Orbitalization ellipse via singular value decomposition

The geometric structure of light beam oscillations in the Orbital Angular Momentum (OAM) space, i.e., the space spanned by the spiral basis <i>e</i>^{<i>i</i><i>l</i><i>ϕ</i>}, <i>l</i>, and <i>ϕ</i> being the OAM index and azimuthal angle, respectively, is determined by the orbitalization ellipse [Opt. Lett.50, 391 (2025) 10.1364/OL.545613]. Here, we show that applying singular value decomposition to a matrix constructed from the beam's OAM components provides an insightful numerical framework for finding the shape and orientation of the ellipse, hence benefiting the optimization of optical systems leveraging OAM-carrying beams of high dimensions.