The properties of the polarization ellipse are deduced in terms of the ratio of the Cartesian components of the complex electric vector of a beam of radiation by utilizing the Argand representation of a real two-dimensional vector as a complex number.
The two components of a beam that are accepted and rejected by a polarizer or a radio antenna are considered as orthogonal components in the directions of two complex orthonormal vectors. The intensities of the corresponding components of a polarized beam are derived and represented on the Poincaré sphere. The methods are then applied to the important radio case of the Faraday effect in a uniform magneto-ionic medium. Finally, the measurable quantities characterizing a beam of partially polarized radiation are obtained from a diagonalization of the complex polarization tensor that specifies the beam.
K. C. WESTFOLD, "New Analysis of the Polarization of Radiation and the Faraday Effect in Terms of Complex Vectors," J. Opt. Soc. Am. 49, 717-723 (1959)