A two-dimensional inverse problem of geometrical optics

In the framework of geometrical optics we consider the inverse problem consisting of obtaining refractive indexes n = n ( x , y ) of a two-dimensional transparent heterogeneous isotropic (dispersive or not) medium from a known (observed or given) family f ( x , y ) = c 0 of planar light rays of a definite colour. We establish a first-order linear partial differential equation relating the assigned family of light rays with all possible refractive indexes compatible with this family. Using this equation we derive certain criteria to check whether a given family of rays can be traced in the presence of a refractive index, which we assume in advance to be either radial or homogeneous of any degree m . We give appropriate examples for the two special cases and also an example for the general case.