case when EEHG scheme uses an electron bunch with Gaussian temporal profile and a seed laser pulse with flat-top profile in time across the duration of the electron bunch. As demonstrated in e.g. [@EETOL3], the generated bunching is not sensitive to the peak current. Therefore, EEHG can operate with a nonuniform electron bunch profile. In the next paragraph we will demonstrate that in any case, due to non-linear (self-phasing) effects in the Ti:Sa laser system and in the post-laser optics system, the seed laser must have flat-top profile in time with very little temporal variation. Therefore, the model of a seed pulse with flat-top profile and of an electron bunch with Gaussian profile is consistent with the EEHG scheme. Now, if the phase of the seed laser is chirped, the microbunching chirp is simply multiplied by the frequency multiplication factor $N$. This can be seen by looking at the harmonic contents of the current density found in Eq. (\[ffft6\]). That expression includes $\bar{f}_{\zeta}$, which in the case of $\phi_1=\phi_2 = \alpha \zeta^2/(2 \sigma_\zeta^2)$ is given by (see Eq. (\[fbarz\])):

$$\begin{aligned}
&&\bar{f}_{\zeta}(m-1-\Omega)=\int_{-\infty}^{\infty} d\zeta
f_{0\zeta}(\zeta) \exp[i (m-1-\Omega)\zeta] \exp[-i  \phi_1 + i m
\phi_2]\cr && =
\frac{1}{\sqrt{2\pi}\sigma_\zeta}\int_{-\infty}^{\infty} d\zeta
\exp[i (m-1-\Omega)\zeta]\exp\left[-\frac{\zeta^2}{2
\sigma_\zeta^2}\right]\exp\left[i (m-1) \frac{ 