High-frequency (whispering-gallery mode)-to-beam conversion on a perfectly conducting concave-convex boundary

When a well-confined high-frequency (HF) whispering-gallery (WG) mode launched from the concave side of a concave-convex perfectly conducting boundary propagates toward the convex side, it becomes successively less confined and eventually detaches completely to form a radiated beam field. Detailed characterization of the intricate WG mode-to-beam conversion (here considered for the two-dimensional case) poses a challenging problem in HF wave propagation and asymptotics. This paper begins with a review of the accomplishments and shortcomings in previous investigations that dealt with the analytic-asymptotic aspects as well as the construction of numerical reference solutions to validate the HF approximations. We then develop algorithms based upon the earlier results, but with inclusion of new refinements. These yield a comprehensive asymptotic theory for the induced surface currents on the boundary as well as the near and far fields generated by them, well-matched to the WG mode detachment phenomenology and numerically accurate when compared with reference data obtained from a new hybrid analytic-numerical code. The asymptotic analysis makes use of Kirchhoff, physical optics, and spectral integral representations, WG modes on the concave side, modal ray tracing, replacing modal and/or ray caustics by equivalent induced source distributions, creeping waves on the convex side, etc., with self-consistent blending of nonuniform and locally uniform asymptotics through transition regions where transformations of the affected wavefields occur.