POLARIZATION RAY TRACING by Garam Yun Copyright © Garam Yun 2011 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 Garam Yun entitled Polarization Ray Tracing and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy _______________________________________________________Date: Nov. 21, 2011 Russell A. Chipman _______________________________________________________Date: Nov. 21, 2011 Arthur Gmitro _______________________________________________________Date: Nov. 21, 2011 J. Scott Tyo 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: Nov. 21, 2011 Dissertation Director: Russell A. 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. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the copyright holder. SIGNED: Garam Yun 4 ACKNOWLEDGEMENTS I would like to thank my parents who have encouraged me to come to the States and pursue my intellectual thirst in science. Without their support, I would not have been here. I would like to thank my sister for being my best friend and a supporter. Our phone conversations were always my source of energy and laughter. I would like to thank my husband, Stefano Young, for being the first person I met in Tucson, for planning all the great trips we did, for being open to learn my culture and family, and for encouraging me to find my ultimate goals in life. I would like to thank my advisor, Professor Russell Chipman who has challenged me with most interesting polarization problems. Without him, I would not have been able to complete this dissertation. Professor Chipman has continuously given me opportunities and pushed me to be a better researcher, a better public speaker, and a better leader. I would like to thank Professor Jaisoon Kim, for planting the seed, the idea of coming to the college of optical sciences for graduate school and learning Optics, in my senior year in college. I would like to thank Dr. Art Gmitro and Dr. J. Scott Tyo for serving on my committee. I would like to thank Dr. Greg Smith for sharing great ideas in programming and skills in problem solving during many of our long discussions. I would like to thank Dr. Stephan McClain and Dr. Karlton Crabtree who provided helpful discussions and ideas. I would like to thank Hannah Noble, Stacey Sueoka, Anna-Britt Mahler, Paula Smith, and Tiffany Lam for their friendship; you made coming to work more rewarding. 5 DEDICATION To my parents, 윤철수 and 이상옥, and my husband, Stefano with all my heart and love. 6 TABLE OF CONTENTS LIST OF FIGURES ............................................................................................................................. 10 LIST OF TABLES ............................................................................................................................... 16 ABSTRACT ……… ........................................................................................................................... 17 CHAPTER 1 HISTORY OF POLARIZATION RAY TRACING ................................................................ 20 1.1 Polarization Ray Tracing in two-dimensions .................................................................. 20 1.2 Polarization Ray Tracing in three-dimensions ............................................................... 23 CHAPTER 2 DEFINITION OF POLARIZATION RAY TRACING MATRIX ............................................. 25 2.1 Definition of Polarization Ray Tracing Matrix, P ............................................................ 25 2.1.1 2.2 Formalism of Polarization Ray Tracing Matrix using Orthogonal Transformation ........ 32 2.2.1 2.3 Polarization States as Three-element Electric Field Vectors ................................. 29 Retarder Polarization Ray Tracing Matrix Examples .............................................. 35 Conclusion ...................................................................................................................... 37 CHAPTER 3 CALCULATION OF DIATTENUATION ........................................................................... 38 3.1 Diattenuation Calculation using Singular Value Decomposition ................................... 38 3.2 Example – A Hollow Corner Cube .................................................................................. 41 3.3 Conclusion ...................................................................................................................... 47 CHAPTER 4 CALCULATION OF RETARDANCE ................................................................................ 48 4.1 Introduction ................................................................................................................... 48 4.2 Purpose of the Proper Retardance Calculation ............................................................. 50 4.3 Geometrical Transformations ........................................................................................ 51 4.3.1 Local Coordinate Rotation: Polarimeter Viewpoint ............................................... 51 4.3.2 Parallel Transport of Vectors ................................................................................. 55 4.3.3 Parallel Transport Ray Tracing Matrix, Q ............................................................... 57 4.4 Proper Retardance Calculations..................................................................................... 62 4.4.1 Definition of the Proper Retardance...................................................................... 62 4.4.2 Separating Local Coordinate Transformation from P ............................................ 62 4.4.3 The Proper Retardance Algorithm for P, Method 1 ............................................... 63 4.4.4 The Proper Retardance Algorithm for P, Method 2 ............................................... 65 7 TABLE OF CONTENTS - Continued 4.4.5 4.5 Retardance Range .................................................................................................. 66 Examples ........................................................................................................................ 67 4.5.1 Ideal Reflection at Normal Incidence ..................................................................... 67 4.5.2 Brewster’s Angle Analysis ...................................................................................... 70 4.5.3 An Aluminum Coated Three-fold Mirror System ................................................... 76 4.6 Conclusion ...................................................................................................................... 81 CHAPTER 5 POLARIZATION ABERRATION ..................................................................................... 83 5.1 Jones Pupil ..................................................................................................................... 83 5.2 Polarization Aberration Function ................................................................................... 89 5.2.1 5.3 Coherent Beam Combination ................................................................................ 95 Conclusion ...................................................................................................................... 96 CHAPTER 6 A SKEW ABERRATION ................................................................................................. 97 6.1 Definition ....................................................................................................................... 97 6.2 Skew Aberration Algorithm............................................................................................ 98 6.3 Example ........................................................................................................................ 102 6.3.1 Skew Aberration at the Exit Pupil ........................................................................ 103 6.3.2 Skew Aberration’s Effect on Point Spread Function and Modulation Transfer Function ............................................................................................................................. 106 6.4 Statistics – Code V Patent Library ................................................................................ 109 6.5 Skew Aberration in Paraxial Ray Trace ........................................................................ 110 6.6 Conclusion .................................................................................................................... 114 CHAPTER 7 UNDERSTANDING APPARENT RETARDANCE DISCONTINUITIES .............................. 115 7.1 Retardance Calculation for Jones Matrices ................................................................. 116 7.2 Retarder Space ............................................................................................................. 119 7.3 Trajectories of Jones Retarder Matrices as the Polarization State Analyzer Rotates .. 122 7.4 Phase Unwrapping for Homogeneous Retarder Systems using Dispersion Model ..... 125 7.4.1 Dispersion Model ................................................................................................. 125 7.4.2 Phase Unwrapping of the Homogeneous Retarder System ................................ 127 8 TABLE OF CONTENTS - Continued 7.5 Discontinuity in Phase Unwrapped Retardance Values for Compound Retarder Systems of Arbitrary Alignment ............................................................................................................. 137 7.6 Conclusion .................................................................................................................... 163 CHAPTER 8 COHERENCE MATRIX AND POLARIZATION RAY TRACING TENSOR.......................... 165 8.1 Introduction ................................................................................................................. 165 8.2 The Coherence Matrix ................................................................................................. 166 8.3 Projection of the Coherence Matrix onto Arbitrary Planes ......................................... 169 8.4 A Definition of Polarization Ray Tracing Tensor .......................................................... 170 8.5 A Polarization Ray Tracing Tensor for a Non-depolarizing Ray Intercept .................... 174 8.5.1 A Polarization Ray Tracing Tensor from Surface Amplitude Coefficients ............ 175 8.5.2 A Polarization Ray Tracing Tensor from the three-by-three Polarization Ray Tracing Matrix P ................................................................................................................... 181 8.5.3 8.6 Example Polarization Ray Tracing Tensor Calculation ......................................... 183 A Polarization Ray Tracing Tensor for a Ray Intercept with Scattering ....................... 185 8.6.1 Example Polarization Ray Tracing Tensor Calculation ......................................... 188 8.7 Example Polarization Ray Tracing Tensor and Combination of Tensors...................... 190 8.8 Conclusion .................................................................................................................... 206 CHAPTER 9 THREE-DIMENSIONAL (3D) STOKES PARAMETERS .................................................. 207 9.1 Definition of 3D Stokes Parameters ............................................................................. 207 9.2 Example Incoherent Additions of 3D Stokes Parameters ............................................ 210 9.3 2D Stokes Parameters to 3D Stokes Parameters ......................................................... 217 9.4 3D Mueller Matrix ........................................................................................................ 217 9.4.1 9.5 Example 3D Mueller Calculation .......................................................................... 226 Conclusion .................................................................................................................... 230 CHAPTER 10 CONCLUSIONS AND FUTURE WORK ................................................................ 231 10.1 Summary ...................................................................................................................... 231 10.2 Future Work ................................................................................................................. 235 10.3 Conclusion .................................................................................................................... 237 APPENDIX A USA PATENT 2,896,506 LENS PARAMETERS............................................................ 239 9 TABLE OF CONTENTS - Continued APPENDIX B DEPOLARIZATION INDEX OF A POLARIZATION RAY TRACING TENSOR ................... 241 APPENDIX C WATER DROPLET SIZES FOR THE CLOUD EXAMPLE ................................................ 243 REFERENCES ................................................................................................................................. 244 10 LIST OF FIGURES Figure 1.1 (a) Longitude and latitudes lines as the local x and y bases. (b) Singularity at the North Pole. .................................................................................................................. 22 Figure 2.1 The polarization states before (Eq-1) and after (Eq) the qth optical interface are related by the matrix Pq. .................................................................................................... 26 Figure 2.2 The major axis orientation θ measured from v̂1 to a . The axis vector is shown in red arrow............................................................................................................ 31 Figure 3.1 Top view of a hollow aluminum coated corner cube. Three surfaces of the corner cube are perpendicular to each other. .................................................................... 42 Figure 3.2 Top and side views of a corner cube. Propagation vectors are shown in solid black, local s coordinate vectors in solid red, and local p coordinate vectors in dashed blue.................................................................................................................................... 43 Figure 3.3 (a) The exiting state from the corner cube with the maximum intensity and corresponding incident state. (b) The state with the minimum intensity and corresponding incident state. All represented in local coordinates where 2D polarization vectors are defined. Each propagation vector comes out of the page. ............................. 46 Figure 3.4 (a) The exiting state from the corner cube with the maximum intensity and corresponding incident state. (b) The state with the minimum intensity and corresponding incident state. All states are represented in global coordinates (x-y plane) looking into the corner cube; propagation vectors are anti-parallel for the incident and exiting electric field vectors. ............................................................................................. 47 Figure 4.1 A polarimeter measuring a sample retarder with the polarization state analyzer (PSA) (a) aligned with the polarization state generator (PSG) and (b) rotated to an arbitrary orientation. By rotating the PSA, the exiting local coordinates for the Jones matrix are also rotated. The measured retardance of the sample now includes a “circular retardance” component of 2 as well as the proper retardance. ...................................... 50 Figure 4.2 Measured retardance values as a function of rotation angle θ for circular retarders with retardance between zero and one wave of retardance are shown in different colors. Retardance of a half wave circular retarder is independent of the PSA orientation. ........................................................................................................................................... 53 Figure 4.3 Measured retardance values as a function of rotation angle θ for linear retarders with retardance between zero and one wave of retardance are shown in different colors. Retardance of a half wave linear retarder is independent of the PSA orientation. ........................................................................................................................................... 54 11 LIST OF FIGURES - Continued Figure 4.4 (a) The evolution of a local coordinate pair {xˆ A , yˆ A} (green) through a system of three fold-mirrors. The exiting local coordinates (dashed red) undergo a 90° rotation from the initial local coordinates (solid green). (b) A three fold-mirror system. When a collimated beam enters the system along the z-axis the beam exits along the z-axis. ...... 56 Figure 4.5 Incident, reflected, and transmitted local coordinates calculated from parallel transport matrices. {xˆ , yˆ , kˆ } are the right handed incident local coordinates, L ,0 L ,0 0 {xˆ L ,r ,1 , yˆ L ,r ,1 , kˆ r ,1} are the left handed reflected local coordinates, and {xˆ L,t ,1 , yˆ L,t ,1 , kˆ t ,1} are the right handed transmitted local coordinates. ................................................................ 59 Figure 4.6 A two mirror system. The red solid lines show the s vector at the first mirror and its geometric transformation along each ray segment using Q. The blue dashed lines show the p vector in object space and its geometric transformations. .............................. 60 Figure 4.7 An ideal reflection at normal incidence with the incident and exiting righthanded local coordinates, {xˆ L,0 , yˆ L,0 } and {xˆ L,1 , yˆ L,1} . In this particular choice of local coordinates, the xˆ L vector was flipped after the reflection. .............................................. 69 Figure 4.8 Fresnel external (top) and internal (bottom) reflection coefficients for spolarization (dashed) and p- polarization (solid) are shown as the angle of incidence changes. ............................................................................................................................. 71 Figure 4.9 The relative phase shift for external (top) and internal (bottom) reflection as a function of the angle of incidence. .................................................................................... 72 Figure 4.10 The incident s-polarized states (dotted) and reflected (solid) polarization states as angles of incidence change from normal incidence to glazing incidence. Red arrow indicates the reflected electric field at Brewster’s angle. ....................................... 74 Figure 4.11 The incident p-polarized states (dotted) and corresponding reflected polarization states (solid) as the angle of incidence changes. Red arrow indicates the reflected electric field vector at the Brewster’s angle. At the Brewster’s angle, Fresnel reflection coefficient for p-polarization is zero. ............................................................... 75 Figure 4.12 Incident electric field vectors at 45º between s and p-polarizations (dotted) and corresponding reflected (solid) electric field vectors are plotted as the angle of incidence changes. Red lines are for the incident and reflected pair at the Brewster’s angle; the top view shows that at Brewster’s angle, the reflected light (solid red line) is spolarized. ........................................................................................................................... 76 Figure 4.13 An aluminum coated three fold-mirror system. ............................................ 77 12 LIST OF FIGURES - Continued Figure 4.14 Local coordinate transformation using Qq's for the three fold-mirror system. Incident xˆ L,0 state (solid red) exits as -xˆ polarized and incident yˆ L,0 state (dashed blue) exits as ŷ polarized after three reflections due to the geometric transformation. ............ 79 Figure 5.1 A field vector on the image plane and the exit pupil vector. ........................... 83 Figure 5.2 Optical system layout of USA patent 2896506 from Code V library has seven lens elements. The system is defined with three field angles. .......................................... 85 Figure 5.3 Wavefront aberration maps at the exit pupil using Code V are plotted for the on-axis field. For on-axis field, 0.3 waves of spherical aberration is the dominant aberration. ......................................................................................................................... 86 Figure 5.4 Amplitude part of Jones Pupil at the edge of field of USA patent 2896506 shows variation across the pupil as well as off-diagonal components, which often appear for systems with high NA. ................................................................................................ 87 Figure 5.5 Jones pupil local x and y coordinates. ............................................................. 88 Figure 5.6 Wavefront aberration maps at the exit pupil using Code V are plotted for a point source at the edge of the field demonstrating three waves of astigmatism. ............ 89 Figure 5.7 Amplitude part of a polarization aberration function ( PTotal ,s ) at the exit pupil of USA patent 2896506. ................................................................................................... 93 Figure 5.8 Phase part of a polarization aberration function ( PTotal ,s ) at the exit pupil of USA patent 2896506. ........................................................................................................ 94 Figure 6.1 (a) A double pole grid of reference vectors on a unit sphere viewed along the chief ray’s propagation vector and (b) oblique view. ..................................................... 100 Figure 6.2 Optical system layout of USA patent 2896506 from Code V library has seven lenses. The system is defined with three field angles. .................................................... 103 Figure 6.3 (a) Skew aberration of a ray grid with 32º field angle is evaluated at the exit pupil of the patent 2896506. The maximum skew aberration is +7.01º and the minimum is -7.01º (ray A). Both extreme occur from skew rays at the edge of the pupil. (b) Horizontal cross section (indicated in orange dashed line in part (a)) of the skew aberration exit pupil map has zero skew aberration for the center ray, which is the chief ray. .................................................................................................................................. 104 Figure 6.4 Skew ray A's skew aberration contribution from each lens surface sums to 7.01º through the system. ................................................................................................ 105 Figure 6.5 A point spread matrix (PSM) of the example system calculated from a discrete Fourier transform of the parallel transport matrix of the system at the exit pupil. The elements are elliptical due to three waves of astigmatism. ............................................. 107 Figure 6.6 An optical transfer matrix (OTM) of the example calculated by a discrete Fourier transform of the PSM. ........................................................................................ 108 13 LIST OF FIGURES - Continued Figure 6.7 Histogram of the maximum skew aberration evaluated for 2383 non-reflecting optical systems in Code V’s library of patented lenses. ................................................. 109 Figure 6.8 Paraxial skew ray at point A's skew aberration contribution from each lens surface sums to -2.00º through the system. ..................................................................... 113 Figure 7.1 A range of total retardance δ calculated from Eq. (7.1.6). ............................ 119 Figure 7.2 A retarder space with δH, δ45, and δR as axes where δ is the distance from the origin to a point which has a magnitude of retardance δ. ............................................... 120 Figure 7.3 Two groups of Mueller matrices (A and B) with the same retardance modulo to 2π are shown in the retarder space. Each point in the groups is 2π away from each other and shares the same fast and slow axes. ................................................................ 122 Figure 7.4 Trajectories of horizontal fast axis linear retarders in the retarder space as the PSA rotates by θ from zero to 2π. Each trajectory repeats twice for 0≤ θ ≤ 2π. ........... 124 Figure 7.5 (a) The principal retardance of the first retarder and its fast axis orientation as a function of wavelength. (b) The principal retardance of the second retarder and its fast axis orientation with respect to the horizontal axis as a function of wavelength. .......... 128 Figure 7.6 Retardance plots as a function of wavelength for two horizontal fast axis linear retarders (HLR) with different thicknesses (green and blue) and a combination of two HLRs with a shared horizontal fast axis (red). The principal retardance has the horizontal fast axis for downward sloping regions and has the vertical fast axis for upward sloping regions. ............................................................................................................................ 129 Figure 7.7 The principal retardance of the combination of two HLRs and the system’s fast axis orientation with respect to the horizontal axis as a function of wavelength. .... 130 Figure 7.8 Each segment of the principal retardance has a mode number q to apply phase unwrapping algorithm. Starting from the right side of the graph, blue segments have odd mode numbers and the red segments have even mode numbers. ................................... 131 Figure 7.9 Principal retardance vector trajectories are shown in the retarder space as the wavelength changes. Each figure corresponds to a different mode number starting from the longest wavelength (mode 1) to the shortest wavelength. ........................................ 134 Figure 7.10 The principal retardance trajectory in the retarder space as the wavelength of the ray changes. Discontinuity occurs on a sphere of radius π. ..................................... 134 Figure 7.11 (a) Principal retardance trajectory of a two-aligned HLR system in the retarder space as the wavelength gets shorter. (b) Retardance trajectory of the same system after the phase unwrapping. ................................................................................ 136 Figure 7.12 Phase unwrapped retardance as a function of wavelength for two HLRs (green and blue) and a system of two-aligned HLR (red). ............................................. 137 14 LIST OF FIGURES - Continued Figure 7.13 Total retardance for a system with two half-wave linear retarders is plotted as the fast axis orientation (θ) of the second retarder changes with respect to the first retarder’s fast axis orientation. ........................................................................................ 139 Figure 7.14 The principal retardance of JTotal , a system of two HRLs misaligned by π/16, is plotted as a function of wavelength. Green dotted circles in the top figure indicate the area where the principal retardance changes its slope without going down to zero. ...... 143 Figure 7.15 Principal retardance vector trajectories are plotted in the retarder space as the wavelength reduces. Each figure corresponds to a segment of the trajectory from the longer wavelength to the shorter wavelength as the retardance vector approaches the π sphere. ............................................................................................................................. 145 Figure 7.16 The top view of the principal retardance vector trajectories in the retarder space as the wavelength reduces. .................................................................................... 146 Figure 7.17 Phase unwrapped retardance plotted as a function of wavelength for the aligned (red) and forced phase unwrapping for a misaligned (blue) two-HLR system. . 147 Figure 7.18 Phase unwrapped Major (top) and Minor (bottom) plotted as a function of wavelength. The phase unwrapped Major is the linear addition of 1 and 2 of the each linear retarder. ................................................................................................................. 148 Figure 7.19 The principal retardance plots as a function of wavelength for two HLRs (green and blue) and a system of two misaligned HLRs (red). Two HLRs have different retardances and the angle between two fast axes is θ. .................................................... 151 Figure 7.20 A system of two HLRs misaligned by θ. Four output beams exiting the system with four different optical path lengths are shown offset for clarity. ................. 152 Figure 7.21 The principal retardance as a function of wavelength for the system with two HLRs with the fast axes misaligned by π/4. The principal retardance has minima other than zero. ......................................................................................................................... 