Correcting the Polarization Leakage Phases and Amplitudes Throughout the Primary Beam of an Interferometer
TT o appear in R adio S cience . Preprint typeset using L A TEX style emulateapj v. 08 / / CORRECTING THE POLARIZATION LEAKAGE PHASES AND AMPLITUDES THROUGHOUT THE PRIMARY BEAMOF AN INTERFEROMETER
R. I. R eid , A. D. G ray , T. L. L andecker , and A. G. W illis
Dominion Radio Astrophysical Observatory, Herzberg Institute of Astrophysics, National Research Council, P.O. Box 248,Penticton, BC, Canada, V2A 6J9.
To appear in Radio Science.
ABSTRACTPolarimetric observations are a ff ected by leakage of unpolarized light into the polarization channels, in away that varies with the angular position of the source relative to the optical axis. The o ff -axis part of theleakage is often corrected by subtracting from each polarization image the product of the unpolarized map anda leakage map, but it is seldom realized that heterogeneities in the array shift the loci of the leaked radiationin a baseline-dependent fashion. We present here a method to measure and remove the wide-field polarizationleakage of a heterogeneous array. The process also maps the complex voltage patterns of each antenna, whichcan be used to correct all Stokes parameters for imaging errors due to the primary beams. Subject headings: instrumentation: interferometers — instrumentation: polarimeters — techniques: interfero-metric — techniques: polarimetric INTRODUCTION
The hardware typically used in radio telescopes has thegreat benefit of observing the Stokes Q , U , and V parameterssimultaneously with Stokes I , but always allows some mixingbetween the polarization channels, as in Figs. 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 smallfraction of the total intensity, and therefore easily swampedby similarly strong leakages from I .A radio interferometer uses two or more antennas to mea-sure the amplitudes and phases of the electric field imping-ing on their receivers. The measurements are stored as “vis-ibilities”, which are the correlations of the receiver voltages.Given some conditions which this article will assume to havebeen met, the visibilities sample the Fourier transform of thesky multiplied by the primary beam (directional sensitivityfunction) of the antennas (Clark 1999; Thompson 1999).Mathematically, the e ff ect of a pair of antennas A and B onthe visibilities they observe, V obs , AB , is conveniently expressedusing the Hamaker-Bregman-Sault (Hamaker et al. 1996) for-malism, where the four polarizations are combined into a col-umn vector. The true visibilities are multiplied on the left bya set of Jones matrices, each one the outer product of Jonesmatrices for antennas A and B , i.e. D AB = D A ⊗ D ∗ B represents the on-axis mixing between the nominally orthog-onal polarization channels, often called the “ D terms”. Theouter product of two matrices M and N , M ⊗ N , is formedby multiplying each entry of M with all of N , and is used,along with a complex conjugation of the second factor, tobring together the elements from each antenna in a corre-lation. Hamaker et al. (1996) explain the algebraic proper-ties, including coordinate transformations, of Jones matricesand the outer product in more detail. As in Bhatnagar et al.(2006), direction dependent e ff ects can also be included, but Electronic address: [email protected] Now at the National Radio Astrophysical Observatory, 520 EdgemontRd., Charlottesville, VA, 22903, USA
C pp q q DD V obsAB Ψ,Γ
OAb H P B W B lm A n F ig . 1.— Conceptual diagram of polarization leakage in an interferometer.Each antenna measures two nominally orthogonal polarizations p and q , butthey are partially mixed before entering the correlator C . b : Baseline (separation) between antennas. OA : The optical axis (i.e. pointing direction).( l , m ): Longitudinal and latitudinal o ff sets perpendicular to OA . n : An arbitrary o ff set in ( l , m ). Γ A : Voltage pattern of antenna A , factored to exclude polarizationleakage.HPBW: Half Power Beam Width. D B : n independent factor of the polarization leakage of antenna B . Ψ : n dependent factor of the polarization leakage. V obs , AB : Observed vector of visibilities in each polarization.Although the diagram places Ψ and Γ above the receiver and D below, eachincludes e ff ects from the feed, reflector surface, and receiver support struts. a r X i v : . [ a s t r o - ph ] J a n they must go inside the Fourier integral: V obs AB = D AB (cid:90) Ψ AB ( n ) Γ AB ( n ) S I S ( n + n c ) e i n · b AB d n (1)where n is a direction on the sky relative to the “phase track-ing center”, n c . n c is set electronically, but usually it is cho-sen to coincide with the pointing direction of the antennas. S is the Stokes matrix, which transforms the sky’s Stokes pa-rameters, I S = ( I , Q , U , V ), into the observational polarizationbasis, typically correlations of either circular or linear polar-izations. Ψ AB and Γ AB are respectively the wide-field leak-age pattern and primary beam for the correlation of antennas A and B . They are sometimes multiplied together to