Fourth order perturbative expansion for the Casimir energy with a slightly deformed plate

TL;DR

This paper develops a fourth-order perturbative method to calculate the Casimir energy between a flat and a slightly deformed mirror, improving accuracy over previous approximations for corrugated surfaces.

Contribution

It introduces a fourth-order perturbative expansion for the Casimir energy with small surface deformations, extending previous second-order approaches.

Findings

Derived explicit formulas up to fourth order in deformation

Applied the expansion to corrugated mirror configurations

Reevaluated the proximity force approximation in this context

Abstract

We apply a perturbative approach to evaluate the Casimir energy for a massless real scalar field in 3+1 dimensions, subject to Dirichlet boundary conditions on two surfaces. One of the surfaces is assumed to be flat, while the other corresponds to a small deformation, described by a single function $\eta$, of a flat mirror. The perturbative expansion is carried out up to the fourth order in the deformation $\eta$, and the results are applied to the calculation of the Casimir energy for corrugated mirrors in front of a plane. We also reconsider the proximity force approximation within the context of this expansion.