# Approaching the Attosecond Frontier of Dynamics in Matter with the Concept of X-ray Chronoscopy

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## Abstract

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## Featured Application

**Herein, an innovative methodology, called X-ray chronoscopy, is proposed for exploration of ultrafast processes in matter with attosecond precision using current XFEL sources. The method is based on measuring the change in an X-ray pulse temporal profile induced by interaction with a medium.**

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Principle of THz Streaking for Time-Dependent X-ray Pulse Analysis

#### 2.2. Time-Dependent Studies of Saturable Absorption at Hard X-ray Energies

^{2}. It was demonstrated that X-ray-induced transmission in the aluminum can be increased by up to 65% at the highest applied pulse fluences. From the experiment, it was concluded that the first part of the pulse ionizes L-shell electrons, and the created L-shell holes are filled at lower decay rates than ionization yields. The relatively long lifetime of the L-shell hole (40 fs) in aluminum allows for the strong depletion of L-states when using 15 fs X-ray pulse leading to the transparency increase.

_{2}O

_{3}nanoparticle system in water solution [42]. It has been demonstrated that while the 1s core-hole lifetime is relatively short (below 1 fs) compared to the incidence pulse length (30 fs), the secondary processes and decay cascades are of importance for SA strength. Theoretical simulations confirmed that the return to the ground state after a single photoionization event is longer than the 35 fs-pulse length employed in the experiment.

## 3. X-ray Pulse Transmission in Saturable Absorption Conditions

^{−2}s

^{−1}), $\sigma $ is the interaction cross section (in cm

^{2}), dominated by the photoabsorption component at photon energy close to the K-shell binding energy, and $n$ is the concentration of the target atoms (in cm

^{−3}). At a low intensity of the incident X-ray beam (low $I\left(x,t\right)$), atoms have enough time to return to the ground state before arrival of another photon; thus, the atom concentration $n$, being effectively the concentration of capable absorbing centers, remains constant across the target depth and over the course of the pulse. At a higher intensity, atoms are excited with a higher rate, which makes a significant part of them unable to interact with many arriving photons. This phenomenon is referred to as saturable absorption. It necessitates consideration of a variable concentration of active absorbers $n\left(x,t\right)$, dependent on both $x$ and $t$ through the $\left(x,t\right)$-dependent photon flux and the dynamics of ground state reestablishment.

^{−3}) and ${N}_{1}\left(x,t\right)$, a dimensionless value in the range (0,1), is the average ground state population per atom at depth $x$ and moment $t$. With this relation, Equation (2) takes the following form

^{−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 $x\in \left[0\text{}\mathsf{\mu}\mathrm{m},\text{}20\text{}\mathsf{\mu}\mathrm{m}\right]$ the populations of initial states were ${N}_{1}\left(x,t=0\right)=1$ and ${N}_{2}\left(x,t=0\right)=0$, 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., $I\left(x=0\text{}\mathsf{\mu}\mathrm{m},t\right)$, 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 µm, 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 × 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.

^{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 Kα fluorescence radiation trapped in the sample, similar to the radiation trapping effect observed in the optical regime [48]. The Kα 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 Kβ 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.

^{17}–10

^{19}W/cm

^{2}, our two-level model underestimates the saturable absorption rates as compared to the XATOM results. It can be explained by the effect of competing photoionization and decay processes other than the K-shell photoionization, which are covered by the XATOM calculation, while in our model, only the K-shell photoionization is considered, as it is highly dominating over others (e.g., the ratio of partial photoionization cross sections σ

_{K}/(σ

_{L1}+σ

_{L2}+σ

_{L3}) ≈ 9 [44]). For X-ray fluxes above 10

^{19}W/cm

^{2}, the two models show a similar increase in the sample transparency as well as the photoabsorption saturation. Nevertheless, our model still well explains the experimental data.

## 4. Summary

^{18}cm

^{2}, is expected to be around 6% for 50 incident SASE pulses. Based on the known XFEL performances and published THz streaking data, one can thus estimate that the X-ray chronoscopy experiment should be accomplished with good quality within a few hours by means of the proposed X-ray chronoscopy setup. Worth noting is the ongoing development of time diagnostics methods at XFELs, providing higher precision in temporal characterization of X-ray pulses and bringing X-ray chronoscopy closer to realization at XFELs. For instance, a pulse profile determination using a velocity map imaging (VMI) spectrometer has been demonstrated for attosecond soft X-ray pulses [10]. The fast, robust and simple derivation of pulse duration based on the center-of-energy shifts in the THz-streaked ionized electron spectra is also promising [21]. Precise pulse duration measurement before and after the target sample could be a prelude to realization of X-ray chronoscopy at XFELs.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic concept of X-ray chronoscopy experiment using two terahertz streaking spectrometers. The first THz setup is used to determine time distribution of the incident pulse (${I}_{0}\left(t\right)$ measurement), and the second one provides a measure of time distribution of X-ray pulse after interaction with the sample (${I}_{1}\left(t\right)$ measurement). Both spectrometers work in shot-to-shot mode, giving the possibility of either single-shot or cumulative measurement. The ratio of the measured time distribution curves will provide details on nonlinear processes induced by X-rays in the sample.

**Figure 2.**The streaking of an X-ray-induced photoelectron pulse in a THz vector potential. The X-ray pulse temporal profile is encoded in the energy spectrum of the measured photoelectron spectrum.

**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 ${I}_{1}\left(t-d/c\right)/{I}_{0}\left(t\right)$ of time distributions of the transmitted pulse ${I}_{1}\left(t\right)=I\left(x=\mathrm{d},t\right)$ and of the incident pulse ${I}_{0}\left(t\right)=I\left(x=0\text{}\mathsf{\mu}\mathrm{m},t\right)$, with the target thickness $d$ and the speed of light $c$, for different incident pulse average intensities $\overline{{I}_{0}\left(t\right)}$ × 7130 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 ${I}_{0}\left(t\right)$ and ${I}_{1}\left(t-d/c\right)$ (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 ${I}_{0}\left(t\right)$ and ${I}_{1}\left(t-d/c\right)$ (not to scale) for low intensity (blue) and high intensity (red).

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**MDPI and ACS Style**

Błachucki, W.; Wach, A.; Czapla-Masztafiak, J.; Delcey, M.; Arrell, C.; Fanselow, R.; Juranić, P.; Lundberg, M.; Milne, C.; Sá, J.;
et al. Approaching the Attosecond Frontier of Dynamics in Matter with the Concept of X-ray Chronoscopy. *Appl. Sci.* **2022**, *12*, 1721.
https://doi.org/10.3390/app12031721

**AMA Style**

Błachucki W, Wach A, Czapla-Masztafiak J, Delcey M, Arrell C, Fanselow R, Juranić P, Lundberg M, Milne C, Sá J,
et al. Approaching the Attosecond Frontier of Dynamics in Matter with the Concept of X-ray Chronoscopy. *Applied Sciences*. 2022; 12(3):1721.
https://doi.org/10.3390/app12031721

**Chicago/Turabian Style**

Błachucki, Wojciech, Anna Wach, Joanna Czapla-Masztafiak, Mickaël Delcey, Christopher Arrell, Rafał Fanselow, Pavle Juranić, Marcus Lundberg, Christopher Milne, Jacinto Sá,
and et al. 2022. "Approaching the Attosecond Frontier of Dynamics in Matter with the Concept of X-ray Chronoscopy" *Applied Sciences* 12, no. 3: 1721.
https://doi.org/10.3390/app12031721