The Double-cusp Unfolding of the 0 X 9 Diffraction Catastrophe

Abstract

We define the double-cusp unfolding to be that four-dimensional unfolding of the codimension-8 unimodal singularity ⁰X₉ for which the generating function is separable into a sum of two cusp catastrophe generating functions, one in each of the two state variables. We have made an experimental and theoretical study of the double-cusp unfolding, ψ_dc, of the ⁰X₉ diffraction catastrophe by photographing plane sections of ψ_dc produced in light refracted by the water surface of a ripple tank, and comparing them with computer simulations from numerical evaluations of the diffraction integral. Experiment and theory are in excellent agreement. Computation shows that the intensity maximum is not at the geometrical focus, and we prove this result analytically. The product structure of ψ_dc has direct consequences for the wavefront dislocations, which are two orthogonal families of parallel straight lines. We also report observations of a five-dimensional unfolding of the ⁰X₉ diffraction catastrophe, which we relate to ψ_dc dc and to the underlying geometrical caustics.