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First, this study obtained the fields of an Airy beam (AiB) with optical vortex (OV) for a Fourier transform (FT) system and a fractional Fourier transform (fractional FT) system; thereafter, their intensity and phase patterns were simulated numerically. The splitting on each line of the phase pattern indicates the position of an OV. The results show that the OV position will change when the power of the fractional FT (p) changes. Moreover, the uniformity of the spot beam disappears for the beam with OV. Further, the characteristics of an AiB such as number, width, height, uniformity of the spot beam and the effective beam size will change when there is a change in the values of p and z.

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.

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In this paper, we investigated the propagation of the Mainardi, cos-Mainardi and cosh-Mainardi beams under the Fractional Fourier transforms (FrFT) and Fourier transforms (FT). The results show that with the change of b, q, p, and z, which are the hyperbolic parameter, the fractional order of Mittag-Leffler function, the fractional Fourier transform order, and the distance propagation, respectively, the properties of the Mainardi beam with and without optical vortex can be controlled. The results show that with changing b and q parameters, the beam size, number, width and direction of the beam changes. These results can be useful in signal processing and beam manipulation. Mainardi beam similar to AiB holds promise for wide applications because of its unique features of being diffraction-free, self-accelerating, and self-healing.