  It is shown that the vector-vortex Laguerre-Gauss modes with
polarization-orbital angular momentum (OAM) entanglement are the vector
solutions of the Maxwell equations in a graded-index medium. Focusing of
linearly and circularly polarized vortex light beams with nonzero azimuthal and
radial indices in a cylindrical graded-index medium is investigated. The wave
shape variation with distance taking into account the spin-orbit and
nonparaxial effects is analyzed. Effect of long-term periodical revival of wave
packets due to mode interference in a graded-index cylindrical optical
waveguide is demonstrated. High efficiency transfer of a strongly focused spot
through an optical waveguide over large distances takes place with a period of
revival.
