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Nonlinear optical response of triple-mode silicon photonic crystal microcavities coupled to single channel… Schelew, Ellen N. 2017

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Nonlinear optical response oftriple-mode silicon photonic crystalmicrocavities coupled to single channelinput and output waveguidesbyEllen N. SchelewB.A.Sc., Queen’s University, 2009M.A.Sc., The University of British Columbia, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2017c© Ellen N. Schelew 2017AbstractOptical and opto-electronic components play important roles in both classical and quantum infor-mation processing technologies. Despite fundamental differences in these technologies, both standto benefit greatly from moving away from bulky, individually packaged components, toward a scal-able platform that supports dense integration of low power consumption devices. Planar photoniccircuits, composed of devices etched in a thin slab of high refractive index material, are consideredan excellent candidate, and have been used to realize many key components, including low-losswaveguides, light sources, detectors, modulators, and spectral filters. In this dissertation, a noveltriple-microcavity structure was designed, externally fabricated, and its linear and nonlinear opticalproperties were thoroughly characterized. The best of the structures exhibited both high four-wavemixing conversion efficiencies and low threshold optical bistability, which are relevant to frequencyconversion and all-optical switching applications.The device consisted of three coupled photonic crystal (PC) microcavities with three nearlyequally spaced resonant frequencies near telecommunication wavelengths (λ ∼ 1.5 µm), with highquality factors (∼ 105, 104 and 103). The microcavity system was coupled to independent input andoutput PC waveguides, and the cavity-waveguide coupling strengths were engineered to maximizethe coupling of the input waveguide to the central mode, and the output waveguide to the twomodes on either side.A novel and sophisticated measurement and analysis protocol was developed to characterize thedevices. This involved measuring and modelling the linear and nonlinear transmission characteris-tics of each of the modes separately with a single tunable laser, as well as the frequency conversionefficiency (via stimulated four-wave mixing) when two tunable lasers pumped two of the modes,and the power generated in the third mode was monitored.Comparisons of the entire set of model and experimental results led to the conclusion that thisiiAbstractstructure can be used to achieve both low-power-threshold optical switching and high efficiencyfour-wave-mixing-based frequency conversion. The advantages of this structure over others inthe literature are its small footprint, multi-mode functionality and independent input and outputchannels. The main disadvantage that requires further refinement, has to do with its sensitivity tofabrication imperfections.iiiLay SummaryThe transmission infrastructure of the Internet consists of a global network of glass optical fibresthat carry light signals, encoded with information, between data centers. In modern data centres,information processing is performed in microelectronic chips, and relatively bulky equipment isused to interface the optical and electronic signals. In the field of “integrated optics”, research anddevelopment is aimed towards realizing compact and efficient interfacing components, integrated inlight-based “photonic” microchips, much like their electronic counterparts, to help meet the grow-ing demands of the ever-expanding Internet. One class of photonic technologies aims to achieveall-optical information processing functionalities by controlling light with light in the microchip. Inthis dissertation, a compact and efficient microchip device of this class is presented that exhibits anumber of potentially useful functionalities. In order to understand and predict its performance,a complex analysis protocol is developed, that uses a novel combination of complimentary experi-mental probes, to characterize the device behaviour.ivPrefaceIdentification and design of the research program was a collaborative effort between myself andmy research supervisor, Dr. Jeff Young. I was the primary contributor to the device design,measurements and analysis. The device fabrication was facilitated by Dr. Lukas Chrostowski, andconducted at the University of Washington Microfabrication/Nanotechnology User Facility, by Dr.Richard Bojko and Shane Patrick.One conference publication has arisen from this work:• E. N. Schelew and J. F. Young, “Stimulated Four-Wave Mixing in a Heterostructure PhotonicCrystal Triple Microcavity,” in Advanced Photonics 2016 (IPR, NOMA, Sensors, Networks,SPPCom, SOF), OSA technical Digest (online) (Optical Society of America, 2016), paperIW2B.5.The work presented in this publication overlaps with the content of this thesis, namely Chapters 3,4 and 5, however it is not directly reproduced in this dissertation. My contributions to this workare the same as those described above.The nonlinear coefficients in Table 5.3 were calculated based on microcavity mode profilessimulated numerically using Lumerical FDTD Solutions. All other calculations associated with theleast-squares analysis in Chapter 5 were completed using MATLAB code that I wrote.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxviiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Photonic integrated circuit basics . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Thesis motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Nonlinear processes in silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.1 Bulk silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.2 Applications of nonlinear processes in silicon . . . . . . . . . . . . . . . . . . 101.3.3 Efficient and compact nonlinear devices . . . . . . . . . . . . . . . . . . . . . 131.3.4 Nonlinear device design and performance . . . . . . . . . . . . . . . . . . . . 16viTable of Contents1.4 Photonic crystal structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.5 Dissertation overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 General overview of linear and nonlinear measurements . . . . . . . . . . . . . . 252.1 Transmission in linear regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1.1 Fabry-Perot cavity transmission . . . . . . . . . . . . . . . . . . . . . . . . . 292.1.2 Microcavity transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.1.3 Technical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.2 Nonlinear transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2.1 Nonlinear transmission lineshape . . . . . . . . . . . . . . . . . . . . . . . . . 382.2.2 Nonlinear transmission data analysis . . . . . . . . . . . . . . . . . . . . . . 402.2.3 Technical consideration: time scales . . . . . . . . . . . . . . . . . . . . . . . 412.3 Stimulated four-wave mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3.1 Technical consideration: four-wave mixing set-up . . . . . . . . . . . . . . . 453 Design and fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.1 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.1.1 Microcavity simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.1.2 Bandstructure simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Device design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2.1 Microcavity design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2.2 Input and output ports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.3 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.3.1 Fabrication layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.3.2 Post-fabrication processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734 Survey results to identify the best devices for full FWM analysis . . . . . . . . . 764.1 Survey of devices across microchips . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.2 Stimulated four-wave mixing spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 804.3 Analysis of survey results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80viiTable of Contents5 Nonlinear characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.1 Measurement and modelling results . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.2 Derivation of model functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2.1 Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.2.2 Nonlinear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.3 Impact of microcavity parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.4 Analysis procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.4.1 Calculations of the sum of the squared differences, X2 . . . . . . . . . . . . 1025.4.2 Least-squares analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.4.3 Best fit parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.4.4 Consistency checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.1 Triple microcavity performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.1.1 Four-wave mixing efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.1.2 Kerr effect input power threshold . . . . . . . . . . . . . . . . . . . . . . . . 1226.2 Best fit parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.2.1 Thermal resistance Rth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.2.2 Saturated free-carrier lifetime τcarrier . . . . . . . . . . . . . . . . . . . . . . . 1266.2.3 Linear absorption quality factor Qabs . . . . . . . . . . . . . . . . . . . . . . 1276.2.4 Waveguide coupling efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.3 Novelty of the nonlinear characterization procedure . . . . . . . . . . . . . . . . . . 1307 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327.1 Suggestions for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A Nonlinear measurement considerations . . . . . . . . . . . . . . . . . . . . . . . . . 151viiiTable of ContentsA.1 Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151A.2 Four-wave mixing excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152A.2.1 Signal: high Q mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152A.2.2 Signal: low Q mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154B Finite-difference time-domain analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 156B.1 The finite-difference time-domain method . . . . . . . . . . . . . . . . . . . . . . . . 156C Grating coupler design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159D Transmission efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162D.1 Linear and nonlinear transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162D.2 Four-wave mixing idler power calibration . . . . . . . . . . . . . . . . . . . . . . . . 164D.3 Integrated component transmission efficiencies . . . . . . . . . . . . . . . . . . . . . 165D.3.1 Calculations of transmission efficiencies gin(λ) and gout(λ) . . . . . . . . . . 166D.3.2 Fabry-Perot model comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 169D.3.3 Channel-to-photonic crystal waveguide transmission . . . . . . . . . . . . . . 171D.4 Filter transmission efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171D.5 Single photon detector efficiency and dead-time . . . . . . . . . . . . . . . . . . . . 172E Nonlinear model function derivations . . . . . . . . . . . . . . . . . . . . . . . . . . 177E.1 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177E.2 Nonlinear lifetimes and frequency shifts . . . . . . . . . . . . . . . . . . . . . . . . . 183E.2.1 Cavity lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183E.2.2 Nonlinear frequency shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186F Nonlinear coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189G X2 minimization plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191ixList of Tables2.1 Summary of the linear and nonlinear microcavity parameters. The subscript m labelsthe microcavity modes, of which there are three. The number of unknown parametersof each type are listed in the “Unknown” column. The nonlinear parameters thatare known for bulk silicon have no entry in this column. The number of unknownparameters that enter the model functions for the linear transmission (LT), nonlineartransmission (NLT), and stimulated four-wave mixing (FWM) measurements areindicated, and check marks indicate which of the known parameters are also included.The NLT model contains parameters for m = 1 and 2 only, as only these two modesare measured. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1 Design parameters for the triple photonic crystal microcavity device coupled to inputand output waveguides. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Summary of the quality factors simulated for triple photonic crystal microcavitydevices with input and output waveguides. The quality factors Qinm, Qoutm , Qothermcorrespond to coupling to the input waveguide, the output waveguide, and otherloss channels including scattering absorption, respectively, while Qm is the totalquality factor, for modes m = 1, 2, 3. The probability that photons generated in thecavity will couple to the output waveguide is given by poutm = Qm/Qoutm . The relativetransmission from the input to the output waveguide is given by T = 4Q2m/(QinmQoutm ).The total quality factors for the microcavity structure without input and outputwaveguides, Qnowgm , are also included. . . . . . . . . . . . . . . . . . . . . . . . . . . 50xList of Tables3.3 Summary of the quality factors simulated for triple photonic crystal microcavitydevices with input and output waveguides, and hole radii rwg,sym = 0.912r for theholes are labelled in yellow in Fig. 3.1 (i.e. swg,sym = 0.912). The quality factors Qinm,Qoutm , Qotherm correspond to coupling to the input waveguide, the output waveguide,and other loss channels including scattering absorption, respectively, while Qm is thetotal quality factor, for modes m = 1, 2, 3. The probability that photons generatedin the cavity will couple to the output waveguide is given by poutm = Qm/Qoutm .The relative transmission from the input to the output waveguide is given by T =4Q2m/(QinmQoutm ). The total quality factors for the microcavity structure without inputand output waveguides, Qnowgm , are also included. . . . . . . . . . . . . . . . . . . . 653.4 Summary of device parameters bracketed across the layout. The set of values isgiven for each bracket parameter. The “Bracket name” describes how each bracketis referred to in the text, usually followed by a number (e.g. Group 1 correspondsto devices with sgc = 0.91), with the exception of “Type” where they are labelledas Type I (swg,sym = 1) and II (swg,sym = 0.912). The “Affected” column describeswhich of the microcavity and reference devices are affected by this parameter. . . . . 723.5 Recipe for post-fabrication undercutting. . . . . . . . . . . . . . . . . . . . . . . . . 744.1 Comparison of experimental results averaged over “good” candidate devices, andsimulation results for the designed structures. λ2 is the resonant wavelength of thecenter mode, ∆λms is the mode separation, and Qm are the total quality factorsfor modes m = 1, 2, 3. Type II microcavities have four holes that are scaled byswg,sym = 0.912. These holes are not scaled in Type I microcavities (swg,sym = 1). . 795.1 Design parameters for the triple microcavity devices studied in this thesis. The factorsh scales the holes radii r, rmid, rmid,ms, rinwg, routwg , and rwg,sym described in Table 3.1and illustrated in Fig. 3.1. The hole shift hms is also presented therein. The factorswg,sym applies an additional scaling factor to rwg,sym. . . . . . . . . . . . . . . . . . 825.2 Summary of the cavity lifetimes and nonlinear frequency shifts. The summationindices m′, l, l′ are over {1, 2, 3}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94xiList of Tables5.3 Summary of nonlinear coefficients derived from perturbation theory. χ(3)Si is thediagonal element of the χ(3) tensor for silicon. . . . . . . . . . . . . . . . . . . . . . 955.4 Summary of linear and nonlinear silicon material constants. Dispersive constant aregiven near λ = 1540 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.5 Best fit parameters found from an analysis involving the minimization of X2tot, andfrom a separate analysis where an iterative approach is taken to find X2T and X2FWM.Rth is the thermal resistance, τcarrier is the effective saturated free-carrier lifetimeand Qabs is the quality factor associated with linear material absorption. . . . . . . 1125.6 Typical parameter space initially tested for nonlinear transmission analysis. Qabs =ωmτabs/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.7 Summary of mean best fit values found from analyses of linear transmission, nonlin-ear transmission and four-wave mixing measurements for Devices 2 to 4. . . . . . . . 1156.1 Four-wave mixing idler power efficiency, ηPFWM = Pi/(P2pPs). For the devices inthis work, the ηPFWM listed are reported for the maximum experimental FWM withM3 as the signal mode (first), then M1 as the signal mode (second). The modequality factors for each structure is also given. When the Q’s of the three modesare approximately equal, a single Q is reported. For the first and third structuresreported, the Q’s are listed in order of signal, pump then idler. For the devices inthis thesis, the Q’s are listed in the order of M1, M2 and M3. Estimates of the deviceareas are also provided. The structures are made in silicon, unless noted by ‡. . . . 1236.2 Estimates of threshold input power, Pth, required to excite a bistable response ofthe microcavity. The threshold power is measured relative to the input 2D photoniccrystal waveguide, unless noted by ‡. The quality factor, Q, effective mode volume,Veff , and thermal resistance, Rth, of the bistable mode are also given. The Pth andQ’s reported for the devices in this work are listed for M1, then M2. . . . . . . . . . 1256.3 Thermal resistance, Rth reported for suspended photonic crystal (PC) structures. Itis noted whether the value was found experimentally or from modelling. . . . . . . 126xiiList of Tables6.4 Effective free-carrier lifetimes, τcarrier reported based on experimental findings forsuspended photonic crystal (PC) structures. . . . . . . . . . . . . . . . . . . . . . . . 127D.1 Summary of the transmission efficiencies for components in the transmission andfour-wave mixing (FWM) set-ups illustrated in Fig. 2.5 and Fig. 2.11 . . . . . . . . 163E.1 Polarization terms for four-wave mixing. For each mode, m, K(−ωm;ω1m, ω2m, ω3m)and electric field terms are listed for each distinct set for ωm. . . . . . . . . . . . . . 181F.1 Nonlinear coefficients calculated from FDTD simulations. . . . . . . . . . . . . . . . 189xiiiList of Figures1.1 Schematic example of information sending and receiving. Three electronic signals,produced in one microelectronic chip, are transmitted to a remote microelectronicchip via a single optical fibre. The electronic signals (black dashed lines) are appliedto modulators that encode the information in light arriving from continuous wave(CW) lasers via optical fibres (solid coloured lines). Each laser has a different opticalfrequency, and the three optical signals leaving the modulators are directed intoa single optical fibre using a spectral multiplexer. The optical fibre carries themultiplexed signal to a remote destination, where it is separated into the threeoptical frequencies using a demultiplexer. The individual optical signals then arriveat photodetectors, that translate them into electronic signals, which are then directedto a second microelectronic chip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Schematic of PIC components. (a) An example waveguide that is composed of ahigh index channel that confines light by total internal reflection. The waveguidesits on a low index substrate, and is otherwise surrounded by air. An example ofthe cross-sectional mode profile for a PIC waveguide is shown on the right, wherethe white lines contour the channel and substrate. (b) A beam-splitter (directionalcoupler), that transfers light between two waveguides by evanescent coupling. (c) Acompact spectral filter formed by a ring resonator coupled to two waveguides. Inthis example, the input excitation at λ1, is resonant with the ring and is transmittedthrough to the top waveguide, while the other wavelengths are non-resonant and aredirectly transmitted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3xivList of Figures1.3 Schematic of a photonic integrated circuit, where optical components are defined in ahigh refractive index material, that sits on a slab of lower index index material (gray).Passive components including grating couplers, beam-splitters and spectral filters areshown, in addition to active components including phase-shifters and detectors. Thedotted arrows show that light is routed elsewhere in the photonic circuit. . . . . . . 41.4 (a) Schematic of spontaneous four-wave mixing. Two “pump” photons with fre-quency ωp are converted to a pair of “signal” and “idler” photons with frequenciesωs and ωi, respectively. Energy conservation requires that 2ωp = ωs +ωi. (b) Sponta-neous photon pair generation is implemented a heralded single photon source, whenpumped with light at ωp. The detection of a signal photon, by a single photondetector, heralds the presence of the idler photon. . . . . . . . . . . . . . . . . . . . 71.5 Schematic of stimulated four-wave mixing (FWM) signal processing applications.The signal, pump and idler frequencies are ωs, ωp and ωi, where 2ωp = ωs + ωi.(a) The signal-to-noise ratio (SNR) in a modulated amplitude signal (high and lowamplitudes represent 1’s and 0’s, respectively) is reduced through the FWM processby translating the data signal from the pump light to idler light. The SNR is im-proved in the idler amplitude owing to the quadratic dependence of the idler poweron the pump power. The schematic shows that the amplitude-modulated pumplight undergoes frequency mixing with the unmodulated signal (ωs) light, to producemodulated light at the idler frequency ωi. Here the solid lines show the modulatedsignals, that sit above the no-power baselines (dashed-lines). (b) The timing jitter ina return-to-zero data stream (each bit is separated by a low amplitude time interval)is reduced by modulating the signal on the same clock, such that idler power canonly be generated when the signal excitation is on. . . . . . . . . . . . . . . . . . . 11xvList of Figures1.6 Proposed photonic crystal structure that supports multiple nonlinear functionalities.(a) Scanning electron microcavity image of a fabricated structure. The microcavityand waveguide regions are indicated. (b)-(d) Schematics of the microcavity (cir-cle) coupled to input and output waveguides (rectangles) under different excitationschemes. The red, green and blue arrows show the passage of light at three differentfrequencies. (b) Spontaneous four-wave mixing. Pump light at ωp (green) entersthe input waveguide and resonantly excites a mode of the microcavity. Photon pairsare generated at signal and idler frequencies ωs (blue) and ωi (red), respectively,through spontaneous four-wave mixing. Energy conservation requires 2ωp = ωs +ωi,and the conversion process is only efficient where ωs and ωi pairs coincide with mi-crocavity resonant modes. Photons leave the microcavity through coupling to theoutput waveguide, the input waveguide, and radiation, as indicated by the arrows.The microcavity photons are also absorbed by the silicon. (c) Stimulated four-wavemixing. Pump and signal excitations at ωp (green) and ωs (blue), respectively, res-onantly excite two modes of the microcavity and photons are generated at the idlerfrequency ωi (red). (c) Kerr effect. One mode of the microcavity is resonant probedwith a single laser and the induced nonlinear changes to the microcavity refractiveindex affect transmission spectrum lineshape through resonant frequency shifts anddecreased peak transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17xviList of Figures1.7 Planar photonic crystal defined in a silicon-on-insulator (SOI) wafer. (a) A hexagonalperiodic lattice of holes is etched in the top silicon slab of the SOI, called the devicesilicon, which sits on an oxide layer and a silicon base. (b) Light is contained inthe device silicon with respect to the zˆ direction by total internal reflection (TIR)occuring at the air and SiO2 oxide interfaces. The planar photonic crystal supportstransverse-electric (TE) and transverse-magnetic (TM) modes. (c) Bandstructure ofa PC, plotted as a function of the three reciprocal lattice vectors, in the first Brillouinzone. The TE and TM modes are shown as red and blue lines, respectively. TheTE bandgap is indicated (shaded green), and the light cone is also indicated (shadedyellow). Modes that fall below of the light cone (solid lines) are bound to the devicesilicon through TIR, while those falling within the light cone are partially unbound(dashed lines). This figure is reproduced and modified, with permission, from Ref.[24]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.8 Photonic crystal (PC) defect structures defined in silicon-on-insulator. (a) Photoniccrystal waveguide introduced to the PC by omitting a row of holes. The Ey fieldprofiles in the center slab of the silicon for two different transverse electric waveguidemodes are plotted, and the PC holes are outlined in white. (b) Bandstructure dia-gram for the line defect, plotted as a function of the wavevector along the waveguideaxis, kx, in units of 2pi/a, where a is the lattice spacing of the PC. The continuum ofbulk PC modes are shown in gray. Two waveguide modes that exist in the bandgapare plotted as the solid blue and dashed red lines. The light cone is shown in paleyellow. The field profiles of the two modes at kx = pi/a edge, indicated by solidand dotted arrows, are shown in (a), as the left and right plots, respectively. (c)Photonic crystal microcavity introduced to the PC by omitting three holes in a row,called an “L3” microcavity. The field intensity of the fundamental mode supportedby the defect is plotted, and the holes are outlined in white. (d) Examples of cou-pling geometries between PC waveguides (WG) marked with red arrows and an L3microcavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20xviiList of Figures2.1 Triple photonic crystal (PC) microcavity coupled to input and ouput waveguidesstudied in this thesis. (a) Schematic diagram of the cavity modes coupling to var-ious channels. The cavity is represented by the circle, while the waveguides arerepresented by rectangles. The coupling lifetimes for mode m are labelled, where τ inmand τoutm are the coupling lifetimes to the input and output waveguides, respectively,while τ scattm is the scattering lifetime and τabsm is the linear absorption lifetime. (b)-(d)Electric field intensity plots of modes M1, M2 and M3 (in that order). The PC holesare shown with white contours. The input and output waveguides are labelled is“IN” and “OUT”. (e) Example of an experimental transmission spectrum measuredin the linear regime for a triple microcavity device. . . . . . . . . . . . . . . . . . . 262.2 Fabry-Perot cavity. (a) Schematic of a Fabry-Perot cavity with two mirrors sepa-rated by a distance L, containing a medium with refractive index n. The mirrorshave field transmission and reflection coefficients ri and ti. (b) Example relativetransmission spectrum of a lossless Fabry-Perot cavity with identical mirrors. Theresonant frequencies, ωm are labelled, along with the linewidth δωm and maximumtransmission Tlinm for mode m. The free spectral range (FSR) is also labelled. (c)Relative transmission spectra for lossless cavity with identical mirrors (solid line),a cavity with loss and identical mirrors (dashed line), and a losses cavity with non-identical mirrors (dash-dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Simplified schematic of transmission measurement of photonic crystal (PC) triplemicrocavity device. Light from a tunable laser is focused on the input grating couplerand diffracted into a parabolic waveguide that narrows to a single-mode channelwaveguide. This waveguide is interfaced with the input PC waveguide, which couplesthe microcavity. Light is extracted from the microcavity through the ouput PCwaveguide, which is coupled to channel and parabolic waveguides terminated by theoutput grating coupler. The extract light is measured by a photodetector. . . . . . 32xviiiList of Figures2.4 Single photonic crystal linear three hole defect (L3) microcavity with input and out-put waveguides. (a) Schematic of the microcavity, where black circles indicate holesetched in the silicon. The red arrows show the passage of light through the de-vice for transmission measurements. (b) Transmission spectrum for the fundamentalmode of an L3 cavity simulated with a finite-difference time-domain simulation. Theresonant wavelength, λ1, linewidth, δλ1 and maximum transmission Tlin1 are labelled. 332.5 Transmission set-up. Light from a tunable laser is coupled into an optical fibre, andthe out-coupled light passes through a polarizer (yˆ-polarized) then is focused bytwo aspherical lenses onto the input grating coupler of a device on the sample (seeFig. 2.3), at an angle θ. Light leaving the output grating coupler is collected by anelliptical mirror (also at angle θ) that directs light through the output polarizer alsoyˆ-polarized) and focuses it in a plane of an iris, and only the output grating lightis passed onto a photodetector. Alternatively, the iris is left open and the sample isimaged on the CCD Camera. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.6 Nonlinear transmission spectra measured for the center mode of a photonic crystal(PC) triple cavity, where the transmission is relative to the input/output PC waveg-uides. (a) The absolute nonlinear transmission spectra measured with a forwardsweep (wavelength swept from low to high) are plotted for input powers rangingfrom 16 to 250 µW . The arrow indicates the minimum threshold power, Pth, wherethe sharp drop in the transmission spectrum begins to appear. (b) Same as in (a) butfor the relative transmission. (c) Same as in (b) but for a backward sweep (wave-length swept from high to low). (d) Example forward (blue) and backward (red)spectra taken at two 16 and 250 µW . The bistability is present in the higher powerspectra, as the forward and backward sweep have a range of wavelengths where thespectra are non-overlapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37xixList of Figures2.7 Nonlinear behaviour a triple microcavity under (a)-(b) single frequency excitation,and (c)-(d) dual frequency excitation when cross-coupling of nonlinear effects be-tween modes is present. (a) Energy loaded in mode M2 as a function of sweepwavelength for an input power of Pin = 300 µW. (b) Resonant wavelength of thecavity mode as a function of sweep wavelength (solid line). The dashed line showsthe sweep wavelength, for reference. (c) Nonlinear resonant wavelength shift of modeM1, ∆λNL1 , as a function of the energies U1 and U2 loaded in modes M1 and M2,respectively, for the same microcavity device in (a) and (b). The color scale is inunits of nanometer. (d) Same as (c) but the resonant wavelength shift of M2, ∆λNL2 ,is plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.8 Example experimental data extracted from nonlinear transmission spectra taken as afunction of input power for Modes M1 and M2. (a) Peak transmission. (b) Resonantwavelength shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.9 (a) Simplified schematic of the stimulated four-wave mixing (FWM) set-up. Pumpand signal lasers tuned to wavelengths λp and λs are coupled into the input waveguideof the multimode microcavity and the light collected is sent through a spectral filter,to extract the idler photons at wavelength λi, before measurement by a single photondetector. The green and blue arrows represent path the pump and signal continuouswave excitations, respectively, and the red arrows represent the path of the idlerphotons generated through FWM. (b) Transmission spectrum for a triple photoniccrystal microcavity. The three frequencies involved in FWM are indicated withdashed lines, and the three microcavity modes are labelled M1, M2 and M3. . . . . . 43xxList of Figures2.10 Examples of stimulated four-wave mixing (FWM) idler powers experimentally mea-sured for triple photonic crystal microcavity devices. The red and blue markersindicate that the FWM configuration implemented resulted in idler photons gener-ated near Mode 3 and Mode 1, respectively. Circles and triangles indicate that thepower measured as a function of signal and pump power, respectively. The solid linesare estimates of what the idler powers are predicted to be in the absence of nonlinearlosses (linear/quadratic for signal/pump power dependencies). (a) Stimulated FWMresults for a triple cavity probed largely in the low power limit. The signal powersweep has a fixed pump power of 29 µW , and the pump power sweep has a fixedsignal power = 3.6 µW . Nonlinear loss effects result in sub-linear and sub-quadraticpower dependencies near the ends of the sweeps. (b) A different triple cavity deviceprobed largely in the high power limit. The signal power sweep has a fixed pumppower of 44 µW , and the pump power sweep has a fixed signal power = 44 µW . . 442.11 Optical set-up used to measure four-wave mixing. Light from the pump and signallasers are coupled into a single mode fibre and focused on the input grating couplerof a triple microcavity device. Light that leaves the sample surface is collected andfocused by an elliptical mirror, and an iris in the image plane is used to transmitonly the light leaving the output grating. A lens is then used to focus the light intoan optical fibre, that is connected to two JDSU TB9 Tunable Grating Filter, tunedto the idler wavelength, such that idler photons are detected by the id210 avalanchephotodetector. See Fig. 2.3 for more details on unlabelled components. . . . . . . . 45xxiList of Figures3.1 Triple photonic crystal (PC) microcavity with input and output waveguides. (a)Schematic of the microcavity with design parameters highlighted. The heterostruc-ture is composed of photonic crystal regions labelled I, II, and III, and separated byred lines, where the lattice spacings in the x direction are a1, a2 and a3, respectively,while the row spacing of the bulk PCs is√3a1/2 throughout. The widths of the threeline defects are wwg. The solid black circles represent holes with radius r, while thecircles coloured in gray have radius rmid. The input and output waveguides are la-belled, and the holes lining the waveguides are coloured cyan and green, and haveradii rinwg and routwg , respectively. The 12 holes that finely control the mode spacingare coloured purple, and have radius rmid,ms and are shifted outward from the linedefect by a distance hms. The four yellow holes have radius rwg,sym. (b) Scanningelectron microscope image of a fabricated triple microcavity device. . . . . . . . . . 483.2 Finite-difference time-domain (FDTD) microcavity simulations using Lumerical FDTD[79]. (a) The simulation layout of the triple microcavity device is shown in the x− y(top) and x − z (bottom) planes. The thick orange regions outline the simulationvolume and the perfectly matched layers (PML). The blue region in the bottom plotsshow the symmetric boundary condition applied in zˆ. The thin orange lines in the topimage show the boundaries of the heterostructure mesh override regions. A sourcepolarized in yˆ is placed at an antinode in the top defect region to excite the modes.A time monitor (yellow cross) is placed at the location of an antinode in the bottomdefect. Two-dimensional planar frequency monitors are placed at the outputs of thetwo waveguides (WGs) to measure waveguide transmission (yellow lines in the topimage) and a box of 2D monitors encloses the full simulation power to measure allpower leaving the volume. (b) Fourier transform of the Ey field measured by thetime monitor (blue), apodized to remove the source excitation. A Gaussian spectralfilter around the center mode is shown as a dashed red line. (c) Natural logarithmof Ey decay envelope for center mode, found from the inverse Fourier transmissionof the filter spectrum. The dashed red lines is the line of best fit. . . . . . . . . . . . 51xxiiList of Figures3.3 Bandstructure simulations using Lumerical FDTD Solutions [79]. (a) The simulationlayout for photonic crystal line defect is shown in the x−y (top) and x− z (bottom)planes. The x− y plane near the center axis is expanded in the image to the right.The simulation volume spans one lattice spacing in xˆ, ax and is outlined in orange.Symmetric boundary conditions are applied in zˆ, as indicated by the coloured blueregion in the bottom image, and Bloch boundary conditions are applied in xˆ. Dipolesources (blue double arrows) are randomly placed to excite the Bloch mode. Timemonitors (yellow crosses) are also randomly placed to measure the field excited inthe silicon. (b) Example of the bandstructure measured for a line defect. The base10 logarithm of the intensity spectrum extracted from simulations with fixed kx areplotted. The light line is shown with the white line. . . . . . . . . . . . . . . . . . . 543.4 Single photonic crystal (PC) heterostructure microcavity. (a) Schematic of the struc-ture, where the different PC regions are highlighted in different shades of gray andlabelled I, II and III. The horizontal lattice spacings are exaggerated for clarity. (b)Electric field intensity of the mode profile. The PC heterostructure boundaries arehighlighted with yellow lines and the PC holes are outlined in white. The horizontallattice spacings are a1 = 410 nm, a2 = 418 nm and a3 = 425 nm, respectively. (c)Bandstructure calculation for the PC line defects (missing row of holes) in RegionsI, II and III. The wavevector on the x axis is given in terms of the horizontal latticespacing a for each region considered (i.e. a = a1, a2, a3). The solid lines indicate thePC waveguide bands. The continuum of bulk PC bands are colored in gray, and areoutlined for each region with dotted lines. The black dashed line shows the resonantcavity mode frequency, and the light cone is shaded in yellow. . . . . . . . . . . . . . 553.5 Simulation results for a triple heterostructure photonic crystal (PC) microcavity withhole sizes r = 124 nm throughout and waveguide width wwg = 693 nm. Electricfield intensity mode profiles are plotted for modes (a) M1, (b) M2 and (c) M3. Theheterostructure boundaries are shown with yellow lines and the PC holes are outlinedin white. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57xxiiiList of Figures3.6 Process followed to design the triple photonic crystal (PC) microcavity for spon-taneous four-wave mixing applications. The microcavity parameters are defined inTable 3.1 and highlighted in Fig. 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 583.7 Triple photonic crystal (PC) microcavity designs as a function of waveguide widthwwg. (a) Bandstructure calculation for the line defect of Region I (outermost PC)of the heterostructure. The bulk PC mode continuum is shown in gray, and thePC waveguide (WG) modes are shown as thick dash-dotted, solid, and dashed bluelines for structures with wwg = 675, 693 and 710 nm, respectively. The resonantfrequencies for each of these structures are shown as thin lines blue, green and redlines for modes M1, M2 and M3 respectively. The highlighted yellow shows theregion above the air light-line. (b) Total quality factors for M1, M2 and M3. . . . . 593.8 Resonant frequencies of the triple microcavity structures as a function of mode spac-ing tuning parameters described in Fig. 3.1. (a) The resonant frequencies found fromFDTD simulations for M1 (blue circles), M2 (green triangles) and M3 (red squares)are plotted as a function of rmid. (b) Frequency offset, ∆foff = f1 + f2 − 2f2, as afunction of rmid (yellow diamonds) calculated with FDTD simulations, and rmid,ms(empty diamonds) with fixed rmid = 126 nm calculated with perturbation theory,where fm are the mode resonant frequencies. (c) Mode frequencies found from FDTDsimulations plotted as a function of the shift hms [markers same as in (a)]. The largeopen markers show the mode frequencies calculated using perturbation theory. (d)Frequency offsets plotted for the resonant frequencies presented in (c), found fromFDTD simulations (filled yellow diamonds) and perturbation theory (empty dia-monds). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60xxivList of Figures3.9 Input and output photonic crystal waveguides (PC WG) and the coupling to single-mode channel waveguides. (a) Bandstructure plots for the input and output PC WGmodes. The bulk PC mode continuum is shown in gray for the Region I PC with r =124nm. The mode frequencies are plotted for modes M1, M2 and M3 respectively.The region above the air light-line is shaded in pale yellow. (b) Finite-difference time-domain (FDTD) simulation layout [79] of the impedance matching region betweenthe PC and single-mode waveguides. The simulation volume is outlined in orange.A mode source is used to launch the TE channel waveguide mode , and a planar two-dimensional monitor is used to measure the transmission into the PC WG. The regioncontained in the blue box is expanded to the right, where the design parameters ofthe impedance matching region are labelled. (c) Transmission spectra measuredby the simulation layout in (b), for the input and output waveguides. The dashedand solid vertical lines show the bottom of the input and output waveguide bands,respectively. The dashed-dotted lines show the resonant frequencies of the triplemicrocavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.10 Triple photonic crystal (PC) microcavity waveguide coupling geometries. The het-erostructure PC region boundaries are shown with red lines and are labelled I, II andIII. Input waveguides are introduced in (a) and (b), and the holes lining the waveg-uide are highlighed in cyan. The coupling geometries are (a) (Xwgin , Ywgin ) = (3, 7)and (b) (Xwgin , Ywgin ) = (4, 7). Output waveguide are introduced in (c) and (d), andthe holes lining the waveguide are highlighed in green. The coupling geometries are(c) Xwgout = 7 and (d) Xwgout = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.11 Input and output quality factors for waveguide coupling. (a) Input waveguide qualityfactors, Qinm are shown for modes M1 (blue circles), M2 (green triangles) and M3(red squares) simulated for coupling geometries (Xwgin , Ywgin ) = (3, 7) (left) and (4, 7)(right). (b) Output waveguide quality factors, Qoutm are shown simulated for couplinggeometries Xwgout = 7 (left) and Xwgout = 8 (right). . . . . . . . . . . . . . . . . . . . . . 64xxvList of Figures3.12 Input/output port for the microcavity device. The single-mode channel waveguideis expanded from 0.5 µm to 20 µm by a parabolic waveguide, which is terminatedby a grating coupler. The black regions show where silicon is removed to define thestructure. The grating coupler is defined by an array of slots with apodized widths,designed for coupling light between the waveguide and light propagating backwardsin free-space at an angle θ = 45◦. Input and output coupling are shown in the topdiagram with pairs of solid blue and dashed red the arrows, respectively. . . . . . . 663.13 Apodized grating coupler design and simulation results. (a) Schematic of the apodizedgrating coupler, where black regions show where the silicon is removed. (b) Scanningelectron microscope image of an apodized grating coupler. (c) Simulated input andoutput transmission efficiencies between free-space and the single mode waveguide(via the grating coupler and parabolic waveguide). (d) Far field electrical field in-tensity projection on a 15 cm hemisphere of the field monitored 1 µm off the gratingsurface when the single-mode waveguide is excited at λ = 1530 nm. The collectionarea of the elliptical mirror, that is situated on the hemisphere at an angle of −45◦from the vertical, is outlined in black. (e) Same as in (d) but for light launched inthe single mode waveguide at λ = 1565 nm. (f) Simulated input and output, andtotal transmission efficiencies. The output transmission efficiency shown in this plotaccounts for the collection efficiency of the elliptical mirror [unlike the plot in (c)]. . 683.14 Fabrication layout of the devices studied in this thesis. There are 168 microcavitydevices, and 48 reference devices. Markings on the chip are used to identify the Groupnumber and Set number of the devices adjacent. Devices in the same group haveidentical grating couplers, while devices in the same set have identical hole size scalefactors. Alignment marks are also included for the post-fabrication photolithographyprocess. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70xxviList of Figures3.15 Reference devices included in the fabrication layout that contain input and outputgrating couplers and parabolic waveguides. In (a) and (b) PCWGin and PCWGoutdevices are shown that contain photonic crystal (PC) waveguides with structuralparameters identical to those of the input and output PC waveguides of the triplemicrocavity device, respectively. In (c) and (d), the PC waveguides are omitted andthe channel waveguides are directly connected to form G2Gout and G2Gout referencesdevices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.16 Optical microscope images of the silicon microchip at various stages in undercuttingprocess. (a) Image taken post-photolithography, such that the developed photoresistis intact. Light pink lines where the bare silicon is exposed. (b) Same as in (a), butwith higher magnification. (c) Image taken after the hydrofluoric acid etch and thephotoresist removal. The pink hallow around the photonic crystal region outlineswhere the oxide is removed beneath the silicon. . . . . . . . . . . . . . . . . . . . . 754.1 Transmission measurements for G2Gin devices in Groups 1 to 4 of Chip A (bottom totop), (a) taken with a −41◦ coupling angle, (b) taken with coupling angles between−42◦ and −37◦ (labelled on plot) to achieve peaks near λ = 1545 nm. . . . . . . . . 774.2 (a) Transmission measurements of Type I microcavity devices on Chip A, with modespacing parameter sms = 0 nm, for the three hole size brackets labelled on the plot.(b) Transmission measurements of Type II microcavity devices on Chip A, withsh = 0.97 and mode spacing parameters labelled on the plot. . . . . . . . . . . . . . 77xxviiList of Figures4.3 Microcavity mode spacing offsets, ∆λoff = λ1 + λ3 − 2λ2 found from transmissionmeasurements, plotted as a function of device number in the sh = 0.97 set, where λmare the resonant wavelength, for Type I and Type II microcavities. (a)-(d) Resultsfor Groups 1 to 4 (top to bottom) devices on Chip A. (e)-(h) Results for Groups 1to 4 (top to bottom) devices on Chip B. In Groups 1 to 3, each device has a differentmode spacing parameter, with sms = 8, 4, 2, 0,−2,−4 nm for Devices 1 to 7, wheresms is the position shift of a select group of holes. In Group 4, the mode spacingis modified by scaling a select group of holes by sms = 0.968 to 1.02, linearly overthe seven devices. Data is omitted in cases where three modes are not observedin the transmission spectrum. The dashed lines show the simulated results (fromperturbation theory) for Type I and II microcavities. The gray region has a widthof 0.5 nm, which is a typical linewidth for the M3 mode (the lowest Q mode). . . . 784.4 (a) Linear transmission for a microcavity structure studied with four-wave mixing(FWM). Resonant wavelengths λ1, λ2, and equally spaced λFWM3 = 2λ2 − λ1, areplotted as the dashed blue, green and red lines, respectively. (b)-(d), Four-wavemixing results, measured using pump and signal powers 38.5 µW and 22.4 µW re-spectively. The idler power is reported relative to the output PC waveguide (withoutaccounting for the output spectral filtering). (b) Stimulated FWM idler power asa function of the filter center wavelength for λsignal = 1543.01 nm (circles). Thedashed line shows the wavelength λFWM3 . (c) Same as (b) but with λsignal = 1547.58nm. The shaded regions in (b)-(c) are the scaled filter transmission spectra, andthe triangles show the background power when the signal laser is turned off and thepump laser remains on. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.1 Linear transmission spectra for (a) Device 1, (b) Device 2, (c) Device 3 and (d)Device 4. The transmission is calculated using the best fit Fabry-Perot parameters,φin and φout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83xxviiiList of Figures5.2 Linear transmission results for the four devices characterized: (a) resonant wave-lengths, λm, (b) total quality factors, Qlinm , and (c) peak transmission, Tlinm . Theresults are plotted for modes M1, M2 and M3. The T linm are calculated using the bestfit Fabry-Perot parameters, φin and φout. . . . . . . . . . . . . . . . . . . . . . . . . 835.3 Nonlinear transmission and four-wave mixing analysis results for Devices 1 to 4(top to bottom rows). Columns 1 and 2 contain the peak transmission and reso-nant wavelength shifts, respectively, for Mode 1 (blue) and Mode 2 (green), foundexperimentally (filled circles), and using the model with best fit parameters (openmarkers). Column 3 shows the experimental FWM idler power in Mode 1 (blue) andMode 3 (red), as a function of pump power (triangles) for fixed signal power at Ps(labelled on the plots) and as a function signal power (circles) for fixed pump powerat Pp (also labelled). The fixed pump power The predicted idler powers are shownwith thick black lines. The dashed lines show the predicted power when nonlinearabsorption is ignored. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.4 Nonlinear transmission results for Devices 1 to 4 (top to bottom rows), for Modes 1and 2 (left and right columns). Bottom plots show the experimental data and topplots show the spectra predicted using the model with best fit parameters, whereboth are plotted on absolute transmission scales, shifted relative to each other. Theinput powers for each spectrum corresponds power for each marker in the nonlineartransmission data plots of Fig. 5.3 (first two columns). . . . . . . . . . . . . . . . . . 86xxixList of Figures5.5 Nonlinear transmission and four-wave mixing analysis results for Devices 1 to 4 (topto bottom rows) when Rth, Qabs and τcarrier are held fixed at their average values.Columns 1 and 2 contain the peak transmission and resonant wavelength shifts,respectively, for Mode 1 (blue) and Mode 2 (green), found experimentally (filledcircles), and using the model with best fit parameters (open markers). Column 3shows the experimental FWM idler power in Mode 1 (blue) and Mode 3 (red), as afunction of pump power (triangles) when the signal power is fixed at Ps (labelled) andsignal power (circles) when the pump power is fixed at Pp (labelled). The predictedidler power are shown with thick black lines. The dashed lines show the predictedpower when nonlinear absorption is ignored. . . . . . . . . . . . . . . . . . . . . . . . 