  Focusing with a 4$\pi$ parabolic mirror allows for concentrating light from
nearly the complete solid angle, whereas focusing with a single microscope
objective limits the angle cone used for focusing to half solid angle at
maximum. Increasing the solid angle by using deep parabolic mirrors comes at
the cost of adding more complexity to the mirror's fabrication process and
might introduce errors that reduce the focusing quality. To determine these
errors, we experimentally examine the focusing properties of a 4$\pi$ parabolic
mirror that was produced by single-point diamond turning. The properties are
characterized with a single $^{174}$Yb$^{+}$ ion as a mobile point scatterer.
The ion is trapped in a vacuum environment with a movable high optical access
Paul trap. We demonstrate an effective focal spot size of 209 nm in lateral and
551 nm in axial direction. Such tight focusing allows us to build an efficient
light-matter interface. Our findings agree with numerical simulations
incorporating a finite ion temperature and interferometrically measured
wavefront aberrations induced by the parabolic mirror. We point at further
technological improvements and discuss the general scope of applications of a
4$\pi$ parabolic mirror.
