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Highly efficient thermo-optic switches on silicon-on-insulator Murray, Kyle 2015

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Highly Efficient Thermo-Optic Switches onSilicon-On-InsulatorbyKyle MurrayB. Eng., McMaster University, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of Applied ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Electrical and Computer Engineering)The University of British Columbia(Vancouver)August 2015c© Kyle Murray, 2015AbstractWe analyze and demonstrate the performance of dense dissimilar waveguide rout-ing as a method for increasing the efficiency of thermo-optic phase shifters ona silicon-on-insulator platform. Optical, mechanical, and thermal models of thephase shifters are developed and used to propose metrics for evaluating device per-formance. The lack of cross-coupling between dissimilar waveguides allows highlydense waveguide routing under heating elements and a corresponding increase inefficiency. We demonstrate a device with highly dense routing of 9 waveguides un-der a 10 µm wide heater and, by thermally isolating the phase shifter by removalof the silicon substrate, achieve a low switching power of 95 µW, extinction ratiogreater than 20 dB, and less than 0.1 dB ripple in the through spectrum. The devicehas a footprint of less than 800 µm × 180 µm. The increase in waveguide densityachieved by using dissimilar waveguide routing is found not to negatively impactthe switch response time.iiPrefaceI am the main author of the Optics Express journal paper titled “Dense dissimi-lar waveguide routing for highly efficient thermo-optic switches on silicon” [1], acoauthor of the submitted journal paper titled “Michelson interferometer thermo-optic switch on SOI with a 50 microwatt power consumption” [2], and a contributorto Chapter 4.1 of the book titled “Silicon Photonics Design: From Devices to Sys-tems” [3].My supervisor, Dr. Lukas Chrostowski, requested I perform simulations ofdissimilar waveguide coupling to appear in [3], and the model and calculationmethods that I developed to perform these simulations were later expanded intothe coupled mode model of light propagation in folded waveguides that appears in[1] and Section 2.2 of this thesis.I conceived the idea of utilizing dissimilar waveguide routing to increase theefficiency of thermo-optic phase shifters, developed models of the devices, de-signed the devices, measured the performance of the devices, and wrote the OpticsExpress paper. This work is the basis of the paper [1], and appears in Chapter 1,Section 2.2, and in Chapters 3 and 4 of this thesis. H. Jayatilleka assisted with thelayout of the folded waveguide structures. H. Jayatilleka and Z. Lu assisted withediting of [1].Z. Lu is the main author of the paper [2], for which I provided simulationsof dissimilar waveguide coupling, advice on the design of the folded waveguides,and editing of the draft. The thermal simulations in [2], performed by Z. Lu, wereexpanded upon by myself to form Section 2.4 of this thesis.Dr. Lukas Chrostowski guided my research by encouraging further explorationof dissimilar waveguide routing, by providing insight in to the design and meaure-iiiment of devices, and by editing drafts of the publications and of this thesis.The following is a copyright notice regarding the content reproduced from [1]:c©2015 Optical Society of America. One print or electronic copy may be madefor personal use only. Systematic reproduction and distribution, duplication of anymaterial in this paper for a fee or for commercial purposes, or modifications of thecontent of this paper are prohibited.The complete list of publications are:[1] K. Murray, Z. Lu, H. Jayatilleka, and L. Chrostowski, “Dense dissimilar waveg-uide routing for highly efficient thermo-optic switches on silicon,” Opt. Express23, 1957519585 (2015).[2] Z. Lu, K. Murray, H. Jayatilleka, and L. Chrostowski, “Michelson interferome-ter thermo-optic switch on SOI with a 50 microwatt power consumption,” Photon.Technol. Lett., IEEE pp. 11 (2015).[3] L. Chrostowski and M. Hochberg, Silicon Photonics Design: From Devices toSystems (Cambridge University Press, 2015). Chap. 4.1.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Current State of Silicon Photonic Switches . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Objective of this Thesis . . . . . . . . . . . . . . . . . . . . . . . 51.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Theoretical Device Analysis . . . . . . . . . . . . . . . . . . . . . . . 72.1 General Switch Structure . . . . . . . . . . . . . . . . . . . . . . 72.2 Optical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Coupled Mode Theory of Dissimilar Waveguides . . . . . 92.2.2 Folded Waveguide Structures . . . . . . . . . . . . . . . . 132.3 Mechanical Model . . . . . . . . . . . . . . . . . . . . . . . . . 28v2.4 Thermal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5 Chapter Summary and Conclusions . . . . . . . . . . . . . . . . . 413 Device Design and Experimental Investigation . . . . . . . . . . . . 423.1 Device Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.1.1 A 2×2 Switch . . . . . . . . . . . . . . . . . . . . . . . 423.1.2 A 4×4 Switch . . . . . . . . . . . . . . . . . . . . . . . 453.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . 463.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 473.4 Chapter Summary and Conclusions . . . . . . . . . . . . . . . . . 604 Conclusions and Suggestions for Future Work . . . . . . . . . . . . 624.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . 624.2 Suggestions for Future Work . . . . . . . . . . . . . . . . . . . . 63Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65A Path Integral Approach for a Directional Coupler . . . . . . . . . . 73B A Lower Bound for Switch Figure of Merit . . . . . . . . . . . . . . 76viList of TablesTable 3.1 Device parameters . . . . . . . . . . . . . . . . . . . . . . . . 45Table 3.2 Switch routing states . . . . . . . . . . . . . . . . . . . . . . . 47Table 3.3 Tuning efficiency of Mach-Zehnder interferometer (MZI) switches(mW/pi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Table 3.4 10-90 Response times of MZI switches (µs) . . . . . . . . . . 55Table 3.5 Figures of merit of MZI switches (nJ) . . . . . . . . . . . . . . 57viiList of FiguresFigure 1.1 a) Cross-section of a phase shifter with a metal heater. b)Cross-section of a phase shifter with a doped silicon heater. . . 3Figure 2.1 a) A schematic of an MZI. b) A 50-50 Splitter . . . . . . . . . 8Figure 2.2 Dissimilar waveguide structure and horizontal electric field pro-file of its modes. Modes |1〉 and |2〉 are the modes of the two-waveguide structure. Modes |A0〉 and |B0〉 are the modes whenonly waveguide A or waveguide B are present, respectively.c©Optical Society of America, 2015, by permission. . . . . . 10Figure 2.3 Maximum crosstalk between 220 nm thick waveguides with1 µm pitch. c©Optical Society of America, 2015, by permission. 13Figure 2.4 Schematic diagram of folded waveguide structure. c©OpticalSociety of America, 2015, by permission. . . . . . . . . . . . 14Figure 2.5 a) Calculated spectra of the folded waveguide structure for 9identical waveguides with gaps gI = 500 nm, 750 nm, and1000 nm, and alternating dissimilar waveguides with widths of500 nm and 600 nm with gap gD = 500 nm, b) Minimum trans-mission of the folded waveguide structure for similar (solid)and dissimilar (dashed) waveguides. c©Optical Society of Amer-ica, 2015, by permission. . . . . . . . . . . . . . . . . . . . . 17viiiFigure 2.6 Intensity of forward and backward travelling modes as a func-tion of position along the folded waveguide structure, mea-sured as the distance along the path with no cross-coupling.The 9 waveguides are all of 500 nm width, 90 µm length, andthe gap is 500 nm. The On state is at 1553.16 nm and the Offstate is at 1556 nm. . . . . . . . . . . . . . . . . . . . . . . . 18Figure 2.7 a) Folded waveguide structure with next-to-nearest neighbourcoupling. b) An equivalent racetrack resonator. . . . . . . . . 20Figure 2.8 A Feynman diagram representing one term in an infinite sumdescribing electron-electron scattering through the electromag-netic field. The electrons, e−, interact via the exchange of avirtual photon, γ . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 2.9 Folded waveguide structure divided into M subsections. . . . 22Figure 2.10 a) An example of a typical path through the structure, b) a paththat has a large contribution for high coupling, and c) a paththat has a large contribution for low coupling. . . . . . . . . . 23Figure 2.11 Calculated group delay of the folded waveguide structure for9 identical waveguides with gaps gI = 500 nm, 750 nm, and1000 nm, and alternating dissimilar waveguides with widthsof 500 nm and 600 nm with gap gD = 500 nm. . . . . . . . . 25Figure 2.12 A set of paths that are related by a small deformation describedby the parameter δ . . . . . . . . . . . . . . . . . . . . . . . . 27Figure 2.13 Top-down view of suspended glass structure a) without supportbridges, and b) with two pairs of support bridges undergoingan acceleration, a, oriented out of the page. . . . . . . . . . . 29Figure 2.14 Stress distribution in a suspended heater structure under uni-form 1g acceleration along the direction normal to the chip fora) a 90 µm long structure with no support bridges, and b) a290 µm long structure with two pairs of support bridges. . . . 30Figure 2.15 Maximum von Mises stress in suspended structure. . . . . . . 31Figure 2.16 Switching power required for devices with 9 waveguides andvarying number of support bridges. . . . . . . . . . . . . . . 34ixFigure 2.17 Normalized switching power required for devices as the widthis varied. The number of waveguides for each width is themaximum number of waveguides that can fit within the struc-ture with a 1 µm pitch. . . . . . . . . . . . . . . . . . . . . . 35Figure 2.18 Temperature distribution in 90 µm long, 12 µm wide, sus-pended structure. a) overhead view, and b) cross-sectionalview. Note the different scale in b). The black rectangles indi-cate the positions of the silicon waveguides. . . . . . . . . . . 36Figure 2.19 Temperature distribution in 290 µm long suspended structurewith two pairs of support bridges. a) overhead view, and b)cross-sectional view in the center of one of the pairs of sup-port bridges. The black rectangles indicate the positions of thesilicon waveguides. . . . . . . . . . . . . . . . . . . . . . . . 37Figure 2.20 Finite element simulated time domain response of a) unetchedand b) underetched heaters of different lengths. The 90 µmlong device has no support bridges, and the 90 µm long de-vice has two pairs of support bridges. A power of 1 mW iscontinuously supplied beginning at time 0. . . . . . . . . . . . 38Figure 3.1 a) Schematic of thermally tunable MZI switch. Black traces:WGs, Red: Oxide openings define underetched region, Purple:Routing metal, and Green: Heater metal. Inset: Waveguidetaper region. b) An optical micrograph of a fabricated device.c) Thermal phase shifter cross-section before underetching. d)Thermal phase shifter cross-section after underetching. In c)and d), silicon dioxide is blue, silicon is tan, the metal heateris grey, and air is white. c©Optical Society of America, 2015,by permission. . . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 3.2 A schematic of a 4×4 switch. The 2×2 switches I through VIroute light from the inputs A,B,C, and D to the outputs 1,2,3,and 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46xFigure 3.3 A schematic diagram of the experimental setup. Solid anddashed lines indicate optical and electrical connections, re-spectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Figure 3.4 a) A schematic of the adiabatic splitter. The tan is 200 nm thicksilicon, and the orange is 90 nm thick silicon slab. b) Measuredtransmission through an adiabatic splitter. . . . . . . . . . . . 48Figure 3.5 Transmission to the two output ports of an implementation ofdevice 2 in the through and cross states. The numbers 1 and 2indicate the upper and lower arms, respectively, of the switchinput and output. . . . . . . . . . . . . . . . . . . . . . . . . 49Figure 3.6 Measured spectra of the underetched versions of a) device 3and b) device 5. c©Optical Society of America, 2015, by per-mission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 3.7 Normalized transmission functions of the a) short (devices 2and 4) unetched, b) long (devices 3 and 5) unetched, c) shortunderetched, and d) long underetched MZI switches. c©OpticalSociety of America, 2015, by permission. . . . . . . . . . . . 54Figure 3.8 Temporal response of a) unetched, and b) underetched MZIswitches. c©Optical Society of America, 2015, by permission. 56Figure 3.9 Comparisons of the a) switching powers, b) characteristic risetimes, and c) figures of merit of thermo-optic switches overthe past decade. The asterisk markers represent works relatedto this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . 