## 3 Echo Enabled Harmonic Generation Seeding

Table 3: Nominal electron beam parameters for seeding operations at different spectral regions. at such a high harmonic can be achieved. Moreover, recent experiments have shown EEHG bunching up to the 100 th harmonic [81]. However, continuous EEHG operation of a seeded FEL at such a high harmonic has not yet been demonstrated. For this reason, detailed FEL simulations have been carried out for the case of the shortest anticipated wavelength (4 nm). For the presented setup in the last section, an optimized EEHG at 4 nm with an ideal beam can produce more than 5% initial bunching at the radiator entrance. Such a level of bunching is high enough to overcome the beam shot noise and allow a proper FEL amplification of the coherent signal along the full length of the radiator (Figure 4). With the possibility of using up to 11 undulators, FEL saturation power can be exceeded with the proper tapering of the undulator parameter K. Currently, simulations with ideal beams show that a peak power with optimal tapering of more than 2 GW can be extracted. The use of a very long radiator (\(\sim\)10 gain lengths) with a well bunched beam requires a proper undulator tapering to exceed the FEL saturation power (Figure 4b) while maintaining a well-defined and narrow frequency spectrum (Figure 4d). However, the extra power extracted from the electron beam after saturation (at \(\sim\)17 m) comes with a slightly increased FEL pulse length with respect to the optimal case (Figure 4c). From ongoing studies, it is also evident that a different optimization of the tapering could be exploited to emphasize the leading super radiant spike and generate very short (a few fs) pulses [85]. Expected FEL parameters for the upgraded FEL confirmed by Genesis [84] FEL simulations are reported in Table 6. Results based on Genesis [84] simulations do not account for the possible sources of bunching deterioration occurring before the FEL process starts. Figure 4: Genesis [83; 84] simulation of (**a**) the FEL power amplification of the EEHG bunching in the FLASH1 radiator (grey and white areas represent active undulator and drift, respectively); (**b**) temporal and (**d**) spectral profiles of the FEL at the exit of the radiator. (**c**) Evolution of the FEL temporal profile (normalized) along the radiator length. Only negligible modulations of the electron beam phase space are expected based on recent micro bunching studies along the FLASH2020+ linac [86]. However, previous works have shown that EEHG bunching at very high harmonic can be smeared by intra-beam scattering, incoherent synchrotron radiation, and coherent synchrotron radiation occurring in between the first modulator and the radiator [87; 88; 89]. In contrast to other schemes [88], in our setup a second seed laser is used that is much shorter than the electron beam and the first seed laser beam. Consequently, the sensitivity to these effects is significantly reduced and is expected not to affect the bunching content and coherent properties dramatically [90]. Essential studies of the possible parameter fluctuations have started considering the high sensitivity of the final FEL performance on the fine adjustment of the tapering along the radiator beamline. At longer wavelengths, EEHG can easily generate bunching as high as 10% or more. Consequently, parameters need to be adjusted to keep the bunching at a 5% level to avoid quick over-bunching in the long radiator. Moreover, since the gain length is significantly shorter at the long-wavelength (\(\sim\)1.2 m at 10 nm), FEL amplification develops much faster in the radiator and makes the undulator tapering crucial [91]. For the 10 nm case, Genesis simulations show that FEL saturation is reached in approximately four radiator sections (Figure 5). Optimization of post saturation FEL dynamics requires an accurate adjustment of the undulator tapering. Otherwise, the pulse profile departs from a Gaussian profile in both the temporal and the spectral domains.