Hypergeometric laser beams in a parabolic waveguide

Expression for a complex amplitude of paraxial hypergeometric laser beams, propagated in a gradient parabolic waveguide is obtained. Wide class of mode solves of Helmholtz equation in cylindrical coordinates for a gradient parabolic medium is received. This solves are proportion to Kummer functions, but only those have limited energy (physical able), which is equal to Laguerre-Gauss modes. A part of gradient parabolic fiber is considered as a parabolic micro lens, equations for numerical aperture and focal spot full width half maximum are obtained for it. Focal spot diameter can be less then a diffraction limit in a medium with a refraction number is equal to the diffraction number of the lens optical axis. Strict formulation for rings radii of a binar lens, which approximates gradient parabolic lens is obtained.