Polarization Model and Control in Fiber-Based Bidirectional Systems with Reflections

Transcription

1 Polarization Model and Control in Fiber-Based Bidirectional Systems with Reflections by William La A thesis submitted in conformity with the requirements for the degree of Masters of Applied Science Department of Electrical and Computer Engineering University of Toronto Copyright by William La 2010

2 Polarization Model and Control in Fiber-Based Bidirectional Systems with Reflections Abstract William La Masters of Applied Science Department of Electrical and Computer Engineering University of Toronto 2010 We present, for the first time, methods to model and control the polarization of the output lightwave of a bidirectional fiber-optic system, in which the lightwave propagates through polarization control elements in both directions. Using the dynamic eigenstate (DES) principle, we built a model to simulate the behavior of the polarization evolution. In a bidirectional system with one control element, we extracted system parameters from experimental data and achieved less than 3% angular deviation between modeled and experimental state of polarization (SOP). The theory was further validated by varying the input SOP to the bidirectional system. Our method can be extended to predict the SOP of a system with multiple actuators. Furthermore, combining our deterministic control method and a feedback loop, we are able to control the output SOP to be within a mean angular deviation of 5.5% from the target SOP, with as few as three iterations. ii

3 Acknowledgments This work would not have been successful if it were not for the individuals that provided support along the way. I would like to thank my colleagues and my friends who always lent an ear to my problems and offered useful suggestions whenever I encountered a roadblock. Without them, I would not have discovered new routes for this thesis. I am grateful to my supervisor, Professor Qian, for her guidance and advice that helped me along the path to my thesis goal and for her endless patience for the many farfetched ideas that I came up with during this work. Without the many discussions that we had, I would not have been able to come up with the ideas and solutions that I needed. And always, thank you to my parents for their continued support. With them, I found the motivation to complete this work. William La Toronto, June 2010 iii

4 Table of Contents Acknowledgments... iii Table of Contents... iv List of Tables... vii List of Figures... viii List of Appendices... xix Chapter 1 Motivation for Polarization Control Examples of Polarization Control in Optical Systems Polarization Control in Optical Communication Systems Polarization Control in Optical Sensing Systems Polarization Control in Optical Arbitrary Waveform Generation Systems Polarization Control in Fiber Lasers Common Configurations for Polarization Control Scope and Goal of the Thesis Thesis Outline Chapter 2 Review of Polarization Modeling and Control SOP Representation Principal States of Polarization DES Transmission Matrix Approach to Polarization Evolution A Geometrical Approach to Polarization Evolution Geometric Description of the Output SOP for Multiple Control Elements Deterministic Control Algorithm for Polarization Control iv

5 Chapter 3 Bidirectional Model with One Control Element System Model and Considerations Transmission Matrix of the FBG Relationship between Forward and Reverse Transmission Matrices System Simplification Geometric Model for Bidirectional Configuration Geometric Model Properties and System Parameters From Experimental Data to Theoretical Model Chapter 4 Bidirectional Model with Multiple Control Elements Two Actuator Model Cascaded System Effect on the Effective Input SOP Cascaded System Effect on DES Vectors Multiple Actuator Analysis New Effective Input Equivalent DES Vectors A Nodal Approach to Modeling Multiple Actuators Chapter 5 Single Polarization Control Element Model Validation Experimental Setup Modeling Experimental Data with One Actuator Validity of One Actuator Model Experimental Limitations Limits of the DES Model Hysteresis of Piezoelectric Material Response Time of Piezoelectric Material FFT Error Additional Sources of Error v

6 Chapter 6 Multiple Polarization Control Elements Model Validation Experimental Data with Multiple Actuators Modeling Experimental Data with Multiple Actuators Predicting Curves through System Parameters Only Modified Method of Modeling with System Parameters State of Polarization Coverage on Poincaré Sphere Chapter 7 Bidirectional Polarization Control Controlling to Target SOP Offline Bidirectional Control System Characterization Algorithm Implement Results Online Control through Dithering Combined Offline and Online Control Scheme Comparison of all Polarization Control Methods Chapter 8 Conclusion Future Work References vi

7 List of Tables Table 5-1: System components used for the experiment. Index corresponds to components in Figure Table 5-2: Parameters necessary to characterize the system as was determined from the characterization steps Table 5-3: The mean system parameters as extracted from each of the curves in Figure Table 5-4: A set of voltages applied to the actuator in the following sequence Table 6-1: The system parameters for two actuators in the PPC Table 7-1: The twenty target SOP Table 7-2: The actuators DES Table 7-3: Performance data of all three systems vii

8 List of Figures Figure 1-1: A fiber based local air turbulence measurement system [7]... 3 Figure 1-2: A fiber based setup for an optical sensor network that applies frequency shifted interferometry for interrogating a FBG sensor array [8] Figure 1-3: A free space dynamic pulse shaping system that delivers a shaped pulse with fiber optics [10] Figure 1-4: An all fiber dynamic pulse shaping system. Output waveforms are formed by spectrally controlling the amplitude and phase of the input spectral lines. Amplitude control is a two step process that involves controlling the polarization and then resolving the amplitude through a polarizer [11] Figure 1-5: A multiwavelength brillouin erbium fiber laser with partially reflecting FBG [13].. 6 Figure 1-6: A fiber based mode-locked laser with a semiconductor optical amplifier as the gain region and electroabsorption modulator as the active modulator inside the resonator [14]... 7 Figure 1-7: A typical fiber based configuration for a polarization control system in a unidirectional configuration... 8 Figure 1-8: A polarization control system for a bidirectional configuration... 9 Figure 2-1: Poincaré sphere used for mapping Stokes parameters. The 2D representations of the SOP are shown in blue at the major axes to show how they correspond to the SOP mapped on the Poincaré sphere Figure 2-2: The system block diagram that depicts an optical field as transmitted through a fiber medium and the resulting in an output optical field Figure 2-3: Basic system transmission matrix that shows how the PC alters an input SOP, to an output SOP through a transmission matrix. The system transmission matrix depends viii

9 on a control variable which is an applied voltage for a piezoelectric actuator that will alter the birefringence of the fiber Figure 2-4: The evolution of the output SOP is a circular trace on the surface of the Poincaré sphere when a field propagates through a single polarization control element that varies the local birefringence of a unidirectional system Figure 2-5: A cascaded birefringent network that represents how the measured output is related to the immediate output of the PC through a fixed fiber rotation Figure 2-6: At the immediate output of the PC, is the red circular trace. The blue circular trace is the measured SOP at the polarimeter and is denoted. The blue circular trace is rotated form of the red circular trace due to the fiber connecting the PC to the polarimeter Figure 2-7: The effective system block in which the fiber rotation is accounted for in the effective rotation matrix and the effective input SOP, Figure 2-8: The geometrical representation of the output SOP and the corresponding vectors Figure 2-9: The method used by Wang et al demonstrating how polarization control can be achieved. The two circles are the traces formed by the waveplates in the commercial PC. is the input SOP and is the required output SOP, while is an intermediate SOP. After a coordinate system transformation, represents the actual output SOP state at the output of their polarization control system. [21] Figure 2-10: The system block diagram for a system with two piezoelectric actuators, where is the DES that characterizes actuator A, and is the DES that characterizes actuator B Figure 2-11: The output SOP trajectories as a result of using multiple actuators. The trajectories (a) are concentric when the actuator closer to the input is operated first followed by a sweep of the birefringence by the proceeding actuator while (b) the trajectories do not share the same DES when an actuator closer to the output is operated first followed by a sweep of the birefringence by the actuator closer to the input Figure 2-12: The DES of each circular trace from Figure 2-11b) (colored lines) form part of a larger circle, (), that models the new effective DES vector. The center of rotation of the ix

10 larger circle, represents the DES of actuator B which is the actuator that is closer to the output Figure 2-13: The SOP traces formed by two actuators. The red is the trace formed from the initial output SOP, and the actuator s DES,. The blue trace is the trace formed from the target output and the actuator s DES,. The black trace is the trajectory that the SOP must be transformed along to go from the initial SOP to the target SOP Figure 2-14: An example algorithm for unidirectional polarization control Figure 3-1: The basic system with one polarization control element. The input signal propagates through the fiber into the PC. The signal exits the PC into the fiber and travels to the FBG. The FBG reflects the signal back to PC where it exits the system Figure 3-2: The system block diagram of a signal propagating through a PC with an FBG Figure 3-3: Equivalent system block diagram detailing how an output signal is transformed as it propagates through each component. Note: there is only one actuator but the signal passes through the actuator twice in different directions Figure 3-4: In addition to the system components, a polarimeter is used to measured the output SOP. In order to measure the output SOP, fiber is connected between the polarimeter and the PC which will rotate the output SOP, to an SOP, Figure 3-5: Forward and reverse transmission in Jones Formulation Figure 3-6: The simplified system block diagram with the effects of the mirror included into the first actuator and the input Figure 3-7: Sample SOP traces from the bidirectional system configuration that resemble a (a) lemniscate-type shape and a (b) limacon-type shape Figure 3-8: The individual frequency components of Equation (3.23) being plotted onto the Poincaré sphere. The fundamental harmonic forms an ellipse (blue), while the second harmonic x

11 forms a circle (red) and the DC component is a vector (green). The curve,, (black) can be constructed from the sum each of the individual components Figure 3-9: The second harmonic component of Equation (3.23) plotted in Stokes Space. and are equal in magnitude and are orthogonal, and thus form a circle. The rotational axis to the circle is Figure 3-10: When multiplying the output curve with the rotation matrix with a DES of, and the resultant curve is the output of the first system block diagram Figure 3-11: A 4016 point FFT of the first Stokes Component of the curve shown in Figure 3-8 resulting in (a) the amplitude, and (b) the phase Figure 4-1: The effect of the fixed-voltage actuator on the effective input,. represents the fixed angular setting on the actuator. and are the DES vectors of the fixed-voltage actuator. All other actuators are inactive; i.e. the other actuators rotation matrices are the identity matrix. is the new effective input after passing through the fixed-voltage actuators. 52 Figure 4-2: The system block diagram for a fixed-voltage actuator, A, preceding a variablevoltage actuator, B. This system has already accounted for the effects caused by the FBG Figure 4-3: The simplified system with the combined effects of the fixed-voltage actuator incorporated into the variable-voltage actuator s rotation matrices Figure 4-4: The output SOP traces as simulated for different fixed-voltage settings when the input signal enters the fixed-voltage actuator before the variable-voltage actuator. The black circle represents a fixed nodal point that is common for all curves Figure 4-5: The system block diagram for when the variable-voltage actuator, B, precedes the fixed-voltage actuator C Figure 4-6: The output SOP traces as predicted by simulation for different fixed-voltage settings when the input signal enters the variable-voltage actuator before the fixed-voltage actuator. The black circle represents a fixed nodal point that is common for all curves xi

12 Figure 4-7: A system with multiple actuators. The variable-voltage actuator (yellow-box) has a forward-transmission DES vector,, and reverse-transmission DES vector,. There are actuators (red boxes) that precede the variable-voltage actuator with forward transmission DES vector, and reverse transmission DES vector,,, where [, ]. There are -actuators (green) that follow the variable-voltage actuator with forward transmission DES vector, and reverse transmission DES vector,, where [, ]. The various refer to the fixed-rotation angle as a result of the different applied voltages applied to the fixed-voltage actuators Figure 4-8: The equivalent system when we applied the simplification techniques on Figure 4-5; i.e. all fixed-voltage rotations have been integrated into, and Figure 4-9: The system when we ignore the fiber rotation effects between the actuator and the FBG Figure 4-10: The equivalent system with an equivalent fiber rotation,, from the output of the PC to the polarimeter to ensure that this system is equivalent to Figure Figure 5-1: Experimental setup of the bidirectional configuration Figure 5-2: The experimental SOP trace from varying the applied voltage to a piezoelectric actuator from 0 to 67.5V. 30 data points were collected and plotted on the Poincare sphere Figure 5-3: The (a) amplitude and (b) phase information from a 384-point FFT applied to the first Stokes parameter, Figure 5-4: The Stokes parameters versus angle of rotation with the trigonometric fit Equation (5.1) applied as determined from the FFT Figure 5-5: The curve as determined from the FFT fitting (blue) as plotted with the experimental data Figure 5-6: The percent angular deviation of the fitted curve with the experimental trace. No corrections have been made to correct a discrepancy between the applied voltage and the angular rotation xii

13 Figure 5-7: The curve that represents the output of the forward transmission matrix in Figure 3-6 experimentally Figure 5-8: (a) The experimental data and modeled data, Equation (5.5), using the extracted DES vectors. (b) The percent error between the experimental datapoint and the modeled datapoint as determined by the ratio of surface distance over the maximum distance that two points can be from each other on the sphere Figure 5-9: The assumed angular response to the applied voltage (red) compared with an adjusted angular response to the applied voltage (blue) Figure 5-10: The percent angular deviation between the experimental and modeled data with the angle correction Figure 5-11: Different experimental SOP traces (red) and the modeled data (blue) for varying the input SOP to the bidirectional system. The effective input of each trace are: (a) =... (b) =..., (c) =... (d) =... (e) = Figure 5-12: The percent angular deviation between the experimental and modeled data for each varying with the corrections Figure 5-13: The forward DES vectors, (blue) and backward DES vectors, (cyan) of the experimental curves in Figure Figure 5-14: The (a) experimental output SOP trace,, in a unidirectional configuration. The curve initially has a DES, but evolves to at high applied voltages and spirals outwards. As a result the the centre vector of the output SOP trace, also shifts to. The theoretical circular curve based on the initial DES is shown in (b) Figure 5-15: An experimental SOP trace for the bidirectional configuration when the spiraling effect is pronounced Figure 5-16: The Stokes components versus the applied voltage when the applied voltage is increasing from 0V to 67.5V (red) and when the voltage is decreasing from 67.5V to 0V (blue). xiii

14 There appears to be a phase shift between the two curves as evident in the (a), (b), and (c) Stokes parameters. The data was collected from 10 data runs Figure 5-17: The response of the Stokes parameter over a period of time after the actuator has been reset Figure 5-18: Resetting the actuator before applying a voltage (red) in comparison to the characterization trace (blue) for the (a) (b) and (c) Stokes parameters. There appears to no phase shift between the two curves. The data was taken over 10 data runs Figure 5-19: The Stokes components (a), (b), and (c) versus the applied voltage when the applied voltage is increasing from 0V to 67.5V (red) and when the voltage is decreasing from 67.5V to 0V (blue). Prior to applying a decreasing voltage, a reset is done on the actuator. The data was collected from 10 data runs Figure 5-20: A set of voltages applied to the PPC in comparison with the characterization trace when the actuator is first (a) reset and (b) and when actuator is not reset. The error bars signify the standard deviation of the data over 10 datasets Figure 5-21: The percent deviation between the SOP as obtained from the randomly chosen applied voltages and the SOP as obtained from the characterization curve when a reset is applied prior to applying the voltage (red) compared to when no reset is applied prior to applying a voltage (blue) Figure 5-22: The FFT of the component in Figure 5-3(a) over the entire range when the window size covers (a) 54V compared with the FFT whose window size covers (b) 64.8V Figure 6-1: The experimental output SOP curves for different fixed-voltage settings when the input signal enters the fixed-voltage actuator before the variable-voltage actuator. The different color lines represent different voltages applied to the fixed-voltage actuator. The black circle is the nodal SOP of the set of curves Figure 6-2: The experimental output SOP traces for different fixed-voltage settings when the input signal enters the variable-voltage actuator before the fixed-voltage actuator. The different xiv

