Focussing of vortex laser beams
The phenomenon of diffraction is one of the manifestations of the wave nature of light. When the light beam propagates along the optical axis from the initial plane to the observation plane (both planes are parallel to each other and perpendicular to the optical axis), the light fields coming from different parts of the initial plane interfere with each other, and a diffraction pattern appears in the observation plane the intensity distribution in which is generally different from the distribution in the initial plane. The propagation of a light beam in a homogeneous medium is described by the Helmholtz equation and its paraxial approximation – a Schrödinger type equation. Despite diffraction, these equations have solutions that describe light fields that are free from this phenomenon. First of all, these are the well-known traditional and recently discovered Bessel [1, 78, 88], Mathieu  and Airy  asymmetric beams. The Bessel beams propagate without diffraction in the three-dimensional space, and the Airy beams – in the two-dimensional space. Also, the plane waves do not have diffraction, since the intensity distribution of them does not change as it propagates from one plane to another. In the three-dimensional case, any light field does not have diffraction, the angular spectrum of plane waves of which is non-zero on an infinitely thin ring . Along with the Bessel and Airy beams, paraxial structurally stable beams that are invariant to propagation are of interest. Such beams are not diffraction-free, but during propagation the structure of their intensity distribution in the transverse plane does not change, only the scale changes. The best-known such beams are the Hermite–Gauss and Laguerre–Gauss 195beams , hypergeometric modes . In a recent paper , Pearcey beams were considered as three-dimensional analogues of the Airy beams. The distribution of the complex amplitudes of such beams is described by the Pearcey function [190, 191], defined as the integral of the complex exponent, whose argument is a polynomial (like the Airy function). The angular spectrum of such beams is a phase modulated parabola. These beams have the autofocus feature and are restored after distortion by obstacles. A recent paper  proposed a virtual source that forms the Pearcey beam.