Approximate close-form expression for Gaussian beams passing through an ABCD optical system with hard-edge aperture

The propagation of Gaussian beams through a paraxial ABCD optical system with hard-edge aperture is studied. In the treatment the rectangular function is expanded into a sum of complex Gaussian functions, thus the approximate analytical propagation expression can be derived. As an application example, numerical calculations are performed for Gaussian beams propagating in free space with hard-edge aperture. The advantage of greatly reducing the computing time and computation errors are discussed. Further extension to plan waves and flattened Gaussian beams is made to confirm the wide applicability of this method.