Characterizing the Accuracy of ALMA Linear-polarization Mosaics

When an astronomical source is not observed at the pointing center, off-axis errors in linear and circular polarization will affect the resulting observations of polarized emission. This is true for all telescopes. In the case of telescopes with on-axis receiver feeds, off-axis errors in the linear polarization appear. In those with receiver feeds whose axes are offset with respect to the axis of the reflector, off-axis errors in both linear and circular polarization become evident (Chu & Turrin 1973); and indeed, the vast majority of telescope with multiple receiver bands fall into this latter category, including, e.g., the Atacama Large Millimeter/submillimeter Array (ALMA) and the Karl G. Jansky Very Large Array (VLA).

In addition to the traditional method of full-polarization holographic imaging (e.g., Harp et al. 2011; Perley 2016; Jagannathan et al. 2017), another method of characterizing the wide-field polarization errors across the primary beam of an interferometer is to use the full array to perform a grid of observations, where all antennas simultaneously observe a polarized point source (e.g., a quasar or blazar) in many offset positions. Here we report results of 11 × 11 observations by ALMA at Bands 3, 5, 6, and 7 (3 mm, 1.6 mm, 1.3 mm, and 0.87 mm, respectively) toward the highly linearly polarized blazar 3C 279.¹³ Since the wide-field polarization response of an antenna manifests itself in the antenna's frame of reference, in all tests the offset pointings were evenly spaced in the azimuth-elevation (Azimuth, Elevation) frame, not in the (RA, DEC) frame like a typical observation.

A primary reason for characterizing the wide-field polarization performance of ALMA antennas is to understand the effect that wide-field polarization errors have in a polarization mosaic, where many pointings are stitched together. Performing wide-field polarization science using an image made from a single ALMA pointing is not advisable because the polarization performance far from the center of the primary beam is sub-optimal due to residual off-axis errors. However, mosaicking an image alleviates this problem, because many pointings are combined, and the on-axis emission (i.e., emission located at or very near the pointing center) in any giving pointing is more heavily weighted than the off-axis emission. On the other hand, in contrast to a non-mosaicked, on-axis, single-pointing observation, a mosaic contains some off-axis emission from one or more adjacent pointings in every location in the image: see, for example, the black shaded region in Figure 1, which indicates the regions of a standard Nyquist mosaic that do not coincide with the inner $\tfrac{1}{3}$ FWHM of any given pointing.¹⁴ This emission from off-axis regions of multiple neighboring pointings could corrupt the final image, and thus here we also characterize the error in a mosaicked image by analyzing linear-polarization observations of an extended, highly linearly polarized source with ALMA at Bands 3 and 6 (3 mm and 1.3 mm, respectively).

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Our target is the Kleinmann–Low Nebula in Orion (Orion-KL), which lies in the OMC-1 region at the center of the "integral-shaped filament," a filamentary cloud with a length of ≳7 pc that lies in the northern portion of the Orion A star-forming cloud (Johnstone & Bally 1999). Our observations of Orion-KL comprise a mosaic centered on the iconic Source I (see, e.g., Plambeck et al. 2009, and references therein), and extending NNE and SSW to cover the filamentary structure, including the "Northern Ridge" (Johnstone & Bally 1999; Hull et al. 2014) that lies ∼25'' to the NE of Source I. Later in this work, we compare our ALMA mosaics with a similar 1.3 mm linear-polarization mosaic of Orion-KL performed by Hull et al. (2014) using the Combined Array for Research in Millimeter-wave Astronomy (CARMA).

Single-dish polarization observations using the POL2 camera on the James Clerk Maxwell Telescope (JCMT) by Pattle et al. (2017) yielded a high-sensitivity image of the magnetic field in Orion A, which is roughly perpendicular to the filament's long axis, as frequently seen in star-forming clouds whose magnetic field is thought to be dynamically important (e.g., Fissel et al. 2016; Planck Collaboration et al. 2016). These JCMT results, along with the results that we present here and in a companion paper (P. C. Cortes et al. 2020, in preparation) are recent contributions to a large body of literature comprising single-dish and interferometric studies of millimeter, submillimeter, and far-infrared polarization toward the Orion-KL region and its environs. The first detection of polarization toward Orion-KL in this wavelength regime was made by a balloon-borne polarimeter (Cudlip et al. 1982); numerous later detections used polarimeters on the Kuiper Airborne Observatory (KAO), the Berkeley-Illinois-Maryland Association (BIMA) array, the JCMT, the Caltech Submillimeter Observatory (CSO), the Submillimeter Array (SMA), CARMA, the Stratospheric Observatory for Infrared Astronomy (SOFIA), and the IRAM 30 m telescope (Hildebrand et al. 1984; Novak et al. 1989; Rao et al. 1998; Schleuning 1998; Plambeck et al. 2003; Girart et al. 2004; Houde et al. 2004; Matthews et al. 2009; Tang et al. 2010; Hull et al. 2014; Ritacco et al. 2017; Chuss et al. 2019). See Pattle et al. (2017) for a more exhaustive list of references.

Below we begin with a description of the observations, calibration, and imaging procedures for the 11 × 11 observations of 3C 279 (Section 2), after which we discuss the results from these tests (Section 3). Similarly, we then present the mosaicked polarization observations toward Orion-KL (Section 4), and the results from those mosaic tests (Section 5); the latter section includes a comparison between ALMA and CARMA observations toward Orion-KL (Section 5.2). We end by offering our conclusions (Section 6).

All of the data we present here, and in a companion paper by P. C. Cortes et al. (2020, in preparation), were collected as part of the ALMA Extension and Optimization of Capabilities (EOC) program. The EOC program is based at the Joint ALMA Observatory (JAO) in Santiago, Chile—with collaboration from scientists both at the ALMA Regional Centers (ARCs) as well as at external institutions—and is focused on using science-quality data to test, verify, and open new observing modes at ALMA.

