Geometric mechanics of ray optics as particle dynamics: refraction index with cylindrical symmetry

Abstract

Starting from the Fermat principle of geometrical optics, we analyse the ray dynamics in a graded refractive index system device with cylindrical symmetry and a refractive index that decreases parabolically with the radial coordinate. By applying Hamiltonian dynamics to the study of the ray path we obtain the strict equivalence of this optical system with the dynamics of a particle with an equivalent mass moving in a potential function that may exhibit a well, depending on the value of some associated parameters. We analyse the features of this potential function as well as the energy values and the symmetries of the system and see that both the azimuthal and axial components of the optical conjugate momentum are two constants of motion. The phase space relation for the momentum radial component is obtained analytically, and then we can obtain the components of the momentum vector at any point, given the value of the radial coordinate, and from this we have the direction of the ray. We discuss the optical path length as an action functional and we can evaluate this stationary path, with initial and final arbitrary points, as a line integral of the optical momentum, by showing that this momentum is a conservative vector field. We integrate the equations of motion numerically and obtain different ray paths which depend on the initial conditions. We believe that with this work the physics student will appreciate very clearly the close connection between geometrical optics and particle Hamiltonian dynamics.