form asingle Jones matrix which is a generalization of the primarybeam, but the magnitudes of the e ff ects are more easily as-sessed if they are kept separate. With the separation, Γ AB isdiagonal since it does not mix polarizations in the observa-tional basis, and the diagonal elements of Ψ AB are all one.The on-axis portion of the leakage, D AB , is dealt with bystandard polarimetric calibration techniques, but the leakagevaries with direction, growing worse toward the edges of theprimary beam, as in Fig. 2. This paper is concerned with thewide-field polarization leakage, Ψ AB , and will assume that V obs , AB has already been corrected by multiplication with D − AB .If all of the antennas in an array are identical, the e ff ects ofthe primary beam and leakage patterns can be removed in theimage plane. At the Dominion Radio Astrophysical Observa-tory (DRAO) we previously corrected the wide-field contami-nation or polarization by multiplying the Stokes I image with“leakage maps”, and subtracting the results from the mea-sured Q and U images, as in Fig. 4. The leakage maps weremeasured by observing the apparent Q / I and U / I of an intrin-sically unpolarized source in a grid of o ff sets from the primarybeam center (Peracaula 1999). This correction is performedcompletely in the image plane, so we call it the “image-based”leakage removal method. It was immediately applicable forDRAO’s Synthesis Telescope (ST, Landecker et al. (2000))since its antennas are equatorially mounted and thus its leak-age patterns never rotate relative to the sky. The leakage pat-terns of an altitude-azimuth mounted telescope such as theVery Large Array (VLA) rotate relative to the sky over thecourse of an observation, but the image-based method can stillbe applied if the data are first broken up into a series of snap-shots (Cotton 1994).Unfortunately, there are di ff erences, known or unknown,between the antennas of any real interferometer. The ar-ray may be a combination of antennas from originally sep-arate telescopes, such as the Combined Array for Researchin Millimeter-Wave Astronomy (CARMA, Bock (2006)), andmost very long baseline interferometers. It could also be in atransition period where only some antennas have been mod-ified, like the partially Enhanced Very Large Array, and / orhave serious surface errors as at (sub)mm wavelengths. TheST is an example of an array where the antennas are simi-lar to each other, but with known di ff erences between them.The two outermost antennas have 9.14 m diameters with fourmetal struts supporting their receivers, while the other fiveare 8.53 m in diameter with three struts, made of either metalor fiberglass. The di ff erences in antenna diameter obviouslycreate di ff erences in the half-power beamwidths (HPBWs),which at 1420 MHz are 101.8 (cid:48) for the two outer antennasand 108.8 (cid:48) for the rest. The variation in the number andcomposition of the struts a ff ects the scattering of incoming light, which is an important component of polarization leak-age (most of the rest comes from the feeds).In polarization images the di ff erences between antennas areseen as mismatches between the standard point spread func-tion (PSF, or “dirty beam”) and the PSF of the leakage. Whenthere are phase di ff erences between the leakages of the an-tennas, the e ff ective PSF of the leakage is asymmetric (Ek-ers 1999) and o ff set from the peak of the unpolarized emis-sion. The e ff ective PSF of the leakage also varies across thefield, meaning that subtracting a multiplication of the Stokes I map with a leakage map cannot fully correct the polarizationleakage of a heterogeneous array. Additionally, the responseof each antenna in an array, both in leaked and true radia-tion, depends on the scale of the source(s). Resolved featureshave less power at high spatial frequencies, so antennas thatonly participate in long baselines will contribute little leakageto them. Unresolved features have no such attenuation withbaseline length, and elicit an equally weighted mix of leak-age from all antennas. Usually leakage maps are measuredusing a bright unresolved object, so in the case of a heteroge-neous array their corrections are only accurate for unresolvedsources. Fig. 4 exhibits both of these problems, as can beseen in comparison with Fig. 5, the same image corrected withthe method described in Section 2. The main change for theunresolved sources is the presence or lack of surrounding arcs,but the supernova remnant also shows a strong di ff erence inthe on-source residual leakage.The work described here aims to improve polarizationimaging from the DRAO ST. The telescope is engaged in anextensive survey of the major constituents of the InterstellarMedium, the Canadian Galactic Plane Survey (CGPS, Tayloret 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 theleakage patterns for each antenna of the ST (similar to a holo-gram measurement, but with finer spacing over a smaller area)and have started using them to correct its Q and U observa-tions, as seen in Fig. 5. REMOVING LEAKAGE FROM LINEAR POLARIZATION FOR AHETEROGENEOUS ARRAY
