UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Grating coupler design based on silicon-on-insulator Wang, Yun 2013

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
24-ubc_2013_spring_wang_yun.pdf [ 4.74MB ]
Metadata
JSON: 24-1.0073806.json
JSON-LD: 24-1.0073806-ld.json
RDF/XML (Pretty): 24-1.0073806-rdf.xml
RDF/JSON: 24-1.0073806-rdf.json
Turtle: 24-1.0073806-turtle.txt
N-Triples: 24-1.0073806-rdf-ntriples.txt
Original Record: 24-1.0073806-source.json
Full Text
24-1.0073806-fulltext.txt
Citation
24-1.0073806.ris

Full Text

Grating Coupler Design Based on Silicon-On-Insulator by Yun Wang  B.Sc., Shenzhen University, 2011  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The Faculty of Graduate Studies (Electrical and Computer Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2013 c Yun Wang 2013  Abstract Silicon-on-insulator has become a promising platform for high-density integrated photonics circuits. The large refractive index contrast between the functional silicon layer and its cladding raises a coupling issue between an optical fibre and on-chip devices. Grating coupler provides a compact and efficient way to tackle the coupling issue between the optical fibre and silicon waveguide. In this thesis, a universal design methodology, which accommodates various etch depths, silicon thicknesses, and cladding materials has been demonstrated and verified by both FDTD simulation and measurement results. A fully etched grating coupler with a sub-wavelength grating structure has been proposed to reduce the large back reflection of existing fully etched grating couplers. Back reflection of the proposed fully etched grating coupler has been reduced from more than 20% to about 5%. The insertion loss and bandwidth of the proposed structure have also been improved. In addition, a bidirectional grating coupler for vertical coupling has been proposed to improve the insertion loss and bandwidth of the traditional grating coupler. A simulated insertion loss of -1.5dB with a 3dB bandwidth of more 100nm has been achieved with the proposed structure.  ii  Preface I am the first author on two conference papers and titled “Fully Etched Grating Coupler With Low Back Reflection” and “ Universal Grating Coupler Design” respectively. I also co-authored five papers, including two journal papers [44, 45] and three conference papers. In addition, I am the main author of a book chapter [9]. During the last year, I proposed a universal design methodology to design grating couplers for various fabrication processes and applications. This design flow has been implemented in Pyxis to automatically generate the desired grating coupler with user-specified input parameters. The validity of this design method has been verified by both theoretical calculation and measurement results. I also proposed a fully etched grating coupler with sub-wavelength grating structure to reduce the back reflection of the existing fully etched grating coupler. In addition, I proposed a bidirectional grating coupler structure for vertical coupling, the potential of which has been verified by theoretical calculation and numerical simulation. My complete list of publications are: 1. Yun Wang, Jonas Flueckiger, Charlie Lin, and Lukas Chrostowski, “ Fully Etched Grating Coupler With Low Back Reflection ” Photonics North 2013 (accepted); 2. Yun Wang, Jonas Flueckiger, Charlie Lin, and Lukas Chrostowski, “ Universal Grating Coupler Design ” Photonics North 2013 (accepted); 3. Wei Shi, Han Yun, Wen Zhang, Charlie Lin, Ting Kai Chang, Yun Wang, Nicolas A. F. Jaeger, and Lukas Chrostowski .“ Ultra-Compact, High-Q Silicon Micodisk Reflectors ”, Optics Express, Vol.20, Issue 20, pp.21846(2012) iii  Preface 4. Wei Shi, Han Yun, Charlie Lin, Mark Greenburg, Xu Wang, Yun Wang, Sahba Talebi Fard, Jonas Flueckiger, Nicolas A. F. Jaeger, and Lukas Chrostowski, “ Ultra-compact, flat-top demultiplexer using antireflection contra-directional couplers for CWDM networks on silicon ” Optics Express, (accepted), http://www.opticsinfobase.org/oe/upcomingissue.cfm 5. Wei Shi, Ting Kai Chang, Han Yun, Wen Zhang, Yun Wang, Charlie Lin, Nicolas A. F. Jaeger, and Lukas Chrostowski, “ Differential Measurement of Transmission Losses of Integrated Optical Components Using Waveguide Ring Resonators ” Proc. SPIE 8412, Photonics North 2012, 84120R (October 23, 2012); doi:10.1117/12.2001409 6. Wei Shi, Han Yun, Charlie Lin, Xu Wang, Yun Wang, Jonas Flueckiger, Nicolas A. F. Jaeger, and Lukas Chrostowski, “ Silicon CWDM Demultiplexers Using Contra-Directional Couplers ”, CLEO, 2013 7. Han Yun, Wei Shi, Yun Wang, Lukas Chrostowski, and Nicolas A.F. Jaeger “2x2 Adiabatic 3-dB Couplign on Silicon-on-insulator Rib Waveguides ”, Photonics North 2013 (accepted); 8. Lukas Chrowstowski and Michael Hochberg, “ Silicon Photonics Design ” (Chapter 6), 2013, ISBN: 9781105948749  iv  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  List of Tables  vi  Preface  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Dedication  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1  Silicon Photonics . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.2  Challenge of Coupling Light Into Nanophotonic Waveguide .  3  1.3  Grating Coupler . . . . . . . . . . . . . . . . . . . . . . . . .  5  1.4  Slab Waveguide and Channel Waveguide  . . . . . . . . . . .  6  1.5  Polarization of Waveguide Modes  . . . . . . . . . . . . . . .  7  1.6  State-of-the-art Grating Couplers  . . . . . . . . . . . . . . .  8  1.7  Measurement Setup . . . . . . . . . . . . . . . . . . . . . . .  12  2 Theory And Numerical Methods . . . . . . . . . . . . . . . .  14  2.1  2.2  Bragg Condition . . . . . . . . . . . . . . . . . . . . . . . . .  14  2.1.1  Bragg’s Law  14  2.1.2  The Bragg Condition for Grating Coupler  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  15  Effective Index Method . . . . . . . . . . . . . . . . . . . . .  17 v  Table of Contents 2.3  Finite-Difference Time-Domain Method . . . . . . . . . . . .  3 Detuned Grating Coupler 3.1  3.2  21  . . . . . . . . . . . . . . . . . . . .  25  Detuned Shallow Etched Grating Coupler . . . . . . . . . . .  25  3.1.1  Initial Condition . . . . . . . . . . . . . . . . . . . . .  26  3.1.2  Design Parameters . . . . . . . . . . . . . . . . . . . .  27  3.1.3  Optimization of the Grating Coupler  . . . . . . . . .  34  3.1.4  Design Stability  . . . . . . . . . . . . . . . . . . . . .  38  Universal Grating Coupler Design  . . . . . . . . . . . . . . .  41  3.2.1  Design Approach  . . . . . . . . . . . . . . . . . . . .  41  3.2.2  Simulation Results . . . . . . . . . . . . . . . . . . . .  43  3.2.3  Mask Layout . . . . . . . . . . . . . . . . . . . . . . .  49  3.2.4  Measurement Results . . . . . . . . . . . . . . . . . .  50  4 Fully Etched Grating Coupler . . . . . . . . . . . . . . . . . .  56  4.1  Regular Fully Etched Grating Coupler . . . . . . . . . . . . .  56  4.1.1  Bottlenecks of Regular Fully Etched Grating Couplers  57  4.1.2  Optimization of the Regular Fully Etched Grating Couplers  . . . . . . . . . . . . . . . . . . . . . . . . . . .  60  Improved Fully Etched Grating Coupler . . . . . . . . . . . .  64  4.2.1  Design Approach  . . . . . . . . . . . . . . . . . . . .  65  4.2.2  Simulation Results . . . . . . . . . . . . . . . . . . . .  67  4.2.3  Measurement Results . . . . . . . . . . . . . . . . . .  68  5 Vertical Grating Coupler . . . . . . . . . . . . . . . . . . . . .  70  4.2  5.1  Regular Vertical Grating Couplers . . . . . . . . . . . . . . .  70  5.2  Bidirectional Grating Coupler  . . . . . . . . . . . . . . . . .  73  5.2.1  Device Layout . . . . . . . . . . . . . . . . . . . . . .  73  5.2.2  Design and Simulation  . . . . . . . . . . . . . . . . .  74  . . . . . . . . . . . . . . . . . .  78  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  81  6 Discussion And Future Work  vi  Table of Contents  Appendices A FDTD code to generate universal grating coupler model .  89  B Pyxis code for universal grating coupler design . . . . . . . 101  vii  List of Tables 1.1  State-of-the-art grating couplers  . . . . . . . . . . . . . . . .  11  3.1  Initial values . . . . . . . . . . . . . . . . . . . . . . . . . . .  26  3.2  Tuning coefficients of various parameters . . . . . . . . . . . .  32  3.3  Input paramters . . . . . . . . . . . . . . . . . . . . . . . . .  44  3.4  Tuning coefficient of different parameters . . . . . . . . . . .  51  6.1  Comparison of the published results and the result of the grating coupler generated by the universal design methodology 79  viii  List of Figures 1.1  Schematic of SOI wafer . . . . . . . . . . . . . . . . . . . . .  2  1.2  Schematic of grating coupler . . . . . . . . . . . . . . . . . . .  4  1.3  Schematic of slab waveguide and channel waveguide in SOI .  6  1.4  (a) The amplitude of the electric field of the first order TE-like mode in a rectangular channel waveguide; (b) The amplitude of the magnetic field of the first order TE-like mode in a rectangular channel waveguide [9] . . . . . . . . . . . . . . . .  1.5  9  (a)The amplitude of the electric field of the first order TM-like mode in a rectangular channel waveguide; (b)The amplitude of the magnetic filed of the first order TM-like mode in a rectangular channel waveguide [9] . . . . . . . . . . . . . . . .  10  1.6  (a) Illustration of the automated setup; (b) automated setup  12  1.7  Fibre array ribbon, ribbon holder and ribbon arm [26] . . . .  13  2.1  Bragg’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . .  15  2.2  Schematic of a grating coupler . . . . . . . . . . . . . . . . .  16  2.3  Diagram for Bragg condition . . . . . . . . . . . . . . . . . .  17  2.4  Cross section of silicon-on-insulator waveguide . . . . . . . . .  18  2.5  Schematic of a SOI strip waveguide . . . . . . . . . . . . . . .  18  2.6  Schematic of the effective index of a strip waveguide . . . . .  19  2.7  Effective index of TE and TM modes . . . . . . . . . . . . . .  21  2.8  FDTD mesh for a grating coupler . . . . . . . . . . . . . . . .  22  2.9  (a) Schematic of simulation structure for input grating coupler; (b) Schematic of simulation structure for output grating coupler  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  24  ix  List of Figures 3.1  Insertion loss and back reflection to the waveguide of the initial grating coupler . . . . . . . . . . . . . . . . . . . . . . . .  27  3.2  Variations of grating period . . . . . . . . . . . . . . . . . . .  28  3.3  Variations of duty cycle . . . . . . . . . . . . . . . . . . . . .  29  3.4  Variations of etch depth . . . . . . . . . . . . . . . . . . . . .  30  3.5  Variations of incident angle . . . . . . . . . . . . . . . . . . .  31  3.6  Schematic of reflections at different interfaces . . . . . . . . .  32  3.7  Variations for BOX thickness . . . . . . . . . . . . . . . . . .  33  3.8  Variations for cladding thickness . . . . . . . . . . . . . . . .  34  3.9  Grating coupler design fabricated through OpSIS-IME . . . .  35  3.10 Images of fibre ribbon . . . . . . . . . . . . . . . . . . . . . .  36  3.11 Impacts of the gap between fibre ribbon tip and photonic chip on the insertion loss and bandwidth of the grating coupler . .  37  3.12 Spectra of simulation results with different gap distance and measurement results . . . . . . . . . . . . . . . . . . . . . . .  38  3.13 The insertion losses of the same grating coupler design at different positions of the chip . . . . . . . . . . . . . . . . . .  39  3.14 The 3dB bandwidths of the same grating coupler design at different positions of the chip . . . . . . . . . . . . . . . . . .  40  3.15 The central wavelengths of the same grating coupler design at different positions of the chip . . . . . . . . . . . . . . . . .  40  3.16 Flow chart of the universal design method . . . . . . . . . . .  42  3.17 Universal grating couplers with 10 degree incident angle for TE mode wave with oxide cladding . . . . . . . . . . . . . . .  46  3.18 Universal grating couplers with 10 degree incident angle for TM mode wave with oxide cladding  . . . . . . . . . . . . . .  46  3.19 Universal grating coupler with 10 degree incident angle for TE mode wave with air cladding . . . . . . . . . . . . . . . .  47  3.20 Universal grating coupler with 10 degree incident angle for TM mode wave with air cladding . . . . . . . . . . . . . . . .  47  3.21 Comparison of designs generated by universal grating coupler model and the optimized design for 1550nm TE wave with 10 degree incident angle . . . . . . . . . . . . . . . . . . . . . . .  48 x  List of Figures 3.22 Mask layout of a grating coupler with focusing grating curve  50  3.23 Measurement vs. simulation of universal grating couplers with 10 degree incident angle . . . . . . . . . . . . . . . . . .  52  3.24 Measurement vs. simulation of universal grating couplers with 15 degree incident angle . . . . . . . . . . . . . . . . . .  52  3.25 Measurement vs. simulation of universal grating couplers with 20 degree incident angle . . . . . . . . . . . . . . . . . .  53  3.26 (a) Peak power of simulation and measurement results with 10 incident angle; (b)comparison of simulated and measured wavelength mismatch with 10 degree incident angle . . . . . .  54  3.27 (a)Peak power of simulation and measurement results with 15 incident angle; (b)comparison of simulated and measured wavelength mismatch with 15 degree incident angle . . . . . .  54  3.28 (a)Peak power of simulation and measurement results with 20 incident angle; (b)comparison of simulated and measured wavelength mismatch with 20 degree incident angle . . . . . .  55  4.1  Schematic of fully etched grating coupler . . . . . . . . . . . .  57  4.2  Schematic of diffraction, reflection and penetration of a regular fully etched grating coupler  4.3  . . . . . . . . . . . . . . . .  58  Insertion loss of grating couplers with 20 degree incident angle for TE operation wave as function of the thickness of the buried oxide. . . . . . . . . . . . . . . . . . . . . . . . . . . .  4.4  59  Directionality, insertion loss and reflection to waveguide of a general fully-etched grating coupler . . . . . . . . . . . . . . .  61  4.5  Mask layout of a fully etched grating coupler test structure .  62  4.6  Comparison of measurement result and simulation result of fully-etched grating coupler . . . . . . . . . . . . . . . . . . .  4.7  Comparison of back reflections between shallow-etched grating coupler and fully-etched grating coupler . . . . . . . . . .  4.8  63 64  Schematic of a fully etched grating coupler with sub-wavelength gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  66  xi  List of Figures 4.9  Comparison of regular fully etched grating coupler and the fully etched grating coupler with minor sub-wavelength gratings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  67  4.10 Measurement results of the regular fully etched grating coupler and the fully-etched grating coupler with minor subwavelength gratings . . . . . . . . . . . . . . . . . . . . . . .  68  5.1  Diagram of wave vectors for vertical grating coupler . . . . .  71  5.2  Insertion loss and back reflection of a regular vertical grating coupler  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  72  5.3  Schematic of bidirectional grating coupler for vertical coupling 73  5.4  Cross section of a bidirectional grating coupler . . . . . . . .  5.5  The insertion loss of the optimized bidirectional grating coupler 75  5.6  Insertion of the bidirectional vertical grating coupler as function of wavelength for different offset . . . . . . . . . . . . . .  5.7  74  76  Schematic of a bidirectional grating coupler with incident wave off from the centre of the grating. . . . . . . . . . . . . .  77  xii  Acknowledgements I would like to thank my supervisor Dr. Lukas Chrostowski. For the care he provided to help me settle down when I first came to the University of British Columbia as an international student, for the patient guidance and help he provided when I was faced with difficulties and for the friendly and warm atmosphere he has created within our research group. I would like also to thank Dr. Nicolas Jaeger for the helpful instruction and kindly help with my research. Also, I would like to thank my colleagues Wei Shi, Xu Wang, Han Yun, Jonas Flueckiger, Samantha Grist, Charlie Lin, and Sahba Talebi Fard. Special thanks to Wei Shi for his instructive discussions and kind help. In addition, I would also like to acknowledge the nano-fabrication center at University of Washington, CMC Microsystems and OpSIS IME for the fabrication of my devices and Lumerical Solutions Inc. for their technical support.  xiii  Dedication To my father Junping Wang and my mother Miaoling Zhang, who have shown me what love is. Specifically, the courage, determination and passion from my father to help me find what I truly love and to follow my own heart, and the rationality from my mother, which helped me to think independently when I met obstacles in my life. To my fiance Ge Shi, for her dedication and love.  xiv  Chapter 1  Introduction 1.1  Silicon Photonics  After dominating the electronics industry for decades, silicon is on the verge of becoming the material of choice for the photonics industry [21]. The early work in the area of silicon photonics dates back to the late 1980s and the early 1990s [12, 20, 43, 46, 51, 52, 52, 53, 62]. The motivation for silicon photonics is its compatibility with the silicon Integrated Circuit (IC) manufacturing process, which represents the most spectacular convergence of technological sophistication and economies of scale. The industry is able to produce microprocessors with hundreds of millions of components, all integrated onto a thumb-size chip, and to offer them at extremely low price. Another motivation is the availability of high-quality silicon-on-insulator (SOI) wafers, an ideal platform for creating planar waveguide circuits. A schematic of an SOI wafer is shown in Fig. 1.1. An SOI wafer consists of three layers: a silicon substrate at the bottom (of Fig. 1.1) for mechanical support, a buried oxide (BOX) layer (in the middle) acting as the insulator layer, and another silicon layer (on the top) acting as the functional wave guiding layer. There may be another cladding layer on top of the silicon layer for protection. The strong optical confinement offered by the high index contrast between Si (n=3.45) and SiO2 (n = 1.45) makes it possible to make photonics devices on the scale of a few hundred nanometers. Such small dimensions are required for compatibility with the IC processing. The Smart Cut [2] technology commercialized by Soitec also propelled the commercial applications of SOI into exponential growth and entered the mainstream of Ultra Large Scale Integration (ULSI).  