The truncated second-order moments and generalized M<sup>2</sup> factor (M<sub>G</sub><sup>2</sup> 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<sub>G</sub><sup>2</sup> factor take forms similar to those for the nontruncated case. The closed-form expression for the M<sub>G</sub><sup>2</sup> factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams is derived that depends on the truncation parameter β and beam order <i>N</i>. For N → ∞, the M<sub>G</sub><sup>2</sup> factor equals 4/∛ corresponding to the value of truncated plane waves, which guarantees consistency of the formalism.
© 2004 Optical Society of America
Baida Lü and Shirong Luo, "Parametric characterization of rotationally symmetric hard-edged diffracted beams," J. Opt. Soc. Am. A 21, 193-198 (2004)