# Modeling Non-Equilibrium Dynamics and Saturable Absorption Induced by Free Electron Laser Radiation
## 3 Results and Discussion
### Transmission by Ultrafast Pulse of Photons with 33.3 and 37 eV for Al Foil  
Another EUV transmission experiment [8] was performed at the TIMEX end station [19; 20] of the Elastic and Inelastic Scattering (EIS) beamline [21], using the FERMI@Elettra FEL-1 source, a seeded FEL providing clean, tunable [2], and intense subpicosecond (100 fs FWHM) photon pulses in the 19-62 eV photon energy range. Accurate transmission measurements were carried out using unsupported self-standing 100-nm-thick Al foils. The maximum energy per pulse delivered by the FEL-1 source were 180 \(\upmu\)J and 130 \(\upmu\)J range at 33.3 and 37 eV photon energy, respectively. The actual incoming fluence \(F\) at sample position was estimated by measuring the area of the focal spot (\(\sigma^{2}\approx 100\)\(\upmu\)m\({}^{2}\)) and accounting for mirror reflectivity and beamline transmission. Data were collected over several decades of fluence \(F\) (0.01-20 J/cm\({}^{2}\)) through the combined use of filters and a gas attenuator. Appreciable damage of the sample was observed after irradiation with a single FEL pulse with \(F>0.1\) J/cm\({}^{2}\), and single-shot transmittance measurements were carried out in the high fluence regime. EUV transmission data were measured with unprecedented accuracy, and evidence for a nonmonotonic trend as a function of incoming fluence was found. In particular, an increased transmittance at higher fluence of about 5-10 J/cm\({}^{2}\)--associated with saturable absorption effects--was accompanied by an initial decrease at intermediate fluences, as shown in Figure 5.  
As we have seen in previous sections, this decrease can not be reproduced by a pure single-photon optical effect. Two-photon effects can decrease the observed transmittance under suitable conditions. However, we have found that experimental data showing a non-monotonic behaviour cannot be reproduced by realistic TPA models. The cross-section \(\sigma_{2}\) for two-photon absorption should be at least on the order of \(10^{-48}\) cm\({}^{4}\)s, two orders of magnitude larger than expected. In a recent work done by [PERSON] et al. [14], detailed calculations of the cross-section for selected photon energy and electron temperatures for aluminium were reported. These calculations show clear changes in photon absorption due to the electron distribution, and in particular, a peculiar increase of absorption in an intermediate range of electron temperatures (1-10 eV) was found. We introduced  
Figure 5: Experimental (expt.) EUV transmission data at 37 eV (**left panel**) and 33.3 eV (**right panel**) compared with different calculations, including: only optical saturation phenomena without accounting for temperature effects in the cross-section (Sat. only); only electron heating but neglecting saturation effects (El. heat. only); both electron heating and saturation effects (Sat.+El. heat.). the results of the calculations through a temperature-dependent absorption coefficient \(\sigma(T)\) with effective electron temperature \(T_{e}\). For each slice of \(\Delta z\) for the simulation, we estimated the local electron temperature from a Maxwell-Boltzmann model as \(k_{B}\Delta T_{e}=\frac{2}{3n_{e}}\frac{\Delta F}{\Delta z}\), where \(n_{e}\) is a number of valence electron and \(\Delta F\) is deposited energy per unit surface. Figure 5 shows experimental (expt.) EUV transmission data at 37 eV (left panel) and 33.3 eV (right panel) compared with different calculations, including: only optical saturation phenomena without accounting for temperature effects in the cross section (Sat. only); only electron heating but neglecting saturation effects (El. heat. only); both electron heating and saturation effects (Sat.+El. heat.). We see that the Sat. only model (no account for electron temperature) does not reproduce the decrease of the transmission between 1 to 5 J/cm\({}^{2}\). The El. heat. only (temperature-dependent, linear) model shows the decrease in transmittance, but the increase at high fluence is not reproduced. Finally, the Sat.+El. heat.