ALMA Band-to-band Phase Referencing: Imaging Capabilities on Long Baselines and High Frequencies

The Atacama Large Millimeter/submillimeter Array (ALMA) is a very powerful instrument to investigate emissions in millimeter/submillimeter waves with very high angular resolutions (Bachiller & Cernicharo, 2008). With the 16 km ALMA configuration, the spatial resolutions will be 12, 7, and 5 milliarcseconds (mas) at the observing frequencies of 400, 650, and 850 GHz, respectively. With these high angular resolutions, ALMA is expected to reveal unseen astrophysical properties in a variety of celestial objects in submillimeter waves.

The long-baseline capabilities of ALMA have been tested since ALMA early science operation started in 2011 (Matsushita et al., 2012, 2016; Asaki et al., 2016). ALMA long-baseline campaigns (LBCs) were organized in 2014 and 2015 to accomplish the most extended array configuration with 16 km baselines in order to make science verification observations in Bands 3, 6, and 7 (ALMA Partnership et al., 2015a, b, c, d). ALMA has opened 16 km baseline observations to the user community up to Band 7 frequencies in Cycle 7 ($\geq 0.8$ mm wavelength, or $\leq 373$ GHz), with the ability to regularly achieve resolutions higher than $\sim 20$ mas. For example, ALMA imaged scenes of star formation in a distant gravitationally lensed galaxy SDP.81 at $z=3.042$ with the angular resolution of $\sim 23$ mas in Band 7, corresponding to a spatial scale of $\sim 180$ pc (ALMA Partnership et al., 2015c). ALMA allows us to address planet formation around very young solar-type stars in the Milky Way through observations of protoplanetary disks, where the dust temperature distribution, the nature of dust particles, and the gas dynamics can be investigated in unprecedented detail. For example, ALMA revealed a protoplanetary dust disk with a pattern of bright rings and dark gaps surrounding HL Tau with the angular resolution of 25 mas (ALMA Partnership et al., 2015b) and TW Hya with the angular resolution of 20 mas in Band 7 (Andrews et al., 2016), corresponding to 3.5 and 1 au, respectively, at the target distance. It is also expected to image a few au scale circumplanetary disks around massive planets in such protoplanetary disks (e.g. Isella et al., 2019; Tsukagoshi et al., 2019). Recently the magnetic field structure around a very young protostar, MMS 6, in the Orion Molecular Cloud-3 was studied using dust emission polarization with the angular resolution of $\sim 25$ mas, corresponding to $\sim 10$ au (Takahashi et al., 2019). For evolved stars, ALMA captured a bright spot on the asymmetric photosphere of Betelgeuse with the 13 mas angular resolution, corresponding to 2 au (O’Gorman et al., 2015), and detected the circumstellar mass-loss gas with the angular resolution of 18 mas in Band 7 (Kervella et al., 2018). ALMA is also an excellent tool to study solar system objects: an asteroid, 3 Juno, was observed with the angular resolution of 42 mas in Band 6, corresponding to 60 km at 1.97 au and was able to resolve thermal emission from the surface of the main belt asteroid, to measure the asteroid geometric shape, rotational period, and soil characteristics (ALMA Partnership et al., 2015d).

