Ray paths in a nonuniform axially symmetric medium

The general equation for rays in a nonuniform medium is expanded in full for an axially symmetric refractive index in which the rays are confined to a selected co-ordinate surface. A solubility condition arises which limits the general solution, and the simplest form discovered is a spherical medium with both radial and angular variations. The ray equations for this case are integrable, giving a generalisation of Snell´s law, and the ray paths for arbitrary surface refractive-index laws. A class of surface functions is derived for which the rays from a source on the equator refocus at points around the equator. This process can be continued into the equatorial plane giving 3-dimensional refocusing around a circle. The refractive-index law, and its analogue as a potential, have several possible applications in particle dynamics, terrestrial radio propagation and acoustics. As an optical-guiding medium it could be used for curved fibres with a small radius of curvature.