156 Figure 7.22 Phase unwrapped retardance of the first (green) and second (blue) retarders with the fast axis orientation ( fast ) of the compound system. Xs mark wavelengths where individual plates have integer waves of retardance, and don’t contribute to the axis of the retarder. ................................................................................................................. 157 Figure 7.23 Green plot shows the retardance of the HLR, δ1(λ) as a function of wavelength. Blue plot is the retardance of the 45° fast axis linear retarder, δ2(λ). Red plot shows the retardance of the system of the two HLRs with π/4 misalignment between two fast axes orientation, δTotal(λ). .................................................................................. 158 Figure 7.24 Phase unwrapped Minor plotted as a function of wavelength. .................... 159 15 LIST OF FIGURES - Continued Figure 7.25 A principal retardance trajectory of the system with two misaligned HLRs at 45° in the retarder space as the wavelength reduces. When the trajectory reaches the boundary of π, the trajectory moves to the opposite point on the π sphere and the fast axis changes to the orthogonal state. ...................................................................................... 162 Figure 7.26 Phase unwrapped retardance of the compound system as the wavelength changes with the corresponding points in Figure 7.25. .................................................. 163 Figure 8.1 A triplet followed by a lens barrel. A collimated grid of rays propagates through the triplet and scatters off from the lens barrel before it reaches the detector plane on the right. ..................................................................................................................... 173 Figure 8.2 A ray propagating along the z axis reflects from an Aluminum coated surface. ......................................................................................................................................... 184 Figure 8.3 A volume of water droplets in the air which scatters the incident collimated beam of light. The incident beam of light is plotted in dark red arrows and some of the individual scattering ray paths are shown in different colors. ........................................ 191 Figure 8.4 s (red) and p (blue) polarization reflection coefficients as a function of scattering angle calculated from the MiePlot program. .................................................. 193 Figure 8.5 A nine-by-nine grid of the detector with its surface normal along the x-axis is shown. Each false color corresponds to the summation of the polarization ray tracing tensors along the x-axis over the same color mapped water droplets. ............................ 194 Figure 8.6 3D degree of polarization is calculated for each exiting coherence matrix at the detector. The x-axis indicates the pixel number. 3D DOP shows that the exiting light is mostly unpolarized. ......................................................................................................... 200 Figure 8.7 2D degree of polarization of each 2D Stokes vector is calculated at each pixel on the detector. The values indicate that the exiting light is mostly unpolarized. ......... 201 Figure 8.8 S0 components of the exiting 2D Stokes vectors at each pixel...................... 202 Figure 8.9 Diattenation of the scattered light calculated from the Mie Plot program. For positive values s polarization has the greater scattering amplitudes than p polarization.203 Figure 8.10 S2 components of the exiting 2D Stokes vectors at each pixel.................... 204 Figure 8.11 Exiting light polarization vectors on the detector plane are shown in red whereas linearly polarized light at 45º are shown in dashed blue. ................................. 205 Figure 9.1 Three plane wavefronts propagating along the x, y, and z axis are meeting at the origin. ........................................................................................................................ 211 16 LIST OF TABLES Table 2.1 Polarization ray tracing matrix for a horizontal fast axis linear quarter wave retarder without beam deviation for three different propagation directions, along z-axis, yaxis, and x-axis. The Jones matrices are specified in a symmetric phase convention where the fast axis polarization state is advanced by an eighth of a wave and the slow axis is delayed by an eighth of a wave. ............................................................................. 35 Table 3.1 Propagation vectors, local coordinate basis vectors, surface normal vectors and polarization ray tracing matrices associated with a ray path through aluminum coated hollow corner cube. ........................................................................................................... 44 Table 3.2 The maximum intensity of output and associated incident electric field, the minimum intensity of transmitted electric field and associated incident electric field, and the diattenuation from the ray through the corner cube system. ....................................... 45 Table 4.1 Propagation vectors, Pq’s, and Qq’s for a ray propagating through the aluminum coated three fold-mirror system ........................................................................................ 77 Table 8.1 A polarization ray tracing tensor in global coordinates as a function of amplitude coefficients in local coordinates. Each shows a three-by-three matrix component of the tensor. ................................................................................................. 180 Table 9.1 3D Stokes vectors from an incoherent addition of three electric field vectors with different polarization states, measured on xz-plane are shown. All the amplitudes are set to 1.0 for the simplicity. ............................................................................................ 213 Table 9.2 A nine-by-nine rotation matrix RotM( elements are shown. The RotM( matrix rotates to . ............................................................................................... 225 17 ABSTRACT A three-by-three polarization ray tracing matrix method is developed to calculate the polarization transformations associated with ray paths through optical systems. The relationship between the three-by-three polarization ray tracing matrix P method and the Jones calculus is shown in Chapter 2. The diattenuation, polarization dependent transmittance, is calculated via a singular value decomposition of the P matrix and presented in Chapter 3. In Chapter 4 the concept of retardance is critically analyzed for ray paths through optical systems. Algorithms are presented to separate the effects of retardance from geometric transformations. The parallel transport of vectors is associated with nonpolarizing propagation through an optical system. A parallel transport matrix Q establishes a proper relationship between sets of local coordinates along the ray path, a sequence of ray segments. The proper retardance is calculated by removing this geometric transformation from the three-by-three polarization ray trace matrix. Polarization aberration is wavelength and spatial dependent polarization change that occurs as wavefronts propagate through an optical system. Diattenuation and retardance of interfaces and anisotropic elements are common sources of polarization aberrations. Two representations of polarization aberration using the Jones pupil and a polarization ray tracing matrix pupil, are presented in Chapter 5. In Chapter 6 a new class of aberration, skew aberration is defined, as a component of polarization aberration. Skew aberration is an intrinsic rotation of 18 polarization states due to the geometric transformation of local coordinates; skew aberration occurs independent of coatings and interface polarization. Skew aberration in a radially symmetric system primarily has the form of a tilt plus circular retardance coma aberration. Skew aberration causes an undesired polarization distribution in the exit pupil. A principal retardance is often defined within (- , ] range. In Chapter 7 an algorithm which calculates the principal retardance, horizontal retardance component, 45° retardance component, and circular retardance component for given retarder Jones matrices is presented. A concept of retarder space is introduced to understand apparent discontinuities in phase unwrapped retardance. Dispersion properties of retarders for polychromatic light is used to phase unwrap the principal retardance. Homogeneous and inhomogeneous compound retarder systems are analyzed and examples of multi-order retardance are calculated for thick birefringent plates. Mathematical description of the polarization properties of light and incoherent addition of light is presented in Chapter 8, using a coherence matrix. A three-by-three-bythree-by-three polarization ray tracing tensor method is defined in order to ray trace incoherent light through optical systems with depolarizing surfaces. The polarization ray tracing tensor relates the incident light’s three-by-three coherence matrix to the exiting light’s three-by-three coherence matrix. This tensor method is applicable to illumination systems and polarized stray light calculations where rays at an imaging surface pixel have optical path lengths which vary over many wavelengths. 19 In Chapter 9 3D Stokes parameters are defined by expanding the coherence matrix with Gell-Mann matrices as a basis. The definition of nine-by-nine 3D Mueller matrix is presented. The 3D Mueller matrix relates the incident 3D Stokes parameters to the exiting 3D Stokes parameters. Both the polarization ray tracing tensor and 3D Mueller matrix are defined in global coordinates. In Chapter 10 a summary of my work and future work are presented followed by a conclusion. 20 CHAPTER 1 HISTORY OF POLARIZATION RAY TRACING 1.1 Polarization Ray Tracing in two-dimensions The objective of polarization ray tracing is to calculate the evolution of the polarization state through an optical system and to determine the polarization properties, such as diattenuation and retardance, associated with ray paths through the system, and assess the impact of polarization aberration on image formation. By tracing many rays, the polarization aberrations associated with an optical system can be assessed, and the behavior of a particular optical and coating design compared with the optical system’s polarization specifications. One of the most common descriptions of polarization state used in polarization analysis is the Jones vector Ex J Ey (1.1.1) where Ex and Ey are complex amplitudes 1 . The Jones vector specifically refers to a monochromatic plane wave, describing the electric field and the polarization ellipse with respect to an x-y coordinate system in the transverse plane. If the plane wave is not propagating along the z-axis, then the x-y coordinates are referred to as “local coordinates” associated with a particular transverse plane. Most optical design software use two-bytwo Jones matrices to describe the optical elements since, unlike Mueller matrices, Jones matrices preserve the phase information. Jones matrices describe polarization effects 21 such as the polarization elements used to transform between polarization states1. For paraxial beams or beams with small numerical aperture where the wavefront is not very curved and the z-component of the field is small, Jones matrix pupil functions2, the Jones matrix as a function of exit pupil coordinates, are widely used. However, to use Jones vectors and matrices in optical design for the ray tracing of highly curved beams, local coordinate systems are required for each ray, and each of its ray segments, to define the direction of the Jones vector’s x- and y- components in space, and these local coordinate systems lead to complications due to the intrinsic singularities of local coordinates. For example, to define a Jones matrix between an incident spherical wave and an exiting wave two pairs of local coordinates are necessary and this pair of local coordinates is different for each of the wavefront’s rays. Jones vectors and matrices are readily adapted to sequences of ray segments with beam deviation by carrying along a local coordinate system. One algorithm the author and others have used for arbitrary propagation direction is to use lines of longitudes and latitudes to establish a local x and y-basis for Jones vectors, as shown in Figure 1.1 (a). Consider the unit propagation vector k̂ puncturing a unit sphere demarcated with latitude and longitude arcs; the sphere’s poles might be located at {0, 0, ±1}. The latitude specifies xˆ Local and longitude specifies yˆ Local for any ray, although there is an obvious problem for propagation along the z-axis. This leads to the following problem. Using any latitude/longitude algorithm for specifying Jones vector local coordinates results in two singularities at the north and 22 south poles where the local coordinates are undefined. (Figure 1.1 (b)) Near the poles, the local coordinates vary rapidly with small change in propagation direction. The workaround that polarization ray tracing computer programs use is to trap propagation close to the poles (z in this case) and handle the associated quantities as a special case via separate rules. In the experience of the author, working in Jones vector local coordinates leads to a cascade of minor complications, both in handling rays near the singularities and in describing high numerical aperture beams. Figure 1.1 (a) Longitude and latitudes lines as the local x and y bases. (b) Singularity at the North Pole. Such issues are intrinsic to any choice of local coordinates. According to the Winding Number Theorem3, it is impossible to define a continuous and differentiable vector field constrained to lie on the surface of a sphere over the entire sphere without at least two zeros in the field; a set of latitude vectors or conversely a set of longitude vectors provide two examples, where the zeros occur at the poles. All local coordinate choices have such singularities. 23 1.2 Polarization Ray Tracing in three-dimensions This problem of singularities in local coordinates can be avoided and systematized by generalizing a two-by-two Jones matrix into a three-by-three polarization ray tracing matrix to handle arbitrary propagation directions. Polarization ray tracing matrix methods using Jones matrices have been in use in optical design for at least twenty years. Knowlden used a polarization ray tracing technique to analyze the instrumental polarization effects caused by coatings on several nonplanar optical surfaces 4 . Polarization ray tracing was used to integrate thin-film design and crystal optics into optical design5. Waluschka merged traditional lens design and analysis computer codes and thin-film codes6 in order to construct a polarization ray trace algorithm. Polarization ray tracing, which calculates the Jones matrix associated with an arbitrary ray path through an optical system, was introduced to calculate the polarization aberration function 7 . Polarization ray tracing of “thick” thin-film optical coatings (thicker than several 's ) was used to calculate the phase shift caused by the physical thickness of the film8. From Chapter 2 to Chapter 6, except in Chapter 7, polarization effects at each ray intercept are described by a three-by-three polarization ray tracing matrix, P. Polarization effects are propagated along ray paths through optical systems by matrix multiplication of the P matrices for each ray intercept. In image space the P matrix can then be used to determine a full three element electric field vector. Three-dimensional 24 polarization ray tracing methods have been mentioned in several manuscripts9,10,11,12 and one of the methods is contained in a Code V macro. Three-dimensional polarization ray tracing algorithms in [13] and [14] are the basis of the polarization ray tracing code, “Polaris-M”15,16, developed and in use at our Polarization Laboratory at the University of Arizona. Chapter 2 defines the three-by-three polarization ray tracing matrix P and Chapter 3 and Chapter 4 provide diattenuation and retardance algorithms using the P matrix. Chapter 5 discusses polarization aberration analysis using the P matrix and Chapter 6 defines a skew aberration, which is a component of polarization aberration. Materials in Chapter 2 to Chapter 5 are based on reference [13] and [14] and Chapter 6 is based on reference [17]. Chapter 7 addresses a discontinuity in retardance values that occurs in the simulation and measurement of compound retarders, and provides a phase unwrapping method using Jones matrices. No systematic development of this method has yet been presented in the literature. Chapter 8 introduces a coherence matrix and defines a polarization ray tracing tensor, which provides a polarization ray tracing method through depolarizing optical systems. Chapter 9 further extends the coherence matrix to threedimensional (3D) Stokes parameters and defines a three-dimensional Mueller matrix. Chapter 10 summarizes the previous chapters and states conclusions. 25 CHAPTER 2 DEFINITION OF POLARIZATION RAY TRACING MATRIX In this section, the polarization ray tracing matrix is defined. It is shown that the relationships between all pairs of incident and exiting electric field vectors do not provide enough constraints to uniquely specify all nine elements of the polarization ray tracing matrix, so an additional constraint involving operation on the propagation vector is added to the matrix definition. 2.1 Definition of Polarization Ray Tracing Matrix, P The polarization ray tracing matrix P characterizes the change in a three-element electric field vector due to interaction with an optical element, a sequence of optical elements, or an entire optical system. Consider the evolution of the polarization state of a ray through an optical system with N interfaces labeled by index q. Further, if all the materials are isotropic, then polarization changes will occur only at interfaces. (This restriction will be removed later.) At each interface q, the propagation vectors kˆ q -1 and kˆ q may be different due to reflection or refraction. Eq-1 and Eq before and after the interaction are related by the polarization ray tracing matrix for the qth ray intercept, as shown in Figure 2.1, Ex , q Eq E y ,q Pq Eq-1 . Ez , q (2.1.1) 26 Note Pq is associated with specific incident and exiting propagation vectors, kˆ q -1 and kˆ q . Figure 2.1 The polarization states before (Eq-1) and after (Eq) the qth optical interface are related by the matrix Pq. A ray interacting with a series of optical elements is represented by cascading the Pq matrices for each ray intercept yielding a net polarization ray tracing matrix PTotal which represents the entire ray path, PTotal PN PN -1 Pq P2 P1 1 P q N ,-1 q . (2.1.2) The optical path lengths between the optical system entrance and exit pupils are summed to calculate a ray’s effect to the wavefront aberration (Section 5.2). The polarization ray tracing matrix describes the polarization dependent transmission and polarization dependent corrections to the optical path length. If the optical system includes anisotropic or birefringent media, the propagation portions cannot be modeled as identity matrices, but will take the form of retarder 27 matrices for birefringent media and/or diattenuation matrices for dichroic media. The propagation effect from ray interface q to q+1 is denoted as Aq +1,q . With the inclusion of polarization ray tracing matrices for anisotropic materials, Eq. (2.1.2) for the polarization ray tracing matrix for a ray through an optical system becomes PTotal PN A N , N -1PN -1 A3,2 P2 A 2,1P1 PN 1 q N -1,-1 A q +1,q Pq . (2.1.3) This formulation works well for stress birefringent and weakly anisotropic materials. In strongly birefringent materials like calcite and rutile, birefraction between the two modes (ordinary and extraordinary) causes ray doubling and two separated rays will continue to the exit port of the optical system. In this case, each of the rays refracting into a birefringent material has a separate polarization ray tracing matrix in the form of a polarizer. This polarizer matrix selects the incident state that couples into the specified mode. Further comments on ray tracing in anisotropic materials are beyond the scope of the present section. Pq as defined in Eq. (2.1.1) is under-constrained. Eq. (2.1.1) is equivalent to the Jones matrix equation, Ex ,q wq J q w q -1 E y ,q (2.1.4) except that Eq. (2.1.1) is formulated in global coordinates. In Eq. (2.1.4), Jones vectors, wq-1 and wq, are constrained to the transverse plane; so only two linearly independent polarization states are required to form a basis for all possible wq-1, for example wa and 28 wb. In Eq. (2.1.1) the transformation of all polarization states can be described as linear combinations of the transformations of wa and wb Ea Pq Ea , Eb Pq Eb . (2.1.5) The relationship in Eq. (2.1.5) yields six equations, one for each row, but Pq has nine elements. So Eq. (2.1.1) does not fully constrain Pq . In order to uniquely define Pq , an additional set of three constraints is applied, Pq kˆ q -1 kˆ q (2.1.6) The choice of is arbitrary, but only two values, either 0 or 1, allow Pq to be repeatedly cascaded and maintain the value of . Both choices of describe the same polarization effects at the ray intercept with minor differences as described below. With = 0, Pq is always singular and so Pq1 never exists. One of the singular values of Pq will always be zero, as will one of the eigenvalues. With = 1, only ideal polarizers have singular matrices. Thus = 1 was elected for this manuscript. This gives Pq the additional property that Pq kˆ q -1 kˆ q . (2.1.7) By adding this relationship between the incident and exiting propagation vectors, Pq is now uniquely defined by nine constraints, Eq.(2.1.1) and (2.1.7). 29 2.1.1 Polarization States as Three-element Electric Field Vectors For a plane wave propagating in an arbitrary direction k̂ , the electric field e r,t is e r, t Re E e where k 2 n i kkˆ r - t , (2.1.8) , n is the refractive index, and E has units of volts/meter. E is perpendicular to the propagation vector in isotropic and linear media so E kˆ 0. (2.1.9) For linearly polarized light, the electric field magnitude goes to zero twice per period i.e., all three elements of the time-dependent polarization vector in Eq. (2.1.8) become zero simultaneously. The condition for linearly polarized light is that the phases of all three elements of E are equal or 180° out of phase. From Eq. (2.1.8), e r 0, t 0 Re E Er . The electric field at t / 2 is e r 0, t / 2 Re Ee-i / 2 Im E Ei . (2.1.10) If the electric field is circularly polarized, the electric field vector at time zero and the electric field vector at quarter of the full cycle are perpendicular to each other and have the same magnitude. Thus, E describes circularly polarized light if Er Ei 0 and | Er || Ei |, (2.1.11) since the electric field for circularly polarized light should have the same amplitude at any given time. Otherwise the light is in an elliptical state. 30 In this presentation we adopt the convention that when a circularly polarized electric field vector rotates clockwise at a fixed observation plane (for example, r 0 ) looking into the beam, the electric field is right circularly polarized. If the vector rotates counterclockwise the electric field is left circularly polarized. For a unit amplitude circularly polarized electric field vector with a propagation vector along the z-axis, the cross product of electric field vectors at t 0 and t / 2 is {0, 0, -1}, e 0, 0 e 0, / 2 -zˆ . (2.1.12) In general, the cross product of the electric field vector at time t = 0 and t / 2 is anti-parallel to the propagation vector for right circularly polarized light, and parallel for left circularly polarized light. The same relation holds for elliptically polarized light. Thus the handedness of the elliptically polarized or circularly polarized light is determined by the sign of {e 0, 0 e 0, / 2 } kˆ . (2.1.13) If Eq. (2.1.13) is positive, the electric field is rotating counter clockwise (i.e. left circular) and if Eq. (2.1.13) is negative, the electric field is rotating clockwise (i.e. right circular). For elliptically polarized light, the major axis orientation can be calculated by calculating the electric field vector defined in local coordinates in a plane perpendicular to k̂ . For a given k̂ , two normalized real-valued vectors ( vˆ 1 and vˆ 2 ) that are orthogonal 31 to each other and orthogonal to k̂ can be calculated. vˆ 1 and vˆ 2 can be any normalized three-element vectors which satisfy vˆ 1 vˆ 2 , kˆ vˆ 1 , and kˆ vˆ 2 . (2.1.14) Then any electric field vector propagating along k̂ can be written as a superposition of vˆ 1 and vˆ 2 , i kkˆ r - t e r, t Re{Ee i kkˆ r - t } Re{[(E vˆ 1 ) vˆ 1 (E vˆ 2 ) vˆ 2 ]e i kkˆ r - t Re{( Ev1eiv1 vˆ 1 Ev 2eiv 2 vˆ 2 )e , } } (2.1.15) where Evi are real amplitudes along vˆ i . The major axis orientation is arctan( Ev 2 ), Ev1 (2.2.1) where is measured from v̂1 to v̂ 2 as shown in Figure 2.2. Figure 2.2 The major axis orientation θ measured from v̂1 to a . The axis vector is shown in red arrow. 32 The major axis vector a in global coordinates is a cos( ) vˆ 1 sin( ) vˆ 2 . 2.2 (2.2.2) Formalism of Polarization Ray Tracing Matrix using Orthogonal Transformation Ray tracing calculations using the polarization ray tracing calculus involve frequent transformations between the global coordinates of the polarization ray tracing matrix and local coordinates where the physics of polarization elements, anisotropic materials, thin film interfaces, anisotropic materials, diffraction gratings, reflection, refraction, and other phenomena are formulated. Orthogonal transformations between different coordinate systems, such as s-p coordinates, are straightforward and ubiquitous. This section explains the coordinate transformation notation. Orthogonal matrices, also known as real unitary matrices, describe rotations of orthogonal coordinate systems. In our case, orthogonal matrices transform between a local coordinate basis selected for a calculation at an interface and the global coordinate basis and vice versa. A separate pair of basis vectors is needed before and after the interface due to the change of ray direction. For reflection and refraction from surfaces, the s and p-polarization states along with the propagation vector form a natural basis, the {sˆ, pˆ , kˆ } basis. For simple and 33 isotropic media, sˆ and pˆ are defined as being perpendicular and parallel to the plane of incidence and thus are the eigenpolarizations for the Fresnel equations. The surface local coordinates {sˆ q , pˆ q , kˆ q -1} before and {sˆq , pˆ q , kˆ q } after the qth surface are, sˆ q kˆ q -1 kˆ q , pˆ q kˆ q -1 sˆ q , and sˆq sˆ q , pˆ q kˆ q sˆ q . ˆ ˆ | k k | q -1 (2.2.3) q The sˆ q vector is the same before and after the surface; only the pˆ q vector changes. The orthogonal matrices are O In ,q sˆx ,q sˆy ,q sˆz , q pˆ x ,q pˆ y ,q pˆ z ,q kˆx ,q -1 kˆy ,q -1 , kˆz ,q -1 O out ,q sˆx ,q sˆy ,q sˆz ,q pˆ 'x ,q pˆ ' y ,q pˆ 'z ,q kˆx ,q kˆy ,q . kˆz ,q (2.2.4) Oin1,q operates on Eq-1 in global coordinates and calculates the {sˆ q , pˆ q , kˆ q -1} basis Es ,q -1 components for the incident light, E p ,q -1 , which is a projection of Eq-1 onto the incident 0 local coordinates. Oout ,q rotates the surface local vectors {sˆq , pˆ q , kˆ q } back to the global coordinates {xˆ , yˆ , zˆ} . Reflection and refraction at dielectric, metal, and multilayer coated interfaces are described in terms of {sˆ , pˆ } components. Pq for a refraction or reflection can be derived by using J t ,q and J r ,q , which are defined in local ŝ and p̂ basis, and Eq. (2.2.4) 34 J t ,q 0 s ,t , q 0 p ,t , q 0 0 0 0 s ,r ,q 0 and J r ,q 0 p,r ,q 0 1 0 0 0, 1 (2.2.5) Subscript t indicates refraction, r indicates reflection, s for s-polarization and p for ppolarization. s ,t ,q , p ,t ,q are s and p-amplitude transmission coefficients and s ,r ,q , p,r ,q are reflection coefficients. For an uncoated interface between two isotropic media, the coefficients are calculated from the Fresnel equations. For coated interfaces, the coefficients are calculated from multi-layer coating calculations18,19,20. The polarization ray tracing matrix for refraction or reflection is Pq Oout ,q J qOin1,q . (2.2.6) The Jones matrices for gratings, holograms, sub-wavelength gratings, and other non-isotropic interfaces have off-diagonal elements. The Pq matrix for a ray intercept is Pq O out ,q J q O 1 in , q j11 where J q j21 0 j12 j22 0 0 0 , 1 (2.2.7) where J q represents the Jones matrix for the interaction in a local coordinate system. Oin,q and Oout ,q transform the local coordinate basis vectors associated with J q into the global coordinates. Different diffraction orders would naturally have different J q . For interactions that don’t change the ray direction, for example sheet polarization elements, the surface local coordinates are arbitrarily chosen to be perpendicular to the propagation vector and Eq. (2.2.7) becomes 35 Pq Oin,q J qOin1,q . 