Applicability of the position-dependent diffusion approach to localized transport through disordered waveguides

The diffusive description of wave transport in random media has a long history [1]. This macroscopic approach describes the ensemble-averaged intensity of the wave on scales longer than the transport mean free path . As such, the diffusive description has a practical advantage compared to the direct solution of the wave equation for each statistical realization of disorder and subsequent averaging over the ensemble of solutions. The diffusion coefficient can become renormalized [2] due to the wave localization phenomenon [3]. In three dimensions, for sufficiently strong disorder, diffusion vanishes for an infinitely large system [4]. In practice, however, one deals with transport through finite systems. Both the self-consistent theory (SCT) of localization [2,5] and supersymmetric (SUSY) theory [6] have predicted [7,8] that the diffusive-like description can also be applied to the finite systems that exhibit localized transport, in particular, to low-dimensional systems. In such a description, the diffusion coefficient becomes dependent on position, system size [9–11], and geometry [12]. In quasi-one-dimensional (quasi-1D) or 1D lossless media both SCT and SUSY lead to the following equation for the ensemble-averaged intensity 〈I (z,z′)〉 in the presence of a point source J0 at z′: − ∂ ∂z [ D(z) ∂