The distribution of cross polarization across the main beam and near sidelobes of a reflector antenna is calculated. Results are expressed in terms relevant to imaging in radio astronomy, using Stokes parameters, as plots of instrumental polarization Q/I, U/I, and V/I, showing conversion of total intensity of a signal which is unpolarized into apparent linear and circular polarization. The calculations use GRASP8, software that is based on physical optics and the physical theory of diffraction. For purposes of calculation, the symmetrical paraboloidal reflector (diameter ∼40 wavelengths) is fed at the prime focus with a linearly polarized signal. Computed radiation patterns at a number of feed orientations are averaged to establish the antenna response to an unpolarized radio astronomy signal. The results of the computations are consistent with measurements of instrumental polarization of the Dominion Radio Astrophysical Observatory Synthesis Telescope at 1420 MHz made using unpolarized radio sources. For this telescope, the dominant source of instrumental polarization across the field is the cross polarization of the feed. The next most significant effect is scattering by the feed struts; both three-strut and four-strut configurations are examined. Struts affect performance in linear polarization but also introduce some instrumental circular polarization. The contribution to instrumental polarization from the reflector itself is comparatively small. Roughness of the reflector surface has relatively little effect in the main beam in Q and U but introduces V and also randomizes the polarization of the sidelobes. In all cases considered, the computations show that the first and subsequent sidelobes are highly polarized, with levels of instrumental polarization up to 50%.
 The polarization properties of an antenna modify the signals it receives. In communications antennas the principal effect is cross polarization, the transfer of energy from the desired polarization into the orthogonal one. This also occurs in polarimetry in radio astronomy, where the challenge is the accurate measurement of a low level of polarization, either linear or circular, in a signal which is predominantly randomly polarized (we use the term randomly polarized to describe a broadband signal whose time average has no net polarization). The radio telescope receives two orthogonal components of the randomly polarized incident field; the instrumental polarization enhances one of the components at the expense of the other, and thus converts a randomly polarized signal into an apparently polarized one. In an aperture synthesis telescope, which makes high-resolution images across a field of view defined by its antennas, the instrumental polarization across the entire beam is significant. This paper demonstrates a technique for analysis of the polarization properties across the main beam and near sidelobes of reflector antennas used in radio astronomy.
 The Stokes parameters, I, Q, U, and V, provide a concise and useful description of the polarization state of an electromagnetic signal [Tinbergen, 1996]. I represents the total power of the wave. Q and U represent its linearly polarized component, and specify the polarized intensity and the angle of the plane of polarization, χ. The linearly polarized intensity is P = , and the angle is χ = arctan (). Note that the measures Q and U are separated by 45°. The fractional polarization, p = P/I, is usually low in radio astronomy signals. V represents circular polarization, with V > 0 corresponding to right hand circular polarization (RHCP) and V < 0 to left (LHCP). The instrumental polarization of the radio telescope converts some of the power in the incoming signal into apparent Q, U, and V, and this paper presents calculations of those effects. For correct measurement of Q, U, and V of the incoming signal, the instrumental contributions must be corrected.
 We analyze the polarization properties of 9-m antennas used in the Dominion Radio Astrophysical Observatory (DRAO) Synthesis Telescope [Landecker et al., 2000]. The telescope uses seven such antennas for radio continuum imaging at 408 and 1420 MHz, and for spectral line imaging of atomic hydrogen at 1420 MHz (the H I line). Polarization images are produced at 1420 MHz [Smegal et al., 1997a], where the scientific goal is the detection of linear polarization of synchrotron emission; the fraction of circular polarization in synchrotron emission is very low [Pacholczyk, 1970]. The astronomical applications of the telescope are based on its wide field of view and its excellent sensitivity to extended structure. Its major project is the Canadian Galactic Plane Survey [Taylor et al., 2003] and wide-field images of the polarized Galactic emission at 1420 MHz are part of that survey [e.g., Uyanıker et al., 2003].
