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 e i l ϕ , l , and ϕ 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.