875.6 Schematic of coupled-mode theory for cavity supporting three modes. Cavity modem has field amplitude am and energy |am|2. The mode is coupled with lifetime τ im tochannel i. The optical power of light propagating toward the microcavity is |sinm+|2,and the optical powers exiting through the input and output waveguides are |sinm−|2and |soutm−|2, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.7 Nonlinear effects in a typical triple microcavity, as a function of energy loaded inthe microcavity mode M2, under single frequency excitation. (a) The inverse of thenonlinear quality factors due to two-photon absorption (TPA),QTPA, and free-carrierabsorption (FCA), QFCA, along with the linear and total quality factors, Qlin andQtot (dashed), respectively. The quality factors are related to the lifetimes through Q= ωτ/2. (b) Nonlinear wavelength shifts are plotted for contributions including theKerr effect, ∆λKerr, free-carrier dispersion (FCD), ∆λFCD, and thermal effects due topower absorption through the linear material absorption ∆λth,abs, TPA, ∆λth,TPA,and FCA, ∆λth,FCA. The total shift is also plotted ∆λth,tot . The shifts ∆λth,abs and∆λth,abs have similar values for the range of energies studied, and ∆λth,abs is dottedfor clarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91xxxList of Figures5.8 Numerical solutions for the nonlinear transmission predicted for triple microcavityDevice 3, Mode 2. (a) Solutions for input powers P in2 = 100 and 300 µW. Circlesshow where a single solution is found, while the top, middle and bottom branchesin the bistable regime are marked by +, 2 and ×, respectively. (b) Solutions forP in2 = 300 µW are shown as in (a), and the transmission spectra predicted for forward(solid black) and backward (dashed red) wavelength sweeps are also shown. . . . . . 975.9 Nonlinear transmission spectral features calculated for an excitation power of 300µW, using best fit parameter values for M2 of Device 3 (given in Table 5.7 andFig. 5.17), while one parameter is free to vary on the x-axis. (a)-(b) Peak relativetransmission, TNL. (c)-(f) Peak wavelength shift, ∆λNL. ηwg2 = τout2 /τin2 and Qabs =ωτabs/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.10 Four-wave mixing idler power calculated for pump and signal excitations of 84 µWand 43 µW, respectively, using best fit parameter values for Device 3 (given in Table5.7 and Fig. 5.17), while one parameter is free to vary on the x-axis. Two excitationconfigurations are included: M1/M3 are the signal/idler modes (red), and M3/M1are the signal/idler modes (blue). ηwgm = τoutm /τinm . . . . . . . . . . . . . . . . . . . . . 1015.11 Four-wave mixing idler power calculation flowchart. The solid arrows indicate thestep sequence and the dashed arrow indicates an iterative loop that continues untilconvergence is met. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.12 Nonlinear transmission analysis example results for Device 2. The base 10 logarithmX2T is plotted across the parameter set {ηwg1 , ηwg2 , Qabs, Rth, τcarrier, φin, φout}. Foreach parameter pair on the x and y axes, the minimum log 10(X2T) across all otherparameters is plotted. Blank results show where no viable parameter sets exist dueto a lack of energy conservation. Each plot has the same color scale (shown to theright of each row). There is a total of 13 degrees of freedom in the X2T calculation. 108xxxiList of Figures5.13 Four-wave mixing example results for Device 2. The X2FWM is plotted across theparameter set {ηwg1 , ηwg2 , ηwg3 }, while Qabs, τcarrier, φin and φout are held fixed at thevalues that minimize X2T. For each parameter pair on the x and y axes, the minimumX2FWM across the other parameter is plotted. Blank results show where no viableparameter sets exist due to a lack of energy conservation. Each plot has the samecolor scale (shown to the right). There are 55 degrees of freedom in the X2FWMcalculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.14 Nonlinear transmission results for Device 2. Plots (a)-(c) show X2tot, X2FWM , andX2T. The parameter on each x axis is fixed and the minimum X2 over all otherfit parameters involved in the respective analysis is shown. X2T is calculated as afunction of ηwg1 , ηwg2 , Rth, τcarrier, Qabs, φin and φout. X2FWM is calculated as a functionof ηwg1 , ηwg2 , ηwg3 , while τcarrier, Qabs, φin and φout are held fixed at the values thatminimize X2T. X2tot is the sum over X2T and X2FWM. Plots (d)-(e) show X2tot and theX2 minimization that is used to determine the fit parameter on the x axis using theiterative fitting process, where X2T in (d) is calculated for fixed ηwg2 and X2FWM in(e) and (f) is calculated with fixed ηwg1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.15 Nonlinear transmission analysis example results for Device 2. The X2T is plottedacross the parameter set {ηwg1 , Qabs, Rth, τcarrier, φin, φout}, when ηwg2 = 3 is heldfixed at the best fit values from the four-wave mixing analysis. For each parameterpair on the x and y axes, the minimum X2T across all other parameters is plotted.Each plot has the same color scale (shown to the right), where X2T > 100 at set themaximum of the color scale range. White lines contour X2T = min(X2T) + 1. Thereis a total of 12 degrees of freedom in the X2T calculation. . . . . . . . . . . . . . . . . 111xxxiiList of Figures5.16 Nonlinear characterization results. (a)-(h) Best fit parameters (markers), with ±1σuncertainties (rectangles). Parameters that minimize the X2 are shown with blackmarkers, and those that do not are red. (i)-(j) The minimum X2= X2/(N−k) fromthe nonlinear transmission and the four-wave mixing analyses are shown, where Nis the number of data points in the fit and k is the number of fit parameters. Fitswhere X2 does not reach a minimum with respect to all parameters are marked asred. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.17 Nonlinear characterization results, for both the original individual devices analyses(circles), for when Rth, Qabs and τcarrier are fixed to the average values in the modelfunctions (squares). (a)-(c) Best fit parameters (markers), with ±1σ uncertainties(gray rectangles for original fit parameters, and yellow rectangles for the new fitparameters). Parameters that minimize the X2tot are shown with black markers, andthose that do not are red. (d) The minimum X2tot = X2tot/(Ntot − ktot) is shown,where Ntot is the number of data points in the fit and ktot is the number of fitparameters. Fits where X2tot is not fully minimized are marked as red. . . . . . . . . 1155.18 Nonlinear transmission results for Device 2 when Rth, Qabs and τcarrier are held fixedat the mean values. The parameter on each x axis is fixed and the minimum X2tot,X2T, and X2FWM, over all other fit parameters involved in the respective analysis isshown. X2T is calculated as a function of ηwg1 , ηwg2 . X2FWM is calculated as a functionof ηwg1 , ηwg2 , ηwg3 , while φin and φout are held fixed at the best fit values from the X2Tminimization. X2tot is the sum over X2T and X2FWM. . . . . . . . . . . . . . . . . . . . 116xxxiiiList of Figures5.19 Nonlinear characterization results for the best fit analysis (filled circles) and for theconverged results (empty squares) when TNL,modelm,i is used to calculate the Um,i in themodel functions [Eqns. (5.37),(5.38) and (5.40)]. The minimum X2T = X2T/(NT−kT)from the nonlinear transmission is shown in (a), where NT is the number of datapoints in the fit and kT is the number of fit parameters. Fits where X2T does notreach a minimum with respect to all parameters are marked as red. (b)-(e) Best fitvalues with ±1σ uncertainties (gray rectangles) are shown, along with the convergedparameter values. Parameters that minimize the X2T are shown with black markers,and those that do not are red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.20 Effective free-carrier lifetime calculated for both M1 (circles) and M2 (triangles) ofDevices 1 to 4 (top to bottom). The best fit τcarrier are drawn as dashed lines. . . . . 1186.1 Four-wave mixing efficiency ηPFWM = Pi/(P2pPs) for (a) Device 1, (b) Device 2, (c)Device 3 and (d) Device 4. The experimental FWM idler powers for idler photonsin Mode 1 and Mode 3 are plotted as a function of pump power (triangles) when thesignal power is fixed at Ps (labelled) and signal power (circles) when the pump poweris fixed at Pp (labelled). The idler powers predicted using the model function withbest fit parameters found from the least-squares analysis when Rth, Qabs and τcarrierare held fixed at their average values are shown with thick solid and dashed-dottedblack lines for the pump and signal sweeps. The dashed lines show the predictedpower when nonlinear absorption is ignored. . . . . . . . . . . . . . . . . . . . . . . . 1216.2 Quality factors for Devices 1 to 4, found by the nonlinear characterization (NLC)performed with the average values for Rth, τcarrier and Qabs, and by simulations, areplotted for modes (a) M1, (b) M2, and (c) M3. The total quality factor Qtot, theinput waveguide quality factor, Qin and the output waveguide quality factor Qoutare all plotted, as is summarized in the legend. . . . . . . . . . . . . . . . . . . . . . 128xxxivList of Figures6.3 Mode field profiles for heterostructure photonic crystal (PC) microcavities. Electricfield Re(Ey) mode profiles in the center plane of the silicon are plotted for modes (a)M1, (b) M2 and (c) M3 of a triple microcavity, in the absence of input and outputwaveguides. The heterostructure boundaries are shown with black lines and the PCholes are outlined in white. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129B.1 Schematic of a Yee cell [107]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157C.1 Apodized grating coupler design. (a) FDTD simulation layout for in-coupling of lightfrom a Gaussian beam with a 1/e2 width of 10 µm at θ = −45◦ into the silicon slabvia a 1D grating. (b) Grating spacing, as, that optimizes in-coupling transmission(θ = −45◦), as a function of the slot width, ws, for a uniform grating. The line ofbest fit for a linear function is shown (red dashed). (c) FDTD simulation layoutfor out-coupling light from the silicon slab mode to free-space above the grating.(d) Out-coupling radiation profile measured 1 µm above the uniform grating couplerwith ws = 100 nm and as = 474 nm (blue), when the silicon slab mode is launched.The best fit exponential decay curve (dash-dotted yellow line), and the profile of theinput Gaussian beam (dashed red) are also included. (e) Decay rates as a functionof ws, for uniform gratings with as found from the line of best fit in (b). (f) Out-coupling radiation profile measured 1 µm above the apodized grating coupler (blue),when the silicon slab mode is excited. The predicted decay profile (yellow dashdotted) and the excitation Gaussian beam for in-coupling profile (red dashed), asalso plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160xxxvList of FiguresD.1 Transmission spectra for reference devices in Group 3 of Chip A, with sms = 0.97.(a) Transmission from directly before the input grating coupler to directly after col-lection by the elliptical mirror, for the G2Gout (blue), G2Gin (red), PCWGout (pur-ple) and PCWGin (yellow) devices. (b) Transmission efficiencies, ηoutref,GC (blue) andηinref,GC (red) from found transmission measurements of G2Gout and G2Gin respec-tively. (c) Transmission efficiencies, ηoutref,PCwg (purple) and ηinref,PCwg (yellow) fromfound transmission measurements of PCWGout and PCWGin, respectively. (d). Thegreen lines in (b) and (c) are ηmeanref,X (λ), found by filtering ηref,X(λ) spectra to removethe fast oscillations. The black lines are the best fit g(λ, φ) functions. (d) The scaledcoupling efficiencies for (bottom to top) G2Gin (blue), G2Gout and PCWGin (blue),PCWGout. (e) The input (yellow) and output (purple) PC waveguide coupling effi-ciencies. Black lines plot the best fit sinusoidal functions. . . . . . . . . . . . . . . . 167D.2 Spectral filter transmission efficiencies measured by fixing the filter center wavelengthat λfilter = 1545 nm and sweeping the laser wavelength. (a) Transmission efficienciesof filters A (blue), B (red) and C (yellow) on a linear scale. (b) Same as (a) but on alog scale. (c) Product of the transmission efficiencies for filters A and B, when theirpeak wavelengths are aligned, plotted on a linear scale. (d) Same as (c) but on a logscale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173D.3 Single photon detector characterization, when set with a dead time of 100 µs andefficiency 10 %. (a) Detected power calculated with a dead-time τDT = 99.9788 µsfor high-power (blue circles) and low-power (orange triangles) measurements, andcalculated with the detector setting τDT = 100 µs is also plotted (large gray circles).The line of best fit for τDT = 99.9788 µs is plotted (black line), with the slope givingdetection efficiency ηD = 0.116. (b) Chi squared based on the square differencesbetween the calculated detected power (dependent on τDT) and the linear best fit tothe data. The X2 is minimized for τDT = 99.9788 µs. (c) The detection efficienciescalculated with τDT = 99.9788 µs for high-power (blue circles) and low-power (orangetriangles) measurements. Error bars show uncertainty in the measurement. . . . . . 175xxxviList of FiguresG.1 Device 1 X2 minimization plots. The parameter on each x axis is fixed and theminimum X2 over all other parameters is shown. The straight black lines show themin(X2) + 1. The shaded region in (g) indicates ηwg2 values outside of the range ofpossible values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191G.2 Device 2 X2 minimization plots. The parameter on each x axis is fixed and theminimum X2 over all other parameters is shown. The straight black lines show themin(X2) + 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192G.3 Device 3 X2 minimization plots. The parameter on each x axis is fixed and theminimum X2 over all other parameters is shown. The straight black lines show themin(X2) + 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193G.4 Device 4 X2 minimization plots. The parameter on each x axis is fixed and theminimum X2 over all other parameters is shown. The straight black lines show themin(X2) + 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194G.5 Device 1 X2 minimization plots, when Rth, Qabs and τcarrier are fixed to the averagevalues in the model functions. The parameter on each x axis is fixed and the mini-mum X2 over all other free parameters is shown. The straight black lines show themin(X2) + 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194G.6 Device 2 X2 minimization plots, when Rth, Qabs and τcarrier are fixed to the aver-age values in the model functions. The parameter on each x axis is fixed and theminimum X2 over all other parameters is shown. The straight black lines show themin(X2) + 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195G.7 Device 3 X2 minimization plots, when Rth, Qabs and τcarrier are fixed to the aver-age values in the model functions. The parameter on each x axis is fixed and theminimum X2 over all other parameters is shown. The straight black lines show themin(X2) + 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195G.8 Device 4 X2 minimization plots, when Rth, Qabs and τcarrier are fixed to the aver-age values in the model functions. The parameter on each x axis is fixed and theminimum X2 over all other parameters is shown. The straight black lines show themin(X2) + 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195xxxviiList of AbbreviationsCW Continuous waveEBL Electron beam lithographyFCD Free-carrier dispersionFDTD Finite-difference time-domainFWHM Full width at half maximumFWM Four-wave mixingG2G Grating to grating reference deviceM1 Cavity mode 1 (with lowest resonant wavelength)M2 Cavity mode 2 (with center resonant wavelength)M3 Cavity mode 3 (with highest resonant wavelength)NLC Nonlinear characterizationPC Photonic crystalPCWG Photonic crystal waveguide reference devicePIC Photonic integrated circuitQ Quality factorQI Quantum informationSEM Scanning electron microscopeTCMT Temporal coupled mode theoryTIR Total internal reflectionTMC Triple microcavity deviceTPA Two-photon absorptionxxxviiiList of SymbolsβFWM Four-wave mixing conversion coefficientβTPA Two-photon absorption coefficient∆λNLm Resonant wavelength shift of mode m∆ωNLm Resonant frequency shift of mode mζFCD Free-carrier dispersion nonlinear parameterηPFWM Four-wave mixing conversion efficiencyηwgm Waveguide coupling ratio of mode m, τoutm /τinmλm Resonant wavelength of mode mσFCA Free-carrier absorption cross-sectionτm Total microcavity lifetime of mode mτabs Linear material absorption lifetimeτcarrier Effective free-carrier lifetimeτ inm Input waveguide coupling lifetime of mode mτ linm Total linear microcavity lifetime of mode mτoutm Output waveguide coupling lifetime of mode mτ scattm Scattering lifetime of mode mτFCAm Free carrier absorption lifetime of mode mτTPAm Two-photon absorption lifetime of mode mφin Fabry-Perot phase shift of input portφout Fabry-Perot phase shift of output portχ(3) Third order nonlinear susceptibilityX2 Sum of the squared differencesωm Resonant frequency of mode mxxxixList of Symbolsdn/dT Refractive index temperature dependencenSi Refractive index of siliconn2,Si Nonlinear index coefficientN Free-carrier densitypoutm Probably that cavity photon in mode m couples to output waveguidePth Kerr effect input power thresholdQm Total quality factor of mode mQabs Linear material absorption quality factorQinm Input waveguide coupling quality factorQoutm Output waveguide coupling quality factorRth Thermal resistanceTlinm Linear peak relative transmissionTNLm Noninear peak relative transmissionvg Group velocityxlAcknowledgementsFirst and foremost, I would like to sincerely thank Dr. Jeff Young for his guidance and supportduring my graduate studies. I thank him for sharing his vast knowledge, for challenging me, andfor providing a wide variety of learning opportunities. I leave this lab knowing that no piece of labequipment is too old to resurrect.Thanks to the members of the Nanolab for creating a friendly and supportive environment.A special thanks to fellow graduate students Jonathan Massey-Allard, Dr. Hamed Mirsadeghi,Dr. Charles Foell and Xiruo Yan, it’s been a pleasure learning and laughing with them over theyears, and I thank them for all the stimulating discussions and for all their help. Thanks to JingdaWu, Megan Nantel, and Lisa Rudolph for being great lab-mates during the end of my time in theNanolab.Thanks to Dr. Moshen Akhlaghi for his mentorship and for sharing with me his systematicproblem solving techniques. Thanks to Dr. Mario Beaudoin and Dr. Alina Kulpa for their help inthe cleanroom, and to Dr. Lukas Chrostowski, Dr. Richard Bojko and Shane Patrick for their helpwith device fabrication. Thanks to Dr. Ovidiu Toader, I don’t think there is any computer relatedproblem he can’t solve.I also thank Dr. James Day, Andrew Macdonald, Nathan Evetts, and the rest of the AMPELbuilding lunch crew for the light-hearted discussions about science and well beyond.I am also very grateful for the support I received from the Natural Sciences and EngineeringResearch Council (NSERC), the Silicon Electronic-Photonic Integrated Circuits (Si-EPIC) Programand the Quantum Electronic Science and Technology (QuEST) Program, as well as the meaningfulexperiences they have afforded me. I’d like to thank the people at Lumerical for the opportunity todo an internship, and specifically Dr. James Pond and Dylan McGuire for sharing their expertiseand for patiently teaching me.xliAcknowledgementsThanks to my family and friends for all their love and support during this journey. Finally,an immeasurable thank you to my partner Dan, for his unwavering support and encouragementthrough it all.xliiChapter 1Introduction1.1 ContextOver the last two decades, the rapid growth of the Internet has lead to increasing demands for highbandwidth information processing and transmission. Large amounts of information are processedand stored in data centers using microelectronic chips. For long-haul transmission, the informationis encoded in light, and launched into glass optical fibres that form a global network. The elec-tronic and optical information is interfaced using optical and electro-optical components includinglasers, photodetectors, modulators, and spectral multiplexers and demultiplexers, as illustrated inFig. 1.1. On the transmission end, electronic signals are encoded in laser light using modulators,meanwhile on the receiving end, optical signals are converted to electronic signals using photode-tectors. In transmission, multiple signals are transmitted in a single optical fibre by multiplexing(and demultiplexing) optical signals with different carrier frequencies. In modern day data centers,many of these interfacing components are bulky individual units, with high power consumption,that are installed in racks. This is in stark contrast to the microelectronic chips, where millions ofcompact electronic components are densely integrated.One vision in “integrated optics” is to replace these racks of optical components with compactand efficient photonic integrated circuits (PICs) [48, 49, 83, 87] that support the same functional-ities. This PIC approach has the potential to revolutionize data centers, and substantial progresshas been made, by both academic and industry parties, toward developing the necessary integratedcomponents. In PICs, the “wires” that transport light through the circuit are waveguides definedin planar 2D geometries, where total internal reflection confines the light to the high refractiveindex guiding channels, as shown in Fig. 1.2(a). Optical components like beam-splitters [54] andspectral filters [12, 100] are realized in compact geometries, as shown in Fig. 1.2(b) and (c). A11.1. ContextMicroelectronic chipModulatorLaserModulatorLaserModulatorLaserMultiplexerDemultiplexerMicroelectronic chipPhotodetectorsFigure 1.1: Schematic example of information sending and receiving. Three electronic signals, produced inone microelectronic chip, are transmitted to a remote microelectronic chip via a single optical fibre. Theelectronic signals (black dashed lines) are applied to modulators that encode the information in light arrivingfrom continuous wave (CW) lasers via optical fibres (solid coloured lines). Each laser has a different opticalfrequency, and the three optical signals leaving the modulators are directed into a single optical fibre usinga spectral multiplexer. The optical fibre carries the multiplexed signal to a remote destination, where itis separated into the three optical frequencies using a demultiplexer. The individual optical signals thenarrive at photodetectors, that translate them into electronic signals, which are then directed to a secondmicroelectronic chip.long-standing goal has been to greatly reduce conversions between optical and electric signals byperforming processing tasks like switching and routing in PICs, where light is controlled with light[2, 47, 60–62]. This will require devices that operate in the nonlinear optical regime, where one lightsignal perturbs the optical material through a nonlinear interaction, thus affecting the response ofother light signals as they pass through the device.In parallel, quite independently, while this PIC research and development has been evolving,so has the field of quantum information (QI). In one of the most application-ready areas of QI,namely quantum communication, manipulation of light at the single photon level is essential [28]. Inaddition, one of the many schemes being considered for realizing a full quantum computer involveslaunching and routing single photons through complex paths of beam-splitters, phase-shifters anddetectors [36, 45, 57]. Just as with the Internet example above, current demonstrations have reliedmostly on bulky components [77, 90, 108], but the potential advantages of using PICs for QIapplications, chiefly scalability and manufacturability, are driving the development of integrated21.1. Context-20 -10 0 10 20x ( µm)-505y (µm)-0.5 0 0.5y ( µm)-0.500.5z (µm)(a)(b)(c)channelsubstratechannelsubstrateFigure 1.2: Schematic of PIC components. (a) An example waveguide that is composed of a high indexchannel that confines light by total internal reflection. The waveguide sits on a low index substrate, and isotherwise surrounded by air. An example of the cross-sectional mode profile for a PIC waveguide is shown onthe right, where the white lines contour the channel and substrate. (b) A beam-splitter (directional coupler),that transfers light between two waveguides by evanescent coupling. (c) A compact spectral filter formedby a ring resonator coupled to two waveguides. In this example, the input excitation at λ1, is resonant withthe ring and is transmitted through to the top waveguide, while the other wavelengths are non-resonant andare directly transmitted.waveguide circuits that incorporate single photon sources and detectors [1, 17, 29, 36, 57, 64, 68,78, 98].1.1.1 Photonic integrated circuit basicsIn PICs, optical components are linked by channel waveguides that are interfaced with optical fibresto bring light in and out of the photonic circuit, as is schematically shown in Fig. 1.3. An ideal31.1. Contexthost-material for PICs supports low-loss passive components, like waveguides, beam-splitters andspectral filters, as well as active components like light sources (lasers, single-photon), detectors,and modulators, in which the material’s optical properties are controlled electrically or optically.DetectorPhase-shifterRing resonator spectral lterDirectional coupler beam-splitterGrating coupler input/output portFigure 1.3: Schematic of a photonic integrated circuit, where optical components are defined in a highrefractive index material, that sits on a slab of lower index index material (gray). Passive componentsincluding grating couplers, beam-splitters and spectral filters are shown, in addition to active componentsincluding phase-shifters and detectors. The dotted arrows show that light is routed elsewhere in the photoniccircuit.There are a number of different host-materials that have been explored for classical and quantumPICs, including most-significantly, silica-on-silicon, compound III-V semiconductors and silicon-on-insulator. Silica-on-silicon PICs have been widely employed, particularly for passive deviceslike waveguides and spectral filters (multiplexers and demultiplexers) [43], and to date, most PICdemonstrations of QI-relevant circuits have been in this platform [36, 69]. In this material system,silica is thermally grown on a silicon base and optical components are defined by doping the silicato change the refractive index. The most-significant advantage of this host-material is the smallpropagation loss that can be achieved with its relatively large cross section waveguides, that alsoaffords efficient coupling to optical fibres. One primary disadvantage is that because the refractiveindex contrast is small, the critical angle for total internal reflection (TIR) is also small, which limitsthe bending radius of the waveguides, resulting in large device footprints (chip sizes on the order41.1. Contextof several cm2). In addition, these devices have limited active functionality. Index of refractiontuning, essential for phase-shifters in QI-related circuits, has been realized by integrating heatingelements, however this process is inefficient and slow, due to the low thermo-optic coefficient of silica[36, 57]. Light sources and detectors are not viable in this platform without hybrid integration ofmore suitable active materials [33].Compound III-V semiconductors, like InP, GaAs, GaInP, AlGaAs and InAs, are used to formcompact devices with a number of active and passive functionalities. Many different types of de-vices have been successfully realized in this platform including lasers, detectors, modulators, singlephoton sources, photon pair sources, and optical switches [22, 35, 49, 62]. These semiconductormaterials have direct electronic bandgaps, that make them excellent candidates for both classicaland single-photon sources, as well as detectors [15, 42, 99]. The semiconductor crystal symme-try also supports a second order nonlinear susceptibility such that relatively efficient high speedelectro-optic modulators are achievable. Many third order nonlinear processes, like four-wave mix-ing, have also been demonstrated, with relatively low nonlinear losses. These III-V PICs are inmany ways the most versatile and successful to date, but in the context of high-volume manufac-turing, of sophisticated opto-electronic circuits, they suffer from high material costs and a lack ofwell-developed foundries for mass manufacturing. The latter issue is related to another practicaldisadvantage, that large-scale microelectronic circuits are not manufactured in this platform, thusdirect integration of optical and electronic components is not simple.In contrast, silicon-on-insulator is highly attractive due to the central role that silicon playsin the semiconductor industry [37, 86, 87, 91], and the advanced complementary metal-oxide-semiconductor (CMOS) fabrication technologies that now exist for large scale manufacturing forboth microelectronic and photonic circuits. Silicon-on-insulator (SOI) wafers contain a silicondevice layer where optical components are defined by lithography and etching, that is supportedby an oxide layer that is supported by a silicon base. Owing to the strong index contrast betweensilicon and the surrounding materials (oxide and air), TIR is strong, and compact silicon photonicdevices are realized. There is a large academic and industrial research effort toward developingsilicon PIC components, and a wide range of passive and active components have been realized [12,78, 91, 97, 104]. However, the main drawback of working with silicon is its indirect bandgap, which51.2. Thesis motivationmakes it challenging to realize good sources and detectors. In addition, the centrosymmetry of thesilicon prohibits second order nonlinear effects that are typically used for fast electro-optic switchingin other host-materials. Alternative approaches are explored to overcome these issues. Silicon hasa relatively strong third order nonlinear susceptibility, and third order nonlinear processes havebeen employed for all-optical signal processing devices and photon sources. Hybrid integrationwith other semiconductor materials has also seen some success, where laser sources and detectorsmade from III-V direct bandgap materials are wafer bonded to the silicon [91]. Ultra-sensitivedetectors have also been realized by depositing and patterning superconducting materials on thesilicon surface [1, 68].1.2 Thesis motivationThe original motivation behind this work was to develop a photon source for quantum informationprocessing in silicon photonic microchips. In quantum photonic technologies, a source that gener-ates single photons either on demand, or at known times, is critical for photon state preparationand processing. One avenue for realizing an on-demand single photon source involves the deposi-tion of quantum emitters, like quantum dots (semiconductor crystals with ∼ 5 nm dimensions), onthe silicon surface. A quantum dot acts like an “artificial atom” that emits a single photon whenoptically or electrically excited with a pulse [15]. This type of source is considered to be “deter-ministic”, as there is control over when the photons are emitted. There are a number of technicalchallenges associated with ensuring the quantum emitter efficiently emits into the silicon microchip,often requiring a combination sophisticated device engineering and/or complex chemistry.An alternative approach, the one taken in this thesis, takes advantage of the strong thirdorder nonlinearity of silicon to spontaneously generate pairs of photons through a process calledspontaneous four-wave mixing [17]. When the silicon device is excited with a single continuous-wave(CW) pump laser, signal and idler pairs of photons are spontaneously generated at two differentfrequencies, equally spaced above and below the pump frequency such that energy is conserved,as illustrated in Fig. 1.4(a). This process is proposed to realize “heralded” single photon sources,where the detection of one of the photons from a pair signals (or “heralds”) the presence of the61.3. Nonlinear processes in silicon(a) (b)Single photon detectorFigure 1.4: (a) Schematic of spontaneous four-wave mixing. Two “pump” photons with frequency ωpare converted to a pair of “signal” and “idler” photons with frequencies ωs and ωi, respectively. Energyconservation requires that 2ωp = ωs +ωi. (b) Spontaneous photon pair generation is implemented a heraldedsingle photon source, when pumped with light at ωp. The detection of a signal photon, by a single photondetector, heralds the presence of the idler photon.other, as shown in Fig. 1.4(b). This source is non-determinstic, in that there is no control over whenthe pairs are generated. However, proposals have been suggested for achieving near-deterministicoperation by multiplexing multiple sources and incorporating routing components like optical delaysand switches [19, 101, 102]. The photon pair sources have also been used to realize quantummechanically entangled photons, which also have applications in quantum information processing[29, 78, 97].A silicon PIC was designed and fabricated for this purpose. In the process of characterizing itsperformance, specifically in comparing its nonlinear behaviour to model predictions, it was realizedthat owing to slight differences in the fabricated devices, as compared to the corresponding designs,many of the key model parameters had to be extracted from a non-trivial set of linear and nonlinearexperiments, none directly associated with spontaneous pair generation. Some of these ancillaryexperiments are pertinent to other all-optical signal processing functions (alluded to above). Inthe end, the novel and sophisticated characterization scheme developed for this generic nonlinearstructure became the main theme of the thesis.1.3 Nonlinear processes in siliconThe nonlinear processes of interest to this thesis are first presented in the context of bulk siliconin Section 1.3.1. Applications based on these nonlinear processes are then discussed in Section1.3.2, and examples of photonic structures that can be used to reduce the operating power and thefootprint of nonlinear optical devices are reviewed in Section 1.3.3. The photonic structure pursued71.3. Nonlinear processes in siliconin this work is presented in Section 1.3.4.1.3.1 Bulk siliconStimulated and spontaneous four-wave mixing (generating light at frequencies otherthan the driving frequency(s))The nonlinear device presented in this thesis is specifically designed to promote pair generationthrough spontaneous four-wave mixing. Spontaneous four-wave mixing is a quantum mechanicalnonlinear process, which is closely related to its classical counterpart, stimulated four-wave mixing.To help explain these third order nonlinear processes, classical nonlinear frequency mixing is firstconsidered, using a simplified picture of nonlinear interactions (a more rigorous derivation forstimulated FWM is presented in Chapter 5).In classical nonlinear frequency mixing, excitations at two or more different optical frequen-cies induce a nonlinear polarization density in the material that generates light at one or morenew frequencies. For example, when excitations at optical frequencies ω1 and ω2 are presentin material with a second order nonlinearity, the induced second order polarization density isP (2)(t) = 0χ(2)E(t)2, where E(t) is the total electric field present and is approximately equal toE1 cos(ω1t) + E2 cos(ω2t) (ignoring nonlinear field contributions) [14]. The polarization density inthe medium thus contains contributions that go as ∼ cos(ω1t) cos(ω2t), that acts as a source thatgenerates new electromagnetic fields at mixing frequencies |ω1 ± ω2|.In third order nonlinear materials, the polarization density depends on the field through P (3)(t) =0χ(3)E(t)3. When excitations at two optical frequencies (derived from incident laser fields at ω1 andω2) are present, the polarization density has contributions with |ωx±ωy±ωz|, where ωx, ωy, ωz = ω1or ω2. In stimulated four-wave mixing, light is generated at the new frequency ω3 = |±ω2±ω2∓ω1|through the third order nonlinear polarization density, where ω1, ω2 and ω3 are commonly called thesignal, pump and idler frequencies, respectively. In this process, two pump photons at the centerfrequency (ω2) are mixed with one signal photon (at the other excitation frequency, ω1) to generatean idler photon at the mixing frequency ω3, where energy conservation requires ω2 = (ω1 + ω3)/2(i.e. the three optical frequencies are equally spaced). The photon generation rate (i.e. power) at81.3. Nonlinear processes in siliconthe idler frequency has a quadratic dependence on the pump power, and a linear dependence onthe signal power.When two continuous-wave lasers tuned to ω1 and ω2 are launched along the same axis into thebulk nonlinear material, the idler power depends on the degree to which the propagating opticalmodes involved in the conversion process (at ω1, ω2 and ω3) are phase-matched.1 This can besatisfied fairly naturally for FWM, by choosing |ω2 − ω1| such that |ω2 − ω1| << ω2 and suchthat the bandwidth for the FWM process is narrow enough that the linear material dispersion isminimal.In the spontaneous version of this process, only the pump excitation is present (ω2), and vacuumfluctuations facilitate the conversion of two ω2 pump photons to a pair of photons with frequenciesω1 and ω3. Alternatively, when two excitations are present at frequencies ω1 and ω3, two singlephotons at each of these frequencies are spontaneously converted to two photons at ω2.Kerr effect and two-photon absorption (modifying propagation properties at thedriving frequency)The Kerr nonlinearity is a classical third order nonlinear process which causes changes to boththe real and imaginary parts of the refractive index of silicon, dependent on the field intensity.In contrast to the stimulated frequency mixing processes described above, where the third orderpolarization density oscillates at a new frequency, the Kerr nonlinearity arises due to polarizationdensity contributions that oscillate at the excitation frequencies, through mixing processes like| ± ω1 ∓ ω1 ± ω1| = ω1. In the linear limit, the refractive index is n0 =√1 + χ(1), where P (1) =0χ(1)E(t). When excitation at a single frequency is injected in the silicon, with electric fieldE(t) = E1 cos(ω1t), the Kerr nonlinearity introduces a third order polarization contribution thatis proportional to both E(t) and the field intensity I, such that the total polarization density isP (t) ∼ 0[χ(1) + 3χ(3)20cn0I]E(t), and the nonlinear refractive index is nNL ' n0 + 3χ(3)4n200cI.In silicon, the real part of the χ(3), at the frequencies of interest, is positive such that the real1Phase matching means that the polarization density with oscillation frequency ω3, which in general has a spatialdependence of exp(ikP3 z), where z is the propagation direction, will only efficiently generate electromagnetic fields atω3 if kP3 matches the wavevector that light with frequency ω3 propagates in the medium, k3 = ω3n(ω3)/c, where n(ω3)is the refractive index at ω3. The polarization density that arises from FWM is kP3 = 2k2−k1, where k1 = ω1n(ω1)/cand k2 = ω2n(ω2)/c, such that phase matching requires ω3n(ω3) = 2ω2n(ω2)− ω1n(ω1).91.3. Nonlinear processes in siliconpart of nNL increases with intensity, thus introducing nonlinear dispersion. The imaginary partof the silicon χ(3) is associated with two-photon absorption, a process in which two photons, withtotal energy ωtot ∼ 2.4 × 1015 rad/s (in units of ~), are absorbed to excite a free-carrier abovethe silicon electronic bandgap (∆ωSi = 1.68× 1015 rad/s). The absorption process is mediated byphonons to achieve momentum conservation across the indirect bandgap. Two-photon absorptionhas implications beyond directly introducing loss through the χ(3)-dependent refractive index. Thefree-carriers generated cause dispersion by lowering the real part of the refractive, and they alsointroduce further loss by absorbing radiation, thus adding yet another nonlinear contribution to theimaginary part of the refractive index. Furthermore, the power absorbed in both of these nonlinearprocesses result in thermal dispersion, such that the real part of the refractive index increases withheating. These “knock-on” effects can be described in terms of their contributions to an effectivedegenerate χ(3).In the discussions above, the χ(3) is assumed to be a scalar, while in practice, it is a tensor thatdepends on the frequencies involved in the mixing processes. As such, the non-degenerate frequencymixing χ(3) tensor, is different from degenerate nonlinear propagation χ(3) tensor. However, for thecases considered in this thesis, the differences in the non-degenerate frequencies are so small thatthe χ(3) tensor is effectively the same as that for degenerate mixing.1.3.2 Applications of nonlinear processes in siliconFrequency mixingThe original motivation of this work was to use spontaneous FWM in silicon to generate photonpairs for QI-related applications, including non-deterministic heralded single photon sources. It isinteresting to note that second order nonlinear photon pair sources are currently the workhorsesof bulk optical QI implementations, and also widely used for testing QI-related photonic circuits.These sources efficiently generate photon pairs through a second order process called “spontaneousparametric down-conversion” (SPDC), where a single pump photon is converted to two lower en-ergy signal and idler photons [41]. Spontaneous FWM sources cannot, in general, serve as directreplacements for these sources, as SPDC sources can be operated in configurations where they101.3. Nonlinear processes in silicongenerate photon pairs in quantum mechanically entangled states, which are of great interest inQI applications. It is not possible for spontaneous FWM sources to directly produce entangledphoton pairs, however entangled states can be achieved by passing the photons through networksof beam-splitters and other optical components [29, 78, 97].PumpSignalIdlerIdler(a)(b)PumpSignalFigure 1.5: Schematic of stimulated four-wave mixing (FWM) signal processing applications. The signal,pump and idler frequencies are ωs, ωp and ωi, where 2ωp = ωs +ωi. (a) The signal-to-noise ratio (SNR) in amodulated amplitude signal (high and low amplitudes represent 1’s and 0’s, respectively) is reduced throughthe FWM process by translating the data signal from the pump light to idler light. The SNR is improved inthe idler amplitude owing to the quadratic dependence of the idler power on the pump power. The schematicshows that the amplitude-modulated pump light undergoes frequency mixing with the unmodulated signal(ωs) light, to produce modulated light at the idler frequency ωi. Here the solid lines show the modulatedsignals, that sit above the no-power baselines (dashed-lines). (b) The timing jitter in a return-to-zero datastream (each bit is separated by a low amplitude time interval) is reduced by modulating the signal on thesame clock, such that idler power can only be generated when the signal excitation is on.Stimulated four-wave mixing has applications in classical quantum information processing, forvarious all-optical signal processing tasks [25, 74]. For example, this process can be used to dis-tribute information over multiple frequency channels for parallel processing, as illustrated in Fig.1.5(a). Information encoded in the modulated amplitude of an optical signal with carrier frequencyωp (the pump), is translated to another frequency, ωi (the idler), by activating a signal laser atωs = 2ωp − ωi which stimulates the four-wave mixing conversion process. Four-wave mixing is alsoused for improving the quality of optical signals by increasing the signal-to-noise and reducing thetiming jitter, as illustrated in Fig. 1.5(a) and (b), respectively. In Fig. 1.5(a), when stimulatedFWM is performed with an amplitude modulated pump excitation, and an unmodulated signal ex-citation, the idler light generated has a higher signal-to-noise than the pump due to the quadratic111.3. Nonlinear processes in silicondependence on the pump power. The timing jitter is reduced when the signal is modulated on aclock that coincides with that of the pump, such that idler power can only be generated over thedurations of time when the signal is active, as shown in Fig. 1.5(b). This approach is appropriatefor “return-to-zero” data streams where there are time intervals between each bit when the signalis held in a low state.The spontaneous and stimulated four-wave mixing applications are adversely affected by non-linear absorption introduced by the Kerr nonlinearity. Input powers are raised to compensate forlosses in the pump/signal excitations. In addition, absorption reduces the idler power in stimulatedFWM applications and reduces pair generation rates in spontaneous four-wave mixing applications.In the context of heralded photon sources, the latter problem is two-fold, as the absorption of asingle photon in a pair leads to a greater number of occurrences when the heralding photon triggersthe detector in the absence of a partner photon [35].Nonlinear dispersionNonlinear dispersion effects are employed to realize all-optical signal processing components like all-optical switches, all-optical bit memories and optical logic functions [3, 60, 63]. One basic approachtaken involves using strong “pump” and weak “probe” excitations. The transmission response ofthe weak probe through the silicon device depends on the strength of the pump excitation thatnonlinearly modifies the real part of the refractive index. When the transmission response functionhas a very strong dependence on the refractive index, low pump powers can achieve the desiredcontrol of the probe transmission. Optical resonators that contain nonlinear optical materials arecommonly employed for this reason, as the resonant transmission frequencies are very sensitive tothe refractive index.In a high finesse optical resonator that contains a nonlinear optical material, high intensities oflight build up inside the resonator when it is resonantly excited, such that nonlinear interactions canbe achieved even with relatively low excitation powers. At very low incident powers the resonatoreffectively acts like a spectral filter, such that it only transmits light (and correspondingly onlyallows light within the resonator) with optical frequencies lying within the narrow-bandwidths ofthe resonant modes it supports. A probe excitation tuned to a resonant frequency is transmitted121.3. Nonlinear processes in siliconthrough the device, however its transmission is affected when a pump excitation, tuned near adifferent resonant mode frequency, is activated. The nonlinear change of the refractive index withinthe resonator induced by the pump causes the resonator mode frequencies to shift, such that theprobe becomes non-resonant and its transmission significantly drops. This is the basic premise ofall-optical switching.There is also an “optical bistability” phenomenon that arises due to nonlinear dispersion inresonator structures, and plays an important role in a number of all-optical processing devices.Optical bistability is effectively a hysteresis in the resonator response function. In other words,the resonator response depends on the history of how it was probed. For example, when thetransmission of light through the resonator is measured as a function of the excitation frequency,the resulting spectrum is different when the frequency is swept from low-to-high, as compared towhen its swept high-to-low, and the nature of this altered spectrum depends on the incident power.1.3.3 Efficient and compact nonlinear devicesFor the bulk silicon nonlinear processes described above, the third order nonlinear polarizationdensity is large when the local excitation intensity is high. A strong third order nonlinear responseis achieved in bulk silicon when high excitation intensities are maintained over large volumes.Consider the case where the excitation is derived from laser light launched into the bulk silicon. Atrivial way to increase the excitation intensity in the bulk silicon, is to simply increase the laserpower. Alternatively, the laser can be tightly focused, such that a strong localized intensity isachieved. The highest intensity reached in a Gaussian beam is inversely proportional to the beamwaist radius (at the narrowest point) squared. Away from the waist, the beam width diverges dueto diffraction. The length over which the width is maintained below a factor of√2 of the waistradius, called the Raleigh length, is proportional to the beam waist squared. As a result, for focusedlasers in bulk silicon, there is a trade-off between peak intensity and the volume over which a highintensity is maintained, that ultimately limits the achievable total nonlinear response for a givenlaser power. This means that in practice, nonlinear experiments in bulk silicon are almost alwaysdone with pulsed laser sources that have much higher peak power than CW sources.The waveguides used to route light in PICs offer a simple and very effective means of avoiding131.3. Nonlinear processes in siliconthis diffraction limit. Their cross-sectional areas are on the order of a squared wavelength, closeto the diffraction limited area of a focused laser beam, but their Raleigh length is effectively thelength of the waveguide itself. The effective distance of propagation over which the nonlinearinteractions occur is thus increased over the bulk case by roughly the ratio of the waveguide lengthto the Raleigh length associated with a diffraction limited beam waist: this ratio can be huge. Infour-wave mixing applications in silicon, CW pump powers of < 10 mW are launched into 1.0 cmlong waveguides, and signal to idler conversion efficiencies of ∼ −35 dB are observed [26]. As inthe bulk case, it is critical that the three waveguide modes involved in the nonlinear process (atthe pump, signal and idler frequencies) have propagation phases compatible with phase-matching.In the waveguides, the related mode dispersion is due to a combination of material dispersion andso called “waveguide dispersion” associated with the index contrast and waveguide dimensions.The footprint of such a device can be dramatically reduced by turning a short segment ofwaveguide into a circular loop (radius ∼ 5 − 10 µm), such that light circulates many times be-fore eventually coupling out into a connecting waveguide. The