## 3 Linear Streaking Measurements at Free-Electron Lasers
### Experimental implementation
In the second step of the evaluation process one uses this vector potential as a calibration factor to convert the measured spectral width of each streaked photoelectron wave packet into an upper limit for the duration of the corresponding X-ray pulse. To illustrate the principle of evaluation, have a look at an example from the data, sketched in blue in Figure 11a. In this case a streaked spectrum ranging from 915 eV to 945 eV was measured, corresponding to a width of the streaked spectrum of \(\delta_{\mathrm{s}}\) = 30 eV and a maximal energy shift in relation to the unstreaked photoline at 920 eV of \(\Delta\varepsilon\) = (945 \(-\) 920) eV = 25 eV. Following Ref. [99], one assumes the vector potential \(A_{\mathrm{NIR}}(t)\) of the streaking field to be approximately linear in time,  
\[A_{\mathrm{NIR}}(t)\approx-E_{\mathrm{peak}}\times\left(t-t_{\mathrm{peak}} \right), \tag{2}\]  
with a slope given by the peak electric field \(E_{\mathrm{peak}}=E(t_{\mathrm{peak}})\) of the half-cycle that the photoelectrons are overlapping with, which occurs at the moment \(t_{\mathrm{peak}}\). The period of the streaking laser was \(t_{\mathrm{period}}=8\) fs, setting the time interval for the vector potential to rise from a zero-crossing to a maximum (a quarter period) to 8 fs/4 = 2 fs. This value corresponds to the above calculated maximum shift \(\Delta\varepsilon\) = 25 eV in the energy domain. Now one can use the measured width of the streaked spectrum, \(\delta_{\mathrm{s}}\) = 30 eV, as a ruler for the pulse length, resulting in an upper limit for the X-ray pulse duration:  
\[\tau_{\mathrm{X-ray}}=\frac{\delta_{S}}{\Delta\varepsilon}\times\frac{t_{ \mathrm{period}}}{4}, \tag{3}\]  
which is (30/25) \(\times\) (8/4) = 2.4 fs in this case, represented by the blue X-ray pulse on the left in Figure 11a. There is, in addition, a residual ambiguity in the analysis method that has to be considered. Figure 11a shows schematically a second XFEL pulse (green, in the middle) with a different duration that nevertheless leads to nearly the same measured width of the streaked spectrum. Since it was assumed that the X-ray pulse partly overlaps with a local maximum of the vector potential, one cannot discern streaking effects caused by the right or the left edge of this vector potential curve. To get a correct upper limit for the possible X-ray pulse duration, one therefore must double the above calculated value to 4.8 fs. All pulse lengths between 2.4 fs and 4.8 fs are compatible with the measurement. Figure 11, panels (b)-(e), shows measured streaked photoelectron spectra generated by four different X-ray shots (solid blue curves) and their derived pulse duration upper limits (red bars on the right). From the maximum shifted peak in panel (b) a maximal energy shift \(\Delta\varepsilon\) = \(\mid-\)32 \(\mid\) eV is inferred, i.e., the laser half-cycle of 4 fs is mapped onto an energy range of 2 \(\times\)\(\Delta\varepsilon\) = 64 eV. The complete spectrum spans from \(-\)32 eV to 27 eV, setting the overall width to \(\delta_{\mathrm{s}}\) = 59 eV and corresponding to a pulse duration of 3.7 fs. In a similar manner one derives upper limits for the other pulse durations of 3.4 fs in panel (c), 2.8 fs in panel (d) and 2.6 fs in panel (e), respectively. Considering the ambiguity of the energy-to-time mapping, all these shots are in principle also compatible with the doubled pulse lengths. Owing to the statistical nature of the SASE generated XFEL pulses originating from noise, the X-ray pulse profiles differ greatly from shot to shot. For the first time, this variation in the intensity substructure of XFEL pulses can be shown directly in the time domain. In Figure 11b, an XFEL pulse consisting of only one single peak overlaps with the zero crossing of the vector potential and is broadened compared to the photoline.