to $$\begin{aligned}
E_{\rm PFA} = E_{\rm PP}+\frac{1}{2}\frac{\partial^{2}E_{\rm PP}}{\partial L^{2}}\int \frac{{\rm d}^{2} {\bf r}}{A}(h_{1}({\bf r})+h_{2}({\bf r}))^{2}.  \label{PFAgen}\end{aligned}$$ The trivial first-order term has been discarded, assuming that the deformations have zero spatial averages $\int {\rm d}^{2}{\bf r}h_{i=1,2}({\bf r}) / A =0$.

The evaluation of the effect of geometry through the PFA, based on a summation procedure over local contributions assumes some additivity property of the Casimir effect, whereas the Casimir force is known not to be additive. The PFA can only be accurate for surfaces which can be considered as nearly plane with respect to other scales such as the separation distance $L$ [@GenetEPL03]. For example, it allows one to calculate the Casimir force in the plane-sphere (PS) configuration as $$\begin{aligned}
F_{\rm PS}=\frac{2\pi R}{A}E_{\rm PP},  \ \ \textrm{with} \ \ L\ll R, \label{PFAPS}\end{aligned}$$ where $E_{\rm PP}$ is the Casimir energy in the plane-plane (PP) geometry. Most recent experiments are performed in the plane-sphere geometry which is much simpler to control than the plane-plane configuration. The PFA is here expected to be valid provided the radius $R$ of the sphere is much larger than the distance $L$ of closest approach.

But the PFA certainly fails for describing more general surface profiles. As far 