Manual 21027781

Manual 21027781
MUELLER MATRIX ROOTS
by
Hannah Dustan Noble
=
BY:
A Dissertation Submitted to the Faculty of the
COLLEGE OF OPTICAL SCIENCES
In Partial Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
2011
2
THE UNIVERSITY OF ARIZONA
GRADUATE COLLEGE
As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Hannah Dustan Noble entitled Mueller Matrix Roots and
recommend that it be accepted as fulfilling the dissertation requirement for the
Degree of Doctor of Philosophy.
Date: 25 July 2011
Russell Chipman
Date: 25 July 2011
Scott Tyo
Date: 25 July 2011
Stephen McClain
Final approval and acceptance of this dissertation is contingent upon the candidate’s
submission of the final copies of the dissertation to the Graduate College.
I hereby certify that I have read this dissertation prepared under my direction and
recommend that it be accepted as fulfilling the dissertation requirement.
Date: 25 July 2011
Dissertation Director: Russell Chipman
3
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for an
advanced degree at the University of Arizona and is deposited in the University
Library to be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission,
provided that accurate acknowledgment of source is made. This work is licensed
under the Creative Commons Attribution 3.0 United States License. To view a
copy of this license, visit http://creativecommons.org/licenses/by/3.0/us/ or send a
letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California,
94105, USA.
SIGNED:
Hannah D. Noble
4
ACKNOWLEDGEMENTS
I would like to thank my parents, who have always seen the best in me and encouraged me to live an intellectually fulfilling life. Without their unfailing support
I wouldn’t be here. They have taught me to develop the life of the mind, which
has largely shaped my worldview and thirst for challenge. I would like to thank my
brother for always being a good friend - our late night technical conversations have
inspired me throughout my graduate career.
I would like to thank my advisor, Professor Russell Chipman, without whom
this dissertation wouldn’t have been possible. The day I walked into his office to
discuss a graduate research position was a pivitol moment in my academic and professional life. Since that day, Professor Chipman has shared a tremendous amount
of knowledge, dared me to push my intellectual boundaries, and given me continuous opportunities to spend my time working on some of the most interesting,
mind-twisting technical problems imaginable.
I would like to thank Professor John Essick for believing in me as an undergraduate, and sparking my interest in the field of optics.
I would like to thank Dr. Stephen McClain, who has spent hours and hours
discussing the painstaking details of my dissertation with me. This guidance has
been critical. Many thanks to Dr. Greg Smith, who is always available to contribute
a creative idea or programming solution, as needed. I would like to thank my
colleagues and fellow Polarization Lab graduate students (and alumni) - Garam Yun,
Paula Smith, Stacey Sueoka, Karlton Crabtree, Tiffany Lam, Wei-Liang Hsu, Tyson
Ririe, Michihisa Onishi, Brian Daugherty, and Anna-Britt Mahler. You’ve all helped
with paper editing, technical and programming discussions and collaborations, but
your most important contributions have been your friendship. Thanks to Garam,
Miena, and Stacey for all of the great Cafe Luce breaks.
I would like to thank my partner, Houssine Makhlouf, for his constant love and
support throughout this long road to the PhD.
5
DEDICATION
To my parents and grandmother with Love.
6
TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
CHAPTER 1 METHODS TO CALCULATE THE PROPERTIES
MUELLER MATRICES VIA MATRIX DECOMPOSITION . . . . . .
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Polar decomposition . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Lu-Chipman decomposition . . . . . . . . . . . . . . . . . . . . .
1.4 Symmetric decomposition . . . . . . . . . . . . . . . . . . . . . .
1.5 Additive decompositions . . . . . . . . . . . . . . . . . . . . . . .
1.6 Mueller matrix root decomposition . . . . . . . . . . . . . . . . .
1.7 Pauli spin matrices . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Jones N-matrices and the Mueller matrix roots decomposition . .
OF
. .
. .
. .
. .
. .
. .
. .
. .
. .
18
18
19
20
23
25
26
27
28
CHAPTER 2 THE MUELLER MATRIX ROOTS DECOMPOSITION ALGORITHM AND COMPUTATIONAL CONSIDERATIONS . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Matrix roots decomposition . . . . . . . . . . . . . . . . . . . . . . .
2.3 pth Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Principal matrix root algorithms . . . . . . . . . . . . . . . .
2.4 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Polarizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Half wave retarders . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Depolarizing non-uniform Mueller matrices . . . . . . . . . . .
2.5 Numerical accuracy and root order . . . . . . . . . . . . . . . . . . .
2.5.1 Choice of p . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Algorithm and flow chart . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Statistical algorithm implementation . . . . . . . . . . . . . . . . . .
2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
33
35
38
39
40
40
42
45
47
47
51
53
56
CHAPTER 3 INTERPRETATION OF MUELLER MATRIX ROOTS DECOMPOSITION PARAMETERS . . . . . . . . . . . . . . . . . . . . . . . 57
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7
TABLE OF CONTENTS – Continued
3.2
3.3
3.4
3.5
3.6
3.7
Mueller matrix roots decomposition . . . . . . . . . . . . . . . . . .
Definition of diagonal, phase, and amplitude depolarization . . . . .
Degree of polarization maps of depolarizing Mueller matrices . . . .
Depolarization generation and cyclic permutations . . . . . . . . . .
3.5.1 Cyclic permutations . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Generation of depolarization via averaging of non-depolarizing
Mueller matrices . . . . . . . . . . . . . . . . . . . . . . . .
Experimental samples . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Ground glass . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.2 Pencil at 505 nm . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
58
60
63
67
67
.
.
.
.
.
68
70
71
75
80
CHAPTER 4 ADDITIONAL MATRIX ROOTS GENERATOR PROPERTIES AND APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1 Convergence to the identity matrix . . . . . . . . . . . . . . . . . . . 81
4.2 Generator properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2.1 Trajectories of non-depolarizing generators . . . . . . . . . . . 84
4.2.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2.3 Depolarization index . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.4 Unitary transformations and depolarization classification . . . 91
4.2.5 Conditions for physicality of depolarizing generators . . . . . . 96
4.3 Interpretation of MM root parameters . . . . . . . . . . . . . . . . . 99
4.3.1 Nondepolarizing matrix root parameters . . . . . . . . . . . . 99
4.3.2 Depolarizing matrix roots parameters . . . . . . . . . . . . . . 103
4.3.3 The diagonal depolarization tetrahedron . . . . . . . . . . . . 105
4.4 Applications for depolarization in optical design . . . . . . . . . . . . 106
4.4.1 Diattenuation and retardance defocus . . . . . . . . . . . . . . 108
4.4.2 Depolarization effects from diattenuation and retardance defocus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
CHAPTER 5 POLARIZATION SYNTHESIS BY COMPUTER GENERATED HOLOGRAPHY USING ORTHOGONALLY POLARIZED AND
CORRELATED SPECKLE PATTERNS . . . . . . . . . . . . . . . . . . . 115
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 PCGH illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3 PCGH diffuser design . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.4 PCGH fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.5 PCGH measurement and reconstruction . . . . . . . . . . . . . . . . 121
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
8
TABLE OF CONTENTS – Continued
CHAPTER 6 SQUARE WAVE RETARDER FOR POLARIZATION COMPUTER GENERATED HOLOGRAPHY . . . . . . . . . . . . . . . . . . . 125
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.2 Experimental geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.3 Calcite square-wave retarder design and fabrication . . . . . . . . . . 132
6.4 Measured polarization properties of the reconstructed image . . . . . 136
6.5 Alignment and illumination errors . . . . . . . . . . . . . . . . . . . . 142
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
APPENDIX A JONES N -MATRIX ALGORITHM . . . . . . . . . . . . . . 147
APPENDIX B THE EFFECT OF REFLECTION ON DEPOLARIZATION
PARAMETERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
APPENDIX C INFERRING THE ORIENTATION OF TEXTURE FROM
POLARIZATION PARAMETERS . . . . . . . . . . . . . . . . . . . . . . 151
C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
C.2 Aluminum measurements and analysis . . . . . . . . . . . . . . . . . 152
C.2.1 Aluminum sample . . . . . . . . . . . . . . . . . . . . . . . . . 152
C.2.2 Mueller matrix measurements . . . . . . . . . . . . . . . . . . 153
C.3 Diattenuation and retardance . . . . . . . . . . . . . . . . . . . . . . 155
C.3.1 Determining orientation from diattenuation and retardance . . 156
C.4 Depolarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
C.4.1 Lu-Chipman depolarizing Mueller matrix and depolarization
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
C.4.2 Mueller matrix roots depolarizing parameters . . . . . . . . . 165
C.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
9
LIST OF FIGURES
1.1
Taking the root of a uniform Mueller matrix is analagous to slicing
it into very thin identical pieces. . . . . . . . . . . . . . . . . . . . . . 30
2.1
Taking the root of a uniform Mueller matrix is analogous to slicing
it into very thin identical pieces. . . . . . . . . . . . . . . . . . . . .
The error of the root calculation vs. root order for a simple retarder
converges to a minimum relative error just beyond the 105 th root. .
The norm of the depolarizing retarder’s diagonal depolarization parameters (D13 , D14 , and D15 ) converges to a steady value of approximately 3.03 beyond the 104 th root. . . . . . . . . . . . . . . . . . .
The norm of the depolarizing retarder’s matrix roots retardance parameters (D4 , D5 , and D6 ) converges to a steady value of approximately 1.07 beyond the 104 th root. . . . . . . . . . . . . . . . . . .
Relative error for Mathematica’s matrix root calculation. . . . . . .
Matrix Roots algorithm flow chart. . . . . . . . . . . . . . . . . . .
A histogram of the matrix root amplitude depolarization values for
76,336 randomly generated physical Mueller matrices. . . . . . . . .
A histogram of the matrix root phase depolarization values for 76,336
randomly generated physical Mueller matrices. . . . . . . . . . . . .
A histogram of the matrix root diagonal depolarization values for
76,336 randomly generated physical Mueller matrices. . . . . . . . .
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.1
3.2
3.3
3.4
3.5
3.6
Taking the root of a uniform Mueller matrix is analagous to slicing
it into very thin identical pieces. . . . . . . . . . . . . . . . . . . . .
Degree of polarization maps for depolarizing Mueller matrices M7,15
through M15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measurement geometry for the University of Arizona scattering infrared polarimeter. . . . . . . . . . . . . . . . . . . . . . . . . . . .
The magnitude of the matrix roots parameters D0 through D15 of
the ground glass sample for (a) the specular angle pair and (b) the
non-specular angle pair. . . . . . . . . . . . . . . . . . . . . . . . .
The Mueller matrix for a graphite and wood pencil, measured at
505nm with a Mueller matrix imaging polarimeter. . . . . . . . . .
The non-depolarizing matrix roots parameters for a graphite and
wood pencil, measured at 505nm with a Mueller matrix imaging polarimeter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 34
. 48
. 49
. 50
. 51
. 52
. 54
. 55
. 55
. 59
. 64
. 71
. 73
. 76
. 77
10
LIST OF FIGURES – Continued
3.7
The depolarizing matrix roots parameters for a graphite and wood
pencil, measured at 505nm with a Mueller matrix imaging polarimeter. 79
4.1
A histogram of the randomly generated Mueller matrix determinants
shows the statistical distribution of the determinant values. . . . . . .
4.2 This histogram shows increasing convergence towards the identity
matrix as the root order p increases. . . . . . . . . . . . . . . . . . . .
4.3 A linear diattenuator with diattenuation of 0.2 approaches the identity matrix as it is taken to the 10p th root. . . . . . . . . . . . . . . .
4.4 The trajectory of generator G2 with d2 = 2 ∗ 10−10 as the matrix
is taken to the power of 10p approaches a linear diattenuator with
diattenuation of 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 The trajectory of an elliptical retarder with δ45 = π/4 and δR = π/10
approaching the identity matrix as the matrix is taken to the 10p th
root. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 The trajectory of the generator product G5 (d5 = π/4∗10−9 )·G6 (d6 =
π/10 ∗ 10−9 ) approaching an elliptical retarder as the matrix is taken
to the power of 10p . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 The diference between π/2 and the vector angle between G1 (d1 = d)
and G2 (d2 = d) is plotted as a function of d, showing that generators
G1 and G2 are not orthogonal in general. . . . . . . . . . . . . . . . .
4.8 The diference between π/2 and the vector angle between G11 (d11 = d)
and G12 (d12 = d) is plotted as a function of d, showing that generators
G11 and G12 are not orthogonal in general. . . . . . . . . . . . . . . .
4.9 The four eigenvalues for the coherency matrices associated with the
nine depolarization generators. . . . . . . . . . . . . . . . . . . . . . .
4.10 The physical region for each depolarizing generator G7 through G14
multiplicatively combined with G15 is shown in the white region. One
axis represents d15 and the other axis represents the other d-parameter
of a corresponding generator. G7 through G9 and G10 through G12
are shown on the same plot since their physical regions are identical. .
4.11 Physical Mueller matrix diagonal depolarization space forms a tetrahedron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.12 Diattenuation defocus shifts along the y-axis and off-axis in x and
y, illustrating the effect of tilt on diattenuation defocus in the pupil.
Retardance defocus looks identical to diattenuation defocus, but the
effect is different. Diattenuation defocus affects the transmission of
polarization states, whereas retardance defocus adds a given retardance to the transmitted polarization state. . . . . . . . . . . . . . .
82
83
85
86
87
88
90
91
97
98
107
110
11
LIST OF FIGURES – Continued
4.13 With diattenuation defocus, diagonal depolarization parameters D14
and D15 decrease in magnitude as the pupil center is shifted off-axis. .
4.14 With retardance defocus, diagonal depolarization parameters D14 and
D15 increase in magnitude as the pupil center moves off-axis. . . . . .
4.15 Circular phase depolarization D12 and diagonal depolarization D13
vary with pupil shift x0 . . . . . . . . . . . . . . . . . . . . . . . . . .
4.16 Circular phase depolarization D12 and diagonal depolarization D13
vary with pupil shift x0 . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
5.2
5.3
5.4
5.5
6.1
6.2
6.3
6.4
The interlaced PCGH is illuminated with shifted cross-polarized
Ronchi rulings, and its reconstruction and polarization distribution
are measured. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x- and y-polarized components add to form a tangentially polarized
annulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flowchart of the PCGH design method. . . . . . . . . . . . . . . . .
Normalized linear Stokes vector components demonstrate polarization synthesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Polarization orientation rotates around the annulus. . . . . . . . . .
Interlaced x and y-polarized components add to form a tangentially
polarized annulus. Overlap of these two coherent states produces
~ o with a continuously varying
the ideal tangentially polarized ring U
polarization orientation. . . . . . . . . . . . . . . . . . . . . . . . .
A polarization compensator system (enclosed in the dashed line region) provides an arbitrary polarization state incident on the etched
calcite square wave retarder. A 4f optical system images the square
wave retarder to illuminate the interlaced CGH, providing alternating
stripes of x and y polarization. A CCD camera is used at the back
focal plane of an imaging lens to view the reconstructed image. . . .
The calcite square wave retarder generates alternating stripes of orthogonal linear polarizations when illuminated with a linearly polarized plane wave oriented at 45o with respect to the crystals fast axis.
Grooves are etched into the substrate to the base thickness level and
are filled with index matching oil to produce half-wave plate ridges
which rotate the incident linear polarization by 90o . . . . . . . . . .
An SEM cross-section of the fabricated calcite for the square-wave
retarder. The grooves are smooth and uniform, although there is
some rounding of the corners. . . . . . . . . . . . . . . . . . . . . .
112
113
113
114
. 117
. 118
. 120
. 122
. 122
. 126
. 129
. 133
. 136
12
LIST OF FIGURES – Continued
6.5
Retardance of a 1 mm wide horizontal cross section at the center of
the calcite square wave retarder sample was measured at 632.8 nm
using the Mueller matrix imaging polarimeter. Retardance associated
with the half wave plate ridges varies between approximately 160o and
the ideal 180o in the center of the sample. . . . . . . . . . . . . . .
6.6 The measured Stokes vector for the PCGH reconstruction shows that
the polarization is linearly tangential with some circular polarization
dispersed throughout the annulus. S3 shows that the presence of
circular polarization is most notable in the synthesis regions. S1 , S2 ,
and S3 are normalized to S0 , which remains un-normalized. . . . . .
6.7 The orientation ϕ of the major axis of the polarization ellipse around
the annulus is calculated from the normalized Stokes vector. The
measured polarization orientation of the PCGH reconstruction is tangential. Its orientation differs by several degrees from the desired
orientation at the sides of the annulus. . . . . . . . . . . . . . . . .
6.8 The pixel-by-pixel degree of linear polarization (DoLP) for the PCGH
reconstruction. The average DoLP over the annulus is 0.81. The
synthesis regions have a lower DoLP than the vertical and horizontal
regions of the annulus because the DoLP in these regions is dependent on a high degree of speckle correlation from the two orthogonal
polarizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9 Pixel-by-pixel ratio of tangential polarization (RoP) to flux for the
PCGH reconstruction. The RoP is noticeably lower in the 45o and
225o regions of the annulus and in the dim regions between speckles.
Average RoP around the ring is 84%. . . . . . . . . . . . . . . . . .
6.10 Simulation of square-wave retarder output. (a) In an ideal optical
system, the polarization orientation exiting the calcite square wave
retarder alternates between x and y according to the grooves and
ridge stripes; (b) An extraordinary axis orientation error of 10o with
respect to the etched stripes causes an orientation error of 20o in the
y polarized component exiting the calcite. . . . . . . . . . . . . . .
6.11 Properties of residual wedge in the calcite. (a) MMIP measurement
of retardance as a function of position in an ungrooved substrate
exhibiting a small residual wedge; and (b) Conceptual simulation
illustrating transmitted polarization from a square-wave retarder in
the presence of a 0.3o residual wedge. . . . . . . . . . . . . . . . . .
. 137
. 138
. 139
. 140
. 142
. 143
. 144
A.1 Algorithm flow-chart for calculation of diattenuation and retardance
vectors using Jones N-Matrix method. . . . . . . . . . . . . . . . . . 148
13
LIST OF FIGURES – Continued
A.2 Retardance parameters of a linear retarder with a retardance of π/4
plotted as a function of θ. . . . . . . . . . . . . . . . . . . . . . . . . 148
A.3 Diattenuation parameters of a linear diattenuator plotted as a function of θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
C.1 Surface profile of sanded aluminum sample. White regions are areas
of low signal where signal-to-noise ratio is poor. The white regions are
likely areas of high slope where the reflected light is not substantially
captured by the instrument. . . . . . . . . . . . . . . . . . . . . . . .
C.2 Autocovariance of sanded aluminum sample. Units of x-profile are in
millimeters, units of y-profile are in microns. . . . . . . . . . . . . . .
C.3 Normalized Mueller matrix for −60o incident, 60o scattering specular angle (solid line) and −60o incident, 30o scattering off-specular
angle (dashed line) with dotted-line fits. The x-axis represents texture orientation angle of aluminum in degrees, y-axis represents the
magnitude of each Mueller matrix element. . . . . . . . . . . . . . . .
C.4 Goniometric polarimeter coordinate system. All angles are measured
from sample normal. Negative angles are in the region of backscattered light, and positive angles are in the direction of forward scatter.
C.5 The two measurement configurations analyzed in this section: −60o
incident, 60o scattering and −60o incident, 30o scattering. . . . . . . .
C.6 Normalized average irradiance from light reflected off of the sample
as the groove orientation angle changes. The x-axis lists the orientation angle of the aluminum grooves; the y-axis represents the average
irradiance reflected from the sample. The specular measurement is
plotted with a solid line, and the off-specular measurement is plotted
with the dashed line. The sinusoidal fit from equation (C.1) is plotted
with the dotted lines. . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.7 Matrix root diattenuation parameters with fit (dotted lines) for the
specular angle (solid line) and the off-specular angle (dashed line). . .
C.8 Matrix roots retardance parameters in radians with fit (dotted lines)
for the specular angle (solid line) and the off-specular angle (dashed
line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.9 Aluminum sample groove orientation calculated with the specular
matrix roots retardance parameters demonstrate the estimation of
orientation from a single Mueller matrix measurement following calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
158
159
159
160
160
161
161
161
162
14
LIST OF FIGURES – Continued
C.10 Normalized depolarization Mueller matrix from the Lu-Chipman decomposition with fit (dotted line) for specular measurement (solid
line) and off-specular measurement (dashed line). . . . . . . . . . .
C.11 Depolarization index vs. orientation angle with fit (dotted line)
for specular measurement (solid line) and off-specular measurement
(dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.12 Amplitude depolarization parameters D7 , D8 , and D9 (from left to
right) vs. orientation angle with fit (dotted line) for specular measurement (solid line) and off-specular measurement (dashed line). .
C.13 Phase depolarization parameters D10 , D11 , and D12 (from left to
right) vs. orientation angle with fit (dotted line) for specular measurement (solid line) and off-specular measurement (dashed line). .
C.14 Diagonal depolarization parameters D13 , D14 , and D15 (from left to
right) vs. orientation angle with fit (dotted line) for specular measurement (solid line) and off-specular measurement (dashed line). .
C.15 Aluminum sample groove orientation calculated with the off-specular
linear phase depolarization parameters D10 and D11 , demonstrating
the estimate of orientation from a single Mueller matrix measurement
following calibration. . . . . . . . . . . . . . . . . . . . . . . . . . .
. 164
. 165
. 167
. 167
. 168
. 168
15
LIST OF TABLES
2.1
The sixteen polarization properties of the Mueller matrix given by
the Mueller matrix roots decomposition. . . . . . . . . . . . . . . . . 38
3.1
The non-depolarizing Mueller matrix generators G1 (d1 ) through
G6 (d6 ) and their first order Taylor series approximations. . . . . . . .
The depolarizing Mueller matrix generators G7 (d7 ) through G15 (d15 )
and their first order Taylor series approximations. . . . . . . . . . . .
Notation for the basis diattenuator and retarder Mueller matrices
oriented along the three Stokes axes (horizontal/vertical, 45o /135o
and right/left circular), as well as an attenuating identity matrix. q
and r are the maximum and minimum transmission, q 6= r, δ is the
magnitude of the retardance vector, and α is the attenuating coefficient.
Depolarization properties (shown by parameters D7 through D15 )
produced by averaging two non-depolarizing Mueller matrices. . . . .
Non-diagonal depolarization properties (shown by parameters D7
through D12 ) produced by averaging two non-depolarizing Mueller
matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The matrix roots parameters from a ground glass sample for specular
{−70o , 70o } and non-specular angle pairs {−70o , 10o }. . . . . . . . . .
3.2
3.3
3.4
3.5
3.6
61
62
69
70
70
75
4.1
4.2
4.3
Effect of the rotation transformation for G10 through G12 about R1 (θ). 94
Effect of rotation transformation for G10 through G12 about R2 (θ). . 95
Effect of rotation transformation for G13 (d13 ) through G15 (d15 ) about
R2 (θ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.1
RoP and DoLP are measured as a function of the horizontal translation of the calcite and calcite rotation about the optical axis. . . . . . 145
C.1 Sinusoidal fit coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . 163
16
ABSTRACT
This dissertation is comprised of two separate topics within the domain of polarization optical engineering. The first topic is a Mueller matrix roots decomposition,
and the second topic is polarization computer generated holography.
The first four chapters of the dissertation are on the topic of the Mueller matrix
roots decomposition. Recently, an order-independent Mueller matrix decomposition
was proposed in an effort to organize the nine depolarization degrees of freedom.
Chapter 1 discusses relevant Mueller matrix decomposition prior art and the motivation for this work. In chapter 2, the critical computational issues involved in
applying this Mueller matrix roots decomposition are addressed, along with a review of the principal root and common methods for its calculation. The choice of
the pth root is optimized at p = 105 , and computational techniques are proposed
which allow singular Mueller matrices and Mueller matrices with a half-wave of
retardance to be evaluated with the matrix roots decomposition. A matrix roots
algorithm is provided which incorporates these computational results. In chapter
3, the Mueller matrix roots decomposition is reviewed and a set of Mueller matrix
generators are discussed. The parameterization of depolarization into three families, each with three degrees of freedom is explained. Analysis of the matrix roots
parameters in terms of degree of polarization maps demonstrates that depolarizers
fall into two distinct classes: amplitude depolarization in one class, and phase and
diagonal depolarization in another class. It is shown that each depolarization family
and degree of freedom can be produced by averaging two non-depolarizing Mueller
17
matrix generators. This is extended to provide further insight on two sample measurements, which are analyzed using the matrix roots decomposition. Chapter 4
discusses additional properties of the Mueller matrix roots generators and parameters, along with a pupil aberration application of the matrix roots decomposition.
Appendix C, adapted from a conference proceedings paper, presents an application
of the matrix roots depolarization parameters for estimating the orientation of a
one-dimensionally textured object.
The last two chapters are on the topic of polarization computer generated
holography. In chapter 5, an interlaced polarization computer-generated hologram
(PCGH) is designed to produce specific irradiance and polarization states in the
image plane. The PCGH produces a tangentially polarized annular pattern with
correlated speckle, which is achieved by a novel application of a diffuser optimization
method. Alternating columns of orthogonal linear polarizations illuminate an interlaced PCGH, producing a ratio of polarization of 88% measured on a fabricated sample. In chapter 6, an etched calcite square-wave retarder is designed, fabricated, and
demonstrated as an illuminator for an interlaced polarization computer generated
hologram (PCGH). The calcite square-wave retarder enables alternating columns of
orthogonal linear polarizations to illuminate the interlaced PCGH. Together, these
components produce a speckled, tangentially polarized PCGH diffraction pattern
with a measured ratio of polarization of 84% and a degree of linear polarization of
0.81. An experimental alignment tolerance analysis is also reported.
18
CHAPTER 1
METHODS TO CALCULATE THE PROPERTIES OF MUELLER MATRICES
VIA MATRIX DECOMPOSITION
1.1 Motivation
In recent years, Mueller matrices have become increasingly easy to measure both
quickly and accurately. But in order to be useful, a Mueller matrix must be analyzed
and broken down in terms of its basic properties, which fall into three general families
- diattenuation, retardance, and depolarization.
Diattenuation and retardance each have three degrees of freedom, which are
all well-understood. Despite many efforts within the polarization community (Ossikovski et al., 2008), the nine depolarization degrees of freedom (Chipman, 2005b)
remain difficult to address. This work seeks to develop new ways to analyze and interpret depolarization, using an order-independent approach called a Mueller matrix
roots decomposition, which is described in detail in chapters 2, 3, and 4.
A profound understanding of depolarization within the Mueller matrix, from a
phenomenological perspective, opens the door to new ways of approaching optical
scattering analysis and optical design. This dissertation begins with a survey of the
relevant prior art in the remaining sections of this chapter. Chapter 2 discusses the
Mueller matrix decomposition algorithm, along with important mathematical considerations involved in using the matrix roots decomposition. Chapter 3 addresses
the matrix roots decomposition’s parameterization of Mueller matrices, the character of these parameters, their properties, and how they arise. Because chapters 2
19
and 3 are reformatted from manuscripts for publication, some relevant properties
(which were not included in the publications) are discussed in chapter 4. Chapter 4
also includes the application of this method to analysis of polarization aberrations
within the context of optical design.
1.2 Polar decomposition
In linear algebra, polar decomposition has its origin in the familiar polar form of a
complex number, where λ = λ0 eiγ , λ0 ≥ 0 and 0 ≤ γ < 2π. The theorem governing
polar decomposition states that any matrix A ∈ Fn×n can be represented in the
form
A = HU,
(1.1)
where Hij ≥ 0 and U is unitary. The matrix H is unique and Hermitian, and can
therefore be expressed as H = (AA∗ )1/2 .
Nondepolarizing Mueller matrices transform a completely polarized Stokes vector
to a completely polarized one and can be derived from Jones matrices (Lu and
Chipman, 1998). Within the field of polarimetry, polar decomposition is used as a
data reduction operation to factor an arbitrary non-depolarizing Mueller matrix M
into a retarder Mueller matrix MR and a diattenuator Mueller matrix MD1 or MD2
(Gil and Bernabeu, 1987),
M = MD1 MR = MR MD2 .
(1.2)
This is done to better undestand M. Here, MR is a unitary matrix and both MD1
and MD2 are Hermitian Mueller matrices. MD1 and MR do not commute in general,
20
but MD1 and MD2 are related as follows:
MD2 = MTR MD1 MR and MD1 = MR MD2 MTR .
(1.3)
This method was applied to the analysis of inhomogeneous nondepolarizing Mueller
matrices in (Lu and Chipman, 1994).
1.3 Lu-Chipman decomposition
The Lu-Chipman Mueller matrix decomposition (Lu and Chipman, 1996; Goldstein,
2003; Chipman, 2009) is a Mueller matrix data reduction method based on polar
decomposition. It is one of the most commonly used methods for analyzing measured
Mueller matrices, so a brief summary of this method is included here. The LuChipman method decomposes an arbitrary Mueller matrix into the products of
a diattenuator, a retarder, and a depolarizing matrix. The matrix properties of
diattenuation, retardance, and the depolarization index can then be retrieved from
these three factors:
M = M∆ MR MD .
(1.4)
The results of this decomposition allow one to define and calculate the diattenuation and retardance vectors for a given Mueller matrix. The three Lu-Chipman
decomposition factors are defined as follows.
Diattenuation is defined as
D≡
|Tmax − Tmin |
|Tmax + Tmin |
(1.5)
where Tmax and Tmin are the maximum and minimum transmittances over all possi-
21
ble incident polarization states, and D can have a value between 0 and 1. From the
definition in equation (1.5), we see that diattenuation changes the transmission (of
intensity) of an incident state of polarization. If we define the diattenuation vector
with the same structure as the Stokes parameters, we have a vector with 3 degrees
of freedom defined as


