DESIGN OF A FULLY AUTOMATED POLARIMETRIC IMAGING SYSTEM FOR REMOTE CHARACTERIZATION OF SPACE MATERIALS. A Thesis. Presented to

Transcription

1 DESIGN OF A FULLY AUTOMATED POLARIMETRIC IMAGING SYSTEM FOR REMOTE CHARACTERIZATION OF SPACE MATERIALS A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Master of Science Jeff Petermann May, 2012

2 DESIGN OF A FULLY AUTOMATED POLARIMETRIC IMAGING SYSTEM FOR REMOTE CHARACTERIZATION OF SPACE MATERIALS Jeff Petermann Thesis Approved: Accepted: Advisor Dr. George Giakos Dean of the College Dr. George K. Haritos Committee Member Dr. Arjuna Madanayake Dean of the Graduate School Dr. George R. Newkome Committee Member Dr. Kye-Shin Lee Department Chair or School Director Dr. Alex De Abreu-Garcia Date. ii

3 ABSTRACT The work done in this study supported the development of the United States Air Force Research Laboratory (AFRL) Polarimetric Multifunction Imaging Platform. Polarimetry can be used for identification, classification, and analysis of a material s optical properties using polarization as the key discriminator. Polarimetry has great potential to be used in space applications because it has the useful ability to interrogate objects at great distance using relatively low power. In a polarimetric system, the determination of material properties are not dependent upon the magnitude of the returned signal, rather the transformation of the polarization that the incident light has undergone after interacting with the object. The presented system is an automated full polarimetric system utilizing liquid crystal devices which can be controlled electronically with no moving parts. In this study several materials typically used in space applications were tested for their full Mueller Matrix at varying aspect angles. Additionally, Mueller Matrices were decomposed into constituent matrices representing key material characteristics and several figures-of-merit were calculated. The outcome of this study provides a wealth of information in terms of system design, calibration techniques, testing, and applying remote measuring methodologies. iii

4 DEDICATION This work is dedicated to my family. My wife Kari has endured countless hours of my absence and has been my biggest supporter. I would truly not have been able to complete this without her efforts. I also dedicate this to my children Edward, Elise, and my third that is on the way. Even though they do not realize the sacrifice they have made, they have made it nonetheless. My mother-in-law has covered us too many times to count when our schedules were impossible. My extended family and friends have shared in the effort as well through a wide variety of means. I owe them all my eternal gratitude for their patience and support. iv

5 ACKNOWLEDGEMENTS First and foremost I would like to acknowledge the patience and guidance of Dr. George Giakos. I appreciate the opportunity to work on this project and for his continued support throughout the last two years. I have taken away a great deal of knowledge that I might not have been exposed to otherwise. His dedication to research has been an inspiration. I also thank Peter Crabtree, R H. Picard, Phan D. Dao and Patrick McNicholl (AFRL Space Vehicle Directorate) for offering me this distinct opportunity to participate in the development of their system. I would also like to thank Dr. Sheffer in the biomedical department at the University of Akron for his assistance with statistics. I have had the good fortune to work with some excellent people in our group. Thank you to my fellow researchers for their team attitude, support, and enthusiasm for the systems we have been designing. Specifically Stefanie, Chaya, Divya, and Suman it has been an absolute pleasure to work together. v

6 TABLE OF CONTENTS Page LIST OF TABLES LIST OF FIGURES. viii x CHAPTER I. INTRODUCTION Overview Technical Challenges Objectives of the Study Literature Review II. FUNDAMENTALS OF LIGHT Overview Electromagnetic Optics and Polarization Photon Optics Polarization Optics Stokes Parameters Mueller Matrices Data Reduction Technique Decomposition of Mueller Matrices Stokes Analysis. 32 vi

7 III. EXPERIMENTAL PROCEDURES AND TECHNIQUES Overview System Components and Calibration System Testing NI Labview Interface / Automation 51 IV. RESULTS AND DISCUSSION Overview Material Samples Mueller Matrix Results Decomposition Results Organized by Material Decomposition Parameters Organized Property Stokes Analysis Statistical Analysis Evaluation of System Noise V. CONCLUSION. 91 REFERENCES APPENDICES. 98 APPENDIX A. FULL MUELLER MATRIX CALCULATIONS USED IN CALIBRATION APPENDIX B. COMPLETE MATRIX DATA FOR ALL TESTS APPENDIX C. MATLAB CODE FOR ONE-WAY ANOVA TESTS vii

8 LIST OF TABLES Table Page 2.1 Electromagnetic Spectrum and Key Parameters Examples of Degenerate Forms of the Polarization Ellipse MM Elements and the Optical Properties They Describe Meadowlark LC Rotator Specifications Summary of Required LC Device Parameters for the Generator Summary of Required LC Device Parameters for the Analyzer Statistical Analysis of Calibration Results Statistical Analysis of Individual MM Elements Summary of Automation Routine with System Inputs/Outputs The Breakdown of a Single States Measurement Cycle Standard ANOVA Table ANOVA Results for all Materials/Properties (All Angles) ANOVA Results for Properties at Selected Angles (All Materials) Noise Summary by Source B1 Mueller Matrices for Amorphous Silicon B2 Diattenuation Matrices for Amorphous Silicon 106 B3 Depolarization Matrices for Amorphous Silicon B4 Retardance Matrices and Decomposed Coefficients for AS viii

9 B5 Intensity by System State for Amorphous Silicon B6 Mueller Matrices for Poly-Silicon B7 Diattenuation Matrices for Poly-Silicon B8 Depolarization Matrices for Poly-Silicon B9 Retardance Matrices and Decomposed Coefficients for PS 113 B10 Intensity by System State for Poly-Silicon B11 Mueller Matrices for Mylar B12 Diattenuation Matrices for Mylar B13 Depolarization Matrices for Mylar B14 Retardance Matrices and Decomposed Coefficients for Mylar 118 B15 Intensity by System State for Mylar B16 Mueller Matrices for Kapton B17 Diattenuation Matrices for Kapton. 121 B18 Depolarization Matrices for Kapton B19 Retardance Matrices and Decomposed Coefficients for Kapton. 123 B20 Intensity by System State for Kapton ix

10 LIST OF FIGURES Figure Page 2.1 Vertically Polarized Wave Propagating Along the Z Axis Linear Vertical Polarization in the Ex-Ey plane Circularly Polarized Wave Propagating Along the Z Axis Circular Polarization in Ex-Ey Plane Diagram of an Object s MM Altering the Incident Polarization Optical Train of Mueller Matrices Optical Train of MM s divided into a Generator and Analyzer Rotated Polarization Ellipse by an angle ψ Operational Principle of LC Devices System Arrangement Used for Calibration Meadowlark LC Rotator Generator Calculated Vs. Actual Intensities Analyzer Calculated Vs. Actual Intensities (P, M, and V) Analyzer Calculated Vs. Actual Intensities (H, RCP, and LCP) Experimental Arrangement for Qualification Tests Simplified Block Diagram of the Measurement Procedure Experimental Arrangement for Space Material Testing Photo of the Experimental Arrangement for Space Material Testing.. 58 x

11 4.3 Additional System Images Experimental Mueller Matrices of Materials at 0⁰ Incidence MM Elements for Amorphous Silicon at Selected Angles MM Elements for Poly Silicon at Selected Angles MM Elements for Mylar at Selected Angles MM Elements for Kapton at Selected Angles Decomposition Parameters vs. Object Angle (AS) Decomposition Parameters vs. Object Angle (PS) Decomposition Parameters vs. Object Angle (MY) Decomposition Parameters vs. Object Angle (KA) Diattenuation Magnitude vs. Object Rotation Angle for all Materials Net Depolarization vs. Object Rotation Angle for all Materials Total Retardance vs. Object Rotation Angle for all Materials DOP vs. Object Rotation Angle for all Materials DOLP vs. Object Rotation Angle for all Materials DOCP vs. Object Rotation Angle for all Materials Ellipticity and Eccentricity vs. Object Rotation Angle for all Materials Major Axis of Polarization Ellipse vs. Object Rotation Angle Histogram Examples for Co-polarized Generator-Analyzer Standard Deviations of MM Elements for Amorphous Silicon at 0⁰ Standard Deviations of MM Elements for Poly Silicon at 0⁰ Standard Deviations of MM Elements for Mylar at 0⁰ Standard Deviations of MM Elements for Kapton at 0⁰ xi

12 4.26 Box Plot for Diattenuation Results at -0.07⁰ Box Plot for Diattenuation Results at Normal Incidence Box Plot for Diattenuation Results at +0.07⁰ Box Plot for Depolarization Results at -0.07⁰ Box Plot for Depolarization Results at Normal Incidence Box Plot for Depolarization Results at +0.07⁰ Box Plot for Total Retardance Results at -0.07⁰ Box Plot for Total Retardance Results at Normal Incidence Box Plot for Total Retardance Results at +0.07⁰ Histogram of Avg. Pk-Pk System Noise (laser off) by Detector Gain Signal to Noise Ratio of HH State by Material vs. Object Angle xii

13 CHAPTER I INTRODUCTION 1.1 Overview The purpose of this study is the design, development, calibration, and testing of the United States Air Force Research Laboratory (AFRL) Polarimetric Multifunction Imaging Platform aimed to assist in the remote characterization of space materials. Imaging of the polarization states of light offers distinct advantages for a wide range of detection and classification problems, due to the intrinsic potential of optical backscattering to provide high-contrast for different polarization components of the backscattered light. Polarized imaging can yield high-specificity images under low-light conditions, in scattering media, or cluttered target environment [1], [12-25]. A large variety of intensity patterns can be observed by varying the polarization state of the incident laser light and changing the analyzer configuration. The acquired polarimetric signals provide imaging information related to the object s material composition, object geometry, and molecular and chemical composition of the target. Most natural or man-made objects exhibit significant depolarization due to diffuse and non-specular reflection [1, 17]. Depolarization is defined as the mechanism of transforming polarized light into 1

14 unpolarized. This can be quantified in terms of a decrease in the degree of polarization (DOP). Therefore, the concept of DOP, which is used to measure the randomness of the polarization of a light beam, has been applied to material characterization in this study for depolarizing cases. One of the unique features of the optical polarimetric platform is its operation under single-pixel detection principles with varying the aspect angle of the object. Single-pixel detection allows one to focus on only a few pixels of the target, obtaining information, with high scatter rejection, decoupled from any interfering signals (noise) that may arise from the adjacent pixels of the object. Therefore, it can be a very effective method to detect polarimetric signatures from cluttered or unresolved objects, with high sensitivity and high background rejection. On the other hand, detection of objects under different apect angles provide enfanced identification and discriminaton signatures. This study consists of the following phases: i) Design, development, calibration and testing of the AFRL multifunctional polarimetric platform operating at 1065nm. ii) Electronic control of the liquid crystal-based polarimeter, system automation, and detailed calibration procedures. iii) Active backscattered Mueller matrix analysis of space materials under different aspect angles. 2

