# Manual 21027781

MUELLER MATRIX ROOTS by Hannah Dustan Noble = BY: A Dissertation Submitted to the Faculty of the COLLEGE OF OPTICAL SCIENCES In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY In the Graduate College THE UNIVERSITY OF ARIZONA 2011 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Hannah Dustan Noble entitled Mueller Matrix Roots and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy. Date: 25 July 2011 Russell Chipman Date: 25 July 2011 Scott Tyo Date: 25 July 2011 Stephen McClain Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. Date: 25 July 2011 Dissertation Director: Russell Chipman 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. This work is licensed under the Creative Commons Attribution 3.0 United States License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/us/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA. SIGNED: Hannah D. Noble 4 ACKNOWLEDGEMENTS I would like to thank my parents, who have always seen the best in me and encouraged me to live an intellectually fulfilling life. Without their unfailing support I wouldn’t be here. They have taught me to develop the life of the mind, which has largely shaped my worldview and thirst for challenge. I would like to thank my brother for always being a good friend - our late night technical conversations have inspired me throughout my graduate career. I would like to thank my advisor, Professor Russell Chipman, without whom this dissertation wouldn’t have been possible. The day I walked into his office to discuss a graduate research position was a pivitol moment in my academic and professional life. Since that day, Professor Chipman has shared a tremendous amount of knowledge, dared me to push my intellectual boundaries, and given me continuous opportunities to spend my time working on some of the most interesting, mind-twisting technical problems imaginable. I would like to thank Professor John Essick for believing in me as an undergraduate, and sparking my interest in the field of optics. I would like to thank Dr. Stephen McClain, who has spent hours and hours discussing the painstaking details of my dissertation with me. This guidance has been critical. Many thanks to Dr. Greg Smith, who is always available to contribute a creative idea or programming solution, as needed. I would like to thank my colleagues and fellow Polarization Lab graduate students (and alumni) - Garam Yun, Paula Smith, Stacey Sueoka, Karlton Crabtree, Tiffany Lam, Wei-Liang Hsu, Tyson Ririe, Michihisa Onishi, Brian Daugherty, and Anna-Britt Mahler. You’ve all helped with paper editing, technical and programming discussions and collaborations, but your most important contributions have been your friendship. Thanks to Garam, Miena, and Stacey for all of the great Cafe Luce breaks. I would like to thank my partner, Houssine Makhlouf, for his constant love and support throughout this long road to the PhD. 5 DEDICATION To my parents and grandmother with Love. 6 TABLE OF CONTENTS LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 CHAPTER 1 METHODS TO CALCULATE THE PROPERTIES MUELLER MATRICES VIA MATRIX DECOMPOSITION . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Polar decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Lu-Chipman decomposition . . . . . . . . . . . . . . . . . . . . . 1.4 Symmetric decomposition . . . . . . . . . . . . . . . . . . . . . . 1.5 Additive decompositions . . . . . . . . . . . . . . . . . . . . . . . 1.6 Mueller matrix root decomposition . . . . . . . . . . . . . . . . . 1.7 Pauli spin matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Jones N-matrices and the Mueller matrix roots decomposition . . OF . . . . . . . . . . . . . . . . . . 18 18 19 20 23 25 26 27 28 CHAPTER 2 THE MUELLER MATRIX ROOTS DECOMPOSITION ALGORITHM AND COMPUTATIONAL CONSIDERATIONS . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Matrix roots decomposition . . . . . . . . . . . . . . . . . . . . . . . 2.3 pth Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Principal matrix root algorithms . . . . . . . . . . . . . . . . 2.4 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Polarizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Half wave retarders . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Depolarizing non-uniform Mueller matrices . . . . . . . . . . . 2.5 Numerical accuracy and root order . . . . . . . . . . . . . . . . . . . 2.5.1 Choice of p . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Algorithm and flow chart . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Statistical algorithm implementation . . . . . . . . . . . . . . . . . . 2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 33 35 38 39 40 40 42 45 47 47 51 53 56 CHAPTER 3 INTERPRETATION OF MUELLER MATRIX ROOTS DECOMPOSITION PARAMETERS . . . . . . . . . . . . . . . . . . . . . . . 57 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7 TABLE OF CONTENTS – Continued 3.2 3.3 3.4 3.5 3.6 3.7 Mueller matrix roots decomposition . . . . . . . . . . . . . . . . . . Definition of diagonal, phase, and amplitude depolarization . . . . . Degree of polarization maps of depolarizing Mueller matrices . . . . Depolarization generation and cyclic permutations . . . . . . . . . . 3.5.1 Cyclic permutations . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Generation of depolarization via averaging of non-depolarizing Mueller matrices . . . . . . . . . . . . . . . . . . . . . . . . Experimental samples . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Ground glass . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Pencil at 505 nm . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 60 63 67 67 . . . . . 68 70 71 75 80 CHAPTER 4 ADDITIONAL MATRIX ROOTS GENERATOR PROPERTIES AND APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.1 Convergence to the identity matrix . . . . . . . . . . . . . . . . . . . 81 4.2 Generator properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2.1 Trajectories of non-depolarizing generators . . . . . . . . . . . 84 4.2.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2.3 Depolarization index . . . . . . . . . . . . . . . . . . . . . . . 90 4.2.4 Unitary transformations and depolarization classification . . . 91 4.2.5 Conditions for physicality of depolarizing generators . . . . . . 96 4.3 Interpretation of MM root parameters . . . . . . . . . . . . . . . . . 99 4.3.1 Nondepolarizing matrix root parameters . . . . . . . . . . . . 99 4.3.2 Depolarizing matrix roots parameters . . . . . . . . . . . . . . 103 4.3.3 The diagonal depolarization tetrahedron . . . . . . . . . . . . 105 4.4 Applications for depolarization in optical design . . . . . . . . . . . . 106 4.4.1 Diattenuation and retardance defocus . . . . . . . . . . . . . . 108 4.4.2 Depolarization effects from diattenuation and retardance defocus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 CHAPTER 5 POLARIZATION SYNTHESIS BY COMPUTER GENERATED HOLOGRAPHY USING ORTHOGONALLY POLARIZED AND CORRELATED SPECKLE PATTERNS . . . . . . . . . . . . . . . . . . . 115 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2 PCGH illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.3 PCGH diffuser design . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.4 PCGH fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.5 PCGH measurement and reconstruction . . . . . . . . . . . . . . . . 121 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8 TABLE OF CONTENTS – Continued CHAPTER 6 SQUARE WAVE RETARDER FOR POLARIZATION COMPUTER GENERATED HOLOGRAPHY . . . . . . . . . . . . . . . . . . . 125 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.2 Experimental geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.3 Calcite square-wave retarder design and fabrication . . . . . . . . . . 132 6.4 Measured polarization properties of the reconstructed image . . . . . 136 6.5 Alignment and illumination errors . . . . . . . . . . . . . . . . . . . . 142 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 APPENDIX A JONES N -MATRIX ALGORITHM . . . . . . . . . . . . . . 147 APPENDIX B THE EFFECT OF REFLECTION ON DEPOLARIZATION PARAMETERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 APPENDIX C INFERRING THE ORIENTATION OF TEXTURE FROM POLARIZATION PARAMETERS . . . . . . . . . . . . . . . . . . . . . . 151 C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 C.2 Aluminum measurements and analysis . . . . . . . . . . . . . . . . . 152 C.2.1 Aluminum sample . . . . . . . . . . . . . . . . . . . . . . . . . 152 C.2.2 Mueller matrix measurements . . . . . . . . . . . . . . . . . . 153 C.3 Diattenuation and retardance . . . . . . . . . . . . . . . . . . . . . . 155 C.3.1 Determining orientation from diattenuation and retardance . . 156 C.4 Depolarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 C.4.1 Lu-Chipman depolarizing Mueller matrix and depolarization Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 C.4.2 Mueller matrix roots depolarizing parameters . . . . . . . . . 165 C.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 9 LIST OF FIGURES 1.1 Taking the root of a uniform Mueller matrix is analagous to slicing it into very thin identical pieces. . . . . . . . . . . . . . . . . . . . . . 30 2.1 Taking the root of a uniform Mueller matrix is analogous to slicing it into very thin identical pieces. . . . . . . . . . . . . . . . . . . . . The error of the root calculation vs. root order for a simple retarder converges to a minimum relative error just beyond the 105 th root. . The norm of the depolarizing retarder’s diagonal depolarization parameters (D13 , D14 , and D15 ) converges to a steady value of approximately 3.03 beyond the 104 th root. . . . . . . . . . . . . . . . . . . The norm of the depolarizing retarder’s matrix roots retardance parameters (D4 , D5 , and D6 ) converges to a steady value of approximately 1.07 beyond the 104 th root. . . . . . . . . . . . . . . . . . . Relative error for Mathematica’s matrix root calculation. . . . . . . Matrix Roots algorithm flow chart. . . . . . . . . . . . . . . . . . . A histogram of the matrix root amplitude depolarization values for 76,336 randomly generated physical Mueller matrices. . . . . . . . . A histogram of the matrix root phase depolarization values for 76,336 randomly generated physical Mueller matrices. . . . . . . . . . . . . A histogram of the matrix root diagonal depolarization values for 76,336 randomly generated physical Mueller matrices. . . . . . . . . 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.1 3.2 3.3 3.4 3.5 3.6 Taking the root of a uniform Mueller matrix is analagous to slicing it into very thin identical pieces. . . . . . . . . . . . . . . . . . . . . Degree of polarization maps for depolarizing Mueller matrices M7,15 through M15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement geometry for the University of Arizona scattering infrared polarimeter. . . . . . . . . . . . . . . . . . . . . . . . . . . . The magnitude of the matrix roots parameters D0 through D15 of the ground glass sample for (a) the specular angle pair and (b) the non-specular angle pair. . . . . . . . . . . . . . . . . . . . . . . . . The Mueller matrix for a graphite and wood pencil, measured at 505nm with a Mueller matrix imaging polarimeter. . . . . . . . . . The non-depolarizing matrix roots parameters for a graphite and wood pencil, measured at 505nm with a Mueller matrix imaging polarimeter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 . 48 . 49 . 50 . 51 . 52 . 54 . 55 . 55 . 59 . 64 . 71 . 73 . 76 . 77 10 LIST OF FIGURES – Continued 3.7 The depolarizing matrix roots parameters for a graphite and wood pencil, measured at 505nm with a Mueller matrix imaging polarimeter. 79 4.1 A histogram of the randomly generated Mueller matrix determinants shows the statistical distribution of the determinant values. . . . . . . 4.2 This histogram shows increasing convergence towards the identity matrix as the root order p increases. . . . . . . . . . . . . . . . . . . . 4.3 A linear diattenuator with diattenuation of 0.2 approaches the identity matrix as it is taken to the 10p th root. . . . . . . . . . . . . . . . 4.4 The trajectory of generator G2 with d2 = 2 ∗ 10−10 as the matrix is taken to the power of 10p approaches a linear diattenuator with diattenuation of 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 The trajectory of an elliptical retarder with δ45 = π/4 and δR = π/10 approaching the identity matrix as the matrix is taken to the 10p th root. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 The trajectory of the generator product G5 (d5 = π/4∗10−9 )·G6 (d6 = π/10 ∗ 10−9 ) approaching an elliptical retarder as the matrix is taken to the power of 10p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 The diference between π/2 and the vector angle between G1 (d1 = d) and G2 (d2 = d) is plotted as a function of d, showing that generators G1 and G2 are not orthogonal in general. . . . . . . . . . . . . . . . . 4.8 The diference between π/2 and the vector angle between G11 (d11 = d) and G12 (d12 = d) is plotted as a function of d, showing that generators G11 and G12 are not orthogonal in general. . . . . . . . . . . . . . . . 4.9 The four eigenvalues for the coherency matrices associated with the nine depolarization generators. . . . . . . . . . . . . . . . . . . . . . . 4.10 The physical region for each depolarizing generator G7 through G14 multiplicatively combined with G15 is shown in the white region. One axis represents d15 and the other axis represents the other d-parameter of a corresponding generator. G7 through G9 and G10 through G12 are shown on the same plot since their physical regions are identical. . 4.11 Physical Mueller matrix diagonal depolarization space forms a tetrahedron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Diattenuation defocus shifts along the y-axis and off-axis in x and y, illustrating the effect of tilt on diattenuation defocus in the pupil. Retardance defocus looks identical to diattenuation defocus, but the effect is different. Diattenuation defocus affects the transmission of polarization states, whereas retardance defocus adds a given retardance to the transmitted polarization state. . . . . . . . . . . . . . . 82 83 85 86 87 88 90 91 97 98 107 110 11 LIST OF FIGURES – Continued 4.13 With diattenuation defocus, diagonal depolarization parameters D14 and D15 decrease in magnitude as the pupil center is shifted off-axis. . 4.14 With retardance defocus, diagonal depolarization parameters D14 and D15 increase in magnitude as the pupil center moves off-axis. . . . . . 4.15 Circular phase depolarization D12 and diagonal depolarization D13 vary with pupil shift x0 . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16 Circular phase depolarization D12 and diagonal depolarization D13 vary with pupil shift x0 . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 5.2 5.3 5.4 5.5 6.1 6.2 6.3 6.4 The interlaced PCGH is illuminated with shifted cross-polarized Ronchi rulings, and its reconstruction and polarization distribution are measured. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x- and y-polarized components add to form a tangentially polarized annulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flowchart of the PCGH design method. . . . . . . . . . . . . . . . . Normalized linear Stokes vector components demonstrate polarization synthesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polarization orientation rotates around the annulus. . . . . . . . . . Interlaced x and y-polarized components add to form a tangentially polarized annulus. Overlap of these two coherent states produces ~ o with a continuously varying the ideal tangentially polarized ring U polarization orientation. . . . . . . . . . . . . . . . . . . . . . . . . A polarization compensator system (enclosed in the dashed line region) provides an arbitrary polarization state incident on the etched calcite square wave retarder. A 4f optical system images the square wave retarder to illuminate the interlaced CGH, providing alternating stripes of x and y polarization. A CCD camera is used at the back focal plane of an imaging lens to view the reconstructed image. . . . The calcite square wave retarder generates alternating stripes of orthogonal linear polarizations when illuminated with a linearly polarized plane wave oriented at 45o with respect to the crystals fast axis. Grooves are etched into the substrate to the base thickness level and are filled with index matching oil to produce half-wave plate ridges which rotate the incident linear polarization by 90o . . . . . . . . . . An SEM cross-section of the fabricated calcite for the square-wave retarder. The grooves are smooth and uniform, although there is some rounding of the corners. . . . . . . . . . . . . . . . . . . . . . 112 113 113 114 . 117 . 118 . 120 . 122 . 122 . 126 . 129 . 133 . 136 12 LIST OF FIGURES – Continued 6.5 Retardance of a 1 mm wide horizontal cross section at the center of the calcite square wave retarder sample was measured at 632.8 nm using the Mueller matrix imaging polarimeter. Retardance associated with the half wave plate ridges varies between approximately 160o and the ideal 180o in the center of the sample. . . . . . . . . . . . . . . 6.6 The measured Stokes vector for the PCGH reconstruction shows that the polarization is linearly tangential with some circular polarization dispersed throughout the annulus. S3 shows that the presence of circular polarization is most notable in the synthesis regions. S1 , S2 , and S3 are normalized to S0 , which remains un-normalized. . . . . . 6.7 The orientation ϕ of the major axis of the polarization ellipse around the annulus is calculated from the normalized Stokes vector. The measured polarization orientation of the PCGH reconstruction is tangential. Its orientation differs by several degrees from the desired orientation at the sides of the annulus. . . . . . . . . . . . . . . . . 6.8 The pixel-by-pixel degree of linear polarization (DoLP) for the PCGH reconstruction. The average DoLP over the annulus is 0.81. The synthesis regions have a lower DoLP than the vertical and horizontal regions of the annulus because the DoLP in these regions is dependent on a high degree of speckle correlation from the two orthogonal polarizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Pixel-by-pixel ratio of tangential polarization (RoP) to flux for the PCGH reconstruction. The RoP is noticeably lower in the 45o and 225o regions of the annulus and in the dim regions between speckles. Average RoP around the ring is 84%. . . . . . . . . . . . . . . . . . 6.10 Simulation of square-wave retarder output. (a) In an ideal optical system, the polarization orientation exiting the calcite square wave retarder alternates between x and y according to the grooves and ridge stripes; (b) An extraordinary axis orientation error of 10o with respect to the etched stripes causes an orientation error of 20o in the y polarized component exiting the calcite. . . . . . . . . . . . . . . 6.11 Properties of residual wedge in the calcite. (a) MMIP measurement of retardance as a function of position in an ungrooved substrate exhibiting a small residual wedge; and (b) Conceptual simulation illustrating transmitted polarization from a square-wave retarder in the presence of a 0.3o residual wedge. . . . . . . . . . . . . . . . . . . 137 . 138 . 139 . 140 . 142 . 143 . 144 A.1 Algorithm flow-chart for calculation of diattenuation and retardance vectors using Jones N-Matrix method. . . . . . . . . . . . . . . . . . 148 13 LIST OF FIGURES – Continued A.2 Retardance parameters of a linear retarder with a retardance of π/4 plotted as a function of θ. . . . . . . . . . . . . . . . . . . . . . . . . 148 A.3 Diattenuation parameters of a linear diattenuator plotted as a function of θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 C.1 Surface profile of sanded aluminum sample. White regions are areas of low signal where signal-to-noise ratio is poor. The white regions are likely areas of high slope where the reflected light is not substantially captured by the instrument. . . . . . . . . . . . . . . . . . . . . . . . C.2 Autocovariance of sanded aluminum sample. Units of x-profile are in millimeters, units of y-profile are in microns. . . . . . . . . . . . . . . C.3 Normalized Mueller matrix for −60o incident, 60o scattering specular angle (solid line) and −60o incident, 30o scattering off-specular angle (dashed line) with dotted-line fits. The x-axis represents texture orientation angle of aluminum in degrees, y-axis represents the magnitude of each Mueller matrix element. . . . . . . . . . . . . . . . C.4 Goniometric polarimeter coordinate system. All angles are measured from sample normal. Negative angles are in the region of backscattered light, and positive angles are in the direction of forward scatter. C.5 The two measurement configurations analyzed in this section: −60o incident, 60o scattering and −60o incident, 30o scattering. . . . . . . . C.6 Normalized average irradiance from light reflected off of the sample as the groove orientation angle changes. The x-axis lists the orientation angle of the aluminum grooves; the y-axis represents the average irradiance reflected from the sample. The specular measurement is plotted with a solid line, and the off-specular measurement is plotted with the dashed line. The sinusoidal fit from equation (C.1) is plotted with the dotted lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . C.7 Matrix root diattenuation parameters with fit (dotted lines) for the specular angle (solid line) and the off-specular angle (dashed line). . . C.8 Matrix roots retardance parameters in radians with fit (dotted lines) for the specular angle (solid line) and the off-specular angle (dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.9 Aluminum sample groove orientation calculated with the specular matrix roots retardance parameters demonstrate the estimation of orientation from a single Mueller matrix measurement following calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 159 159 160 160 161 161 161 162 14 LIST OF FIGURES – Continued C.10 Normalized depolarization Mueller matrix from the Lu-Chipman decomposition with fit (dotted line) for specular measurement (solid line) and off-specular measurement (dashed line). . . . . . . . . . . C.11 Depolarization index vs. orientation angle with fit (dotted line) for specular measurement (solid line) and off-specular measurement (dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.12 Amplitude depolarization parameters D7 , D8 , and D9 (from left to right) vs. orientation angle with fit (dotted line) for specular measurement (solid line) and off-specular measurement (dashed line). . C.13 Phase depolarization parameters D10 , D11 , and D12 (from left to right) vs. orientation angle with fit (dotted line) for specular measurement (solid line) and off-specular measurement (dashed line). . C.14 Diagonal depolarization parameters D13 , D14 , and D15 (from left to right) vs. orientation angle with fit (dotted line) for specular measurement (solid line) and off-specular measurement (dashed line). . C.15 Aluminum sample groove orientation calculated with the off-specular linear phase depolarization parameters D10 and D11 , demonstrating the estimate of orientation from a single Mueller matrix measurement following calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 . 165 . 167 . 167 . 168 . 168 15 LIST OF TABLES 2.1 The sixteen polarization properties of the Mueller matrix given by the Mueller matrix roots decomposition. . . . . . . . . . . . . . . . . 38 3.1 The non-depolarizing Mueller matrix generators G1 (d1 ) through G6 (d6 ) and their first order Taylor series approximations. . . . . . . . The depolarizing Mueller matrix generators G7 (d7 ) through G15 (d15 ) and their first order Taylor series approximations. . . . . . . . . . . . Notation for the basis diattenuator and retarder Mueller matrices oriented along the three Stokes axes (horizontal/vertical, 45o /135o and right/left circular), as well as an attenuating identity matrix. q and r are the maximum and minimum transmission, q 6= r, δ is the magnitude of the retardance vector, and α is the attenuating coefficient. Depolarization properties (shown by parameters D7 through D15 ) produced by averaging two non-depolarizing Mueller matrices. . . . . Non-diagonal depolarization properties (shown by parameters D7 through D12 ) produced by averaging two non-depolarizing Mueller matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The matrix roots parameters from a ground glass sample for specular {−70o , 70o } and non-specular angle pairs {−70o , 10o }. . . . . . . . . . 3.2 3.3 3.4 3.5 3.6 61 62 69 70 70 75 4.1 4.2 4.3 Effect of the rotation transformation for G10 through G12 about R1 (θ). 94 Effect of rotation transformation for G10 through G12 about R2 (θ). . 95 Effect of rotation transformation for G13 (d13 ) through G15 (d15 ) about R2 (θ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.1 RoP and DoLP are measured as a function of the horizontal translation of the calcite and calcite rotation about the optical axis. . . . . . 145 C.1 Sinusoidal fit coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . 