Parametric characterization of Mathieu-Gauss beams

In recent years, much attention has been paid to the parametric characterization of laser beams based on the irradiance moments. Thus, many parameters and factors have been introduced to describe different laser beams in practice. Of them, we found the M_G ² factor which is useful to predict the laser beam quality [1, 2]. And the kurtosis parameter K which is useful to describe the degree of flatness (or sharpness) of laser beam irradiance distribution [2, 3]. While the M_G ² factor is invariant for beam propagating through any paraxial optical system, the kurtosis parameter depends on the axial coordinate along the beam axis z except for a few beams, for example fundamental Gaussian beams and Hermite-Gaussian beams. The M_G ² factor is written in terms of second-irradiance moments and the kurtosis is written in terms of second- and fourth-irradiance moments. In the present contribution, we project to calculate M_G ² factor of truncated MGBs. Also, we use the propagation equation of non-truncated Mathieu-Gauss beams (MGBs) [4] through paraxial optical system to study the propagation of kurtosis parameter of these beams in polar coordinate. In this work, we adopt the expansion of MGBs in terms of Bessel-Gauss beams (BGBs) to calculate the irradiance moments. Finally, some numerical calculations are performed and discussions are given.