  In this paper we show how to recast the results of the semiclassical method
of Baier, Katkov & Strakhovenko for pair production, including the possibility
of specifying all the spin states and photon polarization, in a form that is
suitable for numerical implementation. In this case, a new type of integral
appears in addition to the ones required for the radiation emission process. We
compare the resulting formulas with those obtained for a short pulse plane wave
external field by using the Volkov state. We investigate the applicability of
the local constant field approximation for the proposed upcoming experiments at
FACET II at SLAC and LUXE at DESY. Finally, we provide results on the
dependence of the pair production rate on the relative polarization between a
linearly polarized laser pulse and a linearly polarized incoming high energy
photon. We observe that even in the somewhat intermediate intensity regime of
these experiments, there is roughly a factor of $2$ difference between the pair
production rates corresponding to the two relative photon polarizations, which
is of interest in light of the vacuum birefringence of QED.
