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Propagation of Laguerre-Gaussian and Bessel-Gaussian scalar beams in an effective anisotropic background

This paper investigates the propagation of Laguerre-Gaussian and Bessel-Gaussian scalar beams in an effective anisotropic background inspired by the Standard-Model Extension, demonstrating how a single dimensionless parameter $\lambda$ systematically redistributes radial intensity and alters beam structure, such as shifting dominant peaks or modifying ring broadening, while preserving axial symmetry.

Original authors: C. A. Escobar, Román Linares, E. Plácido-Flores

Published 2026-08-20

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CC BY 4.0

Original authors: C. A. Escobar, Román Linares, E. Plácido-Flores

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Light is more than just a beam that illuminates a room; it is a structured field that can be shaped into complex patterns, carrying information and energy in specific ways. Scientists have long used special types of light, such as those with ring-like shapes or swirling phases, to perform delicate tasks like moving tiny particles or sending data through the air. These patterns are not just visual tricks; they are robust, meaning they can travel long distances without losing their shape, which makes them incredibly useful for technology. However, when light travels through materials that are not uniform in all directions—such as certain crystals or engineered environments—its behavior changes in ways that depend on how the material is oriented. Understanding exactly how these structured beams reshape themselves in such environments is crucial for designing better optical devices and communication systems.

A team of researchers has now explored how two specific families of these structured light beams change as they travel through a simplified, theoretical environment that mimics a material with directional properties. Instead of studying a real crystal with all its complex physical details, the scientists created a mathematical model that isolates one specific effect: how the light's intensity spreads out or squeezes together in a radial direction, moving from the center of the beam outward. They focused on two common types of beams: one that looks like a series of concentric rings with a bright center, and another that resembles a hollow tube of light. By adjusting a single number in their model, they could simulate a material that either pushes the light outward or pulls it inward, observing how the beams responded as they moved forward.

The researchers began by setting up a precise starting point for their light beams, defining exactly what the pattern looked like at the moment it was launched. They then used a sophisticated mathematical method to predict how these beams would evolve as they traveled through their simulated environment. This approach allowed them to verify that their starting conditions were perfectly preserved at the beginning of the journey, ensuring that any changes they saw later were truly caused by the environment and not by errors in their setup. They tested their model with different values for the directional parameter, ranging from negative to positive, to see how the beams would behave under different conditions.

When they examined the first type of beam, which features a bright center surrounded by rings, they found that the direction of the material's influence dramatically altered the beam's fate. If the parameter was set to a negative value, the bright center of the beam remained dominant for a longer distance, resisting the urge to spread out. However, as the parameter became positive, the central bright spot began to fade more quickly, and the energy shifted to the surrounding rings. The rings grew brighter and moved closer to the center, effectively taking over as the main feature of the beam. This transition happened faster as the positive value increased, showing that the material could actively force the light to reorganize itself from a central spot into a hollow ring structure.

The second type of beam, which naturally resembles a hollow ring, showed a different but related behavior. In this case, the directional parameter acted like a lens that either widened or narrowed the ring. When the parameter was negative, the ring expanded, becoming broader and less intense at the center. When the parameter was positive, the ring compressed, becoming tighter and sharper. Remarkably, this ring-shaped beam was more resistant to change than the first type; it held onto its original structure for a longer distance before the directional influence caused it to shift its main brightness away from the center. This suggests that the ring-like shape is inherently more stable against these specific types of directional changes.

The study did not rely on physical experiments with lasers and crystals but was conducted entirely through numerical simulations based on the laws of physics. The results provide a clear, controlled picture of how a single type of directional influence can reshape light. The researchers found that the effect is not a simple blurring or a rigid shift of the entire pattern, but a genuine redistribution of energy. The light does not just get bigger or smaller; it actively moves its weight from the center to the sides, or vice versa, depending on the nature of the environment it travels through. This work isolates the radial behavior of light, offering a baseline understanding that can be used later when scientists add more complex factors, such as polarization, to their models. By understanding this fundamental reshaping, engineers and physicists can better predict how structured light will behave in the advanced materials of the future.

Technical Summary: Propagation of Laguerre-Gaussian and Bessel-Gaussian Scalar Beams in an Effective Anisotropic Background

Problem Statement
This work investigates the propagation of structured scalar optical beams—specifically Laguerre-Gaussian (LG) and Bessel-Gaussian (BG) families—through an effective anisotropic background. Unlike studies involving full vectorial Maxwell treatments in crystals or bianisotropic media, which depend on polarization, dielectric tensors, and optical axis orientation, this paper addresses a deliberately reduced scalar problem. The model is inspired by the scalar sector of the Standard-Model Extension (SME), a framework used to parametrize controlled departures from Lorentz invariance. The central problem is to determine how a single dimensionless parameter, $\lambda$ , representing a reduced radial anisotropy, reorganizes the radial intensity distribution of structured beams when a finite aperture profile is prescribed at a launch plane ( $z=0$ ). The study seeks to distinguish generic radial reshaping effects from polarization-dependent phenomena.

