Polarimetric observations are affected by leakage of unpolarized light into the polarization channels, in a way that varies with the angular position of the source relative to the optical axis. The off-axis part of the leakage is often corrected by subtracting the product of the unpolarized map and a leakage map from each polarization image, but it is seldom realized that heterogeneities in the array shift the loci of the leaked radiation in a baseline-dependent fashion. We present here a method to measure and remove the wide-field polarization leakage of a heterogeneous array. The process also maps the complex voltage patterns of each antenna, which can be used to correct all Stokes parameters for imaging errors due to the primary beams.
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 The hardware typically used in radio telescopes has the great benefit of observing the Stokes Q, U, and V parameters simultaneously with Stokes I, but always allows some mixing between the polarization channels, as in Figures 1 and 2. This “leakage” is particularly troublesome when it goes from I into Q, U, or V, since the polarized signals are usually a small fraction of the total intensity, and therefore easily swamped by similarly strong leakages from I.
 A radio interferometer uses two or more antennas to measure the amplitudes and phases of the electric field impinging on their receivers. The measurements are stored as “visibilities”, which are the correlations of the receiver voltages. Given some conditions which this article will assume to have been met, the visibilities sample the Fourier transform of the sky multiplied by the primary beam (directional sensitivity function) of the antennas [Clark, 1999; Thompson, 1999].
 Mathematically, the effect of a pair of antennas A and B on the visibilities they observe, VobsAB, is conveniently expressed using the Hamaker-Bregman-Sault [Hamaker et al., 1996] formalism, where the four polarizations are combined into a column vector. The true visibilities are multiplied on the left by a set of Jones matrices, each one the outer product of Jones matrices for antennas A and B, i.e.,
represents the on-axis mixing between the nominally orthogonal polarization channels, often called the “D terms”. The outer product of two matrices and , ⊗ , is formed by multiplying each entry of with all of , and is used, along with a complex conjugation of the second factor, to bring together the elements from each antenna in a correlation. Hamaker et al.  explain the algebraic properties, including coordinate transformations, of Jones matrices and the outer product in more detail. As in the work by Bhatnagar et al. , direction-dependent effects can also be included, but they must go inside the Fourier integral:
where n is a direction on the sky relative to the “phase tracking center”, nc. nc is set electronically, but usually it is chosen to coincide with the pointing direction of the antennas. is the Stokes matrix, which transforms the sky's Stokes parameters, IS = (I, Q, U, V), into the observational polarization basis, typically correlations of either circular or linear polarizations. ΨAB and ΓAB are the wide-field leakage pattern and primary beam for the correlation of antennas A and B, respectively. They are sometimes multiplied together to form a single Jones matrix which is a generalization of the primary beam, but the magnitudes of the effects are more easily assessed if they are kept separate. With the separation, ΓAB is diagonal since it does not mix polarizations in the observational basis, and the diagonal elements of ΨAB are all one.
 The on-axis portion of the leakage, AB, is dealt with by standard polarimetric calibration techniques, but the leakage varies with direction, growing worse toward the edges of the primary beam, as in Figure 2. This paper is concerned with the wide-field polarization leakage, ΨAB, and will assume that VobsAB has already been corrected by multiplication with AB−1.
 If all of the antennas in an array are identical, the effects of the primary beam and leakage patterns can be removed in the image plane. At the Dominion Radio Astrophysical Observatory (DRAO) we previously corrected the wide-field contamination or polarization by multiplying the Stokes I image with “leakage maps”, and subtracting the results from the measured Q and U images, as in Figure 2b. The leakage maps were measured by observing the apparent Q/I and U/I of an intrinsically unpolarized source in a grid of offsets from the primary beam center [Peracaula, 1999]. This correction is performed completely in the image plane, so we call it the “image-based” leakage removal method. It was immediately applicable for DRAO's Synthesis Telescope (ST) [Landecker et al., 2000] since its antennas are equatorially mounted and thus its leakage patterns never rotate relative to the sky. The leakage patterns of an altitude-azimuth mounted telescope such as the Very Large Array (VLA) rotate relative to the sky over the course of an observation, but the image-based method can still be applied if the data are first broken up into a series of snapshots [Cotton, 1994].
