positioned at random. Ideally, one can describe a balance between these two
competing effects.

For a general SIS lens, de Freitas et al. (2018) derived analytic,
perturbative solutions for the source-plane cross-sections $\sigma_{\mu}$—or
the solid angle of the region with magnifications exceeding some threshold
$\mu_{\rm thr}$—as a piecewise function of $R_{0}$:

$\displaystyle\sigma_{\mu}$ $\displaystyle=\begin{cases}\frac{4\pi}{\mu_{\rm
thr}^{2}}+\frac{\pi}{4}R_{0}^{2},&R_{0}<R_{J}\\\ \pi(4R_{0}^{2}-\mu_{\rm
thr}R_{0}^{3})\\\ \hskip
28.45274pt-\frac{1}{R_{0}^{2}}\big{(}\frac{3\pi-8}{(4-8/\pi)^{2}}\big{)}\\\
\hskip 28.45274pt\times(4R_{0}^{2}-\mu_{\rm thr}R_{0}^{3})^{2},&R_{0}\geq
R_{J}\end{cases}$ (8)

where the joining point $R_{J}=2.15/\mu_{\rm thr}$ marks a transitional point
to the Einstein ring regime. As $\sigma_{\mu}$ is relative to the Einstein
radius, it can be multiplied by $R_{0}^{2}$ to obtain an absolute solid angle
(e.g. in arcsec2). This treatment implies that the peak area for $\mu_{\rm
thr}=5$ is approximately $R_{0}=0.6$ (i.e., source size 60% of the Einstein
radius).

Figure 12:  Top panel: Magnifications vs. the ratio $R_{\rm e}/\theta_{\rm
Ein}$, or source-plane effective radius (in arcsec) to Einstein radius (also
in arcsec). This ratio can serve as a proxy for the compactness of the
background object relative to the foreground lens. There is an apparent
negative correlation between this ratio and magnification, suggesting possibly
that smaller background objects can align more precisely with the highest-
magnification regions in the source plane (which increase in size with
Einstein radius). The redshift uncertainty of the foreground for 3 objects
does not strongly influence these values, as $\theta_{\rm Ein}$ is computed in
angular units. Bottom panel: The predicted source-plane cross sections
$\sigma_{\mu}$ as a function of the same ratio, for magnification thresholds
of $\mu_{\rm thr}=5,10,20$, using the analytic perturbative solutions
calculated by de Freitas et al. (2018). The relative solid angle
$\sigma_{\mu}$ can be converted to absolute angular units by multiplying by
$\theta_{\rm Ein}^{2}$. Dotted vertical lines indicate the optimal source size
for given magnification thresholds (shown as the conjoined horizontal lines in
the upper panel) to maximize the cross-sectional area. These maxima are
located at $\mu\approx 2.9522/R_{0}$, which is shown as a black solid line in
the top panel. The color gradient in the top panel shows the distribution of
$\sigma_{\mu}$ by computing these curves as a function of $R_{0}$ for a range
of continuous $\mu$ values. This indicates that smaller magnifications have
larger cross sections, as expected, but there is also a thin diagonal band of
elevated cross sections around the black solid line.

These analytic curves are shown in the bottom panel of Fig. 12. For given
magnification $\mu$, the cross-sectional area $\sigma_{\mu}$ essentially
describes the probability of a given $R_{0}$ ratio. We find that the peaks for
the selected magnification thresholds ($\mu_{\rm thr}=5,10,$ and 20) are
approximately concordant with the negative correlation seen in the upper panel
of Fig. 12 for the PASSAGES sample, as indicated by the conjoined
vertical/horizontal dotted lines. Specifically, the local maxima of
$\sigma_{\mu}$ in Equation 8 are found to occur at $\mu(R_{0})\approx
2.9522/R_{0}$, shown as a solid black line in Fig. 12. There is perhaps
greater deviation from this relation for smaller values of $R_{0}$, but there
are also fewer objects occupying this regime, so the deviation is not a robust
conclusion.

The relative probability of lower-magnification lenses is higher, as
$\sigma_{\mu}\sim\mu_{\rm thr}^{-2}$, which would suggest that they would be
more numerous. However, this will not be fully reflected for the PASSAGES
sample, as the selection is not uniformly sensitive to all magnifications.
This is because selection by apparent flux serves to maximize the
multiplicative product of magnification and intrinsic luminosity (the latter
of which is itself correlated with source-plane size, as discussed in §4.3.2).
For this reason, the recovered distributions of intrinsic luminosities,
magnifications, Einstein radii, and source-plane sizes are all interconnected
for PASSAGES and other lensing samples defined by flux.

In future work (Englert at al., in prep.), we intend to place the PASSAGES
sample in context with other studies that have used magnification and Einstein
radius distributions as constraints on the halo mass function and other
cosmological parameters (e.g., Eales 2015; Amvrosiadis et al. 2018). This is
an analysis which benefits immensely from a large sample with consistent
selection effects.

##  5 Conclusions

We have presented detailed lens modeling for 15 members of the PASSAGES
(Planck All-Sky Survey to Analyze Gravitationally-lensed Extreme Starbursts)
sample of dusty star-forming galaxies ($z=1.1-3.3$) that were introduced in
Harrington et al. (2016) and Berman et al. (2022). To do this, we gather
complementary information from an extensive set of recent multi-wavelength
sub-arcsecond continuum imaging with ALMA, JVLA, HST, and Gemini-S. We
implement parametric lens modeling with lenstool, where priors on the nature
of the foreground mass distribution are established by optical/near-IR
imaging, and multiple image positions of background objects (from imaging at
all wavelengths) provide modeling constraints. In analyzing the results of
this modeling and the resultant source-plane reconstructions, we find:

  * •

The 1 mm and 6 GHz magnification factors of the DSFGs modeled in this sample
range from $\mu=2-28$, with more than half at lower magnification, $\mu<10$.
Correcting the apparent infrared luminosities from Harrington et al. (2016)
and Berman et al. (2022) with these magnification factors, we find intrinsic
values of $L_{\rm IR}=0.2-5.9\times 10^{13}~{}L_{\odot}$ (median $1.4\times
10^{13}~{}L_{\odot}$), placing them solidly within the regime of hyperluminous
infrared galaxies, or HyLIRGs (Fig. 3). The corresponding inferred star
formation rates (also corrected for lensing) span $200-6300~{}M_{\odot}~{}{\rm
yr}^{-1}$ (median $1500~{}M_{\odot}~{}{\rm yr}^{-1}$). Their extreme, rare
properties are likely to be a direct result of the all-sky selection method
for PASSAGES. While initially assumed to be starbursts based on their large
star formation rates, evidence from their molecular gas depletion times and
star formation rate surface densities indicates that—while nearly all above
the center of the star-forming main sequence—they largely are not predicted to
surpass a $\Delta{\rm MS}>0.6$ dex threshold typically imposed to define
starbursts. Future work to attempt to constrain stellar masses is thereby
warranted in order to directly establish their relation to the main sequence.

  * •

Despite being among the most IR-luminous galaxies ever discovered in the
Universe, owing to their advantageous large-area selection, their
submillimeter magnification factors are largely consistent with those of
lensed DSFGs identified by the likes of Herschel, SPT, and ACT (Bussmann et
al. 2012, 2013, 2015; Hezaveh et al. 2013; Dye et al. 2014, 2018; Spilker et
al. 2016; Enia et al. 2018; Rivera et al. 2019; Liu et al. 2022) and other
Planck-selected lenses not in PASSAGES (Cañameras et al. 2015, 2018a); see
Fig. 4.

