Random amplifying medium: statistics of super-reflectance and time-delay fluctuations

A light wave incident on a random amplifying medium (RAM) may be reflected with a reflection coefficient (r) exceeding unity. This super-reflectance and the associated time-delay will be non-self-averaging, and show sample-specific fluctuations characteristic of coherent transport through random media. The statistics of these random fluctuations involve a synergetic inter-play of disorder (localization) and coherent amplification (non-resonant distributed feedback), which is disallowed for its electronic (fermionic) counterpart. In this work, we review and report on some of our recent analytical results for these statistics for the case of a long one-dimensional (1D) RAM that physically corresponds to a single-channel optical fibre having gain and disorder, and operating in the reflection mode. Our main result is that the coherent amplification makes the probability density of the super-reflection peak for some r>1 with a long tail, while the otherwise infinite mean delay-time is rendered finite. Interestingly, our treatment of amplification provides a natural counter to monitor the time-delay for the corresponding passive (e.g., electronic) problem as well. Our approach, based on the invariant imbedding technique, is valid below the threshold for lasing, i.e., for the localization length < the gain-length.