User:Jantognini/van Cittert-Zernike theorem

The Van Cittert-Zernike theorem is a formula in coherence theory that states that under certain conditions the Fourier transform of the mutual coherence function of a distant, incoherent source is equal to its complex visibility. This implies that the wavefront from an incoherent source will appear mostly coherent at large distances. Intuitively, this can be understood by considering the wavefronts created by two incoherent sources. If we measure the wavefront immediately in front of one of the sources, our measurement will be dominated by the nearby source. If we make the same measurement far from the sources, our measurement will no longer be dominated by a single source; both sources will contribute almost equally to the wavefront at large distances.

This reasoning can be easily visualized by dropping two stones in the center of a calm pond. Near the center of the pond, the disturbance created by the two stones will be very complicated. As the disturbance propagates towards the edge of pond, however, the waves will smooth out and will appear to be nearly circular.

The van Cittert-Zernike theorem has important implications for radio astronomy. With the exception of pulsars and masers, all astronomical sources are spatially incoherent. Nevertheless, because they are observed at distances large enough to satisfy the van Cittert-Zernike theorem, all astronomical objects exhibit a non-zero degree of coherence at different points in the imaging plane. By measuring the degree of coherence at different points in the imaging plane (the so-called "visibility function") of an astronomical object, a radio astronomer can thereby reconstruct the source's brightness distribution and make a two-dimensional map of the source's appearance.

Statement of the theorem
If $$\Gamma_{12}(u,v,0)$$ is the mutual coherence function between two points on a plane perpendicular to the line of sight, then

$$\Gamma_{12} (u,v,0) = \int \!\! \int I(l,m) e^{-2\pi i(ul+vm)} dl \, dm$$