Casimir interaction between a sphere and a grating

TL;DR

This paper derives an explicit formula for the Casimir energy between a sphere and a 1D grating, and numerically investigates how geometry, material properties, and separation influence the force, highlighting the sphere's potential as a local force sensor.

Contribution

It provides a new explicit expression for the Casimir energy between a sphere and a grating, valid for arbitrary materials and geometries, and compares numerical results with the proximity force approximation.

Findings

Casimir energy depends on grating height and filling factor.

PFA overestimates the force in sphere-grating geometry.

PFA applied to both objects yields better estimates due to error compensation.

Abstract

We derive the explicit expression for the Casimir energy between a sphere and a 1D grating, in terms of the sphere and grating reflection matrices, and 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 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 show how the sphere can be used as a local sensor of the Casimir force geometric features. To this purpose we mostly concentrate on separations and sphere radii of the same order of the grating parameters (here of the order of one micrometer). 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, 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 PFA is applied to both sphere and 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.