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Silicon ring resonator add-drop multiplexers Boeck, Robert 2011

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Silicon Ring Resonator Add-Drop Multiplexers  by Robert Boeck B.A.Sc., The University of British Columbia, 2009  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  Master of Applied Science in THE FACULTY OF GRADUATE STUDIES (Electrical and Computer Engineering)  The University Of British Columbia (Vancouver) October 2011 c Robert Boeck, 2011  Abstract Wavelength-division multiplexing (WDM) using silicon-on-insulator (SOI) waveguides have become an attractive area of research to decrease the footprint of optical interconnects as well as to ensure high speed data transmission. Specifically, research into using SOI ring resonator add-drop filters for WDM applications have been increasingly pursued. A ring resonator coupled on both sides by straight waveguides enables one to add (multiplex) or drop (demultiplex) wavelengths. Using series-coupled ring resonators, with each resonator having a different length, enables better spectral performance than single ring resonators. In this thesis, we have analyzed the properties of SOI strip waveguides and directional couplers. We have compared different spectral properties of single and series-coupled ring resonators and showed the advantages of using series-coupled ring resonators. SOI strip waveguide series-coupled racetrack resonators exhibiting the Vernier effect were designed by us and fabricated at a leading edge foundry. The free spectral range was 36 nm, which is comparable to the span of the optical C-band. The drop port response showed interstitial peak suppression between 9 dB and 17 dB and minimal resonance splitting.  ii  Preface I am the main author of the conference paper and Optics Express journal paper titled, “Experimental Demonstration of the Vernier Effect using Series-Coupled Racetrack Resonators”[6] and “Series-Coupled Silicon Racetrack Resonators and the Vernier Effect: Theory and Measurement,”[7] respectively. Over a span of eight months, I proposed the design of the Vernier effect device, modelled and simulated the device using analytical and numerical methods, and optimized its performance. I then created a design layout that was fabricated at IMEC. I characterized the device using an experimental set-up created by Dr. Nicolas Rouger. My supervisors, Drs. Nicolas Jaeger and Lukas Chrostowski, guided my research by providing important insights into the design of the device as well as editing drafts of the conference paper, poster, and journal paper. The complete list of publications are: 1. R. Boeck, N. A. F. Jaeger, and L. Chrostowski, “Experimental Demonstration of the Vernier Effect using Series-Coupled Racetrack Resonators, in 2010 International Conference on Optical MEMS & Nanophotonics, Sapporo, Japan, Aug. 9-12, 2010. 2. R. Boeck, N. A. Jaeger, N. Rouger, and L. Chrostowski, “Series-coupled silicon racetrack resonators and the Vernier effect: theory and measurement,” Optics Express, 18(24): 2515125157, Nov. 2010.  iii  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vii  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xi  Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xii  1  2  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1  Current State of Optical Interconnects . . . . . . . . . . . . . . .  1  1.2  Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . .  3  1.3  Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8  1.4  Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . .  11  Silicon-On-Insulator Ring Resonators . . . . . . . . . . . . . . . . .  12  2.1  Silicon-On-Insulator Strip Waveguides . . . . . . . . . . . . . . .  13  2.1.1  Effective Index and Group Index . . . . . . . . . . . . . .  18  2.1.2  Directional Waveguide Couplers . . . . . . . . . . . . . .  20  2.2  Signal Flow Graphs and Mason’s Rule . . . . . . . . . . . . . . .  27  2.3  Single Ring Resonator Add-Drop Filters . . . . . . . . . . . . . .  30  2.3.1  30  Single Ring Resonator Add-Drop Filter Transfer Functions  iv  2.3.2 2.4  Spectral Characteristics of the Single Ring Resonator AddDrop Filters . . . . . . . . . . . . . . . . . . . . . . . . .  32  Series-Coupled Ring Resonator Add-Drop Filters . . . . . . . . .  45  2.4.1  Series-Coupled Ring Resonator Add-Drop Filter Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . .  2.4.2  Spectral Characteristics of Series-Coupled Ring Resonator Add-Drop Filters . . . . . . . . . . . . . . . . . . . . . .  3  45 48  Series-Coupled Silicon Racetrack Resonators and the Vernier Effect: Theory and Measurement . . . . . . . . . . . . . . . . . . . . .  65  3.1  Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  65  3.2  Experimental Results . . . . . . . . . . . . . . . . . . . . . . . .  70  3.3  Curve-Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . .  74  Conclusion and Suggestions for Future Work . . . . . . . . . . . . .  78  4.1  Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  78  4.2  Suggestions for Future Work . . . . . . . . . . . . . . . . . . . .  79  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  82  4  Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A  Derivation of the Free Spectral Range . . . . . . . . . . . . . . . . .  B  Derivation of the Full-Width-at-Half-Maximum and the Quality  C  89  Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  91  Derivation of the Resonance Splitting Wavelengths . . . . . . . . .  94  v  List of Tables Table 1.1  Performance of commercial single-channel DWDM filters. . .  3  Table 2.1  Propagation losses versus waveguide width [8]. . . . . . . . .  15  vi  List of Figures Figure 1.1  Example of a three-channel DWDM system. . . . . . . . . .  Figure 1.2  (a) shows how a ring resonator can be used as a demultiplexer and (b) shows how it can be used as a multiplexer . . . . . . .  Figure 1.3  4  (a) Image of cascaded SOI disc resonators exhibiting the Vernier effect and (b) the experimental drop port response [32]. c Optical Society of America, 2006, adapted by permission. . . . . . . .  Figure 1.4  2  9  (a) SEM of series-coupled SOI racetrack resonators exhibiting the Vernier effect and (b) the experimental drop port response [57]. c Chinese Optics Letters, 2009, adapted by permission.  10  Figure 2.1  General structure of an SOI strip waveguide. . . . . . . . . .  14  Figure 2.2  Refractive index of (a) silicon and (b) silicon dioxide versus λ  16  Figure 2.3  Refractive index of (a) silicon and (b) silicon dioxide versus T  17  Figure 2.4  (a) ne f f versus λ and (b) ng versus λ for various waveguide widths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  19  Figure 2.5  (a) ne f f and (b) ng versus T . . . . . . . . . . . . . . . . . . .  20  Figure 2.6  Schematic of a directional coupler. . . . . . . . . . . . . . . .  21  Figure 2.7  Schematic showing the cross-section of a directional coupler. .  22  Figure 2.8  Normalized electric field (Ex) of the (a) even and (b) odd supermode for a gap distance of 200 nm . . . . . . . . . . . . .  Figure 2.9  23  Difference between the effective index of the even and of the odd supermodes versus wavelength for various waveguide widths 24  Figure 2.10 |κ|2 and |t|2 versus Lc for a gap distance of 200 nm . . . . . .  26  Figure 2.11 |κ|2 versus λ for various waveguide widths . . . . . . . . . .  26  vii  Figure 2.12 Example of a signal flow graph with two forward paths and one feedback loop (adapted from [39]). . . . . . . . . . . . .  27  Figure 2.13 Reduction method used to determine the transfer function x4 /x1 where (a) shows feedback transformation, (b) shows cascade transformation, and (c) shows the multipath transformation. .  28  Figure 2.14 (a) Schematic of ring resonator add-drop filter and (b) shows the feedback loop and forward paths. . . . . . . . . . . . . .  31  Figure 2.15 ne f f versus λ and curve-fit for a waveguide width and height of 500 nm and 220 nm, respectively . . . . . . . . . . . . . .  33  Figure 2.16 (a) shows the through port and drop port spectral characteristics of a single ring resonator add-drop filter and (b) shows a zoom in of the resonance peak of the drop port response. . . . Figure 2.17 ILdrop versus |κ1  |2  = |κ2  |2  for (a) various propagation losses  and (b) for various resonator lengths . . . . . . . . . . . . . . |2  Figure 2.18 OBRR versus |κ1 = |κ2  |2  ERthrough versus |κ1 |2 = |κ2 |2  38  for (a) various propagation losses  and (b) for various resonator lengths . . . . . . . . . . . . . . |2  36  for (a) various propagation losses  and (b) for various resonator lengths . . . . . . . . . . . . . . Figure 2.19  34  40  |2  Figure 2.20 ∆λFW HM versus |κ1 = |κ2 for (a) various propagation losses and (b) for various resonator lengths . . . . . . . . . . . . . . Figure 2.21 Q versus |κ1  |2  = |κ2  |2  42  for (a) various propagation losses and  (b) for various resonator lengths . . . . . . . . . . . . . . . .  44  Figure 2.22 (a) shows the schematic of series-coupled ring resonator adddrop filter and (b) shows the feedback loops. . . . . . . . . .  46  Figure 2.23 (a), (b), (c), and (d) show the forward paths for the seriescoupled ring resonator add-drop filter. . . . . . . . . . . . . .  47  Figure 2.24 Comparison between the drop port response of a single and series-coupled ring resonator add-drop filter . . . . . . . . . .  49  |2  Figure 2.25 Drop port response sensitivity to various values of |κ2 with a fixed value for |κ|2 . . . . . . . . . . . . . . . . . . . . . . . Figure 2.26  |κcdrop |2  versus  |κ|2  51  for (a) various propagation losses and (b)  for various resonator lengths . . . . . . . . . . . . . . . . . .  viii  52  Figure 2.27 (a) shows the resonance splitting wavelengths versus |κ2 |2 for a fixed |κ|2 = 0.001 and resonator lengths equal to 100 µm using equations 2.47 and 2.48 and the closely matched values determined graphically from the drop port response. (b) shows the resonance splitting wavelengths versus |κ|2 for a fixed |κ2 |2 = 0.3. The increasing error in using equations 2.47 and 2.48 is clearly shown for large values of |κ|2 compared to the values determined from the drop port response . . . . . . . . . . . . Figure 2.28  OBRR2ring at κcdrop  versus |κ|2  54  for (a) various propagation losses  and (b) for various resonator lengths . . . . . . . . . . . . . . Figure 2.29 Through port response sensitivity for various values of |κ2  56  |2  when |κ|2 is fixed at 0.5 . . . . . . . . . . . . . . . . . . . .  57  |2  Figure 2.30 Through port insertion loss at β L = 2πm versus |κ2 for various values of |κ|2 . . . . . . . . . . . . . . . . . . . . . . . . Figure 2.31 Comparison between  |κcthru |2  and  |κcdrop |2  versus  |κ|2 .  . . . .  59 60  Figure 2.32 Comparison of single and series-coupled ring resonator adddrop filters based on (a) the through port ER and (b) the drop port OBRR . . . . . . . . . . . . . . . . . . . . . . . . . . .  63  Figure 2.33 Comparison of the drop port response of series-coupled ring resonator add-drop filters with identical and different ring resonator lengths that have FSRs greater than the span of the C-band. 64 Figure 3.1  Theoretical drop port response of the un-optimized device illustrating the twin peaks, extended FSR, and minimum interstitial peak suppression, and (b) the main resonance peak splitting [7]. c Optical Society of America, 2010, by permission .  Figure 3.2  68  (a) Theoretical drop port response of the optimized device illustrating large interstitial peak suppression and (b) no main resonance splitting [7]. c Optical Society of America, 2010, by permission . . . . . . . . . . . . . . . . . . . . . . . . . .  Figure 3.3  69  SEM of (a) fabricated series-coupled racetrack resonators, and (b) coupling region [7]. c Optical Society of America, 2010, by permission . . . . . . . . . . . . . . . . . . . . . . . . . . ix  71  Figure 3.4  (a) Experimental and curve-fit drop port response for seriescoupled racetrack resonators, (b) shows the minimal main resonance splitting (zoom in of Figure 3.4(a)), and (c) shows the straight waveguide transmission response used for calibration. (Although minor modifications were made to the above figures, permission was obtained to reproduce figures from [7]). c Optical Society of America, 2010, by permission. . . . . .  Figure 3.5  73  (a) Main resonance intensity, (b) minimum interstitial peak suppression, and (c) main resonance splitting depth versus |κ1 |2 , |κ2 |2 , and |κ3 |2 . Purple circles indicate values used for curvefitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Figure 3.6  75  Experimental drop port response for single racetrack resonators with a radius of 6.545 µm and 4.225 µm. (Although minor modifications were made to the above figure, permission was obtained to reproduce the figure from [7]). c Optical Society of America, 2010, by permission. . . . . . . . . . . . . . . .  Figure 4.1  77  Schematic of a C-band demultiplexer using series-coupled ring resonators and on-chip Ge PIN detectors. . . . . . . . . . . .  x  81  Acknowledgments I would like to thank my supervisors, Drs. Nicolas Jaeger and Lukas Chrostowski for the help they provided when I was faced with difficult problems in my research. I am also very thankful for the warm and friendly atmosphere they have created within the research groups. In particular, I would like to thank Dr. Nicolas Jaeger for acting as a mentor to me. Special thanks to all my colleagues for their support and friendship especially, ´ Miguel Angel Guill´en Torres, Wei Shi, Dr. Behnam Faraji, Raha Vafaei, Dr. Nicolas Rouger, Xu Wang and Dr. Mark Greenberg. I would also like to thank Fr. William Ashley and Fr. Fernando Mignone for their guidance and friendship. Also, I would like to thank Raha Vafaei and Stephanie Flynn for obtaining SEM images of my Vernier effect device. I gratefully acknowledge the NSERC Canada (CGS-M) for supporting my research. In addition, I would also like to acknowledge ePIXfab at IMEC for the fabrication of my device as well as CMC Microsystems and Lumerical Solutions Inc. for supporting my research.  