Surface modes of a sphere coupled to a semi-infinite medium

The surface modes of a sphere coupled to a semi-infinite medium are studied. The author obtains a relationship which describes the dependence of frequency on the distance between the sphere and the semi-infinite medium. His results reduce to previously reported results in the limit of a very large separation between the sphere and the semi-infinite medium. Two surface modes for each order n are shown to exist. The theory is applied to three cases: first, a metallic sphere in vacuum coupled to a semi-infinite metal of the same material; second, a metallic sphere embedded into a semi-infinite metal bounded by vacuum; third, a spherical cavity inside a semi-infinite metal bounded by vacuum.