structure described here is a ringresonator, that supports travelling wave resonant modes, such that only excitations with opticalfrequencies near the resonant mode frequencies are allowed to propagate in the ring. Over thephase-matching bandwidth of the waveguide, the modes frequencies are nearly equally spaced and,the mode profiles are spatially overlapping. This resonator structure is suitable for four-wave mix-ing applications, where the signal, pump and idler frequencies coincide with resonant modes. Itis also suitable for the all-optical switching applications described above, as it supports multiplemodes, and has a resonant response that is sensitive to nonlinear dispersion promoted in the ring.Other types of travelling wave resonators are also considered for four-wave mixing applications,including pedestal microdisks and microtoroidal resonators [44, 72]. Light travels along the innerrim of the circular structure, which is suspended in air by a pedestal at its center, and is evanescentlycoupled to a tapered optical fibre for excitation and extraction. Silica microtoroidal whisperinggallery mode resonators support very high quality factor modes (Q > 108) and demonstrate efficientfrequency conversion spanning large bandwidths (cascaded over many modes), to create frequencycombs [44]. However, these structures are not easily integrated into photonic circuits.In general, resonator structures have the drawback of being narrowband (they only work if141.3. Nonlinear processes in siliconthe excitation wavelengths correspond to the resonant mode frequencies). An alternative way toenhance the effective nonlinear interaction strength in a non-resonant manner is by engineering thedispersion of a straight waveguide to greatly reduce the group velocity of the relevant modes. Byslowing the propagation speed of light, the relevant interaction time can be achieved over a muchshorter propagation distance [9]. A common way to achieve “slow-light” is using photonic crystalwaveguide structures [19, 56, 103]. In these structures, coherent scattering of submicron featuresetched in the silicon results in net propagation speeds of ∼ 30 times smaller than the speed of light[9]. Photonic crystal waveguides ∼ several 100 µm long have been found to yield spontaneous four-wave mixing pair generation efficiencies ∼ 20 times higher than centimeter long wire waveguides[35]. These compact structures are less amenable to the nonlinear dispersion applications discussedabove as they are non-resonant.The ultimate limit for reducing the footprint of nonlinear devices is reached by stopping lightpropagation. This is achieved with resonator structures that confine light for thousands or millionsof optical cycles in cavities with dimensions on the order of a cubic wavelength. Section 1.4 explainshow such resonators can be fabricated in silicon PICs using photonic crystal-based microcavities.These types of ultra-compact resonators have a footprint on the order 100 squared wavelengths,compared to a ring resonator, which typically occupies 1000 squared wavelengths. For opticalswitching applications, the smaller mode volume of these PC microcavities compared to ring res-onators allow them to perform optical switching functions at reduced power levels. However, forFWM applications, a single PC microcavity can’t be used since it typically supports only a singlehigh Q mode.One way to take advantage of these compact microresonators for FWM applications is to fabri-cate three copies in close enough proximity so that their nearly degenerate modes couple and formthree distinct modes of a triple-cavity structure. A triple microcavity structure, was found to havespontaneous four-wave mixing photon generation rates 100 times greater than a ring resonator [8].To realize efficient, low power, nonlinear devices based on optical resonators, like ring resonatorsand PC microcavities, it is critical that light is efficiently loaded in the microcavity, and unloadedinto PIC waveguides. This requires careful engineering of the waveguide coupling. Single ringresonator structures are evanescently side coupled channel waveguides, with limited flexibility. In151.3. Nonlinear processes in siliconcontrast, 2D PC microcavities offer a platform for flexible waveguide coupling, which adds to thethe benefits of realizing nonlinear devices in this platform [58].1.3.4 Nonlinear device design and performanceA photonic structure is proposed in this thesis, originally for use as an efficient photon pair source,but it is also well-suited for all-optical frequency conversion, and all-optical switching applications.A scanning electron microscope (SEM) image of the structure is shown in Fig. 1.6(a). It incorpo-rates three coupled PC microcavities and two waveguides, and has a device area of ∼ 50 µm2. Thestructure is defined in an SOI wafer with a 220 nm device layer of silicon, that sits on 3 µm thicklayer of SiO2 that lies on the silicon base.Schematics of the excitation schemes used to measure the structure’s nonlinear functionalitiesare shown in Figs. 1.6(b)-(d). Figure 1.6(b) illustrates that spontaneous four-wave mixing mea-surements involve resonantly coupling pump light from a single laser into the PC microcavity viathe input waveguide, resulting in the generation of signal and idler photons in separate microcav-ity modes. These photons, along with pump microcavity photons, are coupled into the outputwaveguide, and directed off-chip for detection. Photons also leave the microcavity through othermechanisms, including coupling back into the input waveguide, scattering, and absorption in thesilicon. The major challenge that the proposed design sought to overcome was to achieve preferen-tial coupling of the signal and idler photons to the output waveguide, and preferential coupling ofthe pump photons to the input waveguide. In this work, attempted measurements of spontaneousFWM did not yield clear indications of photon pair generation, and future studies are required tounderstand the results from these measurements.The frequency conversion is observed by resonantly exciting two modes of the microcavityusing light from two lasers, such that idler photons are generated in a third resonant mode throughstimulated four-wave mixing, as shown in Fig. 1.6(c). Across the four devices measured, the bestfrequency conversion efficiency (idler power divided by the product of the signal power and thesquared pump power) is found to be ηPFWM = 1.4 × 10−8 µW−2, which is an order or magnitudehigher than a comparable triple nanobeam photonic crystal microcavity [8] and ring resonatorstructures [5, 7, 92, 109].161.3. Nonlinear processes in siliconFigure 1.6: Proposed photonic crystal structure that supports multiple nonlinear functionalities. (a) Scan-ning electron microcavity image of a fabricated structure. The microcavity and waveguide regions areindicated. (b)-(d) Schematics of the microcavity (circle) coupled to input and output waveguides (rect-angles) under different excitation schemes. The red, green and blue arrows show the passage of light atthree different frequencies. (b) Spontaneous four-wave mixing. Pump light at ωp (green) enters the inputwaveguide and resonantly excites a mode of the microcavity. Photon pairs are generated at signal and idlerfrequencies ωs (blue) and ωi (red), respectively, through spontaneous four-wave mixing. Energy conservationrequires 2ωp = ωs + ωi, and the conversion process is only efficient where ωs and ωi pairs coincide withmicrocavity resonant modes. Photons leave the microcavity through coupling to the output waveguide, theinput waveguide, and radiation, as indicated by the arrows. The microcavity photons are also absorbedby the silicon. (c) Stimulated four-wave mixing. Pump and signal excitations at ωp (green) and ωs (blue),respectively, resonantly excite two modes of the microcavity and photons are generated at the idler frequencyωi (red). (c) Kerr effect. One mode of the microcavity is resonant probed with a single laser and the in-duced nonlinear changes to the microcavity refractive index affect transmission spectrum lineshape throughresonant frequency shifts and decreased peak transmission.The nonlinear Kerr effect, which is critical to all-optical switching applications, is demonstratedby exciting the microcavity with a single laser and measuring the transmission spectrum as a171.4. Photonic crystal structuresfunction of the laser power, as illustrated in Fig. 1.6(d). When probed at sufficiently high power,the nonlinear Kerr effect causes changes to the real and imaginary parts of the refractive index thatresults in shifts of the resonant wavelength and reduced transmission, such that the transmissionlineshape becomes distinctly different from the Lorentzian lineshape observed at low powers. Basedon the transmission spectra, it is possible to extract the minimum power threshold figure-of-merit,above which the Kerr effect is strong enough for the structure to be considered for all-opticalprocessing applications. For the structures presented in this thesis, power thresholds as low as 17µW are measured which compares well with those reported in the literature for 2D PC structures,which range from ∼ 10− 200 µW [11, 63, 84, 94].1.4 Photonic crystal structuresA photonic crystal (PC) is a periodic arrangement of dielectric materials, and a planar two-dimensional (2D) PC is commonly realized by etching a periodic array of holes in a dielectricslab[61], as shown in Figs. 1.7(a) and (b). Planar photonic crystals are designed so that there isa range of frequencies over which light bound to the slab is prohibited from propagating in thecrystal, called the photonic bandgap, which is analogous to the electronic band gap for an electronin an atomic lattice. The photonic bandgap plays important roles in common PC-based devices,slow-light waveguides and compact microcavities that strongly promote light-matter interactions.Figure 1.7(c) shows the photonic bandstructure of a planar 2D PC, consisting of a high-indexslab perforated by an hexagonal array of holes and surrounded by a uniform low-index regionabove and below. The bandstructure describes the electromagnetic modes supported by the PCat frequency ω, which obey Bloch’s theorem, Ek(r, ω) = exp(ik · r)uk(r, ω), where uk(r, ω) is aperiodic function, with lattice vector a in the plane of the slab, such that uk(r + a, ω) = uk(r, ω).This is alternatively stated as, Ek(r+a, ω) = exp(ik·a)Ek(r, ω) [38]. Here modes are plotted for thethree different axes of symmetry of the hexagonal PC structures. The k labels an electromagneticmode that satisfies the Bloch equation with that k vector. There are two types of modes inthis symmetrically clad slab, which appear in the bandstructure: transverse electric (TE) andtransverse magnetic (TM). Transverse electric and magnetic modes have electric and magnetic181.4. Photonic crystal structuresdevice silicon, n = 3.47SiO2, n = 1.44air or vacuum, n = 1.0TE modes: TM modes: MKM KTE bandgap0.05.04.03.01.02.0light conezˆyˆ xˆ(b)(c)(a)Bloch wavevectorNormalized frequencyFigure 1.7: Planar photonic crystal defined in a silicon-on-insulator (SOI) wafer. (a) A hexagonal periodiclattice of holes is etched in the top silicon slab of the SOI, called the device silicon, which sits on an oxidelayer and a silicon base. (b) Light is contained in the device silicon with respect to the zˆ direction by totalinternal reflection (TIR) occuring at the air and SiO2 oxide interfaces. The planar photonic crystal supportstransverse-electric (TE) and transverse-magnetic (TM) modes. (c) Bandstructure of a PC, plotted as afunction of the three reciprocal lattice vectors, in the first Brillouin zone. The TE and TM modes are shownas red and blue lines, respectively. The TE bandgap is indicated (shaded green), and the light cone is alsoindicated (shaded yellow). Modes that fall below of the light cone (solid lines) are bound to the device siliconthrough TIR, while those falling within the light cone are partially unbound (dashed lines). This figure isreproduced and modified, with permission, from Ref. [24].field components that lie completely in the plane of the slab at its midpoint, respectively.The bandstructure in Fig. 1.7(c) shows that the TE modes have a large photonic bandgap inthin slabs containing a periodic array of etched holes, which makes them ideal for device engineering.It is important to note that for 2D planar PCs, the bandgap is never a full bandgap, as there arealways radiation modes that exist where ω ≥ c|k|/ns, with ns being the index of the materialsurrounding the slab (typically air and/or SiO2). The presence of radiation modes goes hand inhand with the limitations of TIR in confining light to the slab. It is common to indicate the presence191.4. Photonic crystal structuresof the radiation modes in the bandstructure by including a “light-line” or “light-cone”, as shownin Fig. 1.7(c). Modes that fall within the light cone (above the light-line) are called “quasi-bound”modes, while those that fall below the light-line are called “bound modes” as they are rigorouslyconfined to the slab.-1 0 1x ( µm)-2-1012y (µm)-1 0 1x ( µm)-2-1012y (µm)1-10(a) Photonic crystal waveguide0 0.1 0.2 0.3 0.4 0.5160180200220240260Frequency (THz)kx(2π/a)(b)(d)-2 0 2x ( µm)-2-1012y (µm)10WGWGWGBulk PC continuum of modesBulk PC continuum Waveguide bands(c) Photonic crystal microcavityFigure 1.8: Photonic crystal (PC) defect structures defined in silicon-on-insulator. (a) Photonic crystalwaveguide introduced to the PC by omitting a row of holes. The Ey field profiles in the center slab of thesilicon for two different transverse electric waveguide modes are plotted, and the PC holes are outlined inwhite. (b) Bandstructure diagram for the line defect, plotted as a function of the wavevector along thewaveguide axis, kx, in units of 2pi/a, where a is the lattice spacing of the PC. The continuum of bulk PCmodes are shown in gray. Two waveguide modes that exist in the bandgap are plotted as the solid blue anddashed red lines. The light cone is shown in pale yellow. The field profiles of the two modes at kx = pi/a edge,indicated by solid and dotted arrows, are shown in (a), as the left and right plots, respectively. (c) Photoniccrystal microcavity introduced to the PC by omitting three holes in a row, called an “L3” microcavity. Thefield intensity of the fundamental mode supported by the defect is plotted, and the holes are outlined inwhite. (d) Examples of coupling geometries between PC waveguides (WG) marked with red arrows and anL3 microcavity.When a defect is introduced to the uniform photonic crystal structure, quasi-bound and boundmodes can be introduced within the PC bandgap. A PC waveguide is created in a two-dimensional201.4. Photonic crystal structures(2D) planar PC by a omitting a line of holes [9, 53], as shown in Fig. 1.8(a). An example ofthe bandstructure along the waveguide axis is shown Fig. 1.8(b). In PC waveguides, light istransversely confined to the waveguide due to the absence of modes supported in the bulk photoniccrystal. The guided light coherently scatters off the crystal as it propagates, making it possibleto engineer very low group velocities, as exemplified by the flat dispersion curve. Photonic crystalwaveguides are routinely used to achieve propagation speeds of ∼ c/30, for applications requiringstrong light-matter interactions.Photonic crystal structures are not only used to slow light, but also to stop light. Microcav-ity resonators supporting standing-wave modes are realized in photonic crystals by perturbing thephotonic crystal in a localized volume, like the example shown in Fig. 1.8(c). The microcavitiesprovide a means to confine light to small volumes [∼ (λ/n)] for many optical cycles, creating anideal environment for enhancing light-matter interactions. PC microcavities are considered forapplications in a number of different fields included nonlinear optics, cavity quantum electrody-namics, optical trapping and optomechanics. Various designs have been proposed including linearhole defect structures [like the “L3” cavity in Fig. 1.8(c)] where one or more holes are omitted,heterostructure structures where the lattice spacing is perturbed, and mode gap structures whereholes are shifted to create a defect region.The oxide layer beneath the planar 2D PC microcavities is typically removed, in a process called“undercutting”, to increase the photon cavity lifetime (and quality factor). The removal of SiO2increases the index contrast at the boundary (nSi = 3.47, nSiO2 = 1.44, nair = 1), resulting in astronger TIR condition (or equivalently, making the light-cone smaller), such that radiation lossesare reduced. This process has been found to improve quality factors by over an order of magnitude.Light is coupled into and out of PC microcavities using PC waveguides that are brought inclose enough proximity that their mode fields overlap. This is achieved using a variety of waveguidecoupling schemes, like those shown for an L3 cavity in Fig. 1.8(d).Ring resonators and nanobeam microcavities are two alternative compact microresonators con-sidered for similar applications as 2D PC microcavities. Ring resonators typically have radii of∼ 5−10 µm and Q ∼ 104, and they support multiple modes separated by a free-spectral range, muchlike Fabry-Perot cavities. The free-spectral range is dependent on the waveguide mode/material211.5. Dissertation overviewdispersion. While the mode volumes of ring resonators are larger than 2D PC structures, theyare competitive candidates for nonlinear applications owing to the multiple modes supported andthe relative ease of fabrication. Nanobeam microcavities are another strong candidate. These arestructures formed by patterning a one dimensional PC along a channel waveguide. These micro-cavities also benefit strongly from undercutting, and high Q undercut cavities with small modevolumes have been observed, comparable to those found for 2D PC structures. Two disadvantagesof the nanobeam structure compared to the 2D PC structures is that they are less structurallystable after undercutting, and they offer less flexibility for waveguide coupling.1.5 Dissertation overviewThis thesis describes an efficient nonlinear integrated optical device structure that is thoroughlycharacterized using a novel combination of complementary experimental probes and associatednonlinear models. The device exhibits optical bistable behaviour at low excitation powers, andhigh efficiency frequency conversion (via stimulated four-wave mixing).The primary focus of this work is on the device characterization, which is nontrivial in thisstructure primarily due to the presence of the two distinct waveguides coupled to the microcavity.The waveguide coupling strengths play important roles in controlling how much energy is coupledinto the microcavity, and thus affect the magnitude of the nonlinear response. For a microcavitycoupled to a single waveguide, the waveguide coupling strength is the same for both directions (as-suming perfect fabrication), and so can be found from a straight-forward measurement, taken in thelinear regime. As a result, the linear behaviour of the microcavity is reasonably well characterized,and the nonlinear behaviour can be studied directly [11, 94]. When a second microcavity-coupledwaveguide is introduced, as is done here, it becomes impossible to determine both waveguidecoupling strengths from the same, relatively simple, linear measurement. Numerical simulationsprovide good estimates for waveguide coupling strengths, so long as fabrication imperfections donot strongly affect them. Unfortunately, the current state of the art for SOI PC device fabricationdoes not satisfy this condition. As a result, it was necessary to develop a complex analysis protocolthat takes into account, and relies on, a complete set of linear and nonlinear transmission, and221.5. Dissertation overviewstimulated FWM data.Chapter 2 contains a basic overview of the linear transmission, nonlinear transmission (Kerreffect/bistability), and stimulated four-wave mixing spectroscopy measurement used in this work.The device design and fabrication considerations are described in Chapter 3. The systematicprocedure followed to design the photonic crystal structure, for spontaneous four-wave mixing, isoutlined, and the numerical simulations that guide the design process are described. The design ofthe other circuit components, namely the input and output ports is also presented. An overview isalso given of the fabrication layout, where critical device parameters are varied across the layout inan effort to ensure that at least a subset of the actual fabricated devices are similar to the designeddevices. The “reference devices” included in the layout, which ultimately make it possible to esti-mate the contribution of the input and output ports to the total device response, are also described.Finally, the post-fabrication procedure used to undercut the fabricated devices is presented.In Chapter 4, the initial measurements used to characterize the basic properties of the micro-cavity devices and lay the groundwork for the nonlinear measurements and characterization arepresented. For example, to identify “good” microcavity candidates for four-wave mixing, whichsupport three nearly equally spaced modes, the linear transmission spectra are surveyed for 56devices across two microchips. The basic measurement scheme and results for stimulated four-wavemixing are presented for a good candidate device.The main results of this work are presented in Chapter 5. Here the linear and nonlineartransmission results are reported, along with the four-wave mixing results, for four different triple-mode microcavity devices. The analysis procedure used to characterize the devices, and find the15 microcavity parameters for each device that ultimately determine the device behaviour in theregimes of interest, is described. The best fit parameters are presented, and experimental data iscompared to the model functions that contain these parameters.In Chapter 6, the results presented in the previous chapter are discussed. The best fit parametersresulting from the analysis are compared to the literature and to numerical simulations. Twoperformance metrics for the nonlinear device are also compared to those in the literature: thestimulated four-wave mixing efficiency and the Kerr effect threshold power.Finally, the conclusions of this work are presented in Chapter 7. The impact of this work is231.5. Dissertation overviewsummarized here, and future work is proposed.There are a number of appendices that are referenced within the body of the work. Theseappendices provide supplementary information on a range of different components of this work,including design, theory and analysis.24Chapter 2General overview of linear andnonlinear measurementsMeasurements of the linear and nonlinear functionalities of fabricated triple microcavity photoniccrystal structures are central to this thesis. The measurements results not only help quantify thestructure’s suitability for frequency conversion and all-optical processing applications, but they areactually required in the characterization protocol used to extract the optical parameters necessaryfor creating a model that describes all nonlinear functionalities.There are 15 unknown parameters that are sought after, which are summarized in Table 2.1. Inthe “linear” regime, when the microcavity is probed at low enough power that nonlinear effects arenegligible, the parameters that dictate the response include the resonant wavelengths of the threeresonant modes of interest, λm, and the lifetimes of different scattering and absorption processesthat determine how light couples into and out of the microcavity, shown schematically in Fig.2.1(a). The circle represents cavity mode m which has a standing wave pattern, and the rectanglesrepresent the input and output PC waveguides. The black arrows show that light couples betweenthe microcavity mode and the input and output waveguide modes with coupling lifetimes τ inmand τoutm , respectively. The coupling strengths depend on the overlap between the cavity modeand the waveguide modes supported at the mode resonant frequency. Each mode has differentcoupling strengths to the input/output waveguides due to their distinct mode profiles, as shownin Figs. 2.1(b)-(d). The gray arrow shows that light can also couple out of the cavity throughother mechanisms, like out-of-plane scattering and material absorption, with lifetimes τ scattm andτabs, respectively. The absorption lifetime is not taken to be mode-dependent in this study, as itis expected to be very similar for each of the three modes, owing to the similar surface to volume25Chapter 2. General overview of linear and nonlinear measurementsratios. Throughout this thesis, modes m = 1, 2, 3 refer to the lowest, middle and highest resonantwavelengths, respectively, and are also referred to as M1, M2 and M3.1540 1545 1550Wavelength (nm)00.51Transmission(c) (d)(b)(a)INPUT WAVEGUIDEOUTPUT WAVEGUIDE-4 -2 0 2 4x ( µm)-4-202y (µm)-4 -2 0 2 4x ( µm)-4-202y (µm)-4 -2 0 2 4x ( µm)-4-202y (µm)00.20.40.60.81(e)INOUTM1M2M3Figure 2.1: Triple photonic crystal (PC) microcavity coupled to input and ouput waveguides studied in thisthesis. (a) Schematic diagram of the cavity modes coupling to various channels. The cavity is representedby the circle, while the waveguides are represented by rectangles. The coupling lifetimes for mode m arelabelled, where τ inm and τoutm are the coupling lifetimes to the input and output waveguides, respectively,while τ scattm is the scattering lifetime and τabsm is the linear absorption lifetime. (b)-(d) Electric field intensityplots of modes M1, M2 and M3 (in that order). The PC holes are shown with white contours. The input andoutput waveguides are labelled is “IN” and “OUT”. (e) Example of an experimental transmission spectrummeasured in the linear regime for a triple microcavity device.In the “nonlinear” regime, the optical Kerr effect leads to changes in both the real and imaginaryparts of the refractive index, that impact the resonator response. The effective free-carrier lifetime,τcarrier, and the thermal resistance, Rth, are two microcavity parameters that determine the changein free carrier density and sample temperature as a function of excitation power. Excitation-power-depedent free-carrier density and microcavity temperature contribute to the nonlinear shifts of thecavity resonant frequency and Q value. There are other important parameters related directly orindirectly to the nonlinear response that are known for bulk silicon, and therefore do not haveto be found in the characterization process. They include: the two-photon absorption coefficient,βTPA, the free-carrier absorption cross-section, σFCA, the electron and hole free-carrier dispersion26Chapter 2. General overview of linear and nonlinear measurementsTable 2.1: Summary of the linear and nonlinear microcavity parameters. The subscript m labels the mi-crocavity modes, of which there are three. The number of unknown parameters of each type are listed inthe “Unknown” column. The nonlinear parameters that are known for bulk silicon have no entry in thiscolumn. The number of unknown parameters that enter the model functions for the linear transmission (LT),nonlinear transmission (NLT), and stimulated four-wave mixing (FWM) measurements are indicated, andcheck marks indicate which of the known parameters are also included. The NLT model contains parametersfor m = 1 and 2 only, as only these two modes are measured.Parameter Description Unknown LT NLT FWMλm Resonant wavelength 3 3 2 3τ inm Input waveguide coupling lifetime 3 3 2 3τoutm Output waveguide coupling lifetime 3 3 2 3τ scattm Scattering lifetime 3 3 2 3τabs Linear material absorption lifetime 1 1 1 1Rth Thermal resistance 1 1τcarrier Effective free-carrier lifetime 1 1 1βTPA Two-photon absorption coefficient 3 3σFCA Free-carrier absorption cross-section 3 3ζFCD Free-carrier dispersion nonlinear parameter 3dn/dT Refractive index temperature dependence 3βFWM† FWM conversion coefficient 3Total unknown parameters 15 13 11 14† Calculated using χ(3) for silicon (known) and numerical simulations.parameter, ζe,hFCD, and temperature dependent index change of bulk silicon, dn/dT . Four-wavemixing also depends on the conversion coefficient βFWM, which captures the strength of the thirdorder susceptibility, χ(3), and the overlap between the three modes involved in the process, and iswell-estimated using numerical simulations.Linear transmission spectra, measured at low power, directly and almost trivially provide ninepieces of information about the linear microcavity response: the resonant wavelength, λm, the peakrelative transmission (normalized to the input power), Tlinm , and the total lifetime, τlinm , for all threemodes. However, the linear transmission model depends on 13 linear parameters [see the “LT”(linear transmission) column of Table 2.1]. So while the three λm are directly extracted from thelinear transmission spectra, there are only 6 pieces of information about the unknown 10 lifetimes.27Chapter 2. General overview of linear and nonlinear measurementsIt is evident that this measurement alone is not sufficient for extracting all unknown microcavityparameters. Supplemental measurements in the nonlinear regime are required to provide necessaryadditional information on the linear parameters, and to extract the effective free carrier lifetimeand thermal resistance parameters.Measurements of the optical Kerr effect and stimulated frequency conversion together provideenough supplemental information about the optical response to reliably extract all unknown pa-rameters. The optical Kerr effect is observed by measuring the transmission spectrum as a functionof the input laser power. Specifically, in the nonlinear regime, the resonant wavelength undergoesa shift, ∆λNLm , and the peak transmission, TNLm , decreases. For the structures studied in this the-sis, two of the three modes have sufficiently high Qs that enough energy is loaded in the modesto induce observable nonlinear effects (for the laser powers accessible). The nonlinear transmis-sion ∆λNLm and TNLm for the two modes involved in the analysis are modelled using 11 unknownparameters, as shown in the “NLT” (nonlinear transmission) column of Table 2.1.Frequency conversion is measured by resonantly exciting two modes of the microcavity usingtwo lasers, and the power generated in the third mode through stimulated four-wave mixing isrecorded as a function of one of the input powers, while the other is held fixed. There are twodifferent excitation configurations (excite M1 and M2, or excite M2 and M3), resulting in a totalof four data sets. The model that simultaneously describes both excitation configurations involves14 unknown parameters, as shown in the “FWM” column of Table 2.1.Conceptually, and in practice, the data obtained from the linear transmission spectra of eachof the three modes, the nonlinear transmission of the two high Q modes, and the stimulated FWMpower-dependent conversion efficiencies can be, and are, all combined into a single, least-squaresanalysis, where the squared differences between the experimental results and their respective modelfunctions are globally minimized to extract a unique set of device parameters that adequatelyexplain all linear and nonlinear behaviours. This approach was in fact only used after following aslightly different way of separately and iteratively applying a least-squares analysis to the nonlineartransmission and stimulated FWM data. While both approaches yield the same final result, thelatter was used originally because it was more intuitive, and helped reveal useful information aboutcorrelations that exist between various parameters when focused on just one or the other of these282.1. Transmission in linear regimenonlinear datasets.2.1 Transmission in linear regimeIn linear transmission measurements, light from a single laser is coupled into the microcavity, andout-coupled light is studied as a function of the laser wavelength. This is in many ways similarto the transmission measurement of a conventional macroscopic Fabry-Perot cavity, which is firstreviewed here, before considering a more complex microcavity system.2.1.1 Fabry-Perot cavity transmission00.51Relative Transmission00.51Relative Transmission(a) (b) (c)Figure 2.2: Fabry-Perot cavity. (a) Schematic of a Fabry-Perot cavity with two mirrors separated by adistance L, containing a medium with refractive index n. The mirrors have field transmission and reflectioncoefficients ri and ti. (b) Example relative transmission spectrum of a lossless Fabry-Perot cavity withidentical mirrors. The resonant frequencies, ωm are labelled, along with the linewidth δωm and maximumtransmission Tlinm for mode m. The free spectral range (FSR) is also labelled. (c) Relative transmissionspectra for lossless cavity with identical mirrors (solid line), a cavity with loss and identical mirrors (dashedline), and a losses cavity with non-identical mirrors (dash-dotted line).The simple Fabry-Perot cavity consists of two parallel flat mirrors with field reflectance r andtransmittance t, as shown in Fig. 2.2(a), separated by distance L, containing a dielectric mediumwith refractive index n. In the transmission measurement, the light from single a continuous-wave(CW) laser is incident through one of the cavity mirrors and the light transmitted through thesecond mirror at the output of the cavity is measured. An example transmission spectrum isplotted as the solid line in Figs. 2.2(b) and (c), for the case where two mirrors are identical andthere is no loss inside the cavity. This is a relative transmission spectrum, calculated by dividing292.1. Transmission in linear regimethe transmitted power after the second mirror by the power incident on the first mirror.The peaks in the spectrum correspond to the resonant modes characteristic of the cavity, wherethe round-trip phase of light circulating the cavity is an integer multiple of 2pi. The resonant modesare separated by the free spectral range,FSR =picngL(2.1)where c is the speed of light in vacuum, ng is the group velocity of light the cavity dielectric medium,and L is the length of the cavity. In the following, dispersion is ignored such that ng = n = constant,and ωm = mFSR, where m is the mode number.Each resonant peak has three important features, which are labelled in Fig. 2.2(b): the resonantfrequency, ωm, the linewidth δωm, and the maximum relative transmission, Tlinm . These parameterscontain information about the cavity geometry, material, and mirrors.The linewidth, δωm, is the full-width at half-maximum (FWHM) of the resonant transmissionpeak, and is given by,δωm =FSRF(2.2)where F = pi/(1 − √R1R2) is the cavity finesse, and R1 = |r1|2 and R2 = |r2|2 are the intensityreflection coefficients for the first and second mirrors, respectively 2. When the mirror reflectivitiesare high, such that F is large, the linewidth is small because light trapped in the cavity undergoesmany optical cycles before leaking out, so only a small detuning of the laser frequency off-resonanceleads to a large amount of destructive interference within the cavity, resulting in a low steady-stateenergy in the cavity and a drop in the transmission. The linewidth is related to the lifetime ofphotons in the cavity, τm, by,τm =2FFSR=2δωm, (2.3)The photon lifetime is also expressed in units of the oscillation period using the quality factor, Q,Qm = τmωm/2 =δωmωm= mF . (2.4)2Here the cavity is assumed to be a “good” resonator with R1R2 ' 1, such that it is appropriate to express thefinesse F = pi(R1R2)1/4/(1−√R1R2) to the lowest order of (1−√R1R2).302.1. Transmission in linear regimeBoth F and Q are useful dimensionless parameters used to describe the strength of a resonator.For macroscopic cavities, it is sometimes more practical to work with F because m can be verylarge for the resonant modes in the laser wavelength range (m ∼ 105).For the transmission spectrum in Fig. 2.2(c), the maximum transmission relative to the inputpower outside of the first mirror, Tlinm , is unity. This is a signature of a lossless cavity with equalmirror reflectivities. When losses are introduced in the cavity, due to material absorption, scatteringand other effects, Tlinm is reduced along with τm, Qm and Fm, resulting in a broadening of theresonant linewidth, δωm, as shown in the dashed line in Fig. 2.2(c). The transmission peak alsofalls below unity when the mirrors have different reflectivities, as is shown by the dash-dotted linein Fig. 2.2(c). This can be qualitatively understood by considering that when the reflectivitiesare equal, light resonantly reflected from the cavity completely destructively interferes with thenon-resonantly (directly) reflected light off the first mirror, resulting in no reflected light. Whenthe mirror reflectivities are unbalanced, this destructive interference condition is no longer met dueto an imbalance in the resonantly and non-resonantly reflected light, resulting in some net reflectionon resonance.While the intensity of the output wave can never exceed the intensity of the input wave (i.e.Tlinm ≤ 1), the intensity of light circulating in the cavity when excited on resonance can be signif-icantly larger than the input intensity. For example, consider a lossless cavity with equal mirrorreflectivities, probed on resonance. Given that the cavity transmission is unity, the right-wardtravelling wave inside the microcavity has intensity Iright = Iout/T = Iin/T , where Iin and Ioutare the input and output wave intensities (outside of the cavity), and T = |t|2 is the transmissionof the mirror. When T is very small, as true for a high finesse cavity, Iright >> Iin, and in thelimit of high finesse, Iright ' (Fm/2pi)Iin [85]. In a high finesse cavity, there are both right-wardand left-ward travelling waves with close to equal amplitudes, such that a sinusoidal standing-wavepattern is formed, and the total intracavity intensity as a function of position z is,Icav(z) '∣∣∣√I exp(ikz)−√I exp(−ikz)∣∣∣2 = 4I sin2(kz) (2.5)where I = Iright ' Ileft, and k = ωn/c is the propagation wavevector inside the cavity. This results312.1. Transmission in linear regimeTunable laserOUTPUTINPUTPhotodetectorFigure 2.3: Simplified schematic of transmission measurement of photonic crystal (PC) triple microcavitydevice. Light from a tunable laser is focused on the input grating coupler and diffracted into a parabolicwaveguide that narrows to a single-mode channel waveguide. This waveguide is interfaced with the input PCwaveguide, which couples the microcavity. Light is extracted from the microcavity through the ouput PCwaveguide, which is coupled to channel and parabolic waveguides terminated by the output grating coupler.The extract light is measured by a photodetector.in a maximum cavity intensity of ' (2Fm/pi)Iin. The cavity enhancement of the field intensityplays an important role in promoting light-matter interactions for the nonlinear processes exploredin this thesis.Despite some major physical differences between a planar three-dimensional photonic crystalmicrocavity and the macroscopic Fabry-Perot cavity, both transmission spectra have the same threemain features discussed above (ωm, δωm, and Tlinm ).2.1.2 Microcavity transmissionA simplified schematic of the measurement scheme used to study the transmission through micro-cavities is illustrated in Fig. 2.3. A more detailed schematic and description is given later, withother technical considerations in Section 2.1.3. The PC waveguides adjacent to the microcavityare coupled to channel waveguides that are expanded parabolically and terminated by grating cou-plers, which play the role of input/output ports. In transmission measurements, light from a singletunable CW laser is focused on the input grating coupler to launch light into the waveguides, whilelight leaving the output grating coupler is collected and sent to a photodetector.The transmission spectrum for the single PC microcavity structure in Fig. 2.4(a), shown in Fig.2.4(b), has a Lorentzian lineshape, and λm, δλm, and Tlinm are labelled. The transmission spectrumis obtained by dividing the spectrum at the output PC waveguide (the point labelled A in Fig.322.1. Transmission in linear regime1530 1532 1534 1536 1538Wavelength (nm)00.10.20.30.40.5Relative Transmission(a) (b)ABFigure 2.4: Single photonic crystal linear three hole defect (L3) microcavity with input and output waveg-uides. (a) Schematic of the microcavity, where black circles indicate holes etched in the silicon. The redarrows show the passage of light through the device for transmission measurements. (b) Transmission spec-trum for the fundamental mode of an L3 cavity simulated with a finite-difference time-domain simulation.The resonant wavelength, λ1, linewidth, δλ1 and maximum transmission Tlin1 are labelled.2.4(a)) by the spectrum at the input PC waveguide (the point labelled B in Fig. 2.4(a)), such thatthe effects of all other optical and device components are excluded. The way in which these twospectra are determined is described in Section 2.1.3.When contrasting the microcavity linear transmission spectrum in Fig. 2.4(a) with the Fabry-Perot transmission spectrum in Fig. 2.2(b), one superficial difference is that the microcavity trans-mission spectrum is plotted as a function of wavelength λ, as opposed to frequency, ω, in order tobe consistent with the majority of the literature considered in this thesis. As such, the linewidthis labelled as δλ, and the Q is well approximated by Qm = δλm/λm, so long as δλm << λm, whichis true for the microcavities studied here.One important difference between the microcavity and Fabry-Perot spectra is that the formercontains only a single mode of interest. While multiple modes can be supported by these micro-cavities, the free spectral range is enormous compared to typical macroscopic resonator cavities,owing to the short microcavity ”length” that is on the order of λ. As a result, one typically workswith a microcavity mode with mode number m = 1, instead of m ∼ 105 for macroscopic cavities.Given the low order of the modes considered for microcavities, it is common practice to quote theQ, instead of the finesse F . When m = 1, Q serves as a good estimate of the cavity enhancement,where max(Icav) ' (2F/pi)Iin ' QIin.332.1. Transmission in linear regimeThe large free-spectral range of this single wavelength-scale microcavity makes this resonatorstructure a poor candidate of four-wave mixing processes, which involve frequency mixing betweenlight simultaneously in three resonant modes separated by a tiny fraction of the centre frequency.Ideally, all three modes are close to λ = 1545 nm, nearly equally spaced in wavelength, and withinthe laser tuning range. This can be achieved using these low-order microcavity modes by includingthree such cavities in close proximity, as shown in Fig. 2.1. In this structure, three nearly identicalcavities are fabricated in close proximity to form a coupled-resonator system. The coupled cavitysystem modes in Fig. 2.1 are a result of degenerate mode splitting. Transmission spectra of coupled-cavity devices show peaks associated with each of the three modes, with an example shown in Fig.2.1(e). Each mode has a distinct set of λm, δλ and Tlinm , as determined by the various rates atwhich energy in the three distinct modes dissipates to the environment.2.1.3 Technical considerationsTransmission set-upThe transmission set-up used to measure the triple cavity devices is illustrated in Fig. 2.5. Lightfrom a Venturi TLB 6600 Swept-Wavelength Tunable Laser is coupled into a single mode fibre(mode-field diameter 10.4 µm), and light exiting the fibre passes through a yˆ-polarized linearpolarizer (consistent with transverse electric excitation of the grating coupler) then is focused onthe silicon chip surface using a pair of aspherical lenses (Thorlabs AL2550-C: focal length 50 mm,diameter 25 mm, numerical aperture 0.230). The focusing optics are arranged to produce one-to-one imaging, such that the spot on the sample (chip) is approximately 10 µm (verified usinga knife edge measurement [10]). Light is collected off the chip surface using an elliptical mirror(numerical aperture ∼ 0.06) placed 15 cm away from the sample that focuses light scattering fromthe chip surface at 1.5 m from the mirror, resulting in a 10× magnification factor. The collectedlight is reflected off Mirror 1, and is then sent through a linear polarizer (also yˆ-polarized) and aniris before it’s incident the Newport 818-G photodetector. Mirror 1 is adjusted to direct only thelight from output grating coupler through the iris. While the elliptical mirror remains fixed, boththe excitation optics (lenses and input polarizer) and the sample are mounted on x, y, z translation342.1. Transmission in linear regimeFigure 2.5: Transmission set-up. Light from a tunable laser is coupled into an optical fibre, and the out-coupled light passes through a polarizer (yˆ-polarized) then is focused by two aspherical lenses onto the inputgrating coupler of a device on the sample (see Fig. 2.3), at an angle θ. Light leaving the output gratingcoupler is collected by an elliptical mirror (also at angle θ) that directs light through the output polarizeralso yˆ-polarized) and focuses it in a plane of an iris, and only the output grating light is passed onto aphotodetector. Alternatively, the iris is left open and the sample is imaged on the CCD Camera.352.1. Transmission in linear regimestages to facilitate alignment. They are also mounted on separate rotation stages, making it possibleto send/collect light over a range of angles, θ. This angular flexibility is important for gratingcoupling, as discussed in Section 3.2.2. In the initial alignment stages, the photodetector is replacedby an Electrophysics CCD camera (model 7290), such that the sample image is viewed.Provided that each mode is spaced at least a couple linewidths away from neighbouring modes,it is possible to identify the three main spectral features distinct for each mode. The measurementapparatus and device components are typically optimized to work within a range of wavelengths,so ideally the modes also are spaced close enough together such that they all can be probed usingonly one measurement configuration.Transmission normalizationOne of the key pieces of information extracted from the relative transmission spectra is the abso-lute value at resonance. This quantity requires knowledge of the power in the input and outputwaveguides directly before and after the microcavity, respectively, which are not directly accessible.In typical transmission measurements, power measurements occur outside of the device, either infree-space or using optical fibers. There are often a number of optical elements between wherethe input/output powers are measured and the input/output waveguides. Some of these elementsbelong to the optical set-up (e.g. mirrors, lenses, polarizers), others are in the device itself (grat-ing couplers, waveguides, impedance matching waveguide couplers that interface PC and channelwaveguides, etc), as shown in Fig. 2.5. These components cause losses in the system that need tobe accounted for to properly determine the power at the input/output PC waveguides. Some ofthese components also have spectral dependences that need to be carefully measured, and includedin the normalization of the transmission spectrum.It is possible to characterize the optical components that lie outside of the device on an individualbasis. It is typically very difficult or impossible to characterize the intermediate components withinthe actual device of interest, so it is common practice to use reference devices, that contain isolated(or grouped) identical components that are studied separately. Of course, this method is onlysuitable so long as fabrication inconsistencies are minimal, and the reference components representthe actual device components well. This is generally true for the devices studied in this thesis,362.2. Nonlinear transmissionwith the exception of one issue. Fabry-Perot interference between different circuit elements leadsto parasitic sinusoidal modulations of the transmitted light intensity, which are phase-shifted fromdevice to device. To deal with this issue, two unknown phase shift parameters, φin and φout, enterthe least-squares analysis and are extracted as fit parameters, for each device. The transmissionnormalization procedure is described in greater detail in Appendix D.2.2 Nonlinear transmission1545.2 1545.4 1545.6 1545.8Wavelength (nm)00.51Relative Transmission1545.2 1545.4 1545.6Wavelength (nm)00.51Relative Transmission(a)Bistable1545.2 1545.4 1545.6Wavelength (nm)050100150Absolute Transmission (µW)1545.2 1545.4 1545.6Wavelength (nm)00.51Relative Transmission(b)(c) (d)Forward sweep Forward sweepBackward sweepFigure 2.6: Nonlinear transmission spectra measured for the center mode of a photonic crystal (PC) triplecavity, where the transmission is relative to the input/output PC waveguides. (a) The absolute nonlineartransmission spectra measured with a forward sweep (wavelength swept from low to high) are plotted forinput powers ranging from 16 to 250 µW . The arrow indicates the minimum threshold power, Pth, wherethe sharp drop in the transmission spectrum begins to appear. (b) Same as in (a) but for the relativetransmission. (c) Same as in (b) but for a backward sweep (wavelength swept from high to low). (d)Example forward (blue) and backward (red) spectra taken at two 16 and 250 µW . The bistability is presentin the higher power spectra, as the forward and backward sweep have a range of wavelengths where thespectra are non-overlapping.Transmission spectra are measured in the nonlinear regime where the laser power is sufficientlyhigh that changes to the silicon refractive index are induced through the Kerr effect. In this regime,the transmission lineshape become distinctly non-Lorentzian, as shown in Figure 2.6 where examples372.2. Nonlinear transmissionof the absolute and relative transmission are plotted as a function of probe power. Changes inducedin the real part of the refractive index result in a shift in the resonant wavelength of the microcavitymode, and changes in the imaginary part of the index result in losses, which lower the total cavitylifetime and reduce the transmission. These effects result in complex spectral behaviour that isdependent on the sweep direction as well as the input power level, and contains information aboutwaveguide coupling, and the nonlinear properties of the material and cavity structure.The transmission spectra are also used to quantify the strength of the Kerr effect using a powerthreshold figure-of-merit. This is the minimum power where the sharp drop in the transmissionspectrum begins to appear (see the spectrum labelled the arrow in Fig. 2.6(a)), which signifiesthat the structure can be considered for all-optical processing applications like all-optical switching[31, 63].2.2.1 Nonlinear transmission lineshapeEach spectrum in Fig. 2.6 