POLARIZATION SIGNATURES in VECTOR SPACE Dissertation

Home , Jones calculus, Stokes parameters

POLARIZATION SIGNATURES IN VECTOR SPACE

Dissertation

Submitted to

The School of Engineering of the

UNIVERSITY OF DAYTON

In Partial Fulfillment of the Requirements for The Degree of Doctor of Philosophy in Electro-Optics

By Diane Krupp Beamer, B.S., M.S., M.B.A.

UNIVERSITY OF DAYTON Dayton, Ohio August, 2018

POLARIZATION SIGNATURES IN VECTOR SPACE

Name: Beamer, Diane Krupp

APPROVED BY:

______Partha P. Banerjee, Ph.D. Joseph W. Haus, Ph.D. Advisory Committee Chairman Committee Member Department Chair Professor Department of Electro-Optics and Photonics Department of Electro-Optics and Photonics

______Dean R. Evans, Ph.D. Samir Bali, Ph.D. Committee Member Committee Member Air Force Research Laboratory Professor Wright Patterson Air Force Base Physics Department Graduate Faculty Miami University of Ohio Department of Electro-Optics and Photonics

______Robert J. Wilkens, Ph.D., P.E. Eddy M. Rojas, Ph.D., M.A., P.E. Associate Dean for Research and Innovation Dean Professor School of Engineering School of Engineering

Diane Krupp Beamer

2018

iii

ABSTRACT

POLARIZATION SIGNATURES IN VECTOR SPACE

Name: Beamer, Diane Krupp University of Dayton

Advisor: Dr. Partha P. Banerjee

The use of polarimetric information to enhance the identification process for unresolved and unknown targets in GEO orbit is quite rare. The majority of observations made of objects at that distance are spectral measurements, without the polarization information that is available in the same stream of photons being captured. It is proposed to use polarimetric signatures to enhance the speed and accuracy of the identification of unknown objects.

Three methods of polarization signatures are proposed. First, the standard Stokes signature plotted as a function of solar phase angle is recommended. But beyond that, two other methods are proposed to help qualitative and quantitative assessment of the unknown target. The second method is the use of vector space, a ratio of one Stokes parameter to another in all combinations possible (six spaces), plotted against the solar phase angle in a similar manner to the Stokes signatures. The third method is novel, using a technique borrowed from astronomers who characterize distant stars based on color-color analysis of the intensity ratios of celestial objects. A statistical method of analyzing the resulting

object characterization is also proposed, which is the Bhattacharyya distance, a non-

Euclidean method to characterize the clusters produced by the third method.

The experiments were conducted comparing simple geometries with one another to determine if polarization could differentiate simple geometries. Next different materials and surfaces were compared with one another to test polarization as a differentiator of the materials. Third, two complex models made up of three simple geometries that are representative of satellite construction were built and tested by the same three methods, to differentiate them from each other based on polarization signatures. Last, a third composite, different than both previous composites was built, and the method tested to see if it would identify the geometry most dissimilar to the new object so that a selection process could be created.

Each of the polarization methods tested was successful in differentiating geometry, material, surface configuration, and complex composite geometries made up of the simpler geometries tested. Polarization signatures are ubiquitous and consistent. It was found that the peaks and valleys observed in the signatures of compound objects were caused by the geometry and material of the simpler parts making up the complex object. A selection procedure was demonstrated based on identifying the most dissimilar object of two choices, and eliminating it. Polarization signatures were capable of differentiating which objects were most like and most dissimilar to the test object.

In conclusion, using polarimetry as an analytical tool to help identify unresolved objects in space is potentially an excellent process that would complement current methods of spectral analysis.

DEDICATION

This work is dedicated first to the Lord, Jesus Christ, King of kings and Lord of all lords, at Whose command I undertook this effort. I have set my face as flint and did as instructed. This work is also dedicated to my husband, Denny Beamer, who listened throughout the years and encouraged me, and to my children, Laura Dawson and Christian

Dawson, who cheered and encouraged me the whole way. Last but not least, this work is dedicated to my friend Judy Vineyard, for all the prayer support and encouragement she supplied to me. Without all of you, I could not have done it. My deepest thanks and heartfelt appreciation are offered to you.

ACKNOWLEDGEMENTS

First of all I would like to thank my advisor Dr. Partha Banerjee for his guidance, wisdom, insight and understanding applied to this process from start to finish. He supplied a listening ear, good questions, and many good ideas on my research. I appreciate all his support. I would also like to thank Dr. Joseph Haus for his support when I needed someone to listen to new ideas. I thank Dr. Dean Evans for the loan of his

AOM for several months during early experimentation. I thank Dr. Samir Bali for his extensive feedback and attention to my proposal and dissertation.

My special thanks go to Dr. Ujitha Abeywickrema for reading and editing all my papers, for his endless support in the lab, for his ready smile and the many times when we all laughed out loud. I also thank (soon to be Dr.) Rudra Gnawali for his friendship and encouragement.

I thank the UD physics department for the 18 month loan of their halogen lamp as a light source for the polarization experiments. I would especially like to thank the EO staff for continuous support and help overcoming the day to day problems and making the department a great place to work.

vii

TABLE OF CONTENTS

ABSTRACT ...... iv

DEDICATION ...... vi

ACKNOWLEDGEMENTS ...... vii

LIST OF FIGURES ...... xii

LIST OF ABBREVIATIONS ...... xix

LIST OF SYMBOLS ...... xxi

CHAPTER 1 INTRODUCTION AND OBJECTIVES ...... 1

CHAPTER 2 THE SCIENCE OF POLARIZATION ...... 5

2.1 Introduction ...... 5

2.2 The Electric Vector ...... 8

2.3 Special Cases – Linear and Circular Polarization ...... 9

2.4 Polarization by Reflection...... 12

2.5 What Polarimetry Adds...... 13

CHAPTER 3 POLARIZATION ALGEBRA AND PROPAGATION ...... 18

3.1 Jones Calculus ...... 18

viii

3.2 Stokes Vectors ...... 22

3.3 The Poincaré Sphere: the Basis for Vector Space ...... 27

3.4 Propagation of the Polarization Vector ...... 31

3.5 Conclusion ...... 38

CHAPTER 4 ANALYTICAL METHODS ...... 40

4.1 Introduction and Motivation ...... 40

4.2 Traditional Polarization Signatures...... 42

4.3 Intensity Analysis Methods from Other Disciplines ...... 45

4.4 Application of Color Space Concept to Polarimetry ...... 48

4.5 A Computational Approach to Identifying Targets ...... 54

4.6 Statistical Approach ...... 55

4.7 Conclusions ...... 73

CHAPTER 5 RELATED WORK ON POLARIMETRY OF SPACE OBJECTS ...... 74

5.1 Polarization Signatures of Space Objects ...... 74

5.2 Polarization Detection Through Haze and Clouds...... 82

CHAPTER 6 EXPERIMENTAL SETUP FOR STOKES VECTOR

DETERMINATION ...... 85

6.1 Materials and Methods ...... 85

6.1.1 Selection of model geometries and materials ...... 86

6.2 Experimental Setup ...... 89

6.2.1 Active vs. passive illumination ...... 93

6.3 Sources of Error and Calibration ...... 93

CHAPTER 7 EXPERIMENTAL RESULTS ...... 103

7.1 Polarization Signatures of Simple Geometries ...... 103

7.1.1 Cylinder...... 104

7.1.2 Model of cuboid bus ...... 109

7.1.3 Comparison of Stokes parameters for cylinder and cube ...... 109

7.1.4 Comparison of vector spaces for cylinder and cube ...... 112

7.1.5 Vector-vector spaces of cube and cylinder ...... 114

7.2 Polarization Signatures of Textured Surfaces ...... 117

7.2.1 Stokes parameters of textured surfaces ...... 119

7.2.2 Vector spaces ...... 119

7.2.3 Vector-vector spaces ...... 120

7.3 Polarization Signatures of Solar Panels ...... 122

7.3.1 Stokes parameters of the solar panels ...... 122

7.3.2 Vector spaces of the solar panels ...... 123

7.3.3 Vector-vector spaces of solar panels...... 124

7.4 Polarization Signatures of Composite Models ...... 126

7.4.1 Component signatures vs. composite signatures ...... 127

7.4.2 Comparison of Composite1 signatures with Composite2

signatures ...... 130

7.5 Comparison of Unknown Body Signature to Known Composites ...... 133

7.5.1 Stokes parameters ...... 135

7.5.2 Vector space comparison of unknown body ...... 136

7.5.3 Vector-vector spaces ...... 138

7.6 Vector Spaces of Powered Solar Panels ...... 141

7.7 Vector Spaces of Sphere vs. Cube ...... 145

7.8 Vector Spaces of Mylar Dish vs. Painted Dish ...... 148

7.9 Solar Panel Orientation Comparisons ...... 150

CHAPTER 8 CONCLUSIONS AND FUTURE WORK ...... 154

8.1 Conclusions ...... 154

8.2. Future Work ...... 156

REFERENCES ………….……………………………………………………………...157

LIST OF FIGURES

Figure 2.1: Representation of EM wave electric and magnetic fields ...... 6

Figure 2.2: The plane of polarization, propagation out of the page...... 7

Figure 2.3: Changing direction of the electric vector over time ...... 8

Figure 2.4: Spin and orbital angular momentum density ...... 11

Figure 2.5: (a) Visible imagery, (b) long wave IR imagery, (c) long wave IR polarimetry ...... 15

Figure 2.6: Polarization imaging produces increased detail: (a) unpolarized,

(b) linearly polarized ...... 16

Figure 3.1: The Poincaré Sphere, on or within which the three Stokes vectors S1, S2,

and S3 (or Q, U, V) are plotted in Cartesian coordinates...... 29

Figure 3.2: Polarization States on the Poincaré Sphere...... 30

Figure 3.3: Propagation through a bulk medium (BM) assumed to be anisotropic

(region b), surrounded by isotropic regions a and c...... 35

Figure 4.1: Computer generated images of objects in Earth orbit that are currently

being tracked ...... 41

Figure 4.2: Spectral signature (S0) and polarimetric signature of AMC-15 (S1);

UTC time plotted on horizontal axis ...... 43

Figure 4.3: Example of comparison of two objects based on their Stokes S1 signatures. 44

Figure 4.4: Color-color plot for planets and unknown stellar object ...... 46

xii

Figure 4.5: Color-color diagram of two objects...... 48

Figure 4.6: Polarization vector space in the Poincaré sphere; view looking along the S3 parameter axis...... 50

Figure 4.7: Example of Vector space S1/S0 signature as a function of reflection angle. .. 51

Figure 4.8: Example of vector-vector space for cubic bus S1/S0 on horizontal axis,

S2/S0 on vertical axis...... 52

Figure 4.9: S1/S0 vector signatures for cubic bus and fiberglass wire dish...... 53

Figure 4.10: Vector-vector space of cubic bus and wire dish antenna S1/S0 plotted on horizontal axis, S2/S0 plotted on the vertical axis...... 54

Figure 4.11: A cluster of measurements with the mean; mean indicated by red circle. ... 61

Figure 4.12: Two object clusters and their cluster means...... 63

Figure 4.13: Overlapping clusters with differing shapes...... 64

Figure 4.14: Two clusters of observations in feature space with three unknown objects. 65

Figure 4.15: Two clusters of data and their overlap in the test functional space...... 66

Figure 4.16: Relationship between distance and error...... 67

Figure 4.17: Distance in context of data scatter: (a) uncorrelated data; (b) correlated

data...... 70

Figure 5.1: Image of DTV-4S and SES-1 ...... 75

Figure 5.2: S0 vs. time for DTV-4 and SES-1, with three nights of data overlaid ...... 76

Figure 5.3: S1 vs. time for DTV-4s and SES-1, with three nights of data overlaid ...... 77

Figure 5.4: Image of Directv-10 and 12 ...... 77

Figure 5.5: Comparison of S0 for DTV-10 and DTV-12, with three nights of data

overlaid ...... 78

xiii

Figure 5.6: S1 Stokes vector comparison of DTV-10 and DTV-12, 3 nights of data overlaid ...... 78

Figure 5.7: Comparison of Stokes parameter S0 of six different satellites SPA plotted in blue ...... 80

Figure 5.8: Comparison of Stokes parameters S1 of the same satellites...... 81

Figure 5.9: Stokes parameters S0 and S1 for DTV-10 and DTV-12 – strongest polarization signatures at dusk and dawn...... 82

Figure 5.10: (a) Polarization sum image through haze (b) polarization difference

image through haze ...... 83

Figure 6.1: Examples of cuboid satellite buses; (a) SwissCube; (b) RadarSat2...... 86

Figure 6.2: Composite geometries: (a) Composite 1, comprising painted cuboid bus, monocrystalline solar panel and painted dish antenna; (b) Composite 2, comprising

Kapton covered bus, polycrystalline solar panel and fiberglass wire dish antenna...... 87

Figure 6.3: Experimental setup with unpolarized light source and light shielded detector...... 90

Figure 6.4: Rotational frames of reference of satellite body and solar panel...... 92

Figure 6.5: Stray light contamination (a) entering measurement optic; (b) blocked from re-entry by rotated element...... 96

Figure 7.1: Stokes parameters S1, S2, S3 of a painted cylinder as a baseline...... 106

Figure 7.2: Spectral and Stokes intensity of a painted cylinder. Right side axis shows

S1, S2 ...... 107

xiv

Figure 7.3: Normalized vector spaces of a painted cylinder - (a) linear:

S1/S0, S2/S0, S2/S1; (b) Circular S3/S1, S3/S0, S3/S2. Note: scales for S2/S1; and S3/S2; plotted on right hand vertical axes...... 107

Figure 7.4: Vector-vector space for painted cylinder S2/S0 vs. S1/S0 ...... 108

Figure 7.5: Stokes parameters for painted cube...... 109

Figure 7.6: S1 signatures for cube and cylinder...... 110

Figure 7.7: S2 signatures for cube and cylinder...... 111

Figure 7.8: S3 signatures for cube and cylinder...... 112

Figure 7.9: S1/S0 vector space signature comparison, cube and cylinder...... 113

Figure 7.10: S3/S1 vector spaces for cube and cylinder...... 114

Figure 7.11: Cylinder and cube models used in the comparison, (a) oblique view;

(b) view from detector position, cylinder left, cube right...... 115

Figure 7.12: Vector-vector space of cube and cylinder, S3/S0 vs. S1/S0...... 116

Figure 7.13: Vector-vector space of cube and cylinder, S2/S0 vs. S1/S0...... 117

Figure 7.14: Dish antennae tested: (a) painted surface; (b) fiberglass wire mesh on top of painted surface...... 118

Figure 7.15: Stokes parameters comparing fiberglass wire dish and painted dish in signature format: (a) S1; (b) S2...... 119

Figure 7.16: Vector spaces for fiberglass wire and painted dishes: (a) S1/S0; (b) S2/S0 .. 120

Figure 7.17: Vector-vector spaces for fiberglass wire mesh and painted dishes:

(a) S2/S0 vs. S1/S0; (b) S2/S1 vs. S1/S0...... 121

Figure 7.18: Vector-vector spaces for fiberglass wire mesh and painted dishes:

(a) S3/S1 vs. S1/S0; (b) S3/S2 vs. S1/S0 ...... 121

Figure 7.19: Solar panels separated from composite models: (a) AMX3d 5v, 30 mA;

(b) Sunnytech 5v, 100 mA...... 122

Figure 7.20: Stokes parameters for solar panels: (a) S1; (b) S2...... 123

Figure 7.21: Vector spaces for two solar panels: (a) S2/S1; (b) S3/S1...... 124

Figure 7.22: Vector-vector spaces of solar panels: (a) S2/S0 vs. S1/S0; (b) S3/S0 vs. S1/S0...... 125

Figure 7.23: Additional vector-vector space comparisons for solar panels:

(a) S2/S1 vs S1/S0; (b) S3/S1 vs. S1/S0...... 126

Figure 7.24: Component drivers of Stokes Composite 1 signature: (a) S1 comparison of Composite 1 and its dish; (b) S1 comparison of Composite 1 and

its bus. These graphs are not normalized...... 128

Figure 7.25: Component drivers of Stokes Composite 2 signature: (a) S1 comparison of Composite 2 and bus; (b) S1 comparison of Composite 2 and dish.

Graphs are not normalized...... 129

Figure 7.26: Composites 1 and 2 Stokes parameters (horizontal axis is reflection angle). The vertical axes denote a) S1/S0, b) S2/S0...... 131

Figure 7.27: Vector spaces comparing Composite 1 and Composite 2: (a) S1/S0 ;

(b) S2/S0 ...... 131

Figure 7.28: Vector-vector spaces with highest pair-wise distance between clusters:

(a) S2/S0 vs. S1/S0; (b) S2/S1 vs. S2/S0...... 133

Figure 7.29: Mylar covered dish antenna for the unknown composite body...... 134

Figure 7.30: Comparison of Stokes S1 of unknown body to (a) Composite 1 and (b) Composite 2...... 135

xvi

Figure 7.31: Comparison of Stokes S2 of unknown with (a) Composite 1 and (b) Composite 2...... 135

Figure 7.32: Comparison of S1 vector space of unknown with (a) Composite 1

and (b) Composite 2...... 136

Figure 7.33: Comparison of S2 vector spaces of unknown object with (a) Composite

1 and (b) Composite 2...... 137

Figure 7.34: S2/S1 vector space comparison of unknown object to (a) Composite 1 and (b) Composite 2...... 138

Figure 7.35: S3/S0 vs. S1/S0 vector-vector space comparison for the unknown object with (a) Composite 1 and (b) Composite 2...... 139

Figure 7.36: S2/S0 – S1/S0 vector space comparison of unknown object with

(a) Composite 1 and (b) Composite 2...... 140

Figure 7.37: Stokes parameter S1 measured for powered and unpowered solar panel. 142

Figure 7.38: S1/S0 vector space comparison for powered vs. unpowered solar panels. .. 142

Figure 7.39: S2/S1 vector space comparison of powered vs. unpowered solar panels. .. 143

Figure 7.40: S2/S0 vector space for powered vs. unpowered solar panels...... 144

Figure 7.41: Vector-vector space comparison of powered and unpowered solar panel. 144

Figure 7.42: Stokes S1 and S2 for Sphere vs. Cube...... 146

Figure 7.43: S1/S0 comparison of Sphere and Cube...... 147

Figure 7.44: S3/S0 vs. S1/S0 - for Sphere and Cube...... 147

Figure 7.45: Stokes parameters for Mylar dish and painted dish a) S1; b) S2...... 148

Figure 7.46: Vector spaces for Mylar and painted dishes; (a) S1/S0; (b) S2/S1...... 149

xvii

Figure 7.47: Vector-vector spaces for Mylar and painted dishes; (a) S2/S0 vs. S1/S0

linear; (b) S3/S1 vs. S2/S0 circular...... 149

Figure 7.48: Solar panel with vertical stringers...... 150

Figure 7.49: Stokes parameter S1 for orthogonal orientations of solar panel stringers. 151

Figure 7.50: Stokes vector space S1/S0 for horizontal and vertical stringer orientation of solar panels...... 152

Figure 7.51: Vector-vector space S2/S1 vs. S1/S0 for a solar panel in two orthogonal orientations...... 153

xviii

LIST OF ABBREVIATIONS

1D One dimensional

2D Two dimensional

AM Angular momentum

B, V, R, I Blue, Visible, Red, Infrared

EM Electromagnetic waves

BD Bhattacharyya distance

B-V Blue-Visible

BM Bulk medium

BMM Berreman matrix method

B-R Blue-Red

B-I Blue-Infrared

DTV DirecTV

EM Electromagnetic waves

GEO Geosynchronous orbit

GSM Gaussian Schell model

LEO Low earth orbit

LP Linear polarizer

LWIR Long Wave Infrared

xix

MDM Mueller deviation matrix

PDF Probability density function

Pe Probability of error

QWP Quarter wave plate

R-I Red-Infrared

RSO Resident space object

SNR Signal to noise ratio

SP1 Solar panel from Composite 1

SP2 Solar panel from Composite 2

SPA Solar phase angle

SSA Space situational awareness

SAM Spin angular momentum

SPH Solar panel with horizontal stringers

SPV Solar panel with vertical stringers

TLE Two-line element

TMM Transfer matrix method

V-R Visible-Red

V-I Visible- Infrared

LIST OF SYMBOLS

푬 Electric field

푬풙 X component of electric field

푬풚 Y component of electric field

푬ퟎ Initial value of electric field

∗ 푬풙 The complex conjugate of electric field in the x direction

흉 Phase of component measured on each axis

휹 Change in phase component measured

흃 Ellipticity of polarized light a Major axis of light ellipse b Minor axis of light ellipse

φ Angle between the x axis and the major axis of the ellipse of polarization

D Dielectric displacement vector field

ε Absolute permittivity

ε0 Electric constant

휺풓 Permittivity of dielectric medium, relative permittivity

푷̅ Dielectric polarization density

푱̅ Total angular momentum of field

xxi

푴̅ Orbital angular momentum density

풅풕 Derivative with respect to time

δ The net phase of the electromagnetic wave, 훿 = 훿푥 − 훿푦

δx The phase of the x component of wave

δy The phase of the y component of wave

푱 The Jones matrix

푱풊풊 Component of Jones matrix in position ii

푱풓 Jones matrix for rotator

푱푸푾푷 Jones matrix for quarter wave plate

푱푯푾푷 Jones matrix for half wave plate

푺ퟎ The first Stokes parameter, total intensity

푺ퟏ The second Stokes parameter, horizontal minus vertical intensity

푺ퟐ The third Stokes parameter, diagonal intensity at 45° minus diagonal at 135°

푺ퟑ The fourth Stokes parameter, right circularly polarized intensity minus left circularly polarized intensity

푺 The Stokes vector

푰ퟎ Initial intensity value

푰(풏ퟎ, 풎ퟎ) Intensity measurement with polarization pass axis at 푛0, and retardance of 푚0 i The imaginary component, √(-1)

푰 The expectation value of the sum of measurable intensities in two orthogonal directions.

xxii

Q The expectation value of the difference of measurable intensities of horizontal and vertical directions.

U The expectation value of the difference of measurable intensities of diagonal at 45° and diagonal at 135° directions

V The expectation value of the difference of measurable intensities of right circularly and left circularly polarized directions.

ω Angular frequency

BD Bhattacharyya distance

MD Mahalanobis distance

xxiii

CHAPTER 1

INTRODUCTION AND OBJECTIVES

Polarimetry has surfaced as a potent means to augment the information being extracted from remote sensing applications throughout the world [1]. Polarization is one of the real, measurable attributes of electromagnetic (EM) waves, whether they are visible light, long radio waves or short ultraviolet waves. The polarization of an EM field indicates the direction of the electric field vector, which may vary with time as the wave travels. The amplitude of the components of the electric field are, in general, complex constants. This means that these field components carry phase information [2]. Polarimetry can be as simple as recording the same wavefront with several different polarization screens and analyzing the intensity of the wave as it varies with polarization, or it can involve complex instruments with precision motor driven data collection.

Polarimetry is a natural application for the monitoring of manmade objects and debris near the earth [3]. As the number of orbiting objects continues to increase, scientists continue to search for methods of improving the capability and speed of identifying deep space objects, including natural and man-made debris [4]. Therefore it is important to provide baseline features for characterizing orbiting objects in low earth orbit [LEO] and geosynchronous orbit [GEO], and to support object identification in deep space in order to aid in the recognition and characterization of space objects for improved overall situational

awareness [5, 6]. Several cases of space debris colliding with existing satellites in GEO and LEO orbits have been documented and have resulted in thousands of additional debris products creating a cascading threat concern [7]. It is imperative to gain mastery of this situation for the safety and economic health of all space-faring nations, since one satellite can cost millions of dollars [8]. Our scientific community is searching for non-imaging techniques to determine identification and status of unresolved orbiting bodies [9].

In almost every nation on earth, telescopes observe these objects of interest in space. In most cases, the information recorded is the total intensity, and may be analyzed by wavelength or wavelength band [6]. Information on polarization states is contained in the same beam of light that is being captured by observing telescopes. The information carried by polarization is unrelated to the image information being extracted, so it is an increase to the amount of information that can be extracted from the same signal.

Maximizing the information extracted is best from a cost perspective, and since the telescope is already acquiring the signal, the analysis should include extraction of the information carried by the polarization signal.

Polarimetry appears to be a naturally complementary process to spectral imaging since the same light beam providing the spectral information can be further analyzed to extract additional information [10]. While traditional analysis of light gathered through a telescope focuses on the wavelength of light, optical resolution, and tracking of point object orbit and velocity, polarimetry is focused on measurement of the vector nature of the optical field being observed. The spectral data can be analyzed for material composition and color, but the polarization will provide detailed information regarding shape, roughness, edges, shading and other identifying surface features [11].

