Numerical simulation of propagation of light through random disordered media to model branched flow phenomena

Abstract

Wave branching occurs during propagation in a gently disordered medium. It appears in many different physical situations involving diverse length scales: from electron waves refracted in semiconductors to ocean waves deflected by surface eddies. Instead of producing completely random speckle patterns, the slowly varying disordered potential gives rise to focused filaments that divide to form a branched pattern. This very general phenomenon of branched flow is tightly connected to the formation of random caustics. Very recently, this phenomenon has been observed in light by studying the propagation of laser beams in soap films, opening an exciting field of research where the entire machinery of structured light can be brought to bear. The fundamental equation which describes the propagation of a beam through a varying refractive index medium is the Helmholtz equation. Here, we develop computational routines to solve this second order differential equation in order to simulate and characterize the branched flow of light propagating through two-dimensional inhomogeneous media. We present the effects of varying the correlation length of the scattering medium, the influence of shaping the input beam, and the statistical features depicting the branching of light by analysing the variance of the wave intensity as a function of the propagation distance from the source. The numerical method chosen to solve the Helmholtz equation is the Crank-Nicolson method, which uses finite differences analysis and this scheme is unconditionally stable if the refractive index is independent of x and z.