=
\exp\left[-\frac{\sigma_\zeta^2}{2}(m-1-\Omega)^2 \right]~.\cr &&
\label{depzeta2}\end{aligned}$$

In this case, the adiabatic approximation can be simply enforced imposing that $\sigma_\zeta \gg 1$. Finally, it should be noted that the initial electron density distribution and laser phases $\phi_1$ and $\phi_2$ are not only functions of $\zeta$, but also of the transverse position $\vec{r}$. It should be understood that the transverse direction can be factorized, which is a simplifying but not principal assumption, and that therefore, all the expressions above are considered valid at any fixed transverse position.

To conclude, let us consider our initial question, whether the general principle of the frequency multiplier chains is valid or not for EEHG. The answer is affirmative, and can be seen by inspecting Eq. (\[ffft6x\]) and Eq. (\[fbarz\]). In the case when $\phi_1=\phi_2$ such principle can be applied strictly. In case $\phi_1$ and $\phi_2$ differ, but are still of the same order of magnitude, we can conclude that, since $n=-1$ and $m$ is large, only $\phi_2$ is important and the principle is applicable with accuracy roughly $1/N$.

\[sec:tre\] Temporal quality of the seed laser beam
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Nowadays, high peak power laser systems are capable of producing very high intensities, thus fulfilling the requirements for many high 