Degenerate Band Edge Resonators in Silicon Photonics

Transcription

1 Degenerate Band Edge Resonators in Silicon Photonics DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Justin R. Burr M.S. B.S. Graduate Program in Electrical and Computer Science The Ohio State University 2015 Dissertation Committee: Prof. Ronald M. Reano, Advisor Prof. George J. Valco Prof. Fernando L. Teixeira

2 Copyright by Justin R. Burr 2015

3 Abstract Photonic band gaps form in infinitely periodic dielectric structures. The propagation of light for frequencies close to the edge of a photonic band gap is highly dispersive and at the band edge the group velocity is zero. In finite length periodic dielectric structures, resonances form when forward and backward propagating modes constructively interfere. On resonance, the modal fields extend over the entire periodic structure. These large and distributed modal fields have been exploited for applications in light emission, optical switching and nonlinear optics. However, transmission resonances near a regular band edge are limited by quality factors that scale only as the third power of the number of periods. Resonances near a degenerate band edge can scale to the fifth power of the number periods. This work is the first experimental demonstration of Quality factor scaling to the fifth power. Transmission resonances near a degenerate band edge are realized in silicon strip waveguides with one-dimensional periodicity. The design and simulation of one dimensional periodic strip DBE cavities is presented. Experimental results include measurement of quality factor of 27,000 in a 35 period cavity. Transmission measurements show Fano resonances with a sharp transmission peak to bandgap extinction ratio of 20 db. ii

4 Dedication This document is dedicated to my family. iii

5 Acknowledgments I would like to acknowledge and thank Professor Ronald M. Reano for the opportunity to be part of his research group at The Ohio State University. My time spent here has been an invaluable learning experience. Professor Reano has cultivated a rich learning environment where students are able to pursue exciting research in the area of integrated photonics. His enthusiasm for teaching and research is motivating and is evidenced by the success of his students. The group of researchers trained by Professor Reano is also first class: Dr. Alexander Ruege, Dr. Peng Sun, Dr. Galen Hoffman, Dr. Li Chen, Michael Wood, Qiang Xu, Tyler Nagy, Jiahong Chen, and Ryan Patton. Office discussions, help in the cleanroom with fabrication, or time spent in the optics lab with these friends and colleagues have been immeasurably beneficial. I chuckle when I think back and recall the hectic weeks that always precede a conference deadline. It is late at night and the entire clean room at Nanotech West is empty except for four or five of Professor Reano s students. I would also like to thank the dissertation committee: Professor George J. Valco and Professor Fernando L. Teixeira. Their time is valuable and I am grateful for their willingness to serve on this committee and as well as my candidacy exam. iv

6 Vita B.S. Electrical Engineering, Michigan State University 2007 to present... Graduate Research Associate, The Ohio State University Publications J. R. Burr, N. Gutman, C. M. de Sterke, I. Vitebskiy, and R. M. Reano, "Degenerate band edge resonances in coupled periodic silicon optical waveguides," Opt. Express 21(7), (2013). M. G. Wood, L. Chen, J. R. Burr, and R. M. Reano; Optimization of electron beam patterned hydrogen silsesquioxane mask edge roughness for low-loss silicon waveguides, J. Nanophoton. 8(1), (2014). J. R. Burr and R. M. Reano, " Zero-coupling-gap degenerate band edge resonators in silicon photonics," Opt. Express 23(24), (2015). Fields of Study Major Field: Electrical and Computer Engineering v

7 Table of Contents Abstract... ii Dedication... iii Acknowledgments... iv Vita... v Publications... v Fields of Study... v Table of Contents... vi List of Figures... ix Chapter 1: Introduction Background Cavities in Periodic Dielectric Structures Defect and Band Edge Cavities Dispersion Near the Band Edge DBE Research by Others One Dimensional Stacks with Anisotropic Layers Fiber Gratings Coupled Microstrip Lines Coupled Periodic Dielectric Waveguides Contributions of This Dissertation Organization of the Dissertation... 8 Chapter 2: Degenerate Band Edge Resonances in Coupled Periodic Silicon Optical Waveguides Chapter Introduction Design vi

8 2.3 Transmission and Resonance Properties near the DBE Spectral Location of Resonances Chapter Conclusion Chapter 3: Zero-coupling-gap Degenerate Band Edge Resonators in Silicon Photonics Chapter Introduction Complex Dispersion Relationships with Degenerate Band Edge Conversion of a RBE to a DBE Comparison of DBE Designs with and without a Coupling Gap Transmission Resonances in Finite Length Structures Spatial Distribution of Electric Field Dependence of Resonance Frequency and Q-factor on Number of Periods Transmission Properties Through zcgdbe Resonator Chapter Conclusion Chapter 4: DBE Fabrication Overview Chapter Introduction Low Loss Silicon Waveguide Fabrication Top Cladding Nominal Device Design Port Terminations Design Variations Fabrication Results Deviating from Nominal Design Chapter Conclusion Chapter 5: Measurement and Characterization of DBE Resonators Optical Transmission Measurement Procedure Group Velocity Measurement Procedure Transmission Measurement Results Demonstration of Quality Factor Scaling to the Fifth Power vii

9 5.5 Demonstration of Quartic Dispersion Chapter Conclusion Chapter 6: Conclusions and Future Outlook Dissertation Conclusion Future Outlook Bibliography viii

10 List of Figures Figure 1.1. Scanning electron microscope images of three different periodic dielectric structures with a photonic bandgap. (a) Fiber Bragg grating [16] patterned on 1.8 μm diameter silica fiber with focused ion beam milling. (b) Three dimensional photonic crystal [17] formed in polycrystalline silicon with band gap between 10.0 μm and 14.5 μm. (c) Cross section of one dimensional mirror[18] with alternating layers of GaAs and Al 0.9 Ga Figure 1.2. Examples of defect cavities. (a) Modified 3 hole defect cavity in a 2D hexagonal lattice [33]. (b) 1D periodic cavity formed by broken translational symmetry at center [34] Figure 1.3. The two panels highlight the differences between RBE and DBE structures. The frequency ω is plotted in red as a function of the wave vector k. The unit cells of both periodic structures have at least one anisotropic dielectric layer. Because of the anisotropy, the dispersion is dependent on the incident polarization. Therefore, we observe two bands of propagating modes at frequencies below the first bandgap. The bandgap begins at frequency ω 0 and wave vector k 0. (a) Dispersion diagram of periodic structure with RBE. Unit cell has one anisotropic layer A1 and one isotropic layer B. (b) Dispersion diagram of periodic structure with DBE. Unit cell has two anisotropic layers A1 & A2 and one isotropic layer B Figure 1.4. Examples of RF structures designed to have a DBE. (a) 100 periods of form birefringent ABS [44] with a band edge near 10 GHz. (b) Microstrip antenna [50] based on DBE unit cell with resonance at 2.59 GHz. (c) 3 periods of low loss form birefringent ceramic laminates [43]. (d) Coupled periodic waveguides with arrays of dielectric pillars [52] with resonances near 9 GHz... 7 Figure 2.1. Coupled periodic silicon optical waveguides designed to exhibit a DBE at telecommunications wavelengths; (a) Top view; (b) Cross-section showing waveguides are completely encapsulated in SiO Figure 2.2 (a)-(f) Dispersion diagrams that show frequency as a function of wavevector. Only propagating modes with real valued k are shown. In (c) (f) the bandgap is indicated by a gray rectangular patch. The complete ix

11 bandgap is not shown. (a) Single mode waveguide; (b) Two coupled single mode waveguides; (c) Single periodic dielectric waveguide; (d) Two coupled periodic dielectric waveguides; (e) Two coupled periodic waveguides with a 0.5a lateral shift; (f) Lateral shift of 0.37a to realize a DBE; (g) Frequency difference versus wavenumber difference on loglog scale for RBE; (h) Frequency difference versus wavenumber difference on loglog scale for DBE (orange) and the lower spectral branch (blue) Figure 2.3. Outgoing power from all ports for excitation at port 1 only (N = 65). (a) Outgoing power from ports 1 and 3. (b) Outgoing power from ports 2 and 4. (c) The band diagram shows the two lowest order modes. The transmission resonances are near the band edge of the second mode (indicated in orange). The second mode exhibits a DBE. The yellow region indicates the frequency range of the plotted s-parameters Figure 2.4. Outgoing power from all ports for excitation at port 3 only (N = 65). (a) Outgoing power from ports 1 and 3. (b) Outgoing power from ports 2 and Figure 2.5. Outgoing power from all ports at the fundamental resonance ( = nm), for excitation at both ports 1 and 3, parameterized by α 2 and ϕ (N = 65). (a)&(c) Reflected power from ports 1 and 3, respectively. (b)&(d) Transmitted power from ports 2 and 4, respectively Figure 2.6. Outgoing power versus wavelength for excitation at both ports 1 and 3 (N=65). Red lines correspond to the excitation condition for maximum transmission from ports 2 and 4 (condition 1). Blue lines correspond to the excitation condition for maximum reflection from ports 1 and 3 (condition 2). (a) Sum of the reflected power from ports 1 and 3. (b) Sum of the transmitted powers from ports 2 and Figure 2.7. (a) Quality factor scaling as a function of number of periods for 50 nm gap DBE. Fitted dashed line corresponds to RBE-like behavior. Fitted solid line corresponds to DBE-like behavior. (b) Quality factor scaling as a function of the number of periods for two DBE structures with 50 and 150 nm gaps. True DBE-like behavior is only observed for the 50 nm gap structure in the plotted region Figure 2.8: (a) The envelopes of the fields inside the waveguide at the resonance frequency for N = 65. The E x component (black line) is decomposed into two evanescent Bloch modes (yellow and blue lines) and two propagating Bloch modes (green and red lines); (b)-(c) Resonance frequency versus the number of periods fitted for DBE and RBE, respectively x

12 Figure 3.1. Coupling-gap degenerate band edge (cgdbe) periodic structure realized in silicon-on-insulator. (a) Top-down view, (b) Cross-section Figure 3.2. Zero-coupling-gap DBE (zcgdbe) periodic structure realized in silicon-on-insulator. (a) Top-down view, (b) Cross-section Figure 3.3. Complex dispersion relationships calculated from 3D FEM. (a) Strip waveguide, (b) Strip waveguide with holes, (c) Coupling-gap RBE, (d) Coupling-gap DBE, (e) Zero-coupling-gap RBE, (f) Zero-coupling-gap DBE Figure 3.4. Comparison of quartic dispersion regimes for DBE designs with and without a coupling gap. (a) cgdbe and (b) zcgdbe Figure 3.5. Finite length zcgdbe resonator. (a) Transmission resonances, (b) Electric field magnitude, (c) Bloch mode decomposition for first resonance. For definiteness, the number of periods is N = 30 and the period a = 380 nm Figure 3.6. (a) Frequency of first resonance, relative to the band edge, versus number of periods for cgdbe and zcgdbe resonators. (b) Q-factor of first resonance versus the number of periods Figure 3.7. Top view of zcgdbe resonator with y-junction terminations Figure 3.8. Port 1 excitation of zcgdbe resonator with Z L as parameter. (a) Outgoing power at Port 1, (b) Outgoing power at port 4, (c) Electric field distribution for first resonance. For definiteness, the number of periods is N = 30 and the period a = 380 nm Figure 4.1: Device schematic. (a) Angled view illustration of DBE cavity. Top cladding is not shown. (b) Cross-sectional view of periodic region. The silica top cladding has two layers. Bottom layer is 540 nm of spin-onglass. Top layer is 500 nm of PECVD SiO Figure 4.2: (a) Cross-section of two parallel Si waveguides separated by 225 nm gap. SiH 4 based PECVD SiO 2 top cladding deposited at 9.5 Torr 350 C and 175 Watts RF power. (b) Cross-section of two parallel Si waveguides separated by 225 nm gap with TEOS based PECVD SiO 2 deposited by Noel Tech Figure 4.3 Two Si waveguides separated by 75 nm are planarized with 540 nm of FOx-15 SOG xi

