The equation for the transport of electromagnetic polarization

The traditional mathematical instrument to track the evolution of the polarization traversing an optical medium, is the Poincaré sphere. In this paper, we follow an alternative strategy which does not rely on spherical trigonometry or geodesic trajectories on this sphere. We show how a rigorous first order differential equation can be implemented as wave equation for polarized light. Its derivation is inspired by the Dirac treatment of the relativistic quantum wave equation of the electron. As generic example, we will study the polarization transport along the optical axis in a gyrotropic medium. The Jones differential N-matrix is thereby identified as major constituent term in the transport equation. This leads to define the Gouy phase for non-Gauss-Laguerre modes and to the relation with the geometric phase. The N-matrix is also shown to be related to the Berry connection. Finally, the methodology shows that propagation of the angular spectrum is related to the Jones M-matrix and its eigenpolarizations.