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

Abstract

We investigate the propagation of structured scalar optical beams in an effective anisotropic background inspired by the scalar sector of the Standard-Model Extension and controlled by a single dimensionless parameter $λ$. The physically relevant configuration is a transverse radial director field that modifies the radial part of the Helmholtz operator while preserving axial symmetry. Starting from the Green-function representation, we cast the propagation problem as an initial-value spectral reconstruction of a prescribed finite-aperture entrance profile at $z=0$ and verify that this profile is recovered at the launch plane across the values of $λ$ used in the analysis, within small numerical error. We use the Laguerre--Gaussian mode $L_3$ as the representative vortex-free Laguerre case, retain $L_4$ only as a quantitative benchmark for radial-order dependence, and compare both with a Bessel--Gaussian beam of input order $m=0$. The effective anisotropy produces a systematic redistribution of radial intensity, determines whether the central peak remains dominant or is overtaken by off-axis maxima as propagation advances, and controls the radial displacement of the dominant side lobes. For the finite-aperture Bessel--Gaussian beam, the same parameter quantifies how the approximately diffraction-resistant ring structure broadens for negative $λ$ and compresses for positive $λ$ during propagation.