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Arecibo Observatory Technical Memo AOTM 99-02 THE LBW FEED: POINTING ACCURACY, BEAMWIDTH, BEAM SQUINT, BEAM SQUASH
Carl Heiles

ABSTRACT
During February-March 1999 we made extensive observations to obtain a complete characterization of the polarization properties of the LBW feed over the frequency range 1420:4 6:25 MHz. In a previous memo AOTM 99-01 we presented our scheme of Mueller matrices. Here we present the polarized beam patterns and, as a by product, pointing accuracy. The executive summary of this report is a succinct six sentences: 1 The system temperature in each polarization channel increases with zenith angle and ranges from about 36 to 40 Kelvins. 2 The pointing errors have a systematic o set in zenith angle and a random component. The total standard deviation, including both components, is 8:900 . This is astronomically negligible. 3 The beam shape is elliptical, with ellipticitychanging quadratically with za; the ellipse is aligned with az ; z a. For za =11 , HPBWza is minimum at 3:660 and HPBWaz maximum at 3:140 . At za = 20 these numbers become 3:920 and 2:870 , respectively. 4 The beam squint in Stokes V is about 1:300 for za 16 and rises steeply to 3:200 at za =20 . Its direction also changes with za, with a total range 60 . Beam squint is a problem in Zeeman splitting measurements; this is more than acceptably small. The beam squash in Stokes V is negligible, but the Stokes V HPBW is systematically larger than the Stokes I HPBW by0:6100 0:1300 . 5 The beam squint in Stokes Q and U is about 1:800 . The directions di er by about 100 , and they both rotate with za over a total range of about 60 . Beam squint in linear polarization is unexpected. 6 The beam squash in Stokes Q and U is about 300 , independent of za; the position angles di er by 45 , as expected. There is also a systematic di erence in HPBW, independent of position angle, with the HPBW in Stokes U larger by about 1:600 .


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1. GENERAL METHODOLOGY
This memo observationally derives the properties of the LBW beam. The observations consist of scans oriented at four position angles across continuum sources. We assume an elliptical Gaussian beam at arbitrary position angle, so the Stokes I scans should follow a Gaussian whose o set and beamwidth re ect the pointing error and beam shape. In the other Stokes parameters the beam is described by beam squint and squash, whichwe de ne below. Section 2 develops the expected behavior of the measured Stokes parameters along a scan and section 3 the measurement technique. Section 4 presents the results.

2. BEAM PARAMETERS 2.1. De nition of beam parameters and coordinate system
The beam properties include pointing error, beamwidth, squint, and squash. Pointing error and beamwidth are familiar concepts and we won't de ne them; we de ne the others below. All the beam properties have a magnitude and direction, and we will assume that they are xed with respect to azimuth and zenith angle.
ZA

PAscan

PAsquint

+
AZ

+

Fig. 1.| Schematic illustration of beam squint. The ZA axis is the zenith angle direction and corresponds physically to the azimuth arm. PAsquint refers to the position angle of the squint and PAscan to that of a line along which observations are taken. Clearly, observed e ects change as cosPAscan , PAsquint . A similar diagram and equation apply for pointing error. We will discuss observations taken along a line oriented at an angle PAscan with respect to


3 to the azimuth arm, measured towards increasing azimuth towards the East. This is illustrated explicitly in Figure 1. Azimuth increases towards the east, which is the same sense as PAscan , so for the position angle on the sky these two angles add. Roughly the azimuth arm has PAaz az , 180 . Thus the position angle of our scan on the sky is PAsky = PAaz + PAscan az + PAscan , 180 . In our calculations we use the exact equation, not the approximation.
ZA