 DH

~ = D
D
 45

DC





(1.6)
where DH is horizontal diattenuation, D45 is 45◦ linear diattenuation, and DC is the
circular diattenuation. The magnitude of the diattenuation vector is
D=
q
2
2
DH
+ D45
+ DC2 ,
(1.7)
~
and {1, D/D}
identifies the Stokes vector of highest transmission.
A pure diattenuator has a symmetric Mueller matrix, which has the form

T

~
 1 D 
MD = Tu 

~
D mD
√
√
T
mD =
1 − D2 I + (1 − 1 − D2 )D̂D̂ ,
(1.8)
(1.9)
~
where I is a three by three identity matrix, D̂ is the (normalized) unit vector for D,
and Tu is the transmittance for unpolarized light.
We can set up a retardance vector and Mueller matrix in a similar fashion.
Retardance is defined as the difference in phase changes between the two eigenpolarizations,
δ = |δq − δr |,
(1.10)
22
and the retardance vector ~δ is defined as


 δH

~δ ≡  δ
 45

δC





(1.11)
in a similar fashion to the diattenuation vector. The magnitude of the retardance
vector is
q
2
2
+ δ45
+ δC2 ,
δ = δH
(1.12)
and {1, ~δ/δ} is the Stokes vector of the retarder’s fast-axis. The ideal retarder has
a Mueller matrix given by


~
 1 0T 
MR = 

~0 mR
(1.13)
(mR )ij = δij cos δ + αi αj (1 − cos δ) +
3
X
ijk ak sin δ
(1.14)
k=1
i, j = 1, 2, 3
where ijk is the Levi-Cività permutation symbol, ~0 is a three-element zero vector,
and δij is the Kronocker delta.
The depolarizing factor combines the depolarizing part of the Mueller matrix
with polarizance, and is written as

 1
M∆ ≡ 
P~∆

~0T


m∆
(1.15)
~ ∆ is the polarizance vector for a depolarizer,
where m∆ is a symmetric matrix. P
23
defined as

 PH

~∆ ≡ P
P
 45

PR


 m10



1

=
 m00  m20


m30



.


(1.16)
The Lu-Chipman polar decomposition is very useful for calculating the nondepolarizing properties of a Mueller matrix, but yields less insight into the the
depolarizing properties of a Mueller matrix.
1.4 Symmetric decomposition
The symmetric decomposition (Ossikovski, 2009) method decomposes a depolarizing
Mueller matrix into a sequence of five factors: a diagonal depolarizer M∆d in between
two retarder and diattenuator pairs (MR and MD ). The resulting decomposition is
written out as follows, where
M = MD2 MR2 M∆d MR1 MD1 .
(1.17)
One advantage of this decomposition method is that it is relatively easy to calculate,
since it is a singular value decomposition (SVD) based method. The first step is to
solve these eigenvalue equations for S1 and S2 :
MT GMG S1 = d20 S1
MGMT G S2 = d20 S2
(1.18)
(1.19)
24
where G = diag(1, −1, −1, −1) is the Minkowski metric matrix,


 1 
S1 = 
 , and
~
D1


 1 
S2 = 
.
~2
D
(1.20)
(1.21)
~ 1 and D
~ 2,
The second step is to construct the two diattenuator matrices from D
where

~T

 1 D 
MD = Tu 
 and
~ mD
D
√
√
mD = 1 − D2 I + (1 − 1 − D2 )D̂D̂T .
(1.22)
(1.23)
The third step is to calculate an intermediate matrix M0 , which has null polarizance
and diattenuation vectors:


~T
 d0 0 
−1
M0 = M−1
MM
=

.
D2
D1
~0 m0
(1.24)
The fourth step is to perform a SVD on the three by three block m0 of M0 to
construct the retarder matrices. This is possible because of M0 s zero polarizance
and diattenuation vectors:
m0 = mR2 m∆d mTR1 .
(1.25)
25
MR1 and MR2 are constructed from mR1 and mR2 as follows, where


~T
 1 0 
MR = 
.
~0 mR
(1.26)
This yields all the terms from equation 1.17.
The symmetry of this decomposition is an obvious advantage in addition to
the ease of calculation. The form is somewhat similar to that of a Mueller matrix
polarimeter with some kind of depolarizing sample measured, if we were to replace
the two diattenuators on each edge with polarizers. It is perhaps most useful in
analyzing a scattering process that undergoes retardance and diattenuation before
the scattering and again after the scattering. One disadvantage is that it can’t
explicitly quantify the overall diattenuation or retardance of a system.
1.5 Additive decompositions
According to the Cloude additive decomposition (Cloude, 1989; Ossikovski et al.,
2008), any depolarizing Mueller matrix M can be written as a weighted sum of four
non-depolarizing Mueller matrices,
M = λ1 M1 + λ2 M2 + λ3 M3 + λ4 M4 ,
(1.27)
where the λ0i s are the 4 eigenvalues of M’s covariance matrix C. The decomposition
is performed as follows. The first step is to calculate the covariance matrix C from
the elements mij of M and the Pauli spin matrices σi where
C=
4
X
i=1,j=1
mij (σi × σj∗ ).
(1.28)
26
The second step is to calculate the eigenvalues λi and normalized eigenvectors ei of
C, and calculate the covariance matrices Ci for the non-depolarizing Mueller matrix
components Mi . The covariance matrices Ci are calculated from the eigenvectors
ei ,
Ci = ei e†i .
(1.29)
The third and final step is to calculate each Mi from its Ci by inverting equation
1.28.
This decomposition is useful for visualizing the behaviour of the coherency matrix
eigenvalues, especially for mildly non-physical measured Mueller matrices. Physically, the decomposition corresponds to a system with multiple beams having different paths such as a focal plane with significant ghost images or a divided aperture
polarization state generator.
1.6 Mueller matrix root decomposition
While the polar decomposition methods discussed in the previous sections have
been proven to be viable methods for Mueller matrix analysis, they have two major
shortcomings. The first is the lack of order independence. A positive consequence
of order independence is that the properties should not change according to the
order of the decomposition. However, it is often useful to model a particular system
such that there is an intuitive sequence of optical elements that are not in reality
order independent. The more significant pitfall of the established decomposition
methods is that none of these methods fully address the nine depolarization degrees
of freedom.
Jones developed an order independent representation for Jones matrices (or nondepolarizing Mueller-Jones matrices) using what’s referred to as a Jones N -matrix
27
representation (Jones, 1948). He used the analogy of propagation through a dichroic
and birefringent anisotropic crystal, divided into very short lengths. He represented
the N -matrices differentially. We approximate this differential representation as
NJ ≈
p
n
MJ ,
(1.30)
where NJ is the Jones matrix for a weak polarization element and MJ is an arbitrary
Jones matrix.
Our method is based on the idea of using infinitesimal Mueller matrices (or
Mueller matrices expressing infinitesimal properties) to analyze and physically interpret the fundamental Mueller matrix properties. Taking the nth root of a Mueller
matrix simplifies its properties as n → ∞ and the Mueller matrix becomes a weak
element. The properties of the Mueller matrix near the identity matrix are discussed
in chapter 3.
The Jones N -matrix method is the building block for our Mueller matrix roots
method. Implementation of the Jones N -matrix method is discussed in section
1.8. Section 1.7 describes the Pauli spin matrices, the basis for the Jones N-Matrix
method.
1.7 Pauli spin matrices
The Pauli spin matrices are used in quantum mechanics in order to describe angular
momentum. They are written as








 1 0 
 1 0 
 0 1 
 0 −i 
σ0 = 
 , σ1 = 
 , σ2 = 
 , σ3 = 
.
0 1
0 −1
1 0
i 0
(1.31)
28
The Pauli spin matrices are a useful basis for Jones matrices. Within the Jones
calculus, σ1 represents a horizontal retarder, σ2 represents a 45o linear retarder,
σ3 represents a circular half-wave retarder, and σ0 (the identity matrix) represents
empty space. Any Jones matrix J can be written in terms of the sum
J=
3
X
ck σk .
(1.32)
k=0
Here, the constants c0 , c1 , c2 , and c3 are complex Pauli coefficients for the Jones
matrix. If the Jones matrix J is written in terms of the Pauli basis, the coefficients
ck can be written in short form as
1
ck = T r (J • σi ) ,
2
(1.33)
where T r is the sum of the diagonal elements of the matrix.
1.8 Jones N-matrices and the Mueller matrix roots decomposition
Jones’ N-matrix decomposition simplifies the properties of Jones matrices by dividing them into infinitesimaly small segments which approach the identity matrix.
The N -matrices can then be expressed as a sum of matrices, where each matrix’s
real and imaginary part in the sum corresponds to one fundamental property. For
any non-depolarizing physical Mueller-Jones matrix which can be converted to a
Jones matrix, this approach provides accurate and useful analytical results for experimentally or computationally generated Mueller matrices.
Jones N-matrices are very useful for isolating the different non-depolarizing polarization effects. For the weak polarization element NJ from equation 1.30, the
polarization properties can be determined from the Pauli basis, where first order
29
approximations lead to the relationship (Chipman, 2009)
Dx + iδx
D45 + iδ45
DR + iδR
NJ ≈ ρ0 exp iδ0 σ0 +
σ1 +
σ2 +
σ3 .
2
2
2
(1.34)
This expression is correct to first order in D and δ. In this expression, DH and
δH are the horizontal and vertical diattenuation and retardance, D45 and δ45 are
the 135o and 45o diattenuation and retardance, and DR and δR are the right and
left circular diattenuation and retardance. ρ0 is the amplitude, and δ0 is the global
phase. Equation 1.34 allows for order-independent analysis of Jones matrices, or
non-depolarizing Mueller matrices that can be converted to Jones matrices. This is
particularly useful for non-homogeneous Jones matrices.
The concept of N -matrices can be readily extended to Mueller matrices by dividing a Mueller matrix M into p infinitesimal slices by calculating its root:
N=
√
p
M.
(1.35)
This extension arises naturally as follows. A train of n optical elements is described
by multiplying their Mueller matrices from right to left, starting with the first element’s Mueller matrix M1 and ending with the last element’s Mueller matrix Mn ,
M = Mn Mn−1 ...M2 M1 ,
(1.36)
for n = 1, 2, ....p. If each component on the right hand side of equation 1.36 is
identical, it becomes
M = Mn Mn−1 ...M2 M1 = (Mn )p .
(1.37)
30
M
p
M
( M)
p
p
Figure 1.1: Taking the root of a uniform Mueller matrix is analagous to slicing it
into very thin identical pieces.
Since compounding identical polarization elements in the Mueller matrix formalism
is akin to raising its Mueller matrix to a power, taking a small slice of a polarization
element is akin to taking its root, as illustrated in Fig. 1.1.
The pth root of a large class of Mueller matrices approaches the identity matrix
I in the limit as p becomes very large,
lim
p→∞
√
p
M = I.
(1.38)
Chapter 2 addresses the optimal choice of p; values on the order of p ∼
= 105 work well.
In this study, the class of Mueller matrices that obey equation 1.38 is referred to as
uniform Mueller matrices. Our method is only applicable to this subset of Mueller
31
matrices. This dissertation does not address which subset of Mueller matrices is
uniform, though Mueller matrices with negative real eigenvalues are not uniform
(Higham, 2008).
The polarization properties of
√
p
M for large p simplify as its properties become
infinitesimal. Properties of Mueller matrices with infinitesimally small properties are
considerably less complex than a Mueller matrix with non-infinitesimal properties.
Performing a Taylor series expansion on the Mueller matrix forms for elliptical
diattenuators and retarders provides insight into the symmetry and structure of
weak polarization elements.
The first order terms in the Taylor series expansion of the general equation for an
elliptical retarder (Chipman, 2009) yield the following expression for a weak retarder
in terms of the three-dimensional retardance vector {δH , δ45 , δR } (Lu and Chipman,
1996):


1
0
0
0




0 1
δR −δ45 


2
2
ER(δH , δ45 , δR ) ∼
, δ45
, δR2 ).
=
 + O(δH
0 −δ

1
δH 

R


0 δ45 −δH
1
(1.39)
Similarly, the first order Taylor expansion terms for an elliptical diattenuator with
diattenuation vector dH , d45 , dR and average transmision Tavg are


1
d
d
d
H
45
R




 dH 1
0
0


ED(dH , d45 , dR ) ∼
= Tavg 
 + Tavg O(d2H , d245 , d2R ).
d
1
0
 45 0



dR 0
0
1
(1.40)
Combining the weak diattenuator and retarder yields an order independent equation
32
for a weak non-depolarizing Mueller matrix,


1
d
d
d
H
45
R




 dH
1
δR −δ45 
ER · ED + ED · ER


= Tavg 
.
d

2
−δ
1
δ
 45
R
H 


dR δ45 −δH
1
(1.41)
The matrix is correct to first order in d and δ. The symmetry of the weak nondepolarizing matrices is striking - weak diattenuators are symmetric in the top row
and first column, weak retarders are anti-symmetric between the off diagional lower
and upper right three by three elements. Deviation from these symmetries indicates
the presence of depolarization.
This symmetry for infinitesimal Mueller matrix properties motivates what we
refer to as Mueller matrix generators. Within the field of polarization, generators
have been discussed in relating the Jones and Mueller matrix formalism to the concept of the Lie group (Takenaka, 1973). Generators have also been used to define
possible expressions of an arbitrary nonsingular Mueller matrix, but not necessarily in 16-dimensional space (Devlaminck and Terrier, 2008). Here, Mueller matrix
generators are used to express infinitesimal properties of the Mueller matrix (Chipman, 2009). Our generators provide an order-independent representation of uniform
Mueller matrices to provide additional insights into the polarization properties of
Mueller matrices. The Mueller matrix generators and their properties are discussed
in greater detail in chapters 3 and 4.
33
CHAPTER 2
THE MUELLER MATRIX ROOTS DECOMPOSITION ALGORITHM AND
COMPUTATIONAL CONSIDERATIONS
This chapter is reformatted from a manuscript which has been submitted to Optics
Express.
2.1 Introduction
Polarization elements and their associated Mueller matrices are typically decomposed and analyzed in terms of the polarization properties of diattenuation, retardance, and depolarization. For Mueller matrices with a mixture of all three, the
properties are distributed among the matrix elements in a complex manner. Lu
and Chipman described a Mueller matrix data reduction method based on polar
decomposition (Lu and Chipman, 1996). It decomposes the Mueller matrix into
an order-dependent product of a depolarizer, a retarder, and a diattenuator. Ossikovski’s symmetric decomposition method decomposes a depolarizing Mueller matrix into a sequence of five factors: a diagonal depolarizer between two retarder and
diattenuator pairs (Ossikovski, 2009). The diattenuation, retardance, and depolarization parameters calculated by these methods depend on the arbitrary order of
the decomposition. The Cloude additive decomposition (Cloude, 1989) separates a
Mueller matrix into the sum of four non-depolarizing Mueller matrices which are
scaled by eigenvalues of the Mueller matrix covariance matrix. Since this is an additive decomposition, its terms are order-independent. None of these methods clearly
34
M
p
M
( M)
p
p
Figure 2.1: Taking the root of a uniform Mueller matrix is analogous to slicing it
into very thin identical pieces.
elucidate the nine degrees of freedom associated with depolarization.
Recently, a Mueller matrix roots decomposition was introduced by Chipman in
(Chipman, 2009) to provide an order-independent description of polarization properties and provide a clear analysis of the nine depolarization degrees of freedom. The
Mueller matrix roots decomposition extends the concept of the Jones N -matrices
(Jones, 1948) to Mueller matrices by dividing the Mueller matrix M into p infinitesimal slices:
N=
√
p
M,
(2.1)
as illustrated in Fig. 2.1, where N is the Mueller matrix for one small slice. When p
√
becomes very large, the polarization properties of p M separate as the diattenuation,
retardance, and depolarization become very small near the identity matrix. In the
limit as p approaches infinity, the principal pth root of a large class of Mueller
35
matrices approaches the identity matrix I,
lim
p→∞
√
p
M = I.
(2.2)
In this study, the class of Mueller matrices that obey equation 3.2 are called uniform
Mueller matrices. The Mueller matrix roots decomposition from (Chipman, 2009) is
applicable to this subset of Mueller matrices, with the exception of the non-uniform
special cases highlighted in section 2.4.
The calculation of the principal pth root of square matrices is an extensively
studied subject (Higham, 2008; Bini et al., 2005; Guo, 2010), and numerical accuracy
and noise are well understood within the field of numerical computing (Skeel and
Keiper, 1993). In this study, these concepts are applied to the calculation of the
Mueller matrix roots for the purpose of Mueller matrix decomposition. This paper
reviews and updates the Mueller matrix roots decomposition from (Chipman, 2009),
addresses several computational issues, and highlights several common non-uniform
special cases that arise. Considerations for these computational issues and special
cases are implemented in a matrix roots decomposition algorithm, which is applied
in a statistical analysis of a large quantity of randomly generated physical Mueller
matrices.
2.2 Matrix roots decomposition
The goal of the Mueller matrix roots decomposition is to calculate the magnitudes of
sixteen distinct properties of the Mueller matrix, including the nine depolarization
degrees of freedom, for the set of uniform Mueller matrices.
To calculate the matrix roots decomposition of a Mueller matrix M, a Mueller
36
matrix N with infinitesimal polarization properties is first calculated from the pth
principal root of M, where p is some large integer (typically 105 ),
N=
√
p
M.
(2.3)
The appropriate choice of p is discussed in detail in section 2.5.
The infinitesimal polarization parameters d0 through d15 are defined from the
symmetric and antisymmetric parts of N:


d1 + d7
d2 + d8
d3 + d9 
 1


d1 − d7 1 − f13
d6 + d12 −d5 + d11 


N = d0 
,
d − d −d + d

1
−
f
d
+
d
 2
8
6
12
14
4
10 


d3 − d9 d5 + d11 −d4 + d10 1 − f15
(2.4)
where d13 , d14 , and d15 are solved from the first-order generator products in terms
of the parameters f13 , f14 , and f15 :
f14 − f13
2
2
= √ (−2f15 + f14 + f13 )
3
1
= √ (f15 + f14 + f13 ).
6
d13 =
(2.5)
d14
(2.6)
d15
(2.7)
The infinitesimal polarization parameters d0 through d15 are rescaled by p to produce
the matrix roots parameters D0 through D15 :
Dn = p ∗ dn , n=0,1,2,...,15.
D0 through D15 parameterize the sixteen degrees of freedom of M.
(2.8)
37
There are three matrix roots parameters for diattenuation, D1 , D2 , D3 , and
three matrix roots parameters for retardance, D4 , D5 , D6 . The three degrees of
freedom for each property correspond to the axes in the Stokes/Mueller formalism
(horizontal/vertical, 45o /135o , right/left circular).
Nine more parameters, D7 through D15 , describe depolarizing effects - one for
each of the nine depolarizing degrees of freedom described in (Chipman, 2009).
There are three families of depolarizing parameters, each similarly divided into horizontal/vertical, 45o /135o , and right/left circular components. The first order terms
for amplitude depolarization, D7 , D8 , and D9 , share the same matrix elements as
the parts of the Mueller matrix associated to first order with diattenuation, on the
horizontal/vertical, 45o /135o , and right/left circular axes (Chipman, 2009). They
are named amplitude depolarization because they depolarize and affect the flux of an
incident Stokes vector. The first order terms for phase depolarization, D10 , D11 , and
D12 , correspond with the parts of the Mueller matrix associated to first order with
retardance on the horizontal/vertical, 45o /135o , and right/left circular axes. They
do not affect the flux of an incident Stokes vector. D13 , D14 , and D15 are named
diagonal depolarization because they lie on the matrix diagonal. D15 expresses the
overall isotropic depolarizing power, since D15 reduces the degree of polarization of
all incident Stokes vectors equally, independent of the Stokes vector’s location on
the Poincare sphere. D13 expresses the relative strength between the diagonal depolarization on the two linear axes (horizontal/vertical and 45o /135o ). D14 expresses
the relative strength between linear and circular diagonal depolarization. The diagonal depolarization parameters have been modified from (Chipman, 2009) so that
only the diagonal depolarization parameter D15 changes the depolarization index
(Chipman, 2005a), while D0 through D14 do not. Table 2.1 lists the parameters D1
38
through D15 and categorizes them into their corresponding families and axes.
Table 2.1: The sixteen polarization properties of the Mueller matrix given by the
Mueller matrix roots decomposition.
Property
Diattenuation
Retardance
Amplitude Depolarization
Phase Depolarization
Diagonal Depolarization
Horizontal/Vertical
45o /135o
D1
D2
D4
D5
D7
D8
D10
D11
Relative Linear
Relative Linear or Circular
D13
D14
Right/Left Circular
D3
D6
D9
D12
Isotropic
D15
2.3 pth Root
This section discusses the definition of the principal pth matrix root along with
relevant examples and common methods of calculating the pth matrix root.
Matrices have multiple roots. For a nonsingular matrix A ∈ Cn×n (in complex
space) with s distinct eigenvalues, there are precisely ps pth roots (Higham, 2008).
So long as the matrix A ∈ Cn×n has no negative, real eigenvalues, there is a unique
pth root of A whose eigenvalues’ arguments lie between −π/p and π/p, and that
unique root is defined as the principal root of A (Higham, 2008). If A is real, then
its principal root A1/p is real.
Singular matrices (such as polarizer Mueller matrices) do not have principal matrix roots. Nonetheless, matrix roots of singular Mueller matrices near the identity
matrix can still be found. Methods for calculating roots of singular Mueller matrices
(such as linear polarizers) are discussed in section 2.4.1. Methods for calculating
roots of non-depolarizing and depolarizing Mueller matrices with π retardance are
discussed in section 2.4.2.
39
2.3.1 Principal matrix root algorithms
This section provides a brief summary of common methods for calculating principal
matrix roots, including the Schur method, Newton’s method, and the Schur-Newton
algorithm. This provides a starting point for the reader who is interested in applying
these methods to the calculation of Mueller matrix roots.
Schur methods form a Schur decomposition of A and compute a pth root of the
resulting upper triangular factor using various (stable) recursive formulae (Higham,
1997). Newton’s method calculates the pth root of A using an iterative approach
(Higham, 2008). The Newton method is largely considered to have poor convergence
and stability properties (Smith, 2002), and the Schur method from (Smith, 2002) is
the numerically stable benchmark against which other methods are often compared
(Bini et al., 2005). The Schur-Newton algorithm applies iterative computations to
the upper triangular matrices from the Schur decomposition (Higham, 2008). Many
of these algorithms are available in the MATLAB Matrix Computation Toolbox
(Bini et al., 2005).
Mathematica has built-in routines that diagonalize a matrix to easily calculate
its root, so long as the matrix is diagonalizable. If the matrix is not diagonalizable,
its algorithm performs a singular value decomposition, which factors the matrix A
into the product of three matrices,
A = Zdiag(λi )Z−1 ,
(2.9)
where λi are the eigenvalues of A, the columns of Z are its eigenvectors, and diag(λi )
is a diagonal matrix with its ith diagonal element equal to λi . Then the matrix root
40
of A is calculated as
A1/p = Zdiag(λi )1/p Z−1 .
(2.10)
This method often (but not always) yields the principal root of a Mueller matrix,
so long as its principal root exists. However if A is real and has some complex
eigenvalues, then the computed A1/p , which should be real, may acquire a tiny
imaginary part due to computational rounding errors. This imaginary part should
be discarded. However, numerical instability can produce a large spurious imaginary part, so diagonalization-based computations of matrices with any imaginary
eigenvalues should be treated with care (Higham, 2008).
2.4 Special cases
2.4.1 Polarizers
The Mueller matrix for a polarizer is singular, regardless of ellipticity or orientation.
Therefore it is not uniform, and the procedure described in section 2.2 will not yield
meaningful matrix roots parameters when applied to a polarizer Mueller matrix.
However, the perturbation treatment proposed in this section allows for this special
case to be analyzed using the Mueller matrix roots decomposition.
Because an ideal homogeneous polarizer (Lu and Chipman, 1994) is always idempotent, the matrix roots decomposition will break down at the point where the pth
root is calculated. For example, the formula for a homogeneous linear polarizer as
41
a function of orientation θ is

cos 2θ
sin 2θ 0
 1


1
2

cos
2θ
cos
2θ
sin
4θ
0
1


2
LP(θ) = 
.
2  sin 2θ 1 sin 4θ sin2 2θ 0


2


0
0
0
0
(2.11)
LP(θ)LP(θ) = LP(θ),
(2.12)
LP(θ)1/p = LP(θ).
(2.13)

