## 4 Outlook and Conclusions  
In this paper, we investigated frequency mixing generation at the SASE3 line of the European XFEL. We implemented it by using the first part of SASE3 to generate two frequencies \(\omega_{1}\) and \(\omega_{2}\) in alternating K configuration, by subsequently obtaining a large bunching at \(\omega_{2}-\omega_{1}\) using a few non-resonant segments as dispersive elements and finally by generating radiation at the target frequency in a last, short part of SASE3 used as a radiator; see Figure 1. We demonstrated, for the first time, frequency mixing in the X-ray region, between 500 eV and 1100 eV. We showed that the bunching can be amplified in the radiator, and that six SASE3 segments dedicated to amplification allow reaching 4.5 mJ at 500-600 eV, see Table 1 and Figure 7. We studied the actual amplification bandwidth by means of K-parameter scans in the radiator; see Figure 4. Finally, we demonstrated easy tuneability over a range of 100 eV (see Figure 3) and showed that the frequency mixing mode of operation only needs a single XGM in order to be established and operated. Nevertheless, in some of the experiments we also acquired spectra, see Figures 5 and 6, and transverse profile distribution, see Figure 7 of the frequency mixed signal. It should be noted that together with the bunching at the difference frequency \(\omega_{2}-\omega_{1}\) one automatically generates bunching at the sum frequency \(\omega_{2}+\omega_{1}\) as well. The sum frequency generation is certainly an interesting phenomenon to consider, and it might be useful to reach shorter wavelengths than allowed by the baseline mode of operation of an FEL when a suitable radiator is available. However, in this study we limited ourselves to the difference frequency generation. Figure 6: Frequency resolved studies during one of the experiments, Exp. 2 (see Table 1). Upper plot: color-coded spectra as a function of the K parameter; inset: maximum of the spectrometer signal as a function of the K parameter. Lower plot: pulse energy as a function of the K parameter. Each point is found by averaging over 50 single SASE3 pulses (5 s at 10 Hz). This is not only of interest as a phenomenon pertaining FEL physics, but has the practical relevance of a method to provide low photon energies for operation at high electron energies. In fact, due to constraints posed by the parallel operation of three FEL lines (SASE1 and SASE2, enabling hard X-rays and SASE3, in the soft X-rays range), it is usually preferred to operate the accelerator at relatively high electron energies. To be specific, European XFEL usually operates with an electron energy of 14 GeV and a charge of 250 pC. For this electron energy, the lowest photon energy in the SASE3 undulator is 660 eV. This is limited by the maximum value of the undulator parameter \(K\simeq 9\), with a period of 6.8 cm. The generation of pulses with lower photon energy is possible, but only at lower electron energies, which poses issues in the planning of simultaneous experimental activities at SASE1, SASE2, and SASE3. Consider now the addition of a short radiator reaching lower photon energies than those achievable by the main FEL undulator at the fundamental and at a fixed electron energy. Using a such radiator, frequency mixing allows generating intense FEL pulses at lower photon energies than those permitted in standard SASE mode. This offers an interesting alternative for operating the facility at the same time for very soft and very hard x-ray radiation using a single electron energy. An Apple-X afterburner made of four segments with 22 periods each and a period length of 9 cm will soon be installed after the main SASE3 undulator. At 14 GeV, it will allow to be resonant down to 440 eV in circular or linear horizontal/vertical polarization mode. Therefore, one can use the frequency mixing mode to generate bunching between 440 eV and 660 eV in the main SASE3 undulator, as difference of allowed energies, and subsequently exploit the Apple-X afterburner to actually radiate at those photon energies.