 We present calculations of the instrumental polarization of the antennas at 1420 MHz (a wavelength, λ, of 21.1 cm) using a variety of software tools, and discuss the influence of various antenna parameters on polarization performance. The calculations are compared with measurements of actual telescope performance. The work described here was undertaken in order to understand instrumental polarization across the field, and on that basis to correct for it more effectively, improving wide-field imaging. However, our methods and conclusions are relevant to polarization imaging with all aperture synthesis telescopes, and to the design of radio telescopes in general.
 The engineering literature contains abundant information on the cross polarization of reflector antennas on boresight, but very few measurements of the instrumental polarization across the main beam and sidelobes. In this paper we present measurements of the instrumental response of the antennas of the DRAO Synthesis Telescope in Q and U over a substantial part of the main beam. The other data available in the literature also pertain to radio telescopes. Troland and Heiles  made extensive measurements of the circularly polarized sidelobes of a 25-m paraboloidal reflector. Heiles et al.  measured the instrumental polarization in Q, U, and V of the main beam and first sidelobe of the Arecibo radio telescope, and describe the response in terms of a small number of parameters. Cornwell  presents data in all Stokes parameters for the 25-m antennas of the Very Large Array.
 The Synthesis Telescope accepts both senses of circular polarization. The LHCP and RHCP channels are not ideal, nor are they perfectly orthogonal. On-axis instrumental polarization, which arises largely in waveguide components attached to the feed, is routinely measured using unpolarized compact radio sources, and all data are appropriately corrected (the calibration process is described by Smegal et al. [1997a]. We will not discuss this component of instrumental polarization.
 The Synthesis Telescope uses seven paraboloidal antennas of two slightly different sizes. Diameters are 8.53 m (five antennas) and 9.14 m (two antennas), with focal lengths and opening angles of 3.66 and 3.81 m and 60.5° and 61.9° respectively. The feed is supported on the 8.53-m antennas by three struts attached at the reflector rim, and on the 9.14-m antennas by four struts, also attached at the rim. Two antennas are equipped with feed support struts of triangular cross section, designed to reduce scattering of ground radiation into the aperture [Landecker et al., 1991]. The antennas are on equatorial mounts.
2.2. Feed and Its Radiation Properties
 The dual-mode prime focus feed is based on the design of Scheffer ; it illuminates the aperture approximately uniformly to an angle from the feed axis of θ′ ≈ 35°, dropping to a level of −16 dB at the reflector edge (θ′ ≈ 60°). Figure 1 shows the copolar and cross-polar radiation patterns of the feed at 1420 MHz, computed using the MICRO-STRIPES software package [Flomerics, Ltd., 2002]. These computed patterns correspond closely to measurements (see Trikha et al.  for measurements of the copolar response and Smegal et al. [1997b] for measurements of the cross-polar response).
3. Calculation of Instrumental Polarization
 We used the program GRASP8 [Pontopiddan, 2002] which calculates the radiated field from an antenna on the basis of physical optics (PO) and the physical theory of diffraction (PTD). PO derives currents on the surface of the reflector from the incident field. PTD models the current near the reflector rim; these currents are strong determinants of cross-polarization performance [Bach and Viskum, 1986].
 GRASP8 calculates the radiation from an antenna system comprising feed, reflector, and any scattering structures in the aperture. Figure 2 shows the coordinate system used in the calculations. The program proceeds in four steps to calculate the copolarized and cross-polarized components of the far field.
 1. The currents induced on the struts by the spherical wave emanating from the feed are calculated.
 2. The currents on the reflector are calculated, considering both the direct spherical wave from the feed and the reradiation from strut currents.
 3. The radiation from the reflector then illuminates the struts, and the scattered fields from reflector and struts are calculated.
 4. Finally, scattered fields from the reflector and the struts are summed.
 We simulated random polarization by calculating the response of the antenna to linear polarization, and averaging over many values of the incident polarization angle (because we are considering the antenna as a transmitter we actually changed the feed orientation). The number of angles in the averaging process determined the accuracy of the result; for reasons explained below we averaged calculations made at eight values of feed orientation, from 0° to 157.5° in steps of 22.5°.