reflects the steady-state transmission for each individual wavelength,which means that each time the sweep wavelength is adjusted, there is a delay before the mea-surement is taken, to allow any transients in the temperature, free-carrier distribution and cavityenergy to pass. To understand the transmission lineshape, it is useful to first consider what ishappening to the resonant wavelength of the microcavity as a function of sweep wavelength andhow that affects the transmission. In this example microcavity, it is assumed that the resonantwavelength shift is dominated by thermal effects, resulting in a red-shift as a function of energyloaded in the cavity. Figures 2.7(a) and (b) show the energy loaded in the microcavity and resonantwavelength as a function of the sweep wavelength, respectively.For the forward wavelength sweep shown here, the laser wavelength starts blue-detuned offresonance, such that little energy is loaded into the cavity and the resonant wavelength is essentiallyequal to the cold cavity (linear) λm. As the wavelength is increased toward λm, more energy is loadedinto the cavity and the resonant wavelength begins red-shift. As the sweep wavelength increases,more energy continues to be loaded into the cavity and the resonant wavelength continues to red-shift, until the sweep wavelength eventually catches the resonant wavelength. Just beyond thispoint is where the steady-state transmission undergoes a sudden drop due to an unstable cycle:382.2. Nonlinear transmission1545.4 1545.5 1545.6 1545.7Wavelength (nm)00.20.40.60.81Energy (J)×10 -141545.4 1545.5 1545.6 1545.7Wavelength (nm)1545.41545.51545.61545.7Resonant Wavelength (nm)0 0.5 1U1 (J)00.51U2 (J)10-1400.5110-14x0 0.5 1U1 (J)00.51U2 (J)00.51x 10-14x10-14x(a) (b)(c) (d)Figure 2.7: Nonlinear behaviour a triple microcavity under (a)-(b) single frequency excitation, and (c)-(d)dual frequency excitation when cross-coupling of nonlinear effects between modes is present. (a) Energyloaded in mode M2 as a function of sweep wavelength for an input power of Pin = 300 µW. (b) Resonantwavelength of the cavity mode as a function of sweep wavelength (solid line). The dashed line shows thesweep wavelength, for reference. (c) Nonlinear resonant wavelength shift of mode M1, ∆λNL1 , as a functionof the energies U1 and U2 loaded in modes M1 and M2, respectively, for the same microcavity device in (a)and (b). The color scale is in units of nanometer. (d) Same as (c) but the resonant wavelength shift of M2,∆λNL2 , is plotted.the microcavity energy decreases due to the excitation being off-resonance, causing the resonantwavelength to blue-shift as the heating reduces, such that the excitation is increasingly furtheroff-resonance. Beyond the drop, the relative transmission follows the low power spectral lineshape.The transmission lineshape for the backward wavelength sweep (laser starts red-detuned fromλm and ends blue-detuned) is different from the forward sweep and is illustrated in Fig. 2.6(c).For the backward sweep, as the wavelength is swept toward λm, energy is loaded into the cavityand the resonant wavelength is red-shifted toward the sweep wavelength. The sharp increase in thesteady-state transmission occurs when an unstable cycle is initiated: the cavity energy is increasing,causing the resonant wavelength to approach the sweep wavelength, causing more energy to be392.2. Nonlinear transmissionloaded. Beyond this point, the wavelength is tuned increasingly off resonance and the energy in thecavity reduced. The wavelength where the transmission jumps is generally not at the same pointwhere the power drops in a forward sweep 2.6(d), due to how energy gets loaded as a function ofwavelength. The microcavity is in a bistable state for the range of wavelengths where the forwardand backward sweeps yield different transmission, as labelled in Fig. 2.6(d).Nonlinear losses also play a role in defining the nonlinear transmission lineshape. For bothforward and backward sweeps, the transmission is reduced due to free-carrier nonlinear losses whenthe energy loaded in the cavity is high, as seen in Figs. 2.6(b) and (c), respectively. This results ina decrease in the maximum relative transmission as a function of sweep power.2.2.2 Nonlinear transmission data analysisAll of this detailed spectral information obtained at a number of input power settings (see Fig. 2.6)can be reduced, for the purposes of modelling and parameter extraction, to two data sets: the peaktransmission, TNLm , and peak resonant wavelength shift, ∆λNLm , versus input power, both obtainedin the forward sweep direction, like the example shown in Fig. 2.8. These data sets are obtainedfor the two high Q modes, M1 and M2, while the low Q mode (M3) is not studied, as insufficientenergy is loaded into the cavity to induce significant nonlinear effects, even when the laser is at itsmaximum power.0 200 400Input Power ( W)00.51Transmission0 200 400Input Power (µW)00.050.10.150.2λ(nm)NLµM2M1M2M1Figure 2.8: Example experimental data extracted from nonlinear transmission spectra taken as a functionof input power for Modes M1 and M2. (a) Peak transmission. (b) Resonant wavelength shift.The model used to predict the wavelength shifts and transmission maximum values for M1 andM2 includes 11 unknown microcavity parameters: λ1, λ2, τin1 , τin2 , τout1 , τout2 , τscatt1 , τscatt2 , τabs, τcarrier,402.2. Nonlinear transmissionand Rth. In the least-squares approach taken in this work, there are six fit parameters that aredirectly extracted (ηwg1 = τout1 /τin1 , τabs, τcarrier, Rth, φin, φout), while the rest of the parameters areinformed from the linear transmission and four-wave mixing analysis results.2.2.3 Technical consideration: time scalesTransmission spectra considered in this thesis are composed of sequences of steady-state measure-ments. This requires that all transient effects have passed when each step measurement is taken.There are a number of different processes at play during nonlinear transmission measurements, eachwith their own time scales. The time scales most relevant to the device operation are: the effectivefree-carrier lifetime, the effective thermal relaxation lifetime and the optical cavity lifetime.The effective free-carrier lifetime, τcarrier, depends on a number of different factors, includingthe carrier density and the microcavity structure. Single microcavities comparable to the thoseinvolved in the triple cavity have been found to have saturated effective carrier lifetimes around0.5 − 1.8 ns [11, 95, 105], and unsaturated lifetimes on the order of 10’s of nanoseconds[11]. Thesaturated lifetime is expected to be limited by surface recombination effects [89, 95]. The thermalrelaxation rate is related to the thermal resistance. A higher thermal resistance results in slowerdiffusion of heat from the microcavity. The time scale for single photonic crystal microcavitieshas been found to be around 100 ns [63]. The time scale for light loading and unloading from themicrocavity depends on the total lifetime of the mode. For the triple cavity, the total lifetimes aredifferent for each of the modes. The lowest Q modes have lifetimes ∼ 3 ps (Q ∼ 2000), while thehighest Q modes have lifetimes ∼ 300 ps, (Q ∼ 2× 105).The overall time scale required for the microcavity to reach steady-state is therefore limitedby the thermal relaxation, to ∼ 100 ns. In nonlinear transmission measurements, a delay ofmicroseconds between the wavelength step and the measurement is sufficient for ensuring that thesystem has reached steady-state. A relatively large delay of 50 ms was employed in the experimentsin this thesis, to safely allow time for the necessary electronic communications to occur, and forthe laser to change the wavelength.412.3. Stimulated four-wave mixing2.3 Stimulated four-wave mixingIn four-wave mixing frequency conversion measurements, the microcavity is excited by pump andsignal lasers, at wavelengths λpump and λsignal, respectively, and photons are generated at the idlerwavelength λidler, as illustrated in the simplified schematic in Fig. 2.9(a). The full measurementset-up is described in Section 2.3.1. The microcavity idler photons are coupled into the output PCwaveguide, along with unconverted signal and pump photons, and are sent off-chip where spectralfilters are used to transmit only the idler photons to a single photon detector. The idler, pump andsignal microcavity photons also couple to the other possible waveguide and loss channels.Conservation of energy requires that the signal and idler frequencies are equally spaced fromthe pump frequency. For photons with wavelengths near 1550 nm, the wavelength spacing is alsoessentially equal [(λpump − λsignal/idler) << λpump]. Ideally the pump, signal and idler wavelengthscoincide with resonant wavelengths of the microcavity. The triple PC cavities considered in thisthesis are generally not perfectly equally spaced, and support two relatively high Q neighbouringmodes (Q >∼ 30, 000), and a low Q mode (Q ∼ 3000), as in the example given in Fig. 2.9(b). Forthe devices measured in this thesis, the pump wavelength coincides with the central high Q mode,and one of the signal/idler wavelengths coincides with the outer high Q mode, while the thirdwavelength (idler/signal) falls within a linewidth of the low Q mode peak. The details regardinghow the laser and filter wavelengths are chosen for optimal generation rates are discussed morethoroughly in Appendix A.The idler power is generally reported with respect to the output waveguide, and is found bydividing the photon count rate measured by the single photon detector by the losses between theoutput waveguide and the detector, and multiplying by the energy of an idler photon, ~ωidler. Thiscalculation includes the same optical and device components considered in the relative transmissionnormalization, as well as additional sources of loss like the spectral filter and the detector efficiency.The normalization procedure is described in Appendix D. The idler steady-state photon generationrate is measured as a function of the pump or signal power, while the other input power is keptfixed. This is done for both signal/idler mode excitation configurations.When the pump and signal powers are relatively low such that nonlinear losses are negligible,422.3. Stimulated four-wave mixing1542 1543 1544 1545 1546 1547 1548 1549Wavelength (nm)00.51Relative Transmission(a)(b)Pump laserOUTPUTINPUTSingle photon detectorTunable spectral lterSignal laserCouplerM1 M2 M3Figure 2.9: (a) Simplified schematic of the stimulated four-wave mixing (FWM) set-up. Pump and signallasers tuned to wavelengths λp and λs are coupled into the input waveguide of the multimode microcavityand the light collected is sent through a spectral filter, to extract the idler photons at wavelength λi, beforemeasurement by a single photon detector. The green and blue arrows represent path the pump and signalcontinuous wave excitations, respectively, and the red arrows represent the path of the idler photons generatedthrough FWM. (b) Transmission spectrum for a triple photonic crystal microcavity. The three frequenciesinvolved in FWM are indicated with dashed lines, and the three microcavity modes are labelled M1, M2 andM3.the idler power depends quadratically on the pump power, and linearly on the signal power, asis illustrated in Fig. 2.10(a). These dependencies are somewhat intuitive, as the FWM processinvolves two pump photons and one signal photon that get converted to one idler photon.When the pump and/or signal powers are sufficiently high, nonlinear absorption losses affectall three modes, and cause reductions in the idler power. Free-carrier effects reduce the totalmicrocavity cavity lifetimes of all three modes, resulting in a reduction in the amount of energythat can be optimally loaded into the pump and signal modes, and absorption of the generatedphotons. The nonlinear absorption effects also cause the resonant wavelengths to shift due tochanges in the real part of the refractive index. It is common practice to adjust the pump, signaland idler wavelengths as a function of pump/signal power to track the mode peaks and maintain432.3. Stimulated four-wave mixing10 1 10 2Input Power ( µW)10 -610 -410 -2Idler Power (µW)10 0 10 1 10 2Input Power (µW)10 -810 -6Idler Power (µW)(a) (b) LegendIdler Mode 3, pump sweep, exp.Model without NL absorptionIdler Mode 3, signal sweep, exp.Idler Mode 1, pump sweep, exp.Idler Mode 1, signal sweep, exp.Figure 2.10: Examples of stimulated four-wave mixing (FWM) idler powers experimentally measured fortriple photonic crystal microcavity devices. The red and blue markers indicate that the FWM configurationimplemented resulted in idler photons generated near Mode 3 and Mode 1, respectively. Circles and trianglesindicate that the power measured as a function of signal and pump power, respectively. The solid lines areestimates of what the idler powers are predicted to be in the absence of nonlinear losses (linear/quadraticfor signal/pump power dependencies). (a) Stimulated FWM results for a triple cavity probed largely inthe low power limit. The signal power sweep has a fixed pump power of 29 µW , and the pump powersweep has a fixed signal power = 3.6 µW . Nonlinear loss effects result in sub-linear and sub-quadratic powerdependencies near the ends of the sweeps. (b) A different triple cavity device probed largely in the highpower limit. The signal power sweep has a fixed pump power of 44 µW , and the pump power sweep has afixed signal power = 44 µW .equal spacing. The modes typically shift very close to the same amount as they have similar modefield distributions relative to silicon and the surrounding materials (air, oxide).Nonlinear losses cause the FWM idler power pump dependence to become sub-quadratic andthe signal dependence to become sub-linear, as shown in Fig. 2.10(b). A subtle but importantconsequence is that at low pump or signal sweep powers, the generation rate does not necessarilyreturn to the nonlinear loss-free quadratic or linear trend, respectively, as the trends may be shiftedto overall lower powers due to nonlinear losses induced by the other (fixed) input excitation.The least-squares analysis for each device involves four FWM data sets: the idler powers mea-sured separately as a function of pump and signal powers, for both excitation configurations (sig-nal/pump/idler modes as M1/M2/M3 and M3/M2/M1). In the presence of nonlinear losses, themodel function used to predict the idler powers depends on 14 microcavity parameters, as indicatedin Table 2.1. There are only two fit parameters for this analysis, ηwg2 = τout2 /τin2 and ηwg3 = τout3 /τin3 ,and the rest of the model parameters are informed from the linear and nonlinear transmission mea-surements.442.3. Stimulated four-wave mixing2.3.1 Technical consideration: four-wave mixing set-upFigure 2.11: Optical set-up used to measure four-wave mixing. Light from the pump and signal lasers arecoupled into a single mode fibre and focused on the input grating coupler of a triple microcavity device.Light that leaves the sample surface is collected and focused by an elliptical mirror, and an iris in the imageplane is used to transmit only the light leaving the output grating. A lens is then used to focus the light intoan optical fibre, that is connected to two JDSU TB9 Tunable Grating Filter, tuned to the idler wavelength,such that idler photons are detected by the id210 avalanche photodetector. See Fig. 2.3 for more details onunlabelled components.The transmission set-up for four-wave mixing measurements is shown in Fig. 2.11. The basecomponents in this set-up are the same as the ones in the transmission set-up, shown in Fig. 2.5,however there are modifications to the excitation and detection schemes. Two lasers, a VenturiTLB 6600-H-CL and a Venturi TLB 6600-L-CL are included in this set-up to provide the pumpand signal excitations. The Venturi TLB 6600-H-CL is a higher power laser with a noise level of−43 dB below the continuous-wave power, due to spontaneously emitted photons. A JDSU TB9Tunable Grating Filter is placed along the optical path of this laser to reduce the noise level to∼ −90 dB. This noise reduction is necessary to maximize the signal to noise ratio obtained whenmeasuring the photon power generated by FWM. The noise level of the Venturi TLB 6600-L-CLlower power laser is ∼ −80 dB, so no filtering is required. A 50/50 coupler is used to couple the452.3. Stimulated four-wave mixinglight from both lasers into a single optical fibre.The detection scheme is modified such that in lieu of putting a photodetector or CCD camerain the image plane, the output light propagates to a lens where it is focused into a single modefibre. The fibre is connected to two tunable JDSU filters that are used to filter out the pump andsignal photons, and transmit the idler photons, by providing a total of ∼ 100 dB rejection at ∼ 2.5nm away from the center wavelength, as reported in Appendix D. The idler photons are then sentto a ID Quantique id210 single photon detector, which is an avalanche photodetector (APD).Stimulated four-wave mixing measurements are critically dependent on the pump and signalwavelengths, as well as the idler filter wavelength. Special care is taken to choose the wavelengthsthat optimize the generated photon rate. This presents an added challenge when pump and/orsignal excitations introduce nonlinear absorption effects. In this regime, resonant wavelengths shiftas a function of both the pump and signal powers. This is exemplified in Figs. 2.7(c) and (d)where the nonlinear resonant wavelength shifts of modes M1 and M2, respectively, are plotted asa function of the energies simultaneously loaded into these modes. In Appendix A, the proceduresused to align the pump, signal and idler wavelengths are discussed for the triple cavity, wherethe modes are unequally spaced and have different Q’s. Different strategies are used for optimalalignment, depending on whether the signal is exciting the high Q outer mode, or the low Q mode.46Chapter 3Design and fabricationIn this chapter, the triple microcavity structure central to this thesis is described in detail. Theoriginal design goal was to engineer a resonator structure suitable for spontaneous FWM in silicon-on-insulator, that supports three high Q modes that are spatially overlapping, and equally spacedin frequency. Ideally, the center mode couples preferentially to the input waveguide and the outermodes preferentially couple to the output waveguide, to enable efficient loading of the pump photonsand efficient unloading of the signal and idler photons spontaneously generated.The full microcavity structure for spontaneous FWM, that includes three side-coupled het-erostructure cavities, and input and output waveguides, is plotted in Fig. 3.1(a) and a scanningelectron microscope (SEM) image of a fabricated structure is shown in Fig. 3.1(b). The microcavitystructure is defined in an SOI wafer with a silicon device layer thickness of tSi = 220 nm and aburied oxide thickness of tSiO2 = 3 µm, however after fabrication the oxide beneath the microcavityis removed, to leave the silicon device layer suspended (also called “undercut”). The optimizedstructure is defined by 19 design parameters, listed in Table 3.1.The quality factors numerically simulated for the optimized design structure are reported inTable 3.2, for each of the three resonant modes of interest. Also reported here are the probabilitiesthat a photon generated in a microcavity mode couples to the output waveguide, and the peakrelative transmission values.3.1 Numerical simulationsNumerical simulations play a very important role in both the design and analysis procedures. Dueto the complexity of the dielectric structures considered, it is not practically possible to studythe electromagnetic responses analytically, and numerical approaches are required. The numerical473.1. Numerical simulationsOUTPUTINPUTIII IIIII I(a) (b)Figure 3.1: Triple photonic crystal (PC) microcavity with input and output waveguides. (a) Schematic ofthe microcavity with design parameters highlighted. The heterostructure is composed of photonic crystalregions labelled I, II, and III, and separated by red lines, where the lattice spacings in the x direction area1, a2 and a3, respectively, while the row spacing of the bulk PCs is√3a1/2 throughout. The widths of thethree line defects are wwg. The solid black circles represent holes with radius r, while the circles coloured ingray have radius rmid. The input and output waveguides are labelled, and the holes lining the waveguidesare coloured cyan and green, and have radii rinwg and routwg , respectively. The 12 holes that finely control themode spacing are coloured purple, and have radius rmid,ms and are shifted outward from the line defect bya distance hms. The four yellow holes have radius rwg,sym. (b) Scanning electron microscope image of afabricated triple microcavity device.approach that is primarily used in this thesis is the finite-difference time-domain (FDTD) method,which effectively evaluates Maxwell’s equations on a spatial discretized grid, at discrete time steps.The method is described in greater detail in Appendix B. All FDTD simulations are done usingLumerical FDTD Solutions software [79].3.1.1 Microcavity simulationsSimulation set-up considerationsA typical microcavity simulation studied in this thesis shown in Fig. 3.2(a). This simulation isused to extract the mode frequencies, Q’s and mode profiles of the triple cavity shown in Fig.3.1. The approach taken here is generally applicable to 2D planar PC microcavities. The silicondevice layer is shown in red and the etched holes defining the PC microcavity are gray. Theorange box shows the boundaries of the simulations volume. The blue double arrow shows a dipolesource, polarized in the yˆ direction. This is a passive “soft” source that introduces a polarization483.1. Numerical simulationsTable 3.1: Design parameters for the triple photonic crystal microcavity device coupled to input and outputwaveguides.Parameters Value DescriptionPC microcavitya1 410 nm Lattice spacing for Region Ia2 418 nm Region II lateral lattice spacing fora3 425 nm Region III lateral lattice spacing forwwg 693 nm Waveguide widthNy,cav 3 Inter-cavity separation (# of rows)r 124 nm Radius of holes in top and bottom PCsrmid 126 nm Radius of holes in middle two PCsrmid,ms variable Radius of holes that finely tune mode spacinghms variable Shift of holes that finely tune mode spacingPC waveguiderwg,sym 115 nm Hole radiusrinwg 111 nm Radius of holes lining input PC waveguideroutwg 108 nm Radius of holes lining output PC waveguideX inwg 7 Starting X position of input waveguideY inwg 4 Starting Y position of input waveguideXoutwg 7 Starting X position of input waveguidevin 14 nm Input PC to channel waveguide impedancematching parametervout 14 nm Output PC to channel waveguide impedancematching parametertin 341 nm Input PC to channel waveguide impedancematching parametertout 346 nm Output PC to channel waveguide impedancematching parameterhin 146 nm Input PC to channel waveguide impedancematching parameterhout 166 nm Output PC to channel waveguide impedancematching parameterdensity pulse to one rectangle of the spatial grid, and allows scattered fields to pass through thesource unperturbed. The source pulse length is typically ∼ 9 fs, resulting in a spectral bandwidthδf = 50 THz that is centered near the cavity resonant mode frequencies, such that these modes493.1. Numerical simulationsTable 3.2: Summary of the quality factors simulated for triple photonic crystal microcavity devices with inputand output waveguides. The quality factors Qinm, Qoutm , Qotherm correspond to coupling to the input waveguide,the output waveguide, and other loss channels including scattering absorption, respectively, while Qm is thetotal quality factor, for modes m = 1, 2, 3. The probability that photons generated in the cavity will coupleto the output waveguide is given by poutm = Qm/Qoutm . The relative transmission from the input to the outputwaveguide is given by T = 4Q2m/(QinmQoutm ). The total quality factors for the microcavity structure withoutinput and output waveguides, Qnowgm , are also included.Mode, mParameter Description 1 2 3Qm Total quality factor 2.8× 104 1.3× 105 3.7× 103Qinm Input coupling quality factor 1.5× 106 1.5× 105 1.4× 105Qoutm Output coupling quality factor 3.2× 104 2.6× 106 4.1× 103Qotherm Scattering/other losses quality factor 3.2× 105 9.7× 105 4.5× 104Qnowgm Total quality factor in absence of waveguides 2.2× 106 1.4× 106 8.1× 105poutm Probability of output waveguide coupling 0.89 0.89Tm Relative transmission 0.065 0.16 0.096are transiently excited, and then gradually leak out of the cavity region after the excitation pulsehas completely disappeared from the simulation volume. The yellow cross is a time monitor thatrecords the fields at the nearest mesh cell as a function of time. A number of planar 2D frequencymonitors are also included (outlined in yellow). Two of these monitors lie at the ends of the PCwaveguides, and six form a box around the simulation volume. These record the fields at each ofthe mesh points within the monitor, in the frequency domain, by implementing a numerical Fouriertransform. The power transmission through the monitors is calculated based on the fields. It isalso possible to include a 3D frequency monitor that records the fields over a volume (not shown).There are a number of considerations that are involved in setting up the simulation. The sim-ulation boundaries are chosen to be “perfectly matched layers” (PML), which absorb all outgoingradiation. In the zˆ direction, a symmetric boundary condition is applied to the “undercut” sus-pended microcavity (indicated by the blue shading in the x− z view), that effectively reduces thesimulation volume and run-time by half. The symmetry is known a priori to be consistent withthe symmetry of the modes of interest (here the base silicon layer of the SOI is ignored). It isimportant that the hole pitch is an integer multiple of the mesh size to maintain the periodicity503.1. Numerical simulations195.5 196 196.5Frequency (THz)00.51|Ey|20 10 20 30Time (ps)-3-2.5-2(|Ey|2)(b)(c)xzln(a)2D monitor(input WG)2D monitor(output WG)Time monitorSourceBox of2D monitorsxyI III IIIIIPC RegionFigure 3.2: Finite-difference time-domain (FDTD) microcavity simulations using Lumerical FDTD [79]. (a)The simulation layout of the triple microcavity device is shown in the x−y (top) and x− z (bottom) planes.The thick orange regions outline the simulation volume and the perfectly matched layers (PML). The blueregion in the bottom plots show the symmetric boundary condition applied in zˆ. The thin orange lines inthe top image show the boundaries of the heterostructure mesh override regions. A source polarized in yˆ isplaced at an antinode in the top defect region to excite the modes. A time monitor (yellow cross) is placedat the location of an antinode in the bottom defect. Two-dimensional planar frequency monitors are placedat the outputs of the two waveguides (WGs) to measure waveguide transmission (yellow lines in the topimage) and a box of 2D monitors encloses the full simulation power to measure all power leaving the volume.(b) Fourier transform of the Ey field measured by the time monitor (blue), apodized to remove the sourceexcitation. A Gaussian spectral filter around the center mode is shown as a dashed red line. (c) Naturallogarithm of Ey decay envelope for center mode, found from the inverse Fourier transmission of the filterspectrum. The dashed red lines is the line of best fit.of the PC. For the “heterostructure” cavity studied here, which has five regions with differentPC lattice spacings (labelled with I,II, and III in Fig. 3.1(a), and discussed in greater detail inSection 3.2), five mesh override regions are appropriately defined. The mesh override regions areoutlined in orange in Fig. 3.2(a), and have ∆x = ax/15, ∆y = a1√3/30 and ∆z = tSi/20 = 110nm, where ax is the horizontal lattice spacing of the heterostructure region, and a1 is the latticespacing in Region I PC. The dipole source and time monitor are also deliberately located at twoantinodes of all three cavity modes of interest and the dipole polarization direction (yˆ) is consistentwith the field direction at the source antinode. These decisions are informed by knowledge fromprevious simulations involving randomly placed monitors and dipoles, the latter with randomized513.1. Numerical simulationspolarization directions. The simulation time is 35 ps, which is not long enough to fully resolvemodal spectral properties, due to the high Q’s of the modes, but it is sufficiently long to accuratelycapture the information of interest.Analysis of simulation resultsModal frequencies and lifetimes The mode frequencies appear as peaks in the spectrum foundby taking the Fast Fourier Transform (FFT) of the Ey field recorded by the time monitor. Thelinewidths of the peaks in the FFT spectrum are not resolved due to the short simulation time,resulting in the sinc functions plotted in Fig. 3.2(b). The quality factors (and actual linewidths)are extracted from the decay characteristics of the time monitor signal. After the excitation haspassed, the cavity has the following time dependence,E(t) = A sin(−i2pifmt) exp(−t/τm) (3.1)where fm is the mode frequency and τm is the total lifetime of the mode. The total quality factor,Qm = pifmτm, is extracted using a multi-step process: i) a Gaussian filter is applied around the“positive frequency peak” of the mode of interest in the FFT data, as illustrated in Fig. 3.2(b),ii) the inverse FFT of the filtered signal is computed, iii) the absolute value is taken to remove thee−i2pifmt carrier signal time dependence, leaving the envelope of the mode time decay, clear of thefast carrier oscillations and the responses of other modes, iv) the natural logarithm is taken and isfit with a line, as shown in Fig. 3.2(c), where the slope is −1/τm, and the Q is computed. Thisprocess is repeated for each of the three modes of interest.The two-dimensional planar frequency monitors at the ends of the input and output PC waveg-uides measure the power transmitted through the monitor, Pin(λ) and Pout(λ), respectively, and thebox of monitors around the whole simulation volume monitor the total power leaving the volume,Pbox(λ). The individual quality factors for the PC waveguides are found by studying the relativepower transmitted on resonance through these monitors, where Qim = QmPboxm /Pim, with i = “in”,“out” and “other”, and P otherm = Pboxm − P inm − P outm . This relationship between the transmittedpower and the quality factors is found by considering that the energy in the cavity mode m decays523.1. Numerical simulationsas Um ∼ exp(−2t/τm), such that the power lost to each channel is P im ∼ (−2/τ im) exp(−2t/τ im),resulting in P im(fm) ∝ 1/τ im ∝ 1/Qim.Modal field distributions To find the mode field distributions in the centre of the silicon slab,Em(x, y, z = 0; fm), a 2D planar frequency monitor is placed in the x−y plane at z = 0 (not shownin Fig. 3.2(a)), and the recorded fields are evaluated at the mode frequencies, fm. The time signalthat enters the FFT begins after the source excitation has passed, such that only the resonant fieldis involved in the calculation.For some calculations, it is necessary to extract the mode field distributions over the microcav-ity volume, Em(r; fm), in which case a 3D frequency monitor is implemented. For example, 3Dfrequency monitors are used to find the four-wave mixing coefficient βFWM, introduced in Chapter2, based on volume overlap integrals involving the three mode field distributions and the silicon.The 3D modal field distributions are also used in a perturbative approach to study changes inthe mode frequencies due to small perturbations in the dielectric environment, by calculating [39],δfmfm= −12∫drδ(r)|Em(r)|2∫dr(r)|Em(r)|2 , (3.2)where (r) is the original permittivity, Em(r) is the original mode profile for mode m, and δ(r) =new(r)− (r), with new(r) as the new permittivity. While there are some known issues with thisapproach [39], it appears to give sufficiently accurate results for the cases tested here, and is usedto quickly test microcavity geometries.3.1.2 Bandstructure simulationsNumerical FDTD simulations are also used to extract the mode bandstructures for periodic dielec-tric structures, which is useful for identifying the photonic band gap of the host 2D PCs and thewaveguide modes supported by PC waveguides. A typical simulation for a PC waveguide is shownas an example in Fig. 3.3(a). The simulation span in x is equal to the lateral pitch ax, while thespan in y encompasses 7 rows on both sides of the waveguide. “Bloch” boundary conditions areapplied for the x boundaries, while PML is applied for the y and z boundaries, in addition to a533.1. Numerical simulationsxyxz(a)0 0.1 0.2 0.3 0.4 0.5kx(2 /a)050100150200250Frequency (THz)01(b)Figure 3.3: Bandstructure simulations using Lumerical FDTD Solutions [79]. (a) The simulation layout forphotonic crystal line defect is shown in the x − y (top) and x − z (bottom) planes. The x − y plane nearthe center axis is expanded in the image to the right. The simulation volume spans one lattice spacingin xˆ, ax and is outlined in orange. Symmetric boundary conditions are applied in zˆ, as indicated by thecoloured blue region in the bottom image, and Bloch boundary conditions are applied in xˆ. Dipole sources(blue double arrows) are randomly placed to excite the Bloch mode. Time monitors (yellow crosses) are alsorandomly placed to measure the field excited in the silicon. (b) Example of the bandstructure measured fora line defect. The base 10 logarithm of the intensity spectrum extracted from simulations with fixed kx areplotted. The light line is shown with the white line.symmetric boundary condition applied in z to study the transverse electric modes of interest only.The Bloch boundaries apply the following condition:Fkx(x+ ax, y, z) = exp(ikxax)Fkx(x, y, z), (3.3)where Fkx(x + ax, y, z) is the electric or magnetic field for a mode labelled by kx, where kx isthe wavevector in the xˆ direction set for the simulation. A cluster of broadband dipole sourcesoriented in the plane are used to excite the modes and time monitors record the fields at variouslocations. The Fourier transforms of the time signals are computed after apodizing the signal toremove the source pulse, and their squared magnitudes are summed together. The spectra areplotted in Fig.3.3(b), on a logarithmic (base 10) color scale, as a function of kx, from 0 to pi/ax.The air light line, ω = ckx, where c is the speed of light in vacuum, is also included as the whiteline. The mode frequencies correspond to the peaks in the spectra, and these shift as a functionof kx, revealing the mode dispersion. The mode shapes associated with different bands can beextracted by including a 2D planar frequency monitor.543.2. Device design3.2 Device design3.2.1 Microcavity design(a) III IIIII I-4 -2 0 2 4x ( µm)-202y (µm)01(b) III IIIII Ixy(c)0 0.1 0.2 0.3 0.4 0.5kx(2 /a)180190200210220Frequency (THz)Region I bandsRegion II bandsRegion III bandsBulk PC  continuum of bandsFigure 3.4: Single photonic crystal (PC) heterostructure microcavity. (a) Schematic of the structure, wherethe different PC regions are highlighted in different shades of gray and labelled I, II and III. The horizon-tal lattice spacings are exaggerated for clarity. (b) Electric field intensity of the mode profile. The PCheterostructure boundaries are highlighted with yellow lines and the PC holes are outlined in white. Thehorizontal lattice spacings are a1 = 410 nm, a2 = 418 nm and a3 = 425 nm, respectively. (c) Bandstructurecalculation for the PC line defects (missing row of holes) in Regions I, II and III. The wavevector on the xaxis is given in terms of the horizontal lattice spacing a for each region considered (i.e. a = a1, a2, a3). Thesolid lines indicate the PC waveguide bands. The continuum of bulk PC bands are colored in gray, and areoutlined for each region with dotted lines. The black dashed line shows the resonant cavity mode frequency,and the light cone is shaded in yellow.The microcavity structure in Fig. 3.1(a) is composed of three nearly identical PC heterostruc-ture cavities located in close proximity (along the y direction), allowing them to couple, lifting thedegeneracy of the three modes’ frequencies. An example of a single heterostructure cavity, known tosupport high Q, small volume modes [81], is illustrated in Fig. 3.4(a). The regions highlighted I,IIand III are derived from three different hexagonal photonic crystals, with lattice constants a1 = 410553.2. Device designnm, a2 = 418 nm and a3 = 425 nm, respectively, and hole radius r = 124 nm. The modificationto the lattice is that the row separation in yˆ is held constant at ay =√3a1/2 throughout all PCregions. In Fig. 3.4(a), the spacings a2 and a3 are exaggerated for clarity. There is a line defect ofomitted holes along the center of the microcavity, with width wwg =√3× 400 nm = 693 nm. Theintensity profile of the fundamental mode of this cavity is plotted in Fig. 3.4(b).The mechanism by which this structure localizes light in region III can be understood byconsidering how the dispersion of the 1D waveguide modes supported in the line defect (missingrow of holes), at frequencies within the band gap of the bulk PC, varies between Regions I, II, andIII. As shown in Fig. 3.4(c), there is a range of frequencies where waveguide modes can propagatein Regions II and III, but not in Regions I. The Regions I therefore act as “potential barriers”,or more specifically as mirrors that define a Fabry-Perot like cavity between them. Regions II areintroduced, instead of directly interfacing Regions I and III, in order to make the transition moregentle, such that scattering is minimized and high Q’s are achieved. The separation of the “mirrors”(the effective width of Regions II and III) affects the cavity mode frequency. It is important thatall of the relevant 1D bands involved in all regions exist below the light line, to avoid intrinsicout-of-plane scattering losses.In order to achieve three equally spaced modes, three heterostructure cavities are side-coupled,as shown in Fig. 3.5, where the mode intensity profiles are plotted for the triple cavity version of thestructure presented in Fig. 3.4. Three rows of holes are chosen to separate the cavities, Ny,cav = 3,resulting in mode spacings of ∆fms ' (f3 − f1)/2 ' 0.4 THz [∆λms ' (λ3 − λ1)/2 ' 3 nm]. Incontrast, when the cavities are spaced by a single row (i.e. Ny,cav = 1), the modes are very stronglycoupled and the spacing is ∆fms ' 8 THz (∆λms ' 65 nm). With such high frequency shifts, thehigh energy mode is no longer reflected by Regions I, and the low energy mode gets pushed belowthe bulk PC band-edge of Region III: the Q values then degrade and the mode separation becomesasymmetric. For an intercavity spacing Ny,cav = 2, the x positions of omitted the holes are differentin the outer cavity line defects as compared to the center line defect, and only two distinct cavitymodes appear in the spectra.The design process to determine the microcavity parameters in Table 3.1, is outlined in Fig.3.6. The waveguide width wwg = 693 nm is chosen to appropriately shift the waveguide band away563.2. Device design(c)-4 -2 0 2 4x ( µm)-202y (µm)III IIIII I-4 -2 0 2 4x ( µm)-202y (µm)-4 -2 0 2 4x ( µm)-202y (µm)(a) (b)III IIIII I III IIIII I010101(c)M2M3M1Figure 3.5: Simulation results for a triple heterostructure photonic crystal (PC) microcavity with hole sizesr = 124 nm throughout and waveguide width wwg = 693 nm. Electric field intensity mode profiles areplotted for modes (a) M1, (b) M2 and (c) M3. The heterostructure boundaries are shown with yellow linesand the PC holes are outlined in white.from the lower band edge of the host PC while maintaining resonant frequencies within the lasertuning range (λlaser = 1520 to 1610 nm). Figure 3.7(a) shows the waveguide dispersion curves as afunction of the waveguide width, where dashed-dotted, solid and dashed blue lines correspond towaveguide modes supported when wwg = 675, 693 and 710 nm, respectively. The resonant modefrequencies f1 (blue), f2 (green) and f3 (red) are also plotted for each of the three triple microcavitystructures. As expected, the waveguide and resonant cavity modes shift to lower energies as thewidth is increased, due to an increase in the amount of the mode energy concentrated in the highindex silicon, as opposed to the low index air 3. The Q’s for modes 1,2 and 3 are plotted inFig. 3.7(b), as circles, triangles and squares as a function of wwg. The Q’s decrease when wwgis increased from 693 nm to 710 nm because the waveguide band become closer to the bulk PCband-edge resulting in greater leakage into the bulk PC. For wwg = 675 nm, the cavity modes f13This is effectively due to the variational principle for electromagnetic modes. The energy functional, Uf =(∫d3r|∇×E(r, fm)|2)/(∫d3r(r)E(r, fm)|2), is minimized for each mode [38]. Modes concentrated in high dielectricconstant regions have relatively low mode frequencies (energies).573.2. Device designFind the waveguide wdith,        , that optimizes the mode Q's Find the hole radius,        , that optimizes the resonant mode frequency spacing, such that   * For the fabrication layout consider ne tuning the mode spacing using                 and     Choose radii          and           for the holes lining the input and output PC waveguides, such that the waveguide modes are outside of the slow light regime at the microcavity resonant frequencies.  Determine  input and output waveguide-microcavity coupling geometries  that result in preferential coupling of M1 and M3 to the output waveguide, and M2 to the input waveguide, while maintaining high Qs. Figure 3.6: Process followed to design the triple photonic crystal (PC) microcavity for spontaneous four-wavemixing applications. The microcavity parameters are defined in Table 3.1 and highlighted in Fig. 3.1.and f2 are closer to the waveguide band, leading to a reduction in Q1 and Q2. The wwg = 693 nm ischosen to achieve modes with relatively high Q’s (Q1 = 2.3× 105, Q2 = 1.4× 105, Q3 = 7.2× 104 ),and frequencies within the laser tuning range. At this point, the mode frequencies are f1 = 195.72,f2 = 195.46 and f3 = 194.94 THz (λ1 = 1531.77, λ3 = 1533.75 and λ2 = 1537.83 nm), which iswithin ∼ 1.5 THz (∼ 10 nm) of the laser tuning range edge.The resonant frequencies presented above for this structure are not equally spaced: there is afrequency offset of ∆foff = f3 − fFWM3 = f1 + f3 − 2f2 = −0.26 THz, where fFWM3 = 2f2 − f1is the four-wave mixing frequency. The mode profiles in Fig. 3.5(b) show that the center modehas relatively little of its field in the center cavity region, thus if the structure is perturbed in thisregion, one would expect it to affect this mode less than the others, thus making it possible to tunethe outer mode frequencies relative to the center mode frequency. The hole radii for the center twoblocks of the structure (with three rows each), gray holes in Fig. 3.1, are tuned from rmid = 124 to128 nm, and the resulting resonant frequencies are plotted in Fig. 3.8(a). The ∆foff is plotted inFig. 3.8(b) as yellow diamonds. This tuning method is effective and rmid = 126 nm is chosen, toreduce the offset to ∆foff = 0.085 THz (∆λoff = λ1 + λ3 − 2λ2 = −0.65 nm).Fine tuning of the mode spacing is incorporated across the fabrication layout, to ensure thata subset of devices are near equally spaced. In the tuning method present above, ∆foff changes583.2. Device design0 0.1 0.2 0.3 0.4180190200210220Frequency (THz)(a)kx(2π/a)6750.511.522.5Q×10 60.5Waveguidewg(nm)(b)width, w693 710Bulk PC  continuum of bandsLegendCavity modes M1M2M3LegendWG675 nm 693 nm 710 nmM1M2M3Figure 3.7: Triple photonic crystal (PC) microcavity designs as a function of waveguide width wwg. (a)Bandstructure calculation for the line defect of Region I (outermost PC) of the heterostructure. The bulkPC mode continuum is shown in gray, and the PC waveguide (WG) modes are shown as thick dash-dotted,solid, and dashed blue lines for structures with wwg = 675, 693 and 710 nm, respectively. The resonantfrequencies for each of these structures are shown as thin lines blue, green and red lines for modes M1, M2and M3 respectively. The highlighted yellow shows the region above the air light-line. (b) Total qualityfactors for M1, M2 and M3.by ∼ 0.17 THz per nanometer increase in the hole radius (∆λoff changes by ∼ −1.3 nm pernanometer), which is large relative to the typical linewidth of the lowest Q mode of the fabricateddevices, δf3 ' 0.065 THz (δλ3 ' 0.5 nm). To achieve finer tuning of the mode spacing, the holeradius is tuned for only 12 holes in the middle two PC blocks, highlighted in purple in Fig. 3.1, asopposed to all of them. When rmid = 126 nm and the 12 hole radii are tuned from rmid,ms = 122 to128 nm, the resulting ∆foff change by ∼ 0.056 THz per nanometer increase in radius, as shown bythe empty diamonds in Fig. 3.8(b), simulated using perturbation theory. Alternatively, the modespacing is tuned by shifting these 12 holes. Figures 3.8(c) and (d) show the mode frequencies and∆foff calculated both full FDTD simulations (filled markers) and using perturbation theory (emptymarkers), for rmid = 126 nm and hole shifts hms = −4 to 8 nm (outward from the waveguide isa “positive” shift), in respectively. Both simulation approaches yields similar results, with ∆foffchanging by ∼ 0.028 THz per nanometer shift. The fine tuning methods, using rmid,ms and hms,are both applied in the fabrication layout to bracket the mode spacing.At this stage, the microcavity structure with Ny,cav = 3, wwg = 693 nm, r = 124 nm andrmid = 126 nm, supports three nearly equally spaced modes with high Qs. The input and outputwaveguides are now introduced.593.2. Device design122 123 124 125 126 127 128Hole radius (nm)-0.200.20.4∆f off(THz)-4 -2 0 2 4 6 8Shift (nm)-0.200.20.4∆f off(THz)-4 -2 0 2 4 6 8Shift (nm)195195.5196196.5197Resonant frequency (THz)124 125 126 127 128Hole radius, rmid(nm)195195.5196196.5197Resonant frequency (THz)(a) (b)(c) (d)M1M2M3M1M2M3SimulationPerturbation theory SimulationPerturbation theory Figure 3.8: Resonant frequencies of the triple microcavity structures as a function of mode spacing tuningparameters described in Fig. 3.1. (a) The resonant frequencies found from FDTD simulations for M1 (bluecircles), M2 (green triangles) and M3 (red squares) are plotted as a function of rmid. (b) Frequency offset,∆foff = f1 + f2 − 2f2, as a function of rmid (yellow diamonds) calculated with FDTD simulations, andrmid,ms (empty diamonds) with fixed rmid = 126 nm calculated with perturbation theory, where fm are themode resonant frequencies. (c) Mode frequencies found from FDTD simulations plotted as a function ofthe shift hms [markers same as in (a)]. The large open markers show the mode frequencies calculated usingperturbation theory. (d) Frequency offsets plotted for the resonant frequencies presented in (c), found fromFDTD simulations (filled yellow diamonds) and perturbation theory (empty diamonds).Input and output photonic crystal waveguidesThe input and output photonic crystal waveguides defined in the microcavity structure are labelledin Fig. 3.1. The output waveguide is introduced by reducing the radii of holes lining the centerline defect (highlighted in green), and the input waveguide is defined by introducing a line defecton the diagonal and reducing the radii of holes lining it (highlighted in cyan).Photonic crystal waveguide structure The first step in designing the input and output waveg-uides is to ensure that the waveguides support modes at the cavity resonant frequencies. The603.2. Device design188 190 192 194 196Frequency (THz)00.20.40.60.81Transmission0 0.1 0.2 0.3 0.4 0.5180190200210220Frequency (THz)kx(2π/a)(a) (c)(b)thvOutput PC WGInput PC WGMode M2Mode M1Mode M1Output PC WGInput PC WGMode M2Mode M1Mode M1Cavity mode frequenciesBand edgesSpectraInput PC WGOutput PC WGLegendBulk PC continuum of modesFigure 3.9: Input and output photonic crystal waveguides (PC WG) and the coupling to single-mode channelwaveguides. (a) Bandstructure plots for the input and output PC WG modes. The bulk PC mode continuumis shown in gray for the Region I PC with r = 124nm. The mode frequencies are plotted for modes M1,M2 and M3 respectively. The region above the air light-line is shaded in pale yellow. (b) Finite-differencetime-domain (FDTD) simulation layout [79] of the impedance matching region between the PC and single-mode waveguides. The simulation volume is outlined in orange. A mode source is used to launch the TEchannel waveguide mode , and a planar two-dimensional monitor is used to measure the transmission intothe PC WG. The region contained in the blue box is expanded to the right, where the design parametersof the impedance matching region are labelled. (c) Transmission spectra measured by the simulation layoutin (b), for the input and output waveguides. The dashed and solid vertical lines show the bottom of theinput and output waveguide bands, respectively. The dashed-dotted lines show the resonant frequencies ofthe triple microcavity.waveguide band energies are lowered, to intersect with the resonant mode frequencies, by reducingthe hole radii lining the input and output waveguides to rinwg = 108 nm and routwg = 111 nm, as isplotted in Fig. 3.9(a) with dashed and solid thick lines, respectively. These radii are chosen so thatthe bands intersect the resonant frequencies above the “slow light” region where the bands are flatand propagation losses are high due to enhanced interaction with the photonic crystal waveguidestructure scattering off non-uniformities. The hole radii are different because the waveguides havedifferent widths and the bulk PC holes have different radii as the input waveguide is defined in the613.2. Device designbulk PC in Region I, while the output waveguide is defined in the line defect (input wwg = 710 nm,r = 124 nm; output wwg = 693 nm, rmid = 126 nm).Impedance matching region between photonic crystal and channel waveguides Thecoupling efficiencies between the PC waveguide modes and the single mode waveguides is brieflyvisited before taking a closer look at the resonant cavity mode coupling lifetimes. Figure 3.9(b)shows the simulation set-up employed to study the impedance matching region between the twotypes of waveguides, which is parameterized by three labelled properties, h, t and v [59]. A modesource is used to excite the fundamental mode of the channel waveguide at f = 196 THz andthe power transmitted into the PC waveguide is monitored 7 lattice spacings away from the edge.Lumerical FDTD Solutions’s optimization algorithm is used to find the set of properties thatoptimizes the transmission. The optimal transmission efficiency at f = 196 THz is 0.93 for bothwaveguides, achieved with hin = 146 nm, tin = 341 nm, vin = 9 nm and hout = 166 nm, tout = 346nm, vout = 14 nm for the input and output PC waveguides