Singularities of wave fronts at the boundary between two media (Q1896961)

This paper is in substance an abstract of the authors's more extensive work [Algebra Anal. 5, No. 4, 149-169 (1993; Zbl 0799.58011)]. This abstract contains basic notations and definitions from contact geometry, which are used in the description of wave fronts and caustics. When a system of optical rays normal to the initial wave front passes through the smooth boundary between two media, a caustic arises at the point of complete reflection and the refracted wave front becomes singular. The typical singularities of such fronts are studied. Let us consider the propagation of a two-dimensional front in three- dimensional space, which is divided by a smooth surface \(\mathcal S\) onto two homogeneous and isotropic media. The author proves that for generic initial front and boundary \(\mathcal S\), the wave front at a typical time in some neighborhood of the complete reflection point is diffeomorphic with the discriminant of some cubic polynomial.