 Like all antenna software, GRASP8 makes approximations to speed computation, and there are differences between our computer model and the physical antennas. Most notable is that the struts in the model do not touch the reflector surface. Gaps are needed to avoid numerical problems which arise in the calculation of surface currents induced by very close scatterers [Pontopiddan, 2002]. These gaps are small, and are in a region where the illumination is strongly reduced by the aperture taper, so errors should be small.
4. Results of Calculations
 The plots in the left-hand column of Figure 3 shows the calculated performance of a reflector with no aperture blockage (i.e., no struts) illuminated by a feed with no cross polarization (the copolar component shown in Figure 1 was used to illuminate the reflector but the cross-polar component was set to zero). Polarization performance is perfect on boresight, but conversion of I, the randomly polarized total intensity, into Q and U, which represent linear polarization, increases with radius. The circular feature is the first null (at a radius of ∼2.4°), and Q/I and U/I rise to high values at that radius. Everything beyond the first null is a sidelobe response, and the sidelobes are highly polarized, with Q/I and U/I reaching values as high as 50%. Because of symmetry there is no conversion of I into V, and V/I is zero within the numerical precision of the calculations, and is therefore not shown in Figure 3. The conversion of I into Q and U seen here is produced by the reflector alone. Currents near the reflector rim dominate cross-polarization performance [Bach and Viskum, 1986], and polarization performance could be improved by reducing the edge illumination.
 In the plots in the right-hand column of Figure 3 we show the performance of the same reflector, now illuminated by the actual feed, including its cross polarization. The characteristic of the instrumental polarization changes substantially, with the orientation of the lobes within the main beam rotated through 90°: it is evident that feed cross polarization dominates that arising from the reflector. The calculated performance now begins to resemble that of the actual telescope (see Figure 6). We note that the cross polarization of the feed, as seen in Figure 1, is relatively high: other feeds for prime focus application can have cross-polarization levels 10 dB lower [see, e.g., Olver et al., 1994]. It is well known that a feed consisting of a Huygens source (an orthogonal electric and magnetic dipole with equal intensity) produces precisely linear illumination of a paraboloidal reflector, thereby eliminating cross polarization due to the reflector, at least in principle [Cutler, 1947; Jones, 1954; Koffman, 1966]. In fact cross polarization due to effects at the reflector rim remains. Of course, real feeds can only approximate the ideal Huygens feed.
 Because of symmetry, V/I is expected to be zero in both cases shown in Figure 3. We increased the number of feed orientations used to approximate random polarization until this was attained. Averaging two orientations the maximum value of V/I was −0.44. Four orientations gave essentially zero within the main beam region, but a peak of −0.15 outside it. For eight orientations the peak value was below 10−8 over the entire 12° × 12° region shown in Figure 3. We used eight orientations for all data shown in this paper.
 There is also central blockage of the aperture due to the feed itself, but the feed has complex structure and is difficult to model accurately as a scatterer. Central blockage by a simple disk produced little change in the computed main beam performance, but did change details of the sidelobes. This is to be expected, since the relatively small central blockage translates into a broad, smooth effect in the secondary pattern. We therefore did not include scattering by the feed, but nevertheless consider that our computations show the main beam performance accurately and the performance in the sidelobes with sufficient precision for assessing the impact of their polarization properties on observations. Our computed results compare well with measurements within the main beam (see below) suggesting that we have not ignored any important factors.
 Small differences in the antenna, including details of the struts and other aperture blockage, produce big changes in the sidelobes, but they remain highly polarized under all conditions. The strong polarization of the antenna sidelobes can be readily understood. On axis the instrumental polarization is low because of the circular symmetry of the antenna structure; at angles off axis the antenna becomes progressively more asymmetrical, and its polarization performance degrades.
 The significance of the high polarization level of the sidelobes is put into perspective when the absolute levels of the response in those directions are considered. The (total power) sidelobe level at a radius of 3° is about −25 dB, while at a radius of 6° it is closer to −35 dB. Spurious polarized signals from the sidelobes will be produced only by strong emitters. Sources of significance for a radio telescope will include very strong radio sources, interfering signals, and emission from the ground.