Note that
and thus
The linear polarizer Mueller matrix is its own root to all orders, and the roots do
not approach the identity matrix.
A homogeneous polarizer has two orthogonal, physical eigenpolarizations with
eigenvalues (Tmax , 0), where Tmax is the maximum transmission, and the minimum
transmission Tmin is 0. The polarizer can be perturbed to a nearby uniform Mueller
matrix by adjusting the maximum and minimum transmission by a small number . The perturbed matrix is now uniform and it is a diattenuator with the
same orthogonal eigenpolarizations, but the eigenvalues associated with its physical
eigenpolarizations are (Tmax − , ).
This is accomplished by calculating the latitude η and longitude θ of the state
of maximum transmission on the Poincare sphere from the polarizer’s three dimensional diattenuation vector {dH , d45 , dR } (Lu and Chipman, 1996; Chipman, 2009),
42
where
1
dH
arctan
, and
2
d45
dR
η = arcsin p 2
.
dH + d245 + d2R
θ=
(2.14)
(2.15)
η and θ are then plugged in to the general formula for an elliptical diattenuator
(Goldstein, 2003; Chipman, 2009). For an ideal polarizer, Tmax = 1 and Tmin = 0.
Perturbing Tmax and Tmin by some small number yields a nonsingular, uniform
Mueller matrix. Values of = 10−7 work well with Mathematica’s default numerical
accuracy. The pth root of this perturbed polarizer approaches the identity matrix
for large p, and the Mueller matrix roots decomposition algorithm can proceed as
in the general case.
2.4.2 Half wave retarders
Mueller matrix roots for retarders are easily calculated and understood. For two
linear retarders with the same fast-axis orientation, retardance is additive,
δ = δ1 + δ2 .
(2.16)
Thus the square root of a linear retarder is a retarder with half the retardance
and the same fast-axis orientation. This is equivalent to cutting a wave plate in
half. Similarly, the pth root of an ideal homogeneous linear retarder LR(δ, θ) with
retardance δ at orientation θ is
LR(δ, θ)1/p = LR(δ/p, θ).
(2.17)
43
A retarder with a half-wave of retardance has negative, real eigenvalues, and therefore no principal root, so the half-wave retarder must be treated as a special case.
For example, the half-wave linear retarder oriented at 0o ,

1

0

o
LR(π, 0 ) = 
0


0

0

1 0
0

,
0 −1 0 


0 0 −1
0
0
(2.18)
has negative real eigenvalues (λ = {−1, −1, 1, 1}), so the half wave retarder has no
principal pth root. For any value of p, two eigenvalues of HWR1/p are (−1)p with
arguments of −π/p and π/p, which lie on the edge of the principal root segment
defined in section 2.3. While the desired solution in this case is not a principal root,
it is the solution which approaches the identity matrix. The half wave retarder’s
uniform pth root can be calculated analogously to the polarizer case, by means of a
small perturbation.
Angle θ and latitude η can be calculated from the three-dimensional retardance
vector {δH , δ45 , δR } (Lu and Chipman, 1996; Chipman, 2009) as follows,
δH
1
arctan
, and
2
δ45
δR
η = arcsin p 2
.
2
δH + δ45
+ δR2
θ=
(2.19)
(2.20)
The general form for an elliptical retarder with retardance δ, orientation θ, and
latitude η can be written in terms of three linear retarders as
ER(η, θ, δ) = LR(η, θ + π/4)LR(δ, θ)LR(η, θ − π/4),
(2.21)
44
where the general form for the linear retarder LR(δ, θ) with retardance δ and an
orientation of θ is (Chipman, 2009)


0
0
0

1


2
2
δ
2
0 cos 2θ + cos δ sin 2θ
− sin δ sin 2θ
sin 2 sin 4θ


.
LR(δ, θ) = 
(2.22)

0
2 δ
2
2
sin 2 sin 4θ
cos δ cos 2θ + sin 2θ cos 2θ sin δ 



0
sin δ sin 2θ
− cos 2θ sin δ
cos δ
After θ and η are calculated from the retardance vector, the half-wave retarder is
perturbed by some small number ( ≈ 10−7 ) to a nearby elliptical retarder ER0 of
retardance (π − ) by using equation 2.21:
ER0 (η, θ, δ = π − ) = LR(η, θ + π/4)LR(π − , θ)LR(η, θ − π/4).
(2.23)
This is a unitary transformation - therefore any higher order root of a half wave
retarder can be calculated as follows:
1
1
ER(η, θ, δ = π) p = LR(η, θ + π/4)LR((π − ), θ) p LR(η, θ − π/4).
(2.24)
After following this procedure, the matrix roots of the perturbed half-wave retarders
can be calculated without issue.
It is also possible to calculate the root in equation 2.24 without perturbation of
the retardance,
1
π
HWR(η, θ) p = LR(, θ + π/4)LR( , θ)LR(η, θ − π/4),
p
(2.25)
but this method cannot be extended to a depolarizing Mueller matrix with a half-
45
wave of retardance. This is discussed further in the following section.
2.4.3 Depolarizing non-uniform Mueller matrices
When half-wave retarders or polarizers are combined with other polarization properties, the resulting Mueller matrix is also non-uniform, and therefore cannot be
analyzed with the standard Mueller matrix roots decomposition. Using the procedure discussed in this section, they can be perturbed to nearby uniform Mueller
matrices.
A depolarizing Mueller matrix that also has a half-wave of retardance has negative, real eigenvalues and is therefore non-uniform. For example, the product of a
half wave retarder oriented at 0o and a partial diagonal depolarizer with positive,
real a, b, and c

1

0


0


0

0  1


1 0
0
 0


0 −1 0 
 0

0
0 0 −1
0
0
 
0 0 0  1
 

a 0 0
 0
=

0 b 0
 0
 
0
0 0 c

0

a 0
0

,
0 −b 0 


0 0 −c
0
0
(2.26)
has eigenvalues λ = {1, a, −b, −c}, and therefore no principal root.
To perturb a half-wave retarder mixed with depolarization and/or diattenuation
properties to a nearby uniform Mueller matrix, the Lu-Chipman decomposition (Lu
and Chipman, 1996) is first calculated as an intermediate step,
M = M∆ MR MD .
(2.27)
M∆ is a depolarizing Mueller matrix, MR is a pure retarder, and MD is a pure
46
diattenuator. The half-wave retarder found in the MR term is perturbed to the
nearby retarder M0R with (π − ) retardance as shown in equation 2.24. Then the
terms are recombined to form M0 , where
M0 = M∆ M0R MD .
(2.28)
The uniform pth root of the perturbed depolarizing half-wave retarder from equation
2.28 can be calculated from M0 , since its eigenvalues are no longer negative, and its
Mueller matrix roots decomposition parameters can be found.
Because the product of a polarizer and any other Mueller matrix is also singular,
a polarizer multiplied by a depolarizer or retarder (or even a diattenuator) in any
order is also singular. These Mueller matrices can be identified by testing for a
diattenuation vector D of magnitude D = 1 and a depolarization index less than 1.
To perturb a polarizer with a mixture of other polarization properties to a nearby
uniform Mueller matrix, the Lu-Chipman decomposition (Lu and Chipman, 1996)
is first calculated as an intermediate step,
M = M∆ MR MD .
(2.29)
The polarizer’s maximum transmission can be perturbed by to the nearby uniform
diattenuator according to the procedure in section 2.4.1. The modified M0D is substituted into equation 2.29 in place of MD , and the three matrices are recombined
to form a modified, non-singular matrix M0 :
M0 = M∆ MR M0D .
(2.30)
47
Then the pth root of M0 can be calculated, and its Mueller matrix roots decomposition parameters can be found.
2.5 Numerical accuracy and root order
2.5.1 Choice of p
The choice of root order p is an important consideration when calculating the matrix
roots parameters. p must be large enough so that the Mueller matrix elements
become sufficiently small for the polarization properties to separate. However, the
magnitude of p should also be as small as possible so as to minimize unnecessary
loss of numerical precision from the calculation.
2.5.1.1 Accuracy
Because the Mueller matrix roots of retarders are straightforward and wellunderstood, they provide a good reference from which to study the convergence
of Equation 2.8. The pth root of a retarder Mueller matrix with retardance δ results
in a pth principal matrix root with retardance δ/p. Therefore a Mueller matrix
of an ideal elliptical retarder with retarder vector {δH , δ45 , δR } should have matrix
roots parameters of D4 = δH , D5 = δ45 , and D6 = δR . In order to evaluate the
correct choice of p in consideration of this criteria, the matrix roots retardance
parameters (D4 through D6 ) were calculated (using Mathematica’s default doubleprecision machine arithmetic) for a pure elliptical retarder with retardance parameters {δH , δ45 , δR } = {0.294, 0.302, 0.997} for different values of root order p. The
relative error ∆x was then calculated according to the following expression,
p
(δH − D4 )2 + (δ45 − D5 )2 + (δR − D6 )2
∆x =
.
δ
(2.31)
48
Convergence HPure RetarderL
Relative Error HLogL
-12
-14
-16
-18
-20
-22
-24
-26
0
2
4
6
8
10
12
Log10 HpL
Figure 2.2: The error of the root calculation vs. root order for a simple retarder
converges to a minimum relative error just beyond the 105 th root.
Fig. 2.2 shows the relative error between the input retardance vector {δH , δ45 , δR }
and the corresponding matrix roots retardance vector D4 through D6 for different
choices of p for the pth root. In Fig. 2.2, the relative retardance error converges
to a minimum value just beyond the 105 th root. For large p, the relative error increases, due to numerical rounding and the loss of precision associated with machine
arithmetic.
An equally important factor to consider is the convergence of the Mueller matrix
root parameter values. When the choice of p is sufficiently large, the D-parameters
converge to values that are independent of p. In order to demonstrate this convergence, a more complex Mueller matrix was generated by multiplying an elliptical
retarder of randomly generated input retardance vector by a partial depolarizer PD
of the form

1

0

PD(a, b, c) = 
0


0

0 0 0

a 0 0

.
0 b 0


0 0 c
(2.32)
49
Diagonal Depolarization Vector Norm
Convergence HDepolarizing RetarderL
3.10
3.05
3.00
2.95
2.90
2.85
0
2
4
6
8
10
12
Log10 HpL
Figure 2.3: The norm of the depolarizing retarder’s diagonal depolarization parameters (D13 , D14 , and D15 ) converges to a steady value of approximately 3.03 beyond
the 104 th root.
The randomly generated elliptical retarder with retardance vector {δH , δ45 , δR } =
{0.210, 0.033, 1.003} was multiplied by PD(0.1, 0.2, 0.3), generating the Mueller matrix


0.494 0.010 0.003 0.419


0.104 0.002 0.0 0.029


ED(0.210, 0.033, 1.003) · PD(0.1, 0.2, 0.3) = 
.
0.016 0.0

0.0
0.005




0.495 0.010 0.003 0.142
(2.33)
Its Mueller matrix root decomposition parameters were then calculated for varying
values of p. Fig. 2.4 shows the convergence of the norm of the matrix roots retardance parameters (D4 , D5 , and D6 ). The norm of the matrix roots retardance
parameters converges to a steady value of approximately 1.07 following the 104 th
root. The convergence of the norm of the matrix roots diagonal depolarization parameters (D13 , D14 , and D15 ) behaves in a strikingly similar manner, as shown in
Fig. 2.3.
50
Convergence HDepolarizing RetarderL
Retardance Vector Norm
1.10
1.05
1.00
0.95
0.90
0
2
4
6
8
10
12
Log10 HpL
Figure 2.4: The norm of the depolarizing retarder’s matrix roots retardance parameters (D4 , D5 , and D6 ) converges to a steady value of approximately 1.07 beyond
the 104 th root.
2.5.1.2 Numerical precision
This section addresses issues related to the numerical precision of the matrix root
calculation, using Mathematica’s built-in commands. Computational software programs such as Mathematica and Matlab use floating-point numbers with machine
precision by default (Skeel and Keiper, 1993). The value of machine precision that
produced the results included here is 15.96 digits, which corresponds to a 53 digit
binary double precision number with a mantissa (Skeel and Keiper, 1993). Rounding errors have been shown to increase linearly so long as small-scale oscillations are
overlooked (Skeel and Keiper, 1993).
Fig. 2.5 shows the relative error (on a log scale) of the matrix root calculation
on a randomly generated Mueller matrix MR as a function of root order p = 10k ,
for k between 1 and 14. This relative error was calculated as follows,
∆root
p
√
kMR − p MR k
.
=
2kMR k
(2.34)
The relative error increases linearly with each operation, so it is recommended to
51
Matrix Root Computational Error
Relative Error HLogL
0
-5
-10
-15
-20
-25
-30
2
4
6
8
10
12
14
Log10 p
Figure 2.5: Relative error for Mathematica’s matrix root calculation.
balance this with the optimization of the convergence properties discussed in section
2.5.1.1. The linear increase in error with each numerical operation is independent
of the choice of Mueller matrix.
Based on the behaviors documented in Fig. 2.2 through 2.5, a good choice for p
is on the order of 105 . This choice balances the relative error generated from the root
calculation while achieving convergence of its matrix root polarization properties.
Relative error of 5 · 10−12 is achieved with p = 105 for the randomly generated
Mueller matrix MR .
2.6 Algorithm and flow chart
An algorithm to calculate the matrix roots decomposition incorporating the results
from Sections 2.4 and 2.5 follows.
Fig. 2.6 shows a flow chart for the algorithm, beginning with an input Mueller
matrix. If the Mueller matrix is not physical, the decomposition parameters will
not be meaningful. Since many measured Mueller matrices are slightly nonphysical
(Goldstein, 2003), if the matrix is nonphysical, it is recommended to use a forcephysical routine (Barakat, 1987; Twietmeyer et al., 2008) in Step 2. Step 3 tests
52
step 1: input Mueller matrix step 2: force physical step 8: mul:ply d0 through d15 by p=105 to obtain D0 through D15 step 3: check for special cases* step 4: check for yes nega:ve eigenvalues abort step 7: solve for d0 through d15 using equa:on 2 no step 5: calculate 105th matrix root step 6: drop spurious imaginary parts *Address linear polarizers and half-­‐wave retarders according to sec:on 4 and proceed to step 4. Figure 2.6: Matrix Roots algorithm flow chart.
53
for the special cases discussed in section 2.4 - Mueller matrices with a half wave of
retardance and singular Mueller matrices. If the Mueller matrix tests positive it is
perturbed according to the procedure in section 2.4, and the algorithm resumes. Step
4 tests for negative real eigenvalues, and if present, the decomposition is aborted.
Step 5 calculates the 105 th matrix root of the Mueller matrix. (This choice of
p = 105 was discussed in section 2.5.) The matrix root can be calculated using a
built-in matrix root algorithm (such as Mathematica’s matrix power function), or
with any principal matrix root algorithm such as those discussed in section 2.3.1.
The principal root algorithms discussed in section 2.3.1 can fail to converge to a
solution, particularly for such a high root order. As discussed in section 2.3.1,
for any matrix with imaginary eigenvalues, computational rounding errors can lead
to spurious imaginary parts. These may be discarded so long as they are small
enough not to affect the accuracy of the calculation. Step 6 discards these spurious
imaginary parts. In step 7, the parameters d0 through d15 are determined from
the principal root according to equation 2.4. Step 8 rescales the infinitesimal roots
parameters by p = 105 , resulting in the desired matrix roots parameters D0 through
D15 .
2.7 Statistical algorithm implementation
The matrix roots decomposition algorithm from section 2.6 was implemented on
76,336 randomly generated, non-singular, physical Mueller matrices with no real
negative eigenvalues. By definition, all of these Mueller matrices have principal
roots. The statistical analysis resulting from this implementation yields information
about the range in values of the depolarizing matrix roots parameters for a large
quantity of random Mueller matrices.
54
Amplitude Depolarization Matrix Root Parameters
8000
6000
D7
D8
D9
4000
2000
0
-2
-1
0
1
2
Parameter Value
Figure 2.7: A histogram of the matrix root amplitude depolarization values for
76,336 randomly generated physical Mueller matrices.
The Mueller matrices are generated by a brute-force numerical method. A fourby-four matrix is generated by setting the m0,0 element to a value of one, and the
other fifteen matrix elements are uniformly randomly distributed between negative one and one. If the matrix is nonphysical or has negative real eigenvalues, it
is discarded. The Mueller matrix roots parameters were calculated for all of the
remaining matrices. 76,336 physical, non-singular Mueller matrices with no real
negative eigenvalues were found from the set of 109 randomly generated matrices.
Fig. 2.7 through 2.9 show the distribution of the depolarizing matrix roots
parameters. The distributions for the amplitude depolarization parameters (D7 , D8 ,
and D9 ) are entirely overlapping and largely lie within the range of -1 to 1, with a fullwidth half maximum (FWHM) of 0.6. The distributions for the phase depolarization
parameters (D10 , D11 , and D12 ) also overlap entirely and range mostly between 2 and 2, with a FWHM of 0.8. The distributions of the diagonal depolarization
55
Phase Depolarization Matrix Root Parameters
10 000
8000
6000
D10
D11
D12
4000
2000
0
-4
0
-2
2
4
Parameter Value
Figure 2.8: A histogram of the matrix root phase depolarization values for 76,336
randomly generated physical Mueller matrices.
Diagonal Depolarization Matrix Root Parameters
10 000
8000
6000
D13
D14
D15
4000
2000
0
-4
-2
0
2
4
6
Parameter Value
Figure 2.9: A histogram of the matrix root diagonal depolarization values for 76,336
randomly generated physical Mueller matrices.
56
parameters D13 and D14 overlap and have a very similar distribution to the phase
depolarization parameters, with a range primarily between -2 and 2 and FWHM of
0.8. D15 has a distinct distribution. It has a hard limit at zero, as it cannot have a
negative value. Its distribution cuts off near 4, with a FWHM of 1.1. Out of all the
depolarizing matrix roots parameters, it has the only non-symmetric distribution.
2.8 Conclusion
Computational issues involved in applying the Mueller matrix roots decomposition
have been addressed. The definition of the pth matrix root is reviewed, along with a
brief discussion of the most common methods of calculating the pth principal matrix
root. Our study indicates that the decomposition is optimized around p = 105 in
consideration of numerical accuracy and noise as well as parameter convergence.
Practical values for the roots of singular Mueller matrices can be obtained through
perturbing them to nearby diattenuating matrices. Similarly, Mueller matrices with
a half wave of retardance can be evaluated by perturbing their retardance from a
half wave, without changing the retarder form. An algorithm is provided which
incorporates the computational considerations involved in calculating the matrix
roots decomposition. Finally, the algorithm is implemented to perform a statistical
analysis on a large set of randomly generated Mueller matrices in order to yield
insight on the typical ranges of the matrix roots parameters for physical Mueller
matrices.
57
CHAPTER 3
INTERPRETATION OF MUELLER MATRIX ROOTS DECOMPOSITION
PARAMETERS
This chapter is reformatted from a manuscript which has been submitted to Applied
Optics.
3.1 Introduction
Polarization elements and their associated Mueller matrices are typically decomposed and analyzed in terms of the polarization properties of diattenuation, retardance, and depolarization. For Mueller matrices with a mixture of all three, the
properties are distributed among the matrix elements in a complex manner. Lu and
Chipman described a Mueller matrix data reduction method based on polar decomposition (Lu and Chipman, 1996). It decomposes the Mueller matrix into an orderdependent product of a depolarizer, a retarder, and a diattenuator. Ossikovski’s
symmetric decomposition method decomposes a depolarizing Mueller matrix into a
sequence of five factors: a diagonal depolarizer between two retarder and diattenuator pairs (Ossikovski, 2009). The diattenuation, retardance, and depolarization
parameters calculated by these methods depend on the arbitrary order of the decomposition. The Cloude additive decomposition (Cloude, 1989) separates a Mueller
matrix into the sum of four non-depolarizing Mueller matrices which are scaled by
eigenvalues of the Mueller matrix covariance matrix. Since this is an additive decomposition, its terms are order-independent. None of these methods clearly elucidate
58
the nine degrees of freedom associated with depolarization. Recently, a Mueller matrix roots decomposition was introduced by Chipman in (Chipman, 2009) to provide
an order-independent description of polarization properties and a clear analysis of
the nine depolarization degrees of freedom. Chipman’s matrix roots decomposition
classifies depolarizing Mueller matrices into three families of depolarization, each
with three degrees of freedom.
A brief background of the Mueller matrix roots decomposition from (Chipman,
2009) is presented in section 3.2, followed by a definition of its nine depolarizing
parameters in section 3.3. In section 3.4, degree of polarization maps are used to
differentiate the three families of depolarization and demonstrate the unity between
two of these families. Section 3.5 discusses the generation of depolarization via the
averaging of non-depolarizing Mueller matrices, which is then extended in section
3.6 to provide insight on the forms of depolarization which occur in two different
scattering measurements.
3.2 Mueller matrix roots decomposition
Jones developed an order-independent representation for Jones matrices based on
his N -matrix representation. He used the analogy of propagation through a dichroic
and birefringent anisotropic crystal, divided into very short lengths, where the N matrices represented differential components (Jones, 1948). Recently, Ossikovski
extended Azzam’s differential matrix formalism (Azzam, 1978) to include depolarizing media (Ossikovski, 2011).
We extend the concept of the Jones N -matrices to Mueller matrices by dividing
59
M
p
M
( M)
p
p
Figure 3.1: Taking the root of a uniform Mueller matrix is analagous to slicing it
into very thin identical pieces.
a Mueller matrix M into p infinitesimal slices:
N=
√
p
M,
(3.1)
as illustrated in Fig. 3.1, where N is the Mueller matrix for one small slice. The
principal pth root of a large class of Mueller matrices approaches the identity matrix
I in the limit as p becomes very large,
lim
p→∞
√
p
M = I.
(3.2)
We describe the class of Mueller matrices that obey equation 3.2 as uniform Mueller
matrices. Our decomposition method is applicable to this subset of Mueller matrices.
√
For large p, the matrix root p M of a uniform Mueller matrix approaches identity
matrix, and its polarization properties separate. Mueller matrices near the identity
60
matrix are parameterized in terms of fifteen infinitesimal Mueller matrix generators and parameters, G1 (d1 ) through G15 (d15 ), and infinitesimal transmittance d0
(Chipman, 2009):
√
p
M = e−d0
15
Y
!
Gi (di ) .
(3.3)
i=1
Thus a uniform Mueller matrix M can be decomposed to yield its polarization
properties in terms of the sixteen matrix roots parameters D0 through D15 :
M = e−D0
15
Y
i=1
Gi
Di
p
!p
.
(3.4)
D0 is a transmittance factor, D1 through D3 are diattenuation parameters and D4
through D6 are retardance parameters. D7 through D15 are the nine depolarization
parameters, which are discussed in more detail in section 3.3. The fifteen Mueller
matrix generators Gi and their first order Taylor series approximations are listed in
Tables 3.1 and 3.2.
3.3 Definition of diagonal, phase, and amplitude depolarization
Nine depolarization parameters, D7 through D15 , describe depolarizing effects one for each of the nine depolarizing degrees of freedom (Lu and Chipman, 1996).
The depolarizing parameters are divided into three families. Each family has three
degrees of freedom corresponding to the Stokes parameter axes - horizontal/vertical,
45o /135o , and right/left circular.
The first order terms for amplitude depolarization, D7 , D8 , and D9 , share the
same off-axis matrix elements as the parts of the Mueller matrix associated with diattenuation on the horizontal/vertical, 45o /135o , and right/left circular axes (Chipman, 2009). They are named amplitude depolarization because they depolarize and
61
Table 3.1: The non-depolarizing Mueller matrix generators G1 (d1 ) through G6 (d6 )
and their first order Taylor series approximations.
Number

1
2
3
4
5
6
Generator

d1
0
0

1 p 0
0


0
1 − d21 p 0
1 − d21
0
0

0
d
0
2
p

0
1 − d22 0


0
1 p 0
0
0
1 − d22

0
d3
p 0
1 − d23 p 0
0

2
1 − d3 0 
0
0
0
1
1 0
0
0
0 1
0
0 


0 0 cos d4 sin d4 
0 0 − sin d4 cos d4 
1
0
0
0
0 cos d5 0 − sin d5 


0
0
1
0 
0 sin d5 0 cos d5 
1
0
0
0
0 cos d6 sin d6 0


0 − sin d6 cos d6 0
0
0
0
1
1
d1

0
0

1
0

d2
0

1
0

0
d
3
First-Order Form


1 d1 0 0
d1 1 0 0


 0 0 1 0
0 0 0 1


1 0 d2 0
 0 1 0 0


d2 0 1 0
0 0 0 1


1 0 0 d3
0 1 0 0


0 0 1 0
d 0 0 1
 3

1 0 0
0
0 1 0
0


0 0 1 d4 
0 0 −d4 1 
1 0 0 0
0 1 0 −d5 


0 0 1 0 
0 d5 0 1 
1 0
0 0
0 1 d6 0


0 −d6 1 0
0 0
0 1
affect the flux of an incident Stokes vector. The first order terms for phase depolarization, D10 , D11 , and D12 , share the same off-axis elements as the parts of the
Mueller matrix associated with retardance on the horizontal/vertical, 45o /135o , and
right/left circular axes. They do not affect the flux of an incident Stokes vector.
D13 , D14 , and D15 are named diagonal depolarization because they lie on the matrix diagonal. D15 expresses the overall isotropic depolarizing power. D15 reduces
62
Table 3.2: The depolarizing Mueller matrix generators G7 (d7 ) through G15 (d15 ) and
their first order Taylor series approximations.
Number
Generator

