Statistical Properties of the Reflectance and Transmittance of an Amplifying Random Media

TL;DR

This paper analytically explores the statistical behavior of transmittance and reflectance in amplifying disordered media, revealing unique properties such as infinite average transmittance and reflectance due to resonant phenomena.

Contribution

It provides a novel analytical investigation into the statistical properties of amplifying disordered layers, highlighting differences from absorbing media and discussing physical implications.

Findings

Average transmittance and reflectance are infinite for finite layer thickness.

Typical transmittance decreases exponentially with layer thickness.

Resonant realizations dominate the statistical tail of the distribution.

Abstract

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