When the polarization leakage (or primary beam) varieswith both direction and baseline (i.e. antenna pair), there isno way to isolate their e ff ects to one of either the image or u v planes. Multiplying V obs , AB on the left by Ψ − AB ( n ) does notwork as it does for D AB , because n must be marginalized awayby integrating with I S . I S is the true intensity distribution ofthe sky, which is unfortunately unknown. A set of Stokes I CLEAN (H¨ogbom 1974) components makes an acceptablesubstitute, however, both in the replacement of the true sky byCLEAN components, and the temporary neglect of Q , U , and V .Since the correction for a heterogeneous array must beadded directly to the visibilities, the I model used must matchthe true I visibility function within the sampled part of the u v plane. Specifically, it should not be tapered by any sort ofsmoothing in the image plane, and the almost certain discrep-ancies between the CLEAN components and true visibilityfunction outside the sampled part of the u v plane are imma-terial for this purpose. The CLEANed I image should havesmall enough pixels to avoid quantization errors in the com- Even then, only if the images are made with the same baseline weightingas used for the leakage map. F ig . 2.— An example of leakage from Stokes I into Stokes U in a CGPS fieldcontaining the supernova remnant IC443. The thin circle is the 75 (cid:48) radius(24% power) cuto ff of the usable part of the beam in polarization. The imagehas not been corrected for the sensitivity dropo ff of the primary beam, andonly includes ST data. Note that it has been CLEANed, so the arcs are mostlyleakage. The grayscale goes from -5 (black) to 5 (white) mJy / beam. F ig . 4.— Fig. 2 (same grayscale) after image-based leakage correction. The“on-source” correction of unresolved sources is accurate to 1% of I , close tothe theoretical precision of the measured leakage map, but the arcs aroundstrong leakage remain una ff ected. Leakage amplitude di ff erences betweenantennas produce rings, and phase di ff erences produce asymmetric arcs.F ig . 3.— The sources of the I radiation that leaked into Fig. 2. The grayscalegoes from -12 (white) to 350 (black) mJy / beam. F ig . 5.— Fig. 2 (same grayscale) after correcting leakage using measuredpatterns for each antenna. Leakage measurements were made only insidethe circle, but they have been extrapolated to the edge of the image, whichworks well for clearing up the arcs of sources slightly outside the limit. Theremaining arcs are primarily due to di ff erences between the primary voltagepatterns of the antennas. ponent positions, and be CLEANed to at least a moderatelyfaint level. Very faint I emission does not need to be includedsince it will be multiplied by the leakage, typically less thana few percent, and it tends to have many more components,which would considerably slow down the calculation of thecorrection. Leakage from such emission could be quickly andadequately removed by the image-based leakage map method,using the CLEAN residual image as the I map. Calculatingthe correction for both Q and U of a CGPS field, with a vari-able number, on the order of several thousand, of CLEANcomponents, and 1 . × visibilities per polarization, takesfrom 15 minutes to overnight on a 2 GHz personal computer.Assuming that I s = ( I , , ,
0) in correcting the wide-field leakage of Eq. 1 requires some care, since its validity de-pends on what Stokes parameters are wanted, and whetherthe feeds are circularly or linearly polarized. In general eachmeasured Stokes parameter is nominally the true Stokes pa-rameter, plus first order leakage from two of the other Stokesparameters, plus second order leakage from the remainingone. This comes from the leakage Jones matrices for eachantenna having only ones on diagonal, with the leakage termso ff -diagonal. As a rule of thumb, the true Q and U can bethought of as fractions of I , and V as an even smaller frac-tion (i.e. second order). With circularly polarized feeds theleakage of I into V is second order, and thus possibly of thesame magnitude as the leakage from linear polarization, butthe fact that V = ( RR − LL ) / V with one of Q or U in a similar situation. Ifnecessary, multiple Stokes parameters can be CLEANed toform an estimate of I S , to be iteratively improved using theprocedure below.The visibilities are corrected using the set of I S CLEANcomponents V C by subtracting L b AB ( t ) = (cid:88) j Ψ AB ( n j ) V C , b AB ( t ) , j (2)from the visibilities in each polarization at baseline b AB ( t ). V C , b AB ( t ) , j is the set of visibilities in each polarization for an-tennas A and B at time t for the j th CLEAN component. Weprefer to use V C , b AB ( t ) , j in the form of Stokes parameters in-stead of feed correlations since usually only one image ( I )needs to be CLEANed before applying the correction. Ψ AB istherefore transformed into Stokes form, Ψ S , AB : Ψ S , AB = S − Ψ AB S . (3)For the ST, with its circularly polarized feeds, the correctionis only applied to Stokes Q and U and second order