Analytical solution for an anomalous hollow beam in a fractional Fourier transforming optical system with a hard aperture

Based on the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, the approximate analytical expressions for the output field distribution of an anomalous hollow beam (AHB) passing through an apertured fractional Fourier transform (FRT) system are derived. By using the approximate analytical formulae and diffraction integral formulae, the propagation properties of an AHB in circular and rectangular apertured FRT system are studied numerically. The results show that this method provides a convenient tool for studying the propagation properties of an AHB passing through apertured FRT system, and the apertured FRT system can be applied to laser beam shaping conveniently.