xi  Dedication To my mother, Ildiko, and my father, Max, whose unconditional love has given me the strength to overcome obstacles and to succeed in so many aspects and areas of my life. Specifically, the faith and courage my mother showed and the care my father provided for her during her six year battle with cancer has given me a newfound gratitude for each day, a new passion for my faith, and above all, a greater understanding and appreciation for life, commitment and love.  xii  Chapter 1  Introduction 1.1  Current State of Optical Interconnects  Optical interconnects are currently being deployed in many supercomputers such as the Roadrunner (first petaflop supercomputer). This is due to the decrease in the cost of optical fibers as well as the limitations of electrical interconnects as bitrates increase beyond 5 Gb/s [28]. It is predicted that by 2020 the first exaflop machine will be created which will, based on current technology, cost approximately $1011 and consume approximately 2 GW. Therefore, new and improved optical interconnects must be created to reduce this predicted cost and power consumption [28]. One possibility is to use on-chip silicon photonics which is expected to significantly reduce power consumption [28]. Silicon photonics would enable cost reductions due to the use of pre-exisiting CMOS manufacturing techniques [28]. Also, higher interconnect density may be achieved if silicon photonics coupled with wavelength-division multiplexing (WDM) were used [28]. Also, the demand for high speed data transmission to end-users has increased and fiber-to-the home (FTTH) has become an attractive solution to satisfy this demand [60]. As of 2008, worldwide users of FTTH was 23 million [60]. Here also, WDM silicon photonics is seen as a possible way to reduce cost and power consumption and to increase bitrate [60]. The typical link to transmit data is shown in Figure 1.1. In this case, there are three optical transmitters (OTXs) that transmit data at three different wavelengths 1  that are spaced 1.6 nm apart - a spacing of 1.6 nm is one of the spacings that is used for dense-wavelength multiplexing (DWDM) systems. The three wavelengths are multiplexed onto the same fiber and transmitted to the specified location. The optical signals at the three different wavelengths are then demultiplexed and converted to electrical signals using three optical receivers (ORXs). A few of the currently available commercial, single-channel DWDM filter performance specifications are shown in Table 1.1. These filters have been designed so that they can add (multiplex) or drop (demultiplex) wavelengths at the through port and drop port, respectively. The out-of-band rejection ratio (OBRR) is an important parameter of the drop-port which tells us the contrast between the pass-band and the stop-band [30]. The bandwidth (BW) of a filter is important because it partly determines the maximum single-channel data rate [54]. The drop port insertion loss (ILdrop ) measures the amount of loss that occurs when an optical signal is demultiplexed [30]. The extinction ratio (ER) of the through port is the value of the difference in intensity between the pass-band and the stop-band of the through port. This becomes important if we have a multi-channel filter in which an optical signal is dropped by the first filter and then another signal at the same resonance wavelength is added later on by the second filter [30]. If the ER is not large enough, the residual power at the through port may cause errors to the added optical signal [30]. OTX 1  !1  !1  !  ORX 1  fiber !2 OTX 2  MUX  ! 1+! 2+! 3  DEMUX  !2 ORX 2  !3 OTX 3  !3  Figure 1.1: Example of a three-channel DWDM system.  2  ORX 3  Table 1.1: Performance of commercial single-channel DWDM filters. OBRR (dB)  ER (dB)  0.5-dB BW (nm)  ILdrop (dB)  ≥ 40  ≥ 12  ≥ 0.5  ≤ 0.8 [1]  ≥ 35  −  ≥ 0.5  ≤ 0.6 [3]  ≥ 40  ≥ 12  ≥ 0.5  ≤ 0.9 [2]  1.2  Literature Review  Multiplexing and demultiplexing wavelengths can be achieved using a ring waveguide resonator that is coupled to two straight waveguides. Figure 1.2(a) shows how a ring resonator can be used to demultiplex wavelengths from the input port. In this example, the optical signal at the input port consists of two wavelengths, λ1 and λ2 . To separate the two wavelengths, one and only one of the wavelengths, say λ2 , should satisfy the resonance condition of the ring resonator, i.e. that the round-trip phase be equal to 2πm where m (also called the mode number) is an integer. If the resonance condition is met, constructive interference will occur [30]. Successive resonances occur at successive integer values of m and the wavelength span between two successive resonances is called the free spectral range (FSR). At resonance, each round-trip for λ2 results in constructive interference which results in a build-up of optical power within the resonator whereas a decrease in optical power at the through port occurs due to destructive interference with that portion of the light that couples to the through port, as explained below [30]. At the drop port, a certain portion of light will couple out of the resonator. In each cycle, the drop port will see an increase in optical power [30]. In contrast, the through port will initially transmit a large amount of power that will diminish its intensity as a result of destructive interference with the light coupled back to the through port after consecutive round-trips within the resonator [30]. However, the increase at the drop port and decrease at the through port will eventually reach a steadystate value at which maximum power and minimum power is achieved at the drop port and through port, respectively [30]. However, it should be mentioned that at wavelengths very near λ2 , constructive interference can still occur but maximum output power at the drop port is lower than the output power at λ2 [30]. Thus, the 3  !  same holds true for the through port, except that destructive interference occurs for wavelengths near λ2 . Figure 1.2(b) shows the case in which we wish to multiplex an optical signal with a wavelength of λ2 to the through port. In this case, we inject the optical signal at the add port and steady state is reached, in which the through !  port sees a maximum added power at λ2 . Since ring resonators can be used as a multiplexer by adding a signal to the through port and as a demultiplexer by dropping them to the drop port, this structure is commonly called the ring resonator add-drop filter.  !1+!2!  a)  !1 ! through port  input port  DEMUX  !2 !  !  !2 ! drop port  b)  !1 !  !1+!2! through port  input port  !2 !  MUX !  !2 ! add port  Figure 1.2: (a) shows how a ring resonator can be used as a demultiplexer (DEMUX) and (b) shows how it can be used as a multiplexer (MUX). 4  Researchers have been fabricating optical ring resonators since the 1980s using various types of materials. One of the first ring resonators created used three meters of single-mode optical fiber [55]. Nevertheless, here we are interested in technologies for on-chip applications. In the 1990s, advances in fabrication techniques enabled Little et al. [36] to fabricate add-drop microring resonators using 500 nm polySi strip waveguides with radii of 3 µm, 4 µm, and 5 µm [36]. The FSRs of their devices ranged from 20 nm to 30 nm [36]. However, large insertion losses were present in the through port responses [36] which is undesirable for WDM systems. Also, they do not provide experimental results for the drop port response, thus an evaluation of the quality of their devices for demultiplexing applications cannot be made. Further advances in more realistic modelling of ring resonators as well as advances in the quality of fabrication have improved the quality of sub-micrometer SOI waveguide ring resonator add-drop filters. A single microring resonator using SOI strip waveguides was fabricated with a radius of 2.5 µm, an FSR of 32 nm, propagation losses of approximately 45 dB/cm, an OBRR of approximately 30 dB, and a 3-dB bandwidth of approximately 0.8 nm [63]. Although the FSR is quite large, the propagation losses are considerable. Also, it should be mentioned at this point that microring resonators have a very small coupling region which is undesirable since the coupling of light into the ring resonator will be very sensitive to fabrication errors. Also, the modelling of such a small coupling region is quite difficult since the light is being coupled from a straight waveguide into a very small bent section of the ring resonator. Since in WDM applications more than one channel will be required to be multiplexed and/or demultiplexed, consistency between each channel is important; the use of microring resonators increases the difference in performance between each resonator due to the very small coupling region which is sensitive to fabrication errors. Thus, inserting straight waveguide sections into the ring resonator in the coupling regions will decrease the sensitivity as well as enable the ability to increase the gap distances and thereby achieve the same amount of coupling as in the microring resonator case. Increasing the gap distances also enables us to reduce fabrication proximity effects, such as the waveguide width reduction within the coupling region [18]. This type of resonator is commonly called the racetrack resonator add-drop filter. It is true that the FSR will decrease due to the increase in resonator length, however, 5  as will be shown later on, the Vernier effect can be used to extend the FSR beyond the span of the C-band using series-coupled racetrack resonators with different resonator lengths. The extended FSR is due to the fact that each resonator has its own FSR, thus, at a certain integer multiple of each FSR, the two rings will resonate at the same wavelengths causing a maximum peak in the spectrum [10]. However, at any other resonance wavelength there is a mismatch between the resonances of the rings which suppress the resonance peaks of the system [10]. Recently, multi-channel 1st order SOI ring resonator add-drop filters have been fabricated [17, 33, 52, 66]. A thermally tunable, two-channel add-drop filter using SOI racetrack resonators that have a radii of 4 µm and coupling lengths of 2 µm was fabricated [17]. It had an FSR of 19 nm, an OBRR of approximately 22-30 dB, and a 3-dB bandwidth of approximately 0.2 nm. Zheng et al. [66] have fabricated thermally tunable, four-channel add-drop filters using SOI microring resonators that have a radii of 12 µm, an FSR of 8.2 nm, an OBRR of approximately 21 dB, and a 3-dB bandwidth of approximately 0.4 nm. Unlike Dong et al. [17] and Zheng et al. [66] , Shen et al. [52] fabricated a thermally tunable, eight-channel add-drop filter using microring resonators with radii that are all different from eachother such that each resonates at a different wavelength. The radius of the first resonator was 5 µm and had an FSR of 16 nm. The radius of each additional ring was 5 µm+8 nm×(x+1) where x is the channel number. Unfortunately, only the through port response was shown in the paper, thus the bandwidth and OBRR cannot be determined. However, an important observation they made was, that even though they tried to design each resonator for a specific resonant wavelength, thermal tuning was still needed to align each resonator to the desired resonance wavelength due to fabrication errors [52]. Lee et al. [33] fabricated a nine-channel add-drop filter using SOI microring resonators with radii ranging from 5 µm to 5.08 µm and a polymer upper cladding. This multi-channel add-drop filter had a 3-dB bandwidth ranging from 0.12 nm to 0.18 nm and an OBRR of approximately 40 dB [33]. Although, in most cases mentioned above, the OBRR is quite large, the drop port response has a Lorentzian shape which is undesirable for DWDM applications. A box-like drop port response is desirable in which there is a fast transition from the pass-band to the stop-band while ensuring a large enough 3-dB bandwidth [31]. Also, a flat drop port resonance peak is desirable. However, these spectral charac6  teristics cannot be obtained using 1st order ring resonators. If we serially couple multiple ring resonators with identical resonator lengths together to form an adddrop filter, these desirable spectral characteristics can be obtained as well as an improvement in the OBRR [31]. A twenty-channel 2nd order series-coupled SOI microring resonator add-drop filter was fabricated by Dahlem et al. [15]. They gave the drop port response before and after thermally tuning eleven of these channels which illustrates how crucial it is to use thermal tuning to align each resonator’s resonant wavelength to the designed resonance wavelength. The radius of each ring was 6.735 µm for the first channel and the ring radius and width were changed for each subsequent channel [15]. The FSR was approximately 16 nm, the 3-dB bandwidth was approximately 0.16 nm, and the OBRR was approximately 44 dB [15]. Park et al. [43] have fabricated sixteen-channel and thirty-two-channel, 3rd order series-coupled SOI racetrack resonator add-drop filters with a radii of 9 µm and coupling lengths of 2 µm. The FSR was 12.8 nm and the 3-dB bandwidth was approximately 0.56 nm [43]. However, the OBRR was not given. Popovic et al. [45] fabricated the first thermally tunable 4th order series-coupled SOI microring resonator add-drop filter. The ring radii was 7 µm, the FSR was 16 nm, the OBRR was greater than 32 dB, the 3-dB bandwidth was approximately 0.53 nm, and the pass-band of the drop port was very flat [45]. Lastly, Xia et al. [62] fabricated a 5th order series-coupled SOI racetrack resonator add-drop filter. The radius of each resonator was 4 µm and the coupling length was 3.5 µm [62]. The drop port response of this device showed an FSR of 18 nm, an OBRR greater than 30 dB, a 3-dB bandwidth of approximately 2.5 nm, and a flat-top response [62]. Although using multiple ring resonators in series can improve the flattness of the drop port response, increase the OBRR (while ensuring the bandwidth is large enough), the FSR remains the same as that of the single ring resonator. Ideally, one would like to obtain an FSR greater than the span of the C-band but using single and identical series-coupled ring resonators cannot achieve this due to bending losses which limit the achievable FSR. However, using series-coupled ring resonators that have different resonator lengths exhibiting the Vernier effect can substantially increase the FSR [10].  