15 The experimental results of this study indicate that enhanced discrimination signatures can be obtained by analyzing polarimetric properties such as depolarization, diattenuation, and retardance of the sample materials. In addition, several space and man-made materials are shown to exhibit distinct depolarization signatures which can be used to characterize, classify, and identify those materials. In summary, the experimental results presented in this study indicate that space objects and man-made materials exhibit distinct depolarization signatures, which can be used to characterize, classify, and identify those materials. The outcome of this study will advance analytic and predictive space remote-sensing capabilities that can be introduced to enhanced LADAR detection and image quality measures, as well as introduce unique detection and imaging methodologies. 1.2 Technical Challenges Polarimetry utilizes the polarization of electromagnetic radiation as a metric for characterizing radiation-matter interaction. To conduct full Mueller matrix polarimetric measurements several issues must be overcome such as: i) Acquisition of a minimum of 16 measurements with ideally stationary object conditions. Therefore, it is of paramount significance to design a system without optical moving parts offering immunity to vibrations or mechanical movements of the optical components. 3

16 ii) Typical polarimetric systems often use optics that must be physically rotated or added to the system which reduces measurement speed [15]. iii) Often the angle of incidence and hence the object s aspect angle is unknown with sufficient precision. iv) Detailed calibration procedures and enhanced statistics require a large volume of acquired data with long processing times. Therefore, complete automatization of the data acquisition and processing functions is of extreme significance. 1.3 Objectives of the Study: The design used in this study implements several strategies for efficient polarimetric measurements and addresses the challenges previously mentioned. The system used in this study can be classified as a complete sample measuring system [31]. Sample measuring polarimeters relate scattered or reflected light to the incident light by evaluating material properties through determination of the object s Mueller Matrix. In a complete system, the full Stokes vector can be determined as well. Incomplete polarimeters cannot determine the full Stokes vector or Mueller Matrix of the object. The issues of vibration and the need for non-rotating optical components are addressed by utilizing liquid crystal (LC) devices. Several successful polarimeters have been constructed using the Dual Rotating Retarder method, however as the name implies the retarders are driven by motors [22, 32, 33]. In contrast, LC devices are electronically controlled. Other systems may also 4

17 require manual adjustment or physical insertion/removal of physical components from the system which can increase measurement time significantly [15]. The system utilizes a National Instruments (NI) Labview and a Compact DAQ system to interface with the LC devices and the optical receiver. Electronic control also reduces the data acquisition time of each measurement. In this study, measurement time was limited by the response time of the LC device, desired statistics, and the user-selected sampling rate. Several space materials were tested to determine if unique signatures could be obtained for remote identification. Algorithms for identification of unknown objects from databases of known materials is not addressed in this study directly, however further detail can be found in references [1]. In a typical complete polarimetric system, the material characteristics of an object are defined by its Mueller Matrix (MM). In this study, additional post-processing was used to extract more specific information related to several key material properties through Mueller Matrix decomposition [16]: i) Diattenuation: Absorption of specific polarizations of light. ii) Retardance: Phase shift introduced to specific polarizations of light after interaction with the sample material. iii) Depolarization: Loss of the original polarization of the incident optical field after interaction with the sample material. The above parameters were tested not only for the case of normal incidence, but also when the object undergoes small angular rotations (aspect angle). In 5

18 addition, the MM itself contains material information itself in specific elements or groups of elements such as isotropic absorption and polarizance. 1.4 Literature Review To lay the proper groundwork for discussing the results of this study, first a look at existing systems and theoretical development is appropriate. From an applications standpoint, space-borne remote sensing applications have traditionally involved radars, or more specifically advanced synthetic aperture radars (SAR) [2]. Within this scope, RF Polarimetry has proven effective for characterizing Earth features such as hydrological process, forecasting, agricultural activities planning, and climate model validation [3]. Some work has even been done to investigate the use of Polarimetry in X-ray bands [4]. The focus in this study however, is in the optical spectral regions. Links between RF Polarimetry and Optical Polarimetry including the formalism for carrying out measurements in each are detailed by Boerner and Colin [5, 6]. Many notable advances of optical polarimetry for remote sensing applications have been achieved in the area of image enhancement. In Hooper et al. an airborne imaging system consisting of multiple cameras could take images remotely in several spectral bands and using three polarization states [7]. This method did not utilize full polarimetric measurements; however Degree of Linear Polarization (DOLP) images were shown to highlight features of geological interest with greater specificity. Similarly, Wang et al. utilized polarimetric 6

19 properties to estimate refractive indexes of interrogated objects [8]. As a further enhancement, algorithms were explored to evaluate the angle of incidence within the framework of the pbrdf, or polarization bi-directional reflectance distribution function, commonly used in many geophysics studies. Hyde et al. provides greater details of the pbrdf used in that study [9]. In an earlier study, Giakos demonstrated image enhancement that utilized a combination of both multispectral imaging and the DOLP to enhance a single image [10]. Further work suggested the possibility of depth resolution from this technique in biomedical applications [11]. Recently, Giakos et al. [35-37] explored the pbrdf principles as part of their efforts on the development of efficient multispectral space surveillance Ladar sensor design architectures for enhanced object detection and characterization. The same authors [35-37] introduced new polarimetric wavelet detection principles applied to the backscattering characteristics of space materials in the near infrared. This resulted in remote characterization of space materials and structures with enhanced discrimination, localization, and high-dynamic range while maintaining uncompromised sensitivity. A novel feature of their study is the introduction of a new formalism, where the Mueller Matrix elements as well as the Mueller Matrix decomposition are expressed as a function of the aspect angle, so that an angular Mueller Matrix description of the object, and corresponding wavelet analysis, can be achieved. Specific to system level development, several studies have contributed great theoretical significance to this study. First, Liu et al. developed a system 7

20 that utilized LC devices, Labview, PC Control, and CCD Images [12]. In this study the utility of polarimetry was demonstrated by extracting details not seen by non-polarization-sensitive imaging. The system described consisted of a sample, generator, and analyzer which were all fixed with 45⁰ between the generator and analyzer but object rotation was not studied. Another system implemented by Yao applied to Optical Computed Tomography (OCT) [13]. In this work, the incident and reflected signal were along the same path instead of 45⁰ as in Liu and others. Two other LC systems were discussed in Baba et al. and Bueno [14, 15]. The focus of their work was related to system calibration and accuracy. Both studies conducted tests on known samples such as air, reflectors, and linear polarizers. Both were helpful in establishing benchmarks for calibration in this study. The Baba system was fixed at 45⁰ with respect to the generator and analyzer and the Bueno system required manual insertion of two components. A powerful technique used in this study is the Lu-Chipman algorithm which breaks MM data down into constituent matrices, each representing specific properties [16]. Shortly after the Lu-Chipman publication, Brehonnet et al. tested materials for isotropic and anisotropic depolarization using decomposition [17]. The net depolarization, also referred to as the depolarization index, was capable of material identification among a group of materials consisting of dielectrics, granite, polished steel, and nylon. Identical materials of varying roughness demonstrated a lower depolarization index, which supports findings in this study. 8

21 Later, work done by Twietmeyer-Chipman and Ghosh et al. further demonstrated the usefulness of the algorithm by analysis of retinal scans and known scattering structures, respectively [18, 19]. To gain insight into the physical results represented by the decomposed parameters as well as the individual Mueller Matrix elements, a very complete theoretical summary is given in two works by Jenson and Schellman [20, 21]. Considerable significant research efforts have been undertaken by D. Goldstein (AFRL) who investigated the use of Polarimetry for material classification [1, 22]. The subject of [1] is actually a summary of five years worth of research. Several design issues and results in that research can be related to work in this study such as studying polarimetry in a monostatic geometry, reflectance data for varying surface roughness, identification of polarizance behavior in relation to angle of incidence, and the development of algorithms that can be used to identify materials under test from polarimetric data. 9

22 CHAPTER II FUNDAMENTALS OF LIGHT 2.1 Overview To provide further framework for discussion of the results obtained in this study, properties of light propagation and interaction with materials will be presented. Light is fundamentally electromagnetic radiation so its propagation can be described by Maxwell s equations. Analysis in this context is often referred to as electromagnetic optics. Since the focus of this work is on space materials plane-wave descriptions of light are appropriate since propagation in free space yields plane wave solutions. Furthermore, polarization of light can be intuitively described in the context of electromagnetic optics, which will be fundamental to methodology used in this study. In other situations, describing light based on energy levels is very useful. This approach is generally referred to as Photon Optics or Quantum Optics [23]. Photon optics treats light as particles (photons) of discrete energy packets or quantums. Unlike photon optics, electromagnetic optics descriptions of light are mostly used when the wavelength of light is larger than features of the object. In many cases, photon optics can simplify analysis by avoiding rather complicated and lengthy expressions associated with wave propagation. Table (2.1) shows the Electromagnetic Spectrum with several key parameters, namely 10

23 frequency, photon energy, and generally accepted classifications of bands and/or sub-bands [23]. Table 2.1: Electromagnetic spectrum and key parameters. Another approach of describing light interaction with matter is referred to as polarization optics [23]. This is the key methodology used in this study and describes light based upon its polarization. Stokes parameters describe energy fluxes of specific polarizations and can completely quantify the associated timeaveraged electric fields [28]. An advantage of polarization optics is that complicated calculations can be carried out in an efficient manner by arranging the Stokes parameters into a 4x1 vector called the Stokes vector [28]. Materials can be modeled using Mueller Matrices (MM) which describes the transformation of incident to resultant light after interaction with the object [28]. Generally the MM of a material is unknown and needs to be determined. One framework for determining the MM is the data-reduction technique [31] which can be applied to a wide range of experimental set-ups. Finally, further insight into material properties can be obtained by decomposing the MM into 11