1 d7
0
0
−d7 1

0
0
p


 0

0
1 − d27 p 0
0
0
0
1 − d27


1 p 0
d8
0
2

 0
0
0
1
−
d
8



−d8
0
1 p 0
0
0
0
1 − d28


1 p 0
0
d9
 0
0
1 − d29 p 0


 0
0
1 − d29 0 
−d
0
0
1
 9

1 0
0
0
0 1

0
p 0


0 0
1 − d210 p d10 
2
0 0
d10
1 − d10


1 p 0
0
0
2
0
1 − d11 0
d11 


0

0
1 p 0
2
0
d11
0
1 − d11


1 p 0
0
0
2
0
1 − d12 p d12
0


0
d12
1 − d2 0

7
8
9
10
11
12
12
0
0
0
1

1
0
0
p0
0 −d13 + 1 − d213 /2

0


p0
0

0
d13 + 1 − d213 /2 p 0
2
0
0
0
1 − d13

0
0
p0
d√14
2

+ 1 − 2d14 /3
0
0

3
p

d√14
2
0
+ 1 − 2d14 /3
0

3
p
2
√14 +
0
0
− 2d
1
−
2d
/3
14
3


1
0
0
0
q


2
0
0
0 1 − 3 d15



q


2
0

0
1 − 3 d15
0


q
0
0
0
1 − 23 d15

13
14
15

1
0


0
0

1
0

0
0
First-Order Form


1 d7 0 0
−d7 1 0 0


 0
0 1 0
0
0 0 1


1 0 d8 0
 0 1 0 0


−d8 0 1 0
0 0 0 1


1 0 0 d9
 0 1 0 0


 0 0 1 0
−d9 0 0 1


1 0 0
0
0 1 0
0


0 0 1 d10 
0 0 d10 1


1 0 0 0
0 1 0 d11 


0 0 1 0 
0 d11 0 1


1 0
0 0
0 1 d12 0


0 d12 1 0
0 0
0 1
0
0
1 − d13
0
0
1 + d13
0
0

0
0

0
1


1
0 √
0
0
0 1 + d14 / 3

0 √
0


0
0
1 + d14 / 3
0 √ 
0
0
0
1 − 2d14 / 3


1
0
0
0
q


2
0
0
0 1 − 3 d15



q


2
0

0
1 − 3 d15
0


q
0
0
0
1 − 23 d15
the degree of polarization of all incident Stokes vectors equally, independent of the
Stokes vector’s location on the Poincare sphere. All depolarizing Mueller matrices
parameterized by equation 3.4 must contain a component of isotropic depolarization D15 . D13 expresses the relative strength between the diagonal depolarization
on the two linear axes (horizontal/vertical and 45o /135o ). D14 expresses the relative
63
strength between linear and circular diagonal depolarization. These properties arise
from the structure of G13 , G14 , and G15 in Table 3.2. Isotropic depolarization is
explained in more detail in section 3.4.
3.4 Degree of polarization maps of depolarizing Mueller matrices
Degree of polarization (DoP) maps provide a tool to understand the characteristics
of the three families of depolarization. They represent the variation in DoP of the
exiting polarization state as a function of the incident polarization state’s orientation
θ and latitude φ on the Poincare sphere (Deboo et al., 2004):

m0,0

m1,0

DoP(MS(θ, φ)) = DoP 
m
 2,0

m3,0


m0,1 m0,2 m0,3  
1



cos 2θ cos φ
m1,1 m1,2 m1,3 



 .



m2,1 m2,2 m2,3   sin 2θ cos φ 



m3,1 m3,2 m3,3
sin φ
0≤θ≤π
−
π
π
≤φ≤
(3.5)
2
2
Typically, a depolarizing Mueller matrix depolarizes different incident polarization
states by different amounts (Chipman, 2009). Comparison of the DoP maps for
each form of depolarization indicates both the differences and similarities between
the nine depolarization degrees of freedom.
Fig. 3.2 shows the DoP maps for nine physical, depolarizing Mueller matrices
- three amplitude depolarizer Mueller matrices, three phase depolarizer Mueller
matrices, and three diagonal depolarizer Mueller matrices.
The Mueller matrix for an isotropic diagonal depolarizer, M15 , was generated by
64
Figure 3.2: Degree of polarization maps for depolarizing Mueller matrices M7,15
through M15 .
65
taking the isotropic diagonal depolarization generator G15 to the pth power,
M15
p
Di
= G15
.
p
(3.6)
M15 has the effect of uniformly decreasing the DoP of an incident Stokes vector,
independent of its location on the Poincare sphere, moving the exiting Stokes vector
uniformly towards the center of the Poincare sphere. For this reason, M15 ’s DoP
map in Fig. 3.2 has a constant DoP, which varies with D15 .
Amplitude, phase, and non-isotropic diagonal depolarizer Mueller matrices were
generated by multiplying the depolarizing generators G7 through G14 with the
isotropic depolarization generator G15 and taking their product to the pth power,
Mi,15
p
Di
Di
= Gi
G15
.
p
p
i = 7, 8, ...14
(3.7)
Without some isotropic depolarization D15 , the depolarizing Mueller matrices
M7−14,15 would not be physical. Therefore the values for Di (i = 7, 8, ...14) were
arbitrarily selected to be Di = 0.2, and D15 was selected to be D15 = 0.5.
The amplitude depolarizers (M7−9,15 ) have two types of behavior distinct from
phase or diagonal depolarizers. First, since they have components in the top row of
the Mueller matrix, they alter the output intensity in addition to affecting the degree
of polarization. Second, their DoP maps have one maximum and one minimum
at orthogonal polarization states, lying on opposite ends of an axis through the
Poincare sphere. Three degrees of freedom are needed to establish this behavior,
including one for the ratio of minimum to maximum degree of polarization and two
to establish a point on the sphere.
66
The phase and non-isotropic diagonal depolarizers (M10−14,15 ) have three behaviors that distinguish them from the amplitude depolarizers. First, they do not
affect the intensity of a transmitted beam. Second, their DoP maps each have two
maxima and two minima, with the exception of M14,15 . The DoP map for M14,15
has two minima - one at the top and bottom of the Poincare sphere, at right and
left circular polarizations. Its maximum spans the entire equator of the Poincare
sphere. Third, points lying on opposite ends of any axis of the Poincare sphere (or
any pair of orthogonal polarization states) have equal degree of polarization.
The non-isotropic diagonal depolarizers M13,15 and M14,15 represent the three
degrees of freedom for an arbitrary amount of depolarization along each of the three
Stokes axes on the Poincare sphere. G13 and G14 account for two degrees of freedom, while the third degree of freedom originates from G15 . The axes formed by
the non-isotropic depolarizers along the Stokes axes of the Poincare sphere can be
called the Principal Depolarization Axes. These three degrees of freedom are independent, though they are restricted by additive inequalities in order to be physically
realizable. The phase depolarizers M10−12,15 add the degrees of freedom necessary
to move the Principal Depolarization Axes to arbitrary locations on the Poincare
sphere. Two degrees of freedom are required to specify the point of maximum DoP,
and the remaining degree of freedom specifies the axis of minimum degree of polarization. Therefore we can assert that phase and diagonal depolarization can result
from the same physical processes, occuring at different orientations.
67
3.5 Depolarization generation and cyclic permutations
3.5.1 Cyclic permutations
In section 3.5.2, depolarization is analyzed via the averaging of non-depolarizing
Mueller matrices, and the three families are shown to be governed by cyclic permutations. This section introduces these cyclic permutations.
The Pauli spin matrices and identity matrix σ0








 0 −i 
 0 1 
 1 0 
 1 0 
σ0 = 

 σ3 = 
 σ2 = 
 σ1 = 
i
0
1 0
0 −1
0 1
(3.8)
form a basis for the Jones matrices. As Jones matrices, σ1 represents a horizontal
half-wave retarder, σ2 represents a 45o linear half-wave retarder, σ3 represents a
circular half-wave retarder, and σ0 represents empty space. The Pauli spin matrices
obey the recursion relation
σi σj =
i,j,k
ii,j,k σk .



+1 (i, j, k)
=
(1, 2, 3), (3, 1, 2), (2, 3, 1)



=
−1 (i, j, k)
=
(1, 3, 2), (3, 2, 1), (2, 1, 3)




 0
i=j j=k k=i
(3.9)
For example, a muliplicative cascade of a horizontal half-wave retarder σ1 and the
45o half-wave retarder σ2 results in the circular half-wave retarder σ3 .
A similar cyclic permutation occurs for multiplicative combinations of nondepolarizing Mueller matrices. For example, multiplying the Mueller matrix for
a pure linear, horizontal, half-wave retarder (RH/V (δ = π)) and a pure linear,
45o , half-wave retarder (R45o /135o (δ = π)) yields a pure circular half-wave retarder
68
(RR/L (δ = π)):
RR/L (π) = R45o /135o (π) · RH/V (π).
(3.10)
Other sequences of retarder Mueller matrices oriented along the Stokes axes obey a
similar cyclic permutation.
3.5.2 Generation of depolarization via averaging of non-depolarizing Mueller matrices
In this section, the families of depolarization are analyzed in terms of sums of nondepolarizing Mueller matrices.
Given two non-depolarizing Mueller matrices M1 and M2 and β between 0 and
1, the matrix averaging operation
MA = βM1 + (1 − β)M2
(3.11)
represents a divided aperture, and generates a depolarizing matrix MA . These averaging combinations show a relationship between non-depolarizing Mueller matrices
and the different families of depolarization, and provides insight on the forms of
depolarization which occur in scattering processes.
Consider the averaging operation on all forty-nine combinations of two nondepolarizing matrices from Table 3.3, followed by a Mueller matrix roots decomposition to identify the resulting forms of depolarization. Table 3.4 shows the
depolarizing matrix roots parameters D7 through D15 which result from the linear combinations of our basis set of non-depolarizing Mueller matrices. Every linear
combination generates diagonal depolarization, emphasizing the relative importance
of these degrees of freedom. Table 3.5 shows only the amplitude and phase depo-
69
Table 3.3: Notation for the basis diattenuator and retarder Mueller matrices oriented
along the three Stokes axes (horizontal/vertical, 45o /135o and right/left circular),
as well as an attenuating identity matrix. q and r are the maximum and minimum
transmission, q 6= r, δ is the magnitude of the retardance vector, and α is the
attenuating coefficient.
αI
DH/V (q, r)
D45o /135o (q, r)
DR/L (q, r)
RH/V (δ)
R45o /135o (δ)
RR/L (δ)
Attenuator
Horizontal/Vertical Linear Diattenuator
45o /135o Linear Diattenuator
Circular Diattenuator
Horizontal/Vertical Linear Retarder
45o /135o Linear Retarder
Circular Retarder
larization generated by these linear combinations. Both tables are symmetric about
the diagonal, since matrix addition commutes.
For example, averaging the diattenuator DH/V (q, r) and retarder R45o /135o (δ)
yields D9 (right/left circular amplitude depolarization). Conversely, circular amplitude depolarization D9 is also generated by averaging RH/V (δ)) and D45o /135o (q, r).
Note that the coupling into depolarization follows the cyclic permutation rules. Averaging a diattenuator oriented along one Stokes axis with a retarder oriented along
a different Stokes axis yields amplitude depolarization (D7 , D8 , D9 ) oriented along
the third Stokes axis.
Alternatively, averaging the two diattenuators DH/V (q, r) and D45o /135o (q, r)
yields circular phase depolarization D12 , as does the average of the two retarders
RH/V (δ) and R45o /135o (δ). Averaging two diattenuators or two retarders oriented
along two different Stokes axes generates phase depolarization (D10 , D11 , or D12 )
oriented along the third Stokes axis. The resulting phase depolarization observes
the same cyclic permutation rules.
Many depolarizing optical components and systems are optically equivalent to
70
parallel combinations of non-depolarizing optical systems (Gil and Bernabeu, 1985).
While the combinations summarized in Tables 3.4 and 3.5 are not unique, they
describe rules for the generation of depolarization from spatially inhomogeneous
surfaces, where different regions have different polarization properties. These spatial
inhomogeneities of polarization properties produce depolarization (Gil, 2000).
Table 3.4: Depolarization properties (shown by parameters D7 through D15 ) produced by averaging two non-depolarizing Mueller matrices.
Matrix
αI
DH/V (q, r)
D45o /135o (q, r)
DR/L (q, r)
RH/V (δ)
R45o /135o (δ)
RR/L (δ)
αI
D15 ,D14 ,D13
D15 ,D14 ,D13
D15
D15 ,D14 ,D13
D15 ,D14 ,D13
D15
DH/V (q, r)
D15 ,D14 ,D13
D15 ,D14 ,D12
D15 ,D14 ,D13 ,D11
D15 ,D14 ,D13
D15 ,D14 ,D13 ,D9
D15 ,D14 ,D13 ,D8
D45o /135o (q, r)
D15 ,D14 ,D13
D15 ,D14 ,D12
D15 ,D14 ,D13 ,D10
D15 ,D14 ,D13 ,D9
D15 ,D14 ,D13
D15 ,D14 ,D13 ,D7
DR/L (q, r)
D15
D15 ,D14 ,D13 ,D11
D15 ,D14 ,D13 ,D10
D15 ,D14 ,D13 ,D8
D15 ,D14 ,D13 ,D7
D15
RH/V (δ)
D15 ,D14 ,D13
D15 ,D14 ,D13
D15 ,D14 ,D13 ,D9
D15 ,D14 ,D13 ,D8
D15 ,D14 ,D12
D15 ,D14 ,D13 ,D11
R45o /135o (δ)
D15 ,D14 ,D13
D15 ,D14 ,D13 ,D9
D15 ,D14 ,D13
D15 ,D14 ,D13 ,D7
D15 ,D14 ,D12
RR/L (δ)
D15
D15 ,D14 ,D13 ,D8
D15 ,D14 ,D13 ,D7
D15
D15 ,D14 ,D13 ,D11
D15 ,D14 ,D13 ,D10
D15 ,D14 ,D13 ,D10
Table 3.5: Non-diagonal depolarization properties (shown by parameters D7 through
D12 ) produced by averaging two non-depolarizing Mueller matrices.
Matrix
αI
DH/V (q, r)
D45o /135o (q, r)
DR/L (q, r)
RH/V (δ)
R45o /135o (δ)
RR/L (δ)
αI DH/V (q, r)
D12
D11
D9
D8
D45o /135o (q, r)
DR/L (q, r)
D12
D11
D10
D10
D9
D7
D8
D7
RH/V (δ)
R45o /135o (δ) RR/L (δ)
D9
D9
D8
D12
D11
D7
D12
D8
D7
D11
D10
D10
3.6 Experimental samples
Two sample measurements are analyzed with the Mueller matrix roots decomposition, providing insight into their depolarization properties. The samples were both
71
10o Incident Angle Sca7ering Angles -­‐70o 70o 90o -­‐90o Sample Plane Figure 3.3: Measurement geometry for the University of Arizona scattering infrared
polarimeter.
measured with Mueller matrix imaging polarimeters in a scattering configuration.
Their depolarization properties are consistent with those which can be generated by
the combinations of non-depolarizing spatial inhomogeneities from Tables 3.4 and
3.5.
3.6.1 Ground glass
The first example is the Mueller matrix for a sample of ground glass. Polarized light
incident onto a sample of ground glass experiences multiple reflections at unpredictable angles due to its rough, irregular surface, and scattering from the ground
glass results in depolarization effects.
The ground glass was measured at 1550 nm with the University of Arizona
scattering infrared polarimeter (Noble et al., 2007). The measurement configuration
72
labels the surface normal as 0o . Positive scattering angles are measured from the
surface normal in the forward scattering direction. Negative incident angles are
measured from the surface normal in the backscattering direction. The measurement
geometry is illustrated in Fig. C.5. Two angle combinations are analyzed here - one
specular angle pair and one non-specular angle pair. The specular angle pair is
illuminated at −70o and its scattering angle is 70o . The non-specular angle pair
has the same illumination angle of −70o and a scattering angle of 10o , and was
selected because it is far from specular. Because this measurement was performed
in reflection mode, the data was corrected for the geometric 180o phase change via
right-multiplication by the Mueller matrix for a horizontal half wave linear retarder
(DeBoo et al., 2005),

1

0

RH (δ = π) = 
0


0

0

1 0
0

.
0 −1 0 


0 0 −1
0
0
(3.12)
The Mueller matrix for the non-specular angle pair,


0.004




−0.400 0.781 −0.058 −0.02


Mn = 
,
−0.019 0.053

0.216
0.179




0.001 −0.015 −0.180 0.138
1
−0.437
0.032
(3.13)
73
and the specular angle pair,


0.003




−0.214 0.947 −0.026 −0.035


Ms = 
,
−0.010 0.037 −0.610 0.370 




0.000 −0.037 −0.339 −0.598
1
−0.245
0.010
(3.14)
yields information about the polarization effects due to scattering. Mn has smaller
diagonal terms than Ms , particularly in m2,2 and m3,3 , which implies that Mn is
depolarizing in the 45o /135o linear and circular axes. Ms has larger diagonal terms,
and its m2,2 and m3,3 terms are negative. Ms is close to the form of equation
3.12, which implies the presence of significant retardance. Mn also exhibits more
diattenuation than Ms , as seen in m0,1 , m0,2 , and m0,3 .
Matrix Root Parameters for Specular Angle
Matrix Root Parameters for Non-Specular Angle
ÈDi È
3.0
ÈDi È
3.0
2.5
2.5
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5
D0 D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 D13 D14 D15
D0 D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 D13 D14 D15
Figure 3.4: The magnitude of the matrix roots parameters D0 through D15 of the
ground glass sample for (a) the specular angle pair and (b) the non-specular angle
pair.
The matrix roots decomposition parameters for both angle pairs are shown in
table 3.6. Figs.s 3.4(a) and 3.4(b) plot the absolute value of each matrix root
parameter for both angle pairs. For each case, the diattenuation lies predominantly
on the vertical linear axis (D1 < 0), and the non-specular angle pair exhibits larger
74
diattenuation than the specular angle pair. The specular angle pair has a large
retardance of 2.61 radians predominantly in the linear horizontal direction (D4 ).
The non-specular angle pair has only 0.78 radians of retardance in the horizontal
direction, but this axis still dominates over the other orientations. For both angle
pairs, the dominant non-depolarizing effects are diattenuation and retardance along
the horizontal/vertical Stokes axis. According to Tables 3.4 and 3.5, averaging these
effects can lead to diagonal depolarization D13 , D14 , and D15 .
Indeed, diagonal depolarization is the dominant depolarizing effect in the scattering measurements for both angle pairs. The non-specular angle pair exhibits more
diagonal depolarization than the specular angle pair by several orders of magnitude.
We expect the specular angle pair to exhibit less depolarization than the angle pair
further from specular.
The diagonal depolarization terms D13 , D14 , and D15 yield the following information. The negative magnitude of D13 for both cases means that more diagonal
depolarization takes place along the linear horizontal/vertical axis than the linear
45o /135o axis. The magnitude of D13 for the non-specular angle is approximately
triple that of the specular angle, which means that the differences between the two
linear depolarization axes is more dramatic for the non-specular angle. The positive
magnitude of D14 for both angles implies that the diagonal depolarization effects
are predominantly linear in nature (both on the horizontal/vertical and 45o /135o
axes), as opposed to circular. This does not mean that the circular diagonal depolarization term is negligible - it simply means that the linear diagonal effects on
all axes combined are more significant than the circular diagonal depolarization.
This effect is more dramatic in the non-specular angle. D15 is the overall (isotropic)
diagonal depolarization power, and as expected, the D15 value of 1.169 is larger for
75
the non-specular angle than the specular angle, with a D15 of 0.292.
Roots Parameter
D0
D1
D2
D3
D4
D5
D6
D7
D8
D9
D10
D11
D12
D13
D14
D15
Specular Angle {−70o , 70o }
0.029
-0.243
0.002
-0.006
2.612
0.007
-0.075
-0.018
0.001
0
0.108
0.029
0.007
-0.149
0.065
0.292
Non-Specular Angle {−70o , 10o }
0.096
-0.470
0.006
-0.008
0.785
0.005
-0.119
-0.059
0.016
-0.001
-0.024
-0.004
-0.006
-0.430
0.459
1.169
Table 3.6: The matrix roots parameters from a ground glass sample for specular
{−70o , 70o } and non-specular angle pairs {−70o , 10o }.
3.6.2 Pencil at 505 nm
The second example is the Mueller matrix for a wood-graphite pencil. Polarized
light scatters very differently for the two materials - the graphite is smoother, with
less depolarization upon scattering, while the wood is more textured, with more
depolarization upon scattering.
Fig. 3.5 shows the Mueller matrix image for the graphite pencil measured in
reflection mode with the University of Arizona Mueller matrix imaging polarimeter
(MMIP) (DeBoo et al., 2005) at 505nm. The images have been median filtered to
reduce noise and corrected for reflection using the procedure described in section
3.6.1. Inspection of the Mueller matrix reveals that the pencil is predominantly
a diagonal depolarizer. The wood is more depolarizing than the smooth graphite
76
Figure 3.5: The Mueller matrix for a graphite and wood pencil, measured at 505nm
with a Mueller matrix imaging polarimeter.
surface, particularly in the m3,3 image. Other than these gross features, it is difficult
to see the magnitudes of the depolarizing properties from the Mueller matrix image
alone.
The matrix roots decomposition yields more insight into both the nondepolarizing and depolarizing properties of the pencil. Fig. 3.6 shows the pencil’s
non-depolarizing matrix roots parameters. The pencil tip exhibits a small amount
of vertical diattenuation on the order of -0.09 in the wooden region and -0.12 on the
graphite tip in D1 , and minimal 45o or circular diattenuation in D2 and D3 . The
pencil tip exhibits some retardance on all three axes in D4 , D5 , and D6 . In D4 ,
the horizontal/vertical retardance term, the wooden region of the pencil exhibits a
minimum of 0.2 radians of retardance and a maximum of 0.6 radians of retardance.
D5 , the 45o /135o retardance term, exhibits between 0 radians and -0.6 radians of
77
D1
D2
D3
D4
D5
D6
Figure 3.6: The non-depolarizing matrix roots parameters for a graphite and wood
pencil, measured at 505nm with a Mueller matrix imaging polarimeter.
78
retardance in the wooden region of the pencil tip. The negative retardance value
means that the fast-axis of the retardance is closer to 135o than 45o . The graphite
region of the pencil exhibits trace amounts of horizontal retardance and slightly
more circular retardance, while the wooden region of the pencil exhibits between
0.2 and 0.4 radians of circular retardance, as shown in the D6 parameter.
The graphite tip of the pencil has very little depolarization. This is consistent
throughout all the degrees of freedom of the three depolarization families (amplitude,
phase, and diagonal depolarization). The lack of depolarization in the graphite
pencil tip is not surprising, as the graphite is a relatively smooth, shiny surface,
especially compared to the wooden region of the pencil.
The wooden region of the pencil exhibits no significant linear horizontal/vertical
amplitude depolarization D7 , almost no circular amplitude depolarization D9 , and a
45o /135o amplitude depolarization of D8 = 0.05 in the upper half of the wood. The
phase depolarization (D10 , D11 , and D12 ) is most significant in the left circular direction, with the negative magnitude of D12 approaching -0.3 in the maximum regions.
These results are consistent with the results of averaging the spatial inhomogeneities
in diattenuation and retardance from Table 3.5. In this case, the presence of D10 ,
D11 , and D12 are consistent with the averaging of spatial inhomogeneities along all
three retardance axes (two at a time), and 45o /135o amplitude depolarization D8 is
consistent with the averaging of horizontal/vertical diattenuation D1 and circular
retardance D6 .
The diagonal depolarization parameter D13 is a measure of the relative strength
between the diagonal Mueller matrix m1,1 term and the diagonal m2,2 term. The
magnitude of the entire D13 image is very small, which means that the wooden
region of the pencil is almost equally depolarizing along the horizontal/vertical and
79
D7
D8
D9
D10
D11
D12
D13
D14
D15
Figure 3.7: The depolarizing matrix roots parameters for a graphite and wood
pencil, measured at 505nm with a Mueller matrix imaging polarimeter.
80
45o /135o axes. D14 is a measure of the relative strength between the linear diagonal
terms (m1,1 and m2,2 ) and the circular diagonal term (m3,3 ). The positive magnitude of D14 throughout the wooden region of the pencil means that the diagonal
depolarization in the wooden region of the pencil is predominantly linear. D15 is a
measure of the overall diagonal depolarization power - it reaches a maximum of 4
at the top edge of the pencil where the incident and scattering angles in the imaging measurement are steepest, and a minimum of approximately 1.8 in the central
region of the wood.
3.7 Conclusion
The Mueller matrix roots decomposition from (Chipman, 2009) and the associated
families of depolarization (amplitude depolarization, phase depolarization, and diagonal depolarization) were explored. Degree of polarization maps show the unity
between phase and diagonal depolarization; amplitude depolarization remains a distinct class. The nine depolarization parameters can be generated via the averaging
of two non-depolarizing Mueller matrices, and the orientation of the resulting depolarization follows the cyclic permutations of the Pauli spin matrices. Averaging two
non-depolarizing Mueller matrices of the same class (retarders with retarders or diattenuators with diattenuators) results in phase depolarization, while averaging two
non-depolarizing Mueller matrices of different classes (retarders with diattenuators)
results in amplitude depolarization. Scattered light is often a combination of many
incoherent states that result from spatial inhomogeneities. These nine forms of depolarization were applied to Mueller matrices from two scattering measurements a sample of ground glass and a graphite and wood pencil tip.
81
CHAPTER 4
ADDITIONAL MATRIX ROOTS GENERATOR PROPERTIES AND
APPLICATIONS
4.1 Convergence to the identity matrix
The matrix roots decomposition is useful for the subset of uniform Mueller matrices.
The principal root of a uniform matrix converges to the identity matrix I as the root
order p approaches infinity:
lim
p→∞
√
p
M = I.
(4.1)
To show that the majority of physical Mueller matrices are uniform, the proportion of physical Mueller matrices with a negative determinant is estimated. The
proportion of physical Mueller matrices which have a root that converges to the
identity matrix as the root order p approaches infinity is examined.
To this end, a brute-force numerical calculation method was applied. A four
by four Mueller matrix is generated with random numbers (uniformly distributed
between negative one to one), and its m0,0 element is set to a value of one. Thus
the 15 element space within which normalized Mueller matrices reside is uniformly
sampled. Each matrix is tested for physical realizability by checking that the eigenvalues of its coherency matrix are all greater than or equal to zero. If the matrix is
physical, then its determinant is calculated and the Mueller matrix’s determinant
is classified to be either positive or negative. (Mueller matrices with negative determinants are non-uniform because their roots are complex.) Approximately one
hundred fifteen thousand physical Mueller matrices were generated in this manner
82
Determinants of Random Mueller Matrices
Number of Mueller Matrices
3000
2500
2000
1500
1000
500
0
0.00
0.05
0.10
Determinant
Figure 4.1: A histogram of the randomly generated Mueller matrix determinants
shows the statistical distribution of the determinant values.
from a set of 109 matrices, and some statistics of this set were calculated. Approximately one in eight Mueller matrices (12.11%) from a sample of 115,218 physical
Mueller matrices was found to have a negative determinant. Out of these 13,951
negative determinant matrices, the most negative determinant was -0.02837. The
most negative determinant of a physically realizable Mueller matrix is −1/27, or 0.037 (Chipman, 2009). The histogram in Fig. 4.1 shows the statistical distribution
of the randomly generated Mueller matrices’ determinants, and indicates the rapid
reduction of the volume of Mueller matrices as the determinant moves away from
zero in both directions.
The second goal of this numerical experiment was to demonstrate that the majority of physical, positive determinant Mueller matrices converge towards the identity
matrix as the pth root becomes very large. The 10n th matrix root with n = 4,
through n = 6 for each physical positive determinant Mueller matrix was calcu-
83
Root Convergence to Identity Matrix
Number of Mueller Matrices
80 000
60 000
10-4
10-5
40 000
10-6
20 000
0
0.00000
0.00002
0.00004
0.00006
0.00008
0.00010
0.00012
0.00014
RMS Divergence
Figure 4.2: This histogram shows increasing convergence towards the identity matrix
as the root order p increases.
lated and subtracted from the identity matrix. The root-mean-square (RMS) of
this difference was then calculated,
RM S =
3
1 X
16 i,j=0
√
10n
mi,j − Ii,j
2
.
n = 4, 5, 6 (4.2)
A histogram of the RMS for the five values of root-order p = 10n in Fig. 4.2 shows
that as the value of p increases from p = 104 incrementally to p = 106 , the RMS
shrinks rapidly. 671 out of 101,267 positive determinant Mueller matrices do not converge to the identity matrix. Because the majority of positive determinant Mueller
matrices do converge towards the identity matrix as their root order approaches a
very large number, this matrix roots method is applicable to the majority of these
cases.
84
4.2 Generator properties
4.2.1 Trajectories of non-depolarizing generators
Non-depolarizing generators, except for the singular and non-uniform cases discussed
in section 2.4, are uniform. Two simple examples in figures 4.3 and 4.5 illustrate
this behavior.
In figure 4.3, the root of a linear diattenuator with diattenuation of 0.2 at 0o
approaches the identity matrix as the root order increases. Figure 4.4 shows the
10p
trajectory of the generator (G2 (d2 = 2 ∗ 10−10 ))
approaching the linear diattenu-
ator from figure 4.3 when p = 0 as the root order 10p increases. These figures are
complementary - figure 4.3 shows the path from the diattenuator to its infinitesimal
generator, and 4.4 shows the path from infinitesimal generator to diattenuator.
Similarly, figure 4.5 shows the trajectory of the root of an elliptical retarder with
retardance vector {δH = 0, δ45 = π/4, δR = π/10} approaching the identity matrix
as the matrix is taken to the 10p th root. Figure 4.6 shows the trajectory of the
product G5 (d5 = π/4 ∗ 10−9 ) · G6 (d6 = π/10 ∗ 10−9 ) approaching the macroscopic
retarder from figure 4.5 at p = 0 as the matrix is taken to the 10p th power.
4.2.2 Orthogonality
The generator trajectories for G1 through G15 were constructed to be orthogonal
around the identity matrix. We introduce the following notation, where a Mueller
matrix is flattened into a one-dimensional array and drop the first element. For a
generic Mueller matrix M with elements m0,0 through m3,3 , the Mueller matrix is
85
M1,0
M0,0
M3,0
M2,0
1.0000
1.0
0.020
1.0
0.9995
0.5
0.015
0.5
0.9990
2
4
6
8
10-p -0.5
2
4
6
8
10-p
4
6
8
10
-0.5
2
M0,2
4
6
8
10-p
2
0.005
2
4
6
8
4
6
8
10-p
0.5
2
4
4
6
8
10-p
-0.5
-0.5
-1.0
-1.0
-1.0
8
M3,2
0.5
2
4
6
8
10
4
10-p
-p -0.5
-1.0
M3,3
1.0000
0.9995
2
-0.5
6
10-p
-0.5
M2,3
1.0
8
2
1.0
2
0.5
6
10
0.9990
1.0
4
8
-1.0
M1,3
10-p
6
0.9995
0.5
2
4
-p
-0.5
1.0
8
0.5
-0.5
10-p
-1.0
M0,3
6
10-p
M3,1
1.0000
0.5
8
-1.0
M2,2
1.0
0.010
10-p
-1.0
M1,2
6
1.0
2
-1.0
0.015
8
0.5
0.9990
0.9985
0.020
6
1.0
0.9995
-p
4
M2,1
1.0000
0.5
4
-0.5
2
M1,1
1.0
2
0.005
-1.0
M0,1
2
0.010
4
6
8
10-p
0.9990
0.9985
2
4
6
8
10-p
Figure 4.3: A linear diattenuator with diattenuation of 0.2 approaches the identity
matrix as it is taken to the 10p th root.
86
M0,0
M2,0
M1,0
M3,0
1.020
1.0
0.20
1.0
1.015
0.5
0.15
0.5
1.010
2
1.005
4
6
8
10 p
4
6
8
10 p
-1.0
2.0
0.5
1.5
2
4
6
8
10 p
8
0.15
4
6
1.020
0.5
1.015
2
6
8
10 p
10
4
6
8
10
8
6
8
10 p
-1.0
M3,2
1.0
0.5
2
4
10 p
-0.5
1.005
-1.0
6
4
-0.5
p 1.010
-0.5
2
4
M1,3
M0,3
2
M2,2
0.10
0.05
8
-1.0
1.0
8
-1.0
p
-0.5
6
6
10 p
0.5
2
4
8
M3,1
0.5
10 p
6
1.0
M1,2
0.20
8
1.0
2
M0,2
4
6
1.0
-1.0
2
4
10 p
M2,1
0.5
-0.5
4
-0.5
2
M1,1
M0,1
1.0
2
0.05
-0.5
2
0.10
6
8
10 p
-1.0
M3,3
M2,3
2.0
1.0
1.0
1.0
0.5
0.5
2
4
6
8
10
1.5
0.5
p
2
4
6
8
10
p
2
-0.5
-0.5
-0.5
-1.0
-1.0
-1.0
4
6
8
10
p
1.0
0.5
2
4
10 p
Figure 4.4: The trajectory of generator G2 with d2 = 2∗10−10 as the matrix is taken
to the power of 10p approaches a linear diattenuator with diattenuation of 0.2.
87
M0,0
M3,0
M1,0
2.0
M2,0
1.0
1.0
1.5
1.0
0.5
1.0
2
0.5
4
6
8
10-p
-0.5
2
4
6
8
10-p
2
0.5
2
4
6
8
-0.5
2
4
4
6
6
8
-0.5
8
1.5 2.0 2.5 3.0 3.5
2
-0.02
0.5
4
6
8
10-p
-0.04
-0.5
-0.06
-1.0
-0.08
2.0
2.5
3.0
4
10-p
0.07
0.06
0.05
0.04
0.03
0.02
1.2
10-p
2
6
10-p
4
8
10
6
8
2
4
1.6
1.8
10-p
0.012
0.010
0.008
0.006
0.004
10-p 0.002
1.5
6
2.0
2.5
10-p
M3,3
-p
-0.002
-0.004
-0.006
-0.008
-0.010
1.4
M3,2
M2,3
M1,3
1.0
2
1.5
-0.005
-0.010
-0.015
-0.020
10-p -0.025
-0.030
1.0000
0.9999
0.9998
0.9997
0.9996
M0,3
8
M3,1
M2,2
0.030
0.025
0.020
0.015
10-p 0.010
0.005
-1.0
6
-1.0
M1,2
M0,2
0.5
8
4
M2,1
1.0000
0.9995
0.9990
0.9985
-p 0.9980
10 0.9975
0.9970
2
1.0
6
2
-0.5
-1.0
M1,1
-1.0
4
10-p
-0.5
-1.0
M0,1
1.0
0.5
0.5
8
10-p
1.0000
0.9995
0.9990
0.9985
0.9980
0.9975
2
4
6
8
10-p
Figure 4.5: The trajectory of an elliptical retarder with δ45 = π/4 and δR = π/10
approaching the identity matrix as the matrix is taken to the 10p th root.
88
M0,0
M3,0
M2,0
M1,0
1.0
2.0
1.0
1.0
1.5
0.5
0.5
1.0
2
0.5
2
4
6
8
4
0.5
2
4
6
8
-0.5
10 p -1.0
-1.0
4
-1.0
1.0
0.5
4
6
8
-0.5
0.5
-1.0
6
8
-0.1
-0.2
-0.3
10 p -0.4
-0.5
-0.6
-0.7
6
8
10 p
4
10 p
6
8
10 p
8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
7.5
2
4
8.0
8.5
9.0
10 p
6
8
0.12
0.10
0.08
0.06
0.04
p
10 0.02
M2,3
10 p
6
M3,2
0.97
2
4
M2,2
0.98
4
2
-0.5
7.0
M1,3
1.0
4
10 p
9.0
0.99
2
2
8
-0.05
-0.10
-0.15
-0.20
-0.25
8.5
1.00
M0,3
-0.5
6
0.25
0.20
0.15
p
10 0.10
0.05
-1.0
10
8
M3,1
8.0
M1,2
M0,2
6
M2,1
1.00
0.95
0.90
0.85
p 0.80
10
0.75
2
4
-1.0
M1,1
-0.5
2
8
2
-0.5
M0,1
1.0
6
10 p
0.5
p
6
7
8
9
10 p
M3,3
0.12
0.10
0.08
0.06
0.04
0.02
2
4
6
8
10
1.00
0.95
0.90
0.85
p 0.80
0.70
2
4
6
8
10 p
Figure 4.6: The trajectory of the generator product G5 (d5 = π/4 ∗ 10−9 ) · G6 (d6 =
π/10 ∗ 10−9 ) approaching an elliptical retarder as the matrix is taken to the power
of 10p .
89
transformed from a four by four matrix to a fifteen element vector Mv as follows:

