of variations of the characteristics of output radiation when the FEL is seeded by a laser with non-ideal properties, including effects such as linear and nonlinear frequency chirp and wavefront distortion.

A description of the impact of phase chirp of the EEHG (or HGHG) FEL output can be made without numerical simulation codes. In fact, as is well-known, if the phase of the seed laser is chirped, the chirp is simply multiplied by the frequency multiplication factor $N$. In this case, the method used to describe the output field perturbation is independent of the specific kind of harmonic generation technique: it only depends on the frequency multiplication factor $N$. It follows that both EEHG and HGHG FELs starting from a Ti:Sa laser with a wavelength around $800$ nm can produce transform-limited radiation down to wavelengths of $1$ nm only when the relative discrepancy of the time-bandwidth product of the compressed $800$ nm laser pulse from the ideal, transform-limited performance is no more than one in a million, roughly corresponding to the squared of the harmonic number. However, pulses from a commercially available Ti:Sapphire chirped pulse amplifiers are usually limited in such discrepancy to around $1\%$, due to non-ideal effects. Therefore, 