Equivalence properties of lenslike media with parabolic index profiles are analyzed on the basis of geometrical optics. By equivalent media we mean here those with the same ray-transfer matrix representation. An equivalence theorem is presented that provides a simple method of equivalent transformation of parabolic media. The practical importance of the equivalent transformation is that it leads to flexibility in the design of distributed optical structures, such as gas lenses and inhomogeneous optical fibers.
S. Yamamoto and T. Makimoto, "Equivalence Relations in a Class of Distributed Optical Systems–Lenslike Media," Appl. Opt. 10, 1160-1163 (1971)