Disordered Wave - guides with Absorption : Transmittances Distributions at the Diffusive - Localized Crossover

Propagation and localization of waves of various kinds in quasi-one dimensional disordered media have been studied intensively for several decades. Examples of particular interest are electron transport in disordered solid and electromagnetic wave propagation in random dielectric media. In contrast to electronic systems, where the total flux is conserved, in optical systems the transmittance may be affected as a result of absorption. The conductance (transmittance) distribution in electronic system P(T) is known to evolve from a Gaussian distribution P(T) (deep in the diffusive regime) to a lognormal distribution (deep in the localization regime) [1]. The analysis of the conductance distribution for disordered wires in the diffusive-localization crossover shows a non-obvious transition of the distribution between both regimes [2]. The properties of the transmittance of a multi-channel random wave-guide with absorption have been computed within the Random Matrix framework [3,4]. In the limit of very strong absorption compared with the disorder, statistical properties have been calculated in the diffusive limit (the transmittance distribution is Gaussian) and the localized limit (lognormal distribution) [4]. The understanding of the transmittance properties in the crossover region between these limits and in systems where the disorder strength can be comparable with the absorption may be important in determining the onset of localization in photonic systems [5-8]. In this work disordered wave-guides have been modeled, both with and without absorption. The transport properties (distributions) of the wave-guides have been calculated as their length it is varied, covering the ballistic, diffusive and localization regimens and the crossovers between them. Absorption considered is strong compared with the disorder used but it is far from the limit considered in ref. [4]. In spite of this consideration and the few channel considered in our calculations, our results show a similar behavior to experimental findings of ref. [8]. Distributions are obtained from numerical calculations from the well-known "tight-binding" model. To illustrate our finding, the results of two sets of wave-guides (with and without disorder) are discussed. Both sets have a width of N=7 and disorder strength W=1. The guides with absorption have an absorption length of L a =39. The localization lengths are ξ=108 for the case of wave-guides with absorption (L a <ξ, but far for the limit considered in ref [4] where L a /ξ is used as a small parameter) and ξ=161 if there is not absorption. Note that a non-zero imaginary component of the on-site energy produces, as …