Amendment of the numerical dispersion in particle-in-cell methods for evaluation of charges of self-injected electron bunches in the laser wakefield acceleration

Total charge and energy evaluations for the electron beams generated in the laser wakefield acceleration (LWFA) is the primary step in the determination of the required target and laser parameters. Particle-in-cell (PIC) simulations is an efficient numerical tool that can provide such evaluations unless the effect of numerical dispersion is not diminished. The numerical dispersion, which is specific for the PIC modeling, affects not only the dephasing lengths in LWFA but also the total amount of the self-injected electrons. A numerical error of the order of $10^{-4}-10^{-3}$ in the calculation of the speed of light results in a significant error in the total injected charge and energy gain of the accelerated electron bunches. In the standard numerical approach, the numerical correction of the speed of light either requires infinitely small spatial grid resolution (which needs large computation platform) or force to compromise with the numerical accuracy. A simple and easy to implement numerical scheme is shown to suppress the numerical dispersion of the electromagnetic pulse in PIC simulations even with a modest spatial resolution, and without any special treatments to the core structure of the numerical algorithm. Evaluated charges of the self-injected electron bunches become essentially lower owing to the better calculations of the wake phase velocity.