Geometrical theory of diffracted rays, orbiting and complex rays

Abstract

In this article, the ray tracing method is studied beyond the classical geometrical theory. The trajectories are here regarded as geodesics in a Riemannian manifold, whose metric and topological properties are those induced by the refractive index (or, equivalently, by the potential). First, we derive the geometrical quantization rule, which is relevant to describe the orbiting bound-states observed in molecular physics. Next, we derive properties of the diffracted rays, regarded here as geodesics in a Riemannian manifold with boundary. A particular attention is devoted to the following problems: (i) modification of the classical stationary phase method suited to a neighborhood of a caustic; (ii) derivation of the connection formulae which enable one to obtain the uniformization of the classical eikonal approximation by patching up geodesic segments crossing the axial caustic; (iii) extension of the eikonal equation to mixed hyperbolic-elliptic systems, and generation of complex-valued rays in the shadow of the caustic. By these methods, we can study the creeping waves in diffractive scattering, describe the orbiting resonances present in molecular scattering beside the orbiting bound-states, and, finally, describe the generation of the evanescent waves, which are relevant in the nuclear rainbow.Comment: 30 pages, 2 figure

Transition from resonances to surface waves in pi^+-p elastic scattering

“…Furthermore, there is an additional phaseshift between the rays (counterclockwise and clockwise) which explains the factor −i = e −iπ/2 for the counterclockwise rays and the factor i 2 = −1 for the clockwise ones (see Ref. [33] for a detailed mathematical analysis). The term ν 1 is given by:…”

Width estimation of a subsurface water-filled crack using internal multiple reflections from electromagnetic scattering model of a cylinder

“…The electromagnetic (EM) scattering of a dielectric cylinder with circular cross-section by means of transient time-domain technique has been made great progress [11][12][13][14][15]. However, for the case of a dielectric cylinder with electric size comparable to the working wavelength, the geometrical theory of diffraction interpretation and internal multiple reflections should be also considered in order to obtain a physical point of view of the scattering characteristics of such target [16][17][18][19]. An excellent research on such problem of EM scattering from a dielectric cylinder lying in a dielectric medium was studied by Ludwig and Leviatan [11].…”