This work proposes to demonstrate that analysis of the polarization of light passively reflected from satellites can contribute to the body of knowledge crucial to space situational awareness (SSA).

The hypotheses to be tested by this work are:

1) polarimetry differentiates simple geometries and materials from one another by

producing a signature that can be quantified statistically;

2) polarimetry differentiates complex objects from one another using statistical

tools applied to signature analysis;

3) polarimetric signatures of complex objects are strongly influenced by the

polarimetric signatures of the individual components (signatures based on full

Stokes parameters);

4) polarimetric signatures can be used to correctly associate an unknown object

with one of two clusters of data representing two complex objects.

Methods by which this will be accomplished are:

1) extrapolation of astronomy’s analytical method of analysis using color-color

space to the intensity analysis of the vector space of polarization intensity; and

2) analysis of the vector space data using existing mathematical and statistical

techniques that lend themselves to computational methods.

The organization of the paper is as follows. Chapter 2 provides a synopsis of the science of polarization and how it is useful for SSA. Chapter 3 provides the mathematical description of polarization, the Stokes vector, the Poincare sphere, and a theoretical discussion of the propagation of the polarization vector. Chapter 4 presents the current uses of polarization and signatures that it generates, introduces the astronomical tool, color-

color analysis that is adapted to polarization and the description of the novel application of vector analysis to the problem of identification of unresolved objects in space, as well as the statistical approach proposed to allow hi speed information analysis. Chapter 5 contains a brief discussion of previous related work. Chapter 6 presents the experimental setup, the sources of error and the calibration of the equipment. Chapter 7 provides the experimental results and Chapter 8 the conclusions.

CHAPTER 2

THE SCIENCE OF POLARIZATION

2.1 Introduction

Light is an EM wave with electric and magnetic fields that are vectorial in nature, but displays measurable scalar aspects. Most people are more familiar with the scalar properties such as intensity, which is a measure of the brightness or luminance of a source of light, or color, the human standard for determining the wavelength of visible light. At the same time, light clearly has a vector nature, which can be detected by human senses noting the appearance of shadows that are sharp and distinct.

Even the simplest EM wave has vector properties that carry significant information.

The most straightforward EM wave is a plane wave, illustrated in Figure 2.1, which can be described by the direction of propagation of the wavefront along the z axis, and the electric and magnetic fields, which are oscillating fields that are perpendicular to the direction of propagation. It is described as a plane wave because the wavefront, or the locus of constant phase is situated in a plane that is transverse to the direction of propagation. The electric field is normally the field of interest in the study of polarization, not the magnetic field.

Figure 2.1: Representation of EM wave electric and magnetic fields [12].

Further study of the electric vector motion will clarify the concept of polarization.

Consider for a moment the path traced by a time harmonic wave propagating in the z direction. If a momentary stretch of time is examined that allows the electric vector to make one circuit of the axis of propagation, we would observe the pattern shown in Figure

2.2. The tip of the electric vector traces out a pattern that can be represented by the magnitude of the two components

퐸푥 = 푎1 cos (휃) (2.1) and

퐸푦 = 푎2 cos (휃 + 훿), (2.2)

where the arguments (휃) and (휃 + 훿) of the cosine terms represent the phase of the component; the angle 휑 푖푛 Figure 2.2 denotes the angle between the 푥 axis and the major axis (a) of the ellipse of polarization. In more general terms, 휃 = 푘0푧 − 휔0 푡, where

2휋 푘 = and 휔 = 2 휋 푓. 0 휆 0

Figure 2.2: The plane of polarization, propagation out of the page.

This shape as it is traced out is always in the plane of polarization. Note that the time elapsed for the electric vector to trace out the polarization ellipse is on the order of

10-15 seconds, too fast to actually be observed. The polarization ellipse is therefore the locus of the tip of the electric field vector at a certain location during propagation. [13].

Recall that the wave is propagating in the z direction, while the electric vector is rotating azimuthally around the z axis as it travels. The net effect is a wave that is spiraling around the z axis in time, as shown in Figure 2.3.

Figure 2.3: Changing direction of the electric vector over time [14].

2.2 The Electric Vector

The polarization of light is determined by the behavior of the electric vector in space. Polarization can be considered to mean orientation, and refers to the idealized orientation of the electric vector in the field transverse to the direction of propagation. For simplicity, z-polarization has been ignored, since the magnitude is extremely small, and only the transverse field will be considered in this work. In general, polarization is a description of the amplitude, orientation and oscillation of the electric vector around the optical axis, which is coincident with the direction of propagation along the z axis. When the amplitudes of the 푥 and 푦 components are unequal and non-zero, the polarization is elliptical, and is the most general form of polarization.

In order to define the polarization of an electric wave, the ellipticity must be determined, and it is defined as

푏 휉 = ± , (2.3) 푎

where 푎 and 푏 are the major and minor axes of the ellipse, as shown in Figure 2.2.

Given this description, it is always possible to define a frame of reference in which the 푥 and 푦 axes are coincident with the major and minor axes of the ellipse defined by the motion of the electric vector. This general convention will be followed, with propagation along the +푧 axis.

2.3 Special Cases – Linear and Circular Polarization

As the relative values of 퐸푥 and 퐸푦 change, some special configurations occur that will be considered well-defined and which will be the basis for measurements and characterization of polarization. When 퐸푦 goes to 0, this is defined as linear polarization.

Any polarization state in which the electric vector oscillation is restricted to one plane, regardless of its orientation to the reference frame, is an example of linear polarization.

On the other hand, when 퐸푥 and 퐸푦 are of equal amplitude, the electric vector will trace a circle instead of an ellipse, and this is defined as circular polarization. The vector shown in Figure 2.3 is an example of circular polarization.

The complete description of the state of polarization of a light beam requires the ellipticity, the orientation of the ellipse, and a third important characteristic, the handedness of the wave. A rotation that is counter-clockwise is defined as right handed, or positive rotation, and a rotation that is clockwise is left handed or negative. This convention was

chosen to correspond to the quantum mechanical definition of the angular momentum of a photon [13].

Another way to determine the direction of handedness is to look at the equation for

퐸푥 and 퐸푦, where 퐸푦 contains the term 훿 (see Eq. 2.2). Handedness coincides with the sign of the 푠푖푛훿. As 훿 changes, the orientation of the ellipse and the direction of rotation changes.

This form of angular momentum relates to spin angular momentum (SAM), which is only a portion of total angular momentum (AM). The SAM is a quantum property of photons and elementary force carriers in nature, and does not have a classical counterpart.

Each photon carries a linear momentum of ћ 푘0 per photon, where ћ is Planck’s constant divided by 2휋 and 푘0 is the magnitude of the k number at that moment. The SAM of a photon is physically expressed in the form of circularly polarized light, which can be produced by a birefringent optical plate [15]. Spin is capable of producing a torque on birefringent materials as demonstrated in the 1936 Beth experiments [16]. The torque from spin is produced because the dielectric “constant”, ε is a tensor:

퐷̅ = 휀 ̿ 퐸̅ = 휀0퐸̅ + 푃̿, (2.4) where ε̿ = ε0 ε푟 , 퐷̅ is the electric displacement vector, and where

푑퐽̅ = (퐷̅ × 퐸̅), (2.5) 푑푡 with 퐽̅ is the total angular momentum. In other words, the angular momentum exists because 퐸̅ is not parallel to 퐷̅ (and 푃̅̅̅). On the other hand, there is an additional component of angular momentum, in the form of orbital angular momentum density, for which the expression is

푀̅ = 휀0푟̅ × (퐸̅ × 퐵̿) (2.6) where 퐵̅ is the magnetic induction vector, and 푟 is distance from the optic axis.

Theoretically, plane waves cannot carry an axial component of angular momentum.

However, real beams are changed by the way they are measured and carry a small axial component of angular momentum [17]. It has been demonstrated that Laguerre-Gaussian beams possess both spin and orbital angular momentum [15]. An illustration may be helpful in demonstrating the difference between SAM and OAM as shown in Figure 2.4, where OAM can be thought of as the direction of the Poynting vector as the photon propagates, or the direction of the paraxial beam, whose angular plane wave components are near to but not quite axial in direction.

Figure 2.4: Spin and orbital angular momentum density [18]

OAM is related to the spatial distribution of the beam, and such states have an azimuthal phase term that is proportional to 푒푖푙훼, where 푙 is topological charge and 훼 is azimuthal angle. By this quality, any value of 푙 will produce a beam with a helical phase front that is orthogonal to any other phase front with a different value for 푙 [19]. These beams have helical wave fronts with the number of intertwined wave fronts defined by the

magnitude of 푙, and its sign provides the direction of rotation. The Poynting vector, which is parallel to the surface normal of the phase fronts, has an azimuthal component around the beam which produces the OAM along the axis of propagation. All helically phased beams carry an OAM that can be quantified in terms of ћ, and is in fact equal to a value of

푙 ћ per photon [17]. The SAM and OAM are in general distinct from one another. This work will consider SAM only, and not OAM.

2.4 Polarization by Reflection

The source of light used in this study is a halogen light source intended to replicate sunlight, which is unpolarized, yet the study concerns observing and measuring polarization. Light can become polarized when interacting with objects and their surfaces by reflection, refraction, and scattering [20]. A simple explanation for this is the fact that reflection coefficients for s and p polarization for a given angle of incidence are, in general different. With the distances involved in measuring light from space-borne objects, it is the reflected light that is of most interest. The angle of reflection is therefore of great interest in this study.

By analyzing characteristics of reflected waves, it has been observed that at angles of reflection above 25°, there is a significant difference in reflectivities of s and p polarization for conductors and dielectrics. Also, as the angle of reflection changes, the polarization changes as well. Since this study primarily considers sunlight reflected from space objects, the angles of incidence and reflection are important. It has been noted that for GEO satellites, the incident angle of the incoming light must be greater than 20° or the satellite will be in the earth shadow and not visible [21]. For most space-borne objects,

this angle is measured by the solar phase angle (SPA), which is the sun-satellite-observer angle with the satellite at the vertex of the angle. Note that this angle is twice the angle usually measured from the normal, since it includes the angle of incidence and the angle of reflection. For GEO satellites, the SPA has a component that varies every night with the position of the sun relative to the satellite, which appears stationary over the observer, and another component that varies seasonally [22]. This study will be mostly concerned with the nightly variation, and thus the experiments are set up to measure polarization based on the angle of reflection between the source and detector, with the target at the vertex.

2.5 What Polarimetry Adds

The problem of distant object recognition, particularly for objects in GEO orbit, is one of great complexity and difficulty. Objects at this distance are unresolved, which may be interpreted to mean that shape characteristics of the object have been lost. Yet in the study of vision as applied to object recognition, it has been found that recognition by shape is probably the most established aspect of object recognition, and it is the first characteristic that is used to distinguish an object when it is resolved [23] .

Traditional imagery has focused on wavelength and intensity measurements, and the visible optical field, which can provide information about color and the nature of the material from which it was reflected [21]. Polarization, on the other hand, measures the vector nature of the optical field [13] captured by a reflected beam of light, and provides information primarily about geometry: shapes, edges, special features, and surface finish.

It provides information that is generally uncorrelated with the spectral information that is usually being measured. From this it is easy to deduce that man-made objects, with the

regularity of surfaces and sharp edges, will be more readily discernable from a natural background using a polarimetry measurement than an intensity measurement.

Image analysis is also highly dependent on contrast, and polarization analysis can significantly enhance contrast in any scene. This can be accomplished by simple one- dimensional (1D) polarimetry by adjusting the polarization to maximize the contrast between an object and the background [11]. When a second polarization measurement is made, polarization difference imaging can be performed by adding and subtracting the intensities of the two different states [24]. These provide 푆0 and 푆1, the first two of the four Stokes parameters [13]. Two-dimensional (2D) polarimetry is useful in clutter rejection, contrast improvement, and in reducing noise from random media, again significantly improving imaging edge enhancements. 2D polarization discrimination has been demonstrated to increase the range at which desired targets can be recognized by a distance factor of two to three times greater than spectral analysis alone [24].

Algorithms have also been written to use 2D polarimetry to improve the resolution of images to more than twice the diffraction limit [25]. Atmospheric turbulence, which distorts visual images, is also mitigated by using polarization, which is relatively insensitive to turbulence [26]. Polarization therefore has the capability to strongly improve image quality and increase terrain situational awareness when viewing the earth from space. An example can provide context showing how useful polarimetry can be in detecting man-made objects within terrain and lighting conditions that make traditional imaging far less effective. Figure 2.5 compares a visible wavelength image of man-made

objects in a field, with the same picture taken using long wave IR intensity, and the final image using long wave IR polarimetry.

Figure 2.5: (a) Visible imagery, (b) long wave IR imagery, (c) long wave IR polarimetry [11].

Hidden within all three pictures are man-made objects, but they are very difficult to detect in the visible and IR imagery. When viewing Figure 2.5(a) without being told there are two trucks parked in the shade, it is almost impossible to detect the vehicles due to the poor contrast provided by the vehicle location in the shaded portion of the picture.

Figure 2.5(b) is somewhat better, but without prior knowledge it would still be very difficult to identify the existence of the trucks. However, in Figure 2.5(c) the trucks are clearly visible, but the vegetation, which exhibits random polarization, is almost completely lost. The shape, edges, and surface orientation of the vehicles are clearly discernable by polarimetric imaging, while other methods are far less effective.

Another study using polarization in the long wave infrared (LWIR) provides a much higher probability of detecting targets, extremely low false alarms, and is based on the idea that facets found on targets that are man-made show a much higher degree of polarization, and are therefore highly likely to produce polarization signatures [27]. One of the conclusions of that study is that polarization is helpful under less than optimal

atmospheric conditions. This topic will be addressed later in the discussion of the current state of research.

Another example of polarization vision improvement is shown in Figure 2.6 where a normal intensity image is provided of fish swimming in murky water. The outlines of the fish are barely discernable in Figure 2.6 (a) the unpolarized image, but clearly discernable in an image that is partially linearly polarized. Note that in (a) there are two fish that can be discerned, while in Figure 2.6 (b) four fish can be identified. In this example, the reddish color of the water indicates horizontal polarization, while the fish display enhanced contrast relative to the background. The murkiness of the water has almost no power to diminish contrast in (b), improving the ability to discern living things from background such as water or atmosphere. Similar contrast enhancement can be used to examine damage to metal structures such as aircraft or ships [28].

Figure 2.6: Polarization imaging produces increased detail: (a) unpolarized, (b) linearly polarized [28].

From this brief review, it can be seen that polarization is a profoundly useful tool in discerning the normally indiscernible. Non-random surfaces, edges, planes and surface finishes impact the state of polarization of the light reflected from them. The problem

requiring solution is identification of a point source object an extremely long distance away. These clues should be an indication that there is more in an unresolved point than color, and that information needs to be exploited. Chapter 3 will provide the mathematical tools that can aid in this analysis.

CHAPTER 3

POLARIZATION ALGEBRA AND PROPAGATION

3.1 Jones Calculus

The vector nature of light can be described mathematically. One of the tools that is helpful in understanding polarization is Jones Calculus, created by RC Jones in 1941.

This method was introduced to describe the interaction of polarized light with optical devices using very compact mathematical notation [29]. One of the limitations of Jones calculus is that it can only be used for completely polarized light, but there are other tools available to describe mixed states of polarization. Mueller matrices are capable of analyzing partially polarized light, and some of the Jones calculus equations can be used in a Mueller matrix, so this format will be presented as the simplest and most easily understood.

Some examples of Jones calculus will be useful in comparing a full mathematical expression to the Jones vector method. If the following assumptions are made about the beam:

1) quasi-monochromatic light,

2) propagation along the z-axis,

3) propagation in the form of a plane wave,

4) time harmonic traveling nature,

then the envelope of the electric field can be resolved into its transverse components as:

푖 훿푥 퐸푥 = 퐸0푥푒 , (3.1)

푖 훿푦 퐸푦 = 퐸0푦푒 , (3.2) where the 훿푠 represent the phases of the x and y components. Equations (3.1) and (3.2) can be arranged in a 2 x 1 column vector representing the electric field, 퐸̅:

퐸 퐸 푒푖 훿푥 퐸̅ = ( 푥) = ( 0푥 ), (3.3) 푖 훿푦 퐸푦 퐸0푦 푒 called the Jones vector for elliptically polarized light. The difference between the phase of the 퐸푥 and 퐸푦 phase components will often be abbreviated by the symbol 훿 where

훿 = 훿푥 − 훿푦. (3.4)

It can also be readily seen that 푒푖 훿 = 푖 , when 훿 = 휋/2, and 푒i δ = −1 when 훿 =

±휋, etc. By factoring the common portions, an abbreviated form of the normalized Jones vector can then be utilized with the standard states of polarization defined as follows:

1 Linearly polarized in the x direction 퐸̅ = ( ), (3.5a) 0

0 Linearly polarized in the y direction 퐸̅ = ( ), (3.5b) 1

1 1 Linearly polarized at +45⁰ 퐸̅ = ( ), (3.5c) √2 1

1 1 Linearly polarized at -45⁰ 퐸̅ = ( ), (3.5d) √2 −1

1 1 Right circularly polarized light 퐸̅ = ( ), (3.5e) √2 푖

1 1 Left circularly polarized light 퐸̅ = ( ), (3.5f) √2 −푖

Another important property of polarization that can be easily expressed using this notation is the orthogonality of the vectors. Orthogonality of the vectors, say 퐸̅1,2 are defined in terms of their inner product:

푎 ̅† ̅ ∗ 2 퐸1 ∙ 퐸2 ≡ (푎1 푏1)  ( ) = 0, (3.6) 푏2 where †refers to the adjoint operation or the transpose conjugate, and * denotes the complex conjugate. This can be tested with horizontally and vertically polarized light, viz.,

0 (1 0)∗ ( ) = 0, (3.7) 1 which explains why crossed polarizers allow no light to pass: the polarizers are orthogonal to each other, light is only allowed in the plane specified, and cancel when the dot product is taken.

Similarly for right and left circularly polarized light:

1 (1 푖)∗ ( ) = 0. (3.8) −푖

Jones calculus can be used for the superposition of waves of different polarizations, such as horizontally and vertically polarized light:

1 0 1 퐸̅ = 퐸̅ + 퐸̅ = ( ) + ( ) = ( ), (3.9) 푥 푦 0 1 1 which is the equation for light linearly polarized at +45⁰.

The transformation of the polarization states is effected through the Jones matrix which can be used to represent optical elements such as polarizers (diattenuators), retarders

(wave plates), and rotators. If the Jones matrix J is represented as

퐽푥푥 퐽푥푦 퐽̿ = ( ) (3.10) 퐽푦푥 퐽푦푦 then

퐸̅표푢푡 = 퐽̿ 퐸̅푖푛. (3.11)

In general, the Jones matrix for a polarizer is

푝푥 0 퐽푝̿ = ( ). (3.12) 0 푝푦

For example, a horizontal polarizer has a Jones matrix:

1 0 퐽̿ = ( ). (3.13) ℎ 0 0

An ideal linear polarizer will pass light that is polarized in the orientation of the polarizer and block all other light. If horizontally polarized light represented as

1 퐸̅ = ( ), (3.14) 푖푛 0 is passed through a horizontal polarizer, all of the light should pass through:

1 0 1 1 퐸̅ = ( )  ( ) = ( ), (3.15) 표푢푡 0 0 0 0 yielding horizontally polarized light. On the other hand, if horizontally polarized light is passed through a vertical polarizer with the Jones matrix

0 0 퐽̿ = ( ), (3.16) 푣 0 1 none should pass, and it will all be blocked, as shown through the transformation:

0 0 1 0 퐸̅ = ( )  ( ) = ( ). (3.17) 표푢푡 0 1 0 0

Using this notation, very complex systems of polarizers, wave plates, retarders and other polarizing elements can be represented mathematically, such as [13]:

Rotator:

푐표푠 휃 푠푖푛 휃 퐽̿ = ( ); (3.18) 푟 −푠푖푛 휃 푐표푠 휃

Rotated linear polarizer

2 퐽̿ = ( 푐표푠 휃 푠푖푛 휃 푐표푠 휃 ); (3.19). 휃 푠푖푛 휃 푐표푠 휃 푠푖푛2 휃

Quarter wave plate (QWP), slow axis vertical:

1 0 퐽̿ = 푒−푖 휋/4 ( ); (3.20). 푄푊푃푉 0 푖

QWP, slow axis horizontal:

1 0 퐽̿ = 푒푖 휋/4 ( ) (changes handedness); (3.21). 푄푊푃ℎ 0 −푖

Half wave plate [HWP]

1 0 퐽̿ = 푒−푖 휋/2 ( ). (3.22). 퐻푊푃 0 −1

3.2 Stokes Vectors

As mentioned earlier, Jones Calculus is limited to describing totally polarized light, but there are other tools available that allow measurement of both fully polarized and partially polarized light. The Stokes vectors are one such tool. In 1852, Sir George Gabriel

Stokes found that the polarization of light could be quantified by four experimental quantities that are now named after him[13]. The first parameter, designated 푆0, represents the total intensity of the optical field. The other three parameters describe the horizontal/vertical linear polarization (푆1), diagonal linear polarization (푆2), and the circular polarization (푆3), of the field.

Stokes was successful where others had not been because he used an experimental definition of parameters based on intensity measurements and produced an equivalent mathematical characterization of polarized and unpolarized light. Using the experimental approach of measuring intensity instead of the analytical concept of wave amplitude, he

was able to derive a suitable mathematical description of the Fresnel-Arago laws [30] that

Fresnel and his successors had not been able to do [13]. These laws of polarized light are:

1) Two linearly but orthogonally polarized coherent waves cannot interfere with each

other.

2) Two linearly polarized waves with the same polarization will interfere with each

other.

3) Two components of natural light that are orthogonal to each other cannot interfere

with each other [31].

Almost a hundred years later, a Nobel laureate, S. Chandrasekhar recognized the importance of Stokes’ work [31]. It should be noted that if one takes the experimental time averaged intensity taken in the correct orientations, one would obtain exactly the four

Stokes parameters. The experimental and mathematical representations of the four Stokes parameters follow, from Collett’s text [13].

The first Stokes parameter, 푆0, can be experimentally obtained by measuring and adding linear horizontally polarized light intensity and linear vertically polarized light intensity in the optical field.

0 0 0 0 푆0 = 퐼(0 , 0 ) + 퐼(90 , 0 ), (3.23) where the format 퐼(휃, 훿) indicates the intensity at the polarization angle, 휃, and the phase delay, 훿, between the two polarizations. Note that when the phase delay 훿 equals 00, the

푥 component and 푦 component are in phase with each other and the light is linearly polarized. When 훿 equals 900, and if the component amplitudes are equal, the light is circularly polarized. Analytically 푆0 can be represented as

∗ ∗ 푆0 ∝ 퐸푥퐸푥 + 퐸푦퐸푦, (3.24)

where the * represents the complex conjugate.

The next Stokes parameter, 푆1, can be experimentally obtained by measuring linear horizontally polarized light intensity and subtracting linear vertically polarized light intensity in the optical field.

0 0 0 0 푆1 = 퐼(0 , 0 ) − 퐼(90 , 0 ). (3.25).

Once again, light is linearly polarized because the phase delay δ equals 0° Analytically this can be represented as

∗ ∗ 푆1 ∝ 퐸푥퐸푥 − 퐸푦퐸푦. (3.26)

푆1 reflects the important concept of polarization differencing, and it is useful in enhancing image contrast.

The next Stokes parameter, 푆2, can be experimentally obtained by measuring linearly polarized light intensity measured at +45⁰ and subtracting linearly polarized light intensity measured at 135⁰:

0 0 0 0 푆2 = 퐼(45 , 0 ) − 퐼(135 , 0 ). (3.27).

Analytically this can be represented as

∗ ∗ 푆2 ∝ 퐸푥퐸푦 + 퐸푦퐸푥. (3.28).

Finally, the last Stokes parameter, 푆3, can be experimentally obtained by measuring circularly polarized light intensity measured at +45⁰ and subtracting circularly polarized light intensity measured at 135⁰:

0 0 0 0 푆3 = 퐼(45 , 0 ) − 퐼(135 , 90 ). (3.29).

It is circularly polarized because it can be seen from the designation 퐼(휃0, 900), that there is a 90⁰ phase delay between the horizontal and vertical components of the beam.

Analytically this can be represented as

∗ ∗ 푆3 ∝ 푖(퐸푥퐸푦 − 퐸푦퐸푥). (3.30).