leakagesare ignored since the leakage from I to Q and U is first order.That reduces the used portion of Ψ S , AB to linear combinationsof elements of Ψ A and Ψ ∗ B , allowing the leakages of I into Q or U for a given baseline to be easily calculated on the flyfrom combinations of leakage maps for the individual anten-nas instead of storing leakage maps for each combination ofantennas: l PAB = l PA + l ∗ PB (4)where P is Q or U . Note that the imaginary part would becancelled out if A and B were identical. The Jones matrices ofindividual antennas are in circular coordinates ( p = R , q = L ),so l QA = Ψ A , − Ψ A , , and (5) l UA = − i (cid:0) Ψ A , + Ψ A , (cid:1) . (6)These are the leakage patterns that are shown in Figs. 6 to 9.Note that the 12 and 21 subscripts refer to the o ff -diagonalelements of the Jones matrix, not baselines between antennas1 and 2. SIMULATED LEAKAGE MAPS
Ng et al. (2005) calculated theoretical leakage voltage pat-terns for the ST’s three and four metal strut antennas. Ap-plying them to correcting polarization leakage in the CGPS(Taylor et al. 2003) confirmed that heterogeneity in the STwas having a noticeable e ff ect on the CGPS polarization im-ages that was not being corrected by subtracting the Stokes I images multiplied by leakage maps. The correction stillleft significant residuals, however, which was not surprisingsince the simulated patterns were based on an overly simplis-tic model of the ST. Some of the three-strut antennas havefiberglass supports for their receivers. Treating those as zerostrut antennas would be incorrect because each receiver boxhas cables running along one of its supporting struts. Theunknown e ff ective blockage of those cables, along with thepartial transparency of the fiberglass struts, made measuringthe actual leakage patterns essential. ANTENNA PATTERN MEASUREMENTS
If one antenna, A , in an interferometer points directly ata bright isolated source while the others look at it askew, A will not have any o ff -axis leakage or primary beam attenua-tion ( Ψ A ( ) = Γ A ( ) = ), and the e ff ective leakage and pri-mary beam patterns will be those of the other antennas alone.Such o ff set observations with one antenna on axis are oftendone for hologrammatic measurements of antenna surface er-rors, and with two modifications the hologram scheme can beadapted to measure the leakage and primary complex voltagepatterns of each antenna.The first modification is to compress the sampling grid ofo ff sets. Since there is a Fourier transform relationship be-tween the physical features of an antenna and its angularpower pattern, hologram measurements need to sample a widesection of the celestial sphere to resolve small scale errors (i.e.a misadjusted panel or smaller) on an antenna. In an antennapattern measurement, however, it is more important to samplethe main lobe well, so we confined the sampling grid to withinthe first null. In theory the antenna patterns should not varyany faster with angle than the primary beam. For the ST thatmeans its patterns should be fairly smooth on scales smallerthan approximately a degree, so the measurements were madeon a grid with 25 (cid:48) spacing out to a maximum distance of 75 (cid:48) from the beam center (the extent of beam used for polarizationmosaics).The second modification is only in software, in that theantenna patterns come directly from the measured visibili-ties, instead of requiring a Fourier transform like surface errormeasurements. 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 : V obs AB = (cid:0) Ψ A ( ) ⊗ Ψ ∗ B ( n ) (cid:1) (cid:0) Γ A ( ) ⊗ Γ ∗ B ( n ) (cid:1) S I S . (7)Since the source is e ff ectively I δ ( ) the integral of Eq. 1 wasreadily evaluated for Eq. 7. It can be further simplified by not-ing that Ψ A ( ) and Γ A ( ) are identity matrices, and that (un-surprisingly, given the physics it represents) the outer prod-uct has the redistribution property (Eq. 5 of Hamaker et al.(1996)): ( M A ⊗ M B ) ( N A ⊗ N B ) = ( M A N A ) ⊗ ( M B N B ) . Eq. 7 becomes: V obs AB = (cid:104) v s ⊗ ( Ψ B ( n ) Γ B ( n ) v s ) ∗ (cid:105) (8) = Γ ∗ B , ( n ) (cid:10) p s p ∗ s (cid:11) + Γ ∗ B , ( n ) Ψ ∗ B , ( n ) (cid:10) p s q ∗ s (cid:11) − Γ ∗ B , ( n ) Ψ ∗ B , ( n ) (cid:10) p s p ∗ s (cid:11) + Γ ∗ B , ( n ) (cid:10) p s q ∗ s (cid:11) Γ ∗ B , ( n ) (cid:10) q s p ∗ s (cid:11) + Γ ∗ B , ( n ) Ψ ∗ B , ( n ) (cid:10) q s q ∗ s (cid:11) − Γ ∗ B , ( n ) Ψ ∗ B , ( n ) (cid:10) q s p ∗ s (cid:11) + Γ ∗ B , ( n ) (cid:10) q s q ∗ s (cid:11) v s is ( p s , q s ), the voltages that s nominally imposes on thefeeds. s is unpolarized, so (cid:10) p s q ∗ s (cid:11) = (cid:10) q s p ∗ s (cid:11) =
0, and (cid:10) p s p ∗ s (cid:11) = (cid:10) q s q ∗ s (cid:11) , reducing Eq. 8 to V obs AB = (cid:10) p s p ∗ s (cid:11) + (cid:10) q s q ∗ s (cid:11) Γ ∗ B , ( n ) − Γ ∗ B , ( n ) Ψ ∗ B , ( n ) Γ ∗ B , ( n ) Ψ ∗ B , ( n ) Γ ∗ B , ( n ) . Both the simulations of Ng et al. (2005) and the more intuitive realizationthat objects smaller than the antenna diameter, such as struts, produce featuresbroader than the primary beam.
