Some comments on the descriptions of the polarization states of waves

TLDR

Procedures are developed to specify the polarization characteristics of n‑dimensional waves, especially three‑dimensional geophysical waves. The authors construct degree‑of‑polarization measures from the spectral matrix S’s scalar invariants, avoid diagonalization, derive a unitary polarization vector directly from S, express it as u = r₁ + i r₂, and compute ULF magnetic wave parameters using a single set of five equations. They show that a pure or totally polarized wave can be represented by a single unitary vector, that elliptically polarized waves have major/minor axes r₁ and r₂ with ellipticity equal to their magnitude ratio, and that the computed ULF magnetic wave parameters agree with established techniques. Summary.

Abstract

Summary. Procedures are developed for specifying the polarization characteristics of n-dimensional waves, and in particular three-dimensional waves of geophysical interest. We show that when a wave is in a pure state or is totally polarized, all the polarization information can be represented by a single vector u in an n-dimensional unitary space. Simple measures of the degree of polarization of the wave are constructed from the characteristic equation of the spectral matrix S. These measures are functions only of the scalar invariants of S and consequently S need not be diagonalized. If S represents a purely polarized wave, the unitary vector which contains the polarization information about the wave can be obtained directly from S using any 2n– 1 equations of n2 possible equations. By multiplying by a phase-factor this unitary vector can be written in the form u=r1+ir2 where r1 and r2 are orthogonal vectors in a real space. For an elliptically polarized wave, r1 and r2 locate the major and minor axes of the ellipse, and the ellipticity is given by the ratio of their magnitudes. The polarization parameters of ULF magnetic waves at the Earth's surface are computed from one set of five equations (n= 3) and compared with parameters calculated using established techniques.