Study on the Mainardi beam through the fractional Fourier transforms system

Abstract

In this paper, we introduced the Mainardi beam and indicated its attributes under the Fractional Fourier transform for power variations of Fractional Fourier transform. The results represent that the behavior of the Mainardi beam is similar to that of the Airy beam. The obtained formula is a very powerful tool to describe propagation of a Mainardi beam through the FFT and the FrFT systems. An analytical expression of the Mainardi beam passing through an Fractional Fourier transform system presented. The influences of the Fractional Fourier transform, rational order of the Mittag-Leffler function (Fourier transform of the Mainardi function) on the normalized intensity distribution and characteristics of the Mainardi beam in the Fractional Fourier transform system examined. Power of the Fractional Fourier transform (p) and rational order of the Mittag-Leffler function (q) control characteristics of the Mainardi beam such as effective beam size, number, width, height, and orientation of the beam spot.

Comparison of Mainardi, cos-Mainardi and cosh-Mainardi beams with and without optical vortex in FT and FrFT systems

“…Then, we substituted equation (1) in Fresnel integral (equation ( 2)). By calculating different steps [50], we can rewrite equation (2) as follows [51].…”

“…In 2018, the Mainardi beam and its propagation were investigated under FrFT. It was observed that by changing the fractional order of the Mittag-Leffler function that is the general state of the Airy function (see appendix), the degree of freedom to control the properties of these beams can be increased [51,52]. The function M is the second kind of the Wright function, denoted by W λ,μ and defined as −1 < λ < 0.…”

“…In this paper, we studied the properties of beam propagation through FrFT optical systems. Based on Collins diffraction integral and Lohmann optical system, the electric field analytical expressions of the Mainardi, cosh-Mainardi and cos-Mainardi beams propagating through FrFT optical systems were obtained theoretically (method has been reported in [51]). The cosh-Mainardi beam with an optical vortex in Cartesian coordinates is as follows.…”