2.2.1 (2.2.8) Retarder Polarization Ray Tracing Matrix Examples The matrix P is a polarization ray tracing matrix for a single ray since orthogonal transformation matrices as in Eq. (2.2.4) are different for each ray (unless the beam of light is collimated and all surfaces are plane surfaces). The P matrix in Eq. (2.2.7) is not only dependent on the corresponding Jones matrix but also on the propagation vector. Jq e-i / 4 0 e-i / 4 0 0 i / 4 e 0 ei / 4 kˆ q -1 kˆ q 0 0 1 0 0 1 0 0 1 1 0 0 -i / 4 e 0 1 1 i / 4 e 0 0 0 0 0 Table 2.1 Polarization ray tracing matrix for Pq matrix e-i / 4 0 0 0 e i / 4 0 0 0 1 e-i / 4 0 0 1 0 0 0 0 ei / 4 0 0 1 - i / 4 0 0 e 0 0 ei / 4 a horizontal fast axis linear quarter wave retarder without beam deviation for three different propagation directions, along z-axis, y-axis, and x-axis. The Jones matrices are specified in a symmetric phase convention where the fast axis polarization state is advanced by an eighth of a wave and the slow axis is delayed by an eighth of a wave. Because of the many subtleties involved in transforming between Jones and polarization ray tracing matrices, several example P matrices are provided. Since P is unique for each ray, P can be different for the same optical element (i.e. Jones matrix) depending on the ray’s propagation direction. Table 2.1 shows P matrices for a quarter- 36 wave retarder with different propagation vectors at normal incidence. The Jones matrix e-i / 4 for a quarter-wave retarder oriented along x- and y-axes is 0 0 . e i / 4 The corresponding P matrix for a ray propagating along the z-axis is different from P for a ray propagating along the y-axis or x-axis. Note that Pq for a ray propagating along the z-axis is same as the Jones matrix padded with zeros and a single one. But in general, P matrices are different from Jones matrices. Pq’s in Table 2.1 relate the phase to the corresponding component of the electric field in global coordinates. For a more complicated example, the P matrix for a quarter wave linear retarder with a fast axis along {1, 0, 0} , surface normal {0,sin , cos } and propagation vector parallel to the surface normal vector will be e-i / 4 0 0 e cos + sin cos sin (1- e ) . cos sin (1- ei / 4 ) cos 2 + ei / 4 sin 2 0 i / 4 2 0 2 i / 4 (2.2.9) Since kˆ q -1 is parallel to the surface normal vector of the retarder kˆ q {0,sin , cos } . Eq (2.2.9) advances eigenpolarization {1, 0, 0} by / 4 and retards eigenpolarization {0, cos , -sin } by / 4 . 37 2.3 Conclusion These three-by-three polarization ray tracing matrices perform ray tracing in global coordinates, which provide an easy basis to interpret polarization properties for most systems. The apparent rapid variation of polarization states and properties around local coordinate singularities is avoided. It remains straightforward to convert results into other local coordinate bases. Formalism for polarization ray tracing using three-by-three matrices has been demonstrated. The relationship to the Jones calculus has been shown. Algorithms for reflection, refraction, and polarization elements are summarized with specific examples. 38 CHAPTER 3 3.1 CALCULATION OF DIATTENUATION Diattenuation Calculation using Singular Value Decomposition One objective of a polarization ray trace is to understand the polarization properties associated with the polarization changes induced by the optical system. Analyzing the polarization properties of Jones matrices is well established in the literature21, 22, 23. This section derives an algorithm for the diattenuation associated with P. Here, it is assumed the ray begins and ends in air or vacuum with refractive index of one; these results are readily generalized to other object and image space refractive indices. The diattenuation, D , depends on the difference of the maximum Imax and minimum Imin intensity transmittances considered over all incident polarization states as D I max I min and 0 D 1 . I max + I min (3.1.1) An ideal polarizer has diattenuation equal to one; one incident polarization state is completely discarded. Eigenvectors of the P matrix do not generally represent polarization states because in general light rays enter and exit in different directions24. The diattenuation of the P matrix can be calculated by using the singular value decomposition (SVD)25, 26. kˆx ,Q P U D V † kˆy ,Q ˆ k z ,Q u x ,1 u y ,1 u z ,1 u x ,2 1 0 u y ,2 0 1 u z ,2 0 0 * kˆ x ,0 * v x ,1 2 v* x ,2 0 0 kˆ*y ,0 v* y ,1 v* y ,2 kˆ*z ,0 v* z ,1 , (3.1.2) v* z ,2 39 which decomposes P into unitary matrices U and V , and a diagonal matrix D. The diagonal elements of D are the non-negative real singular values i 27 1 2 0 (3.1.3) V † (V* )T . (3.1.4) and † indicates Hermitian adjoint The diagonal elements of D are the singular values of P. Since P was constructed such that P kˆ q -1 kˆ q (Eq. (2.1.7)), one of P’s singular values is always one, and the associated column of V is k̂ 0 . Eq. (3.1.2) places the one in the first column. The other two columns of V , v1 and v2 , are two special polarization vectors in the incident transverse plane that generate the maximum and minimum transmitted flux. Similarly, the columns of U are the exiting propagation vector kˆ Q and two orthogonal polarization vectors u1 and u2 in the exiting transverse plane. The relationship between P, its singular values and these special polarization vectors, is P v1 1u1 , P v 2 2u 2 , and P kˆ 0 kˆ Q . (3.1.5) v1 and v2 are in general the only two orthogonal incident polarization states that remain orthogonal when they emerge from P as u1 and u2 . So these orthogonal states form a canonical basis for incident and exiting polarization states. 40 An arbitrary normalized incident polarization state E can be expressed as a linear combination of v1 and v2 as E v1 v2 , where and are complex, (3.1.6) | |2 | |2 1 . The transmitted electric field vector after P is P E . Therefore the flux of the transmitted electric field is ITrans | P E |2 E† P† P E (3.1.7) Using Eq. (3.1.5) and (3.1.7), it can be shown that the flux of the transmitted electric field is ITrans | |2 12 | |2 22 | |2 (12 22 ) 22 . (3.1.8) Since both | |2 (12 22 ) and 22 are positive since 1 2 by construction, the maximum intensity transmittance occurs when the incident state is v1, and the minimum intensity occurs when the incident state is v2 i.e. I 12 if | |2 1 , ITrans max 2 2 I min 2 if | | 0 (3.1.9) for any polarization ray tracing matrix P . Thus the diattenuation of P is D 12 22 , 12 22 (3.1.10) 41 and v1 vmax and v2 vmin are the incident polarization states for which P gives the maximum and minimum transmittance. P v1 and P v 2 are the corresponding exiting polarization states. For the case when the beam is undeviated ( kˆ 0 kˆ Q ) the two unitary matrices, U and V , have the same eigenstates; thus the diattenuation can be calculated from the eigenvalues of the P matrix as well as from Eq. (3.1.10). 3.2 Example – A Hollow Corner Cube A hollow aluminum coated corner cube provides an example of an inhomogeneous polarization component, an element in which the diattenuation and retardance are not aligned. Corner cubes are commonly used as retroreflectors, and their polarization properties have been studied in several manuscripts28, 29, 30. As shown in Figure 3.1, the hollow corner cube consists of three mutually perpendicular aluminum surfaces. A refractive index of 0.77 + 6.06i is assumed for aluminum at 500nm. There are six different ray paths for a collimated beam of light that enters the corner cube31. Figure 3.1 shows one of the ray paths with a set of propagation vectors (black arrows). The incident and the exiting propagation vectors are anti-parallel. 42 Figure 3.1 Top view of a hollow aluminum coated corner cube. Three surfaces of the corner cube are perpendicular to each other. Our example ray is incident along the symmetry axis, kˆ 0 {0, 0, -1} . Figure 3.2 shows the corner cube with the propagation vectors in black, s-local coordinate vectors in solid red and p-local coordinate vectors in dashed blue. The figure shows how the local coordinate bases change as the ray propagates through the corner cube. 43 Figure 3.2 Top and side views of a corner cube. Propagation vectors are shown in solid black, local s coordinate vectors in solid red, and local p coordinate vectors in dashed blue. The reflecting surface configuration is specified in Table 3.1 by the surface normal vectors, Pq’s, and the various vectors for each ray intercept. Pq is calculated using Eq.(2.2.4), (2.2.6), and the Fresnel equations for aluminum. 44 1 kˆ q1 kˆ q pˆ q pˆ q sˆ q Surface Normal Pq 0 0 -1 -2 2 3 0 -1 3 -1 0 0 -1 3 0 2 2 3 0 -1 0 2 3 0 -1 3 0 0.94 0.26 0.16i 0 -0.96 0.18 i 0 -0.75 - 0.46i 0 0.33 -1 6 3 2 2 3 5 6 1 2 3 2 3 -1 2 3 1 2 2 3 -1 6 -1 2 -1 3 0.25 - 0.08i 0.44 0.33i 0.7 0.24i -0.95 - 0.05i -0.04 0.08i 0.23 0.14i -0.15 0.72 0.27i -0.57 - 0.01i 1 6 -1 2 3 2 2 3 1 2 - 3 2 0 - 3 2 -1 2 0 -1 6 1 2 -1 3 -0.65 - 0.09i -0.53- 0.15i 0.37 0.23i -0.53- 0.15i -0.04 0.08i -0.65 - 0.4i -0.47 0.82 0.33 2 -2 2 - 2 3 0 -1 3 3 - 2 3 2 3 1 3 3 2 3 1 3 0 0 1 Table 3.1 Propagation vectors, local coordinate basis vectors, surface normal vectors and polarization ray tracing matrices associated with a ray path through aluminum coated hollow corner cube. The net polarization ray tracing matrix Pcc (cc for corner cube) for this ray path is calculated by cascading the three P matrices in Table 3.1, 0.39 + 0.78i 0.01+ 0.02i 0 Pcc P3 P2 P1 -0.02i 0.40 + 0.78i 0 . 0 0 -1 The singular value decomposition of Pcc gives (3.2.1) 45 0 0.63 + 0.15i 0.74 0.17i U cc 0 0.37 - 0.66i -0.32 0.57i , 1 0 0 0 0 1 Dcc 0 0.88 0 , 0 0 0.87 0 0.43 - 0.49i 0.47 - 0.6i Vcc 0 -0.41- 0.64i 0.38 0.52i . -1 0 0 (3.2.2) As shown in Eq. (3.1.2), Vcc and Ucc have the incident and exiting propagation vectors as their first columns. Table 3.2 lists the maximum and minimum intensity transmittances assuming the incident electric field’s intensity is one and the diattenuation of the corner cube is calculated from the singular values of the Pcc matrix. I max v1 I min v2 D -0.91i 0.65e-0.85i 0.65e1.30i 0.757 0.76e 0.76 0.014 -2.15i -0.91i -2.15i 0.94 i 1.85i 0.76e e 0.65e e 0.76 0.65e 0 0 0 0 Table 3.2 The maximum intensity of output and associated incident electric field, the 0.774 minimum intensity of transmitted electric field and associated incident electric field, and the diattenuation from the ray through the corner cube system. Figure 3.3 shows the polarization states associated with the maximum and the minimum intensity transmittances; the last two columns of Vcc and Ucc represent two incident polarization states ( v1 , v 2 ) and exiting states ( u1 , u2 ) with the maximum and the minimum intensity transmittances. v1 , v2 and u1 , u2 are elliptically polarized, so this path 46 through the corner cube acts as a weak elliptical diattenuator with a diattenuation of 0.014. v1 and v2 are the only pair of two orthogonal incident polarization states which remain orthogonal upon exit. Incident polarization states ( v1 , v 2 ) are defined in the incident local coordinates and exiting polarization states ( u1 , u2 ) are defined in the exiting local coordinates. (a) (b) Figure 3.3 (a) The exiting state from the corner cube with the maximum intensity and corresponding incident state. (b) The state with the minimum intensity and corresponding incident state. All represented in local coordinates where 2D polarization vectors are defined. Each propagation vector comes out of the page. Using local coordinate systems in describing polarization vectors with opposite propagation directions complicates the discussion of the polarization state and its transformation. In global coordinates, as shown in Figure 3.4, the direction of rotation of the electric field is in the same direction for the corresponding incident and exiting states; 47 v1 and u1 have the same direction of rotation and v2 and u2 have the same direction of rotation due to the anti-parallel propagation vectors, kˆ 0 and kˆ 3 . (a) (b) Figure 3.4 (a) The exiting state from the corner cube with the maximum intensity and corresponding incident state. (b) The state with the minimum intensity and corresponding incident state. All states are represented in global coordinates (x-y plane) looking into the corner cube; propagation vectors are anti-parallel for the incident and exiting electric field vectors. 3.3 Conclusion The calculation of the diattenuation is achieved via the singular value decomposition. Unitary matrices provide two canonical polarization states that are orthogonal to each other and related by the singular values. The incident propagation vector and the exiting propagation vector are related by a singular value of unity due to Eq. (2.1.7). The method was illustrated on a hollow aluminum corner cube. 48 CHAPTER 4 CALCULATION OF RETARDANCE 4.1 Introduction When describing rays propagating through optical systems, the effects of coordinate system changes on refraction can masquerade as circular retardance; this is shown in Section 4.3.1. Similarly, coordinate system changes on reflection can masquerade as a half wave of linear retardance. This section’s objectives are: (1) explore the local coordinate transformation associated with parallel transport of transverse vectors along ray paths through optical systems, and (2) present an algorithm for the calculation of proper retardance in polarization ray tracing using the three-by-three polarization raytracing calculus13. This algorithm separates the part of the polarization ray-tracing matrix that describes proper retardance from the part that describes non-polarizing rotations. Examples highlight the associated subtleties. The term “retardance” refers to a physical property by which optical path length accumulation depends on the incident polarization state. The classic retarder is a crystalline waveplate that divides a beam into two modes having two distinct polarization states and optical path lengths 1, 32, 33. The retardance, measured in radians, is the phase difference that accrues corresponding to that optical path difference, a difference in transit time. For more complex optical systems the optical path length can be multivalued (such as for sequences of crystals) or even undefined (such as for thin-film coated surfaces). The Jones matrix has two eigenpolarizations. When the eigenpolarizations are 49 orthogonal, the retardance is calculated to be the difference in the phases of the associated eigenvalues. For the case of inhomogeneous elements, which have nonorthogonal eigenpolarizations, the element is expressed as a product of a pure retarder with a pure diattenuator, and the retardance is well-defined to be that of the pure retarder, as described by Lu21. Often skew rays through optical systems are slightly inhomogeneous. The action of a retarder on polarization states can be depicted by the rotation of the Poincaré sphere by the ideal retarder; the rotation angle is the retardance and the Poincaré rotation axis identifies the fast and slow axes34. Knowledge of the distribution of an optical system’s retardance provides a partial description of the polarization dependence of the exiting wavefronts35. There are complications in extending the concept of retardance to threedimensional polarization ray trace matrices. Since the entering and exiting rays need not be collinear, the eigenpolarizations of the polarization ray tracing matrix may not represent actual electric field states. This can be solved by dropping back into a local coordinate system, but then the calculated retardance will depend on the local coordinate system selected. Section 4.4 presents well-defined retardance calculation algorithms for the three-by-three polarization ray tracing matrix, which require using the entire set of propagation vectors associated with all ray segments. 50 4.2 Purpose of the Proper Retardance Calculation The polarization dependent phase change associated with a ray path through an optical system has two components: (1) the proper retardance; the phase retardation (optical path difference) arising from physical processes, such as propagation through birefringent materials or reflection or refraction from a surface, and (2) a geometric transformation due to the local coordinate selection used for determining the phase. Figure 4.1 is a graphical representation of a Jones matrix polarimeter performing a calibration run in air. By rotating the polarization state analyzer (PSA) by θ, the exiting local coordinates for the Jones matrix also rotate by θ. Therefore, the polarimeter measures this empty compartment as a circular retarder with retardance of 2θ. Rotating the exiting local coordinates does not introduce an optical path difference between right and left circularly polarized light. Here the “proper” retardance is zero. The retardance value measured by polarimeters depends on the relative choice of incident and exiting local coordinates. Figure 4.1 A polarimeter measuring a sample retarder with the polarization state analyzer (PSA) (a) aligned with the polarization state generator (PSG) and (b) rotated to an arbitrary orientation. By rotating the PSA, the exiting local coordinates for the Jones matrix are also rotated. The measured retardance of the 51 sample now includes a “circular retardance” component of 2 as well as the proper retardance. Our goal is to develop a retardance calculation algorithm which separates the geometric transformation, an “optical activity-like” geometric rotation and/or inversion, from the proper retardance. A parallel transport matrix, Q, described in Section 4.3.3, identifies canonical pairs of local coordinate systems for general sequences of ray paths, thus characterizing the geometric transformation. Q is the tool which separates retardance from geometric transformation. 4.3 Geometrical Transformations The local coordinates which are necessary to specify Jones vectors propagating in arbitrary directions may be rotated and/or inverted between object and image space10 in ways which are not associated with any retardance. This section provides a definition of a parallel transport matrix and describes how it keeps track of geometric transformations of the local coordinates. 4.3.1 Local Coordinate Rotation: Polarimeter Viewpoint Jones matrices are defined with respect to local coordinates in the transverse plane; one set is associated with the incident Jones vector and another with the exiting Jones vector. Retardance of a Jones matrix is calculated from eigenvalues of the matrix arg(1 ) arg(2 ), (4.3.1) 52 where J w1 1w1 and J w2 2 w2 36. First, consider a Jones matrix polarimeter measurement of an empty compartment, which has the identity Jones matrix. If the exiting local coordinates are rotated by an angle θ with respect to the incident local coordinates by rotating the PSA by θ, the measured Jones matrix is a rotation matrix instead of the identity matrix, cos J ( ) R ( ) I sin -sin 1 0 cos cos 0 1 sin -sin , (4.3.2) cos with eigenvalues 1 exp(i ), 2 exp(-i ) (4.3.3) and right and left circularly polarized polarizations 1 1 w1 , w 2 . (4.3.4) -i i Note the similarity of J ( ) to the form of a circular retarder. Unless the exiting local coordinate orientation is parallel to the incident local coordinate orientation, a nonpolarizing element, which should be described by the identity matrix, appears to have a “circular retardance” of arg(1 ) arg(2 ) 2 where is the local coordinate rotation. (4.3.5) This is an example of the geometric transformation, a result of the choice of local coordinates in the description of the Jones matrix. 53 Now consider a retarder Jones matrix measured by this polarimeter with rotated PSA. Eq. (4.3.6) shows the measured Jones matrix, J ( ) from this polarimeter cos J ( ) sin sin j11 cos j21 j12 . j22 (4.3.6) Retardance calculated from Eq. (4.3.1) and (4.3.6) changes as the rotation angle θ changes. For circular retarders with retardance of δ, the measured retardance from the polarimeter as a function of the PSA angle is Measured 2 arg(cos( ) i | sin( ) |) . 2 2 (4.3.7) Figure 4.2 is the plot of Eq. (4.3.7) for different retardance ( ) of the circular retarder in [0, 2 ] in different colors. For a half wave circular retarder ( ), retardance is independent of the PSA orientation and remains as a half wave of retardance as shown as a blue line in the figure. Figure 4.2 Measured retardance values as a function of rotation angle θ for circular retarders with retardance between zero and one wave of retardance are shown in 54 different colors. Retardance of a half wave circular retarder is independent of the PSA orientation. For a linear retarder with retardance of , retardance as a function of the PSA angle is Measured 1-(cos cos )2 2 2Arctan( ) . cos cos 2 (4.3.8) Figure 4.3 shows the plot of Eq. (4.3.8) different retardance ( ) of the linear retarder in [0, 2 ] in different colors. Again, retardance of half wave linear retarders is independent of the PSA orientation. Figure 4.3 Measured retardance values as a function of rotation angle θ for linear retarders with retardance between zero and one wave of retardance are shown in different colors. Retardance of a half wave linear retarder is independent of the PSA orientation. 55 4.3.2 Parallel Transport of Vectors We now explore this geometric transformation for skew ray paths through optical systems using parallel transport of local coordinate vectors. Parallel transport of a vector over a sphere is the process of moving the vector along a series of great circle arcs such that the angle between the vector and each arc is constant. At a vertex where the path transitions from the first to the second arc (and so on), the angle at the vertex is maintained along the second arc, and so on for the entire series of arcs. Consider tracing a single skew ray through an optical system with N interfaces. First we consider a ray where the incident propagation vector kˆ incident (first ray segment) is parallel to the exiting propagation vector kˆ exit (last ray segment), but the skew ray has changed its propagation direction kˆ q many times while traversing the optical system. This set of kˆ q can be represented as points on the unit sphere connected by great circle arcs. Since kˆ incident kˆ exit , the arcs form a closed spherical polygon. Let a set of orthogonal local coordinate vectors be defined in the transverse plane for the first ray segment. When this set of local coordinates is carried through the system by parallel transport, it rotates by an angle in radians equal to the spherical polygon’s solid angle 37. This rotation is equivalent to the Pancharatnam phase38,39 or the Berry phase40. This rotation is shown in Figure 4.4 (a), which depicts the parallel transport of an arbitrarily selected pair of incident local basis vectors {xˆ A , yˆ A} (solid green) through an 56 optical system where the propagation vector changes its direction from point A B C D . Only one of the pair of basis vectors is labeled in Figure 4.4. The exiting local basis vectors {xˆ D , yˆ D } (dashed red) are rotated from {xˆ A , yˆ A} by / 2 radian, equivalent to the solid angle of / 2 steradians of the associated spherical triangle. Figure 4.4 (a) The evolution of a local coordinate pair {xˆ A , yˆ A} (green) through a system of three fold-mirrors. The exiting local coordinates (dashed red) undergo a 90° rotation from the initial local coordinates (solid green). (b) A three fold-mirror system. When a collimated beam enters the system along the z-axis the beam exits along the z-axis. A simple example of parallel transport and the Pancharatnam phase is the three fold-mirror system shown in Figure 4.4 (b). Three mirrors are aligned so that the angle of incidence for a collimated beam at each mirror is 45°. Let each reflection be an ideal non-polarizing reflection so that the incident polarization ellipse enters and exits the 57 optical system with the same ellipticity. The incident propagation vector k̂ 0 is {0, 0,1} and after three reflections, the exiting propagation vector k̂ 3 is also {0, 0,1} where kˆ 1 {1, 0, 0} and kˆ 2 {0,1, 0} . Thus for this system, propagation vectors are mapped to points A, B, and C on a unit sphere. One might naively select the same local coordinates for the incident space and exiting space since kˆ 0 kˆ 3 for this system. However, this ray path has a geometric transformation of a 90° rotation, the solid angle subtended by the spherical triangle ABC. Thus if the initial local coordinates and the exiting local coordinates for a Jones matrix are chosen to be parallel to each other, the system appears to have “circular retardance”. Associated with this rotation of the local coordinates, incident x-polarized light (solid green) in Figure 4.4 (b) exits the system as y-polarized light (dashed red), even though in this model the mirrors are non-polarizing. If this non-polarizing system is measured by a polarimeter with parallel polarization state generator and polarization state analyzer, the polarimeter will measure 180° of circular retardance and inversion resulting from the odd number of reflections, the retardance arising purely from the local coordinate transformation. To measure the proper retardance, the analyzer should be rotated 90° from the generator. 4.3.3 Parallel Transport Ray Tracing Matrix, Q The parallel transport matrix Qq at qth ray intercept is defined as a real unitary three-bythree ray tracing matrix, calculated by assuming that each ray intercept is non-polarizing. 58 Qq for refraction is a rotation about kˆ q -1 kˆ q (the ŝ vector) by the angle the ray is deviated which is equivalent to sliding basis vectors around a unit sphere via parallel transport of vectors. Qq for refraction rotates all incident polarization ellipses transverse to kˆ q -1 to the same ellipses transverse to kˆ q . Qq for reflection is an inversion with respect to the reflecting surface which is normal to kˆ q -1 kˆ q . Other interactions such as diffraction and scattering can use either reflection or refraction Qq matrices depending on the geometry of the system; if both kˆ q -1 and kˆ q are in the same medium, the reflection algorithm is used and if the two propagation vectors are in different media, the refraction algorithm is used. If an incident electric field vector is aligned with a basis vector, a sequence of Qq ’s performs the vector parallel transport into the corresponding exiting basis vector. Other than this function, Qq has no polarization effects. The cumulative parallel transport ray tracing matrix for a ray through a system with N interfaces is QTotal 1 Q q N ,-1 q Q N Q N -1 Q q Q 2Q1 . (4.3.9) Depending on the number of reflections in the system, {xˆ L ,q , yˆ L ,q , kˆ q } can form either a right-handed coordinate system or a left-handed coordinate system. Figure 4.5 shows the incident ( {xˆ L,0 , yˆ L,0 }), reflected ( {xˆ L,r ,1 , yˆ L,r ,1} ), and transmitted ( {xˆ L,t ,1 , yˆ L,t ,1} ) local coordinate pairs calculated from Q1 where the incident local coordinates are 59 xˆ L,0 kˆ 0 kˆ 1 , yˆ L,0 kˆ 0 xˆ L,0 . ˆ ˆ | k 0 k1 | (4.3.10) Figure 4.5 Incident, reflected, and transmitted local coordinates calculated from parallel transport matrices. {xˆ L,0 , yˆ L,0 , kˆ 0 } are the right handed incident local coordinates, {xˆ L ,r ,1 , yˆ L ,r ,1 , kˆ r ,1} are the left handed reflected local coordinates, and {xˆ L,t ,1 , yˆ L,t ,1 , kˆ t ,1} are the right handed transmitted local coordinates. Note that {xˆ L ,0 , yˆ L,0 , kˆ 0 } and {xˆ L ,t ,1 , yˆ L ,t ,1 , kˆ t ,1} form a right-handed set while {xˆ L ,r ,1 , yˆ L ,r ,1 , kˆ r ,1} form a left-handed set. Subscript r stands for reflection and t for transmission. For reflection and transmission xˆ L,0 is the s-basis vector and yˆ L,0 is the pbasis vector. 60 For birefringent interfaces, grating diffraction, and other similar cases, xˆ L,0 and yˆ L,0 do not generally correspond to s and p. Figure 4.6 shows the s and p vectors in object space and their geometric transformation along each ray segment using a set of Qq . Figure 4.6 A two mirror system. The red solid lines show the s vector at the first mirror and its geometric transformation along each ray segment using Q. The blue dashed lines show the p vector in object space and its geometric transformations. The explicit formula for refraction Qq is A2 B (1 B) D 2 1 Qq 2 F [(1 B) D L] A D[(1 B)C L] F [(1 B) D G ] D[(1 B)C H ] A2 B (1 B) F 2 C[(1 B) F H ] C[(1 B) F G ] A2 B (1 B)C 2 (4.3.11) where A Norm[kˆ q -1 kˆ q ], B kˆ q -1 kˆ q , {C , D, F } kˆ q -1 kˆ q , G A 1 B 2 A2 F 2 C2 F 2 ,H A 1 B 2 A2 C 2 C 2 D2 (4.3.12) ,L A 1 B 2 A2 D 2 D2 F 2 . 