Statistical properties of the reflectance and transmittance of an amplifying random medium

Statistical properties of the transmittance $(T)$ and reflectance $(R)$ of an amplifying layer with one-dimensional disorder are investigated analytically within the random phase approximation. Whereas the transmittance at typical realizations decreases exponentially with the layer thickness L just as it does in absorbing media, the average $〈T〉$ and $〈R〉$ are shown to be infinite even for finite L due to the contribution of low-probability resonant realizations corresponding to the non-Gaussian tail of the distribution of lnT. This tail differs drastically from that in the case of absorption. The physical meaning of typical and resonant realizations is discussed.