A B C D -75-50-250255075 -75-50-250255075 -75-50-250255075 m ( ) -75-50-250255075 -75-50-250255075 -75-50-250255075
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75 75 50 25 0 - - - l ( ) F ig . 6.— Real parts of the voltage leakage from I into Q of antennas 1 (top)to 7 (bottom) for bands A (left) to D (right). The colorscale goes from -0.05(blue) to 0.05 (red). A B C D -75-50-250255075 -75-50-250255075 -75-50-250255075 m ( ) -75-50-250255075 -75-50-250255075 -75-50-250255075
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75 75 50 25 0 - - - l ( ) F ig . 7.— Imaginary parts of the voltage leakage from I into Q of antennas 1(top) to 7 (bottom) for bands A (left) to D (right). The colorscale goes from-0.05 (blue) to 0.05 (red). The o ff -diagonal elements of B ’s leakage Jones matrix are Ψ B , ( n ) = ( V obs AB , qp / V obs AB , qq ) ∗ , and Ψ B , ( n ) = ( V obs AB , pq / V obs AB , pp ) ∗ , which completely specifies Ψ B , since the diagonal elementsare 1.Measuring the primary voltage patterns requires knowing (cid:10) p s p ∗ s (cid:11) (cid:0) = (cid:10) q s q ∗ s (cid:11)(cid:1) . Their diagonal entries (the only nonzeroones) can be estimated from a regular on-axis observation (cid:16) i.e. (cid:68) p A ( ) p ∗ B ( ) (cid:69)(cid:17) , so Γ B , ( n ) (cid:39) (cid:68) p A ( ) p ∗ B ( n ) (cid:69)(cid:68) p A ( ) p ∗ B ( ) (cid:69) ∗ , and Γ B , ( n ) (cid:39) (cid:68) q A ( ) q ∗ B ( n ) (cid:69)(cid:68) q A ( ) q ∗ B ( ) (cid:69) ∗ . To within the noise, since the e ff ects of the primary voltage patterns aredefined to be whatever is left after on-axis calibration. The 1420 MHz feeds of the ST are not o ff set from the cen-tral axes of the antennas, so there should be no di ff erence be-tween its Γ B , ( n ) and Γ B , ( n ) because of beam squint. Wetherefore collapse its primary voltage patterns from Jones ma-trices to a scalar for each antenna: g o ff -axis , B ( n ) (cid:39) (cid:68) p A ( ) p ∗ B ( n ) (cid:69)(cid:68) p A ( ) p ∗ B ( ) (cid:69) + (cid:68) q A ( ) q ∗ B ( n ) (cid:69)(cid:68) q A ( ) q ∗ B ( ) (cid:69) / . This approach can even be useful for telescopes with o ff setfeeds, such as the VLA, if care is taken to perform all cali-bration and self-calibration with I = ( pp + qq ) / pp and / or qq individually (conversation with J. Uson, 2006).In practice there is some error introduced for wide-field po-larimetry by approximating Γ with a scalar, since although Γ does not mix polarizations in the observational basis, it typi-cally does in the Stokes basis. For circularly polarized feedssquint mixes I and V for directions away from the pointingcenter. This does not greatly contaminate I since V is almostalways ∼
0, but is a serious problem for measuring V , espe-cially for continuum observations where spectroscopic tech-niques cannot help. The ST does have 1-2% leakage from I into V at the half-power level of the primary beam, and al- A B C D -75-50-250255075 -75-50-250255075 -75-50-250255075 m ( ) -75-50-250255075 -75-50-250255075 -75-50-250255075
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75 75 50 25 0 - - - l ( ) F ig . 8.— Real parts of the voltage leakage from I into U of antennas 1 (top)to 7 (bottom) for bands A (left) to D (right). The colorscale goes from -0.05(blue) to 0.05 (red). though it could be interpreted as squint the direction of the ap-parent squint sweeps through 180 ◦ as the frequency goes fromband A to D. The Robert Byrd Telescope at Green Bank alsosees a change in the direction of the apparent squint with fre-quency (Heiles et al. 2003). Such a variance with frequency isinconsistent with the geometrical e ff ect that a ff ects the VLA.The ST has only been used to measure V for exceptional caseslike pulsars and the Sun, that have strong circular polarization.Observations that need to measure V o ff -axis for more weaklypolarized sources, especially in continuum, will need to applya more extensive treatment. Similarly, when using linearlypolarized feeds (Sault & Ehle 1996) squint mixes I with Q instead of V , making the Γ ( n ) = Γ ( n ) approximation lessattractive.Using g , the primary beam B s , t ( n ) for a baseline formed bycorrelating antennas s and t is then B s , t ( n ) = g o ff -axis , s ( n ) g ∗ o ff -axis , t ( n ) . 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 bemuch smaller than the angular scale of variations in the pri- A B C D -75-50-250255075 -75-50-250255075 -75-50-250255075 m ( ) -75-50-250255075 -75-50-250255075 -75-50-250255075
