Ray tracing in a general nonhomogeneous cylindrical medium

The complete ray equations are derived for a generalised cylindrical co-ordinate system obtained by the translation of a plane orthogonal system parallel to the cylinder axis. The condition that a ray is confined to a surface of a constant co-ordinate value is shown to lead to complete integrability of the reduced equations. Ray traces can then be made in specific refractive-index media, such as the parabolic. Alternatively, specified ray behaviour such as coherent refocusing, gives the required refractive index. In the latter case, this is shown to be a generalisation of the sech law for a circularly symmetrical medium.