Our first set of observations comprise 11 × 11 grids of offset pointings toward 3C 279 at Bands 3, 5, 6 and 7, the goal of which is to characterize the off-axis polarization errors across the field of view of single-pointing ALMA observations at several ALMA bands. The 11 × 11 grid pointings are evenly spaced in (Azimuth, Elevation), and extend out to approximately the FWHM of the primary beam at each observing frequency.¹⁵ Figure 2 shows the pattern of the 121 pointings. Each set of observations of 3C 279 includes three separate executions spaced by ∼1 hr, which yields sufficient parallactic angle coverage to allow us to solve for the leakage terms (D-terms) and the cross-hand phase (XY-phase); we discuss the derivation of both of these quantities later in this section. We list the observation dates and data identification codes (or UIDs, each corresponding to an individual ALMA execution/observation) in Table 1. We chose to observe 3C 279 because, at the time of observation in 2015/2016, it was both extremely bright and had an extremely high linear polarization fraction. All observations were taken using the standard correlator setup for continuum polarization observations, which includes four 1.875 GHz-wide spectral windows with 64 channels each. See Table 2 for the on-axis results, and for the observational setup at each band.

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Table 2.  Observational Setup and on-axis Results (11 × 11)

Band	ν	θ	Source	I	Q	U	Pfrac	χ
	(GHz)							(deg)
3	97.479	331 × 249	3C 279	1	0.0404	0.1074	0.115	34.7
5	183.261	103 × 061	3C 279	1	–0.0005	0.1095	0.110	45.1a
6	233.000	115 × 088	3C 279	1	0.0398	0.1142	0.121	35.4
7	343.479	082 × 055	3C 279	1	0.0373	0.1158	0.122	36.1

Note. Results from the 11 × 11 observations, listed in the same order as in Table 1. ν is the average frequency of the observations. θ is the synthesized beam (resolution element) of the images. All fluxes are peak flux values, and are scaled relative to a normalized peak Stokes I flux of 1. The absolute Stokes I flux densities of 3C 279 were ∼14.7 Jy (Band 3, 2015 March), ∼6.8 Jy (Band 5, 2016 November, interpolated using the available Band 3 and Band 7 fluxes from the ALMA Calibrator Source Catalogue:https://almascience.eso.org/sc), ∼7.5 Jy (Band 6, 2015 May), and ∼5.6 Jy (Band 7, 2015 May).

^aThe change in polarization angle of 3C 279 from ∼35° in 2015 March–May (when the Band 3, 6, 7 data were taken) to ∼45° in 2016 November (when the Band 5 data were taken) can be seen in all frequency bands; see http://www.alma.cl/~skameno/AMAPOLA.

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Note that, as no other calibrators besides 3C 279 were observed in the 11 × 11 tests, we perform every step of the reduction process using 3C 279. We first split out the scans where 3C 279 was located at the pointing center (i.e., the central point of the 11 × 11 grid) and concatenate them into a separate data set; we use these on-axis data to derive the on-axis polarization calibration via the current ALMA polarization calibration scheme. For further details about standard, on-axis reduction of ALMA polarization data, see Nagai et al. (2016) as well as the 3C 286 Polarization CASA Guide.¹⁶

We first perform an initial phase-versus-time gain calibration, which we apply on-the-fly when solving for the bandpass solution. After solving for the bandpass, we perform gain calibration (amplitude and phase), assuming an initial unpolarized source model for 3C 279. Since ALMA has crossed-linear feeds, the linear polarization properties of the source manifest themselves in the gain amplitude solutions when an initial, unpolarized model is used. To make a first estimate of the polarization of 3C 279, we use the task qufromgain from almapolhelpers.py. Next, we solve for the cross-hand delay using the scans where the XX/YY polarization ratio is changing the fastest (this can be seen by viewing the gain-amplitude calibration table). We then solve for the XY-phase, which is the phase difference between the X and the Y polarizations of the reference antenna. This solution can have angle ambiguities, which we resolve by using the task xyamb; this task uses both the initial XY-phase solution as well as the previously derived initial model of the source polarization to solve for the final polarization model of 3C 279. With the correct model of 3C 279 in hand,¹⁷ we then re-derive the gains (amplitude and phase) using the model in order to remove the polarization response from the gain amplitudes. Finally, we derive the D-terms. After doing so, we apply all of the calibration tables (the bandpass; the modified gains after removing the source polarization; the cross-hand delay; the XY-phase; and the D-terms) to our on-axis data set.

After applying the standard on-axis calibration to all of the on- and off-axis data, we perform one round of phase-only self-calibration (selfcal) to correct the residual phase errors as a function of time for both the on-axis data and each off-axis pointing. With the exception of the selfcal phase solutions, all calibration tables that we apply to the off-axis data are solved for using on-axis data only—this includes D-terms and XY-phase. We do this to mimic the standard calibration method of ALMA polarization data.

After completing the data reduction process, we produce I, Q, U, V images using the task clean from CASA version 5.6.1 with standard imaging parameters, including a Briggs visibility weighting of robust = 2.0 (i.e., natural weighting). After making the images, we extract the I, Q, U, V peak values using the CASA task imfit; these are the values that we analyze and report in Section 3. Finally, we use these Stokes maps to produce maps of the linearly polarized intensity P, linear polarization fraction ${P}_{\mathrm{frac}}$, and linear polarization angle χ:

$$\begin{eqnarray}&&P=\sqrt{{Q}^{2}+{U}^{2}}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{P}_{\mathrm{frac}}=\displaystyle \frac{P}{I}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\chi =\displaystyle \frac{1}{2}\,\arctan 2\,\left(\displaystyle \frac{U}{Q}\right).\end{eqnarray} \tag{ 3 }$$

The dynamic range (peak flux value divided by the rms noise value) in the Stokes I continuum maps ranges from ∼10,000 to 100,000, depending on the observing frequency (lowest dynamic range at Band 7, highest at Band 3).¹⁸ The dynamic range in the polarized intensity (P) maps is always a factor of ∼10 lower, since 3C 279 had a polarization fraction of ∼10% in all of our observations (see Table 2). Nevertheless, the dynamic range of our P maps is always >1000; in a region where the dynamic range of P is 1000, the statistical error of the polarization angle (${\sigma }_{\chi }=0.5\,{\sigma }_{P}/P$) is ∼003, which is much smaller than the on-axis uncertainty of ∼04 in the polarization angle calculated by Nagai et al. (2016) (Nagai et al. note that the exact value depends on the number of antennas used in each observation). Therefore, changes in the image properties of the I, Q, and U Stokes parameters as a function of offset pointing are not caused by sensitivity limits, but rather by systematic polarization variations across the primary beam.