15 color lines represent the different voltages applied to the fixed voltage actuator. The black circle represents the nodal SOP Figure 6-3: Using the method of characterization as developed in Section 3.7 to model each curve in both configurations. The first configuration (a) is where the signal enters the fixedvoltage actuator first followed by the variable-voltage actuator while in (b) the signal enters the variable voltage actuator first followed by the fixed-voltage actuator Figure 6-4: (a) The resultant modeled curves (red) are plotted against the experimental curves (black) using the method in Section 3.7 and the theory in Section The signal enters the fixed-voltage actuator before entering the variable-voltage actuator. (b) The percent error between the experimental data and the predicted curve. The error gradually increases. The maximum error is at % which occurs when the applied voltage to the fixed-voltage actuator is highest Figure 6-5: (a) The resultant modeled curves (red) are plotted against the experimental curves (black) using the method in Section 3.7 and the theory in Section (b) The percent error between the experimental data and the predicted curve. The error gradually increases. The maximum error is at % which occurs when the applied voltage to the fixed-voltage actuator is highest Figure 6-6: (a) The resultant modeled curves (blue) are plotted against the experimental curves (black) using partially the theory in Section , supplemented by the nodal SOP as an additional parameter in the prediction process. The signal enters the fixed-voltage actuator before entering the variable-voltage actuator. (b) The percent error between the alternative method of predicting the experimental curve when the applied voltage to the fixed-voltage actuator is changed. The maximum error is 30.73% which occurs when the voltage to the variable-voltage actuator is at its highest Figure 6-7: (a) The resultant modeled curves (blue) are plotted against the experimental curves (black) using partially the theory in Section , supplemented by the nodal SOP as an additional parameter in the prediction process. The signal enters the variable-voltage actuator before entering the fixed-voltage actuator. (b) The percent error between the alternative method xv

16 of predicting the experimental curve when the applied voltage to the fixed-voltage actuator is changed. The maximum error is 12.82% Figure 6-8: The SOP coverage by using three different combinations of actuators as labeled by the cyan, magenta and yellow traces. The DES vectors were taken from experimental data Figure 6-9: Three separate views of the Poincaré sphere for (a) the plane, (b) plane and the (c) plane. The nonoverlapping areas suggest regions where certain actuator combinations cannot reach Figure 6-10: The experimental data for stepping and varying three different combinations of actuators Figure 6-11: The experimental data as collected from operating two actuators at a time for three separate views of the Poincaré sphere: (a) the plane, (b) plane and the (c) plane. The nonoveralapping areas suggest regions where certain actuator combinations cannot reach Figure 7-1: Target SOP as plotted on the Poincaré sphere Figure 7-2: Flow chart for offline polarization control. The algorithm needs to characterize the actuators first for the system parameters before a user can input a target SOP. The algorithm then calculates the necessary angular rotation on which actuators to reach the target SOP. Finally the system converts the angular rotations to voltage values Figure 7-3: Flow chart for the characterization process of one actuator Figure 7-4: SOP evolution for the three actuators as determined from the initial characterization step Figure 7-5: The angular rotation versus applied voltage for the three actuators Figure 7-6: Flow chart that outlines how the algorithm determines the actuator settings to reach the target SOP Figure 7-7: Percent deviation between target SOP and implemented SOP xvi

17 Figure 7-8: Flow chart for online polarization control Figure 7-9: The percent deviation from the final SOP state and the target SOP for the twenty different target SOP in Table 7-1 through the dithering algorithm when the actuators are initially in their reset state Figure 7-10: The number of updates to the applied voltages of the actuators Figure 7-11: The percent deviation from the final SOP state and the target SOP for the twenty different target SOP in Table 7-1 through the dithering algorithm when the actuators start initially from = and = Figure 7-12: The number of voltage updates to the actuators in order to reach their final SOP state Figure 7-13: The flow chart for the combined offline and online algorithm Figure 7-14: The percent deviation from the final SOP state and the target SOP for the twenty different target SOP in Table 7-1 using the combined deterministic and dithering algorithm Figure 7-15: The number of voltage updates to actuators to reach the final SOP state using the combined deterministic and dithering algorithm Figure 7-16: The comparison of (a) the angular deviation between target SOP and final SOP and (b) the number of voltage updates to reach the final SOP of each of the control systems studied Figure A-1: The and vectors as plotted on the Poincare sphere which are used to characterize the system for the DES vectors and the angular response to the applied voltage.. A-2 Figure B-1: System with two polarization rotational elements that transform the input SOP to an output SOP... B-1 Figure B-2: The simplified system with only one system component. All other components have been integrated into the one system block and the input has to be changed to reflect the effect of the existing system component.... B-2 xvii

18 Figure B-3: A system with three rotational polarization components and a new output... B-2 Figure B-4: The effective system block diagram after combining and together from Figure B-3... B-3 Figure B-5: The effective system after integrating into the other two system components. B-3 Figure B-6: The vectors as plotted on the Poincare sphere used to characterize the system for the DES vectors and the angular response to the applied voltage.... B-4 xviii

19 List of Appendices Appendix A: Unidirectional Polarization Control... A-1 A.1. Unidirectional System Calibration... A-1 A.2. Unidirectional System Algorithm... A-4 A.2.1. Polarization Control with One Actuator... A-4 A.2.2.Polarization Control with Multiple Actuators... A-5 Appendix B: System Simplification Techniques... A-1 B.1. Simplification Techniques for Cascaded Components... B-1 Appendix C Bidirectional Geometric Model Derivation... B-1 C.1. Derivation... C-1 Appendix D: Equivalent System Transformation for Nodal SOP Method of System Parameter Extraction... C-1 D.1. Equivalent System Proof... D-1 xix

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21 Chapter 1 Motivation for Polarization Control 1 The oscillation orientation of an electromagnetic wave, or its polarization, has significant effects on the operation of many engineering systems. Optical polarization has pronounced effects on a variety of fiber-optic applications that range from imaging tools to telecommunication systems. Thus, there is a need to control the state of polarization (SOP) in many of these fiber-optic systems. 1.1 Examples of Polarization Control in Optical Systems Polarization Control in Optical Communication Systems In optical communication systems, polarization control has been applied to 1) enhancing the optical signal to noise ratio (OSNR) at the receiver in coherent detection systems [1, 2]; 2) mitigating polarization mode dispersion (PMD) in high bit-rate fiber-optic communications systems [3]; and 3) developing coding schemes to increasing the available capacity of a fiberoptic link [4]. Coherent detection has been important in increasing spectral efficiency within a channel s bandwidth [4]. However, the polarization mismatch between the reference local oscillator (LO) and the incoming signal results in a low mixing efficiency which degrades the OSNR at the receiver end. Hence, it is important to match the SOP of the LO and the incoming signal. In addition to polarization matching for coherent detection schemes, there is also a lot of research going into PMD compensation. With higher bit-rate systems, the power penalty increases due to the mismatch in propagation vectors between the two orthogonal polarization modes resulting in a mean time delay of the pulses as it propagates through fiber [5]. Methods such as polarization 1

22 2 CHAPTER 1 MOTIVATION FOR POLARIZATION CONTROL diversity, polarization scrambling and polarization control have been applied to compensate PMD [6]. Furthermore, there is an increasing interest in polarization division multiplexing. Optical fiber allows for two orthogonal modes of polarization. In an intensity modulated link, signals can be encoded in two modes and transmitted through fiber without interfering with each other [4]. Hence, it is possible to increase the channel capacity by at least a factor of two. From coherent detection, PMD compensation and enhanced coding schemes, polarization control is a very active research area in optical communication systems Polarization Control in Optical Sensing Systems In addition to optical communication systems, many optical sensing systems rely on polarization control such as air turbulence measurements [7], temperature, gas composition, and stress and strain sensing [8][9]. Interferometers used in optical sensing require matched SOPs in order to achieve maximum signal to noise ratio (SNR). For example, local optical turbulence intensity measurements are important for monitoring the fluctuations in the refractive index of the atmosphere which can hinder the performance of ground based imaging and laser systems [7]. These optical turbulence intensity measurements can be done with a fiber based Mach-Zehnder interferometer (see Figure 1-1). Light in one arm of the interferometer is used to sense the atmosphere, while light in the other arm of the interferometer is the reference source. The fluctuating refractive index of the air will phase shift the light in the sensing arm. To determine this phase shift, a piezoelectric fiber stretcher will phase shift the reference signal so that both the sensing light and reference light interfere at the coupler. Polarization control is important in this type of sensing system as the air may also change the polarization of the sensing light. The polarization change can compromise the SNR at the detector. To ensure there is maximum signal contrast, a polarization controller (PC) has to rotate the polarization of one of the signals such that both the sensing and reference light SOPs are matched.

23 1.1 EXAMPLES OF POLARIZATION CONTROL IN OPTICAL SYSTEMS 3 Figure 1-1: A fiber based local air turbulence measurement system [7] Another example of polarization control in optical sensing is the use of frequency shifted interferometry (FSI) for interrogating fiber Bragg grating (FBG) sensor arrays (see Figure 1-2) [8]. In FSI, an acousto-optic modulator (AOM) is placed asymmetrically in a Sagnac interferometer. This configuration allows a user to make measurements at different locations by adjusting the frequency of the AOM, and examining the differential interference signal s intensity at the balanced detector. Like the local optical turbulence measurement system, a PC is present in order to ensure that the SNR is not affected. Unlike the local optical turbulence measurement system, the signal reflected back from the FBG sensors have to pass the PC again before reaching the detector. This system has particular importance to this thesis which will be discussed in Section 1.2.

24 4 CHAPTER 1 MOTIVATION FOR POLARIZATION CONTROL Figure 1-2: A fiber based setup for an optical sensor network that applies frequency shifted interferometry for interrogating a FBG sensor array [8] Polarization Control in Optical Arbitrary Waveform Generation Systems Another application area for polarization control is the design of novel devices. Many devices are polarization sensitive. For example, the diffraction gratings in most free-space pulse shaping systems require a specific SOP in order to diffract light efficiently (Figure 1-3). Figure 1-3: A free space dynamic pulse shaping system that delivers a shaped pulse with fiber optics [10].

25 1.1 EXAMPLES OF POLARIZATION CONTROL IN OPTICAL SYSTEMS 5 The input pulse is delivered to the diffraction grating by fiber which rotates the input SOP, so the signal s SOP has to be adjusted with polarization control in order to effectively separate the different spectral lines to a liquid crystal matrix that is used to shape the pulse [10]. Another pulse shaping system that relies on polarization in its operation is an all-fiber dynamic arbitrary waveform shaping system (Figure 1-4) [11]. Figure 1-4: An all fiber dynamic pulse shaping system. Output waveforms are formed by spectrally controlling the amplitude and phase of the input spectral lines. Amplitude control is a two step process that involves controlling the polarization and then resolving the amplitude through a polarizer [11]. The system uses FBGs as frequency dependent reflectors to separate each spectral line. Each spectral line can be amplitude and phase controlled. Phase control is done with fiber stretchers which change the optical path length that a signal travels between the FBG and the output. Amplitude control is a two-step process. First, the spectral lines have their polarization rotated

26 6 CHAPTER 1 MOTIVATION FOR POLARIZATION CONTROL by a polarization control, and secondly, amplitude resolution is achieved when the signal passes through an inline fiber polarizer. By controlling the fiber stretchers and PCs, any output waveform can be achieved Polarization Control in Fiber Lasers The gain medium in fiber amplifiers and lasers are also sensitive to SOP fluctuations. Rare earth fiber amplifiers and lasers suffer from polarization dependent gain (PDG) which is primarily a result of polarization hole burning (PHB) [12]. To compensate for these effects, PCs are used to compensate for the polarization effects. For example in a ring cavity multiwavelength brillouinerbium fiber laser with a partially reflecting FBG, a PC is used to compensate for the PHB that would result in the erbium doped fiber gain section (see Figure 1-5) [13]. Figure 1-5: A multiwavelength brillouin erbium fiber laser with partially reflecting FBG [13] In addition, mode-locked fiber lasers also require PCs. For example, a PC is used to optimize the output powers for a mode-locked semiconductor laser for the polarization dependent devices such as the electroabsorption modulator which serves as a gate for the pump source and assists in locking the laser modes together, and a polarization dependent semiconductor optical amplifier (see Figure 1-6) [14].

27 1.2 COMMON CONFIGURATIONS FOR POLARIZATION CONTROL 7 Figure 1-6: A fiber based mode-locked laser with a semiconductor optical amplifier as the gain region and electroabsorption modulator as the active modulator inside the resonator [14] The need for polarization control is evident in the number of fiber-optic systems that depend on it. As a result there have been many methods of understanding how polarization evolves in fiber, and how to control the polarization of a signal. 1.2 Common Configurations for Polarization Control Controlling the SOP to a specific target SOP is essentially an automatic control problem, where the use of a feedback system is required to obtain a stable output SOP. Polarization control, polarization stabilization and polarization tracking are interchangeable terms that refer to this automatic control problem [4]. A polarization control system consists of three components: 1) a PC that transforms an input SOP to any arbitrary output SOP; 2) a device to monitor or infer the polarization state; 3) and an intelligent controller that produces the appropriate control signal for the PC based on the SOP of the monitored signal. Figure 1-7 shows a common setup for a polarization control system in fiber optics.

28 8 CHAPTER 1 MOTIVATION FOR POLARIZATION CONTROL Figure 1-7: A typical fiber based configuration for a polarization control system in a unidirectional configuration There are many existing methods for polarization modeling and control for optical communication systems. However, all existing models and approaches have only considered polarization control in a unidirectional fiber-optic system; i.e. lightwave propagates through the PC only in one direction. Another common configuration in optical sensing systems is with PCs in a bidirectional configuration with reflective structures, such as fiber Bragg gratings (FBGs). The signals in these systems will travel through a PC twice such as seen in Figure 1-8.

29 1.3 SCOPE AND GOAL OF THE THESIS 9 Figure 1-8: A polarization control system for a bidirectional configuration The model that describes the effect of polarization control with a reflector has not been studied, hence no control methodologies exists. This configuration with a reflector is important in a number of fiber sensing systems [8] and other optical devices [11], [15]; some of which have already been discussed in the previous section. Therefore, there is a need to develop models for PCs in such bidirectional systems. 1.3 Scope and Goal of the Thesis The goal of this thesis is to address how polarization can be controlled in a bidirectional configuration. To achieve polarization control in a bidirectional configuration, we must also address how the SOP evolution can be modeled in the bidirectional configuration, which has not been done prior to this work.

