Demonstration of the Nontrivial Boundary Dependence of the Casimir Force

The Casimir force between an aluminum-coated plate with small sinusoidal corrugations and a large sphere was measured for surface separations between 0.1 and 0.9 mm using an atomic force microscope. The measured force shows significant deviation from the perturbative theory. The measured Casimir force between the same sphere and flat plate shows good agreement with the same theory in the limit of zero amplitude of corrugation. These together demonstrate the nontrivial boundary dependence of the Casimir force. [S0031-9007(99)09229-7] PACS numbers: 12.20.Fv, 61.16.Ch Casimir [1] predicted an attractive force between two neutral metal plates. The force results from the alteration by the metal boundaries of the zero point electromagnetic energy E › P ‘ s1y2d ¯ hvn, where ¯ hvn is the photon energy in each allowed photon mode n [1‐ 4]. Initially the Casimir force was thought to be a simple extension of the van der Waals (vdW) force which is an attractive force between two neutral molecules [2]. Lifshitz [5] generalized the vdW force between two extended bodies as the force between fluctuating dipoles induced by the zero point electromagnetic fields and obtained the same result as Casimir for two perfectly reflecting flat plates. However, it was realized that the Casimir force is a strong function of geometry and that between two halves of thin metal spherical shells is repulsive [6]. The sign and value of the Casimir force becomes even more interesting for complex topologies such as encountered with a torus [4]. Thus the Casimir force explores the dependence of the vacuum fluctuations on the geometry of the boundary. The Casimir force has been demonstrated between two flat plates [7] and a large sphere and a flat plate [8] and its value shown to be in agreement with the theory to an average deviation of 1% [9]. For dielectric bodies the resulting force has been measured with reasonable agreement to the theory [10]. Here we report the first experimental demonstration of the nontrivial boundary dependence by measuring the Casimir force between a large sphere and plate with periodic uniaxial sinusoidal corrugations (PUSC) for surface separations between 0.1 and 0.9 mm using an atomic force microscope (AFM). The amplitude of the corrugation is only 59.4 nm and is much smaller than the separation. Yet the measured force shows significant deviations from a perturbative theory which takes into account the small periodic corrugation of the plate in the surface separation (the results of the theory correspond to the trivial boundary dependence). Such a deviation can be expected due to the changes in zero point photon modes from diffraction off the periodic corrugation. We also compare the measured Casimir force between the same sphere and identically coated flat plate and show that it agrees well with the same theory in the limit of zero amplitude of corrugation. The results together demonstrate the nontrivial boundary dependence of the Casimir force. The boundary dependence of the Casimir force can be easily obscured by errors in the measurement of the surface separation [4]. To eliminate this ambiguity we use the electrostatic force to determine the exact surface separation and establish procedures for consistent comparison to theory. The regularized zero point energy per unit area given two parallel plates of infinite conductivity a distance z apart is given by [2 ‐ 5] Uszd › 2 p 2 ¯ hc