13 Figure 4.4: Ellipsometry measurements of FOx-15 (blue squares) and PECVD SiO 2 (orange circles) films on silicon substrate after rapid thermal anneal. (a) Thickness of thin film as a function of RTA temperature. (b) Film refractive index at 632 nm as a function of RTA temperature Figure 4.5 (a) Optical micrograph of ports 2 & 4. Port 2 is terminated with a matched load where a 6.5 μm inverse silicon taper converts the silicon strip waveguide mode to a SiO 2 slab mode. Port 4 is terminated with a standard cantilever coupler for coupling to a tapered fiber. (b) Inset shows angled view of finite-difference time-domain simulation. In the schematic only, the SiO 2 top cladding is made transparent for half of the simulation domain revealing the underlying silicon waveguide. Reflected power at the simulation input is measured and presented Figure 4.6 The hole radius design parameter is considered. Each point on the curve represents a DBE design. (a) Plot of period versus hole offset. (b) Plot of period versus band edge wavelength. (c). Plot of offset versus band edge wavelength. Parameters not specified in the plots are identical to those given in Chapter 3 for the zcgdbe Figure 4.7 The waveguide width design parameter is considered. Each point on the curve represents a DBE design. (a) Plot of period versus hole offset. (b) Plot of period versus band edge wavelength. (c) Plot of offset versus band edge wavelength. Parameters not specified in the plots are identical to those given in Chapter Figure 4.8 (a) Top-down optical micrograph of DBE resonator showing all four ports. (b) Angled-view SEM micrograph of etched DBE device. The DBE device is patterned on a silicon substrate that is co-processed with SOI DBE resonators. Dashed box in (a) correspond to image in (b) Figure 4.9 (a) SEM cross-section of etched silicon waveguide co-processed with SOI DBE resonator. The waveguide was designed to have a width of 730 nm. Sidewall angle of the etched waveguide is 85. The widths at the top and bottom of the waveguide are w top = 724 and w bottom = 734. (b) SEM cross-section of etched silicon waveguide co-processed with SOI DBE resonator. The waveguide was designed to have a width of 730 nm and 220 nm gap. Measured gap dimensions are used to approximate dimensions of etched holes. Gap dimensions are 2r top = 2r bottom = 232 nm Figure 4.10 The propagating mode closest to the band edge is plotted for seven different periodic structures that are close to a DBE. The structures are discussed in detail section 4.6. Their dispersion is compared against the zcgdbe structure presented in Chapter 3. The zcgdbe is plotted in black xii

14 Figure 4.11 Using the dispersion data presented in Figure 4.10, the wave number difference versus frequency difference is plotted. Six design variations are compared to the zcgdbe of Chapter 3 and its quartic fitting. One of the design variations (solid blue line) also has a DBE. The other design variations are classified as a RBE or SBE Figure 5.1 Optical device measurement and characterization setup. Light from a tunable laser source is modulated with a lithium niobate modulator at 4GHz. A polarization controller is used to maximize transmission through the modulator. A second polarization controller converts light from the modulator to quasi-te polarization. Light is coupled into and out of the device under test (DUT) with tapered SMF-28 fiber aligned to cantilever couplers on chip. The light out of the DUT is directed to both a low speed photodetector and a high speed photodetector connected to a vector network analyzer. Our setup allows for measurement of both carrier amplitude and modulated RF signal amplitude/phase Figure 5.2 Measured group delay. Measured transmission response from port 1 to port 4 of DBE resonator with N = 35, w = 730 nm, r = 110 nm and a = 390 nm. The carrier is modulated at f mod = 4 GHz. (a) Amplitude of modulated RF signal recorded by vector network analyzer is plotted in orange. The corresponding amplitude of the optical carrier is plotted in blue. (b) The phase of the modulated RF signal is plotted in orange. The phase ϕ (in degrees) of the RF signal is related to the group delay τ g = - ϕ/(360 f mod ). The group delay is plotted on resonance for the first five resonances as black triangles Figure 5.3: Measured transmission response from port 1 to port 4 of DBE resonator with N = 35, w = 730 nm, r = 110 nm and a = 390 nm. Fano lineshapes are fit to the first four resonances Figure 5.4: Quality factor versus number of periods. The Quality factor is plotted on a logarithmic scale as a function of the number of periods for both 3D finite difference time domain simulations and measured DBE cavities where w = 730 nm, r = 110 nm and a = 390 nm. Quality factor scaling is shown for the first three resonances. Lines of best fit corresponding to Q N 3 and Q N 5 are fit to the first resonance. Quality factor scaling to the fifth power is observed for N > 25 periods for the first resonance Figure 5.5 Two measurements demonstrating quartic dispersion. Experimental results correspond to resonator with design parameters N = 35, w = 730 nm, r = 110 nm and a = 390 nm. (a) Fano lineshapes are fit to the optical transmission response and from the fitting the resonance wavelength is extracted from the seven resonances closest to the band edge. The xiii

15 wavevector for each resonance is given approximately as ka/(2 ) 1/2 m/(2n) where N is the number of periods, m is the resonance order and m = 1 corresponds to the resonance closest to the band edge. The resonance wavelength is plotted against the approximate wave vector as black squares. Experimental dispersion is compared to PWE simulation where the quartic band edge mode is plotted in orange. (b) The optical carrier is modulated at f mod = 4 GHz. The group velocity is calculated from the phase shift of the RF signal and plotted as black triangles for the five resonances closest to the band edge. The phase delay was too small to measure when m > 5. Experimental results are compared with PWE simulation where the quartic DBE is plotted in orange. Dimensions used in PWE simulation correspond to SEM cross-sections presented in Figure 4.9 and ellipsometry measurements. The hole radius is 116 nm. The waveguide width at the top is 724 nm and 734 nm at the bottom. The period is a = 390 nm. The FOx-15 layer thickness is 540 nm and the refractive index is The BOX layer is 1 μm thick and its refractive index is The PECVD SiO 2 is 500 nm thick and its refractive index is xiv

16 Chapter 1: Introduction 1.1 Background Physical properties such as refractive index and absorption determine how light propagates through and interacts with a material. Engineered artificial materials allow access to unique electromagnetic properties not observed in nature [1] [9]. The periodic dielectric structure [10] is one important class of engineered artificial material. Analogous to an electronic band structure [11], band gaps form in materials with a modulated refractive index. Light cannot propagate at frequencies inside the band gap. Although the term photonic crystal has been used in literature to describe any periodic dielectric structure with a photonic band gap, we use the more generic term periodic dielectric structure to include structures with one-dimensional periodicity [12]. Examples of periodic dielectric structures with a photonic band gap include but are not limited to fiber Bragg gratings [13], dielectric mirrors[14], and photonic crystals[15]. Scanning electron microscope images of a few representative structures are shown in Figure

17 Figure 1.1. Scanning electron microscope images of three different periodic dielectric structures with a photonic bandgap. (a) Fiber Bragg grating [16] patterned on 1.8 μm diameter silica fiber with focused ion beam milling. (b) Three dimensional photonic crystal [17] formed in polycrystalline silicon with band gap between 10.0 μm and 14.5 μm. (c) Cross section of one dimensional mirror[18] with alternating layers of GaAs and Al 0.9 Ga Cavities in Periodic Dielectric Structures Defect and Band Edge Cavities Cavities or resonators formed in periodic dielectric structures have many exciting applications [19] ranging from telecommunications [20] to quantum mechanics [21] to sensors on a chip [22] [23]. Cavities based on periodic structures can be categorized into two classes, namely, defect cavities and band edge cavities. Defect cavities are formed by breaking the translational symmetry of a periodic structure [24] [26]. Mode localization occurs due to the defect and the resonance frequencies are inside the band 2

18 gap [10]. Small modal volume and high quality factor are characteristics of defect cavities. Examples of defect cavities are shown in Figure 1.2 (a) & (b). On the other hand, band edge cavities are favorable for applications requiring a larger modal field distribution [27]. Mode localization occurs due to the finite length of an otherwise periodic structure and the resonance frequencies are near the band edge but outside the band gap. Suitable applications of band edge cavities include light emission [28] [30], optical switching [31] and nonlinear optics [32]. Examples of band edge cavities with unbroken translational symmetry are shown in Figure 1.1 (a)-(c). Figure 1.2. Examples of defect cavities. (a) Modified 3 hole defect cavity in a 2D hexagonal lattice [33]. (b) 1D periodic cavity formed by broken translational symmetry at center [34] Dispersion Near the Band Edge A regular band edge (RBE) can be approximated by a quadratic dispersion relationship, ω ω 0 ~ (k k 0 ) 2, where is the frequency and k is the wave vector. Parameters ω 0 and k 0 designate the location of the stationary point in the dispersion at the band edge. An example of a RBE is shown in Figure 1.3 (a). Quadratic dispersion results in resonator quality factors, Q, that scale to the third power of the number of 3

19 periods N. Band edge cavities with similar Q can be significantly miniaturized if the resonance is located near a degenerate band edge (DBE) [35], [36]. DBE resonators can be approximated by a quartic dispersion relationship, ω ω 0 ~ (k k 0 ) 4 (see Figure 1.3 (b) ). The Q-factor scales to the fifth power of the number of periods near the band edge [37]. A DBE resonator with N periods can have an internal intensity comparable to an RBE resonator with N 2 periods[35], which is a significant difference with respect to miniaturization. In other words, DBE designs can be N times smaller than RBE designs with the same internal intensity. (a) (b) A 1 B A 1 A 2 B (ω ω 0 ) ~ (k k 0 ) 2 (ω ω 0 ) ~ (k k 0 ) 4 Frequency ω ω (ω 0, k 0 ) Frequency ω ω (ω 0, k 0 ) Wave vector k Wave vector k Figure 1.3. The two panels highlight the differences between RBE and DBE structures. The frequency ω is plotted in red as a function of the wave vector k. The unit cells of both periodic structures have at least one anisotropic dielectric layer. Because of the anisotropy, the dispersion is dependent on the incident polarization. Therefore, we observe two bands of propagating modes at frequencies below the first bandgap. The bandgap begins at frequency ω 0 and wave vector k 0. (a) Dispersion diagram of periodic structure with RBE. Unit cell has one anisotropic layer A1 and one isotropic layer B. (b) Dispersion diagram of periodic structure with DBE. Unit cell has two anisotropic layers A1 & A2 and one isotropic layer B. 4

20 Realization of a quartic DBE requires a geometry that supports four eigenmodes at a single frequency. Considering structures that are periodic in the z-direction, the eigenmodes are a pair of z directed propagating modes with real eigenvalues and a pair of z directed evanescent modes with complex eigenvalues. At the band edge (ω 0, k 0 ), there is only one eigenvector with repeated eigenvalue, k 0, of multiplicity four [38]. For a small range of wave vectors near the band edge, however, the propagating and evanescent modes are nearly parallel to each other and the eigenvalues are approximately degenerate. The approximately parallel eigenvectors allow both propagating and evanescent modes to contribute to cavity miniaturization through the formation of standing waves with large field amplitudes on resonance in finite length resonators. Evanescent modes also contribute to mode matching at the interface between the periodic cavity and the feed waveguide, allowing for high transmission on resonance [39]. Further away from the band edge, the eigenmodes become sufficiently distinct and thus it is no longer appropriate to approximate the dispersion as quartic. 1.3 DBE Research by Others One Dimensional Stacks with Anisotropic Layers The DBE was first theoretically described in the context of one dimensional (1D) layered periodic structures with a high degree of birefringence (anisotropy) [35]. While useful for theoretical analysis [36] [38], [40] [42] the 1D birefringent configuration is not practical for physical realization at optical frequencies. However, results including transmission response and internal field measurements have been reported [43] for form birefringent structures designed to have a DBE near 10 GHz (see Figure 1.4 (a) ). Also, 5