7  1.3  Objectives  In the following section, publications, prior to our published work, on various combinations of resonators exhibiting the Vernier effect will be discussed. The goals of our work will be outlined and experimental results presented. Lastly, papers on SOI ring resonators exhibiting the Vernier effect, published after our work, are discussed. For simplicity, ring resonator refers to both microring and racetrack configurations, unless otherwise stated. Single ring resonators have a small OBRR and FSR. Series-coupled ring resonators in which the two rings have the same length enable an increase in the OBRR, enable a flatter resonance peak of the drop port response, and enable a faster transition from pass-band to stop-band (box-like drop port response) compared to single resonators. However, the FSR remains the same. As previously mentioned, one solution to the limited FSR is to use series-coupled ring resonators with each resonator having a different length. Also, this ring resonator configuration has a box-like drop port response. However, the interstitial peak suppression is a serious problem which can negate the benefits of the Vernier effect. Prior to our work, a few papers had been published on the theoretical aspects of series-coupled ring resonators that exhibit the Vernier effect, see for example [10, 21, 50]. These papers discussed losses, to some degree, but neglected losses in their simulations of Vernier effect devices consisting of two ring resonators, making them non-realistic. Others have fabricated series-coupled ring resonators [30, 41, 61, 64], however, silicon nitride [30, 61], TiO2 − SiO2 [41], and Ta2 O5 − SiO2 [64] waveguides were used to create these Vernier effect devices. Also, cascaded ring [12, 25, 57] and disc resonators [11, 32] exhibiting the Vernier effect had been fabricated using silicon [12, 25, 32, 57] and InGaAsP [11] waveguides. An example of cascaded disc resonators is shown in Figure 1.3.  8  a)  b)  b)  Figure 1.3: (a) Image of cascaded SOI disc resonators exhibiting the Vernier effect and (b) the experimental drop port response [32]. c Optical Society of America, 2006, adapted by permission. To the best of our knowledge, prior to our work there had been only one research group that had fabricated SOI series-coupled racetrack resonators exhibiting the Vernier effect as shown in Figure 1.4 [57]. However, their Vernier effect device showed weak interstitial peak suppression and large splitting of the main resonance peak at the drop port.  9  b)  a)  b)  Figure 1.4: (a) SEM of series-coupled SOI racetrack resonators exhibiting the Vernier effect and (b) the experimental drop port response [57]. c Chinese Optics Letters, 2009, adapted by permission. Hence, the goal of our project was to demonstrate that it was possible to use SOI series-coupled racetrack resonators exhibiting the Vernier effect that showed better spectral characteristics at the drop port: 1) large interstitial peak suppression; 2) minimal resonance peak splitting; 3) a free spectral range as large as the span of the C-band. We have successfully created such devices which exhibit an interstitial 10  peak suppression between 9 dB and 17 dB, minimal resonance peak splitting, and a free spectral range of 36 nm [6, 7]. Since we first published our work, numerous papers have appeared on SOI cascaded ring resonators exhibiting the Vernier effect at the drop port [13, 24, 26, 27]. However, all of these papers show minimal interstitial peak suppression and use significantly larger ring lengths, which increases the foot-print of the device compared to our devices. However, it should be noted that these papers are using the Vernier effect devices as sensors, so larger ring lengths and minimal interstitial peak suppression may be desirable. Recently, Mancinelli et al. [37] used ePIXfab at IMEC to fabricate series-coupled SOI racetrack resonators exhibiting the Vernier effect. Their through port response which showed an FSR of approximately 20 nm. Also, their device has an on-resonance extinction ratio of approximately 4.2 dB which is comparable to that of our device (5.43 dB). Unfortunately, they do not show the drop port response so a comparison between interstitial peak suppression cannot be made.  1.4  Thesis Organization  This thesis consists of four chapters. The first chapter describes the current state of optical interconnects and the possible improvements that can be made using silicon photonics and WDM. Also, a literature review on SOI ring resonators is presented. The objectives of this thesis are given and papers on series-coupled ring resonators, published before and after our work, are discussed. In Chapter 2, an explanation of the theory and modelling of SOI strip waveguides and directional couplers are presented. Also, single and series-coupled ring resonator add-drop filters are analyzed using signal flow graphs and Mason’s rule. In Chapter 3, the theory of series-coupled racetrack resonators exhibiting the Vernier effect is provided. The experimental results of a series-coupled racetrack resonator Vernier effect device as well as the responses of individual racetrack resonators with the same radii as those found in our Vernier effect device are presented. Lastly, Chapter 4 provides conclusions and suggestions for future work.  11  Chapter 2  Silicon-On-Insulator Ring Resonators In this chapter, we present an overview of the characteristics of SOI strip waveguides such as mode confinement, losses, wavelength and temperature dependency of silicon and silicon dioxide. Also, the theory and simulation results of the wavelength, width, and temperature dependency of the effective index and group index are presented. Directional couplers are an important component of ring resonators in that they determine the amount of light that is coupled into the resonator. Thus, we discuss supermode theory and how one uses it to determine the fraction of light coupled into the ring. We also present simulation results showing the wavelength and waveguide width dependency of the power coupling factor. Since many optical circuits, such a series-coupled ring resonators, are quite complex, a simple method to derive the transfer functions is necessary. Thus, we introduce signal flow graphs and Mason’s rule. We then use Mason’s rule to derive the transfer functions of the single ring resonator add-drop filter. Next, we present the derivations and simulation results of important spectral characteristics of single ring resonator add-drop filters, such as the FSR, OBRR, on-resonance insertion loss, quality factor, and through port ER. Then, we derive the transfer functions of the series-coupled ring resonator add-drop filter using Mason’s rule. Next, we present the derivations and simulation results of important spectral characteristics of series-coupled ring resonator add-drop filters, such as the drop port insertion loss, the critical field cou12  pling factors for the drop port and through port response, the two resonance splitting wavelengths, the OBRR at critical coupling, the through port insertion loss, and the through port ER. Also, a comparison of the through port ER and OBRR of single ring and series-coupled ring resonator add-drop filters is presented. Lastly, a comparison of the drop port responses of series-coupled ring resonators having identical and different resonator lengths is presented.  2.1  Silicon-On-Insulator Strip Waveguides  The ability to transmit light from one location to another using an optical waveguide relies on the choice of the materials and the geometry of the waveguide [8] which must also be chosen for the specific application and the level of signal integrity that is desired. For example, the typical SOI strip waveguide has propagation losses greater than 3 dB/cm making it only suitable for small optical devices that are less than a few centimeters in length. SOI strip waveguides with sub-micrometer dimensions minimize losses due to the high refractive index contrast between the core and the cladding. The high refractive index contrast enables strong confinement of the optical mode [8]. The general structure of an SOI strip waveguide is shown in Figure 2.1. The core of the waveguide is often made of crystalline silicon on top of a layer of silicon dioxide. The layer of silicon dioxide is used to reduce leakage losses that are dependent on the proximity of the waveguide core to the silicon substrate [9, 58].  13  Si SiO2 Si  Figure 2.1: General structure of an SOI strip waveguide. To ensure minimal substrate leakage losses, the height of the silicon dioxide layer should be at least 1 µm to ensure that the intensity of the mode exponentially decays enough before the substrate is reached [9, 51, 58]. To ensure single mode propagation and low propagation loss, the waveguide core height and width must be carefully chosen. If the waveguide width is too large, there can be multiple guided modes. However, the higher order modes leak into the silicon dioxide layer because the higher order modes have an effective index closer to that of the silicon dioxide. If the waveguide width is too small, none of the modes can be guided and all of the light radiates into the surrounding media [8]. Bogaerts [8] measured the propagation losses versus waveguide width with a fixed height of 220 nm as shown in Table 2.1. For a width of 500 nm, the losses are 2.4 dB/cm±1.6 dB/cm and the calculated the substrate leakage loss is 1.1 dB/cm when the height of the lower cladding layer is 1 µm [8]. Also, propagation losses exponentially increase with decreasing waveguide width primarily due to weaker confinement of the optical mode. The strong confinement of light in sub-micrometer SOI waveguides allows sharp bends to be created due to low bend losses which is essential for the creation of small radii ring resonators. However, below a certain bend-radius, bending losses can increases substantially [51]. For example, in an SOI strip waveguide with a width of 500 nm bending losses increase from 0.02 dB/90o to 0.071 dB/90o for a decrease in bend-radius from 3 µm to 1 µm, respectively [51].  14  Table 2.1: Propagation losses versus waveguide width [8]. Width (nm)  Propagation Loss (dB/cm)  Theoretical substrate leakage (dB/cm)  400  34 ± 1.7  3.6  440  9.5 ± 1.8  1.8  450  7.4 ± 0.9  1.7  500  2.4 ± 1.6  1.1  The difference in the refractive indices of silicon and silicon dioxide offer many benefits when creating optical waveguides as discussed above. However, to accurately model optical waveguide devices, the wavelength, λ , and temperature, T, dependence of the refractive indices must be known. Material dispersion and temperature effects of bulk silicon and silicon dioxide have been modeled using curve-fitted experimental data [35, 56]. Figure 2.2 shows the wavelength dependence of the refractive index at a specific temperature for both silicon and silicon dioxide using the models determined by Li [35] and Tan and Arndt [56]. In both cases, the slope is negative within the wavelength range of 1300 nm to 1700 nm. Also, the wavelength dependence of silicon dioxide is significantly less than that of silicon. Figure 2.3 shows the temperature dependence of the refractive index at a wavelength of 1550 nm. In both cases, the refractive index increases with an increase in temperature. However, the temperature dependence of silicon dioxide is significantly less than that of silicon.  15  3.51  a)  3.505 3.5  n  3.495 3.49 3.485 3.48 3.475 3.47 1.3  1.35  1.4  1.45 1.5 1.55 Wavelength ( µm)  1.4  1.5 Wavelength (µm)  1.6  1.65  1.7  b) 1.448 1.447  n  1.446 1.445 1.444 1.443 1.3  1.6  1.7  Figure 2.2: Refractive index of (a) silicon and (b) silicon dioxide versus λ at T = 298.15 K and 296.65 K, respectively (data from [35, 56]).  16  3.496  a)  3.494 3.492 3.49  n  3.488 3.486 3.484 3.482 3.48 3.478 3.476  280 290 300 310 320 330 340 350 360 370 Temperature (K)  1.4448 b) 1.4447 1.4446  n  1.4445 1.4444 1.4443 1.4442 1.4441 1.444 300  310  320  330 340 350 Temperature (K)  360  370  Figure 2.3: Refractive index of (a) silicon and (b) silicon dioxide versus T at λ = 1550 nm (data from [35, 56]).  17  2.1.1  Effective Index and Group Index  The propagation constant of the confined mode is proportional to the effective index of the mode and inversely proportional to the wavelength [65]. Since the silicon and silicon dioxide refractive indices are temperature and wavelength dependent (material dispersion) and the optical mode is dependent on the height and width of the waveguide as well as the wavelength, the effective index will be dependent on the temperature, wavelength, height and width of the waveguide [48]. Thus, the total dispersion (dne f f /dλ ) of a guided mode is the summation of the material dispersion and the waveguide dispersion, where the waveguide dispersion represents the variation of the effective index with wavelength regardless of the refractive index dependency on wavelength [65]. For a waveguide height of 220 nm, the strip waveguide becomes single mode for TE polarization when the waveguide width is between 550 nm and 285 nm in the wavelength range around 1550 nm [8]. Total dispersion is larger in high index contrast waveguides as compared to low index contrast waveguides [18]. Therefore, the effective index tends to be smaller than the group index [18]. The group index is especially important when trying to determine the free spectral range of a ring resonator add-drop filter. The group index, ng , is defined by, ng (λ , T, w, h) = ne f f (λ , T, w, h) − λ  dne f f (λ , T, w, h) dλ  (2.1)  where ne f f is the effective index, T is temperature, and w and h are the width and height of the core of the waveguide, respectively [47]. In order to obtain the effective index and group index, one can use a 2D FD mode solver [20]. Since the temperature dependency of the silicon dioxide layer is negligible, the temperature is fixed at 296.65 K in the silicon dioxide model. Also, the waveguide height is fixed at 220 nm in all simulations. Figure 2.4 shows the wavelength and width dependencies of the effective index and group index. The total dispersion is negative and is slightly dependent on the wavelength. Thus, the group index increases with an increase in wavelength as shown in Figure 2.4(b). The effective index decreases with an increase in wavelength for two main reasons: 1) the refractive index of the material decreases with wavelength and 2) the confinement of the optical mode becomes less as the wavelength increases [65]. The effective index increases with 18  an increase in waveguide width whereas the group index decreases because the optical mode confinement increases with an increase in waveguide width. Figure 2.5 shows that the effective and group indices increase when there is an increase in temperature.  