The characteristic polarization states and the equi-power curves

Characteristic polarization state theory is restudied for the symmetric coherent Sinclair scattering matrix case. First, the geometric relations of the characteristic polarization states on the Poincaré sphere are derived. Based on these relations, simple formulas are given for all of the characteristic polarization states of this Sinclair matrix in Stokes vector form. From the formulation, it is clear that the CO-POL Nulls are fundamental characteristic polarization states for the symmetric coherent Sinclair scattering matrix case, in that the others can straightforwardly be obtained from the Stokes vectors of the CO-POL Nulls. For further study of the characteristic polarization state and the distribution of the received powers on the Poincaré sphere, we introduce the concept of the equi-power curve. It is defined as the curve on the Poincaré sphere on which the received powers in some defined channel have the same value. We deal with the characteristics of the equi-power curves for various special cases. In addition, we show how the characteristic polarization states are generated by the equi-power curves. It is demonstrated that the characteristic polarization states can usually be regarded as the points of contact of the Poincaré sphere and a conicoid representing a power-related quadratic form. This leads to a new method to introduce the characteristic polarization states. The new method provides a geometric interpretation for visualization of changes in polarized states on the Poincaré sphere.