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

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.