Asymptotic of the Green function for the diffraction by a perfectly conducting plane perturbed by a sub-wavelength rectangular cavity

This work is aimed at understanding the amplification and confinement of electromagnetic fields in open sub-wavelength metallic cavities. We present a theoretical study of the electromagnetic diffraction by a perfectly conducting planar interface which contains a sub-wavelength rectangular cavity. We derive a rigorous asymptotic of the Green function associated to the Helmholtz operator when the width of the cavity shrinks to zero. We show that the limiting Green function is that of a perfectly conducting plane with a dipole in place of the cavity. We give an explicit description of the effective dipole in terms of the wavelength and of the geometry of the cavity.