  The focus of this article is on providing compact analytical expressions for
the differential number of polarization flipped signal photons constituting the
signal of vacuum birefringence in the head-on collision of x-ray free electron
(XFEL) and optical high-intensity laser pulses. Our results allow for
unprecedented insights into the scaling of the effect with the waists and pulse
durations of both laser beams, the Rayleigh range of the high-intensity beam,
as well as transverse and longitudinal offsets. They account for the decay of
the differential number of signal photons in the far-field as a function of the
azimuthal angle measured relative to the beam axis of the probe beam in forward
direction, typically neglected by conventional approximations. Moreover, they
even allow us to extract an analytical expression for the angular divergence of
the perpendicularly polarized signal photons. We expect our formulas to be very
useful for the planning and optimization of experimental scenarios aiming at
the detection of vacuum birefringence in XFEL/high-intensity laser setups, such
as the one put forward at the Helmholtz International Beamline for Extreme
Fields (HIBEF) at the European XFEL.