PAscan
-

PA

squash

+
AZ

+

-

Fig. 2.| Schematic illustration of beam squash. The ZA axis is the zenith angle direction and corresponds physically to the azimuth arm. PAsquash refers to the position angle of the squash and PAscan to that of a line along which observations are taken. Clearly, observed e ects change as cos 2PAscan , PAsquash. A similar diagram and equation apply for telescope beamwidth. One can usually obtain nearly perfect polarization performance for a point source at beam center by deriving and applying the Mueller matrix. A previous memo AOTM 99-01 derived this matrix for the LBW feed. However, away from beam center various symmetries of the telescope are broken, and this produces polarization within the main beam. There are two kinds of beam polarization that are predicted theoretically: 1 Beam squint occurs when the two circular polarizations point in di erent directions by angle . This makes the beam of the Stokes V parameter look like the rst derivative of the Stokes I beam. It produces a two-lobed" pattern, one positive and one negative, as we show schematically in Figure 1. One measures beam squintby observing Stokes V along a line, the scan direction; beam squint produces a ,=+ pattern with amplitude cosPAscan , PAsquint . 2 Beam squash occurs when the two linear polarizations have di erent beamwidths by amount . This makes the beams of the Stokes Q and U parameters look roughly like the second derivative of the Stokes I beam. These are four-lobed" beams, in which two lobes on


4 opposite sides of the beam center have the same sign and two lobes rotated 90 have the opposite sign, as we show schematically in Figure 2. One measures beam squash by observing Stokes Q or U along a line, the scan direction; beam squash produces a +=0=+ pattern with amplitude cos 2PAscan , PAsquash. Practice does not always follow theory, and this is true for Arecibo's beam: there is signi cant beam squint in the linear polarizations. We know of no theories that explicitly address polarization outside of the main beam. However, as we move further away from beam center the symmetries are broken more strongly. Thus we expect the di raction sidelobes to be increasingly polarized with distance from beam center. In addition, there are additional sidelobes that are inherently unpredictable because they result mainly from telescope surface roughness and scattering o of the telescope support structures.

2.2. The de nitions of Stokes Q, U, V
In this memo we use the astronomically-de ned Stokes parameters Sastr, whose derivation is discussed in AOTM 99-01. These Stokes parameters are de ned so that position angle 0 is aligned with the azimuth arm. That is, they are consistent with the de nition of PAscan in Figures 1 and 2. This means that Stokes Q, as used herein, is the product of the two linear probes and U is the di erence; this is in contrast to the de nition used by the datataking software. Stokes V is de ned in the usual way, as LC P , RC P from the sky. In the terms of AOTM 99-01, this means that we have applied the total matrix correction in equation 13 of that memo. Wehave veri ed these de nitions astronomically.

2.3. Mathematical description of the Stokes I beam parameters
Let RI be the directional response of the beam to Stokes parameter I ; for the other Stokes parameters we use appropriate subscripts. The parameters of RI include a pointing error, which has two-lobed symmetry and is squintlike, and an elliptical beam, which has four-lobed symmetry and is squashlike. The beam is Gaussian, so for observations along a line at position angle PAscan we expect the observed Stokes I to be
scan , PA2 ; RI = IsrcG = Isrc exp , PA2 "

1

where Isrc is the Stokes I of the source and scan is the angle along the scan line measured from the nominal beam center and increasing towards the direction of PAscan . Later we use the symbol


5

G as shorthand for the Gaussian. PA depends on PA because the beam is elliptical. We
expect the pointing o set and beamwidth to vary as PA= pointing cosPAscan , PApointing PA= 0 +beam cos 2PAscan , PAbeam : Here pointing is the total pointing o set and PApointing its position angle. The HPBW is numerically related to by the usual factor 2ln 21=2 ,so wehave for example 2a 2b

HP B W HP B W

=1:6650 , beam cos 2PAbeam za direction =1:6650 +beam cos 2PAbeam
az direction

3a 3b

Note this important point: RI varies as cos2PA instead of cosPA. This means that if the beam is not aligned with the az ; z a directions, then observations taken in these directions alone will not reveal the misalignment. As it happens, the alignment is good, but perhaps not perfect. We take data along a line, called a scan, at position angle PAscan. For each scan, we least-square t equation 1 plus a constant to obtain PA and PA. We have many scans at four di erent PA's. We then least square t all these data with equation 2 to obtain pointing ;P Apointing and, also, the HPBW and its PA.