  * •

In characterizing their de-lensed, spatial extents in the source plane at
rest-frame far-IR and radio, we find that the PASSAGES DSFGs are generally
more extended than typical DSFGs (both lensed and unlensed). Since they appear
to be consistent with size-luminosity scaling relations for unlensed submm-
bright objects (Fujimoto et al. 2017), we interpret their larger sizes to be a
consequence of their higher star formation rates (or vice versa); see Figs. 6
and 7. They are consistent with a framework of some DSFGs being the result of
active growing of and accretion of gas onto a star-forming disk, with a wider
distribution of star formation than low-$z$ ULIRGs, which form stars in
compact nuclear regions.

  * •

We compute the radio-FIR correlation parameter $q_{\rm FIR}$ by assuming a
single radio spectral index; the range of values for $q_{\rm FIR}$ is in line
with those from Giulietti et al. (2022) for Herschel-selected lenses, but we
find no clear evolution with redshift, as predicted by some recent works. We
also find that PASSAGES objects do not show signs of significant contributions
from AGN, which might be indicated by a radio excess, $q_{\rm FIR}<1.8$.

  * •

In comparing their radio vs. sub-mm sizes, we find some weak dependence on
$L_{\rm IR}$, $\Sigma_{\rm SFR}$, and the radio-FIR correlation parameter
$q_{\rm FIR}$, with the more luminous and more densely star-forming galaxies
having preferentially more compact 1 mm sizes (or more extended 6 GHz halos).
Relative to $q_{\rm FIR}$, we find that galaxies closer to being AGN-powered
(i.e. showing a radio excess) had generally smaller radio sizes, which might
be indicating that deeply-buried AGN are leading to more compact half-light
radii. At present, our conclusions are limited by the small number of 10
objects with both radio and FIR sizes measured.

  * •

The larger sizes of PASSAGES galaxies (relative to the broader population of
DSFGs) may also owe to their concentration around Cosmic Noon, $z<3.5$, in
contrast to higher-$z$ DSFGs like Spilker et al. (2016). Nonetheless, we
observe no clear trend in size as a function of redshift for the narrow range
covered by these PASSAGES objects (Fig. 10). This agrees with recent results
by Wang et al. (2022) for the FIR size of stacked main-sequence galaxies and
Jiménez-Andrade et al. (2019) for the radio size of star-forming galaxies,
which evolves only shallowly with redshift.

  * •

There may be some size bias present in the sample (Fig. 5), by which more
compact objects are capable of producing higher lensing magnifications
(Hezaveh et al. 2012; Spilker et al. 2016). We parameterize this effect by
comparing magnifications with the ratio of the angular size of background
objects to the Einstein radius of the foreground, and find a modest downward
trend as expected (Fig. 12), which is in line with predictions for isothermal
lenses (e.g. de Freitas et al. 2018). That the PASSAGES sample is not fully
coincident with the distribution of lensing cross-sections $\sigma_{\mu}$ as a
function of the size ratio and magnification is an indication of the selection
function at play, as we are not uniformly sensitive to all magnifications.
Given the selection of these objects by flux, they are likely of an optimal
size that is simultaneously sufficiently large to host high intrinsic
luminosities and sufficiently small to align with higher-magnification regions
of the source plane.

  * •

Since the spatial extents of their star-forming regions are proportionally
larger to match their extreme luminosities, the star-formation surface
densities for PASSAGES are significantly sub-Eddington on global scales (in
agreement with most DSFGs for which this measurement has been made). There is
still certainly a possibility that their rapid stellar mass assembly is
substantially regulated by radiation pressure, but this effect must be on sub-
kpc scales not captured by these current $0.4^{\prime\prime}$ (image-plane)
resolution ALMA images, perhaps over short timescales that are not
contemporaneous between sites of star formation. For these observations, ALMA
does not significantly resolve the far-IR continuum, and so the peak values of
$\Sigma_{\rm SFR}$ in each object are not much larger than the respective
globally-averaged values. In the future, we hope to repeat this analysis with
higher angular resolution from ALMA to properly assess the densities of star
formation, which we expect may be clumpy and extended beyond just nuclear
regions.

##  6 Acknowledgments

.

PSK would like to thank Belén Alcalde Pampliega and Lilah Mercadante for
helpful discussions related to this work, and the scientific staff at the NRAO
Array Operations Center for assistance with data reduction. PSK gratefully
acknowledges support from the NRAO Student Observing Support (SOS) award
SOSPA7-019 and the Massachusetts Space Grant Consortium. This research is
based on observations made with the NASA/ESA Hubble Space Telescope obtained
from the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS 5—26555. These observations are associated with programs GO-14653
and GO-14223. Support for program GO-14653 was provided by NASA through a
grant from the Space Telescope Science Institute. Some of the data presented
in this paper were obtained from the Multimission Archive at the Space
Telescope Science Institute (MAST). This paper makes use of the following ALMA
data: ADS/JAO.ALMA #2017.1.01214.S, #2015.1.01518.S, #2019.1.01636.S. 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. The National Radio
Astronomy Observatory is a facility of the National Science Foundation
operated under cooperative agreement by Associated Universities, Inc. This
work is based in part on observations obtained at the international Gemini
Observatory, a program of NSF’s NOIRLab (acquired through the Gemini
Observatory Archive and processed using the Gemini IRAF package), which is
managed by the Association of Universities for Research in Astronomy (AURA)
under a cooperative agreement with the National Science Foundation on behalf
of the Gemini Observatory partnership: the National Science Foundation (United
States), National Research Council (Canada), Agencia Nacional de Investigación
y Desarrollo (Chile), Ministerio de Ciencia, Tecnología e Innovación
(Argentina), Ministério da Ciência, Tecnologia, Inovações e Comunicações
(Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea).
This research has made use of NASA’s Astrophysics Data System, of APLpy, an
open-source plotting package for Python (Robitaille & Bressert 2012), and of
Astropy, a community-developed core Python package for Astronomy (Astropy
Collaboration et al. 2013).

##  Appendix A Notes on individual objects

Section 5 of Harrington et al. (2016) and Appendix B of Berman et al. (2022)
provide notes on each member of the larger PASSAGES sample (in addition to
Table 1 of Harrington et al. 2021). In this section, we review the information
necessary for our lens models, including spectroscopic redshifts of the
background DSFGs, photometric/spectroscopic redshifts of foreground objects,
and multiple image morphologies, summarized in Table 1.

PJ011646. This galaxy-scale lens system appears to consist of multiple
background components lensed by an elliptical galaxy with a spectroscopic
redshift of $z_{\rm spec}=0.555$ determined with VLT MUSE observations
(Kamieneski et al., in prep.). There is a smaller, secondary foreground galaxy
of unknown redshift to the northeast, but it has minimal impact on the lensing
morphology and so we exclude it from our model. The foreground elliptical is
modeled with our standard approach: an SIE profile with centroid, ellipticity,
orientation, and velocity dispersion all kept as free parameters. The DSFG,
with a CO(3-2) detection at $z=2.125$ by LMT/RSR (Berman et al. 2022), is
detected as two primary arcs in ALMA and JVLA imaging and appears to spatially
coincide with a bright double image detected by HST (labeled $2ab$ in Figure
1). Another image family (labeled $1abcd$) is a clear quad image, nearly
spatially coincident with family $2ab$ but noticeably bluer in the RGB image
with Gemini $r^{\prime}$ and $z^{\prime}$ (Fig. 1). It is not entirely clear
if this image family is located at the same redshift as $2ab$, but since the
two families have very similar Einstein radii and appear to be lensed by
virtually the same foreground mass profile, then this should dictate that the
background objects lie at similar redshifts. Within the Einstein radius
uncertainty for this object (see discussion in Section 3.5), $\theta_{\rm
Ein,eq}=2.34-2.41^{\prime\prime}$, and holding other quantities fixed, this
corresponds to a redshift uncertainty of $z\approx 2.0-2.4$ (assuming $z_{\rm
lens}=0.555$). For the purposes of deriving our lens models, it is sufficient
to assume that they lie at the same redshift. Given the proximity of this quad
component and the doubly-imaged DSFG component in the source plane ($\lesssim
1^{\prime\prime}$, 8.5 kpc at the CO-derived redshift), it seems most likely
that these two components may be interacting (or else a very serendipitous
alignment of two objects around $z\sim 2$). We also note that these multiple
components are aligned with the foreground in such a way as to cover most of
the caustic curve region so that there is a very complex set of concentric,
partial Einstein ring structures in the image plane, offering excellent
constraints on the mass of the foreground elliptical, $(1.54\pm 0.03)\times
10^{12}M_{\odot}$ within the Einstein radius at $z_{\rm lens}=0.555$ (see
Table 2).