61 Qq for reflection is 2(k x ,q -1 k x ,q ) 2 1 A B Qq A C A B A 2(k y ,q -1 k y ,q ) 2 1 A D A D A 2(k z ,q -1 k z ,q ) 2 1 A C A (4.3.13) where A Norm[kˆ q -1 kˆ q ], B 2(k x ,q -1 k x ,q )(k y ,q -1 k y ,q ), C 2(k x ,q -1 k x ,q )(k z ,q -1 k z ,q ), D 2(k y ,q -1 k y ,q )(k z ,q -1 k z ,q ). (4.3.14) 1 reverses the geometric transformation contained within PTotal . Therefore, QTotal for an optical system with N interfaces, QTotal xˆ L,0 xˆ L, N 1 QTotal xˆ L, N xˆ L,0 QTotal yˆ L ,0 yˆ L , N 1 QTotal yˆ L, N yˆ L,0 QTotal kˆ 0 kˆ N (4.3.15) 1 QTotal kˆ N kˆ 0 where {xˆ L ,0 , yˆ L,0 , kˆ 0 } are the incident local coordinates and {xˆ L, N , yˆ L , N , kˆ N } are the parallel local coordinates in the exit space transformed by QTotal . The vectors xˆ L,0 and yˆ L,0 here are assumed to be an arbitrary pair of orthogonal vectors in the transverse plane of the first ray segment calculated by Eq. (4.3.10). 62 4.4 4.4.1 Proper Retardance Calculations Definition of the Proper Retardance The “proper retardance” or just “retardance” is the accumulation of polarization dependent optical path difference from physical processes associated with the ray path. Retardance is generated by mechanisms which cause a polarization dependent phase change, such as s and p phase differences in reflection or refraction, propagation through a waveplate, birefringent material, or diffraction grating. Retardance is invariant with respect to the selection of local or global coordinates. The following sections present retardance algorithms for ray paths through optical systems that are represented by P matrices; this algorithm calculates retardance which does not contain any geometric transformation. 4.4.2 Separating Local Coordinate Transformation from P A method to keep track of the local coordinate transformation for a ray path through an optical system is presented in this section. Using the parallel transport matrix allows the retardance to be uniquely defined despite the fact the incident and exiting propagation vectors are different. QT o t a (Eq. l (4.3.9)) provides a well-defined relationship between local coordinates in the two transverse planes; if an arbitrary polarization state represented by an electric field vector is specified along the first ray segment, the corresponding electric field vector for a nonpolarizing system is determined along the exiting ray segment. 1 reverses the QTotal 63 geometric transformation so that the coordinate system in exit space is reverted back to the initial coordinate system. The operation 1 ΜTotal QTotal PTotal (4.4.1) yields ΜTotal , a polarization ray tracing matrix with the exiting electric field vectors rotated about kˆ 0 kˆ N and/or reflected in the surface so the incident and exiting transverse planes are parallel, and both are orthogonal to k̂ 0 . Note that ΜTotal is not a Mueller matrix but a three-by-three matrix. Because calculation of Q requires knowledge of all ray segments, the proper retardance cannot be separated for an unknown “black box” system. 4.4.3 The Proper Retardance Algorithm for P, Method 1 Calculating proper retardance consists of applying the polar decomposition to ΜTotal and computing the eigenvalues of the unitary retarder matrix. Two methods may be used. The polar decomposition can be applied directly to the three-by-three ΜTotal , or the twoby-two Jones matrix can be retrieved from ΜTotal and formulas from ref[24] applied. Here we follow the first approach. We present the second approach in the next section. The polar decomposition of ΜTotal yields a unitary matrix and a nonnegative definite Hermitian matrix , D ΜTotal , R , ΜTotal ΜTotal , R ΜTotal , D ΜTotal (4.4.2) 64 where ΜTotal , R is a retarder (unitary) matrix and ΜTotal , D and ΜTotal , D are diattenuator (nonnegative definite Hermitian) matrices. The retardance of ΜTotal is the retardance of ΜTotal , R . ΜTotal , R has three eigenvectors v1 , v 2 , kˆ 0 , (4.4.3) 1 , 2 , 3 . (4.4.4) and three associated eigenvalues One of the eigenvalues, 3 1 , relates the incident propagation vector k̂ 0 to the rotated exiting propagation vector. The retardance ( ) is calculated from the two eigenvalues associated with the transverse plane, 1 and 2 , as the difference in their phases, arg(2 ) arg( 1 ) (4.4.5) assuming arg(2 ) arg(1 ) . The fast axis orientation is along the eigenpolarization v1 which is defined in the object space of the system. When PTotal is homogeneous, applying the polar decomposition to ΜTotal is unnecessary; ΜTotal and ΜTotal , R have the same eigenvalues and eigenpolarizations. Thus, Eq. (4.4.5) gives the retardance of PTotal where 1 and 2 are eigenvalues of ΜTotal . 65 4.4.4 The Proper Retardance Algorithm for P, Method 2 In this section the second approach is presented; a two-by-two Jones matrix is retrieved from an inhomogeneous ΜTotal , and retardance is calculated from the Jones matrix. The first step of retrieving a two-by-two Jones matrix is rotating ΜTotal , 0 J S R U ΜTotal U 0 . 0 0 1 † (4.4.6) U is a unitary three-by-three rotation matrix which rotates all vectors by cos1 (kˆ 0 zˆ ) counterclockwise about the kˆ 0 zˆ axis so that k̂ 0 is rotated to ẑ . For kˆ 0 {kˆx , kˆy , kˆz } U is k x2 cos + k y2 1 k x k y (cos -1) H H k x sin k x (cos - k y ) - H k x sin k x2 + k y2 cos H k y sin - H k y sin , (4.4.7) H cos where H = k x2 + k y2 . The upper two-by-two submatrix of S R serves as a Jones matrix J in the following equation for the retardance of the PTotal matrix, | trJ 2 cos 1 ( det J trJ † | | det J | 2 tr(J † J ) 2 | det J | ) . (4.4.8) 66 The unitary matrix (retarder) of polar decomposed J ( J R in ref [24]) has two eigenpolarizations {w1 , w2 } . The fast axis orientation of the retarder is along the eigenpolarization which has the smaller eigenvalue argument since the author follows the decreasing phase convention. These eigenpolarizations can be written as three-element electric field vectors, which provide a canonical basis set in the incident space, v1 U† w1 , v 2 U† w2 , kˆ 0 , (4.4.9) where w1 {w x,1 , w y ,1 ,0} and w2 {w x,2 , w y ,2 ,0} . In exit space, the canonical basis set is v1 Q v1 Q U† w1 , v2 Q v 2 Q U† w2 , kˆ N . 4.4.5 (4.4.10) Retardance Range Optical path difference (OPD) and retardance may assume any value between 0 and infinity. However, just as with the Jones calculus and Mueller calculus, the retardance algorithms in the previous section return a retardance of less than a wave. This is similar to the phase of the electric field which is usually represented modulo 2π while the optical path length can assume any value. Further discussion of methods to extend the retardance calculation beyond 2π by phase unwrapping or other methods is beyond the scope of this section41,42 and will be discussed in Chapter 7. 67 4.5 Examples In this section simple examples are presented to elucidate retardance algorithms. All the examples have homogeneous polarization ray tracing matrices thus eigenvalues of ΜTotal were used directly. 4.5.1 Ideal Reflection at Normal Incidence Consider ideal (100%) reflection from a mirror at normal incidence. Since the mirror itself is non-polarizing, the retardance should be zero. The P matrix of this system with the incident propagation vector along the z-axis, is -1 0 0 P 0 -1 0 . 0 0 -1 (4.5.1) P demonstrates that x and y electric fields reflect without a differential phase change, but the propagation vector direction flips from z to -z. The upper diagonal elements have -1 due to a π phase shift upon external reflection for x and y component of the electric field vector. When right circularly polarized light enters, -1 0 0 1 i 0 -1 0 -i e 0 0 -1 0 1 -i 0 (4.5.2) the same electric field vector exits, but since the propagation vector changed to {0,0,-1}, this light is left circularly polarized. Similarly when linearly polarized light enters, 68 -1 0 0 cos cos i 0 -1 0 sin e sin 0 0 -1 0 0 (4.5.3) the same electric field vector exits. The electric field is oscillating in the same global plane, but in the Jones matrix local coordinates the incident angle θ is mapped into -θ. In Eq (4.5.3) no relative phase changes have been introduced between x and y component of the electric field. This is very different from the standard Jones matrix for reflection 43 ,44 , Jf = -1 0 . Jf appears to include a relative π phase shift between x and y polarization 0 1 components. In Jf this phase shift serves two purposes: (1) it reflects right circularly polarized light into left circularly polarized and vice versa, and (2) it changes the orientation of incident linearly polarized light from θ to -θ, which is appropriate when maintaining right-handed {xˆ L , yˆ L , kˆ } coordinates after reflection. In order to keep all the Jones matrix local coordinates right-handed (as shown in Figure 4.7), the Jones matrix for reflection has to contain -1 in one of the diagonal elements. Thus the Jones reflection matrices have the form of half wave linear retarders. The minus sign does not indicate a physical half wave linear retardance; it indicates a local coordinate system change. 69 Figure 4.7 An ideal reflection at normal incidence with the incident and exiting right-handed local coordinates, {xˆ L,0 , yˆ L,0 } and {xˆ L,1 , yˆ L,1} . In this particular choice of local coordinates, the xˆ L vector was flipped after the reflection. The Q matrix for this system reveals the local coordinate transformation clearly, 1 0 0 Q 0 1 0 . 0 0 -1 (4.5.4) The minus sign is now associated with the propagation vector, where it properly belongs. Using Eq. (4.4.5) and (4.5.4), zero retardance is calculated from an ideal retro-reflection. 70 4.5.2 Brewster’s Angle Analysis This section extends the normal incidence reflection analysis to reflections at any angles of incidence. Unlike reflection from normal incidence, when the p-polarized light reflects from a non-absorbing dielectric material, a physical phase change occurs as angle of incidence approaches Brewster’s angle; the p Fresnel reflection coefficient is zero at Brewster’s angle. If unpolarized light is incident on a surface at the Brewster’s angle, the reflected light is rendered linearly polarized with the electric vector transverse to the plane of incidence; it is s-polarized. The transmitted light is partially polarized while the reflected beam is completely polarized. Since the p Fresnel reflection coefficient changes its sign before and after Brewster’s angle ( B ), there will be π phase shift upon reflection for the incident angle larger than B . This change in sign of the coefficient is not discontinuous since the p-polarization element has very small amplitude reflection coefficient near B and becomes zero at B . Figure 4.8 shows Fresnel s and p reflection coefficients for external and internal reflections as a function of angle of incidence for an air-glass (n=1.5) interface33. The Fresnel coefficients are defined in the incident and exiting local coordinates which are both right-handed; at normal incidence s and p reflection coefficients have the opposite sign due to this choice of local coordinates. 71 External Reflection Reflection Coefficient 0.2 0.5 1.0 1.5 0.2 Angle of Incidence 0.4 0.6 0.8 1.0 Internal Reflection Reflection Coefficient 1.0 0.8 0.6 0.4 0.2 0.5 1.0 1.5 Angle of Incidence 0.2 Figure 4.8 Fresnel external (top) and internal (bottom) reflection coefficients for spolarization (dashed) and p- polarization (solid) are shown as the angle of incidence changes. Figure 4.9 shows relative phase shifts in reflection coefficients for external and internal reflections as a function of angle of incidence for an air-glass (n=1.5) interface. B is the Brewster’s angle and C is the critical angle. Again this result is based on the right-handed incident and exiting local coordinate choice. 72 Figure 4.9 The relative phase shift for external (top) and internal (bottom) reflection as a function of the angle of incidence. Jones matrices use Fresnel reflection or refraction coefficients to describe rays reflecting from or refracting into isotropic media. If the Jones matrix is defined in {s, p} local coordinates, the matrix is a diagonal matrix. Thus the relative phase shifts in Fresnel coefficients are the relative phase shifts in the Jones calculus. For angles of incidence smaller than B , Jones calculus shows π phase shift due to the right-handed 73 local coordinate choice before and after the reflection. This value is not the proper retardation that electric field experiences upon reflection but is the geometric transformation of the local coordinates described. The retardance algorithm using the P matrix yields zero retardance for internal reflections at B and π retardance for reflections at B C . Following three figures plot the normalized incident electric field vector and corresponding internal reflected electric field vector calculated from P matrices at an air-glass interface as the angle of incidence ( ) changes from 0º to 90º. By plotting electric field vectors in global {x, y, z} coordinates, one can truly understand reflection. All three figures follow the following rules: the surface normal vector of the glass surface is along the z axis, {0, 0, -1}. All the propagation vectors lie in the x-z plane. Each incident electric field vector (dotted line) and corresponding reflected electric field vector (solid line) pair is plotted in the same color. The arrows represent the oscillations of the electric field vector in time. The length of the arrow represents the amplitude of the electric field vector. For better visualization, the first two figures show s and ppolarization states separately and the third figure shows the combination of s and ppolarization states. Figure 4.10 shows s-polarized incident electric field vector (dotted) and corresponding reflected electric field vector (solid) for the external reflection as the angle of incidence changes. Figure 4.10 shows a side view (left) and a top view (right) of the external reflection for better visualization. Note that the reflected electric field vector has s pol reflection 74 s pol reflection smaller amplitude than the corresponding incident electric field. The s-component of the electric field reflects more as the angle of incidence increases. Side View Top View Figure 4.10 The incident s-polarized states (dotted) and reflected (solid) polarization states as angles of incidence change from normal incidence to glazing incidence. Red arrow indicates the reflected electric field at Brewster’s angle. Similarly, Figure 4.11 shows p-polarized incident electric field vector (dotted) and corresponding reflected electric field vector (solid). Since the p-polarization figures would overlap for different angles of incidence, each incident and reflected pair has been spatially translated for better visualization. The p-component of the electric field changes its sign after Brewster’s angle and has zero reflectance at the Brewster’s angle. This is the origin of π retardance for reflections at B C . p pol reflection 75 p pol reflection rp = 0 at Brewster’s Angle Figure 4.11 The incident p-polarized states (dotted) and corresponding reflected polarization states (solid) as the angle of incidence changes. Red arrow indicates the reflected electric field vector at the Brewster’s angle. At the Brewster’s angle, Fresnel reflection coefficient for p-polarization is zero. Figure 4.12 shows incident electric field vectors at 45º between the s and p– polarizations (dotted) and corresponding reflected electric field vectors (solid) as the angle of incidence changes. At the Brewster’s angle, reflection of p-polarization is zero; the reflected electric field vector is s-polarized. 76 45° pol reflection 45° pol reflection Reflected beam at Brewster’s Angle Side View Top View Figure 4.12 Incident electric field vectors at 45º between s and p-polarizations (dotted) and corresponding reflected (solid) electric field vectors are plotted as the angle of incidence changes. Red lines are for the incident and reflected pair at the Brewster’s angle; the top view shows that at Brewster’s angle, the reflected light (solid red line) is s-polarized. In summary, polarization ray tracing matrix calculus yields zero retardance upon reflection at normal incidence and π phase shift for B C . The retardance changes rapidly for the angle of incidence greater than C . 4.5.3 An Aluminum Coated Three-fold Mirror System The aluminum coated three fold-mirror system is analyzed. In Figure 4.13 two polarization states are followed through the optical system. Each mirror is aligned so that the collimated incident light has a 45° angle of incidence. Mirrors are coated with aluminum with a refractive index of 0.77 6.06i assumed. 77 Figure 4.13 An aluminum coated three fold-mirror system. Table 4.1 contains the propagation vector, the polarization ray tracing matrix, and the Q matrix for each surface. The exiting propagation vector k̂ 3 is the same as the incident propagation vector k̂ 0 , and both are along the z-axis. q kˆ q Pq Qq 1 1 0 0 0 0 1 0 -0.947 + 0.219i 0 -0.849 + 0.415i 0 0 0 0 1 0 1 0 1 0 0 2 0 1 0 0 0 -0.849 + 0.415i 0 0 1 0 0 -0.947 + 0.219i 0 1 0 1 0 0 0 0 1 3 0 0 1 0 -0.947 + 0.219i 0 0 0 -0.849 + 0.415i 0 1 0 1 0 0 0 0 1 0 1 0 Table 4.1 Propagation vectors, Pq’s, and Qq’s for a ray propagating through the aluminum coated three fold-mirror system 78 The diattenuation for this ray is 0.0285. The diattenuation of the 2nd and 3rd mirrors are equal but 90° apart and thus cancel each other; thus the total diattenuation is equal to the first mirror’s contribution. For the diattenuation calculation algorithm see Section 3.1. The system’s P matrix and Q matrix are PTotal 0 -0.549 + 0.705i 0 0 1 0 -0.365 + 0.788i 0 0 , QTotal 1 0 0 . 0 0 1 0 0 1 (4.5.5) PTotal demonstrates that x-polarized incident light exits as y-polarized and y-polarized incident light exits as x-polarized while the incident and exiting propagation vectors are the same. 79 Figure 4.14 Local coordinate transformation using Qq's for the three fold-mirror system. Incident xˆ L,0 state (solid red) exits as -xˆ polarized and incident yˆ L,0 state (dashed blue) exits as ŷ polarized after three reflections due to the geometric transformation. Figure 4.14 shows how each reflection transforms the incident local coordinates, {xˆ L,0 , yˆ L,0 }(red and blue arrows). The incident local coordinates are {xˆ L ,0 , yˆ L ,0 , kˆ 0 } {-yˆ , xˆ , zˆ} (4.5.6) and the corresponding local coordinates in exit space are {xˆ L,3 , yˆ L,3 , kˆ 3} {-xˆ , yˆ , zˆ} (4.5.7) 80 where {xˆ , yˆ , zˆ} are global coordinates. Thus one proper pairing of Jones matrix basis vectors between entrance and exit space would be {-yˆ , xˆ } and {-xˆ , yˆ } , a result of the 90° rotation after the parallel transport of the initial local coordinates through the system as shown in Figure 4.4 (a) and an inversion from an odd number of reflections. Note that the exiting local coordinates, {xˆ L ,3 , yˆ L ,3 , kˆ 3 } are left-handed; xˆ L,0 yˆ L,0 in the incident space gives k̂ 0 but xˆ L,3 yˆ L,3 in the exiting space gives -kˆ 3 . Local coordinates transformed by Q change handedness if a system has an odd number of reflections and maintain their handedness for an even number of reflections. {xˆ L ,3 , yˆ L ,3 , kˆ 3 } are the proper set of local coordinates for the polarization state analyzer (as described in Section 4.3.1) for measuring the proper retardance of the system. 1 Multiplying PTotal by QTotal cancels the geometric transformation . ΜTotal of the system is ΜTotal Q 1 Total PTotal 0 0 -0.365 + 0.788i 0 -0.549 + 0.705i 0 . 0 0 1 (4.5.8) Since PTotal is homogeneous, the retardance of the system is found by calculating eigenvalues of the ΜTotal . The eigenvalues of Eq. (4.5.8) are 1 0.868ei 2.005 , 2 0.8938ei 2.232 , 3 1 and the eigenpolarization states associated with the eigenvalues are (4.5.9) 81 v1 {1, 0, 0}, v 2 {0,1, 0}, v3 kˆ 0 {0, 0,1}. (4.5.10) The retardance of the system is arg(2 ) arg(1 ) 0.227 (4.5.11) with the fast axis orientation along the global x̂ -axis. This value does not contain any effects from the geometric transformation. Similar to the cancelation of diattenuation, this proper retardance is equal to the first mirror’s contribution, since the last two mirror retardances cancel. The retardance calculated from the Jones matrix of the first mirror is 1 0.945e-i 0.455 , 2 0.972ei 2.914 arg(2 ) arg(1 ) 3.369 193.0 (4.5.12) with the fast axis orientation along the y L,0 -axis which is the global x̂ -axis. The retardance from the Jones calculus and the one from the polarization ray tracing matrix differ by π since Jones calculus uses right-handed local coordinates for data reduction. 4.6 Conclusion This section presented a critical analysis of retardance. To calculate the true polarization dependent phase change, the retardance, the geometric transformation needs to be removed. P describes all polarization state changes due to diattenuation, retardance, and geometric transformations. The parallel transport matrix Q describes the associated nonpolarizing optical system and thus keeps track of the geometric transformation. 82 Μ Q1 P is a fundamental equation for calculating retardance without spurious circular retardance arising from a poor choice of local coordinates. Μ also clarifies the meaning of the troublesome minus sign in the Jones matrix for reflection. The difference in eigenvalue arguments of Μ R , the unitary part of the polar decomposed Μ , gives proper retardance. However, proper retardance cannot be assigned to a black box whose ray propagation vectors are unknown. 83 CHAPTER 5 POLARIZATION ABERRATION A polarization aberration is variation of the polarization properties of an optical system with wavelength, pupil, and object coordinates. Rays propagating through systems with optical coatings and birefringent materials experience diattenuation, polarization dependent transmission, and retardance, polarization dependent optical path length. In this section two ways of representing polarization aberration- Jones pupil and polarization aberration function- are contrasted. 5.1 Jones Pupil Many optical systems with no depolarizing elements or scattering elements can be described by a Jones matrix. The Jones matrix of the system is a function of pupil and field coordinates as shown in Figure 5.1. For a given field point, the Jones matrix as a function pupil coordinates is called a Jones pupil. Figure 5.1 A field vector on the image plane and the exit pupil vector. 84 When a grid of rays gets traced through a system and reaches the exit pupil surface, each ray’s Jones matrix gets calculated, and a grid of Jones matrices is obtained. Each Jones matrix is associated with a particular pupil coordinate; the grid of Jones matrices is the Jones pupil for that object. The Jones pupil is usually decomposed into apodization pupil, wavefront aberration pupil, diattenuation pupil, and retardance pupil. Apodization and wavefront pupils are scalar functions, and diattenuation and retardance pupils have magnitude and orientation45. Each ray’s Jones matrix is defined in its exit pupil local coordinates; each Jones matrix on the Jones pupil is defined on a different transverse plane perpendicular to the propagation vector of the corresponding ray. Jones pupil at the edge of field of USA patent 289650646 from Code V47 library is calculated and the lens parameters are in Appendix A. The system has f/1.494, maximum FOV of 32º and is rotationally symmetric. Figure 5.2 shows the layout of the system with seven lens elements; three field angles are shown. 85 Figure 5.2 Optical system layout of USA patent 2896506 from Code V library has seven lens elements. The system is defined with three field angles. Figure 5.3 shows two types of wavefront aberration maps at the exit pupil of the USA patent 2896506 for the on-axis field. For the on-axis field, the dominant wavefront aberration is the spherical aberration. This figure was generated by Code V. 86 WAVEFRONT ABERRATION USA PATENT 2896506 AZUMA Waves .01087 -.0945 -0.200 Field = ( 0.000, 0.000) Degrees Wavelength = 656.0 nm Defocusing = 0.000000 mm Figure 5.3 Wavefront aberration maps at the exit pupil using Code V are plotted for the on-axis field. For on-axis field, 0.3 waves of spherical aberration is the dominant aberration. Figure 5.4 shows amplitude part of Jones pupil at the edge of field of the system. The shape of the pupil is due to vignetting. As the rays propagate through this high NA and wide FOV optical system, each ray experiences different Fresnel refraction coefficients. Thus, the diagonal elements have amplitude variation across the pupil. Systems with only paraxial rays have zero off-diagonal components. Non-zero offdiagonal components of Jones pupil, as in Figure 5.4, show that the y-component of the incident electric field effects the x-component of the exiting electric field and the xcomponent of the incident electric field effects the y-component of the exiting electric field. 87 Figure 5.4 Amplitude part of Jones Pupil at the edge of field of USA patent 2896506 shows variation across the pupil as well as off-diagonal components, which often appear for systems with high NA. The exit pupil local coordinates where the Jones pupil is defined is shown in Figure 5.5. Red arrows are the local x vectors in the exit pupil and the blue arrows are the local y vectors. Depending on the local pupil coordinates, Jones Pupil changes its values. 88 Figure 5.5 Jones pupil local x and y coordinates. 16:51:12 Wavefront aberration maps at the edge of the field calculated from Code V are shown in Figure 5.6. For this field, the dominant wavefront aberration is astigmatism. 4 Waves X: 0.00 DEGREES X 128 4 Waves ID SIZE 128 USA PATENT 2896506 AZUMA POSITION ORA 1 11-Oct-11 WAVE ABERRATION FIELD ANGLE - Y: 32.00 DEGREES X: 0.00 DEGREES DEFOCUSING: 0.000000 MM WAVELENGTH: 656.00 NM HORIZONTAL WIDTH REPRESENTS GRID SIZE 128 X 128 89 Figure 5.6 Wavefront aberration maps at the exit pupil using Code V are plotted for a point source at the edge of the field demonstrating three waves of astigmatism. 5.2 Polarization Aberration Function In this section, polarization ray tracing matrices are extended to construct a polarization aberration function to characterize the transformations of a wavefront described by a grid of rays, as is common in optical design programs7, 9. A polarization aberration function P(r ) generalizes the wavefront aberration function in optical design, where r represents the position vector on the terminal surface of the ray trace, typically the exit pupil. The wavefront’s polarization information is contained in the polarization aberration function P(r ) and the optical path length function OPL(r ) . The polarization aberration function will give an amplitude transmission map, a phase map within less than one wave limit, the wavefront diattenuation D (r ) , which is 90 a polarization dependent apodization map and the wavefront retardance (r ) , which is a polarization dependent optical path difference (OPD) map. The exiting electric field can be calculated from the polarization aberration function and the incident electric field, E(r) P(r) E0 (r)exp(-i2 / OPL(r)) , (5.2.1) P(r ) contains corrections to OPL(r ) from coatings and other polarization effects. When coatings on optical elements change, P(r ) changes but OPL(r ) does not change. Coatings frequently add small amounts of defocus and astigmatism. An optical system without OPD wavefront aberration can still have a polarizationinduced wavefront aberration arising, for example, from differences in the Fresnel reflection and transmission coefficients for s-polarized light and p-polarized light. A ray tracing program can only sample P(r ) by tracing grids of rays, not calculate it at an infinite number of locations. Assume an optical system for which a grid of L M rays has been traced yielding an L M grid or matrix of propagation vectors, positions, and optical path lengths 91 kˆ 1,1 kˆ 1,2 kˆ 1, M kˆ s ˆ ˆ kˆ L , M k L ,1 k L ,2 r1,1 r1,2 r1, M rs , r r rL, M L ,1 L ,2 OPL1,1 OPL1,2 OPL s OPL OPL L ,1 L,2 , L M propagation vector grid L M exit pupil position grid (5.2.2) OPL1, M , L M optical path length grid. OPLL , M where s stands for sampled quantity. The sampled polarization aberration function is also an L M grid of polarization ray tracing matrices for each ray position, PTotal , s P1,1 P1,2 P2,1 P2,2 P P L ,1 L ,2 P1, M P2, M PL , M (5.2.3) where Total stands for ray tracing through an optical system from entrance to exit pupil. If the system has a pupil, such as a typical circular pupil, the sampled polarization aberration function will have three-by-three matrices filled with zeros outside of the pupil, represented here by the digit zero, 92 PTotal , s 0 0 0 0 0 0 0 0 P2,2 0 0 P3,2 0 0 P N -2,2 0 0 0 PN -1,2 0 0 0 0 0 0 0 0 0 P2, M -1 0 0 0 P3, M -1 0 0 . PN -2, M -1 0 0 PN -1, M -1 0 0 0 0 0 0 0 (5.2.4) The elements of PTotal ,s are complex matrices and the argument of a complex number is always less than 2π. Therefore, the Optical Path Length ( OPL s ) is calculated separately from Pi , j in order to keep track of optical path length which is greater than 2π (or one wavelength). All the wavefront polarization information can be extracted from the polarization aberration function PTotal ,s which is a grid of polarization ray tracing matrices sampled by rays. Figure 5.7 shows an example polarization aberration function at the exit pupil of the USA patent 2896506, the same system as in Figure 5.4. 