Localization in a Disordered Multi-Mode Waveguide with Absorption or Amplification

Abstract

An analytical and numerical study is presented of transmission of radiation through a multi-mode waveguide containing a random medium with a complex dielectric constant $ε= ε'+iε''$. Depending on the sign of $ε''$, the medium is absorbing or amplifying. The transmitted intensity decays exponentially $\propto\exp(-L/ξ)$ as the waveguide length $L\to\infty$, regardless of the sign of $ε''$. The localization length $ξ$ is computed as a function of the mean free path $l$, the absorption or amplification length $|σ|^{-1}$, and the number of modes in the waveguide $N$. The method used is an extension of the Fokker-Planck approach of Dorokhov, Mello, Pereyra, and Kumar to non-unitary scattering matrices. Asymptotically exact results are obtained for $N\gg1$ and $|σ|\gg1/N^2l$. An approximate interpolation formula for all $σ$ agrees reasonably well with numerical simulations.