to a rich variety of behaviours in the Casimir physics [@Balian7778; @Plunien86; @Balian0304]. Recent advances make it possible to explore this interplay, both from experimental and theoretical point of views. This also offers new possibilities for tailoring the Casimir force through specific designs [@EmigEPL03].

Force and energy evaluations between non planar mirrors are commonly obtained using the so-called proximity-force approximation (PFA) [@Derjaguin68; @Langbein71]. This approximation amounts to an averaging of plane-plane contributions over the distribution of local interplate separations defined by the chosen geometry. For the energy, the PFA leads to $$\begin{aligned}
E_{\rm PFA} = \int\frac{{\rm d}^{2}{\bf r}}{A}E_{\rm PP}\left(\ell)\right)  \ , \ 
\ell \equiv L-h_{1}({\bf r})-h_{2}({\bf r}), \label{PFArough}\end{aligned}$$ with $h_{1}({\bf r})$ and $h_{2}({\bf r})$ the surface profiles of each mirrors. Such profiles can be described by their spectra evaluated over the surface $A$ of the mirrors $$\begin{aligned}
\int \frac{{\rm d}^{2} {\bf r}}{A}h_{i}({\bf r})h_{j}({\bf r})=\int\frac{{\rm d}^{2}{\bf k}}{(2\pi)^{2}}h_{i}[{\bf k}]h_{j}[-{\bf k}] \ , \ i,j=1,2   \end{aligned}$$ with $h_{i}[{\bf k}]$ the Fourier transform of $h_{i}({\bf r})$, and by the associated correlation lengths $\ell_{\rm C}$. When they are smaller than the other length scales, the amplitudes of deformations can be considered as perturbations. A second order expansion in the profiles can thus be performed leading 