corrugation profiles, induces a contrast with the case of roughness where the effect was associated with quadratic terms $h_{i=1,2}^{2}$. It implies that the evaluation of the lateral force only involves first-order non-specular amplitudes calculated on each mirror separately. The full calculation gives the second-order correction to the Casimir energy induced by the corrugations $$\begin{aligned}
\delta E_{\rm PP}=A\frac{a_{1}a_{2}}{2}\cos (k b) G_{\rm C}[k].
\label{secorder}\end{aligned}$$ The function $G_{\rm C}[{\bf k}]$ is given in [@RodriguesPRL2006] and does only depend on the modulus $k$ of ${\bf k}$. Here again, the PFA regime is recovered in the $k\rightarrow 0$ limit, as a consequence $$\begin{aligned}
G_{\rm C}[k\rightarrow 0]=\frac{1}{2} \frac{\partial^{2}E_{\rm PP}}{\partial L^{2}}.\end{aligned}$$ This theorem is ensured, for any specific model of the material medium, by the fact that $G_{\rm C}$ is given for $k\rightarrow 0$ by the specular limit of non-specular reflection amplitudes [@MaiaNetoPRA05].

In order to compare with experiments, we consider the expression of the lateral force in the plane-sphere configuration. It is derived from the plane-plane configuration using the PFA, reliable as long as $L\ll R$. In fact, there is no interplay between curvature and corrugation provided $RL\gg\lambda_{\rm C}^{2}$, a condition met in the experiment reported in [@ChenPRL02].

From Eq.(\[PFAPS\]), the lateral force in the plane-sphere 