North-Holland Lie-Poisson description of Hamiltonian ray optics

We express classical Hamiltonian ray optics for light rays in axisymmetric f bers as a Lie-Poisson dynamical system defined in R 3, regarded as the dual of the Lie algebra sp(2, •). The ray-tracing dynamics i interpreted geometrically as motion in ~3 along the intersections of two-dimensional level surfaces of the conserved optical Hamiltonian and the skewness invariant (the analog of angular momentum, conserved because of the axisymmetry of the medium). In this geometrical picture, a Hamiltonian level surface is a vertically oriented cylinder whose cross section describes the radial profile of the refractive index, and a level surface of the skewness function is a hyperboloid ofrevolution around a horizontal axis. Points of tangency of these surfaces are equilibria, which are stable when the Gaussian curvature of the Hamiltonian level surface (constrained by the skewness function) is negative definite at the equilibrium point. Examples are discussed for various radial profiles of the refractive index. This discussion places optical ray tracing in fibers into the geometrical setting of Lie-Poisson Hamiltonian dynamics and provides an example of optical ray trapping within separatrices (homoclinic orbits). 1. Optical phase space The phase space of geometrical optics is four-dimensional. Referred to a standard planar screen, a ray is determined by two position coor-dinates q = (qx, qy) defining its intersection with the screen, and two momentum coordinates p = (Px, Py) that cue the projection onto the screen of a three-vector ff tangent o the ray, whose length n(q) is the refractive index of the medium at that point. See fig. 1. Only rays perpendicular to the optical axis cannot be parametrized in this man-ner. The coordinate normal to the screen, z, ex-tends along the optical axis. The projection along the optical axis of the vector if(q, z)-wh ich in general is allowed to depend on z- is given b