  The two-color above-threshold ionization (ATI) of atoms and ions is
investigated for a vortex Bessel beam in the presence of a strong near-infrared
(NIR) light field. While the photoionization is caused by the photons from the
weak but extreme ultra-violet (XUV) vortex Bessel beam, the energy and angular
distribution of the photoelectrons and their sideband structure are affected by
the plane-wave NIR field. We here explore the energy spectra and angular
emission of the photoelectrons in such two-color fields as a function of the
size and location of the target (atoms) with regard to the beam axis. In
addition, analogue to the circular dichroism in typical two-color ATI
experiments with circularly polarized light, we define and discuss seven
different dichroism signals for such vortex Bessel beams that arise from the
various combinations of the orbital and spin angular momenta of the two light
fields. For localized targets, it is found that these dichroism signals
strongly depend on the size and position of the atoms relative to the beam. For
macroscopically extended targets, in contrast, three of these dichroism signals
tend to zero, while the other four just coincide with the (standard) circular
dichroism, similar as for Bessel beams with small opening angle. Detailed
computations of the dichroism are performed and discussed for the $4s$
valence-shell photoionization of Ca$^+$ ions.
