Page numbers followed by f and t indicate figures and tables, respectively.
Aberration, 17; see also Polarization aberrations
Fresnel, 545
paraxial, see Paraxial polarization aberrations
skew, see Skew aberration
wavefront, 543, 572, 580, 653, 663
Aberration-free system
ARM of, 602
MPSM of, 604
Absentee layer, 488
Absent mode, 702
Absolute phase, 580
about, 166
change, 133
of Jones matrix, 129
of light, 33
Absorption, 365
axis, 187
factor, 148
Absorption coefficient, 365
Achromatic retarders, 715
Active materials, optically, 169
Addition form, of polarization ray tracing matrix, 344–347
Addition of circularly polarized beams (example), 50
Addition of Mueller matrices, 201–202
Addition of polarized beams; see also Interference of polarized light
Gaussian wave packet example, 109–112, 112t
of polarized light of two different frequencies, 103–105
of polychromatic beams, 105–109
Adjoint of vector, 36
Aerosol, 260
polarimeter, 8
Aerosol polarimeter sensor (APS) instrument, 261
Airborne multi-angle spectro polarimetric imager (AirMSPI), 258, 261f, 262f
Air passing, collimated plane wave, 724
Air to calcite, ray parameters refract and reflect, 757
patterns, 773
Algorithm
multilayer thin films, 487–488
Alignment layer, 843
Aluminum beam splitters, 485–486
Aluminum-coated three-fold mirror system, 647–649
Aluminum diattenuation vs. angle of incidence, 944
Aluminum oxide, 500
Aluminum retardance
in radians vs. angle of incidence, 945
Aluminum wire grid polarizer, 822
Amplitude aberration, 382, 545, 653
Amplitude coefficients, 298, 299–300, 383–385, 544, 550; see also Fresnel equations
coating and, 479
multilayer coatings, 487, 490, 499–500
Amplitude depolarization, 200
Amplitude distribution, 92
Amplitude grating, 812
structure, 812
Amplitude response matrix (ARM), 593, 594, 623, 774, 937, 938, 954, 958
of aberration-free system, 602–603
absolute value of, 954
amplitude of, 775
of Cassegrain telescopes, 472, 473f, 602–603
components, 903
critical angle CCR and, 619–621
encircled energy (EE), 775
overview, 601
parameters, imaging calculation, 955
polarization structure of images and, 607
polarized pupil with unpolarized object, 610–611
skew aberration and, 664
of uncoated single-element lens, 452–453, 453f
Amplitude transmission, 382
Amplitude transmittances, 141
Analytical manipulation, 494
Analyzer vector
about, 178
calculation, 251
Angle dependence; see also Polarization problems in optical systems
of polarizers, 12
of retarders, 13
Angle of incidence (AOI), 614
Angle of incidence, paraxial, 544, 550–553, 562, 572, 586
variation of, 551
Angle of incidence maps, 562–563
cell phone lens, 362–363, 362f
Angle of linear polarization (AoLP), 262
Angle of major axis, 68
Angle of polarization (AoP), 68
Angles of incidence, 940
Angular dispersion, 815
Angular momentum, 137
Anisotropic devices, 718
Anisotropic interfaces, 480, 725
Anisotropic material, 11, 36, 297
Anisotropic media; see also Birefringent materials
classes of, 670f
electromagnetic waves in, 672–673
optical properties of, 669
Anisotropic plate assembly, 726
Annealing, 15
Anti-commutative, Pauli matrices as, 510
Anti-pole, 430
Antireflection-coated interfaces, 21
Antireflection coatings, 8, 20, 23, 482–485, 546, 931
single-layer, 485
Antireflection grating’s groove orientation, 826
Antireflection subwavelength antireflection coatings (SWG), 825–827
intensity transmission of, 827
AOI, see Angle of incidence
AOI map, 860
Aperture function, 598–599, 937
Apodization, 382, 545, 599, 653, 786, 937
Approximately linear and quadratic function, 449
Arbitrary shape gratings, 813
ARM, see Amplitude response matrix
ARMOS model, 733
Artifacts in polarimetric images; see also Polarimetry
pixel misalignment, 281
Associative rule for matrix multiplication, 165, 167
Astigmatism, 362, 558, 572, 583, 786, 950
into on-axis beam, 25
Atmospheric polarization
about, 259
Average degree of polarization, 195
Average transmission, 179
Axometrics AxoStep Muller matrix imaging polarimeter, 580
Baker–Hausdorff–Campbell formula, 531–532, 533
Bandpass filters, 491
Bare aluminum mirror
diattenuation, 944
reflection coefficients, 943
Basis functions, 567
Basis polarization states, 38, 51t, 65t, 75t
Basis vectors, 656
Beam combination
illumination pattern, 718
with polarization ray tracing matrices, 715–736
principles, 716
wavefronts, pupil of imaging systems, 718
Beam overlapping, 725
coatings, 479
blocks, ray splitting, 706–707
Biaxial cube, reflection inside, 707–710
Biaxial multilayer films, 17, 843
Bidirectional reflectance distribution function, 935
Binodal polarization, 558
Birefringence, 13, 118, 122, 675–676
of a ray, 745
Birefringent components, 716
Birefringent filters, 345
Birefringent interface
mode labeling for rays, 671t
parameters, 671t
polarization ray tracing matrices for, 695–705
reflections and refractions at, 681–693
Birefringent materials
interface, see Birefringent interface
KTP crystal, ray tracing, 670, 672, 686–693
optical properties of, 669
optic axis of, 744
ray doubling, data structure for, 694–695
Birefringent plate
collimated wavefront passes, 717
Birefringent ray tracing complications, 934
anisotropic materials, 934
birefringent interface, 934
ray doubling, 934
spatially varying birefringence medium, 934
wavefronts, 934
Birefringent retarders, 168
Birefringent-to-birefringent interface, 703
Birefringent-to-isotropic interface, 701–703
Birefringent waveplates, 13
Blaze angle, 819
Blazed grating, 813
Blaze wavelength, 818
Blazing, 818
Blue phase (BP) mode, 855
double twist cylinder, 855
double twist cylinders stacking, 856
Brewster’s angle, 68f, 305–306, 636n, 787, 935; see also Fresnel equations
Brightness
response time of liquid crystal, 865
CAD file, 891
Calcite, 11
A-plate, 271
Calcite CaCO3 crystal structure
with vertical optic axis, 743
Calcite quarter waveplate
polarization aberrations of, 726
Calibration, 279
Canonical basis, for polarization states, 336
Canonical forms, 509
Jones matrix, 546
Canonical local coordinates, 640–642
Canonical summation form, Pauli matrices, 514–515
Cartesian form, 500
Cartesian rotation matrix, 42, 137, 511
Cassegrain telescopes, 24, 25, 469–473, 470f, 598, 942, 943
diattenuation pupil maps, 470, 471f
specifications, 470t
Cavity Q-factor, 111
CCD detector, 715
c-Coefficients, Pauli, 512, 512f
CCRs, see Corner cube retroreflectors
angle of incidence maps, 362–363, 362f
angle of incidence (AoI) variation, 362
optical path length (OPL) variation, 361–362
prescription, 416–417, 417t–418t
wavefront aberration of, 404–405
Characterized matrix, 487
Chief ray angle, 660
of incidence, 552
Chief rays, 372, 372f–373f, 585, 658
Chromatic aberration, 394, 499
Circular diattenuation, 942
Circular polariscope, 266–267; see also Polariscopes
Circular polarized light, 636
Circular retardance, 265, 629, 636, 663, 666
tilt, 659
Circular retarder, 122, 137, 169, 184, 632
Jones matrix, 619
Circularly polarized basis, 49, 50
Circularly polarized light, 22, 35, 43–45, 71f, 104
Coating polarization, 940
Coatings, 667
Co-cross-polarized systems
extinction ratio map, 804
CODE V’s U.S. patent library, 658, 666–667; see also Skew aberration
Coherence simulation, 935
partially coherent/incoherent light, 935
Coherent combination, of polarization ray tracing matrices, 344–347
Coherent monochromatic light, 935
Coherent ray tracing, 407–409, 936
Color displays, 843
Combining P matrices for interferometers, 346–347
Combining wavefronts process, 716
Commutator, 149
Commuting matrix, 123
Complete analyzer, 254
Complete polarimeters, 222
Complex Pauli coefficients, 508, 510–511, 548–550
Complex pulses, 109
Complex vector, 34
Complex wavefront aberrations, 786
Compound retarder, 185, 186f, 267n
Jones matrix decomposition, 917–919
principal retardance trajectory, 924
Conoscope, 13, 270–271, 693, 694; see also Polariscopes
Conservation of momentum, 297
Constant degree of circular polarization, 83
Constitutive relation, 676
Contrast ratio, 845
Conventional optical design, 17, 20
Conversion into Jones vectors (example), 52
ConvolutionMPSM, 604
Coordinate vectors, 640
Co-propagating overlapped wavefronts, 722
Co-propagating wavefront combination, 718
beam combining configuration, 718
incident beam yields, 718
Corner cube retroreflectors (CCRs), 275, 614–618
system, 653
C-plate, 14
crystal waveplates, aberrations of, 769–771
twisted nematic (TN cell), 848
Critical angle, 306–307; see also Fresnel equations
corner cube retroreflector, 618–622, 655
Crossed linear polariscope, 264, 265
Crossed mirror configuration, 464–469
Crossed polarizers, 14, 15f, 22
Crystal, anisotropic, 11
Crystal axes, 673
Crystalline quartz, 677
Crystalline solid structure, 838
Crystal polarizers, 715, 785, 786
field of view (FOV), 785
Glan–Taylor polarizer, see Glan–Taylor polarizer
materials for, 786
anisotropic, 785
Crystal prisms, 778
uniaxial crystal blocks, 778
Crystal’s index ellipsoid, 751
Crystal’s principal indices, 754
Crystal waveplates, aberrations of, 765–771
birefringent material, 765
field of view (FOV)
refracts into uniaxial plane parallel plate, 766
converging beam, 768