PJ014341. This compact galaxy-scale lens is one of the lowest-redshift members
of our sample, with a CO(2-1) detection from the background DSFG at $z=1.096$.
The lensing foreground galaxy is detected spectroscopically at $z_{\rm
spec}=0.594$ from SDSS (Berman et al. 2022). Its small Einstein radius of
$\approx 0.5^{\prime\prime}$ (on the lower end of our sample, see Table 2) is
due primarily to the small separation in redshift space between the lens and
source planes. The enclosed foreground mass is $(1.3\pm 0.3)\times
10^{11}M_{\odot}$. The redshift geometry also has the consequence of making
lensed image identification difficult, as they are likely heavily blended with
foreground light in the optical (Fig. 1), and only marginally resolved beyond
the beam size in the interferometric ALMA/JVLA images. Nonetheless, the image
morphology is consistent with a classic 4-image fold-caustic geometry, with
two isolated images and an additional merging pair (Fig. 2). The radio
morphology is similar (Fig. 2) but more difficult to interpret due to its
lower signal-to-noise, so it is not included as a constraint in this iteration
of the model. Given the compactness of the lensing configuration, the source-
plane area subtended by the caustic curve is likely smaller than or similar in
size to the DSFG, so slight offsets between radio and far-IR emitting regions
can result in very different image morphologies. There appear to be other
foreground objects nearby, but given the compactness of the lensing
configuration, we only parameterize the primary deflector with our standard
SIE model. The best-fit model finds a high ellipticity,
$e=0.77^{+0.15}_{-0.21}$, which suggests these other foreground deflectors may
contribute non-trivially. A refined model with more available constraints in
the future might allow for a more complex mass distribution.

PJ020941. Also known as 9io9 or the “Red Radio Ring", PJ020941 has been
studied at length by Geach et al. (2015, 2018), Harrington et al. (2016,
2019), Rivera et al. (2019), Doherty et al. (2020), and Liu et al. (2022). A
comparison of our lens model with existing models is provided in Appendix D.
The target was first revealed by the gravitational lens citizen science
project Space Warps (Geach et al. 2015; Marshall et al. 2016; More et al.
2016) before it was identified independently as a strong sub-millimeter source
in surveys by Planck (Harrington et al. 2016), the Atacama Cosmology Telescope
(Su et al. 2017; Gralla et al. 2020), and Herschel (Viero et al. 2014). Near-
infrared $i$-band and $J/K_{s}$-band imaging from the VISTA-CFHT Stripe 82
survey (Geach et al. 2017) and CFHT-MegaCam Stripe 82 survey (Moraes et al.
2014) revealed a nearly complete, red Einstein ring (Geach et al. 2015),
confirmed with high-resolution $H$-band HST imaging (Geach et al. 2018,
Lowenthal et al. in prep.). The 1.4 GHz eMERLIN and 5 GHz JVLA radio images
(Geach et al. 2015) show a partial Einstein ring that largely coincides
spatially with the near-IR emission, which is also borne out by ALMA continuum
and spectral line images (Geach et al. 2018; Doherty et al. 2020; Berman et
al. 2022; and this work).

PJ020941 has a CO(3-2) detection by LMT/RSR at $z=2.553$ (Harrington et al.
2016, 2021). It is strongly lensed by an elliptical galaxy (SDSS
J020941.27+001558.5) at $z_{\rm fg}=0.202$, and by a secondary galaxy assumed
to lie at a similar redshift (which is expected to have a non-negligible
influence on the lens, Rivera et al. 2019). Both are included in our model,
with a standard SIE approach for the primary elliptical, and an SIS model
(with position held fixed to the centroid of the luminous component) for the
secondary object.

The overall lens morphology consists of a double image, with an extended arc
to the west and a less-magnified counter-image to the northeast. Recent Cycle
7 ALMA $\theta\sim 0.15^{\prime\prime}$ Band 6 (1.1 mm) continuum imaging
(2019.1.01197.S, PI: P. Kamieneski; to be presented in forthcoming work by
Kamieneski et al. in prep.) provides further information. Embedded inside the
extended arc are 3 distinct quad-imaged regions, suggestive of a cusp-caustic
configuration, which can be identified by matching clumps and using parity
information. This is a commonly-observed lensing configuration where some
portion of a galaxy crosses into the inner region of the caustic, which is
readily apparent from the morphology in Fig. 2. This quadruply-imaged portion
is also evident from subtle surface brightness peaks in the HST image (Fig. 1)
that coincide with the 1 mm peaks.

PJ022633. This partial Einstein ring consists of a background object lensed
primarily by a group of three foreground objects, assumed to lie at the same
redshift. The DSFG CO(3-2) detection is at $z=3.120$ (Berman et al. 2022),
with the brightest lensing galaxy at $z=0.41$ (Wen & Han 2015). Obvious
lensing features include a bright, extended partial Einstein ring to the
southeast (images $1bc$) and a counter-image to the northeast (image $1a$),
with faint emission connecting the two (more apparent in the optical image).
There is some evidence that the lensed image expected to be located on the
western side of the primary deflector (opposite the semi-ring) is split into
two images by the secondary foreground galaxy to the northwest, visible
faintly in $H$-band and at 6 GHz (images $1de$, Figs. 1 and 2). The radio
images are nearly spatially coincident with an extended double-lobed radio jet
emanating from the primary foreground elliptical, and the HST image is
affected by a bright star nearby, so confirmation of this proposed morphology
is still pending. However, this interpretation is motivated by the image
predictions of a simple zeroth-order lens model including only the mass of the
primary and secondary ellipticals. The primary is modeled as an SIE profile,
with the secondary as an SIS model, with its position allowed to vary to a
small amount.

PJ030510. This object is a galaxy-scale (or small group-scale) lens, with a
CO(3-2) detection of the DSFG at $z=2.263$ (Harrington et al. 2021; Berman et
al. 2022). A preliminary red cluster sequence analysis (Gladders & Yee 2000)
yields a foreground redshift of $z_{\rm phot}=0.4-0.5$. Near-IR imaging by HST
and Gemini reveal ring-like features surrounding two foreground galaxies in a
dense environment. Our model parameterizes the combined mass contribution from
both as an SIE model. ALMA 1 mm continuum imaging has a slightly larger beam
size of $0.85\times 0.59^{\prime\prime}$, which leaves the DSFG only
marginally resolved. It appears to consist of two lensed images, one of which
is notably fainter, on opposite sides of the foreground pair. It is possible
that higher-resolution imaging might reveal more of a cusp-like quadruple-
image morphology, but at present, a double-image configuration is the best
explanation.