59xiGlossarySOI silicon-on-insulatorMZI Mach-Zehnder interferometerCMOS complimentary metal-oxide-semiconductorMEMS micro-electro-mechanical systemsWDM wavelength division multiplexingQFT quantum field theoryxiiAcknowledgementsFirst, I would like to thank my supervisor, Dr. Lukas Chrostowski, for his inspira-tion, support, and for all of the opportunities he has provided. I am also thankful tomy other committee members, Dr. Nicolas Jaeger and Dr. Sudip Shekhar, for theirtime, useful feedback, and insight.I would like to thank all of my colleagues for their support and friendship,in particular M. Caverley, M. Guille´n-Torres, H. Jayatilleka, Z. Lu, and S. TalebiFard, for their assistance and collaboration.I offer thanks to Ranovus Inc. for providing an internship opportunity duringmy studies.I gratefully acknowledge CMC Microsystems, Huawei, and the Natural Sci-ences and Engineering Research Council of Canada, including the Si-EPIC pro-gram, for financial support. I would also like to thank Lumerical Solutions Inc.and Mentor Graphics Inc. for the design tools used in this work.xiiiChapter 1Introduction1.1 Current State of Silicon Photonic SwitchesSilicon photonics has recently generated a great deal of interest for telecommuni-cations applications. Due to the compatibility of silicon-on-insulator (SOI) plat-forms with mature complimentary metal-oxide-semiconductor (CMOS) electronicsfabrication technologies, silicon photonics offers the potential for inexpensive inte-gration with the electronics needed for their operation. Numerous types of deviceshave been implemented on SOI including filters [4–6], switches [7], modulators[8, 9], detectors [10–12], sensors [13, 14], and polarization splitters and/or rotators[15, 16]. Several companies have invested into silicon photonics research, includ-ing IBM, who have demonstrated monlithic integration of 8×8 silicon photonicswitches with CMOS control [17], and Intel, who have demonstrated a 4-channelsilicon photonics wavelength division multiplexing (WDM) links with integratedelectronics [18]. Despite the recent interest in developing large systems, it is stillimportant to improve performance at the device level.Early demonstrations of thermo-optic switches on SOI required large powerconsumption due to the inefficiency of the phase shifters used [19]. As the numberof switches in a network is increased, the power consumption of each individualswitch will become more important as the waste heat produced requires increas-ingly complex thermal management [20].Switches on an SOI platform can be implemented using a number of interfer-1ometric structures, including Mach Zehnder interferometers (MZIs) [19, 21, 22],Michelson interferometers [23], and micro-ring resonators [24, 25]. These types ofswitches typically require the use of optical phase shifters to implement the switch-ing action. Phase shifters find fundamental applications in many silicon photonicdevices including switches [26, 27], modulators [28, 29], and tunable filters [30].Two common ways of implementing a phase shifter rely on the plasma dispersioneffect [31], in which a change in the density of charge carriers affects the refractiveindex of silicon, and the thermo-optic effect [32], in which a temperature changeaffects the refractive index. Plasma dispersion-based phase shifters, while offeringfast operation, often require large footprints or high operating voltages and have anoptical loss modulation associated with the phase shift [31]. Thermo-optic phaseshifters can achieve large phase shifts in small footprints with low operating volt-ages and without introducing optical loss modulation. However, they have slowerresponse times and typically require more power for switching [33, 34].As an alternative to interferometry-based switches, micro-electro-mechanicalsystems (MEMS), in which switching is achieved by electrically modulating the mo-tion of waveguides, have been proposed as potential alternatives to interferometricswitching requiring phase shifters. Both experimental and theoretical analysis ofMEMS-based switches have been presented [35, 36]. MEMS-based switches offerboth low switching power and high switching speed. However, they require mov-ing parts and so are generally expected to have shorter lifetimes than devices withno moving parts [37]. MEMS-based switches using various operating principleshave been demonstrated. For example, the modulation of the distance betweenparallel waveguides has been used to modulate the amount of evanescent coupling,and thus the total cross-coupling, in a directional coupler switch [35, 38]. Alterna-tively, modulation of the air gap between concatenated waveguides has been used tomodulate the amount of transmission between the waveguides [39]. MEMS-basedswitches have been used to create 50×50 switch networks [40].The focus of this thesis is on interferometry-based switches, and in particular,on the design of highly efficient thermo-optic phase shifters.26L6XEVWUDWH2[LGH&ODGGLQJ,ĞĂƚĞƌ6L:DYHJXLGH(a)6L6XEVWUDWH2[LGH&ODGGLQJ0HWDO&RQWDFW+HDYLO\'RSHG6L/LJKWO\'RSHG6L:DYHJXLGH(b)Figure 1.1: a) Cross-section of a phase shifter with a metal heater. b) Cross-section of a phase shifter with a doped silicon heater.31.2 Literature ReviewThe heating element in thermo-optic phase shifters has been implemented in mul-tiple ways. Figures 1.1 a) and b) show schematic cross-sections of heaters imple-mented using metallic heaters, and using doped silicon heaters, respectively. Mostdemonstrations of thermo-optic switches have utilized metallic heaters. Theseheaters are typically positioned several microns above a waveguide to be heated,in order to prevent absorption of light by the heater. Since the rate of heat loss toconvection at the cladding surface is typically smaller than the rate of heat loss toconduction to the silicon substrate, due to the large thermal conductivity of silicon,the need for the metal heater to be kept away from the waveguide does not dramat-ically decrease efficiency. A drawback of using such metal heaters is that, due tofabrication limits, the heater must usually be significantly wider than the waveg-uides. This results in the heating of a large volume of cladding oxide that does notcontribute to a phase shift. Heaters formed by doped silicon have the advantageof heating the silicon directly. However, a silicon slab is needed to make electricalcontact to the doped silicon. The conduction of heat away from the waveguidealong the slab region results in a degraded thermal efficiency. Additionally, theneed to dope the silicon results in an increase in optical loss. Phase shifters utiliz-ing doped heaters have been demonstrated in [41, 42].Thermo-optic phase shifters on SOI have shown orders of magnitude reductionin switching power over the last decade [2]. This has been a result of a numberof methods for increasing the efficiency of thermo-optic phase shifters being pro-posed. The most effective of these methods include improving thermal isolationby the removing the material surrounding the phase shifters [43–45], and folding awaveguide many times under a heater to increase the optical interaction length withthe heated region [7, 46]. When folding a waveguide under a heater, the waveguidespacing between each fold is limited by the evanescent coupling of light betweenadjacent waveguides. When adjacent waveguides are identical, the coupling ofpower between them is resonant, and a complete transfer of power occurs over acharacteristic coupling length [47]. The coupling length is strongly dependent onthe waveguide spacing, and so the spacing must be chosen such that the power cou-pling over the length of the device is sufficiently small for a desired application.4This need for a sufficiently large spacing limits the achievable density of waveg-uide routing and, therefore, limits the number of times a waveguide can be foldedunder a heater and its power efficiency. In this work we propose utilizing differ-ent waveguide widths in each fold of a phase shifter to overcome this limit. Theevanescent coupling between dissimilar waveguides does not achieve phase match-ing. Therefore, the power coupling between waveguides is not complete [48]. Fora given waveguide spacing, if the mismatch between adjacent waveguide widths issufficiently large, then the power coupling can be made negligibly small over anycoupling length. Without the need to have a large spacing, the density of waveguidefolding under a heating element can be increased dramatically, and the efficiencyof thermal heaters can be correspondingly improved.Prior to this work, the use of dissimilar waveguides to increase waveguide rout-ing density in photonic circuits was proposed [3, 49], and an in depth analysis inthis context has been performed [50]. More recently, dissimilar waveguide rout-ing has also been proposed for dense mode division multiplexing, with gaps be-tween adjacent waveguides as small as 100 nm [51]. After the submission of [,1] Mrejen et al. demonstrated control of the coupling between two waveguides bycontrolling the refractive index of an intermediate dissimilar waveguide [52]. Wehave recently demonstrated a Michelson interferometer using dissimilar waveg-uide routing, showing that the technique suggested herein can be extended to otherswitching architectures to achieve extremely low switching power [2].1.3 Objective of this ThesisIn this thesis, the use of dense dissimilar waveguide routing in folded waveguidethermo-optic phase shifters is proposed, with the objective of reducing the powerconsumption required for switching. The optical properties of such folded waveg-uide structures are modelled to assess their feasibility, to identify important con-siderations in their design, and to provide metrics for evaluating their performance.Thermal and mechanical models of the structure are also developed in order tooptimize the structure’s performance, and to provide predictions against which ex-perimental results can be compared.51.4 OverviewThis thesis consists of four chapters. In this chapter, an introduction to the uses ofsilicon photonic switches, as well as an overview of the state of the art and cur-rent challenges in implementing these switches, with particular attention made tothermo-optic phase shifters, is presented. A review of the literature is also pre-sented, and the objectives of this thesis are outlined. In Chapter 2, models ofthe optical, mechanical, and thermal properties of the phase shifters designed inthis work are developed. In Chapter 3, the design of the switches that were fab-ricated are described in detail, and the results of experiments are presented. Theexperimental results are compared against those of previous works. In Chapter 4,conclusions are drawn and suggestions for future work are made.6Chapter 2Theoretical Device Analysis2.1 General Switch StructureFigure 2.1 a) shows a schematic of an ideal 2×2 MZI. The MZI consists of a 50-50beam splitter, two phase arms, and a second 50-50 beam splitter. The electric fieldsat the upper and lower ports of the input coupler are denoted by EIn1 and EIn2,respectively, while the electric fields at the output coupler are similarly denoted byEOut1 and EOut2. Figure 2.1 b) shows a schematic of the 50-50 beam splitter. Theelectric fields at the upper and lower ports of the input are denoted by EA and EB,respectively, while the electric fields at the two output ports are denoted by EC andED. The relationship between the input and output fields of the coupler dependson the particular design of the coupler. For the adiabatic coupler used in this work,the fields are related as [53][ECED]=1√2[1 1−1 1][EAEB]≡ T50-50[EAEB]. (2.1)Supposing that light with unit electric field amplitude is input into the upperport of the input coupler of the MZI, the fields at the MZI output are then given by[EOut1EOut2]= T−150-50[eiφ1 00 eiφ2]T50-50[10]≡ TMZI[10], (2.2)7,Q,Q2XW2XW6SOLWHU6SOLWHU,QSXW 2XWSXW(a),QSXW 2XWSXW6SOLWHU(b)Figure 2.1: a) A schematic of an MZI. b) A 50-50 Splitterwhere φ1 and φ2 are the phases acquired along the upper and lower arms of theMZI, respectively. The fields at the output then satisfy|EOut1|2 = cos2(φ1−φ22)(2.3)and|EOut2|2 = sin2(φ1−φ22). (2.4)Thus, if the phase difference between the arms can be changed from 0 to pi , thenthe intensity at an output arm can be changed from a maximum to a minimumvalue, or vice versa. This effect can be used to make switches, but also provides8a straightforward way to characterize the performance of a phase shifter. The fol-lowing sections consist of an analysis of a particular implementation of a phaseshifter.2.2 Optical ModellingIn many applications a switch is, ideally, expected to simply change which di-rection light exits the switch without modifying any of its other properties. Forexample, an ideal switch should not introduce light reflection at the switch input,introduce additional loss, or introduce dispersion. In this section we introduce twomodels of light propagation in the folded waveguide structure that forms the inter-ferometer arms of the switches discussed in Section 2.1. The first model, based oncoupled mode theory gives an analytical expression of the transmission and disper-sion of the folded waveguide structure. However, this model does not provide anyintuitive explanation of the behaviour of light in the structure. The second modeldeveloped, based on a formal path integral approach provides this intuitive expla-nation, although it is computationally infeasible to implement in the general case.Together, the models provide both quantitative