General formulae are obtained for the transmission and reflection of highfrequency harmonic electromagnetic waves by a curved interface between two homogeneous dielectrics. If the surface is concave to the source there are, in addition to the standard refracted rays, tunnelling rays which are emitted from an evanescent region when total internal reflection would be expected to prevent radiation. Uniformly valid formulae covering the transition from refraction to tunnelling in both transmission and reflection are derived. The theory is more general than previous investigations in that coupling between the field components at the interface is permitted.
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