a)  w = 450 nm w = 500 nm w = 550 nm w = 600 nm  2.55 2.5  neff  2.45 2.4 2.35 2.3 2.25 1.5  1.52  1.54 1.56 Wavelength ( µm)  4.55 b)  1.58  1.6  w = 450 nm w = 500 nm w = 550 nm w = 600 nm  4.5 4.45  ng  4.4 4.35 4.3 4.25 4.2 4.15 1.5  1.52  1.54 1.56 Wavelength ( µm)  1.58  1.6  Figure 2.4: (a) ne f f versus λ and (b) ng versus λ at T = 298.15 K for waveguide widths of 450 nm, 500 nm, 550, and 600 nm.  19  2.41  a)  neff  2.405  2.4  2.395  2.39 280  300  320 340 Temperature (K)  360  300  320 340 Temperature (K)  360  b) 4.375  ng  4.37 4.365 4.36 4.355 4.35  280  Figure 2.5: (a) ne f f and (b) ng versus T for waveguide width of 500 nm at λ = 1550 nm.  2.1.2  Directional Waveguide Couplers  Directional waveguide couplers typically consist of two waveguides that are placed in close proximity to one another to allow a certain portion of the light injected into one of the waveguides to be coupled to the other waveguide [22], as shown in Fig-  20  ure 2.6. The transfer of light between two waveguides is an essential function for numerous optical devices, such as the ring resonator add-drop filter. The operation of ring resonator add-drop filters depends critically on the amount of light that is coupled into the ring.  y  a -j!  x  b z=0  z  z=Lc  Lc  Figure 2.6: Schematic of a directional coupler. For two identical, parallel waveguides, the transfer of light from one waveguide to the other can be represented by the superposition of the even and odd supermode electric fields, EeUe (x, y)e− jβe z and EoUo (x, y)e− jβo z , respectively, of the coupled waveguide structure, d d d Ea − , 0, z = EeUe (− , 0)e− jβe z + EoUo (− , 0)e− jβo z 2 2 2 Eb  d d d , 0, z = EeUe ( , 0)e− jβe z + EoUo ( , 0)e− jβo z 2 2 2  (2.2)  (2.3)  ! where Ea and Eb are the electric fields of waveguide a and b, respectively, Ee and  Eo are the magnitudes of the even and odd supermode fields, respectively, Ue (x, y) and Uo (x, y) are their respective normalized transverse mode shapes, and βe and βo are their respective supermode propagation constants. The coordinates of the even and odd supermodes are defined in the cross-sectional view of the coupled waveguide structure that is shown in Figure 2.7. The coordinates (x, y) = (− d2 , 0) and (x, y) = ( d2 , 0) correspond to the centers of waveguide a and b, respectively.  21  y waveguide a  d Si -  d 2  waveguide b Si  gap distance SiO2  !  x  d 2  Figure 2.7: Schematic showing the cross-section of a directional coupler. The coupling length, Lπ , at which 100% power transfer occurs, is the length of the coupling region over which the relative difference in the phases of the even and odd supermodes becomes π. This can be expressed by, Lπ =  π λ π = = (βe − βo ) βdi f f 2(nee f f − noe f f )  (2.4)  where nee f f and noe f f are the even and odd supermode effective indices [42] that can be calculated using a 2D FD mode solver [20, 48]. For simplicity, all simulations from this point onwards will be for a fixed waveguide width and height of 500 nm and 220 nm, respectively. Figure 2.8 shows the positive x-component of the electric fields (normalized to the peak value) of the even and odd supermodes for a gap distance of 200 nm and a wavelength of 1550 nm. Figure 2.9 shows the difference between the effective indices of the even and odd supermodes versus wavelength for gap distances of 200 nm, 300 nm, 400 nm, and 500 nm. As expected, the difference between nee f f and noe f f decreases with increasing gap distance. Also, nee f f is greater than noe f f since the odd mode is less confined. Once the waveguides are sufficiently far apart, the difference between nee f f and noe f f becomes effectively zero, which corresponds to an infinite Lπ . When Lπ is infinite, there is no coupling of light.  22  a) 2  !5  air  !10  y ( µm)  1.5  !15 !20  Si  1  !25 !30  0.5  !35 SiO2  !40  0.2 0.4 0.6 0.8 1 x ( µm)  b) 2  !5  air  !10  y ( µm)  1.5  1  !15 !20  Si  !25 !30  0.5  !35 SiO  !40  2  0.2 0.4 0.6 0.8 1 x ( µm)  Figure 2.8: Normalized electric field (Ex) (in dB) of the (a) even and (b) odd supermode for λ = 1550 nm, waveguide width = 500 nm, gap distance = 200 nm, and T = 298.15 K.  23  0.02 0.018 0.016  neeff  noeff  0.014  gap distance = 200 nm gap distance = 300 nm gap distance = 400 nm gap distance = 500 nm  0.012 0.01 0.008 0.006 0.004 0.002 1.5  1.52  1.54 1.56 Wavelength ( µm)  1.58  1.6  Figure 2.9: Difference between the effective index of the even and the odd supermodes versus wavelength for gap distances of 200 nm, 300 nm, 400 nm, and 500 nm. To determine the field coupling factor, κ, and field transmission factor, t, equation 2.2 and equation 2.3 are solved for the following initial conditions, d Ea − , 0, 0 = ao , Eb 2  d , 0, 0 = 0. 2  (2.5)  EeUe (− d2 , 0) and EoUo (− d2 , 0) are determined to be equal to ao /2. EeUe ( d2 , 0) and EoUo ( d2 , 0) are determined to be equal to ao /2 and −ao /2, respectively. After some basic manipulations of equation 2.2 and equation 2.3, κ and t are described by the following equations, Eb  d 2,  0, Lc  ao  = − jκ = − j sin  βdi f f πLc Lc e(− jβavg Lc ) = − j sin 2 2Lπ  e(− jβavg Lc ) (2.6)  Ea − d2 , ao  0, Lc  = t = cos  βdi f f πLc Lc e(− jβavg Lc ) = cos 2 2Lπ  e(− jβavg Lc ) (2.7)  where Lc is the length of the coupling region and βavg is the average propagation constant of the even and odd supermodes ((βe +βo )/2). Equations 2.6 and 2.7 are  24  similar to those found in Cusmai et al. [14]. The j in front of the κ in equation 2.6 represents a 90o relative difference in the phases of the outputs of the two waveguides, a and b, which is important for ring resonator add-drop filters. At resonance, each consecutive round-trip of the optical signal within the ring will constructively interfere with the input optical signal. However, the input light is shifted by 90o with respect to the light at the through port. Also, at resonance, each consecutive round-trip of the optical signal within the ring will destructively interfere with the input optical signal at the through port since the light that is coupled back to the through port from the ring resonator will have an additional relative phase change of 90o which will result in the two optical signals being 180o out of phase and, therefore, result in destructive interference. If Lc = Lπ , total transfer of light from one waveguide to the other is achieved. If we assume no losses in the coupling region, we can write, |t|2 + |κ|2 = 1  (2.8)  and |t|2 and |κ|2 are the power transmission factor and power coupling factor, respectively [14]. Figure 2.10 shows the power coupling factor and power transmission factor versus coupling length for a fixed gap distance of 200 nm. At a coupling distance n×52.4 µm, where n is an odd integer, total transfer of light occurs. If the gap distance decreases, the length at which total power transfer occurs will decrease. Figure 2.11 shows the gap distance and wavelength dependency of the power coupling factor for a fixed coupling length of 15 µm. The power coupling factor increases with an increase in wavelength due to the mode being less confined. Also, the power coupling factor increases when the gap distance decreases.  25  1  | |2 |t|2  Relative Power  0.8  0.6  0.4  0.2 L 0  0  20  40  60 Lc ( µm)  80  100  Figure 2.10: |κ|2 and |t|2 versus Lc at T = 298.15 K, gap distance = 200 nm, and λ = 1550 nm.  0.25  | |2  0.2 gap distance = 200 nm gap distance = 300 nm gap distance = 400 nm gap distance = 500 nm  0.15 0.1 0.05 0 1.5  1.52  1.54 1.56 Wavelength ( µm)  1.58  1.6  Figure 2.11: |κ|2 versus λ at T = 298.15 K, Lc = 15 µm, and gap distances of 200 nm, 300 nm, 400 nm, and 500 nm.  26  2.2  Signal Flow Graphs and Mason’s Rule  Signal flow graphs have been extensively used to graphically represent the dependency of the variables in a set of linear equations [23, 38, 39, 53]. An example of a set of linear equations is represented by [39], ax1 + ex3 = x2 ,  (2.9)  bx2 = x3 ,  (2.10)  dx1 + cx3 = x4 .  (2.11)  Equations 2.9 - 2.11 can be expressed in a graphical form by inspection as shown in Figure 2.12. First, we should define some terms. A node is a connection between two different branches [39]. A source node and a sink node have only outgoing and incoming branches, respectively [39]. In Figure 2.12, the nodes are x1 , x2 , !  x3 , and x4 where x1 and x4 are a source and sink node, respectively. A path is a succession of branches in the direction of propagation, and a forward path is a path from a source to a sink in which every node is only encountered once [39]. A feedback loop encounters the same node once per cycle, beginning and ending at the same node [39]. A forward path gain and feedback loop gain are the product of all of the branches within the forward path and feedback loop, respectively [39]. In Figure 2.12, the two forward path gains are abc and d between nodes x1 and x4 .  e source  x1  a  b x2  c x3  sink  x4  d Figure 2.12: Example of a signal flow graph with two forward paths and one feedback loop (adapted from [39]).  27  To determine the transfer function x4 /x1 , the signal flow graph can be reduced to a single branch between x1 and x4 with a forward path gain representing the transfer function as shown in Figure 2.13(c). To find the transfer function for Figure 2.12, three transformations can be applied: feedback, cascade, and multipath !  transformations [38]. Figure 2.13(a) shows the feedback transformation between nodes x2 and x3 . Figures 2.13(b) and (c) show the final two reduction steps using a cascade transformation and a multipath transformation, respectively. Thus, the transfer function, x4 /x1 is [39], x4 abc abc + d(1 − be) =d+ = . x1 1 − be 1 − be  (2.12)  !  a)  x1  b/(1-be)  a x2  c x3  x4  d !  b)  x1  abc/(1-be)  x4  d c)  x1  d+abc/(1-be)  x4  Figure 2.13: Reduction method used to determine the transfer function x4 /x1 where (a) shows feedback transformation, (b) shows cascade transformation, and (c) shows the multipath transformation.  28  In the 1950’s Samuel J. Mason derived a simplified method called Mason’s rule to determine the transfer function of linear systems using signal flow graphs without explicitly using the reduction technique previously mentioned [39]. Mason’s rule states that between any source and sink, the transfer function is given by, G=  ∑ki=1 Gi ∆i ∆  (2.13)  where Gi is the gain of the ith forward path, ∆ is the determinant of the entire signal flow graph and is given by, ∆ = 1 − ∑ Pm1 + ∑ Pm2 − ∑ Pm3 + ... m  m  (2.14)  m  where Pmr is the gain product of the mth possible combination of r non-touching loops [39]. Two loops are said to be non-touching if they do not share any nodes [39]. ∆i is the value of ∆ after removing loops touching the ith forward path and is called the co-factor [39]. If no non-touching loops remain, ∆i = 1 [39]. The proof of equations 2.13 and 2.14 can be found in numerous papers [23, 39, 40]. To show the equivalence of Mason’s rule to the reduction method, the example shown in Figure 2.12 will be solved using Mason’s rule. To determine the transfer function using equation 2.13, the following five steps should be performed [53]: 1) find all the forward path gains; 2) find all the loop gains; 3) determine the co-factor for the corresponding forward path; 4) determine the determinant of the entire system; and 5) substitute steps 1 - 4 into equation 2.13. The following provides an example of using Mason’s rule to determine the transfer function for the signal flow graph shown in Figure 2.12. There are two forward path gains given by, G1 = abc, G2 = d.  (2.15)  There is only one feedback loop which has a loop gain given by, P11 = be.  (2.16)  Since there are two forward paths, there are two co-factors as shown in equa-  29  tion 2.17. The first co-factor is equal to 1 since the only feedback loop touches this forward path. The second co-factor is equal to 1 − be since the feedback loop does not touch the branch with gain d. ∆1 = 1, ∆2 = 1 − be  (2.17)  The determinant for the entire graph is given by, ∆ = 1 − be.  (2.18)  Therefore, substituting equations 2.15 - 2.18 into equation 2.13 yields the transfer function of the system shown in Figure 2.12 [39], G=  x4 abc + d(1 − be) = x1 1 − be  (2.19)  (which is equivalent to the transfer function given by equation 2.12).  2.3 2.3.1  Single Ring Resonator Add-Drop Filters Single Ring Resonator Add-Drop Filter Transfer Functions  Mason’s rule has been used to determine the transfer functions of complex optical circuits in a straight-forward way compared to other methods such as the transfer matrix method [5, 10, 30]. Since we will be looking at the Vernier effect which uses series-coupled resonators, Mason’s rule will enable us to determine the transfer functions in a simple step-by-step process without the need for matrix manipulations. The following shows the detailed derivation of the through port and drop port transfer functions of a single ring resonator add-drop filter using Mason’s rule [10, 30]. Figure 2.14 shows the schematic of a typical ring resonator add-drop filter as well as the feedback loop and the three forward paths.  30  a) Ein  t1  Ethrough  -j! 1  e(-" L-j# L)/2  e(-" L-j# L)/2  -j! 2  Edrop  t2 Eadd G3  b) !  G2  Ein  Ethrough !  PP1111  Edrop  Eadd G1  Figure 2.14: (a) Schematic of ring resonator add-drop filter and (b) shows the feedback loop and forward paths. From Figure 2.14(b), it is clear that there are three forward paths with the corresponding gains given by [10], G1 = −κ1 κ2 X 1/2 , G2 = −κ12t2 X, G3 = t1  (2.20)  where X = exp(−αL − jβ L), L is the total length of the ring resonator, κ1 and κ2 are the symmetric (real) point field coupling factors to the bus waveguides, t1 and t2 are the field transmission factors, α [/m] is the field loss coefficient, and 31  β =2πne f f (λ )/λ . Since there is only one ring resonator, there is only one loop gain given by [10], P11 = t1t2 X.  (2.21)  The co-factors for the corresponding forward paths are [10], ∆1 = 1, ∆2 = 1, ∆3 = 1 − t1t2 X.  (2.22)  The determinant of the add-drop filter is found by using equation 2.14 and equation 2.21 [10], ∆ = 1 − t1t2 X.  (2.23)  Thus, the add-drop filter transfer functions for the through-port and drop-port can be determined by substituting equations 2.20 - 2.23 into equation 2.13 [10, 30], Gthrough =  G2 ∆2 + G3 ∆3 t1 − t2 X = , ∆ 1 − t1 t2 X  Gdrop =  2.3.2 1 To  G1 ∆1 −κ1 κ2 X 1/2 = . ∆ 1 − t1 t2 X  (2.24)  (2.25)  Spectral Characteristics of the Single Ring Resonator Add-Drop Filters  determine the through port and drop port intensity responses of a single ring  ∗ resonator add-drop filter, Gthrough Gthrough and Gdrop G∗drop must be determined as  shown in equations 2.26 and 2.27. ∗ Gthrough Gthrough = Ithrough =  t12 + t22 e−2αL − 2t1t2 e−αL cos(β L) 1 + t12t22 e−2αL − 2t1t2 e−αL cos(β L)  (2.26)  Gdrop G∗drop = Idrop =  κ12 κ22 e−αL 1 + t12t22 e−2αL − 2t1t2 e−αL cos(β L)  (2.27)  The simulation results performed here are based on similar parameter variations as used by Klein [30]. However, Klein [30] used silicon nitride waveguides 1 Equations  2.26, 2.27, 2.29, 2.30, 2.32, 2.33, 2.34 were derived by us and are consistent to those derived by Klein [30]  32  whereas we have chosen to use silicon waveguides. Figure 2.16 shows an example of a single ring resonator add-drop filter intensity response of the through port and drop port using the curve-fitted effective index equation shown in Figure 2.15. The total length of the resonator is 100 µm, the propagation loss is 3 dB/cm and the power coupling factors are both 0.05. The most important spectral characteristics are shown in Figure 2.16(a) and (b) which are the FSR, the extinction ratio of the through port (ERthrough ), the OBRR, the on-resonance drop port insertion loss (ILdrop ), the resonance wavelength (λr ), and the full-width-at-half-maximum bandwidth (∆λFW HM ). 2.55  2.5  neff(!) = ! 0.0012692! + 4.3614  n  eff  2.45  2.4  2.35  2.3 1500  1520  1540 1560 Wavelength (nm)  1580  1600  Figure 2.15: ne f f versus λ (red) and curve-fit (blue/dash) for a waveguide width and height of 500 nm and 220 nm, respectively.  33  FSR 0 a)  Insertion Loss (dB)  !5 !10  ERthrough  OBRR  !15 !20 !25 !30 1534 1536 1538 1540 1542 1544 1546 1548 Wavelength (nm) 0 b)  ILdrop  Insertion Loss (dB)  !0.5 !1 !1.5 !2 !2.5 !3 "!FWHM  !3.5 1541.5  !r 1541.58 1541.54 Wavelength (nm)  1541.62  Figure 2.16: (a) shows the through port and drop port spectral characteristics of a single ring resonator add-drop filter and (b) shows a zoom in of the resonance peak of the drop port response.  34  If we assume a first order dispersion approximation (as shown in Figure 2.15), ng will be constant. Therefore, the free spectral range can be defined by, FSR = λr+1 − λr =  λr λr+1 ng L  (2.28)  where λr+1 is the resonance wavelength adjacent to λr and λr+1 > λr , which is the same as found in [30, 47] except that the authors assumed that λr λr+1 ≈λr2 which does not hold when the FSR is large as compared to the wavelength. The derivation of equation 2.28 can be found in Appendix A. The drop port on-resonance insertion loss, ILdrop , is determined by finding the max , which occurs when the round-trip phase, β L, maximum drop port intensity, Idrop max is given by, is 2πm. Thus, Idrop  max Idrop =  κ12 κ22 e−αL (1 − t1t2 e−αL )2  ,  (2.29)  and ILdrop is given by, max ILdrop = −10log10 (Idrop ).  (2.30)  Figure 2.17(a) shows the sensitivity of ILdrop to a change in propagation loss and power coupling factor for a fixed resonator length of 100 µm. As the power coupling factor increases, the insertion loss decreases exponentially. Also, an increase in the propagation loss increases the insertion loss. However, the insertion loss becomes sufficiently small regardless of the propagation losses when the power coupling factor is above about 0.2. Figure 2.17(b) shows the sensitivity of the ILdrop to a change in the resonator length and power coupling factor for a fixed propagation loss of 3 dB/cm. The sensitivity to a change in the length of the resonator is similar to a change in the propagation losses, since as the length of the resonator increases, the insertion loss increases due to the increase in the round-trip loss.  35  30  a)  loss = 20 dB/cm loss = 10 dB/cm loss = 3 dB/cm  25  ILdrop (dB)  20 15 10 5 0 10  20  3  10  2  2  2  | 1| = | 2|  10  1  b)  L = 50 µm L = 100 µm L = 200 µm  15 ILdrop (dB)  0  10  10  5  0 10  3  10  2  2  2  | 1| = | 2|  10  1  0  10  Figure 2.17: ILdrop versus |κ1 |2 = |κ2 |2 for (a) propagation losses of 3 dB/cm, 10 dB/cm, and 20 dB/cm with a fixed resonator length of 100 µm and (b) for resonator lengths of 50 µm, 100 µm, and 200 µm with a fixed propagation loss of 3 dB/cm.  36  The out-of-band rejection ratio, OBRR, for the single ring resonator add-drop filter is determined by finding the maximum and minimum drop port intensities that occur when the round-trip phase, β L, is 2πm and π+2πm, respectively. Thus, min is given by, Idrop min Idrop =  κ12 κ22 e−αL  .  (2.31)  (1 + t1t2 e−αL )2 . (1 − t1t2 e−αL )2  (2.32)  (1 + t1t2 e−αL )2  The OBRR is given by, OBRR = 10log10  max Idrop min Idrop  = 10log10  Figure 2.18(a) and (b) show the OBRR versus power coupling factor for a variation in the propagation losses and total length of the resonator, respectively. In both cases, as the power coupling factor decreases, the OBRR increases. Also, as the power coupling factor decreases, the sensitivity to a change in propagation loss and total length of the resonator increases. We also see that an increase in the propagation loss and total length of the resonator will decrease the OBRR for small power coupling factors. A large OBRR is desirable but the sensitivity to fabrication errors will increase as one decreases the power coupling factors which will cause discrepancies between the design of the device and the fabricated device. Thus, there is a trade-off between designing for a large OBRR and ensuring minimal discrepancy between experimental and theoretical OBRR values.  37  55  a)  loss = 3 dB/cm loss = 10 dB/cm loss = 20 dB/cm  50  OBRR (dB)  45 40 35 30 25 20 15 10 10  3  10  2  2  | | =| | 1  60  2  10  2  b)  L = 50 µm L = 100 µm L = 200 µm  50 OBRR (dB)  1  40 30 20 10 10  3  10  2  2  2  | 1| =| 2|  10  1  Figure 2.18: OBRR versus |κ1 |2 = |κ2 |2 for (a) propagation losses of 3 dB/cm, 10 dB/cm, and 20 dB/cm with a fixed resonator length of 100 µm and (b) for resonator lengths of 50 µm, 100 µm, and 200 µm with a fixed propagation loss of 3 dB/cm.  38  The through port extinction ratio, ERthrough , for the single ring resonator adddrop filter is determined by finding the maximum and minimum through port intensities that occur when the round-trip phase, β L, is π+2πm and 2πm, respectively. max min Thus, Ithrough and Ithrough are,  max Ithrough =  t1 + t2 e−αL  =  (2.33)  (1 + t1t2 e−αL )2  and min Ithrough  2  t1 − t2 e−αL  2  (1 − t1t2 e−αL )2  .  (2.34)  Therefore, ERthrough is given by,  ERthrough = 10log10  max Ithrough min Ithrough  = 10log10  t1 + t2 e−αL 1 − t1t2 e−αL (1 + t1t2 e−αL ) (t1 − t2 e−αL )  2  . (2.35)  Figure 2.19(a) and (b) show the ERthrough versus power coupling factor for a variation in propagation losses and total length of the resonator, respectively. As the power coupling factor decreases, ERthrough decreases. Also, ERthrough decreases with a increase in propagation loss and total length of the resonator which is a similar trend to those of the ILdrop and the OBRR. However, we can see that another trade-off occurs since it is desirable to have both the OBRR and ERthrough be large; if we decrease the power coupling factors, the OBRR will increase but the ERthrough will decrease.  39  50  a)  loss = 3 dB/cm loss = 10 dB/cm loss = 20 dB/cm  ERthrough (dB)  40 30 20 10 0 10  50  3  10  2  2  | 1| =| 2|  b)  10  1  L = 50 µm L = 100 µm L = 200 µm  45 40 ERthrough (dB)  2  35 30 25 20 15 10 5 10  3  10  2  2  2  | 1| =| 2|  10  1  Figure 2.19: ERthrough versus |κ1 |2 = |κ2 |2 for (a) propagation losses of 3 dB/cm, 10 dB/cm, and 20 dB/cm with a fixed resonator length of 100 µm and (b) for resonator lengths of 50 µm, 100 µm, and 200 µm with a fixed propagation loss of 3 dB/cm.  40  The full-width-at-half-max bandwidth, ∆λFW HM is given by,  ∆λFW HM =  λr2 arccos  −(1−4t1 t2 e−αL +t12 t22 e−2αL ) 2t1 t2 e−αL  πng L  (2.36)  which is derived in Appendix B. Figure 2.20(a) and (b) show the ∆λFW HM versus power coupling factor for a variation in propagation losses and total length of the resonator, respectively. As the power coupling factor decreases, ∆λFW HM decreases. Also, as the propagation losses increase, ∆λFW HM increases. However, the opposite is true for an increase in the total length of the resonator due to ∆λFW HM being inversely proportional to the total length of the resonator.  41  a)  loss = 3 dB/cm loss = 10 dB/cm loss = 20 dB/cm  0  FWHM  (nm)  10  1  10  2  10  10  3  10  2  2  | 1| =| 2|  b)  FWHM  (nm)  10  10  10  1  L = 50 µm L = 100 µm L = 200 µm  0  10  2  1  2  10  3  10  2  2  2  | 1| =| 2|  10  1  Figure 2.20: ∆λFW HM versus |κ1 |2 = |κ2 |2 for (a) propagation losses of 3 dB/cm, 10 dB/cm, and 20 dB/cm with a fixed resonator length of 100 µm and (b) for resonator lengths of 50 µm, 100 µm, and 200 µm with a fixed propagation loss of 3 dB/cm.  42  The quality factor, Q, is defined by, Q=  λr ∆λFW HM  πng L  = λr arccos  −(1−4t1 t2 e−αL +t12 t22 e−2αL ) 2t1 t2 e−αL  (2.37)  which is in accordance with the findings of Vorckel et al. [59] except that we have taken into account the wavelength dependency of the effective index. The derivation of equation 2.37 can be found in Appendix B. Figure 2.21(a) and (b) show Q versus power coupling factor for a variation in the propagation losses and the total length of the resonator, respectively. As the power coupling factor decreases, Q increases. Also, as the propagation losses increase, Q decreases. However, the opposite trend is true for an increase in the total length of the resonator due to Q being proportional to the total length of the resonator.  43  a)  loss = 3 dB/cm loss = 10 dB/cm loss = 20 dB/cm  5  Q  10  4  10  10  3  10  2  2  2  | 1| =| 2|  10  1  b)  L = 50 µm L = 100 µm L = 200 µm  5  Q  10  4  10  3  10  10  3  10  2  2  2  | 1| =| 2|  10  1  Figure 2.21: Q versus |κ1 |2 = |κ2 |2 for (a) propagation losses of 3 dB/cm, 10 dB/cm, and 20 dB/cm with a fixed resonator length of 100 µm and (b) for resonator lengths of 50 µm, 100 µm, and 200 µm with a fixed propagation loss of 3 dB/cm.  44  2.4  Series-Coupled Ring Resonator Add-Drop Filters  2.4.1  Series-Coupled Ring Resonator Add-Drop Filter Transfer Functions  The following shows the detailed derivation of the through port and drop port transfer functions of a series-coupled ring add-drop filter obtained using Mason’s rule [10, 30]. Figure 2.22 shows the schematic of a typical series-coupled ring resonator add-drop filter as well as the three feedback loops. The feedback loop, P31 , is a unique feature of series-coupled rings which can cause resonance splitting when the two rings are brought close together [10, 16, 34, 46, 49]. Figure 2.23 shows the four forward paths. The gains of these forward paths are described by [10], 1/2  1/2  G1 = t1 , G2 = jκ1 κ2 κ3 Xa Xb , G3 = κ12 κ22t3 Xa Xb , G4 = −t2 κ12 Xa  (2.38)  where Xn = exp(−αn Ln − jβn Ln ), La and Lb are the total lengths of the first and second ring, respectively, κ1 and κ3 are the symmetric (real) point field coupling factors to the bus waveguides, κ2 is the (real) point inter-ring field coupling factor, t1,2,3 are the field transmission factors, αa and αb are the total field loss coefficients for the first and second ring, respectively, and βa and βb are the propagation constants of the first and second ring, again respectively.  45  a)  t1  Ein  Ethrough  -j! 1  e(-" aLa - j# aLa)/2  e(-" aLa - j# aLa)/2 t2 -j! 2  e(-" bLb - j# bLb)/2  e(-" bLb - j# bLb)/2 t3 -j! 3  Eadd  Edrop  b) !  Ein  Ethrough  !  PP1111  !  P31 P31  ! 21 PP21  Eadd  Edrop  Figure 2.22: (a) shows the schematic of series-coupled ring resonator adddrop filter and (b) shows the feedback loops.  46  !  a)  GG1 1  !  b)  Ein  !  Ethrough  Ein  Ethrough  G2  !  !  P31  Eadd !  Eadd  Edrop  c)  Edrop !G 2  d) !  Ein  P31  Ethrough  Ein  Ethrough  G 3 !  !G G44  G3  !  Eadd  !  P31  Edrop  P31  Eadd  Edrop  Figure 2.23: (a), (b), (c), and (d) show the forward paths for the seriescoupled ring resonator add-drop filter. Since there are two ring resonators coupled in series, there are four loop gains: two loop gains corresponding to the individual resonators, one loop gain corresponding to the gain product of two non-touching loops, and an extra loop gain due to the coupling of the two resonators that forms a figure eight loop gain [10], P11 = t1t2 Xa , P21 = t2t3 Xb , P31 = −t1t3 κ22 Xa Xb , P12 = t1t22t3 Xa Xb .  47  (2.39)  The co-factors for the corresponding forward paths are [10], ∆1 = 1 − t1t2 Xa − t2t3 Xb + t1t3 Xa Xb , ∆2 = 1, ∆3 = 1, ∆4 = 1 − t2t3 Xb .  (2.40)  The determinant of the series-coupled ring resonator is found by using equation 2.14 and equation 2.39 [10], ∆ = 1 − t1t2 Xa − t2t3 Xb + t1t3 Xa Xb .  (2.41)  Thus, the series-coupled ring resonator add-drop filter transfer functions for the through port and drop port can be determined by substituting equations 2.38 - 2.41 into equation 2.13 [10], Gthrough =  G1 ∆1 + G3 ∆3 + G4 ∆4 t1 − t2 Xa − t1t2t3 Xb + t3 Xa Xb = , ∆ 1 − t1t2 Xa − t2t3 Xb + t1t3 Xa Xb 1/2  Gdrop =  2.4.2 2  (2.42)  1/2  jκ1 κ2 κ3 Xa Xb G2 ∆2 = . ∆ 1 − t1t2 Xa − t2t3 Xb + t1t3 Xa Xb  (2.43)  Spectral Characteristics of Series-Coupled Ring Resonator Add-Drop Filters  The drop port spectral response of a series coupled ring resonator add-drop filter,  as shown in Figure 2.24, can be determined by multiplying the drop port transfer function, Gdrop , by its complex conjugate, G∗drop , to give the drop port intensity drop function I2ring . The following simulations assume that La = Lb = L, κ1 = κ3 =  κ, t1 = t3 = t, αa = αb = α, and βa = βb = β . Figure 2.24(a) and (b) show the advantages of using series-coupled ring resonator add-drop filters: larger out-ofband-rejection ratio, flatter resonance peak and steeper roll-off from the pass-band to the stop-band which is in accordance with the theory presented in Yanagase et al. [64]. 