30 10 CHAPTER 1 MOTIVATION FOR POLARIZATION CONTROL For this work on polarization control, we will not be addressing some of the practical limitations such as reset-free operation and endless polarization control of PCs which have been an emphasis for many previous works [1, 2, 4, 16, 17]. Reset-free operation refers to not resetting the control settings on the PC. A reset operation can perturb the output SOP. Endless polarization control refers to tracking the SOP without being limited by the operational range of the PC. Both of these concepts are more for optical communication systems in which the input SOP is fluctuating on a microsecond to millisecond timescale and the output SOP has to be stabilized [4]. These operational considerations are not within the scope of this work. The scope of this work will include a model that can predict the SOP evolution in a bidirectional configuration consisting of multiple polarization control elements. We will also assess the limits of our model and examine some of the practical limitations such as hysteresis effects and material responses of experimental PCs. In addition, the scope of this work also includes a polarization control system based on our model that can accurately determine the control signals to reach a target SOP. 1.4 Thesis Outline This thesis will cover how polarization control can be achieved in a fiber optic system in the bidirectional configuration. In Chapter 2, we will review some of the existing concepts and methods for modeling polarization in the unidirectional configuration. A control scheme for the unidirectional model will also be presented. From the unidirectional system review, Chapter 3 will detail the model development for the bidirectional system in which one polarization control element is present in the system. Each component in the bidirectional configuration will be modeled. Simulation results for the expected SOP evolution will be shown. In addition, a method for extracting the system parameters from an experimental data will be shown. In Chapter 4, the behavior of the SOP evolution under multiple polarization control elements will be modeled and analyzed. Chapter 5 will present the results for validating the theoretical model through experimental results when one polarization control element is operated. In Chapter 6, the results for multiple control elements will be presented. Chapter 7 will cover some of the methods for polarization control in the bidirectional configuration and experimental results from implementing those methods in the experimental system. Finally, we conclude with Chapter 8 and present prospective research areas that can stem from this work.

31 Chapter 2 Review of Polarization Modeling and Control 2 Polarization control for unidirectional fiber-optic systems has been developed based on the understanding of SOP evolution in such systems [4]. To understand the SOP evolution of a signal propagating in a bidirectional configuration, it is necessary to examine the basic principles of modeling and control in the unidirectional configuration. We will first review the SOP representations used for polarization control. This will be followed by a review on a concept for modeling the SOP. Finally, a method of polarization control in the unidirectional configuration will be discussed. 2.1 SOP Representation There are many methods for visualizing polarization. With Jones representation, the SOP is described by a C column vector that represents the two orthogonal electric field amplitudes of the propagating wave and their relative phase. A birefringent component that transforms an input SOP to an output SOP without any losses can be described by a complex unitary transformation matrix, : = (2.1) Stokes representation is the other common representation for polarization. Stokes parameters are useful for describing the degree of polarization of a signal. Typically, Stokes parameters are represented by R vectors. The first component,, represents the intensity of the lightwave. The other three components,, describe the type of polarization the signal has, which can 11

32 12 CHAPTER 2 REVIEW OF POLARIZATION MODELING AND CONTROL be mapped onto a unit sphere in 3-dimensional space known as the Poincaré sphere as shown in Figure 2-1. Figure 2-1: Poincaré sphere used for mapping Stokes parameters. The 2D representations of the SOP are shown in blue at the major axes to show how they correspond to the SOP mapped on the Poincaré sphere. With polarized light 1, we can ignore the component; only,, matter and we can describe the SOP by a R vector. With polarized light, the transformation matrix from an input SOP to a final output SOP is represented by a R orthogonal rotation matrix,, which will be discussed in detail in Section 2.3. Like many other studies [4], we will be using Stokes space as the Poincaré sphere allows us to map any SOP onto its surface, and the transformation of a SOP from one point on the sphere to another point is easily visualized. Jones vectors can be more cumbersome when an inappropriate choice of bases is used and there is a lack of intuition from the 2-dimensional representation of the SOP. Consequently, Jones representation will only 1 To determine the degree of polarization, we consider the ratio ( + + )/. However, ( + + ) =, since we only consider polarized light. Furthermore, in our analysis,,,, will always be normalized with respect to, such that ( + + ) =1.

33 2.2 PRINCIPAL STATES OF POLARIZATION 13 be reserved for deriving certain relationships which are then converted to Stokes representation. To convert a Jones vector, to a Stokes vector,, we apply the transformation 2 : = (2.2) where is the Pauli Spin vector, =(,, ) (2.3) where the vector is composed of three matrices: = , = , = 0 0 (2.4) The Stokes Space rotation matrix,, can be related back to the Jones transmission matrix,, through the Pauli Spin matrices, in Equation (2.3). = (2.5) where is the Hermitian transpose of Equation (2.1). To develop a control algorithm, typically only two variables are needed to describe any SOP on the Poincaré sphere the azimuth angle and the elevation. This implies that the solution to this control problem is a minimum on a surface representation; i.e. it is a double-variable optimization algorithm [4]. We will show there is a geometrical interpretation to the SOP evolution, and will derive an algorithm from the geometrical interpretation of the SOP evolution in both the unidirectional and bidirectional configurations. However, we will first look at a common method for modeling polarization in fiber optics. 2.2 Principal States of Polarization The principal states of polarization (PSP) are a property of birefringent components, and are useful for describing the SOP evolution through a complex birefringent system. The PSPs are the eigenstates of a birefringent system. When an optical pulse is launched with the same polarization as the PSP, we observe that the pulse is insensitive to polarization mode dispersion 2 The bra-ket notation is only a short form notation

34 14 CHAPTER 2 REVIEW OF POLARIZATION MODELING AND CONTROL (PMD). PMD is a dispersion effect that causes an optical pulse to broaden due to the birefringence of the system and is especially detrimental in high-bit rate systems. Typically, to quantify or compensate the amount of PMD in a system, scientists and engineers try to model how the polarization will evolve in a system [18]. We will be using similar modeling tools and the PSP concept to model the bidirectional configuration. However, to understand why the PSP concept is important as a modeling tool, we will first examine why the PSP exists in physical systems, and how the PSP is related to the system parameters that we will be considering in this work. To model the evolution of the polarization in fiber, the fiber can be thought of as a sequence of random birefringent sections; the birefringence axes and magnitudes change randomly along the fiber. When an input signal s frequency is varied and its SOP is fixed, the output signal s SOP will change according to the varying frequency. For an optical pulse with finite bandwidth, the birefringence will cause the optical pulse to broaden as it propagates through the fiber. However, there is a pair of orthogonal SOPs that are not affected by a change in frequency of the input signal to first order [5]. This pair of orthogonal states is known as the PSP which can be derived from theory. Before we derive the PSP from theory using the approach from [5], we need to mention that the model being considered is an idealized model. The signals are from an ideal monochromatic CW source with a fixed and stable input SOP; there are no losses including coupling losses and polarization dependent losses (PDL) through any components including the fiber; and the system is under a controlled environment such that there are no external stimuli acting on any component. Under the above assumptions, consider a monochromatic optical field,, that is transmitted through fiber which produces an optical field after a certain length of propagation like in Figure 2-2 [5].

35 2.2 PRINCIPAL STATES OF POLARIZATION 15 Figure 2-2: The system block diagram that depicts an optical field as transmitted through a fiber medium and the resulting in an output optical field The relationship between the two fields is: = () () (2.6) where () is the phase accumulation as the wave propagates down the fiber and () is the transmission matrix described by Equation (2.1). Both terms are dependent on the optical frequency. The two complex field vectors, and can be expressed as: = = (2.7) = = (2.8) where,, are the amplitudes and, are the phases of the two fields respectively. and are the 2D complex unit vectors that specify the polarizations of the two fields. Consider an arbitrary but fixed input SOP with a varying frequency. The output SOP will vary with the frequency of the input wave. At an operation frequency, a set of two mutually orthogonal input SOPs exist such that the corresponding output SOPs are independent of frequency to first order. To show this, consider the derivative of the output field with respect to. = (2.9) The primes represent differentiation with respect to frequency. In addition to Equation (2.9), consider the derivative of the field as related to the input field.

36 16 CHAPTER 2 REVIEW OF POLARIZATION MODELING AND CONTROL We can then combine both Equation (2.9) and (2.10), = (Φ + ) (2.10) = ( ) (2.11) where is: = + Φ 1 (2.12) We set / of Equation (2.11) to zero to determine the conditions for a nonchanging output SOP. ( ) =0 (2.13) This is an eigenvalue problem. The eigenvalues can be expressed in terms of the transmission matrix components: ± =± + (2.14) and the corresponding eigenvectors are: ± ± = 2 ± ( ± Im( + ) ± 2 ± ( ± Im( + ) (2.15) where is an arbitrary phase and ± are the input eigenvectors resulting in nonchanging output SOP states to first order. ± are also known as our input PSPs. If we launch a wave with polarization or, and vary the frequency, the output SOPs will not change to the first order. To determine the output SOP eigenvectors, we use Equation (2.6) and use the input PSPs as our input fields. Furthermore, we can extend the PSP concept for Jones space into Stokes space with the Jones to Stokes conversion method described in Section 2.1. = (2.16) so, the components of is [19]:

37 2.2 PRINCIPAL STATES OF POLARIZATION 17 = 2 + = 2 Im = 2 Re (2.17) (2.18) (2.19) where is the differential group delay (DGD) given by: =2 + (2.20) However, typically we do not have the transmission matrix for the birefringent components, which means we cannot express the PSP in terms of the elements of the transmission matrix. We will show that the PSP physically manifest itself in the results when we characterize a system with a component that varies the birefringence. The physical importance behind the PSP is with long fiber sections that have a randomly changing birefringence. There still exists a set of orthogonal SOPs that do not result in pulse broadening. The PSPs depend on the collective birefringence of the fiber medium. Furthermore, the PSP will change when additional birefringent components are attached to the fiber section [18]. It is important to note that with multiple birefringent components, there will still be only one pair of PSPs associated with the optical frequency of the input lightwave. The importance of these results to our model is when the PSP is generalized. The output SOP is not only dependent on the optical frequency, but also on other physical parameters such as temperature and pressure that would change the birefringence [20]. Consequently, generalized PSP can be found from varying physical parameters other than frequency. The generalized PSP are also called the dynamic eigenstates (DES) and will be the term used in this thesis, as we are studying the effects of how the output SOP changes with respect to physical control variables that directly alter the birefringence. Similar to the PSP, there is only one pair of DES vectors associated with one varying physical parameter. For a system with multiple controllable birefringent components (e.g. polarization controllers), there is one pair of DES vectors

38 18 CHAPTER 2 REVIEW OF POLARIZATION MODELING AND CONTROL associated with the control parameter of each of the controllable birefringent components, provided only one control parameter is varied at a given time. We can model these birefringent components with a transmission matrix that depend on the DES which will be discussed in the next section. 2.3 DES Transmission Matrix Approach to Polarization Evolution The DES approach allows us to specify a transmission matrix in Stokes space of how an input SOP is transformed to an output SOP. A birefringent component can be modeled as a system block as seen in Figure 2-3: Figure 2-3: Basic system transmission matrix that shows how the PC alters an input SOP, to an output SOP through a transmission matrix. The system transmission matrix depends on a control variable which is an applied voltage for a piezoelectric actuator that will alter the birefringence of the fiber. The input SOP is related to the output SOP by a transmission matrix : = (2.21) describes a rotation of the input SOP about the DES vector of the system, :

39 2.3 DES TRANSMISSION MATRIX APPROACH TO POLARIZATION EVOLUTION 19 ((), ) cos + (1 cos ) (1 cos) + sin (1 cos) sin = (1 cos) sin cos + (1 cos) (1 cos) + sin (1 cos) + sin (1 cos) sin cos + (1 cos ) (2.22) where,, are the components of one of the DES vectors denoted by. is dependent on the changes to the local birefringence and it relates directly to the trajectory of the output SOP; i.e. depends on the control variable,, that is applied to the birefringent component. The local birefringence of the fiber can be changed through an applied stress such as that from a piezoelectric actuator. If the output SOP was mapped onto the Poincaré sphere, the trajectory of the output SOP appears to follow a circular path (Figure 2-4) in response to the applied stress. The axis of rotation corresponds to two diametrically opposed vectors which are the DES vectors [18]. When the output SOP is aligned with either of the DES vectors, the output SOP will not change to first order when the control parameter is varied. If the output SOP is not aligned with the DES vectors, the output SOP will rotate around the DES vectors. Maximum change will occur when the output SOP is orthogonal to the DES vectors; a small change in will result in a large change in the output SOP.

40 20 CHAPTER 2 REVIEW OF POLARIZATION MODELING AND CONTROL Figure 2-4: The evolution of the output SOP is a circular trace on the surface of the Poincaré sphere when a field propagates through a single polarization control element that varies the local birefringence of a unidirectional system. Equation (2.21) described how an input SOP will change when it passes through a single birefringent component. Any birefringent component such as waveplates and PCs can be described by the transmission matrix. Additional birefringent components can be connected in series to form a birefringent network. A black-box model for such a birefringent network can be developed to model the relationship between the input SOP and output SOP. For example, we mentioned that Equation (2.21) describes the SOP relationship at the immediate input and output of the PC. However, in practice, the immediate input and output are usually not accessible. Instead, the output SOP is typically measured with a polarimeter which is connected to the output port of the PC via a certain length of fiber as shown in Figure 2-5.

41 2.3 DES TRANSMISSION MATRIX APPROACH TO POLARIZATION EVOLUTION 21 Figure 2-5: A cascaded birefringent network that represents how the measured output is related to the immediate output of the PC through a fixed fiber rotation. The relationship between a fixed input SOP, and the output SOP measured at the polarimeter, becomes: = (2.23) where can also be characterized by a rotation matrix. The input SOP in a practical system is generally not known and not measured; therefore, it is more convenient to use the initial output SOP, as a reference SOP to mark the starting point of the output SOP evolution. corresponds to the output SOP when the PC is inactive, i.e. =0 in Equation (2.22) which results in an identity matrix for. Therefore, = (2.24)

42 22 CHAPTER 2 REVIEW OF POLARIZATION MODELING AND CONTROL Figure 2-6: At the immediate output of the PC, is the red circular trace. The blue circular trace is the measured SOP at the polarimeter and is denoted. The blue circular trace is rotated form of the red circular trace due to the fiber connecting the PC to the polarimeter. The measured SOP is just a rotation from the initially measured output SOP (see Figure 2-6). The rotation matrix that corresponds to the rotated circular trace is: = (2.25) is the rotation matrix that includes the SOP rotation caused by. The relation between and is just a change in the rotation axis: (, ) = (, ) (2.26) Equation (2.25) indicates that the evolution of the SOP can be determined from the reference SOP, and the rotated central axis, only. The system described in Figure 2-5 can be effectively simplified to:

43 2.4 A GEOMETRICAL APPROACH TO POLARIZATION EVOLUTION 23 0 Figure 2-7: The effective system block in which the fiber rotation is accounted for in the effective rotation matrix and the effective input SOP,. Note, is the initial output SOP, which can also be treated as the effective input to the simplified system shown in Figure 2-7. A more detailed discussion on combining system components is discussed in Appendix B. Zhengyong et al [21] used a basic DES transmission matrix method to model the SOP evolution in a system with a piezoelectric polarization controller (PPC). The use of the DES allowed them to accurately predict and control the output SOP to any other output SOP with an average error deviation of 1.51 on the Poincaré sphere. Furthermore, their deterministic method allowed them to accurately reach the output SOP within 50 without any feedback loop. Thus, the DES transmission matrix method allows us to model the SOP evolution as a blackbox with multiple birefringent components and the method has been experimentally proven for polarization control. A more intuitive model that takes advantage of the SOP behavior and will complement the DES transmission matrix model is shown in the next section. 2.4 A Geometrical Approach to Polarization Evolution One of the observation that researchers have used [21, 22] to control the polarization is that the SOP evolves as a circle on the surface of a Poincaré sphere when a PC varies the birefringence (Figure 2-4). A geometric model can be developed to describe this circular evolution of the SOP on the Poincaré sphere. This geometric model and the DES model are essential in describing the SOP evolution in the bidirectional system, and both models are used to determine the system parameters that characterize the bidirectional system. In this section, the geometric model will

44 24 CHAPTER 2 REVIEW OF POLARIZATION MODELING AND CONTROL be developed for the unidirectional system, which can then be expanded to model the bidirectional system. As mentioned, a circular trace is formed on the Poincaré sphere when a polarization control element varies the birefringence. The DES vectors manifest themselves as the central axes of rotation of the SOP evolution. In order to obtain, the central axis is determined from the circular trace which will form a plane in Stokes space. Hence, is obtained by taking the cross product of any two vectors in that plane (Figure 2-8). = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (2.27) Once the DES vector is determined, the output SOP can be modeled by: () = cos +, sin + (2.28) where = ( ) (2.29) which represents the center vector of the circle. The is the inner product operation. is the initial output SOP. In other words, the center vector is determined by the projection of the initial output SOP, on to the DES vector,. in Equation (2.28) is a vector defined by: = (2.30) which is a vector pointing from the center vector to the initial output SOP,., is a vector perpendicular to :, = (2.31)

45 2.5 GEOMETRIC DESCRIPTION OF THE OUTPUT SOP FOR MULTIPLE CONTROL ELEMENTS 25 Figure 2-8: The geometrical representation of the output SOP and the corresponding vectors. Equation (2.28) describes the SOP evolution from the initial reference SOP. A target SOP can be determined to be reachable by a polarization control element, if it lies on the trace formed by the polarization control element. In fact, a target SOP,, is reachable if it lies on the same plane as the SOP circle; i.e.: = = ( ) (2.32) This statement says that if both the target SOP and the initial reference SOP have the same center vector, then a polarization control element can transform the output SOP to reach the target SOP. By using the initial reference SOP, from the output SOP trace, there is no need for knowledge of the input SOP,. However, if the target SOP does not lie on the SOP trace, then more than one PC element is required to reach the target SOP.