163 16 ABSTRACT This dissertation is comprised of two separate topics within the domain of polarization optical engineering. The first topic is a Mueller matrix roots decomposition, and the second topic is polarization computer generated holography. The first four chapters of the dissertation are on the topic of the Mueller matrix roots decomposition. Recently, an order-independent Mueller matrix decomposition was proposed in an effort to organize the nine depolarization degrees of freedom. Chapter 1 discusses relevant Mueller matrix decomposition prior art and the motivation for this work. In chapter 2, the critical computational issues involved in applying this Mueller matrix roots decomposition are addressed, along with a review of the principal root and common methods for its calculation. The choice of the pth root is optimized at p = 105 , and computational techniques are proposed which allow singular Mueller matrices and Mueller matrices with a half-wave of retardance to be evaluated with the matrix roots decomposition. A matrix roots algorithm is provided which incorporates these computational results. In chapter 3, the Mueller matrix roots decomposition is reviewed and a set of Mueller matrix generators are discussed. The parameterization of depolarization into three families, each with three degrees of freedom is explained. Analysis of the matrix roots parameters in terms of degree of polarization maps demonstrates that depolarizers fall into two distinct classes: amplitude depolarization in one class, and phase and diagonal depolarization in another class. It is shown that each depolarization family and degree of freedom can be produced by averaging two non-depolarizing Mueller 17 matrix generators. This is extended to provide further insight on two sample measurements, which are analyzed using the matrix roots decomposition. Chapter 4 discusses additional properties of the Mueller matrix roots generators and parameters, along with a pupil aberration application of the matrix roots decomposition. Appendix C, adapted from a conference proceedings paper, presents an application of the matrix roots depolarization parameters for estimating the orientation of a one-dimensionally textured object. The last two chapters are on the topic of polarization computer generated holography. In chapter 5, an interlaced polarization computer-generated hologram (PCGH) is designed to produce specific irradiance and polarization states in the image plane. The PCGH produces a tangentially polarized annular pattern with correlated speckle, which is achieved by a novel application of a diffuser optimization method. Alternating columns of orthogonal linear polarizations illuminate an interlaced PCGH, producing a ratio of polarization of 88% measured on a fabricated sample. In chapter 6, an etched calcite square-wave retarder is designed, fabricated, and demonstrated as an illuminator for an interlaced polarization computer generated hologram (PCGH). The calcite square-wave retarder enables alternating columns of orthogonal linear polarizations to illuminate the interlaced PCGH. Together, these components produce a speckled, tangentially polarized PCGH diffraction pattern with a measured ratio of polarization of 84% and a degree of linear polarization of 0.81. An experimental alignment tolerance analysis is also reported. 18 CHAPTER 1 METHODS TO CALCULATE THE PROPERTIES OF MUELLER MATRICES VIA MATRIX DECOMPOSITION 1.1 Motivation In recent years, Mueller matrices have become increasingly easy to measure both quickly and accurately. But in order to be useful, a Mueller matrix must be analyzed and broken down in terms of its basic properties, which fall into three general families - diattenuation, retardance, and depolarization. Diattenuation and retardance each have three degrees of freedom, which are all well-understood. Despite many efforts within the polarization community (Ossikovski et al., 2008), the nine depolarization degrees of freedom (Chipman, 2005b) remain difficult to address. This work seeks to develop new ways to analyze and interpret depolarization, using an order-independent approach called a Mueller matrix roots decomposition, which is described in detail in chapters 2, 3, and 4. A profound understanding of depolarization within the Mueller matrix, from a phenomenological perspective, opens the door to new ways of approaching optical scattering analysis and optical design. This dissertation begins with a survey of the relevant prior art in the remaining sections of this chapter. Chapter 2 discusses the Mueller matrix decomposition algorithm, along with important mathematical considerations involved in using the matrix roots decomposition. Chapter 3 addresses the matrix roots decomposition’s parameterization of Mueller matrices, the character of these parameters, their properties, and how they arise. Because chapters 2 19 and 3 are reformatted from manuscripts for publication, some relevant properties (which were not included in the publications) are discussed in chapter 4. Chapter 4 also includes the application of this method to analysis of polarization aberrations within the context of optical design. 1.2 Polar decomposition In linear algebra, polar decomposition has its origin in the familiar polar form of a complex number, where λ = λ0 eiγ , λ0 ≥ 0 and 0 ≤ γ < 2π. The theorem governing polar decomposition states that any matrix A ∈ Fn×n can be represented in the form A = HU, (1.1) where Hij ≥ 0 and U is unitary. The matrix H is unique and Hermitian, and can therefore be expressed as H = (AA∗ )1/2 . Nondepolarizing Mueller matrices transform a completely polarized Stokes vector to a completely polarized one and can be derived from Jones matrices (Lu and Chipman, 1998). Within the field of polarimetry, polar decomposition is used as a data reduction operation to factor an arbitrary non-depolarizing Mueller matrix M into a retarder Mueller matrix MR and a diattenuator Mueller matrix MD1 or MD2 (Gil and Bernabeu, 1987), M = MD1 MR = MR MD2 . (1.2) This is done to better undestand M. Here, MR is a unitary matrix and both MD1 and MD2 are Hermitian Mueller matrices. MD1 and MR do not commute in general, 20 but MD1 and MD2 are related as follows: MD2 = MTR MD1 MR and MD1 = MR MD2 MTR . (1.3) This method was applied to the analysis of inhomogeneous nondepolarizing Mueller matrices in (Lu and Chipman, 1994). 1.3 Lu-Chipman decomposition The Lu-Chipman Mueller matrix decomposition (Lu and Chipman, 1996; Goldstein, 2003; Chipman, 2009) is a Mueller matrix data reduction method based on polar decomposition. It is one of the most commonly used methods for analyzing measured Mueller matrices, so a brief summary of this method is included here. The LuChipman method decomposes an arbitrary Mueller matrix into the products of a diattenuator, a retarder, and a depolarizing matrix. The matrix properties of diattenuation, retardance, and the depolarization index can then be retrieved from these three factors: M = M∆ MR MD . (1.4) The results of this decomposition allow one to define and calculate the diattenuation and retardance vectors for a given Mueller matrix. The three Lu-Chipman decomposition factors are defined as follows. Diattenuation is defined as D≡ |Tmax − Tmin | |Tmax + Tmin | (1.5) where Tmax and Tmin are the maximum and minimum transmittances over all possi- 21 ble incident polarization states, and D can have a value between 0 and 1. From the definition in equation (1.5), we see that diattenuation changes the transmission (of intensity) of an incident state of polarization. If we define the diattenuation vector with the same structure as the Stokes parameters, we have a vector with 3 degrees of freedom defined as DH ~ = D D 45 DC (1.6) where DH is horizontal diattenuation, D45 is 45◦ linear diattenuation, and DC is the circular diattenuation. The magnitude of the diattenuation vector is D= q 2 2 DH + D45 + DC2 , (1.7) ~ and {1, D/D} identifies the Stokes vector of highest transmission. A pure diattenuator has a symmetric Mueller matrix, which has the form T ~ 1 D MD = Tu ~ D mD √ √ T mD = 1 − D2 I + (1 − 1 − D2 )D̂D̂ , (1.8) (1.9) ~ where I is a three by three identity matrix, D̂ is the (normalized) unit vector for D, and Tu is the transmittance for unpolarized light. We can set up a retardance vector and Mueller matrix in a similar fashion. Retardance is defined as the difference in phase changes between the two eigenpolarizations, δ = |δq − δr |, (1.10) 22 and the retardance vector ~δ is defined as δH ~δ ≡ δ 45 δC (1.11) in a similar fashion to the diattenuation vector. The magnitude of the retardance vector is q 2 2 + δ45 + δC2 , δ = δH (1.12) and {1, ~δ/δ} is the Stokes vector of the retarder’s fast-axis. The ideal retarder has a Mueller matrix given by ~ 1 0T MR = ~0 mR (1.13) (mR )ij = δij cos δ + αi αj (1 − cos δ) + 3 X ijk ak sin δ (1.14) k=1 i, j = 1, 2, 3 where ijk is the Levi-Cività permutation symbol, ~0 is a three-element zero vector, and δij is the Kronocker delta. The depolarizing factor combines the depolarizing part of the Mueller matrix with polarizance, and is written as 1 M∆ ≡ P~∆ ~0T m∆ (1.15) ~ ∆ is the polarizance vector for a depolarizer, where m∆ is a symmetric matrix. P 23 defined as PH ~∆ ≡ P P 45 PR m10 1 = m00 m20 m30 . (1.16) The Lu-Chipman polar decomposition is very useful for calculating the nondepolarizing properties of a Mueller matrix, but yields less insight into the the depolarizing properties of a Mueller matrix. 1.4 Symmetric decomposition The symmetric decomposition (Ossikovski, 2009) method decomposes a depolarizing Mueller matrix into a sequence of five factors: a diagonal depolarizer M∆d in between two retarder and diattenuator pairs (MR and MD ). The resulting decomposition is written out as follows, where M = MD2 MR2 M∆d MR1 MD1 . (1.17) One advantage of this decomposition method is that it is relatively easy to calculate, since it is a singular value decomposition (SVD) based method. The first step is to solve these eigenvalue equations for S1 and S2 : MT GMG S1 = d20 S1 MGMT G S2 = d20 S2 (1.18) (1.19) 24 where G = diag(1, −1, −1, −1) is the Minkowski metric matrix, 1 S1 = , and ~ D1 1 S2 = . ~2 D (1.20) (1.21) ~ 1 and D ~ 2, The second step is to construct the two diattenuator matrices from D where ~T 1 D MD = Tu and ~ mD D √ √ mD = 1 − D2 I + (1 − 1 − D2 )D̂D̂T . (1.22) (1.23) The third step is to calculate an intermediate matrix M0 , which has null polarizance and diattenuation vectors: ~T d0 0 −1 M0 = M−1 MM = . D2 D1 ~0 m0 (1.24) The fourth step is to perform a SVD on the three by three block m0 of M0 to construct the retarder matrices. This is possible because of M0 s zero polarizance and diattenuation vectors: m0 = mR2 m∆d mTR1 . (1.25) 25 MR1 and MR2 are constructed from mR1 and mR2 as follows, where ~T 1 0 MR = . ~0 mR (1.26) This yields all the terms from equation 1.17. The symmetry of this decomposition is an obvious advantage in addition to the ease of calculation. The form is somewhat similar to that of a Mueller matrix polarimeter with some kind of depolarizing sample measured, if we were to replace the two diattenuators on each edge with polarizers. It is perhaps most useful in analyzing a scattering process that undergoes retardance and diattenuation before the scattering and again after the scattering. One disadvantage is that it can’t explicitly quantify the overall diattenuation or retardance of a system. 1.5 Additive decompositions According to the Cloude additive decomposition (Cloude, 1989; Ossikovski et al., 2008), any depolarizing Mueller matrix M can be written as a weighted sum of four non-depolarizing Mueller matrices, M = λ1 M1 + λ2 M2 + λ3 M3 + λ4 M4 , (1.27) where the λ0i s are the 4 eigenvalues of M’s covariance matrix C. The decomposition is performed as follows. The first step is to calculate the covariance matrix C from the elements mij of M and the Pauli spin matrices σi where C= 4 X i=1,j=1 mij (σi × σj∗ ). (1.28) 26 The second step is to calculate the eigenvalues λi and normalized eigenvectors ei of C, and calculate the covariance matrices Ci for the non-depolarizing Mueller matrix components Mi . The covariance matrices Ci are calculated from the eigenvectors ei , Ci = ei e†i . (1.29) The third and final step is to calculate each Mi from its Ci by inverting equation 1.28. This decomposition is useful for visualizing the behaviour of the coherency matrix eigenvalues, especially for mildly non-physical measured Mueller matrices. Physically, the decomposition corresponds to a system with multiple beams having different paths such as a focal plane with significant ghost images or a divided aperture polarization state generator. 1.6 Mueller matrix root decomposition While the polar decomposition methods discussed in the previous sections have been proven to be viable methods for Mueller matrix analysis, they have two major shortcomings. The first is the lack of order independence. A positive consequence of order independence is that the properties should not change according to the order of the decomposition. However, it is often useful to model a particular system such that there is an intuitive sequence of optical elements that are not in reality order independent. The more significant pitfall of the established decomposition methods is that none of these methods fully address the nine depolarization degrees of freedom. Jones developed an order independent representation for Jones matrices (or nondepolarizing Mueller-Jones matrices) using what’s referred to as a Jones N -matrix 27 representation (Jones, 1948). He used the analogy of propagation through a dichroic and birefringent anisotropic crystal, divided into very short lengths. He represented the N -matrices differentially. We approximate this differential representation as NJ ≈ p n MJ , (1.30) where NJ is the Jones matrix for a weak polarization element and MJ is an arbitrary Jones matrix. Our method is based on the idea of using infinitesimal Mueller matrices (or Mueller matrices expressing infinitesimal properties) to analyze and physically interpret the fundamental Mueller matrix properties. Taking the nth root of a Mueller matrix simplifies its properties as n → ∞ and the Mueller matrix becomes a weak element. The properties of the Mueller matrix near the identity matrix are discussed in chapter 3. The Jones N -matrix method is the building block for our Mueller matrix roots method. Implementation of the Jones N -matrix method is discussed in section 1.8. Section 1.7 describes the Pauli spin matrices, the basis for the Jones N-Matrix method. 1.7 Pauli spin matrices The Pauli spin matrices are used in quantum mechanics in order to describe angular momentum. They are written as 1 0 1 0 0 1 0 −i σ0 = , σ1 = , σ2 = , σ3 = . 0 1 0 −1 1 0 i 0 (1.31) 28 The Pauli spin matrices are a useful basis for Jones matrices. Within the Jones calculus, σ1 represents a horizontal retarder, σ2 represents a 45o linear retarder, σ3 represents a circular half-wave retarder, and σ0 (the identity matrix) represents empty space. Any Jones matrix J can be written in terms of the sum J= 3 X ck σk . (1.32) k=0 Here, the constants c0 , c1 , c2 , and c3 are complex Pauli coefficients for the Jones matrix. If the Jones matrix J is written in terms of the Pauli basis, the coefficients ck can be written in short form as 1 ck = T r (J • σi ) , 2 (1.33) where T r is the sum of the diagonal elements of the matrix. 1.8 Jones N-matrices and the Mueller matrix roots decomposition Jones’ N-matrix decomposition simplifies the properties of Jones matrices by dividing them into infinitesimaly small segments which approach the identity matrix. The N -matrices can then be expressed as a sum of matrices, where each matrix’s real and imaginary part in the sum corresponds to one fundamental property. For any non-depolarizing physical Mueller-Jones matrix which can be converted to a Jones matrix, this approach provides accurate and useful analytical results for experimentally or computationally generated Mueller matrices. Jones N-matrices are very useful for isolating the different non-depolarizing polarization effects. For the weak polarization element NJ from equation 1.30, the polarization properties can be determined from the Pauli basis, where first order 29 approximations lead to the relationship (Chipman, 2009) Dx + iδx D45 + iδ45 DR + iδR NJ ≈ ρ0 exp iδ0 σ0 + σ1 + σ2 + σ3 . 2 2 2 (1.34) This expression is correct to first order in D and δ. In this expression, DH and δH are the horizontal and vertical diattenuation and retardance, D45 and δ45 are the 135o and 45o diattenuation and retardance, and DR and δR are the right and left circular diattenuation and retardance. ρ0 is the amplitude, and δ0 is the global phase. Equation 1.34 allows for order-independent analysis of Jones matrices, or non-depolarizing Mueller matrices that can be converted to Jones matrices. This is particularly useful for non-homogeneous Jones matrices. The concept of N -matrices can be readily extended to Mueller matrices by dividing a Mueller matrix M into p infinitesimal slices by calculating its root: N= √ p M. (1.35) This extension arises naturally as follows. A train of n optical elements is described by multiplying their Mueller matrices from right to left, starting with the first element’s Mueller matrix M1 and ending with the last element’s Mueller matrix Mn , M = Mn Mn−1 ...M2 M1 , (1.36) for n = 1, 2, ....p. If each component on the right hand side of equation 1.36 is identical, it becomes M = Mn Mn−1 ...M2 M1 = (Mn )p . (1.37) 30 M p M ( M) p p Figure 1.1: Taking the root of a uniform Mueller matrix is analagous to slicing it into very thin identical pieces. Since compounding identical polarization elements in the Mueller matrix formalism is akin to raising its Mueller matrix to a power, taking a small slice of a polarization element is akin to taking its root, as illustrated in Fig. 1.1. The pth root of a large class of Mueller matrices approaches the identity matrix I in the limit as p becomes very large, lim p→∞ √ p M = I. (1.38) Chapter 2 addresses the optimal choice of p; values on the order of p ∼ = 105 work well. In this study, the class of Mueller matrices that obey equation 1.38 is referred to as uniform Mueller matrices. Our method is only applicable to this subset of Mueller 31 matrices. This dissertation does not address which subset of Mueller matrices is uniform, though Mueller matrices with negative real eigenvalues are not uniform (Higham, 2008). The polarization properties of √ p M for large p simplify as its properties become infinitesimal. Properties of Mueller matrices with infinitesimally small properties are considerably less complex than a Mueller matrix with non-infinitesimal properties. Performing a Taylor series expansion on the Mueller matrix forms for elliptical diattenuators and retarders provides insight into the symmetry and structure of weak polarization elements. The first order terms in the Taylor series expansion of the general equation for an elliptical retarder (Chipman, 2009) yield the following expression for a weak retarder in terms of the three-dimensional retardance vector {δH , δ45 , δR } (Lu and Chipman, 1996): 1 0 0 0 0 1 δR −δ45 2 2 ER(δH , δ45 , δR ) ∼ , δ45 , δR2 ). = + O(δH 0 −δ 1 δH R 0 δ45 −δH 1 (1.39) Similarly, the first order Taylor expansion terms for an elliptical diattenuator with diattenuation vector dH , d45 , dR and average transmision Tavg are 1 d d d H 45 R dH 1 0 0 ED(dH , d45 , dR ) ∼ = Tavg + Tavg O(d2H , d245 , d2R ). d 1 0 45 0 dR 0 0 1 (1.40) Combining the weak diattenuator and retarder yields an order independent equation 32 for a weak non-depolarizing Mueller matrix, 1 d d d H 45 R dH 1 δR −δ45 ER · ED + ED · ER = Tavg . d 2 −δ 1 δ 45 R H dR δ45 −δH 1 (1.41) The matrix is correct to first order in d and δ. The symmetry of the weak nondepolarizing matrices is striking - weak diattenuators are symmetric in the top row and first column, weak retarders are anti-symmetric between the off diagional lower and upper right three by three elements. Deviation from these symmetries indicates the presence of depolarization. This symmetry for infinitesimal Mueller matrix properties motivates what we refer to as Mueller matrix generators. Within the field of polarization, generators have been discussed in relating the Jones and Mueller matrix formalism to the concept of the Lie group (Takenaka, 1973). Generators have also been used to define possible expressions of an arbitrary nonsingular Mueller matrix, but not necessarily in 16-dimensional space (Devlaminck and Terrier, 2008). Here, Mueller matrix generators are used to express infinitesimal properties of the Mueller matrix (Chipman, 2009). Our generators provide an order-independent representation of uniform Mueller matrices to provide additional insights into the polarization properties of Mueller matrices. The Mueller matrix generators and their properties are discussed in greater detail in chapters 3 and 4. 33 CHAPTER 2 THE MUELLER MATRIX ROOTS DECOMPOSITION ALGORITHM AND COMPUTATIONAL CONSIDERATIONS This chapter is reformatted from a manuscript which has been submitted to Optics Express. 2.1 Introduction Polarization elements and their associated Mueller matrices are typically decomposed and analyzed in terms of the polarization properties of diattenuation, retardance, and depolarization. For Mueller matrices with a mixture of all three, the properties are distributed among the matrix elements in a complex manner. Lu and Chipman described a Mueller matrix data reduction method based on polar decomposition (Lu and Chipman, 1996). It decomposes the Mueller matrix into an order-dependent product of a depolarizer, a retarder, and a diattenuator. Ossikovski’s symmetric decomposition method decomposes a depolarizing Mueller matrix into a sequence of five factors: a diagonal depolarizer between two retarder and diattenuator pairs (Ossikovski, 2009). The diattenuation, retardance, and depolarization parameters calculated by these methods depend on the arbitrary order of the decomposition. The Cloude additive decomposition (Cloude, 1989) separates a Mueller matrix into the sum of four non-depolarizing Mueller matrices which are scaled by eigenvalues of the Mueller matrix covariance matrix. Since this is an additive decomposition, its terms are order-independent. None of these methods clearly 34 M p M ( M) p p Figure 2.1: Taking the root of a uniform Mueller matrix is analogous to slicing it into very thin identical pieces. elucidate the nine degrees of freedom associated with depolarization. Recently, a Mueller matrix roots decomposition was introduced by Chipman in (Chipman, 2009) to provide an order-independent description of polarization properties and provide a clear analysis of the nine depolarization degrees of freedom. The Mueller matrix roots decomposition extends the concept of the Jones N -matrices (Jones, 1948) to Mueller matrices by dividing the Mueller matrix M into p infinitesimal slices: N= √ p M, (2.1) as illustrated in Fig. 2.1, where N is the Mueller matrix for one small slice. When p √ becomes very large, the polarization properties of p M separate as the diattenuation, retardance, and depolarization become very small near the identity matrix. In the limit as p approaches infinity, the principal pth root of a large class of Mueller 35 matrices approaches the identity matrix I, lim p→∞ √ p M = I. (2.2) In this study, the class of Mueller matrices that obey equation 3.2 are called uniform Mueller matrices. The Mueller matrix roots decomposition from (Chipman, 2009) is applicable to this subset of Mueller matrices, with the exception of the non-uniform special cases highlighted in section 2.4. The calculation of the principal pth root of square matrices is an extensively studied subject (Higham, 2008; Bini et al., 2005; Guo, 2010), and numerical accuracy and noise are well understood within the field of numerical computing (Skeel and Keiper, 1993). In this study, these concepts are applied to the calculation of the Mueller matrix roots for the purpose of Mueller matrix decomposition. This paper reviews and updates the Mueller matrix roots decomposition from (Chipman, 2009), addresses several computational issues, and highlights several common non-uniform special cases that arise. Considerations for these computational issues and special cases are implemented in a matrix roots decomposition algorithm, which is applied in a statistical analysis of a large quantity of randomly generated physical Mueller matrices. 2.2 Matrix roots decomposition The goal of the Mueller matrix roots decomposition is to calculate the magnitudes of sixteen distinct properties of the Mueller matrix, including the nine depolarization degrees of freedom, for the set of uniform Mueller matrices. To calculate the matrix roots decomposition of a Mueller matrix M, a Mueller 36 matrix N with infinitesimal polarization properties is first calculated from the pth principal root of M, where p is some large integer (typically 105 ), N= √ p M. (2.3) The appropriate choice of p is discussed in detail in section 2.5. The infinitesimal polarization parameters d0 through d15 are defined from the symmetric and antisymmetric parts of N: d1 + d7 d2 + d8 d3 + d9 1 d1 − d7 1 − f13 d6 + d12 −d5 + d11 N = d0 , d − d −d + d 1 − f d + d 2 8 6 12 14 4 10 d3 − d9 d5 + d11 −d4 + d10 1 − f15 (2.4) where d13 , d14 , and d15 are solved from the first-order generator products in terms of the parameters f13 , f14 , and f15 : f14 − f13 2 2 = √ (−2f15 + f14 + f13 ) 3 1 = √ (f15 + f14 + f13 ). 6 d13 = (2.5) d14 (2.6) d15 (2.7) The infinitesimal polarization parameters d0 through d15 are rescaled by p to produce the matrix roots parameters D0 through D15 : Dn = p ∗ dn , n=0,1,2,...,15. D0 through D15 parameterize the sixteen degrees of freedom of M. (2.8) 37 There are three matrix roots parameters for diattenuation, D1 , D2 , D3 , and three matrix roots parameters for retardance, D4 , D5 , D6 . The three degrees of freedom for each property correspond to the axes in the Stokes/Mueller formalism (horizontal/vertical, 45o /135o , right/left circular). Nine more parameters, D7 through D15 , describe depolarizing effects - one for each of the nine depolarizing degrees of freedom described in (Chipman, 2009). There are three families of depolarizing parameters, each similarly divided into horizontal/vertical, 45o /135o , and right/left circular components. The first order terms for amplitude depolarization, D7 , D8 , and D9 , share the same matrix elements as the parts of the Mueller matrix associated to first order with diattenuation, on the horizontal/vertical, 45o /135o , and right/left circular axes (Chipman, 2009). They are named amplitude depolarization because they depolarize and affect the flux of an incident Stokes vector. The first order terms for phase depolarization, D10 , D11 , and D12 , correspond with the parts of the Mueller matrix associated to first order with retardance on the horizontal/vertical, 45o /135o , and right/left circular axes. They do not affect the flux of an incident Stokes vector. D13 , D14 , and D15 are named diagonal depolarization because they lie on the matrix diagonal. D15 expresses the overall isotropic depolarizing power, since D15 reduces the degree of polarization of all incident Stokes vectors equally, independent of the Stokes vector’s location on the Poincare sphere. D13 expresses the relative strength between the diagonal depolarization on the two linear axes (horizontal/vertical and 45o /135o ). D14 expresses the relative strength between linear and circular diagonal depolarization. The diagonal depolarization parameters have been modified from (Chipman, 2009) so that only the diagonal depolarization parameter D15 changes the depolarization index (Chipman, 2005a), while D0 through D14 do not. Table 2.1 lists the parameters D1 38 through D15 and categorizes them into their corresponding families and axes. Table 2.1: The sixteen polarization properties of the Mueller matrix given by the Mueller matrix roots decomposition. Property Diattenuation Retardance Amplitude Depolarization Phase Depolarization Diagonal Depolarization Horizontal/Vertical 45o /135o D1 D2 D4 D5 D7 D8 D10 D11 Relative Linear Relative Linear or Circular D13 D14 Right/Left Circular D3 D6 D9 D12 Isotropic D15 2.3 pth Root This section discusses the definition of the principal pth matrix root along with relevant examples and common methods of calculating the pth matrix root. Matrices have multiple roots. For a nonsingular matrix A ∈ Cn×n (in complex space) with s distinct eigenvalues, there are precisely ps pth roots (Higham, 2008). So long as the matrix A ∈ Cn×n has no negative, real eigenvalues, there is a unique pth root of A whose eigenvalues’ arguments lie between −π/p and π/p, and that unique root is defined as the principal root of A (Higham, 2008). If A is real, then its principal root A1/p is real. Singular matrices (such as polarizer Mueller matrices) do not have principal matrix roots. Nonetheless, matrix roots of singular Mueller matrices near the identity matrix can still be found. Methods for calculating roots of singular Mueller matrices (such as linear polarizers) are discussed in section 2.4.1. Methods for calculating roots of non-depolarizing and depolarizing Mueller matrices with π retardance are discussed in section 2.4.2. 