Electromagnetic polarization matrix and its physical interpretation

Abstract: Although electric and magnetic fields are intrinsically coupled in random stationary light, a complete and physically consistent second-order description of polarization requires a joint electromagnetic treatment. The 6x6 electromagnetic polarization matrix introduced here generalizes the conventional electric 3x3 matrix by incorporating both electric and magnetic contributions together with their mutual correlations. It consists of diagonal 3x3 blocks, representing the electric and magnetic polarization matrices, and off-diagonal 3x3 blocks that encode the full structure of electric-magnetic cross-correlations. The information contained in this matrix can be interpreted through physically meaningful quantities such as active and reactive energy fluxes, in-phase and quadrature alignment matrices, and global indices describing electric-magnetic coupling. The formalism is applied to a field composed of two orthogonally propagating plane waves sharing a common linear electric polarization (a simple yet physically realizable configuration) that demonstrates the need for a general combined electric-magnetic representation even in free space. This approach provides a comprehensive and unified framework for characterizing electromagnetic polarization beyond the electric-field description alone, bridging classical statistical optics with quantum-like density-matrix interpretations.