Analytic solutions are found for skew ray paths in inhomogeneous cylindrical systems such as GRIN rods and gas lenses, which focus rays by virtue of a gradual radial decrease in refractive index. The solutions are obtained by employing digital computers to carry out analytic, as opposed to numerical, calculations.
K. B. Paxton and W. Streifer, "Analytic Solution of Ray Equations in Cylindrically Inhomogeneous Guiding Media. Part 2: Skew Rays," Appl. Opt. 10, 1164-1171 (1971)