An analytical and numerical study of transmission of radiation through
a multi-mode waveguide containing a random medium with a complex diel
ectric constant epsilon = epsilon' + i epsilon '' is presented. Depend
ing 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 th
e mean free path (, the absorption or amplification length \sigma\(-1)
, and the number of modes in the waveguide N. The method used is an ex
tension 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 reaso
nably well with numerical simulations.