Three-dimensional inverse problem of geometrical optics: A mathematical comparison between Fermat's principle and the eikonal equation

In the framework of geometrical optics, we consider the following inverse problem: given a two-parameter family of curves (congruence) (i.e., f (x,y,z) = c1,g(x,y,z) = c2), construct the refractive-index distribution function n = n(x,y,z) of a 3D continuous transparent inhomogeneous isotropic medium, allowing for the creation of the given congruence as a family of monochromatic light rays. We solve this problem by following two different procedures: 1. By applying Fermat's principle, we establish a system of two first-order linear nonhomogeneous PDEs in the unique unknown function n = n(x,y,z) relating the assigned congruence of rays with all possible refractive-index profiles compatible with this family. Moreover, we furnish analytical proof that the family of rays must be a normal congruence. 2. By applying the eikonal equation, we establish a second system of two first-order linear homogeneous PDEs whose solutions give the equation S(x,y,z) = const. of the geometric wavefronts and, consequently, all pertinent refractive-index distribution functions n = n(x,y,z). Finally, we make a comparison between the two procedures described above, discussing appropriate examples having exact solutions.