negative uniaxial, 767
y-oriented calcite, 767
principal section with incident angles, 768
Cumulative mode label, 939
Cumulative P matrix, for reference and test paths, 342–344
Curvature, 585
Dark fringes, 271
Dark state, 846
Data reduction, see Mueller matrices
Data structure for ray doubling, 694–695
Daughter rays, 695
Decreasing phase convention, 379
Defocus, chromatic aberration, 497–499
Defocus (quadratic), polarization aberrations, 543, 544f, 557
diattenuation, 553–556, 557, 564–566
one wave of, with a circular aperture, 599
Degree of circular polarization (DoCP), 68
Degree of linear polarization (DoLP), 67, 244, 259, 262f, 280
Degree of polarization (DoP), 66–70, 80, 105, 193, 493, 871, 936
Degrees of freedom (DoF)
about, 200
Jones vector, 38
in polarization matrices, 194
Density matrices, 109
Depolarization, 871, 936; see also Mueller matrix
addition of Mueller matrices, 201–202
defined, 193
degree of polarization surfaces/degree of polarization maps, 196–198
depolarization index/average degree of polarization, 195
non-depolarizing Mueller matrices, 192
physically realizable Mueller matrices, testing for, 198–200
weak depolarizing elements, 200–201
Depolarization index (DI), 195
Depolarized beam, 193
Depolarizers, 6
DFT, see Discrete Fourier transform
Diagonal depolarization, 200
Diagonal depolarizer Mueller matrix, 194
Diagonal elements, of ARM, 602
Diagonal Jones matrix, 143
Diagonal matrix, 335
Diattenuation, 937
aberration, 653
about, 6
components, 140
defined, 119
higher-order polarization aberrations, 578–580
matrix properties of Jones matrices, 144–145
of metal at non-normal incidence, 315–316
microscope, 581
for multi-element lens, 23
parameter, defined, 178
paraxial, 553
polarization-dependent transmittance, 130
as polarization properties, 117, 119–120, 630, 649
in polarization ray tracing, 20, 21
seven-element lens system, 561–562, 564–567
vector, 178
weakly polarizing ray intercepts, 549
Diattenuation calculation of P matrix, 334–337
singular value decomposition (SVD), 335–337
Diattenuation defocus, 949
paraxial, 553–556, 557, 562, 564–566; see also Defocus (quadratic), polarization aberrations
Diattenuation maps
crossed mirror configurations, 465–469, 466f, 467t, 469f
Diattenuation piston, 557, 558f, 562, 564–566, 949; see also Piston (constant), polarization aberrations
Diattenuation space
inhomogeneous Jones matrices, 533–534, 534f–535f
Diattenuation tilt, 557, 562, 949; see also Tilt (linear), polarization aberrations
Diattenuators
Jones matrix, 139
Mueller matrix, 182; see also Polarizer/diattenuator Mueller matrices
operation of, 187
polarization element/polarization properties, 5–6
optical element, 120
polarizer, 180–182; see also Poincaré sphere operations
Dichroic and birefringent materials, 118; see also Jones matrices
Dichroic dyes, 243
Dichroic polarizer, 12, 118, 179
Dielectric anisotropy, 840
Dielectric refraction, 308–309; see also Fresnel refraction/reflection
Dielectric tensors, 669, 673–674
Diffracted wavevectors, 816
Diffraction gratings, 252, 935, 936
Diffraction-limited MPSM, 654
Diffraction limited wavefront, 405
Diffractive optical elements (DOEs), 93, 811
application of, 811
Dipole coordinates, 424, 426–430
âLOC, 429
defined, 426
local to global polarization state, 427–428
wavefront aberration function, 440–411
Dipole electromagnetic wave, 56
Dipole model, 56
Dipole oscillators, 306
Dipole radiation, 56
Dipole radiator model, 424
Dipoles, 423
Direction cosines, 32
Direction of energy propagation, 752
Director distributions, with splay, 844
Disclinations, 862
Discotic LC molecules, 868
Discotic molecules
compensating film layer, 870
schematics for, 840
uniform film of, 870
Discontinuities, see Retardance discontinuities
Discrete Fourier transform (DFT), 238, 594–597
2D DFT, 596
MPSM of patent lens system, 666
Discrete ray data yielding, 731
Disk-shaped molecules, 17
Dispersion, 13
Division of amplitude, 93
Division-of-amplitude polarimetry (DOAP), 241, 246–248; see also Simultaneous polarimetric measurement
Division-of-aperture polarimeter, 241
Division-of-aperture polarimetry, 242; see also Simultaneous polarimetric measurement
Division-of-focal-plane polarimetry, 242–245; see also Simultaneous polarimetric measurement
Division of wavefront, 92
Double-crossed mirror system, 464–469
Double image, 664
Double pole coordinates, 424, 430–436, 937
âLOC, 436
defined, 430
parabolic reflector and, 431, 434
wavefront aberration function, 440–411
Double pole local coordinates, 598
Double pole polarization, 656
Double pole spherical wave, 57
Double refraction, 670, 679, 741
through a calcite rhomb, 680f
through a ulexite biaxial block, 680f
through three biaxial crystal blocks, 706–707
Double twist cylinder, 855
Dove prism
parallel transport matrix for, 639–640
Dual rotating retarder Mueller matrix polarimeter, 274–275; see also Mueller polarimetry configurations
Dual tetrahedron Mueller polarimeter (example), 284
Earth’s energy balance, 260
Earth Venture Instrument program, 258
Eccentricity, 74
Effective depth, multilayer film, 491, 492f
eiei-mode, 695
eie-mode, 695
Eigenmodes
birefringent materials, 669, 679–681
subaperture, 616f
unitary matrix and, 630
Eigenpolarizations, 480, 841; see also Eigenmodes
defined, 120
Eigenvalues
for Pauli matrix sum, 513
for Pauli matrix sum, 513
eio-mode, 695
Elastic constant, 843
Electric field aberrations, 568–572
Electric field amplitude, 727, 728, 728
Electric field intensity, 802
Electric field of unpolarized light, 65
Electric field vectors, 33f
Electromagnetic waves in anisotropic media, 672–673
Ellipsoid of wave normal, 750
Ellipsometers, 930
in microlithography industry, 12
sample-measuring polarimeter, 8
Elliptical diattenuators, 931
Elliptically polarized light, 35, 45–48
Elliptical polarization parameters, 73–74
Elliptical polarizer, 130
Elliptical retarders, 122, 141–143, 142, 916, 931; see also Jones matrices
Mueller matrix, 171
Energy density, 749
Entrance pupil, 370, 370f, 654
Entrance Pupil Diameter (EPD), 370
EPD, see Entrance Pupil Diameter
Equi-rectangular projection, 83
Even permutation, Pauli matrices, 510
Exit pupil, 370–372, 371f, 500, 550, 551, 564, 580, 581, 582, 594, 598–601, 654, 663
Exponential form of polarization components, 525
Exponential Pauli coefficients, 509
Exponentiation of Jones matrices, 518–519
Extended Jones matrix method, 856–857
birefringent plates with twisting and tilting directors, 857
External reflection, 306, 309–310; see also Fresnel refraction/reflection
Extinction ratio, 120
distribution, 802
Extraordinary principal axis, 743
Extraordinary ray, 742
Fabry–Perot interferometers, 717
Faraday effect, 677
Faraday rotation, 169
unchanged convention, 134, 135t, 136
Fast-mode, birefringent interfaces, 681–682
Fast single-element lens, see Uncoated single-element lens
Fast–slow–fast-mode, ray trace parameter, 707t
Fiber optic pulse measurement, 718
Field curvature, 583
Field distribution, 717
Field Guide to Geometrical Optics (Greivenkamp), 583
Field of view (FOV)
diattenuation aberrations across, 556–557
retardance aberrations across, 556–557
skew aberration and, 654, 658, 667
Finite differences, 498
Finite difference time domain algorithms (FDTD), 940
Finite element modeling (FEM), 889
Fixed analyzer polarimeter, 251–253
Fixed polarizer polarimeter, 254–255
Flange, 892
Floquet’s condition, 830
Flowcharts, coherent/incoherent imaging, 594
Flux, 37
Flux components, finding (example), 77
Flux vector, 224
Focusing grating, 813
Fold mirrors, 455–460, 567, 941
parallel transport matrix for, 639
Forward problem, 117
Foucault tests, 929
Four-dimensional phase space, 374
Fourier coefficients, 252
function, 830
Fourier optics, 18
Fourier transform, 18, 71, 664
Fourier transform algorithms, 719
Fourier transformation operation, 500
Fourier transform spectrometers, 718
Fourth-order wavefront aberrations, 583
FOV, see Field of view
Fréedericksz cell, 841, 846, 858
degree of polarization map, 871
director orientations, 847
dispersion measurement, 867
retardance modulator for adjustable linear retardance, 847
retardance of, 868
single-pixel, 864
variation of director orientation, 841
Frequency domain analysis, 238
Fresnel aberrations, 447–475, 545, 940
Cassegrain telescopes, 469–473
uncoated single-element lens, 448–455
Fresnel amplitude coefficients, 496
Fresnel coefficients, 811
for aluminum (example), 313
approximate representations of
Taylor series for Fresnel coefficients, 317–318
Fresnel equations, 21, 21f, 22, 447, 448, 544, 550
amplitude coefficients, 299–300
Fresnel coefficients, approximate representations of
Taylor series for Fresnel coefficients, 317–318
Fresnel refraction/reflection
dielectric refraction, 308–309
intensity coefficients, 301–303
intensity/phase change with incident angle, 307–308
Jones matrices with Fresnel coefficients, 308
propagation of light
homogeneous/isotropic interfaces, 297
light propagation in media, 297–298
plane of incidence, 296
plane waves/rays, 296
s- and p-polarization components, 298–299
Fresnel refraction/reflection
dielectric refraction, 308–309
