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Spectral tailoring of silicon integrated Bragg gratings Cheng, Rui 2020

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Spectral Tailoring of Silicon IntegratedBragg GratingsbyRui ChengB.Sc., Shenyang Ligong University, 2013M.A.Sc., Huazhong University of Science and Technology, 2016A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)March 2020c© Rui Cheng 2020The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the dis-sertation entitled:Spectral Tailoring of Silicon Integrated Bragg Gratingssubmitted by Rui Cheng in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin Electrical and Computer EngineeringExamining Committee:Lukas Chrostowski, Electrical and Computer EngineeringSupervisorNicolas Jaeger, Electrical and Computer EngineeringSupervisory Committee MemberKeng Chou, Department of ChemistryUniversity ExaminerMu Chiao, Mechanical EngineeringUniversity ExaminerAdditional Supervisory Committee Members:Sudip Shekhar, Electrical and Computer EngineeringSupervisory Committee MemberiiAbstractIntegrated Bragg gratings (IBGs) developed on the silicon-on-insulator (SOI)platform, owning to their high spectral flexibility, have become key compo-nents in photonic integrated circuits. Despite the rapid development ofsilicon IBG devices, there still lacks a comprehensive design methodologyto achieve arbitrary, sophisticated, complex (amplitude and phase) spectralresponses on IBGs. In addition, various problems also exist in practicaldesigns and implementations of IBGs. The objectives of this thesis are toaddress these issues and, thus, to facilitate and improve the spectral tailoringof silicon IBG devices.A comprehensive and sophisticated design methodology of IBGs to achievearbitrary spectral responses has been developed, and each individual step ofthe design and implementation process has been elaborated in detail. Fur-thermore, to address the IBG modeling and apodization issues existing inthe design process, we have proposed (1) a highly efficient and reliable IBGmodeling method by directly synthesizing the physical structure of the grat-ings; and (2) a high-performance apodization technique for IBGs based onperiodic phase modulation.Multichannel photonic Hilbert transformers (MPHTs) based on com-plex synthesized IBGs have been designed, fabricated, and experimentallycharacterized. The realizations of these MPHTs are based on using the com-prehensive IBG design methodology developed in this thesis. MPHTs with atotal wavelength channel number of up to 9 and a single channel bandwidthof up to 625 GHz have been successfully achieved.The impacts of apodization phase errors (APE) on the spectral responsesof apodized silicon IBGs have been characterized. The characterization re-sults show that APE can largely distort the spectral responses of apodizedIBGs from the designed ones. Then, to address this issue, a methodology tocompensate and thus to eliminate APE of an apodized IBG to correct thedistorted response has been developed and experimentally validated.A novel apodization profile [κ(z)] amplification technique for IBGs hasbeen proposed. Using this κ(z) amplification technique for designing IBGscan bring about significant improvements in the apodization performence foriiithe given fabrication constraints. Therefore, this technique can largely over-come the current apodization limitations of silicon IBGs due to fabricationconstraints, thus facilitating their spectral tailoring applications.ivLay SummaryA silicon integrated Bragg grating (IBG) is essentially a silicon waveguidestructure whose waveguide width is changed periodically along its length.Due to such periodic waveguide width variations, an IBG will reflect certainwavelengths of light, and thus can act a wavelength-specific mirror. Thereflection responses of IBGs can be easily modified by modulating the pa-rameters of their periodic width variation profiles, such as the amplitudeand phase. Such high spectral flexibility makes them attractive in many ap-plications, such as optical telecommunications and optical signal processing.Despite many studies of silicon IBGs, there still lacks a comprehensivedesign methodology to achieve arbitrary reflection responses on IBGs. Inaddition, various problems also exist in practical designs and implementa-tions of IBGs. This thesis is a comprehensive study on spectral designs ofsilicon IBGs, which aims to address these issues and, thus, to facilitate andimprove the spectral tailoring of silicon IBG devices.vPrefaceThe content of this thesis is mostly based on the publications listed below,which resulted from collaborations with other researchers.Section 2.6.2 and Chapter 5 have been published in• R. Cheng, Y. Han, and L. Chrostowski, “Characterization and com-pensation of apodization phase noise in silicon integrated Bragg grat-ings,” Optics Express, vol. 27, 9516-9535 (2019).I conceived the idea, conducted the device design, performed the mea-surements and data analysis, and drafted the manuscript. Y. Hanprovided useful suggestions regarding the device design and assistedin editing the manuscript. L. Chrostowski supervised the project andedited the manuscript.Chapter 3 has been published in• R. Cheng and L. Chrostowski, “Multichannel photonic Hilbert trans-formers based on complex modulated integrated Bragg gratings,” Op-tics Letters, vol. 43, 1031-1034 (2018).I conceived the idea, conducted the device design, performed the mea-surements and data analysis, and drafted the manuscript. L. Chros-towski supervised the project and edited the manuscript.Chapter 4 has been published in• R. Cheng and L. Chrostowski, “Apodization of silicon integrated Bragggratings through periodic phase modulation,” IEEE Journal of Se-lected Topics in Quantum Electronics, vol. 26, no. 2, pp. 1-15,March-April 2020.I conceived the idea, conducted the device design, performed the mea-surements and data analysis, and drafted the manuscript. L. Chros-towski supervised the project and edited the manuscriptviChapter 6 has been published in• R. Cheng, H. Yun, S. Lin, Y. Han, and L. Chrostowski, “Apodiza-tion profile amplification of silicon integrated Bragg gratings throughlateral phase delays,” Optics Letters, vol. 44, 435-438 (2019).I conceived the idea, conducted the device design, performed the mea-surements and data analysis, and drafted the manuscript. H. Yun, S.Lin and Y. Han helped with many discussions and edited the manuscript.L. Chrostowski supervised the project and edited the manuscript.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . xxiiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Silicon Integrated Bragg Gratings . . . . . . . . . . . . . . . 21.2 Applications of Response Tailored IBGs . . . . . . . . . . . . 41.2.1 Versatile optical filters . . . . . . . . . . . . . . . . . 41.2.2 Routers and multi/demultiplexers . . . . . . . . . . . 51.2.3 All-optical signal processing . . . . . . . . . . . . . . 61.2.4 Microwave photonics signal processing . . . . . . . . 71.3 Existing Issues . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 About This Thesis . . . . . . . . . . . . . . . . . . . . . . . . 111.4.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.2 Thesis organization . . . . . . . . . . . . . . . . . . . 112 Design of Silicon IBGs to Achieve Arbitrary Spectral Re-sponses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Overall Design Flow . . . . . . . . . . . . . . . . . . . . . . . 142.2 Conversion of a Target Response to be Physically Realizable 16viii2.3 Calculation of Required Grating Profiles from a Target Re-sponse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Fundamental Grating Structural Parameter Determination . 222.4.1 Corrugation shape . . . . . . . . . . . . . . . . . . . . 222.4.2 Grating period . . . . . . . . . . . . . . . . . . . . . . 242.4.3 Corrugation width . . . . . . . . . . . . . . . . . . . . 242.5 Grating Apodization . . . . . . . . . . . . . . . . . . . . . . 272.5.1 Corrugation width modulation . . . . . . . . . . . . . 282.5.2 Duty-cycle modulation . . . . . . . . . . . . . . . . . 282.5.3 Lateral misalignment modulation . . . . . . . . . . . 292.5.4 Cladding modulation . . . . . . . . . . . . . . . . . . 302.5.5 Sinusoidal phase modulation . . . . . . . . . . . . . . 312.6 Grating Modeling . . . . . . . . . . . . . . . . . . . . . . . . 322.6.1 Various grating modeling methods . . . . . . . . . . . 322.6.2 Structure synthesis-based transfer matrix method . . 333 Multichannel Photonic Hilbert Transformers based on Com-plex Synthesized IBGs . . . . . . . . . . . . . . . . . . . . . . . 413.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . 473.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 Apodization of Silicon IBGs through Periodic Phase Mod-ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.1.1 Basic theory . . . . . . . . . . . . . . . . . . . . . . . 544.1.2 Relationship between the phase amplitude A and F0for several common periodic functions . . . . . . . . . 554.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 584.2.1 Determination of the total phase profile from the grat-ing strength and phase profiles . . . . . . . . . . . . . 594.2.2 Phase-modulated grating structures . . . . . . . . . . 624.3 Design Considerations . . . . . . . . . . . . . . . . . . . . . . 674.3.1 Impact of different phase periods on the spectral per-formance . . . . . . . . . . . . . . . . . . . . . . . . . 684.3.2 Comparison of grating robustness against fabricationconstraints for different periodic phase functions . . . 714.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . 744.4.1 Gaussian apodized IBGs . . . . . . . . . . . . . . . . 74ix4.4.2 Single-channel flattop filters . . . . . . . . . . . . . . 754.4.3 3-channel flattop filters . . . . . . . . . . . . . . . . . 774.4.4 Flattop dispersion compensating filters . . . . . . . . 784.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 Characterization and Compensation of Apodization PhaseErrors in Silicon IBGs . . . . . . . . . . . . . . . . . . . . . . . 815.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Impact of APE on Apodized IBG Responses . . . . . . . . . 825.3 APE Compensation and Spectral Correction . . . . . . . . . 875.3.1 APE distribution extraction . . . . . . . . . . . . . . 875.3.2 IBG structure corrections for APE compensation . . 905.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . 925.4.1 Gaussian-apodized IBGs . . . . . . . . . . . . . . . . 935.4.2 Square-shaped filters . . . . . . . . . . . . . . . . . . 955.4.3 Photonic Hilbert transformers . . . . . . . . . . . . . 965.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986 Apodization Profile Amplification of Silicon IBGs . . . . . 1006.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.2 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 1026.3.1 κ(z) amplification in ∆W -modulated IBGs . . . . . . 1026.3.2 κ(z) amplification in phase-modulated IBGs . . . . . 1046.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . 1086.4.1 Gaussian-apodized ∆W -modulated IBGs . . . . . . . 1086.4.2 Phase-modulated IBGs . . . . . . . . . . . . . . . . . 1096.5 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . 1117 Conclusion and Future Work . . . . . . . . . . . . . . . . . . 1127.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117xList of Tables4.1 Spectral parameters of single-channel flattop filters . . . . . . 774.2 Average channel performances of the 3-channel flattop filters 785.1 Spectral parameters of the Gaussian-apodized IBGs . . . . . 955.2 Spectral parameters of the square filters based on ∆L-modulatedIBGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.1 Average channel performances of the 5-channel dispersionlesssquare filters based on phase-modulated IBGs. . . . . . . . . 110xiList of Figures1.1 (a) Cross-sectional view of silicon-on-insulator (SOI) wafer.(b) Common waveguides in silicon photonics; (left) strip waveg-uide and (right) rib waveguide. Reprinted with permissionfrom Springer Nature [15]. . . . . . . . . . . . . . . . . . . . . 21.2 (a) and (b) Schematic illustrations of a typical sidewall Bragggrating developed on a strip silicon waveguide; W : waveguidewidth; ∆W : corrugation width; ΛG: grating period; N : num-ber of the periods. (c) Scanning electron microscope (SEM)image of a representative IBG developed on a strip siliconwaveguide. (d) Typical reflection and transmission responsesof an IBG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Various optical filters achieved on apodized silicon IBGs. (a)Bandpass filter with a high side-lobe suppression ratio. (b)Square-shaped filter with little in-band dispersion, or a flatin-band group-delay response. (c) Square-shaped filter witha linear in-band group-delay response, which can be usedfor dispersion compensating purposes. (d) 3-channel square-shaped filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 (a)-(b) Schematic of a lateral misalignment-modulated, Gaus-sian apodized, grating-assisted, contra-directional coupler (CDC).(a) and (b) are 3D view and top view of the device, respec-tively. (c) SEM image of the fabricated device. Reprintedfrom [28]. Copyright 2018, IEEE. . . . . . . . . . . . . . . . 61.5 Schematic illustration of using a complex synthesized IBG formicrowave photonics signal processing. Adapted from [19].Copyright 2013, Optical Society of America. . . . . . . . . . 71.6 Schematic illustrations of various fabrication issues for sili-con IBGs. (a)-(d) are lithography writing jitters, lithographysmoothing effect, quantization errros, and minimum realiz-able feature size/spacing, respectively. . . . . . . . . . . . . . 10xii1.7 Simulated spectral responses of a Gaussian-apodized, corru-gation width-modulated, SOI IBG with and without consider-ing waveguide width and thickness variations along the grat-ing length (∆W and ∆H) due to manufacturing variations. 112.1 Schematic comparison of an IBG and a finite impulse response(FIR) filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Design flow of a silicon IBG to achieve an arbitrary complexspectral response, where the design of a microwave photonicsHilbert transformer is used as an example. . . . . . . . . . . 142.3 (a) Ideal amplitude (blue) and phase (red) spectral responsesof the designed MWP Hilbert transformer. (b) Truncated andspatially shifted impulse response, where the inset shows theoriginal impulse response. (c) Physically realizable spectralresponse of the designed MWP Hilbert transformer. . . . . . 162.4 Spectral performance evolution as a function of the grat-ing length for a IBG-based MWP Hilbert transformer. Thedashed curves represent the ideal responses, while the solidcurves are the simulated grating responses where the waveg-uide thickness and width variations along the grating lengthdue to fabrication nonuniformity are taken into account. . . 172.5 Discrete model of a grating used in the LPA. the forward-propagating and backward-propagating fields are denoted byuj and vj , respectively. Reprinted from [50]. Copyright 2001,IEEE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.6 (a) Complex response of the designed MWP Hilbert trans-former. (b) Grating strength and phase profiles, κ(z) andφG(z), required by the design, calculated by using the LPA. 222.7 (a) Schematic representations of rectangular and sinusoidalIBGs. (b) Experimentally extracted grating strength as afunction of grating corrugation width for rectangular and si-nusoidal IBGs. (c) Spectral comparison of lateral misalign-ment modulated, Gaussian-apodized rectangular and sinu-soidal IBGs, calculated by using the structure synthesis-basedtransfer matrix method (SS-TMM) under the same samplinginterval of 6 nm. . . . . . . . . . . . . . . . . . . . . . . . . . 232.8 Reflection spectra of silicon IBGs with different grating peri-ods (ΛG). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.9 Illustration of choosing ∆W according to the maximum valueof the grating strength profile. . . . . . . . . . . . . . . . . . 24xiii2.10 (a) Comparison of typical band structures for uniform me-dia and 1-D photonic crystals. (b) Schematic illustration ofhow the band structure is related to the reflection band of aBragg grating. (c) FDTD band structure analysis in Lumeri-cal FDTD Solutions. (d) Band structure diagram of a waveg-uide Bragg grating with ∆W of 50 nm. (e) Fourier transform(magnitude) of the time domain signals at the band edge fromthe FDTD simulation for Bragg gratings with different ∆W ;the frequency spacing between the two peaks corresponds tothe band gap width. Reprinted with permission from SpringerNature [15]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.11 Comparison of the relationships between κ and ∆W obtainedfrom the band structure simulations and from the experimen-tal results. Adapted with permission from Springer Nature[15]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.12 Schematic representation of a Gaussian-apodized IBG viamodulating ∆W . . . . . . . . . . . . . . . . . . . . . . . . . 282.13 (a) Schematic of grating duty-cycle modulation (left) andthe relationship curve between duty-cycle and κ (right). (b)Schematic of side-wall misalignment (∆L) modulation (left)and the relationship curve between ∆L/ΛG and κ (right).Reprinted with permission from Springer Nature [15]. . . . . 292.14 Schematic representation of a cladding-modulated Bragg grat-ing; ΛG: period of the silicon cylinders; W : diameter ofthe silicon cylinder; d: distance of the silicon cylinders fromthe edge of the waveguide. Reprinted with permission fromSpringer Nature [15]. . . . . . . . . . . . . . . . . . . . . . . 31xiv2.15 (a) Schematic flow showing the process of the SS-TMM mod-eling. (b) The upper figure plots the normalized Gaussianapodization profile (blue, left axis) and the translated lat-eral misalignment-to-period ratio ∆L/ΛG (red, right axis)along the grating; the bottom diagrams illustrate the grat-ing structures at the different positions (1-3), whose loca-tions are indicated in the upper plot. (c) Schematic illus-trations of spatial sampling of the cell structure of a lateralmisalignment-modulated IBG in different cases of (i) rectan-gular and (ii) sinusoidal grating shapes, where the bottomfigures plot the corresponding ∆Ws(k) profiles; ∆Ws(k) isdefined as the width variation of the kth segment from theunperturbed waveguide width [denoted as W in (c)]. (d) Il-lustration of the transfer matrices describing the wave prop-agation through an interface (left) and through a uniformsection (right); the upper diagrams show the original grat-ing structures while the lower ones illustrate the equivalentmultiplayer structures used in the SS-TMM modeling. . . . . 342.16 Comparison of the implementation process in conventionalCMT-TMM and SS-TMM. . . . . . . . . . . . . . . . . . . . 382.17 Comparison of the grating spectra calculated by the CMT-TMM and SS-TMM and the experimental results for Gaussian-apodized IBGs through (a) misalignment and (b) duty-cyclemodulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1 Ideal spectral responses of (a) traditional single channel and(b) multichannel photonic HTs. . . . . . . . . . . . . . . . . 423.2 (a) Ideal amplitude (blue) and phase (red) spectral responseof the designed 5-channel MPHT with B = 3 nm. (b) Trun-cated and time-shifted impulse response, where the inset showsthe original impulse response. (c) Realizable spectral re-sponse of the designed MPHT. . . . . . . . . . . . . . . . . . 433.3 (a) The grating strength and phase profiles required by thetarget spectral response, which are calculated by using theLPA. (b) Phase profile of the designed grating for the apodiza-tion; the phase period Λφ is 2.2 µm. . . . . . . . . . . . . . . 44xv3.4 (a) Schematic flow showing the process of mapping the apodiza-tion profile into grating structure. (b) Schematic diagrams il-lustrating the grating cell structures for different phase differ-ences between the adjacent periods. (c) Phase differences be-tween the neighboring periods (upper), and the calculated dis-tances between adjacent corrugations (lower). (d) Schematicof the overall (left) and zoomed-in portion (right) of the spiralphase-modulated grating. . . . . . . . . . . . . . . . . . . . . 463.5 (a) SEM images of a sample 5-channel MPHT based on aspiral IBG. (b) Measured amplitude (blue) and phase (red)spectral responses of the IBG. . . . . . . . . . . . . . . . . . . 483.6 Required κ(z) and φG(z) profiles (left figures) and measuredspectral responses (right figures) of MPHTs with differentbandwidths and channel numbers; N: total number of gratingperiods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.7 (a) Building block used in the numerical analysis to verifythe potential of MPHTs as multichannel SSB filters. (b) Cal-culated transfer functions for different ports when using theexperimental S parameters of (left) 5 channel MPHT withB = 3 nm, and (right) 7 channel MPHT with B = 3 nm; thedash lines indicate different SSB filtering channels . . . . . . 514.1 (a)-(c) Different periodic phase functions (left) and the cor-responding relationships between the phase amplitude A andthe 0th Fourier coefficient, F0 (right). . . . . . . . . . . . . . 564.2 Illustration of two different definitions of normalized gratingcoupling strength profiles, κn(z). κn(z) in (a) is defined torepresent only the grating strength while the grating phasedistribution is represented by a separate function [φG(z)]; thisdefinition is chosen to be used in this thesis. κn(z) in (b) isdefined to include both the grating strength and phase infor-mation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3 Schematic flow showing the design process of a phase-modulatedIBG to achieve a target response. . . . . . . . . . . . . . . . 594.4 (a) Normalized Gaussian apodization profile. (b) Phase am-plitude profiles along the grating for different periodic phasefunctions. (c)-(e) Overall (left) and zoomed-in (right) gratingphase modulation profiles in cases of different periodic phasefunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60xvi4.5 (a) Spectrum of the designed flattop filter. (b) Grating strengthκ (blue, left axis) and phase φG (grey, right axis) profiles re-quired by the designed spectral response, calculated via theLPA. (c) Phase amplitude profile along the grating, A(z), incase of square phase functions, calculated from the normalizedκ(z) profile via Eq. 4.14. (d) Phase profile, φκ(z) (yellow),and its amplitude profile, A(z) (red); the top-left inset showsthe enlarged view of the highlighted range of φκ(z). (e) To-tal grating phase profile, φtol(z), which is the superpositionof φκ(z) and φG(z). (f) Reconstructed spectrum of an IBGphase-modulated by φtol(z) in (e), calculated via the CMT-based TMM. . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.6 (a)-(b) are schematic illustrations of two different phase-to-structure conversion schemes, denoted as Schemes 1 and 2,respectively. (a) A typical phase-modulated IBG structuregenerated using Scheme 1. (b) (Upper) schematic flow show-ing the implementation process of Scheme 2, and (lower) atypical phase-modulated IBG structure created using Scheme2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.7 (a)-(c) Typical grating structures modulated by different pe-riodic phase functions. For each figure, the upper plot showsthe phase function while the lower diagram is the correspond-ing grating structure, created based on Scheme 1. . . . . . . 644.8 (a) Square phase functions with different amplitudes and pe-riods. (b) Comparison of grating structures corresponding tothe different phase functions shown in (a), generated basedon Scheme 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.9 (a) Schematic illustration of the developed basic computa-tional lithography model; the upper and bottom diagramsshow the original and lithography-simulated grating struc-tures, respectively; the positions labeled by 1 and 2 in theupper grating structure indicate feature spacing and size thatare smaller than the minimum realizable values and thusare not resolved in the lithography-simulated grating. (b)Schematic illustration of the SS-TMM for IBG modeling; theupper diagram schematically illustrates the spatial samplingprocess of an IBG, while the bottom one shows the equivalentoptical multiplayer structure. . . . . . . . . . . . . . . . . . . 67xvii4.10 (a) Simulated responses of flattop filters developed on saw-tooth phase-modulated IBGs using different phase period (Λφ)of 12 and 8 µm; the displayed wavelength range is as largeas 120 nm to include the side-resonances. (b)-(d) Compari-son of simulated responses of the flattop grating filters withdifferent Λφ over a narrow wavelength range. (e) Side-lobe-suppression-ratio (SLSR) of the grating response as a functionof Λφ for the flattop filter (blue, left axis) and the Gaussian-apodized grating (red, right axis). (f) SLSR as a function ofΛφ for the two filters in the ideal case, i.e., without consider-ing the fabrication limitations. The grid size of the lithogra-phy and the minimum feature spacing/size in the lithographymodel used for the calculations in (a)-(e) are set to be 6 nmand 60 nm, respectively. . . . . . . . . . . . . . . . . . . . . 704.11 (a) Response of the designed 3-channel flattop filter. (b)Grating strength (κ) and phase (φG) profiles required by thedesign. (c) Grating responses for different periodic phasefunctions in the ideal case (without considering the fabrica-tion limitations) calculated by the CMT-based TMM. . . . . 714.12 (a)-(c) are comparisons of grating responses for different peri-odic phase functions in cases of low Λφ, a large minimum real-izable feature size/spacing and a low lithography resolution,respectively; the responses are predicted by the SS-TMM to-gether with a computational lithography model to take intoaccount the fabrication constraints. . . . . . . . . . . . . . . 734.13 Measured responses of Gaussian-apodized phase-modulatedIBGs using different periodic phase functions and based ondifferent structure determination schemes of Schemes 1 and 2. 754.14 (a) Measured responses (blue) of the flattop grating filters us-ing different periodic phase functions with the same Λφ of 1.7µm, where the ideal responses (grey) are included for com-parison. (b) and (c) are comparisons of measured responsesof flattop filters based on square phase-modulated IBGs usingdifferent Λφ. . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.15 Measured responses of 3-channel flattop filters developed onphase-modulated IBGs using different periodic phase func-tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77xviii4.16 (a) Reflection (blue, left axis) and group delay (grey, rightaxis) responses of the designed flattop dispersion-compensatingfilters. (b) Grating strength (blue, left axis) and phase (grey,right axis) profiles required by the design, calculated by usingthe LPA. (c) Phase profile, φκ(z) (yellow), and its amplitudeprofile, A(z) (red). (d) Total grating phase profile, φtol(z)(yellow), which is the superposition of φκ(z) and φG(z) (grey).(e) and (f) are the measured complex responses of the gratingfilters using square and sawtooth phase functions, respectively. 795.1 (a) Calibrated model for ∆neff -versus-∆W used in the SS-TMM for IBGs developed on 500 nm (wide) × 220 nm (high)SOI strip waveguides. (b) Reflection and transmission spec-tra of the ∆L-modulated Gaussian-apodized IBG calculatedby the SS-TMM using a sampling interval of 6 nm. (c) Com-parison of the calculated reflection spectra of the IBG usingdifferent sampling intervals; the black curve represents theideal reflection spectrum calculated via the CMT-TMM. . . 835.2 (a) Schematic illustration of the DC modulation in an IBG(left) and the relationship between the DC and normalizedgrating strength (right). (b) Normalized grating strengthprofile (blue, left axis) and the converted DC distributionalong the grating (red, right axis). (c) Spectrum of the DC-modulated Gaussian-apodized IBG calculated by the SS-TMM(blue), and the ideal spectrum (black). . . . . . . . . . . . . 845.3 (a) Spectrum of the designed square filter, which has beenmodified to be physically realizable. (b) Grating strength(upper) and phase (lower) profiles required by the designedspectral response, calculated via the LPA. The blue curves in(c) and (d) are the SS-TMM predicted spectra of the ∆L- andDC-modulated IBGs, respectively, where the black curves arethe ideal spectra calculated by using the CMT-TMM. . . . . 855.4 (a) Complex spectral response of the designed photonic Hilberttransformer, which has been modified to be physically realiz-able. (b) Grating strength (upper) and phase (lower) profilesrequired by the design. (c) Complex spectral response of the∆L-modulated IBG calculated by the SS-TMM. . . . . . . . 86xix5.5 Design process of an IBG with the APE compensation in-cluded; the procedures enclosed by the red dashed line arefor the extraction of the APE distribution. LPA: layer peel-ing algorithm [50]. . . . . . . . . . . . . . . . . . . . . . . . . 885.6 (a) Grating strength (blue, left axis) and phase (red, rightaxis) profiles used for creating the temporary IBG structurethat is dedicated for the APE extraction purpose. (b) Am-plitude response of the temporary IBG calculated by the SS-TMM. (c) Grating strength (blue, left axis) and phase (red,right axis) profiles of the temporary IBG synthesized fromits complex response using the LPA. (d) Phase differencesbetween neighboring periods. . . . . . . . . . . . . . . . . . . 895.7 Schematic illustrations of the correction of (a) grating periodΛG and (b) distance between adjacent grating corrugations daccording to different values of ∆φC(i) in a uniform IBG. . . 915.8 (a) and (c) show the ΛG and d correction profiles, respectively,for the designed square filter based on the ∆L-modulatedIBG. (b) and (d) present the SS-TMM calculated spectra ofthe ΛG- and d-corrected IBGs, respectively, with the idealspectrum (black) included in each figure for comparison. . . 925.9 (a)-(c) Experimental data of the ∆L-modulated Gaussain-apodized IBG. (a) Measured (blue) and SS-TMM predicted(red) spectra of the originally designed IBG. (b) ΛG-correctionprofile of the IBG. (c) Measured spectrum of the ΛG-correctedIBG (blue). (d)-(f) Experimental data of the DC-modulatedGaussain-apodized IBG. (d) Measured (blue) and SS-TMMpredicted (red) spectra of the originally designed IBG. (e)ΛG-correction profile of the IBG. (f) Measured spectrum ofthe ΛG-corrected IBG (blue). The black curves in (a), (c),(d) and (f) are the ideal spectra for comparison. . . . . . . . 935.10 Experimental data of the square filter based on the ∆L-modulatedIBG. (a) Measured (blue) and SS-TMM predicted (red) spec-tra of the originally designed IBG. (b) ΛG-correction profileof the IBG. (c) Measured spectrum of the ΛG-corrected IBG(blue). The black curves in (a) and (c) are the ideal spectrafor comparison. . . . . . . . . . . . . . . . . . . . . . . . . . 95xx5.11 (a) Measured (blue) and SS-TMM predicted (red) amplitudeand phase responses of the photonic Hilbert transformer basedon the original ∆L-modulated IBG. (b) ΛG-correction pro-file of the IBG. (c) Measured complex response of the ΛG-corrected IBG (blue), and the ideal response (black). (d)Building block used in the numerical analysis for achievinga single side-band filtering response. (e) Calculated transferfunctions of the building block when using the experimentalS parameters of the original (red) and corrected (blue) IBGs. 976.1 (a) Schematic flow illustrating the implementation process ofthe κ(z)-amplification technique in a ∆W -modulated IBG.(b) Relationship curve between AF and ∆φ described by Eq.6.3. (c) Original and scaled-up Gaussian κ(z)/∆W (z) pro-files. (d) Center portions of the (left) original and (right)κ(z)-amplified (AF = 5) IBGs in the layout. . . . . . . . . . 1036.2 (a) Amplitude and group-delay responses of the designed 5-channel dispersionless square filter. (b) Normalized gratingstrength and phase profiles required by the design. . . . . . . 1046.3 (a) Schematic flow showing the process of determining theκ(z)-amplified phase-modulated IBG structure. (b) Originaland amplified κn(z) profiles, where the dashed lines indicatethe divided grating regions. (c) (Upper) lateral phase de-lay and (lower) compensation phase distribution along thegrating. (d) d(i) profiles of the original and κ(z)-amplifiedIBGs. (e) Structures of (upper) the κ(z)-amplified IBG inthe AF ≈ 23 region (z = 60.5 µm), and (lower) the unam-plified IBG at the same position; ∆W have been exaggeratedfor illustrative purposes. . . . . . . . . . . . . . . . . . . . . 