 Unfortunately, there are differences, known or unknown, between the antennas of any real interferometer. The array may be a combination of antennas from originally separate telescopes, such as the Combined Array for Research in Millimeter-Wave Astronomy (CARMA) [Bock, 2006], and most very long baseline interferometers. It could also be in a transition period where only some antennas have been modified, like the partially Enhanced Very Large Array, and/or have serious surface errors as at (sub)mm wavelengths. The ST is an example of an array where the antennas are similar to each other, but with known differences between them. The two outermost antennas have 9.14 m diameters with four metal struts supporting their receivers, while the other five are 8.53 m in diameter with three struts, made of either metal or fiberglass. The differences in antenna diameter obviously create differences in the half-power beam widths (HPBWs), which at 1420 MHz are 101.8′ for the two outer antennas and 108.8′ for the rest. The variation in the number and composition of the struts affects the scattering of incoming light, which is an important component of polarization leakage (most of the rest comes from the feeds).
 In polarization images the differences between antennas are seen as mismatches between the standard point spread function (PSF, or “dirty beam”) and the PSF of the leakage. When there are phase differences between the leakages of the antennas, the effective PSF of the leakage is asymmetric [Ekers, 1999] and offset from the peak of the unpolarized emission. The effective PSF of the leakage also varies across the field, meaning that subtracting a multiplication of the Stokes I map with a leakage map cannot fully correct the polarization leakage of a heterogeneous array. Additionally, the response of each antenna in an array, both in leaked and true radiation, depends on the scale of the source(s). Resolved features have less power at high spatial frequencies, so antennas that only participate in long baselines will contribute little leakage to them. Unresolved features have no such attenuation with baseline length, and elicit an equally weighted mix of leakage from all antennas. Usually leakage maps are measured using a bright unresolved object, so in the case of a heterogeneous array their corrections are only accurate for unresolved sources. (Even then, only if the images are made with the same baseline weighting as used for the leakage map.) Figure 2b exhibits both of these problems, as can be seen in comparison with Figure 2d, the same image corrected with the method described in section 2. The main change for the unresolved sources is the presence or lack of surrounding arcs, but the supernova remnant also shows a strong difference in the on-source residual leakage.
 The work described here aims to improve polarization imaging from the DRAO ST. The telescope is engaged in an extensive survey of the major constituents of the Interstellar Medium, the Canadian Galactic Plane Survey (CGPS) [Taylor et al., 2003] which includes imaging in Stokes parameters Q and U at 1420 MHz along the plane of the Milky Way.
 We recently measured the real and imaginary parts of the leakage patterns for each antenna of the ST (similar to a hologram measurement, but with finer spacing over a smaller area) and have started using them to correct its Q and U observations, as seen in Figure 2d.
2. Removing Leakage From Linear Polarization for a Heterogeneous Array
 When the polarization leakage (or primary beam) varies with both direction and baseline (i.e., antenna pair), there is no way to isolate their effects to one of either the image or uv planes. Multiplying VobsAB on the left by ΨAB−1(n) does not work as it does for AB, because n must be marginalized away by integrating with IS. IS is the true intensity distribution of the sky, which is unfortunately unknown. A set of Stokes I CLEAN [Högbom, 1974] components makes an acceptable substitute, however, both in the replacement of the true sky by CLEAN components, and the temporary neglect of Q, U, and V.