Fresnel transmission, 727
coefficients, 727
Fresnel zone patterns, 813
Fringes, 935
Fringe visibility, 91, 97, 935
Full sky polarimeter, 259
Gaussian function, 594
Gaussian wave packet example, 109–112, 112t
Geometrical ray tracing, 359; see also Polarization ray tracing (PRT)
Geometrical transformation, 618, 629, 649
non-polarizing optical systems, 633–634
parallel transport matrix, 636–640
parallel transport of vectors, 634–636
polarization ray tracing matrix, separating from, 642–645
rotation of local coordinates, 631–633
Glan–Foucault prisms, 778
Glan–Taylor polarizer, 669, 742, 785, 786, 787
aberrations of, 797
configurations for calculating extinction, 800
co-polarized system to cross-polarized system, 799
FOV of, 800
light incident on hypotenuse calcite/air interface, 787
multiple polarized wavefronts, 793–795
multiple potential ray paths, 789–793
pairs of, 799
extinction ratio, 799
performance of, 800
polarized wavefronts exiting, 796–797
Zernike polynomial fitting, 797
Glan–Thompson prism polarizers, 179, 280, 742, 786
Glan–Thompson prisms, 778
Glan-type polarizers, 694, 778
Global coordinates, 325, 631, 674
Gold
phase discontinuities in, 500, 501f
Grating
equation, 814
in-plane, 814
out-of-plane diffraction, 814
free spectral range, 818
parameters and notation definitions, 814
resolving power, 818
vector, 816
Greivenkamp, John, 583
Grid tracing, of rays emerging, 716
Half linear wave retarders, 134
Half wave circular retarder, 184
Half wave elliptical retarders (HWR), 172, 173
Half wave linear retarder (HWLR), 6, 121, 136, 169, 170t, 333–334; see also Linear retarders
Jones matrix for, 337
Half wave retarders, 169
Hermitian adjoint, 335
for Pauli matrix sum, 513
Hermitian matrices, 144–145, 148, 150, 182
Hermitian matrix, 523–525, 643
Hexabenzocoronene
discotic molecules, 869
Higher-order aberrations, 498
Higher-order basis functions, 567
Higher-order polarization aberrations
electric field aberrations, 568–572
Fourier transform and, 594
vector Zernike polynomials, 568–572, 569t–570t, 571f
High numerical aperture wavefronts, 436–437
Hollow aluminum corner cube, 347–351
Hologram, 102
Holography, defined, 102
Homogeneous diattenuator, 130
Homogeneous interfaces, 297, 480
Homogeneous Jones matrices, 523–527, 632
examples, 532
Homogeneous matrices, 631
Homogeneous retarder system, retardance of, 912–914
Horizontal fast axis linear retarder (HLR), 911
trajectories of, 912
trajectory of, 911
Horizontal linear polarizer, 128
Hyperspectral imaging, 260
Ideal depolarizer, 195
Mueller matrix, 194
Ideal diattenuators, 181
Ideal horizontal linear polarizer, 182
Ideal polarization element matrices, notation, 127t
Ideal reflection at normal incident, 631
Ideal retarder, 120
Identities, Pauli matrices, 509–510, 510t
Identity matrix, 165, 167, 201
Illumination
extinction ratio map, 803
Illumination pattern, 718
Image formation, through A-plate, 771–777
Image formation, with polarization aberrations
amplitude response matrix, 593, 594, 601–603, 605–607, 610–611, 615, 617, 619–621
critical angle corner cube retroreflector, 618–622
discrete fourier transformation, 594–597
Mueller Point Spread Matrix, 582, 593, 594, 602, 603–607, 611–614, 620–622
optical transfer matrix, 593, 608–610, 621–622
polarization structure of images, 607
polarized pupil with unpolarized object, 610–614
skew aberration and, 663
Image quality of lens, 17
Imaging polarimeters, 241
Imaging systems, 17
Impermeability tensor, 750
Incident, 716
Incident angle
radian and diattenuation, 728
Incident beam
diffracts into multiple orders, 816
Incident light descriptions, 831
Incident plane wave, 299
Incoherent imaging systems, 608
Incoherent light, 91
Incoherent ray tracing, 407–409, 936
Incomplete polarimeters, 222
Index ellipse, 751
Index indicatrix, 750
Indium tin oxide (ITO), 843
glass substrate, 861
Inhibited reflection, 692
Inhomogeneous interface, 297, 480
Inhomogeneous Jones matrices, 531–532
diattenuation space, 533–534, 534f–535f
example, 534
retardance spaces, 534, 534f–535f
Inhomogeneous matrices, 126, 149, 631
Inhomogeneous polarizer, 188
Injection-molded lens, 890
stress tensor coefficient, 892
In-plane switching (IPS), 846
LC cell, 852
advanced S-IPS (AS-IPS), 853
electrodes, 853
Mueller matrix spectropolarimetric test, 874
transmittance through, 853
well-aligned, retardance eigenstates of, 874
Instrumental polarization, see Polarization aberrations
Intensity, in radiometry, 37
Intensity distribution, 96
Intensity gradients, 280
Intensity interference pattern, 102
Intensity modulation, 843
Intensity/phase change with incident angle, 307–308
Intensity reflection coefficients, 311
Intensity transmission coefficients, 21, 311
Intensity transmittances, 130
Interference colors, 267–269; see also Polariscopes
Interference fringes, 91
Interference of horizontally/vertically polarized light (example), 98–99
Interference of polarized light; see also Polarized light
addition of polarized beams
addition of polarized light of two different frequencies, 103–105
addition of polychromatic beams, 105–109
Gaussian wave packet example, 109–112, 112t
holography, polarization in, 102
of nearly parallel monochromatic plane waves, 95–100
plane waves at large angles, interference of, 100–101
Interference of right/left circularly polarized light (example), 99–100
Interferometer(s), 93–95, 630, 655, 715, 880; see also Interference of polarized light
beam splitter coatings in, 479
combining P matrices for, 346–347
Interferometer with PBS, 337–344
analyzer, 341
cumulative P matrix for, 342
cumulative P matrix for, 342–344
Internal reflection, 306, 311–312; see also Fresnel refraction/reflection
Interpolation method, 716, 732
Interrupted sinusoidal projection, 83
Inverse-distance weighted interpolation algorithm, 733
Inverse problem, 117
IPA technology, 853
Irregular pupil shapes, 594
Isoplanatic patch, 594
Isotropic interfaces, 297, 480
Isotropic material, 297
Isotropic media, 669, 674t; see also Birefringent materials
classes of, 670f
propagation in, 36; see also Polarized light
Isotropic modes, 670
Isotropic-to-biaxial interface, 705
Jones calculus, 425
Jones exit pupil, 550, 551, 564, 580, 581, 582, 594, 598–601
Jones matrices, 323, 545, 630, 832, 911, 918, 924, 935, 936, 941; see also Jones vectors
about, 21
canonical form, 546
circular retarder, 619
component calculation, 207
decomposition
in decreasing/increasing phase conventions, 154t–155t
diattenuator, 521–523, 528–529
dichroic and birefringent materials, 118
elliptical diattenuator, 140–141
elliptical retarders and retarder space, 529–531
with Fresnel coefficients, 308
for HLR, 921
increasing phase sign convention, 153
Jones matrix notation, 127, 127t
for linear diattenuator (example), 131
for linear diattenuator, derivation (example), 131
linear diattenuator Jones matrices, 130–132
matrix properties of
Hermitian matrices, diattenuation, 144–145
polar decomposition, separating retardance from diattenuation, 149–152
unitary matrices/unitary transformations, retarder, 145–149
for monochromatic beams, 912
to Mueller matrices (conversion) using Pauli matrices, 207
into Mueller matrices (transformation) using tensor product, 202–206
to Mueller matrix, converting (numerical example), 205–206
non-polarizing Jones matrices for amplitude/phase change, 143–144
objectives, 123
Pauli matrices and, see Pauli matrices
polarizer Jones matrices, 128–130, 129t
for ray intercept, 546
for reflection and refraction, 329
retarder Jones matrices
circular retarder Jones matrices, 137
linear retarder Jones matrices, 133–137, 135t
rotation of Jones matrices, 127
sequences of polarization elements, 515–518
3D generalization of, see Polarization ray tracing matrix (P)
weak polarization elements, 508, 535
weak polarization interactions, 546–548
Jones pupils, 21, 404, 424, 545, 550, 567, 580, 594, 598–599, 937, 940, 948; see also Local coordinates
artifacts, 438
converting P pupils to, 437–439
elements, value of, 946
polarization ray tracing, 949
polarized pupil with unpolarized object, 610–614
Jones rotation matrix, 127, 146
Jones vectors, 323–324, 568, 631, 921; see also Polarized light
elements, 124
elliptically polarized (example), 47
with local coordinates, see Local coordinates
method for polarized light calculations, 32t
for normalized linearly polarized light, 43
for polarization states, 5
rotation of, 42
rotation operation, 43
sign conventions, 75
and Stokes parameters, conversions between, 75–78
into Stokes parameters, converting (example), 78
Kriging interpolation, 730, 734
KTP (potassium titanyl phosphate) crystal, ray tracing, 670, 672, 686–693
Lagrange invariant (H), 373–374, 661
Laser beams, 104
Laser light, 70
Lasers, high-power, 257
Laser systems
high-power, 786
Latitude, 423, 424; see also Local coordinates; Longitude
Least square fit, 495
Left circularly polarized beam, 44
Left-handed coordinate system, 641
Left hand rule, 43
Lens design, 17
Light
defined, 2
monochromatic, 2
Light beams, 368
Light distribution, 728
Light frequency
in radians per second, 751
Light intensity, 686
Light-measuring polarimeter, 8, 221, 222