21 high gain antenna measurements [44] have been shown for an antenna assembly of a slot coupled microstrip line combined with a six layer stack of form birefringent ceramic. (see Figure 1.4 (c) ) Fiber Gratings Another platform more suitable for creating a DBE at optical wavelengths is a multimode optical fiber with multiple gratings [39], [45], [46]. A complication with the physical realization of DBEs in fibers is that it requires precise control of the period and relative amplitude of each individual grating Coupled Microstrip Lines Propagation through layered anisotropic media can be emulated with coupled microstrip lines [47] [48]. There has been no experimental demonstration of a coupled microstrip line DBE. However, a band diagram of a magnetic photonic crystal (closely related to a DBE) has been measured [49] near 3 GHz. Miniaturization of a loop antenna formed by two unit cells of a coupled microstrip DBE has also been shown [50] (see Figure 1.4 (b) ) Coupled Periodic Dielectric Waveguides A fourth option is a system of two coupled periodic dielectric waveguides [51]. We refer to this design as a coupling-gap DBE (cgdbe). An example is shown in Figure 1.4 (d), where two parallel rows of dielectric posts, with period of 14 mm, are separated by a coupling gap of 14 mm [52]. An RBE results when longitudinal shift is zero. The RBE is converted into a DBE at a particular non-zero value of longitudinal shift. 6

22 Figure 1.4. Examples of RF structures designed to have a DBE. (a) 100 periods of form birefringent ABS [44] with a band edge near 10 GHz. (b) Microstrip antenna [50] based on DBE unit cell with resonance at 2.59 GHz. (c) 3 periods of low loss form birefringent ceramic laminates [43]. (d) Coupled periodic waveguides with arrays of dielectric pillars [52] with resonances near 9 GHz 1.4 Contributions of This Dissertation The work presented in this dissertation describes the design, simulation, fabrication, and experimental validation of DBE resonances realized in a silicon on insulator platform for operations at optical telecommunication wavelengths. Three key contributions that are unique to this work include: 1) The study and analysis of transmission resonances near a DBE in finite length cavities. Full wave three-dimensional finite-difference time-domain simulations are used to study transmission resonances near a DBE. Specifically we look at the number of periods versus resonance frequency and number of periods versus resonance quality factor. It is observed that for the 7

23 coupling gap design at least 65 periods are needed to see quality factor scaling to the fifth power. 2) The design and analysis of an improved coupling-gap design where the coupling gap is zero nm. Compared to a coupling-gap design, the zerocoupling-gap has two fewer sidewalls. Designs that reduce sidewall surface area per unit length would result in less sidewall surface scattering. More importantly, simulations show that only 25 periods are required to observe quality factor scaling to the fifth power. 3) Characterization and validation of a fabricated DBE cavity designed to operate at telecommunication wavelengths. This work is the first demonstration of quality factor scaling to the fifth power of periods. This is also the first demonstration of a band edge with quartic dispersion. Fifth power quality factor scaling and quartic dispersion are defining characteristics of a DBE, but have not been demonstrated before this work at any frequency. The work contains the first experimental results reported for a DBE at frequencies larger than 10 GHz. 1.5 Organization of the Dissertation In Chapter 2, full three-dimensional analysis is used to show that coupled periodic optical waveguides can exhibit a giant slow light resonance associated with a DBE. We consider the silicon-on-insulator material system for implementation in silicon photonics at optical telecommunications wavelengths. It is shown that the coupling of the resonance mode with the input light can be controlled continuously by varying the input power ratio 8

24 and the phase difference between the two input arms. Near unity transmission efficiency through the degenerate band edge structure can be achieved, enabling exploitation of the advantages of the giant slow wave resonance. In Chapter 3, we present the design and analysis of zero-coupling-gap degenerate band edge resonators. Complex band diagrams are computed for the unit cell with periodic boundary conditions that convey characteristics of propagating and evanescent modes. Dispersion features of the band diagram are used to describe changes in resonance scaling in finite length resonators. Resonators with non-zero and zero coupling gap are compared. Analysis of quality factor and resonance frequency indicates significant reduction in the number of periods required to observe fifth power scaling when degenerate band edge resonators are realized with zero-coupling-gap. High transmission is achieved by optimizing the waveguide feed to the resonator. Compact band edge cavities with large optical field distribution are envisioned for light emitters, switches, and sensors. In Chapter 4, we discuss the fabrication of DBE resonators in SOI. Additional changes or tweaks are made to the design as it transitions from a computer model to a working device. Unused ports are terminated with matched loads. Light is coupled into and out of the resonator using compact cantilever couplers. We characterize how dimensions change during fabrication. To observe DBE resonance in lab, we make 18 different design variations. The DBE sensitivity to changes in design parameters is considered. 9

25 In Chapter 5, we report the details of the first experimental demonstration of Quality factor scaling to the fifth power. Transmission resonances near a degenerate band edge are realized in silicon strip waveguides with one-dimensional periodicity. Quality factors of 27,000 in a 35 period cavity are observed. Transmission measurements show Fano resonances with a sharp transmission peak to bandgap extinction ratio of 20 db. Conclusions and a discussion of future work are treated in Chapter 6. 10

26 Chapter 2: Degenerate Band Edge Resonances in Coupled Periodic Silicon Optical Waveguides 2.1 Chapter Introduction Slow wave resonance occurs in the vicinity of transmission band edges of many finite periodic structures including: photonic crystals, stratified media, periodic waveguides, and chains of coupled resonators. For this reason, a slow wave resonance is often referred to as a transmission band edge resonance. A regular slow wave resonance occurs near a regular transmission band edge (RBE), where the dispersion relation, ω(k), can be approximated as ω ω 0 ~ (k k 0 ) 2. The defining characteristic of a giant slowwave resonance is that it occurs near a degenerate transmission band edge (DBE), where the dispersion curve can be approximated as ω ω D ~ (k k D ) 4. Detailed theoretical analysis of this and related phenomena can be found in [35] [37], [39], [45], [46], [51], [53], and references therein. A regular slow wave resonance is usually a simple standing wave composed of two Bloch eigenmodes with equal and opposite real-valued wave numbers. By contrast, a DBE-related giant slow wave resonance cannot be thought of this way [37], because contribution of the evanescent modes to the resonance field inside the photonic structure becomes equally important. The energy stored in a regular slow wave resonator is proportional to N 2, where N is the number of unit cells in the periodic 11

27 structure. By comparison, the energy stored in a giant slow wave resonator is proportional to N 4, which for sufficiently large N, is a much larger amount of energy for the same size of resonator (see, for example, [37]). The latter circumstance makes the giant slow wave resonance particularly attractive for applications in lasing [27], [29], sensing [54], and switching [31], [55]. The possibility of a giant transmission resonance in a structure depends on the presence of a DBE in the Bloch dispersion relation. An essential element is that the structure has two modes which are coupled. In periodic layered structures, a DBE can exist because of the coupling between two polarization modes when each unit cell contains layers with sufficiently strong and misaligned birefringence [35], [37]. A more practical alternative is provided by using waveguides. The two modes that are essential for a DBE can be created by tuning the waveguide width. DBEs were shown to exist in different waveguiding structures such as optical fiber with multiple gratings [39], [45], [46] and coupled periodic waveguides [51], [53]. A complication with the physical realization of DBEs in fibers is that it requires precise control of the period and relative amplitude of each individual grating. The more realistic option is the formation of a DBE-related giant transmission resonance in a pair of coupled periodic optical waveguides [51], [53]. Referring to Figure 2.1(a), waveguide modes are coupled by cylindrical holes punched in coupled strip waveguides. The dispersion relationship can be controlled by tuning the longitudinal shift between the coupled waveguides. The platform is attractive for realizing a DBE in compact integrated optics. Before this work, numerical studies of the dispersion relationship of 12

28 coupled periodic dielectric waveguides for the realization of a DBE have only been performed with two-dimensional simulations. Furthermore, finite length waveguides capable of developing DBE-related giant transmission resonances are yet to be analyzed. In this chapter we present the dispersion engineering of coupled periodic silicon planar waveguides exhibiting a DBE at telecommunications wavelengths. We choose the silicon-on-insulator (SOI) material system for applications in silicon photonics [56], [57]. Using 3D finite difference time domain simulations, we fully characterize the scattering matrix of four-port DBE periodic structure, shown in Figure 2.1, with two input ports and two output ports. The optical reflection and transmission properties of the structure can be controlled continuously by varying the input power ratio and the relative phase difference between the two input arms. The ability to tune input coupling through input amplitude and phase control enables high transmission efficiency devices that can exploit the extraordinary internal field intensities associated with the DBE-related giant transmission resonance. This chapter is organized as follows. In section 2.2, the design of the coupled periodic dielectric waveguides exhibiting a DBE is presented. The transmission and resonance properties of finite length DBE structures are described in section 2.3. The quality factor scaling of the DBE resonance is also shown as a function of the number of periods and the gap width between waveguides. Section 2.4 discusses the location of the resonances and section 2.5 provides a conclusion. 13

29 2.2 Design The structure under consideration is shown in top view in Figure 2.1(a) and in cross-sectional view in Figure 2.1(b). Two dielectric strip waveguides which have two periodic arrays of cylindrical holes are brought into close proximity. The core material of the waveguides is silicon. The cladding and the material in the cylindrical holes is silicon dioxide (SiO 2 ). The cross-sectional width and height of an individual waveguide are 450 nm and 250 nm respectively. These values are chosen so that the feed strip waveguides into the periodic structure are single mode for TE optical polarization (dominant electric field component in the x direction). We denote the gap separation between the waveguides with parameter x 0 and the longitudinal offset between waveguides with parameter z o. The longitudinal period between the cylindrical holes is denoted with parameter a. The coupled structure is a four-port network. The ports are labeled as indicated in Figure 2.1(a). (a) (b) a 1 2 x 3 z 4 0 x nm Silicon SiO 2 z x y x nm Figure 2.1. Coupled periodic silicon optical waveguides designed to exhibit a DBE at telecommunications wavelengths; (a) Top view; (b) Cross-section showing waveguides are completely encapsulated in SiO 2 14

30 Dispersion relationships for the infinitely long periodic structures were calculated using the freely available MIT Photonic Bands (MPB) software package that implements the plane-wave expansion method in three-dimensions [58]. We use a refractive index of 3.48 for the silicon and 1.45 for the silicon dioxide. The dispersion engineering of a DBE in two coupled waveguides is explained in Figure 2.2. Figure 2.2(a) shows the band structure, in the first Brillouin zone, of a reference single mode feed waveguide with no holes. Figure 2.2(b) shows the splitting of the single mode by the coupling between two waveguides in close proximity. The artificial folding of the dispersion about the point ka/(2 ) = 0.5 observed in Figure 2.2(a) and (b) is a consequence of the plane wave expansion. Although the structures are not periodic, MPB applies an artificial periodicity of length a. The introduction of holes in the strip waveguides causes a band gap (gray rectangular patch) to open as shown in Figure 2.2(c) and Figure 2.2(d). A non-zero longitudinal offset between the coupled periodic dielectric waveguides leads to a split band edge, shown in the top band in Figure 2.2(c), and a DBE, shown in Figure 2.2(f) (orange). For the DBE in Figure 2.2(f), both the group velocity and group velocity dispersion are zero (i.e. / k = 2 / k 2 = 0 ) at ka/(2 ) = 0.5. In order to scale the DBE wavelength to be near 1500 nm, the period is chosen to be a = 325 nm. Parameter values are chosen by considering the limits of our electron-beam lithography (EBL) process. The gap between the waveguides is set to x 0 = 50 nm, the EBL process minimum feature size. In the following chapter, the importance of minimizing x 0 is discussed. The EBL software discretizes all design features with a 5 nm resolution. The discretization is most severe in holes with a small 15

