Paraxial propagation of Mathieu beams through an apertured ABCD optical system

Based on the relationship between separable solutions of the Helmholtz equation, we expanded the field's amplitude associated with Mathieu beams in terms of Bessel beams. By using the expansion of the circ function into a finite sum of complex Gaussian functions, we derived an approximated analytical expression of the amplitude distribution of Mathieu beams at the output plane of any apertured paraxial ABCD optical system. Some numerical calculations are performed and it is shown that our approach can substitute the usual numerical integration of the Helmholtz equation or the numerical calculation of the diffraction integral.