39 2.3.1 Principal matrix root algorithms This section provides a brief summary of common methods for calculating principal matrix roots, including the Schur method, Newton’s method, and the Schur-Newton algorithm. This provides a starting point for the reader who is interested in applying these methods to the calculation of Mueller matrix roots. Schur methods form a Schur decomposition of A and compute a pth root of the resulting upper triangular factor using various (stable) recursive formulae (Higham, 1997). Newton’s method calculates the pth root of A using an iterative approach (Higham, 2008). The Newton method is largely considered to have poor convergence and stability properties (Smith, 2002), and the Schur method from (Smith, 2002) is the numerically stable benchmark against which other methods are often compared (Bini et al., 2005). The Schur-Newton algorithm applies iterative computations to the upper triangular matrices from the Schur decomposition (Higham, 2008). Many of these algorithms are available in the MATLAB Matrix Computation Toolbox (Bini et al., 2005). Mathematica has built-in routines that diagonalize a matrix to easily calculate its root, so long as the matrix is diagonalizable. If the matrix is not diagonalizable, its algorithm performs a singular value decomposition, which factors the matrix A into the product of three matrices, A = Zdiag(λi )Z−1 , (2.9) where λi are the eigenvalues of A, the columns of Z are its eigenvectors, and diag(λi ) is a diagonal matrix with its ith diagonal element equal to λi . Then the matrix root 40 of A is calculated as A1/p = Zdiag(λi )1/p Z−1 . (2.10) This method often (but not always) yields the principal root of a Mueller matrix, so long as its principal root exists. However if A is real and has some complex eigenvalues, then the computed A1/p , which should be real, may acquire a tiny imaginary part due to computational rounding errors. This imaginary part should be discarded. However, numerical instability can produce a large spurious imaginary part, so diagonalization-based computations of matrices with any imaginary eigenvalues should be treated with care (Higham, 2008). 2.4 Special cases 2.4.1 Polarizers The Mueller matrix for a polarizer is singular, regardless of ellipticity or orientation. Therefore it is not uniform, and the procedure described in section 2.2 will not yield meaningful matrix roots parameters when applied to a polarizer Mueller matrix. However, the perturbation treatment proposed in this section allows for this special case to be analyzed using the Mueller matrix roots decomposition. Because an ideal homogeneous polarizer (Lu and Chipman, 1994) is always idempotent, the matrix roots decomposition will break down at the point where the pth root is calculated. For example, the formula for a homogeneous linear polarizer as 41 a function of orientation θ is cos 2θ sin 2θ 0 1 1 2 cos 2θ cos 2θ sin 4θ 0 1 2 LP(θ) = . 2 sin 2θ 1 sin 4θ sin2 2θ 0 2 0 0 0 0 (2.11) LP(θ)LP(θ) = LP(θ), (2.12) LP(θ)1/p = LP(θ). (2.13) Note that and thus The linear polarizer Mueller matrix is its own root to all orders, and the roots do not approach the identity matrix. A homogeneous polarizer has two orthogonal, physical eigenpolarizations with eigenvalues (Tmax , 0), where Tmax is the maximum transmission, and the minimum transmission Tmin is 0. The polarizer can be perturbed to a nearby uniform Mueller matrix by adjusting the maximum and minimum transmission by a small number . The perturbed matrix is now uniform and it is a diattenuator with the same orthogonal eigenpolarizations, but the eigenvalues associated with its physical eigenpolarizations are (Tmax − , ). This is accomplished by calculating the latitude η and longitude θ of the state of maximum transmission on the Poincare sphere from the polarizer’s three dimensional diattenuation vector {dH , d45 , dR } (Lu and Chipman, 1996; Chipman, 2009), 42 where 1 dH arctan , and 2 d45 dR η = arcsin p 2 . dH + d245 + d2R θ= (2.14) (2.15) η and θ are then plugged in to the general formula for an elliptical diattenuator (Goldstein, 2003; Chipman, 2009). For an ideal polarizer, Tmax = 1 and Tmin = 0. Perturbing Tmax and Tmin by some small number yields a nonsingular, uniform Mueller matrix. Values of = 10−7 work well with Mathematica’s default numerical accuracy. The pth root of this perturbed polarizer approaches the identity matrix for large p, and the Mueller matrix roots decomposition algorithm can proceed as in the general case. 2.4.2 Half wave retarders Mueller matrix roots for retarders are easily calculated and understood. For two linear retarders with the same fast-axis orientation, retardance is additive, δ = δ1 + δ2 . (2.16) Thus the square root of a linear retarder is a retarder with half the retardance and the same fast-axis orientation. This is equivalent to cutting a wave plate in half. Similarly, the pth root of an ideal homogeneous linear retarder LR(δ, θ) with retardance δ at orientation θ is LR(δ, θ)1/p = LR(δ/p, θ). (2.17) 43 A retarder with a half-wave of retardance has negative, real eigenvalues, and therefore no principal root, so the half-wave retarder must be treated as a special case. For example, the half-wave linear retarder oriented at 0o , 1 0 o LR(π, 0 ) = 0 0 0 1 0 0 , 0 −1 0 0 0 −1 0 0 (2.18) has negative real eigenvalues (λ = {−1, −1, 1, 1}), so the half wave retarder has no principal pth root. For any value of p, two eigenvalues of HWR1/p are (−1)p with arguments of −π/p and π/p, which lie on the edge of the principal root segment defined in section 2.3. While the desired solution in this case is not a principal root, it is the solution which approaches the identity matrix. The half wave retarder’s uniform pth root can be calculated analogously to the polarizer case, by means of a small perturbation. Angle θ and latitude η can be calculated from the three-dimensional retardance vector {δH , δ45 , δR } (Lu and Chipman, 1996; Chipman, 2009) as follows, δH 1 arctan , and 2 δ45 δR η = arcsin p 2 . 2 δH + δ45 + δR2 θ= (2.19) (2.20) The general form for an elliptical retarder with retardance δ, orientation θ, and latitude η can be written in terms of three linear retarders as ER(η, θ, δ) = LR(η, θ + π/4)LR(δ, θ)LR(η, θ − π/4), (2.21) 44 where the general form for the linear retarder LR(δ, θ) with retardance δ and an orientation of θ is (Chipman, 2009) 0 0 0 1 2 2 δ 2 0 cos 2θ + cos δ sin 2θ − sin δ sin 2θ sin 2 sin 4θ . LR(δ, θ) = (2.22) 0 2 δ 2 2 sin 2 sin 4θ cos δ cos 2θ + sin 2θ cos 2θ sin δ 0 sin δ sin 2θ − cos 2θ sin δ cos δ After θ and η are calculated from the retardance vector, the half-wave retarder is perturbed by some small number ( ≈ 10−7 ) to a nearby elliptical retarder ER0 of retardance (π − ) by using equation 2.21: ER0 (η, θ, δ = π − ) = LR(η, θ + π/4)LR(π − , θ)LR(η, θ − π/4). (2.23) This is a unitary transformation - therefore any higher order root of a half wave retarder can be calculated as follows: 1 1 ER(η, θ, δ = π) p = LR(η, θ + π/4)LR((π − ), θ) p LR(η, θ − π/4). (2.24) After following this procedure, the matrix roots of the perturbed half-wave retarders can be calculated without issue. It is also possible to calculate the root in equation 2.24 without perturbation of the retardance, 1 π HWR(η, θ) p = LR(, θ + π/4)LR( , θ)LR(η, θ − π/4), p (2.25) but this method cannot be extended to a depolarizing Mueller matrix with a half- 45 wave of retardance. This is discussed further in the following section. 2.4.3 Depolarizing non-uniform Mueller matrices When half-wave retarders or polarizers are combined with other polarization properties, the resulting Mueller matrix is also non-uniform, and therefore cannot be analyzed with the standard Mueller matrix roots decomposition. Using the procedure discussed in this section, they can be perturbed to nearby uniform Mueller matrices. A depolarizing Mueller matrix that also has a half-wave of retardance has negative, real eigenvalues and is therefore non-uniform. For example, the product of a half wave retarder oriented at 0o and a partial diagonal depolarizer with positive, real a, b, and c 1 0 0 0 0 1 1 0 0 0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 1 a 0 0 0 = 0 b 0 0 0 0 0 c 0 a 0 0 , 0 −b 0 0 0 −c 0 0 (2.26) has eigenvalues λ = {1, a, −b, −c}, and therefore no principal root. To perturb a half-wave retarder mixed with depolarization and/or diattenuation properties to a nearby uniform Mueller matrix, the Lu-Chipman decomposition (Lu and Chipman, 1996) is first calculated as an intermediate step, M = M∆ MR MD . (2.27) M∆ is a depolarizing Mueller matrix, MR is a pure retarder, and MD is a pure 46 diattenuator. The half-wave retarder found in the MR term is perturbed to the nearby retarder M0R with (π − ) retardance as shown in equation 2.24. Then the terms are recombined to form M0 , where M0 = M∆ M0R MD . (2.28) The uniform pth root of the perturbed depolarizing half-wave retarder from equation 2.28 can be calculated from M0 , since its eigenvalues are no longer negative, and its Mueller matrix roots decomposition parameters can be found. Because the product of a polarizer and any other Mueller matrix is also singular, a polarizer multiplied by a depolarizer or retarder (or even a diattenuator) in any order is also singular. These Mueller matrices can be identified by testing for a diattenuation vector D of magnitude D = 1 and a depolarization index less than 1. To perturb a polarizer with a mixture of other polarization properties to a nearby uniform Mueller matrix, the Lu-Chipman decomposition (Lu and Chipman, 1996) is first calculated as an intermediate step, M = M∆ MR MD . (2.29) The polarizer’s maximum transmission can be perturbed by to the nearby uniform diattenuator according to the procedure in section 2.4.1. The modified M0D is substituted into equation 2.29 in place of MD , and the three matrices are recombined to form a modified, non-singular matrix M0 : M0 = M∆ MR M0D . (2.30) 47 Then the pth root of M0 can be calculated, and its Mueller matrix roots decomposition parameters can be found. 2.5 Numerical accuracy and root order 2.5.1 Choice of p The choice of root order p is an important consideration when calculating the matrix roots parameters. p must be large enough so that the Mueller matrix elements become sufficiently small for the polarization properties to separate. However, the magnitude of p should also be as small as possible so as to minimize unnecessary loss of numerical precision from the calculation. 2.5.1.1 Accuracy Because the Mueller matrix roots of retarders are straightforward and wellunderstood, they provide a good reference from which to study the convergence of Equation 2.8. The pth root of a retarder Mueller matrix with retardance δ results in a pth principal matrix root with retardance δ/p. Therefore a Mueller matrix of an ideal elliptical retarder with retarder vector {δH , δ45 , δR } should have matrix roots parameters of D4 = δH , D5 = δ45 , and D6 = δR . In order to evaluate the correct choice of p in consideration of this criteria, the matrix roots retardance parameters (D4 through D6 ) were calculated (using Mathematica’s default doubleprecision machine arithmetic) for a pure elliptical retarder with retardance parameters {δH , δ45 , δR } = {0.294, 0.302, 0.997} for different values of root order p. The relative error ∆x was then calculated according to the following expression, p (δH − D4 )2 + (δ45 − D5 )2 + (δR − D6 )2 ∆x = . δ (2.31) 48 Convergence HPure RetarderL Relative Error HLogL -12 -14 -16 -18 -20 -22 -24 -26 0 2 4 6 8 10 12 Log10 HpL Figure 2.2: The error of the root calculation vs. root order for a simple retarder converges to a minimum relative error just beyond the 105 th root. Fig. 2.2 shows the relative error between the input retardance vector {δH , δ45 , δR } and the corresponding matrix roots retardance vector D4 through D6 for different choices of p for the pth root. In Fig. 2.2, the relative retardance error converges to a minimum value just beyond the 105 th root. For large p, the relative error increases, due to numerical rounding and the loss of precision associated with machine arithmetic. An equally important factor to consider is the convergence of the Mueller matrix root parameter values. When the choice of p is sufficiently large, the D-parameters converge to values that are independent of p. In order to demonstrate this convergence, a more complex Mueller matrix was generated by multiplying an elliptical retarder of randomly generated input retardance vector by a partial depolarizer PD of the form 1 0 PD(a, b, c) = 0 0 0 0 0 a 0 0 . 0 b 0 0 0 c (2.32) 49 Diagonal Depolarization Vector Norm Convergence HDepolarizing RetarderL 3.10 3.05 3.00 2.95 2.90 2.85 0 2 4 6 8 10 12 Log10 HpL Figure 2.3: The norm of the depolarizing retarder’s diagonal depolarization parameters (D13 , D14 , and D15 ) converges to a steady value of approximately 3.03 beyond the 104 th root. The randomly generated elliptical retarder with retardance vector {δH , δ45 , δR } = {0.210, 0.033, 1.003} was multiplied by PD(0.1, 0.2, 0.3), generating the Mueller matrix 0.494 0.010 0.003 0.419 0.104 0.002 0.0 0.029 ED(0.210, 0.033, 1.003) · PD(0.1, 0.2, 0.3) = . 0.016 0.0 0.0 0.005 0.495 0.010 0.003 0.142 (2.33) Its Mueller matrix root decomposition parameters were then calculated for varying values of p. Fig. 2.4 shows the convergence of the norm of the matrix roots retardance parameters (D4 , D5 , and D6 ). The norm of the matrix roots retardance parameters converges to a steady value of approximately 1.07 following the 104 th root. The convergence of the norm of the matrix roots diagonal depolarization parameters (D13 , D14 , and D15 ) behaves in a strikingly similar manner, as shown in Fig. 2.3. 50 Convergence HDepolarizing RetarderL Retardance Vector Norm 1.10 1.05 1.00 0.95 0.90 0 2 4 6 8 10 12 Log10 HpL Figure 2.4: The norm of the depolarizing retarder’s matrix roots retardance parameters (D4 , D5 , and D6 ) converges to a steady value of approximately 1.07 beyond the 104 th root. 2.5.1.2 Numerical precision This section addresses issues related to the numerical precision of the matrix root calculation, using Mathematica’s built-in commands. Computational software programs such as Mathematica and Matlab use floating-point numbers with machine precision by default (Skeel and Keiper, 1993). The value of machine precision that produced the results included here is 15.96 digits, which corresponds to a 53 digit binary double precision number with a mantissa (Skeel and Keiper, 1993). Rounding errors have been shown to increase linearly so long as small-scale oscillations are overlooked (Skeel and Keiper, 1993). Fig. 2.5 shows the relative error (on a log scale) of the matrix root calculation on a randomly generated Mueller matrix MR as a function of root order p = 10k , for k between 1 and 14. This relative error was calculated as follows, ∆root p √ kMR − p MR k . = 2kMR k (2.34) The relative error increases linearly with each operation, so it is recommended to 51 Matrix Root Computational Error Relative Error HLogL 0 -5 -10 -15 -20 -25 -30 2 4 6 8 10 12 14 Log10 p Figure 2.5: Relative error for Mathematica’s matrix root calculation. balance this with the optimization of the convergence properties discussed in section 2.5.1.1. The linear increase in error with each numerical operation is independent of the choice of Mueller matrix. Based on the behaviors documented in Fig. 2.2 through 2.5, a good choice for p is on the order of 105 . This choice balances the relative error generated from the root calculation while achieving convergence of its matrix root polarization properties. Relative error of 5 · 10−12 is achieved with p = 105 for the randomly generated Mueller matrix MR . 2.6 Algorithm and flow chart An algorithm to calculate the matrix roots decomposition incorporating the results from Sections 2.4 and 2.5 follows. Fig. 2.6 shows a flow chart for the algorithm, beginning with an input Mueller matrix. If the Mueller matrix is not physical, the decomposition parameters will not be meaningful. Since many measured Mueller matrices are slightly nonphysical (Goldstein, 2003), if the matrix is nonphysical, it is recommended to use a forcephysical routine (Barakat, 1987; Twietmeyer et al., 2008) in Step 2. Step 3 tests 52 step 1: input Mueller matrix step 2: force physical step 8: mul:ply d0 through d15 by p=105 to obtain D0 through D15 step 3: check for special cases* step 4: check for yes nega:ve eigenvalues abort step 7: solve for d0 through d15 using equa:on 2 no step 5: calculate 105th matrix root step 6: drop spurious imaginary parts *Address linear polarizers and half-‐wave retarders according to sec:on 4 and proceed to step 4. Figure 2.6: Matrix Roots algorithm flow chart. 53 for the special cases discussed in section 2.4 - Mueller matrices with a half wave of retardance and singular Mueller matrices. If the Mueller matrix tests positive it is perturbed according to the procedure in section 2.4, and the algorithm resumes. Step 4 tests for negative real eigenvalues, and if present, the decomposition is aborted. Step 5 calculates the 105 th matrix root of the Mueller matrix. (This choice of p = 105 was discussed in section 2.5.) The matrix root can be calculated using a built-in matrix root algorithm (such as Mathematica’s matrix power function), or with any principal matrix root algorithm such as those discussed in section 2.3.1. The principal root algorithms discussed in section 2.3.1 can fail to converge to a solution, particularly for such a high root order. As discussed in section 2.3.1, for any matrix with imaginary eigenvalues, computational rounding errors can lead to spurious imaginary parts. These may be discarded so long as they are small enough not to affect the accuracy of the calculation. Step 6 discards these spurious imaginary parts. In step 7, the parameters d0 through d15 are determined from the principal root according to equation 2.4. Step 8 rescales the infinitesimal roots parameters by p = 105 , resulting in the desired matrix roots parameters D0 through D15 . 2.7 Statistical algorithm implementation The matrix roots decomposition algorithm from section 2.6 was implemented on 76,336 randomly generated, non-singular, physical Mueller matrices with no real negative eigenvalues. By definition, all of these Mueller matrices have principal roots. The statistical analysis resulting from this implementation yields information about the range in values of the depolarizing matrix roots parameters for a large quantity of random Mueller matrices. 54 Amplitude Depolarization Matrix Root Parameters 8000 6000 D7 D8 D9 4000 2000 0 -2 -1 0 1 2 Parameter Value Figure 2.7: A histogram of the matrix root amplitude depolarization values for 76,336 randomly generated physical Mueller matrices. The Mueller matrices are generated by a brute-force numerical method. A fourby-four matrix is generated by setting the m0,0 element to a value of one, and the other fifteen matrix elements are uniformly randomly distributed between negative one and one. If the matrix is nonphysical or has negative real eigenvalues, it is discarded. The Mueller matrix roots parameters were calculated for all of the remaining matrices. 76,336 physical, non-singular Mueller matrices with no real negative eigenvalues were found from the set of 109 randomly generated matrices. Fig. 2.7 through 2.9 show the distribution of the depolarizing matrix roots parameters. The distributions for the amplitude depolarization parameters (D7 , D8 , and D9 ) are entirely overlapping and largely lie within the range of -1 to 1, with a fullwidth half maximum (FWHM) of 0.6. The distributions for the phase depolarization parameters (D10 , D11 , and D12 ) also overlap entirely and range mostly between 2 and 2, with a FWHM of 0.8. The distributions of the diagonal depolarization 55 Phase Depolarization Matrix Root Parameters 10 000 8000 6000 D10 D11 D12 4000 2000 0 -4 0 -2 2 4 Parameter Value Figure 2.8: A histogram of the matrix root phase depolarization values for 76,336 randomly generated physical Mueller matrices. Diagonal Depolarization Matrix Root Parameters 10 000 8000 6000 D13 D14 D15 4000 2000 0 -4 -2 0 2 4 6 Parameter Value Figure 2.9: A histogram of the matrix root diagonal depolarization values for 76,336 randomly generated physical Mueller matrices. 56 parameters D13 and D14 overlap and have a very similar distribution to the phase depolarization parameters, with a range primarily between -2 and 2 and FWHM of 0.8. D15 has a distinct distribution. It has a hard limit at zero, as it cannot have a negative value. Its distribution cuts off near 4, with a FWHM of 1.1. Out of all the depolarizing matrix roots parameters, it has the only non-symmetric distribution. 2.8 Conclusion Computational issues involved in applying the Mueller matrix roots decomposition have been addressed. The definition of the pth matrix root is reviewed, along with a brief discussion of the most common methods of calculating the pth principal matrix root. Our study indicates that the decomposition is optimized around p = 105 in consideration of numerical accuracy and noise as well as parameter convergence. Practical values for the roots of singular Mueller matrices can be obtained through perturbing them to nearby diattenuating matrices. Similarly, Mueller matrices with a half wave of retardance can be evaluated by perturbing their retardance from a half wave, without changing the retarder form. An algorithm is provided which incorporates the computational considerations involved in calculating the matrix roots decomposition. Finally, the algorithm is implemented to perform a statistical analysis on a large set of randomly generated Mueller matrices in order to yield insight on the typical ranges of the matrix roots parameters for physical Mueller matrices. 57 CHAPTER 3 INTERPRETATION OF MUELLER MATRIX ROOTS DECOMPOSITION PARAMETERS This chapter is reformatted from a manuscript which has been submitted to Applied Optics. 3.1 Introduction Polarization elements and their associated Mueller matrices are typically decomposed and analyzed in terms of the polarization properties of diattenuation, retardance, and depolarization. For Mueller matrices with a mixture of all three, the properties are distributed among the matrix elements in a complex manner. Lu and Chipman described a Mueller matrix data reduction method based on polar decomposition (Lu and Chipman, 1996). It decomposes the Mueller matrix into an orderdependent product of a depolarizer, a retarder, and a diattenuator. Ossikovski’s symmetric decomposition method decomposes a depolarizing Mueller matrix into a sequence of five factors: a diagonal depolarizer between two retarder and diattenuator pairs (Ossikovski, 2009). The diattenuation, retardance, and depolarization parameters calculated by these methods depend on the arbitrary order of the decomposition. The Cloude additive decomposition (Cloude, 1989) separates a Mueller matrix into the sum of four non-depolarizing Mueller matrices which are scaled by eigenvalues of the Mueller matrix covariance matrix. Since this is an additive decomposition, its terms are order-independent. None of these methods clearly elucidate 58 the nine degrees of freedom associated with depolarization. Recently, a Mueller matrix roots decomposition was introduced by Chipman in (Chipman, 2009) to provide an order-independent description of polarization properties and a clear analysis of the nine depolarization degrees of freedom. Chipman’s matrix roots decomposition classifies depolarizing Mueller matrices into three families of depolarization, each with three degrees of freedom. A brief background of the Mueller matrix roots decomposition from (Chipman, 2009) is presented in section 3.2, followed by a definition of its nine depolarizing parameters in section 3.3. In section 3.4, degree of polarization maps are used to differentiate the three families of depolarization and demonstrate the unity between two of these families. Section 3.5 discusses the generation of depolarization via the averaging of non-depolarizing Mueller matrices, which is then extended in section 3.6 to provide insight on the forms of depolarization which occur in two different scattering measurements. 3.2 Mueller matrix roots decomposition Jones developed an order-independent representation for Jones matrices based on his N -matrix representation. He used the analogy of propagation through a dichroic and birefringent anisotropic crystal, divided into very short lengths, where the N matrices represented differential components (Jones, 1948). Recently, Ossikovski extended Azzam’s differential matrix formalism (Azzam, 1978) to include depolarizing media (Ossikovski, 2011). We extend the concept of the Jones N -matrices to Mueller matrices by dividing 59 M p M ( M) p p Figure 3.1: Taking the root of a uniform Mueller matrix is analagous to slicing it into very thin identical pieces. a Mueller matrix M into p infinitesimal slices: N= √ p M, (3.1) as illustrated in Fig. 3.1, where N is the Mueller matrix for one small slice. The principal pth root of a large class of Mueller matrices approaches the identity matrix I in the limit as p becomes very large, lim p→∞ √ p M = I. (3.2) We describe the class of Mueller matrices that obey equation 3.2 as uniform Mueller matrices. Our decomposition method is applicable to this subset of Mueller matrices. √ For large p, the matrix root p M of a uniform Mueller matrix approaches identity matrix, and its polarization properties separate. Mueller matrices near the identity 60 matrix are parameterized in terms of fifteen infinitesimal Mueller matrix generators and parameters, G1 (d1 ) through G15 (d15 ), and infinitesimal transmittance d0 (Chipman, 2009): √ p M = e−d0 15 Y ! Gi (di ) . (3.3) i=1 Thus a uniform Mueller matrix M can be decomposed to yield its polarization properties in terms of the sixteen matrix roots parameters D0 through D15 : M = e−D0 15 Y i=1 Gi Di p !p . (3.4) D0 is a transmittance factor, D1 through D3 are diattenuation parameters and D4 through D6 are retardance parameters. D7 through D15 are the nine depolarization parameters, which are discussed in more detail in section 3.3. The fifteen Mueller matrix generators Gi and their first order Taylor series approximations are listed in Tables 3.1 and 3.2. 