Generalized M2 factor of hard-edged diffracted controllable dark-hollow beams

On the basis of the truncated second-order moments method on the cylindrical coordinate systems, an analytical expression of the generalized beam propagation factor (MG2 factor) of hard-edged diffracted controllable dark-hollow beams is derived and illustrated numerically. It is shown that the MG2 factor of truncated controllable dark-hollow beams is dependent on the beam truncation parameter δ and the beam parameters N and ε. The result can be reduced to that for the non-truncated case as the truncation parameter approaches to be infinite. The power fraction is also discussed analytically and numerically.