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THE CASE OF A CENTRAL POINT SOURCE

For a given interferometer, the measured visibilities for a point source in the centre of the field! can be described by a 4-element `coherency vector' $\vec{v}\,$, which is related to the so-called `Stokes visibility vector' $\vec{s}_{ij}^{}$ = (I, Q, U, V)_ij^T of the observed source by a matrix operation,

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

Here subscripts p and q represent the two polarisation channels measured by each antenna. (NB: They are named X and Y for WSRT and ATCA, and R and L for the VLA). The subscripts i and j represent the antenna numbers, and $\otimes$ represents the matrix direct product (also called the tensor product, or Kronecker product).

The Stokes visibility vector depends on the brightness distribution and on the length and orientation of baseline ij. The 4 x 4 matrix $\vec{S}\,$ converts it into a coherency vector (ignoring instrumental effects). $\vec{S}\,$ is unitary, except for a normalising constant: $\vec{S}^{-1}_{}$ = 2$\vec{S}^{*T}_{}$. It cannot be decomposed into antenna-based parts.

$$\vec{S}\, {\textstyle\frac{1}{2}} \left(\vphantom{\begin{array}{rrrr} 1&1&\;0&0\\ 0&0&1&i\\ 0&0&1&-i\\ 1&-1&0&0 \end{array}}\right. \begin{array}{rrrr} 1&1&\;0&0\\ 0&0&1&i\\ 0&0&1&-i\\ 1&-1&0&0 \end{array} \left.\vphantom{\begin{array}{rrrr} 1&1&\;0&0\\ 0&0&1&i\\ 0&0&1&-i\\ 1&-1&0&0 \end{array}}\right)$$ (2)

The 4-element vector $\vec{A}_{ij}^{}$ represents additive interferometer-based effects. Examples are receiver noise, and correlator offsets. The 4 x 4 diagonal matrix $\vec{M}_{ij}^{}$ represents multiplicative interferometer-based effects, which cannot be factored into antenna-based contributions. Examples are decorrelations, which usually give diagonal matrices with identical elements. Fortunately, the elements of $\vec{A}_{ij}^{}$ and $\vec{M}_{ij}^{}$ tend to be small or close to unity in practice, and will be ignored here.

Thus it is assumed that, in the case of a central point source, all instrumental effects can be factorised into antenna-based contributions. The 4 x 4 interferometer response matrix $\vec{J}^{\null}_{i}$ $\otimes$ $\vec{J}^{\ast}_{j}$ then consists of a direct matrix product of two 2 x 2 antenna-based response matrices.¹ The reader will note that this is the polarimetric generalisation of the familiar `Selfcal assumption'.

This antenna response matrix $\vec{J}_{i}^{}$ can be decomposed into a product of matrices, each of which models a specific instrumental effect in the signal path.²

$$\vec{J}_{i}^{} \vec{G}_{i}^{} \vec{D}_{i}^{} \vec{C}_{i}^{} \vec{B}_{i}^{} \vec{P}_{i}^{} \vec{F}_{i}^{}$$ (3)

in which,

• $\vec{G}_{i}^{}$ represents complex gain (amplitude, phase) per IF channel. It includes the effects of all the electronics after the feed (amplifiers, mixers, LO, cables etc), excluding the correlator of course. It is usually also assumed to includes atmospheric gain effects, even though $\vec{G}_{i}^{}$ does not necessarily commute with $\vec{D}_{i}^{}$ (see below).

  $$\vec{G}_{i}^{} \left(\vphantom{\begin{array}{cc}g_{i\rm p} & 0\\ 0 & g_{i\rm q} \end{array}}\right. \begin{array}{cc}g_{i\rm p} & 0\\ 0 & g_{i\rm q} \end{array} \left.\vphantom{\begin{array}{cc}g_{i\rm p} & 0\\ 0 & g_{i\rm q} \end{array}}\right)$$ (4)

  NB: It is wrong to identify IF-channels with receptors, like the X and Y dipoles at WSRT or ATCA. In a VLA antenna, the circularly polarised IF-signals are a combination of the signals of the two linear dipoles. Another example: in the ATCA and the new WSRT frontends, the signals from the dipoles may be rapidly switched between IF-channels, for calibration purposes.