Light polarization, 745
Light propagation, 787; see also Fresnel equations
homogeneous/isotropic interfaces, 297
plane of incidence, 296
plane waves/rays, 296
four-dimensional phase space of, 374
Light reflection
diattenuation of, 823
Light–surface interaction, 719
Light transmission
diattenuation of, 823
Light waves, 92–93; see also Interference of polarized light
LiNbO3 modulators
in fiber optics, 865
Linear diattenuation (LD), 139–140, 179, 180, 545, 572, 573; see also Jones matrices
followed by retarder (example), 189
Linear diattenuators, 546
followed by retarder (example), 189
Linearly polarized light, 34, 42–43, 67
Linearly polarized Stokes parameters, 73
Linearly polarized wavefront, 423
Linear polariscope, 263–266; see also Polariscopes
Linear polarization sensitivity, 179
Linear polarization state, 122, 168
Linear polarizer
defined, 120
on eigenpolarization states (example), 129
on eigenpolarization states, operation of (example), 129
polarization elements, 5
uncoated lens surface, 22
Linear retardance, 545, 572, 573
Linear retarders, 546
defined, 122
with horizontally polarized fast axis, 134, 169
and linear polarized components, 6
Linear retardersP matrix for, 331–332, 332t
Linear skew aberration, 654, 655, 659, 660
Line spread function, 818
Liquid crystal display (LCD), 12, 16–17, 843; see also Polarization problems in optical systems
configuration, 843
device, 717
image quality, 837
monitors, 847
TVs, 873
Liquid crystal on silicon (LCoS), 275
Liquid crystal on silicon (LCoS) cells, 854
schematic of, 854
Liquid crystals (LCs), 273
polymers, 940
Liquid crystals (LCs) cell, 837, 838, 839, 841, 845
cell configuration, 843
cell performance, limitations, 864
exiting polarization state, spectral variation of, 867
polarization aberrations with biaxial films, 868–870
retardance with angle of incidence, 867–868
cell technologies, 846
compensating film, 876
configurations of
in-plane switching (IPS), 852–854
liquid crystal on silicon (LCoS) cells, 854–855
super twisted nematic cell (STN), 849–850
90° twisted nematic cell, 847–849
vertically aligned nematic (VAN), 850–852
oscillating square wave voltage, 863
spacers, 861
disclinations, 872
display, high contrast ratio intensity modulation, 845
distributions of directors, 841, 842
driving voltages, 863
limitations, 864
linear retardance magnitude plot, 869
mean linear retardance magnitude and peak-to-peak values, 869
molecules, temperature influences, 839
Mueller matrix spectropolarimeter, 872
projector, 845
retardance and orientation, 872
retardance magnitude, 864
TV sets, 845
types of, 838
Liquid crystal (LC) variable linear retarders, 254, 254f
Liquid crystal (LC) variable retarders, 259
Lissajous figure, 103
Littrow monochromator, 819, 821
Local basis vectors, 636
Local coordinates, 323–324, 325, 629, 649
converting P pupils to Jones pupils, 437–439
dipole coordinates, see Dipole coordinates
double pole coordinates, see Double pole coordinates
for entrance and exit pupils, 423–424
high numerical aperture wavefronts, 436–437
Jones matrices, 665
objective of, 424
orthogonal transformations between, 325–331
right-handed, 632
Logarithms of Jones matrices, 519–520
Longitude, 423, 424; see also Latitude; Local coordinates
Lu–Chipman decomposition, 630
Luneburg lens, 17n
Mach–Zehnder interferometer, 93, 94, 94f, 95f
MacNeille, S. M., 492
MacNeille beam splitter, 492–493, 493f–494f
MacNeille-type cube beam splitter, 855
Magnesium fluoride (MgfO2), 483–484, 485, 489–490, 493, 500
vectors, 33f
Magnetic induction, 676
Magnetic permeability tensor, 676
Major axis, orientation of, 45, 47, 83
Maltese cross, 22
Cassegrain telescopes, 472
uncoated single-element lens, 451–452, 451f
Marginal rays, 372, 372f–373f, 585, 658
angle, 660
Mathematica, 734
Mathematics of polarimetric measurement/data reduction
Mueller data reduction matrix, 226–230
Mueller matrix elements, measuring, 225–226
null space/pseudoinverse, 230–240
MATLAB, 734
Matrix cube, for Pauli matrix sum, 514
Matrix exponential, 518; see also Exponentiation of Jones matrices
Matrix inverse, for Pauli matrix sum, 513
Matrix logarithm, 518; see also Logarithms of Jones matrices
Matrix multiplication, 123
Matrix product of Mueller matrices, 201
Matrix properties of Jones matrices
Hermitian matrices, diattenuation, 144–145
polar decomposition, separating retardance from diattenuation, 149-152
unitary matrices/unitary transformations, retarder, 145–149
Matrix square roots, 151
for Pauli matrix sum, 514
Matrix transpose, for Pauli matrix sum, 513
Matrix vector multiplication, 41
Maxwell’s equations, 296, 672, 830, 940
for periodic boundary conditions, 818
Maxwell’s fisheye lens, 17n
Measurements, polarization aberrations, 580–583
Mechanical quality factors, 255
Media, light propagation in, 297–298
Media refractive index, 815
Meridional plane, 667
Merit function, 941
Metal reflection; see also Fresnel equations
normal incidence reflectance of metals, 315
retardance/diattenuation of metal at non-normal incidence, 315–316
Metal reflectors, 490
Metric thicknesses, 486
Michael–Levy interference color chart, 268f, 271
Michelson interferometers, 718
optics, 667
Micro-polarizer array polarimeter method, 242–243
Mie theory, 260
Mirror element
retardance maps, 950
Mode combination surface
coordinate system, 731
Mode locked lasers, 111
Modulated polarimeters, 240
example, 234
Modulation transfer functions (MTFs)
polarization, 904
Modulation transfer matrix (MTM), 594
Mollweide projection, 83
Monochromatic light, 2
Monochromator/spectrometer
linear dispersion of, 818
Monte Carlo methods, 936
Monte Carlo routines, 935
Motion blur, due to slow response time, 865
MOTM, see Mueller matrix optical transfer matrix
MPSM, see Mueller point spread matrix
MSPI/MAIA imaging polarimeters, 258–259
MTM, see Modulation transfer matrix, see Mueller modulation transfer matrix
Mueller coherence matrix, 198
Mueller–Jones matrices, 192
Mueller matrices, 21, 774, 832, 871, 874, 875, 909, 935, 936
Cassegrain telescope, 472, 473f
columns, meaning of, 164
conversion of Jones matrices to Mueller matrices using Pauli matrices, 207
depolarization
addition of Mueller matrices, 201–202
degree of polarization surfaces/degree of polarization maps, 196–198
depolarization index/average degree of polarization, 195
physically realizable Mueller matrices, testing for, 198–200
weak depolarizing elements, 200–201
for horizontal linear polarizer (example), 174
incoherent ray tracing, 407–409
Jones matrices into Mueller matrices using tensor product, transforming, 202–206
for linear polarizer, 175
multiplication (example), 174
non-depolarizing Mueller matrices, 192
non-polarizing Mueller matrix, 165–166
overview, 163
for physicality, testing (example), 199
Poincaré sphere operations
indicating polarization properties, 188
of polarizers and diattenuators, 187
of retarders on Poincaré sphere, 182–185
of rotating linear retarder, 186
point spread matrix (PSM), 955
polarization elements, sequences of, 165
polarizer/diattenuator Mueller Matrices
basic polarizers, 174–177, 176t
polarizance, 180
transmittance and diattenuation, 177–180
ray tracing with
about, 211
Mueller matrices for reflection, 212–213
Mueller matrices for refraction, 212
retarder Mueller matrices, 168–174, 170t
rotating polarization elements about light direction, 166–167
rotation matrix for, 72
transforming into Jones matrices, 207–210
weakly polarized (example), 191
weak polarization elements, 190–192
Mueller matrix image, 868
of poor-quality polarizing film, 871
Mueller matrix imaging polarimeter, 580–582
measurements, 910
Mueller matrix optical transfer matrix (MOTM), 617
Mueller matrix polarimeters, 867, 872, 913, 930, 936
testing, 872
Mueller matrix polarimetry
principal retardance spectra, 913
Mueller matrix spectrum, 873
Mueller modulation transfer matrix (MTM), 608
Mueller optical transfer matrix, 608
Mueller point spread matrix (MPSM), 582, 593, 594, 602, 623, 654, 774, 937
of aberration-free system, 604
of Cassegrain telescope, 604–605
critical angle CCR and, 620–622
diffraction-limited, 654
o- and e-wavefronts, 775
overview, 603
polarization structure of images and, 607
polarized pupil with unpolarized object, 611–614
for U.S. Patent 2,896,506, 665–666
Mueller polarimetry configurations, 272–276; see also Polarimetry
dual rotating retarder Mueller matrix polarimeter, 274–275
polarimetry near retroreflection, 275–276
Mueller rotation matrix, 166
Multi-angle imager for aerosols (MAIA), 257, 258, 259
Multi-angle spectro polarimetric imager (MSPI), 258
Multi-domain vertically aligned (MVA) cell, 851
configuration for, 852
LCD pixel, 876
Multilayer thin films, 486–496
polarizing beam splitters, 492–496
quarter and half wave films, 488–489
reflection-enhancing coatings, 489–492
Multi-order retardance, 933; see also Retardance
compound retarder, 933
misaligned two-element compound linear retarder, 934
principal retardance component, 934
retardance discontinuities, 933
retardance space, 933
unwrapped retardance, 933
Multiple birefringent plates
wavefront exiting systems of, 723
