Generation of ultrarelativistic polarized positrons during interaction of an ultrarelativistic electron beam with a counterpropagating two-color petawatt laser pulse is investigated theoretically. Our Monte Carlo simulation based on a semi-classical model, incorporates photon emissions and pair productions, using spin-resolved quantum probabilities in the local constant field approximation, and describes the polarization of electrons and positrons for the pair production and photon emission processes, as well as the classical spin precession in-between. The main reason of the polarization is shown to be the spin-asymmetry of the pair production process in strong external fields, combined with the asymmetry of the two-color laser field. Employing a feasible scenario, we show that highly polarized positron beams, with a polarization degree of $zetaapprox 60%$, can be produced in a femtosecond time scale, with a small angular divergence, $sim 74$ mrad, and high density $sim 10^{14}$ cm$^{-3}$. The laser-driven positron source, along with laser wakefield acceleration, may pave the way to small scale facilities for high energy physics studies.
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