Astigmatic transformation of a fractional-order edge dislocation

It is shown theoretically that an astigmatic transformation of an edge dislocation (straight line of zero intensity) of the $\nu$-th order ($\nu=n+\alpha~$ is a real positive number, $n$ is integer, $0<\alpha<1$ is the fractional part of the number) forms at twice the focal length from a cylindrical lens $n$ optical elliptical vortices (screw dislocations) with a topological charge of $-1$, located on a straight line perpendicular to the edge dislocation. Coordinates of these points are zeros of the Tricomi function. At some distance from these vortices and on the same straight line, another additional vortex with a topological charge of $-1$ is also generated, which moves to the periphery if $\alpha$ decreases to zero, or approaches $n$ vortices if $\alpha$ tends to $1$. In addition, at the periphery in the beam cross-section, a countable number of optical vortices (intensity zeros) are formed, all with a topological charge of $-1$, which are located on diverging curved lines (such as hyperbolas) equidistant from a straight line on which the main $n$ intensity zeros are located. These “accompanying” vortices approach the center of the beam, following the additional “passenger” vortex, if $0<\alpha<0.5$, or move to the periphery, leaving the “passenger” next to the main vortices, if $0.5<\alpha<1$. At $\alpha=0$ and $\alpha=1$, the “accompanying” vortices are situated at infinity. The topological charge of the entire beam at fractional $\nu$ is infinite. The numerical simulation confirms theoretical predictions.