  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.