The highest angular resolutions in radio interferometry can be realized with the combination of the longest baseline possible and highest observing frequencies available. To achieve the highest angular resolutions with ALMA, interferometer phase stability for the 16 km baselines in high-frequency (HF) bands (Bands 8–10, or the frequency range of 385–950 GHz) is crucial. Atmospheric fluctuations increase the interferometer phase error as a function of baseline length (Matsushita et al., 2017), so that the longer the baseline, the worse the phase stability, and thus the synthesized image becomes blurred (Carilli & Holdaway, 1999). The interferometric phase error can usefully be translated into a coherence factor. Assuming that the residual phase variations in the visibility data are pseudo-random, the coherence factor (percentage decrease of the peak intensity of the image) can be evaluated by $\exp{({-{\sigma_{\Phi}}^{2}/2})}$, where $\sigma_{\Phi}$ is the phase rms in rad (Thompson et al., 2001). This value is directly related to the image performance of radio interferometers. The phase RMS has to be decreased to 0.84 rad in order to achieve coherence values above 70%. Since the interferometer phase is proportional to a product of an electrical path length error of the electromagnetic wave and observing frequency, the corresponding path length fluctuation is 100, 61, and 47 $\mu$m RMS at 400, 650, and 850 GHz, respectively. Reducing the atmospheric phase fluctuations is therefore a critical requirement for any calibration technique in order to achieve successful ALMA HF long-baseline observations.

To reduce the atmospheric phase fluctuations, phase referencing is a frequently adopted technique for interferometer phase correction (Beasley & Conway, 1995; Asaki et al., 2007). The general implementation of phase referencing is to frequently observe a phase calibrator, typically a bright point-like quasar (QSO), close to the target source at the same frequency. This is referred to as in-band phase referencing and is the common phase correction method used for ALMA and other radio and millimeter interferometers. ALMA has been preparing a millimeter-wave bright QSO catalog (ALMA calibrator source catalogue¹¹1 /https://almascience.nrao.edu/alma-data/calibrator-catalogue) for in-band phase referencing since the early science operations. During the ALMA mission concept stage and through various site tests, it was realized that calibration of the longer baselines with a standard scheme would require nearby calibrators within a few degrees, limited by both potential antenna slew speeds and intrinsic site phase stability. It was also predicted that a few thousand calibrators would be available on the sky at a wavelength of 3 mm and thus could easily be found close to any target (e.g. Holdaway, 1992, 2001). However, the flux density of a typical QSO is proportional to $\nu^{-0.8}$ in the millimeter/submillimeter-wave regimes (Hardcastle & Worrall, 2000) and the system noise (both from the sky and receivers) increases, meaning that the integration time to achieve sufficient signal-to-noise ratio (S/N) becomes too large. This combination makes it unlikely to find a bright enough nearby phase calibrator for a randomly located target (Asaki et al., 2020).

In order to mitigate the difficulty in finding a phase calibrator at HF, the so-called band-to-band (B2B) phase referencing provides an alternative technique, employing observations of a sufficiently close phase calibrator at a low-frequency (LF) Band. The use of a phase calibrator closer to the target than would otherwise be possible with in-band phase correction at HF will provide more accurate phase referencing (Holdaway & D’Addario, 2004). This B2B phase referencing technique was firstly demonstrated using the Nobeyama Millimeter Array between 148 and 19.5 GHz (Asaki et al., 1998) and has since been used successfully at various other facilities. The Korean very long baseline interferometry (VLBI) network (KVN) can observe 22, 43, 86, and 129 GHz simultaneously with a quasi-optical receiver system (Han et al., 2013) and achieves multiwave B2B phase referencing, which is called source frequency phase referencing. It has been demonstrated in VLBI observations of active galactic nuclei (Rioja et al., 2014) and astrophysical maser sources (Dodson et al., 2014). A similar multifrequency phase correction has been successfully applied to Combined Array for Research in Millimeter-wave Astronomy (CARMA) observations at 227 GHz where a subset of antennas positioned close to the main array observed at 30 GHz and were used to correct the 227 GHz target phase using the 30 GHz delay measurements in the CARMA paired antenna calibration system (C-PACS; Pérez et al., 2010; Zauderer et al., 2016).

