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 $\epsilon ={\epsilon }^{\prime }+i{\epsilon }^{\prime \prime }$. Depending on the sign of ${\epsilon }^{\prime \prime }$, the medium is absorbing or amplifying. The transmitted intensity decays exponentially $\propto \exp (-L/\xi )$ as the waveguide length $L\to \infty$, regardless of the sign of ${\epsilon }^{\prime \prime }$. The localization length $\xi$ is computed as a function of the mean free path $l$, the absorption or amplification length $|\sigma {|}^{-1}$, 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 $N\gg 1$ and $|\sigma |\gg 1/N^{2}l$. An approximate interpolation formula for all $\sigma$ agrees reasonably well with numerical simulations.

13 pages, RevTeX, 1 postscript figure