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arXiv:2409.15643 · Applied Optics

Revealing the propagation dynamic of a Laguerre-Gaussian beam with two Bohm-like theories

Pengfei Huang, Ya Xiao, Shanchuan Dong, Yongjian Gu

Published Sep 4, 2024• 1 citation
arXivHTMLPDF

Abstract

By employing x -Bohm theory and p -Bohm theory, we construct the position and momentum trajectories of single-mode and superposed-mode Laguerre-Gaussian (LG) beams. The dependence of divergence velocity and rotation velocity on the initial position and propagation distance is quantified, indicating that LG beams exhibit subluminal effects, even in free space. Additionally, we clarify the formation of the petal-shaped intensity distribution of the superposed-mode LG beam in terms of motion trajectory, where the particle-like trajectory and wave-like interference are “simultaneously” observed. Our work provides an intuitive way to visualize the propagation characteristics of LG beams and deepen the comprehension of Bohm-like theory.

DOI
10.1364/ao.533017

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By employing x -Bohm theory and p -Bohm theory, we construct the position and momentum trajectories of single-mode and superposed-mode Laguerre-Gaussian (LG) beams. The dependence of divergence velocity and rotation velocity on the initial position and propagation distance is quantified, indicating that LG beams exhibit subluminal effects, even in free space. Additionally, we clarify the formation of the petal-shaped intensity distribution of the superposed-mode LG beam in terms of motion trajectory, where the particle-like trajectory and wave-like interference are “simultaneously” observed. Our work provides an intuitive way to visualize the propagation characteristics of LG beams and deepen the comprehension of Bohm-like theory.
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The paraxial wave equation is formally equivalent to the Schrödinger equation, enabling quantum-mechanical concepts to be applied to classical optical fields. Within this framework, we investigate the Bohmian dynamics of superpositions of Laguerre-Gaussian beams. We derive the velocity field and quantum potential and analyze the resulting trajectory dynamics. Interference between modes with different orbital angular momentum generates vortex formation, trajectory exclusion from nodal regions, and spiral motion around phase singularities. Remarkably, despite the complexity of the Bohmian trajectories, the vortex cores follow nearly linear paths along propagation. The dynamics are governed by the amplitude ratio and OAM indices, while the relative phase only induces global rearrangements. These results establish a coherent trajectory-based description of structured light and highlight vortex cores as robust topological features.
The propagation of a Laguerre-Gaussian (LG) beam through a dispersive medium is investigated. The effect of the doughnut-like intensity profile of the probe LG beam on the group velocity is studied. We find an analytical expression for the group velocity out of the optical axis and compare with its projection onto the propagation axis. It is turned out that the group velocity vector is along the optical axis at the waist of the beam and the Rayleigh range. We numerically and analytically investigate the effect of the helical phase front of the coupling LG field on the group velocity of the probe LG field in a four-level double V-type quantum system. Our analysis predicts a strange behavior for the group velocity of the probe LG beam inside a normal dispersive medium in the gain region so that it can exceed the speed of light in free space, leads to the gain-assisted superluminal light propagation in normal dispersion. Such an unusual propagation of the LG beam results from the distortion of its helical phase front via the classical interference of the planar and LG fields. The obtained results may find some potential applications in increasing the velocity of the information transmission in optical communications.
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Beam splitters allow us to superpose two continuous single mode quantum systems. To study the behaviour of beam splitters' strongly mode mixing dynamics we consider variable beam splitters acting on Wigner's phase space distribution, W , the evolution of which is governed by the continuity-equation {\partial τ} W = - {\nabla} J. We derive the form of the corresponding Wigner current, J. J's form allows us to use a classical trajectories-approach to analyze the influence of the two modes on each other. We show that the dynamics for variable beam splitters amounts to a rotation confined within the plane of the two positions together with the same simultaneous rotation confined within the plane of the two momenta. In this way explicit and very transparent expressions for the rotated Wigner distributions and Wigner currents can be given in terms of classical trajectories. This helps us to gain deeper insights and perform geometrical analyses of the mixing of modes at beam splitters.
In this work, we show that the Madelung–Bohm trajectories determined by any solution, Ψ( ξ , ζ ), of the paraxial wave equation in an arbitrary two-dimensional optical medium, where ζ is the propagation coordinate and ξ the transversal one, are given by the contours of the function G ( ξ , ζ )≡∫|Ψ( ξ , ζ )| 2 d ξ . Furthermore, we remark that an analogous result follows directly for the one-dimensional Schrödinger equation for a particle evolving in an arbitrary classical potential, where ξ gives the position of the particle at time ζ . We apply this result to several examples in free and quadratic optical media. They include the Airy beam, Hermite–Gaussian beams, and some of their superpositions. Additionally, this work provides an alternative derivation of Hermite–Gaussian (HG) beams in free space by employing the quantum potential equations and the framework of quantum trajectories. By analyzing known trajectories, we demonstrate that the fundamental parameters characterizing these beams—the Gouy phase, beam radius, and wavefront curvature—emerge naturally within the quantum potential formalism.