The dye-laser equation with saturation and its local linearization

Abstract

The single-mode dye laser has been a generic example for the study of stochastic systems driven by colored noise, in part because of its experimental accessibility. Furthermore, the evolution of the field amplitude can be described by a stochastic differential equation that can be easily simulated and therefore used to test approximate theories. In a previous paper we tested one of these theories, the “best Fokker-Planck equation” (BFPE) method against direct simulations of the dye-laser equation with saturation. The BFPE can be solved analytically when the laser gain parameter is much greater than the loss parameter, i.e. far from the resonance condition. The theorical steady-state distribution was found to reproduce the simulated results extremely accurately for wide ranges of parameter values. In this paper we consider the situation near resonance, a parameter regime that is closer to existing experimental results. The BFPE method does not yield analytic results in this regime, so instead we use a geometrical treatment that is particularly useful in the absence of additive white noise (spontaneous emission). In this case the resulting laser description is equivalent to that obtained by the “local linearization” method. Once again, the approximate theoretical steady-state distribution is found to agree extremely well with the results obtained from the direct simulation of the laser equation. With these twp approximations it has therefore been possible to describe the steady-state operation of a single-mode dye laser analytically and extremely accurately over wide ranges of parameter values.