Exact and geometrical optics energy trajectories in Bessel beams via the quantum potential

Abstract

The aim of the present work is to show that any monochromatic solution to the scalar wave equation in free space defines a conservative Hamiltonian system, describing a particle of mass m = 1 / 2 and energy E = 1 , under the influence of the so-called quantum potential. We remark that the integral curves of its Poynting vector, exact optics energy trajectories, define a particular subset of solutions to the corresponding Hamilton equations. Furthermore, we introduce the zero quantum potential straight lines concept, as the family of tangent lines to the integral curves of the Poynting vector at the zeros of the quantum potential. These general results are applied to a family of plane waves and to Bessel beams. In the case of Bessel beams, we present a detailed study of the trajectories determined by the corresponding Hamiltonian system, and we show that the zero quantum potential straight lines coincide with the geometrical light rays, geometrical optics energy trajectories. Furthermore, we show that the areal velocity, determined by the exact optics energy trajectories, for non-zero order Bessel beams is not a constant of motion. However, its projection along the z ^ direction is a constant of motion because L z is a constant.