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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.

Several optical schemes performing the complex-order fractional Fourier transform are considered. It is shown that these schemes, containing only Gaussian apertures or their combination with lenses, have eigenbeams represented by Hermite—Gaussian modes with transverse indices m, n≤1 and Laguerre—Gaussian modes with p=0 and l=1. The wave front of the eigenbeams is, as a rule, spherical.

The evolution of the cross-spectral density of an initially spatially incoherent beam propagating through an iterated sequence of spherical lenses and Gaussian apertures is studied. The global degree of spatial coherence, the spot size and the radius of wavefront curvature are calculated as functions of the number of transits. Steady-state values of the beam parameters are also given in a closed form. The dependence of the results on the aperture width and the focal length is discussed.