van Cittert–Zernike theorem

The van Cittert–Zernike theorem, named after physicists Pieter Hendrik van Cittert and Frits Zernike,[1] is a formula in coherence theory that states that under certain conditions the Fourier transform of the intensity distribution function of a distant, incoherent source is equal to its complex visibility.

This reasoning can be easily visualized by dropping two stones in the center of a calm pond.

As the disturbance propagates towards the edge of the pond, however, the waves will smooth out and will appear to be nearly circular.

Nevertheless, because they are observed at distances large enough to satisfy the van Cittert–Zernike theorem, these objects exhibit a non-zero degree of coherence at different points in the imaging plane.

By measuring the degree of coherence at different points in the imaging plane (the so-called "visibility function") of an astronomical object, a radio astronomer can thereby reconstruct the source's brightness distribution and make a two-dimensional map of the source's appearance.

[5] This theorem was first derived by Pieter Hendrik van Cittert[6] in 1934 with a simpler proof provided by Frits Zernike in 1938.

[7] This theorem will remain confusing to some engineers or scientists because of its statistical nature and difference from simple correlation or even covariance processing methods.

A good reference (which still might not clarify the issue for some users, but does have a great sketch to drive the method home) is Goodman, starting on page 207.

Far away from the sources, however, we should expect the mutual coherence function to be relatively large because the sum of the observed fields will be almost the same at any two points.

This small correction can therefore be neglected, further simplifying our expression for the cross-correlation of the electric field at

, the square roots may be Taylor expanded, yielding, to first order, which, after some algebraic manipulation, simplifies to Now,

The van Cittert–Zernike theorem rests on a number of assumptions, all of which are approximately true for nearly all astronomical sources.

is the characteristic size of the observation area (e.g. in the case of a two-dish radio telescope, the length of the baseline between the two telescopes) then Using a reasonable baseline of 20 km for the Very Large Array at a wavelength of 1 cm, the far field distance is of order

But if the source has a large angular extent, we cannot neglect this third direction cosine and the van Cittert–Zernike theorem no longer holds.

Because most astronomical sources subtend very small angles on the sky (typically much less than a degree), this assumption of the theorem is easily fulfilled in the domain of radio astronomy.

This requirement implies that a radio astronomer must restrict signals through a bandpass filter.

Because radio telescopes almost always pass the signal through a relatively narrow bandpass filter, this assumption is typically satisfied in practice.

In the case of a heterogeneous medium one must use a generalization of the van Cittert–Zernike theorem, called Hopkins's formula.

In practice, however, variations in the refractive index of the interstellar and intergalactic media and Earth's atmosphere are small enough that the theorem is approximately true to within any reasonable experimental error.

Suppose we have a situation identical to that considered when the van Cittert–Zernike theorem was derived, except that the medium is now heterogeneous.

Following a similar derivation as before, we find that If we define then the mutual coherence function becomes which is Hopkins's generalization of the van Cittert–Zernike theorem.

The primary advantage of Hopkins's formula is that one may calculate the mutual coherence function of a source indirectly by measuring its brightness distribution.

By applying the van Cittert–Zernike theorem, the astronomer can then take the inverse Fourier transform of the visibility function to discover the brightness distribution of the source.

In practice, radio astronomers rarely recover the brightness distribution of a source by directly taking the inverse Fourier transform of a measured visibility function.

Because the brightness distribution must be real and positive everywhere, the visibility function cannot take on arbitrary values in unsampled regions.

Thus, a non-linear deconvolution algorithm like CLEAN or Maximum Entropy may be used to approximately reconstruct the brightness distribution of the source from a limited number of observations.

An AO system must make a number of different corrections to remove the distortions from the wavefront.

By measuring the size and separation of the fringes, the AO system can determine phase differences along the wavefront.

[13] The van Cittert–Zernike theorem implies that the mutual coherence of an extended source imaged by an AO system will be the Fourier transform of its brightness distribution.

The van Cittert–Zernike theorem can be used to calculate the partial spatial coherence of radiation from a free-electron laser.