Relation between the surface impedance and the extinction theorem on a rough surface

The surface-impedance (SI) expression that was recently obtained for a corrugated metal surface, up to second order in the penetration depth, is reconsidered in connection with the extinction-theorem equations for the electromagnetic field. Instead of using a coordinate transformation, which stretches the corrugated surface, we obtain here the same SI expression from Green’s theorem in a more natural and easy way. In addition, this method allows us to evaluate SI corrections of the third order in the penetration depth. We also show that for a one-dimensional corrugation our result reduces to the SI expression given recently by Maradudin.