93 Figure 5.7 Amplitude part of a polarization aberration function ( PTotal ,s ) at the exit pupil of USA patent 2896506. Each point on the Jones pupil has different local coordinates i.e., the Jones pupil is a collection of different transverse planes. PTotal ,s is defined in global coordinates; thus PTotal ,s shows polarization effects along the optic axis component as well as the transverse plane components in x, y, z global coordinates. Figure 5.8 shows the phase part of the 94 polarization aberration function of the USA patent 2896506. Since all the lenses in this system are uncoated, PTotal ,s has little retardance due to different phases coming from Fresnel refraction coefficients for different rays; Fresnel refraction coefficients for airglass interface are real valued. This is why the phase map in Figure 5.8 has mostly zero or π phase values across the pupil. Figure 5.8 Phase part of a polarization aberration function ( PTotal ,s ) at the exit pupil of USA patent 2896506. 95 5.2.1 Coherent Beam Combination Many optical analysis simulations involve beams of collimated rays propagating through one or more plane parallel crystals or optical elements. Consider an example where a ray enters a calcite plate, ray splitting occurs, and two parallel rays, the ordinary and extraordinary rays, exit the plate. In the region where the exiting beams overlap, it is desired to calculate the combined polarization properties, which can be done by combining the two P matrices from the ordinary and extraordinary rays coherently. The individual P matrices represent polarizers, since only one linear polarization propagates into each mode. The combined P matrix represents a retarder, a more useful description of the calcite plate. Similar optical design calculations arise when simulating the beams exiting an interferometer if the beams have been adjusted to exit parallel to one another. Pcombined , the coherent combination of P matrices for a pair of rays with shared incident propagation vectors, is not merely the sum of the individual P matrices. Each P must be multiplied by the corresponding phase factors ( exp(-i 2 / OPL) ) to account for its different optical path lengths. Further modification is required in order for Pcombined to correctly transform the incident propagation vector ( kˆ In ) to the exiting propagation vector ( kˆ Exit ) as in Eq.(2.1.7); kˆ Exit Pcombined kˆ In . The dyad k D is the outer product matrix of kˆ q and kˆ q 1 , and represents the propagation vector components in P 96 kˆx , In kˆx , Exit k D Outer[kˆ Exit , kˆ In ] kˆx , In kˆy , Exit ˆ ˆ k x , In k z , Exit kˆy , In kˆx , Exit kˆ kˆ y , In y , Exit kˆy , In kˆz , Exit kˆz , In kˆx , Exit kˆz , In kˆy ,Exit . kˆz , In kˆz , Exit (5.2.5) The following equation for Pcombined takes these effects into account: Pcombined (P1 k D )exp(-i2 / OPL1 ) (P2 k D )exp(-i2 / OPL2 ) k D . (5.2.6) The dyad k D is subtracted from each Pi before the coherent addition so that kˆ Exit Pcombined kˆ In . 5.3 Conclusion Jones pupil contains the polarization effects of the system as a function of exit pupil coordinates. Similarly, a polarization aberration function PTotal ,s represents the spatial variation of the polarization ray tracing matrix over the exit pupil. It is a generalized wavefront aberration function that characterizes the polarization dependent transformations of a wavefront. Under certain circumstances, the matrices for rays can be coherently combined by a modified form of matrix addition. 97 CHAPTER 6 A SKEW ABERRATION Aberrations can be considered as deviations from the mapping of spherical waves with uniform amplitude and polarization into spherical waves with uniform amplitude and polarization, i.e., the ideal behavior of imaging optical systems. The main categories are wavefront aberration 48 , apodization (amplitude aberration) 49 , 50 , 51 , and polarization aberration7. Wavefront aberration is the variation in optical path length, and all the commercial ray tracing programs calculate the wavefront aberration. Apodization is an amplitude aberration; different rays have different transmittances due to reflection losses and absorption. Polarization aberration, a non-uniform polarization change across wavefronts is divided into diattenuation aberration, which is polarization dependent transmission, and retardance aberration, which is polarization dependent optical path length difference. In this section, a definition of skew aberration is given and an algorithm which calculates the skew aberration is presented. The algorithm is applied to 2383 non-reflecting optical systems in the Code V47 patent library and the statistics of the skew aberration is demonstrated. The skew aberration of an example system from the library is further analyzed. 6.1 Definition This work introduces another component of polarization aberration, “skew aberration.” Skew aberration is a rotation of each ray’s polarization state between the entrance pupil 98 space and the exit pupil space due to intrinsic geometric transformation of local coordinates via parallel transport of vectors. Skew aberration is independent of the incident polarization state or the coatings applied to the optical interface. Skew aberration is categorized as a class of polarization aberration that is distinct from diattenuation and retardance, since its origin arises from purely geometric effects. Skew aberration does not arise from polarization properties of optical elements. Skew aberration occurs even for rays propagating through ideal, aberration-free, and nonpolarizing optical systems. The definition of ideal and non-polarizing optical systems is discussed at length in ref [14]. Thus, if a ray has non-zero geometric transformation of local coordinates, its skew aberration is also non-zero. The skew aberration is determined solely by the ray’s propagation path, i.e., its sequence of normalized propagation vectors {k In , k1 , k 2 ,...k j ,..., k Exit } . k In is a propagation vector of a ray at the entrance pupil, k j is a propagation vector after the j th surface, and k Exit is a propagation vector of a ray at the exit pupil. 6.2 Skew Aberration Algorithm A skew aberration map in the exit pupil elucidates whether the polarization distribution of the exiting wavefront after a non-polarizing optical system is identical to the polarization distribution of the incident wavefront. This change in polarization distribution is independent of the incident polarization states. Therefore a grid of 99 reference vectors ( g In,i ), defined on the entrance pupil, is traced through an ideal and non-polarizing optical system, and compared with another grid of reference vectors ( g Exit ,i ), defined on the exit pupil. Sub index i stands for i th ray on a grid. The only requirements for the reference vectors are, g In,i k In,i and g Exit ,i k Exit ,i . (6.2.1) Our preferred choice of reference grids are calculated by the following steps. First, define a vector ( gC ) which is perpendicular to the center ray’s propagation vectors on the entrance and exit pupil gC k In,C k Exit ,C , (6.2.2) and use gC as a reference vector of the center ray on the entrance and exit pupil. g In,i is a counterclockwise rotation of gC along axisIn,i by In,i and g Exit ,i is a counterclockwise rotation of gC along axisExit ,i by Exit ,i , g In,i R( In,i , axisIn,i ) gC g Exit ,i R( Exit ,i , axisExit ,i ) gC (6.2.3) where In,i arccos(k In,i k In,C ) , axisIn,i k In,C k In,i and Exit ,i arccos(k Exit ,i k Exit ,C ) , axisExit ,i k Exit ,C kExit ,i , and R ( , axis ) is a 3D rotation matrix for a counterclockwise rotation around the 3D vector axis by . 100 This rotation method is analogous to parallel transporting gC along a great circle arc on a unit sphere which connects points k In,C and k In,i . Therefore, the resulting grid has radial symmetry. We call this grid a “double-pole grid.” The double-pole grid is often used to describe linearly polarized grid of rays on a spherical wavefront. Figure 6.1 shows an example double-pole grid on a spherical entrance pupil in two different views. Figure 6.1 (a) A double pole grid of reference vectors on a unit sphere viewed along the chief ray’s propagation vector and (b) oblique view. Once g In,i and g Exit ,i are calculated for an optical system, the system’s geometric transformation needs to be calculated in order to trace g In,i through ideal, non-polarizing optical system. Each surface in the system has a certain amount of geometric transformation of local coordinates due to the change in ray propagation direction from 101 k j -1 to k j . A parallel transport matrix Q j 14 of the j th surface calculates the geometric transformation of the surface. Q j for refracting surface j is equivalent to sliding vectors from a point k j -1 to a point k j on a unit sphere following the great circle arc which connects two points. Q j for reflecting surface j is equivalent to inverting vectors on a point k j -1 about k j -1 k j and then moving them to a point k j by parallel transport. Equations are shown below and derivations can be found in the reference14. Q j for a refracting surface j is A2 B (1 B) D 2 1 F [(1 B) D L] A2 D[(1 B)C L] F [(1 B) D G ] D[(1 B)C H ] A2 B (1 B) F 2 C[(1 B) F H ] C[(1 B) F G ] A2 B (1 B)C 2 (6.2.4) where A Norm[k j -1 k j ], B k j -1 k j ,{C, D, F} k j -1 k j , and {G, H , L} A 1 B2 C F 2 2 { A2 F 2 , A2 C 2 , A2 D2 }. Q j for a reflecting surface j is A 2(k x , j -1 k x , j ) 2 1 B A C B A 2(k y , j -1 k y , j ) D 2 2 A 2(k z , j -1 k z , j ) C D (6.2.5) where A Norm[k j 1 k j ], B 2(kx, j 1 kx, j )(k y , j 1 k y , j ), C 2(kx, j -1 kx, j )(kz , j -1 k z , j ), and D 2(k y , j -1 k y , j )(kz , j -1 kz , j ). 102 QTotal ,i determines the i th ray’s geometric transformation QTotal ,i In Q j Exit j Q Exit Q j Q1 Q In . (6.2.6) Tracing g In,i though the non-polarizing system yields gExit ,i QTotal ,i g In,i . The i th (6.2.7) ray’s skew aberration is defined as the angle between g Exit ,i and gExit ,i . If gExit ,i is the counter clockwise rotation of the g Exit ,i looking into the beam, the ray has positive skew aberration. 6.3 Example Optical systems with high numerical aperture (NA) and wide field of view (FOV) tend to experience larger skew aberration. A skew aberration analysis is presented on USA patent 289,650,646, which has a high skew aberration compared to other systems in the Code V patent library. The system has f/1.494, maximum FOV of 32ºand is rotationally symmetric. Figure 6.2 shows the layout of the system with seven lenses and three field angles. 103 Figure 6.2 Optical system layout of USA patent 2896506 from Code V library has seven lenses. The system is defined with three field angles. 6.3.1 Skew Aberration at the Exit Pupil Figure 6.3 shows the skew aberration on the exit pupil for a grid of rays with 32º field angle. The pupil coordinates are in mm. The maximum skew aberration of 7.01º rotation occurs for a skew ray passing the edge of the exit pupil. A skew ray at the opposite end of the pupil (point A) has -7.01º of skew aberration. Rays of rotationally symmetric optical systems are classified as meridional rays and skew rays. Skew aberration only occurs for skew rays but not for meridional rays since meridional rays remain on a plane containing the optical axis. A meridional fan of rays through the center of the exit pupil has zero skew aberration as shown in Figure 6.3 (a). 104 Figure 6.3 (a) Skew aberration of a ray grid with 32º field angle is evaluated at the exit pupil of the patent 2896506. The maximum skew aberration is +7.01º and the minimum is -7.01º (ray A). Both extreme occur from skew rays at the edge of the pupil. (b) Horizontal cross section (indicated in orange dashed line in part (a)) of the skew aberration exit pupil map has zero skew aberration for the center ray, which is the chief ray. 105 Skew aberration of this example has a form of a circular retardance tilt (linear pupil dependence) plus coma (cubic pupil dependence) aberration; skew aberration has the same Jones matrix as a circular retarder, both of them cause a rotation of the polarization state. For the radially symmetric lenses in our study the pupil dependence is linear near the center of the field with an increasing cubic component towards the edge of the object. Skew aberration contribution of ray A (Figure 6.3 (a)) at each lens surface is shown in Figure 6.4. Figure 6.4 Skew ray A's skew aberration contribution from each lens surface sums to -7.01º through the system. 106 6.3.2 Skew Aberration’s Effect on Point Spread Function and Modulation Transfer Function When there is a variation of skew aberration across the exit pupil of the wavefront, there are changes in polarization character of the wavefront. This variation of skew aberration across the pupil creates undesired polarization components in the exit pupil; typically cross polarized satellites form around the PSF45, 52, 53. Thus the point spread function (PSF) and an optical transfer function (OTF) changes and image quality can be degraded. In the presence of polarization aberration, the scalar PSF can be generalized to a four-byfour point spread matrix (PSM) in Mueller matrix notation and an OTF can be generalized to a four-by-four optical transfer matrix (OTM)53. Following figures show a PSM and OTM of the example system. Figure 6.5 is the PSM of the example system calculated by discrete Fourier transform of the parallel transport matrix of the system at the exit pupil. The shape of the peaks in the figure is not radially symmetric since the optical system is not isoplanatic for this maximum field angle of 32º. The diagonal elements are the dominant terms; the exiting wavefront after the ideal, nonpolarizing ray tracing is almost the same as the incident wavefront polarization. The offdiagonal elements introduce polarization mixing; the exiting wavefront has polarization components that did not exist in the incident wavefront due to rotation that are introduced by skew aberration. Since different skew rays have different skew aberration, the exiting 107 wavefront not only has undesired polarization components but also has some variation across the pupil. Figure 6.5 A point spread matrix (PSM) of the example system calculated from a discrete Fourier transform of the parallel transport matrix of the system at the exit pupil. The elements are elliptical due to three waves of astigmatism. Figure 6.6 shows the MTF of the example system calculated by a discrete Fourier transform of the PSF in Figure 6.5. Each term is normalized so that the maximum is one. 108 The off-diagonal elements show the additional effects due to polarization components that are presents on the exit pupil, which skew aberration creates. Figure 6.6 An optical transfer matrix (OTM) of the example calculated by a discrete Fourier transform of the PSM. 109 Due to the variation of skew aberration across the pupil, the deviation of PSM and OTM from the ideal PSM and OTM also has variation across the pupil. And this can degrade the image quality of the system. 6.4 Statistics – Code V Patent Library To understand the typical magnitude of skew aberration and its importance in the families of aberrations, the skew aberration of 2383 non-reflecting optical systems in Code V’s US patent library was calculated and shown in Figure 6.7. The mean is 0.89º and the standard deviation is 1.37º. The maximum skew aberration is 17.45ºand the minimum is -11.33º. Figure 6.7 Histogram of the maximum skew aberration evaluated for 2383 nonreflecting optical systems in Code V’s library of patented lenses. 110 When all polarization states of an incident wavefront experience the same amount of rotation, the exiting wavefront polarization is simply a rotation of the incident wavefront polarization. This is analogous to having a constant phase over the entire wavefront such as piston aberration. Neither piston nor uniform polarization rotation should degrade image quality or the point spread function (PSF). It is the variation of skew aberration across the exit pupil of the wavefront that changes the polarization character of the wavefront. 6.5 Skew Aberration in Paraxial Ray Trace Paraxial optics is a method of determining the first-order properties of a radially symmetric optical system that assumes all ray angles and angles of incidence are small54. A paraxial ray trace or first-order ray trace is a linearized approximation of real ray behavior. The first order properties of radially symmetric systems, such as focal lengths, magnifications, principal planes and others are defined in terms of the paraxial ray trace. The paraxial ray trace provides an invaluable coordinate system for the description of aberrations. Although the computer now makes it easy to trace real rays, the linearity of paraxial ray slopes and coordinates makes them useful for solves, which specify thickness or curvature indirectly in terms of paraxial ray properties. Paraxial refraction and transfer equations at the qth ray intercept are yq 1 yq wq q yq 1 yq wq q wq 1 wq yqq wq 1 wq yqq (6.5.1) 111 where q tq / nq , wq nquq , tq is the distance between the qth and the q+1th surface vertices along the axis, nq is the refractive index of the material following the qth surface, y is the marginal ray height, y is the chief ray height, u is the marginal ray angle, and u is the chief ray angle. The Code V paraxial ray trace table traces two rays, the paraxial marginal ray (from center of object to edge of entrance pupil) and full-field paraxial chief ray (from top of object to center of entrance pupil)55. All paraxial rays can be calculated from linear combinations of these two rays. For the paraxial skew ray from the top of the object and edge of the pupil the paraxial marginal ray height at each surface is the xcoordinate of the skew ray, the paraxial chief ray height is the y-coordinate of the skew ray, and the vertex of each surface is the z-coordinate of the skew ray. Thus, the propagation vector k q after the qth ray intercept is along { yq1 yq , yq1 yq , tq } . Since the skew aberration calculation uses the normalized propagation vectors, further manipulation provides a normalized k q kq {wq , wq ,1} wq2 wq2 1 . (6.5.2) For paraxial ray trace, spherical polygons that the parallel transport of skew rays trace reduces to a polygon on the transverse plane that is perpendicular to {0, 0,1} . Thus, the paraxial ray trace skew aberration is proportional to the area of the polygon. By 112 dropping the z-component of the propagation vectors, 2D propagation vectors which form the polygon can be calculated {wq , wq } k 2 D ,q wq2 wq2 1 . (6.5.3) An area of a triangle that connects the origin, k 2 D,q , and k 2 D ,q1 is Areaq wq wq 1 wq 1wq 1 . 2 2 wq wq2 1 wq21 wq21 1 (6.5.4) Further manipulations using Eq. (6.5.1) results Areaq q 2 H 2 wq yq wq yq wq2 wq2 1 wq21 wq21 1 (6.5.5) q wq2 wq2 1 wq21 wq21 1 where H wq yq wq yq is the Lagrange invariant of the system. Therefore, the skew aberration of the system is proportional Lagrange invariant, and is closely related to the sum of the individual surface powers (Eq. (6.5.5)) Total Area H 2 q q w w 1 wq21 wq21 1 2 q 2 q . (6.5.6) The optical system of Section 6.3 is used to calculate the skew aberration in paraxial ray trace. A paraxial skew ray at point A in Figure 6.3 (a) is created by paraxial 113 ray traced marginal ray height and chief ray height. The skew aberration of -2.00º rotation occurs for a paraxial ray at point A. The skew aberration is calculated by following the algorithm in Section 6.2. Skew aberration contribution of a paraxial ray at point A at each lens surface is shown in Figure 6.8. Figure 6.8 Paraxial skew ray at point A's skew aberration contribution from each lens surface sums to -2.00º through the system. The existence of skew aberration in paraxial regime shows the possibility to further describe skew aberration using series expansion method, which is one of the directions for future work. 114 6.6 Conclusion Skew aberration is a component of polarization aberration that originates from pure geometric effects. Pupil variation of skew aberration affects PSF and degrades image quality. The skew aberration of a chief ray serves as a piston-like aberration i.e., overall polarization rotation across the pupil. Skew aberration is typically a small effect in lenses but it could be a concern in microlithography optics and other polarization sensitive systems with high NA and large FOV. 115 CHAPTER 7 UNDERSTANDING APPARENT RETARDANCE DISCONTINUITIES Issues in the measurement and modeling of the order of retarders are addressed. When measuring the retardance spectra of compound retarders, the retardance can appear to “turn around” and avoid integer numbers of waves of retardance. In this chapter, studies on phase unwrapping of the principal retardance for polychromatic light are presented in order to explain the behavior of retardance of compound retarders. One approach is using a dispersion model to describe the retardance behavior. The other approach is considering multiple wavefronts exiting the compound retarder system i.e., multi-valued optical path length (OPL). A common definition describes a retarder as a device that divides a beam into two orthogonal modes and introduces a relative phase difference δ1. Another view of retarders is provided by the Mueller calculus and Poincaré sphere. A retarder rotates polarization states on the Poincaré sphere by retardance δ; as light propagates through a retarder the incident state on the Poincaré sphere is rotated about a retardance axis to another state. In this Mueller / Poincaré picture, cascading retarders is equivalent to cascading rotations of the Poincaré sphere. This view of retarders and its implementation with Mueller matrices have ambiguities of 2 n where n are integers describing the order of the retarder. In the Mueller picture, the final polarization state always ends up in the 116 right place but we only know the retardance modulo 2 . Similarly, the retardance axis orientation is modulo to 90 . In the following section an algorithm for finding total retardance, horizontal retardance, 45° retardance, and circular retardance for given retarder Jones matrices is presented. In Section 7.2 a retarder space concept is introduced to better understand phase unwrapped retardance and compound retarder systems with multiple retarder elements. Dispersion properties of retardance for homogeneous and inhomogeneous compound retarders are analyzed. 7.1 Retardance Calculation for Jones Matrices In this section an algorithm calculating retardance values – the total retardance ( ), a horizontal component of the retardance ( H ), a 45º component of the retardance ( 45 ), and a circular component of the retardance ( R ) - from the Jones matrix of a pure retarder is presented. Any Jones matrix J can be expanded with Pauli spin matrices j J 11 j21 Where j12 c0 c1 c0σ 0 c1σ1 c2σ 2 c3σ3 j22 c2 ic3 c2 ic3 , c0 c1 (7.1.1) 1 0 1 0 0 1 are complex numbers, σ 0 , σ1 , σ2 , and 0 1 0 1 1 0 0 i σ3 . For any given pure retarder Jones matrix, calculating ci is simple, i 0 117 c0 j11 j22 j j j j i( j j ) , c1 11 22 , c2 12 21 , c3 12 21 . 2 2 2 2 (7.1.2) The Jones matrix of a pure retarder (no amplitude, specified phase) can be expressed as the exponential of a sum of Pauli spin matrices, J exp(i( H σ1 45σ 2 R σ 3 ) / 2) H σ1 45σ 2 R σ 3 ), (7.1.3) σ 0 cos( ) i sin( )( 2 2 where H2 452 R2 . Eq. (7.1.1) can be written as J c0 (σ 0 c c1 c σ1 2 σ 2 3 σ3 ) c0 (σ 0 d1σ1 d2σ 2 d3σ3 ), c0 c0 c0 (7.1.4) where c0 is the polarization independent part of the Jones matrix i.e., the absolute amplitude and phase change. Comparing Eq. (7.1.1) and Eq. (7.1.4) d1 i tan( ) H , d 2 i tan( ) 45 , d3 i tan( ) R , 2 2 2 (7.1.5) and using H2 452 R2 2arctan( -d12 - d 22 - d32 ) H i d3 i d1 i d 2 , 45 , R , tan( / 2) tan( / 2) tan( / 2) (7.1.6) 118 with the fast and slow axes along a fast { H , 45 , R } and aslow {- H , - 45 , - R }. If 1 , Eq. (7.1.6) converges to H 2id1, 45 2id2 , R 2id3 , using lim 0 (7.1.7) (7.1.8) 2. tan( / 2) When c0 is zero, Eq. (7.1.1) becomes J c1σ1 c2σ2 c3σ3 (7.1.9) and cos( ) in Eq. (7.1.3) should be zero. Thus this Jones matrix is a half wave retarder. 2 Eq. (7.1.3) becomes J i sin( H σ1 45σ 2 R σ 3 i )( ) ( H σ1 45σ 2 R σ 3 ). 2 Using Eq. (7.1.9) and (7.1.10), H i c1, 45 i c2 , R i c3 , (7.1.11) and using c12 c22 c32 - sin 2 ( / 2) 2arcsin( -c12 - c22 - c32 ). (7.1.12) (7.1.10) 119 The total retardance δ from Eq. (7.1.6) has π as its maximum as shown in Figure 7.1. The red dotted line indicates the retardance value , which is the value that the total retardance approaches. Figure 7.1 A range of total retardance δ calculated from Eq. (7.1.6). Retardance magnitude less π than is called the “principal retardance”. 7.2 Retarder Space Retarders can be represented as points in a retarder space which has horizontal component of the retardance ( H ), 45º component of the retardance ( 45 ), and circular component of the retardance ( R ) as axes. The Figure 7.2 shows a corresponding location of a retarder with total retardance δ in the retarder space. The coordinate of the point is { H , 45 , R } , where H is the horizontal component of the retarder, 45 is the 45° 120 component of the retarder, R is the right circular component of the retarder, and H2 452 R2 . R 45 H Figure 7.2 A retarder space with δH, δ45, and δR as axes where δ is the distance from the origin to a point which has a magnitude of retardance δ. Points in the H - 45 plane represent linear retarders while points along the R axis represent pure circular retarders. The corresponding Jones eigenpolarizations for a retarder with retardance H , 45 , and R are, H H v F 45 i R , v S 45 i R , (7.2.1) 1 1 and Stokes eigenpolarizations are 121 H - H SF ,S , 45 S - 45 R - R (7.2.2) where F and S stand for fast and slow modes. The ellipticity of the eigenpolarizations is 1 2 arctan( R H2 452 ), (7.2.3) and the orientation of major axis is 1 2 f arctan( 45 ), s f . H 2 (7.2.4) In the retarder space, there is no limit on the range of retardance; retardance value can have any magnitude. Points on a sphere of radius 2π with its center at the origin represent retarders with one wave of retardance and points on a sphere of radius 4π represent retarders with two waves of retardance, etc. Figure 7.3 shows two groups of identical retarder Mueller matrices with different absolute phases in the retarder space. Each sphere represents retarders with retardance n . The origin and spheres of radius (retardance) 2n are identity Mueller matrices. Red points ( A and A ) are identical retarder Mueller matrices and are symmetrically located about the origin. Similarly, green points ( B and B ) show another set of identical Mueller matrices in the retarder space. 122 45 B A B A B I B A A 2 I H B B 4 I Figure 7.3 Two groups of Mueller matrices (A and B) with the same retardance modulo to 2π are shown in the retarder space. Each point in the groups is 2π away from each other and shares the same fast and slow axes. For each Mueller matrix with retardance there is a series of Mueller matrices with the retardance 2n with the same fast axes { H , 45 , R } and with the retardance 2n with the orthogonal fast axes {- H , - 45 , - R } . 7.3 Trajectories of Jones Retarder Matrices as the Polarization State Analyzer Rotates As mentioned in section 4.3.1 rotation of the polarization state analyzer (PSA) affects the retardance value measured from the Jones matrix polarimeter. Figure 7.4 shows two 123 views of trajectories of horizontal fast-axis linear retarder (HLR) with retardance δ0 in the retarder space as the PSA is rotated by θ from zero to 2π. Since retarders have redundancy in every nπ, points in the retarder space repeat as the PSA rotates; an initial point for each δ0 is on the H -axis and moves toward positive R -axis along the trajectory. When the trajectory reaches the boundary it moves to the origin symmetric point and comes back to the starting point as θ reaches 2π. During 2π rotation of θ the trajectory repeats twice. 