1  1.1. Silicon Photonics  Figure 1.1: Schematic of SOI wafer  The main application envisioned for silicon photonics is optical interconnects for CMOS electronics [34]. Conventional wisdom holds that optical interconnects are much better suited than copper interconnects in handling high data rates. The physical benefits of changing the technology used for interconnects to silicon photonic chips in computing and switching systems have been detailed in [33]. With the advantages of design simplification, architectural advantages and physical benefits such as reduction of power dissipation in interconnects and voltage isolation etc., light beams that have dominated long-distance communication are progressively taking over in shorter distance networks. Luxtera announced the world’s first 40 Gigabit Optical Active Cable (OAC) in 2007 [28]. Intel demonstrated their 4x12.5 Gbps CWDM silicon photonics link using integrated hybrid silicon lasers [25, 37] in 2010. IBM unveiled the holey optochip to transfer information at terabit per second speed in 2012 [10]. All those advancess show the potential of silicon photonics as the future solution for optical interconnects. Another important application of silicon photonics is bio-sensing. A disposable mass-produced sensor would be attractive as it could grow the market for biosensors. Sensor applications are somewhat different from optical communication ones as there are other low cost optical technologies that  2  1.2. Challenge of Coupling Light Into Nanophotonic Waveguide compete in this space [23]. One likely application area for silicon photonics is the so-called lab-on-a-chip, in which both reaction and analysis are performed in a single device. In the future, this could be extended to include electronic intelligence and wireless communications to create intelligent sensor networks for environmental monitoring [21]. In addition, silicon has material properties that are important for a new class of mid-infrared (IR) photonics devices. These include the linear and non-linear optical properties in the mid-wave IR spectrum. The high optical intensity arising from the large index contrast (between Si and SiO2 ) makes it possible to observe nonlinear optical interactions, such as Raman and Kerr effects, in chip-scale devices. Far from being limited to the near-IR data communication band, from 1.26 to 1.67 microns, silicon also has a low loss wavelength window extending from 1.1 to nearly 7 microns [22]. But oxide limits its use in SOI to 3 microns. Silicon has high thermal conductivity (10x higher than GaAs), high optical damage threshold (10x higher than GaAs), and high third-order optical nonlinearities [21].  1.2  Challenge of Coupling Light Into Nanophotonic Waveguide  Due to the large refractive index contrast between the silicon core (n≈ 3.47 at 1550 nanometers) and the silicon dioxide cladding (n≈1.5 at 1550 nanometers), propagation modes are highly confined within the waveguide with a dimension on the order of a few hundred nanometers, which enables largescale integration. However, the small feature size of the waveguide raises the problem of huge mode mismatch between the optical mode within an optical fibre and the mode within the waveguide. The cross-sectional area of an optical fibre-core (with a diameter of 10 microns) is almost 3,000 times larger than that of a silicon waveguide (with dimensions of 500 nanometers x 220 nanometers). Several approaches have been demonstrated to tackle the problem of mode mismatch. Edge coupling using spot size converters and lensed fi-  3  1.2. Challenge of Coupling Light Into Nanophotonic Waveguide bres is one solution used to address this, and high-efficiency coupling with an insertion loss below 0.5 dB has been demonstrated [30]. However, this approach can be only used at the edge of the chips, and the implementation of such designs requires complicated post-processes, which increase the packaging cost dramatically. The alignment of such devices during the measurement requires high accuracy, which often takes a lot of effort. The grating couplers are an alternative solution to tackle the issue of mode mismatch. Compared to the edge coupling, grating couplers have several advantages: alignment to grating couplers during measurement is much easier than alignment to edge couplers; the fabrication of grating couplers does not require post-processing, which reduces the fabrication cost; grating couplers can be put anywhere on a chip, which provides flexibility in the design as well as enabling wafer scale automated testing. Both academic and industrial research groups have demonstrated high efficiency grating couplers [32, 35, 41].  Figure 1.2: Schematic of grating coupler  4  1.3. Grating Coupler  1.3  Grating Coupler  Figure 1.2 is a schematic of a shallow-etched grating coupler design in siliconon-insulator. The thickness of the functional Si layer and the thickness of the buried oxide (BOX) layer are determined by the wafer type. A cladding layer is often employed to protect the functional silicon layer, as shown in Fig. 1.2. However, for some uses such as bio-sensing, air cladding is required. In Fig. 1.2 : • Λ denotes the grating period, which represents the length of the periodic pattern; • W denotes duty cycle of the grating coupler, which is the width of the grating tooth; • f f denotes fill factor, which is defined as the ratio of the grating period and the duty cycle, i.e, f f = Λ/W ; • θ denotes the incident angle of the grating coupler, which is the angle between the incident wave and the normal to the grating surface; • ed denotes the heights of the grating teeth, which are defined as etch depth. In addition to the geometric variables defined above, some other terms are often used when discussing grating couplers: 1. Directionality: the ratio between the power diffracted upwards (Pup ) and the input power from the waveguide (Pwg ) [35], which is usually expressed in decibels (dB) as 10 · log10 (Pup /Pwg ); 2. Insertion loss (coupling efficiency): the ratio between the power coupled into the fundamental mode of the fibre (Pfund ) and the input power from the waveguide (Pwg ), which is usually expressed in dB as 10 · log10 (Pfund /Pwg ); 3. Penetration loss: the ratio between the power lost in the substrate (Psub ) and the input power from the waveguide, which is usually expressed in dB as 10 · log10 (Psub /Pwg ); 5  1.4. Slab Waveguide and Channel Waveguide 4. Back reflection to the waveguide: due to the refractive index contrast between the silicon wire waveguide and the grating, part of the input light from the waveguide will be reflected back into the waveguide. The ratio between the reflected power and the input power from the waveguide is called back reflection to the waveguide. It is usually expressed in dB or in percentage. This back reflection is unwanted because it will cause Fabry-Perot oscillations by reflecting back and forth between the input and output grating couplers [35, 55].  1.4  Slab Waveguide and Channel Waveguide  At angles of incidence above the critical angle it is possible to achieve total internal reflection from a dielectric interface given that the incident electromagnetic wave is in the medium with the higher refractive index. So, it should be possible to place a slab of dielectric with a high refractive index between two media with lower refractive indices and confine a plane wave to propagate in the high index dielectric slab [19]. Such a waveguide is called the slab waveguide. A schematic of a slab waveguide in SOI is shown in the left side of Fig. 1.3. The slab is such that it extends to infinity in the x and z directions. However, the waveguide used in photonic systems are the ones with two-dimensional refractive index profiles, which is shown in the right side of Fig. 1.3. Such a waveguide is called the channel waveguide.  Figure 1.3: Schematic of slab waveguide and channel waveguide in SOI  6  1.5. Polarization of Waveguide Modes  1.5  Polarization of Waveguide Modes  If we orient our coordinate system so that the interface between the silicon and the cladding and the interface between the silicon and the buried oxide lie in the xz plane (shown in Fig. 1.3) and the guided mode propagates along the z axis, then the wave vectors for the incident, reflected, and transmitted waves are all contained in the yz plane. Thus the yz plane is called the plane of incidence [19]. Transverse electric (TE) modes are defined as these modes having their electric fields perpendicular to the plane of incidence (i.e. x axis here) and transverse magnetic (TM) modes are defined as these modes having their magnetic fields perpendicular to the plane of incidence. The amplitude of the electric field and magnetic field of the fundamental (first order or lowest order) TE-like mode are shown in Fig. 1.4. The amplitude of the electric field and magnetic field of the fundamental TM-like mode are shown in Fig. 1.5. By comparing the amplitude of the fundamental TE-like mode and the fundamental TM-like mode, we can see that the TE-like mode is better confined than the TM-like mode. In all subsequent sections of this thesis, channel waveguides are assumed, and we will drop the -like extension and refer to the modes as simple TE and TM modes. The propagation constant, β, is employed to represent the behaviour of different modes within the channel waveguide and is defined as [19]: β=  2·π · neff λ0  (1.1)  where λ0 denotes the operation wavelength and neff denotes the effective refractive index of the mode. The effective refractive index is introduced to describe and compare the confined modes within the channel waveguide and is defined as: neff =  β c = vp k0  (1.2)  where c is the speed of light in vacuum, vp is the phase velocity of the mode, and k0 denotes the wave vector in free space, i.e., k0 = 2π/λ.  7  1.6. State-of-the-art Grating Couplers  1.6  State-of-the-art Grating Couplers  Some state-of-the-art grating couplers are listed in Table 1.1. Important parameters such as central wavelength, polarization, insertion loss, and bandwidth, for comparison purpose, are shown in the table. Corresponding fabrication details are also listed because the performance of a grating coupler is highly dependent on how it is made. The insertion loss of the grating coupler is mainly determined by the thickness of the top silicon layer and the thickness of the buried oxide (BOX), because these two values affect the phase conditions of different wavelengths at the interface between the grating layer and the buried oxide layer. Bottom distributed Bragg reflectors (DBR) [31] and top overlays [56] have been used to decrease the insertion loss, but nonstandard wafers and fabrication processes are required for these approaches. Both shallow etched and fully etched grating couplers are shown in the table and the central wavelengths are around 1550 nm and 1310 nm, which are in the two most commonly used optical windows in telecommunications. These grating couplers listed are mainly designed for TE mode operation.  8  1.6. State-of-the-art Grating Couplers  0.4 0.8  0.3  0.7  0.2  0.6  micron  0.1 0.5  0 0.4  −0.1 0.3  −0.2  0.2  −0.3 −0.4 −0.5  0.1  0 micron  0.5  (a) −5  0.4  x 10 7  0.3  6  0.2  5  micron  0.1 4  0 3  −0.1 2  −0.2 1  −0.3 −0.4 −0.5  0 micron  0.5  (b) Figure 1.4: (a) The amplitude of the electric field of the first order TE-like mode in a rectangular channel waveguide; (b) The amplitude of the magnetic field of the first order TE-like mode in a rectangular channel waveguide [9]  9  1.6. State-of-the-art Grating Couplers  0.4 0.8  0.3  0.7  0.2  0.6  micron  0.1 0.5  0 0.4  −0.1 0.3  −0.2  0.2  −0.3 −0.4 −0.5  0.1  0 micron  0.5  (a) −5  x 10 4.5  0.4  4  0.3  3.5  0.2  3  micron  0.1 2.5  0 2  −0.1 1.5  −0.2  1  −0.3 −0.4 −0.5  0.5  0 micron  0.5  (b) Figure 1.5: (a)The amplitude of the electric field of the first order TM-like mode in a rectangular channel waveguide; (b)The amplitude of the magnetic filed of the first order TM-like mode in a rectangular channel waveguide [9]  10  [48] [56] [3] [35] [60] [31] [13] [58]  Polarization TE TE TE TE TE TE TM TE  Insertion Loss -5.1 dB -1.6dB -1.2dB -3dB -4.4dB -0.75 dB -3.7dB -2.29dB  Bandwidth 40nm (1dB) 80nm (3dB) N/A 58nm 45nm (1.5dB) N/A 60nm 60nm  Process 220nm Si, 1um BOX, shallow etch amorphous Si overlay, shallow etch 340nm Si, 2um BOX, shallow etch 400nm Si, shallow etch 220nm Si, 2um BOX, shallow etch shallow etch, bottom DBR full etch full etch, 250 Si ,1um BOX  Table 1.1: State-of-the-art grating couplers  1.6. State-of-the-art Grating Couplers  2006 2010 2010 2011 2012 2012 2010 2012  Wavelength 1550nm 1530nm 1530nm 1310nm 1550nm 1490nm 1550nm 1550nm  11  1.7. Measurement Setup  1.7  Measurement Setup  (a)  (b) Figure 1.6: (a) Illustration of the automated setup; (b) automated setup  12  1.7. Measurement Setup The automated setup used for the measurement of the devices shown in this thesis are built by Charlie Lin [26] and Jonas Flueckiger. The illustration and physical map of the automated setup are shown in Fig. 1.6 (a) and Fig. 1.6(b), respectively. The chip is placed on a platform which is on top of an angle rotator. An X-axis motorized stage and a Y-axis stage are seated below the angle rotator, which forms a cross configuration. The fibre array ribbon is held by a custom-made aluminum fibre ribbon holder, which is suspended on top of the chip platform. An aluminum arm is used to hold the fibre ribbon holder and is attached to an angle rotator (shown in Fig. 1.7). The angle rotator is fixed onto a Z-axis actuator that is bolted to a raised platform so the fibre ribbon height can be manually adjusted accordingly. The ribbon-to-chip image is captured two microscopes; one microscope shows the top view and is used for alignment purpose; the other microscope is angled from the side to display the height displacement between the fibre array and the chip to prevent crashing the fibre array into the chip during alignment. The light source for the microscopes illuminates the chip platform at an angle from behind the fibre ribbon [26].  Figure 1.7: Fibre array ribbon, ribbon holder and ribbon arm [26]  13  Chapter 2  Theory And Numerical Methods In this chapter, we will give a short overview on the fundamental theories and numerical methods related to grating coupler design. We will start from Bragg’s Law, from which we will derive the most fundamental theory to deal with periodic structures, i.e., the Bragg condition. Also, the effective Index Method (EIM) will be introduced to obtain an approximation of the effective index of refraction of a slab waveguide. Finally, the Finite-Difference Time Domain (FDTD) method is introduced to obtain the optimized solution for the grating coupler design and to predict the performance.  2.1 2.1.1  Bragg Condition Bragg’s Law  A schematic of Bragg’s Law is shown in Fig. 2.1. Periodic dots are seated in the air with a period of d in the y direction. A plane wave is incident on the periodic structure and is scattered by each plane periodic dots in such a way that the portion scattered from the second plane of dots undergoes an extra length of 2dsin(θ), as compared to the portion scattered by the first plane of dots. Depending on the phase condition, either constructive interference or destructive interference occurs. Constructive interference occurs when the extra length is equal to an integer multiple of the wavelength of the incident wave, i.e.: 2 · d · sinθ = m · π  (2.1)  14  2.1. Bragg Condition where m is an integer, λ is the wavelength of the incident wave, and θ is the scattering angle.  Figure 2.1: Bragg’s Law  2.1.2  The Bragg Condition for Grating Coupler  The most fundamental formula concerning periodic structures is Bragg condition. The grating coupler discussed in this thesis is a one-dimensional periodic structure, as shown in Fig. 2.2. In Fig. 2.2 we assume that the wave incident on the grating is a guided wave propagating in a slab waveguide, and its direction of propagation is in the same plane as the grating and is normal to the grating teeth. The general form of the Bragg condition can be expressed as: β − kz = m · K  (2.2)  where β denotes the wave vector of the input wave, i.e., β = 2πnwg /λ, nwg denotes the effective index of the incident wave, kz denotes the component of the wave vector of the diffracted wave in the direction of the incident wave, where kz = 2πnc /λ; and K = 2π/Λ, which is determined by the periodicity 15  2.1. Bragg Condition of the structure. This relation can also be depicted by a diagram shown in Fig. 2.3, which is easier to understand.  