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.