Visibility Function

In aperture synthesis and radio astronomy, the visibility function (or simply visibility) is the complex correlation coefficient measured between the signals received by a pair of antennas. Formally, it is the Fourier transform of the sky brightness distribution, sampled at a spatial frequency determined by the baseline vector between the two antennas. The visibility encodes both the amplitude and phase of the spatial frequency component, and a sufficient collection of visibilities — measured across many baselines and orientations — enables the reconstruction of the source image through Fourier inversion.

The visibility is not a direct observable of the sky. It is an observable of the interferometer: the product of voltage measurements at two points in space, averaged over a coherence time during which the atmospheric and instrumental phase fluctuations are small. This distinction is crucial. The visibility is a cross-correlation in the spatial domain, not a measurement of the source itself. The source is inferred; the correlation is measured.

In the uv-plane — the coordinate system of spatial frequencies — each baseline samples a single point. The Earth's rotation changes the projected baseline orientation, tracing an ellipse in the uv-plane and providing multiple samples from a fixed antenna pair. This is the principle of earth-rotation synthesis, the technique that made modern radio interferometry practical.

The visibility's phase is particularly fragile. Atmospheric turbulence, clock errors, and instrumental phase shifts corrupt the phase before the correlation is computed. For this reason, phase closure techniques — using the sum of phases around closed triangles of baselines — were developed to recover calibration-independent phase information. The amplitude is more robust but still subject to gain fluctuations that require calibration against known sources.

The visibility function is where the ontology of interferometry becomes explicit. We do not image the sky. We image the correlations between measurements of the sky, and the sky itself exists for us only as the inverse Fourier transform of those correlations. This is not instrumentalism. It is the practical recognition that in aperture synthesis, the instrument and the image are inseparable.