#  Swift J1727.8-1613 has the Largest Resolved Continuous Jet Ever Seen in an
X-ray Binary

Callan M. Wood International Centre for Radio Astronomy Research, Curtin
University, GPO Box U1987, Perth, WA 6845, Australia  James C. A. Miller-Jones
International Centre for Radio Astronomy Research, Curtin University, GPO Box
U1987, Perth, WA 6845, Australia  Arash Bahramian International Centre for
Radio Astronomy Research, Curtin University, GPO Box U1987, Perth, WA 6845,
Australia  Steven J. Tingay International Centre for Radio Astronomy Research,
Curtin University, GPO Box U1987, Perth, WA 6845, Australia  Steve Prabu
International Centre for Radio Astronomy Research, Curtin University, GPO Box
U1987, Perth, WA 6845, Australia  Thomas D. Russell INAF, Istituto di
Astrofisica Spaziale e Fisica Cosmica, Via U. La Malfa 153, I-90146 Palermo,
Italy  Pikky Atri ASTRON, Netherlands Institute for Radio Astronomy, Oude
Hoogeveensedĳk 4, 7991 PD Dwingeloo, The Netherlands  Francesco Carotenuto
Astrophysics, Department of Physics, University of Oxford, Keble Road, Oxford,
OX1 3RH, UK  Diego Altamirano School of Physics and Astronomy, University of
Southampton, University Road, Southampton SO17 1BJ, UK  Sara E. Motta INAF -
Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate, Italy
Lucas Hyland Mathematics & Physics, School of Natural Sciences, University of
Tasmania, Private Bag 37, Hobart, Tasmania, 7001  Cormac Reynolds CSIRO
Astronomy and Space Science, P.O. Box 1130, Bentley, WA 6102, Australia
Stuart Weston  Space Operations New Zealand Ltd, Hargest House, PO Box 1306,
Invercargill 9840, New Zealand  Rob Fender  Astrophysics, Department of
Physics, University of Oxford, Keble Road, Oxford, OX1 3RH, UK  Elmar Körding
Department of Astrophysics/IMAPP, Radboud University, P.O. Box 9010, 6500 GL
Nijmegen, The Netherlands  Dipankar Maitra Department of Physics and
Astronomy, Wheaton College, Norton, MA 02766, USA  Sera Markoff Anton
Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904,
1098 XH Amsterdam, The Netherlands  Gravitation and Astroparticle Physics
Amsterdam Institute, University of Amsterdam, Science Park 904, 1098 XH 195
196 Amsterdam, The Netherlands  Simone Migliari  Aurora Technology, Calle
Principe de Vergara, 211, 1-B, E-28002 Madrid, Spain  David M. Russell Center
for Astrophysics and Space Science (CASS), New York University Abu Dhabi, P.O.
Box 129188, Abu Dhabi, UAE  Craig L. Sarazin Department of Astronomy,
University of Virginia, 530 McCormick Road, Charlottesville, VA 22904-4325,
USA  Gregory R. Sivakoff Department of Physics, University of Alberta, CCIS
4-181, Edmonton AB T6G 2E1, Canada  Roberto Soria INAF - Osservatorio
Astrofisico di Torino, Strada Osservatorio 20, 10025 Pino Torinese, Italy
College of Astronomy and Space Sciences, University of the Chinese Academy of
Sciences, Beijing 100049, People’s Republic of China  Sydney Institute for
Astronomy, School of Physics A28, The University of Sydney, Sydney, NSW 2006,
Australia  Alexandra J. Tetarenko Department of Physics and Astronomy,
University of Lethbridge, Lethbridge, Alberta, T1K 3M4, Canada  Valeriu Tudose
Institute for Space Sciences, National Institute for Laser, Plasma and
Radiation Physics, Atomistilor 409, PO Box MG-23, 077125 Bucharest-Magurele,
Romania

###### Abstract

Multi-wavelength polarimetry and radio observations of Swift J1727.8-1613 at
the beginning of its recent 2023 outburst suggested the presence of a bright
compact jet aligned in the north-south direction, which could not be confirmed
without high angular resolution images. Using the Very Long Baseline Array and
the Long Baseline Array, we imaged Swift J1727.8-1613 during the hard/hard-
intermediate state, revealing a bright core and a large, two-sided,
asymmetrical, resolved jet. The jet extends in the north-south direction, at a
position angle of $-0.60\pm 0.07\degree$ East of North. At 8.4 GHz, the entire
resolved jet structure is $\sim 110(d/2.7\,\text{kpc})/\sin i$ AU long, with
the southern approaching jet extending $\sim 80(d/2.7\,\text{kpc})/\sin i$ AU
from the core, where $d$ is the distance to the source and $i$ is the
inclination of the jet axis to the line of sight. These images reveal the most
resolved continuous X-ray binary jet, and possibly the most physically
extended continuous X-ray binary jet ever observed. Based on the brightness
ratio of the approaching and receding jets, we put a lower limit on the
intrinsic jet speed of $\beta\geq 0.27$ and an upper limit on the jet
inclination of $i\leq 74\degree$. In our first observation we also detected a
rapidly fading discrete jet knot $66.89\pm 0.04$ mas south of the core, with a
proper motion of $0.66\pm 0.05$ mas hour-1, which we interpret as the result
of a downstream internal shock or a jet-ISM interaction, as opposed to a
transient relativistic jet launched at the beginning of the outburst.

††facilities: LBA, MAXI, NRAO, VLBA††software: AIPS (Wells, 1985; Greisen,
2003), ADS (https://ui.adsabs.harvard.edu/), Arxiv (https://astrogeo.org/),
Astrogeo (https://astrogeo.org/), Astropy (Astropy Collaboration et al., 2013,
2018, 2022), CDS (Simbad; Wenger et al., 2000), Cmasher (van der Velden,
2020), Corner (Foreman-Mackey, 2016), Dynesty (Speagle, 2020), eht-imaging
(Chael et al., 2018), Jupyter (Kluyver et al., 2016), Matplotlib (Hunter,
2007), Numpy (Harris et al., 2020), Scipy (Virtanen et al., 2020)

##  1 Introduction

Black hole low-mass X-ray binaries (BH LMXBs) are ideal systems to study the
launching of relativistic jets and their connection to the processes of
accretion, due to their proximity and variability on human timescales. The
properties of relativistic jets are strongly coupled to the properties of the
accretion inflow, which both evolve dramatically and rapidly during bright
outbursts.

Typical BH LMXB outbursts begin in a rising hard state, where the radio
emission is dominated by a compact, steady, partially self-absorbed,
synchrotron-emitting, continuous jet (e.g. Corbel et al., 2000; Fender, 2001).
These hard-state jets111Throughout this letter, we refer to these continuous
jets as ‘hard-state jets’ due to their historical association with the hard
state, although they can persist into the hard-intermediate state, but never
into the soft state. become quenched as the source moves towards the
transition into the soft state, via intermediate states, where bright,
discrete transient jet ejecta are often launched (see e.g. Fender et al.,
2004, for a review of jets and X-ray binary outbursts). The hard-state jets
are often called compact jets, due to their appearance as bright, compact
point sources in high angular resolution observations. Resolved hard-state
jets have only been observed in a few black hole X-ray binaries: the high-mass
X-ray binary Cyg X-1 (Stirling et al., 2001); and the low-mass X-ray binaries
GRS 1915+105 (Dhawan et al., 2000; Ribó et al., 2004); MAXI J1836-194 (Russell
et al., 2015); and MAXI J1820+070 (Tetarenko et al., 2021).

