M² factor of controllable dark-hollow beams through a multi-apertured ABCD optical system

Under an optical system with multiple hard-edged apertures in a cylindrical coordinate system, the recurrence propagation expression is derived for the controllable dark-hollow beams (CDHBs) by expanding the hard-aperture function into a finite sum of complex Gaussian functions. Given the recurrence propagation expression, we deduce the approximate analytical expressions of the beam propagation factor M² in terms of the generalized truncated second-order moments. This provides a fast algorithm for the evaluation of the beam propagation quality for CDHBs through complicated optical trains with a series of apertures. The propagation of CDHBs through a two-aperture-lens ABCD optical system serves as the special case of multi-apertured ABCD optical systems. Our numerical results suggest that a one-aperture-lens optical system reduces the beam propagation quality of CDHBs, and a two-aperture-lens optical system improves the beam propagation quality of CDHBs by selecting appropriate beam parameters and aperture parameters.