1066.4 (a) SEM image of the center portion of the fabricated κ(z)-amplified IBG (AF = 5). (b) Measured reflection spectraof the original and κ(z)-amplified Gaussian-apodized ∆W -modulated IBGs, along with the ideal spectrum. . . . . . . . 1086.5 SEM images of the fabricated κ(z)-amplified, phase-modulatedIBG in (a) AF ≈ 23 region and (b) the boundary portion be-tween AF = 1 and AF ≈ 3 regions. . . . . . . . . . . . . . . 1096.6 (a) Measured (left) normalized reflection and (right) groupdelay responses of the κ(z)-amplified IBG. (b) Measured re-flection response of the unamplified IBG. . . . . . . . . . . . 110xxi7.1 (a) Layouts of periodic phase-modulated IBG; (upper) as-designed geometry, and (lower) geometry after performing193 nm deep-ultraviolet lithography simulations. (b) Simu-lated spectra of original and lithography modeled IBGs; ∆Wof the IBG to be lithography modeled has been set to be 20nm, which is much larger than that of 6 nm in the originaldesign to compensate for the strong smoothing effects of pho-tolithography. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147.2 Schematic illustration of a grating-assisted contra-directionalcoupler (CDC). . . . . . . . . . . . . . . . . . . . . . . . . . . 115xxiiList of AbbreviationsAF Amplification factorAPE Apodization phase errorsCDC Contral-diectional couplerCMOS Complementary metal-oxide-semiconductorCMT-TMM Coupled mode theory-based transfer matrix methodDWDM Dense wavelength-division multiplexingFDTD Finite-difference time-domainFIR Finite impulse responseHT Hilbert transformerIBG Integrated Bragg gratingLPA Layer peeling algorithimMPHT Multichannel photonic Hilbert transformerMWP Microwave photonicsRF Radio frequencySEM Scanning electron microscopeSLSR Side-lobe suppression ratioSOI Silicon-on-insulatorSSB Single side-bandSS-TMM Structure synthesis-based transfer matrix methodTMM Transfer matrix methodWDM Wavelength-division multiplexingxxiiiAcknowledgmentsI would like to thank my family, especially my Mom and Dad, for their loveand continuous support throughout my time in Canada. I could not havedone it without them.I would like to thank my supervisor, Prof. Lukas Chrostowski, for thekind support, guidance and advice that he has provided me throughout myPhD studies. I particularly appreciate the research freedom he has given me,and the plenty of fabrication opportunities he provided, which has greatlyfacilitated my research. He is a humorous and knowledgeable supervisor,and one of the smartest people I know. I hope that I could be as lively,enthusiastic, and energetic as him.I would like to thank Prof. Nick Jaeger for his mentorship and beingpart of my PhD supervisory committee, and for his generous help on myscientific writing. His rigorous scholarship has been an important influenceon me and will benefit my future career. I also appreciate Prof. SudipShekhar for being part of my PhD supervisory committee.I acknowledge the University of British Columbia for providing stablefinancial support (Four Year Fellowship) throughout my PhD studies.I would like to thank my colleagues, Dr. Huiye Qiu, Ya Han, Stephen HLin, Han Yun, Enxiao Luan, Hossam Shoman, Mustafa Hammood, AjayMistry, Minglei Ma, Jaspreet Jhoja, Abdelrahman Afifi, and MohamedShemis, for their friendship and help.xxivChapter 1IntroductionThe rapid development of integrated optics has enabled numerous free-spaceoptical systems and components to be monolithically integrated into singlechips [1]. These integrated optical systems possess many appealing advan-tages compared with their discrete free-space versions, including low-cost,small-footprint, higher immunity to mechanical vibration, scalability, lowpower consumption and potential of mass-production with low cost [2]. Op-tical integration has benefited a wide range of applications, such as spec-troscopy [3], sensors and lab-on-chip for medical diagnostics [4], imagery [5],quantum computing [6], and especially, optical communication systems [7].Among various integrated platforms, silicon-on-insulator (SOI) platformsare particularly promising [2]. This is mainly due to the high index con-trast of the SOI waveguide and, more importantly, the compatibility withmature complementary metal-oxide-semiconductor (CMOS) manufacturingprocesses. The latter allows the use of electronics fabrication facilities tomake photonic circuitry, and offers the possibility of integration with large-scale electronic circuits to provide increased functionality [8].In this background, there has been a strong motivation to develop var-ious critical free-space optical components on SOI platforms, such as ringresonators [9], Fabry-Pe´rot (FP) etalons [10], Mach-Zehnder interferometers[11], and Bragg gratings [12, 13]. Among these optical components, inte-grated Bragg gratings (IBGs) are extremely useful devices and have receiveda great deal of attention from integrated optics community. For example,they are used to make mirrors for silicon integrated lasers as commercial-ized by Intel [14]. SOI-based IBGs have excellent advantages including highextinction ratio, easy fabrication, low insertion loss, and most importantly,extremely high spectral flexibility. In principle, any physically-realizablecomplex (amplitude and phase) spectral response can be realized on an IBGby modulating the grating strength and phase profiles [15]. Owning to theseexcellent features, IBG devices have become fundamental building blocks ofphotonic integrated circuits and have been extensively used in diverse ap-plications, including optical telecommunications [16], large-scale quantumphotonic systems [17], sensing [18], microwave photonics [19], and optical1signal processing [15]. In this chapter, we will review the typical structuresof silicon-based sidewall IBGs, several important applications of responsetailored IBGs, and the existing issues in the practical designs and imple-mentations of them.1.1 Silicon Integrated Bragg GratingsAn optical Bragg grating is a waveguide structure whose effective refractiveindex is changed periodically along the length. These components can befabricated on optical fibers by exposing photosensitive fibers to ultravioletlight beam [20]. In this way, an optical interference pattern can be im-posed on the fiber, thereby producing a periodic effective refractive indexchange along the fiber. Such in-fiber Bragg gratings have developed intocritical components in many applications such as fiber communications andfiber sensing networks [21]. Recently, significant interest has also gone torealizing Bragg grating devices in SOI waveguides [2, 12]. The first siliconwaveguide-based Bragg grating was demonstrated in 2001 by Murphy et al.[22]. As the SOI platform is promising for ultra-dense on-chip integration ofphotonic and electronic circuitry, building Bragg grating devices on SOI isof great interest for many on-chip applications, such as wavelength-divisionmultiplexing (WDM) [23] and photonic signal processing [19].Figure 1.1: (a) Cross-sectional view of silicon-on-insulator (SOI) wafer. (b)Common waveguides in silicon photonics; (left) strip waveguide and (right)rib waveguide. Reprinted with permission from Springer Nature [15].Figure 1.1(a) shows the cross-sectional view of a typical SOI wafer, whichconsists of a ∼700 µm of silicon substrate, 2 µm of oxide (buried oxide,or BOX), and 220 nm of crystalline silicon. There are different types ofwaveguides that Bragg gratings can be developed on. The most commonone is the strip waveguide, as shown in the left of Fig. 1.1(b). The typicalcross-sectional dimension for a single-mode strip waveguide is 500 nm wideand 220 nm high. Note that all of the IBGs studied in this thesis will be2developed on such typical, single-mode, strip silicon waveguides. IBGs havealso be fabricated on rib waveguides [13]. A rib waveguide is schematicallyshown in the right of Fig. 1.1(b). The rib waveguide (also known as ridgewaveguide) is another important waveguide geometry in silicon photonics.They typically have larger cross-sectional dimensions, thus allowing higherfabrication tolerance [13]. Rib waveguides are also usually used for electro-optic devices, such as modulators, since they allow for electrical connectionsto be made to the waveguide [24]. In addition to strip and rib waveguide,IBGs can also be developed on slot silicon waveguides [25]. These slot-waveguide-based IBGs exhibit particular advantages in sensing applications[18], because the electric field for a slot waveguide is concentrated in thelow-index slot region and, thus, is more sensitive to refractive index changesof the cladding.(a) (b)1535 1540 1545 1550 1555-30-24-18-12-60Intensity (dB)Wavelength (nm) R T(c)(d)GGFigure 1.2: (a) and (b) Schematic illustrations of a typical sidewall Bragggrating developed on a strip silicon waveguide; W : waveguide width; ∆W :corrugation width; ΛG: grating period; N : number of the periods. (c) Scan-ning electron microscope (SEM) image of a representative IBG developed ona strip silicon waveguide. (d) Typical reflection and transmission responsesof an IBG.The most common IBGs are formed by sidewall modulation along thewaveguide. Figures 1.2(a) and 1.2(b) present schematic representations ofsidewall IBGs developed on strip waveguides, where ∆W is the grating cor-rugation width, and ΛG is the period of the sidewall modulation, i.e., thegrating period. Such a sidewall modulation can lead to an equivalent peri-odic perturbation of the effective refractive index along the waveguide, thus3forming a waveguide Bragg grating. ∆W and ΛG will determine the grat-ing strength and the Bragg wavelength (i.e., the center wavelength of thegrating), respectively. The Bragg wavelength, λB, is related to ΛG via thefirst-order Bragg conditionλB = 2neff(λB)ΛG (1.1)where neff(λB) is the effective refractive index of the waveguide as a functionof the wavelength. The effective index of the fundamental TE mode forcommon 500 nm (wide) × 220 nm (high) single-mode strip SOI waveguides is∼2.44 at wavelengths around 1.55 µm. Thus, to obtain a Bragg resonance at∼1.55 µm, the period of a strip SOI waveguide-based IBG should be designedto be near 317 nm. Figures 1.2(c) and 1.2(d) present a scanning electronmicroscope (SEM) image of a representative sidewall IBG developed on astrip silicon waveguide, and typical reflection and transmission responses ofan IBG, respectively.1.2 Applications of Response Tailored IBGsSilicon IBGs have many advantages, such as easy fabrication, low insertionloss, high extinction ratio, and compactness. But the most distinguish-ing one is the flexibility to achieve desired amplitude and phase spectralresponses [15, 26, 27]. This spectral flexibility makes them particularlyattractive for applications where versatile filters or customized spectral re-sponses of the devices are of great interest, such as optical telecommunica-tions, WDM networks, optical signal processing, and microwave photonics.In this section, we will review several important applications of responsetailored IBGs.1.2.1 Versatile optical filtersBragg grating devices are inherently wavelength specific mirrors, and thuscan naturally act as optical filters. Furthermore, the high spectral flexibilitymakes them ideally suited as versatile filters in various applications such asoptical telecommunications and WDM systems. To illustrate the spectralflexibility of IBGs, Fig. 1.3 shows four various optical filters achieved onapodized IBGs. Figure 1.3(a) presents the spectral response of a bandpassfilter with a high side-lobe suppression ratio (SLSR), which is achieved on aGaussian-apodized silicon IBG. Figures 1.3(b)-1.3(d) show the responses ofa dispersion-less square filter, a dispersion-compensating square filter, and4a 3-channel square filter, respectively, which are realized based on complex-synthesized silicon IBGs. The designs of these useful filters use the compre-hensive IBG design methodology that will be presented in the next chapter.-12 -9 -6 -3 0 3 6 9 12-60-45-30-150Reflection (dB)∆λ (nm)(a) High-SLSR filter-4 -2 0 2 4-25-20-15-10-5(b) Square dispersion-less filterReflection (dB)∆λ (nm)-4 -2 0 2 4-30-24-18-12-60(c) Square dispersion-compensating filterReflection (dB)∆λ (nm) -6 -4 -2 0 2 4 6-30-24-18-12-6(d) Multichannel square filterReflection (dB)∆λ (nm)-3003060Group delay (ps)06121824Group delay (ps)Figure 1.3: Various optical filters achieved on apodized silicon IBGs. (a)Bandpass filter with a high side-lobe suppression ratio. (b) Square-shapedfilter with little in-band dispersion, or a flat in-band group-delay response.(c) Square-shaped filter with a linear in-band group-delay response, whichcan be used for dispersion compensating purposes. (d) 3-channel square-shaped filter.1.2.2 Routers and multi/demultiplexersIntegrated optical routers (or add-drop filters) and multi/demultiplexers arecritical devices in photonic integrated circuits [29]. There are a number ofintegrated components that can achieve these functionalities, including mi-croring resonators [30], photonics crystal waveguides [31], Mach-Zehnderinterferometers [32], and IBG-based devices [33, 34]. Compared with theothers, using IBG-based devices for realizing these functionalities can allowfor a much higher spectral flexibility. For example, in applications of addingand dropping optical signals, grating-based devices can be apodized to pro-vide high SLSRs to ensure high channel isolation ratios [28, 33], be designedto have flattop responses for high-speed signals which have large bandwidths5Figure 1.4: (a)-(b) Schematic of a lateral misalignment-modulated, Gaussianapodized, grating-assisted, contra-directional coupler (CDC). (a) and (b)are 3D view and top view of the device, respectively. (c) SEM image of thefabricated device. Reprinted from [28]. Copyright 2018, IEEE.[35], or even be designed to act as mode multi/demultiplexers [36]. Whenused for add-drop filtering and signal multi/demultiplexing, grating-assisted,contra-directional couplers (CDCs) are usually used instead of conventionalIBGs [34]. Different from traditional IBGs, a grating-assisted CDC drops thereflected signals into another waveguide through contra-directional coupling.As such, no circulator or directional is required for the add-drop operation,which allows for monolithic integration with other photonic components,and also decreases the insertion loss. Grating-assisted CDCs inherit thespectral flexibility advantage of traditional IBGs. A grating-assisted CDCwith a Gaussain apodization profile apodized by modulating the lateral mis-alignment to achieve a high SLSR is schematically illustrated in Figs. 1.4(a)and 1.4(b), reprinted from [28].1.2.3 All-optical signal processingThe high spectral flexibility of IBGs make them ideal candidates for per-forming on-chip all-optical signal processing. By arbitrarily designing the6amplitude and phase responses of an IBG, one can in principle achieve anyall-optical signal processor. Response tailored IBGs have already played akey role in some important all-optical signal processing applications, such asoptical temporal Fourier transformations [37], ultra-fast pulse shaping [38],optical signal modulation [16], and optical delay lines [39].1.2.4 Microwave photonics signal processingFigure 1.5: Schematic illustration of using a complex synthesized IBG formicrowave photonics signal processing. Adapted from [19]. Copyright 2013,Optical Society of America.Microwave photonics (MWP), a field which combines photonics andradio-frequency (RF) engineering, has received a great deal of attention inrecent years [40]. The key idea of this field is to deal with RF signals in thephotonic domain to overcome the existing electronic bottlenecks, as well asto attain advantages that can not be realized in the electrical domain. Specif-ically, various RF signal processing functionalities have been accomplishedin the photonic domain, such as Hilbert transformers [41], integrators [42],and differentiators [43]. The recent advances in integrated optics have alsoenabled MWP signal processing to be performed on integrated platforms[44]. Integrated MWP signal processing has an even brighter future owingto the advantages of compactness, low power consumption and large-scalephotonic integration.The fact that the amplitude and phase responses of IBGs can be freelytailored makes them ideally suited for integrated MWP signal processing.In theory, an IBG can be designed to act as an arbitrary MWP processor.7Such a complex designed IBG can be used in the microwave photonics linkto process RF signals, as illustrated in Fig. 1.5, adapted from [19]. Vari-ous MWP signal processing functionalities have already be demonstrated insilicon IBGs, including Hilbert transformers [27, 41], integrators and differ-entiators [45], and MWP filters [46]. These IBG-based MWP signal proces-sors have shown unprecedented advantages including ultra-high speed andbandwidth, high time-bandwidth product, small sizes, and reconfigurabilityand programmability.1.3 Existing IssuesAlthough significant advance has been made on silicon IBGs in recent years,there are still issues existing in the practical designs and implementations ofthem, which prevent from achieving high-performence sophisticated opticalfilters, MWP and all-optical signal processors. These issues are as below.• There still lacks a comprehensive methodology for designing and imple-menting IBGs to achieve arbitrary, sophisticated, complex (amplitudeand phase) spectral responses.• The most important part of spectral tailoring of Bragg gratings is thegrating apodization, i.e., controlling the grating strength. Therefore,the apodization performance is of the utmost importance for designingIBGs to achieve desired responses. However, reliable and accurateapodization for silicon IBGs still remains challenging compared withthat in fiber Bragg gratings, with two main issues, as below.– Apodization of silicon IBGs usually suffers from a relatively lowprecision, resolution, and dynamic range [12]. This is because themodulation of the grating strength in silicon IBGs is generallyrealized by modifying the physical waveguide grating structure,such as the corrugation width. Due to the small dimension of thesingle-mode SOI strip waveguide, the resolution and precisionof the grating strength control in practice are usually limitedby the fabrication constraints, such as lithography quantizationerrors. Also, the high index contrast of SOI waveguide placesan inherent difficulty to achieve weak grating strengths underfabrication limitations [13], which also constrains the apodizationdynamic range.8– Physical waveguide structure modifications for apodization of sil-icon IBGs could introduce unintended grating phase variations.These phase variations mainly come from (1) feature positionmodulations, which are involved in some particular apodizationtechniques such as lateral misalignment modulations [47], and (2)average waveguide effective index variations due to the waveguidestructure modifications. Such apodization-induced phase varia-tions or errors, as we shall see in Chapter 5, can significantly dis-tort the apodized IBG responses, especially when sophisticatedspectral responses are designed.• The realization of a sophisticated spectral response on a silicon IBGtypically requires a long grating length (usually > 1 mm). Such along grating length will increase the device size, and also make thedevice susceptible to fabrication issues, such as chip nonuniformityand lithography stitching errors.• Spectral modeling of IBGs is an essential part in the IBG design pro-cess. However, there still lacks an efficient and reliable spectral model-ing tool suitable for long, complex-synthesized IBGs. The traditional,fast modeling method for apodized Bragg gratings is the coupled-modetheory-based transfer matrix method (CMT-TMM) [48]. This methodhas been widely employed for characterizing apodized fiber Bragg grat-ings. However, the method is not a structure-aware grating modelingmethod and, thus, is not suitable and accurate for apodized IBGs.This is because, first, the CMT-TMM uses the grating strength andphase of each divided section for the field propagation calculation,which can not be directly extracted from an apodized IBG. More im-portantly, as the CMT-TMM is not structure-aware, the modelingdoes not take into account the unintended waveguide property changesinduced by the structure modifications for the apodization, such asthe aforementioned apodization-induced grating phase errors. Thismakes the method no longer reliable when modeling apodized IBGs.Conventional structure-aware grating modeling methods include 3-Dfinite-difference time-domain (3D-FDTD) and bidirectional eigenmodeexpansion methods. Despite their high accuracy, these simulation ap-proaches are computationally intensive, and when modeling a long,complex synthesized IBG, it is impractical to simulate the whole grat-ing structure to calculate its spectral response.9(a) (b)(c) (d)Figure 1.6: Schematic illustrations of various fabrication issues for siliconIBGs. (a)-(d) are lithography writing jitters, lithography smoothing effect,quantization errros, and minimum realizable feature size/spacing, respec-tively.• Fabrication imperfections can largely degrade the IBG spectral per-formances. There are various fabrication issues, including lithogra-phy quantization errors, lithography writing jitters, minimum realiz-able feature size/spacing, lithography smoothing effect, and fabrica-tion nonuniformity or variations [2]. Some of these fabrication imper-fections are schematically illustrated in Fig. 1.6. Among these issues,fabrication nonuniformity can have a great impact on the actual grat-ing responses, especially for long gratings. Specifically, the thicknessand width of a silicon waveguide can vary from region-to-region withina chip due to fabrication variations [49], which can lead to effective re-fractive index variations along the waveguide grating that can degradethe IBG spectral performance. To show the impact of such fabrica-tion variations on IBG responses, Fig. 1.7 compares simulated spec-tral responses of a Gaussian-apodized, corrugation width-modulated,SOI IBG with and without considering waveguide width and thicknessvariations (denoted by ∆W and ∆H, respectively). The standard de-viations, σ, for ∆W and ∆H are chosen to be 3.855 nm and 1.316nm, respectively, which are statistical results extracted from a 200-mm-wafer fabricated through a 248-nm deep ultraviolet lithographyprocess in [49]. The corrugation width of the Gaussian-apodized grat-101536 1540 1544 1548 1552 1556-60-40-200Reflection (dB)Wavelength (nm)  Ideal  W and H includedFigure 1.7: Simulated spectral responses of a Gaussian-apodized, corruga-tion width-modulated, SOI IBG with and without considering waveguidewidth and thickness variations along the grating length (∆W and ∆H) dueto manufacturing variations.ing increases from 0 nm at both grating ends to a maximum value of10 nm at the grating center. We can see from Fig. 1.7 that the simu-lated spectral side-lobe suppression ratio of the grating when consid-ering ∆W and ∆H is much smaller compared with that of the idealspectrum. This result clearly shows the significant adverse impact offabrication nonuniformity on grating spectral performance.1.4 About This Thesis1.4.1 ObjectiveThe first research objective of this thesis was to develop a comprehensivedesign methodology that allows us to achieve arbitrary complex spectralresponses on silicon IBGs. The next goal was to further improve andstrengthen the developed methodology, by addressing the existing issuesin the IBG design and implementation, and proposing novel techniques tofurther improve the response tailoring of IBGs. In addition, the work alsoaimed to achieve various leading-edge IBG devices based on the developeddesign methodology, including optical filters and signal processors1.4.2 Thesis organizationIn Chapter 2, a comprehensive methodology for designing and implementingIBGs to achieve arbitrary, complex spectral responses is developed. Then,11the implementation details of individual steps of the design process are elab-orated.In Chapter 3, by using the developed IBG design methodology, wedemonstrate the first, multichannel, photonic Hilbert transformers (MPHTs)based on complex synthesized silicon IBGs. MPHTs with up to nine wave-length channels and a single-channel bandwidth of up to ∼625 GHz areexperimentally achieved. The potential of using the devices for multichan-nel single-sideband signal modulation is also explored.In Chapter 4, a high-performance apodization technique for IBGs basedon periodic phase modulation is proposed and demonstrated. This chapteralso presents the general implementation process of the proposed apodiza-tion technique to realize a desired response on an IBG, and then furtherexplores the design considerations of periodic phase-modulated IBGs.In Chapter 5, the impacts of apodization phase errors (APE) on thespectral responses of apodized IBGs are studied, by characterizing varioussilicon IBGs apodized by lateral misalignment and duty-cycle modulationsand designed with different responses. Then, a methodology to compensatethe APE and thus to correct the distorted grating responses is developed.Finally, the developed APE compensation and spectral correction method-ology is experimentally demonstrated.In Chapter 6, a novel apodization profile [κ(z)] amplification techniqueto overcome the apodization limitations of silicon IBGs due to fabricationconstraints is proposed and demonstrated. The technique is first imple-mented in a corrugation width-modulated Gaussian-apodized silicon IBG,and then used in a periodic phase-modulated silicon IBG to achieve a 5-channel dispersionless flattop filter. The experimental results are presented,and the measured spectra of the original (unamplified) and κ(z)-amplifiedIBGs are compared.Finally, in Chapter 7, the conclusion of the work and the outlook forfuture research are given.12Chapter 2Design of Silicon IBGs toAchieve Arbitrary SpectralResponsesThe basic principle of tailoring the reflection response of an IBG resemblesthe design of a finite impulse response (FIR) filter, as illustrated in Fig. 2.1.An IBG can be considered as distributed mirrors which provide distributedoptical feedback, and its reflection response is ultimately determined by thesummation of these feedback. This is similar to a multi-tap FIR filter wherethe filter response is decided by the weighted sum of the multiple samples. Inthe design of an IBG filter, modulating the grating strength (i.e., apodizingthe grating) and phase along the grating length essentially serve to changethe weights and delays of the distributed feedback, respectively, thereby,modifying the grating reflection responses. In principle, any physically-realizable complex response can be achieved on an IBG. In this chapter,we will present a comprehensive design methodology to achieve arbitrarycomplex reflection responses on silicon IBGs. In particular, we will firstIBG (distributed optical feedback)  Finite impulse response filterFigure 2.1: Schematic comparison of an IBG and a finite impulse response(FIR) filter.describe the overall design flow, and then focus on individual steps of thedesign and implementation process. Note that we have assumed that thereis only a single reflection event happening in each period of the grating,which is reasonable for weak IBGs (gratings with the maximum reflectivity13of less than -3 dB). Therefore, the methodology presented below will not besuitable for designing strong IBGs.2.1 Overall Design FlowTarget responseGrating strength (κ) and phase (ΦG) profiles1b) Layer peeling algorithm (LPA).Physical grating structure2b) Grating strength and phase modulation according to κ(z) and ΦG(z), respectively. 33) Structure-aware grating modeling & spectral comparison.4) Fabrication & measurement.5) Comparison with the target response.45Measurement results2a) Determination of fundamental grating structural parameters, including ΔW, ΛG, etc.  2b2a1a) Transfer the target response to be physically realizable.1b1aFigure 2.2: Design flow of a silicon IBG to achieve an arbitrary complexspectral response, where the design of a microwave photonics Hilbert trans-former is used as an example.The basic design flow of a silicon IBG to achieve an arbitrary complexspectral response is illustrated in Fig. 2.2, where the design of a microwavephotonics (MPW) Hilbert transformer is used as an example. A MWPHilbert transformer can be considered as a square filter with a pi-phasejump in the phase response at the central wavelength. The design startsfrom determining the target complex spectral response. The initial targetresponse, however, is usually not physically realizable and must be convertedto be physically realizable before the subsequent grating design. Once thephysically realizable response is obtained, one can use the layer peeling al-gorithm (LPA) to calculate the grating strength and phase profiles [denotedas κ(z) and φG(z), respectively] required by the target response. The LPAis a powerful inverse scattering method and has long been used for Bragggrating synthesis [50]. Note that the required φG(z) can either be discrete,i.e., only having discrete pi phase shifts [see Fig. 3.3(a) as an example], or be14continuous [see Fig. 4.16(b) as an example], which correspond to the caseswhen the designed IBG is non-chirped or chirped, respectively.After the κ(z) and φG(z) profiles required by the target response areacquired, the next step is to create the physical grating structure accordingto κ(z) and φG(z). Before the structure determination, several fundamen-tal grating parameters should be first decided, including corrugation width(∆W ), corrugation shape (rectangular or sinusoidal) and grating period(ΛG). Then, the grating structure can be created based on modulating thegrating strength and phase along the grating length according to κ(z) andφG(z), respectively. There are various methods for controlling the gratingstrength, i.e., various apodization methods, for silicon IBGs, such as ∆W ,lateral misalignment, and duty-cycle modulation methods [15], each of whichhas its own advantages and drawbacks. We will also propose a high perfor-mance apodization method based on periodic phase modulation in Chapter4, which is particularly suitable for implementing complex-synthesized IBGsfor achieving sophisticated spectral responses.Once the whole physical grating structure is created, it is importantto simulate the grating structure to check if the corresponding spectral re-sponse is consistent with the target one. This can be achieved by usinga structure-aware grating modeling method, i.e., modeling approaches thatare performed directly based on the grating physical structures. Variousstructure-aware grating modeling methods can be used, including 3-D finitedomain time difference (3D-FDTD), bidirectional eigenmode expansion, andthe structure synthesis-based transfer matrix method (SS-TMM) that willbe proposed in Section 2.6.2. Compared with the other two methods, theSS-TMM is much more efficient while offering a sufficient accuracy in mostcases and, thus, is more suitable for modeling long, complex-synthesizedIBGs. After the spectral verification, the grating mask layouts are created,which are then sent for fabrication. Finally, when the chip is back, the grat-ings are experimentally characterized, and the results are compared with thetarget response, which completes an entire design cycle of an IBG. This cyclemay need to be repeated until satisfying experimental results are obtained.Below, the individual steps of the design and implementation process ofIBGs will be elaborated in detail.152.2 Conversion of a Target Response to bePhysically RealizableIn practical design, the initial target spectral response is usually not phys-ically realizable. This is because the corresponding impulse response is in-finite in length and non-causal, i.e., with elements in the negative spatialdomain. In this case, one must first convert the target spectral responseto be physically realizable before the IBG design. This can be achieved bytruncating its impulse response, and then shifting the truncated impulse re-sponse to positive spatial domain. The physically realizable version of thetarget response can be obtained by performing an inverse Fourier transformto the new impulse response.