Simulation of Fresnel diffraction of output beams of unstable resonators

The solution is given of the problem of diffraction of a spherical wave by an aperture in the form of a doubly connected region with a rectangular or circular contour, followed by the propagation of this wave across an optical system characterized by an arbitrary ray matrix. The exact expression is obtained for the Fresnel-Kirchhoff integral in terms of Fresnel integrals or Lommel functions of two variables. An investigation is made of the evolution of the effective beam width when its effective length and the shape of the aperture altered. A simple method is proposed for applying this solution in approximate simulation of the distribution of the field in laser output beams generated in unstable optical resonators.