Evaluation of cuspoid and umbilic diffraction catastrophes of codimension four

The evaluation of the two diffraction catastrophes of codimension four, namely, the butterfly and the parabolic umbilic, is here proposed by means of a simple computational approach developed in the past to characterize the whole hierarchy of the structurally stable diffraction patterns produced by optical diffraction in three-dimensional space. In particular, after expanding the phase integral representations of butterfly and parabolic umbilic in terms of (slowly) convergent power series, the retrieving action of the Weniger transformation on them is investigated through several numerical experiments. We believe that the methodology and the results presented here could also be of help for the dissemination of catastrophe optics to the widest scientific audience.