24 constituent matrices with specific physical meanings. The following sections highlight the mathematical treatment of light based upon the preceding methodologies. 2.2 Electromagnetic Optics and Polarization Electromagnetic optics is a description of light based on Maxwell s equations and treats light in terms of waves. Rearrangement of Maxwell s equations results in wave equations. Light propagation in free space can be expressed through a second order homogeneous wave equation, as shown in Eq. 2.1: The solution of this equation, in terms of the electric field waves, propagating in z direction, with amplitude oscillating in the x-y plane is shown using cartesian coordinates in Equation (2.2). Note that electromagnetic waves consist of both electric and magnetic field waves. Typically, solutions assume propagation of a single (monochromatic) timeharmonic frequency (ω); however in reality light can consist of multiple frequencies, where the superposition principle applies [23]. Polarization describes the elliptical pattern of the electric field s orientation as it propagates in time when looking into the beam along the axis of 12

25 propagation. The magnitudes (E 0X and E 0Y ) and relative phases (Φ X and Φ Y ) determine the polarization. Figure (2.1) and Figure (2.2) show linear polarization in the vertical direction. Figure 2.1: Vertically Polarized Wave propagating along the z axis. Figure 2.2: Linear Vertical Polarization in the Ex-Ey plane. 13

26 Figure (2.3) and Figure (2.4) show circularly polarized light for normalized electric fields as they propagate along the z axis and as viewed in the Ex-Ey plane, respectively. Figure 2.3: Circularly Polarized Wave propagating along the z axis. Figure 2.4: Circular Polarization in the Ex-Ey plane. 14

27 In general, polarization patterns follow the equation of an ellipse, where linear and circular polarization states are simply degenerate forms of the ellipse. Table 2.2 lists further examples of polarization along with their associated parameters. Electric field components are normalized to unity while the phase difference is listed in multiples of radians for one wavelength and repeats thereafter [28, 29]. Table 2.2: Examples of degenerate forms of the polarization ellipse. Polarization E 0X E 0Y δ=δy-δx Linear Vertical 0 1 Any Linear Horizontal 1 0 Any Linear +45⁰ 1 1 π Linear -45⁰ Right Circular 1 1 π/2 Left Circular 1 1 3π/2 Another important quantity is the Poynting Vector which indicates the flow of power given in Equation (2.3) and the intensity is the magnitude of the Poynting Vector. For optical experiments, the intensity of the wave is a key parameter. However, H is related to E by the impedance of the medium, η. By taking the time average and magnitude, the intensity becomes Equation (2.4) [23]. 15

28 2.3 Photon Optics In some cases, the behavior of light interacting with matter can best be described by treating light as discrete quantities of energy. In this case, light is physically described as being made up of particles called photons. This description of light found success in explaining certain phenomena that electromagnetic optics had failed to describe previously such as blackbody radiation, photoelectric effects, and Compton Effects [24]. The term photon optics is a subset of a larger classification often referred to as Quantum Mechanics [23]. The photon energy (Ep) is given in Equation (2.5) where h is plank s constant, which is x [Js], and v is the frequency. This can also be written in terms of the intrinsic speed of light (c) and the wavelength (λ) [25]. Different styles of photonic detectors used in experiments all read voltages that are proportional to the intensity incident on the detector. Equation (2.6) shows how this intensity can be related to Equation (2.5). The photon energy for one photon, divided by time, represents power. Then multiplying this by the # of photons divided by area yields a power density which is the intensity (I). A model for propagation through materials often used in conjunction with photon optics is the Beer-Lambert law shown in Equation (2.7), which describes the linear absorption of a photon by a material [26]. The initial intensity, Io, is 16

29 reduced exponentially as photon energy is absorbed by a material. The experimentally found coefficient k(v) is the molar extinction coefficient of a material and is generally frequency dependent with units of [L/(mol*cm)]. The variable c is the molar concentration with units of [mol/cm], which is different than the c used in Equation (2.5). Since the absorption is frequency dependent, the intensity at a point z along the path of propagation is also frequency dependent. 2.4 Polarization Optics The key methodology used in this study is Polarization Optics. Polarization Optics provides an efficient means of describing light based on energy fluxes along specific axis. The polarization can be uniquely defined by three specific polarizations which are described by the Stokes vector and their relative intensity with respect to the total intensity. There are two key advantages inherent in polarization optics. First, the Stokes vector can accurately describe partially polarized light as well as the ideal case of completely polarized light. Secondly, since polarization is the basis, intensity plays a decreased role in analysis. Stokes vectors are normalized to the total intensity and also represent only relative intensities along specific polarizations. This is advantageous in many experiments where it is not practical to account for all optical power in the system, for example when only a fractional backscattered signal is detected. Hence valuable information regarding the object is maintained in the detected signal. 17

30 2.4.1 Stokes Parameters As stated, the Stokes vectors characterize light in terms of polarization. Yet polarization is of course dependent on field quantities, and hence the Stokes vectors can be derived from electric fields. The premise of using polarization as a basis comes from the landmark paper by George Gabriel Stokes in which he stated that any particular polarization can be constructed from the combination of other polarizations and is optically equivalent [27]. To derive the Stokes vector from electric fields [28], consider a plane wave in a Cartesian basis with an electric field defined by Equation (2.8). Φ Φ The derivation begins by rewriting Equation (2.8) using the identity cos(α+β) = cos(α)cos(β)-sin(α)sin(β) as shown in Equation (2.9). Φ Φ Φ Φ Next, Equation (2.9a) is solved for cos(ωt) and sin(ωt) and substituted into Equation (2.9b). The result in Equation (2.10) is found using the identities: i) cos(α)sin(β) = (sin(α+β)-sin(α-β))(1/2) ii) sin(-α) = -sin(α) 18

31 Φ Φ Φ Φ Φ Φ Φ Φ Equation (2.11) is obtained by squaring both sides of each equation in (2.10) and then summing them together where Φ=Φ y -Φ x : Φ Φ This is the equation of the polarization ellipse. Since instantaneous values of Ex(t) and Ey(t) cannot be reasonably measured, the time average is taken. For example the time average of Ex(t) is (1/2)*E 0X^2. More generally, this is stated in Equation (2.12) where < > indicates the time average. In addition every term is multiplied by 4*E 0X^2*E 0Y^2. This leads to an expression of perfect squares and is shown in Equation (2.13). Φ Φ The Stokes parameters are the terms inside the parenthesis and are labeled as shown in Equation (2.14). The Stokes vector is constructed by arranging the Stokes parameters into a column matrix referred to as the Stokes vector, where S represents the source [29]. 19

32 Φ Φ Hence the Stokes parameters define the polarization ellipse, and are based upon time averaged measureable quantities. It is clear that circular and linear polarizations are simply special cases of elliptically polarization. Summarizing the four Stokes parameters: S 0 is the total intensity of the light. S 1 describes the amount of linear horizontal or vertical polarization. S 2 describes the amount of ±45 polarization. S 3 describes the amount of right/left circular polarization within the beam. The Stokes vectors for some special degenerate forms of the polarization ellipse are listed below where the subscripts H, V, P, M, R, L refer to Horizontal, Vertical, +45⁰, -45⁰, Right-Circular, and Left Circular respectively [29]. 20

33 2.4.2 Mueller Matrices Using Stokes vectors for modeling light-material interactions involves representing objects by a Mueller Matrix (MM). The MM represents the transformation of the incident light to the resulting light in terms of polarization. Suppose that a source S, contacts a new barrier, medium, or material. Some light may be reflected, transmitted, or scattered. The polarization of S will have undergone a transformation that can be modeled with a 4x4 matrix, M, which is the MM. S is the reflected, scattered, or transmitted light which has been modified. Equation (2.14) shows this mathematically and schematically in Figure (2.5) for the case of transmission. y S S y z Object (M) z x Figure 2.5: Diagram of an Object s MM altering the incident polarization state. For actual applications there are generally many components that light will pass through between the source and the detector, each with its own MM. To 21

34 account for the total transformation of the light incident on the detector these MM s will need to be multiplied together and the proper order of multiplication is required [24] as demonstrated in Figure (2.6). y S S S y z z x M1 M2 Figure 2.6: Optical Train of Mueller Matrices. The incident light is S. After interacting with the first object (M1) the light is transformed to S by Equation (2.16). S exits M 1 and interacts with the second object (M 2 ) where it is transformed to S by Equation (2.17). Inserting Equation (2.16) into Equation (2.17) yields the total transformation of light from input to output and can be written as Equation (2.18). This can be extended to any number of objects and MM s. 22

35 It should be noted that experimentally found MM s should be tested to ensure that they are physically realizable. Numerous constraints have been developed, however three are shown in Equation (2.19) as examples [28, 30]. A further explanation behind the physical meaning of individual Mueller Matrix Elements will be detailed in Section Data Reduction Technique Generally, the Mueller Matrix of an object is unknown and needs to be solved. To derive all 16 elements of a MM, a minimum of sixteen linearly independent equations are needed. A typical detector only measures the first element of the Stokes vector, S 0, which is the total intensity. As a result a minimum of sixteen measurements will need to be taken. Efficient calculation of the MM can be done using a framework called the data reduction technique [31]. It is useful to separate the optical train of components into two parts, namely the generator and analyzer. The generator consists of all components, including the source prior to interaction with the object while the analyzer refers to all components after the object including the detector. By altering the generator and analyzer to known polarization configurations in a sequence of at least sixteen measurements, a suitable system of equations can be generated. This is shown schematically in Figure (2.7). 23

36 y S Source z Object M ob x M G1 M G2 M A1 M A2 Detector Generator Analyzer Figure 2.7: Optical Train of MM s divided into a Generator and Analyzer. The generator consists of the source and components M G1 and M G2 while the analyzer consists of M A1, M A2, and the detector. The generator is defined by a Stokes vector representing its output while an analyzer vector is defined for the particular polarization that it measures maximally [31]. For example, an analyzer that outputs maximum intensity for an incident linear horizontal (H) polarization would be [1, 1, 0, 0] T. Equivalently, a generator that outputs H can be described by the same vector. The total system can then be represented as Equation (2.20) for each state q. Each q-state intensity measurement typically results from a unique generatoranalyzer combination. Note that the value Pq refers to the intensity (S 0 ) since typical detectors cannot measure S 1 -S 3. 24

37 Since the detector only measures the intensity(s 0 ), it is possible to rewrite Equation (2.20) as Equation (2.21) where the generator analyzer combination is reordered in a polarimetric measurement matrix W q along with a flattened Mueller Vector M (qx1) [31]. Note that under other circumstances this would not be permitted since it changes the order of matrix multiplication. Also, the result of an Analyzer-Generator combination is the direct product of A q,j and G q,k. Hence, W would be 16x16 for the case of q=16. In that scenario, P q would be a 16x1 vector and the MM has been rearranged into a 16x1 Mueller Vector. As long as W contains a minimum of 16 linearly independent rows, all 16 elements of the objects MM can be determined taking the inverse of W as shown in Equation (2.22). For an over-determined system where W is not square, the pseudo-inverse can be taken as in Equation (2.23) [31]. The primary advantage of this method is that it is quite general and independent of specific types of sources and detectors. That is, they can be identified by a particular Stokes vector that is either ideal or non-ideal as determined by calibration procedures. Also Generator-Analyzer combinations 25

