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arXiv:1912.01133 · Applied Sciences

Bohmian-Based Approach to Gauss-Maxwell Beams

Ángel S. Sanz, Milena Davidović, Mirjana Božić

Published Mar 6, 2020• 11 citations
arXivHTMLPDF

Abstract

Usual Gaussian beams are particular scalar solutions to the paraxial Helmholtz equation, which neglect the vector nature of light. In order to overcome this inconvenience, Simon et al. (J. Opt. Soc. Am. A 1986, 3, 536–540) found a paraxial solution to Maxwell’s equation in vacuum, which includes polarization in a natural way, though still preserving the spatial Gaussianity of the beams. In this regard, it seems that these solutions, known as Gauss-Maxwell beams, are particularly appropriate and a natural tool in optical problems dealing with Gaussian beams acted or manipulated by polarizers. In this work, inspired in the Bohmian picture of quantum mechanics, a hydrodynamic-type extension of such a formulation is provided and discussed, complementing the notion of electromagnetic field with that of (electromagnetic) flow or streamline. In this regard, the method proposed has the advantage that the rays obtained from it render a bona fide description of the spatial distribution of electromagnetic energy, since they are in compliance with the local space changes undergone by the time-averaged Poynting vector. This feature confers the approach a potential interest in the analysis and description of single-photon experiments, because of the direct connection between these rays and the average flow exhibited by swarms of identical photons (regardless of the particular motion, if any, that these entities might have), at least in the case of Gaussian input beams. In order to illustrate the approach, here it is applied to two common scenarios, namely the diffraction undergone by a single Gauss-Maxwell beam and the interference produced by a coherent superposition of two of such beams.

