A Higher-Order Poincar\'e Ellipsoid representation for elliptical vector beams

The Higher-Order Poincar\'e Sphere (HOPS) provides a powerful geometrical tool for representing vector beams as points on the surface of a unitary sphere. Since a particular position on the surface represents any spatial mode regardless of its shape, this representation cannot be used to discern between the spatial modes geometries of vector modes. For instance, Laguerre- and Ince-Gauss vector beams are ambiguously represented using the same unitary sphere, even though their spatial profiles are circular and elliptical, respectively. As such, in this manuscript, we propose a generalisation of the HOPS that we call the Higher-Order Poincar\'e Ellipsoid (HOPE). Our approach allows an unambiguous representation of helical Ince-Gauss vector modes of ellipticity $\varepsilon$ onto the surface of an ellipsoid of eccentricity $\bf e$, providing a unique way to visualise elliptically-shaped vector modes. We provide a transformation that links the ellipticity $\varepsilon$ of helical Ince-Gauss vector modes to the eccentricity $\bf e$ of an ellipsoid, such that the HOPS is recovered for $\varepsilon=0$. Since this representation preserves the Stokes parameters formalism, the transition from the HOPS to the HOPE is straightforward, thus making its implementation appealing for the structured light community. We anticipate the concepts outlined here will pave the path toward the representation of structured light beams' properties using other geometrical objects.