1.2Polarization States and the Poincaré Sphere
1.3Polarization Elements and Polarization Properties
1.4Polarimetry and Ellipsometry
1.6Typical Polarization Problems in Optical Systems
1.6.1Angle Dependence of Polarizers
1.6.2Wavelength and Angle Dependence of Retarders
1.6.3Stress Birefringence in Lenses
1.6.4Liquid Crystal Displays and Projectors
1.7.2Polarization Aberrations of Lenses
1.7.3High Numerical Aperture Wavefronts
1.8Comment on Historical Treatments
1.9Reference Books on Polarized Light
2.1The Description of Polarized Light
2.3Properties of the Polarization Vector
2.4Propagation in Isotropic Media
2.5Magnetic Field, Flux, and Polarized Flux
2.10Circularly Polarized Light
2.11Elliptically Polarized Light
2.16Converting Polarization Vectors into Jones Vectors
2.17Decreasing Phase Sign Convention
2.18Increasing Phase Sign Convention
2.19Polarization State of Sources
3Stokes Parameters and the Poincaré Sphere
3.1The Description of Polychromatic Light
3.2Phenomenological Definition of the Stokes Parameters
3.4Partially Polarized Light and the Degree of Polarization
3.6Rotation of the Polarization Ellipse
3.7Linearly Polarized Stokes Parameters
3.8Elliptical Polarization Parameters
3.9Orthogonal Polarization States
3.10Stokes Parameter and Jones Vector Sign Conventions
3.11Polarized Fluxes and Conversions between Stokes Parameters and Jones Vectors
3.12The Stokes Parameters’ Non-Orthogonal Coordinate System
3.14Flat Mappings of the Poincaré Sphere
4Interference of Polarized Light
4.4Interference of Nearly Parallel Monochromatic Plane Waves
4.5Interference of Plane Waves at Large Angles
4.6Polarization Considerations in Holography
4.7The Addition of Polarized Beams
4.7.1Addition of Polarized Light of Two Different Frequencies
4.7.2Addition of Polychromatic Beams
4.7.3A Gaussian Wave Packet Example
5Jones Matrices and Polarization Properties
5.2Dichroic and Birefringent Materials
5.3Diattenuation and Retardance
5.4.3Rotation of Jones Matrices
5.5Polarizer and Diattenuator Jones Matrices
5.5.2Linear Diattenuator Jones Matrices
5.6.1Linear Retarder Jones Matrices
5.6.2Circular Retarder Jones Matrices
5.7General Diattenuators and Retarders
5.8Non-Polarizing Jones Matrices for Amplitude and Phase Change
5.9Matrix Properties of Jones Matrices
5.9.1Hermitian Matrices: Diattenuation
5.9.2Unitary Matrices and Unitary Transformations: Retarder
5.9.3Polar Decomposition: Separating Retardance from Diattenuation
5.10Increasing Phase Sign Convention
6.3Sequences of Polarization Elements
6.4Non-Polarizing Mueller Matrices
6.5Rotating Polarization Elements about the Light Direction
6.7Polarizer and Diattenuator Mueller Matrices
6.7.2Transmittance and Diattenuation
6.8.1The Operation of Retarders on the Poincaré Sphere
6.8.2The Operation of a Rotating Linear Retarder
6.8.3The Operation of Polarizers and Diattenuators
6.8.4Indicating Polarization Properties
6.10Non-Depolarizing Mueller Matrices
6.11.1The Depolarization Index and the Average Degree of Polarization
6.11.2Degree of Polarization Surfaces and Maps
6.11.3Testing for Physically Realizable Mueller Matrices
6.11.4Weak Depolarizing Elements
6.11.5The Addition of Mueller Matrices
6.12Relating Jones and Mueller Matrices
6.12.1Transforming Jones Matrices into Mueller Matrices Using Tensor Product
6.12.2Conversion of Jones Matrices to Mueller Matrices Using Pauli Matrices
6.12.3Transforming Mueller Matrices into Jones Matrices
6.13Ray Tracing with Mueller Matrices
6.13.1Mueller Matrices for Refraction
6.13.2Mueller Matrices for Reflection
6.14The Origins of the Mueller Matrix
7.2What Does the Polarimeter See?