Figure 4 shows the influence of surface roughness on the polarization performance. For these calculations GRASP8 added a random component to the surface with RMS amplitude 0.005 m (0.024λ) and characteristic length 0.5 m (2.4λ). Results are shown for two different starting points for generating the random component of the surface.
 Within the main beam there is little influence on Q/I or U/I, but the sidelobe responses are effectively randomized. The level of instrumental polarization in the sidelobe region is slightly reduced, and this is perhaps one of the few benefits of an imprecise reflector surface. In a synthesis telescope, the sidelobes of each antenna, although polarized, will be different in detail. Exact calculations of instrumental polarization in the sidelobe region are then of little practical value, since all real antennas have some surface imprecision, and from here forward we will display a smaller range of angle, since our interest is primarily in performance within the main beam. Imaging of polarized emission with the DRAO Synthesis Telescope rejects data beyond a radius of 75′ from the center of the main beam, where the level is 0.24, 6.1 dB down from the peak response. The level of conversion of I into Q and U remains high in the sidelobe region under all circumstances.
 We see from Figure 4 that surface roughness introduces V/I, even within the main beam. Instrumental circular polarization arises from the loss of perfect symmetry. The pattern of V/I seen in Figure 4, with positive values on one side of the beam and negative values on the other, is the characteristic signature of a displacement between the main beams in the two hands of circular polarization. Such a displacement occurs in offset reflector systems [Chu and Turrin, 1973], and that is probably the correct interpretation here. The best fit paraboloid to the rough surface will not necessarily coincide with the perfect paraboloid, and the feed is therefore likely to be slightly offset from the best fit focus. This interpretation suggests that V/I will increase with increasing surface roughness. The orientation of the beam displacement is orthogonal to the line in the main beam where V/I is zero, and this varies in the two examples shown in Figure 4. Within the half-power beam width (diameter 1.75°) the level of V/I reaches −35 dB in one case shown in Figure 4 and −39 dB in the other. The two cases give slightly different results because the offset from the best fit focus varies slightly. In both cases the level increases further as radius increases.
Ghobrial  has shown that surface roughness introduces axial cross polarization in reflector antennas. This amounts to moving power from one linear polarization to the orthogonal one. The on-axis instrumental polarization from our computations (e.g., those in Figure 4) are consistent with Ghobrial's results.
 In Figure 5 we show the influence of feed support struts on instrumental polarization. Surface roughness has not been included in these calculations but feed cross polarization has. The three-strut configuration introduces an asymmetry into the Q/I and U/I responses which closely resembles that seen in the measured performance of the telescope (Figure 6): the calculation was made with strut orientation set to match that on the telescope. The four-strut configuration has a higher degree of symmetry, and this is seen in the Q/I and U/I performance. The conversion of I into Q and U is somewhat smaller in the main beam.
 It is clear that blockage by the struts generates V/I, where there was none in the unblocked aperture. However, the effect may be no greater than that attributable to surface roughness.
5. Measurements of Instrumental Polarization
 Data on the instrumental polarization across the field of view have been derived by observing strong unpolarized small-diameter sources (3C147 and 3C295) on a grid of positions across the primary beam of the telescope antennas [Peracaula, 1999; Taylor et al., 2003]. All measurements were corrected for the on-axis cross polarization, which arises in imperfections in the polarization transducer which connects the feed to the receiver. Such corrections are a routine part of polarization observations, and are made by observing 3C147 or 3C295 at the center of the field (for details, see Smegal et al. [1997a]). Telescope response to the unpolarized source was measured, placing the source successively on 99 points on a 15′ grid across the primary beam to a maximum radius of 90′.
 The results are shown in Figure 6a as plots of Q and U as a percentage of I. These plots represent the net effect of the instrumental polarization of the seven antennas, which have slight differences in construction (see section 2.2). We have not made measurements of the polarization performance of the individual antennas. Polarization images from the telescope are routinely corrected for instrumental polarization using the data shown in Figure 6.