Since the time when these data were obtained 4–5 yr ago, there have been no major changes in the receiver optics. The sub-reflectors in each antenna also remain un-tilted. A number of ALMA 12 m antennas were found to be astigmatic, which requires a corrective scheme to be applied to the surface of the dishes; however, as of the time of publication, this scheme has not yet been implemented. If the antenna astigmatism introduces wide-field polarization errors, then the errors reported in this work should improve once the correction scheme is applied. In the future, ALMA is planning new receivers, which will potentially change the optics (e.g., additional external mirrors) and will also use the tilting capabilities of the sub-reflector to improve the alignment with the receiver feeds. These changes will most likely improve the errors that we report here, but future tests will be needed to confirm this.

3.1. Error in Q, U, ${P}_{\mathrm{frac}}$, and χ

In Figure 3 (Band 3), and in Appendix A (Bands 5, 6, and 7; see Figures A.1, A.2, and A.3, respectively), we show the errors across the primary beam in Stokes Q, Stokes U, χ, and ${P}_{\mathrm{frac}}$ that we derived from the observations toward the highly linearly polarized blazar 3C 279, which had ${P}_{\mathrm{frac}}\approx 12 \%$ at the time of observation. Assuming a circular, azimuthally symmetric primary beam in Stokes I, we can remove the primary beam response (i.e., the sensitivity fall-off with radius) in the Stokes Q and U images by dividing the off-axis Stokes Q_off and U_off values by the value of I_off in the corresponding pixels:

$$\begin{eqnarray}&&{Q}_{\mathrm{off},\mathrm{norm}}={Q}_{\mathrm{off}}/{I}_{\mathrm{off}}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{U}_{\mathrm{off},\mathrm{norm}}={U}_{\mathrm{off}}/{I}_{\mathrm{off}}.\end{eqnarray} \tag{ 5 }$$

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We then subtract the on-axis Stokes Q_on or U_on value from the normalized value in each pixel, leaving the residual errors δQ and δU:

$$\begin{eqnarray}&&\delta Q={Q}_{\mathrm{off},\mathrm{norm}}-{Q}_{\mathrm{on}}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\delta U={U}_{\mathrm{off},\mathrm{norm}}-{U}_{\mathrm{on}}.\end{eqnarray} \tag{ 7 }$$

We define the off-axis ${P}_{\mathrm{frac},\mathrm{off}}$ to be

$$\begin{eqnarray}&&{P}_{\mathrm{frac},\mathrm{off}}=\displaystyle \frac{\sqrt{{Q}_{\mathrm{off}}^{2}+{U}_{\mathrm{off}}^{2}}}{{I}_{\mathrm{off}}}.\end{eqnarray} \tag{ 8 }$$

As all of our Stokes I maps are normalized to I_on = 1, this simplifies the equations for P_frac:

$$\begin{eqnarray}&&{P}_{\mathrm{frac},\mathrm{on}}=\sqrt{{Q}_{\mathrm{on}}^{2}+{U}_{\mathrm{on}}^{2}}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{P}_{\mathrm{frac},\mathrm{off}}=\sqrt{{Q}_{\mathrm{off},\mathrm{norm}}^{2}+{U}_{\mathrm{off},\mathrm{norm}}^{2}}.\end{eqnarray} \tag{ 10 }$$

Finally, we calculate the error in the position angle δχ and in the polarization fraction δP_frac as the differences in the on- versus off-axis pointings:

$$\begin{eqnarray}&&\delta \chi ={\chi }_{\mathrm{off}}-{\chi }_{\mathrm{on}}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\delta {P}_{\mathrm{frac}}={P}_{\mathrm{frac},\mathrm{off}}-{P}_{\mathrm{frac},\mathrm{on}}.\end{eqnarray} \tag{ 12 }$$

In the error maps of χ and P_frac, we subtract the central value from each pixel.

After analyzing the observations of the highly linearly polarized blazar 3C 279, we find that the systematic errors in the polarization fraction δP_frac and in the position angle δχ are similar for all observing frequencies. Within the inner $\tfrac{1}{3}$ FWHM, the residual errors in P_frac and χ are ≲0.001 (≲0.1% of Stokes I) and ≲ 1°, respectively. Near the FWHM, the errors increase to ∼0.003–0.005 (∼0.3%–0.5% of Stokes I) and ∼1–5°, respectively, as shown in Figures 3, 10, 11, and 12.

3.2. V Beam Shape (Beam Squint)

Each Stokes V map exhibits a double-lobed "beam squint" pattern at the 1%–2% level of the on-axis Stokes I value: see Figure 4 (note that the Stokes V values have not been primary-beam corrected). Squint arises when a receiver is not aligned with the telescope's optical axis (Chu & Turrin 1973; Adatia & Rudge 1975; Rudge & Adatia 1978).¹⁹ This is the case for all ALMA receivers, which are installed in a single dewar along several concentric circles, all of which are offset from the optical axis (Lamb et al. 2001). When dual-polarization receivers are offset, the response to left- and right-circular polarization (LCP and RCP, respectively) are slightly displaced from one another (this is true regardless of whether the receivers are crossed-linear like those at ALMA, or circular like, e.g., the 1.3 mm receivers at CARMA). Since Stokes V ≡ RCP—LCP (IEEE 1997), this offset in the circular-polarization response of the two separate receivers results in a double-lobed Stokes V pattern.