3.3 Definition of diagonal, phase, and amplitude depolarization Nine depolarization parameters, D7 through D15 , describe depolarizing effects one for each of the nine depolarizing degrees of freedom (Lu and Chipman, 1996). The depolarizing parameters are divided into three families. Each family has three degrees of freedom corresponding to the Stokes parameter axes - horizontal/vertical, 45o /135o , and right/left circular. The first order terms for amplitude depolarization, D7 , D8 , and D9 , share the same off-axis matrix elements as the parts of the Mueller matrix associated with diattenuation on the horizontal/vertical, 45o /135o , and right/left circular axes (Chipman, 2009). They are named amplitude depolarization because they depolarize and 61 Table 3.1: The non-depolarizing Mueller matrix generators G1 (d1 ) through G6 (d6 ) and their first order Taylor series approximations. Number 1 2 3 4 5 6 Generator d1 0 0 1 p 0 0 0 1 − d21 p 0 1 − d21 0 0 0 d 0 2 p 0 1 − d22 0 0 1 p 0 0 0 1 − d22 0 d3 p 0 1 − d23 p 0 0 2 1 − d3 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 cos d4 sin d4 0 0 − sin d4 cos d4 1 0 0 0 0 cos d5 0 − sin d5 0 0 1 0 0 sin d5 0 cos d5 1 0 0 0 0 cos d6 sin d6 0 0 − sin d6 cos d6 0 0 0 0 1 1 d1 0 0 1 0 d2 0 1 0 0 d 3 First-Order Form 1 d1 0 0 d1 1 0 0 0 0 1 0 0 0 0 1 1 0 d2 0 0 1 0 0 d2 0 1 0 0 0 0 1 1 0 0 d3 0 1 0 0 0 0 1 0 d 0 0 1 3 1 0 0 0 0 1 0 0 0 0 1 d4 0 0 −d4 1 1 0 0 0 0 1 0 −d5 0 0 1 0 0 d5 0 1 1 0 0 0 0 1 d6 0 0 −d6 1 0 0 0 0 1 affect the flux of an incident Stokes vector. The first order terms for phase depolarization, D10 , D11 , and D12 , share the same off-axis elements as the parts of the Mueller matrix associated with retardance on the horizontal/vertical, 45o /135o , and right/left circular axes. They do not affect the flux of an incident Stokes vector. D13 , D14 , and D15 are named diagonal depolarization because they lie on the matrix diagonal. D15 expresses the overall isotropic depolarizing power. D15 reduces 62 Table 3.2: The depolarizing Mueller matrix generators G7 (d7 ) through G15 (d15 ) and their first order Taylor series approximations. Number Generator 1 d7 0 0 −d7 1 0 0 p 0 0 1 − d27 p 0 0 0 0 1 − d27 1 p 0 d8 0 2 0 0 0 1 − d 8 −d8 0 1 p 0 0 0 0 1 − d28 1 p 0 0 d9 0 0 1 − d29 p 0 0 0 1 − d29 0 −d 0 0 1 9 1 0 0 0 0 1 0 p 0 0 0 1 − d210 p d10 2 0 0 d10 1 − d10 1 p 0 0 0 2 0 1 − d11 0 d11 0 0 1 p 0 2 0 d11 0 1 − d11 1 p 0 0 0 2 0 1 − d12 p d12 0 0 d12 1 − d2 0 7 8 9 10 11 12 12 0 0 0 1 1 0 0 p0 0 −d13 + 1 − d213 /2 0 p0 0 0 d13 + 1 − d213 /2 p 0 2 0 0 0 1 − d13 0 0 p0 d√14 2 + 1 − 2d14 /3 0 0 3 p d√14 2 0 + 1 − 2d14 /3 0 3 p 2 √14 + 0 0 − 2d 1 − 2d /3 14 3 1 0 0 0 q 2 0 0 0 1 − 3 d15 q 2 0 0 1 − 3 d15 0 q 0 0 0 1 − 23 d15 13 14 15 1 0 0 0 1 0 0 0 First-Order Form 1 d7 0 0 −d7 1 0 0 0 0 1 0 0 0 0 1 1 0 d8 0 0 1 0 0 −d8 0 1 0 0 0 0 1 1 0 0 d9 0 1 0 0 0 0 1 0 −d9 0 0 1 1 0 0 0 0 1 0 0 0 0 1 d10 0 0 d10 1 1 0 0 0 0 1 0 d11 0 0 1 0 0 d11 0 1 1 0 0 0 0 1 d12 0 0 d12 1 0 0 0 0 1 0 0 1 − d13 0 0 1 + d13 0 0 0 0 0 1 1 0 √ 0 0 0 1 + d14 / 3 0 √ 0 0 0 1 + d14 / 3 0 √ 0 0 0 1 − 2d14 / 3 1 0 0 0 q 2 0 0 0 1 − 3 d15 q 2 0 0 1 − 3 d15 0 q 0 0 0 1 − 23 d15 the degree of polarization of all incident Stokes vectors equally, independent of the Stokes vector’s location on the Poincare sphere. All depolarizing Mueller matrices parameterized by equation 3.4 must contain a component of isotropic depolarization D15 . D13 expresses the relative strength between the diagonal depolarization on the two linear axes (horizontal/vertical and 45o /135o ). D14 expresses the relative 63 strength between linear and circular diagonal depolarization. These properties arise from the structure of G13 , G14 , and G15 in Table 3.2. Isotropic depolarization is explained in more detail in section 3.4. 3.4 Degree of polarization maps of depolarizing Mueller matrices Degree of polarization (DoP) maps provide a tool to understand the characteristics of the three families of depolarization. They represent the variation in DoP of the exiting polarization state as a function of the incident polarization state’s orientation θ and latitude φ on the Poincare sphere (Deboo et al., 2004): m0,0 m1,0 DoP(MS(θ, φ)) = DoP m 2,0 m3,0 m0,1 m0,2 m0,3 1 cos 2θ cos φ m1,1 m1,2 m1,3 . m2,1 m2,2 m2,3 sin 2θ cos φ m3,1 m3,2 m3,3 sin φ 0≤θ≤π − π π ≤φ≤ (3.5) 2 2 Typically, a depolarizing Mueller matrix depolarizes different incident polarization states by different amounts (Chipman, 2009). Comparison of the DoP maps for each form of depolarization indicates both the differences and similarities between the nine depolarization degrees of freedom. Fig. 3.2 shows the DoP maps for nine physical, depolarizing Mueller matrices - three amplitude depolarizer Mueller matrices, three phase depolarizer Mueller matrices, and three diagonal depolarizer Mueller matrices. The Mueller matrix for an isotropic diagonal depolarizer, M15 , was generated by 64 Figure 3.2: Degree of polarization maps for depolarizing Mueller matrices M7,15 through M15 . 65 taking the isotropic diagonal depolarization generator G15 to the pth power, M15 p Di = G15 . p (3.6) M15 has the effect of uniformly decreasing the DoP of an incident Stokes vector, independent of its location on the Poincare sphere, moving the exiting Stokes vector uniformly towards the center of the Poincare sphere. For this reason, M15 ’s DoP map in Fig. 3.2 has a constant DoP, which varies with D15 . Amplitude, phase, and non-isotropic diagonal depolarizer Mueller matrices were generated by multiplying the depolarizing generators G7 through G14 with the isotropic depolarization generator G15 and taking their product to the pth power, Mi,15 p Di Di = Gi G15 . p p i = 7, 8, ...14 (3.7) Without some isotropic depolarization D15 , the depolarizing Mueller matrices M7−14,15 would not be physical. Therefore the values for Di (i = 7, 8, ...14) were arbitrarily selected to be Di = 0.2, and D15 was selected to be D15 = 0.5. The amplitude depolarizers (M7−9,15 ) have two types of behavior distinct from phase or diagonal depolarizers. First, since they have components in the top row of the Mueller matrix, they alter the output intensity in addition to affecting the degree of polarization. Second, their DoP maps have one maximum and one minimum at orthogonal polarization states, lying on opposite ends of an axis through the Poincare sphere. Three degrees of freedom are needed to establish this behavior, including one for the ratio of minimum to maximum degree of polarization and two to establish a point on the sphere. 66 The phase and non-isotropic diagonal depolarizers (M10−14,15 ) have three behaviors that distinguish them from the amplitude depolarizers. First, they do not affect the intensity of a transmitted beam. Second, their DoP maps each have two maxima and two minima, with the exception of M14,15 . The DoP map for M14,15 has two minima - one at the top and bottom of the Poincare sphere, at right and left circular polarizations. Its maximum spans the entire equator of the Poincare sphere. Third, points lying on opposite ends of any axis of the Poincare sphere (or any pair of orthogonal polarization states) have equal degree of polarization. The non-isotropic diagonal depolarizers M13,15 and M14,15 represent the three degrees of freedom for an arbitrary amount of depolarization along each of the three Stokes axes on the Poincare sphere. G13 and G14 account for two degrees of freedom, while the third degree of freedom originates from G15 . The axes formed by the non-isotropic depolarizers along the Stokes axes of the Poincare sphere can be called the Principal Depolarization Axes. These three degrees of freedom are independent, though they are restricted by additive inequalities in order to be physically realizable. The phase depolarizers M10−12,15 add the degrees of freedom necessary to move the Principal Depolarization Axes to arbitrary locations on the Poincare sphere. Two degrees of freedom are required to specify the point of maximum DoP, and the remaining degree of freedom specifies the axis of minimum degree of polarization. Therefore we can assert that phase and diagonal depolarization can result from the same physical processes, occuring at different orientations. 67 3.5 Depolarization generation and cyclic permutations 3.5.1 Cyclic permutations In section 3.5.2, depolarization is analyzed via the averaging of non-depolarizing Mueller matrices, and the three families are shown to be governed by cyclic permutations. This section introduces these cyclic permutations. The Pauli spin matrices and identity matrix σ0 0 −i 0 1 1 0 1 0 σ0 = σ3 = σ2 = σ1 = i 0 1 0 0 −1 0 1 (3.8) form a basis for the Jones matrices. As Jones matrices, σ1 represents a horizontal half-wave retarder, σ2 represents a 45o linear half-wave retarder, σ3 represents a circular half-wave retarder, and σ0 represents empty space. The Pauli spin matrices obey the recursion relation σi σj = i,j,k ii,j,k σk . +1 (i, j, k) = (1, 2, 3), (3, 1, 2), (2, 3, 1) = −1 (i, j, k) = (1, 3, 2), (3, 2, 1), (2, 1, 3) 0 i=j j=k k=i (3.9) For example, a muliplicative cascade of a horizontal half-wave retarder σ1 and the 45o half-wave retarder σ2 results in the circular half-wave retarder σ3 . A similar cyclic permutation occurs for multiplicative combinations of nondepolarizing Mueller matrices. For example, multiplying the Mueller matrix for a pure linear, horizontal, half-wave retarder (RH/V (δ = π)) and a pure linear, 45o , half-wave retarder (R45o /135o (δ = π)) yields a pure circular half-wave retarder 68 (RR/L (δ = π)): RR/L (π) = R45o /135o (π) · RH/V (π). (3.10) Other sequences of retarder Mueller matrices oriented along the Stokes axes obey a similar cyclic permutation. 3.5.2 Generation of depolarization via averaging of non-depolarizing Mueller matrices In this section, the families of depolarization are analyzed in terms of sums of nondepolarizing Mueller matrices. Given two non-depolarizing Mueller matrices M1 and M2 and β between 0 and 1, the matrix averaging operation MA = βM1 + (1 − β)M2 (3.11) represents a divided aperture, and generates a depolarizing matrix MA . These averaging combinations show a relationship between non-depolarizing Mueller matrices and the different families of depolarization, and provides insight on the forms of depolarization which occur in scattering processes. Consider the averaging operation on all forty-nine combinations of two nondepolarizing matrices from Table 3.3, followed by a Mueller matrix roots decomposition to identify the resulting forms of depolarization. Table 3.4 shows the depolarizing matrix roots parameters D7 through D15 which result from the linear combinations of our basis set of non-depolarizing Mueller matrices. Every linear combination generates diagonal depolarization, emphasizing the relative importance of these degrees of freedom. Table 3.5 shows only the amplitude and phase depo- 69 Table 3.3: Notation for the basis diattenuator and retarder Mueller matrices oriented along the three Stokes axes (horizontal/vertical, 45o /135o and right/left circular), as well as an attenuating identity matrix. q and r are the maximum and minimum transmission, q 6= r, δ is the magnitude of the retardance vector, and α is the attenuating coefficient. αI DH/V (q, r) D45o /135o (q, r) DR/L (q, r) RH/V (δ) R45o /135o (δ) RR/L (δ) Attenuator Horizontal/Vertical Linear Diattenuator 45o /135o Linear Diattenuator Circular Diattenuator Horizontal/Vertical Linear Retarder 45o /135o Linear Retarder Circular Retarder larization generated by these linear combinations. Both tables are symmetric about the diagonal, since matrix addition commutes. For example, averaging the diattenuator DH/V (q, r) and retarder R45o /135o (δ) yields D9 (right/left circular amplitude depolarization). Conversely, circular amplitude depolarization D9 is also generated by averaging RH/V (δ)) and D45o /135o (q, r). Note that the coupling into depolarization follows the cyclic permutation rules. Averaging a diattenuator oriented along one Stokes axis with a retarder oriented along a different Stokes axis yields amplitude depolarization (D7 , D8 , D9 ) oriented along the third Stokes axis. Alternatively, averaging the two diattenuators DH/V (q, r) and D45o /135o (q, r) yields circular phase depolarization D12 , as does the average of the two retarders RH/V (δ) and R45o /135o (δ). Averaging two diattenuators or two retarders oriented along two different Stokes axes generates phase depolarization (D10 , D11 , or D12 ) oriented along the third Stokes axis. The resulting phase depolarization observes the same cyclic permutation rules. Many depolarizing optical components and systems are optically equivalent to 70 parallel combinations of non-depolarizing optical systems (Gil and Bernabeu, 1985). While the combinations summarized in Tables 3.4 and 3.5 are not unique, they describe rules for the generation of depolarization from spatially inhomogeneous surfaces, where different regions have different polarization properties. These spatial inhomogeneities of polarization properties produce depolarization (Gil, 2000). Table 3.4: Depolarization properties (shown by parameters D7 through D15 ) produced by averaging two non-depolarizing Mueller matrices. Matrix αI DH/V (q, r) D45o /135o (q, r) DR/L (q, r) RH/V (δ) R45o /135o (δ) RR/L (δ) αI D15 ,D14 ,D13 D15 ,D14 ,D13 D15 D15 ,D14 ,D13 D15 ,D14 ,D13 D15 DH/V (q, r) D15 ,D14 ,D13 D15 ,D14 ,D12 D15 ,D14 ,D13 ,D11 D15 ,D14 ,D13 D15 ,D14 ,D13 ,D9 D15 ,D14 ,D13 ,D8 D45o /135o (q, r) D15 ,D14 ,D13 D15 ,D14 ,D12 D15 ,D14 ,D13 ,D10 D15 ,D14 ,D13 ,D9 D15 ,D14 ,D13 D15 ,D14 ,D13 ,D7 DR/L (q, r) D15 D15 ,D14 ,D13 ,D11 D15 ,D14 ,D13 ,D10 D15 ,D14 ,D13 ,D8 D15 ,D14 ,D13 ,D7 D15 RH/V (δ) D15 ,D14 ,D13 D15 ,D14 ,D13 D15 ,D14 ,D13 ,D9 D15 ,D14 ,D13 ,D8 D15 ,D14 ,D12 D15 ,D14 ,D13 ,D11 R45o /135o (δ) D15 ,D14 ,D13 D15 ,D14 ,D13 ,D9 D15 ,D14 ,D13 D15 ,D14 ,D13 ,D7 D15 ,D14 ,D12 RR/L (δ) D15 D15 ,D14 ,D13 ,D8 D15 ,D14 ,D13 ,D7 D15 D15 ,D14 ,D13 ,D11 D15 ,D14 ,D13 ,D10 D15 ,D14 ,D13 ,D10 Table 3.5: Non-diagonal depolarization properties (shown by parameters D7 through D12 ) produced by averaging two non-depolarizing Mueller matrices. Matrix αI DH/V (q, r) D45o /135o (q, r) DR/L (q, r) RH/V (δ) R45o /135o (δ) RR/L (δ) αI DH/V (q, r) D12 D11 D9 D8 D45o /135o (q, r) DR/L (q, r) D12 D11 D10 D10 D9 D7 D8 D7 RH/V (δ) R45o /135o (δ) RR/L (δ) D9 D9 D8 D12 D11 D7 D12 D8 D7 D11 D10 D10 3.6 Experimental samples Two sample measurements are analyzed with the Mueller matrix roots decomposition, providing insight into their depolarization properties. The samples were both 71 10o Incident Angle Sca7ering Angles -‐70o 70o 90o -‐90o Sample Plane Figure 3.3: Measurement geometry for the University of Arizona scattering infrared polarimeter. measured with Mueller matrix imaging polarimeters in a scattering configuration. Their depolarization properties are consistent with those which can be generated by the combinations of non-depolarizing spatial inhomogeneities from Tables 3.4 and 3.5. 3.6.1 Ground glass The first example is the Mueller matrix for a sample of ground glass. Polarized light incident onto a sample of ground glass experiences multiple reflections at unpredictable angles due to its rough, irregular surface, and scattering from the ground glass results in depolarization effects. The ground glass was measured at 1550 nm with the University of Arizona scattering infrared polarimeter (Noble et al., 2007). The measurement configuration 72 labels the surface normal as 0o . Positive scattering angles are measured from the surface normal in the forward scattering direction. Negative incident angles are measured from the surface normal in the backscattering direction. The measurement geometry is illustrated in Fig. C.5. Two angle combinations are analyzed here - one specular angle pair and one non-specular angle pair. The specular angle pair is illuminated at −70o and its scattering angle is 70o . The non-specular angle pair has the same illumination angle of −70o and a scattering angle of 10o , and was selected because it is far from specular. Because this measurement was performed in reflection mode, the data was corrected for the geometric 180o phase change via right-multiplication by the Mueller matrix for a horizontal half wave linear retarder (DeBoo et al., 2005), 1 0 RH (δ = π) = 0 0 0 1 0 0 . 0 −1 0 0 0 −1 0 0 (3.12) The Mueller matrix for the non-specular angle pair, 0.004 −0.400 0.781 −0.058 −0.02 Mn = , −0.019 0.053 0.216 0.179 0.001 −0.015 −0.180 0.138 1 −0.437 0.032 (3.13) 73 and the specular angle pair, 0.003 −0.214 0.947 −0.026 −0.035 Ms = , −0.010 0.037 −0.610 0.370 0.000 −0.037 −0.339 −0.598 1 −0.245 0.010 (3.14) yields information about the polarization effects due to scattering. Mn has smaller diagonal terms than Ms , particularly in m2,2 and m3,3 , which implies that Mn is depolarizing in the 45o /135o linear and circular axes. Ms has larger diagonal terms, and its m2,2 and m3,3 terms are negative. Ms is close to the form of equation 3.12, which implies the presence of significant retardance. Mn also exhibits more diattenuation than Ms , as seen in m0,1 , m0,2 , and m0,3 . Matrix Root Parameters for Specular Angle Matrix Root Parameters for Non-Specular Angle ÈDi È 3.0 ÈDi È 3.0 2.5 2.5 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 D0 D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 D13 D14 D15 D0 D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 D13 D14 D15 Figure 3.4: The magnitude of the matrix roots parameters D0 through D15 of the ground glass sample for (a) the specular angle pair and (b) the non-specular angle pair. The matrix roots decomposition parameters for both angle pairs are shown in table 3.6. Figs.s 3.4(a) and 3.4(b) plot the absolute value of each matrix root parameter for both angle pairs. For each case, the diattenuation lies predominantly on the vertical linear axis (D1 < 0), and the non-specular angle pair exhibits larger 74 diattenuation than the specular angle pair. The specular angle pair has a large retardance of 2.61 radians predominantly in the linear horizontal direction (D4 ). The non-specular angle pair has only 0.78 radians of retardance in the horizontal direction, but this axis still dominates over the other orientations. For both angle pairs, the dominant non-depolarizing effects are diattenuation and retardance along the horizontal/vertical Stokes axis. According to Tables 3.4 and 3.5, averaging these effects can lead to diagonal depolarization D13 , D14 , and D15 . Indeed, diagonal depolarization is the dominant depolarizing effect in the scattering measurements for both angle pairs. The non-specular angle pair exhibits more diagonal depolarization than the specular angle pair by several orders of magnitude. We expect the specular angle pair to exhibit less depolarization than the angle pair further from specular. The diagonal depolarization terms D13 , D14 , and D15 yield the following information. The negative magnitude of D13 for both cases means that more diagonal depolarization takes place along the linear horizontal/vertical axis than the linear 45o /135o axis. The magnitude of D13 for the non-specular angle is approximately triple that of the specular angle, which means that the differences between the two linear depolarization axes is more dramatic for the non-specular angle. The positive magnitude of D14 for both angles implies that the diagonal depolarization effects are predominantly linear in nature (both on the horizontal/vertical and 45o /135o axes), as opposed to circular. This does not mean that the circular diagonal depolarization term is negligible - it simply means that the linear diagonal effects on all axes combined are more significant than the circular diagonal depolarization. This effect is more dramatic in the non-specular angle. D15 is the overall (isotropic) diagonal depolarization power, and as expected, the D15 value of 1.169 is larger for 75 the non-specular angle than the specular angle, with a D15 of 0.292. Roots Parameter D0 D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 D13 D14 D15 Specular Angle {−70o , 70o } 0.029 -0.243 0.002 -0.006 2.612 0.007 -0.075 -0.018 0.001 0 0.108 0.029 0.007 -0.149 0.065 0.292 Non-Specular Angle {−70o , 10o } 0.096 -0.470 0.006 -0.008 0.785 0.005 -0.119 -0.059 0.016 -0.001 -0.024 -0.004 -0.006 -0.430 0.459 1.169 Table 3.6: The matrix roots parameters from a ground glass sample for specular {−70o , 70o } and non-specular angle pairs {−70o , 10o }. 3.6.2 Pencil at 505 nm The second example is the Mueller matrix for a wood-graphite pencil. Polarized light scatters very differently for the two materials - the graphite is smoother, with less depolarization upon scattering, while the wood is more textured, with more depolarization upon scattering. Fig. 3.5 shows the Mueller matrix image for the graphite pencil measured in reflection mode with the University of Arizona Mueller matrix imaging polarimeter (MMIP) (DeBoo et al., 2005) at 505nm. The images have been median filtered to reduce noise and corrected for reflection using the procedure described in section 3.6.1. Inspection of the Mueller matrix reveals that the pencil is predominantly a diagonal depolarizer. The wood is more depolarizing than the smooth graphite 76 Figure 3.5: The Mueller matrix for a graphite and wood pencil, measured at 505nm with a Mueller matrix imaging polarimeter. surface, particularly in the m3,3 image. Other than these gross features, it is difficult to see the magnitudes of the depolarizing properties from the Mueller matrix image alone. The matrix roots decomposition yields more insight into both the nondepolarizing and depolarizing properties of the pencil. Fig. 3.6 shows the pencil’s non-depolarizing matrix roots parameters. The pencil tip exhibits a small amount of vertical diattenuation on the order of -0.09 in the wooden region and -0.12 on the graphite tip in D1 , and minimal 45o or circular diattenuation in D2 and D3 . The pencil tip exhibits some retardance on all three axes in D4 , D5 , and D6 . In D4 , the horizontal/vertical retardance term, the wooden region of the pencil exhibits a minimum of 0.2 radians of retardance and a maximum of 0.6 radians of retardance. D5 , the 45o /135o retardance term, exhibits between 0 radians and -0.6 radians of 77 D1 D2 D3 D4 D5 D6 Figure 3.6: The non-depolarizing matrix roots parameters for a graphite and wood pencil, measured at 505nm with a Mueller matrix imaging polarimeter. 78 retardance in the wooden region of the pencil tip. The negative retardance value means that the fast-axis of the retardance is closer to 135o than 45o . The graphite region of the pencil exhibits trace amounts of horizontal retardance and slightly more circular retardance, while the wooden region of the pencil exhibits between 0.2 and 0.4 radians of circular retardance, as shown in the D6 parameter. The graphite tip of the pencil has very little depolarization. This is consistent throughout all the degrees of freedom of the three depolarization families (amplitude, phase, and diagonal depolarization). The lack of depolarization in the graphite pencil tip is not surprising, as the graphite is a relatively smooth, shiny surface, especially compared to the wooden region of the pencil. The wooden region of the pencil exhibits no significant linear horizontal/vertical amplitude depolarization D7 , almost no circular amplitude depolarization D9 , and a 45o /135o amplitude depolarization of D8 = 0.05 in the upper half of the wood. The phase depolarization (D10 , D11 , and D12 ) is most significant in the left circular direction, with the negative magnitude of D12 approaching -0.3 in the maximum regions. These results are consistent with the results of averaging the spatial inhomogeneities in diattenuation and retardance from Table 3.5. In this case, the presence of D10 , D11 , and D12 are consistent with the averaging of spatial inhomogeneities along all three retardance axes (two at a time), and 45o /135o amplitude depolarization D8 is consistent with the averaging of horizontal/vertical diattenuation D1 and circular retardance D6 . The diagonal depolarization parameter D13 is a measure of the relative strength between the diagonal Mueller matrix m1,1 term and the diagonal m2,2 term. The magnitude of the entire D13 image is very small, which means that the wooden region of the pencil is almost equally depolarizing along the horizontal/vertical and 79 D7 D8 D9 D10 D11 D12 D13 D14 D15 Figure 3.7: The depolarizing matrix roots parameters for a graphite and wood pencil, measured at 505nm with a Mueller matrix imaging polarimeter. 