Understanding radio polarimetry IV. The full-coherency analogue of scalar self-calibration: Self-alignment, dynamic range and polarimetric fidelity

Paper II of this series studied the calibration process in mostly qualitative terms. In developing the underlying mathematics this paper completes that analysis and extends it in several directions. It exploits the analogy between scalar and matrix algebras to reformulate the self-calibration method in terms of 2 x 2 Jones and coherency matrices. The basic condition that the solutions must satisfy in either case is developed and its consequences are investigated. The fourfold nature of the matrices and the non-commutativity of their multiplication are shown to lead to a number of new effects. In the same way that scalar selfcal leaves the brightness scale undefined, matrix selfcal gives rise to a more complicated indeterminacy. The calibration is far from complete: self-alignment describes more properly what is actually achieved. The true brightness is misrepresented in the image obtained by an unknown brightness-scale factor las in scalar selfcal) and an undefined poldistortion of the Stokes brightness. The latter is the product of a polrotation of the polvector (Q, U, V) and a polconversion between unpolarized and polarized brightness. The relation of these concepts to conventional "quasi-scalar" calibration methods is demonstrated. Like scaler selfcal, matrix self-alignment is shown to suppress spatial scattering of brightness in the image, which is a condition for attaining high dynamic range. Poldistortion of the brightness is an in-place transformation but must be controlled in order to obtain polarimetric fidelity. The theory is applied to reinterpret the quasi-scalar methods of polarimetry including those of Paper II, and to prove two major new assertions: (a.) An instrument calibrated on an unpolarized calibrator measures the degree of polarization correctly regardless of poldistortion; (b.) Under the usual a priori assumptions, a heterogeneous instrument (i.e. one with unequal feeds) can be completely calibrated without requiring a phase-difference measurement