Controlling Semiconductor Optical Amplifiers for Robust Integrated Photonic Signal Processing by Scott B. Kuntze A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto Copyright c 2009 by Scott B. Kuntze Abstract Controlling Semiconductor Optical Amplifiers for Robust Integrated Photonic Signal Processing Scott B. Kuntze Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto 2009 How can we evaluate and design integrated photonic circuit performance systematically? Can active photonic circuits be controlled for optimized performance? This work uses control theory to analyze, design, and optimize photonic integrated cir- cuits based on versatile semiconductor optical amplifiers (SOAs). Control theory provides a mathematically robust set of tools for system analysis, design, and control. Although control theory is a rich and well-developed field, its application to the analysis and design of photonic circuits is not widespread. Following control theoretic methods already used for fibreline systems we derive three interrelated state-space models: a core photonic model, a photonic model with gain compression, and a equivalent circuit optoelectronic model. We validate each model and calibrate the gain compression model by pump–probe experiments. We then linearize the state-space models to design and analyze SOA controllers. We apply each linearized model to proof-of-concept SOA control applications such as suppressing interchannel crosstalk and regulating output power. We demonstrate the power of linearized state-space models in controller design and stability analysis. To illustrate the importance of using the complete equivalent circuit model in con- troller design, we demonstrate an intuitive bias-current controller that fails due to the dynamics of the intervening parasitic circuitry of the SOA. We use the linearized state- space models to map a relationship between feedback delay and controller strength for stable operation, and demonstrate that SOAs pose unusual control difficulties due to their ultrafast dynamics. Finally, we leverage the linearized models to design a novel and successful hybrid controller that uses one SOA to control another via feedback (for reliability) and feedfor- ward (for speed) control. The feedback controller takes full advantage of the equivalent ii circuit modelling by sampling the voltage of the controlled SOA and using the error to drive the bias current of the controller SOA. Filtering in the feedback path is specified by transfer function analysis. The feedforward design uses a novel application of the linearized models to set the controller bias points correctly. The modelling and design framework we develop is entirely general and opens the way to the robust optoelectronic control of integrated photonic circuits. iii Preface: Light, the Internet, and everyone I suppose the first time I really appreciated the combination of light and control was as an adolescent watching Genesis concert videos of the 1980s and 90s with their impres- sive light shows—hundreds of robotic Varilites sweeping and changing colour in perfect synchronization with the musical dynamics. I had never seen anything like it and I was enthralled by the visual impact of the shows. My fascination with the Internet began a little later in 1994 when I realized I could find information and discussion on the obscure “progressive rock” music I loved as a teenager. Just waiting for me out there were heated debates, recommendations, reviews, and catalogues. Suddenly, the Internet was personal and useful for me. During the last three years I have been enthralled and fascinated by the control of another kind of light—the invisible kind that pulses through modern telecommunication circuits and that conveys high-definition concert video around the world via the Internet in the blink of an eye. My work would not have been possible with a number of key people. My Doctorate supervisors Stewart Aitchison and Lacra Pavel have been so very supportive throughout the life of this project. They allowed me to initiate this joint photonics–control project and encouraged me at every turn. Together, they comple- mented each other ideally as thesis advisors and I am fortunate to have found their perfect mix for my background and interests. My Masters supervisor Ted Sargent taught me to be a researcher and was so support- ive over changes of direction. I owe much of my research process and work flow training to my time working with him. Along the way we had many discussions that were always helpful and encouraging. iv Preface v My colleagues Aaron Zilkie and Baosen Zhang helped me to push my work beyond a couple of obstacles with their contributions. Prof. John