Symmetry Requirements for Helicity Eigenstates of Guided Light

When a transformation generated by an operator is continuous, states are the eigenstates of the corresponding operator, such as translation symmetry for momentum eigenstates and rotational symmetry for total angular momentum eigenstates. Here, we show that the duality symmetry, the transformation generated by the helicity operator in the electric-magnetic representation, is insufficient to ensure that light forms a helicity eigenstate. As required by the symmetry of helicity, the eigenvectors of the helicity operator should be isomorphic to a two-dimensional irreducible representation, determined by the geometrical symmetry of the waveguides. When the symmetry of modes is met, but the duality symmetry of the waveguide is broken, helicity eigenstates only approximately exist in the paraxial limit of a guided mode. This reveals that transverse confinement is the underlying physical origin of duality symmetry breaking for generic guided modes.