38 are arbitrary and can occur in any order provided they generate linearly independent measurement matrices. Lastly, a measurement system is not restricted to a specific number of states and can accommodate an overdetermined system which may increase the accuracy of measurements Decomposition of Mueller Matrices Since an object s Mueller Matrix represents a transformation of light from one state of polarization to another, each element represents optical properties of the material. Table (2.3) on the following page lists each MM element along with its physical meaning [20, 21, 30, 38]. Table 2.3: MM Elements and the optical properties they describe. M11 Isotropic Absorption M12 -Linear Dichroism(0) M13 -Linear Dichroism(45) M14 Circular Dichroism M21 -Linear Dichroism(0) M22 Isotropic Absorption M23 + Circular Birefringence M24 + Linear Birefringence(45) M31 -Linear Dichroism(45) M41 Circular Dichroism M32 - Circular Birefringence M42 - Linear Birefringence(45) M33 Isotropic Absorption M43 + Linear Birefringence(0) M34 - Linear Birefringence(0) M44 Isotropic Absorption Isotropic absorption describes the attenuation of the incident light s intensity after interacting with an object. Dichroism, or diattenuation, represents unequal absorption of orthogonally polarized components of the electric field [16]. For many materials, absorption is not equal in all directions. MM elements in the first 26

39 row and column, excluding M 11, relate absorption to specific Stokes vector elements, namely linear vertical (V), linear +45⁰ (P), and circular polarizations. In addition, objects can induce phase shifts to electric field components which are generally called retardation. Birefringence refers to unequal retardation and refractive indexes of specific polarizations along the same principle axis [16, 26]. A powerful tool in further evaluation these parameters involves decomposing the object s MM into constituent matrices of diattenuation and retardance. In addition, different materials exhibit the behavior of depolarizing light. For example, light from a common bulb or the sun is randomly polarized. Its electric field demonstrates no particular preference in its orientation. If a perfectly polarized light interacts with a depolarizing object, its electric field begins to oscillate randomly either completely or more commonly for some percentage of time. The algorithm for splitting a MM into these three constituents is called the Lu-Chipman Algorithm and is represented by Equation (2.24) where M, M R, and M D represent depolarization, retardance, and diattenuation respectively [16]. One method to derive the constituent matrices, adapted from Goldstein, begins with the diattenuation magnitude (D) in Equation (2.25) using transmittances (T) [28]. 27

40 To relate transmittances to Mueller Matrix elements, consider the simple case described in Equation (2.15) where the resultant transmittance is related to the first row of the MM and the Stokes vector as shown in Equation (2.26) where on the right-hand side M 2,3,4 and S 1,2,3 with the over-bar refer to a portion of the MM row and Stokes vector respectively [28]. Since the operation in the numerator of the rightmost term is a dot product, the minimum and maximum are M (-S 0 ) and M (S 0 ) respectively which gives the minimum and maximum for the transmittance. After inserting this result into Equation (2.26) and rearranging, D is solved in terms of MM elements as shown in Equation (2.27). This is the diattenuation magnitude which is a useful coefficient representing a material characteristic of the object [16, 19]. The result from Equation (2.27) can be rewritten as a column vector as shown in Equation (2.28) which represents diattenuation along specific polarizations [28]. To solve for the complete diattenuation matrix, the possibility of an object polarizing incident unpolarized light must be accounted for. This behavior is termed polarizance and can be arranged in a vector (P) as shown in Equation 28

41 (2.29). The Polarizance Vector is the first column of the diattenuation matrix, which can be written in the general form of Equation (2.30) [28]. The solution of M D is not trivial but it can be shown that the full diattenuation matrix is equal to the expression in Equation (2.31) where the coefficients a and b are given in Equation (2.32) and Equation (2.33) respectively and D= D [28]. With the diattenuation matrix known, the retardance matrix can be solved by Equation (2.34). This result is written explicitly in Equation (2.35) in terms of MM elements [28]. 29

42 The total retardance is another useful coefficient that will be used in analysis later. Its solution is shown in Equation (2.36), or which there are two cases [28]. To consider the effects of depolarization rearrange Equation (2.24) to isolate M R and M as shown in Equation (2.37) where M represents the joint behavior the depolarization and retardance matrices [28]. Then the task is to decompose M and solve for M. M can be written in terms of retardance and depolarization as shown in Equation (2.38) [28]. The vector P can be derived using the definitions in Equation (2.28) and Equation (2.29) and is shown in Equation (2.39) where m is the lower 3x3 submatrix (does not include the first row and column) of the MM. The product of the two 3x3 matrices is simplified with the notation in Equation (2.40). 30

43 In Equation (2.40) m is known since it is the lower 3x3 submatrix of M which was determined in Equation (2.37). The three diagonal elements of m are depolarizing coefficients for horizontal or vertical, +45⁰, and circularly polarized light [16, 19]. The net depolarization ( ) is then given by Equation (2.41). An alternative means of calculating this is given in Equation (2.42) [30]. If the eigenvalues of m (m ) T are λ 1, λ 2, and λ 3 then the eigenvalues of m are the square roots of λ 1, λ 2, and λ 3. Then, using the Cayley-Hamilton Theorem an expression for m can be expressed as Equation (2.43) [28]. Thus all components have been derived to construct the constituent matrices. The full derivations are rather involved and complete details can be found in references [16, 18, 19, 28]. 31

44 2.4.5 Stokes Analysis The system developed in this study is known as a complete polarimetric system. In other words all sixteen elements of the objects MM are derived. Conversely, incomplete polarimeters do not solve all sixteen parameters. The goal of an incomplete system may be to only calculate particular elements related to specific physical properties, or to derive a portion of the Stokes vector received at the analyzer. The advantage of this may be a simpler or less expensive system. Whatever the case, valuable information may still be determined from the Stokes vector itself since it describes the polarization ellipse. These will be described qualitatively since it is much more intuitive than the full geometric derivation which was already done in Section (2.4.1). The equation for the Stokes vector in terms of the electric field is repeated in Equation (2.44) where Φ = Φy-Φx [29]. Φ Φ Stokes analysis includes several figures-of-merit which can be understood by bearing in mind Equation (2.24) and the physical meaning of each Stokes element. For example, representing the polarization as a ratio of Stokes parameters (S 1, S 2, and S 3 ) and total intensity (S 0 ) is the Degree of Polarization (DOP) since the total intensity may or may not be completely polarized. The DOP in terms of the Stokes parameters is given by Equation (2.45) [31]. 32

45 This can also be calculated as a function of particular polarizations such as linear or circular polarizations, namely the Degree of Linear Polarization (DOLP) shown in Equation (2.46) or the Degree of Circular Polarization (DOCP) shown in Equation (2.47) [31]. In addition, Figure (2.8) shows the general polarization ellipse, rotated by an angle ψ adapted from Goldstein [28]. The Ellipticity is described as the ratio of the minor axis to the major axis, which is given in terms of Stokes parameters in Equation (2.48). Y Y 2E oy b X a ψ X 2E ox Figure 2.8: Rotated Polarization Ellipse by an angle ψ [28] (w/o permission). 33

46 Another closely related parameter, Eccentricity, is given in Equation (2.49) where e is the Ellipticity. Eccentricity ranges from 0, which describes circular polarization, to 1 which describes linear polarization. Furthermore, the rotation of the major axis can be described by Equation (2.50) [31]. In the case of a complete polarimeter these relations are particularly easy to calculate once the MM has been determined by using Equation (2.14) which is repeated here in Equation (2.51) where S is the received Stokes vector, M is the Mueller Matrix of the object, and S is the incident Stokes vector. Of course for a complete polarimeter, a minimum of sixteen flux measurements will be recorded in determining the MM of the object. Therefore, each measurement state will have its own set of received Stokes vectors. When using the data-reduction technique the full received Stokes vector is not calculated for each state per say, but can be solved for mathematically when the MM is known. Other methods exist for solving only the received Stokes vectors such as the Fourier analysis with a rotating quarter-wave retarder, the Dual rotating retarder method, and the Classical method using a quarter-wave retarder and a polarizer which are detailed in references [28, 32, 33]. 34

47 CHAPTER III EXPERIMENTAL PROCEDURES AND TECHNIQUES 3.1 Overview Traditional optical experiments can be time consuming to execute using traditional rotation mounts and physical optics components. This is especially true for complete MM measurement since a minimum of 16 linearly independent states need to be created. The use of liquid crystal (LC) devices allow for electronic control of experiments which substantially reduce measurement time. One of the goals of this system was to achieve a high degree of automation. The Development of the US AFRL Liquid Crystal Polarimetric Multifunctional Imaging Module is articulated in four stages: i) Calibration: Alignment and calibration of LC components. ii) Testing: Initial measurements and material characterization. iii) Automation: Improved statistical tracking of results and reduced measurement, post-processing, and analysis times. iv) Platform: Next stage prototype. 35

48 3.2 System Components and Calibration To achieve electronic control, Meadowlark LC rotators and retarders were used to control the polarization states of the generator and analyzer optical trains. Table 3.1 details the LC rotator s specifications [39]. Retarder Material Table 3.1: Meadowlark LC Rotator Specifications [39] Nematic liquid crystal with birefringent silica Substrate Material Optical quality synthetic fused silica Wavelength nm (specify) Polarization Rotation 180⁰ or more Polarization Purity 150:1 average Transmittance > 92% with polarized input Transmitted Wavefront Distortion (@ 632.8nm) λ/4 Surface Quality scratch and dig Beam Deviation 2 arc min Reflectance (per surface) 0.5% at normal incidence Diameter Tolerance ±0.005 in. Temperature Range 0⁰ C to 50⁰ C Recommended Safe Operating Limit 500 W/cm 2, CW 300mJ/cm 2, 10ns, visible Electronic control is achieved by applying a potential across a chamber filled with liquid crystal material. With no potential applied, the crystals align themselves as in Figure (3.1a). An applied potential causes the crystals to rotate within the liquid as shown in Figure (3.1b). As they rotate, the index of refraction along principal axis changes (birefringence). The phase shifts that are introduced as a result can be used to create different polarization states [39]. 36