In relations (3.24), (3.26), (3.28) and (3.30), the proportional to sign has been used to designate that a constant proportionality factor has been omitted. This factor, which is the same for all relations, is due to the time averaging of the electric fields needed to determine the respective intensities in (3.23), (3.25), (3.27) and (3.29), as well as to accommodate for the difference between the units of intensity and the square of the electric field, which are related through the characteristic impedance of the medium.

It can be shown that:

2 2 2 2 푆0 ≥ 푆1 + 푆2 + 푆3 , (3.31) where the ‘greater than’ sign is applicable to partially polarized light, and the ‘equals to’ sign is applicable to completely polarized light [13].

The Stokes parameters are often experimentally designated by the different symbols 퐼, 푄, 푈, and 푉, respectively [13].

Also, the Stokes parameters can be recast in terms of the magnitudes 퐸0푥, 퐸0푦 of the complex electric fields 퐸푥, 퐸푦 (see Eq. (3.3)) according to

2 2 푆0 = 퐸0푥 + 퐸0푦, (3.32)

2 2 푆1 = 퐸0푥 − 퐸0푦, (3.33)

푆2 = 2퐸0푥퐸0푦푐표푠훿, (3.34)

푆3 = 2퐸0푥퐸0푦푠푖푛훿. (3.35)

Here, 훿 is the phase difference between 퐸푥 and 퐸푦 as previously noted. Also, in writing Eqs. (3.32) - (3.35), the proportionality sign has been omitted, for simplicity.

Observe that if 푆0, 푆1 푎푛푑 푆2 are measured, 푆3 can be calculated, so that a linear polarimeter could be used for full Stokes Polarimetry with some accommodation.

Standard Stokes parameters for the optical elements reviewed for Jones Calculus are listed below [13]:

• Linear horizontally polarized light (퐸0푦 = 0)

2 2 푆0 = 퐸0푥, 푆1 = 퐸0푥, 푆2 = 0, 푆3 = 0. (3.36)

• Linear vertically polarized light (퐸0푥 = 0):

2 2 푆0 = 퐸0푦, 푆1 = −퐸0푦, 푆2 = 0, 푆3 = 0. (3.37)

• Linearly polarized light at +45⁰:

2 2 푆0 = 2퐸0 , 푆1 = 0, 푆2 = 2 퐸0 , 푆3 = 0. (3.38)

• Linearly polarized light at -45⁰:

2 2 푆0 = 2퐸0 , 푆1 = 0, 푆2 = −2퐸0 , 푆3 = 0. (3.39)

• Right circularly polarized light:

2 2 푆0 = 2퐸0 , 푆1 = 0, 푆2 = 0, 푆3 = 2퐸0 . (3.40)

• Left circularly polarized light is defined as

2 2 푆0 = 2퐸0 , 푆1 = 0, 푆2 = 0, 푆3 = −2퐸0 . (3.41)

The four Stokes parameters are often written in a shorthand notation called the

Stokes vector where elliptically polarized light is represented as

퐸2 + 퐸2 푆0 0푥 0푦 2 2 ̅ 푆1 퐸0푥 − 퐸0푦 푆 = ( ) = . (3.42) 푆2 2퐸0푥퐸0푦푐표푠훿 푆 3 (2퐸0푥퐸0푦푠푖푛훿)

The shorthand version for special cases of elliptical polarization is written as

±|푆0| ±|푆1| 푆̅ = 퐼0 ( ). (3.43) ±|푆2| ±|푆3|

For example, horizontally (vertically) polarized light is

1 푆̅ = 퐼 (±1). (3.44) 0 0 0

Linearly ±45⁰ polarized light is

1 푆̅ = 퐼 ( 0 ). (3.45) 0 ±1 0

Finally, right (left) circularly polarized light is

1 푆̅ = 퐼 ( 0 ). (3.46) 0 0 ±1

3.3 The Poincaré Sphere: the Basis for Vector Space

As mentioned earlier, the Jones calculus and Mueller matrices were developed for studying polarization in the early 1940’s, but polarization had been studied long before that. The French mathematician and mathematical physicist Henri Poincaré devised a tool to analyze the effect of optical elements on a polarized beam of light in 1892, when he published a paper introducing the Poincaré Sphere. Poincaré understood the implications

of geometry on polarization, and provided a tool which projected the state of polarization onto a sphere using stereographic projection. The technique of stereographic projection extends a line from the pole of a sphere to any other point on the sphere and then to the plane of projection. Poincaré realized that the polarization ellipse could be represented on the complex plane, and he reversed the process used in the stereographic method to project the ellipse onto a sphere. The sphere thus generated contains a one to one correspondence with all possible states of polarization of a beam of light being represented on the sphere.

Poincare’s unique insight provides a visualization tool by which an intuitively geometric view can be obtained of the change of state of polarization made by the interaction of a polarized beam of light with an optical element or system. Since matrix calculus was not generally available during his time, he was able to demonstrate that the use of his sphere would simplify many of the problems associated with calculation of the effects of polarization [31].

The Poincaré sphere is still generating new articles and new applications in optics and is quite a useful tool in understanding polarization. Among its most remarkable properties is that the three Stokes vectors, 푆1, 푆2, and 푆3, representing the two Stokes linear polarization parameters and the Stokes circular polarization parameter are the orthogonal basis of the Poincaré space, as shown in Figure 3.1. Any point, 푃, located on the surface of the sphere represents the intersection of the three Stokes vectors which characterize the polarization as fully elliptically polarized. When an optical element interacts with the existing state of polarization, it can be represented by a rotation of the Poincaré Sphere, and the new point on the sphere can be broken down into the three vectors representing the new polarization state. If the light is only partially polarized, the same relationships exist,

but the initial and final points are located inside the sphere instead of on the surface of the sphere. Many polarimeter readouts use the Poincaré sphere to graphically indicate the polarization states, so it is useful to understand its general meaning.

Figure 3.1: The Poincaré Sphere, on or within which the three Stokes vectors 푆1, 푆2, and 푆3 (or 푄, 푈, 푉) are plotted in Cartesian coordinates.

The Poincaré sphere is constructed in Cartesian coordinates with axes equivalent to the vectors 푆1, 푆2, and 푆3 normalized values. The sphere is constructed with unit radius, so that

2 2 2 |푆0| = 푃 = √푆1 + 푆2 + 푆3 , (3.47) and is representative of the total intensity of a fully polarized beam of light.

The angle 휃 represents the azimuthal rotation about the center vectors 푆3 axis, and

휀 represents the ellipticity of the electric vector tip’s path, and is equivalent to the angle

measured from the horizontal to the point of interest on the sphere (elevation). Others measure the angle 2휒 which is the compliment to 2휀.

Some interesting characteristics of the Poincaré sphere, illustrated in Figure 3.2, include the following [30]:

i. Any two antipodal points on the sphere are orthogonal to each other.

ii. Right circular polarization is represented by the north pole of the sphere.

iii. Left circular polarization is represented by the south pole of the sphere.

iv. Linear polarization states are located on the equator of the sphere.

v. The vertical great circle represents elliptical states that are of constant

azimuth (axis orientation about the direction of propagation).

vi. The intersection of a horizontal plane with the sphere represents states of

constant ellipticity.

vii. The center of the sphere is completely unpolarized, a singularity, while the

surface represents completely polarized light.

Figure 3.2: Polarization States on the Poincaré Sphere.

3.4 Propagation of the Polarization Vector

The question arises as to whether the polarization of the electromagnetic beam changes as it propagates through free space or the atmosphere. If it is changing, is the change stochastic or can it be modeled as a function of the distance it has traveled? These are excellent questions that have often been addressed in the literature [32, 33]. They are relevant to the work in this study. If the polarization is changing during propagation, does the concept of a signature have meaning if it is compared between detection from the top of a mountain in Hawaii as compared to a valley in mid-continent United States?

Historically, it has been assumed that the state of polarization of a light beam is invariant as it propagates in free space, whether it is fully or partially polarized. However, it has been known for quite some time that the state of a partially polarized beam may change as it propagates. The change can be as been hypothesized to be a function of wavelength, distance from the optical axis, and propagation distance, 푧 [34], and was shown by Wolf to also be related to the coherence of the light [35]. However, experimental studies negate the first hypothesis, where Saleh [36] demonstrated that the loss of linear polarization in a beam that propagated 2.6 km was 10-9 radians. Keep in mind that earth’s weather is contained in a layer of approximately 9-17 km, and the rest of the propagation is in extremely thin atmosphere or free space where there is no turbulence [37]. Additional studies reveal that if the beam propagates into the far field, the degree of polarization returns to its original state at the point defined as the source plane, so that as 푘푧 → ∞, the degree of polarization as a function of distance from the axis, frequency and propagation distance are approximately equal to their values at the source plane [38]. Since the area of

concern in this paper involves propagation of the beam from space and normal incidence on the detector, the far field is certainly far enough away to assume that the signature analysis will be similar regardless of where on the earth the measurements are taken. This does raise the question about what to expect in the near field or Fresnel region, and since this study may be of interest to others besides those studying satellites, it is interesting to consider this problem, keeping in mind the minute change experimentally determined by

Saleh.

The primary tool used in this work to analyze polarization is the Stokes vector and its components or Stokes parameters, which are an experimentally defined quantities, although they have analytical representations in terms of the components of the electric field, as shown in the previous Section. Keep in mind that Stokes parameters do not propagate and represent a momentary measurement correlation between the Cartesian components of the fluctuating electric field vector [35]. They are specific measurements of the state of polarization in terms of only the linear polarization and the circular polarization. Any other state of polarization is reclassified into one of the primary states

[39]. Wolf also states that attempting to represent the changes of polarization using the

Stokes vector is severely limited due to the fact that the Stokes parameters only provide properties of the correlation of components of the fluctuating EM at a single point in space at the same instant of time [35]. Any change in the properties or even location as the wave propagates cannot be predicted using the Stokes parameters, so an alternate method is needed and here proposed.

Partially polarized light can be considered in terms of a Gaussian Schell model

(GSM) source, since it may be characterized by a correlation matrix whose elements have

the same form as the mutual intensity of a GSM [33]. A propagating beam can be considered as a collection of random EM fields that is statistically stationary (that is, the probability density function is invariant with respect to time) and which is propagated close to the optical axis but beyond the limits of paraxial propagation along the 푧 axis [40], so that some degree of oblique incidence onto a normal plane may be observed. The 푥 and 푦 axes chosen are orthogonal and perpendicular to the 푧 axis. If the angles of incidence and diffraction are small, the electric field may be represented by a statistical ensemble,

{퐸̅(푟, 휔)} defined as the ensemble of the angular plane wave components of frequency ω at the source plane [35]. Since the light is considered to be propagating as a beam in the 푧 direction, the 푧 components are ignored in the analysis [41]. It is also valid to assume the atmosphere may be modeled as layers of anisotropic media, and much work has been done in this area [42].

Many ways of analyzing the properties of a deterministic propagating EM wave have been proposed, and some of the standard methods applied use the transfer matrix method (TMM) and the Berreman matrix method (BMM) [43]. In BMM, Maxwell’s equations in time harmonic form are first represented in a 6 x 6 matrix that includes Faraday rotation and optical activity. Again, using Maxwell’s equations, the 6x6 matrix can be reduced to a 4 x 4 matrix useful for studying propagation in stratified anisotropic media.

This will be used to analyze the propagation of the polarization vector in this work. In general, the Berreman matrix is given by

2 휖zx 휇yz휇zy 푘푥 휖zy 휇yz 휇yz휇zx 푘푥 휔0(−휇yy + + 2 ) 푘푥( − ) 휔0(휇yx − ) 휖zz 휇zz 휔0 휖zz 휖zz 휇zz 휇zz 휖xz휖zx 휖13 휖xz휖zy 휔0(−휖xx + ) 푘푥 휔0(−휖xy + ) 0 휖zz 휖33 휖zz 푀퐵 = −푖 . (3.48) 휇xz휇zy 휇xz 휇xz휇zx 0 휔0(휖휇xy − ) 푘푥 휔0(−휇xx + ) 휇33 휇zz 휇zz 2 휖yz휖zy 휖yz 휇zy 휖yz휖zy 푘푥 휇zx 휔0(−휖yx + ) 푘푥( − ) 휔0(−휖yy + + 2 ) −푘푥 [ 휖zz 휖zz 휇zz 휖zz 휔0 휇zz 휇zz ]

In deriving Eq. (3.48), it has been assumed that the EM wave propagates with an angle in the x-z plane in the bulk medium (see Figure 3.3). The spatial frequency 푘푥 is indicative of the angle and is given by 푘푥 = 푘0푠푖푛휃푖 where 휃푖 is the angle of incidence onto the bulk medium (BM) and 푘0 is the propagation constant in the surrounding medium, considered to be another layer of air, with characteristics that differentiate it from the layer under consideration. The angular frequency is 휔0, and the 휖푖푗’s and 휇푖푗’s denote the components of the permittivity and permeability tensors. Propagation of the transverse

EM field phasor components 퐸푝푥, 퐸푝푦, 퐻푝푥 and 퐻푝푦 can be found by solving the system of ordinary differential equations written as

퐸 퐸 푝푥 푝푥 퐻 퐻 휕 푝푦 푝푦 = 푀퐵 . (3.49) 휕푧 퐸푝푦 퐸푝푦 [−퐻푝푥] [−퐻푝푥]

The solution of Eq. (3.49) can be symbolically expressed as

퐸 (푧) 퐸 (0) 푝푥 푝푥 퐻푝푦(푧) 퐻푝푦(0) = [푒푥푝푀 푧] . (3.50) 퐸 (푧) 퐵 퐸 (0) 푝푦 푝푦 [−퐻푝푥(푧)] [−퐻푝푥(0)]

Figure 3.3: Propagation through a bulk medium (BM) assumed to be anisotropic (region b), surrounded by isotropic regions a and c.

Some simplifying assumptions can be made by letting the magnetic permeability be isotropic, and setting the off-diagonal elements of the permittivity tensor to 0.

Furthermore, assuming a uniaxial BM so that 휖푥푥 = 휖푦푦 ≠ 휖푧푧 the Berreman matrix takes the simplified form of

2 푘푥 0 −휔0휇 + 0 0 휔0휖푧푧 −휔 휖 0 0 0 푀 = −푖 0 푥푥 . (3.51) 0 −휔0휇 0 0 2 푘푥 0 0 −휔0휖푥푥 + 0 [ 휔0휇 ]

A simplified result assuming unidirectional propagation can be found without exponentiation of the Berreman matrix. To achieve this, note that with 푀퐵 defined as in

Eq. (3.51), the system of differential equations represented by Eq. (3.49) can be explicitly written as:

2 푑 푘푥 퐸푥 = −푖 ( − 휇휔0)퐻푦, (3.52) 푑푧 휖푧푧휔0

푑 퐻 = 푖 휖 휔 퐸 , (3.53) 푑푧 푦 푥푥 0 푥

푑 퐸 = −푖 휇 휔 퐻 , (3.54) 푑푧 푦 0 푥

2 푑 푘푥 퐻푥 = 푖 ( − 휖푥푥 휔0)퐸푦. (3.55) 푑푧 휇 휔0

Differentiating Eq. (3.52) w.r.t. 푧 and substituting Eq. (3.53) into the resulting equation yields

2 2 푑 2 푘푥 2 퐸푥 + 휔0 휖푥푥 (휇 − 2 ) 퐸푥 = 0. (3.56) 푑 푧 휔0 휖푧푧

A solution propagating along the +푧 axis takes the form:

2 푘푥 퐸푥(푧) = 퐸푥(0) 푒푥푝 푖 (휔0√휖푥푥휇√1 − 2 푧). (3.57) 휔0 휖푧푧휇

To obtain a similar solution for 퐸푦, Eqs. (3.54) and (3.55) are used, yielding

2 푘푥 퐸푦(푧) = 퐸푦(0) 푒푥푝 푖 (휔0√휖푥푥휇√1 − 2 푧). (3.58) 휔0 휖푥푥휇

It is remarked that the phasor fields are related to the corresponding envelopes (used in the development of the Jones vectors and the Stokes vectors) through the relations

퐸푥,푦(푧) = 퐸푥,푦(0) exp 푖푘푧푧, (3.59) where 푘푧 is the propagation constant in the BM.

If the atmosphere is completely isotropic, the case would be that 휖푥푥 = 휖푧푧, and the exponentials in Eq. (3.57) and Eq. (3.58) are identical. If atmospheric conditions such as water vapor or aerosols are encountered, anisotropy can result [38]. In this case, we may choose to model the atmosphere by layers of anisotropic conditions that are stratified. This is an ideal setup for application of Berreman’s approach.

The Stokes parameters have been defined earlier in this Chapter (see Section 3.2) and are an excellent description of the state of polarization of the ensemble. However, the

Stokes parameters are experimentally obtained, and although they have an analytical equivalent, they do not change with time. They are defined as sums and differences of measurements of linear and circular polarization at one instant in time, and because of this definition, they do not propagate.

Since Stokes parameters traditionally only depend on the envelopes of the electric fields and do not have distance as a parameter, we have defined polarization parameters

푇푖(푧), 푖 = 0,1,2,3 in a fashion similar to Stokes parameters, but which are expressed in terms of the electric field phasors as:

∗ ∗ 푇0 ∝ 퐸푝푥퐸푝푥 + 퐸푝푦퐸푝푦, (3.60)

∗ ∗ 푇1 ∝ 퐸푝푥퐸푝푥 − 퐸푝푦퐸푝푦, (3.61)

∗ ∗ 푇2 ∝ 퐸푝푥퐸푝푦 + 퐸푝푦퐸푝푥, (3.62)

∗ ∗ 푇3 ∝ 푖(퐸푝푥퐸푝푦 − 퐸푝푦퐸푝푥). (3.63)

It is clear that for propagation in an isotropic BM, 푇푖(푧), 푖 = 0,1,2,3 will not change during propagation, i.e., 푇푖(푧) = 푇푖(0), 푖 = 0,1,2,3, as evidenced by using Eqs. (3.58) and

(3.59) with 휖푥푥 = 휖푧푧. However, for anisotropic propagation, 푇푖(푧) = 푇푖(0), 푖 = 0,1. The variations of 푇푖(푧), 푖 = 2,3 can be determined by using Eqs. (3.58) and (3.59). Considering

2 푘푥 paraxial propagation, i.e., 2 ≪ 1, Eqs. (3.58) and (3.59) can be approximated to 휔0 휖푥푥,푧푧휇

2 푘푥 퐸푝푥(푧) ≈ 퐸푝푥(0) 푒푥푝 푖 휔0√휖푥푥휇 (1 − 2 ) 푧, (3.64) 2휔0 휖푧푧휇

2 푘푥 퐸푝푦(푧) ≈ 퐸푝푦(0) 푒푥푝 푖 휔0√휖푥푥휇 (1 − 2 ) 푧. (3.65) 2휔0 휖푥푥휇

Then if δ is defined as the difference in the phase of the two components, δ = δ푥 −

δ푦, with 훿0 equal to the initial value (at 푧 = 0) , then

2 푘푥√휖푥푥/휇 1 1 훿(푧) = ( [ − ] 푧 + 훿0). (3.66) 2휔0 휖푥푥 휖푧푧

For the special case where

|퐸푥(0)| = |퐸푦(0)|, 퐸푥(0) = |퐸푥(0)|, 퐸푦(0) = |퐸푥(0)| 푒푥푝 − 푗훿0, (3.67)

2 2 푘푥√휖푥푥/휇 1 1 푇2(푧) = 2|퐸푥(0)| 푠푖푛 ( [ − ] 푧 + 훿0), (3.68) 2휔0 휖푥푥 휖푧푧

2 2 푘푥√휖푥푥/휇 1 1 푇3(푧) = 2|퐸푥(0)| 푐표푠 ( [ − ] 푧 + 훿0). (3.69) 2휔0 휖푥푥 휖푧푧

In the general case, 푇2(푧) and 푇3(푧) can be determined from the initial angular plane wave spectrum of the electric fields and the ensuing phase during propagation by directly using Eqs. (3.62) and (3.63), respectively. This shows that the polarization parameters, and hence the polarization vector which can be defined akin to the Stokes vector, changes during propagation in an anisotropic BM, and depends on the angular plane wave spectrum of the initial beam profile.

3.5 Conclusion

It has been shown that BMM results for an anisotropic material provides a succinct description of the optical characteristics of the atmosphere if it is modeled as a series of layers of an isotropic bulk medium surrounded by other layers that differ from the BM initially considered. Beginning with Maxwell’s equations in time harmonic form, they are simplified them into a compact four by four matrix developed by Berreman. Following the solution detailed by Eq. (3.50), multiplying through with the simplified Berreman

matrix, and differentiating with respect to z, one obtains equations for both linear and circular polarization vectors that indicate that polarization vectors can change during propagation through an anisotropic BM.

In Chapter 4, the statistical methods selected to analyze the experimental data will be discussed.

CHAPTER 4

ANALYTICAL METHODS

4.1 Introduction and Motivation

In Chapter 3, the theoretical aspects of the Stokes vector and the generalized polarization vector is presented and discussed. In this chapter, the method of analyzing the polarization data taken from each experimental model is provided. In particular, the problem is framed by the very large increase in populations of objects in the GEO orbit.

The first satellites were launched to geosynchronous orbits in 1963.

Geosynchronous satellite communication has since offered many benefits in technology

[44]. However, because of the value of a satellite in one of those positions, the level of activity and use of the orbit have been remarkable. Thousands upon thousands of objects are orbiting in those locations, and many are non-functional. Those objects are primarily payloads and the stages that were used to deliver the satellite into its coveted position. A well-known graphic is shown in Figure 4.1, where 95% of the orbiting objects are no longer functioning spacecraft.

Figure 4.1: Computer generated images of objects in Earth orbit that are currently being tracked [44].

With the ever-increasing congestion in orbit as well as the potential for debris collision, space authorities are looking for technology that will allow rapid and shrewd identification of critical events in space. The focus is characterization of space objects and the identification of specific events of interest. Due to the large numbers of items that need to be processed and tracked, satellite identification must use highly efficient approaches to identify an unknown object.

The issue at hand is therefore recognition of objects quickly and efficiently.

Characterization of complex objects using their shapes is fast becoming a major tool in computer vision and image recognition [45]. As mentioned before, recognition by shape properties is probably the most customary aspect of recognition, and shape is very often the first aspect that is used to identify an object when it is resolved [23]. However, in the case of satellites in GEO orbits, the image cannot be resolved, leading to the traditional approach of identification by the predominance of spectral characteristics by telescope.

This is why analyzing by polarization is a step in the right direction, since it is sensitive to shape: edges, planes, smooth or rough surface finish, material, and many of the other characteristics we look at to recognize an object from a distance. Although spectral information is the primary source of current information used in satellite identification, color is not necessarily the best differentiator. We recognize an evergreen tree even when it is blue, because of the shape characteristics [23]. Comparison to a known set of images or parameters becomes much less useful when attempting to discriminate between objects of the same class, with high similarity. It has been shown that polarization has been capable of a very high level of discrimination between two similar objects. Speicher et al. demonstrated that polarization intensity analysis could be used to distinguish between two identical satellites launched eighteen months apart [46].

4.2 Traditional Polarization Signatures

The reflection of sunlight from resident space objects produces light that is partially polarized [13]. There are many ways of measuring and characterizing polarization, but the

Stokes vector has been one of the most useful since it is effective in representing partially polarized light [47]. However, in the case of satellite observations, the solar phase angle is constantly changing, so one observation is insufficient to characterize the satellite, even if all Stokes parameters are captured. This has led to the classic form of satellite signatures, represented by measuring various parameters and plotting them as they vary with solar phase angle. This has been demonstrated by Speicher et al., [46], who have compared polarization signatures with spectral signatures as shown in

Figure 4.2. These signatures have been taken from the AMC-15 satellite on the night of

March 21, 2015.

Figure 4.2 (taken from data provided aby Speicher et al.), [46] contains the spectral signature of the satellite (in blue) as well as a polarization signature (in gold). Note that

UTC time is plotted on the horizontal axis, but no meaningful units are available for the vertical axis. The reader is to observe the shape of these two signatures to determine which might be easier to use to identify a target. The spectral signature shown is characteristic of almost every satellite spectral signature, containing one primary peak at or near the minimum of the solar phase angle. The polarization signature 푆1 however, displays several peaks and valleys which may be indicative of various geometric or material characteristics of the satellite, but the peaks have not yet been associated with the physical characteristic driving the changing parameters. With four peaks and a valley, as well as a very high vertical polarization (a negative 푆1 Stokes parameter) achieved as the solar phase angle approaches 90° (near dawn), there may be more information conveyed in the polarimetric signature than in the spectral signature. What the meanings of the various peaks are remains to be researched and resolved.