Multiple scattering, 260
Multivalued optical path length (OPL), 711, 909
misaligned two-element compound linear retarder, 910
NA, see Numerical aperture
Nearly parallel monochromatic plane waves, interference of, 95–100
Nearly singular Mueller polarimeter (example), 285
Negative uniaxial crystals, 674
birefringence spectra of, 745
Nematic liquid crystals, 839
Nematic molecules
schematics for, 840
FOV for, 778
Nilpotent matrix (example), 144
Nodal aberration theory, 400
Non-co-propagating wavefront combination, 728
Non-depolarizing Mueller matrices, 192, 195, 202
Non-ideal polarization elements, 221
Non-orthogonal coordinate system of Stokes parameters, 78–79, 166
Nonorthogonal igenpolarizations, 149
Non-periodic grating, 813
Non-polarizing beam splitter, 93, 486
Non-polarizing Mueller matrix, 165–166
Non-polarizing optical element, 165
Non-polarizing optical systems, 633–634
Non-polarizing reflection, 639
Non-polarizing rotations, 629
Non-sequential ray tracing, 407
Normal (ray intercept), 586
Normal incidence, 304–305; see also Fresnel equations
reflectance of metals, 315
Normalized coordinates, wavefront analysis, 393
Normalized flux, 41
Normalized Jones vector, 38, 39t
Normalized Pauli summation, 549
Normalized propagation vectors, 95
Normalized Stokes parameters
defined, 178
North pole, 4
Null space, 220, 225, 230–240, 232, 282
Numerical aperture (NA), 23, 25, 26f, 373
skew aberration and, 654, 658, 667
wavefronts, high, 25–26; see also Polarization problems in optical systems
Object plane, 369
Odd permutations, Pauli matrices, 510
Off-axis telescopes, 567
oie-mode, 695
oio-mode, 695
Open set, Jones matrices, 523
Optical activity, 137
Optical coherence tomography, 718
Optical design
polarization ray tracing, 20–21
programs, 20
Optical detectors, 37
Optical elements, 941
Optical indicatrix, 749
Optical isolators, 715
Optically active materials, 676
Optical path difference (OPD), 182, 307, 642, 645, 724, 766, 912
contribution, 724
Optical path length (OPL), 17, 20, 120, 143, 166, 359, 545, 630, 716, 858, 931
A-plate, 768
retardance magnitude through the quarter wave, 769
birefringent materials, 684–685
within birefringent plate, 724
compound retarder, 910
decreasing phase convention, 379
eigenmode propagating, in anisotropic material, 718
eigenstates, 722
grid locations, on spherical surface, 734
intermediate matrices, 720
interpolated function, 733, 735
misaligned retarder, 910
misaligned two-element compound linear retarder, 910
multi-order retarder, 910
multivalued, 711
off-axis collimated wavefront
birefringent plate, 724
polarization state distributions, 730
polarization states, 723
ray grids, 729
rays and ray segments, 718
retarder space, 910
tangential and sagittal rays, 771
waveplate, 724
Optical power, 585
Optical rotatory power, 676
Optical systems
defined, 364
etendué, see Etendué
interfaces, 367
Lagrange invariant (H), 373–374
marginal and chief rays, 372, 372f–373f
numerical aperture (NA), 373
object plane, 369
specifications of, 364–368, 365f
surface equations, 366
surface parameters, 365t
wavelength, 365
Optical thicknesses, 486
Optical transfer function (OTF), 500, 593, 937
Optical transfer matrix (OTM), 593, 608–610, 623, 937
of Cassegrain telescope, 608–609
critical angle CCR and, 621–622
polarized pupil with unpolarized object, 611–614
Optical tweezers, 139
Optical waveplates
wavefront aberrations, 742
Order-dependent terms, 546
Order of the retarder, 645
Ordinary refractive index, 741
Orientation, plane of incidence, 551, 552–553
Orientor basis set, 573
defined, 573
Orthogonality of two Jones vectors (example), 48
Orthogonal Jones vectors, 48–49, 48f
Orthogonal polarization states, 74–75, 75t, 336
Orthogonal unit vectors, 168
Orthonormal polarization states, 50
OSA Handbook of Optics, 785
Oscillating electric dipole, 430
Oscillating element polarimetry, 220
OTF, see Optical transfer function
OTM, see Optical transfer matrix
Out-of-plane diffraction grating equation, 816
Out-of-plane grating equation, 817
graphic depiction, 817
Over driving circuit (ODC) technologies, 865
Overlapping wavefronts, see Wavefronts, overlapping
Pancharatnam-design retarders, 916
Parallel polarizer linear polariscope, 265
matrix, 631, 636–640, 642, 654
Parallel transport of vectors, 634–636
Paraxial angle of incidence, 544, 550–553, 562
Paraxial calculation, 562
Paraxial limit, 660
Paraxial marginal ray, 586, 660
Paraxial optics, 497, 543, 583–585, 657, 660, 664
Paraxial polarization aberrations
binodal polarization, 558
diattenuation, 553
over series of surfaces, 558–560
overview, 550
retardance, 553
seven-element lens system, 560–567
tilt, 557
Paraxial ray trace, 550, 586–587, 633
equations, 544
polarization aberrations, 550–560
Paraxial skew aberration, 662–663
Paraxial skew rays, 588
Partial coherence calculations, 935
Partial depolarizer Mueller matrix, 194
Partially linearly polarized light, 67
Partially polarized light, 66–70, 73
Mueller matrix for, 180
Patterned vertical-aligned cell (PVA), 851
director distribution, 852
Pauli c-coefficients, 512, 512f
Pauli coefficients
combination of weak diattenuations, 550f
Pauli logarithm coefficients, 523–525
Pauli matrices, 543, 547, 573, 917; see also Jones matrices
as anti-commutative, 510
canonical summation form, 514–515
Cartesian rotation matrix, 511
eigenvalues, 513
eigenvectors, 513
even permutation, 510
odd permutations, 510
polarization element rotated about optical axis, 511–512
sign convention, 511
Periodic grating, 812
Phase change, 143
Phase convention, 133
Phase depolarization, 200
Phase discontinuities, in thin films, 499–501
Phase distribution, 92
Phase grating, 812
structure, 813
Phase matching, 297
Phase modulation polarimetry, 220
Phase of light, 76
Phase retardation (optical path difference), 631
Phase sign convention
increasing, 55–56, 153, 154t–155t
Phase transfer matrix (PTM), 594, 609
Phase unwrapping operation, 500
Photoelasticity, 880
Photoelastic modulator (PEM), 254, 259
Photon, polarization of, 65
Physically realizable Mueller matrices, testing for, 198–200
Piezoelectric transducer, 255
Piston (constant), polarization aberrations, 543, 544f, 557, 558f
diattenuation, 557, 562, 564–566
Pixel gain, variation in, 259
Pixel misalignment, 281; see also Artifacts in polarimetric images
Planar aligned nematic cell, 846
Planar wavefront incident
birefringent plate, 721
Plane of incidence
ray intercepts, 547
Plane of polarization, 25
Plane waves at large angles, interference of, 100–101
Plate birefringence, 777
Poincarè sphere, 4–5, 252, 630, 845, 867, 873, 909, 932
Poincaré sphere operations
indicating polarization properties, 188
of polarizers and diattenuators, 187
of retarders on Poincaré sphere, 182–185
of rotating linear retarder, 186
Point spread function (PSF), 18–20, 500, 582, 593, 603, 611, 654, 818, 937, 958
of Cassegrain telescope, 472, 473f
cross-section profiles, 958
DoP variation, 956
log10 scale, 957
shape of, 959
skew aberration effect on, 663–665
of uncoated single-element lens, 453–455, 454f
Point spread matrix (PSM), 453–455, 956
Mueller matrix, 955
polarization fluctuations, 955
shape of, 959
Polarcor, 180
Polar Decomposition Example, 151
Polar decompositions, 149–152, 643
Polarimeters, 630, 631–633, 910
classes of
division-of-amplitude polarimeter, 241
division-of-aperture polarimeters, 241
imaging polarimeters, 241
modulated polarimeters, 240
time-sequential polarimeter, 240
complete/incomplete, 222
with four polarizers (example), 224
light-measuring, 221
polarization generators/analyzers, 222
remote sensing, 259
sample-measuring, 221
Polarimetric data reduction equation, 223, 228
Polarimetric data reduction matrix, 223, 224, 253
Polarimetric measurement equation, 223
Polarimetric measurement matrix, 223, 228, 236, 252
Polarimetric measurement vector, 227
Polarimetry, 580
artifacts in polarimetric images
pixel misalignment, 281
mathematics of polarimetric measurement/data reduction
Mueller data reduction matrix, 226–230
Mueller matrix elements, measuring, 225–226
null space/pseudoinverse, 230–240
Mueller matrix image interpretation, 276–279
polarimeter
complete/incomplete, 222
light-measuring, 221
polarization generators/analyzers, 222
sample-measuring, 221
polarimeter calibration, 279–280
polarimeter optimization, 282–286
polarimeters, classes of
division-of-amplitude polarimeters, 241
division-of-aperture polarimeters, 241
imaging polarimeters, 241
modulated polarimeters, 240
time-sequential polarimeter, 240
polarization images, 220
sample-measuring polarimeter
about, 262
Mueller polarimetry configurations, 272–276
Stokes polarimeter configurations
atmospheric polarization images (example), 259–262
MSPI/MAIA imaging polarimeters, 258–259
photoelastic modulator (PEM) polarimeters, 255–258
rotating element polarimetry, 249–253
simultaneous polarimetric measurement, 242–248
variable retarder/fixed polarizer polarimeter, 254–255
Polariscopes, 880, 898; see also Polarimetry; Sample measuring polarimeter
CD substrate, 898
defined, 14
plastic glasses, 899
plastic lens, 899
plastic tape dispenser, 898
