Temporal resolution criterion for correctly simulating relativistic electron motion in a high-intensity laser field

Particle-in-cell codes are now standard tools for studying ultra-intense laser-plasma interactions. Motivated by direct laser acceleration of electrons in sub-critical plasmas, we examine temporal resolution requirements that must be satisfied to accurately calculate electron dynamics in strong laser fields. Using the motion of a single electron in a perfect plane electromagnetic wave as a test problem, we show surprising deterioration of the numerical accuracy with increasing wave amplitude $a_0$ for a given time-step. We go on to show analytically that the time-step must be significantly less than $\lambda/c a_ 0$ to achieve good accuracy. We thus propose adaptive electron sub-cycling as an efficient remedy.