Polarization effects for electromagnetic wave propagation in random media

We study Maxwell's equations in random media with small fluctuations of the electric permittivity. We consider a setup where the waves propagate along a preferred direction, called range. We decompose the electromagnetic wave field in transverse electric and transverse magnetic plane waves with random amplitudes that model cumulative scattering effects in the medium. Their evolution in range is described by a coupled system of stochastic differential equations driven by the random fluctuations of the electric permittivity. We analyze the solution of this system with the diffusion limit theorem and obtain a detailed asymptotic characterization of the electromagnetic wave field in the long range limit. In particular, we quantify the loss of coherence of the waves due to scattering, by calculating the range scales (scattering mean free paths) on which the mean amplitudes of the transverse electric and magnetic plane waves decay. We also quantify the loss of polarization induced by scattering, by analyzing the Wigner transform (energy density) of the electromagnetic wave field. This analysis involves the derivation of transport equations with polarization. We study in detail these equations and connect the results with the existing radiative transport literature.