are defined with respect to reference mirror planes separated by the mean distance $L$. We assume that profiles have zero averages and show no cross-correlations. We also suppose that the area $A$ of each plate is large enough to include many correlation areas ($A\gg \ell_{\rm C}^{2}$), so that surface averages are identical to statistical averages. Up to second order in the profiles, the correction to the Casimir energy may thus be written as follows $$\begin{aligned}
\delta E_{\rm PP}=\int\frac{{\rm d}^{2}{\bf k}}{(2\pi)^{2}}G_{\rm rough}[{\bf k}]\sigma[{\bf k}].  \label{beyondPFArug}\end{aligned}$$ Here $\sigma[{\bf k}]$ corresponding to the roughness spectrum added over the two plates. $G_{\rm rough}[{\bf k}]$ is a spectral sensitivity to roughness of the Casimir energy. Due to cylindrical symmetry with respect to rotations in the transverse plane, it only depends on $k=|{\bf k}|$. This dependence reveals that the roughness correction does not only depend on the root-mean-square (rms) roughness, but also on the spectral distribution of the roughness. Fig.(\[fig:3\]) displays $G_{\rm rough}[k]$ normalized by $E_{\rm PP}$ as it has been calculated for Au-covered mirrors and for various interplate distances.

The rich behaviours of $G_{\rm rough}[k]$ as a function of the length scales is discussed in [@MaiaNetoEPL05].

What we want to stress here is that this 