6. Comparison of Calculations With Measured Performance
 There is a strong resemblance of the measured performance of the telescope, shown in Figure 6a, with the calculations presented in Figure 5, but for a detailed comparison we need to derive the polarization response of a telescope consisting of an array of five three-strut and two four-strut antennas (cf. section 2.1). The telescope includes ten interferometers (antenna pairs) composed of two three-strut antennas, one composed of two four-strut antennas, and ten composed of a three-strut antenna combined with a four-strut antenna. The responses of interferometers made up of pairs of three-strut antennas and pairs of four-strut antennas are those depicted in Figure 5. However, calculating the response of the interferometers which combine a three-strut antenna with a four-strut antenna is more involved. We used the complex electric field vectors computed by GRASP8. For each antenna, x and y components of the field at each pixel within the field of view are available. Stokes parameters characterizing the interferometer response were calculated for each pixel by appropriately combining the field vectors. This was done for the eight feed position angles, and the results averaged to obtain the interferometer response to unpolarized radiation. The response of the Synthesis Telescope as a whole was found from an appropriately weighted sum of interferometer responses. The result is shown in Figure 6b. Comparison of Figures 6a and 6b shows good agreement. The measured pattern clearly shows the signature of the three-strut configuration, seen particularly in the asymmetry of the Q/I pattern. A point-to-point comparison between measured and calculated patterns shows that the calculated response is strongly similar in form to the measured performance, but the absolute magnitudes of the calculated Q/I and U/I responses are, on average, about 10% higher than the measured values across the main beam. Overall, we can conclude that we have correctly modelled the polarization performance of the feed and the reflector system.
 There remains one apparent difference between Figures 6a and 6b: the measured patterns appear to be rotated by ∼4° relative to their theoretical counterparts. This is not attributable to mechanical rotation of any part of the antenna structure, and must arise from mixing between the Q and U response patterns (note that the Q pattern is essentially the U pattern rotated by 45°). This mixing could be produced by coupling between LHCP and RHCP in the antenna feed, or by an error in the calibration measurements used to define the observational (Q, U) coordinate system. The effects of coupling within the feed are removed using calibration observations of unpolarized point sources. High levels of feed coupling can produce leakage between Q and U, but this is a second-order effect [Smegal et al., 1997a] and ∼25% coupling in all feeds would be required to produce the apparent rotation, far exceeding the average coupling of 5% and the worst case of 10%. On the other hand, there is an error in the polarization angle of ∼2° caused by (time variable) ionospheric Faraday rotation (for which no correction has been attempted) occurring during observations of the polarization angle calibrator, the extragalactic source 3C286. Since polarization angle is 0.5 arctan (U/Q), a 2° error in calibration of angles produces a 4° rotation of the Q and U responses. The sense of the rotation is consistent with this interpretation.
7. Polarization Effects of Struts
 We have seen that our method of calculating instrumental polarization produces results which closely resemble the measured performance of the antennas. It is clear that one of the determining influences is blockage by the struts. In this section we aim to understand the effects of struts and to compare our results with other experimental data.
 Struts produce conical sidelobes passing through the main beam, as reported by many authors [e.g., Rusch et al., 1982; Landecker et al., 1991]. Troland and Heiles  measured circularly polarized sidelobes of a large (∼120λ) paraboloidal reflector equipped with four feed support struts. The conical strut sidelobes are clearly seen in their measured V response [Troland and Heiles, 1982, Figure 1]. The central part of their measured pattern resembles that seen in the V/I plots in Figure 5, where the main beam is surrounded by sidelobes of alternating sign.
 It is hardly surprising that strut blockage affects polarization performance. Each strut is a long, thin object: in the case we are considering the struts are more than 20λ long and only 0.75λ in diameter. Such a structure can be expected to have markedly different scattering properties for radiation whose polarization is parallel to the long dimension and one that is orthogonal to it. This difference leads to instrumental effects seen in linear polarization, which appear in our Q/I and U/I images. If the effects of the strut on the two orthogonal linear components is different in phase as well as amplitude, then the strut blockage will also produce spurious circular polarization, as we see in V/I results.
