Statistical properties of a single-mode laser driven by additive and multiplicative coloured noises with a coloured cross-correlation for different correlation times

We study a system for a single-mode laser driven by additive and multiplicative coloured noises with a coloured cross-correlation. The analytical expression of the stationary intensity distribution (SID) for the laser is derived in the case of three different correlation times. The influences of each stochastic parameter on the SID are discussed, the effects of the noise "colour" and the cross-correlation of noises on the mean intensity \langle I\rangle, the variance, \lambda_2(0), and the skewness, \lambda_3(0) of the single-mode laser are investigated. We find that there are colourful phase transitions for the SID above a threshold, and re-entrant transitions induced by the "colour" of the additive noises. Further research of the curves of \langle I\rangle -\tau_i, \lambda_2(0)-\tau_i, and \lambda_3(0)-\tau_i (i=1,2,3) (where \tau_i is the correlation time) shows that the output intensity not only increases with the additive noise correlation time \tau_2 and the cross-correlation time \tau_3, but also the quality of the output of laser beams is optimized.