High angular resolution observations of resolved hard-state jets can provide
independent probes of their fundamental jet properties, complementing other
methods such as radio and infrared timing studies (e.g. Kalamkar et al., 2016;
Malzac et al., 2018; Tetarenko et al., 2019, 2021), astrometry and core shift
measurements (e.g. Miller-Jones et al., 2021; Prabu et al., 2023), and broad-
band spectral modelling and spectral break studies (e.g. Markoff et al., 2001;
Corbel et al., 2002; Chaty et al., 2011; Gandhi et al., 2011; Russell et al.,
2014a; Péault et al., 2019; Russell et al., 2020; Echiburú-Trujillo et al.,
2024).

###  1.1 Swift J1727.8-1613

Swift J1727.8-1613 (hereafter J1727) was first detected on 2023 August 24 by
Swift/BAT (Page et al., 2023), with follow-up X-ray observations revealing it
to be a new candidate black hole low-mass X-ray binary in the hard-state at
the beginning of a bright outburst (Kennea & Swift Team, 2023; Negoro et al.,
2023a, b; Nakajima et al., 2023; O’Connor et al., 2023a, b; Castro-Tirado et
al., 2023). Optical observations suggested that J1727 has a black hole primary
with an early K-type companion star, with an orbital period of $\sim 7.6$
hours at a distance of $d=2.7\pm 0.3$ kpc.

Optical and near-infrared observations of J1727 during the rising hard-state
at the beginning of the outburst showed that the source was reddening,
possibly due to the onset of a compact hard-state jet (Baglio et al., 2023).
Radio observations from the Submillimetre Array (SMA), the Karl G. Jansky Very
Large Array (VLA), the enhanced Multi Element Remotely Linked Interferometer
(eMERLIN), and the Allen Telescope Array (ATA), showed a bright, flat-
spectrum, unresolved source (Vrtilek et al., 2023; Miller-Jones et al., 2023;
Baglio et al., 2023; Williams-Baldwin et al., 2023). Polarisation observations
at X-ray wavelengths with the Imaging X-ray Polarimetry Explorer (IXPE), at
optical wavelengths with the 60 cm Tohoku telescope, at 230 GHz with the SMA,
and at 5.5 and 9 GHz with the Australian Telescope Compact Array (ATCA)
revealed J1727 to be polarised across the electromagnetic spectrum, with a
position angle consistent with being aligned in the north-south direction
(Dovciak et al., 2023; Kravtsov et al., 2023; Veledina et al., 2023; Vrtilek
et al., 2023; Ingram et al., 2023). These observations suggested that J1727
had a bright, compact, hard-state jet aligned in the north-south direction.

In this letter we present four VLBI observations with the Very Long Baseline
Array (VLBA) and the Long Baseline Array (LBA), which reveal the highly
extended, resolved, north-south jet of J1727 during the hard/hard-intermediate
state, as well as an apparently disconnected discrete jet knot downstream. In
Section 2 we detail our observations, calibration, and imaging procedures, and
in Section 3 we present our images and perform analysis on the jet, which we
discuss in Section 4.

##  2 Observation, Calibration, and Imaging

Following the beginning of the outburst and the detection of a bright radio
counterpart (Miller-Jones et al., 2023), we observed J1727 with the VLBA
(project code BM538) at 8.37 GHz on 2023 August 30, as part of the Jet
Acceleration and Collimation Probe Of Transient X-Ray Binaries (JACPOT XRB;
Miller-Jones et al., 2011b) program. Following that observation, we observed
$~{}\sim 4$ hours later with the LBA (project code V456) at 8.44 GHz, and then
twice more on 2023 September 04 and September 06. Figure 1 shows the timing of
our observations during the beginning of the outburst as seen by the Monitor
of All-sky X-ray Image (MAXI222http://maxi.riken.jp/; Matsuoka et al., 2009)
mission. The observation details can be found in Table 1. Further VLBI
observations of the evolution of J1727 throughout its outburst will be
presented in a future paper (Wood et al. in prep.).

Table 1: VLBI observation log of the Swift J1727.8-1613 hard-state jet.

Label | Date | Time | MJD | Telescope | Observation | Frequency | Bandwidth | Stations†
---|---|---|---|---|---|---|---|---
| (2023) | (UTC) | (Midpoint) |  | Code | (GHz) | (MHz) |
VLBA | 30 Aug | 00:32:46–03:20:37 | 60186.08 | VLBA | BM538A | 8.37 | 512 | FD,HN,KP,LA,
|  |  |  |  |  |  |  | MK,NL,OV,SC
LBA1 | 30 Aug | 07:05:56–12:46:35 | 60186.41 | LBA | V456H | 8.44 | 64 | CD,HB,KE,MP,
|  |  |  |  |  |  |  | PA,WW
LBA2 | 04 Sep | 06:35:56–12:19:59 | 60191.39 | LBA | V456I | 8.44 | 64 | AT,CD,HB,KE,
|  |  |  |  |  |  |  | MP,PA,WW
LBA3 | 06 Sep | 06:36:02–12:19:59 | 60193.39 | LBA | V456J | 8.44 | 64 | AT,CD,MP,PA,
|  |  |  |  |  |  |  | WW

† LBA Stations: CD=Ceduna, HB=Hobart 12m, KE=Katherine, MP=Mopra, PA=Parkes,
WW=Warkworth 30m, AT=ATCA

Figure 1: MAXI light curve of the evolution of Swift J1727.8-1613 at the
beginning of its 2023 outburst. The time of our four VLBI observations are
marked by the vertical lines, and their details are given in Table 1.

For the VLBA, we used ICRF J174358.8-035004 (J1743-0350) as a fringe finder,
ICRF J172134.6-162855 (J1721-1628) as a phase reference source, and ICRF
J172446.9-144359 (J1724-2914) as a check source (Charlot et al., 2020). We
observed geodetic blocks (Reid et al., 2009) for $\sim$30 minutes at the
beginning and end of the observation to improve astrometric calibration. For
the LBA, we used ICRF J192451.0-291430 (B1921-293; Charlot et al., 2020) as a
fringe finder. In order to maximise signal-to-noise on the long baselines for
phase calibration, we swapped the phase reference and check sources for the
LBA. The data were correlated using the DiFX software correlator (Deller et
al., 2007, 2011), and calibrated according to the standard procedures within
the Astronomical Image Processing System (aips, version 31DEC22; Wells, 1985;
Greisen, 2003). After the standard external gain calibration, we performed
several rounds of hybrid mapping of the phase reference source to derive the
time-varying phase, delay, and rate solutions, which we interpolated to J1727.
We also performed a single round of amplitude self-calibration to get most
accurate time-varying amplitude gain calibration, which we applied to J1727.
To match the flux density scales of the VLBA and the LBA, we used the VLBA map
of J1724-2914 to derive a global amplitude gain solution for each LBA antenna
IF, and polarisation to scale the a priori amplitude gains approximated from
the zenith system equivalent flux densities. We used the VLBA and LBA scans of
J1721-1628 to confirm that the LBA flux density scale matched the VLBA to
within 5%.