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75 75 50 25 0 - - - l ( ) F ig . 9.— Imaginary parts of the voltage leakage from I into U of antennas 1(top) to 7 (bottom) for bands A (left) to D (right). The colorscale goes from-0.05 (blue) to 0.05 (red). mary beam, to avoid smearing the pattern samples.With an interferometric array the patterns can be simulta-neously measured for all of the antennas except the referenceantenna (i.e. B is anything but A in the above equations) bykeeping only the reference antenna pointed at the source whilethe other antennas look at it with the same grid of o ff sets. Thepatterns of the antenna used as a reference in that set of obser-vations can be measured by repeating the observations with adi ff erent antenna as the reference. OBSERVED LEAKAGE MAPS
In order to minimize any e ff ects from interference orcrosstalk the antennas were placed so that the distances be-tween them were no smaller than 47 m. Observations weremade of 3C 147, an unresolved bright source with a flux den-sity of 22 Jy at 1420 MHz.The beams were sampled on a square grid with 25 (cid:48) spac-ing out to a maximum radius of 75 (cid:48) from the beam center.The time spent on each spot was varied to achieve approxi-mately the same uncertainty for each leakage measurement,by making the integration intervals inversely proportional tothe nominal value of the primary beam: t int ( n ) ∝ cos − (cid:32) − / )HPBW | n | (cid:33) . (9)The on-axis pointing was observed longer because it was ob-servationally convenient and it is relatively important since itis used to normalize the patterns.The entire grid was observed twice, once with antenna 1 asthe reference antenna, and then again with antenna 7 as thereference antenna. That allowed the leakage maps and pri-mary voltage patterns of all antennas in the ST to be measuredwithout requiring a separate reference antenna.The leakage patterns were sampled out to 75 (cid:48) away fromthe beam center, because that is the portion of the beam usedby the CGPS. Beyond that limit (the 24% power level of thebeam) the leakages are expected to be large, and require longintegration times to measure with the same accuracy. To helpremove errors that extend within the 75 (cid:48) from objects just out-side it, the leakage pattern measurements are extrapolated, us-ing a nearest-neighbor method, as far as 120 (cid:48) away from thebeam center. The leakage patterns are also interpolated withcubic splines to a grid with 0.20 (cid:48) spacing to match the pix-els of the CLEAN component images. An example correctionwith the measured patterns of leakage from I into U is shownin Fig. 5. QUALITY OF LEAKAGE CORRECTION
Since the form of primary beam used in Equation 9 is notnecessarily the correct one, the uncertainty in the primaryvoltage pattern for antenna A , g o ff -axis , A ( n ), is calculated as: (cid:32) σ g o ff -axis , A ( n ) g o ff -axis , A ( n ) (cid:33) = n samps , A ( n ) (cid:32) σ I , , A ( n ) I A ( n ) (cid:33) + (cid:32) σ I A ( ) I A ( ) (cid:33) ,σ g o ff -axis , A ( n ) = (cid:32) n samps , A ( ) n samps , A ( n ) + (cid:12)(cid:12)(cid:12) g o ff -axis , A (cid:12)(cid:12)(cid:12) ( n ) (cid:33) / σ I A ( ) I A ( ) .g o ff -axis , A ( n ) gets its name from acting like direction dependentfactor of A ’s gain. n samps , A ( n ) is the number of samples forantenna A in direction n . l QA ( n ) is calculated (for an antenna A that comes before thereference antenna, B ) as l QA ( n ) = (cid:68) R A ( n ) L ∗ B ( ) (cid:69)(cid:68) L A ( n ) L ∗ B ( ) (cid:69) + (cid:68) L A ( n ) R ∗ B ( ) (cid:69)(cid:68) R A ( n ) R ∗ B ( ) (cid:69) = (cid:10) ( R + Ψ A , L )( n ) L ( ) (cid:11) (cid:104) L ( n ) L ∗ ( ) (cid:105) + (cid:10) ( L + Ψ A , R )( n ) R ∗ ( ) (cid:11) (cid:104) R ( n ) R ∗ ( ) (cid:105) = (cid:34)(cid:32) (cid:104) R ( n ) L ∗ ( ) (cid:105)(cid:104) L ( n ) L ∗ ( ) (cid:105) (cid:33) nl + Ψ A , + (cid:32) (cid:104) L ( n ) R ∗ ( ) (cid:105)(cid:104) R ( n ) R ∗ ( ) (cid:105) (cid:33) nl + Ψ A , (cid:35) . Note that the reference antenna is observing on-axis, so it hasno leakage. The uncertainty in l QA comes from the noise inthe receivers: (cid:12)(cid:12)(cid:12) σ l QA (cid:12)(cid:12)(cid:12) = (cid:88) S = RR ∗ , LL ∗ , RL ∗ , LR ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ l QA ∂ S σ S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) The source is intrinsically unpolarized, so the crosscorrela-tions