Conversion of circular beams by a spiral phase plate: Generation of Generalized Humbert beams

Generalized Humbert beams (GHBs) are a novel donut beams family which created by converting of circular beams (CiBs) passing through a paraxial ABCD optical system with a spiral phase plate (SPP). The amplitude field of CiBs can be described by either the Whittaker or the confluent hypergeometric functions. The transformation and propagation of CiBs by the use of a SPP after propagating the considered optical system is presented and derived in detail. Some special cases of GHBs as standard, elegant and generalized Laguerre-Gaussian beams; Hypergeometric-Gaussian beams; Fractional order elegant Laguerre-Gaussian beams; Bessel-Gauss beams; quadratic Bessel-Gauss beams; and Whittaker-Gaussian beams are deduced in this paper. Numerical simulations are performed to study the transformation of the Whittaker-Gaussian beam by a SPP in free space as a particular example in this work.