PJ105353. This object was discovered independently by Harrington et al. 2016
and Cañameras et al. 2015 (also referred to as “the Ruby" or
PLCK_G244.8+54.9), and has also been studied by Cañameras et al. (2017a, b,
2021). We compare our modeling results with theirs in Appendix D. The lensing
galaxy is at a higher redshift than most other PASSAGES objects, $z=1.525$,
with a source redshift of $z=3.005$. It is modeled as an SIE profile, with
position, ellipticity, orientation, and velocity dispersion as free
parameters. As we discuss in Appendix D, CO(4-3) imaging presented in
Cañameras et al. (2017a) reveals a complicated image-plane structure,
dominated by an apparent quadruply-imaged component with no clear counterpart
in rest-frame optical (see also Cañameras et al. 2017a, b), likely due to
blending with the foreground as suggested by Frye et al. (2019). High-
resolution ($0.07^{\prime\prime}$) 1 mm imaging with ALMA (PID 2015.1.01518.S,
PI: N. Nesvadba), which was not directly included in the model by Cañameras et
al. (2017a), leads us to a slightly different conclusion on the quad-image
configuration. For this work, we utilize the Band 7 ($870\mu$m) continuum
image to derive the dust magnification, as the observing setup (resolution
$\gtrsim 0.15^{\prime\prime}$) is closer to that of the other objects.

PJ112713. This object has not been imaged by AzTEC or ALMA, but a CO(2-1) line
was detected by RSR at $z=1.303$ (Harrington et al. 2021; Berman et al. 2022).
A photometric redshift for the lens was derived to be $z_{\rm phot}=0.42$
(Lowenthal et al., in prep.) using SDSS. JVLA imaging indicates two arcs
($1ab$) separated by $\sim 1^{\prime\prime}$, assumed to be the lensed DSFG,
as there are no other bright radio sources within 1′. There is also indication
of an incomplete Einstein ring in HST imaging, surrounding a single foreground
object, with an Einstein radius that approximately matches the 6 GHz arcs,
$\theta_{\rm Ein}=0.6^{\prime\prime}$. The foreground galaxy is modeled with
the standard SIE approach of this work. Intriguingly, there are two bright
peaks in the optical Einstein ring (denoted $2ab$) that roughly correspond to
minima in the partial radio ring. We interpret this as possibly the result of
radio continuum tracing sites of heavily dust-obscured active star formation,
whereas the optical peaks arise from less-extincted sightlines. Both $1ab$ and
$2ab$ are utilized as constraints for our lens model.

PJ113805. A single CO(2-1) line was detected by RSR and interpreted to lie at
$z=2.019$ (Harrington et al. 2021; Berman et al. 2022). ALMA imaging reveals a
compact galaxy-scale lens that is not resolved into multiple components (due
in part to a lower-resolution synthesized beam, $1.03\times
0.66^{\prime\prime}$). The system is however resolved into two primary
components by JVLA 6 GHz imaging (Fig. 2). These double images appear to
nearly coincide with faint arcs detected by HST ($1ab$), although the latter
are heavily contaminated by light from the foreground lensing elliptical,
which has a photometric redshift from SDSS of $z_{\rm phot}=0.52$. Since the
radio peaks are slightly offset from the optical images, we add them as
independent constraints ($2ab$), although they are likely to provide only
minimal new information on the foreground mass. Given the sparse number of
model constraints, this object is modeled only as an SIS profile, with
position and velocity dispersion as free parameters.

PJ113921. This object was discovered independently by Cañameras et al. (2015),
identified as PLCK_G231.3+72.2, and later reported in Berman et al. (2022).
Our model-derived magnification is compared with that of (Cañameras et al.
2018a) in Appendix D. The RSR CO(3-2) line is interpreted from photometric
support to be at $z=2.858$, in agreement with additional spectroscopic
coverage (Cañameras et al. 2018a; Nesvadba et al. 2019; Harrington et al.
2021). ALMA partially resolves what appears to be a quad-image in 260 GHz
continuum ($1abcd$), centered around a foreground elliptical with a
photometric redshift of $z=0.57$ (Berman et al. 2022), but likely perturbed by
at least one other nearby galaxy. Our model here includes the primary lens as
an SIE, and the deflecting galaxy to the west as a simple SIS profile. A faint
red feature in HST is possibly affiliated with the background DSFG, as it is
close to the ALMA continuum emission, or it may be associated with an apparent
foreground spiral galaxy to the southeast. In this iteration of the model, we
do not include this feature in constraining our lens model.

PJ132630. This object was discovered independently through our sample (Berman
et al. 2022) and in the H-ATLAS survey (Bussmann et al. 2013; Yang et al.
2017), known also as NAv1.195. A single CO(3–2) line was detected with RSR at
$z=2.951$ (confirmed with spectroscopic follow-up by Yang et al. 2017 and
Harrington et al. 2021). Bussmann et al. 2013 published a lens model for 340
GHz imaging by the SMA ($\mu_{\rm 340GHz}=4.1\pm 0.3$), with a lens redshift
of $z_{\rm fg}=0.786$. We compare their results with our model in Appendix D.
ALMA and JVLA imaging reveals that the DSFG is doubly-imaged, with images
separated by 3.7′′ in the image plane ($3ab$). An optical counterpart appears
to separate into two components that surround the long-wavelength emission; we
label these image families $1ab$ and $2ab$ and include them separately in the
model. There is evidence for additional background sources lensed by the
foreground elliptical, which we model using a standard SIE, but they are at an
unknown redshift and are thus not considered as constraints.

PJ133634. Two CO lines ($J=3-2$ and $J=4-3$) were detected by RSR at $z=3.254$
(Harrington et al. 2021; Berman et al. 2022). A pair of primary lensing
galaxies separated by 0.9′′ have an SDSS photometric redshift of $z=0.26$. The
6 GHz continuum reveals a clumpy, partial Einstein ring, centered $\approx
0.2^{\prime\prime}$ east of the center of the foreground elliptical (likely
due to a small perturbation by the secondary). This object is too northern to
be visible to ALMA, but the 1.1 mm continuum from AzTEC (8.5′′ beam) coincides
with the radio continuum, which we thus interpret to originate from the Planck
DSFG. The radio ring consists of multiple clumps larger than the synthesized
beam, and we identify 3 doubly-imaged families in the eastern counter-image
and western semi-ring to be used in the model ($1ab$, $2ab$, and $3ab$),
making use of the expected parity flip between the counter-image and more
extended arc. The foreground mass is modeled as a single SIE profile, as the
effect of the secondary perturber is not large enough to break the degeneracy
with the primary lens.

PJ144653. This lower-redshift member of our sample has a CO(2-1) line detected
by RSR at $z=1.084$ (Harrington et al. 2021; Berman et al. 2022). The lensing
galaxy has an SDSS spectroscopic redshift of $z=0.493$, and has the appearance
of a face-on spiral galaxy. However, some of the spiral-like structure may
actually be the result of the multiply-imaged background DSFG. At present, it
is not feasible to make this distinction without imaging in an additional
filter with comparable quality to the HST image. ALMA and JVLA images are
consistent with each other and reveal the DSFG to follow a fold-caustic
lensing configuration, with two isolated images and an additional merging
image pair towards the southeast (labeled $1abc$). The images form a partial
Einstein ring with a radius of $\approx 0.9^{\prime\prime}$. The foreground is
modeled with a single SIE profile. The optimized model has a high ellipticity,
$e=0.65^{+0.18}_{-0.25}$, possibly because of the small number of available
constraints currently available, which necessitates a perhaps too-simplistic
model.

PJ144958. This cluster lens has a CO(3-2) line detection at $z=2.153$
(Harrington et al. 2021; Berman et al. 2022) with a preliminary foreground
cluster redshift measured at $z_{\rm phot}=0.4$. Optical emission probed by
HST and Gemini appears to be more extended than at 1 mm and 6 GHz, which
reveal a set of more point-source-like images. However, there appears at first
to be four distinct images all on one side of a foreground galaxy group, with
no obvious counter-image. We consider it likely that the middle pair of images
($1bc$) is in fact a merging pair on either side of the critical curve (see
Fig. 2). The northernmost image $1d$ also appears to be split into four images
by another intervening cluster member ($4abc$), visible only with HST’s higher
resolution and unresolved by ALMA and JVLA. As constraints on our model, we
include images $1abcd$ (JVLA/ALMA locations), $2abc$ for the peaks of the HST
images, and $3abc$ as the fainter tail visible only in optical (adjacent to
$2abc$). Lastly, we include images $4abc$ for the point-like sources
surrounding the perturbing foreground. The lensing distribution is
parameterized by a cluster-scale NFW profile, 3 smaller SIS profiles
describing the group of elliptical galaxies interior to the arc, and a fourth
SIS profile for the perturbing galaxy at the northern end of the arc.