-4 -2 0 2 4-100-80-60-40-200  Reflection (dB) -2 0 2-0.060.000.06  -4 -2 0 2 4-40-30-20-100Physically realizable response (a) (c)Reflection (dB)∆λ (nm)0.0 0.4 0.8 1.2 1.6-0.06-0.030.000.030.06(b)Windowed & Spatially-shiftedIntensity (a.u)z (mm)Ideal response-0.50.00.5Phase (pi)-0.50.00.51.01.5Phase (pi)Figure 2.3: (a) Ideal amplitude (blue) and phase (red) spectral responses ofthe designed MWP Hilbert transformer. (b) Truncated and spatially shiftedimpulse response, where the inset shows the original impulse response. (c)Physically realizable spectral response of the designed MWP Hilbert trans-former.Figure 2.3 illustrates the conversion of an ideal target spectral responseto be physically realizable when designing a MWP Hilbert transformer. Anideal MWP Hilbert transformer is essentially a bandpass filter in the am-plitude response with a pi-phase jump in the phase response at the centralwavelength. The initial target response is shown in Fig. 2.3(a). The band-width of the Hilbert transformer is about 6 nm, and the maximum reflectiv-ity is 0.5. Such an ideal target response is not physically realizable becausethe corresponding impulse response is infinite in length and non-causal, asshown in the inset of Fig. 2.3(b). To modify the target response to be phys-ically realizable, we can truncate the impulse response, and then shift thetruncated impulse response to the positive spatial domain. The modifiedimpulse response, as plotted in Fig. 2.3(b), now becomes finite in lengthand causal. Next, we can perform an inverse Fourier transform to the new16impulse response, which will produce the physically realizable version of thetarget response, as shown in Fig. 2.3(c). As can be seen, the realizable am-plitude response is degraded by (1) in-band ripples and side-lobes, and (2)a spectral notch at the central wavelength, which arises due to the pi-phasejump in the phase response. The realizable phase response, however, is veryclose to the ideal one, with no considerable degradation. The obtained phys-ically realizable complex response can be used for the subsequent design ofthe IBG. One can find that the realizable phase response has a differentpi-phase jump direction to that for the initial target response. This phaseinconsistency is due to the inherent phase ambiguities and does not suggestany physical difference.-4 -2 0 2 4-25-20-15-10-50Ideal              PredictedReflection (dB)∆λ (nm)L ≈0.4 mm-1012Phase (pi)-4 -2 0 2 4-25-20-15-10-50 L ≈0.5 mmReflection (dB)∆λ (nm)-1012Phase (pi)-4 -2 0 2 4-25-20-15-10-50 L ≈0.6 mmReflection (dB)∆λ (nm)-1012Phase (pi)-4 -2 0 2 4-25-20-15-10-50 L ≈0.9 mmReflection (dB)∆λ (nm)-1012Phase (pi)-4 -2 0 2 4-25-20-15-10-50 L ≈1.3 mmReflection (dB)∆λ (nm)-1012Phase (pi)-4 -2 0 2 4-25-20-15-10-50 L ≈1.7 mmReflection (dB)∆λ (nm)-1012Phase (pi)Figure 2.4: Spectral performance evolution as a function of the gratinglength for a IBG-based MWP Hilbert transformer. The dashed curves rep-resent the ideal responses, while the solid curves are the simulated gratingresponses where the waveguide thickness and width variations along thegrating length due to fabrication nonuniformity are taken into account.It should be noted that the truncation window length of the impulseresponse will directly determine the length of the grating. A longer windowlength can lead to a smaller spectral deviation of the produced realizableresponse from the ideal one. However, this can also increase the gratinglength and make the device more susceptible to manufacturing variabilityissues. Thus, the window length or the grating length needs to be chosencarefully to make a good compromise between these two aspects. Figure 2.417illustrates the spectral performance evolution as a function of the gratinglength for a MWP Hilbert transformer based on a corrugation-width modu-lated IBG. The dashed curves represent the ideal responses, while the solidcurves show the simulated grating responses where the waveguide thicknessand width variations along the grating length due to fabrication nonunifor-mity are taken into account. The standard deviations for the thickness andwidth variations are chosen to be 3.855 nm and 1.316 nm, respectively, whichare statistical results taken from [49]. It can be seen in Fig. 2.4 that whenthe grating length is increased from 0.4 to 0.9 mm, the grating complex re-sponse, especially the phase response, becomes closer to the ideal. However,when the grating length is further increased from 0.9 to 1.7 mm, the gratingresponse becomes worse with stronger ripples and side-lobes. This spectraldegradation with increasing the grating length is because, on the one hand,the window length is already long enough to include the majority of theinformation of the initial target response. On the other hand, as the gratingis longer, it becomes more subject to fabrication nonuniformity. Therefore,the spectral degradation due to the fabrication nonuniformity now outweighsthe spectral improvement due to a longer grating length. These results showthat the designers need to carefully choose the grating length to optimizethe practical grating performance.2.3 Calculation of Required Grating Profilesfrom a Target ResponseOnce the realizable target spectral response is obtained, one needs to calcu-late the required grating strength and phase profiles, κ(z) and φG(z), fromthe target response. This problem is essentially equal to finding the gratingproperties (grating strength and phase distributions) required to achieve aspecified, complex spectral response. This can be accomplished by usingthe layer peeling algorithm (LPA) [50]. The LPA is an inverse scatteringmethod, which has long been used as a powerful tool for synthesizing Bragggratings. In the LPA, the grating is divided into many thin layers, each as-sumed to have a uniform profile. Then, the structure of the layered mediumis determined layer by layer, by using the causality principle. Below, wewill briefly review the principle of the LPA. Note that the LPA may beperformed based on either continuous or discrete model, but we will usediscrete model-based LPA in this thesis, as it is faster and more stable thanthe continuous one [50].The LPA will start from using the well-known transfer matrix of a Bragg18Figure 2.5: Discrete model of a grating used in the LPA. the forward-propagating and backward-propagating fields are denoted by uj and vj ,respectively. Reprinted from [50]. Copyright 2001, IEEE.grating which connects the field at z + ∆ with the field at z, which can beexpressed as [50][u(z + ∆, δ)v(z + ∆, δ)]=[cosh(γ∆) + i δγ sinh(γ∆)qγ sinh(γ∆)q∗γ sinh(γ∆) cosh(γ∆)− i δγ sinh(γ∆)][u(z, δ)v(z, δ)](2.1)where u(z, δ) and v(z, δ) represent the amplitudes of the forward and back-ward propagating fields, respectively; γ2 = |q|2−δ2, q = q(z) is the couplingcoefficient of the grating; and δ is the wavenumber detuning compared withthe Bragg wavenumber, which can be expressed asδ = 2ping(1λ− 1λB) (2.2)where ng is the group index of the waveguide, and λB is the Bragg wave-length of the grating.We can make further approximation by replacing the matrix in Eq. 2.1with the product of two transfer matrices, with one representing a discretereflector (Tρ) and the other describing the pure propagation of the fields(T∆), which are given byTρ = (1− |ρ|2)(−1/2)[1 −ρ∗−ρ 1](2.3)19andT∆ =[exp(iδ∆) 00 exp(−iδ∆)](2.4)where ρ represents the discrete, complex reflection coefficient, which isρ = − tanh(|q|∆) q∗|q| (2.5)The entire grating thus can be modeled as a series of discrete, com-plex reflectors spaced by ∆, as illustrated in Fig. 2.5. The task now is toreconstruct the complex reflector amplitudes (ρj , j = 1, 2, ...N) from a tar-get, realizable complex reflection response, r1(δ). The field before the firstsection can be expressed as [u1(δ)v1(δ)]=[1r1(δ)](2.6)Using the causality we can find the complex amplitude of the first re-flector. Due to that the light does not have enough time to propagate toand from the reflectors ρj with j > 1, the impulse response of the reflectorstack for the time t = 0 is independent of the reflectors ρj with j > 1.Thus, when looking at the impulse response of the stack for t = 0, we cansolve the response as if only the first reflector was present. ρ1 therefore canbe computed from the inverse Fourier transform of r1(δ) = v1(δ)/u1(δ) att = 0. Once ρ1 is obtained, we can use T∆Tρ1 to transfer the fields to thenext section. This way, the first layer is essentially “peeled off”, and wecan repeat this procedure until the complex reflection coefficients of all thereflectors are determined.From Eqs. 2.3 and 2.4 the task of transferring the fields by T∆Tρ1 canbe expressed in terms of the local reflectivities asr2(δ) = exp(−i2δ∆) r1(δ)− ρ11− ρ∗1r1(δ)(2.7)where rj(δ) = vj(δ)/uj(δ).To show how ρ1 can be calculated from the inverse Fourier transformof r1(δ), we note that r1(δ) can be expressed as a discrete-time Fouriertransform of the impulse response h1(τ)r1(δ) =∞∑τ=0h1(τ) exp(iδτ2∆) (2.8)20because the impulse response is discrete with the sample period 2∆, whichis equal to the “round-trip” propagation length of one layer. τ = t/(2∆) hasbeen defined as the discrete time variable with t as the normalized time. Asthe impulse response at τ = 0 can be calculated as if only the first reflectorexisted, ρ1 is simply the 0th order Fourier coefficient of Eq. 2.8, and thuscan be calculated fromρ1 = h1(0) =1MM∑m=1r1(m) (2.9)where r1(m) is a discrete version of the target spectrum [r1(δ)] in the range|δ| ≤ pi2∆ , andM is the number of the detuning points (or wavelength points).Now, we can summarize how to use the LPA algorithm as below [50]:1) convert the target ideal spectral response to be physically realizable;2) compute ρ1 from Eq. 2.9;3) propagating the fields using Eq. 2.7;4) repeat step 2) until the complex reflection coefficients of the entiregrating structure (ρj , j = 1, 2, ...N) is determined.5) calculate the complex coupling coefficients (qj) of the entire gratingstructure according to the obtained ρj through Eq. 2.5. The amplitudeand phase elements of qj will represent the grating strength and phase,respectively.Figure 2.6 shows the use of the LPA to calculate κ(z) and φG(z) profilesrequired to achieve the MWP Hilbert transformer designed in Fig. 2.3. Thetarget physically realizable response is shown in Fig. 2.6(a). The detuningwindow (δW ) is set to ∼547 mm−1, which translates to a wavelength windowof ∼50 nm at wavelengths around 1550 nm. The layer thickness (∆) is alsodependent on δW through∆ =piδW(2.10)Thus, the corresponding ∆ is ∼5.7 µm. The total number of the detuningpoints (or wavelength points) is 1000.The κ(z) and φG(z) profiles calculated by using the LPA are plotted inFig. 2.6(b). As can be seen, the κ(z) profile is complicated, which indicatesthat a high-performance and reliable IBG apodization method is neededto implement the grating. Also, from φG(z), multiple pi-phase shifts arerequired to be applied along the grating.21-0.50.00.51.01.501-200 -100 0 100 200-50-40-30-20-100(a)Reflection (dB)δ (mm-1)Phase (pi)0.0 0.2 0.4 0.6 0.802468101214(b)κ (mm-1 )z (mm) φ G (pi)-20 -10 0 10 20∆λ (nm)Figure 2.6: (a) Complex response of the designed MWP Hilbert transformer.(b) Grating strength and phase profiles, κ(z) and φG(z), required by thedesign, calculated by using the LPA.2.4 Fundamental Grating Structural ParameterDeterminationOnce the grating strength and phase profiles required by the target responseare obtained, the next step is to map the grating profiles into a physical grat-ing structure. Before this grating structure mapping process, several funda-mental grating structural parameters need to be first decided, including thecorrugation shape, grating period, and corrugation width.2.4.1 Corrugation shapeThe corrugation shape of a silicon IBG can be set to be either rectangularor sinusoidal, as illustrated in Fig. 2.7(a). Although rectangular corruga-tion shapes are more commonly used in SOI-based IBGs, our recent studiessuggested that sinusoidal IBGs exhibit some advantages compared with rect-angular IBGs. The first advantage of sinusoidal IBGs is that they allow fora higher grating strength than rectangular ones, which also suggests a largerdynamic range when controlling κ. Figure 2.7(b) plots the experimentallyextracted grating strength as a function of the grating corrugation width(∆W ) for rectangular and sinusoidal IBGs. These IBGs were developed onSOI waveguides with cross sections of 500 nm (wide) × 220 nm (high), andwere fabricated through e-beam lithography. The grating strengths were ex-tracted from the bandwidths of the measured spectra of the IBGs. It can beseen that for the rectangular IBG, when ∆W is larger than 90 nm, the grat-ing strength begins to saturates and even starts to decrease as ∆W further22(a) (b) (c)30 60 90 120 150020406080100120140-10 -5 0 5 10-50-40-30-20-100Grating strength (mm-1)Corrugation width nm) Rectangular SinusoidalReflection (dB) (nm) Rectangular SinusoidalFigure 2.7: (a) Schematic representations of rectangular and sinusoidalIBGs. (b) Experimentally extracted grating strength as a function of gratingcorrugation width for rectangular and sinusoidal IBGs. (c) Spectral compar-ison of lateral misalignment modulated, Gaussian-apodized rectangular andsinusoidal IBGs, calculated by using the structure synthesis-based transfermatrix method (SS-TMM) under the same sampling interval of 6 nm.increases. This could be due to that the large ∆W makes the fundamentalwaveguide mode no longer in the perturbative regime. This saturation ef-fect, however, happens in a much larger value of ∆W (> 130 nm) for thesinusoidal IBG. This indicates that higher grating strengths can be realizedin sinusoidal IBGs than those in rectangular ones.Another advantage for sinusoidal IBGs is that they are less subject toquantization errors due to limited lithography grid size compared with rect-angular IBGs. This advantage is illustrated in Fig. 2.7(c), which comparesthe reflection spectra of lateral misalignment modulated, Gaussian-apodizedIBGs with rectangular and sinusoidal corrugation shapes, simulated by us-ing the structure synthesis-based transfer matrix method (SS-TMM) [whichwill be presented in Section 2.6.2] using the same sampling interval of 6 nm.Here, the sampling interval can be considered as the lithography grid sizeused in the fabrication. One can find that the spectrum of the sinusoidalIBG has weaker ripples compared with that of the rectangular one, due tothe fact that sinusoidal grating profiles are less subject to the discretizationissue than rectangular grating profiles. Note that a sinusoidal grating shapewill also lead to a smaller grating strength and thus a lower grating reflec-tivity than a rectangular grating shape when using the same ∆W , as alsoindicated in Fig. 2.7(c).231530 1540 1550 1560 1570-14-12-10-8-6-4-20Reflection (dB)Wavelength (nm) ΛG = 312 nm ΛG = 314 nm ΛG = 316 nm ΛG = 318 nm ΛG = 320 nmFigure 2.8: Reflection spectra of silicon IBGs with different grating periods(ΛG).2.4.2 Grating periodThe grating period will determine the Bragg wavelength, or wavelengthcenter of the IBG spectrum. For IBGs developed on single-mode strip SOIwaveguides with cross sections of 500 nm (wide) × 220 nm (high), to obtaina Bragg wavelength of around 1550 nm, we need the grating period to beabout 316 nm. An evolution of the reflection spectrum against the gratingperiod for a silicon IBG is presented in Fig. 2.8.2.4.3 Corrugation widthΔWΔWFigure 2.9: Illustration of choosing ∆W according to the maximum value ofthe grating strength profile.For many apodization methods (e.g., phase modulation [12], duty-cyclemodulation, and lateral misalignment modulation [51] methods), the corru-gation width, ∆W , is constant along the grating. In these cases, ∆W willdetermine the maximum value of the grating strength distribution along the24apodized grating. Thus, ∆W should be selected according to the maximumvalue of the κ(z) profile required by the target response, as illustrated inFig. 2.9. This requires one to know the accurate relationship between ∆Wand κ for the specific waveguide geometry where the gratings are based on.This relationship can be extracted from the experimental results, i.e., devel-oping an empirical model regarding κ versus ∆W by fabricating and testinga series of IBGs with different ∆W [47]. Despite the high reliability and ac-curacy of this method, performing these additional fabrication and tests todevelop the empirical model can be time-consuming and expensive. Further-more, the empirical model needs to be rebuilt when a different waveguidegeometry is used to implement IBGs.The relationship between κ and ∆W for an IBG developed on a spe-cific waveguide geometry can also be calculated by using the 3-D FDTDsimulations. However, the FDTD simulation is computationally intensive,and if the grating is very long, it can be impractical to run the simulationover the entire grating structure to calculate its response. Wang et al. in[47] proposed a way to obtain the coupling coefficient by calculating theband gap of IBG band structures using the 3-D FDTD. Because this calcu-lation is implemented based on only a unit cell of the grating using Blochboundary conditions, it is much more efficient than full FDTD simulations.Photonic band structures are similar to electronic band structures in solid-state physics, which describe the range of energies that an electron withinthe solid may have (called energy bands) and ranges of energy that it maynot have (called band gaps). Similarly, for the photonic band structures,they indicate different frequency ranges of the photonic material in whichthe light can transmit or be reflected. The photonic band structure canbe calculated by determining the optical frequencies f , as a function of theBloch wave vector k for all the Bloch modes in a given frequency range.The left of Fig. 2.10(a) shows a typical band structure of a uniform op-tical guiding medium. It is seen that the two Bloch modes are crossing witheach other without any gap, which means that light at all the frequenciescan transmit through this uniform medium. The right plot of Fig. 2.10(a) isa typical band structure for 1-D photonic crystals, where a gap between thetwo mode lines can be seen. This gap corresponds to the frequency rangein which the light is reflected and can not transmit through the medium.Bragg gratings can be considered as 1-D photonic crystals, and the bandgap in such cases can be interpreted as the reflection band, as illustratedin Fig. 2.10(b). The location and width of the gap will indicate the centerwavelength and bandwidth (∆λ) of the Bragg grating, respectively. Once∆λ is obtained from the band gap, κ can be calculated through the following25Figure 2.10: (a) Comparison of typical band structures for uniform mediaand 1-D photonic crystals. (b) Schematic illustration of how the band struc-ture is related to the reflection band of a Bragg grating. (c) FDTD bandstructure analysis in Lumerical FDTD Solutions. (d) Band structure dia-gram of a waveguide Bragg grating with ∆W of 50 nm. (e) Fourier transform(magnitude) of the time domain signals at the band edge from the FDTDsimulation for Bragg gratings with different ∆W ; the frequency spacing be-tween the two peaks corresponds to the band gap width. Reprinted withpermission from Springer Nature [15].equation [47]κ = ping∆λ/λ20 (2.11)where ng is the group index of the waveguide.Some commercial optical analysis software, such as Lumerical, can pro-vide a powerful tool for the band structure analysis. Figure 2.10(c) showsthe band structure analysis for a grating unit cell taken from LumericalFDTD Solutions. By sweeping the Bloch vector k while Fourier transferringthe time-domain signals collected by the monitors [yellow X marks in Fig.2.10(c)] put in the FDTD simulation region, the band structure diagram canbe constructed, as plotted in Fig. 2.10(d). A gap at the band edge can beseen in the diagram. Figure 2.10(e) plots the Fourier transfer results of the26time domain signals at the band edge for gratings with different ∆W . Thefrequency spacing between the two peaks corresponds to the band gap. Onecan find that the width of the gap increases with the ∆W , which meansthat a grating with a bigger ∆W has a larger ∆λ and thus a higher κ.The band structure method for κ determination is accurate, can be usedfor various grating structures, and is suitable for both strongly and weaklycoupled Bragg gratings. Also, it is very efficient, as the analysis is based onanalyzing a single grating cell unit. More theories regarding photonic bandstructures and the relevant calculation methods can be found in [52, 53].10 20 30 40 50 60 705×104  1052×104κ (m-1)Corrugation width (nm) Bandstructure simulation Experiment104Figure 2.11: Comparison of the relationships between κ and ∆W ob-tained from the band structure simulations and from the experimental re-sults. Adapted with permission from Springer Nature [15].To investigate the accuracy of the band structure method, the calculatedgrating strengths are compared with the experimental results, which wereextracted from the measured spectra of IBGs fabricated by e-beam lithog-raphy. The comparison results are shown in Fig. 2.11. We can find thatthe results obtained from the FDTD band-structure calculations agree wellwith the experimental values, demonstrating the high reliability of the bandstructure approach for assessing κ of waveguide Bragg gratings.2.5 Grating ApodizationOnce the fundamental grating structural parameters have been decided, onethen needs to modulate the grating strength and phase according to κ(z)and φG(z), thereby, creating the physical grating structure. Modulating thegrating strength, i.e., apodizing the grating according to κ(z), is the most im-portant part in the spectral tailoring of IBGs. The apodization performance27is critically important and will directly determine the spectral performanceof the designed IBG. In this section, we review various apodization methodsand discuss their advantages and drawbacks.2.5.1 Corrugation width modulation0 2 4 6 8 10 12 14 16 180.20.40.60.81.0Norm. Period No.ΔWFigure 2.12: Schematic representation of a Gaussian-apodized IBG via mod-ulating ∆W .The most straightforward method to control grating strength is to simplychange the corrugation width, ∆W , of the grating. Figure 2.12 schematicallyillustrates a Gaussian-apodized IBG via modulating ∆W . The advantagesof the ∆W modulation method are the simplicity and easy implementationin practice. However, due to the strong coupling nature of silicon IBGsdeveloped on high-index contrast SOI waveguides and the small dimensionsof typical single-mode strip SOI waveguides, even a small ∆W variation ofseveral nanometers can significantly change the grating strength. As such,this apodization scheme is typically subject to low dynamic range, resolutionand precision due to fabrication constraints, such as limited lithographyresolution. Thus, the method has been limited to achieving apodized IBGswith simple apodization profiles, such as Gaussian profiles [33, 54], and is notsuitable for implementing complex synthesized IBGs in which the involvedapodization profiles are typically complicated.2.5.2 Duty-cycle modulationThe grating strength can be modulated by changing the grating duty-cycle,as schematically illustrated in Fig. 2.13(a). Here, the duty-cycle, DC, isdefined as the ratio of the corrugation length d to the grating period ΛG,i.e., DC = d/ΛG. In this modulation scheme, κ is related to the duty-cyclethrough the following relationship [55]κ ∝ sin(pi ×DC) (2.12)28For this method, the lower and upper limit of the practically achievableduty-cycle is determined by the minimum realizable feature size and spacing,respectively [55]. Compared with the conventional ∆W modulation method,the duty-cycle modulation method possesses an improved apodization res-olution, accuracy, and dynamic range. The drawback of this approach isthat varying the duty-cycle can also change the local effective refractive in-dex of the waveguide, which can introduce grating phase errors that canbroaden or distort the spectral response. These index variations, however,can be compensated for by modifying the grating structure, which will bedescribed later in Chapter 5.ΔLdʌG Duty cycle = d / ʌG (a) Duty cycle modulation Lateral misalignment modulation(b))sin( DC  )cos(GL 0.0 0.1 0.2 0.3 0.4 0.50.00.20.40.60.81.0  Normalized Duty cycle0.0 0.1 0.2 0.3 0.4 0.50.00.20.40.60.81.0NormalizedL / G Figure 2.13: (a) Schematic of grating duty-cycle modulation (left) and therelationship curve between duty-cycle and κ (right). (b) Schematic of side-wall misalignment (∆L) modulation (left) and the relationship curve be-tween ∆L/ΛG and κ (right). Reprinted with permission from Springer Na-ture [15].2.5.3 Lateral misalignment modulationThe grating strength can be controlled by intentionally maligning the cor-rugations on the two sidewalls [12, 47], as illustrated in Fig. 2.13(b). Tounderstand the principle of this apodization method, a laterally-misalignedIBG can be treated as a superposition of two identical sub-gratings formedon either side of the waveguide with one of the sub-gratings phase-delayed by∆φ. The effective refractive index modulation of the overall Bragg gratingthen can be expressed as∆n(z) ∝ Re{exp j2piΛGz + exp j(2piΛGz + ∆φ)}(2.13)29After simplification, Eq. 2.13 reduces to∆n(z) ∝ cos(∆φ2)Re{exp j(2piΛGz +∆φ2)}(2.14)We can see in this equation that the index modulation amplitude of theoverall grating, i.e., the overall grating strength, is now scaled by a factor ofcos(∆φ2 ). Therefore, the grating strength can be controlled by modulating∆φ, which is practically realized by modulating the lateral misalignmentbetween the two sub-gratings, ∆L. Using the relationship ∆L = ΛG∆φ/(2pi)[28], the grating strength as a function of ∆L can be written byκ ∝ cos(pi∆LΛG) (2.15)One can also find in Eq. 2.14 that an additional phase variation ∆φ2 isintroduced in the grating due to the lateral phase shift. This phase termwill vary along the grating length for an IBG with a spatially varying lateralmisalignment, which will generate unintended grating phase variations orerrors that can distort the grating response, as will be seen in Chapter 5.In theory, the unwanted phase term ∆φ2 in Eq. 2.14 can be eliminated ifthe apodization is performed by phase-delaying both two sub-gratings in acomplementary manner, i.e., one is phase-delayed by ∆φ2 while the other isphase-delayed by −∆φ2 . This will modify Eq. 2.13 and will ultimately causethe phase term ∆φ2 in Eq. 2.14 to be canceled. This can be practicallyachieved by position modulations of both sides of the grating features in acomplementary manner. This scheme, however, may be more difficult tobe practically implemented in complicated grating devices, such as those in[28, 56] and the κ(z)-amplified IBGs that will be shown in Chapter 6.2.5.4 Cladding modulationTan et al. in [57] proposed novel cladding-modulated IBGs. Such cladding-modulated Bragg gratings are formed by placing periodic silicon cylindersin the cladding at a fixed distance (denoted as d) from a single-mode siliconwaveguide, as illustrated in Fig. 2.14. The grating strength of such grat-ing structures can be controlled by changing the diameters of the cylinders(denoted as W ), or d. Due to the smaller overlap between the light modeand the index perturbation structures compared with conventional sidewallBragg gratings, cladding-modulated gratings can easily achieve weak κ andthus narrow-band spectra. Also, as the grating strength can be varied by30adjusting the distance of the cylinders to the waveguide, the control of κ canbe achieved with an improved resolution. Grating strengths differing by 1order of magnitude for cladding-modulated gratings were demonstrated in[57]. Also, a reflection bandwidth as small as 0.6 nm and a high extinctionratio of 21.4 dB in the transmission were demonstrated in [58] by using thecladding modulation method. Nevertheless, this method has not been usedfor the modulation of complicated apodization profiles to realize sophisti-cated spectral responses. Also, the small dimensions of the silicon cylindersmay not be fabricated repeatably and precisely in practice, which bringschallenges to accurately control the amplitude and phase spectral responseof the grating. Silicon cylinder arrayWaveguideΛG: period WdFigure 2.14: Schematic representation of a cladding-modulated Bragg grat-ing; ΛG: period of the silicon cylinders; W : diameter of the silicon cylinder;d: distance of the silicon cylinders from the edge of the waveguide. Reprintedwith permission from Springer Nature [15].2.5.5 Sinusoidal phase modulationSimard et al. in [26] proposed a promising sinusoidal phase modulationapodization method for silicon IBGs. The basic idea of the apodizationmethod is to add a sinusoidally varying phaseφκ(z) = A(z) sin(2pizΛφ) (2.16)with a z-dependent amplitude A(z) and a period of Λφ, into the effectiveindex change along the grating to control the coupling coefficient. Thenormalized grating strength (to 1) , κn(z), will finally be related to thephase amplitude A(z) viaA(z) = J−10 (κn(z)) (2.17)31where J0 is the 0th-order Bessel function. Thus, applying a specific κ(z)profile on a Bragg grating is achieved by phase-modulating the grating withφκ(z) that can be calculated from Eqs. 2.16 and 2.17.This sinusoidal phase modulation-based apodization method shows greatpromise for implementing complex-synthesized IBGs to achieve sophisti-cated spectral responses [26, 27]. This is because the method has highapodization precision, resolution, and dynamic range. Furthermore, thegrating phase modulation introduces little additional grating phase errors,thus allowing the amplitude and phase responses of an IBG to be preciselycontrolled. Finally, the implementation of this phase apodization method ismuch simpler than other methods (such as DC and ∆L modulations) whenimplementing sophisticated spectral responses. This is because a compli-cated grating phase profile, which is usually required by a sophisticatedtarget response, can be directly incorporated into the final grating phaseprofile to determine the grating structure. This eliminates the need for anyadditional complicated grating phase modulation process.In Chapter 4, we will extend this promising sinusoidal phase modula-tion apodization technique to arbitrary periodic phase functions. We willdemonstrate, both theoretically and experimentally, that apodization of anIBG can be achieved by modulating the grating phase using any periodicfunction. This will provide an additional degree of freedom for the designand optimization of phase-modulated IBGs. In addition, we will propose ageneral implementation process of the periodic phase modulation apodiza-tion technique to realize a desired response on a silicon IBG. Also, we willstudy the limiting factors of the apodization performance, design trade-offsand optimization, and grating robustness against fabrication constraints fordifferent periodic phase functions.2.6 Grating Modeling2.6.1 Various grating modeling methodsOnce the grating structure is created, the created IBG structure needs tobe simulated to check if the corresponding spectral response is consistentwith the design. This can be realized by using a structure-aware model-ing method, i.e., a modeling method that is performed directly based onthe physical IBG structure. The structure-aware property of the modelingmethod is essential for modeling apodized silicon IBGs where the apodiza-tion is typically achieved via grating structure modifications. This ensuresthat the modeling results can take into account the structure modification-32induced unintended grating property changes, such as average waveguideeffective refractive index variations and grating phase variations due to cor-rugation shifts. Various structure-aware modeling methods are available,including 3-D FDTD and bidirectional eigenmode expansion. They canusually be accessed from commercial software, such as Lumerical. However,these methods are computationally intensive, and it is impractical to usethem to simulate the whole structure of a long, complex-synthesized IBG toacquire its spectral responses.Another type of modeling method for arbitrarily-apodized Bragg grat-ings is based on transfer matrix method (TMM). TMM-based modelingmethods, compared with those based on full Maxwell solver simulations, aremuch more efficient. The traditional TMM for modeling Bragg gratings isbased on the coupled-mode theory (CMT), in which the standard transfermatrix solution is derived from the CMT equations [48]. Such a CMT-basedTMM has been widely used for modeling apodized fiber Bragg gratings. Thismethod, however, is not a structure-aware modeling method, since in thefield propagation calculation, each grating section is identified by the gratingstrength and phase while the physical structure of the grating is ignored.This makes the CMT-TMM no longer accurate when modeling apodized sil-icon IBGs as it cannot take into account the structure modification-inducedgrating property changes.2.6.2 Structure synthesis-based transfer matrix methodIn this section, we propose a powerful modeling tool for apodized siliconIBGs which is performed by directly synthesizing the physical structure ofthe grating with the assistance of the TMM. Such a structure synthesis-basedTMM (SS-TMM), which is a structure-aware modeling method, is more ac-curate than conventional CMT-based modeling methods for apodized siliconIBGs. Furthermore, as the method is based on TMM, it is much more effi-cient than 3-D FDTD and bidirectional eigenmode expansion methods.Principle and implementationWe illustrate the implementation of the SS-TMM by modeling a siliconIBG apodized by modulating the lateral misalignment (∆L). The IBG isdeveloped on a single-mode SOI strip waveguide with a cross section of 500nm (wide) × 220 nm (high), and is designed for the fundamental quasi-TEmode. The grating period ΛG is 316 nm, corresponding to a Bragg resonanceof around 1550 nm. A Gaussian apodization profile [blue in the upper plot330 200 400 600 8000.00.20.40.60.81.0n (a.u)Period No.0.00.10.20.30.40.5L/nk nk+1κ(z) & ΦG (z)IBG structureCoupling coefficient and phase profiles ΔWs (k)Spectral response(a)Δneff (k)Width variations along the segmentsneff variations along the segmentsTMM Waveguide sampling ApodizationW to neff mapping(b) (c)(i) (ii)(d)nkB1 B2A2A1A1 A2B1 B2132Tk(k+1) TkΔW123ΔLk k+1 kWΔLΔL = 0G1 2 3 4 5 6 7 8 9 10 11 12WsSegment No. kW0W1 2 3 4 5 6 7 8 9 10 11 12WsSegment No. kW0WFigure 2.15: (a) Schematic flow showing the process of the SS-TMM mod-eling. (b) The upper figure plots the normalized Gaussian apodization pro-file (blue, left axis) and the translated lateral misalignment-to-period ratio∆L/ΛG (red, right axis) along the grating; the bottom diagrams illustratethe grating structures at the different positions (1-3), whose locations areindicated in the upper plot. (c) Schematic illustrations of spatial samplingof the cell structure of a lateral misalignment-modulated IBG in differentcases of (i) rectangular and (ii) sinusoidal grating shapes, where the bottomfigures plot the corresponding ∆Ws(k) profiles; ∆Ws(k) is defined as thewidth variation of the kth segment from the unperturbed waveguide width[denoted as W in (c)]. (d) Illustration of the transfer matrices describingthe wave propagation through an interface (left) and through a uniform sec-tion (right); the upper diagrams show the original grating structures whilethe lower ones illustrate the equivalent multiplayer structures used in theSS-TMM modeling.34of Fig. 2.15(b)] is applied to the grating to minimize the spectral side-lobes. The total period number is 800, which translates to a grating lengthof ∼0.25 mm. The bottom schematic diagrams of Fig. 2.15(b) illustratethe ∆L-modulated grating structures at the different positions (1-3), whoselocations are indicated in the upper plot of Fig. 2.15(b). In the gratingimplementation, ∆L is controlled by modulating the positions of one side ofthe grating features with the other side of the grating remaining unchanged.As discussed in Section 2.5.3, this implementation scheme will introduceadditional grating phase variations or errors that can distort the gratingspectrum from the ideal spectrum, as we will see in the experimental results.The existence of these phase errors, however, will also allow us to validate thecapability of the SS-TMM to take into account such structure modification-induced grating property changes, by comparing the experimental and SS-TMM-predicted grating spectra.Figure 2.15(a) is a schematic flow showing the process of SS-TMM model-ing. The first step is to determine the whole IBG physical structure from thegrating strength and phase profiles, which are κ(z) and φG(z) respectively.Note that there is no phase modulation required by the current Gaussian-apodized IBG, i.e., φG(z) = 0. Phase modulations of the gratings are usuallyrequired when sophisticated spectral responses are designed. Several funda-mental grating physical parameters are needed to be decided first, includingthe corrugation width, the grating period (which has been chosen to be 316nm), and the corrugation shape (rectangular or sinusoidal). Then, the wholegrating structure can be mapped from the normalized apodization profile,κn(z), based on the rule of the chosen apodization scheme. Here, κn(z)is translated to a lateral misalignment as a function of the grating periodnumber, ∆L(i), through the following relationship [47]∆L(i) =ΛG · cos−1 (κn (i))pi(2.18)The calculated ∆L(i)ΛG profile is plotted as the red curve in the upper plot ofFig. 