The relation between the spherical aberration of a lens and the spun cusp diffraction catastrophe

Abstract

A lens with spherical aberration, illuminated with an axial plane wave, produces a rotationally-symmetric cusped caustic together with an axial caustic line. Both caustics are truncated by the finite aperture of the lens, and they are decorated by diffraction. One may pass continuously from the limit of small aperture, where the diffraction pattern consists simply of Airy rings around the focus, to the limit of infinite aperture, where the diffraction pattern is that of the three-dimensional spun cusp. This contains ring zeros both inside and outside the cusped caustic. The rings are structurally stable phase singularities (wave dislocations), whose progress out of the focal plane can be traced as the aperture is enlarged. In any axial plane the dislocations are points. Before reaching their final destinations these dislocation points invariably trace out spirals, whose detailed form may be deduced by a perturbation theory. Apart from this, their trajectories, births and deaths are different from those encountered in the analogous case of the two-dimensional Pearcey pattern.