without leakage, (cid:104) RL ∗ (cid:105) nl and (cid:104) LR ∗ (cid:105) nl , are zero, and thusso are the derivatives of l QA with respect to RR ∗ and LL ∗ . The uncertainty of l QA reduces to (cid:12)(cid:12)(cid:12) σ l QA ( n ) (cid:12)(cid:12)(cid:12) = σ Q (cid:104) L ( n ) L ∗ ( ) (cid:105) + (cid:104) R ( n ) R ∗ ( ) (cid:105) = σ Q | g o ff -axis , A ( n ) | I ( )since antenna A is the o ff -axis one. σ l UA has the same form,and in our case is identical since σ Q = σ U .The uncertainties are roughly independent of n because ofthe time weighting, with an average value for antennas 2 to 6of 0.0012. The beam centers are an exception, with averageuncertainties for antennas 2 to 6 of 6 × − . Antennas 1 and 7were each used as reference antennas half of the time, so theiruncertainties are worse by a factor of nearly √ DISCUSSION
The measured leakage patterns, Figs. 6 to 9, show that al-though there is some overall consistency in the patterns, theirdetails are unpredictable, both from antenna to antenna andfrom band to band in frequency. Most noticeably, the an-tennas with quadrupod receiver supports, 1 and 7, are struc-turally nearly identical, but their leakage patterns do not showany more similarity to each other than they do to those ofthe tripod antennas. Likely this is because most of the leak-age comes not from the struts, but from the feeds. The feedsare nominally identical, and their individual flaws are neithereasily apparent to visual inspection nor tied to the type of an-tenna they are mounted on. This suggests that wide-field po-larimetry with even nominally homogeneous arrays requiresmeasuring the leakage patterns of each antenna, if the neededfidelity warrants it.Variation of the leakage patterns from band to band isprominent in the real parts of the leakage patterns. This rapidchange with frequency seems surprising at first glance: onemight expect properties of a waveguide feed to vary quiteslowly with frequency, and hardly at all across a band thatis only 2% of the center frequency. The cause appears to bethe probes used to feed the reflector at 408 MHz; they arehoused within the 1420 MHz feed (Veidt et al. 1985). Com-puted simulations (B.G. Veidt, private communication) indi-cate that these probes cause some fine structure in the perfor-mance at 1420 MHz.The primary voltage patterns, Figs. 10 and 11, reassuringlyexhibit only the expected dependence on wavelength; namelytheir angular scales are proportional to the observing wave-length. Their apparent tight link to antenna structure suggeststhat primary voltage pattern errors are more amenable to cor-rection by adjusting the antennas, as is often done using holo-grams. Once the primary voltage patterns are known, theire ff ect can also be reduced post-observation, even for an in-homogeneous array (Bhatnagar et al. 2006). Currently sucherrors are attacked with direction dependent self-calibration(modcal, (Willis 1999), also called peeling), which is vulner-able to confusing true features on the sky with unwanted ar-tifacts. Measuring the antenna patterns with a bright unre-solved calibration source instead of through self-calibrationwith a potentially complicated fainter science target removesthat vulnerability.Although we have only tested heterogeneous array leak-age correction with equatorially mounted antennas, in prin-ciple it would be even easier to adapt it to antennas onaltitude-azimuth mounts than the image-based leakage map A B C D -75-50-250255075 -75-50-250255075 -75-50-250255075 m ( ) -75-50-250255075 -75-50-250255075 -75-50-250255075
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75 75 50 25 0 - - - l ( ) F ig . 10.— Real parts of the primary voltage patterns of antennas 1 (top) to 7(bottom) for bands A (left) to D (right). The grayscale goes from 0.4 (white)to 1.0 (black), and the contours go from 0.4 to 0.9 in steps of 0.1. A B C D -75-50-250255075 -75-50-250255075 -75-50-250255075 m ( ) -75-50-250255075 -75-50-250255075 -75-50-250255075