respectively. The transmission spectrafound for the input and output waveguides using the optimal impedance matching regions areplotted as dashed and solid black lines in Fig. 3.9(c), respectively. Also plotted are the waveguideband-edge frequencies as dashed and solid thick lines, and the mode resonant frequencies (blue,green and red thin lines). The coupling efficiencies increase from near zero to > 0.75 quickly forresonant frequencies approaching the waveguide band-edges, and then they level out at ∼ 0.93.Input and output waveguide coupling geometries Input and output waveguides are firstindependently introduced to the microcavity structure, and two different geometries for each areinvestigated, as is illustrated in Fig. 3.10. The coupling strengths are studied using the procedureoutlined in Section 3.1. The best combination of four possibilities is identified, and then the fullmicrocavity structure, with both input and output coupling is simulated.Figures 3.10(a) and (b) show the two input waveguide geometries, where the diagonal waveguidebegins on the third and fourth rows, respectively. In both cases the waveguide begins seven holesaway from the center such that the geometries are defined as (Xwgin , Youtwg ) = (7, 3) and (7, 4). Theinput coupling quality factors, Qinm, are summarized in Fig. 3.11(a) for all three modes. The Qinm is623.2. Device designINPUTIII IIIII I(a)INPUTIII IIIII IIII IIIII IOUTPUTIII IIIII IOUTPUT(b)(c) (d)Figure 3.10: Triple photonic crystal (PC) microcavity waveguide coupling geometries. The heterostructurePC region boundaries are shown with red lines and are labelled I, II and III. Input waveguides are introducedin (a) and (b), and the holes lining the waveguide are highlighed in cyan. The coupling geometries are (a)(Xwgin , Ywgin ) = (3, 7) and (b) (Xwgin , Ywgin ) = (4, 7). Output waveguide are introduced in (c) and (d), and theholes lining the waveguide are highlighed in green. The coupling geometries are (c) Xwgout = 7 and (d) Xwgout= 8.approximately an order of magnitude larger for the second configuration.Figures 3.10(c) and (d) show the two output waveguide geometries, where the in-line waveguidebegins seven and eight holes from the center, such that Xoutwg = 7 and 8, respectively. The outputcoupling quality factors, Qoutm , are summarized in Fig. 3.11(b) for all three modes. For M1, Qoutmis approximately the same for both geometeries, while for M2 and M3, Qoutm increases by over one,and three order(s) of magnitude when Xoutwg goes from 7 to 8, respectively. The Xoutwg = 8 couplingscheme is clearly not an option, as Qout3 is so high that M3 cannot preferentially coupling to theoutput waveguide, for either of the input waveguides chosen.The design waveguide is chosen to have (Xwgin , Youtwg ) = (7, 4) and Xoutwg = 7. The former ischosen in order to promote preferential coupling of M1 to the output waveguide, as Qin1 is over633.2. Device designan of magnitude larger than Qout1 for (Xwgin , Youtwg ) = (7, 4), while for the other input couplingscheme, they are much closer. The quality factors for the microcavity with both input and outputwaveguides are summarized in Table 3.2. The probability that a photon couples to the outputchannel is calculated as poutm = Qm/Qoutm , and is included in the table for the signal/idler modes M1and M3. Both pout1 and pout3 are equal to 0.89, and thus meet the qualitative design requirement forpreferential coupling to the output waveguide. The predicted transmission from the input to theoutput waveguide, T = 4Q2m/(QinmQoutm ), is also included in the table. The pump mode transmissionis 0.16, which is fairly low due to the weak coupling to the output channel, and also qualitativelymeets the design requirement.It is also interesting to compare Qotherm to the total quality factors of the microcavity structurewithout input and output waveguides, Qnowgm , which are also included in Table 3.2. The Qnowgm arefound to be higher than Qotherm , which suggests that the waveguides introduce additional scatteringlosses (i.e. light scattered off the waveguides radiates out of the microcavity, instead of coupling toa waveguide mode).At this stage, the main design for the microcavity has been described. Preferential waveguidecoupling of the signal and idler modes to the output waveguide is achieved, and the center modetransmission is low. Overall, the Q’s are high, relative to the previously reported nanobeam design,where Q2 is 7 times larger than that of the nanobeam, Q1 is 4.6 times larger, and Q3 is on par.1 2 3Mode103104105106107108109Qin1 2 3Mode103104105106107108109Qout(a) (b)Figure 3.11: Input and output quality factors for waveguide coupling. (a) Input waveguide quality factors,Qinm are shown for modes M1 (blue circles), M2 (green triangles) and M3 (red squares) simulated for couplinggeometries (Xwgin , Ywgin ) = (3, 7) (left) and (4, 7) (right). (b) Output waveguide quality factors, Qoutm are shownsimulated for coupling geometries Xwgout = 7 (left) and Xwgout = 8 (right).643.2. Device designSymmetrizing holes One final microcavity design parameter is introduced, swg,sym, in an effortto lower Qout1 , so that it’s closer to Qout3 , for better symmetry. To achieve this, the four holes alongthe center axis, opposite the output waveguide, are perturbed to have radii rwg,sym = swg,sym × r,where swg,sym is a scaling factor. These holes are labelled in yellow in Fig. 3.1, and the new qualityfactors are included in Table 3.3. The output waveguide coupling is strengthened (Qout1 is reduced),however the scattering is also increased (Qother1 is reduced). The increased scattering is somewhatsurprising as Qnowg1 is roughly the same for the waveguide-less microcavities with swg,sym = 0.912and swg,sym = 1 (unperturbed). Both types of microcavities (swg,sym = 1, 6= 1) are ultimatelyincluded in the fabrication layout, as is described below in Section 3.3.1.Table 3.3: Summary of the quality factors simulated for triple photonic crystal microcavity devices withinput and output waveguides, and hole radii rwg,sym = 0.912r for the holes are labelled in yellow in Fig. 3.1(i.e. swg,sym = 0.912). The quality factors Qinm, Qoutm , Qotherm correspond to coupling to the input waveguide,the output waveguide, and other loss channels including scattering absorption, respectively, while Qm is thetotal quality factor, for modes m = 1, 2, 3. The probability that photons generated in the cavity will coupleto the output waveguide is given by poutm = Qm/Qoutm . The relative transmission from the input to the outputwaveguide is given by T = 4Q2m/(QinmQoutm ). The total quality factors for the microcavity structure withoutinput and output waveguides, Qnowgm , are also included.Mode, mParameter Description 1 2 3Qm Total quality factor 5.4× 103 1.3× 105 4.1× 103Qinm Input coupling quality factor 1.6× 106 1.5× 105 1.6× 105Qoutm Output coupling quality factor 5.9× 103 2.9× 106 4.6× 103Qotherm Scattering/other losses quality factor 6.7× 104 9.9× 105 4.8× 104Qnowgm Total quality factor in absence of waveguides 2.0× 106 1.4× 106 6.2× 105poutm Probability of output waveguide coupling 0.91 0.89Tm Relative transmission 0.013 0.14 0.0913.2.2 Input and output portsThe input and output ports of the microcavity devices consist of 20 µm wide grating couplersthat terminate 150 µm long parabolic waveguides that narrow to 500 nm wide single mode channelwaveguides, as is shown in Fig. 3.12. The relatively large grating width is designed to accommodatethe free-space coupling scheme used in this thesis, shown in Fig. 2.3, where light from a single mode653.2. Device designFigure 3.12: Input/output port for the microcavity device. The single-mode channel waveguide is expandedfrom 0.5 µm to 20 µm by a parabolic waveguide, which is terminated by a grating coupler. The black regionsshow where silicon is removed to define the structure. The grating coupler is defined by an array of slotswith apodized widths, designed for coupling light between the waveguide and light propagating backwardsin free-space at an angle θ = 45◦. Input and output coupling are shown in the top diagram with pairs ofsolid blue and dashed red the arrows, respectively.optical fibre is focused on the input grating coupler with a 1/e2 intensity Gaussian beam diameterof ∼ 10 µm, and light from the output grating coupler is collected by an elliptical mirror with anumerical aperture, NA= 0.06.The grating couplers are designed to efficiently couple light near the microcavity resonant fre-quencies into and out of device at a −45◦ angle, as is illustrated in Fig. 3.12. The negative angleis chosen so that uncoupled incident light is scattered away from the rest of the device, such thatbackground noise is reduced. The 45◦ angle is chosen to be compatible with another optical-setup(ultimately not used in this thesis) where the sample is placed within a cryostat with windows at−45◦ and 45◦.The grating coupler is composed of a series of slots, with widths and spacings that are carefullyengineered so that the spatial profile and direction of light coupled out of the grating roughlymatches the spatial profile and direction of the excitation beam. The better the overlap betweenthese two profiles, the higher the coupling efficiency. Another aim of the design is to keep reflectionsbetween the parabolic section and the grating coupler at a minimum. The design of the gratingcoupler is outlined in Appendix C. A schematic of the apodized grating design is shown in Fig.3.13(a), along with an SEM image of a fabricated grating, in 3.13(b).663.2. Device designThe parabolic waveguide has a full width that is given by,w(x) =√w2i −[w2i − w2f] xL, (3.4)where wf = 20 µm is the final width, wi = 500 nm is the initial width, L = 150 µm is the length,and x goes from 0 to L. The length of the parabolic waveguide is chosen to be long enough toreduce propagation losses, and short enough to be convenient for the fabrication and measurementprocesses. The final width, wf , is chosen to be small enough to minimize propagation losses, andalso large enough that its wider than the input excitation beam diameter, and also sufficiently largethat out-coupled light diffracts over a small enough angle for efficient collection by the ellipticalmirror.The transmission efficiency of light from free-space into the single-mode channel waveguideis estimated using a combination of FDTD simulations and simulations with Lumerical’s MODEsolver [80]. A similar approach is taken to estimate the transmission efficiency of light from thesingle-mode waveguide, to beyond the elliptical collection mirror. These are described in AppendixC.The in-coupling and out-coupling transmission efficiencies between the single-mode waveguideand free-space directly above the grating-coupler (elliptical mirror collection not included here)are plotted in Fig. 3.13(c). These are plotted as a function of wavelength to be consistent withmeasurement results presented later in this thesis. The in-coupling spectrum peaks around λ = 1535nm (f = 195.4 THz), while the out-coupling exhibits modulations due to multiple reflectionsoccuring in the 3 µm thick buried oxide. The lack of a peak is because the transmission is takento be the integral over all outgoing radiation, such that all diffraction angles are accepted.Figures 3.13(d) and (e) shows a plot of the far field distribution at λ = 1530 and 1565 nm overthe surface of a half sphere of radius R = 15 cm, along with a contour of the mirror collection areaon this surface. Here it is shown that light at λ = 1530 nm diffracts at an angle close to −45◦ andis collected by the mirror, while that at λ = 1565 nm diffracts away from the mirror. The collectionefficiency of the elliptical mirror is estimated by considering the fraction of light intensity that fallswithin the elliptical mirror, and is explained in further detail in Appendix C. The total output673.2. Device design-10 0 10x (cm)-10010y (cm)(a) (b)(d)0145o90o1500 1600Wavelength (nm)00.20.40.6Transmission efficiency(f)1500 1600Wavelength (nm)00.20.40.60.8Transmission efficiency-10 0 10x (cm)-10010y (cm)45o90o(e)InputOutputInputOutputTotal(c)Figure 3.13: Apodized grating coupler design and simulation results. (a) Schematic of the apodized gratingcoupler, where black regions show where the silicon is removed. (b) Scanning electron microscope image ofan apodized grating coupler. (c) Simulated input and output transmission efficiencies between free-spaceand the single mode waveguide (via the grating coupler and parabolic waveguide). (d) Far field electricalfield intensity projection on a 15 cm hemisphere of the field monitored 1 µm off the grating surface whenthe single-mode waveguide is excited at λ = 1530 nm. The collection area of the elliptical mirror, that issituated on the hemisphere at an angle of −45◦ from the vertical, is outlined in black. (e) Same as in (d) butfor light launched in the single mode waveguide at λ = 1565 nm. (f) Simulated input and output, and totaltransmission efficiencies. The output transmission efficiency shown in this plot accounts for the collectionefficiency of the elliptical mirror [unlike the plot in (c)].efficiency is plotted in Fig. 3.13(f), along with the input efficiency and the product of in-couplingand out-coupling spectra. The bandwidth of the full transmission spectrum is 35 nm.The ratio between the in-coupling and out-coupling efficiencies, which is not possible to exper-imentally measure, is required for the nonlinear analysis. For the grating coupler described above,the peak input efficiency is 1.96 times larger than the peak output efficiency. The calculationpresented here is repeated for each of the four grating coupler structures used in the nonlinearcharacterization analysis.683.3. Fabrication3.3 FabricationThe devices are fabricated by electron beam lithography (EBL) with a 100 keV JEOL JBX-6300FSwriting system at the University of Washington Microfabrication/Nanotechnology User Facility. Inthis EBL process, the “5th lens” writing mode is employed, with a low beam current of 500pA,and a shot pitch of 2 nm (minimum beam translation step distance). The electon beam writesthe pattern by selectively exposing the ZEP-520 A positive resist (Nippon-Zeon Co. Ltd.) thatcovers the SOI. The exposed resist is chemically removed, then the exposed silicon is etched usingan Oxford PlasmaLab System 100 with chlorine gas. The layout pattern submitted for fabricationis composed of the geometric shapes that define the etched silicon areas.Two fabrication iterations were done as part of this project, where the second iteration producedthe Chips A and B studied in this thesis. The first iteration is not discussed in detail here, howeverthe results from this iteration were used to guide the design second iteration, and is acknowledgedin this context.3.3.1 Fabrication layoutThe fabrication layout for each chip is shown schematically in Fig. 3.14. It contains 168 microcavitydevices, and 48 reference devices, with different device parameters systematically varied over a rangethat includes the nominal design value.Reference devicesThe four different types of reference devices included in the layout are illustrated in Fig. 3.15.The reference devices are designed to mimic either the input or output components of the triplemicrocavity (TMC) devices. The reference device in Fig. 3.15(a), contains PC waveguide prop-erties identical to the input waveguide of the TMC, and is referred to here as PCWGin. The PCwaveguide length is twice as long as that in the TMC device such that the full PCWGin device,including input and ouput parabolic waveguides and grating couplers, is effectively a mirrored ver-sion of the TMC input components. Similarly, the reference device in Fig. 3.15(b), PCWGout,contains a mirrored version of the TMC output components, including the appropriate output PC693.3. FabricationGroup #Set #Alignment MarksFigure 3.14: Fabrication layout of the devices studied in this thesis. There are 168 microcavity devices,and 48 reference devices. Markings on the chip are used to identify the Group number and Set numberof the devices adjacent. Devices in the same group have identical grating couplers, while devices in thesame set have identical hole size scale factors. Alignment marks are also included for the post-fabricationphotolithography process.waveguide. The devices in Fig. 3.15(c) and (d) are grating-to-grating devices G2Gin and G2Gout,respectively, that are the same as PCWGin and PCWGout, however without the PC waveguides.The total length of each device is made the same by choosing appropriate lengths of the singlemode channel waveguides. This makes it possible to layout devices in columns with their input andouput grating couplers aligned, which makes measurements within a column convenient. Trans-mission measurements of the reference devices are used to determine the efficiencies of input andoutput grating couplers, parabolic waveguides and the PC waveguides, that affect the TMC deviceresponse.BracketingIn anticipation of deviations between the design and the actual devices, a number of device param-eters are varied across the layout. This increases the probability that there is a subset of devices onthe chip that have similar properties as the target. The device parameters that are varied across thechip are summarized in Table 3.4 and are discussed below. The ranges over which each parameter703.3. Fabrication(a) PCWGin(b) PCWGout(c) G2Gin(d) G2GoutFigure 3.15: Reference devices included in the fabrication layout that contain input and output gratingcouplers and parabolic waveguides. In (a) and (b) PCWGin and PCWGout devices are shown that containphotonic crystal (PC) waveguides with structural parameters identical to those of the input and output PCwaveguides of the triple microcavity device, respectively. In (c) and (d), the PC waveguides are omitted andthe channel waveguides are directly connected to form G2Gout and G2Gout references devices.is varied are chosen based on the results from the first iteration of fabrication.The grating coupler groove widths are bracketed over the layout to ensure that light can becoupled into and out of at least a subset of microcavity devices. Each quadrant of the layout inFig. 3.14 contains groups of gratings, where Groups 1 to 4 are indicated by the number of squaresin the first column of markers to the left of each set of devices. For Groups 1, 2 and 3, the gratingslot widths are scaled by factors sgc = 0.91, 0.94 and 0.97, respectively. The devices within eachof these groups are otherwise identical. Group 4 contains gratings with sgc = 0.94, and modifiedmicrocavity devices, addressed shortly. These sgc are chosen because, in the previous iteration, thetransmission of the sgc = 1 grating peaked at f ' 197.4 THz (f ' 1520 nm), which is in goodagreement with the simulated result and at higher energy than the target resonant mode centerwavelength f2 ' 196 THz (λ2 ' 1530 nm). The scaling factors of sgc = 0.91, 0.94 and 0.97 aresimulated to give a peaks near λ ' 193.5, 194.8 and 196 THz, respectively. The choice is madeto err on the side of small hole sizes, because the grating coupling peaks can be shifted to higherfrequencies by decreasing the magnitude of the coupling angle from 45◦ down to 10◦ (shift is ∼ 0.5THz per degree), while the angle can only be increased to ∼ 50◦, limiting the lowest achievablepeak frequencies.713.3. FabricationTable 3.4: Summary of device parameters bracketed across the layout. The set of values is given for eachbracket parameter. The “Bracket name” describes how each bracket is referred to in the text, usually followedby a number (e.g. Group 1 corresponds to devices with sgc = 0.91), with the exception of “Type” wherethey are labelled as Type I (swg,sym = 1) and II (swg,sym = 0.912). The “Affected” column describes whichof the microcavity and reference devices are affected by this parameter.Parameters Bracket name Affected Descriptionsgc = 0.91, 0.94, 0.97, 0.94 Group All Grating groove width scalingfactorsh = 0.91, 0.94, 0.97, 0.94 Set TMC,PCWGin,PCWGoutPC hole radius scaling factorhms = −4,−2, 0, 2, 4, 6, 8 nm Number TMC† Mode spacing tuning holeshiftsms = 0.97 to 1.02∗ Number TMC‡ Mode spacing tuning hole ra-dius scaling factorswg,sym = 1, 0.912 Type TMC Symmetrizing hole radiusscaling factor† Only affects TMC devices in Groups 1 to 3∗ Linearly spaced over 7 devices‡ Only affects TMC devices in Group 4Within each group, the overall hole sizes are scaled by three different factors, sh = 0.91, 0.94and 0.97, for Sets 1, 2 and 3, indicated by the number of squares in the second column of markersto the left of each set. The scaling factors are applied to all hole radii, including r, rmid, rwg,sym, rinwgand routwg . These scaling factors are chosen because, in the first iteration, the microcavity modeenergies for sgc = 1 were ∼ 197.3 THz, which is 1.4 THz higher than the simulated result, suchthat many of the mode frequencies coincided with the high energy end of the laser tuning range(197.3 THz). Simulations show that decreasing the hole radii by a 3% shifts the modes by ∼ −1.4THz, so the bracket used incrementally shifts the modes away from the laser tuning limit.For Groups 1 to 3, the mode spacing of the microcavities is fine-tuned by shifting the targetedholes near the center of the microcavity. The shifts are hms = −4,−2, 0, 2, 4, 6 and 8 nm, for deviceNumbers 1 to 7. The two nanometer increment is the minimum shift possible given that the shotpitch is 2 nm for this EBL process. In Group 4, instead of shifting holes, the mode spacing istuned by scaling the radii by sms = 0.97 to 1.02 nm over 7 devices (rmid,ms = 122 to 128 nm, when723.3. Fabricationsh = 1). The mode spacing parameters are chosen because, the “identical” TMC target devices onthe first iteration chip have ∆foff between −0.10 to 0.18 THz, and the chosen hms and sms resultin mode spacing tuning roughly over this range, as shown in Fig. 3.8(d) and (c), respectively. Theshift is chosen as the primary tuning method (Groups 1 to 3) because it was anticipated to be morereliably reproduced than the hole size tunings of ∆r ∼ 1 nm.Finally, for each of the brackets described above, two types of microcavity devices are includedin the layout. Type I devices are defined by setting rwg,sym = rmid for the holes that aim toprovide symmetry around the output waveguide, such that the scaling factor applied to these holesis swg,sym = 1. In Type II devices, symmetrization is applied and swg,sym = 0.912. Within each setof the layout, the seven Type I and II devices are laid out, paired as the top two devices in eachcolumn, and the bottom two. The center device of each column is a reference device (G2Gout,G2Gin,PCWGin,PCWGout, from left to right).Alignment marks are included in the fabrication layout (crosses in Fig. 3.14) to aid the pho-tolithography process described in the next section.3.3.2 Post-fabrication processingOnce the silicon-on-insulator chips are fabricated, the buried oxide layer (silicon dioxide, SiO2) isremoved from underneath the microcavity regions defined in the silicon device layer in order toachieve high Q modes. This processing involves photolithography and a hydrofluoric acid (HF)etch.The full undercutting procedure is outlined in Table 3.5. First, photoresist is spin-coated on thefabricated microchip. The photoresist is selectively exposed with ultra-violet radiation by using amask that covers all but the regions around the microcavities and a mask aligner. The exposedresist is removed using a developer and the microchip is submerged in HF to etch the silicon dioxide.Finally, remaining photoresist is removed from the sample.An optical microscope images of Chip B, after the photoresist is developed is shown in Figs.3.16(a) and (b). The light pink strips show where the photoresist has been removed. Figure 3.16(c)shows TMC device after the undercutting process is complete. The darker pink hallow roughlyoutlines where the silicon dioxide is removed beneath the silicon device layer.733.3. FabricationTable 3.5: Recipe for post-fabrication undercutting.ProcessStep Procedure Time Duration1 Cover with HMDS Primer2 Ramp up to 500 rpm/s3 Spin at 1700 rpm 40s4 Spin down at 500 rpm5 Let HMDS dry 1 min6 Cover with AZ P4110 photoresist7 Repeat steps 2 to 48 Soft bake at 90◦C 10 minExposure9 Align mask10 Expose with 320 nm light at 22 W/cm2 45 sDevelopment11 Emerse in AZ P4110 developer solution(4:1, developer: DI water)90 s12 Rinse with DI water and dry with N2 gas13 Hard bake at 120◦C 10 minEtch14 Emerse in HF solution (10 parts 40% NH4Fto 1 part 49% HF)15-20 min15 Rinse with DI water and dry with N2 gas743.3. Fabrication(a)(b)(c)Figure 3.16: Optical microscope images of the silicon microchip at various stages in undercutting process.(a) Image taken post-photolithography, such that the developed photoresist is intact. Light pink lines wherethe bare silicon is exposed. (b) Same as in (a), but with higher magnification. (c) Image taken after thehydrofluoric acid etch and the photoresist removal. The pink hallow around the photonic crystal regionoutlines where the oxide is removed beneath the silicon.75Chapter 4Survey results to identify the bestdevices for full FWM analysisIn this chapter, a survey of the linear transmission spectra for devices across the microchip isdescribed that was used to identify the specific structures used for the nonlinear characterizationreported in the following chapter.4.1 Survey of devices across microchipsThe transmission spectra of the grating couplers are first studied, in order to ensure that light isappropriately coupled into and out of the devices. As discussed in Section 3.3.1, the grating couplergroove widths are bracketed with three different scaling factors, where sgc = 0.91, 0.94, 0.97 and0.94, for Groups 1,2,3 and 4, respectively. Figure 4.1 plots the raw transmission spectra measuredfor grating-to-grating G2Gin devices in each of the four groups on Chip A, that directly link inputand output gratings with parabolic and channel waveguides. In Fig. 4.1(a), the angle of incidenceis θ = −41◦ for all devices measured, and in (b) the angles of incidence, −37◦, −39◦, −42◦ and−39◦, are chosen for Groups 1 to 4, to achieve peak coupling at ∼ 1545 nm, which is near thetarget microcavity resonant wavelengths. The peaks shift by ∼ 4 nm/degree, and by adjusting thegrating coupling angle (adjustable between ∼ 10◦− 50◦), it is possible to shift the grating couplingpeaks over the full tuning range of both lasers (λ = 1520 nm to 1610 nm). This means that anymicrocavity devices with resonant wavelengths within the tuning range of the lasers are accessiblewith these gratings. Similar results are found for the grating couplers on Chip B.There is good agreement between the transmission spectra measured for “identical” (same lay-out) grating-to-grating reference devices within the same group: the transmission efficiencies vary764.1. Survey of devices across microchipswithin ±3 %, while the peak wavelengths vary within ±1 nm. This indicates that the referencedevices can be used to reliably estimate the input and output coupling efficiencies of the micro-cavity devices. However, the fast sinusoidal modulations in the spectra (see Fig. 4.1), caused bymultiple reflections off the grating and parabolic waveguide components, have phase-shifts thatvary significantly from device to device. The approach to dealing with this issue is discussed ingreater detail in Appendix D.1520 1540 1560 1580 1600 1620Wavelength (nm)00.511.522.5Transmission (arb. units)1520 1540 1560 1580 1600 1620Wavelength (nm)00.511.522.5Transmission (arb. units)(a) (b)Figure 4.1: Transmission measurements for G2Gin devices in Groups 1 to 4 of Chip A (bottom to top), (a)taken with a −41◦ coupling angle, (b) taken with coupling angles between −42◦ and −37◦ (labelled on plot)to achieve peaks near λ = 1545 nm.1540 1545 1550 1555Wavelength (nm)00.511.52Transmission (arb. units)1540 1550 1560 1570Wavelength (nm)00.511.52Transmission (arb. units)(a) (b)Figure 4.2: (a) Transmission measurements of Type I microcavity devices on Chip A, with mode spacingparameter sms = 0 nm, for the three hole size brackets labelled on the plot. (b) Transmission measurementsof Type II microcavity devices on Chip A, with sh = 0.97 and mode spacing parameters labelled on the plot.The microcavity transmission is studied by adjusting the grating coupler angle appropriatelysuch that the coupling peak is near the resonant mode wavelengths. Only a subset of all fabri-774.1. Survey of devices across microchips1 2 3 4 5 6 7-10123∆λoff(nm)1 2 3 4 5 6 7-10123∆λoff(nm)1 2 3 4 5 6 7-10123∆λoff(nm)1 2 3 4 5 6 7Device-10123∆λoff(nm)1 2 3 4 5 6 7-10123∆λoff(nm)1 2 3 4 5 6 7-10123∆λoff(nm)1 2 3 4 5 6 7Device-10123∆λoff(nm)1 2 3 4 5 6 7-10123∆λoff(nm)(a) (e)(b) (f)(c) (g)(d) (h)Chip A Chip BGroup 1Group 2Group 3Group 4LegendType I, Exp. Type II, Exp. Type I, Sim. Type II, Sim. Figure 4.3: Microcavity mode spacing offsets, ∆λoff = λ1 +λ3−2λ2 found from transmission measurements,plotted as a function of device number in the sh = 0.97 set, where λm are the resonant wavelength, for TypeI and Type II microcavities. (a)-(d) Results for Groups 1 to 4 (top to bottom) devices on Chip A. (e)-(h)Results for Groups 1 to 4 (top to bottom) devices on Chip B. In Groups 1 to 3, each device has a differentmode spacing parameter, with sms = 8, 4, 2, 0,−2,−4 nm for Devices 1 to 7, where sms is the position shiftof a select group of holes. In Group 4, the mode spacing is modified by scaling a select group of holes bysms = 0.968 to 1.02, linearly over the seven devices. Data is omitted in cases where three modes are notobserved in the transmission spectrum. The dashed lines show the simulated results (from perturbationtheory) for Type I and II microcavities. The gray region has a width of 0.5 nm, which is a typical linewidthfor the M3 mode (the lowest Q mode).cated microcavity devices are expected to have resonant mode wavelengths near the target, due tobracketing of the PC hole sizes across the microchip. The overall hole size is bracketed with scalingfactors, sh = 0.91, 0.94 and 0.97, for Sets 1 to 3, and the resonant wavelengths are red-shifted by 10nm per 3% decrease in hole size, as shown in the transmission spectra in Fig. 4.2(a). Transmissionspectra for the last set, with the largest holes (sh = 0.97), have resonant wavelengths near ∼ 1545nm, and there are three well defined resonant peaks for most devices in Set 3. Devices from this set784.1. Survey of devices across microchipsare best suited for FWM measurements and are the ones used for the studies reported in Chapter5.Within each set with sh = 0.97, the mode spacing tuning parameter, hms, is bracketed over sevendifferent values, for the two types of microcavity devices, resulting in a total of 14 devices per set.The raw transmission spectra for the third, fifth (target), and seventh Type II devices of a sms = 0.97bracket on Chip A are plotted in Fig. 4.2(b). The mode spacing offsets, (∆λoff = λ1 + λ3 − 2λ2),extracted from the transmission spectra of all devices in the four sets of each chip are summarizedin Figs. 4.3(a)-(d) and 4.3(e)-(h) for Chips A and B respectively. The mode spacings roughly followthe simulated results (using perturbation theory), which are shown as dashed lines4. There are 10and 12 devices on Chips A and B, respectively, that have mode spacing offsets within roughly alinewidth of the third mode (typical linewidth is ∼ 0.5 nm), and are “good” candidates for four-wavemixing.In Table 4.1, the average experimental results for “good” candidate devices are compared tosimulation results for the designed Type I and II microcavity structures (with the layout holescaling, sh = 0.97). The center mode (M2) resonant wavelengths are fairly close to those simulated,however the simulated mode spacings are larger. The quality factors of the lowest Q mode (Q3)agrees well with those simulated, while the higher Q’s don’t agree as well.Table 4.1: Comparison of experimental results averaged over “good” candidate devices, and simulationresults for the designed structures. λ2 is the resonant wavelength of the center mode, ∆λms is the modeseparation, and Qm are the total quality factors for modes m = 1, 2, 3. Type II microcavities have four holesthat are scaled by swg,sym = 0.912. These holes are not scaled in Type I microcavities (swg,sym = 1).Experiment SimulationParameter Type I Type II Type I Type IIλ2 (nm) 1546 1547 1541 1541∆λms (nm) 2.4 2.4 3.1 3.1Q1 73,000 74,000 59,000 16,000Q2 30,000 37,000 102,000 102,000Q3 3,200 3,100 3,800 3,3004The simulated results are evaluated for discrete hms and sms, and the dashed lines in Fig. 4.3 are actually theconnections between the discrete results.794.2. Stimulated four-wave mixing spectrum4.2 Stimulated four-wave mixing spectrumStimulated four-wave mixing measurement results for a typical “good” candidate device, are pre-sented in Fig. 4.4. The linear transmission spectrum is shown in Fig. 4.4(a), where the dashedblue and green lines coincide with the resonant wavelengths of M1 and M2, while the dashed redline coincides with λFWM3 = 2λ2−λ1. Figures 4.4(b) and (c) show the stimulated FWM idler power(circles) as a function of the output filter center wavelengths when signal wavelength is set to λ1and λFWM3 , respectively, and the pump wavelength is tuned to λ2. The pump and signal powers are38.5 µW and 22.4 µW, respectively. The idler power spectra are essentially a convolution betweenthe filter transmission spectrum, ηfilter(λ), and the actual spectrum of idler power in the outputPC waveguide. The filter transmission spectrum lineshape is plotted as the gray shaded regions inthese figures. The idler power closely follows the filter lineshape near the peak, which means thatthe linewidth over which the idler photons are actually generated is much much smaller than thelinewidth of the filter (0.17 nm). Also included in Fig. 4.4(b) and (c) is the background powermeasured (triangles) when the signal laser is turned off, but the pump power remains active. Whilein Chapter 1, this excitation configuration was suggested for measuring spontaneous FWM, it isnot currently clear if the background signal here is due to spontaneous nonlinear FWM, or perhapssome other processes. This is discussed further in Chapter 7.4.3 Analysis of survey resultsAfter surveying the linear transmission data from a large number of devices, examples of whichwere provided above, it was clear that the fabricated structures behaved only qualitatively as thesimulations done during their design. While the absolute resonant wavelengths were within ∼ 10nm of those expected, and their dependences on systematic, bracketed parameter sweeps agreedwell, many other properties did not. In particular, the absolute Q values of M1 and M2 were off(some higher, some lower) by as much as a factor of 5, and the absolute on-resonant transmissionvalues were often off by factors of 3. The wavelength separation of the modes was also off by asmuch as 1 nm. Clearly the original simulations of various device parameters can not be used toaccurately predict or explain the nonlinear behaviour of the actual samples.804.3. Analysis of survey results1542 1544 1546 1548 1550Wavelength (nm)00.51Transmission1547 1548 1549Filter Wavelength (nm)10 -810 -710 -610 -510 -4Idler Power (µW)1542.6 1543 1543.4Filter Wavelength (nm)10 -910 -810 -710 -610 -5Idler Power (µW)(a) (b)(c)Stim. FWM M3 idler powerBackground power (signal o)Stim. FWM M1 idler powerFilterFilterFiltermode M1 mode M2 mode M3 Background power (signal o)Figure 4.4: (a) Linear transmission for a microcavity structure studied with four-wave mixing (FWM).Resonant wavelengths λ1, λ2, and equally spaced λFWM3 = 2λ2 − λ1, are plotted as the dashed blue, greenand red lines, respectively. (b)-(d), Four-wave mixing results, measured using pump and signal powers 38.5µW and 22.4 µW respectively. The idler power is reported relative to the output PC waveguide (withoutaccounting for the output spectral filtering). (b) Stimulated FWM idler power as a function of the filtercenter wavelength for λsignal = 1543.01 nm (circles). The dashed line shows the wavelength λFWM3 . (c) Sameas (b) but with λsignal = 1547.58 nm. The shaded regions in (b)-(c) are the scaled filter transmission spectra,and the triangles show the background power when the signal laser is turned off and the pump laser remainson.The bulk of this thesis work addressed a solution to this problem. By quantitatively andthoroughly measuring and modelling the nonlinear transmission of modes M1 and M2, and thepower dependence of the stimulated FWM response measured using both M1 and M3 as separatesignal channels, a least-squares minimization of the modelled and actual data yields well-definedvalues for all relevant linear and nonlinear device parameters of the actual devices. This overallcharacterization protocol for a triple-microcavity nonlinear structure therefore represents a novelmeans of extracting detailed linear and nonlinear device parameters that cannot be obtained usinglinear transmission spectroscopy alone.81Chapter 5Nonlinear characterizationThree types of optical measurements are used to characterize the triple microcavity performanceand develop a model to predict its nonlinear response: linear transmission, nonlinear transmissionand stimulated four-wave mixing measurements. Least-squares analyses of the nonlinear experi-mental results are used to determine the 15 unknown microcavity parameters, and two Fabry-Perotparameters, described in Chapter 2, that enter the nonlinear response models.5.1 Measurement and modelling resultsFour devices with nearly identical designs are measured and modelled. The microcavity structuresare differentiated by only small perturbations. The design parameters for these four devices aresummarized in Table 5.1. Three are from Chip A and one is from Chip B.Table 5.1: Design parameters for the triple microcavity devices studied in this thesis. The factor sh scalesthe holes radii r, rmid, rmid,ms, rinwg, routwg , and rwg,sym described in Table 3.1 and illustrated in Fig. 3.1. Thehole shift hms is also presented therein. The factor swg,sym applies an additional scaling factor to rwg,sym.Device Chip sh hms swg,sym1 A 0.97 0 nm 12 A 0.97 -2 nm 0.9123 A 0.97 0 nm 0.9124 B 0.97 0 nm 0.912The linear transmission spectra are plotted in Fig. 5.1. All peaks are resolved in these plots.The resonant wavelengths λm, total quality factors Qlinm and maximum transmissions Tlinm extractedfrom these results are summarized in Figs. 5.2(a), (b) and (c), respectively. The quality factor isestimated as Q = δλm/λm, where δλm is the full-width at half maximum. It is unknown why the825.1. Measurement and modelling resultslow Q mode, M3, is not Lorentzian for Devices 2 to 4, as FDTD simulations predict a Lorentzianlineshape.1540 1545 1550Wavelength (nm)00.51Transmission1540 1545 1550Wavelength (nm)00.51Transmission1540 1545 1550Wavelength (nm)00.51Transmission1540 1545 1550Wavelength (nm)00.51Transmission(a) (b)(c) (d)Device 1 Device 2Device 3 Device 4Figure 5.1: Linear transmission spectra for (a) Device 1, (b) Device 2, (c) Device 3 and (d) Device 4. Thetransmission is calculated using the best fit Fabry-Perot parameters, φin and φout.1 2 3 4Device00.20.40.60.81Tlin m1 2 3 4Device154015451550λm(nm)1 2 3 4Device10 210 310 410 510 6Qlin m(a) (c)(b)LegendM1M2M3Figure 5.2: Linear transmission results for the four devices characterized: (a) resonant wavelengths, λm, (b)total quality factors, Qlinm , and (c) peak transmission, Tlinm . The results are plotted for modes M1, M2 andM3. The T linm are calculated using the best fit Fabry-Perot parameters, φin and φout.Figure 5.3 shows the experimental results for nonlinear transmission and stimulated four-wavemixing measurements for the four devices (results for Devices 1 to 4 are shown in Rows 1 to 4 (top835.1. Measurement and modelling results10 1 10 2Input Power ( µW)10 -610 -410 -2Idler Power (µW)0 200 400Input Power ( W)00.51Transmission0 200 400Input Power ( µW)00.050.10.150.2λ(nm)10 1 10 2Input Power ( µW)10 -610 -410 -2Idler Power (µW)0 250 500Input Power ( µW)00.51Transmission0 100 200Input Power ( µW)00.51Transmission0 100 200Input Power ( µW)00.050.10.150.2λ(nm)0 200 400Input Power ( µW)00.51Transmission0 200 400Input Power ( µW)00.050.10.15λ(nm)10 0 10 1 10 2Input Power ( µW)10 -810 -6Idler Power (µW)10 1 10 2Input Power ( µW)10 -610 -410 -2Idler Power (µW)0 250 500Input Power ( µW)00.20.40.6λ(nm)NLNLNLNLDEVICE 1DEVICE 2DEVICE 3DEVICE 4Mode 1, exp.Mode 1, modelLegendNonlinear transmissionMode 2, exp.Mode 2, modelStimulated FWMIdler Mode 3, pump sweep, exp.Model with NL absorptionModel without NL absorptionIdler Mode 3, signal sweep, exp.Idler Mode 1, pump sweep, exp.Idler Mode 1, signal sweep, exp.µFigure 5.3: Nonlinear transmission and four-wave mixing analysis results for Devices 1 to 4 (top to bottomrows). Columns 1 and 2 contain the peak transmission and resonant wavelength shifts, respectively, forMode 1 (blue) and Mode 2 (green), found experimentally (filled circles), and using the model with best fitparameters (open markers). Column 3 shows the experimental FWM idler power in Mode 1 (blue) and Mode3 (red), as a function of pump power (triangles) for fixed signal power at Ps (labelled on the plots) and asa function signal power (circles) for fixed pump power at Pp (also labelled). The fixed pump power Thepredicted idler powers are shown with thick black lines. The dashed lines show the predicted power whennonlinear absorption is ignored.to bottom), respectively). The data sets presented here are used in the least-squares analyses foreach device, where the experimental data is compared against model functions. The model function845.2. Derivation of model functionspredictions, that apply the extracted best fit values, are also plotted in Fig. 5.3. The first twocolumns show nonlinear transmission results, where the experimental results for Modes 1 and 2 areblue and green filled circles, respectively, while the model predictions are upward and downwardopen triangles. The last column shows four-wave mixing results, where circles and triangles showexperimental idler powers measured as a function of the signal and pump power, respectively, whilethe other input power is fixed. The red filled markers are for idler photons generated in M3, andthe blue markers are for idler photons generated in M1. The black lines show the predicted idlerpowers.Figure 5.4 shows the experimental nonlinear transmission spectra, from which the peak trans-mission and wavelength shifts in Fig. 5.3 are extracted. The spectra predicted based on the modelfunctions are also plotted, where results for Devices 1 to 4 are shown in Rows 1 to 4 (top to bottom),respectively. The left and right columns show the transmission spectra for M1 and M2, respectively.The results presented in Fig. 5.3 are replotted in Fig. 5.5, where new model functions applymicrocavity parameters extracted from a modified analysis procedure. In the modified procedure,a subset of the parameters (Rth, τcarrier, and τabs) are held fixed at the average best fit valuesfound across Devices 2 to 4, instead of entering the analyses as fit parameters. These parametersare expected to be roughly the same for all devices as they are relatively insensitive to smallperturbations in the photonic structures (introduced by design or due to fabrication imperfections).The Device 1 best fit parameters are excluded from the average because they couldn’t be reliablyextracted, as the sum of the least-squares wasn’t minimized across all parameters.5.2 Derivation of model functionsThe models used in the microcavity least-squares analyses are derived using temporal coupled modetheory (TCMT). In this theory, localized microcavity modes and propagating waveguide modes areweakly coupled through a small perturbation. Conservation of energy is used to calculate thesteady-state energy in the microcavity modes for a given continuous-wave waveguide excitationscheme. This method is appropriate for studying resonant modes with Q & 30 [38], as is the casehere. The triple microcavity is modelled as a single resonator that supports multiple modes. The855.2. Derivation of model functions1543.5 1543.54 1543.58Wavelength (nm)00.050.10.150.2Transmission1545.8 1546 1546.2Wavelength (nm)00.511.52Transmission1542.82 1542.86 1542.9Wavelength (nm)00.050.10.150.2Transmission1545.1 1545.2 1545.3 1545.4Wavelength (nm)00.511.52Transmission1543.15 1543.2 1543.25 1543.3Wavelength (nm)00.10.20.30.4Transmission1545.4 1545.6 1545.8Wavelength (nm)00.511.52Transmission1540.2 1540.25 1540.3Wavelength (nm)00.050.10.150.2Transmission1542.8 1543 1543.2 1543.4Wavelength (nm)00.511.52Transmission(a) (b)(c) (d)(e) (f)(g) (h)DEVICE 1DEVICE 2DEVICE 3DEVICE 4exp.model modelexp.modelexp.modelexp.modelexp.exp.modelexp.modelexp.modelMODE 1 MODE 2Figure 5.4: Nonlinear transmission results for Devices 1 to 4 (top to bottom rows), for Modes 1 and 2 (leftand right columns). Bottom plots show the experimental data and top plots show the spectra predicted usingthe model with best fit parameters, where both are plotted on absolute transmission scales, shifted relativeto each other. The input powers for each spectrum corresponds power for each marker in the nonlineartransmission data plots of Fig. 5.3 (first two columns).865.2. Derivation of model functions0 200 400Input Power ( µW)00.51Transmission0 200 400Input Power ( µW)00.050.10.150.2λ(nm)0 200 400Input Power ( µW)00.51Transmission0 200 400Input Power ( µW)00.050.10.15λ(nm)10 1 10 2Input Power ( µW)10 -610 -410 -2Idler Power (µW)0 250 500Input Power ( µW)00.51Transmission0 100 200Input Power ( µW)00.51Transmission0 100 200Input Power ( µW)00.050.10.150.2λ(nm)10 1 10 2Input Power (µW)10 -610 -410 -2Idler Power (µW)10 1 10 2Input Power ( µW)10 -610 -410 -2Idler Power (µW)0 250 500Input Power ( µW)00.20.40.6λ(nm)10 0 10 1 10 2Input Power ( µW)10 -810 -6Idler Power (µW)Mode 1, exp.Mode 1, modelLegendNonlinear transmissionMode 2, exp.Mode 2, modelStimulated FWMIdler Mode 3, pump sweep, exp.Model with NL absorptionModel without NL absorptionIdler Mode 3, signal sweep, exp.Idler Mode 1, pump sweep, exp.Idler Mode 1, signal sweep, exp.NLNLNLNLDEVICE 1DEVICE 2DEVICE 3DEVICE 4Figure 5.5: Nonlinear transmission and four-wave mixing analysis results for Devices 1 to 4 (top to bottomrows) when Rth, Qabs and τcarrier are held fixed at their average values. Columns 1 and 2 contain the peaktransmission and resonant wavelength shifts, respectively, for Mode 1 (blue) and Mode 2 (green), foundexperimentally (filled circles), and using the model with best fit parameters (open markers). Column 3shows the experimental FWM idler power in Mode 1 (blue) and Mode 3 (red), as a function of pump power(triangles) when the signal power is fixed at Ps (labelled) and signal power (circles) when the pump poweris fixed at Pp (labelled). The predicted idler power are shown with thick black lines. The dashed lines showthe predicted power when nonlinear absorption is ignored.equations of motion are decoupled in the linear regime and become coupled in the nonlinear regime,where the refractive index of the silicon can, in general, be dependent on the electromagnetic energy875.2. Derivation of model functionsstored in any of the three modes.The TCMT is first considered in the case where nonlinear effects are negligible. Next, nonlinearfrequency conversion and absorption effects are introduced and included in the TCMT model, andthe nonlinear transmission and FWM formulations are reported.5.2.1 Linear modelThe cavity modes considered here are coupled to one input and output waveguide, as well as losschannels, including radiation and material absorption, as illustrated in Fig. 5.6. The electric fieldof light in the cavity mode is Em(r, t) = Re[am(t)E˘m(r)/√∫d3x12ε(r)|E˘m(r)|2], where E˘m(r) isthe unnormalized electric field mode profile and |am(t)|2 is the energy stored in the cavity. Here,am(t) is a classical amplitude, not a quantum operator. In the linear regime, the equation of motionof the amplitude, am(t), for cavity mode m is [38],maINPUT WAVEGUIDE OUTPUT WAVEGUIDEFigure 5.6: Schematic of coupled-mode theory for cavity supporting three modes. Cavity mode m has fieldamplitude am and energy |am|2. The mode is coupled with lifetime τ im to channel i. The optical power oflight propagating toward the microcavity is |sinm+|2, and the optical powers exiting through the input andoutput waveguides are |sinm−|2 and |soutm−|2, respectively.a˙m(t) = −(iωm + τ−1m )am(t) +√2τ jmsinm+(t) (5.1)where ωm is the mode resonance frequency, τm−1 =∑i(τ im)−1 where τ im are the coupling lifetimesfor channel i = {in, out, scatt, abs}, and |sinm+(t)|2 is the power that is launched into the inputwaveguide from an external source (here the output waveguide is not excited). The total qualityfactor of the mode is Qm = ωmτm/2. The power leaving via channel i, |sim−|2, is described by,sim−(t) = ζsim+(t) +√2τ imam(t). (5.2)885.2. Derivation of model functionswhere sinm+(t) 6= 0 and soutm+(t) = sscattm+ (t) = sabsm+(t) = 0. The first term on the right hand side is acontribution from input light that is directly (non-resonantly) reflected from the microcavity, whilethe second term is a contribution from light that leaks into the ith channel from energy stored inthe mth mode of the microcavity. Time reversal symmetry requires that ζ = −1 [38].In the linear transmission measurements studied in this thesis, continuous-wave excitation islaunched into the input waveguide, and the power transmitted to the output waveguide is measuredin steady-state. The drive amplitude of the input excitation is sinm+(t) = sinm+e−iωdt, and in steady-state, the electric field in the cavity oscillates at the drive frequency, such that am(t) = ame−iωdt,where the steady-state amplitude am is described by,0 = iωdmam − iωmam − τ−1m am +√2τ inmsinm,+, (5.3)andsim− = −sim+ +√2τ imam, (5.4)where soutm+ = sscattm+ = sabsm+ = 0. The steady-state energy stored in the cavity is,Um = |am|2 = 2τinm−1P inm(ωdm − ωm)2 + τ−2m, (5.5)where P inm = |sinm+|2, which has a Lorentzian lineshape. The transmission spectrum of power in theoutput channel is, directly proportional to Um,Tm(ωd) =2τoutmUm =4τ inm−1τoutm−1P inm(ωdm − ωm)2 + τ−2m, (5.6)thus it also has a Lorentzian lineshape.5.2.2 Nonlinear modelIn this section, nonlinear absorption effects are included in the model, along with nonlinear fre-quency conversion through the third order nonlinearity, χ(3). These effects are discussed here andderived in Appendix E.895.2. Derivation of model functionsThe equations of motion that include nonlinear absorption and frequency conversion are givenby [52, 71, 110],a˙1(t) =− i[ω1 + ∆ωNL1 (|a1|2, |a2|2, |a3|2)]a1(t)− τ1(|a1|2, |a2|2, |a3|2)−1a1(t) (5.7)+ iω1β1[a2(t)]2a∗3(t) +√2τ in1sin1+(t),a˙2(t) =− i[ω2 + ∆ωNL2 (|a1|2, |a2|2, |a3|2)]a2(t)− τ2(|a1|2, |a2|2, |a3|2)−1a2(t) (5.8)+ iω2β2a1(t)a3(t)a∗2(t) +√2τ in2sin2+(t),a˙3(t) =− i[ω3 + ∆ωNL3 (|a1|2, |a2|2, |a3|2)]a3(t)− τ3|a1|2, |a2|2, |a3|2)−1a3(t) (5.9)+ iω3β3[a2(t)]2a∗1(t) +√2τ in3sin3+(t).andsoutm−(t) =√2τoutmam(t). (5.10)Three main modifications are made to the linear equation of motion in Eqn. (5.1) to arrive at Eqns.