Casimir energy due to inhomogeneous thin plates

TL;DR

This paper develops a theoretical framework to compute Casimir energy between inhomogeneous, planar mirrors using a Lifshitz-like formula, and explores perturbative and derivative expansion methods for smooth property variations.

Contribution

It introduces a novel Lifshitz-like formula for Casimir energy with inhomogeneous mirrors and applies perturbative and derivative expansion techniques for practical calculations.

Findings

Derived a Lifshitz-like formula for inhomogeneous mirror systems

Applied perturbative methods for nearly constant properties

Showed in some cases, inhomogeneities mimic non-planar mirror effects

Abstract

We study the Casimir energy due to a quantum real scalar field coupled to two planar, infinite, zero-width, parallel mirrors with non-homogeneous properties. These properties are represented, in the model we use, by scalar functions defined on each mirror's plane. Using the Gelfand-Yaglom's theorem, we construct a Lifshitz-like formula for the Casimir energy of such a system. Then we use it to evaluate the energy perturbatively, for the case of almost constant scalar functions, and also implementing a Derivative Expansion, under the assumption that the spatial dependence of the properties is sufficiently smooth. We point out that, in some particular cases, the Casimir interaction energy for non-planar perfect mirrors can be reproduced by inhomogeneities on planar mirrors.