Figure 4.2: Spectral signature (푆0) and polarimetric signature of AMC-15 (푆1); UTC time plotted on the horizontal axis. Data are nominal and normalized. [46].

This approach to signature analysis is the first type of signature we study in this dissertation. A key question that needs to be answered is whether the peaks and valleys noted in the Stokes signature can be physically associated with a geometric feature or material of construction of the object being measured. An example of a Stokes signature looks like the data in Figure 4.3. The signature has the reflection angle on the horizontal axis and Stokes parameter 푆1 on the vertical axis. The two objects clearly have different signatures with peaks and valleys that can potentially be used to identify the object by comparison to a known signature.

Figure 4.3: Example of comparison of two objects based on their Stokes 푆1 signatures.

There is another method of signature development that could be deduced from the work completed by Speicher et al., by the normalization of his data. Normalization of polarization data is performed by dividing the second, third, and fourth Stokes parameters by the first parameter, 푆0 [13], the total intensity.

4.3 Intensity Analysis Methods from Other Disciplines

One of the best ways to generate out-of-the-box technical approaches that have not yet been exhausted is to take the methods used in one technical field and apply them to problems in another technical field [48] [49]. This cross fertilization of ideas and methods has often produced excellent new technical approaches.

Much study of light intensity has been conducted in astronomy, and spectroscopy has often been used to analyze satellites, looking for “signatures” that will distinguish one satellite from another [50]. One such technique, called astronomical photometry, uses color indices and color-color plots to analyze celestial objects. The basis of the color index is a measurement of the intensity in a color bandwidth such as the visible wavelengths, compared to the intensity in another wavelength band, such as the blue wavelengths [51].

This method of analysis has long been used in astronomy to determine the stellar characteristics of celestial objects, identifying the mass, density, temperature, and chemical constituents of the stars. It was noted that color-color analysis of satellites provided similar patterns to such stellar tracks, indicating that the analysis could provide a unique tool to help discriminate between unknown orbiting objects as well based on their spectral characteristics. Even when the wavelength is not in the visible range, the same techniques are quite useful in characterizing the IR and UV wavelengths [52].

An example of how the color spaces are used to study unknown astronomical objects is shown in Figure 4.4. This was a comparison of the albedo in the ultraviolet-blue

(UV-B) spectrum plotted against blue-visible (B-V). Albedo in this case represents the directional integration of reflectance by the body over all solar phase angles, and is representative of the entire cluster of measurements. The planets differentiate themselves

as they occupy different spaces in the graph. The unknown object appears to be more like

Titan, a moon, than like the larger planets based on its distance from the planetary positions.

0.8

0.6 Venus Uranus Neptune

Visible Jupiter - 0.4 Saturn

0.2 Titan Albedo for Blue Blue Albedofor HD189733b

0 0 0.2 0.4 0.6 0.8 Albedo for Ultraviolet-Blue

Figure 4.4: Color-color plot for planets and unknown stellar object [52].

The same method of color photometry has been used to study the spectral characterization of GEO satellites in order to classify and characterize their signatures, an admittedly difficult task given their faint, unresolved images. The process of target identification begins with satellite classification, and proceeds to identification, status monitoring and detection of anomalies [53].

The method of spectral photometry comprises a set of filters which will isolate a category of wavelengths or color, measure their incoming spectral intensity, and create ratios of those measurements by which to characterize them. As an example, the ratio of the green intensity relative to a measurement of the intensity in the blue spectrum would indicate which of the two bands was being preferentially reflected. This provides the

intensity magnitude difference or intensity excess, which is defined as the color index for

B-V. Using the Johnson photometry bandwidths of the four well-known wavelengths, blue

(B), visible (V), red (R), and infrared (I), this provides six unique combinations of color in a scalar analysis of intensity: B-V, B-R, B-I, V-R, V-I, and R-I. The intensity in all color bands is taken simultaneously, followed by a similar periodic measurement taken throughout one observation session. Once a satellite’s intensity in the various bands has been measured and these scalar color index values determined, further analysis can be performed in “color-color” space, where the intensity excess in B-V for instance, is plotted against the intensity excess in B-R [9]. The method produces fifteen separate combinations of color-color space that can be analyzed in this manner. This tool is used to find distinguishing spectral characteristics that will separate one satellite from another. When viewed in a graph, the points plotted tend to cluster in groups associated with a specific satellite, and in locations that are physically separate from one another. When large amounts of data are available, this technique has proven successful in singling out anomalies from the main population – that is, something that distinguishes that object from others.

One example of what this analysis would look like is displayed in Figure 4.5, where two objects are represented in color-color space, indicated by different color symbols. One color magnitude ratio is plotted on the horizontal scale (blue-red is abbreviated B-R), and another scalar ratio is plotted on the vertical scale (red-infrared is abbreviated R-I).

Figure 4.5: Color-color diagram of two objects.

The two groups of data cluster together in clearly identifiable 2D space, with a small amount of overlap of the data, but with a specific locus in color-color space that can be associated with each group, indicated as Regions 1 or 2. A straight line can be drawn that represents the space for object 1 and the space for object 2, with only a few overlap points. The existence of points from object 2 in Region 1 and from object 1 in Region 2 indicate the uncertainty associated with the division of space. Once an object has been characterized in each of the fifteen color-color spaces, this data can be used to uniquely distinguish it from similar objects.

In applying this method to satellite identification, the concept can be further explained by comparing the color signatures thus developed from multiple satellites with each other. The method has been to plot existing known objects onto the same graph, and analyze the data to separate physical objects from one another [9].

4.4 Application of Color Space Concept to Polarimetry

Since the concepts produced by color analysis have been so useful in characterizing stellar objects as well as satellites, the author concluded that the same methods could yield

very useful information if applied to polarization intensity. While considering how the concept of color analysis could be applied to polarization, it was observed that there are four well-known intensity measurements, scalar in nature, but often projected into vector space on the Poincaré sphere. These are the three normalized Stokes parameters viz., 푆1 =

푆1 푆2 푆3 , 푆2 = , 푆3 = ; so there can be three unique combinations, e.g., 푆2 vs. 푆1; 푆3vs. 푆1, 푆0 푆0 푆0 and 푆3 vs. 푆2. Each of these is a scalar intensity measurement ratio. It is remarked that in this study we will be looking at partially polarized light, which when considered as part of the Poincaré sphere, will always be located within the sphere. Only fully polarized light vectors reach the surface of the sphere. By plotting these normalized intensities on two axes, two-dimensional vector spaces can been created instead of the three-dimensional vector space on the Poincaré sphere. As an example, consider a simultaneous measurement of two Stokes parameters. For 푆1, one needs to measure the difference between intensities for horizontal linear polarization and vertical linear polarizations, which are located on the positive and negative 푠1-axes of the Poincaré sphere (see Figure 4.6). The 푆2 parameter is determined by subtracting the 135° linear diagonal polarization intensity from the 45° linear diagonal intensity. These orthogonal diagonal linear polarizations are located on the vertical 푠2 axis of the Poincaré sphere. In order to visualize one polarization vector space,

푆 2 , four measurements must be taken, horizontal, vertical, 45° linear diagonal and 135° 푆1 linear diagonal (see Figure 4.6).

Figure 4.6: Polarization vector space in the Poincaré sphere; view looking along the S3 parameter axis.

Figure 4.7 provides a view of a polarization vector signature in vector space. The intensity data for a cubic bus was measured at thirty points with progressively higher solar phase angle, representing the movement of a satellite with the earth’s rotation. The question then arises as to whether this type of analysis can help differentiate between two satellites. Although no one has previously proposed using vector-vector space for satellite identification, both Stokes parameters and vector space graphs have been used to analyze satellite signatures [5]. Measurements were taken in our lab of the cubic bus to view it for signature potential.

S1 / S0 0.00

-0.04

-0.08

-0.12 Intesnity Ratio Intesnity -0.16

-0.20 10 30 50 70 Reflection Angle [°] Cubic Bus

Figure 4.7: Example of Vector space 푆1/푆0 signature as a function of reflection angle.

Note the positive and negative peaks in Figure 4.7 that may be indicative of a distinguishing feature of the geometry being viewed in the signature analysis. This approach, as well as plotting a Stokes parameter such as 푆1 on the vertical axis with the reflection angle on the horizontal axis produces a signature that is familiar in form and nature. The reflection angle changes in a manner similar to solar phase angle, which changes with respect to time. Figure 4.7 therefore provides a view of what the object signature would like with respect to time as well as solar phase angle. The Stokes parameter 푆1 plotted this way would be identical in form, but would have a different scale on the vertical axis.

On the other hand the vector-vector spaces do not conform to the traditional satellite signature format, since they are a two dimensional space where time or solar phase angle is not a parameter. For these spaces, we are interested in the cluster of all points in a space defined by any two of the three vector spaces created with the normalized Stokes parameters, as well as ratios of these three vector spaces. An example of the plot for a

vector-vector space is shown in Figure 4.8. As expected, the points tend to cluster in a specific location in space, with a few outliers that will tend to be associated with a peak or valley on the vector space graph. These outliers will tend to be useful in identifying specific geometries or materials of the object being tested.

0.04

0.02

0.00

/S 2 S -0.02

-0.04

-0.06 -0.20 -0.15 -0.10 -0.05 0.00

S1/S0 Cubic Bus

Figure 4.8: Example of vector-vector space for cubic bus, 푆1/푆0 on horizontal axis, 푆2/푆0 on vertical axis.

In order to use either vector space or vector-vector space as a tool for discriminating between objects, it must be demonstrated that another object that is measured in the same manner will be distinct from the first object that was measured. If it is not, the hypothesis being tested will be falsified. In the previous examples, a cubic bus was measured and plotted. To demonstrate that a dissimilar object will be differentiated from a cubic bus, one of the antenna profiles, a fiberglass wire dish, was selected and plotted in the same vector-vector spaces previously selected as examples.

The vector-space signatures of the two components are displayed in Figure 4.9.

The two signatures are quite distinct from one another, and each displays numerous peaks

and valleys that may prove to be useful in identifying the component as part of a larger composite. Note that all the readings are in the negative, indicative of a predominance of vertical polarization over horizontal polarization. Also note that the vertical signal seems to strengthen on both objects as the reflection angle mimics the dawn solar phase angle.

The clear differentiation between objects of differing geometry was an early encouragement in this work that the hypothesis made would not be falsified, and the technique could be useful for identifying significant additional information regarding signature analysis of composite bodies that simulate the structures of satellites.

0.00

-0.04

-0.08

-0.12 Intesnity Ratio Intesnity -0.16

-0.20 10 30 50 70 Reflection Angle [°]

Cubic Bus Wire Dish

Figure 4.9: 푆1/푆0vector signatures for cubic bus and fiberglass wire dish.

Applying the same process to the vector-vector space plot, we observe the results plotted in Figure 4.10. Both objects occupy a specific portion of the two dimensional space, and the clusters representing each object do not overlap with the cluster points of the other.

A clean straight line could be drawn between the two objects so that if a new object known to be one of these two classes of object were to appear, it’s cluster location on the graph

would clearly identify which object it was. Note that the cluster means have been added to the graph to help visualize the non-Euclidean distance between the clusters.

Once the method for distinguishing between two classes is determined, the knowledge gained will be used to apply to a third unknown object, to determine if this approach will correctly associate the unknown with the class that it is most like.

0.04

0.02

0.00

/S 2 S -0.02

-0.04

-0.06 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15

S1/S0 Cubic bus Bus Mean Wire Dish Dish Mean

Figure 4.10: Vector-vector space of cubic bus and wire dish antenna: 푆1/푆0 plotted on horizontal axis, 푆2/푆0 plotted on the vertical axis.

4.5 A Computational Approach to Identifying Targets

The next issue to address is how to increase the speed of discrimination between two objects being analyzed. The figures above provide visual clues that distinguish two objects, but recalling the quantity of objects orbiting in the GEO layer, as shown in Figure

4.1, there must be a way to quantitatively approach this problem to speed the effort to identify the nature of a new target. The obvious conclusion is that the process of

discrimination must be one that lends itself to quantification and discrimination by an algorithm.

Stokes vectors are primarily a visual comparison, but they can be adapted to a computational analysis of their properties. It has been found that there are four major parameters that can affect object recognition in space: viewing position, photometric effects such as the light source, object setting (such as background clutter), and changing shape [23]. Depending on which of these issues is predominant, two approaches could be opted, either an image processing approach or a statistical approach. Of the four influencing factors, the viewing position will have the most effect for an orbiting object.

This is the conclusion since the value of the solar phase angle and its impact on the object being viewed change with the observer position (e.g., whether the observer is located in

Hawaii or the east coast of the US), the object may present a different geometry profile because it is being viewed from a different observation point. The other factors are not considered to be of significant impact since they are less likely to impact the signature analysis, for these reasons: a) the passive light source in space is always the sun, so photometric effects do not come into play; b) the object setting is quite the same (the GEO orbit in space) from any viewpoint and can be easily compensated where small differences exist; and c) at least so far, changing shape has not been a problem although it may become one as satellite capabilities increase.

4.6 Statistical Approach

This work is intended to advance the body of knowledge being used in SSA. The large number of bodies occupying the GEO orbit requires a constant sharpening of the analytical tools used to understand the evolving arena of resident space objects (RSOs).

An important objective of satellite recognition is to classify the objects being observed into meaningful categories, that can be parsed in some useful way. The process of recognition is a complex one at best, and the tools we have to recognize objects that are located 26,000 miles away are relatively elementary. The scientific methods usually employed include, but are not limited to detecting and analyzing colors, textures, shapes, movement, and location [54]. One of the first methods to recognize an orbital object of interest is by its movement and location. A preliminary effort using satellite orbital characteristics such as two-line elements (TLEs) would be used to determine whether an object is orbiting and is of interest or is not. This work does not address the orbital elements of recognition, but once the body has been determined to have those orbital characteristics that make it of interest, the methods described herein become relevant.

Consider for a moment the problem of recognition of an object that is in the observer’s far field. Recognition occurs through two primary methods: shape and color.

In fact, recognition by shape properties is one of the primary methods by which recognition normally occurs, and is the first aspect of a distant object that is used to discriminate it from other objects [23]. A distant tree can be distinguished from a distant bush primarily by its shape. It could be argued that color is the first method by which we begin to discriminate distant objects, in that we first distinguish a living object from the mountain on which it is growing by its color. Be that as it may, both geometry and color are the primary tools by which we identify distant objects.

However, as one considers objects in the GEO orbit, it is noted that these objects remain unresolved in most cases, and without resolution, it seems that the concept of shape or geometry is lost as a method of identification of an object. What is left is color, which

has become the primary method by which RSOs are recognized. Observations by telescope most often use spectral analysis to discriminate objects being studied [6]. Both the eye and the camera pixel can be used to determine the color of the light being observed. But the problem of object discrimination is significantly impeded without knowledge of geometry, as in the first example: the tree and bush are difficult to distinguish from one another based on color alone, and geometry is a key element in the process of recognition.

This is where the vector nature of the optical field becomes relevant. Much work has been done to demonstrate the persistence of polarization through atmospheric turbulence, and this will be covered in Chapter 5. The evidence exists that polarization of reflected light from an object from which the light is scattered persists even through atmospheric turbulence [55]. Since polarization is sensitive to the geometry of edges, planes, surface finishes, and other forms of geometry, it is a tool that lends itself readily to the problem of target identification. It is therefore judicious to measure the polarization information that is being received by the telescopes observing the RSO’s.

This brings us back to the problem of thousands of RSO’s, and the need to be able to recognize a target of interest quickly and efficiently, and therefore the need to introduce a statistical approach to the problem. However, as indicated in Chapter 1, this work is focused on a subset of the recognition problem, and that is whether polarization can be used to discriminate between two objects and whether the peaks and valleys observed in polarization signatures can be related to the geometry of the objects being observed.

There are many statistical approaches that can be taken, the approach that is selected will be clarified here. The research covered by this work was conducted in a lab, and focused on the recognition problem, but in order to apply the work to the real world,

actual satellites need to be observed and data recorded. Under those conditions, there will be differences in observation times, intervals between observations that are required by weather variability, atmospheric turbulence and aerosol differences, and many more variables that will influence the ability to distinguish the characteristics being sought, which are not present in the lab setting. In addition, observations may be made from distant observatory locations, adding perspective to the recognition problem. Satellites may not look the same from one side as from the other, just as the dark side of the moon is not the same as the side observed from earth. In addition to these innate issues, there is also the time element that must be addressed, because the magnitude and intensity of any satellite observation is also a function of time. These factors add disparity to the statistical properties of the satellite observations being made, producing two major challenges for the statistical approach chosen:

1) The problem is a three dimensional one with an expected baseline that varies with

time;

2) The observation data will need to be correlated over time and space, and the objects

being recorded and the positions from which the recordings take place are

nonstationary processes with inhomogeneous and disparate atmospheric influences

[56]. The satellite is moving relative to the earth, the source of illumination is

moving with respect to both satellite and earth, the locations of observation may be

far distant from one another and the environmental conditions may be entirely

different.

Because of these factors, the statistical approaches developed to accommodate for the climate, the earth, and the environment were considered to be far more relevant to the

problem at hand than a more theoretical approach might be for lab work. However, the approach used in the lab may need to be modified when applied to actual satellite data.

Therefore it is not the intent of this dissertation to derive a new statistical technique based on first principles, but to demonstrate that existing statistical methods may be used to facilitate the solution of the satellite recognition problem whether it involves objects in the lab or in orbit. The references used to select the statistical methods herein therefore tend to come from work being applied to the statistical complexities of our environment and from other broad, complex fields, as well as from work applied to satellite recognition.

Statistics can be divided into two broad areas, descriptive statistics and inferential, or Bayesian statistics. Descriptive statistics are those which are related to the organization and summation of data. Inferential statistics consists of methods and procedures used to infer conclusions regarding underlying processes that generate the data. Either of these could be relevant to satellite discrimination and uncertainty underlies both approaches.

Descriptive statistics were developed to understand games of chance, and are useful when there are extremely large collections of observations. Inferential statistics on the other hand take into account that not everything is random, and that cause can be inferred. The frequency of an event may be completely predictable in a card game with one deck, but historical frequency may not be comprehensive in predicting the win rate next season for your favorite sports teams. In the second case, the subjective determination is that probability represents a degree of belief, or quantified judgment about the occurrence of an uncertain event [57]. This work is more concerned with the second approach than the first.

Observations of satellites will however involve great batches of information that will need to be processed and understood. It is definitely a nontrivial task to make sense of the data. The intention of this analytical work is to demonstrate a novel method of analysis that will produce groups of related observations in a dataset. Cluster analysis is the search for groups of related observations in a dataset. The purpose of cluster analysis is to identify groups of data with previously unknown structure to which statistical methods can be applied to find hidden relationships [58]. With large amounts of data that could be considered, the approach will be to produce data clusters of two objects, construct a binary hypothesis structure to apply to the satellite model data, characterize it in the fifteen spaces previously identified in the discussion of vector-vector space, and choose the best characterization of the differences or dissimilarities of the two objects. Once these are characterized statistically, a third unknown object can be added to determine to which class it most likely belongs. Therefore the problem will be structured in such a way as to be given an observed object, characterize it, and compare with two competing probability models to determine to which it most likely belongs.

There may be some reservations that this will be sufficient to help resolve the problem of identification of an unknown satellite type, since there are many different types of satellites of which an unknown could be a member. However the binary structure just sets up the method of problem solution. Any recognition problem could be considered a series of comparisons between two objects [59]. The unknown can be compared to two classes of objects, and the one most like it is determined by finding the one that is most dissimilar and eliminating it. Then the remaining class can be paired with another new class, and the unknown can be compared to find the most dissimilar class of the current

two classes, and eliminate that class. In this way, the binary hypothesis can be used in a multi-comparison structure with several classes of satellites. In an environment that has almost unlimited computer capacity, this is a realistic and often used method to approach this kind of decision making [60, 61].

The proposed method selected to differentiate one object from another is as follows.

Observations of an object to be analyzed are made and recorded. The two-dimensional space in which the measurements are plotted, which has been selected to test the clustering of the data will be designated as a ‘feature space,’ since it remains undefined at this point.

These data are collected and plotted in the feature space and the clustering observed. An example cluster point distributions is shown in Figure 4.11. Axes labels x and y are nominal and experimental charts can be seen in Chapter 7.

Figure 4.11: A cluster of measurements with the mean; mean indicated by red circle.

The intent of applying statistical analysis to data sets in a binary hypothesis test is to attempt to minimize the error by preventing false positives (Type I error) or false negatives (Type II error) [62]. When observing such a distribution, there are several questions that can arise regarding the distribution, such as:

1) How far is it from the mean to each point?

2) Is there a wide variation in these distances?

3) Is there a bias along one axis or the other, that is, is the distribution spherical or

is it elongated along one of the axes? Are the variables of the space correlated?

Each of these questions could be useful in characterizing a cluster of measurements, so answers to these questions would be useful in selecting a statistical tool to use for analysis [63].

However, the problem at hand is not quite that simple. There are many different kinds of satellites, and the process chosen needs to address the reality of that situation. To continue with a description of the approach chosen, it will be assumed that there are two types of satellites, and it will need to be determined how similar or dissimilar they are. The measure will need to be quantitative so that a statistical test can be devised. Two example clusters with their means are shown in Figure 4.12. Axes labels are nominal, and experimental distributions can be viewed in Chapter 7.

Figure 4.12: Two object clusters and their cluster means.

When considered with multiple clusters, more questions arise:

1) Are the clusters centered in the same place in the feature space? It is desired that

they are not.

2) How far apart are the clusters?

3) Is the mean a good representation of the cluster and can the distances be measured

in simple Euclidean space?

Another example is shown in Figure 4.13 where the distributions completely overlap but also have different spatial form. Distinguishing between these two clusters would be much more difficult than distinguishing between the two shown in Figure 4.12.

But note that not only are they on top of each other, they do not have the same shape. The orange distribution is more evenly distributed in both horizontal and vertical directions, but the blue distribution is more elongated along the horizontal axis. Is a simple Euclidean distance appropriate to evaluate such a condition? And clearly, the center of the cluster

location in the feature space will need to be quantified to take it into account when differentiating between objects. Axes labeled x and y are nominal. See Chapter 7.

Figure 4.13: Overlapping clusters with differing shapes.

Yet this still does not capture the complete problem at hand, which requires multiple classes of satellites and unknowns to compare with them. Consider the problem presented in Figure 4.14, where two clusters are shown with three unknown objects.

Figure 4.14: Two clusters of observations in feature space with three unknown objects.

It can be easily observed that there is some overlap in the two clusters in Figure

4.14, but not as much as in Figure 4.13. In an ideal situation, the clusters would be far enough apart in the selected feature space so that they would not overlap at all. When some unidentified objects are added, it is not always clear with which cluster to group the new point. For example, in Figure 4.14, the association of the red star can be estimated to be clustered with the blue points with some certainty, and the gold star can be estimated to be clustered with the orange points. But it is not at all clear to which cluster the cyan star should be associated. This uncertainty could be the source of misclassification of a new object. Some method to reduce the uncertainty must be selected.

Another way to look at the clusters is to look at their probability density functions,

(PDFs) which better represents the concept of distance. Two distributions are shown in

Figure 4.15, one in purple and one in red. These represent the physical distribution of points in the 2D feature space chosen to test as a discriminator. Here it is easily discernable

that there is a small probability of overlapping space for the two clusters. If a third object is located in this feature space, a reasonable test must be applied to see which distribution function it is more likely to be a member.

Figure 4.15: Two clusters of data and their overlap in the test functional space.

How is the question of distance applied to the PDF, and how does it affect the association of an unknown object with one of the clusters? In this way, the concept of distance between clusters and between clusters and points begins to have meaning. The greater the distance between the two clusters, the less likely it is that an error will be made when comparing a new point to determine to which cluster it belongs [64]. If the clusters are on top of each other, it makes determining the class of the new object much harder, and so separation, or distance is a measure of the dissimilarity between the objects being compared. The more dissimilar two clusters are, the higher the distance measure between them should be. This concept is illustrated in Figure 4.16.

Figure 4.16: Relationship between distance and error.

If the distance between the two peaks were measured, it can be seen that as the distance increases, the chance of overlap decreases, and therefore the likelihood of misclassifying an object decreases. In Figure 4.16, 푑2 is greater than 푑1. When the distance between the clusters increases in feature space, but the cluster itself has not changed, the tails of the distributions no longer overlap as much. There is much less likelihood of an identification error occurring when 푑2 is true. So it can be seen that distance between clusters is another factor that needs to be taken into account as a test is determined to differentiate between objects.

In summary, to perform the cluster analysis, a statistical tool needs to be found that takes into account the cluster locations in the feature space, the shape of the clusters, the variance of the distributions, and their separation from each other.

