  In this work, we derived formulae concerning the electric and magnetic field
characteristics of a focused radially polarized Gaussian vector beam. Such a
beam is consistent with Maxwell's equations contrary to plane waves having
uniform field distribution. Hence a realistic picture is provided of the
focused field distributions having importance before designing applications
such as particle acceleration. For focusing a perfectly reflecting large
numerical aperture on-axis parabolic mirror was supposed to have practical
importance. The computation technique was based on the Stratton-Chu vector
diffraction method. We pointed out that this offers a unique opportunity in the
long wavelength regime, where the Richards-Wolf theory becomes unreliable. In
the terahertz frequency range longitudinal electric field component with an
amplitude of $\sim$160 $\text{MV}/\text{cm}$ was predicted, which is ideal for
particle acceleration applications. Based on the field characteristics
experienced as a function of the focusing angle, the possibility of using a
paraboloid ring for particle acceleration was suggested. Its advantage is
reflected not only in the strong available longitudinal field but also in
ensuring the unobstructed transfer of particles as a practical point of view.
The axial and radial distributions of the longitudinal electric field component
for different incident beam divergences were analyzed in detail. It was found
that the shift of the physical focus relative to the geometrical focus along
the longitudinal direction shows a linear dependence on the divergence. The
effect of the divergence angle on the field enhancement factor was also
studied.