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It is essential to correct for the known squint profile before attempting wide-field circular polarization observations (either mosaics or single pointings) with ALMA. This is particularly true because unlike the more extended "squash" error pattern in the linearly polarized Q and U maps (see Section 3.3), the squint profile is compact, manifesting itself well within the FWHM at all bands. Direction-dependent errors such as squint can be removed from VLA observations using the awproject keyword in the CASA task tclean (Bhatnagar et al. 2008); however, this has not yet been implemented for ALMA observations. Efforts to implement full-polarization voltage-pattern corrections (i.e., primary beam models) at ALMA are underway (S. Bhatnagar et al., in preparation), and will allow us in the future to calibrate out the wide-field polarization errors that we analyze in this paper.

3.3. Q and U Errors (Beam Squash)

We observed Orion-KL using the 12 m array at Bands 3 and 6. The data were taken in the standard sessions mode used for ALMA polarization observations (Remijan et al. 2019), and had the standard correlator setup for continuum polarization observations, which includes four 1.875 GHz-wide spectral windows with 64 channels each. The Band 3 and Band 6 sessions comprised three and two executions, respectively. We list the details in Table 3.

Table 3.  Observational Details (Orion-KL)

Band	ν	Obs. date	Source	Config.	θ	Nant	UID
	(GHz)	(UTC)
3	97.479	2018 Mar 17	Orion-KL	C-4	$1\buildrel{\prime\prime}\over{.} 43\times 0\buildrel{\prime\prime}\over{.} 95$	39	A002_Xca795f_X50f
		2018 Mar 18					A002_Xca795f_Xcbd
		2018 Mar 18					A002_Xca795f_Xd0
6	233.000	2019 Apr 12	Orion-KL	C-3 → C-4	065 × 048	43	A002_Xdab261_X13a0a
		2019 Apr 12					A002_Xdab261_X1448a

Note. Observations of Orion-KL. ν is the average frequency of the observations. "Config." is the ALMA antenna configuration during which the observations were performed. θ is the synthesized beam (resolution element) of the images. N_ant is the number of antennas in the observation. The UIDs each refer to an individual execution/observation. Note that all executions, per band, were consecutive and were performed as part of a session.

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We calibrate the data using the standard procedures for processing ALMA polarization observations (see, e.g., Cortes et al. 2016; Nagai et al. 2016; Hull et al. 2017, 2018). We use J0522–3627 as the polarization calibrator, J0423–0120 as the bandpass calibrator, and J0529–0519 as the gain calibrator. The same calibration sources were used in both the Band 3 and 6 observations. A bandpass scan was performed only in the first execution of the observing session. We derive the bandpass solutions and the flux scaling from both the bandpass calibrator (J0423–0120) and the polarization calibrator (J0522–3627), whereas we correct the amplitude and phase by deriving complex gains from the gain calibrator (J0529–0519). The polarization calibrator (J0522–3627) was observed every  ∼35 minutes, and had a flux of 8.2 Jy and P_frac ≈ 3.4% during the Band 3 observations, and a flux of 3.8 Jy and P_frac ≈1.7% during the Band 6 observations.

The mosaicked observations of Orion-KL were set up to allow us to compare different mosaic patterns, which we label according to their different densities of pointings: Hyper-Nyquist (separation between the pointings of ∼1/4 of the FWHM; i.e., over-sampled by a factor of ∼2 in each dimension relative to a standard Nyquist-sampled mosaic), Super-Nyquist (separation of ∼1/3 of the FWHM; i.e., over-sampled by a factor of ∼1.5), Nyquist (separation of ∼1/2 of the FWHM; i.e., standard sampling), and a single field (the reference pointing, centered on Source I itself).

The primary beam of each mosaic pointing is assumed (in CASA) to be a 2D Gaussian peaked at the center of the pointing. Figure 5 shows the full 3 × 5 pointing rectangular mosaic pattern of our observations (i.e., the Hyper-Nyquist pattern) at Band 3, and the similar 5 × 9 pattern at Band 6. In the case of Band 3, the field of view of the single pointing includes most of the filament, which allows us to compare the single pointing with the various ALMA mosaics. In contrast, the ALMA Band 6 field of view is too small to allow us to comparison the single pointing and the mosaics.

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We image the data using the task tclean from CASA version 5.4.0, using standard imaging parameters and a Briggs visibility weighting of robust = 0.5, resulting in maps with synthesized beams (resolution elements) of 143 × 095 at Band 3 and 065 × 048 at Band 6. We independently image all four cases (i.e., Hyper-Nyquist, Super-Nyquist, Nyquist, and single field) in order to compare them with one another. We perform the imaging of all of these cases by specifying the fields that belong to each of the mosaic sampling patterns (see Table 4). For Band 3, all of the mosaicking patterns include the center field. In the Band 6 data, however, the Nyquist and Super-Nyquist mosaics do not include the center field. Consequently, we re-grid these latter mosaicked images to the Hyper-Nyquist frame using the CASA task imregrid in order to match the coordinates.

For each of the four patterns, we clean the Stokes I, Q, and U maps separately. Note that we do not perform self-calibration on any of the images of Orion-KL. Furthermore, note that spectral lines were not flagged prior to making the continuum images (the default Band 6 spectral setup avoids major lines of interest). We primary-beam correct the final images using the CASA task impbcor to the 20% sensitivity level of the Stokes I primary beam model. We then use these Stokes maps to produce maps of P, P_frac, and χ. The mosaicked images of Orion-KL at both Bands 3 and 6 have median levels of P_frac ≈ 6%–7% across the maps. While the signal-to-noise ratio (S/N) of the P images can be >30 at the peaks of polarized emission, the median S/N values across the P maps at Bands 3 and 6 are between 7–8.

In order to assess the performance of the ALMA's linear-polarization mosaicking mode in a typical observational scenario, we performed standard (R.A., Decl.) observations of the Orion-KL star-forming region at Bands 3 and 6, and compare the images made using a single pointing versus the three different mosaic patterns (Hyper-Nyquist, Super-Nyquist, and Nyquist). Table 5 lists the peak fluxes and rms noise levels in the I, Q, U, and P maps for both bands and all sampling patterns.

