POLARIZATION RAY TRACING

POLARIZATION RAY TRACING
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.
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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
Ea  Pq Ea , Eb  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 Pq1 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{( Ev1eiv1 vˆ 1  Ev 2eiv 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 {sˆq , pˆ q , kˆ q } after the qth surface are,
sˆ q 
kˆ q -1  kˆ q
, pˆ q  kˆ q -1  sˆ q , and sˆq  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)
Oin1,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 {sˆq , 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 qOin1,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 qOin1,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 q1
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 ΜTotal , 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   cos1 (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† w2 , kˆ 0 ,
(4.4.9)
where w1  {w x,1 , w y ,1 ,0} and w2  {w x,2 , w y ,2 ,0} .
In exit space, the canonical basis set is
v1  Q v1  Q U† w1 , v2  Q v 2  Q U† w2 , 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  Q1 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
gExit ,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 gExit ,i . If
gExit ,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  yqq
wq 1  wq  yqq
(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 { yq1  yq , yq1  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 ,q1 is
Areaq 
wq wq 1  wq 1wq
1
.
2
2 wq  wq2  1 wq21  wq21  1
(6.5.4)
Further manipulations using Eq. (6.5.1) results
Areaq 
q
2
H

2
wq yq  wq yq
wq2  wq2  1 wq21  wq21  1
(6.5.5)
q
wq2  wq2  1 wq21  wq21  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 wq21  wq21  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 ()  21 ()  31 () .
(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  Q1 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
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