The aim of the present work is to give a geometrical characterization of Durnin’s 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
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.
The main contribution of the present work is to use the probability density of an Airy beam to identify its maxima with the family of caustics associated with the wavefronts determined by the level curves of a one-parameter family of solutions to the Hamilton–Jacobi equation with a given potential. To this end, we give a classical mechanics characterization of a solution of the one-dimensional Schrödinger equation in free space determined by a complete integral of the Hamilton–Jacobi and Laplace equations in free space. That is, with this type of solution, we associate a two-parameter family of wavefronts in the spacetime, which are the level curves of a one-parameter family of solutions to the Hamilton–Jacobi equation with a determined potential, and a one-parameter family of caustics. The general results are applied to an Airy beam to show that the maxima of its probability density provide a discrete set of: caustics, wavefronts and potentials. The results presented here are a natural generalization of those obtained by Berry and Balazs in 1979 for an Airy beam. Finally, we remark that, in a natural manner, each maxima of the probability density of an Airy beam determines a Hamiltonian system.
From a geometric perspective, the caustic is the most classical description of a wave function since its evolution is governed by the Hamilton–Jacobi equation. On the other hand, according to the Madelung–de Broglie–Bohm equations, the most classical description of a solution to the Schrödinger equation is given by the zeros of the Madelung–Bohm potential. In this work, we compare these descriptions, and, by analyzing how the rays are organized over the caustic, we find that the wave functions with fold caustic are the most classical beams because the zeros of the Madelung–Bohm potential coincide with the caustic. For another type of beam, the Madelung–Bohm potential is in general distinct to zero over the caustic. We have verified these results for the one-dimensional Airy and Pearcey beams, which, according to the catastrophe theory, have stable caustics. Similarly, we introduce the optical Madelung–Bohm potential, and we show that if the optical beam has a caustic of the fold type, then its zeros coincide with the caustic. We have verified this fact for the Bessel beams of nonzero order. Finally, we remark that for certain cases, the zeros of the Madelung–Bohm potential are linked with the superoscillation phenomenon.
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