Nonlinear cusp diffraction catastrophe and vortex quadrupoles from a smooth initial beam

Self-action effects such as self-defocusing and self-bending are well-known nonlinear optical phenomenon dating back to the 1960s. Until recently the so-called blooming or lensing effect of a symmetric beam has been a main topic of investigation. A smaller number of experiments using asymmetric beams were reported (see, for example Khoo, Golub, and Swartzlander.) Whereas many early experiments revealed complex patterns, an explanation of the observed phenomena predated modern mathematical tools and paradigms for describing complex pattern formations, e.g. the theory of optical catastrophes. In recent years the formation of optical vortices has guided much work in nonlinear refractive optics (see, for example, Refs. 18, 57,