Amplification in one-dimensional random active medium near the lasing threshold

Abstract

We studied the transmittance of a random amplifying medium near the lasing threshold by using the convergence criterion proposed by Nam and Zhang [Phys. Rev. B 66 73101, 2002] that allows separating the physical solutions of the time-independent Maxwell equations from the unphysical ones and describing critical size L _c of a random system in statistical terms. We found that the dependence of the critical gain ∈ ^″ _c (at which the lasing threshold occurs) as a function of number of layers is configuration-dependent and thus the lasing condition for random media is described by means of the probability of finding of physical solutions and evaluated by averaging over the ensemble of random configurations. By employing this approach we inspect the validity of the two-parameter scaling model by Zhang [Phys. Rev. B 52 7960, 1995], according to which the behavior of the random system with gain is described by relation 1/ξ = 1/ξ ₀ + 1/l _g, where ξ and ξ ₀ are localization length with and without gain, respectively, and l _g = 2/ω∈ ^″, is gain length, where ∈ ^″ is imaginary part of the dielectric constant that represents gain. We show that the range of validity of this relation depends on the ratio of both lengths and also affects the slope of the ln Λ_c(q) (where Λ_c ≡ L _c/ξ ₀ is normalized critical size and q ⁻¹ ≡ l _g/ξ ₀ is dimensionless gain length). We extend the study of the relation for the critical size L _c by inspecting the dependence of the slope of the ln Λ_c(q) on the strength of the randomness. We interpret its behavior in terms of the statistical properties of the localized states. Namely, by studying of the variance of the Lyapunov exponent λ (the inverse of the localization length ξ ₀) we have found that the slope of the ln Λ_c(q)) reflects the transition between two different regimes of localization with Anderson and Lifshits-like behavior that is known to be indicated by peak in var(λ). We discuss the generalization of two-parameter scaling model by implementing revisited single parameter scaling (SPS) theory by Deych et al. [Phys. Rev. Lett. 84 2678, 2000] which allows describing non-SPS regime in terms of a new scale l _s.