31 radius. As the hole radius increases, discretization is less observable. The decision to set hole radius equal to 100 nm is a compromise between maximizing hole radius to reduce discretization effects and avoiding a hole boundary that is too close the waveguide edge. The lateral offset needed for a DBE is z 0 = 0.37a = 120 nm and is determined by sweeping z 0 from 0 to a/2 to find a value that minimizes group velocity near ka/(2 ) = 0.5. A plot of frequency difference versus wavenumber difference from the degenerate band edge, plotted on a loglog scale in Figure 2.2(h), shows a slope of 4, verifying the quartic nature of the band gap. The spectral region where the lowest order of the band edge has a quartic dependence is approximately a/(2 c) < 10-4 and if fitted to D k kd 4 / 4 (2-1) yields 4 = /(a 3 c) for degenerate band edge frequency D a/(2 c) = For comparison, frequency difference versus wavenumber difference for a periodic single waveguide is plotted in Figure 2.2(g). On the loglog scale, the slope is 2 (quadratic), as expected. 16

32 Figure 2.2 (a)-(f) Dispersion diagrams that show frequency as a function of wavevector. Only propagating modes with real valued k are shown. In (c) (f) the bandgap is indicated by a gray rectangular patch. The complete bandgap is not shown. (a) Single mode waveguide; (b) Two coupled single mode waveguides; (c) Single periodic dielectric waveguide; (d) Two coupled periodic dielectric waveguides; (e) Two coupled periodic waveguides with a 0.5a lateral shift; (f) Lateral shift of 0.37a to realize a DBE; (g) Frequency difference versus wavenumber difference on loglog scale for RBE; (h) Frequency difference versus wavenumber difference on loglog scale for DBE (orange) and the lower spectral branch (blue). 2.3 Transmission and Resonance Properties near the DBE Having established that a DBE can be realized in the infinitely long periodic structure, we next calculate the transmission and resonance properties for wavelengths close to the DBE for a finite length waveguide of L = an, where N is the number of 17

33 periods. Spectral resonances are the result of the finite length of the periodic structure. A commercial-grade simulator [59] based on the three-dimensional finite-difference timedomain method [60] [63] was used to perform the calculations and obtain the transmission response. The simulated structure is the 4-port structure of Figure 2.1. A uniform mesh grid of nm, equivalent to 20 points per period, is applied to the periodic section of the structure. At 20 points per period, the spectral response of the FDTD simulation is consistent with the DBE band diagram shown in Figure 2.2(f) calculated via plane-wave expansion method. Outgoing power is normalized by the total power injected into the system. We are interested in the transmission properties of the structures for arbitrary excitation input into ports 1 and 3. The spectral properties are characterized by a 4 4 scattering matrix (S-matrix) [64]. The entire 4 4 S-matrix is obtained from four FDTD simulations, where a single simulation determines or defines one column of the S-matrix. The first simulation defines column 1 of the S-matrix. The excitation is at port 1 while the other ports are terminated with a matched load via a perfectly matched layer (PML) boundary condition [65] [67]. In general, the nth column is determined by exciting the nth port and terminating the remaining ports with a matched load. Outgoing waves from the 4 port network are related to the input excitation via the scattering matrix as follows E E E E S S S S S S S S S S S S S S S S (2-2) where 18

34 2 j 1 e, 0 1, 0 2 (2-3) and E i is the outgoing electric field at the i th port. In this notation, the magnitude of the power input into ports 1 and 3 is 2 and β 2, respectively. The phase difference between port 1 and 3 excitations is. Figure 2.3 shows the outgoing power from all ports for N = 65 and excitation at port 1 only (i.e. β = 0). The reflected power from port 1 is denoted S 11 2 and the outgoing power from port 3 is denoted S We observe three resonance peaks corresponding to the three lowest order resonances of the finite periodic structure. The narrowest resonance, near µm, is the fundamental resonance closest to the band edge. On the fundamental resonance, the reflected power from port 1 is relatively low. Off resonance, the reflected power is relatively large. Figure 2.3(b) shows that the outgoing power from ports 2 and 4 are relatively low for all wavelengths. At the fundamental resonance, there is nearly equal outgoing power from both ports 2 and 4. Figure 2.4 shows the outgoing power from all ports for the case of excitation at port 3 only. In this case, the reflected power from the excitation port, S 33 2, shows a smaller value of extinction ratio for the fundamental resonance and larger values of extinction ratio for the second and third resonances, when compared to S The extinction ratio differences between Figure 2.3 and Figure 2.4 are due to the asymmetry of the physical structure. In each excitation case the evanescent and propagating modes are coupled with different intensities, which affect the total transmission and extinction ratio. Furthermore, larger transmission peaks are observed for the outgoing power through port 2, for the second and third resonances, 19

35 while the fundamental resonance is slightly smaller. Finally, the S 13 2 is observed to be equal to S 31 2, as expected, since the 4-port network is reciprocal. Figure 2.3. Outgoing power from all ports for excitation at port 1 only (N = 65). (a) Outgoing power from ports 1 and 3. (b) Outgoing power from ports 2 and 4. (c) The band diagram shows the two lowest order modes. The transmission resonances are near the band edge of the second mode (indicated in orange). The second mode exhibits a DBE. The yellow region indicates the frequency range of the plotted s- parameters. To scan the possible transmissions and reflections for arbitrary input into both ports 1 and 3, Figure 2.5 shows the outgoing power from all ports given a parametric sweep of α 2 (input power into port 1) and ϕ (the phase difference between the excitations at port 1 and port 3) for a 65 period DBE at the fundamental resonance. The corresponding input power into port 3 is β 2 = 1 - α 2. We observe two regions on the color plot. The first region corresponds to maximum outgoing power (transmission) at ports 2 and 4. The second region is characterized by maximum outgoing power (reflection) at the excitation ports 1 and 3. The maximum transmission state occurs when α 2 = , 2 = , and = o (condition 1). Likewise, the minimum transmission state (maximum reflection) is observed for α 2 = , 2 = , and 20

36 = o (condition 2). The observation that conditions 1 and 2 are complementary can be understood from the analysis in [45]. Figure 2.4. Outgoing power from all ports for excitation at port 3 only (N = 65). (a) Outgoing power from ports 1 and 3. (b) Outgoing power from ports 2 and 4. Figure 2.5. Outgoing power from all ports at the fundamental resonance ( = nm), for excitation at both ports 1 and 3, parameterized by α 2 and ϕ (N = 65). (a)&(c) Reflected power from ports 1 and 3, respectively. (b)&(d) Transmitted power from ports 2 and 4, respectively. 21

37 We also plot the sum of the outgoing power from ports 1 and 3 (Figure 2.6(a)) and ports 2 and 4 (Figure 2.6(b)) as a function of wavelength when the excitation corresponds to either maximum combined transmission out of ports 2 and 4 (labeled as condition 1) or minimum combined transmission out ports 2 and 4 (labeled as condition 2). The excitation is at both ports 1 and 3. Red lines correspond to condition 1 and blue lines correspond to condition 2. At condition 1, the peak transmission at the fundamental resonance approaches unity. Figure 2.6. Outgoing power versus wavelength for excitation at both ports 1 and 3 (N=65). Red lines correspond to the excitation condition for maximum transmission from ports 2 and 4 (condition 1). Blue lines correspond to the excitation condition for maximum reflection from ports 1 and 3 (condition 2). (a) Sum of the reflected power from ports 1 and 3. (b) Sum of the transmitted powers from ports 2 and 4. The quality factor of the fundamental resonance as a function of the number, N, of unit cells is shown in Figure 2.7(a). It is not until N is greater than 65 that the quality factor becomes proportional to N 5, which is characteristic of a DBE-related transmission resonance (see, for example, [37]). For N < 30, the quality factor Q is proportional to N 3, which is characteristic of a RBE-related transmission resonance. Results from a second DBE structure with weaker coupling between waveguides is shown in Figure 2.7(b). The 22

38 second structure has gap width x 0 = 150 nm and longitudinal offset z 0 = 145 nm. Both cases exhibit quality factor scaling to a power greater than three for N greater than 40. However, we do not observe scaling to the fifth power for the 150 nm gap when N is less than 90 periods. Smaller gap widths are therefore observed to be necessary in order to realize N 5 scaling in more compact structures. Figure 2.7. (a) Quality factor scaling as a function of number of periods for 50 nm gap DBE. Fitted dashed line corresponds to RBE-like behavior. Fitted solid line corresponds to DBE-like behavior. (b) Quality factor scaling as a function of the number of periods for two DBE structures with 50 and 150 nm gaps. True DBE-like behavior is only observed for the 50 nm gap structure in the plotted region. 2.4 Spectral Location of Resonances In close vicinity of a DBE, inside the transmission band, there are four Bloch eigenmodes a pair of propagating modes with equal and opposite real wave numbers, and a pair of evanescent modes with complex conjugate wave numbers [35] [37], [39], 23

39 [45], [53]. The frequency dependence of the four Bloch wave numbers can be approximated by the following expression k j k D i j 4 4 D D k k e 2, j 0,1,2,3, (2-4) where k 4 4 D. The wave numbers of the propagating and evanescent modes are, respectively k kd k and k1, 3 k i. (2-5) 0,2 D k At resonance frequency, we can decompose the E x component of electric field into the four Bloch modes [68]. The amplitude of each of the Bloch modes is shown in Figure 2.8(a) together with the total amplitude (black curve). Figure 2.8(a) shows that although the evanescent modes (blue and yellow curves) play an important role at the interfaces of the waveguide, they decay toward the center and their contribution become insignificant compared to that of the propagating Bloch modes. In the middle of the waveguide, the two propagating modes (red and green curves) are dominant, and their amplitudes are larger than that of the incident wave by a factor which scales as N 4. This distinguishes the giant DBE related transmission resonance from regular RBE-related transmission resonances, where the maximum amplitude in the middle of the periodic structure is proportional to N 2. In the case of a RBE, the resonance field inside the periodic structure is similar to a standing Bloch wave with the minima at the boundaries and the maximum in the middle of the periodic structure. By contrast, for a DBE, the propagating components of the resonance field do not form minima at the boundaries. Instead, the evanescent components play an equally important role by interfering destructively with 24

40 the propagating components. For this reason, the most effective coupling of the incident light with the DBE resonance occurs when the amplitudes of the evanescent and propagating components at either interface are comparable in magnitude. If, on the other hand, the parameters α and are such that only the evanescent (or only the propagating) waves are excited, the giant DBE resonance does not occur at all. The role of the parameters α and here is similar to that of the incident light polarization in the case of DBE resonance in a layered structure [35], [37]. At the resonance frequency, the forward and the backward propagating modes constructively interfere in the middle of the waveguide. For this to happen a round trip of a propagating wave inside the cavity must be equal to a multiple of 2π. The phase gathered by a propagating wave inside the structure is equal to kl, where L is the length of the cavity. Further, at the interfaces, when the propagating mode is reflected it gathers a phase φ. At the resonance, the total phase gathered in a round trip must be equal to 2 m 2 2kL, (2-6) where m is an integer. According to Eq. (2-4), k = k D + Δk, where k D is the edge of the Brillouin zone and is equal to π/a. The length is taken to be an integer, N, periods multiplied by a single period length, a. Hence, 2k D an is always equal to a multiple of 2π, so 4 2 1/. 2 m 2 4 an (2-7) The resonance frequency versus the number of periods follows D 4 m 4 / ( an). (2-8) D 4 25

