Generalized M2-factor of hollow Gaussian beams through a hard-edge circular aperture

Based on the generalized truncated second-order moments, the generalized M2-factor (MG2-factor) of three-dimensional hollow Gaussian beams (HGBs) through a hard-edge circular aperture is studied in cylindrical coordinate system analytically and numerically. The closed-form expression for the MG2-factor of the truncated HGBs, which is dependent on the truncation parameter β and the beam order n, can be simplified to that of the truncated, the untruncated Gaussian beams and the untruncated HGBs. Also, the power fraction is demonstrated analytically and numerically, which shows that the area of the dark region across the HGBs increases as n increasing.