Polaris-M polarization analysis program, 560, 615, 670
Polaris-M polarization ray tracing program, 827
Polarizance, 180
Polarizance vector, defined, 180
Polarization
about, 2
components, exponential form of, 151
critical optical systems, 1
Polarization aberration function, 21, 323, 394–395, 395f, 397–398, 452, 455, 599
Polarization aberrations, 323, 447, 448, 786; see also Aberration
amplitude response matrix, 954–955
analysis, 944
conventional optical design, 20
defined, 543
diattenuation at center of pupil, 951
ghost PSF in XY- and YX-components, 953–954
linear variation of diattenuation, 952
linear variation of retardance, 952
polarization-dependent astigmatism, 952–953
PSF shear between XX- and YY-components, 952
retardance at center of pupil, 951–952
diattenuation aberration, 653
diattenuation/retardance contributions, 949–950
division, 653
functions, 773, 774, 775, 827, 937
higher-order, see Higher-order polarization aberrations
aberration expression, 946–949
and Jones vector, 318
of lenses, 22–25; see also Polarization problems in optical systems
liquid crystal displays/projectors, 16
mitigation, 940
analyzing polarization ray tracing output, 941–942
Mueller matrix point spread matrices, 955–958
paraxial, see Paraxial polarization aberrations
polarization aberration expression, 943–945
polarization ray trace, 945–946
polarization ray tracing, comparison of, 942
PSF image components, location of, 958–959
retardance aberration, 653
second-order, 557
skew aberration, 653, 667; see also Skew aberration
understanding of, 630
weakly polarizing ray intercepts, 548–550
weak polarization interactions, 546–548
Polarization and Directionality of Earth’s Reflectances (POLDER), 260
Polarization beam splitter (PBS), 95f
cubes, 492
Polarization bidirectional reflectance distribution function (PBRDF), 935
models, 936
Polarization dependent loss (PDL), 179
Polarization economy, 837
Polarization elements, 940
about, 2
about light direction, rotating, 166–167
defined, 117
depolarizers, 6
optical properties, 118
in polarization ray tracing, 20
retarders, 6
sequences of, 165; see also Mueller matrix
Polarization ellipse, 2, 3, 22, 34, 38, 40, 74, 103, 572
Polarization engineering, 1, 929
Polarization fringes, 91
Polarization generator, 221, 222f
Polarization images, 220
Polarization-independent amplitude change, 525
Polarization-independent phase change, 525
Polarization independent transmission, 120
Polarization leakage, 15
Polarization matrix propagation method, 21
Polarization models, 856
extended Jones matrix method, 856–857
liquid crystal cell ZLI-1646, 859–861
multilayer interference models, 859
retardance of single-pixel polarization controller, 862
single pass, with polarization ray tracing matrices, 857–858
Polarization piston, 557, 558f
Polarization problems in optical systems
angle dependence of polarizers, 12
high numerical aperture wavefronts, 25–26
liquid crystal displays/projectors, 16–17
polarization aberrations of lenses, 22–25
stress birefringence in lenses, 14–15
wavelength and angle dependence of retarders, 13
Polarization properties; see also Jones matrices
homogeneous Jones matrices, 523–527
inhomogeneous Jones matrices, 531–532
polarization elements and, 5–7
of polarization elements and optical elements, 117
Polarization quality, wavefront analysis, 406
Polarization ray tracing (PRT), 1, 20, 118, 323–324, 550, 615, 629, 642, 649, 665, 723, 892–895
additional information obtained from, 381–382, 381t
amplitude transmission, 382
for birefringent interface, 695–705
cell phone lens, see Cell phone lens
hollow aluminum corner cube, 347–351
interferometer with PBS, 337–344
with P matrix, see Polarization ray tracing matrix (P)
proper retardance algorithm for, 643–545
Polarization ray tracing complications, 930
elliptical polarization properties, of ray paths, 931
geometrical transformation, 933
linear skew aberration transforms, 933
optical path length/phase, 931
antireflection coatings, 931
constructively interfere, 931
spherical wavefront, from direct light through the system, 932
optical system description complications, 930
diffraction gratings, 930
liquid crystal cells, 930
reasonable distribution, of stress birefringence, 931
stress birefringence, 930
thin films, 930
parallel transport, 933
proper retardance, 933
retardance, 932
Polarization ray tracing (PRT) matrices, 324–334, 385–392, 932, 935, 936
diattenuation calculation, 334–337
Jones pupils, converting into, 437–439
orthogonal transformations, 325–331
for refraction and reflection, 330
two gold fold mirrors, 387–390
Polarization ray tracing methods, 937
alternative simulation methods, 940
cell phone lens system, 938
Jones matrices, 937
Jones pupil, 937
optical transfer matrix (OTM), 937
P matrix, 937
point spread function (PSF), 937
Polarization rotation, 655
Polarization state analyzer (PSA), 263
in Jones matrix, 911
rotates, 911
Polarization state generator (PSG), 263f
Polarization states
exiting, 164
time-varying, 103
variations of, 17
Polarization structure, of images, 607
Polarization transformations, 184
Polarization vector, 672
defined, 32
to Jones vectors, converting, 51–54
method for polarized light calculations, 32t
Polarization vortices, 138
Polarized flux, 36–37, 66, 75–78
Polarized incident beam, 722
Polarized incident light, 220
Polarized light, 1
change of basis, 49
circularly polarized light, 43–45
elliptically polarized light, 45–48
Jones vectors
rotation of, 42
linearly polarized light, 42–43
magnetic field/flux/polarized flux, 36–37
methods for calculations, 32t
and optical systems, 721
overall phase, evolution of, 41–42
phase sign convention
polarization state of sources, 56–59
polarization vector
to Jones vectors, converting, 51–54
polarized flux components, 50–51, 51t
propagation in isotropic media, 36
Polarized light of two different frequencies, addition of, 103–105
Polarized part of flux, 76
Polarized sunglasses, 219
Polarizer; see also Poincaré sphere operations
about, 2
angle dependence of, 12; see also Polarization problems in optical systems
characterized by diattenuation, 177
defined, 119
ideal, 174
operation of, 187
over aperture (example), 201
tests, 132
Polarizer/diattenuator Mueller matrices
basic polarizers, 174–177, 176t
polarizance, 180
transmittance and diattenuation, 177–180
Polarizer Jones matrices, 129t
Polarizer Mueller matrices for basis polarization states, 176t
Polarizing beam splitters (PBS), 6, 6f, 93, 246, 247, 247f, 479, 729; see also Beam splitters coatings
collimated wavefronts, 729
multilayer thin films, 492–496
two collimated wavefronts, 729
Polarizing films, 870
Polycarbonate (PC), 882
Polychromatic beams, addition of, 105–109
Polychromatic light
defined, 63
Polychromatic unpolarized beam, measurement of, 105
Polymethylmethacrylate (PMMA), 882
Polynomial curve fitting, 495–497
Polystyrene (PS), 882
Positive uniaxial crystals, 674
birefringence spectra of, 744
Potassium titanyl phosphate (KTP) crystal, ray tracing, 670, 672, 686–693
Poynting vectors, 37, 368, 716, 720, 752, 777
mapping, 720
Primary cloudbow, 261
Principal refractive indices, 673
Probability amplitude, 932
Product notation, 124
Projection screens, 6
Propagation sphere, 425, 425f; see also k-sphere
Propagation vectors, 17, 32, 324, 325f, 655, 660, 662, 667, 672, see Snell’s law
Propagation vector sphere, 663
Proper retardance
definition of, 642
identifying, 633
parallel transports for reflection and, 636
4’-propyl-bicyclohexyl-4-carboxylic acid 3,4,5-trifluoro-phenyl ester (C22H29F3O2)
chemical structure of, 839
PRT, see Polarization ray tracing
Pseudo electric field, 702
Pseudoinverse, 230–240, 233, 240, 496
PSF, see Point spread function
PTM, see Phase transfer matrix
Pulse trains, 111
Pure diattenuator, 130
Pure retarder, 168
Quadratic diattenuation coefficients, 561
Quadratic radially oriented retardance, 600–601
Quadratic retardance coefficients, 562
Quality, wavefront analysis
polarization, 406
wavefront, 405
Quantum optics, 109
Quarter wave coating, 931
Quarter wave layers, 489
Quarter wave linear retarder (QWLR), 331, 331f
Jones matrices for, 134, 338–339, 516
Mueller matrices for, 169, 170t, 171
Poincaré sphere, 183
sequence, 516
spinning (example), 202
Quarter waveplates, 766
Quarter wave retarder (QWR), 169, 247f
Quarter wave thick magnesium fluoride coatings, 8
Quartz HLR, 919
Quartz retardance, 913
principal retardance, 914
function of wavelength, 923
unwrapped retardance, 923
Quasi-monochromatic light, 70
Radially symmetric optical system, 658
Radially symmetric systems, 550
Radiometry with polarization elements, 221
Ray diffracts
into multiple diffraction, 812
angular variation of retardance, 727
birefringent plate, 725
corresponding modes, 731
cross- and co-polarized systems, 801
interpolation functions, 733
irregular grids, combination, 730
inverse-distance weighted interpolation, 732–734
at locations, 732
point source on spherical wavefronts, 717
positions of ray, 730
propagating, 728
quarter wave calcite A-plate with optic axis, 772
algorithm for, 378
Jones matrix for, 546
multiplicity with surface, 378
