Asymptotics of evanescence

Abstract

The wave from an isotropic point source in two and three dimensions is separated into its homogeneous (H) and evanescent (E) parts, with respect to a distinguished direction z, and the far field r θ evaluated asymptotically as a function of polar angle θ. Only in the ‘forward needle’ θ = 0 (three dimensions) and the transverse directions θ = π/2 (two and three dimensions) is the E wave of comparable strength to the H wave. Uniform asymptotic approximations (in terms of Bessel functions and Fresnel integrals) accurately interpolate between these directions of significant evanescence and all other angles (where the amplitude of E relative to H decays as 1/r¹/²). The forward needle in three dimensions is analogous to glory scattering.