Noncontact gears. I. Next-to-leading order contribution to the lateral Casimir force between corrugated parallel plates

Abstract

We calculate the lateral Casimir force between corrugated parallel plates, described by -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.

Calculating Casimir interactions for periodic surface-relief gratings using theCmethod

“…The dashed line represents the next to leading order expansion as calculated in Refs. [24]. For larger amplitudes, the perturbative expansion dramatically underestimates the lateral Casimir force, not shown in the figure.…”

“…We emphasize that in all previous work, in order to find the perturbative corrections to the reflection coefficients the Rayleigh hypothesis was assumed. In addition to the scattering approach, other techniques were employed to calculate the Casimir forces for perfect materials [19,20,24,25]. These calculations were also done under the assumption of the Rayleigh hypothesis.…”

“…The first theoretical calculation of the Casimir force between geometrically patterned surfaces not using the PFA was reported in 2001 [19,20], which described the normal and lateral forces between two aligned corrugated surfaces in terms of a perturbative expansion in the profile height. Since then, this perturbative approach has been expanded to include real material properties and unaligned corrugations [21][22][23][24][25]. More recently the scattering method has been used to obtain the Casimir forces in periodic systems [26][27][28].…”

Field theoretic approach to roughness corrections

“…This problem has to some extent been addressed in the calculation of lateral Casimir forces for corrugated plates [17,19], but regular, one dimensional corrugations are very special. Lateral Casimir forces are automatically finite and vanish if one of the two parallel plates is flat.…”

“…When the plate separation a is large compared to the correlation length ℓ of the profile one recovers translational invariance and the reflection coefficient becomes diagonal in the transverse momentum. The Casimir energy per unit area, E(a), in this case is given by a dimensionally reduced gtgt-formula [14,17],…”