80 45o /135o axes. D14 is a measure of the relative strength between the linear diagonal terms (m1,1 and m2,2 ) and the circular diagonal term (m3,3 ). The positive magnitude of D14 throughout the wooden region of the pencil means that the diagonal depolarization in the wooden region of the pencil is predominantly linear. D15 is a measure of the overall diagonal depolarization power - it reaches a maximum of 4 at the top edge of the pencil where the incident and scattering angles in the imaging measurement are steepest, and a minimum of approximately 1.8 in the central region of the wood. 3.7 Conclusion The Mueller matrix roots decomposition from (Chipman, 2009) and the associated families of depolarization (amplitude depolarization, phase depolarization, and diagonal depolarization) were explored. Degree of polarization maps show the unity between phase and diagonal depolarization; amplitude depolarization remains a distinct class. The nine depolarization parameters can be generated via the averaging of two non-depolarizing Mueller matrices, and the orientation of the resulting depolarization follows the cyclic permutations of the Pauli spin matrices. Averaging two non-depolarizing Mueller matrices of the same class (retarders with retarders or diattenuators with diattenuators) results in phase depolarization, while averaging two non-depolarizing Mueller matrices of different classes (retarders with diattenuators) results in amplitude depolarization. Scattered light is often a combination of many incoherent states that result from spatial inhomogeneities. These nine forms of depolarization were applied to Mueller matrices from two scattering measurements a sample of ground glass and a graphite and wood pencil tip. 81 CHAPTER 4 ADDITIONAL MATRIX ROOTS GENERATOR PROPERTIES AND APPLICATIONS 4.1 Convergence to the identity matrix The matrix roots decomposition is useful for the subset of uniform Mueller matrices. The principal root of a uniform matrix converges to the identity matrix I as the root order p approaches infinity: lim p→∞ √ p M = I. (4.1) To show that the majority of physical Mueller matrices are uniform, the proportion of physical Mueller matrices with a negative determinant is estimated. The proportion of physical Mueller matrices which have a root that converges to the identity matrix as the root order p approaches infinity is examined. To this end, a brute-force numerical calculation method was applied. A four by four Mueller matrix is generated with random numbers (uniformly distributed between negative one to one), and its m0,0 element is set to a value of one. Thus the 15 element space within which normalized Mueller matrices reside is uniformly sampled. Each matrix is tested for physical realizability by checking that the eigenvalues of its coherency matrix are all greater than or equal to zero. If the matrix is physical, then its determinant is calculated and the Mueller matrix’s determinant is classified to be either positive or negative. (Mueller matrices with negative determinants are non-uniform because their roots are complex.) Approximately one hundred fifteen thousand physical Mueller matrices were generated in this manner 82 Determinants of Random Mueller Matrices Number of Mueller Matrices 3000 2500 2000 1500 1000 500 0 0.00 0.05 0.10 Determinant Figure 4.1: A histogram of the randomly generated Mueller matrix determinants shows the statistical distribution of the determinant values. from a set of 109 matrices, and some statistics of this set were calculated. Approximately one in eight Mueller matrices (12.11%) from a sample of 115,218 physical Mueller matrices was found to have a negative determinant. Out of these 13,951 negative determinant matrices, the most negative determinant was -0.02837. The most negative determinant of a physically realizable Mueller matrix is −1/27, or 0.037 (Chipman, 2009). The histogram in Fig. 4.1 shows the statistical distribution of the randomly generated Mueller matrices’ determinants, and indicates the rapid reduction of the volume of Mueller matrices as the determinant moves away from zero in both directions. The second goal of this numerical experiment was to demonstrate that the majority of physical, positive determinant Mueller matrices converge towards the identity matrix as the pth root becomes very large. The 10n th matrix root with n = 4, through n = 6 for each physical positive determinant Mueller matrix was calcu- 83 Root Convergence to Identity Matrix Number of Mueller Matrices 80 000 60 000 10-4 10-5 40 000 10-6 20 000 0 0.00000 0.00002 0.00004 0.00006 0.00008 0.00010 0.00012 0.00014 RMS Divergence Figure 4.2: This histogram shows increasing convergence towards the identity matrix as the root order p increases. lated and subtracted from the identity matrix. The root-mean-square (RMS) of this difference was then calculated, RM S = 3 1 X 16 i,j=0 √ 10n mi,j − Ii,j 2 . n = 4, 5, 6 (4.2) A histogram of the RMS for the five values of root-order p = 10n in Fig. 4.2 shows that as the value of p increases from p = 104 incrementally to p = 106 , the RMS shrinks rapidly. 671 out of 101,267 positive determinant Mueller matrices do not converge to the identity matrix. Because the majority of positive determinant Mueller matrices do converge towards the identity matrix as their root order approaches a very large number, this matrix roots method is applicable to the majority of these cases. 84 4.2 Generator properties 4.2.1 Trajectories of non-depolarizing generators Non-depolarizing generators, except for the singular and non-uniform cases discussed in section 2.4, are uniform. Two simple examples in figures 4.3 and 4.5 illustrate this behavior. In figure 4.3, the root of a linear diattenuator with diattenuation of 0.2 at 0o approaches the identity matrix as the root order increases. Figure 4.4 shows the 10p trajectory of the generator (G2 (d2 = 2 ∗ 10−10 )) approaching the linear diattenu- ator from figure 4.3 when p = 0 as the root order 10p increases. These figures are complementary - figure 4.3 shows the path from the diattenuator to its infinitesimal generator, and 4.4 shows the path from infinitesimal generator to diattenuator. Similarly, figure 4.5 shows the trajectory of the root of an elliptical retarder with retardance vector {δH = 0, δ45 = π/4, δR = π/10} approaching the identity matrix as the matrix is taken to the 10p th root. Figure 4.6 shows the trajectory of the product G5 (d5 = π/4 ∗ 10−9 ) · G6 (d6 = π/10 ∗ 10−9 ) approaching the macroscopic retarder from figure 4.5 at p = 0 as the matrix is taken to the 10p th power. 4.2.2 Orthogonality The generator trajectories for G1 through G15 were constructed to be orthogonal around the identity matrix. We introduce the following notation, where a Mueller matrix is flattened into a one-dimensional array and drop the first element. For a generic Mueller matrix M with elements m0,0 through m3,3 , the Mueller matrix is 85 M1,0 M0,0 M3,0 M2,0 1.0000 1.0 0.020 1.0 0.9995 0.5 0.015 0.5 0.9990 2 4 6 8 10-p -0.5 2 4 6 8 10-p 4 6 8 10 -0.5 2 M0,2 4 6 8 10-p 2 0.005 2 4 6 8 4 6 8 10-p 0.5 2 4 4 6 8 10-p -0.5 -0.5 -1.0 -1.0 -1.0 8 M3,2 0.5 2 4 6 8 10 4 10-p -p -0.5 -1.0 M3,3 1.0000 0.9995 2 -0.5 6 10-p -0.5 M2,3 1.0 8 2 1.0 2 0.5 6 10 0.9990 1.0 4 8 -1.0 M1,3 10-p 6 0.9995 0.5 2 4 -p -0.5 1.0 8 0.5 -0.5 10-p -1.0 M0,3 6 10-p M3,1 1.0000 0.5 8 -1.0 M2,2 1.0 0.010 10-p -1.0 M1,2 6 1.0 2 -1.0 0.015 8 0.5 0.9990 0.9985 0.020 6 1.0 0.9995 -p 4 M2,1 1.0000 0.5 4 -0.5 2 M1,1 1.0 2 0.005 -1.0 M0,1 2 0.010 4 6 8 10-p 0.9990 0.9985 2 4 6 8 10-p Figure 4.3: A linear diattenuator with diattenuation of 0.2 approaches the identity matrix as it is taken to the 10p th root. 86 M0,0 M2,0 M1,0 M3,0 1.020 1.0 0.20 1.0 1.015 0.5 0.15 0.5 1.010 2 1.005 4 6 8 10 p 4 6 8 10 p -1.0 2.0 0.5 1.5 2 4 6 8 10 p 8 0.15 4 6 1.020 0.5 1.015 2 6 8 10 p 10 4 6 8 10 8 6 8 10 p -1.0 M3,2 1.0 0.5 2 4 10 p -0.5 1.005 -1.0 6 4 -0.5 p 1.010 -0.5 2 4 M1,3 M0,3 2 M2,2 0.10 0.05 8 -1.0 1.0 8 -1.0 p -0.5 6 6 10 p 0.5 2 4 8 M3,1 0.5 10 p 6 1.0 M1,2 0.20 8 1.0 2 M0,2 4 6 1.0 -1.0 2 4 10 p M2,1 0.5 -0.5 4 -0.5 2 M1,1 M0,1 1.0 2 0.05 -0.5 2 0.10 6 8 10 p -1.0 M3,3 M2,3 2.0 1.0 1.0 1.0 0.5 0.5 2 4 6 8 10 1.5 0.5 p 2 4 6 8 10 p 2 -0.5 -0.5 -0.5 -1.0 -1.0 -1.0 4 6 8 10 p 1.0 0.5 2 4 10 p Figure 4.4: The trajectory of generator G2 with d2 = 2∗10−10 as the matrix is taken to the power of 10p approaches a linear diattenuator with diattenuation of 0.2. 87 M0,0 M3,0 M1,0 2.0 M2,0 1.0 1.0 1.5 1.0 0.5 1.0 2 0.5 4 6 8 10-p -0.5 2 4 6 8 10-p 2 0.5 2 4 6 8 -0.5 2 4 4 6 6 8 -0.5 8 1.5 2.0 2.5 3.0 3.5 2 -0.02 0.5 4 6 8 10-p -0.04 -0.5 -0.06 -1.0 -0.08 2.0 2.5 3.0 4 10-p 0.07 0.06 0.05 0.04 0.03 0.02 1.2 10-p 2 6 10-p 4 8 10 6 8 2 4 1.6 1.8 10-p 0.012 0.010 0.008 0.006 0.004 10-p 0.002 1.5 6 2.0 2.5 10-p M3,3 -p -0.002 -0.004 -0.006 -0.008 -0.010 1.4 M3,2 M2,3 M1,3 1.0 2 1.5 -0.005 -0.010 -0.015 -0.020 10-p -0.025 -0.030 1.0000 0.9999 0.9998 0.9997 0.9996 M0,3 8 M3,1 M2,2 0.030 0.025 0.020 0.015 10-p 0.010 0.005 -1.0 6 -1.0 M1,2 M0,2 0.5 8 4 M2,1 1.0000 0.9995 0.9990 0.9985 -p 0.9980 10 0.9975 0.9970 2 1.0 6 2 -0.5 -1.0 M1,1 -1.0 4 10-p -0.5 -1.0 M0,1 1.0 0.5 0.5 8 10-p 1.0000 0.9995 0.9990 0.9985 0.9980 0.9975 2 4 6 8 10-p Figure 4.5: The trajectory of an elliptical retarder with δ45 = π/4 and δR = π/10 approaching the identity matrix as the matrix is taken to the 10p th root. 88 M0,0 M3,0 M2,0 M1,0 1.0 2.0 1.0 1.0 1.5 0.5 0.5 1.0 2 0.5 2 4 6 8 4 0.5 2 4 6 8 -0.5 10 p -1.0 -1.0 4 -1.0 1.0 0.5 4 6 8 -0.5 0.5 -1.0 6 8 -0.1 -0.2 -0.3 10 p -0.4 -0.5 -0.6 -0.7 6 8 10 p 4 10 p 6 8 10 p 8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 7.5 2 4 8.0 8.5 9.0 10 p 6 8 0.12 0.10 0.08 0.06 0.04 p 10 0.02 M2,3 10 p 6 M3,2 0.97 2 4 M2,2 0.98 4 2 -0.5 7.0 M1,3 1.0 4 10 p 9.0 0.99 2 2 8 -0.05 -0.10 -0.15 -0.20 -0.25 8.5 1.00 M0,3 -0.5 6 0.25 0.20 0.15 p 10 0.10 0.05 -1.0 10 8 M3,1 8.0 M1,2 M0,2 6 M2,1 1.00 0.95 0.90 0.85 p 0.80 10 0.75 2 4 -1.0 M1,1 -0.5 2 8 2 -0.5 M0,1 1.0 6 10 p 0.5 p 6 7 8 9 10 p M3,3 0.12 0.10 0.08 0.06 0.04 0.02 2 4 6 8 10 1.00 0.95 0.90 0.85 p 0.80 0.70 2 4 6 8 10 p Figure 4.6: The trajectory of the generator product G5 (d5 = π/4 ∗ 10−9 ) · G6 (d6 = π/10 ∗ 10−9 ) approaching an elliptical retarder as the matrix is taken to the power of 10p . 89 transformed from a four by four matrix to a fifteen element vector Mv as follows: m0,0 m1,0 M= m 2,0 m3,0 m0,1 m0,2 m0,3 m1,1 m1,2 m1,3 −→ m2,1 m2,2 m2,3 m3,1 m3,2 m3,3 Mv = (m0,1 , m0,2 , m0,3 , m1,0 , m1,1 , m1,2 , m1,3 , m2,0 , m2,1 , m2,2 , m2,3 , m3,0 , m3,1 , m3,2 , m3,3 ) . (4.3) The direction of a generator trajectory from the identity matrix is the derivative of a space curve which can be calculated as the limit I − Gi (di ) . di →0 di (4.4) lim The vector angle between the generators is found to be π/2 in the limit surrounding the identity matrix: −1 θ = lim cos di ,dj →0 (I − Gi (di ))v · (I − Gj (dj ))v k(I − Gi (di ))v kk(I − Gj (dj ))v k = π 2 i, j = 1, 2, ...15 (i 6= j) (4.5) While the generators are orthogonal to each other in the limit near the identity matrix, they are not orthogonal in general. This is straightforward to show by counterexample. Figures 4.7 and 4.8 show the distance from orthogonality (π/2 − θ) between G1 (d1 ) and G2 (d2 ), and G11 (d11 ) and G12 (d12 ) (respectively), plotted as a function of d1 = d2 = d and d11 = d12 = d. In figures 4.7 and 4.8, the vector angle 90 Non-Orthogonality Between G1 and G2 0.0012 Radians 0.0010 0.0008 0.0006 0.0004 0.0002 0.0000 0.00 0.02 0.04 0.06 0.08 0.10 d Figure 4.7: The diference between π/2 and the vector angle between G1 (d1 = d) and G2 (d2 = d) is plotted as a function of d, showing that generators G1 and G2 are not orthogonal in general. between the generators is calculated and then subtracted from π/2. 4.2.3 Depolarization index The depolarization index (DI) is a numerical metric for characterizing the depolarization of a Mueller matrix (Gil and Bernabeu, 1985). The DI is defined as P DI(M) = 3 2 i,j=0 (mi,j ) √ 2 − (m0,0 ) 3m0,0 1/2 . (4.6) The DI equals one for non-depolarizing Mueller matrices and equals zero for an ideal depolarizer. It is the distance of a normalized Mueller matrix from the ideal √ depolarizer, divided by 3. For G1 (d1 ) through G14 (d14 ), the DI is equal to one, for any value of di . In and of themselves, none of these generators change the depolarization index - all of √ the generator trajectories lie on a hypersphere of radius 3. However, as G7 (d7 ) 91 Non-Orthogonality Between G11 and G12 0.0012 Radians 0.0010 0.0008 0.0006 0.0004 0.0002 0.0000 0.00 0.02 0.04 0.06 0.08 0.10 d Figure 4.8: The diference between π/2 and the vector angle between G11 (d11 = d) and G12 (d12 = d) is plotted as a function of d, showing that generators G11 and G12 are not orthogonal in general. through G14 (d14 ) are not physical on their own, they must be paired with some amount of G15 (d15 ) to produce a physical Mueller matrix (as discussed in section 4.2.5). Of all the generators, only G15 (d15 ) has DI < 1: r DI (G15 (d15 )) = 1 − 2 d15 . 3 (4.7) p This implies that d15 has a range between zero and 3/2, where d15 = 0 corresponds p to a DI of one (at the identity matrix) and d15 = 3/2 corresponds to a DI of zero, p or an ideal depolarizer. Increasing d15 between zero and 3/2 increases the DI linearly. 4.2.4 Unitary transformations and depolarization classification Performing rotation operations on the depolarizing generators G7 through G15 yields insight into the relationship between the different families of depolarization. It 92 is shown that phase depolarization and diagonal depolarization are related via a coordinate rotation, while amplitude depolarization remains distinct from the other two families under coordinate rotation. Matrix rotations are performed with a real, orthogonal matrix transformation such that Mo = OT MO, (4.8) where O is an orthogonal, real unitary matrix. As Mueller matrices these arre elliptical retarders. The columns of O form an orthonormal basis in real space. This orthogonal transformation on a Mueller matrix M results in a rotation between two or more axes of the Stokes coordinate system intrinsic to M. The two classes of matrices for rotation used here are R1 R2 1 0 0 0 0 cos 2θ − sin 2θ 0 = 0 sin 2θ cos 2θ 0 0 0 0 1 0 0 0 1 0 cos2 2θ cos 2θ sin 2θ − sin 2θ = , 0 cos 2θ sin 2θ sin2 2θ cos 2θ 0 sin 2θ − cos 2θ 0 (4.9) (4.10) where R1 rotates an incident Stokes vector between the S1 and S2 axes (about the S3 axis) and R2 rotates an incident Stokes vector between the S1 , S2 , and S3 axes. (R2 is equivalent to a quarter-wave linear retarder with fast-axis orientation θ.) If a rotation is performed on a depolarizing generator and the resulting matrix has been rotated into another family of depolarization, this implies that the two 93 families are related by rotation. To demonstrate this effect, the orthogonal rotation transformation from equation 4.8 is performed on the depolarizing generators G7 (d7 ) through G15 (d15 ) about the Stokes axes for the two rotations. Because these matrices are generators, they are expressed in terms of the (depolarizing) infinitesimal matrix roots parameters d7 through d15 . Therefore after the rotation operation is performed, the resulting matrix can be solved for the infinitesimal matrix roots parameters. Because d7 through d15 are infinitesimal, any resulting second order terms are dropped. G15 (d15 ) is invariant under all of the rotation transformations discussed here, since it depolarizes equally along each Stokes axis. Amplitude depolarization generators G7 (d7 ), G8 (d8 ), and G9 (d9 ) are rotated among themselves, forming a depolarization family. They do not rotate into any other families of depolarization. Table 4.1 shows that the rotation of G10 (d10 ) and G11 (d11 ) by R1 (θ) results in a sinusoidal rotation between d11 and d10 with a period of 2θ. Because d11 and d10 are different (linear) axes of the same phase depolarization family, this is an obvious and expected result. The same transformation around R1 (θ) on G12 (d12 ) results in a sinusoidal rotation between circular phase depolarization d12 and diagonal depolarization terms d13 and d14 : d12 = d12 cos 4θ (4.11) d13 = −d12 sin 4θ (4.12) d14 = d12 sin 4θ. (4.13) The rotation of circular phase depolarization d12 into diagonal depolarization demonstrates the relationship between phase and diagonal depolarization. 94 Table 4.2 shows the effect of rotating G10 (d10 ), G11 (d11 ), and G12 (d12 ) about R2 (θ). G10 (d10 ), G11 (d11 ), and G12 (d12 ) all rotate between the three phase depolarization axes d11 and d12 and diagonal depolarization d13 . Table 4.3 shows the effect of rotating G13 (d13 ), G14 (d14 ), and G15 (d15 ) about R2 (θ). This rotation of G13 (d13 ) is unique among all of the depolarizing generators in that d13 rotates into all three axes of phase depolarization (d10 , d11 , and d12 ) as well as diagonal depolarization d14 . This method can be extended by applying many other varieties of rotation transformations (such as more complex elliptical rotations), though their closed form solutions are too complex to include here. These unitary rotation operations show that phase depolarization and diagonal depolarization are related via simple coordinate rotations, while amplitude depolarization remains distinct. This approach has uniquely demonstrated that depolarizers fall into two distinct classes: amplitude depolarization in one class, and phase and diagonal depolarization in another class. Table 4.1: Effect of the rotation transformation for G10 through G12 about R1 (θ). d7 d8 d9 d10 d11 d12 d13 d14 d15 G10 (d10 ) RT1 (θ)G10 (d10 )RT1 (θ) 0 0 0 0 0 0 d10 d10 cos 2θ 0 d10 sin 2θ 0 0 0 0 0 0 0 0 G11 (d11 ) 0 0 0 0 d11 0 0 0 0 RT1 (θ)G11 (d11 )R1 (θ) G12 (d12 ) 0 0 0 0 0 0 −d11 sin 2θ 0 d11 cos 2θ 0 0 d12 0 0 0 0 0 0 RT1 (θ)G12 (d12 )R1 (θ) 0 0 0 0 0 d12 cos 4θ −d12 sin 4θ d12 sin 4θ 0 95 Table 4.2: Effect of rotation transformation for G10 through G12 about R2 (θ). d7 d8 d9 d10 d11 d12 d13 d14 d15 G10 (d10 ) 0 0 0 d10 0 0 0 0 0 RT2 (θ).G10 (d10 ).R2 (θ) 0 0 0 −d10 cos2 2θ 1 d sin 4θ 2 10 −d10 cos 4θ sin 2θ −d10 sin 2θ sin 4θ 0 0 G11 (d11 ) 0 0 0 0 d11 0 0 0 0 RT2 (θ).G11 (d11 ).R2 (θ) G12 (d12 ) RT2 (θ).G12 (d12 ).R2 (θ) 0 0 0 0 0 0 0 0 0 1 d sin 4θ 0 d cos 4θ sin 2θ 11 12 2 −d11 sin2 2θ 0 d12 cos 2θ cos 4θ 1 −d11 cos 2θ sin 4θ d12 d sin2 4θ 2 12 −2d11 cos2 2θ sin 2θ 0 −d sin 8θ/4 √ 12 3 0 0 d sin 4θ 2 12 0 0 0 Table 4.3: Effect of rotation transformation for G13 (d13 ) through G15 (d15 ) about R2 (θ). d7 d8 d9 d10 d11 d12 d13 d14 d15 G13 (d13 ) 0 0 0 0 0 0 d13 0 0 RT2 (θ).G13 (d13 ).R2 (θ) 0 0 0 d13 sin 2θ sin 4θ 2d13 cos2 2θ sin 2θ − 14 d13 sin 8θ d√13 cos2 4θ − 23 d13 cos 4θ 0 G14 (d14 ) 0 0 0 0 0 0 0 d14 0 RT2 (θ).G14 (d14 ).R2 (θ) G15 (d15 ) RT2 (θ).G15 (d15 ).R2 (θ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 √ 1 3d sin 4θ 0 0 14 2 √ − 23 d14 cos 4θ 0 0 −d14 /2 0 0 0 3d15 /2 3d15 /2 96 4.2.5 Conditions for physicality of depolarizing generators Cloude presented a method for evaluating the physical realizability of a Mueller matrix in terms of its coherency matrix (Cloude, 1989). The coherency matrix C of a Mueller matrix M is calculated from the elements mij of M and the Pauli spin matrices σi , C= 3 X mij (σi × σj∗ ), (4.14) i=0,j=0 where in our notation the Pauli spin matrices are 0 −i 0 1 1 0 1 0 σ0 = . , σ3 = , σ2 = , σ1 = i 0 1 0 0 −1 0 1 (4.15) A depolarizing Mueller matrix is physical so long as all four eigenvalues of the Mueller matrice’s coherency matrix are greater than zero (Ossikovski et al., 2008; Twietmeyer et al., 2008). The non-depolarizing generators G1 (d1 ) through G6 (d6 ) are all physical Mueller matrices - in each case, three coherency matrix eigenvalues are zero, and the fourth is real and positive. Individually, G7 (d7 ) through G14 (d14 ) are not physical Mueller matrices unless combined with G15 (d15 ). To construct a physical (macroscopic) Mueller matrix, we combine them as p di d15 Gi G15 . p p (4.16) The coherency matrix eigenvalues are plotted within the range of negative one to one for each value of di because the generators are only real within this range. Figure 4.9 graphs the four eigenvalues for −1 < di < 1. The coherency matrix eigenvalues for G7 (d7 ) through G12 (d12 ) are identical, and are plotted together. G15 (d15 ) is the only 97 Eigenvalues Λ1 G7-G12 -1.0 Λ3 Λ2 0.4 0.4 0.9 0.2 0.2 0.3 0.8 0.2 0.7 0.5 1.0 d7-12 -1.0 -0.4 0.5 -0.5 -0.2 1.0 d7-12 0.1 Λ2 Λ1 -1.0 0.6 -0.5 -0.5 -0.02 0.5 1.0 0.4 0.2 -0.06 0.2 0.5 1.0 d13 -0.08 Λ1 -0.5 -0.1 -1.0 -0.5 d14 -1.0 -0.5 Λ1 1.0 d14 -1.0 -0.5 0.1 0.5 1.0 -1.0 1.0 0.7 d13 -0.5 -0.1 1.0 -1.0 0.5 -0.1 -0.1 -0.2 -0.2 -0.2 0.5 1.0 0.5 1.0 d13 1.0 d14 -0.5 d14 Λ4 0.5 -0.5 -0.1 1.0 Λ4 1.6 1.4 1.2 1.0 0.8 0.6 0.1 d15 0.5 1.00 0.95 0.90 0.85 0.80 0.75 Λ3 0.5 -0.5 0.5 0.2 0.1 d15 0.8 -1.0 0.2 G15 -0.5 0.5 d7-12 0.9 Λ3 Λ2 0.2 -1.0 -0.2 -0.4 1.0 Λ4 0.4 0.3 0.2 0.1 0.6 0.4 0.2 1.0 -0.2 0.5 -0.5 1.0 -1.0 Λ2 0.5 -1.0 0.6 -0.04 -0.2 1.0 d7-12 Λ3 d13 0.4 0.4 0.3 0.2 0.1 -1.0 0.5 -0.5 -0.10 G14 0.6 -0.4 -1.0 -1.0 1.0 0.4 -0.5 -0.2 G13 Λ4 0.5 1.0 d15 -1.0 -0.5 d15 Figure 4.9: The four eigenvalues for the coherency matrices associated with the nine depolarization generators. generator with a region for which all four coherency eigenvalues are greater than zero. G7 (d7 ) through G14 (d14 ) do not have any region with coherency eigenvalues greater than zero, indicating its intrinsic physicality. The depolarizing generators G7 (d7 ) through G14 (d14 ) become physical when combined with G15 (d15 ) as Gi (di )G15 (d15 ) for i = 7, ...14. This yields a single region of di and d15 for which the Mueller matrix is physical, because all four eigenvalues are greater than zero. These regions are shown in figure 4.10 for combinations of d7 through d14 with d15 . The physical regions are plotted in white, and non-physical regions are shown in black. Because the maps for G7 , G8 , and G9 combined with G15 are identical, they have been combined on one map. Similarly, the maps for G11 and G12 combined with G15 98 Figure 4.10: The physical region for each depolarizing generator G7 through G14 multiplicatively combined with G15 is shown in the white region. One axis represents d15 and the other axis represents the other d-parameter of a corresponding generator. G7 through G9 and G10 through G12 are shown on the same plot since their physical regions are identical. are identical. The angle of this region was calculated analytically for each of the depolarizing generators combined with G15 as follows. First, the two generator matrices are multiplied, Gi,15 = Gi (di ) · G15 (d15 ), i = 7, 8, ...14. (4.17) Then, for each value of i, the coherency matrix for Gi,15 is calculated and solved for its four eigenvalues. The outer boundaries of the regions of physicality shown in figure 4.10 are defined by two eigenvalues. As an example, the two eigenvalues for 99 the combination G7,15 which limit its physical region are √ √ ~λ7,15 = { 1 (−6d7 + 6d15 (1 + d7 )), 1 (− 6d15 (d7 − 1) + 6d7 )}. 12 12 (4.18) Then these equations are set to zero and solved for d7 , √ d+ 7 d− 7 6d15 , 6d15 − 6 √ 6d15 . = −√ 6d15 − 6 = √ (4.19) (4.20) The angle of the physical region in this plane for G7,15 is calculated from the vector − angle between the two curves formed by d+ 7 and d7 , in the limit as d15 approaches zero. This procedure can be applied to the other generator combinations shown in figure 4.10. The angle at the origin for the the generator combinations Gi=7−13,15 is the √ arctan(2 6/5), or approximately 44.415o . The angle at the origin for the generator √ combination G14,15 is the arctan(4 2/7), or approximately 38.94o . These angles suggest that the physical region of generator space in the vicinity of the identity matrix is cone-like in 15 dimensions with a half angle of 44.415o in most of its cross sections. 4.3 Interpretation of MM root parameters 4.3.1 Nondepolarizing matrix root parameters The non-depolarizing Mueller matrix roots parameters are D0 through D6 . D0 is the optical density, D1 , D2 , and D3 are the matrix roots diattenuation parameters, and D4 , D5 , and D6 are the matrix roots retardance parameters. 100 4.3.1.1 Retardance parameters The Mueller matrix roots parameters for retarders are straightforward to calculate and interpret, because retardance is additive for a sequence of retarders with the same fast-axis orientation or the same eigenpolarization. Thus the pth root of a retarder with retardance δ results in a retarder with the same fast-axis orientation and retardance δ/p. This is equivalent to cutting the retarder into p pieces. Calculating the matrix roots retardance parameters D4 , D5 , and D6 for a retarding Mueller matrix (either homogeneous or non-homogeneous) yields an order independent representation of the three dimensional retardance vector {δH , δ45 , δR } (Goldstein, 2003), D4 ≈ δH (4.21) D5 ≈ δ45 (4.22) D6 ≈ δR , (4.23) where δH , δ45 , and δR are expressed in radians. The order independence of these parameters is an important point to highlight. For pure retarders, equations 4.21 through 4.23 are equal to the retardance vector as defined in (Goldstein, 2003). Because the matrix roots decomposition yields an order-independent representation, the roots decomposition of a retarder mixed with other polarization properties will yield different values for D4 , D5 , and D6 than the retardance vector yielded by the Lu-Chipman decomposition because the Lu-Chipman decomposition is order-dependent (Ossikovski et al., 2008). 101 4.3.1.2 Diattenuation parameters Like retardance, optical density is additive for a sequence of absorbers with the same transmision-axis orientation. Therefore the matrix roots parameters for average transmission and diattenuation, D0 , D1 , D2 , and D3 are defined analagously to optical density. In the infinitesimal case they become equal to diattenuation. The definition of diattenuation D as a function of maximum transmission Tmax and minimum transmission Tmin is D= Tmax − Tmin . Tmax + Tmin (4.24) Given absorption coefficients αmax and αmin associated with Tmax and Tmin , diattenuation varies with distance z as D(z) = z e−αmax z − e−αmin z = − tanh (α − α ) . max min e−αmax z + e−αmin z 2 (4.25) Because diattenuation does not behave linearly when cascaded (in z), the matrix roots parameters D0 through D3 do not correspond to the traditional definition of average transmission and the diattenuation vector. The parameters D1 , D2 , and D3 are in units of optical density like decibels, except the matrix roots diattenuation parameters are defined using the natural log. For a linear diattenuator with transmission Tmax and Tmin oriented at angle θ, the transmission and diattenuation 102 parameters in the limit as the root-order p approaches infinity are 1 D0 = − ln(Tmax Tmin ) 2 1 Tmax D1 = cos 2θ ln 2 T min Tmax D2 = cos θ sin θ ln Tmin (4.27) D3 = 0. (4.29) (4.26) (4.28) The diattenuation parameters are expressed as the natural log of the ratio between the maximum and minimum transmission. This is additive for aligned diattenuators, and subtracts for crossed diattenuators. For this case of an ideal homogeneous diattenuator, Tmax = e−D0 +D1 sec 2θ = e−D0 +D2 csc 2θ and (4.30) Tmin = e−D0 −D1 sec 2θ = e−D0 −D2 csc 2θ . (4.31) The diattenuation is related to the matrix roots parameters D0 , D1 , and D2 as D= eD2 csc θ sec θ − 1 e2D1 sec 2θ − 1 = . e2D1 sec 2θ + 1 eD2 csc θ sec θ + 1 (4.32) For a circular diattenuator with maximum transmittance Tmax and minimum trans- 103 mission Tmin , the rescaled roots parameters are 1 D0 = − ln Tmax Tmin 2 (4.33) D1 = 0 (4.34) D2 = 0 (4.35) D3 = ln Tmax . Tmin (4.36) Solving for maximum and minimum transmission using D0 and D3 results in Tmax = e−D0 +D3 /2 and (4.37) Tmin = e−D0 −D3 /2 , (4.38) and diattenuation D= eD3 − 1 . eD3 + 1 (4.39) 4.3.2 Depolarizing matrix roots parameters 4.3.2.1 The diagonal depolarizer The diagonal partial polarizer is an ideal starting point for the interpretation of the depolarizing Matrix roots parameters. The most common form of a diagonal partial depolarizer is 1 0 PD(a, b, c) = 0 0 0 0 0 a 0 0 , 0 b 0 0 0 c where a, b, and c are all positive real numbers. (4.40) 104 Because PD(a, b, c) is a diagonal depolarizer it only has diagonal depolarization matrix roots parameters D13 , D14 , and D15 . In the limit as p approaches infinity, D0 through D12 are zero-valued, and the diagonal depolarization parameters are 1 (log a − log b) 2√ 3 (log a + log b − 2 log c) = − 6 1 = − √ (log a + log b + log c). 6 D13 = (4.41) D14 (4.42) D15 (4.43) D13 is a logarithmic measure of the relative strength between linear horizontal/vertical and linear 45o /135o diagonal depolarization. A positive D13 results from a sample that is more diagonally depolarizing along the linear horizontal/vertical axis. A negative D13 results from a sample that is more diagonally depolarizing along the linear 45o /135o axis. D14 is a logarithmic measure of the relative strength between the two linear diagonal depolarization axes and circular diagonal depolarization. Negative D14 results from a sample that is more diagonally depolarizing along the circular axis, and positive D14 results from a sample that is more linearly depolarizing along the diagonal. D15 is a logarithmic measure of the total depolarizing power on all three axes. It should always be positive-valued. For an ideal depolarizer, when a = b = c = 0, D15 = ∞, and D13 = D14 = 0. 