Polarization Calibration of the FAST L-band 19-beam Receiver: II. Beam Measurements of Full Stokes Parameters

To determine the beam structures, their dependence on zenith angle, and time stability, multiple observations were conducted over different sources and epochs. We chose the VLA polarization calibrators¹¹1https://science.nrao.edu/facilities/vla/docs/manuals/obsguide /modes/pol J0854+2006 and J1407+2827 as our sources. Among the VLA polarization calibrators, J0854+2006 is the one closest to the zenith of FAST, and OTF observations over 1.7 hours during the transit of J0854+2006 were carried out on 2020/09/10, 2020/09/11, 2021/10/06, and 2021/10/07. J1407+2827 is known for low polarization at L-band and hence is an ideal target to map the Stokes $Q$, $U$, and $V$ beams. The 2.1-hour OTF observations during the transit of J1407+2827 were carried out on 2022/09/19 and 2022/09/21. Raster scans using the MultiBeamOTF mode of the telescope with a scan spacing of 1 arcmin and a scanning speed of 15 arcsec/sec were performed in either the declination (Dec) or right ascension (RA) direction, covering a sky region of approximately $20^{\prime}\times 20^{\prime}$, with a consistently low zenith angle ($\leq 18.7^{\circ}$) for all observations. Offsets in RA and Dec relative to the equatorial coordinates of all sources were computed according to § 2.2.2 of Jiang et al. (2020). The full-polarization correlation products of the signals were simultaneously recorded using the ROACH backend²²2https://casper.berkeley.edu/wiki/ROACH-2_Revision_2, featuring 65,536 spectral channels in each polarization. The spectral bandwidth, centered at the frequency of the 21-cm HI line, was 31.25 MHz, with a channel spacing of $\sim$477 Hz. Data acquisition employed a sampling rate of 0.1 s and noise diode injection cadence of 0.2 s/1.8 s for on/off switching. Figure 1 illustrates an example of the central beam track for a raster scan targeting J0854+2006 in 2021.

We used the IDL RHSTK package³³3http://w.astro.berkeley.edu/heiles/ developed by C. Heiles and T. Robishaw to reduce the FAST raw data and generate the Stokes $I$, $Q$, $U$, and $V$ spectra. Widely adopted for Arecibo and GBT polarization data reduction, this package excels in executing gain and phase calibrations for the two polarization paths, bandpass calibrations for the four correlated spectra, and polarization calibrations. The process of polarization calibration used for the data in this work was provided in Paper I. Based on the Stokes $I$, $Q$, $U$, and $V$ spectra calibrated using the RHSTK package, the Asymmetrically Reweighted Penalized Least Squares smoothing algorithm, as proposed by Baek et al. (2015), was employed to eliminate the baselines of the spectra. With a pixel size of $0.5^{\prime}$, beam maps were generated using a Gaussian smoothing kernel with a full width at half maximum (FWHM) of $7.9^{\prime\prime}$ to account for the uncertainty in the pointing accuracy of FAST (Jiang et al., 2020).

The Stokes $I$, $Q$, $U$, and $V$ beam patterns of the FAST telescope were modeled using the method described in Du et al. (2022). First, the radio telescope was simulated as a transmitting antenna using the commercial software CST Studio Suite (2024) and GRASP (2024). The far-field radiation pattern of the feed horn was simulated using CST. Only one feed horn (instead of a 19-horn array) was simulated, as mutual-coupling between the horns is minimal (Smith et al., 2017). The reflector antenna was then simulated in GRASP by placing the simulated feed pattern at the corresponding positions of the 19 horns on the receiver, one at a time. The results are two sets of 19 radiation patterns of the telescope, one for each linear polarization, in the form of far-field electric fields. For each of the 19 beams, the 2 × 2 Jones matrix of the telescope (in transmitting mode) can be constructed using these simulated far-field electric fields. The Jones matrix in the receiving mode was then deduced from the transmitting-mode Jones matrix using the theorem of reciprocity (Potton, 2004), and further converted to the 4 × 4 Mueller matrix. For each of the 19 feed horns, 16 beam patterns are produced—one for each of the Mueller matrix elements—that fully characterize the telescope’s response to unpolarized, linearly-polarized, and circularly-polarized incoming waves. Among them, the Stokes $I$, $Q$, $U$, and $V$ beam patterns presented in this paper describe the telescope’s response to unpolarized incoming light.