41 We fit Eq. (2-8) to the resonance frequencies calculated via 3D FDTD [59] (Figure 2.8(b)) and find φ = 0.18(2π) for the DBE. For comparison, we also plot (Figure 2.8(c)) the resonance frequencies versus the number of periods for the RBE depicted in Figure 2.2(c). Figure 2.8: (a) The envelopes of the fields inside the waveguide at the resonance frequency for N = 65. The E x component (black line) is decomposed into two evanescent Bloch modes (yellow and blue lines) and two propagating Bloch modes (green and red lines); (b)-(c) Resonance frequency versus the number of periods fitted for DBE and RBE, respectively. The quality factor scaling shown in Figure 2.7(a), together with the decomposition of the field in Figure 2.8(a), demonstrate that a slow light giant DBE resonance can be created in waveguides, which can be effectively integrated. Thus slow 26

42 light resonances can potentially have very large quality factors that scale as N 5, which can be used instead of reflector based cavities, for such application as delay lines, sensors, light amplification [69], and nonlinear enhancement [70], where large distributed electric fields are important. Further benefit in using waveguides for giant DBE resonances is the ability to create higher order DBEs, such as sextic [ω - ω D ~ (k - k D ) 6 ], for which the quality factor will scale as N 7 [45], with the possibility of achieving higher quality factors in comparatively shorter waveguides. 2.5 Chapter Conclusion We presented a full three-dimensional design and analysis of coupled periodic dielectric optical waveguides exhibiting giant DBE-related transmission resonances at optical telecommunications wavelengths in the silicon-on-insulator material system. The design provides a continuous control of the coupling between the input light and the resonance mode. The coupling can be changed from zero to near unity transmission by adjusting the input power ratio and the phase shift between the two input arms. The proposed design paves a route towards on-chip applications that can exploit the extraordinary internal field intensities associated with the DBE-related giant transmission resonance. 27

43 Chapter 3: Zero-coupling-gap Degenerate Band Edge Resonators in Silicon Photonics 3.1 Chapter Introduction The DBE was first theoretically described in the context of one dimensional (1D) layered periodic structures with a high degree of birefringence [35]. While useful for theoretical analysis, the 1D birefringent configuration is not practical for physical realization at optical frequencies. Another platform more suitable for creating a DBE at optical wavelengths is a multimode optical fiber with multiple gratings [39], [45], [46]. In the last chapter we described a system of two coupled periodic waveguides in planar integrated optics for realizing a DBE. We refer to this design as a coupling-gap DBE (cgdbe). As shown in Figure 3.1, two parallel periodic dielectric waveguides, with period a, are separated by coupling gap g. An RBE results when longitudinal offset z 0 is zero. The RBE is converted into a DBE at a particular non-zero value of longitudinal offset. Although the structure supports a DBE, quality factor scaling to the fifth power requires a large number of periods for resonances to exist within the quartic dispersion regime [71]. Techniques that reduce the required number of periods for the onset of fifth power scaling would enable more compact cavities. Furthermore, the cgdbe requires four sidewalls and is therefore susceptible to large propagation losses from sidewall 28

44 surface roughness [72]. Designs that reduce sidewall surface area per unit length would result in less sidewall surface scattering. In this chapter, we introduce the zero-coupling-gap DBE (zcgdbe), illustrated in Figure 3.2, where the coupling-gap [53], [73] has been entirely removed, to achieve fifth power quality factor scaling with fewer periods and lower sidewall surface scattering. The design is cast in the silicon-on-insulator (SOI) material system with a view towards silicon photonic integration [56], [57], [74] [76]. Complex band diagrams are computed for the unit cell with periodic boundary conditions to determine longitudinal offset, z 0, that converts a RBE into a DBE and to determine the role of both propagating and evanescent modes. Dispersion features of the band diagram are used to describe changes in resonance scaling in finite length resonators. Resonators with non-zero and zero coupling gap are compared. Analysis of quality factor and resonance frequency indicates significant reduction in the number of periods required to observe fifth power scaling when degenerate band edge resonators are realized with zero-coupling-gap. Finally, by optimizing the waveguide feed to the resonator, we show that high transmission through the structure can be achieved on resonance. (a) a (b) 2r cg g g h cg y x z 0 z Silicon SiO 2 z y x w cg Figure 3.1. Coupling-gap degenerate band edge (cgdbe) periodic structure realized in silicon-on-insulator. (a) Top-down view, (b) Cross-section. 29

45 (a) a (b) x 0 2r zcg h zcg x y y z z 0 Silicon SiO 2 z x w zcg Figure 3.2. Zero-coupling-gap DBE (zcgdbe) periodic structure realized in siliconon-insulator. (a) Top-down view, (b) Cross-section. 3.2 Complex Dispersion Relationships with Degenerate Band Edge Conversion of a RBE to a DBE The evolution of a RBE to a DBE is presented in Figure 3.3(a)-(f) using complex dispersion relations computed using three-dimensional finite element method (3D FEM [77], [78]). The weak expressions defining the quadratic eigenvalue problem are implemented using the commercial FEM program COMSOL Multiphysics following the procedure described in [8]. The geometries presented in Figure 3.3(a)-(d) are identical to the geometries in Figure 2.2 (a),(c),(d) & (f), respectively. However, the dispersion relations presented in Figure 2.2 show only propagating modes i.e. the k-vector is purely real. Whereas, Figure 3.3 shows both propagating modes and evanescent modes with complex-valued k-vector. Also, Figure 3.3 shows the complete first bandgap. Calculations are performed using infrared refractive indices of 3.48 for silicon and 1.45 for silicon dioxide. We consider only modes with quasi transverse electric (TE) optical polarization (dominant electric field component in the x-direction). Since the dispersion relationship is symmetric about the Brillouin zone edge, only +z directed propagating and evanescent modes are shown. 30

46 Our starting point is a single uniform strip waveguide of width w cg = 1.38a and height h cg = 0.77a that is translationally symmetric along the z-direction. The lowest order modes of the structure are shown in Figure 3.3(a). The introduction of a periodic array of holes, with radius r cg = 0.31a, produces a bandgap with regular band edge, as shown in Figure 3.3(b). For ω ω 0, there exists one propagating mode under the light line with purely real k. For frequencies in the bandgap, the mode is evanescent with complex valued wave vector. The structure of Figure 3.3(b) does not support a DBE which requires both propagating and evanescent modes at a particular frequency. Propagating and evanescent modes at a given frequency can be realized with the addition of a second parallel periodic waveguide. A RBE design with two coupled periodic waveguides separated by a distance of g = 0.15a is shown in Figure 3.3(c). The single mode of Figure 3.3(b) splits into two modes which are even and odd. The degree of mode splitting increases with shrinking coupling gap between the parallel waveguides. A zero-coupling-gap RBE design is shown in Figure 3.3(e) where the waveguide width w zcg = 1.95a, height h zcg = 0.66a, hole radius r zcg = 0.30a, and distance between waveguide center and hole center x 0 = 0.40a. Waveguide height is set to be the same for both coupling-gap and zero-coupling-gap designs. Width w zcg is constrained so that a uniform strip waveguide of dimensions w zcg h zcg supports only two TE modes. Radius r zcg and period a are adjusted to shift the band edge towards telecommunication wavelengths, since larger mode splitting in the zero-coupling-gap design moves the location of the band edge. Parameters for both designs are chosen to maximize mode- 31

47 splitting. Although both designs in Figure 3.3(c) and Figure 3.3(e) produce propagating and evanescent modes, the wave vectors are different at frequency ω 0. We break the mirror symmetry about the y-z plane for dispersion engineering of both the propagating and evanescent modes. The symmetry breaking is accomplished by translating one row of holes along the z-axis by an offset 0 < z 0 < 0.5a. There exists one value of z 0 that will transform the RBE of Figure 3.3(c) into a DBE. When z 0 is 0.37a, the bandgap of Figure 3.3(c) widens and the RBE flattens into a DBE as shown in Figure 3.3(d). Likewise, for z 0 equal to 0.24a, the RBE of Figure 3.3(e) becomes a DBE as shown in Figure 3.3(f). For ω ω 0, the lowest order mode is evanescent near the band edge. At ω 0, the imaginary part of the wave vector for the lowest order mode is zero and the dispersion in real k is quartic. Comparing the dispersion diagrams of the zcgdbe to the cgdbe, we observe that the flatness of the orange colored band of the zcgdbe extends over a wider range of k-vectors, indicating reduced group velocity. Designs that maximize the separation between the two lowest order modes of the two RBE structures in Figure 3.3(c) and Figure 3.3(e), indicated in orange and blue, will maximize the flatness of the DBE also. 32

48 Figure 3.3. Complex dispersion relationships calculated from 3D FEM. (a) Strip waveguide, (b) Strip waveguide with holes, (c) Coupling-gap RBE, (d) Coupling-gap DBE, (e) Zero-coupling-gap RBE, (f) Zero-coupling-gap DBE Comparison of DBE Designs with and without a Coupling Gap To compare DBE designs with and without a coupling gap, we employ a Taylor series expansion about the band edge (ω 0, k 0 ). Since the dispersion is symmetric about the band edge, we express the expansion using terms in even powers as 33

49 ω ω ω ( k k0) ( k k0) 4 4! 4 k k0 k0 ω 2! 2 k... (3-1) Near the band edge, we consider only the lowest order term in the expansion. For a RBE, the lowest order term produces a quadratic dispersion approximation 2 0 D2 ( k k0) ω ω, (3-2) where D 2 2 ω/ k 2 /2!. For a DBE, ω/ k = 2 ω/ 2 k = 0 [36]. Therefore, the lowest order non-zero term in Eq. (3-1) produces a quartic dispersion approximation 4 0 D4 ( k k0) ω ω, (3-3) where D 4 4 ω/ k 4 /4!. In Figure 3.4(a) and Figure 3.4(b), the wave number difference, k k 0, versus frequency difference, ω ω 0, is shown for the second lowest order mode of the cgdbe and the zcgdbe designs, respectively, calculated using three-dimensional plane wave expansion using a freely available software package [58] (3D PWE). PWE is used, instead of FEM, because we are interested in only real values of k near and at the band edge. Equation (3-3) is fit to the PWE result using a least squares method. A quartic dispersion regime is defined as the region of wave number difference where the PWE result and the quartic fitting differ by less than one percent. For the cgdbe, quartic dispersion occurs for normalized frequency difference less than and normalized wave number difference less than For the zcgdbe, quartic dispersion occurs for normalized frequency difference less than and wave number difference less than The wave number difference quartic dispersion regime is larger for the 34

50 zcgdbe. As discussed in the following sections, a larger quartic dispersion regime is accompanied by a smaller number of periods required to observe Q factor scaling to the fifth power in finite length structures. (a) y x z (b) y x z Figure 3.4. Comparison of quartic dispersion regimes for DBE designs with and without a coupling gap. (a) cgdbe and (b) zcgdbe. 3.3 Transmission Resonances in Finite Length Structures Spatial Distribution of Electric Field Truncating the periodic structure to a finite number of periods creates resonances in the transmission spectrum due to reflections at the interfaces. Finite length structures with N number of periods are characterized using the three-dimensional finite difference time domain method (3D FDTD [59]) at infrared wavelengths. In Figure 3.5(a), the first three resonances closest to the bandgap are shown for a finite length zcgdbe cavity with N equal to 30 periods. The Q-factor is largest for the resonance closest to the bandgap and decreases for the second and third resonances. The vector magnitude of the electric 35

