$\omega_0$ and the quadratic term $\Gamma_i t^2$. The parameter $\Gamma_i$ is given by: $$\label{gamma}
\Gamma_i = \pm\frac{1}{4\sigma_t^2}\sqrt{4\sigma_t^2 \sigma_{\omega}^2-1},$$ $\sigma_t$ and $\sigma_{\omega}$ being the standard deviations of the temporal intensity profile and the spectrum, respectively. The sign of $\Gamma_i$ corresponds to the sign of the induced frequency dispersion. In the setup adopted at FERMI@Elettra, the frequency dispersion of the seed pulse may be controlled in three different (but eventually combined) ways: (1) by propagation of the seed through different materials; (2) by passage through a dedicated set of gratings; and (3) by increasing the seed intensity, which causes self-phase modulation [@Diels]. It is worth noting that in the presence of self-phase modulation, the chirp cannot be considered as being linear at or near the edges of the pulse. Moreover, the self-modulation process is associated with a spectral broadening. This broadening has been measured and taken into account in the present study. We also note that, for the experiments reported in this paper, the Gaussian shape of the pulse is preserved over the full range of seed energies.

The instantaneous frequency of the pulse is the derivative of the temporal phase (see Eq. \[Efield\_t\]): $$\label{frequency}
\omega(t) = \omega_0 + 2\Gamma_i t.$$ Positive 