49 Figure 3.1: Operational principle of LC devices [39] (w/o permission). The specific Meadowlark LC Rotator and Retarder models used were the LPR-100 and LRC-100 respectively. They also include a control unit that interfaces with NI Labview (Version 9.0) and supplies the necessary variable square-wave drive voltage to alter the retardation. The laser used was a Pilas 1065nm pulsating laser with collimating optics and its own control unit capable of modulating the pulses. The detector used was a New Focus #2151. All calibration measurements were taken with a Lecroy 7300A oscilloscope. To describe rotations and polarization, the coordinate system used throughout this document places zero degrees along a right-handed cartesian system s x-axis, looking into the beam along the z-axis, where positive rotation is counter-clockwise, and propagation is along the z-axis. Hence, looking into the beam, zero degrees is always at the 3 O clock position and ninety degrees would be at the 12-O clock position. 37

50 A schematic showing the calibration configuration is shown in Figure (3.2). The enclosures for the LC devices are marked with a line indicating the orientation of the device as noted in Figure (3.2). Laser Laser: Pilas, 1065nm Polarizer(G) Θ=45 LC Rotator(G) Fast Axis = 135⁰ (Marking Line) GENERATOR Fast Axis = 45⁰ (Marking Line) LC Retarder(G) Fast Axis = 45⁰ (Marking Line) LC Retarder(A) LC Rotator(A) Fast Axis = 135⁰ (Marking Line) ANALYZER Polarizer(A) Θ=-45 Detector Figure 3.2: System arrangement used for calibration. The MM s representing these devices is as follows. Equation (3.1) is the general MM for a variable retarder where θ is the fast axis location while φ is the retardance. φ φ φ φ φ φ φ φ φ Equation (3.2) is the MM for a polarizer where θ is the maximum transmission axis. The Meadowlark LC rotator is constructed internally by a variable retarder followed by a fixed quarter-wave retarder. The full general equation for this is shown in Equation (3.3) along with a diagram of the device in Figure (3.3) where 38

51 F and S refer to the Fast and Slow Axis. The marking line on the outside of the case refers to the fast axis of the internal fixed quarter-wave retarder as specified by Meadowlark. S variable =90⁰ F fixed = 135⁰ S fixed = 45⁰ F variable(θ2) =0⁰ Lead Wire Meadowlark LC Rotator Meadowlark Marking Line Figure 3.3: Meadowlark LC Rotator The fast and slow axis are fixed at 135⁰ and 90⁰ respectively for the fixed retarder when using the pre-tapped holes for mounting. The variable retarder s fast axis (θ2) is fixed at 0⁰ (always +45⁰ from fixed retarder s fast axis) while the retardance remains variable (φ2). With these values the Mueller Matrix reduces to Equation (3.5) where φ is φ2 from Equation (3.4). Note that the actual angle of rotation is one half of the variable retardance and rotates in the clockwise direction. The operation of the Meadowlark LC rotator is thus completely defined using Equation (3.4) by specifying the variable retardance and the fast axis of the fixed retarder (Meadowlark marking line) which is φ2 and θ 1 respectively. From here on the rotator will be specified with φ, which is φ2 in Equation (3.4), since θ 1 remains fixed in this system. 39

52 φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ (3.4) (3.5) To calibrate the system components are added one at a time starting with the components in the generator in a face-to-face configuration as shown in Figure (3.1). As each component is added the appropriate voltages are applied until the proper rotation or retardation is achieved. Then once the proper generator states are obtained, the analyzer components are added and the process is repeated to determine their proper voltages. For both the generator and analyzer, the following states were determined: V, H, P, M, RCP, and LCP. To verify the state of polarization at the detector, polarizer(a) was always included in the arrangement [28]. One common way to calibrate generator and analyzer states is to use the null intensity method where polarizer(a) is cross-polarized with polarizer(g) and the voltage is adjusted to find a null intensity for a desired state [14]. In theory this is ideal since looking for maximum intensities will always indicate the signal plus the noise. In contrast, with a null intensity only the noise is ideally present. This method is problematic due to the presence of system noise, however small, because it obscures the location of the null. What is actually observed in the 40

53 presence of noise is the intensity dips below the noise floor at some position θ of polarizer(a) and then reappears after a small change in θ when the intensity becomes greater than the system noise again. Since it is nearly impossible to eliminate all system noise the true null is difficult to observe. As a result, a second technique called the method of swings was used in conjunction with the null intensity method [34]. This involves setting polarizer(a) at some angle +/- dθ about the suspected null. When the intensities are identical for both the +/- dθ s, the true location of the null is known to be the θ that is equidistant to the +/- values. Note that this method works even if a slight ellipticity exists in linear polarization states. A summary of the required generator retardances and their corresponding voltages are listed in Table (3.2). The laser s polarization remains vertical while polarizer(g) s transmission axis is fixed at 45⁰, or P-State. The LC retarder(g) serves the purpose of correcting for slight ellipticities introduced by each device in the optical train resulting in values close to 0⁰ retardance for linear states. Table 3.2: Summary of required LC device parameters for the generator. 41

54 On the following page Figure (3.4) shows the results of measured intensities obtained by rotating the maximum transmission axis of polarizer(a) versus calculated values. Rotating polarizer(a) in this fashion is only done to confirm the calibration as detailed in Goldstein [28]. Detailed values used for calculation using Stokes vectors and Mueller Matrices can be found in Appendix A. Calculations shown there leave the solution as a function of polarizer(a) s maximum transmission axis. The transmission axis was rotated from 0 to 360⁰ in increments of 22⁰. Intensities were recorded at each point and compared to calculated values of the same angle. Note also that for circular states shown in Figure (3.4C), the intensity is the same regardless of polarizer(a) s transmission axis since the input to it is a circular polarization. Using this method accurately describes the shape of the polarization ellipse in the Ex-Ey plane provided sufficient positions of the transmission axis are chosen, but in the case of circular states it does not indicate differences between right (RCP) and left (LCP) circular states. Separate tests using a retarder where conducted to confirm the state RCP and LCP. 42

55 (A) (B) (C) Figure 3.4: Generator Calculated Vs. Actual Intensities. (A) P and M States (B) V and H States (C) Right Circular and Left Circular Polarization. 43

56 Once the generator has been calibrated, then the analyzer can be calibrated. Table (3.3) summarizes the parameters necessary to obtain the required analyzer states. In the actual system, polarizer(a) will remain fixed at - 45 degrees so for each input state, so the task of the analyzer will be to convert the incoming Stokes vector to M. For calibration, the input to the analyzer will be the previously calibrated generator states. There is an inherent advantage in doing this because if a slight rotation exists in generator states as a whole, the analyzer can compensate for this since it is essentially calibrated to the generator s coordinate system. To test the accuracy of the analyzer states, polarizer(a) was rotated in the same fashion as was done previously for the generator for calibration purposes only. The results are shown in Figure (3.5) and Figure (3.6) on the following pages, while detailed matrix calculations are given in Appendix A. Table 3.3: Summary of required LC device parameters for the analyzer. 44

57 (A) (B) (C) Figure 3.5: Analyzer Calculated Vs. Actual Intensities. (A) P-State. (B) M- State. (C) V-State. 45

58 (A) (B) (C) Figure 3.6: Analyzer Calculated Vs. Actual Intensities. (A) H-State. (B) RCP. (C) LCP-State. 46

59 The following statistics were generated from the data to evaluate quantitatively the accuracy of the calibration. As mentioned, polarizer(a) was rotated for the purpose of testing the calibrations [28] for six polarizations: H, V, P, M, RCP, and LCP. The actual measured intensities are I j where j takes on integer values from one to sixteen which represents positions of polarizer(a). The expected values (μ j ) are the results of calculations assuming ideal components. The variance of a j-position is then given by Equation (3.6). The variance of a system state (q) is given by Equation (3.7) where N = 16 measurements while the standard deviation is given by Equation (3.8) for all measurements under consideration [40]. This is calculated both individually for the generator/analyzer and the system as a whole. The depolarization ratio is calculated by Equation (3.9) when the measurements include linear co and cross-polarized conditions This has no meaning when evaluating the generator s circular states since the nominal output despite Polarizer(A) s position is always the same value. To evaluate the calibration, the average value is compared to the maximum variation as shown in Equation (3.10).. 47

60 Since all values are normalized to 1, the standard deviation can be interpreted as percentage. Overall, the measurements were within 4.5% of the calculated values assuming ideal components. Table (3.4) summarizes the statistics from the calibration procedure. If the components were ideal, Equation (3.9) could be used as an estimate of the signal to noise ratio for linear states. This parameter is listed in Table 3.4 along with results from Equation (3.10) for circular states. Table 3.4: Statistical Analysis of Calibration Results. ARM GENERATOR ANALYZER COMBINED STATE ROTATOR [V] RETARDER [V] STATISTICS VARIANCE S/N (σq)^2 V E H E P E M E RCP E LCP E STANDARD DEVIATION (σ): 32.2E-3 AVG S/N RATIO: 205 V E H E P E M E RCP E LCP E STANDARD DEVIATION (σ): 31.8E-3 AVG S/N RATIO: 171 TOTAL STND. DEVIATION (σ): 45.3E-3 AVG S/N RATIO:

61 3.3 System Testing Before the acquisition of backscattered polarimetric measurements from materials of interest, several tests were conducted for objects with relatively known Mueller Matrices. These qualification tests were conducted in the line-ofsight configuration used during calibration as shown in Figure (3.7). Three conditions were tested: Air, a Polarizer with its maximum transmission axis oriented in the vertical direction (LVP, 90⁰), and a Polarizer with its maximum transmission axis oriented in the horizontal direction (LHP, 0⁰). Laser Laser: Pilas, 1065nm Polarizer(G) Θ=45 LC Rotator(G) Fast Axis = 135⁰ (Marking Line) GENERATOR Fast Axis = 45⁰ (Marking Line) LC Retarder(G) OBJECT Fast Axis = 45⁰ (Marking Line) LC Retarder(A) LC Rotator(A) Fast Axis = 135⁰ (Marking Line) ANALYZER Polarizer(A) Θ=-45 Detector Figure 3.7: Experimental arrangement for qualification tests. The ideal MM for Air (A), the LVP, and the LHP versus the test results are shown in Equation (3.11), Equation (3.12), and Equation (3.13), respectively. 49

