Coherency matrix description of optical polarization singularities

The polarization figures of strictly monochromatic, elliptically polarized light are true ellipses only at particular points in time; at other times, and in other fields, the figures differ from ellipses, and can often exhibit a bewildering variety of complex shapes. A modified real coherency matrix of the field, M, is shown to provide a convenient representation of the polarization figures in arbitrary monochromatic, and arbitrary polychromatic, two dimensional, and three dimensional, not necessarily stationary, electromagnetic fields that are observed at arbitrary times for arbitrary intervals. The eigenvalues of M permit a simple classification of the polarization figures, and their singularities; the eigenvectors of M define the important line of sight down which the singularities are properly viewed; the zeros of the discriminant of the characteristic equation of M track the path of the singularities through space. Instructive examples are discussed that illustrate the utility of this particular matrix approach to optical polarization.