On the extraordinary transmission through sub-wavelength apertures in perfectly conducting sheets

Extraordinary transmission of electromagnetic power through sub-wavelength apertures is usually observed in a narrow bandwidth range and the transmission outside this range is low. In this paper, it is shown that the transmission through periodic apertures in an otherwise perfectly electric conducting sheet satisfies a sum rule that relates wavelength regions with transmission above a threshold with the polarizability of the complementary structure. The theoretical results are illustrated with a numerical example for split ring apertures.