Proximity force approximation and specular reflection: Application of the WKB limit of Mie scattering to the Casimir effect

Abstract

The electromagnetic Casimir interaction between two spheres is studied within the scattering approach using the plane-wave basis. It is demonstrated that the proximity force approximation (PFA) corresponds to the specular-reflection limit of Mie scattering. Using the leading-order semiclassical WKB approximation for the direct reflection term in the Debye expansion for the scattering amplitudes, we prove that PFA provides the correct leading-order divergence for arbitrary materials and temperatures in the sphere-sphere and the plane-sphere geometry. Our derivation implies that only a small section around the points of closest approach between the interacting spherical surfaces contributes in the PFA regime. The corresponding characteristic length scale is estimated from the width of the Gaussian integrand obtained within the saddle-point approximation. At low temperatures, the area relevant for the thermal corrections is much larger than the area contributing to the zero-temperature result.

Role of diffraction in the Casimir effect beyond the proximity force approximation

“…to which the two polarizations contribute equally. As shown in [19], the PFA result corresponds to the leading term of the saddle-point approximation which arises from rayoptical specular reflection at the point of the sphere closest to the plate, i.e. scattering channel (i) in Fig.…”

“…was shown to result from the leading order of a saddle-point evaluation of the round-trip decomposition (8) [19]. Here, we will focus on the evaluation of the constant β 1 .…”

“…Within PFA, the Casimir energy is obtained from Lifshitz's formula for parallel planes by averaging over the local distance between the surfaces [18]. PFA provides the correct leading asymptotics for large sphere radius R when considering general materials and arbitrary temperatures [19]. As reviewed in [20], typical experiments are close to the validity range of PFA since they correspond to aspect ratios R/L > 10 2 , where L is the minimal distance between the surfaces as indicated in Fig.…”

Advancing numerics for the Casimir effect to experimentally relevant aspect ratios

“…The proximity force approximation can be generalized to account for arbitrary electromagnetic response of sphere and plate as well as arbitrary temperatures by using the corresponding expression for the force in the plane-plane geometry. Recently, it has been proven for the sphere-sphere geometry that the proximity force approximation yields the leading small-distance behaviour for arbitrary temperatures and materials [8]. In particular, the proximity force approximation for the sphere-plane geometry is obtained by taking one sphere radius to infinity.…”

“…Furthermore, Ref. [8] has established a connection between the specularreflection limit of Mie scattering and the proximity force approximation.…”