The combined log-normal and Gaussian distribution approach was applied to analyze 19 beams over three years (for the central beam, both the Gaussian-only and log-normal-and-Gaussian methods are applied). The analysis included determining fitting parameters and computing characteristics such as the position of the beam center, beam size, the relative beam efficiency, and the ratio of the sidelobe intensity contribution to the main lobe intensity ($P_{\rm slb}/P_{\rm beam}$). Regarding the elongated asymmetrical axis of the main beam as the major axis, we calculate the FWHM for the major and minor axes using the following formulas:

Subsequently, the beam size is derived as the geometric mean of $\mathrm{FWHM_{major}}$ and $\mathrm{FWHM_{minor}}$, given by:

The relative beam efficiency of the off-center beams with respect to the central beam is determined by comparing the ratio of the total intensity of each beam to that of the central beam. The ratio $P_{\rm slb}/P_{\rm beam}$ is obtained by comparing the total intensity of the sidelobes to the total intensity of the main lobe. All the fitted parameters are listed in Table 1, while other parameters calculated based on the fitted values (e.g., beam center, beam size) are shown in Table 2.

Figure 6 displays Stokes $I$ maps for each individual beam based on 2021 observations of J0854+2006, while Figure 7 presents the corresponding simulated results. The coma and sidelobes are clearly displayed and perfectly fitted, with the outer beams showing more prominent coma and sidelobes compared to the inner beams. Additionally, the position angles of the beams follow a regular pattern, with the orientation of the major axis of each main lobe and the orientation of each sidelobe being largely consistent. Furthermore, the coma all extend outward from the central beam, indicating a systematic variation in the position angle of the main lobe. Overall, the observational results closely align with the simulations, particularly regarding the structures of the coma and sidelobes. Figure 8 presents the fitted results (left) and the simulations (right) for the combined response of the 19 beams, further demonstrating a high degree of consistency. The outer sidelobe structures reported by Jiang et al. (2020) are absent. This is because the raster scan mapping is a mirrored image of the beam pattern. So each of the 19 maps should be mirrored about RA and Dec before combining them together.

In general, the 19 beams of the FAST telescope can be categorized into four groups based on the exact angular distance of each beam’s center from that of the central beam M01, denoted as $d_{\rm M01}$. For simplicity, we classify the beams into three groups: (1) the central beam M01; (2) inner beams with relatively small $d_{\rm M01}$ ($\leq$ 7^′), comprising M02, M03, M04, M05, M06, and M07; (3) outer beams with larger $d_{\rm M01}$ ($\geq$ 7^′), comprising M08, M09, M10, M11, M12, M13, M14, M15, M16, M17, M18, and M19. For beams M01, M06, M17, and M18, each characterized by distinct $d_{\rm M01}$ values, we present the distribution maps of total intensity with fitting results using a logarithmic-scale plot to explore the relationships between asymmetrical structures and $d_{\rm M01}$, as depicted in Figure 9. Three years of observations yield consistent results: For beam M01, no asymmetrical structures were detected. In contrast, the asymmetrical features of the sidelobes and the coma begin to appear in M06 and become more prominent in M17 and M18, indicating that the asymmetrical structures become more pronounced as $d_{\rm M01}$ increases. However, the position angle of the M06 sidelobe in 2022 differs significantly from the results in 2020 and 2021. This variation may be due to the different zenith angles of the two sources or a time variability of the feed from 2021 to 2022. The coma and sidelobes are oriented in opposite directions. Since these features are intrinsic to the laterally displaced horn, they are expected to remain stable over time. These results are in agreement with the simulation, indicating that the farther from the focus, the more pronounced the coma and sidelobes become.