m0,0

m1,0

M=
m
 2,0

m3,0

m0,1 m0,2 m0,3 

m1,1 m1,2 m1,3 

 −→
m2,1 m2,2 m2,3 


m3,1 m3,2 m3,3
Mv = (m0,1 , m0,2 , m0,3 , m1,0 , m1,1 , m1,2 , m1,3 , m2,0 , m2,1 , m2,2 , m2,3 , m3,0 , m3,1 , m3,2 , m3,3 ) .
(4.3)
The direction of a generator trajectory from the identity matrix is the derivative of
a space curve which can be calculated as the limit
I − Gi (di )
.
di →0
di
(4.4)
lim
The vector angle between the generators is found to be π/2 in the limit surrounding the identity matrix:
−1
θ = lim cos
di ,dj →0
(I − Gi (di ))v · (I − Gj (dj ))v
k(I − Gi (di ))v kk(I − Gj (dj ))v k
=
π
2
i, j = 1, 2, ...15
(i 6= j) (4.5)
While the generators are orthogonal to each other in the limit near the identity
matrix, they are not orthogonal in general. This is straightforward to show by
counterexample. Figures 4.7 and 4.8 show the distance from orthogonality (π/2 − θ)
between G1 (d1 ) and G2 (d2 ), and G11 (d11 ) and G12 (d12 ) (respectively), plotted as a
function of d1 = d2 = d and d11 = d12 = d. In figures 4.7 and 4.8, the vector angle
90
Non-Orthogonality Between G1 and G2
0.0012
Radians
0.0010
0.0008
0.0006
0.0004
0.0002
0.0000
0.00
0.02
0.04
0.06
0.08
0.10
d
Figure 4.7: The diference between π/2 and the vector angle between G1 (d1 = d)
and G2 (d2 = d) is plotted as a function of d, showing that generators G1 and G2 are
not orthogonal in general.
between the generators is calculated and then subtracted from π/2.
4.2.3 Depolarization index
The depolarization index (DI) is a numerical metric for characterizing the depolarization of a Mueller matrix (Gil and Bernabeu, 1985). The DI is defined as
P
DI(M) =
3
2
i,j=0 (mi,j )
√
2
− (m0,0 )
3m0,0
1/2
.
(4.6)
The DI equals one for non-depolarizing Mueller matrices and equals zero for an
ideal depolarizer. It is the distance of a normalized Mueller matrix from the ideal
√
depolarizer, divided by 3.
For G1 (d1 ) through G14 (d14 ), the DI is equal to one, for any value of di . In
and of themselves, none of these generators change the depolarization index - all of
√
the generator trajectories lie on a hypersphere of radius 3. However, as G7 (d7 )
91
Non-Orthogonality Between G11 and G12
0.0012
Radians
0.0010
0.0008
0.0006
0.0004
0.0002
0.0000
0.00
0.02
0.04
0.06
0.08
0.10
d
Figure 4.8: The diference between π/2 and the vector angle between G11 (d11 = d)
and G12 (d12 = d) is plotted as a function of d, showing that generators G11 and G12
are not orthogonal in general.
through G14 (d14 ) are not physical on their own, they must be paired with some
amount of G15 (d15 ) to produce a physical Mueller matrix (as discussed in section
4.2.5). Of all the generators, only G15 (d15 ) has DI < 1:
r
DI (G15 (d15 )) = 1 −
2
d15 .
3
(4.7)
p
This implies that d15 has a range between zero and 3/2, where d15 = 0 corresponds
p
to a DI of one (at the identity matrix) and d15 = 3/2 corresponds to a DI of zero,
p
or an ideal depolarizer. Increasing d15 between zero and 3/2 increases the DI
linearly.
4.2.4 Unitary transformations and depolarization classification
Performing rotation operations on the depolarizing generators G7 through G15 yields
insight into the relationship between the different families of depolarization. It
92
is shown that phase depolarization and diagonal depolarization are related via a
coordinate rotation, while amplitude depolarization remains distinct from the other
two families under coordinate rotation.
Matrix rotations are performed with a real, orthogonal matrix transformation
such that
Mo = OT MO,
(4.8)
where O is an orthogonal, real unitary matrix. As Mueller matrices these arre
elliptical retarders. The columns of O form an orthonormal basis in real space. This
orthogonal transformation on a Mueller matrix M results in a rotation between two
or more axes of the Stokes coordinate system intrinsic to M.
The two classes of matrices for rotation used here are

R1
R2

1
0
0
0




0 cos 2θ − sin 2θ 0


= 

0 sin 2θ cos 2θ 0




0
0
0
1


0
0
0 
1


0
cos2 2θ
cos 2θ sin 2θ − sin 2θ


= 
,
0 cos 2θ sin 2θ
sin2 2θ
cos 2θ 




0
sin 2θ
− cos 2θ
0
(4.9)
(4.10)
where R1 rotates an incident Stokes vector between the S1 and S2 axes (about the
S3 axis) and R2 rotates an incident Stokes vector between the S1 , S2 , and S3 axes.
(R2 is equivalent to a quarter-wave linear retarder with fast-axis orientation θ.)
If a rotation is performed on a depolarizing generator and the resulting matrix
has been rotated into another family of depolarization, this implies that the two
93
families are related by rotation. To demonstrate this effect, the orthogonal rotation transformation from equation 4.8 is performed on the depolarizing generators
G7 (d7 ) through G15 (d15 ) about the Stokes axes for the two rotations. Because these
matrices are generators, they are expressed in terms of the (depolarizing) infinitesimal matrix roots parameters d7 through d15 . Therefore after the rotation operation
is performed, the resulting matrix can be solved for the infinitesimal matrix roots
parameters. Because d7 through d15 are infinitesimal, any resulting second order
terms are dropped.
G15 (d15 ) is invariant under all of the rotation transformations discussed here,
since it depolarizes equally along each Stokes axis. Amplitude depolarization generators G7 (d7 ), G8 (d8 ), and G9 (d9 ) are rotated among themselves, forming a depolarization family. They do not rotate into any other families of depolarization.
Table 4.1 shows that the rotation of G10 (d10 ) and G11 (d11 ) by R1 (θ) results in
a sinusoidal rotation between d11 and d10 with a period of 2θ. Because d11 and d10
are different (linear) axes of the same phase depolarization family, this is an obvious
and expected result. The same transformation around R1 (θ) on G12 (d12 ) results in
a sinusoidal rotation between circular phase depolarization d12 and diagonal depolarization terms d13 and d14 :
d12 = d12 cos 4θ
(4.11)
d13 = −d12 sin 4θ
(4.12)
d14 = d12 sin 4θ.
(4.13)
The rotation of circular phase depolarization d12 into diagonal depolarization demonstrates the relationship between phase and diagonal depolarization.
94
Table 4.2 shows the effect of rotating G10 (d10 ), G11 (d11 ), and G12 (d12 ) about
R2 (θ). G10 (d10 ), G11 (d11 ), and G12 (d12 ) all rotate between the three phase depolarization axes d11 and d12 and diagonal depolarization d13 .
Table 4.3 shows the effect of rotating G13 (d13 ), G14 (d14 ), and G15 (d15 ) about
R2 (θ). This rotation of G13 (d13 ) is unique among all of the depolarizing generators
in that d13 rotates into all three axes of phase depolarization (d10 , d11 , and d12 ) as
well as diagonal depolarization d14 .
This method can be extended by applying many other varieties of rotation transformations (such as more complex elliptical rotations), though their closed form
solutions are too complex to include here. These unitary rotation operations show
that phase depolarization and diagonal depolarization are related via simple coordinate rotations, while amplitude depolarization remains distinct. This approach has
uniquely demonstrated that depolarizers fall into two distinct classes: amplitude
depolarization in one class, and phase and diagonal depolarization in another class.
Table 4.1: Effect of the rotation transformation for G10 through G12 about R1 (θ).
d7
d8
d9
d10
d11
d12
d13
d14
d15
G10 (d10 ) RT1 (θ)G10 (d10 )RT1 (θ)
0
0
0
0
0
0
d10
d10 cos 2θ
0
d10 sin 2θ
0
0
0
0
0
0
0
0
G11 (d11 )
0
0
0
0
d11
0
0
0
0
RT1 (θ)G11 (d11 )R1 (θ) G12 (d12 )
0
0
0
0
0
0
−d11 sin 2θ
0
d11 cos 2θ
0
0
d12
0
0
0
0
0
0
RT1 (θ)G12 (d12 )R1 (θ)
0
0
0
0
0
d12 cos 4θ
−d12 sin 4θ
d12 sin 4θ
0
95
Table 4.2: Effect of rotation transformation for G10 through G12 about R2 (θ).
d7
d8
d9
d10
d11
d12
d13
d14
d15
G10 (d10 )
0
0
0
d10
0
0
0
0
0
RT2 (θ).G10 (d10 ).R2 (θ)
0
0
0
−d10 cos2 2θ
1
d sin 4θ
2 10
−d10 cos 4θ sin 2θ
−d10 sin 2θ sin 4θ
0
0
G11 (d11 )
0
0
0
0
d11
0
0
0
0
RT2 (θ).G11 (d11 ).R2 (θ) G12 (d12 ) RT2 (θ).G12 (d12 ).R2 (θ)
0
0
0
0
0
0
0
0
0
1
d
sin
4θ
0
d
cos
4θ
sin 2θ
11
12
2
−d11 sin2 2θ
0
d12 cos 2θ cos 4θ
1
−d11 cos 2θ sin 4θ
d12
d sin2 4θ
2 12
−2d11 cos2 2θ sin 2θ
0
−d
sin 8θ/4
√ 12
3
0
0
d
sin 4θ
2 12
0
0
0
Table 4.3: Effect of rotation transformation for G13 (d13 ) through G15 (d15 ) about
R2 (θ).
d7
d8
d9
d10
d11
d12
d13
d14
d15
G13 (d13 )
0
0
0
0
0
0
d13
0
0
RT2 (θ).G13 (d13 ).R2 (θ)
0
0
0
d13 sin 2θ sin 4θ
2d13 cos2 2θ sin 2θ
− 14 d13 sin 8θ
d√13 cos2 4θ
− 23 d13 cos 4θ
0
G14 (d14 )
0
0
0
0
0
0
0
d14
0
RT2 (θ).G14 (d14 ).R2 (θ) G15 (d15 ) RT2 (θ).G15 (d15 ).R2 (θ)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
√
1
3d
sin
4θ
0
0
14
2 √
− 23 d14 cos 4θ
0
0
−d14 /2
0
0
0
3d15 /2
3d15 /2
96
4.2.5 Conditions for physicality of depolarizing generators
Cloude presented a method for evaluating the physical realizability of a Mueller
matrix in terms of its coherency matrix (Cloude, 1989). The coherency matrix C
of a Mueller matrix M is calculated from the elements mij of M and the Pauli spin
matrices σi ,
C=
3
X
mij (σi × σj∗ ),
(4.14)
i=0,j=0
where in our notation the Pauli spin matrices are








0 −i
0 1 
1 0 
1 0
σ0 = 
.
 , σ3 = 
 , σ2 = 
 , σ1 = 
i 0
1 0
0 −1
0 1
(4.15)
A depolarizing Mueller matrix is physical so long as all four eigenvalues of the
Mueller matrice’s coherency matrix are greater than zero (Ossikovski et al., 2008;
Twietmeyer et al., 2008).
The non-depolarizing generators G1 (d1 ) through G6 (d6 ) are all physical Mueller
matrices - in each case, three coherency matrix eigenvalues are zero, and the fourth
is real and positive. Individually, G7 (d7 ) through G14 (d14 ) are not physical Mueller
matrices unless combined with G15 (d15 ). To construct a physical (macroscopic)
Mueller matrix, we combine them as
p
di
d15
Gi
G15
.
p
p
(4.16)
The coherency matrix eigenvalues are plotted within the range of negative one to one
for each value of di because the generators are only real within this range. Figure 4.9
graphs the four eigenvalues for −1 < di < 1. The coherency matrix eigenvalues for
G7 (d7 ) through G12 (d12 ) are identical, and are plotted together. G15 (d15 ) is the only
97
Eigenvalues
Λ1
G7-G12
-1.0
Λ3
Λ2
0.4
0.4
0.9
0.2
0.2
0.3
0.8
0.2
0.7
0.5
1.0
d7-12
-1.0
-0.4
0.5
-0.5
-0.2
1.0
d7-12
0.1
Λ2
Λ1
-1.0
0.6
-0.5
-0.5
-0.02
0.5
1.0
0.4
0.2
-0.06
0.2
0.5
1.0
d13
-0.08
Λ1
-0.5
-0.1
-1.0
-0.5
d14 -1.0
-0.5
Λ1
1.0
d14
-1.0
-0.5
0.1
0.5
1.0
-1.0
1.0
0.7
d13
-0.5
-0.1
1.0
-1.0
0.5
-0.1
-0.1
-0.2
-0.2
-0.2
0.5
1.0
0.5
1.0
d13
1.0
d14
-0.5
d14
Λ4
0.5
-0.5
-0.1
1.0
Λ4
1.6
1.4
1.2
1.0
0.8
0.6
0.1
d15
0.5
1.00
0.95
0.90
0.85
0.80
0.75
Λ3
0.5
-0.5
0.5
0.2
0.1
d15
0.8
-1.0
0.2
G15
-0.5
0.5
d7-12
0.9
Λ3
Λ2
0.2
-1.0
-0.2
-0.4
1.0
Λ4
0.4
0.3
0.2
0.1
0.6
0.4
0.2
1.0
-0.2
0.5
-0.5
1.0
-1.0
Λ2
0.5
-1.0
0.6
-0.04
-0.2
1.0
d7-12
Λ3
d13
0.4
0.4
0.3
0.2
0.1
-1.0
0.5
-0.5
-0.10
G14
0.6
-0.4
-1.0
-1.0
1.0
0.4
-0.5
-0.2
G13
Λ4
0.5
1.0
d15
-1.0
-0.5
d15
Figure 4.9: The four eigenvalues for the coherency matrices associated with the nine
depolarization generators.
generator with a region for which all four coherency eigenvalues are greater than zero.
G7 (d7 ) through G14 (d14 ) do not have any region with coherency eigenvalues greater
than zero, indicating its intrinsic physicality. The depolarizing generators G7 (d7 )
through G14 (d14 ) become physical when combined with G15 (d15 ) as Gi (di )G15 (d15 )
for i = 7, ...14.
This yields a single region of di and d15 for which the Mueller
matrix is physical, because all four eigenvalues are greater than zero. These regions
are shown in figure 4.10 for combinations of d7 through d14 with d15 . The physical
regions are plotted in white, and non-physical regions are shown in black. Because
the maps for G7 , G8 , and G9 combined with G15 are identical, they have been
combined on one map. Similarly, the maps for G11 and G12 combined with G15
98
Figure 4.10: The physical region for each depolarizing generator G7 through G14
multiplicatively combined with G15 is shown in the white region. One axis represents
d15 and the other axis represents the other d-parameter of a corresponding generator.
G7 through G9 and G10 through G12 are shown on the same plot since their physical
regions are identical.
are identical.
The angle of this region was calculated analytically for each of
the depolarizing generators combined with G15 as follows. First, the two generator
matrices are multiplied,
Gi,15 = Gi (di ) · G15 (d15 ), i = 7, 8, ...14.
(4.17)
Then, for each value of i, the coherency matrix for Gi,15 is calculated and solved
for its four eigenvalues. The outer boundaries of the regions of physicality shown in
figure 4.10 are defined by two eigenvalues. As an example, the two eigenvalues for
99
the combination G7,15 which limit its physical region are
√
√
~λ7,15 = { 1 (−6d7 + 6d15 (1 + d7 )), 1 (− 6d15 (d7 − 1) + 6d7 )}.
12
12
(4.18)
Then these equations are set to zero and solved for d7 ,
√
d+
7
d−
7
6d15
,
6d15 − 6
√
6d15
.
= −√
6d15 − 6
= √
(4.19)
(4.20)
The angle of the physical region in this plane for G7,15 is calculated from the vector
−
angle between the two curves formed by d+
7 and d7 , in the limit as d15 approaches
zero. This procedure can be applied to the other generator combinations shown in
figure 4.10.
The angle at the origin for the the generator combinations Gi=7−13,15 is the
√
arctan(2 6/5), or approximately 44.415o . The angle at the origin for the generator
√
combination G14,15 is the arctan(4 2/7), or approximately 38.94o . These angles
suggest that the physical region of generator space in the vicinity of the identity
matrix is cone-like in 15 dimensions with a half angle of 44.415o in most of its cross
sections.
4.3 Interpretation of MM root parameters
4.3.1 Nondepolarizing matrix root parameters
The non-depolarizing Mueller matrix roots parameters are D0 through D6 . D0 is
the optical density, D1 , D2 , and D3 are the matrix roots diattenuation parameters,
and D4 , D5 , and D6 are the matrix roots retardance parameters.
100
4.3.1.1 Retardance parameters
The Mueller matrix roots parameters for retarders are straightforward to calculate
and interpret, because retardance is additive for a sequence of retarders with the
same fast-axis orientation or the same eigenpolarization. Thus the pth root of a
retarder with retardance δ results in a retarder with the same fast-axis orientation and retardance δ/p. This is equivalent to cutting the retarder into p pieces.
Calculating the matrix roots retardance parameters D4 , D5 , and D6 for a retarding Mueller matrix (either homogeneous or non-homogeneous) yields an order independent representation of the three dimensional retardance vector {δH , δ45 , δR }
(Goldstein, 2003),
D4 ≈ δH
(4.21)
D5 ≈ δ45
(4.22)
D6 ≈ δR ,
(4.23)
where δH , δ45 , and δR are expressed in radians.
The order independence of these parameters is an important point to highlight.
For pure retarders, equations 4.21 through 4.23 are equal to the retardance vector
as defined in (Goldstein, 2003). Because the matrix roots decomposition yields
an order-independent representation, the roots decomposition of a retarder mixed
with other polarization properties will yield different values for D4 , D5 , and D6
than the retardance vector yielded by the Lu-Chipman decomposition because the
Lu-Chipman decomposition is order-dependent (Ossikovski et al., 2008).
101
4.3.1.2 Diattenuation parameters
Like retardance, optical density is additive for a sequence of absorbers with the same
transmision-axis orientation. Therefore the matrix roots parameters for average
transmission and diattenuation, D0 , D1 , D2 , and D3 are defined analagously to
optical density. In the infinitesimal case they become equal to diattenuation.
The definition of diattenuation D as a function of maximum transmission Tmax
and minimum transmission Tmin is
D=
Tmax − Tmin
.
Tmax + Tmin
(4.24)
Given absorption coefficients αmax and αmin associated with Tmax and Tmin , diattenuation varies with distance z as
D(z) =
z
e−αmax z − e−αmin z
=
−
tanh
(α
−
α
)
.
max
min
e−αmax z + e−αmin z
2
(4.25)
Because diattenuation does not behave linearly when cascaded (in z), the matrix
roots parameters D0 through D3 do not correspond to the traditional definition of
average transmission and the diattenuation vector. The parameters D1 , D2 , and
D3 are in units of optical density like decibels, except the matrix roots diattenuation parameters are defined using the natural log. For a linear diattenuator with
transmission Tmax and Tmin oriented at angle θ, the transmission and diattenuation
102
parameters in the limit as the root-order p approaches infinity are
1
D0 = − ln(Tmax Tmin )
2
1
Tmax
D1 =
cos 2θ ln
2
T
min Tmax
D2 = cos θ sin θ ln
Tmin
(4.27)
D3 = 0.
(4.29)
(4.26)
(4.28)
The diattenuation parameters are expressed as the natural log of the ratio between
the maximum and minimum transmission. This is additive for aligned diattenuators,
and subtracts for crossed diattenuators. For this case of an ideal homogeneous
diattenuator,
Tmax = e−D0 +D1 sec 2θ = e−D0 +D2 csc 2θ and
(4.30)
Tmin = e−D0 −D1 sec 2θ = e−D0 −D2 csc 2θ .
(4.31)
The diattenuation is related to the matrix roots parameters D0 , D1 , and D2 as
D=
eD2 csc θ sec θ − 1
e2D1 sec 2θ − 1
=
.
e2D1 sec 2θ + 1
eD2 csc θ sec θ + 1
(4.32)
For a circular diattenuator with maximum transmittance Tmax and minimum trans-
103
mission Tmin , the rescaled roots parameters are
1
D0 = − ln Tmax Tmin
2
(4.33)
D1 = 0
(4.34)
D2 = 0
(4.35)
D3 = ln
Tmax
.
Tmin
(4.36)
Solving for maximum and minimum transmission using D0 and D3 results in
Tmax = e−D0 +D3 /2 and
(4.37)
Tmin = e−D0 −D3 /2 ,
(4.38)
and diattenuation
D=
eD3 − 1
.
eD3 + 1
(4.39)
4.3.2 Depolarizing matrix roots parameters
4.3.2.1 The diagonal depolarizer
The diagonal partial polarizer is an ideal starting point for the interpretation of the
depolarizing Matrix roots parameters. The most common form of a diagonal partial
depolarizer is

