Paraxial Coupling of Electromagnetic Waves in Random Media

We consider the propagation of temporally pulsed electromagnetic waves in a threedimensional random medium. The main objective is to derive effective white-noise paraxial equations from Maxwell’s equations. We address the scaling regime in which (1) the carrier wavelength is small compared to the incident beam radius, which itself is small compared to the propagation distance; (2) the correlation length of the fluctuations of the random medium is of the same order as the beam radius, and the typical amplitude of the fluctuations is small. In this regime we prove that the wave field is characterized by a white-noise paraxial wave equation that has the form of a Schrödinger-type equation driven by a Brownian field. We identify the covariance function of the Brownian field in terms of the two-point statistics of the fluctuations of the dielectric permittivity and the magnetic permeability of the medium. We also study the case in which a strong interface is embedded in the random medium and study the reflected wave, which again is characterized by a Schrödinger-type equation. We discuss applications to enhanced backscattering, time reversal, and imaging.