62 The standard deviation of MM elements was.029,.043, and.016 for Air, LVP, and LHP respectively which demonstrates the high quality of the acquired measurements. Note that the specified wavelengths of the polarizer used to measure LVP and LHP was of 700nm to 1100nm which is near the Pilas Laser s specified range. Despite this, test results agree well with theoretical values. Throughout the system s development process great care was taken to improve calibration and measurement quality. Standard deviations were decreased by 50% or more from initial testing. The final values are shown in Table (3.5). Table 3.5: Statistical analysis of individual MM elements. MM ij Ideal Mueller Matrix Measured Mueller Matrix Element Variances Air LVP LHP Air LVP LHP Air LVP LHP m m E E E-6 m E-3 3.3E E-6 m E E E-6 m E-6 1.0E E-6 m E-6 1.6E E-6 m E-6 3.3E E-6 m E E E-6 m E-3 9.2E E-6 m E-6 8.8E E-6 m E E E-6 m E E-6 1.2E-6 m E-3 1.0E E-6 m E-6 6.1E E-6 m E E-6 4.8E-6 m E E E-6 STANDARD DEVIATION: 29.9E E E-3 50

63 3.4 NI Labview Interface / Automation A high level of automation is desirable in order to achieve improved statistics and decreased measurement and data acquisition times. Automation was achieved by using NI Labview software. Automated calculations were performed using Matlab and several Excel templates created specifically for these experiments. Table (3.6) on the following page shows an algorithm of the steps executed by the system along with input and output signals. The system inputs are handled by a graphical user interface (GUI) on the host PC. The GUI displays real-time data pertaining to the current measurements including the current sampled waveform, averaged peak-to-peak amplitudes, current settings input by user, current voltages being applied to the LC devices, and LC device controller information. 51

64 Table 3.6: Summary of automation routine with system inputs/outputs. STEP DESCRIPTION SYSTEM INPUTS SYSTEM OUTPUTS 1 GUI Interface: User-Selected Parameters Delay Time for LC Devices Sampling Rate for DAQ. Measurement Time /State Datapoints to Collect /State Max Waveform Data Size # of Times To Repeat Test Noise Threshold Modulation Frequency LC Device Voltages /State 2 Run Test NA 3 4 Post-Processing Routine #1 Post-Processing Routine #2 Optional: Enter Laser Modulation Frequency to improve appearance of plots. Text Files recorded from all angles of object rotation. NA Raw Pk-Pk Volts/State Average Pk-Pk /State Store Waveform Sample *Repeat (User-selected) Avg. Intensity (Normalized) Avg. Intensity (Un- Normalized) Mueller Matrix Elements Diattenuation Matrix Elements Retardance Matrix Elements Depolarization Matrix Elements Diattenuation Magnitude Net Depolarization Histogram of amplitudes Waveform Plot Avg. values from all Repetitions Standard Deviations Summarizes ALL data by angle of object rotation from smallest to largest. Identifies maximum specular reflection angle. Plots all results by angle. 52

65 For calculating the average peak-peak amplitudes a special algorithm is implemented that disregards peak-to-peak values that are less than the amplitude specified by the user in the voltage threshold field of the GUI. If the system noise is determined prior to testing, it is entered as the threshold, which can remove some influence of noise from the results. In addition, the system will require a minimum number of datapoints above the threshold dictated by Equation (3.14). An explanation of this formula is that the Laser Modulation Frequency (F Mod ) is converted to time. Next 10% of this time is calculated and divided by the sampling time (Ts) to acquire the minimum quantity of datapoints required. This method ensures that spurious noise spikes unrelated to the signal are not included in averaged results provided that their duration is less than 10% of the modulation frequency. To further understand the delays mentioned in Table (3.6), the measurement cycle can be divided into four distinct periods for one particular system state. A system state is defined as a particular generator-analyzer combination. This is shown in Table (3.7) where states T1, T2, and T3 can be adjusted by the user via the GUI. Obviously there is a trade-off between measurement speed and the number of peak-peak measurements that are averaged together. 53

66 Table 3.7: The breakdown of a single system states measurement cycle TIME T0 T1 T2 T3 DESCRIPTION Apply Voltages to LC Devices to Control Polarization States. Delay Built in to allow stabilization of LC s. The minimum time is approximately 15mS and is limited by the LC Device specifications. Measurement Time Can be altered to sample more data and average more results. (User Defined, Results utilized 750mS) Delay Time before next measurement In our system this was chosen to be approximately 750mS however it can be 0 if maximum speed is desired. A block diagram of the NI Labview code and measurement process is shown in Figure (3.8). Additional Matlab scripts were written to parse the data for all aspect angles and combine the results into a single spreadsheet in predetermined ranges. The spreadsheet automatically plots the results from the predetermined ranges for analysis. 54

67 1 Completed All States? Measurement State [q]: * Get LC Device Voltage for Current State. Note: q=1, 2, 3..N 0 START GUI: User Entered Test Parameters Verify LC and DAQ Hardware LC Delay Complete? 0 1 Output LC Device Voltages Write Waveform Sample to Text File. Sample Peak-Peak Amplitudes Display Current Waveform and Results to GUI. Write Data to Text Files: * Raw Pk-Pk Amplitudes * Average Pk-Pk Update GUI 1 State Clock Timeout? 0 Max Data Limit? 0 *Calculate Averages *Store Totals 1 Test Repeat? 0 Matlab: Post Processing END 1 Figure 3.8: Simplified block diagram of the measurement procedure. 55

68 CHAPTER IV RESULTS AND DISCUSSION 4.1 Overview Recall that all Mueller Matrices are a function of material properties, wavelength, and orientation. Given this dependence, it is desirable for the polarimetric system to have scanning capability such that the MM of an object is measured with varying aspect angle. Then, by the use of a suitable discriminator, the polarimetric system can identify this case. This scenario was investigated using the LC system shown in Figure (4.1) where the object was equivalently rotated instead of disturbing the position of the polarimetric system. Note: Dimensions not drawn to scale. Depicted angles are exaggerated for clarity (See Actual Dimensions Below). B OBJECT +Φ (varied) -Φ (varied) Dimensions: AB = cm BC = cm DE = 5.08 cm FG = 6.35 cm Angle α: 2.46⁰ (fixed) Angle Φ: -0.2 to 0.2⁰ α (fixed) Fast Axis = 45⁰ (Marking Line) LC Retarder(G) LC Retarder(A) Fast Axis = 45⁰ (Marking Line) ANALYZER (Slight Tilt) GENERATOR LC Rotator(A) Fast Axis = 135⁰ (Marking Line) Fast Axis = 135⁰ (Marking Line) LC Rotator(G) Polarizer(A) Θ=-45 Figure 4.1: Experimental arrangement for space material testing. Polarizer(G) Θ=45 Laser: Pilas, 1065nm C A D E Detector F G Laser 56

69 4.2 Material Samples Several materials were tested with the LC polarimeter using the calibration values previously discussed in Section (3.0). The MM of the material was derived for many known angles spanning a range of +/- Φ about the on-axis location. From Figure (4.1) observe that the generator and analyzer were placed very close together. Additionally, a large enough distance to the object was used so that specular reflection angle (α) approaches 0⁰. A very slight tilt was introduced to the analyzer to further approach 0⁰. This approximates the conditions in which the device would need to operate in to be deployed in an actual scenario. The material being tested is identified as the object in Figure (4.1). A positive Φ indicates the object was rotated off the axis of maximum specular reflection counter-clockwise while a negative Φ indicates clockwise rotation. Rotation was achieved by way of a stepper motor with 0.01⁰ resolution. In the following results, the angle of maximum intensity for a co-polarized state (i.e. HH or VV) was considered to be the on-axis condition or 0⁰. Figure (4.2) on the following page shows a labeled photo of the experimental set-up. Also on the following page, Figure (4.3) shows additional images of the system and material samples. 57

70 Figure 4.2: Photo of the experimental arrangement for space material testing. A B C D E F Figure 4.3: Additional System Images. (A) System Looking Towards Object. (B) Looking towards system from the object. (C) Amorphous Silicon (Solar Panel). (D) Poly-Silicon (Solar Panel). (E) Layered Kapton Tape on rigid backing. (F) Mylar Film on rigid backing. 58

71 The two types of silicon are typical solar panels which are used widely in space applications shown in Figures (4.3C) and (4.3D). Kapton is a lightweight dielectric with excellent thermal and electrical insulating properties. The sample was made from tape which was layered to a rigid surface as shown in Figure (4.3E). Mylar is a trade name for Biaxially-Oriented Polyethylene Terephthalate which has the properties of high tensile-strength, thermal insulation, electrical insulation, and is also highly reflective. The high reflectivity is evident from the photo of Figure (4.3F) where the reflection of the lab bench can be seen in the photo. A special note regarding the Mylar sample though is that the material had some residual roughness due to packaging in the form of small creases of approximately 1mm width. Efforts were made to interrogate smooth portions of the sample; however some variability could be seen in the data. The Mylar was secured to a rigid plexiglass surface. 4.3 Mueller Matrix Results To begin to characterize materials based on their optical properties, a review of MM data for each material is presented. First a comparison of MM s at normal incidence is shown in Figure (4.4) where AS, PS, MY, and KA represent Amorphous Silicon, Poly-Silicon, Mylar, and Kapton respectively. It is interesting to note the significant difference between the dielectrics versus the two forms of silicon as evidenced by the diagonal of the matrix, which represents Isotropic Absorption. For a description of each Mueller Matrix Elements Phenomenology refer to Section 2.3.4, Table (2.3). 59

72 Mueller Matrices of Materials at 0⁰ Incidence Figure 4.4: Experimental Mueller matrices of materials at 0⁰ incidence. Next, MM s are compared at various angles of rotation. The following graphs show Mueller Matrix Element Intensities at selected angles due to the large amount of data collected. Full MM element values for all materials and angles are listed in Appendix B. Figure (4.5) shows the results for Amorphous Silicon. The bars in the graph are ordered from negative to positive angles through zero degrees. Each MM element shows a relatively high degree of symmetry regardless of the direction of rotation. Figure (4.6), Figure (4.7), and Figure (4.8) show the MM s for Poly-Silicon, Mylar, and Kapton respectively. A major observation from plotting the results in this manner is that magnitude of the polarizance, which is defined by MM elements M 21, M 31, and M 41, increases for off-axis measurements. This observation is supported by results obtained in previous studies [1]. 60

73 Figure 4.5: MM Elements of Amorphous Silicon at Selected Angles. Figure 4.6: MM Elements of Poly Silicon at Selected Angles. 61

