Test of the Proximity Force Approximation

We study the geometrical corrections to the simple Proximity Force Approximation (PFA) for the non-retarded Casimir force. We extend traditional PFA in two ways: We take the whole surfaces of the objects facing each other into account, not just the curvatures at the point of closest distance; we take the thickness of the coating of coated objects into account in the formalism. We present analytical and numerical results for a sphere above a substrate, for a spherical shell above a substrate, and for two interacting spheres. We compare the results to those from a multi-polar expansion method, a method based on a more solid foundation.