Light amplification and absorption in a random medium

Using an invariant imbedding approach, we derive exact expressions for the distributions of reflection and transmission amplitude coefficients and phases of light backscattered coherently from an amplifying or absorbing random medium in one dimension. The length L of the system is assumed to be small compared to the wavelength, which is itself smaller than the localization length Lc of the light. These distributions and their statistical moments are analyzed in various length ranges defined by the characteristic absorption and amplification lengths. The results for the statistics of the reflection and transmission coefficients are contrasted with analogous results obtained previously for the strong localization domain, LLc, by assuming the random phases to be uniformly distributed. The case where L is larger than the wavelength but small compared to Lc, where the disorder is weak, is also briefly analyzed.