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In this study, we investigate Airy, cos-Airy, and cosh-Airy beam lasers with optical vortex (OV). Then, taking into account the Fourier transform, we examine and compare the propagation of these beams in a strong nonlinear system. The results indicate that by changing the input power p 0 and the parameters b and propagation length, the location of OV displaces. It is shown that nonlinearity is one of the parameters that control OV in optical systems used in these types of lasers. An important property in these beams is their direction variation due to the change in the mentioned parameters, which can be used in applications such as beam switching. [ABSTRACT FROM AUTHOR]

We rigorously investigate the rapidly oscillating contributions in the sinc-function representation of the Dirac delta function and the Fourier transform of the Coulomb potential. Beginning with a derivation of the standard integral representation of the Heaviside step function, we examine the representation of the Dirac delta function that contains a rapidly oscillating sinc function. By contour integration, we prove that the representation satisfies the properties of the Dirac delta function, although it is a function divergent at nonzero points. This is a good pedagogical example demonstrating the difference between a function and a distribution. In most textbooks, the rapidly oscillating contribution in the Fourier transform of the Coulomb potential into the momentum space has been ignored by regulating the oscillatory divergence with the screened potential of Wentzel. By performing the inverse Fourier transform of the contribution rigorously, we demonstrate that the contribution is a well-defined distribution that is indeed zero, even if it is an ill-defined function. Proofs are extended to exhibit that the Riemannâ€"Lebesgue lemma can hold for a sinc function, which is not absolutely integrable. [ABSTRACT FROM AUTHOR]