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.
Highlights
One of the most appealing features of geometrical optics, and a remarkable and convenient advantage, is perhaps the fact that this model relies on the well-defined and very intuitive concept of ray
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
The above experiments typically involve Gaussian beams and, coherent superpositions of such beams, which include a given polarization state. This leads to a series of natural questions—If a Gaussian beam is assigned a particular polarization state, is there a paraxial vector description for such a beam in the same way there is for a scalar one? If so, how does the flux associated with such a beam propagate along the optical axis and spread across the transverse directions? Or, how does the polarization state influence the interference between two of such beams while they evolve along the optical axis? With the purpose to provide an answer to these questions, here we develop a hydrodynamic description for Gauss-Maxwell electromagnetic beams, developed in 1989 by Simon et al [28], in terms of electromagnetic energy flow lines or rays
Summary
One of the most appealing features of geometrical optics, and a remarkable and convenient advantage, is perhaps the fact that this model relies on the well-defined and very intuitive concept of ray. This can readily be done if the role of the probability density is identified with the electromagnetic energy density, and the quantum density current or quantum flux [6] with the Poynting vector [7,8,9] This prescription, where the corresponding electromagnetic streamlines or rays describe the paths along which (electromagnetic) flows, allows to describe the wave phenomena accounted for by Maxwell’s equations on an event-by-event basis [8,10,11,12,13] in compliance with what one experimentally finds in low-intensity experiments [14,15], facilitating the understanding of the statistical results typically obtained in quantum optics in the large photon-count limit [16] without the need to involve Fock states in the description.
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