Propagation of strongly focused light beams in graded-index planar waveguide and cylindrical fiber is investigated using exact analytical solutions of the wave equation. It is shown that high efficiency transfer of a strongly focused subwavelength spot through optical waveguide over large distances takes place with a period of revival (regeneration). Intensity distributions of a focused light beam and focusing efficiency are simulated. Super-oscillatory hot-spots with the sizes which are beyond the conventional Abbe diffraction limit can be observed at large distances from the source. This can provide the possibility to detect optical super-resolution information in the far-field without any evanescent waves. Far-field super-resolution imaging capabilities of a graded-index waveguide are also analyzed. 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 cylindrical waveguide. The wave shape variation with distance taking into account the spin-orbit interaction 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 also demonstrated.