• $\vec{D}_{i}^{}$ represents the signal `leakage' between the two receptors in a feed, due to deviations from nominal. The real part corresponds roughly to a dipole alignment error, and the imaginary part to ellipticity.

  $$\vec{D}_{i}^{} \left(\vphantom{\begin{array}{cc}1 & d_{i\rm p}\\ -d_{i\rm q} & 1 \end{array}}\right. \begin{array}{cc}1 & d_{i\rm p}\\ -d_{i\rm q} & 1 \end{array} \left.\vphantom{\begin{array}{cc}1 & d_{i\rm p}\\ -d_{i\rm q} & 1 \end{array}}\right)$$ (5)

  NB: Other processes may also contribute to leakage. For example, cross-talk between the two IF-channels can be described by adding a non-zero mutual coupling factor c_i to the off-diagonal terms.

• $\vec{C}_{i}^{}$ represens the nominal feed configuration, which includes rotation of the dipoles w.r.t. the antenna (e.g. WSRT) , and conversion from linear to circular polarisation (e.g. VLA). See section 3 for examples.

• $\vec{B}_{i}^{}$ represents the antenna beam response. For the important case of a dominating compact source in the centre of the field (and an axially symmetric antenna-feed combination), it is the unit matrix. See also section 4.

• $\vec{P}_{i}^{}$ represents parallactic angle $\phi$, i.e. rotation of the antenna w.r.t. the sky. For an equatorial antenna like the WSRT, it is a unit matrix. For an alt-az antenna, $\phi$ varies smoothly with Hour Angle, as a function of Latitude and Declination: ( $\phi$(t) = arctan[cosLATsinHAcosDEC  -  sinLATcosHAsin DEC] (or something like that):

  $$\vec{P}_{i}^{alt-az} \left(\vphantom{\begin{array}{cc}\cos\phi(t) & \sin\phi(t)\\ -\sin\phi(t) & \cos\phi(t) \end{array}}\right. \begin{array}{cc}\cos\phi(t) & \sin\phi(t)\\ -\sin\phi(t) & \cos\phi(t) \end{array} \left.\vphantom{\begin{array}{cc}\cos\phi(t) & \sin\phi(t)\\ -\sin\phi(t) & \cos\phi(t) \end{array}}\right)$$ (6)

• $\vec{F}_{i}^{}$ represents (ionospheric) Faraday rotation.

  $$\vec{F}_{i}^{} \left(\vphantom{\begin{array}{cc}\cos\psi & \sin\psi\\ -\sin\psi & \cos\psi \end{array}}\right. \begin{array}{cc}\cos\psi & \sin\psi\\ -\sin\psi & \cos\psi \end{array} \left.\vphantom{\begin{array}{cc}\cos\psi & \sin\psi\\ -\sin\psi & \cos\psi \end{array}}\right)$$ (7)

It should be noted that these matrices generally do not commute, so that their order is important! For instance, a diagonal matrix like $\vec{G}_{i}^{}$ does not commute with a matrix with non-zero elements off the diagonal, like $\vec{D}_{i}^{}$. Thus, one should be a little careful in sweeping all complex gain effects into a single matrix $\vec{G}_{i}^{}$, irrespective of where they occur in the signal path.

Matrices $\vec{G}_{i}^{}$, $\vec{P}_{i}^{}$ and $\vec{F}_{i}^{}$ are Cartesian coordinate transforms, and thus unitary: $\vec{A}^{-1}_{}$ = $\vec{A}^{*T}_{}$. Matrix $\vec{C}_{i}^{}$ is not unitary if it represents a conversion from linear to circular polarisation (..?). Do they commute??