# Bohmian-Based Approach to Gauss-Maxwell Beams

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

**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.

## 1. Introduction

## 2. Standard Scalar Gaussian Beams

#### 2.1. General Aspects for Monochromatic Scalar Fields

- This relation enables a direct switch from propagation in time of an extended wave, namely the scalar field $\mathsf{\Psi}\left(\mathbf{r}\right)$, to space diffusion along the longitudinal (axial) direction of a particular “slice” of such a wave. More specifically, if we consider a transverse section or plane of the full wave [consider, for instance, that such a wave describes a pulse with amplitude $\psi \left(\mathbf{r}\right)$] within the $XY$ plane for a given value ${z}_{0}$ (in other words, the input plane $z={z}_{0}$, analogous to considering ${t}_{0}$ in a time-propagation), we shall obtain its spatial redistribution or accommodation to the corresponding boundaries at subsequent planes, with z increasing. Consequently, if the pulse has an extension along the z direction, considering different “slices” of the pulse (i.e., different ${z}_{0}$ planes), we may easily determine the shape of the pulse at a further distance by just combining all the resulting “slices” ${z}_{f}$. Notice that this fact also allows to establish the validity limit for the approximation, which is going to remain correct provided dispersion along the z direction can be assumed to be negligible (i.e., as long as all “slices” travel with nearly the same speed).
- On the other hand, such a relation enables a simple, direct link between wave optics and matter-wave optics, which arises from the formal relation established between Schrödinger’s equation and the Helmholtz equation in paraxial form, both being parabolic differential equations describing the transport of the quantity $\mathsf{\Psi}$ (regardless of the nature of such a quantity, that is, whether it describes an electromagnetic amplitude or a probability amplitude) with an imaginary diffusion constant (this complex valuedness is precisely the fundamental trait that allows interference in the solution of these equations, but not in the heat equation, although it is also a differential equation of the parabolic class). This is also a rather convenient issue both analytically and computationally, because it explicitly shows that optical and matter-wave problems ruled by the same equation form have the same solution and, eventually, the same interpretations [33].

#### 2.2. Gaussian Beam Propagation

## 3. Gauss-Maxwell Beams

#### 3.1. General Considerations on the Propagation Procedure

- First, there is a space translation, from ${\mathbf{r}}_{\perp}\mathbb{I}$ to ${\mathbb{R}}_{\perp}$, by an effective amount ${k}^{-1}{\mathbb{G}}_{\perp}$, proportional to $\lambda $.
- Then, the beam undergoes a boost, accounted for by the action of the momentum operator ${\widehat{\mathbf{p}}}_{\perp}\mathbb{I}$, while it is freely propagating along the z-direction.

#### 3.2. Linearly Polarized Gauss-Maxwell Beams

- If $z\ll {z}_{R}$, then ${\sigma}_{z}\approx {\sigma}_{0}$ and ${\phi}_{z}\approx z/{z}_{R}\ll \pi /2$. So, in the near field regime, we have$$\mathbf{E}({\mathbf{r}}_{\perp};z)\approx {E}_{0}\psi ({\mathbf{r}}_{\perp};z)\left(\widehat{\mathbf{x}}+\frac{\Delta x}{{z}_{R}}\phantom{\rule{4pt}{0ex}}{e}^{-i\pi /2}\widehat{\mathbf{z}}\right),$$$$\frac{{E}_{z}}{{E}_{x}}=\frac{\Delta x}{{z}_{R}}\phantom{\rule{4pt}{0ex}}{e}^{-i\pi /2}.$$As it can be noticed, only at the center of the beam there is horizontal polarization (parallel to the x transverse coordinate); anywhere else, the z-component of the polarization increases linearly with the distance with respect to the center of the beam. Since this a regime where the expansion of the beam is still negligible, this polarization effect should be particularly relevant for distances (from the center) of the order of the beam waist, ${w}_{0}$, where the fast decrease of the intensity might complicate its detection (particularly, if we consider typical values used in diffraction experiments).
- On the other hand, for $z\gg {z}_{R}$, in the far field regime, we have ${\sigma}_{z}\approx z{\sigma}_{0}/{z}_{R}$ and ${\phi}_{z}\approx \pi /2$. Accordingly, the electric field component reads as$$\mathbf{E}({\mathbf{r}}_{\perp};z)\approx {E}_{0}\psi ({\mathbf{r}}_{\perp};z)\left(\widehat{\mathbf{x}}-\frac{\Delta x}{z}\phantom{\rule{4pt}{0ex}}\widehat{\mathbf{z}}\right),$$$$\frac{{E}_{z}}{{E}_{x}}=-\frac{\Delta x}{z},$$

