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

## TL;DR

This paper links the proximity force approximation (PFA) to the specular reflection limit of Mie scattering, using the WKB approximation to analyze the Casimir effect between spheres and planes, confirming PFA's validity at leading order.

## Contribution

It demonstrates that PFA corresponds to the specular reflection limit of Mie scattering and uses semiclassical WKB approximation to justify PFA's leading-order divergence for various materials and temperatures.

## Key findings

- PFA matches the specular reflection limit of Mie scattering.
- WKB approximation confirms PFA's leading-order divergence.
- Thermal corrections involve a larger area at low temperatures.

## 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.