Fresnel Diffraction of a Shifted Incident Beam on Double Fork-Shaped Gratings

Diffraction of shifted incident beam on double fork-shaped gratings with unit charges has been numerically simulated by the using of the Fresnel diffraction integral. The trajectories of singularities dependance on the position of the incident Gaussian beam have been investigated. For the case of same topological charges of dislocations it is shown that the shift of incident beam along the horizontal axis leads to the shift of singularities trajectories along the vertical axis, and vice versa. It is shown, that these results can be qualitatively predicted by using the hydrodynamic approach. For the case of diffraction on double fork-shaped grating which have dislocations with topological charges of opposite signs, the effect of the incident beam center shift depends on the initial distance between dislocations $2r_d$ as well as the initial positions of positive and negative topological charges. The aforementioned findings shows that the result of diffraction on the double fork-shaped gratings highly depends on the position of the incident beam center, and therefore, it provides the valuable information that can enrich our understanding of the spatial dynamics of optical vortices.