2015/09/23 by Liliana Borcea, Borcea, Liliana, Josselin Garnier +1 · 1 citation
Engineering · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Microwave Imaging and Scattering Analysis #Numerical methods in inverse problems #Random lasers and scattering media #math.AP
paper · pdf · doi:10.48550/arxiv.1509.06960
33 pages
openalex publication_date 2015/09/23 · arxiv created 2016/01/02 · arxiv updated 2016/01/05 · openalex created_date 2022/11/16 · openalex updated_date 2026/07/28
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.