Attenuation and focusing of electromagnetic surface waves rounding gentle bends

Abstract

We study electromagnetic waves in and near dielectric surfaces S whose radii of curvature R are large compared with the surface wavelength 2n/K,, for cases when the dielectric constant c < - 1. If such gentle bends are convex towards the vacuum the wave is not perfectly bound-it attenuates with decay length K,-', the lost energy being radiated to infinity. The radiation appears to emerge tangentially into vacuum from the height z, above S where the wave changes from evanescent to oscillatory. We obtain analytic formulae for z, and K, in terms of K,R and c. When S is concave to the vacuum we argue that there should be no attenuation. On general gently bent surfaces the wave energy travels along rays that are geodesics on S. We discuss the focusing of families of rays on S and show that the imperfect focus of a plane wave incident on a general circularly symmetric hill is a cusped caustic with two rainbows as asymptotes. Perfect focusing is also possible and we calculate the shape of the 'geodesic lens' that would produce this. Finally, we suggest some experiments to test the theory.