From Airy rings to the elliptic umbilic diffraction catastrophe

The universal three-dimensional diffraction patterns associated with generic optical caustics are built on the loci of their line zeros—features variously known as wave dislocations, phase singularities, or optical vortices. All such patterns may be regarded as natural evolutions from the Fraunhofer diffraction pattern of concentric Airy rings surrounding the point focus of a lens. This paper considers as an example the elliptic umbilic caustic; it describes in detail, from numerical computation, how the simple planar system of Airy rings can change continuously into the much more complicated three-dimensional dislocation pattern decorating the caustic. The change is accomplished by the repeated use of five elementary changes in topology, which are treated analytically.