2.4. Mathematical description of the Stokes Q, U, V beam parameters
Let DP be the directional response of the beam to a particular polarization P ; here P can be one of the circular polarizations or a linear polarization. These combine in pairs to form Stokes parameter beams, whichwe designate by the letter R. Here we write equations only for circular polarization, but similar equations apply to the other polarizations. For observations along a line at position angle PAscan the beam squint and squash distort the beam as follows:
0 scan + PA 2 2 DLC P = LC Psrc exp , PA+ PA 2 " PA 2 0 2 DRC P = RC Psrc exp , scan , PA PA , 2
"

2


4a 4b

2


6
0 Here, for each scan scan = scan , PA, with PA and PA determined following section 2.3. The Stokes I beam RI is the sum, in which the beam squint and squash drop out to rst order. The Stokes V beam is the di erence; because the beam squint and squash are small quantities linear expansions are adequate, so wehave

RV 

scan

Vsrc R + 1  @RI PA+ @RI PA ; = I I 2 @PA @ PA src

5

sr where Vsrcc is the fractional circular polarization. As with the Stokes I parameter, we expect I

PA= V cosPAscan , PAV PA= V cos 2PAscan , PAV : We treat the polarized Stokes parameter data we rst obtain the beam parameters and Isrc from equation 5 plus a constant and least-squares t for four di erent PA's. We then least square t these their associated position angles. as we did the Stokes the Stokes I t. We PA and PA. data with equation 6

6a 6b

use those parameters in We have many scans at to obtain V ; V , and

I data. For each scan,

3. MEASUREMENT TECHNIQUE 3.1. The chcross" observing procedure and its data reduction
We observed a good sample of sources using the chcross" observing procedure, which samples 7 points in each of two orthogonal directions on the sky. The central point is the peak of the @G source. Its nearest neighbors are the o sets where the rst derivatives of the Gaussian @ are @G maximum. The next neighbors are where derivatives @ are maximum. The outermost points are far enough away to act as suitable o positions. In observing we assumed HPBW of 3:50 for all PAscan and used o sets of :425;:736; 1:27HPBW. Our crosses were oriented both aligned with az ; z a and also at 45 , i.e. at PAscan = 0 ; 45 ; 90 ; 135 . We repetitively measured these crosses over the full tracking range of each source. We used the correlator in Stokes mode. Before processing any data we rst corrected with the Mueller matrix described in our previous memo. For each 7-point set we rst t the Gaussian in equation 1, plus a constant, to the Stokes I . The constant is the o -source system temperature and the Gaussian parameters are the source de ection, position error, and beamwidth. Denote this Gaussian by RI , as in equation 1; note that it is an observed quantity with 7 points, and because it is derived from the t to Stokes I it


7 is derived variation. The coe produced with high precision but its intensity scale is inaccurate because of the zenith angle gain We then t the 7 points in the other Stokes parameters with equation 5 plus a constant. cient of the constant term is an o set with units of Kelvins; it is the polarized emission by the losses or re ections in the feed.

3.2. Astronomical source selection
We observed 15 sources with the chcross procedure: 3C33, 3C41, 3C98, 3C138, P0736+01, 3C223, 3C234, 3C274.1, 3C286, P1414+11, 3C336, 3C386, 3C399.1, CTA102, 3C454.3. 3C336 was inadvertently omitted from all of our data reduction. We wanted to include as many sources as possible to increase the az ; z a coverage. However, some are partially resolved in some Stokes parameters which made them unsuitable for deriving some beam parameters; we comment on this in each section below.

Fig. 3.| Stokes I system temperature versus za.


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4. THE BEAM PROPERTIES IN STOKES I 4.1. O -source system temperature in Stokes I
Figure 3 exhibits the o -source Stokes I system temperature versus zenith angle; squares refer to the az direction PAscan =90 and crosses to za PAscan =0 . This gure includes only those sources with ux density 10 Jy to avoid a small nonlinear calibration problem. The crosses exhibit systematically smaller system temperatures. This behavior is mystifying because the crosses are derived from za scans. The za HPBW is larger than the az HPBW, so one would expect the az o -source points to have a smaller residual source contribution and thus a smaller system temperature, contrary to the observations. One should estimate a lower envelope in Figure 3 to avoid contributions from the sources themselves. The Stokes I system temperature decreases monotonically with za, reaching a minimum of about 72 Kelvins and rising to about 80 Kelvins at za =20 . These are twice the system temperature in each polarization channel.

