Cylindrical paraxial optical beams described by the incomplete gamma function

Abstract

An analytic formula for a certain type of a cylindrical beam, which might be called a $\ensuremath{\gamma}$ beam, has been derived directly from the paraxial equation and independently using the method of the Hankel transform formulated in our previous work [T. Rado\ifmmode \dot{z}\else \.{z}\fi{}ycki, Opt. Laser Technol. 147, 107670 (2022)]. The fundamental properties of this beam are analyzed and the parameters characterizing the beam shape are identified. The connection with Gaussian and elegant Laguerre-Gauss beams is demonstrated. In the plane perpendicular to the propagation axis, this beam is shown to display an expanding ring of high-energy concentration. At large radial distances the spatial profile exhibits a power-law falloff. The phase of the wave is also studied in this paper. Close to the symmetry axis it is shown to be typical of modes that exhibit vortex character, such as Gaussian beams of $n\mathrm{th}$ order, but at large distances it reveals a peculiar behavior distinct from Gaussian-type beams.