High-Frequency Scattering by a Transparent Sphere. II. Theory of the Rainbow and the Glory

Abstract

The treatment, initiated in Paper I [J. Math. Phys. 10, 82 (1969)], of the high‐frequency scattering of a scalar plane wave by a transparent sphere is continued. The main results here are an improved theory of the rainbow and a theory of the glory. The modified Watson transformation is applied to the third term of the Debye expansion of the scattering amplitude in terms of multiple reflections. Only the range 1<N<√2 , where N is the refractive index, is considered. In the geometrical‐optic approximation, this term is associated with rays transmitted after one internal reflection, and there are three angular regions, corresponding to one ray, two rays, or no ray (shadow) passing through each direction. Together with transition regions, this leads to six different angular domains. In the 1‐ray and 2‐ray regions, geometrical‐optic terms are dominant. Correction terms corresponding to the 2nd‐order WKB approximation are also evaluated. In the 0‐ray region, the amplitude is dominated by complex rays and surface waves. The 1‐ray/2‐ray transition is a Fock‐type region. The rainbow appears in the 2‐ray/0‐ray transition region. The extension of the method of steepest descents due to Chester, Friedman and Ursell is applied. The result is a uniform asymptotic expansion for the scattering amplitude. It reduces to Airy's theory in the lowest‐order approximation, but its domain of validity is considerably greater, both with regard to size parameter and to angles. The glory is an example of strong ``Regge‐pole dominance'' of the near‐backward scattering amplitude. Van de Hulst's conjecture that surface waves are responsible for the glory is confirmed. However, besides surface waves taking two shortcuts through the sphere, higher‐order terms in the Debye expansion must also be taken into account. By considering also the effect of higher‐order surface‐wave contributions, all the features observed in the glory (apart from the polarization) are explained. Resonance effects associated with nearly‐closed paths of diffracted rays lead to large, rapid, quasiperiodic intensity fluctuations. The same effects are responsible for the ripple in the total cross‐section. Similar fluctuations appear in any direction, but their amplitude increases with the scattering angle, becoming a maximum near the backward direction, where they are dominant. They can also be interpreted as a collective effect due to many nearly‐resonant partial waves in the edge domain. The dominant surface‐wave contributions can also be summed to all orders for N < 1, leading to a renormalization of the propagation constants of surface waves.