2.15(b).Once the entire grating structure is determined, the grating structureis spatially sampled into many short uniform segments, as schematicallyillustrated in Fig. 2.15(c). Then, the width variation of each grating seg-ment from the unperturbed waveguide width [W in Fig. 2.15(c)] can beobtained, which is defined as ∆Ws(k), where k is the index number of thegrating segment. Figures 2.15(c)-i and 2.15(c)-ii schematically illustrate thediscretized cell structures of ∆L-modulated IBGs with rectangular and si-nusoidal corrugation shapes, respectively, while the bottom plots show the35corresponding ∆Ws(k) profiles. The sampling interval can be regarded asthe fabrication resolution (or grid size of the lithography) of the IBG. Sincethe effective refractive index of the waveguide is related to its width, the∆Ws(k) profile can be translated to the effective refractive index changesalong the grating segments, ∆neff (k). To obtain the relationship between∆W and ∆neff that can be used for this translation, eigenmode analysis isfirst used to calculate the real ∆neff -versus-∆W relationship of the waveg-uide, and the scale of the obtained curve is then reduced by a factor of ∼2.3.This ∆neff -versus-∆W curve with a reduced scale is finally used in our SS-TMM for translating the ∆Ws(k) into ∆neff (k) profile, and will be shownin Fig. 5.1(a). The reason for scaling down the curve is that the effectiverefractive index variations due to different waveguide widths here are nottotally equivalent to the refractive index changes along an optical multilayerstructure, and the actual mode coupling strength here is smaller than thatin a real multi-layer structure [15]. The scaling factor of ∼2.3 used herehas been estimated based on our previous experimental results of SOI-basedIBGs fabricated in e-beam lithography. It is worth noting that the rela-tionship between ∆W and ∆neff can also be obtained from experimentalresults [47].Once the ∆neff (k) profile is obtained, the discretized IBG can be treatedas an optical multilayer structure which consists of a stack of layers withdifferent refractive indices. The spectral characterization of the IBG thencan be regarded as the general problem of the wave propagation in a multi-layer structure [59]. Thus, the TMM used in optical multilayer structuresnow can be borrowed for characterizing the current discretized IBG, whichwill be elaborated on in the following.The transfer matrix that relates the complex amplitudes of forward andbackward propagating waves (defined as A and B respectively) is defined as[A1B1]=[T11 T12T21 T22][A2B2](2.19)Two basic individual transfer matrices can be built: one is for wave prop-agation through an interface between neighboring grating segments [Tk(k+1)in left of Fig. 2.15(d)], and another one is for wave propagation througha uniform grating segment [Tk in right of Fig. 2.15(d)]. These two basictransfer matrices can be derived using Fresnel equations to be [59]Tk(k+1) = nk+nk+12√nknk+1 nk−nk+12√nknk+1nk−nk+12√nknk+1nk+nk+12√nknk+1 (2.20)36andTk =[eβd 00 e−βd](2.21)where nk is the effective refractive index of the kth segment, which can befound from the ∆neff (k) profile; β =2pinkλ is the propagation constant forthe field; and d is the length of each grating segment, which is equal to thesampling interval.The total transfer matrix for the wave propagation through the wholediscretized IBG can be obtained by the multiplication of all individual trans-fer matricesTtot =[Ttot−11 Ttot−12Ttot−21 Ttot−22]= T1T12T2...T(N−1)T(N−1)N (2.22)This complete transfer matrix relates the incident and reflected waves atthe input port with the incident and reflected waves at the output port forthe grating. Therefore, it can be used to extract the grating reflection andtransmission coefficients, defined as r and t, respectively, which will be [59]r =Ttot−21Ttot−11(2.23a)andt = (Ttot−11)−1 (2.23b)By repeating the above TMM calculation for each wavelength, the reflectionand transmission spectrum of the apodized IBG can be finally obtained. Itis important to note that the dependence of the effective refractive index onwavelength, i.e., the waveguide dispersion, must be considered when repeat-ing the calculation at different wavelengths.Now, we can highlight the difference between CMT-TMM and SS-TMM.Figure 2.16 compares the implementation processes of the two modelingmethods. Both of them start from using the target grating profiles, κ(z)and φG(z). For CMT-TMM, it will first discretize the grating profiles forthe subsequent TMM calculation. Then, the transfer matrix calculation isperformed directly based on the discretized grating profiles, where the basictransfer matrix of the mth profile segment is based on the general transfersolution of the coupled-mode equations [60]:Mm =[cosh(smlm) + i∆β2smsinh(smlm) iκmsmsinh(smlm)−iκ∗msm sinh(smlm) cosh(smlm)− i∆β2smsinh(smlm)](2.24)37Spatial samplingκ(z) & ΦG(z) IBG structureProfile discretization & TMMSS-TMMCMT-based transfer matrix solution [Eq. (2.24)] Spectral resultsApodizationDiscretized IBG structureTMMFresnel equation-based transfer matrix solution [Eqs. (2.20) and (2.21)] κ(z) & ΦG(z)Spectral resultsCMT-TMMFigure 2.16: Comparison of the implementation process in conventionalCMT-TMM and SS-TMM.where κm is the coupling constant for the mth segment, lm is the lengthof the mth segment, ∆β = 2β − 2piΛ is the wavenumber mismatch, andsm =√|κm|2 − (∆β2 )2. As CMT-TMM only uses the ideal grating profilesfor the spectral calculation, the obtained result will be the ideal spectralresponse led by specific grating profiles. For the SS-TMM, however, it willfirst convert the grating profiles into a physical IBG structure, followedby the spatial sampling process to discretize the IBG structure. Then, thetransfer matrix calculation is performed along all the grating segments of thediscretized IBG structure, using the transfer matrices based on the transfersolutions derived from the Fresnel equations [Eqs. 2.20 and 2.21].It should be noted that the SS-TMM may not be accurate for char-acterizing strong Bragg gratings, as it cannot predict unexpected effectsthat could happen in strong Bragg gratings, such as the saturation effect ofthe grating strength when using a large corrugation width [shown in Fig.2.7(b)]. The SS-TMM would also not be accurate for modeling multimodewaveguide-based Bragg gratings, since it cannot consider interactions andcoupling between modes of different orders. In these cases, the 3D-FDTDmethod will provide more reliable spectral results.Experimental resultsThe SS-TMM, which is a structure-aware modeling method and thus canconsider the structure modification-induced grating property changes, ismore accurate than conventional CMT-TMM when characterizing apodizedIBGs. This is demonstrated in Fig. 2.17, which compares the calculatedgrating spectral results from the CMT-TMM and SS-TMM with the ex-perimental results. The spectra shown in Figs. 2.17(a) and 2.17(b) aremeasurement results of Gaussian-apodized gratings through ∆L and duty-38-6 -3 0 3 6-30-20-100  Reflection (dB)∆λ (nm)M i s a l i g n m e n t - m o d u l a t e d  I B G  Measurement SS-TMM CMT-TMM(a)-6 -3 0 3 6-30-20-100(b) D u t y  c y c l e - m o d u l a t e d  I B G  Reflection (dB)∆λ (nm)Figure 2.17: Comparison of the grating spectra calculated by the CMT-TMM and SS-TMM and the experimental results for Gaussian-apodizedIBGs through (a) misalignment and (b) duty-cycle modulations.cycle (DC) modulations, respectively. These gratings are developed on SOIwaveguides with cross sections of 500 nm (wide) × 220 nm (high), and werefabricated by e-beam lithography. As can be seen, for both gratings, theresults predicted by the SS-TMM agree much better with the experimentalresults compared with those calculated by the CMT-TMM, validating thehigh accuracy and reliability of the SS-TMM. Note that in Chapter 5, the ac-curacy of SS-TMM will be further examined in various complex synthesizedIBGs.As mentioned before, the result obtained by the CMT-TMM actuallyrepresents the ideal spectral response produced by specific grating strengthand phase profiles. As seen in Fig. 2.17, the measured spectral responsesfor both gratings are wider than the ideal ones. The spectral broadening forthe two gratings are caused by the apodization-induced phase errors, whichwill be discussed in greater detail in Chapter 5.DiscussionAs the SS-TMM is a structure-aware grating modelling method, it is possi-ble to include manufacturing imperfections and variations into the gratingstructure for a better spectral prediction. For example, the grating struc-ture to be modelled can include waveguide width variations and waveguideroughness, and be modified via a lithography model which can take lithog-raphy smoothing effects and minimum realizable feature size/spacing intoaccount [2]. After these, such a ”virtually fabricated” grating can be re-simulated by the SS-TMM to obtain a better prediction of the forthcoming39experimental results. This concept will be used in Section 4.3 to characterizethe spectral performance evolution of phase-modulated IBGs against designparameters under fabrication constraints.In addition, the SS-TMM in the current work is for modelling siliconIBGs based on strip waveguides. There will, however, be benefits in someapplications to develop IBGs on other silicon waveguides such as ridge andslot waveguides, as discussed in Section 1.1. Thus, one direction of futureresearch is to extend the SS-TMM model to IBGs based on those differ-ent silicon waveguides. The SS-TMM implementation in those cases maybe similar to that used in current strip waveguide-based IBGs. However,the effective refractive index of each layer of the discretized grating willno longer be simply related to its waveguide width, which would need tobe solved through other mathematical techniques, such as effective indexmethod [61] and the photonic band-structure calculations, or be extractedfrom experimental results.40Chapter 3Multichannel PhotonicHilbert Transformers basedon Complex SynthesizedIBGs3.1 IntroductionMicrowave signal processing in the photonic domain can offer operationbandwidths and speeds far beyond their electronic counterparts, owing totheir capability to overcome the inherent bottlenecks caused by limited elec-tronic sampling speeds [40]. Microwave photonics (MWP) signal processingbased on integrated optics devices is even more promising, due to the highcompactness and low power consumption of integrated optics [44]. Manykey microwave functions have been demonstrated on photonic chips, includ-ing Hilbert transformers [41], differentiators [43], and integrators [45]. A lotof work has also been dedicated to improve the processing performance interms of bandwidth, reconfigurability, power efficiency, etc., with some re-markable features realized, such as THz processing bandwidths [62], sub-voltcontrol [63], and a fully programmable capability [45].While considerable efforts have been made to facilitate the advancementof MWP signal processing technology, most work so far has focused on pro-cessors operating at a single optical wavelength. The motivation for thework presented in this chapter is, by exploiting the high spectral flexibil-ity of IBGs, to develop MWP signal processors that can simultaneouslyand independently operate on multiple wavelength channels. Such novelmultichannel MWP processors will possess significantly improved capabili-ties. Furthermore, one can envisage that they can be integrated in currentlywidespread WDM systems to allow for a new variety of useful and advancedmultichannel processing applications.In this chapter, by using the comprehensive design methodology of IBGs41developed in Chapter 2, we demonstrate the first, multichannel, photonicHilbert transformers (MPHTs) based on complex synthesized IBGs. The si-nusoidal phase modulation apodization method (described in Section 2.5.5)is used for implementing the designed IBGs. By modulating the phase ofeach grating period, the complicated grating strength profiles required byMPHT spectral responses are precisely applied on the grating. In addition,the designed IBGs are implemented on highly compact spiral shapes to al-leviate fabrication issues such as chip nonuniformity and stitching errors.MPHTs with total wavelength channels of up to 9, and a single channelbandwidth of up to 625 GHz, are successfully achieved. The work rep-resents an important step in developing a new variety of promising mul-tichannel MWP processors. The studies also offer important insight intothe design and implementation of complex synthesized Bragg gratings fordifferent advanced applications.3.2 DesignThe design of IBGs to achieve MPHT responses follows the overall designand implementation flow presented in Chapter 2. Hilbert transformation(HT) is a fundamental processing function with many applications rangingfrom microwave engineering to image processing [62]. A traditional photonicHT is essentially a bandpass filter with a single discrete pi-phase shift atthe central wavelength in the phase response [Fig. 3.1(a)]. To design aphotonic HT that can operate simultaneously and independently at multiplewavelengths, we require multiple discrete phase shifts distributed within thebandpass range. Such a spectral response will form multiple photonic HTFigure 3.1: Ideal spectral responses of (a) traditional single channel and (b)multichannel photonic HTs.channels centered at different wavelengths, as illustrated in Fig. 3.1(b),42where a 5 channel MPHT is used for the illustration. Phase shift pointswill be at the centers of different HT channels, and RF signals carried atdifferent HT centers with bandwidths smaller than the channel bandwidthcan be processed simultaneously and independently. The number of the HTchannels will be equal to the number of the discrete phase shifts, and thechannel bandwidth will be determined by the wavelength spacing betweenneighboring phase shifts.0.00.20.40.6  Reflection (n.u)B = 3 nm-20 0 20-404  -9 -6 -3 0 3 6 9-50-40-30-20-100  (b)(a) (c)Reflection (dB)∆λ (nm)0 5 10 15-404Windowed & Time-shiftedIntensity (a.u)t (ps)-9 -6 -3 0 3 6 9-4-2024Phase (pi)∆λ (nm)0246810Phase (pi)Figure 3.2: (a) Ideal amplitude (blue) and phase (red) spectral responseof the designed 5-channel MPHT with B = 3 nm. (b) Truncated andtime-shifted impulse response, where the inset shows the original impulseresponse. (c) Realizable spectral response of the designed MPHT.A 5-channel MPHT with a channel bandwidth B of 3 nm, is first de-signed, with the initial target spectral response plotted in Fig. 3.2(a). Theamplitude response (blue) shows a bandpass filtering feature, while the phaseresponse (red) exhibits a staircase-like behavior from −2.5pi to 2.5pi with fivepi phase jumps spaced by 3 nm. This ideal spectral response, however, is notphysically realizable, because the corresponding impulse response is infiniteand non-causal (i.e., with elements located in the negative time or spatialdomain), as shown in the inset of Fig. 3.2(b). Thus, the spectral responseshould be first modified to be physically realizable. As described in Section2.2, this can be achieved by first truncating its impulse response and thenshifting the truncated impulse response to the positive time (or spatial) do-main, as illustrated in Fig. 3.2(b). The physically realizable version of thetarget response can be obtained by performing an inverse Fourier transformto the new impulse response. The realizable spectral response obtained isshown in Fig. 3.2(c). As can be seen, the amplitude response (blue) is de-graded by (1) spectral ripples, which are caused by the temporal truncation,and (2) spectral notches at the phase shift wavelengths, which, similar totraditional single-channel photonic HTs, arise from the pi phase jumps in430.0 0.2 0.4 0.6 0.8 1.0012κ (m-1 × 104)Length (mm)0.0 0.2 0.4 0.6 0.8 1.0-3-2-10123(b)Φκ (rad)Length (mm)(a) 0 1000 2000 3000 4000Period number01ΦG (pi)Figure 3.3: (a) The grating strength and phase profiles required by thetarget spectral response, which are calculated by using the LPA. (b) Phaseprofile of the designed grating for the apodization; the phase period Λφ is2.2 µm.the phase response [62]. The phase response (red), however, is very close tothe ideal one, which has correct phase jumps and no noticeable ripples.One may notice that the realizable phase response in Fig. 3.2(c) hasdifferent pi phase shift directions from the target one in Fig. 3.2(a), whichis only a mathematical rather than physical difference. Due to the inherentphase ambiguities [64], there is no essential difference between positive andnegative pi phase shifts in our case. Similar inconsistent phase jump direc-tions due to phase ambiguities can also be found in experimentally measuredphase responses, as we shall see later. Such phase ambiguities will be elim-inated when the phase response is constrained between its principal valueeither (−pi, pi] or (0, 2pi].Next, we can perform the design of the IBG based on the obtainedphysically realizable spectral response. The first step is to calculate thegrating strength and phase profiles, κ(z) and φG(z), that are required toachieve the target spectral response. As described in Section 2.3, this canbe accomplished by using the LPA on the realizable spectral response, andthe calculated results are shown in Fig. 3.3(a). As can be seen, the requiredκ(z) profile is highly complicated, which indicates that a high-performanceapodization method is essential for implementing the designed IBG. φG(z)is a discrete profile with many pi-phase shifts. The length of the grating,which is determined by the window length of the impulse response, is ∼1.1mm.Now, our task is to apodize the IBG according to κ(z). Current apodiza-tion approaches for grating strength modulation with high resolution and44accuracy include sidewall corrugation misalignment and sinusoidal phasemodulation [12]. In this work, the latter one is chosen for apodizing of ourgrating. The basic principle of the sinusoidal phase modulation apodiza-tion method has been briefly described in Section 2.5.5. The basic ideaof this apodization method is to add a sinusoidally varying phase φκ(z) =A(z) sin(2pizΛφ ), with a z-dependent amplitude A(z) and a period of Λφ, intothe effective index change along the grating to control the grating strength[12]. The normalized coupling coefficient, κn(z), will finally be related tothe phase amplitude A(z) via A(z) = J−10 (κn(z)), where J0 is the 0th-orderBessel function. Note that in the next chapter, this sinusoidal phase modu-lation apodization technique will be extended into arbitrary periodic phasefunctions. The grating phase profile φκ of the designed MPHT calculatedfrom κ(z) is plotted in Fig. 3.3(b). The phase spatial period Λφ has beenselected to be 2.2 µm. As will be discussed in the next chapter, a smallervalue of Λφ will give a higher apodization spatial resolution, but will alsoimpose a higher requirement on the fabrication. Thus, Λφ needs to be chosencarefully to make a good compromise between the apodization performanceand the grating fabrication requirement.In addition to the grating strength modulation, the phase profile requiredby the target response, φG(z) [gray curve in Fig. 3.3(a)], is also needed tobe applied into the grating. This is accomplished by inserting an additionhalf-period long grating segment at each pi phase shift position of φG(z). Aswill be shown in the next chapter, this can also be realized by incorporatingφG(z) into φκ(z) and then using the superimposed phase profile [φG(z) +φκ(z)] for the subsequent phase-to-structure mapping process.The process of determining the grating structure from the phase profileφκ(z) is illustrated in Fig. 3.4(a). Instead of imposing the phase profile di-rectly on IBGs, we first calculate the phase difference between each adjacentpair of periods, which are then considered as discrete phases shifts betweenadjacent periods to determine the grating cell structure. Specifically, we firstsample φκ(z) with a sampling interval equal to the grating period, whichwill produce the phase distribution, φκ(i), as a function of the grating pe-riod number i. Then, the phase shifts between neighbouring periods canbe obtained, via ∆φκ(i) = φκ(i + 1) − φκ(i). Finally, the distance betweenneighbouring corrugations, d(i), can be determined from the phase shiftsd(i) =ΛG2(1− ∆φκ(i)pi) (3.1)as illustrated in Fig. 3.4(b). ΛG is 317 nm for all the gratings designed here,and will lead to a Bragg resonance around 1550 nm.45d(i)ΔΦk(i) = 0d(i) = ɅG/2ΔΦk(i) = π/3d(i) = ɅG/3ΔΦk(i) = π/2d(i) = ɅG/4ΔΦk(i) = πd(i) = 0ith period (i+1)th periodκ (z) φk (z)Grating strength profile Phase distribution along gratingΔφk (i)Distance between adjacent corrugationsPhase difference between adjacent periodsGrating structure (a)(b) (c)(d)ΔWɅG/2d (i)52 um62.5 um-2-1012  (rad)0 1000 2000 3000100200d (nm)Period No.Figure 3.4: (a) Schematic flow showing the process of mapping the apodiza-tion profile into grating structure. (b) Schematic diagrams illustrating thegrating cell structures for different phase differences between the adjacentperiods. (c) Phase differences between the neighboring periods (upper),and the calculated distances between adjacent corrugations (lower). (d)Schematic of the overall (left) and zoomed-in portion (right) of the spiralphase-modulated grating.46The upper and bottom figures in Fig. 3.4(c) plot ∆φκ(i), which is ob-tained from the phase profile in Fig. 3.3(b), and the calculated d(i), respec-tively. It can be seen that d(i) exhibits an oscillating behavior around ΛG/2(158.5 nm) with a minimum level of ∼55 nm. These are obtained basedon a phase period Λφ of 2.2 µm. A smaller Λφ, which can lead to a higherapodization spatial resolution, will bring about a higher overall oscillationamplitude and thus a smaller minimum level of the d(i) profile. Specifically,d(i) can reach as low as 40 nm and 30 nm if Λφ is chosen to be 1.9 µmand 1.7 µm respectively. Those would be beyond the fabrication capabilitywhere a feature spacing of 50-60 nm is typically the minimum for e-beamlithography. Thus, the fabrication limits the minimum Λφ that can be usedand thus the apodization spatial resolution.The last step of our design is to translate the grating structure into acompact spiral shape [Fig. 3.4(d)], to increase the compactness of the deviceand also to alleviate fabrication issues including manufacturing nonunifor-mity and stitching errors [65]. The spacing between the closest spiral waveg-uides is 3.6 µm, and the radius of curvature increases from the minimum of52 um at the center to 62.5 um at the outmost of the spiral. An S-shape curveis used at the center to connect the inward and outward spirals. It is crucialto correctly transfer the distance between corrugations from a straight intospiral grating according to the local radius, to maintain the average Braggwavelength throughout the grating to avoid parasitic chirping.3.3 Experimental ResultsThe designed MPHT based on the above has been fabricated via e-beamlithography, with hydrogen silsesquioxane (HSQ) resist and a single etchof an SOI wafer with 220 nm thick silicon on a 3 µm thick buried oxidelayer. The resolution grid of the lithography was 6 nm, and the siliconwaveguide width was 500 nm. The scanning electron microscope (SEM)images of a sample 5-channel MPHT based on a spiral IBG are shown inFig. 