Euler‐Top Gaussian Modes: Structured Beams From Quadratic Angular Momentum Dynamics

We propose a new family of paraxial Gaussian modes, the Euler‐top Gaussian modes. Motivated by SU(2) angular momentum spin algebra, these modes are defined as eigenfunctions of a natural quadratic combination of spin‐like Fradkin–Stokes parameters K=L²−M₊². Together with Hermite–Gaussian, Laguerre–Gaussian and Ince–Gaussian modes, these new modes complete the natural (quadratic) families of Gaussian eigenmodes, up to SU(2) transformations. Euler‐top Gaussian modes with positive eigenvalues have a real Laguerre–Gaussian appearance, and those with negative eigenvalues have a coupled Hermite–Gaussian appearance with fourfold symmetry. Certain other modes quantise a separatrix state between the two regimes. Their structure is motivated by paths on the Gaussian ray‐orbital Poincaré sphere, analogous to the polhode curves describing the angular momentum of an Euler top, that is, a classical rigid body with three different axes of inertia. Various properties of these new modes are considered, including their analogy to Lipkin–Meshkov–Glick modes from nuclear physics.