Generalization of the transformation

of Flat-Topped Mathieu-Gauss beams

by paraxial optical systems


L. Ez-zariy, B. Zouhal, H. Nebdi, A. Belafhal*

Laboratoire de Physique Nucléaire, Atomique et Moléculaire.

B.P. 20, Département de Physique, Faculté des sciences,

Université Chouaïb Doukkali, 24000 El Jadida. Morocco,

* Corresponding author. E-mail:

Received: 20 March 2012; revised version accepted: 05 May 2012



     Characteristics of the propagation of Li’s flattened Mathieu-Gaussian (FTMG) beams traveling a free paraxial ABCD optical system with: annular aperture, circular aperture and black screen are studied theoretically in this work, respectively. By means of expanding of a hard aperture function into a finite sum of complex Gaussian functions, and based on the Collins integral formula, a main approximate analytical expression of propagation of FTMG beams through an annular apertured ABCD optical system is developed. From this general result, we investigate the propagation of FTMG beams through an ABCD optical system with a circular aperture system, a circular black screen and unapertured optical system as particular cases.  The actual work generalizes also the above studies concerning the Mathieu, Mathieu-Gauss, flattened Bessel, Bessel-Gauss, pure Bessel, flattened Gaussian, and pure Gaussian beams passing through the considered optical systems. Some numerical calculations of the cases of propagation of FTMG through an annular apertured ABCD optical system and with a circular black are given and discussed in the present study.


Keywords: Collins formula; Flattened Mathieu-Gauss beam; Paraxial ABCD optical systems; Annular aperture; Unapertured system; Circular aperture; Circular black screen.

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