3.5(a). Light was injected and extracted using sub-wavelength gratingcouplers (GCs), which have a wide 1-dB bandwidth up to 90 nm [66], anda polished polarization maintaining single-mode optical fiber array (PLCConnections Inc.). A directional coupler was placed between the input grat-ing coupler and the IBG to direct the reflected light back. In our case,only the reflected light was of interest and measured, and the transmissionlight was terminated at the end of the IBGs. The corrugation width (∆W )was selected to be 18 nm. An optical vector analyzer (OVA, Luna Innova-47tions) was used to measure the S parameters of the IBG. The wavelengthrange was from 1529 to 1571 nm with a spectral resolution of 2.5 pm. Atime-domain filter provided by the OVA was used to eliminate the backreflections from GCs and fiber interfaces. The OVA essentially uses swept-wavelength coherent interferometry to measure the Jones matrix, therebyacquiring both amplitude and phase responses of the tested devices [67].Note that there are also other methods available to measure a photonic de-vice’s phase response without relying on a commercial OVA, such as thephase-shift approach [68], the interferometry approach [69], and the opticalsingle-sideband modulation-based optical vector network analyzer [70, 71].20 umTape out(a)(b)Light in400 nmReflected lightFigure 3.5: (a) SEM images of a sample 5-channel MPHT based on a spiralIBG. (b) Measured amplitude (blue) and phase (red) spectral responses ofthe IBG.The insertion loss and phase responses of the 5-channel MPHT capturedby the OVA are shown in Fig. 3.5(b). The linear portion of the phaseresponse caused by the length of the devices has been removed, and thephase of Channel 1 (centred around 1550 nm) has been offset to zero. Themeasured complex response shows an excellent agreement with the designedresponse, validating our IBG design and implementation methodology. Thedeviations of the phase jumps from pi for all the channels have been measured48to be ≤ 0.05pi. One may notice that the measured insertion loss level isrelatively high (> 30 dB). Such a high level of loss, however, is mainlycontributed by other components in the circuit instead of the IBG, includingGCs, directional coupler and the long waveguide before the IBG. For theIBG, we calculate according to its ∆W that the reflectivity is ∼50% , whichcorresponds an insertion loss of ∼3 dB.A MPHT with a larger channel bandwidth B = 5 nm, correspondingto a frequency of ∼625 GHz, has also be designed and fabricated. The re-quired κ(z) and φG(z) calculated by using the LPA are shown in the leftplot of Fig. 3.6(a). The IBG has a relatively short length of ∼0.66 mmand is actually a spatially compressed version of the previous 3 nm band-width 5-channel MPHT [Fig. 3.3(a)]. This means that the grating strengthvaries more rapidly compared with that of the previous narrower bandwidthMPHT, which also suggests a higher requirement for spatial resolution ofthe apodization. Nevertheless, the measured results [right Fig. 3.6(a)],demonstrate the successful achievement of such a broader band MPHT. Nonoticeable spectral degradation compared with the previous case is found,suggesting that the channel bandwidth may be further increased towardsone THz based on our IBGs.The feasibility of achieving higher channel-count MPHTs has also beeninvestigated, where a 7-channel MPHT with B = 3 nm [Fig. 3.6(b)], and a9-channel MPHT with B = 1.6 nm [Fig. 3.6(c)] are designed and fabricated.Compared with those of the previous 5-channel MPHTs, the required grat-ing profiles for these two higher channel-count MPHTs [left figures in Figs.3.6(b) and 3.6(c)] become even more complex, with more high-frequencyoscillations in κ(z) and more pi-phase jumps in φG(z). This suggests thathigher resolution and dynamic range of the apodization are required forimplementing higher channel-count MPHTs. The measured results of thetwo MPHTs are shown in the right plots of Figs. 3.6(b) and 3.6(c), whichare again in excellent agreement with the designed spectral responses. Theamplitude response of the 9-channel MPHT exhibits higher ripples thanother cases, which could be due to that some high-frequency elements ofκ(z) have not been fully realized with the current apodization spatial res-olution, which, as discussed before, is determined by Λφ. Nevertheless, theachievement of the 9-channel MPHT with B = 1.6 nm, corresponding to afrequency bandwidth of ∼200 GHz, indicates that MPHTs with more than10 channels with a narrower channel bandwidth of around 100 GHz can beapproached. MPHTs with such wavelength features can be readily incorpo-rated in current dense WDM (DWDM) systems.An important application of traditional single-channel photonic HTs is490.0 0.2 0.4 0.60.00.61.21.8κ (m-1  × 104)Length (mm) 1544 1552 1560 1568-80-70-60-50-409 Chan. ;   = 1.6 nm; N = 4047; W = 14 nm7 Chan. ;  = 3 nm; N = 3391; W = 18 nm5 Chan. ;   = 5 nm; N = 2089;  W = 16 nmInsertion loss (dB)Wavelength (nm)(a)0.0 0.3 0.6 0.9012(c)(b)κ (m-1  × 104)Length (mm) 1548 1554 1560 1566-70-60-50-40Insertion loss (dB)Wavelength (nm)02468Phase (pi)0.0 0.5 1.00.00.51.01.5κ (m-1  × 104)Length (mm) 1540 1544 1548 1552-80-70-60-50-40Insertion loss (dB)Wavelength (nm)-8-6-4-20Phase (pi)0246Phase (pi)01ΦG(z)01ΦG(z)01ΦG(z)Figure 3.6: Required κ(z) and φG(z) profiles (left figures) and measuredspectral responses (right figures) of MPHTs with different bandwidths andchannel numbers; N: total number of grating periods.50for single-side band (SSB) signal generations [62]. For the currently designedMPHTs, one can expect that they can be used for achieving novel multichan-nel SSB signal generations for WDM communication links [72]. To verifythis, we calculate the transfer function of a typical building block using HTfor SSB modulation [Fig. 3.7(a)] [62], using the experimental S parametersmeasured from our MPHTs. The calculation method is similar to that in[62]. The building block has a MZI structure, with one arm consisting ofoptical S parameters operating at reflection mode, and the other having anoptical delay and attenuator to match the first branch. This building blockS11  S12S21  S22S-ParameterΔφ (a)Delay Attenuator12(b)Figure 3.7: (a) Building block used in the numerical analysis to verify thepotential of MPHTs as multichannel SSB filters. (b) Calculated transferfunctions for different ports when using the experimental S parameters of(left) 5 channel MPHT with B = 3 nm, and (right) 7 channel MPHT withB = 3 nm; the dash lines indicate different SSB filtering channelsessentially exploits the pi-phase shift between the left- and right-side wave-length bands of a photonic Hilbert transformer to realize a SSB filter. Inparticular, when the phase of the light traveling through the bottom armis tuned to match that of the left (right)-side wavelength band of the pho-tonic Hilbert transformer, a constructive interference will occur on the left(right)-side band while a destructive interference will happen on the right(left)-side band due to the pi-phase shift between them, which finally formsa SSB filtering behaviour on the transfer function of the system. Figure3.7(b) shows the calculated transfer functions of the circuit when using theS parameters from (left) the 5-channel MPHT with B = 3 nm, and (right)the 7-channel MPHT with B = 3 nm. The transfer functions of differentports suggest that such a building block with our MPHTs can be used aseither optical channel interleavers, or multichannel SSB filters as indicatedby the dashed lines. In the later case, the channel number is equal to that ofthe used MPHT, and the average sideband suppression ratio is higher than5110 dB, with steep rejection slopes at the band-edges.3.4 ConclusionTo summarize, in this chapter, MPHTs based on sinusoidal phase-modulatedspiral IBGs have been designed, fabricated, and experimentally character-ized. The realizations of these MPHTs are based on the comprehensivedesign methodology of IBGs developed in Chapter 2. MPHTs with up to9 wavelength channels and a single channel bandwidth of up to ∼625 GHzhave been achieved. The potential of the devices for multichannel single-side band signal generations has also been suggested. Such MPHTs can bedesigned and implemented with either a wide channel bandwidth of poten-tially > 1 THz for broadband multichannel processing, or a high channelnumber of > 10 for applications involving dense wavelength channels, suchas DWDM systems. The work offers a new possibility of utilizing wavelengthas an extra degree of freedom in designing MWP signal processors. Suchmultichannel processors are expected to possess improved capacities and apotential to greatly benefit current widespread WDM systemsThere are several avenues for future work. First, using higher order IBGsshould considerably relax the fabrication constraints imposed by minimumrealizable feature spacing/size and quantization errors due to limited lithog-raphy resolution [12]. This would enable a higher apodization performance,and hence broader band and higher channel-count MPHTs. Also, a sourceof the ripples in the complex responses and the side-lobes in the amplituderesponses could be the accumulated grating phase noise due to waveguidesidewall roughness, especially for the current long IBGs. It was suggestedthat this issue can be significantly alleviated by developing IBGs on widermultimode waveguides [26, 73]. Finally, it can be expected that the samemethodology can also be used to develop other novel MWP processors, suchas multichannel integrators and differentiators.52Chapter 4Apodization of Silicon IBGsthrough Periodic PhaseModulationAs discussed in Section 2.5, among various apodization methods for siliconIBGs, the sinusoidal phase modulation approach shows great promise forimplementing complex-synthesized IBGs to realize sophisticated spectral re-sponses. This is mainly due to two reasons. Firstly, this phase modulationtechnique offers high apodization precision/accuracy and resolution whileintroducing little unwanted index variations and thus grating phase errors,which is essential for a precise control of the grating response. In addi-tion, a complicated grating phase profile [φG(z)] that is usually required bya sophisticated target response can be directly incorporated into the finalgrating phase profile to create the grating structure, thus largely simplifyingthe implementation. This apodization technique has already enabled sev-eral sophisticated optical filters and processors to be achieved for the firsttime on silicon IBGs, such as narrow-band single- [26] and multi-channel [74]dispersion-less flattop filters, and the multichannel photonic Hilbert trans-formers that were demonstrated in Chapter 3.In this chapter, we extend this promising sinusoidal phase modulationapodization technique to arbitrary periodic phase functions. We demon-strate, both theoretically and experimentally, that apodization of a siliconIBG can be achieved by modulating the grating phase using any periodicfunction. Apodization properties of several common periodic functions, in-cluding square, sawtooth, triangle, and sinusoidal phase functions, are de-rived and analyzed. The study shows that using different periodic phasefunctions can lead to different apodization characteristics, grating physicalstructures, actual apodized grating performances, etc. The work providesan additional degree of freedom for the design and optimization of phase-modulated IBGs, and will drive future exploration of novel periodic functionsthat offer new useful apodization features.53In addition, this chapter also proposes a general implementation processof the periodic phase modulation apodization technique to realize a desiredresponse on a silicon IBG. Then, the limiting factors of the apodization per-formance, design trade-offs and optimization, and grating robustness againstfabrication constraints for several common periodic functions are studied.The grating characterization tool used in the study is based on a basiccomputational lithography model, which is used to consider non-ideal fab-rication effects, together with the SS-TMM grating emulator proposed inSection 2.6.2. The work will have important implications to practical im-plementations of the apodization technique for spectral tailoring of siliconBragg grating devices.4.1 Principle4.1.1 Basic theoryAssuming a periodic phase function, φκ(z), is applied on a Bragg grating,the effective refractive index change along the grating length, z, then canbe expressed as∆n(z) = ∆nRe{exp(j2pizΛG) · exp(jφκ(z))}(4.1)where ∆n is the constant index modulation amplitude and ΛG is the gratingperiod. The term exp(jφκ(z)) will also be periodic with the same period asφκ(z), and thus can be expanded into the Fourier series:exp(jφκ(z)) =+∞∑m=−∞Fm exp(j2mpizΛφ) (4.2)where Fm is the mth Fourier coefficient, and Λφ is the period of the phasefunction.Taking Eq. 4.2 into Eq. 4.1, Eq. 4.1 can be rewritten as∆n(z) =∞∑m=−∞Fm∆nRe{exp(j(2pizΛG+2pimzΛφ))}(4.3)It can be seen that due to the periodic phase modulation, the grating indexprofile now has multiple frequency components. These extra frequency com-ponents in the grating spectrum will be represented as multiple resonanceslocated on the two sides of the original or center Bragg resonance (m = 0).54The ±mth resonance strength is proportional to the mth Fourier coefficientof the periodic term exp(jφκ(z)), Fm. The spacing between neighboringresonances, ∆λ, can be derived to be∆λ = λB(2ngΛφλB+ 1)−1 (4.4)where λB is the center Bragg wavelength of the grating, and ng is the groupindex of the waveguide. Equation 4.4 indicates that use of a shorter phaseperiod (Λφ) will lead to a larger spacing between adjacent resonances.In this work, we consider that the center Bragg resonance (m = 0) istailored. The index change profile for the center Bragg resonance can beexpressed according to Eq. 4.3 as∆n0th(z) = F0∆nRe{exp(j2pizΛG)}(4.5)It can be seen that the strength of the center Bragg resonance is nowproportional to the 0th Fourier coefficient of the periodic term exp(jφκ(z)),F0, which can be derived throughF0 =1Λφ∫ Λφ0exp(jφκ(z))dz (4.6)It is clear that F0 depends on the parameters of φκ(z). In particular, F0will vary with the amplitude of φκ(z), as we will show later. Now, one canexpect that if φκ(z) has a slowly varying amplitude along the grating length,z, then, F0 will also change against z. This way, we can design φκ(z) to havean amplitude profile, A(z), such that the corresponding F0(z) is equal to thenormalized target apodization profile, κn(z). This offers the basic principleof the general periodic phase modulation apodization method.4.1.2 Relationship between the phase amplitude A and F0for several common periodic functionsIn the following, we will derive the relationship between the phase amplitude,A, and the 0th Fourier coefficient of exp(jφκ(z)), F0, for several commonperiodic functions. As the strength of the Bragg resonance will be propor-tional to F0, the obtained curves can also be regarded as the relationshipsbetween A and the normalized grating strength, κn.550 1 2 3 40-APhase (rad)z/ΛφSinusoidalA0.0 0.5 1.0 1.5 2.00.00.51.00th order Bessel functionF 0Phase amplitude A (rad)≈ 2.40480 1 2 3 40Phase (rad)z/ΛφSquare(b)A-A0.0 0.5 1.0 1.5 2.0 2.5 3.0-1.0-0.50.00.51.0pipiCosine functionF 0Phase amplitude A (rad)pi/20 1 2 3 40Phase (rad)z/Λφ(c) Triangle and sawtooth Triangle      Sawtooth    A-A0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.51.0 Sinc functionF 0Phase amplitude A (rad)0 10 2001A (rad) F 0(a)0 10 2001A (rad) F 0Figure 4.1: (a)-(c) Different periodic phase functions (left) and the corre-sponding relationships between the phase amplitude A and the 0th Fouriercoefficient, F0 (right).Sinusoidal phase functionsA sinusoidal phase function with an amplitude A and a period of Λφ [Fig.4.1(a), left] can be written asg(z) = A sin(2pizΛφ) (4.7)F0 can be derived from Eq. 4.6, which will be a 0th order Bessel functionof AF0 = J0(A) (4.8)The right plot of Fig. 4.1(a) shows the 0-th order Bessel function profile overthe range 0 < A < 2.4048, where F0 (or κn) decreases from 1 to approxi-mately the first zero. This range is sufficient for full control of the normalizedgrating strength, and thus can be used for the apodization purpose. Thetop-right inset shows the relationship curve over a broader range.56Square phase functionsA square function, with a period of Λφ, an amplitude of A, and a duty cycleof 50% [Fig. 4.1(b), left] can be defined over the first period [(0,Λφ]] asg(z) ={A, 0 < z ≤ Λφ/2−A, Λφ/2 < z ≤ Λφ, (4.9)By using Eq. 4.6, F0 in this case can be derived to be a cosine functionof the phase amplitudeF0 = cos(A) (4.10)An interesting feature provided by use of square phase functions for theapodization is that F0 or κn can be controlled from 1 to -1 by varying thephase amplitude from 0 to pi, as shown in the right plot of Fig. 4.1(b).Different signs of κn (positive or negative) actually means that there is api-phase shift between them, and can happen if κn(z) is defined to includeboth the grating strength and phase information [as shown in Fig. 4.2(b)].This definition of κn(z) can be used if the grating phase profile containsonly discrete pi-phase shifts, which is typically the case when the designedgrating is not chirped. In this case, using square phase functions will allowfull control of κn(z) from 1 to -1 by varying A from 0 to pi. However, tonot lose generality and to make it easier to understand physically, for allof the IBGs designed in this thesis, κn(z) is defined to only represent thegrating strength while the grating phase distribution is represented by aseparate function [φG(z)], as illustrated in Fig. 4.2(a). Therefore, κn(z)here will always be positive and located between 0 and 1, and thus A forsquare functions should be tuned only from 0 to pi/2 to control κn from 1 to0. Figure 4.2 compares κn(z) profiles required by a flattop filter expressedbased on the two different definitions.Triangle and sawtooth phase functionsA triangle function with a period of Λφ and an amplitude or maximum valueof A [Fig. 4.1(c), left] can be defined over the first period asg(z) ={A(2zt − 1), 0 < z ≤ tA( −2zΛφ−t +Λφ+tΛφ−t), t < z ≤ Λφ, (4.11)where t is the position of the maximum value in the first period. Anothersimilar type of periodic functions are sawtooth functions, which can be con-sidered as a special case of triangle functions with t = Λφ, and can be570 500 1000 1500 20000.00.20.40.60.81.0κn (a.u)Period No. 0 500 1000 1500 2000-0.20.00.20.40.60.81.0(b)κn (a.u)Period No.(a)01ΦG (pi)Figure 4.2: Illustration of two different definitions of normalized gratingcoupling strength profiles, κn(z). κn(z) in (a) is defined to represent onlythe grating strength while the grating phase distribution is represented bya separate function [φG(z)]; this definition is chosen to be used in this the-sis. κn(z) in (b) is defined to include both the grating strength and phaseinformation.expressed over the first period asg(z) =2AzΛφ−A (4.12)It can be derived using Eq. 4.6 that for both triangle and sawtoothfunctions, F0 versus A will follow a sinc functionF0 =sin(A)A(4.13)Figure 4.1(c) (right) plots the sinc function over the range 0 < A < pi,where F0 decreases from the initial value of 1 to the first zero. This rangeis sufficient for the apodization purpose. The top-right inset shows therelationship curve over a wider range.4.2 ImplementationFigure 4.3 is a schematic flow showing the design process of a phase-modulatedIBG to achieve a desired response. The first step is to determine the requiredgrating strength and phase profiles [κ(z) and φG(z), respectively] from thephysically realizable target response, which can be achieved through theLPA. The obtained κ(z) and φG(z) profiles are subsequently used to calcu-late the total phase profile [φtot(z)] of the IBG. Then, the physical structureof the IBG is translated from φtot(z). The created IBG structure can be58simulated to check if the corresponding spectral response will be consistentwith the design. This can be realized by using a structure-aware modelingmethod, such as the SS-TMM method proposed in Section 2.6.2. Finally, thegratings are sent for fabrication, followed by measurement and test. Below,we will focus on two key implementation steps: 1) determining the totalphase profile of the grating, φtot(z), and 2) mapping φtot(z) into an IBGphysical structure.κ(z) & φG(z) Grating strength and phase profiles Φtol(z)Physical grating structureTotal phase profiler(λ)Simulation & Spectral ComparisonFabricationLPATarget spectral responseLayoutΦtol(z) = ΦΚ(z)  + ΦG(z)  Figure 4.3: Schematic flow showing the design process of a phase-modulatedIBG to achieve a target response.4.2.1 Determination of the total phase profile from thegrating strength and phase profilesLet us start from a simple case of a Gaussian-apodized IBG. The normalizedgrating strength profile [κn(z)] is plotted in Fig. 4.4(a). The grating periodis 316 nm, and the total period number is 800. As discussed before, applyinga specific κn(z) profile on an IBG can be achieved via phase-modulating thegrating with a periodic function φκ(z) having an amplitude profile of A(z)such that F0(z) = κn(z). Thus, when the type of the periodic phase functionhas been selected, the first step is to calculate A(z) from κn(z), viaA(z) = f−1 (κn (z)) , A ∈ [0, B] (4.14)where f represents the mapping function from A to F0, which, as derivedabove, are the 0-th order Bessel, cosine and sinc functions in cases of si-nusoidal, square and triangle/sawtooth phase functions, respectively; and[0, B] is the range of A where F0 changes from 1 to the first zero. B asindicated in Fig. 4.1 will be ∼2.4048, pi/2 and pi for sinusoidal, square andtriangle/sawtooth phase functions, respectively. As an example, when asquare phase function is used, A(z) can be calculated from κn(z) viaA(z) = cos−1 (κn (z)) , A ∈ [0, pi/2]Once A(z) is obtained, φκ(z) can be easily calculated viaφκ(z) = A(z)g(z) (4.15)590 200 400 600 8000.00.20.40.60.81.0(a)κn (a.u)Period No. 0 200 400 600 8000123 Square Sinusoidal Sawtooth(b)APeriod No.0 200 400 600 800-2-1012  A(z)    φ(z): A(z)×g(z)  (c)Phase (rad)Period No.Square0 200 400 600 800-3-2-10123 Sinusoidal(d)Phase (rad)Period No.0 200 400 600 800-4-2024 Sawtooth(e)Phase (rad)Period No.200 250 300-101Phase (rad)Period No.200 250 300-2-1012Phase (rad)Period No.200 250 300-202Phase (rad)Period No.Figure 4.4: (a) Normalized Gaussian apodization profile. (b) Phase ampli-tude profiles along the grating for different periodic phase functions. (c)-(e)Overall (left) and zoomed-in (right) grating phase modulation profiles incases of different periodic phase functions.where g(z) represents the basic periodic function and are expressed in Eqs.4.7, 4.9, 4.11 and 4.12 for different types of the periodic functions. In thecurrent Gaussian-apodized grating, as there is no phase modulation requiredby the design, i.e., φG(z) = 0, φκ(z) will be the total phase profile of thedesigned grating. The obtained A(z) profiles via Eq. 4.14 for differentperiodic functions are plotted in Fig. 4.4(b). It can be seen that in all cases,A is higher (or the phase modulation is stronger) when κn is weaker, andwhen κn is the highest (i.e., equal to 1), the phase amplitude is 0, meaningthat there is no phase modulation applied on the grating. The calculatedphase modulation profiles [φκ(z)] for different periodic phase functions areplotted in the left plots of Figs. 4.4(c)-4.4(e), while the right figures ofFigs. 4.4(c)-4.4(e) show the enlarged views of the phase profiles. The phaseperiods, Λφ, are 3 µm. Note that Λφ is an important parameter and should60010 600 1200 1800 24000.00.51.01.5κ (m-1 ×104)Period No.0 600 1200 1800 24000.00.40.81.21.6(b)APeriod No.A(z)0 600 1200 1800 2400-2024φtol(z): φκ(z) + φG(z) Phase (rad)Period No.ΦG(pi)0 600 1200 1800 2400-202 (f)(d)  φκ(z): A(z)×g(z)      A(z) Phase (rad)Period No.(a)-6 -4 -2 0 2 4 6-30-20-100Reflection (dB)∆λ (nm)3.8 nm1060 1110-101 (e)(c)-6 -4 -2 0 2 4 6-30-20-100Reflection (dB)∆λ (nm) Reconstructed spectrum Ideal Figure 4.5: (a) Spectrum of the designed flattop filter. (b) Grating strengthκ (blue, left axis) and phase φG (grey, right axis) profiles required by thedesigned spectral response, calculated via the LPA. (c) Phase amplitudeprofile along the grating, A(z), in case of square phase functions, calculatedfrom the normalized κ(z) profile via Eq. 4.14. (d) Phase profile, φκ(z)(yellow), and its amplitude profile, A(z) (red); the top-left inset shows theenlarged view of the highlighted range of φκ(z). (e) Total grating phaseprofile, φtol(z), which is the superposition of φκ(z) and φG(z). (f) Recon-structed spectrum of an IBG phase-modulated by φtol(z) in (e), calculatedvia the CMT-based TMM.be chosen carefully: a smaller Λφ can lead to a better spectral performance,but will also increase the fabrication requirement of the IBGs. This trade-offwill be studied later.Next, we illustrate how to determine the total grating phase profile in amore complex case where a flattop grating filter is designed. The designedfiltering response is shown in Fig. 4.5(a), which has been modified to bephysically realizable. The passband width is ∼3.8 nm. The grating strengthand phase profiles, κ(z) and φG(z), required by the design are calculated viathe LPA, and the results are plotted in Fig. 4.5(b). The grating period is316 nm, and the total period number is 2486. We can see that a complicatedphase profile φG(z) containing multiple discrete pi-phase shifts is requiredby the design. The total phase profile of the designed IBG in the currentcase is the superposition of φκ(z) and φG(z)φtol(z) = φκ(z) + φG(z) (4.16)61φκ(z) can be obtained from κn(z) via Eqs. 4.14 and 4.15 as described before.We here use a square phase function as the example. Figure 4.5(c) is thephase amplitude profile A(z). Figure 4.5(d) plots the φκ(z) profile (yellow),which has an envelope of A(z) (red), and the top-left inset is a zoomed-inview of the highlighted range of φκ(z). The phase period has been selectedto be 1.7 µm. Figure 4.5(e) plots the total grating phase profile, φtol(z),obtained via Eq. 4.16. To confirm if the obtained φtol(z) profile will leadto the target flattop passband response, we perform the CMT-based TMM[60] to calculate the spectrum of an IBG that is phase-modulated by φtol(z)in Fig. 4.5(e) and has a constant κ of 1.43 × 104 m−1 throughout thegrating [i.e., the maximum value of the required κ(z) profile]. The calculatedspectrum is shown in Fig. 4.5(f), where the spectral result agrees well withthe design, validating our phase determination process.4.2.2 Phase-modulated grating structuresPhase-to-structure conversion rulesΦtot(z) Φtot(i) d(i)ΔΦtot(i)d(k-1) d(k) d(k+1)ΛG/2SamplingDistance between adjacent corrugations(b)(a)zPhase difference between adjacent periodsPhase -versus- grating period No.Grating structureΔWW)(zff ub Scheme 1Scheme 2 222)(WzzsWzf totGu  Figure 4.6: (a)-(b) are schematic illustrations of two different phase-to-structure conversion schemes, denoted as Schemes 1 and 2, respectively. (a)A typical phase-modulated IBG structure generated using Scheme 1. (b)(Upper) schematic flow showing the implementation process of Scheme 2,and (lower) a typical phase-modulated IBG structure created using Scheme2.In this section, we describe how to map a total grating phase profile62φtot(z) into an IBG physical structure. Before the phase-to-structure map-ping process, several fundamental physical parameters of the IBG should befirst decided, including ΛG, corrugation shape (rectangular or sinusoidal),and ∆W . The determination of these fundamental parameters has beendiscussed in Section 2.4. Once the fundamental physical parameters aredecided, the structure of the IBG can be translated from the total gratingphase profile φtol(z). This can be performed through two different struc-ture determination schemes. The two schemes will produce different gratingstructures, while leading to the same spectral response, which will be elab-orated on below.In the first scheme, which will be referred to as Scheme 1, the total phaseprofile φtot(z) is directly taken into the spatial edge functions of the IBG,as illustrated in Fig. 4.6(a). The spatial functions of the upper and bottomedges of the IBG modulated by φtol(z) can be written asfu(z) =∆W2s(2pizΛG+ φtol (z))+W2(4.17a)fb(z) = −fu(z) (4.17b)where s(z) is the grating structural profile, which will be sin(z) and sgn (sin (z))for IBGs with sinusoidal- and rectangular-shaped corrugations, respectively,and W is the unperturbed waveguide width. This scheme is easy andstraightforward, and thus should be preferred in most cases. The possi-ble disadvantage is that as continuous spatial functions of the gratings areused, a sufficiently high sampling rate of the spatial functions and thus alarge number of vertices are required to fully construct the phase-modulatedgrating structure, especially when the phase period is small.The second scheme (Scheme 2) is schematically illustrated in Fig. 4.6(b),which can only be used for IBGs with rectangular corrugations. In thisscheme, the phase profile is first converted to the phase differences betweeneach pair of adjacent periods, which are then considered as discrete phaseshifts between adjacent periods to determine the grating cell structures.The process flow of the scheme is shown in the upper part of Fig. 4.6(b).To calculate the phase differences between adjacent periods, φtot(z) is firstsampled with an interval of ΛG to be expressed as a function of the gratingperiod number, φtot(i). Then, the phase differences between the (i − 1)thand the ith periods can be obtained via ∆φtot(i) = φtot(i) − φtot(i − 1).Finally, the distance between the (i − 1)th and the ith corrugations, d(i),can be calculated from ∆φtot(i) viad(i) =ΛG2(1− ∆φtot(i)pi) (4.18)63A typical phase-modulated IBG structure created by this scheme is illus-trated in the lower diagram of Fig. 4.6(b). Note that the corrugation lengthof ΛG/2 remains unchanged throughout the grating. The advantage of thisscheme is that the grating structure can be constructed by a stack of boxes,hence requiring fewer vertices than Scheme 1. The drawback is that the cal-culation of d(i) from φtot(z) increases the complexity of the implementation.Note that the structure determinations of the phase-modulated IBG-basedMPHTs presented in Chapter 3 were actually based on Scheme 2.Typical grating structures modulated by different periodic phasefunctions0.0 0.5 1.0 1.5 2.0 2.5-2-1012Phase (rad)z/0.0 0.5 1.0 1.5 2.0 2.5-101Phase (rad)z/0.0 0.5 1.0 1.5 2.0 2.5-3.0-1.50.01.53.0Phase (rad)z/Phase-shifting point(a)(b)(c)AGG1'1Figure 4.7: (a)-(c) Typical grating structures modulated by different pe-riodic phase functions. For each figure, the upper plot shows the phasefunction while the lower diagram is the corresponding grating structure,created based on Scheme 1.Figures 4.7(a)-4.7(c) illustrate typical grating structures modulated bydifferent periodic phase functions, where the phase-to-structure conversionis based on Scheme 1 [Fig. 4.6(a)]. For each of Figs. 4.7(a)-4.7(c), the upper64plot is the phase profile while the lower diagram illustrates the correspondinggrating structure. Figure 4.7(a) shows the case of an IBG modulated by a si-nusoidal phase profile. As shown in the bottom diagram, the grating periodexhibits an oscillating behavior along the length. This is because the phasemodulation can also be regarded as the modulation of the spatial period (orfrequency) of the grating, which will vary according to the first derivative ofthe phase profile. As the first derivative of a sinusoidal function still followsa sinusoidal behavior, the grating period thus oscillates sinusoidally alongthe length. The oscillation amplitude of the grating period will increase withthe amplitude of the phase modulation profile, A. Figure 4.7(b) shows thesituation where the grating is modulated by a square phase function. Thegrating phase in this case changes discretely rather than continuously. Thus,the corresponding grating is uniform except in the phase-shifting positions(indicated by the dashed lines), where abrupt changes of the grating struc-tural profile can occur which can lead to small features or feature spacing.The case of a grating modulated by a sawtooth phase function is presentedin Fig. 4.7(c). Within the linear region of the sawtooth function, the grat-ing is uniform but with a “new” period that is shorter than the originallydesigned one. This is because the first derivative of a linear phase functionis a constant that is equal to its slope, thus leading to a constant deviationof the spatial period. The new period (denoted as Λ′G) is related with thesawtooth function parameters (Eq. 4.12) via1Λ′G=1ΛG+ApiΛφ. (4.19)Therefore, Λ′G is smaller when the amplitude (or slope) of the sawtoothfunction is larger, or when the phase period Λφ is smaller. In the phase-shifting positions, similar to the case of square functions, abrupt structuralprofile changes can happen which can form small features or feature spacing.Impact of the phase amplitude and phase period on the gratingstructureIn this section, we analyze how different phase amplitude (A) and phaseperiod (Λφ) can affect a phase-modulated grating structure, so as to in-vestigate their impacts on the fabrication requirement of the grating. Weuse square phase functions for the current investigation. Figure 4.8(a) plotsthree different square phase profiles (Profiles 1-3). Profile 2 (red) has thesame amplitude with Profile 1 (blue) but with a smaller spatial period, while65123(b)(a)0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-1.5-1.0-0.50.00.51.01.5Phase (rad)z (um)1: =3.6 um; A=/4  2:=0.6 um; A=/4  3:=0.6 um; A=/2  Figure 4.8: (a) Square phase functions with different amplitudes and periods.