Electrostatic and electromagnetic surface shape resonances

The homogeneous integral equations that give the electromagnetic field in the vicinity of a protuberance or a depression on the otherwise planar interface with vacuum of a semi-infinite dielectric medium or a thin dielectric film on a semi-infinite substrate have been obtained. The Rayleigh hypothesis, the vectorial equivalent of the Kirchhoff integral, and the extinction theorem have been used for this purpose. The assumption that the perturbation of the vacuum dielectric interface has cylindrical symmetry about the normal to the nominal surface allows a significant simplification of these integral equations to be carried out. We have used Gaussian quadrature schemes to convert the resulting integral equations into matrix equations, and have obtained the frequencies of the shape resonances by equating to zero the determinants of the matrices obtained. Calculations have been carried out for Gaussian (x ₃=Aexp(–x ² /R ²)) and exponential (x ₃=Aexp(–x /R)) surface profiles, and convergent results obtained for values ofA/R of the order of unity.Dedicated to B. Mühlschlegel on the occasion of his 60th birthday