PJ160722. This member of our sample has a CO(2-1) line observed by RSR at
$z=1.482$, confirmed with photometric information and subsequent spectroscopic
follow-up (Harrington et al. 2016, 2021). HST imaging reveals two apparent
foreground lensing objects with a photometric redshift of $z=0.65$, determined
from the optical to mid-IR SED in (Harrington et al. 2016). A bright radio AGN
is detected in the northern foreground (Fig. 2). Both near-IR and radio reveal
extended ring-like arcs tightly surrounding the foreground. Four peaks
embedded with the surrounding ring are apparent in the radio and, to a lesser
extent, in the optical image, which we interpret as a quadruply-imaged region
of the source plane. The lensing mass distribution is represented by a single
SIE model, given the proximity of the two foregrounds (and therefore strong
degeneracies in optimal parameters between the two).

PJ231356. This object has a CO(3-2) line observed by RSR at $z=2.215$,
confirmed by spectroscopic follow-up (Harrington et al. 2021; Berman et al.
2022). HST/WFC3 1.6µm imaging appears to show a primary lensing galaxy, with a
secondary galaxy to the north that appears to have a substantial impact on the
lensing morphology, leading to a surprisingly straight arc towards the north.
Numerous background arcs are visible in HST, some of which coincide with the
arcs in ALMA 260 GHz continuum. Most of these multiple images have not been
identified definitively, but the quadruple-image family $1abcd$ is a
sufficient constraint for the model at present. The lensing mass distribution
is parameterized by the primary elliptical (SIE) and the secondary perturber
to the north (SIS). While the $x$-position of the primary galaxy is left as a
free parameter, we hold fixed its $y$-position to avoid degeneracy with the
secondary lens.

##  Appendix B Lensed multiple image systems used as constraints

In §3.1, we discuss our approach to identifying multiply-imaged features of
the background DSFGs using HST, Gemini, JVLA, and ALMA observations. In Table
5, we summarize these features that were used as constraints for our models,
which are also marked on Fig. 1.

Table 5:  Multiple image positions used as constraints in the lensing models
(presented in Table 6), as per §3.1. In some rare cases, the location and
orientation of caustic curves as inferred by lensing morphologies are used as
constraints.

Field | Image # | RA | Dec. | $z$
---|---|---|---|---
|  | [deg.] | [deg.] |
PJ011646 | 1a | 19.195512 | -24.617222 | $2.125$
| 1b | 19.194428 | -24.617920 | —
| 2a | 19.195611 | -24.617580 | $2.125^{a}$
| 2b | 19.195158 | -24.616568 | —
| 2c | 19.194320 | -24.617293 | —
| 2d | 19.194704 | -24.617876 | —
PJ014341 | 1a | 25.921762 | -1.790685 | $1.096$
| 1b | 25.921923 | -1.7905189 | —
| 1c | 25.921752 | -1.7904362 | —
| 1d | 25.921651 | -1.7905364 | —
PJ020941 | 1a | 32.422486 | 0.26643019 | 2.554
| 1b | 32.421489 | 0.26561695 | —
| 2a | 32.422507 | 0.26633282 | —
| 2b | 32.421262 | 0.26661786 | —
| 3a | 32.422500 | 0.26639240 | —
| 3b | 32.421370 | 0.26574041 | —
| 3c | 32.421216 | 0.26607387 | —
| 3d | 32.421181 | 0.26628788 | —
| 4a | 32.422470 | 0.26642723 | —
| 4b | 32.421480 | 0.26561101 | —
PJ022633 | 1a | 36.642144 | 23.758811 | $3.120$
| 1b | 36.642601 | 23.757341 | —
| 1c | 36.642373 | 23.757083 | —
| 1d | 36.640842 | 23.75822 | —
| 1e | 36.640552 | 23.758414 | —
PJ030510 | 1a | 46.294327 | -30.608571 | $2.263$
| 1b | 46.294248 | -30.608259 | —
(PA $=$ 114∘) | critic1 | 46.294451 | -30.608419 | —
(PA $=$ 49∘) | critic2 | 46.294136 | -30.608439 | —
PJ105353 | 1a | 163.47142 | 5.9387914 | 3.005
| 1b | 163.47121 | 5.9385352 | —
| 1c | 163.47158 | 5.9386185 | —
| 1d | 163.47144 | 5.9383833 | —
| 2a | 163.47152 | 5.9386507 | —
| 2b | 163.47128 | 5.9383962 | —
PJ112713 | 1a | 171.80585 | 46.156730 | 1.303
| 1b | 171.80619 | 46.156579 | —
| 2a | 171.80595 | 46.156529 | —
| 2b | 171.80630 | 46.156730 | —
| 3a | 171.80619 | 46.156825 | —
| 3b | 171.80596 | 46.156528 | —

Note. — † Lensed images of PJ231356 are blended with a foreground double-lobed
radio jet at 6 GHz, so the 6 GHz flux and effective radius of the lensed DSFG
cannot be determined. The reported flux includes significant contribution from
the bright foreground object.

Table 5:  (continued)

Field | Image # | RA | Dec. | $z$
---|---|---|---|---
|  | [deg.] | [deg.] |
PJ113805 | 1a | 174.52314 | 32.965895 | 2.019
| 1b | 174.523 | 32.965711 | —
| 2a | 174.52318 | 32.965891 | —
| 2b | 174.52302 | 32.96573 | —
PJ113921 | 1a | 174.84046 | 20.414097 | 2.858
| 1b | 174.84068 | 20.414272 | —
| 1c | 174.84023 | 20.414594 | —
| 1d | 174.84063 | 20.414802 | —
PJ132630 | 1a | 201.62551 | 33.735859 | 2.951
| 1b | 201.62637 | 33.735124 | —
| 2a | 201.62552 | 33.735968 | —
| 2b | 201.62625 | 33.735165 | —
| 3a | 201.62549 | 33.735844 | —
| 3b | 201.62636 | 33.735133 | —
PJ133634 | 1a | 204.14616 | 49.220425 | 3.254
| 1b | 204.14534 | 49.220706 | —
| 2a | 204.14607 | 49.220635 | —
| 2b | 204.14515 | 49.220373 | —
| 3a | 204.14586 | 49.220788 | —
| 3b | 204.14545 | 49.220180 | —
PJ144653 | 1a | 221.72176 | 17.876174 | $1.084$
| 1b | 221.72135 | 17.875992 | —
(PA $=$ 35∘) | critic1 | 221.72186 | 17.875843 | —
PJ144958 | 1a | 222.49332 | 22.642215 | $2.153$
| 1b | 222.49347 | 22.643774 | —
| 1c | 222.49374 | 22.644154 | —
| 1d | 222.49416 | 22.644439 | —
| 2a | 222.49329 | 22.642269 | $2.153$
| 2b | 222.49346 | 22.643802 | —
| 2c | 222.49371 | 22.644152 | —
| 3a | 222.49331 | 22.642477 | $2.153$
| 3b | 222.49333 | 22.643417 | —
| 3c | 222.4939 | 22.644268 | —
| 4a | 222.49389 | 22.644384 | $2.153$
| 4b | 222.49402 | 22.644521 | —
| 4c | 222.49415 | 22.644548 | —
PJ160722 | 1a | 241.84517 | 73.783702 | 1.484
| 1b | 241.84558 | 73.784165 | —
| 1c | 241.84418 | 73.784227 | —
| 1d | 241.84422 | 73.783829 | —
PJ231356 | 1a | 348.48594 | 1.1542249 | 2.217
| 1b | 348.48654 | 1.1545733 | —
| 1c | 348.48615 | 1.1550713 | —
| 1d | 348.48519 | 1.1548946 | —

Note. — a Assumed to be located at the same redshift as the DSFG.