2 Equations 2.44, 2.46, 2.51, 2.52, 2.54 were derived by us and are consistent to those derived by Kato and Kokubun [29].  48  0 a)  two rings one ring  Insertion Loss (dB)  5  10  15  20  1534  1536  1538 1540 Wavelength (nm)  0 b)  1544  two rings one ring  0.5 Insertion Loss (dB)  1542  1 1.5 2 2.5 3 1541  1541.2 1541.4 1541.6 1541.8 Wavelength (nm)  1542  1542.2  Figure 2.24: Comparison between the drop port response of a single and series-coupled ring resonator add-drop filter. Each resonator has a total length of 100 µm, propagation loss of 3 dB/cm, |κ|2 = 0.5, and |κ2 |2 = 0.111. 49  The resonance condition for a single ring resonator is β L = 2πm, which also holds true for series-coupled ring resonators when κ2 is below a certain critical value to be determined further down. The drop port intensity at β L = 2πm can be found by, drop |β L=2πm = I2ring  κ 4 κ22 e2αL . (−2tt2 eαL + t 2 + e2αL )2  (2.44)  To determine the critical coupling point at which no resonance splitting occurs and drop at which a maximally flat resonance peak is obtained, the derivative of I2ring |β L=2πm drop with respect to κ2 is equal to zero, which is the point at which I2ring |β L=2πm is  maximum (here we have used t2 = (1 − κ22 )1/2 ), drop dI2ring |β L=2πm  dκ2  =  2κ 4 κ22 e2αL (2teαL − t2 + t2 κ 2 − t2 e2αL ) = 0. (2tt2 eαL − 1 + κ 2 − e2αL )3t2  (2.45)  The solution to equation 2.45 gives the critical coupling field factor, κcdrop , with t2 replaced by 1 − κ22  1/2  ,  −2teαL κcdrop = κ2 = 1 − 2 t + e2αL  2  1 2  =  e2αL − t 2 . e2αL + t 2  (2.46)  Figure 2.25 shows an example of the five coupling cases for the drop port: undercoupled (orange and blue lines), critically coupled (green line), and over-coupled (red and pink lines). Resonance splitting occurs when |κ2 |2 is greater than |κcdrop |2 . In this case, |κcdrop |2 is 0.1132. Also, Figure 2.25 shows that as |κ2 |2 decreases drop below |κcdrop |2 , I2ring |β L=2πm also decreases in value which is shown by the blue  and orange lines. When |κ2 |2 is equal to |κcdrop |2 , no resonance splitting occurs drop and I2ring |β L=2πm has a maximum value which is shown by the green line. If |κ2 |2 drop increases above |κcdrop |2 , resonance splitting occurs and I2ring |β L=2πm decreases in  value which is shown by the red and pink lines.  50  0  | |2 = 0.005 2  5  | |2 = 0.01 2  2  Idrop (dB) 2ring  | 2| = 0.1132  10  | |2 = 0.3 2  | |2 = 0.5 2  15 20 25 1536 1537 1538 1539 1540 1541 1542 1543 1544 Wavelength (nm)  Figure 2.25: Drop port response sensitivity to |κ2 |2 equal to 0.005, 0.01, 0.1132, 0.3, and 0.5 with |κ1 |2 = |κ3 |2 = |κ|2 fixed at 0.5, propagation loss of 3 dB/cm, and the total length of each resonator equal to 100 µm. In Figure 2.26(a) and (b) we plot |κcdrop |2 versus |κ|2 for a variation in propagation losses as well as the total length of the resonators, respectively. In both variation cases, |κcdrop |2 decreases with a decrease in |κ|2 . For an increase in propagation losses as well as an increase in the total length of the resonators, |κcdrop |2 increases.  51  10  0  a) 1  10  2  10  3  |  drop 2 | c  loss = 3 dB/cm loss = 10 dB/cm loss = 20 dB/cm  10  4  10  5  10  10  3  10  2  2  10  1  0  10  | | 10 10  10  b)  L = 200 µm L = 100 µm L = 50 µm  1  2  3  |  drop 2 | c  10  0  10 10 10  4  5  6  10  3  10  2  2  10  1  0  10  | |  Figure 2.26: |κcdrop |2 versus |κ|2 for (a) propagation losses of 3 dB/cm, 10 dB/cm, and 20 dB/cm with the resonator lengths fixed at 100 µm and (b) for resonator lengths of 50 µm, 100 µm, and 200 µm with a fixed propagation loss of 3 dB/cm. 52  The resonance splitting wavelengths are dependent on κ and κ2 . However, the splitting of the main resonance wavelength is primarily due to κ2 . Thus, when κ << 1, the resonance splitting wavelengths can be described by, λsplit1 =  2πne f f (λsplit1 )L 2πne f f (λsplit1 )L = , 2πm + βdi f f Lc /2 2πm + arcsin(κ2 )  (2.47)  λsplit2 =  2πne f f (λsplit2 )L 2πne f f (λsplit2 )L = 2πm − βdi f f Lc /2 2πm − arcsin(κ2 )  (2.48)  where ne f f (λsplit1 ) and ne f f (λsplit2 ) are the effective indices at the resonance split wavelengths, λsplit1 and λsplit2 , respectively, which are in accordance with Schwelb [49]. The derivations of equations 2.47 and 2.48 can be found in Appendix C. The derivations assume that the bus waveguides are removed. Figure 2.27(a) shows the resonance splitting wavelengths versus |κ2 |2 for a fixed |κ|2 = 0.001 and ne f f = 2.4 using equations 2.47 and 2.48. Since |κ|2 = 0.001 is <<1, the resonance splitting wavelengths determined from the spectrum (red and blue dots) match the values given by the resonance splitting equations. It is clearly seen that as |κ2 |2 increases, the separation between the two resonance splitting wavelengths is increasing. Neglecting κ is equivalent to not having the bus waveguides, which is clearly not the case for add-drop filters. Therefore, it is of interest to understand the effect of such an approximation compared to the actual values determined from the drop port response. Figure 2.27(b) shows the resonance splitting wavelengths versus |κ|2 for a fixed |κ2 |2 = 0.3 and ne f f = 2.4 using equations 2.47 and 2.48. The blue and red data points correspond to the resonance splitting values determined from the drop port spectrum. As we can clearly see, when |κ|2 approaches one, the error between the resonance splitting wavelength values determined by equations 2.47 and 2.48 increases compared to the actual values determined from the graph.  53  1540  a)  (nm)  1539.5 1  2  split  ,  , equation 2.48  split  2  1538.5  , graphically  split  1  split  , equation 2.47  split  1539  1  1538  split  , graphically  2  1537.5 1537 0  1539.4  0.1  0.2  | 2|2  0.3  0.4  0.5  b)  1539.2 1539 (nm) 2  split  , equation 2.48  2  1538.6  split  1538.4  , graphically  1  1  split  , equation 2.47  1  1538.8  split  ,  split  1538.2  split  , graphically  2  1538 1537.8 1537.6 10  3  10  2  2  10  1  | |  Figure 2.27: (a) shows the resonance splitting wavelengths versus |κ2 |2 for a fixed |κ|2 = 0.001 and resonator lengths equal to 100 µm using equations 2.47 and 2.48 and the closely matched values determined graphically from the drop port response. (b) shows the resonance splitting wavelengths versus |κ|2 for a fixed |κ2 |2 = 0.3. The increasing error in using equations 2.47 and 2.48 is clearly shown for large values of |κ|2 compared to the values determined from the drop port response.  54  The out-of-band rejection ratio, OBRR2ring , at critical coupling can be deterdrop drop mined by taking the ratio I2ring |β L=2πm to I2ring |β L=π+2πm and substituting equa-  tion 2.46 for κ2 . The minimum drop port intensity can be determined when β L = π + 2πm, drop I2ring |β L=π+2πm  where  tcdrop  = 1−  κcdrop  2  κ 4 κcdrop =  2  e2αL (2.49)  (2ttcdrop eαL + t 2 + e2αL )2  1 2  . Therefore, OBRR2ring is defined by,  OBRR2ring = 10log10  2ttc eαL + t 2 + e2αL −2ttc eαL + t 2 + e2αL  2  .  (2.50)  Figure 2.28(a) and (b) show the OBRR2ring at κcdrop versus |κ|2 for a variation in propagation losses and total length of the resonators, respectively. The trends seen here for the series-coupled ring resonator case are very similar to those of the OBRR of the single ring resonator add-drop filter shown in Figure 2.18. Fortunately, the OBRR2ring at critical coupling is greater than the OBRR for all values of |κ|2 equal to the power coupling factors of the single ring resonator thus showing one of the important advantages of using series-coupled ring resonators as compared to single ring resonators.  55  140 a)  loss = 3 dB/cm loss = 10 dB/cm loss = 20 dB/cm  OBRR2ring (dB)  120 100 80 60 40 20 10  3  10  2  2  10  1  10  0  | | 140 b)  L = 200 µm L = 100 µm L = 50 µm  OBRR2ring (dB)  120 100 80 60 40 20 10  3  10  2  2  10  1  10  0  | |  Figure 2.28: OBRR2ring at κcdrop versus |κ|2 for (a) propagation losses of 3 dB/cm, 10 dB/cm, and 20 dB/cm with the resonator lengths fixed at 100 µm and (b) for resonator lengths of 50 µm, 100 µm, and 200 µm with a fixed propagation loss of 3 dB/cm.  56  The through port spectral response of a series coupled ring resonator add-drop filter as shown in Figure 2.29 can be determined by multiplying the drop port trans∗ , to give the through port fer function, Gthrough , by its complex conjugate, Gthrough through . Also, Figure 2.29 shows the five coupling cases for intensity function I2ring  the through port: under-coupled (blue and orange lines), critically coupled (green line), and over-coupled (pink and red lines). The critically coupled case occurs when |κ2 |2 = |κcthru |2 = 0.1111. Resonance splitting occurs when |κ2 |2 is greater than |κcthru |2 .  0  Ithrough (dB) 2ring  20 40  | 2|2 = 0.5 | |2 = 0.3  60  2  | |2 = 0.1111 2  80  | 2|2 = 0.08 100  | 2|2 = 0.05 1540.5  1541  1541.5 1542 Wavelength (nm)  1542.5  Figure 2.29: Through port response sensitivity to |κ2 |2 equal to 0.5, 0.3, 0.1111, 0.08, and 0.05 with |κ|2 fixed at 0.5, propagation loss of 3 dB/cm, and the total length of each resonator equal to 100 µm. Resonance splitting occurs when |κ2 |2 is above the critical power coupling factor. In this case, the critical power coupling factor is 0.1111. The through port intensity at β L = 2πm can be found by, through I2ring |β L=2πm  t − t2 e−αL − t 2t2 e−αL + te−2αL = 1 − 2tt2 e−αL + t 2 e−2αL 57  2  .  (2.51)  The critical field coupling factor to create a minimum through port response at which no resonance splitting occurs (as shown in Figure 2.29) can be determined by setting the numerator of equation 2.51 to zero, solving for t2 , and then finding κ2 which is equal to the critical field coupling factor, κcthru , κcthru = κ2 =  1 − t 2 e−2αL e−2αL − t 2 e−2αL (1 + t 2 )2  1 2  .  (2.52)  Since |κcthru |2 must be greater than 0 and less than 1, exp(−2αL) must be greater than |t|2 . When the condition for |t|2 is not met, critical coupling does not exist through [29]. Figure 2.30 shows I2ring |β L=2πm versus |κ2 |2 for various values of |κ|2 .  The minimum value of each graph corresponds to |κ2 |2 =|κcthru |2 . It can be clearly seen from Figure 2.30 that as |κ|2 increases, so does the value of |κcthru |2 . Also, if |κ2 |2 <|κcthru |2 there is only one resonance peak. If |κ2 |2 >|κcthru |2 , resonance through splitting occurs and I2ring |β L=2πm increases in value with increasing |κ2 |2 which is through undesirable. However, a certain amount of splitting is acceptable if I2ring |β L=2πm  is low enough.  58  0  Ithrough | 2ring  L=2 m  20  | |2 = 0.5 | |2 = 0.3  40  | |2 = 0.1  60 80 100 120 10  4  10  3  2  10 | 2|2  10  1  10  0  Figure 2.30: Through port insertion loss at β L = 2πm versus |κ2 |2 for |κ|2 = 0.1, 0.3, and 0.5 with a fixed propagation loss of 3 dB/cm and the total length of each resonator equal to 100 µm. If we compare the critical coupling conditions for the drop port and through port as shown in equations 2.46 and 2.52, the two equations are clearly not the same. Thus, the optimal condition for the through port and drop port will not be simultaneously met. However, Figure 2.31 shows that |κcdrop |2 and |κcthru |2 have similar values when |κ|2 is sufficiently large.  59  10  10  1  |  thru 2 | c  |  drop 2 | c  2  10  3  |  drop 2 | c  ,|  thru 2 | c  10  0  10  10  4  5  10  2  1  0  10 | |2  10  Figure 2.31: Comparison between |κcthru |2 and |κcdrop |2 versus |κ|2 for a fixed propagation loss of 3 dB/cm and the total length of each resonator equal to 100 µm. However, if propagation losses are neglected (α = 0), then the critical field coupling factors for the through port and drop port are the same and equations 2.46 and 2.52 reduce to, κcdrop = κcthru =  1 − t2 . 1 + t2  (2.53)  The through port extinction ratio, ER2ring , can be determined by taking the ratio through I2ring |β L=π+2πm  through to I2ring |β L=2πm . The maximum through port intensity can be  determined when β L = π + 2πm, through I2ring |β L=π+2πm =  t + t2 e−αL + t 2t2 e−αL + te−2αL 1 + 2tt2 e−αL + t 2 e−2αL  60  2  .  (2.54)  Therefore, ER2ring is defined by, ER2ring = 10log10  through |β L=π+2πm I2ring through I2ring |β L=2πm  .  (2.55)  Figure 2.32(a) and (b) show a comparison between the extinction ratio and out-ofband rejection ratio for single and series-coupled ring resonators for a fixed loss of 3 dB/cm and a resonator length of 100 µm. Figure 2.32(a) clearly shows that there is a dramatic increase in the extinction ratio for the series-coupled ring resonator for a |κ2 |2 close to |κcthru |2 = 0.1111. Also, Figure 2.32(b) shows that the out-ofband rejection ratio for the series-coupled ring resonator is larger for all values of |κ|2 compared to that of the single ring resonator. Although using series-coupled ring resonator add-drop filters can improve the flattness and the roll-off from pass-band to stop-band of the drop port response (as shown in Figure 2.24), as well as increase the OBRR (as shown in Figure 2.32(b)), the FSR remains the same as that of the single ring resonator. Ideally, one would like to obtain an FSR greater than the span of the C-band. However, the use of series-coupled ring resonators with each resonator having the same length would require very small resonator lengths which would result in the inability to design the resonators in a racetrack configuration due to large bending losses. However, using series-coupled ring resonators with each resonator having a different length exhibiting the Vernier effect can substantially increase the FSR while ensuring minimal bending losses [10]. Figure 2.33(a) and (b) show a comparison between the drop port response of series-coupled ring resonators with each resonator having the same length (blue, dashed) and a different length (red) with a fixed propagation loss of 3 dB/cm. The series-coupled ring resonator shown by the blue dashed line has identical lengths of 14 µm, |κ|2 = 0.024, propagation loss of 3 dB/cm, and |κ2 |2 = 0.00016. The series-coupled ring resonator shown by the red line has lengths equal to 56 µm and 70 µm, |κ|2 = 0.1, propagation loss of 3 dB/cm, and |κ2 |2 = 0.0033. Figure 2.33 (b) clearly shows that both series-coupled ring resonator configurations result in a flat peak in the pass-band. Although, it can be clearly seen that the OBRR of the series-coupled ring resonator, with each resonator having the same length, is significant compared to the range of the interstitial peak  61  suppression (taken as the difference between the intensity of the largest resonance peak and the second largest resonance peak as well as the difference between the largest resonance peak and the smallest resonance peak), the resonator lengths are 14 µm which requires a microring configuration since a racetrack configuration would require a ring radius of much less than 2 µm. Even in a microring configuration, besides the increased sensitivity to fabrication errors, a length of 14 µm corresponds to a radius of 2.23 µm which has large bending losses. In the next chapter, there will be an in-depth discussion of series-coupled racetrack resonators exhibiting the Vernier effect.  