Fig. 4.| Pointing errors in the az squares and za crosses directions versus za.


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4.2. Pointing accuracy
We included all sources except 3C33, even the partially resolved ones, because their centroid positions are well de ned. Figure 4 exhibits the pointing errors versus zenith angle; squares refer to the az direction and crosses to za. The squares might show some za dependence at the level of a few arcsec. For all squares the mean is ,1:4 0:600 and the standard deviation is 5:100 . The crosses are systematically negative, particularly at low za. The mean of all crosses is not representative because it is highly weighted by the higher za's, where the position error becomes less negative; this mean is ,5:100 0:500 and the standard deviation is 5:000 . We emphasize that these standard deviations do not include the contributions from 1 the catalog positions, which are from BDFL 1972, and 2 the observations themselves; they are therefore upper limits. Adding all pointing errors quadratically, including the systematic o sets, yields a standard deviation of 8:900 ; excluding the systematic o sets yields 7:100 . Again, we emphasize that these are upper limits. Our main purpose is not to determine accurate pointing information so we are not inclined to delve into these details. The important statement is the astronomical one: these pointing errors are negligible!

Fig. 5.| HPBWs in the az squares and za crosses directions versus za; solid lines are the ts of equation 7.


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Fig. 6.| HPBWs at 45 to the az ; z a directions; solid lines are the ts of equation 7. These HPBWs are almost equal, showing that the elliptical beam is well-aligned with these directions.

4.3. HPBW
We excluded partially resolved sources, leaving us with the following: 3C41, 3C138, P0736+01, 3C286, 3C336, 3C399.1, CTA102, 3C454.3. Figure 5 exhibits the HPBW versus zenith angle in the cardinal az ; z a directions. Squares refer to the az direction PAscan =90 and crosses to za PAscan =0 . Both have a quadratic-like za dependence, with the az beamwidth decreasing and the za beamwidth increasing at low and high za. Figure 6 exhibits the HPBW versus zenith angle at 45 to the cardinal az ; z a directions; the near-equality of these HPBWs shows that the beam is well-aligned with the cardinal directions. Moreover, the opposite za dependences of the HPBWs in Figure 5 conspire to make these HPBWs more nearly independentof za. We least-square t the data to equation 2b, with each coe cient being a second-degree polynomial in in za10 =za , 10 . We found that jPAbeam j 7 ; although the departure from zero is formally larger than the errors, the incomplete coverage in az ; z a makes this departure questionable so we take PAbeam =0 for simplicity. With this, the HPBW is well represented by the solid lines in Figures 5 and 6 and the following equations:


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HP B W =3:396 0:004 + H za cos2PAscan H za= 0:262 0:009 , 0:067 10 0:019 za10 + 0:331000:03 za2 : 10

7a 7b

The narrowest HPBWza occurs at za =11 , where the HPBWs are 3:1360 ; 3:6550 for az ; z a.

Fig. 7.| Stokes V beam squint in the az squares and za crosses directions versus za; solid lines are the ts of equation 8.

5. THE BEAM PROPERTIES IN STOKES V
Astronomical sources exhibit very little circular polarization and all sources were useful for deriving the Stokes V squint and squash.


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5.1. Squint
Figure 7 exhibits the Stokes V squint versus zenith angle; squares are for the az direction with PAscan = 90 and crosses the za direction with PAscan = 0 . The za squint is roughly independentof za. The az squint is roughly constant for za 16 ; then it increases sharply. This behavior is expected theoretically. First, recall the situation for a prime-focus feed: if the feed points away from the center of symmetry in a particular direction, beam squint is produced in the orthogonal direction. At Arecibo, for high za the dish illumination pattern becomes asymmetric in the za direction, so this should cause a squintinthe az direction.

