external seed chirp was created both during the propagation through different optical components such as lenses and windows, and by self-phase modulation due to high intensity. We typically measured a seed bandwidth $\sigma_{\lambda} \approx 0.45\,$nm and a duration $\sigma_t \approx 90\,$fs (determined by cross-correlating the seed pulse with an infrared pulse, extracted from the oscillator of the Ti:Sapphire laser chain, whose FWHM duration $\approx 80\,$fs). This gives $\Gamma_i \approx 6.2 \cdot 10^{-5}\,$fs$^{-2}$, leading to the retrieval of the theoretical separation between the two sub-pulses, defined in Section \[description\]. From Eqs. \[dw\] and \[dwn\], we obtain: $$\label{dt}
\Delta t = \frac{\Delta\omega}{2\Gamma_i} = \frac{n\pi c}{\Gamma_i\lambda^2} \cdot \Delta\lambda _n,$$ where $c$ is the speed of light. For an harmonic order $n=6$, $\Delta t \approx 230\,$fs (equal to $2.5 \sigma_t$). The accuracy of this result will in principle be affected by the intrinsic chirp induced in the radiation during the FEL process [@Wu] and by any quadratic energy chirp on the electron bunch. However, the size of the former is usually quite small relative to other chirps and the contribution of the latter is minor for our conditions, as will be confirmed in Section \[simulations\].

Up to now, we mainly focused on the effect 