B2B phase referencing is currently being implemented at ALMA. The motivation of the research presented in this paper is to verify the technical feasibility of B2B phase referencing and evaluate its performance in synthesized images with the high angular resolutions. In this paper, we report on HF long-baseline (14–16 km) image capability tests for a point source QSO J2228$-$0753 in Band 7 (289 GHz) and Band 8 (405 GHz) observed together with a closely located phase calibrator, J2229$-$0832, in Band 3 (96 GHz) and Band 4 (135 GHz), respectively, to demonstrate a quasi end-to-end ALMA science observation. The data were obtained during the ALMA HF LBC (HF-LBC-2017; Asaki et al., 2020). In Section 2, the basic strategy of B2B phase referencing for HF long-baseline observations is mentioned. In Sections 3 and 4, we outline our ALMA B2B phase referencing observation experiments and data reduction, respectively. Section 5 provides the results of the experiments, which are discussed in Section 6. We summarize this feasibility study in Section 7. A summary of the HF-LBC-2017, which these tests are part of, is presented by Asaki et al. (2020), while further tests investigating calibrator separation angles and switching cycle times are presented by Maud, L. T. et al. (in preparation).

The main goal of the interferometer phase correction is to remove systematic phase errors due to instrumental electrical path length errors and a priori antenna position errors, atmospheric delay models over the array, and short-term atmospheric delay variations. An important delay correction produced by water vapor over each 12 m antenna is obtained by water vapor radiometers (WVRs; Nikolic et al., 2013, and references therein) on each antenna. The water vapor emission in the direction of the radio source measured with the WVR is converted into delay determined every second. The WVR phase correction is always applied to the ALMA data and typically removes the majority of the water vapor delay (Matsushita et al., 2017; Maud et al., 2017), although it is somewhat dependent on the quantity of the water vapor content in the atmosphere as precipitable water vapor (PWV), the depth of the water in a column of the atmosphere to the zenith.

In 2017, HF-LBC-2017 was organized to test the submillimeter-wave (wavelength of shorter than 1.1 mm) imaging capability of ALMA using the longest baselines up to 16 km (Asaki et al., 2020). One focus of the campaign was to extensively test the B2B phase referencing method and begin providing appropriate observing scripts and data reduction scripts. In B2B phase referencing, a close phase calibrator is observed at an LF. The LF phase of the phase calibrator is multiplied by the HF/LF ratio. Asaki et al. (2020) describe the realization and implementation of B2B phase referencing in ALMA in detail.

An important observation parameter in phase referencing (in-band or B2B) is the switching cycle time $t_{\mathrm{swt}}$ that is measured by the time difference between two calibrator scans. For ALMA observations with the longer baselines, the typical switching cycle time is 72 s. In many cases shorter switching cycle times provide more accurate phase referencing and are investigated. Our experiments adopted a much faster switching cycle time of 20 s because HF observations may require more frequent visits to a phase calibrator to compensate for the atmospheric phase fluctuations. The ALMA antennas can quickly change their position by several degrees in a few seconds to accommodate such a quick source change, which is most important at HF, where the atmospheric fluctuations are most variable.

Use of B2B phase referencing requires an additional phase correction to remove the instrumental phase offset difference between the two respective Bands. This can be achieved through a cross-band calibration, which is referred to as differential gain calibration (DGC). In the DGC, a bright QSO is observed at both an HF and LF, using frequency switching, such that the phase offset difference can be solved. If the phase offset difference is time constant, a single DGC measurement is enough. One important question at the ALMA system level is whether the temporal instrumental phase stability is sustained in the case where the HF phase is corrected by the LF phase. This uncertainty is due to the fact that the highly stabilized local oscillator (LO) signal does not pass through identical signal paths in the instrumentation for the LF and HF data during B2B phase referencing. Intrinsically the HF phase to be corrected flows through the HF receiver, while the correcting phase at the LF passes though the LF receiver. Thus, it is important to confirm that the phase offset differences between the respective Bands are stable with time during B2B phase referencing as has been confirmed for continuous single (in-band) observations (e.g. Matsushita et al., 2012). In addition, the atmospheric phase variations add a pseudo-random phase difference between HF and LF in DGC; hence, the frequency switching should be as fast as possible.

