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

An analytical and numerical study is presented of transmission of radiation through a multi-mode waveguide containing a randommedium 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 ∝ exp(−L/ξ) as the waveguide length L → ∞, 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 ≫ 1 and |σ| ≫ 1/N2l. An approximate interpolation formula for all σ agrees reasonably well with numerical simulations. PACS numbers: 78.45.+h, 42.25.Bs, 72.15.Rn, 78.20.Ci Typeset using REVTEX