Derivative expansion for the electromagnetic and Neumann-Casimir effects in 2+1 dimensions with imperfect mirrors

Abstract

We calculate the Casimir interaction energy in d = 2 spatial dimensions between two (zero-width) mirrors -one flat and the other slightly curved- upon which imperfect conductor boundary conditions are imposed for an electromagnetic (EM) field. Our main result is a second-order derivative expansion (DE) approximation for the Casimir energy, which is studied in different interesting limits. In particular, we focus on the emergence of a nonanalyticity beyond the leading-order term in the DE, when approaching the limit of perfectly conducting mirrors. We also show that the system considered is equivalent to a dual one, consisting of a massless real scalar field satisfying imperfect Neumann conditions (on the very same boundaries). Therefore, the results obtained for the EM field hold true also for the scalar field model.