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The connection between Poincar\'e spheres for polariz-ation and Gaussianbeams is explored, focusing on the interpretation of elliptic polarization interms of the isotropic 2-dimensional harmonic oscillator in Hamiltonianmechanics, its canonical quantization and semiclassical interpretation. Thisleads to the interpretation of structured Gaussian modes, the Hermite-Gaussian,Laguerre-Gaussian and Generalized Hermite-Laguerre Gaussian modes aseigenfunctions of operators corresponding to the classical constants of motionof the 2-dimensional oscillator, which acquire an extra significance asfamilies of classical ellipses upon semiclassical quantization.

philosophical transactions - royal society. mathematical, physical and engineering sciences/philosophical transactions - royal society. mathematical, physical and engineering sciences

Swings and roundabouts: Optical Poincar\'e spheres for polarization and Gaussian beams