Approximate analytical expression of flattened Gaussian beams passing through a paraxial optical system with an annular aperture

Propagation characteristics of flattened Gaussian beams through an annular apertured optical system remain unknown. Here we derive the approximate analytical expressions of flattened Gaussian beams passing through a paraxial optical system with an annular aperture based on the Collins integral and the expansion of the hard aperture function into a finite sum of complex Gaussian functions. Meanwhile, the corresponding closed-forms for the unapertured or circular apertured or circular black screen cases are also given. Some numerical examples are given to illustrate the propagation properties of flattened Gaussian beams through an optical system with an annular aperture. the numerical examples would be useful for some engineering applications.