Normal and lateral Casimir forces between deformed plates

The Casimir force between macroscopic bodies depends strongly on their shape and orientation. To study this geometry dependence in the case of two deformed metal plates, we use a path-integral quantization of the electromagnetic field which properly treats the many-body nature of the interaction, going beyond the commonly used pairwise summation (PWS) of van der Waals forces. For arbitrary deformations we provide an analytical result for the deformation induced change in the Casimir energy, which is exact to second order in the deformation amplitude. For the specific case of sinusoidally corrugated plates, we calculate both the normal and the lateral Casimir forces. The deformation induced change in the Casimir interaction of a flat and a corrugated plate shows an interesting crossover as a function of the ratio of the mean plate distance H to the corrugation length $\lambda :$ For $\lambda \ll H$ we find a slower decay $\sim H^{-4},$ compared to the $H^{-5}$ behavior predicted by PWS which we show to be valid only for $\lambda \gg H.\mathrm{}$ The amplitude of the lateral force between two corrugated plates which are out of registry is shown to have a maximum at an optimal wavelength of $\lambda \approx 2.5H.\mathrm{}$ With increasing $H/\lambda \gtrsim 0.3$ the PWS approach becomes a progressively worse description of the lateral force due to many-body effects. These results may be of relevance for the design and operation of novel microelectromechanical systems (MEMS) and other nanoscale devices.