  Remote focusing of light in a graded-index medium via mode interference is
demonstrated using exact analytical solutions of the wave equation. Strong
focusing of light occurs at extremely long distances and it revivals
periodically with distance due to mode interference. High efficiency transfer
of a strongly focused subwavelength spot through optical waveguide over large
distances takes place with a period of revival. 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.