We imaged J1727 within aips using the CLEAN algoritm (Högbom, 1974) with
natural weighting to maximise sensitivity, and we performed multiple rounds of
phase-only self-calibration followed by a single round of amplitude self-
calibration to obtain the final images.

##  3 Images and Analysis

Figure 2: First VLBI image of Swift J1727.8-1613 with the VLBA. The contours
mark $\pm\sigma\times\sqrt{2}^{n}$ mJy beam-1 where $n=3,4,5,...$, and
$\sigma$ is the rms noise shown in the lower left of the image. The ellipse in
the lower right corner shows the synthesised beam. The image shows a bright
core with highly resolved ($\sim 40$ mas) apparently asymmetric bipolar jets
extending in the north-south direction, and a discrete jet knot to the south
at a separation of $66.7\pm 0.2$ mas from the core at a position angle of
$179.4\pm 0.15$° East of North.

Figure 2 shows the VLBA image of J1727 from 2023 August 30, which reveals a
bright core with a highly resolved, asymmetric jet extending in the north-
south direction, and a discrete jet knot to the south at a separation of
$66.7\pm 0.2$ mas from the core (which we later refined with visibility
modelling). The extended continuous structure is a total of $\sim 40$ mas in
length, with the southern and northern jets extending $\sim 30$ and $\sim 10$
mas from the core, respectively.

We measured the position of the core in the VLBA image by fitting a point
source to the brightest region of the jet using the aips task JMFIT, prior to
applying any phase self-calibration. This gave a position of

$\text{R.A.}=17^{\text{h}}27^{\text{m}}43^{\text{s}}.3135784\pm 0.0000003,$
$\text{Dec.}=-16\degree 12^{\prime}19^{\prime\prime}.180424\pm 0.000011,$

where the errors are the $1\sigma$ statistical errors reported by AIPS
(assuming the position of J1721-1628 to be
$\text{R.A.}=17^{\text{h}}22^{\text{m}}56^{\text{s}}.498932\pm
0.000077,\text{Dec.}=-16\degree 30^{\prime}19^{\prime\prime}.26363\pm 0.00072$
in the ICRS reference frame and J2000 equinox).

By fitting a constant position angle to the locations of the positive flux
density CLEAN components in the VLBA observations, we measured the position
angle of the jet to be $-0.60\pm 0.07\degree$ East of North. This is
consistent with the position angle from the core to the southern discrete jet
knot.

In Figure 3 we show all four observations of the resolved jet of J1727. These
observations similarly show a bright core with an extended jet at the same
position angle, however the LBA images have poorer angular resolution than the
VLBA image, and thus we only resolve the southern jet. We do not see the
southern discrete jet knot in any of the LBA observations. We summarise the
observed peak and integrated flux densities of the observations in Table 2.
The integrated flux density of the jet decreased from the first observation to
the final two observations.

The LBA was slightly more sensitive to diffuse emission than the VLBA, which
explains why we recovered slightly more emission at the tip of the southern
jet in the first LBA observation than with the VLBA. By imaging the VLBA
observation with only the shortest baselines, we were unable to detect any
more diffuse emission between the extended jet and the discrete jet knot. We
note that the VLBA was missing the Pie Town station, and therefore lacked a
few key short baselines, which could explain why we only partially detect the
diffuse emission at the tip of the southern continuous jet, causing its knotty
appearance. The measured flux densities from our first two observations are
consistent with the observations from eMERLIN and the ATA on 2023 August 29
and August 30, respectively (Williams-Baldwin et al., 2023; Bright et al.,
2023), to within the $\sim$10% amplitude calibration uncertainties, suggesting
that we have recovered most of the emission of the jet of J1727 in our
observations.

Figure 3: VLBI images of the resolved jet of Swift J1727.8-1613 during the hard state. The contours mark $\pm\sigma\times\sqrt{2}^{n}$ mJy beam-1 where $n=3,4,5,...$, and $\sigma$ is the rms noise shown in the lower left of each image. The ellipse in the lower right corner of each image shows the synthesised beam. The observation parameters can be found in Table 1 and the image parameters can be found in Table 2. The jet is resolved over multiple epochs during the hard state, but appears less extended as the source moves towards the hard-intermediate state. Table 2: Summary of the jet image parameters from Figure 3. The core flux density is calculated by fitting a point source to the core. Here we add a 10% amplitude calibration error in quadrature with the $1\sigma$ statistical errors reported by aips. The statistical error in the integrated flux density is given by $\sigma\sqrt{N_{\text{beam}}}$, where $\sigma$ is the rms noise in the image and $N_{\text{beam}}$ are the number of independent beams in the integrated area. Observation | Integrated Flux Density | Core Flux Density | Beam Dimensions | Beam Position Angle
---|---|---|---|---
| (mJy) | (mJy beam-1) | (mas) x (mas) | (° East of North)
VLBA | $101\pm 10$ | $46\pm 5$ | 3.9 x 0.9 | -15.7
LBA1 | $90\pm 16$ | $64\pm 6$ | 7.3 x 3.3 | 17.7
LBA2 | $71\pm 8$ | $58\pm 6$ | 7.6 x 4.7 | 14.0
LBA3 | $75\pm 8$ | $65\pm 7$ | 9.9 x 1.7 | 6.5

For symmetric, continuous, steady-state jets inclined to the line of sight,
the approaching and receding jets should be approximately symmetric, however
the apparent surface brightness of these jets will be asymmetric due to
relativistic aberration. The flux density ratio of approaching and receding
continuous jets is given by

$\frac{S_{a}}{S_{r}}=\left(\frac{1+\beta\cos i}{1-\beta\cos
i}\right)^{2-\alpha},$ (1)

where $S_{a}$ and $S_{r}$ are the flux densities of the jets at an equal
angular separation from the core, and $\alpha$ is the spectral index of the
jets ($S_{\nu}\propto\nu^{\alpha}$; Ryle & Longair, 1967; Scheuer & Readhead,
1979; Mirabel & Rodríguez, 1999). The emission from the receding jet will fall
below the noise limit of an observation at a smaller separation than the
approaching component, which explains the asymmetry seen in our images if the
northern jet is receding and the southern jet is approaching.