(b) Comparison of grating structures corresponding to the different phasefunctions shown in (a), generated based on Scheme 1.Profile 3 (yellow) contains the same period as Profile 2 but has a larger am-plitude. Figure 4.8(b) compares the IBG structures corresponding to thethree different square functions, created based on Scheme 1. As describedbefore, for a grating modulated by a square phase function, the gratingis uniform except for the phase-shifting positions where abrupt structuralchanges can occur that can lead to small feature size and spacing. To resolvethem in the fabrication needs both a high lithography resolution and a smallminimum realizable feature size/spacing. When the square phase functionhas a smaller Λφ, phase-shifting and thus small feature size/spacing willappear more frequently along the grating, as compared by Figs. 4.8(b)-1and 4.8(b)-2. On the other hand, if the square phase function has a largeramplitude, this leads to larger values of the discrete phase shifts that arecloser to pi, which are more likely to form smaller feature size/spacing, ascompared by Figs. 4.8(b)-2 and 4.8(b)-3.Thus, it can be concluded from the above that smaller Λφ and largerA will increase the fabrication requirement of the IBG. Although this pointis made based on a square phase-modulated IBG with rectangular corru-gations, similar trends can also be found for other periodic phase functions66and sinusoidal corrugation-based IBGs. As realizing lower κ needs a higherphase modulation amplitude as discussed before, the minimum achievable κ,or the dynamic range of the apodization, in practice is ultimately limited bythe fabrication capability, including the lithography resolution and minimumachievable feature size/spacing. Also, Λφ should be small enough to ensuresufficient spacing between the side-resonances and the Bragg resonance asindicated by Eq. 4.4. Low Λφ is also required to obtain a high spectral qual-ity and low noise floor as we will show later. Hence, Λφ needs to be chosencarefully to make a good compromise between the apodization performanceand the grating fabrication requirement, which will be discussed later.4.3 Design ConsiderationsLithography model 200 nm1 2n1 n2 n3n13n14Structure synthesis-based TMM Structure samplingEquivalent multiplayer structure (a) (b)Figure 4.9: (a) Schematic illustration of the developed basic computationallithography model; the upper and bottom diagrams show the original andlithography-simulated grating structures, respectively; the positions labeledby 1 and 2 in the upper grating structure indicate feature spacing and sizethat are smaller than the minimum realizable values and thus are not re-solved in the lithography-simulated grating. (b) Schematic illustration ofthe SS-TMM for IBG modeling; the upper diagram schematically illustratesthe spatial sampling process of an IBG, while the bottom one shows theequivalent optical multiplayer structure.In this section, we will further investigate the design considerations ofphase-modulated IBGs. Specifically, we will first explore the spectral perfor-mance evolution of a phase-modulated IBG against the phase period, andthen compare the robustness of the gratings modulated by different peri-odic phase functions against fabrication constraints. All the IBGs designedand characterized in this section are developed on single-mode strip silicon67waveguides with cross sections of 500 nm (wide) × 220 nm (high) and de-signed for the fundamental quasi-TE mode; the grating periods are 316 nm,leading to Bragg wavelengths of ∼1550 nm; the corrugation shapes are rect-angular; and the phase-to-structure determinations are based on Scheme1. The characterization tool of IBGs used in our study consists of a ba-sic computational lithography model and the SS-TMM IBG emulator. Thegrating structure to be simulated by the SS-TMM will be first applied bythe lithography model so that non-ideal fabrication effects can be includedin the modeling results of the SS-TMM. This basic lithography model, asschematically illustrated in Fig. 4.9(a), can take into account several funda-mental non-ideal lithography effects, including lithography resolution limit(or grid resolution) and minimum realizable feature size/spacing. The con-sideration of a limited lithography resolution will take into account errorsdue to discretization of the grating structural profiles to the manufacturinggrid, and will also decrease the steepness or sharpness of the grating struc-ture. The grid quantization of the grating structure will be performed onlyalong the z direction to avoid unexpected ∆W variations which are irrelevantto the current investigations. The inclusion of minimum achievable featuresize/spacing will clip the features and feature spacings that are smaller thanthe minimum realizable value [as illustrated in positions 1 and 2 in Fig.4.9(a)]. The SS-TMM modeling method, as schematically illustrated in Fig.4.9(b), is performed by directly synthesizing the physical structure of thegrating. The principle and implementation details of the SS-TMM have beenpresented in Section 2.6.2. The developed basic lithography model and theSS-TMM will be used to predict the general trend of spectral performanceagainst design parameters under fabrication constraints.4.3.1 Impact of different phase periods on the spectralperformanceA smaller Λφ will lead to a larger spacing between side-resonances and Braggresonance [as indicated by Eq. 4.4], which will bring about a lower noise flooraround the center spectrum, as we will see later. A lower Λφ will also leadto a higher spatial apodization resolution, which is important for achievinga complicated κn(z) profile to realize a sophisticated target response. How-ever, since a smaller Λφ will also increase the grating fabrication requirementas discussed before, the resulting performance improvement may be offset bythe spectral degradation due to fabrication constraints. To study this trade-off, we characterize the previously deigned flattop filter [Fig. 4.5] developedon sawtooth phase-modulated IBGs with different Λφ. The grating corruga-68tion width ∆W is 8 nm. The implementation of the phase-modulated IBGsis performed as described before. The created grating structure is modifiedby the lithography model, and is then modeled by the SS-TMM to obtainthe spectral response. The grid size of the lithography and the minimumrealizable feature spacing/size in the lithography model are set to be 6 nmand 60 nm, respectively.Figure 4.10(a) presents the modeling results of the designed IBGs withdifferent Λφ of 12 and 8 µm over a large wavelength range of 120 nm. Side-resonances due to periodic phase modulation can be seen on the two sides ofthe Bragg resonance. As expected, the spacing between the Bragg resonanceand the side-resonances (∆λ) is larger for a smaller Λφ. Specifically, ∆λ arefound to be ∼24 and ∼35 nm for Λφ of 12 and 8 µm, respectively, whichagree well with 23.4 and 34.8 nm predicted by Eq. 4.4. The grating responsefor Λφ = 8 µm also has a significantly lower noise floor near the centerwavelength than that for Λφ = 12 µm. This is because for a smaller Λφ,the side-resonances are located farther from the Bragg resonance, and thustheir tails have less impacts on the spectrum near the Bragg wavelength.Figure 4.10(b) shows the grating responses for different Λφ of 12, 8 and 3µm over a narrower wavelength range (10 nm) for a better comparison of thespectral performance. As Λφ becomes smaller, in addition to a lower noisefloor and a higher side-lobe suppression ratio (SLSR), the in-band ripplealso becomes weaker and the response is closer to the designed flattop shape.This is because a smaller Λφ can bring about a higher spatial apodizationresolution, thus leading to a higher spectral quality of the apodized grating.Figure 4.10(c) compares the grating responses for smaller Λφ of 2.2, 1.6 and 1µm. The grating responses for these different Λφ values show similar spectralqualities and noise floors. This is because when Λφ falls within such a range,the spectral improvement due to lower Λφ is less significant and begins tobe offset by spectral degradation due to the increased grating fabricationrequirement. This can be understood by noting that ∆λ is already largeenough and hence decreasing Λφ to further increase ∆λ will have less impacton the noise floor around the Bragg wavelength. Also, Λφ is already smalland the resulting apodization spatial resolution is sufficient for modulationof the current κ(z) profile. Therefore, the spectral performance saturateswhen varying Λφ in such a range. Figure 4.10(d) compares the responses ofthe IBGs with even smaller Λφ of 0.7, 0.4 and 0.3 µm. An opposite trendis observed in this case compared with that for the larger Λφ values shownin Fig. 4.10(b): when Λφ is smaller, the grating performance is worse witha lower SLSR and higher in-band ripple. This is because the fabricationrequirement is well beyond the fabrication capability. Thus, as Λφ further69-60 -40 -20 0 20 40 60-40-30-20-100 the +1st resonance(a)  Λφ: 12 um    Λφ: 8 um  (b)the -1st resonanceReflection (dB)∆λ (nm)the +2nd resonance-4 -2 0 2 4 Λφ: 0.7 um    Λφ: 0.4 um Λφ: 0.3 um  ∆λ (nm)-4 -2 0 2 4-30-20-100Λφ: 12 um    Λφ: 8 um Λφ:3 um (e)Reflection (dB)∆λ (nm)Ideal case103 104181522 Square filterSLSR (dB) Λφ (nm)-4 -2 0 2 4(d) Λφ: 2.2 um    Λφ: 1.6 um Λφ:1 um  ∆λ (nm)11192735 Gaussian-apodzied gratingSLSR (dB)103 1045121926(c)(f)SLSR (dB) Λφ (nm) Square filter17283950 Gaussian-apodzied gratingSLSR (dB)Figure 4.10: (a) Simulated responses of flattop filters developed on sawtoothphase-modulated IBGs using different phase period (Λφ) of 12 and 8 µm;the displayed wavelength range is as large as 120 nm to include the side-resonances. (b)-(d) Comparison of simulated responses of the flattop gratingfilters with different Λφ over a narrow wavelength range. (e) Side-lobe-suppression-ratio (SLSR) of the grating response as a function of Λφ forthe flattop filter (blue, left axis) and the Gaussian-apodized grating (red,right axis). (f) SLSR as a function of Λφ for the two filters in the idealcase, i.e., without considering the fabrication limitations. The grid size ofthe lithography and the minimum feature spacing/size in the lithographymodel used for the calculations in (a)-(e) are set to be 6 nm and 60 nm,respectively.70decreases, the spectral performance improvement is far outweighed by theincreased spectral degradation due to the fabrication limitations.Figure 4.10(e) (blue, left axis) plots the SLSR of the flattop filter as afunction of Λφ, which highlights the trends shown in Figs. 4.10(b)-4.10(d).The SLSR here serves as an indicator of the spectral performance of theapodized grating. It can be seen that Λφ should be selected within thesaturation region to ensure the best performance of the apodized gratingin practical design. The SLSR versus Λφ in case of a Gaussian-apodziedsawtooth phase-modulated grating designed according to Fig. 4.4(a) witha corrugation width of 8 nm is also calculated and plotted in Fig. 4.10(e)(red, right axis). A similar overall trend is observed. However, the gratingperformance begins to decrease at a higher Λφ value than that for the flattopfilter. This is because the Gaussian apodization profile is much simplerwith a lower spatial frequency, and thus does not require a high spatialapodization resolution and thus a low Λφ as in the flattop filter. Hence,the turning point happens at a higher value of Λφ. The result suggeststhat the selected Λφ should be larger when the target response and thus therequired apodization profile are simpler. Figure 4.10(f) also shows SLSRversus Λφ for the two grating filters in the ideal case, i.e., without consideringthe fabrication limitations. For both gratings, the spectral performanceinitially increases as Λφ decreases, and then saturates at smaller Λφ values,as expected.4.3.2 Comparison of grating robustness against fabricationconstraints for different periodic phase functions-4 -2 0 2 4-30-20-1000 1000 2000 30000.00.61.21.801-6 -4 -2 0 2 4 6-30-20-1003 nm∆λ (nm)Reflection (dB)1.8 nm (b)Period No.κ (m-1×104 )(a)ΦG (pi)(c) Square   Sawtooth   Sinusoidal∆λ (nm)Reflection (dB)Figure 4.11: (a) Response of the designed 3-channel flattop filter. (b) Grat-ing strength (κ) and phase (φG) profiles required by the design. (c) Gratingresponses for different periodic phase functions in the ideal case (withoutconsidering the fabrication limitations) calculated by the CMT-based TMM.71In this section, we compare the robustness of gratings modulated bydifferent periodic phase functions against fabrication constraints. To studythis, we implement and characterize a more complicated 3-channel flattopfilter based on phase-modulated IBGs using different periodic phase func-tions. The target response is shown in Fig. 4.11(a): the bandwidth ofeach channel is ∼1.8 nm, and the channel spacing is ∼3 nm. The requiredgrating strength and phase profiles, which are calculated by using the LPA,are plotted in Fig. 4.11(b). The total number of the grating periods is3168, corresponding to a grating length of ∼1 mm. The grating structureis modified by the lithography model and is then modeled by the SS-TMM.It should be noted that in the ideal case, i.e., without any fabrication lim-itation, different periodic phase functions should lead to almost the samespectral performance of the gratings. This can be seen in Fig. 4.11(c) whichcompares the grating responses for different periodic phase functions in theideal case with the same Λφ of 0.4 µm, calculated by the CMT-based TMM.However, as different periodic phase profiles can produce largely differentgrating structures as shown before, it is expected that they should have dif-ferent robustness to fabrication constraints, which will be explored below.First, we compare the susceptibility of the grating response to a smallphase period (Λφ) for different periodic phase functions under fabricationconstraints. A very small Λφ can be required when a high apodization reso-lution is needed for achieving a highly complicated κ(z) profile to realize anadvanced spectral response, or in applications where a very large neighbour-ing resonance spacing is demanded to eliminate possible channel crosstalkdue to side-resonances of the grating devices, such as WDM systems. Weemploy a very low Λφ of 0.09 µm in the design while using relatively re-laxed fabrication constraints in the lithography model to minimize theirinfluence on the comparison: the minimum realizable feature spacing/sizeis 20 nm, and the lithography resolution is 1 nm. The modeling results ofthe phase-modulated gratings using different periodic phase functions areshown in Fig. 4.12(a). The gratings modulated by square and sawtoothphase functions exhibit comparable spectral performances and similar levelsof agreement with the design, while that modulated by a sinusoidal phasefunction shows the worst spectral performance. These results suggest thatsinusoidal phase-modulated gratings should be more susceptible to low Λφunder fabrication limitations, and thus should not be chosen when a verylow Λφ is required and used in practice.Then, we explore the robustness of the grating response against thefabrication limitation of minimum realizable feature spacing/size for differ-ent periodic phase functions. We set the minimum realizable feature spac-72-4 -2 0 2 4-20-15-10-50Norm. Reflection (dB)∆λ (nm)S q u a r e-4 -2 0 2 4-20-15-10-50 S a w t o o t h ∆λ (nm) -4 -2 0 2 4-20-15-10-50 S i n u s o i d a l ∆λ (nm)-4 -2 0 2 4-20-15-10-50(b) Litho. modeled grating      Ideal  Norm. Reflection (dB)∆λ (nm)S q u a r e  Λφ = 0.8 um; Min. feature size/spacing = 85 nm; Litho. Res. = 1 nm(a)-4 -2 0 2 4-20-15-10-50Λφ = 0.09 um; Min. feature size/spacing = 20 nm; Litho. Res. = 1 nmS a w t o o t h ∆λ (nm) -4 -2 0 2 4-20-15-10-50 S i n u s o i d a l ∆λ (nm)-4 -2 0 2 4-20-15-10-50Norm. Reflection (dB)∆λ (nm)S q u a r e-4 -2 0 2 4-20-15-10-50Λφ = 0.8 um; Min. feature size/spacing = 0 nm; Litho. Res. = 90 nm(c)∆λ (nm)S a w t o o t h-4 -2 0 2 4-20-15-10-50∆λ (nm)S i n u s o i d a lFigure 4.12: (a)-(c) are comparisons of grating responses for different peri-odic phase functions in cases of low Λφ, a large minimum realizable featuresize/spacing and a low lithography resolution, respectively; the responsesare predicted by the SS-TMM together with a computational lithographymodel to take into account the fabrication constraints.ing/size to be 85 nm in the lithography model, while selecting Λφ and thelithography resolution to be 0.8 µm and 1 nm, respectively. The modelingresults of the gratings using different periodic phase functions are plottedin Fig. 4.12(b). The response of the square phase-modulated grating inthis case shows the highest agreement with the design, whereas that of thesinusoidal phase-modulated grating again presents the worst performanceand is greatly distorted from the design. The results imply that gratingsmodulated by square phase functions should be less sensitive to minimumrealizable feature size/spacing constraint, and thus should be preferred forfabrication process with a large minimum realizable feature size/spacing.Finally, the immunity of the grating response to a low lithography res-olution for different periodic phase functions is explored. In this case, the73lithography resolution is set to be 90 nm, with Λφ and the minimum featurespacing/size chosen to be 0.8 µm and 0 nm. The calculated responses fordifferent phase functions are shown in Fig. 4.12(c). The three different grat-ings in this case show similar spectral qualities, suggesting that there is nosignificant difference between the robustness of the gratings modulated bydifferent periodic phase functions against a low lithography grid resolution.Although only several common periodic functions have been analyzedand compared here, the dependence of the grating performance on the peri-odic function used offers an extra degree of freedom to optimize the apodizedgrating performance.4.4 Experimental ResultsPhase-modulated silicon IBGs designed for different target responses andusing different periodic phase functions were fabricated and experimentallytested to demonstrate the extended periodic phase modulation apodizationtechnique. The fabrication was based on e-beam lithography, using a singleetch process on an SOI wafer with 220 nm thick silicon on a 3 µm thickburied oxide layer. A 2 µm thick silicon dioxide cladding layer was depositedon the etched sample. The silicon waveguide width was 500 nm. The gridsize of the lithography was 6 nm, and the minimum feature size/spacingwas conservatively established as a design rule of 60 nm. For all IBGs, thegrating period was 316 nm, leading to a Bragg wavelength of ∼1550 nm, andthe corrugation shape was rectangular. An OVA was employed to measurethe reflection spectra of the gratings. A time-domain filter provided by theOVA was used to eliminate the back reflections from the GCs and fiberinterfaces.4.4.1 Gaussian apodized IBGsGaussian-apodzied phase-modulated IBGs using different periodic phasefunctions and based on different phase-to-structure translation schemes werefabricated and measured. The apodization profile is the same as that shownin Fig. 4.4(a). As the current Gaussian apodization profile is simple andthus does not need a low Λφ to achieve a high spatial apodization resolu-tion, a moderate Λφ of 2.4 µm was used in these IBGs. This value of Λφ alsoensures that the side-resonances are far enough from the center wavelength[∆λ > 100 nm according to Eq. 4.4] such that they have little impact on thecenter spectrum and would not cause any problem in potential applicationssuch as filtering in WDM systems. The corrugation widths of the gratings74-10 -5 0 5 10-30-20-100  20 dB∆λ (nm)Sinusoidal19 dB-10 -5 0 5 10-30-20-100 Scheme 1Scheme 2IdealNorm. Reflection (dB) 23 dB25 dBSquare ∆λ (nm) -10 -5 0 5 10-30-20-100   21 dB26 dBSawtooth∆λ (nm)Figure 4.13: Measured responses of Gaussian-apodized phase-modulatedIBGs using different periodic phase functions and based on different struc-ture determination schemes of Schemes 1 and 2.were 8 nm. The measured results are plotted in Fig. 4.13, where the idealspectrum (grey) calculated by the CMT-based TMM is included in eachfigure for comparison. All the Gaussian-apodized phase-modulated grat-ings present high SLSRs >19 dB, validating the extended phase modulationapodization technique and the proposed structure determination schemes.The response shapes and bandwidths of all the gratings are in close agree-ment with the ideal one, suggesting little apodization phase errors resultingfrom the grating phase modulation. This is an important advantage of thephase modulation apodization method compared with other approaches suchas the duty-cycle modulation scheme. The small differences in spectral per-formance for the gratings modulated by different periodic phase functionsare attributed to fabrication un-certainties, such as width and thicknessvariations along the grating lengths.4.4.2 Single-channel flattop filtersWe also fabricated and tested single-channel flattop filters designed in Figs.4.5(a) and 4.5(b) based on phase-modulated IBGs using different periodicphase functions and different Λφ. The grating corrugation widths, decidedaccording to the maximum value of κ(z), were 8 nm. The phase-to-structureconversion is based on Scheme 1. Figure 4.14(a) shows the measured re-sponses of the gratings modulated by different periodic phase functions withthe same Λφ of 1.7 µm. Excellent flattop filtering responses are obtained forall cases, and the spectral bandwidths and shapes are in good agreement withthe design. Figure 4.14(b) compares the measured responses of the squarephase-modulated gratings with different Λφ of 1.7 and 10 µm over a broad75-10 -5 0 5 10-21-14-70 Ideal        Measured    (b)∆λ (nm)Sinusoidal-10 -5 0 5 10-21-14-70   Sawtooth∆λ (nm)-10 -5 0 5 10-21-14-70Norm. Reflection (dB)   Square∆λ (nm)-20 -10 0 10 20-21-14-70Λφ: 10 um          Λφ: 1.7 um Norm. Reflection (dB)∆λ (nm)(a)-10 -5 0 5 10-21-14-70   ∆λ (nm) -10 -5 0 5 10-21-14-70       Λφ: 0.1 um         Λφ: 1.7 um  (c)∆λ (nm)Figure 4.14: (a) Measured responses (blue) of the flattop grating filtersusing different periodic phase functions with the same Λφ of 1.7 µm, wherethe ideal responses (grey) are included for comparison. (b) and (c) arecomparisons of measured responses of flattop filters based on square phase-modulated IBGs using different Λφ.(left plot) and narrow (right plot) wavelength range. As expected, the grat-ing response for Λφ = 10 µm exhibits a significantly higher noise floor andlarger in-band ripple compared to that with Λφ = 1.7 µm. This is due to thesmaller neighbouring resonance spacing and lower spatial apodization reso-lution for larger Λφ. Figure 4.14(c) also compares the grating response usinga low Λφ of 0.1 µm to that with a Λφ of 1.7 µm. The grating response forΛφ = 0.1 µm presents a worse spectral performance than that for Λφ = 1.7µm. This is because, as discussed before, such a small Λφ of 0.1 µm will leadto a very complicated grating structure with a high spatial frequency thatcannot be accurately fabricated under the actual fabrication constraints,hence resulting in considerable spectral degradation. These comparison re-sults are consistent with the study performed above (Fig. 4.10). Table 4.1summarizes the extracted 1) 5 dB bandwidths (BW5 dB), 2) rising band-widths (Rising BW), defined as the bandwidth over which the reflectiongoes from -1 dB to -10 dB, which can be used to measure the steepness atthe edges and thus the quality of the flattop response, 3) SLSRs (measuredwithin ±10 nm from the center wavelength), and 4) standard deviations76Table 4.1: Spectral parameters of single-channel flattop filtersGrating Λφ(µm)BW5 dB(nm)RisingBW (nm)SLSR(dB)SD of In-bandRipples (dB)Ideal 4.4 0.6 24.8 0.1Square0.1 3.1 4.9 6.9 2.01.7 4.6 0.8 13.4 0.310 4.1 2.1 10.2 0.7Sawtooth 1.7 4.7 0.7 11.8 0.3Sinusoidal 1.7 4.7 1.2 11.7 0.2(SD) of in-band ripple measured within the 3.8 nm bandwidth (i.e., the de-signed passband width) for all the measured grating filters, with the idealspectral parameters included for comparison.4.4.3 3-channel flattop filters-6 -4 -2 0 2 4 6-18-12-60Measured        Ideal  Norm. Reflection (dB)∆λ (nm)Square-6 -4 -2 0 2 4 6-18-12-60   Sawtooth∆λ (nm) -6 -4 -2 0 2 4 6-18-12-60   Sinusoidal∆λ (nm)Figure 4.15: Measured responses of 3-channel flattop filters developed onphase-modulated IBGs using different periodic phase functions.This section presents the measurement results of the 3-channel flattoppassband filters designed based on Figs. 4.11(a) and 4.11(b). The gratingcorrugation widths were 10 nm, and the grating structures were mappedfrom the phase using Scheme 1. As the apodization profile in this caseis more complicated than the previous cases and thus requires a higherspatial apodization resolution, a smaller Λφ of 0.4 µm is chosen in the design.The use of such a small value of Λφ could also be used to validate theresult of the previous study [Fig. 4.12(a)] that sinusoidal phase-modulated77gratings could have a lower actual spectral performance when using a smallΛφ compared with those modulated by square and sawtooth phase functions.The measured results for the gratings modulated by different phase functionsare presented in Fig. 4.15. The spectral performances of the gratings withsquare and sawtooth phase functions are of a similar level, while that ofthe sinusoidal-phase modulated grating is slightly worse than the other two,with an appreciably lower SLSR and higher in-band ripple. This can alsobe seen in Table 4.2 which compares the average channel performances ofthe gratings. The comparison result is consistent with that predicted in Fig.4.12(a).Table 4.2: Average channel performances of the 3-channel flattop filtersGrating SLSR (dB) SD of In-band Ripple (dB)Square 14 0.30Sawtooth 16.3 0.35Sinusoidal 11.9 0.624.4.4 Flattop dispersion compensating filtersFinally, we demonstrate the capability of using the periodic phase mod-ulation apodization technique to tailor a complex (amplitude and phase)grating response. We implemented flattop dispersion-compensating filtersbased on phase-modulated IBGs. The target complex response is shown inFig. 4.16(a). The passband width is ∼4 nm, and the in-band group delaychanges linearly with a slope of ∼-2.9 ps/nm. The required grating strengthand phase profiles [κ(z) and φG(z)] are calculated through the LPA andthe results are plotted in Fig. 4.16(b). The total number of the gratingperiod is 3393, corresponding to a grating length of ∼1.07 mm. We cansee that the required grating phase profile changes continuously, instead ofalternating between 0 and pi as in those required by the previous single- andmulti-channel flattop filters [shown in Fig. 4.5(b) and Fig. 4.11(b), respec-tively]. This continuous behaviour of the grating phase indicates the changeof the grating period along the length, or the grating chirp. The currentdesign again highlights the simpler implementation of the phase modulationapodization method by noting that such a continuous phase profile requiredby the design can be directly added into the total grating phase profilefor the grating structure determination. In contrast, for other apodization78-25-20-15-10-500 1000 2000 30000.00.20.40.60.8κ (m-1×104)Period No.0 1000 2000 3000-20-15-10-50(b) φG(z)    φtol(z)   φtol(z): φκ(z) + φG(z) Phase (rad)Period No.φ G (rad)0 1000 2000 3000-4-2024(c) (d) A(z)     φκ(z) Phase (rad)Period No.(a)-4 -2 0 2 4-30-20-100Slope ≈ -2.9 ps/nm Reflection (dB)∆λ (nm)0612182430 Group delay (ps)4 nm1350 1400 1450-101-4 -2 0 2 4-15-10-50(e)Norm. Reflection (dB)∆λ (nm)Slope ≈ -3 ps/nm Square1580159016001610Group Delay (ps)-4 -2 0 2 4-15-10-50(f) SawtoothSlope ≈ -2.9 ps/nm Norm. Reflection (dB)∆λ (nm)1580159016001610Group Delay (ps)Figure 4.16: (a) Reflection (blue, left axis) and group delay (grey, right axis)responses of the designed flattop dispersion-compensating filters. (b) Grat-ing strength (blue, left axis) and phase (grey, right axis) profiles required bythe design, calculated by using the LPA. (c) Phase profile, φκ(z) (yellow),and its amplitude profile, A(z) (red). (d) Total grating phase profile, φtol(z)(yellow), which is the superposition of φκ(z) and φG(z) (grey). (e) and (f)are the measured complex responses of the grating filters using square andsawtooth phase functions, respectively.approaches, such as lateral misalignment modulations, applying such a con-tinues phase profile on the grating can be much more complex and mayneed to modulate the grating period along the length according to the firstderivative of the phase profile. The phase-modulated grating is implementedbased on the procedures described before. The phase amplitude profile A(z)is first calculated from the normalized grating strength profile κn(z) via Eq.4.14, which is then used to obtain φκ(z) using Eq. 4.15. The total phaseprofile φtol(z) is then determined by superposing φG(z) and φκ(z). Theφκ(z) and φtol(z) profiles in case of a sawtooth phase function are shown inFigs. 4.16(c) and 4.16(d), respectively, where Λφ has been selected to be 1.9µm. Finally, the grating structure is mapped from φtol(z) based on Scheme791. The OVA was used to measure the reflection and group delay responsesof the gratings. The corrugation widths of the gratings were selected to be5 nm according to the maximum value of the κ(z) profile. The designedIBGs based on square and sawtooth phase functions were fabricated, andthe measurement results are shown in Figs. 4.16(e) and 4.16(f). A flattopreflection response with a linear in-band group delay response is observedfor both cases. The fitted slopes of the in-band group delay responses forthe gratings with square and sawtooth phase functions are around -3 and-2.9 ps/nm, respectively, which are in good agreement with the design.4.5 ConclusionIn this chapter, we have extended the promising sinusoidal phase modulationapodization technique to any periodic phase function for silicon IBGs. It hasalso been shown that the apodization characteristic, physical grating struc-ture and actual apodized grating performance are related to the periodicfunction used. The extended periodic phase modulation apodization tech-nique has been demonstrated by fabricating and measuring a series of differ-ently designed phase-modulated silicon IBGs, including Gaussian-apodizedgratings, single- and multi-channel flattop filters and flattop dispersion-compensating filters, using different periodic phase functions. The workprovides an extra degree of freedom that can potentially allow us to opti-mize the apodized grating performance in a specific case by designing theperiodic function. It will also motivate future exploration of novel periodicfunctions that offer new useful apodization features.Besides, this chapter has given a general implementation process of theperiodic phase modulation apodization technique for achieving a desired re-sponse on an IBG, and explored the limiting factors of the apodization per-formance, design trade-offs and optimization, and grating robustness levelsagainst fabrication constraints for different periodic phase functions. Thestudy will aid practical implementations of the apodization technique forspectral engineering of silicon Bragg grating devices for a variety of ap-plications such as WDM systems, optical signal processing and microwavephotonics.80Chapter 5Characterization andCompensation ofApodization Phase Errors inSilicon IBGs5.1 IntroductionAn important apodization issue for silicon IBGs is that the physical waveg-uide structure modifications for controlling κ in silicon IBGs can also intro-duce unwanted phase variations or errors in the gratings for some particularapodization schemes, including lateral misalignment and grating duty-cyclemodulations. Such apodization phase errors (APE) mainly come from twosources. The first one is the average effective refractive index variationsalong the waveguide grating due to the waveguide structure modifications[54, 55, 75]. These index changes will vary the local effective period of thegrating and, thus, lead to phase variations. Also, unwanted phase changescan arise from feature position modulations involved in some apodizationschemes, such as lateral misalignment modulations and similar methods withgrating corrugation shifts [28, 41, 76]. This can be seen by noting that vari-ations of the feature displacements with respect to the period centers couldintroduce grating phase shifts [77]. APE due to these two factors can actu-ally lead to significant distortions of the complex response of an apodizedIBG, as will be shown later in this chapter. However, as yet, there has beenlittle study to characterize the impact of APE on the responses of apodizedIBGs, and to eliminate the APE to correct the distorted grating responsesto be consistent with the designs. APE is difficult to be included in con-ventional coupled-mode theory (CMT) based modeling methods, where themodeled grating is identified by the strength and phase profiles only whilethe physical grating structure is ignored [48].Although the phase modulation apodization technique proposed in Chap-81ter 4 can introduce little APE, this technique may not be used in someparticular cases. Specifically, in fabrication environments with large valuesof minimum realizable feature size/spacing (such as deep ultraviolet pho-tolithography), long phase periods (Λφ) may have to be used to implementphase-modulated gratings to make them manufacturable. The use of longΛφ will also cause undesired spectral side-resonances to be close to the cen-ter spectra of phase-modulated gratings, which would be an issue for someapplications such as filtering in WDM systems. In these cases, alternativeapodization methods that are side-resonance-free, such as lateral misalign-ment modulations, may be preferred.In this chapter, by using the SS-TMM proposed in Section 2.6.2, we firstcharacterize the impact of APE on spectral responses of apodized IBGs.We model a series of different silicon IBGs apodized by modulating lat-eral misalignment (∆L) and duty-cycle (DC) and designed with differentresponses. The modeling results show that the responses of apodized IBGsdue to the APE can be significantly distorted from the target responses,especially when elaborate spectral responses are designed. Then, to addressthis issue, we propose a methodology to compensate/eliminate the APE ofan apodized IBG to correct the distorted grating response. The original andAPE-compensated silicon IBGs designed were fabricated and measured ex-perimentally. The accuracy of the SS-TMM is examined by comparing themeasured grating spectra with those predicted by the SS-TMM, with goodagreement obtained for each case. Then, spectral corrections are demon-strated in Gaussian-apodized gratings based on ∆L- and DC-modulatedIBGs and a square-shaped filter based on a ∆L-modulated IBG. Finally,we achieve a complex (amplitude & phase) spectral correction of a photonicHilbert transformer developed on a ∆L-modulated IBG.5.2 Impact of APE on Apodized IBG ResponsesIn this section, we will perform spectral modeling of several different apodizedsilicon IBGs to investigate the impact of APE on apodized IBG spectral re-sponses, using the SS-TMM proposed in Section 2.6.2. The ∆L-modulatedGaussian-apodized IBG designed in Fig. 2.15(b) is first characterized. Re-call that in this grating, ∆L is controlled by position modulation of oneside of the grating features with the other side remaining unchanged. Thissingle-side feature position modulation scheme will be used in all of the ∆L-modulated IBGs designed and characterized in this chapter. The gratingshape is rectangular, and the corrugation width is 8 nm. We use Matlab to82-10 -5 0 5 10-50-40-30-20-100-15 -10 -5 0 5 10 15-12-8-404812-10 -5 0 5 10-50-40-30-20-100 (c)  Reflection (dB)∆λ (nm) Sampling interval = 6 nm Sampling interval = 2 nm Ideal (b)  ∆n eff (×10−3 )∆W (nm)∆             ∆W             	∆W(a)   Reflection  Reflection (dB)∆λ (nm)-8-6-4-20  TransmissionTransmission(dB)Figure 5.1: (a) Calibrated