Since the basic problem to be resolved is object recognition, it seemed appropriate to review methods by which recognition has been achieved by others. In his signal work on high level vision with its complexities of interpretation, Ullman mentions ‘statistical

distance’ as an elementary approach to distinguishing between objects [23]. Simple distance measures mentioned include the Hamming distance, which captures points of dissimilarity in an image, and the L2-Norm, a vector norm which sums the squared differences in intensity values at corresponding points of an image. Since the equipment used in the subject lab work does not involve imaging, and instead involves an integrated total scene measurement, these distance techniques are not as useful as others. However,

Ullman goes on the say that these distance concepts lend themselves to more abstract measurements, particularly when represented as vector with specific properties for comparison [23].

One of the easiest concepts of distance to recognize is Euclidean distance, which corresponds to our everyday, intuitive understanding of distance. Euclidean distance can be generalized to three or more dimensions, and is a consequence of the Pythagorean theorem. In one dimension, it can be stated as

̅ 푘 2 ‖푥̅ − 푥′‖ = √∑푘=1(푥푘 − 푥′푘) (4.1)

It is often also convenient to work in terms of squared distances with no loss of integrity, and at the same time simplifying the calculations. It is a monotonic and invertible transformation, and is often used in statistical distances [57].

However, Euclidean distance treats separation of points in space equally regardless of the orientation and distribution of the points relative to one another. It is relevant to consider further the concept of a statistical distance which differs from Euclidean distance, and which can be considered as measures of dissimilarity or unusualness, and thus we consider non-Euclidean distance measures [65]. Non-Euclidean distance measures take

the dissimilarity of the cluster shape into account, viz., a long narrow distribution is different than a tall narrow distribution. From the statistical perspective, distance is defined as a quantitative degree of how far apart two objects, or clusters are [66]. The following discussion is intended to elucidate the concept of dissimilarity and statistical distance.

This dissimilarity is manifested in the joint probability distribution for the data points which is characterized by the scatter of a sample as shown in Figure 4.17. Note points A and B in each plot. The distributions are centered at the origin, but the scatter of points is much flatter in the y direction than it is in the x direction in Figure 4.17(a).

Because of the difference in the magnitude of the dispersion along the two axes, the position of A is much more unusual than the position of B. From this perspective, the statistical distance of A from the origin is much larger than the statistical distance to B.

The scatter dispersion along one axis is also an indicator of an absence of correlation in the data scatter [63]. When the points are scattered as shown in Figure 4.17(b), it is an indication of a strong correlation, but in this case as well, point B is closer to the origin than point A is from the perspective of statistical distance because it is less unusual. In other words, the more dissimilar a point or object is, the larger the non-Euclidian distance will be. This is a characteristic that would be desirable in a distance measure to discriminate two objects. If they are unlike each other, or are dissimilar, the distance would be larger. If they are similar to each other, the distance is smaller. This is a characteristic that facilitates the choice of the statistical tool used in this study.

Figure 4.17: Distance in context of data scatter: (a) uncorrelated data; (b) correlated data.

One of the widely used non-Euclidean distance measures in statistics is the MD2- statistic of Mahalanobis Distance (MD) [57], developed about 1925. The MD was developed after the Pearson C2 statistic because it provided a measure of the magnitude of divergence or dissimilarity between two groups. Pearson’s statistic was a test, and not a measure, so Mahalanobis’ statistic provides more than the Pearson test in that it is a measure of dissimilarity [67].

Mahalanobis distance, MD for a sample of data may be calculated as follows [67]:

−1 훴̿ +훴̿ 푀퐷 = 퐷2(푥, 휇) = 푎 ∗ (휇̅ − 휇̅ )푇 ( 1 2) (휇̅ − 휇̅ ) (4.2) 1 2 2 1 2 where 푎 is a constant, 휇1̅ = (푥휇1, 푦휇1) is the mean vector of the first class of vectors 푉̅1 =

{푥̅푖1, 푦̅푖1} being compared, and 휇̅2 = (푥휇2, 푦휇2) is the mean vector of the second class 푉̅2 =

{푥̅푖2, 푦̅푖2}, each representing the “center of mass” of the respective class68. 훴̿1 is a 2×2 covariance matrix of the measurement vectors 푉̅1 and its elements are defined as follows:

훴11 = 퐸{(푥푖1 − 푥휇1)(푥푖1 − 푥휇1)}; (4.3)

훴12 = 훴21 = 퐸{(푥푖1 − 푥휇1)(푦푖1 − 푦휇1)}; (4.4)

훴22 = 퐸{(푦푖1 − 푦휇1)(푦푖1 − 푦휇1)}. (4.5).

In multivariate cluster analysis, such distance measures as MD are applied to outlier detection, pattern recognition, sample selection techniques and methods that compare the successful representation of the data sample to the whole.

The MD is not the only non-Euclidean distance measure, and others have interesting properties in addition to what exists in MD, including usefulness in signal selection, which is a way to characterized targets for identification. Another distance measure of interest is the Bhattacharyya Distance (BD) which has a property quite useful for signal selection [64]. If for two sets of clusters, α and β, the BD(α) > BD(β), then the probability of an error (푃푒) through misidentification with 푃푒(훼) < 푃푒( 훽) [64] where

푃푒(훼) is the probability of an error through misidentification. This is important because it means that when the BD between two clusters increases, the likelihood of a Type I error goes down. This is the same concept illustrated in Figure 4.16, where the tails of the distributions overlap less as the distance between the clusters increases. If the overlap decreases, the likelihood of making an error of classification decreases, and this is an important discriminant that indicates BD may be useful for the problem at hand.

Several forms of the BD equation exist, with one of the first versions defining BD as:

퐵퐷 = −푙푛(∫ √푝1(푥)푝2(푥)) (4.6) where 푝1(푥) is the probability density function (PDF) of the first cluster and 푝2(푥) is the

PDF of the second cluster. The quantity within the parenthetical is labeled the

Bhattacharyya coefficient while the entire equation is the BD. Later in the dissertation, when BD is evaluated for a multivariate probability distribution, another term is added

which creates a form similar to later variations of BD. Other findings presented in the same paper [69] include:

1) for a Gaussian distribution, BD provides better results than MD, (where satellite

intensity has been found to demonstrate a Gaussian distribution);

2) at low signal-to-noise ratios (SNR), MD and BD are relatively equivalent, but at

high SNR, BD provides better results than MD;

3) if the two clusters being compared are indicated by two vectors from the origin to

their means, the Bhattacharyya coefficient can be considered to be equal to the cosine

of the angle between the two vectors. The importance of this is the relationship of the

distance between the clusters in feature space that is chosen. As the cosine between

the two vectors increases, the distributions begin to separate, decreasing the probability

of error in distinguishing between the two. Therefore, as the BD increases, the

likelihood of error decreases, as previously stated.

Included in early forms of the BD was an equation that combined the MD and BD into one, as discussed in connection with the use of BD to evaluate color-color statistics

[70]. All of these combined forms are similar, but the form used our work is as follows:

−1 1 푇 훴̿1+훴̿2 1 |훴̿1+훴̿2| 퐵퐷 = (휇̅1 − 휇̅2) ( ) (휇1̅ − 휇̅2) + 푙푛 ( ) (4.7) 8 2 2 2|훴̿1∗훴̿2|

Note that the first term is related to the MD. In the last term on the RHS of Eq. (4.7), the notation |퐴̿| represents the Frobenius norm of the matrix 퐴̿ [71]. The Frobenius norm is chosen over the determinant because the former gives a better estimate of the maximum excursion of a vector during a linear transformation using the matrix, even if the determinant of the matrix is, say, zero. The BD is also included in a paper summarizing

distance metrics, and is classified as being a member of the “fidelity” family or squared chord family, where squared chord distance is related to the Frobenius norm [66].

Equation (4.7) is the distance metric that has been chosen to evaluate cluster distances from each other, particularly since it includes the cosine of the angle between the vectors from the origin to the clusters being compared, and has a term that is closely related to the MD. The MD itself will be used to estimate the distance from a new unidentified target to the clusters it is being compared with for identification.

4.7 Conclusions

This chapter presents the concepts that are applied in this dissertation to the polarization data, and the method of analysis selected with the rationale. Chapter 5 will briefly discuss related work that contained concepts valid for use in this work.

CHAPTER 5

RELATED WORK ON POLARIMETRY OF SPACE OBJECTS

5.1 Polarization Signatures of Space Objects

It has been definitively demonstrated that satellites display polarization signatures that are more distinctive than the intensity signature alone. The results of analysis of solar light reflected from orbiting geosynchronous satellites was provided by Speicher [72], where it was demonstrated that satellites could be clearly distinguished from one another by the signatures made up of two of the four Stokes parameters. These signatures were obtained by recording polarization states of reflected sunlight using a subset of 푆0, total polarization, and one of the linear Stokes vectors components, 푆1.

This experiment used a two channel polarimeter that detected horizontal and vertical polarizations by using an Edmund Optics polarizing beamsplitter which sent each beam to an Apogee F47 back illuminated CCD. The targets for analysis were all western manufactured commercial satellites such as DirecTV-4S, (DTV) AMC-15, and SES-1.

Eight targets were selected, some with different bus structures, and some identical except for launch date, to determine the effectiveness of polarization to distinguish between geometric configurations as well as to test the ability of polarization to differentiate between more subtle differences, such as those caused by space aging [72]. The reason polarization would be more effective than wavelength (color) is that polarization is

responsive to geometry – planes, vertices, cylindrical surfaces, etc. Since the mission of the satellite changes the equipment used and basic shape of the satellite, there should be some correlation between detected geometry and the purpose and source of the object, and therefore an indicative change in the polarization. This was successfully demonstrated using partial Stokes polarimetry [72].

The graphs and images that follow are all taken from the same source [72]. Tests were conducted to determine the effectiveness of polarization in distinguishing between geometric configurations of different satellites. The graphs below demonstrate the success of these tests. For example, DTV-4S and SES-1 were each sampled on three different nights. These satellites have different bus structures, with DTV-4S, manufactured by

Boeing, having the BSS-601HP bus and the SES-1, manufactured by Orbital Sciences

Corporation, having a Star-2.4 bus, as shown in Figure 5.1. Both of these are cuboid bus geometries.

Figure 5.1: Image of DTV-4S and SES-1 [72].

The horizontal and vertical polarization intensities were recorded, allowing calculation of the 푆0 and 푆1 Stokes parameters. These parameters were then plotted against the solar phase angle (SPA), which is related to the time and orientation of the observer,

satellite and sun, equivalent to the angle of the reflected light. These plots were visually compared to determine if signatures exist that differentiate one satellite from another. The

푆0 parameter comparison was presented first, with readings from three nights overlaid one upon another, as shown in Figure 5.2. There is clearly a difference in the 푆0 signatures, with intensity varying much more on DTV-4S than on SES-1, so the , 푆0 parameter could be used to differentiate between the two in this example.

Figure 5.2: 푆0 vs. time for DTV-4 and SES-1, with three nights of data overlaid [72].

Next, the Stokes 푆1 parameter signatures are shown in Figure 5.3. Note that the vertical scale for the polarization signal goes from more negative to less negative, indicating that the highest absolute magnitude 푆1 polarization readings were taken at dawn and dusk, consistent with other observations. These graphs again demonstrate a clear difference between the two satellites, with the difference in polarizations much more irregular for DTV-4S than for the SES-1 satellite. DTV-4S had lower negative readings

(preference for vertical polarization), but more consistent readings. In this case, both 푆0 and 푆1 could be used individually or in combination to distinguish between the two satellites.

Figure 5.3: 푆1 vs. time for DTV-4s and SES-1, with three nights of data overlaid [72].

Another comparison was made between DTV-10 and DTV-12, which are both manufactured by Boeing, with the BSS-702 bus structure, but were launched at different times. The DTV satellite configuration is shown in Figure 5.4.

Figure 5.4: Image of Directv-10 and 12 [72].

These satellites were also sampled on three different nights. Comparison of the 푆0 parameters is shown in Figure 5.5. The peak intensity corresponds very closely with the lowest solar phase angle. There are slight differences between the two configurations, but it would be difficult to distinguish between the two satellites based on intensity as measured by 푆0 vs. time.

Figure 5.5: Comparison of 푆0 for DTV-10 and DTV-12, with three nights of data overlaid [72].

In comparison, Figure 5.6 displays the 푆1 polarization difference signatures of the same satellites collected on the same three nights. As can be seen by examining these identical satellite signatures, there is a clear difference detected by the Stokes 푆1 parameter between the two, which have identical configurations but are different in age by eighteen months. The difference is attributed to space weather aging. If aging smooths corners and edges, the rounded pattern seen in the DTV-10 would be a logical representation of an aged

DTV-12.

Figure 5.6: 푆1 Stokes vector comparison of DTV-10 and DTV-12, 3 nights of data overlaid [72].

Another way to consider these results is to analyze polarization information by comparing Stokes parameters of the same kind to each other, as shown in Figure 5.7. Note

that Stokes parameter 푆0, is a measure of total intensity, the same measurement made for a color signature. These signatures show the distinct pattern of one peak at or near minimum solar phase angle (near midnight, when the sun is opposite the satellite.) Note that each satellite shows one distinct peak in 푆0, correlated very closely with the minimum solar phase angle each night, with peaks occurring directly at the minimum phase angle or just before the minimum. This makes using intensity alone as measured by 푆0 to distinguish between satellites a less than optimal tool, since there are only slight differences between the signatures. Figure 5.7 shows that the 푆0 signature of all six satellites are very similar to one another, and would make detecting the difference a discouraging task using the sum of intensities, the equivalent of a photometric image.

Figure 5.7: Comparison of Stokes parameter 푆0 of six different satellites SPA plotted in blue [72]

In comparison, the Stokes parameter 푆1 differencing measurement is shown in

Figure 5.8. Each of the polarization differencing plots is distinguished from all the others and would provide a better method of identifying each of the satellites and differentiating it from the others.

Figure 5.8: Comparison of Stokes parameters 푆1 of the same satellites [72].

A third method to use the Stokes parameters is to combine them, using 푆0 and 푆1 to generate a signature that distinguishes one from all others. This approach is quite common [46] and is shown in Figure 5.9. This demonstrates that 푆1 clearly distinguishes very small changes between otherwise identical satellites – in this case the age of otherwise identical satellites. Note that the polarization scale is in the negative range, implying a preference for vertical polarization, but also that the strongest polarization signals are at dusk and dawn, while the strongest intensity signal is located at the lowest solar angle near midnight.

Several key conclusions were drawn in the paper by Speicher [46]. One finding was that Stokes polarization differencing vector , 푆1 is negatively correlated to 푆0

indicating vertical (or P-polarization) is correlated more strongly with 푆0 than horizontal polarization.

Figure 5.9: Stokes parameters 푆0 and 푆1 for DTV-10 and DTV-12 – strongest polarization signatures at dusk and dawn [72].

Another finding was that the strongest polarization signal is obtained when the weakest intensity signal is obtained, and vice versa, making polarimetry a strong complementary tool to photometry signatures. It was found that 푆1 had excellent repeatability from night to night in the general shape of the curve; and that 푆1 could differentiate both geometrical shape differences and variances in material [72] and color, as demonstrated by detecting the difference in identical satellites of diverse ages.

5.2 Polarization Detection Through Haze and Clouds

Perhaps a little more applicable to target identification are those applications of polarization differencing that view objects through a hazy atmosphere. It has been found recently that degradation of visibility caused by clouds and haze can be significantly reduced by filtering with polarization. [73], [74]. So it can be seen that improvements in imaging can be obtained by using polarization [75]. The images in Figure

5.10 were taken of a naval yard (public attraction) from an aircraft through haze that

precluded any visual perception of the image, but application of polarization filters improved the detection of important information that could not have be acquired from a conventional image.

Figure 5.10: (a) Polarization sum image through haze (b) polarization difference image through haze [75].

The next work that is worthy of noting on this subject is one of the early studies of using a polarimeter to detect satellites and determine if a polarization signature existed

[22]. Along with successful measurement of repeatable characterization of the satellites,

Sanchez et al., [22] noted the advantage of having varying weather conditions during their measurements. The worst conditions occurred first, when measurements were obtained of calibration stars through thick cirrus clouds, through thin cirrus clouds, and in clear conditions, but the weather was never characterized in any way while collecting the data, other than the visual observation of seeing conditions as just described.

Objects in a LEO orbit can be detected and tracked with radar, and optical systems can acquire images, but resolution is not high. But for those objects located in GEO orbits and beyond, other tools need to be employed, because resolution of spectral data at those distances is quite a challenge, and tools such as spectral analysis have been employed [76].

As the size of satellites gets reduced further, the physics of light will prevent obtaining resolved images of these smaller objects, with a ten meter telescope aperture required to resolve a 2.2 meter object in GEO orbit [72]. Even with the most sophisticated tools available, a large amount of time can elapse between detection and identification of a target, sometimes as much as 7 to 30 days, estimated in 2011 [72]. These trends begin to push us from the comfortable approach of spectral analysis into territory that is less well explored: polarimetry for identifying satellites. If polarimetry can be shown to provide more comprehensive clues to the identity of an unknown object in a timely manner, the identification process can be improved. In addition to polarimetry’s usefulness, polarization data acquisition has been shown to be low cost, fast and practical, making it a tool that is readily applied to difficult problems

CHAPTER 6

EXPERIMENTAL SETUP FOR STOKES VECTOR DETERMINATION

Polarization measurements can be used to differentiate unresolved objects of different geometry and material. In this work, it is proposed and experimentally demonstrated that polarization can detect geometric shape and material differences of unresolved objects. Chapter 6 provides the experimental method, the model construction, sources of error and calibration methods.

6.1 Materials and Methods

The objects of interest in space are often satellites. They generally have three major divisions of their construction geometry, comprising the bus, the solar panels and the antennae [77]. Preliminary work has been performed using the tools described to demonstrate that simple bus geometries of a cylinder and a cube could be differentiated using polarimetry and vector spaces [78]. A large number of simple geometries and materials we compared before selecting the satellite model geometries to be tested. These are presented in Chapter 7. But satellites are primarily a composite of the three geometries mentioned above, as well as a few others that may contribute to the state of polarization, and it is the composite geometry that is the focus of this work.

6.1.1 Selection of model geometries and materials

Two combinations of these three complex geometries are selected and models are constructed to serve as a representation of satellites that are to be analyzed by polarimetry.

In the selection of the makeup for the composite geometries, each one was chosen to have a contrasting type for the three geometric elements. Descriptions of each geometry are described below in this Section.

6.1.1.1 Bus geometries

Bus geometries commonly are in the form of either a cuboid [a three-dimensional box] or a cylinder. After review of many satellite geometries, two cuboid geometries, viz., a cubic bus (similar to the SwissCube satellite) and a rectangular bus (as in Canada’s

RadarSat2) are selected for this study. An artist’s rendering of the satellites is shown in

Figure 6.1 for reference. An examination of many satellite buses will reveal that the instruments and connections mounted on them are often covered with a layer of Kapton sheeting. Kapton is a dielectric polyimide film developed by DuPont in the late 1960s [79] that remains stable across a wide range of temperatures, from −269 to +400 °C. Kapton is often used as a thermal blanket on spacecraft, satellites, and various space instruments [80].

Figure 6.1: Examples of cuboid satellite buses; (a) SwissCube [81]; (b) RadarSat2 [82].

For our experiments, the first model is constructed with a painted cube similar to

SwissCube for the bus structure, as shown in Figure 6.2(a). Throughout this work, this model comprising the cube, solar panel and antenna, will be referred to as Composite 1.

The second model constructed, referred to as Composite 2, uses a rectangular bus structure covered in Kapton sheeting, similar to RadarSat2, as shown in Figure 6.2(b), along with the solar panel and antenna.

Figure 6.2: Composite geometries: (a) Composite 1, comprising painted cuboid bus, monocrystalline solar panel and painted dish antenna; (b) Composite 2, comprising Kapton covered bus, polycrystalline solar panel and fiberglass wire dish antenna.

6.1.1.2 Solar Panels

Only a modest amount of information is available on the construction of the solar panels used on satellites, but general information on solar panels and their construction methods is widely available. For our models two types of solar panels are used. The most common forms of commercial panels used today are monocrystalline silicon [83], polycrystalline silicon, and thin film amorphous panels, ordered by their efficiency.

Monocrystalline silicon is not light reflective, as shown in the picture for Composite 1

Figure 6.2(a), while polycrystalline silicon is light reflective, as evident from the picture for Composite 2 (Figure 6.2(b)). The monocrystalline panel used for Composite 1 is a 5 volt, 30 mA Micro Mini Power Solar cell produced by AMX3d in Illinois, with a thin rectangular, non-reflective surface. The wire stringers which conduct the electrons are visible in the picture, and are encapsulated in epoxy resin to protect from oxidation and weather. The polycrystalline cell for Composite 2 is a 5 volt, 100 mA mini Solar Panel

GP80*80-10A100 solar cell produced by Sunnytech in New Zealand. It is also epoxy coated and the stringers are visible in the photo.

The difference in size and material can help differentiate between and classify two geometric elements of the same type, as is to be expected if these elements are produced in different fabrication facilities, possibly in different countries. The use of real solar cells is intended to replicate actual satellite construction and polarimetric response as closely as possible.

6.1.1.3 Dish Antennae

Several types of antennae are used on satellites, but the most common antenna geometry is a paraboloid of revolution [84], usually referred to as a dish. Most dish antennae are made from a fiberglass composite including a reflective metallic material, or a metal like aluminum, with small structures at the center of the dish. When the dishes are metallic, the aluminum plate is perforated with holes of the correct diameter to receive the wavelength and power of the signal required. A paint powder coating is then applied, and the dish heated to melt the powder. Another method of dish construction 85 is the assembly

of mesh petals mounted on ribs that attach to the hub. The mesh is lighter weight and allows debris to pass through.

Several models of dish antennae are tested in this work. Those having a coating of aluminum foil are not selected for the composites since the reflection from them overwhelm all other intensity measurements. For final comparisons, the selected dishes have a reflective paint for composite 1 and a fiberglass wire mesh for Composite 2. The sizes of the dishes has been reduced to keep from overwhelming other reflections, and to keep relative dimensions proportional to actual satellite construction.

All the model geometries have been compacted to fit within the working beam diameter in the experimental setup, which is approximately two inches.

6.2 Experimental Setup

The experimental setup shown in Figure 6.3 is made of an unpolarized light source and a polarization analyzer by which the intensities used to determine the Stokes parameters are measured from the reflected light. A halogen light source with a parabolic mirror to collimate the beam is used as the source of unpolarized light, intended to reproduce nearly parallel broadband, unpolarized light from the sun. The primary targets for this portion of the experiment are the two composite objects previously described. The light shielded polarization analyzer contains the polarization analyzer made up of a quarter wave plate, linear polarizer and power meter to measure intensity, as shown in Figure 6.3.

A Newport achromatic quarter wave plate (Model 46-588), followed by a Thorlabs broadband wire grid linear polarizer for visible light (model WP25M-VIS) are used to analyze the Stokes parameters. A Newport power meter (model 2936-C) is used to detect

the intensity, as shown in the inset in Figure 6.3. The optics are mounted on a rail to maintain alignment, and oriented at 3° from the axial direction to reduce unwanted reflections. All optics are enclosed in a light shielded enclosure (black cardboard box lined with black felt) to further reduce reflections or stray light contamination. A small aperture of about one square inch, facing the composite object is made in the enclosure at the same height as the light source. The mounting fixture for the target object and any nearby objects that might reflect light are covered in black felt and draped with black cloth to prevent stray, unwanted reflections.

Figure 6.3: Experimental setup with unpolarized light source and light shielded detector.

Each composite object is composed of the three elements mentioned earlier, the bus, the solar panel, and the antenna. To understand the influence of each part of the composite object, a full set of readings are taken on each individual part of the composite as well as the entire object, without moving any instrumentation. A black matte drape

material is used to cover portions of the composite model that are not being analyzed, and all required readings are taken for each geometric element to acquire all four Stokes parameters. In this way, all intensity readings are taken on the composite and each individual component without any movement of equipment. The only change made between polarization state measurements are the positions of the quarter wave plate (QWP) and the linear polarizer (LP). When measuring linear parameters, the QWP is removed.

These analyzer elements are mounted in Thorlabs RSP1X225 indexing rotation mounts.

Once all measurements for each component and the composite are made, the angular position of the detector/analyzer is changed, and the process repeated.

The composite objects are placed in the halogen beam and the detector/analyzer observation angle is placed at 30 different reflection angles between 15° and 83° from the source beam, as shown in Figure 6.3. This change in angular orientation replicates the change in angle of illumination that occurs with changing solar phase angle relative to an orbiting object. For a satellite at midnight, the solar phase angle is approximately 20°, since the sun cannot be directly behind the earth relative to the satellite, or the satellite would be in earth shadow and could not be seen. As the earth rotates, the solar phase angle increases to the point where it approaches 80° - 90°, near dawn.

