Kepler's law for optical beams

It is well known that optics and classical mechanics are intimately related. One of the most important concepts in classical mechanics is that of a particle in a central potential that leads to the Newtonian description of the planetary dynamics. Within this, a relevant result is Kepler's second law that is related to the conservation of orbital angular momentum, one of the fundamental laws in physics. In this paper, we demonstrate that it is possible to find the conditions that allow us to state Kepler's second law for optical beams with orbital angular momentum by analyzing the streamlines of their energy flow. We find that the optical Kepler's law is satisfied only for cylindrical symmetric beams in contrast to the classical mechanics situation that is satisfied for the other conic geometries, namely, parabolic, elliptical and hyperbolic. We propose a novel approach to confirm our analytic results: we observe the propagation of the Arago's spot created by a beam with orbital angular momentum as a local "light-tracer" instead of looking at the propagation of the whole beam. The observed patterns fully agree with the prediction of our formalism.