Sinusoidal Gaussian optical vortex as a superposition of two hypergeometric beams

We analyze the propagation of hypergeometric beams with a parabolic initial wavefront in a homogeneous medium. While hypergeomentric beams have a central amplitude singularity in the initial plane and are of infinite energy, superposition of two such beams has no singularity and is of finite energy. A particular case of such a superposition we study in detail is a sinusoidal Gaussian beam with a unit topological charge. This beam belongs to the class of elegant laser beams since it is described by the same complex-argument function both in the initial plane and in the Fresnel diffraction zone. The diameter of the first light ring of the sinusoidal Gaussian beam is almost independent of the Gaussian beam waist radius.