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 <i>SU</i>(2) angular momentum spin algebra, these modes are defined as eigenfunctions of a natural quadratic combination of spin-like Fradkin-Stokes parameters <i>K</i> = <i>L</i>² − <i>M</i>²₊. To gether with Hermite-Gaussian, Laguerre-Gaussian and Ince-Gaussian modes, these new modes complete the natural (quadratic) families of Gaussian eigenmodes, up to <i>SU</i>(2) transformations. Euler-top Gaussian modes with positive eigenvalues have real Laguerre-Gaussian appearance, and negative eigenvalues, coupled Hermite-Gaussian appearance with fourfold symmetry. Certain other modes quantize a separatrix state between the two regimes. Their structure is motivated by paths on the Gaussian ray-orbital Poincaré sphere, analogous to the polhodes curves describing the angular momentum of an Euler top, i.e. 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.