45 2 0 2 2 0 2 2 0 H 2 R 124 H 20 2 0 2 2 R 0 0 0 34 0 4 2 2 0 0 0 2 45 Figure 7.4 Trajectories of horizontal fast axis linear retarders in the retarder space as the PSA rotates by θ from zero to 2π. Each trajectory repeats twice for 0≤ θ ≤ 2π. For δ0 = 0 retarder, which is an identity matrix, the trajectory is a single line and the retarder behaves as a circular retarder as θ increases. As δ0 increases trajectories start to curve and form a spiral. For δ0 = π, which is a half wave retarder, the trajectory stays in H - 45 plane, as we expected, and repeats around the circle twice keeping the total retardance to π. A family of parameters - rotation angle θ between the incident and exiting local coordinates, retardance δ, eigenpolarization trajectories on the retardance space, and 125 eigenpolarizations of a Jones matrix - are needed to calculate correct retardance of a Jones matrix. For simple cases, such as single retarder with rotating analyzer or s-p reflection, there is no complexity. However, calculating correct retardance from multiple skew reflections through compound inhomogeneous systems requires a more thorough analysis. 7.4 Phase Unwrapping for Homogeneous Retarder Systems using Dispersion Model 7.4.1 Dispersion Model For monochromatic light we cannot distinguish matrices that are identical with different absolute phase terms; for example, 0 0 1 0 2 i 1 4 i 1 J1 , J2 e , J3 e , (7.4.1) 0 1 0 1 0 1 J1 , J 2 , and J3 are the same polarization elements for monochromatic light. Thus the absolute phase term is often ignored for monochromatic light source and all three matrices are treated as an identity matrix. In the Michelson interferometer, using polychromatic light the location of the zero optical path difference can be found. So polychromatic light can be applied with care to distinguish absolute phases in cases such as Eq. (7.4.1). 126 When a polychromatic wavefront propagates through a waveplate, each wavelength of the light experiences different retardance. This is the origin of the dispersion model for retarders. A plane wave can be written as E(r , t ) E0 ei (k r t ) E0e i( 2 n ˆ k r t ) (7.4.2) where k̂ is the normalized propagation direction, is the wavelength of the plane wave, n is the refractive index of the material, is the frequency of the plane wave. When this plane wave propagates through a waveplate, the distance that the plane wave propagates within the waveplate is d kˆ r (7.4.3) and the wavelength dependent retardance, which comes from the phase of the plane wave, is ( ) 2 nd 0 , (7.4.4) assuming the refractive index of the waveplate is not dependent on the wavelength. Eq. (7.4.4) is the dispersion model of the retardance. This model is used for the phase unwrapping of the principal retardance for homogeneous and inhomogeneous retarder systems. 127 7.4.2 Phase Unwrapping of the Homogeneous Retarder System In this section, a phase unwrapping algorithm of the principal retardance using the dispersion model is introduced. For a given Jones or Mueller matrix, retarder order at is determined using the dispersion model and pieces of principal retardance from different orders are rearranged. Unwrapped retardance is the true retardance of the system for a given wavelength. An exam system consisting of two horizontal linear retarders (HLR) with shared fast axes is used to understand the phase unwrapping algorithm. When two HLRs are aligned to have the same fast or slow retarder axis, they form a homogeneous retarder system since the eigenpolarizations are orthogonal to each other. In this section two HLRs made from the same material with different thicknesses are aligned along a shared fast axis. The retardance of each plate as a function of wavelength has a linear relationship to each other, i.e., 2 ( ) K 1 ( ) . Consider the case when the thickness of the second retarder is twice of the first retarder i.e., K 2 . Figure 7.5 shows the first and the second retarders’ principal retardance and the fast axis orientation as a function of wavelength ranging from 200nm to 4μm. The principal retardance is calculated by following the algorithm explained in Section 7.1 and the fast axis orientation is calculated using Eq. (7.1.7). The orientation of the fast axis is along the horizontal axis for downward sloping regions and is along the vertical axis for upward sloping regions. 128 rad rad 3.0 3.0 2.5 2.5 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 500 1000 1500 2000 2500 3000 3500 4000 nm fast axis 500 1000 1500 2000 2500 3000 3500 4000 nm fast axis 1.5 1.5 1.0 1.0 0.5 0.5 1000 2000 3000 4000 nm 1000 (a) 2000 3000 4000 nm (b) Figure 7.5 (a) The principal retardance of the first retarder and its fast axis orientation as a function of wavelength. (b) The principal retardance of the second retarder and its fast axis orientation with respect to the horizontal axis as a function of wavelength. The total retardance is the sum of each retarder’s retardance Total ( ) 1 ( ) 2 ( ) 1 () 21 () 31 () . (7.4.5) Figure 7.6 shows the retardance of the first retarder 1 ( ) in green, the retardance of the second retarder 2 ( ) in blue, and the total retardance Total ( ) of the system in red as a function of wavelength in one plot. All the retardance values are principal 129 retardances with horizontal fast axis for downward sloping regions and with vertical fast axis for upward sloping regions. rad 3.0 2.5 2.0 1.5 1.0 0.5 1000 2000 3000 4000 nm Figure 7.6 Retardance plots as a function of wavelength for two horizontal fast axis linear retarders (HLR) with different thicknesses (green and blue) and a combination of two HLRs with a shared horizontal fast axis (red). The principal retardance has the horizontal fast axis for downward sloping regions and has the vertical fast axis for upward sloping regions. The principal retardance and the fast axis of the combination of two HLRs as a function of wavelength are shown in Figure 7.7. 130 rad 3.0 2.5 2.0 1.5 1.0 0.5 500 1000 1500 2000 2500 3000 3500 4000 nm fast axis 1.5 1.0 0.5 1000 2000 3000 4000 nm Figure 7.7 The principal retardance of the combination of two HLRs and the system’s fast axis orientation with respect to the horizontal axis as a function of wavelength. The principal retardance value oscillates between zero and π as the wavelength gets shorter. Thus, a phase unwrapping is necessary to recover the real retardance values at a given wavelength. This phase unwrapping method requires knowledge of the 131 wavelength dependence of a retarder; the phase unwrapping algorithm uses the retardance dispersion model shown in Eq. (7.4.4). Coming from the right side of the principal retardance plot in Figure 7.7, the retardance increases to π with the horizontal fast axis, then decreases with the vertical fast axis. When the retardance reaches zero, then it increases with the horizontal fast axis. In order to phase unwrap the principal retardance, a mode number q is assigned to each segment of the principal retardance with different fast axis orientation; odd q’s are for the horizontal fast axis and even q’s are for the vertical fast axis. Figure 7.8 shows segments with odd mode numbers in blue and segments with even mode numbers in red coming from the right side of the graph (longer wavelength). For this example, q = 1, 2, …, 16. Figure 7.8 Each segment of the principal retardance has a mode number q to apply phase unwrapping algorithm. Starting from the right side of the graph, blue segments have odd mode numbers and the red segments have even mode numbers. 132 Figure 7.9 show the trajectory of the principal retardance within the π sphere as the wavelength reduces; the upper left corner figure corresponds to the mode 1 in Figure 7.8. The red color indicates the principal retardance vector { H , 45 , R } for the longest wavelength. As the wavelength reduces, color changes to yellow → green → blue → purple → magenta. The R axis is coming out of the page and each segment shows the points along the same fast axis. The upper left corner figure has the horizontal fast axis (along H axis) and the retardance is increasing. Once the retardance reaches the π sphere, its fast axis changed to the vertical direction (along - H axis) and moves to the origin symmetric point. The next trajectory is continued in the upper middle figure and so on. The origin {0, 0, 0} is equivalent to the identity matrix which is a full wave retarder or 2nπ retarder for an integer n. As the wavelength reduces, the fast axis orientation changes 15 times alternating along the horizontal and vertical directions. 133 134 Figure 7.9 Principal retardance vector trajectories are shown in the retarder space as the wavelength changes. Each figure corresponds to a different mode number starting from the longest wavelength (mode 1) to the shortest wavelength. Figure 7.10 shows the principal retardance trajectory as the wavelength changes; the system’s principal retardance gets larger as the wavelength gets shorter. When the trajectory reaches a sphere with radius π (point x in the figure) it goes to x’, which is the origin symmetric point of x and spirals to x’ within the sphere, instead of continuing to x. R X X ' ' 45 X X 2 H Figure 7.10 The principal retardance trajectory in the retarder space as the wavelength of the ray changes. Discontinuity occurs on a sphere of radius π. 135 The phase unwrapping algorithm maintains the fast axis orientation as the wavelength reduces and calculates the true retardance at a given wavelength without the upper limit. The basic assumption of this method is that at long enough wavelengths, for retarders under consideration will have less than a half wave of retardance, which is the true retardance of the retarder at that wavelength. Therefore when q = 1, the principal retardance is the true retardance. For even q’s the true retardance is q principal while the fast axis orientation is kept along the horizontal axis. For odd q’s the true retardance is (q 1) principal with the fast axis along the horizontal axis, i.e., unwrapped principal when q 1 q principal when q even . (q 1) principal when q odd (7.4.6) Thus the phase unwrapped retardance range has no upper limit. Figure 7.11 (a) shows the principal retardance trajectory in the retarder space for two-aligned HLR system as the wavelength reduces; when the principal retardance reaches the boundary value π the trajectory moves to the origin symmetric point on the sphere and changes its fast axis to the orthogonal direction. Figure 7.11 (b) shows the retardance trajectory of the same system after the phase unwrapping; the horizontal retardance increases continuously keeping the fast axis orientation along the horizontal direction. As shown in Figure 7.2, the distance from the origin to a point in the retarder space is the total retardance of the system. 136 R 45 B’ A’ B A C H (a) R 45 3 A B 5 C H (b) Figure 7.11 (a) Principal retardance trajectory of a two-aligned HLR system in the retarder space as the wavelength gets shorter. (b) Retardance trajectory of the same system after the phase unwrapping. Retardance plots in Figure 7.6 can be phase unwrapped to Figure 7.12. 137 rad 40 30 20 10 500 1000 1500 2000 2500 3000 3500 4000 nm Figure 7.12 Phase unwrapped retardance as a function of wavelength for two HLRs (green and blue) and a system of two-aligned HLR (red). In phase unwrapped figure, the total retardance value always follows the relationship in Eq. (7.4.5) for all the wavelengths. For homogeneous retarder systems, there is no discontinuity in phase unwrapped retardance values. 7.5 Discontinuity in Phase Unwrapped Retardance Values for Compound Retarder Systems of Arbitrary Alignment For two retarders with the same fast axes, the total retardance is the sum of individual retardance and the retardance is linear. Many interesting phenomena are associated with 138 phase unwrapping and order determination of compound linear retarders whose fast axes are neither parallel nor perpendicular. Such retarders may result from misalignment or be intentionally at arbitrary angles. Sequences of retarders whose axis are neither parallel nor perpendicular are in general elliptical. We will explore phenomena in which the retardance itself appears to be discontinuous and explain why. If the fast axis of the second retarder is slightly misaligned from that of the first retarder, as it always will be in practice, the total principal retardance has slightly different behaviors from the aligned system. For example, if two horizontal fast axis linear retarders with retardance 1 ( ) and 2 ( ) respectively, are misaligned by θ, the Jones matrix of the system is JTotal ( [ ( ] ) [ [ ] ] ( ( ) [ ] [ ] [ ]) ) (7.5.1) Using the algorithm explained in Section 7.1, the retardance of the systems is . (7.5.2) From this equation, it is clear that the retardance of the system is not only dependent on individual retarder’s retardance but also dependent on the angle between the fast axes orientations. If both retarders are half-wave retarders with horizontal fast axes, the total retardance will be (full wave) when two fast axes are aligned and the total retardance 139 will be zero when the fast axes are orthogonal to each other. Figure 7.13 shows how the total retardance of the system varies as the second retarder’s fast axis orientation changes (θ) with respect to the horizontal direction, which is the first retarder’s fast axis orientation. rad 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 rad Figure 7.13 Total retardance for a system with two half-wave linear retarders is plotted as the fast axis orientation (θ) of the second retarder changes with respect to the first retarder’s fast axis orientation. Note that in the figure, zero total retardance when θ = π/2 and two end points when θ = 0 and π imply different total retardance. Since the plot shows the principal retardance, two end points are when the total retardance is 2π and the middle zero is when the total retardance is zero. First, the origin of apparent discontinuities in phase unwrapped retardance of compound retarder systems is explained by separating the Jones matrix JTotal of the 140 compound retarder system in Eq. (7.5.1) into two parts, θ dependent Jones matrix J Major and θ independent Jones matrix J Minor . Two linear retarders in Eq. (7.5.1) can be written as a sum of Pauli spin matrices LRJ 2 ( 2 , ) cos( 1 )σ 0 i sin( 1 )σ1 2 2 LRJ1 (1 , 0) cos( 2 2 )σ 0 i cos(2 )sin( 2 2 )σ1 i sin(2 )sin( 2 2 (7.5.3) )σ 2 , where LRJ ( , ) is a linear retarder with a principal retardance and the fast axis along with respect to the horizontal axis. Thus Eq. (7.5.1) becomes J Total LRJ 2 ( 2 , ) LRJ1 (1 , 0) J Major J Minor cos( 1 2 2 )σ 0 i sin( 2 1 2 2 )σ1 J Major , independent part 2 1 2 2i sin ( ) sin( ) cos( )σ1 2 2 , dependent part J 2 1 Minor 2i sin( ) cos( ) sin( ) cos( )σ 2 2 2 2i sin( ) cos( ) sin( 2 ) sin( 1 )σ 3 . 2 2 2sin 2 ( ) sin( ) sin( 1 )σ 0 2 2 (7.5.4) The J Major matrix is a horizontal linear retarder with retardance Major 1 2 ( LRJ(1 2 ,0) ) and J Minor can be written as J Minor c0 (σ 0 d1σ1 d 2σ 2 d3σ 3 ) 2sin 2 ( )sin( 2 )sin( 1 )[σ 0 i cot( 1 )σ1 i cot( ) cot( 1 )σ 2 i cot( )σ 3 ]. 2 2 2 2 (7.5.5) 141 Using Eq. (7.1.6), the retardance of J Minor is Minor cos 2 ( 1 ) sin 2 ( 1 ) cos 2 ( ) 2 2 2 arctan( ). 2 1 2 sin ( )sin ( ) 2 (7.5.6) The discontinuities, which occur when the principal retardance in Eq. (7.5.6) is phase unwrapped, originate from the J Minor matrix. The J Major matrix always has continuous phase unwrapped retardance. The discontinuities get greater as amplitude of Minor gets greater as gets larger since the gets larger. Let’s consider a system consists of two linear retarders – a HLR with retardance δ1 and a HLR with retardance δ2 but misaligned by π/16 from the horizontal axis. The retardance ratio is δ1: δ2 = 2 nd1 2 nd 2 : = 1.1: 2 and both retarders are made from the same material where d1 and d 2 are the thickness of the retarders as in the dispersion model (Eq. (7.4.4)). The misaligned retarder system has a Jones matrix, JTotal LRJ 2 ( 2 , /16) LRJ1 (1,0) ( ). Using the dispersion model for each retarder’s retardance and Eq. (7.5.2) with (7.5.7) = π/16, the principal retardance of the combination ( JTotal ) is calculated as a function of wavelength. 142 Figure 7.14 shows principal retardance of JTotal and the fast axis orientation as a function of wavelength. Coming from the right side, the principal retardance increases to π (half wave), then decreases and the axis rotates to vertical. The principal retardance only decreases to 0.7 before turning around and increasing to π, the second maximum from the right. At this half wave point the retardance (δ) must be 3π/2, corresponding to the second maxima in the top figure in Figure 7.7. The retardance of the system never has the transformation which would correspond to δ = 2π. 143 Figure 7.14 The principal retardance of JTotal , a system of two HRLs misaligned by π/16, is plotted as a function of wavelength. Green dotted circles in the top figure indicate the area where the principal retardance changes its slope without going down to zero. 144 A full wave retarder or 2nπ corresponds to the origin {0, 0, 0} in the retarder space. When the retardance vectors of the compound system are plotted in the retarder system, the trajectory misses the origin as the wavelength reduces. Figure 7.15 and Figure 7.16 show two different views of the trajectories in the retarder space as the wavelength reduces. Two views are shown for the clarity and both figures follow the same color scheme as Figure 7.9; red → yellow → green → blue → purple → magenta. Each segment shows the trajectory as it approaches the π sphere boundary. For example, the upper middle figure starts at and follows the trajectory to as the retardance vector reaches the π sphere. 145 Figure 7.15 Principal retardance vector trajectories are plotted in the retarder space as the wavelength reduces. Each figure corresponds to a segment of the trajectory from the longer wavelength to the shorter wavelength as the retardance vector approaches the π sphere. 146 Figure 7.16 The top view of the principal retardance vector trajectories in the retarder space as the wavelength reduces. Unlike the aligned system’s retarder space trajectory (Figure 7.9), the compound system’s retarder space trajectory misses the origin, i.e., the phase unwrapped retardance of the compound system increases from π to 3π without passing through 2π point, the origin. 147 Using the principal retardance of JTotal and the fast axis orientation, the true retardance can be calculated by using the phase unwrapping algorithm, Eq. (7.4.6). Discontinuities are clearly visible when phase unwrapped retardances for the misaligned (blue) and aligned (red) system are plotted together as shown in Figure 7.17. The blue plot has similar values as the red plot since the misalignment is small. However, the blue plot has discontinuities whenever the retardance value crosses 2nπ boundaries, which are plotted as horizontal blue dashed lines. Figure 7.17 Phase unwrapped retardance plotted as a function of wavelength for the aligned (red) and forced phase unwrapping for a misaligned (blue) two-HLR system. 148 Figure 7.18 shows phase unwrapped Major of the compound system in the top and Minor in the bottom. Figure 7.18 Phase unwrapped Major (top) and Minor (bottom) plotted as a function of wavelength. The phase unwrapped Major is the linear addition of 1 and 2 of the each linear retarder. 149 The phase unwrapped Major follows the exact same curve as the red plot in Figure 7.17 since Major is a linear addition of 1 and 2 of the each linear retarder. The discontinuities in Figure 7.17 occur due to the non-linear retardance coming from the J Minor matrix. Figure 7.19 shows the principal retardance as a function of wavelength for a system of two misaligned HLRs with different misalignment amounts θ; the angle between two fast axes orientation θ varies from π/16 to π/2. Green and blue lines are principal retardance of the first and the second retarders. Red lines show the principal retardance of the misaligned system while purple lines indicate a half wave of retardance, the boundary value. 150 151 Figure 7.19 The principal retardance plots as a function of wavelength for two HLRs (green and blue) and a system of two misaligned HLRs (red). Two HLRs have different retardances and the angle between two fast axes is θ. Whenever the principal retardance of the misaligned HLR system has minima other than zero, the phase unwrapped retardance has discontinuity. At 2 two HLRs are orthogonal to each other thus, the total retardance (red) is Total ( ) 1 ( ) 2 ( ) 2 ( ) 2 2 ( ) 2 ( ) 2 , (7.5.8) and the fast axis of the combined system is along the fast axis of the second retarder. Retarders have two modes, fast and slow modes. For misaligned compound retarder systems which consists of two retarders, multiple modes, typically more than two, exit the system. They are F1 F2, F1 S2, S1 F2, and S1 S2 where the F and S stand for the fast and slow modes and 1 and 2 stand for the first and the second retarders. Two of the modes (F1 F2 and S1 S2) have most of the intensity and the other two modes (F1 S2 and S1 152 F2) have fairly small intensity. For example, F1 F2 and S1 S2 modes for JTotal in Eq. (7.5.7) have 96% ( ) of the intensity and the other two modes have 4% of the intensity. Therefore, the phase unwrapped retardance has a similar behavior as the aligned system in Section 7.4. However, whenever the phase difference between F1 F2 and S1 S2 modes are multiple of 2π (full waves of retardance), effects from the other two modes become more noticeable and thus discontinuities occur as shown in Figure 7.17. Consider a misaligned compound retarder system as shown in Figure 7.20; the system has two HLRs, one with retardance δ1 and the other with retardance δ2, and the fast axes are misaligned by . When a ray enters the system, there are four modes, not two, exiting the system with different optical path lengths (OPLs); F1F2, F1S2, S1F2 and S1S2 have different OPLs. 1 ,0 2 , F1F2 F1S2 S1F2 S1S2 Figure 7.20 A system of two HLRs misaligned by θ. Four output beams exiting the system with four different optical path lengths are shown offset for clarity. 153 A HLR can be represented as sum of horizontal and vertical polarizers with absolute phases of optical path lengths along fast and slow axes, respectively HLR = e i 2 OPL f 1 1 0 i 2 OPLs1 0 0 e , 0 0 0 1 (7.5.9) and a HLR at is a sum of linear polarizers at and / 2 with absolute phases of optical path lengths along fast and slow axes, respectively LR[ ] = e i 2 OPL f 2 2 sin(2 ) e 1 cos(2 ) 1 cos(2 ) sin(2 ) i 2 OPLs 2 2 1 cos(2 ) sin(2 ) . sin(2 ) 1 cos(2 ) (7.5.10) Therefore, each mode from the system of two linear retarders at can be calculated from the multiplication of two linear polarizers with associated OPLs as absolute phases. For example, the first mode ( FF 1 2 ) is the eigenstate of a system of a horizontal linear polarizer followed by a linear polarizer at with associated absolute phases, X1 F1 F2 e i ( OPL f 1 OPL f 2 ) Similarly, the other three modes are cos( ) . sin( ) (7.5.11) 154 cos( ) sin( ) i ( OPL f 1 OPLs 2 ) sin( ) X1 F1S2 e cos( ) sin( ) X 2 S1S2 e i (OPLs1 OPLs 2 ) . cos( ) X 2 S1 F2 e i ( OPLs 1 OPL f 2 ) (7.5.12) Red X marks indicate the exiting mode from the second retarder is the fast mode and blue X marks indicate the exiting mode from the second retarder is the slow mode. A conventional definition of retardance takes the difference in optical path lengths of two modes exiting the retarder. Since there are more than two modes exiting the system with the same intensity, the compound system retardance value cannot be calculated from the conventional eigen-analysis. Multiple modes exiting the compound retarder system is the origin of the discontinuities in phase unwrapped retardance. For simplicity in calculation, set OPLs 0 . This does not affect retardance of the system since the retardance is the optical path difference. Using the dispersion model (Eq. (7.4.4)) each mode’s phase is arg( X1 ) OPL f 1 OPL f 2 2 n1d1 arg( X 2 ) OPLs1 OPL f 2 0 arg( X1 ) OPL f 1 OPLs 2 2 n2 d 2 2 n2 d 2 2 n1d1 0 arg( X 2 ) OPLs1 OPLs 2 0 0 0. (7.5.13) 155 where ni is the refractive index of the ith retarder and di is the distance that ray propagates within the ith retarder. The retardance is the optical path difference (OPD) between four modes. When the misalignment is small, most of the intensity of the exiting light is in X1 and X2 . Therefore, the retardance of the compound system as a function of wavelength follows the curve of the OPD between X1 and X2 , major OPL f 1 OPL f 2 OPLs1 OPLs 2 2 (n1d1 n2 d 2 ) , (7.5.14) where major indicates the overall tendency of the compound system’s retardance behavior; retardance curve of both aligned and misaligned retarder systems follow 1 curve. However, whenever the retardance of one of the retarders becomes 2nπ for an integer n, effects from the OPLs of the other two modes ( X2 and X1 ) become larger and the retardance of the compound system deviates from the major curve. An example compound retarder system, two HLRs with the π/4 misalignment in fast axes, is used for further studies. The retardance ratio between two retarders is 2 / 1 4 / 2.2 1.82 and two retarders are made from the same material. Due to multi-valued OPL for this system, interesting polarization artifacts occur; sometimes the principal retardance of the system avoids certain values such as 2 n . The principal retardance is puzzling and discontinuity occurs due to characteristics of 156 ArcTan(λ). Figure 7.21 shows the principal retardance as a function of wavelength for this system using algorithms in Section 7.1. The principal retardance of the system has minima other than zero which are shown in blue dotted areas; this is the origin of the discontinuities in phase unwrapped retardance values near 2nπ. Arbitrary Wavelength Units Figure 7.21 The principal retardance as a function of wavelength for the system with two HLRs with the fast axes misaligned by π/4. The principal retardance has minima other than zero. To better understand the behavior of the fast axis orientation of the compound system, Figure 7.22 shows the first (green) and second (blue) retarders’ retardance as functions of wavelength along with the fast axis orientation of the compound system, θfast. When the retardance of the second retarder becomes 2nπ, the fast axis of the system is 157 along θfast = 0. When the retardance of the first retarder becomes 2nπ, the system fast axis is along θfast = π/4, the misalignment amount between two HLRs. Figure 7.22 Phase unwrapped retardance of the first (green) and second (blue) retarders with the fast axis orientation ( fast ) of the compound system. Xs mark wavelengths where individual plates have integer waves of retardance, and don’t contribute to the axis of the retarder. Using Eq. (7.4.6), the principal retardance can be phase unwrapped assuming that the overall behavior of the retardance is 1/λ. Figure 7.23 shows the phase unwrapped retardance of the first HLR (green), a 45° fast axis linear retarder (blue), and the combined system (red). Orange dotted lines indicate 2nπ. 158 Unlike the compound system with misalignment, each mode in this 16 system has 25% of the total intensity. Therefore, the discontinuities in phase unwrapped retardance are more apparent than the one in Figure 7.14. Figure 7.23 Green plot shows the retardance of the HLR, δ1(λ) as a function of wavelength. Blue plot is the retardance of the 45° fast axis linear retarder, δ2(λ). Red plot shows the retardance of the system of the two HLRs with π/4 misalignment between two fast axes orientation, δTotal(λ). Note that the discontinuities in red plot occur whenever the retardance of one of the retarders is 2nπ; when one of the retarders has multiple waves of retardance, interferences 159 between (S1F2 , F1F2 ) and ( F1S2 , S1S2 ) become dominant while major is defined by the OPD between ( F1F2 , S1S2 ) . Figure 7.24 shows phase unwrapped Minor of the compound system. The phase unwrapped Major follows the exact same curve as the Major in Figure 7.18 since Major is a linear addition of 1 and 2 of the each linear retarder. The amplitude of the phase unwrapped Minor is greater than the one in Figure 7.18. This explains the greater discontinuities in Figure 7.23 compared to Figure 7.17, where the misalignment in fast axes was /16 . Figure 7.24 Phase unwrapped Minor plotted as a function of wavelength. Figure 7.25 shows the principal retardance trajectory of the compound system in the retarder space as the wavelength reduces. A side and top views are shown for clarity. 