Propagation of Hermite-Laguerre-Gaussian beams through a paraxial optical ABCD system with rectangular hard-edged aperture

Based on a recurrence approach the analytical equations of the propagation of Hermite-Laguerre-Gaussian (HLG) beams through a paraxial optical ABCD system with rectangular hard-edged aperture are derived. The results are strict and can be used to study the propagation of HLG beams of any orders. The effect of the aperture on the propagation of HLG beams and the focal shifts are analyzed with numerical examples.