APPLICATION TO EXISTING TELESCOPES

To allow the reader to get some intuitive `feeling' for the matrix formalism, it will be made explicit in this section for some practical cases. This is done for the simplified case of a central point source (rather than for the full Measurement Equation described in section 4), because:

It should be noted that the formalism makes it very easy to calculate the response of an interferometer that consists of two antenna's which can each be described by an antenna-based response matrix $\vec{J}_{i}^{}$ , even if the constituent antenna's are quite different. This is particularly important for VLBI, with its collection of often quite dissimilar antenna's. Another potentially important example is the integration of one or two prototype antenna's for the Square Km Array (SKAI) in the WSRT. An example of a type of antenna that cannot be easily described by its own $\vec{J}_{i}^{}$ seems to be a tied array (see section 3.2).

WSRT, VLA, ATCA, GMRT

An (equatorial) WSRT antenna, with its linear dipoles rotated w.r.t. the antenna over $\alpha$ degrees. Special cases are `parallel' (+) dipoles, i.e. rotated over $\alpha$ = 0, and `crossed' (X) dipoles, i.e. rotated over $\alpha$ = - 45.

$\displaystyle \vec{J}_{i}^{WSRT}$ = $\displaystyle \vec{G}_{i}^{}$ $\displaystyle \vec{D}_{i}^{}$ $\displaystyle \left(\vphantom{\begin{array}{cc}\cos\alpha & \sin\alpha\\ -\sin\alpha & \cos\alpha \end{array}}\right.$ $\displaystyle \begin{array}{cc}\cos\alpha & \sin\alpha\\ -\sin\alpha & \cos\alpha \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{cc}\cos\alpha & \sin\alpha\\ -\sin\alpha & \cos\alpha \end{array}}\right)$ $\displaystyle \left(\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right.$ $\displaystyle \begin{array}{cc}1 & 0\\ 0 & 1 \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right)$ $\displaystyle \left(\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right.$ $\displaystyle \begin{array}{cc}1 & 0\\ 0 & 1 \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right)$ $\displaystyle \vec{F}_{i}^{}$

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The resulting WSRT interferometer response matrix is of course equivalent to the equations derived by Weiler [Weiler72]. In the new WSRT frontends (due in 1997/8), the dipoles cannot be rotated anymore, so they will always be `parallel' ( $\alpha$ = 0):

$\displaystyle \vec{J}_{i}^{WSRT+}$ = $\displaystyle \vec{G}_{i}^{}$ $\displaystyle \vec{D}_{i}^{}$ $\displaystyle \left(\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right.$ $\displaystyle \begin{array}{cc}1 & 0\\ 0 & 1 \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right)$ $\displaystyle \left(\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right.$ $\displaystyle \begin{array}{cc}1 & 0\\ 0 & 1 \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right)$ $\displaystyle \left(\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right.$ $\displaystyle \begin{array}{cc}1 & 0\\ 0 & 1 \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right)$ $\displaystyle \vec{F}_{i}^{}$

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An (alt-az) ATCA antenna, with fixed linear dipoles. The parallactic angle in matrix $\vec{P}_{i}^{alt-az}$ (see equ 6) varies smoothly with Hour-Angle (time):

$\displaystyle \vec{J}_{i}^{ATCA}$ = $\displaystyle \vec{G}_{i}^{}$ $\displaystyle \vec{D}_{i}^{}$ $\displaystyle \left(\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right.$ $\displaystyle \begin{array}{cc}1 & 0\\ 0 & 1 \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right)$ $\displaystyle \left(\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right.$ $\displaystyle \begin{array}{cc}1 & 0\\ 0 & 1 \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right)$ $\displaystyle \vec{P}_{i}^{alt-az}$ $\displaystyle \vec{F}_{i}^{}$

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An (alt-az) VLA antenna, with circularly polarised receptors. The C-matrix descibes the transformation of the signals from the fixed linear dipoles into Right and Left circularly polarised R and L signals, by means of a `hybrid'. Any instrumental (e.g. phase) effects in this hybrid could be modelled too. The matrix $\vec{B}_{i}^{}$ is not a unit matrix, even for a central point source, because the system is not axially symmetric and produces some instrumental polarisation even in the centre of the field.

$\displaystyle \vec{J}_{i}^{VLA}$ = $\displaystyle \vec{G}_{i}^{}$ $\displaystyle \vec{D}_{i}^{}$ $\displaystyle {\frac{1}{\sqrt{2}}}$ $\displaystyle \left(\vphantom{\begin{array}{cc}1 & i\\ 1 & -i \end{array}}\right.$ $\displaystyle \begin{array}{cc}1 & i\\ 1 & -i \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{cc}1 & i\\ 1 & -i \end{array}}\right)$ $\displaystyle \vec{B}_{i}^{}$ $\displaystyle \vec{P}_{i}^{alt-az}$ $\displaystyle \vec{F}_{i}^{}$

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For a central point source (i.e. when the beam $\vec{B}_{i}^{}$ reduces to a unit matrix), an (alt-az) GMRT antenna is mathematically equivalent to an ATCA antenna (see expression 10 above):

The same is true for the (alt-az) single dish Effelsberg radio telescope, with fixed linear dipoles:

The GBT is not axially symmetric, so that the beam $\vec{B}_{i}^{}$ has to be taken into account even for a central point source.

