## 3 Linear Streaking Measurements at Free-Electron Lasers
### Experimental implementation
This is evident when looking at the spectra presented in Figure 10a,d, which show 50 simulated (a) and measured (d) streaked spectra obtained with X-ray pulses stretched to around 25 fs (by changing the compression of the electron bunch) and a streaking field with a wavelength of 2 \(\upmu\)m. For the X-ray pulses at optimized compression and inserting the spoiler [32], one can see in Figure 10b,e that when employing the same 2 \(\upmu\)m streaking laser, the situation is globally unchanged, whereas when using a 2.4 \(\upmu\)m wavelength NIR laser, one observes the apparition of a clear streaking regime [21,66] (Figure 10c,f). This is clear evidence that the XFEL pulses under these experimental conditions are at least shorter than 8 fs (one period at 2.4 \(\upmu\)m wavelength) and one can apply the streaking concept for the X-ray pulse duration acquisition. **Single-shot X-ray pulse duration measurement**  
As described above, one of the major challenges for these types of correlation measurements is the arrival time jitter of each single XFEL pulse with respect to the NIR laser (see Figure 9). Due to the statistical nature of the SASE process the actual pulse shape and thus the duration of consecutive XFEL pulses also varies from shot to shot, necessitating a single-shot measurement technique for its determination. That means that one has to adopt a different method to extract the X-ray pulse duration from the measured spectra than what is conventionally used in streaking experiments with precisely synchronized gas-phase high harmonic sources. Figure 10: Streaked X-ray photoelectron spectra (top: Simulation, bottom: Experiment). Six color-coded intensity plots for photoelectron spectra of 50 consecutive shots, for two different X-ray pulse durations (\(\tau_{\mathrm{X-ray}}\approx\) 25 fs and \(\tau_{\mathrm{X-ray}}\approx\) 4 fs) streaked by optical lasers with two different central wavelengths (\(\lambda\) = 2 \(\upmu\)m and \(\lambda\) = 2.4 \(\upmu\)m). Panels **(a–c)** are simulations, while **(d–f)** are constructed from the corresponding measurements taken at the LCLS during our experiment. The long 25 fs X-ray pulses (panels **(a,d)**) exhibit nearly no spectral shift but rather uniform smearing of the photoelectron (PE) spectra, revealing that for these long pulses the measurement method breaks down. A similar behavior can also be observed for the short X-ray pulses (<5 fs) when streaked with the 2 \(\upmu\)m NIR field, shown in the middle in panels **(b,e)**: the ratio of the period of one optical cycle at this wavelength to the X-ray pulse duration is still not optimal, resulting in a high probability to find the spectral center near the unstreaked value. In contrast, for the 2.4 \(\upmu\)m streaking field, right panels **(c,f)**, the whole spectrum is hugely shifted away from the mean, as is expected when the PEs ‘see’ only part of the NIR vector potential, a clear demonstration of the onset of the streaking regime. Note that the overall shift of the central energy in panels **(e,f)** from 930 eV to 920 eV is due to an accordingly lower X-ray photon energy (1790 eV instead of 1800 eV) for this measurement and is not a streaking effect. (Figure from [96]). As the relative phase between the X-ray pulse and the streaking field is not known a priori, one has, in a first step, to determine the value of the streaking vector potential for each single XFEL pulse. This fact is critical for the analysis and restricts the characterization to the determination of upper limits for the X-ray pulse durations. The peak of each single-shot spectrum that is pushed farthest away from the unstreaked center is identified and is assumed to stem from an overlap of the X-ray pulse with a local maximum of the vector potential. This sets an experimental limit for the local electric field strength of the streaking laser for each specific X-ray shot and thus for the gradient of the vector potential, which will be underestimated on average.