Table 5.  Image Statistics from Orion-KL Observations

Pattern	Band	Ipeak	Irms	Qpeak	Qrms	Upeak	Urms	Ppeak	Prms
		$\left(\tfrac{\mathrm{mJy}}{\mathrm{beam}}\right)$	$\left(\tfrac{\mathrm{mJy}}{\mathrm{beam}}\right)$	$\left(\tfrac{\mathrm{mJy}}{\mathrm{beam}}\right)$	$\left(\tfrac{\mathrm{mJy}}{\mathrm{beam}}\right)$	$\left(\tfrac{\mathrm{mJy}}{\mathrm{beam}}\right)$	$\left(\tfrac{\mathrm{mJy}}{\mathrm{beam}}\right)$	$\left(\tfrac{\mathrm{mJy}}{\mathrm{beam}}\right)$	$\left(\tfrac{\mathrm{mJy}}{\mathrm{beam}}\right)$
Hyper-Nyquist	3	112	0.90	–3.39	0.100	3.73	0.102	3.79	0.157
Super-Nyquist	3	112	0.93	–3.40	0.102	3.72	0.107	3.78	0.164
Nyquist	3	113	0.96	–3.47	0.118	3.75	0.120	3.81	0.180
Single field	3	115	1.31	–3.46	0.150	3.34	0.139	3.35	0.243
Hyper-Nyquist	6	415	2.08	11.3	0.522	12.3	0.495	13.5	0.485
Super-Nyquist	6	418	2.11	11.4	0.535	12.1	0.508	13.5	0.519
Nyquist	6	418	2.40	11.4	0.556	12.2	0.524	13.6	0.755

Note. The peak intensities and the rms noise levels in the I, Q, U, and P maps derived from the mosaicked and single-field (for Band 3) maps of Orion-KL.

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We first focus our analysis on comparing mosaicked images with the inner $\tfrac{1}{3}$ FWHM region of the single pointings, where the effects of residual off-axis errors are minimal. For both the Band 3 and Band 6 data, we analyze several individual pointings that are common to all three of the mosaics (see Table 4) and that show emission in I, Q, and U. We first produce difference maps (i.e., Hyper-Nyquist minus single pointing, Super-Nyquist minus single pointing, and Nyquist minus single pointing) for I, Q, and U using the CASA task immath. We perform our comparison both with and without a cutoff, where the cutoff limits the difference images to the region(s) where the emission in the single pointing images is 5× the rms noise level. We then use the CASA task imsubimage to extract the inner $\tfrac{1}{3}$ FWHM regions from the difference maps in order to quantify the impact of overlapping pointings in the mosaics, under the initial assumption that by stitching together pointings to make a mosaic, the emission in the mosaic should be "polluted" by the off-axis contributions from neighboring pointings (however, as we see later, we are not actually able to detect the effects of the off-axis pointings in the on-axis data).

We do not show the distributions of the differences in the Stokes I, Q, and U maps here because all three difference histograms (for example, the three histograms of the differences in Stokes Q between the chosen single pointing versus the Hyper-Nyquist, Super-Nyquist, and Nyquist mosaics) have similar structure, and have standard deviations that tend to be well below the rms noise level in the single-pointing images. This suggests that we can consider all of the mosaicked images to be the same when compared with the single pointing. Note also that the maps are dynamic range limited; in no case have we reached the thermal noise limit.²¹ These findings allow us to draw two main conclusions: first, the errors in the inner $\tfrac{1}{3}$ FWHM of the Q and U (and also I) maps are primarily caused by imaging artifacts stemming from the inability of our ALMA observations to recover emission at large spatial scales in Orion-KL, rather than by residual off-axis polarization errors; if the latter were dominant, we should expect to see a change in the width of the Q and U difference histograms with increased mosaic packing, since increased packing should reduce the contribution from off-axis errors. Second, while packing the pointings in the mosaics more closely should in theory improve the accuracy of the polarization images, we are not able to detect these incremental improvements among the Nyquist, Super-Nyquist, and Hyper-Nyquist mosaics. Thus, based on our dynamic-range-limited Orion-KL images (and for other sources with similarly complex, multi-scale structure), we cannot recommend using a mosaic packing that is tighter than the standard Nyquist pattern.

When analyzing the Orion-KL results, we choose not to analyze the fractional polarization errors, as the fractional polarization is equal to $\sqrt{{Q}^{2}+{U}^{2}}/I;$ with respect to Stokes I, Stokes Q and U have different physical origins and thus different spatial structure (including different distributions of power as a function of spatial scale) when the source is resolved (for example, see the polarized intensity P versus Stokes I emission in high-resolution ALMA polarization maps in, e.g., Maury et al. 2018; Cortes et al. 2019; Hull et al. 2020; Le Gouellec et al. 2019). Finally, the Stokes Q and U maps have very different dynamic range limitations compared with the Stokes I maps. Consequently, characterization of polarization fraction errors is better performed with point sources, as we have done in the 11 × 11 observations of 3C 279 (see Section 3.1).

We thus choose only to use the polarization position angle χ to characterize the effects of residual off-axis errors in the Orion-KL mosaics. Below we analyze χ only for the lower-resolution Band 3 data, because of the simplicity afforded by the larger field of view relative to Band 6, and because the large-scale structure in the Band 3 data is significantly less spatially filtered than in the higher resolution Band 6 data, resulting in lower-level imaging artifacts.²²

5.1. Comparison of χ in the Mosaics versus the Inner Region ($r\,\lt \,\tfrac{1}{3}$ FWHM) and Outer "Donut" Region ($\tfrac{1}{3}$ FWHM  < r < FWHM) of Single Pointings

We compute difference maps of the polarization position angle χ in the Hyper-Nyquist mosaic versus single-pointing images in the same manner as described above. The three single pointings we use are those in the Band 3 Nyquist mosaic; the setup of the test can be seen in Figure 6. We use a conservative (5 σ_P) cutoff in the χ maps in order to avoid considering any spurious points at the edge of the maps in our statistics.

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The histograms of the χ differences within the inner $\tfrac{1}{3}$ FWHM of the single pointings can be seen in Figure 7. They all have standard deviations of <13, revealing that at the ∼1° level, the regions of the mosaic corresponding to the inner $\tfrac{1}{3}$ FWHM region of a given pointing are not, in fact, "polluted" by the off-axis regions of the neighboring pointings, as we had initially assumed.

