powerful coherent light source (e.g., a low-order harmonic of a Ti:Sapphire laser). This interaction coherently modulates the electron energy which then leads to a longitudinally-regular charge distribution after the electrons pass through a chromatically-dispersive magnet. Such a spatial modulation, called bunching, has nonzero spectral components at both the seed wavelength and at its higher harmonics. After the dispersive section, electron beam enters one or more additional undulators (called the radiators), whose normalized magnetic deflection parameters $K$ are tuned for fundamental FEL resonance to occur at an integral harmonic of the seed wavelength: $$\label{resonance}
\lambda_n=\frac{\lambda_u}{2 n \gamma^2}\, (1+\frac{K^2}{2}).$$ Here $\lambda_u$ is the undulator period, $n$ the harmonic order and $\gamma$ is the relativistic Lorentz factor characterizing the mean energy of the electron beam.

The resultant FEL radiation, which is highly temporally coherent due to the periodic bunching, can undergo quadratic (low-gain regime) or exponential (high-gain regime) amplification. For a sufficiently long radiator, the FEL process will reach saturation. The main parameters of the FERMI@Elettra configuration relevant for our experimental studies are listed in Table \[Table\_FEL1\_parameters\] and its caption. One should note that on FERMI@Elettra the seed pulses are much shorter than the electron beam. This is a critical point both for 