# Modeling Non-Equilibrium Dynamics and Saturable Absorption Induced by Free Electron Laser Radiation
## 3 Results and Discussion
### Simulations for Two Photon Absorption  
Another source of non-linear optical phenomena in FEL absorption studies is represented by the presence of two photon absorption (TPA) effects. The presence of TPA can be obtained at extremely large fluences, but the order of two photon absorption cross-section is predicted to be rather small, with typical values in the \(10^{-50}\) cm\({}^{4}\)s range. In order to estimate the size of this effect quantitatively, we apply the following set of equations:  
\[\frac{dN_{1}(z,t)}{dt}=\frac{g(z,t)I(z,t)}{h\
u}+\frac{g_{2}(z,t)I^{2}(z,t)}{ h\
u}+\frac{N_{2}(z,t)}{\tau_{21}}+\frac{N_{T}(z,t)}{\tau_{\rm TI}} \tag{11}\]  
Figure 2: Time dependence of the density of occupation numbers (normalized by the total number \(N\)) at the end face of Al thin film. The conditions are the same as in Figure 1. The fluences of incoming Gaussian pulses are 1.5, 15 and 150 J/cm\({}^{2}\) from the top to the bottom of three figures, respectively. The solid curves and dotted lines are estimated by the three-channel model and _long pulse_ model, respectively. The red color indicates the ground state \(|1\rangle\), the blue one the excited state \(|2\rangle\) and the green one the relaxed state \(|3\rangle\). After some time, the curves asymptotically approach the steady state of the _long pulse_ model. In the region of the light blue shade, the system has reached a steady state. After about 30 fs (the total pulse width), the excited state decays exponentially. The averaged experimental data are indicated by red points with the errors. \[\frac{dN_{2}(z,t)}{dt}=-\frac{g(z,t)I(z,t)}{h\
u}-\frac{N_{2}(z,t)}{\tau_{21}} \tag{12}\]  
\[\frac{dN_{T}(z,t)}{dt}=-\frac{g_{2}(z,t)I^{2}(z,t)}{h\
u}-\frac{1}{\tau_{T1}}N_ {T}(z,t) \tag{13}\]  
\[\frac{dI(z,t)}{dz}+\frac{1}{c}\frac{dI(z,t)}{dt}=g(z,t)I(z,t)+g_{2}(z,t)I^{2}(z,t) \tag{14}\]  
\[g(z,t)=\sigma\left(N_{2}(z,t)-N_{1}(z,t)\right) \tag{15}\]  
\[g_{2}(z,t)=\frac{\sigma_{2}}{h\
u}\left(N_{T}(z,t)-N_{1}(z,t)\right) \tag{16}\]  
\[N=N_{1}(z,t)+N_{2}(z,t)+N_{T}(z,t) \tag{17}\]  
where \(\sigma_{2}\) is the two photon absorption cross-section (in \(10^{-50}\,\mathrm{cm}^{4}\mathrm{s}\)). The new term \(g_{2}(z,t)I^{2}(z,t)\) in Equation (14) contributes to the increase (or decrease) of the intensity through the two photon absorption and stimulated emission. We introduced a new state \(|T\rangle\) which is accessible by TPA. This state is only connected to the ground state \(|1\rangle\) by two photon absorption and stimulated emission. The lifetime of state \(|T\rangle\) to the ground state \(|1\rangle\), \(\tau_{T1}\) is set to be the same as \(\tau_{21}\), which is 40 fs. In order to see the effects of TPA only, we did not introduce the relaxed state. The ionization energy of the ionic state with a hole in 2p orbital is 93 eV [7], which is higher than the photon energy, 92 eV, so we do not need to include the sequential two photon absorption. We report calculations with four different values of \(\sigma_{2}\); namely 0, \(10^{-50}\), \(10^{-49}\) and \(10^{-48}\,\mathrm{cm}^{4}\mathrm{s}\). \(\sigma_{2}=0\) reduces to a single photon absorption two-channel model. For a limit of infinitely long pulse, it corresponds to the _long pulse_ model without relaxed state (green line) in Figure 1. \(\sigma_{2}=10^{-49}\) and \(10^{-48}\,\mathrm{cm}^{4}\mathrm{s}\) represent large cross-sections for TPA and are probably not realistic, but we show the results to check the tendency for TPA effects in X-ray transmission. In Figure 3, we show the transmission of X-ray pulse for Al thin film with the same condition as in Figure 2:0 Application of the present TPA models does not seem to reproduce the experimental results. As we see also from the ground state asymptotic model (steady-state TPA, Appendix C, Equation (A19)), TPA plays an important role at high photon intensity, decreasing the total transmission. In the high fluence regime, TPA and saturable absorption are both factors affecting the transmission.