Numerical simulations of X-rays Free Electron Lasers (XFEL)

TL;DR

This paper investigates a nonlinear Schrödinger equation modeling electron behavior in X-ray Free Electron Lasers (XFEL), demonstrating through numerical simulations that a time-averaged approximation is effective even in complex, near-blowup scenarios.

Contribution

It provides the first systematic numerical analysis of the convergence of the XFEL model to its time-averaged form in complex regimes, extending previous analytical results.

Findings

Time-averaged model is effective even near blowup conditions.

Operator splitting pseudo-spectral method accurately captures model behavior.

Convergence holds in energy subcritical and supercritical cases with periodic lattice.

Abstract

We study a nonlinear Schr\"odinger equation which arises as an effective single particle model in X-ray Free Electron Lasers (XFEL). This equation appears as a first-principles model for the beam-matter interactions that would take place in an XFEL molecular imaging experiment in \cite{frat1}. Since XFEL is more powerful by several orders of magnitude than more conventional lasers, the systematic investigation of many of the standard assumptions and approximations has attracted increased attention. In this model the electrons move under a rapidly oscillating electromagnetic field, and the convergence of the problem to an effective time-averaged one is examined. We use an operator splitting pseudo-spectral method to investigate numerically the behaviour of the model versus its time-averaged version in complex situations, namely the energy subcritical/mass supercritical case, and in the presence of a periodic lattice. We find the time averaged model to be an effective approximation, even close to blowup, for fast enough oscillations of the external field. This work extends previous analytical results for simpler cases \cite{xfel1}.