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Vacuum polarization (VP) is investigated for the interaction of a polarized
$\gamma$-ray beam of GeV photons with a counterpropagating ultraintense laser
pulse. In a conventional setup of a vacuum birefringence measurement, a VP
signal is the emerging small circular (linear) polarization of the initially
linearly (circularly) polarized probe photons. The pair production via the
nonlinear Breit-Wheeler process in such a high-energy environment eliminates
part of the $\gamma$-photons in the outgoing $\gamma$-beam, increasing the
statistical error and decreasing the accuracy of this VP signal. In contrast,
we investigate the conversion of the emerging circular polarization of
$\gamma$-photons into longitudinal polarization of the created positrons,
considering the latter as the main VP signal. To study the VP effects in the
highly nonlinear regime, where the Euler-Heisenberg effective Lagrangian method
breaks down, we have developed a Monte-Carlo simulation method, incorporating
vacuum birefringence and dichroism via the one-loop QED probabilities in the
locally constant field approximation. Our Monte Carlo method will enable the
study of VP effects in strong fields of arbitrary configuration. With 10~PW
laser systems, we demonstrate the feasibility of detecting the fermionic signal
of the VP effect at the 5$\sigma$ confidence level with a few hours of
measurement time.
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