46 26 CHAPTER 2 REVIEW OF POLARIZATION MODELING AND CONTROL 2.5 Geometric Description of the Output SOP for Multiple Control Elements Controlling the output SOP to any state on the Poincaré sphere requires usually 2 to 3 polarization control elements. A PC typically consists of multiple actuators or dynamic waveplates that control the birefringence of the fiber. By controlling each actuator or waveplate independently, they will each form a different circle on the Poincaré sphere. Wang et al [22] suggests that only two circles are needed to go from one SOP to another SOP (Figure 2-9) and with the use of coordinate system transformation, they can relate the input SOP to the PC to the output SOP of the PC. Their method allows them to move from one output SOP to any arbitrary output SOP within 50ms. However, we have already shown that coordinate system transformations are not necessary for controlling the output SOP to any other arbitrary output SOPs and the knowledge of the input SOP is not required. All we need is knowledge of the initial output SOP,, which is obtainable from output monitoring. Figure 2-9: The method used by Wang et al demonstrating how polarization control can be achieved. The two circles are the traces formed by the waveplates in the commercial PC. is the input SOP and is the required output SOP, while is an intermediate SOP. After a coordinate system transformation, represents the actual output SOP state at the output of their polarization control system. [22]

47 2.5 GEOMETRIC DESCRIPTION OF THE OUTPUT SOP FOR MULTIPLE CONTROL ELEMENTS 27 As already mentioned, when each polarization control element is operated independently, the response of the system to the control signals will yield different circular traces with different DES vectors that characterize each circle. Thus, each control variable has a pair of output DES vectors associated with it. When multiple control elements are operated simultaneously, the DES might be changing dynamically. Instead, we model how the DES of each control variable is affected by considering the operation of one control variable at a time. To see this more clearly, consider two piezoelectric actuators, actuators A and B, connected in series (Figure 2-10). They change the birefringence of the fiber when a voltage is applied to the actuator. Figure 2-10: The system block diagram for a system with two piezoelectric actuators, where is the DES that characterizes actuator A, and is the DES that characterizes actuator B. A is the actuator closer to the input while B is the actuator closer to the output. There are two actuator states to be considered: fixed and variable-state. The actuator in the fixed state (referred to as the fixed-actuator) will only change the initial output SOP to another output SOP on its trace. The actuator in the variable state (referred to as the variable-actuator) will sweep through an operational range to yield a circular trace on the Poincaré sphere. With these two states, there are two cases to be considered: 1) when the fixed-actuator is actuator A and the variable-actuator is actuator B, and 2) when the fixed-actuator is actuator B and the variable-actuator is actuator A. In both cases, the fixed-actuator is operated first followed by the variable actuator. In the first case, actuator A is the fixed-actuator and the variable-actuator is actuator B. Each time the operational setting of the fixed-actuator is changed, a new circular trace is formed by the variable actuator. The traces formed will yield concentric circular traces that have the same DES vector (Figure 2-11a). The DES vector associated with actuator B is not affected when actuator

48 28 CHAPTER 2 REVIEW OF POLARIZATION MODELING AND CONTROL A is operated first. Actuator A only changes the input to actuator B. The response of the system will still yield circles in which the DES vector is the same. In the second case, actuator B is the fixed-actuator and the variable-actuator is actuator A. When we sweep through an operational range of actuator A, the circular SOP traces formed are not concentric to that of its initial circular trace (Figure 2-11b). Instead, it appears that the DES to the system is changing. Nonetheless, the actual DES of that control parameter has not changed, but due to the new birefringence in the fiber section as applied by actuator B, the DES of A has appeared to have rotated. (= 4 ) (= 3 ) (= 2 ) (= 1 ) (a) (b) Figure 2-11: The output SOP trajectories as a result of using multiple actuators. The trajectories (a) are concentric when the actuator closer to the input is operated first followed by a sweep of the birefringence by the proceeding actuator while (b) the trajectories do not share the same DES when an actuator closer to the output is operated first followed by a sweep of the birefringence by the actuator closer to the input. The output circular traces that are observed in (Figure 2-11b) are not random; they can be modeled. The DES of each circular trace in (Figure 2-11b) is different from each trace. In fact, the trace formed by the tips of the DES vectors all output SOP circles show in (Figure 2-11b) is another circle, rotating around the DES of actuator B as shown in Figure 2-12 if a trace is made with the DES of each output circle observed in Figure 2-11b, the centre of rotation of that circle would be the DES of actuator B.

49 2.5 GEOMETRIC DESCRIPTION OF THE OUTPUT SOP FOR MULTIPLE CONTROL ELEMENTS 29 (= 3 ) (= 4 ) (= 2 ) () (= 1 ) (=0) = Figure 2-12: The DES of each circular trace from Figure 2-11b) (colored lines) form part of a larger circle, (), that models the new effective DES vector. The center of rotation of the larger circle, represents the DES of actuator B which is the actuator that is closer to the output. Furthermore, by applying a set voltage to actuator A, the angular change that is made is proportional to the rotation of actuator B s DES from its original position. To model this, the same model that was derived for the output SOP is used. The center of rotation of the DES vectors is first determined = (2.33) Using this center vector, the same procedure that was used to determine the new SOP when the voltage was varied to an actuator is applied to find the new DES that actuator A will appear to have. = (2.34)

50 30 CHAPTER 2 REVIEW OF POLARIZATION MODELING AND CONTROL, = (2.35) () = cos +, sin + (2.36) Equation (2.36) is used to find the new SOP, which then can be used in Equation (2.28). This method is applied only to determine the change of the DES of an actuator if the proceeding actuators are already operational. A target SOP that is not on any of the initial actuators SOP trace can still be reached with two actuators. The trajectory that the output SOP follows to reach a target SOP can be calculated by modeling the SOP traces that would be formed by each actuator. For example, in Figure 2-13, the red circular SOP trace is modeled using the initial output SOP,, and the DES associated with actuator A,, using the method for obtaining Equation (2.28) Another circular SOP trace (blue) is modeled using the target output SOP,, and a separate actuator, actuator B, with DES. A pair of SOPs, and, characterize the intersection points between the two modeled circles. If we form a trajectory from the initial output SOP,, to either pair of SOPs, or, then we can then use actuator B to reach the target SOP,. The target SOP,, is not reachable from a reference SOP,, when no intersection points exists between the two circular SOP traces.

51 2.6 DETERMINISTIC CONTROL ALGORITHM FOR POLARIZATION CONTROL 31 0 Figure 2-13: The SOP traces formed by two actuators. The red is the trace formed from the initial output SOP, and the actuator s DES,. The blue trace is the trace formed from the target output and the actuator s DES,. The black trace is the trajectory that the SOP must be transformed along to go from the initial SOP to the target SOP. 2.6 Deterministic Control Algorithm for Polarization Control The algorithm that was developed in this work, which utilized the results of the DES concept, is shown in the following flowchart.

52 32 CHAPTER 2 REVIEW OF POLARIZATION MODELING AND CONTROL Figure 2-14: An example algorithm for unidirectional polarization control There are three main modules in this unidirectional polarization control system: calibration, algorithm and implementation.

53 2.6 DETERMINISTIC CONTROL ALGORITHM FOR POLARIZATION CONTROL 33 The calibration module is to determine the behavior of the actuators. In a PPC, there are typically three piezoelectric actuators. Each actuator will rotate the output SOP around a different DES. The calibration step is to determine the DES that characterizes each actuator. After the calibration is done, the user can input their target SOP. Once the user has entered their target SOP, the algorithm runs to determine the angular trajectory of the SOP from its initial state to the target state. The algorithm is divided into two steps. The first step is to determine if only one actuator can reach a target SOP. This can be done by applying Equation (2.32). If this step shows that none of the actuators can reach a target SOP, then two actuators are used instead. The algorithm runs to determine the two SOP circles that need to be formed for the SOP to travel from the input SOP to the target SOP (see Figure 2-8). The third module is to translate the angular trajectory into voltages necessary for the PPC. The voltages are then applied to the PPCs, which will then transform the initial SOP to the final SOP. Combining the three modules, we can demonstrate a unidirectional polarization control system that can accurately reach the target SOP within 2.7%. The calibration and implementation module can be customized for any physical polarization control system while the algorithm is universal to any unidirectional polarization control system. For a more detail explanation of how the algorithm and the implementation modules were implemented, see Appendix A. With the basic description of how the unidirectional configuration can be modeled and controlled, we can now apply those same modeling tools to the bidirectional configuration.

54 Chapter 3 Bidirectional Model with One Control Element 3 The unidirectional system provided modeling tools that can be extended to the bidirectional configuration. To develop a model for the bidirectional configuration, we will in this Chapter focus on a system (shown in Figure 3-1) with one polarization control element. A model for the system consisting of multiple control elements will be developed in Chapter System Model and Considerations Figure 3-1: The basic system with one polarization control element. The input signal propagates through the fiber into the PC. The signal exits the PC into the fiber and travels to the FBG. The FBG reflects the signal back to PC where it exits the system. Each of the physical components in Figure 3-1 will be modeled as a system block and an equivalent transmission matrix. A signal with input SOP,, propagating through a PC with reflections in the system can be visualized as a system block diagram Figure

55 3.1 SYSTEM MODEL AND CONSIDERATIONS 35 Figure 3-2: The system block diagram of a signal propagating through a PC with an FBG The unidirectional model was developed under ideal conditions as stated in Section 2.2. Those assumptions are also applied to the bidirectional model. Some additional assumptions include 1) the FBG is a frequency dependent mirror that is completely reflective and is not polarization dependent; and 2) the PC is an ideal piezoelectric actuator that only alters the birefringence on a point on the fiber. The system block diagram in Figure 3-2 can be difficult to analyze, so another approach is to treat the FBG as a system component with a transmission matrix [23]. The components in front of the FBG can be mirrored and the signal can be visualized as propagating through the system in one direction instead of two as seen in Figure 3-3. Figure 3-3: Equivalent system block diagram detailing how an output signal is transformed as it propagates through each component. Note: there is only one actuator but the signal passes through the actuator twice in different directions. This method allows us to apply a transmission matrix approach to the cascaded system. Even though Figure 3-3 depicts a signal passing through another PC and another fiber section, it should be noted that physically these two components are still the same. However, the

56 36 CHAPTER 3 BIDIRECTIONAL MODEL WITH ONE CONTROL ELEMENT transmission matrices for these components in the forward and reverse directions are different, which will be discussed in the next section. With the aid of the system block diagram in Figure 3-3, the output SOP, in relation to the input SOP can be described as: =,,,, (3.1) Like the unidirectional system, in order to measure the output SOP,, a polarimeter is used which means that an additional fiber section is connected to the output of the PC as seen in Figure 3-4: Figure 3-4: In addition to the system components, a polarimeter is used to measured the output SOP. In order to measure the output SOP, fiber is connected between the polarimeter and the PC which will rotate the output SOP, to an SOP,. The fiber connecting the polarimeter to the output of the PC will rotate the output SOP. Therefore the measured SOP, is related to the output SOP by: =, (3.2) In the next section, the transmission matrix for the reflective element will be discussed, followed by a discussion on the relationship between the forward and backward components. 3.2 Transmission Matrix of the FBG The FBG is a fixed optical component and can be represented with a transmission matrix. Since the FBG acts like a mirror, it can be described with the transmission matrix for a mirror. From [23], the Jones matrix that describes the effect of a mirror is: = 1 0 (3.3) 0 1

57 3.3 RELATIONSHIP BETWEEN FORWARD AND REVERSE TRANSMISSION MATRICES 37 Here, we assume 100% transmission of the FBG. Even if the FBG transmission is less than 100%, as long as the polarization dependence on transmission is negligible, one can still use Equation (3.3) with a scaling factor, which would not change the result derived in this section other than the scaling factor. So, for the purpose of our theoretical model, the assumption that the FBG is a perfect mirror is reasonable. Furthermore, this assumption simplifies the development of the model. In order to apply the transmission matrix of the mirror in this model, Equation (3.3) needs to be converted to Stokes Space. Equation (2.5) is applied to : =0 1 0 (3.4) Relationship between Forward and Reverse Transmission Matrices As mentioned, the matrices that describe the forward and reverse transmission matrices of the PC,,, and the forward and reverse transmission matrices of the fiber,,, is in the form of Equation (2.22). As it was alluded to Section 3.1, the transmission matrices in the forward and reverse propagation direction are different. However, there is a method to relate the transmission matrix in the forward direction to the reverse direction.,,,,,,,, Figure 3-5: Forward and reverse transmission in Jones Formulation The direction of propagation and its effect on the transmission matrix has been studied in Jones space [23]. In Jones space, the transmission matrix s effect on an input SOP in the forward direction is:,, =,, (3.5)

58 38 CHAPTER 3 BIDIRECTIONAL MODEL WITH ONE CONTROL ELEMENT where and describe the and components of the electric field in the forward direction and the subscripts and denote that the components are the input and output Jones vectors respectively. is the Jones transmission matrix. In the reverse direction, the transmission matrix does not have to be changed provided the order of operations follows this [19]: [,, ] =[,, ] (3.6) where and describe the and components of the electric field in the reverse direction. The components that describe the field are a row vector instead of a column vector. To put this back in the traditional order of operations, a transpose 3 of Equation (3.6) is applied.,, =,, (3.7) Hence, the transmission matrix in the reverse direction is just the transpose of the transmission matrix in the forward direction. Consider the following forward transmission matrix with the following components: = (3.8) with DES, = (3.9) To obtain the transmission matrix for the reverse direction in Stokes space, we convert (3.8) to Jones space through Equation (2.5) and determine the reverse transmission matrix in Jones space. After converting back to Stokes space, the transmission matrix for the reverse directions is: = (3.10) 3 Note: This is not the Hermitian transpose but the normal matrix transpose

