Consider a refracted rather than reflected beam of light. Recall that Snell's law of refraction of light crossing the boundary between two media (i.e. our curve) tells that for a given pair of media, n₁*sin(θ₁)=n₂*sin(θ₂), where n₁ and n₂ are the indices of refraction of the media and θ₁ and θ₂ are the angles of incidence and refraction (angles between the corresponding beams and the normal to the curve). Make a model of such a beam and explore the caustics you obtain.