descriptions of the folded waveg-uide structure’s optical properties and a useful way of thinking about the structure.2.2.1 Coupled Mode Theory of Dissimilar WaveguidesIn this section we compute the crosstalk between a pair of dissimilar waveguides.To achieve this, we define a notion of the power in a waveguide by projectingthe optical field of the two-waveguide system onto the field of a single waveguidemode. We perform a change of basis from the two-waveguide eigenmode basis, inwhich the propagation is simple to describe, to a basis in which the power in eachwaveguide is simple to compute. In this basis the propagation is more complicateddue to the appearance of coupling between modes.Consider two parallel waveguides, denoted as waveguides A and B, of thick-ness t and widths wA and wB separated by a gap, g, as shown in Fig. 2.2. The twowaveguide system has transverse electric (TE) eigenmodes |1〉 and |2〉, each nor-malized to unit power, with propagation constants k1 and k2 respectively. Waveg-uides A and B considered in isolation have eigenmodes |A0〉 and |B0〉, respectively.9ǁ ǁƚŐFigure 2.2: Dissimilar waveguide structure and horizontal electric field pro-file of its modes. Modes |1〉 and |2〉 are the modes of the two-waveguidestructure. Modes |A0〉 and |B0〉 are the modes when only waveguide Aor waveguide B are present, respectively. c©Optical Society of America,2015, by permission.With the inner product [48]:〈ψ1|ψ2〉=14[∫E1×H∗2 ·dS+∫E∗2×H1 ·dS], (2.5)where Ei and Hi, i = 1,2, are the transverse electric and magnetic field profilesof any two modes |ψi〉 and S is the plane normal to the propagation direction,we can decompose the single waveguide state |A0〉 in terms of the two-waveguideeigenmodes:|A〉= 〈1|A0〉 |1〉+ 〈2|A0〉 |2〉 . (2.6)The difference between |A〉 and |A0〉 is due to not including the complete set ofradiation modes in the mode decomposition. Define the power normalized state∣∣A¯〉=|A〉√〈A|A〉, (2.7)and |B¯〉 similarly. A general superposition in the |1〉, |2〉 basis is then denoted as a10vector with components a and b:V =[ab]= a |1〉+b |2〉 . (2.8)Evolution along the propagation direction, z, is given by:dVdz= i[k1 00 k2]V≡ iPV. (2.9)Performing a change to the∣∣A¯〉, |B¯〉 basis with components c and d,V¯ =[cd]= c∣∣A¯〉+d |B¯〉 (2.10)V =〈1|A0〉√〈A|A〉〈1|B0〉√〈B|B〉〈2|A0〉√〈A|A〉〈2|B0〉√〈B|B〉V¯≡MV¯, (2.11)the new evolution follows:dV¯dz= iM−1PMV¯≡ iP¯V¯. (2.12)It should be noted that since in general M is not unitary the inner product isV†V = V¯†M†MV¯ 6= V¯†V¯, (2.13)where † indicates the conjugate transpose operation, so the sum of the squares ofthe norms of the components of V¯ is not in general a conserved quantity. Nev-ertheless, we will identify the squares of the norms of the components of V¯ withthe informal notion of the power contained in each waveguide. More precisely, thesquare of the norm of the first component of V¯ is the power that would be trans-mitted in to waveguide A if waveguide B were abruptly terminated and the squarednorm of the second component has a similar interpretation.If we consider a situation where at z = 0 the waveguide system is excited inthe state∣∣A¯〉, then one can consider the power coupled to waveguide B over some11length L as the squared norm of the amplitude of |B¯〉 at z = L. In the specialcase where waveguides A and B are identical, the power is transferred completelyfrom waveguide A to waveguide B over a characteristic length, Lc = pi/(k1− k2),depending on the dimensions of the waveguides and their separation [48]. Thus, ifone wishes to limit the crosstalk between the waveguides over their length, then theseparation between the waveguides must be made large enough such that Lc >> L.If the two waveguides are not identical, then the power is still periodicallycoupled between the waveguides, but the transfer of power is incomplete [48].The maximum crosstalk, CT, can then be computed as the maximum value of thesquared norm of the second component of V¯ in the solution to Eq. (2.12):V¯(z) = M−1eiPzM V¯(0) = M−1eiPzM[10](2.14)CT = maxz(| [0 1] V¯(z)|2)= maxz∣∣∣∣∣[0 1]M−1eiPzM[10]∣∣∣∣∣2 (2.15)= 4 ·〈B|B〉〈A|A〉·| 〈1|A0〉〈2|A0〉 |2| 〈1|A0〉〈2|B0〉−〈1|B0〉〈2|A0〉 |2Figure 2.3 shows the computed maximum crosstalk for waveguides with a fixedcenter to center separation of 1 µm and thickness 220 nm as the widths of thewaveguides are varied for a wavelength of 1550 nm [3]. The modes and propaga-tion constants were computed using a numerical mode solver. It can be seen thatby making the waveguide widths sufficiently different the crosstalk can be limitedfor small separations regardless of the length of the coupler. The asymmetry of thecrosstalk under interchange of waveguide A and B is due the difference betweenfirst exciting state |A〉, then later measuring state |B〉, and first exciting state |B〉,then later measuring state |A〉. This difference is due to the non-orthogonality of|A〉 and |B〉 for dissimilar waveguides.120D[LPXP&URVVWDONG%Figure 2.3: Maximum crosstalk between 220 nm thick waveguides with1 µm pitch. c©Optical Society of America, 2015, by permission.2.2.2 Folded Waveguide StructuresA Coupled Mode ApproachFigure 2.4 shows a schematic of the folded waveguide structure consisting of awaveguide folded N times, with widths wm, m = 1,2, ..N. Light is injected intowaveguide 1 and the transmitted light is measured at waveguide N. We consideronly an odd number of waveguides so that the input and transmitted light are trav-elling in the same direction. To model the propagation, we utilize a tight-bindingcoupled mode model where we consider coupling only between nearest neighbourwaveguides. The propagation is described by the differential equations:dV¯+mdz= amV¯+m +bmV¯+m−1 + cmV¯+m+1 (2.16)dV¯−mdz=−amV¯−m −bmV¯−m−1− cmV¯−m+1, (2.17)where V¯+m and V¯−m , m = 1,2, ...N, are the amplitudes of the modes in waveguidem travelling in the positive and negative z directions, respectively. Here am, bm,13ǁϭǁϮǁϯ ǁEǁEͲϭ͘͘/ŶƉƵƚKƵƚƉƵƚϭϮEͲϭnj>Figure 2.4: Schematic diagram of folded waveguide structure. c©Optical So-ciety of America, 2015, by permission.and cm are the pairwise self and cross coupling coefficients computed as describedin equation (2.12). Specifically, am are the self coupling coefficients found as thediagonal elements of iP¯, and bm and cm are the cross-coupling coefficients foundas the off-diagonal elements of iP¯. Further, the system adheres to the boundaryconditions:V¯+0 (0) = 1 (2.18)V¯−N (L) = 0 (2.19)V¯−m (L) = V¯+i−1(L)eiφm−1V¯+m (L) = V¯−m−1(L)e−iφm−1}for m even (2.20)V¯−m (0) = V¯+m−1(0)eiφm−1V¯+m (0) = V¯−m−1(0)e−iφm−1}for m odd, m > 1, (2.21)14where φm is the phase associated with the bend connecting waveguide m withwaveguide m+1. To solve the boundary value problem described by equations2.16 and 2.17, subject to the boundary conditions of equations 2.18-2.21, we con-struct vectors V¯+ and V¯− with components V¯+m and V¯−m , respectively. Equations2.16 and 2.17 can then be written in matrix form asdV¯+dz= SV¯+ (2.22)dV¯−dz=−SV¯−, (2.23)where the propagation matrix, S, isS =a1 c1 0 0 ... 0b2 a2 c2 0 ... 00 b3 a3 c3 ... 0. .. .. .0 0 ... 0 bN aN. (2.24)Equations 2.16 and 2.17 have general solutionsV¯+(z) = eSzV¯+(0) (2.25)V¯−(z) = e−SzV¯−(0). (2.26)Evaluating these equations at z = 0 and z = L, the boundary conditions given byequations 2.18-2.21 constitute a set of 2N coupled linear equations for the initialvalues of the components of V¯+(0) and V¯−(0), which are readily solved by amatrix inversion. The field reflection and transmission are then V¯−1 (0) and V¯+N (L),respectively.The system of equations (2.16, 2.17) was numerically solved for N = 9, L= 90 µm,and identical waveguides with a thickness of 220 nm, a width of 500 nm, and gapsbetween the waveguides of g = 500 nm, 750 nm, and 1 µm. Additionally, the15system of equations was solved for a system consisting of alternating waveguideswith 500 nm and 600 nm widths separated by a gap of 500 nm for N = 9, andL = 90 µm. The phases φm from the bends were all set to zero for simplicity. Theresults as a function of wavelength are presented in Figure 2.5 a). It is clear that inthe case of identical waveguides there is already significant ripple in the spectrumfor a gap of 750 nm, and that a stop band appears for a gap of 500 nm. With the dis-similar waveguides, however, the ripple in the spectrum for a gap of 500 nm is lessthan that for the identical waveguides at a gap of 1 µm. The shape of the spectrumdepends strongly on the phases φ j, however, the degree of ripple in the spectrumdoes not. This would generally hold true for more realistic, wavelength dependent,phases associated with the bends as well. The degree of ripple was characterizedby computing the minimum transmission of the folded waveguide structure as thegap between the waveguides was varied. The results are presented in Fig. 2.5b). It can be seen that for identical waveguides, the ripple in the spectrum causesthe transmission to rapidly fall off for waveguide separations less than 1 µm. Onthe other hand, the reduction in crosstalk between the dissimilar waveguides effec-tively keeps the spectrum from developing significant ripple until the waveguideseparation is less than 500 nm.Figure 2.6 shows the calculated intensity of the forward and backward trav-elling light along the length of a device, following the waveguides without cross-coupling, for a folded waveguide structure with 9 identical waveguides. The waveg-uides have a width of 500 nm, and a gap between them of 500 nm. The transmis-sion of this structure is shown in Figure 2.5 a), which shows that the transmissionis high for some wavelengths, which will be called the On state, and low for somewavelengths, which will be called the Off state. In Figure 2.6, it can be seen thatin the Off state the amount of forward travelling light decays quickly along thelength of the structure as the light cross-couples into a backward travelling mode.In contrast, in the On state the behaviour is much more interesting. First, perhapscontrary to expectation, the backward travelling mode has significant intensity overmost of the structure, only decaying to zero in the first and last waveguides. Sec-ondly, there is a resonant buildup of intensity in the forward travelling mode overthe entire structure, excepting the input and output. Supposing that the resonantbuildup of intensity relies on a ring resonator-like closed path for light in the struc-161545 1550 1555 1560 156500.20.40.60.81Wavelength (nm)Transmission  gI = 500 nmgI = 750 nmgI = 1000 nmgD = 500 nm(a)0 250 500 750 1000 1250 1500−2−1.5−1−0.500.5Gap (nm)Minimum Transmission (dB)  w = 500 nm w = 500 , 600 nm(b)Figure 2.5: a) Calculated spectra of the folded waveguide structure for 9 iden-tical waveguides with gaps gI = 500 nm, 750 nm, and 1000 nm, andalternating dissimilar waveguides with widths of 500 nm and 600 nmwith gap gD = 500 nm, b) Minimum transmission of the folded waveg-uide structure for similar (solid) and dissimilar (dashed) waveguides.c©Optical Society of America, 2015, by permission.170 200 400 600 80000.20.40.60.811.2Position (µm)Normalized IntensityForward OnBackward OnForward OffBackward OffFigure 2.6: Intensity of forward and backward travelling modes as a functionof position along the folded waveguide structure, measured as the dis-tance along the path with no cross-coupling. The 9 waveguides are allof 500 nm width, 90 µm length, and the gap is 500 nm. The On state isat 1553.16 nm and the Off state is at 1556 nm.ture, like in an example that is discussed below, this observation can help explainthe observation about the backward travelling mode. This is because, as it is sim-ple to convince oneself of, any closed path in the structure necessarily requiressome portion of the path laying in a backward travelling mode. These results showthat a structure as simple as a folded waveguide can exhibit rich and non-obviousbehaviour.A simple argument shows that the neglect of next-to-nearest neighbour cou-pling, and more generally any coupling across an even number of waveguides, doesnot effect the degree of ripple in the through spectrum for lossless waveguides. Tosee this, consider a hypothetical system where nearest-neighbour coupling is neg-ligible but coupling to next-to-nearest neighbour waveguides is not, which can beapproximately implemented with waveguides of alternating widths. In this case,due to the geometry of the folded waveguides, no amount of coupling can result inlight propagating backwards. Thus, there can be no reflected signal. Due to con-18servation of energy, one can conclude that at all wavelengths the transmission mustbe unity. In the case where nearest-neighbour coupling is also present, the effectof next-to-nearest neighbour coupling would be to change the wavelengths whereconstructive and destructive interference occur, but not the degree of interference.A toy model of next-to-nearest waveguide coupling in lossy waveguidesHere, a toy model of next-to-nearest neighbour coupling in lossy waveguides ispresented to explore its effects.Consider a folded waveguide structure with N = 3, where the first and thirdwaveguides are identical and the second waveguide is dissimilar from the othertwo. Coupling can then be assumed to occur only between the first and thirdwaveguides. Figure 2.7 a) shows a schematic of the structure. Let the phase ac-quired along each of the two identical