Coherently amplifying random medium: Statistics of super-reflection

Abstract

Light-wave propagation in a randomly scattering but uniformly and coherently amplifying optical medium is analysed for the statistics of the coefficient of reflectionr(l) that may now exceed unity, hence super-reflection. Uniform coherent amplification is introduced phenomenologically through a constant negative imaginary part added to the otherwise real dielectric constant of the medium, assumed random. The probability densityp(r,l) for the reflection coefficient, calculated in a random phase approximation, tends to an asymptotic formp(r, ∞) having a long tail with divergent mean 〈r〉 in the limit of weak disorder but with the sample lengthl(measured in units of the localization length in the absence of gain) ⪢1. This super-reflection is attributed to a synergetic effect of localization and coherent amplification, as distinct from the classically diffusive path-length prolongation. Our treatment is based on the invariant-imbedding method for the one-channel (single-mode) case, and is in the spirit of the scattering approach to wave transport as pioneered by Landauer. Generalization to the multichannel case is pointed out. Relevance to random lasers, and some recent results for a sub-meanfree path sample are also briefly discussed.