1

0

PD(a, b, c) = 
0


0

0 0 0

a 0 0

,
0 b 0


0 0 c
where a, b, and c are all positive real numbers.
(4.40)
104
Because PD(a, b, c) is a diagonal depolarizer it only has diagonal depolarization
matrix roots parameters D13 , D14 , and D15 . In the limit as p approaches infinity,
D0 through D12 are zero-valued, and the diagonal depolarization parameters are
1
(log a − log b)
2√
3
(log a + log b − 2 log c)
= −
6
1
= − √ (log a + log b + log c).
6
D13 =
(4.41)
D14
(4.42)
D15
(4.43)
D13 is a logarithmic measure of the relative strength between linear horizontal/vertical and linear 45o /135o diagonal depolarization. A positive D13 results from
a sample that is more diagonally depolarizing along the linear horizontal/vertical
axis. A negative D13 results from a sample that is more diagonally depolarizing
along the linear 45o /135o axis. D14 is a logarithmic measure of the relative strength
between the two linear diagonal depolarization axes and circular diagonal depolarization. Negative D14 results from a sample that is more diagonally depolarizing
along the circular axis, and positive D14 results from a sample that is more linearly depolarizing along the diagonal. D15 is a logarithmic measure of the total
depolarizing power on all three axes. It should always be positive-valued.
For an ideal depolarizer, when a = b = c = 0, D15 = ∞, and D13 = D14 = 0.
105
4.3.3 The diagonal depolarization tetrahedron
The diagonal partial depolarizer,


1
0
0
0




0 m11 0
0 


PD(m11 , m22 , m33 ) = 
,
0 0 m

0


22


0 0
0 m33
(4.44)
is only physical under certain conditions. The eigenvalues of its coherency matrix
must all be greater than or equal to zero in order for it to be a physical Mueller
matrix. This leads to the following restrictions on m11 , m22 , and m33 :
1 + m11 − m22 − m33 ≥ 0
(4.45)
1 − m11 + m22 − m33 ≥ 0
(4.46)
1 − m11 − m22 + m33 ≥ 0
(4.47)
1 + m11 + m22 + m33 ≥ 0.
(4.48)
These restrictions on the ranges of m11 , m22 , and m33 define a physical region formed
by a regular three-dimensional tetrahedron made of four planes,
m33 = 1 + m11 − m22
(4.49)
m33 = 1 − m11 + m22
(4.50)
m33 = m11 + m22 − 1
(4.51)
m33 = −(1 + m11 + m22 ).
(4.52)
106
The tetrahedron, shown in figure 4.11, has the following characteristics. Its top
point is (m11 = 1, m22 = 1, m33 = 1), the identity matrix in Mueller matrix space.
The three base points, point A=(m11 = −1, m22 = 1, m33 = −1), point B=(m11 =
1, m22 = −1, m33 = −1), and point C=(m11 = −1, m22 = −1, m33 = 1) all form
retarders in Mueller matrix space. Point A is a half wave linear retarder oriented
at 45o . Point B is a linear half wave retarder oriented at 0o . Point C is a circular
retarder with half a wave of retardance. All interior angles of the tetrahedron faces
are 60o .
The central point at the base of the tetrahedron, point D=(m11 = −1/3, m22 =
−1/3, m33 = −1/3) represents the minimum possible negative determinant for a
physical Mueller matrix. In Mueller matrix space, this matrix is also the average of
the Mueller matrices which form points A, B, and C on the tetrahedron:
LR(π, 0o ) + LR(π, 45o ) + CR(π)
= PD(−1/3, −1/3, −1/3).
3
(4.53)
The Euclidian distance between the minimum determinant point D and any corner
p
at the base of the tetrahedron (A, B, or C) is 2 2/3, while the Euclidian distance between the the minimum determinant point D and the center edge of one
p
tetrahedron wall is 2/3.
4.4 Applications for depolarization in optical design
One goal of this work is to provide tools for analyzing depolarization in optical
design. This section studies the depolarization resulting from the most common diattenuation and retardance polarization aberrations (McGuire and Chipman, 1994;
Chipman, 1985) in radially symmetric optical systems. These methods might be
107
1.0
m22 0.5
0.0
-0.5
-1.0
1.0
0.5
m330.0
-0.5
-1.0
-1.0
-0.5
0.0
m11
0.5
1.0
Figure 4.11: Physical Mueller matrix diagonal depolarization space forms a tetrahedron.
108
used to direct an optimization process towards minimizing these depolarization effects.
4.4.1 Diattenuation and retardance defocus
When a plane wave is incident on a spherical lens, there is a range in angles of
incidence along the surface of the lens. These angles of incidence form a rotationally
symmetric pattern, and due to the Fresnel and thin film equations, they produce a
variation in diattenuation and retardance as a function of pupil radius, which has
a quadratic term as its lowest order in an aberration expansion. This variation in
diattenuation and retardance with pupil radius is analagous to the defocus term
traditionally used in scalar aberration theory. Therefore these affects are separately
categorized as diattenuation defocus and retardance defocus.
The symbolic Mueller matrix for a linear diattenuator LD [q, r, θ] with maximum
transmission q and minimum transmission r at orientation θ is


(q
−
r)
cos
2θ
(q
−
r)
sin
2θ
0




√
√
2
2

(q + r − 2 qr) cos 2θ sin 2θ
0 
1 (q − r) cos 2θ (q − r) cos 2θ + 2 qr sin 2θ


.
√
√
2 (q − r) cos 2θ (q − r − 2 qr) cos 2θ sin 2θ (2 qr cos2 2θ + (q + r) sin2 2θ) 0 




√
0
0
0
qr
(4.54)
q+r
The Mueller matrix for diattenuation defocus across a pupil with axes x and y is
constructed from equation 4.54 by adding quadratic variation to q along x and y by
setting r = 1 − D0 (x2 + y 2 ), and writing the orientation in terms of x and y, where
θ = arctan (x, y). Then linear diattenation defocus can be written in function form
as
LD q = 1, r = 1 − D0 x2 + y 2 , θ = arctan(x, y) .
(4.55)
109
Tilt and piston occur as the center of the pupil moves off-axis, and are added by
shifting the quadratic terms and orientation by x0 and yo ,
LD q = 1, r = 1 − D0 (x − x0 )2 + (y − y0 )2 , θ = arctan(x − x0 , y − y0 ) . (4.56)
Similarly, the Mueller matrix for a linear retarder with retardance δ at orientation
θ is


0
0
0
1



2
2
δ
0 cos2 2θ + cos δ sin 2θ

sin
sin
4θ
−
sin
δ
sin
2θ


2
LR[δ, θ] = 
,
 (4.57)

0
2 δ
2
2
sin 2 sin 4θ
cos δ cos 2θ + sin 2θ cos 2θ sin δ 



0
sin δ sin 2θ
− cos 2θ sin δ
cos δ
so the Mueller matrix for retardance defocus is
LR δ = R0 x2 + y 2 , θ = arctan(x, y) .
(4.58)
Again, tilt and piston are added by shifting the quadratic terms and orientation by
x0 and y0 ,
LR δ = R0 (x − x0 )2 + (y − y0 )2 , θ = arctan(x − x0 , y − y0 ) .
(4.59)
Figure 4.12 illustrates the quadratic variation of the diattenuation defocus from
the center of the pupil outwards, as well as the shifted diattenuation defocus which is
a result of adding tilt terms x0 and y0 . For an incident horizontally polarized planewave, the on-axis diattenuation defocus pupil will transmit the highest percentage
of the incident polarization along the horizontal axis and the minimum percentage
110
Diattenuation Defocus
On-Axis
Linear Shift Along y-Axis
Off-Axis in x and y
Figure 4.12: Diattenuation defocus shifts along the y-axis and off-axis in x and y, illustrating the effect of tilt on diattenuation defocus in the pupil. Retardance defocus
looks identical to diattenuation defocus, but the effect is different. Diattenuation
defocus affects the transmission of polarization states, whereas retardance defocus
adds a given retardance to the transmitted polarization state.
of the incident horizontal polarization along the vertical axis.
A pupil line map of retardance defocus looks identical to the diattenuation defocus shown in figure 4.12, except the lines represent the fast-axis. While diattenuation
defocus affects the transmission and orientation of incident polarization states, retardance defocus adds polarization dependent phase and ellipticity to the transmitted
polarization state. On-axis, the retardance increases quadratically towards the edge
of the pupil and is rotationally symmetric. Off-axis, the retardance across the pupil
is assymmetric. Because the retardance increases quadratically from the center to
the edge of the pupil, the off-axis case is more retarding on the far edge of the pupil.
4.4.2 Depolarization effects from diattenuation and retardance defocus
The depolarization effects and their Mueller matrix roots parameters were studied
for these fundamental polarization aberrations. These polarization aberrations cause
polarization variations in the point spread function (PSF), derived by (McGuire
and Chipman, 1994). When a polarimeter measures the entire PSF on one pixel, it
111
averages the Stokes vectors over the PSF. Thus averaging the Mueller matrix over
the pupil relates input to measured Stokes parameters and describes the associated
depolarization. Numerical integration of diattenuation or retardance defocus across
the pupil provides insight on the cumulative effects of these polarization aberrations
in an optical system.
The Mueller matrices for diattenuation and retardance defocus (from equations
4.56 and 4.59 are each integrated across a unit circular pupil,
Z
1
√
Z
√
− 1−x2
−1
Z
1
−1
1−x2
√
Z
LD 1, 1 − D0 (x − x0 )2 + (y − y0 )2 , arctan(x − x0 , y − y0 ) dydx
(4.60)
1−x2
√
LR R0 (x − x0 )2 + (y − y0 )2 , arctan(x − x0 , y − y0 ) dydx, (4.61)
− 1−x2
where D0 = 0.3 and R0 = 1.
To understand the depolarization for off-axis fields, this operation is repeated for
a series of shifts in x0 and y0 in the upper right quadrant, moving the center of the
pupil off-axis. Figure 4.13 shows the diagonal depolarization parameters D14 and
D15 from the diattenuation defocus pupil, integrated at varying pupil shifts off-axis.
The field on the vertical axis of figure 4.13 is calculated from the radius of the shift
p
in x0 and y0 , x20 + y02 . The magnitude of the diagonal depolarization terms D14
and D15 generated by the diattenuation defocus decreases as the pupil shifts off-axis.
The depolarization is at its maximum when the pupil is centered on-axis because
the difference in transmission between the different regions of the pupil are at their
maximum. The depolarization is at its minimum when the pupil is centered off-axis
because the variation in diattenuation orientation is at its minimum.
Figure 4.14 shows the diagonal depolarization parameters D14 and D15 from the
retardance defocus pupil, integrated at varying off-axis pupil shifts. The magnitude
112
Diagonal Depolarization from Diattenuation Defocus
1.0
0.8
Field
0.6
D15
0.4
D14
0.2
0.0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Diagonal Depolarization
Figure 4.13: With diattenuation defocus, diagonal depolarization parameters D14
and D15 decrease in magnitude as the pupil center is shifted off-axis.
of the diagonal depolarization D14 and D15 is at its minimum when the pupil is
on-axis, and increases as the pupil shifts off-axis. The depolarization increases offaxis because the distribution of retardance in the pupil becomes asymmetric offaxis, and the fast axis varies less. Additionally, the amount of retardance increases
quadratically with the radius of the pupil. When the pupil is shifted so that its
center is almost at the edge of the aperture, the effective radius is larger, generating
this quadratic increase in retardance across the pupil. These two effects lead to
a roughly quadratic increase in magnitude of depolarization as the pupil moves
off-axis.
Figures 4.15 and 4.16 show cross-sections of the circular phase depolarization
D12 and diagonal depolarization D13 generated from shifting the pupil off-axis in
x0 .
Integrating diattenuation defocus and retardance defocus across the pupil only
results in diagonal depolarization (D13 , D14 , and D15 ) and circular phase depolarization (D12 ). This result is consistent with the averaging operations in Section
113
Diagonal Depolarization from Retardance Defocus
1.0
0.8
Field
0.6
D15
0.4
D14
0.2
0.0
0.0
0.5
1.0
1.5
2.0
Diagonal Depolarization
Figure 4.14: With retardance defocus, diagonal depolarization parameters D14 and
D15 increase in magnitude as the pupil center moves off-axis.
Depolarization from Diattenuation Defocus HCross-SectionL
0.6
0.5
x0
0.4
D12
0.3
0.2
D13
0.1
0.0
-0.004
-0.002
0.000
0.002
0.004
Depolarization
Figure 4.15: Circular phase depolarization D12 and diagonal depolarization D13 vary
with pupil shift x0 .
114
Depolarization from Retardance Defocus HCross-SectionL
1.0
0.8
0.6
x0
D12
0.4
D13
0.2
0.0
-0.025
-0.020
-0.015
-0.010
-0.005
0.000
0.005
0.010
Depolarization
Figure 4.16: Circular phase depolarization D12 and diagonal depolarization D13 vary
with pupil shift x0 .
3.5.2. In the diattenuation defocus integration, linear diattenuation was averaged
with linear diattenuation at varying orientations, so we expect diagonal depolarization and circular phase depolarization to result. Similarly, when retardance defocus
was integrated across the pupil, linear retardance was averaged with linear retardance at varying orientations. This produces circular phase depolarization as well.
In a realistic optical system, a surface would experience both diattenuation defocus
and retardance defocus, which would produce diagonal depolarization and circular
amplitude depolarization D9 .
115
CHAPTER 5
POLARIZATION SYNTHESIS BY COMPUTER GENERATED
HOLOGRAPHY USING ORTHOGONALLY POLARIZED AND CORRELATED
SPECKLE PATTERNS
This chapter is adapted from H. Noble et. al., “Polarization synthesis by computer
generated holography using orthogonally polarized and correlated speckle patterns,”
Opt. Lett. 35, 3423-3425 (2010).
5.1 Introduction
Computer-generated holograms (CGHs) can efficiently produce bandwidth-limited
arbitrary irradiance in reconstructed images (Lohmann and Paris, 1967),Poon
(2006). This Letter addresses application of CGHs to the formation of a specific irradiance and polarization distribution in the image. Several authors have discussed
polarization selective CGHs, where orthogonal illumination polarization states act
independently. For example, Hossfeld et al. proposed a system with a birefringent
crystal for this purpose (Hossfeld et al., 1993). Xu et al. demonstrated polarization
selective CGHs that apply an independent phase profile to orthogonal polarization
states using birefringent substrates joined face to face (Xu et al., 1995), as well as
form birefringent nanostructures (Xu et al., 1996). These methods fail to produce
arbitrary polarization and irradiance in the image, because the phase relationship
of the orthogonal images is not considered in the design. One method for producing
arbitrary polarization and irradiance is image segmentation, where different por-
116
tions of the image are generated from separate CGHs illuminated with the desired
polarization state of each image portion (Tsuji, September 4 2007). But in order
to realize a continuously changing polarization direction, infinite sets of sub-CGHs
are required. Therefore, a more elegant method is to synthesize an arbitrary polarization distribution from two orthogonally polarized CGH images. The advantage
of this approach is its simplicity. Todorov et al. showed that analog recording of
two holograms in a photoanisotropic medium can produce a coherent combination
of polarization states in the overlap region of the diffracted images, depending on
the polarization state of the illuminating beam (Todorov et al., 1985). However,
Todorovs analog recordings did not demonstrate computer design of an arbitrary
image pattern, and the image polarization state did not vary in the overlap region of
orthogonal polarization states. This Letter demonstrates design, construction, and
measurement of a synthesized tangentially polarized annulus using an interlaced
polarization CGH (PCGH) that is illuminated with alternating columns of x- and
y-polarized light. A synthesized hologram consisting of a correlated speckle pattern
is produced from a coherent combination of these two orthogonal linear polarization
states in the image.
5.2 PCGH illumination
The system illumination is a 15 mm diameter spatially uniform plane wave from
a linearly polarized 633 nm He-Ne laser. As shown in Fig. 5.1, the laser power
is divided equally between x- and y-polarized channels by placing a polarization
beam splitter (PBS) after a polarizer oriented at 45o . Beams reflect from 45o fold
mirrors and transmit through offset 50% duty cycle Ronchi rulings with 67.2 µm
periods (which contain one dark and one light stripe). A PBS recombines the
117
Figure 5.1: The interlaced PCGH is illuminated with shifted cross-polarized Ronchi
rulings, and its reconstruction and polarization distribution are measured.
orthogonally polarized channels. The optical path length of one channel is adjusted
with a piezoelectric transducer on the fold mirror, so that without the Ronchi rulings,
the combined beams interfere to form a linearly polarized state oriented at 45o to
the x-axis. The Ronchi rulings are used as arrays of masks and apertures that are
imaged onto the PCGH with a 4f system. The rulings are offset from each other by
half a period to provide alternating stripes of x and y polarization, which correspond
to the interlaced channels of the PCGH. A 12-bit CCD camera is used at the back
focal plane of the imaging lens to view the reconstructed image. The camera is used
at a plane conjugate to the PCGH and Ronchi rulings to view and align them. A
linear polarizer is rotated before the camera to analyze the reconstructed images
polarization properties.
The target image was selected to be a tangentially polarized ring, as shown in Fig.
5.2, because it is a continuously varying polarization. The ring can be decomposed
118
Figure 5.2: x- and y-polarized components add to form a tangentially polarized
annulus.
into x- and y-polarized components, where
0
Ux0 = A0x eiφx = −gs sin θ,
0
Uy0 = A0y eiφy = gs cos θ.
(5.1)
(5.2)
Ax0 and Ay0 are amplitude functions, φ0x and φ0y are phase functions, and gs is a
common structure function that consists of a binary ring with an inside diameter
of 80% of the outer edge. All of these functions depend on r and θ. Overlap of the
~ 0.
two ideal coherent states produces the ideal tangentially polarized ring U
5.3 PCGH diffuser design
A flowchart of the modified Gerchberg-Saxton design method (Gerchberg and Saxton, 1972) is shown in Fig. 5.3. j is the iteration number, where N iterations are
performed. Diffusers for Fourier CGHs generally have the advantage of reducing
CGH- modulus dynamic range, but they alter polarization. For our application,
119
using a common diffuser for the two po- larization channels allows dynamic-range
reduction without affecting polarization. An initial common diffuser is defined as an
array of uniformly random phase pixels over [−π, π]. In the initial step (j = 1) of
the calculation loop, ideal images defined by Eqs. 6.1 and 6.2 are multiplied by the
random diffuser to create two parallel channels (x and y) for the calculation. In Step
2, Fourier transforms are applied to each channel, yielding the encoded wavefronts
for the PCGHs. In Step 3, the modulus of each transform is set to unity, and each
transform is interlaced according to the alternating stripes of x- and y-polarizations
from the Ronchi illumination. If j = N , the calculation is stopped, and individual
phases are used to fabricate the interlaced PCGH. In Step 4, inverse Fourier transforms are applied to the phase distributions of the results from Step 3, bringing the
calculation back into the space of the image. Ideal object phases φ0x and φ0y , derived
from Eqs. (1) and (2), are then removed from image estimates Ux1 and Uy1 in Step
5 to extract the diffuser-only phase. In Step 6, the resulting complex amplitudes
~ 1 . This step contains a novel departure from Gerchbergare averaged to produce U
Saxton algorithms that consider only independent channels. The common diffuser
~ 1 . For the next loop (j = 2), φ1 is used
phase φ1D is simply the phase angle of U
D
as the starting point for the diffuser in Step 1. After typically N = 50 iterations,
the diffuser converges to produce acceptable characteristics of image estimates Uxj
and Uyj , when the phases calculated in Step 3 are used as the individual components of the PCGH. Application of an aperture function or introduction of phase
quantization can be performed at Step 3.
Averaging complex amplitudes in Step 6 assures that the target image for the
next iteration contains correlated speckle between the two channels on a microscopic
scale. Although multiplication of the ideal objects in Step 1 of the next iteration
120
Figure 5.3: Flowchart of the PCGH design method.
does not guarantee that correlation is maintained in the PCGH, a sufficiently high
number of iterations N consistently shows that speckle correlation can be produced
with sufficient quality.
5.4 PCGH fabrication
The PCGH is fabricated in polyimide photoresist on a fused silica substrate. The
photoresist is gray-scale patterned with the University of Arizona Maskless Lithography Tool, a direct laser writer that scans the photoresist with a focused UV light
spot of varying power (Tamkin et al., 2003). After the photoresist is exposed, it is
developed in a 3 : 1 solution of deionized water and developer for 2 to 4 minutes.
Gray-scale exposure of the positive resist results in topographical features after development, producing phase changes in transmission through the PCGH when used
in the reconstruction setup. The samples are postbleached with high-intensity halogen light after fabrication to stabilize the refractive index.
121
PCGH matrices for x- and y- polarizations are interlaced as shown in Fig. 5.1,
where the gray scale corresponds to PCGH phase (0 =black, 2π =white). Each
interlace stripe is 33.6 µm wide, which corresponds to four 8.4 µm pixels of the
component PCGH. The 67.2 µm periods of the Ronchi rulings are designed to overlap
exactly with y-oriented columns of the PCGH. x- (y-) polarized light illuminates only
the x- (y-) component columns of the PCGH. A total of 512 × 512 pixels in the
PCGH yields a side dimension of approximately 4.3 mm. The rectangular PCGH
interlacing produces diffracted orders in the image. In this analysis, only zero-order
properties are considered. The zero-order diffraction pattern is a ring diverging at
0.5o full width. Higher diffraction orders lie outside the zero-order ring.
5.5 PCGH measurement and reconstruction
The linear Stokes vector {S0 , S1 , S2 }T of the image is measured by rotating a linear
analyzer to orientations of 0o , 45o , 90o , and 135o in front of the CCD camera and
applying the following equations (Goldstein, 2003):



S0 = 41 (I90o + I0o + I45o + I135o )



