Optical caustics and their modelling as singularities

Abstract

Optical caustics are bright patterns, formed by the local focalization of light rays. They are caused, for instance, by the reflection or the refraction of the sun rays through a wavy water surface. In the absence of an appropriate mathematical frame, their main characteristics have remained unrecognized for a long time and the caustics appeared in the literature under different names: evolutes, envelopes, focals, etc. The creation of the singularity theory in the middle of the 20th century radically changed the situation. Caustics are now understood as physical realizations of Lagrangian singularities. In this modelling, one predicts their local classification into five stable types (R. Thom, V. Arnold): folds, cusps, swallowtails, elliptic umbilics and hyperbolic umbilics. This local classification is indeed observed in experiments. However the global properties of the caustics are only partially taken into account by the Lagrangian model. In fact, it has been proved by Yu. Chekanov that the special form of the eikonal equation governing the propagation of the optical wave fronts imposes the existence of a topological constraint on the singular set (representing the caustic in the phase space) and restricts the number of possible bifurcations. Our experiments on caustics produced by bi-periodic structures in liquid crystals confirm the existence of the topological constraint, and validate the modelling of the caustics as special types of Lagrangian singularities.