We conducted the observing experiments in 2017 October and November with the maximum baseline lengths of 16 and 14 km, respectively, as shown in Figure 1 with 40–50 12 m antennas. Overall four experiments were conducted, two in October (9th and 10th) and two in November (2nd and 3rd). We arranged B2B phase referencing experiments at two specific frequency pairs. One is at 289 GHz in Band 7 for the target with the phase calibrator observed at 96 GHz in Band 3 (Band 7-3 experiment), while the other is at 405 GHz in Band 8 for the target with the phase calibrator at 135 GHz in Band 4 (Band 8-4 experiment). Each epoch consists of one Band 7-3 and one Band 8-4 experiment. For convenience, the first- and second-epoch Band 7-3 experiments are referred to as Band 7-3(1) and Band 7-3(2), respectively, while the Band 8-4 experiments are denoted Band 8-4(1) and Band 8-4(2). Details are listed in Table 1.

The observed sources are listed in Table 2. We selected the QSO J2228$-$0753 as a target at the HF, while the QSO J2229$-$0832, with a separation angle of $0^{\circ}$.7, was observed as a phase calibrator at the LF. For DGC, we selected a very bright QSO J2253$+$1608 located $25^{\circ}$ away from the target. Note that the phase offset difference measured by a DGC source is independent of the QSO location in the sky, since it is predominately determined by internal electronic phase differences between the HF and LF (Asaki et al., 2020). We switch between the HF and LF using the same frequency of the photonic LO signal (Shillue et al., 2012) in order to minimize overhead times for the frequency switching (Asaki et al., 2020). This allows us to change frequency with only a $\sim 2$ s delay.

ALMA science operations have two criteria to assess go/no-go of a given user observation depending on weather conditions. First is the amount of the water vapor (PWV) because of the absorption of the radio signal. A loss of 20% absorption corresponds to a PWV of 1 and 0.7 mm at frequencies of 350 and 450 GHz, respectively, and these are guidelines of whether observations should be started. The PWV is also associated with short-term phase variations over each antenna. The variations are somewhat correlated with the PWV value but also depend on the water vapor clumpiness and wind velocity. Second is an estimate of the interferometer phase stability determined from the phase behavior in the previous observation, and/or a short observation known as go/no-go. The nominal criterion used from the above test is that the phase RMS with the time interval of 2 minutes for baselines about half the maximum length is less than $\sim 1$ rad. At the start of all our experiments, the two criteria were passed. However, note that the conditions can change during an observation and over the sky, so these go/no-go criteria are only approximate.

In order to demonstrate general ALMA science observations, we prepared experimental Scheduling Blocks (SBs; Nyman et al., 2010) containing all the observation information required for B2B phase referencing and DGC, as well as normal observing sequences such as system noise temperature measurements, pointing calibration, and flux calibration at the HF. A DGC block begins and ends with an LF scan and consists of eight HF and nine LF scans executed by turn. The experimental SBs for the second epoch are depicted in Figure 2. A single observation consists of repetitions of the B2B phase referencing block for the target and phase calibrator and regular visits to the DGC block where we only switch the frequency while pointing at the DGC source. The B2B phase referencing block between the target and phase calibrator lasts 12 minutes, after which a 2.5 minute DGC block is inserted. Such a frequent insertion of the DGC block was made not only to check the DGC solution repeatability and stability but also to avoid a heavy load to the antenna control computers by a long sequence of antenna pointing changes.