Figure 2.2: Schematic of a grating coupler  The diffractions of a grating coupler can be observed in those directions where constructive interference is achieved. Each value of m that results in diffraction is referred to as the m-th order diffraction: neff · Λ − nc · Λ · sinθ = m · λ  (2.3)  where neff denotes the effective index of the grating, nc denotes the effective index of the fibre mode in the cladding, Λ denotes the period of the grating, θ denotes the diffraction angle, λ denotes the wavelength of the incident wave (or out-coupled wave), and m is an integer denoting the diffraction order. The diffraction order normally used for coupling is the first order (i.e., m = 1), so the Bragg Condition for a grating coupler can be further simplified to: neff − nc · sin(θ) =  λ Λ  (2.4)  It should be noted that the Bragg condition is only exact for infinite gratings, i.e., one-dimensional grating with infinite grating period. In a  16  2.2. Effective Index Method  Figure 2.3: Diagram for Bragg condition  finite grating, there is not a discrete k-vector for which diffraction occurs, but a range of k-vectors around the one predicted by the Bragg condition [49].  2.2  Effective Index Method  The Effective Index Method (EIM) was initially proposed for the analysis of dielectric waveguides with rectangular cores [24]. The basic idea of this method is to replace a two-dimensional waveguide with a one-dimensional one with an effective index derived from the geometry and refractive index of the original structure [7]. The EIM has been applied to various structures such as optical waveguides [17, 29], optical fibres [7], and waveguide arrays [8]. Figure 2.4 shows the schematic of a silicon-on-insulator waveguide. It is a three-layer structure with SiO2 (n=1.444) in the middle and Si (n=3.47) layers on the top and bottom. Air (n=1) has been employed as the cladding in this structure. The procedure for calculating the effective index for silicon17  2.2. Effective Index Method  Figure 2.4: Cross section of silicon-on-insulator waveguide  on-insulator waveguide with a two-dimensional cross-section is shown in Fig. 2.5 and Fig. 2.6. The basic approach is to solve the mode condition for a particular mode type in one dimension and find the propagation constant. The effective index can be derived from the propagation constant, neff = β/k0 , and then applied to the other dimension of the structure [19].  Figure 2.5: Schematic of a SOI strip waveguide  18  2.2. Effective Index Method  Figure 2.6: Schematic of the effective index of a strip waveguide  To begin with, we divide the structure into three regions. These three regions are x < −w/2, −w/2 < x < w/2, and w/2 < x, where w denote the width of the silicon waveguide. The cladding of the waveguide is air, which has a refractive index of 1. Next we decide on the mode type that we wish to solve for. For the lowest order TE-like mode, the T E0 mode, we solve the mode condition for TE modes in the y-direction of a slab waveguide with the refractive index profile in region I. The mode condition for TE modes within region I can be expressed by [19]: q p ht = mπ + tan−1 ( ) + tan−1 ( ); m = 0, 1, 2, ... h h  (2.5)  h=  k02 n22 − β 2  (2.6)  p=  β 2 − k02 n23  (2.7)  q=  β 2 − k02 n21  (2.8) 19  2.2. Effective Index Method where m is the mode order, n1 , n2 , n3 , are the refractive index of the cladding, silicon core and SiO2 , respectively. β is the propagation constant of the mode supported by the slab waveguide, which is defined as k0 neffI , where neffI is the effective index of the mode within the slab waveguide in region I. Now we create the one-dimensional waveguide structure shown in Fig. 2.6, in which our calculated neffI is used for the central slab, i.e., for region I. The refractive index chosen for region II depends on the actual structure; nevertheless, for SOI strip waveguides with air cladding, one can use refractive index of the air, i.e., n=1. In the x-direction, the mode condition is solved for the appropriate TM mode of the structure. The condition for TM modes can be expressed as [19]: q¯ p¯ ht = mπ + tan−1 ( ) + tan−1 ( ); m = 0, 1, 2, ... h h  (2.9)  p¯ =  n22 ·p n23  (2.10)  q¯ =  n22 ·q n21  (2.11)  Using the EIM, we have now successfully converted a two-dimensional waveguide into a one-dimensional structure, and then solved for that structure. By doing this, the EIM enables us to simulate three-dimensional structures as two-dimensional ones, which saves significant computational effort and time. Following on this idea, we can take complex three-dimensional waveguides and reduce them to two-dimensional systems in which each waveguide is replaced by its effective index in a plane that is parallel to the the interface of the Si layer and the BOX layer. The effective index of refraction of a strip waveguide, with a dimension of 220 nm by 500 nm, for both T E0 and T M0 modes, as a function of wavelength, are shown in Fig. 2.7. The T E0 mode has a larger effective index than the T M0 mode because the TE mode is better confined within the waveguide. In this thesis, we only use the first step of the EIM to find the effective index of the slab waveguide which forms the grating coupler. 20  2.3. Finite-Difference Time-Domain Method 3 TE TM  2.8 2.6  effective index  2.4 2.2 2 1.8 1.6 1.4 1.2 1 1500  1520  1540 1560 wavelength (nm)  1580  1600  Figure 2.7: Effective index of TE and TM modes  2.3  Finite-Difference Time-Domain Method  The components used in photonic integrated circuits are normally complicated three-dimensional structures such as gratings, rings, waveguide couplers, etc. It is not possible to obtain the exact analytical solutions for these structures, except for some special cases. In practice, we use numerical methods to obtain the solutions for such structures. Finite-Differential Time-Domain (FDTD) method is a very popular numerical method used for obtaining solutions to two-dimensional and three-dimensional structures. When Maxwell’s differential equations are examined, it can be seen that the change in the E-field in time (the time derivative) depends on the change in the H-field in space (the curl). This results in the basic FDTD timestepping relation that, at any point in space, the updated value of the E-field in time depends on the stored value of the E-field and the numerical curl of the local distribution of the H-field in space[59]. The H-field is time-stepped in a similar manner. At any point in space, the updated value of the H-field in time is dependent on the stored value of the H-field and the numerical 21  2.3. Finite-Difference Time-Domain Method curl of the local distribution of the E-field in space. Iterating the E-field and H-field updates results in a “marching-in-time” process, wherein sampleddata analogs of the continuous electromagnetic waves under consideration propagate in a numerical grid stored in the computer memory[18].  Figure 2.8: FDTD mesh for a grating coupler  During the simulation, the structure is discretized using a uniform grid as shown in Fig. 2.8. A brute-force calculation of Maxwell’s equations on each mesh point will be operated within the time domain. The simulation accuracy is highly dependent on the grid size. A smaller grid size is required to get more accurate results. The advantage of the FDTD method is that it can deal with arbitrarily complicated structures. However, the drawback of this method is long calculation times and large computational memory is required for accurate simulations. The FDTD method may not be appropriate for simulating extra long structures, but it is ideal for simulating compact structures such as grating couplers. FDTD Solutions, a commercial product from Lumerical Solutions Inc., was employed as the simulation tool for all grating couplers in this thesis.  22  2.3. Finite-Difference Time-Domain Method Two-dimensional simulations are generally used to simulate grating couplers because it takes much less computational memory and simulation time. After designing a grating using 2D simulations, 3D simulations are used to verify the behaviours of the final design. The schematic shown in Fig. 2.9 depicts a general grating coupler structure: a Si layer on the bottom for mechanical strength, a functional Si layer on top of 2-um buried oxide, and a top oxide layer for protection. The orange rectangle defines the simulation region and Perfectly Matched Layer (PML) boundary is used so that radiation appears to propagate out of the computational area and therefore, does not interfere with the fields inside. The yellow lines shown in the graphs represent frequency-domain power monitors which collect highaccuracy power flow information in the frequency domain from simulation results across spatial regions within the simulation. The green area denotes the fibre, the purpose of which will be explained below. Two types of simulation structures are often employed to simulate a grating coupler. Figure. 2.9 (a) is used to simulate the input grating coupler and Fig. 2.9 (b) is used to simulate the output grating coupler. A polished optical fibre is presented on top of the cladding, with a light green area indicating the fibre core and a dark green area indicating the cladding of the fibre. For an input grating coupler, a fundamental TE mode is launched from the fibre core and coupled into the waveguide by the grating. Power monitors are used to record the insertion loss and reflection to the fibre of the grating coupler. For an output grating coupler, a fundamental TE mode is launched from the waveguide and out-coupled into the fibre. A mode expansion monitor is used to calculate the power that goes into the fundamental mode of the fibre [18]. This is essential for a coupler because not all of the power is coupled into the fundamental mode due to the mode mismatch.  23  2.3. Finite-Difference Time-Domain Method  (a)  (b) Figure 2.9: (a) Schematic of simulation structure for input grating coupler; (b) Schematic of simulation structure for output grating coupler  24  Chapter 3  Detuned Grating Coupler The most important properties for a grating coupler are the insertion loss, the back reflection to the waveguide, and the bandwidth. Compared to the fully etched grating couplers, shallow etched grating couplers have the advantages of high coupling efficiency and low back reflection to the waveguide. So it is more popularly employed in the integrated photonics circuits. A small angle is often employed between the incident wave and the normal of the grating surface, so that the large second order Bragg reflection can be avoided, which is called the detuned case. In this chapter, the design flow for detuned shallow etched grating couplers will be presented, and the influences of various factors on the properties of the grating couplers will be discussed in detail. In addition, a universal grating coupler design methodology will be introduced to generate shallow etched grating couplers, using analytic calculations instead of numeric simulations.  3.1  Detuned Shallow Etched Grating Coupler  Designing a grating coupler follows a procedure. The first step is to get the initial condition of the desired grating coupler using theoretical calculations, and the second step is to optimize the performance of the initial design by sweeping various parameters such as grating period, duty cycle, and incident angle.  25  3.1. Detuned Shallow Etched Grating Coupler  3.1.1  Initial Condition  Designing a grating coupler should follow some restrictions: some of the parameters are determined by the wafer type we use, such as the thickness of the silicon layer and the thickness of the buried oxide; some of the parameters are determined by the fabrication process, such as the cladding material, the etch depth, and the minimum feature size; and some of the parameters are decided by the application and tunability of the measurement setup, such as the central wavelength and the incident angle. In our case, all the known initial values are listed in Table 3.1. Si 220 nm  SiO2 2um  Cladding air  Etch Depth 70 nm  λ 1550 nm  θ 20 degree  Table 3.1: Initial values  Given the initial values listed in Table 3.1, we can calculate the effective index of refraction of the grating. We used the Finite-difference time-domain (FDTD) method to calculate the effective index of refraction of the grating. The effective index of refraction of the shallow etched slab waveguide, neff1 , is calculated to be 2.534, and the effective index of refraction of the unetched slab waveguide, neff2 , is calculated to be 2.848. Thus, the overall effective index of refraction of the grating neff can be calculated from the following equation: neff = neff1 · f f + neff2 · (1 − f f )  (3.1)  where f f denotes the fill factor of the grating coupler. With the effective index of refraction of the grating, we can get the grating period, Λ, from the Bragg condition: nc · sinθ = neff −  λ Λ  (3.2)  where nc denotes the effective index of the fibre mode, θ denotes the incident angle, neff denotes the effective index of refraction of the grating, and λ denotes the desired central wavelength. In our case, the grating period was 26  3.1. Detuned Shallow Etched Grating Coupler calculated to be 660 nanometers. The insertion loss and back reflection of the initial grating coupler are shown in Fig. 3.1. The central wavelength of the initial grating coupler design is 1553nm, with an insertion loss of -3.1dB and a 3dB bandwidth of 70nm. 0 −5  power (dB)  −10 −15  InsertionLoss Reflection  −20 −25 −30 −35 1400  1450  1500 1550 1600 wavelength (nm)  1650  1700  Figure 3.1: Insertion loss and back reflection to the waveguide of the initial grating coupler  3.1.2  Design Parameters  Once we got the initial condition, optimization of the design can be achieved by sweeping different design parameters such as the grating period, the duty cycle, the etch depth, and the incident angle. The impacts that each of the parameters has on the grating coupler are described in the following sections. Grating Period Grating period influences the performance of a grating coupler through the following equation: Λ=  λ neff − nc · sin(θ)  (3.3) 27  3.1. Detuned Shallow Etched Grating Coupler 0 −5  power (dB)  −10 −15 −20 −25  620nm 640nm 660nm 680nm 700nm  −30 −35 1400  1450  1500 1550 1600 wavelength (nm)  1650  1700  Figure 3.2: Variations of grating period  where Λ denotes the period of the grating coupler, λ denotes the central wavelength, neff denotes the effective index of the grating, nc denotes the effective refractive index of the fibre mode and θ denotes the incident angle. As shown in the equation, the central wavelength of the grating coupler is proportional to the grating period. Figure. 3.2 shows the simulation results for varying the grating period. We kept nc = 1, θ = 20◦ , and f f = 0.5. As we varied the grating period from 620 nanometers to 700 nanometers, the central wavelength of the grating coupler shifted from 1500 nanometers to 1612 nanometers. The redshift shown in the central wavelength is consistent with our analytical calculation. And the tuning coefficient of the grating period, which is defined as δλ/δΛ, was calculated to be 1.4 nm/nm.  28  3.1. Detuned Shallow Etched Grating Coupler Duty Cycle Duty cycle affects the performance of a grating coupler through its impact on the effective index of refraction of the grating: neff = nc · sin(θ) +  λ Λ  (3.4)  where neff denotes the effective index of refraction of the grating, θ is the incident angle and λ is the central wavelength. For a given grating period, the effective index of refraction of the grating is proportional to the duty cycle of the grating. Figure. 3.3 shows the simulation results for varying duty cycle. We kept grating period constant at 660 nanometers and varied the duty cycle from 230 nanometers to 430 nanometers. The central wavelength shifted from 1536 nanometers to 1579 nanometers as we varied the duty cycle. The tuning coefficient of the duty cycle, which is defined as  δλ δW ,  was  calculated to be 0.215 nm/nm. By comparing Fig. 3.3 and Fig. 3.2, we note that grating period has a stronger impact on the central wavelength of the grating than the duty cycle does. 0 −5  power (dB)  −10 −15 −20 −25  230nm 280nm 330nm 380nm 430nm  −30 −35 1400  1450  1500 1550 1600 wavelength (nm)  1650  1700  Figure 3.3: Variations of duty cycle  29  3.1. Detuned Shallow Etched Grating Coupler Etch Depth Etch depth of a grating coupler also influences the performance of the grating coupler through its impact on the effective index of refraction of the grating: neff = nc · sin(θ) +  λ Λ  (3.5)  where neff denotes the effective index of refraction of the grating, θ denotes the incident angle, and λ denotes the central wavelength. As the etch depth increases, the effective index of refraction of the shallow etched area decreases, thus neff decreases. The effective index of refraction is proportional to the central wavelength of the grating, so the etch depth of the grating coupler is inversely proportional to the central wavelength of the grating. Figure. 