Propagation of hypergeometric laser beams in a medium with a parabolic refractive index

An expression to describe the complex amplitude of a family of paraxial hypergeometric laser beams propagating in a parabolic-index fiber is proposed. A particular case of a Gaussian optical vortex propagating in a parabolic-index fiber is studied. Under definite parameters, the Gaussian optical vortices become the modes of the medium. This is a new family of paraxial modes derived for the parabolic-index medium. A wide class of solutions of nonparaxial Helmholtz equations that describe modes in a parabolic refractive index medium is derived in the cylindrical coordinate system. As the solutions derived are proportional to Kummer's functions, only those of them which are coincident with the nonparaxial Laguerre-Gaussian modes possess a finite energy, meaning that they are physically implementable. A definite length of the graded-index fiber is treated as a parabolic lens, and expressions for the numerical aperture and the focal spot size are deduced. An explicit expression for the radii of the rings of a binary lens approximating a parabolic-index lens is derived. Finite-difference time-domain simulation has shown that using a binary parabolic-index microlens with a refractive index of 1.5, a linearly polarized Gaussian beam can be focused into an elliptic focal spot which is almost devoid of side-lobes and has a smaller full width at half maximum diameter of 0.45 of the incident wavelength.