Complex polarizations of ionospheric characteristic waves: A revisit in terms of real representation

The polarizations of ionospheric characteristic waves (CWs) are determined by the two solutions of the quadratic equation in the complex polarization ratio R within the magneto-ionic theory. This paper revisits these solutions and their relationship from the perspective of real representation, given that various real polarization parameter sets are also commonly used and play important parts in characterizing wave polarization. Multiple two-dimensional (2D) real representations are analyzed by re-examining several real two-parameter sets mathematically under a unified framework to prevent possible confusion from examining different parameters in separate frameworks, and by graphically representing the polarizations of the two CWs using 2D polarization plane plots constructed from paired real parameters. A three-dimensional (3D) real representation as the extension of two dimensions is further considered by the mathematical re-examination of the three normalized Stokes parameters for the two CWs and the application of the 3D Poincaré sphere visualization to overcome the non-uniform polarization distribution on the plane plot and the non-unique correspondence between plane point and polarization state. Each real parameter in these real representations is expressed as a function of R. Importantly, the relationship between the same real parameters of the two CWs is derived. Numerical examples presented through the polarization plane or Poincaré sphere plot demonstrate how these real parameters vary with certain ionosphere medium-related quantities and validate the correctness of the derived relationships. The comparison of each real representation with and without collisions reveals the impact of collisions on the real parameters and polarization orthogonality of the two CWs.