(5.7) to (5.9). Nonlinear frequency shifts, ∆ωNL(|a1|2, |a2|2, |a3|2), are introduced, which depend onthe energies |am|2 loaded in each of the modes. These shifts arise to due nonlinear effects that per-turb the real part of the refractive index of the silicon. The total mode lifetimes τm(|a1|2, |a2|2, |a3|2)are also modified to include the mode energy dependent perturbations to the imaginary part of therefractive index of the silicon. In steady-state, the frequency shifts and lifetimes depend on thesteady-state mode energies, U1, U2 and U3. Finally frequency conversion/mixing terms involvingthe conversion coefficient βm are included. Each of these nonlinear effects is now briefly describedin more detail.The steady-state nonlinear total cavity lifetime, τm(U1, U2, U3) is given by,τm(U1, U2, U3)−1 = τ inm−1+ τoutm−1+ τ scattm−1+ τabsm−1+ τTPAm (U1, U2, U3)−1+ τFCAm (U1, U2, U3)−1,(5.11)where the first four terms are linear contributions discussed above, τTPAm (U1, U2, U3) is the two-photon absorption lifetime, and τFCAm (U1, U2, U3) is the free-carrier absorption lifetime. As discussed905.2. Derivation of model functions0 0.5 1Energy (J) ×10 -140241/Q×10 -50 0.5 1Energy (J) ×10 -14-6-4-202ω(rad/s)×10 11Figure 5.7: Nonlinear effects in a typical triple microcavity, as a function of energy loaded in the microcavitymode M2, under single frequency excitation. (a) The inverse of the nonlinear quality factors due to two-photon absorption (TPA), QTPA, and free-carrier absorption (FCA), QFCA, along with the linear and totalquality factors, Qlin and Qtot (dashed), respectively. The quality factors are related to the lifetimes through Q= ωτ/2. (b) Nonlinear wavelength shifts are plotted for contributions including the Kerr effect, ∆λKerr, free-carrier dispersion (FCD), ∆λFCD, and thermal effects due to power absorption through the linear materialabsorption ∆λth,abs, TPA, ∆λth,TPA, and FCA, ∆λth,FCA. The total shift is also plotted ∆λth,tot . Theshifts ∆λth,abs and ∆λth,abs have similar values for the range of energies studied, and ∆λth,abs is dotted forclarity.in Chapter 2, two photon absorption occurs when two photons are absorbed in the silicon to excitean electron above the electronic bandgap of silicon. This process is associated with the imaginarypart of the third order susceptibility, χ(3)(r) of silicon. In addition, the free-carriers excited throughTPA absorb photons. Both absorption processes depend on the energy loaded in the cavity, andreduce the total mode lifetimes. Figure 5.7(a) shows an example of the energy dependent qualityfactors (proportional to lifetimes) when mode M2 of a triple microcavity is excited by a singleexcitation frequency.915.2. Derivation of model functionsThe steady-state nonlinear frequency shift, ∆ωNL(U1, U2, U3), is given by,∆ωNLm (U1, U2, U3) =∆ωKerrm (U1, U2, U3) (5.12)+ ∆ωFCD,em (U1, U2, U3) + ∆ωFCD,hm (U1, U2, U3)+ ∆ωthermalm (U1, U2, U3)where ∆ωKerrm (U1, U2, U3) is the Kerr nonlinearity shift, ∆ωFCD,em (U1, U2, U3) and ∆ωFCD,hm (U1, U2, U3)are free-carrier dispersion shifts for electrons and holes, respectively, and ∆ωthermalm (U1, U2, U3) isthe thermal shift due to heating of the silicon. The Kerr shift is associated with the real partof the χ(3) susceptibility of silicon. Free-carrier dispersion is caused by the free-carriers excitedthrough TPA. The thermal shift depends on the total power absorbed through linear materialabsorption (τabsm ), TPA [τTPAm (U1, U2, U3)] and FCA [τFCAm (U1, U2, U3)]. Figure 5.7(b) shows theenergy dependent frequency shifts for the example introduced above.It is important to note that both free-carrier absorption and dispersion depend on the densityof free-carriers present, which in turn depends on the lifetime of the free-carriers in the vicinity ofthe cavity. The lifetime is sensitive to the carrier density, proximity of carriers to surfaces and thenature of the surfaces. An effective free-carrier lifetime, τcarrier, is often considered that capturesthe net lifetime across the microcavity, and has been found to saturate at effective free-carrierdensities & 1016 cm−3 [11, 50]. The saturated τcarrier is one of the key parameters sought after inthe characterization scheme.The frequency conversion terms in Eqns. (5.7) to (5.9) arise due to the presence of light in thethree microcavity modes (and only these three modes), which undergo mixing through the thirdorder nonlinear χ(3) of silicon. The factors involving the βm introduce nonlinear driving terms toeach equation of motion. The conversion coefficient βm is based on an overlap integral between themode profiles of the three modes and the silicon.Model functions that describe the nonlinear effects are summarized in Tables 5.2 and 5.3. In Ta-ble 5.2, expressions are given for the various contributions to ∆ωNLm (U1, U2, U3) and τm(U1, U2, U3).These include coefficients that depend on material parameters and overlap integrals between thesilicon and the mode fields, which are given in Tables 5.3 and 5.4. The frequency conversion925.2. Derivation of model functionscoefficients βm are also given in Table 5.3. The model functions are derived using perturbationtheory.In the perturbation theory, weak changes to the real and imaginary parts of dielectric constantof silicon, δεNLm (r), at mode frequency ωm, result in a complex valued change δωNLm to the resonantfrequencies ωm, where [39, 71],δωNLmωm= −12∫d3xδεNLm (r)|Em(r)|2∫d3xε(r)|Em(r)|2 . (5.13)The linear equation of motion [Eqn. (5.1)], is modified such that ωm → ωm + δωNLm , and thenonlinear equations of motion [Eqns . (5.8) to (5.9)] result. The perturbed resonant frequenciesare,δωNLm = δωKerrm + δωthermalm + δωFCDm + δωTPAm + δωFCAm + δωFWMm , (5.14)and are alternatively expressed as,δωNL1 =∆ωNL1 − i(τTPA1 (U1, U2, U3)−1 + τFCA1 (U1, U2, U3)−1)− ω1β1(a2)2(a3)∗/a1, (5.15)δωNL2 =∆ωNL2 − i(τTPA2 (U1, U2, U3)−1 + τFCA2 (U1, U2, U3)−1)− ω2β2a1a3(a2)∗/a2, (5.16)δωNL3 =∆ωNL3 − i(τTPA3 (U1, U2, U3)−1 + τFCA3 (U1, U2, U3)−1)− ω3β3(a2)2(a1)∗/a3. (5.17)The derivations for the nonlinear contributions are in Appendix E. In this appendix, the sub-set of perturbations that arise directly due to the third order polarization, P(3)m (r), through thesusceptibility χ(3), are found by re-expressing the perturbation equation above as,δω(3)mωm= −12∫d3xP(3)m (r) ·E∗m(r)∫d3xε(r)|Em(r)|2 . (5.18)where P(3)m (r) = δε(3)m (r)Em(r), and δω(3)m = δωKerrm + δωFWMm + δωTPAm .Nonlinear transmissionIn nonlinear transmission measurements, light from a CW laser excites the microcavity throughthe input waveguide, and the steady-state transmission for each fixed wavelength is recorded. A935.2. Derivation of model functionsTable 5.2: Summary of the cavity lifetimes and nonlinear frequency shifts. The summation indices m′, l, l′are over {1, 2, 3}.Cavity lifetimeLinear τ inmτoutmτ scattmτabsTwo-photon absorption τTPAm−1= ρ0∑m′αm,m′Um′Free-carrier absorption τFCAm−1= κ0∑l,l′κFCAm,l,l′UlUl′Resonant frequency shiftKerr effect ∆ωKerrm = −α0∑m′αm,m′Um′Free-carrier dispersion ∆ωFCD,em = νFCD,e0∑l,l′κFCAm,l,l′UlUl′∆ωFCD,hm = νFCD,h0∑l,l′κFCD,hm,l,l′ (UlUl′)0.8Thermal effects ∆ωthermalm = −2ΓthmnSidndTRthωm∑m′(τabsm′−1+ τTPAm′−1+ τFCAm′−1)Um′model for the nonlinear transmission is found by considering the equation of motion for a singlecavity mode in the presence of excitation sinm+(t),a˙m(t) = −i[ωm + ∆ωNLm (|am|2)]am(t)− τm(|am|2)−1am(t) +√2τ inmsinm+(t). (5.19)This is obtained from the general equations of motion [Eqns. (5.7) to (5.9)] by assuming there isexcitation at a single frequency, such that no frequency mixing occurs, and the terms with βm areset to 0. The steady-state equations for CW excitation sinm+(t) = sinm+e−iωdmt are,0 = iωdmam − i[ωm + ∆ωNLm (Um)]am − τm(Um)−1am +√2τ inmsinm+, (5.20)andsoutm− =√2τoutmam. (5.21)945.2. Derivation of model functionsTable 5.3: Summary of nonlinear coefficients derived from perturbation theory. χ(3)Si is the diagonal elementof the χ(3) tensor for silicon.α0 =ωm4 ε0 Re(χ(3)Si )ρ0 =ε0 Im(χ(3)Si )ωm4κ0 =σFCAτcarrier Im(χ(3)Si )ε0c8~nSiνFCD,e0 =ωmζeSiτcarrierε0 Im(χ(3)Si )4~nSiνFCD,h0 =ωmnSi(ζhSiτcarrierε0 Im(χ(3)Si )4~)0.8αm,m =∫Si d3x[(E˘∗m·E˘∗m)(E˘m·E˘m)+2|E˘m|4](∫d3xε(r)|E˘m|2)2αm,m′ = 2∫Si d3x[(E˘m·E˘m′ )(E˘∗m·E˘∗m′ )+(E˘m·E˘∗m′ )(E˘∗m·E˘m′ )+|E˘m|2|E˘m′ |2](∫d3xε(r)|E˘m|2)(∫d3xε(r)|E˘m′ |2)κFCAm,l,l =∫Si d3x[(E˘∗l ·E˘∗l )(E˘l·E˘l)+2|E˘l|4]ε|E˘m|2(∫d3xε|E˘m|2)(∫d3xε|E˘l|2)2κFCAm,l,l′ = 2∫Si d3x[(E˘l·E˘l′ )(E˘∗l ·E˘∗l′ )+(E˘l·E˘∗l′ )(E˘∗l ·E˘l′ )+|E˘l|2|E˘l′ |2]ε|E˘m|2(∫d3xε|E˘m|2)(∫d3xε|E˘l|2)(∫d3xε|E˘l′ |2)κFCD,hm,l,l =∫Si d3x[(E˘∗l ·E˘∗l )(E˘l·E˘l)+2|E˘l|4]0.8ε|E˘m|2(∫d3xε|E˘m|2)(∫d3xε|E˘l|2)1.6κFCD,hm,l,l′ = 20.8∫Si d3x[(E˘l·E˘l′ )(E˘∗l ·E˘∗l′ )+(E˘l·E˘∗l′ )(E˘∗l ·E˘l′ )+|E˘l|2|E˘l′ |2]0.8ε|E˘m|2(∫d3xε|E˘m|2)(∫d3xε|E˘l|2)0.8(∫d3xε|E˘l′ |2)0.8Γthm =∫Si d3xε(r)|Em(r)|2∫d3xε(r)|Em(r)|2β2 =12∫Si d3xε0 Re(χ(3)Si )[(E˘∗2·E˘∗2)(E˘1·E˘3)+2(E˘∗2·E˘1)(E˘∗2·E˘3)](∫d3xε(r)|E˘2|2)(∫d3xε(r)|E˘1|2)1/2(∫d3xε(r)|E˘3|2)1/2 = 2β∗1 = 2β∗3The steady-state energy stored in the cavity is,Um =2τ inm−1P inm(ωdm − [ωm + ∆ωNLm (Um)])2 + τm(Um)−2, (5.22)and the absolute cavity transmission spectrum is,Tm(ωdm) =2τoutmUm =4τ inm−1τoutm−1P inm(ωdm − [ωm + ∆ωNLm (Um)])2 + τm(Um)−2. (5.23)955.2. Derivation of model functionsTable 5.4: Summary of linear and nonlinear silicon material constants. Dispersive constant are given nearλ = 1540 nm.Parameter Value Units SourcenSi 3.478 [67]n2,Si 4.4× 10−18 m2W−1 [11, 23]βTPASi 8.4× 10−12 mW−1 [11, 23]Re(χ(3)Si ) 1.88× 10−19 m2V−2 Calculated†Im(χ(3)Si ) 4.4× 10−20 m2V−2 Calculated†σFCASi 14.5× 10−22 m2 [21, 82]ζeSi 8.8× 10−28 m3 [21, 82]ζhSi 4.6× 10−28 m3 [21, 82]dnSi/dT 1.86× 10−4 K−1 [18]† χ(3)Si =4ε0c2nSi(ω)23ω(ωc n2,Si(ω) +iβTPASi (ω)2)[51]The nonlinear transmission in Eqn. (5.23) depends on the cavity energy Um, which is found bynumerically solving Eqn. (5.22). For a certain range of driving frequencies and input powers, thereare three solutions to Um, and the cavity exhibits bistable behaviour, otherwise there is a singlesolution.The nonlinear transmission solutions for the microcavity studied in Section 2.2 are plotted as afunction of wavelength, for two different input powers in Fig. 5.8(a). For the higher input power,the spectrum has a range of wavelengths where three solutions exist, marked by +, 2 and ×.While the top (+) and bottom (×) branches are stable, the middle branch is not (2) [20]. Thetransmission spectrum depend on the sweep direction, as shown in Fig. 5.8(b). In this example,the forward sweep transmission follows the top stable branch and then drops down to the stablesingle solutions (◦) at higher wavelengths. The backward sweep transmission follows the bottomstable branch until it jumps up the to the stable single solutions at lower wavelengths.Stimulated four-wave mixingIn stimulated four-wave mixing measurements, the pump and signal CW lasers excite two of themicrocavity modes, and idler photons are generated in the third mode. Here, temporal coupled-965.2. Derivation of model functions1545.4 1545.5 1545.6 1545.7Wavelength (nm)050100150200Transmission (µW)1545.4 1545.5 1545.6 1545.7Wavelength (nm)050100150200Transmission (µW)(a) (b)Figure 5.8: Numerical solutions for the nonlinear transmission predicted for triple microcavity Device 3,Mode 2. (a) Solutions for input powers P in2 = 100 and 300 µW. Circles show where a single solution isfound, while the top, middle and bottom branches in the bistable regime are marked by +, 2 and ×,respectively. (b) Solutions for P in2 = 300 µW are shown as in (a), and the transmission spectra predicted forforward (solid black) and backward (dashed red) wavelength sweeps are also shown.mode theory is used to predict the idler generation rate in mode M3 in steady-state when a cavity isexcited by continuous-wave pump tuned near M2 and a signal tuned near M1. The other signal/idlerconfiguration (M3/M1) is described by the same theory, but with 1→ 3 and 3→ 1. The equationsof motion that describe this excitation scheme in the nonlinear regime are given by,a˙1(t) =− i[ω1 + ∆ωNL1 (|a1|2, |a2|2, |a3|2)]a1(t)− τ1(|a1|2, |a2|2, |a3|2)−1a1(t) (5.24)+ iω1β1[a2(t)]2a∗3(t) +√2τ in1sin1+(t),a˙2(t) =− i[ω2 + ∆ωNL2 (|a1|2, |a2|2, |a3|2)]a2(t)− τ2(|a1|2, |a2|2, |a3|2)−1a2(t) (5.25)+ iω2β2a1(t)a3(t)a∗2(t) +√2τ in2sin2+(t),a˙3(t) =− i[ω3 + ∆ωNL3 (|a1|2, |a2|2, |a3|2)]a3(t)− τ3(|a1|2, |a2|2, |a3|2)−1a3(t) (5.26)+ iω3β3[a2(t)]2a∗1(t).For CW signal and pump driving amplitudes sin1+e−iωd1 t and sin2+e−iωd2 t, respectively, the steady-state cavity M1 and M2 amplitudes oscillate at their respective drive frequencies such that a1(t) =a1e−iωd1 t and a2(t) = a2e−iωd2 t. This results in idler amplitude a3(t) = a3e−iωd3 t, where ωd3 = 2ωd2−ωd1 ,which comes from the photon generation term (∼ (a2)2a∗1). The steady-state field amplitudes are975.2. Derivation of model functionsdescribed by,0 = iωd1a1 − i[ω1 + ∆ωNL1 (U1, U2, U3)]a1 − τ1(U1, U2, U3)−1a1 +√2τ in1sin1+ (5.27)0 = iωd2a2 − i[ω2 + ∆ωNL2 (U1, U2, U3)]a2 − τ2(U1, U2, U3)−1a2 +√2τ in2sin2+ (5.28)0 = iωd3a3 − i[ω3 + ∆ωNL3 (U1, U2, U3)]a3 − τ3(U1, U2, U3)−1a3 + iω3β3(a2)2(a1)∗. (5.29)and Eqn. (5.21). It is important to note that here the undepleted pump approximation has beenapplied to remove the frequency conversion terms from the M1 and M2 equations of motion. Thisassumes that the number of photons generated or lost in the four-wave mixing process is only verysmall compared to the total number of photons in the cavity, which is a good assumption for themeasurements studied here. This assumption is crucial to being able to find a solution for thesteady-state idler mode amplitude a3.The steady-state idler output power, P out3 = |soutm−|2, is given by,P out3 =2τout3ω23|β3|2(U2)2U1(ωd3 − [ω3 + ∆ωNL3 (U1, U2, U3)])2 + τ3(U1, U2, U3)−2, (5.30)where U1 and U2 are the cavity mode energies given by,Um′ =2τ inm′−1P inm′(ωdm′ − [ωm′ + ∆ωNLm′ (U1, U2, U3)])2 + τm′(U1, U2, U3)−2, (5.31)for m′ = {1, 2}. The M1 and M2 cavity mode energies are essentially the same as those found forthe nonlinear transmission in Eqn. (5.22), except here ∆ωNLm′ and τm′ depend on all mode energies,although here the dependence on U3 is ignored as the idler cavity energy is much smaller than thesignal and pump cavity energies. It is possible to calculate the idler power P out3 once U1 and U2 arefound. While there is no analytical solution for these, they are found numerically using an iterativemethod, that achieves consistency between U1, U2,∆ωNLm′ (U1, U2) and τm′(U1, U2).In the simplified case where the microcavity modes have equally spaced resonant frequencies,and the signal and pump powers are low enough that nonlinear absorption is negligible, such that985.3. Impact of microcavity parametersωd1 = ω1, ωd2 = ω2 and ωd3 = ω3 = 2ω2 − ω1, the idler power is,P out3 = 256|β3|2ω3ω22ω1(Q2)4(Q1)2(Q3)2(P in2 )2P in1(Qin2 )2Qin1 Qout3, (5.32)The output power of generated photons is optimized for high Q cavities with efficient loading intothe pump and signal modes, as well as efficient unloading from the idler (generated photon) mode.5.3 Impact of microcavity parametersThe least-squares fitting procedure used to extract the best fit microcavity parameters involvesmodel functions of the experimentally measured quantities that are based on the theory presentedin Section 5.2. The sensitivities of the measured quantities on key microcavity parameters areexplored here, to help develop intuition for the competing effects at play, before presenting themodel functions, analysis procedure and results in Section 5.4. Figures 5.9 and 5.10 include plots ofthe nonlinear transmission spectral features and the FWM idler power, respectively, as a function ofone parameter, while the rest of the microcavity parameters in the calculation are held fixed at thebest fit values for Device 3. The best fit parameters values extracted for Device 3 are reported inTable 5.7 and Fig. 5.17. The parameters are also described in Table 2.1 and discussed in Chapter2. The competing parameter dependencies shown in this section illustrate why it is difficult tountangle the various contributions to extract parameter values from experimental data.In Figs. 5.9(a) and (b), the peak relative transmission is plotted for excitation of M2, withan input power of 300 µW. The transmission decreases with increasing waveguide coupling ratioηwg2 = τout2 /τin2 because light is more efficiently loaded in the microcavity when ηwg2 is large, leadingto greater nonlinear absorption. The transmission also decreases with increasing effective free-carrier lifetime, τcarrier, because the steady-state carrier density is higher, leading to greater free-carrier absorption. The peak wavelength shift is plotted in Figs. 5.9(c)-(f). The shift increases afunction of both ηwg2 and Rth, due to increased heating of the silicon. The shift is fairly insensitiveto the quality factor associated with linear material absorption, Qabs = ωτabs/2, because thisabsorption mechanism is weak relative to the others at play. The shift is also fairly insensitive toτcarrier because the blue-shift of free-carrier dispersion competes against the red-shift of thermal995.3. Impact of microcavity parameters0 5 10Qabs ×10600.20.4λNL(nm)0 20 40 60Rth(K/mW)00.20.4λNL(nm)0.5 1 1.5 2ηwg200.20.4λNL(nm)0.5 1 1.5 2ηwg20.40.50.60.70.8TNL((((((0.5 1 1.5τcarrier(ns)00.20.4λNL(nm)0.5 1 1.5τcarrier(ns)0.40.50.60.70.8TNLFigure 5.9: Nonlinear transmission spectral features calculated for an excitation power of 300 µW, usingbest fit parameter values for M2 of Device 3 (given in Table 5.7 and Fig. 5.17), while one parameter isfree to vary on the x-axis. (a)-(b) Peak relative transmission, TNL. (c)-(f) Peak wavelength shift, ∆λNL.ηwg2 = τout2 /τin2 and Qabs = ωτabs/2.dispersion from free-carrier absorption (see Fig. 5.7).In Fig. 5.10, the FWM idler powers are calculated as a function of key parameters, with pumpand signal powers of 84 µW and 43 µW, respectively, for two configurations, one where M1/M3correspond to signal/idler modes, and the other where M3/M1 correspond. In Fig. 5.10(a), theidler power generated in idler mode M3 increases with ηwg1 , while that in M1 decreases. This isbecause a large ηwg1 loads energy efficiently into signal mode M1, and unloads energy inefficientlyout of idler mode M1. An equivalent argument can be made for ηwg3 , where M1 and M3 areswapped. Meanwhile, a large ηwg2 loads the pump energy efficiently, thus increases the idler power1005.4. Analysis procedure0 0.5 1 1.5 2ηwg310 -510 -410 -3Idler Power (µW)0 0.5 1 1.5 2ηwg110 -510 -410 -3Idler Power (µW)0.5 1 1.5τcarrier(ns)10 -510 -410 -3Idler Power (µW)0.5 1 1.5 2ηwg210 -510 -410 -3Idler Power (µW)(a) (b)(c) (d)Idler M3Idler M1Idler M3Idler M1Idler M3Idler M1Idler M3Idler M1Figure 5.10: Four-wave mixing idler power calculated for pump and signal excitations of 84 µW and 43µW, respectively, using best fit parameter values for Device 3 (given in Table 5.7 and Fig. 5.17), whileone parameter is free to vary on the x-axis. Two excitation configurations are included: M1/M3 are thesignal/idler modes (red), and M3/M1 are the signal/idler modes (blue). ηwgm = τoutm /τinm .for both FWM configurations. The idler power is relatively insensitive to τcarrier because free-carrierabsorption is weak at these excitation powers.5.4 Analysis procedureAs discussed in Chapter 2, the model functions that describe linear transmission, nonlinear trans-mission and stimulated four-wave mixing involve 15 unknown microcavity parameters (see Table2.1) that are found using least-squares calculations. The model functions are based on the temporalcoupled-mode theory Eqns. [(5.7)-(5.9)].This analysis process requires that the nonlinear coefficients in Table 5.3 (α, κFCA, κFCD,h,Γthmand β) are first calculated using finite-difference time domain simulations. The simulated coeffi-cients used for Devices 1 to 4 are summarized in Appendix F.1015.4. Analysis procedure5.4.1 Calculations of the sum of the squared differences, X2Nonlinear transmission X2TIn the nonlinear transmission analysis, the experimental wavelength shifts, ∆λNLm , and peak trans-mission values, TNLm , measured as a function of input power for Modes 1 and 2, with forwardwavelength sweeps, are involved in the least-squares fitting scheme. The model function that simul-taneously predicts ∆λNL1 , ∆λNL2 , TNL1 and TNL2 depends on 13 parameters: λ1, λ2, τin1 , τin2 , τout1 , τout2 ,τ scatt1 , τscatt2 , τabs, τcarrier, Rth, φin and φout. However, given the information gained from the lineartransmission measurements, these are reduced to seven unknown parameters: ηwg1 , ηwg2 , Qabs, Rth,τcarrier, φin, and φout, where ηwgm = τoutm /τinm and Qabs = ωτabs/2 (ω = ω2 ' ω1 ' ω3).The sum of the square differences, X2T, is calculated as,X2T =∑m∑i[(∆λNLm,i −∆λNL,modelm,iδ∆λNLm,i)2+(TNLm,i − TNL,modelm,iδTNLm,i)2](5.33)where the first sum is over both modes m = {1, 2} , and the second sum is over the number ofinput powers considered in each data set. Only input powers that are sufficiently high, such thatthe free-carrier lifetime is saturated, are considered. The uncertainties δTNLm /TNLm,i = 0.04, andδ∆λm(Pinm,i) = 0.002 for m = 1, and δ∆λNLm,i = 0.005 for m = 2. The relative uncertainty in TNLm,i isdiscussed in Appendix D, and the uncertainty in the wavelength shift is the sweep step size.For each set of these seven parameters, the following procedure is followed to calculate X2T:1. The experimental linear transmission data[Tlin1 (φin, φout), Tlin2 (φin, φout)]and nonlinear trans-mission data[TNL1 (φin, φout), TNL2 (φin, φout)]are calculated (see Appendix D).2. The following lifetimes are calculated as,τ inm =√4(τ linm )2Tlinm (φin, φout)ηwgm, (5.34)τoutm =√4(τ linm )2ηwgmTlinm (φin, φout), (5.35)1025.4. Analysis procedureandτ scattm =(τ linm−1 − τabs−1 −√Tlinm (φin, φout)4(τ linm )2ηwgm(ηwgm + 1))−1. (5.36)3. The following model functions for peak transmission and the wavelength shift are calculatedfor both modes (m = 1, 2), at each input power “i”, as,TNL,modelm,i =4τ inm−1τoutm−1(τ linm−1+ ρ0αm,mUm,i + τcarrierσFCA Im(χ(3)TPA)ε0c8~nSi κFCAm,m,m(Um,i)2)2 , (5.37)and ∆λNL,modelm,i = λm∆ωNL,modelm,i /ωm, where∆ωNL,modelm,i =− α0αm,mUm,i (5.38)+ τcarrierωmζeSiε0 Im(χ(3)TPA)4~nSiκFCAm,m,m(Um,i)2+ (τcarrier)0.8ωmnSi(ζhSiε0 Im(χ(3)TPA)4~)0.8κFCD,hm,m,m(Um,i)1.6− 2ΓthmnSidndTRth(τabs−1 + ρ0αm,mUm,i (5.39)+ τcarrierσFCA Im(χ(3)TPA)ε0c8~nSiκFCAm,m,m(Um,i)2)Um,i,andUm,i =τoutm2TNLm,i. (5.40)Equations (5.37) and (5.38) are based the nonlinear shifts and lifetimes summarized in Table5.2, and the nonlinear transmission in Eqn. (5.23) taken on resonance. Here the experimentaltransmission is used to calculate the microcavity energy that enters the model calculation.Four-wave mixing X2FWMThere are four sets of experimental four-wave mixing data measured for each device that areinvolved in the least-squares analysis, as shown in the final column of Fig. 5.3. Here the idlerpower is measured separately as a function of pump power and signal power, for both of thetwo signal/idler excitation configurations. The model function that describes the idler powers1035.4. Analysis proceduredepends on 16 parameters: λm, τinm , τoutm , τscattm , τabs, τcarrier, φin and φout, where m = 1, 2 and 3.However, with information gained from linear transmission measurements, this set is reduced toseven parameters: ηwg1 , ηwg2 , ηwg3 , Qabs, τcarrier, φin, and φout.The sum of the square differences is calculated as,X2FWM =∑q∑i(P exp,idlerq,i − Pmodel,idlerq,iδP exp,idlerq,i)2(5.41)where q is the data set number (q = {1, 2, 3, 4} here), i is ith data point in set q, and P exp,idlerq,iand Pmodel,idlerq,i are the ith experimental and calculated idler powers in data set q, respectively.The calculation of the relative uncertainty in the idler power, δP exp,idlerq,i /Pexp,idlerq,i is describedin Appendix D. The following procedure is followed to calculate X2FWM for a given set of sevenparameters:1. The experimental input pump and signal powers are appropriately calculated given the φin,and the idler power is appropriately calculated using both φin and φout (see Appendix D).2. The τ inm , τoutm and τscattm lifetimes are calculated using Eqns. (5.34), (5.35) and (5.36).3. The idler power model function is calculated is based on Eqn. (5.30), which is rewritten as,Pmodel,idler =2τoutidler(ωidler)2|βidler|2(Upump)2Usignal(δωidler)2 + τidler(Usignal, Upump)−2, (5.42)where “pump”, “signal”, and “idler” label the modes excited by pump, signal and idler pho-tons, and δωidler is the detuning between the idler frequency and the idler resonant modefrequency. For each excitation scheme, the idler power is calculated by first determining thepump and signal energies, Upump and Usignal, which are then used to calculate the energydependent idler mode lifetime, τidler(Usignal, Upump). The Usignal and Upump energies are cal-culated with an iterative process, for each input power pair of a data set, in order to accountfor the interplay between the nonlinear effects of the two modes. This process is describedbelow and is summarized in the flowchart in Fig. 5.11:(a) The pump and signal cavity peak energies, U tmp,1signal and Utmp,1pump are first estimated as the1045.4. Analysis procedureSolve forCalculateSolve forSolve forCalculateCalculateConverged?yesnoFigure 5.11: Four-wave mixing idler power calculation flowchart. The solid arrows indicate the step sequenceand the dashed arrow indicates an iterative loop that continues until convergence is met.energies that would be loaded by independent probing of the pump and signal, and arefound by numerically solving Eqn. (5.22),U tmp,1m =2τ inm−1P inm(δωm)2 + τm(Utmp,1m )−2, (5.43)where δωpump = 0, and δωsignal = 0 when the signal mode is the high Q M1 mode, andδωsignal = 2ω2 − ω1 − ω3 when it is the low Q M3 mode. This is consistent with theexperimental wavelength tuning scheme described in Section 2.3.(b) Using the U tmp,1pump and Utmp,1signal found in the previous step, the energies are recalculated bynumerically solving for U tmp,2pump and Utmp,2signal below:U tmp,2pump =2τ inpump−1P inpump(δωpump)2 + τpump(Utmp,2pump , Utmp,1signal , Uidler = 0)−2 , (5.44)1055.4. Analysis procedureandU tmp,2signal =2τ insignal−1P insignal(δωsignal)2 + τsignal(Utmp,2signal , Utmp,1pump , Uidler = 0)−2, (5.45)withτm(Utmp,xpump , Utmp,xsignal , Uidler = 0)−1= τ linm−1+ ρ0∑m′αm,m′Um′ + κ0∑l,l′κFCAm,l,l′UlUl′ ,where m′, l, l′ = {1, 2, 3}, and the superscripts “x” in U tmp,xpump and U tmp,xsignal in Eqn. (5.46)are substituted with the appropriate numbers that appear in Eqns. (5.44) and (5.45).(c) Step 2 is iterated until Upump and Usignal have converged. Typically only one or twoiterations are necessary.(d) The total lifetime of the idler mode is calculated, given the presence of nonlinear lossesinduced by the energy in the pump and signal modes. It is given by,τidler−1 = τ linidler−1+ τTPAidler−1+ τFCAidler−1(5.46)= τ linidler−1+ ρ0∑m′αidler,m′Um′ + κ0∑l,l′κFCAidler,l,l′UlUl′ ,where the sums of m′, l and l′ are done over only the pump and signal modes. Thisassumes that the idler mode contributions to τTPAm and τFCAm are negligible.(e) The idler power is calculated using Eqn. (5.42), with δωidler = 2ω2 − ω1 − ω3 when theidler corresponds to the low Q mode, and δωidler = 0 when it corresponds to the high Qmode. The use of δω3 = 2ω2−ω1−ω3 here and in Step 1 assumes that the detuning doesnot change, even in the presence of nonlinear absorption at excitation high powers. Thisassumption is valid if all three modes shift by the same amount, and provided laser andfilter wavelengths appropriately track the high Q mode (and idler) resonant wavelengths,as is done experimentally (see Appendix A). The assumption of equally shifting modesappears to be valid to first order, as a uniform change in the refractive index is predictedto shift all modes, to within 0.3 % of each other. This was verified by examining Γthmgiven in Appendix F.1065.4. Analysis procedure5.4.2 Least-squares analysisOne approach to extracting the 15 unknown microcavity parameters and the 2 unknown Fabry-Perot parameters from the nonlinear experimental data would be to find the parameter set {ηwg1 ,ηwg2 , ηwg3 , Qabs, τcarrier, Rth, φin, φout} that minimizes the total squared differenceX2tot = X2T+X2FWM.These 8 fit parameters, along with nine pieces of information extracted from the linear transmissionmeasurements complete the characterization. Unfortunately, this approach is not practical becauseeach computation of X2FWM takes a relatively large amount of time as the model idler powers arenot calculated directly, instead they require that multiple equations are solved numerically.Alternatively, to reduce the computation time, a more practical approach is taken where X2FWMis evaluated over only ηwg1 , ηwg2 and ηwg3 , which have the greatest impact on the idler power model.In this approach, for each set of {ηwg1 , ηwg2 ,ηwg3 }, the remaining parameters that enter the model(Qabs, τcarrier, φin, φout), are evaluated at the parameter values that minimize X2T for this {ηwg1 , ηwg2 }pair. The total X2tot = X2T +X2FWM is then minimized to extract the set of best fit parameters.In the main approach taken in this work, the computation time is reduced further by separatelyminimizing X2T and X2FWM, in an iterative process that achieves consistency between the twocalculations. This iterative process was conceived after careful consideration of the X2T and X2FWMtopographies across the relevant parameter spaces.Figure 5.12 plots the base 10 logarithm of the X2T for Device 2 as a function of fit parametersfrom the set {ηwg1 , ηwg2 , Qabs, Rth, τcarrier, φin, φout}. These results are generally representative of theresults for all devices. For each plot, the parameters on the x and y axis are held fixed, and theminimum X2T is found across all other fit parameters for each (x, y) pair. The same color scaleis used for each plot (shown to the right of each row), and the color is left blank (white) whenthere are no viable parameter sets (energy conservation limits the ranges of the cavity lifetimes).Figures 5.12(a) and (b) show that Rth decreases with increasing ηwg1 and ηwg2 , respectively. Thesecorrelations arise due to the dependence of the nonlinear wavelength shifts and peak transmissionson the energy loaded in the cavity. The microcavity energy increases with ηwgm , and Rth must bereduced to maintain the same nonlinear wavelength shifts. Similarly, τcarrier is also reduced forincreasing ηwgm , in order to maintain both the same shifts and peak transmissions, as shown in Figs.1075.4. Analysis procedure22.533.50 1 2 3Qabs x 10630405060Rth (W/mK)(a) (b)(e) (g)(f)(c) (d)22.533.5(K/mW)Figure 5.12: Nonlinear transmission analysis example results for Device 2. The base 10 logarithm X2T isplotted across the parameter set {ηwg1 , ηwg2 , Qabs, Rth, τcarrier, φin, φout}. For each parameter pair on the xand y axes, the minimum log 10(X2T) across all other parameters is plotted. Blank results show where noviable parameter sets exist due to a lack of energy conservation. Each plot has the same color scale (shownto the right of each row). There is a total of 13 degrees of freedom in the X2T calculation.5.12(c) and (d). As a result of these correlations, ηwg2 and ηwg1 are also correlated, as shown in Fig.5.12(e). The minimization of X2T as a function of Qabs, φin and φout is relatively weak comparedto the other parameters, as is shown in Figs. 5.12(f) and (g), where the minimum X2T vary by lessthan an order of magnitude over the majority of the parameters spaces, in contrast to Figs. 5.12(a)and (e), where the minimum X2T vary by ∼ 2 orders of magnitude over the parameter spaces.Figures 5.13 shows the X2FWM as a function of parameters ηwg1 , ηwg2 and ηwg3 . For each (ηwgx , ηwgy )pair in the plots, the minimum X2FWM across the third parameter is shown. The values forτcarrier, Qabs, φin and φout are held fixed at the values that minimize X2T in Fig. 5.12. Figure 5.13(a)shows a strong correlation between ηwg3 and ηwg1 . The physics of this correlation is explained byfirst considering the data sets where M1 is the signal mode and M3 is the idler mode. When ηwg1is increased, the signal energy in the microcavity is increased, and in order to compensate, ηwg3 is1085.4. Analysis procedure(a) (b)Figure 5.13: Four-wave mixing example results for Device 2. The X2FWM is plotted across the parameterset {ηwg1 , ηwg2 , ηwg3 }, while Qabs, τcarrier, φin and φout are held fixed at the values that minimize X2T. For eachparameter pair on the x and y axes, the minimum X2FWM across the other parameter is plotted. Blankresults show where no viable parameter sets exist due to a lack of energy conservation. Each plot has thesame color scale (shown to the right). There are 55 degrees of freedom in the X2FWM calculation.also increased such that coupling of the generated idler photons to the output channel is weakened.The same arguments apply when the signal and idler modes are exchanged. Figure 5.13(b) showsthat X2FWM is minimized as a function of ηwg2 . Only a very weak correlation exists between ηwg2and ηwg3 , which is unsurprising as both idler powers scale with ηwg2 in the linear regime and thereis no symmetry between the pump and the signal/idler.It is important to note that there is no clear minimum for X2T, nor X2FWM, when they areconsidered separately. However, their sum, X2tot, does have a clear minimum, which is why it iscrucial that both nonlinear transmission and four-wave mixing data are included in this analysis.The X2tot is plotted as a function of ηwg1 , ηwg2 , and ηwg3 in Figure 5.14, along with X2T and X2FWM.The minimization of X2tot can be understood by first noting that X2FWM in Fig. 5.13(b) is minimizedas a function of ηwg2 . Given the correlations in X2T shown in Figs. 5.12(a)-(e), the minimizationas a function of ηwg2 is sufficient to minimize X2T over the rest of the parameters involved in thisanalysis. In turn, given the correlations in X2FWM shown in Fig. 5.13(a), the minimization of X2Tas a function of ηwg1 results in a minimization of X2FWM over ηwg3 .The iterative method employed in this thesis that accounts for these correlations and involvesindependent minimizations of X2T and X2FWM, yields essentially identical results as the X2tot mini-mization (calculated with the reduced χ2FWM parameter space). It is outlined here:1. The set of seven parameters {ηwg1 , ηwg2 , Rth, τcarrier, Qabs, φin, φout} that minimize X2T are found1095.4. Analysis procedure(a) (b) (c)(d) (e) (f)LegendLegendFigure 5.14: Nonlinear transmission results for Device 2. Plots (a)-(c) show X2tot, X2FWM , and X2T. Theparameter on each x axis is fixed and the minimum X2 over all other fit parameters involved in the respectiveanalysis is shown. X2T is calculated as a function of ηwg1 , ηwg2 , Rth, τcarrier, Qabs, φin and φout. X2FWM iscalculated as a function of ηwg1 , ηwg2 , ηwg3 , while τcarrier, Qabs, φin and φout are held fixed at the values thatminimize X2T. X2tot is the sum over X2T and X2FWM. Plots (d)-(e) show X2tot and the X2 minimization thatis used to determine the fit parameter on the x axis using the iterative fitting process, where X2T in (d) iscalculated for fixed ηwg2 and X2FWM in (e) and (f) is calculated with fixed ηwg1 .from the nonlinear transmission least-squares analysis. There is an element of randomness tothis, as X2T is not well minimized (ηwg2 is not yet constrained).2. The best fit parameters found in Step 1 are applied to the FWM analysis as fixed values,excluding ηwg2 , which is introduced as a free fit parameter, along with ηwg3 . The (ηwg2 , ηwg3 )pair that minimize X2FWM is found.3. The nonlinear transmission analysis is repeated with ηwg2 held fixed at the FWM best fitvalue, and the set of six parameters {ηwg1 , Rth, τcarrier, Qabs, φin, φout} that minimize X2T arefound.4. Steps 2 and 3 are repeated until the best fit ηwg2 from the FWM least-squares converges. This1105.4. Analysis proceduretypically only requires one or two iterations.Figure 5.15 illustrates how fixing the value of ηwg2 in the X2T analysis of Device 2 results in aminimization across the remaining fit parameters. In this example, ηwg2 = 3, which is the best fitvalue extracted from the FWM analysis. The white contour lines show where X2T = min(X2T) + 1.The extremes of the contours in the x and y directions show the limits of the ±1σ uncertaintyrange for the parameters on the respective axes.A correlation between Rth and Qabs is also revealed in Fig. 5.15(c), where an increase in Rthresults in an increase in Qabs, in order to maintain the same thermal shift (larger Qabs results inless linear material absorption). Beyond a certain point, the Rth essentially becomes independentof Qabs (where the X2T trough flattens towards horizontal), because at some point Qabs is so highthat linear material absorption is insignificant compared to nonlinear absorption through the TPAand FCA processes.1 2 3Qabs x10635404550Rth (W/mK)406080100(a) (c)(b) (d)(K/mW)Figure 5.15: Nonlinear transmission analysis example results for Device 2. The X2T is plotted across theparameter set {ηwg1 , Qabs, Rth, τcarrier, φin, φout}, when ηwg2 = 3 is held fixed at the best fit values from thefour-wave mixing analysis. For each parameter pair on the x and y axes, the minimum X2T across all otherparameters is plotted. Each plot has the same color scale (shown to the right), where X2T > 100 at set themaximum of the color scale range. White lines contour X2T = min(X2T) + 1. There is a total of 12 degreesof freedom in the X2T calculation.The Xtot minimization curves for Device 2 are compared to those found using this iterativemethod in Fig. 5.14(d)-(f). In Fig. 5.14(d), the minimization of X2T with respect to ηwg1 is shown,while in Figs. 5.14(d) the minimization of X2FWM with respect to ηwg2 and ηwg3 is shown. There isclose agreement between the values of ηwg1 , ηwg2 and ηwg3 that minimize X2tot and the X2’s from theiterative process (X2T or X2FWM).1115.4. Analysis procedureThe resulting Device 2 best fit values forRth, Qabs and τcarrier, given in Table 5.5 are consequentlyvery close as well. In the iterative method, the ±1σ uncertainties in ηwg1 , Rth, Qabs, τcarrier, φinand φout are all estimated by finding the parameter ranges where X2T < min(X2T) + 1, while theuncertainties in ηwg2 and ηwg2 are estimated from min(X2FWM) + 1. For the X2tot method examplegiven above, the uncertainties in ηwg1 , ηwg2 and ηwg3 are estimated from min(X2tot) + 1, while thosefor Rth, Qabs, τcarrier, φin are estimated from looking at min(X2T) + 1 when ηwg1 , ηwg2 are limited tobe within the 1σ uncertainty.Table 5.5: Best fit parameters found from an analysis involving the minimization of X2tot, and from a separateanalysis where an iterative approach is taken to find X2T and X2FWM. Rth is the thermal resistance, τcarrieris the effective saturated free-carrier lifetime and Qabs is the quality factor associated with linear materialabsorption.X2tot method Iterative methodBest fit −1σ +1σ Best fit −1σ +1σRth (K/mW) 40.7 39.3 42.0 40.7 39.8 43.0τcarrier (ns) 0.79 0.783 0.828 0.78 0.754 0.804Qabs(×106) 1.16 1.0 1.34 1.22 1.1 1.77For the analyses of the four devices, the typical parameter space used for the iterative least-squares minimization approach is shown in Table 5.6. In order to satisfy energy conservation, giventhe linear transmission τ linm and Tlinm , ηwgm is constrained to be between,min ηwgm = 1√Tlinm−√1Tlinm− 12 (5.47)andmax ηwgm = 1√Tlinm+√1Tlinm− 12 . (5.48)Parameter sets that result in a negative τ scattm calculated by Eqn. (5.36) are discarded. Once theminimum χ2T and χ2FWM have been identified, the parameter space is typically narrowed and refinedto achieve better resolution.1125.4. Analysis procedureTable 5.6: Typical parameter space initially tested for nonlinear transmission analysis. Qabs = ωmτabs/2.Parameter Min Max # of pointsηwg1 min ηwg1 5 20ηwg2 min ηwg2 max ηwg2 20φin −pi pi 7φout −pi pi 7Rth 5 K/mW 100 K/mW 20Qabs 1× 105 4× 106 20τcarrier 0.3 ns 1.5 ns 205.4.3 Best fit parametersThe best fit parameters and the±1σ uncertainties are plotted in Fig. 5.16, where the gray rectanglesshow the±1σ uncertainties (see Appendix G forX2 minimization plots for all devices). Red markersindicate the parameters for which the relevant X2 (X2T or X2FWM) is not minimized. The minimumX2T = X2T/(NT − kT) and X2FWM = X2FWM/(NFWM − kFWM) are also plotted for the four devicesin Fig. 5.16, where NT and NFWM are the total number of data points in the fits, and kT = 6 andkFWM = 2 are total number of fit parameters.These characterization results show that the parameters Rth, Qabs and τcarrier are generallyconsistent for Devices 2 to 4, while for Device 1, Rth and Qabs are significantly higher than theothers. Given that each device is processed the same way and nominally support similar modes(mode wavelengths and quality factors), these three parameters are expected to be close for eachdevice. In contrast, the waveguide coupling ratios differ more from device to device, as they aremore sensitive to structural non-uniformities. The inconsistency in the Rth and Qabs parametersof Device 1 is likely related to the fact that X2FWM does not reach a minimum as a function of ηwg2 ,which is a key parameter in this iterative approach.Table 5.7 reports the weighted means of Rth, Qabs and τcarrier, across Devices 2 to 4, where theweights are taken to be 1/[(δXupper − δXlower)/2]2. This ignores asymmetries in the parameteruncertainties, which is not expected to have a large impact, given that the unweighted mean valuesare close to the weighted values (Rth = 41.4 K/mW, Qabs = 1.5× 106 and τcarrier = 0.97 ns).1135.4. Analysis procedure1 2 3 4Device0204060Rth(K/mW)1 2 3 4Device0123η2wg1 2 3 4Device00.020.040.06η3wg1 2 3 4Device0246Qabs10 61 2 3 4Device012η1wg1 2 3 4Device012φin(π)1 2 3 4Device012φout(π)1 2 3 4Device00.51τcarrier(ns)(g) (h)(a) (b)(d)(e) (f)(c)1 2 3 4Device02468min(χ2 T)1 2 3 4Device0510152025min(χ2 FWM)(i) (j)XXFigure 5.16: Nonlinear characterization results. (a)-(h) Best fit parameters (markers), with±1σ uncertainties(rectangles). Parameters that minimize the X2 are shown with black markers, and those that do not are red.(i)-(j) The minimum X2= X2/(N − k) from the nonlinear transmission and the four-wave mixing analysesare shown, where N is the number of data points in the fit and k is the number of fit parameters. Fits whereX2 does not reach a minimum with respect to all parameters are marked as red.To check for consistency between the devices, the least-squares analyses are repeated withRth, Qabs and τcarrier held fixed at the average values. The fit results from these analyses arecompared to the original results in Fig. 5.17, and the agreement for Devices 2 to 4 is generallyvery good, while that for Device 1 is poor. For this analysis, best fit values for ηwg1 , ηwg2 and ηwg3are extracted by minimizing the total X2tot instead of using the iterative method, and φin and φout1145.4. Analysis procedureTable 5.7: Summary of mean best fit values found from analyses of linear transmission, nonlinear transmissionand four-wave mixing measurements for Devices 2 to 4.Parameter Weighted MeanRth 40.5± 0.4 K/mWQabs 1.26± 0.04× 106τcarrier 0.92± 0.02 nsare held fixed at the original best fit values. The reduced parameter space makes it possible torun this calculation over a practical time scale. Figure 5.18 shows plots of the X2tot (solid), X2T(dash-dotted) and X2FWM (dashed) as a function of the fit parameters for Device 2.1 2 3 4Device0204060min(χ2 tot)1 2 3 4Device00.020.040.060.08η3wg1 2 3 4Device012345η1wg1 2 3 4Device0123η2wg(c) (d)(a) (b)XFigure 5.17: Nonlinear characterization results, for both the original individual devices analyses (circles),for when Rth, Qabs and τcarrier are fixed to the average values in the model functions (squares). (a)-(c) Bestfit parameters (markers), with ±1σ uncertainties (gray rectangles for original fit parameters, and yellowrectangles for the new fit parameters). Parameters that minimize the X2tot are shown with black markers,and those that do not are red. (d) The minimum X2tot = X2tot/(Ntot − ktot) is shown, where Ntot is thenumber of data points in the fit and ktot is the number of fit parameters. Fits where X2tot is not fullyminimized are marked as red.5.4.4 Consistency checksEnergy Um,i calculationThe nonlinear transmission ∆λNL,modelm and TNL,modelm data plotted in the first two columns of Fig.5.15 are calculated using the model functions in Eqns. (5.38) and (5.37), and by calculating the1155.4. Analysis procedure(a) (b) (c)LegendFigure 5.18: Nonlinear transmission results for Device 2 when Rth, Qabs and τcarrier are held fixed at themean values. The parameter on each x axis is fixed and the minimum X2tot, X2T, and X2FWM, over all otherfit parameters involved in the respective analysis is shown. X2T is calculated as a function of ηwg1 , ηwg2 . X2FWMis calculated as a function of ηwg1 , ηwg2 , ηwg3 , while φin and φout are held fixed at the best fit values from theX2T minimization. X2tot is the sum over X2T and X2FWM.resonant microcavity energy Um,i by numerically solving,Um,i = 2τinm−1P inm,iτm(Um,i)2. (5.49)In the X2T calculation, the microcavity energy Um,i is directly calculated based on the experimentalnonlinear transmission using Eqn. (5.40), instead of calculating it based on the input power. Thismethod is computationally much faster, and is more practical for the large parameters spaces tested.A consistency test is done by repeating the X2T minimization using the TNL,modelm,i values in the Um,icalculation in Eqn. (5.40). This is iterated until convergence is met in the best fit parameters(typically between 2 to 4 iterations). Figure 5.19 shows that the resulting minimum X2T and bestfit parameters are not significantly different from those found using the original fitting scheme, forthe four devices studied here.Saturated free-carrier lifetimeA consistency test is also done to ensure that the effective free-carrier lifetime does indeed appearto be saturated for the data used in the nonlinear transmission fitting analysis (which is a subsetof the full data set). The effective free-carrier lifetime is estimated by solving Eqn. (5.37) forτcarrier, where all other best fit parameters are applied, TNL,modelm,i is replaced by TNLm,i, and Um,i is1165.4. Analysis procedure1 2 3 4Device00.51τcarrier(ns)1 2 3 4Device012η1wg1 2 3 4Device0246Qabs10 61 2 3 4Device02468min(χ2 T)(a) (b)(c) (d)(e)1 2 3 4Device0204060Rth(K/mW)Figure 5.19: Nonlinear characterization results for the best fit analysis (filled circles) and for the converged re-sults (empty squares) when TNL,modelm,i is used to calculate the Um,i in the model functions [Eqns. (5.37),(5.38)and (5.40)]. The minimum X2T = X2T/(NT− kT) from the nonlinear transmission is shown in (a), where NTis the number of data points in the fit and kT is the number of fit parameters. Fits where X2T does not reacha minimum with respect to all parameters are marked as red. (b)-(e) Best fit values with ±1σ uncertainties(gray rectangles) are shown, along with the converged parameter values. Parameters that minimize the X2Tare shown with black markers, and those that do not are red.calculating by Eqn. (5.40). The effective free-carrier density, N , is estimated using Eqns. (E.33)and (E.40) as,N =2τFCAm σFCAvg=2κ0κm,m,mU2m,iσFCAvg(5.50)Figure 5.20 shows a plot of the resulting τcarrier(N) for the M1 data (circles) and M2 data (triangles)of Devices 1 to 4. The best fit τcarrier’s are drawn as dashed lines. The effective free-carrier lifetimeappears to be relatively constant over almost all carrier densities. Deviations of the calculatedcarrier lifetimes from the best fit carrier lifetimes increase for densities N . 