7.3.1Light-Measuring Polarimeters
7.3.2Sample-Measuring Polarimeters
7.3.3Complete and Incomplete Polarimeters
7.3.4Polarization Generators and Analyzers
7.4Mathematics of Polarimetric Measurement and Data Reduction
7.4.2Measuring Mueller Matrix Elements
7.4.3Mueller Data Reduction Matrix
7.4.4Null Space and the Pseudoinverse
7.5.1Time-Sequential Polarimeters
7.6Stokes Polarimeter Configurations
7.6.1Simultaneous Polarimetric Measurement
7.6.1.1Division-of-Aperture Polarimetry
7.6.1.2Division-of-Focal-Plane Polarimetry
7.6.1.3Division-of-Amplitude Polarimetry
7.6.2Rotating Element Polarimetry
7.6.2.1Rotating Analyzer Polarimeters
7.6.2.2Rotating Analyzer Plus Fixed Analyzer Polarimeter
7.6.2.3Rotating Retarder and Fixed Analyzer Polarimeters
7.6.3Variable Retarder and Fixed Polarizer Polarimeter
7.6.4Photoelastic Modulator Polarimeters
7.6.5The MSPI and MAIA Imaging Polarimeters
7.6.6Example Atmospheric Polarization Images
7.7Sample-Measuring Polarimeters
7.7.1.4Polariscope with Tint Plate
7.7.2Mueller Polarimetry Configurations
7.7.2.1Dual Rotating Retarder Polarimeter
7.7.2.2Polarimetry Near Retroreflection
7.8Interpreting Mueller Matrix Images
7.10Artifacts in Polarimetric Images
8.2.3Homogeneous and Isotropic Interfaces
8.2.4Light Propagation in Media
8.3.1s- and p-Polarization Components
8.3.8Intensity and Phase Change with Incident Angle
8.3.9Jones Matrices with Fresnel Coefficients
8.4Fresnel Refraction and Reflection
8.4.4.1Normal Incidence Reflectance
8.4.4.2Retardance and Diattenuation of Metal at Non-Normal Incidence
8.5Approximate Representations of Fresnel Coefficients
8.5.1Taylor Series for the Fresnel Coefficients
9Polarization Ray Tracing Calculus
9.1Definition of Polarization Ray Tracing Matrix, P
9.2Formalism of Polarization Ray Tracing Matrix Using Orthogonal Transformation
9.3Retarder Polarization Ray Tracing Matrix Examples
9.4Diattenuation Calculation Using Singular Value Decomposition
9.5Example—Interferometer with a Polarizing Beam Splitter
9.5.1Ray Tracing the Reference Path
9.5.2Ray Tracing through the Test Path
9.5.3Ray Tracing through the Analyzer
9.5.4Cumulative P Matrix for Both Paths
9.6The Addition Form of Polarization Ray Tracing Matrices
9.6.1Combining P Matrices for the Interferometer Example
9.7Example—A Hollow Corner Cube
10.3Specification of Optical Systems
10.4Specifications of Light Beams
10.5.3Entrance and Exit Pupils
10.5.4Importance of the Exit Pupil
10.5.6Numerical Aperture and Lagrange Invariant
10.6.2Multiplicity of Ray Intercepts with a Surface
10.6.4Reflection and Refraction
10.6.5Polarization Ray Tracing
10.6.7Amplitude Coefficients and Interface Jones Matrix
10.6.8Polarization Ray Tracing Matrix
10.7.2Wavefront Aberration Function
10.7.3Polarization Aberration Function
10.7.4Evaluation of the Aberration Function
10.7.5Seidel Wavefront Aberration Expansion
10.9Coherent and Incoherent Ray Tracing
10.9.1Polarization Ray Tracing with Mueller Matrices
10.10The Use of Polarization Ray Tracing
10.11Brief History of Polarization Ray Tracing
10.14Appendix: Cell Phone Lens Prescription
11The Jones Pupil and Local Coordinate Systems
11.1Introduction: Local Coordinates for Entrance and Exit Pupils
11.5High Numerical Aperture Wavefronts
11.6Converting P Pupils to Jones Pupils
11.7Example: Cell Phone Lens Aberrations
11.8Wavefront Aberration Function Difference between Dipole and Double Pole Coordinates
12.2Uncoated Single-Element Lens
12.4Combination of Fold Mirror Systems
13.2.2Ideal Single-Layer Antireflection Coating
13.3.2Quarter and Half Wave Films
13.3.3Reflection-Enhancing Coatings
13.3.4Polarizing Beam Splitters
13.4Contributions to Wavefront Aberrations
13.7Appendix: Derivation of Single-Layer Equations
14Jones Matrix Data Reduction with Pauli Matrices
14.2Pauli Matrices and Jones Matrices
14.2.2Expansion in a Sum of Pauli Matrices
14.2.4Pauli Coefficients of a Polarization Element Rotated about the Optical Axis
14.2.5Eigenvalues and Eigenvectors and Matrix Functions for the Pauli Sum Form
