Closed laser-beam trajectories in plano-spherical resonators with Gaussian apertures

The closed trajectories of a Gaussian beam in plano-spherical degenerate and nondegenerate resonators with two Gaussian apertures (with two pump regions with the Gaussian distribution of the gain) are described analytically and the existence conditions are found for such trajectories. It was assumed that Gaussian apertures (gain regions) were located in the plane of a flat mirror at equal distances on both sides of the resonator axis and the size of apertures restricted their action by the corresponding half-plane. (beams, resonators)