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 $SU\left(2\right)$ angular momentum spin algebra, these modes are defined as eigenfunctions of a natural quadratic combination of spin-like Fradkin-Stokes parameters $K=L^2-M_{+}^2$ . Together with Hermite-Gaussian, Laguerre-Gaussian and Ince-Gaussian modes, these new modes complete the natural (quadratic) families of Gaussian eigenmodes, up to $SU\left(2\right)$ 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.