Use of Diffraction Gratings with Curved Lines to Study the Optical Catastrophes D⁺₆ and D⁻₆

Abstract

Plane diffraction gratings having curved lines can be used to produce wavefronts of predetermined shape. This makes it possible to generate experimentally optical caustics corresponding to any specified catastrophe. The method, invented by Lee, is applied here to two catastrophes of codimension 5 that have not previously been studied experimentally, the second hyperbolic umbilic (D⁺ ₆) and the second elliptic umbilic (D⁻ ₆). By varying the choice of two parameters, E and S, controlling the shape of the lines on the grating, two-dimensional sections of all the unfoldings, with their diffraction patterns, may be produced. One or three umbilic points are present on the first-order wavefront emerging from the grating, and as the parameters E and S are changed they move and react together. It is shown that the wavelength dependence of a diffraction catastrophe can be studied by keeping the radiation the same but observing the different orders of diffraction.