3.4 shows the simulation results for varying the etch depth. We 0  −5  power (dB)  −10  −15  −20 60nm 65nm 70nm 75nm 80nm  −25  −30 1400  1450  1500 1550 1600 wavelength (nm)  1650  1700  Figure 3.4: Variations of etch depth  kept grating period, duty cycle, incident angle constant and varied the etch depth of the grating coupler from 60 nanometers to 80 nanometers. The central wavelength shows a blueshift as the etch depth increases, which is  30  3.1. Detuned Shallow Etched Grating Coupler consistent with our analytical calculation. The tuning coefficient of the etch depth, which is defines as δλ/δed, was calculated to be 1.9 nm/nm. 0 −5  power (dB)  −10 −15 −20 −25  15 17.5 20 22.5 25  −30 −35 1400  1450  1500 1550 1600 wavelength (nm)  1650  1700  Figure 3.5: Variations of incident angle  Incident Angle The incident angle of a grating coupler is defined as the angle between the incident wave (or out-coupled wave) and the normal to the grating surface. A positive angle indicates the case in which the incident wave and the coupled wave in the waveguide propagate in the same direction and a negative angle indicates the case in which the incident wave and the coupled wave in the waveguide propagate in opposite directions. The incident angle influences the central wavelength of the grating coupler through the following equation: sin(θ) =  neff − nc  λ Λ  (3.6)  where neff denotes the effective index of refraction of the grating, λ is the central wavelength, Λ is the grating period and nc denotes the effective index 31  3.1. Detuned Shallow Etched Grating Coupler of the wave incident on the grating. Simulation results for varying incident angle are shown in Figure 3.5. We kept the grating period, duty cycle and etch depth of the grating constant. As we increased the incident angle from 15 degree to 25 degree, the central wavelength shifted from 1594 nanometers to1524 nanometers. So the tuning coefficient, which defined as δλ/δθ, was calculated to be 7 nm/degree. From the simulation results shown above, we can see that grating period, duty cycle, etch depth and incident angle all have impacts on the central wavelength of the grating coupler, but the tuning coefficient is different. Table 3.2 shows the comparison of the tuning coefficients of various parameters. Parameter Tuning coefficient  period 1.4 nm/nm  duty cycle 0.215 nm/nm  etch depth 1.9 nm/nm  incident angle 7 nm/degree  Table 3.2: Tuning coefficients of various parameters  Figure 3.6: Schematic of reflections at different interfaces  32  3.1. Detuned Shallow Etched Grating Coupler Cladding and Buried Oxide The thickness of the buried oxide and the thickness of the cladding are two important factors that have impacts on the insertion loss of the grating coupler. An illustration of different reflections at various interfaces of the grating coupler is shown in Figure. 3.6. The phase conditions between those reflections determine the insertion loss of the grating coupler. Minimum insertion loss can be achieved when Preflection1 and Preflection2 result in destructive interference and Preflection3 and Preflection4 result in destructive interference. Simulation results for varying buried oxide are shown in Figure. 3.7. As we varied the thickness of the buried oxide from 1 um to 3 um, the insertion loss of the grating coupler changed in a sinusoidal way, which is determined by the phase condition between Preflection1 and Preflection2 . The thickness of the buried oxide for a particular wafer type is chosen to achieve constructive interference between Preflection1 and Preflection2 , therefore, low insertion loss can be obtained. −3 −4  power (dB)  −5 −6 −7 −8 −9 −10 1  1.5  2 2.5 thickness of the BOX (um)  3  Figure 3.7: Variations for BOX thickness  33  3.1. Detuned Shallow Etched Grating Coupler Similarly, the phase condition of Preflection3 and Preflection4 varies as the thickness of the cladding changes. We define the thickness of the cladding to be the height from the interface of the silicon and buried oxide to the top surface of the cladding. Figure. 3.8 shows the simulation results for varying the cladding thickness. Minimum insertion loss achieved where destructive interference occurs and maximum insertion loss achieved where constructive interference occurs. Depending on the incident angle and the central wavelength, the optimal thickness of cladding changes. By comparing Fig. 3.7 and Fig. 3.8 we note that the thickness of the buried oxide has a larger impact on the insertion loss of the grating. This is the case because the reflection coefficients of Preflection3 and Preflection4 are larger than those of Preflection1 and Preflection2 . −2 −2.2  power (dB)  −2.4 −2.6 −2.8 −3 −3.2 −3.4 1  1.5 2 2.5 thickness of the cladding (um)  3  Figure 3.8: Variations for cladding thickness  3.1.3  Optimization of the Grating Coupler  Optimization of a grating coupler involves simulation sweeps of various parameters such as grating period, duty cycle, and incident angle, etc, but  34  3.1. Detuned Shallow Etched Grating Coupler not all of parameters are variable. The thickness of different layers are determined by the wafer type, whereas the etch depth and the cladding are normally determined by the fabrication process. So the design variables are the grating period, duty cycle, and incident angle. The optimization process requires a lot of simulation sweeps on those design variables. Figure. 3.9 shows the simulation and measurement results of the grating coupler designed for the fabrication process provided by Opsis-IME [1]. The designed grating coupler has a grating period of 650 nanometers with a duty cycle of 350 nanometers. The simulated results show an insertion loss of -2.74dB with a 3dB bandwidth of 79.8nm, and the measurement shows an insertion loss of -4.64dB with a 3dB bandwidth of 74.9nm. The simulation results shown in Fig. 3.9 is obtained from the model shown in Fig. 2.9 (a) with the assumption that the distance between the fibre tip and the chip is negligible. −2  power (dB)  −4 −6 −8 Simulation Measurement  −10  −12 1500  1520  1540 1560 wavelength (nm)  1580  1600  Figure 3.9: Grating coupler design fabricated through OpSIS-IME  From Fig. 3.9 we note that the measured insertion loss is smaller than the simulated insertion loss. In addition, the bandwidth of the measured spectrum is narrower than that of the simulated one. The mismatches in insertion loss and bandwidth mainly results from the gap between the fibre 35  3.1. Detuned Shallow Etched Grating Coupler ribbon and the photonics chip. The images of the fibre ribbon used in our measurement setup are shown in Fig. 3.10. The tip of the fibre ribbon is polished at 20.3 degrees. According to Snell’s Law, if the fibre tip is parallel to the photonics chip, it will give us a 30 degree incident angle in the air. In order to get a 20 degree incident angle in air, we need to rotate the fibre ribbon (as shown in the right image of Fig. 3.10). In this case, there will be a gap between the tip of the fibre ribbon and the photonic chip. In addition, extra space is left intentionally to protect the chip from scratching by the ribbon. So a large air gap exists between the ribbon and the photonic chip. The mode profile becomes larger as it propagates between the fibre ribbon and the chip, which introduces extra loss and narrows the bandwidth.  Figure 3.10: Images of fibre ribbon  Figure. 3.11 shows the impact of the gap between the fibre ribbon and the measured photonic chip. The blue curve indicates the change in insertion loss and the green curve indicates the change in bandwidth. For each value of the gap, i.e., the distance from the fibre to the chip, we optimized the XY position of the fibre to achieve lowest insertional loss. As we varied the gap from 0 to 200 nanometers, the insertion loss of the grating coupler decrease from -2.47dB to -6.08 dB and the bandwidth decreased from 77 nanometers to 44.5 nanometers.  36  3.1. Detuned Shallow Etched Grating Coupler According to the simulation results shown in Fig. 3.11, we revised our simulation model and simulated the grating coupler design shown in Fig. 3.9 with a 25 um air gap between the fibre ribbon tip and the chip. The comparison of the simulation results and measurement results of the grating coupler are shown in Fig. 3.12. The blue curve denotes the simulation results of the grating structure with 1um air gap between the fibre ribbon tip and the chip, the green curve denotes the simulation results with a 25 um air gap between the grating and the chip, and the red curve denotes the measurement results for the same gating design. As we take the air gap between the fibre ribbon tip and the grating into consideration, a closer match between the simulation results and the measurement results have been achieved. The remaining mismatch of insertion loss mainly comes from the optical connections of the measurement system, such as the connections between the fibre  0  80  −5  60  −10 0  50 100 150 Gap between fiber ribbon and chip (um)  Bandwidth (nm)  Insertion Loss (dB)  and the laser and the connection between the fibre and the detector.  40 200  Figure 3.11: Impacts of the gap between fibre ribbon tip and photonic chip on the insertion loss and bandwidth of the grating coupler  37  3.1. Detuned Shallow Etched Grating Coupler −2 −3 −4  power (dB)  −5 −6 −7 −8 −9 −10 Gap=1um Gap=25um Measurement  −11 −12 1500  1520  1540 1560 wavelength (nm)  1580  1600  Figure 3.12: Spectra of simulation results with different gap distance and measurement results  3.1.4  Design Stability  To examine the stability of the grating coupler design at different positions on the same photonic chip, we measured the performance of the same grating coupler design at different positions of the chip. The fabrication of the grating couplers were done by electron beam lithography at University of Washington. The grating coupler is designed based on a wafer with 2um buried oxide and 220 nm top Si layer. A shallow etch layer with an etch depth of 70 nm was used and air was employed as the cladding. The insertion losses, 3dB bandwidths, and the central wavelengths of the same design at 10 different positions were measured and the comparison of these results are shown in Fig. 3.13, Fig. 3.14, and Fig. 3.15. Fig. 3.13 shows the insertion losses of the same grating coupler design at different positions of the chip. The measured insertion losses are between -7.58dB and -8.24 dB. In order to protect the chip from scratching, we intentional left a gap between the chip the fibre tip. Therefore, the insertion loss of the measurement results is  38  3.1. Detuned Shallow Etched Grating Coupler much larger than the simulation results. However, the stability in insertion loss of the grating coupler has been observed. Fig. 3.14 shows the measurement results of 3dB bandwidths of the same grating coupler at 10 different positions. The measured 3dB bandwidths of the grating coupler is between 47.5 nm and 42 nm. The chip may not perfectly aligned horizontally, which results in different gap distances between the fibre ribbon tip and the chip at different positions. The variations in gap distances is the main source for differences in bandwidth. Fig. 3.15 shows the central wavelengths of the same grating coupler design at different positions of the chip. Most of the measured grating couplers have a central wavelength of about 1550 nm, which is the designed central wavelength. The small discrepancy in central wavelength indicates the high accuracy of the fabrication process. −7.5 −7.6  Insertion loss (dB)  −7.7 −7.8 −7.9 −8 −8.1 −8.2 −8.3 0  2  4  6  8  10  Device ID  Figure 3.13: The insertion losses of the same grating coupler design at different positions of the chip  39  3.1. Detuned Shallow Etched Grating Coupler 48  3dB bandwidth (nm)  47 46 45 44 43 42 41 0  2  4  6  8  10  Device ID  Figure 3.14: The 3dB bandwidths of the same grating coupler design at different positions of the chip  1551.5 1551  central wavelength (nm)  1550.5 1550 1549.5 1549 1548.5 1548 1547.5 1547 0  2  4  6  8  10  Device ID  Figure 3.15: The central wavelengths of the same grating coupler design at different positions of the chip  40  3.2. Universal Grating Coupler Design  3.2  Universal Grating Coupler Design  Depending on the fabrication process and applications, various grating couplers are required [16, 32, 35, 44, 56]. The traditional way of designing grating couplers involves brute-force simulations on all of the design parameters, which is very time-consuming and a lot of computational memory is required. Here we present a methodology for designing grating couplers based on theoretical calculations instead of numeric simulations. This methodology is enabled by the combination of Bragg condition and Effective Index Method (EIM). It is has been validated for the wavelengths from 1260 nm to 1675 nm for both TE and TM mode waves. Based on this method, we also generated a script to draw the mask layout for grating couplers having only the central wavelength and the incident angle as inputs. The script accommodates various etch depths, silicon thickness (e.g.,220nm, 300nm), and cladding material (e.g., silicon oxide or air). This methodology has been verified by both FDTD simulations and measurement results.  3.2.1  Design Approach  The design flow of the proposed design method is shown in the flow chart on the next page. For any specific grating coupler there are two types of input parameters: process determined parameters and design intent parameters. The process determined parameters include the etch depth, the cladding material, and the thickness of each layer, which are determined by the fabrication process and wafer type. The design intent parameters include central wavelength, λ, incident angle, θ, and the polarization of the operational wave.  41  3.2. Universal Grating Coupler Design  Figure 3.16: Flow chart of the universal design method  42  3.2. Universal Grating Coupler Design Knowning the cladding material, the thickness of the top silicon layer, and the etch depth, we can obtain the effective index of the grating teeth and the effective index of the grating slots using the EIM, under the assumption that the grating has a infinite width. This assumption holds under the condition that grating couplers for an optical fibre normally have widths of more than 12 microns, which is much larger than the central wavelength. If we denote the effective index of the grating tooth to be neff1 , and the effective index of refraction of the grating slots to be neff2 , then the effective index of the grating can be expressed as: neff = f f · neff1 + (1 − f f ) · neff2  (3.7)  where f f denotes the fill factor of the grating, which is defined as the ratio of the duty cycle and the grating period, i.e., f f = w/Λ. After obtaining the effective index neff of the grating, the Bragg condition is employed to calculate the grating period: nc · sinθ = neff − m ·  λ Λ  (3.8)  where nc denotes the effective index of the fibre mode in the cladding, θ is the incident angle, λ is the desired central wavelength, and m denotes the diffraction order. For a specific grating coupler nc , λ, and θ are known. In addition, the effective index of the grating neff can be calculated by the EIM. Therefore, we can get the period of the grating from Equation 3.18. This method is used to design a one-dimensional coupler, which can be simulated by 2D simulations in FDTD.  3.2.2  Simulation Results  Simulation models have been built using FDTD Solutions (Scripts shown in Appendix A). Given the process determined parameters and design intent parameters, our model can generate the desired grating couplers and simulate them. The input parameters used in our case are listed in Table 3.3. Extensive simulations have been done for the wavelengths ranging from  43  3.2. Universal Grating Coupler Design Top Si thickness 220 nm  Etch depth 70nm  Cladding air/oxide  λ 1550 nm  θ 10 degree  Polarization TE/TM  Table 3.3: Input paramters  1260nm to 1675nm, which covers all of the six optical bands. Air cladding and SiO2 cladding have been examined for both TE and TM waves to verify the accuracy of the universal design method. The input parameters used in our case are shown in Table 3.3. Figure 3.17 shows the simulation results of the grating couplers for TE light with a 10 degree incident angle and SiO2 as the cladding. The x-axis denotes the designed central wavelength, λ. The left y-axis, indicated by the blue curve, denotes the insertion loss of the grating couplers generated by the model, and the right y-axis, indicating by the green curve, denotes the wavelength mismatches. The wavelength mismatch, δλ, is defined as the difference between the actual central wavelength, λreal , obtained from FDTD simulations and the design intent wavelength, λ, i.e., δλ = λreal − λ. The insertion loss of the grating couplers varies as the wavelength changes, which results from the phase condition changes between Preflection3 and Preflection4 as shown in Fig. 3.6. Constructive interferences is obtained between Preflection3 and Preflection3 around 1310nm and 1550nm, therefore, the insertion losses of the grating couplers around these wavelengths are smaller. However, destructive interference between Preflection3 and Preflection4 is obtained around 1400nm, therefore, the insertion loss of the grating couplers around these wavelengths are larger. For most wavelengths, the simulated central wavelengths are close to the designed central wavelengths, especially around the two most commonly used optical windows around 1310 nm and 1550 nm. The wavelength mismatch can be calibrated by adjusting the incident angle during the measurement or by compensating for the wavelength during the design stage. Given that the tuning coefficient is 7 nm/degree for TE light and 10 nm/degree for TM light, the wavelength mismatches are within 2 degrees in most cases. Figure. 3.18 shows the simulation results of the grating couplers for TM light with a 10 degree incident angle and SiO2 as the cladding. The wave44  3.2. Universal Grating Coupler Design length mismatches of grating couplers for TM light are larger than that of the grating couplers for TE light at some wavelengths. This is the case because the TM modes are less confined within the waveguides, therefore, the central wavelengths of the grating couplers for TM light are more sensitive to refractive index change than the central wavelengths of the grating couplers for TE light. For comparison purposes, we simulated the grating couplers generated by the universal design model with a 10 degree incident angle and employed air as the cladding. Figure 3.19 shows the simulation results of the grating couples for TE operation with a 10 degree incident angle and air as the cladding. Figure 3.20 shows the simulation results of the grating couplers for TM operation with a 10 degree incident angle and air as the cladding. The insertion loss of the grating couplers with SiO2 as the cladding are smaller than the ones with air cladding. This is the case because the refractive index contrast between the air and the top Si layer is reduced by employing the cladding layer, therefore, more light can be coupled into the grating by choosing the right thickness for the cladding layer. Also, the cladding layer mitigates the refractive index change of the grating as the period and the duty cycle of the grating varies, which results in the central wavelength being less sensitive to the changes in the duty cycle and grating period.  