# Approaching the Attosecond Frontier of Dynamics in Matter with the Concept of X-ray Chronoscopy
## 3 X-ray Pulse Transmission in Saturable Absorption Conditions
The linear attenuation coefficient \(\mu\) is empirically determined by measuring the radiation beam intensity before and after interaction with the target sample, and in a pulse-specific approach it can be calculated using the relation  
\[\mu=\frac{-ln\Big{(}\frac{\int\frac{I_{1}(t)\ \mathrm{d}t}{I\ 0(t)\ \mathrm{d}t}}{ \mathrm{d}t}\Big{)}}{\mathrm{d}}=\frac{-ln(T)}{d}, \tag{8}\]  
where the integration is performed over the range covering the entire pulse temporal envelope. The higher transmission \(T\) expected for higher incident intensity will be reflected in smaller measured values of \(\mu\). Figure 4a presents the intensity dependence of the \(\mu/\mu_{\mathrm{cold}}\) ratio, where the index \"cold\" refers to low incident intensity, i.e., to linear absorption conditions. As shown, the attenuation coefficient starts decreasing at intensities over \(10^{17}\) W/cm\({}^{2}\), and the decrease strongly depends on the excited state lifetime. The saturation intensity, i.e., the intensity at which \(\mu=0.5\ \mu_{\mathrm{cold}}\), is above \(10^{19}\) W/cm\({}^{2}\), grows with a decreasing lifetime. The saturation intensity grows much faster for lifetimes smaller than the pulse duration. Turning the iron foil transparent is illustrated in Figure 4b, showing hundreds-times increase in transmission in the intensity range studied. The results simulated using the simple two-level model described in this work correspond with the experimental and calculated data reported elsewhere [38]; however, a certain discrepancy can be observed that is not well understood due to the complexity of the decay dynamics at higher incident intensities. The discrepancy may be explained for example by the intense secondary Fe K\(\alpha\) fluorescence radiation trapped in the sample, similar to the radiation trapping effect observed in the optical regime [48]. The K\(\alpha\) radiation would resonantly sustain the excited state of the part of the iron atoms leading to the elongation of an effective excited state lifetime and turning the target transparent at an intensity lower than the one indicated by the calculations. The discrepancy might also originate from other ionization events occurring in parallel with the K-shell photoionization and initiated by the electron-impact ionization by the secondary photo- and Auger electrons. It has been shown that at high X-ray intensities, even with ultrashort XFEL pulses, a considerable fraction of Fe atoms probed via K\(\beta\) decay had already interacted with the energetic electrons released through the photoionization of the atom's surrounding [49]. It implies that at higher intensities, certain Fe atoms may have their M- and L-shells modified upon the incidence of the 7130 eV-photon, which would make them incapable of K-shell photoionization and would boost the photoabsorption saturation. Figure 3: (**a**) Temporal evolution of transmission simulated for low- (blue) and high-intensity (red) pulses of 7130 eV photons through a 20 μm thick Fe foil. Ratio \(I_{1}(t-d/c)/I_{0}(t)\) of time distributions of the transmitted pulse \(I_{1}(t)=I(x=\mathrm{d},t)\) and of the incident pulse \(I_{0}(t)=I(x=0\ \upmu,t)\), with the target thickness \(d\) and the speed of light \(c\), for different incident pulse average intensities \(\overline{I_{0}(t)}\times 7130\ \mathrm{eV}\). The plotted ratio shows the effect of pulse-target interaction on photon time distribution around the temporal center of mass of the incident pulse, hence the subtraction \(t-d/c\). (**b**) Gaussian-like time envelopes \(I_{0}(t)\) and \(I_{1}(t-d/c)\) (not to scale) for low intensity (blue) and high intensity (red). Gaussian shape is delineated with the white dashed line. The transmitted pulse time envelope is shifted in time phase and loses the original Gaussian shape. (**c**) SASE-like time envelopes \(I_{0}(t)\) and \(I_{1}(t-d/c)\) (not to scale) for low intensity (blue) and high intensity (red).