The caustic structure near a grazing point in the plane

A grazing point of a concave mirror illuminated by a large beam of rays is a point of the mirror's edge where the incident ray is parallel to the mirror. In the neighborhood of such point the rays are reflected a great number of times. We show that an ordered series of caustics passes through the grazing point, each caustic corresponding to a fixed number of reflections by the mirror. We study, in the framework of planar geometrical optics, the structure of this remarkable set of caustics. Our main result is a formula giving the curvature of the caustic curves at the grazing point as a function of the number of reflections. This sequence is universal in the sense that it is independent of the shape of the incident wavefront. A grazing point in the plane is an unstable point and we show how the caustic structure is modified under the effect of a small perturbation of the optical system.