In Table 2 I compare the quasi-scalar and matrix methods as they stand. Due to the difference between the two approaches, the comparison is not always straightforward, but the table does bring out the important differences. In the longer term, the disadvantages of the matrix method may disappear as the practical knowledge and the ingenuity of observers are brought to bear upon it. The linearising assumptions limit the validity of the quasi-scalar approach. It is not clear to me whether, within these limits, linearisation helps to constrain the poldistortion. Clearly, if the degree of polarization is assumed to be small in the source and found to be small in the image, polconversion must be small as well, but I have not succeeded in casting this argument in a convincing mathematical form.
matrix to represent the coherency results in an analysis whose form
is an exact replica of the scalar one. Since the algebra of matrices follows almost
the same rules as that of scalars, many expressions of scalar interferometry remain
valid when we reinterpret the variables as matrices. In this respect, the matrix
representation of coherency turns out to be preferable over the vector form of Paper
I. The close analogy allows us to
retain most of our familiar ways of thinking and continue to reap the fruits of half
a century of theoretical and instrumental developments.
There is one important exception to the conformity: matrix multiplication is
non-commutative. This means that factors in a multiple product cannot be arbitrarily
merged; for example, the factors 
and
in Eq. (17) donot cancel,
even though their product equals
I. Non-commutativity together with the fourfold
content of Jones and coherency matrices gives rise to important effects that have no
scalar counterpart.
The crux of these is that the matrix analogue of self-calibration fails to actually calibrate but only aligns an observation. The indeterminacy that remains is the matrix analogue of the unknown scale factor in scalar selfcal. It entails, in addition to a similar scale factor, an unknown transformation of the brightness distribution: the poldistortion.
Further analysis showed that the latter is the product of a. a polrotation of the Stokes polvector (Q,U,V) in its three-dimensional vector space; and b. a polconversion between this polvector and Stokes I, the total brightness. Both of these are in-place transformations: Like scalar selfcal for a scalar (i.e. unpolarized) source, matrix self-alignment should be highly effective in suppressing spatial scattering of the matrix brightness and thereby produce images with a high dynamic range.
Methods for eliminating the poldistortion follow the pattern established in quasi-scalar polarimetry:
Intensity self-alignment on unpolarized reference sources suppresses the polconversion. This has an interesting consequence that had not been recognised before. Indeed, the remaining polrotation leaves both the total intensity and the length of the polvector invariant: although not fully calibrated, an intensity-aligned array measures the degree of polarization correctly. As in the quasi-scalar method, prior knowledge of the average feed characteristics can be applied to suppress two cartesian components of the polrotation. In conventional homogeneous arrays, a (phase) measurement is necessary to eliminate the remaining one.
Situations in which no unpolarized reference sources can be used are problematic in either context. In the quasi-scalar context, the variation of parallactic angle in an alt-az-mounted antenna has been advanced as a means to separate poldistortion from true source structure; it seems to provide for at least a partial solution. One may hope that it can be put to good use in matrix form as well, but this remains to be shown.
In addition to shedding a different light on polarimetric calibration, the matrix formalism allows us to consider heterogeneous arrays. I see several possible applications:
An interesting property of a heterogeneous array is that receiver phase is coupled to the feed parameters in such a way that aligning the latter to their nominal average has the effect of aligning the phases at the same time. No additional phase measurement is needed.
To become a practical reality, matrix-based interferometry demands an entirely new set of matrix-based computer programs. These must incorporate procedures for matrix self-alignment and the treatment of polconversion and polrotation, along with collateral ones e.g. for the extraction of polarized source models. Having endorsed Paper I as the basis of its data and processing model, AIPS++ (1998) is the obvious environment in which such software can be developed. A project is underway at NFRA using a special observation made with the Westerbork Telescope in a heterogeneous configuration.
Handicapped by a scalar foundation that is fundamentally incorrect, radio astronomy has been remarkably successful in producing meaningful results. Now at last, we can transplant our accumulated understanding and experience into a conceptual environment that does full justice to the basic vector nature of electromagnetic radiation, without sacrificing what we have learnt in the scalar domain. What remains to be seen is not if, but only when the transition will actually happen.
Acknowledgements
This paper has been thoroughly revised and extended after an anonymous referee pointed out a fundamental flaw in my original formulation of the selfcal problem. In various stages A. van Ardenne, W.N. Brouw, D. Gabuzda and particularly J.D. Bregman and J. Tinbergen made important contributions toward a clearer presentation. The Netherlands Foundation for Research in Astronomy (NFRA) is operated with financial support from the Netherlands Organisation for Scientific Research (NWO).
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