Model orientation to the light source and detector is very important in this experiment. The models must replicate the light conditions of actual satellites, with have two rotational frames of reference which must be reproduced. The satellite configuration has the bus and antennae, which are the satellite’s payload, and they are all nadir pointing.

As a result, the observer always sees the same portion of the satellite bus and antennae,

which is helpful for signature analysis. Thus, the bus and antennae change their orientation with respect to the sun as the earth turns and the satellite rotates to remain nadir facing.

The solar panels on the other hand are always oriented fully toward the sun, so that the solar panel never changes its orientation with respect to the sun, but changes constantly with respect to the observer. This is illustrated in Figure 6.4, where it can be observed that the bus and antennae always face the observatory, and the solar panels remain in an unchanging orientation with respect to the sun.

Figure 6.4: Rotational frames of reference of satellite body and solar panel.

The experimental setup is such that the same dynamics are observed as the reflection angle changes. The models or their components that represent a bus or antenna are mounted on a rail with the detector, so that it moves as a unit, changing orientation with respect to the light source. The solar panels, on the other hand, are mounted separately and

do not move during any of the testing and remain facing the light source. As the detector moves, the angle of the solar panel relative to the observer changes.

6.2.1 Active vs. passive illumination

This study is based on observation of reflected light from an unpolarized light source. This arrangement most closely resembles actual conditions for observing space objects, whether debris or satellites. The question may arise as to whether or not active illumination of space objects with a coherent beam might yield more positive results, but the legal ramifications of illuminating a spacecraft and anything flying that happens to pass between the light source and its target can be extreme. To observe a satellite using an active beam requires written permission from the owner of the satellite. In addition, illuminating air space with a visible laser beam is illegal unless special written permission has been received from the FAA and FDA [86].

6.3 Sources of Error and Calibration

There are two types of experimental error, random and systematic. Random error will shift a measurement away from its true value by a random amount and in a random direction. Random error cannot be calibrated out. Systematic error on the other hand, will shift a measurement from its true value by the same amount and in the same direction repeatedly. Systematic error can be calibrated out of the experiment, which is the purpose of calibration [87]. Random error in this experiment can come from instrumental noise, differences in placement position of targets or sensors, and inaccuracies in channel selection settings. Since random error cannot be calibrated out of the system, care is taken in equipment selection and placement to minimize it. It is estimated that random

experimental error can be as much as 2.5%, and total error at 3.5%. Error is shown on one point on Figure 7.4.

Random error minimization is accomplished as follows. Instrumentation and the target for each experiment are mounted on a rail to maintain axial alignment. Positions for the rail are measured and marked on the optics table to make angular placement repeatable.

The distance between the target and the sensor was about 18”, as determined by the placement on the rail, so a 1° error in azimuthal target placement would be equivalent to more than 0.3 inches. The placement position is much closer than that. The use of the rail prevents radial errors from occurring, since the equipment is always mounted at the same location on the rail. Variations in intensity with changes in position are measured and were used to estimate total random error.

Measurement errors are minimized by removing the potential for random error in the setting of the LP and QWP by use of Thorlabs RSP1X225 indexing rotation mounts.

These rotation mounts lock into an indexing position, such as 90°, and two indexing clicks rotate the analyzer by 45° with repeatability estimated to be within 5 arcminutes at each setting. An azimuthal error of this amount had created a measured error in intensity of about 0.25% in intensity.

Regardless of the small amount of the error associated with replacement of the equipment, random error is further reduced by taking all required measurements without any movement of the equipment except that required to change the polarization channel being measured. As mentioned earlier, all component measurements and composite measurements that are compared to one another are taken at one time for each composite

object. Data for comparisons between one composite and another are taken on different days with all minimization of error previously discussed.

Sources of systematic error include instrumentation error and contamination by stray reflections. These can be calibrated out, and minimized with the setup. Polarization measurements are sensitive to stray reflections, since all light contains polarization which can contaminate the values measured. For instance, the mount for the polarizer optic and the optic itself can reflect light back from the instrumentation to the light source, then reflect again to the sensor, contaminating the reading as shown in Figure 6.5. This is quite obvious when using a laser with optical elements, since the reflections are focused and easily discerned by eye. With incoherent light, it is less obvious to the eye, but nonetheless reflections do exist and can be measured. The reader should note that the light source may be the halogen light itself or the target object from which reflected light is received. To prevent back reflections all optics are mounted at a 3° angle to the optic axis, as shown in

Figure 6.5. As long as the polarization optic is rotated by less than 5°, it remains within the linear portion of the optic and does not cause a measurement error [88]. The inside of the light shielding enclosure is lined with black felt material to absorb any stray light that does enter. When polarization generator elements are in place outside the light shielded detector, the mounts are also covered by light absorbing materials to prevent back reflections.

Figure 6.5: Stray light contamination (a) entering measurement optic; (b) blocked from re-entry by rotated element.

Systematic error from instrumentation is not so easily overcome, and system calibration must be performed. Systematic equipment errors can include noise in electronic components such as the power meter, which is a part of the polarization analyzer used. The

Newport power meters are factory calibrated, but may still be a source of error from a pedestal of counts that exist when no light source is available (dark counts). To remove this error, a set of readings is periodically taken in a dark room with the meter inside the light shielded detector, and the dark pedestal is subtracted from all raw intensity readings taken; the final step of the radiometric portion of the calibration is to normalize the channel responses to the same value, since the incoming light is unpolarized, producing a radiometrically calibrated measurement. The pedestal is measured in picoWatts, so the factory calibration is very good. With instrumentation such as a camera, integration time can be calibrated as well, but the power meter is self-integrating. Therefore, as all intensity readings were taken, it is observed that there is a momentary fluctuation in the reading for a second or two after the optic is changed and a new reading needs to be taken. All intensity readings were taken after the fluctuation had settled down.

The remaining errors in the equipment can cause a bias in the readings, and Mueller calculus can be used to calibrate the system [89]. As mentioned earlier, the Stokes vector

can accurately characterize the polarization of the incoming light, whether it is partially or fully polarized. The instruments and optics and their use as an analyzer introduce error into the measurement process. Theoretically, the Mueller matrices of the cascaded optics can be multiplied to produce the Mueller matrix operator that will transform the incoming light. The Mueller matrix for different polarizers is listed below.

Ideal Polarizer

. (6.1)

Diattenuator, axis 0°, intensity transmittance q, r:

. (6.2).

Linear Retarder, fast axis 0°, retardance δ:

. (6.3)

Cascaded diattenuator and retarder, fast axis 0°, retardance δ, intensity transmittance

q, r:

. (6.4).

Eqn. 6.4 is a theoretical analyzer matrix. The response of the system could be calculated using these theoretical models, but the theory does not capture all of the errors resident in the physical system. Therefore, the instrumentation system matrix is fully determined experimentally.

Any system with polarization can be represented by a system response matrix that relates the incoming Stokes vector 푆̅ to an N-element output channel measurement. There are many sources of this information, but the process described here is from the Persons paper on polarimetric calibration [88]. The goal of calibration is to accurately determine the system response matrix, 푊̿ ,

퐿̅ = 푊̿ ∙ 푆 ̅ , (6.5) where 퐿̅ is the measured vector, and may consist of many measurements.

Although the final system matrix is a 4 x 4 matrix that includes circular polarization, the process described here is the linear portion, for the sake of simplicity. Since 퐿̅ and 푆̅ are vectors, Eqn. 6.3.5 can be rewritten as shown in 6.3.6:

̅ ̿ ̅ [퐿̅1 퐿̅2 … 퐿푀] = 푊 ∙ [푆1̅ 푆2̅ … 푆푀], (6.6)

̅ ̅ ̿ where [퐿̅1 퐿̅2 … 퐿푀] and [푆1̅ 푆2̅ … 푆푀] are now matrices, henceforth referred to as 퐿 and

푆̿ , respectively. Equation 6.5 now expands to

퐿̿ = 푊̿ ∙ 푆.̿ (6.7)

An accurate measurement must be attained for the incoming Stokes vectors, which is accomplished by using a polarization generator, located outside the light shielded detector/analyzer, and placed at the exit of the unpolarized halogen light source. The system response matrix can be calculated by inverting equation 6.7:

푊̿ = 퐿̿ ∙ 푆̿+ (6.8)

where 푆+ is the pseudo-inverse of the matrix 푆̿. The pseudo-inverse is required because only a square matrix can have an inverse [90]. 퐿̿ can have any number of polarization state measurements, and in this case will be over-determined with twenty-four measurements per polarization channel. The data reduction matrix 푅̿ is calculated as the pseudo-inverse of the system response matrix, 푊̿ . Once 푅̿ is obtained, it can be multiplied by any radiometrically calibrated channel measurement vector 퐿̅, to determine the calibrated measured input Stokes vector 푆̅′:

푆̅′ = 푅̿ ∙ 퐿̅, (6.9) where 푅̿ = 푊̿ + is the pseudo-inverse of 푊̿ .

The method of determining a calibrated input vector requires the elimination of any reflected or emitted light, but for visible light this reduces to eliminating the reflected light from the source by mounting the polarization optics at an angle to the optic axis, as

previously discussed, and removing any other light contamination. The linear portion of the calibration method is detailed below for clarification, and similar calculations are performed for the circular calibration. Measurements are made at the four measurement settings of 0°, 45°, 90°, and 135°. A Stokes input vector, 푆̅휃 is created for each of the four channels, of the form

퐿퐿 휃 푆̅ (퐿, 휃) = [퐷 ∙ 퐿퐿푐표푠[2휃]] (6.10) 퐷 ∙ 퐿퐿푠푖푛[2휃] where 퐿퐿 denotes the linear polarization measurements, 퐷 is the measured diattenuation of the polarizer, and 휃 is the orientation of the measurement axis, one of the four listed previously. Since the four terms (0°, 45°, 90°, 135°) are evaluated at 2휃 in Eq. (6.10) the second and third terms simplify to either zero or one (푐표푠[2휃] or 푠푖푛[2휃]). Four input vectors are created in this manner, and combined into the 푆̿ matrix to calculate 푊̿ as follows:

(6.11) where 퐿퐿 is the average of all the intensity measurements made for the input vector 푆̅, and the channel values are averaged. The pseudo-inverse 푆+ of matrix S is calculated at this point. This system of equations can be solved using the linear least squares method.

If the system is ideal, Eq. (6.11) can then be rewritten as

퐿 1 퐿 [퐿̅0 + 퐿̅45 + 퐿̅90 + 퐿̅135] = 푊 ∙ [ 0 ] (6.12) 4 푖 푖 푖 푖 0

100

Because the uncalibrated channel responses are equalized, equation 6.3.11 can be rewritten

(6.13)

When solved under ideal conditions, a system response matrix with the first column equal to ones is produced. Using all the measurements taken, the first column of W can be solved, and when solved under ideal conditions, should produce a matrix with the first column equal to a vector of 1’s. Any deviation is due to non-linearities, but it is exactly true at the calibration points. It is therefore allowable to set the first column of the calculated W matrix to be equal to a column of ones, and does not transfer any error to the other columns

[88].

, (6.14) whereas the ideal should be

. (6.15).

Once all of the system deviations have been identified, they can all be transferred into a single Mueller matrix. This can be stated using 퐿 the output channel measurement as:

퐿 = 푊. 푆 ≡ 푊푖푑푒푎푙 . 푀훥 . 푆, (6.16)

101

where 푀훥 is the Mueller deviation matrix (MDM). Solving for 푀훥, it can be seen that

+ 푀훥 = 푊푖푑푒푎푙 . 푊. (6.17).

Multiplying the MDM by any measured Stokes vector provides a calibrated measurement of the incoming vector. Solving for the MDM of the experimental system produces the linear 푀훥 matrix:

1 −0.1137 −0.1093 푀훥 = [0 1.175 0.1870 ]. (6.18) . 0 0.1418 1.046

An ideal, perfectly aligned polarimeter would have an 푀훥 matrix that is equal to the identity matrix.

A similar set of calculations is performed for the calibration of the circular polarization.

Chapter 6 provided the experimental setup, model composition, and the sources of error and calibration method. Chapter 7 details the experimental results.

102

CHAPTER 7

EXPERIMENTAL RESULTS

Chapter 7 provides the experimental results, beginning with the components of the two composite models chosen for comparison, and proceeding to the composite models.

In addition, a third composite model is constructed that is identical to Composite 1 except that the dish is covered by a mylar coating instead of a painted surface. The “unknown” composite model signature elements are compared to the two known composites, and a finding that the unknown composite is correctly found to be most like Composite 1, demonstrating the method of target identification by comparison to known targets.

7.1 Polarization Signatures of Simple Geometries

A review of the hypotheses to be addressed by this experimental work would be helpful in the review of the results. These are:

1) Polarimetry will differentiate simple geometries and materials from one another

by producing a signature that can be quantified statistically.

2) Polarimetry will differentiate complex objects from one another using statistical

tools applied to signature analysis. Polarimetric signatures of complex objects

are strongly influenced by the polarimetric signatures of the individual

components, (signatures based on full Stokes parameters.

103

3) Polarimetric signatures can be used to correctly associate an unknown object

with one of two clusters of data representing two complex objects.

A thorough study of Stokes parameters and one-dimensional and two-dimensional vector spaces of the composites and their components has been performed. A great number of geometries, materials, and composites are tested as part of this work effort. Since the data were collected over a period of months, the analyses are normalized to account for small variations in illumination or other slight variations in experimental setup. Any analyses that are not normalized are called out in the text. After considerable testing of various combination of geometry and material, it is concluded that the best approach to a solution would be to begin with the simplest of geometries or materials, and to change one thing to compare the element for which a signature was sought. Geometries will be reviewed first, followed by materials, then composite objects.

It should be noted that the vector-vector space constructed here is based on the

Poincare Sphere, where the basis vectors consist of 푆1/푆0, 푆2/푆0, and 푆3/푆0. The space is

Euclidean and orthogonal, but the 푆0 space is not orthogonal to the basis vectors, since it represents the distance to any point on the unit sphere. It does however provide a fourth variable that can be used in vector-vector space to help differentiate between the objects being compared.

7.1.1 Cylinder

Recall that the experimental geometry is such that the rotation angle changes with respect to an axis perpendicular to the optics table. No vertical variation was measured.

104

In order to set a baseline for comparison, for which there would be almost no signature, a vertically oriented cylinder was chosen. As the cylinder rotates about its vertical axis, the only change measured by the analyzer is the amount of reflected light as the reflection angle changes. No visible change in geometry is presented, and the expectation may be that there is no change in polarization. This simple object, with a diffuse painted surface is used as a comparator for other objects and materials, beginning with the Stokes parameters. The paint applied to the wood surface of the cylinder was Rust-Oleum Stops

Rust Vintage Metallic Warm Gold Protective Enamel Spray Paint, applied in two coats to evenly cover the surface of the wooden component.

The three Stokes parameters, 푆1, 푆2, and 푆3, for the diffusely painted cylinder is shown in Figure 7.1. Immediately it can be seen that the “uninteresting” baseline displays interesting qualities. Some of the areas that are notable are the values of each parameter relative to zero. 푆1 is negative, indicating a predominance of vertically polarized light, while 푆2, and 푆3 remain near zero. However, 푆1, 푆2, and 푆3, all display sharp changes in relative value at about the 30° reflection angle, with 푆2 displaying a peak and valley near the 30° reflection angle. 푆3 has the same characteristics, but is quite small relative to 푆1 and 푆2, and is difficult to display. Following charts will not display the 푆3 data unless a specific point about it is being made, and it will be scaled to be visible. So it seems even the baseline geometry has a polarization signature that may be useful in recognizing it.

105

Figure 7.1: Stokes parameters 푆1, 푆2, 푆3.of a painted cylinder as a baseline.

Spectral signatures often display a “glint” like this with a very high intensity peak.

It is interesting to compare the polarimetric peaks and valleys to the spectral intensity measured at the same position. This is shown in Figure 7.2 where the normalized absolute values of 푆1,and 푆2 are compared to the normalized spectral intensity. Recall that normalization is performed by dividing by the magnitude of 푆0. Notice the activity in all three measurements at the same location. This peak and valley combination are seen in other objects as well.

106

Figure 7.2: Spectral and Stokes intensity of a painted cylinder. Right side axis shows 푆1, 푆2

The next method of signature analysis is to study the six vector spaces made by combining the Stokes parameters into intensity ratios. These are displayed in Figure 7.3.

Figure 7.3: Normalized vector spaces of a painted cylinder - (a) linear: 푆1/푆0, 푆2/푆0, 푆2/푆1; (b) Circular: 푆3/푆1, 푆3/푆0, 푆3/푆2. Note: scales for 푆2/푆1; and 푆3/푆2; plotted on right hand vertical axes.

The six spaces have been plotted in two graphs to make the comparisons easier.

The three spaces that are made up of the linear Stokes parameters, 푆1/푆0, 푆2/푆0, and 푆1/푆2

107

are plotted in the first graph Figure 7.3(a), and the three spaces that depend on circular polarization, 푆3/푆0, 푆3/푆1, and 푆3/푆2 are plotted in Figure 7.3(b). Again, the peak activity near the 30° reflection angle is evident in all six spaces, indicating that the values are real and potentially significant in terms of signature. The peaks, valleys, and general shape of the light curve will be compared to the same phenomena on other objects to determine if polarization has significant markers that will distinguish between them.

The third method of signature analysis will use the vector-vector space plots.

Recall that there are fifteen possible combinations, and one such plot is shown in Figure

7.4, where 푆2/푆0is plotted against 푆1/푆0. These plots will be used to perform a statistical distance analysis between clusters from one object and clusters of a second object. Without the comparative object, these plots are only of interest to note the shapes of the clusters.

Figure 7.4: Vector-vector space for painted cylinder 푆2/푆0 푣푠. 푆1/푆0.

The cylinder is a good representation of a satellite cylindrical bus. Older satellites with spin stabilized buses display this geometry. Newer satellites are three-axis stabilized and are cuboid in geometry. Thus, the next geometry that is examined is a painted cube.

108

7.1.2 Model of cuboid bus

Recall that the setup for any bus-like object is that it rotates with the observer, so that the detector will always face the same aspect of the object, but the object will rotate with respect to the light source, as shown in Figure 6.4. The first signature analysis is again comprised of the three Stokes parameters. 푆1, 푆2,, and 푆3, for the flat painted cube are plotted in Figure 7.5. It is evident from studying this graph that 푆1 has its highest absolute value (for its predominantly vertically polarized intensity) when the detector is at a low angle relative to the source. As the cube face turns away from the source, the detector sees a decreasing absolute value of 푆1. However, 푆2, and 푆3 remain fairly flat throughout the rotation with the exception of the deep valley in 푆2 at the same position as it was observed on the cylinder.

Figure 7.5: Stokes parameters for painted cube.

7.1.3 Comparison of Stokes parameters for cylinder and cube

Comparing the Stokes parameters for cylinder and cube, it is clear that 푆1 distinguishes between the flat face of the cube and the curved surface of the cylinder as

109

shown in Figure 7.6. The cube displays a constantly decreasing magnitude of 푆1 while the cylinder fluctuates slightly around a trendline that tends toward a predominance of vertical polarization. This difference is exactly the pattern that will validate the hypothesis of this work if it is consistent for comparisons with other objects. The variation from object to object is what creates the signature. When there is no variation, or little variation, no signature element will be available from that parameter. For the signature analysis, 푆1 is a good differentiator between the cube and cylinder.

Figure 7.6: 푆1 signatures for cube and cylinder.

The 푆2 signatures, on the other hand, are very similar to each other, demonstrating the same fluctuations at the 31° reflection angle, as shown in Figure 7.7. The cube has one valley without the corresponding peak, but both have a valley of vertical polarization at the same place. For the signature analysis, 푆2 is not a good differentiator between the cube and cylinder, since the signatures are so close to each other in the location of peak variations.

110

Figure 7.7: 푆2 signatures for cube and cylinder.

The 푆3 comparison of the Stokes parameters shows almost no variation in circular polarization for the cube, but there are potential signature elements for the cylinder. These are compared in Figure 7.8. 푆3 can therefore be a potential signature element to differentiate between a cylindrical bus and a cubic bus, along with 푆1. In the first case, measuring a very subtle difference, viz., a flat surface and a curved surface, Stokes parameters can detect the difference and display it in two of the three measurements. The collection of each of the differentiating measurements, along with a quantification of a difference, can go a long way toward creating a signature that will help identify an unknown object. This is the first and simplest of the signature comparisons, and it is highly successful.

111

Figure 7.8: 푆3 signatures for cube and cylinder.

7.1.4 Comparison of vector spaces for cylinder and cube

The (normalized) vector space 푆1/푆0 provides an excellent signature differentiating the cube from the cylinder, as shown in Figure 7.9. To form it, the 푆1 data are divided by a smoothly decreasing spectral intensity, which causes the smaller deviations to be emphasized. When presented in this format, the cube has distinct features which differentiate it from the cylinder. The 푆1/푆0 vector space would be a good addition to a signature analysis to differentiate these two simple geometries. Note that both signatures are consistently negative but the spread between them increases as the reflection angle increases. It is clear to see how geometry can help create such distinctive features with the help provided by the vector behavior of the reflected light.

112

Figure 7.9: 푆1/푆0 vector space signature comparison, cube and cylinder.

Of the other five vector spaces, the 푆3/푆1 vector space has some interesting points at which differentiation occurs. Figure 7.10 displays the signatures for 푆3/푆1 where the cylinder and cube peaks are opposite to one another, or slightly offset. These are the indications of the potential for measuring even slight differences between objects with diverse geometry.

113

Figure 7.10: 푆3/푆1 vector spaces for cube and cylinder.

7.1.5 Vector-vector spaces of cube and cylinder

Recall that the only difference the detector can see is a flat surface or a curved surface, with light reflecting from different angles. The edge of the cube does not play into this comparison since the cube is nadir facing with the flat face parallel to the detector, as shown in Figure 7.11. Although the photo of the models was taken with perspective to show the geometry, the detector is set at the same height as the target, with the side of the cube or cylinder toward the detector. The two geometries have identical surface treatment, the end faces of the objects do not reflect any light back to the detector, nor do the edges, as shown in Figure 7.11(b). Only the curvature of the faces of the objects are different, and the difference is quite subtle.

114

Figure 7.11: Cylinder and cube models used in the comparison, (a) oblique view; (b) view from detector position, cylinder left, cube right.

The vector-vector spaces provide a computational method to compare these two simple objects. BDs were calculated, and the graphs with the highest pair-wise distance were selected for review. In Figure 7.12, 푆3/푆0 (the vector space for circular polarization) is plotted vs. (vector space for linear polarization) the cube and cylinder, including a linear space and a circular space. Each point is the value for the corresponding object of 푆3/푆0 on the vertical axis and 푆1/푆0 on the horizontal axis. The means of each cluster are illustrated on the graph as a visual aid in assessing the distance between the clusters, but the means are not used as the distance. Note that the points for the cylinder and cube cluster in different locations within the vector-vector space, enabling the use of statistical analysis to separate the clusters from one another.

115

Figure 7.12: Vector-vector space of cube and cylinder, 푆3/푆0 vs. 푆1/푆0.

The BD distance is displayed in the upper right corner of the graph. Since all the data has been normalized, the magnitude of the value is not relevant, but comparison between graphs to select the best differentiators is important. The vertical distance spanned is about 0.02. Note that the clusters are dispersed in both horizontal and vertical directions, and are fairly well separated, with outliners for the cube along the horizontal axis and outliers for the cylinder along the vertical axis. There are a few overlapping points in each cluster, but the great majority are separate. This graph would be a good signature element.

The graph with the second largest BD is shown in Figure 7.13 comparing the linear space 푆2/푆0 vs. 푆1/푆0. The dispersion or spread in this graph is much greater along the horizontal axis, with the cube displaying far outliers on the both axes, and the cylinder displaying outliers along the vertical axis. The cluster means are again well separated. For measuring such a subtle difference, polarization has done quite well.

116

Figure 7.13: Vector-vector space of cube and cylinder, 푆2/푆0 vs. 푆1/푆0.

In summary, signature elements that distinguish well between a cube body and bus body would include the Stokes 푆1 and 푆3signatures (two of three), the 푆3/푆1 and 푆1/푆0 vector spaces (two of six selected), and the 푆3/푆0 vs. 푆1/푆0, the 푆2/푆0 vs. 푆1/푆0 vector- vector spaces (two of fifteen). Each of these elements provides information that can help identify subtle differences between targets based solely on geometry.

