# Approaching the Attosecond Frontier of Dynamics in Matter with the Concept of X-ray Chronoscopy
## 2 Materials and Methods  
### Principle of THz Streaking for Time-Dependent X-ray Pulse Analysis  
THz and infrared streaking experiments have been used for nearly two decades to measure the temporal and spatial structure of high-harmonic generated (HHG) EUV pulses [22; 23], and significant strides have been made to characterize soft X-ray pulse lengths at FEL facilities, e.g., at FLASH [17; 18]. In addition, at the Linac Coherent Light Source (LCLS), [PERSON] et al. used streaking experiments to measure the temporal structure of soft X-ray pulses directly in the time domain, and concluded that the shortest X-ray pulses available were no longer than 4.4 fs [26]. More importantly, they validated that the method is non-invasive, independent of photon energy, decoupled from machine parameters, and provides an upper bound on the X-ray pulse duration. The next step toward the method adaptation to hard X-ray FELs was the design of PALM (Photon Arrival and Length Monitor) at the SwissFEL, which was tested on HHG sources and the hard XFEL at SACLA [4; 5; 27].  
The working principle of THz or IR streaking is based on the interaction of X-rays with a gas target, resulting in the generation of photoelectrons, which may gain or lose energy if the photoionization mechanism takes place in an external electric field. If the field amplitude is changing as a function of time, the energy change of the photoelectron from its normal photoionization energy is a way to establish when the electron was generated. The mapping of the temporal profile of the pulse to an electron energy spectrum can be approximated with the following equation [28]:  
\[K_{f}=K_{0}\pm\sqrt{K_{0}U_{p}}\cos(\vartheta)\sin(\omega t), \tag{1}\]  
where \(U_{p}=e^{2}{E_{0}}^{2}/\left(2m_{e}\omega^{2}\right)\) is the ponderamotive potential generated by the THz/IR beam, \(K_{f}\) and \(K_{0}\) are the final and initial kinetic energies of the electrons, respectively, \(\vartheta\) is the angle between the path of the electron and the polarization of the THz/IR beam, \(\omega\) is the angular frequency of the THz/IR pulse, \(t\) is time, and \(e\) and \(m_{e}\) are the electron charge and mass, respectively. For X-ray FEL pulses coming close to \(t=0\), the dependence of \(K_{f}\) on arrival time is nearly linear, with \(\Delta K=K_{f}-K_{0}\approx\sin(\omega\Delta t)\) or \(\Delta K\approx sK_{0}\Delta t\), where \(s=\mathrm{d}K/\mathrm{d}T\) is the slope of energy versus time. Schematic principle of streaking of photoelectrons generated by X-ray pulse in a THz field is shown in Figure 2.  
The restrictions on the method are that the X-ray temporal pulse length has to be shorter than the linear region around \(t=0\). For short pulses, as presented here, one could use 20-40 THz frequencies fairly easily, and if the jitter between the arrival time of the streaking field and the X-ray is small enough, one could even use 1600 nm wavelengths for the streaking effect. Such measurements have been performed already in laser laboratories [29], where the \(K_{0}\) from the HHG photoionization was fairly small and the accuracy was sub-100 as. With the schemes available for dealing with the likely arrival of a time jitter problem at XFELs [4; 27], the path to sub-fs precision in hard X-ray pulse duration measurements is clear.