Influence of the radial number of Laguerre-Gaussian vortex beams on their propagation in a turbulent medium

TL;DR

This work analyzes the propagation of Laguerre-Gaussian beams in a turbulent medium by decomposing LG modes $LG_{p,\ell}$ into a Zernike basis $Z_n^m$ and evaluating coefficients $C_{n}^{\,p,\ell}$. It demonstrates that the stability of the beam depends on the radial index $p$ and the azimuthal index $\ell$, with $|C_n^{p,\ell}|=0$ for $n<|\ell|$ and generally smaller coefficients for larger $p$. The authors use a split-step propagation model to simulate turbulence effects and show that increasing $p$ enhances transverse-profile stability, suggesting radial-mode encoding as a robust alternative to azimuthal OAM multiplexing under atmospheric fading. The results inform practical design of free-space optical links by highlighting the trade-off between robustness and beam diameter, and propose radial-mode coding as a route to improved resilience.

Abstract

The paper considers the propagation of Laguerre-Gaussian beams in a turbulent gas medium. As demonstrated by numerical modelling and the decomposition of Laguerre-Gaussian modes according to the basis of orthogonal Zernike polynomials, disparities in the stability of Laguerre-Gaussian modes with differing radial and azimuthal numbers are evident. The demonstration is made of the minimization of the distortion of the transverse beam profile with an increase in the radial number.

Figures (5)

• Figure 1: Numerical simulation of the interference pattern of LG modes with a Gaussian reference beam. The first index is the radial number $p$, the second is the azimuthal number $\ell$.
• Figure 2: Summary table of the results of numerical simulation of the passage of LG modes $LG_{0,0}, LG_{0,1}, LG_{1,1}, LG_{0,2}, LG_{2,2}, LG_{0,3}, LG_{3,3}$ through a turbulent gas medium; $r_0$ --- the Fried parameter characterizing the value turbulence of the medium (see Appendix). Images of the intensity of the transmitted beam and the result of interference with the reference Gaussian beam are presented.
• Figure 3: The absolute values of the coefficients of expansion in a series of Zernike polynomials $|C_{n}^{p,\ell}|$ for fixed values of radial numbers $p$.
• Figure 4: The absolute values of the coefficients of expansion in a series of Zernike polynomials $C_{n}^{p,\ell}$ for fixed values of the azimuthal number $\ell$.
• Figure 5: Coefficients of the Zernike polynomial expansion $C_{n}^{p,\ell}$ as functions of $p$ in cases where the azimuthal number $\ell$ is equal to the index $n$.