Ray-transfer matrix for GRIN lenses: application to the crystalline lens

We present a Hamiltonian optics approach to determine the paraxial propagation of rays in gradient-index (GRIN) media. A derivation of the evolution equation for the corresponding ray-transfer matrix is given. An analytical solution can be found for a certain class of GRIN media. The proposed matrix is shown to be a generalization of the well-known matrices for planar and cylindrical GRIN lenses. We provide the accuracy results when applying our analytical method to arbitrary refractive indices and verify its validity by comparing it to other matrices in the literature with satisfactory results.