waveguides be φC, and the phase acquiredalong the bends and the intermediate wavguide be φR. Furthermore, assume thatthe intermediate waveguide is lossy, with a forward field amplitude transmission a.Lastly, let the field through and cross-coupling coefficients of the coupler formedby the identical waveguides be t and κ , respectively. The field at the output, Eo, forunit input can then be written asEo = iκeiφC + t2aei(2φC+φR)∞∑j=0(iκaei(φC+φR)) j, (2.27)where first term represents the light that immediately crosses to the output alongthe blue arrow in Figure 2.7 a), and the jth term in the sum represents the light thatcrosses through the red arrow in Figure 2.7 a) j times. Summing the geometricseries and simplifying using coupler power conservation, κ2 + t2 = 1, givesEo = eiφC iκ+aei(φC+φR)1− iκaei(φC+φR). (2.28)This transmission function is of the same form as a that of a first order racetrackresonator with the through coupling and cross-coupling coefficients interchanged[54]. This could also have been seen by identifying the schematic in Figure 2.7 a)with the schematic of such a racetrack resonator in Figure 2.7 b), where equivalent19(RDLțLțW W(a)(RDLțLțWW(b)Figure 2.7: a) Folded waveguide structure with next-to-nearest neighbourcoupling. b) An equivalent racetrack resonator.20paths in each device are coloured the same to help see the equivalence.From equation 2.28, one can see that if κ ≈ a, then at the resonance condition,φC + φR = 3pi/2+ 2mpi for integer m, the transmission can be expected to dropto near zero, in stark contrast to the prediction above for the lossless case. Thus,if one wants to prevent significant ripple in the through spectrum due to resonantnext-to-nearest neighbour coupling, then the coupling coefficient must be made tobe much smaller than a. Since for reasonably short and low loss waveguides a≈ 1,this is not a stringent requirement.A Path Integral ApproachPath integral formulations of physical theories are often used in the context ofquantum theory. Although non-relativistic quantum mechanics can be formulatedin terms of path integrals [55, 56], most elementary presentations do not take thisapproach [57, 58]. On the other hand, in quantum field theory (QFT) the path inte-gral approach is ubiquitous, and is a convenient starting point for perturbation the-ory [59–61]. The path integral approach, remarkably, has even been applied to theclassical mechanics of particles [62], and has shown a deep relationship betweenquantum mechanics and statistical mechanics [63]. Feynman diagrams, introducedas a shorthand for writing down transition amplitudes in perturbative QFT as devel-oped through the path integral approach [64], have become the basis for popularintuitive descriptions of quantum mechanical phenomenon. For example, Figure2.8 shows a Feynman diagram for electron-electron scattering, which gives rise tothe popular description of the Coulomb force as being the result of the exchangeof virtual photons. In this section, it is the usefulness of the path integral approachfor providing intuition that will be exploited, with ray-like paths through the foldedwaveguide structure playing a role analogous to that of Feynman diagrams in QFT.Here we develop a formal path integral model of the folded waveguide structurewhich, while not computationally feasible to implement, provides a useful heuristicfor thinking about how light propagates in the structure. Again, for simplicity weconsider only nearest neighbour coupling.Consider breaking the folded waveguide structure with N identical waveguidesinto M equal length subsections, as shown in Figure 2.9, where in propagating21HHHHȖFigure 2.8: A Feynman diagram representing one term in an infinite sum de-scribing electron-electron scattering through the electromagnetic field.The electrons, e−, interact via the exchange of a virtual photon, γ .ϭϮ͘͘͘͘͘͘DFigure 2.9: Folded waveguide structure divided into M subsections.22(a) (b)(c)Figure 2.10: a) An example of a typical path through the structure, b) a paththat has a large contribution for high coupling, and c) a path that has alarge contribution for low coupling.23across each section the light acquires a phase, φ(M), has a field transmission co-efficient, t(M), and field cross-coupling coefficient, κ(M), to the same waveguideand to each of its adjacent waveguides, respectively. One can imagine inputtinglight into the structure, after which it can take many paths to get to the output,some of which are illustrated for the case N = 5 in Figure 2.10. Since the onlypaths that end at the output waveguide are those that cross-couple an even numberof times, these are the only paths that need to be considered. The field at the output,Eo, for a unit input field, can be expressed as a weighted sum over all paths,Eo = tMeiMφ +(i2)κ2 ∑s∈P(M,2)tη(s)ei(η(s)+2)φ +(i4)κ4 ∑s∈P(M,4)tη(s)ei(η(s)+4)φ + ...(2.29)Eo =∞∑l=0∑s∈P(M,2l)[(−κ2e2iφ )ltη(s)eiη(s)φ], (2.30)where P(M,2l) is the set of paths through the folded structure discretized into Msegments with l pairs of cross-couplings, and η(s) is the number of segments thatthe light is transmitted through while propagating through the path s. Formally tak-ing the limit as M tends to infinity would give the true transmission of the structure,and the two sums in equation 2.30 would become an integral over all paths. Thereflected field can be expressed in a similar manner as a sum over all paths withan odd number of cross-couplings. As an illuminating example, the path integralapproach for a simple directional coupler is derived in Appendix A and is shownto give the correct result.Analysing equation 2.30 in full generality is unlikely to provide any more in-sight into the problem than the computationally simple coupled mode model dis-cussed above. In the limiting cases of high and low coupling, however, a picture ofwhich paths are important can be gleaned.For the case of high coupling, that is when κ is ’large’ and t is ’small’, the mostimportant consideration that determines the strength of the contribution of a pathis the number of segments through which the light is transmitted, since the tη(s)term exponentially damps the contribution of paths as this number is increased.For a fixed l, this means that paths are exponentially damped with increasing pathlength. From this, one can conclude that the most important terms in the sum will241545 1550 1555 1560 156505101520253035Wavelength (nm)Group Delay (ps)  gI = 500 nmgI = 750 nmgI = 1000 nmgD = 500 nmFigure 2.11: Calculated group delay of the folded waveguide structure for 9identical waveguides with gaps gI = 500 nm, 750 nm, and 1000 nm,and alternating dissimilar waveguides with widths of 500 nm and 600nm with gap gD = 500 nm.be those for which coupling occurs at almost every segment, like that shown inFigure 2.10 b). In the case of low coupling, that is when κ is ’small’ and t is’large’, then the contributions of paths are exponentially damped by the number ofcross-couplings. Thus, to an approximation the sum over l can be truncated afterthe first few terms. An example of a significant path for low coupling is shown inFigure 2.10 c). Of course, as M goes to infinity and the segment length approacheszero the coupling coefficient must approach 0 and the transmission coefficient mustapproach 1. ’Small’ and ’large’ coupling coefficients then refer to the rate at whichthese coefficients change with increasing M.Figure 2.11 shows the group delay, dφdω , where φ is the phase at the outputof the structure and ω is the optical angular frequency, for the structures whosetransmission were calculated in Figure 2.5. For low coupling one can see that thegroup delay is constant, which can be interpreted as the light following the pathwith no cross coupling and thus having a constant propagation time through the25structure given by t = NLn/c, where n is the average refractive index and c isthe speed of light in vacuum. Here we have neglected the wavelength dependenceof n for simplicity. For high coupling, however, it can be seen that there are somewavelengths for which the group delay is less than the low coupling group delay,and some wavelengths for which the group delay is larger than that for the lowcoupling case. This can be interpreted by there being certain wavelengths wherethe light takes paths like those in 2.10 b) that are shorter than the low couplingpath, and wavelengths where light takes paths containing closed loops, like thosediscussed in the toy model of a 3 waveguide system discussed above, that make thepath longer than the low coupling path, respectively.The Geometrical LimitIf we consider paths that are close to each other, that is paths that are related by asmall deformation, and have the same amplitude in the sum, that is have the samenumber of cross-couplings and transmissions, then one can discuss the geometricaloptics limit of the structure. An example of a set of paths that are close togetheris shown in Figure 2.12, for which the deformation is parameterized by a smalldistance, δ . If for small δ the phase acquired along each of these paths, relative tothe path with δ = 0, is proportional to kδ then for kL >> 2pi the sum over thesepaths for all δ will be an infinite sum of complex numbers with the same amplitudeand phase uniformly distributed over [0,2pi], and will sum to zero. That is to saythat any path with this property can be ignored in the sum. Thus, the only pathsthat contribute to the sum are those for which the phase acquired is stationary withrespect to small deformations in the path. This is a precise statement of Fermat’sprinciple [65], which more loosely says that a ray of light will follow the path ofshortest optical length. Applied to this structure, we can then express the waveoptics solution, 2.30, as stating that the light follows all geometrically allowedpaths and interferes at the structure output.A closer look at the propagation in the structure reveals that, since transmis-sion and cross-coupling commute (ie. tκ = κt), the phase is stationary to smalldeformations for all paths. Thus, every path is geometrically allowed and Fermat’sprinciple does not offer any simplification to equation 2.30. From this observa-26Figure 2.12: A set of paths that are related by a small deformation describedby the parameter δ .tion we can conclude that, for non-zero coupling, propagation through the foldedwaveguide structure is intrinsically wave-like and its transmission cannot be welldescribed in the language of ray optics. That is, there is no meaningful way todescribe any one path that light follows through the structure. Although describingthese types of devices in terms of the paths that light takes is conceptually help-ful, one should be careful to not take explanations using this type of language tooseriously.The ability to discuss the propagation of light in waveguides in terms of pathintegrals, a technique most often encountered in quantum mechanics, is not a co-incidence. The correspondence between the two types of systems is made clearby noting that equation 2.9 is of the form of the Schro¨dinger equation when po-sition is interchanged with time, and the momentum operator, P, is interchangedwith the Hamiltonian operator, H, in natural units (h¯ = 1) [57]. Due to this cor-respondence, a number of authors have made analogies between light propagationin N coupled waveguides to the time evolution of a particle in a system with Nenergy eigenstates [52, 66, 67]. Under this analogy, what is the quantum systemcorresponding to the folded waveguide structure? In the folded waveguide struc-27ture the light propagates in both the forward and backward directions. The moststraightforward way to map this to a quantum system is to consider two particlesin N level atoms, where one of the particles is evolving backwards in time. Theboundary conditions imposed on the waveguides, equations 2.18-2.21, would thenrepresent conditions relating the state of both particles at some intitial and finaltimes. Clearly, this is not a physically relevant system. An alternative way to mapfrom the waveguide picture to the quantum mechanical picture is to suppose thatthere are two particles, both evolving forward in time, in N level atoms where oneof the particles has negative energy states (ie. is bound) and the other particle haspositive energy states (ie. is free), where the energy eigenvalues of both systemshave the same magnitude. Like in the previous example, the boundary conditionsdo not have an obvious physical implementation. Thus, it does not seem likely thatthe analogy to quantum mechanics will prove to be insightful in this case.2.3 Mechanical ModelThe underetching of the silicon substrate to form a suspended structure raises con-cerns for the mechanical stability of the phase shifter arms. During processingusing liquid reactants (eg., etching, cleaning, etc.) large surface tension forces canarise that may put damaging stresses on the suspended structure, or lead to stic-tion between the suspended structure and nearby surfaces [37]. Further, in prac-tical applications the structure may be subject to vibrations or large accelerations.Vibrations in particular could cause time-varying stress-induced refractive indexchanges [68] in the waveguides that could lead to increased noise in data switchingapplications. In this section we investigate a simple