In this paper, we consider using a binary curved fork grating not only for the simultaneous formation of several Bessel vortex beams but also for detecting the presence of angular harmonics (optical vortices) in an analyzed beam outside the focal plane. In this case, we use the effect of a longitudinal shift of focus (or correlation peak) when the lens is supplemented with an axicon. The combination of a lens and an axicon provides, in the focal plane, the generation of narrow light rings whose radii do not depend on the vortex phase component. Thus, the detection of an optical vortex cannot be carried out in the focal plane, but only at a certain distance from it, which provides additional security for information transmission. In addition, the use of these gratings makes it possible to simultaneously manipulate microparticles in various planes, and they can be used for fabricating complex three-dimensional metastructures as well as the single-shot pulse laser processing of transparent materials or polymers.

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A theoretical and experimental study of the propagation of vortex laser beams in a random aerosol medium is presented. The theoretical study is based on the extended Huygens-Fresnel principle with the generation of a random field, using the fast Fourier transform. The simulation shows that the stability of vortex beams to fluctuations of an optical medium falls with rising order of optical vortices. Moreover, a coherence length (radius) of the random medium is of great importance. The coherence radius extension affects adversely the conservation of a beam structure in the random medium. During further free-space propagation, increasing coherence enables reduction of the negative effects of fluctuations for beams with high-value topological charges. Experimental studies in the random aerosol medium have shown that at small distances vortex beams mostly demonstrate lower stability than a Gaussian beam. However, at considerable distances, vortex beams start to demonstrate greater stability that may be explained by their capacity to be regenerated after they passed obstacles.

In this paper we consider the use of a one-dimensional fractional Fourier transform for gradient-index optical waveguides. We calculate eigenfunctions of the transform in view of a limited range in the spatial and spectral domain.

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