160 The trajectory starts at a point A at and stays inside a sphere with radius π; once the trajectory reaches the boundary of the sphere (point B) it moves to the opposite point (point B′) on the π sphere and the fast axis changes to the orthogonal state. Thus, point B and B′ are the same retarder within the π sphere. Each figure shows part of the principal retardance trajectory in a continuous fashion; point A corresponds to the retardance for the longest wavelength and point J corresponds to the retardance for the shortest wavelength. Trajectory follows A→B→ B′→C→ C′→D→ D′→E→ E′→F→ F′→G→ G′→H→ H′→I→ I′→J. 161 162 Figure 7.25 A principal retardance trajectory of the system with two misaligned HLRs at 45° in the retarder space as the wavelength reduces. When the trajectory reaches the boundary of π, the trajectory moves to the opposite point on the π sphere and the fast axis changes to the orthogonal state. 163 Figure 7.26 shows points in Figure 7.25 on phase unwrapped retardance plot as a function of wavelength. Dotted horizontal lines indicate nπ for an integer n. Whenever the phase unwrapped retardance crosses nπ lines, points in the retarder space moves to the opposite points on the π sphere. Figure 7.26 Phase unwrapped retardance of the compound system as the wavelength changes with the corresponding points in Figure 7.25. 7.6 Conclusion Using Pauli spin matrices as basis the principal retardance, horizontal retardance ( H ), 45º retardance ( 45 ) and circular retardance ( R ) can be calculated. These components 164 can be mapped on a retarder space which has H , 45 , and R components as axes. Each point {1 , 2 , 3} in the retarder space represents a retarder with retardance 12 22 32 which is the distance from the origin to that point. The retarder space provides an insightful tool to understand the retardance trajectories. A retardance dispersion model ( ) 0 is used to phase unwrap the principal retardance. Systems consisting of two horizontal linear retarders with shared and nonparallel fast axes were used as examples to demonstrate the principal retardance and phase unwrapped retardance behaviors. For the misaligned two-waveplate system, discontinuities in phase unwrapped retardance were identified; sometimes the principal retardance of the system avoids certain values such as 2 n . The origin of discontinuities is the multi-valued optical path lengths for compound retarder systems; for the misaligned two-waveplate system, four modes ( F1F2 , S1F2 , F1S2 , and S1S2 ) have different OPLs. As the wavelength reduces, individual retarder’s retardance as well as the fast axis of the compound system changes. For N waveplates, there will be 2N different optical path lengths. Although this chapter only explained two linear retarder systems, mathematics for N waveplates is similar to two-waveplate system. 165 CHAPTER 8 COHERENCE MATRIX AND POLARIZATION RAY TRACING TENSOR 8.1 Introduction This chapter considers polarization ray tracing with incoherent light and presents a method that is suitable for stray light calculation. For non-polarizing systmes, eight independent parameters – amplitude, phase, three for diattenuation, and three for retardance – are required to describe polarization characteristics of the media. The general case, with depolarizing elements or scattering, requires sixteen independent parameters – amplitude, three for diattenuation, three for retardance, and additional nine degrees of freedom for depolarization – for a complete polarization characterization of the system. To provide mathematical description of the polarization properties of light, a coherence matrix of an electric field vector in global coordinates is used. A polarization ray tracing tensor is defined. Algorithms to calculate the tensor from surface amplitude coefficients defined in local coordinates, a Mueller matrix defined in its local coordinates, and a three-by-three polarization ray tracing matrix defined in global coordinates, are derived. The polarization ray tracing tensor is defined in global coordinates and is used to ray trace incoherent light through optical systems with depolarizing surfaces. The polarization ray tracing tensor operates on the incident electric field’s coherence matrix 166 and returns the exiting coherence matrix in global coordinates. Therefore, this method is suitable for scattered light ray tracing and incoherent addition of light. For the case of a collimated beam of light, polarization ray tracing tensors can be added to get the exiting coherence matrix. Therefore, the combined polarization ray tracing tensor is defined for a specific incident propagation vector but not restricted by the exiting propagation vector. Polarization ray tracing tensor calculus through a volume of scattering particles is presented as an example. 8.2 The coherence matrix The Coherence Matrix of a light beam contains all the measurable 2nd order correlation information about the state of polarization, including intensity, of an ensemble of electromagnetic waves at a point 56, 57 . This positive semidefinite Hermitian 3 x 3 matrix is defined as ( ( ) ) (8.2.1) 167 where ; is the instantaneous electric field vector; stands for the Kronecker product; is the transpose conjugate of is the complex conjugate of ; ; the brackets indicate the time average of the components . ∫ Under the assumption that (8.2.2) are stationary and ergodic, the brackets can alternatively be considered as ensemble averaging of . In general, the time of measurement T is much larger than the coherence time for the partially coherent electromagnetic waves. Therefore, is suited to describe coherency of quasimonochromatic partially polarized light. Conventional, two-dimensional (2D) Stokes parameters are defined on a plane perpendicular to the propagation vector of the light, using specific local coordinates. For a plane wave propagating along the z-axis ( ). (8.2.3) Thus the Stokes parameters associated with this plane wave are ( and the degree of polarization for this plane wave is ), (8.2.4) 168 √ √ . (8.2.5) Similar relations can be developed for plane waves propagating in other directions. The three-dimensional degree of polarization57 is defined by using the coherence matrix |√ ( ‖ ‖ ‖ ‖ )|, (8.2.6) where the Euclidean norms are ‖ ‖ ( ‖ ‖ √(∑ ) , | | ). The 3D degree of polarization (8.2.7) takes into account not only the degree of polarization of the mean polarization ellipse but also the stability of the plane that contains the instantaneous components of the electric field of the wave. Unpolarized light with a fixed propagation vector direction has 2D degree of polarization whereas . Further discussions on can be found in the ref [57]. 169 8.3 Projection of the Coherence Matrix onto Arbitrary Planes To understand the 2nd order correlations that result when two wavefronts with two different electric fields overlap at a point, each electric field vector is converted to a coherence matrix , , . (8.3.1) Coherence matrices can be added since the addition operator and integral operator commute. Therefore, the total coherence matrix of the two wavefronts is . The advantage of (8.3.2) is it provides incoherent addition of two wavefronts in global coordinates with complete polarization information along all three axes. Therefore, this method is particularly useful for incoherent addition of multiple wavefronts with different propagation directions. Scattered light wavefronts do not follow law of reflection, refraction or diffraction but have various distributions of propagation directions depending on the type of scattering at the ray intercept. Since each coherence matrix is defined in global coordinates, simple summation of coherence matrices provides the incoherent addition of wavefronts with different propagation directions. 3D electric field vectors or the coherence matrices contain full information along x, y, and z direction. However, polarization state is defined on a 2D plane and majority of polarization analysis or intensity measurements are done on a 2D plane. Therefore, an 170 algorithm to find a projection of the coherence matrix onto an arbitrary plane is necessary. Projecting onto an arbitrary plane of interest is done by using proper local coordinates on the plane ̂ and ̂ , which are perpendicular to the plane’s surface normal ̂ ̂ ̂ ̂ ̂ ̂ . (8.3.3) onto a plane spanned by ̂ Then, projected ̂ (̂ ̂ ̂ ̂ ̂ ̂ ̂ and ̂ is ). (8.3.4) This matrix is written in its local coordinates as Jones matrices are often written in s and p local coordinates. Using Eq. (8.5.4), can be written in global coordinates, . where ̂ 8.4 ̂ (8.3.5) ̂ . A Definition of Polarization Ray Tracing Tensor The coherence matrix is defined in global coordinates and thus allows incoherent addition of light by simple addition. Therefore, an operator which deals with in 171 global coordinates can provide a new tool for incoherent ray tracing through depolarizing optical systems. A Polarization Ray Tracing Tensor ( T ti , j ,k ,l ) describes a depolarizing or nondepolarizing polarization element or an interaction at a ray intercept ( ) ( ( ( ) ( ) ) ( ) ( ) ) ( ) ( ) ) ( .(8.4.1) operates on the incident coherence matrix which is defined in global coordinates ∑ (8.4.2) yielding the output coherence matrix where i, j, k, l = x, y, z, i.e., ( ) ( The main advantage of the polarization ray tracing tensor ray tracing matrix P is that ). (8.4.3) over a polarization can describe depolarizing optical systems. Therefore having an index that indicates how depolarizing a given tensor is will be meaningful. 172 Analogous to how the Mueller depolarization index 58, 59 is defined, the depolarization index of the can be defined when the tensor is associated with a single exiting propagation vector. However, since the definition of depolarization index for the tensor is not complete, it is included in Appendix B. For a ray propagating through an optical system with multiple surfaces, each surface in the system contributes a polarization ray tracing tensor. To get a cumulative polarization ray tracing tensor for that particular ray, cascading the tensors is necessary. Figure 8.1 shows an example optical system with a triplet followed by a lens barrel. A collimated grid of rays enters the optical system, propagates through the triplet, and scatters off the lens barrel before it reaches the detector. Each ray in the grid has a polarization ray tracing tensor at each ray intercept. In order to ray trace through the entire system, each ray’s polarization ray tracing tensors need to be cascaded to get a grid of cumulative polarization ray tracing tensors at the detector. 173 Figure 8.1 A triplet followed by a lens barrel. A collimated grid of rays propagates through the triplet and scatters off from the lens barrel before it reaches the detector plane on the right. Considering the primary property of the polarization ray tracing tensor shown in Eq. (8.4.2) and its dimensions ( and ), cascading two polarization ray tracing tensors is ∑ , and the exiting coherence matrix after the ( ∑ ) is ( ∑ ) ∑ Similarly, the (i, j (8.4.4) . ) component of the (8.4.5) for a ray propagating through N ray intercepts can be calculated by cascading summations, ∑ ∑ ∑ ∑ When a collimated N-by-N grid of rays with a propagation vector ̂ (8.4.6) enters an optical system with N surfaces, each ray’s cumulative polarization ray tracing tensor can be added to get the exiting coherence matrix for the incoherent addition of the exiting rays, 174 ( ) ∑ (∑ ) ( where q stands for the ray index, have the same then applied to tensor (∑ ) , (8.4.7) . Since all the rays in the incident grid , all the cumulative polarization ray tracing tensors can be added and to get . Note that the combined polarization ray tracing ) is defined for a single ̂ but is not restricted for the exiting propagation vector direction. Thus, the combined polarization ray tracing tensor can accommodate multiple exiting propagation vector directions, and this is one of the main advantages of using the tensor method in stray light calculus. If the incident grid of rays were not collimated, then the exiting coherence matrix of each ray needs to be calculated using Eq. (8.4.2) and then added incoherently in order to get the incoherent addition of the exiting rays, ∑ 8.5 ∑ ( ) . (8.4.8) A Polarization Ray Tracing Tensor for a Non-depolarizing Ray Intercept Although the main purpose of using the polarization ray tracing tensor tracing through depolarizing optical systems, is incoherent ray can be used for the incoherent ray tracing through non-depolarizing optical systems. In this section, two ways of calculating the tensor are presented; one is using amplitude coefficients, defined in the surface local 175 coordinates. The other is using the three-by-three polarization ray tracing matrix, defined in global coordinates. 8.5.1 A Polarization Ray Tracing Tensor from Surface Amplitude Coefficients If surface reflection or transmission coefficients are given in ̂ as in Section 2.2 for the light propagating along ̂ ̂ local coordinates , the exiting electric field vector projected onto the local coordinate plane perpendicular to ̂ EOut , s ss ps EIn , s . EOut , p sp pp EIn , p is (8.5.1) Therefore, the exiting coherence matrix in this local coordinates ( ) is , , , (8.5.2) , , where Eq. (8.5.2) can be written in terms of the polarization ray tracing tensor in local {̂ ̂ ̂ } and { ̂ ̂ ̂ } coordinates, , 176 ( ) ( ∑ ) ( ) ( ( ) ) ∑ ( ( ) ( ) ) ( ) (8.5.3) ( where ). The polarization ray tracing tensor in global coordinates ( ) can be calculated by applying proper coordinate transformation from the local coordinates to global coordinates using rotation matrices. Using ̂ ̂ ̂ vectors defined in Eq. (2.2.3) as basis vectors of the rotation matrices, the incident coherence matrix ( ) in global coordinates is , where ( {̂ ̂ ̂ } . Thus, the incident coherence matrix in local coordinates ’s can be written as a function of ( (8.5.4) ) . (8.5.5) 177 Similarly, the exiting coherence matrix in local coordinates ( , {̂ where ̂ ̂ ) is (8.5.6) }. Inserting Eq. (8.5.5) and (8.5.6) to Eq. (8.5.3), ∑ . (8.5.7) Therefore, the exiting coherence matrix in global coordinates is [ {∑ ( ) } ] . (8.5.8) Comparing Eq. (8.4.2) and (8.5.8), the components of the polarization ray tracing tensor in global coordinates ( ) are the corresponding coefficients of for using Eq. (8.5.8). Table 8.1 shows the polarization ray tracing tensor ( ) in global coordinates as a function of amplitude coefficients in local coordinates and the incident and exiting local coordinate basis vectors for, ̂ ̂ } ̂ { { ( { } ̂ ) } { } 178 ( ) ( ) ( ) ( ) ( ) ( ) (8.5.9) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 179 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 180 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Table 8.1 A polarization ray tracing tensor in global coordinates as a function of amplitude coefficients in local coordinates. Each shows a three-by-three matrix component of the tensor. In Mathematica, the tensor can be found by using the following command [ ] (8.5.10) 181 where is calculated from Eq. (8.5.8). 8.5.2 A Polarization Ray Tracing Tensor from the three-by-three Polarization Ray Tracing Matrix P This section presents an algorithm to calculate the polarization ray tracing tensor T that corresponds to a three-by-three polarization ray tracing matrix P. This conversion is straightforward since the P matrix is already defined in global coordinates. The exiting electric field vector from the P matrix is ( ) ( )( ) ( ). (8.5.11) Using Eq. (8.2.1), can be calculated and comparing the result with Eq. (8.4.2), the relationship of the polarization ray tracing tensor to the P matrix is , (8.5.12) where i, j, k, l = 1, 2, 3. Again, a set of Ts can be added for a collimated grid of incident rays using Eq. (8.4.7). If a polarization ray tracing tensor T is associated with a single { ̂ ̂ pair and is representing a non-depolarizing incoherent ray trace, associated three-by-three polarization ray tracing matrix P can be uniquely defined to within an unknown absolute phase . Since the tensor T is associated with intensity values of the electric field 182 vector while the P matrix is associated with the amplitude values of the electric field vector, the absolute phase of the P matrix is lost when transforming P into T. First, the norm of each element in P matrix is calculated. Then, the phase of each element is calculated relative to . Expressing P’s elements in polar coordinates, the matrix becomes ( ) ( ). (8.5.13) From Eq. (8.5.12), diagonal elements in the tensor gives the norm of the P matrix elements | | , (8.5.14) therefore, √ The phase of . (8.5.15) can be calculated by choosing a reference; if particular i and j, its phase can be set to the absolute phase, . And all the other phases are defined relative to the absolute phase. From Eq. (8.5.12) [ For example, if { ( )} . for a (8.5.16) 183 [ { ( )} . (8.5.17) Using Eq. (8.5.15) and (8.5.16), the P matrix is uniquely defined with the absolute phase . 8.5.3 Example Polarization Ray Tracing Tensor Calculation This section provides an example tensor calculation where the incident light reflects from an Aluminum coated surface with the following parameters (shown in the Figure 8.2): ̂ ̂ { √ } where n is Aluminum’s refractive index at 500nm. reflection coefficients ( ) ( , (8.5.18) The corresponding amplitude ’s are the Fresnel reflection coefficients, ) ( ). (8.5.19) 184 Figure 8.2 A ray propagating along the z axis reflects from an Aluminum coated surface. The polarization ray tracing tensor in local coordinates is ( ) ( ( ) ) ( ) ( , ) √ ( ) . ( √ (8.5.20) ) Since the incident propagation vector is along the z-axis, any incident electric field vector can be written as . Therefore, the incident coherence matrix in local coordinates and the one in global coordinates are the same ( ) (8.5.21) Using Eq. (8.5.8), ( ).(8.5.22) 185 Therefore, the polarization ray tracing tensor in global coordinates is ( ) ( ( ( ) ( ) ) ( ( ) ( ) ( ) ) . ( ) ) (8.5.23) Using the algorithm in Section 2.2, a three-dimensional polarization ray tracing matrix can be calculated for this example, ( ), (8.5.24) and the exiting electric field vector is ( which gives the same 8.6 ), (8.5.25) as in Eq. (8.5.22). A Polarization Ray Tracing Tensor for a Ray Intercept with Scattering In order to ray trace through optical systems with scattering surfaces or depolarizing surfaces, Fresnel coefficients or amplitude coefficients do not provide sufficient 186 information to describe polarization characteristics of such interactions. In general, a Mueller matrix or a Mueller matrix BRDF is used to describe depolarizing optical surfaces. In this section, a method to transform the Mueller matrix into a polarization ray tracing tensor when the incident and exiting propagation vector, a Mueller matrix, and its local coordinates for the incident and exiting space are given. The logic is analogous to the previous section and Eq. (8.4.2) and (8.5.10) still holds. The only differences are the intermediate steps in getting a relationship between and . As shown in Eq. (8.2.4), 2D Stokes parameters are related to the coherence matrix elements. Similar to Jones vectors, 2D Stokes parameters are defined in its local coordinates (Section 9.3). Therefore, the tensor can be calculated by representing the incident and exiting Stokes vectors in global coordinate coherence matrix elements. 2D Mueller calculus shows ( ( ) ) . ( (8.6.1) ) The incident Stokes vector has a coherence matrix in the incident local coordinates { ̂ ̂ ̂ }, ( ), (8.6.2) 187 and the exiting Stokes vector has a coherence matrix in the exiting local coordinates {̂ ̂ ̂ }, ( ) ( where ) ( ( ) (8.6.3) ( ) ( ) ) ( ) ( , ( ) ( ) ) ( ) , and each local coordinate basis vectors form right-handed local coordinates, ̂ ̂ ̂ ̂ ̂ where i = In, Out. Using the inverse of Eq. (8.5.4) can be written as a function of ’s in global coordinates , {̂ where ’s. Similarly, ̂ ̂ } . (8.6.4) Eq. (8.6.4) provides relationship between ’s and 188 , {̂ where ̂ ̂ }. Using Eq. (8.6.2) and (8.6.4), using Eq. (8.6.3), (8.6.5) can be written as a function of can be written as a function of and k, l = 0, 1, 2, 3. Then can be written as can be written as a function of and and ’s. Then where i, j = x, y, z by using Eq. (8.6.5) i.e., . Again, the components of the tensor are the coefficients of for . 8.6.1 Example Polarization Ray Tracing Tensor Calculation In this section, the example in Section 8.5.3 is revisited; the incident and exiting propagation vectors and a Mueller matrix, which is defined in the incident and exiting local coordinates are the given parameters, ̂ ̂ { √ }, ( ) √ ( ( √ ) Using Eq. (8.6.4) ). (8.6.6) 189 ( ) ( ). (8.6.7) Thus, (8.6.8) ( ) (( )) From the Mueller matrix and Eq. (8.6.8) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) , (8.6.9) )) and ( ( ) ) Using Eq. (8.6.5), the exiting coherence matrix in global coordinates is (8.6.10) 190 ( ),(8.6.11) and the polarization ray tracing tensor is ( ) ( ( ( ) ( ) ) ( ) ( ( ) ( ) ) , ( ) ) (8.6.12) which are the same as Eq. (8.5.22) and (8.5.23). 8.7 Example Polarization Ray Tracing Tensor and Combination of Tensors Most clouds have various particles with different scattering properties, sizes, refractive indices, etc. In this section, a simple and tractable but also realistic cloud model is set up in order to understand some aspects of the complex phenomena of cloud polarization. The polarization ray tracing tensor calculus is implemented to ray trace through an example cloud model and incoherently added for the data analysis. The scattering particles are spherical water droplets with refractive index of in air with refractive index of 1.0002857; this is the simplified model of cubical cloud. Mie scattering is assumed and the s and p polarization reflection coefficients at various 191 scattering angles are calculated from MiePlot60. The black body radiation from the sun with the spectrum between 380nm to 700nm is assumed for the light source. The size of the water droplets has a normal distribution with mean of 5μm and 5% standard deviation. Scattered light intensity is calculated by averaging over 30 different wavelengths within the spectrum and 50 different water droplet sizes from the normal distribution as the scattering angle changes. The 50 sample sizes are shown in Appendix C. The geometry of the volume scattering calculation is shown in Figure 8.3 in two different views. The darker red array of arrows indicates the incident beam of light and different colored thicker arrows indicate sample single and double scattered ray paths from the incident light to the detector. Figure 8.3 A volume of water droplets in the air which scatters the incident collimated beam of light. The incident beam of light is plotted in dark red arrows and some of the individual scattering ray paths are shown in different colors. 192 Twenty seven scattering volumes of water droplets are positioned in a cubic grid. The position vectors are , , and droplet is numbered from 1 to 27 starting from to . Each water . A collimated beam of light from the sun is incident on the scattering volume along kˆ In . A polarimeter views the volume along the x-axis ( kˆ Out ) which is chosen to allow easy summation of many different ray paths, kˆ In {1,1,1}/ 3, kˆ Out {1, 0, 0}. (8.7.1) No absorption or extinction is assumed along the ray paths. The majority of the ray paths experience two scattering events (red, green, and orange ray paths in Figure 8.3) and the rest experiences single scattering event (blue ray path in Figure 8.3). Ray paths with single scattering event are called path1 and ray paths with two scattering events are called path2 . There are 27 path1 and 702 path2 ray paths. By fixing the viewing angle of the polarimeter along kˆ Out , only the scattered light along kˆ Out after the first scattering for path1 , and after the second scattering for path2 , get detected by the polarimeter. A polarization ray tracing tensor is calculated for each scattering event and for each ray path using the reflection coefficients calculated from the Mie scattering function at a given scattering angle as shown in Figure 8.4. Each tensor has subscripts q and r where r stands for the first water droplet and q stands for the second water droplet from which each ray path scatters; . When , represents the single scattering tensor from the qth water droplet. Scattering angles are in degrees and the s 193 polarization reflection coefficients are plotted in red whereas the p polarization reflection coefficients are plotted in blue. The scattering angle is the angle between the incident and scattered light propagation vectors. If the scattering angle is less than 90º, the interaction is forward scattering since the propagation vectors are along the same direction and if the scattering angle is greater than 90º, the interaction is backward scattering. Figure 8.4 s (red) and p (blue) polarization reflection coefficients as a function of scattering angle calculated from the MiePlot program. Then tensors representing ray paths that scatter from the qth water droplet toward the polarimeter are added ∑ where . Each (8.7.2) now contains depolarization effects from scattering. 194 The last step is adding the tensors from the same colored water droplets (along the x-axis) and calculating , ∑ where (8.7.3) . This step creates a nine-by-nine grid of polarization ray tracing tensors as shown in Figure 8.5. The false color in the figure is coded to match the color for each column of water droplets. Figure 8.5 A nine-by-nine grid of the detector with its surface normal along the xaxis is shown. Each false color corresponds to the summation of the polarization ray tracing tensors along the x-axis over the same color mapped water droplets. The polarization ray tracing tensors corresponding to detector pixels are 195 196 197 (8.7.4) For ( an electric field vector ), the exiting coherence matrix ( with a coherence matrix ) after the propagating through a volume of scatterers and detected at jth pixel can be calculated from Eq. (8.4.2) and in Eq. (8.7.4). For example, is , 198 ( ) ( ) ( ), , ( ( ) ( ), ) ( ) , ( ) ( ) (8.7.5) The coherence matrix of unpolarized light is ( and the ) (8.7.6) . This light is equivalent to a 2D Stokes vector with the propagation vector kˆ In . The exiting coherence matrix at each pixel on the detector is calculated from tensors in Eq. (8.7.4) ( ), ( ), 199 ( ), ( ), ( ), ( ), ( ), ( ), ( ). (8.7.7) Note that none of the exiting coherence matrices have x-electric field components since the exiting propagation vector is along the x-axis. The 3D degree of polarization of each exiting coherence matrix indicates that the exiting light is mostly unpolarized. Figure 8.6 show for each pixel number 1, 2,…, 9. 200 Figure 8.6 3D degree of polarization is calculated for each exiting coherence matrix at the detector. The x-axis indicates the pixel number. 3D DOP shows that the exiting light is mostly unpolarized. Each exiting coherence matrix can be reduced to 2D Stokes vectors in its local coordinates on the detector plane, ( ), ( ), ( ), ( ), ( ), ( ), 201 ( ), ( ), ( ), (8.7.8) where the local coordinates are xˆ Loc 0 0 1 ˆ ˆ 0 , y Loc 1 , k Out 0 . -1 0 0 (8.7.9) 2D degree of polarization can be calculated for each 2D Stokes vector and is plotted in Figure 8.7. Again, 2D degree of polarization indicates that the exiting light is mostly unpolarized. Figure 8.7 2D degree of polarization of each 2D Stokes vector is calculated at each pixel on the detector. The values indicate that the exiting light is mostly unpolarized. 202 Both 2D and 3D degree of polarization of pixels 1, 2, 3, 4, and 7 are the same values. The pixels 1, 2, and 3 are aligned with the bottom row (the smallest z-value) of the water droplets and the pixels 3, 4, and 7 are aligned with the left most column (the smallest y-values) of the water droplets, which are the closest water droplets from the collimated incident plane wave. Therefore, most of the path2 exiting from the water droplets that are aligned with pixels 1, 2, 3, 4, and 7 come from the backward scattering (scattering angles > 90º) whereas other water droplets have more forward scatterings than backward scatterings. All the single scattering paths path1 are equally distributed among nine pixels. As shown in Figure 8.4 backward scattering reflection coefficients are smaller than the forward scattering reflection coefficients. Therefore, S0 components of the 2D Stokes vectors in Eq. (8.7.8) are smaller for the pixels 1, 2, 3, 4, and 7 than other pixels as shown in Figure 8.8. Figure 8.8 S0 components of the exiting 2D Stokes vectors at each pixel. 