Since the receding jet was only resolved in the VLBA observation, we used this
observation to constrain the flux density ratio. The CLEAN components for the
northern jet terminated at a separation of $6.5$ mas from the core, and so we
integrated the flux density of the CLEAN components along the northern and
southern jets from 3-6.5 mas separation from the core. We chose the lower
limit of 3 mas to avoid the emission of the unresolved core (see Appendix A).
This yielded a flux density ratio of $4.8\pm 0.4$, which gave $\beta\cos
i=0.29\pm 0.03$, assuming a spectral index of $\alpha=-0.6\pm 0.2$ (since the
extended jet should be optically thin beyond the core).

###  3.1 Modelling the Southern Knot

By separately imaging the first and second halves of the VLBA observation, we
found that the southern knot appeared to be moving away from the core. We were
able to precisely measure this motion by implementing the time-dependent
model-fitting procedure described in Wood et al. (2023), where we fit a time-
evolving model directly to the measured visibilities using the Bayesian
inference algorithm nested sampling (Skilling, 2006) implemented in the
dynesty333https://github.com/joshspeagle/dynesty Python package (Speagle,
2020). Before modelling the knot, we subtracted the CLEAN components of the
extended jet from the visibilities to create a residual observation containing
only the southern knot. We then fit an analytical model directly to these
residual visibilities, consisting of a circular Gaussian with fixed size
moving away from the core at a fixed position angle ($\theta$) with a constant
speed ($\dot{r}$). We allowed the flux density ($F$) of this component to vary
linearly within the observation. The flux density evolved as

$F(t)=F_{0}+\dot{F}(t-t_{0}),$ (2)

and the position of the knot relative to the core as

$\Delta x(t)=(r_{0}+\dot{r}(t-t_{0}))\sin\theta,$ (3)

and

$\Delta y(t)=(r_{0}+\dot{r}(t-t_{0}))\cos\theta,$ (4)

where $t_{0}$ is the reference time chosen as the approximate midpoint of the
observation (02:00:00 UTC on 2023 August 30th), $r_{0}$ is the separation of
the jet knot from the core (in mas) at the reference time, and $\Delta x$ and
$\Delta y$ are in the directions of positive R.A. and Dec., respectively (in
mas). We placed Gaussian priors on the separation of the jet knot at the
reference time, and its position angle, based on the location of the knot in
the image. We placed uniform priors on all other parameters. The prior
distributions are listed in Table 3 along with the posterior estimates. We
also tried models that included linear expansion of the jet knot, but found
that the expansion speed was consistent with zero. Similarly, we were unable
to measure any intra-observational deceleration. We also found that models
with exponential or power-law flux density decays were indistinguishable from
a linear decay model on the timescale of the $\sim 3$ hour observation.

Table 3: Prior distributions and posterior estimates for the moving jet knot in the VLBA observation in Figure 2, using equations 2-4. The reference time is defined as 02:00:00 (UTC) on 2023 August 30th, which is approximately the midpoint of the observation. We report the median of the marginal posterior distributions as the best-fit parameters, and the 16th and 84th percentiles as the uncertainties. Parameter | Prior Distribution | Posterior Estimate
---|---|---
$F_{0}$ (mJy) | $\mathcal{U}(\text{min}=0,\text{max}=10)$ | $4.60\pm 0.10$
$\dot{F}$ (mJy hour-1) | $\mathcal{U}(\text{min}=-10,\text{max}=10)$ | $-1.11\pm 0.09$
FWHM (mas) | $\mathcal{U}(\text{min}=0,\text{max}=10)$ | $1.92\pm 0.05$
$r_{0}$ (mas) | $\mathcal{N}(\mu=65,\sigma=5)$ | $66.89\pm 0.04$
$\dot{r}$ (mas hour-1) | $\mathcal{U}(\text{min}=0,\text{max}=10)$ | $0.66\pm 0.05$
$\theta$ (° East of North) | $\mathcal{N}(\mu=180,\sigma=5)$ | $179.24\pm 0.02$

We found that the knot was clearly resolved, and was moving away from the core
at a proper motion of $\dot{r}=0.66\pm 0.05$ mas hour-1, at a position angle
consistent with the extended jet position angle. We found that the jet knot
was rapidly decaying over the observation, which explains why we do not see
the knot in the first LBA observation $\sim 4$ hours later. Using the
separation of the knot and its ballistic speed, we calculated an ejection date
of MJD $60181.8\pm 0.2$, although in Section 4.3 we argue that this knot is
not a discrete transient jet. The separation and FWHM size of the jet knot in
the VLBA observation (see Table 3) give a projected half-opening angle of
$0.822\pm 0.021$°.

To ensure that the process of self-calibration was not corrupting or inducing
any false variability in the data, we performed the subtraction and modelling
process on versions of the observation without any self-calibration, after
phase-only self-calibration, and after amplitude and phase self-calibration.
We found that the position and motion of the knot was consistent between all
three self-calibration scenarios, and that the size and flux density changed
slightly after amplitude self-calibration. We also found marginal evidence of
expansion in the observation containing amplitude self-calibration, however we
only report the model parameters from the fit to the observation containing
phase-only self-calibration, since we could not confirm that the amplitude
self-calibration was not inducing false structure and variability in the jet
knot.

##  4 Discussion

We have observed the highly resolved jet of Swift J1727.8-1613 over four
epochs with the VLBA and the LBA. The jet is continuous, extended, apparently
asymmetrical, and aligned in the north-south direction. In the first
observation there was also a discrete jet knot moving away from the core to
the south, which does not appear in the subsequent observations. We were able
to constrain the motion and flux density variability of the jet knot with
time-dependent visibility model-fitting.

###  4.1 Extended Jet

The VLBA image of J1727 (Figure 2) is the most resolved image of an X-ray
binary hard-state jet. Cyg X-1 and GRS 1915+105 are the only other X-ray
binaries with a hard-state jet that has been resolved in an image over
multiple synthesised beams (Stirling et al., 2001; Dhawan et al., 2000; Ribó
et al., 2004). Assuming a distance to J1727 of $2.7\pm 0.3$ kpc (Mata Sánchez
et al., 2024), the size of the entire resolved jet structure (including the
receding jet) was $\sim 110/\sin i$ AU, and the extent of the approaching jet
was $\sim 80/\sin i$ AU, or $5\times 10^{8}/\sin i\,r_{g}$ (assuming a black
hole mass of $8\,M_{\odot}$ Kreidberg et al., 2012). X-ray polarisation
observations of J1727 with IXPE suggested that the inclination of the inner
accretion flow is between $\sim 30\degree-60\degree$ (Veledina et al., 2023).
Therefore the physical extent of the approaching resolved jet of J1727 on 2023
August 30th is between $\sim 95-160$ AU, or $\sim(0.6-1)\times 10^{9}\,r_{g}$.

