Numeric corrections to the proximity-force approximation for lateral Casimir forces

We report a numeric investigation on the proximity-force approximation (PFA) for lateral Casimir forces between a sphere and a grating. Near-unity force correlations are found between the approximated force and the exact values, due to geometric effects. A minimal model yields a best-fit expression of the numeric correction to the PFA, for gratings in the dilute limit. Our results are not restricted to specific material of the sphere, and allows simple estimation of Casimir interactions for micro-scale spheres, and thus shall be useful in relevant experimental and engineering Casimir applications.