##  Appendix C Best-fit lens model parameters

In Table 6, we summarize the best-fit lens models for each object. In most
cases, we list “median," “best," and “mode" solutions, which represent the set
of parameters consisting (respectively) of the median of the posterior
distribution, the highest-likelihood solution recovered from the MCMC
iterations, and the mode of the posterior distribution. The latter solution is
usually employed when the posterior distribution is bimodal, especially in
terms of position angle where there may be solutions offset by $90^{\circ}$.

Table 6:  Optimized gravitational lens model parameters. Unless otherwise
noted, all potentials are parameterized with singular isothermal ellipsoid (or
spheroid) models. For NFW potentials, scale radius in arcsec ($r_{s}$)
replaces velocity dispersion as a parameter. While the median of the posterior
distribution for each parameter is typically the preferable statistical
estimator, multi-modal distributions and highly non-Gaussian distributions may
be better summarized by the mode of the posterior or the highest-likelihood
(“best") sample. The first solution set listed (median, best, or mode) is
taken to be the preferred model, but others are listed for completeness.

Field |  |  |  |  |  |
---|---|---|---|---|---|---
Model description | RA (deg.) | Dec. (deg.) | RMS${}_{\rm im}(^{\prime\prime})$ | $\sigma_{\rm position}(^{\prime\prime})$ | $\nu$ | $\chi_{\nu}^{2}$
potential $1,2,...,n$ |  $\Delta\alpha$ (′′) |  $\Delta\delta$ (′′) | $e$ |  $\theta$ (∘) |  $\sigma$ (km s-1) | $z$
PJ011646 |  |  |  |  |  |
Median | 19.194989 | -24.617376 | — | 0.10 | 3 | 2.0
potential 1 | $-0.12^{+0.03}_{-0.04}$ | $0.29^{+0.03}_{-0.03}$ | $0.37^{+0.04}_{-0.04}$ | $15.8^{+1.2}_{-1.2}$ | $364.0^{+3.0}_{-3.0}$ | $0.555^{*}$
Best | — | — | 0.05 | — | 3 | 0.5
potential 1 | $-0.12$ | $0.30$ | $0.36$ | $15.7$ | $363.8$ | $0.555^{*}$
PJ014341 |  |  |  |  |  |
Median | 25.921794 | -1.7905377 | — | 0.17 | 3 | 1.7
potential 1 | $-0.06^{+0.08}_{-0.07}$ | $-0.03^{+0.06}_{-0.06}$ | $0.77^{+0.15}_{-0.21}$ | $3.6^{+16.1}_{-11.2}$ | $209.7^{+11.4}_{-10.8}$ | $0.594^{*}$
Best | — | — | 0.04 | — | 3 | 0.6
potential 1 | $-0.06$ | $-0.02$ | $0.55$ | $-1.7$ | $202.1$ | $0.594^{*}$
PJ020941 |  |  |  |  |  |
Median | 32.42192 | 0.26621843 | — | 0.15 | 6 | 0.8
potential 1 | $-0.28^{+0.13}_{-0.12}$ | $0.12^{+0.04}_{-0.05}$ | $0.36^{+0.08}_{-0.08}$ | $-14.3^{+1.0}_{-0.9}$ | $312.0^{+5.9}_{-8.5}$ | $0.202^{*}$
potential 2 (SIS) | $0.22^{*}$ | $2.59^{*}$ | — | — | $82.6^{+42.2}_{-48.4}$ | $0.202^{*}$
Best | — | — | 0.04 | — | 6 | 0.1
potential 1 | $-0.26$ | $0.12$ | $0.33$ | $-14.6$ | $314.7$ | $0.202^{*}$
potential 2 (SIS) | $0.22^{*}$ | $2.59^{*}$ | — | — | $67.6$ | $0.202^{*}$
PJ022633 |  |  |  |  |  |
Median | 36.641654 | 23.757872 | — | 0.40 | 1 | 5.9
potential 1 | $-0.08^{+0.23}_{-0.26}$ | $-0.12^{*}$ | $0.25^{+0.11}_{-0.09}$ | $-9.0^{+6.8}_{-63.9}$ | $376.3^{+10.9}_{-11.5}$ | $0.41^{*}$
potential 2 (SIS) | $2.94^{+0.43}_{-0.33}$ | $2.08^{+0.32}_{-0.55}$ | — | — | $208.3^{+25.6}_{-26.3}$ | $0.41^{*}$
Best | — | — | 0.10 | — | 1 | 0.3
potential 1 | $-0.08$ | $-0.12^{*}$ | $0.10$ | $-13.6$ | $383.0$ | $0.41^{*}$
potential 2 (SIS) | $3.17$ | $1.88$ | — | — | $170.7$ | $0.41^{*}$
PJ030510 |  |  |  |  |  |
Median | 46.294291 | -30.608356 | — | 0.14 | 1 | 2.0
potential 1 | $-0.04^{+0.13}_{-0.09}$ | $-0.48^{+0.19}_{-0.14}$ | $0.39^{+0.32}_{-0.28}$ | $3.5^{+56.9}_{45.1}$ | $169.6^{+13.6}_{-11.8}$ | $0.4^{*}$
Best | — | — | $<0.01$ | — | 3 | 0.002
potential 1 | $-0.01$ | $-0.39$ | $0.07$ | $-2.1$ | $165.0$ | $0.4^{*}$
PJ105353 |  |  |  |  |  |
Median | 163.4714 | 5.9385728 | — | 0.10 | 3 | 2.4
potential 1 | $0.05^{+0.05}_{-0.04}$ | $0.06^{+0.03}_{-0.02}$ | $0.35^{+0.15}_{-0.11}$ | $-6.8^{+3.9}_{-3.9}$ | $283.0^{+7.2}_{-7.6}$ | $1.525^{*}$
Best | — | — | 0.05 | — | 3 | 0.5
potential 1 | $0.03$ | $0.03$ | $0.20$ | $-5.9$ | $281.7$ | $1.525^{*}$
PJ112713 |  |  |  |  |  |
Median | 171.80606 | 46.15670 | — | 0.20 | 3 | 0.8
potential 1 (SIS) | $0.00^{+0.08}_{-0.09}$ | $-0.32^{+0.08}_{-0.08}$ | — | — | $182.7^{+7.4}_{-6.8}$ | $0.415^{*}$
Best | — | — | 0.08 | — | 3 | 0.3
potential 1 (SIS) | $-0.03$ | $-0.28$ | — | — | $179.2$ | $0.415^{*}$
PJ113805 |  |  |  |  |  |
Mode | 174.52305 | 32.965806 | — | 0.20 | 1 | 0.3
potential 1 (SIS) | $0.04^{+0.14}_{-0.14}$ | $-0.15^{+0.20}_{-0.17}$ | — | — | $145.5^{+11.2}_{-9.6}$ | $0.52^{*}$
Best | — | — | 0.03 | — | 2 | 0.6
potential 1 (SIS) | $-0.01$ | $-0.14$ | — | — | $146.1$ | $0.52^{*}$
PJ113921 |  |  |  |  |  |
Modea | 174.84041 | 20.414479 | — | 0.39 | 2 | 1.3
potential 1 | $0.00^{*}$ | $0.01^{*}$ | $0.65$ | $28.6$ | $230.5$ | $0.57^{*}$
potential 2 (SIS) | $2.56^{*}$ | $0.58^{*}$ | — | — | $179.0$ | $0.57^{*}$
Median | — | — | — | — | 2 | 1.7
potential 1 | $0.00^{*}$ | $0.01^{*}$ | $0.49^{+0.15}_{-0.19}$ | $31.6^{+14.0}_{-12.4}$ | $232.7^{+17.3}_{-18.8}$ | $0.57^{*}$
potential 2 (SIS) | $2.56^{*}$ | $0.58^{*}$ | — | — | $191.5^{+58.9}_{-57.3}$ | $0.57^{*}$
