# Circularly Symmetric Light Waves: An Overview

## TL;DR

This paper provides a comprehensive analysis of vortex solutions to the Helmholtz equation, including scalar and vector beams, emphasizing their angular momentum properties and propagation characteristics.

## Contribution

It introduces a rigorous operator-based framework for classifying and analyzing vortex solutions, extending understanding of OAM beams and their polarization-spatial coupling.

## Key findings

- Characterization of scalar and vector vortex solutions
- Analysis of propagation properties of OAM beams
- Connection with two-dimensional rotation group

## Abstract

Orbital Angular Momentum (OAM) waves were first recognized as those specific vortex solutions of the paraxial Helmholtz equation for which the orbital contribution to the total angular momentum of the beam yields an integer multiple of $\hbar$ along the propagation direction. However, this class of solutions can be generalized to include more sophisticated vector vortex waves with coupled polarization and spatial complexity, that are eigenfunctions of the third component of the angular momentum operator. In this work, a rigorous framework is proposed for the analysis of all the possible families of vortex solutions to the homogeneous Helmholtz equation. Both the scalar and vector cases are studied in depth, making use of an operator approach which emphasizes their intimate connection with the two-dimensional rotation group. Furthermore, a special focus is given to the characterization of the propagation properties of the most popular families of paraxial OAM beams.