2.0× 1016 cm−3, andτcarrier for N & 2.0 × 1016 cm−3 appear to be saturated. In the analysis for Device 2, the fittingdata included the right-most (highest power) four data points for M1 and the right-most six datapoints for M2, which appear to be consistent with a saturated τcarrier.1175.4. Analysis procedure0 2 4 6 8 10 12 14×10 1610 -1010 -910 -810 -7carrier(s)0 2 4 6 8 10 12 14×10 1610 -1010 -910 -810 -7carrier(s)0 2 4 6 8 10 12 14×10 1610 -1010 -910 -810 -7carrier(s)0 2 4 6 8 10 12 14Carrier density, N (cm -3) ×10 1610 -1010 -910 -810 -7carrier(s)Device 1Device 2Device 3Device 4M1 M2 Best fitFigure 5.20: Effective free-carrier lifetime calculated for both M1 (circles) and M2 (triangles) of Devices 1to 4 (top to bottom). The best fit τcarrier are drawn as dashed lines.Roles of φin, φout, Qabs and τcarrier in the FWM analysisAs described above φin and φout are not included as free parameters in the FWM analysis, despitethe dependence of the experimental idler powers on these parameters. Much like the nonlineartransmission analysis, these two parameters do not play a crucial role in determining the topographyof X2FWM, thus these parameters are held fixed to limit the computational time required to completethe analysis. For example, the best fit parameters were found to vary insignificantly when the φinvalue for Device 2 was shifted by pi, and the analysis was repeated. The Qabs parameter is also heldfixed in the FWM least-squares analysis. This is appropriate as this idler powers do not directlydepend on this parameter, given that thermal shifts are ignored. The only impact that it has is onlimiting the range of possible ηwgm that give positive (physical) τscatt by Eqn. (5.36).1185.4. Analysis procedureThe free-carrier lifetime τcarrier is also held fixed in this analysis. This parameter is not generallyexpected to play a significant role in determining the true X2FWM topography (when included as afree parameter) because the majority of FWM measurements are taken in the low power regime (dueto technical challenges of performing FWM measurements in the bistable limit), where nonlinearabsorption plays only a small role in determining the idler power, as compared to the waveguidecoupling ratios ηwgm . In addition, for the low power measurements, the effective free-carrier lifetimeis not in the saturated limit, and is power-dependent, thus τcarrier is not a reliable fit parameterfor this analysis. The current analysis scheme is a compromise, where τcarrier is held fixed at thesaturated value found from the nonlinear transmission analysis. This results in an underestimateof τcarrier at low powers, which has the potential to push the best fit ηwg2 to slightly lower values,however the minimization of X2tot with respect to τcarrier is expected to be dominated by X2T, thusthis effect is expected to be very small.119Chapter 6DiscussionIn this chapter, the nonlinear performance and nonlinear characterization results for the triplemicrocavity devices are discussed.6.1 Triple microcavity performance6.1.1 Four-wave mixing efficiencyThe four-wave mixing results presented in Chapter 5 demonstrate that the triple microcavitystructure has potential to be considered for frequency conversion applications. A common met-ric used to quantify the frequency conversion performance is the FWM idler power efficiency,ηPFWM = Pi/(P2pPs), where Pi, Pp and Ps are the idler, pump and signal powers in the input waveg-uide. Here ηPFWM is reported in units of µW−2. When evaluated in the linear regime, the efficiencypredicts the idler power for a given pump and signal power. Devices with higher ηPFWM are favor-able for most applications, as lower input powers are required to achieve the same idler power. Analternative definition for four-wave mixing efficiency is sometimes given as ηsigFWM = Pi/Ps, which isa unitless parameter that describes the conversion of signal photons to idler photons. This metricrequires knowledge of the pump power to put it in context. Due to this lack of generality, the choiceis made to report ηPFWM instead of ηsigFWM in the following.The experimental four-wave mixing idler power efficiencies are plotted in Fig. 6.1 for Devices 1to 4. Data is shown for idler photon generation in M1 (blue) and M3 (red), measured as a functionof the pump power (triangles) for fixed signal power at Ps (labelled on the plots), and as a functionof signal power (circles) for fixed pump power at Pp (also labelled). The efficiencies predictedbased on the model function, with the best fit parameters from the least-squares analysis withfixed average value Rth, τcarrier and Qabs, are also plotted as thick solid and dash-dotted black lines1206.1. Triple microcavity performance10 1 10 2Input Power (µW)10 -1010 -9FWMP(µW-2)10 0 10 1 10 2Input Power ( µW)10 -1010 -9FWMP(µW-2)10 1 10 2Input Power (µW)10 -1010 -910 -8FWMP(µW-2)(a) (b)(d)10 1 10 2Input Power (µW)10 -1010 -9FWMP(µW-2)LegendIdler Mode 3, pump sweep, exp.Model with NL absorption,pump sweepModel without NL absorptionIdler Mode 3, signal sweep, exp.Idler Mode 1, pump sweep, exp.Idler Mode 1, signal sweep, exp.Model with NL absorption,signal  sweep(c)Device 1 Device 2Device 4Device 3Figure 6.1: Four-wave mixing efficiency ηPFWM = Pi/(P2pPs) for (a) Device 1, (b) Device 2, (c) Device 3 and(d) Device 4. The experimental FWM idler powers for idler photons in Mode 1 and Mode 3 are plotted as afunction of pump power (triangles) when the signal power is fixed at Ps (labelled) and signal power (circles)when the pump power is fixed at Pp (labelled). The idler powers predicted using the model function withbest fit parameters found from the least-squares analysis when Rth, Qabs and τcarrier are held fixed at theiraverage values are shown with thick solid and dashed-dotted black lines for the pump and signal sweeps.The dashed lines show the predicted power when nonlinear absorption is ignored.for the pump and signal dependences, respectively. Also shown is the model function prediction forthe efficiency when nonlinear absorption is disabled (thin dashed line). For the majority of datasets, the efficiency decreases as a function of power, due to nonlinear absorption effects. For a givenidler mode, the efficiencies of the two data sets (pump and signal power sweeps) do not necessarilyreturn to the same value at low powers. This is because the fixed power for these sweeps is in somecases large enough to induce nonlinear absorption, and as a result, the data set is entirely taken ina nonlinear absorption regime.The highest experimental four-wave mixing efficiencies for the data sets shown in Fig. 6.1 aresummarized and compared to those in the literature in Table 6.1. The ηPFWM efficiencies for the M3idler mode data are 1.4−14×10−9 µW−2, while those for the M1 idler mode data are substantiallylower, 6.8×10−11−3.2×10−10 µW−2. This large difference is qualitatively explained by considering,1216.1. Triple microcavity performancethat in the linear regime, ηPFWM ∝ (Qtotp )4(Qtots )2(Qtoti )2/[(Qinp )2Qins Qouti ] for equally spaced modes,as is found from Eqn. (5.42), where “p”, “s” and “i” subscripts are for the pump, signal and idlermodes. While the total quality factors and Qinp are the same for both of the pumping configurations,Qins and Qouti are different. The efficiency is much higher when M3 is the idler mode because Qout3is over two orders of magnitudes smaller than Qout1 , while Qin3 is approximately an order magnitudesmaller than Qin1 .As shown in Table 6.1, the highest measured FWM efficiency, 1.4× 10−8 µW−2 (Device 2 withM3 as the idler mode) is an order of magnitude higher, or more, than the efficiencies found for atriple nanobeam cavity [8], a triple microring resonator [109], a coupled travelling wave resonator[5] and microrings with radii R = 5 µm [7] and R = 10 µm [92]. The triple nanobeam structure’sefficiency is likely lower as it supports modes with lower total Q’s than the 2D PC triple cavitypresented in this work. While the triple microring resonator, the coupled travelling wave resonatorand the R = 10 µm microring resonator support three high Q modes, the mode volumes arerelatively large, which results in weaker nonlinear interactions. The microring with radius R = 5µm has an even smaller conversion efficiency, despite the smaller mode volume, due to its relativelylow Q caused by the TIR condition on the tight waveguide bend. An additional PC structure,with 10 coupled microcavities, is also included in Table 6.1 [55], which is made in Gallium IndiumPhosphide (GaInP), and has a very high ηPFWM = 3×10−6 µW−2, owing to the small mode volume,high Q’s and the Kerr coefficient, n2, that is nearly twice as large as that for silicon.6.1.2 Kerr effect input power thresholdThe nonlinear transmission spectra presented in Chapter 5 for modes M1 and M2 of the triplemicrocavity demonstrate that the Kerr effect is significant when probed at sufficiently high power.The minimum power at which the Kerr effect is strong enough for a structure to be consideredfor all-optical processing applications, like all-optical switching, is called the power threshold, Pth[63]. This figure-of-merit coincides with the minimum power where bistable behaviour is observed.Microcavities with lower Pth, can operate more efficiently, and are better candidates for scalableintegration. In general, a low Pth is achieved by microcavities with high Q, low mode volumes [46],Veff , and high efficiency loading of light into the microcavity. High thermal resistances and effective1226.1. Triple microcavity performanceTable 6.1: Four-wave mixing idler power efficiency, ηPFWM = Pi/(P2pPs). For the devices in this work, theηPFWM listed are reported for the maximum experimental FWM with M3 as the signal mode (first), then M1as the signal mode (second). The mode quality factors for each structure is also given. When the Q’s ofthe three modes are approximately equal, a single Q is reported. For the first and third structures reported,the Q’s are listed in order of signal, pump then idler. For the devices in this thesis, the Q’s are listed inthe order of M1, M2 and M3. Estimates of the device areas are also provided. The structures are made insilicon, unless noted by ‡.Structure ηPFWM (µW−2) Q Device area (µm2) SourceTriple nanobeamcavity1.2× 10−93× 1034× 1036× 10310 [8]Triple microringresonator8× 10−10 ∼ 5× 104 200 [109]Coupled travelling waveresonator2.2× 10−11 ∼ 1× 104† 130 [5]Microring resonator(R = 10µm)7.8× 10−10 2.3× 104 310 [92]Microring resonator(R = 5µm)1.2× 10−10 7.9× 103 80 [7]Coupled PC microcavities‡ 3× 10−62.4× 1048.3× 1041.4× 105150 [55]Triple microcavity,Device 12.9× 10−9, 6.8× 10−111.6× 1052.6× 1043.0× 10350 This workTriple microcavity,Device 21.4× 10−8, 3.2× 10−101.5× 1054.8× 1041.2× 10350 This workTriple microcavity,Device 32× 10−9, 1.2× 10−109.7× 1043.5× 1042.8× 10350 This workTriple microcavity,Device 41.4× 10−9, 8× 10−111.5× 1052.9× 1041.4× 10350 This work† Estimated from spectrum.‡ Made in a suspended Gallium Indium Phosphide membrane.1236.2. Best fit parametersfree-carrier lifetimes also lead to low Pth, but typically result in a trade-off with the switchingspeed. The resonant wavelength shifts observed for the triple microcavities studied in this thesisare dominated by thermal effects, where the primary source of power absorption comes from free-carrier absorption, as is illustrated in Fig. 5.7. The Pth for these devices are compared to those inthe literature in Table 6.2.The Pth for the triple microcavities entries in Table 6.2 are estimated by looking for the sharpdrop in the nonlinear transmission spectra, which is a signature of the bistable regime. Here Pth istaken to be the input power for the first (lowest power) spectrum where bistable features appear,and is found separately for M1 and M2. The triple microcavity Pth ranges from 17−172 µW, whichis in range of the Pth reported for other suspended silicon 2D PC microcavities in the literature.The total quality factors and effective mode volumes (estimated from simulations) are also given,as these both play important roles in the nonlinear interactions that give rise to the bistable state.For the 2D PC microcavities, differences in Pth for modes with similar Q’s and mode volumes arelikely attributed to differences in the coupling efficiencies of light into the cavities, as the thermalresistance and effective free-carrier lifetimes are likely to be similar. In the case of the 1D PCnanocavity, the Pth is significantly lower, due to its higher thermal resistance.6.2 Best fit parametersThe model functions resulting from the analysis procedure generally do a good job of simultaneouslydescribing the linear and nonlinear functionalities measured in this thesis. One way to assesswhether the parameters extracted from the analysis procedure are reasonable (physically sensible),is to compare them to those reported in the literature, for comparable photonic structures insilicon. These comparisons can be made for Rth, τcarrier and Qabs, as they are generally consistentfor structures with similar geometries and mode profiles (surface to volume ratios).6.2.1 Thermal resistance RthThe average best fit thermal resistance found in this work is Rth = 40± 0.4 K/mW. This compareswell with other Rth reported in the literature for similar suspended 2D planar photonic crystal1246.2. Best fit parametersTable 6.2: Estimates of threshold input power, Pth, required to excite a bistable response of the microcavity.The threshold power is measured relative to the input 2D photonic crystal waveguide, unless noted by ‡.The quality factor, Q, effective mode volume, Veff , and thermal resistance, Rth, of the bistable mode are alsogiven. The Pth and Q’s reported for the devices in this work are listed for M1, then M2.Structure Pth (µW) Q Veff (λ/n)3 Rth (K/mW) Source2D PC microcavity 200 3.8× 104 0.9 15− 35 [11, 84]2D PC microcavity 40 3.3× 105 1.1 [63]2D PC microcavity 10− 28† 2.3× 105 1.2 50 [94]1D PC nanocavity 1.6‡ 2.5× 105 1.4 550 [31]Triple microcavityDevice 117, 172 1.6× 105, 2.6× 104 1.7, 2.1 40.5 This workTriple microcavityDevice 234, 33 1.5× 105, 4.8× 104 1.6, 2.1 40.5 This workTriple microcavityDevice 3105, 102 9.7× 104, 3.5× 104 1.7, 2.1 40.5 This workTriple microcavityDevice 493, 150 1.5× 105, 2.9× 104 1.7, 2.1 40.5 This work† Estimated from spectra.‡ Pth is relative to the channel waveguide.structures in silicon, where Rth ' 16 − 50 K/mW [4, 11, 30, 31, 75], as is summarized in Table6.3. The thermal resistance describes how easily heat is transfered from the microcavity to thesurrounding environment, and is found to be reduced when the buried oxide layer beneath thesilicon is left intact, due to the difference in the relative thermal conductivities of air and SiO2(SiO2: 1.05 K/mW, air: 0.026 W/mK, silicon: 149 K/mW) [65]. The suspended 2D PC thermalresistances are lower than those estimated for 1D PC stack and ladder microcavities, 250 K/mWand 550 K/mW respectively [31]. The 1D PC stack microcavity consists of isolated boxes of silicon(1.45 µm ×0.18 µm ×200 nm) that sit on SiO2, such that a significant amount of heat diffusesthrough the oxide. In contrast, the ladder microcavity is a 1D nanobeam microcavity that isbridged in air, such that heat primarily diffuses through the silicon contact points at the endsof the nanobeam. In both of these examples, the thermal resistance is high because heat diffusesprimarily in one direction, unlike the 2D PC microcavity, where it diffuses over the planar slab. For1256.2. Best fit parameterssimilar reasons, a microdisk resonator (R = 1.16 µm) supported by an oxide pedestal, surroundedby a sunflower-type circular PC, is found to have a high thermal resistance, Rth = 530 K/mW,due to the limited directionality of diffusion [88]. Devices with large thermal resistances have lowthermo-optic bistable switching powers, however, this comes at the cost of lower switching speeds(which are typically limited by thermal effects), as it takes longer for heat to diffuse.Table 6.3: Thermal resistance, Rth reported for suspended photonic crystal (PC) structures. It is notedwhether the value was found experimentally or from modelling.Structure Rth (K/mW) Method SourcePC microcavity 15-35 Exp. [11]PC microcavitywith metallic pads16.8 Modelling [30]PC microcavity 20 Modelling [31]PC microcavity 50† Exp. [4]PC waveguide 18.5 Exp. [75]Triple PC microcavity 40.5 Exp. This work†Used as a model parameter in Refs. [94, 111], and good agreement is found with therespective experimental data presented.6.2.2 Saturated free-carrier lifetime τcarrierThe average best fit saturated effective free-carrier lifetime is 0.92±0.02 ns. This compares well withother reports in the literature for suspended PC photonic crystal cavities, where τcarrier ' 0.5− 1.8ns [11, 95, 105], as is summarized in Table 6.4. The effective free-carrier lifetime is expected to bedominated by recombination at surfaces, and is sensitive to the surface conditions [89, 95]. Thefree-carrier lifetime for an unsuspended 2D PC microcavity, lying on the burried oxide is found tobe 0.12 ns [65]. These lifetimes can be compared to the lifetimes for bulk silicon (∼ 1− 10 µs [66]),submicron waveguides (1 ns [66]), and a microring resonator (∼ 0.5 ns [2]).The saturation characteristics of the effective free-carrier lifetime are plotted in Fig. 5.20. Thelifetime appears to be relatively constant for carrier densitiesN > 2×1016 cm−3. BelowN ' 2×1016cm−3, the scatter of the calculated lifetimes from the best fit values increases. The uncertainties1266.2. Best fit parametersalso increase, as the free-carrier lifetime plays a relatively small role at low carrier densities, thusit is more difficult to extract from fits accurately. There appears to be a general trend of τcarrierincreasing for low densities, where the highest τcarrier are roughly an order of magnitude larger thanthe saturated value (with the exception of Device 2, where the scatter is fairly evenly distributed).This is in comparison to other studies of the free-carrier lifetime that show an increase of 1-2 ordersof magnitude, for densities N < 1× 1016 cm−3, such that the lifetime is saturated for N & 1× 1016cm−3 [11, 50].Table 6.4: Effective free-carrier lifetimes, τcarrier reported based on experimental findings for suspendedphotonic crystal (PC) structures.Structure τcarrier (ns) SourcePC microcavity 0.5† [11]PC microcavity 0.5 [105]PC microcavity 1.4-1.8 [95]Triple PC microcavity 0.92 This work†Used as a model parameter in Refs. [94, 111], and good agreement is found withthe respective experimental data presented.6.2.3 Linear absorption quality factor QabsThe average best fit linear material absorption quality factor was found to be Qabs = 1.26± 0.04×106, however the X2T doesn’t generally change significantly when it is increased, which impliesthat the linear material absorption plays a relatively small role compared to the TPA and FCAabsorption in the measurements reported here. In the literature, the absorption quality factor hasbeen estimated to be Qabs ∼ 4 × 104 − 2.7 × 105 [11, 13] and Qabs ' 1.4 × 106 [40, 106, 111]. Ina number of different studies Qabs is not included in optical bistability analyses likely due to itssmall role relative to other effects [88, 94, 105].6.2.4 Waveguide coupling efficienciesThe waveguide coupling quality factors extracted from the least-squares analysis, performed withthe fixed averaged Rth, τcarrier and Qabs values, are compared to those found from FDTD simulations1276.2. Best fit parametersof the microcavity structures. The waveguide quality factors Qin and Qout are calculated using thebest fit waveguide coupling efficiencies, ηwgm , and Eqns. (5.34) and (5.35). These are plotted,along with the total Q’s (labelled Qtot), in Figs. 6.2(a), (b) and (c) for modes M1, M2, andM3, respectively. Closed and open markers are used for the nonlinear characterization (NLC) andsimulated results, respectively. Similar trends are found from device to device. Generally, there isreasonably good agreement between the NLC and simulated input coupling Qin values. However,there is very poor agreement in the output coupling Qout values for M1 and M2, while the M3values agree reasonably well.1 2 3 4Device103104105106107Q1 2 3 4Device103104105106107Q1 2 3 4Device103104105106107Q(a) (b) (c)NLC QinNLC QoutNLC QtotSim. QinSim. QoutSim. QtotLegendFigure 6.2: Quality factors for Devices 1 to 4, found by the nonlinear characterization (NLC) performedwith the average values for Rth, τcarrier and Qabs, and by simulations, are plotted for modes (a) M1, (b) M2,and (c) M3. The total quality factor Qtot, the input waveguide quality factor, Qin and the output waveguidequality factor Qout are all plotted, as is summarized in the legend.To gain insight on the variable agreement in the NLC and simulated Qin and Qout, the Re(Ey)field profiles for modes M1, M2 and M3 of the triple microcavity in the absence of waveguides areplotted in Figs. 6.3(a),(b) and (c), respectively. The waveguide coupling strengths depend on thefield overlaps between the cavity modes and the waveguide mode. For M2, there is essentially no fieldin the center defect, and the uppermost and lowermost defects localize fields with opposite parity,such that the mode is anti-symmetric, and coupling to the output waveguide mode (symmetricabout y = 0) is suppressed. The simulated coupling strength isn’t perfectly zero because the1286.2. Best fit parameters-4 -2 0 2 4x ( µm)-202y (µm)-1-0.500.51-4 -2 0 2 4x ( µm)-202y (µm)-1-0.500.51-4 -2 0 2 4x ( µm)-202y (µm)-1-0.500.51(a) (b)(c)Amplitude AmplitudeAmplitudeM2M3M1Figure 6.3: Mode field profiles for heterostructure photonic crystal (PC) microcavities. Electric field Re(Ey)mode profiles in the center plane of the silicon are plotted for modes (a) M1, (b) M2 and (c) M3 of a triplemicrocavity, in the absence of input and output waveguides. The heterostructure boundaries are shown withblack lines and the PC holes are outlined in white.diagonal input waveguide weakly breaks the symmetry of the mode. This suggests that Qout forM2 is likely sensitive to fabrication imperfections that further break the mode symmetry, and mightexplain why the experimental Qout are significantly lower than the simulated values.The coupling of M1 to the output waveguide is likely also sensitive to fabrication imperfections,as the field localized to the outer defects destructively interferes with the field localized to thecenter defect, where the fields extend into the output waveguide region. The experimental Qout arehigher than those simulated, which suggests that the fabricated devices support greater destructiveinterference. It is interesting to note that the simulated Qout for M1 of Device 1, is an order ofmagnitude higher than those for Devices 2 to 4. For these three devices, the radius of four holesalong the center line defect, opposite the waveguide, are shrunk by scaling factor swg,sym = 0.912(see Fig. 3.1), while for Device 1 they are not shrunk (swg,sym = 1). It is unclear why there isa significant difference in the Qout, as inspection of the simulated mode profiles does not reveal1296.3. Novelty of the nonlinear characterization proceduresignificantly different relative weightings of the field concentrations in the outer and inner defects.However, this further supports the hypothesis that Qout for M1 is very sensitive to perturbationsin the structure.In contrast, for M3, the field parity in the outer defects is opposite to the center defect such thatconstructive interference happens in the output waveguide, rendering this mode less sensitive tofabrication imperfections. This is consistent with the good agreement found between experimentaland simulated Qout for M3.The input waveguide mode primarily overlaps with the fields localized to the lowermost defect.This coupling scheme is not particularly sensitive to the mode symmetries, and thus is consistentwith relatively good agreement between experiment and simulation for Qin.6.3 Novelty of the nonlinear characterization procedureIn the nonlinear characterization method presented here, the 15 unknown microcavity parameters,and the two Fabry-Perot parameters, are extracted for each of the four devices by analyzing acombination of results from linear transmission, nonlinear transmission and four-wave mixing mea-surements. This characterization procedure was generally successful, despite the relative complexityof the analysis, and technical challenges associated with the nonlinear measurements of the triplemicrocavity devices. For three of the four devices studied here (Device 2 to 4), the X2 was wellminimized across the parameters of interest, with the minimum X2tot per degree of freedom . 10,and the best fit results agreed well with those in the literature. The analysis of Device 1 was lesssuccessful, where X2 was not minimized as a function of the key microcavity parameter ηwg2 , withinthe range of possible values. While the source of this issue is unclear, the nonlinear transmissionand four-wave mixing physics is still reasonably well captured using the average parameter results.As discussed above, the Rth, τcarrier and Qabs parameters extracted agree well with those in theliterature.The method used here to extract the parameters is unique from those presented in the literaturefor a number of reasons. To the best of our knowledge, this is the first characterization presentedfor the linear and nonlinear parameters of a coupled-cavity device with distinct, independent input1306.3. Novelty of the nonlinear characterization procedureand output waveguides. The closest microcavity characterization reported is for one mode of a2D PC microcavity coupled to just one PC waveguide [11], which was completed using only linearand nonlinear reflection data. The use of only reflection data in Ref. [11] was possible becausewhen a microcavity is coupled to a single waveguide, there is one less unknown parameter and thewaveguide coupling efficiency is extracted directly from the linear reflection data (or transmissiondata for side-coupled geometries). With the waveguide coupling lifetime as a known parameter,the nonlinear transmission X2T analysis for a single mode is much better minimized as a functionof the rest of the microcavity parameters, and supplementary data and analysis is not required toextract the best fit values.In other reports in the literature, experiments are designed to characterize a single nonlinearparameter, for example either Rth [4, 30, 88] or τcarrier [2, 65, 95, 105], but not all parameterssimultaneously. In some studies, previously published nonlinear parameters are used to populatemodel functions that are compared to experimental data, in efforts to qualitatively seek consistenciesin the nonlinear behaviour [94, 111].As described above, four-wave mixing measurements are not typically used to directly charac-terize the linear and nonlinear parameters, because when the waveguide coupling efficiencies areknown for linear measurements, the four-wave mixing analysis is not required to complete the anal-ysis. However, stimulated and spontaneous four-wave mixing measurements commonly observenonlinear absorption effects, where the results are often studied in the context of how these effectsdegrade the device performance [34, 35], as opposed to as a characterization tool.131Chapter 7ConclusionA triple photonic crystal microcavity device with independent input and output waveguides was de-signed, externally fabricated, and its linear and nonlinear responses were thoroughly characterizedand modelled. The experimental results demonstrate that this structure offers nonlinear function-ality at least on par with, and in some cases better than, other integrated nonlinear optical devicesstructures. The four-wave mixing is potentially relevant in all-optical processing, including wave-length conversion [25] and signal generation [92]. The bistable behaviour has potential relevanceto all-optical switching [63] and all-optical memories [62]. More generally, the triple microcavitydevice may be considered for other applications that require strong light-matter interactions, likecavity quantum-electrodynamics (cQED), owing to the high Q modes and small mode volumes [96].The best device studied in this work exhibits a four-wave mixing conversion efficiency that isover an order of magnitude higher than microring based structures [6, 7, 92, 109], owing to thesmaller mode volumes. The top efficiency measured here is also 10 times higher than that of thetriple nanobeam cavity in Ref. [8]. The 2D PC triple microcavity from this work has greaterpotential for reaching high Q’s than the nanobeam (and consequently high conversion efficiencies)in future iterations, as this structure can be suspended, while the nanobeam structure requires anoxide layer below the silicon. There is also greater flexibility in terms of waveguide coupling thanksto the 2D PC structure. One benefit of the current design is that it provides a good compromisebetween achieving high Q’s for high efficiency conversion, while maintaining one lower Q mode witha large enough linewidth that the requirement for equally spaced modes is relaxed. As a result,the fabrication yield for “good” FWM candidates (FWM frequency within a linewidth of the lowQ mode resonance frequency) is fairly high, as most groups of 7 devices over which the modespacing is bracketed have at least one good FWM candidate, as shown in Fig. 4.3. Given that the132Chapter 7. Conclusiongood candidates fairly consistently fall close to the center of the bracket, the bracket range couldarguably be reduced, therefore increasing the yield. In contrast, for the triple microring structuresupporting only high Q modes, microheaters are required to tune resonant frequencies [109].The inclusion of both input and output waveguides makes it possible to use relatively straight-forward measurement techniques, as compared to many related devices reported in the literature. Insome cases, the dual-waveguide coupling also makes the device more eligible for scalable integration.For example, the FWM coupled PC cavity structure in Ref. [55] has a single end-coupled waveguide,such that a free-space circulator is required to deal with the input and output light exchanged withthe waveguide. For the bistability measurements presented in [11], a technically challenging fibretaper coupling scheme is used to manage excitation of, and collection from, a single PC waveguideend-coupled to the microcavity. Other microcavity structures implement a single side-coupledwaveguide, such that non-resonant light is transmitted and resonant light is reflected [111], howeverthis coupling scheme has limited design flexibility as the input and output coupling rates are thesame.For applications where two modes of a microcavity are used for bistable switching, the triplemicrocavity with independent input and output coupling has some potential advantages over single2D PC cavity devices [63]. In this application, a strongly modulated pump excitation is tuned tothe control mode resonant wavelength, and a weak unmodulated probe excitation is tuned near theprobe mode. Excitation of the control mode induces nonlinear effects in the microcavity, that inturn shift the resonant wavelength of the probe mode, and cause modulations in the transmissionof probe light. Ideally, the pump mode is efficiently loaded with energy, while the probe modeis efficiently transmitted. Not only does the triple microcavity support two high Q modes withlow mode volumes, the mode profiles are spatially distinct across the three defects, which makes itpossible to engineer unique waveguide coupling for the two modes. This is in contrast to the single2D PC microcavity geometry, where the modes are roughly localized to the same defect region,making it more difficult to customize coupling. Devices 1 and 2 are potentially good candidatesfor this application as M1 has a high Q (> 1× 104) low bistability power threshold, which is goodfor the control mode, while M2 has a relatively high “cold” transmission, > 0.6 (in the absence ofnonlinear loss), as well as a high Q (> 1× 104), which is good for the probe mode.133Chapter 7. ConclusionThere are however some disadvantages of working with this triple microcavity structure. Theprimary disadvantage is that the waveguide coupling strengths to the output waveguide for M1 andM2 are critically dependent on symmetries in the mode profiles, and as a result are sensitive tofabrication imperfections. This means that reproducibility is poor, as two devices with the samelayout (like Devices 3 and 4 in this study), can have significantly different actual coupling strengthsto the output waveguide (for example M1 in Fig. 6.2). It also means that there are large deviationsfrom the simulation results, which make it difficult to engineer the waveguide coupling. In recentwork, where 2D PC triple microcavities are proposed for frequency conversion applications, usingstructures that involve nontrivial coupling geometries, Monte Carlo simulations are used to modelfabrication imperfections [58]. While this may make it possible to plan for the worst, there arelikely to be unavoidable trade-offs between device performance and fabrication yield.With respect to four-wave mixing applications, the other main disadvantage of this structure isthat it promotes strong nonlinear absorption effects. While these effects make the triple microcavitya good candidate for bistable switching, the same effects put an upper limit on the conversionefficiency, as shown in Fig. 6.1. Given that the free-carrier lifetime is approximately the samefor 2D PC structures and microring resonators, the structure itself is not the fundamental issue.The main issue is the large TPA coefficient for silicon, at wavelengths near 1550 nm. A commonsolution proposed to help resolve this problem is to install p-i-n junctions across the structure andapply a bias to sweep out the carriers [70, 73, 93]. The p-i-n junction is compatible with CMOSfabrication methods such that the proposed devices are still leveraged by the benefits of workingwith silicon. Alternatively, amorphous hydrogenated silicon, a-Si:H, is also being explored as itexhibits a lower TPA coefficient and a higher Kerr nonlinearity, which both improve the nonlinearbehaviour, however these devices are plagued by linear losses that limit their performance [34].The input and output ports of the microcavity devices also proved to be problematic dueto the relatively high reflections between components, that caused sinusoidal modulations in thetransmitted intensity, which in turn complicated the analysis process. These effects can be avoidedusing alternative coupling schemes, for example, low reflection parabolic grating couplers [1] thatfocus light into the 500 nm wide channel waveguide over a relatively short distance of ∼ 20 µm,instead of 150 µm (parabolic waveguide length used in this work), such that the Fabry-Perot1347.1. Suggestions for future workmodulation is minimized, free-spectral range is longer, and the structure response is less sensitiveto fabrication imperfections.Beyond the study of these specific triple microcavity devices, the nonlinear characterizationresults presented in this thesis are also of potential interest to the nonlinear photonics communityin general, as there are currently limited publications citing measurements of Rth and τcarrier formicrocavity structures. The values reported here are in agreement with those previously reported,further supporting this literature. Additionally, as increasingly complex microcavity structures,variously and non-trivially coupled to waveguides, are designed for applications in classical andquantum photonics [58], sophisticated characterization methods will be required to understand themicrocavity physics, increasing the demand for robust characterization methods like that presentedhere.7.1 Suggestions for future workAs alluded to above, the basic concept of a triple coupled PC microcavity coupled to indepen-dent input and output waveguides does have a number of potential nonlinear optical applications.The work in this thesis suggests the need to explore different types of microcavities and couplinggeometries that would retain the good qualities identified in the current design, namely the threespatially overlapping high Q modes with nearly equally spaced resonant frequencies and small modevolumes, but be less sensitive to the fabrication imperfections. As a guide, it would be ideal toavoid using coupled microcavity modes with waveguide coupling strengths that critically depend onmode symmetries with respect to the waveguide axis. The problematic output waveguide couplinggeometry presented here was a consequence of the attempt to suppress the pump mode transmis-sion down to ∼0.1 in the original design. In hindsight, since approximately 10 orders of magnitudepump suppression is ultimately required for the pump mode transmission, it will have to be ac-complished external to the cavity, so this design constraint on the microcavity can easily be liftedgoing forward. Pump suppression of over 95 dB has been achieved on-chip using Bragg reflectorsand ring resonators [32].was originally designed to suppress the pump mode transmission down to ∼0.1, in an effort to1357.1. Suggestions for future workreduce the need for external spectral filters around the idler frequencies. However given that ap-proximately 10 orders of magnitude pump suppression is ultimately required for four-wave mixingmeasurements, even the best-case scenario suppression was hardly worth the fabrication imperfec-tion sensitivity introduced.From a scientific point of view, further studies of spontaneous four-wave mixing in this devicewould be of great interest to the quantum photonics community. This device was originally de-signed for applications as a photon pair source, however, this was not ultimately pursued due tounexplained nonlinear behaviour observed, that rendered the device ineligible as a photon pairsource. When the microcavity is excited with a single pump laser tuned to M2, one would expectphoton pairs to be generated over the same bandwidths (near M1 and M3), where the bandwidthof the joint density of states for this nonlinear interaction is expected to be limited by the narrowM1 bandwidth. Here, photon generation is observed over the entire bandwidths of both the highQ M1 mode and the low Q M3 mode, as can be seen by observing the background spectra (tri-angles) in Fig. 4.4. Correlation measurements between the photons generated at the signal andidler frequencies were attempted, to study the photon pair statistics (not reported in this thesis),however they failed, possibly due to the large spectral mismatch between the photons generated inthe high Q M1 (∼ 0.01 nm bandwidth), and the photons generated in M3 (∼ 0.5 nm bandwidth),which could only be resolved with a large bandwidth filter (∼ 0.17 nm). Even if a narrow (∼ 0.01nm) filter were available to better match the spectral bandwidths of the signal and idler photonsin the correlation measurement, the signals would be very weak and it would be unlikely that acorrelation signal could be detected above the avalanche single photon detector noise. However, theuse of superconducting single photon detectors, which are known to have much lower dark countrates, may make it possible to study these correlations.It is possible that other types of nonlinear processes contribute to, and even dominate, thebackground spectrum. For example, silicon photonic crystal microcavity structures have beenfound to exhibit broadband photoluminescence, covering the wavelength range of interest (λ ∼1500 to 1600 nm), when excited with light above the silicon electronic bandgap (e.g. λ = 532 nm)[76]. This photoluminescence occurs due to optically active defects that are introduced when thesilicon-on-insulator wafer is manufactured. If this is the case, the above-bandgap excitation may1367.1. Suggestions for future workarise here due to two-photon absorption, or second and third harmonic generation. 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When smaller steps are taken, the nonlinear effects incrementally strengthento the steady-state values, and the laser wavelength continues to chase the ever red-shifting reso-nant wavelength. In this thesis, typical wavelength steps are between ∆λ = 0.002 nm and 0.005nm. The small wavelength step also provides good resolution for the transmission spectra. Giventhat the time step is ∼ 50 ms, and measurements are typically taken over ∼ 0.5 nm, the time torun each spectrum is less than 15 s.Instabilities in the measurement environment also cause the transmission to drop in the bistableregime. For example, a drop may be caused by a fluctuation of the power launched in the inputwaveguide. While the lasers employed in this experimental work are stable enough to avoid problemslike this, displacement of the excitation beam on the input grating coupler can lead to a reduction inpower. This displacement can be caused by a disturbance in the optical set-up, like an accidental151A.2. Four-wave mixing excitationnudge of the optical table, or even natural vibrations (in more extreme cases). Changes in thetemperature of the room can also cause a drop due to the change it induces in the index of refraction,which causes the mode to shift off resonance. Alignment and temperature fluctuations do nottypically cause premature drops in the transmission during nonlinear transmission measurements.They are more likely to be problematic when the aim is to maintain a microcavity in a bistablestate over some period of time. This is the case for stimulated four-wave mixing measurements.The likelihood that these fluctuations cause an unwanted drop partly depends on how close thelaser wavelength is tuned to the edge of the bistability. The closer to the edge, the system is lessstable, it takes less of a disturbance to cause a drop.A.2 Four-wave mixing excitationThere are two excitation configurations for four-wave mixing that are employed in this thesis. Inboth configurations, the center M2 mode is pumped on-resonance. In one configuration, the highQ M1 mode is excited on-resonance by the signal laser, and the output filter is centred to the idlerwavelength near the low Q M3. In the other configuration, signal wavelength is tuned near M3,but not necessarily on-resonance, so that idler photons are generated on-resonance with M1. Thedifferent procedures are followed to optimally tune the pump, signal excitation wavelengths, andthe spectral filter idler wavelength.A.2.1 Signal: high Q modeIn this excitation configuration, the goal is to optimally load pump and signal light into the twohigh Q microcavity modes, by tuning the wavelengths to the resonant wavelengths. The idler filterwavelength is then set so that the signal and idler wavelengths are equally spaced from the pumpwavelength. The idler wavelength is easily estimated with a quick calculation, and then refined bysweeping the filter center wavelength until there is a peak in the idler photon generation rate. Theidler wavelength does not necessarily coincide with the resonant frequency of the low Q mode, asthe microcavity modes are generally not perfectly spaced.When both the pump and signal powers are sufficiently low that nonlinear absorption effects are152A.2. Four-wave mixing excitationnegligible (as in the linear limit), the laser wavelengths are simply tuned to the cold cavity resonantwavelengths λm of the two respective modes. The resonant wavelengths are found by sequentialwavelength sweeps of the two excitation lasers to find the appropriate resonant peak wavelengthsin the transmission spectra. With the optical set-up implemented in this experimental work, itis easy to go between doing the transmission measurements required wavelength alignment, andthe four-wave mixing measurements, as the transmission set-up is simply a subset of the four-wavemixing measurement set-up.In the nonlinear regime, when one or both of the pump and signal excitations induce nonlinearabsorption effects, the aim remains to tune to the pump and signal lasers to the resonants wave-lengths of the high Q modes, however this becomes significantly more technically difficult than inthe linear limit due to the changes refractive index of the silicon that simultaneously affect bothmodes as energy is loaded in the microcavity. As an example, Fig. 2.7(c) shows the nonlinear reso-nant wavelength shift of mode M1 as a function of the energies loaded in both this mode, and M2.Meanwhile, Fig. 2.7(d) shows the resonant wavelength shift of M2, as a function of the energies inM1 and M2.When only the pump laser is sufficiently strong to induce nonlinear effects, the following strategyis used for optimal loading. The pump excitation is first turned on, while the signal laser remainsoff. The optimal wavelength of the pump is that which maximizes the nonlinear transmission at thegiven pump power. A full forward sweep is run to identify the optimal wavelength (assuming red-shift dominates, otherwise backward sweep would be used), then the sweep is redone but stoppedjust short of the large drop in transmission, so that the high transmission is achieved. The pumpis fixed at this wavelength. The signal laser is then turned on and swept to its peak transmissionwavelength, and fixed at this wavelength. At this point, both microcavities are optimally loaded.It’s important to note that the peak resonant wavelength for the signal mode is shifted from its coldcavity λm, and the maximum transmission is lower than the cold cavity peak, due to the energyloaded in the pump mode that is causing changes to the real and imaginary parts of the refractiveindex, respectively. This makes the order of the sweeps important, because if the signal mode wereto be sought and fixed first, then the signal laser wavelength would be off resonance (suboptimal)after the pump energy is subsequently loaded.153A.2. Four-wave mixing excitationWhen both of the mode excitations are in bistable limits, it becomes even more challenging tooptimally load both cavity modes, and an iterative process is required. One laser is turned is onand a forward sweep is used to identify the peak wavelength (again, assuming red-shift dominates).The laser wavelength is fixed near this wavelength (just before the large drop in transmission) andthen the second laser is turned on, swept to find its peak wavelength, and swept once more toreach the peak, without going too far. At this stage, the modes are likely not optimally loaded, asthe nonlinear effects introduced by the second mode have likely caused the peak wavelength of thefirst mode to red-shift, leaving the laser for the first mode detuned. To reach simultaneous optimalloading for both cavities, the wavelength for each laser is iteratively increased in very small steps(∼ 0.002 nm), taking turns between the two lasers, to account for the red-shifting of both modes.This iterative process is terminated when the transmission ceases to increase (or the increase isvery small compared to the increases observed in previous iterations). The modes are then veryclose to optimally loaded. This is a very unstable excitation, and is prone to sudden drops in thepower loaded in both cavities.A.2.2 Signal: low Q modeIn the second excitation configuration, the signal wavelength excites the low Q mode, and theidler wavelength coincides with the outer high Q mode resonant wavelength. The most importantdifference between this configuration and the previous one is that, in this case, the goal is not totune the signal wavelength such that it optimizes the signal light loaded in the low Q mode, thegoal is actually to optimize the wavelength so that the idler photons generated are on resonancewith the high Q idler mode. The pump wavelength tuning process does remain similar however, itis tuned to optimally load energy in the center mode.The signal wavelength optimization is challenging because it cannot be guided by transmissionmeasurements. Instead, the signal wavelength is tuned by directly optimizing the idler generationrate, with the idler filter wavelength set to the idler mode resonant wavelength. For each signalwavelength tested in the optimization process, the single photon detector is observed after waitingsufficient time for the average count rate to settle. Sometimes it requires many of these wavelengthsteps to complete the optimization, because a small wavelength step is used due to the small154A.2. Four-wave mixing excitationlinewidth of the high Q mode. The count rate only surpasses the noise once the idler wavelength iswithin a couple linewidths of the high Q mode resonant wavelength (typical linewidth < 0.05 nm).155Appendix BFinite-difference time-domain analysisB.1 The finite-difference time-domain methodThe finite-difference time-domain method is an exact numerical approach to solving the macroscopicMaxwell’s equations, which in differential form are,∇×E(r, t) = −∂B(r, t)∂t(B.1)∇×H(r, t) = J(r, t) + ∂D(r, t)∂t(B.2)∇ ·D(r, t) = ρ(r, t) (B.3)∇ ·B(r, t) = 0 (B.4)where E(r, t) is the electric field, H(r, t) is the magnetic field, D(r, t) is the displacement field,B(r, t) is the magnetic flux, ρ is the free electric charge density (not including bound charges) andJ(r, t) is the free current density (not including bound polarization and magnetization currents).In this thesis, all FDTD simulations are done in the linear regime where D(r, t) = (r)E(r, t)and H(r, t) = µ(r)B(r, t), where (r) and µ(r) are material electric permittivity and magneticpermeability, respectively. For the materials considered here µ(r) = µ0, where µ0 is the permeabilityof free space.Lumerical FDTD Solutions software implements the Yee Algorithm approach [79, 107], suchthat the simulation volume is discretized into Yee cells, illustrated in Fig. B.1. The xˆ, yˆ and zˆelectric and magnetic field components are evaluated at different positions, staggered over the grid.The time step between when the fields are evaluated is ∆t. In this approach, the electric fieldcalculations occur at times offset by ∆t/2 from the magnetic field calculations. For example, the156B.1. The finite-difference time-domain methodFigure B.1: Schematic of a Yee cell [107].Ex(r, t) fields are evaluated at,t→ n∆t (B.5)x→ (mx + 12)∆x (B.6)y → my∆y (B.7)z → mz∆z (B.8)where n is the integer time index, ∆x,∆y and ∆z are the mesh step sizes in the xˆ, yˆ, and zˆdirections, and mx,my and mz are the x, y and z coordinates of the Yee cell in terms of the integerstep number, respectively. The basic update equations, in the absence of free currents and charges,are,En+1 = En +∆t∇×Hn+ 12 (B.9)Hn+3/2 = Hn+12 − ∆tµ0∇×En+1 (B.10)(B.11)where the spatial curls are evaluated at each of the field component positions. For example, the157B.1. The finite-difference time-domain methodEx update is,Exn+1mx+12,my ,mz=Exnmx+12,my ,mz(B.12)+∆t(Hzn+ 12mx+ 12 ,my+ 12 ,mz −Hzn+ 12mx+ 12 ,my− 12 ,mz∆y(B.13)−Hyn+ 12mx+12,my ,mz+12−Hzn+12mx+12,my ,mz− 12∆z)(B.14)158Appendix CGrating coupler designThe first step of the design procedure considers gratings with uniform slot widths, ws, and spacings,as. In 2D FDTD simulations, a Gaussian beam with 1/e2 diameter d0 = 10 µm is incident on theuniform grating at −45◦ with f = 196 THz (near the simulated resonant mode frequencies) and thelight coupled into the fundamental mode of the 20 µm wide waveguide is monitored, as shown inFig. C.1(a). The simulation spans the entire depth of the SOI, including the buried oxide and thetop of the base silicon layer. The slot spacing that optimizes the transmission for each grating widthtested is determined, and is plotted in Fig. C.1(b). It approximately follows: as = 0.58ws + 415nm.In the next step, light with f = 196 THz is launched into the fundamental mode of the waveg-uide, and the spatial profile of out-coupled light 1 µm above the uniform grating coupler is studiedusing the 2D simulations shown in Fig. C.1(c). The spatial profile for the uniform grating with ws= 100 nm and as = 474 nm (optimized pair) is shown in Fig. C.1(d), where the trend shows thatthe intensity exponentially decays following ∼ exp(−αx). The decay rates, α(ws), are plotted inFig. C.1(e), where for each ws, as = 0.58ws + 415 nm is applied. The exponential decay does notmimic that profile of the Gaussian beam (incident at 45◦), also plotted in Fig. C.1(d), thus theuniform grating is not optimal.An apodization scheme is then determined that helps match the profiles of the out-coupledlight and the excitation beam. The slot widths are chosen so that the decay rate starts small, tominimize radiation immediately at the beginning of the grating, then increases quickly to matchthe drop of Gaussian excitation. At this stage, the design chosen has the following composition: 7slots with ws = 100 nm, 15 slots over which the width is linearly varied between ws = 100 nm and300 nm, followed by 300 nm slots out to the simulation boundary, all separated by the appropriate159Appendix C. Grating coupler design-10 0 10 20x ( µm)00.51Intensity (a.u.)