Geometric interpretation of the characteristic polarizations

The scattering mechanism can be analyzed from the geometrical point of view using the polar decomposition which decomposes the normalized scattering matrix in a unitary matrix and in a Hermitian (positive definite) one. Each of them can be built using parameters (polar parameters) such as vectors, angles and scalars which have geometrical meaning in the Stokes space. In this work it is shown the relation between them and some of the characteristic polarizations of a normalized, symmetric scattering matrix: the copolar null and the copolar maxima. Moreover, the inverse relations are found: the parameters which characterize the unitary and the Hemitian matrices can be expressed as function of the copolar nulls only. Using these relations, the problem to find the polarizations such that the received voltage is maximized (or minimized) for a given S, is reduced to apply the polar decomposition and compute the polar parameters. Furthermore, the scattering matrix is parameterized by the characteristic polarization states. Since it exists a direct connection between a unitary matrix and a rotation and between a Hermitian matrix and a boost, the Mueller (or Kennaugh) matrix can be built using the same parameters. This is the first step towards the study of the characteristic polarization states in the more general case of received partially polarized waves, the stochastic case.