105 4.3.3 The diagonal depolarization tetrahedron The diagonal partial depolarizer, 1 0 0 0 0 m11 0 0 PD(m11 , m22 , m33 ) = , 0 0 m 0 22 0 0 0 m33 (4.44) is only physical under certain conditions. The eigenvalues of its coherency matrix must all be greater than or equal to zero in order for it to be a physical Mueller matrix. This leads to the following restrictions on m11 , m22 , and m33 : 1 + m11 − m22 − m33 ≥ 0 (4.45) 1 − m11 + m22 − m33 ≥ 0 (4.46) 1 − m11 − m22 + m33 ≥ 0 (4.47) 1 + m11 + m22 + m33 ≥ 0. (4.48) These restrictions on the ranges of m11 , m22 , and m33 define a physical region formed by a regular three-dimensional tetrahedron made of four planes, m33 = 1 + m11 − m22 (4.49) m33 = 1 − m11 + m22 (4.50) m33 = m11 + m22 − 1 (4.51) m33 = −(1 + m11 + m22 ). (4.52) 106 The tetrahedron, shown in figure 4.11, has the following characteristics. Its top point is (m11 = 1, m22 = 1, m33 = 1), the identity matrix in Mueller matrix space. The three base points, point A=(m11 = −1, m22 = 1, m33 = −1), point B=(m11 = 1, m22 = −1, m33 = −1), and point C=(m11 = −1, m22 = −1, m33 = 1) all form retarders in Mueller matrix space. Point A is a half wave linear retarder oriented at 45o . Point B is a linear half wave retarder oriented at 0o . Point C is a circular retarder with half a wave of retardance. All interior angles of the tetrahedron faces are 60o . The central point at the base of the tetrahedron, point D=(m11 = −1/3, m22 = −1/3, m33 = −1/3) represents the minimum possible negative determinant for a physical Mueller matrix. In Mueller matrix space, this matrix is also the average of the Mueller matrices which form points A, B, and C on the tetrahedron: LR(π, 0o ) + LR(π, 45o ) + CR(π) = PD(−1/3, −1/3, −1/3). 3 (4.53) The Euclidian distance between the minimum determinant point D and any corner p at the base of the tetrahedron (A, B, or C) is 2 2/3, while the Euclidian distance between the the minimum determinant point D and the center edge of one p tetrahedron wall is 2/3. 4.4 Applications for depolarization in optical design One goal of this work is to provide tools for analyzing depolarization in optical design. This section studies the depolarization resulting from the most common diattenuation and retardance polarization aberrations (McGuire and Chipman, 1994; Chipman, 1985) in radially symmetric optical systems. These methods might be 107 1.0 m22 0.5 0.0 -0.5 -1.0 1.0 0.5 m330.0 -0.5 -1.0 -1.0 -0.5 0.0 m11 0.5 1.0 Figure 4.11: Physical Mueller matrix diagonal depolarization space forms a tetrahedron. 108 used to direct an optimization process towards minimizing these depolarization effects. 4.4.1 Diattenuation and retardance defocus When a plane wave is incident on a spherical lens, there is a range in angles of incidence along the surface of the lens. These angles of incidence form a rotationally symmetric pattern, and due to the Fresnel and thin film equations, they produce a variation in diattenuation and retardance as a function of pupil radius, which has a quadratic term as its lowest order in an aberration expansion. This variation in diattenuation and retardance with pupil radius is analagous to the defocus term traditionally used in scalar aberration theory. Therefore these affects are separately categorized as diattenuation defocus and retardance defocus. The symbolic Mueller matrix for a linear diattenuator LD [q, r, θ] with maximum transmission q and minimum transmission r at orientation θ is (q − r) cos 2θ (q − r) sin 2θ 0 √ √ 2 2 (q + r − 2 qr) cos 2θ sin 2θ 0 1 (q − r) cos 2θ (q − r) cos 2θ + 2 qr sin 2θ . √ √ 2 (q − r) cos 2θ (q − r − 2 qr) cos 2θ sin 2θ (2 qr cos2 2θ + (q + r) sin2 2θ) 0 √ 0 0 0 qr (4.54) q+r The Mueller matrix for diattenuation defocus across a pupil with axes x and y is constructed from equation 4.54 by adding quadratic variation to q along x and y by setting r = 1 − D0 (x2 + y 2 ), and writing the orientation in terms of x and y, where θ = arctan (x, y). Then linear diattenation defocus can be written in function form as LD q = 1, r = 1 − D0 x2 + y 2 , θ = arctan(x, y) . (4.55) 109 Tilt and piston occur as the center of the pupil moves off-axis, and are added by shifting the quadratic terms and orientation by x0 and yo , LD q = 1, r = 1 − D0 (x − x0 )2 + (y − y0 )2 , θ = arctan(x − x0 , y − y0 ) . (4.56) Similarly, the Mueller matrix for a linear retarder with retardance δ at orientation θ is 0 0 0 1 2 2 δ 0 cos2 2θ + cos δ sin 2θ sin sin 4θ − sin δ sin 2θ 2 LR[δ, θ] = , (4.57) 0 2 δ 2 2 sin 2 sin 4θ cos δ cos 2θ + sin 2θ cos 2θ sin δ 0 sin δ sin 2θ − cos 2θ sin δ cos δ so the Mueller matrix for retardance defocus is LR δ = R0 x2 + y 2 , θ = arctan(x, y) . (4.58) Again, tilt and piston are added by shifting the quadratic terms and orientation by x0 and y0 , LR δ = R0 (x − x0 )2 + (y − y0 )2 , θ = arctan(x − x0 , y − y0 ) . (4.59) Figure 4.12 illustrates the quadratic variation of the diattenuation defocus from the center of the pupil outwards, as well as the shifted diattenuation defocus which is a result of adding tilt terms x0 and y0 . For an incident horizontally polarized planewave, the on-axis diattenuation defocus pupil will transmit the highest percentage of the incident polarization along the horizontal axis and the minimum percentage 110 Diattenuation Defocus On-Axis Linear Shift Along y-Axis Off-Axis in x and y Figure 4.12: Diattenuation defocus shifts along the y-axis and off-axis in x and y, illustrating the effect of tilt on diattenuation defocus in the pupil. Retardance defocus looks identical to diattenuation defocus, but the effect is different. Diattenuation defocus affects the transmission of polarization states, whereas retardance defocus adds a given retardance to the transmitted polarization state. of the incident horizontal polarization along the vertical axis. A pupil line map of retardance defocus looks identical to the diattenuation defocus shown in figure 4.12, except the lines represent the fast-axis. While diattenuation defocus affects the transmission and orientation of incident polarization states, retardance defocus adds polarization dependent phase and ellipticity to the transmitted polarization state. On-axis, the retardance increases quadratically towards the edge of the pupil and is rotationally symmetric. Off-axis, the retardance across the pupil is assymmetric. Because the retardance increases quadratically from the center to the edge of the pupil, the off-axis case is more retarding on the far edge of the pupil. 4.4.2 Depolarization effects from diattenuation and retardance defocus The depolarization effects and their Mueller matrix roots parameters were studied for these fundamental polarization aberrations. These polarization aberrations cause polarization variations in the point spread function (PSF), derived by (McGuire and Chipman, 1994). When a polarimeter measures the entire PSF on one pixel, it 111 averages the Stokes vectors over the PSF. Thus averaging the Mueller matrix over the pupil relates input to measured Stokes parameters and describes the associated depolarization. Numerical integration of diattenuation or retardance defocus across the pupil provides insight on the cumulative effects of these polarization aberrations in an optical system. The Mueller matrices for diattenuation and retardance defocus (from equations 4.56 and 4.59 are each integrated across a unit circular pupil, Z 1 √ Z √ − 1−x2 −1 Z 1 −1 1−x2 √ Z LD 1, 1 − D0 (x − x0 )2 + (y − y0 )2 , arctan(x − x0 , y − y0 ) dydx (4.60) 1−x2 √ LR R0 (x − x0 )2 + (y − y0 )2 , arctan(x − x0 , y − y0 ) dydx, (4.61) − 1−x2 where D0 = 0.3 and R0 = 1. To understand the depolarization for off-axis fields, this operation is repeated for a series of shifts in x0 and y0 in the upper right quadrant, moving the center of the pupil off-axis. Figure 4.13 shows the diagonal depolarization parameters D14 and D15 from the diattenuation defocus pupil, integrated at varying pupil shifts off-axis. The field on the vertical axis of figure 4.13 is calculated from the radius of the shift p in x0 and y0 , x20 + y02 . The magnitude of the diagonal depolarization terms D14 and D15 generated by the diattenuation defocus decreases as the pupil shifts off-axis. The depolarization is at its maximum when the pupil is centered on-axis because the difference in transmission between the different regions of the pupil are at their maximum. The depolarization is at its minimum when the pupil is centered off-axis because the variation in diattenuation orientation is at its minimum. Figure 4.14 shows the diagonal depolarization parameters D14 and D15 from the retardance defocus pupil, integrated at varying off-axis pupil shifts. The magnitude 112 Diagonal Depolarization from Diattenuation Defocus 1.0 0.8 Field 0.6 D15 0.4 D14 0.2 0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Diagonal Depolarization Figure 4.13: With diattenuation defocus, diagonal depolarization parameters D14 and D15 decrease in magnitude as the pupil center is shifted off-axis. of the diagonal depolarization D14 and D15 is at its minimum when the pupil is on-axis, and increases as the pupil shifts off-axis. The depolarization increases offaxis because the distribution of retardance in the pupil becomes asymmetric offaxis, and the fast axis varies less. Additionally, the amount of retardance increases quadratically with the radius of the pupil. When the pupil is shifted so that its center is almost at the edge of the aperture, the effective radius is larger, generating this quadratic increase in retardance across the pupil. These two effects lead to a roughly quadratic increase in magnitude of depolarization as the pupil moves off-axis. Figures 4.15 and 4.16 show cross-sections of the circular phase depolarization D12 and diagonal depolarization D13 generated from shifting the pupil off-axis in x0 . Integrating diattenuation defocus and retardance defocus across the pupil only results in diagonal depolarization (D13 , D14 , and D15 ) and circular phase depolarization (D12 ). This result is consistent with the averaging operations in Section 113 Diagonal Depolarization from Retardance Defocus 1.0 0.8 Field 0.6 D15 0.4 D14 0.2 0.0 0.0 0.5 1.0 1.5 2.0 Diagonal Depolarization Figure 4.14: With retardance defocus, diagonal depolarization parameters D14 and D15 increase in magnitude as the pupil center moves off-axis. Depolarization from Diattenuation Defocus HCross-SectionL 0.6 0.5 x0 0.4 D12 0.3 0.2 D13 0.1 0.0 -0.004 -0.002 0.000 0.002 0.004 Depolarization Figure 4.15: Circular phase depolarization D12 and diagonal depolarization D13 vary with pupil shift x0 . 114 Depolarization from Retardance Defocus HCross-SectionL 1.0 0.8 0.6 x0 D12 0.4 D13 0.2 0.0 -0.025 -0.020 -0.015 -0.010 -0.005 0.000 0.005 0.010 Depolarization Figure 4.16: Circular phase depolarization D12 and diagonal depolarization D13 vary with pupil shift x0 . 3.5.2. In the diattenuation defocus integration, linear diattenuation was averaged with linear diattenuation at varying orientations, so we expect diagonal depolarization and circular phase depolarization to result. Similarly, when retardance defocus was integrated across the pupil, linear retardance was averaged with linear retardance at varying orientations. This produces circular phase depolarization as well. In a realistic optical system, a surface would experience both diattenuation defocus and retardance defocus, which would produce diagonal depolarization and circular amplitude depolarization D9 . 115 CHAPTER 5 POLARIZATION SYNTHESIS BY COMPUTER GENERATED HOLOGRAPHY USING ORTHOGONALLY POLARIZED AND CORRELATED SPECKLE PATTERNS This chapter is adapted from H. Noble et. al., “Polarization synthesis by computer generated holography using orthogonally polarized and correlated speckle patterns,” Opt. Lett. 35, 3423-3425 (2010). 5.1 Introduction Computer-generated holograms (CGHs) can efficiently produce bandwidth-limited arbitrary irradiance in reconstructed images (Lohmann and Paris, 1967),Poon (2006). This Letter addresses application of CGHs to the formation of a specific irradiance and polarization distribution in the image. Several authors have discussed polarization selective CGHs, where orthogonal illumination polarization states act independently. For example, Hossfeld et al. proposed a system with a birefringent crystal for this purpose (Hossfeld et al., 1993). Xu et al. demonstrated polarization selective CGHs that apply an independent phase profile to orthogonal polarization states using birefringent substrates joined face to face (Xu et al., 1995), as well as form birefringent nanostructures (Xu et al., 1996). These methods fail to produce arbitrary polarization and irradiance in the image, because the phase relationship of the orthogonal images is not considered in the design. One method for producing arbitrary polarization and irradiance is image segmentation, where different por- 116 tions of the image are generated from separate CGHs illuminated with the desired polarization state of each image portion (Tsuji, September 4 2007). But in order to realize a continuously changing polarization direction, infinite sets of sub-CGHs are required. Therefore, a more elegant method is to synthesize an arbitrary polarization distribution from two orthogonally polarized CGH images. The advantage of this approach is its simplicity. Todorov et al. showed that analog recording of two holograms in a photoanisotropic medium can produce a coherent combination of polarization states in the overlap region of the diffracted images, depending on the polarization state of the illuminating beam (Todorov et al., 1985). However, Todorovs analog recordings did not demonstrate computer design of an arbitrary image pattern, and the image polarization state did not vary in the overlap region of orthogonal polarization states. This Letter demonstrates design, construction, and measurement of a synthesized tangentially polarized annulus using an interlaced polarization CGH (PCGH) that is illuminated with alternating columns of x- and y-polarized light. A synthesized hologram consisting of a correlated speckle pattern is produced from a coherent combination of these two orthogonal linear polarization states in the image. 5.2 PCGH illumination The system illumination is a 15 mm diameter spatially uniform plane wave from a linearly polarized 633 nm He-Ne laser. As shown in Fig. 5.1, the laser power is divided equally between x- and y-polarized channels by placing a polarization beam splitter (PBS) after a polarizer oriented at 45o . Beams reflect from 45o fold mirrors and transmit through offset 50% duty cycle Ronchi rulings with 67.2 µm periods (which contain one dark and one light stripe). A PBS recombines the 117 Figure 5.1: The interlaced PCGH is illuminated with shifted cross-polarized Ronchi rulings, and its reconstruction and polarization distribution are measured. orthogonally polarized channels. The optical path length of one channel is adjusted with a piezoelectric transducer on the fold mirror, so that without the Ronchi rulings, the combined beams interfere to form a linearly polarized state oriented at 45o to the x-axis. The Ronchi rulings are used as arrays of masks and apertures that are imaged onto the PCGH with a 4f system. The rulings are offset from each other by half a period to provide alternating stripes of x and y polarization, which correspond to the interlaced channels of the PCGH. A 12-bit CCD camera is used at the back focal plane of the imaging lens to view the reconstructed image. The camera is used at a plane conjugate to the PCGH and Ronchi rulings to view and align them. A linear polarizer is rotated before the camera to analyze the reconstructed images polarization properties. The target image was selected to be a tangentially polarized ring, as shown in Fig. 5.2, because it is a continuously varying polarization. The ring can be decomposed 118 Figure 5.2: x- and y-polarized components add to form a tangentially polarized annulus. into x- and y-polarized components, where 0 Ux0 = A0x eiφx = −gs sin θ, 0 Uy0 = A0y eiφy = gs cos θ. (5.1) (5.2) Ax0 and Ay0 are amplitude functions, φ0x and φ0y are phase functions, and gs is a common structure function that consists of a binary ring with an inside diameter of 80% of the outer edge. All of these functions depend on r and θ. Overlap of the ~ 0. two ideal coherent states produces the ideal tangentially polarized ring U 5.3 PCGH diffuser design A flowchart of the modified Gerchberg-Saxton design method (Gerchberg and Saxton, 1972) is shown in Fig. 5.3. j is the iteration number, where N iterations are performed. Diffusers for Fourier CGHs generally have the advantage of reducing CGH- modulus dynamic range, but they alter polarization. For our application, 119 using a common diffuser for the two po- larization channels allows dynamic-range reduction without affecting polarization. An initial common diffuser is defined as an array of uniformly random phase pixels over [−π, π]. In the initial step (j = 1) of the calculation loop, ideal images defined by Eqs. 6.1 and 6.2 are multiplied by the random diffuser to create two parallel channels (x and y) for the calculation. In Step 2, Fourier transforms are applied to each channel, yielding the encoded wavefronts for the PCGHs. In Step 3, the modulus of each transform is set to unity, and each transform is interlaced according to the alternating stripes of x- and y-polarizations from the Ronchi illumination. If j = N , the calculation is stopped, and individual phases are used to fabricate the interlaced PCGH. In Step 4, inverse Fourier transforms are applied to the phase distributions of the results from Step 3, bringing the calculation back into the space of the image. Ideal object phases φ0x and φ0y , derived from Eqs. (1) and (2), are then removed from image estimates Ux1 and Uy1 in Step 5 to extract the diffuser-only phase. In Step 6, the resulting complex amplitudes ~ 1 . This step contains a novel departure from Gerchbergare averaged to produce U Saxton algorithms that consider only independent channels. The common diffuser ~ 1 . For the next loop (j = 2), φ1 is used phase φ1D is simply the phase angle of U D as the starting point for the diffuser in Step 1. After typically N = 50 iterations, the diffuser converges to produce acceptable characteristics of image estimates Uxj and Uyj , when the phases calculated in Step 3 are used as the individual components of the PCGH. Application of an aperture function or introduction of phase quantization can be performed at Step 3. Averaging complex amplitudes in Step 6 assures that the target image for the next iteration contains correlated speckle between the two channels on a microscopic scale. Although multiplication of the ideal objects in Step 1 of the next iteration 120 Figure 5.3: Flowchart of the PCGH design method. does not guarantee that correlation is maintained in the PCGH, a sufficiently high number of iterations N consistently shows that speckle correlation can be produced with sufficient quality. 5.4 PCGH fabrication The PCGH is fabricated in polyimide photoresist on a fused silica substrate. The photoresist is gray-scale patterned with the University of Arizona Maskless Lithography Tool, a direct laser writer that scans the photoresist with a focused UV light spot of varying power (Tamkin et al., 2003). After the photoresist is exposed, it is developed in a 3 : 1 solution of deionized water and developer for 2 to 4 minutes. Gray-scale exposure of the positive resist results in topographical features after development, producing phase changes in transmission through the PCGH when used in the reconstruction setup. The samples are postbleached with high-intensity halogen light after fabrication to stabilize the refractive index. 121 PCGH matrices for x- and y- polarizations are interlaced as shown in Fig. 5.1, where the gray scale corresponds to PCGH phase (0 =black, 2π =white). Each interlace stripe is 33.6 µm wide, which corresponds to four 8.4 µm pixels of the component PCGH. The 67.2 µm periods of the Ronchi rulings are designed to overlap exactly with y-oriented columns of the PCGH. x- (y-) polarized light illuminates only the x- (y-) component columns of the PCGH. A total of 512 × 512 pixels in the PCGH yields a side dimension of approximately 4.3 mm. The rectangular PCGH interlacing produces diffracted orders in the image. In this analysis, only zero-order properties are considered. The zero-order diffraction pattern is a ring diverging at 0.5o full width. Higher diffraction orders lie outside the zero-order ring. 5.5 PCGH measurement and reconstruction The linear Stokes vector {S0 , S1 , S2 }T of the image is measured by rotating a linear analyzer to orientations of 0o , 45o , 90o , and 135o in front of the CCD camera and applying the following equations (Goldstein, 2003): S0 = 41 (I90o + I0o + I45o + I135o ) S1 = I0o − I90o S2 = I45o − I135o (5.3) The measured linear Stokes vector image in Fig. 5.4 displays a tangential orientation, demonstrating the achievement of polarization synthesis. Figure 5.5 shows that the linear polarization orientation is tangential to the annulus, with horizontal orientation at the top of the annulus and vertical orientation at the sides of the annulus. The measured polarization orientation is nonuniform, differing by several degrees from the desired orientation at the sides of the annulus. 122 Figure 5.4: Normalized linear Stokes vector components demonstrate polarization synthesis. Figure 5.5: Polarization orientation rotates around the annulus. 123 Degree of linear polarization (DoLP) is determined in terms of the Stokes vector as DoLP = (S12 +S22 )1/2 /S0 . DoLP=1 is linearly polarized light and DoLP=0 is either unpolarized or circularly polarized. The average DoLP over the annulus is 0.88, with a standard deviation of 12.8%. (The DoLP distribution is non-symmetric about the mean.) Unpolarized regions result from uncorrelated speckle patterns, while circularly or elliptically polarized regions result from correlated speckle patterns that are out of phase with each other. The algorithm outlined in Fig. 5.3 generates correlated speckle patterns from each polarization component of the PCGH, but experimental constraints preclude an ideal result. The synthesis regions (at ±45o and ±135o around the ring) have a lower DoLP than the vertical and horizontal regions of the annulus because the DoLP in these regions is dependent on correlation of speckle from the two orthogonal polarizations. The ratio of polarization (RoP) is a metric to evaluate the quality of tangential polarization around the annulus as a fraction of flux in the reconstructed hologram, where 1 IT = RoP = S0 2 S2 S1 cos 2θ + sin 2θ . 1+ S0 S0 (5.4) IT is the transmission through a properly oriented tangential analyzer, orientation angle with respect to the x-axis is θ, and S0 is total irradiance. Average RoP around the ring is 88%, with a standard deviation of 6%. 5.6 Conclusions In conclusion, the first demonstration, to the best of our knowledge, of computerdesigned polarization synthesis is achieved with a tangentially polarized annulus that combines two linearly polarized images from interlaced CGHs. This new type of hologram is called a PCGH. The design algorithm and fabrication produced cor- 124 related speckle in the experimental image. The correlated speckle resulted in a high degree of properly oriented linear polarization. 125 CHAPTER 6 SQUARE WAVE RETARDER FOR POLARIZATION COMPUTER GENERATED HOLOGRAPHY This chapter is reformatted from H. Noble et. al., “Square-wave retarder for polarization computer generated holography,” Appl. Opt. 50, 3703-3710 (2011). 6.1 Introduction Computer-generated holograms (CGHs) can efficiently produce bandwidth-limited arbitrary irradiance in reconstructed images (Lohmann and Paris, 1967; Dallas, 2006; Schnars and Jueptner, 2005). Several authors have discussed polarization selective CGHs, where orthogonal illumination polarization states remain spatially distinct. For example, Hossfeld et al. proposed a system with a birefringent crystal for this purpose (Hossfeld et al., 1993). Xu et al. demonstrated polarization selective CGHs that apply an independent phase profile to orthogonal polarization states using birefringent substrates joined face to face (Xu et al., 1995) as well as formbirefringent nanostructures (Xu et al., 1996). It has also recently been shown that polarization-selective CGHs can be fabricated with arbitrary amplitude and phase of each pixel by recording a birefringent structure with a defined retardance and orientation of the fast-axis into a photopolymer film (Fratz et al., 2009). However, these methods fail to produce arbitrary polarization and irradiance in the image because the phase relationship of the orthogonal images is not considered in the design. One method for producing arbitrary polarization and irradiance is image 126 Figure 6.1: Interlaced x and y-polarized components add to form a tangentially polarized annulus. Overlap of these two coherent states produces the ideal tangentially ~ o with a continuously varying polarization orientation. polarized ring U segmentation, where different portions of the image are generated from separate CGHs illuminated with the desired polarization state of each image portion (Tsuji, September 4 2007). But in order to realize a continuously changing polarization direction, infinite sets of sub-CGHs are required. A more elegant method that avoids this limitation is to synthesize an arbitrary polarization distribution from two orthogonally polarized CGH images. The advantage of the synthesis approach is its simplicity. In recent work, a new type of hologram was demonstrated. The hologram achieved a synthesized tangentially polarized annulus using an interlaced CGH illuminated with alternating columns of x- and y-polarized light (Noble et al., 2010). This new type of hologram is called a polarization computer generated hologram (PCGH). A synthesized hologram consisting of a correlated speckle pattern was produced from a coherent combination of these two orthogonal linear polarization states in the image. 