The two-sided jets of both GRS 1915+105 and Cyg X-1 have been resolved at 8.4
GHz (Dhawan et al., 2000; Ribó et al., 2004; Miller-Jones et al., 2021). The
approaching jet of Cyg X-1 has been detected out to an extent of $\sim 50$ AU
(Stirling et al., 2001; Miller-Jones et al., 2021). In 1997, the approaching
jet of GRS 1915+105 was detected out to $\sim 16$ AU (updated with the most
recent distance constraint; Dhawan et al., 2000; Reid & Miller-Jones, 2023),
and in 2003 the jet was measured to be $\lesssim 150$ AU (Ribó et al., 2004,
with the upper limit due to interstellar scatter broadening). The approaching
jet of MAXI J1836-194 (which has a poorly constrained distance and
inclination) was marginally resolved at 8.4 GHz during the decay of its 2011
outburst, with an extent of $\sim 7-115$ AU (Russell et al., 2014a, b, 2015).
The approaching hard-state jet of MAXI J1820+070 was marginally resolved at 15
GHz, corresponding to a physical extent of $\sim 3$ AU at 8.4 GHz (scaling the
size as $z\propto 1/\nu$; Blandford & Königl, 1979; Tetarenko et al., 2021).

The hard-state jet of J1727 was detected out to a further extent than the
hard-state jets of Cyg X-1, GRS 1915+105 in its 1997 flare, and MAXI
J1820+070. Depending on inclination, the resolved hard-state jet of J1727 may
also have been larger than the jet of GRS 1915+105 during its 2003 flare, and
the decaying hard-state jet of MAXI J1836-194, and therefore J1727 may have
had the most extended hard-state jet ever observed in an X-ray binary. The
extent of a resolved a hard-state jet is the distance from the core at which
the surface brightness of the expanding jet material falls below the noise
limit. This depends on both the physical properties of the jet (e.g. opening
angle, jet speed, jet content, magnetic field strength, and jet internal
structure), as well as the observation parameters, particularly the angular
resolution, the overall sensitivity, and the sensitivity to diffuse structure.

We measured the location of the core in the VLBA image, which is likely
consistent with the location of the central black hole. J1727 contains an
early K-type dwarf companion (Mata Sánchez et al., 2024), and thus the jet is
not absorbed by a strong stellar wind, as in Cyg X-1, where the distance
between the black hole and the synchrotron photosphere is around $2\times
10^{13}$ cm at 8.4 GHz (Zdziarski et al., 2023), which would correspond to an
angular separation of tens of micro-arcseconds at the distance of J1727 (Mata
Sánchez et al., 2024). The distance between the black hole and the synchrotron
photosphere in J1727 is therefore likely much smaller than the size of the
pixels in our images.

Our measured position angle of both the extended jet and the jet knot is
consistent with the radio, mm, and X-ray polarisation position angles of
J1727, which we compare in Figure 4. These measurements are also consistent
with the more poorly constrained optical polarisation orientation (Kravtsov et
al., 2023). These measurements suggest that structures in the accretion flow,
the jet base, and the downstream jet are aligned to within a few degrees in
the plane of the sky.

Figure 4: Comparison of polarisation angles and the position angle of the
resolved jet of Swift J1727.8-1613. The background image is the same as shown
in Figure 2. We plot the polarisation measured by the Sub-Millimeter Array at
1.3mm (SMA; Vrtilek et al., 2023), the Australia Telescope Compact Array at
5.5 and 9 GHz (ATCA; Ingram et al., 2023), and the Imaging X-ray Polarimetry
Explorer (IXPE; Dovciak et al., 2023; Veledina et al., 2023; Ingram et al.,
2023). The shaded arcs represent the reported $1\sigma$ error intervals, with
the lines showing their nominal values. The polarisation observations span the
hard and hard-intermediate states and are consistent with the north-south
orientation of the resolved jet in our VLBI observations.

Between the observations on 2023 August 30th and September 4th and 6th, the
extended continuous jet became fainter and less extended, while the core
remained relatively constant. We note that the noise level in the observations
changes, and thus the distance from the core to which we could detect an
identical jet in each observation is different. Our LBA observations did not
have sufficient angular resolution and sensitivity to resolve the counter jet,
and so we cannot comment on how the jet speed changed as the compact jet
faded. We cannot use the lack of a detected counter jet to place a lower limit
on the brightness ratio, since the northern jet in the VLBA image can be
completely contained within the beam of the subsequent LBA observations. We
also saw no evidence of a systematic shift in the position of the core as the
jet faded, although we had to correct for a slight offset due to the different
phase calibrators for the VLBA and LBA. Given the changing resolution of the
observations, we also cannot meaningfully compare the peak intensity of the
unresolved core across the epochs.

###  4.2 Intrinsic Jet Speeds

Without observing a northern counterpart to the southern jet knot, we cannot
uniquely constrain the intrinsic speed and inclination of the source, however
we can constrain possible combinations of $\beta$ and $i$. The proper motion
of the approaching jet knot, $\mu$, is related to the intrinsic speed of the
jet knot, $\beta$, by,

$\mu=\frac{\beta\sin i}{1-\beta\cos i}\frac{c}{d},$ (5)

where $i$ is the inclination angle of the jet to the line of sight, $d$ is the
distance to J1727 ($2.7\pm 0.3$ kpc; Mata Sánchez et al., 2024), and $c$ is
the speed of light (Mirabel & Rodríguez, 1999). We cannot use the non-
detection of a receding counterpart to set an upper limit on the flux density
ratio to constrain $\beta\cos i$, since we need to measure the flux densities
when the jets are at equal separations (i.e. at the same age) since the jets
vary in brightness as they move downstream (Miller-Jones et al., 2004).

In Figure 5 we plot the possible values of $\beta$ and $i$ for both the
extended jet and the jet knot, using equations 1 and 5, respectively. These
constraints place a lower limit on the intrinsic jet speed of both the
continuous and discrete jet knot of $\beta\geq 0.27$ and $\beta\geq 0.2$,
respectively. The brightness ratio also gives an upper limit on the jet
inclination of $i\leq 74\degree$. Although they do not have to share the same
intrinsic speed, assuming that the discrete jet knot and the continuous
extended jet are at the same inclination (i.e. assuming no rapid large-scale
precession of the jet axis, as in V404 Cygni; Miller-Jones et al., 2019), then
their intrinsic speeds are consistent within $1\sigma$ in the interval $i\sim
27-37\degree$. In the inclination range favoured by X-ray polarisation
measurements ($i\sim 30\degree-60\degree$; Veledina et al., 2023), the jet
knot is moving at the same speed or slower than the continuous jet.

Figure 5: Constraints on the intrinsic speed and inclination of the extended
and discrete jets seen in the VLBA observation in Figure 2. The solid lines
show the nominal values and the shaded areas show the $1\sigma$ uncertainties.
We derive these constraints from the brightness ratio of the approaching and
receding arms of the extended jet (equation 1) and the proper motion of the
southern jet knot (equation 5). The brightness ratio puts an upper limit on
the jet inclination and a lower limit on the intrinsic speed of the continuous
jet.

