van Cittert-Zernike Theorem Fundamentals of Radio Interferometry, Section 4.5 Griffin Foster SKA SA/Rhodes University NASSP 2016

What is the Fourier transform of the sky? NASSP 2016

Important Points: The van Cittert-Zernike Theorem is at the heart of aperture synthesis. Fourier transforms are essential to synthesis imaging. NASSP 2016

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 NASSP 2016

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. NASSP 2016

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. NASSP 2016

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. NASSP 2016

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Hand-wavy Derivation Image of the Sky NASSP 2016

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). NASSP 2016

Hand-wavy Derivation 2. We measure the electric field at two spatially separated points r1, r2. NASSP 2016

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. NASSP 2016

Hand-wavy Derivation Geometric Delay: Phase: 4. For any relative spatial difference there is this phase difference is constant. NASSP 2016

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. NASSP 2016

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. NASSP 2016

See Interferometry and Synthesis in Radio Astronomy (TMS) Chapter 14 Real Derivation See Interferometry and Synthesis in Radio Astronomy (TMS) Chapter 14 NASSP 2016

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. NASSP 2016

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: NASSP 2016

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. NASSP 2016

What is the Fourier transform of the sky? NASSP 2016

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