Figure 10 provides a comparative analysis of Stokes $I$ parameters across various observations. Figure 10(a) illustrates the distribution of $\sigma_{1}$, where larger values of $\sigma_{1}$ indicate greater deviation from the Gaussian distribution, corresponding to more pronounced coma and higher levels of asymmetry. Over the three-year epochs, outer beams consistently exhibit higher levels of asymmetry compared to inner beams. This is consistent with the results of Jiang et al. (2020), who proposed that notable features become more significant with increasing $d_{\rm M01}$ for the outer beams. Figure 10(b) delineates the variation in the relative beam efficiency across all observations. The majority of inner beams (M01 to M07) exhibit high efficiencies, averaging $\geq 95\%$, while outer beams demonstrate relatively lower efficiencies, averaging between $95\%$ to $90\%$. In Figure 10(c), the beam size of all beams is depicted. The beam size estimated based on observations of J1407+2827 consistently exceeds that based on J0854+2006. Inner beams consistently demonstrate relatively smaller beam sizes compared to outer beams, with a difference not exceeding 0.1 arcmin. Additionally, noticeable disparities in beam size are observed when observing different sources, while temporal variations in beam size are negligible when observing the same source. Moreover, the average difference between the major and minor beam widths is 0.1/0.2 arcmin for inner/outer beams, contrasting with Sun et al. (2021), who proposed that the difference is very small. These findings underscore the superior performance of inner beams in terms of both beam efficiency and size relative to outer beams. Interestingly, in the case of the same source J0854+2006, when comparing data between 2020 and 2021, it is noted that the asymmetry level and beam size of the outer beams are systematically smaller in the 2021 results, which indicates that the performance of the outer beams in 2021 may have surpassed that of 2020.

Disparities in position angle across different observations are illustrated in Figure 10(d). The position angle distribution shows a high degree of consistency over three years of observations. For the inner beams, the position angles follow a linear distribution with a slope of approximately 60.7 degrees, while the outer beams exhibit a linear distribution with a slope of approximately 29.2 degrees. This result confirms the stability of the FAST system, which is designed such that the outer beams are spaced 30 degrees apart and the inner beams are spaced 60 degrees apart, as shown by the black lines in Figure 8. To better understand the relationship between sidelobes and beams, we calculate the angle difference ($\phi_{\rm dif}=\phi-\phi_{c}$), representing the angle between the major axis of the beam and the symmetry axis of the sidelobe. Figure 10(e) shows the distribution of $\phi_{\rm dif}$, with most beams exhibiting $\phi_{\rm dif}\leq 20^{\circ}$, indicating a strong correlation between the location of sidelobes and the position angle of the beam. This correlation is also evident in Figure 8, where black lines intersect the center of sidelobes. These results indicate that the coma and sidelobes largely determine the position angle of each beam, making the position angle stable and solely dependent on the design of the lateral offset of the horns in FAST, without varying over time.

Of particular importance is the quantification of the percentage of flux contributed by sidelobes. By successfully modeling sidelobes of each beam, we can now reliably quantify this parameter, denoted by $P_{\rm slb}/P_{\rm beam}$, representing the ratio of the total intensity of sidelobes to the total intensity of the main lobe, as depicted in Figure 10(f). sidelobes of inner beams contribute nearly 2% of the total flux of the main lobe, while this contribution increases to 5% for outer beams, peaking at 6.8% for M16. These results indicate that sidelobes may significantly impact related studies, such as the determination of the position of Fast Radio Bursts (FRBs) and the HI intensity mapping survey. Further research is warranted to quantify these impacts stemming from sidelobes.

In Figure 10, the black dashed, blue dash-dotted, and red solid lines indicate the average values of the corresponding parameters in 2020, 2021, and 2022, respectively. In most panels, the black dashed lines and blue dash-dotted lines have similar values, but the deviations of the red solid lines from the black dashed lines and blue dash-dotted lines are statistically significant compared to the scattering of the data, especially the inner and outer beams of Figure 10(c). These deviations of the 2022 results from the 2020 and 2021 results may indicate a dependence of beam structure on source declination or a time variability of the receiving system.

