at port 3 was below thenoise floor for both circulators. Figure C.1 shows the time-domain outputpower signals measured at port 2 of these circulators, and the correspondingFast Fourier Transforms (FFTs). The rationale behind the obtention of theFFTs was to identify the frequency range of deleterious signals producingnoise or drift.211Appendix C. Time-Domain Measurements in Selected Components0 10 20 30 40 50 60 70 8010−310−210−1100101102103time [s]P out [µW]  Red circ. port 1Red circ. port 3(a) Red circulator, time domain10−4 10−2 100 102 10410−1010−810−610−410−2100102104Frequency [Hz]|P OUT(f)|  Red circ. port 1Red circ. port 3(b) Red circulator, frequency domain0 2 4 6 8 10 1210−610−410−2100102104(c) Blue circulator, time domain (d) Blue circulator, frequency domainFigure C.1: Linear power vs. time and FFT spectra for two PM circulators.(a, b) Red (first) circulator. (c, d) Blue (second) circulator.212Appendix DTransfer Functions ofResonators withThermally-TuneableCouplersTo obtain the tuneable coupler transfer matrix of the device shown in Fig.3.33, Ktc, we will assume that the input signal enters through port C1 andpropagates towards ports F1 and F2

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