An Introduction to Single-Antenna Radio Astronomical Polarimetry

TL;DR

This work provides a rigorous, unified treatment of single-antenna radio polarimetry by tying the propagation and instrumental chain to transformations in both the electric field and Stokes parameter spaces. Using Jones and Mueller formalisms, along with axis-angle decompositions and polarization-transfer tensors, it clarifies how gains, phases, feed nonidealities, and Faraday rotation reshape polarization states, and it prescribes calibration and model-selection criteria that ensure physicality and numerical stability. The framework bridges the classical polarization ellipse geometry with modern four-dimensional Stokes/Mueller formalisms, enabling robust inference of intrinsic source polarization from complex instrumental effects. Its emphasis on physical motivation, surjectivity/injectivity, and stability provides practical guidance for designing, calibrating, and validating polarimetric observations in both single-dish and interferometric contexts, with implications for high-precision pulsar timing and imaging near strong magnetic-field sources.

Abstract

This tutorial reviews the mathematical foundations of single-antenna radio polarimetry with the aim of fostering a conceptual understanding of the relationships between a physical description of signal propagation (gain, delay, reflection, down-conversion, etc.), the corresponding transformations of the electric field vector, and the equivalent operations on the Stokes parameters. The adopted framework is based on the work of Britton (2000) and Hamaker (2000) and applied to analyze the signal path described by Hamaker et al. (1996) with additional corrections for phase convention and reflection. Some objective criteria for selecting a model of the instrumental response are introduced and discussed, along with some practical guidelines that facilitate polarimetric calibration. Further relevant background material and lengthier mathematical proofs are included in the appendix, which introduces the vector, matrix, and tensor notation and concepts of linear algebra used in this work. The appendix also reviews some of the basics of analog and digital signal processing that are relevant to radio astronomy, and discusses some numerical instabilities that arise when modeling observations.

Figures (10)

• Figure 1: A left-handed ellipse is traced by the electric field vector on each cycle of a monochromatic wave travelling in the positive $z$ direction, as observed when looking toward the source. The position angle, $-\pi/2 \le \psi \le \pi/2$, describes the orientation of the semi-major axis of the ellipse with respect to the $x$-axis; it is positive when the $x'$ and $y'$ axes are rotated in a counter-clockwise direction with respect to the $x$ and $y$ axes. The tangent of the ellipticity angle, $-\pi/4 \le \chi \le \pi/4$, is equal to the ratio between the semi-minor axis and the semi-major axis; it is positive for a left-hand polarized wave and negative for a right-hand polarized wave. The size of the ellipse is defined by $r$, which is equal to the length of the hypotenuse of the right triangle that contains $\chi$.
• Figure 2: The spherical coordinates of the Stokes polarization vector, $\underline{\hbox{\boldmath $S$}}=(S_1, S_2, S_3)=(Q,U,V)$, include the vector length $|\underline{\hbox{\boldmath $S$}}|=Ip$, longitude $2\psi$, and latitude $2\chi$. Points in the $S_1$--$S_2$ plane ($S_3 = 0$) represent linearly polarized states, points above this plane ($S_3 > 0$) represent left-hand elliptically polarized states and the positive $S_3$ pole represents left-hand circular polarization.
• Figure 3: The polarization ellipses inscribed by the electric field vector for the six special cases of monochromatic light described in Section \ref{['sec:polarization_ellipse']}.
• Figure 4: The radio wave signal path, from free space propagation on the right to detection on the left. Also from right to left are Faraday rotation F due to free electrons permeated by a magnetic field (represented by blue lines with arrows); the coordinate projection P between the $x$ and $y$ axes of the receiver and the North and East basis vectors of the celestial reference frame; the nominal configuration C of an ideal antenna and feed; unintended deviations D from that ideal; and the gains G and phase convention $\underline{\underline{{\bf \Phi}}}$ of the down-conversion system and instrumentation used to detect the Stokes parameters.
• Figure 5: In the circular basis, the direction of the Stokes polarization vector, $\underline{\hbox{\boldmath $S$}}=(S_1, S_2, S_3)=(V,Q,U)$, is defined by the polar angles, $\xi$ and $\zeta$. These angles together define a right triangle on the surface of the Poincaré sphere with hypotenuse equal to the colatitude, $\theta=\pi/2 - 2\chi$, measured with respect to the pole defined by positive $S_1$ (Stokes V). Also depicted is the longitude $2\psi$, measured with respect to positive $S_2$ (Stokes Q) toward positive $S_3$ (Stokes U).

Theorems & Definitions (34)

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