7.2 Polarization Signatures of Textured Surfaces

The first goal of this research is reiterated: polarimetry will differentiate simple geometries and materials from one another by producing a signature that can be quantified statistically.

So far, the objects examined have been related to geometry, but every physical object has a texture or finish that is a large part of object recognition. A black and white photo of the skins of an orange and an apple would differentiate between the two. To test this

117

differentiation with polarimetry, an experimental comparison needs to be made. Again, the difference between the two objects is intended to be as little as possible, with only one thing being changed, so that the measurement differences will be easy to attribute to the desired change. For this purpose, two antennae, both dishes, were selected for evaluation.

Both dishes are structurally the same, were constructed the same way and to the same size.

One dish has a silver painted surface, a common finish for actual satellite dishes, and the second has a fiberglass wire grid across the surface, on top of a silver painted finish. They are identically constructed except for the layer of fiberglass wire mesh, as shown in

Figure 7.14.

Figure 7.14: Dish antennae tested: (a) painted surface; (b) fiberglass wire mesh on top of painted surface.

Having completed a comparison of geometries, the same sequence of comparisons will be provided for the two surface textures.

118

7.2.1 Stokes parameters of textured surfaces

The Stokes parameters offer good potential for differentiation between these two textures. The graphs of Stokes 푆1 and 푆2 for each of the dishes is shown in Figure 7.15. the signatures are plotted on the same scale for purposes of comparison.

Figure 7.15: Stokes parameters comparing fiberglass wire dish and painted dish in signature format: (a) 푆1; (b) 푆2.

The many geometric opportunities for reflection from the fiberglass wire grid would portend a less regular signature, which is exactly the observation. The curvilinear surface of the smoothly painted dish is reflected in its signature for both 푆1 and 푆2. 푆3 in this case provides similar information but not enough to warrant adding it to the signature elements. Notably, both types of dishes portray the same trendline, but the roughness of the surface is captured by the Stokes parameters.

7.2.2 Vector spaces

The two best vector spaces for the two dishes are shown in Figure 7.16. Again, the smoothness of the surface of the painted dish is reflected in these signatures, which are significantly displaced from one another in magnitude for a portion of the 푆2 signature and

119

for all of the 푆1 signature. Both of these linear measures would be useful in identifying these targets or differentiating the materials. It appears from observation that geometry drives the overall curved shape of the vector spaces, and the unusual addition of the fiberglass wire mesh decreases the vertical polarization and produces more dispersion than the smooth surfaced form. The signatures are plotted on the same scale.

Figure 7.16: Vector spaces for fiberglass wire and painted dishes: (a) 푆1/푆0; (b) 푆2 /푆0.

7.2.3 Vector-vector spaces

The vector-vector spaces clearly separate and identify these two surface textures, which drive the polarization signatures we have seen earlier. The top two vector-vector spaces for the two textures are displayed in Figure 7.17. These two graphs display clusters that are almost entirely separate from one another, with BDs greater than one. The materials of construction could be very significant in an analysis separating satellites that are perhaps, constructed in different nations. It would be expected that surface materials, although similar, may be distinctive enough to make a noteworthy signature, as indicated by these two graphs. The horizontal scales are the same, but they are placed in different portions of the graph, and graph (a) signature is virtually flat if the vertical scales are the

120

same. The BD in this case is related to the separation on the both the horizontal and vertical scales on graph (b) with BD of graph (a) of 0.183 and BD of graph (b) of 1.53.

Figure 7.17: Vector-vector spaces for fiberglass wire mesh and painted dishes: (a) 푆2/푆0 vs. 푆1/푆0; (b) 푆2/푆1 vs. 푆1/푆0.

The next two vector-vector spaces with large BD are circular in source data, but are just as effective in differentiating between the two surfaces. These are displayed in Figure

7.18. Again the clusters based on circular vector spaces are widely separated on the horizontal scale, with BD >1, and with almost no overlap of cluster points in the vector- vector spaces, all very positive indicators of meaningful signatures.

Figure 7.18: Vector-vector spaces for fiberglass wire mesh and painted dishes: (a) 푆3/푆1vs. 푆1/푆0; (b) 푆3/푆2vs. 푆1/푆0.

121

7.3 Polarization Signatures of Solar Panels

The third component of normal satellite geometry is some form of solar panel to power the satellite. Just as different surface materials reflect different polarization states, it would be expected that solar panels produced by different manufacturers would provide a signature that would help differentiate one target from another. The solar panels described in Chapter 6 as components of the composite models were individually evaluated for signatures as well. For reference, a photo of the solar panels is shown in Figure 7.19.

Figure 7.19: Solar panels separated from composite models: (a) AMX3d 5v, 30 mA; (b) Sunnytech 5v, 100 mA.

7.3.1 Stokes parameters of the solar panels

The first two Stokes parameter comparisons are shown in Figure 7.20. Although the solar panel from Composite 1 (SP1) shows a distinct signature with increasing vertical polarization as the reflection angle increases, the solar panel from Composite 2 (SP2)

122

remains near zero with little to differentiate it from any other object. A Stokes parameter near zero means there is little difference in the intensity measured for its two orthogonal components, horizontal and vertical for 푆1 and the diagonal linear components for 푆2. An examination of the photo of the two panels should provide an explanation, since SP2 is a polycrystalline cell with randomly oriented cell structure visible. It is not surprising that the random crystal orientation would also randomize the polarization produced upon reflection. The Stokes parameters for SP1 would be a good differentiator, but for SP2, when it is combined with other components, it is likely to be quite “invisible” from a polarimetric perspective. But even this information can be used to differentiate two targets, as Figure 7.20 demonstrates well.

Figure 7.20: Stokes parameters for solar panels: (a) 푆1; (b) 푆2.

7.3.2 Vector spaces of the solar panels

The vector spaces provide another perspective on the solar panels. Instead of selecting the same spaces we have looked at previously on other components, the third and fourth vector spaces are selected as signature elements for their capability in displaying the detail of polarization on SP2, 푆2/푆1 and 푆3/푆1. From Figure 7.21, it can be seen in both

123

graphs that there is rapid transition from positive to negative, indicating that the orthogonal components of these parameters are changing with the rapidly changing cell structure of

SP2. For SP1, there is little to differentiate it since the net polarization remains near zero for this component, implying that the diagonal intensities are nearly equal for SP1. The vector spaces add significant information to the signature analysis. Note that the vertical scales are not the same in Figure 7.21.

Figure 7.21: Vector spaces for two solar panels: (a) 푆2/푆1; (b) 푆3/푆1.

7.3.3 Vector-vector spaces of solar panels

The clustering of the data in the first two vector spaces (푆2/푆0 vs. 푆1/푆0and 푆3/푆0 vs. 푆1/푆0.) provides clear differentiation between these two types of solar panels. In all the vector-vector spaces, SP1 displays dispersion along a linear axis, and only in the circular vector-vector spaces does SP2 display significant dispersion. As can be seen in

Figure 7.22, SP2 forms a tight cluster with little dispersion, while SP1 is dispersed diagonally in the 푆2/푆0 - 푆1/푆0 space, and horizontally in the 푆3/푆0 - 푆1/푆0space. Again the polycrystalline nature of SP2 would explain little difference in the linear Stokes

124

parameter. These vector-vector spaces would provide an excellent source of information when looking at the solar panels and distinguishing one type from another.

Figure 7.22: Vector-vector spaces of solar panels: (a) 푆2/푆0vs. 푆1/푆0; (b) 푆3/푆0 vs. 푆1/푆0.

Interestingly, when the other vector-vector spaces are examined, some dispersion in SP2 is also observed, which is the benefit of having fifteen spaces by which to examine and compare objects. Shown in Figure 7.23 are the 푆2/푆1 - 푆1/푆0space, and a circular space, 푆3/푆1 - 푆1/푆0, which each display dispersion in SP2, and which could be used to help identify this type of panel. All four of these vector-vector spaces could help differentiate the solar panels being examined.

125

Figure 7.23: Additional vector-vector space comparisons for solar panels: (a) 푆2/푆1vs. 푆1/푆0; (b) 푆3/푆1 vs. 푆1/푆0.

The first hypothesis, that polarization can differentiate geometries and materials has been demonstrated thoroughly with these examples. There are many more and some of these graph comparisons can be found in later in this chapter.

7.4 Polarization Signatures of Composite Models

The first hypothesis has been experimentally confirmed. The second and third hypotheses are:

2) Polarimetry will differentiate complex objects from one another using statistical

tools applied to signature analysis.

3) Polarimetric signatures of complex objects are strongly influenced by the

polarimetric signatures of the individual components, (signatures based on full

Stokes parameters).

Individual components with different geometry and material can be clearly identified by judicious use of polarization feature spaces, as demonstrated in Section 7.2.

126

The question then arises as to how effective polarimetry can be in differentiating between two complex objects made up of these same simper objects, and whether the signature peaks and valleys may be associated with physical features of the composite object, representative of the composite geometries of a satellite, that would help identify it.

The third hypothesis refers to Stokes parameter signatures which have been displayed and discussed in other papers [9, 46]. Since these are the simpler tools, this hypothesis will be addressed first.

7.4.1 Component signatures vs. composite signatures

It is found that the association of a physical feature observable in both a component and in the composite signature is consistent in the signatures of the composite objects tested in the lab. As an example, Composite 1 is compared to its components, as seen below in

Figure 7.24. The bus signature has a strong vertical element between 10° and 30° (negative

푆1), which increases the vertical strength of the composite signature in the same area. But as the bus face turns away from the light source with increasing reflection angle, and the bus intensity approaches zero, the Composite 1 signature more closely mirrors the dish signature. These graphs have not been normalized to better demonstrate the similarity between component and composite signature.

127

Figure 7.24: Component drivers of Stokes Composite 1 signature: (a) 푆1 comparison of Composite 1 and its Dish; (b) 푆1 comparison of Composite 1 and its Bus. These graphs are not normalized.

On the other hand, Composite 2 signature is more driven by the Kapton covered bus than the dish signature, as can be observed in Figure 7.25. In Figure 7.25(a) the

Composite 2 푆1 signature is plotted with the 푆1 signature of the Kapton covered rectangular bus. The Composite 2 signature clearly mirrors the bus signature peaks in the 50°-60° range of reflection angle, with two peaks visible in both signatures at the same locations.

The Composite 2 signature is more negative (indicating a stronger vertical polarization component) than the bus itself, because the other components also contribute to the vertical polarization intensity. This can be seen in Figure 7.25(b) where the dish signature is increasing in negative 푆1 (the vertical polarization) at the same reflection angle, but not nearly to the extent of the bus signature at the 60° peak.

128

Figure 7.25: Component drivers of Stokes Composite 2 signature: (a) 푆1 comparison of Composite 2 and bus; (b) 푆1 comparison of Composite 2 and dish. Graphs are not normalized.

However, as the face of the rectangular bus face turns away from the light source and the bus polarization signal goes to zero, it can be seen that the dish drives 푆1 further to the negative, with the composite signature tracking the dish, which has become the strongest component signal in the latter half of the signature. The dish therefore increases the peak size at the 60° reflection angle, and begins to drive the composite signature only as the bus signature fades. Based on these two comparisons, the composite signature is indeed a sum of the component signatures, and the size and location of the peaks in the composite signature could potentially be used to identify a geometry or material signature that is being sought on an unidentified target.

This finding is significant for application to existing polarimetric signatures, indicating that further research may identify the geometric or material components that produce peaks or valleys at specific locations within this form of the signature. The components displayed have identifying peaks or overall shapes that are easy to observe, demonstrating that the component signatures do influence the overall composite signature.

129

Based on the review of the two composite Stokes signatures, it is thus concluded that the component signatures are the sources for the peaks and valleys observed in composite signatures. Further analysis on actual satellites is required to determine the location and direction of these identifying peaks and valleys.

7.4.2 Comparison of Composite1 signatures with Composite2 signatures

The two composite models are each comprised of a bus, a dish antenna, and a solar panel. The combination of the three components with only slight differences between the components is an excellent test of the second hypothesis that polarimetry will differentiate complex objects from one another using statistical tools applied to signature analysis. As usual, the Stokes parameters are checked first.

7.4.2.1 Stokes parameters

The normalized 푆1 and 푆2 Stokes parameters comparing Composite 1 and

Composite 2 clearly differentiates between the two composites.

Readers are reminded that there are six unique spaces that constitute our vector spaces. The vector spaces can be plotted on the vertical axis while the horizontal axis is the experimental reflection angle, the equivalent of a solar phase angle on a satellite. This creates a traditional signature style representation of the vector space, so that the vector spaces for each composite can be contrasted to one another. Two of the six feature spaces are displayed in Figure 7.26.

Note that Composite 1 and Composite 2 are clearly distinct from one another visually, indicating the discriminatory nature of the Stokes parameters. This is indicative of a good potential for these parameters to be used as part of signature analysis.

130

Figure 7.26: Composites 1 and 2 Stokes parameters (horizontal axis is reflection angle). The vertical axes denote a) 푆1/푆0, b) 푆2/푆0.

7.4.2.2 Vector spaces

The vector spaces provide very similar information to the Stokes parameter signatures. Two of the vector spaces are displayed in Figure 7.27, where it can be seen that the two objects are clearly distinguishable from one another, even though they are each comprised of bus, antenna, and solar panel. Note that the overall trend at higher reflection angles is toward a negative 푆1, the equivalent of a predominance of vertically polarized light.

Figure 7.27: Vector spaces comparing Composite 1 and Composite 2: (a) 푆1/푆0; (b) 푆2/푆0.

131

It is of interest to note that in the vector space 푆1/푆0 which can also be considered to be normalized 푆1, Speicher [72] has found that several satellites have decreasing 푆1/푆0 at dusk and dawn, reflecting increasing vertical polarization relative to horizontal polarization. The reflection angle used as the x-axis in the lab represents the same set of angles at which the sun reflects from the satellite between midnight and dawn (solar phase angle). As the reflection angle approaches 90°, we would expect to see similar polarization states as would be seen at dawn (or dusk) on observed satellites. As expected, we do observe the same increase in vertical polarization in the lab experiments. It can be seen in the upper left graph of Figure 7.27 both Composite 1 and Composite 2 have a decreasing

푆1, intensity ratio as the reflection angle goes to 85°. This is indicative of the Stokes parameter 푆1 definition as horizontally polarized minus vertically polarized. The increasing negative value indicates increasing vertical polarization.

7.4.2.3 Vector-vector spaces

The vector spaces for the composite objects also separate and provide a computational tool to find the most separation. Two vector-vector spaces are selected,

푆2/푆0vs. 푆1/푆0, and 푆2/푆1vs. 푆2/푆0, shown in Figure 7.28. In each graph, the individual point clusters are shown, along with the mean for each cluster as a way of visualizing the separation of the clusters from one another, as displayed previously. Both graphs present clusters occupying distinct areas of the feature space, with some overlap. However it is clear that the composites are differentiated by the vector-vector spaces as well as the other signature methods reviewed. The horizontal scale is the same but the vertical scale is fitted to the data.

132

Figure 7.28: Vector-vector spaces with highest pair-wise distance between clusters: (a) 푆2/푆0 푣푠. 푆1/푆0; (b) 푆2/푆1푣푠. 푆2/푆0.

The graphs in this section demonstrate the proof of the second hypothesis, that polarimetry can differentiate between two complex objects and they can be statistically separated.

7.5 Comparison of Unknown Body Signature to Known Composites

The purpose of analyzing the polarimetry of the components and composite models is to have a reference to which an unknown target can be compared to determine which it is most like. This is the basis of the proposed identification process, in which an initial estimate of type is made, and the target is compared to the two classes of satellites most like the supposition. The class that is most unlike the target is eliminated and another class is compared and the process repeated until the class that is most like the target has been determined. This was articulated as the fourth hypothesis to be tested and was stated thus:

4) Polarimetric signatures can be used to correctly associate an unknown object

with one of two clusters of data representing two complex objects.

133

In order to test the hypothesis a third composite object is required that is clearly like one of the two composites, but not the same. A third composite was therefore constructed that is comprised of the same bus and solar panel of Composite 1, but with a different dish antenna. The antenna of the “unknown” body is covered with a Mylar coating, but other than that, the composite object is the same as Composite 1. Composite

2 is therefore different in all three aspects of its construction, and the polarimetry and computation methods proposed should correctly identify the unknown with the class comprised of Composite1. The Mylar covered dish is shown in Figure 7.29.

Figure 7.29: Mylar covered dish antenna for the unknown composite body.

The unknown composite is then tested in the same manner as the previous composites. The testing method is to compare each of the polarimetry signatures of the unknown object to Composite1 and Composite 2 signatures to determine which the new composite is most like. Following are the comparisons.

134

7.5.1 Stokes parameters

The first evaluation is to compare the Stokes parameters of the unknown to the two known objects. 푆1 is the first chosen Stokes parameter and is shown in Figure 7.30.

Figure 7.30: Comparison of Stokes 푆1 of unknown body to (a) Composite 1 and (b) Composite 2.

The unknown object does not compare directly with either class it is being compared with, which means that the polarization signatures are detecting a different object. Although neither class to which the unknown is compared is identical, the unknown does not have the sharp valley in 푆1 located near 57° that Composite 2 has. It has a more sharply declining 푆1 (increasing vertical polarization) than either of the two comparison classes. 푆2 is the next comparison, shown in Figure 7.31.

Figure 7.31: Comparison of Stokes 푆2 of unknown with (a) Composite 1 and (b) Composite 2.

135

The Stokes 푆2 parameter makes the signature analysis clearer. The overall shape of the 푆2 signature for the unknown is much closer to Composite 1 than Composite 2, the some divergence at the highest reflection angles. After these two comparisons, the Stokes parameters lean toward Composite 1 as the closest class of objects being compared.

7.5.2 Vector space comparison of unknown body

The vector spaces often provide more comparative detail than the Stokes parameter signatures alone. The space for 푆1/푆0 comparing both classes to the unknown target is shown in Figure 7.32. Evaluating (a), the comparison with Composite 1, the two objects are similar for the first 45°, then begin to diverge. The same overall trend of decreasing 푆1 is visible in both, but some of the peaks are reversed from Composite 1. In looking at (b), the two curves are immediately divergent through the first 45° of comparison, but may appear to be converging in the latter half. A negative peak in the unknown may possibly coincide with the negative peak of Composite 2. Another space is required to assess the situation.

Figure 7.32: Comparison of 푆1 vector space of unknown with (a) Composite 1 and (b) Composite 2.

136

The 푆2 space of the unknown is compared to the 푆2 spaces of the known composites in Figure 7.33. The unknown object diverges from both known composites, and is yet unclear as to which class it is most like. Another vector space will be compared.

Figure 7.33: Comparison of 푆2 vector spaces of unknown object with (a) Composite 1 and (b) Composite 2.

The vector space chosen to compare next is the 푆2/푆1 space, as shown in Figure

7.34. Finally, the unknown tracks closely with Composite 1, but remains separated from

Composite 2 throughout the entire set of measurements. The vector space evaluation correctly finds Composite 1 more like the target than Composite 2.

137

Figure 7.34: 푆2/푆1 vector space comparison of unknown object to (a) Composite 1 and (b) Composite 2.

7.5.3 Vector-vector spaces

The final test for the unknown object is performed by plotting the clusters in vector- vector space, calculating the distances, and determining which class the unknown object is statistically most like. The MD is used to determine distance between a new object and a cluster, so that is the method that will be used.

Recall that the unknown object is known to be more like Composite 1 than

Composite 2. Therefore, when comparing vector-vector spaces it would be expected that comparison with Composite 1 would be less conclusive since the spaces should overlap more for Composite 1 than Composite 2. It should also be noted that the largest pairwise distance between the unknown and Composite 1 will not be the found in the same vector- vector space as the largest pairwise distance for Composite 2. This is the first test. After evaluating MDs for Composite 1 and the unknown, it is determined that the best differentiator of the two for Composite 1 is the 푆3/푆0vs 푆1/푆0 graph, as shown in Figure

7.35 with an MD between Composite 1 and the unknown of 0.664. This space is a poor

138

differentiator for Composite 2 since nine of the fifteen spaces have longer MDs between the unknown and Composite 2. The graphs are not conclusive, and the populations of both clusters overlap, an indication that Composite 1, with its best distance graph, is very similar to the unknown.

Figure 7.35: 푆3/푆0 vs. 푆1/푆0 Vector-vector space comparison for the unknown object with (a) Composite 1 and (b) Composite 2.

The best distance graph for Composite 2 should be compared next. After reviewing the MD table for the unknown and Composite 2, the largest pairwise distance in vector- vector space is 푆2/푆0 - 푆1/푆0 as shown in Figure 7.36. This is at best a mediocre differentiator for the unknown with Composite1, since five of the fifteen spaces have longer MD distances. Scales are the same for these spaces.

139

Figure 7.36: 푆2/푆0 -. 푆1/푆0 vector space comparison of unknown object with (a) Composite 1 and (b) Composite 2.

However, this comparison of vector-vector spaces is definitive from the perspective of Composite 2. The MD between Composite 2 and the unknown is 1.7, far higher than the 0.664 between Composite 1 and the unknown. In examining the graph, the clusters for

Composite 2 and the unknown are completely separate, with no overlap at all. The highest pair-wise distance as evaluated by cluster analysis and MD indicates that the unknown object is less like Composite 2 than Composite 1, a correct conclusion.

This evaluation using the proposed vector-vector spaces provides the clearest answer both qualitatively (visually) and quantitatively using the statistical evaluation tool.

This is a successful demonstration that the fourth hypothesis, which states polarimetric signatures can be used to correctly associate an unknown object with one of two clusters of data representing two complex objects, has been successfully shown to be true. This completes the formal presentation of results. However there are additional comparisons that follow, for the interested reader.

140

7.6 Vector Spaces of Powered Solar Panels

Every pair of components compared with different geometries and materials have clearly been differentiated by their polarization signatures. To be completely thorough, it was determined to check the signatures of powered solar panels vs. unpowered solar panels, to see if the polarization signature changed with power.

The halogen lamp is a Platinum FCR Bulb, 100 W, 12 V, GY6.35 and the solar panel is the Sunnytech 0.5W, 5V, 100 mA power source. It was mounted the same distance from the light source as for other models being compared, and placed in the center of the beam. The panel was connected through a breadboard to a resistor and an LED, 3v, 20mA.

The LED was shielded from the detector so that it would not influence the intensity readings. As with other comparisons, none of the equipment was moved during the data generation except for the analyzer settings, so that all readings could be taken. Readings were taken of the polarization intensity with the solar panel powered, and then unpowered.

The results are interesting. In Figure 7.37 the Stokes parameter 푆1 as measured on the powered and unpowered cells is shown. Although this did measure a difference, it would be difficult to tell them apart with this tool.

141

Figure 7.37: Stokes parameter 푆1 measured for powered and unpowered solar panel.

It may be observed that the unpowered panel has a more positive polarization

(horizontal greater than vertical) than the unpowered panel until about 45°, at which point the powered panel becomes more positive (horizontally polarized.)

The difference is a little clearer when viewing the vector spaces of the two panel states. As shown in Figure 7.38, the separation is distinctly visible in the 푆1/푆0 vector space.

Figure 7.38: 푆1/푆0vector space comparison for powered vs. unpowered solar panels.

142

The 푆1/푆0 vector space is not the only one that can detect the difference between the powered and dark panel. The 푆2/푆1vector space comparison is shown in Figure 7.39, where the large peak on the unpowered panel is not visible on the powered panel, and changes in direction and location of other peaks are visible.

Figure 7.39: 푆2/푆1vector space comparison of powered vs. unpowered solar panels.

When the graph for 푆2/푆1 is viewed in vector space, it can be observed that there is a distinct shift toward the negative has occurred, indicating a shift toward 135° diagonal linear polarization as shown in Figure 7.40. This vector space is shown for comparison to one of its vector-vector spaces in Figure 7.41.

143

Figure 7.40: 푆2/푆0 vector space for powered vs. unpowered solar panels.

Figure 7.41: Vector-vector space comparison of powered and unpowered solar panel.

144

When the vector-vector spaces are examined, it is clear that the polarization measurements detected some change. The clusters have separated themselves, and moved apart, with as much differentiation as other objects tested. Note that the powered cluster is much tighter, having compacted on both axes, as well as taking a shift toward more 135° polarization and a small shift toward diagonal vs. horizontal polarization. The shift in the vector spaces may be more distinct to most, but the shift in vector-vector spaces can be computed and determined analytically.

