Open Access

Benjamin Spreng

• We study the Casimir interaction in the plane-sphere and sphere-sphere geometry within the scattering formalism. A widely used to tool to estimate the Casimir interaction in those geometries is the proximity-force approximation (PFA). Within the PFA, the Casimir interaction is obtained by averaging the interaction between parallel planes over the local distances, thus simplifying the numerical evaluation. In this thesis, we develop an analytical and numerical method based on plane waves for calculating the Casimir interaction beyond the PFA. Our analytical method is based on an asymptotic expansion of the scattering formula when the sphere radii are large compared to the surface-to-surface distance between the interacting objects. We find the PFA as the asymptotic leading order term in the expansion and interpret it in terms of geometic optics. The correction to the PFA is calculated and it has its largest contribution from modifications of the geometric optical picture due toWe study the Casimir interaction in the plane-sphere and sphere-sphere geometry within the scattering formalism. A widely used to tool to estimate the Casimir interaction in those geometries is the proximity-force approximation (PFA). Within the PFA, the Casimir interaction is obtained by averaging the interaction between parallel planes over the local distances, thus simplifying the numerical evaluation. In this thesis, we develop an analytical and numerical method based on plane waves for calculating the Casimir interaction beyond the PFA. Our analytical method is based on an asymptotic expansion of the scattering formula when the sphere radii are large compared to the surface-to-surface distance between the interacting objects. We find the PFA as the asymptotic leading order term in the expansion and interpret it in terms of geometic optics. The correction to the PFA is calculated and it has its largest contribution from modifications of the geometric optical picture due to diffraction. Our results are valid for arbitrary temperatures and dielectric materials of the interacting objects. Thus, we are able to provide formulas beyond the PFA which have not been given in the literature before. Our numerically exact method is built on a Nyström discretization of the continuous plane-wave modes. We show that our new approach exhibits far superior convergence properties when compared to approaches using spherical multipolar waves. We demonstrate that our efficient methods can be used on standard hardware to make beyond-PFA predictions for typical experimental setups. We apply our numerical method to study the Casimir interaction in aqueous colloidal systems. For the example of polystyrene bodies in water, we show that the PFA performs worse than assumed in the literature. Moreover, our numerical approach allows us to assess the accuracy of our analytical formula obtained by the asymptotic expansion. We find that it may serve as an accurate short-distance formula for comparison with experimental results not just within colloidal systems, but also for typical Casimir experiments involving gold surfaces interacting across vacuum.…