The van Cittert-Zernike theorem relates the spatial coherence function to the distribution of intensity of the incoming radiation, . It shows that the spatial correlation function depends only on and that if all the measurements are in a plane, then
Let us assume that the source is distant and can be approximated as a brightness distribution on the celestial sphere of radius (see Figure 2.2). Let the electric field2.5 at a point at the source be given by . The field at the observation point is given by2.6
is the distance between and .
Similarly if is the field at some other observing point
cross-correlation between these two fields is given by
If we further assume that the emission from the source is spatially incoherent,
, then we have
where is the intensity at the point . Since we have assumed that the source can be approximated as lying on a celestial sphere of radius we have , , and ; () are called ``direction cosines''. It can be easily shown2.7 that and that . We then have:
Putting this back into equation 2.4.7 we get
Note that since , the two directions cosines are sufficient
to uniquely specify any given point on the celestial sphere, which is why the intensity
has been written out as a function of only.
It is customary to measure distances in the observing plane in units of the
wavelength , and to define ``baseline co-ordinates'' such that
. The spatial correlation function
referred to as the ``visibility''
. Apart from the
constant factor (which we will ignore hence forth)
equation 2.4.14 can then be written as
This fundamental relationship between the visibility and the source intensity distribution is the basis of radio interferometry. In the optical literature this relationship is also referred to as the van Cittert-Zernike theorum.
Equation 2.4.15 resembles a Fourier transform. There are two situations
in which it does reduce to a Fourier transform. The first is when the observations
are confined to a the plane, i.e. when . In this case we have
i.e. the visibility
is the Fourier transform of the modified brightness
. The second situation is
when the source brightness distribution is limited to a small region of the sky. This
is a good approximation for arrays of parabolic antennas because each antenna responds
only to sources which lie within its primary beam (see Chapter 3).
The primary beam is typically , which is a very small area of sky.
In this case
. Equation 2.4.15 then
or if we define a modified visibility