ABSTRACT
Polarization ray tracing, which calculates the Jones matrix associated with an arbitrary ray path through an optical system has been used over twenty years. The Jones matrix deals with Jones vectors that specifically refer to a monochromatic plane wave, describing the electric field and the polarization ellipse with respect to a local coordinate system in the transverse plane. However, to use Jones vectors and matrices in optical design for highly curved beams, frequent coordinate conversion is required to define the direction of the Jones vector’s x- and y- components in space, and this leads to complications due to the intrinsic singularities of such coordinate conversions.
This chapter develops a 3×3 polarization ray tracing matrix and associated algorithms which systematizes polarization ray tracing with a three-dimensional polarization ray tracing matrix, the P matrix, a generalization of the Jones matrix into three-dimension. A major advantage of the P matrix is its definition in global coordinates; it solves deep problems with Jones matrices and local coordinates due to singularities and non-uniqueness. As a result, anyone who ray traces an optical system with P will get the same matrix, unlike a Jones or Mueller matrix calculation where the answer depends on the sequence of local coordinates selected. Algorithms are provided to calculate diattenuation and retardance using P. The polarization ray tracing matrix describes the polarization-dependent transmission and phase contributions to the optical path length due to coated and uncoated interfaces, diffraction gratings, holographic elements, and other polarization effects. An interferometer with a polarizing beam splitter is used as an example for the polarization ray tracing calculus demonstrating the utility for calculations where beams repeatedly change directions and coherently interfere.