 Since the correction for a heterogeneous array must be added directly to the visibilities, the I model used must match the true I visibility function within the sampled part of the uv plane. Specifically, it should not be tapered by any sort of smoothing in the image plane, and the almost certain discrepancies between the CLEAN components and true visibility function outside the sampled part of the uv plane are immaterial for this purpose. The CLEANed I image should have small enough pixels to avoid quantization errors in the component positions, and be CLEANed to at least a moderately faint level. Very faint I emission does not need to be included since it will be multiplied by the leakage, typically less than a few percent, and it tends to have many more components, which would considerably slow down the calculation of the correction. Leakage from such emission could be quickly and adequately removed by the image-based leakage map method, using the CLEAN residual image as the I map. Calculating the correction for both Q and U of a CGPS field, with a variable number, on the order of several thousand, of CLEAN components, and 1.2 × 105 visibilities per polarization, takes from 15 min to overnight on a 2 GHz personal computer.
 Assuming that Is = (I, 0, 0, 0) in correcting the wide-field leakage of equation (1) requires some care, since its validity depends on what Stokes parameters are wanted, and whether the feeds are circularly or linearly polarized. In general each measured Stokes parameter is nominally the true Stokes parameter, plus first-order leakage from two of the other Stokes parameters, plus second-order leakage from the remaining one. This comes from the leakage Jones matrices for each antenna having only ones on diagonal, with the leakage terms off-diagonal. As a rule of thumb, the true Q and U can be thought of as fractions of I, and V as an even smaller fraction (i.e., second order). With circularly polarized feeds the leakage of I into V is second order, and thus possibly of the same magnitude as the leakage from linear polarization, but the fact that V = (RR − LL)/2 means it is more likely corrupted by errors in the right and left gains. Linearly polarized feeds replace V with one of Q or U in a similar situation. If necessary, multiple Stokes parameters can be CLEANed to form an estimate of IS, to be iteratively improved using the procedure below.
 The visibilities are corrected using the set of IS CLEAN components VC by subtracting
from the visibilities in each polarization at baseline bAB(t). is the set of visibilities in each polarization for antennas A and B at time t for the jth CLEAN component. We prefer to use in the form of Stokes parameters instead of feed correlations since usually only one image (I) needs to be CLEANed before applying the correction. ΨAB is therefore transformed into Stokes form, ΨS,AB:
 For the ST, with its circularly polarized feeds, the correction is only applied to Stokes Q and U and second-order leakages are ignored since the leakage from I to Q and U is first order. That reduces the used portion of ΨS,AB to linear combinations of elements of ΨA and ΨB*, allowing the leakages of I into Q or U for a given baseline to be easily calculated on the fly from combinations of leakage maps for the individual antennas instead of storing leakage maps for each combination of antennas:
where P is Q or U. Note that the imaginary part would be cancelled out if A and B were identical. The Jones matrices of individual antennas are in circular coordinates (p = R, q = L), so
These are the leakage patterns that are shown in Figures 3 and 4. Note that the 12 and 21 subscripts refer to the off-diagonal elements of the Jones matrix, not baselines between antennas 1 and 2.
3. Simulated Leakage Maps
Ng et al.  calculated theoretical leakage voltage patterns for the ST's three and four metal strut antennas. Applying them to correcting polarization leakage in the CGPS, Taylor et al.  confirmed that heterogeneity in the ST was having a noticeable effect on the CGPS polarization images that was not being corrected by subtracting the Stokes I images multiplied by leakage maps. The correction still left significant residuals, however, which was not surprising since the simulated patterns were based on an overly simplistic model of the ST. Some of the three-strut antennas have fiberglass supports for their receivers. Treating those as zero strut antennas would be incorrect because each receiver box has cables running along one of its supporting struts. The unknown effective blockage of those cables, along with the partial transparency of the fiberglass struts, made measuring the actual leakage patterns essential.
4. Antenna Pattern Measurements
 If one antenna, A, in an interferometer points directly at a bright isolated source while the others look at it askew, A will not have any off-axis leakage or primary beam attenuation (ΨA(0) = ΓA(0) = 1), and the effective leakage and primary beam patterns will be those of the other antennas alone. Such offset observations with one antenna on axis are often done for hologrammatic measurements of antenna surface errors, and with two modifications the hologram scheme can be adapted to measure the leakage and primary complex voltage patterns of each antenna.
