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We calculate the lateral Casimir force between corrugated parallel plates,described by $\delta$-function potentials, interacting through a scalar field,using the multiple scattering formalism. The contributions to the Casimirenergy due to uncorrugated parallel plates is treated as a background from theoutset. We derive the leading- and next-to-leading-order contribution to thelateral Casimir force for the case when the corrugation amplitudes are small incomparison to corrugation wavelengths. We present explicit results in terms offinite integrals for the case of the Dirichlet limit, and exact results for theweak-coupling limit, for the leading- and next-to-leading-orders. Thecorrection due to the next-to-leading contribution is significant. In the weakcoupling limit we calculate the lateral Casimir force exactly in terms of asingle integral which we evaluate numerically. Exact results for the case ofthe weak limit allows us to estimate the error in the perturbative results. Weshow that the error in the lateral Casimir force, in the weak coupling limit,when the next-to-leading order contribution is included is remarkably low whenthe corrugation amplitudes are small in comparison to corrugation wavelengths.We expect similar conclusions to hold for the Dirichlet case. The analogouscalculation for the electromagnetic case should reduce the theoretical errorsufficiently for comparison with the experiments.

Non-contact gears: I. Next-to-leading order contribution to lateral Casimir force between corrugated parallel plates