Propagation of Laguerre-Gaussian beams through an optical system with hard edge aperture

An approximate analytical propagation equation of Laguerre Gaussian beams passing through a paraxial ABCD optical system with hard edge aperture is derived,where the circa function is expanded into a finite sum of complex Gaussian functions.Numerical examples are given to illustrate the application of the propagation equation and discrepancies in use of the approximate formula are discussed.