model for ∆neff -versus-∆W used in the SS-TMM for IBGs developed on 500 nm (wide) × 220 nm (high) SOI stripwaveguides. (b) Reflection and transmission spectra of the ∆L-modulatedGaussian-apodized IBG calculated by the SS-TMM using a sampling intervalof 6 nm. (c) Comparison of the calculated reflection spectra of the IBG usingdifferent sampling intervals; the black curve represents the ideal reflectionspectrum calculated via the CMT-TMM.generate the vertices of the apodized IBG structure and perform the waveg-uide sampling to obtain the ∆Ws(k) profile, which, as defined in Section2.6.2, represents the width variation of the kth segment from the unper-turbed waveguide width. Then, ∆Ws(k) is converted to ∆neff (k) basedon a calibrated model for ∆neff -versus-∆W plotted in Fig. 5.1(a), which,as described in Section 2.6.2, is obtained from eigenmode analysis with thescale of the curve decreased by a factor of ∼2.3. Figure 5.1(b) shows thereflection and transmission spectra of the ∆L-modulated Gaussian-apodizedIBG calculated by the SS-TMM using a sampling interval of 6 nm. Figure5.1(c) compares the calculated reflection spectra of the IBG using differentsampling intervals of 2 nm and 6 nm, where the ideal reflection spectrumobtained from the CMT-based TMM is also included for comparison. Thegrating spectra predicted by the SS-TMM are significantly broader than theideal one. Such a spectral broadening is caused by the APE, which leads toa chirp-like effect along the grating. The APE here, as discussed in Section2.5.3, mainly comes from the lateral shifts of single-side grating features.One can also find in Fig. 5.1(c) that the use of different sampling intervals,which can be considered as different fabrication resolutions of the gratings,also results in small differences in the spectral results. The calculated spec-trum for the larger sampling interval of 6 nm has stronger ripple on the twosides of the spectrum than that with the smaller sampling interval of 2 nm.This indicates that a high fabrication resolution would decrease the spectralripples of the IBG. Note that as the e-beam lithography writing resolution83used in the fabrication will be 6 nm, the sampling intervals have been chosento be 6 nm for all the SS-TMM calculation results shown in the below.Note that according to the analysis in Section 2.5.3, there is actually anexplicit relationship between the APE [essentially represented by ∆φ2 in Eq.2.14] and ∆L for the ∆L-modulated IBG characterized above, and thus theAPE distribution can be directly obtained from the ∆L distribution alongthe grating. Also, the APE can be eliminated if ∆L is modulated by positionmodulations of both sides of the grating features in a complementary man-ner. However, the APE could not always be explicitly obtained in variousapodization methods involving feature position modulations. Furthermore,as also discussed in Section 2.5.3, position modulations of both sides of thegrating features in a complementary manner may be more difficult to bepractically implemented for more complicated grating devices. Therefore,by choosing to implement ∆L-modulated IBGs by position modulation ofsingle-side grating features, the current studies aim to provide a generalsolution to characterize and compensate such feature position modulation-induced grating phase errors.Duty cycle = d / ʌG (a) (b) (c)dʌG )sin( DCn  0.0 0.1 0.2 0.3 0.4 0.50.00.20.40.60.81.0  n (a.u)Duty cycle-12 -8 -4 0 4 8 12-50-40-30-20-1000 200 400 600 8000.00.20.40.60.81.0 nn (a.u) Period No. SS-TMM IdealReflection (dB)(nm)0.00.10.20.30.40.5 DCDCFigure 5.2: (a) Schematic illustration of the DC modulation in an IBG(left) and the relationship between the DC and normalized grating strength(right). (b) Normalized grating strength profile (blue, left axis) and theconverted DC distribution along the grating (red, right axis). (c) Spectrumof the DC-modulated Gaussian-apodized IBG calculated by the SS-TMM(blue), and the ideal spectrum (black).Next, we use the SS-TMM model to characterize an IBG apodized viathe duty-cycle (DC) modulation [55]. This apodization technique uses thecoupling strength dependence on the DC, defined as the ratio of the cor-rugation length to the grating period, as schematically illustrated in Fig.5.2(a). The relationship between the normalized grating strength and theDC follows a sinusoidal functionκn = sin(pi ×DC) (5.1)84For this apodization technique, the main source of the APE could be ex-pected to be the average effective refractive index variations between dif-ferent grating periods with different DCs. The same Gaussian apodizationprofile as the previous IBG is used and is plotted in Fig. 5.2(b) (blue, leftaxis), where the DC distribution obtained based on Eq. 5.1 is also shown(red, right axis). The fundamental grating parameters, including the gratingcorrugation shape, ΛG and ∆W , are also the same as those of the previous∆L-modulated IBG. Figure 5.2(c) shows the SS-TMM modeling result andthe ideal spectrum. Due to the grating phase errors arising from the spatiallyvarying DC, the spectrum of the apodized IBG is considerably broadenedand distorted from the ideal one.0 500 1000 1500 2000 250001-6 -4 -2 0 2 4 6-30-20-100(a)D C  m o d u l a t i o nReflection (dB)∆ λ (nm) L  m o d u l a t i o n4 nm0.00.61.21.8  κ (m-1 ×104)(b)-6 -4 -2 0 2 4 6-25-20-15-10-50(c)Reflection (dB)∆λ (nm) Apodized IBG Ideal-6 -4 -2 0 2 4 6-25-20-15-10-50 Apodized IBG Ideal(d)Reflection (dB)∆λ (nm)φ G (pi)Period No.Figure 5.3: (a) Spectrum of the designed square filter, which has been mod-ified to be physically realizable. (b) Grating strength (upper) and phase(lower) profiles required by the designed spectral response, calculated viathe LPA. The blue curves in (c) and (d) are the SS-TMM predicted spectraof the ∆L- and DC-modulated IBGs, respectively, where the black curvesare the ideal spectra calculated by using the CMT-TMM.Having characterized the Gaussian-apodized IBGs above, we now use theSS-TMM to model more complicated IBGs designed with elaborate spectralresponses. We design a square filter on both ∆L- and DC-modulated siliconIBGs. The designed square filtering response is shown in Fig. 5.3(a), whichhas a flattop bandwidth of ∼4 nm. Note that the spectrum in Fig. 5.3(a)has been modified to be physically realizable. The required κ(z) and φG(z)calculated by using the LPA are shown as a function of the period num-85ber in Fig. 5.3(b). The total grating period number is 2486. The physicalstructures of the apodized IBGs are then created based on the κ(z) andφG(z) profiles. The corrugation widths of the IBGs are 10 nm. The gratingperiod is 316 nm, and the corrugation shapes are rectangular. The normal-ized κ(z) profile is first converted to ∆L and DC values as a function of thegrating period number for the ∆L- and DC-modulated IBGs, respectively.The phase profile φG(z) is then applied into the gratings by inserting anadditional waveguide section with a length of half the grating period in eachpi-phase shift position. By this way, the whole apodized IBG structures canbe determined, and the SS-TMM can be performed on them to calculatetheir spectral responses. The spectral results obtained from the SS-TMMfor the ∆L- and DC-modulated IBGs are shown in Figs. 5.3(c) and 5.3(d)respectively, with the ideal grating spectrum calculated via the CMT-TMMincluded in each figure for comparison. For both two apodization techniques,the spectrum of the apodized IBG is largely broadened and distorted fromthe ideal one, indicating a failure of the spectral tailoring. This shows thatthe spectral distortion due to the APE can be even larger when an elaboratespectral response is designed.0 1000 2000 3000010.00.51.01.5   κ (m-1×104 ) φ G (pi)Period No.-4 -2 0 2 4-30-20-100 (c)(b) Reflection (dB)∆λ (nm)(a) 5.8 nm-4 -2 0 2 4-30-20-100 Reflection (dB)∆λ (nm) L - m o d u l a t e d  I B G0123Phase (pi)0123Phase (pi)Figure 5.4: (a) Complex spectral response of the designed photonic Hilberttransformer, which has been modified to be physically realizable. (b) Grat-ing strength (upper) and phase (lower) profiles required by the design. (c)Complex spectral response of the ∆L-modulated IBG calculated by the SS-TMM.Finally, we investigate the impact of APE on the complex (amplitude& phase) spectral response of an apodized IBG. We design and model aphotonic Hilbert transformer based on a ∆L-modulated IBG for the investi-gation. A photonic Hilbert transformer is essentially a bandpass filter witha discrete pi-phase shift at the central wavelength in the phase response [41].The complex spectral response of the designed photonic Hilbert transformer86is shown in Fig. 5.4(a), which has been modified to be physically realizable.The phase response has been wrapped into the range (0, 2pi]. The band-width of the transformer is ∼5.8 nm. A spectral notch exists at the centralwavelength due to the pi-phase jump. The κ(z) and φG(z) profiles requiredby the design are calculated using the LPA and the results are shown inFig. 5.4(b). The total grating period number is 3053. The whole physicalstructure of the ∆L-modulated IBG is created based on the κ(z) and φG(z)profiles and the created grating structure is then modeled by the SS-TMM.The modeling result is shown in Fig. 5.4(c), where we can see that both theamplitude and phase responses are distorted due to the APE. The phaseresponse is considerably deviated from the designed staircase-like behavior.Such a distorted complex spectral response may not provide a sufficient per-formance for photonic Hilbert transforming applications. These results showthe considerable impact of APE on both the amplitude and phase responsesof an apodized IBG.5.3 APE Compensation and Spectral CorrectionIn this section, we describe our proposed methodology for the APE compen-sation and spectral correction of apodized IBGs. The whole design processof an IBG with the APE compensation included is shown in Fig. 5.5. Themain concept of the methodology is to first extract the APE distributionalong the apodized grating by using the SS-TMM or 3D-FDTD, with therelevant procedures enclosed by the red dashed line in Fig. 5.5. Then, theIBG physical structure is modified accordingly to compensate the APE andthus to correct the spectral response. Next, we will illustrate how to per-form the APE compensation for the previously designed square filter in Figs.5.3(a) and 5.3(b) developed on a ∆L-modulated IBG.5.3.1 APE distribution extractionAs mentioned before, we will first extract the APE distribution along theapodized grating. For this purpose, we create a temporary grating that isdedicated for the APE extraction. The structure of the temporary gratingis determined based on only the κn(z) profile required by the design whilediscarding the phase profile φG(z), i.e., setting the phase to be a constantalong the grating. Such a temporary IBG will be modeled by the SS-TMM,and the obtained spectral result will be synthesized to retrieve the grat-ing phase and thus the APE distribution. The reason for discarding φG(z)when creating the temporary IBG is that the use of only κn(z) is sufficient87Temporary IBG structureSS-TMM or 3D-FDTDAPE-impaired spectral responseLPAAPE distributionCorrected IBG structureFabricationCompensation phase ΔΦC  APE distribution extractionSS-TMM or 3D-FDTDTarget spectral responseComparisonLPAκn(z)κn (z) & ΦG (z) Figure 5.5: Design process of an IBG with the APE compensation included;the procedures enclosed by the red dashed line are for the extraction of theAPE distribution. LPA: layer peeling algorithm [50].88to extract the APE distribution from the spectral response. This is becausethe APE distribution along the grating is only related to the spatial distri-bution of the apodizaiton physical parameter [∆L(i) here] which is solelydetermined by the κn(z) profile. Also, if φG(z) is imposed on the temporarygrating, the grating phase profile synthesized from its spectral response willbe superimposed by both the APE and φG(z), making it difficult for theAPE extraction. Note that in cases where there is no phase modulationrequired by the design, such as Gaussian-apodized gratings, the temporarygrating will be equivalent to the designed grating.-30 -20 -10 0 10 20 30-60-40-200∆Φ'G = Φ'G(i) - Φ'G(i-1)(d)(c)(b)Reflection (dB)∆ λ (nm)T e m p o r a r y  g r a t i n g  s p e c t r u m0 500 1000 1500 2000 25000.00.20.40.60.81.0κn (a.u)Period No. κn0 500 1000 1500 2000 25000.00.20.40.60.81.0  κnκn (a.u)Period No.(a)0 500 1000 1500 2000 2500-0.06-0.04-0.020.000.020.04∆Φ' G (rad)Period No.-1258 PhaseΦ' G (rad)-101 PhaseΦ' G(rad)Figure 5.6: (a) Grating strength (blue, left axis) and phase (red, right axis)profiles used for creating the temporary IBG structure that is dedicated forthe APE extraction purpose. (b) Amplitude response of the temporary IBGcalculated by the SS-TMM. (c) Grating strength (blue, left axis) and phase(red, right axis) profiles of the temporary IBG synthesized from its complexresponse using the LPA. (d) Phase differences between neighboring periods.The amplitude response of the temporary grating calculated by the SS-TMM is shown in Fig. 5.6(b). Note that the wavelength range of thespectrum is as large as 70 nm to obtain a high spatial resolution of thespectral synthesis result using the LPA [50]. As the SS-TMM model hastaken the APE into account, it is possible to extract the APE distributionby synthesizing the calculated complex response of the temporary grating.Thus, we use the LPA on the amplitude and phase (not shown) responses of89the temporary grating to retrieve its strength and phase (φ′G) profiles. Onecan expect that if there is no phase variation due to the apodization, theretrieved phase should be a constant along the grating as designed. However,as can be seen in Fig. 5.6(c), due to the existence of the APE, the retrievedgrating phase now varies considerably along the grating. These unexpectedphase variations along the grating thus represent the APE. To facilitate thefollowing phase-to-structure translation, we calculate the phase differencebetween adjacent periods, i.e, ∆φ′G(i) = φ′G(i) − φ′G(i − 1). The obtained∆φ′G(i), shown in Fig. 5.6(d), will be regarded as the APE distribution andbe used for the APE compensation process described next.5.3.2 IBG structure corrections for APE compensationNow, with the APE distribution, ∆φ′G(i), along the apodized IBG deter-mined, we can introduce an additional phase profile along the IBG to com-pensate or eliminate the APE. The compensation phase as a function of thegrating period number, ∆φC(i), can be easily determined to be∆φC(i) = −∆φ′G(i) (5.2)The APE compensation task now becomes to translate this compensationphase profile into the modifications/corrections of the IBG physical struc-ture. Such a phase difference profile along the grating can be interpretedby either of the two physical meanings. The first one is the first derivativeof the grating phase, which corresponds to the effective spatial frequency ofthe grating index modulation profile, or the effective grating period. Thesecond physical interpretation of ∆φC(i) is discrete phase shifts betweenneighboring grating periods. These two interpretations will lead to differentphase-to-structure translation schemes, which will be elaborated on in thefollowing.For the first interpretation, the compensation phase profile, ∆φC(i), isconsidered as the variations of the effective grating periods, and thus shouldbe converted to the corrections of the local periods. The period modificationfor the ith period, ∆ΛG(i), is related with ∆φC(i) via∆ΛG(i) = −ΛG × ∆φC(i)2pi(5.3)Figure 5.7(a) schematically illustrates the modification of the period accord-ing to different values of ∆φC(i) in a uniform IBG.In the second scheme, the phase difference profile is regarded as discretephase shifts between adjacent grating periods. Such phase shifts will be90(a)ΔΦC(i) = 0 ΔΦC(i) = -p/3 ΔΦC(i) = p/3 ΔΦC(i) ΔΛG(i) ΔΦC(i) Δd(i)ΛG/2(b)d(i)ΛG(i)ΔΛG(i) Δd(i)Figure 5.7: Schematic illustrations of the correction of (a) grating periodΛG and (b) distance between adjacent grating corrugations d according todifferent values of ∆φC(i) in a uniform IBG.translated to the corrugation position shifts, or equivalently, the variationsof the distances between adjacent corrugations, denoted as ∆d(i) for thedistance modification between the (i − 1)th and ith corrugations. ∆d(i) isrelated to ∆φC(i) via [27]∆d(i) = −ΛG × ∆φC(i)2pi(5.4)Figure 5.7(b) illustrates the modification of d(i) in cases of different valuesof ∆φC(i). Note that the original value of d is half the grating period (158nm)Simulations are then performed to validate the APE compensation method-ology. We correct the apodized IBG structure based on both the ΛG andd correction schemes. Figures 5.8(a) and 5.8(c) show, respectively, the ΛGand d correction profiles of the designed square grating filter, which are cal-culated from the compensation phase profile ∆φC(i) using Eqs. 5.3 and 5.4,respectively. One may notice from Eqs. 5.3 and 5.4 and Figs. 5.8(a) and5.8(c) that ∆d(i) and ∆ΛG(i) profiles required by a certain ∆φC(i) profilewill be exactly the same. This will lead to the same total lengths of theIBGs corrected by the two different phase-to-structure mapping schemes,indicating the physical equivalence of the two schemes. The corrected IBGstructures are then modeled by the SS-TMM to obtain their new spectralresponses. Figures 5.8(b) and 5.8(d) shows the calculated spectral resultsof the ΛG- and d-corrected IBGs, respectively, where the ideal grating spec-trum (black) is also included for comparison. As can be seen, the spectral910 500 1000 1500 2000 2500-3-2-1012∆ΛG (nm)Period No.(a)-4 -2 0 2 4-25-20-15-10-50(b)  ΛG-corrected IBG Ideal   Reflection (dB)∆λ (nm)0 500 1000 1500 2000 2500-3-2-1012(c)∆d (nm)Period No. -4 -2 0 2 4-25-20-15-10-50(d)   d-corrected IBG  Ideal   Reflection (dB)∆λ (nm)Figure 5.8: (a) and (c) show the ΛG and d correction profiles, respectively,for the designed square filter based on the ∆L-modulated IBG. (b) and (d)present the SS-TMM calculated spectra of the ΛG- and d-corrected IBGs,respectively, with the ideal spectrum (black) included in each figure forcomparison.responses of both two corrected IBGs now are in excellent agreement withthe ideal one. This clearly shows that the APE has been nearly elimi-nated, validating our spectral correction methodology. Note that as periodcorrections are more straightforward and easier to implement in differentapodization schemes, this correction scheme is chosen for the validation ofour APE compensation methodology shown below.5.4 Experimental ResultsTo experimentally validate the impact of APE on apodized IBG spectral re-sponses predicted by the SS-TMM and the spectral correction methodology,we fabricated and tested the previously designed silicon IBGs. Both theoriginal and APE-compensated gratings were fabricated and characterizedfor each design. The fabrication was performed based on e-beam lithogra-phy, using a single etch process on an SOI wafer with 220 nm thick siliconon a 3 µm thick buried oxide layer. A 2 µm thick silicon dioxide claddinglayer was deposited on the etched sample. The silicon waveguide width was500 nm. The resolution grid of the lithography was 6 nm, and the minimum92feature size/spacing was conservatively established as a design rule of 60 nm.An OVA was used to measure the reflection spectra of the gratings. A time-domain filter provided by the OVA was used to eliminate the back reflectionsfrom the grating couplers, directional coupler, and fiber interfaces.5.4.1 Gaussian-apodized IBGs-6 -3 0 3 6-30-20-100(f)(e)(d)(c)(b)   Experiment SS-TMM Ideal Norm. Reflection (dB)∆λ (nm)(a) O r i g i n a l l y  d e s i g n e d  I B G0 200 400 600 800-0.4-0.20.00.20.4  ∆ΛG (nm)Period No.-6 -3 0 3 6-30-20-100 C o r r e c t e d  I B G   Experiment IdealNorm. Reflection (dB)∆λ (nm)27.3 dB-6 -3 0 3 6-30-20-100  Experiment SS-TMM Ideal O r i g i n a l l y  d e s i g n e d  I B G  Norm. Reflection (dB)∆λ (nm)0 200 400 600 800-0.8-0.6-0.4-0.20.00.20.4  ∆ΛG (nm)Period No.-6 -3 0 3 6-30-20-10016.5 dB Experiment IdealC o r r e c t e d  I B G Norm. Reflection (dB)∆λ (nm)Figure 5.9: (a)-(c) Experimental data of the ∆L-modulated Gaussain-apodized IBG. (a) Measured (blue) and SS-TMM predicted (red) spectraof the originally designed IBG. (b) ΛG-correction profile of the IBG. (c)Measured spectrum of the ΛG-corrected IBG (blue). (d)-(f) Experimentaldata of the DC-modulated Gaussain-apodized IBG. (d) Measured (blue) andSS-TMM predicted (red) spectra of the originally designed IBG. (e) ΛG-correction profile of the IBG. (f) Measured spectrum of the ΛG-correctedIBG (blue). The black curves in (a), (c), (d) and (f) are the ideal spectrafor comparison.We first show the experimental results of the Gaussian-apodized IBGsmodulated by the ∆L and DC which are designed based on Fig. 2.15(b) andFig. 5.2(b), respectively. The corrugation widths of the IBGs are 8 nm. Fig-ures 5.9(a) and 5.9(d) compare the measured (blue) and SS-TMM predicted(red) spectra for the original ∆L- and DC-modulated IBGs, respectively,with the ideal spectrum (black) included in each figure for comparison. As93can be seen, for each apodization technique, the measured spectrum of theoriginally designed IBG due to the APE is largely deviated from the idealone but is in close agreement with that predicted by the SS-TMM. The ΛG-correction profiles for the ∆L- and DC-modulated IBGs are plotted in Figs.5.9(b) and 5.9(e), respectively, which are obtained based on the proceduresdescribed in the last section. One may notice that the ΛG-corrections aremuch smaller than the resolution grid of the lithography (6 nm). These pe-riod corrections below the lithography resolution, however, can be resolvedin the actual fabrication. This can be explained if we consider the lithogra-phy as a spatial sampling process of the grating structure. Thus, these ΛGvariations, even much smaller than the sampling interval, can be still cap-tured provided that the sampling frequency is sufficiently large comparedwith the spatial frequency of the grating structure.Figures 5.9(c) and 5.9(f) show the measured spectra of the ΛG-corrected∆L- and DC-modulated IBGs, respectively, together with the ideal spec-tra (black). It can be seen that the spectra of both two corrected IBGshave been fully corrected, which now are in close agreement with the idealone, validating our APE compensation and spectral correction methodology.Furthermore, the side-lobe suppression ratios (SLSRs) are as high as 27.3dB and 16.5 dB for the ∆L- and DC-modulated IBGs, respectively. Thisindicates that the APE has been almost exactly compensated with littlephase error introduced. The lower SLSR of the corrected DC-modulatedIBG compared with that of the corrected ∆L-modulated one should be dueto the lower apodization dynamic range of the DC modulation scheme, whichprevents the achievement of the entire Gaussian apodization profile over thegrating. This can be seen by noting that realizing low κ for the DC mod-ulation scheme requires small feature spacings in the grating structure [asillustrated in the diagrams of Fig. 5.2(a)], which is limited by the minimumachievable feature spacing in the fabrication. Hence, the feature spacingsnear both the ends of the DC-modulated grating would be too small to beresolved in the fabrication, which truncates the applied apodization profileand limits the maximum achievable SLSR. Due to the larger apodizationdynamic range of the ∆L modulation method, this apodization scheme hasbeen chosen for implementing more complicated IBGs that will be presentedlater. Table 5.1 summarizes the 10 dB and 20 dB bandwidths (BW10 dB andBW20 dB) and SLSRs of the original and corrected Gaussian-apodized IBGs,where the parameters of the ideal spectrum calculated from the CMT-TMMis also included for comparison.94Table 5.1: Spectral parameters of the Gaussian-apodized IBGsBW10 dB(nm)BW20 dB(nm)SLSR (dB)Ideal 4.7 6.1 60.7Original ∆L-modulated 6.1 9.8 25Corrected ∆L-modulated 4.2 5.6 27.3Original DC-modulated 7.1 11.1 6.8Corrected DC-modulated 4.4 6.1 16.55.4.2 Square-shaped filters-6 -4 -2 0 2 4 6-20-15-10-50O r i g i n a l l y  d e s i g n e d  I B G  Experiment SS-TMM IdealNorm. Reflection (dB)∆λ (nm)0 500 1000 1500 2000 2500-3-2-1012  ∆ΛG (nm)Period No.-6 -4 -2 0 2 4 6-20-15-10-50 Experiment IdealC o r r e c t e d  I B G(c)(b) Norm. Reflection (dB)∆λ (nm)(a)Figure 5.10: Experimental data of the square filter based on the ∆L-modulated IBG. (a) Measured (blue) and SS-TMM predicted (red) spectraof the originally designed IBG. (b) ΛG-correction profile of the IBG. (c)Measured spectrum of the ΛG-corrected IBG (blue). The black curves in(a) and (c) are the ideal spectra for comparison.Now, we show the measurement results of the square grating filterswhich are designed based on Figs. 5.3(a) and 5.3(b) and developed on∆L-modulated IBGs. ∆W of the IBGs are 10 nm. Figure 5.10(a) showsthe experimental (blue) and SS-TMM predicted (red) spectra of the original∆L-modulated IBGs, with the ideal spectrum (black) included for compar-ison. Again, the measured spectrum of the originally designed IBG due tothe APE is greatly distorted from the ideal one, but shows good agreementwith that predicted by the SS-TMM. Figure 5.10(b) plots the ΛG-correctionprofile for the IBG, while Fig. 5.10(c) shows the measured spectrum ofthe ΛG-corrected IBG. After the APE compensation, the spectrum of theIBG has been corrected and now agrees well with the ideal one. This val-idates the capability of the methodology for correcting complicated IBGs95Table 5.2: Spectral parameters of the square filters based on ∆L-modulatedIBGsBW5 dB(nm)Rising BW(nm)SLSR (dB)Ideal 4.6 0.72 18.2Original grating 5.7 4.6 2Corrected grating 4.7 0.72 12.4designed with elaborate responses. Table 5.2 summarizes 5 dB bandwidths(BW5 dB), rising bandwidths and SLSRs of the original and corrected IBGs,where those of the ideal spectral response are also included for comparison.The rising bandwidth here is defined as the bandwidth over which the re-flection goes from -0.5 dB to -10 dB, and is used to measure the steepness atthe edges of the square response. A smaller rising bandwidth means sharperspectral edges and thus a higher quality of the square filtering response. Itcan be seen from the table that the corrected IBG has a bandwidth closer tothat of the ideal, a much smaller rising bandwidth and a much higher SLSRcompared with those of the originally designed IBG. These results highlightthat in addition to correcting the apodized IBG response to be closer tothe design, the APE compensation can also lead to an improved spectralperformance due to the elimination of the phase errors.5.4.3 Photonic Hilbert transformersFinally, we demonstrate the correction of the complex spectral responseof the previously designed photonic Hilbert transformer [Figs. 5.4(a) and5.4(b)] based on the ∆L-modulated IBG. ∆W of the IBG is 7 nm. Theamplitude and phase spectral responses of the IBGs were measured usingthe OVA. The linear portions of the measured phase responses shown inFigs. 5.11(a) and 5.11(c) due to the length of the testing circuit have beenremoved. Also, all the phase responses presented in Fig. 5.11 have beenwrapped into the range (0, 2pi] and offset to have the same baseline for com-parison purpose. The measured and SS-TMM predicted complex responsesof the originally designed IBG are plotted in Fig. 5.11(a). Both the mea-sured amplitude and phase responses exhibit good agreement with thosepredicted by the SS-TMM. Figure 5.11(b) shows the period correction pro-file along the IBG. Figure 5.11(c) compares the measured complex spectralresponse of the corrected IBG with the ideal response. We can see that96S11  S12S21  S22S-ParameterΔφ (d)Delay Attenuator(e)4.5 dB14.5 dBInput Output(a) (b) (c)-2 0 2-20-15-10-50  Intensity (dB)(nm)Original IBGCorrected IBGFigure 5.11: (a) Measured (blue) and SS-TMM predicted (red) amplitudeand phase responses of the photonic Hilbert transformer based on the origi-nal ∆L-modulated IBG. (b) ΛG-correction profile of the IBG. (c) Measuredcomplex response of the ΛG-corrected IBG (blue), and the ideal response(black). (d) Building block used in the numerical analysis for achieving asingle side-band filtering response. (e) Calculated transfer functions of thebuilding block when using the experimental S parameters of the original(red) and corrected (blue) IBGs.97both the amplitude and phase responses have been corrected and exhibitexcellent agreement with the ideal one, which demonstrates the capabilityof the proposed methodology for correcting complex spectral responses ofapodized IBGs.The corrected complex response of the photonic Hilbert transformer willlead to a corresponding improvement in the related overall system perfor-mance. To verify this, we calculate the transfer function of a typical buildingblock using a photonic Hilbert transformer for achieving a single side-band(SSB) filter [Fig. 5.11(d)] [62]. We use the experimental S parameters mea-sured from the IBGs in the calculation. Note that similar numerical anal-ysis has been performed in Section 3.3 [Fig. 3.7] for multichannel photonicHilbert transformers. Figure 5.11(e) shows the calculated transfer functionsof the building block when using the experimental S parameters of the orig-inal (red) and corrected (blue) IBGs. We can see that the obtained SSBfiltering response for the corrected IBG has a much higher extinction ratio(14.5 dB) compared with that of the original IBG (4.5 dB). This higher ex-tinction ratio for the corrected IBG is mainly because of the higher qualityof the grating phase response which is much closer to the desired staircase-like behaviour, thus leading to a better SSB filtering response. The resultclearly shows that the APE compensation to correct the spectral response ofan IBG will bring about an improvement in the related system performance.5.5 ConclusionIn summary, in this chapter we have characterized the impacts of apodiza-tion phase errors (APE) on the spectral responses of apodized silicon IBGs,and proposed an APE compensation methodology to correct the APE-distorted grating responses. By using the SS-TMM to model different siliconIBGs, it has been shown that the APE can greatly distort both the ampli-tude and phase responses of an apodized IBG such that the responses canbe significantly deviated from the designed ones. The APE compensationmethodology is implemented by first extracting the APE distribution ofthe apodized IBG, and then correcting the grating structure accordingly tocompensate the APE. The designed silicon IBGs were fabricated and exper-imentally characterized. The measured spectra for the originally designedIBGs are largely deviated from the designs due to the APE but exhibit goodagreement with those predicted by the SS-TMM. For the APE-compensatedIBGs, their experimental spectra are successfully corrected and agree wellwith the designs. Finally, we have achieved a complex (amplitude & phase)98spectral correction of an IBG-based photonic Hilbert transformer. For someapplications, a perfect control of the grating response shape may be notcritical and thus the APE compensation may not be essential, such as IBG-based sensors and high extinction ratio filters. However, there are manyapplications that will benefit from our work where the complex responseof a designed grating is required to be consistent with the target one forachieving the required functionality. Those include all-optical signal pro-cessors, multi-channel flattop filters in WDM networks, dispersion compen-sators/controllers in optical communications, etc.99Chapter 6Apodization ProfileAmplification of Silicon IBGs6.1 IntroductionGrating apodization is the most important part in spectral tailoring of Bragggratings, and the apodization performance will directly determine the spec-tral performance of the designed gratings. However, compared with apodiza-tion in fiber Bragg gratings (FBGs), apodization in SOI-based IBGs stillsuffers from a relatively low performance with limited resolution/precisionand dynamic range, which is due to two main factors. First, different withthat in FBGs, control of the grating strength κ in silicon IBGs is gener-ally realized by modifying the physical waveguide grating geometry, such asthe corrugation width [54]. Therefore, the resolution and precision of the κmodulation are inherently limited because of fabrication constraints, suchas limited lithography resolution [57]. This is particularly true for com-mon IBGs developed on single-mode silicon strip waveguides which havesmall cross-sectional dimensions. In addition, the high index contrast ofSOI waveguides places an inherent