Theoretical investigation of vortex Gaussian beams focusing along the axis of the crystal

In this paper we investigate analytically and numerically sharp focusing of uniformly polarized laser Gaussian beams with a vortex phase along the axis of an anisotropic crystal. Two models are used for the analysis: geometrical optics, implemented in the software product ZEMAX, and wave optics based on the expansion in plane waves. An analytical expression is obtained in the frame of nonparaxial wave optics for a complex amplitude in focusing a vortex Gaussian beam in an anisotropic medium. It is shown that when focusing is weak ordinary and extraordinary beams are mixed and the beam formed has a mixed "spiral" type of polarization. In case of sharp focusing two focuses corresponding to the ordinary and extraordinary beams are formed along the crystal axis. If a first-order vortex phase is present in an incident beam with circular polarization cylindrical vector distributions with azimuthal polarization for the ordinary beam and those with radial polarization for the extraordinary beam occur in these focuses. Analytical calculations are illustrated by the results of numerical simulation. Both the intensity distribution for components of the generated laser fields and their polarization states are shown in detail. The studies completed are useful for the development of devices that perform polarization conversion.