are also affected by the quality of the electron beam. In particular, effects due to an energy chirp (i.e., the mean electron energy varies with longitudinal position) can couple with those from the frequency chirp carried by the seed laser. This interplay may have an impact on the process of formation of the two peaks, as well as on their spectral and temporal separations. In order to better explain the process of production of the double peak, we will now provide a description of the seed laser and electron beam chirps, with particular reference to the case of FERMI@Elettra.

Description of seed and electron beam chirps {#description}
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Seed pulse: frequency chirp {#seed-pulse-frequency-chirp .unnumbered}
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The seed pulse used at FERMI@Elettra is the third harmonic of a Ti:Sapphire laser; both the temporal and spectral profiles are well fit by a Gaussian dependence [@seedFERMI]. The presence of a linear frequency chirp can be modelled by a quadratic term in the phase of the laser electric field: $$\label{Efield_t}
E(t) = A(t) e^{i(\omega_0 t + \Gamma_i t^2)}.$$ Here A(t) stands for the temporal envelope while the exponential term carries the phase of the oscillations, including the rapidly-oscillating term at the central angular frequency 