Spectral representation of the Casimir Force Between a Sphere and a Substrate

We develop a spectral representation formalism to calculate the Casimir force, in the non-retarded limit, or van der Waals force between a spherical particle and a substrate, both with arbitrary local dielectric properties. The spectral formalism allows to study the system as a function of its geometrical properties separately from its dielectric properties. The calculated force is attractive, and at a small separations it is orders of magnitude larger for nanometric-size spheres than for micrometer particles. We also found that the force depends more on the dielectric properties of the sphere than of the substrate. Recent advances in micro and nano devices have opened the possibility of studying quantum phenomena that occur at these length scales. Such is the case of the Casimir force [1] that is a macroscopic manifestation of the quantum vacuum fluctuations, as predicted by quantum electrodynamics. The textbook example [2, 3, 4] consists of two parallel neutral conducting plates which attract each other. The first experimental measurements were done in 1951 using dielectric materials [5], and in 1958 using conductors [6]. These measurements have large errors, and up to recently, it was possible to perform measurements with about 15% of precision on truly parallel metal surfaces [7]. The difficulty of keeping the two plates parallel at separations of few nanometers makes it easier to measure the Casimir force between a sphere and a plane [8, 9, 10, 11, 12]. In this case, the Casimir theory for parallel plates can be extended using the proximity theorem [5]. The approximation is valid when the minimum separation between the sphere and the plane is much smaller than the radius of the sphere. This theorem was employed to corroborate experimental measurements of the