Motivation

Rogers (1983) pointed out in the context of the VLBA, that the polarization leakage cause closure errors even in nominally co-polar visibilities. Massi et al. (1997) have carried out a detailed study of this effect for the telescopes of the European VLBI Network (EVN). The motivation behind this word was that the current single sideband GMRT correlator uses the so called Indian mode of the VLBA MAC chips to produce only the co-polar visibilities. Also, the planned Walsh switching has not yet been implemented at the GMRT and in any case, would not eliminate leakage generated before the switching point. Tests done using strong point source dominated fields show unaccounted closure errors at a few percent level. The motivation behind developing an algorithm to solve for gains and leakages simultaneously, using only the co-polar visibilities was to determine if the measured closure errors could be due to polarization leakage in the system. Estimates of leakage can then be used in the primary calibration to remove the effects of polarization leakage. This is where this work differs from the earlier work of HBS which is about the calibration using the full Stokes visibility function, needed for observations of polarized sources. The polarization leakage in some of the EVN antennas corrupts the co-polar visibilities at a level visible as a reduction in the dynamic range of the maps (Massi & Aaron1997a; Massi & Aaron1997b; Massi et al.1998). Thus such a method can also be used in imaging data from the EVN and other telescopes affected by such closure errors.

Let ${g_i^{\it p}}$ represent the complex gain for the ${\it p}$-polarization channel of the $i^{th}$ antenna and ${\alpha_i^{\it q}}$ represent the leakage^10.1 of the q-polarization signal into the p-polarization channel. The electric field measured by antenna $i$ can then be written as

$$E^{\it p}_i=g^{\it p}_i {E^{\it p}_{i,\circ}}+ {\alpha_i^{\it q}}{E^{\it q}_{i,\circ}}$$ (10.1)

where ${E^{\it p}_{i,\circ}}$ and ${E^{\it q}_{i,\circ}}$ are the responses of an ideal antenna to the incident radiation in the ${\it p}$- and ${\it q}$-polarization states respectively. For an unpolarized source of radiation, $\langle {E^{\it p}_{i,\circ}}{E^{{\it q}\star}_{j,\circ}}\rangle = 0$. The co-polar visibility for such a source, measured by an interferometer using two antennas denoted by the subscripts $i$ and $j$, is given by

$$\begin{split}{\rho_{ij}^{{\it p}{\it p}}}<tex2html_comment_ma... ...{\rho_{ij,\circ}^{{\it q}{\it q}}}+ \epsilon_{ij} \end{split}$$ (10.2)

where $\epsilon_{ij}$ is independent gaussian random baseline based noise and ${\rho_{ij,\circ}^{{\it p}{\it p}}}=\langle {E^{\it p}_{i,\circ}}{E^{{\it p}\star}_{j,\circ}}\rangle$ and ${\rho_{ij,\circ}^{{\it q}{\it q}}}=\langle {E^{\it q}_{i,\circ}}{E^{{\it q}\star}_{j,\circ}}\rangle$ are the two ideal co-polar visibilities. $\epsilon_{ij}$ usually represents the contribution to ${\rho_{ij}^{{\it p}{\it p}}}$ which cannot be separated into antenna based quantities. $\epsilon_{ij}$ therefore is a measure of the intrinsic closure errors in the system and is usually small.

For an unpolarized point source $\langle {E^{\it p}_{i,\circ}}{E^{{\it p}\star}_{j,\circ}}\rangle = \langle {E... ...}}{E^{{\it q}\star}_{j,\circ}}\rangle = {\rho_{ij,\circ}^{{\it p}{\it p}}}= I/2$ where $I$ is the total flux density. Writing $X_{ij}^{{pp}}={\rho_{ij}^{{\it p}{\it p}}}/{\rho_{ij,\circ}^{{\it p}{\it p}}}$ we get

$$X_{ij}^{{pp}}= {g_i^{\it p}}{g_j^{\it p}}^\star + {\alpha_i^{\it q}}{\alpha_j^{\it q}}^\star + \epsilon_{ij}$$ (10.3)

where $\epsilon_{ij}$ now refers to the baseline based noise in $X_{ij}^{{pp}}$.

Assuming ${\alpha_i^{\it q}}$s to be negligible, the usual antsol algorithm estimates ${g_i^{\it p}}$s such that $\sum_{{i,j}\atop{i \ne j}} \left\vert X_{ij}^{{pp}} - {g_i^{\it p}}{g_j^{\it p}}^\star \right\vert^2$ is minimized (see section 7.3). Normally, Walsh switching (Thompson et al.1986) is used to eliminate the polarization leakage due to cross-talk between the signal paths, such that ${\alpha_i^{\it q}}{\alpha_j^{\it q}}^\star \ll \epsilon_{ij}$. However, ${\alpha_i^{\it q}}$s can also be finite due to mechanical imperfections in the feed or the cross-polar primary beam, which cannot be eliminated by Walsh switching.

In the case of significant antenna based polarization leakage (compared to $\sqrt{\epsilon_{ij}}$), the second term in Equation 7.3 involving ${\alpha_i^{\it q}}$s will combine with the closure noise $\epsilon_{ij}$. The polarization leakage therefore manifests itself as increased closure errors (see Section 7.5 for a geometric explanation on the Poincaré sphere). This has also been pointed out by Rogers (1983) in the context of VLBA observations. However, as written in Equation 7.3, the leakages and gains are actually antenna based quantities and can be solved for, using only the co-polar visibilities.