51 field corresponding to each of the resonances is shown in Figure 3.5(b). The spatial field distribution involves the interference of propagating and evanescent waves. The average spatial envelope of the first resonance is characterized by a single lobe. Two and three lobes characterize the second and third resonances, respectively. In Figure 3.5(c), the field distribution for the first resonance is decomposed into Bloch modes [68]. The Bloch modes consist of two propagating modes and two evanescent modes as expected from the dispersion analysis. (a) 1st 2nd 3rd (b) 1st E (db) +20 2nd 0 3rd (c) +z propagating Bloch mode x z y +z evanescent Bloch mode -20 E (db) +20 -z propagating Bloch mode -z evanescent Bloch mode 0 x y z -20 Figure 3.5. Finite length zcgdbe resonator. (a) Transmission resonances, (b) Electric field magnitude, (c) Bloch mode decomposition for first resonance. For definiteness, the number of periods is N = 30 and the period a = 380 nm Dependence of Resonance Frequency and Q-factor on Number of Periods For large number of periods N, the wave number distance of the first transmission resonance from the band edge is approximately [36] 36

52 k (an) (3-4) Thus, increasing the number of periods pushes resonances towards the band edge where the group velocity decreases and the Q-factor increases. When the resonance frequency is in the quartic dispersion regime, the quality factor scales to the fifth power of the number of periods. Combining equations (3-2) and (3-4) yields an expression for the location of the first resonance relative to the band edge,, in terms of the number of periods for a RBE 2 2 ( ( an) ) D, (3-5) ω RBE Likewise, combining equations (3-3) using (3-4) results in an analogous expression for a DBE ω DBE D 4 4 ( ( an) ), (3-6) The location of the first resonance is quadratic in 1/N for an RBE and quartic in 1/N for a DBE. For small number of periods N, the location of the first transmission resonance is determined from 3D FDTD [59] computations. We consider the dependence of the frequency difference Δω r = (ω r ω 0 ) on 1/N, where ω r is the resonance frequency and ω 0 is the band edge frequency. In Figure 3.6(a), the frequency difference is shown for the cgdbe and zcgdbe designs. Also shown in Figure 3.6(a) are lines of best fit corresponding to quadratic and quartic dependencies on 1/N. For N less than 40 periods, the cgdbe resonant frequency difference scales quadratically in 1/N. Near 40 periods, the frequency difference scaling begins to deviate from quadratic dependence to higher 37

53 order dependence. For N less than 40, Δω r a/(2πc) is greater than Since quartic dispersion occurs for (ω ω 0 )a/(2πc) less than , there are too few periods for resonances to fall within the quartic regime. In contrast, for the zcgdbe design, quadratic dependence in 1/N is observed only for N less than 10 and quartic scaling is observed for N greater than 25. For N equal to 25, Δω r a/(2πc) is , which is inside the boundary for the quartic dispersion regime established as (ω ω 0 ) a/(2πc) < Designs that maximize the extent of the quartic dispersion regime will minimize the number of periods required for DBE scaling. In Figure 3.6(b), the Q-factor is plotted against the number of periods N for both the cgdbe and the zcgdbe designs. For the cgdbe design, quality factor scaling to the third power is observed out to N equal to 40. Quality factor scaling to the fifth power is not observed until 65 periods [71]. For the zcgdbe design, we observe quality factor scaling to the third power for N less than 7. Quality factor scaling to the fifth power is observed for N greater than 25 periods, consistent with the frequency difference scaling of Figure 3.6(a). The zcgdbe facilitates the realization of high quality factor cavities in much smaller geometries because fifth power quality factor scaling begins at 40 fewer periods. For example, at 40 periods, the quality factor of zcgdbe is approximately 40,000 while the quality factor of the cgdbe design is only 6,

54 Figure 3.6. (a) Frequency of first resonance, relative to the band edge, versus number of periods for cgdbe and zcgdbe resonators. (b) Q-factor of first resonance versus the number of periods. 3.4 Transmission Properties Through zcgdbe Resonator A zcgdbe resonator with input and output y-junction terminations is shown in Figure 3.7. High transmission through the resonator is possible when both propagating and evanescent Bloch modes are properly excited in the cavity [39]. The y-junctions are composed of two single mode strip waveguides that converge to one multimode waveguide. The width of the single mode and multimode waveguides is 370 nm and 740 nm, respectively. Parameter Z L is the length of the multimode waveguide z L w zcg x z y 4 Silicon SiO 2 Figure 3.7. Top view of zcgdbe resonator with y-junction terminations. 39

55 The transmission properties are characterized using a scattering (S) matrix [64]. The four port structure is represented by a 4 4 S-matrix. Each column of the matrix corresponds to excitation at a single port and matched loads at the remaining three ports. Incoming and outgoing electric fields are related to the S-matrix, employing Einstein summation convention, by E i = S ij E + j where indices i and j are 1, 2, 3, or 4. We consider ports 1 and 3 on the left end of the DBE resonator to be input ports and ports 2 and 4 on the right end of the DBE resonator to be output ports so that E + 1 = α, E + 2 = 0, E + 3 = β, and E + 4 = 0, We relate to and introduce parameter as 2 j 1 e, 0 1, 0 2. (3-7) In this notation, the magnitude of the power launched into ports 1 and 3 is α 2 and β 2, respectively. The phase difference between port 1 and 3 excitations is ϕ. The outgoing power from all four ports as a function of the input power α 2 at port 1 and the phase difference ϕ between ports 1 and 3 is calculated for Z L equal to 1.0 m and N equal to 30. The power at port 3 is β 2 = 1 α 2. The resonance wavelength is nm, determined by choice of period a equal to 380 nm. Two particular excitation conditions are observed. The first excitation condition corresponds to maximum power reflection at the excitation ports 1 and 3. We refer to this as condition 1. Condition 1 corresponds to α 1 2 = , β 1 2 = , and ϕ 1 = o. The second condition corresponds to maximum power out of ports 2 and 4. Condition 2 is realized for α 2 2 = , β 2 2 = , and ϕ 2 = o. We note that the vectors [α 1, β 1 ] and [α 2, β 2 ] are orthogonal. 40

56 Transmission can be optimized given excitation only at port 1 by tuning length Z L. Condition 2 is shifted to α 2 2 = 0.97 for Z L equal to 1.5 m. The reflection, S 11 2, is shown in Figure 3.8(a) for Z L equal to 1.00 μm, 1.25 μm, 1.50 μm and 1.75 μm. On resonance, there is no significant change in the reflection properties for each value of Z L. The transmission, S 41 2, is shown in Figure 3.8(b). Changing Z L from 1.00 μm to 1.50 μm increases the transmission from 0.51 to The dependence of transmission on Z L is due to the phase velocity difference between the modes in the multimode region. The magnitude of the electric field is presented for the optimized design in Figure 3.8(c). The structure performs largely as a 2-port network. (a) (b) (c) y x z E (db) Figure 3.8. Port 1 excitation of zcgdbe resonator with Z L as parameter. (a) Outgoing power at Port 1, (b) Outgoing power at port 4, (c) Electric field distribution for first resonance. For definiteness, the number of periods is N = 30 and the period a = 380 nm. 3.5 Chapter Conclusion Zero-coupling-gap DBE resonators are designed and analyzed in the silicon-oninsulator integrated optics platform. Propagating and evanescent modes, internal to the 41

57 DBE cavity, interfere to produce large and distributed spatial electric fields. Resonator quality factor can scale to the fifth power of the number of periods, enabling significant miniaturization. Scaling to the fifth power occurs, however, only for structures with sufficient number of periods. The minimum number of required periods is related to the extent of the quartic dispersion regime. Fewer periods are required for fifth power scaling in DBE designs with zero-coupling-gap, compared to DBE designs with non-zero coupling gap. A zero-coupling-gap DBE resonator in silicon photonics can exhibit quality factor scaling to the fifth power with as few as 25 periods. Evanescent modes also contribute to mode matching at the interface between the periodic cavity and the feed waveguide, allowing for high transmission on resonance. Since mode volume scales linearly with number of periods, DBE resonators are capable of large Purcell factors. Chip scale degenerate band edge resonators provide a route to realizing compact cavities with large spatial field distribution for future applications in light emission, switching, and sensing. 42

58 Chapter 4: DBE Fabrication Overview 4.1 Chapter Introduction The realization of a DBE device is critically dependent upon the quality of the fabrication process. Transmission band edge resonances form in finite length periodic structures when forward and backward propagating modes constructively interfere. However, the formation of resonances is conditional upon the uniformity of the periodic structure. The fabrication process introduces structural disorder into the design and changes the nature of light propagation[79] [84]. Disorder in the periodic structure reduces resonance quality factor[85]. In [86], we presented the optimization of electron beam patterned hydrogen silsesquioxane (HSQ)[87] masks for low-loss silicon waveguides. HSQ is an attractive resist for the patterning of silicon photonic devices[88] because it allows patterning of sub 5nm features [89]. Additionally HSQ can also be directly used as a dry etch mask [90]. In this chapter we describe the fabrication of DBE resonators in a silicon-oninsulator platform. The DBE fabrication process is based upon our low-loss silicon waveguide fabrication process [86]. However, we change the top cladding from plasma-enhanced chemical vapor deposition (PECVD) SiO 2 to a combination of spin-onglass and PECVD SiO 2 as shown in Figure 4.1. The spin-on glass allows for complete 43

59 filling of etched holes. Due to the 5 nm minimum step size of our EBL tool, we also made one change to the design dimensions given in the previous chapter. We expect the final dimensions of any fabricated device to vary from the nominal design. Therefore we include a strategy for design perturbations about the nominal design. Finally we share results of the fabrication process and discuss the consequences of final dimensions differing from the nominal design. (a) a 1 2 x w 3 z 0 Z L 4 Buried x Silicon SiO z 2 y z (b) PECVD SiO 2 Spin On Glass oxide x 0 2r h y Figure 4.1: Device schematic. (a) Angled view illustration of DBE cavity. Top cladding is not shown. (b) Cross-sectional view of periodic region. The silica top cladding has two layers. Bottom layer is 540 nm of spin-on-glass. Top layer is 500 nm of PECVD SiO Low Loss Silicon Waveguide Fabrication DBE devices are fabricated on a silicon-on-insulator (SOI, SOITEC) substrate with 250 nm silicon on top of 1 μm buried oxide (BOX). An 80 nm thin film of 4% HSQ (Dow-Corning XP1541-4) is spin-coated onto the substrate. Samples are then cured in an N 2 purged oven at 50 C for 40 min. Features are written into the HSQ resist as the electron-beam (e-beam) of a Vistec EBPG 5000 rasters across the surface in 5nm steps. The beam acceleration voltage is 100 kv and the exposure dose is 5000 μc/cm 2. After e-beam writing, the HSQ resist is developed in 25% TMAH at room temperature. Resist 44

