2 Coherency-matrix formulation of interferometry

   
2.1 The scalar/matrix analogy

The algebraic properties of scalars and matrices are very similar. Every elementary property of scalars has an immediate matrix counterpart, with one very important exception, viz. that matrix multiplication is non-commutative.

The analogy extends further. There is, for example, a matrix counterpart of the polar representation $a = \vert a\vert \exp{i\alpha}$ of complex numbers. An overview is given in Table 1.

   
2.2 The coherency matrix

As in the previous papers of this series, I represent the electric field in cartesian coordinates by a vector

$$\begin{eqnarray*}\vec{e}= \left( \begin{array}{l} e_{x \vphantom{'}}\\ e_{y \vphantom{'}}\end{array} \right) . \end{eqnarray*}$$

The equivalent of the scalar visibility is the coherency tensor. In mathematical terms, it is a complex-valued 2-dimensional tensor of rank 2 ([Landau & Lifshitz 1995]). In Paper I, we represented it in the form of the coherency vector and the Stokes vector. Here I shall use yet another representation, the coherency matrix. It is composed of the same four elements as the coherency vector, but arranged in the form of a $2 \times 2\$matrix

 

$$\vec{E}_{jk \vphantom{'}}= \,< \vec{e}_{j \vphantom{'}}\,\vec{e}_... ...tom{'}}^*> & <e_{j y \vphantom{'}}e_{k y \vphantom{'}}^*> \end{array} \right) .$$ (1)

All of these forms are entirely equivalent representations of one and the same underlying tensor. Both the form (vector, matrix or other) and the coordinate system (e.g. cartesian or circular, cf. Paper I) of the representation are a matter of convenience. I will use geometric xy coordinates throughout.

   
2.3 The interferometer equation

I recall from Paper I that the elements in the signal path in one antenna transform the electric field or voltage vector:

$$\begin{eqnarray*}\vec{w}_{j \vphantom{'}}= \vec{J}_{j \vphantom{'}}\vec{e}_{j \vphantom{'}} \end{eqnarray*}$$

where $\vec{J}_{j \vphantom{'}}$ is a Jones matrix. It is then readily seen that an interferometer with Jones matrices $\vec{J}_{j \vphantom{'}}$ and $\vec{J}_{k \vphantom{'}}$ transforms the coherency matrix according to

 

$$\vec{W}_{jk \vphantom{'}}= \vec{J}_{j \vphantom{'}}\vec{E}_{jk \vphantom{'}}\vec{J}_{k \vphantom{'}}^{\dagger}.$$ (2)

Perhaps superfluously, I reiterate that this is no more than another representation of the basic underlying transformation of the coherency tensor by an interferometer. The advantage over the coherency-vector representation of Paper I is that both coherencies and antenna/receiver systems are now represented by $2 \times 2\$matrices and we need only one type of multiplication operator. This leads to a complete formal analogy between scalar and matrix selfcal, which will allow us to extrapolate our knowledge of the former in trying to understand the latter.

Coherency and Jones matrices having the same form, it should be clear from the context which is which, just as in the scalar domain. In addition, note that Jones matrices carry the single index of an antenna whereas the coherency matrices have a double, interferometer index. This difference will remain also when we later add another index t for sampling time. I note in passing that in the particular case of a single dish, j=k and Eq. (2) reduces to

 

$$\vec{W}= \vec{J}\vec{E}\vec{J}^{\dagger}$$ (3)

which is known as a congruence transformation. We will see in Sect. 4 that this same transformation describes a "self-aligned'' synthesis array.

   
2.4 Matrix and Stokes brightnesses

In its original form, the van Cittert-Zernike theorem (Paper I Appendix C; [Thompson et al. 1986]; [Perley et al. 1994]; [Born & Wolf 1964]) establishes a spatial Fourier-transform relation between a scalar visibility function $e(\vec{r})$ of baseline $\vec{r}$ and a scalar brightness function $B(\vec{l})$ of sky position $\vec{l}$.

In an observation, we measure the visibility at discrete times t; our observables are the samples

$$\begin{eqnarray*}e_{jkt \vphantom{'}}= e(\vec{r}_{jkt \vphantom{'}}) . \end{eqnarray*}$$

In self-calibration theory, the sampling times are assumed to coincide for all interferometers. Also discretising the brightness, we approximate the Fourier integral by a sum

 

$$e(\vec{r}_{jkt \vphantom{'}}) = \sum w(\vec{r}_{jkt \vphantom{'}},\vec{l}) B(\vec{l}).$$ (4)

The theorem can be readily generalised to show that each element of the coherency matrix is the Fourier transform of the corresponding element of a brightness matrix. In the same approximation

 

$$\begin{array}{rcl} \vec{E}_{jkt \vphantom{'}}&=& \displaystyle \s... ...\,(\vec{l}) \\ &&\qquad j,k = 1, \ldots, N,\quad t = 1, \ldots, M. \end{array}$$ (5)

The four elements of the brightness matrix correspond to those of the coherency matrix. A more enlightening representation is provided by the Stokes brightness (I,Q,U,V). It is another function of $\vec{l}$, defined by the transformation

 

$$\vec{B}= \left( \begin{array}{cc} I+Q &U-iV \\ U+iV & I-Q\end{array} \right) = I\,\mathbf{I}+ Q\,\mathbf{Q}+U\,\mathbf{U}+V\,\vec{V}$$ (6)

where

 

$$\begin{array}{lcl} \mathbf{I}&=&\left( \begin{array}{cc} 1& 0\\ ... ...\vec{V}= \left( \begin{array}{rr} 0&-i\\ i& 0\end{array} \right) . \end{array}$$ (7)

The matrix constants I, Q, U and $\vec{V}$ are known in physics as the Pauli (spin) matrices.

The Stokes parameter I is the total brightness or intensity. For (Q,U,V) a proper name is the "polarized-brightness vector''; more conveniently, I shall call it the polvector. The dichotomy between I and the other Stokes parameters can be understood as a consequence of the Pauli matrix I being the identity matrix. The domain of the polvector is closely related to that of the Poincaré sphere ([Born & Wolf 1964]; [Cornbleet 1976]; [Simmons & Guttman 1970]). It is convenient to introduce a shorthand for Eq. (6):

 

$$\vec{B}(\vec{l}) = [\, I(\vec{l}) + \vec{p}(\vec{l}) \,]$$ (8)

where $\vec{p}\equiv (Q,U,V)$ is the polvector.

   
2.5 Quaternions

The transformation Eq. (6) or Eq. (8) does not depend on $\vec{B}$ being a brightness matrix. It can be applied to an arbitrary $2 \times 2\$matrix $\vec{A}$:

 

$$\vec{A}= [\, a + \vec{a}\,].$$ (9)

The entity in square brackets is known as a quaternion. Quaternions were invented and named by Hamilton in the middle of the nineteenth century in a mathematical quest for generalisations of the concept of complex numbers. Physicists of the time ignored them in favour of the vector algebra that was developed at the same time ([Hestenes 1986]). In the analysis to be presented here they prove to be extremely useful, because they can be added and multiplied in exactly the same way that matrices can: in mathematical terms the "quaternion group is isomorphous with the group of $2 \times 2\$matrices''. ([Korn & Korn 1961]).

In the same way that the Stokes vector is preferable because of its physical content, the quaternion form of equations such as the interferometer equation Eq. (2) can be analysed in a more meaningful way than the corresponding matrix equations. The analysis is an essential part of this paper, but I have chosen to present it in an appendix. In the main text, I concentrate on the results and their physical interpretation.

The notation for quaternions is not standardised. The form Eq. (9) is an ad-hoc choice of my own.