Fig. 8.| Stokes V beam squint from equation 8 with parameters in the form of equation 6a. We also obtained chcross data at 45 to the az ; z a directions. We least-square t all data to equation 6a actually, to its cos + sin counterpart, with each coe cient being a third-degree polynomial in za10 =za , 10 . The Stokes V squintisshown by the solid lines in Figure 8 and given by units are arcseconds V =cosza cosPAscan + sinza sinPAscan cos =,0:88 0:04 + 0:39 10 0:13 za10 + 0:111000:15 za2 + ,0:11 0:27 za3 10 10 1000 8a 8b


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sin =0:96 0:04 + ,0:3810 0:13 za10 + 0:581000:15 za2 + 1:96 0 10 1000 Figure 8 exhibits this dependence, as parameterized in the form of equation 6a. is typically 1:400 , smaller than for any other telescope ever used for measuring of the 21-cm line Heiles 1996; after correction, its e ects will be unimportant.

:27 za3 10

8c

The beam squint Zeeman splitting

5.2. Squash
There is no statistically signi cant beam squash for Stokes V . However, wewere surprised to nd the Stokes V HPBW to be systematically larger than the Stokes I HPBW by 0:6100 0:1300 , independentof PAscan .

Fig. 9.| Stokes Q beam squintat 45 to the cardinal az ; z a directions versus za. Dashed lines are the ts of equation 9. The see-saw behavior of points for az 12 is an intrinsic e ect of 3C138 see text.


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6. THE BEAM PROPERTIES IN STOKES Q AND U
Beam measurements for Stokes Q and U are greatly complicated by the fact that astronomical sources have not only signi cant linear polarization, but also angular structure in linear polarization. We found many sources to have signi cant intrinsic squint and squash. We went to a great deal of trouble to examine each source individually to determine its suitability as a calibrator. Stokes Q is derived from the di erence between the two linear polarizations. The usual opinion is that it is therefore inaccurate and noisy. However, we found it to be accurate and only somewhat noisier than Stokes U . This bespeaks the accuracy of our Mueller matrix correction and, in addition, the stability of the calibration noise diodes.

Fig. 10.| Stokes U beam squintat 45 to the cardinal az ; z a directions versus za. Dashed lines are the ts of equation 10.

6.1. Squint
We excluded sources that exhibited deleterious intrinsic squint. Some exhibited benign intrinsic squint, meaning that it was recognizable but the data were consistent with the other


15 data. We used sources 3C41, 3C138, P0736+01, 3C286, 3C399.1, CTA102, 3C454.3. Figure 9 exhibits the Stokes Q and U squint versus zenith angle for scans in the cardinal directions; Figure 10 is for the 45 directions. We least-square t the data to equation 6b actually, to its cos + sin counterpart, with each coe cient being a rst-degree polynomial in za10 =za , 10 . The ts are shown by the dashed lines in Figures 9 and 10. For Stokes Q we have units are arcseconds Q =Qcosza cosPAscan +Qsinza sinPAscan Qcos =,0:80 0:09 + 0:87 10 0:16 za10 Qsin =,1:73 0:09 + ,0:3310 0:16 za10 9a 9b 9c

Fig. 11.| Stokes Q beam squint from equation 9 with parameters in the form of equation 6a. and for Stokes U U =U cosza cosPAscan + U sinza sinPAscan 10a


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U cos =1:62 0:08 + 0:33 10 0:14 za10 U sin =,0:30 0:08 + 1:46 10 0:14 za10

10b 10c

Fig. 12.| Stokes U beam squint from equation 10 with parameters in the form of equation 6a. Figures 11 and 12 exhibit this dependence, as parameterized in the form of equation 6a. Theoretically, one doesn't expect beam squint in linear polarization. One might argue that it somehow arises from passage through the telescope support structure; however, if this were the case one would expect it to depend on telescope azimuth and or zenith angle. We found no discernible dependence on either coordinate but we must comment that our az ; z a coverage is insu ciently complete to rule out such dependence. Similarly, one expects an e ect like this from the angular polarization structure of continuum sources. In fact, these e ects are clear for some of our sources; they vary as cosPAsky instead of cos2PAsky , which means that the parallactic angle doesn't change enough to get a reliable t. An example is in gure 9: the see-saw behavior of points for az 12 are 3C138 on opposite sides of transit, which have PAsky di erent by 160 . These intrinsic source e ects contribute to the scatter in the gures, but the instrumental e ect dominates. Thus, the linear polarization beam squint is tied mainly to the receiver, not to the sky or the telescope support structure.


