numerical codes [@perseo]. Figure \[Figure\_fork\_seed\_exp\_vs\_Perseo\] shows a comparison of the spectral pulse splitting between the experimental data and 1D Perseo code simulations. As it can be seen, the agreement is very satisfactory. Note that in the simulations, the relative power before splitting is slightly higher than the one measured experimentally.

Let us now focus on one single measurement. Figure \[Figure\_output\_Perseo\] shows the predicted spectrum and temporal profiles, obtained for simulation parameters that allow a good reproduction of the dashed-line spectrum reported in Fig. \[Figure\_R56\_experiment\]. The simulations have been carried out both for an ideal (i.e., flat in current and in energy) electron beam (dashed line), and the “real” electron-beam energy distribution shown in Fig. \[Figure\_ebeam\_measurement\] (solid line). As shown in the left panel of Fig. \[Figure\_output\_Perseo\], the positions of the spectral peaks are similar for the two cases: a slight reduction of the pulse separation on the order of $0.01\,$nm can be observed in the “real" case. Such a small contribution may be neglected in our conditions and, in any case, the double peak formation is not affected. Moreover, if we calculate using Eq. \[dt\] the theoretical separation between the two sub-pulses in the spectrum represented in Fig. \[Figure\_R56\_experiment\], we 