to multiples $n+m$ of the laser frequency. In [@EEHG2] it is reported that, in order to maximize the modulus of the bunching factor one should impose $n=\pm 1$. This can be seen directly by inspecting the right hand side of Eq. (\[uno\]). In fact, for values of $\Omega$ near to $n+m$, the argument in the exponential function can be written as $p^2 ( B_1 n +  B_2 (n+m))^2/2$. When $n=-1$ and $m$ is positive and large for example, one sees that that $B_1 n$ is large and negative, while $B_2 m$ is large and positive. Therefore, $m$ can be chosen such that $ - B_1 + B_2 (m-1) \simeq
0$. This is guarantees remarkable up-frequency conversion efficiency, almost independently on the energy spread and constitutes one of the great advantages of the EEHG scheme. We will restrict our investigation to the case $n=-1$ and $m>0$, thus obtaining

$$\begin{aligned}
\bar{\rho}(\Omega,m-1) && = \bar{f}_{\zeta}(m-1-\Omega)
J_{-1}\left[-\Omega A_1(B_1+B_2)+m A_1 B_1\right]\cr && \times
J_m\left[-\Omega A_2 B_2\right] \exp[(\Omega (B_1+B_2)-m B_1)^2/2]
~.  \label{ffft6x}\end{aligned}$$

Note that if the laser phases would not depend on $\zeta$, which is not true in general, one could separately calculate

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