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In this article, the ray tracing method is studied beyond the classicalgeometrical theory. The trajectories are here regarded as geodesics in aRiemannian manifold, whose metric and topological properties are those inducedby the refractive index (or, equivalently, by the potential). First, we derivethe geometrical quantization rule, which is relevant to describe the orbitingbound-states observed in molecular physics. Next, we derive properties of thediffracted rays, regarded here as geodesics in a Riemannian manifold withboundary. A particular attention is devoted to the following problems: (i)modification of the classical stationary phase method suited to a neighborhoodof a caustic; (ii) derivation of the connection formulae which enable one toobtain the uniformization of the classical eikonal approximation by patching upgeodesic segments crossing the axial caustic; (iii) extension of the eikonalequation to mixed hyperbolic-elliptic systems, and generation of complex-valuedrays in the shadow of the caustic. By these methods, we can study the creepingwaves in diffractive scattering, describe the orbiting resonances present inmolecular scattering beside the orbiting bound-states, and, finally, describethe generation of the evanescent waves, which are relevant in the nuclearrainbow.

Geometrical theory of diffracted rays, orbiting and complex rays