Repulsive Casimir interaction between conducting and permeable corrugated plates

Abstract

We consider the Casimir interaction between perfectly conducting and infinitely permeable corrugated parallel plates. We first study smooth plates which face each other at the distance $H$ with mixed boundary conditions (BCs) and show that the Casimir force is repulsive between them. Then we consider the effect of corrugation on the Casimir interaction. We investigate two geometries with mixed BCs, one with a corrugated plate and a flat plate (cf) and another with two corrugated surfaces. For the geometry of cf, at small distances the correction to the Casimir energy is ${E}_{\mathrm{cf}}^{\mathrm{N},\mathrm{D}}\ensuremath{\sim}\frac{+\ensuremath{\hbar}c{a}^{2}}{{H}^{5}}$, while at large distances ${E}_{\mathrm{cf}}^{\mathrm{N},\mathrm{D}}\ensuremath{\sim}\frac{+\ensuremath{\hbar}c{a}^{2}}{{H}^{4}\ensuremath{\lambda}}$, where $a$ and $\ensuremath{\lambda}$ are amplitude and wavelength of the sinusoidal corrugated surface respectively. On the other hand, when both surfaces are sinusoidally corrugated with a wavelength of $\ensuremath{\lambda}$, they experience a periodic sinusoidal lateral casimir force ${F}_{l}\mathrm{sin}(2\ensuremath{\pi}l/\ensuremath{\lambda})$, where $l$ is the lateral displacement of the plates with respect to each other. In the limit of $\frac{H}{\ensuremath{\lambda}}\ensuremath{\ll}1$, ${F}_{l}\ensuremath{\sim}\ensuremath{-}\frac{\ensuremath{\hbar}c{a}^{2}}{{H}^{5}\ensuremath{\lambda}}$ while at the region of $\frac{H}{\ensuremath{\lambda}}\ensuremath{\gg}1$, ${F}_{l}$ decays exponentially as ${F}_{l}\ensuremath{\sim}\ensuremath{-}\frac{\ensuremath{\hbar}c{a}^{2}}{H{\ensuremath{\lambda}}^{5}}{e}^{\ensuremath{-}2\ensuremath{\pi}H/\ensuremath{\lambda}}$. For fixed $H$ and $\ensuremath{\lambda}$, stable and unstable mechanical equilibrium states of the system occur when $l=n\ensuremath{\lambda}$ and $l=(n+\frac{1}{2})\ensuremath{\lambda}$ respectively, where $n$ is an integer number.