More exotic cases

A dipole array, with North-South and East-West linear dipoles in the horizontal plane. This is one of the possible concepts for an element of the Square Km Array (SKAI) planned by NFRA. It is assumed that the dipole array is internally calibrated, i.e. that it behaves like a single antenna. The parallactic rotation of the EW dipoles w.r.t. the sky differs from that of the NS dipoles: $\phi_{EW}^{}$ (t) = arctan(tanHAsin DEC), and $\phi_{NS}^{}$ (t) = f (HA, DEC, latitude)^...?. The beam properties described by $\vec{B}_{i}^{}$ change as a function of the same parameters.

$\displaystyle \vec{J}_{i}^{SKAI}$ = $\displaystyle \vec{G}_{i}^{}$ $\displaystyle \vec{D}_{i}^{}$ $\displaystyle \left(\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right.$ $\displaystyle \begin{array}{cc}1 & 0\\ 0 & 1 \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right)$ $\displaystyle \vec{B}_{i}^{}$ (t) $\displaystyle \left(\vphantom{\begin{array}{cc}\cos\phi_{NS}(t) & \sin\phi_{NS}(t)\\ -\sin\phi_{EW}(t) & \cos\phi_{EW}(t) \end{array}}\right.$ $\displaystyle \begin{array}{cc}\cos\phi_{NS}(t) & \sin\phi_{NS}(t)\\ -\sin\phi_{EW}(t) & \cos\phi_{EW}(t) \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{cc}\cos\phi_{NS}(t) & \sin\phi_{NS}(t)\\ -\sin\phi_{EW}(t) & \cos\phi_{EW}(t) \end{array}}\right)$ $\displaystyle \vec{F}_{i}^{}$

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The description of the Arecibo antenna resembles that of a dipole array ^...:

$\displaystyle \vec{J}_{i}^{Arecibo}$ = $\displaystyle \vec{G}_{i}^{}$ $\displaystyle \vec{D}_{i}^{}$ $\displaystyle \left(\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right.$ $\displaystyle \begin{array}{cc}1 & 0\\ 0 & 1 \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right)$ $\displaystyle \vec{B}_{i}^{}$ (t) $\displaystyle \left(\vphantom{\begin{array}{cc}\cos\phi_{NS}(t) & \sin\phi_{NS}(t)\\ -\sin\phi_{EW}(t) & \cos\phi_{EW}(t) \end{array}}\right.$ $\displaystyle \begin{array}{cc}\cos\phi_{NS}(t) & \sin\phi_{NS}(t)\\ -\sin\phi_{EW}(t) & \cos\phi_{EW}(t) \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{cc}\cos\phi_{NS}(t) & \sin\phi_{NS}(t)\\ -\sin\phi_{EW}(t) & \cos\phi_{EW}(t) \end{array}}\right)$ $\displaystyle \vec{F}_{i}^{}$

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A free-floating space VLBI antenna can be rotated over an arbitrary parallactic angle $\phi$ , and does not suffer from ionospheric Faraday rotation. Assuming that it has fixed linear dipoles and is axially symmetric, we get:

$\displaystyle \vec{J}_{i}^{Space}$ = $\displaystyle \vec{G}_{i}^{}$ $\displaystyle \vec{D}_{i}^{}$ $\displaystyle \left(\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right.$ $\displaystyle \begin{array}{cc}1 & 0\\ 0 & 1 \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right)$ $\displaystyle \left(\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right.$ $\displaystyle \begin{array}{cc}1 & 0\\ 0 & 1 \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right)$ $\displaystyle \left(\vphantom{\begin{array}{cc}\cos\phi & \sin\phi\\ -\sin\phi & \cos\phi \end{array}}\right.$ $\displaystyle \begin{array}{cc}\cos\phi & \sin\phi\\ -\sin\phi & \cos\phi \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{cc}\cos\phi & \sin\phi\\ -\sin\phi & \cos\phi \end{array}}\right)$ $\displaystyle \left(\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right.$ $\displaystyle \begin{array}{cc}1 & 0\\ 0 & 1 \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right)$

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A tied array, in which the signals of a number of identical WSRT (+) antenna's are added coherently. This case is important for VLBI, where arrays like WSRT and VLA are used as tied arrays. It does not seem possible to describe such a system with a single Jones antenna-based matrix $\vec{J}_{i}^{}$ . Therefore, we write down the interferometer response equation of an interferometer between two tied arrays. The indices i run over the antenna's that make up the first tied array, and the j run over antenna's of the second. Thus, the overall output is the sum of all the constituetnt interferometers:

$\displaystyle \left(\vphantom{\begin{array}{c} v_{\rm pp}\\ v_{\rm pq}\\ v_{\rm qp}\\ v_{\rm qq} \end{array}}\right.$ $\displaystyle \begin{array}{c} v_{\rm pp}\\ v_{\rm pq}\\ v_{\rm qp}\\ v_{\rm qq} \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{c} v_{\rm pp}\\ v_{\rm pq}\\ v_{\rm qp}\\ v_{\rm qq} \end{array}}\right)_{ij}^{}$ = $\displaystyle \sum_{i=1}^{n}$ $\displaystyle \sum_{j=1}^{m}$ ( $\displaystyle \vec{J}^{\null}_{i}$ $\displaystyle \otimes$ $\displaystyle \vec{J}^{\ast}_{j}$ ) $\displaystyle \vec{S}\,$ $\displaystyle \left(\vphantom{\begin{array}{c} I\\ Q\\ U\\ V \end{array}}\right.$ $\displaystyle \begin{array}{c} I\\ Q\\ U\\ V \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{c} I\\ Q\\ U\\ V \end{array}}\right)_{ij}^{}$

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