Analytical expressions of generalized M2 factor of hard-edged diffracted Laguerre-Gaussian beams

On the basis of the truncated second-order moments method in the cylindrical coordinate systems, exact analytical expressions of the generalized beam propagation factor (MG2 factor) of hard-edged diffracted standard and elegant Laguerre-Gaussian beams are derived and illustrated numerically. It is shown that the MG2 factor of truncated standard and elegant Laguerre-Gaussian beams is dependent on the beam orders m and p and the beam truncation parameter δ. The result can also be reduced to that for the non-truncated case as the truncation parameter approaches to be infinite, and to that for truncated Gaussian beams if the mode order of Laguerre polynomial is zero.