Finite beaming effect on QED cascades

The quantum electrodynamic (QED) theory predicts the photon emission and pair creation involved in QED cascades occur mainly in a forward cone with finite angular spread $\Delta\theta \sim 1/\gamma_{i}$ along the momenta of incoming particles. This finite beaming effect has been assumed to be negligible because of the particles' ultra-relativistic Lorentz factor $\gamma_{i}\gg1$ in laser-driven QED cascades. We develop an energy- and angularly resolved particle-tracking code, resolving both the energy spectra and the momentum profile of the outgoing particles in each QED event, which improves substantially the agreement between the simulation and exact QED results. We investigate QED cascades driven by two counter-propagating circularly polarized laser pulses, and show that the narrow beaming could be accumulated to effectively suppress the long-term growth of cascades, even though it can hardly affect the early formation of cascades. For QED cascades longer than $10$ laser cycles, the finite beaming effect could decrease the final pair yield, especially at ultrahigh intensities $\xi>600$, by more than $10\%$.