Generalized Casimir Energies for systems with arbitrary ($\epsilon(z)$) planar geometry

TL;DR

This paper derives a general relation connecting Casimir forces between planar systems with arbitrary dielectric profiles to their transmission and reflection coefficients, unifying known results like Lifshitz theory.

Contribution

It introduces a simple, general framework to compute Casimir energies for systems with arbitrary planar dielectric profiles based on their optical coefficients.

Findings

Derives a universal relation linking Casimir force to transmission and reflection coefficients.

Recovers known Lifshitz force as a special case.

Provides a basis for calculating Casimir energies in complex planar systems.

Abstract

The Casimir force between parallel plates of arbitrary kind is shown to be simply related to the plates transmission and reflection coefficient. A trivial application of this general relation leads to the known Lifshitz force between dielectrics as well as its generalizations.