# Modeling Non-Equilibrium Dynamics and Saturable Absorption Induced by Free Electron Laser Radiation
## 3 Results and Discussion  
### Transmission by Ultrashort Pulse of Photons with 92 eV for Al Foil  
As mentioned above, saturable absorption using FEL pulses was first observed in a pioneering experiment by [PERSON] et al. [7], in which single-shot transmission data of a 53 nm Al foil were collected using 92 eV ultra-short (15 fs) photon pulses up to fluences in the 200 J/cm\({}^{2}\) range. FEL pulses are thus able to excite Al 2p core electrons (binding energies are 72.7 and 73.1 eV), and the lifetime of the excited core-hole state is 40 fs [18].  
Several approximations can be used for calculating X-ray transmission using the model reported in the previous section. In particular, fast calculations can be performed considering two opposite limits, namely: (i) a _long pulse_ limit in which the pulse width is much longer than the lifetime of the excited state; (ii) a _short pulse_ limit in which the lifetime is much longer than the pulse width.  
In both limits, the system under the photon field is assumed to be in a steady-state. Hence, the time dependency is neglected; that is to say, the time derivative of the occupation numbers and the intensity (see Equation (5)) become zero. The details of the models are described in Appendixes A and B.  
We have performed calculations of Al transmission comparing the results of the long and short pulse limits with those of the three-channel model including the dynamics of the pulse-matter interaction. In Figure 1, we show the comparison between the different models and experimental data [7]. For the dynamical three-channel and _long pulse_ models, we included the ground state \(|1\rangle\), excited state \(|2\rangle\) and relaxed state \(|3\rangle\). We assumed that the lifetime of the relaxed state \(\tau_{31}\) is the same as the one of the excited state \(\tau_{21}\) (namely, 40 fs), while the relaxation time to the relaxed state \(\tau_{32}\) is 1 fs, corresponding to the electron mean free path [7]. We also tested a _short pulse_ model, which represents just the opposite limit of the _long pulse_ model. This model assumes that the pulse width is much shorter than the lifetime of the excited state, so that decay of excited state is neglected. Without decay process, the state cannot access the relaxed state, thus the model reduces to a two channel model.  
As shown in Figure 1, all four models show an increase in transmission with the fluence until reaching the saturation at extremely high fluence above 200 J/cm\({}^{2}\). In the low-fluence limit, the calculations show the convergence to the linear transmission (Lambert-Beer law). The four models result in a completely different transmission at intermediate fluences, and the _long pulse_ with and without the relaxed state and _short pulse_ limits fail to reproduce the experimental data. These static models imply remarkably increased transmission in the 2-100 J/cm\({}^{2}\), anticipating the onset of saturable absorption of at least one order of magnitude. On the other hand, the dynamical three-channelmodel--which explicitly considers the time evolution of the occupation numbers along the pulse direction--is found to be in very good agreement with the experimental data.  
As will be shown in the next Section, the steady condition of the _long pulse_ model is not achieved at the beginning and end of the incident pulse, while a \"steady\" condition can be obtained only above a threshold intensity of the pulse taking account of its shape.