  We study x-ray photon scattering in the head-on collision of an XFEL pulse
and a focused high-intensity laser pulse, described as paraxial
Laguerre-Gaussian beam of arbitrary mode composition. For adequately chosen
relative orientations of the polarization vectors of the colliding laser
fields, this gives rise to a vacuum birefringence effect manifesting itself in
polarization flipped signal photons. Throughout this article the XFEL is
assumed to be mildly focused to a waist larger than that of the high-intensity
laser beam. As previously demonstrated for the special case of a fundamental
paraxial Gaussian beam, this scenario is generically accompanied by a
scattering phenomenon of x-ray energy signal photons outside the forward cone
of the XFEL beam, potentially assisting the detection of the effect in
experiment. Here, we study the fate of the x-ray scattering signal under
exemplary deformations of the transverse focus profile of the high-intensity
pump.
