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.
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