-10 0 10 20x ( µm)00.51Intensity (a.u.)0.1 0.2 0.3ws(µm)00.51(µm-1)0.1 0.15 0.2ws(µm)0.450.50.55as(µm)(a) (c)(b)(d) (f)(e)aswsmonitor monitorFigure C.1: Apodized grating coupler design. (a) FDTD simulation layout for in-coupling of light from aGaussian beam with a 1/e2 width of 10 µm at θ = −45◦ into the silicon slab via a 1D grating. (b) Gratingspacing, as, that optimizes in-coupling transmission (θ = −45◦), as a function of the slot width, ws, for auniform grating. The line of best fit for a linear function is shown (red dashed). (c) FDTD simulation layoutfor out-coupling light from the silicon slab mode to free-space above the grating. (d) Out-coupling radiationprofile measured 1 µm above the uniform grating coupler with ws = 100 nm and as = 474 nm (blue), whenthe silicon slab mode is launched. The best fit exponential decay curve (dash-dotted yellow line), and theprofile of the input Gaussian beam (dashed red) are also included. (e) Decay rates as a function of ws, foruniform gratings with as found from the line of best fit in (b). (f) Out-coupling radiation profile measured1 µm above the apodized grating coupler (blue), when the silicon slab mode is excited. The predicted decayprofile (yellow dash dotted) and the excitation Gaussian beam for in-coupling profile (red dashed), as alsoplotted.as(ws). The decay intensity pattern predicted for this grating apodization scheme is plotted in Fig.C.1(f), along with the simulated profile and the excitation Gaussian. The reflection is estimated tobe 0.072 and is reduced to 0.041 when the first grating slot is replaced by one with ws = 80 nm.A schematic of the final apodized grating design, extended to two-dimensions, is shown in Fig.3.13(a). As described in Section 3.2.2, the transmission efficiencies between free-space and thesingle-mode waveguide (via the parabolic waveguide), plotted in Fig. 3.13(c) and (f) are estimatedusing a combination of FDTD and MODE eigen-mode expansion (EME) simulations, which arenow described.The in-coupling efficiency, from free-space to the single-mode channel waveguide is found by firstsimulating the coupling of light into the opening of the parabolic using FDTD simulation, then thatlight is propagated in the parabolic waveguide to the channel waveguide using the MODE eigenmodesolver[80]. In the in-coupling FDTD simulation, the beam is centred 4 µm from the front slot, andthe field profile at the entrance to the parabolic waveguide is saved, along with the transmission160Appendix C. Grating coupler designthrough the planar monitor. The field profile is then launched as a source in an EME simulationof the parabolic waveguide and finally the transmission into the mode of the 500 nm wide channelwaveguide is found, and is plotted in Fig. 3.13(c). The transmission of light through the parabolicwaveguide is found to be ∼ 0.85, while the reflection is found to be 0.0062 for the grating studiedhere. In EME simulation, the parabolic waveguide is divided into 90 cells, and the modes supportedby waveguide cross-section at each cell interface are calculated. The transmission through the fullwaveguide is calculated based on scattering matrices found by considering the boundary conditionsat each interface such that a bi-direction (transmission and reflection) estimate for light propagationis achieved. This method is not exact, like the FDTD method, however it yields almost identicalresults and requires significantly less computational resources.For out-coupling simulations, the fundamental mode is launched in the single-mode channelwaveguide, and the field profile at the end of the parabolic waveguide is calculated with the EMEsolver and is saved, along with the transmission efficiency. It is then used in an FDTD simulationas a source directed toward the grating, and transmission of light through a monitor lying 1 µmabove the grating surface is monitored, as is plotted in Fig.3.13(c).The collection efficiency of the elliptical mirror is accounted for by considering the distributionof light in the far field, as illustrated in Figs. 3.13(d) and (e). The far field is calculated usingLumerical FDTD’s “farfield3d” function, which calculates the Fraunhofer diffraction pattern. Thecollection efficiency of the mirror is calculated by dividing the integrated field intensity within themirror surface area by the total integrated field intensity, both found using the “farfield3dintegrate”function. The total out-coupling efficiency is found by multiplying the collection efficiency by thefree-space out-coupling transmission efficiency.161Appendix DTransmission efficienciesThe analyses of experimental data in this thesis require knowledge of the optical powers in the inputand output PC waveguides. These powers are calculated by considering the raw powers measuredin the experiment, and the transmission efficiencies of components along the optical path.D.1 Linear and nonlinear transmissionIn transmission measurements, the power coupled into the output PC waveguide is,T (λ) =Traw(λ)ηouttot (λ), (D.1)where ηtot(λ) is the total transmission efficiency of all free-space and integrated components on theoptical path between the output PC waveguide and the photodetector, where the raw transmissionTraw(λ) is measured. The power in the input PC waveguide is,P in(λ) =P inraw(λ)ηintot(λ)ηinm2, (D.2)where ηintot(λ) is the total transmission efficiency of all integrated components on the optical pathbetween the input PC waveguide and free-space directly off the chip surface, and ηinm2 is the trans-mission efficiency of the mirror (“Mirror 2”) that is placed directly before the chip to redirect lighttowards a photodetector that measures the raw input power P inraw(λ). The relative transmission is,T (λ) =T (λ)P in(λ). (D.3)162D.1. Linear and nonlinear transmissionThe free-space and integrated component transmission efficiencies that compose ηouttot and ηintot aredescribed the following, and are summarized in Table D.1.Table D.1: Summary of the transmission efficiencies for components in the transmission and four-wave mixing(FWM) set-ups illustrated in Fig. 2.5 and Fig. 2.11Set-up Component Transmission efficiencyTransmission Set-upElliptical mirror + Mirror 1 ηemηm1 = 0.864± 0.007Output Polarizer ηop = 0.529± 0.005Mirror 2 ηm2 = 0.922± 0.01FWM Set-upFibre coupling ηfibre = 0.27± 0.02Filter max[ηfilter(λ)] = 0.355± 0.010SPD ηD = 0.116± 0.005Integrated ComponentsInput grating coupling ηinGC(λ)Output grating coupling ηoutGC(λ)Input parabolic waveguide ηinpar(λ)Output parabolic waveguide ηoutpar(λ)Input PC waveguide ηinPCwg(λ)Output PC waveguide ηoutPCwg(λ)The total transmission efficiency along the collection path is,ηouttot (λ) = ηemηm1ηopgout(λ), (D.4)where ηem, ηm1 and ηop are the measured transmission efficiencies of the elliptical mirror, Mirror 1and the output polarizer, respectively, along free-space collection path of the transmission set-up inFig. 2.5. The gout(λ) function contains the transmission efficiencies of the integrated components,gout(λ) = ηoutGC(λ)ηoutpar(λ)ηoutPCwg(λ), (D.5)where ηoutGC(λ), ηoutpar(λ) and ηoutPCwg(λ) are the transmission efficiencies for the output grating coupler,parabolic waveguide and the channel waveguide-to-PC waveguide transition region, respectively.Similarly, the total transmission efficiency along the input path, which includes only the input163D.2. Four-wave mixing idler power calibrationintegrated components, is,ηintot(λ) = gin(λ) = ηinGC(λ)ηinpar(λ)ηinPCwg(λ). (D.6)The free-space component efficiencies, ηem, ηm1 and ηop, have only very weak wavelength de-pendencies over the tuning range of the lasers and are taken to be constants. Here, the ellipticalmirror transmission efficiency accounts for loss and scattering from the mirror surface, and doesnot account for the geometrical collection efficiency (this is considered part of the grating couplercharacterization, discussed shortly). The polarizer transmission efficiency is reported here for in-cident light polarized along the transmission axis, which is normal to the plane of Fig. 2.3 to beconsistent with the s-polarized grating modes of interest.The relative uncertainty in the relative microcavity transmission, δT/T , is estimated to be 4%.This is based on the uncertainties in the optical component transmission efficiencies (see Table D.1),and fluctuations in the transmitted powers measured for the reference devices and the microcavitydevice.D.2 Four-wave mixing idler power calibrationIn four-wave mixing measurements, fibre-based components are added to the transmission set-up,including spectral filters and a single photon detector (shown in Fig. 2.11), with transmissionefficiencies ηfilter(λ) and ηD respectively. The values for these efficiencies are reported in TableD.1, along with the coupling efficiency of free-space light to the single mode fibre, ηfibre. Fibrecomponents are also added to the excitation path, including a spectral filter and a 50/50 splitter,however these do not affect the input power calculation, as the power is measured downstream oftheir locations.The idler power in the output PC waveguide, Pidler is calculated from the raw idler power, Prawidler,measured by the detector (after accounting for the dead-time, as described in Appendix D), using,Pidler =P rawidlerηouttot (λidler, φout)ηfibre max[ηfilter(λ)]ηD. (D.7)164D.3. Integrated component transmission efficienciesThe input pump and signal (“p/s”) powers in the input PC waveguide are calculated as,P inp/s =P inraw,p/s(λ)ηintot(λp/s, φin)ηinm2, (D.8)The relative uncertainty in the idler power δPidler/Pidler, is estimated based on two contribu-tions that are added in quadrature: 1) the uncertainty arising from the transmission efficiencycalculations, which includes the optical components transmission efficiency uncertainties (see TableD.1), and fluctuations in the transmitted powers measured for the reference devices, and 2) theuncertainty due to fluctuations in the count rate measured by the single photon detector. Theformer is estimated to be 5%, while the former is estimated for each measurement individually.D.3 Integrated component transmission efficienciesThe transmission efficiencies of the integrated components in the microcavity devices are addressedhere. To calculate the power in the input PC waveguide, the transmission of the Gaussian beam ex-citation through to the PC waveguide, gin(λ), must be found. This transmission efficiency capturesthe passage of light through the input grating coupler, the parabolic waveguide, and finally, intothe PC waveguide. Similarly, to calculate the power in the output PC waveguide, the transmissionefficiency gout(λ), must be found, that captures passage of light from the output PC waveguide,through the parabolic waveguide and output grating coupler, and finally the collection by the el-liptical mirror. The gin(λ) and gout(λ) have two main differences: 1) the input and output PCwaveguides have different structures in the microcavity device, and 2) the spectrum associatedwith coupling the Gaussian beam excitation into the input channel waveguide via the input grat-ing coupler, is different from the spectrum associated with the elliptical mirror collection of lightleaving output channel waveguide via the output grating coupler. These two points are consideredin the following.To address the first point, gin(λ), is estimated by analyzing the transmission spectrum of refer-ence device PCWGin, which has the PC waveguide structure that matches that of the microcavityinput waveguide, as described in Section 3.3.1. Meanwhile, gout(λ) is estimated by analyzing thetransmission spectrum of PCWGout, whose PC waveguide structure matches that out the micro-165D.3. Integrated component transmission efficienciescavity output waveguide.The second point is less straight-forward to deal with. If input and output coupling efficiencieswere the same, then it would be possible to find the efficiencies by taking the square roots of thePCWGin and PCWGout transmission spectra. For example, this would be an appropriate approachto take if the transmission measurements involved direct coupling of light between the gratingcouplers and optical fibres. In the free-space transmission measurements taken in this thesis,with non-symmetric excitation and collection optics, there is no way of knowing the individualcontributions from input and output coupling spectra. As a result, simulation results are used toestimate the ratio between the two.Finally, one added challenge in determining gin(λ) and gout(λ) is that oscillations appear inthe transmission spectra due to effective Fabry-Perot cavities formed in the parabolic waveguides.The peaks of these oscillations are shifted from device to device, which makes the reference devicetransmission spectrum unreliable in this regard.D.3.1 Calculations of transmission efficiencies gin(λ) and gout(λ)The relative transmission spectra, taken from directly before the input grating coupler to directlyafter the elliptical mirror collection, T˜ (λ), are shown in Fig. D.1 for the four reference devices inthe sgc = 0.97 set of Group 3 on Chip A, measured with θ = −41◦. Here, the raw transmissionspectra measured by the photodetector, Traw(λ), are used to calculate T˜ (λ) using the following,T˜ (λ) =T (λ)P in(λ)=Traw(λ)P inraw(λ)ηm2ηemηm1ηop, (D.9)where T (λ) = Traw(λ)/(ηemηm1ηop) is the transmitted power estimated after collection by theelliptical mirror (however with non-geometrical mirror losses removed), and P in(λ) = P inraw(λ)/ηm2is the input power estimated at the chip surface. The transmission spectra peaks are centred nearλ = 1542 nm, with bandwidths ' 30 nm. The spectral lineshapes agree well with the product ofthe simulated input and output grating coupling spectra for sgc = 0.92 and θ = −41◦, where thesimulated peak wavelength is 1540 nm and the bandwidth is 34 nm. The difference in experimentaland simulated sgc implies that the actual grating slots are likely smaller than intended (in the166D.3. Integrated component transmission efficiencies1520 1540 1560 1580Wavelength (nm)00.10.20.30.40.5ηref,GC1520 1540 1560 1580Wavelength (nm)00.10.20.30.40.5ηref,PCwg1520 1530 1540 1550Wavelength (nm)00.20.40.6(ηref-ηrefmean)/ηrefmean1535 1540 1545 1550Wavelength (nm)00.20.40.60.81ηPCwg1520 1540 1560 1580Wavelength (nm)00.020.040.060.080.10.12Transmission,T(λ)(a) (b)(c) (d)(e)~Figure D.1: Transmission spectra for reference devices in Group 3 of Chip A, with sms = 0.97. (a) Trans-mission from directly before the input grating coupler to directly after collection by the elliptical mirror,for the G2Gout (blue), G2Gin (red), PCWGout (purple) and PCWGin (yellow) devices. (b) Transmissionefficiencies, ηoutref,GC (blue) and ηinref,GC (red) from found transmission measurements of G2Gout and G2Ginrespectively. (c) Transmission efficiencies, ηoutref,PCwg (purple) and ηinref,PCwg (yellow) from found transmissionmeasurements of PCWGout and PCWGin, respectively. (d). The green lines in (b) and (c) are ηmeanref,X (λ),found by filtering ηref,X(λ) spectra to remove the fast oscillations. The black lines are the best fit g(λ, φ)functions. (d) The scaled coupling efficiencies for (bottom to top) G2Gin (blue), G2Gout and PCWGin (blue),PCWGout. (e) The input (yellow) and output (purple) PC waveguide coupling efficiencies. Black lines plotthe best fit sinusoidal functions.167D.3. Integrated component transmission efficiencieslayout). In addition, the simulated transmission efficiency is 1.9 times greater than that foundexperimentally. It is unclear why this large difference exists, but is possibly due to fabricationimperfections in the grating couplers or the long parabolic waveguides.From the reference device transmission spectra in Fig. D.1(a), it is possible to estimate thefollowing transmission efficiencies,ηinref,GC(λ) = ηinGC(λ)ηinpar(λ) =√fGCT˜ inG2G(λ), (D.10)ηoutref,GC(λ) = ηoutGC(λ)ηoutpar(λ) =√T˜ outG2G(λ)fGC, (D.11)ηinref,PCwg(λ) = ηinGC(λ)ηinpar(λ)ηinPCwg(λ) =√fGCT˜ inPCwg(λ), (D.12)andηoutref,PCwg(λ) = ηoutGC(λ)ηoutpar(λ)ηoutPCwg(λ) =√T˜ outPCwg(λ)fGC, (D.13)where T˜ inG2G(λ) and T˜outG2G(λ) are the transmission spectra for G2Gin and G2Gout devices, andT˜ inPCwg(λ) and T˜outPCwg(λ) are those for PCWGin and PCWGout devices, respectively. Here fGC =ηinGC/ηoutGC, is the ratio between input and output coupling, as found using simulations (see Section3.2.2). For the grating coupler structure studied here, and this coupling angle of θ = −41◦, fGCis estimated to be 1.75. The ηref,GC(λ) [Eqns. (D.10) and (D.11)] are plotted in Fig. D.1(b) andηref,PCwg(λ) [Eqns. (D.12) and (D.13)] are plotted in Fig. D.1(c).The gin(λ) and gout(λ) transmission efficiency spectra are essentially given by ηinref,PCwg(λ) andηoutref,PCwg(λ). These spectra capture the slow underlying peak, associated with grating coupling[i.e. ηGC(λ)] and coupling between the PC and channel waveguides [i.e. ηPCwg(λ)], in addition tothe fast oscillations that arise to due an effective Fabry-Perot cavity formed inside each parabolicwaveguide as a result of reflections off the grating coupler on one end, and off the waveguidetransition to a narrow single mode waveguide on the other. The fast oscillations are problematicbecause, from device to device, the oscillations peaks occur at different wavelengths due to smalldifferences in the roundtrip phase for light propagating in the long parabolic waveguide. This168D.3. Integrated component transmission efficienciesmeans that reference measurements can’t reliably capture the phase of the sinusoid, and as a resultthe φin and φout parameters are introduced to the transmission efficiency functions as unknowns,that are ultimately fit parameters in the nonlinear analysis. This parameterization is importantbecause the oscillation amplitude is large enough that phase errors in the efficiency functions couldpotentially result in an unphysical resonant mode relative transmission. For example, modes withnear unity relative transmission may be found to have Tmax> 1.Here g(λ) function associated with the reference device X (any one of the four) are given by,gX(λ) = ηmeanref,X (λ)[1 +A cos(2pi∆λFSRλ+ φ)], (D.14)where ηmeanref,X (λ) is the slowly varying spectrum that results when the fast oscillations are filtered outof ηref,X(λ) calculated by Eqns. (D.10) to (D.13). The ηmeanref,X (λ) are plotted for each spectrum inD.1(b) and (c) as a green centerline. To find oscillation amplitude, A, and the period, ∆λFSR, theexperimental data in D.1(b) and (c) are replotted in Figure D.1(d) as [ηref,X(λ)−ηmeanref,X (λ)]/ηmeanref,X (λ),and are fit with the function A cos(2pi∆λFSRλ+ φ). The amplitudes are found to be A ' 0.075 andthe periods are ∆λFSR ' 2.06 nm, while the phase shifts varying between 0 to 2pi. The best fitcosine functions are plotted as black lines in Figure D.1(d), and the g(λ, φ) functions are plottedas solid black lines in D.1(b) and (c).It is important to note that the gin(λ) and gout(λ) functions, determined from ηinref,PCwg(λ)and ηoutref,PCwg(λ) respectively, must be found for each set of devices studied with different sgc andsh pairs, as the grating coupling spectral features depend on sgc, and the PC waveguide spectralfeatures depend on sh.D.3.2 Fabry-Perot model comparisonThe relationship between the A,∆λFSR and φ parameters, and the properties of the parabolicwaveguide Fabry-Perot cavity, are now discussed. The transmission of light through a generalFabry-Perot cavity has the following form [85],TFP(λ) =Tmax(λ)1 +(2Fpi)2sin2(2pingLλ) (D.15)169D.3. Integrated component transmission efficiencieswhere F = pi(R1R2)1/4/(1−√R1R2) is the finesse for the weak cavity, and Tmax(λ) = T1T2/(1−√R1R2) is the maximum transmission. This is used as an approximate model for the Fabry-Perotcavity formed by the parabolic waveguide. The grating coupler is taken to be the first reflector,with transmission T1 = TGC and R1 = RGC, and other end of the parabolic waveguide, where theconstriction is fastest, is considered to be the other reflector with T2 = Tpar and R2 = Rpar. Thephase shifts accumulated on reflection are ignored here. The group index is taken to be constantwithin the parabolic waveguide, which is an oversimplification, as multiple different modes areexcited throughout the waveguide as it expands. Despite these approximations, this model providesa foundation to study some of the basic physics of this effective cavity. In the limit of low finesse,the cavity transmission is,TFP(λ) 'Tmax(λ)[1−(2Fpi)2sin2(2pingLλ)](D.16)=Tmax(λ)[1−B +B cos(4pingLλ)], (D.17)where B = 2(F/pi)2. Using a Taylor Series expansion for 1/λ around λ0 = 1545 nm, it is approxi-mated as,TFP(λ) ' Tmax(λ)[1−B +B cos(2pi∆λFSRλ+ φ(λ0))], (D.18)where ∆λFSR = λ20/(2ngL) is the free spectral range5, and φ(λ0) is a phase shift. The formof the model function g(λ, φ) presented above in Eqn. (4.4), is matched to TFP(λ) by settingηmeanref,X (λ) = Tmax(λ)(1−B) and A = B/(1−B).Finite-difference time-domain simulations predict that the parabolic waveguide reflection isapproximately independent of wavelength, with Rpar ' 0.0008, while the grating reflection (frominside the parabolic waveguide) varies between RGC ' 0.02 to 0.03 over λ = 1530 nm to 1580 nm.The group index for the fundamental mode of the 500 nm wide channel waveguide is ng = 4.22 nearλ = 1545 nm, while that for the 20 µm on the other end of the parabolic waveguide is ng = 3.59.These simulated parameters result in an estimated A ' 0.01, which is within an order of magnitudefrom that found experimentally, and ∆λFSR ' 1.9− 2.2 nm, which is very close to experiment.5This FSR is consistent with that found in Eqn. 2.1 in Section 2.1.1, where FSR/ω0 ' ∆λFSR/λ0.170D.4. Filter transmission efficiencyThe locations of the sinusoidal peak wavelengths in the experimental Fabry-Perot spectra inFig. D.1 are similar for three of the four devices in the set studied here, while one is shifted by∼ ∆λFSR/3. In general, there can be variation in the peak locations across a set of devices dueto fabrication imperfections in the relatively long parabolic waveguides (L ' 100λ). For example,Fabry-Perot peaks near λ = 1545 nm shift by approximately one ∆λFSR if the group index changesby 0.5%.D.3.3 Channel-to-photonic crystal waveguide transmissionFinally, the coupling efficiencies between the channel waveguides and the input/output PC waveg-uides are found by taking ηPCwg(λ) = ηmeanref,PCwg(λ)/ηmeanref,GC(λ), where the appropriate “in”/“out”superscripts are applied across the efficiencies. The PC waveguide coupling efficiencies are plottedin Figure D.1(e). Near λ = 1545 nm, ηinPCwg(λ) = 0.86 and ηoutPCwg(λ) = 0.78, which are lower thanthose simulated, ηsim,inPCwg(λ) = 0.93 and ηsim,outPCwg (λ) = 0.92. The efficiencies here are plotted only overthe range of wavelengths where microcavity modes appear, however by inspecting Fig. D.1(a) and(e), there appears to be a large drop in the PC waveguide coupling efficiencies at long wavelengths,near λ = 1564, and λ = 1554, for input and output PC waveguides, respectively. This is dueto the presence of the low-energy waveguide mode band edge. The efficiency drops observed insimulations of the waveguide coupling region occur at λ = 1587 nm, and λ ' 1576 nm, respectively.This large ∼ 24 nm difference between the experimental and simulated results is surprising giventhat the cavity resonant modes compare quite closely (within several nanometers), and might alsobe related to the difference between the experimental and simulated coupling efficiencies.D.4 Filter transmission efficiencyThree fibre-coupled spectral filters (JDSU TB9 Tunable Grating Filter) are included in the opticalset-up for four-wave mixing. Two of which are placed before single photon detector, allowingidler photons to pass while rejecting pump and signal photons, while the third is placed after thehigh power tunable laser (Venturi TLB 6600-H-CL) to suppress noise. The transmission efficiencyspectrum for each grating filter, plotted in Figs. D.2(a) and (b) on linear and log scales respectively,171D.5. Single photon detector efficiency and dead-timeis measured by sweeping the wavelength of the low power laser (Venturi TLB 6600-L-CL) whilethe filter centre wavelength is held fixed at λ = 1545 nm 6. The peak transmission efficiencies are0.63, 0.56 and 0.65 for filters A, B and C, respectively and the FWHM widths are 0.235 nm, 0.243nm and 0.231 nm, which are close to the specifications (peak transmission > 0.4, and bandwidth0.25 ± 0.15 nm). The wavelength calibration is slightly varied between different filters and thelaser, as shown by the small offsets in the peak wavelengths from 1545 nm. Each filter has > 40 dBrejection for wavelengths > 1 nm away from the center, as seen in the log scale plot in Fig. D.2(b).The transmission efficiency spectrum for filters A and B paired together is plotted in Figs.D.2(c) and (d) on linear and log scales, respectively. Here the center wavelengths for filter B areoffset such that the peak transmission for A and B are aligned. The peak transmission efficiencyis ηfilter = 0.355 ± 0.010, the bandwidth is 0.166 nm. The uncertainty in ηfilter is primarily due tothe uncertainty in the filter wavelength alignment during FWM measurements (∼ ±0.02 nm). Therejection at approximately one mode spacing (∼ 2.4 nm) away from the peak is ∼ 100 dB.The filter C is placed at the output of the high power laser suppresses the noise from -43 dBdown to ∼ −90 dB, which is sufficient for practical four-wave mixing measurements, as describedin Section 2.3.1.D.5 Single photon detector efficiency and dead-timeThe single photon detector is implemented in “free-running mode”, where the avalanche photodiodeis biased above the breakdown voltage threshold, such that the absorption a photon triggers anavalanche that produces effectively an infinite gain, resulting in sufficient amplification to detectthe single photon. The avalanche is quenched after a detection event (i.e. the diode is biasedbelow the breakdown voltage) for the duration of the “dead time”. The efficiency of the detectoris improved by increasing the bias voltage above the breakdown, however, this comes at the costdecreasing the signal to noise, due to carriers generated in the diode junction that also trigger theavalanche response, resulting in dark counts. Unwanted counts are also generated by an effect called“after pulsing”, when carriers generated and trapped during an avalanche are released and retrigger6This measurement is easier to implement than holding the laser wavelength fixed and scanning the filter centerwavelength. Both approaches give very close to the same results.172D.5. Single photon detector efficiency and dead-time1544.5 1545 1545.5Laser Wavelength (nm)00.10.20.30.4Transmission Efficiency1540 1545 1550Laser Wavelength (nm)-120-100-80-60-40-200Transmission Efficiency (dB)1544.5 1545 1545.5Laser Wavelength (nm)00.10.20.30.40.50.6Transmission Efficiency1540 1545 1550Laser Wavelength (nm)-60-50-40-30-20-100Transmission Efficiency (dB)(a) (b)(c) (d)Figure D.2: Spectral filter transmission efficiencies measured by fixing the filter center wavelength at λfilter =1545 nm and sweeping the laser wavelength. (a) Transmission efficiencies of filters A (blue), B (red) and C(yellow) on a linear scale. (b) Same as (a) but on a log scale. (c) Product of the transmission efficiencies forfilters A and B, when their peak wavelengths are aligned, plotted on a linear scale. (d) Same as (c) but ona log scale.the avalanche. After-pulsing effects are reduced by using a longer dead time. The efficiency anddeadtime settings used in this thesis are 0.10 and 100 µs.To determine the actual average rate of photons incident on the detector (Rph) based on thetotal count rate (Rtot) measured, the detector efficiency (ηD), dead-time (τDT), and dark countrate (RDC) must all be considered. The dark count rate, RDC, is measured by blocking the lightthat enters the detector. The raw count rates Rtot and RDC reported by the detector represent theaverage number of counts per time, where this “per time” includes when the detector is blanked(during the dead-time). The true detection rate, which represents the number of counts per active173D.5. Single photon detector efficiency and dead-timedetector time, are calculated by,Ractive(R, τDT) = (R−1 − τDT)−1, (D.19)where R = Rtot or RDC. The average rate of photons incident on the detector is,Rph =Ractive(Rtot, τDT)−Ractive(RDC, τDT)ηD. (D.20)To enable accurate calculations of Rph, ηD and τDT are measured by strongly attenuating lightfrom a laser and sending it into the detector. Figure D.3(a) is a plot of the detected power as afunction of input power for measurements with two different attenuation schemes. The detectedpower is calculated as PD = ~ω0[Ractive(Rtot, τDT) − Ractive(RDC, τDT)], where ω0 is the laser fre-quency. The gray circles show the calculated detected power when τDT = 100 µs is applied. Alinear relationship is expected between the input and detected power (i.e. constant detection effi-ciency), away from the detector saturation, however this is not observed with τDT = 100 µs (see theblack line for reference). The nonlinear trend is arises due to the deviation of the actual dead-timefrom the detector setting value. Even a small deviation causes a large error for high count rates,when R−1tot approaches τDT. The actual dead-time is estimated using a least squares calculationthat involves minimizing the squared difference between the calculated PD(Pin, τDT) and the linearbest fit to PD(Pin, τDT). The X2 for this calculation is given byX2(τDT) =∑Pin[PD(Pin, τDT)− hDT(Pin, τDT)]2σ2P(D.21)where Pin is the input power, σP is the measurement uncertainty in PD(Pin, τDT), and hDT(Pin, τDT)is the best linear fit to PD(Pin, τDT). The X2 is plotted as a function of τDT in Fig. D.3(b), andis clearly minimized for τDT = 99.9788 µs. The detected power, calculated with the best fitτDT = 99.9788 µs, is plotted for the two data sets (blue circles and orange triangles) in Fig. D.3(a),along with the best linear fit trend (black line referred to earlier). For the highest input powertested here, Pin = 13.3 pW, there is a 25% difference between PD(13.3 pW, 100 µs) and PD(13.3pW, 99.9788 µs), which is substantial given the small difference in the τDT applied. Based on the174D.5. Single photon detector efficiency and dead-timeslope of the line of best fit, the detector efficiency is ηD = 0.116± 0.005. The detector efficiencies,PD(Pin, 99.9788 µs)/Pin, are plotted in Fig. D.3(c), where the error bars show the uncertainty inthe measurement. Based on the detector specifications, the efficiency is expected to be vary by lessthan 0.003 over the wavelength range of interest.0 2 4 6 8 10 12 14Input Power (pW)0.1050.110.1150.120.125Detection Efficiency10 -1 10 0 10 1Input Power (pW)10 -210 -110 0Detected Power (pW)99.94 99.96 99.98 100Dead time (µs)10 110 210 310 410 52(a) (b)(c)Figure D.3: Single photon detector characterization, when set with a dead time of 100 µs and efficiency10 %. (a) Detected power calculated with a dead-time τDT = 99.9788 µs for high-power (blue circles) andlow-power (orange triangles) measurements, and calculated with the detector setting τDT = 100 µs is alsoplotted (large gray circles). The line of best fit for τDT = 99.9788 µs is plotted (black line), with theslope giving detection efficiency ηD = 0.116. (b) Chi squared based on the square differences between thecalculated detected power (dependent on τDT) and the linear best fit to the data. The X2 is minimizedfor τDT = 99.9788 µs. (c) The detection efficiencies calculated with τDT = 99.9788 µs for high-power (bluecircles) and low-power (orange triangles) measurements. Error bars show uncertainty in the measurement.The stimulated FWM idler power is ultimately found by subtracting off the background photoncounts, measured when the signal laser is turned off, while the pump laser remains active. The175D.5. Single photon detector efficiency and dead-timeidler power is calculated as,Pidler =~ω0ηouttot ηfibre max[ηfilter(λ)]ηD(Ractive(RFWMtot , τDT)−Ractive(RFWMDC , τDT) (D.22)− [Ractive(RBGtot , τDT)−Ractive(RBGDC, τDT)]),where “BG” indicates the background count rate.176Appendix ENonlinear model function derivationsIn this Appendix, the perturbative approach used to introduce nonlinearities to the coupled modeequations presented in Chapter 5 is reviewed. The nonlinear frequency shifts and lifetimes in Table5.2 are derived, along with the nonlinear coefficients in Table 5.3.E.1 Perturbation theoryAs discussed in Chapter 5, the nonlinearity is introduced to the coupled mode equations throughweak complex-valued changes in the mode resonance frequencies, ωm → ωm + δωNLm , due to pertur-bations of the dielectric constant δεNLm (r), where [39],δωNLmωm= −12∫d3xδεNLm (r)|Em(r)|2∫d3xε(r)|Em(r)|2 (E.1)and Em(r) = amE˘m(r)/√∫d3x12ε(r)|E˘m(r)|2 is the mth steady-state mode field. There are anumber of contributions to δωNLm :δωNLm = δωKerrm + δωFWMm + δωTPAm + δωFCAm + δωFCDm + δωthermalm , (E.2)which are associated with the Kerr effect, four-wave mixing, two-photon absorption, free-carrierabsorption, free-carrier dispersion, and thermal dispersion. The first three contributions, δω(3)m =δωKerrm + δωFWMm + δωTPAm , are directly a result of the third order nonlinear susceptibility χ(3) ofsilicon, while other effects are indirectly associated with these nonlinearities.The perturbation due to the third order nonlinearity is rewritten in terms of the third order177E.1. Perturbation theorypolarization, P(3)m (r) = δε(3)m (r)Em(r), to give [71],δω(3)mωm= −12∫d3xP(3)m (r) ·E∗m(r)∫d3xε(r)|Em(r)|2 . (E.3)In this model for FWM, it is assumed that the electric field is composed of monochromatic wavesat the frequencies {ω′i} = {±ωm} (m = 1, 2, 3) such that [16],E(r, t) =12∑ω′i≥0[Eω′i(r)e−iω′it + E−ω′i(r)eiω′it](E.4)where E−ω′i(r) = E∗ω′i(r) and Eωm(r) = Em(r) above. Similarly,P(3)(r, t) =12∑ω′i≥0[P(3)ω′i(r)e−iω′it + P(3)−ω′i(r)eiω′it]. (E.5)The frequency dependent electric field and polarization are found by taking the fourier transformsof (E.4) and (E.5) respectively, resulting in,E(r, ω) =12∑ω′i≥0[Eω′i(r)δ(ω − ω′i) + E−ω′i(r)δ(ω + ω′i)](E.6)P(3)(r, ω) =12∑ω′i≥0[P(3)ω′i(r)δ(ω − ω′i) + P(3)−ω′i(r)δ(ω + ω′i)]. (E.7)In general, the nonlinear polarization is given by,P(3)(r, ω) = ε0∫dω1∫dω2∫dω3χ(3) · · ·E(r, ω1)E(r, ω2)E(r, ω3)δ(ω − ω1 − ω2 − ω3) (E.8)where “ · · · ” indicates multiplication over three tensor dimensions. This is alternatively written as,(P (3)(r, ω))α = ε0∑ijk∫dω1∫dω2∫dω3χ(3)αijkEi(r, ω1)Ej(r, ω2)Ek(r, ω3)δ(ω−ω1−ω2−ω3) (E.9)to explicitly show the tensor multiplication, where i, j and k are each summed over {x, y, z}.178E.1. Perturbation theorySubstituting (E.6) into (E.8) yields (3 × 2)3 terms inside of the summation, although not allof which are unique. For example, consider a set of three frequencies, ωx : {ω1x, ω2x, ω3x} whereωx = ω1x + ω2x + ω3x and each ωix is one of the ±ω′i terms. The polarization amplitude for ωx willhave terms that look like,(P (3)ωx )α = 2ε0∑ijk[χ(3)αijk(−ωx;ω1x, ω2x, ω3x)12(Eω1x)i12(Eω2x)j12(Eω3x)k (E.10)+χ(3)αijk(−ωx;ω2x, ω1x, ω3x)12(Eω2x)i12(Eω1x)j12(Eω3x)k (E.11)+...].Here the r-dependence has be dropped for convenience. The terms in (E.10) and (E.11) are iden-tical, as is found by relabelling dummy indices (i, j) in (E.11) to (j, i), then applying permutationsymmetry χ(3)α,i,j.k(−ωx;ω1x, ω2x, ω3x) = χ(3)α,j,i,k(−ωx;ω2x, ω1x, ω3x)). The number of identical terms thatappear depends how many distinguishable frequencies ωix there are in the ωx set, as sets with twoor more identical ωix will result in fewer identical terms. The polarization is written as,(P (3)ωx )α = ε0∑ijkK(−ωx;ω1x, ω2x, ω3x)χ(3)αijk(−ωx;ω1x, ω2x, ω3x)(Eω1x)i(Eω2x)j(Eω3x)k (E.12)whereK(−ωx;ω1x, ω2x, ω3x) =(14)3!Nax !Nbx!Ncx!. (E.13)The right-most fraction in K(−ωx;ω1x, ω2x, ω3x) is the number of permutations of {ωix}, where Nax ,N bx, Ncx are the number of times each unique ωix (labelled a, b, c..) appear in {ωix}.In general, the same ωx can be achieved with distinct sets. For example, when ω1, ω2 and ω3are equally spaced (as is the case for FWM), then two possible distinct sets are ω1 : {ω2, ω2,−ω3}and ω1 : {ω1, ω2,−ω2}. The polarization amplitude in (E.12) is generalized to sum of distinct sets179E.1. Perturbation theoryof ωx,(P (3)ωx )α = ε0∑distinct {−ωx;ω1x,ω2x,ω3x}∑ijkK(−ωx;ω1x, ω2x, ω3x)χ(3)αijk(−ωx;ω1x, ω2x, ω3x)(Eω1x)i(Eω2x)j(Eω3x)k.(E.14)In evaluating the sum over i, j, k, it is useful to consider the crystalline properties of the nonlinearmaterial. In this study, the device layer is silicon, which is a centrosymmetric 3m3 crystal withnon-zero χ(3) tensor terms [14]:χ(3)xxxx = χ(3)yyyy = χ(3)zzzzχ(3)xyxy = χ(3)yxyx = χ(3)xzxz = χ(3)zxzx = χ(3)yzyz = χ(3)zyzyχ(3)xxyy = χ(3)yyxx = χ(3)xxzz = χ(3)zzxx = χ(3)yyzz = χ(3)zzyyχ(3)xyyx = χ(3)yxxy = χ(3)xzzx = χ(3)zxxz = χ(3)yzzy = χ(3)zyyz (E.15)and χ(3)xxxx = 3χ(3)xyxy = 3χ(3)xxyy = 3χ(3)xyyx. In the following, the notation for χ(3)iiii is simplified to χ(3)Sifor silicon, as is typical in the literature.The non-zero χ(3) tensor terms are evaluated in (E.14), and applied to P(3)m (r) · E∗m(r) in theperturbation equation (E.3), to give,P(3)ωm ·E−ωm =ε03∑distinct {−ωm;ω1m,ω2m,ω3m}K(−ωm;ω1m, ω2m, ω3m)χ(3)Si (−ωm;ω1m, ω2m, ω3m)[(Eω1m ·Eω2m)(Eω3m ·E−ωm) + (Eω1m ·Eω3m)(Eω2m ·E−ωm)+(Eω1m ·E−ωm)(Eω2m ·Eω3m)](E.16)A summary of the K(−ωm;ω1m, ω2m, ω3m) values and the electric field terms for each distinct setat each of the mode frequencies is listed in Table E.1. The simplified notation for the electric fieldprofile, Em(r) = Eωm(r), is used in this table and in the following.The perturbed mode frequencies are found by using the terms in Table E.1 to insert (E.16) into(E.3), and by substituting, Em(r) = amE˘m(r)/√∫d3x12ε(r)|E˘m(r)|2,180E.1. Perturbation theoryTable E.1: Polarization terms for four-wave mixing. For each mode, m, K(−ωm;ω1m, ω2m, ω3m) and electricfield terms are listed for each distinct set for ωm.m Distinct sets for ωm K Electric field terms1 {ω2, ω2,−ω3} 34 (E2 ·E2)(E∗1 ·E∗3) + 2(E2 ·E∗1)(E2 ·E∗3){ω1, ω1,−ω1} 34 (E1 ·E1)(E∗1 ·E∗1) + 2|E1|4{ω1, ω2,−ω2} 32 (E1 ·E2)(E∗1 ·E∗2) + (E1 ·E∗2)(E∗1 ·E2) + |E1|2|E2|2{ω1, ω3,−ω3} 32 (E1 ·E3)(E∗1 ·E∗3) + (E1 ·E∗3)(E∗1 ·E3) + |E1|2|E3|22 {ω1, ω3,−ω2} 32 (E∗2 ·E∗2)(E1 ·E3) + 2(E∗2 ·E1)(E∗2 ·E3){ω2, ω1,−ω1} 32 (E2 ·E1)(E∗2 ·E∗1) + (E2 ·E∗1)(E∗2 ·E1) + |E2|2|E1|2{ω2, ω2,−ω2} 34 (E2 ·E2)(E∗2 ·E∗2) + 2|E2|4{ω2, ω3,−ω3} 32 (E2 ·E3)(E∗2 ·E∗3) + (E2 ·E∗3)(E∗2 ·E3) + |E2|2|E3|23 {ω2, ω2,−ω1} 34 (E2 ·E2)(E∗1 ·E∗3) + 2(E2 ·E∗1)(E2 ·E∗3){ω3, ω1,−ω1} 32 (E3 ·E1)(E∗3 ·E∗1) + (E3 ·E∗1)(E∗3 ·E1) + |E3|2|E1|2{ω3, ω2,−ω2} 32 (E3 ·E2)(E∗3 ·E∗2) + (E3 ·E∗2)(E∗3 ·E2) + |E3|2|E2|2{ω3, ω3,−ω3} 34 (E3 ·E3)(E∗3 ·E∗3) + 2|E3|4δω(3)1ω1= −β1(a2)2(a3)∗/a1 − (α0 + iρ0)(α11|a1|2 + α12|a2|2 + α13|a3|2) (E.17)δω(3)2ω2= −β2a1a3(a2)∗/a2 − (α0 + iρ0)(α22|a2|2 + α21|a1|2 + α23|a3|2)δω(3)3ω3= −β3(a2)2(a1)∗/a3 − (α0 + iρ0)(α33|a3|2 + α32|a2|2 + α31|a1|2)whereβ2 =12∫Si d3xε0 Re(χ(3)Si )[(E˘∗2 · E˘∗2)(E˘1 · E˘3) + 2(E˘∗2 · E˘1)(E˘∗2 · E˘3)](∫d3xε(r)|E˘2|2)(∫d3xε(r)|E˘1|2)1/2(∫d3xε(r)|E˘3|2)1/2, (E.18)β1 = β3 = β∗2/2, (E.19)181E.1. Perturbation theoryαm,m =∫Si d3x[(E˘∗m · E˘∗m)(E˘m · E˘m) + 2|E˘m|4](∫d3xε(r)|E˘m|2)2, (E.20)αm,m′ = 2∫Si d3x[(E˘m · E˘m′)(E˘∗m · E˘∗m′) + (E˘m · E˘∗m′)(E˘∗m · E˘m′) + |E˘m|2|E˘m′ |2](∫d3xε(r)|E˘m|2)(∫d3xε(r)|E˘m′ |2), (E.21)α0 =ωm4ε0 Re(χ(3)Si), (E.22)andρ0 =ωm4ε0 Im(χ(3)Si). (E.23)Here∫Si indicates integration over the volume occupied by silicon, and χ(3)Si is the complex diagonalelement of the third order nonlinear susceptibility tensor, with units [m2/V2], where the explicitfrequency dependence of χ(3)Si is removed. This is appropriate for the nonlinear processes considered,as ω1 ' ω2 ' ω3 thus the same χ(3)Si (−ω;±ω,±ω,±ω) is applied for the mixing processes studied.The explicit dependence of E˘m on position r in the nonlinear coefficients is removed to simplifynotation.The individual contributions to Eqn. (E.17) are identified as,δωFWM1ω1= −β1(a2)2(a3)∗/a1 (E.24)δωFWM2ω2= −β2a1a3(a2)∗/a2 (E.25)δωFWM3ω3= −β3(a2)2(a1)∗/a3 (E.26)δωKerrmωm= α0∑m′αmm′ |am′ |2 (E.27)δωTPAmωm= iρ0∑m′αmm′ |am′ |2 (E.28)where m′ = {1, 2, 3}. The first three terms contribute nonlinear photon generation (βFWM = β1in Chapter 2 ). Equations (E.27) and (E.28) result in a frequency shift due to the nonlinear Kerreffect ∆ωKerrm , two photon absorption lifetime τTPAm , respectively.182E.2. Nonlinear lifetimes and frequency shiftsE.2 Nonlinear lifetimes and frequency shiftsIn this section, the effects of two-photon absorption (TPA), free-carrier absorption (FCA) andthermal absorption on the cavity mode lifetimes and resonant frequencies are derived, building offof the perturbative approach presented above.E.2.1 Cavity lifetimeWith the inclusion of nonlinear losses, the total cavity lifetime for each mode m, is given by,τm(U1, U2, U3)−1 = τ inm−1+ τoutm−1+ τ scattm−1+ τabsm−1+ τTPAm (U1, U2, U3)−1+ τFCAm (U1, U2, U3)−1,(E.29)where τTPAm (U1, U2, U3) and τFCAm (U1, U2, U3) are derived below.Two-photon absorption (TPA) Two-photon absorption introduces an imaginary part to theχ(3)Si tensor, such that Im(χ(3)Si ) = 2βTPA0c2nSi2/(3ωm), where βTPA is the TPA coefficient withunits [mW−1], and nSi is the index of refraction of the nonlinear material for a single frequency, inthis case silicon [51], both found in Table 5.4. The perturbation equation is,δωTPAmωm= −iρ0∑m′αm,m′ |am′ |2 ≡ − iωm1τTPAm(E.30)such that the TPA lifetime is,1τTPAm= ρ0∑m′αTPAm,m′Um′ (E.31)The m′ = m contribution to the TPA lifetime arises from the absorption of two ωm photons,whereas the m′ 6= m contributions arise from absorption of one ωm photon and one ωm′ photon.The latter process is often referred to as “cross TPA”, or XTPA.Free-carrier absorption The free-carriers excited in the TPA process present an additional lossmechanism. The Drude model predicts that light propagating in a bulk material, in the presence183E.2. Nonlinear lifetimes and frequency shiftsof free-carriers, is attenuated with loss per unit length,αFCA = σFCAN(r), (E.32)where N(r) is the density of electron hole pairs, and σFCA = σFCAe + σFCAh is the free-carrier cross-section with units of [m2], taken to be the sum of the electron and hole cross-sections, and is foundin Table 5.4. The loss per unit length is related to the local absorption rate of the energy densitythrough the group velocity vg, such thatγFCAm (r) = σFCAvgN(r). (E.33)The material dispersion is typically small in bulk measurements, so the group index ng ' nSi, aswill be used in the following.The time dependence of N(r, t) follows [11, 51],dN(r, t)dt= G(r)− N(r, t)τcarrier(N(r, t)), (E.34)where G(r) is the local carrier generation rate, and τcarrier(N(r, t)) is the effective free-carrierlifetime, including recombination, diffusion and drift effects. The temperature dependence of τcarrieris neglected here, as the temperature changes due to nonlinear absorption are relatively small. Whilethe local free-carrier lifetime depends on the proximity to surfaces, the τcarrier(N(r, t)) consideredhere represents a mean lifetime for the carrier distribution.In steady-state,N(r) = τcarrier(N(r))G(r). (E.35)The local generation rate is approximated to be,G(r) =pTPA(r)2~ωm, (E.36)184E.2. Nonlinear lifetimes and frequency shiftswhere pTPA(r) is time-averaged the power absorbed through TPA given by,pTPA(r) = −12Re(iωmP(3)m,TPA(r) ·E∗m(r)) (E.37)The lifetime τFCAm is expressed through,−iωm1τFCAm=δωFCAmωm=−i2ωm∫Si d3xγFCAm (r)ε(r)|Em(r)|2∫d3xε(r)|Em(r)|2 , (E.38)withγFCAm (r) = −σFCAvgτcarrier(N(r))4~ωmRe(iωmP(3)m,TPA(r) ·E∗m(r)). (E.39)The effective energy absorption rate, γFCAcarrier,m = 2/τFCAm , is essentially a weighted average ofthe local absorption rate γFCAm (r). This is derived from perturbation theory by setting δεFCA =2nε0δnFCA(r) = iγFCAm (r)n2ε0/(ωm) in Eqn. (E.3). The dot product is evaluated using Eqn.(E.16), where only the imaginary part of χ(3)Si is applied, i Im(χ(3)Si ). The fields are normalized bysetting Em(r) = amE˘m(r)/√∫d3x12ε(r)|E˘m(r)|2 to give,1τFCAm= κ0∑l,l′κFCAm,l,l′UlUl′ (E.40)where for l = l′,κFCAm,l,l =∫Si d3x[(E˘∗l · E˘∗l )(E˘l · E˘l) + 2|E˘l|4]ε|E˘m|2(∫d3xε|E˘m|2)(∫d3xε|E˘l|2)2, (E.41)and for l′ 6= l,κFCAm,l,l′ = 2∫Si d3x[(E˘l · E˘l′)(E˘∗l · E˘∗l′) + (E˘l · E˘∗l′)(E˘∗l · E˘l′) + |E˘l|2|E˘l′ |2]ε|E˘m|2(∫d3xε|E˘m|2)(∫d3xε|E˘l|2)(∫d3xε|E˘l′ |2). (E.42)withκ0 =σFCAτcarrier Im(χ(3)Si )ε0c8~nSi(E.43)185E.2. Nonlinear lifetimes and frequency shiftsE.2.2 Nonlinear frequency shiftThe total nonlinear resonance frequency shift is∆ωNLm = ∆ωKerrm + ∆ωFCDm + ∆ωthermalm , (E.44)where ∆ωKerrm is the shift due to the Kerr effect, which has already been considered for the model,∆ωFCDm is the shift due to free-carrier dispersion, and ∆ωthermalm is the thermal shift resulting fromthe energy absorbed that heats the nonlinear material and changes its refractive index.Kerr nonlinearity The frequency shift due to the Kerr nonlinearity is given as∆ωKerrm = δωKerrm = −α0∑m′αm,m′Um′ (E.45)Free-carrier dispersion The local index of refraction changed induced by free-carriers predictedby a Drude model is,δnFCD(r) = −ζN(r) (E.46)where ζ = ζe + ζh is a free-carrier dispersion nonlinear material parameter with units [m3], whichincludes both electron and hole effects. Based on experimental results [82], the Drude model ismodified for silicon:δnFCD(r) = −[ζeSiNe(r) +(ζhSiNh(r))0.8](E.47)The frequency shift predicted by perturbation theory is∆ωFCDmωm=δωFCDmωm= −∫d3x δnFCD(r)nSi(r) ε(r)|Em(r)|2∫d3xε(r)|Em(r)|2 . (E.48)where ∆ωFCDm = ∆ωFCD,em + ∆ωFCD,hm . Assuming electrons and holes are generated in pairs, suchthat N e = Nh = N , the frequency shift due to electron free-carriers has the same form as Eqn.186E.2. Nonlinear lifetimes and frequency shifts(E.38), except with γFCAm (r)/2 = σFCAvgN(r)/2 replaced by −δnFCDm /nSi = ζeSiN(r)/nSi to yield,∆ωFCD,em =2ωmζSiσFCAcκ0∑l,l′κFCAm,l,l′UlUl′ . (E.49)The shift due to hole-carriers requires revisions to the integrals to account for the power of 0.8 inthe modified Drude model. The total resulting FCD shift is given by,∆ωFCDm = νFCD,e0∑l,l′κFCAm,l,l′UlUl′ + νFCD,h0∑l,l′κFCD,hm,l,l′(UlUl′)0.8, (E.50)where for l = l′,κFCD,hm,l,l =∫Si d3x[(E˘∗l · E˘∗l )(E˘l · E˘l) + 2|E˘l|4]0.8ε|E˘m|2(∫d3xε|E˘m|2)(∫d3xε|E˘l|2)1.6, (E.51)and for l′ 6= l,κFCD,hm,l,l′ = 20.8∫Si d3x[(E˘l · E˘l′)(E˘∗l · E˘∗l′) + (E˘l · E˘∗l′)(E˘∗l · E˘l′) + |E˘l|2|E˘l′ |2]0.8ε|E˘m|2(∫d3xε|E˘m|2)(∫d3xε|E˘l|2)0.8(∫d3xε|E˘l′ |2)0.8, (E.52)withνFCD,e0 =ωmζeSiτcarrierε0 Im(χ(3)Si )4~nSi, (E.53)andνFCD,h0 =ωmnSi(ζhSiτcarrierε0 Im(χ(3)Si )4~)0.8. (E.54)Thermal effect The mean index change based on a thermal shift is,δnthermal =dndT∆T =dndTRthPabs (E.55)where ∆T is the mean temperature change of the nonlinear material due to the power absorbedby nonlinear processes, Pabs. The thermo-optic coefficient, dn/dT , is known a priori for the non-linear material, while the thermal resistance, Rth = dT/dPabs, is unknown. Assuming that the187E.2. Nonlinear lifetimes and frequency shiftstemperature is evenly distributed [11], then the thermal frequency shift is given by,∆ωthermalmωm=δωthermalmωm= −ΓthmnSidndTdTdPabsPabs (E.56)wherePabs =∑m′(2τabsm′+2τTPAm′+2τFCAm′)Um′ (E.57)andΓthm =∫Si d3xε(r)|Em(r)|2∫d3xε(r)|Em(r)|2 . (E.58)188Appendix FNonlinear coefficientsThe integral nonlinear coefficients in Table 5.3 are calculated based on FDTD simulation results.A subset of the nonlinear coefficient values are reported in Table F.1.Table F.1: Nonlinear coefficients calculated from FDTD simulations.Coefficient Units Device 1 Device 2 Devices 3 & 4|β2| [1/J ] 1.29× 108 1.21× 108 1.19× 108Γth1 0.936 0.936 0.936Γth2 0.939 0.939 0.939Γth3 0.939 0.938 0.938ΓTPA1 0.983 0.982 0.981ΓTPA2 0.983 0.983 0.983ΓTPA3 0.982 0.982 0.982ΓFCA1 0.991 0.991 0.990ΓFCA2 0.991 0.991 0.991ΓFCA3 0.990 0.990 0.990V TPA1 (λ1/nSi)3 7.95 7.48 7.80V TPA2 (λ2/nSi)3 6.83 6.83 6.84V TPA3 (λ3/nSi)3 9.51 9.75 9.16V FCA1 (λ1/nSi)3 5.57 5.20 5.44V FCA2 (λ2/nSi)3 5.39 5.39 5.40V FCA3 (λ3/nSi)3 7.14 7.33 6.75V eff1 (λ1/nSi)3 1.69 1.60 1.66V eff2 (λ2/nSi)3 2.08 2.06 2.09V eff3 (λ3/nSi)3 2.33 2.34 2.10189Appendix F. Nonlinear coefficientsIn Table F.1, β2,Γthm and Veffm are reported and are given by,β2 = 2β∗1 = 2β∗3 =12∫Si d3xε0 Re(χ(3)Si )[(E˘∗2 · E˘∗2)(E˘1 · E˘3) + 2(E˘∗2 · E˘1)(E˘∗2 · E˘3)](∫d3xε(r)|E˘2|2)(∫d3xε(r)|E˘1|2)1/2(∫d3xε(r)|E˘3|2)1/2(F.1)andΓthm =∫Si d3xε(r)|Em(r)|2∫d3xε(r)|Em(r)|2 , (F.2)andV effm =∫d3xε(r)|E˘m|2max[ε(r)|E˘m|2] . (F.3)The coefficients αm,m and κFCAm,m,m are indirectly reported through ΓTPAm , ΓFCAm , VTPAm , andV FCAm , where,αm,m =∫Si d3x[(E˘∗m · E˘∗m)(E˘m · E˘m) + 2|E˘m|4](∫d3xε(r)|E˘m|2)2=3ΓTPAmV TPAm (εSi)2, (F.4)andκFCAm,m,m =∫Si d3xε(r)|E˘m|2[(E˘∗m · E˘∗m)(E˘m · E˘m) + 2|E˘m|4](∫d3xε(r)|E˘m|2)3=3ΓFCAm(V FCAm )2(εSi)2. (F.5)The reported coefficients ΓTPAm , ΓFCAm , VTPAm and VFCAm are commonly quoted in the literature forvarious PC microcavity structures [11, 94, 106, 111]. They are given by,ΓTPAm =13∫Si d3xε(r)2[(E˘∗m · E˘∗m)(E˘m · E˘m) + 2|E˘m|4]∫d3xε(r)2|E˘m|4, (F.6)V TPAm =(∫d3xε(r)|E˘m|2)2∫d3xε(r)2|E˘m|4, (F.7)ΓFCAm =13∫Si d3xε(r)3|E˘m|2[(E˘∗m · E˘∗m)(E˘m · E˘m) + 2|E˘m|4]∫d3xε(r)3|E˘m|6, (F.8)and (V FCAm)2=(∫d3xε(r)|E˘m|2)3∫d3xε(r)3|E˘m|6. (F.9)190Appendix GX2 minimization plotsIn this appendix, the X2 minimizations are plotted as a function of the 8 fit parameters found fromthe nonlinear transmission analysis. Figures G.1 to G.4 are plots for the independent characteriza-tions of Devices 1 to 4, while Figs. G.5 to G.8 are for the minimizations calculated with Rth, τcarrierand Qabs held fixed at the mean values.0 5 10Qabs×10630405060240 50 60 70Rth3040506020.8 1 1.2τcarrier(ns)304050602-1 0 1φin(π)3035402-1 0 1φout(π)30354022.4 2.6 2.8ηwg13040506020.35 0.4ηwg2400600800100020.044 0.046ηwg38048068088108122T T T TT T FWMFWM(K/mW)Figure G.1: Device 1 X2 minimization plots. The parameter on each x axis is fixed and the minimum X2over all other parameters is shown. The straight black lines show the min(X2) + 1. The shaded region in(g) indicates ηwg2 values outside of the range of possible values.191Appendix G. X2 minimization plots1 2 3Qabs ×1063040506021.8 2 2.2ηwg130405060242242342442542620.024 0.025ηwg3422423424425426238 42 46Rth3040506020.7 0.8 0.9τcarrier(ns)3040506020 1 2φin(π)3040506020 1 2φout(π)304050602T T T TTT FWMFWM(K/mW)2.95 3 3.05ηwg2Figure G.2: Device 2 X2 minimization plots. The parameter on each x axis is fixed and the minimum X2over all other parameters is shown. The straight black lines show the min(X2) + 1.192Appendix G. X2 minimization plots0 2 4Qabs×1069095100235 40 45Rth909510020.8 0.9 1τcarrier(ns)90951002-1 0 1φin(π)8590951001052-1 0 1φout(π)85909510010520.4 0.42 0.44ηwg1909510020.66 0.68 0.7ηwg254054555020.019 0.021ηwg35405455502 FWMFWMTT T TTT(K/mW)Figure G.3: Device 3 X2 minimization plots. The parameter on each x axis is fixed and the minimum X2over all other parameters is shown. The straight black lines show the min(X2) + 1.193Appendix G. X2 minimization plots2 4 6Qabs×1069010011012020.8 0.86ηwg190100110120240 45Rth9010011012021.1 1.15ηwg241424344454621.2 1.3 1.4τcarrier(ns)9010011012020 1 2φin(π)10015020025020 1 2φout(π)1001502002502T T TT TTFWM0.038 0.042ηwg34142434445462 FWM(K/mW)Figure G.4: Device 4 X2 minimization plots. The parameter on each x axis is fixed and the minimum X2over all other parameters is shown. The straight black lines show the min(X2) + 1.4.2 4.4 4.6ηwg1457045754580χ20.45 0.5 0.55ηwg2457045754580χ20.065 0.075ηwg3457045754580χ2(a) (c)(b)tottottotFigure G.5: Device 1 X2 minimization plots, when Rth, Qabs and τcarrier are fixed to the average values in themodel functions. The parameter on each x axis is fixed and the minimum X2 over all other free parametersis shown. The straight black lines show the min(X2) + 1.194Appendix G. X2 minimization plots1.8 2 2.2ηwg1525530535540545550χ20.022 0.026ηwg3525530535540545550χ23 3.1 3.2ηwg2525530535540545550χ2(a) (c)(b)tottottotFigure G.6: Device 2 X2 minimization plots, when Rth, Qabs and τcarrier are fixed to the average values inthe model functions. The parameter on each x axis is fixed and the minimum X2 over all other parametersis shown. The straight black lines show the min(X2) + 1.0.35 0.4 0.45ηwg1635640645650χ20.67 0.68 0.69ηwg2635640645650χ20.018 0.021ηwg3635640645650χ2(a) (c)(b)tottottotFigure G.7: Device 3 X2 minimization plots, when Rth, Qabs and τcarrier are fixed to the average values inthe model functions. The parameter on each x axis is fixed and the minimum X2 over all other parametersis shown. The straight black lines show the min(X2) + 1.0.9 0.95 1ηwg1300310320330χ21.25 1.3 1.35ηwg2300310320330χ20.05 0.06ηwg3300310320330χ2(a) (c)(b)tottottotFigure G.8: Device 4 X2 minimization plots, when Rth, Qabs and τcarrier are fixed to the average values inthe model functions. The parameter on each x axis is fixed and the minimum X2 over all other parametersis shown. The straight black lines show the min(X2) + 1.195

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