Parametric characterization of rotationally symmetric hard-edged diffracted beams

The truncated second-order moments and generalized M2 factor (M(G)2 factor) of two-dimensional beams in the Cartesian coordinate system are extended to the case of three-dimensional rotationally symmetric hard-edged diffracted beams in the cylindrical coordinate system. It is shown that the propagation equations of truncated second-order moments and the M(G)2 factor take forms similar to those for the nontruncated case. The closed-form expression for the M(G)2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams is derived that depends on the truncation parameter beta and beam order N. For N --> infinity, the M(G)2 factor equals 4/square root of 3 corresponding to the value of truncated plane waves, which guarantees consistency of the formalism.