Wavefronts, actions and caustics determined by the probability density of an Airy beam

Abstract

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.

Classical Characterization of quantum waves: Comparison between the caustic and the zeros of the Madelung-Bohm potential

“…Additionally, we find that the physical phase for the Airy beam can be obtained from a geometric perspective since the physical phase of the beam and the geometrical wavefronts coincide over the caustic. We conjecture that these results are linked with the observations made by Espíndola-Ramos et al [4], González-Juárez and Silva-Ortigoza [33], for the Airy and Bessel beams; they found a family of caustics which characterize in qualitative and quantitative form all the maxima of the Probability Density Function. On the other hand, we find that if there are rays that cross the caustic in a non-tangent form, then a non-uniform interference pattern is generated near to the caustic, so that The zeros of the Madelung-Bohm potential (dashed red lines) and the caustic (blue curve).…”

“…If S(x, P, t) is a solution to both the Hamilton-Jacobi equation and Laplace equation, then Φ(x, t) satisfies the Schrödinger equation for an arbitrary real function O(P ) [4,26]. Under these conditions, the rays are determined from the stationary points of S(x, P, t), that is to say…”

“…In recent works, Espíndola-Ramos et al [4], González-Juárez and Silva-Ortigoza [33], showed that the Airy and Bessel beams can be completely described from a geometrical perspective (or semiclassical perspective). The Hamilton-Jacobi theory suggests a generalization for the rays Eq.…”

“…The geometric description of a wavefunction is a practical tool to study wave propagation in quantum mechanics and optics, and even in the general relativity background. The knowledge of the set of rays and geometrical wavefronts associated with the wavefunction enables us to study a series of properties and approximations such as self-reconstruction [1,2], the classical description of the beam [3,4], image formation, lens design and ronchigrams [5,6,7,8], and even an approximation for the field associated with the refraction phenomena [9]. Nowadays, the research on the engineering of structured beams has an astonishing impulse due to its applications [10,11], from nanoparticle manipulation and nanotechnology applications [12,13,14,15,16] to the enhancement of communication systems [17,18,19], high-resolution observation [20,21] and metrology [22,23,24], to name a few.…”

Geometrical characterization of nondiffracting beams: geometric optical current and geometric vorticity for Bessel beams

“…On this line of research, it has been shown that from a geometrical optics point of view, each nondiffracting beam is characterized by a particular family of wavefronts and a caustic, which are invariant under a translation along the axis of propagation [15,16]. In the case of the Bessel, parabolic and Airy beams [17,18], the caustic provides a qualitative description of the maximum of the intensity, and although this is not true in general for the Mathieu beams [19], it is possible that this geometric analysis gives information for other known types of nondiffracting beams, such as hypergeometric beams [20], generalization of classical beams [21][22][23] or even beams with no analytical representation [24,25]. Therefore, we conclude that the geometrical optics approximation to an optical field is good enough if the caustic gives a qualitative description of the maximum of the intensity pattern.…”