 The first modification is to compress the sampling grid of offsets. Since there is a Fourier transform relationship between the physical features of an antenna and its angular power pattern, hologram measurements need to sample a wide section of the celestial sphere to resolve small-scale errors (i.e., a misadjusted panel or smaller) on an antenna. In an antenna pattern measurement, however, it is more important to sample the main lobe well, so we confined the sampling grid to within the first null. In theory the antenna patterns should not vary any faster with angle than the primary beam. (Both the simulations of Ng et al.  and the more intuitive realization that objects smaller than the antenna diameter, such as struts, produce features broader than the primary beam.) For the ST that means its patterns should be fairly smooth on scales smaller than approximately a degree, so the measurements were made on a grid with 25′ spacing out to a maximum distance of 75′ from the beam center (the extent of beam used for polarization mosaics).
 The second modification is only in software, in that the antenna patterns come directly from the measured visibilities, instead of requiring a Fourier transform like surface error measurements. The primary voltage pattern of an antenna B comes from a observation with an on-axis reference antenna A of an unpolarized and unresolved source s:
Since the source is effectively Iδ(0) the integral of equation (1) was readily evaluated for equation (7). It can be further simplified by noting that ΨA(0) and ΓA(0) are identity matrices, and that (unsurprisingly, given the physics it represents) the outer product has the redistribution property [Hamaker et al., 1996, equation 5]:
vs is (ps, qs), the voltages that s nominally imposes on the feeds. s is unpolarized, so 〈psqs*〉 = 〈qsps*〉 = 0, and 〈psps*〉 = 〈qsqs*〉 reducing equation (8) to
 The off-diagonal elements of B's leakage Jones matrix are
which completely specifies ΨB, since the diagonal elements are 1.
 Measuring the primary voltage patterns requires knowing 〈psps*〉 (=〈qsqs*〉). Their diagonal entries (the only nonzero ones) can be estimated (to within the noise, since the effects of the primary voltage patterns are defined to be whatever is left after on-axis calibration) from a regular on-axis observation (i.e., 〈pA(0)pB*(0)〉), so
 The 1420 MHz feeds of the ST are not offset from the central axes of the antennas, so there should be no difference between its ΓB,11(n) and ΓB,22(n) because of beam squint. We therefore collapse its primary voltage patterns from Jones matrices to a scalar for each antenna:
This approach can even be useful for telescopes with offset feeds, such as the VLA, if care is taken to perform all calibration and self-calibration with I = (pp + qq)/2 instead of pp and/or qq individually (J. Uson, personal conversation, 2006). In practice there is some error introduced for wide-field polarimetry by approximating Γ with a scalar, since although Γ does not mix polarizations in the observational basis, it typically does in the Stokes basis. For circularly polarized feeds squint mixes I and V for directions away from the pointing center. This does not greatly contaminate I since V is almost always ∼ 0, but is a serious problem for measuring V, especially for continuum observations where spectroscopic techniques cannot help. The ST does have 1–2% leakage from I into V at the half-power level of the primary beam, and although it could be interpreted as squint the direction of the apparent squint sweeps through 180° as the frequency goes from band A to D. The Robert Byrd Telescope at Green Bank also sees a change in the direction of the apparent squint with frequency [Heiles et al., 2003, available at http://www.local.gb.nrao.edu/∼rmaddale/GBT/Commissioning/Polarization/PolarizationCalibration.pdf]. Such a variance with frequency is inconsistent with the geometrical effect that affects the VLA. The ST has only been used to measure V for exceptional cases like pulsars and the Sun, that have strong circular polarization. Observations that need to measure V off-axis for more weakly polarized sources, especially in continuum, will need to apply a more extensive treatment. Similarly, when using linearly polarized feeds [Sault and Ehle, 1996, available at http://www.atnf.csiro.au/observers/memos/d97793∼1.pdf] squint mixes I with Q instead of V, making the Γ11(n) = Γ22(n) approximation less attractive.
