# FEL Pulse Duration Evolution along Undulators at FLASH
## 4 Discussion
[19] that the fluctuations in pulse duration and pulse energy are inversely proportional to the square root of the number of modes, as is also indicated by the scaling in Figure 1a. Therefore, we expect less fluctuations for longer pulses (i.e., larger numbers of modes), as can be seen in Figure 4a,c. The 1D simulation was calculated for three different mode numbers, while the 3D simulation was only conducted for 5.1 modes, due to the larger computational load. Indeed, the 3D simulation results lay between the 1D simulation for four and eight numbers of modes for the pulse duration fluctuations and in the exponential gain regime for the energy fluctuations. However, itis important to note that the 3D simulation predicts significantly less energy fluctuations as the 1D simulations in the saturation regime, which we could not verify experimentally due to technical reasons. Since the measured XUV pulse durations were in the range of \(\tau_{el}\) = 15-20, as shown in Table 1, they have to be compared mainly to the 1D simulation with \(\tilde{\tau}_{el}=16\). The experimental values show essentially the same trend as the simulation calculated with, however, significantly higher fluctuations than the simulation predicts for the inherent SASE fluctuations. This observation was already described in [19] and can be attributed to additional fluctuations, including the measurement uncertainties, fluctuations in the energy gain, and compression of the electron bunches (due to acceleration field phase instability). An interesting property for experiments is the radiation power (pulse energy divided by the pulse duration). In Figure 5a, we compare how the power scales with \(z\). It turns out that the achievable power of the FEL pulses increases for all experimental settings continuously along the undulators, even in saturation without a hint for a local maximum. The chirped 3D simulation predicts the same behavior and shows a very good agreement with the experimental data. The nonchirped 3D simulation exhibits a similar trend in the linear regime and a small local maximum at the saturation point, which is not reproduced by the experimental data. In contrast, in the 1D simulation, the power reaches a distinct maximum saturation and grows again only for deeply saturated operation. By comparing  
Figure 4: Shown are the fluctuations as functions of the undulator coordinate \(z\) for (**a**) the pulse duration and (**c**) the pulse energy. For each setting, several thousand FEL pulses were measured, leading to a broad, Gaussian distribution as shown in (**b**) the pulse duration and (**d**) the pulse energy. The example histograms in (**b**,**d**) are corresponding to the data point shown with the dashed circle. The values for the fluctuations shown are the rms values of the measured pulse energies and pulse durations which are equivalent to the \(1/e\) width of the histogram (for the nearly Gaussian distribution). In addition, the theoretically determined values for the fluctuations (only taking the SASE-induced effects into account) are plotted for the chirped and nonchirped 3D simulation and 1D simulation for the cases of \(\tau_{el}\) = 4, 8, and 16. The fact that the measured fluctuations are larger than the theoretically expected values (only for SASE) can be attributed to measurement uncertainty and additional technical fluctuations in the accelerator, as was discussed in Ref. [19]. the results from the simulation methods, the chirped 3D simulation shows a much better agreement with the experimental data and should be used as a basis for the future prediction of FLASH radiation parameters. Looking at Figure 5, one sees a continuous power increase along the undulator length. However, as pointed out in Refs. [14; 15; 35], considering the spatial and spectral properties of the FEL beam, it turns out that the maximum brilliance of the radiation is rather achieved in the very beginning of the nonlinear regime.