Majorana constellations for optical scalar beams and vector fields

We explore the Majorana stellar representation on the Riemann sphere for optical scalar beams and vector fields, utilizing the spin and orbital angular momentum of light. In this framework, star constellations for a scalar Laguerre-Gaussian beam basis appear at the poles of the sphere, while those for vector fields correspond to well-defined circular polarizations. Leveraging the $\text{su}(2)$ symmetry of angular momentum, we populate the sphere through generalized unitary rotations of basis elements, producing optical analogs of Bloch and generalized $\text{su}(2)$ coherent states for both scalar beams and vector fields. These rotations position constellations for scalar Hermite-Gaussian beams and vector Hermite-Gaussian partial Poincar\'e fields, restricted to linear polarization, along the equator of the sphere. We also address the inverse problem, reconstructing optical scalar beams and vector fields from their stellar constellations. Additionally, we explore scalar and vector analogs of cat codes and kings of quantumness, contributing to the visualization and characterization of complex optical beams and fields. These insights suggest potential applications in optical communications.