\alpha \zeta^2}{2
\tau_\zeta^2}\right]\cr && \label{fbarz2}\end{aligned}$$

The last phase factor under integral shows that the laser phase is indeed multiplied by $N=m-1$.

We will define the frequency chirp in the seed laser pulse only across the target duration of the electron bunch, and use the same time normalization as for the beam microbunching. The complex field envelope of a laser pulse with stepped profile and linear frequency chirp is given by

$$\begin{aligned}
E(\zeta) = E_0 \exp\left[i \frac{\alpha \zeta^2}{2
\sigma_\zeta^2}\right]~, \label{Et}\end{aligned}$$

where $E_0$ is a constant. As discussed above, the frequency multiplication yields a complex “microbunching” envelope with carrier frequency $\omega_0 = (m-1) \omega_l$

$$\begin{aligned}
a(\zeta) = a_0 \exp\left[- \frac{\zeta^2}{2 \sigma_\zeta^2}\right]
\exp\left[i \frac{\alpha_m \zeta^2}{2 \sigma_\zeta^2}\right]
\label{Et}\end{aligned}$$

where $\rho_0$ is a constant, and $\alpha_m = N\alpha$ is the microbunching chirp parameter. Note that what we loosely defined as “microbunching” is, more formally, the slowly-varying amplitude of the electron density modulation with carrier frequency $\Omega=N$. It follows from the previous analysis that the EEHG scheme can produce nearly transform-limited microbunching only under the restriction $\alpha_m \lesssim 1$, meaning that the laser chirp parameter must obey $\alpha \lesssim 1/N$. The EEHG seed laser is assumed to be a Ti:Sa laser. The actual seed laser beam 