14.2.6Canonical Summation Form
14.3Sequences of Polarization Elements
14.4Exponentiation and Logarithms of Matrices
14.4.1Exponentiation of Matrices
14.4.5Polarization Properties of Homogeneous Jones Matrices
14.5Elliptical Retarders and the Retarder Space
14.6Polarization Properties of Inhomogeneous Jones Matrices
14.7Diattenuation Space and Inhomogeneous Polarization Elements
14.7.1Superposing the Diattenuation and Retardance Spaces
14.8Weak Polarization Elements
15Paraxial Polarization Aberrations
15.2.1Interaction of Weakly Polarizing Jones Matrices
15.2.2Polarization of a Sequence of Weakly Polarizing Ray Intercepts
15.3Paraxial Polarization Aberrations
15.3.1Paraxial Angle and Plane of Incidence
15.3.2Paraxial Diattenuation and Retardance
15.3.4Diattenuation Defocus and Retardance Defocus
15.3.5Diattenuation and Retardance across the Field of View
15.3.6Polarization Tilt and Piston
15.3.8Summation of Paraxial Polarization Aberrations over Surfaces
15.4Paraxial Polarization Analysis of a Seven-Element Lens System
15.5Higher-Order Polarization Aberrations
15.5.1Electric Field Aberrations
15.5.3Diattenuation and Retardance
15.6Polarization Aberration Measurements
15.8.2Setting Up the Optical System
15.8.4Reduced Thicknesses and Angles
16Image Formation with Polarization Aberration
16.2Discrete Fourier Transformation
16.3Jones Exit Pupil and Jones Pupil Function
16.4Amplitude Response Matrix (ARM)
16.5Mueller Point Spread Matrix (MPSM)
16.6The Scale of the ARM and MPSM
16.7Polarization Structure of Images
16.8Optical Transfer Matrix (OTM)
16.9Example—Polarized Pupil with Unpolarized Object
16.10Example—Solid Corner Cube Retroreflector
16.11Example—Critical Angle Corner Cube Retroreflector
16.12Discussion and Conclusion
17Parallel Transport and the Calculation of Retardance
17.1.1Purpose of the Proper Retardance Calculation
17.2Geometrical Transformations
17.2.1Rotation of Local Coordinates: Polarimeter Viewpoint
17.2.2Non-Polarizing Optical Systems
17.2.3Parallel Transport of Vectors
17.2.4Parallel Transport of Vectors with Reflection
17.2.5Parallel Transport Matrix, Q
17.3Canonical Local Coordinates
17.4Proper Retardance Calculations
17.4.1Definition of the Proper Retardance
17.5Separating Geometric Transformations from P
17.5.1The Proper Retardance Algorithm for P, Method 1
17.5.2The Proper Retardance Algorithm for P, Method 2
17.6.1Ideal Reflection at Normal Incidence
17.6.2An Aluminum-Coated Three-Fold Mirror System Example
18.2Definition of Skew Aberration
18.4Lens Example—U.S. Patent 2,896,506
18.5Skew Aberration in Paraxial Ray Trace
18.6Example of Paraxial Skew Aberration
18.7Skew Aberration’s Effect on PSF
18.8PSM for U.S. Patent 2,896,506
18.9Statistics—CODE V Patent Library
19.1Ray Tracing in Birefringent Materials
19.2Description of Electromagnetic Waves in Anisotropic Media
19.3Defining Birefringent Materials
19.4Eigenmodes of Birefringent Materials
19.5Reflections and Refractions at Birefringent Interface
19.6Data Structure for Ray Doubling
19.7Polarization Ray Tracing Matrices for Birefringent Interfaces
19.7.1Case I: Isotropic-to-Isotropic Intercept
19.7.2Case II: Isotropic-to-Birefringent Interface
19.7.3Case III: Birefringent-to-Isotropic Interface
19.7.4Case IV: Birefringent-to-Birefringent Interface
19.8Example: Ray Splitting through Three Biaxial Crystal Blocks
19.9Example: Reflections Inside a Biaxial Cube
20Beam Combination with Polarization Ray Tracing Matrices
20.3Co-Propagating Wavefront Combination
20.4Non-Co-Propagating Wavefront Combination
20.5Combining Irregular Ray Grids
20.5.1General Steps to Combine Misaligned Ray Data
20.5.2Inverse-Distance Weighted Interpolation
21Uniaxial Materials and Components
21.1Optical Design Issues in Uniaxial Materials
21.2Descriptions of Uniaxial Materials
21.3Eigenmodes of Uniaxial Materials
21.4Reflections and Refractions at a Uniaxial Interface
21.5Index Ellipsoid, Optical Indicatrix, and K- and S-Surfaces
