*Astron. Astrophys. Suppl. Ser.* **143**, 515-534

**J.P. Hamaker**

**Send offprint request: **J.P. Hamaker,

e-mail: hamaker@nfra.nl

Netherlands Foundation for Research in Astronomy, Postbus 2, 7990 AA Dwingeloo, The Netherlands

Received September 23, 1998; accepted January 13, 2000

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 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 (as 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 scalar 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.

**Key words: ** instrumentation: interferometers -- instrumentation: polarimeters --
methods: analytical -- methods: observational -- techniques: interferometric --
techniques: polarimetric

- 1 Introduction
- 2 Coherency-matrix formulation of interferometry
- 3 Scalar self-calibration
- 4 Matrix self-alignment
- 5 The poldistortion
- 6 Polrotation and intensity self-alignment
- 7 Quasi-scalar methods
- 8 Heterogeneous arrays
- 9 The general polconversion problem
- 10 Conclusion
- 11 Appendix: Mathematical theory
- A. Quaternion algebra
- B. Special matrices
- C. Congruence transformations
- D. Matrix solution techniques
- References

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