Quantum theory of the linewidth of a laser with a saturable absorber: phase diffusion

A quantum theory of the spectral width of a laser with a saturable absorber is presented. We perform a density matrix calculation yielding the off-diagonal element p_n,n+1(t) which is proportional to the average value of the electric field <E(f)> associated with the laser field having a large photon number n. The general linewidth theory is applied to any general laser which has saturable absorber features for a portion of the building of the laser field. We have found that the saturable absorber action contribution to the linewidth of such a laser can be substantial.