1. van Cittert-Zernike Theorem
Fundamentals of Radio Interferometry, Section 4.5
Griffin Foster
SKA SA/Rhodes University
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2. What is the Fourier transform of the sky?
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3. 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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4. van Cittert-Zernike Theorem
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
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5. 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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6. Thus, the Visibility function is in the uvw-space.
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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7. 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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9. Hand-wavy Derivation
Image of the Sky
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10. 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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11. Hand-wavy Derivation
2. We measure the electric field at two spatially separated points r1, r2.
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12. 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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13. Hand-wavy Derivation Geometric Delay: Phase:
4. For any relative spatial difference there is this phase difference is constant.
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14. Hand-wavy Derivation Geometric Delay: Fringe Pattern Phase:
5. This fringe pattern results in a sinusoidal wave in the 2-D mutual spatial coherence space.
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15. Hand-wavy Derivation Visibility Function Image of the Sky
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.
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16. See Interferometry and Synthesis in Radio Astronomy (TMS) Chapter 14
Real Derivation
See Interferometry and Synthesis in Radio Astronomy (TMS) Chapter 14
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17. 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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18. Approximations to reduce to a 2D Fourier transform
Small Angle/Narrow Field of View Approximation:
From positional astronomy, the sky instensity is distributed on the celestial (unit) sphere such that for any position (l,m,n):
If we are only interested in a small area on the sky then the extant of (l,m) is small:
Then the van Cittert-Zernike theorem can be approximated as:
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19. 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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20. What is the Fourier transform of the sky?
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