  The electron dynamics in counter-propagating laser waves is investigated by
employing a novel approach, where the new Hamiltonian is time-independent when
the perturbative laser wave is absent. The physical picture of stochastic
electron dynamics is clearly revealed and the threshold values of the amplitude
of the perturbative laser field for triggering stochastic electron acceleration
are derived for different laser polarization directions and initial electron
momentum. It demonstrates that the dephasing rate (new Hamiltonian) between the
electron and the dominant laser can be randomly reduced if the amplitude of the
perturbative laser is above the threshold such that the electron could be
accelerated by the dominant laser well beyond the ponderomotive energy scaling.
The impact of a superluminal phase velocity is examined, which slightly changes
the stochastic region in Hamiltonian space if the superluminal phase velocity
is under a threshold value but significantly decreases the maximum electron
kinetic energy. All the analytic considerations are confirmed by the numerical
simulations.