Each Band amplifies two linear polarizations ($X$ and $Y$) separately at each 12 m antenna. The amplified signal is split into four intermediate-frequency signals, which are referred to as basebands (BBs) with the bandwidth of 2 GHz. Two of the four BBs are transmitted from the upper side band (USB) of the ALMA front-end receiver, and the other two are transmitted from the lower side band (LSB). The four BBs with the $X$ and $Y$ polarizations are digitized at the antenna and transferred to the 12 m array correlator (Remijan et al., 2020). In our experiments, the digitized BBs were filtered out to have a 1.875 GHz bandwidth each with 128 frequency channels for two polarization pairs of $XX$ and $YY$ to form a spectral window (SPW). The center frequencies of the HF SPWs are 282.5 (LSB), 284.4 (LBS), 294.5 (USB), and 296.5 (USB) GHz in Band 7, and 398.0 (LSB), 399.9 (LSB), 410.0 (USB), and 412.0 (USB) GHz in Band 8. The integration time used for recording each of the four SPWs at both the HF and the LF respectively was 1.01 s.

For the B2B phase referencing blocks, a typical science target scan length was 8 s at the HF, while the phase calibrator scan length at the LF was 6 s. For the DGC blocks, the above frequency switching sequence was adopted, and the scan length for the HF and LF DGC scan was typically 8 and 6 s, respectively. The antenna slew and/or the frequency switch are done simultaneously and take 2–3 s. The total length for the one switching cycle is $\sim 20$ s in both the B2B phase referencing sequence and DGC blocks.

Since the selected DGC source is one of the brightest sources in the sky, we also used DGC scans for a bandpass calibration at both the LF and HF, so a separate bandpass scan was not prepared in the SBs. We also used the HF DGC scans for the flux calibration as well in the data reduction, while flux calibration at the LF is not required, as we are only interested in the transfer of the phase solutions. During operations, it is envisaged that separate bandpass and flux calibrators may be observed.

Data reduction was carried out using the Common Astronomy Software Applications (CASA) software (McMullin et al., 2007), and we have developed a semi-automatic CASA data reduction python script for our experiments. The offline WVR phase correction (WVRGCAL) (Nikolic et al., 2012) was applied for all the experiment data. The obtained antenna-based water vapor brightness is translated into an antenna-based phase solution. The obtained PWVs in the course of WVRGCAL are listed in Table 3. WVRGCAL could reduce the atmospheric phase fluctuations even in the low PWV conditions ($<1$ mm), with a reduction of the phase RMS between 4% and 35% on the longest baselines.

Most of the target scan lengths enclosed by the phase calibrator scans have the switching cycle time of 20 s, while some of the target scans have longer switching cycles (typically, $\sim 1$ minute) because the ALMA control software inserts a $\sim 20$ s system noise temperature measurement just after the first phase calibrator scan in the target-phase calibrator block. Since one of the purposes of our experiments is to see the phase correction efficiency with a short switching cycle time, the data reduction script flagged out the target scans with such a longer switching cycle time (at most 6–7 scans of the $\sim 100$ during the observations).

There are two phase transfer processes in the data reduction: the first is for the DGC, in which J2253$+$1608’s HF scan phase is corrected using its LF scan phase. The second is for the target J2228$-$0753 whose phase is corrected using the phase calibrator J2229$-$0832. The DGC phase transfer is performed prior to the target–phase calibrator process. In the DGC, we first obtain the SPW-dependent LF phase offset by averaging the LF DGC scans in time for each SPW. The obtained LF phase offset is self-applied to the LF scans of the DGC source, as well as to the phase calibrator. After removing the LF phase offset from the LF scans, the LF DGC SPWs are averaged for phase-up, and the phase transfer is conducted to remove the atmospheric phase fluctuations from the HF DGC scans using LF DGC scans temporally closest. The HF DGC phase can be then averaged in time for each SPW at each DGC block to obtain the SPW-dependent HF phase offset. In this paper, the HF DGC phase offset is referred to as the DGC solution. One DGC solution is obtained by averaging the eight HF DGC scans from a DGC block after the phase transfer. To carry out the phase transfer from the phase calibrator to the target, the SPWs of the LF phase calibrator are averaged for the phase-up and then applied to the HF target phase for each SPW together with the aforementioned DGC solution. One of the most essential parts in B2B phase referencing is the temporal stability of the DGC solution, which is described in Section 5.2. In the course of the WVR phase correction, antenna position calibration, bandpass calibration, flux calibration, and B2B phase referencing, significant outliers in either of the phase or amplitude were flagged out from the visibility data.

