# Modal Majorana sphere and hidden symmetries of structured-Gaussian beams

## TL;DR

This paper introduces the modal Majorana sphere as a comprehensive representation of structured-Gaussian beams, revealing hidden symmetries and geometric phases through constellation symmetries, with experimental verification.

## Contribution

It generalizes the modal Poincaré sphere to higher-order modes and uncovers hidden symmetries and geometric phases in structured-Gaussian beams.

## Key findings

- Representation of beams by Majorana constellation points
- Identification of symmetries leading to geometric phases
- Experimental confirmation of symmetries and phases

## Abstract

Structured-Gaussian beams are shown to be fully and uniquely represented by a collection of points (or constellation) on the surface of the modal Majorana sphere, providing a complete generalization of the modal Poincar\'e sphere to higher-order modes. The symmetries of this Majorana constellation translate into invariances to astigmatic transformations, giving way to continuous or quantized geometric phases. The experimental amenability of this system is shown by verifying the existence of both these symmetries and geometric phases.