45  −2  5  −3  0  −4  −5  −5  −10  −6  −15  −7 1200  1300  1400 1500 wavelength (nm)  1600  wavelength mismatch (nm)  power (dB)  3.2. Universal Grating Coupler Design  −20 1700  −2  50  −4  0  wavelength mismatch (nm)  power (dB)  Figure 3.17: Universal grating couplers with 10 degree incident angle for TE mode wave with oxide cladding  −6 −50 1250 1300 1350 1400 1450 1500 1550 1600 1650 1700 wavelength (nm)  Figure 3.18: Universal grating couplers with 10 degree incident angle for TM mode wave with oxide cladding  46  20  −4  0  −6  −20  power (dB)  −2  −8 1200  1300  1400 1500 wavelength (nm)  1600  wavelength mismatch (nm)  3.2. Universal Grating Coupler Design  −40 1700  −3  10  −4  0  −5  −10  −6  −20  −7  −30  −8  −40  −9 1200  1300  1400 1500 wavelength (nm)  1600  wavelength mismatch (nm)  power (dB)  Figure 3.19: Universal grating coupler with 10 degree incident angle for TE mode wave with air cladding  −50 1700  Figure 3.20: Universal grating coupler with 10 degree incident angle for TM mode wave with air cladding  47  3.2. Universal Grating Coupler Design −2 −3  power (dB)  −4 −5 −6 −7 −8 −9 1500  Optimized UGC 1520  1540 1560 wavelength (nm)  1580  1600  Figure 3.21: Comparison of designs generated by universal grating coupler model and the optimized design for 1550nm TE wave with 10 degree incident angle  In order to see the accuracy of the universal grating coupler design method, we picked the most commonly used wavelength in telecommunications, i.e., 1550 nm, to compare the spectrum of the grating coupler generated by the universal design model and the spectrum of the grating coupler design optimized by FDTD simulation. Figure. 3.21 shows the comparison of the simulation results of the grating coupler generated by the universal design method and the grating coupler optimized by FDTD simulations. The green dash curve denotes the spectrum of the grating coupler generated by the universal design method, which has an insertion loss of -2.79 and a 3dB bandwidth of 78.8 nm, and the blue curve denotes the spectrum of the grating coupler design optimized by FDTD simulations, which has an insertion loss of -2.74 and a 3dB bandwidth of 79.8 nm. The central wavelength mismatch between the two grating couplers is as small as 3 nanometers and the insertion loss only improved by 1% after hundreds of simulation sweeps for different design variables. In this thesis, the optimization of the grat48  3.2. Universal Grating Coupler Design ing coupler we use only considered uniform gratings. The improvements in the insertion loss can be obtained by more sophisticated designs with non-uniform gratings [50].  3.2.3  Mask Layout  With the physical parameters calculated from the Bragg condition, we can generate the mask layout of the grating coupler. So far, we have only addressed the grating couplers with straight gratings. For those grating couplers, two dimensional tapers are required to convert the mode from the grating into the waveguide. Typically, the width of the grating is about 12 microns, and the width of the waveguide is only about 500 nanometers. To achieve lossless conversion between the grating and the waveguide, two dimensional tapers with a length on the order of a few hundred microns are required [56], which is not space efficient. To tackle this issue, curved grating designs have been employed to make the grating couple more compact [54]. According to [54], compact grating structure can be obtained by curving the grating lines following the equation: qλ0 = neff  y 2 + z 2 − znt cos(θc )  (3.9)  where q is the integer indicating each grating lines, θc denotes the angle between the fibre and the chip surface, nt denotes the refractive index of the environment, λ0 denotes the central wavelength, and neff denotes the effective index of the grating. The expression of Bragg condition for the curved gratings can be also converted to a cylindrical system [32]: ne kr = nc krsin(θ)cos(φ) + 2πN  (3.10)  where ne denotes the effective of the mode propagating inside the grating, k denotes the wave vector in free space, nc denotes the effective index of the wave in the cladding, θ denotes the incident angle, and φ denotes the angle subtended by an arbitrary points on the grating and to the z-axis. We generated a universal script in Pyxis (shown in Appendix B), a commercial  49  3.2. Universal Grating Coupler Design software from Mentor Graphics, to draw the mask layout of the desired grating couplers with curved gratings. Our script is based on the cylindrical expression of the curved grating as shown by Equation. 3.10. An example of the mask layout of the grating coupler generated by our model is shown in Fig. 3.22.  Figure 3.22: Mask layout of a grating coupler with focusing grating curve  3.2.4  Measurement Results  Test structures of the grating couplers generated by the universal design method have been fabricated through the electron beam lithography at University of Washington using the JEOL JBX-6300FS E-Beam Lithography system. Standard SOI wafer with a 220-nanometer top silicon layer and a 2-micron buried oxide were used for the fabrication. Silicon oxide was employed as the cladding layer to protect the functional structures. The etch depth of the shallow etched grating couplers is 70 nanometers. The grating couplers were designed for TE mode wave with incident angles of 10 degree, 15 degree and 20 degree. The target central wavelengths are 1500nm, 1520nm, 1540nm, 1560nm, 1580nm, and 1600nm. Figure 3.23 shows the simulated and measured spectra of the grating couplers generated by the universal design method with a 10 degree incident 50  3.2. Universal Grating Coupler Design angle. The simulation and measurement results of the grating couplers with the same central wavelength are indicated by the same colour. Due to the misalignment, inevitable system loss and fabrication errors, the measured insertion loss of the grating couplers are larger than the simulated results. In addition, the bandwidths of the measured grating coupler are narrower than the simulated ones, which is hypothesized to be originating from the large air gap. Similarly, Figure 3.24 shows the simulated and measured spectra of the grating couplers generated by the universal design method with a 15 degree incident angle, and Fig. 3.25 shows the simulated and measured spectra of the grating couplers generated by the universal design method with a 20 degree incident angle. The spectra of the measurement results follows the same trend as the simulation results but with the addition of extra insertion loss. The mismatch in central wavelength between the simulation results and measurement results may caused by the fabrication errors,e.g., the possible fabrication errors in grating period, duty cycle, and the thickness variation of the Si layer across the samples. The sensitivity of central wavelength of the design as function of design parameters such as grating period, duty cycle etc have been shown in Table 3.4, which has been explained in the previous section. Parameters Tuning coefficient  period 1.4  duty cycle 0.215  etch depth 1.9  incident angle 7 nm/degree  Table 3.4: Tuning coefficient of different parameters  51  3.2. Universal Grating Coupler Design −2 −4  Power (dB)  −6 −8 −10 −12 −14 −16 −18 1450  1500  1550 Wavelength (nm)  1600  1650  Figure 3.23: Measurement vs. simulation of universal grating couplers with 10 degree incident angle  −2 −4  Power (dB)  −6 −8 −10 −12 −14 −16 1450  1500  1550 Wavelength (nm)  1600  1650  Figure 3.24: Measurement vs. simulation of universal grating couplers with 15 degree incident angle  52  3.2. Universal Grating Coupler Design −2 −4 −6 Power (dB)  −8 −10 −12 −14 −16 −18 −20 1450  1500  1550 Wavelength (nm)  1600  1650  Figure 3.25: Measurement vs. simulation of universal grating couplers with 20 degree incident angle  Further comparison of simulation and measurement results of the grating couplers generated by the universal design method are shown in Fig. 3.26 to Fig. 3.28. Figure. 3.26 (a) shows the comparison of the simulated and measured insertion loss of the grating couplers with 10 degree incident angle. Figure. 3.26 (b) shows the comparison of the simulated and measured central wavelength mismatch. Similarly, the comparison between simulation and measurement results of the grating couplers with 15 degree incident angle and 20 degree incident angle are shown in Fig. 3.27 to Fig. 3.28. Obviously, the insertion loss of the simulation results and measurement results follows the same trends. However, the central wavelength mismatch of the measurement results shows a random behaviour, which is not consistent with the simulation results. The discrepancy in central wavelength is mainly caused by the fabrication error because the possible fabrication errors in grating period, duty cycle, etch depth all have impacts on the central wavelength of the grating coupler.  53  3.2. Universal Grating Coupler Design 6  −3  4  −3.5  2  wavelength mismatch (nm)  power (dB)  −2.5  −4 −4.5 −5 −5.5 −6 −6.5  1520  1540 1560 wavelength (nm)  0 −2 −4 −6 −8 −10  measurement simulation 1500  measurement simulation  1580  −12 1500  1600  1520  1540 1560 wavelength (nm)  1580  1600  (a) (b) Figure 3.26: (a) Peak power of simulation and measurement results with 10 incident angle; (b)comparison of simulated and measured wavelength mismatch with 10 degree incident angle −2.5  10  −3  8 wavelength mismatch (nm)  −3.5  power (dB)  −4 −4.5 −5 −5.5 −6 −6.5 −7  1520  1540 1560 wavelength (nm)  6 4 2 0 −2 −4  measurement simulation 1500  measurement simulation  1580  1600  −6 1500  1520  1540 1560 wavelength (nm)  1580  1600  (a) (b) Figure 3.27: (a)Peak power of simulation and measurement results with 15 incident angle; (b)comparison of simulated and measured wavelength mismatch with 15 degree incident angle  54  3.2. Universal Grating Coupler Design −2.5  6  −3 wavelength mismatch (nm)  −3.5  power (dB)  −4 −4.5 −5 −5.5 −6 −6.5 −7  measurement simulation  4  1520  1540 1560 wavelength (nm)  0 −2 −4 −6 −8 −10  measurement simulation 1500  2  1580  1600  −12 1500  1520  1540 1560 wavelength (nm)  1580  1600  (a) (b) Figure 3.28: (a)Peak power of simulation and measurement results with 20 incident angle; (b)comparison of simulated and measured wavelength mismatch with 20 degree incident angle  55  Chapter 4  Fully Etched Grating Coupler All of the grating couplers mentioned so far are shallow etched grating couplers, and the fabrication of these grating couplers requires two lithography steps. Shallow etched grating couplers have the advantage of smaller back reflection to the waveguide and lower insertion loss than the fully etched grating couplers. However, the additional lithography step also increases the fabrication cost and complexity. Fully etched grating couplers offer a fast and economic experimental solution for the coupling issue, especially for quick prototyping through electron beam lithography. All of the fundamental building blocks of the photonics integrated circuits can be fabricated in just one step by implementing the fully etched grating coupler. In this chapter, the performance of the traditional fully etched grating couplers will be presented first, and the drawbacks of which will be pointed out. Then, fully etched grating couplers with refractive index engineered subwavelength structures will be introduced to improve the performance of the fully etched grating couplers.  4.1  Regular Fully Etched Grating Coupler  Figure 4.1 is a schematic of a regular fully etched grating coupler in SOI with uniform gratings. The grating teeth are formed by fully etching through the functional silicon layer instead of partially etching. The thicknesses of the functional silicon layer and buried oxide used in our case are 2 um and 220 nm, respectively. Air is employed as the cladding for the grating coupler design. 56  4.1. Regular Fully Etched Grating Coupler  Figure 4.1: Schematic of fully etched grating coupler  4.1.1  Bottlenecks of Regular Fully Etched Grating Couplers  The disadvantages of the regular fully etched grating couplers include the poor insertion loss and the large back reflection to the waveguide. The poor insertion loss is mainly caused by the mode mismatch and penetration loss, and the large back reflection to the waveguide results from the large refractive index contrast between the waveguide and the grating. When considering the grating coupler as an output coupler (shown in Fig. 4.1), the power in the waveguide will be exponentially decaying due to the presence of the grating [49]: Pwg (z) = Pwg (z = 0)exp(−2αz)  (4.1)  where 2α is the coupling strength or leakage factor of the grating, Pwg (z) denotes the power of the mode at z. The inverse of the coupling strength is defined as the coupling length, i.e., Lc = (2α)−1 . For shallow etched grating couplers, the coupling strength α is small, and the coupling length Lc of the grating couplers is similar to the diameter of the mode from an optical fibre. Therefore, a large mode overlap between the exponential mode from the grating and the Gaussian mode of a fibre can be achieved. However, 57  4.1. Regular Fully Etched Grating Coupler the coupling strength α of fully etched grating couplers is much larger than that of the shallow etched ones, therefore the coupling length Lc of the fully etched grating couplers is much smaller than that of the shallow etched ones, which results in a larger mode mismatch between the grating and the optical fibre.  Figure 4.2: Schematic of diffraction, reflection and penetration of a regular fully etched grating coupler  The large penetration loss is another reason leading to the poor insertion loss of the fully etched grating couplers. Figure 4.2 shows the diffraction, reflection, and penetration behaviours of a fully etched grating coupler. As the incident wave impinges on the gratings, a portion of the light is diffracted upwards to the air and a potion of the light is scattered downward to the substrate. At the interface of the buried oxide and the Si substrate, part of the light scattered downward from the grating is reflected back to the grating and the other part of the light penetrates into the substrate. At the interface of the functional Si layer and the buried oxide, the reflected light from the lower interface interferes with the portion of light diffracted upwards. Depending on the phase condition, either constructive interference or destructive interference can be obtained. Lowest insertion loss can be achieved when constructive interference is obtained. 58  4.1. Regular Fully Etched Grating Coupler −3  Insertionl Loss (dB)  −4 −5 −6 −7 −8 −9  Fully etched GC Shallow etched GC  −10 1  1.5 2 2.5 Thickness of the buried oxide (um)  3  Figure 4.3: Insertion loss of grating couplers with 20 degree incident angle for TE operation wave as function of the thickness of the buried oxide.  The phase condition between the diffracted wave, i.e., Pdiffraction , and the reflected wave, i.e., Preflected , depends on the thickness of the buried oxide. Figure 4.3 shows the insertion loss of grating couplers with 20 degree incident angles for TE operation wave as function of the thickness of the buried oxide. As we vary the thickness of the buried oxide from 1 um to 3 um, the insertion loss of the grating couplers varies as the phase conditions changes. The standard SOI wafers normally have a 2-um thick buried oxide, which is chosen for the shallow etched grating couplers, but not ideal for fully etched grating couplers. The large back reflection to the waveguide of the fully etched grating couplers results from the Fresnels reflection. The reflection coefficient for TE wave can be expressed as: rT E =  n1 · cos(θi ) − n2 · cos(θt ) n1 · cos(θi ) + n2 · cos(θt )  (4.2)  59  4.1. Regular Fully Etched Grating Coupler where n1 , n2 are the refractive indices of corresponding layers. And θi , θt are the incident angle and transmission angle, respectively. In our case, n1 equals to the refractive index of silicon, i.e., n1 = 3.47, and n2 equals to the refractive index of cladding material. The Fresnels reflection coefficient is as high as 0.414 in the case with silica as cladding material, which means more than 17% of the power will be reflected back into the waveguide. The back reflection to waveguide is even larger in the case with air as cladding since air has a lower refractive index than silica.  