M² factor of nonparaxial truncated flat-topped beams

By using the second-order moment method of the power density and a new model of flat-topped beams, the waist width，far-field divergence angle and beam quality factor of nonparaxial truncated flat-topped beams are derived analytically. It is shown that the beam quality factor of nonparaxial truncated flat-topped beams depends on the truncation parameter, the beam order and the initial waist-width to wavelength ratio. The far-field divergence angle approaches an asymptotic value of 63.435° when the truncation parameter approaches to 0. For the special case of 1 order beam and the truncation parameter approaching to +∞, the results reduce to those of nonparaxial truncated Gaussian beams and nonparaxial untruncated flat-topped beams, respectively. It is also shown that the second-order mome