59 3.4 SYSTEM SIMPLIFICATION 39 Without requiring a conversion from Stokes space to Jones space to determine the new reverse transmission matrix for an optical component, a transformation in Stokes Space that describes the new transmission matrix was determined from inspection as seen: = (3.11) As a result, the DES of the backward transmission matrix is slightly different. Analyzing the elements of Equation (3.10) and relating it back to the Equation (3.8), it can be shown that the DES is: = (3.12) With the knowledge of each component in Equation (3.1), it is possible to describe how the system will react to any input SOPs. In addition, this model can be further simplified using the techniques discussed in the next section. 3.4 System Simplification Though it is possible to determine the output SOP using the 3 3 Stokes matrices and the relationship found in Equation (3.1), it can be difficult to use this approach for polarization control, especially when multiple actuators are involved. A couple of simplifications are made to the model in Figure 3-4, from our understanding of the system components and the techniques that are described in Appendix B. The overall effect will be that all fixed-rotation components will be combined into the variable-rotation components leaving a system block diagram that looks like:

60 40 CHAPTER 3 BIDIRECTIONAL MODEL WITH ONE CONTROL ELEMENT 1 = 2 = Figure 3-6: The simplified system block diagram with the effects of the mirror included into the first actuator and the input where,. =, (, ) =,, (3.13) (3.14) and,, =, (, ) (3.15) = (3.16) and the new effective input to the system is: =,, (3.17) which is also the initial output of the system; i.e. when, and, are inactive and represented by an orthogonal matrix. Note, is an experimentally obtainable parameter. The derivation of Equations (3.13) - (3.17) is based on the simplification technique described in Appendix B. When simplifying the transmission matrix relationships, the effect of fixed-fiber rotations can be effective taken account of by rotating the DES vectors of the matrices. To account for the effects of,,,, and that precede,, we rotate by those rotation matrices to

61 3.5 GEOMETRIC MODEL FOR BIDIRECTIONAL CONFIGURATION 41 yield (Equation (3.14)). Similarly, since precedes,, we account for by rotating to yield (Equation (3.16)). With these simplifications, the evolution of the output SOP,, from a reference SOP, can be described by =, (, ), (, ) (3.18) The relationship between and the control variable is known, and and can be found using the geometric model described in the next section. Therefore,, can be predicted through Equation (3.18). 3.5 Geometric Model for Bidirectional Configuration In this section, we develop a geometric model that can describe the SOP evolution in a bidirectional configuration. Also, in this section, the DES parameters, and and the output have been relabeled as, and respectively (see Figure 3-6). When a sweep of, from =0 to =2 is applied, the output SOP,, traces a circle on the Poincaré sphere with a rotational axis along the DES similar to Figure 2-8 and can be described with Equation (2.28). The output is then used as the input to the next actuator,,. Therefore, Equation (2.28) is used to model but with as the input and as the new DES of, to get. () = cos +, sin + (3.19) where, () = ( () ) () = () (), () = () () (3.20) (3.21) (3.22) Since, each of the individual parameters of is a function of, a further expansion can take place (see Appendix C for the derivation). The final equation will have the form: () = cos 2 + sin 2 + cos + sin + (3.23)

62 42 CHAPTER 3 BIDIRECTIONAL MODEL WITH ONE CONTROL ELEMENT where: = 1 2 ( ( ) ( ) + ( )( ) ) (3.24) = 1 2 ( + ) ( )( ) ( ) = ( ) 2( )( ) + ( ) = ( )( ) +( ) (3.25) (3.26) (3.27) = ( )( ) ( ( ) ( ) + ( )( ) + ) (3.28) Also, there are three main frequency components associated with Equation (3.23) one at 2, and at DC if we take = (3.29) where relates to the applied step voltage to the PC and is a constant that converts the applied voltage to angle.

63 3.6 GEOMETRIC MODEL PROPERTIES AND SYSTEM PARAMETERS 43 The general form of Equation (3.23) can be visualized as a lemniscate or a limacon [24] that has been projected onto the surface of a sphere as seen in Figure 3-7. (a) (b) Figure 3-7: Sample SOP traces from the bidirectional system configuration that resemble a (a) lemniscate-type shape and a (b) limacon-type shape. Thus, Equation (3.23) is a geometrical method that allows us to explain the behavior of the SOP under a bidirectional configuration. This model also allows us to experimentally extract the DES and determine if a point is on the SOP trace, which will be explained in the next section. 3.6 Geometric Model Properties and System Parameters Each of the individual frequency components of Equation (3.23) can be plotted in Stokes space as seen in Figure 3-8.

64 44 CHAPTER 3 BIDIRECTIONAL MODEL WITH ONE CONTROL ELEMENT 2() = 1 cos sin cos + 2 sin + 1 cos sin 2 1 cos + 2 sin Figure 3-8: The individual frequency components of Equation (3.23) being plotted onto the Poincaré sphere. The fundamental harmonic forms an ellipse (blue), while the second harmonic forms a circle (red) and the DC component is a vector (green). The curve,, (black) can be constructed from the sum each of the individual components. From inspection, the component at the second harmonic, 2, forms a circle; the component at the fundamental harmonic,, forms an ellipse; and the DC component is a vector. All three components are in Stokes space, but they do not lie on the Poincaré sphere. Instead, the sum of all three components will generate the polarization curve that will lie on the Poincaré sphere. The ellipse at the fundamental can be thought of as the envelope of the curve upon which the circular component of the 2 nd harmonic rotates around to form the general shape. The DC component can be considered the center of mass of the entire shape and is what puts the entire shape onto the surface of the Poincaré sphere.

65 3.6 GEOMETRIC MODEL PROPERTIES AND SYSTEM PARAMETERS 45 1 cos sin Figure 3-9: The second harmonic component of Equation (3.23) plotted in Stokes Space. and are equal in magnitude and are orthogonal, and thus form a circle. The rotational axis to the circle is. One of the interesting aspects of the second harmonic component is the cross product of and of Equation (3.23) gives the direction of the DES, : = (3.30) To get, we make use of the model in Figure 3-6 and Equation (3.18). Rearranging Equation (3.18):, (, ) =, (, ) = (3.31) which is really the output of the first system block,. The resultant curve from applying this operation is seen in Figure 3-10.

66 46 CHAPTER 3 BIDIRECTIONAL MODEL WITH ONE CONTROL ELEMENT 1 1 Figure 3-10: When multiplying the output curve with the rotation matrix with a DES of, and the resultant curve is the output of the first system block diagram. By determining the central axis of the circle by taking the cross product of two vectors in the plane formed from the circle will yield. If we need to determine if an SOP, is on the SOP trace, then we will need to investigate some properties of. We first look at the dot product of Equations (3.24) to (3.28) with. Applying these operations would result in some of the terms to cancel out. First, we will examine the output projected onto : = cos 2 + sin 2 + cos + sin + (3.32) Simplifying the results: = cos + sin + (3.33) Rearranging Equation (3.33):

67 3.7 FROM EXPERIMENTAL DATA TO THEORETICAL MODEL 47 =cos tan (3.34) + To test if is on, we substitute into in Equation (3.34). However, from the math, Equation (3.34) will yield two possible solutions for on the interval [0,2]. In order to check whether is on the SOP trace,, we substitute both solutions of into Equation (3.23). If both yields SOPs that are different from the SOP,, then does not lie on the SOP trace,. If one of the solutions yields the target SOP, then lies on the SOP trace. 3.7 From Experimental Data to Theoretical Model To arrive at Equation (3.23) requires knowledge of the initial output SOP and the DES. Therefore, starting in reverse from the experimental data to obtain the initial output SOP and the DES is necessary; i.e. it is necessary to characterize the system. The experimental data consists of a set of SOPs, where each SOP is characterized by the three Stokes parameters. Each SOP is measured when the control parameter to a PC is varied. The initial output SOP is available from the dataset when the PC is in its initial state, but the DES vectors are not available which are essential in describing the SOP evolution, especially when multiple actuators are used. To obtain the DES vectors, Equation (3.23) is written in its equivalent form first:, () = cos(2 +, )+ cos +, + (3.35) where is the,, components of the vectors that correspond to the,, Stokes parameters and, = =,,, +, +, +, (3.36)

68 48 CHAPTER 3 BIDIRECTIONAL MODEL WITH ONE CONTROL ELEMENT = =,,, +, +, +, (3.37) tan,,, =, = tan,,, tan,, (3.38) tan,,, =, = tan,,, tan,, (3.39) and remains the same as Equation (3.23). Equation (3.23) and Equation (3.35) are equivalent. Equation (3.35) is used instead since it is obtainable from the experimental data. The data can be split into three separate datasets based off the three Stokes components. Then, an FFT is applied to each individual dataset. This will reveal three frequency components one at DC, one at the fundamental frequency component, and another at the second harmonic, 2 as seen in Figure 3-11.

69 3.7 FROM EXPERIMENTAL DATA TO THEORETICAL MODEL X: Y: Amplitude, T x X: 0 Y: X: Y: Frequency (Ω) 2 (a) X: Y: X: Y: Phase, T x X: 0 Y: Frequency (Ω) (b) Figure 3-11: A 4016 point FFT of the first Stokes Component of the curve shown in Figure 3-8 resulting in (a) the amplitude, and (b) the phase. To obtain each component,, and, the amplitude information of the FFT is used in the following way: =2 (2), =2 (2), =2 (2) (3.40) =2 (), =2 (), =2 () (3.41) = (0), = (0), = (0) (3.42)

70 50 CHAPTER 3 BIDIRECTIONAL MODEL WITH ONE CONTROL ELEMENT where,, are the FFT of the individual Stokes component. and are obtained directly from the phase data of the FFT plots., = (2),, = (2),, = (2), = (),, = (),, = () (3.43) (3.44) With each of the parameters in Equation (3.35) determined, the output SOP can be described. To obtain DES, and are used to get and : cos2 +, = cos, cos(2) sin, sin(2) (3.45) Where =,,, = cos, (3.46), = sin, (3.47) And by applying Equation (3.30), can be obtained. Equation (3.31) can be used to obtain and since we already have the initial output SOP from the experimental data, the system can thus be modeled by applying the transmission matrix model and with the necessary parameters extracted from experimental data using the geometric model.

71 Chapter 4 Bidirectional Model with Multiple Control Elements 4 The basic bidirectional system with one polarization control element was completely modeled with the DES transmission and geometric model in Chapter 3. However, typically a PC has more than one polarization control element. In this section, we will be considering a PC with multiple actuators and how multiple actuators will affect the SOP evolution. 4.1 Two Actuator Model To understand the effect of how multiple actuators affect the output SOP curves, we only need to consider two actuators at a time like we did in Section 2.5. One actuator will be applied with a fixed voltage (i.e. 0) and will be referred to as the fixed-voltage actuator. The other actuator will be swept through its controllable range = [0,2] which will generate all available output SOPs that can result from that actuator; i.e. this actuator creates a bidirectional output SOP trace as was seen in Figure 3-7. This actuator will be referred to as the variablevoltage actuator. Operating two actuators at a time has two effects: 1) it affects the effective input SOP and 2) it affects one or both of the effective DES vectors of the variable-voltage actuator depending on the physical order of the actuators. The goal will be to simplify the system by integrating the fixed-voltage actuator s effect into the rest of the system Cascaded System Effect on the Effective Input SOP The fixed-voltage actuator will affect the effective input, when the rotations caused by the fixed-voltage actuator are being integrated with the rest of the system. Consider the case where the variable-voltage actuator is initially set with =0; i.e. the variable-voltage actuator s forward and reverse transmission matrices will be the identity matrix. As a result, the order of 51

72 52 CHAPTER 4 BIDIRECTIONAL MODEL WITH MULTIPLE CONTROL ELEMENTS the fixed-voltage actuator and the variable-voltage actuator does not matter. In addition, the effective system will only have one active actuator as seen below: 0 0, 1, 2 Figure 4-1: The effect of the fixed-voltage actuator on the effective input,. represents the fixed angular setting on the actuator. and are the DES vectors of the fixed-voltage actuator. All other actuators are inactive; i.e. the other actuators rotation matrices are the identity matrix. is the new effective input after passing through the fixed-voltage actuators. Thus, the new effective input, to the system is: = (4.1) Furthermore, will be observed at the polarimeter as the initial output when the variablevoltage actuator is inactive. In fact, if we examine the SOP trace that would form if we performed a sweep of over the range of [0,2], then we will observe that this is just the SOP trace that would be formed by the fixed voltage actuator Cascaded System Effect on DES Vectors As in the unidirectional model, the order of the actuators matter when we consider how the rotations of the fixed-voltage actuator affect the DES vectors. Therefore, the second effect of multiple actuators in rotating the effective DES has to be analyzed with two cases: 1) when the fixed-voltage actuator precedes the variable actuator (Figure 4-2) or 2) the fixed-voltage actuator follows the variable-voltage actuator (Figure 4-5).

73 4.1 TWO ACTUATOR MODEL Fixed-voltage actuator precedes the variable-voltage actuator 0 Input SOP Actuator A Actuator B Actuator B Actuator A (Forward) (Forward) (Reverse) (Reverse), (, 1 ),, 1,, 2, (, 2 ) Output SOP Variable Voltage Fixed Voltage Figure 4-2: The system block diagram for a fixed-voltage actuator, A, preceding a variablevoltage actuator, B. This system has already accounted for the effects caused by the FBG. In this scenario, the fixed-voltage actuator, actuator A, is what the signal encounters first as seen in Figure 4-2. This is followed by the next actuator, actuator B, which is the variable-voltage actuator; i.e. Actuator B is being swept through its controllable range = [0,2]. This model can be simplified, as was done in the one actuator scenario, by considering the effects of the fixed-voltage actuator. With the simplification techniques, we can create an equivalent system that integrates the rotation effects of the fixed-voltage actuator into the parameters of the variable-voltage actuator. The resultant system will only have two effective blocks (see Figure 4-3) instead of four as in Figure 4-2. With the configuration shown in Figure 4-2, the effects of the reverse transmission matrix of actuator A,,, can be accounted for by rotating the effective DES, and. =, (, ) (4.1) =, (, ) (4.2) Thus, the new forward and reverse transmission matrices of the variable-voltage actuator will have the above DES. The effects of the forward transmission matrix of actuator A,, has already been accounted for in the effective input as was explained in the previous section., does not affect the DES vectors of actuator B, because the forward transmission matrix of actuator A, precedes actuator B s transmission matrices. The resultant simplified system looks like below with the new effective DES,, and the new input SOP.

