Raman and the mirage revisited: confusions and a rediscovery

Abstract

Raman argued that in a continuously varying layered medium, such as air above a hot road, a ray that bends so as to become horizontal must remain so, implying that the reflection familiar in the mirage cannot be explained by geometrical optics. This is a mistake, as standard ray curvature arguments demonstrate. But a simple limiting process, in which the smoothly varying refractive index is approximated by a stack of thin discrete layers, is not quite straightforward because it involves a curious singularity, related to the level ray envisaged by Raman. In contrast to individual rays, families of rays possess caustic (focal) singularities. These can be calculated explicitly for two families of rays that are relevant to the mirage. Only exceptionally does the locus of reflection (lowest points on the rays) coincide with the caustics. Caustics correspond to the ‘vanishing line’, representing the limiting height of objects that can be seen by reflection. For these two families, the waves that decorating mirage caustics are described by the universal Airy function, and can be calculated exactly.