Paraxial optical fields whose intensity pattern skeletons are stable caustics

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

“…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…”

“…For this reason the geometrical wavefronts associated with Φ(x, t) are defined as the level curves of S(x, t). If the wavefunction propagates on free space, the set of singular points of the geometrical wavefronts is the caustic, which does not only characterizes the region of maximum contribution to the Probability Density Function but also determines the stability of the beam [25] and it represents the most classical description of the wavefunction Φ(x, t) since its evolution is governed by the Hamilton-Jacobi equation [26]. It is worth noting that the caustic represents the region of transition on which the number of rays that intersect on each point of space-time, changes.…”

“…( 14), the Madelung-Bohm potential is zero over the caustic, and the Hamilton-Jacobi equation is recovered at those points. Thus, the wavefunctions with this type of caustic are the most classical beams, because the evolution of the caustic, and then the evolution of the zeros of the Madelung-Bohm potential, are governed by the Hamilton-Jacobi equation [26]. Furthermore, the evolution in time of the maxima and minima points of the Probability Density Function travels with the caustic, forming a uniform interference pattern on the space-time framework (see, for example, figure 3).…”

“…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. In all these cases the caustic (that is, the envelope of the rays) plays an important role since it does not only characterizes the region of maximum contribution to the intensity but also it determines the stability of the beam [25] and is the most classical description of the wavefunction since its evolution is governed by the Hamilton-Jacobi equation [26]. On the other hand, the Madelung-de Broglie-Bohm equations of quantum mechanics also enables us to associate a classical description to a wavefunction through a modified Hamilton-Jacobi equation, which contains an additional potential (in comparison to the well known Hamilton-Jacobi equation) that we call as Madelung-Bohm potential [27,28].…”

Structural stability of open vortex beams

“…The connection between the ray and wave picture of light is given by the eikonal equation at the high-frequency limit 20 . Such a theoretical consistency is sometimes referred to as the ray-wave duality, which provides useful ideas and techniques for tailoring new types of structured laser beams, e.g., ray-optical Poincaré spheres for structured beams [21][22][23] , propagation-invariant light with shaped caustics 24 , SU(2) geometric modes generated from cavities 25 and holograms 26 , high-dimensional classically entangled light 27 , and new interpretations of self-accelerating beams 28,29 . Inevitably, vortex beams, as an essential class of structured light, can be analyzed in the framework of ray-wave duality.…”