CittertZernike

The Cittert–Zernike theorem, often cited as the Cittert–Zernike theorem, is a foundational result in optical coherence theory that relates the spatial coherence of light from an incoherent, distant source to the source’s angular brightness distribution. It was independently derived in the 1930s by Anton van Cittert and Frits Zernike and has since become central to the practice of astronomical interferometry and aperture synthesis.

In its standard form, the theorem states that for a quasi-monochromatic, spatially incoherent source observed in the far field, the complex degree of spatial coherence (or visibility) between two points in an optical or radio interferometer is proportional to the Fourier transform of the normalized intensity distribution of the source on the sky. In practical terms, the measured visibility V(u,v) at baseline coordinates (u,v) is the two-dimensional Fourier transform of the source brightness I(l,m), evaluated over the celestial sphere. The relationship is often written as Gamma(u,v) = ∫∫ I(l,m) exp[-2πi(ul + vm)] dl dm divided by ∫∫ I(l,m) dl dm, where l and m are direction cosines.

Applications of the theorem include reconstruction of an object's image from interferometric data, enabling aperture synthesis in radio and optical astronomy. It also provides a fundamental framework for interpreting measurements of spatial coherence. Limitations arise from bandwidth (polychromatic light), finite field of view, polarization effects, and departures from the assumptions of incoherence and far-field propagation, which require generalizations or corrections in practical setups.