Poincaré sphere representation of the Wigner distribution function of Ince–Gauss beams

The analytical expressions for mapping the Wigner distribution function (WDF) of even, odd, and helical Ince-Gauss (IG) beams onto the Poincaré sphere are presented. The analogy between the paraxial wave equation and the two-dimensional quantum harmonic oscillator allows us to use the SU(2) formalism for expressing the WDF in terms of quadratic invariants of the harmonic oscillator. This representation reveals the internal symmetries and patterns that characterize the phase-space structure of IG beams. Numerical simulations confirm the precision of the analytical formulas and illustrate the continuous transition between the Hermite, Ince, and Laguerre-Gauss families as the ellipticity parameter varies.