9 - The Debye expansion

Summary

The first mirrored the next, as though it were

Rainbow from rainbow

(Dante, Paradiso XXXIII, 118)

We now begin to apply the CAM method to Mie scattering. The penetration of the field and the resulting interactions within the scatterer yield a rich structure, where all semiclassical diffraction effects are found. One consequence is that a preliminary transformation of the scattering amplitude is required before applying the Poisson representation to obtain fast convergence.

This transformation, first employed by Debye (1908) in scattering by a circular cylinder, is analogous to the multiple internal reflection treatment of the Fabry–Perot interferometer (Born & Wolf 1959). In a geometrical-optic approximation, it corresponds to the ray-tracing procedure illustrated in fig. 2.1: the scattering amplitude is decomposed into an infinite series of terms representing the effects of successive internal reflections.

In the present chapter, we discuss the results obtained by the CAM method for the first two terms of this series, that represent the effects of direct reflection from the surface and direct transmission through the sphere, employing transitional approximations (Nussenzveig 1969a, Khare 1975). One new feature found is the penetration of diffracted rays within the scatterer, taking ‘shortcuts’ through it before reemerging as surface waves.

The effective potential

The Mie scattering amplitudes were given in Sec. 5.1. Unless otherwise stated, we assume that the sphere is perfectly transparent, so that the relative refractive index N is real. This is the hardest case to treat: absorption damps out contributions from all but the shortest paths through the sphere and tends to greatly improve convergence; the CAM treatment remains valid for complex N.