17 It is conceivable that this e ect arises from a di erence in physical location of the phase centers of the two linear probes in the feed. With Arecibo's focal length 500 m, a 100 angle corresponds to a linear distance of only 1.2 mm in the focal plane. It is clear that this would produce a beam squint in Stokes U , whichisthe di erence between the two probes. But it isn't clear to us, at any rate what such a di erence should produce in Stokes parameters Q and V , or for beam squash. We note that the di erence PAQ , PAU varies from 86 ! 118 , whichis fairly close to 90 .

Fig. 13.| Stokes Q beam squash in the az squares and za crosses directions versus za.

6.2. Squash
All but three sources exhibited unacceptably large intrinsic squash in linear polarization. 3C386 was particularly extreme with an intrinsic squash 500 . The preponderance of intrinsic squash is undoubtedly a re ection of the structure of typical radio galaxies: two highly linearly polarized lobes are the very de nition of squash. The only sources we included here are 3C138, P0736+01, 3C286. Figures 13 and 14 exhibit the squash in Stokes Q and U . Figure 13 is for the chcross procedure at 45 with respect to az ; z a, while Figure 14 is for chcross aligned. We pick these


18 to illustrate the four-lobed Stokes parameters. The di shown and goes to zero for the other directions, which of PAscan. symmetry of the squash: the four lobes rotate 45 between these two erence in squash between squares and crosses peaks for the directions the other directions. However, the squash itself doesn't go to zero for means that there is a constant o set in the squash that is independent

Fig. 14.| Stokes U beam squash in the az +45 squares and za +45 crosses directions versus za. We least-squares t data at all PAscan to equation 2b actually, to its cos + sin counterpart, with each coe cient being a constant, independentof za. The ts are shown by the dashed lines in Figures 13 and 14. For Stokes Q wehave units are arcseconds Q =,1:23 0:07 + 3:46 0:13 cos 2 PAscan , 0:3 0:7 U =0:36 0:11 + 2:59 0:13 cos 2 PAscan , 44:9 2:1 11a 11b

The directions are as expected. Consider linear polarization near the wall of circular waveguide or a circular horn. On a radius perpendicular to E- eld, the E- eld is shorted out by the wall, but 90 away it is not. Looking end-on at the circular waveguide the E- eld amplitude is


19 roughly elliptical, with the narrower size in the direction perpendicular to E. This translates into a broader illumination of the dish surface and a narrower HPBW. Thus the wider HPBW lies in the direction parallel to the E- eld, meaning that the Stokes Q HPBW should be wider towards PAscan =0 . That is, the Stokes Q squash should be oriented towards PAscan =0 and the U squash towards PAscan =45 , as observed. One would expect coe cients of the cos2PAscan term to be equal for the Stokes Q and U ; they are not. Moreover, one would not expect the constant terms to be statistically signi cant; they are. The constant terms mean that the Stokes Q beam is systematically larger than the Stokes U beam at all position angles. We found a similar e ect for Stokes V . As we remarked earlier, observation does not always correspond to theory. It is a pleasure to thank Phil Perillat for providing the excellent observing software and critical, stimulating comments; and MikeDavis for general assistance and advice.

REFERENCES
Bridle, A.H., Davis, M.M., Fomalont, E.B., and Lequeux, J. 1972, AJ, 77, 405. Heiles, C. 1996, ApJ, 466, 224. Heiles, C. and Fisher, R. 1999, Green Bank Tech Memo, in preparation.

A This preprintwas prepared with the AAS L TEX macros v4.0.