7 Quasi-scalar methods

It is interesting to see how the concepts revealed by the above analysis appear in the usual quasi-scalar methods of polarization calibration. Unlike in the rest of this paper, Stokes parameters in this section represent visibilities.

It is assumed that the feed errors and degree of polarization are both small; authors typically state a few percent as the upper limit. It is further required that all antennas have the same type of feed. In the interferometer equation Eq. (2), second-order products of order 10^-3 are dropped and errors at a few times this level are accepted.

The linearised equations can be found in Paper II (and in almost every paper on polarimetric calibration, e.g. Thompson et al. (1986), Perley et al. (1994) and the papers quoted in Sect. 7.2). I show them here in an elementary form for feeds with left (L) and right (R) circularly-polarized receptors:

 

$$\begin{array}{lcrcl} I'_{jkt \vphantom{'}}&+& V'_{jkt \vphantom{'... ...Q_{jkt \vphantom{'}}- iU_{jkt \vphantom{'}}+ D_{kj \vphantom{'}}I). \end{array}$$ (19)

(For linearly polarized feeds, Q, U and V must be cyclically interchanged in these equations and in the following discussion, cf. Papers I and II.)

   
7.1 Intensity calibration

The starting point is again an intensity calibration. The procedure is discussed at length in Paper II. The case where such calibrators are not available has been taken up only recently (cf. Sect. 7.2). In Paper II a point source is assumed, but this is not necessary.

Usually V' is assumed to be zero. Scalar selfcal is applied to the first two lines of Eq. (19) to obtain a good I' image and complex R and L antenna-channel gains, each with an unknown phase.

In the second pair of equations, the difference between these phases enters through the g and g^* factors; its effect is a polrotation in the Q,U plane of polvector space. The trailing terms in the equations describe the leakage from I to Q and U: a polconversion. As a consequence of the linearisation, polrotation and polconversion appear in a sum rather than a product.

For an unpolarized source, the leakage terms can be measured per interferometer because Q and U are zero. The Q,U-plane polrotation can only be eliminated by a measurement of some sort. Several approaches, each with their own uncertainties, are discussed in Paper II.

As with the matrix technique, unpolarized foreground or background sources can be used as in-field calibrators. An example is the observation by Wardle et al. (1998) of weak circular polarization in a quasar whose strong core is assumed to be unpolarized.

   
7.2 Calibration without an intensity calibrator

A quite serious problem arises if there are no unpolarized sources that can serve as a reference. This situation is actually occuring in VLBI: observed fields are so small that they donot contain any background sources to speak of, and sources strong enough to serve as calibrators (either in the target field or elsewhere) tend to be relatively strongly polarized at VLBI resolutions, particularly at the higher frequencies. For these reasons, VLBI observers ([Cotton 1993]; [Roberts et al. 1994]; [Leppänen et al. 1995]) have pioneered quasi-scalar polarization calibration without the use of an intensity calibrator.

The basic idea in all these papers is to exploit the effect of parallactic-angle variations in alt-az antennas to distinguish the leakage terms in Eq. (19) from the true source visibilities. In discussing this method, I assume that the parallactic-angle rotation is corrected for beforehand: thus Eq. (19) refers to the visibities in sky coordinates and the leakage terms include the inverse of the rotation.

Cotton (1983) mentions the method without giving details.

Roberts et al. (1994) consider two types of calibrator: either an unpolarized one that may be resolved (i.e. my case of an intensity calibrator) or an unresolved one that may be polarized. In the latter case, the true Q and U visibilities in Eq. (19) are constants. The paper illustrates the difference between them and the rotating leakage terms graphically for a couple of interferometers. The leakage terms and the visibilities are well separable by a model fit, provided the parallactic angles cover a sufficiently broad range.

Leppänen et al. (1995) use the empty-sky principle of Sects. 3.1 and 4 to separate true linear source polarization from leakage effects: the true polarized source brightness is necessarily confined to the support of the total intensity, and this provides a strong constraint that the rotated leakage terms have difficulty satisfying. A polarized-source model obeying the constraint is used to estimate the leakage terms and the procedure can be iterated.

The authors introduce the concept of a "leakage beam'' to characterise the transfer of brightness from total to polarized brightness. Its value at the origin represents the polconversion, its sidelobes a polconverted spatial scattering: due to the fragmented character of the quasi-scalar method, polconversion and scattering are not neatly separated and the latter is not completely suppressed. In a simulation Leppänen et al. (1995) find the leakage beam to peak at a few .1% close to the origin, in accord with the accuracy expected under quasi-scalar assumptions.

Unlike Roberts et al. (1994), these authors make no assumptions on the source. In the more comprehensive perspective of matrix theory I doubt that the information they do use is sufficient to suppress polconversion and obtain a unique solution, -- although it may be that the quasi-scalar assumptions constrain it well enough for astrophysically acceptable image fidelity.