Caustic produced by multiple reflections on a parabolic mirror

The reflection of a light beam by a mirror always produces a caustic, which is the envelope of the reflected rays. However, the concrete realization of this optical phenomenon always imposes some constraints, like the finite size of the mirror or the multiple reflections on the mirror, that must be taken into account. We analyze this problem in the case of a parabolic mirror. We show that the complete caustic is not a simple curve, but rather a complex ensemble of real and virtual parts that are connected on the mirror by special types of points, the grazing point and abutting points. We show that our results are not linked to the special shape of the parabolic mirror but are also valid for general caustics of the plane produced by multiple reflections