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.