###  4.3 Southern Jet Knot

While the nature of the jet knot is unclear, there are multiple potential
scenarios. The most likely explanation is that the jet knot is the result of
in-situ particle acceleration in the downstream continuous jet which produced
synchrotron emission. Although the approaching extended jet expanded and faded
beyond detection at a separation of $\sim 30$ mas, the jet material will
continue to propagate downstream until it loses its energy and momentum to the
surrounding medium. An internal shock could be the result of the collision of
fast-moving jet material with previously ejected slower-moving jet material.
Internal shock models have been used in the past to explain both compact
steady jets and discrete transient jets (Jamil et al., 2010; Malzac, 2014;
Malzac et al., 2018). A short lived internal shock could explain why the jet
knot was fading rapidly and was not seen in subsequent observations. This
model only requires a change in the jet speed of the continuous jet as it
extends outwards during the rise of the outburst. In this scenario, the jet
knot does not have to be travelling at the same speed as the continuous jet.

Alternatively, the particle acceleration could be due to a collision between
the continuous jet and a dense interstellar medium (ISM). The discrete knot
may have then been the ‘leading edge’ of the extended jet as it advanced
outwards through the ISM. Under this scenario, and assuming ballistic motion,
the continuous jet would have began extending outwards from the core on MJD
$60181.8\pm 0.2$, which coincides with the peak of the initial hard-state
X-ray rise. If the jet knot was decelerating, then the ’launch date’ would
have occurred later. In-situ particle acceleration due to the interaction
between transient jets and the ISM has been observed to cause downstream
rebrightening in multiple sources (e.g. Corbel et al., 2002, 2005; Migliori et
al., 2017; Espinasse et al., 2020), as well as downstream deceleration of the
jet at both arc-second and mas scales (e.g. Corbel et al., 2002; Yang et al.,
2010; Miller-Jones et al., 2011a; Russell et al., 2019; Espinasse et al.,
2020; Carotenuto et al., 2022; Bahramian et al., 2023), which may be the case
here if the inclination of the jet axis is $\gtrsim 37\degree$. This
interaction between the continuous jet and the ISM would have required a dense
local environment, which can be probed by observations of the motions of
transient ejecta launched later in the outburst. This explanation does not
account for the rapid fading of the jet knot during the VLBA observation, as
the steady continuous jet should have continued to interact with the dense ISM
across all of the observations.

A less likely explanation is that knot was a discrete transient jet, which
travelled ballistically away from the core after being ejected on MJD
$60181.8\pm 0.2$ (or later if the knot was decelerating), similar to those
that are often seen in other BH LMXBs (e.g. Mirabel & Rodríguez, 1994; Tingay
et al., 1995; Hjellming & Rupen, 1995; Miller-Jones et al., 2012; Bright et
al., 2020; Carotenuto et al., 2022). These transient jets fade and become
optically thin as they expand, which could explain the fading of the jet knot
observed here. In other X-ray binaries, these types of jets are ejected at the
peak of the outburst, during the state transition (Fender et al., 2004, 2009).
In this scenario, this would be the earliest time in a BH LMXB outburst that a
transient jet has been seen to have been ejected, occurring at the peak of the
hard state (see Figure 1). Furthermore, these ejection events have previously
been associated with specific X-ray signatures that correspond to changes in
the inner accretion flow such as bright X-ray (and accompanying radio) flares,
as well as a dramatic change in the X-ray spectral and timing properties (e.g.
Fender et al., 2009; Miller-Jones et al., 2012; Russell et al., 2019; Homan et
al., 2020; Wood et al., 2021). J1727 did exhibit many of these behaviours
later in the outburst during the state transition, however, despite relatively
intensive monitoring, no such clear signature of ejection was identified at
the beginning of the outburst. An explanation for the jet knot that involves
the re-acceleration of material in the out-flowing continuous jet does not
require the ejection of a discrete transient jet knot during the hard/hard-
intermediate state, and is more consistent with the current understanding of
jet evolution during BH LMXB outbursts.

## acknowledgments

We respectfully acknowledge the significant contributions made to this
longstanding collaboration by Tomaso Belloni, who sadly passed away during our
observing campaign. His insights and wealth of knowledge are sorely missed by
his colleagues.

The National Radio Astronomy Observatory is a facility of the National Science
Foundation operated under cooperative agreement by Associated Universities,
Inc. This work made use of the Swinburne University of Technology software
correlator, developed as part of the Australian Major National Research
Facilities Programme and operated under licence. The Long Baseline Array is
part of the Australia Telescope National Facility (https://ror.org/05qajvd42)
which is funded by the Australian Government for operation as a National
Facility managed by CSIRO. This work was supported by resources provided by
the Pawsey Supercomputing Research Centre with funding from the Australian
Government and the Government of Western Australia. This research has made use
of the MAXI data provided by RIKEN, JAXA and the MAXI team. This work made use
of the Warkworth 30m telescope as part of the LBA (Woodburn et al., 2015).
From 2023 July 1, operation of Warkworth was transferred from Auckland
University of Technology (AUT) to Space Operations New Zealand Ltd, who
continue to make the facilities available for VLBI out of goodwill.

CMW acknowledges financial support from the Forrest Research Foundation
Scholarship, the Jean-Pierre Macquart Scholarship, and the Australian
Government Research Training Program Scholarship. FC acknowledges support from
the Royal Society through the Newton International Fellowship programme
(NIF/R1/211296). RF and SM acknowledge support from a European Research
Council (ERC) Synergy Grant “BlackHolistic” (grant No. 101071643). DMR is
supported by Tamkeen under the NYU Abu Dhabi Research Institute grant CASS.
AJT acknowledges the support of the Natural Sciences and Engineering Research
Council of Canada (NSERC; funding reference number RGPIN-2024-04458). GRS is
supported by NSERC Discovery Grant RGPIN-2021-0400. VT acknowledges support
from the Romanian Ministry of Research, Innovation and Digitalization through
the Romanian National Core Program LAPLAS VII – contract no. 30N/2023. The
authors wish to recognize and acknowledge the very significant cultural role
and reverence that the summit of Maunakea has always had within the indigenous
Hawaiian community. We are most fortunate to have the opportunity to conduct
observations from this mountain. We also wish to acknowledge the Gomeroi,
Gamilaroi, and Wiradjuri people as the traditional custodians of the LBA
observatory sites.

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##  Appendix A Hard-state Jet Profile

To investigate the brightness ratio of the extended jet in Figure 2, we
extracted intensity profiles down the northern and southern jets. We first
rotated the image of the jet so that it aligns vertically, and then took one
dimensional cross sections of the northern and southern jets every 1.05 mas
downstream from the core (7 pixels). We fit these cross sections with a one
dimensional Gaussian profile to measure the intensity of the approaching and
receding jets as a function of separation from the core, which we show in
Figure 6. We fitted the Gaussian profiles to the pixel intensities using non-
weighted least squares regression with
scipy.optimize.curve_fit444https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html,
where we report the intensity of the jet as the amplitude of the Gaussian
profile. To calculate the uncertainty in the intensity, we added in quadrature
the reported $1\sigma$ statistical uncertainty from the fit (extracted from
the diagonal elements of the covariance matrix), the rms noise in the image,
and a 10% calibration error. In Figure 6, we also show the contribution of the
unresolved core by taking the one dimensional profile of the restoring beam
along the jet axis multiplied by the fit intensity of the core. The
contribution from the unresolved core drops below the $5\sigma$ limit at a
separation of $\sim 3$ mas. We note that the measurements are correlated along
the jet and across the core, since the true intensity profile of the jet is
convolved with the Gaussian restoring beam. This explains why the decay of the
northern jet is much steeper than the southern jet, since at smaller
separations the convolution with the beam across the core causes the intensity
to be biased upwards.

