# Opportunities for Two-Color Experiments in the Soft X-ray Regime at the European XFEL
## 4 Numerical Simulations for Two-Color Generation and Transport
### Generation and Transport of fs-Long Two-Color Pulses in the Low-Charge Mode of Operation
The final step consists in performing wavefront propagation simulations through the SQS beamline. In order to efficiently simulate radiation propagation through the optical beamline, properties of the FEL radiation sources were analized by backpropagation of the simulated radiation distribution with a dedicated module of Ocelot [31]. The position of the source is defined as the point of the maximum photon density. A full FEL pulse propagation through the optical beamline is possible, but this procedure does not add particular insight to the final results. Therefore, in order to simplify the propagation analysis, here we model the source with a Gaussian beam of the same divergence. Figure 6: Radiation properties from U1 tuned at the fundamental photon energy of 630 eV: distribution of FEL shots with different energies per pulse (left subfigure), power along the FEL pulses (middle) and their spectra (right). The mean energy per pulse is 50 \(\upmu\)J (\(5\times 10^{11}\) photons) and the mean pulse duration is about 1.5 fs FWHM. We model the SQS beamline with two offset mirrors OM1 and OM2 and two Kirpatrick-Baez (KB) mirrors KB1 and KB2 as illustrated on Figure 8. For simplicity, in this example we chose the same incident angle of 9 mrad for all mirrors, though other angles are possible for the offset mirrors. The total reflectivity of four mirrors is 71% at 250 eV and 80% at 630 eV, assuming 1 nm RMS height errors for the mirror surfaces. The finite length of the KB focusing mirrors (80 cm) may be affecting the photon density on the sample via clipping the radiation pulse and increasing the Rayleigh length of the image. After passing through the entire beamline, the photon beam is focused at the sample position. One issue concerning the optimal focusing is related to the presence of two spatially separated sources of the two pulses: in fact, the first pulse originates from the last closed cells of U1, and the second pulse from the last cells of U2. In our case study, the distance between the two sources, labelled \(S1\) and \(S2\), amounts to 15 undulator segments, corresponding to a length of about 90 m. Therefore, only one of the two sources, either at 630 or at 250 eV, can be effectively focused at the sample position. Moreover, the two KB mirrors have different focal lengths. Therefore, if we choose to obtain an image of \(S2\) in both transverse planes, S1 would be imaged at different positions in the horizontal and vertical planes. In other words, if we decide to tune the KB mirrors to image one of the two sources at one particular position, the image of the other will not only appear shifted in space, but will also be a subject to astigmatism. Figure 8: Optical elements relevant to SQS and their distances from the end of the SASE3 undulator. Four indicated optical elements are critical for beam transport and were simulated. Reprinted from [32]. Figure 7: Radiation properties from U2 tuned at the fundamental photon energy of 250 eV. Notation the same as on Figure 6. The mean energy per pulse is 70 \(\upmu\)J (\(1.7\times 10^{12}\) photons) and the mean pulse duration is about 3.3 fs FWHM. In order to solve this problem we suggest to re-image on the sample an intermediate, imaginary source located between \(S1\) and \(S2\), labelled \(\tilde{S}\), as in Figure 9 by properly tuning the KB mirror system. In this case, the image of \(\tilde{S}\) appears at the position denoted \(\tilde{I}\). This yields a good compromise in terms of beam sizes at the sample, Figure 10. Based on our simulations, the resulting photon fluxes would be sufficient to conduct the experiment proposed in Section 3.