74 Figure 4.7: MM Elements of Mylar at Selected Angles. Figure 4.8: MM Elements of Kapton at Selected Angles. 62

75 4.4 Decomposition Results Organized By Material In this section, decomposition parameters obtained by the Lu-Chipman Algorithm [16] are grouped by material and analyzed as a function of object rotation angle. The maximum measured intensity for a co-polarized generatoranalyzer combination is plotted as a reference since this is a good indicator of maximum specular reflection and hence normal incidence. This condition for all materials was obtained with a generator-analyzer combination of HH or VV, although HH was chosen as the reference for plotting. Data for full decomposed matrices are listed in Appendix B in tables for diattenuation, retardance, and depolarization. To summarize and compare materials, the coefficients for total behavior will be used next, specifically the diattenuation magnitude, retardance in radians, and the net depolarization. Under normal circumstances, diattenuation would not be defined at non-normal angles. In this study however, the goal is to find useful discriminators for characterization of the material and its orientation. In this scenario, numerical values of diattenuation may not be critical so examination of its general behavior is justifiable. Figure (4.9) shows the results for Amorphous Silicon. Since a large amount of data is plotted simultaneously, note the color coding of the scales for easier comparisons. The vertical axis on the left side serves as a scale for three parameters with the same color as was used in the legend. The vertical axis on the right only refers to retardance (purple). Figure (4.10), Figure (4.11), and Figure (4.12) follow for Poly-Silicon, Mylar, and Kapton respectively. 63

76 Figure 4.9: Decomposition parameters vs. object angle (AS). Figure 4.10: Decomposition Parameters vs. object angle (PS). 64

77 Figure 4.11: Decomposition Parameters vs. object angle (MY). Figure 4.12: Decomposition Parameters vs. object angle (KA). 65

78 The preceding graphs indicate several useful pieces of information that can be related to material characteristics and/or aid in evaluating behavior at nonnormal incidence. i) Intensity for a co-polarized generator-analyzer configuration can be used as a discriminator for determining the angle of normal incidence. ii) Minimum Depolarization correlated well to the angle of normal incidence and can provide additional confidence in making this determination. The data for Mylar suggests that depolarization becomes less reliable for this purpose when a surface is highly reflective. iii) As the object is rotated off-axis, depolarization increased in three of the four materials. iv) The order of materials from least depolarizing to most depolarizing is: Mylar, Amorphous Silicon, Poly-Silicon, and Kapton. This agrees with general knowledge of the material s surface roughness for the particular samples used. Greater depolarization occurs for less smooth surfaces. v) When comparing Poly-Silicon and Amorphous Silicon, the depolarization decreases at a faster rate versus object rotation angle. vi) The Retardance stayed very constant in general. However for smoother surfaces it was much more constant than rougher surfaces. The findings of points (i), (ii), (iv), and (v) are supported by findings from previous studies which compared reflectance, object rotations, and depolarization [17, 22]. 66

79 4.5 Decomposition Parameters Organized By Optical Properties For further analysis, Decomposition Coefficients will be compared by property. More specifically, all materials will be plotted on the same graph for each individual coefficient. Note that the scale used some of the following plots is decreased to exaggerate the behavior of the property. First, the property of Diattenuation (Magnitude) is shown in Figure (4.13). The same pattern of behavior can be seen to a varying degree for all materials, namely the two peaks on that occur as the object is rotated by some angle both plus and minus. However for the more irregular surfaces, Poly-Silicon and Kapton, it is much more pronounced. Figure (4.14) shows the Net Depolarization for all materials versus object rotation angle which highlights point (iv) in the preceding section. Figure 4.13: Diattenuation magnitude vs. object rotation angle for all materials. 67

80 Figure 4.14: Net Depolarization vs. object rotation angle for all materials. Figure (4.15) shows the total retardance versus the object rotation angle. This further highlights point (vi) from the preceding section. The results for Kapton indicate less consistent values than the other materials. The cause for this may need to be investigated further, although one important note is that the signal to noise ratio for Kapton was the least since the received signal was much lower overall. 68

81 Figure 4.15: Total Retardance vs. object rotation angle for all materials. 4.6 Stokes Analysis As mentioned in Section 2.3.5, additional information can be gained by analysis of the Stokes vector received at the Analyzer. In some respects this is redundant since the polarimetric system is complete, and therefore measures the object s full MM. The MM in turn represents the full transformation resulting from interaction of the incident light with the object. However, to demonstrate the utility of these methods in comparison with the complete system, the associated figures-of-merit were calculated. 69

82 Since the data-reduction method was used to calculate the MM, the full Stokes vector for each measurement state was not calculated at the time of the experiment. Recall that each measurement state consists of a particular generator-analyzer combination. The HH state was chosen for analysis of the Stokes parameters since this represents the case of a co-polarized generatoranalyzer combination. This would be the natural condition likely used for an incomplete system. Figures (4.16), (4.17), (4.18), (4.19), and (4.20) show the DOP, DOLP, DOCP, Ellipticity and Eccentricity, and the Major Axis location respectively. Figure 4.16: DOP vs. object rotation angle for all materials. 70

83 Figure 4.17: DOLP vs. object rotation angle for all materials. Figure 4.18: DOCP vs. object rotation angle for all materials. 71

84 Figure 4.19: Ellipticity and Eccentricity vs. object rotation angle for all materials. Figure 4.20: Major axis of polarization ellipse vs. object rotation angle. 72

85 The following observations can be made from the Stokes analysis plots in the preceding figures: i) DOP and DOLP behavior (Figures 4.16 and 4.17) agrees with the Net Depolarization parameters calculated from the full decomposition method using the Lu-Chipman algorithm in Section 4.3 and 4.4. ii) The DOCP and Ellipticity behavior at angles near zero, but not zero, mimics the diattenuation magnitude behavior in Section 4.3 and 4.4. It would be premature to correlate these two but the results warrant future investigation. iii) Given the relatively flat and low value of total retardance seen from the full decomposition, it is reasonable to expect the values of Ellipticity and Eccentricity shown in Figure Specifically Ellipticity is ideally zero for linearly polarized light and the incident light was linearly horizontal for the Stokes analysis. Likewise the Eccentricity remained close to one, which would ideally indicate linearly polarized light. 4.7 Statistical Analysis A wide range of statistics were collected to support the findings in this study. First, the results of continuous peak-peak amplitudes will be shown for each material at normal incidence. A more concise way to summarize the results is to look at standard deviations of individual MM elements. Next, a oneway ANOVA will be shown using the decomposed parameters, which were calculated from the object s MM. Finally, the signal to noise (SN) was calculated 73

86 for each material and angle. Recall that some materials required different detector gain settings, which effect noise levels and amplitudes. In addition, some material s backscattered signal intensity at the detector was lower than others. The photon flux hitting the detector generates a voltage that is proportional to the intensity incident on the detector. These are recorded by calculating peakpeak amplitudes from the CompactDAQ unit. For a given measurement state, or generator-analyzer combination, these measurements are continuous over a duration specified by the user when operating the GUI. Typically around 150 amplitudes were averaged to obtain the recorded intensity for the results in this study. In practice, this value can change depending on the amount of noise present since an algorithm is used to reject amplitudes of insufficient magnitude or pulse duration. Note that the Pilas 1065nm laser outputs fast pulses via the laser controller rather than the sinusoidal modulation achieved by a typical rotating chopper. The system outputs this data in the form of histograms for analysis. The amount of data is quite large, namely four materials encompassing 67 angular measurements, multiplied by 16 states each, and then multiplied by 150 samples or so for the raw amplitudes. As a result, examples of flux averages will be presented to give an indication of general behavior, while more concise metrics will be presented for actual matrix results that give a better indication of total repeatability. 74

87 Figure (4.21) shows the histograms for Amorphous Silicon, Poly Silicon, Mylar, and Kapton under the conditions of a co-polarized generator-analyzer state (HH) at normal incidence. The box size is small, hence the low counts scale for each but the important point is that all amplitude results fall within a narrow range of values, typically within approximately 10% of the average. (A) (B) (C) (D) Figure 4.21: Histogram examples for co-polarized generator-analyzer (HH, state q1) for (A) Amorphous Silicon, (B) Poly Silicon, (C) Mylar, and (D) Kapton. Histograms represent data from continuous peak-peak amplitude measurements. The MM of the material for a given angle is calculated from a series of flux measurements totaling sixteen for the present system. The system however computes each MM several times as dictated by a user-entered field on the GUI. 75

88 All the results obtained in this study repeated each MM measurement a total of six times. Therefore the standard deviation of each repetition gives an excellent metric for the repeatability of the results. The standard deviation of a particular MM element for a given angle (σ ij ) is calculated by Equation (4.1) where MM ij refers to a particular element of the Mueller Matrix and n is the number of times the test was repeated. Figures (4.22), (4.23), (4.24), and (4.25) show standard deviations of Mueller Matrices by element at normal incidence for Amorphous Silicon, Poly Silicon, Mylar, and Kapton respectively. Note that all MM s are normalized to M 11 resulting in a deviation of zero for M 11. In all cases, the standard deviation was less than 1.5%. 76

89 Figure 4.22: Standard Deviations of MM Elements for Amorphous Silicon at 0⁰. Figure 4.23: Standard Deviations of MM Elements for Poly Silicon at 0⁰. 77

90 Figure 4.24: Standard Deviations of MM Elements for Mylar at 0⁰. Figure 4.25: Standard Deviations of MM Elements for Kapton at 0⁰. 78

91 One method to compare data arising from distinct groups within a population is to compare the variances within each group and compare it with the variance of the whole population [40]. This statistical method is referred to as ANOVA (Analysis of Variance) testing [40]. A one-way ANOVA was used to compare results based on both angle and property (specifically diattenuation, depolarization, and total retardance). Recall that MM s are a function of wavelength, incidence angle, and material properties. The wavelength in this study remained fixed so this is not a concern. The source of variations in the data when grouped by aspect angle and material property however should be independent of variations of the entire population. In an ANOVA test, each sub-population is a level while the response would be trial results. In this case, levels are chosen to be diattenuation, depolarization, and total retardance coefficients while responses are the six repeated test results for each property. One result from this comparison is a confidence interval regarding the hypothesis that the means are equal for each angle. A standard ANOVA table is shown in Table 4.1[40]. Table 4.1: Standard ANOVA Table Source SS df MS F p-value Columns SSTr k-1 MSTr MSTr/MSE p Error SSE n-k MSE Total SST n-1 The iterator k is the number of angles and n is the number of tests. The SSTr and SSE define the sum of squares between samples and within samples respectively defined in Equation (4.2) and Equation (4.3). 79