The evidence of the vector spaces and vector-vector spaces indicates that the polarization of the solar panel field and the light reflected from the panel face are changed when current is flowing through the wires of the panel. Recall that the solar panel is 0.5 W,

5V, 100 mA, so the power being generated is very small, yet the polarization tools were able to detect it. The usefulness of this in monitoring spacecraft health is evident. If one solar panel can be detected, it is likely that two will generate a different signature than one would, and certainly a different signature than an unpowered panel would. If a solar panel stopped working, it would be detectable, and could be a cost-effective way to monitor health of expensive satellites. There is at least enough evidence for this phenomenon to investigate further.

7.7 Vector Spaces of Sphere vs. Cube

Initial comparisons of elementary shapes with subtle differences were important in learning how sensitive polarization was to detect shape and texture differences. The painted cube and painted sphere was one of the first comparisons. It can be seen in Figure

7.42 (a) that the sphere 푆1 signature remains flat but reverses with several peaks and

145

valleys not seen in the cube. The flat signature would be expected since a sphere should always present the same face to the detector, no matter the orientation. In Figure 7.42(b) the peaks and valleys are more pronounced and consistent throughout the signature when the scale is enlarged. The scales have been set the same for purposes of comparison, and both signatures appear fairly flat at this scale. This set of objects was not used further since the sphere is not similar to any current satellites.

Figure 7.42: Stokes 푆1 and 푆2 for Sphere vs. Cube.

In Figure 7.43 the vector space 푆1/푆0 shows a flattened cube and the sphere demonstrating increased vertical polarization at high reflection angles. The cube’s characteristic peaks are not repeated on the sphere.

146

Figure 7.43: 푆1/푆0 comparison of Sphere and Cube.

Figure 7.44 shows the vector-vector space of 푆3/푆0 with a distinct separation. Again recall that the shape difference is fairly subtle, comparing the geometry of a square vs. a circle, but the analysis clearly shows a separation of the clusters. A notation is made that with this spherical surface, the vector-vector spaces with the largest separation and the larges BDs were those involving circular polarization as a part of the analysis.

Figure 7.44: 푆3/푆0vs. 푆1/푆0 for Sphere and Cube.

147

7.8 Vector Spaces of Mylar Dish vs. Painted Dish

Polarization signatures have detected differences between every pair of objects that were compared. The painted dish was compared to the fiberglass wire dish in Chapter 7, here the painted dish is compared to the Mylar coated dish, as shown in Figure 7.45 where

푆1 is shown in (a) and 푆2 in (b). The geometry was identical, only the surface was finished differently. The mylar provides much sharper peaks at small observation angles, but goes almost to zero at larger angles, showing that the mylar surface is very near specular, with almost all reflection at smaller angles. Note that the vertical scales are not the same, so that the comparison between the cube and sphere is more visible.

Figure 7.45: Stokes parameters for Mylar dish and painted dish a) 푆1; b) 푆2.

Vector spaces show a similar trend, as can be observed in Figure 7.46, where the painted dish signature is flat and smooth relative to the jagged peaks of the Mylar dish. At larger angles, the two signatures diverge most significantly, creating a flag that would signal the Mylar surface.

148

Figure 7.46: Vector spaces for Mylar and painted dishes; ; (a) 푆1/푆0; (b) 푆2/푆1.

The vector-vector spaces are shown in Figure 7.47 where the graph in (a) shows the linear spaces 푆2/푆0 vs. 푆1/푆0, separated distinctly from one another. The graph in (b) is

푆3/푆1vs. 푆2/푆0, including the circular space. Note that in graph (a) the cluster is stretched out along the horizontal axis, but in (b) it is stretched in the vertical direction. This provides a strong visual separation and an indicator of the difference between the two.

Figure 7.47: Vector-vector spaces for Mylar and painted dishes; (a) 푆2/푆0 vs. 푆1/푆0 linear; (b) 푆3/푆1 vs. 푆2/푆0 circular.

149

7.9 Solar Panel Orientation Comparisons

Polarization and vector-vector space has been so successful in detecting even the smallest of geometric differences, it seemed appropriate to investigate solar panels even further. Solar panels are constructed with “stringers” running through them, which are the wires that carry the current generated by the solar cells. An example of the panel stringers are shown in Figure 7.48 with stringers in a vertical orientation.

Figure 7.48: Solar panel with vertical stringers.

It had been observed that solar panels produce a much more predominant vertical polarization as the reflection angle increases, but it had not always been observed. After some thought, it was decided to test the orientation of the solar panels (and therefore of the stringers running through them) to see if the orientation of the stringers would produce different results, which is what was found. Figure 7.49 shows the Stokes 푆1 parameter measuring the difference between the panel with the stringers in the horizontal orientation

150

(SPH) with the stringers in the vertical orientation (SPV). This panel was powered during both orientation tests, and the data was taken the same day. The only difference in the set up was a 90° change in the orientation of the stringers.

Figure 7.49: Stokes parameter 푆1 for orthogonal orientations of solar panel stringers.

Although both orientations produce vertically polarized signatures, the panel with horizontal stringers produces a much stronger vertical signal than the panel with vertical stringers at lower reflection angles, as well as more variability. This result makes sense if you consider the stringers to be operating in the same way as a wire grid polarizer, which is an absorptive polarizer, passing only light that is perpendicular to the wires. In the same way, the stringers could be operating as an absorbing mechanism, passing the vertically polarized light more than the horizontal light. The horizontal stringers therefore produce a stronger vertical signature, and 푆1 becomes more negative.

151

Figure 7.50 displays the 푆1/푆0vector space, demonstrating clear separation between the two orientations, particularly at larger reflection angles.

Figure 7.50: Stokes vector space 푆1/푆0 for horizontal and vertical stringer orientation of solar panels.

Figure 7.51 is the Vector-vector space demonstrating another clear separation of the clusters from one another. Both axes are linear spaces, with all the movement along the horizontal axis. It can be seen that the horizontal oriented panel has a stronger vertical polarization signal, and the vertically oriented panel signature has moved to the right, or more positive in horizontal polarization. Almost no separation of the diagonal polarized light has been observed, which strengthens the hypothesis that the strings are absorbing polarized light that is oscillating in the same orientation as the wires.

152

Figure 7.51: Vector-vector space 푆2/푆1vs. 푆1/푆0 for a solar panel in two orthogonal orientations.

This subtle change in the geometry of the solar panels is again indicative of what a strong tool polarization signatures are.

153

CHAPTER 8

CONCLUSIONS AND FUTURE WORK

8.1 Conclusions

In this dissertation, the capability of developing polarimetric signatures that can successfully discriminate between simple geometries, diverse surfaces and materials, and complex objects has been studied for application in the identification of unresolved targets in space. In addition, using polarimetry to associate an unknown object with a class of satellites most likely to be similar to it has also been investigated. Polarization signatures have been found to be ubiquitous and distinct. Every comparison made produced a distinct signature that would be useful in identifying and separating objects into classifications.

Specifically, in Chapter 4, the color-color method of discrimination between stars used by the community of astronomers was modified and proposed as a method to identify and classify otherwise unknown objects in space. In addition, various statistical approaches were evaluated, and two non-Euclidean distance methods were selected, the

BD to help determine the dissimilarity between clustered data from measured objects, and the MD to evaluate the distance between a new object and the clusters to which it was being compared. This combination of methodology from another discipline applied polarimetry

154

the statistical evaluation has not previously been proposed and is a novel but extremely useful method to evaluate the growing numbers of objects in the GEO orbit.

In Chapter 6, the experimental method for answering the questions raised as the subject of this work were addressed, and the sources of error and equipment calibration methods were described.

In Chapter 7, experimental results were analyzed and presented demonstrating successful discrimination of simple geometries from one another, discrimination of textures and materials from one another, and discrimination of complex objects made up of the same simple objects explored earlier were presented. In every case examined, polarization signatures are ubiquitous and can differentiate the objects being examined regardless of the subtlety of the differences being examined. Polarization signature techniques, including the classical Stokes signature, vector space signatures, and vector- vector spaces with statistical analysis proved unerring in making the distinction desired. In addition, the same methods were successfully used to associate an unknown composite geometry with the class of composite objects that was most similar to it based on the makeup of the three simple geometries comprising its structure.

The combination of these polarimetric elements into an algorithm that would test each of these parameters and provide qualitative, visual feedback from Stokes signatures and vector spaces, as well as quantitative analysis and specific associations based on the statistical evaluation of vector-vector spaces would provide a strong enhancement to the ongoing spectral analysis currently in use to perform the same task. The opportunity to

155

examine so many more aspects as well as the quantitative approach would strengthen the current methods of unknown object identification.

8.2. Future Work

In this Section, future work to continue this thought process is proposed. During the many experiments conducted, numerous anomalies were observed which could be further investigated. Some of these questions are:

1) Why do some materials such as solar panels and dishes display a stronger

vertical polarization (relative to horizontal) near dusk and dawn? This would

require material testing as well as a theoretical discussion of the source of this

polarization behavior.

2) When actual satellite polarizations are measured, an anomalously high circular

polarization has been measured. For this reason, the full Stokes parameters are

recommended to be tested to determine if this is real or just the repeated

coincidental series of mismeasurements that all reported the same phenomena.

There is no theoretical explanation for the presence of high circular polarization

from unpolarized light reflected from physical surfaces. This could be treated

both theoretically and experimentally in the future.

3) Finally, a polarimeter should be mounted on a telescope and data captured on

the polarization signatures of the existing orbiting satellites. A library of the

signature of these objects can be developed for use in comparing to unknown

objects as they are found through existing observation techniques.

156

REFERENCES

[1] M. Cegarra Polo, A. Alenin, I. Vaughn, and A. Lambert, "GEO satellite characterization through polarimetry using simultaneous observations from nearby optical sensors," Adv. Maui Opt. Space Surveillance Tech. Conf., Maui, HI (2016).

[2] M. Heald, and J. Marion, Classical electromagnetic radiation, Courier

Corporation, (2012).

[3] M. Pasqual, C. Cahoy, and E. Hines, "Active polarimetry for orbital debris identification," Adv. Maui Opt. Space Surveillance Tech. Conf., Maui HI, (2014).

[4] L. McMackin, P. Zetocha, and C. Sparkman, "Intelligent optical polarimetry development for space surveillance missions," ESA SP 440, 199-204 (1999).

[5] S. Bruski, S. Harms, M. Jones, N. Thomas, and S. Dahlke, “Determination of

Satellite Characteristics through Visible Light Intensity Analysis,” Dept. of Astronomy,

Air Force Academy, Colorado Springs, CO (2012).

[6] K. Bush, G. Crockett, and C. Barnard, "Satellite discrimination from active and passive polarization signatures: simulation predictions using the TASAT satellite model,"

Int. Symp. on Opt. Sci. and Tech., Int. Soc. for Opt. and Phot. (2002).

157

[7] D. J. Kessler, and B. Cour‐Palais, "Collision frequency of artificial satellites: The creation of a debris belt," J. of Geoph. Res.: Space Phys. 83.A6, 2637-2646 (1978).

[8] D. Kessler, N. Johnson, J. Liou, and M. Matney, "The Kessler syndrome: implications to future space operations," Adv. in the Astr. Sci. 137.8 (2010).

[9] T. Payne, S. Gregory, N. Houtkooper, and T. Burdullis, "Analysis of multispectral radiometric signatures from geosynchronous satellites," Astro. Data Analysis II. 4847.

Int. Soc. Opt. and Phot. (2002).

[10] Y. Zhao, L. Zhang, D. Zhang, and Q. Pan, "Object separation by polarimetric and spectral imagery fusion," Comp. Vis. Image Understanding 113.8, 855-866 (2009).

[11] J. Tyo, D. Goldstein, D. Chenault, and D. Shaw, "Review of passive imaging polarimetry for remote sensing applications," App. Optics 45, 22, 5453-5469 (2006).

[12] https://en.wikipedia.org/wiki/Polarization_(waves), retrieved 7/20/16.

[13] E. Collett, Polarized light: Fundamentals and applications, Optical Engineering,

New York: Dekker, (1992).

[14] https://en.wikipedia.org/wiki/Circular_polarization, retrieved 7/20/16.

[15] L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys.

Rev. A 45.11, 8185 (1992).

[16] R. Beth, "Mechanical detection and measurement of the angular momentum of light," Phys. Rev. 50.2, 115 (1936).

158

[17] A. Yao, and M. Padgett, "Orbital angular momentum: origins, behavior and applications," Adv. in Opt. and Phot. 3.2, 161-204 (2011).

[18] https://en.wikipedia.org/wiki/Angular_momentum_of_light, retrieved 1/15/18.

[19] G. Rui, B. Gu, Y. Cui, and Q. Zhan, "Detection of orbital angular momentum using a photonic integrated circuit," Sci. Rep. 6, 28262 (2016).

[20] M. Born, and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light, Elsevier, (1980).

[21] J. Stryjewski, D. Hand, D. Tyler, S. Murali, M. Roggemann, and N. Peterson,

"Real time polarization light curves for space debris and satellites," Adv. Maui Opt.

Space Surveillance Tech. Conf., Maui, HI (2010).

[22] D. Sanchez, S. Gregory, S. Storm, T. Payne, and C. Davis, "Photopolarimetric measurements of geosynchronous satellites," Int. Symp. on Opt. Sci. and Tech., Int. Soc. for Opt. and Phot. (2001).

[23] S. Ullman, High-level vision: Object recognition and visual cognition, 2, MIT press, Cambridge, MA (1996).

[24] J. Tyo, M. Rowe, E. Pugh, and N. Engheta, "Target detection in optically scattering media by polarization-difference imaging," Appl. Opt. 35, 11, 1855-1870

(1996).

[25] S. James and S. Cain, “Beyond Diffraction Limited Seeing Through Polarization

Diversity,” Air Force Inst. Tech., Wright-Patterson AFB OH, (2009).

159

[[26] D. Glenar, J. Hillman, B. Saif and J. Bergstralh, "Acousto-optic imaging spectropolarimetry for remote sensing," Appl. Opt. 33 31, 7412-7424 (1994).

[27] Y. Aron and Y. Gronau,, "Polarization in the LWIR: a method to improve target acquisition," Def. and Sec., Int. Soc. for Opt. and Phot. (2005).

[28] L. Wolff, "Polarization vision: a new sensory approach to image understanding,"

Image and Vision Comp. 15.2, 81-93 (1997).

[29] D. Clarke and J. Grainger, “Polarized light and optical measurement:iInternational series of monographs in natural philosophy,” 35, Elsevier, (2013).

[30] https://en.wikipedia.org/wiki/Fresnel%E2%80%93Arago_laws, retrieved July 28,

2016.

[31] C. Brosseau, Fundamentals of polarized light: a statistical optics approach,

Wiley-Interscience, (1998).

[32 ] E. Wolf, "Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation," Opt. Lett.

28.13, 1078-1080 (2003).

[33] F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello and R. Simon,

"Partially polarized Gaussian Schell-model beams," J. of Optics A: Pure and App. Opt. 3.1,

1 (2001).

[34 ] D. James, "Change of polarization of light beams on propagation in free space,"

JOSA A 11.5, 1641-1643 (1994).

160

[35] E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312.5-6, 263-267 (2003).

[36] A. Saleh, "An investigation of laser wave depolarization due to atmospheric transmission," IEEE J. Quantum Elec. 3, 540-543 (1967).

[37] C. Ahrens and C. Donald, Meteorology today: an introduction to weather, climate, and the environment, Cengage Learning (2012).

[38] O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt.

Comm. 233.4-6, 225-230 (2004).

[39] E. Collett, Field guide to polarization, 15 SPIE Press, Bellingham (2005).

[40] V. Tatarskii, "The effects of the turbulent atmosphere on wave propagation,"

Jerusalem: Israel Program for Scientific Translations (1971).

[41] E. Wolf, "Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation," Opt. Lett.

28.13, 1078-1080 (2003).

[42] M. Yao, I. Toselli and O. Korotkova, "Propagation of electromagnetic stochastic beams in anisotropic turbulence," Opt. Exp. 22.26, 31608-31619 (2014).

[43] Gnawali, Rudra, P. Banerjee, J. Haus and D. Evans, "Optical propagation in anisotropic metamaterials," Physics and Simulation of Optoelectronic Devices XXV.

10098, Int. Soc. for Opt. and Phot. (2017).

161

[44] M. Skinner, "Orbital debris: What are the best near-term actions to take? A view from the field," J. of Space Safety Eng., 4.2, 105-111 (2017).

[45] T. Payne, S. Gregory, D. Sanchez, T. Burdullis and S. Storm, "Color photometry of geosynchronous satellites using the SILC filters," Multifrequency Electronic/Photonic

Devices and Systems for Dual-Use Applications, 4490, Int. Soc. For Opt. and Phot. (2001).

[46] J. Speicher, “Identification of geostationary satellites using polarization data from unresolved images,” Diss. University of Denver, (2015).

[47] E. Compain, S. Poirier, and B. Drevillon, "General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers," Appl. Opt. 38 (16) 3490-3502 (1999).

[48] D. Pennington, "Cross-disciplinary collaboration and learning," Ecology and

Society 13 2 (2008).

[49] D. Pennington, "Cross-disciplinary collaboration and learning," Ecology and

Society 13 2 (2008).

[50] C, Elachi and J. Van Zyl, “Introduction to the physics and techniques of remote sensing,” 28, John Wiley & Sons, (2006).

[51] C. Lada and F. Adams, "Interpreting infrared color-color diagrams-Circumstellar disks around low-and intermediate-mass young stellar objects," The Astr. Journal 393,

278-288 (1992).

162

[52] T. Evans, F. Pont, D. Sing, S. Aigrain, J. Barstow, J. Deseri, N. Gibson, K. Heng,

H. Knutson, A. Lecavelier Des Etangs, "The deep blue color of HD 189733b: Albedo measurements with Hubble Space Telescope/space telescope imaging spectrograph at visible wavelengths,” The Astr. J. Lett 772.2, L16 (2013).

[53] T. Payne, S. Gregory, D. Sanchez, T. Burdullis and S. Storm, "Color photometry of geosynchronous satellites using the SILC filters," Multifrequency Electronic/Photonic

Devices and Systems for Dual-Use Applications, 4490, Int. Soc. For Opt. and Phot.

(2001).

[54] A. Srivastava, J. Shantanu, W. Mio and X. Liu, "Statistical shape analysis: clustering, learning, and testing," IEEE Transactions on pattern analysis and machine intelligence 27.4, 590-602 (2005).

[55] W. Cheng,J. Haus, and Q. Zhan, "Propagation of vector vortex beams through a turbulent atmosphere," Opt. Exp. 17.20, 17829-17836 (2009).

[56] R. Madden and R. Katz, "Applications of Statistics to Modeling the Earth's

Climate System," Colloquium organized by the National Center for Atm. Res. Stat.

Project, Boulder, Colorado (1994).

[57] D. Wilks, Statistical methods in the atmospheric sciences, 100, Academic press,

(2011).

[58] C. Fraley, and A. Raftery, "Model-based clustering, discriminant analysis, and density estimation," J. Am. Stat. Assoc. 97.458, 611-631 (2002).

163

[59] D. Pados, K. Halford, D. Kazakos and P. Papantoni-Kazakos, "Distributed binary hypothesis testing with feedback," IEEE transactions on systems, man, and cybernetics

25.1, 21-42 (1995).

[60] H. Delic, P. Papantoni-Kazakos and D. Kazakos, "Fundamental structures and asymptotic performance criteria in decentralized binary hypothesis testing," IEEE transactions on communications 43.1, 32-43 (1995).

[61] H. Chernoff, "A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations," Annals of Math. Stat., 493-507 (1952).

[62] A. Banerjee, U. Chitnis, S. Jadhav, J. Bhawalkar and S. Chaudhury, “Hypothesis testing, type I and type II errors,” Ind. Psych. J., 18(2), 127 (2009).

[63] S. Kachigan, Multivariate statistical analysis: A conceptual introduction, Radius

Press, (1991).

[64] T. Kailath, "The divergence and Bhattacharyya distance measures in signal selection," IEEE transactions on communication technology 15.1, 52-60 (1967).

[65] R. De Maesschalck, D. Jouan-Rimbaud and D. Massart, "The Mahalanobis distance," Chemometrics and Intelligent Laboratory Systems 50.1, 1-18 (2000).

[66] S-H. Cha, “Comprehensive survey on distance/similarity measures between probability density functions”, International Jour. of Math. Models and Meth. In Appl.

Sci. 4, 300-307 (2007).

164

[67] L. Zhao, Z. Lu, W. Yun and W. Wang, “Validation metric based on Mahalanobis distance for models with multiple correlated responses,” Rel. Eng. & Sys. Safety, 159,

80-89 (2017).

[68] R. De Maesschalck, D. Jouan-Rimbaud and D. Massart, "The Mahalanobis distance," Chemometrics and Intelligent Laboratory Systems 50.1, 1-18 (2000).

[69] T. Moulsley, and E. Vilar, "Experimental and theoretical statistics of microwave amplitude scintillations on satellite down-links," IEEE transactions on Antennas and

Propagation 30.6, 1099-1106 (1982).

[70] P. Dao, P. McNicholl, J. Brown, J. Cowley, M. Kendra, P. Crabtree, A.

Dentamaro, E. Ryan and W. Ryan, “Space object characterization with 16-visible-band measurements at Magdalena Ridge Observatory,” Air Force Research Lab Kirtland AFB,

(2008).

[71] G. Golub, and C. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins,

Baltimore, MD (1996).

[72] A. Speicher, M. Matin, R. Tippets, F. Chun and D. Strong, "Results from an experiment that collected visible-light polarization data using unresolved imagery for classification of geosynchronous satellites," Airborne Intelligence, Surveillance,

Reconnaissance (ISR) Systems and Applications XII, 9460, Int. Soc. for Opt. and

Photonics (2015).

165

[73] Namer, E., and Schechner, Y.Y., "Advanced visibility improvement based on polarization filtered images," Optics & Photonics 2005, International Society for Optics and Photonics (2005).

[74] Sassen, K., Cho, B.S., "Subvisual-thin cirrus lidar dataset for satellite verification and climatological research," Journal of Applied Meteorology 31.11 (1992): 1275-1285.

[75] Liang, T., Sun, X., Wang, H., Ti, R., Shu, C., "Airborne polarimetric remote sensing for atmospheric correction," Journal of Sensors 2016 (2015).

[76] Blake, T., Sanchez, M., Krassner, J., Sundbeck, S., “Space domain awareness,”

Defence Advanced Research Projects Agency, Arlington, VA, (2012).

[77] T. Payne, S. Gregory, and K. Luu, "Electro-optical signatures comparisons of geosynchronous satellites," IEEE Aerospace Conference, 6 (2006).

[78] D. Beamer, U. Abeywickrema, and P. Banerjee, "Polarization vector signatures for target identification," Polarization Science and Remote Sensing VIII, 10407, Int. Soc.

Opt. and Phot., (2017).

[79] http://www.dupont.com/products-and-services/membranes-films/polyimide- films/brands/kapton-polyimide-film.html, downloaded 12/17/17.

[80] X-F. Navick, M. Carty, M. Chapellier, G. Chardin, C. Goldback, R. Granelli, S.

Herve, M. Karolak, G. Nollez, F. Nizery, C. Riccio, P. Starzynski, V. Villar, "Fabrication of ultra-low radioactivity detector holders for Edelweiss-II," Nuclear Instr. and Methods in Phys. Res., Section A, 520 (1) 189-192 (2004).

166

[81] http://space.skyrocket.de/doc_sdat/swisscube.htm, downloaded Dec. 1, 2017.

[82] http://space.skyrocket.de/doc_sdat/radarsat-2.htm, downloaded Dec. 1, 2017.

[83] https://www.solarreviews.com/solar-energy/pros-and-cons-of-monocrystalline-vs- polycrystalline-solar-panels/#advantagesMonocrystalline, downloaded August 12, 2017.

[84] http://www.geosats.com/antennas.html, downloaded December 9, 2017.

[85] http://www.madehow.com/Volume-1/Satellite-Dish.html#ixzz4f6LfjOlC, downloaded April 23, 2017.

[86] DoD Instruction 3100.11, Management or laser illumination of objects in space, https://fas.org/irp/doddir/dod/i3100_11.pdf, downloaded 13 April, 2017.

[87] J. Taylor, Introduction to error analysis, the study of uncertainties in physical measurements, University Science Books, Herndon, VA (1997).

[88] C. Persons, M. Jones, C. Farlow, L. Morell, M. Gulley and K. Spradley, "A proposed standard method for polarimetric calibration and calibration verification," Proc.

SPIE. 6682 (2007).

[89] J. Tinbergen, Astronomical polarimetry, (Cambridge University Press, 2005).

[90] R. Penrose, "A generalized inverse for matrices," Mathematical proceedings of the Cambridge philosophical society. 51. 3. Cambridge University Press, New York, NY

(1955).

167