Fundamentals of the Physical Theory of Diffraction, 2nd Edition

6.4 Bodies of Revolution with Nonzero Gaussian Curvature: Backscattered Focal Fields

In this section we study symmetrical scattering at bodies of revolution whose illuminated side is an arbitrary smooth convex surface with nonzero Gaussian curvature. A generatrix of such a surface and related geometry are shown in Figure 6.13. The incident plane wave (6.1) propagates in the positive direction of the z-axis, which represents the symmetry axis of a scattering object. We use two systems of coordinates: cylindrical coordinates ρ, φ, z and spherical coordinates r, ϑ, φ. The generatrix is given as a function ρ = ρ(z). It is assumed that d²ρ/dz² ≠ 0 for the illuminated side (0 ⩽ z ⩽ l) of the object. This condition ensures that the Gaussian curvature of this surface is not zero. We also utilize the following notation related to the edge points (ρ = a): dρ/dz = tan ω for the illuminated side (z = l − 0) and dρ/dz = −tan Ω for the shadowed side (z = l + 0). The shadowed side is an arbitrary smooth surface with 0 ⩽ Ω ⩽ π − ω. In the limiting case Ω = π − ω, the scattering object is an infinitely thin (but still perfectly reflecting) screen ρ = ρ(z) with 0 ⩽ z ⩽ l.

Figure 6.13 Generatrix of the body of revolution.

The principal radii of curvature of the scattering surface are determined according to the differential geometry (Bronshtein and Semendyaev, 1985):

(6.121)

where ρ' = ...