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

TL;DR

This paper analyzes how radiation transmits through a disordered multi-mode waveguide with absorption or amplification, deriving the localization length and validating results with numerical simulations.

Contribution

It extends the Fokker-Planck approach to non-unitary scattering matrices, providing analytical and numerical insights into localization in complex media.

Findings

Transmission decays exponentially with waveguide length

Localization length depends on mean free path, absorption/amplification length, and number of modes

Approximate formula matches numerical simulations

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 $\epsilon= \epsilon'+i\epsilon''$. Depending on the sign of $\epsilon''$, 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 $\epsilon''$. 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 $N\gg1$ and $|\sigma|\gg1/N^2l$. An approximate interpolation formula for all $\sigma$ agrees reasonably well with numerical simulations.