Figure 6: Intensity profile along the northern and southern jet as a function
of separation from the core in the VLBA observation of Swift J1727.8-1613
shown in Figure 2. The intensity profiles are calculated by fitting 1D
Gaussians perpendicular to the jet axis at intervals of $\sim 1$ mas. The
dashed line shows the contribution from the core, calculated by multiplying
the intensity of the core by the profile of the restoring beam along the jet
axis, with the shaded region showing its uncertainty (derived from the fit).


# Circular dichroism induction in WS2 by a chiral plasmonic metasurface

Fernando Lorén  1,2 Cyriaque Genet  3 and Luis Martin-Moreno1,2  1Instituto de
Nanociencia y Materiales de Aragón (INMA), CSIC-Universidad de Zaragoza, 50009
Zaragoza
2Departamento de Física de la Materia Condensada, Universidad de Zaragoza,
50009 Zaragoza
3University of Strasbourg and CNRS, CESQ & ISIS (UMR 7006), 8, allée G. Monge,
67000 Strasbourg, France<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

We investigate the interaction between a monolayer of WS2 and a chiral
plasmonic metasurface. WS2 possesses valley excitons that selectively couple
with one-handed circularly polarised light. At the same time, the chiral
plasmonic metasurface exhibits spin-momentum locking, leading to a robust
polarisation response in the far field. Using a scattering formalism based on
the coupled mode method, we analyse various optical properties of the WS2
monolayer. Specifically, we demonstrate the generation of circular dichroism
in the transition metal dichalcogenide (TMD) by harnessing the excitation of
surface plasmon polaritons (SPPs) in the metasurface. Moreover, we observe the
emergence of other guided modes, opening up exciting possibilities for further
exploration in TMD-based devices.

##  1 Introduction

Transition metal dichalcogenides (TMDs), such as MoS2, WS2, and MoSe2, have
garnered significant attention due to their unique electronic, optical, and
mechanical properties [1, 2, 3, 4, 5, 6, 7]. TMDs consist of atomically thin
layers held together by weak van der Waals forces. The monolayers of these
materials exhibit fascinating characteristics that make them highly desirable
for a range of applications [8, 9], including electronics [10],
optoelectronics [11, 12], energy storage [13] and sensing [14]. Furthermore,
circularly polarised light is particularly relevant in this context, as it
enables the selective excitation and manipulation of the characteristic valley
excitons of the TMDs [15, 16, 17].

On the other hand, plasmonic metasurfaces have demonstrated a wide range of
applications [18, 19, 20] such as sensing [21] or imaging [22, 23]. Chiral
structures, in particular, have emerged due to their capacity to manipulate
and control the polarisation of light waves [24, 25, 26, 27, 28, 29, 30].
Furthermore, they can exhibit spin-momentum locking (SML), which refers to the
coupling between the polarisation and momentum of the light waves involved
[31]. The SML of the metasurface enables a particularly compelling application
in the field of valleytronics [32, 33, 34, 35, 36, 37], where the spin angular
momentum of emitted light can be used to excite and detect valley excitons
selectively.

Although the valley-selective circular dichroism has already been reported in
TMDs [38], there is no proof of circular dichroism ($CD$) in this system yet.
In this work, as proof of concept, we study the system composed of a WS2
monolayer over a chiral plasmonic metasurface to analyse the $CD$ induced in
the WS2. Finally, we show that guided modes can enhance the CD in the presence
of a dielectric spacer between the TMD and the metasurface. Unlike SPPs, these
modes have a transverse electric (TE) field, but we observe that, mutatis
mutandis, its excitation satisfies the 3-step model derived in [39].

##  2 Theoretical description

We consider a WS2 monolayer deposited over a chiral plasmonic metasurface,
with a dielectric spacer in between (see Figure 1(a)), which is based on [32].
As in the experiments in Ref. [32], the metasurface is characterised by a
periodic array of $N=6$ rectangular dimples step-wisely rotated with a winding
number of $n_{w}=1/2$, which is the number of complete $2\pi$ rotations along
the whole unit cell. The unit cell (see Fig. 1(b)) is periodically repeated in
both $\vec{u}_{x}$ and $\vec{u}_{y}$ directions.

Figure 1: a) Scheme of the system studied in the article, with the
corresponding defining magnitudes. b) Unit cell with $N=6$ and $n_{w}=1/2$.
Based on [32].

We use the coupled-mode method (CMM) to derive the system’s electromagnetic
(EM) fields. This method has been extensively employed to study EM properties
in plasmonic arrays [40, 41, 42, 39]. The CMM expands the EM fields in plane
waves in the free space regions and waveguide modes inside the dimples and
finds the electric field amplitudes by properly matching the EM fields at the
interfaces. The excitation of our system consists of an electromagnetic plane
wave impinging our metasurface with an in-plane wavevector
$\vec{k}^{in}=k_{x}^{in}\vec{u}_{x}+k_{y}^{in}\vec{u}_{y}$ and an incident
polarisation $\sigma^{in}$. The goal is to compute the coefficients associated
with the Bragg orders, which define the EM fields in the different regions.

We do not present the full equations resulting from the CMM method, as they
have been given in a similar structure without the TMD in [39]. The effect of
the 2D material is straightforwardly included by considering an additional
interface and satisfying the in-plane magnetic field discontinuity that arises
due to the conductivity $\alpha$.

The accuracy of the CMM relies on the number of considered Bragg modes in the
simulations. In our particular case, we have considered $21$ modes in the
$\vec{u}_{x}$ direction and $3$ modes in the $\vec{u}_{y}$ direction, being
$\vec{G}_{x}=2\pi/(6L)\,\vec{u}_{x}$ and $\vec{G}_{y}=2\pi/L\,\vec{u}_{y}$ the
reciprocal lattice or Bragg vectors of the unit cell. We have checked that
adding more Bragg modes introduces less than $1\%$ variation in the obtained
results.

