Kummer laser beams with a transverse complex shift

Abstract

Here, we study asymmetric Kummer beams, whose complex amplitude is proportional to the Kummer function (a degenerate hypergeometric function). These beams are exact solutions of the paraxial equation (Schrödinger-type equation) and are obtained from the conventional symmetric hypergeometric beams by complex shift of the transverse coordinates. On propagation, the asymmetric Kummer beams change their intensity weakly and rotate around the optical axis. These beams are an example of vortex laser beams with a fractional orbital angular momentum (OAM), which depends on four parameters: the vortex topological charge, the shift magnitude, the logarithmic axicon parameter, and the degree of the radial factor. Changing these parameters, it is possible to control the beam OAM, continuously increasing or decreasing it.