74 54 CHAPTER 4 BIDIRECTIONAL MODEL WITH MULTIPLE CONTROL ELEMENTS 0,, 1,, 2 Figure 4-3: The simplified system with the combined effects of the fixed-voltage actuator incorporated into the variable-voltage actuator s rotation matrices. As mentioned in the previous section, the initial output becomes the effective input to the system. The fixed-voltage actuator s effect on the existing DES and the resultant output SOP traces for different fixed voltage settings were simulated and mapped onto the Poincaré sphere:

75 4.1 TWO ACTUATOR MODEL 55 Figure 4-4: The output SOP traces as simulated for different fixed-voltage settings when the input signal enters the fixed-voltage actuator before the variable-voltage actuator. The black circle represents a fixed nodal point that is common for all curves. The resultant curves appear to rotate around a point on the Poincaré sphere. This is analogous to how the system operates in the unidirectional model when a fixed-voltage actuator follows the variable-voltage actuator resulting in circles that rotated around the Poincaré sphere, which was seen in Figure 2-11b). One of the observations made with the simulation is that there is a nodal SOP that is common to all curves in Figure 4-4. The SOP trace goes through this nodal SOP twice from the [0,2] range Fixed voltage actuator follows the variable-voltage actuator 0,, 1,, 1,, 2,, 2 Figure 4-5: The system block diagram for when the variable-voltage actuator, B, precedes the fixed-voltage actuator C. In this case, the input signal will enter actuator B first, which is the variable-voltage actuator as seen in Figure 4-5. Actuator C, the fixed-voltage actuator, can have its transmission matrix integrated into the preceding system components. Like in the preceding case, the fixed-voltage transmission matrix affects the DES by: =,,,(, ) (4.2) = (4.3) The first effective DES is rotated twice by the two effective DES of the fixed-voltage actuator. The second effective DES of the variable-voltage actuator is not affected since all the fixed transmission matrix components of Actuator C precede the reverse component of the variable-

76 56 CHAPTER 4 BIDIRECTIONAL MODEL WITH MULTIPLE CONTROL ELEMENTS voltage actuator. With the simplifications, the new system is just like Figure 4-3, except the DES components are modeled by Equation (4.2) and Equation (4.3). Simulations were done for different operation voltages of the fixed-voltage actuator followed by a sweep of the variablevoltage actuator. A set of curves are formed as seen in Figure 4-6. Figure 4-6: The output SOP traces as predicted by simulation for different fixed-voltage settings when the input signal enters the variable-voltage actuator before the fixed-voltage actuator. The black circle represents a fixed nodal point that is common for all curves. The resultant curves all appear to stem from a central point and expand outwards. This is analogous to the unidirectional model where circles are concentric about a central DES, when a fixed voltage actuator followed the variable-voltage actuator which was seen in Figure 2-11a). Similar to Figure 4-4, a nodal SOP is also apparent in Figure 4-6 that is common to SOP traces. We will describe a method of using the node for modeling the DES of multiple actuators in the following section.

77 4.2 MULTIPLE ACTUATOR ANALYSIS Multiple Actuator Analysis 0 1, ( 1, 1,1 ) 2, ( 2, 1,2 ), (, 1, ), 1 1, ( 1, 1,1 ) 2, ( 2, 1,2 ), (, 1, ), (, 2, ) 2, ( 2, 2,2 ) 1, ( 1, 2,1 ), 2 1, ( 1, 2,1 ) 2, ( 2, 2,2 ), (, 2, ) Figure 4-7: A system with multiple actuators. The variable-voltage actuator (yellow-box) has a forward-transmission DES vector,, and reverse-transmission DES vector,. There are actuators (red boxes) that precede the variable-voltage actuator with forward transmission DES vector, and reverse transmission DES vector,,, where [, ]. There are -actuators (green) that follow the variable-voltage actuator with forward transmission DES vector, and reverse transmission DES vector,, where [, ]. The various refer to the fixed-rotation angle as a result of the different applied voltages applied to the fixed-voltage actuators. We can extend the simplification techniques that were described in the previous section to more than two actuators. Consider the system in Figure 4-7, where we have multiple actuators. Note that there is only one variable-voltage actuator distinguished by its variable rotation angle,, and DES vectors and which are the forward and reverse DES vectors of the variablevoltage actuator (see yellow boxes in Figure 4-7). Even if there are multiple variable-voltage actuators, we can create an equivalent system where there is only one variable-voltage actuator being considered at a time. The variable-voltage actuator in Figure 4-7 follows number of fixed-voltage actuators with forward transmission DES vector,, and reverse transmission DES vector,, where [1, ]. Following the variable-voltage actuator are number of fixed-voltage actuators with forward transmission DES vector,, and reverse transmission DES vector,, where [1, ]. will refer to the fixed-angles as caused by an applied voltage to a fixed-voltage actuator. It is especially important to note that the actuators being considered have been activated with an applied voltage. If the applied voltage to an actuator is 0, then the actuator s transmission matrices are identity. Consequently, inactive actuators are not considered in Figure 4-7.

78 58 CHAPTER 4 BIDIRECTIONAL MODEL WITH MULTIPLE CONTROL ELEMENTS The goal will be to integrate all fixed-voltage actuator rotations into the variable-voltage actuator s rotation matrices. Thus, we would like apply the simplification techniques on Figure 4-7 to create a system that looks like Figure , 1, 2 Figure 4-8: The equivalent system when we applied the simplification techniques on Figure 4-5; i.e. all fixed-voltage rotations have been integrated into, and. Similar to the two-actuator model, the steps to creating this simplified system are 1) to determine the new effective input SOP, and 2) to determine the new DES vectors of the variable-voltage actuator New Effective Input For the new initial effective SOP,, we consider the effects of all the fixed-voltage actuators like we discussed in Section The new effective input, is a product of all the fixedvoltage transmission matrices: = (), (),,() (), (),,(),,,,,, (4.4) will also be the new initial output that will be observed at the output Equivalent DES Vectors The next step is to determine the new DES vectors. As already mentioned, the DES vectors of the variable-voltage actuator depend on those actuators that precede and follow them. We can apply the steps shown in Section for the configuration in which the variable-voltage actuator follows the fixed-variable actuators and for the configuration in which the variable-

79 4.3 A NODAL APPROACH TO MODELING MULTIPLE ACTUATORS 59 voltage actuator precedes the fixed-variable actuators. The results of following those steps, yields DES vectors that look like: = (), (),,() (4.5) (), (),,(),,, = (), (,(),,() ) (4.6) where we have accounted for the effects of the fixed-voltage actuator rotations. Again, the reverse transmission matrix DES vector, of the variable-voltage actuator is only affected by the reverse transmission rotations caused by the fixed-voltage actuators that precede the variablevoltage actuator. Conversely, the forward transmission matrix DES vector, of the variablevoltage actuator is not affected by the forward transmission rotations of the fixed-voltage actuators that precede the variable-voltage actuator. However, the variable-voltage actuator s forward transmission matrix DES vector,, is affected by all other rotations of the fixedvoltage actuators. 4.3 A Nodal Approach to Modeling Multiple Actuators In Section 4.1.2, we provided the model that explains how the SOP traces will evolve with multiple actuators. In this section, an alternative method for modeling multiple actuators is provided in order to reduce some of the experimental limitations that will be discussed in Chapter 5 and Chapter 6. We will observe that the extracted experimental DES vectors using the method described in Section 4.2 will predict SOP traces for the multiple actuators scenario that do not have fixed nodal SOP. This is contrary to what was observed in our simulations and what we will observe from experimental data. This nodal SOP modeling approach will try to resolve this limitation. We observe that the nodal SOP remains stationary regardless of which actuator is the fixedvoltage actuator or the variable-voltage actuator (see Figure 4-4 and Figure 4-6). This

80 60 CHAPTER 4 BIDIRECTIONAL MODEL WITH MULTIPLE CONTROL ELEMENTS observation leads us to use the nodal SOP as an additional parameter when predicting any SOP traces formed by operating both actuators. In a system with multiple actuators, we determine the DES vectors associated with each actuator independently using the procedure discussed in Section 3.7. In addition to determining the DES vectors associated with each actuator, we can determine the nodal SOP. The nodal SOP will serve as a reference SOP that we will use to determine the forward transmission matrix DES vector which we will discuss in this section. Another observation that was made from simulations is that the nodal SOP appears to be dependent on the input SOP, ; the nodal SOP changed its position when we vary. This effect will be verified experimentally in Chapter 5 (Figure 5-11).,,,, Figure 4-9: The system when we ignore the fiber rotation effects between the actuator and the FBG. In Figure 3-3, we had to include the effects of the fiber between the PC and the FBG. When we ignored all the fiber rotation effects between the FBG and the PC (Figure 4-9), and repeat the simulations, we observed a relationship between the input SOP and the nodal SOP. For example, if the input to the system is given by: 1 = 2 (4.7) 3 Then from simulations, the nodal SOP is related to the parameters by: 1 = 2 3 (4.8) and when the actuators are inactive, the initial output,, would be:

81 4.3 A NODAL APPROACH TO MODELING MULTIPLE ACTUATORS = 2 (4.9) 3 In this section, the tilde is used to denote SOP parameters and rotation matrices that describe a system where the effects of the fiber rotations between the PC and the FBG have been integrated with the rest of the system such that the system has an output SOP curve that is equivalent to the experimental SOP curve (see Appendix D). Equation (4.7)-(4.9) are similar and only differ by their signs. Furthermore, the equations that related to were shown in Equation (3.9) and Equation (3.12). If we combined the mirror s effect onto = (4.10) Comparing Equation (4.10) and (3.12), we notice this relationship describing the DES vectors is similar to the relationship between and. However in a real system, the relationship between the DES vectors does not hold due to the fiber between the FBG and the actuator. If we use the techniques in Appendix D, then the DES vectors, and, would have the direct relationship found in Equation (4.10) and Equation (3.12). The goal to derive an equivalent system in which the relationships between the DES vectors, and the input and nodal SOP would hold. In this manner, we can then generate SOP curves with a specific nodal SOP. However, we need to somehow relate these observations back to the experimental data. Through the derivation in Appendix D, we can integrate the fiber rotation between the PC and the FBG and only consider the rotation caused by the fiber section from the output of the PC to the polarimeter via the circulator. The effect of this fiber rotation from the output of the PC to the polarimeter can be estimated.,,,,, Figure 4-10: The equivalent system with an equivalent fiber rotation,, from the output of the PC to the polarimeter to ensure that this system is equivalent to Figure 3-4.

82 62 CHAPTER 4 BIDIRECTIONAL MODEL WITH MULTIPLE CONTROL ELEMENTS =,, (4.11) where the tilde represents an equivalent system parameter. is still the actual experimental output SOP as obtained through Equation (4.11). As mentioned, the fiber rotation, will conceal the direct relationship between the effective input, and the nodal SOP,, and the DES vectors and. In order to make use of these relations, we use the experimental parameters and and relate it back to and. The relationship between and is related to the equivalent fiber rotation : By superposition, we can subtract Equation (4.13) from (4.12) and with Equation (4.8) and Equation (4.9), we get: = (4.12) = (4.13) = (4.14) 0 = 2 0 (4.15) If has the form of Equation (2.22), with DES vector = [ ] and arbitrary angular parameter,, then we can simplify Equation (4.15) to: (1 cos) + sin =2 cos + (1 cos ) (1 cos) sin (4.16) As mentioned,, 0 are obtained experimentally. In Equation (4.16), there are 5 unknowns. We can determine the value of by applying the fact that the fiber rotation matrix is orthogonal: = = 2 (4.17) Since we are uncertain if is negative or positive, both cases will have to be considered when we try to solve for the parameters for the fiber rotation matrix. After applying Equation (4.17), there are still 4 unknown parameters and 3 equations. We also know the DES vector,, has a norm of unity:

83 4.3 A NODAL APPROACH TO MODELING MULTIPLE ACTUATORS =1 (4.18) Using a least-square fit, an estimate of the parameters for the rotation matrix can then be determined. With, we can determine the DES vector of the equivalent forward transmission matrix 1 by finding 2 through Equation (3.12) and Equation (4.10). We first use the estimated rotation matrix,, on the experimentally determined to obtain the equivalent system parameter. = (4.19) = (4.20) From Equation (3.12) and Equation (4.10), we then relate 2 back to 1. We can then estimate with the rotation matrix. = (4.21) Since we use the nodal SOP to determine the rotation matrix, and the relationship between the nodal SOP and initial output is similar to the relationship between two DES vectors, we can obtain system parameters that will yield SOP traces with the experimental nodal SOP. The system in Figure 4-10 represents a one actuator system. However, this alternative method with the nodal SOP can be extended to the multiple-actuator configuration. In order to extend to the multiple-actuator configuration, we would create an equivalent system from the multipleactuator configuration that integrated all the fixed-voltage rotations first into a system that resembled Figure 4-3. Since the system in Figure 4-3 is similar to a one-actuator system, we can transform the one-actuator system to the system shown in Figure Then, we can apply the steps used to determine the forward transmission matrix DES,.

84 Chapter 5 Single Polarization Control Element Model Validation 5 In Chapter 3, a theoretical model was developed for the polarization control system in the bidirectional configuration for a single polarization control element. In this Chapter, the theoretical model for one actuator in a PC will be verified experimentally through a two-step process: 1) extracting system parameters that characterize experimental SOP trace and 2) using the extracted parameters to model the observed results. The system parameters will then be used to predict the SOP evolution when the input SOP to the bidirectional configuration is varied. Experimental errors that affected the model will also be discussed in this Chapter. 5.1 Experimental Setup The fiber-based bidirectional configuration used to validate the model for one polarization control element is shown in Figure 1-8. The actual components of the system are listed in Table

85 5.1 EXPERIMENTAL SETUP 65 (1) Tunable Laser Source (2) Polarization Controller (Input PC) Input SOP (3) FBG (4) Piezoelectric Polarization Controller (PPC) Legend Fiber Copper Control Variables (5) Computer Measured SOP (6) Polarimeter Figure 5-1: Experimental setup of the bidirectional configuration Table 5-1: System components used for the experiment. Index corresponds to components in Figure 5-1 Index Component Equipment Characteristics (1) Source HP8168F Tunable Laser Source (2) Polarization Controller (Input PC) General Photonics PolaRITE PLC004 : nm : 1.36dBm Manual fiber squeezing and twisting based PC

86 66 CHAPTER 5 SINGLE POLARIZATION CONTROL ELEMENT MODEL VALIDATION (3) Reflective Structure Fiber Bragg Grating 0 : nm (4) Piezoelectric Polarization Controller General Photonics PolaRITE III 3dB Bandwidth: 262 pm Piezoelectric polarization controller (PPC) with a piezoelectric driver board (5) Controller Computer Digital Input Output Card to interface with the PPC s piezoelectric driver board GPIB for reading SOP data from polarimeter Matlab Instrument Control Toolbox for data processing (6) Polarization Analyzer Tektronix PAT9000B (Polarimeter) GPIB output 5.2 Modeling Experimental Data with One Actuator Figure 5-2 shows the SOP evolution when one of the actuators of the piezoelectric polarization controller (PPC) is swept from = [0,2]. The techniques developed in Section 3.6 and Section 3.7 are used to characterize this SOP data and the results will be shown in this section.

87 5.2 MODELING EXPERIMENTAL DATA WITH ONE ACTUATOR 67 Figure 5-2: The experimental SOP trace from varying the applied voltage to a piezoelectric actuator from 0 to 67.5V. 30 data points were collected and plotted on the Poincare sphere. As mentioned in Chapter 3, Equation (3.35) is used to describe the SOP evolution in the bidirectional configuration. Equation (3.35) can be determined from the experimental trace by applying an FFT to each of the three experimental Stokes parameters. Figure 5-3 shows the results of the FFT as applied to the component:

88 68 CHAPTER 5 SINGLE POLARIZATION CONTROL ELEMENT MODEL VALIDATION X: Y: S 1 Amplitude X: 0 Y: X: Y: Frequency (Ω) (a) 4 3 X: Y: S 1 Phase X: Y: Frequency (Ω) (b) Figure 5-3: The (a) amplitude and (b) phase information from a 384-point FFT applied to the first Stokes parameter,. As expected from the theory developed in Chapter 3, there are three frequency components that appear at, 2 and at DC in the FFT as seen in Figure 5-3(a). The amplitude and phase data allows us to model the Stokes parameters against the angle,, as seen in Figure 5-4.