model of the structure wherethe stresses induced by a uniform acceleration are calculated. While this modeldoes not explore all types of possible mechanical modes of failure or noise gener-ation, it gives an indication that with appropriate design these issues can likely beovercome.Here we consider the suspended glass structures shown in Figures 2.13 a) andb). The suspended structures have a length, L, a 12 µm width, and thickness of6 µm. The structure is fixed at both ends. A finite element mechanical simulationwas performed assuming that the structure was undergoing a uniform acceleration,28ȝPȝP)L[HG(QGV6XVSHQGHG*ODVV(a)6XSSRUW%ULGJHȝPȝP)L[HG(QGV6XVSHQGHG*ODVV(b)Figure 2.13: Top-down view of suspended glass structure a) without supportbridges, and b) with two pairs of support bridges undergoing an accel-eration, a, oriented out of the page.a, in the direction normal to the chip surface. The acceleration was modelled bya uniform pressure, P = ma/A, distributed over the upper surface of the structure,where m is the mass of the suspended glass, and A = L×12 µm is the area of theupper surface. We also considered structures where glass support bridges with a6 µm width, 8 µm length, and fixed ends are used to support the structure along itslength. The support bridges are assumed to be equally spaced along the length ofthe structure.Figures 2.14 a) and 2.14 b) show the calculated von Mises stress distribu-tions for a structure with L = 90 µm and a = g with no support bridges,where g = 9.8 ms−2 is the local acceleration due to gravity, and a structure withL = 290 µm and two pairs of support bridges for the same acceleration, respec-291RUPDOL]HGYRQ0LVHV6WUHVV(a)1RUPDOL]HGYRQ0LVHV6WUHVV(b)Figure 2.14: Stress distribution in a suspended heater structure under uniform1g acceleration along the direction normal to the chip for a) a 90 µmlong structure with no support bridges, and b) a 290 µm long structurewith two pairs of support bridges.tively.The maximum von Mises stress was calculated for the suspended structuresdescribed above as their length was varied, and the results are shown in Figure2.15 for an acceleration of 1g. It is clear that adding more support bridges reducesthe maximum stress, and thus will allow for a more robust structure. However,as will be shown in Section 2.4, this comes at the cost reduced thermal isolation,and thus a decrease in device efficiency. The simulations suggest that even for a300 µm long device with no bridges, assuming even a modest yield strength of3050 100 150 200 250 300050100150200Device Length (µm)Maximum von Mises Stress (Pa) No Bridge1 Bridge2 BridgesFigure 2.15: Maximum von Mises stress in suspended structure.20 MPa for glass [69], the structure could withstand an acceleration of 105g beforefailing, while having a maximum displacement due to deformation of only 300 nm.This strongly suggests that for reasonably sized devices structural stability will notpose an issue.2.4 Thermal ModelThe phase shift in thermally actuated switches is achieved through the thermo-optic effect, which is the phenomenon in which the refractive index of a materialis dependent on the temperature, T , of the material. The thermo-optic coefficientof silicon is sufficiently large that small temperature changes can be used to causelarge phase shifts over relatively small lengths [43]. In order to maximize thethermo-optic phase shift in a device for a given power it is important to consider thethermal properties of a design. Specifically, to maximize the thermo-optic phaseshift in a device one must minimize the rate of heat transfer from the device to itssurroundings and ensure that the heated region is localized to the region in whichthe light is propagating. This latter concern is to ensure that power is not wasted31heating a region that does not contribute to a phase shift. Since in some applicationsswitching times may be important in addition to the phase shifter efficiency, theeffect of the methods used to optimize efficiency on respone times should also bepredicted.In this section, the results of finite element simulations of suspended structureson SOI are presented. The dependence of the thermal properties on the geometry ofthe structure are described, and the performance of switches utilizing a suspendedstructure are predicted. Figures of merit for a device’s tradeoff between switchingpower and response time are also proposed.The structure that was simulated consists of a glass structure of length, L,width, w, and 6 µm thickness containing N silicon waveguides and a metal heaterof 10 µm width inside it. The glass structure is fixed at its ends, where it is inthermal contact with the silicon substrate. The structure may also be supportedalong its length from below by the silicon substrate, or underetched to remove thesubstrate and become suspended. Further, the glass structure may have pairs ofsupport bridges of 8 µm length, 6 µm width, and 6 µm thickness, like those con-sidered in Section 2.3. The heater supplies a power of P = 1 mW uniformly overits volume. The bottom of the substrate and the edges of the chip are assumed tobe at 20 ◦C, and all other surfaces are assumed to lose heat by convection. For sim-plicity, the convection coefficient was approximated as that due to free convectionat the surface of a 1 cm×1 cm horizontal plate. That is, the convection coefficientat each surface is approximated as being the same as that which would occur overthe surface of an unpatterned chip. The effect of thermal radiation was neglectedbecause the temperature changes needed for a pi phase shift were small due to thelong lengths of the waveguides used. The thickness of the substrate and distance tothe chip edge in the simulation were determined through convergence testing. Thedevices that were fabricated, as described in Table 3.1, will be frequently used hereas examples.The power required for switching is determined as the power that results in aphase change of pi over the length of the folded waveguides. Specifically,pi = 2piλdneffdTdTdPPNL (2.31)32P =λ2NL1dneff/dT1dT/dP, (2.32)where neff is the effective index of the waveguides. Numerical simulations deter-mined that dneffdT ≈ 1.93× 10−4 K−1 for all of the waveguide widths used in thisstudy.First, the effect of the length of the suspended region was investigated. Figure2.16 shows the simulated switching power, as a function of length, for a 12 µmwide suspended structure with 9 waveguides and various numbers of pairs of sup-port bridges. It is clear that as the length of the device increases the switchingpower decreases dramatically. This is as expected since the rate of heat transfervia conduction along the suspended bridge is proportional to the gradient in tem-perature at its ends [70]. Therefore, if the characteristic length of the structureincreases, then a larger change in temperature is required to dissipate the sup-plied power. Additionally, it can be seen that increasing the number of supportbridges increases the required switching power. This is expected because the sup-port bridges represent additional paths through which heat can be conducted awayfrom the device.Figure 2.17 shows the simulated switching power, as the width is varied, forexample devices consisting of a 90 µm long device with no support bridges, and a290 µm long device with two pairs of support bridges. For each width simulated,the number of waveguides was set to be the maximum number of waveguides thatcould be fit in the width with a 1 µm pitch, while keeping the waveguides at least1 µm away from the edges of the suspended structure. To better visualize thetrends for both examples simultaneously, the switching powers for each device arenormalized to the switching power for that device with a 12 µm width.For the case of the device with no bridges, it can be seen that there is only avery modest increase in switching power with increasing width. A simple argumentsuggests why this is the case. If heat loss is assumed to be dominated by conductionalong the length of the suspended structure, rather than through convection to theair, then the rate of heat loss at a given temperature change is proportional to thecross-sectional area of the suspended structure, and in turn to the width. Thus thepower required to get a given temperature change increases linearly with width.However, the number of waveguides that can fit in the structure also increases3350 100 150 200 250 300050100150200250300Length (µm)Switching Power (µW) No Bridge1 Bridge2 BridgesFigure 2.16: Switching power required for devices with 9 waveguides andvarying number of support bridges.linearly with width. Thus, equation 2.32 predicts that the switching power shouldbe independent of width in this case.On the the other hand, for the device with 2 pairs of support bridges there is asignificant decrease of the switching power with width. This can be understood inmuch the same way as the decrease in switching power observed with increasingdevice length above, but applied to the heat lost due to conduction along the supportbridges. As the width is increased, the average distance for heat to travel along toreach the support bridges is increased. Therefore, for a given temperature change,the rate of heat loss due to conduction along the support bridges is decreased. Thisobservation shows that if support bridges are used, then a decrease in switchingpower can be achieved by increasing the suspended structure width. However, thiscomes at the expense of an increase in device footprint. In fact, a doubling of thestructure width gives only a 25% reduction in switching power, so this trade-off isquite severe.Figures 2.18 and 2.19 shows the temperature distributions for the 90 µm and290 µm long devices, respectively. The normalized temperature change at a point3412 14 16 18 20 22 2400.20.40.60.81Width (µm)Normalized Switching Power90 µm, No Bridge290 µm, 2 BridgesFigure 2.17: Normalized switching power required for devices as the widthis varied. The number of waveguides for each width is the maximumnumber of waveguides that can fit within the structure with a 1 µmpitch.is defined as the ratio of the temperature change at that point to the maximumtemperature change in the structure.To study the effect of underetching the silicon substrate, further simulationswere performed for devices with the substrate not etched. First, the switchingpowers of the underetched and unetched versions of two example devices werecalculated. For a 90 µm long, 12 µm wide device with 5 waveguides and nosupport bridges the simulated switching powers for the underetched and unetchedversions were 6.1 mW an 0.26 mW, respectively. For a 290 µm long, 12 µmwide device with 5 waveguides and two pairs of support bridges, the simulatedswitching powers for the underetched and unetched versions were 3.4 mW and0.14 mW, respectively. It is evident that the underetching results in a dramaticreduction in required switching power. This is due to the removal of the veryefficient conduction path to the substrate with both a large area and short length.Secondly, time domain simulations were carried out. The entire simulation vol-351RUPDOL]HG7HPSHUDWXUH&KDQJH(a)1RUPDOL]HG7HPSHUDWXUH&KDQJH(b)Figure 2.18: Temperature distribution in 90 µm long, 12 µm wide, sus-pended structure. a) overhead view, and b) cross-sectional view. Notethe different scale in b). The black rectangles indicate the positions ofthe silicon waveguides.ume was set to be the same temperature as an initial condition. The power suppliedto the heater was then set to be 1 mW and the temperature change, averaged overthe volume of the waveguides, was recorded as a function of time, t. The resultsfor the unetched and underetched devices are presented in Figures 2.20 a) and b),respectively.The difference in rise time between the long and short devices is much smallerfor the unetched devices than for the underetched devices. To explain this observa-tion it is helpful to explore the dynamics of heating and cooling. Provided that the361RUPDOL]HG7HPSHUDWXUH&KDQJH(a)1RUPDOL]HG7HPSHUDWXUH&KDQJH(b)Figure 2.19: Temperature distribution in 290 µm long suspended structurewith two pairs of support bridges. a) overhead view, and b) cross-sectional view in the center of one of the pairs of support bridges. Theblack rectangles indicate the positions of the silicon waveguides.heat loss to conduction dominates over that due to convection, the rate of changeof average temperature satisfies Newton’s law of cooling [70],CdTdt= P− Q˙, (2.33)where C is the heat capacity of the structure, and Q˙ is the rate of heat transferaway from the device. The heat capacity is proportional to the device volume,and thus to its length. Q˙ is proportional to the temperature, relative to that of370 50 100 150 20000.20.40.60.81Time (µs)Normalized Temperature         ChangeL = 90 µmL = 290 µm(a)0 2000 4000 6000 800000.20.40.60.81Time (µs)Normalized TemperatureChangeL = 90 µmL = 290 µm(b)Figure 2.20: Finite element simulated time domain response of a) unetchedand b) underetched heaters of different lengths. The 90 µm long devicehas no support bridges, and the 90 µm long device has two pairs ofsupport bridges. A power of 1 mW is continuously supplied beginningat time 0.38the structure’s surroundings, the cross-sectional area of the conduction pathways,Ac, and inversely proportional