Cartledge provided an excellent external appraisal of this dissertation and Prof. T.J. Lim offered an excellent external viewpoint on my work; both made valuable contributions the refinement of my thesis through their questions and comments. My parents have always been completely supportive in everything I have done, this lengthy project included. And of course Julia, who offered her patience, support, more patience, understanding, and even more patience as I scrambled to hit deadlines, worked long hours into the night, and occasionally became consumed by simulations, papers, and reports. To these people and to the many other people who factored into my life in some way or another during this time, thank you—you are a part of the pages that follow. Contents Preface iv Contents vi List of Figures x List of Tables xiii 1 The promise of integrated photonics 1 1.1 TheInternetisstilltooslow . .. 1 1.2 Current trends in photonic signal processing . ....... 3 1.2.1 Integration and the versatility of semiconductor optical amplifiers 3 1.2.2 SOA-based integrable functions . 5 1.2.3 True large-scale photonic integration . .... 7 1.3 This work: a theoretical approach to photonic integrated circuit control . 7 1.3.1 The need for photonic regulation: optical amplifier control .... 7 1.3.2 Control theory as a tool for robust photonic design . ..... 9 1.3.3 Bringing control theory to SOA regulator design: an overview . 13 1.4 Conclusion: robust control methods for integrated photonics ....... 17 2 Technical background 18 2.1 Conventions .................................. 18 2.2 Essential semiconductor optical amplifier physics . .......... 19 2.2.1 Principles of architecture and operation . ..... 20 2.2.2 Carrierrateequation . .. .. 24 2.2.3 Opticalpropagationequation . 25 2.3 Controltheorymethods ........................... 27 vi Contents vii 2.3.1 State-spacerealization . 27 2.3.2 State-space model linearization . .. 28 2.3.3 State-spacemodelsolution . 30 2.3.4 Linearmodeltransferfunction. 31 2.3.5 Controllercanonicalform . 32 2.3.6 Usefulsystemproperties . 33 2.3.7 Statefeedback............................. 35 2.3.8 Optimal least-squares state feedback . ... 36 2.3.9 Outputfeedback ........................... 37 2.3.10 Dynamic single-input/single-output controllers . .......... 38 2.3.11 Usefulcalculus ............................ 39 2.3.12 Example: EDFAcontrolformulation . 39 3 Core photonic state-space model 42 3.1 Nonlinearstate-spacemodel . .. 43 3.1.1 Governingequations . 44 3.1.2 Gain.................................. 44 3.1.3 Outputrelations ........................... 47 3.1.4 Stateupdateequation . 48 3.1.5 Nonlinearcontrolformsummary . 48 3.2 Linearizedstate-spacemodel. ... 50 3.3 Feedbackcontrol ............................... 54 3.3.1 State feedback: suppressing cross-talk electronically........ 54 3.3.2 Stateobserver............................. 57 3.3.3 Output Feedback: Suppressing Cross-talk Optically . ....... 58 3.4 Controllingphase ............................... 60 3.5 Conclusion: a new SOA state-space design framework . ....... 64 4 Explicit photonic gain compression state-space model 65 4.1 Governingequations ............................. 66 4.2 Generalstate-spaceform . 67 4.3 Solving the propagation equation with gain compression ......... 69 4.4 State-spacemodel............................... 73 4.5 Modelverification............................... 74 Contents viii 4.5.1 Experimentandsimulation. 74 4.5.2 Deviceandmodelparameters . 76 4.5.3 Experiment–model comparison . 77 4.6 Optical feedback control for constant output power . ........ 79 4.7 Effect of gain compression on required controller strength......... 82 4.8 Conclusion: the first explicit input–output model with gain compression . 84 5 Equivalent circuit dynamic model 85 5.1 Nonlinear state-space equivalent circuit model . ......... 86 5.1.1 State-space realization of the equivalent circuit . ......... 87 5.1.2 SOA active region current I¯(t).................... 89 5.1.3 SOA current–voltage relationship . .. 90 5.1.4 Nonlinear equivalent-circuit space-space model . ........ 90 5.2 Linearized state-space equivalent circuit model . .......... 93 5.3 Conclusion: complete optoelectronic SOA state-space description . 97 6 Impact of feedback delay on closed-loop stability 98 6.1 State feedback into the drive current and system stability......... 99 6.2 The delay margin for feedback stability . ..... 101 6.2.1 Least-squaresoptimalcontrol . 102 6.2.2 Delay margin of the feedback controller . 104 6.3 Hybrid feedforward–feedback controller . ....... 110 6.4 Conclusion: feedback delay constraints are severe for SOAs........ 115 7 Incoherent optoelectronic control of a semiconductor optical