127 The target image in (Noble et al., 2010) was selected to be a tangentially polarized ring, as shown in Fig. 6.1, because it is a continuously varying polarization. The ring can be decomposed into x- and y-polarized components, where 0 Ux0 = A0x eiφx = −gs (r) sin θ, 0 Uy0 = A0y eiφy = gs (r) cos θ. (6.1) (6.2) Ax0 and Ay0 are amplitude functions, φ0x and φ0y are phase functions, and gs is a common structure function that consists of a binary ring with an inside diameter of 80% of the outer edge. Overlap of the two ideal coherent states produces the ideal tangentially polarized ring . In (Noble et al., 2010) orthogonally polarized striped illumination on an interlaced PCGH was generated from two complementary Ronchi-ruling amplitude masks in orthogonally polarized optical paths that were combined with a polarizing beam splitter (PBS). This proof-of-concept experiment provided a basic demonstration of the PCGH concept. Ultimately, the final goal of this research is to construct a PCGH from a single, integrated optical element. The intermediate step, discussed here, is to develop a single illuminator element for the interlaced PCGH. This optical element is called a square-wave retarder, due to its alternating stripes of half-wave/integer-wave retardance. The square-wave retarder replaces the orthogonally polarized Ronchi-ruling amplitude masks from previous work in (Noble et al., 2010), by providing alternating stripes of orthogonally polarized illumination to the PCGH. This paper describes the design, fabrication, and performance of an etched calcite square-wave retarder used as an illuminator for an interlaced PCGH. Together, these components produce a speckled, tangentially polarized PCGH diffraction pattern at 128 the image plane. Section 6.2 describes the experimental geometry and design technique. Section 6.3 discusses the square-wave retarder design and fabrication, and Section 6.4 presents measurement results. Section 6.5 lists alignment and illumination errors, and Section 6.6 contains conclusions from this work. 6.2 Experimental geometry A polarization compensator system is used to provide an arbitrary polarization state incident on the square-wave retarder. As shown in Fig. 6.2, laser power (λ= 633nm) is divided equally between x-polarized and y-polarized channels by placing a polarization beam splitter (PBS) after a polarizer oriented at 45o in the x-z plane. Beams reflect from fold mirrors and recombine at a second PBS, forming a Mach-Zehnder compensator. One fold mirror is placed on a piezoelectric transducer (PZT) for output polarization adjustment. For an ideal square-wave retarder, the polarization state resulting from the compensator is a linearly polarized plane oriented at 45o in the x-y plane. The polarization state incident onto the calcite is adjusted to compensate for thickness and wedge errors in the calcite square-wave retarder, which is arranged such that the crystals fast axis is oriented in the x-direction. Fold mirror tilt corresponds to a linear shift in phase, counteracting any linear change in retardance from residual wedge in the calcite. Mirror translation with the PZT corresponds to adding or subtracting retardance, which balances out excess retardance from nonoptimal calcite thickness. The compensators polarization state is adjusted until the polarization states exiting the square-wave retarders grooves alternate in stripes oriented along the y-axis, which is rotated at 45o with respect to the y-axis. A 4f optical system images transmitted light from the square wave retarder onto 129 Figure 6.2: A polarization compensator system (enclosed in the dashed line region) provides an arbitrary polarization state incident on the etched calcite square wave retarder. A 4f optical system images the square wave retarder to illuminate the interlaced CGH, providing alternating stripes of x and y polarization. A CCD camera is used at the back focal plane of an imaging lens to view the reconstructed image. 130 the interlaced PCGH, providing alternating stripes of x and y polarization. The PCGH is also aligned along the x-y axes so that the interlaced columns are aligned with their corresponding square wave retarder grooves. A 12 bit, 2 Megapixel Olympus Microfire CCD camera with less than 0.5% polarization sensitivity is used at the back focal plane of an imaging lens to view the reconstructed image. The camera is used at a unit-magnification plane conjugate to the PCGH and square-wave retarder to view and align those components. A rotatable linear polarizer (analyzer) and quarter-wave retarder (QWP) are placed before the camera to measure Stokes parameters of the reconstructed image and to align the compensator system. PCGH matrices for x and y polarizations are interlaced, as shown in Fig. 6.2, where gray scale corresponds to PCGH phase (0 = black, 2π = white). Each PCGH interlace stripe is 33.6 µm wide, which corresponds to four 8.4 µm pixels of the component PCGH. The 67.2 µm period of the square-wave retarder overlaps with y-oriented columns of the PCGH. Thus, x-polarized light illuminates only the xcomponent columns of the CGH and y-polarized light illuminates the y-component columns. Overall PCGH dimensions are 4.3mm x 4.3mm with a total of 512 x 512 pixels. Rectangular PCGH interlacing produces diffracted orders in the image. The zero-order diffraction pattern is an annular ring that diverges at a full angle of 0.5o . Higher diffraction orders lie outside the zero-order ring and are not considered in this work. The PCGH diffuser is designed using a modified Gerchberg-Saxton technique (Noble et al., 2010; Gerchberg and Saxton, 1972). An initial common diffuser is defined as an array of uniformly random phase pixels over [−π, π]. In the initial step of the calculation loop, ideal images defined by Eqs. 6.1 and 6.2 are multiplied by the random diffuser to create two parallel channels (x and y) for the calculation. Then, 131 Fourier transforms are applied to each channel, which convert the calculation in to the space of the PCGH. Next, the modulus of each transform is set to unity, and each transform is interlaced according to the alternating stripes of x and y polarizations from the square-wave retarder illumination. The inverse Fourier transforms are applied to the phase distributions of the results, which bring the calculation back into the space of the image. Ideal object phases φ0x and φ0y , derived from Eqs. 6.1 and 6.2 are removed from image estimates in order to extract the diffuser-only ~ 1 . This phase. Next, the resulting complex amplitudes are averaged to produce U step contains a novel departure from Gerchberg-Saxton algorithms that consider only independent channels. The common diffuser phase is simply the phase angle of ~ 1 . For the next iteration loop, this common diffuser phase is used as the starting U point for the diffuser. After typically 50 iterations, the diffuser converges to produce acceptable characteristics of image estimates. Individual phases of the two channels are used to fabricate the interlaced PCGH. For our application, using a common diffuser for the two polarization channels allows dynamic-range reduction without affecting polarization. Using this diffuser algorithm from (Noble et al., 2010) assures that speckle patterns generated from each polarization component of the CGH are correlated in the image. This correlation is critical, because complex fields of the interlaced x and y-polarized components do not add to form a tangentially polarized annulus unless individual speckles from the two components overlap and are in phase to within a fraction of the wavelength. An advantage of this geometry over previously reported work (Noble et al., 2010) is that the calcite square-wave retarder and PCGH components are on a common path for x and y polarizations. Moreover, the 4f system between them is implemented for ease of experimental adjustment and alignment tolerance testing. 132 In a commercial realization, the two components could be combined into a single plate without the 4f system between them. 6.3 Calcite square-wave retarder design and fabrication The ideal square-wave retarder is designed to generate alternating stripes of orthogonal linear polarizations when illuminated with a linearly polarized plane wave oriented at 45o with respect to the crystals fast axis, as illustrated in Fig. 6.2. Figure 6.3 shows the construction of the square-wave retarder. The total substrate thickness corresponds to an integer number of waves of retardance. Grooves are etched into the substrate to the base thickness level in order to produce half-wave plate (HWP) ridges that rotate the incident linear polarization by 90o . The grooves are filled with oil that matches the calcites extraordinary refractive index. The ideal square-wave retarder has no residual wedge in the base. However, the fabricated square-wave retarder has both thickness errors in the base and a small wedge error. In principle, the base thickness of the square wave retarder is arbitrary if the incident polarization state can be tuned with the compensator. One wave of retardance in calcite at normal incidence is generated by each 3.7 µm of material, making it unfeasible to polish to 1/100 of a wave precision. The residual wedge corresponds to a linear phase shift along the un-etched surface of the square-wave retarder, and is typically under 10 µm across the width of the substrate. In order to produce alternating stripes of x and y polarizations, the etched calcite must be illuminated with a matched elliptical polarization state that exhibits a linear phase shift, depending on how many non-integral waves of retardance are produced by the base thickness and the magnitude of the wedge. This elliptical polarization state is produced with the compensator by adjusting the optical path difference between 133 Figure 6.3: The calcite square wave retarder generates alternating stripes of orthogonal linear polarizations when illuminated with a linearly polarized plane wave oriented at 45o with respect to the crystals fast axis. Grooves are etched into the substrate to the base thickness level and are filled with index matching oil to produce half-wave plate ridges which rotate the incident linear polarization by 90o . 134 the two arms and adjusting the fold mirror tilt until polarization states exiting the calcite are optimized for x and y polarization. Quality of the x and y polarization states is determined by observing the square-wave retarder image through the CCD without the PCGH in the system and aligning the analyzer separately to each desired linear state. The QWP is removed for this alignment procedure. The square-wave retarder is made from a naturally formed calcite crystal. A 3 mm thick calcite substrate is cut from the crystal using a TechCut precision sectioning saw. The substrate is mechanically lapped and polished to a base thickness of approximately 500 µm and then chemically polished in a weak solution of HCl and deionized water, which achieves an average surface roughness of 26 nm. The calcite substrate is cleaned in four stages using acetone, isopropanol, deionized (DI) water, and a plasma cleaning chamber. It is then glued onto a fused silica flat substrate with UV epoxy so that the optical axis of the calcite plate is parallel to the diagonal of the rectangular fused silica plate. A binary AZ3312 photoresist is spin coated at 7500 rpm for 30 seconds onto the calcite surface, and the coated calcite is soft baked at 85o C for one minute. The photoresist is written with 128 lines of 33.6 µm width parallel to one side of the fused silica plate using the University of Arizona Maskless Lithography Tool (Tamkin et al., 2003), making a square exposed region with a length of 4.3 mm on each side. The sample is developed in AZ300MIF for 17 seconds. It is wet etched in 1 part 37% HCl to 5000 parts DI water in small steps until the correct groove depth of 1.85 µm (plus the thickness of the photoresist) is achieved, as measured with a Veeco NT9800 profilometer. This processing requires a total etching time of 50 minutes. The final fabrication step is to fill the grooves with an index matching fluid, corresponding to the extraordinary index of the calcite. A fused-silica flat is glued 135 over the fluid filled grooves with UV epoxy. The index matching fluid is necessary so that the orthogonal linearly polarized stripes remain in phase with each other. Without index matching fluid, the phase of the y-polarized output from the square wave retarder is retarded by λ/10 with respect to the x-polarized output, because the optical path length (OPL) of the y-polarized output is larger than the OPL of the x-polarized output. If the orthogonal illumination stripes do not remain in phase, their complex fields will add to form an elliptically polarized field in the PCGH reconstruction, which reduces the percentage of the reconstruction irradiance in the correct polarization state. In order to examine the quality of the etched calcite grooves, a sacrificial square wave retarder was coated with platinum, cleaved perpendicularly to the groove orientation, and examined with a scanning electron microscope (SEM). The SEM images in Fig. 6.4 show some rounding in the groove corners, as well as a slightly deeper etch depth in those corners. Overall, the groove surfaces remain smooth and uniform. A dual rotating retarder Mueller matrix imaging polarimeter (MMIP) is used to measure polarization properties of several 1 mm2 sections over the aperture of the square-wave retarder (Smith, 2002). Figure 6.5 shows the measured retardance over one 1 mm horizontal cross-section from the center of the sample. Retardance associated with the HWP ridges varies between approximately 160o and the ideal 180o . Ridge retardance at the edge of the sample ranges between 140o and 160o , which can be attributed to non-uniform etch rates between the center and edges of the sample. A 160o ridge retardance results in 17.6% ellipticity, while 140o of retardance results in 36.4% ellipticity. These elliptical components contribute to a slightly lower degree of linear polarization (DoLP) in the PCGH illumination compared to ideal conditions. 136 Figure 6.4: An SEM cross-section of the fabricated calcite for the square-wave retarder. The grooves are smooth and uniform, although there is some rounding of the corners. 6.4 Measured polarization properties of the reconstructed image The Stokes vector {S0 , S1 , S2 , S3 }T of the annular PCGH reconstruction is measured by rotating a quarter-wave retarder (QWP) and linear analyzer, which are placed in front of the CCD camera, at several distinct orientations according to the following formula (Goldstein, 2003): S0 S1 S2 S 3 = 1 [I(90o ) 4 + I(0o ) + I(45o ) + I(135o )] = I(0o ) − I(90o ) o (6.3) o = I(45 ) − I(135 ) = 1 [I(45o , 90o ) p2 − I(135o , 90o )] where I(θ) is the intensity measured with a polarizer oriented at θ and I(θ, φ) is the intensity measured with a polarizer orientated at θ and a retarder oriented at φ. 137 Figure 6.5: Retardance of a 1 mm wide horizontal cross section at the center of the calcite square wave retarder sample was measured at 632.8 nm using the Mueller matrix imaging polarimeter. Retardance associated with the half wave plate ridges varies between approximately 160o and the ideal 180o in the center of the sample. 138 Figure 6.6: The measured Stokes vector for the PCGH reconstruction shows that the polarization is linearly tangential with some circular polarization dispersed throughout the annulus. S3 shows that the presence of circular polarization is most notable in the synthesis regions. S1 , S2 , and S3 are normalized to S0 , which remains unnormalized. The transmittance factor of the retarder is p2 , which accounts for absorption and surface reflections. The measured and Stokes vector of the PCGH in Fig. 6.6 shows that the polarization is primarily tangential in orientation. S0 remains unnormalized, while other terms are normalized with respect to S0 . At the top and bottom of the annulus in the s1 = S1 /S0 image, the value is nearly +1, corresponding to horizontally polarized light, while at the left and right sides the s1 value is nearly -1, corresponding to vertically polarized light. Analogously, the s2 = S2 /S0 image shows that the light is 45o polarized in the upper right and lower left corners of the annulus, and 135o polarized in the upper left and lower right corners of the annulus. The image shows that the presence of circular polarization is most notable in the synthesis regions. Figure 6.7 shows the orientation ϕ of the major axis of the polarization ellipse 139 Figure 6.7: The orientation ϕ of the major axis of the polarization ellipse around the annulus is calculated from the normalized Stokes vector. The measured polarization orientation of the PCGH reconstruction is tangential. Its orientation differs by several degrees from the desired orientation at the sides of the annulus. calculated from the normalized Stokes vector: ϕ= S2 1 tan−1 . 2 S1 (6.4) The orientation is tangential, with horizontal orientation at the top of the annulus and vertical orientation at the sides of the annulus. The measured polarization orientation differs by several degrees from the desired orientation at the sides of the annulus, which is visible in Fig. 6.7 and in the Stokes vector element s1 of Fig. 6.6. Simulations verify that the non-uniform orientation around the annulus can be attributed to errors in the orientation of the calcite optical axis with respect to the etched stripes, which is further discussed in Section 6.5. Figure 6.8 shows the degree of linear polarization (DoLP = (S12 + S22 )1/2 /S0 ) over the annulus, where DoLP =1 represents linearly polarized light and DoLP=0 represents either unpolarized or circularly polarized light. The average DoLP over 140 Figure 6.8: The pixel-by-pixel degree of linear polarization (DoLP) for the PCGH reconstruction. The average DoLP over the annulus is 0.81. The synthesis regions have a lower DoLP than the vertical and horizontal regions of the annulus because the DoLP in these regions is dependent on a high degree of speckle correlation from the two orthogonal polarizations. the annulus, weighted by irradiance, is 0.81 with a spatial standard deviation of 20%. The DoLP is less than 1 partly on account of the circularly polarized component s3 = S3 /S0 , as the average degree of polarization (DoP) over the annulus is 0.895. The spatial standard deviation of 20% is due to very low DoLP values in the dim regions between speckles. Unpolarized regions result from uncorrelated speckle, while circularly or elliptically polarized regions result from correlated speckle that are out of phase with each other. The synthesis regions (at ±45o and ±135o around the ring) have a lower DoLP than the vertical and horizontal regions of the annulus, because the DoLP in these regions is dependent on a high degree of speckle correlation between the two orthogonal polarizations. The speckles are well correlated, thus 89.5% DoP is achieved. The PCGH diffuser design algorithm from (Noble et al., 2010) generates correlated speckle patterns from each polarization component of the PCGH, but experimental constraints preclude an ideal result. These constraints are 141 further discussed in Section 6.5. The ratio of polarization (RoP) is a metric defined to calculate the fraction of tangentially polarized light around the annulus relative to total flux in the reconstructed image. RoP=1 is the ideal case (when 100% of the light is tangentially polarized) and RoP=0.5 represents unpolarized or circularly polarized light, half of which is in the desired tangential state. The RoP can be calculated pixel-by-pixel from the measured (un-normalized) Stokes vector and the angular location θ in the annulus, where 1 IT = RoP = S0 2 S1 S2 1+ cos 2θ + sin 2θ . S0 S0 (6.5) IT is the irradiance transmitted through a properly oriented tangential analyzer, S0 is total irradiance, and the orientation angle in the annulus with respect to the x-axis is θ. The measured RoP for each pixel is shown in Fig. 6.9. The RoP is noticeably lower in the synthesis regions of the annulus and in the background regions where the speckle is not present. Average RoP around the ring is 84% with a standard deviation of 6.9%. In previous work (Noble et al., 2010), the PCGH reconstructions average DoLP was 0.88, and the RoP was 88%. These values compare to a DoLP of 0.81 and RoP of 84% in this study, where the square-wave retarder provides orthogonally polarized illumination in place of the orthogonally polarized Ronchi-ruling amplitude masks from (Noble et al., 2010). Systematic errors from the compensator are present in both PCGH illumination systems, but the calcite square wave retarder in this study adds fabrication errors and increased opportunities for optical misalignments. These variables account for the slight decrease in DoLP and RoP compared to (Noble et al., 2010). 142 Figure 6.9: Pixel-by-pixel ratio of tangential polarization (RoP) to flux for the PCGH reconstruction. The RoP is noticeably lower in the 45o and 225o regions of the annulus and in the dim regions between speckles. Average RoP around the ring is 84%. 6.5 Alignment and illumination errors Performance of the PCGH system depends on precise opto-mechanical alignment of the compensator, the square-wave retarder and the CGH. Performance is also affected by defects and fabrication tolerances in the square-wave retarder. Imperfections in the system produce decreased phase correlation in the PCGH reconstructions speckle pattern, which decreases the DoLP, the RoP, and contributes to the presence of circular polarization. A simulation which conceptually illustrates ideal output of the square-wave retarder is shown in Fig.6.10(a). If the square-wave retarders optical axis orientation with respect to the half-wave stripes is not exactly 45o (as shown in Fig. 2), polarization orientation exiting the half-wave stripes rotates, but polarization orientation exiting the full wave stripes does not. An error of ±θ in the calcites crystal axis orientation with respect to the etched stripes results in a ±2θ error in the y component of the polarization transmitted through the calcite, as illustrated in Fig. 6.10(b) for 143 Figure 6.10: Simulation of square-wave retarder output. (a) In an ideal optical system, the polarization orientation exiting the calcite square wave retarder alternates between x and y according to the grooves and ridge stripes; (b) An extraordinary axis orientation error of 10o with respect to the etched stripes causes an orientation error of 20o in the y polarized component exiting the calcite. an exaggerated optical axis misalignment of +10o . Actual fabrication tolerance for this alignment is approximately ±2o . Wedge in the base thickness, shown in Fig. 6.3, produces a linear variation in retardance across the square-wave retarder substrate. This is shown in the measured result of Fig. 6.11(a) for a 1 cm2 area of bare calcite substrate. This linear variation in retardance across the substrate produces a variation in ellipticity across the polarization states transmitted through the square-wave retarder, as shown in the simulation results of Fig. 6.11(b). Fig. 6.11(b) reveals significant variation in the x direction. However, tilt in a compensator mirror can produce an opposite linear phase shift across the field of illumination and introduce opposite ellipticity variations. Therefore the mirror can be used to compensate effects of the residual calcite wedge. Environmental factors such as temperature fluctuations and mechanical vibrations from the laboratory environment can cause temporal fluctuations in alignment. 144 Figure 6.11: Properties of residual wedge in the calcite. (a) MMIP measurement of retardance as a function of position in an ungrooved substrate exhibiting a small residual wedge; and (b) Conceptual simulation illustrating transmitted polarization from a square-wave retarder in the presence of a 0.3o residual wedge. To measure translational sensitivities, the square-wave retarder is translated in 10 µm steps (1.2 CGH pixels, 0.15 square wave grating period cycles) in the x direction up to 20 µm (2.4 CGH pixels, 0.3 square wave grating period cycles) from its optimized location, and the DoLP and RoP of the PCGH reconstruction are measured. The same measurements are performed for rotation of the square-wave retarder around the systems optical axis in 1o increments up to 2o. DoLP is nearly insensitive to both x translation and rotation about the optical axis, and DoLP only varies up to 2% from the aligned condition, which is well-within the precision of the measurement. RoP decreases dramatically with 1o of rotation about the central axis, going from 83.9% to 67.3%, and RoP only decreases to a comparable value of 70.2% after translation of 20 µm (2.4 CGH pixels, 0.3 square wave grating period cycles) along x. The RoP increases from 67.3% to 70.2% between 1o and 2o of rotation because as the rotation angle increases, portions of the grooves will overlap with adjacent grooves - therefore the degradation can experience some mild 145 improvement as a function of continued rotation before further degradation. Results of the translation and rotation tests are summarized in Table 6.1. Allowing for a decrease in RoP of 10%, translation and rotation tolerances can be determined from the translation and rotation tests, assuming linearity over small perturbations. A decrease in RoP of 10% corresponds to a translation of 14.84 µm or rotation of 0.6o . Table 6.1: RoP and DoLP are measured as a function of the horizontal translation of the calcite and calcite rotation about the optical axis. Translation RoP 0 µm 0 pixels 0 period cycles 83.9% 10 µm 1.2 pixels 0.15 period cycles 80.1% 20 µm 2.4 pixels 0.3 period cycles 67.3% Rotation 81% 80.7% 80.0% RoP DoP 0o 83.9% 81% o 1 67.3% 82.7% 2o 70.2% 82.9% 6.6 Conclusions A polarization computer generated hologram (PCGH) is demonstrated with a calcite square-wave retarder illumination geometry, where a square-wave grating pattern has been etched into a calcite plate. Transmission through the grating produces alternating stripes of orthogonally linearly polarized light. The PCGH reconstruction is a tangentially polarized annulus, which synthesizes two orthogonally linearly polarized images from the interlaced PCGH. The PCGH reconstruction produces correlated speckle in the experimental image, which results in a high fraction of properly oriented linear polarization. The RoP of the PCGH reconstruction is 84%, with a DoLP of 0.81 and DoP of 0.895. These results approach those of previously reported work that used different illumination geometry. The small decrease in performance is due to alignment and fabrication imperfections, as verified from 146 experimental and simulation analysis. An adjustable polarization compensator corrects residual wedge and non-integral retardance in the square wave retarder. An experimental alignment tolerance analysis indicates that translation errors must be less than 14.84 µm (1.8 CGH pixels, 0.22 square wave grating period cycles) and rotation errors must be less than 0.6o to maintain a RoP loss of no more than 10%. Although the experimental geometry included separated illumination and PCGH components, these experiments demonstrate that combined single-plate geometries may be successful. 