Effects of saturation in the transient process of a dye laser. III. The case of colored noise with large and small correlation time

Abstract

We have extended the unified-colored-noise approximation proposed by Jung and H\"anggi [Phys. Rev. A 35, 4464 (1987)] to the multidimensional case. The method has been used to obtain the Fokker-Planck equation of two dye-laser models with saturation terms: the loss-noise model (model A) and the gain-noise model (model B). The approximate formulas of the mean, variance, and skewness of first-passage-time distribution (FPTD) for the models are derived. The Fokker-Planck equation we obtained does not suffer from the problem of negative diffusion functions. They are valid for the whole range of the variable x. Based on the feature of the Fokker-Planck equation obtained in this paper, our results of the mean, variance, and skewness of the FPTD for models A and B are not restricted to the small correlation time of the pump fluctuation. It is worth noting that the prefactor of the pump fluctuation term in the model B seems to play a curb role on the increase of the variance and skewness.