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

Abstract

We derive the leading-order correction to the proximity force approximation (PFA) result for the electromagnetic Casimir interaction in the plane-sphere geometry by developing the scattering approach in the plane-wave basis. Expressing the Casimir energy as a sum over round trips between plane and sphere, we find two distinct contributions to the correction. The first one results from the variation of the Mie reflection operator, calculated within the geometric optical WKB approximation, over the narrow Fourier interval associated to specular reflection at the vicinity of the point of closest approach on the spherical surface. The second contribution, accounting for roughly 90% of the total correction, results from the modification of the geometric optical WKB Mie scattering amplitude due to diffraction. Our derivation provides a clear physical understanding of the nature of the PFA correction for spherical surfaces.http://dx.

Casimir Interaction between a Plane and a Sphere: Correction to the Proximity-Force Approximation at Intermediate Temperatures

“…The matrix elements of the reflection operator R S appearing in ( 7) can be written in terms of the standard Mie scattering amplitudes together with coefficients describing the change between the Fresnel and the scattering polarization basis [34,41]. For the evaluation of the leading-order (LO) PFA result and its LO correction for the perfectly reflectors case, the relevant matrix elements effectively reduce to [41] k j , −, TM|R S |k i , +, TM = 2πc ξκ j S 2 k j , −, TE|R S |k i , +, TE = 2πc ξκ j S 1 (8) while the matrix elements involving the coupling between different polarizations do not contribute, i.e., k j , TM|R S |k i , TE = k j , TE|R S |k i , TM = 0. The Mie scattering amplitudes S 1 and S 2 in (8) are given by (A1) of Appendix A.…”

“…Applying the saddle-point approximation (A14) with (A16) and (A18), ( 14) can now be expressed as [41] tr M(ξ…”

“…We will start by discussing the case of positive Matsubara frequencies in Section 3.1 where we make use of results obtained earlier in Ref. [41]. In Section 3.2, we will then derive the asymptotic expansion of the zero-frequency contribution to obtain both the NTLO and the next-to-next-to-leading order (NNTLO) terms.…”

“…It turns out that we can partly use the results obtained for finite frequencies in Ref. [41]. Thus, the expressions for the derivatives of the function f are obtained from (A13) and (A14) of Ref.…”

“…Such condition holds in typical Casimir experiments as λ T ≈ 7.6 µm at T = 300 K. We consider the plane-sphere setup within the perfectly-reflecting model for simplicity. However, our method based on a semiclassical expansion developed in the plane-wave basis [34,41] can also be applied to the sphere-sphere geometry and to real materials.…”