van Cittert-Zernike TheoremFundamentals of Radio Interferometry, Section 4.5

Griffin FosterSKA SA/Rhodes University

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What is the Fourier transform of the sky?

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Important Points:

The van Cittert-Zernike Theorem is at the heart of aperture synthesis.

Fourier transforms are essential to synthesis imaging.

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Simply stated: there exists a relationship between the mutual spatial coherence function and the sky intensity distribution. This relationship can be approximated as:

The mutual spatial coherence function (visibilities) and the sky intensity distribution (image of the sky) are Fourier pairs

van Cittert-Zernike Theorem

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Mutual Spatial Coherence Function

Given a signal, in our case the electric field (E), the mutual spatial coherence function for two point in space

(r1, r2) is the time-averaged correlation of the signal

measured at each point.

The mutual spatial coherence function is a correlation between two points in space, in our case, two radio antennas.

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Mutual Spatial Coherence Function

In aperture synthesis we call the mutual spatial coherence function the Visibility function. Instead of the

absolute position of each point we instead use the spatial difference

Thus, the Visibility function is in the uvw-space.

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Sky Intensity Distribution

The sky intensity distribution is a fancy name for what the sky looks like at a given frequency ν. For the van

Cittert-Zernike theorem we use the direction-cosine reference frame.

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Hand-wavy Derivation

Image of the Sky

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Hand-wavy Derivation

1. Given a source far away, such as an astronomical object, the electric field from that source can be considered a plane wave (i.e. the source is in the far-field).

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Hand-wavy Derivation

2. We measure the electric field at two spatially separated points r1, r2.

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Hand-wavy Derivation

3. For any two point in space in which we measure the electric field there is a phase difference between the measured signals which results in a constructive or destructive interference depending on the relative position of the two measurement points and the observing frequency.

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Hand-wavy Derivation

4. For any relative spatial difference there is this phase difference is constant.

Geometric Delay:

Phase:

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Hand-wavy Derivation

5. This fringe pattern results in a sinusoidal wave in the 2-D mutual spatial coherence space.

Fringe PatternGeometric Delay:

Phase:

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Hand-wavy Derivation

6. The Fourier transform of a sinusoidal function is a delta function at a particular position, i.e. the source of the electric field in the sky.

Visibility Function Image of the Sky

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Real Derivation

See Interferometry and Synthesis in Radio Astronomy (TMS) Chapter 14

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The more complete van Cittert-Zernike Theorem

Our 2-D Fourier relation between the visibility function and the sky is an approximate form of the van Cittert-Zernike theorem:

A more complete version of van Cittert-Zernike is:

Unfortunately, this is not a 2-D Fourier relation.

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Approximations to reduce to a 2D Fourier transform

Small Angle/Narrow Field of View Approximation:

If we are only interested in a small area on the sky then the extant of (l,m) is small:

From positional astronomy, the sky instensity is distributed on the celestial (unit) sphere such that for any position (l,m,n):

Then the van Cittert-Zernike theorem can be approximated as:

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Approximations to reduce to a 2D Fourier transform

Delay term/w-term Approximation:

If our visibility sampling is approximately on a plane, or again we are only interested in a narrow field of view

then the so-called w-term

can be seen as a constant delay term, with a simple correction w=0.

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What is the Fourier transform of the sky?

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