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

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 term in the small-distance limit 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.

Figure 1

Scattering geometry consisting of two spheres with radii $R_1$ and $R_2$ and a surface-to-surface distance $L$. The distance between the spheres' centers is $\mathcal{L}=R_1+R_2+L$. The round trip discussed in the text is displayed between the two spheres.

Figure 2

Geometrical optics limit for the direct reflection by a sphere of radius $R$. Within the WKB approximation, a given scattering angle $\mathrm{\Theta }$ defines the impact parameter $b=Rcos(\mathrm{\Theta }/2).$ Seen from the tangent plane to the sphere, the angle of incidence is given by $(\pi -\mathrm{\Theta })/2$. The missing phase of a ray with frequency $\omega$ reflected on the sphere's surface with respect to a corresponding ray passing via the sphere's center before being deflected amounts to $2(\omega R/c)sin(\mathrm{\Theta }/2)$.

Figure 3

Estimation of the effective area contributing to the Casimir interaction between two spheres. ${\mathbf{K}}_{\text{in}}$ and ${\mathbf{K}}_{\text{rfl}}$ denote the incident and reflected wave vectors, respectively. To be definite, the reflection is shown at the smaller sphere of radius $R_1$. Specular reflection at the tangent plane $\pi \left(\vartheta \right)$ entails that the projections of ${\mathbf{K}}_{\text{in}}$ and ${\mathbf{K}}_{\text{rfl}}$ on $\pi \left(\vartheta \right)$ are equal. On the other hand, the wave-vector projections on the $x-y$ plane are generally different: $\delta k=|{\mathbf{k}}_{\text{rfl}}-{\mathbf{k}}_{\text{in}}|\approx 2\vartheta k_z$ for $\vartheta \ll 1$. We can estimate the angular sector effectively contributing to the Casimir interaction from the width $\delta k$ of the Gaussian integrand in the saddle-point approximation (see text).