# Modeling Non-Equilibrium Dynamics and Saturable Absorption Induced by Free Electron Laser Radiation
## 1 Introduction  
Over the past decade, X-ray and extreme ultraviolet free electron laser (FEL) sources have been developed, providing a source of extremely brilliant and ultrafast photon pulses. The present facilities include FELs in the extreme ultra-violet (EUV) and soft X-ray ranges such as FLASH (Hamburg) [1] and FERMI@Elettra (Trieste) [2], and in the hard X-ray range such asLCLS (Stanford) [3], SACLA (Spring-8) [4], and the European XFEL (presently under construction, Hamburg). Typically, FEL photon pulses show durations in the 10-100 fs range, contain a large number of photons (10\({}^{10}\)-10\({}^{15}\)) with a limited wavelength band width, depending on the pulse generation mechanism. Using suitable optics, the pulse spot dimensions can be reduced to 10-100 \(\upmu\)m\({}^{2}\) or less. In such extreme conditions, pulse fluences can exceed 100 J/cm\({}^{2}\), leading to the observation of non-linear optical processes in condensed matter such as saturation phenomena, two photon absorption, ultrafast electron, and lattice heating (see [5] and the references therein). Non-linear effects are quite familiar in optical laser science, while in the EUV and X-ray energy regime, investigations are still in development both experimentally and theoretically. There are several differences among the optical laser and EUV/X-ray radiation interaction with matter, including the penetration depth, the energy deposited, and the lifetime of the excited states. An important feature is exactly the latter, because for X-ray excitations, the lifetime of the excited state is in the femtosecond range (core-hole lifetime). Intense and ultrashort FEL pulses allow us to perform experiments for which the pulse width is of the same order of the core-hole lifetime at fluences for which non-linear effects are not negligible. Different techniques are used at FEL facilities, including transmission and scattering experiments and pump-and-probe studies using two ultrashort pulses (optical or X-ray). In many cases, simplified approaches are used for modeling the interaction process, basically neglecting the finite time widths and spatial dimensions of the pulses. Although efficient collisional-radiative codes describing dense plasma states are currently available (SCFLY, see [6] and references therein), the complicated dynamical processes associated with the interaction of matter with photon pulses are usually not taken into explicit account. For FEL ultrashort pulses and nanoscale materials, the finite dimensions are expected to play a role, and a detailed modeling of the dynamical pulse-matter interaction appears to be necessary. In particular, reliable theoretical models are needed both for a solid interpretation of the experimental data (transmission, scattering, and so on) and for modeling the evolution of the sample status during the excitation process (transient conditions, local temperature). In principle, transmission measurements are probably the easiest and cleanest experiments that can be performed using FEL radiation, but they also imply several important difficulties to be overcome both for their practical realization and interpretation of results. The importance of developing proper models can be appreciated by looking first at the results of [PERSON] et al. [7], in which saturable soft X-ray absorption (92 eV) of an ultrathin aluminum foil was obtained. Those results were followed by other experiments at lower (23.7-37 eV) [8] and higher photon energy (1540-1870 eV) [9], both indicating the importance of accounting for electron heating phenomena at high fluence. Further experiments in the hard X-ray range (7.1 keV) confirmed the existence of important non-linear phenomena with increasing transmission of a factor of 10 and substantial shifts of the absorption edge in solid iron [10].