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In Figure 7 we also show the histograms of the χ differences in the outer "donut" regions ($\tfrac{1}{3}$ FWHM  < r < FWHM). It is clear that in the two outer pointings (point 6 and point 10), the histograms of the differences in χ in the "donut" regions are significantly wider than in the inner $\tfrac{1}{3}$ FWHM regions, having standard deviations of 36 (point 6) and 27 (point 10). This suggests that we can detect the effects of residual off-axis errors in these single-pointing maps, consistent with what was seen in the 11 × 11 maps.

Mosaicking reduces these off-axis errors as a result of overlapping multiple pointings. A visually simple demonstration of this error-canceling effect can be seen by looking at Figure 6. In all three pointings, the median differences in χ in the high-S/N patches of polarized emission (see the thick contours in Figure 6, which inducate regions where P has an S/N > 15, corresponding to a statistical uncertainty of ∼2°)—when observed by the Hyper-Nyquist mosaic versus when observed on-axis in the single pointings—are always <1°. However, for example, when the patch of emission near the center of point 6 is observed near the FWHM of point 8 (i.e., when the emission is observed off-axis), the median difference with respect to the Hyper-Nyquist mosaic is ∼3°, as expected from the Band 3 11 × 11 tests (see Figure 3). This same effect is also seen when the other patches of high-S/N emission near the centers of points 8 and 10 are observed off-axis: the median position angle difference with respect to the Hyper-Nyquist mosaic when the emission is observed off-axis in the single pointings is always larger than the difference when the same emission is observed on-axis. These simple tests demonstrate again that mosaicking the image reduces the off-axis polarization angle errors.

Note that, in an effort to connect these Orion-KL error results with the errors we derived from the 11 × 11 blazar observations (see Section 3.1), we perform a statistical analysis of the 11 × 11 data. This analysis, which is an attempt to estimate the errors in a mosaic from the 11 × 11 data, can be found in Appendix B.

5.2. Comparison with CARMA

Another mosaicked polarization observation of Orion-KL was performed by Hull et al. (2014) using the CARMA interferometer's full-polarization system (Hull & Plambeck 2015). Here we compare those archival data from an independent instrument with our ALMA results in order to characterize the performance of ALMA linear-polarization mosaics. The 1.3 mm CARMA data were taken in 2013, have a resolution of 307  ×  245, and comprise a seven-pointing mosaic with a field of view similar to that of our ALMA observations. In order to perform the most accurate possible comparison between the CARMA and ALMA data sets, we match the uv-range (using the uvrange keyword in the CASA task clean) and the synthesized-beam size (using the CASA task imsmooth) of the Band 3 and Band 6 ALMA data to those of CARMA. We also re-grid the ALMA images to match the coordinates of the CARMA maps. We use the standard Nyquist mosaic pattern from our ALMA data for the comparison with the CARMA observations, which are also Nyquist sampled.

See Figure 8 for an overlay of the inferred magnetic field from the CARMA data with both the Band 3 and Band 6 ALMA data. We can see that the magnetic field orientation in both the Band 3 and the Band 6 ALMA maps matches the CARMA observations remarkably well, especially in the Northern Ridge. This consistency of inferred magnetic field orientation as a function of frequency is expected in optically thin material (which, we should note, might not be the case toward the very central region near Source I), under the standard assumption that dust-grain alignment via radiative alignment torques (RATs; Lazarian 2007) is the cause of the dust polarization.

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In Figure 9 we show both a map and a histogram of the differences in the polarization position angle χ in the ALMA versus the CARMA data. The distribution of differences peaks around 0°, with a width (defined as the minimum window necessary to encompass 68% of the data) of ±12°. The differences in χ can be as large as ±40° in some locations, and the distribution is somewhat skewed to negative differences, but there is no obvious systematic trend in angle differences in the CARMA versus ALMA maps. The ALMA polarization position angles have a systematic error of <1° on-axis (Nagai et al. 2016). The CARMA polarization position angle measurements have a systematic on-axis error of ±3°, which is the result of systematic errors in the XY-phase correction (including variations in the absolute position angle of the wire grids used to derive the XY-phase passbands; Hull & Plambeck 2015).

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We characterize the wide-field errors in interferometric observations performed using the ALMA array of 12 m antennas. We first measure the errors across the field of view of a single-pointing image by observing the highly linearly polarized blazar 3C 279 in 11 × 11 grids out to the FWHM of primary beam in the (Azimuth, Elevation) frame. Next, in order to characterize ALMA's polarization mosaic performance during a standard-mode observation in the (R.A., Decl.) frame, we performed mosaicked linear-polarization observations at Bands 3 and 6 of the Orion-KL star-forming region using several different mosaic patterns.

These are the main conclusions we draw from the 11 × 11 observations of the highly linearly polarized blazar 3C 279:

• 1.  
  After on-axis calibration of all off-axis pointings, we find that:

  • (a)  
    The systematic errors in polarization fraction (${P}_{\mathrm{frac},\mathrm{err}}\,={P}_{\mathrm{frac},\mathrm{on}}-{P}_{\mathrm{frac},\mathrm{off}}$) for all observed bands (3, 5, 6, 7) are ≲0.001 (≲0.1% of Stokes I) near the beam center, and increase to ∼0.003–0.005 (∼0.3%–0.5% of Stokes I) near the FWHM of the primary beam.
  • (b)  
    The systematic errors in the polarization position angle (${\chi }_{\mathrm{err}}={\chi }_{\mathrm{on}}-{\chi }_{\mathrm{off}}$) are ≲1° near the beam center, and ∼1–5° near the FWHM.
• 2.  
  In all bands, we see the expected double-lobed "beam squint" patterns in the Stokes V maps at the 1%–2% level of Stokes I.
• 3.  
  The off-axis errors in the χ and P_frac maps arise from differences in the shapes of the Q and U beams. Also known as "beam squash," this effect causes cloverleaf patterns of positive and negative lobes in the Q and U error maps. The quadrupolar squash error patterns are approximately twice the linear extent of the double-lobed beam squint patterns, and thus our 11 × 11 observations only sample the inner part of the squash error patterns. Next, we analyze linear-polarization mosaics of the Orion-KL star-forming region observed using several patterns: a single-pointing observation, a standard Nyquist-sampled mosaic, and two more densely packed ("Super-Nyquist" and "Hyper-Nyquist") mosaics. While we conclude that mosaicking improves the accuracy of the polarization images relative to the single pointing images, we are not able to detect incremental improvements in the accuracy among the three mosaics. Thus, based on our Orion-KL images (and for other sources with similarly complex, multi-scale structure), we cannot recommend using a mosaic packing that is tighter than the standard Nyquist pattern. Finally, we compare the ALMA results with archival CARMA data. Our conclusions from these efforts are as follows:
• 4.  
  We compare the Band 3 and Band 6 mosaicked images of Orion-KL with the inner $\tfrac{1}{3}$ FWHM region of several single pointings. We find that the errors in the inner $\tfrac{1}{3}$ FWHM of the Q and U (and also I) maps are primarily caused by imaging artifacts stemming from the inability of our ALMA observations to recover emission at large spatial scales in Orion-KL, rather than by residual off-axis polarization errors. Furthermore, the different spatial structure of the I versus Q and U emission makes it difficult to perform a meaningful analysis of the differences in maps of P_frac. Characterization of errors in P_frac is thus better performed with point sources, as we have done in the 11 × 11 observations of 3C 279.
• 5.  
  The polarization position angle χ is a better quantity for characterizing off-axis errors, as it is independent of I. The difference maps of χ in the (Band 3 only) Hyper-Nyquist mosaic versus the inner $\tfrac{1}{3}$ FWHM regions of three Band 3 single pointings show that the differences are ∼1°, indicating that, at the ∼1° level, the inner regions of a given pointing are not "polluted" by the errors in the off-axis regions of neighboring pointings.
• 6.  
  Next we compare the (Band 3 only) Hyper-Nyquist mosaic with the outer ($\tfrac{1}{3}$ FWHM < r < FWHM) "donut" region of the same three Band 3 single pointings. We find that, in two of the three pointings, the distributions of the differences between χ in the outer "donut" regions of the single pointings versus the mosaic are significantly wider than the differences in the inner $\tfrac{1}{3}$ FWHM of the single pointings versus the mosaic. This suggests that we can detect the effects of off-axis errors in the single-pointing maps, consistent with what was seen in the 11 × 11 tests.
• 7.  
  We perform a simple test using the Band 3 Orion-KL data where we analyze several patches of high-S/N polarized emission. In all three pointings, the median differences in χ in the high-S/N patches of polarized emission—when observed by the Hyper-Nyquist mosaic versus when observed on-axis in the single pointings—are always <1°. However, when the same patches of emission are viewed off-axis in adjacent single pointings, the median position angle differences with respect to the Hyper-Nyquist mosaic are always larger (with typical median values of 2°–4°) than when the same emission is observed on-axis. These larger differences are consistent with the Band 3 11 × 11 tests. These tests demonstrate again that mosaicking reduces off-axis errors.
• 8.  
  Finally, we perform a comparison of both the 3 mm and 1.3 mm ALMA mosaics with a 1.3 mm CARMA mosaic from Hull et al. (2014). After matching the uv-coverage of the ALMA and CARMA data and then smoothing the ALMA data to the coarser CARMA resolution, we find good consistency in the polarization position angles in both bands, especially in the Northern Ridge region of Orion-KL. The distribution of differences in χ in the CARMA versus the ALMA maps has a width of ±12°.

Our results show that, due to the large angular extent of the beam-squash pattern of the ALMA 12 m antennas, the errors introduced by this wide-field effect are modest. We show that mosaicking reduces the magnitudes of these errors because multiple pointings are overlapped, and because the mosaic-imaging algorithm puts a higher weight on emission that is on-axis versus emission that is far from the pointing center. However, ultimately we would like to be able to calibrate out these effects, thus yielding accurate wide-field polarization images even in single-pointing observations. Efforts to implement full-polarization voltage-pattern corrections (i.e., primary beam models) at ALMA are underway, and will allow us to achieve this goal in the future.

The authors thank the anonymous referee for the positive commentary and helpful suggestions. The authors thank Anita Richards for the useful commentary on the manuscript, and for the help preparing the data release. C.L.H.H. acknowledges Sanjay Bhatnagar and Preshanth Jagannathan for the thoughtful discussion. C.L.H.H., P.C.C., S.K., E.B.F., and C.L.B. acknowledge the support of the ALMA Development Program, which funded the Cycle 5 ALMA Development Study, "Full-Mueller Mosaic Imaging with ALMA" (PI: S. Bhatnagar). At the inception of this project, C.L.H.H. was a Jansky Fellow of the National Radio Astronomy Observatory, which is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. C.L.H.H. acknowledges the support of both the NAOJ Fellowship as well as JSPS KAKENHI grants 18K13586 and 20K14527. V.J.M.L.G. acknowledges the support of the ESO Studentship Program. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2011.0.00007.E and ADS/JAO.ALMA#2011.0.00008.E (Orion-KL mosaics), and ADS/JAO.ALMA#2011.0.00009.E (11 × 11 data). ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.

Facilities: ALMA.

Software: APLpy, an open-source plotting package for Python hosted at http://aplpy.github.com (Robitaille & Bressert 2012). CASA (McMullin et al. 2007). Astropy (Astropy Collaboration et al. 2018).

All of the reduction and imaging scripts used to produce the results from this paper are publicly available. They can be found, along with the raw data, by visiting the ALMA Science Archive and searching for the ALMA project codes listed in the acknowledgements section.

We perform a simple statistical analysis based on the the 11 × 11 observations of the highly linearly polarized blazar 3C 279 (see Sections 2 and 3) in order to use those data to estimate the errors in a polarization mosaic. These estimates are an effort to connect the 11 × 11 results with the error results from the Orion-KL mosaic data that we report in Section 5.

