Interferometric Measurements

An array like the GMRT measures the visibility function $V(u,v)$ along baselines which move along tracks in the $u-v$ plane as the earth rotates, For simplicity, let us assume that these measurements have been transferred onto a discrete grid and baselines are measured in units of the wavelength. The sky brightness distribution $I(l,m)$ in the field of view is a function of $l,m$ which are direction cosines of a unit vector to a point on the celestial sphere referred to the $u$ and $v$ axes. The basic relationship between the measured visibility function $V$ and the sky brightness $I$ is a Fourier transform.

$$V(u,v)=\int\int I(l,m) \exp(-2 \pi i (lu+mv)) ~dl ~dm.$$

This expression also justifies the term ``spatial frequency'' to describe the pair $(u,v)$, since $u$ and $v$ play the same role as frequency plays in representing time varying signals.

Many things have been left out in this expression, such as the proper units, polarisation, the primary beam response of the individual antennas, the non-coplanarity of the baselines, the finite observing bandwidth, etc. But it is certainly necessary to understand this simplified situation first, and the details needed to achieve greater realism can be put in later.

Aperture synthesis, as originally conceived, involved filling in the $u-v$ plane without any gaps upto some maximum baseline $b_{max}$ which would determine the angular resolution. Once one accepts this resolution limit, and writes down zeros for visibility values outside the measured circle, the Fourier transform can be inverted. One is in the happy situation of having as many equations as unknowns. A point source at the field centre.(which has constant visibility) would be reconstructed as the Fourier transform of a uniformly filled circular disk of diameter $2b_{max}$. This is the famous Airy pattern with its first zero at $1.22 /(2 b_{max})$. The baseline $b$ is already measured in wavelengths, hence the missing $\lambda$ in the numerator. But even in this ideal situation, there are some problems. Given an array element of diameter $D$ (in wavelengths again!), the region of sky of interest could even be larger than a circle of angular diameter $2/D$. A Fourier component describing a fringe going through one cycle over this angle corresponds to a baseline of $D/2$. But measuring such a short baseline would put two dishes into collision, and even somewhat larger baselines than $D$ run the risk of one dish shadowing the other. In addition, the really lowest Fourier component corresponds to $(u,v)=(0,0)$, the total flux in the primary beam. This too is not usually measured in synthesis instruments Thus, there is an inevitable ``short and zero spacings problem'' even when the rest of the $u-v$ plane is well sampled.