147 APPENDIX A JONES N -MATRIX ALGORITHM The Jones N -matrix algorithm is written according to the flow chart shown in figure A.1. The global phase is extracted from the input Jones matrix and the pth root of the matrix is calculated. Then the infinitesimal diattenuation and retardance parameters (Dx , D45 , DR and δx , δ45 , δR ) will come directly from the real and imaginary parts (respectively) of the second, third, and fourth coefficients arising from the Pauli basis form of the Jones matrix’s nth root as written in equation 1.34. All of these parameters are then rescaled by the matrix power (multiplied by n), yielding the scaled polarization parameters of the input Jones matrix NJ . The Jones matrix for retardance is found by taking the imaginary part of the sum of Pauli basis matrices multiplied by their corresponding rescaled coefficient. Similarly, the Jones for diattenuation is found by taking the real part of the sum of Pauli basis matrices multiplied by their corresponding rescaled coefficients. Two examples of the Jones N -matrix algorithm are shown in figures A.2 and A.3. Figure A.2 illustrates the retardance parameters of a linear retarder with a retardance of π/4 as a function of its orientation θ. Similarly, the diattenuation parameters for a linear diattenuator with a minimum transmission of 0.8 and a maximum transmission of 1 is shown as a function of its orientation θ in figure A.3. In both cases, the three dimensional diattenuation and retardance parameters were calculated using the Jones N -matrix algorithm discussed here. 148 Extract global phase Input Jones matrix Take sum of real Pauli basis elements and rescaled coefficients to calculate diattenuation Jones matrix Calculate pth root of Jones matrix Output diattenuation and retardance Jones matrices Rescale infinitesimal diattenuation and retardance parameters: multiply by p Calculate infinitesimal diattenuation and retardance vectors from Eq. 1.33 Take sum of imaginary Pauli basis elements and rescaled coefficients to calculate retardance Jones matrix Output diattenuation and retardance vectors Figure A.1: Algorithm flow-chart for calculation of diattenuation and retardance vectors using Jones N-Matrix method. HorizontalVertical Linear Retardance RightLeft Circular Retardance 45°135° Linear Retardance Retardance Retardance Retardance 1.0 1.0 1.0 0.5 0.5 0.5 Out[34]= 1 2 3 4 5 6 Orientation 1 2 3 4 5 6 Orientation 1 -0.5 -0.5 -0.5 -1.0 -1.0 -1.0 2 3 4 5 6 Orientation Figure A.2: Retardance parameters of a linear retarder with a retardance of π/4 plotted as a function of θ. HorizontalVertical Linear Diattenuation Diattenuation Out[50]= RightLeft Circular Diattenuation 45°135° Linear Diattenuation Diattenuation Diattenuation 0.15 0.15 0.15 0.10 0.10 0.10 0.05 0.05 1 2 3 4 5 6 Orientation 0.05 1 2 3 4 5 6 Orientation 1 -0.05 -0.05 -0.05 -0.10 -0.10 -0.10 -0.15 -0.15 -0.15 2 3 4 5 6 Orientation Figure A.3: Diattenuation parameters of a linear diattenuator plotted as a function of θ. 149 APPENDIX B THE EFFECT OF REFLECTION ON DEPOLARIZATION PARAMETERS Mueller matrices which have been measured in reflection mode should be corrected for the change in coordinates. This correction is justified because after reflection, the z-component of the light propagation vector ~k is reversed. This assumes that the surface normal is in the z-direction. In order for the coordinate system to remain right-handed, one transverse coordinate must also change sign. After reflection, the coordinates change from (x, y, z) to (x, −y, −z). For this coordinate system, upon reflection, 45o polarized light changes to 135o light, and circularly (or elliptically) polarized light changes helicity (DeBoo and Chipman, 2004). Therefore the S2 and S3 components of the Stokes vector will change sign, with a normalized reflection Mueller matrix R as follows: 0 1 0 0 0 1 0 0 R= 0 0 −1 0 0 0 0 −1 (B.1) As part of the data reduction process, the Mueller matrices (M) are right-multiplied by the reflection matrix R (as shown in equation (B.2)) prior to the calculation of diattenuation, retardance, depolarization parameters, and matrix roots. MR = M ◦ R (B.2) 150 This reflection correction is important to better understand the properties of the material, as opposed to artifacts from the measurement geometry. If the measurement geometry is not corrected before the matrix roots calculations are performed, asymmetries in the Mueller matrix that are a result of reflection may couple retardance and diattenuation into depolarization effects, or vice versa. In the absence of any polarization effects, 45◦ orientation will reflect as 135◦ orientation, and rightcircular orientation will reflect as left-circular orientation. In this reflection case the Mueller matrix is that of a half-wave linear retarder oriented at 0o , even though in reality no polarization effects are present. Multiplying by R removes this half-wave retardance effect, so that small polarization effects vary about 0 instead of a half wave. 151 APPENDIX C INFERRING THE ORIENTATION OF TEXTURE FROM POLARIZATION PARAMETERS This chapter has been modified from an SPIE proceedings paper, H. Noble et. al., “Inferring the orientation of texture from polarization parameters”, Proc. SPIE 7461, 746109 (2009). C.1 Introduction Material properties, surface characteristics, and geometric orientation affect polarization of diffusely reflected light. Depolarization of light by a surface occurs primarily due to multiple scattering events. Thus the depolarization behavior of a surface should provide information about its texture, orientation, material composition, and surface roughness. Much of this information is often below the pixel resolution of an imaging system, which is why polarization analysis of optical scattering can provide more information about surfaces than conventional imaging methods. Early studies of polarization in scattered light were performed by Mie(H.C. van de Hulst, 1981), Beckman and Spizzichino (Beckman and Spizzichino, 1963), and are summarized by Stover (Stover, 1995). Other studies relevant to this work include the polarization dependence of scattering from a one-dimensional rough surface by O’Donnell and Knotts (O’Donnell and Knotts, 1991), ellipsometry and depolarization of rough surfaces by Williams (Williams, 1986), polarization scattering from rough gold and glass surfaces by Hoover et. Al (Hoover et al., 2003), along with 152 others listed in (Germer and Marx, 2004)-(DeBoo and Chipman, 2004). Methods of estimating the orientation of a planar surface using intensity shading information from a single image have been investigated (Babu et al., 1985). C.2 Aluminum measurements and analysis C.2.1 Aluminum sample The polarization light scattering function of a sample with well-defined texture and orientation is studied. The Mueller matrix root parameters are correlated to the orientation of texture as the aluminum is rotated about the surface normal. A block of aluminum, sanded in one direction to produce linear grooves, scatters strongly, has a well-defined texture and orientation, and has aperiodic groove structure. A 4 inch by 3 inch aluminum sample was sanded linearly with 220 count sand paper, generating randomly spaced, roughly parallel grooves, typically 5µm deep. A two-dimensional surface profile measurement is included in figure C.1. The autocovariance graphs in figure C.2 show that the y-profile of the grooves (in the direction of sanding) is relatively uniform, and the x-profile is aperiodic. The polarization properties of specular and diffusely reflected, in-plane light were measured using a Mueller matrix imaging polarimeter operating as a polarization scatterometer, as the sample was rotated about its normal. The polarimeter operates with 5:1 dual rotating retarders (five analyzer rotations for every generator rotation) at a wavelength of 700nm (Smith and Chipman, 2007). The beam size at the sample was 1 cm by 2 cm. Mueller matrix images were acquired at 24 groove orientations from 0◦ orientation to 360◦ . Reference (Smith and Chipman, 2007) contains a more complete description of the polarimeter. Spatial averaging was done to the image profile in its central region in order to 153 average out the scattering behavior. Mueller matrices from the spatially averaged images were multiplied by a reflection matrix to account for the measurement configuration as described in Appendix B – the spatially averaged, reflection corrected Mueller matrices are shown in figure C.3. Polarization properties were calculated from the reflection corrected Mueller matrices. Figure C.4 shows the coordinate system of the polarimeter for in-plane data collection. The reference direction is normal to the sample surface at the point of incidence. The forward scatter side of the surface normal is defined to have positive angles, and the backward scatter side of the surface normal has negative angles. Reference (Noble et al., 2010) contains a more detailed discussion of the measurement coordinate systems. Two configurations are analyzed here, a specular reflection with min 60◦ incident, 60◦ scattering, and an off-specular configuration with −60◦ incident, 30◦ scattering, as shown in figure C.5. A specular beam configuration with −60◦ incident and 60◦ scatter will herewith be referred to as the “specular measurement” and an off-specular beam configuration with −60◦ incident and 30◦ will herewith be referred to as the “off-specular measurement”. C.2.2 Mueller matrix measurements Figure C.3 shows the measured (normalized) Mueller matrices for the sanded aluminum as a function of the mean groove orientation angle (i.e. as the aluminum sample was rotated about its normal). The x-axis represents the aluminum groove orientation in degrees. The y-axis is the magnitude of each normalized Mueller matrix element. The specular and off-specular measurements are shown. All Mueller matrix elements show a nearly sinusoidal variation with groove orientation. Most elements are well fit to the single term Fourier series function with 154 two cycles per 360◦ , f (θ) = a0 + a1 sin 2θ + a2 cos 2θ, (C.1) the only exception bein m12 . This frequency dependence is expected since the sample is nearly invariant under rotations of 180◦ . The normalized average irradiance from the reflections off of the aluminum as the grooves change orientation angle is shown in figure C.6 along with the fit from equation (C.1). The irradiance variation is the least sinusoidal out of all the polarimeter measurements. The average irradiance from the specular reflection is about 20 times larger than that from the diffuse reflection, as seen in figure C.6. The specular and non-specular average reflected irradiance peaks at vertical groove orientations (90◦ and 270◦ ). The light scattered from the sample resembles the light scattered through a dirty windshield (with semi-circular windshield wiper marks), another aperiodic grating with a locally linear structure. For a distant point source viewed through a dirty windshield, a bright line of light is observed, corresponding to the various diffraction orders from the range of spatial frequencies present. Similarly, for a collimated in-plane source, the aluminum sample scatters much of its light into a line of light perpendicular to the grooves. When the grooves are oriented at 90◦ (vertical) this diffracted line lies in the horizontal plane of the detector. When the grooves are oriented vertically, there are more in-plane reflection paths to the detector, as shown in Fig. C.6. When the grooves are oriented horizontally, more light scatters out-of-plane, so the average reflected irradiance is minimum around the horizontal orientations (0◦ and 180◦ ). This scattering behavior provides one method of estimating groove orientation. 155 C.3 Diattenuation and retardance Figure C.7 shows the three matrix roots diattenuation parameters D1 , D2 , and D3 , which characterize the polarization dependence of reflection. The magnitudes of the linear horizontal and 45◦ diattenuation components are larger than the circular diattenuation as expected, since the texture of the aluminum is linearly oriented. The specular horizontal diattenuation (0.35) is approximately twice the off-specular horizontal diattenuation (0.17). For both measurements the 45◦ diattenuation crosses zero near the extrema of the horizonal diattenuation. Later the utility of this will be explored. The maximum magnitude of specular 45◦ diattenuation (0.10) is approximately twice the off-specular (0.04). For the circular diattenuation, specular maxima are approximately twice the off-specular maxima, but with much smaller values (with specular maxima of 0.03). The diattenuation vector demonstrates that for specular reflection, the sanded aluminum behaves as a weak linear diattenuator which rotates with the grooves. The matrix roots retardance parameters D4 , D5 , and D6 are shown in figure C.8 for both the specular and off-specular measurement together with the sinusoid fits. For the horizontal and 45◦ retardance, the specular and off-specular retardance are out of phase by 45o . The specular measurement exhibits a positive horizontal retardance of 0.6 to 0.8 radians which oscillates by approximately 0.1 radians as the grooves rotate. The off-specular angle has a negative horizontal retardance of -0.3 to -0.4. This means that the specular angle has its fast-axis at horizontal, and the off-specular measurement angle has a vertical fast-axis. 156 C.3.1 Determining orientation from diattenuation and retardance Table C.1 lists the sinusoidal fit coefficients from equation (C.1). We now show how the orientation of the sample can be estimated from a single Mueller matrix measurement. To find the orientation ψ using the retardance parameters, we have derived the following mathematical relationship. The method starts with a 3 element polarization parameter, such as diattenuation or retardance. (1) The parameter signals are all centered around zero by subtracting the signal’s dc-offset, the a0 coefficient. (2) Normalize the signal to oscillate between -1 and 1 by dividing the parameter p by a21 + a22 . Since a1 and a2 parameters express the sinusoidal or cosinusoidal amplitude of each signal, dividing by their total magnitude effectively normalizes each oscillating signal. (3) Calculate ψ from the arctangent of the normalized signal. Using the retardance parameters here as an example, q q 2 2 (R − a )/ a (R − a )/ a21,3 + a22,3 + a 2 0,2 3 0,3 1,2 2,2 180 180 o q q arctan = arctan , ψ( ) = 2π 2π (R1 − a0,1 )/ a21,1 + a22,1 (R1 − a0,1 )/ a21,1 + a22,1 (C.2) where ai,j is the coefficient from table 2. The subscript i refers to the fit coefficient term (0,1,2), and the subscript j specifies to which retardance term the fit coefficient term corresponds. In order to calculate the orientation ψ, the numerator of the ArcTangent’s argument should be predominantly sinusoidal, and the denominator of the ArcTangent’s argument must predominantly cosinusoidal. Otherwise the sample orientation cannot be estimated by equation (C.2) or a more general method of function inversion might be applied. As an example, the specular measurement’s retardance parameters provide a good orientation estimate. R1 is dominated by the a2 term, while R2 and R3 are dominated by the a1 term. An example of the application of equation (C.2) is shown in figure C.9, using the off-specular R3 and 157 R1 data. The x-axis indicates groove orientation angle, and the y-axis shows the estimate calculated with equation (C.2). Additionally, this method works effectively with the diattenuation parameters. 158 Figure C.1: Surface profile of sanded aluminum sample. White regions are areas of low signal where signal-to-noise ratio is poor. The white regions are likely areas of high slope where the reflected light is not substantially captured by the instrument. 159 Figure C.2: Autocovariance of sanded aluminum sample. Units of x-profile are in millimeters, units of y-profile are in microns. Figure C.3: Normalized Mueller matrix for −60o incident, 60o scattering specular angle (solid line) and −60o incident, 30o scattering off-specular angle (dashed line) with dotted-line fits. The x-axis represents texture orientation angle of aluminum in degrees, y-axis represents the magnitude of each Mueller matrix element. 160 Figure C.4: Goniometric polarimeter coordinate system. All angles are measured from sample normal. Negative angles are in the region of backscattered light, and positive angles are in the direction of forward scatter. Figure C.5: The two measurement configurations analyzed in this section: −60o incident, 60o scattering and −60o incident, 30o scattering. 161 1.0 0.8 0.6 0.4 0.2 0 45 90 135 180 225 270 Alum. Orien. 315 Figure C.6: Normalized average irradiance from light reflected off of the sample as the groove orientation angle changes. The x-axis lists the orientation angle of the aluminum grooves; the y-axis represents the average irradiance reflected from the sample. The specular measurement is plotted with a solid line, and the off-specular measurement is plotted with the dashed line. The sinusoidal fit from equation (C.1) is plotted with the dotted lines. 0.2 0.2 0.1 0.1 45 90 135 180 225 270 315 Alum. Orien. 0.2 0.1 45 90 135 180 225 270 315 Alum. Orien. 45 -0.1 -0.1 -0.1 -0.2 -0.2 -0.2 -0.3 -0.3 -0.3 -0.4 -0.4 -0.4 90 135 180 225 270 315 Alum. Orien. Figure C.7: Matrix root diattenuation parameters with fit (dotted lines) for the specular angle (solid line) and the off-specular angle (dashed line). 0.5 0.5 45 -0.5 90 135 180 225 270 315 Alum. Orien. 0.5 45 -0.5 90 135 180 225 270 315 Alum. Orien. 45 90 135 180 225 270 315 Alum. Orien. -0.5 Figure C.8: Matrix roots retardance parameters in radians with fit (dotted lines) for the specular angle (solid line) and the off-specular angle (dashed line). 162 Aluminum Orientation Calculated with Retardance 300 Calculated Orien. !°" 250 200 Out[43]= 150 100 50 0 0 50 100 Alum. Orien. !°" 150 200 250 300 350 Figure C.9: Aluminum sample groove orientation calculated with the specular matrix roots retardance parameters demonstrate the estimation of orientation from a single Mueller matrix measurement following calibration. 163 Table C.1: Sinusoidal fit coefficients. Off-Specular Measurement Parameter a0 a1 a2 R1 -0.3666 -0.0026 0.03954 R2 0.0018 0.0471 -0.0609 R3 0.05612 0.0037 0.0040 D1 -0.1022 -0.0029 -0.0002 D2 0.00774 -0.0427 0.00824 D3 0.0736 0.0036 -0.0006 Specular Measurement a0 a1 a2 0.6528 -0.0024 0.0436 -0.0078 0.07552 -0.0195 -0.0917 -0.0069 -0.0001 -0.2413 -0.0039 -0.0008 0.0115 -0.099 -0.0300 0.1059 0.0113 0.0023 164 2.0 1.0 1.0 1.5 0.5 0.5 45 90 135 180 225 270 315 Alum. Orien. 45 90 135 180 225 270 315 Alum. Orien. 1.0 0.5 45 90 Alum. Orien. 135 180 225 270 315 45 0.5 !0.5 !0.5 !0.5 0.0 !1.0 !1.0 !1.0 45 90 135 180 225 270 315 Alum. Orien. !0.005 0.90 !0.010 0.85 !0.015 45 135 180 225 270 315 Alum. Orien. 0.04 0.03 0.02 0.01 0.010 0.005 45 90 135 180 225 270 315 !0.01 !0.02 !0.03 0.015 0.006 0.010 0.004 0.005 0.002 45 90 135 180 225 270 315 Alum. Orien. !0.002 !0.010 !0.004 !0.015 !0.006 0.002 45 90 Alum. Orien. 135 180 225 270 315 90 135 180 225 270 315 Alum. Orien. 90 Alum. Orien. 135 180 225 270 315 0.004 0.002 45 90 135 180 225 270 315 90 135 180 225 270 315 45 90 135 180 225 270 315 Alum. Orien. 0.004 0.002 0.85 !0.002 !0.004 !0.006 !0.008 !0.010 45 !0.006 45 Alum. Orien. !0.002 !0.004 0.90 45 Alum. Orien. 0.004 0.95 Alum. Orien. !0.005 !0.005 90 !0.01 !0.02 !0.03 135 180 225 270 315 0.006 0.04 0.03 0.02 0.01 0.95 90 !0.002 !0.004 !0.006 !0.008 !0.010 Alum. Orien. 0.90 45 Alum. Orien. 90 135 180 225 270 315 0.85 0.80 45 90 135 180 225 270 315 Alum. Orien. Figure C.10: Normalized depolarization Mueller matrix from the Lu-Chipman decomposition with fit (dotted line) for specular measurement (solid line) and offspecular measurement (dashed line). C.4 Depolarization C.4.1 Lu-Chipman depolarizing Mueller matrix and depolarization Index The full Lu-Chipman depolarization matrix is shown in figure C.10. Because the diagonal elements are all in phase, these elements provide little orientation insight. The depolarization index (DI) for both measurements is shown in figure C.11. The DI is defined as 3 P DI(M) = i,j=0 !1/2 m2i,j − m20,0 √ 3m0,0 , (C.3) where the m’s are Mueller matrix elements. A DI of 0 indicates that a sample is completely depolarizing, and a value of 1 indicates that a sample is non-depolarizing 165 0.25 0.20 0.15 0.10 0.05 0 45 90 135 180 225 270 315 Alum. Orien. Figure C.11: Depolarization index vs. orientation angle with fit (dotted line) for specular measurement (solid line) and off-specular measurement (dashed line). for all polarized states (Chipman, 1995). The DI of the specular measurement is approximately 0.05 and hardly varies with groove orientation, indicating polarized light reflects nearly unpolarized. The DI of the off-specular measurement ranges from 0.25 to 0.05, with maxima at 0◦ and 180◦ , and minima at 90◦ and 270◦ . Minima in the depolarization index vs. orientation angle for the off-specular measurement correspond to bright regions in the intensity from Fig. C.6, while maxima correspond to the dark intensity regions. C.4.2 Mueller matrix roots depolarizing parameters The Mueller matrix roots depolarizing parameters (amplitude, phase, and diagonal depolarization) are shown in figures C.12 through C.14, along with their sinusoidal fits. The amplitude depolarization parameters D7 , D8 , and D9 , shown in Fig. C.12, exhibit the following characteristics. Horizontal/vertical amplitude depolarization D7 is noisy, with minimal signal strength. 45o /135o amplitude depolarization D8 has low signal strength (with peak-to-peak amplitude of 0.020 for the off-specular 166 measurement and peak-to-peak amplitude of approximately 0.014 for the specular measurement), but shows a clear sinusoidal variation with the orientation angle of the aluminum. The circular amplitude depolarization D9 of the specular angle shows negligible amplitude, while the off-specular angle has a sinusoidal variation with peak-to-peak amplitude of 0.15. The phase depolarization parameters D10 , D11 , and D12 , shown in Fig. C.13, exhibit the following behavior. The specular angle’s horizontal/vertical phase depolarization D10 shows minimal signal strength and amplitude as the orientation of the aluminum varies, while the off-specular angle has a peak-to-peak amplitude of approximately 0.015, and sinusoidal variation with aluminum orientation. 45o /135o phase depolarization is affected by more noise, but both measurements show the sinusoidal rotation characteristics from Equation C.1. Circular amplitude depolarization D12 shows some modulation within the sinusoidal envelope, which corresponds to the modulation in the m12 term of the Mueller matrix. The diagonal depolarization parameters D13 , D14 , and D15 are shown in Fig. C.14. D13 expresses the relative depolarization between the linear Stokes vector elements S1 and S2 . For both measurements, it oscillates between equal depolarization along the two axes (at D13 = 0) and more depolarization along S1 (when D13 is negative). D14 expresses the relative depolarization between the circular and linear Stokes axes. D14 oscillates between exclusively positive values, which means that it depolarizes more along the circular S3 axis than along the linear Stokes axes. For the off-specular measurement, isotropic diagonal depolarization D15 averages at approximately 0.18, while the off-specular measurement has D15 with an average of 0.05 and almost zero sinusoidal amplitude. The signal strength of the non-depolarizing roots parameters for specular mea- 167 0.010 0.004 45 0.002 90 135 180 225 270 315 Alum. Orien. 0.005 45 90 135 180 225 270 315 -0.005 Alum. Orien. -0.002 45 -0.004 90 135 180 225 270 315 Alum. Orien. -0.010 -0.005 -0.015 -0.006 Figure C.12: Amplitude depolarization parameters D7 , D8 , and D9 (from left to right) vs. orientation angle with fit (dotted line) for specular measurement (solid line) and off-specular measurement (dashed line). 0.005 45 90 135 180 225 270 315 -0.005 -0.010 0.04 Alum. Orien. 45 90 135 180 225 270 315 Alum. Orien. 0.02 45 -0.005 90 135 180 225 270 315 Alum. Orien. -0.02 -0.015 -0.010 -0.04 Figure C.13: Phase depolarization parameters D10 , D11 , and D12 (from left to right) vs. orientation angle with fit (dotted line) for specular measurement (solid line) and off-specular measurement (dashed line). surement was higher than the signal strength of the off-specular measurement. For this reason, the orientation calculated with the retardance parameters used the specular measurement data. However, the off-specular depolarizing matrix roots parameters show a larger sinusoidal amplitude and signal strength than the specular depolarizing matrix roots parameters. This is potentially useful for a measurement application where an optical measurement is only possible at a non-specular or grazing angle, from which we expect more depolarization effects than from a specular angle. Figure C.12 shows the orientation of the scratched aluminum which was estimated from the off-specular linear phase depolarization parameters (D10 and D11 ), using the same method as the specular retardance orientation estimation. 168 0.05 45 90 135 180 225 270 315 0.30 Alum. Orien. 0.04 -0.01 0.25 0.20 0.03 0.15 -0.02 0.02 -0.03 0.01 0.10 -0.04 0.05 45 90 135 180 225 270 315 Alum. Orien. 45 90 135 180 225 270 315 Alum. Orien. Figure C.14: Diagonal depolarization parameters D13 , D14 , and D15 (from left to right) vs. orientation angle with fit (dotted line) for specular measurement (solid line) and off-specular measurement (dashed line). Aluminum Orientation Calculated with Phase Depolarization Calculated Orien. 300 250 200 150 100 50 0 45 90 135 180 225 270 315 Alum. Orien. H°L Figure C.15: Aluminum sample groove orientation calculated with the off-specular linear phase depolarization parameters D10 and D11 , demonstrating the estimate of orientation from a single Mueller matrix measurement following calibration. C.5 Conclusion Diffusely reflected light from a one-dimensionally rough aluminum sample has been analyzed by Mueller matrix light scattering measurements. For a specular and nonspecular pair of beams, polarization terms are identified which vary cosinusoidally and others which vary sinusoidally with respect to the aluminum groove orientation angle. Using this cosinusoidal and sinusoidal modulation information, the orientation may be estimated using a normalizing arctangeant calculation method, as written in equation (C.2). Determining orientation by this method requires pa- 169 rameters exhibiting a sinusoidal variation and others with cosinusoidal information. 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