4.1.2  Optimization of the Regular Fully Etched Grating Couplers  Insertion loss, back reflection to the waveguide, and bandwidth are the most important figures of merit for a grating coupler. The optimization of a regular fully etched grating coupler involves optimizations for different design parameters such as incident angle, grating period, duty cycle, etc. Two efficiencies determine the overall efficiency of the fully etched grating couplers. The first efficiency, η1 , is defined as the ratio of the light scattered upwards and the input power from the waveguide: η1 =  Pup Pwg  (4.3)  where Pup denotes the portion of light scattered upwards and Pwg denotes the input power from the waveguide. η1 is also called the directionality of the grating coupler. Theoretically, η1 can be maximized by employing the optimal thickness of the buried oxide [11, 47, 54]. However, the thickness of the buried oxide is determined by the wafer type, which is not changeable. The second efficiency, η2 , is defined as the ratio of the power coupled into the desired mode, i.e., the fundamental mode of the fibre, and the power scattered upward: η2 =  Pfund Pup  (4.4)  where Pfund denotes the power coupled in the fundamental mode of the fibre, and Pup denotes the power scattered upwards by the grating. This 60  4.1. Regular Fully Etched Grating Coupler efficiency indicates the overlap between the exponential scattering mode from the grating and the Gaussian mode in an optical fibre. The exponential scattering pattern can be engineered by varying the coupling strength α of the grating coupler, and the maximized overlap was reported to be 80% [36], which is consistent with our simulation results shown in Fig. 4.4. The overall coupling efficiency (insertion loss) of the grating coupler can be expressed as the product of the two partial efficiency: η = η1 · η2  (4.5)  −3 Directionality Insertion loss Reflction  −4 −5  power (dB)  −6 −7 −8 −9 −10 −11 −12 −13 1500  1520  1540 1560 wavelength (nm)  1580  1600  Figure 4.4: Directionality, insertion loss and reflection to waveguide of a general fully-etched grating coupler  An optimal design can be found by optimizing the two partial efficiencies η1 and η2 . The thickness of the buried oxide is determined by the wafer type, which is 2um in our case, η1 is fixed with the given central wavelength and incident angle. The simulation results of a fully etched grating coupler with air cladding operated for TE mode at 1550nm have been shown in Fig. 4.4. The blue curve denotes the directionality of the grating, the green curve 61  4.1. Regular Fully Etched Grating Coupler  Figure 4.5: Mask layout of a fully etched grating coupler test structure  denotes the insertion loss of the grating, and the red curve denotes the back reflection to the waveguide of the grating. Due to the large refractive index contrast between the waveguide and the grating, the back reflection of the fully etched grating coupler is huge, which is even larger than the insertion loss of the grating coupler. A mask layout of a fully etched grating coupler test structure is shown in Fig. 4.5. An input coupler and an output coupler are connected by a u-shape silicon wire waveguide. The centre-to-centre distance between the two grating couplers is 127 microns, which is determined by the pitch of the fibre ribbon used in our lab. The comparison of simulation and measurement results for such a test structure is shown in Fig. 4.6. The green curve denotes the simulated insertion loss of the grating pair and the blue curve denotes the measured insertion loss of the same structure. Since the simulation result is obtained from the simulation of a single grating coupler, and the back reflection to the waveguide is not taken into consideration, no oscillations are shown in the simulation spectrum. Strong oscillations have been observed in the measurement spectrum of the grating structure, which is resulting from the large back reflection of the grating coupler. Such big ripples strongly affect the performance of resonator structures such as ring resonators, disks, and Bragg gratings, etc.  62  4.1. Regular Fully Etched Grating Coupler −10 Measurement Simulation −15  −15  −25 power (dB)  power (dB)  −20  −30  −35  −40 1500  −16 −17 −18 −19 1545  1510  1520  1530  1550 wavelength (um) 1540 1550 1560 wavelength (um)  1555  1570  1580  1590  1600  Figure 4.6: Comparison of measurement result and simulation result of fullyetched grating coupler  The extinction ratio of the ripples shown in the measurement spectrum can be calculated by the transmission function of the Fabry-Perot cavity: T =  (1 − R)2 (1 − R)2 + 4 · R · sin2 (δ/2)  (4.6)  where T denotes the transmitted power of a Fabry-Perot cavity, R is the reflectivity of the mirror, i.e., grating couplers in our case, and δ is the round trip phase shift. The reflectivity of the grating coupler is about 20% as shown in Fig. 4.4, so the extinction ratio of the ripples is about 2.8dB which is consistent with our measurement results shown in the inset of Fig. 4.6.  63  4.2. Improved Fully Etched Grating Coupler  4.2  Improved Fully Etched Grating Coupler  As we have discussed in the previous section, the bottlenecks faced by the fully etched grating couplers are the large back reflection to the waveguide and the high insertion loss. The strong back reflection of the fully etched grating couplers results from the large refractive index contrast between the waveguide and the grating. The simulation results of the back reflections of both shallow etched grating couplers and fully etched grating couplers are shown in Fig. 4.7. The green curve denotes the normalized back reflection of a shallow etched grating coupler designed based on a SOI with a 2-um buried oxide, 220 nm top Si layer, and air as the cladding. The blue curve denotes the back reflection of a fully etched grating coupler designed with the same wafer type and cladding. The grating couplers are designed for TE mode with a central wavelength of 1550 nm. The back reflection of the shallow etched grating coupler at the designed central wavelength is much smaller than that of the fully etched grating coupler and a difference of more than 20 dB has been observed. 0 −5  power (dB)  −10 −15  full−etch shallow−etch  −20 −25 −30 −35 1400  1450  1500 1550 1600 wavelength (nm)  1650  1700  Figure 4.7: Comparison of back reflections between shallow-etched grating coupler and fully-etched grating coupler  64  4.2. Improved Fully Etched Grating Coupler Different approaches have been employed to tackle the issues of large back reflection and high insertion loss of fully etched grating couplers. Photonic crystal structures have been implemented in fully etched grating couplers to achieve low insertion loss [27]. However, customized wafers have been used to meet the requirement of the optimized design, which cannot be applied to the general cases with standard SOI wafers. Sub-wavelength structures [13, 14] have been used to reduce the back reflection and improve the insertion loss of the fully etched grating coupler at the same time. However, the reported designs are only applied for TM mode wave. In addition, all of the reported grating couplers are one-dimensional periodic structures and adiabatic tapers on the order of a few hundred micrometers are required to convert the optical mode from the grating to the waveguide, which waste a lot of precious space on the chip. In this section, we will demonstrate a fully etched grating coupler for TE mode wave operated at 1550 nm with reduced back reflection and improved coupling efficiency. The proposed design is based on a SOI wafer with a 2um buried oxide and 220 nm top Si layer. The proposed design also has the potential to be converted to a compact structure with curved gratings, which has a much smaller footprint.  4.2.1  Design Approach  In order to reduce the back reflection of the fully etched grating couplers, we proposed a fully etched grating with effective index areas consisting of air gaps and minor sub-wavelength gratings (show in Fig. 4.8). The major (wider) gratings determine the period and duty cycle of the grating coupler and the minor sub-wavelength gratings are employed to mitigate the refractive index contrast between the major gratings and the original air gaps. Those minor gratings are easier to be fabricated than the holes or rectangles [13, 14, 27]. Another advantage of the proposed design is the potential to be converted to compact structures with curved gratings. Based on the ideal mentioned in [6], one-dimensional gratings can be converted to more compact strcutures with curved gratings, which enables a much  65  4.2. Improved Fully Etched Grating Coupler smaller footprint.  Figure 4.8: Schematic of a fully etched grating coupler with sub-wavelength gratings  According to the effective medium theory (EMT), zeroth-order approximation can be applied to sub-wavelength structures with period-to-wavelength ratio, defined as R = neff · Λ/λ, much smaller than 1: 1 (0) nT E  =[  (1 − f f ) 1/2 ff + ] 2 nL n2si  (4.7)  where n(0)T E denotes the refractive index of the approximated grating region, nL denotes the refractive index of the effective index area with minor subwavelength gratings, nsi denotes the refractive index of the major grating, and f f is defined as the fill factor of the grating. Second-order EMT can be used as a more accurate approximation to explore structures with lateral feature size of the same order as the wavelength in the medium [38]: (0)  (2)  (0)  nT E = nT E [1 +  (0)  n n π2 2 2 R f f (1 − f f )2 (n2L − n2si )2 · ( T M )2 ( T E )4 ]1/2 (4.8) 3 neff nL nsi 66  4.2. Improved Fully Etched Grating Coupler where neff denotes the effective index of the TE mode in the grating region. 0  −5  power (dB)  −10  −15  −20 Directionality InsertionLoss−New ReflectionToWG InsertionLoss−old ReflectionToWG−old  −25  −30 1400  1450  1500 1550 1600 wavelength (nm)  1650  1700  Figure 4.9: Comparison of regular fully etched grating coupler and the fully etched grating coupler with minor sub-wavelength gratings.  4.2.2  Simulation Results  FDTD method has been used to simulate and optimize the grating structure. The simulations were based on the SOI wafer we used for fabrication, with a 2 um buried oxide and 220 nm top Si layer. Silicon oxide was employed as the cladding for protection. Particle Swarm Optimization (PSO), which is an optimization algorithm for electromagnetic optimization problems [39, 61], has been used to optimize the grating design. Design parameters, i.e., grating period, duty cycle, and the width of the minor gratings were optimized to achieve small back reflection as well as low insertion loss. The optimized design has a period of 625 nm, a duty cycle of 190nm, and sub-wavelength gratings with a width of 50 nm. Figure 4.9 shows the simulation results of a regular fully etched grating coupler and the fully etched grating coupler with minor sub-wavelength 67  4.2. Improved Fully Etched Grating Coupler gratings. The dash lines denote the insertion loss and the back reflection of the regular fully etched grating coupler, and the solid lines denote the directionality, insertion loss, and the back reflection of the proposed grating coupler. More than 64% (-1.9 dB) of the input power from the waveguide was diffracted upwards by the grating. However, due to the mode mismatch between the grating coupler and the fibre, only 45% (-3.47 dB) of light from the waveguide was coupled in the fundamental mode of fibre. The back reflection of the grating coupler is reduced from -6 dB to about -18 dB, which is similar to that of a shallow etched grating coupler.  4.2.3  Measurement Results −10  power (dB)  −15 −20 −25 −30 SubGC General FGC  −35 −40 1520  1540  1560 1580 wavelength (nm)  1600  Figure 4.10: Measurement results of the regular fully etched grating coupler and the fully-etched grating coupler with minor sub-wavelength gratings  Test structures, consisting of an input coupler and an output coupler, of the fully etched grating couplers with minor sub-wavelength gratings have been fabricated through electron beam lithography at University of Washington using the JEOL JBX-6300FS E-Beam Lithography system. Test  68  4.2. Improved Fully Etched Grating Coupler structures of regular fully etched grating couplers have also been fabricated for the comparison purpose. Measurement spectra of the regular fully etched grating coupler and the fully etched grating coupler with minor subwavelength gratings are shown in Fig. 4.10. Green curve donates the results of the regular fully etched grating coupler and the blue curve denotes the proposed fully etched grating coupler with minor sub-wavelength gratings. The extinction ratio of the ripples have been reduced from about 3dB to about 0.8 dB, by implementing the minor sub-wavelength gratings. The insertion loss of the grating coupler is also improved by about 2 dB. In addition, the bandwidth of the proposed design is largely improved, which is nearly twice as that of the regular fully etched grating coupler.  69  Chapter 5  Vertical Grating Coupler All of the grating couplers mentioned so far are detuned grating couplers, which indicates the case that a small angle is employed between the incident wave to the normal of the grating surface. The small angle is introduced to avoid the second order Bragg reflection [15], but such a small angle is not desirable for a low-cost packaging process [5]. In this chapter, grating couplers for perfectly vertical coupling will be presented, and a bidirectional grating coupler for vertical coupling will be demonstrated to improve the insertion loss of the vertical grating couplers.  5.1  Regular Vertical Grating Couplers  The main drawback of the vertical grating coupler is the large second order reflection to the waveguide, which can be predicted by the Bragg condition: β · sinθ = β − m · K  (5.1)  where β denotes the wave vector of the propagated mode within the grating, θ denotes the diffraction angle, m is an integer denoting the diffraction order, and K = λ/Λ. In the case where the first order diffraction is perfectly coupled to the fibre, i.e., θ = 0 when m = 1, β = K, i.e., Λ=λ/neff , where neff denotes the effective index of the grating and λ denotes the central wavelength. In this case, the second order diffraction of the grating is in the direction of the input waveguide, i.e., θ = −90 when m = 2. The wave vector diagram of a vertical grating coupler is shown in Fig. 5.1.  70  5.1. Regular Vertical Grating Couplers  Figure 5.1: Diagram of wave vectors for vertical grating coupler  A vertical grating coupler is designed based on a SOI wafer with a 2 um buried oxide and a 220 nm top Si layer. A shallow etch layer with an etch depth of 70 nm is used and SiO2 is employed as the cladding for protection. The designed central wavelength is 1550 nm and FDTD simulations have been used to find the optimized design with low insertion loss and large bandwidth around the desired wavelength. The effective index of the 220 nm Si layer is 2.83 (neff1 ), and the effective index of the shallow etched Si layer is 2.506 (neff2 ). Therefore, the effective index of the grating can be calculated by neff = neff1 /2 + neff2 /2, which is 2.668. Grating period of the desired grating can be calculated from the Bragg condition with the known effective index of the grating, which is 580 nanometers in our case. Further optimization of design parameters such as grating period and duty cycle have been done using FDTD methods. Figure 5.2 shows simulation results of the insertion loss and back reflection to waveguide of an optimized regular vertical grating coupler. The blue curve denotes the insertion loss and the green curve denotes the back reflection to the waveguide. A valley is shown in the spectrum of the insertion loss, which is caused by the large back reflection.  71  5.1. Regular Vertical Grating Couplers 0 −5  power (dB)  −10 −15 −20 −25 −30 −35 1400  Insertion loss Reflection to WG 1450  1500 1550 1600 wavelength (nm)  1650  1700  Figure 5.2: Insertion loss and back reflection of a regular vertical grating coupler  Different approaches have been employed to avoid the large second order reflection of the vertical grating couplers [4, 5, 40, 57]. Asymmetric grating structure [40] has been used to avoid the large Bragg reflection for vertical coupling. However, the extra etch depth required for fabrication increases the complexity as well as the cost. Chirped gratings were also implemented to minimize the mode mismatch between the grating and optical fibre [4, 5]. However, high fabrication accuracy is required to fabricate those chirped grating couplers. Given the fabrication accuracy of the existing technology, the stability and repeatability of the chirped gratings are still doubted. Another drawback of the chided grating coupler is their small bandwidth. The reported 3dB bandwidth of the chirped vertical grating coupler is only about 45 nanometers. Slanted grating coupler was proposed to obtain high coupling efficiency to a vertically positioned optical fibre [57]. However, the technology required for fabrication of the slanted grating slit is too compacted to be compatible with standard CMOS technology, and therefore only suitable for prototyping. 72  5.2. Bidirectional Grating Coupler  5.2 5.2.1  Bidirectional Grating Coupler Device Layout  The regular vertical grating couplers only couple light from one side of the grating. Theoretically, the insertion loss of such vertical grating couplers with uniform gratings can exceed 50 %. Here we introduced a bidirectional grating coupler structure for vertical grating, which breaks the theoretical limitation of regular vertical grating couplers. A schematic of the bidirectional grating coupler is shown in Fig. 5.2. Basically, it is a symmetric structure with the grating in the centre. Two-dimensional tapers are connected to both sides of the grating to convert the mode into the waveguides. Bending waveguides are employed to direct the light from both sides of the grating into a 50-50 coupler. Finally, light from both sides of the grating is combined together with a 50-50 coupler.  Figure 5.3: Schematic of bidirectional grating coupler for vertical coupling  Figure 5.4 shows the cross-section of the bidirectional vertical grating coupler designed in a SOI wafter with a 2 um buried oxide and a 300 nm top Si layer. A shallow etch layer (ed = 155nm) has been employed by the design, and SiO2 is used as the cladding for protection. Λ denotes the grating period and f f denotes the duty cycle of the grating coupler. 