Generalized Jones-Mueller calculus for optical anisotropic media

3-D matrix calculus plays a critical role in describing 3-D polarized transformations. However, existing two 3-D calculi, generalized Jones and Mueller matrix calculus (GJM and GMM), remain flawed and incomplete. First, the global connection between two matrices has not been theoretically explored and established, which results in their independence and inability to form a unified framework (similar to 2-D Jones-Mueller framework). Second, existing GJM calculus models the 3-D interaction with anisotropic media as a paraxial-like system constrained by specific <i>S</i><i>U</i>(3) rotations, neglecting the influence of vectorial light path and defining the differential GJM (dGJM) as inherent and fixed. In this article, we first introduce a Lorentz-like algebra to establish a double-covering homomorphism between <i>S</i><i>L</i>(3,<i>C</i>) and the Lorentz-like group <b>LLG</b>, which enables global and bi-directional mapping between GJM and GMM. Then, we propose a pure-matrix approach covering the vectorial light path to describe the 3-D anisotropic interaction, which enables polarization modeling along arbitrary light paths. The proposed theories further refine the framework of 3-D polarization optics and can be the foundation of a variety of non-paraxial polarization applications.