The measurement of beam shapes for polarized Stokes parameters entails two approaches: observing an unpolarized source or subtracting the polarization emission of a polarized source to emulate an unpolarized source. Unlike the response in Stokes $I$, the telescope’s response in these parameters can be either positive or negative. J1407+2827 is a weakly polarized source⁴⁴4https://science.nrao.edu/facilities/vla/docs/manuals /obsguide/modes/pol. The blazar J0854+2006 has been confirmed to be polarized in the Stokes $Q$ and $U$ parameters but not in Stokes $V$. These two sources can be utilized to measure the Stokes parameters $Q$, $U$, and $V$, which comprise three components: (1) the polarization of the source itself, (2) leakage from Stokes $I$, and (3) features in the Stokes $Q$, $U$, and $V$ beam patterns such as beam squint and beam squash (Robishaw et al., 2021). Beam squash, which appears as a four-lobed cloverleaf pattern, is a linear polarization response that can arise from surface irregularities of the dish or the ellipticity of the feed radiation pattern (Robishaw et al., 2021). In contrast, beam squint refers to the circular polarization response that occurs when a feed antenna is tilted or displaced from the focal point of a reflector. This effect is typically observed in the Stokes $V$ beam pattern and manifests as a two-lobed structure.

To remove the leakage from Stokes $I$ of J0854+2006 and J1407+2827, we used the percentage of the Stokes $I$ intensity profile to represent the leakage contribution. For the linearly polarized source J0854+2006, we further applied a Gaussian profile fit to model its intrinsic linear polarization. In our polarization calibrations conducted in 2020 and 2021, the Stokes $Q$ and $U$ intensities were measured to be approximately 4.6% and 2.4%, respectively. After subtracting these components, the residuals reveal the responses of beam structures in Stokes $Q$, $U$, and $V$. For example, Figure 11 shows the beam structures for the central beam, with the colorbar being labeled with the ratio of the flux to the peak flux of Stokes $I$ (64.44 K, 67.50 K, and 28.85 K for data observed in 2020, 2021, and 2022, respectively) from the central beam, $I_{\rm peak}\%$. The simulated beam patterns for the central beam in Stokes $Q$, $U$, and $V$ are also depicted. As previously mentioned, there is a mirrored relationship between the raster scan image and the beam structure. However, because the beam squash and beam squint patterns in the Stokes $Q$, $U$, and $V$ of the central beam are inherently mirror-symmetric (as shown in the bottom panel of Figure 11), the Stokes $Q$, $U$, and $V$ maps and structures are expected to be consistent without requiring mirror-symmetrization. In Stokes $Q$, all observations exhibit significantly higher noise, which is much stronger than the beam structure and obscures features such as beam squash. Since the noise is observed only in Stokes $Q$ and not in $U$ or $V$, we believe it is more likely intrinsic to the properties of linearly polarized receivers rather than solely attributed to radio frequency interference (RFI). Specifically, for linearly polarized receivers, Stokes $Q$ tends to be considerably noisier than $U$ and $V$ due to time-dependent gain fluctuations in the $E_{X}$ and $E_{Y}$ channels. As described in Heiles (2002) (Section 5.2), for a nonpolarized source, the error in Stokes $Q$ is directly proportional to gain fluctuations, while Stokes $U$ and $V$ remain independent of such errors. Thus, gain errors can introduce noise in $Q$ without affecting $U$ or $V$, because if $E_{X}E_{Y}$ = 0 (as it is for a nonpolarized source), the product remains zero even in the presence of gain fluctuations. In Stokes $U$, the beam squash is evident as a four-lobed cloverleaf pattern, where lobes on opposite sides of the beam center exhibit identical signs. The Stokes $U$ contours at $\pm 0.02$ K, which mark the squash structure, are roughly consistent between the J0854+2006 and J1407+2827 maps. Furthermore, these results align closely with the simulated beam squash pattern. Since beam squash is inherent to the feed pattern ellipticity, it is expected to be constant with time.

Regarding Stokes $V$, all observations consistently reveal a more complex but weaker beam structure. The two-lobed beam squint structure is evident, accompanied with a secondary eight-lobed structure featuring an alternating arrangement of positive and negative lobes. These secondary structures align well with the predictions of the EM simulation. Theoretically, there should not be a squint $V$ pattern in the center beam, as shown in the simulation. The observed beam squint in the center beam suggests that the central horn is either laterally displaced away from the focus or tilted with respect to the focus-vertex line. This can possibly occur from a combination of a few things: (1) the focus cabin is not placed exactly so that the center horn’s phase center is at the focus of the vertex, so a slight positional inaccuracy will incur a squint; (2) the receiver as mounted in the focus cabin could have the plane that contains all the feed horns slightly tilted with respect to the horizontal axis of the focus cabin so that the normal to this receiver plane is not pointed towards the vertex; (3) the receiver is mounted inside the focus cabin so that the feed-horn-plane is aligned with the focus cabin’s horizontal plane, but the focus cabin itself is slightly tilted, which in turn would cause the receiver plane’s normal vector to point off of the vertex of the dish. This may also explain the nearly hundredfold difference in squint structure intensity between the observed results and the model predictions. Despite both J0854+2006 and J1407+2827 being unpolarized in Stokes $V$, the Stokes $V$ beam maps made using J0854+2006 have a higher noise than the map made using J1407+2827 owing to the data quality. For the central beam, responses from beam squints are consistently below 0.1 K, corresponding to $I_{\rm peak}\%\leq 0.3\%$, indicative of robust reliability in detecting Stokes $V$ polarization.