 Using g, the primary beam Bs,t(n) for a baseline formed by correlating antennas s and t is then
Note that the order of s and t matters when antennas s and t are not identical.
 Since the patterns are ratios, the requirement above that s be unresolved can be loosened to requiring that its size be much smaller than the angular scale of variations in the primary beam, to avoid smearing the pattern samples.
 With an interferometric array the patterns can be simultaneously measured for all of the antennas except the reference antenna (i.e., B is anything but A in the above equations) by keeping only the reference antenna pointed at the source while the other antennas look at it with the same grid of offsets. The patterns of the antenna used as a reference in that set of observations can be measured by repeating the observations with a different antenna as the reference.
5. Observed Leakage Maps
 In order to minimize any effects from interference or crosstalk the antennas were placed so that the distances between them were no smaller than 47 m. Observations were made of 3C 147, an unresolved bright source with a flux density of 22 Jy (1 Jy = 10−26 W m−2 Hz−1) at 1420 MHz.
 The beams were sampled on a square grid with 25′ spacing out to a maximum radius of 75′ from the beam center. The time spent on each spot was varied to achieve approximately the same uncertainty for each leakage measurement, by making the integration intervals inversely proportional to the nominal value of the primary beam:
The on-axis pointing was observed longer because it was observationally convenient and it is relatively important since it is used to normalize the patterns.
 The entire grid was observed twice, once with antenna 1 as the reference antenna, and then again with antenna 7 as the reference antenna. That allowed the leakage maps and primary voltage patterns of all antennas in the ST to be measured without requiring a separate reference antenna.
 The leakage patterns were sampled out to 75′ away from the beam center, because that is the portion of the beam used by the CGPS. Beyond that limit (the 24% power level of the beam) the leakages are expected to be large, and require long integration times to measure with the same accuracy. To help remove errors that extend within the 75′ from objects just outside it, the leakage pattern measurements are extrapolated, using a nearest-neighbor method, as far as 120′ away from the beam center. The leakage patterns are also interpolated with cubic splines to a grid with 0.20′ spacing to match the pixels of the CLEAN component images. An example correction with the measured patterns of leakage from I into U is shown in Figure 2d.
6. Quality of Leakage Correction
 Since the form of primary beam used in equation (9) is not necessarily the correct one, the uncertainty in the primary voltage pattern for antenna A, goff-axis,A(n), is calculated as
The variable goff-axis,A(n) gets its name from acting like a direction-dependent factor of A's gain. nsamps,A(n) is the number of samples for antenna A in direction n.
 The variable lQA(n) is calculated (for an antenna A that comes before the reference antenna, B) as
Note that the reference antenna is observing on axis, so it has no leakage. The uncertainty in lQA comes from the noise in the receivers:
The source is intrinsically unpolarized, so the cross correlations without leakage, 〈RL*〉nl and 〈LR*〉nl, are zero, and thus so are the derivatives of lQA with respect to RR* and LL*. The uncertainty of lQA reduces to
since antenna A is the off-axis one. σlUA has the same form, and in our case is identical since σQ = σU.
 The uncertainties are roughly independent of n because of the time weighting, with an average value for antennas 2 to 6 of 0.0012. The beam centers are an exception, with average uncertainties for antennas 2 to 6 of 6 × 10−4. Antennas 1 and 7 were each used as reference antennas half of the time, so their uncertainties are worse by a factor of nearly (ameliorated by their slightly larger diameters).
