Optical components made of anisotropic materials, such as crystal polarizers and crystal waveplates, are widely used in many complex optical system, such as display systems, microlithography, biomedical imaging and many other optical systems, and induce more complex aberrations than optical components made of isotropic materials. The goal of this dissertation is to accurately simulate the performance of optical systems with anisotropic materials using polarization ray trace. This work extends the polarization ray tracing calculus to incorporate ray tracing through anisotropic materials, including uniaxial, biaxial and optically active materials. The 3D polarization ray tracing calculus is an invaluable tool for analyzing polarization properties of an optical system. The 3×3 polarization ray tracing P matrix developed for anisotropic ray trace assists tracking the 3D polarization transformations along a ray path with series of surfaces in an optical system. To better represent the anisotropic light-matter interactions, the definition of the P matrix is generalized to incorporate not only the polarization change at a refraction/reflection interface, but also the induced optical phase accumulation as light propagates through the anisotropic medium. This enables realistic modeling of crystalline polarization elements, such as crystal waveplates and crystal polarizers. The wavefront and polarization aberrations of these anisotropic components are more complex than those of isotropic optical components and can be evaluated from the resultant P matrix for each eigen-wavefront as well as for the overall image. One incident ray refracting or reflecting into an anisotropic medium produces two eigenpolarizations or eigenmodes propagating in different directions. The associated ray parameters of these modes necessary for the anisotropic ray trace are described in Chapter 2. The algorithms to calculate the P matrix from these ray parameters are described in Chapter 3 for anisotropic ray tracing. This P matrix has the following characteristics: (1) Multiple P matrices are calculated to describe the polarization of the multiple eigenmodes at an anisotropic intercept. (2) Each P matrix maps the orthogonal incident basis vectors (Ê_m, Ê_n, Ŝ) before the optical interface into three orthogonal exiting vectors (a_m Ê'_m, a_n Ê'_n, Ŝ') after the interface, where a_m and a_n are the complex amplitude coefficients induced at the intercept. The ray tracing algorithms described in this dissertation handle three types of uncoated anisotropic interfaces isotropic/anisotropic, anisotropic/isotropic and anisotropic/anisotropic interfaces. (3) The cumulative P matrix associated with multiple surface interactions is calculated by multiplying individual P matrices in the order along the ray path. Many optical components utilize anisotropic materials to induce desired retardance. This important mechanism is modeled as the optical phase associated with propagation. (4) The optical path length OPL of an eigenpolarization along an anisotropic ray path is incorporated into the calculation of each P matrix. Chapter 4 presents the data reduction of the P matrix of a crystal waveplate. The diattenuation is embedded in the singular values of P. The retardance is divided into two parts: (A) The physical retardance induced by OPLs and surface interactions, and (B) the geometrical transformation induced by geometry of a ray path, which is calculated by the geometrical transform Q matrix. The Q matrix of an anisotropic intercept is derived from the generalization of s- and p-bases at the anisotropic intercept; the p basis is not confined to the plane of incidence due to the anisotropic refraction or reflection. Chapter 5 shows how the multiple P matrices associated with the eigenmodes resulting from propagation through multiple anisotropic surfaces can be combined into one P matrix when the multiple modes interfere in their overlapping regions. The resultant P matrix contains diattenuation induced at each surface interaction as well as the retardance due to ray propagation and total internal reflections. The polarization aberrations of crystal waveplates and crystal polarizers are studied in Chapter 6 and Chapter 7. A wavefront simulated by a grid of rays is traced through the anisotropic system and the resultant grid of rays is analyzed. The analysis is complicated by the ray doubling effects and the partially overlapping eigen-wavefronts propagating in various directions. The wavefront and polarization aberrations of each eigenmode can be evaluated from the electric field distributions. The overall polarization at the plane of interest or the image quality at the image plane are affected by each of these eigen-wavefronts. Isotropic materials become anisotropic due to stress, strain, or applied electric or magnetic fields. In Chapter 8, the P matrix for anisotropic materials is extended to ray tracing in stress birefringent materials which are treated as spatially varying anisotropic materials. Such simulations can predict the spatial retardance variation throughout the stressed optical component and its effects on the point spread function and modulation transfer function for different incident polarizations. The anisotropic extension of the P matrix also applies to other anisotropic optical components, such as anisotropic diffractive optical elements and anisotropic thin films. It systematically keeps track of polarization transformation in 3D global Cartesian coordinates of a ray propagating through series of anisotropic and isotropic optical components with arbitrary orientations. The polarization ray tracing calculus with this generalized P matrix provides a powerful tool for optical ray trace and allows comprehensive analysis of complex optical system.

A three-by-three polarization ray tracing matrix method is developed to calculate the polarization transformations associated with ray paths through optical systems. The relationship between the three-by-three polarization ray tracing matrix P method and the Jones calculus is shown in Chapter 2. The diattenuation, polarization dependent transmittance, is calculated via a singular value decomposition of the P matrix and presented in Chapter 3. In Chapter 4 the concept of retardance is critically analyzed for ray paths through optical systems. Algorithms are presented to separate the effects of retardance from geometric transformations. The parallel transport of vectors is associated with non-polarizing propagation through an optical system. A parallel transport matrix Q establishes a proper relationship between sets of local coordinates along the ray path, a sequence of ray segments. The proper retardance is calculated by removing this geometric transformation from the three-by-three polarization ray trace matrix. Polarization aberration is wavelength and spatial dependent polarization change that occurs as wavefrontspropagate through an optical system. Diattenuation and retardance of interfaces and anisotropic elements are common sources of polarizationaberrations. Two representations of polarization aberrationusing the Jones pupil and a polarization ray tracing matrix pupil, are presentedin Chapter 5. In Chapter 6 a new class of aberration, skew aberration is defined, as a component of polarization aberration. Skew aberration is an intrinsic rotation of polarization states due to the geometric transformation of local coordinates; skew aberration occurs independent of coatings and interface polarization. Skew aberration in a radially symmetric system primarily has the form of a tilt plus circular retardance coma aberration. Skew aberration causes an undesired polarization distribution in the exit pupil. A principal retardance is often defined within (-π, + π] range. In Chapter 7 an algorithm which calculates the principal retardance, horizontal retardance component, 45° retardance component, and circular retardance component for given retarder Jones matrices is presented. A concept of retarder space is introduced to understand apparent discontinuities in phase unwrapped retardance. Dispersion properties of retarders for polychromatic light is used to phase unwrap the principal retardance. Homogeneous and inhomogeneous compound retarder systems are analyzed and examples of multi-order retardance are calculated for thick birefringent plates. Mathematical description of the polarization properties of light and incoherent addition of light is presented in Chapter 8, using a coherence matrix. A three-by-three-by-three-by-three polarization ray tracing tensor method is defined in order to ray trace incoherent light through optical systems with depolarizing surfaces. The polarization ray tracing tensor relates the incident light’s three-by-three coherence matrix to the exiting light’s three-by-three coherence matrix. This tensor method is applicable to illumination systems and polarized stray light calculations where rays at an imaging surface pixel have optical path lengths which vary over many wavelengths. In Chapter 9 3D Stokes parameters are defined by expanding the coherence matrix with Gell-Mann matrices as a basis. The definition of nine-by-nine 3D Mueller matrix is presented. The 3D Mueller matrix relates the incident 3D Stokes parameters to the exiting 3D Stokes parameters. Both the polarization ray tracing tensor and 3D Mueller matrix are defined in global coordinates. In Chapter 10 a summary of my work and future work are presented followed by a conclusion.