## 1 Introduction
reported [17] the first reconstruction of what they called averaged photon paths from experimental measurements of the photon transverse momentum in a realization of [PERSON]'s two-slit experiment. In principle, according to standard quantum arguments, this is not possible without dramatically affecting the interference diagram. However, by means of a laboratory implementation of the idea of weak measurement [18, 19, 20, 21], it is possible to infer information along the transverse direction with a slight perturbation (weak measurement) prior to the proper detection process--what we typically regard or denote as a strong or von Neumann measurement [22], which gives rise to the irreversible \"collapse\" of the system wave function. With the value of the transverse momentum at different distances from the slits, and taking the Bohmian ideas as a basis, it was possible to reconstruct a set of rays that were in compliance with former calculations based on the above prescription for electromagnetic fields [7, 8]. Thus, although the latter is not a probabilistic approach itself, appropriate to describe single photons in Fock states, in the large-number limit the correspondence with classical electromagnetism shows there is a proportionality between probability distributions and energy densities, and therefore the electromagnetic-flow trajectories will accurately describe the average paths along which photons travel. In the last years, the laboratory implementation of weak measurements, used as an alternative to quantum tomography to determine the photon wave function [23, 24], has also inspired a series of theoretical works concerning the interpretation [9], reconstruction [25], or new observations [26, 27] of Bohmian trajectories. The above experiments typically involve Gaussian beams and, more specifically, 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 [PERSON] et al. [28], in terms of electromagnetic energy flow lines or rays. The motivation behind this analysis is to provide a ray-based description for localized electromagnetic fields (Gaussian-type beams), where both position and polarization degrees of freedom are present, and hence it can readily be applied to the analysis and interpretation of diffraction and interference experiments of the kind mentioned one. As is well known, usual Gaussian beams are exact solutions to the Helmholtz equation in paraxial form, but not to [PERSON]'s equation under paraxial conditions. The approach proposed by [PERSON] et al., though, shows the specific functional form that the beam has to satisfy in order to be an exact solution to [PERSON]'s equations in paraxial form (in vacuum). In this regard, we would also like to mention that there are other approaches in the literature worth exploring with the same methodology (i.e., where the concept of Bohmian trajectory could be exported) in order to determine the corresponding ray equations, such as nondiffracting Helmholtz-Gauss beams [29], in the case of scalar paraxial fields, or vector Helmholtz-Gauss beams and related families [30; 31], in the case of vector paraxial fields, solutions to [PERSON]'s equations.