to the characteristic length, Lc, of the conductionpathways [70]. Thus, with appropriate proportionality parameters a and b,whichare material and geometry dependent, equation 2.33 can be written asdTdt=aL(P−bAcT/Lc). (2.34)To investigate the rise time, the rate of temperature change must be normalizedto the maximum temperature reached, Tmax. Inspection of equation 2.34 givesTmax = PLc/bAc, and in turn1TmaxdTdt=PaLTmax(1−T/Tmax). (2.35)Solving equation 2.35 for an initial temperature of T = 0, relative to the tempera-ture of the surroundings, gives the solutionTTmax= 1− e−t/τc (2.36)τc =LLcabAc, (2.37)where τc is the characteristic rise time of the structure, giving the time required forthe temperature change to reach 1−1/e≈ 63% of its maximum.Using equation 2.36, one can explain the observations made above about theresults in Figures 2.20 a) and b). For the case of the unetched devices the primaryconduction path for heat transfer is expected to be directly to the substrate throughthe botttom of the glass structure. Thus we can identify the area of this surfacewith Ac, and find that it is proportional to the structure length. As for the charac-teristic length for this conduction pathway, it can be identified with the thicknessof the glass structure, which is independent of the device length. Combining theseobservations, and equation 2.4, one can see that the rise time can be expected tobe independent of the device length. This is consistent with the simulation results.For the underetched devices, however, the dominant conduction pathways are ex-pected to be through the ends of the suspended structure and along the support39bridges, if present. The characteristic area Ac can then be associated with the sumof cross-sectional areas of the glass structure and of the support bridges, if present.The characteristic length would then be expected to be proportional to the physicallength of the structure. Thus, the rise time would be expected to scale proportion-ally to L2/Acb. Since the b parameters for the geometries where bridges are andare not present are unknown, this cannot be used to evaluate a predicted ratio forthe rise times to compare with the simulation. However, it does make the usefulprediction that for underetched devices the rise time can be expected to scale withthe square of length.Substituting the expression for Tmax into equation 2.32, in combination withthe scaling arguments presented above for the underetched and unetched devices,gives predictions for the scaling behaviour of the switching power. Specifically,the switching power of unetched devices is predicted to be independent of length,and the switching power of underetched devices is predicted to scale as L−2. Apower law fit to the simulated switching power results for the underetched devicewith no bridges in Figure 2.16 as a function of length gives good agreement witha L−1.7 dependence. The discrepancy between this result and the predicted L−2dependence is most likely due to the neglect of the effects of convection in equation2.32 because the influence of convection, relative to that of conduction, can beexpected to increase with device length.With the above predicted scaling behaviour of the rise time and switchingpower of unetched and underetched devices we can define a figure of merit forcomparing the performance of devices when both speed and efficiency are con-cerns. Considering the product of rise times and switching powers, the above anal-ysis gives that this quantity is predicted to be independent of length for both un-etched and underetched devices. Based on this, the proposed figure of merit, FOMis defined asFOM = τcPpi (2.38)where Ppi is the switching power. This is the same figure of merit used in [34].Of course, convection is neglected in the analysis above so the predicted scalingbehaviour will not be exact. Therefore the figures of merit should be taken as arough comparison metric and not as a definitive predictor of relative performance.402.5 Chapter Summary and ConclusionsIn this chapter an overview of the structure and operating principle of an MZI-based switch was first presented. The remainder of the chapter was devoted tothe development of models of the optical, mechanical, and thermal properties ofsuspended folded waveguide structures.An optical model based on coupled mode theory was used to predict the trans-mission through folded waveguide structures with either identical or dissimilarwaveguides. The ripple in the through spectrum of the structure was shown tobe an appropriate metric for evaluating the extent of crosstalk present, and it waspredicted that a higher waveguide density could be achieved, while maintaininglow crosstalk, with dissimilar waveguides than with identical waveguides. A pathintegral model of the folded waveguide structure was also developed, which pro-vides an intuitive picture for discussing and visualizing how light propagates in afolded waveguide structure when coupling is present. The analogy between thefolded waveguide structure and quantum mechanical systems was discussed, and itwas determined that the analagous systems are not physically relevant.A mechanical model of the structure was used to simulate the stress distributionin a suspended waveguide structure under a uniform acceleration. It was found that,for devices with reasonable cross-sectional dimensions, the suspended structurelikely only needs to be supported at its ends (ie., it does not need support bridges)for lengths up to 300 µm. The addition of support bridges further increases themechanical stability as expected.A thermal model of the folded waveguide structure was used to predict theswitching power and switching speed for devices using different designs, both forunetched and underetched devices. In addition to simulations, simple theoreticalconsiderations were used to predict the scaling behaviour of the switching powerand switching speed with variations of the width or length of the structure. Thepredicted scaling dependencies were used to propose a figure of merit, which quan-tifies the tradeoff between switching power and switching speed, for both unetchedand underetched devices. It was found that to minimize the switching power, oneshould design an underetched device that is long and has no support bridges.41Chapter 3Device Design and ExperimentalInvestigation3.1 Device Design3.1.1 A 2×2 SwitchFigure 3.1 a) shows a schematic diagram of the fabricated devices. Input lightis split by a 50-50 adiabatic splitter [53] and the light traveling along one of theMZI arms passes N times through the thermal phase shifter before recombiningwith the light from the other arm at the device output. Figures 3.1 c) and d) showthe cross-section of the phase shifter region for unetched and underetched devices,respectively. Each of the N waveguides has a thickness of 220 nm, a width, wi,i= 1,2, ...,N, and all waveguides are separated by a common gap, g. A 10 µm wideheater of length L is used to apply a temperature change to the waveguides to inducea thermo-optic phase shift. In the case of the underetched devices, the siliconsubstrate has been removed to form a 12 µm wide suspended bridge to increasethermal isolation. The devices were fabricated using 248 nm optical lithography atthe Institute of Microelectronics (IME), Singapore. The unetched and underetchedversions of the device were measured from different wafers. Figure 3.1 b) showsan optical image of a fabricated device.42>/ŶƉƵƚKƵƚƉƵƚǁϭǁϮ(a),ĞĂƚĞƌhŶĚĞƌĞƚĐŚĞĚZĞŐŝŽŶƐZŽƵƟŶŐDĞƚĂůĚŝĂďĂƟĐϱϬͲϱϬ^ƉůŝƩĞƌ(b)43ϴʅŵ,ĞĂƚĞƌϭϬʅŵϭϮʅŵϲʅŵǁϭ ǁϮǁEͲϭǁEŐ͘͘(c)ϴʅŵ,ĞĂƚĞƌϭϬʅŵϭϮʅŵϲʅŵǁϭ ǁϮǁEͲϭǁEŐ͘͘(d)Figure 3.1: a) Schematic of thermally tunable MZI switch. Black traces:WGs, Red: Oxide openings define underetched region, Purple: Rout-ing metal, and Green: Heater metal. Inset: Waveguide taper region.b) An optical micrograph of a fabricated device. c) Thermal phaseshifter cross-section before underetching. d) Thermal phase shiftercross-section after underetching. In c) and d), silicon dioxide is blue,silicon is tan, the metal heater is grey, and air is white. c©Optical Soci-ety of America, 2015, by permission.44Table 3.1: Device parametersDevice 1 Device 2 Device 3 Device 4 Device 5N 3 5 5 9 9500, 500 500, 400 500, 400 500, 600, 400 500, 600, 400wi (nm) 500 550, 650 550, 650 550, 650, 450 550, 650, 450500 500 600, 400, 500 600, 400, 500g (µm) 3.0 1.0 1.0 0.5 0.5L (µm) 120 90 290 90 290Five different devices were fabricated to study the effect of dense dissimilarwaveguide routing on the tuning efficiency of MZI switches. Device 1 was used asa baseline device using identical waveguides and a gap of 3 µm to ensure no degra-dation of the spectrum due to crosstalk. Devices 2 and 3 used dissimilar waveguiderouting with a gap of 1 µm, and devices 4 and 5 used dissimilar waveguides witha gap of 0.5 µm for the most dense routing. Devices 3 and 5 included two pairsof support bridges with a 6 µm width and 8 µm length. Table 3.1 summarizes theparameters of each device. The widths of the waveguides were picked such that ad-jacent waveguides have a width difference of at least 100 nm and next-to-adjacentwaveguides have a width difference of at least 50 nm to protect against any effect ofnon-nearest neighbour coupling that was not considered in the theoretical analysis[50]. Device 1 was fabricated only in an unetched configuration while devices 2-5were fabricated in both unetched and underetched configurations. The footprintsof device 1, devices 2 and 4, and devices 3 and 4 were approximately 650 µm ×180 µm, 600 µm × 180 µm, and 800 µm × 180 µm, respectively.3.1.2 A 4×4 SwitchFigure 3.2 shows a schematic of a 4×4 switch composed of the switch elementsdescribed above. The switch architecture is the non-blocking switch described in[71]. In addition to the switching elements, the waveguide crossings described in45ϭϯϰϮ,,,,,999,Figure 3.2: A schematic of a 4×4 switch. The 2×2 switches I through VIroute light from the inputs A,B,C, and D to the outputs 1,2,3, and 4.[72] are utilized during routing waveguides between the switches. Due to spaceconstraints, only one 4×4 switch could be fabricated and so the underetched ver-sion of device 5 was used to create the switching elements. Device 5 was chosen toachieve the lowest switching power. By thermally actuating the switches I throughVI it is possible to route light from any of the input ports, labelled by the lettersA-D, to any of the output ports, labelled by the numbers 1-4. Table 3.2 gives thetuning states required of each of the switches for each routing state. The tuningstates given are examples only, as the tuning states of the switches are in generalnot unique.3.2 Experimental ProcedureA schematic of the experimental test setup is shown in Figure 3.3. A tunable lasersource was used to inject 0 dBm of light through an optical fiber into the chipthrough TE grating couplers [73]. After passing through a device the light exitedthe chip through a second fiber grating coupler and the transmitted light was passedto a photodetector. The wavelength of the input light was swept from 1530 nm to1580 nm in 0.1 nm steps and the transmission spectrum of the device was recorded.46Table 3.2: Switch routing statesIn\Out 1 2 3 4A 10X1XX XXX0XX X10101 XX111XB 0XXXXX 1XX10X 1XXX11 11100XC 111X00 XX111X XX1X01 XX0XXXD 11XXX1 X00111 XXXXX0 X01XX1The characters represent the tuning state of switches I through VI in order. 