Paraxial polarized waves in inhomogeneous media

A paraxial equation for electromagnetic wave propagation in a random medium is extended to include the depolarization effects in the narrow-angle, forward-scattering setting. A system of two coupled parabolic equations describes propagation of the polarized wave through random medium. In the Cartesian coordinate formulation the coupling term is related to the second mixed derivative of the refractive index. Closed-form parabolic equation for propagation of the coherence tensor is derived under a Markov random process propagation model. The scattering term in this equation includes a rank-four tensor that contains derivatives of the correlation function of the refractive index up to the fourth order. This equation can be also formulated as vector equations for generalized Stokes or lexicographic vectors. In contrast to the scalar case, these equations do not have an analytical solution. For a general partially coherent and partially polarized beam wave, this equation can be reduced to a system of ordinary differential equations allowing a simple numeric solution. For a special case of statistically homogeneous waves an analytical solution exists. In the Stokes vector formulation this solution is described by a range-dependent Mueller matrix. For isotropic random medium this Mueller matrix is diagonal and describes a pure non-uniform depolarizer. Statistics of the random medium is wrapped in a single parameter - depolarization length which is proportional to the fourth derivative of the covariance function at zero. For propagation through atmospheric turbulence estimates based on the perturbation solution support the common knowledge that the depolarization at the optical frequencies is negligible.