62  120 a)  two rings one ring  100  ER (dB)  80 60 40 20 0 0  120  0.1 0.2 0.3 0.4 | 2|2 (double resonators), | 1|2 (single resonator)  b)  0.5  two rings one ring  OBRR (dB)  100 80 60 40 20 10  3  10  2  2  10  1  | |  Figure 2.32: Comparison of single and series-coupled ring resonator adddrop filters based on (a) the through port ER and (b) the drop port OBRR for a fixed propagation loss of 3 dB/cm, the total length of each resonator equal to 100 µm. The power coupling factor to the bus waveguides of the series coupled resonator is set to 0.5 for determining the ER. 63  0 a)  different lengths identical lengths  10  Insertion Loss (dB)  20 30 40 50 60 70 80 1520  1530  1540  1550 1560 1570 Wavelength (nm)  1580  1590  b)  Insertion Loss (dB)  2 4 6 8 different lengths identical lengths  10 12 1535.1  1535.2  1535.3 1535.4 1535.5 Wavelength (nm)  1535.6  Figure 2.33: Comparison of the drop port response of series-coupled ring resonator add-drop filters with identical and different ring resonator lengths that have FSRs greater than the span of the C-band.  64  Chapter 3  Series-Coupled Silicon Racetrack Resonators and the Vernier Effect: Theory and Measurement 1 In this chapter, we present an explanation of the theory and experimental results of  the Vernier effect using series-coupled racetrack resonators using SOI strip waveguides. We have gathered both theoretical aspects and experimental results within the same paper, coupled to an improvement in a Vernier effect device. For the purpose of comparison, we present measurements made on individual racetrack resonators having the same dimensions as the resonators within the Vernier effect device in order to show that the Vernier effect, in fact, has been achieved.  3.1  Theory  The FSR of a racetrack resonator is inversely proportional to the length of the resonator. If the FSR of a single resonator is comparable to the span of the Cband, the bending losses can become quite large and, thus, reduce the performance of the device. However, the Vernier effect can be used to extend the FSR. The Vernier effect can be created by using series-coupled racetrack resonators. The device consists of two bus waveguides connected to two series-coupled racetrack 1A  version of this chapter has been published in Boeck et al. [7]  65  resonators with different lengths. Ideally, one would like these devices to have low-losses, small foot-prints, extended FSRs that are comparable to the C-band, and to exhibit large interstitial peak suppressions. The drop port transfer function that is used to model the Vernier effect device can be determined by (here we used ng whereas Rabus [47] and Chaichuay et al. [10] used ne f f ), T Fdrop =  jκ1 κ2 κ3 (X1 X2 )1/2 1 − t1t2 X1 − t2t3 X2 + t1t3 X1 X2  (3.1)  where Xi = exp(−αLi − jϕi ), ti is the field transmission factor , ϕi = (2πng Li )/λ , α [/m] is the field loss coefficient, L1 [m] and L2 [m] are the total lengths of the first and second racetrack resonator, respectively, λ is the wavelength, ng is the group index, κ1 and κ3 are the symmetric (real) point field coupling factors to the bus waveguides, and κ2 is the (real) point inter-ring field coupling factor. ng is given by [47], ng (λ , T) = ne f f (λ , T) − λ  δ ne f f (λ , T) δλ  (3.2)  where ne f f is the effective index and T is temperature. ne f f is λ and T dependent since the refractive indices of Si and SiO2 are functions of λ and T. ne f f can be calculated using a mode solver and the λ and T dependency of the refractive indices can be modeled using experimental data [35, 48, 56]. The extended FSR needs to be comparable to the span of the C-band (35 nm) to maximize the number of channels that can be multiplexed. The extended FSR is related to the FSR of each racetrack resonator by [10, 50], FSRextended = m1 FSR1 = m2 FSR2  (3.3)  where m1 and m2 are co-prime integers. The total length of each racetrack resonator is determined by using, m2 L2 = . m1 L1  (3.4)  One must optimize the performance of the device by ensuring: 1) the removal of twin resonance peaks located between the main resonance peaks; 2) that the main resonance peak intensity is high (i.e. ensuring low insertion loss); 3) that there is minimal splitting of the main resonance peak; and 4) that there is large 66  interstitial peak suppression as shown in Figure 3.1. The twin peaks shown in Figure 3.1(a) occur because m1 and m2 are chosen to be 9 and 7, respectively. To remove the occurrence of the twin peaks, m2 should be equal to m1 -1 (assuming m1 >m2 ). The optimized device shown in Figure 3.2(a) has an interstitial peak suppression greater than 41.8 dB which is sufficient for WDM applications. Figure 3.2(b) shows that the optimized device does not have any main resonance peak splitting. If we increase m1 (and m2 ), the interstitial peak suppression decreases. Thus, m1 needs to be small enough to give adequate interstitial peak suppression. The choice of L1 and L2 determines the extended FSR. The power coupling factors need to be optimized to obtain high main resonance intensity, minimal main resonance splitting, and large interstitial peak suppression [10, 50].  67  0  extended FSR interstitial peak twin peaks suppression  a)  Insertion Loss (dB)  !5 !10 !15 !20 !25 1520  1530  1540 1550 1560 Wavelength (nm)  1570  1580  b)  Insertion Loss (dB)  0 !0.2 !0.4 !0.6 !0.8 !1 1568.8  1569 1569.2 1569.4 Wavelength (nm)  1569.6  Figure 3.1: (a) Theoretical drop port response of the un-optimized device illustrating the twin peaks, extended FSR, and minimum interstitial peak suppression, and (b) the main resonance peak splitting. The design parameters are m1 and m2 are 9 and 7, respectively. L1 and L2 are 127.91 µm and 99.487 µm, respectively. |κ1 |2 , |κ2 |2 and |κ3 |2 are 0.35, 0.1, and 0.35, respectively. The waveguide width is 500 nm, the propagation loss is 3 dB/cm, and ng is 4.306 [7]. c Optical Society of America, 2010, by permission.  68  a) !10  Insertion Loss (dB)  !20 !30 !40 !50 !60 !70 !80 !90 1520  !1.4  1530  1540 1550 1560 Wavelength (nm)  1570  1580  b)  Insertion Loss (dB)  !1.6 !1.8 !2 !2.2 !2.4 !2.6 1569.1  1569.15  1569.2 Wavelength (nm)  1569.25  1569.3  Figure 3.2: (a) Theoretical drop port response of the optimized device illustrating large interstitial peak suppression and (b) no main resonance splitting. The design parameters are m1 and m2 are 3 and 2, respectively. L1 and L2 are 42.637 µm and 28.425 µm, respectively. |κ1 |2 , |κ2 |2 and |κ3 |2 are 0.015, 0.00005, and 0.015, respectively. The waveguide width is 500 nm, the propagation loss is 3 dB/cm, and ng is 4.306 [7]. c Optical Society of America, 2010, by permission.  69  3.2  Experimental Results  Our device was fabricated using 193 nm lithography. To couple light into and out of the waveguides, 1-D periodic grating couplers with 400 µm long tapers (to ensure single mode propagation for the TE-polarization) are used [9]. The SOI strip waveguides have a width of 420 nm, a height of 220 nm, and a buried oxide (BOX) thickness of 2 µm. These waveguide dimensions typically result in about 20 dB/cm propagation loss [19]. Although the losses are large compared to a 500 nm width waveguide, the benefit to having a 420 nm waveguide is that the mode is less confined so the gap distances can be increased to reduce the chances of fabrication errors while keeping the power coupling factors relatively the same. We choose to set m2 = 4 and m1 = 5 (ratio equal to 0.8), which turns out to be the same values as given in [10]. The straight sections of the racetrack resonator, Lc , are 15 µm. The radii (defined from the centers of the waveguides), R1 and R2 , of the half-circle sections of the racetrack resonators are 6.545 µm and 4.225 µm, respectively. A small straight section of 50 nm is added to the middle of each half-circle of the racetrack resonator. This straight section was inserted so that several devices with different total resonator lengths (different straight section length) could be fabricated while the power coupling factors remained the same. However, this straight section can increase the scattering losses. The ratio of L2 to L1 is 0.7953. The gap distances for the inter-ring coupling region and the two coupling regions to the bus waveguides are approximately 410 nm and 230 nm, respectively. Figure 3.3 shows scanning-electron micrographs (SEMs) of the fabricated device.  70  a)  Lc k1  k2  R2  k3  R1 10 um  50 nm  50 nm  b)  1 um Figure 3.3: SEM of (a) fabricated series-coupled racetrack resonators, and (b) coupling region [7]. c Optical Society of America, 2010, by permission.  71  Figure 3.4(a) shows the measured drop port response of the series-coupled racetrack resonator. Figure 3.4(c) shows the straight waveguide transmission response that was used for calibration. The extended FSR is 35.96 nm, the quality factor is 2084, the 3-dB bandwidth is 0.74 nm, there is minimal splitting of the main resonance peak (shown in Figure 3.4(b)), and the interstitial peak suppression is between 9.11 dB and 17.11 dB. Thus, we can conclude that using series-coupled racetrack resonators provide a marked improvement in the FSR compared to single racetrack resonator multiplexers. The measured through port response showed an extinction ratio of 5.43 dB. Although the interstitial peak suppression of our device shows a marked improvement compared to other devices reported in literature, the suppression is still insufficient for many WDM applications. In order to improve suppression in future devices, one could increase the gap distances with the inter-ring gap distance having the greatest impact on the suppression. Also, the Vernier effect exhibited in devices such as the one reported here may effect the through port channels due to the increase in dispersion and group delay [44]. A new Vernier scheme proposed by M. Popovic et al. may provide a potential solution to this problem [44].  72  0  a)  Extended FSR  Insertion Loss (dB)  5 curve fit experimental  10 15 20 25 30 35 1500  1  1510  1520 1530 Wavelength (nm)  b)  1550  experimental curve fit  1.5 Insertion Loss (dB)  1540  2 2.5 3 3.5 4 4.5 1541.8  16  1542  1542.2 1542.4 1542.6 1542.8 Wavelength (nm)  1543  c)  Insertion Loss (dB)  18 20 22 24 26 28 30 1500  1510  1520 1530 Wavelength (nm)  1540  1550  Figure 3.4: (a) Experimental and curve-fit drop port response for seriescoupled racetrack resonators, (b) shows the minimal main resonance splitting (zoom in of Figure 3.4(a)), and (c) shows the straight waveguide transmission response used for calibration. (Although minor modifications were made to the above figures, permission was obtained to reproduce figures from [7]). c Optical Society of America, 2010, by permission. 73  3.3  Curve-Fitting  The following presents the details for curve-fitting the results shown in Figure 3.4(a). The device shown in Figure 3.3 has a waveguide width of approximately 420 nm and gap distances of approximately 410 nm and 230 nm and, hence, these are the values used here. Using a 2D FD mode solver [20], the group index is 4.5473. The experimental characteristics that we wish to match with theory are the main resonance intensity, main resonance splitting depth, and interstitial peak suppression. The main resonance intensity (insertion loss), minimum interstitial peak suppression, and main resonance splitting depth are plotted as a function of |κ1 |2 , |κ2 |2 , and |κ3 |2 as shown in Figure 3.5. The best fit between the model and the experiment is for |κ1 |2 = 0.3665, |κ2 |2 = 0.02485 and |κ3 |2 = 0.3665. From these values, we determine the gap distances using supermode analysis to be between 405 nm and 410 nm for the inter-ring coupling region and between 230 nm and 235 nm for the two coupling regions to the bus waveguides, which are very close to the values determined from the SEM images in Figure 3.3(a) and Figure 3.3(b). The curve-fit takes into account the wavelength dependency of the power coupling factors using the 2D FD mode solver. From looking at Figure 3.5, one can clearly see that if |κ1 |2 , |κ2 |2 , and |κ3 |2 are reduced from the curve-fitted values (marked on the plots as purples circles), the interstitial peak suppression increases while ensuring small insertion loss and minimal splitting of the main resonance peak. However, since propagation losses are large for a waveguide width of 420 nm, the insertion loss sensitivity to a decrease in the power coupling factors is large. Nevertheless, if the waveguide width is increased to 500 nm, the propagation losses are reduced and thus the insertion loss sensitivity to a change in the power coupling factors is reduced. Therefore, smaller values of the power coupling factors can be used and interstitial peak suppression can be increased without significant insertion losses.  74  a)  0.2  dB -5  0.15  -10  2  !  ! 2 0.1  -15 -20  0.05 -25  0.1  b)  0.2 0.3 | !1 |2 =| ! 3 |2 !  0.4  0.5  0.2  25 dB  0.15  20  2  !  -30  ! 2 0.1  15 10  0.05  5 0.1  c)  0.2  0.2 0.3 | !1 |2 =| ! 3 |2 !  0.4  0.5  0 dB -5  0.15  -10  2  !  ! 2 0.1 -15 0.05 -20  0.1  0.2 0.3 | !1 |2 =| ! 3 |2 !  0.4  0.5  Figure 3.5: (a) Main resonance intensity, (b) minimum interstitial peak suppression, and (c) main resonance splitting depth versus |κ1 |2 , |κ2 |2 , and |κ3 |2 . Purple circles indicate values used for curve-fitting.  