by

$$\begin{aligned}
p'=p + A_1 \sin (\zeta + \phi_1) ~~, ~~~ \zeta'=\zeta + B_1 p'~,
\label{M1}\end{aligned}$$

where $A_1 = \Delta E_1/\sigma_E$, $\Delta E_1$ being the energy modulation imposed by the seed laser, $\phi_1=\phi_1(\zeta)$ is the phase of the laser pulse, which depends on the time $\zeta$, and $B_1 = R_{56}^{(1)} \sigma_E \omega_l/(E_0 c)$, $R_{56}^{(1)}$ is the strength of the first chicane. Substituting Eq. (\[M1\]) into Eq. (\[f0new\]) one can obtain the distribution after the first modulator and dispersive section. The new phase space variables $(\zeta',p')$ will transform after the passage through the second modulator and dispersive section, to $(\zeta'',p'')$, which are given, in a similar way, by

$$\begin{aligned}
p''=p' + A_2 \sin (\zeta' + \phi_2)~~, ~~~ \zeta''=\zeta' + B_2
p''~, \label{M2}\end{aligned}$$

the subscript $'2'$ referring to the second stage[^2]. Using Eq. (\[f0new\])-(\[M2\]) one can obtain an explicit expression for the phase space distribution after the second stage, $f_2(\zeta'',
p'')$, which will not be reported here. In order to analyze the harmonic composition of the current density we first need to project the phase space distribution onto the real space-time coordinates by performing an integration along $p''$. This leads to the density distribution function $\rho$, that can be Fourier-analyzed further to 