## 5 SASE Performance in the Sub-Angstrom Regime
A summary of the main results at 14 GeV is given in Table 1. Note in Table 1 that the impact of the intersection length between neighbouring undulator sections onto the gain length calculation has been subtracted, and that the numbers given for the power levels by the end of the undulator beamline are corresponding to the peak values of the Gaussian fits to the power distributions as shown in the insets of Figure 8.  
Figure 8: SASE performance in the sub-angström regime using electron bunches produced with a 56.3 MV/m cathode accelerating gradient at a total beam energy of 14 GeV. Pulse energies along the undulator beamline are shown for (**a**) 9.3 keV; (**b**) 20 keV; (**c**) 25 keV; The dashed lines show the exponential fits to the gain curves for estimating the gain lengths. Note that the impact of the intersection length of neighbouring undulator sections are already excluded as calculating the gain length. The inset plots show SASE power distributions at the end of the undulator beamline. The inset numbers indicate saturation pulse energies. For each case, 30 simulations are performed with random noise seeds. The statistical standard deviations of the achievable SASE intensities are up to 5%. Figure 9 shows the FEL performance in the sub-angstrom regime at 17.5 GeV final electron energy by plotting the pulse energy as a function of the longitudinal position along the undulators. The achievable SASE intensity is demonstrated as shortening the photon wavelength from 0.62 A down to 0.27 A. The obtained pulse energy by the end of the undulator beamline can reach 4.2 mJ and 1.7 mJ at photon energies of 20 keV and 30 keV, respectively. At 40 keV and 45 keV, still 0.5 mJ and 0.2 mJ SASE intensity can be achieved. The estimated gain length varies from about 5 m at 20 keV to 16 m at 45 keV. As shown, the gain curves for photon energies at/below 40 keV (black, green and blue curves) have stepped into the saturation regime within the total length of the existing undulators. For the case at 45 keV (red curve), a saturation behavior starts to appear at the end of the beamline; however, the gain curve in this case still tends to grow towards higher pulse energy. Note also that the choice of the accessible photon wavelength is set according to the undulator parameter K via practical gap adjustment for the existing undulators. A summary of the main results at 17.5 GeV is given in Table 2. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline \(\mathbf{E_{e}}\) & \(\mathbf{E_{p}}\) & \(\lambda_{p}\) & \(\mathbf{P}\) & \(\mathbf{I}\) & \(\mathbf{L_{g}}\) & \(\mathbf{E_{cath}}\) \\ \hline
**[GeV]** & **[keV]** & **[nm]** & **[W]** & **[m]** & **[m]** & **[MV/m]** \\ \hline
14 & 9.3 & 0.1333 & \(2.017\times 10^{11}\) & 4.38 & 3.9 & 56.3 \\ \hline
14 & 20 & 0.0620 & \(1.085\times 10^{11}\) & 2.31 & 6.3 & 56.3 \\ \hline
14 & 25 & 0.0496 & \(4.696\times 10^{10}\) & 0.97 & 7.4 & 56.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of SASE performance in the sub-angström regime using low-emittance beams produced with about 56.3 MV/m cathode accelerating gradient at an electron beam energy of 14 GeV. An undulator period of 4 cm is used as that of the existing undulators at the European XFEL. The terms \(\mathbf{E_{e}}\) and \(\mathbf{E_{p}}\) denote electron beam energy and photon energy, respectively. The symbol \(\lambda_{p}\) stands for the resonant wavelength of the photon beam. The notation P, I, \(\mathbf{L_{g}}\) and \(\mathbf{E_{cath}}\) represent saturation power level, photon intensity, gain length and the cathode accelerating gradient, respectively. Figure 9: FEL performance in the sub-angström regime with optimized electron bunches produced at a cathode accelerating gradient of 56.3 MV/m at a final beam energy of 17.5 GeV. Dashed lines correspond to exponential fits to the gain curves. The inset numbers show the obtained pulse energy by the end of the undulator beamline. For each case, 30 simulations are performed with random noise seeds.