Multi-scaled Diffusion-approximation. Applications to Wave Propagation in Random Media

In this paper a multi-scaled diiusion-approximation theorem is proved so as to unify various applications in wave propagation in random media: transmission of optical modes through random planar waveguides; time delay in scattering for the linear wave equation; decay of the transmission coeecient for large lengths with xed output and phase diierence in weakly nonlinear random media. 1. Introduction Wave propagation in random media has become an extensively studied subject. In one-dimensional linear media with random inhomogeneities, lo-calization occurs, which means in particular that the transmitted intensity decays exponentially as a function of the size of the medium. This problem has been analyzed in detail by Carmona et al. (1990). In our paper, we consider wave reeection and transmission from a one-dimensional random slab. Several quantities characterize the reeected wave; here we focus on the reeection coeecient, the phase diierence and the time delay. The analysis puts into evidence the usual scales (see Knapp et al. (1989) and Papanicolaou (1988)): length of the slab, wavelength, amplitude and correlation radius of the random perturbations. We study the asymptotic behavior of the scattered wave in the framework introduced by Papanicolaou based on the separation of these scales. The uctuations of the random coeecients are on a small scale so that we actually deal with diiusion-approximation problems. However we consider here situations where many scaled quantities play a role, so that we need to prove, then to use general multi-scaled diiusion-approximation theorems. Indeed, the study of planar waveguides depends on the above quantities, but also on the thickness of the core. The time delay depends not only on the high carrier frequency of the wave packet, but also on its bandwidth. Finally the amplitude of the nonlinear term is an essential requirement for the study of the behavior of the transmittivity for the nonlinear wave equation. The action of nonlinearity seems to be opposite to that of disorder. Nonlinearity may change the dependence of the transmission coeecient on the length, that still tends to zero as the size of the medium increases, but following a