# Approaching the Attosecond Frontier of Dynamics in Matter with the Concept of X-ray Chronoscopy
## 3 X-ray Pulse Transmission in Saturable Absorption Conditions
For the purpose of testing different parameters, including \(\tau\), we developed a numerical algorithm capable of solving the above system of equations and determining tabulated \(I(x,t)\) and \(N_{1}(x,t)\) with arbitrary steps of time and length. In the simulation, the target was 20 \(\upmu\)m thick iron foil spanning over 0 \(\upmu\)m \(\leq\)\(x\leq\) 20 \(\upmu\)m. The parameters were chosen in order to verify the theoretical simulations with the reported experimental data [38]; thus, we used the X-ray pulse energy of 20 \(\upmu\)J and the cross-section value of 3.73 \(\times\) 10\({}^{-20}\) cm\({}^{2}\) for photoabsorption of 7130 eV photons [44]. Monochromatic X-ray pulses were assumed, thus neglecting the nonresonant excitations caused by the broad beam bandwidth as in [45; 46]. For \(x\in[0\upmu,\;20\upmu]\) the populations of initial states were \(N_{1}(x,t=0)=1\) and \(N_{2}(x,t=0)=0\), and for other values of \(x\), the populations were 0 at all moments. The time distribution of photons before interaction with the target, i.e., \(I(x=0\upmu,t)\), was described with a Gaussian with full width at half maximum (FWHM) of 7 fs. Despite the much more complex character of real time distribution of single XFEL pulses, the simplified Gaussian-like time envelope applied here suffice to evaluate the expected order of signal change. The excited state lifetime was given the value of 1 ns [47], thus distinctly longer than the typical XFEL pulse duration. A simulation was performed for a series of incident pulse intensities, with the time step \(\mathrm{d}t\) of 0.05 fs and the path step size \(\mathrm{d}x=c\times\mathrm{d}t\) (\(c\)--speed of light in vacuum) of 0.015 \(\upmu\)m, as a smaller \(\mathrm{d}t\) and \(\mathrm{d}x\) resulted in an inefficiently long computation time. The speed of light entering the target medium was assumed to remain unchanged; thus, the refractive index was given the value of 1, and its change due to saturable absorption was not considered in the present analysis of pulse transmission. With the values of steps \(\mathrm{d}t\) and \(\mathrm{d}x\), the transmission simulated for the low intensity of 1.5 \(\times\) 10\({}^{9}\) W/cm\({}^{2}\) differed from the transmission calculated with the Beer-Lambert equation by less than 2%, which we find reasonably low to illustrate the message conveyed in the present work. The results of the simulation of the time-dependent X-ray pulse transmission are presented in Figure 3a for different incidence pulse intensities. When the incident pulse has low intensity (blue), linear absorption dominates, and as shown, it results in a constant transmission of about 0.0017 over the course of the pulse. The time distribution of the transmitted pulse preserves the Gaussian behavior of the incident pulse, scaled down by a constant factor, as shown in Figure 3b. In the high-intensity scenario (red), the transmission stays constant only until the moment of about 10 fs, when the photon flux, slowly rising on the Gaussian tail, is still relatively small. After that moment, the transmission undergoes a significant increase, resulting in the transmitted pulse time envelope considerably deviating from that of the incident pulse, including loss of the original Gaussian shape and shift in the time phase (i.e., shift of the pulse temporal center of mass). This effect results from saturable absorption of the photons arriving at earlier moments and turning the target transparent for the later arriving photons [40]. Figure 3a also contains two additional noisy datasets at the lowest and the highest intensities, which were simulated with stochastic SASE time envelopes. These datasets are temporal evolutions of transmission integrated over as few as 50 SASE pulses and precisely follow the chronoscopic signal simulated with Gaussian-like pulse profiles. Modification of time distributions of individual SASE-like pulses, shown in Figure 3c for the high- and low-intensity scenarios, has a similar tendency as for Gaussian-like pulses.