60 not exposed to the e-beam is washed away in the developer. The remaining HSQ serves as a dry etch mask for transferring the device patterns into the silicon layer. Before dry etching, a post-exposure bake is performed in a rapid thermal anneal (RTA) oven at 1000 C for 1 minute in an O 2 ambient. The post-exposure bake densifies the HSQ mask and increases the mask s etch resistance. The silicon is etched by a Cl 2 /O 2 chemistry in a PlasmaTherm SLR770 Inductively Coupled Plasma-Reactive Ion Etcher (ICP-RIE). A post-etch bake in the RTA oven at 1000 C for 1 minute in an O 2 ambient smooths waveguide sidewalls by consuming approximately 6 nm of Si. 4.3 Top Cladding The next step in the fabrication process is the deposition of a 1 μm SiO 2 top cladding. Typically we would deposit PECVD SiO2 using a silane gas (SiH 4 ). However, the silane based oxide film was inadequate at uniformly filling the etched silicon. Voids or keyholes in the film can form in etched trenches [91]. We do not have any method of reliably cleaving a device about the center of an etched hole for cross-section inspection. We assume that filling a 225 nm diameter etched hole is similar to filling a 225 nm wide trench between two parallel silicon waveguides. A cross-section scanning electron microscope (SEM) image is presented in Figure 4.2(a) with 500 nm of PECVD SiO 2 deposited at 9.5 Torr, 350 C, and 175 Watts RF power. Keyholes form 95 nm above the trench bottom. The height at which keyholes form depends on the trench aspect ratio (width versus height). Increasing trench aspect ratios delays the formation of keyholes. Keyhole formation height depends also on deposition process physics. However, we were unable to eliminate keyholes by changing deposition parameters such as pressure, 45

61 temperature or RF power. It has been reported [92], [93] that trench filling is improved when using tetraethyl orthosilicate (TEOS) in place of SiH 4. TEOS PECVD SiO 2 films were deposited by NOEL Tech. Keyhole formation is still observed (see Figure 4.2(b) ). With a TEOS based chemistry, the height at which the keyhole forms is 175 nm. (a) (b) Silane based SiO 2 keyhole TEOS based SiO 2 keyhole Si Si 200 nm 200 nm Figure 4.2: (a) Cross-section of two parallel Si waveguides separated by 225 nm gap. SiH 4 based PECVD SiO 2 top cladding deposited at 9.5 Torr 350 C and 175 Watts RF power. (b) Cross-section of two parallel Si waveguides separated by 225 nm gap with TEOS based PECVD SiO 2 deposited by Noel Tech. Spin-on glass (SOG) is another option for filling etched holes [75][94]. We use Dow Corning s FOx-15 as a top cladding. FOx-15 is an HSQ formulation similar to the e-beam resist XP ; however, FOx-15 has twice the viscosity of XP The higher viscosity makes the FOx-15 formulation ideal for planarizing non-uniform surfaces. We test the planarization capabilities of FOx-15. FOx-15 is spin-coated onto etched silicon waveguides with trench widths ranging from 75 nm to 300 nm. After spin- coating, the samples are baked bake at 400 C for 1 hour in a nitrogen ambient. The 46

62 FOx-15 easily fills the smallest trenches (75 nm) as seen in the SEM cross-section of Figure 4.3. XP Si FOx nm Figure 4.3 Two Si waveguides separated by 75 nm are planarized with 540 nm of FOx-15 SOG. The film thickness and refractive index can be tuned by adjusting the curing temperature and time. Therefore, the transmission response of a resonant cavity can be modified by adjusting the curing conditions [94]. The material properties of the HSQ film are dependent upon the curing conditions. Heating HSQ above 250 C rearranges the intermolecular distribution of Si-O and Si-H bonds [95]. The refractive index is lowest in films cured at 400 C [96]. Curing temperatures above 400 C densify the film; curing temperatures above 650 C convert the film to SiO 2 [95]. The densification of FOx-15 film is shown in Figure 4.4(a). FOx-15 is spin-coated on a silicon substrate and baked at 47

63 400 C for 1 hour in a nitrogen ambient. The film thickness and refractive index is determined via ellipsometry (J.A. Woollam alpha-se Ellipsometer). After the initial measurement, the sample is rapid thermal annealed at 450 C for two minutes. Ellipsometry measurements are repeated. The same sample is again rapid thermal annealed at 500 C for two minutes and ellipsometry measurements are repeated. The process is repeated in increments of 50 C until 800 C. As the film densifies, the refractive index increases as shown in Figure 4.4(b). The refractive index changes from 1.38 to 1.44 at 632 nm, a 4% change. A second Si substrate with a 500 nm PECVD SiO 2 film is processed in parallel to the FOx-15 film. The refractive index of PECVD SiO 2 film changes less than 0.5% in Figure 4.4, Figure 4.4: Ellipsometry measurements of FOx-15 (blue squares) and PECVD SiO 2 (orange circles) films on silicon substrate after rapid thermal anneal. (a) Thickness of thin film as a function of RTA temperature. (b) Film refractive index at 632 nm as a function of RTA temperature. 48

64 4.4 Nominal Device Design Design parameters are chosen to realize DBE resonances near 1550 nm for quasi- TE polarization (dominant electric field in x-direction). The nominal design for our DBE resonator is shown in Figure 4.1 and is based upon the zcgdbe device described in Chapter 3. One minor adjustment to the hole radius dimension was made because of the e-beam 5 nm step size. All dimensions must be a multiple of 5 nm; however, the radius given in Chapter 3 is r equals nm. We round down the radius to r equals 110 nm. The waveguide width w equals 740 nm and height h equals 250 nm. The period a equals 380 nm. Holes are separated in the x direction by distance 2x 0 equals 310 nm. The longitudinal offset in the z direction z 0 equals 90 nm. Light is coupled into and out of the periodic cavity with two y-splitters. Our device is a four port network. A transition length of Z L equal to 1.5 μm maximizes transmission across port one to port four and the resonator effectively operates as a two port structure. The cladding profile is shown in Figure 4.1 (b). 540 nm of FOx-15 fills the etched holes. 500 nm of PECVD SiO 2 is deposited on top of the FOx-15 layer. 4.5 Port Terminations In FDTD [59] simulation, all four ports are terminated with a PML [65] [67] which minimizes reflections back into the cavity. We mimic the simulation s reflectionless boundaries in experiment with compact cantilever couplers [97] at ports 1 and 4. In the lab, light is guided to the DBE resonator with a single-mode optical fiber (SMF). The SMF diameter is tapered to a 1.5 μm diameter tip and butt-coupled to the input of a compact cantilever coupler. The input face of the SiO 2 cantilever coupler is 49

65 2 μm wide 2 μm high. Transverse mode profiles at the SMF tip and the cantilever coupler input are very similar and hence reflection at the fiber-cantilever interface is minimal. Light propagates in the SiO 2 cantilever a distance of 1.5 μm until reaching the silicon waveguide inverse width taper. As the width of the silicon waveguide tapers from 70 nm to 370 nm over a length of 6.5 μm, the guided mode is slowly converted to the mode at port 1. An optical micrograph of a cantilever coupler at port 4 is shown in Figure 4.5(a). Our measurement setup can only align two tapered fibers to our device under test. Therefore we modify the cantilever at ports 2 and 3 to convert the silicon strip waveguide modes to a SiO 2 slab mode. We refer to the modified cantilevers as matched loads. A matched load at port 2 is shown in the optical micrograph of Figure 4.5 (a). In the inset of Figure 4.5 (b), a schematic of the matched load is presented. In the schematic, the top cladding is made transparent for half of the matched load structure and the underlying tapered silicon waveguide is visible. The matched load is identical to the cantilever coupler in all aspects except the matched load lacks a terminating facet. For the matched load structure, the 2 μm 2 μm SiO 2 waveguide connects to a SiO 2 slab waveguide and the strip waveguide mode is converted to a slab waveguide mode. We simulate the matched load structure using FDTD [59]. A quasi-te mode is launched into 370 nm 250 nm silicon strip waveguide at the location labeled input (see inset of Figure 4.5 (b)). The power reflected back into the strip waveguide is plotted in Figure 4.5 (b) as a function of wavelength. Reflected power is less than 35 db over the entire band of our tunable laser (1460 nm to 1580 nm). 50

66 (a) waveguide port 2 tapered waveguide matched load (b) Input SiO 2 air Si air waveguide port 4 air cantilever coupler 8 μm Figure 4.5 (a) Optical micrograph of ports 2 & 4. Port 2 is terminated with a matched load where a 6.5 μm inverse silicon taper converts the silicon strip waveguide mode to a SiO 2 slab mode. Port 4 is terminated with a standard cantilever coupler for coupling to a tapered fiber. (b) Inset shows angled view of finitedifference time-domain simulation. In the schematic only, the SiO 2 top cladding is made transparent for half of the simulation domain revealing the underlying silicon waveguide. Reflected power at the simulation input is measured and presented. 4.6 Design Variations Final dimensions of the fabricated device will vary from the initial design dimensions. The dimension of the HSQ hard mask depend on the exposure dose [98] and the resist development process[99] [100]. influenced by proximity-effects [101][102]. HSQ hard mask dimensions are also The rounding of sharp corners is one example of a proximity-effect. Devices are fabricated on 100 mm diameter SOI wafers from SOITEC. The nominal thickness of the silicon top layer is 250 nm; however, the actual thickness may vary by ±32 nm. The HSQ hard mask erodes during the ICP etch and the etched silicon has a sidewall slope angle (with respect to the substrate) less than 90 [103], [104]. Approximately 6 nm of silicon is consumed during the post-etch RTA 51

67 step. While we could try to characterize the dimension changes after each process step, it is simpler to fabricate multiple design variations. With e-beam lithography, holes can be placed with sub-nanometer precision [79], [105], [106]. Therefore, we assume that hole position parameters z 0 and x 0 can be exactly fabricated. Because the period a is defined with respect to hole position, we also assume the fabricated period is exact. As previously noted, we expect the fabricated dimensions of width, height and hole radius to vary from their nominal value. A perturbation to any single parameter will destroy the DBE. The design variations are chosen by asking, if the width/radius were to change by +10 nm, how must z 0 and a also change to return to a DBE? To answer this question, we calculate the dispersion (using MPB) for a large set of parameters. We consider a set of radius values (105 to 125 nm in steps of 5 nm) and width values (730 to 760 nm in steps of 10 nm). For each value of radius or width in the set, we sweep the period a from 370 to 410 nm in steps of 5 nm. For each value of period in the set, the offset z 0 is adjusted until a DBE is realized. The simulation results form a three-dimensional surface that describes the DBE sensitivity to the different design parameters. However, it is simpler to visualize the surface as a collection of three two-dimensional plots as shown in Figure 4.6 and Figure 4.7. Every point on the curve represents a parameter combination that creates a DBE. In Figure 4.6 (a)-(c) sensitivity curves are shown for the hole radius parameter. Each curve in the three plots represents a different radius ranging from 105 nm to 125 nm. 52

68 Figure 4.6 The hole radius design parameter is considered. Each point on the curve represents a DBE design. (a) Plot of period versus hole offset. (b) Plot of period versus band edge wavelength. (c). Plot of offset versus band edge wavelength. Parameters not specified in the plots are identical to those given in Chapter 3 for the zcgdbe. Figure 4.7 The waveguide width design parameter is considered. Each point on the curve represents a DBE design. (a) Plot of period versus hole offset. (b) Plot of period versus band edge wavelength. (c) Plot of offset versus band edge wavelength. Parameters not specified in the plots are identical to those given in Chapter 3. 53

