Semiclassical stability analysis of a two-photon laser including spatial variation of the cavity field

We investigate the dynamics of a two-photon laser under conditions where the spatial variation of the cavity field along the cavity axis is important. Main attention is paid to linear stability analysis and numerical investigation of a two-photon laser for the Maxwell-Bloch equations. The model assumes pumping to the upper state of the two-photon transition. We consider the Maxwell-Bloch equations on the basis of which we study the stability analysis of the steady state of the system. The system is taken to be contained in a ring-laser cavity. Asymptotic expansion of the eigenvalue and analytic information are obtained in some realistic limits, such as very large reflectivity, very small cavity losses, or very small population relaxation rate. The results are illustrated with an application to a specific atomic system (Potassium) as an amplifying medium.