# X-ray Spectroscopies of High Energy Density Matter Created with X-ray Free Electron Lasers
## 3 Intensity-Dependent EUV and X-ray Transmission  
In optical science, the intensity-dependent absorption or transmission of intense light are typical nonlinear responses of matter. Known as saturable absorption (SA) [26] and reverse saturable absorption (RSA) [27], these nonlinear phenomena are of great important owing to their wide range of applications in advanced laser and photonics technology. For example, SA preceded the invention of the laser; RSA and multi-photon absorption are applied in advanced microscopy, ultrafast optical switching, and laser pulse shaping [28; 29]. In the short wavelength regimes, the advent of FEL has enabled the production of EUV and X-ray intensities high enough to deliver multiple photons to each atom in solid-density samples in tens of femtoseconds. Core-level transitions driven by a large number of X-ray photons can compete with and possibly modulated intrinsic atomic transitions in plasmas, such as Auger processes, fluorescence, and collisions, and nonlinear responses of matter can be expected. In the course of exploring a new regime, aluminum is often considered an ideal element to begin with owing to its simple electronic structure. In ambient conditions, 10 electrons are bound (1s\({}^{2}\)2s\({}^{2}\)2p\({}^{6}\)) and three are delocalized in the conduction band. The density of states (DOS) of these conduction electrons are close to those of free electron gas (\(\sim\)E\({}^{1/2}\)). The first experimental observation of the saturable absorption of EUV photons in aluminum foil was reported by [PERSON] et al. [30]. Photon energy (92 eV) from the FLASH FEL is tuned above the L-edge (73 eV) of cold Al, and the main absorption channel is L-shell photoionization. After an L-electron is ejected from an atom, the L-edge of that particular ion increases to 93 eV, greater than photon energy. The lifetime of an L-hole via Auger decay is approximately 40 fs. Thus, the L-shell photoionization channel for singly ionized atoms will not be restored by the end of the pulse (15 fs). At high intensities, the fraction of such ions is high in the interaction volume, and photons in the later part of the pulse cannot be absorbed. These processes lead to an increase in EUV transmissions, as shown in Figure 3a. In other systems such as fused silica and tin, similar SA effects in the EUV regime have also been observed [31; 32]. Figure 3: (**a**) Transmission of 53-nm-thick aluminum target as a function of 92 eV photon fluence. The circles are the experimental data points, and the solid curve is the theoretical prediction [30]. (**b**) The transmission of 1-\(\upmu\)m-thick aluminum sample for the XFEL beam. The red line shows the experimental trend and the green line is from SCFLY simulations. The blue line indicates the cold transmission [33]. (**c**) The simulated transmission of 1.4-\(\upmu\)m-thick aluminum as a function of X-ray intensity with different photon energies (red: 1487 eV, blue 1501 eV, green: 1515 eV). Experimental measurements at 1487 eV are also plotted via black circles [34]. Data in the figure are taken from [30; 33; 34] and replotted. The observation of the saturable absorption of X-ray photons can also be expected. Given that the core-hole lifetimes decrease with the increasing binding energies, the SA of the X-ray requires higher light intensities. In the soft X-ray regime, the SA of the LCLS pulse with a peak intensity of 10\({}^{17}\) W/cm\({}^{2}\) was observed in aluminum [33]. As an XFEL pulse (1550-1850 eV) propagates through aluminum foil, the system is ionized to high charge states, and the absorption edge is shifted to higher energy. When the X-ray photon energy is tuned to above the K-edge of a specific ionization state, it is possible that in the middle of the pulse, no change state that can absorb a photon for K-shell photoionization occurs. As shown in Figure 3b, the increment of transmission is the most significant with photons just above the edge. High-energy photons can still be absorbed by higher charge states.