Synergistic effects of deep electrostatic potential well and relativistic laser beam on electron acceleration

Electron dynamics in intense laser waves has been the foundation of many early investigations on nonlinear laser plasma interactions. However, recently it was shown that the synergistic effects of deep ($\sim$ few MeVs) electrostatic potential well formed in pre-plasma and laser beam result in a strong increase of electron energy in comparison to a standard ponderomotive scaling. Here we use employ a simple box-like potential well and study numerically the energy gain by electron in the presence of two counter-propagating laser beams. We compare our numerical results with i) the case of electron interaction with single laser beam and box-like potential well and ii) with simplified analytic estimates for the case of electron interactions with two laser beams and box-like potential well. We discuss the physics of synergistic effects of electron interactions with potential well and two laser beams.