92 Analysis of between sample variation and within sample variation is quantified by the F-statistic which compares the mean square for levels (MSTr) and the mean square error (MSE) which is shown in Equation (4.4) and Equation (4.5) respectively. The F-Statistic is shown in Equation (4.6). Using the F-distribution, the F-statistic is used to carry out a hypothesis test and the resulting p-value. The test is for the Null Hypothesis (Ho): μ1 = μ2 = μk which is rejected for low p-values, or probability of the Null Hypothesis [40]. Each MM, and hence each material property, was repeated six times. Table (4.2) on the following page shows a summary of the standard ANOVA table when comparing variances within each angle to the variance of the whole population (material type). The resulting p-values for all tests are extremely low, indicating that the means are different in every case. This supports the expectation that each MM is a function of aspect angle. 80

93 Table 4.2: ANOVA Results for all materials/properties (all angles). Material: Property AS: Diattenuation AS: Depolarization AS: Total Retardance PS: Diattenuation PS: Depolarization PS: Total Retardance MY: Diattenuation MY: Depolarization MY: Total Retardance KA: Diattenuation KA: Depolarization KA: Total Retardance Source SS df MS F p-value Columns E-3 Error E-6 Total Columns E-3 Error E-6 Total Columns E-3 Error E-6 Total Columns E-3 Error E-6 Total Columns E-3 Error E-6 Total Columns E-3 Error E-3 Total Columns E-6 Error E-6 Total Columns E-3 Error E-6 Total Columns E-3 Error E-6 Total Columns E-3 Error E-6 Total Columns E-3 Error E-6 Total Columns E+0 Error E-3 Total E E E E E E E E E E E E-9 81

94 Next, independence of material differences were tested using a one-way ANOVA at three selected angles: -0.07⁰, 0⁰, and +0.07⁰. Table (4.2) shows the F-statistic and the probability only from the ANOVA table. In all cases except for one, the properties are statistically unique for all materials. The case of total retardance for -0.07⁰ yielded a lower confidence level. The results for retardance in Figure (4.15) show greater similarity for materials at this particular angle than compared with other properties, which suggests that this maybe an aberration. Table 4.3: ANOVA results for properties at selected angles (all materials). Property Diattenuation Depolarization Total Retardance Selected From ANOVA Table Angles F Prob>F -0.07⁰ E-14 0⁰ E ⁰ ⁰ E-33 0⁰ E ⁰ E ⁰ E-01 0⁰ E ⁰ E-03 The variances for diattenuation from all six repetitions of the experiment are shown in Figures (4.26), (4.27), (4.28) for angles -0.07⁰, 0⁰, and +0.07⁰ respectively which are box plots where the red line is the median value, the whiskers show the range of values not considered outliers, the top/bottom of each box represent the 75/25% quartile ranges, and any red + symbols represent outliers. This is followed by depolarization in Figures (4.29), (4.30), and (4.31) and finally the retardance (radians) is shown in Figures (4.32), (4.33), and (4.34). The Matlab code for ANOVA analysis is included in Appendix C. 82

95 Figure 4.26: Box plot for diattenuation results at -0.07⁰. Figure 4.27: Box plot for diattenuation results at normal incidence. 83

96 Figure 4.28: Box plot for diattenuation results at +0.07⁰. Figure 4.29: Box plot for depolarization results at -0.07⁰. 84

97 Figure 4.30: Box plot for depolarization results at normal incidence. Figure 4.31: Box plot for depolarization results at +0.07⁰. 85

98 Figure 4.32: Box plot for total retardance results at -0.07⁰. Figure 4.33: Box plot for total retardance results at normal incidence. 86

99 Figure 4.34: Box plot for total retardance results at +0.07⁰. In summary the statistics shows excellent repeatability of peak-peak voltages from the detector. The individual MM elements have standard deviations of less than 1.5%. Lastly a one-way ANOVA comparison suggests that the source of variations across angles and materials are independent of one another. This is consistent with expected behavior of MM s and their fundamental dependencies. 4.8 Evaluation of System Noise During calibration of the LC polarimetric detection system, the noise level was well characterized. The #2151 Detector has two gain settings that were 87

100 used during the course of this study: AC-Low and AC-High. The gain setting has a large impact on the level of noise in the system. The noise can be classified into three main categories: ambient light (not related to the laser), detector noise, and system electrical noise. This is represented in Equation (4.7). The system noise was tested by turning the laser off and allowing the oscilloscope to record one thousand samples (periods between triggers). The Lecroy 7300A has measurement and statistical capabilities which were used to record the peak-to-peak amplitude for each sample. This result was then placed in a histogram where the number of counts is recorded for a particular amplitude value. For uniformity of noise tests, the object was always at 0⁰ with the Generator/Analyzer combination set to HH (Co-Polarized). Next the Detector was turned off and the same process for averaging and histogramming of peak amplitudes was repeated. This represents the electrical noise in the system. The ambient plus detector noise can be determined from Equation (4.7). Figure (4.35) shows the average system noise based on the detector gain settings during calibration. With the detector gain setting at AC-High the system noise is considerably higher. When choosing a gain setting prior to sample testing, the intensity at the detector was measured in both a co and cross-polarized condition at both gain settings. The gain setting that resulted in a higher ratio was used. This was considered to be a representative metric for the signal to noise ratio that could be achieved from the particular gain setting. 88

101 Figure 4.35: Histogram of avg. Pk-Pk system noise (laser off) by detector gain. The average electrical noise (detector off) across all measurements was small in comparison at 3.2mV. Table (4.4) shows a summary of noise amplitudes in millivolts based on the detector gain setting. It can be seen from Figure (4.35) that the noise in the system appeared to be gaussian in nature. Table 4.4: Noise summary by source. Noise (σ) Detector Gain Setting AC-Low[mV] AC-High[mV] σ System σ Ambient+Detector σ Electrical

102 Amorphous Silicon and Mylar were measured on the AC-Low gain setting while Kapton and Poly-Silicon were measured on the AC-High setting. The tradeoff for the higher gain setting is that the noise is amplified as well. Due to the signal strength, it was not possible to measure all the materials with the AC- Low setting. On the other hand, not all materials could be measured with the AC-High setting either because of detector saturation. Before each experiment the system was re-tested for noise levels and this value was used to set the noise threshold used by the Labview routine as detailed in Section (3.4). The signal to noise ratio (SN) varies for each flux measurement since the intensity at the detector varies. Figure (4.36) shows the SN ratio for the generator-analyzer combination HH. The two materials that used the AC-High setting have lower SN ratios. Figure 4.36: Signal to Noise Ratio of HH state by material vs. object angle. 90

103 CHAPTER V CONCLUSION This study was done to support development of the polarimetric functions of the United States Air Force Research Laboratory (AFRL) Polarimetric Multifunction Imaging Platform. Polarimetry is well suited for identifying and characterizing remote objects based upon their material properties, and hence is well suited for many space applications. Four materials were tested using a full polarimetric system, specifically Amorphous Silicon, Poly Silicon, Mylar, and Kapton based on their widespread use in space applications. In polarimetry, light is characterized by Stokes parameters and objects are characterized by Mueller Matrices and their decomposition. Each element of the Mueller Matrix corresponds to intrinsic material properties. In addition to solving for a remote object s Mueller Matrix, several figures of merit were solved for based upon the Stokes parameters. Two other goals of the study were to analyze the behavior of objects as they undergo rotations and to achieve a high level of automation. In remote sensing applications, the angle of incidence may not be known with sufficient precision which impacts the specific Mueller Matrix measured. Several parameters have been identified as useful discriminators for detecting the normal incidence condition. Furthermore, the system achieves a high level of 91

104 automation which reduces testing time drastically and makes it more suitable for deployment in a practical system. Specific results of this study include the following: Solution of an unknown objects Mueller Matrix for material identification. Solution of specific material properties such as absorption, diattenuation, retardance, and depolarization for analyzing the material properties of a material. This is quantified through both constituent matrices of the Mueller Matrix and coefficients representing net behavior. Results indicate several metrics that can be used to identify object rotations and or locate the condition of normal incidence from a scanning system. This supplements quantitative analysis of the Mueller Matrices. Identification of Stokes figures-of-merit that can supplement analysis of material characteristics such as DOP, DOLP, DOCP, Ellipticity, Eccentricity, and Major Axis Rotation of resultant electric fields to analyze material properties. Successful development of an automated polarimetric system with millisecond capabilities. Successful implementation of liquid-crystal devices in a polarimetric system which can be controlled electronically and are free from mechanical movements. This can reduce vibrations in the system which may compromise accuracy. 92

105 A large set of data was acquired to demonstrate the full potential of polarimetric systems. This lays the foundation for future studies to link physical phenomenology and material science to optical testing methods. In addition to these results, the system showed excellent repeatability and accuracy. Included in the development of the automated system was a major effort to collect statistics to support findings confidently. A wide range of parameters were investigated which serves as foundation for future polarimetric development to improve material identification capabilities. The future potential for Polarimetry in a wide range of space and industrial applications has been greatly enhanced in the following two ways. First, the physical phenomenology of materials may be more closely linked to polarimetric observables. Secondly, a framework to handle unknown object rotations has been demonstrated through several parameters included in decomposition of MM s and Stokes analysis. In practical situations, this may have previously been an impediment for many applications. 93

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110 APPENDICES 98

111 APPENDIX A FULL MUELLER MATRIX CALCULATIONS USED IN CALBRATION In Section 3.1, calibration results were presented with actual and theoretical values. The following are the full Mueller Matrix calculations for the generator and analyzer to achieve the theoretical values. In all cases, the final result is in terms of Polarizer(A) s transmission axis θ. The purpose of rotating the transmission axis is a technique useful for the system calibration only [28]. Generator P-State: Generator M-State: 99

112 Generator V-State: Generator H-State: Generator RCP-State: 100

113 Generator LCP-State: Analyzer P-State: Analyzer M-State: 101

114 Analyzer V-State: Analyzer H-State: θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ Analyzer RCP-State: θ 102

115 θ θ θ θ θ θ Analyzer LCP-State: θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ 103

116 APPENDIX B COMPLETE MATRIX DATA FOR ALL TESTS In this study, an extremely large amount of data was recorded and/or processed: (four materials) x (approximately 15 angles) x (16 flux measurements) from which 76 parameters were calculated. This was repeated a total of six times. Selected data was included in the body of this work, however full results are presented in tabular form in this appendix. Tables are organized by material with object angles across the top row. Note however that the values listed in the tables are the averaged result from the six repetitions of the experiment. 104

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