89 5.2 MODELING EXPERIMENTAL DATA WITH ONE ACTUATOR 69 Stokes Parameters S 1 Data S 1 Fit S 2 Data S 2 Fit S 3 Data S 3 Fit θ Figure 5-4: The Stokes parameters versus angle of rotation with the trigonometric fit Equation (5.1) applied as determined from the FFT. The following fit equation was determined from the FFT., () = cos cos (5.1) We can also plot Equation (5.1) and compare it with the experimental SOP trace on the Poincare sphere as seen in Figure 5-5.

90 70 CHAPTER 5 SINGLE POLARIZATION CONTROL ELEMENT MODEL VALIDATION Figure 5-5: The curve as determined from the FFT fitting (blue) as plotted with the experimental data. To calculate the error between the fitted data points and the experimental data points, we look at the percent angular deviation which is given as: % = cos 100% (5.2) The resulting error versus the applied voltage is displayed in Figure 5-6.

91 5.2 MODELING EXPERIMENTAL DATA WITH ONE ACTUATOR Percent Deviation Applied Voltage (V) Figure 5-6: The percent angular deviation of the fitted curve with the experimental trace. No corrections have been made to correct a discrepancy between the applied voltage and the angular rotation. To acquire the system parameters, we need to convert Equation (5.1) to its equivalent form described by Equation (3.23) which is:, () = cos sin cos sin (5.3) To determine the DES vectors, we use Equation (3.30) and (3.31). The parameters to be used in Equation (3.30) are: = , = (5.4)

92 72 CHAPTER 5 SINGLE POLARIZATION CONTROL ELEMENT MODEL VALIDATION and are then used to determine. Equation (3.31) is used to determine the intermediate SOP trace,. From, can be determined with the steps outlined in Section 3.6. The resultant curve from applying Equation (3.31) to Equation (5.3) is shown in Figure 5-7. Figure 5-7: The curve that represents the output of the forward transmission matrix in Figure 3-6 experimentally. Figure 5-7 shows a curve that resembles an oblong circle. From the model, the plot was expected to be a circle on the Poincaré sphere as was seen in Figure This results from the limitations of the DES model which cause the SOP to deviate from an ideal circle. The limitations of the DES model will be discussed in the Section 5.4. We can still use the results from Figure 5-7 to approximate the DES. This parameter is then used in a nonlinear least squares fitting algorithm to refine the value of. Table 5-2 summarizes the DES vectors that were determined.

93 5.2 MODELING EXPERIMENTAL DATA WITH ONE ACTUATOR 73 Table 5-2: Parameters necessary to characterize the system as was determined from the characterization steps DES Vector Value [ ] [ ] Using the model developed in Chapter 3, we input the system parameters into Equations (3.24)- (3.28). The resultant equation for the model is in Equation (5.5)., () = cos sin cos sin (5.5) There is a slight deviation between, and, which is the result of the limitations of the DES model which will be discussed in the Section 5.4. Nonetheless, we plot, with the experimental data collected (Figure 5-8(a)). The percent angular deviation between each experimental datapoint and the modeled datapoint is shown in Figure 5-8(b).

94 74 CHAPTER 5 SINGLE POLARIZATION CONTROL ELEMENT MODEL VALIDATION 10 p(a) 9 8 Percent Deviation Applied Voltage (V) (b) Figure 5-8: (a) The experimental data and modeled data, Equation (5.5), using the extracted DES vectors. (b) The percent error between the experimental datapoint and the modeled datapoint as determined by the ratio of surface distance over the maximum distance that two points can be from each other on the sphere.

95 5.2 MODELING EXPERIMENTAL DATA WITH ONE ACTUATOR 75 There is a slight deviation between the expected angular response and the actual angular response to the applied voltage which is due to the PPC. To correct for these slight deviations, we can shift the in relation to the applied voltage (Figure 5-9) such that the error is minimized (Figure 5-10) Applied Voltage vs θ Theoretical Experimental Theta (θ) Applied Voltage (V) Figure 5-9: The assumed angular response to the applied voltage (red) compared with an adjusted angular response to the applied voltage (blue)

96 76 CHAPTER 5 SINGLE POLARIZATION CONTROL ELEMENT MODEL VALIDATION 3 Error Between Experimental and Modeled Data 2.5 Percent Deviation Applied Voltage (V) Figure 5-10: The percent angular deviation between the experimental and modeled data with the angle correction. The result of these corrections allowed the error between the experimental and modeled data to decrease with a maximum deviation of 2.6%. Additional sources of error will be discussed in Section Validity of One Actuator Model To demonstrate the validity of the model, we vary the input to the system. We use a PC placed before the circulator (Input PC) to vary the input SOP as was shown in Figure 5-1. From theory, the DES of each actuator should remain the same as was characterized in Section 5.2. By varying only the input SOP,, we should be able to use the system parameters from Section 5.2 to predict the experimental data. The results are shown in Figure 5-11

97 5.3 VALIDITY OF ONE ACTUATOR MODEL 77 (a) (b) (c)

98 78 CHAPTER 5 SINGLE POLARIZATION CONTROL ELEMENT MODEL VALIDATION (d) (e) Figure 5-11: Different experimental SOP traces (red) and the modeled data (blue) for varying the input SOP to the bidirectional system. The effective input of each trace are: (a) = [... ] (b) = [... ], (c) = [... ] (d) = [... ] (e) = [... ] There is good correspondence between the predicted curves and the experimental data. The errors between the experimental and predicted curves are shown in Figure 5-14 with the maximum error being 7.7%.

99 5.3 VALIDITY OF ONE ACTUATOR MODEL 79 Percent Deviation (a) (b) (c) (d) (e) Applied Voltage (V) Figure 5-12: The percent angular deviation between the experimental and modeled data for each varying with the corrections. Another way of verifying this model is to characterize each of the individual experimental curves in Figure 5-11 and to extract the DES vectors. Figure 5-13 shows the forward and backward DES vectors plotted on the Poincaré sphere.

100 80 CHAPTER 5 SINGLE POLARIZATION CONTROL ELEMENT MODEL VALIDATION Figure 5-13: The forward DES vectors, (blue) and backward DES vectors, (cyan) of the experimental curves in Figure Table 5-3: The mean system parameters as extracted from each of the curves in Figure DES Mean Vector Value Standard Deviation [ ] 6.67% [ ] 0.82% In Figure 5-13, we can see that the DES of each experimental curve does not change much as expected and in Table 5-3, the standard deviation of the DES vectors are relatively small. These results validate the model that was developed for the bidirectional configuration for a single actuator. However, there are a number of errors that prevented the fit to be as precise as can be. We will discuss those errors in the next section. 5.4 Experimental Limitations As was mentioned in Chapter 2 and 3, the polarization control model was developed under ideal conditions. In practice, there are number of practical limitations. The piezoelectric actuator has some nonideal characteristics such as hysteresis. Furthermore, the limits of the DES model do not hold over the entire range over which the piezoelectric actuator operates. We will discuss

101 5.4 EXPERIMENTAL LIMITATIONS 81 the limits of the DES model, how hysteresis of the piezoelectric material affects polarization tracking, and how resetting the actuator can eliminate some of the nonidealities Limits of the DES Model The DES model holds for a small variation around the physical control variable; i.e. the DES model is a first order approximation to the SOP evolution (see Chapter 2). As a result, the DES model may not hold for high applied voltages, which implies there might be a departure between the model and the experimental results. We present a scenario where the deviation is pronounced by characterizing a piezoelectric actuator in the single pass configuration similar to Figure 1-7. At low applied voltages, the SOP evolution follows a circular trace as was mentioned in Chapter 2 and can be modeled with the method outlined in Section 2.3 and Section 2.4. But at higher applied voltages, the curve deviates away from the circle and a new DES vector can be calculated as result of the deviation as seen in Figure 5-14(a) (a) (b) Figure 5-14: The (a) experimental output SOP trace,, in a unidirectional configuration. The curve initially has a DES, but evolves to at high applied voltages and spirals outwards. As a result the the centre vector of the output SOP trace, also shifts to. The theoretical circular curve based on the initial DES is shown in (b). The spiral shape, rather than a closed circle, is due to the slight change in DES under high stress. Nonetheless, for small magnitudes of applied stress to the fiber, the path can still be approximated by a circular trajectory Figure 5-14(b).

102 82 CHAPTER 5 SINGLE POLARIZATION CONTROL ELEMENT MODEL VALIDATION The DES calculated at lower applied voltages does not hold at higher applied voltages, and this effect will also cause the SOP trace to spiral in the bidirectional configuration as seen in Figure Figure 5-15: An experimental SOP trace for the bidirectional configuration when the spiraling effect is pronounced This spiral deviation in the SOP trace will skew the FFT fit and the accuracy of the DES calculations. Hence, the assumption that the DES vectors do not vary over the 2 range is not entirely valid; i.e. the DES vectors calculated are an average value of the actual DES vectors. It is also this limitation that will affect the SOP prediction for multiple actuators which will be discussed about in Chapter 6. A higher-order DES model might able to resolve some of these limitations which is reserved for future work. For the purpose of this work though, we will show a method in Chapter 7 that will reduce the effects of the DES limitations in predicting the SOP when operating multiple actuators. It should be noted that not all polarization control elements will show the extreme spiraling effect. In Figure 5-2, the spiral effect is less noticeable. The spiral effect depends on the

103 5.4 EXPERIMENTAL LIMITATIONS 83 construction of the piezoelectric actuators, the fiber placement within the actuators, and the birefringence of the fiber, which makes it difficult to predict and model Hysteresis of Piezoelectric Material In addition to the limits of the DES model at higher applied voltages, hysteresis effects are apparent with the piezoelectric actuator. We demonstrate this by first increasing the applied voltage to the PPC which will form one SOP trace, followed by decreasing the applied voltage to the PPC which will form another trace. In Figure 5-16, we observe that the there is a phase shift between the two curves for each of the Stokes parameters. 1 Increasing Voltage Decreasing Voltage 1 Stokes Parameter (S 1 ) Stokes Parameter (S 2 ) Applied Voltage(V) (a) Applied Voltage (V) (b) Stokes Parameter (S 3 ) Applied Voltage (V) (c) Figure 5-16: The Stokes components versus the applied voltage when the applied voltage is increasing from 0V to 67.5V (red) and when the voltage is decreasing from 67.5V to 0V (blue).

104 84 CHAPTER 5 SINGLE POLARIZATION CONTROL ELEMENT MODEL VALIDATION There appears to be a phase shift between the two curves as evident in the (a), (b), and (c) Stokes parameters. The data was collected from 10 data runs. This phase shift can be attributed to the hysteresis response of the piezoelectric material and is a well-documented effect [25] Response Time of Piezoelectric Material After resetting the actuator, a transient response with a long relaxation time was observed in the Stokes parameter as seen in Figure S 1 Stokes Parameters S 2 S Time(s) Figure 5-17: The response of the Stokes parameter over a period of time after the actuator has been reset. The hysteresis and the slow reset response limit endless tracking of the SOP. The slow response will be a source of error and cannot be controlled without sacrificing time to wait for the SOP to settle. However, the phase shift can be dealt with through resetting the actuator; i.e. we apply 0V to the PPC before applying the necessary voltage to get to the target SOP. To demonstrate this, we can compare the SOP trace that is obtained normally through an increasing applied voltage which was done in Section 5.2, to a trace formed from resetting the actuator first and then applying the voltage value to the actuator.

105 5.4 EXPERIMENTAL LIMITATIONS 85 1 Reset No Reset 1 Stokes Parameter (S 1 ) Stokes Parameter (S 2 ) Applied Voltage (V) (a) Applied Voltage (V) (b) Stokes Parameter (S 3 ) Applied Voltage (V) (c) Figure 5-18: Resetting the actuator before applying a voltage (red) in comparison to the characterization trace (blue) for the (a) (b) and (c) Stokes parameters. There appears to no phase shift between the two curves. The data was taken over 10 data runs. Figure 5-18 shows there is a good correspondence between resetting the actuator and not resetting the actuator when the applied voltage is increasing. Similar results are obtained from resetting the actuator before a decreasing voltage is applied to the actuator as shown in Figure 5-19.

106 86 CHAPTER 5 SINGLE POLARIZATION CONTROL ELEMENT MODEL VALIDATION 1 Reset No Reset 1 Stokes Parameter (S 1 ) Stokes Parameter (S 2 ) Applied Voltage (V) (a) Applied Voltage (V) (b) Stokes Parameter (S 3 ) Applied Voltage (V) (c) Figure 5-19: The Stokes components (a), (b), and (c) versus the applied voltage when the applied voltage is increasing from 0V to 67.5V (red) and when the voltage is decreasing from 67.5V to 0V (blue). Prior to applying a decreasing voltage, a reset is done on the actuator. The data was collected from 10 data runs. Resetting the actuator allows us to essentially remove the hysteresis effect. However, since the time between resetting and applying the next voltage is approximately (0.77±0.20)s, there might be a slight perturbation in the output SOP. In systems with a varying input SOP, SOP fluctuations can occur as fast as 100 [26], but more typically they change at a rate of about 5 rads/s on the Poincare sphere [27]. Nevertheless, for the purpose of our model, we will not be

107 5.4 EXPERIMENTAL LIMITATIONS 87 considering the time delays and perturbations incurred which can be suppressed with a faster polarization tracker and a faster controller in which our algorithm can be implemented in. Furthermore, we can compare the accuracies of reaching an SOP state by resetting and not resetting the actuator. The comparison was done by first sweeping the actuator from 0V to 67.5V which corresponds to an angular sweep of =0 to 2. We then do a random sampling of applied voltages (Table 5-4) and plotting that data against the initial data obtained. We expect that the SOP produced by applying a certain voltage will correspond with the curve obtained by applying an increasing voltage to the actuator. The results can be seen in Figure 5-20 (a). Table 5-4: A set of voltages applied to the actuator in the following sequence. Test Voltages (V) [9, 13.5, 33.75, 22.5, , 1.125, 52.5, 30.75, 4.125, 41.25]

108 88 CHAPTER 5 SINGLE POLARIZATION CONTROL ELEMENT MODEL VALIDATION Stokes Parameters S 1 trace S 2 trace S 3 trace Testpoints Applied Voltage (a) Stokes Parameters S 1 trace S 2 trace S 3 trace Testpoints Applied Voltage (b) Figure 5-20: A set of voltages applied to the PPC in comparison with the characterization trace when the actuator is first (a) reset and (b) and when actuator is not reset. The error bars signify the standard deviation of the data over 10 datasets. To demonstrate how resetting the actuator is critical, the same data is collected without resetting the actuator as seen in Figure 5-20(b). The resultant SOPs do not lie on the trace at their

109 5.4 EXPERIMENTAL LIMITATIONS 89 supposed location. The percent angular deviation between resetting and not resetting the actuators with the ten randomly applied voltages are shown in Figure Reset No Reset Percent Deviation Testpoint Index Figure 5-21: The percent deviation between the SOP as obtained from the randomly chosen applied voltages and the SOP as obtained from the characterization curve when a reset is applied prior to applying the voltage (red) compared to when no reset is applied prior to applying a voltage (blue). Therefore, it is necessary to reset the actuator first before applying a voltage as this method will solve some of the hysteresis problems in the PPC. Resetting the actuator first will also be critical when we consider how to control the SOP.

