reduced, e.g. by tuning the chirp of the seed laser. Since the seed and thus the FEL pulses are chirped, one could have expected a significant negative impact on the time-bandwidth product of each sub-pulse. However, this is not the case since each peak is quite narrow, and therefore barely sees the phase curvature of the seed electric field. Moreover, this also indicates that the phase of FEL pulses does not present high-order distortions.

Before concluding, let us consider a situation where the quadratic chirp of the electron beam is no longer negligible. We carried out simulations with same parameters as before for several values of quadratic chirp, but without any linear component (i.e., $\chi_1=0\,$MeV/ps$^2$). In Fig. \[Figure\_quadratic\_energy\_chirp\_Perseo\], we see that the value of $\chi_2$ has almost no effect on the temporal shape of the FEL emission but that, as already stated in Section \[description\], the individual bandwidths and separation of the two spectral peaks are directly affected. Consequently, a quadratic energy chirp of the electron beam acts on the spectrum in a very similar way as does a linear chirp of the seed laser, and thus both may be used individually or in conjunction to modulate the spectral shape 