Topological charge of optical vortices in the far field with an initial fractional charge: optical "dipoles"

In this work, using the Rayleigh-Sommerfeld integral and the Berry formula, the topological charge (TC) of a Gaussian optical vortex with an initial fractional TC is calculated. It is shown that for different fractional parts of the TС, the beam contains a different number of screw dislocations, which determine the TС of the entire beam. With a small fractional part of the TС, the beam consists of the main optical vortex centered on the optical axis with the TС equal to the nearest integer (let be $n$), and two edge dislocations located on the vertical axis (above and below the center). With an increase in the fractional part of the initial TC, a "dipole" is formed from the upper edge dislocation, consisting of two vortices with $TC+1$ and $–1$. With a further increase in the fractional part, the additional vortex with $TC+1$ is displaced to the center of the beam, and the vortex with $TC–1$ is displaced to the periphery. With a further increase in the fractional part of the TC, another "dipole" is formed from the lower edge dislocation, in which, on the contrary, the vortex with $TC–1$ is displaced to the optical axis (to the center of the beam), and the vortex with $TC+1$ is displaced to the beam periphery. When the fractional part of the TC becomes equal to 1/2, the "lower" vortex with $TC–1$, which was displaced to the center of the beam, begins to shift to the periphery, and the "upper" vortex with $TC+1$ moves closer and closer to the center of the beam and merges with the main vortex when the fractional part approaches 1. Such dynamics of additional vortices with upper $TC+1$ and lower $TC–1$ determine the whole TC the beam have ($n$ or $n+1$) for different values of the fractional part from the segment $[n,~n+1]$.