Geometric descriptions for the polarization for nonparaxial optical fields: a tutorial

This tutorial provides an overview of the local description of polarization for nonparaxial light, for which all Cartesian components of the electric field are significant. The polarization of light at each point is characterized by a $3$ component vector in the case of full polarization or by a $3\times3$ polarization matrix for partial polarization. Standard concepts for paraxial polarization like the degree of polarization, the Stokes parameters and the Poincar\'e sphere then have generalizations for nonparaxial light that are either not unique or not trivial. This work aims to clarify some of these discrepancies, present some new concepts, and provide a framework that highlights the similarities and differences with the description for the paraxial regimes. Particular emphasis is placed on geometric interpretations.