Making precise predictions of the Casimir force between metallic plates via a weighted Kramers-Kronig transform

TL;DR

This paper introduces weighted Kramers-Kronig transform techniques to accurately predict the dielectric function of metallic plates, improving Casimir force calculations by reducing low-frequency data extrapolation errors.

Contribution

It develops weighted dispersion relations that better estimate dielectric functions from optical data, enhancing precision in Casimir force predictions for metallic plates.

Findings

Weighted dispersion formulas improve dielectric function estimates.

Application to gold data confirms accuracy of the method.

Reduces need for low-frequency data extrapolation.

Abstract

The possibility of making precise predictions for the Casimir force is essential for the theoretical interpretation of current precision experiments on the thermal Casimir effect with metallic plates, especially for sub-micron separations. For this purpose it is necessary to estimate very accurately the dielectric function of a conductor along the imaginary frequency axis. This task is complicated in the case of ohmic conductors, because optical data do not usually extend to sufficiently low frequencies to permit an accurate evaluation of the standard Kramers-Kronig integral used to compute $\epsilon(i \xi)$. By making important improvements in the results of a previous paper by the author, it is shown that this difficulty can be resolved by considering suitable weighted dispersions relations, which strongly suppress the contribution of low frequencies. The weighted dispersion formulae presented in this paper permit to estimate accurately the dielectric function of ohmic conductors for imaginary frequencies, on the basis of optical data extending from the IR to the UV, with no need of uncontrolled data extrapolations towards zero frequency that are instead necessary with standard Kramers-Kronig relations. Applications to several sets of data for gold films are presented to demonstrate viability of the new dispersion formulae.