Analytical ray tracing based on Hamilton principal function and conjugate variable pairs

A transformation method based on optical Hamilton equations is proposed for 3D ray tracing in axially inhomogeneous gradient-index (GRIN) media with cylindrical symmetry. For a given GRIN field, the optical conjugate variable pairs of physical space can be transformed into a virtual space by applying canonical transformation. The virtual trace can be simply solved as a uniform expression regardless of what the GRIN field is, and one can inversely transform it into the physical space. The transformation is intimately related to a Hamilton principal function, called the function, which simultaneously gives the real ray trace and its conjugate. The "conjugate trace" displays the direction cosines of the ray trace and thus shows the information about propagation direction at every point, and it can be independently derived from the function without knowing the real trace. In addition, as two special dimension-reduction cases, the function is also applicable for 2D structures with only axial inhomogeneity or cylindrical symmetry.