0 is thethrough state, 1 is the cross state, and X indicates that the state of a switch does notmatter.7XQDEOH/DVHU3KRWRGHWHFWRU'HYLFH8QGHU7HVW&XUUHQW6RXUFH6ZLWFKFigure 3.3: A schematic diagram of the experimental setup. Solid and dashedlines indicate optical and electrical connections, respectively.This procedure was repeated while applying several different current levels to thephase shifter heaters and recording the power supplied. Additionally, the adiabaticsplitters used to form the interferometers were characterized by injecting light intoone of their input ports and simultaneously measuring the transmission to theiroutput ports through TE grating couplers.3.3 Results and DiscussionFigure 3.4 a) shows a schematic of the adiabatic splitter used to form the MZI, and3.4 b) shows the measured transmission through the structure. It can be readily477KURXJK&URVV,QSXW(a)1480 1500 1520 1540 1560 1580−50−45−40−35−30−25−20Wavelength (nm)Transmission (dBm)  ThroughCross(b)Figure 3.4: a) A schematic of the adiabatic splitter. The tan is 200 nm thicksilicon, and the orange is 90 nm thick silicon slab. b) Measured trans-mission through an adiabatic splitter.481500 1520 1540 1560 1580−60−50−40−30−20Wavelength (nm)Transmission (dBm)  S11 (Through)S21 (Through)S11 (Cross)S21 (Cross)Figure 3.5: Transmission to the two output ports of an implementation of de-vice 2 in the through and cross states. The numbers 1 and 2 indicate theupper and lower arms, respectively, of the switch input and output.seen that the splitting ratio is both far from 50-50 and strongly wavelength depen-dent. Since the measured splitting ratio in [53] was good for the same design, thissuggests that there may have been a significant fabrication error in producing thesplitter. Similar performance was measured on several chips.Despite the poor splitting ratio measured, the transmission of a switch was in-vestigated to determine if the efficiency of the phase shifters could still be extracted.Figure 3.5 shows the transmission of an implementation of device 2. The throughstate is defined as the state of the switch where the power supplied to the phaseshifter is such that when light is injected into the upper arm of the input splitter, thetransmission to the upper arm of the output splitter, S11, is maximized. Similarly,the cross state is the state where the power supplied to the phase shifter is suchthat the transmission from the upper arm of the input splitter to the lower arm ofthe output splitter, S21, is maximized. It can be seen that the extinction ratio forthrough operation, S11(Through)/S11(Cross), is much less than the extinction ratiofor cross operation, S21(Cross)/S21(Through). Similar behaviour was observed in49all devices tested. The poor performance in through operation is most likely due tothe unequal splitting by the adiabatic splitters. Thus, for all subsequent testing theswitches were tested for cross operation and the on state refers to the cross state,while the off state refers to the through state.In all cases the extinction ratio of the switch was measured to be greater than20 dB. Figures 3.6 a) and 3.6 b) show example optical spectra of the underetchedversions of devices 3 and 5, respectively, in the on and off state. It can be seen thateven for the longest devices tested the more aggressive waveguide routing densityof device 5 compared to device 3 has not had a negative effect on either the extinc-tion ratio of the switch or the ripple in the transmission spectrum, which is main-tained at below 0.1 dB peak to peak. This suggests that the dissimilar waveguideshave successfully prevented cross-coupling of power in the dense routing regionsof the switch. The insertion losses of the switches were estimated to be -0.9 dB,-1 dB, -2.5 dB, -1.2 dB, and -2.9 dB for devices 1-5, respectively. The difference ininsertion loss is due to the difference in propagation loss for the different arm pathlengths. The insertion loss was found not to depend on whether or not the devicewas underetched. The envelope of the transmission spectrum in the on state is dueto the wavelength-dependent coupling efficiency of the grating couplers used [73].The wavelength dependence of the extinction ratio is due to an optical length mis-match between the two arms, which is likely due to variations in the thickness ofthe silicon layer across the wafer [74], as well as due to the wavelength dependenceof the splitter splitting ratio. The period of the variations in extinction ratio couldbe extended to create a more broadband device by designing a switch such that theaverage distance between its arms is smaller, at the expense of an increased thermalcrosstalk between the arms.Figure 3.7 show the normalized transmission functions of the unetched and un-deretched versions of devices 2-5 as functions of the power applied to the thermalphase shifter, along with sinusoidal fits to the data. The wavelength of operationwas 1550 nm. It can be seen that in all cases the devices with more dense waveg-uide routing give higher phase shifter efficiency. The measured efficiencies aregiven in Table 3.3, along with the efficiencies simulated as described in Section2.4. It can be seen that the relative improvement in phase shifter efficiency whenincreasing waveguide routing density is greater for short devices than for long de-501530 1540 1550 1560 1570 1580−70−60−50−40−30−20−10Wavelength (nm)Transmitted Power (dBm)  On StateOff State(a)1530 1540 1550 1560 1570 1580−70−60−50−40−30−20−10Wavelength (nm)Transmitted Power (dBm)  On StateOff State(b)Figure 3.6: Measured spectra of the underetched versions of a) device 3 andb) device 5. c©Optical Society of America, 2015, by permission.51Table 3.3: Tuning efficiency of MZI switches (mW/pi)Device 1 Device 2 Device 3 Device 4 Device 5Measured Unetched 14 8.7 5.9 3.8 4.2Measured Underetched N/A 0.68 0.16 0.20 0.095Simulated Unetched 10 6.1 5.9 3.4 3.3Simulated Underetched N/A 0.26 0.09 0.14 0.048vices, and that the ratio of efficiencies approaches the ratio of waveguide densitiesfor the long devices. Further, the relative change in efficiency when increasingwaveguide routing density is similar for both the unetched and underetched de-vices. The highest efficiency achieved, 95 µW/pi , is to the best of our knowledgethe highest efficiency reported to date for thermally actuated MZI switches. It canbe seen that the simulated efficiencies generally agree with the measured valuesmore closely for the unetched devices than for the underetched devices. The trendsin the measured data, however, are well captured despite a consistent underesti-mation of the efficiencies. One possible explanation of this observation is that thesimple model of convection used is not sufficient to accurately describe the heatloss to the air, which is expected to be more important for underetched devicesthan for unetched devices. The fact that the model correctly predicts the relativeperformance of the different devices suggests that the results of the simulation cannevertheless be a useful tool to make predictions about the relative performance ofpotential future designs.Figure 3.8 shows the temporal response of the MZI switches when the heaterswere driven with a square pulse. The temporal response was found not to dependsignificantly on the waveguide routing density, but only on the heater length andwhether or not the device was underetched. This suggests that the increase in de-vice efficiency with increasing waveguide density does not come at the expenseof a slower response time. The measured response times are summarized in Table3.4, along with the rise times simulated as described in Section 2.4. Simulated falltimes are not included, as they are expected to be the same as the rise times due tothe linearity of the thermal model used. It is clear that the increases in efficiency520 5 10 1500.20.40.60.81Heater Power (mW)Normalized Transmission  Device 2 Device 4 Fit Fit(a)0 1 2 3 4 5 600.20.40.60.81Heater Power (mW)Normalized Transmission  Device 3 Device 5 Fit Fit(b)530 0.1 0.2 0.3 0.4 0.5 0.600.20.40.60.81Heater Power (mW)Normalized Transmission  Device 2 Device 4 Fit Fit(c)0 0.05 0.1 0.1500.20.40.60.81Heater Power (mW)Normalized Transmission  Device 3 Device 5 Fit Fit(d)Figure 3.7: Normalized transmission functions of the a) short (devices 2 and4) unetched, b) long (devices 3 and 5) unetched, c) short underetched,and d) long underetched MZI switches. c©Optical Society of America,2015, by permission.54Table 3.4: 10-90 Response times of MZI switches (µs)Unetched Unetched Etched EtchedDevice 3 Device 5 Device 3 Device 5Measured Rising 40 60 550 750Measured Falling 45 65 550 1200Simulated Rising 35 45 930 2400when underetching devices or increasing device length come with an increase inthe response time due to the improved thermal isolation of the heated region fromits environment. Further, it can be seen that the simulated response times giveagreement with the measured data similar to the agreement achieved between thesimulated and measured phase shifter efficiencies in Table 3.3. Like in Table 3.3,the agreement between measurement and simulation is better for the undetched de-vices than for the underetched devices. Again, we suggest that this may be due tothe simple model of convection used and point out that the relative performanceof the devices can be accurately predicted through simulation. The observed dif-ference between the observed rise and fall times in the experimental results cannotbe explained by the linear thermal model in Section 2.4. One possible explana-tion of the difference observed is that the change in intensity is not proportionalto the change in device temperature due to the non-linear change in the transmis-sion of an MZI when a phase shift is introduced. Thus, for large phase shifts onecan expect to see deviations from the linear model in Section 2.4. This, of course,can also explain some of the discrepancy between the simulated rise times and theexperimental values.Table 3.5 shows the calculated and simulated figures of merit for the unetchedand underetched versions of a short and a long device. For both the long andshort devices, the device with higher waveguide density was chosen to evaluatethe figure of merit because, having lower switching power and similar responsetimes as compared to the less dense waveguide versions, they are expected to havelower figures of merit. Two observations about the data should be noted. Thefirst is that the agreement between the simulated values and the predicted values550 0.5 1 1.5 2−0.200.20.40.60.81Time (ms)Normalized Transmission  Device 2Device 3Signal(a)0 5 10 15−0.200.20.40.60.81Time (ms)Normalized Transmission  Device 4Device 5Signal(b)Figure 3.8: Temporal response of a) unetched, and b) underetched MZIswitches. c©Optical Society of America, 2015, by permission.56Table 3.5: Figures of merit of MZI switches (nJ)Unetched Unetched Etched EtchedDevice 3 Device 5 Device 3 Device 5Measured FOM 102 110 41 30Simulated FOM 90 65 37 50for the underetched devices is generally better than said agreement for either theswitching powers or response times individually. This is because the underestimatein simulated switching power was counteracted by an overestimate in responsetime. Secondly, it can also be seen that the agreement between the simulated andmeasured values is generally better for the short devices than for the long devices.This observation is consistent with the hypothesis that conduction is being poorlymodelled because, since convection is expected to contribute more to the heat lossrelative to conduction as the device length is increased, this inaccuracy should beexpected to cause a greater discrepancy for longer devices.Figures 3.9 a), b), and c) show comparisons of the performance of the under-etched version of device 5, in terms of switching power, Ppi , characteristic rise time,τc, and the figure of merit Ppiτc, respectively, with previously reported experimentalresults [2]. When only 10-90 rise times were given, the characteristic rise time wasestimated using the single pole temporal response described by equation 2.36. Itcan be seen in Figure 3.9 a) that the two results related to this thesis, [1] and [2],have, by utilizing dissimilar waveguides, achieved significantly lower switchingpowers than previously reported results. The result in [2] was achieved by utilizinga similar phase shifter design to that of the underetched version of device 5 in thisthesis in a Michelson interferometer, similarly to the device in [23]. In a Michelsoninterferometer, the light passes through a phase shifter twice, and therefore requireshalf the switching power as compared to an MZI utilizing the same phase shifter.Despite achieving such low switching powers, these devices have response timesthat are lower than some devices with significantly higher switching powers, as isshown in Figure 3.9 b). This observation suggests that the devices in this workdemonstrate a good tradeoff between switching power and response time. This is572002 2004 2006 2008 2010 2012 2014 201610−210−1100101102103   [19]   [21]   [22]   [26]   [41]   [23]   [33]   [46]   [8]   [42]   [44]   [45]   [43]    [7]   [2]   [1]YearP pi (mW)(a)2002 2004 2006 2008 2010 2012 2014 201610−1100101102103104Yearτ c (µs)   [19]   [21]   [22]   [26]   [41]   [23]   [46]    [8]    [42]   [44]   [45]    [43]   [7]   [2]   [1](b)582002 2004 2006 2008 2010 2012 2014 201610−1100101102103104105   [19]   [21]    [22]   [26]   [41]   [23]   [46]    [8]   [42]   [44]   [45]   [43]   [7]   [2]   [1]YearP piτ c (nJ)(c)Figure 3.9: Comparisons of the a) switching powers, b) characteristic risetimes, and c) figures of merit of thermo-optic switches over the pastdecade. The asterisk markers represent works related to this thesis.supported by the results in Figure 3.9 c), which show that the figure of merit of thedevice in this work is competitive with the lowest figures of merit of recent results.Given the derivations in Section 2.4 showing that the figure of merit is independentof length, what is the source of the large difference between the figures of merit inthe literature? The efficiency of heating, in terms of the amount of power that goestowards heating the waveguides as compared to the amount of power that heatstheir surroundings, is the main differentiator. For example, for the devices utilizingfolded waveguides in a suspended structure, the volume of the waveguides as com-pared to the total volume of the suspended structure is greater by approximately afactor of the number of waveguides when compared to a suspended structure of thesame size using only one waveguide. This observation can be used to compute anestimated lower bound on the figure of merit for silicon waveguides as describedin Appendix B. This lower bound is plotted in Figure 3.9 c) for comparison. Thefact that all of the observed values are much greater than the lower bound suggests59that there is room for improvement. The primary way to improve the design ofthe structure in this regard would be to reduce the thickness of the glass in thesuspended structure so that less heat is wasted heating the glass. Reduction in thethickness of the glass is, however, limited by the need for the metal heater to besufficiently