Sets of orthogonal three-dimensional polarization states and their physical interpretation

The spectral and characteristic decompositions of the polarization matrix provide fruitful frameworks for the physical interpretation of three-dimensional (3D) partially polarized light fields. The decompositions are formulated in terms of the three pure eigenstates, which in turn are represented through their associated orthogonal complex 3D Jones vectors. This mathematical orthogonality does not correspond, in general, to orthogonality of the polarization-ellipse planes of the respective eigenstates. Consequently, due to such inherent mathematical complexity, the geometric and physical interpretation of these sets of orthogonal complex vectors, being essential for the best understanding of the structure and properties of partially polarized 3D light, has not been addressed thoroughly. In this work, the geometric and physical features of sets of three orthonormal 3D Jones vectors are identified and analyzed, allowing one to obtain meaningful interpretations of any given mixed (partially polarized) 3D polarization state in terms of either the spectral or the characteristic decompositions. Among other results, it is found that, given a pure polarization state, any plane in space contains the polarization ellipse of a pure state that is orthogonal to it, and the mathematical expressions for the azimuth and ellipticity of such an ellipse are calculated in terms of the angular parameters determining said plane and the ellipticity of the given state. Furthermore, the spin vectors of the three polarization eigenstates are arranged in a peculiar spatial manner, such that they lie in a common plane. Beyond polarization phenomena, the approach presented also has potential applications in areas where 3 x 3 unitary matrices play a key role, like three-level quantum systems and gates for ternary quantum logic circuits.