Singularities in a teacup

Abstract

Standard presentations of optics concentrate on ideal systems made for imaging which bring all rays from a point source to one focus. But, in Nature, or in realistic optical systems with defects, rays do not behave precisely in this way. Rather than the focus simply being blurred, the rays, after reflection or refraction, form beautiful and rather universal patterns of bright lines known as caustics. Mathematically speaking, a family of rays is best viewed as a surface in a higher-dimensional space where we keep track of both the position and direction of rays. The intensity enhancement on approaching the caustic line is a singularity, arising from projection of a smooth surface from higher dimensions to lower dimensions. The universal features of such singularities, which arise in many contexts beyond optics, formed a major theme of Vladimir Arnold’s work after 1965, when he was exposed to René Thom’s vision of ‘catastrophe theory’. Arnold and his school made seminal contributions to singularity theory.