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, de-scribed by a single function η, of a flat mirror. The perturbative expansion is carried out up to the fourth order in the deformation η, 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 (PFA) within the context of this expansion. II- The system Scalar field ϕ satisfying Dirichlet boundary conditions on two surfaces (zero width): L and R: L) x3 = 0 R) x3 = a+ η(x‖) Where η represents a small departure from a flat configuration. The vacuum energy of the system, Evac is Evac = lim T→