75  The slight mismatch between the curve-fit and the experimental results may be due to various assumptions and unknown experimental waveguide characteristics. These assumptions and characteristics include: 1) The waveguides are assumed to be perfectly rectangular (however, cross-sectional SEM images of similar SOI waveguides have shown that the sidewalls can have slopes of approximately 9o [18] and a trapezoidal waveguide structure will change the effective index as well as the gap distances); 2) the group index is assumed to be wavelength independent; 3) the height of the waveguide is assumed to be 220 nm; 4) the waveguide curvature is neglected for ne f f calculations; and 5) the distance between waveguides is not considered for ne f f calculations. To evaluate and confirm the existence of the Vernier effect, individual racetrack resonators with the same structural designs as those that constitute the Verniereffect device were fabricated on the same die. Figure 3.6 shows the experimental drop port response of the two racetrack resonators which was calibrated using a straight waveguide transmission response. The quality factors for the racetrack resonators with a radius of 6.545 µm and 4.225 µm are approximately 1091 and 1157, respectively. The periodic resonance peaks of both racetrack resonators overlap at 1502.44 nm and 1538.24 nm. The difference between these two resonance wavelengths is 35.8 nm, which is close to value of the extended FSR shown in Figure 3.4(a).  76  radius = 6.545 µm radius = 4.225 µm  10  Insertion Loss (dB)  5 0 5 10 15 20 1500  1520 1540 Wavelength (nm)  1560  1580  Figure 3.6: Experimental drop port response for single racetrack resonators with a radius of 6.545 µm and 4.225 µm. (Although minor modifications were made to the above figure, permission was obtained to reproduce the figure from [7]). c Optical Society of America, 2010, by permission.  77  Chapter 4  Conclusion and Suggestions for Future Work 4.1  Conclusion  The analysis of SOI ring resonator add-drop multiplexers has been presented in this thesis. The fundamentals of SOI strip waveguides and their optical mode dependency on temperature, wavelength, and waveguide cross-section have been discussed and analyzed using a 2D FD mode solver [20]. Directional couplers were then analyzed using supermode theory to determine the amount of power that can be coupled from one waveguide to another. To reduce the complexity of determining the transfer functions of optical circuits, a description of signal flow graphs and Mason’s rule was presented. Mason’s rule was used to determine the transfer functions of the through port and drop port of single ring resonators. Next, a thorough analysis of important spectral characteristics of the through port and drop port intensity responses for single ring resonators was given. Then, Mason’s rule was used to determine the transfer functions of the through port and drop port of series-coupled ring resonators. Next, we presented an analysis of important spectral characteristics of through port and drop port intensity responses. The results showed that the through port extinction ratio, OBRR of the drop port, flattness of the drop port pass-band, and roll-off from pass-band to stop-band can be improved by using series-coupled ring resonators as compared to single ring resonators. The 78  FSR of both single and series-coupled ring resonators with resonators having the same lengths have an FSR that is limited due to the inverse relationship between the FSR and resonator length corresponding to a bend-loss limited FSR. However, we have presented the theory and experimental results of using SOI strip waveguide series-coupled racetrack resonators with resonators having different lengths to expand the FSR by using the Vernier effect. The Vernier effect causes the FSR to expand by suppressing all resonances that are not an integer multiple of the FSR of each individual resonator. We have fabricated, at ePIXfab, an SOI series-coupled racetrack resonator exhibiting the Vernier effect that showed better interstitial resonance peak suppression (between 9 dB to 17 dB) and minimal splitting of the main resonance peak compared to the Vernier effect device created by Timotijevic et al. [57]. The FSR of our device is 36 nm, which is comparable to the span of the Cband. Individual ring resonators with the same dimensions as those that were used in the Vernier effect device were also fabricated to confirm the existence of the Vernier effect in the series-coupled racetrack resonator. Ring resonator research is progressing at a rapid pace throughout the world and the Vernier effect may be a very important feature that will enable add-drop multiplexers to span the entire C-band for supercomputers and FTTH related applications.  4.2  Suggestions for Future Work  The long term goal is to create a commercial Vernier effect device using SOI ring resonators that exhibits interstitial peak suppression greater than 35 dB. To increase the interstitial peak suppression, one could increase the gap distances as well as cascade two identical Vernier effect devices together. Another approach might be to use post-processing techniques such as thermal annealing to make the sidewalls of the waveguides smoother and thus have lower scattering losses. Since creating a filter that resonates specifically at the designed wavelength is very difficult due to fabrication errors, the ability to thermally tune the resonators should be included in the designs. One approach is to locate heaters on the chips [17]. Also, a method to permanently adhere the fibers to the grating couplers, while not adversely affecting the response of the grating couplers, must be developed. Recently, researchers at IMEC have developed a method to permanently adhere single-mode fibers to  79  a microring resonator refractive index sensor without affecting its sensitivity [4]. Also, the integration of detectors on the chip is important. Recently, OpSIS at the University of Washington has offered to integrate Ge PIN detectors directly on the chip. Figure 4.1 shows a possible configuration of a demultiplexer for the entire C-band (heaters are neglected in schematic) that could be developed. Since the C-band spans from 1530 nm to 1565 nm, there would be 23 channels that would be spaced 1.6 nm apart. Each channel would be filtered using series-coupled ring resonators exhibiting the Vernier effect with an FSR greater than the span of the C-band. Each demultiplexed wavelength would be received by an on-chip Ge PIN detector. Thus, in total, there would be 46 ring resonators and 23 Ge PIN detectors. A ring resonator demultiplexer, such as the one described here, would enable, for the first time, the ability to demultiplex every channel in the C-band, which has not yet been achieved using ring resonators. Since ring resonator add-drop filters are named due to their ability to multiplex as well as demultiplex wavelengths without changing the configuration of the rings, the same structure shown in Figure 4.1 could be used as a C-band multiplexer by replacing the Ge PIN detectors with optical modulators and lasers. The use of a C-band multiplexer and demultiplexer using series-coupled ring resonators exhibiting the Vernier effect will be beneficial for applications such as supercomputers and FTTH.  80  Ge ORX  !"!  Ge ORX  !'!  Ge ORX  !""!  !"#!$!$!$#!%&! ! !  !%&!  Ge ORX  !"(!  Ge ORX  !"%!  Ge ORX  Figure 4.1: Schematic of a C-band demultiplexer using series-coupled ring resonators and on-chip Ge PIN detectors.  81  Bibliography [1] 200 GHz Single Channel DWDM Add/Drop [Online] (July 16, 2011), . URL http: //www.lighteltech.com/product/43/200-ghz-single-channel-dwdm-add-drop.  [2] DWDM Single Channel Add/Drop Device [Online] (July 16, 2011), . URL http://www.oemarket.com/product info.php?products id=159. [3] DWDM Single Add/Drop Device [Online] (July 16, 2011), . URL http://aoxc.net/DWDM-N1.htm. [4] C. L. Arce, K. D. Vos, T. Claes, K. Komorowska, D. V. Thourhout, and P. Bienstman. Silicon-on-insulator microring resonator sensor integrated on an optical fiber facet. IEEE Photonics Technology Letters, 23(13):890–892, 2011. [5] L. N. Binh, N. Q. Ngo, and S. F. Luk. Graphical representation and analysis of the z-shaped double-coupler optical resonator. Lightwave Technology, Journal of, 11(11):1782–1792, 1993. [6] R. Boeck, N. A. F. Jaeger, and L. Chrostowski. 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A tunable 1x4 silicon CMOS photonic wavelength multiplexer/demultiplexer for dense optical interconnects. Optics Express, 18(5):5151–5160, Mar. 2010.  88  Appendix A  Derivation of the Free Spectral Range The free spectral range of a ring resonator add-drop filter is determined using the equations for ne f f and ng as well as for the round trip phase difference between two adjacent resonance wavelengths, λr and λr+1 , where λr+1 > λr (modified from Dr. Jaeger’s notes). ne f fr = λr  dne f f +b dλ  ne f fr+1 = λr+1 ne f fr − ne f fr+1 = −(λr+1 − λr )  (A.1)  dne f f +b dλ  dne f f + b − b, ne f fr+1 < ne f fr dλ  dne f f dλ 2πne f fr L 2πne f fr+1 L ∆φ = βr L − βr+1 L = − = 2π λr λr+1 ne f fr = ne f fr+1 − (λr+1 − λr )  ne f fr+1 λr+1 − (λr+1 − λr )λr+1  dne f f λr λr+1 − ne f fr+1 λr = dλ L  ne f fr+1 (λr+1 − λr ) − (λr+1 − λr )λr+1 ng = ne f f − λ 89  dne f f dλ  dne f f λr λr+1 = dλ L  (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8)  FSR = λr+1 − λr =  λr λr+1 (ne f fr+1  90  dn − λr+1 dλe f f  = )L  λr λr+1 ng L  (A.9)  Appendix B  Derivation of the Full-Width-at-Half-Maximum and the Quality Factor To determine the full-width-at-half-maximum, ∆λFW HM , the drop port intensity, max (normalized to an input value Idrop , must equal half of the maximum power, Idrop  of 1), Idrop =  max Idrop κ12 κ22 e−αL κ12 κ22 e−αL = = 2 1 + t12t22 e−2αL − 2t1t2 e−αL cos(β L) 2(1 − 2t1t2 e−αL + t12t22 e−2αL ) (B.1)  or κ12 κ22 e−αL κ12 κ22 e−αL = 1 + t12t22 e−2αL − 2t1t2 e−αL cos(β L) 2(1 − 2t1t2 e−αL + t12t22 e−2αL )  (B.2)  which is now solved for β L, β L = arccos  −(1 − 4t1t2 e−αL + t12t22 e−2αL ) = arccos(B) 2t1t2 e−αL  (B.3)  which is the solution when the mode number, m, is zero. However, for m>0, there are two roundtrip phases, β1 L and β2 L, at which half the maximum intensity occurs  91  for a particular m given by, β1,2 L = 2mπ ± r  (B.4)  where r represents the magnitude of the difference in the roundtrip phase between the point of maximum intensity and the point at which half the maximum intensity occurs. Therefore, equation B.3 is given by, β1,2 L = 2mπ ± r = 2πm ± arccos(B).  (B.5)  Thus the difference in the roundtrip phases can be defined by, β1 L − β2 L = 2r = 2 arccos(B) β1 L − β2 L =  2πne f f1 Lv1 2πne f f2 Lv2 − = 2 arccos(B) c c  (B.6) (B.7)  where ne f f1 > ne f f2 , , v1 > v2 , and c is the speed of light or, ne f f 1 v1 − ne f f 2 v2 =  c2 arccos(B) . 2πL  (B.8)  We wish to express the right hand side of equation B.8 in terms of the difference in frequencies, v1 and v2 . The effective index, ne f f1 , can be expressed in terms of ne f f2 by, ne f f1 = ne f f2 + (v1 − v2 )  dne f f dne f f = ne f f2 + (∆v) . dv dv  (B.9)  Equation B.9 can be substituted into equation B.8 and rearranged to give, ∆v ne f f2 + v1  dne f f dv  =  c2 arccos(B) . 2πL  (B.10)  The left-side portion of equation B.10 within the round brackets is known as the group index, ng . Therefore, equation B.10 becomes, ∆vng =  c2 arccos(B) . 2πL  92  (B.11)  The difference between two frequencies can be expressed in terms of the difference in wavelengths by the following approximation, ∆v ≈  c ∆λ . λr2  (B.12)  Therefore, ∆λFW HM can be found from substituting equation B.12 into equation B.11, ∆λFW HM =  λr2 arccos(B) . πng L  (B.13)  Therefore, the quality factor, Q, can be easily determined, Q=  λr ∆λFW HM  =  πng L λr arccos(B)  (B.14)  which is in accordance with the findings of Vorckel et al. [59] except that we have taken into account the wavelength dependency of the effective index.  93  Appendix C  Derivation of the Resonance Splitting Wavelengths The following derivation of the resonance splitting wavelengths is based on similar results shown in previously published papers [16, 34, 46, 49]. The resonance wavelengths of the series-coupled ring resonator add-drop filter can be determined by setting the determinant of the system, ∆, to zero. In this case, the denominator of equation C.1 is the determinant which was found using Mason’s rule. 1/2  1/2  jκ1 κ2 κ3 Xa Xb G2 ∆2 = Gdrop = ∆ 1 − t1t2 Xa − t2t3 Xb + t1t3 Xa Xb  (C.1)  To demonstrate this, the following assumptions are used: La = Lb = L, κ1 = κ3 = κ, t1 = t3 = t, αa = αb = 0, βa = βb = β , Xa = Xb = X. For simplicity, we will remove the bus waveguides (t = 1).  X 2 − 2t2 X + 1 = 0  (C.2)  βdi f f Lc 2  (C.3)  X 2 − 2cos  94  X +1 = 0  βdi f f Lc X = e− jβ L = cos 2  1 2  2  βdi f f Lc ± cos 2  −1  βdi f f Lc 1 1 + cos βdi f f Lc − 1 cos(β L) − jsin(β L) = cos ± 2 2 2 βdi f f Lc j = cos ± √ 1 − cos βdi f f Lc 2 2  βdi f f Lc 2  cos(β L) = cos  −sin(β L) = ±  1 cos βdi f f Lc − 2 2  (C.4)  1 2  (C.5) 1 2  (C.6)  (C.7)  1 2  (C.8)  If β L = βdi f f Lc /2, then -sin(β L) = -sin(βdi f f Lc /2). Thus,  βdi f f Lc 1 cos βdi f f Lc sin(β L) = − = sin 2 2 2 2  2  .  (C.9)  Therefore, the resonance condition for the coupled resonators is different than that of the isolated resonator by a difference equal to βdi f f Lc /2, β L = 2πm ±  βdi f f Lc . 2  (C.10)  To determine the corresponding resonance wavelengths that satisfy equation C.10, we use the well-known relationship between the propagation constant and wavelength, β1,2 =  2πne f f (λsplit1,2 ) λsplit1,2  95  (C.11)  where λsplit1,2 is the first and second resonance splitting wavelengths. After substituting equation C.11 into equation C.10 and rearranging terms, the two resonance wavelengths can be found as shown in equations C.12 and C.13 which is in accordance with the findings of Schwelb [49]. λsplit1 =  2πne f f (λsplit1 )L 2πne f f (λsplit1 )L = 2πm + βdi f f Lc /2 2πm + arcsin(κ2 )  (C.12)  λsplit2 =  2πne f f (λsplit2 )L 2πne f f (λsplit2 )L = 2πm − βdi f f Lc /2 2πm − arcsin(κ2 )  (C.13)  Although we have assumed that t = 1, the resonance splitting wavelengths of the series-coupled ring resonator add-drop filter can be approximated using equations C.12 and C.13 when κ<<1.  96  

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