Determining the carrier-envelope phase of relativistic laser pulses via electron-momentum distribution

The impact of the carrier-envelope phase (CEP) of a long relativistic tightly focused laser pulse on the dynamics of a counterpropagating electron beam has been investigated theoretically in the, so-called, electron reflection regime. Our semiclassical Monte Carlo simulations show that the electrons are reflected at the rising edge of the laser pulse due to the ponderomotive force of the focused laser beam, and an asymmetric electron-momentum distribution emerges along the laser polarization direction, which sensitively depends on the CEP of the driving laser pulse for weak radiative stochasticity effects. The CEP signatures can be determined in the electron-momentum distribution at laser intensities of the order or larger than ${10}^{19}\phantom{\rule{0.16em}{0ex}}\mathrm{W}/{\mathrm{cm}}^{2}$ and for the pulse lengths up to 10 cycles. The CEP detection resolution is proportional to the electron beam density and can achieve approximately $0.{1}^{\ensuremath{\circ}}$ at an electron density of about ${10}^{15}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$. The method is applicable for currently available ultraintense laser facilities with the laser peak power ranging from tens of terawatt to multipetawatt.