geometry is eventually given as [@RodriguesPRL2006] $$\begin{aligned}
F_{\rm PS}^{\rm lat}=-\frac{\partial }{\partial b}E_{\rm PS}^{\rm lat}=\pi a_{1}a_{2}kR\sin (kb )
\int_{\infty}^{L}{\rm d}L^{\prime}G[k,L^{\prime}].\end{aligned}$$ The force is plotted in Fig.(\[fig:5\]) as a function of $k$, with length scales $\lambda_{\rm C}$, $\lambda_{\rm P}$ and $L$ fitting the experimental values [@ChenPRL02]. As the corrugation amplitudes are not small enough in the experiment to meet the perturbation condition, the theory and experiment can unfortunately not be compared directly. The plot on Fig.(\[fig:5\]) nevertheless shows the interesting fact that the length scales taken from the experiment, with $k$ indicated by the vertical dashed line, clearly fall outside the PFA sector in the perturbative calculation. For related implications, we refer the reader to the discussions in [@RodriguesPRA2007].

It appears clearly on the figure that the PFA overestimates the magnitude of the lateral force for arbitrary $k$. We also note that the PFA prediction for the force scales as $k$ when $k$ increases from zero. At larger values of $k$ in contrast, the lateral force decreases. This is due to the one-way propagation factor separating the two first-order non-specular reflections at each plate, given as a decresing exponential $e^{-kL}$ in the high $k$ limit [@RodriguesPRL2006]. It follows that there is a 