73  5.2. Bidirectional Grating Coupler  Figure 5.4: Cross section of a bidirectional grating coupler  5.2.2  Design and Simulation  FDTD method has been used to design and simulate the bidirectional grating coupler design. The grating period of the grating coupler can be predicted by the Bragg condition. Given that the effective index of the 300 nm slab waveguide is 3.052 (neff1 ), and the effective index of the 145 nm slab waveguide is 2.571 (neff2 ), the effective index of the grating (neff ) can be obtained, i.e., neff = 0.5 ∗ neff1 + 0.5 ∗ neff2 , which is 2.812 in our case. The desired central wavelength is 1550 nanometers, so the grating period can be obtained from the Bragg condition, which is about 550 nm in our case. 2D FDTD simulations were used to further optimize the design. Three parameters, i.e., the number of grating, grating period, and the duty cycle were optimized using the Particle Swarm Optimization algorithm to achieve a design with low insertion loss and large bandwidth. The number of grating period determines the length of the coupling region. As the length of the grating region increases, the directionality of the grating increases, which means more power can be diffracted to the fibre, but extra scattering loss will be introduced as the light propagates from the grating to the waveguide. On the other hand, shorter grating region can reduce the scattering loss, but the directionality of the grating will decrease. The grating period along with 74  5.2. Bidirectional Grating Coupler 0 −2 −4  power (db)  −6 −8 −10 −12 −14 −16 −18 −20 1400  1450  1500 1550 1600 wavelength (nm)  1650  1700  Figure 5.5: The insertion loss of the optimized bidirectional grating coupler  the duty cycle of the grating determines the central wavelength of the grating coupler. The optimized design has a grating number of 13, a grating period of 580 nanometers, and a duty cycle of 240 nanometers. The insertion loss of the optimized bidirectional vertical grating couple is shown in Fig. 5.5. The insertion loss is calculated with the assumption that the taper and Y junction are lossless. The assumption holds because theoretically a tape with a length of 200 micron can achieve lossless conversion from 12 um grating to a 500 nm wire waveguide[50], and compact Y junction with low loss has been demonstrated [42, 61]. The insertion loss of the optimized grating coupler is only -1.5 dB, which is much smaller than of that of the regular vertical grating coupler. The 3dB bandwidth of the bidirectional grating coupler is more than 100 nanometers, which is more than twice of the value reported in [5]. The optical waves in different sides of the grating behave like two arms of a MZI. The insertion loss shown in Fig. 5.5 can be only obtained when the incident wave from the fibre impinges in the middle of the grating coupler. In the case when the incident wave is off from the centre of the grating, optical 75  5.2. Bidirectional Grating Coupler 0 −5  power (dB)  −10 −15 −20 dx=0 dx=40nm dx=80nm dx=120nm dx=160nm  −25 −30 −35 1400  1450  1500 1550 1600 wavelength (nm)  1650  1700  Figure 5.6: Insertion of the bidirectional vertical grating coupler as function of wavelength for different offset  waves from the two arms interfere with each other at the junction of the 50-50 coupler. Depending on the deviation of the incident wave, either constructive interference or destructive interference can be obtained at the 50-50 coupler. Figure 5.6 shows the simulation results of the bidirectional grating coupler with various deviation values. The deviation of the input wave, i.e., dx shown in Fig. 5.7, is defined as the distance from centre of the input fibre core to the centre of the grating. The effective index of the channel waveguide used in our case is about 2.4, and the central wavelength of the input wave is 1550nm. According to the phase condition between the propagated wave in the two arm, i.e., neff · 2dx = λ/2, the first destructive interference will obtained be when dx = 160nm. Fig. 5.6 shows the simulation results for various deviation values. As the deviation, dx, goes beyond 160 nm, the insertion loss of the vertical grating coupler will be somewhere in between the maximum value and the minimum value. From the simulation results shown in Fig. 5.6, we note that the insertion loss of the bidirectional vertical grating coupler is very sensitive to the position of the input fibre, which results in 76  5.2. Bidirectional Grating Coupler  Figure 5.7: Schematic of a bidirectional grating coupler with incident wave off from the centre of the grating.  difficulties for alignment of the grating during the measurement. Highly accurate motor controllers are required for the alignment of the bidirectional grating couplers during the measurement.  77  Chapter 6  Discussion And Future Work In this thesis, we mainly presented three different works: 1. A universal design methodology has been presented to design shallow etched grating couples, which accommodates various etch depths, silicon thicknesses, and cladding materials. 2. A fully etched grating coupler with minor sub-wavelength gratings has been demonstrated to improve the insertion loss, back reflection, and the bandwidth of the regular fully etched grating coupler. 3. A bidirectional grating coupler for vertical coupling has been presented. The point of the universal design methodology is that it simplifies the design process of a grating coupler. The traditional way of designing a grating coupler involves brute-force simulations on various design parameters, which requires a lot of computational memory and time. Using the universal design method, time-consuming simulations have been replaced by simple analytical calculations. We also implemented the design process in Pyxis to generate the mask layout of the desired grating coupler with the input parameters determined by the fabrication process and the designer. The analytical calculation is implemented into the scripts so no simulation is ever needed to design a grating coupler using our design methodology. The performance of the grating couplers designed by this method is comparable to the published results, but with a much simplified design flow for system level designers. Table 6.1 shows the comparison between our measurement results and published results. Since the performance of the grating coupler is highly dependent on the wafer type and fabrication process, we only 78  Chapter 6. Discussion And Future Work listed the published results which used similar wafer types and fabrication processes with us. Future work can be done to add a correction coefficient to the existing model, thereby further improving the wavelength mismatch.  2006 [48] 2012 [60] Ours  Pl. TE TE TE  IL -5.1 dB -4.4dB -4.8dB  Bandwidth 40nm (1dB) 45nm (1.5dB) 45nm(3dB)  Process 220nm Si, 1um BOX 220nm Si, 2um BOX 220nm Si, 2um BOX  Table 6.1: Comparison of the published results and the result of the grating coupler generated by the universal design methodology  A fully etched grating coupler with minor sub-wavelength grating has also been proposed to improve the performance of regular fully etched grating coupler. Due to the fully etched slots, the coupling strength of the fully etched grating couplers is very large, which makes high-efficiency coupling more difficult to achieve. In addition, the thickness of buried oxide of the most commonly used SOI wafer is 2 um, which is not ideal for efficient coupling. Large back reflection to the waveguide leads to strong oscillation ripples in the transmission. However, the large back reflection of the fully etched grating coupler has been successfully reduced by employing the minor sub-wavelength gratings between the major gratings. A larger bandwidth has also been achieved by the proposed design. Further effort can be applied to add chirped grating period to the design, therefore the mode overlap between the grating and the optical fibre can be further improved to reduce the insertion loss. In addition, a bidirectional grating coupler for vertical coupling was proposed to reduced the fabrication complexity and improve the insertion loss at the same time. The published vertical grating coupler designs employed non-standard fabrication processes such as slanted grating [57] and extra etch depth [40], which are only suitable for prototyping. The proposed bidirectional grating coupler only requires uniform gratings with a single shallow etch layer, which is compatible with the current CMOS technology. The simulated insertion loss of this optimized design is as low as -1.46 dB 79  Chapter 6. Discussion And Future Work with a 3dB bandwidth of more than 100nm. The 3 dB bandwidth of the bidirectional vertical grating coupler is much larger than the value reported by other research groups. Further work can be done to design a better Y branch with lower insertion loss.  80  Bibliography [1] Tom Baehr-Jones, Ran Ding, Ali Ayazi, Thierry Pinguet, Matt Streshinsky, Nick Harris, Jing Li, Li He, Mike Gould, Yi Zhang, et al. A 25 gb/s silicon photonics platform. arXiv preprint arXiv:1203.0767, 2012. [2] Michel Bruel, Bernard Aspar, and Andre-Jacques Auberton-Herve. Smart-cut: a new silicon on insulator material technology based on hydrogen implantation and wafer bonding. Japanese journal of applied physics, 36(part 1):1636–1641, 1997. [3] X. Chen, C. Li, C.K.Y. Fung, S.M.G. Lo, and H.K. Tsang. Apodized waveguide grating couplers for efficient coupling to optical fibers. Photonics Technology Letters, IEEE, 22(15):1156–1158, 2010. [4] X. Chen, C. Li, and HK Tsang. Characterization of silicon-on-insulator waveguide chirped grating for coupling to a vertical optical fiber. In Optical MEMs and Nanophotonics, 2008 IEEE/LEOS Internationall Conference on, pages 56–57. IEEE, 2008. [5] X. Chen, C. Li, and H.K. Tsang.  Fabrication-tolerant waveguide  chirped grating coupler for coupling to a perfectly vertical optical fiber. Photonics Technology Letters, IEEE, 20(23):1914–1916, 2008. [6] Xia Chen and Hon K Tsang. Polarization-independent grating couplers for silicon-on-insulator nanophotonic waveguides. Optics Letters, 36(6):796–798, 2011. [7] K.S. Chiang. Analysis of optical fibers by the effective-index method. Applied optics, 25(3):348–354, 1986. 81  Bibliography [8] K.S. Chiang. Effective-index method for the analysis of optical waveguide couplers and arrays: An asymptotic theory. Lightwave Technology, Journal of, 9(1):62–72, 1991. [9] Lukas Chrostowski and Michael Hochberg. Silicon Photonics Design. Lulu, 2013. [10] Fuad E Doany, Benjamin G Lee, Alexander V Rylyakov, Daniel M Kuchta, Christopher Jahnes, Christian Baks, Frank Libsch, and Clint L Schow. Terabit/sec 48-channel fiber-coupled optical module based on holey cmos transceiver ic. In Electronic Components and Technology Conference (ECTC), 2012 IEEE 62nd, pages 1499–1504. IEEE, 2012. [11] Robert M Emmons and Dennis G Hall.  Buried-oxide silicon-on-  insulator structures. ii. waveguide grating couplers. Quantum Electronics, IEEE Journal of, 28(1):164–175, 1992. [12] Terry TH Eng, Stephen SY Sin, Sidney C Kan, and George KL Wong. Surface-micromachined movable soi optical waveguides. In Solid-State Sensors and Actuators, 1995 and Eurosensors IX.. Transducers’ 95. The 8th International Conference on, volume 1, pages 348–350. IEEE, 1995. [13] R Halir, P Cheben, JH Schmid, R Ma, D Bedard, S Janz, D-X Xu, A Densmore, J Lapointe, and I Molina-Fern´andez. Continuously apodized fiber-to-chip surface grating coupler with refractive index engineered subwavelength structure. Optics letters, 35(19):3243–3245, 2010. [14] Robert Halir, Pavel Cheben, Siegfried Janz, Dan-Xia Xu, ´I˜ nigo MolinaFern´ andez, and Juan G Wang¨ uemert-P´erez. Waveguide grating coupler with subwavelength microstructures. Optics letters, 34(9):1408–1410, 2009. [15] AMOS Hardy, David F Welch, and William Streifer. Analysis of secondorder gratings. Quantum Electronics, IEEE Journal of, 25(10):2096– 2105, 1989. 82  Bibliography [16] Li He, Yanling He, Andrew Pomerene, Craig Hill, Stewart Ocheltree, Tom Baehr-Jones, and Michael Hochberg. Ultra-thin silicon-oninsulator grating couplers. 2012. [17] GB Hocker and W.K. Burns. Mode dispersion in diffused channel waveguides by the effective index method. Applied optics, 16(1):113– 118, 1977. [18] LumericalSolutions Inc. Fdtd solutions knowledge base, 2013. [19] Nicolas A. F. Jaeger. Notes for eece482. 2011. [20] B Jalali, S Yegnanarayanan, T Yoon, T Yoshimoto, I Rendina, and F Coppinger. Advances in silicon-on-insulator optoelectronics. Selected Topics in Quantum Electronics, IEEE Journal of, 4(6):938–947, 1998. [21] Bahram Jalali and Sasan Fathpour. Silicon photonics. Lightwave Technology, Journal of, 24(12):4600–4615, 2006. [22] Bahram Jalali, Varun Raghunathan, Ramesh Shori, Sasan Fathpour, Dimitrios Dimitropoulos, and Oscar Stafsudd. Prospects for silicon mid-ir raman lasers. Selected Topics in Quantum Electronics, IEEE Journal of, 12(6):1618–1627, 2006. [23] Nan M Jokerst, Martin A Brooke, Sang-Yeon Cho, Mikkel Thomas, Jeffrey Lillie, Daeik Kim, Stephen Ralph, Karla Dennis, Benita Comeau, and Cliff Henderson. Integrated planar lightwave bio/chem oeic sensors on si cmos circuits. In Proc. of SPIE Vol, volume 5730, page 227, 2005. [24] RM Knox and PP Toulios. Integrated circuits for the millimeter through optical frequency range. In Proc. Symp. Submillimeter Waves, volume 20, pages 497–515. Polytechnic Press of Polytechnic Institute of Brooklyn, 1970. [25] Brian R Koch, Andrew Alduino, Ling Liao, Richard Jones, Michael Morse, Brian Kim, Wei-Zen Lo, Juthika Basak, Hai-Feng Liu, Haisheng  83  Bibliography Rong, et al. A 4x12. 5 gbps cwdm si photonics link using integrated hybrid silicon lasers. In CLEO: Science and Innovations. Optical Society of America, 2011. [26] Charlie Lin. Photonic device design flow: from mask layout to device measurement. 2012. [27] Liu Liu, Minhao Pu, Kresten Yvind, and Jørn M Hvam. High-efficiency, large-bandwidth silicon-on-insulator grating coupler based on a fullyetched photonic crystal structure. Applied physics letters, 96(5):051126– 051126, 2010. [28] Luxtera. Luxtera samples world’s first 40 gigabit optical active cable, 2007. [29] E.A.J. Marcatili. Dielectric rectangular waveguide and directional coupler for integrated optics. Bell Syst. Tech. J, 48(7):2071–2102, 1969. [30] Sharee McNab, Nikolaj Moll, and Yurii Vlasov. Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides. Optics Express, 11(22):2927–2939, 2003. [31] Attila Mekis, Sherif Abdalla, Peter M De Dobbelaere, Dennis Foltz, Steffen Gloeckner, Steven Hovey, Steven Jackson, Yi Liang, Michael Mack, Gianlorenzo Masini, et al. Scaling cmos photonics transceivers beyond 100 gb/s. In SPIE OPTO, pages 82650A–82650A. International Society for Optics and Photonics, 2012. [32] Attila Mekis, Steffen Gloeckner, Gianlorenzo Masini, Adithyaram Narasimha, Thierry Pinguet, Subal Sahni, and Peter De Dobbelaere. A grating-coupler-enabled cmos photonics platform. Selected Topics in Quantum Electronics, IEEE Journal of, 17(3):597–608, 2011. [33] David AB Miller. Physical reasons for optical interconnection. International Journal of Optoelectronics, 11(3):155–168, 1997. [34] David AB Miller. Optical interconnects to silicon. Selected Topics in Quantum Electronics, IEEE Journal of, 6(6):1312–1317, 2000. 84  Bibliography [35] N. Na, H. Frish, I.W. Hsieh, O. Harel, R. George, A. Barkai, and H. Rong. Efficient broadband silicon-on-insulator grating coupler with low backreflection. Optics letters, 36(11):2101–2103, 2011. [36] R´egis Orobtchouk, Abdelhalim Layadi, Hamid Gualous, Daniel Pascal, Alain Koster, and Suzanne Laval. High-efficiency light coupling in a submicrometric silicon-on-insulator waveguide.  Applied optics,  39(31):5773–5777, 2000. [37] M Paniccia and R Won. Integrating silicon photonics. Nat. Photonics4 (8), pages 498–499, 2010. [38] Daniel H Raguin and G Michael Morris. Analysis of antireflectionstructured surfaces with continuous one-dimensional surface profiles. Applied optics, 32(14):2582–2598, 1993. [39] Jacob Robinson and Yahya Rahmat-Samii. Particle swarm optimization in electromagnetics. Antennas and Propagation, IEEE Transactions on, 52(2):397–407, 2004. [40] G. Roelkens, D.V. Thourhout, and R. Baets. High efficiency grating coupler between silicon-on-insulator waveguides and perfectly vertical optical fibers. Optics letters, 32(11):1495–1497, 2007. [41] G. Roelkens, D. Vermeulen, S. Selvaraja, R. Halir, W. Bogaerts, and D. Van Thourhout. Grating-based optical fiber interfaces for siliconon-insulator photonic integrated circuits. Selected Topics in Quantum Electronics, IEEE Journal of, (99):1–10, 2011. [42] Atsushi Sakai, Tatsuhiko Fukazawa, and BABA Toshihiko. Low loss ultra-small branches in a silicon photonic wire waveguide. IEICE transactions on electronics, 85(4):1033–1038, 2002. [43] B Schuppert, J Schmidtchen, and K Petermann. Optical channel waveguides in silicon diffused from gesi alloy. Electronics Letters, 25(22):1500– 1502, 1989.  85  Bibliography [44] Wei Shi, Han Yun, Charlie Lin, Mark Greenburg, Xu Wang, Yun Wang, Sahba Talebi Fard, Jonas Flueckiger, Nicolas A. F. Jaeger, and Lukas Chrostowski. Ultra-compact, flat-top demultiplexer using antireflection contra-directional couplers for cwdm networks on silicon. Optics Express, 2013. [45] Wei Shi, Han Yun, Wen Zhang, Charlie Lin, Ting Kai Chang, Yun Wang, Nicolas AF Jaeger, and Lukas Chrostowski. Ultra-compact, high-q silicon microdisk reflectors. Optics Express, 20(20):21840–21846, 2012. [46] Richard Soref and J Larenzo. All-silicon active and passive guidedwave components for λ= 1.3 and 1.6 µm. Quantum Electronics, IEEE Journal of, 22(6):873–879, 1986. [47] Toshiaki Suhara and Hiroshi Nishihara. Integrated optics components and devices using periodic structures. Quantum Electronics, IEEE Journal of, 22(6):845–867, 1986. [48] D. Taillaert, F. Van Laere, M. Ayre, W. Bogaerts, D. Van Thourhout, P. Bienstman, and R. Baets. Grating couplers for coupling between optical fibers and nanophotonic waveguides.  Jpn. J. Appl. Phys,  45(8A):6071–6077, 2006. [49] Dirk Taillaert. Grating couplers as interface between optical fibres and nanophotonic waveguides, 2005. [50] Dirk Taillaert, Peter Bienstman, and Roel Baets. Compact efficient broadband grating coupler for silicon-on-insulator waveguides. Optics letters, 29(23):2749–2751, 2004. [51] PD Trinh, S Yegnanarayanan, F Coppinger, and B Jalali. Siliconon-insulator (soi) phased-array wavelength multi/demultiplexer with extremely low-polarization sensitivity. Photonics Technology Letters, IEEE, 9(7):940–942, 1997.  