Elsevier

Astronomy and Computing

Volume 16, July 2016, Pages 185-188
Astronomy and Computing

Full length article
Simple estimation of all-sky, direction-dependent Jones matrix of primary beams of radio interferometers

https://doi.org/10.1016/j.ascom.2014.11.002Get rights and content

Abstract

The dual-polarized primary beams of imaging radio telescopes are generally not perfectly orthogonal nor have the same gain, the resulting polarimetric images are distorted, in other words, they exhibit instrumental polarization. This is often modeled in radio astronomy using a Jones matrix formalism, and the standard practice is to calibrate (i.e. determine the Jones matrices) using known point source, polarized calibrators. Using point source calibrators on the other hand can be difficult and is ineffective for wide fields-of-view (FoV). Often however, a large portion of the FoV of imaging radio telescopes consists of unpolarized background. In this paper I estimate Jones matrices over the FoV by assuming that most of the background image is unpolarized and then taking the square-root of the brightness matrices. Results from LOFAR LBA data are shown and are consistent with expectation. The usefulness of this particular method, compared to using polarized point source calibrators, is its simplicity and the fact that it can cover most of the FoV.

Introduction

With the advent of low radio frequency phased array telescopes, such as LOFAR (van Haarlem et al., 2013), MWA (Lonsdale et al., 2009), LWA (Ellingson et al., 2009), etc., comes the problem of calibrating these novel telescopes. The basic challenge with these arrays, which consist essentially of many crossed dipole elements, is that their field-of-view (FoV) is very wide–in some cases covering the whole sky–and at the same time there is a lack of good, polarized calibration sources at these low frequencies. This is a problem because the standard approach to polarimetrically calibrating a radio telescope consists of using known point source, polarized calibrators, see  Smirnov (2011). But even if there were useful calibrators, calibration is still more difficult for phased array telescopes than traditional dish telescopes since the beam gain pattern is not generally invariant under changes in pointing, and therefore transferring a calibration done on one point in the sky will not trivially hold at some other point in the sky. This is because pointing in phased array telescopes is done using digital steering, which does not generally compensate for the direction dependent gain of the individual elements, while dishes use mechanical pointing and so physically rotates the element. So in the case of phased arrays, to transfer calibrations done on known sources onto a pointing of interest, one will need to have a priori knowledge of the element gain pattern over the entire hemisphere. This clearly makes the use of source calibrators rather complicated and also risks an infinite regress in the calibration of unknowns.

An alternative way of expressing the main problem facing low frequency phased array telescopes is that they must calibrate for direction dependent effects (DDE) even for narrow fields, the standard scalar gain field calibration used for dish telescopes will no longer work.

In this paper I present an approach to determining the direction dependent Jones matrix over the FoV based on the assumption that most of the background sky is unpolarized. Although this approach does not lead to a unique determination of the Jones matrix of the telescope, the ambiguity is just a unitary ambiguity as with the polarimetric self-cal algorithm (Hamaker, 2000), and is sufficient for a useful partial calibration of polarimetry or an assessment of polarimetric quality.

Section snippets

Background motivation

It is instructive to look at real data to understand the problem addressed in this paper. I show some LOFAR LBA data from the Swedish station called SE607. LBA elements are 4 thin wires that slope downwards from the voltage measurement point and are mutually at right angles as seen from zenith. One linear pair of these wires are the x polarization and the orthogonal pair are the y polarization. The LBA array consists of 96 elements and covers the frequency range 10–90 MHz. A LOFAR station can

Method

As we saw in the previous section the images produced with a LOFAR LBA station were distorted polarimetrically. This is due to the antenna element response which in practice is never really isotropic, but rather the brightness matrix derived from the telescope is related to the true sky brightness matrix via multiplication with the aggregate element array Jones matrix J(l,m). In addition there will be noise from the receivers, so we will use the following as a basic model for the telescope

Results

I apply the Jones matrix estimation algorithm from the previous section to data shown previously in Fig. 1. To visualize the Jones matrix I extract the two effective antenna length vectors from the computed Jones matrices, that is, I define v(x)=(J11J12)v(y)=(J21J22) where v(x) and v(y) are the effective length vectors of the x and y polarization channels of the telescope respectively. I then plot the real parts of these two vectors in (l,m) coordinates. The result is shown in Fig. 2.

As one can

Discussion

In the previous sections a basic method for obtaining direction dependent Jones matrix estimates was sketched. This basic method could be further improved in several ways. Most importantly, it is clear that the assumption that the sky unpolarized in all directions is not realistic or desirable since if one were to naively use the Jones matrix estimated in this way to calibrate data, it would lead to a polarimetric image in which the whole sky was perfectly unpolarized. To remedy this limitation

Conclusion

I have shown how one can estimate direction dependent Jones matrices over the entire hemisphere by assuming that most of the sky is unpolarized and then take the square root of the brightness matrices. The results for international LOFAR LBA station data is consistent with the expectation that the primary beams are low order multipole moments.

Acknowledgment

The author would like to acknowledge discussions with Griffin Foster.

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