Casimir interaction between a sphere and a grating

We derive the explicit expression for the Casimir energy between a sphere and a one-dimensional grating in terms of the sphere and grating reflection matrices. This expression is valid for arbitrary materials, sphere radius, and grating geometric parameters. We then numerically calculate the Casimir energy between a metallic (gold) sphere and a dielectric (fused silica) lamellar grating at room temperature, and we explore its dependence on the sphere radius, grating-sphere separation, and lateral displacement. We quantitatively investigate the geometrical dependence of the interaction, which is sensitive to the grating height and filling factor, and we show how the sphere can be used as a local sensor of the Casimir force geometric features. Toward that end, we mostly concentrate on separations and sphere radii of the same order of the grating parameters (here of the order of 1 mu m). We also investigate the lateral component of the Casimir force, resulting from the absence of translational invariance. We compare our results with those obtained within the proximity force approximation (PFA). When applied to the sphere only, the PFA overestimates the strength of the attractive interaction, and we find that the discrepancy is larger in the sphere-grating than in the sphere-plane geometry. On the other hand, when the PFA is applied to both the sphere and the grating, it provides a better estimate of the exact results, simply because the effect of a single grating is underestimated, thus leading to a partial compensation of errors.