86  Bibliography [52] PD Trinh, S Yegnanarayanan, and B Jalali. Integrated optical directional couplers in silicon-on-insulator. Electronics Letters, 31(24):2097– 2098, 1995. [53] PD Trinh, S Yegnanarayanan, and B Jalali. 5 x 9 integrated optical star coupler in silicon-on-insulator technology. Photonics Technology Letters, IEEE, 8(6):794–796, 1996. [54] Frederik Van Laere, G¨ unther Roelkens, Melanie Ayre, Jonathan Schrauwen, Dirk Taillaert, Dries Van Thourhout, Thomas F Krauss, and Roel Baets. Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides. Lightwave Technology, Journal of, 25(1):151–156, 2007. [55] Diedrik Vermeulen, Yannick De Koninck, Yanlu Li, Emmanuel Lambert, Wim Bogaerts, Roel Baets, and G¨ unther Roelkens. Reflectionless grating coupling for silicon-on-insulator integrated circuits. In Group IV Photonics (GFP), 2011 8th IEEE International Conference on, pages 74–76. IEEE, 2011. [56] Diedrik Vermeulen, S Selvaraja, Pl Verheyen, G Lepage, W Bogaerts, P Absil, D Van Thourhout, and G Roelkens. High-efficiency fiberto-chip grating couplers realized using an advanced cmos-compatible silicon-on-insulator platform. Optics express, 18(17):18278–18283, 2010. [57] B. Wang, J. Jiang, and G.P. Nordin. Embedded slanted grating for vertical coupling between fibers and silicon-on-insulator planar waveguides. Photonics Technology Letters, IEEE, 17(9):1884–1886, 2005. [58] Xiaochuan Xu, Harish Subbaraman, John Covey, David Kwong, Amir Hosseini, and Ray T.Chen. Cmos compatible subwavelength grating couplers for silicon integrated photonics. Photonics Conference, 2012. [59] Kane Yee. Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media. Antennas and Propagation, IEEE Transactions on, 14(3):302–307, 1966. 87  [60] Yi Zhang, Tom Baehr-Jones, Ran Ding, Thierry Pinguet, Zhe Xuan, and Michael Hochberg. Silicon multi-project wafer platforms for optoelectronic system integration. In Group IV Photonics (GFP), 2012 IEEE 9th International Conference on, pages 63–65. IEEE, 2012. [61] Yi Zhang, Shuyu Yang, Andy Eu-Jin Lim, Guo-Qiang Lo, Christophe Galland, Tom Baehr-Jones, and Michael Hochberg. A compact and low loss y-junction for submicron silicon waveguide. Optics Express, 21(1):1310–1316, 2013. [62] CZ Zhao, GZ Li, EK Liu, Y Gao, and XD Liu. Silicon on insulator mach–zehnder waveguide interferometers operating at 1.3 µm. Applied physics letters, 67(17):2448–2449, 1995.  88  Appendix A  FDTD code to generate universal grating coupler model ## Universal Grating Coupler Based on EIM and Bragg Condition ## definition of variables: ## period: grating pitch; ## duty_cycle: the unetched part of a grating in the longitudinal direction; ## fill factor: the ratio of duty_cycle and period; ## etch depth: the length of the etched part of the grating in the vertical direction; ## neff: effective index of the grating region; ## ne1: effective index of the unetched region; ## ne2: effective index of the etched region; ## Si_thickness: the thickenss of the unetched silicon; ##Initialization; Si_thickness=d1=0.22e-6; etch_depth=0.075e-6; theta=20; n_c=1.44; lambda=1.58e-6; switchtolayout; redrawoff; selectall;  89  Appendix A. FDTD code to generate universal grating coupler model delete; # draw silicon substrate; addrect; set(’name’,’Si_sub’); set(’material’,’Si (Silicon) - Palik’); set(’x’,0); set(’x span’, 40e-6); set(’y’,-3e-6); set(’y span’,2e-6); #draw burried oxide; addrect; set(’name’,’BOX’); set(’material’,’SiO2 (Glass) - Palik’); set(’x’,0); set(’x span’, 40e-6); set(’y’,-1e-6); set(’y span’,2e-6); set(’override mesh order from material database’,true); set(’mesh order’,3); set(’alpha’,0.3); #draw waveguide; addrect; set(’name’,’WG’); set(’material’,’Si (Silicon) - Palik’); set(’x min’,0); set(’x max’, 20e-6); set(’y’,0.11e-6); set(’y span’,0.22e-6);  90  Appendix A. FDTD code to generate universal grating coupler model # add simulation region; addfdtd; set(’x max’,5e-6); set(’x min’,-14e-6); set(’y min’,-2.3e-6); set(’y max’,1e-6); if(n_c>1) { select(’BOX’); set(’y max’,1.93e-6); select(’FDTD’); set(’y max’,3e-6); } ## calculating neff for fundamental TE mode in fully etched WG; addmode; set(’name’,’mode’); set(’x’,2e-6); set(’y’,0.5*d1); set(’y span’,1e-6); set(’direction’,’Backward’); set(’center wavelength’,lambda); set(’wavelength span’,0.3e-6); set(’mode selection’,’fundamental mode’); select(’WG’); set(’y’,0.5*d1); set(’y span’,d1); select("mode"); updatesourcemode; neff1=getresult(’mode’,’neff’); ne1=neff1.neff;  91  Appendix A. FDTD code to generate universal grating coupler model # calculating neff for fundamental TE mode in shallow etched WG; select(’WG’); set(’y’,0.5*(d1-etch_depth)); set(’y span’,(d1-etch_depth)); select("mode"); set(’y’,0.5*(d1-etch_depth)); clearsourcedata; updatesourcemode; neff2=getresult(’mode’,’neff’); ne2=neff2.neff; # calculating neff assuming ff is 50%; neff=0.5*(ne1+ne2); ?neff; # change WG strcture to original thickness; select(’WG’); set(’y’,0.11e-6); set(’y span’,0.22e-6); select(’mode’); set(’y’,0.11e-6); ## calculate the period of the grating based on neff; period=lambda/(neff-sin((theta/180)*pi)); ?period; duty_cycle=0.5*period; ## draw uniform GC addrect; set(’name’,’GC_base’); set(’x max’,0); set(’material’,’Si (Silicon) - Palik’); 92  Appendix A. FDTD code to generate universal grating coupler model set(’x min’, -20e-6); set(’y min’,0); set(’y max’,d1-etch_depth); for (i=0:40) { addrect; set(’name’,’GC_tooth’); set(’x min’,-period-i*period); set(’x max’,-duty_cycle-i*period); set(’y’,0.5*d1); set(’y span’,d1); } selectpartial(’tooth’); set(’material’,’Si (Silicon) - Palik’); selectpartial(’GC’); addtogroup(’GC’); ## add Gaussian mode select(’mode’); set(’enabled’,’false’); addgaussian; set(’name’,’gaussian’); set(’x’,-4.5e-6); set(’x span’, 16e-6); set(’direction’,’Backward’); set(’polarization angle’,90); set(’angle theta’,theta); set(’center wavelength’,lambda); set(’wavelength span’,0.1e-6); set(’waist radius w0’,4.5e-6); set(’distance from waist’,10e-6); if (n_c>1) { 93  Appendix A. FDTD code to generate universal grating coupler model set(’y’,2.5e-6); } else { set(’y’,0.5e-6); } ## add monitor; addpower; set(’name’,’r’); set(’monitor type’,’Linear Y’); set(’x’,3e-6); set(’y’,0.5*d1); set(’y span’,1e-6); addpower; set(’name’,’u’); set(’monitor type’,’Linear X’); set(’x’,-4.5e-6); set(’x span’,20e-6); if(n_c>1) { set(’y’,2.8e-6); } else { set(’y’,0.8e-6); } addpower; set(’name’,’d’); set(’monitor type’,’Linear X’); set(’x’,-4.5e-6); 94  Appendix A. FDTD code to generate universal grating coupler model set(’x span’,20e-6); set(’y’,-2.1e-6); run; switchtolayout; redrawoff; selectall; delete; # draw silicon substrate; addrect; set(’name’,’Si_sub’); set(’material’,’Si (Silicon) - Palik’); set(’x’,0); set(’x span’, 40e-6); set(’y’,-3e-6); set(’y span’,2e-6); #draw burried oxide; addrect; set(’name’,’BOX’); set(’material’,’SiO2 (Glass) - Palik’); set(’x’,0); set(’x span’, 40e-6); set(’y’,-1e-6); set(’y span’,2e-6); set(’override mesh order from material database’,true); set(’mesh order’,3); set(’alpha’,0.3); #draw waveguide; addrect; set(’name’,’WG’); set(’material’,’Si (Silicon) - Palik’); 95  Appendix A. FDTD code to generate universal grating coupler model set(’x min’,0); set(’x max’, 20e-6); set(’y’,0.11e-6); set(’y span’,0.22e-6); # add simulation region; addfdtd; set(’x max’,5e-6); set(’x min’,-14e-6); set(’y min’,-2.3e-6); set(’y max’,1e-6); if(n_c>1) { select(’BOX’); set(’y max’,1.93e-6); select(’FDTD’); set(’y max’,3e-6); } ## calculating neff for fundamental TE mode in fully etched WG; addmode; set(’name’,’mode’); set(’x’,2e-6); set(’y’,0.5*d1); set(’y span’,1e-6); set(’direction’,’Backward’); set(’center wavelength’,lambda); set(’wavelength span’,0.3e-6); set(’mode selection’,’fundamental TE mode’); select(’WG’); set(’y’,0.5*d1); set(’y span’,d1); select("mode"); 96  Appendix A. FDTD code to generate universal grating coupler model updatesourcemode; neff1=getresult(’mode’,’neff’); ne1=neff1.neff; # calculating neff for fundamental TE mode in shallow etched WG; select(’WG’); set(’y’,0.5*(d1-etch_depth)); set(’y span’,(d1-etch_depth)); select("mode"); set(’y’,0.5*(d1-etch_depth)); clearsourcedata; updatesourcemode; neff2=getresult(’mode’,’neff’); ne2=neff2.neff; # calculating neff assuming ff is 50%; neff=0.5*(ne1+ne2); ?neff; # change WG strcture to original thickness; select(’WG’); set(’y’,0.11e-6); set(’y span’,0.22e-6); select(’mode’); set(’y’,0.11e-6); ## calculate the period of the grating based on neff; period=lambda/(neff-sin((theta/180)*pi)); ?period; duty_cycle=0.4*period; ## draw uniform GC addrect; set(’name’,’GC_base’); 97  Appendix A. FDTD code to generate universal grating coupler model set(’x max’,0); set(’material’,’Si (Silicon) - Palik’); set(’x min’, -20e-6); set(’y min’,0); set(’y max’,d1-etch_depth); for (i=0:40) { addrect; set(’name’,’GC_tooth’); set(’x min’,-period-i*period); set(’x max’,-duty_cycle-i*period); set(’y’,0.5*d1); set(’y span’,d1); } selectpartial(’tooth’); set(’material’,’Si (Silicon) - Palik’); selectpartial(’GC’); addtogroup(’GC’); ## add Gaussian mode select(’mode’); set(’enabled’,’false’); addgaussian; set(’name’,’gaussian’); set(’x’,-4.5e-6); set(’x span’, 16e-6); set(’direction’,’Backward’); set(’polarization angle’,0); set(’angle theta’,theta); set(’center wavelength’,lambda); set(’wavelength span’,0.3e-6); set(’waist radius w0’,4.5e-6); set(’distance from waist’,10e-6); 98  Appendix A. FDTD code to generate universal grating coupler model if (n_c>1) { set(’y’,2.5e-6); } else { set(’y’,0.5e-6); } ## add monitor; addpower; set(’name’,’r’); set(’monitor type’,’Linear Y’); set(’x’,3e-6); set(’y’,0.5*d1); set(’y span’,1e-6); addpower; set(’name’,’u’); set(’monitor type’,’Linear X’); set(’x’,-4.5e-6); set(’x span’,20e-6); if(n_c>1) { set(’y’,2.8e-6); } else { set(’y’,0.8e-6); } addpower; set(’name’,’d’); set(’monitor type’,’Linear X’); set(’x’,-4.5e-6); 99  Appendix A. FDTD code to generate universal grating coupler model set(’x span’,20e-6); set(’y’,-2.1e-6); run;  100  Appendix B  Pyxis code for universal grating coupler design function EBeam_UGC_double() { local device = $get_device_iobj(); local wl = $get_property_value(device,"wl"); local etch_depth = $get_property_value(device,"etch_depth"); local Si_thickness = $get_property_value(device,"Si_thickness"); local incident_angle = $get_property_value(device,"incident_angle"); local wg_width = $get_property_value(device,"wg_width"); local n_cladding = $get_property_value(device,"n_cladding"); local pl = $get_property_value(device,"pl"); build_EBeam_UGC_double(wl,etch_depth,Si_thickness,incident_angle, wg_width,n_cladding,pl); } function build_EBeam_UGC_double( wl:number {default=1.55}, etch_depth:number {default=0.07}, Si_thickness:number {default=0.22}, incident_angle:number {default=20},wg_width:number {default=0.5}, n_cladding:number{default=1.44},pl:number {default=1}) { //Save Original user settings local selectable_types_orig = $get_selectable_types(); local selectable_layers_orig = $get_selectable_layers(); local autoselect_orig = $get_autoselect();  101  Appendix B. Pyxis code for universal grating coupler design  //Set up selection settings $set_selectable_types(@replace, [@shape, @path, @pin, @overflow,@row, @property_text, @instance, @array, @device, @via_object, @text, @region, @bisector, @channel, @slice], @both); $set_selectable_layers(@replace, ["0-4096"]); $set_autoselect(@true); // calculate the neff of the grating region // local for neff calculation local point=1001; local ii=0; local jj=0; local kk=0; local mm=0; local nn=0; local n0=n_cladding; local n1 = 0; local n2 = 3.473; local n3 = 1.444; local delta=n0-n3; local t = Si_thickness; local t_slot=t-etch_depth; local k0 = 2*3.14159/wl; local b0 = $create_vector(point-1); local te0 = $create_vector(point-1); local te1 = $create_vector(point-1); local tm0 = $create_vector(point-1); local tm1 = $create_vector(point-1); local h0 = $create_vector(point-1); local q0 = $create_vector(point-1); local p0 = $create_vector(point-1); 102  Appendix B. Pyxis code for universal grating coupler design local qbar0 = $create_vector(point-1); local pbar0 = $create_vector(point-1); local mini_TE=0; local index_TE=0; local mini_TE1=0; local index_TE1=0; local mini_TM=0; local index_TM=0; local mini_TM1=0; local index_TM1=0; local nTE=0; local nTE1=0; local nTM=0; local nTM1=0; local ne=0; //...// calculating neff for 0.22 silicon layer if( delta<0) { n1=n3; } else { n1=n0; } for(ii=0;ii<point-1;ii=ii+1) { b0[ii]= n1*k0+(n2-n1)*k0/(point-10)*ii; } for(jj=0;jj<point-1;jj=jj+1) 103  Appendix B. Pyxis code for universal grating coupler design { h0[jj] = sqrt( abs(pow(n2*k0,2) - pow(b0[jj],2))); q0[jj] = sqrt( abs(pow(b0[jj],2) - pow(n0*k0,2))); p0[jj] = sqrt( abs(pow(b0[jj],2) - pow(n3*k0,2))); } for(kk=0;kk<point-1;kk=kk+1) { pbar0[kk] = pow(n2/n3,2)*p0[kk]; qbar0[kk] = pow(n2/n0,2)*q0[kk]; } //calculating neff for TE mode if (pl==1) { for (nn=0;nn<point-1;nn=nn+1) { te0[nn] = tan( h0[nn]*t )-(p0[nn]+q0[nn])/h0[nn]/ (1-p0[nn]*q0[nn]/pow(h0[nn],2)); te1[nn] = tan( h0[nn]*t_slot )-(p0[nn]+q0[nn])/h0[nn]/ (1-p0[nn]*q0[nn]/pow(h0[nn],2)); } local abs_te0=abs(te0); local abs_te1=abs(te1); mini_TE=$vector_min(abs_te0); mini_TE1=$vector_min(abs_te1); index_TE=$vector_search(mini_TE,abs(te0),0); index_TE1=$vector_search(mini_TE1,abs(te1),0); nTE=b0[index_TE]/k0; nTE1=b0[index_TE1]/k0;  104  Appendix B. Pyxis code for universal grating coupler design do { abs_te0[index_TE]=100; mini_TE=$vector_min(abs_te0); index_TE=$vector_search(mini_TE,abs(te0),0); nTE=b0[index_TE]/k0; } while ( nTE<2 || nTE>3); //  $message($format("%4.5f",nTE)); do { abs_te1[index_TE1]=100; mini_TE1=$vector_min(abs_te1); index_TE1=$vector_search(mini_TE1,abs(te1),0); nTE1=b0[index_TE1]/k0; } while ( nTE1<2 || nTE1>3);  //  $message($format("%4.5f",nTE1));  ne=0.55*nTE+0.45*nTE1; $message($format("%4.5f",ne)); } //calculating neff for TM mode else { for (mm=0;mm<point-1;mm=mm+1) { tm0[mm] = tan(h0[mm]*t)-h0[mm]*(pbar0[mm]+qbar0[mm])/ (pow(h0[mm],2)-  pbar0[mm]*qbar0[mm]); 105  Appendix B. Pyxis code for universal grating coupler design tm1[mm] = tan(h0[mm]*t_slot)-h0[mm]*(pbar0[mm]+qbar0[mm])/ (pow(h0[mm],2)-pbar0[mm]*qbar0[mm]); } local abs_tm0=abs(tm0); local abs_tm1=abs(tm1); mini_TM=$vector_min(abs(tm0)); mini_TM1=$vector_min(abs(tm1)); index_TM=$vector_search(mini_TM,abs(tm0),0); index_TM1=$vector_search(mini_TM1,abs(tm1),0); nTM=b0[index_TM]/k0; nTM1=b0[index_TM1]/k0; do { abs_tm0[index_TM]=100; mini_TM=$vector_min(abs_tm0); index_TM=$vector_search(mini_TM,abs(tm0),0); nTM=b0[index_TM]/k0; } while ( nTM<1.5 || nTM>3); //  $message($format("\%4.5f",nTM)); do { abs_tm1[index_TM1]=100; mini_TM1=$vector_min(abs_tm1); index_TM1=$vector_search(mini_TM1,abs(tm1),0); nTM1=b0[index_TM1]/k0; } while ( nTM1<1.5 || nTM1>3);  106  Appendix B. Pyxis code for universal grating coupler design //  $message($format("%4.5f",nTM1));  ne=0.45*nTM+0.55*nTM1; $message($format("%4.5f",ne)); } //...// end of calculating neff //  Start of grating drawing  local period=-1*wl/(sin(rad(incident_angle))*1-ne); local duty_cycle=0; $message($format("%4.5f",period)); if (pl==1) { duty_cycle=period*0.5; $message($format("%4.5f",duty_cycle)); } else { duty_cycle=period*0.45; $message($format("%4.5f",duty_cycle)); } local segnum=75;  //segment number of one grating curve  local seg_points = segnum+1; local arc_vec = $create_vector(2*seg_points+2); local i = 0; local j = 0; local x_r=0; local y_r=0; local x_l=0; local y_l=0; 107  Appendix B. Pyxis code for universal grating coupler design local m1_x=0; local m1_y=0; local m2_x=0; local m2_y=0; local nf=1.44; local e=nf*sin(rad(incident_angle))/ne; local gc_number=$round(21/period); local angle_e=62; local N=$round(18*(1+e)*ne/wl)+1; local gap=period-duty_cycle; for(j=0;j<gc_number;j=j+1) { for(i=0;i<seg_points;i=i+1) { x_r=(N*wl/(ne*(1-e*cos(rad(180-angle_e/2+angle_e/segnum*i))))+ j*period)*cos(rad(180-angle_e/2+angle_e/segnum*i)); y_r=(N*wl/(ne*(1-e*cos(rad(180-angle_e/2+angle_e/segnum*i))))+ j*period)*sin(rad(180-angle_e/2+angle_e/segnum*i)); arc_vec[i] = [x_r,y_r]; m1_x=(N*wl/(ne*(1-e*cos(rad(180+angle_e/2))))+ duty_cycle/2+j*period)*cos(rad(180+angle_e/2)); m1_y=(N*wl/(ne*(1-e*cos(rad(180+angle_e/2))))+ duty_cycle/2+j*period)*sin(rad(180+angle_e/2))-0.1; arc_vec[seg_points] = [m1_x,m1_y]; x_l=(N*wl/(ne*(1-e*cos(rad(180+angle_e/2-angle_e/segnum*i))))+ gap+j*period)*cos(rad(180+angle_e/2-angle_e/segnum*i)); y_l=(N*wl/(ne*(1-e*cos(rad(180+angle_e/2-angle_e/segnum*i))))+ gap+j*period)*sin(rad(180+angle_e/2-angle_e/segnum*i)); arc_vec[seg_points+i+1] = [x_l,y_l];  108  Appendix B. Pyxis code for universal grating coupler design m2_x=(N*wl/(ne*(1-e*cos(rad(180+angle_e/2-angle_e/segnum*i))))+ gap/2+j*period)*cos(rad(180+angle_e/2-angle_e/segnum*i)); m2_y=(N*wl/(ne*(1-e*cos(rad(180+angle_e/2-angle_e/segnum*i))))+ gap/2+j*period)*sin(rad(180+angle_e/2-angle_e/segnum*i))+0.1; arc_vec[2*seg_points+1] = [m2_x,m2_y]; } angle_e = angle_e-0.7; $add_shape(arc_vec,’SiEtch1’); } $add_shape([[-40,13.5],[0,1/2*wg_width],[0,-1/2*wg_width],[-40,-13.5]],’Si’); //$unselect_all(@nofilter); $add_shape([[-1, -wg_width/2], [0, wg_width/2]], "Si", @internal); //Restore original user settings $set_selectable_types(@replace, (selectable_types_orig[0]==void)? []:selectable_types_orig[0],selectable_types_orig[1]); $set_selectable_layers(@replace, selectable_layers_orig); $set_autoselect(autoselect_orig); } function EBeam_UGC_double_parameters( layer:optional number {default=1}, wl:optional number {default=1.55}, etch_depth:optional number {default=0.07}, Si_thickness:optional number {default=0.22},incident_angle:optional number {default=20},wg_width:optional number {default=0.5},n_cladding:optional number {default=1.44},pl:optional number {default=1}) { return [ ["wl",$g(wl)],["etch_depth",$g(etch_depth)],["Si_thickness", $g(Si_thickness)],["incident_angle",$g(incident_angle)], ["wg_width",$g(wg_width)],["n_cladding",$g(n_cladding)],["pl",$g(pl)] ]; }  109  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0073806/manifest

Comment

Related Items