Casimir Effect and Geometric Optics

We propose a new approach to the Casimir effect based on classical ray optics. We define and compute the contribution of classical optical paths to the Casimir force between rigid bodies. We reproduce the standard result for parallel plates and agree over a wide range of parameters with a recent numerical treatment of the sphere and plate with Dirichlet boundary conditions. Our approach improves upon the proximity force approximation. It can be generalized easily to other geometries, other boundary conditions, to the computation of Casimir energy densities, and to many other situations.

Figure 1

Comparison of methods: Casimir energy for a sphere and a plate as a function of $\xi =a/R$. Numerical data from Ref. [4] normalized to unity (stars with error bars); our optical approximation (red triangles); plate based proximity force approximation (PFA) (green diamonds); sphere based PFA (blue squares).

Figure 2

(a) Optical paths for parallel plates. Initial and final points have been separated for visibility; (b) optical paths for plane $+$ sphere. For $n=1$ a reflection off the sphere is shown; for $n=3$ a reflection twice off the sphere is shown.

Figure 3

Local contributions to the optical Casimir energy for a plane and a sphere with $a/R=0.25$. The scale is linear in the hue from red (least) to violet (greatest). The self-energy density given by ${\mathcal{C}}_1$ has been subtracted.