  We present a detailed study of the fields and propagation characteristics
around the focus of ultrashort radially-polarized laser beams (RPLBs) having
low-order spatio-temporal couplings (STCs). The three STCs considered are the
focusing of the different frequencies to different positions along the
longitudinal coordinate, the focusing of the frequencies to different positions
along one transverse coordinate, and the beam waist or Rayleigh range having a
power-law frequency dependence. The STCs considered are deemed low-order
because they are generally linear in frequency. The combination of a low-order
vector beam, ultrashort pulse duration, and the three different STCs shows
promise for exotic applications in dielectric or charged particle manipulation
and potentially other strong-field phenomena. The STCs presented are all
developed in a standard frequency-domain model where each case involves a
different chromatic term. We present the results unique to the vector nature of
the RPLBs and compare them to the linearly polarized cases, opening up
opportunities for control of the electric field around the focus with the
additional element of polarization.
