A generalised Measurement Equation and van Cittert-Zernike theorem for wide-field radio astronomical interferometry

We derive a generalised van Cittert-Zernike (vC-Z) theorem for radio astronomy that is valid for partially polarized sources over an arbitrarily wide field-of-view (FoV). The classical vC-Z theorem is the theoretical foundation of radio astronomical interferometry, and its application is the basis of interferometric imaging. Existing generalised vC-Z theorems in radio astronomy assume, however, either paraxiality (narrow FoV) or scalar (unpolarized) sources. Our theorem uses neither of these assumptions, which are seldom fulfilled in practice in radio astronomy, and treats the full electromagnetic field. To handle wide, partially polarized fields, we extend the two-dimensional electric field (Jones vector) formalism of the standard Measurement Equation of radio astronomical interferometry to the full three-dimensional formalism developed in optical coherence theory. The resulting vC-Z theorem enables all-sky imaging in a single telescope pointing, and imaging using not only standard dual-polarized interferometers (that measure 2-D electric fields), but also electric tripoles and electromagnetic vector-sensor interferometers. We show that the standard 2-D Measurement Equation is easily obtained from our formalism in the case of dual-polarized antenna element interferometers. We find, however, that such dual-polarized interferometers can have polarimetric aberrations at the edges of the FoV that are often correctable. Our theorem is particularly relevant to proposed and recently developed wide FoV interferometers such as LOFAR and SKA, for which direction-dependent effects will be important.