Transformation of spherical nonuniform lenses

The spherically symmetrical nonuniform refractive medium with index variation ¿(r) permits a special form of inversion which transforms ¿(r) into a new spherically symmetrical medium ¿(r) and directly transforms rays in the first medium into rays in the transformed medium. In the general form, the optical line element is scaled by the degree of the inversion. When this is unity, the transformation is the geometrical inversion of the three-dimensional space into itself. The results are illustrated in the cases of the well known archetype nonuniform lenses.