Studies of spectral modification and limitations of the modified paraxial equation in laser wakefield simulations

Laser pulses propagating through plasma undergo spectral broadening through local energy exchange with driven plasma waves. For propagation distances on the order of the energy depletion length, spectral shifts can be comparable to the laser central frequency and wavenumber, a result of approximate action conservation. Here, we examine the local spectral shift, energy depletion, and action conservation of nonlinear laser pulses using the modified paraxial simulation code WAKE. Breakdown of the modified paraxial equation (MPE), which is based on the assumption of slow temporal variation of the pulse envelope, is monitored via consideration of the wave action. Although action is theoretically conserved for the continuous MPE, we observe that for large red shifts, action decays for the discrete implementation. Numerical analysis of the propagation algorithm verified the observed behavior. Increasing resolution improves action conservation up to a time, which is identified to be the validity limit for the MPE. Increased resolution also leads to increased simulation times and eliminates the strength of the MPE solver–efficiency.