 The measured leakage patterns (Figures 3 and 4) show that although there is some overall consistency in the patterns, their details are unpredictable, both from antenna to antenna and from band to band in frequency. Most noticeably, the antennas with quadrupod receiver supports, 1 and 7, are structurally nearly identical, but their leakage patterns do not show any more similarity to each other than they do to those of the tripod antennas. Likely this is because most of the leakage comes not from the struts, but from the feeds. The feeds are nominally identical, and their individual flaws are neither easily apparent to visual inspection nor tied to the type of antenna they are mounted on. This suggests that wide-field polarimetry with even nominally homogeneous arrays requires measuring the leakage patterns of each antenna, if the needed fidelity warrants it.
 Variation of the leakage patterns from band to band is prominent in the real parts of the leakage patterns. This rapid change with frequency seems surprising at first glance: one might expect properties of a waveguide feed to vary quite slowly with frequency, and hardly at all across a band that is only 2% of the center frequency. The cause appears to be the probes used to feed the reflector at 408 MHz; they are housed within the 1420 MHz feed [Veidt et al., 1985]. Computed simulations (B. G. Veidt, private communication, 2007) indicate that these probes cause some fine structure in the performance at 1420 MHz.
 The primary voltage patterns (Figure 5) reassuringly exhibit only the expected dependence on wavelength; namely their angular scales are proportional to the observing wavelength. Their apparent tight link to antenna structure suggests that primary voltage pattern errors are more amenable to correction by adjusting the antennas, as is often done using holograms. Once the primary voltage patterns are known, their effect can also be reduced postobservation, even for an inhomogeneous array [Bhatnagar et al., 2006]. Currently, such errors are attacked with direction-dependent self-calibration (modcal [Willis, 1999], also called peeling), which is vulnerable to confusing true features on the sky with unwanted artifacts. Measuring the antenna patterns with a bright unresolved calibration source instead of through self-calibration with a potentially complicated fainter science target removes that vulnerability.
 Although we have only tested heterogeneous array leakage correction with equatorially mounted antennas, in principle it would be even easier to adapt it to antennas on altitude-azimuth mounts than the image-based leakage map method. Since the leakage voltage pattern method already deals with visibilities on an individual basis, the only modification needed would be make xj and yj in equation (2) functions of time to account for the rotation of the antennas about the optical axis relative to the sky as the Earth turns.
 An implicit, but difficult to avoid, assumption in correcting for the effect of beam patterns is that the patterns do not change with time or observing elevation. The prospect of spending observing time on frequent antenna pattern remeasurements, possibly for a set of elevations and frequencies, is unappealing, so there is considerable pressure to engineer antennas that are stable enough for occasional measurements to capture most of the effects. The ST antennas were not expected to change significantly with time or observing direction, but we confirmed their behavior by comparing recent leakage measurements to the measurements made by Peracaula of the ST's overall leakage amplitude maps at 21 cm wavelength. There was little change over the intervening 10 years, despite some surface modifications to a few of the antennas. Stability is expected to be a more serious problem for larger (as measured in wavelengths) dishes, especially if standing waves create a noticeable resonance effect in the leakage at certain observing frequencies. Interpolation, or theoretical modeling, may be useful for extending the applicability of measured maps to additional elevations and/or frequencies. Alternatively, if an extremely accurate correction is only needed for one bright source within the field of an observation, the antenna patterns could be measured at that spot immediately before and after the science observation, as opposed to mapping the entire main lobe of the antenna patterns.
 The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Incorporated. The Dominion Radio Astrophysical Observatory is operated as a national facility by the National Research Council of Canada. The Canadian Galactic Plane Survey is a Canadian project with international partners. The survey is supported by a grant from the Natural Sciences and Engineering Council (NSERC). We thank the reviewers for their time and helpful comments. R. Kothes kindly pointed out IC443 as a good example of the new correction method's efficacy, and D. Routledge helpfully expanded upon the simulations of Ng et al. We appreciate the assistance of J. E. Sheehan in facilitating the measurements and D. Del Rizzo in partially processing the data. The processing was also assisted by K. Douglas' programming and documentation for antenna surface measurements. R.R. appreciates J. Uson, W. Cotton, C. Brogan, and D. Balser sharing their experience with the VLA and GBT.