Electromagnetic eigenmodes in matter. van der Waals-London and Casimir forces

TL;DR

This paper derives electromagnetic eigenmodes to calculate van der Waals-London and Casimir forces between bodies, highlighting the role of surface modes and retardation effects in these quantum forces.

Contribution

It introduces a method based on eigenmode analysis of electromagnetic fields to derive dispersion forces, incorporating simple models for materials and including retardation effects.

Findings

Van der Waals-London force scales as d^{-3} from surface plasmon zero-point energy.

Casimir force scales as d^{-4} when retardation is included, from surface plasmon-polariton modes.

The same Casimir force result applies under fixed surface boundary conditions.

Abstract

We derive van der Waals-London and Casimir forces by calculating the eigenmodes of the electromagnetic field interacting with two semi-infinite bodies (two halves of space) with parallel surfaces separated by distance d. We adopt simple models for metals and dielectrics, well-known in the elementary theory of dispersion. In the non-retarded (Coulomb) limit we get a d^{-3}-force (van der Waals-London force), arising from the zero-point energy (vacuum fluctuations) of the surface plasmon modes. When retardation is included we obtain a d^{-4}-(Casimir) force, arising from the zero-point energy of the surface plasmon-polariton modes (evanescent modes) for metals, and from propagating (polaritonic) modes for identical dielectrics. The same Casimir force is also obtained for "fixed surfaces" boundary conditions, irrespective of the pair of bodies. The approach is based on the equation of motion of the polarization and the electromagnetic potentials, which lead to coupled integral equations. These equations are solved, and their relevant eigenfrequencies branches are identified.