the spectral quality of the FEL emission and the spectro-temporal pulse-splitting phenomenon.

  -------------------- ----------------------- ----------------------
     Electron beam           Mean energy          $\approx 1\,$GeV
                            Peak current          $\approx 300\,$A
                           Duration (FWHM)           $1.5\,$ps
          Seed           Central wavelength          $261\,$nm
                          Temporal profile            Gaussian
                           Duration (FWHM)       $ \approx 200\,$fs
                               Energy             $10 - 150\,\mu$J
                        Transverse size (RMS)   $\approx 350 \,\mu$m
       Modulator               Period                $100\,$mm
                          Number of periods             $32$
   Dispersive section         $R_{56} $           $20 - 50\,\mu$m
       Radiators               Number                    2
                         Harmonic order $n$              6
                               Period                 $55\,$mm
                          Number of periods             $44$
  -------------------- ----------------------- ----------------------

  : Parameters of FERMI@Elettra relevant for the experiments reported in this paper. The seed transverse size is defined as the standard deviation of the transverse intensity distribution measured at the modulator entrance. The $R_{56}$ parameter is the momentum compaction factor characterizing the strength of the dispersive section [@YuPRA]. The undulators were set so as to generate circularly polarized FEL radiation, with a typical energy/pulse of $20\,\mu$J.[]{data-label="Table_FEL1_parameters"}

As shown in Fig. \[Figure\_double\_peak\_principle\]a, under conditions of moderate seed power and a temporally flat (i.e, both in energy and current) electron beam, the spectro-temporal properties of CHG mimic those of the seed [@PRLSR; @NaturePhot]: starting with a Gaussian seed pulse with a given duration and wavelength $\lambda$, one ends up with an output 