DOI
10.3390/app10051808

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Usual Gaussian beams are particular scalar solutions to the paraxial Helmholtz equation, which neglect the vector nature of light. In order to overcome this inconvenience, Simon et al. (J. Opt. Soc. Am. A 1986, 3, 536–540) found a paraxial solution to Maxwell’s equation in vacuum, which includes polarization in a natural way, though still preserving the spatial Gaussianity of the beams. In this regard, it seems that these solutions, known as Gauss-Maxwell beams, are particularly appropriate and a natural tool in optical problems dealing with Gaussian beams acted or manipulated by polarizers. In this work, inspired in the Bohmian picture of quantum mechanics, a hydrodynamic-type extension of such a formulation is provided and discussed, complementing the notion of electromagnetic field with that of (electromagnetic) flow or streamline. In this regard, the method proposed has the advantage that the rays obtained from it render a bona fide description of the spatial distribution of electromagnetic energy, since they are in compliance with the local space changes undergone by the time-averaged Poynting vector. This feature confers the approach a potential interest in the analysis and description of single-photon experiments, because of the direct connection between these rays and the average flow exhibited by swarms of identical photons (regardless of the particular motion, if any, that these entities might have), at least in the case of Gaussian input beams. In order to illustrate the approach, here it is applied to two common scenarios, namely the diffraction undergone by a single Gauss-Maxwell beam and the interference produced by a coherent superposition of two of such beams.
A companion paper Beetle and Jongewaard de Boer (The paraxial approximation in quantum optics I: henochromatic modes of a scalar field. J Mod Opt. 2025) has argued that the best way to associate single-particle quantum states of a scalar field to the modes of a narrowly collimated beam of classical radiation modelled in the paraxial approximation uses the ‘henochromatic’ states previously introduced by Sudarshan et al. (Paraxial-wave optics and relativistic front description. II. The vector theory. Phys Rev A. 1983;28:2921–2932. doi: 10.1103/PhysRevA.28.2921, Paraxial-wave optics and relativistic front description. I. The scalar theory. Phys Rev A. 1983;28:2933–2942. doi: 10.1103/PhysRevA.28.2921). This paper extends that result to Maxwell fields, again emphasizing the central role of unitarity in defining the association. The principal new technical element in the present discussion has to do with the intertwining of polarization and spatial degrees of freedom in the resulting single-photon states.
Interference is commonly regarded as the most direct manifestation of the superposition principle. This association is natural for the linear Schrödinger equation, where coherent alternatives combine at the level of probability amplitudes. However, the situation becomes less transparent when nonlinear couplings are present, or when the field is only partially coherent. In this work, we argue that a more robust organizing principle is provided by the local flow generated by phase variations. In this sense, phase-induced flow acts as a unifying mechanism for interference-like dynamics in nonlinear and partially coherent Schrödinger systems. The discussion is developed from a hydrodynamic, or Bohmian, perspective, understood here as a practical probing tool rather than as an additional ontology. Three representative situations are considered: interfering Bose--Einstein condensates described by the Gross--Pitaevskii equation, nonlinear Schrödinger dynamics obtained by modifying the quantum-potential contribution, and partially coherent Airy beams described through their cross-spectral density. Although these systems differ in physical origin and mathematical implementation, they share a common dynamical structure: density-related observables are shaped by velocity fields determined by phase, or ensemble-phase, information. From this viewpoint, interference-like traits, localization, self-acceleration and coherence loss can be interpreted in terms of the preservation, deformation or breaking of the symmetries displayed by the underlying flow. This provides a compact way of connecting interference, nonlinear dynamics, classicality, coherence loss, and structured-light propagation within a single trajectory-based framework.
With the extension of the work of the preceding paper, the relativistic front form for Maxwell's equations for electromagnetism is developed and shown to be particularly suited to the description of paraxial waves. The generators of the Poincar\'e group in a form applicable directly to the electric and magnetic field vectors are derived. It is shown that the effect of a thin lens on a paraxial electromagnetic wave is given by a six-dimensional transformation matrix, constructed out of certain special generators of the Poincar\'e group. The method of construction guarantees that the free propagation of such waves as well as their transmission through ideal optical systems can be described in terms of the metaplectic group, exactly as found for scalar waves by Bacry and Cadilhac. An alternative formulation in terms of a vector potential is also constructed. It is chosen in a gauge suggested by the front form and by the requirement that the lens transformation matrix act locally in space. Pencils of light with accompanying polarization are defined for statistical states in terms of the two-point correlation function of the vector potential. Their propagation and transmission through lenses are briefly considered in the paraxial limit. This paper extends Fourier optics and completes it by formulating it for the Maxwell field. We stress that the derivations depend explicitly on the "henochromatic" idealization as well as the identification of the ideal lens with a quadratic phase shift and are heuristic to this extent.
The quantum potential approach provides an intriguing perspective for studying optical beams, offering a more direct description of energy flow than conventional wave optics and connecting naturally to the de Broglie–Bohm interpretation of quantum mechanics. In recent years, this approach has been applied to structured light beams, yet its use in analyzing superposed light fields and interference phenomena remains largely unexplored. In this work, we develop an alternative formulation of the quantum potential that avoids direct calculation of amplitude and phase, and apply it to the interference of Hermite–Gaussian beams. A general analytical expression for the quantum potential of the interference field is derived, in which the total quantum potential is decomposed into a linear superposition term and an interference correction term modulated by the phase difference. Based on this expression, we investigate the mechanisms underlying the formation of optical vortices and interference fringes in Hermite–Gaussian beam interference. We show that the interference term creates sharp potential structures and strong quantum forces that guide the energy flow, leading to the formation of dark fringes, while phase differences capable of inducing angular momentum give rise to optical vortices. We also argue that the term ∂φ∂z\frac{\partial \varphi}{\partial z}∂z∂φ​ should not be neglected in the analysis of interference. Our results provide new theoretical tools for investigating interference phenomena in superposed structured light fields and highlight the decisive role of the quantum potential interference term, with potential applications in structured light, singular optics, and precision optical manipulation.
The quantum potential approach provides an intriguing perspective for studying optical beams, offering a more direct description of energy flow than conventional wave optics and connecting naturally to the de Broglie–Bohm interpretation of quantum mechanics. In recent years, this approach has been applied to structured light beams, yet its use in analyzing superposed light fields and interference phenomena remains largely unexplored. In this work, we develop an alternative formulation of the quantum potential that avoids direct calculation of amplitude and phase, and apply it to the interference of Hermite–Gaussian beams. A general analytical expression for the quantum potential of the interference field is derived, in which the total quantum potential is decomposed into a linear superposition term and an interference correction term modulated by the phase difference. Based on this expression, we investigate the mechanisms underlying the formation of optical vortices and interference fringes in Hermite–Gaussian beam interference. We show that the interference term creates sharp potential structures and strong quantum forces that guide the energy flow, leading to the formation of dark fringes, while phase differences capable of inducing angular momentum give rise to optical vortices. We also argue that the term ∂φ∂z\frac{\partial \varphi}{\partial z}∂z∂φ​ should not be neglected in the analysis of interference. Our results provide new theoretical tools for investigating interference phenomena in superposed structured light fields and highlight the decisive role of the quantum potential interference term, with potential applications in structured light, singular optics, and precision optical manipulation.
In this work we compute, via the quantum potential approach, the Hamiltonian system determined by Hermite–Gaussian beams. Then we show that the integral curves of the Poynting vector, exact optics energy trajectories, conform to a subset of solutions to the corresponding Hamilton equations lying on hyperboloidal surfaces. The geometrical light rays associated with these beams are given by the tangent lines to the integral curves of the Poynting vector at the zeroes of the quantum potential, and the caustic region coincides with the zeroes of quantum potential and quantum force. One of the main contributions of this work is to present the relationship between the physical phase k Φ, the geometrical-optics phase k Φ G , and the quantum potential Q HG in the Hermite–Gaussian beams. Furthermore, note that for any solution to the paraxial wave equation in free space, the tangent lines to the integral curves of the Poynting vector that correspond to the geometric light rays are those that pass through the points where the region determined by zeroes of the quantum potential is tangent to the geometrical caustic determined by the geometric light rays .