#### 3.3. Arbitrarily Polarized Gauss-Maxwell Beams

## 4. Young-Type Interference with Gauss-Maxwell Beams

## 5. Final Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. General Solution to the Paraxial Equation (5)

## Appendix B. Interference with Arbitrarily Polarized Beams

## References

- Feynman, R.P.; Hibbs, A.R. Quantum Mechanics and Path Integrals; McGraw-Hill: New York, NY, USA, 1965. [Google Scholar]
- Born, M.; Wolf, E. Principles of Optics. Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed.; Cambridge University Press: Cambridge, UK, 1999. [Google Scholar]
- Bohm, D. A suggested interpretation of the quantum theory in terms of “hidden” variables. I. Phys. Rev.
**1952**, 85, 166–179. [Google Scholar] [CrossRef] - Holland, P.R. The Quantum Theory of Motion; Cambridge University Press: Cambridge, UK, 1993. [Google Scholar]
- Sanz, A.S. Bohm’s approach to quantum mechanics: Alternative theory or practical picture? Front. Phys.
**2019**, 14, 11301. [Google Scholar] [CrossRef] [Green Version] - Schiff, L.I. Quantum Mechanics, 3rd ed.; McGraw-Hill: Singapore, 1968. [Google Scholar]
- Prosser, R.D. The interpretation of diffraction and interference in terms of energy flow. Int. J. Theor. Phys.
**1976**, 15, 169–180. [Google Scholar] [CrossRef] - Sanz, A.S.; Davidović, M.; Božić, M.; Miret-Artés, S. Understanding interference experiments with polarized light through photon trajectories. Ann. Phys.
**2010**, 325, 763–784. [Google Scholar] [CrossRef] [Green Version] - Bliokh, K.Y.; Bekshaev, A.Y.; Kofman, A.G.; Nori, F. Photon trajectories, anomalous velocities and weak measurements: A classical interpretation. New J. Phys.
**2013**, 15, 073022. [Google Scholar] [CrossRef] [Green Version] - Davidović, M.; Sanz, A.S.; Arsenović, D.; Božić, M.; Miret-Artés, S. Electromagnetic energy flow lines as possible paths of photons. Phys. Scr.
**2009**, 135, 014009. [Google Scholar] [CrossRef] - Božić, M.; Davidović, M.; Dimitrova, T.L.; Miret-Artés, S.; Sanz, A.S.; Weis, A. Generalized Arago-Fresnel laws: The EME-flow-line description. J. Russ. Laser Res.
**2010**, 31, 117–128. [Google Scholar] [CrossRef] [Green Version] - Davidović, M.; Sanz, A.S.; Božić, M.; Arsenović, D.; Dimić, D. Trajectory-based interpretation of Young’s experiment, the Arago-Fresnel laws and the Poisson-Arago spot for photons and massive particles. Phys. Scr.
**2013**, 153, 014015. [Google Scholar] [CrossRef] - Davidović, M.D.; Davidović, M.D.; Sanz, A.S.; Božić, M.; Vasiljević, D. Trajectory-based interpretation of the laser light diffraction by a sharp edge. J. Russ. Laser Res.
**2018**, 39, 438–447. [Google Scholar] [CrossRef] [Green Version] - Dimitrova, T.L.; Weis, A. The wave-particle duality of light: A demonstration experiment. Am. J. Phys.
**2008**, 76, 137–142. [Google Scholar] [CrossRef] [Green Version] - Dimitrova, T.L.; Weis, A. Single photon quantum erasing: A demonstration experiment. Eur. J. Phys.
