Wavefronts and caustic associated with Durnin's beams

The aim of the present work is to give a geometrical characterization of Durnins beams. That is, we compute the wavefronts and caustic associated with the nondiffracting solutions to the scalar wave equation introduced by Durnin. To this end, first we show that in an isotropic optical medium ψ(r, t) = e^i[k0S(r)-wt] is an exact solution of the wave equation, if and only if, S is a solution of both the eikonal and Laplace equations, then from one and two-parameter families of this type of solution and the superposition principle we define new solutions of the wave equation, in particular we show that the ideal nondiffracting beams are one example of this type of construction in free space. Using this fact, the wavefronts and caustic associated with those beams are computed. We find that their caustic has only one branch, which is invariant under translations along the direction of evolution of the beam. Finally, the Bessel beam of order m is worked out explicitly and we find that it is characterized by wavefronts that are deformations of conical ones and the caustic is an infinite cylinder of radius proportional to m. In the case m = 0, the wavefronts are cones and the caustic degenerates into an infinite line.