69 In Figure 4.7(a)-(c), sensitivity curves are shown for the width parameter. Each curve in the three plots represents a width ranging from 730 to 760 nm. The curves in Figure 4.6 and Figure 4.7 are used to choose eighteen design variations where the offset z 0 is fixed at 90 nm. To further reduce the design space, we constrain the period to be between 370 and 390 nm. Then we choose values for radius or width that produce a DBE. From Figure 4.6 (a) we see that a DBE forms if radius equals 110 nm and the period equals 370 nm. If the period equals 390 mm, a larger radius of 115 nm is needed to produce a DBE. The eighteen base designs considered are combinations formed from the sets radius equals 110 nm or 115 nm; width equals 720, 730, or 740 nm; and period equals 370, 380, or 390 nm. For each design variation, multiple devices are fabricated where number of periods is varied from 10 to 100 in increments of Fabrication Results DBE resonators are patterned in silicon following the procedure described in Chapter 4.2. Spin-on glass (FOx-15) is chosen as a top cladding to fill the etched holes. After spin coating, the devices are baked on a hotplate at 200 C for 2 minutes. The spinon-glass layer is densified in an N 2 ambient at 400 C for one hour. Spin-on-glass film thickness and refractive index measured via ellipsometry at nm are 545 nm and 1.38, respectively. An additional 510 nm of PECVD SiO 2 is deposited on top of the spinon-glass. Compact cantilever couplers are patterned at ports 1 and 3 for low loss fiber-tochip coupling [57], [97]. Matched loads are patterned to terminate ports 2 and 4. An 54

70 optical micrograph of a fabricated DBE cavity is shown in Figure 4.8(a). The transition from periodic region to feed waveguides is shown in the SEM image of Figure 4.8(b). Figure 4.8 (a) Top-down optical micrograph of DBE resonator showing all four ports. (b) Angled-view SEM micrograph of etched DBE device. The DBE device is patterned on a silicon substrate that is co-processed with SOI DBE resonators. Dashed box in (a) correspond to image in (b). The dimensions of fabricated DBE devices are estimated based on SEM cross-sections of patterned silicon waveguides. The waveguides are processed on the same SOI substrate as the DBE devices. A SEM of a cross-sectioned etched silicon waveguide is shown in Figure 4.9 (a). From the image, the sidewall angle is measured as 85. The waveguide width is designed to be 730 nm. However, the waveguide width at the top, w top, and the width at the bottom, w bottom, differ from the designed width. We measure widths w top = 724 nm and w bottom = 734 nm. A second etched waveguide with a gap is shown in Figure 4.9 (b). The design dimensions for the waveguide width and gap are 730 and 220 nm, respectively. We assume hole of radius r = 220/2 nm will etch 55

71 similar to a 220 nm wide gap. The sidewall angle in the gap is nearly vertical. The gap width at the top (2 r top ) is the same as the gap width at the bottom of the trench (2 r bottom ). From the SEM we measure 2 r top = 2 r bottom = 232 nm. (a) (b) w top Si w bottom Si 2r top 400 nm SiO nm 2r bottom SiO2 Figure 4.9 (a) SEM cross-section of etched silicon waveguide co-processed with SOI DBE resonator. The waveguide was designed to have a width of 730 nm. Sidewall angle of the etched waveguide is 85. The widths at the top and bottom of the waveguide are w top = 724 and w bottom = 734. (b) SEM cross-section of etched silicon waveguide co-processed with SOI DBE resonator. The waveguide was designed to have a width of 730 nm and 220 nm gap. Measured gap dimensions are used to approximate dimensions of etched holes. Gap dimensions are 2r top = 2r bottom = 232 nm. 4.8 Deviating from Nominal Design In the previous section, it was shown that device dimensions change because of the fabrication process. The waveguide cross-section is not rectangular; it is trapezoidal. Etched holes are larger than their design value. Although the difference between design and actual dimensions is close to 10 nm, the small changes do alter the band diagram. To better understand how the dispersion changes because of fabrication, we numerically analyze six of our design variations (as described in section 4.6) using the freely available MIT Photonic Bands (MPB) software package that implements PWE [58]. For the six 56

72 designs, the period a = 380 nm. The width is 720, 730, or 740 nm. The hole radius is 110 nm or 115 nm. We compare the design variations to the nominal design for the zcgdbe presented in Chapter 3 where the width is 740 nm and the hole radius is nm. The dispersion relationships of the seven periodic structures is shown in Figure 4.10, where we only plot the propagating mode closest to the band gap. As expected, the band edge frequency changes with device dimensions. Increasing the hole radius with respect Figure 4.10 The propagating mode closest to the band edge is plotted for seven different periodic structures that are close to a DBE. The structures are discussed in detail section 4.6. Their dispersion is compared against the zcgdbe structure presented in Chapter 3. The zcgdbe is plotted in black. 57

73 to the nominal design, shifts the band edge to higher frequencies. Likewise, decreasing the hole radius lowers the band edge frequency. The resonances in finite length structures will also shift towards higher/lower frequencies with increasing/decreasing hole radius. To observe the subtle differences between each of the design variations, we plot the wave number difference, k k 0, versus frequency difference, ω ω 0 in Figure If the dispersion of one of the design variations is close to a DBE, it will follow a quartic trend. The solid black line corresponds to the wave number difference versus frequency difference for the zcgdbe defined in Chapter 3. We also plot a quartic line of best fit (dotted black line) to the zcgdbe dispersion. The solid and dotted black lines are identical to the data presented Figure 3.4(b). Using Figure 4.11, we can divide the six design variations into three groups: DBE, RBE or split band edge (SBE) [107], [108]. Both a RBE and DBE have a stationary point, the wave number at which group velocity is minimum, at ka/(2 ) = 0.5. However, the SBE has a stationary point at some wave vector ka/(2 ) < 0.5. All three design variations where the radius is 110 nm can be classified in the SBE group. In Figure 4.11, the stationary point presents as a sharp dip. The stationary point for the 3 SBE designs is located in the range < (k k 0 )a/(2 ) < The RBE group contains the design variations where the width is 730 and 740 nm and the radius is 115 nm. The dispersion for these two devices is closer to a RBE than a DBE. We would expect Q-scaling to the third power for these devices. In the DBE group we have the design variation where width is 720 nm and radius is 115 nm. Its dispersion (plotted as a solid blue line) also follows the quartic 58

74 fit line of the zcgdbe in Chapter 3. However, when (k k 0 )a/(2 ) < , the blue line begins to deviate from the quartic fit. The design represented by the blue line is technically a SBE with a stationary point at some value ka/(2 ) < 0.5. Yet, the stationary point is so close in proximity to ka/(2 ) = 0.5 that the design will have properties of a DBE. We can use Figure 4.11 and Eq. (3-4) to estimate if a finite length device will behave like a DBE. The point where the dispersion deviates from quartic dispersion ( (k k 0 )a/(2 ) ~ ) corresponds to a finite length device of 50 periods. Thus we would expect to see Q-factor scaling to the fifth power up to 50 periods. Figure 4.11 Using the dispersion data presented in Figure 4.10, the wave number difference versus frequency difference is plotted. Six design variations are compared to the zcgdbe of Chapter 3 and its quartic fitting. One of the design variations (solid blue line) also has a DBE. The other design variations are classified as a RBE or SBE. 59

75 4.9 Chapter Conclusion In this chapter, an overview of the fabrication process is given. We discussed the transition from the zcgdbe design given in Chapter 3 to the final design of our fabricated device. Light is coupled to the resonator with compact cantilever couplers at ports 1 and 4. Matched loads terminate ports 2 and 3. Characterization of the fabrication process indicates that fabricated devices have 85 sidewall angles. Also, the etched hole radius is 6 nm larger than design. To account for any dimensional changes during fabrication, we made 18 different design variations. Most of the 18 design variations do not have a DBE, but finite length devices can still have the properties of a DBE if the dispersion is almost quartic. 60

76 Chapter 5: Measurement and Characterization of DBE Resonators On resonance, the modal fields extend over the entire periodic structure. These large and distributed modal fields have been exploited for applications in light [28], [29], optical switching [31] and nonlinear optics [32]. However, transmission resonances near a regular band edge are limited by quality factors that scale only as the third power of the number of periods. Resonances near a degenerate band edge can scale to the fifth power of the number periods [35], [37]. Here, we report the full details of the first [109] experimental demonstration of quality factor scaling to the fifth power. Transmission resonances near a degenerate band edge are realized in silicon strip waveguides with onedimensional periodicity. Quality factors of 27,000 in a 35 period cavity are observed. Transmission measurements show Fano resonances with a sharp transmission peak to bandgap extinction ratio of 20 db. 5.1 Optical Transmission Measurement Procedure A single-mode fiber carries light from a tunable infrared continuous-wave laser to the device. Fiber paddles linearly polarize the light as TE. The tip of the single-modefiber is tapered for mode-matching at the compact cantilever coupler at port 1. Transmitted power is collected at the port 4 cantilever coupler via a second tapered 61

77 single-mode-fiber and measured with a photodetector. Fano lineshapes are fit to the transmission response using nonlinear curve fitting [110]. Quality factors are calculated from the lineshape half-width at half-maximum. Given a transmission response with multiple resonances, it is possible to extract the band edge dispersion relationship. The center wavelength of each resonance is extracted from Fano lineshapes fitted to the transmission response. The resonances are equally spaced in wavevector k, the spacing depends only on the cavity length L[111] as shown in Eq. (5-1). k / L (5-1) The wavenumber at the band edge is k 0 = /a for both a RBE and DBE. The wavenumber of the 1st resonance from the band edge is k 1 = k 0 δk. Likewise for the 2nd resonance the wavenumber k 2 = k 0 2δk. In general, the wavenumber of the mth resonance is given as k m m a an (5-2) for a cavity of length L = an where a is the period and N is the number of periods. From Eq. (5-2) it is seen that as the number of periods N approaches infinity, k m approaches k 0. With Eq. (5-2), an approximate dispersion can be plotted at discrete points on resonance. 5.2 Group Velocity Measurement Procedure The group velocity of light propagating through an optical device can be measured by modulating the optical carrier [112]. Our experimental setup for measuring 62

78 group velocity is shown in Figure 5.1. A vector network analyzer drives a lithium niobate modulator to modulate light from a tunable laser source. The laser wavelength is swept across the range of interest. Low and high speed photodetectors record the transmission response of the cavity at each wavelength. The high speed photodetector is connected to the vector network analyzer and captures the RF magnitude and phase. As an example, the measured RF power and optical power of a DBE resonator with N = 35, w = 730 nm, r = 110 nm and a = 390 nm is presented in Figure 5.2(a). As expected, the measured RF power and optical power agree. The RF phase φ in degrees is related to the group delay g as g 360 f mod (5-3) where f mod is the RF modulation frequency. For our experiment f mod = 4 GHz. In Figure 5.2(b), the RF phase is plotted. Using Eq. (5-3), the group delay is plotted on resonance. The first fives resonances closest to the band edge are discernable. Beyond the fifth resonance, the group delay is too small to distinguish. 63

79 Tunable Laser ( nm) Polarization Controller SMF Vector Network Analyzer Power meter Low speed photodetector Polarization Controller LiNbO 3 Modulator High speed photodetector Power splitter 99:1 DBE Resonator Cantilever coupler Cantilever coupler optical electrical Figure 5.1 Optical device measurement and characterization setup. Light from a tunable laser source is modulated with a lithium niobate modulator at 4GHz. A polarization controller is used to maximize transmission through the modulator. A second polarization controller converts light from the modulator to quasi-te polarization. Light is coupled into and out of the device under test (DUT) with tapered SMF-28 fiber aligned to cantilever couplers on chip. The light out of the DUT is directed to both a low speed photodetector and a high speed photodetector connected to a vector network analyzer. Our setup allows for measurement of both carrier amplitude and modulated RF signal amplitude/phase. The group velocity on resonance is given as v g L g (5-4) where L is the physical length of a cavity. 64