  We study Maxwell's equations in random media with small fluctuations of the
electric permittivity. We consider a setup where the waves propagate toward a
preferred direction, called range. We decompose the electromagnetic wave field
in transverse electric and transverse magnetic plane waves, called modes, 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 Markov 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 modes decay. We also quantify the energy
exchange between the modes, and consequently 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 literature in radiative transport and paraxial wave
propagation.

Analysis of PDEs