203 However, the diattenuation along the s-polarization for the backward scattering is greater than that of the forward scattering as shown in Figure 8.9. Therefore, the backward scattered light is more polarized than the forward scattered light as shown in Figure 8.6 and Figure 8.7. Figure 8.9 Diattenation of the scattered light calculated from the Mie Plot program. For positive values s polarization has the greater scattering amplitudes than p polarization. For rays following path1 , polarization of the scattered light is s-polarized since they experience single scattering. The s-polarization for this example is sˆ kˆ Out kˆ In {0, -1,1}/ 0. (8.7.10) 204 The s-polarization is linearly polarized light at 45º in the detector’s local coordinates. Therefore, S 2 components of the 2D Stokes vectors in Eq. (8.7.8) provide how much of spolarization exists in the exiting light. Figure 8.10 shows the S 2 components of the 2D Stokes vectors at each pixel. Again, the pixels 1, 2, 3, 4, and 7 have the same value. Figure 8.10 S2 components of the exiting 2D Stokes vectors at each pixel. However, SOut at each pixel is incoherent addition of exiting vectors from path1 and path2 . Therefore, the polarization of the exiting light is not purely s-polarized. Figure 8.11 shows the orientation of the exiting light polarization SOut on the detector in red arrows. Blue dashed arrows are linearly polarized light at 45º on the detector plane. SOut is mostly polarized along 45º with little deviations, ( ). (8.7.11) 205 Figure 8.11 Exiting light polarization vectors on the detector plane are shown in red whereas linearly polarized light at 45º are shown in dashed blue. This example can be extended to describe the larger cubical cloud by using more scattering water droplets. Similar example can be setup with different scattering volumes by choosing different refractive index of the scattering particles and the atmosphere. The incident light properties as well as the camera / detector viewing angle can be changed. All the tensor calculation methods that have been used in this example are general and can be modified depending on the assumptions and other conditions of the applications. 206 8.8 Conclusion Algorithms for the incoherent polarization ray tracing through depolarizing optical systems are presented. A coherence matrix of the incident ray’s electric field vector ( ) and a coherence matrix of the exiting ray’s electric field vector ( intercept are related by a polarization ray tracing tensor, ) at qth ray by Eq. (8.4.2). By cascading the polarization ray tracing tensors as shown in Eq. (8.4.6), the electric field vector’s coherence matrix at the exit pupil or a detector can be calculated from the coherence matrix of the incident electric field vector at the entrance pupil or the source. Since and are defined in global coordinates, incoherent addition of the coherence matrices or polarization ray tracing tensors are much less error prone than adding 2D Stokes parameters or Mueller matrices. As shown in Eq. (8.4.7), the polarization ray tracing tensor is not restricted by a single kˆ Out , which is a critical characteristic for dealing with scattering or stray light analysis. , 207 CHAPTER 9 THREE-DIMENSIONAL (3D) STOKES PARAMETERS In Chapter 8 incoherent ray tracing was developed with coherence matrices and polarization ray tracing tensors. In this chapter 3D generalization of Stokes and Mueller matrices are explored for incoherent ray tracing, but it is still under study. At this time we don’t recommend 3D Stokes or Mueller method since our current thinking is that coherence matrices are more straightforward. 9.1 Definition of 3D Stokes Parameters Components of the 3D coherence matrix parameters by expanding define three-dimensional (3D) Stokes with Gell-Mann matrices as a basis. These are eight generators of the SU(3) symmetry group and a 3x3 unit matrix61. The basis matrices are Hermitian, trace orthogonal, and linearly independent ( √ ( ) √ ( √ ( ) ) √ ( ) √ ( √ ( ) ) ) √ ( √ ( For the basis matrices, the following trace-orthogonality equation holds ) ) (9.1.1) 208 ( ) . (9.1.2) The basis matrices in (9.1.1) allow the coherence matrix to be expressed as ∑ (9.1.3) where the nine real coefficients √ , √ ( ) are the 3D Stokes parameters, , √ √ , √ √ √ , √ √ , √ √ , √ √ , √ ( (9.1.4) ) The coherence matrix can be represented using √ ( √ √ √ √ √ √ √ √ √ √ √ is analogous to 2D Stokes parameter intensity of the light. is analogous to of the electric field intensity ( √ √ √ √ √ ). (9.1.5) and both are proportional to the total and shows predominance of the x component ) or of the y component ( ). and are 209 analogous to and on the xy-plane; if projection of the field component on the 45º bisector axis of xy-plane is dominant and if right circular polarization component on the xy-plane is dominant. Similarly interpretation holds for on xz-plane and and ) on yz-plane. and ) represents the intensity in the xy-plane additional to that in the z-direction. 3D Stokes parameters are derived from an ensemble of three-element electric field vectors defined in global coordinates. Therefore, 3D Stokes parameters represents the predominance of x, y, or z component of the field in global coordinates, regardless of the propagation vector direction. Since a polarization state is always defined relative to a propagation vector direction with specific local coordinates, positive 3D Stokes parameter does not always mean the predominance of polarization states along that particular axis. For example, a 3D Stokes vector for a right circularly polarized light √ ) propagating along the y-axis is ( { Note that of √ √ √ }. (9.1.6) is negative for this right circularly polarized 3D Stokes vector propagating along the y-axis. The three-dimensional degree of polarization57 in Eq. (8.2.6) can be written using the 3D Stokes parameters |√ ∑ |. (9.1.7) 210 Since the 3D Stokes parameters are defined in global coordinates, ’s provide complete polarization information in any direction in three-dimensions. Although ’s are real valued quantities which are measurable in real experiments, its basis matrices ( ) have imaginary values. Therefore, finding a projection operator for ’s is not easy. On the other hand, the coherence matrix is a Hermitian matrix defined in global coordinates and projection onto a plane is a simply inner product as shown in Eq. (8.3.5). Therefore, the projected coherence matrix is converted to 3D Stokes parameters ( ) using Eq. (9.1.4) to calculate incoherent addition of 3D Stokes parameters on an arbitrary plane. 9.2 Example Incoherent Additions of 3D Stokes Parameters This section has an example which explains the basic properties of 3D Stokes parameters. In this example, three mutually incoherent plane wavefronts meet at the origin as shown in Figure 9.1. 211 Figure 9.1 Three plane wavefronts propagating along the x, y, and z axis are meeting at the origin. is propagating along the x-axis, is propagating along the y-axis, and is propagating along the z- axis. Each electric field vector relates to a coherence matrix and can be incoherently added ( ), (9.2.1) 212 where If the projection plane is on xz-plane, ̂ and ( ). Using Eq.(8.3.5) , ( As expected, ). (9.2.2) has no y-component. 3D Stokes parameters for this projected field are , √ , , √ , √ √ , √ , (9.2.3) . Table 9.1 shows the result of the incoherent addition of different electric field vector components with equal amplitudes projected onto xz-plane. Incoherent addition of orthogonal polarizations results in an unpolarized 3D Stokes vector ( ) with twice the intensity of individual electric fields. For this example, only x and z components of 213 the electric field vector contributes to 3D Stokes parameters on the xz-plane. Therefore, all the possible combinations of orthogonal polarizations have the same unpolarized 3D Stokes vector, . Incoherent addition of the same polarizations results a fully polarized 3D Stokes vector with twice the intensity of the individual electric field. The diagonal elements are left empty since they represent addition of each electric field vector with itself. For the case of is when is a linearly polarized along 135º axis between x and z axes, propagating along the y-axis. Table 9.1 3D Stokes vectors from an incoherent addition of three electric field vectors with different polarization states, measured on xz-plane are shown. All the amplitudes are set to 1.0 for the simplicity. The 3D Stokes vectors corresponding to the Table 9.1 entries are as follows 214 √ √ √ = ( = ( ( √ ) √ ) √ = √ ) ( (9.2.4) Consider if all three electric fields are right circularly polarized, √ (√ ) ( ) (√ ) √ √ √ (9.2.5) the resulting coherence matrix is ( ), (9.2.6) and the 3D Stokes parameters are √ , √ ( √ ) (9.2.7) . √ ) 215 which has the total intensity of three and three circular Stokes components. Using Eq. (9.1.7), the degree of polarization of the total 3D Stokes vector is 0.5. When is projected onto yz, xz, and xy planes, ( ) ( ) ( ). The 3D Stokes parameters for each √ (9.2.8) are √ √ (9.2.9) √ √ ( √ ) ( √ ) ( √ ) All three projections give total intensity of 2; each projection gets full intensity from one of the electric field vectors and each of the other two electric field vectors contributes ½ of the intensity. The degree of polarization of all three vectors in Eq. (9.2.9) is √ . 216 and and in Eq. (9.2.9) have which is analogous to 2D Stokes parameter which represents the intensity in the xy-plane additional to that in the z-direction. Although Eq. (9.2.8) does not imply any linear polarization components, corresponding 3D Stokes vectors have and . This is the confusing aspect of the 3D Stokes parameters which requires further research. When is projected onto a plane B which is perpendicular to , ( ) and the 3D Stokes vector is √ √ √ . (9.2.11) √ √ ( √ ) The degree of polarization of this vector is . (9.2.10) √ , 217 9.3 2D Stokes Parameters to 3D Stokes Parameters 2D Stokes parameters are defined in local coordinates and 3D Stokes parameters are defined in global coordinates. Therefore, the coherence matrix can be used as an intermediate step between the 2D Stokes parameters and 3D Stokes parameters. Using Eq. (8.6.2) and (8.5.4), the coherence matrix Stokes parameters can be written as a function of 2D . Then using Eq. (9.1.4), 3D Stokes parameters can be calculated from a given 2D Stokes parameters. For example, linearly polarized at 45º in xy-plane with ̂ 2D Stokes representations. √ √ is in 3D Stokes parameter for the same light The vector √ √ polarization propagating along the y-axis while the vector √ is represents the same √ √ √ represents the same polarization propagating along the x-axis. 9.4 3D Mueller Matrix In this section, 3D generalization of Mueller matrix is presented. The main purpose of this chapter is demonstrating the algorithm which calculates a nine-by-nine Mueller matrix ( defined in global coordinates. The nine-by-nine Mueller matrix operates on a 3D Stokes parameters matrix, 218 . ( ) ( (9.4.1) ) Similar to the polarization ray tracing tensor calculation from a given 2D Mueller matrix, the incident and exiting coherence matrices in local coordinates are calculated first and then transformed to 3D Stokes parameters using the following equations: , ( ) ( ) 219 √ √ √ √ √ √ √ √ √ . √ √ √ √ (√ √ √ ) √ (9.4.3) Using Eq. (8.2.4) and , the incident and exiting coherence matrices for a ray propagating along the z-axis, and , can be written as a ’s. This is analogous to the function of the incident 2D Stokes parameters and method used in Section 8.6 and equations are copied for the convenience of the readers ( ( ) ), ( ) ( (8.6.2) ) ( ) 220 ( ) ( ) ( ) ( ) ( ) ( ) . A nine-by-nine matrix satisfies , where (9.4.4) is a flattened coherence matrix in local coordinates into a nine element vector, . ( The (8.6.3) ) ( (9.4.5) ) vector assumes that the propagation vector is along the z-axis. Using the above three equations, components of a nine-by-nine matrix can be calculated ( ) ( ) 221 ( ) ( ) ( ) ( ) ( ) ( ( ) , ( ) , ( ) ) ( , ) ( ), (9.4.6) where ( ). Using a proper rotation matrix, a nine-by-nine matrix coordinates can be calculated, (̂ ) ̂ . (9.4.7) defined in global 222 The matrix represents a surface matrix for a ray propagating along while the matrix represents the corresponding matrix for a ray propagating along ̂ . ( ̂ The ) ( (9.4.8) ) can be any rotation matrix which rotates to ̂ and my preferred choice of rotation matrix is provided in the Table 9.2. This rotation matrix is constructed in nine-by-nine format based on a direction cosine rotation matrix which Mathematica 62 uses. In Mathematica, a three-by-three rotation matrix which rotates to ̂ is √ { } √ ( The matrix rotates ̂ ̂ ̂ to { the rotation, components of √ √ √ . (9.4.9) ) }. By calculating how { ̂ matrix are calculated. } changes after 223 √ 0 0 0 0 √ √ 0 0 0 √ √ √ 0 √ 0 0 √ 0 √ 0 √ 0 √ √ 224 √ √ √ √ √ √ √ 0 √ 0 √ 0 √ √ √ √ √ √ 225 √ √ √ Table 9.2 A nine-by-nine rotation matrix RotM( ̂ RotM( ̂ matrix rotates elements are shown. The to . The inverse of Eq. gives a nine-by-nine matrix which converts 3D Stokes parameters to flattened coherence matrix vectors , ( ) ( ) (9.4.10) 226 √ √ √ √ √ √ √ √ √ . √ √ √ (9.4.11) √ √ √ √ √ ( ) Then Eq.(9.4.8) can be written as , (9.4.12) and therefore . (9.4.13) Thus, the 3D Mueller matrix ( ) is (̂ 9.4.1 ) (̂ ) . (9.4.14) Example 3D Mueller Calculation In this section, the same example used in Section 8.5.3 and 8.6.1 is revisited; the incident and exiting propagation vectors and a Mueller matrix, which is defined in the incident and exiting local coordinates are the given parameters, 227 ̂ ̂ √ { }, ( ). (9.4.15) Using Eq. (9.4.6) the nine-by-nine Mueller matrix for the light propagating along the z-axis is . ( ) (9.4.16) From Table 9.2, (̂ ) ( ) 228 √ √ √ √ √ √ (̂ ) . (9.4.17) √ √ √ √ ( Using Eq. (9.4.7) √ √ ) for this surface reflection is . ( ) (9.4.18) The calculated from Eq. (9.4.8) is 229 , ( which is exactly the same as (9.4.19) ) calculated from the tensor methods with either the amplitude coefficients (Eq. (8.5.22)) or the 2D Mueller matrix (Eq. (8.6.11)). Using Eq. (9.4.14) the 3D Mueller matrix is , ( ) and the exiting 3D Stokes parameters for the incident 3D Stokes parameters are . ( ) (9.4.21) (9.4.20) 230 9.5 Conclusion In this chapter 3D Stokes parameters and 3D Mueller matrix are defined. The 3D Stokes parameters are the expansion coefficients of the coherence matrix when Gell-Mann matrices are used as basis matrices. The 3D Mueller matrix is a generalized 2D Mueller matrix in global coordinates. Similar to three-by-three polarization ray tracing matrix P, the 3D Mueller matrix is designed to incorporate rays propagating in any directions. As mentioned in the beginning of this chapter, further research is required to better understand 3D Stokes parameters and 3D Mueller matrices. I think that coherence matrix calculus is more straightforward and easier to understand than the 3D Stokes parameters and Mueller matrices, but in my mind this remains an open question. I hope that this chapter aids others in understanding the relationship and characteristics of the nine-by-nine matrix calculus and the three-by-three-by-three-by-three tensor calculus. 231 CHAPTER 10 CONCLUSIONS AND FUTURE WORK 10.1 Summary This work contains two major topics; coherent polarization ray tracing and incoherent polarization ray tracing. Unlike Jones or Mueller calculus, both coherent and incoherent polarization ray tracing methods are defined in global coordinates. Thus they provide an easy basis to interpret the polarization properties for most systems. They avoid the apparent rapid variation of polarization states and properties around local coordinate singularities. However, it still remains straightforward to convert results from global coordinates into other local coordinate bases. Coherent polarization ray tracing uses a three-by-three polarization ray tracing matrix P, which is the generalization of the Jones matrix. The three-by-three polarization ray tracing matrix describes all polarization state changes due to diattenuation, geometric transformations, and retardance. The calculation of diattenuation is achieved via singular value decomposition of the three-by-three polarization ray tracing matrix. The parallel transport matrix Q describes the associated non-polarizing optical system and thus keeps track of the geometric transformation. To calculate the true polarization-dependent phase change, also known as the retardance, the geometric transformation needs to be removed. Μ Q1 P is a fundamental equation for calculating retardance without spurious circular retardance arising from a poor choice of local coordinates. Μ clarifies the meaning of the troublesome minus sign in the Jones matrix for reflection. One important and not 232 initially obvious result of my analysis is that the proper retardance cannot be assigned to an optical system inside a black box whose ray propagation vectors are unknown. Two rays with different ray paths through an optical system can have the same polarization ray tracing matrix but different retardances. The three-by-three polarization ray tracing matrix contains corrections to optical path length from coatings and other polarization effects. A polarization aberration function P(r ) is a grid of polarization ray tracing matrices over the exit pupil. P(r ) and OPL(r ) provide a generalized wavefront aberration function that characterizes the polarization-dependent transformations of a wavefront. A skew aberration is a component of polarization aberration which originates from purely geometric effects. Skew aberration of an optical system is calculated from the parallel transport matrix Q. Skew aberration rotates polarization state as rays propagate through optical systems, and this rotation is ray path dependent. Thus, pupil variation of skew aberration creates the undesired polarization components at the exit pupil and thus, affects PSF and degrades image quality. The skew aberration of a chief ray serves as a piston-like aberration, an overall polarization rotation across the pupil. Skew aberration is typically a small effect in lenses but it could be important in microlithography optics and other polarization-sensitive systems with high NA or large FOV. One aspect of the three-by-three polarization ray tracing calculus that continues to trouble me and my colleagues is that the P matrix cannot characterize the phase changes 233 greater than ±λ/2. We suspect that the problem arises due to the intrinsic nature of the polar description of complex numbers. The arguments of complex numbers are located between -π and π so the three-by-three polarization ray tracing matrix can calculate the phase of the exiting light but cannot necessarily calculate the optical path length, if the coating or other contributions exceed the ± λ/2. Similarly, the principal retardance is defined between - π and π. The purpose of optical design simulation is to predict the outcome of measurements by devices such as interferometers, Hartmann-Shack sensors, and the like. A laser interferometer measures phase difference between a reference and test beam. It does not measure optical path length. It cannot distinguish between a wave with zero, or one wave, or two waves of optical path difference. More importantly, it cannot distinguish between partial waves with no uniquely defined optical path lengths. So phase remains the measurable quantity in interferometry and optical path length becomes a difficult concept in systems with multiple partial waves, such as thick multi-layer thin film coatings. In Chapter 7, a system of two non-parallel retarders was used to demonstrate multiple partial waves exiting a system. A phase unwrapping algorithm using the dispersion model was applied to determine the order of the retardance greater than π. Since the P matrix originated from a Jones matrix, it cannot be used by itself to ray trace and calculate the depolarizing effects from depolarizing optical components or surfaces. Therefore, incoherent polarization ray tracing uses a polarization ray tracing tensor T to ray trace through depolarizing optical systems. In this approach, a coherence 234 matrix of an electric field vector is used as a basis. The polarization ray tracing tensor relates the coherence matrix of the incident light and the coherence matrix of the exiting light. This tensor can be derived from amplitude coefficients in local coordinates of the optical surface, a three-by-three polarization ray tracing matrix P in global coordinates, as well as from a Mueller matrix of the surface in its local coordinates. The addition of coherence matrices for the exiting grid of rays represents the incoherent addition of the light. By using the coherence matrix, full polarization information of the light along x, y, and z directions can be calculated from a single polarization ray trace through an optical system, and the polarization information can then be projected onto a plane for further data reduction if necessary. For a collimated grid of incident rays, the polarization ray tracing tensors of each ray can be added. Thus, the combined polarization ray tracing tensor can accommodate multiple exiting propagation vector directions. This is the main advantage of the tensor calculus and why it should be suitable for stray light analysis. When the coherence matrix is expanded by Hermitian, trace-orthogonal, and linearly-independent Gell-Mann matrices, the expansion coefficients are 3D Stokes parameters. The 3D Mueller matrix can also be used for incoherent polarization ray tracing. Both 3D Stokes parameters and 3D Mueller matrices are defined in global coordinates. At the current time, I think the coherence matrix and the polarization ray tracing tensor method is more straightforward than 3D Stokes or 3D Mueller calculus. 235 10.2 Future Work Skew aberration was defined in this dissertation for the first time. In order to complete the analysis of skew aberration’s effects on optical systems, the following goals need to be accomplished: 1) complete understanding of skew aberration’s field and pupil coordinate dependence and 2) thorough skew aberration statistics with larger sets of optical systems. One way of achieving the first goal is using the series expansion of skew aberration in a paraxial ray trace. To complete the second goal, we need to identify further studies on the relationships between skew aberration and skewness of the ray, numerical aperture (NA) of the optical system, and field of view (FOV) of the optical system. Although Code V offers over 2000 optical systems, cutting-edge high NA optical systems with wide FOV are generally not available to regular users. Current micro-lithography optical systems would be the ideal example systems to test the effects of skew aberration on image quality. An example of ray tracing through a volumetric scattering medium was presented in Section 8.7. More accurate scattering models for different materials such as clouds or aerosol particles are necessary to implement the polarization ray tracing tensor method for remote sensing / imaging applications. Since the basic concepts of the polarization ray tracing tensor method are general, the method can be implemented for various studies in polarization ray tracing through the scattering particles in the atmosphere or tissue samples. 236 Definitions of 3D Stokes parameters and the 3D Mueller matrix are presented. However, due to the greater complexity in 3D Stokes parameters than that of the 2D Stokes parameters, a particular analysis devoted to 3D Stokes parameters has not been developed. Further studies on 3D Stokes parameters and the 3D Mueller matrix can be done by implementing the method to ray tracing through systems with volumetric scatterers. Although the coherence matrix and the polarization ray tracing tensor methods are more straightforward, 3D Mueller calculus has advantages of using matrix multiplication over tensor calculations. 237 10.3 Conclusion Several authors [4-12] have described the use of three-by-three polarization ray tracing matrices for optical design and image formation problems. However, none of these references contained details on their implementation of the three-by-three matrix. Now with the conclusion of this dissertation, perhaps we see why. The straightforward and complete implementation of a three dimensional ray tracing matrix method requires attention to a large number of details, such as the necessity of adding additional constraints (Eq. (2.1.7)) to uniquely define the three-by-three matrix P and the difficulties in calculating the retardance from the three-by-three matrix (Chapter 4). Therefore I hope that this work provides a satisfactory treatment of all the essential issues to implement three-by-three extended Jones matrices in polarization ray tracing and optical design. My advisor, Russell Chipman, confesses to beginning work on the three-by-three matrix formulation of polarization ray tracing in 1989 but postponed publication due to the large number of unresolved issues which troubled him until my work. That said, from the present perspective, I feel that the three-by-three P matrix calculus provides a simplification of important issues in polarization ray tracing. Extending the three-by-three P matrix calculus to incoherent polarization ray tracing to describe depolarizing optical systems was one of my long-standing goals in polarization ray tracing. The concepts of 3D Stokes parameters or the coherence matrix are well-known. However, the method of using the polarization ray tracing tensor T for incoherent polarization ray tracing is original and novel. I believe that this method has 238 substantial potential to be used in polarization ray tracing through depolarizing optical systems. 239 APPENDIX A USA PATENT 2,896,506 LENS PARAMETERS The USA patent 2,896,506 lens parameters are shown in PolarisM convention. 240 Shape is the surface type, v is the vertex location, a is the axis of the surface, material 1 and material 2 are the material of the optical surfaces. In PolarisM, all the locations are in global coordinates and the values are in mm. PolarisM calculates the next ray intercept’s material based on in which material that the ray is currently. No coatings were applied to optical surfaces. 241 APPENDIX B DEPOLARIZATION INDEX OF A POLARIZATION RAY TRACING TENSOR The ideal depolarizer matrix ( is a tensor which returns completely unpolarized coherence ) for any incident coherence matrix, where ID stands for ideal depolarizer ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). ) (1) The depolarization index of is the distance between and . The Euclidean distance of a matrix is the maximum singular value of the matrix. For polarization ray tracing tensors its dimension is three-by-three-by-three-by-three and each three-by-three sub-matrix ( ) relates the incident coherence matrix to the i, j component of the exiting coherence matrix as shown in section 8.4. Therefore, it makes the most sense to calculate each ’s Euclidean distance to calculate the distance between the Euclidean distance between and the zero tensor is ( ). (2) and . For example 242 However, having a three-by-three matrix form distance is not as convenient as having a scalar value that tells the distance between two tensors. Euclidean distance between and is the sum of each Euclidean distances of ∑ where , is from ’s, (3) is the maximum singular value of indicates how far Therefore, the . Eq. (3) gives a scalar which . In order to define the depolarization index (DI) between zero and one, an appropriate denominator is required to normalize the distance between most appropriate choice is the Euclidean distance between and . The and the zero tensor. Therefore, the DI is defined as . (4) for the ideal depolarizing tensor. Different definition of the distance between two tensors can be used and as a result, DI will change depending on how one defines the distance between two tensors. However, at the current time, I think that Eq. (3) is the best choice. 243 APPENDIX C WATER DROPLET SIZES FOR THE CLOUD EXAMPLE 50 water droplet sizes from a normal distribution with mean 5μm and deviation 5 are shown in the table. 244 REFERENCES 1. R.C. Jones, “A New Calculus for the Treatment of Optical Systems,” J. Opt. Soc. Am., 31, 488-493, 493-499, 500-503 (1941), 32, 486-493 (1942), 37, 107-110, 110-112 (1947), 38, 671-685 (1948), 46, 126-131 (1956). 2. W. Singer, M. Totzeck, and H. 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