planes of incidence, 547
Rayleigh scattering, 260
Ray paths, 642
Ray propagation, 719
through co-polarized system, 800
Ray splitting, see Double refraction
Ray stop criteria, 695
Ray tracing
algorithms, 729
amplitude coefficients, 383–385
in birefringent materials, see Birefringent materials
coherent and incoherent, 407–410
defined, 1
description, 359
geometrical, 359; see also Polarization ray tracing
interface Jones matrix, 383–385
light beams, 368
with Mueller matrix; see also Mueller matrix
about, 211
Mueller matrices for reflection, 212–213
Mueller matrices for refraction, 212
non-sequential, 407
in optical design, 17
optical path length, see Optical path length
optical system and, see Optical systems
polarization, see Polarization ray tracing
ray intercept, see Ray intercepts
reflection and refraction, 380–381
steps in process of, 360
through birefringent components, 742
wavefront analysis, see Wavefront analysis
Ray tracing DOEs, 818
diffractive retarders, 825
diffractive subwavelength antireflection coatings, 825–829
eigenpolarizations, 818
reflection diffractive gratings, 818–821
Real unitary matrices, 168
Reciprocal ellipsoid, 750
Rectangle function, 594
Reference path, ray tracing, 338–340
cumulative P matrix for, 342
Reference sphere, 370
Reflection, 636, 637f, 638, 646, 649
at birefringent interface, 681–693
diffractive grating, 819
inside a biaxial cube, 707–710
non-polarizing, 639
Reflection-enhancing coatings, 489–492, 497
Reflection gratings
diffractive, 818
Reflection/transmission at glass (example), 302
Reflective coatings, 489; see also Reflection-enhancing coatings
Reflective diffraction gratings, 818
Refraction, 636, 638; see also Reflection; refraction
at birefringent interface, 681–693
Mueller matrices for, 212
Refractive grating
in-plane incident light ray, 815
Refractive index, 20, 36, 297, 365
birefringent material, 675, 683
KTP, aragonite, and mica, 706t
reduced thicknesses and angles, 587–588
skew aberration and, 660
Remote sensing polarimeters, 259
Research Scanning Polarimeter (RSP), 260
Resolution, DFT, 597
Response time
drive voltage and response times, 866
motion blur, due to slow, 865
Response time compensation (RTC), 865
overshoot, 866
Retardance, 932
aberration, 653
with angle, 868
angular variation of, 727
common units for, 121
components, 171
critical angle CCR, 618
from diattenuation, separating, 149–152
higher-order polarization aberrations, 578–580
importance of, 629
measurement technique, 265
of metal at non-normal incidence, 315–316
microscope, 581
for multi-element lens, 23
and optical path lengths, 6
parameters, 173
paraxial, 553
polarization ray tracing, 20, 21, 117
proper, see Proper retardance
of pure retarder Mueller matrix, 168
range, 645
seven-element lens system, 562, 566–567
three-dimensional polarization ray trace matrices, 631
Retardance defocus, 950; see also Defocus (quadratic), polarization aberrations
quadratic radially oriented, 600–601
seven-element lens system, 563–564
Retardance discontinuities, 500
retarder, unwrapped retardance of, 911
Retardance maps, 457
crossed mirror configurations, 465–469, 466f, 467t, 468f, 469f
vs. field of view, 860
Retardance modulation, 845
Retardance modulators, 841, 843
Retardance of P matrices, 337
Retardance piston, 557, 564, 950; see also Piston (constant), polarization aberrations
Retardance space, 933
inhomogeneous Jones matrices, 534, 534f–535f
Retardance tilt, 557, 564, 950; see also Tilt (linear), polarization aberrations
Retardance trajectory, 916
Retardance unwrapping, 645
Retardation plate, 122
Retarders, 6, 118, 120, 121f, 145–149, 168, 630, 909
angle dependence of, 13; see also Polarization problems in optical systems
axis, 183
Jones matrices, 133–137, 520–521, 527
Mueller matrices, 168–174, 170t
multi-order, 909
order of, 645
linear retarders, 331–332, 332t
on Poincaré sphere, operation of, 182–185
Retarder space
Compound retarder at 45°, 922–924
compound retarder’s trajectory, 919–920
half wave linear retarders, 917
homogeneous retarder’s trajectory and retardance unwrapping, 914–916
multiple modes, exit compound retarder system, 920–922
principal retardance, 915, 916, 918, 919, 920
principal retardance trajectory, 916
principal retardance vector trajectories, 915
Quartz HLR, 919
unwrapped retardance, 916, 920
Retroreflection
polarimetry near, 275–276; see also Mueller polarimetry configurations
testing, 275
Retroreflectors, corner cubes as, 347–351
Right circularly polarized light, 43
Right circularly polarized time helix, 44
Right-handed local coordinates, 632
Right hand rule, 43
Rigorous coupled wave analysis (RCWA)
algorithm, 811, 815, 829–832, 830
diffraction efficiency, 820
TE and TM’s, 821
grating/diffractive optical element, 830
simulation, 828
Rochon prisms, 778
Ronchi tests, 929
Rotating element polarimetry, 220; see also Polarimetry
rotating analyzer plus fixed analyzer polarimeter, 250
rotating analyzer polarimeters, 249–250
rotating retarder/fixed analyzer polarimeters, 251–253
Rotating linear retarder, 186
operation of, 186
Rotating quarter wave retarder polarimeter (example), 253
Rotating retarders, 137
imaging polarimeter, 281
Rotation axis, 168
Sag/saggita, 366
Sampled functions, 594
Sample measuring polarimeters, 221, 222
about, 262
Mueller polarimetry configurations, 272–276
polarization properties and, 8
s- and p-components, at ray intercepts, 382–383
Sapphire, aluminum oxide in, 500
Sapphire retarder, 923
Scale
Scattered light, 102
Second-order polarization aberrations, 557
Seidel wavefront aberration expansion, 398–401
Semi-major axis, 47
Semi-minor axis, 47
Sensitive tint plate polariscope, 270
Sequence of polarizers (example), 188
Sequence of retarders, 184
Sequence of rotations of sphere, 184
Sequence of three quarter wave retarders (example), 185
Sequences of polarization elements, 515–518
Seven-element lens system, 560–567
Shifting functions, DFT, 596–597
Simultaneous polarimetric measurement; see also Polarimetry
division-of-amplitude polarimetry (DOAP), 246–248
division-of-aperture polarimetry, 242
division-of-focal-plane polarimetry, 242–245
Simultaneous polarimetric measurement, 242–248
sinc function, 594
Single-element lens, see Uncoated single-element lens
Single-layer antireflection coatings, 485; see also Antireflection coatings
Single-layer thin films, 480–486
antireflection coatings, 482–485
Singularities, of local coordinates, 323–324
Singular matrices, 523
Singular value decomposition (SVD), 150, 282, 335–337
S-IPS mode panels
motion picture image, 851
Skew aberrated local coordinate, 665
Skew aberration, 618, 629; see also Polarization aberrations
CODE V’s U.S. patent library, 658, 666–667
ith ray’s, 657
in paraxial ray trace, 660–661
polarization aberration and, 653, 667
polarization state rotation and, 653–654
U.S. Patent 2,896,506, 658–659
parallel transport, 660
Skew tilt, 663, 664, 667; see also Tilt (linear), polarization aberrations
unchanged convention, 134, 135t
Slow-mode, birefringent interfaces, 681–682
Snell’s law, 17, 754, 306, 360, 480, 583–584, 683, 891
Solid corner cube retroreflectors (CCRs), 614–618
South pole, 4
Spatial filtering operation, 608
Spatially modulated polarimeters, 240
Speckle pattern, 193, 193f, 936
Spectral bandwidth, 2, 70–71; see also Stokes parameters
Spectrally modulated polarimeters, 240
Spectral resolution, 818
Spherical aberration, 498, 572, 583
Spherical interface, 551
Spinning quarter wave linear retarder
about, 186
example, 202
s-/p-polarization components, 298–299
Statistics, skew aberration, 666–667
Stokes image of building (example), 69
Stokes parameter image, 607, 608, 614
Stokes parameter rotation by 45° (example), 72
elliptical polarization parameters, 73–74
example, 65
for Gaussian wave packet spectral components, 112t
and Jones vector, conversions between, 75–78
and Jones vector sign conventions, 75
linearly polarized Stokes parameters, 73
measurement, 104
method for polarized light calculations, 32t
non-orthogonal coordinate system of, 78–79
orthogonal polarization states, 74–75, 75t
partially polarized light/degree of polarization (DoP), 66–70
phenomenological definition of, 64–65, 65t
Poincaré sphere
polarization ellipse, rotation, 72–73
polychromatic light, description of, 63–64
unpolarized light, 65
Stokes polarimeter
configurations; see also Polarimetry
atmospheric polarization images (example), 259–262
MSPI/MAIA imaging polarimeters, 258–259
photoelastic modulator (PEM) polarimeters, 255–258
rotating element polarimetry, 249–253
simultaneous polarimetric measurement, 242–248
variable retarder/fixed polarizer polarimeter, 254–255
with six measurements, 230
Stokes vectors, 64
image, 582
Strehl ratio, 19
Stress birefringence, 880
in lenses, 14–15; see also Polarization problems in optical systems
flange, 892
Stress-induced birefringence, 255, 879; see also Birefringence
computer-aided design (CAD) programs, 879
isotropic glass plate, 881
material’s refractive index, 882, 883
mechanical stress, 880
N-BK7 and polycarbonate (PC), 882
object generate compressive stress and tensile stress, 882