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In single-pointing observations, the image can be divided into the inner region (which we define as $r\,\lt \,\tfrac{1}{3}$ FWHM) and the outer "donut" region ($r\gt \tfrac{1}{3}$ FWHM). The initial ALMA polarization capabilities only allowed observations of objects whose emission falls in the inner region, as polarization errors are known to increase further off-axis, as shown in this paper. However, in a mosaicked observation, neighboring pointings—each of which has varying polarization errors across its respective field of view—are stitched together, and thus the errors combine differently in different parts of the mosaic. While much of a standard Nyquist-mosaicked image will fall within the $\tfrac{1}{3}$ FWHM of a given pointing, there will still be a significant fraction of the image that will fall inside of the FWHM but outside of the $\tfrac{1}{3}$ FWHM region of any given pointing: see the filled black region in Figure 1. In many cases, the wide-field polarization errors in multiple pointings will cancel out, thus reducing the overall error in the mosaicked map; however, the cancellation depends on a number of different factors, including the error patterns in the Stokes Q and U maps, the timing and duration of the observations, the chosen mosaic packing pattern, and the orientation of the source on the sky.

In Section 3 we characterize the average polarized response of the primary beam of the 12 m array at Bands 3, 5, 6, and 7 by observing a blazar in an 11 × 11 grid in the Az/El frame. We derive error maps in Q and U as well as in χ and P_frac. Here we use those results to estimate the average error in the regions of the mosaic that are composed of overlapping outer regions of neighboring pointings. In order to be conservative, in the analysis below we consider all of the data lying in the region $\tfrac{1}{3}$ FWHM < r < FWHM. However, in practice, the data near the FWHM of a given pointing will be substantially down-weighted relative to the data from neighboring pointings, which will be closer to on-axis, since the center of each pointing in a Nyquist mosaic is located at the FWHM of its neighboring pointings.

In order to estimate the error in the regions of a mosaic where multiple pointings overlap, we first use standard error propagation of the values in the maps of δQ and δU. These values, σ_Q and σ_U, are calculated simply as the standard deviation of the values of δQ and δU in the ring that lies outside of the $\tfrac{1}{3}$ FWHM, but inside the FWHM of a given pointing (see Figure B.3 for a diagram of the chosen pointings). These σ values correspond to estimators, in a statistical sense, of the "typical" errors in the off-axis Q and U values. We then average σ_Q and σ_U to estimate the systematic error σ_P in the polarized intensity P; this is justified since, almost always, σ_Q ≈ σ_U (see Table 5). Thus, the error in the polarization fraction σ_Pfrac can be approximated, using error propagation and the fact that σ_Q ≈ σ_U ≡ σ_P, as:

$$\begin{eqnarray}&&{\sigma }_{\mathrm{Pfrac}}={P}_{\mathrm{frac}}\sqrt{{\left(\displaystyle \frac{{\sigma }_{P}}{P}\right)}^{2}+{\left(\displaystyle \frac{{\sigma }_{I}}{I}\right)}^{2}}.\end{eqnarray} \tag{ B1 }$$

Given that I is large relative to P and assuming a circular, azimuthally symmetric primary beam (and thus no position-dependent systematic error ${\sigma }_{I}$), we set the second term to zero. This simplifies the equation to:

$$\begin{eqnarray}&&{\sigma }_{\mathrm{Pfrac}}={P}_{\mathrm{frac}}\displaystyle \frac{{\sigma }_{P}}{P},\end{eqnarray} \tag{ B2 }$$

which is identically equal to the error in the polarized intensity σ_P, since ${P}_{\mathrm{frac}}=P/I$ and we define I to be 1.

The error in the polarization angle χ is:

$$\begin{eqnarray}&&{\sigma }_{\chi }=\displaystyle \frac{0.5}{{P}^{2}}\sqrt{{\left(Q{\sigma }_{U}\right)}^{2}+{\left(U{\sigma }_{Q}\right)}^{2}}.\end{eqnarray} \tag{ B3 }$$

By again assuming that ${\sigma }_{Q}\approx {\sigma }_{U}\equiv {\sigma }_{P}$, we can simplify the equation to:

$$\begin{eqnarray}&&{\sigma }_{\chi }=0.5\displaystyle \frac{{\sigma }_{P}}{P}.\end{eqnarray} \tag{ B4 }$$

Using these equations, we can estimate the typical error in P_frac and χ in the areas indicated in Figure 13. See Table 6 for these values, which have superscript ^a.

Table 6.  Errors in 11 × 11 Maps Toward Highly Linearly Polarized 3C 279

Band	σQ	σU	σP	${\sigma }_{\mathrm{Pfrac}}$ a	${\sigma }_{\chi }$ a	${\sigma }_{\mathrm{Pfrac}}$ b	${\sigma }_{\chi }$ b
					(deg)		(deg)
3	0.0040	0.0032	0.0036	0.0036	0.90	0.0025	1.1
5	0.0020	0.0013	0.0016	0.0016	0.43	0.0013	0.52
6	0.0016	0.0012	0.0014	0.0014	0.33	0.00085	0.43
7	0.0041	0.0032	0.0037	0.0037	0.86	0.0024	1.1

Notes. σ_Q and σ_U are estimates of "typical" errors in the 11 × 11 Q and U maps, calculated as described in this Appendix. σ_P is the average of σ_Q and σ_U.

^aErrors in polarization fraction (σ_Pfrac) and polarization position angle (σ_χ) derived from error propagation of σ_Q and σ_U. ^bErrors estimated by taking the standard deviation of the values in the maps of δP_frac and δχ.

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In addition to error propagation, another way to estimate the typical error in P_frac and χ is to calculate directly the standard deviation of the values in the δP_frac and δχ error maps. See Table 6: these values, denoted by superscript ^b, are the same as the values derived from error propagation to within factors of <40% (σ_Pfrac) and <30% (σ_χ). As mentioned above, the resulting errors are conservatively large, as they are derived from the entire region between the $\tfrac{1}{3}$ FWHM and the FWHM.