Localization in a disordered multi-mode waveguide with absorption or amplification

An analytical and numerical study of transmission of radiation through a multi-mode waveguide containing a random medium with a complex dielectric constant epsilon = epsilon' + i epsilon '' is presented. Depending on the sign of epsilon '', the medium is absorbing or amplifying. The transmitted intensity decays exponentially proportional to exp(-L/xi) as the waveguide length L --> infinity, regardless of the sign of epsilon ''. The localization length xi is computed as a function of the mean free path (, 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 nonunitary scattering matrices. Asymptotically exact results are obtained for N much greater than 1 and \sigma\much greater than 1/N(2)l. An approximate interpolation formula for all sigma agrees reasonably well with numerical simulations.