Polarization in Radio Astronomy

Emission mechanisms which are dominant in low frequency radio astronomy, produce linearly polarized emission. Thus extra-galactic radio sources and pulsars are predominantly linearly polarized, with polarization fractions of typically a few percent. These sources usually have no circular polarization, i.e. $V\sim 0$. Maser sources however, in particular OH masers from galactic star forming regions often have significant circular polarization. This is believed to arise because of Zeeman splitting. Interstellar maser sources also often have some linear polarization, i.e. all the components of the Stokes vector are non zero. In radio astronomy the polarization is fundamentally related to the presence of magnetic fields, and polarization studies of sources are aimed at understanding their magnetic fields.

The raw polarization measured by a radio telescope could differ from the true polarization of the source because of a number of effects, some due to propagation of the wave through the medium between the source and the telescope, (see chapter 16) and the other because of various instrumental non-idealities. Since we are eventually interested in the true source polarization our ultimate aim will be to correct for these various effects, and we will therefore find it important to distinguish between depolarizing and non-depolarizing systems. A system for which the outgoing wave is fully polarized if the incoming wave is fully polarized is called non-depolarizing. The polarization state of the output wave need not be identical to that of the incoming wave, it is only necessary that $P_{out}=1$ if $P_{in}=1$.

The most important propagation effect is Faraday rotation, which is covered in some detail in chapter 16. Here we restrict ourselves to stating that the plane of polarization of a linearly polarized wave is rotated on passing through a magnetized plasma. Faraday rotation can occur both in the ISM as well as in the earth's ionosphere. If the Faraday rotating medium is mixed up with the emitting region, then radiation emitted from different depths along the line of sight are rotated by different amounts, thus reducing the net polarization. This is called Faraday depolarization. If the medium is located between the source and the observer, then the only effect is a net rotation of the plane of polarization, i.e.

$$\mathcal{E}^{'}_x = \mathcal{E}_x \cos\chi +\mathcal{E}_y \... ...l{E}^{'}_y = -\mathcal{E}_x \sin\chi +\mathcal{E}_y \cos\chi ,$$ (15.2.6)

where $\mathcal{E}_x$, $\mathcal{E}^{'}_x$ are the $X$ components of the incident and emergent field respectively and similarly for $\mathcal{E}_y$, $\mathcal{E}^{'}_y$. In terms of the Stokes paramters, the transformation on passing through a Faraday rotating medium is

$$\begin{array}{lcl} I^{'} &=& I\\ V^{'} &=& V \end{array} \h... ...n 2\chi \\ U^{'} &=& -Q\sin 2\chi\ + U\cos 2\chi . \end{array}$$ (15.2.7)

i.e. a rotation of the Stokes vector in the (U,V) plane. The fractional polarization is hence preserved^15.6. Equation (15.2.7) can also be easily obtained from equation (15.1.3) by noting that in a circularly polarized co-ordinate system, the effect of faraday rotation is to introduce a phase difference of $2\chi$ between $\mathcal{E}_r$ and $\mathcal{E}_l$.

Consider looking at an extended source which is not uniformly polarized with a radio telescope whose resolution is poorer than the angular scale over which the source polarization is coherent. In any given resolution element then there are regions with different polarization characteristics. The beam thus smoothes out the polarization of the source, and the measured polarization will be less than the true source polarization. This is called beam depolarization. Beam depolarization cannot in principle be corrected for, the only way to obtain the true source polarization is to observe with sufficiently high angular resolution.

A dual polarized radio telescope has two voltage beam patterns, one for each polarization. These two patterns are often not symmetrical, i.e. in certain directions the telescope response is greater for one polarization than for the other. The difference in gain between these two polarizations usually varies in a systematic way over the primary beam. Because of this asymmetry, an unpolarized source could appear to be polarized, and further its apparent Stokes parameters in general depend on its location with respect to the center of the primary beam. The polarization properties of an antenna are also sharply modulated by the presence of feed legs, etc. and are hence difficult to determine with sufficient accuracy. For this reason determining the polarization across sources with dimensions comparable to the primary beam is a non trivial problem. Given the complexity of dealing with extended sources, most analysis to date have been restricted to small sources, ideally point sources located at the beam center.

Most radio telescopes measure non-orthogonal polarizations, i.e. a channel p which is supposed to be matched to some particular polarization $p$ also picks up a small quantity of the orthogonal polarization $q$. Further, this leakage of the orthogonal polarization in general changes with position in the beam. However, for reflector antennas, there is often a leakage term that is independent of the location in the beam, which is traditionally ascribed to non idealities in the feed. For example, for dipole feeds, if the two dipoles are not mounted exactly at right angles to one another, the result is a real leakage term, and if the dipole is actually matched to a slightly elliptical (and not purely linear) polarization the result is an imaginary leakage term. For this reason, the real part of the leakage is sometimes called an orientation error, and the imaginary part of the leakage is referred to as an ellipticity error^15.7. However, one should appreciate that the actual measurable quantity is only the antenna voltage beam, (i.e. the combined response of the feed and reflector) and this decomposition into `feed' related terms is not fundamental and need not in general be physically meaningful.

The final effect that has to be taken into account has to do with the orientation of the antenna beam with respect to the source. For equitorially mounted telescopes this is a constant, however for alt-az mounted telescopes, the telescope beam rotates on the sky as the telescope tracks the source. This rotation is characterized by an angle called the parallactic angle, $\psi_p$, which is given by:

$$\tan\psi_p = { \cos\mathcal{L} \sin\mathcal{H} \over \sin\m... ...l{L} \cos\delta - \cos\mathcal{L} \sin\delta \sin\mathcal{H}},$$ (15.2.8)

where $\mathcal{L}$ is the latitude of the telescope, $\mathcal{H}$ is the hour-angle of the source, and $\delta$ is the apparent declination of the source. So if one observes a source at a parallactic angle $\psi_p$ with a telescope that is linearly polarized, the voltages that will be obtained at the terminals of the $X$ and $Y$ receivers will be

$$V_x = G_x ( \mathcal{E}_x \cos\psi_p + \mathcal{E}_y \sin\p... ...= G_y (-\mathcal{E}_x \sin\psi_p + \mathcal{E}_y \cos\psi_p ),$$ (15.2.9)

where $G_x$ and $G_y$ are the complex gains (i.e. the product of the antenna voltage gains and the receiver gains) of the $X$ and $Y$ channels.

Footnotes

... preserved^15.6

Note that non-depolarizing only means that $P_{out}=1$ if $P_{in}=1$, and this does not necessarily translate into conservation of the fractional polarization when $P < 1$. Pure faraday rotation is hence not only non-depolarizing, it also preserves the fractional polarization.

... error^15.7

Several telescopes, such as for example the GMRT, use feeds which are sensitive to linear polarization, but by using appropriate circuitry (viz a $\pi/2$ phase lag along one signal path before the first RF amplifier) convert the signals into circular polarization. Non idealities in this linear to circular conversion circuit could also produce complex leakage terms even if the feed dipoles themselves are error free.