S1 = I0o − I90o




 S2 = I45o − I135o
(5.3)
The measured linear Stokes vector image in Fig. 5.4 displays a tangential orientation, demonstrating the achievement of polarization synthesis. Figure 5.5 shows
that the linear polarization orientation is tangential to the annulus, with horizontal
orientation at the top of the annulus and vertical orientation at the sides of the
annulus. The measured polarization orientation is nonuniform, differing by several
degrees from the desired orientation at the sides of the annulus.
122
Figure 5.4: Normalized linear Stokes vector components demonstrate polarization
synthesis.
Figure 5.5: Polarization orientation rotates around the annulus.
123
Degree of linear polarization (DoLP) is determined in terms of the Stokes vector
as DoLP = (S12 +S22 )1/2 /S0 . DoLP=1 is linearly polarized light and DoLP=0 is either
unpolarized or circularly polarized. The average DoLP over the annulus is 0.88, with
a standard deviation of 12.8%. (The DoLP distribution is non-symmetric about
the mean.) Unpolarized regions result from uncorrelated speckle patterns, while
circularly or elliptically polarized regions result from correlated speckle patterns
that are out of phase with each other. The algorithm outlined in Fig. 5.3 generates
correlated speckle patterns from each polarization component of the PCGH, but
experimental constraints preclude an ideal result. The synthesis regions (at ±45o
and ±135o around the ring) have a lower DoLP than the vertical and horizontal
regions of the annulus because the DoLP in these regions is dependent on correlation
of speckle from the two orthogonal polarizations.
The ratio of polarization (RoP) is a metric to evaluate the quality of tangential
polarization around the annulus as a fraction of flux in the reconstructed hologram,
where
1
IT
=
RoP =
S0
2
S2
S1
cos 2θ +
sin 2θ .
1+
S0
S0
(5.4)
IT is the transmission through a properly oriented tangential analyzer, orientation angle with respect to the x-axis is θ, and S0 is total irradiance. Average RoP
around the ring is 88%, with a standard deviation of 6%.
5.6 Conclusions
In conclusion, the first demonstration, to the best of our knowledge, of computerdesigned polarization synthesis is achieved with a tangentially polarized annulus
that combines two linearly polarized images from interlaced CGHs. This new type
of hologram is called a PCGH. The design algorithm and fabrication produced cor-
124
related speckle in the experimental image. The correlated speckle resulted in a high
degree of properly oriented linear polarization.
125
CHAPTER 6
SQUARE WAVE RETARDER FOR POLARIZATION COMPUTER
GENERATED HOLOGRAPHY
This chapter is reformatted from H. Noble et. al., “Square-wave retarder for polarization computer generated holography,” Appl. Opt. 50, 3703-3710 (2011).
6.1 Introduction
Computer-generated holograms (CGHs) can efficiently produce bandwidth-limited
arbitrary irradiance in reconstructed images (Lohmann and Paris, 1967; Dallas,
2006; Schnars and Jueptner, 2005). Several authors have discussed polarization
selective CGHs, where orthogonal illumination polarization states remain spatially
distinct. For example, Hossfeld et al. proposed a system with a birefringent crystal
for this purpose (Hossfeld et al., 1993). Xu et al. demonstrated polarization selective
CGHs that apply an independent phase profile to orthogonal polarization states
using birefringent substrates joined face to face (Xu et al., 1995) as well as formbirefringent nanostructures (Xu et al., 1996). It has also recently been shown that
polarization-selective CGHs can be fabricated with arbitrary amplitude and phase
of each pixel by recording a birefringent structure with a defined retardance and
orientation of the fast-axis into a photopolymer film (Fratz et al., 2009). However,
these methods fail to produce arbitrary polarization and irradiance in the image
because the phase relationship of the orthogonal images is not considered in the
design. One method for producing arbitrary polarization and irradiance is image
126
Figure 6.1: Interlaced x and y-polarized components add to form a tangentially polarized annulus. Overlap of these two coherent states produces the ideal tangentially
~ o with a continuously varying polarization orientation.
polarized ring U
segmentation, where different portions of the image are generated from separate
CGHs illuminated with the desired polarization state of each image portion (Tsuji,
September 4 2007). But in order to realize a continuously changing polarization
direction, infinite sets of sub-CGHs are required. A more elegant method that
avoids this limitation is to synthesize an arbitrary polarization distribution from
two orthogonally polarized CGH images. The advantage of the synthesis approach
is its simplicity.
In recent work, a new type of hologram was demonstrated. The hologram
achieved a synthesized tangentially polarized annulus using an interlaced CGH illuminated with alternating columns of x- and y-polarized light (Noble et al., 2010).
This new type of hologram is called a polarization computer generated hologram
(PCGH). A synthesized hologram consisting of a correlated speckle pattern was
produced from a coherent combination of these two orthogonal linear polarization
states in the image.
127
The target image in (Noble et al., 2010) was selected to be a tangentially polarized ring, as shown in Fig. 6.1, because it is a continuously varying polarization.
The ring can be decomposed into x- and y-polarized components, where
0
Ux0 = A0x eiφx = −gs (r) sin θ,
0
Uy0 = A0y eiφy = gs (r) cos θ.
(6.1)
(6.2)
Ax0 and Ay0 are amplitude functions, φ0x and φ0y are phase functions, and gs is a
common structure function that consists of a binary ring with an inside diameter of
80% of the outer edge. Overlap of the two ideal coherent states produces the ideal
tangentially polarized ring .
In (Noble et al., 2010) orthogonally polarized striped illumination on an interlaced PCGH was generated from two complementary Ronchi-ruling amplitude masks
in orthogonally polarized optical paths that were combined with a polarizing beam
splitter (PBS). This proof-of-concept experiment provided a basic demonstration
of the PCGH concept. Ultimately, the final goal of this research is to construct
a PCGH from a single, integrated optical element. The intermediate step, discussed here, is to develop a single illuminator element for the interlaced PCGH.
This optical element is called a square-wave retarder, due to its alternating stripes
of half-wave/integer-wave retardance. The square-wave retarder replaces the orthogonally polarized Ronchi-ruling amplitude masks from previous work in (Noble
et al., 2010), by providing alternating stripes of orthogonally polarized illumination
to the PCGH.
This paper describes the design, fabrication, and performance of an etched calcite
square-wave retarder used as an illuminator for an interlaced PCGH. Together, these
components produce a speckled, tangentially polarized PCGH diffraction pattern at
128
the image plane. Section 6.2 describes the experimental geometry and design technique. Section 6.3 discusses the square-wave retarder design and fabrication, and
Section 6.4 presents measurement results. Section 6.5 lists alignment and illumination errors, and Section 6.6 contains conclusions from this work.
6.2 Experimental geometry
A polarization compensator system is used to provide an arbitrary polarization state
incident on the square-wave retarder. As shown in Fig. 6.2, laser power (λ= 633nm)
is divided equally between x-polarized and y-polarized channels by placing a polarization beam splitter (PBS) after a polarizer oriented at 45o in the x-z plane. Beams
reflect from fold mirrors and recombine at a second PBS, forming a Mach-Zehnder
compensator. One fold mirror is placed on a piezoelectric transducer (PZT) for
output polarization adjustment. For an ideal square-wave retarder, the polarization
state resulting from the compensator is a linearly polarized plane oriented at 45o in
the x-y plane.
The polarization state incident onto the calcite is adjusted to compensate for
thickness and wedge errors in the calcite square-wave retarder, which is arranged
such that the crystals fast axis is oriented in the x-direction. Fold mirror tilt corresponds to a linear shift in phase, counteracting any linear change in retardance
from residual wedge in the calcite. Mirror translation with the PZT corresponds to
adding or subtracting retardance, which balances out excess retardance from nonoptimal calcite thickness. The compensators polarization state is adjusted until
the polarization states exiting the square-wave retarders grooves alternate in stripes
oriented along the y-axis, which is rotated at 45o with respect to the y-axis.
A 4f optical system images transmitted light from the square wave retarder onto
129
Figure 6.2: A polarization compensator system (enclosed in the dashed line region)
provides an arbitrary polarization state incident on the etched calcite square wave
retarder. A 4f optical system images the square wave retarder to illuminate the
interlaced CGH, providing alternating stripes of x and y polarization. A CCD
camera is used at the back focal plane of an imaging lens to view the reconstructed
image.
130
the interlaced PCGH, providing alternating stripes of x and y polarization. The
PCGH is also aligned along the x-y axes so that the interlaced columns are aligned
with their corresponding square wave retarder grooves. A 12 bit, 2 Megapixel Olympus Microfire CCD camera with less than 0.5% polarization sensitivity is used at
the back focal plane of an imaging lens to view the reconstructed image. The camera is used at a unit-magnification plane conjugate to the PCGH and square-wave
retarder to view and align those components. A rotatable linear polarizer (analyzer)
and quarter-wave retarder (QWP) are placed before the camera to measure Stokes
parameters of the reconstructed image and to align the compensator system.
PCGH matrices for x and y polarizations are interlaced, as shown in Fig. 6.2,
where gray scale corresponds to PCGH phase (0 = black, 2π = white). Each PCGH
interlace stripe is 33.6 µm wide, which corresponds to four 8.4 µm pixels of the
component PCGH. The 67.2 µm period of the square-wave retarder overlaps with
y-oriented columns of the PCGH. Thus, x-polarized light illuminates only the xcomponent columns of the CGH and y-polarized light illuminates the y-component
columns. Overall PCGH dimensions are 4.3mm x 4.3mm with a total of 512 x 512
pixels. Rectangular PCGH interlacing produces diffracted orders in the image. The
zero-order diffraction pattern is an annular ring that diverges at a full angle of 0.5o .
Higher diffraction orders lie outside the zero-order ring and are not considered in
this work.
The PCGH diffuser is designed using a modified Gerchberg-Saxton technique
(Noble et al., 2010; Gerchberg and Saxton, 1972). An initial common diffuser is
defined as an array of uniformly random phase pixels over [−π, π]. In the initial step
of the calculation loop, ideal images defined by Eqs. 6.1 and 6.2 are multiplied by the
random diffuser to create two parallel channels (x and y) for the calculation. Then,
131
Fourier transforms are applied to each channel, which convert the calculation in to
the space of the PCGH. Next, the modulus of each transform is set to unity, and each
transform is interlaced according to the alternating stripes of x and y polarizations
from the square-wave retarder illumination. The inverse Fourier transforms are
applied to the phase distributions of the results, which bring the calculation back
into the space of the image. Ideal object phases φ0x and φ0y , derived from Eqs.
6.1 and 6.2 are removed from image estimates in order to extract the diffuser-only
~ 1 . This
phase. Next, the resulting complex amplitudes are averaged to produce U
step contains a novel departure from Gerchberg-Saxton algorithms that consider
only independent channels. The common diffuser phase is simply the phase angle of
~ 1 . For the next iteration loop, this common diffuser phase is used as the starting
U
point for the diffuser. After typically 50 iterations, the diffuser converges to produce
acceptable characteristics of image estimates. Individual phases of the two channels
are used to fabricate the interlaced PCGH. For our application, using a common
diffuser for the two polarization channels allows dynamic-range reduction without
affecting polarization. Using this diffuser algorithm from (Noble et al., 2010) assures
that speckle patterns generated from each polarization component of the CGH are
correlated in the image. This correlation is critical, because complex fields of the
interlaced x and y-polarized components do not add to form a tangentially polarized
annulus unless individual speckles from the two components overlap and are in phase
to within a fraction of the wavelength.
An advantage of this geometry over previously reported work (Noble et al.,
2010) is that the calcite square-wave retarder and PCGH components are on a
common path for x and y polarizations. Moreover, the 4f system between them is
implemented for ease of experimental adjustment and alignment tolerance testing.
132
In a commercial realization, the two components could be combined into a single
plate without the 4f system between them.
6.3 Calcite square-wave retarder design and fabrication
The ideal square-wave retarder is designed to generate alternating stripes of orthogonal linear polarizations when illuminated with a linearly polarized plane wave
oriented at 45o with respect to the crystals fast axis, as illustrated in Fig. 6.2.
Figure 6.3 shows the construction of the square-wave retarder. The total substrate
thickness corresponds to an integer number of waves of retardance. Grooves are
etched into the substrate to the base thickness level in order to produce half-wave
plate (HWP) ridges that rotate the incident linear polarization by 90o . The grooves
are filled with oil that matches the calcites extraordinary refractive index.
The ideal square-wave retarder has no residual wedge in the base. However, the
fabricated square-wave retarder has both thickness errors in the base and a small
wedge error. In principle, the base thickness of the square wave retarder is arbitrary
if the incident polarization state can be tuned with the compensator. One wave of
retardance in calcite at normal incidence is generated by each 3.7 µm of material,
making it unfeasible to polish to 1/100 of a wave precision. The residual wedge
corresponds to a linear phase shift along the un-etched surface of the square-wave
retarder, and is typically under 10 µm across the width of the substrate. In order
to produce alternating stripes of x and y polarizations, the etched calcite must be
illuminated with a matched elliptical polarization state that exhibits a linear phase
shift, depending on how many non-integral waves of retardance are produced by the
base thickness and the magnitude of the wedge. This elliptical polarization state
is produced with the compensator by adjusting the optical path difference between
133
Figure 6.3: The calcite square wave retarder generates alternating stripes of orthogonal linear polarizations when illuminated with a linearly polarized plane wave
oriented at 45o with respect to the crystals fast axis. Grooves are etched into the
substrate to the base thickness level and are filled with index matching oil to produce
half-wave plate ridges which rotate the incident linear polarization by 90o .
134
the two arms and adjusting the fold mirror tilt until polarization states exiting the
calcite are optimized for x and y polarization. Quality of the x and y polarization
states is determined by observing the square-wave retarder image through the CCD
without the PCGH in the system and aligning the analyzer separately to each desired
linear state. The QWP is removed for this alignment procedure.
The square-wave retarder is made from a naturally formed calcite crystal. A 3
mm thick calcite substrate is cut from the crystal using a TechCut precision sectioning saw. The substrate is mechanically lapped and polished to a base thickness
of approximately 500 µm and then chemically polished in a weak solution of HCl
and deionized water, which achieves an average surface roughness of 26 nm.
The calcite substrate is cleaned in four stages using acetone, isopropanol, deionized (DI) water, and a plasma cleaning chamber. It is then glued onto a fused silica
flat substrate with UV epoxy so that the optical axis of the calcite plate is parallel
to the diagonal of the rectangular fused silica plate. A binary AZ3312 photoresist
is spin coated at 7500 rpm for 30 seconds onto the calcite surface, and the coated
calcite is soft baked at 85o C for one minute. The photoresist is written with 128
lines of 33.6 µm width parallel to one side of the fused silica plate using the University of Arizona Maskless Lithography Tool (Tamkin et al., 2003), making a square
exposed region with a length of 4.3 mm on each side. The sample is developed in
AZ300MIF for 17 seconds. It is wet etched in 1 part 37% HCl to 5000 parts DI
water in small steps until the correct groove depth of 1.85 µm (plus the thickness of
the photoresist) is achieved, as measured with a Veeco NT9800 profilometer. This
processing requires a total etching time of 50 minutes.
The final fabrication step is to fill the grooves with an index matching fluid,
corresponding to the extraordinary index of the calcite. A fused-silica flat is glued
135
over the fluid filled grooves with UV epoxy. The index matching fluid is necessary
so that the orthogonal linearly polarized stripes remain in phase with each other.
Without index matching fluid, the phase of the y-polarized output from the square
wave retarder is retarded by λ/10 with respect to the x-polarized output, because
the optical path length (OPL) of the y-polarized output is larger than the OPL of the
x-polarized output. If the orthogonal illumination stripes do not remain in phase,
their complex fields will add to form an elliptically polarized field in the PCGH
reconstruction, which reduces the percentage of the reconstruction irradiance in the
correct polarization state.
In order to examine the quality of the etched calcite grooves, a sacrificial square
wave retarder was coated with platinum, cleaved perpendicularly to the groove orientation, and examined with a scanning electron microscope (SEM). The SEM images
in Fig. 6.4 show some rounding in the groove corners, as well as a slightly deeper etch
depth in those corners. Overall, the groove surfaces remain smooth and uniform.
A dual rotating retarder Mueller matrix imaging polarimeter (MMIP) is used
to measure polarization properties of several 1 mm2 sections over the aperture of
the square-wave retarder (Smith, 2002). Figure 6.5 shows the measured retardance
over one 1 mm horizontal cross-section from the center of the sample. Retardance
associated with the HWP ridges varies between approximately 160o and the ideal
180o . Ridge retardance at the edge of the sample ranges between 140o and 160o ,
which can be attributed to non-uniform etch rates between the center and edges
of the sample. A 160o ridge retardance results in 17.6% ellipticity, while 140o of
retardance results in 36.4% ellipticity. These elliptical components contribute to
a slightly lower degree of linear polarization (DoLP) in the PCGH illumination
compared to ideal conditions.
136
Figure 6.4: An SEM cross-section of the fabricated calcite for the square-wave retarder. The grooves are smooth and uniform, although there is some rounding of
the corners.
6.4 Measured polarization properties of the reconstructed image
The Stokes vector {S0 , S1 , S2 , S3 }T of the annular PCGH reconstruction is measured
by rotating a quarter-wave retarder (QWP) and linear analyzer, which are placed in
front of the CCD camera, at several distinct orientations according to the following
formula (Goldstein, 2003):



S0





 S1


S2





 S
3
=
1
[I(90o )
4
+ I(0o ) + I(45o ) + I(135o )]
= I(0o ) − I(90o )
o
(6.3)
o
= I(45 ) − I(135 )
=
1
[I(45o , 90o )
p2
− I(135o , 90o )]
where I(θ) is the intensity measured with a polarizer oriented at θ and I(θ, φ) is
the intensity measured with a polarizer orientated at θ and a retarder oriented at φ.
137
Figure 6.5: Retardance of a 1 mm wide horizontal cross section at the center of the
calcite square wave retarder sample was measured at 632.8 nm using the Mueller
matrix imaging polarimeter. Retardance associated with the half wave plate ridges
varies between approximately 160o and the ideal 180o in the center of the sample.
138
Figure 6.6: The measured Stokes vector for the PCGH reconstruction shows that the
polarization is linearly tangential with some circular polarization dispersed throughout the annulus. S3 shows that the presence of circular polarization is most notable
in the synthesis regions. S1 , S2 , and S3 are normalized to S0 , which remains unnormalized.
The transmittance factor of the retarder is p2 , which accounts for absorption and
surface reflections.
The measured and Stokes vector of the PCGH in Fig. 6.6 shows that the polarization is primarily tangential in orientation. S0 remains unnormalized, while other
terms are normalized with respect to S0 . At the top and bottom of the annulus in
the s1 = S1 /S0 image, the value is nearly +1, corresponding to horizontally polarized light, while at the left and right sides the s1 value is nearly -1, corresponding to
vertically polarized light. Analogously, the s2 = S2 /S0 image shows that the light
is 45o polarized in the upper right and lower left corners of the annulus, and 135o
polarized in the upper left and lower right corners of the annulus. The image shows
that the presence of circular polarization is most notable in the synthesis regions.
Figure 6.7 shows the orientation ϕ of the major axis of the polarization ellipse
139
Figure 6.7: The orientation ϕ of the major axis of the polarization ellipse around the
annulus is calculated from the normalized Stokes vector. The measured polarization
orientation of the PCGH reconstruction is tangential. Its orientation differs by
several degrees from the desired orientation at the sides of the annulus.
calculated from the normalized Stokes vector:
ϕ=
S2
1
tan−1 .
2
S1
(6.4)
The orientation is tangential, with horizontal orientation at the top of the annulus
and vertical orientation at the sides of the annulus. The measured polarization
orientation differs by several degrees from the desired orientation at the sides of the
annulus, which is visible in Fig. 6.7 and in the Stokes vector element s1 of Fig.
6.6. Simulations verify that the non-uniform orientation around the annulus can be
attributed to errors in the orientation of the calcite optical axis with respect to the
etched stripes, which is further discussed in Section 6.5.
Figure 6.8 shows the degree of linear polarization (DoLP = (S12 + S22 )1/2 /S0 )
over the annulus, where DoLP =1 represents linearly polarized light and DoLP=0
represents either unpolarized or circularly polarized light. The average DoLP over
140
Figure 6.8: The pixel-by-pixel degree of linear polarization (DoLP) for the PCGH
reconstruction. The average DoLP over the annulus is 0.81. The synthesis regions
have a lower DoLP than the vertical and horizontal regions of the annulus because
the DoLP in these regions is dependent on a high degree of speckle correlation from
the two orthogonal polarizations.
the annulus, weighted by irradiance, is 0.81 with a spatial standard deviation of
20%. The DoLP is less than 1 partly on account of the circularly polarized component s3 = S3 /S0 , as the average degree of polarization (DoP) over the annulus is
0.895. The spatial standard deviation of 20% is due to very low DoLP values in the
dim regions between speckles. Unpolarized regions result from uncorrelated speckle,
while circularly or elliptically polarized regions result from correlated speckle that
are out of phase with each other. The synthesis regions (at ±45o and ±135o around
the ring) have a lower DoLP than the vertical and horizontal regions of the annulus,
because the DoLP in these regions is dependent on a high degree of speckle correlation between the two orthogonal polarizations. The speckles are well correlated,
thus 89.5% DoP is achieved. The PCGH diffuser design algorithm from (Noble et al.,
2010) generates correlated speckle patterns from each polarization component of the
PCGH, but experimental constraints preclude an ideal result. These constraints are
141
further discussed in Section 6.5.
The ratio of polarization (RoP) is a metric defined to calculate the fraction of
tangentially polarized light around the annulus relative to total flux in the reconstructed image. RoP=1 is the ideal case (when 100% of the light is tangentially
polarized) and RoP=0.5 represents unpolarized or circularly polarized light, half of
which is in the desired tangential state. The RoP can be calculated pixel-by-pixel
from the measured (un-normalized) Stokes vector and the angular location θ in the
annulus, where
1
IT
=
RoP =
S0
2
S1
S2
1+
cos 2θ +
sin 2θ .
S0
S0
(6.5)
IT is the irradiance transmitted through a properly oriented tangential analyzer, S0
is total irradiance, and the orientation angle in the annulus with respect to the x-axis
is θ. The measured RoP for each pixel is shown in Fig. 6.9. The RoP is noticeably
lower in the synthesis regions of the annulus and in the background regions where
the speckle is not present. Average RoP around the ring is 84% with a standard
deviation of 6.9%.
In previous work (Noble et al., 2010), the PCGH reconstructions average DoLP
was 0.88, and the RoP was 88%. These values compare to a DoLP of 0.81 and RoP
of 84% in this study, where the square-wave retarder provides orthogonally polarized
illumination in place of the orthogonally polarized Ronchi-ruling amplitude masks
from (Noble et al., 2010). Systematic errors from the compensator are present in
both PCGH illumination systems, but the calcite square wave retarder in this study
adds fabrication errors and increased opportunities for optical misalignments. These
variables account for the slight decrease in DoLP and RoP compared to (Noble et al.,
2010).
142
Figure 6.9: Pixel-by-pixel ratio of tangential polarization (RoP) to flux for the
PCGH reconstruction. The RoP is noticeably lower in the 45o and 225o regions of
the annulus and in the dim regions between speckles. Average RoP around the ring
is 84%.
6.5 Alignment and illumination errors
Performance of the PCGH system depends on precise opto-mechanical alignment
of the compensator, the square-wave retarder and the CGH. Performance is also
affected by defects and fabrication tolerances in the square-wave retarder. Imperfections in the system produce decreased phase correlation in the PCGH reconstructions speckle pattern, which decreases the DoLP, the RoP, and contributes to the
presence of circular polarization.
A simulation which conceptually illustrates ideal output of the square-wave retarder is shown in Fig.6.10(a). If the square-wave retarders optical axis orientation
with respect to the half-wave stripes is not exactly 45o (as shown in Fig. 2), polarization orientation exiting the half-wave stripes rotates, but polarization orientation
exiting the full wave stripes does not. An error of ±θ in the calcites crystal axis orientation with respect to the etched stripes results in a ±2θ error in the y component
of the polarization transmitted through the calcite, as illustrated in Fig. 6.10(b) for
143
Figure 6.10: Simulation of square-wave retarder output. (a) In an ideal optical system, the polarization orientation exiting the calcite square wave retarder alternates
between x and y according to the grooves and ridge stripes; (b) An extraordinary
axis orientation error of 10o with respect to the etched stripes causes an orientation
error of 20o in the y polarized component exiting the calcite.
an exaggerated optical axis misalignment of +10o . Actual fabrication tolerance for
this alignment is approximately ±2o .
Wedge in the base thickness, shown in Fig. 6.3, produces a linear variation in
retardance across the square-wave retarder substrate. This is shown in the measured result of Fig. 6.11(a) for a 1 cm2 area of bare calcite substrate. This linear
variation in retardance across the substrate produces a variation in ellipticity across
the polarization states transmitted through the square-wave retarder, as shown in
the simulation results of Fig. 6.11(b). Fig. 6.11(b) reveals significant variation in
the x direction. However, tilt in a compensator mirror can produce an opposite
linear phase shift across the field of illumination and introduce opposite ellipticity
variations. Therefore the mirror can be used to compensate effects of the residual
calcite wedge. Environmental factors such as temperature fluctuations and mechanical vibrations from the laboratory environment can cause temporal fluctuations in
alignment.
144
Figure 6.11: Properties of residual wedge in the calcite. (a) MMIP measurement of
retardance as a function of position in an ungrooved substrate exhibiting a small
residual wedge; and (b) Conceptual simulation illustrating transmitted polarization
from a square-wave retarder in the presence of a 0.3o residual wedge.
To measure translational sensitivities, the square-wave retarder is translated in
10 µm steps (1.2 CGH pixels, 0.15 square wave grating period cycles) in the x
direction up to 20 µm (2.4 CGH pixels, 0.3 square wave grating period cycles) from
its optimized location, and the DoLP and RoP of the PCGH reconstruction are
measured. The same measurements are performed for rotation of the square-wave
retarder around the systems optical axis in 1o increments up to 2o. DoLP is nearly
insensitive to both x translation and rotation about the optical axis, and DoLP
only varies up to 2% from the aligned condition, which is well-within the precision
of the measurement. RoP decreases dramatically with 1o of rotation about the
central axis, going from 83.9% to 67.3%, and RoP only decreases to a comparable
value of 70.2% after translation of 20 µm (2.4 CGH pixels, 0.3 square wave grating
period cycles) along x. The RoP increases from 67.3% to 70.2% between 1o and
2o of rotation because as the rotation angle increases, portions of the grooves will
overlap with adjacent grooves - therefore the degradation can experience some mild
145
improvement as a function of continued rotation before further degradation. Results
of the translation and rotation tests are summarized in Table 6.1. Allowing for a
decrease in RoP of 10%, translation and rotation tolerances can be determined from
the translation and rotation tests, assuming linearity over small perturbations. A
decrease in RoP of 10% corresponds to a translation of 14.84 µm or rotation of 0.6o .
Table 6.1: RoP and DoLP are measured as a function of the horizontal translation
of the calcite and calcite rotation about the optical axis.
Translation
RoP
0 µm 0 pixels 0 period cycles
83.9%
10 µm 1.2 pixels 0.15 period cycles 80.1%
20 µm 2.4 pixels 0.3 period cycles 67.3%
Rotation
81%
80.7%
80.0%
RoP DoP
0o
83.9% 81%
o
1
67.3% 82.7%
2o
70.2% 82.9%
6.6 Conclusions
A polarization computer generated hologram (PCGH) is demonstrated with a calcite
square-wave retarder illumination geometry, where a square-wave grating pattern
has been etched into a calcite plate. Transmission through the grating produces
alternating stripes of orthogonally linearly polarized light. The PCGH reconstruction is a tangentially polarized annulus, which synthesizes two orthogonally linearly
polarized images from the interlaced PCGH. The PCGH reconstruction produces
correlated speckle in the experimental image, which results in a high fraction of
properly oriented linear polarization. The RoP of the PCGH reconstruction is 84%,
with a DoLP of 0.81 and DoP of 0.895. These results approach those of previously reported work that used different illumination geometry. The small decrease
in performance is due to alignment and fabrication imperfections, as verified from
146
experimental and simulation analysis. An adjustable polarization compensator corrects residual wedge and non-integral retardance in the square wave retarder. An
experimental alignment tolerance analysis indicates that translation errors must be
less than 14.84 µm (1.8 CGH pixels, 0.22 square wave grating period cycles) and
rotation errors must be less than 0.6o to maintain a RoP loss of no more than 10%.
Although the experimental geometry included separated illumination and PCGH
components, these experiments demonstrate that combined single-plate geometries
may be successful.
147
APPENDIX A
JONES N -MATRIX ALGORITHM
The Jones N -matrix algorithm is written according to the flow chart shown in figure
A.1.
The global phase is extracted from the input Jones matrix and the pth root
of the matrix is calculated. Then the infinitesimal diattenuation and retardance
parameters (Dx , D45 , DR and δx , δ45 , δR ) will come directly from the real and
imaginary parts (respectively) of the second, third, and fourth coefficients arising
from the Pauli basis form of the Jones matrix’s nth root as written in equation 1.34.
All of these parameters are then rescaled by the matrix power (multiplied by n),
yielding the scaled polarization parameters of the input Jones matrix NJ . The Jones
matrix for retardance is found by taking the imaginary part of the sum of Pauli basis
matrices multiplied by their corresponding rescaled coefficient. Similarly, the Jones
for diattenuation is found by taking the real part of the sum of Pauli basis matrices
multiplied by their corresponding rescaled coefficients.
Two examples of the Jones N -matrix algorithm are shown in figures A.2 and
A.3. Figure A.2 illustrates the retardance parameters of a linear retarder with a
retardance of π/4 as a function of its orientation θ. Similarly, the diattenuation
parameters for a linear diattenuator with a minimum transmission of 0.8 and a
maximum transmission of 1 is shown as a function of its orientation θ in figure A.3.
In both cases, the three dimensional diattenuation and retardance parameters were
calculated using the Jones N -matrix algorithm discussed here.
148
Extract global
phase
Input Jones matrix
Take sum of real Pauli
basis elements and
rescaled coefficients to
calculate diattenuation
Jones matrix
Calculate pth root
of Jones matrix
Output
diattenuation and
retardance Jones
matrices
Rescale
infinitesimal
diattenuation and
retardance
parameters:
multiply by p
Calculate
infinitesimal
diattenuation and
retardance vectors
from Eq. 1.33
Take sum of imaginary
Pauli basis elements and
rescaled coefficients to
calculate retardance
Jones matrix
Output
diattenuation and
retardance vectors
Figure A.1: Algorithm flow-chart for calculation of diattenuation and retardance
vectors using Jones N-Matrix method.
HorizontalVertical Linear Retardance
RightLeft Circular Retardance
45°135° Linear Retardance
Retardance
Retardance
Retardance
1.0
1.0
1.0
0.5
0.5
0.5
Out[34]=
1
2
3
4
5
6
Orientation
1
2
3
4
5
6
Orientation
1
-0.5
-0.5
-0.5
-1.0
-1.0
-1.0
2
3
4
5
6
Orientation
Figure A.2: Retardance parameters of a linear retarder with a retardance of π/4
plotted as a function of θ.
HorizontalVertical Linear Diattenuation
Diattenuation
Out[50]=
RightLeft Circular Diattenuation
45°135° Linear Diattenuation
Diattenuation
Diattenuation
0.15
0.15
0.15
0.10
0.10
0.10
0.05
0.05
1
2
3
4
5
6
Orientation
0.05
1
2
3
4
5
6
Orientation
1
-0.05
-0.05
-0.05
-0.10
-0.10
-0.10
-0.15
-0.15
-0.15
2
3
4
5
6
Orientation
Figure A.3: Diattenuation parameters of a linear diattenuator plotted as a function
of θ.
149
APPENDIX B
THE EFFECT OF REFLECTION ON DEPOLARIZATION PARAMETERS
Mueller matrices which have been measured in reflection mode should be corrected
for the change in coordinates. This correction is justified because after reflection,
the z-component of the light propagation vector ~k is reversed. This assumes that
the surface normal is in the z-direction. In order for the coordinate system to remain
right-handed, one transverse coordinate must also change sign. After reflection, the
coordinates change from (x, y, z) to (x, −y, −z). For this coordinate system, upon
reflection, 45o polarized light changes to 135o light, and circularly (or elliptically)
polarized light changes helicity (DeBoo and Chipman, 2004). Therefore the S2 and
S3 components of the Stokes vector will change sign, with a normalized reflection
Mueller matrix R as follows:

