IOP PUBLISHING JOURNAL OF OPTICS A: PURE AND APPLIED OPTICS J. Opt. A: Pure Appl. Opt. 10 (2008) 035005 (7pp) doi:10.1088/1464-4258/10/3/035005 Exact and geometrical optics energy trajectories in twisted beams M V Berry 1 and K T McDonald 2 1 H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK 2 Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA Received 18 January 2008, accepted for publication 5 February 2008 Published 28 February 2008 Online at stacks.iop.org/JOptA/10/035005 Abstract Energy trajectories, that is, integral curves of the Poynting (current) vector, are calculated for scalar Bessel and Laguerre–Gauss beams carrying orbital angular momentum. The trajectories for the exact waves are helices, winding on cylinders for Bessel beams and hyperboloidal surfaces for Laguerre–Gauss beams. In the geometrical optics approximations, the trajectories for both types of beam are overlapping families of straight skew rays lying on hyperboloidal surfaces; the envelopes of the hyperboloids are the caustics: a cylinder for Bessel beams and two hyperboloids for Laguerre–Gauss beams. Keywords: Poynting, rays, diffraction 1. Introduction One of several ways to depict optical fields is by the trajectories of energy flow, that is, the lines everywhere tangent to the Poynting vector. Here we will describe light by a scalar wave ψ(r), constructed for example from the vector potential [1, 2], or representing a single field component, or simply regarded as a physical model in which polarization effects are neglected. Then the Poynting vector can be chosen parallel to the current, that is, the expectation value of the local momentum operator, namely P (r) = Im ψ ∗ (r) ∇ψ(r) =|ψ(r)| 2 ∇ arg ψ(r). (1.1) The second equality expresses the fact that in vacuum P (r) is orthogonal to the wavefronts (contour surfaces of phase arg ψ ). P (r) is an important ingredient in calculating the orbital angular momentum of the field [3], and forces on small particles in the field. In quantum physics, the trajectories are the streamlines in the Madelung [4] hydrodynamic interpretation, later regarded as paths of quantum particles in the Bohm–de Broglie interpretation [5]. For optical beams, where there is a well-defined propagation direction z , it is natural to represent the trajectories using z as a parameter, and the field using cylindrical polar coordinates {r,φ, z }. Then, writing P (r) in the form P (r) = F (r)  v r (r) e r + v φ (r) e φ + e z  , (1.2) the trajectories are the integral curves of P , to be obtained by solving the differential equations dr (z ) dz ≡ r ′ (z ) = v r (r (z )) , dφ(z ) dz ≡ φ ′ (z ) = v φ (r (z )) r (z ) . (1.3) Our aim here is to understand the energy trajectories for Bessel and Laguerre–Gauss beams carrying orbital angular momentum (‘twisted beams’), which are of current interest theoretically and experimentally, building on and extending previous studies [6, 7]. We emphasize a fundamental point, central to the understanding of energy flow: the equation (1.1) can be interpreted in two quite different ways, both of which we will employ in the following. In the first (sections 2 and 3), ψ(r) is the exact solution of the relevant wave equation; then P (r) has the advantage that it represents without approximation the flow described by the wave equation. In the second (sections 4 and 5), the lines of P (r) represent the rays of geometrical optics, which although approximate carry the intuitive appeal that their envelopes are the caustic surfaces on which the field is most intense. In this case, ψ(r) represents one of possibly several locally plane waves that are superposed to create the total field. When points in the field are reached by several geometrical rays, the corresponding pattern of trajectories overlap, unlike the exact Poynting trajectories of the total field, which are 1464-4258/08/035005+07$30.00 © 2008 IOP Publishing Ltd Printed in the UK 1