Based on the Collins diffraction formula and by means of the expansion of a hard aperture function\ninto a finite sum of complex Gaussian functions, two analytical approaches of the Finite Olver\nbeams (FOBs) passing through a paraxial ABCD optical system with a circular annular aperture or\na rectangular one are developed in this paper. The propagation properties of the FOBs through an\nun apertured ABCD optical system or through this last with a circular (or rectangular) aperture or\na circular (or rectangular) black screen are deduced, from the main results, as particular cases.\nAlso, the characteristics of Finite ordinary Airy beam passing through the all considered optical\nsystems are derived here that correspond to zeroth-order of the FOBs. According to the predicted\nformulas, computer simulation examples are given to deepen the understanding of the characteristics\nof the FOBs passing through some optical systems of annular aperture basis.
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