The Stokes $Q$, $U$ and $V$ mapping results of J1407+2827 for each individual beam together with the simulation results are depicted in Figures 12, 13, 14, 15, 16, and 17, respectively. As shown in these figures, Stokes $Q$ is much noisier, with no clear beam structure detected in any of the beams. Stokes $U$ exhibits a clear beam squash structure: for the central and inner beams, this structure remains consistent, with positive lobes on either side (in red) clearly separated, while the center region shows negative values. Consequently, the negative lobes on both sides (in blue) form an almost continuous shape, creating a unified structure. In contrast, each outer beam displays a distinct squash pattern. Overall, the beam structure intensity across all beams is below 0.2 K, corresponding to 0.7% of the Stokes $I$ intensity, with no significant intensity difference between inner and outer beams. Notably, beam M12 demonstrates the highest squash intensity. The simulation predicts similar results, indicating that certain lobes in the outer beams’ squash structure are stretched, leading to a pattern distinct from that of the inner beams, while no intensity difference is observed between the inner and outer beams. In Stokes $V$, the two-lobed squint structure and secondary structures are clearly evident in all beams. Comparing with the central beam, inner beams exhibit a more concentrated distribution in one direction that correlates strongly with the position angle. This effect is even more pronounced in the outer beams, where the structures are more compact, resulting in a significant increase in intensity due to the horns being displaced further from the focus. The beam squint orientation is distinctly observed in both the inner and outer beams, with the two-lobed squint structure aligning with the angle of each beam’s position angle. Overall, the Stokes $V$ intensity across all beams remains below 0.1 K, accounting for approximately 0.4% of the Stokes $I$ intensity, with the central beam demonstrating the best performance. These results align closely with the beam squint predictions by the simulation, particularly concerning the stretching of the lobes. For instance, the model of M08 clearly illustrates the positive and negative lobes being stretched into a horizontal configuration, which is in excellent agreement with the observed results. Additionally, the model also indicates that the intensity of the squint structure in the outer beams is significantly greater than that in the inner beams.

Figures 18, 19, and 20 depict the combined response of FAST in the Stokes $Q$, $U$, and $V$ parameters for J1407+2827 together with the theoretical model. Intrinsic noise in Stokes $Q$, likely due to gain fluctuations in linearly polarized receivers, obscures the beam squash. Conversely, the performance of all beams in Stokes $U$ exhibits favorable characteristics, with responses from beam squashes having $I_{\rm peak}\%\leq 1.5\%$, ranging from $-0.5$ K to 0.4 K. As mentioned above, beam M12 displays a more pronounced beam squash compared to other beams. For Stokes $V$, it is observed that the combined response of all 19 beams displays beam squint intensity having $I_{\rm peak}\%\leq 1\%$, ranging from $-0.3$ K to 0.3 K, indicating the highest level of performance among the Stokes $Q$, $U$, and $V$ parameters. Due to data quality issues, the beam squash and squint structures exhibited in the combined response of the 19 beams differ somewhat from the results of the theoretical model; however, both show alternating positive and negative structures in Stokes $U$ and $V$. Furthermore, numerous observed vertical and horizontal structures are present, indicating that these features are not artifacts of the data but rather the result of overlapping squash and squint structures from different beams. These findings affirm the exceptional capabilities of FAST in detecting the Stokes $U$ and $V$ parameters, providing a robust foundation for conducting in-depth investigations into the interstellar magnetic field, particularly concerning Zeeman multi-beam observations.