Electromagnetic semitransparent<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>δ</mml:mi></mml:math>-function plate: Casimir interaction energy between parallel infinitesimally thin plates

Abstract

We derive boundary conditions for electromagnetic fields on a $\ensuremath{\delta}$-function plate. The optical properties of such a plate are shown to necessarily be anisotropic in that they only depend on the transverse properties of the plate. We unambiguously obtain the boundary conditions for a perfectly conducting $\ensuremath{\delta}$-function plate in the limit of infinite dielectric response. We show that a material does not ``optically vanish'' in the thin-plate limit. The thin-plate limit of a plasma slab of thickness $d$ with plasma frequency ${\ensuremath{\omega}}_{p}^{2}={\ensuremath{\zeta}}_{p}/d$ reduces to a $\ensuremath{\delta}$-function plate for frequencies ($\ensuremath{\omega}=i\ensuremath{\zeta}$) satisfying $\ensuremath{\zeta}d\ensuremath{\ll}\sqrt{{\ensuremath{\zeta}}_{p}d}\ensuremath{\ll}1$. We show that the Casimir interaction energy between two parallel perfectly conducting $\ensuremath{\delta}$-function plates is the same as that for parallel perfectly conducting slabs. Similarly, we show that the interaction energy between an atom and a perfect electrically conducting $\ensuremath{\delta}$-function plate is the usual Casimir-Polder energy, which is verified by considering the thin-plate limit of dielectric slabs. The ``thick'' and ``thin'' boundary conditions considered by Bordag are found to be identical in the sense that they lead to the same electromagnetic fields.