as plate deformations are concerned, it can only be valid in the limit $\ell_{\rm C}\gg L$ which corresponds to a trivial specular description of the reflection process on the surfaces [@MaiaNetoPRA05]. For the general case, a description of non specular scattering process on mirrors is available for analyzing the connection between geometry and the Casimir effect [@MaiaNetoPRA05]. An expression for the Casimir energy between parallel mirrors with arbitrary surface profiles has been derived in [@LambrechtNJP06; @RodriguesPRA2007] $$\begin{aligned}
E=\hbar\int\limits_{0}^{\infty}\frac{{\rm d}\xi}{2\pi}{\rm Tr} \ {\rm ln}\left(1-\rm{R}_{1}\left(i\xi\right)e^{-K\left(i\xi\right)L}\rm{R}_{2}\left(i\xi\right)e^{-K\left(i\xi\right)L}\right)\  \  \label{formule}\end{aligned}$$ This expression is based on non-specular reflection matrices $\rm{R}_{1}$ and $\rm{R}_{2}$ associated to each mirror. While the operator $e^{-K\left(i\xi\right)L}$ corresponds to propagation of the field between the two mirrors, and is diagonal in the plane-wave basis with elements given by $K(i\xi) =\sqrt{{\bf k}^2+\xi^2 / c^{2}}$, the two matrices $\rm{R}_{1}$ and $\rm{R}_{2}$ are non-diagonal on plane-waves. This corresponds to a mixing of polarizations and wavevectors, due to non-specularity diffraction on the gratings formed by the profiles on the surfaces of the mirrors.

As it is reviewed below, this formula (\[formule\]) has been used to evaluate the effect of surface roughness [@MaiaNetoPRA05] or corrugations on the Casimir force [@RodriguesEPL2006; @RodriguesPRL2006]. Analytical expressions have been derived through a 