21.6Aberrations of Crystal Waveplates
21.7Image Formation through an A-Plate
22.1Introduction to Crystal Polarizers
22.2Materials for Crystal Polarizers
22.3.2Multiple Potential Ray Paths
22.3.3Multiple Polarized Wavefronts
22.3.4Polarized Wavefronts Exiting from the Polarizer
22.4Aberrations of the Glan–Taylor Polarizer
22.5Pairs of Glan–Taylor Polarizers
23Diffractive Optical Elements
23.3.1Reflection Diffractive Gratings
23.3.4Diffractive Subwavelength Antireflection Coatings
23.4Summary of the RCWA Algorithm
24.3.1Construction of Liquid Crystal Cells
24.3.3Liquid Crystal Display: High Contrast Ratio Intensity Modulation
24.4Configurations of Liquid Crystal Cells
24.4.290° Twisted Nematic Cell
24.4.3Super Twisted Nematic Cell
24.4.4Vertically Aligned Nematic Cell
24.4.6Liquid Crystal on Silicon Cells
24.5.1Extended Jones Matrix Model
24.5.2Single Pass with Polarization Ray Tracing Matrices
24.5.3Multilayer Interference Models
24.5.4Calculation for Liquid Crystal Cell ZLI-1646
24.6Issues in the Construction of LC Cells
24.6.4Oscillating Square Wave Voltage
24.7Limitations on LC Cell Performance
24.7.2Spectral Variation of Exiting Polarization State
24.7.3Variation of Retardance with Angle of Incidence
24.7.4Compensating LC Cells’ Polarization Aberrations with Biaxial Films
24.8Testing Liquid Crystal Cells
24.8.1Twisted Nematic Cell Example
24.8.6Misalignment between Analyzer and Exiting Polarization State
25Stress-Induced Birefringence
25.1Introduction to Stress Birefringence
25.2Stress Birefringence in Optical Systems
25.3Theory of Stress-Induced Birefringence
25.4Ray Tracing in Stress Birefringent Components
25.5Ray Tracing through Stress Birefringence Components with Spatially Varying Stress
25.5.2Refraction and Reflections
25.5.4Polarization Ray Tracing Matrix for Spatially Varying Biaxial Stress
25.5.5Examples of Spatially Varying Stress Function
25.6Effects of Stress Birefringence on Optical System Performance
25.6.1Observing Stress Birefringence Using Polariscope
25.6.2Simulations of Injection-Molded Lens
25.6.3Simulation of a Plastic DVD Lens
26Multi-Order Retarders and the Mystery of Discontinuities
26.2Mystery of Retardance Discontinuity
26.3Retardance Unwrapping for Homogeneous Retarder Systems Using a Simple Dispersion Model
26.3.2Retardance of the Homogeneous Retarder System
26.3.3Homogeneous Retarder’s Trajectory and Retardance Unwrapping in Retarder Space
26.4.1Compound Retarder Jones Matrix Decomposition
26.4.2Compound Retarder’s Trajectory in Retarder Space
26.4.3Multiple Modes Exit the Compound Retarder System
26.4.4Compound Retarder Example at 45°
27.2Polarization Ray Tracing Complications
27.2.1Optical System Description Complications
27.2.2Elliptical Polarization Properties of Ray Paths
27.2.3Optical Path Length and Phase
27.2.4Definition of Retardance
27.2.5Retardance and Skew Aberration
27.2.7Birefringent Ray Tracing Complications
27.3Polarization Ray Tracing Concepts and Methods
27.3.1Jones Matrices and Jones Pupil
27.3.2P Matrix and Local Coordinates
27.3.3Generalization of PSF and OTF
27.3.4Ray Doubling, Ray Trees, and Data Structures
27.3.6Alternative Simulation Methods
27.4Polarization Aberration Mitigation
27.4.1Analyzing Polarization Ray Tracing Output
27.5Comparison of Polarization Ray Tracing and Polarization Aberrations
27.5.1Aluminum Coating and Polarization Aberration Expression
27.5.2Polarization Ray Trace and the Jones Pupil
27.5.3Aberration Expression for the Jones Pupil
27.5.4Diattenuation and Retardance Contributions
27.5.5Design Rules Based on Polarization Aberrations
27.5.5.1Diattenuation at the Center of the Pupil
27.5.5.2Retardance at the Center of the Pupil
27.5.5.3Linear Variation of Diattenuation
27.5.5.4Linear Variation of Retardance, the PSF Shear between the XX- and YY-Components
27.5.5.5The Polarization-Dependent Astigmatism
27.5.5.6The Fraction of Light in the Ghost PSF in XY- and YX-Components
27.5.6Amplitude Response Matrix
27.5.7Mueller Matrix Point Spread Matrices