Abstract

The truncated second-order moments and generalized $M^2$ 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 β and beam order N. For $N\to \infty ,$ the $M_G^2$ factor equals 4/$\sqrt{3}$ corresponding to the value of truncated plane waves, which guarantees consistency of the formalism.

© 2004 Optical Society of America

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