Dislocation lines in the swallowtail diffraction catastrophe

Abstract

The three-dimensional distribution of amplitude and phase represented by the swallowtail catastrophe diffraction integral is based on a network of null lines for amplitude (wave dislocations or optical vortices), where the phase is singular. In the tail region there is four-wave interference, which results in an approximately repeating pattern of amplitude based on the monoclinic space group C 2/ m and also an approximately repeating pattern of wave dislocations based on the black–white monoclinic space group C 2/ m ′. Helical dislocations spring from the plane of symmetry and gradually straighten out to be parallel to the two riblines of the caustic; eventually they become the straight dislocations of the Pearcey pattern for the cusp catastrophe. In the front region, where there are zero or two points of stationary phase, each dark Airy fringe surface associated with the fold surface condenses into a single dislocation in the plane of symmetry.