can be compensated by placing an equal magnitude but negative chirp on the seed pulse [@Graves]. In the context of the subject matter of this paper, namely the formation and control of a double spectro-temporal peak, a quadratic electron beam energy chirp can either enhance or reduce the spectral offset between the two peaks and thus must be carefully taken into account.

A simple linear and quadratic energy chirp can be modelled as: $$\label{energy}
E(t) = E_0 + \chi_1 t + \chi_2 t^2.$$ The exact amplitudes of the $\chi_{1}$ and $\chi_{2}$ coefficients at entrance to the FEL depend upon various propagation effects as the electron beam is transported through the linear accelerator (linac). First of all, the radio-frequency (RF) accelerating fields [@Wangler] of the successive linac sections (generally phased at or very close to crest) induce a negative quadratic chirp. Second, in order to increase the electron beam current (and thus enhance the FEL emission), the electron bunch is time-compressed. On FERMI@Elettra, this is done by first imposing a positive linear chirp component via phasing of two upstream linac sections off crest, and then by sending the electron beam through a magnetic chicane that converts the linear energy chirp into 