Best | — | — | 0.23 | — | 2 | 0.7
potential 1 | $0.00^{*}$ | $0.01^{*}$ | $0.69$ | $-74.8$ | $225.5$ | $0.57^{*}$
potential 2 (SIS) | $2.56^{*}$ | $0.58^{*}$ | — | — | $274.9$ | $0.57^{*}$
PJ132630 |  |  |  |  |  |
Modea | 201.62613 | 33.735272 | — | 0.38 | 1 | 0.4
potential 1 | $-0.03$ | $0.14$ | $0.02$ | $-42.6$ | $331.0$ | $0.786^{*}$
Median | — | — | — | — | 1 | 1.3
potential 1 | $0.01^{+0.14}_{-0.12}$ | $0.15^{+0.13}_{-0.13}$ | $0.19^{+0.18}_{-0.13}$ | $-42.5^{+25.0}_{-29.8}$ | $330.5^{+13.7}_{-12.8}$ | $0.786^{*}$
Best | — | — | 0.02 | — | 1 | 0.02
potential 1 | $0.21$ | $0.31$ | $0.04$ | $-1.8$ | $334.5$ | $0.786^{*}$
PJ133634 |  |  |  |  |  |
Mode | 204.14596 | 49.220455 | — | 0.30 | 5 | 0.7
potential 1 | $0.37$ | $0.43$ | $0.01$ | $29.1$ | $218.7$ | $0.26^{*}$
Median | — | — | — | — | 5 | 0.8
potential 1 | $0.37^{+0.26}_{-0.24}$ | $0.40^{+0.14}_{-0.24}$ | $0.20^{+0.28}_{-0.16}$ | $70.3^{+59.5}_{-41.5}$ | $220.2^{+10.7}_{-8.0}$ | $0.26^{*}$
Best | — | — | 0.08 | — | 5 | 0.1
potential 1 | $0.71$ | $-0.17$ | $0.12$ | $160.6$ | $227.3$ | $0.26^{*}$
PJ144653 |  |  |  |  |  |
Median | 221.72172 | 17.875883 | — | 0.37 | 1 | 3.2
potential 1 | $0.10^{*}$ | $0.47^{*}$ | $0.65^{+0.18}_{-0.25}$ | $95.0^{+14.1}_{-14.0}$ | $262.8^{+18.0}_{-14.7}$ | $0.493^{*}$
Best | — | — | 0.03 | — | 1 | 0.5
potential 1 | $0.10^{*}$ | $0.47^{*}$ | $0.66$ | $96.4$ | $257.8$ | $0.493^{*}$
PJ144958 |  |  |  |  |  |
Median | 222.49514 | 22.642985 | — | 0.50 | 13 | 0.4
potential 1 (NFW) | $-1.11^{+2.56}_{-2.87}$ | $-1.55^{+1.40}_{-1.32}$ | $0.64^{+0.12}_{-0.20}$ | $12.2^{+4.2}_{-5.1}$ | $r_{s}(^{\prime\prime})=26.1^{+4.0}_{-3.5}$ | $0.4^{*}$
potential 2 (SIS) | $3.52^{*}$ | $5.13^{*}$ | — | — | $114.0^{+10.0}_{-11.3}$ | $0.4^{*}$
potential 3 (SIS) | $3.22^{*}$ | $1.73^{*}$ | — | — | $224.5^{+38.9}_{-35.9}$ | $0.4^{*}$
potential 4 (SIS) | $8.44^{*}$ | $7.02^{*}$ | — | — | $89.6^{+76.4}_{-60.2}$ | $0.4^{*}$
potential 5 (SIS) | $-1.63^{*}$ | $-3.80^{*}$ | — | — | $157.8^{+128.8}_{-111.4}$ | $0.4^{*}$
Best | — | — | 0.12 | — | 13 | 0.1
potential 1 (NFW) | $-1.27$ | $-1.96$ | $0.70$ | $10.7$ | $r_{s}(^{\prime\prime})=25.2$ | $0.4^{*}$
potential 2 (SIS) | $3.52^{*}$ | $5.13^{*}$ | — | — | $107.2$ | $0.4^{*}$
potential 3 (SIS) | $3.22^{*}$ | $1.73^{*}$ | — | — | $242.9$ | $0.4^{*}$
potential 4 (SIS) | $8.44^{*}$ | $7.02^{*}$ | — | — | $193.4$ | $0.4^{*}$
potential 5 (SIS) | $-1.63^{*}$ | $-3.80^{*}$ | — | — | $76.4$ | $0.4^{*}$
PJ160722 |  |  |  |  |  |
Best | 241.84491 | 73.784031 | 0.13 | 0.20 | 1 | 1.6
potential 1 | $-0.04$ | $-0.10$ | $0.36$ | $-34.0$ | $273.6$ | $0.65^{*}$
Median | — | — | — | — | 1 | 5.2
potential 1 | $-0.03^{+0.15}_{-0.14}$ | $-0.09^{+0.09}_{-0.07}$ | $0.61^{+0.13}_{-0.20}$ | $-32.1^{+82.6}_{-6.6}$ | $278.1^{+12.0}_{-11.6}$ | $0.65^{*}$
PJ231356 |  |  |  |  |  |
Median | 348.485940 | 1.154663 | — | 0.10 | 1 | 4.0
potential 1 | $-0.10^{+0.07}_{-0.07}$ | $0.04^{*}$ | $0.57^{+0.15}_{-0.09}$ | $117.8^{+12.0}_{-4.0}$ | $306.7^{+10.4}_{-12.9}$ | $0.56^{*}$
potential 2 (SIS) | $0.25^{*}$ | $5.26^{*}$ | — | — | $297.8^{+38.5}_{-43.5}$ | $0.56^{*}$
Best | — | — | 0.03 | — | 1 | 0.4
potential 1 | $-0.11$ | $0.04^{*}$ | $0.62$ | $115.6$ | $308.7$ | $0.56^{*}$
potential 2 (SIS) | $0.25^{*}$ | $5.26^{*}$ | — | — | $290.4$ | $0.56^{*}$

Note. — Fixed parameters are denoted with an asterisk. Asymmetric error bars
indicate $1\sigma$ confidence levels and are determined from the resulting
MCMC posterior distribution. Ellipticity is defined as
$e=(a^{2}-b^{2})/(a^{2}+b^{2})$, for semi-major and semi-minor axes $a$ and
$b$. The position angle $\theta$ is measured counterclockwise from east.
Velocity dispersion $\sigma$ for singular isothermal ellipsoids are
functionally equivalent to physical velocity dispersions. The Right ascension
(RA) and Declination (Dec.) given are reference positions (to which
$\Delta\alpha$ and $\Delta\delta$ are relative), and are typically chosen as
the centroid of the brightest foreground. The positional uncertainty
$\sigma_{\rm position}$ is supplied for each object, and is equal to the size
of the synthesized beam for ALMA or JVLA (whichever is lowest resolution).
RMSim is the RMS deviation between observed and modeled multiple image
positions for each field, calculated in the image plane for the
lowest-$\chi^{2}$ solution. The reduced $\chi_{\nu}^{2}$ is reported for the
given number of degrees of freedom $\nu$. a In some cases, the mode solution
of the optimization is used instead of the median, which is less useful when
the posterior distribution is bimodal or if one parameter’s optimization is
sensitive to the choice of parameter bounds (e.g., when ellipticity is poorly
constrained by the multiple images, the posterior tends to follow more of a
uniform distribution, so the median tends towards the midpoint of the
parameter bounds). For these solutions, the uncertainty in the median is taken
to be representative of the uncertainty in the mode.

