Lateral Casimir force between deeply corrugated dielectric and metallic plates

We study the lateral Casimir force between corrugated dielectric plates. We use the dielectric contrast perturbation theory [R. Golestanian, {\it Phys. Rev. Lett.} {\bf 95}, 230601, (2005)], which allows us to consider arbitrary deformations with large amplitudes. We consider sinusoidal, rectangular, and sawtooth corrugations, for samples made of silicon and gold. We use the plasma and Drude-Lorentz models for the permittivity of gold and silicon, respectively. For these geometries and materials, the lateral Casimir force is {\it not} a sinusoidal function of the relative lateral displacement of plates when the gap between the plates in comparable with the depth of the corrugations. Our results facilitate the design of miniaturized devices based on lateral Casimir forces.