in the introduction, the relevance of the Casimir effect on nanosystems calls for a precise understanding not only of the influence of material optical properties on the Casimir force, but also of the influence of geometrical parameters, such as the thickness of the coatings [@IannuzziPNAS2004; @LisantiPNAS2005] or the thickness of the mirrors themselves. In this context, structures made of silicon, the reference material used in nano-fabrication processes, are particularly interesting to study [@LambrechtEPL2007; @Chen07].

The reflection amplitude corresponding to a slab of finite thickness $D$ is different from the bulk expression and is given through a Fabry-Perot formula $$\begin{aligned}
r_{\bf k}^{p}[i\xi]_{\rm slab}&=&r_{\bf k}^{p}[i\xi]\frac{1-e^{-2\delta}}{1-(r_{\bf k}^{p}[i\xi])^{2}e^{-2\delta}}  \ \ , \nonumber  \\ 
\delta &=&\frac{D}{c}\sqrt{\xi^{2}(\varepsilon[i\xi]-1)+c^{2}\kappa^{2}}.   \label{Rslab}\end{aligned}$$ $r_{\bf k}^{p}[i\xi]$ is the bulk reflection amplitude given by (\[Fresnel\]). Using these reflection amplitudes for calculating the Casimir force between two Si slabs, interesting behaviours have been noted [@LambrechtEPL2007] which differ from the situation of metallic mirrors. In particular, it was shown that the material thickness has a stronger influence on the Casimir force for Si slabs than for Au slabs. For Si, the force decreases as soon as the slab separation $L$ is larger than the slab thickness $D$, as seen on Fig.(\[fig:2\]).

In contrast to metals 