The visibility data of all four HF SPWs and both polarization pairs are combined using multifrequency synthesis imaging in the CLEAN algorithm using a fixed value of 50 CLEANing iterations. We used Briggs weighting with the robustness parameter of 0.5 in imaging and the pixel size of 2 mas for a $512\times 512$ square mas area ($256\times 256$ pixels). In the CLEAN deconvolution, we set a 15-pixel-radius circle mask at the phase tracking center.

We additionally made phase self-calibration (Schwab, 1980) with the solution interval of the target on-source scan length of 8 s for the purpose of investigating the effectiveness of B2B phase referencing: the self-calibrated image provides the synthesized image of the target without phase errors on timescales longer than 8 s, so that comparison of the images with and without the phase self-calibration can inform us of how much B2B phase referencing can be effectively applied for the phase correction. In general, if a target source is bright enough, self-calibration effectively works to correct the interferometer phase without frequent phase calibrator scans (Brogan et al., 2018; Cornwell & Wilkinson, 1981). However, it is not plausible to assume that high S/N is guaranteed for HF observations in general considering the notably higher system noise temperatures and lower antenna efficiencies when compared to the lower-frequency Bands. In addition, target sources observed with the higher angular resolutions are likely to be resolved. Therefore, S/N on the longest baselines could be further reduced, and it may be more difficult to perform self-calibration compared with the point-source case. This means that self-calibration cannot be a default phase correction plan to improve the image quality of the target, so the effectiveness of the self-calibration for ALMA HF long-baseline observations is out of scope in this paper.

ALMA HF long-baseline experiments were conducted by adopting B2B phase referencing with a 20 s switching cycle time in order to demonstrate the ALMA high angular resolution image capability in Bands 7 and 8. In the series of experiments, we observed the target J2228–0753 paired with a phase calibrator J2229–0832 separated by $0^{\circ}.7$. We also observed J2253$+$1608 as the DGC source to remove the instrumental phase offset difference between the HF and LF Band receivers.

The DGC solutions are temporally stable for the receiver pairs of Band 7-3 and Band 8-4 for an hour with the phase RMS of $10^{\circ}$ after the correction of a linear trend. For Bands 7-3 and 8-4, the DGC solutions are so stable that the two DGC blocks are enough to insert at the beginning and end of an observation lasting $\sim 2$ hr.

We achieved an angular resolution of $14\times 11$ mas at 405 GHz for the target using the maximum array configuration with 16 km baselines. We obtained high image coherence above 98% and 88% at 289 GHz and 405 GHz, respectively. One of the experiments indicates a residual path length fluctuation of 27 $\mu$m RMS after B2B phase referencing, almost independent of baseline length, which would have met the requirements to achieve $>$70% image coherence in Bands 9 and 10 using a phase calibrator with the separation angle of $<1^{\circ}$. The measured astrometric positions of the target are consistent with those in the ALMA calibrator source catalogue and the VLBA calibrator source list to within 0.4 mas. Because the separation angle between the target and phase calibrator was so small, that B2B phase referencing effectively reduced the phase error due to the baseline vector uncertainties, as well as the atmospheric phase fluctuations.

We compared the image coherence achieved using the switching cycle times of 20 and 60 s, respectively. At 289 GHz, the image coherence did not significantly degrade, while at 405 GHz the image coherence decreased to 68$-$84%. Observations that impose the condition that the 2-minute phase stability is $<1$ rad satisfy the requirement that the image coherence is $>$70% when using 60 s switching cycle times. At higher frequencies, shorter switching cycle times will likely allow maintaining the coherence even if the 2-minute phase stability exceeds 1 rad.