 There is a more fundamental reason why antennas with struts produce spurious V/I: struts introduce asymmetry into an otherwise symmetrical structure, and asymmetrical reflectors have poor performance for circular polarization. If an asymmetrical reflector is fed from its focus in the two hands of circular polarization, the LHCP and RHCP beams will be offset [Chu and Turrin, 1973]. The offset lies in the plane of the asymmetry. A single strut may then be expected to produce a similar offset between the two circularly polarized beams in the plane orthogonal to the plane of the strut. GRASP8 calculations show just such an effect; one strut produces a pattern of V/I with a positive and negative lobe placed off center. Three struts will produce six sidelobes (and four struts will produce eight) around the edge of the main beam as seen in Figure 5. The signs of these circularly polarized sidelobes alternate.
8. Concluding Discussion
 We have developed a technique for calculating the instrumental polarization of reflector antennas used as radio telescopes, and have shown that the results provide a good match to measured performance. The technique is widely applicable to the study of radio telescope performance.
 In all the antenna configurations that we have modelled, the polarization of the sidelobes is very high when expressed as Q/I or U/I—levels are 50% or higher. The consequence is that signals received in the sidelobes of a radio telescope appear to be highly polarized. This is borne out by experience. In the DRAO Synthesis Telescope, strong sources such as Cas A or the Sun can be received at a significant level through the sidelobes. They create much larger imaging problems in Q and U than in I. Since the sidelobes of the polarized response are strongly affected by surface roughness, radiation from a strong source seen in the sidelobes is likely to enter each interferometer with strongly different amplitude and phase, which presents severe problems for conventional image processing routines such as CLEAN and self-calibration. Image processing techniques to reduce this effect are described by Willis  and by Taylor et al. .
 Assuming as a working hypothesis that the average sidelobe level is 4 dB below isotropic, which is approximately true for a reflector antenna of typical structure regardless of its size, then the contribution of the Sun to antenna temperature is about 2 × 10−6TSun, where TSun is the brightness temperature of the Sun. At λ = 21 cm, TSun ≈ 2 × 105 K even when the solar surface is free of active regions. If conversion of I into Q and U is about 0.5 in the sidelobes, the Sun in the sidelobes will contribute approximately 0.2 K of spurious polarized signal to a single-antenna polarimeter operating at λ = 21 cm, which is comparable to the brightest polarized signal from the Galaxy [Brouw and Spoelstra, 1976]. This simple calculation shows that single-antenna polarimetry is very difficult when the Sun is above the horizon, except with a telescope with very low sidelobe levels.
 If the antennas of a synthesis telescope have equatorial mounts, it will be possible to correct for instrumental polarization across the field of view in the image domain. In a telescope with altitude-azimuth mounts the instrumental polarization response rotates relative to the sky as parallactic angle changes. Deconvolution of the polarized response under these circumstances is more complicated, but is possible [Cornwell, 2003]. Such deconvolution will require accurate knowledge of the instrumental polarization, either measured or computed. Surface roughness of the individual antennas will cause their instrumental polarization to depart from the average, and at some point surface errors will become the limiting factor in polarization imaging with a synthesis telescope. Polarization imaging with a synthesis telescope imposes more stringent requirements on the individual antennas than imaging in total intensity. For Stokes I imaging, a surface accuracy of 0.05λ or even 0.1λ is adequate. For imaging in Stokes Q and U surface accuracy of 0.025λ is preferable, and for imaging in Stokes V an even tighter specification is needed. However, most work involving circular polarization involves sources of very small angular diameter, and on-axis instrumental polarization can then be overcome by calibration using a compact unpolarized source.
 In general it is desirable to design the telescope to have good polarization performance across the beam. Some significant design points can be deduced from our results.
 1. The feed should have low levels of cross polarization.
 2. The struts should be as small in size as possible.
 3. The aperture illumination should taper to the lowest level possible at the edge of the reflector consistent with adequate aperture efficiency.
 4. Surface roughness should be kept as low as possible.
 The Dominion Radio Astrophysical Observatory is operated as a national facility by the National Research Council, 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 Research Council (NSERC). T. Ng has been supported by NSERC discovery grants awarded to T.L.L. and D.R.