optical materials, 880
on optical system performance, 898
glas spannungen, 901
induced retardance magnitude, 903
injection-molded lens, 904
injection-molded lens, CAD image, 902
injection-molded lens, simulations of, 901–903
injection-molded lens, stress images, 902
linear polariscope images, 900
pickups system, optical layout, 903
plane parallel glass plate, 900
plastic DVD lens, simulations of, 903–905
polarized light refracting, point spread functions of, 905
retardance magnitude, 902
tint plate retarder, 901
in optical systems, 881
molding parameters, 881
plastic tape dispenser, 880
diagonal tensor to non-diagonal tensor, 884
finite element modeling (FEM), 889
Jones pupil amplitude, 897
parabolic stress in tempered glass, 895–896
polarization ray tracing matrix, 892–895
reflections, 891
refraction, 891
refraction, through an injection-molded lens, 891
refractive index, 885
rotationally stressed plate, 896–897
spatially varying stress, 889–897
storage of system shape, 890
strain tensor coefficients, 883
stressed plate, linear polariscope images, 897
stress optic tensor coefficients, 885
stress/strain and optical effect, 884
tetrahedron element, 890
varying stress function, 895
residual stress, 880
retardance magnitude, 898
stress distribution, 897
theory of, 881
tolerance, 880
uniaxial stress
compression and tension, 888–889
optic effect, 886
Subwavelength grating (SWG), 813
intensity transmission of, 827
Jones pupil
amplitude of, 828
wavefront aberration, 829
phase shift of, 828
polarization-dependent aberrations, 828
retardance dispersions of, 825
retardance of fused silica, 826
transmitted phase and retardance, 827
Supernumerary bows, 261
Super twisted nematic cell (STN), 849
director orientations, 849
liquid crystal cell’s directors, 850
Surface admittance, 489
Surface equations, optical systems, 366
Surface grating, 812
Surface parameters, optical systems, 365t
K/S-Surfaces, 749
SVD, see Singular value decomposition
Symmetrical phase convention, 134, 135t, 153
Systematic errors, 280
Taylor series
for Fresnel coefficients, 317–318
Tensor product, 203
Testing liquid crystal cells
analyzer and exiting polarization state, 877
Mueller matrix polarimeter testing, 872, 873
sheet retarder defect, 876
twisted nematic cell example, 873–874
VAN cell, 875
Test path, ray tracing, 338, 340–341
cumulative P matrix for, 342–344
Tetrahedron orientation, 282
Thin film equations, 22
phase discontinuities in, 499–501
wavefront aberrations and, 497–499
3D polarization ray tracing, 324
3D TV, 855
Three-fold mirror system, 636, 637f
ThroughputEtendué
Tilt (linear), polarization aberrations, 543, 544f, 557
diattenuation, 557
Time-modulated polarimeters, 240
Time-sequential polarimeter, 240
Time-varying polarization state, 103
Time-varying Stokes parameters, 104
Tint plate, polariscope with, 269–270; see also Polariscopes
Total internally reflected mode, 692
Total internal reflection (TIR), 306–307, 778, 785, 939
o-mode, 778
Transverse electric mode, 299
Transverse electric (TE) modes, 811
diffraction efficiencies, 820
Transverse magnetic (TM) modes, 299, 811
diffraction efficiency, 820
polarization
wire grid polarizer transmission, 824
Trim retarder, 849
Twisted nematic (TN cell), 847
C-plate, 848
dark state, 849
different voltages, 860
director distributions, 848
helix distribution of the liquid crystal directors, 848
polarization ellipses of normal modes, 849
retardance as function applied voltage, 848
retardance eigenstates, 873
tilt test, 874
Twisted nematic liquid crystal cell, 16, 859
Two linear diattenuators with rotation, cascading, 208
Two-retarder polarization controller, 94
Ultra-short laser pulse, 932
Uncoated single-element lens, 448–455
amplitude response matrix (ARM), 452–453, 453f
angle of incidence maps (AoI map), 449, 449f
diattenuation map, 449–451, 450f
point spread matrix (PSM), 453–455
Uniaxial crystal, ray normally incident, 750
Uniaxial interfaces
incident ray refracting, 939
reflections/refractions, 746–749
amplitude coefficients, of transmitted mode, 764
anisotropic materials, 741
birefringence, 744
converging beam, 743
converging beam, polarization ray trace of, 771
descriptions of, 743
dielectric tensor for uniaxial birefringent, 743
double refraction/ray splitting, 745, 746
eigenmodes/ordinary o-mode/extraordinary e-mode, 743
e-mode transmission, 760
exiting polarization ellipses, 771
extraordinary e-mode, 745
extraordinary mode in transmission, 764
imaging doubling through calcite, 742
incident rays with incident angles, 770
index ellipsoid, 755
index ellipsoid of, 751
extraordinary mode in positive uniaxial material, 757
isotropic material, 755
poynting vectors, 754
negative uniaxial crystals, 744
birefringence spectra of, 745
normal incident beam, polarization ray trace of, 777
normally incident ray, 751
normal surface, 753
o- and e-wavefronts, 773
off-axis ray trace from air, 755
o-mode’s P matrices, 772
o-mode transmittion, 759
sagittal and tangential ray bundles, 768
ordinary o-mode, 745
polarization states, 771
polarization structure, 772
positive uniaxial crystals, 744
p-polarized incident light, 763
principal section, 745
ray trace through blocks, 742
ray with propagation vector, 752
reflection matrix, 765
refraction angle, 756
refractive index, 745
refractive index of exiting e-mode, 770
retardance magnitude of spherical wavefront, 771
several possible index ellipses, 752
s-polarized incident light, 762
S-surface, 756
Uniaxial medium, 744
Uniaxial-to-isotropic interface, 704
Unitary change of basis, 49, 127
Unitary Jones matrices, 145
Unitary matrices, 630, 643; see also Orthogonal matrices
Unitary transformation, 146, 147, 149, 168, 181
Unit propagation sphere, 425
Unpolarized generator, 226
Unpolarized incident, 664
Unpolarized light, 65, 71f, 104, 164, 178, 666, 872; see also Stokes parameters
Untwisted nematic cell, 846
Unwrapped retardance, 916
compound retarder Jones matrix decomposition, 917–919
compound retarder systems with arbitrary alignment, 916
U.S. Patent 2,896,506, skew aberration, 658–659
UV photo, 863
Values estimation, by interpolation, 730
VA panels
motion picture image, 851
Variable beam splitter, 102
Variable retarder/fixed polarizer polarimeter, 254–255
Vectors, 572
basis, 656
coordinate, 640
parallel transport of, 634–636
polarization, 672
propagation, 655, 660, 662, 667, 672
Vector Zernike polynomials, 567, 568–572, 569t–570t, 571f, 575–578
Vertically aligned nematic (VAN), 846, 850
cell’s directors at several voltages, 850
dark state, 851
eigenpolarizations in cell without pretilt, 851
with negative dielectric anisotropy, 844
with negative dielectric anisotropy sandwich, 844
retardance versus tilt angle, 876
Vertical quarter wave linear retarder (VQWLR), 170t, 183, 183f
Volume grating, 812
Vortex retarder, 582
Walk-off angle, 777
Walk-off plate, 777
mode index for calcite versus optic axis angle, 777
vs. wavelength for uniaxial materials, 778
aberration function, 719, 937, 941
combination, 730
combination procedures, 716
converges through birefringent plate, 717
eieieiei-wavefront, 801
eie-mode
with zero piston, tilt, and defocus, 799
eie-mode, residual wavefront, 797
encounter beam splitters, 716
incident collimated beam with planar, 721
individual, 726
interfering, 717
local polarization ellipses of, 823
o- and e-modes for uniaxial material, 721
off-axis beam, 724
off-axis collimated, 724
overlapping, 715
overlapping region, polarization of, 726
piston, 667
point source, propagating toward lens, 717
polarization amplitudes, 801
polarization amplitudes
through co-polarized system, 802
polarization variations of, 734–736
quality, 405
quarter wave calcite A-plate, 772
reconstruct, 730
reconstruction, 731
resultant, 725
shear, 723
sheared non-planar, 726
Wavefront aberrations, 17, 18, 19f, 448, 543, 572, 580, 653, 663
thin films contributions to, 497–499
Wavefront aberration function, 11, 360, 545, 593, 598
dipole vs. double pole coordinates, 440–441
aberration function
wavefront, see Wavefront aberration function
normalized coordinates, 393
quality
polarization, 406
wavefront, 405
Seidel wavefront aberration expansion, 398–401
Wavelength and angle dependence of retarders, 13
Wave nature of light, 92
achromatic, 766
first-order quarter, 766
half, 766
polarization ellipse distribution, 736
polarization ellipse on plane, 735
second-order quarter, 766
true zero-order quarter, 766
Waves, retardance magnitude, 829
Weak depolarizing elements, 200–201
Weak diattenuator Mueller matrix, 190
Weakly linearly polarizing interfaces, 548
Weakly polarizing optical elements, 543, 545
Weakly polarizing ray intercepts, 548–550
Weak polarization aberration, surfaces, 558–560
Weak polarization elements, 190–192, 508, 535
Weak polarization interactions, Jones matrix, 546–548
Weak retarder Mueller matrix, 190, 191
Wedges
wavefront exiting systems of, 723
White light beam, 106
Winding number theorem, 324, 428, 429, 430–431
Wire grid polarizers, 12, 242, 786, 822
angle of incidence variation, 823
optimization, 823
transmission
performance comparison of, 825
of TM polarization, 824