far away from the waveguides.When measuring the 4×4 switch shown in Figure 3.2, the currents supplied tothe relevant 2×2 switches for each routing state, as detailed in Table 3.2, were var-ied to obtain either the maximum or the minimum transmission. Due to the poorperformance of the 2×2 switches in through operation as described above, the ex-tinction ratios between the on and the off state were as low as 5 dB for routingbetween some ports. This very poor performance lead to not further investigatingthe performance of the 4×4 switch. Due to space constraints during fabrication,only one 4×4 switch could be fabricated, and therefore there were no switches fab-ricated utilizing different types of 3 dB splitters, such as multimode interferometersplitters [75], which may have given superior performance.3.4 Chapter Summary and ConclusionsIn this chapter, the designs of fabricated switches were described, the experimentalprocedure used for testing was outlined, and the experimental results were pre-sented, compared to results predicted through simulation, and discussed.Five 2× 2 switch designs were fabricated in both unetched and underetchedconfigurations to evaluate the impact of waveguide density and thermal isolationon the performance of the devices. All of the switches utilized adiabatic 3 dBsplitters to form an MZI for testing purposes. The 3 dB splitters were found to havea poor splitting ratio, and so the interferometers only performed well in switchinglight intensity at one of their output arms.It was experimentally shown that the increase in waveguide routing densitynear a heating element achievable by using dissimilar waveguides can be an ef-fective way to improve the efficiency of thermal phase shifters for both undetchedand underetched devices. The switching power was found to be greatly reducedboth by underetching to form suspended structures and by increasing the densityof waveguides in the heated region. Although the dramatic reduction in switch-60ing power with underetching came with a correspondingly large increase in switchresponse time, there was no such tradeoff observed by increasing the waveguidedensity. All of the 2×2 switches fabricated exhibited extinction ratios greater than20 dB, and ripple in the through spectrum of less than 0.1 dB, showing that rout-ing with dissimilar waveguides can effectively suppress crosstalk for even highlyagressive waveguide densities.The agreement of the simulated switching powers and times of the devices withthe measured values was better for the unetched devices than for the underetcheddevices, but the simulations correctly predicted the trends in the data and the rela-tive performance of the devices.The highest efficiency phase shifter fabricated, the underetched version of de-vice 5, was found to require 4 times less switching power than the highest efficiencythermo-optic MZI switch reported to date. Further, this phase shifter demonstrateda figure of merit competitive with previously reported results, indicating that thedevice achieves a good tradeoff between switching power and response time.A 4×4 switch was also analysed and fabricated. Due to the poor splitting ratioof the 3 dB splitters used, however, the 4× 4 switch’s performance was poor andwas not characterized in depth.61Chapter 4Conclusions and Suggestions forFuture Work4.1 Summary and ConclusionsTheoretical models and finite element simulations were used to predict the per-formance of thermo-optic switches based on folded waveguide structures utilizingdense dissimilar waveguides. The dependence of the performance, as characterizedby amount of crosstalk, switching power, switching time, and mechanical stabil-ity, on key device parameters was evaluated. The amount of ripple in a switch’sthrough spectrum was proposed as an appropriate measure of switch crosstalk, anda figure of merit for quantifying the tradeoff between switching power and switch-ing speed was proposed. The limitations of the figure of merit for device compar-isons were discussed. It was experimentally shown that the increase in waveguiderouting density near a heating element achievable by using dissimilar waveguidescan be an effective way to improve the efficiency of thermal phase shifters for bothunetched and underetched devices. Since the increase in device efficiency comesat the expense of a slower response, in applications requiring faster operation it isnot desirable to achieve the highest efficiency possible at the expense of speed. Inlarge switch networks, however, the power required for switching scales with thenumber of switches in the network while the switching speed is independent ofthe number of switches since the switches can be switched simultaneously. Thus,62as the size of switch networks increases it becomes increasingly important to de-sign devices to be no faster than required so as to minimize power consumption.The thermal models developed here can be used to guide the design of switches inthis regard. Utilizing highly dense routing of 9 waveguides under a 10 µm wideheater allowed us to fabricate an MZI switch with ultra-low switching power of95 µW, while maintaining an extinction ratio greater than 20 dB and ripple in thethrough response of less than 0.1 dB. The waveguide routing density was found tonot impact the switch response time, and the switch demonstrated a good tradeoffbetween switching power and response time, competitive with the performance ofpreviously reported devices. The device footprint was less than 800 µm × 180µm.4.2 Suggestions for Future WorkTheoretical considerations suggest some techniques that could be used to furtherincrease the efficiency of the phase shifters designed in this work. First, the me-chanical simulations in Section 2.3 suggested that, for devices as big as 300 µmlong, support bridges are not necessary for mechanical stability. Future work couldconsist of verifying this prediction, and determining the maximum device lengthrequired under various conditions, through mechanical stress testing. Provided thatthe 300 µm long suspended structures are stable without support bridges, the ther-mal simulations summarized in Figure 2.16 predict that by removing the supportbridges from device 5, its switching power could be reduced by a factor of 2.7.This alone would give a predicted switching power of 35 µW, with no increase indevice footprint. Of course, this would come at the expense of switching timeseven longer than the switching times of the already slow device 5. Reducing thelength of the heater relative to the length of the suspended region is another methodof decreasing the switching power that could be explored. Using this method, theheater would only be present in a smaller region in the center of the suspendedstructure, routed to by a metal that is thicker to ensure low resistance. By increas-ing the average distance from the heater to the heat sink formed by the substrate,the thermal isolation would be expected to be improved.The performance of the phase shifters in large switches, like the 4× 4 switch63discussed in Chapter 3, should be investigated. Considerations that would be im-portant here include thermal crosstalk between adjacent switches and the effectsof any small backscattering introduced by the folded waveguides. A theoreticalstudy of the limits on the optical crosstalk-induced spectral ripple and backscatter-ing for various switch specifications could be useful for determining the maximumwaveguide density that can be used for a given application.For the switches used in this work, the purpose of the optical modelling in Sec-tion 2.2 was to guide the design of the folded waveguide structure to achieve thehighest waveguide density possible while suppressing crosstalk. 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Fard,“Impact of fabrication non-uniformity on chip-scale silicon photonicintegrated circuits,” in “Optical Fiber Communication Conference,” (OpticalSociety of America, 2014), p. Th2A.37. → pages 50[75] H. Wei, J. Yu, X. Zhang, and Z. Liu, “Compact 3-dB tapered multimodeinterference coupler in silicon-on-insulator,” Opt. Lett. 26, 878–880 (2001).→ pages 6072Appendix APath Integral Approach for aDirectional CouplerConsider a directional coupler composed of two identical, parallel waveguides thatare sufficiently small so as the directional coupler supports only two eigenmodes,|1〉 and |2〉 with propagation constants k1 and k2, respectively, as in Section 2.2.Define the single waveguide modes∣∣A¯〉and |B¯〉 in the same manner as in Section2.2. If the waveguides are weakly coupled then modal expansion gives [3]∣∣A¯〉= (|1〉+ |2〉)/√2 (A.1)and|B¯〉 = (|1〉− |2〉)/√2. (A.2)Suppose that such a directional coupler, of length L, is initially excited in thestate∣∣A¯〉and the field amplitude of mode∣∣A¯〉at the directional coupler output ismeasured. That is, we are interested in the quantity〈A¯∣∣eiPL∣∣A¯〉, (A.3)where here the momentum operator P is defined in the basis-independent wayd |ψ〉dz= iP |ψ〉 , (A.4)73where z is the propagation distance and |ψ〉 is any state. Rewriting the exponentialin the expression A.3〈A¯∣∣eiPL/M...eiPL/M∣∣A¯〉, (A.5)where there are M products of the exponential, can be interpreted as successivelypropagating over the directional coupler broken down into M subsections. Assum-ing that the structure is lossless, the radiation modes can be ignored and the set{∣∣A¯〉, |B¯〉} is a complete, orthonormal set. Thus, the unit operator can be writtenas the following projection operator:1 = ∑n=A¯,B¯|n〉〈n| . (A.6)Introducing the unit operator M+1 times into the expression A.5 gives the follow-ing expression:∑n1,n2,...nM+1=A¯,B¯〈A¯∣∣nM+1〉〈nM+1|eiPL/M |nM〉〈nM| ... |n2〉〈n2|eiPL/M |n1〉〈n1∣∣A¯〉.(A.7)The expression A.7 is the path integral approach for a directional coupler. Foreach set of the dummy indices, {n1,n2, ...nm}, the term in the expression A.7 con-tains the product of M through or cross-coupling coefficients,〈n j+1∣∣eiPL/M∣∣n j〉,giving a path through the directional coupler. The coefficient is a through couplingcoefficient if n j+1 = n j and a cross-coupling coefficient otherwise. Further, thesum is over all sets of dummy indices, so every possible path contributes to thetransmission amplitude. Since the expression A.7 is equivalent to the expressionA.3 for arbitrary M, the limit as M→ ∞ exists and is equal to A.3.One can convert the expression A.7 to the coupled mode formalism to showthat they are equivalent by noting that the set {|1〉 , |2〉} is also an orthonormal setthat can be used to write down the unit operator. Rewriting the expression A.7using this unit operator gives∑n1,n2,...nM+1=1,2〈A¯∣∣nM+1〉〈nM+1|eiPL/M |nM〉〈nM| ... |n2〉〈n2|eiPL/M |n1〉〈n1∣∣A¯〉.(A.8)Using the fact that |1〉 and |2〉 are the eigenmodes of the momentum operator with74eigenvalues k1 and k2, and that they are orthonormal simplifies expression A.8considerably.∑n1=1,2〈A¯∣∣n1〉eikn1L〈n1∣∣A¯〉(A.9)That is, the amplitude of mode∣∣A¯〉at the output can be obtained by decomposingthe input mode into a sum of the two eigenmodes, propagating the eigenmodesindependently, and then projecting onto∣∣A¯〉. Utilizing the mode decompositionequation A.1 gives12(eik1L+ eik2L). (A.10)The normalized transmission is then given as∣∣∣∣12(eik1L+ eik2L)∣∣∣∣2=12(1+ cos [(k1− k2)L]) . (A.11)This is, of course, the usual result [3].75Appendix BA Lower Bound for SwitchFigure of MeritTo compute a lower bound on a silicon switch figure of merit, here we computethe figure of merit for a silicon wire waveguide phase shifter suspended only byits ends in vacuum and uniformly heated with a total power P. Since all of thepower supplied to the phase shifter goes towards heating the silicon, this geometryrepresents a maximum heating efficiency and should result in a lower bound on thefigure of merit.Consider a straight silicon waveguide with cross-sectional area, A, length, L,specific heat, c, density, ρ , and thermal conductivity, K. Furthermore, assume thatthe waveguide is long enough such that the temperature change corresponding toa pi phase shift is small enough to ignore thermal radiation losses. Denoting thepower supplied per unit volume as Q = P/AL, the temperature distribution, T ,satisfies the heat equation with a source [70]∂T∂ t =Kcρ∂ 2T∂x2 +Qcρ . (B.1)Considering the steady state solution, ∂T∂ t = 0, and imposing the condition thatthe temperature at the ends of the waveguide, x =±L/2, is 0, we get the solutionT =Q2K(L2/4− x2), (B.2)76and the average temperature changeT¯ =PL12AK. (B.3)Thus, by equation 2.32, the switching power isPpi =6AKλL2dneff/dT. (B.4)To calculate the characteristic response time of the device we can solve thetime dependent source free heat equation and determine the fall time constant. Theheat equation then reads∂T∂ t =Kcρ∂ 2T∂x2 . (B.5)Assuming that the solution is separable, T (x, t) = X(x)A(t), we getX ′′X=cρKA′A=−γ, (B.6)where γ is a constant of separation. The lowest order non-zero solution satisfyingthe boundary conditions X(±L/2) = 0 has γ = pi2/L2 and soA′A=−Kpi2cρL2 , (B.7)corresponding to a time constant of τc = cρL2/pi2K. Finally, we have an expressionfor the phase shifter figure of meritPpiτc =6Acρλpi2dneff/dT. (B.8)For a silicon waveguide with cross-section 500 nm×220 nm this gives a figure ofmerit of approximately 0.88 nJ at 1550 nm.77

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