Consider a two element interferometer shown in Figure 4.1.
Two antennas whose (vector) separation is **b**, are directed
towards a point source of flux density S. The angle between the direction to
the point source and the normal to the antenna separation vector is
. The voltages that are produced at the two antennas due to the
electric field from this point source are and
respectively. These two voltages are multiplied together, and then
averaged. Let us start by assuming that
the radiation emitted by the source is monochromatic and has frequency
. Let the voltage at antenna be
.
Since the radio waves from the source have to travel an extra distance
to reach antenna , the voltage there is delayed by
the amount
. This is called the *geometric delay*,
. The voltage at antenna is hence
, where we have assumed that the
antennas have identical gain. , the averaged output of the
multiplier is hence:

where we have assumed that the averaging time T is long compared
to . The
factor hence averages out to 0.
As the source rises and sets, the angle changes. If we
assume that the antenna separation vector, (usually called the *baseline
vector* or just the *baseline*) is exactly east west, and that the
source's declination , then
, (
where is the angular frequency of the earth's rotation) we have:

where is the time at which the source is at the zenith. The output
, (also called the *fringe*), hence varies in a
quasi-sinusoidal form, with its instantaneous frequency being maximum
when the source is at zenith and minimum when the source is either
rising or setting (Figure 4.2).

Now if the source's right ascension was known, then one could compute
the time at which the source would be at zenith, and hence the time at
which the instantaneous fringe frequency would be maximum. If the fringe
frequency peaks at some slightly different time, then one knows that assumed
right ascension of the source was slightly in error. Thus, in principle at
least, from the difference between the actual observed peak time and the
expected peak time one could determine the true right ascension of the source.
Similarly, if the source were slightly extended, then when the
waves from a given point on the source arrive in phase at the two ends
of the interferometer, waves arising from adjacent points on the source
will arrive slightly out of phase. The observed amplitude of the
fringe will hence be less than what would be obtained for a point
source of the same total flux. The more extended the source, the lower
the fringe amplitude^{4.2}. For a sufficiently large source with smooth
brigtness distribution, the fringe amplitude will be essentially
zero^{4.3}. In such circumstances,
the interferometer is said to have *resolved out* the source.

Further, two element interferometers cannot distinguish between sources whose sizes are small compared to the fringe spacing, all such sources will appear as point sources. Equivalently when the source size is such that waves from different parts of the source give rise to the same phase lags (within a factor that is small compared to ), then the source will appear as a point source. This condition can be translated into a limit on , the minimum source size that can be resolved by the interferometer, viz.,

i.e., the resolution of a two element interferometer is . The longer the baseline, the higher the resolution.

Observations with a two element interferometer hence give one information on both the source position and the source size. Interferometers with different baseline lengths and orientations will place different constraints on the source brightness, and the Fourier transform in the van Cittert-Zernike theorem can be viewed as a way to put all this information together to obtain the correct source brightness distribution.

- ... amplitude
^{4.2} - assuming that the source has a uniform brightness distribution
- ...
zero
^{4.3} - This is related to the fact that in the double slit experiment, the interference pattern becomes less distinct and then eventually disappears as the source size is increased (see e.g. Born & Wolf, `Principles of Optics', Sixth Edition, Section 7.3.4). In fact the double slit setup is mathematically equivalent to the two element interferometer, and much of the terminology in radio interferometry is borrowed from earlier optical terminology.