# FEL Pulse Duration Evolution along Undulators at FLASH
## 4 Discussion  
We analyzed the average pulse energies and average pulse durations as well as their respective fluctuations as functions of the number of undulators contributing to the lasing process for six different wavelengths. As a basis, we used the 3D simulation based on the chirped electron bunch. In the first step, the number of undulators was scaled to the undulator coordinate \(z\) such that the onset of measurable SASE (\(>\)0.5 u) coincided with the \(z\) range in which the energy gain became visible in the simulated energy gain curve (\(z/z_{sat}\)\(\sim\)0.7). This determined the undulator axis scaling and already fixed the saturation length \(z_{sat}\). It has to be noted that we took into account that the first two undulator segments at FLASH2 hardly contributed to the lasing due to a slight misalignment, and thus for the experimental data we had to count the first two undulators as 0.5 undulators (i.e., subtracting 1.5 from the actual number of closed undulators). The simulation results were not affected by this technical issue. In the second step, the pulse energy was scaled such that the pulse energy at saturation length \(z=z_{sat}\) was defined as saturation pulse energy \(E_{sat}\) and all energies were normalized by the saturation energy. A similar normalization was applied for the pulse duration. Here, the measured values were scaled to the minimum pulse duration \(\tau_{min}\) at \(z/z_{sat}\)\(\sim\)0.8. It is important to note that the experimental data have been only rescaled and not fitted to the simulation result. Figure 3 shows the experimentally measured pulse duration (red spheres) as well as the 3D simulation including chirp (red line) and the 3D simulation without chirp (red dashed lines). The 3D model without chirp predicts a considerable pulse lengthening when the amplification process enters the nonlinear regime (similar to the 1D models shown in Figure 1), while the chirped simulation shows clearly less pulse lengthening. The experimental pulse duration data also do not indicate a strong lengthening of the XUV pulses in saturation and thus show a good agreement with the 3D simulation including chirp. For the pulse energy, we measured a steady increase in saturation and beyond. This behavior is again well represented by the 3D model including chirp, while the 3D model without chirp predicts less energy increase after the saturation point. The pulse energies as well as the pulse durations were stochastically fluctuating due to the SASE process and fluctuations induced by the acceleration process as well as measurement uncertainties in the actual THz streaking measurement [19], leading to a broad distribution. Example histograms of the fluctuations measured for a specific setting (fixed wavelength and number of undulators) are shown in Figure 4b,d. The rms values of the fluctuations are also used as \"error bars\" in Figure 3, not denoting the measurement uncertainty but showing the range of the measured values. The actual experimental uncertainty for single-shot measurements was determined to be about \(\pm\)20% for pulse duration measurements [31] and about \(\pm\)10% for pulse energy measurements [19]; thus, only a small fraction of the shown \"error bars\" of the averaged values resulted from measurement uncertainties. In Figure 4a,c, the normalized fluctuations are presented as functions of \(z\). The fluctuations are not constant along the amplification process and for fewer closed undulators the fluctuations are much larger. Both the 1D and 3D simulations show that for pulse duration and pulse energy the relative fluctuations are largest in the linear regime and are decreasing strongly in the range of \(z/z_{sat}\) 0.8 to 1 and only decrease slowly after saturation is reached. In contrast to the simulated values of the pulse energy and the pulse duration (Figure 1) which are essentially pulse-length-independent, the respective fluctuations are indeed pulse-duration-dependent. We know from Ref.