Keller's geometrical theory of diffraction

Definition

Let $u$ be a time-harmonic field, a solution of the Helmholtz equation $\Delta u + k^2 n^2 u = 0$ outside an obstacle, with large wavenumber $k$. Keller's geometrical theory of diffraction approximates $u$, as $k \to \infty$, by a sum over rays $u \approx \sum_j A_j \, e^{i k L_j}$ built by these rules.

1. The rays are the incident, reflected and refracted rays of geometrical optics, together with diffracted rays. A diffracted ray starts where a ray hits an edge or a vertex, or grazes a smooth surface.
2. Diffracted rays obey a generalised Fermat's principle: they are paths of stationary optical path length among the paths that touch the edge, vertex or surface.
3. Along each ray the phase is $k L_j$, where $L_j$ is the optical path length. The amplitude $A_j$ changes so that energy is conserved in each narrow tube of rays.
4. Where a diffracted ray starts, its amplitude is the incident field times a diffraction coefficient. This coefficient comes from the asymptotics of an exactly solvable model problem with the same local geometry.