The beam propagation factor of nonparaxial truncated flattened Gaussian beams

By using the second-order moment of the power density, the beam width, far-field divergence angle and M-2 factor of nonparaxial truncated flattened Gaussian (FG) beams are derived analytically. It is shown that the M-2 factor of nonparaxial truncated FG beams depends not only on the truncation parameter delta and beam order N, but also on the initial waist-width to wavelength ratio w(0)/lambda. The far-field divergence angle approaches an asymptotic value of theta(max) = 63.435 degrees when the truncation parameter delta --> 0. For the special cases of N = 0 and delta --> infinity our results reduce to those of nonparaxial truncated Gaussian beams and nonparaxial untruncated FG beams, respectively.