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

TL;DR

This paper calculates the next-to-leading order correction to the lateral Casimir force between corrugated parallel plates, providing explicit results and demonstrating the significance of the correction for accurate theoretical predictions.

Contribution

It introduces a detailed calculation of the next-to-leading order contribution to the lateral Casimir force, improving the accuracy of theoretical models for corrugated plates.

Findings

Next-to-leading order correction is significant.

Weak coupling limit results are exact and numerically evaluated.

Including the correction reduces theoretical error substantially.

Abstract

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 Casimir energy due to uncorrugated parallel plates is treated as a background from the outset. We derive the leading- and next-to-leading-order contribution to the lateral Casimir force for the case when the corrugation amplitudes are small in comparison to corrugation wavelengths. We present explicit results in terms of finite integrals for the case of the Dirichlet limit, and exact results for the weak-coupling limit, for the leading- and next-to-leading-orders. The correction due to the next-to-leading contribution is significant. In the weak coupling limit we calculate the lateral Casimir force exactly in terms of a single integral which we evaluate numerically. Exact results for the case of the weak limit allows us to estimate the error in the perturbative results. We show 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 when the corrugation amplitudes are small in comparison to corrugation wavelengths. We expect similar conclusions to hold for the Dirichlet case. The analogous calculation for the electromagnetic case should reduce the theoretical error sufficiently for comparison with the experiments.