Theory of near-critical-angle scattering from a curved interface.

Abstract

A new type of diffraction effect, different from the standard semiclassical ones (rainbow, glory, forward peak, orbiting), takes place near the critical angle for total reflection at a curved interface between two homogeneous media. A theoretical treatment of this new effect is given for Mie scattering, e.g., light scattering by an air bubble in water; it can readily be extended to more general curved interface problems in a variety of different fields (quantum mechanics, acoustics, seismic waves). The relatively slowly varying Mie diffraction pattern associated with near-critical scattering is obscured by rapid fine-structure oscillations due to interference with unrelated ``far-side'' contributions. These contributions are evaluated and subtracted from the Mie amplitudes to yield the relevant ``near-side'' effects. A zero-order transitional complex angular momentum (CAM) approximation to the near-side amplitude is developed. The most important contributions arise from partial and total reflection, represented by two new diffraction integrals, designated Fresnel-Fock and Pearcey-Fock, respectively. The total reflection contribution is strongly affected by tunneling, giving rise to a generalized version of the Goos-H\anchen shift. In terms of short-wavelength asymptotic methodology, in a generalized Huygens-Fresnel-type integral representation, the new diffraction features arise from nonanalyticity of the integrand amplitude function within the range of a saddle point. Also discussed are the WKB approximation, a known physical-optics approximation, and a modified version of this approximation: they are compared with the ``exact'' near-side Mie amplitude obtained by numerical partial-wave summation, at scatterer size parameters (circumference divided by wavelength) ranging from 1000 to 10 000. It is found that the physical-optics approximations lead to large errors in the near-critical region, whereas the zero-order CAM approximation is in good agreement with the exact solution, accounting for the new diffraction effects in near-critical scattering.