Methodology
The authors employ a nonparaxial scalar-effective treatment based on the massless scalar sector of the minimal SME. The propagation is governed by a modified Helmholtz operator where the radial kinetic term is scaled by $(1-\lambda)$ .

Spectral Implementation: Rather than relying solely on the Green-function kernel, the authors reformulate the problem as an initial-value spectral reconstruction. The field is expanded over regular radial modes adapted to the anisotropy, with spectral amplitudes fixed by the prescribed entrance profile at $z=0$ .
Beam Profiles: The study utilizes axially symmetric ( $m=0$ $m = 0$ ) inputs truncated by a finite aperture ( $0 \le \tilde{\rho} \le 1$ $0 \leq \tilde{ρ} \leq 1$ ).
- Laguerre-Gaussian: The $L_3$ mode serves as the representative case, while $L_4$ is used as a quantitative benchmark for radial-order dependence.
- Bessel-Gaussian: A finite-energy BG beam with input order $m=0$ is used to test the survival of diffraction-resistant ring structures.
Numerical Validation: The spectral reconstruction is validated at the launch plane ( $\tilde{z}=0$ ) across various $\lambda$ values. The relative $L_2$ error between the reconstructed and prescribed profiles is kept below 3.3%, ensuring that observed propagation differences arise from anisotropy-induced evolution rather than initial condition mismatches.
Observables: Key metrics include the dominant peak position ( $\tilde{\rho}_{peak}$ ), on-axis intensity ( $I_{center}$ ), effective radius ( $r_{eff}$ ), and the crossover distance ( $\tilde{z}_{\times}$ ) where off-axis maxima overtake the central peak.

Key Results
The effective anisotropy parameter $\lambda$ systematically redistributes radial intensity, altering the competition between the central lobe and off-axis rings.

Laguerre-Gaussian Beams ( $L_3$ and $L_4$ ):
- Positive $\lambda$ : Accelerates the transfer of intensity from the central lobe to off-axis maxima. As $\lambda$ increases, the crossover to off-axis dominance ( $\tilde{z}_{\times}$ ) occurs earlier, and the dominant side lobes shift to smaller radii.
- Negative $\lambda$ : Delays the off-axis takeover, preserving the central lobe dominance over a longer propagation interval.
- Radial Order: The $L_4$ mode exhibits the same qualitative trends as $L_3$ but with systematically earlier crossover values, indicating that higher radial orders amplify the reshaping mechanism.
Bessel-Gaussian Beams:
- BG beams retain their central structure for longer propagation distances compared to LG beams.
- Negative $\lambda$ : Broadens the central lobe (FWHM increases by ~28% at $\tilde{z}=0.441$ ) and pushes the first radial minimum outward. No off-axis crossover is observed within the computed interval for $\lambda = -0.35$ .
- Positive $\lambda$ : Compresses the ring structure (FWHM decreases) and eventually promotes an off-axis dominant maximum.
- Comparison: The crossover to off-axis dominance occurs later for BG beams than for LG beams, highlighting the robustness of the Bessel ring structure against this specific type of scalar anisotropy.

Significance and Claims
The paper claims that its primary contribution is a nonparaxial scalar-effective analysis that isolates the radial channel of beam propagation without invoking full vectorial medium models.

Novelty: The work does not propose a new constitutive model for specific crystals. Instead, it provides a baseline for understanding how a transverse radial director field reorganizes structured beams. It demonstrates that radial reshaping, side-lobe amplification, and ring displacement can be quantified purely through the scalar channel.
Interpretation of $\lambda$ : The parameter $\lambda$ is interpreted not as a replacement for a dielectric tensor, but as a reduced radial-anisotropy parameter. It measures the relative weight of radial derivatives in the effective Helmholtz operator.
Complementarity: The results are presented as complementary to existing literature on vectorial propagation in uniaxial crystals and bianisotropic media. While those works resolve polarization-sensitive effects (birefringence, walk-off), this study identifies the "baseline radial contribution" against which more complete optical descriptions can be interpreted.
Scope: The authors explicitly state that the model is an approximation valid when propagation dynamics are dominated by a single transverse amplitude, allowing polarization mixing and birefringence to be neglected at leading order. The study concludes by suggesting natural extensions to vortex-carrying modes ( $m \neq 0$ ) and vectorial generalizations, but does not present these results.

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