110 90 CHAPTER 5 SINGLE POLARIZATION CONTROL ELEMENT MODEL VALIDATION FFT Error Amplitude Frequency (Ω) Amplitude Frequency (Ω) (a) (b) Figure 5-22: The FFT of the component in Figure 5-3(a) over the entire range when the window size covers (a) 54V compared with the FFT whose window size covers (b) 64.8V. The FFT is necessary to determine the fit for the SOP curves, and it also used for extracting the DES vectors, and. The window size is essential for the FFT to work. The window size is dependent on a dataset that covers a complete period of its revolution. There are a couple of factors that affect knowledge of the window size and can skew the FFT fit. The largest contribution to the FFT error is the DES model limitation. At higher applied voltages, the curve may start spiraling and if the spiral effect is large, it will affect the overall FFT fitting, as it will no longer be representational of a trigonometric curve with two frequency components. Another smaller factor in the FFT error is the discrete data points. There is a possibility that the data collected does not cover the datapoints of a complete revolution, which will also skew the FFT fit. Based on analysis of the curves with little deviation at higher applied voltages (e.g. Figure 5-2), a complete revolution occurs at approximately 54V. For speed in characterizing the SOP traces, we chose 25 data points spaced out by 2.16V. Thus, N = 25 is the window size used. However, due to the two main factors influencing the FFT, higher frequency components are observable in the FFT (Figure 5-22 (a)) but are largely suppressed. In Figure 5-22 (b), a larger window size is chosen (e.g. N= 30 which corresponds to 64.8V) for more data points, but since N=30 does not correspond to 2, the higher frequency components are more pronounced.

111 5.4 EXPERIMENTAL LIMITATIONS Additional Sources of Error Other errors include the fiber sensitivity to the environment and detector noise. Additionally, in the PPC the actuator s mechanical jaw may not engage or slip with the fiber when the PPC is activated. Furthermore, when a voltage is applied to the piezoelectric actuator, there might be a small discrepancy in the desired voltage and the actual applied voltage to the piezoelectric actuator. The piezoelectric driver board converts a digital voltage value from the computer to an analog voltage value that is applied to the actuator. This digital to analog conversion may have resulted in a slightly fluctuating voltage value that can also be aggravated by a step-up converter that amplifies the voltage value. Nonetheless, all of the above mentioned errors are fairly small between the experimental data and the one actuator model, especially after most of the larger errors were accounted for.

112 Chapter 6 Multiple Polarization Control Elements Model Validation We were able to verify the theoretical model in the single polarization control element case. However, for polarization control, a single polarization control element cannot reach every target SOP; so it is necessary to use multiple control elements to control the output SOP. In this section we will verify the theoretical model for multiple control elements that was developed in Chapter 4. We will also present how characterizing each element in the PC allows us to predict the experimental results Experimental Data with Multiple Actuators In this section we will present the experimental data as obtained from operating multiple actuators in sequence. Figure 6-1 is the set of output SOP traces when operating in the configuration that was presented in Section 4.4.1; the input signal enters the fixed-voltage actuator first, followed by the variablevoltage actuator (see the configuration in Figure 4-2). 92

113 6.1 EXPERIMENTAL DATA WITH MULTIPLE ACTUATORS 93 Figure 6-1: The experimental output SOP curves for different fixed-voltage settings when the input signal enters the fixed-voltage actuator before the variable-voltage actuator. The different color lines represent different voltages applied to the fixed-voltage actuator. The black circle is the nodal SOP of the set of curves. Figure 6-1 resembles the expected behavior for the SOP evolution in this configuration; the curves appear to rotate and appear to have a nodal SOP (black circle). This is similar to what was observed in Figure 4-4. The results for the other configuration, where the input signal enters the variable-voltage actuator first before the fixed-voltage actuator (Section ) are shown in Figure 6-2.

114 94 CHAPTER 6 MULTIPLE POLARIZATION CONTROL ELEMENTS MODEL VALIDATION Figure 6-2: The experimental output SOP traces for different fixed-voltage settings when the input signal enters the variable-voltage actuator before the fixed-voltage actuator. The different color lines represent the different voltages applied to the fixed voltage actuator. The black circle represents the nodal SOP. The results shown are similar to what was predicted in Section The curves appear to expand outwards and are fixed to a nodal SOP as well (black circle), which is the behavior that was observed in the simulations (Figure 4-6). From the curves shown in Figure 4-3, we can characterize and model each of the experimental curves at the different step-angles,, using the method that was outlined in Section 3.7. The resulting modeled data is plotted in Figure 6-3 together with experimental data for comparison.

115 6.2 MODELING EXPERIMENTAL DATA WITH MULTIPLE ACTUATORS 95 (a) (b) Figure 6-3: Using the method of characterization as developed in Section 3.7 to model each curve in both configurations. The first configuration (a) is where the signal enters the fixedvoltage actuator first followed by the variable-voltage actuator while in (b) the signal enters the variable voltage actuator first followed by the fixed-voltage actuator. As can be seen in Figure 6-3, the model agrees with the experimental SOP data quite well. However, instead of characterizing each curve at each step-angle,, it would be more practical if we could predict the behavior of the curves with the system parameters as extracted from operating each actuator independently of the other actuators as developed in the theory in Chapter 4. The next section will show how this behavior can be modeled. 6.2 Modeling Experimental Data with Multiple Actuators With multiple actuators, each individual SOP trace can be characterized as was shown in Figure 6-3. However, we would like to know if characterizing each actuator independently will allow us to predict how the curves will evolve when the actuators are operated together. The first step is to extract the system parameters from the SOP trace formed from sweeping each actuator. This is then followed by using the model developed in Chapter 4 to estimate the curves formed from changing the step-angle,. In section 6.2.1, we will attempt to predict how the curves will evolve through the system parameters as extracted from each SOP curve that is formed by the individual actuators. We will show that the system parameters do not provide an accurate

116 96 CHAPTER 6 MULTIPLE POLARIZATION CONTROL ELEMENTS MODEL VALIDATION description of the SOP evolution through multiple actuators. In section 6.2.2, we will try to compensate for these deviations by using the nodal SOP method as was described in Section Predicting Curves through System Parameters Only We will first attempt to model Figure 6-1(a) with experimentally determined DES vectors associated with the actuators of the PPC. The curves obtained in Figure 6-1(a) was through the configuration shown in Section ; the input signal enters the fixed-voltage actuator first followed by the variable-voltage actuator (Figure 4-2). The fixed-voltage actuator is Actuator A, and the variable-voltage actuator is Actuator B; and their DES vectors are listed in Table 6-1 as determined with the method that was demonstrated in Chapter 5. Table 6-1: The system parameters for two actuators in the PPC Actuator DES Vectors Values 1 [ ] A 2 [ ] B 1 [ ] 2 [ ] And the initial output of the system is: 0 = [ ].

117 6.2 MODELING EXPERIMENTAL DATA WITH MULTIPLE ACTUATORS 97 (a) Percent Error θ step = 0 θ step = π/12 θ step = π/6 θ step = π/4 θ step = π/ Applied Voltage (b) Figure 6-4: (a) The resultant modeled curves (red) are plotted against the experimental curves (black) using the method in Section 3.7 and the theory in Section The signal enters the fixed-voltage actuator before entering the variable-voltage actuator. (b) The percent error between the experimental data and the predicted curve. The error gradually increases. The maximum error is at % which occurs when the applied voltage to the fixed-voltage actuator is highest.

118 98 CHAPTER 6 MULTIPLE POLARIZATION CONTROL ELEMENTS MODEL VALIDATION The percent deviation between the experimental and predicted datasets (Figure 6-4(b)) is increasing at higher applied voltages to the fixed-voltage actuator. Furthermore, the predicted data curves do not appear to have a fixed nodal SOP; instead the nodal SOP appears to rotate with the predicted data as seen in Figure 6-4(a). These errors can be attributed to the limits of the DES model as mentioned in Section If we flip the configuration such that the variable-voltage actuator is the first actuator that the signal enters followed by the fixed-voltage actuator, and apply the theory in Section to predict the curves in Figure 6-1b), then the results are shown in Figure 6-5. The result of the predicted curves follows the general shape (Figure 6-5(a)), but there are a number of deviations in this configuration as well. For example, the curves appear to stray away from the nodal SOP, and at higher applied voltages to the fixed-voltage actuator, the percent error between the predicted SOP and the experimental SOP is also increasing (Figure 6-5(b)). It appears that the DES vectors as extracted with the method discussed in Section lack accuracy when they are used as inputs to the DES model. The modeled curves formed do not agree well with the behavior of multiple actuators in an experimental setting, which is largely due to the DES limitation, and the fluctuating SOP states. An alternative method for predicting the curves with the nodal SOP was provided in Section 4.2 and the results will be detailed in the next section.

119 6.2 MODELING EXPERIMENTAL DATA WITH MULTIPLE ACTUATORS p (a) θ step =0 θ step =π/12 θ step =π/6 20 θ step =π/4 θ step =π/3 Percent Error Applied Voltage (b) Figure 6-5: (a) The resultant modeled curves (red) are plotted against the experimental curves (black) using the method in Section 3.7 and the theory in Section (b) The percent error between the experimental data and the predicted curve. The error gradually increases. The maximum error is at % which occurs when the applied voltage to the fixed-voltage actuator is highest.

120 100 CHAPTER 6 MULTIPLE POLARIZATION CONTROL ELEMENTS MODEL VALIDATION Modified Method of Modeling with System Parameters In this section, we will examine the validity of the nodal method (Section 4.2) for predicting the output SOP traces when multiple actuators are operated. We use the procedure discussed in Section 4.2. The DES vectors remain the same as Table 6-1, but we will only use the reverse transmission matrix DES vector,, for the analysis in this section. and will be determined through the nodal method. The initial output of the system remains as 0 = [ ]. The node is = [ ], which is obtained experimentally. For the configuration when the input signal enters the fixed-voltage actuator first followed by the variable-voltage actuator (Section ), the results are shown in Figure 6-6.

121 6.2 MODELING EXPERIMENTAL DATA WITH MULTIPLE ACTUATORS 101 Percent Error θ =0 step θ =π/12 step θ =π/6 step θ =π/4 step θ =π/3 step (a) Applied Voltage (b) Figure 6-6: (a) The resultant modeled curves (blue) are plotted against the experimental curves (black) using partially the theory in Section , supplemented by the nodal SOP as an additional parameter in the prediction process. The signal enters the fixed-voltage actuator before entering the variable-voltage actuator. (b) The percent error between the alternative method of predicting the experimental curve when the applied voltage to the fixed-voltage actuator is changed. The maximum error is 30.73% which occurs when the voltage to the variable-voltage actuator is at its highest.

122 102 CHAPTER 6 MULTIPLE POLARIZATION CONTROL ELEMENTS MODEL VALIDATION The maximum error in Figure 6-6(b) is 30.73% which is due to the limits of the DES model at high applied voltages as already mentioned in Section At low applied voltages the percent deviation is relatively low. For the configuration when the input signal enters the variable-voltage actuator first followed by the fixed-voltage actuator (Section ), the results are shown in Figure 6-7. The modeled SOP curves appear to agree well with the experimental SOP curves. In addition, the model SOP curves nodal SOPs do not deviate much from the supposed nodal SOP as marked as a purple circle in Figure 6-7(a). The maximum deviation between the predicted and experimental error is 12.82% and the deviation between the predicted and experimental error does not increase significantly at higher step voltages to the fixed-voltage actuator as can be seen Figure 6-7(b). In both cases, the errors are still relatively high. In the first configuration (Figure 6-6), the experimental and predicted data is comparable to the results in Section 6.2.1, but in the second configuration (Figure 6-7), the results held better at higher applied voltage. This can be attributed to the constant reverse transmission matrix DES vector of the variable-voltage actuator,, which from theory does not change when in the configuration described in Section ; we only have to predict how the forward transmission matrix DES vector of the variable-voltage actuator,, will evolve. This is unlike the first configuration, where the signal entered the fixedvoltage actuator first, and both DES vectors of the variable-voltage actuator are rotated. Thus, for polarization control, we will be considering the second configuration; the signal enters the variable-voltage actuator first. The error is still significant with this configuration, but we will show that the errors can be compensated for when we discuss polarization control.

123 6.2 MODELING EXPERIMENTAL DATA WITH MULTIPLE ACTUATORS (a) p θ step = 0 θ = step π/12 θ step = π/6 θ step = π/4 θ = step π/3 Percent Error Applied Voltage (V) (b) Figure 6-7: (a) The resultant modeled curves (blue) are plotted against the experimental curves (black) using partially the theory in Section , supplemented by the nodal SOP as an additional parameter in the prediction process. The signal enters the variable-voltage actuator before entering the fixed-voltage actuator. (b) The percent error between the alternative method of predicting the experimental curve when the applied voltage to the fixed-voltage actuator is changed. The maximum error is 12.82%

124 104 CHAPTER 6 MULTIPLE POLARIZATION CONTROL ELEMENTS MODEL VALIDATION 6.3 State of Polarization Coverage on Poincaré Sphere In the bidirectional configuration, we would like to know if it would be possible to obtain any SOP; i.e. is there a voltage combination within the range of the PPC such that we can reach any target SOP? Unfortunately, as this system does not have a linear model, we cannot use matrix methods for determining if the SOP coverage on the Poincaré sphere [28]. Instead, we can determine SOP coverage by modeling all possible configurations of the system and plotting on the Poincaré sphere. With the DES for the three actuators in this system (Table 6-1), we can predict which SOP can be obtained without having to try all possible combinations experimentally as seen in Figure 6-8. Figure 6-8: The SOP coverage by using three different combinations of actuators as labeled by the cyan, magenta and yellow traces. The DES vectors were taken from experimental data.

125 6.3 STATE OF POLARIZATION COVERAGE ON POINCARÉ SPHERE 105 (a) (b) (c) Figure 6-9: Three separate views of the Poincaré sphere for (a) the plane, (b) plane and the (c) plane. The nonoverlapping areas suggest regions where certain actuator combinations cannot reach. As can be seen from three different views of the Poincaré sphere in Figure 6-9, there are no uncolored regions on the Poincaré sphere that would indicate that a SOP is not reachable. We can attempt to verify this experimentally through the same procedure as in the theoretical method; we step through the control parameters of one actuator, and at each step we vary the

126 106 CHAPTER 6 MULTIPLE POLARIZATION CONTROL ELEMENTS MODEL VALIDATION control parameter of another actuator. This is done for all possible combinations of the PPC. The results are shown in Figure Figure 6-10: The experimental data for stepping and varying three different combinations of actuators.

127 6.3 STATE OF POLARIZATION COVERAGE ON POINCARÉ SPHERE 107 (a) (b) (c) Figure 6-11: The experimental data as collected from operating two actuators at a time for three separate views of the Poincaré sphere: (a) the plane, (b) plane and the (c) plane. The nonoveralapping areas suggest regions where certain actuator combinations cannot reach. In this case, we do observe white spots through the Poincaré sphere on the curves in Figure However, this is due to a sparsely populated dataset; i.e. the datapoints taken were not dense enough to cover the entire Poincaré sphere. However, from the data collected, we can still conclude that these SOPs do appear reachable and that there are no definite uncolored region that would indicate that there are SOPs that are unreachable by the combination of actuators.