8 Heterogeneous arrays

Homogenoeus arrays, i.e. having all identical feeds, have been a natural first choice for obvious engineering reasons and are also required by the quasi-scalar approach. Having removed the latter restriction, we can now take a fresh look.

   
8.1 Coupling of receiver phases and feed errors

Consider the case of an intensity calibrator and perfect feeds represented by known unitary Jones matrices $\vec{F}_{j \vphantom{'}}$. (This assumption is justified in Appendix B.3. The argument becomes slightly more complicated for imperfect feeds, but leads to the same conclusion.) After intensity self-alignment and imposition of our prior knowledge about the feeds, the only error remaining is the receiver phase-difference term of Eq. (18), so the interferometer equation becomes

$$\begin{eqnarray*}\vec{Y}_{\mathbf{q}}(\phi_{j \vphantom{'}}) \vec{F}_{j \vphanto... ... = \vec{F}_{j \vphantom{'}}\ \vec{F}_{k \vphantom{'}}^{\dagger} \end{eqnarray*}$$

in which both $\vec{Y}_{\mathbf{q}}$ terms are diagonal matrices. This equation must hold for all interferometers, which is possible in only two ways:

• For a homogeneous array, $\vec{F}_{j \vphantom{'}}=\vec{F}_{k \vphantom{'}}=\vec{F}$ and $\vec{F}\vec{F}^{\dagger}=\mathbf{I}$. This is the conventional case discussed in Paper II and Sect. 6.3. The one thing that our derivation adds is that it pertains generally to homogeneous arrays with any feed type;
• For a heterogeneous array with $\phi_{j \vphantom{'}}= \phi_{k \vphantom{'}}= 0$.

The latter solution is new: in a heterogeneous array with perfectly known feeds, this knowledge alone suffices to calibrate all receiver phases without an additional measurement. In reality we may at best assume our feed descriptions to be correct on the average, just as for the homogeneous array. The solutions that we find for the individual feed characteristics as well as for the receiver phases will then be imperfect, and perhaps more sensitive to errors in our a priori assumptions and to system noise than in the homogeneous case; this is a matter that needs further study.

The possibility of a priori phase alignment is surprising at first sight. Yet there are several good intuitive reasons to accept it:

Most fundamentally, the nominal receptor characteristics in the heterogeneous case are arbitrarily distributed. There is no natural way to split them into two disjoint groups between which an asymmetry such as the phase difference might arise. The only way out is for the difference not to exist. Another viewpoint is that the homogeneous case is, in physical terms, a degenerate one. The degeneracy results in a decoupling of the feed and phase characteristics which are normally coupled. Mathematically, this decoupling is represented by the product $\vec{F}_{j \vphantom{'}}\vec{F}_{k \vphantom{'}}^{\dagger}$ reducing to the value I. A third viewpoint is that a set of identical feeds defines a preferred direction in polvector space, viz. that of the polvectors to which the two receptors are matched (e.g. the V axis for circular feeds). This results in an asymmetry in the characteristics of the instrument that is directly related to the phase-difference problem. "Randomised'' feed characteristics destroy this preferential direction: the phase difference evaporates and all polarizations can be measured equally well, -- or equally poorly.

An array having altaz mounts is heterogeneous in a sense, since in the course of an observation its feed orientations relative to the source assume a range of values. Yet at any instant it is homogeneous and consequently its long-term heterogeneity does not help in removing the phase error and associated polrotation. As we have seen, it may help in controlling the polconversion (Sect. 7.2) when we have no intensity calibrators.

The only historical example of a truly heterogeneous array is that of the Westerbork Synthesis Radio Telescope (WSRT) in its "crossed-dipole'' configuration. Weiler (1973) analysed this system to show that, in the quasi-scalar approximation, it can be fully calibrated through observations of only one intensity calibrator. His system is too different from modern ones to relate his analysis directly to ours. Yet his work pointed to the potential of heterogeneous arrays as long as a quarter century ago and provided a major motivation for carrying the present investigation to completion.

   
8.2 Minimising the feed errors

The method of Sect. 6.3 for dealing with imperfect feeds is to assume that they are correct on the average. For a heterogeneous array, we may use the equivalent requirement that the sum of the receptor errors squared be minimised.

Following Paper I, we model the antenna Jones matrix in terms of the nominal feed $\vec{F}_{j \vphantom{'}}$ ($\vec{C}$ in Paper I), a feed error $\vec{D}_{j \vphantom{'}}$ and the receiver-gain $\vec{G}_{j \vphantom{'}}$ (a diagonal matrix). After intensity self-alignment we then have

 

$$\vec{J}'_{j \vphantom{'}}= \vec{G}_{j \vphantom{'}}\,\vec{D}_{j \vphantom{'}}\,\vec{F}_{j \vphantom{'}}\,\vec{Y}$$ (20)

and hence we consider values $\vec{D}'_{j \vphantom{'}}$, $\vec{G}'_{j \vphantom{'}}$ and $\vec{Y}'$ that satisfy this equation, or

 

$$\vec{D}'_{j \vphantom{'}}= \,\vec{G}{_{j}^{'-1}} \,\vec{J}'_{j \vphantom{'}}\,\vec{Y}^{\prime -1} \vec{F}_{j}^{-1}.$$ (21)

For an ideal system, the $\vec{D}_{j \vphantom{'}}$ equal I, so we define our best guess at the polrotation $\vec{Y}'$ as the one that minimises the sum of variances (Appendix A.5)

$$\begin{eqnarray*}\sum_{j \vphantom{'}}\mbox{Var}\ (\vec{D}'_{j \vphantom{'}}- \m... ...'_{j \vphantom{'}}\,\vec{Y}'^{-1} \vec{F}_{j}^{-1}-\mathbf{I}), \end{eqnarray*}$$

given the self-aligned Jones matrices $\vec{J}'_{j \vphantom{'}}$ and the nominal feed matrices $\vec{F}_{j \vphantom{'}}$, and under the condition that the gains $\vec{G}'_{j \vphantom{'}}$ are diagonal and $\vec{Y}'$ is unitary.

It may not be obvious that this condition leads to a unique solution. Simulations (Appendix D) using the MATLAB (1997) programming environment show that this is indeed the case. Moreover, the system of equations becomes singular for a homogeneous array; something of this sort was to be expected because of the degeneracy discussed in Sect. 8.1.