Understanding radio polarimetry .1. Mathematical foundations

The measurement of polarized radiation uses entirely different methods at optical and radio wavelengths. As a result, the algebraic analysis of polarimeter performance differs and, in the case of radio interferometry, is unnecessarily complicated. We demonstrate that the mathematical operation of outer matrix multiplication provides the missing link between the two approaches. Within one coherent framework, we then unite the concepts of Stokes parameters and Wolf coherency matrix, the Jones and Mueller calculi from optics, and the techniques of radio interferometry based on multiplying correlators. We relate the polarization performance of a complete radio interferometer to the (matrix) polarization properties of its successive signal processing stages, providing a clear view of how a radio polarimeter works. Our treatment also clarifies the nature of and the relations between the various types of transformations used in optical polarimetry. We develop the analysis from the radio interferometrist's point of view, but include enough background for a wider audience. In a companion paper, we discuss in more detail the application to the calibration of radio interferometer systems; in a third paper we investigate the IAU (1973) radio definition of the Stokes parameters and its precise translation into mathematical form.