Finite-temperature Casimir force between perfectly metallic corrugated surfaces

Abstract

We study the Casimir force between two corrugated plates due to thermal fluctuations of a scalar field. For arbitrary corrugations and temperature $T$, we provide an analytical expression for the Casimir force, which is exact to second order in the corrugation amplitude. We study the specific case of two sinusoidally corrugated plates with corrugation wavelength $\ensuremath{\lambda}$, lateral displacement $b$, and mean separation $H$. We find that the lateral Casimir force is ${F}_{l}(T,H)\mathrm{sin}(2\ensuremath{\pi}b/\ensuremath{\lambda})$. In other words, at all temperatures, the lateral force is a sinusoidal function of the lateral shift. In the limit $\ensuremath{\lambda}\ensuremath{\gg}H,\phantom{\rule{0.28em}{0ex}}{F}_{l}(T\ensuremath{\rightarrow}\ensuremath{\infty},H)\ensuremath{\propto}{k}_{B}T{H}^{\ensuremath{-}4}{\ensuremath{\lambda}}^{\ensuremath{-}1}$. In the opposite limit $\ensuremath{\lambda}\ensuremath{\ll}H,\phantom{\rule{0.28em}{0ex}}{F}_{l}(T\ensuremath{\rightarrow}\ensuremath{\infty},H)\ensuremath{\propto}{k}_{B}T{H}^{\ensuremath{-}1/2}{\ensuremath{\lambda}}^{\ensuremath{-}9/2}{e}^{\ensuremath{-}2\ensuremath{\pi}H/\ensuremath{\lambda}}$.