**2010**, 31, 625–637. [Google Scholar] [CrossRef] [Green Version] - Aspden, R.S.; Padgett, M.J.; Spalding, G.C. Video recording true single-photon double-slit interference. Am. J. Phys.
**2016**, 84, 671–677. [Google Scholar] [CrossRef] [Green Version] - Kocsis, S.; Braverman, B.; Ravets, S.; Stevens, M.J.; Mirin, R.P.; Shalm, L.K.; Steinberg, A.M. Observing the average trajectories of single photons in a two-slit interferometer. Science
**2011**, 332, 1170–1173. [Google Scholar] [CrossRef] [PubMed] - Aharonov, Y.; Albert, D.Z.; Vaidman, L. How the result of a measurement of a component of the spin of a spin-12 particle can turn out to be 100. Phys. Rev. Lett.
**1988**, 60, 1351–1354. [Google Scholar] [CrossRef] [Green Version] - Aharonov, Y.; Vaidman, L. Properties of a quantum system during the time interval between two measurements. Phys. Rev. A
**1990**, 41, 11–20. [Google Scholar] [CrossRef] - Wiseman, H.M. Grounding Bohmian mechanics in weak values and bayesianism. New J. Phys.
**2007**, 9, 165. [Google Scholar] [CrossRef] - Dressel, J.; Malik, M.; Miatto, F.M.; Jordan, A.N.; Boyd, R.W. Colloquium: Understanding quantum weak values: Basics and applications. Rev. Mod. Phys.
**2014**, 86, 307–316. [Google Scholar] [CrossRef] [Green Version] - Von Neumann, J. Mathematical Foundations of Quantum Mechanics; Princeton University Press: Princeton, NJ, USA, 1955. [Google Scholar]
- Lundeen, J.S.; Sutherland, B.; Patel, A.; Stewart, C.; Bamber, C. Direct measurement of the quantum wavefunction. Nature
**2011**, 474, 188–191. [Google Scholar] [CrossRef] [Green Version] - Lundeen, J.S.; Bamber, C. Procedure for direct measurement of general quantum states using weak measurements. Phys. Rev. Lett.
**2012**, 108, 070402. [Google Scholar] [CrossRef] - Schleich, W.P.; Freyberger, M.; Zubairy, M.S. Reconstruction of Bohm trajectories and wave functions from interferometric measurements. Phys. Rev. A
**2013**, 87, 014102. [Google Scholar] [CrossRef] - Matzkin, A. Observing trajectories with weak measurements in quantum systems in the semiclassical regime. Phys. Rev. Lett.
**2012**, 109, 150407. [Google Scholar] [PubMed] [Green Version] - Braverman, B.; Simon, C. Proposal to observe the nonlocality of Bohmian trajectories with entangled photons. Phys. Rev. Lett.
**2013**, 110, 060406. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Simon, R.; Sudarshan, E.C.G.; Mukunda, N. Gaussian-Maxwell beams. J. Opt. Soc. Am. A
**1986**, 3, 536–540. [Google Scholar] - Gutiérrez-Vega, J.C.; Bandres, M.A. Helmholtz-Gauss waves. J. Opt. Soc. Am. A
**2005**, 22, 289–298. [Google Scholar] [CrossRef] [Green Version] - Bandres, M.A.; Gutiérrez-Vega, J.C. Vector Helmholtz-Gauss and vector Laplace-Gauss beams. Opt. Lett.
**2005**, 30, 2155–2157. [Google Scholar] [CrossRef] [Green Version] - Hernández-Aranda, R.I.; Gutiérrez-Vega, J.C.; Guizar-Sicairos, M.; Bandres, M.A. Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems. Opt. Express
**2006**, 14, 8974–8988. [Google Scholar] [CrossRef] - Sanz, A.S.; Campos-Martínez, J.; Miret-Artés, S. Transmission properties in waveguides: An optical streamline analysis. J. Opt. Am. Soc. A
**2012**, 29, 695–701. [Google Scholar] [CrossRef] [Green Version] - Sanz, A.S. Neutron matter-wave diffraction: A computational perspective. In Advances in Neutron Optics; Calvo, M.L., Fernández Álvarez-Estrada, R., Eds.; CRC Press: Boca Ratón, FL, USA, 2020; pp. 79–122. [Google Scholar]
- Sanz, A.S.; Miret-Artés, S. Quantum phase analysis with quantum trajectories: A step towards the creation of a Bohmian thinking. Am. J. Phys.
**2012**, 80, 525–533. [Google Scholar] [CrossRef] [Green Version] - Davis, L.W. Theory of electromagnetic beams. Phys. Rev. A
**1979**, 19, 1177–1179. [Google Scholar] [CrossRef] - McDonald, K.T. Gaussian laser beams with radial polarization. Physics Examples. 2012, pp. 1–12. Available online: http://puhep1.princeton.edu/~mcdonald/examples/axicon.pdf (accessed on 6 March 2020).
- McDonald, K.T. Second-order paraxial Gaussian beam. Physics Examples. 2016, pp. 1–5. Available online: http://puhep1.princeton.edu/~mcdonald/examples/davis_psi2.pdf (accessed on 6 March 2020).
- McDonald, K.T. Reflection of a Gaussian optical beam by a flat mirror. Physics Examples. 2009, pp. 1–12. Available online: http://puhep1.princeton.edu/~mcdonald/examples/mirror.pdf (accessed on 6 March 2020).
- Sanz, A.S.; Miret-Artés, S. Aspects of nonlocality from a quantum trajectory perspective: A WKB approach to Bohmian mechanics. Chem. Phys. Lett.
**2007**, 445, 350–354. [Google Scholar] [CrossRef] [Green Version] - Collett, E. Field Guide to Polarization; SPIE: Bellingham, WA, USA, 2005. [Google Scholar]
- Barnett, S.M.; Loudon, R. The enigma of optical momentum in a medium. Philos. Trans. R. Soc. A
**2010**, 368, 927–939. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sanz, A.S.; Miret-Artés, S. A trajectory-based understanding of quantum interference. J. Phys. A Math. Theor.
**2008**, 41, 435303. [Google Scholar] [CrossRef] [Green Version] - Luis, A.; Sanz, A.S. What dynamics can be expected for mixed states in two-slit experiments? Ann. Phys.
**2015**, 357, 95–107. [Google Scholar] [CrossRef] [Green Version] - Walborn, S.P.; Cunha, M.O.T.; Pádua, S.; Monken, C.H. Double-slit quantum eraser. Phys. Rev. A
**2002**, 65, 033818. [Google Scholar] [CrossRef] [Green Version] - Bohm, D. Quantum Theory; Dover Publications: New York, NY, USA, 1989; First printed by Prentice Hall in 1951. [Google Scholar]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sanz, Á.S.; Davidović, M.D.; Božić, M.
Bohmian-Based Approach to Gauss-Maxwell Beams. *Appl. Sci.* **2020**, *10*, 1808.
https://doi.org/10.3390/app10051808

**AMA Style**

Sanz ÁS, Davidović MD, Božić M.
Bohmian-Based Approach to Gauss-Maxwell Beams. *Applied Sciences*. 2020; 10(5):1808.
https://doi.org/10.3390/app10051808

**Chicago/Turabian Style**

Sanz, Ángel S., Milena D. Davidović, and Mirjana Božić.
2020. "Bohmian-Based Approach to Gauss-Maxwell Beams" *Applied Sciences* 10, no. 5: 1808.
https://doi.org/10.3390/app10051808