Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity

The general problem of mathematical description of the beam diffraction by a thin hologram containing the pattern of interference between a regular wave with smooth wavefront and an optical-vortex beam (“fork” grating) is posed. The hologram output is presented as a sum of discrete diffraction orders; the field of a single order is described as a paraxial beam with current complex amplitude determined by the Kirchhoff formula. For small-angle diffraction, an analytical representation of the output beam through the modified Bessel functions is found which generalizes the Kummer beam representation introduced earlier. On this base, the behavior of beams produced by a “fork” hologram is studied both analytically and numerically in comparison with the standard Laguerre–Gaussian modes. Main distinctions are the “ripple” structures superimposed on the output beam transverse profile, much slower power-law amplitude decay at the beam periphery and higher divergence growing with the order of the optical vortex produced.