On the first-order phase transition in the laser with saturable absorber

First, the usual single-mode laser is considered and a renormalized Fokker-Planck equation (RFPE) for its Glauber quasi-probability distribution P(r, phi, t) is derived, showing that at steady state this FPE is essentially equivalent to a previously proposed time evolution equation which includes derivatives of all orders with respect to r. Then the RFPE is generalized to the laser with saturable absorber; this allows us to give a full description of the fluctuations of this bistable device for all values of the ratio between the saturation parameters of the passive and active atoms. In particular, the role of saturation in the amplitude diffusion term of the RFPE is investigated. Three main effects are found: i) an enhancement to the sharpness of the first-order-like phase transition in the laser with saturable absorber, ii) a large shift in the threshold of this bistable laser towards lower values of the pump of the amplifier, iii) a notable narrowing of the peak of the intensity fluctuations as a function of the pump parameter. (author)