##  Appendix D Comparison with existing lens models

Several of the objects included in this work have already been the subject of
gravitational lens modeling, but we independently derive models here to both
incorporate information from new observations and to ensure consistency
between models for this subsample. In particular, we examine discrepancies in
derived magnifications as a benchmark.

First, PJ020941 (or 9io9) is perhaps the most well-studied DSFG included, with
independent modeling efforts by Geach et al. (2015, 2018); Rivera et al.
(2019), and Liu et al. (2022). The initial model by Geach et al. (2015) was
based on 4 different radio observations and near-IR ground-based imaging
(seeing $\approx 0.8^{\prime\prime}$), and was parameterized by an isothermal
ellipsoid and a satellite isothermal spheroid for the secondary foreground,
with added external shear. With the gravlens software (Keeton 2011), they
modeled the near-IR as a single Sérsic profile and the radio as two Gaussian
components, finding a magnification of $\mu\approx 10$, in agreement with what
we find, $\mu_{\rm 1mm}=10.5\pm 3.7$ and $\mu_{\rm 6GHz}=12.0\pm 3.5$ (Table
2). Revising this model using CO(4-3) ALMA spectro-imaging and a semi-linear
inversion approach (Warren & Dye 2003; Dye et al. 2018), Geach et al. (2018)
found a slightly higher magnification of $\mu_{\rm CO4-3}=14.7\pm 0.3$. Rivera
et al. (2019) incorporated CO(3-2) imaging with NOEMA, with 32 independent
velocity channels as constraints, finding a smooth velocity gradient
consistent with the structure found by Geach et al. (2018). Also using a
2-component lens model with external shear, they found that the magnification
varied strongly with velocity, ranging from $\mu\approx 7-22$, but with a
luminosity-weighted mean of $\langle\mu\rangle\approx 13$. Most recently, Liu
et al. (2022) used newly-collected adaptive optics $H$ and $K_{s}$-band
imaging, alongside HST/WFC3 F125W, SMA 870 $\mu$m, and multi-J CO observations
with ALMA in order to build their own lens model. As with our work, Liu et al.
used lenstool to fit the primary and secondary foreground with an SIE and SIS
profile, constrained by the F125W image. This approach is the one most similar
to our own, so an equivalent comparison is possible. The authors found
optimized parameters that agreed with all in our model (Table 6) within
uncertainties, except for only a small disagreement for position angle (PA
$=-8\pm 2$∘ vs. our $-14\pm 1$∘), and their source-plane reconstruction is
visually very similar to ours (Fig. 2). As a result, Liu et al. found a dust
continuum magnification of $\mu_{\rm dust}=12.8\pm 0.3$ and a stellar
component magnification of $\mu_{\rm star}=13.6\pm 0.4$, consistent with our
value of $\mu_{\rm dust}\approx 10.5$, especially given the different
interferometric configurations of ALMA vs. SMA.

PJ105353 (or G244.8+54.9, the “Ruby") was previously discussed and modeled by
Cañameras et al. (2017a, b); Frye et al. (2019). Cañameras et al. used
lenstool to model the $z=1.525$ foreground with a pseudo-isothermal elliptical
mass density (PIEMD) profile (with the poorly-constrained core and cut radii
held fixed at 0.15 kpc and 100 kpc), based on image locations and parities
from CO(4-3) observations (angular resolution 0.1′′). However, the authors
acknowledged that these image identifications were not completely unambiguous,
but settled ultimately on a pair of source-plane objects imaged 2 and 4 times,
respectively. An additional 0.07′′-resolution Band 6 (1 mm) observation with
ALMA (Program 2015.1.01518.S, PI: N. Nesvadba) was not included as a
constraint, but our interpretation of the image configuration differs
slightly. Whereas Cañameras et al. identify a cusp-like configuration, the 1
mm image appears to instead show more of an Einstein cross, with 4 well-
separated images (nearly aligned north, south, east, and west). However, the
second, doubly-imaged system is in accord with Cañameras et al..
Unfortunately, the non-detection of the background object with HST (as it is
blended with the foreground; Frye et al. 2019) makes confirmation of this
morphology difficult. Cañameras et al. find a dust magnification of $\mu_{\rm
3mm}=21.8\pm 0.6$ (reported in Cañameras et al. 2018a), with magnifications
for individual clumps ranging from $\mu\approx 7-38$. In our case, we find a
significantly different total value of $\mu_{870\mu{\rm m}}=7.6\pm 0.7$ (or
$\mu_{\rm 1mm}\approx 12$). Given the difficulty in deriving robust
constraints on the model, a discrepancy is not particularly unexpected,
especially as both interpretations agree that the source-plane galaxy lies
near to a caustic, where the magnification gradient is steep and highly
sensitive to the model. On the other hand, values like the Einstein radius are
not as sensitive, and our value is in close agreement with that of Cañameras
et al. (2017a).

While a lens model for PJ113921 (G231.3+72.2) has not yet been published,
Cañameras et al. (2018a) provide magnifications of $\mu_{\rm dust}=7.9\pm 0.3$
and $\mu_{\rm gas}=6.0\pm 0.5$, based on SMA 880 $\mu$m continuum and multi-J
CO imaging with the IRAM Plateau de Bure Interferometer (PdBI), which are
broadly consistent with our measurements of $\mu\approx 5-9$ from ALMA and
JVLA.

Lastly, PJ132630 was modeled by Bussmann et al. (2013) based on HST F110W
near-IR and SMA 880 $\mu$m imaging observations, using an interferometric
visibility-based approach. They found an Einstein radius of
$\theta_{E}=1.80\pm 0.02$ (consistent with our $\theta_{\rm Ein}=1.78\pm
0.18$; Table 2), a lens ellipticity of $\epsilon=0.26\pm 0.04$ (similar to our
median solution, $e=0.19\pm 0.16$; Table 6), and a dust continuum
magnification of $\mu_{880\mu{\rm m}}=4.1\pm 0.3$ (in close agreement with our
$\mu_{\rm 1mm}=4.3\pm 0.6$).

While magnifications are generally robust between various lens modeling
approaches, the non-trivial differences between this work and others should
not be discounted. Spilker et al. (2016) computes magnification as flux-
weighted average over the elliptical model components describing the source
plane, whereas Bussmann et al. (2013, 2015) use the ratio of image-plane to
reconstructed source-plane flux. On the other hand, Dye et al. (2018) explore
the sensitivity of magnification to imaging resolution and depth by
calculating magnification as a function of fraction of source-plane flux (i.e.
surface brightness threshold). The authors found that, while variation can be
small, in the vicinity of a caustic in particular, magnification can vary by
up to 25% as a function of interferometric configuration. Our work does not
make explicit assumptions of source-plane structure, but also does not take
advantage of the entirety of information present in the interferometric images
like the visibility-modeling approach of these other works. Future work to
apply a visibility-based modeling approach to these objects will allow for a
more direct comparison. An additional promising method that we hope to explore
in the future involves Regularized Semi-linear Inversion (Warren & Dye 2003;
Nightingale & Dye 2015; Enia et al. 2018), by which the source plane is
tessellated but not assumed to follow a specific parametric form.

In summary, the general agreement for most objects with prior models (derived
from independent constraints) helps to lend credence to their robustness. We
interpret any tensions in results—in particular, for PJ105353—to be more
influenced by different observing setups than by different lens modeling
approaches.

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