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75 75 50 25 0 - - - l ( ) F ig . 11.— Imaginary parts of the primary voltage patterns of antennas 1(top) to 7 (bottom) for bands A (left) to D (right). The colorscale goes from-0.1 (blue) to 0.1 (red). The symmetry of antenna 6’s patterns suggests that itis out of focus. method. Since the leakage voltage pattern method alreadydeals with visibilities on an individual basis, the only modifi-cation needed would be make x j and y j in Equation 2 func-tions of time to account for the rotation of the antennas aboutthe optical axis relative to the sky as the Earth turns.An implicit, but di ffi cult to avoid, assumption in correct-ing for the e ff ect of beam patterns is that the patterns donot change with time or observing elevation. The prospectof spending observing time on frequent antenna pattern re-measurements, possibly for a set of elevations and frequen-cies, is unappealing, so there is considerable pressure to en-gineer antennas that are stable enough for occasional mea-surements to capture most of the e ff ects. The ST antennaswere not expected to change significantly with time or ob-serving direction, but we confirmed their behavior by compar-ing recent leakage measurements to the measurements madeby Peracaula of the ST’s overall leakage amplitude maps at21 cm wavelength. There was little change over the inter-vening 10 years, despite some surface modifications to a fewof the antennas. Stability is expected to be a more seriousproblem for larger (as measured in wavelengths) dishes, espe-cially if standing waves create a noticeable resonance e ff ect in the leakage at certain observing frequencies. Interpolation,or theoretical modeling, may be useful for extending the ap-plicability of measured maps to additional elevations and / orfrequencies. Alternatively, if an extremely accurate correc-tion is only needed for one bright source within the field of anobservation, the antenna patterns could be measured at thatspot immediately before and after the science observation, asopposed to mapping the entire main lobe of the antenna pat-terns.The National Radio Astronomy Observatory is a facility ofthe National Science Foundation operated under cooperativeagreement by Associated Universities, Incorporated. The Do-minion Radio Astrophysical Observatory is operated as a na-tional facility by the National Research Council of Canada.The Canadian Galactic Plane Survey is a Canadian projectwith international partners. The survey is supported by agrant from the Natural Sciences and Engineering Council(NSERC). We thank the reviewers for their time and helpfulcomments.R. Kothes kindly pointed out IC443 as a good example ofthe new correction method’s e ffi cacy, and D. Routledge help-fully expanded upon the simulations of Ng et al. We appre-ciate the assistance of J.E. Sheehan in facilitating the mea-surements and D. Del Rizzo in partially processing the data.The processing was also assisted by K. Douglas’ program- ming and documentation for antenna surface measurements.R.R. appreciates J. Uson, W. Cotton, C. Brogan, and D. Balsersharing their experience with the VLA and GBT. REFERENCESBhatnagar, S., Cornwell, T., & Golap, K. 2006, Correction of errors due toantenna power patterns during imaging, EVLA Memo 100, NationalRadio Astrophysical Observatory 1, 7Bock, D. C.-J. 2006, in Astronomical Society of the Pacific ConferenceSeries, Vol. 356, Revealing the Molecular Universe: One Antenna isNever Enough, ed. D. C. Backer, J. M. Moran, & J. L. Turner, 17– + , ADSlink 1Clark, B. G. 1999, in ASP Conf. Ser. 180: Synthesis Imaging in RadioAstronomy II, ed. G. B. Taylor, C. L. Carilli, & R. A. Perley, 1–, ADSlink 1Cotton, W. 1994, Widefield Polarization Correction of VLA SnapshotImages at 1.4 GHz, AIPS Memo 86, National Radio AstrophysicalObservatory 1Ekers, R. D. 1999, in ASP Conf. Ser. 180: Synthesis Imaging in RadioAstronomy II, ed. G. B. Taylor, C. L. Carilli, & R. A. Perley, 321–334,ADS link 1H¨ogbom, J. A. 1974, A&AS, 15, 417, ADS link 2Hamaker, J. P., Bregman, J. D., & Sault, R. J. 1996, A&AS, 117, 137, ADSlink 1, 4Heiles, C., Robishaw, T., Troland, T., & Roshi, D. A. 2003, A PreliminaryReport: Calibrating the GBT at L, C, and X Bands, GBT CommissioningMemo 23, NRAO 4Landecker, T. L., Dewdney, P. E., Burgess, T. A., Gray, A. D., Higgs, L. A.,Ho ff mann, A. P., Hovey, G. J., Karpa, D. R., Lacey, J. D., Prowse, N.,Purton, C. R., Roger, R. S., Willis, A. G., Wyslouzil, W., Routledge, D., &Vaneldik, J. F. 2000, A&AS, 145, 509, ADS link 1 Ng, T., Landecker, T. L., Cazzolato, F., Routledge, D., Gray, A. D., Reid,R. I., & Veidt, B. G. 2005, Radio Science, 40, 5014 3, 3Peracaula, M. 1999, Instrumental polarization corrections at 1420 MHz forthe DRAO Synthesis Telescope fields, Tech. rep., Dominion RadioAstrophysical Observatory 1Sault, R. J. & Ehle, M. 1996, The ATCA 13-cm Polarimetric Response,ATNF Technical Memo 39.3 //