Wave dislocations in the diffraction pattern of a higher-order optical catastrophe

Abstract

The paper explores a partial unfolding of the canonical three-dimensional diffraction field associated with the optical catastrophe X9 with modulus K = −6. A practical realization would be the focal region of a thin lens created by setting a drop of water on a horizontal glass slide and constraining its perimeter to be square. The pattern of caustics formed around the focus is a twisted and ribbed double trumpet with 4-fold symmetry. Like all diffraction catastrophes the essential structure is based on a pattern of line singularities (wave dislocations or optical vortices) on which the amplitude is zero and the phase is indeterminate. The caustic is encircled on the outside, and in the focal plane, by a highly puckered and non-circular ring and a forest of other dislocations. Far from the axis these are organized by the planar group 3m, despite the 4-fold symmetry. On the inside, the dislocation lines form a curved quasi-periodic lattice of small, nearly planar, nearly circular, rings based on the tetragonal space group I4mm. There are similarities to the pattern for the elliptic umbilic catastrophe, and, just as in that case, far from the focus the inner rings in lines close to the ribs of the caustic eventually join together to become the straight inner dislocations of the Pearcey pattern for the cusp. But the way in which this transition is accomplished, which involves four simultaneous reconnections, is quite different for the two catastrophes. Further, in the elliptic (and hyperbolic) umbilic catastrophes diffraction splits the focal spot longitudinally; in X9 with K = −6 it does not.