## 2 Sideband Method for X-ray Pulse Characterization
The analysis of electron spectra for many individual pulses provides direct information about the temporal stability and can be used as inherent time marker for time-resolved pump/probe experiments. The actual sideband amplitude depends on several parameters. First, good spatial overlap is crucial for the sideband generation. In addition, a meaningful analysis of sideband spectra is only possible when the X-ray focus is significantly smaller compared to the NIR focal spot size. Under such conditions the electrons are created only in the maximal field of the NIR pulse and are all exposed to (almost) the same NIR field strength. Nonetheless, pointing fluctuations of either of the two beams present a severe limitation for the practical use of the method. Second, temporal overlap has to be assured, since the sidebands correspond to free-free transitions in the ionization continuum and both pulses have to interact with the electronic state at the same time. When the intensity conditions are such that a single sideband is generated, the interaction is determined by one X-ray and one NIR photon yielding a linear dependence on the relative overlap of the two pulses for the resulting sideband amplitude [78]. However, at high NIR intensities more than one sideband is present, implying the involvement of several NIR photons in the sideband formation process at the same time. This leads to a nonlinear response of the observed electron spectra, with a strong dependence on the intensity of the driving NIR laser. Thus, the sideband amplitudes are no longer  
Figure 3: Single-shot sideband spectra. (**a**) Single-shot electron spectra in the region of the Ne 2p\({}^{-1}\) photoemission line sorted with respect to different NIR intensities (reprinted with permission from [72], Copyright IOP Publishing, 2010). (**b**) Single-shot electron spectra in the region of the Xe 5p\({}^{-1}\) photoemission line and the high-energy sidebands, taken at the nominal temporal overlap. Due to the relative arrival time jitter the actual timing varies by more than the pulse durations. The sidebands are used to get information about the actual degree of temporal (and spatial) overlap. In addition, a schematic of the temporal overlap between FEL and NIR pulses is shown (reprinted with permission from [75], Copyright AIP Publishing, 2007). linearly connected to the temporal overlap. Therefore, quantitative information about the X-ray pulse duration \(\tau_{\rm X-ray}\) can only be extracted by comparing the sideband features to a theoretical model. The experimental line profiles can be fitted with a simplified analytical model based on the so-called soft-photon approximation (SPA), allowing us to simulate the sideband spectrum [79]. In general, the SPA gives excellent agreement with experimental data and can match the more elaborate non-perturbative methods such as the time-dependent Schrodinger equation (TDSE) when the electron kinetic energy is large compared to the photon energy of the dressing field [79; 80; 51; 81]. The simulated sideband spectra are defined by four parameters: duration, \(\tau_{\rm NIR}\), and intensity \(I_{\rm NIR}\), of the NIR laser pulse as well as duration (and shape) \(\tau_{\rm X-ray}\) of the FEL pulse and, finally, the relative temporal delay between the two pulses \(\delta\). The NIR pulse width and intensity can be determined with good accuracy before the experiment leaving \(\tau_{\rm X-ray}\) and \(\delta\) as parameters to be determined during the measurement [51]. The sideband spectra can be analyzed in two ways. One approach comprises a statistical analysis of the single shot spectra while setting the relative delay to a fixed value, namely at nominal maximum overlap of X-ray and NIR pulses. Due to the inherent arrival time fluctuations the sideband amplitudes and the number of sidebands vary significantly from shot to shot. However, looking at the frequency distribution of the sideband amplitudes (histogram), information about the average temporal jitter and the X-ray pulse duration can be extracted [51; 75].