difficulty to achieve weak κ for siliconIBGs under fabrication limitations. This well-known issue makes it diffi-cult to achieve narrowband IBG filters which require weak grating strengths[13]. More importantly, it constrains the apodization dynamic range andthus makes it challenging to fully shape and control the complex spectralresponse of an IBG. Although various advanced apodization methods, suchas the periodic phase modulation method, have been developed to addressthe issues above, higher apodization performance is still highly needed tofurther improve the grating performance and to achieve more sophisticatedgrating devices.In this chapter, we present a novel apodization profile [κ(z)] amplifica-tion technique to overcome apodization limitations for silicon IBGs. Weshow that a target κ(z) profile of a silicon IBG can be scaled up or ampli-fied greatly by introducing lateral phase delays between the two sides of the100apodized grating. This amplification of κ(z) brings about significant im-provements in the apodization dynamic range and resolution/precision forthe given fabrication constraints while leading to the same spectral responseas that of the original κ(z). Therefore, by using this κ(z) amplification tech-nique, we can largely overcome the apodization limitations for silicon IBGsdue to fabrication constraints to facilitate their spectral tailoring applica-tions. Remarkably, we show that the proposed κ(z) amplification techniquecan also be used in promising periodic phase-modulated IBGs.Below, the basic principle of the technique is first introduced. Then, thetechnique is exploited in a corrugation width-modulated Gaussian-apodizedsilicon IBG, and subsequently employed in a periodic phase-modulated sil-icon IBG to achieve a 5-channel dispersionless flattop filter. Finally, bothoriginally-designed and κ(z)-amplified IBGs for each grating design are fab-ricated and tested, and their experimental results are presented and com-pared. Significant spectral performance improvements brought by the κ(z)amplification are demonstrated in each case.6.2 PrincipleLet us treat a laterally phase-delayed apodized IBG as being formed by twosides of the sub-gratings, which have the same grating parameters includingthe index modulation profile or apodization profile, A(z)/2, phase distribu-tion, φ(z), and period, ΛG, except for a phase delay, ∆φ, between them.The effective refractive index modulation of the overall Bragg grating thencan be expressed as the superposition of those two sub-gratings∆n(z) =Re{A (z)2exp(j(2piΛGz + φ (z)))+A (z)2exp(j(2piΛGz + φ (z) + ∆φ))} (6.1)After simplification, Eq. 6.1 reduces to∆n(z) = cos(∆φ2)A (z) Re{exp(j(2piΛGz + φ (z) +∆φ2))}(6.2)Several important features can be found in Eq. 6.2. First, the indexmodulation profile of the overall grating is scaled by a factor of cos(∆φ2 ).This means that if we apply an amplified/scaled-up κ(z) on the IBG withan amplification factor (AF) equal toAF = cos(∆φ/2)−1 (6.3)101then, such a κ(z)-amplified, laterally phase-delayed IBG will be equivalentto the normal IBG apodized by the original κ(z). This provides the basicprinciple of our κ(z)-amplification concept. Second, a phase shift of ∆φ2 isintroduced in the overall grating, which is similar to that in a misalignment-modulated grating as analyzed in Section 2.5.3. Such a phase componentmust be compensated when different AFs are used within a single grating,as will be detailed later. Last, the introduction of the lateral phase delaypreserves the original phase distribution φ (z) of the IBG. This indicates thatthe amplification concept can also be used in gratings with phase modula-tion, such as chirped gratings, and the promising periodic phase modulatedgratings (as we shall see later). Note that similar operations of laterallymisaligning IBGs have been described before [12, 47]. However, for all theprevious work, the operation was solely used as an apodization approachfor modulating the local grating strength. This is fundamentally differentfrom the key idea here to amplify the required κ(z) profile to facilitatethe subsequent apodization and spectral tailoring process regardless of theapodization method used.6.3 Implementation6.3.1 κ(z) amplification in ∆W -modulated IBGsWe first illustrate the use of the κ(z)-amplification concept in a corrugationwidth (∆W ) modulated Gaussian-apodized SOI-IBG. The implementationprocess is illustrated in Fig. 6.1(a). The ∆W modulation is the most com-mon apodization scheme for IBGs, and Gaussian apodization is often usedfor minimizing the side-lobe suppression ratio (SLSR) of the grating spec-trum. The IBG is developed on a single-mode strip SOI waveguide with across section of 500 nm (wide) × 220 nm (high), and designed for the fun-damental quasi-TE mode. It has a sinusoidal corrugation shape, a periodΛG of 316 nm (leading to a Bragg resonance of ∼1550 nm), and a totalperiod number of 800. The target Gaussian apodization profile κ(z) [bluecurve in Fig. 6.1(c), left axis] has a low central peak value of 1 × 104 m−1to achieve a narrow reflection spectrum with a 3 dB bandwidth of ∼2.6 nm.By assuming a linear relationship between κ and ∆W , the required corruga-tion width distribution along the grating length, ∆W (z), follows the sameGaussian behavior as κ(z), and has a peak value of ∼6 nm at the center.Such small ∆W variations from 0 to 6 nm are difficult to be achieved in thefabrication and are particularly limited by machine grid quantization in e-beam lithography (typically 1 to 6 nm). However, we can apply a scaled-up102(b) (c)ΔL(d)AF = 1 AF = 5 ΔW300 nm1)2cos( AFΔW(z)κ(z)Amplified apodization profile Original IBGAmplified  IBGΔLΔφAmplified ΔW profileFinal IBG(a)0.00 0.25 0.50 0.75 1.00012345Amplification facctor0 200 400 600 800012345   AF=1  AF=3  AF=5Ref.  (m-1104)Period Number0612182430W (nm)AFFigure 6.1: (a) Schematic flow illustrating the implementation process ofthe κ(z)-amplification technique in a ∆W -modulated IBG. (b) Relationshipcurve between AF and ∆φ described by Eq. 6.3. (c) Original and scaled-upGaussian κ(z)/∆W (z) profiles. (d) Center portions of the (left) original and(right) κ(z)-amplified (AF = 5) IBGs in the layout.103κ(z) and ∆W (z) profile on the grating, and then introduce a phase delay∆φ between the two sides of the sub-gratings to make the grating weaker.This will lead to the same spectral response while significantly decreasingthe grating fabrication requirements. Figures 6.1(b) and 6.1(c) show the re-lationship between the AF and ∆φ described by Eq. 6.3, and the differentlyscaled-up κ(z) (left axis) and ∆W (z) (right axis) profiles, respectively. Notethat as the amplified κ(z) profile does not represent the real κ distributionalong the grating and can be considered as a reference profile for the gratingstructure creation, it is represented as ”Ref. κ(z)” in the relevant figures[Fig. 6.1(c) and Fig. 6.3(b)]. The phase delay ∆φ will be converted to aphysical offset (∆L) between the two sub-gratings via∆L = ΛG∆φ/(2pi) (6.4)Figure 6.1(d) compares the center regions of the original and κ(z)-amplified(AF = 5) gratings, to show the largely enlarged grating dimensions for theκ(z)-amplified IBG.6.3.2 κ(z) amplification in phase-modulated IBGs-6 -4 -2 0 2 4 6-20-1000 1000 2000 30000.00.20.40.60.81.0-100-50050100012.3 nm∆λ (nm)Norm. reflection (dB) 1.7 nm (b)Period No. κn (a.u)(a)Group delay (ps) Grating phase φG (pi)Figure 6.2: (a) Amplitude and group-delay responses of the designed 5-channel dispersionless square filter. (b) Normalized grating strength andphase profiles required by the design.Now, we show how to employ the κ(z)-amplification concept in IBGsapodized by the promising periodic phase-modulation method proposed inChapter 4. This phase-based apodization technique is basically operatedby adding a periodical phase function (φκ) with a z-dependent amplitudealong the IBG to control κ. Here, sinusoidal phase functions are used for thephase modulation. We exploit the proposed κ(z)-amplification technique toachieve a 5-channel dispersionless square-shaped filter in a phase-modulatedsilicon IBG. The target amplitude and group-delay responses are shown in104Fig. 6.2(a), which have been modified to be physically realizable. Thisfiltering characteristic can be of great interest for WDM applications. The 3dB bandwidth for each passband is ∼1.7 nm, and the channel spacing is ∼2.3nm. For the phase response, the group delay is constant within the passbandof each channel, so that there is little in-band dispersion. By using the LPA,the required normalized grating strength (κn) and phase (φG) profiles canbe obtained, which are plotted in Fig. 6.2(b). The total period numberis 3577, corresponding to a total grating length of ∼1.13 mm. As seen inFig. 6.2(b), a large number of weak oscillations exist in the κn(z) profile.Precisely mapping these weak portions of κn(z) into the grating to fullycontrol the complex spectral response can be very challenging. However,in the following, we will show that these weak portions can be amplifieddrastically so that they can be more accurately applied on the grating.It is important to note first that different from the ∆W modulationmethod, for some advanced apodization techniques including the phase mod-ulation technique used here, κ is controlled by another relative grating pa-rameter, with ∆W remaining constant throughout the grating. In thesetechniques, a relative κ distribution along the grating is first achieved, whilethe maximum value of the κ distribution is ultimately decided by ∆W .Those techniques include phase, duty cycle and sidewall misalignment mod-ulation methods. For example, κ in the phase modulation technique is re-lated to the ratio of the distance between adjacent corrugations to the periodwhen using the structure determination scheme of Scheme 2 [Fig. 4.6(b)];for the duty-cycle modulation technique, κ is modulated by the ratio of thecorrugation length to the grating period [15, 55]. Consequently, the effectiveamplification of κ(z) in these cases will require an increase in the levels ofthe weak portions separately so as to improve their levels relative to thepeak, instead of scaling up the entire κ(z) profile.Therefore, for effective amplification of κ(z) for the phase-modulatedIBG here, we divide the κn(z) profile into multiple regions according to theirdifferent levels, as indicated by the dashed lines in Fig. 6.3(b). DifferentAFs are imposed on them so that the peak value of each region reaches 1.In the current proof-of-concept demonstration, the κn(z) profile is dividedinto 13 regions, and the boundaries are set at the positions where κn(z) = 0to avoid abrupt change in the profile and its possible adverse impact onthe grating. A finer division of κn(z) can allow for better amplificationsof the different weak oscillations, but will also increase the complexity ofthe grating implementation. Figure 6.3(b) shows the original and amplifiedκn(z), where the AFs for different regions are also indicated. After theamplification, the weak oscillations now become much higher, which will1050 500 1000 1500 2000 2500 3000 35000100200300 Original     Amplified  d (nm)Period No.0 500 1000 1500 2000 2500 3000 35000.00.20.40.60.81.0Period No. Original     Amplified28925431389 Ref.n (a.u)  23 5 20 4 21AF:0 500 1000 1500 2000 2500 3000 35000.00.10.20.30.40.5Period No. c ()0.00.20.40.60.81.0 (e) AF ≈ 23; Δφ ≈ 0.97π Original k(z) apodized(b)(c)(d)κ(z)Amplified apodization profile φ (z)Distance between adjacent corrugations(a)d (i)ΔφRegion n ΔLn ΔLn+1d(i-1)Primary phase-modulated IBG structure Final IBG structure(i) (ii)Region n+1 Region n Region n+1 d(i)cG   Figure 6.3: (a) Schematic flow showing the process of determining the κ(z)-amplified phase-modulated IBG structure. (b) Original and amplified κn(z)profiles, where the dashed lines indicate the divided grating regions. (c)(Upper) lateral phase delay and (lower) compensation phase distributionalong the grating. (d) d(i) profiles of the original and κ(z)-amplified IBGs.(e) Structures of (upper) the κ(z)-amplified IBG in the AF ≈ 23 region(z = 60.5 µm), and (lower) the unamplified IBG at the same position; ∆Whave been exaggerated for illustrative purposes.bring about a greatly improved apodization resolution in these regions. Theupper plot of Fig. 6.3(c) shows the lateral phase delays for different regions,which are related with AFs through Eq. 6.3.Now, recall that according to Eq. 6.2, the overall IBG due to the intro-duced lateral phase delay will have an additional phase component of ∆φ2 .This did not impact the spectral response of the previous ∆W -modulatedgrating, where the entire κ(z) profile was scaled up and thus the phase com-ponent was constant along the grating. However, for the current case, thisphase component will vary for the different regions with different ∆φ, whichwill lead to unwanted phase shifts that can distort the spectral response,and thus must be compensated. If the phase of the first region is defined asa reference (∆φ1), then, the compensation phase for the nth region, denoted106as φc-n, should beφc-n = −(∆φn2− ∆φ12), (6.5)where ∆φn is the lateral phase delay of the nth region. The lower plot ofFig. 6.3(c) plots the required compensation phase distribution φc(z). Tosummarize all the phase terms used, φκ is the sinusoidal phase function formodulating κ, φG is the grating phase required by the designed response[gray in Fig. 6.2(b)], φc is the compensation phase, and ∆φ is the lateralphase delay.We can convert the phase information of the designed κ(z)-amplifiedphase-modulated IBG into a physical grating structure. This is accom-plished by two main steps: 1) determine the primary phase-modulated IBGstructure [(i) in Fig. 6.3(a)]; and then 2) introduce lateral physical offsetsin the different grating regions according to their different ∆φ. For the firststep, the grating is modulated by a superimposed phase profileφ(z) = φκ(z) + φG(z) + φc(z). (6.6)φκ(z) = A(z) sin(2pizΛφ) is a sinusoidal phase function with a spatial pe-riod of Λφ and a z-dependent amplitude of A(z) decided by κn(z) throughA(z) = J−10(κn(z)), where J0 is the 0th-order Bessel function [12]. Λφ is se-lected to be 1.5 um here. Note that as discussed in Section 4.3.1, a smallerΛφ can bring about higher spatial apodization resolution and lower noisefloor, but at the cost of a higher fabrication requirement. The structure de-termination of the phase-modulated grating here is based on Scheme 2 shownin Fig. 4.6(b). In this structure determination scheme, the superimposedphase profile φ(z) is finally translated to the distance between neighbor-ing corrugations against the grating period number, d(i), to determine thephase-modulated grating structure.Figure 6.3(d) plots the obtained d(i) profiles for the κ(z)-amplified andunamplified IBGs. For each case, the profile exhibits oscillating behaviorsaround ΛG/2 (158 nm), and the oscillation is weaker for the region whereκn is higher. Thus, the d(i) oscillation amplitudes for the κ(z)-amplifiedIBG are significantly weaker within the κ-amplified regions than those ofthe unamplified one, as also schematically illustrated in Fig. 6.3(e) whichcompares the structures of the two IBGs at the same position (z = 60.5µm). This reduced d(i) oscillation amplitude along the κ(z)-amplified IBGwill lead to a lower spatial frequency and complexity of the grating structure,which also means a decreased fabrication requirement in terms of both thelithography resolution and minimum feature size/spacing.1076.4 Experimental ResultsThe designed κ(z)-amplified and unamplified IBGs were fabricated via e-beam lithography, based on a single etch of an SOI wafer with a 3 µm thickburied oxide layer. The resolution grid of the lithography was 6 nm, andthe minimum feature size/spacing was conservatively established as a designrule of 60 nm. Light was injected and extracted using grating couplers (GCs)and a polished polarization maintaining single-mode optical fiber array. Adirectional coupler (DC) was placed between the GCs and the IBG to directthe reflected light back. An OVA was used to measure the reflection spectraof the gratings, where a time-domain filter was used to eliminate the backreflections from the GCs, the DC, and fiber interfaces.6.4.1 Gaussian-apodized ∆W -modulated IBGs300 nmΔL ≈ 138 nm(a)(b)-6 -3 0 3 6-36-30-24-18-12-6018.6 dBReflection (dB) (nm) Ideal AF=1 AF=5Figure 6.4: (a) SEM image of the center portion of the fabricated κ(z)-amplified IBG (AF = 5). (b) Measured reflection spectra of the originaland κ(z)-amplified Gaussian-apodized ∆W -modulated IBGs, along with theideal spectrum.The measurement results of the Gaussian-apodized ∆W -modulated IBGsdesigned according to Fig. 6.1 are presented here. An SEM image of theκ(z)-amplified IBG structure with AF = 5 at the grating center is shownin Fig. 6.4(a). Figure 6.4(b) compares the measured reflection spectra ofthe original and κ(z)-amplified IBGs with AF = 5 after calibrating out theinsertion losses from other parts of the circuit, where the ideal spectrumcalculated using the CMT-TMM is also included for comparison. The spec-trum of the κ(z)-amplified IBG exhibits an excellent agreement with theideal one with a high SLSR of ∼18.6 dB. In contrast, the unamplified IBG108presents a much weaker and broader reflection than the design, togetherwith a smaller SLSR (∼9.6 dB) and noticeable spectral distortions, indicat-ing a failure of the apodization. These results are expected, since for most ofthe unamplified IBG (except for the central portion) the waveguide pertur-bation size is below the lithography resolution and is therefore unresolved.Thus, the fabricated grating length is shorter than the design, leading toa wider and weaker reflection. Also, due to its small grating dimensions,the unamplified IBG will be more sensitive to manufacturing issues, such asquantization errors [47], leading to spectral distortions. By comparison, ow-ing to the largely increased waveguide perturbation sizes and ∆W (z) profileamplitude, the κ(z)-amplified IBG is less prone to the fabrication issues andis more accurately patterned, thus providing a higher-quality spectrum thatis in good agreement with the design. Note that by using an alternativeapodization method with a higher apodization resolution here, such as theperiodic phase modulation technique, the unamplified IBG response wouldbe much closer to the ideal response. The point here, however, is to show theapodization performance improvement brought by the κ(z) amplification inthe same apodization scheme and lithography environment.6.4.2 Phase-modulated IBGs(a) 300 nmΔL ≈ 154 nmAF ≈ 23; Δφ ≈ 0.97π AlignedAF = 1; Δφ = 0 AF ≈ 3; Δφ ≈ 0.78π ΔL ≈ 123 nm(b)Figure 6.5: SEM images of the fabricated κ(z)-amplified, phase-modulatedIBG in (a) AF ≈ 23 region and (b) the boundary portion between AF = 1and AF ≈ 3 regions.The κ(z)-amplified and unamplified phase-modulated IBGs that weredesigned as the 5-channel dispersionless square filters were fabricated via thesame fabrication process as the previous Gaussian-apodized IBGs. The OVAwas used again to characterize the complex responses of the IBGs. Figure6.5 presents SEM images of the fabricated κ(z)-amplified IBG at differentpositions. The ∆W was 12 nm. Figure 6.6(a) plots the measured (left)normalized reflection and (right) group delay responses of the κ(z)-amplifiedIBG, along with the design (blue) for the comparison. The obtained spectralresponse shows an excellent agreement with the design, with low in-band109-6 -4 -2 0 2 4 6-20-15-10-50 Experiment (b)∆λ (nm) (a)  Norm. reflection (dB)Design-6 -4 -2 0 2 4 6-40-2002040 ∆λ (nm)Group delay (ps)-6 -4 -2 0 2 4 6-20-15-10-50  Norm. reflection (dB)∆λ (nm)ExperimentDesignFigure 6.6: (a) Measured (left) normalized reflection and (right) group delayresponses of the κ(z)-amplified IBG. (b) Measured reflection response of theunamplified IBG.dispersions. Such a high degree of spectral agreement suggests that the weakoscillations of the κn(z) profile after the amplification have been preciselyimposed on the IBG. The measured spectrum of the unamplified IBG isshown in Fig. 6.6(b), which presents a considerably lower performance: thepassbands are distorted away from the target square shape, with higherin-band ripples and lower SLSRs. This lower spectral performance is dueto the poor apodization performance under the fabrication constraints tofully and precisely map the original κn(z) profile into the IBG. Actually,the incorrect apodization of the weak-κ regions may act as a source of phasenoise which could in turn impair the spectral response. Table 6.1 summarizesTable 6.1: Average channel performances of the 5-channel dispersionlesssquare filters based on phase-modulated IBGs.Grating SD of in-bandripple (dB)SLSR(dB)SD of GD(ps)Original 1.6 8.4 2.6Amplified 0.6 11.2 1.2the average channel performances for the original and κ(z)-amplified phase-modulated IBGs. The standard deviation (SD) of the group delay (GD)within the 1.7 nm bandwidth (the designed 3-dB channel bandwidth) ofeach passband is extracted to compare the dispersion-free performance, andthe in-band ripples are also measured similarly. The κ(z)-amplified IBGpresents a better performance in each parameter, i.e., lower passband ripples,higher SLSRs, and a better dispersion-free performance, highlighting theapodization performance improvements coming from the κ(z) amplification.1106.5 Discussion and ConclusionThe maximum achievable AF (AFmax) is limited by different factors in dif-ferent cases. For example, AFmax for ∆W -modulated IBGs could be limitedby the maximum applicable ∆W : it has been shown in Fig. 2.7(b) that atlarge values of ∆W , κ of an IBG can saturate with increasing ∆W , whichis likely because the large ∆W makes the fundamental waveguide mode nolonger in the perturbative regime [78]. For phase-modulated IBGs, AFmaxcould be related to the minimum resolvable lateral phase delay (∆φmin) inthe fabrication via AFmax = cos((pi −∆φmin)/2)−1. If we consider lithog-raphy as a spatial sampling process of the mask layout, then, in additionto the lithography resolution, ∆φmin could also be affected by the spatialfrequency of the phase-modulated IBG structure, which could be affected byvarious factors including the complexity of κ(z), Λφ, ΛG, etc, and thus is notstraightforward to be determined. Nevertheless, the current work shows thatAFs of several tens of times are achievable for a typical phase-modulatedIBG.In conclusion, we have shown that an apodization profile to realize a spe-cific spectral response for a silicon IBG can be amplified/scaled-up greatly byan order of magnitude by introducing phase delays between two sides of theapodized grating. This amplification brings about significant improvementsin the apodization dynamic range and resolution/precision, thereby facilitat-ing the spectral tailoring of the IBGs. The concept has been first exploitedin a corrugation width-modulated, Gaussian-apodized, silicon IBGs, andthen used in a phase-modulated, silicon IBGs to achieve a 5-channel disper-sionless square-shaped filter. Significant spectral performance improvementsbrought by the amplification have been demonstrated in both cases. Theproposed κ(z)-amplification concept will improve the spectral engineeringof IBGs for many optical applications such as optical signal processing andWDM systems. The concept may also be applicable for fiber Bragg gratings,where two apodization profiles with a phase delay may be superimposed overone another through multiple ultraviolet exposures.111Chapter 7Conclusion and Future Work7.1 ConclusionIn this thesis, we have performed comprehensive studies of spectral tailoringof silicon IBGs to achieve arbitrary, complex spectral responses. These workswill aid practical designs and implementations of silicon IBGs to realize de-sired spectral responses for various applications such as optical telecommuni-cations, WDM networks, optical signal processing, and microwave photonics.The key contributions of this thesis are summarized below.1. Development of a comprehensive methodology for designing and im-plementing silicon IBGs to achieve arbitrary, sophisticated, complexspectral responses.2. Development of an efficient and reliable structure-aware grating mod-eling method based on directly synthesizing the physical structure ofthe gratings with the assistance of the transfer matrix method, namely,SS-TMM. This grating modeling method is much more efficient thanthose based on full Maxwell solver simulations, such as 3D-FDTD,while at the same time providing accurate and reliable spectral re-sults.3. First demonstrations of multichannel photonic Hilbert transformers(MPHTs) based on complex-synthesized IBGs, by using the developedIBG design methodology. MPHTs with total wavelength channels ofup to 9 and a single channel bandwidth of up to 625 GHz have beensuccessfully achieved. The work represents an important step in de-veloping a new variety of promising multichannel microwave photonicsprocessors. Such multichannel processors can offer largely improvedprocessing bandwidth and speed, and can also be integrated in cur-rently widespread WDM systems to allow for new useful multichannelprocessing applications.4. Development and demonstration of an IBG apodization techniquebased on periodic phase modulation. This phase modulation apodiza-112tion technique is particularly promising for implementing complex-synthesized IBGs for realizing sophisticated spectral responses, owingto its high apodization performance (high accuracy, resolution, anddynamic range), little phase error, and simple implementation.5. First studies of impacts of apodization phase errors (APE) on spec-tral responses of apodized silicon IBGs. The modeling and experi-mental results have shown that the responses of apodized IBGs canbe significantly distorted from the designs due to the APE, especiallywhen sophisticated spectral responses are designed. These results haverevealed the importance to compensate or eliminate the APE of anapodized IBG for a precise control of its spectral response.6. Development of an APE compensation methodology to correct thedistorted IBG spectral responses due to the APE. Spectral correctionshave been successfully achieved in Gaussian-apodized IBGs apodizedby the lateral misalignment (∆L) and duty-cycle (DC) modulations,and in a flattop IBG filter apodized by the ∆L modulation. Further-more, a complex (amplitude & phase) spectral correction of a pho-tonic Hilbert transformer developed on a ∆L modulated IBG has beenachieved.7. Development of a novel apodization profile [κ(z)] amplification tech-nique to further improve the spectral tailoring of IBGs. This κ(z) am-plification technique can bring about significant improvements in theapodization dynamic range, resolution and precision, thereby, over-coming the current apodization limitations of silicon IBGs due tofabrication constraints. The technique has been exploited in variousIBGs, and significant spectral performance improvements brought bythe κ(z) amplification have been demonstrated in each case.7.2 Future WorkAll IBGs demonstrated in this thesis were fabricated by e-beam lithogra-phy. E-beam lithography can realize very small feature size and spacing,and also allows for rapid prototyping. These features are particularly use-ful for validations of the IBG design methodology and iterative designs ofcomplex-synthesized IBGs at the current stage. However, e-beam lithogra-phy is not suitable for commercial applications. In contrast, photolithog-raphy, such as 193 nm deep-ultraviolet lithography, which is CMOS com-patible and can provide mass production capabilities, will be much more113As designedLithography modeled(a) (b)-6 -4 -2 0 2 4 6-30-24-18-12-60Norm. reflection (dB)(nm)Original (W = 6 nm)Litho. modeled (W = 20 nm)Figure 7.1: (a) Layouts of periodic phase-modulated IBG; (upper) as-designed geometry, and (lower) geometry after performing 193 nm deep-ultraviolet lithography simulations. (b) Simulated spectra of original andlithography modeled IBGs; ∆W of the IBG to be lithography modeled hasbeen set to be 20 nm, which is much larger than that of 6 nm in the originaldesign to compensate for the strong smoothing effects of photolithography.suited for commercial use of complex-synthesized IBGs [13]. Therefore, fu-ture work is expected to demonstrate the developed IBG design methodol-ogy in photolithography fabrications. Compared with e-beam lithography,photolithography, however, typically has larger minimum realizable featuresize/spacing and stronger smoothing effects. To assess spectral performanceof complex-synthesized IBGs fabricated by photolithography, initial sim-ulations are performed to predict the spectral response of a flattop filterdeveloped on a periodic phase-modulated IBG fabricated by 193 nm deep-ultraviolet lithography. The flattop filter is designed based on those shownFigs. 4.5(a) and 4.5(b). A square phase function with a phase period of2.6 µm is used for the apodization. To take into account the lithographicdistortions in the simulation, the created grating structure is first modi-fied by using a lithography model developed in [79], which is built on teststructures previously fabricated using 193 nm deep-ultraviolet lithographyby IMEA*STAR. A comparison of the as-designed and lithography modeledphase-modulated grating structures is shown in Fig. 7.1(a). The spectrumof the lithography modeled grating is then calculated by using the SS-TMM.The simulation results of the original grating and the lithography modeledgrating are presented in Fig. 7.1(b). No appreciable spectral degradationis observed for the lithography modeled grating compared with the orig-inal grating, showing that the developed design methodology can also beused for IBGs fabricated by photolithography. Note that ∆W of the IBGto be lithography-modeled has been set to be 20 nm, which is much larger114than that of 6 nm used in the original design to compensate for the strongsmoothing effects of photolithography.Figure 7.2: Schematic illustration of a grating-assisted contra-directionalcoupler (CDC).In addition, for the conventional IBGs studied in this thesis, they areessentially 2-port devices and are operated in reflection mode. Thus, anadditional directional coupler or an optical circulator is generally neededto be placed before the IBG to collect the reflected light, which increasesboth insertion loss and complexity of the circuit. Grating-assisted contra-directional couplers (CDCs), as opposed to conventional IBGs, are 4-port de-vices [28, 33, 34], as schematically illustrated in Fig. 7.2. In grating-assistedCDCs, the reflected light is dropped to another waveguide through contra-directional coupling. Such 4-port devices eliminate the need for extra direc-tional couplers or circulators, and are also much easier to be used in variousoptical networks for, for example, serving as wavelength-selective add-dropfilters and routers in WDM systems. Consequently, grating-assisted CDCsare promising devices for large-scale photonic integrated circuits. However,spectral tailoring of these devices to achieve arbitrary complex spectral re-sponses has remained relatively unexplored. Therefore, another directionof future work is to transfer the developed deign methodology of conven-tional IBGs to grating-assisted CDCs. One can expect that grating-assistedCDCs with arbitrarily tailored spectral responses, along with the advantagescoming from their 4-port operation properties, will be highly appealing formany applications including WDM networks, optical signal processing, andmicrowave photonics.Future work could also be directed toward adding electrical tuning ca-pability into IBGs to achieve reconfigurability and programmability of thespectral functionalities. This would enable a variety of reconfigurable de-vices such as all-optical and MWP signal processors, optical filters, androuters. Preliminary demonstrations of this concept have been presented115in [80, 81], where fully reconfigurable IBG-based signal processors that canperform multiple signal processing functions including temporal differentia-tion, microwave time delay, and frequency identification are achieved. Themain concept of those work is to divide a long IBG into multiple shortsub-gratings, each of which incorporates an independent lateral PN junc-tion that is connected to an individual pair of contacts for local refractiveindex tuning. In this way, the index modulation profile of the overall grat-ing can be electrically reconfigured by field programming all the bias volt-ages, thereby controlling the spectral functionality of the grating. 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