The propagation of hypergeometric beams through an annular apertured paraxial ABCD optical system

By means of expanding the hard aperture function into a finite sum of complex Gaussian functions and based on the generalized Huygens–Fresnel diffraction integral, a novel approximate analytical expression of hypergeometric (HyG) beams passing through a paraxial ABCD optical system with an annular aperture is derived. The results could be reduced to the case of a circular aperture or a circular black screen. Some numerical simulations are also performed and illustrated for the propagation characteristics and focusing properties of a HyG beam through a paraxial ABCD optical system with an annular aperture. The results obtained from the approximate analytical formula provide more efficiency than the usual way of using diffraction integral formula directly. (paper)