0
 1 0 0

 0 1 0
0

R=
 0 0 −1 0


0 0 0 −1









(B.1)
As part of the data reduction process, the Mueller matrices (M) are right-multiplied
by the reflection matrix R (as shown in equation (B.2)) prior to the calculation of
diattenuation, retardance, depolarization parameters, and matrix roots.
MR = M ◦ R
(B.2)
150
This reflection correction is important to better understand the properties of the
material, as opposed to artifacts from the measurement geometry. If the measurement geometry is not corrected before the matrix roots calculations are performed,
asymmetries in the Mueller matrix that are a result of reflection may couple retardance and diattenuation into depolarization effects, or vice versa. In the absence of
any polarization effects, 45◦ orientation will reflect as 135◦ orientation, and rightcircular orientation will reflect as left-circular orientation. In this reflection case the
Mueller matrix is that of a half-wave linear retarder oriented at 0o , even though in
reality no polarization effects are present. Multiplying by R removes this half-wave
retardance effect, so that small polarization effects vary about 0 instead of a half
wave.
151
APPENDIX C
INFERRING THE ORIENTATION OF TEXTURE FROM POLARIZATION
PARAMETERS
This chapter has been modified from an SPIE proceedings paper, H. Noble et. al.,
“Inferring the orientation of texture from polarization parameters”, Proc. SPIE
7461, 746109 (2009).
C.1 Introduction
Material properties, surface characteristics, and geometric orientation affect polarization of diffusely reflected light. Depolarization of light by a surface occurs primarily due to multiple scattering events. Thus the depolarization behavior of a surface
should provide information about its texture, orientation, material composition, and
surface roughness. Much of this information is often below the pixel resolution of an
imaging system, which is why polarization analysis of optical scattering can provide
more information about surfaces than conventional imaging methods.
Early studies of polarization in scattered light were performed by Mie(H.C. van
de Hulst, 1981), Beckman and Spizzichino (Beckman and Spizzichino, 1963), and are
summarized by Stover (Stover, 1995). Other studies relevant to this work include
the polarization dependence of scattering from a one-dimensional rough surface by
O’Donnell and Knotts (O’Donnell and Knotts, 1991), ellipsometry and depolarization of rough surfaces by Williams (Williams, 1986), polarization scattering from
rough gold and glass surfaces by Hoover et. Al (Hoover et al., 2003), along with
152
others listed in (Germer and Marx, 2004)-(DeBoo and Chipman, 2004). Methods
of estimating the orientation of a planar surface using intensity shading information
from a single image have been investigated (Babu et al., 1985).
C.2 Aluminum measurements and analysis
C.2.1 Aluminum sample
The polarization light scattering function of a sample with well-defined texture and
orientation is studied. The Mueller matrix root parameters are correlated to the
orientation of texture as the aluminum is rotated about the surface normal. A block
of aluminum, sanded in one direction to produce linear grooves, scatters strongly,
has a well-defined texture and orientation, and has aperiodic groove structure.
A 4 inch by 3 inch aluminum sample was sanded linearly with 220 count sand
paper, generating randomly spaced, roughly parallel grooves, typically 5µm deep. A
two-dimensional surface profile measurement is included in figure C.1. The autocovariance graphs in figure C.2 show that the y-profile of the grooves (in the direction
of sanding) is relatively uniform, and the x-profile is aperiodic.
The polarization properties of specular and diffusely reflected, in-plane light were
measured using a Mueller matrix imaging polarimeter operating as a polarization
scatterometer, as the sample was rotated about its normal. The polarimeter operates
with 5:1 dual rotating retarders (five analyzer rotations for every generator rotation)
at a wavelength of 700nm (Smith and Chipman, 2007). The beam size at the sample
was 1 cm by 2 cm. Mueller matrix images were acquired at 24 groove orientations
from 0◦ orientation to 360◦ . Reference (Smith and Chipman, 2007) contains a more
complete description of the polarimeter.
Spatial averaging was done to the image profile in its central region in order to
153
average out the scattering behavior. Mueller matrices from the spatially averaged
images were multiplied by a reflection matrix to account for the measurement configuration as described in Appendix B – the spatially averaged, reflection corrected
Mueller matrices are shown in figure C.3. Polarization properties were calculated
from the reflection corrected Mueller matrices.
Figure C.4 shows the coordinate system of the polarimeter for in-plane data
collection. The reference direction is normal to the sample surface at the point of
incidence. The forward scatter side of the surface normal is defined to have positive
angles, and the backward scatter side of the surface normal has negative angles. Reference (Noble et al., 2010) contains a more detailed discussion of the measurement
coordinate systems. Two configurations are analyzed here, a specular reflection with
min 60◦ incident, 60◦ scattering, and an off-specular configuration with −60◦ incident, 30◦ scattering, as shown in figure C.5. A specular beam configuration with
−60◦ incident and 60◦ scatter will herewith be referred to as the “specular measurement” and an off-specular beam configuration with −60◦ incident and 30◦ will
herewith be referred to as the “off-specular measurement”.
C.2.2 Mueller matrix measurements
Figure C.3 shows the measured (normalized) Mueller matrices for the sanded aluminum as a function of the mean groove orientation angle (i.e. as the aluminum
sample was rotated about its normal). The x-axis represents the aluminum groove
orientation in degrees. The y-axis is the magnitude of each normalized Mueller
matrix element. The specular and off-specular measurements are shown.
All Mueller matrix elements show a nearly sinusoidal variation with groove orientation. Most elements are well fit to the single term Fourier series function with
154
two cycles per 360◦ ,
f (θ) = a0 + a1 sin 2θ + a2 cos 2θ,
(C.1)
the only exception bein m12 . This frequency dependence is expected since the sample
is nearly invariant under rotations of 180◦ .
The normalized average irradiance from the reflections off of the aluminum as the
grooves change orientation angle is shown in figure C.6 along with the fit from equation (C.1). The irradiance variation is the least sinusoidal out of all the polarimeter
measurements. The average irradiance from the specular reflection is about 20 times
larger than that from the diffuse reflection, as seen in figure C.6. The specular and
non-specular average reflected irradiance peaks at vertical groove orientations (90◦
and 270◦ ). The light scattered from the sample resembles the light scattered through
a dirty windshield (with semi-circular windshield wiper marks), another aperiodic
grating with a locally linear structure. For a distant point source viewed through
a dirty windshield, a bright line of light is observed, corresponding to the various
diffraction orders from the range of spatial frequencies present. Similarly, for a collimated in-plane source, the aluminum sample scatters much of its light into a line of
light perpendicular to the grooves. When the grooves are oriented at 90◦ (vertical)
this diffracted line lies in the horizontal plane of the detector. When the grooves
are oriented vertically, there are more in-plane reflection paths to the detector, as
shown in Fig. C.6. When the grooves are oriented horizontally, more light scatters
out-of-plane, so the average reflected irradiance is minimum around the horizontal orientations (0◦ and 180◦ ). This scattering behavior provides one method of
estimating groove orientation.
155
C.3 Diattenuation and retardance
Figure C.7 shows the three matrix roots diattenuation parameters D1 , D2 , and D3 ,
which characterize the polarization dependence of reflection. The magnitudes of the
linear horizontal and 45◦ diattenuation components are larger than the circular diattenuation as expected, since the texture of the aluminum is linearly oriented. The
specular horizontal diattenuation (0.35) is approximately twice the off-specular horizontal diattenuation (0.17). For both measurements the 45◦ diattenuation crosses
zero near the extrema of the horizonal diattenuation. Later the utility of this will
be explored. The maximum magnitude of specular 45◦ diattenuation (0.10) is approximately twice the off-specular (0.04). For the circular diattenuation, specular
maxima are approximately twice the off-specular maxima, but with much smaller
values (with specular maxima of 0.03). The diattenuation vector demonstrates that
for specular reflection, the sanded aluminum behaves as a weak linear diattenuator
which rotates with the grooves.
The matrix roots retardance parameters D4 , D5 , and D6 are shown in figure C.8
for both the specular and off-specular measurement together with the sinusoid fits.
For the horizontal and 45◦ retardance, the specular and off-specular retardance
are out of phase by 45o . The specular measurement exhibits a positive horizontal
retardance of 0.6 to 0.8 radians which oscillates by approximately 0.1 radians as the
grooves rotate. The off-specular angle has a negative horizontal retardance of -0.3
to -0.4. This means that the specular angle has its fast-axis at horizontal, and the
off-specular measurement angle has a vertical fast-axis.
156
C.3.1 Determining orientation from diattenuation and retardance
Table C.1 lists the sinusoidal fit coefficients from equation (C.1). We now show how
the orientation of the sample can be estimated from a single Mueller matrix measurement. To find the orientation ψ using the retardance parameters, we have derived
the following mathematical relationship. The method starts with a 3 element polarization parameter, such as diattenuation or retardance. (1) The parameter signals
are all centered around zero by subtracting the signal’s dc-offset, the a0 coefficient.
(2) Normalize the signal to oscillate between -1 and 1 by dividing the parameter
p
by a21 + a22 . Since a1 and a2 parameters express the sinusoidal or cosinusoidal
amplitude of each signal, dividing by their total magnitude effectively normalizes
each oscillating signal. (3) Calculate ψ from the arctangent of the normalized signal.
Using the retardance parameters here as an example,
q
q
2
2
(R
−
a
)/
a
(R
−
a
)/
a21,3 + a22,3
+
a
2
0,2
3
0,3
1,2
2,2
180
180
o
q
q
arctan
=
arctan
,
ψ( ) =
2π
2π
(R1 − a0,1 )/ a21,1 + a22,1
(R1 − a0,1 )/ a21,1 + a22,1
(C.2)
where ai,j is the coefficient from table 2. The subscript i refers to the fit coefficient
term (0,1,2), and the subscript j specifies to which retardance term the fit coefficient
term corresponds. In order to calculate the orientation ψ, the numerator of the
ArcTangent’s argument should be predominantly sinusoidal, and the denominator
of the ArcTangent’s argument must predominantly cosinusoidal. Otherwise the
sample orientation cannot be estimated by equation (C.2) or a more general method
of function inversion might be applied. As an example, the specular measurement’s
retardance parameters provide a good orientation estimate. R1 is dominated by
the a2 term, while R2 and R3 are dominated by the a1 term. An example of the
application of equation (C.2) is shown in figure C.9, using the off-specular R3 and
157
R1 data. The x-axis indicates groove orientation angle, and the y-axis shows the
estimate calculated with equation (C.2). Additionally, this method works effectively
with the diattenuation parameters.
158
Figure C.1: Surface profile of sanded aluminum sample. White regions are areas of
low signal where signal-to-noise ratio is poor. The white regions are likely areas of
high slope where the reflected light is not substantially captured by the instrument.
159
Figure C.2: Autocovariance of sanded aluminum sample. Units of x-profile are in
millimeters, units of y-profile are in microns.
Figure C.3: Normalized Mueller matrix for −60o incident, 60o scattering specular
angle (solid line) and −60o incident, 30o scattering off-specular angle (dashed line)
with dotted-line fits. The x-axis represents texture orientation angle of aluminum
in degrees, y-axis represents the magnitude of each Mueller matrix element.
160
Figure C.4: Goniometric polarimeter coordinate system. All angles are measured
from sample normal. Negative angles are in the region of backscattered light, and
positive angles are in the direction of forward scatter.
Figure C.5: The two measurement configurations analyzed in this section: −60o
incident, 60o scattering and −60o incident, 30o scattering.
161
1.0
0.8
0.6
0.4
0.2
0
45
90
135
180
225
270
Alum. Orien.
315
Figure C.6: Normalized average irradiance from light reflected off of the sample as
the groove orientation angle changes. The x-axis lists the orientation angle of the
aluminum grooves; the y-axis represents the average irradiance reflected from the
sample. The specular measurement is plotted with a solid line, and the off-specular
measurement is plotted with the dashed line. The sinusoidal fit from equation (C.1)
is plotted with the dotted lines.
0.2
0.2
0.1
0.1
45
90
135
180
225
270
315
Alum. Orien.
0.2
0.1
45
90
135
180
225
270
315
Alum. Orien.
45
-0.1
-0.1
-0.1
-0.2
-0.2
-0.2
-0.3
-0.3
-0.3
-0.4
-0.4
-0.4
90
135
180
225
270
315
Alum. Orien.
Figure C.7: Matrix root diattenuation parameters with fit (dotted lines) for the
specular angle (solid line) and the off-specular angle (dashed line).
0.5
0.5
45
-0.5
90
135
180
225
270
315
Alum. Orien.
0.5
45
-0.5
90
135
180
225
270
315
Alum. Orien.
45
90
135
180
225
270
315
Alum. Orien.
-0.5
Figure C.8: Matrix roots retardance parameters in radians with fit (dotted lines)
for the specular angle (solid line) and the off-specular angle (dashed line).
162
Aluminum Orientation Calculated with Retardance
300
Calculated Orien. !°"
250
200
Out[43]=
150
100
50
0
0
50
100
Alum. Orien. !°"
150
200
250
300
350
Figure C.9: Aluminum sample groove orientation calculated with the specular matrix roots retardance parameters demonstrate the estimation of orientation from a
single Mueller matrix measurement following calibration.
163
Table C.1: Sinusoidal fit coefficients.
Off-Specular Measurement
Parameter
a0
a1
a2
R1
-0.3666 -0.0026 0.03954
R2
0.0018 0.0471 -0.0609
R3
0.05612 0.0037 0.0040
D1
-0.1022 -0.0029 -0.0002
D2
0.00774 -0.0427 0.00824
D3
0.0736 0.0036 -0.0006
Specular Measurement
a0
a1
a2
0.6528 -0.0024 0.0436
-0.0078 0.07552 -0.0195
-0.0917 -0.0069 -0.0001
-0.2413 -0.0039 -0.0008
0.0115 -0.099 -0.0300
0.1059 0.0113 0.0023
164
2.0
1.0
1.0
1.5
0.5
0.5
45
90
135 180 225 270 315
Alum. Orien.
45
90
135 180 225 270 315
Alum. Orien.
1.0
0.5
45
90
Alum. Orien.
135 180 225 270 315
45
0.5
!0.5
!0.5
!0.5
0.0
!1.0
!1.0
!1.0
45
90 135 180 225 270 315
Alum. Orien.
!0.005
0.90
!0.010
0.85
!0.015
45
135 180 225 270 315
Alum. Orien.
0.04
0.03
0.02
0.01
0.010
0.005
45
90 135 180 225 270 315
!0.01
!0.02
!0.03
0.015
0.006
0.010
0.004
0.005
0.002
45
90 135 180 225 270 315
Alum. Orien.
!0.002
!0.010
!0.004
!0.015
!0.006
0.002
45
90
Alum. Orien.
135 180 225 270 315
90
135 180 225 270 315
Alum. Orien.
90
Alum. Orien.
135 180 225 270 315
0.004
0.002
45
90 135 180 225 270 315
90 135 180 225 270 315
45
90 135 180 225 270 315
Alum. Orien.
0.004
0.002
0.85
!0.002
!0.004
!0.006
!0.008
!0.010
45
!0.006
45
Alum. Orien.
!0.002
!0.004
0.90
45
Alum. Orien.
0.004
0.95
Alum. Orien.
!0.005
!0.005
90
!0.01
!0.02
!0.03
135 180 225 270 315
0.006
0.04
0.03
0.02
0.01
0.95
90
!0.002
!0.004
!0.006
!0.008
!0.010
Alum. Orien.
0.90
45
Alum. Orien.
90 135 180 225 270 315
0.85
0.80
45
90
135 180 225 270 315
Alum. Orien.
Figure C.10: Normalized depolarization Mueller matrix from the Lu-Chipman decomposition with fit (dotted line) for specular measurement (solid line) and offspecular measurement (dashed line).
C.4 Depolarization
C.4.1 Lu-Chipman depolarizing Mueller matrix and depolarization Index
The full Lu-Chipman depolarization matrix is shown in figure C.10. Because the
diagonal elements are all in phase, these elements provide little orientation insight.
The depolarization index (DI) for both measurements is shown in figure C.11.
The DI is defined as
3
P
DI(M) =
i,j=0
!1/2
m2i,j − m20,0
√
3m0,0
,
(C.3)
where the m’s are Mueller matrix elements. A DI of 0 indicates that a sample is
completely depolarizing, and a value of 1 indicates that a sample is non-depolarizing
165
0.25
0.20
0.15
0.10
0.05
0
45
90
135
180
225
270
315
Alum. Orien.
Figure C.11: Depolarization index vs. orientation angle with fit (dotted line) for
specular measurement (solid line) and off-specular measurement (dashed line).
for all polarized states (Chipman, 1995). The DI of the specular measurement is
approximately 0.05 and hardly varies with groove orientation, indicating polarized
light reflects nearly unpolarized. The DI of the off-specular measurement ranges
from 0.25 to 0.05, with maxima at 0◦ and 180◦ , and minima at 90◦ and 270◦ . Minima
in the depolarization index vs. orientation angle for the off-specular measurement
correspond to bright regions in the intensity from Fig. C.6, while maxima correspond
to the dark intensity regions.
C.4.2 Mueller matrix roots depolarizing parameters
The Mueller matrix roots depolarizing parameters (amplitude, phase, and diagonal
depolarization) are shown in figures C.12 through C.14, along with their sinusoidal
fits.
The amplitude depolarization parameters D7 , D8 , and D9 , shown in Fig. C.12,
exhibit the following characteristics. Horizontal/vertical amplitude depolarization
D7 is noisy, with minimal signal strength. 45o /135o amplitude depolarization D8
has low signal strength (with peak-to-peak amplitude of 0.020 for the off-specular
166
measurement and peak-to-peak amplitude of approximately 0.014 for the specular
measurement), but shows a clear sinusoidal variation with the orientation angle
of the aluminum. The circular amplitude depolarization D9 of the specular angle
shows negligible amplitude, while the off-specular angle has a sinusoidal variation
with peak-to-peak amplitude of 0.15.
The phase depolarization parameters D10 , D11 , and D12 , shown in Fig. C.13,
exhibit the following behavior. The specular angle’s horizontal/vertical phase depolarization D10 shows minimal signal strength and amplitude as the orientation of
the aluminum varies, while the off-specular angle has a peak-to-peak amplitude of
approximately 0.015, and sinusoidal variation with aluminum orientation. 45o /135o
phase depolarization is affected by more noise, but both measurements show the sinusoidal rotation characteristics from Equation C.1. Circular amplitude depolarization D12 shows some modulation within the sinusoidal envelope, which corresponds
to the modulation in the m12 term of the Mueller matrix.
The diagonal depolarization parameters D13 , D14 , and D15 are shown in Fig.
C.14. D13 expresses the relative depolarization between the linear Stokes vector
elements S1 and S2 . For both measurements, it oscillates between equal depolarization along the two axes (at D13 = 0) and more depolarization along S1 (when
D13 is negative). D14 expresses the relative depolarization between the circular and
linear Stokes axes. D14 oscillates between exclusively positive values, which means
that it depolarizes more along the circular S3 axis than along the linear Stokes axes.
For the off-specular measurement, isotropic diagonal depolarization D15 averages at
approximately 0.18, while the off-specular measurement has D15 with an average of
0.05 and almost zero sinusoidal amplitude.
The signal strength of the non-depolarizing roots parameters for specular mea-
167
0.010
0.004
45
0.002
90
135
180
225
270
315
Alum. Orien.
0.005
45
90
135
180
225
270
315
-0.005
Alum. Orien.
-0.002
45
-0.004
90
135
180
225
270
315
Alum. Orien.
-0.010
-0.005
-0.015
-0.006
Figure C.12: Amplitude depolarization parameters D7 , D8 , and D9 (from left to
right) vs. orientation angle with fit (dotted line) for specular measurement (solid
line) and off-specular measurement (dashed line).
0.005
45
90
135
180
225
270
315
-0.005
-0.010
0.04
Alum. Orien.
45
90
135
180
225
270
315
Alum. Orien.
0.02
45
-0.005
90
135
180
225
270
315
Alum. Orien.
-0.02
-0.015
-0.010
-0.04
Figure C.13: Phase depolarization parameters D10 , D11 , and D12 (from left to right)
vs. orientation angle with fit (dotted line) for specular measurement (solid line) and
off-specular measurement (dashed line).
surement was higher than the signal strength of the off-specular measurement. For
this reason, the orientation calculated with the retardance parameters used the
specular measurement data. However, the off-specular depolarizing matrix roots
parameters show a larger sinusoidal amplitude and signal strength than the specular
depolarizing matrix roots parameters. This is potentially useful for a measurement
application where an optical measurement is only possible at a non-specular or grazing angle, from which we expect more depolarization effects than from a specular
angle. Figure C.12 shows the orientation of the scratched aluminum which was estimated from the off-specular linear phase depolarization parameters (D10 and D11 ),
using the same method as the specular retardance orientation estimation.
168
0.05
45
90
135
180
225
270
315
0.30
Alum. Orien.
0.04
-0.01
0.25
0.20
0.03
0.15
-0.02
0.02
-0.03
0.01
0.10
-0.04
0.05
45
90
135
180
225
270
315
Alum. Orien.
45
90
135
180
225
270
315
Alum. Orien.
Figure C.14: Diagonal depolarization parameters D13 , D14 , and D15 (from left to
right) vs. orientation angle with fit (dotted line) for specular measurement (solid
line) and off-specular measurement (dashed line).
Aluminum Orientation Calculated with Phase Depolarization
Calculated Orien.
300
250
200
150
100
50
0
45
90
135
180
225
270
315
Alum. Orien. H°L
Figure C.15: Aluminum sample groove orientation calculated with the off-specular
linear phase depolarization parameters D10 and D11 , demonstrating the estimate of
orientation from a single Mueller matrix measurement following calibration.
C.5 Conclusion
Diffusely reflected light from a one-dimensionally rough aluminum sample has been
analyzed by Mueller matrix light scattering measurements. For a specular and nonspecular pair of beams, polarization terms are identified which vary cosinusoidally
and others which vary sinusoidally with respect to the aluminum groove orientation
angle. Using this cosinusoidal and sinusoidal modulation information, the orientation may be estimated using a normalizing arctangeant calculation method, as
written in equation (C.2). Determining orientation by this method requires pa-
169
rameters exhibiting a sinusoidal variation and others with cosinusoidal information.
Thus for a series of similar one-dimensionally rough samples, we expect that after
polarimetric characterization, the orientation of grooves about the normal can be
measured from a single specular or diffuse Mueller matrix measurement. Sample
orientation is calculated using our method with the matrix roots retardance parameters and the matrix roots phase depolarization parameters. This result is shown
for one sample, but the method is applicable to a large class of samples.
Calibration with sinusoidal and cosinusoidal fit parameters can indicate the suitability of this method for any particular sample, including more complex surfaces
than those discussed in this work. Much more study will be required to understand
the types of rough surfaces for which orientation can be recovered by Mueller matrix
scattering polarimetry.
170
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