We treat the metallic structure using the surface impedance boundary
conditions (SIBC) approximation, which considers the actual dielectric
constant of the metal $\epsilon_{M}$ and the penetration of the EM fields into
the metal slab through the surface impedance $z_{s}=1/\sqrt{\epsilon_{M}+1}$,
which leads to the exact dispersion relation of SPPs in a metal-vacuum
interface. The semi-infinite region above the 2D material and the dielectric
spacer is characterised by the dielectric constants $\epsilon_{I}$ and
$\epsilon_{d}$, respectively. Finally, the 2D material is characterised by its
2D conductivity $\alpha$. We obtain it from the experimental dielectric
constant $\epsilon_{2D}$ [43] through $\alpha=w\pi(\epsilon_{2D}-1)/(\lambda
i)$, where $w$ is the width of the 2D material ($w\simeq 1\,nm$), $\lambda$ is
the wavelength of the EM fields and Gaussian units have been used. Moreover,
we fit the conductivity to remove the background, which is associated with
sample imperfections and can be decreased by, for instance, encapsulating the
TMD between graphene monolayers [44, 45].

Once the EM fields are computed, we obtain the total absorptance of the system
as $A_{T}=(W_{inc}-W_{ref})/W_{inc}$, where $W_{inc}$ and $W_{ref}$ are the
incoming and reflected EM energy fluxes, respectively, calculated via the
integration of the Poynting vector over an $x-y$ plane in the semi-infinite
regions above the 2D material. The reflectance is $R=W_{ref}/W_{inc}=1-A_{T}$.
The absorptance of the WS2 monolayer depends on the conductivity tensor
$\overleftrightarrow{\alpha}$ and the in-plane electric field $\vec{E}$ at the
2D material, integrated over the $x-y$ plane:
$A_{WS_{2}}=\left(\int\vec{j}^{*}\vec{E}\,dx\,dy\right)/W_{inc}=\left(\int\vec{E}^{*}\overleftrightarrow{\alpha}\vec{E}\,dx\,dy\right)/W_{inc}$.
In the linear polarisation basis $\overleftrightarrow{\alpha}$ is diagonal,
with diagonal components equal to $\alpha$.

Associated with them, we also analyse the circular dichroism and the g-factor:
$CD=A_{WS_{2}}^{+}-A_{WS_{2}}^{-}$,
$CD_{norm}=CD/(A_{WS_{2}}^{+}+A_{WS_{2}}^{-})$, $g=R^{+}-R^{-}$, and
$g_{norm}=g/(R^{+}+R^{-})$. The superscripts refer to the spin of the incoming
plane waves $\sigma^{in}=\pm$, and we will work with both normalised and not-
normalized magnitudes.

For definiteness, the geometrical parameters considered in this paper are as
in Ref. [32]: each dimple has a short side $a=80\,nm$, a long side $b=220\,nm$
and a depth $d=60\,nm$, and the distance between the centres of nearest
dimples is $L=480\,nm$ (in both $\vec{u}_{x}$ and $\vec{u}_{y}$).

##  3 Circular Dichroism in WS2

To analyse the quantities described in the previous section, we range both the
energy $\omega$ and the momentum $k_{x}^{in}$ of the incoming plane wave.

Figure 2: (a,b) Total absorptance of the system for $\sigma^{in}=\pm$,
respectively. (c,d) Absorptance of the WS2 monolayer for $\sigma^{in}=\pm$,
respectively. (e) Circular dichroism. (f) Normalised circular dichroism. (g)
g-factor. (h) Normalised g-factor. For all the plots, we have considered a
distance $h=10\,nm$ between the metasurface and the WS2, above the 2D material
$\epsilon_{I}=1$ (vacuum), and a spacer dielectric constant
$\epsilon_{d}=2.5$, which corresponds to the typical $SiO_{2}$ spacer.

In Figs. 2(a-d), we observe the well-known behaviour of both the metasurface
and the TMD monolayer. In the upper region of the $A_{T}^{\pm}$ figures, we
identify the large absorption of gold at $\lambda\sim 500\,nm$. Moreover, the
four absorptance figures present plasmonic parabolic features provided by the
metasurface. The SML, produced by the rotation of the dimples along the unit
cell, is also noted through the selective excitation of the left and right
SPPs (which depend on the spin $\sigma^{in}=\pm$ of the incoming plane wave,
respectively [32, 39]). The influence of the WS2 monolayer is also seen in all
absorptance figures. Both exciton bands are apparent: The A-exciton band
appears at $\omega\simeq 2.03\,eV$, and the B-exciton band at $\omega\simeq
2.4\,eV$. Besides, in Figs. 2(c,d), we observe the interplay between the
A-exciton band from the WS2 monolayer and the SPP through the absorptance
enhancement at the band’s crossing points.

The same metasurface was previously studied in [39], where it was shown that
spin-momentum locking is an approximate symmetry. The symmetry breakdown
arises because the EM plane waves are transversal with respect to their
momentum, while the interaction with the surface presents symmetries with
respect to the metasurface normal. Consequently, when circularly polarised
light is projected onto the planar surface, it transforms into an elliptical
polarisation state comprising a combination of $\pm$ spin states, spoiling the
SML effect. This is not apparent in the contour plots in Fig. 2 because of
their limited resolution. However, if we cut Fig. 2(a) at fixed energy, we
would observe two small absorptance peaks at the $k_{x}^{in}$ for which
plasmon with the opposite spin states appear.

Figs. 2(e,f) show the computed $CD$ and the $CD_{norm}$. We observe a
different absorptance in the WS2 depending on the incident spin $\sigma^{in}$.
Notice that, without the chiral metasurface, the $K$ valley would
preferentially absorb $+$ light (in fact, it would only absorb $+$ light at
normal incidence) and the same for the $K^{\prime}$ valley with respect to $+$
light, but this would lead to $CD=0$. The non-zero $CD$ in the TMD monolayer
induced by the chiral plasmonic array is maximal when the spin-momentum locked
SPPs are excited. In Fig. 2(e), around the A-exciton band, we observe a
difference in absorptances of $\sim 0.06$, which is large considering that the
typical absorptance in that band is $\sim 0.3$. Besides, in Fig. 2(f), the
plasmonic parabolas present $|CD_{norm}|\simeq 0.5$, which implies that the
WS2 absorptance for one polarization is three times larger than for the
opposite.

In Figs. 2(g,h), we represent $g$ and $g_{norm}$. Their behaviour is inverse
to those of $CD$ or $A_{T}^{\pm}$ because they are associated with the
reflectance $R^{\pm}$. For example, $g_{norm}$ at the rightmost parabola is
positive because the chiral metasurface reflects the $+$ light while $-$ light
excites an SPP and is thus partially absorbed. In general, $g_{norm}$ is
mainly determined by the chiral plasmonic metasurface, not being strongly
affected by the presence of the TMD.

These results are just a proof of concept of how $CD$ in a 2D material can be
induced by a chiral plasmonic array. Optimising the structure’s geometric
parameters to enhance the circular dichroism or other quantities is
straightforward, given the model’s simplicity. For example, we could explore
varying the distance between the dimples, their size and depth, the distance
between the metasurface and the 2D material, or the dielectric constant of the
spacer, among other parameters.

##  4 Influence of the TE waveguide modes

The second main result of this article is developed in this section, where we
show that the spacer can induce the existence of a set of guided modes with
imprinted chiral properties from the chiral holey surface.

Figure 3: Total absorptance, considering $\epsilon_{I}=1$, $\epsilon_{d}=2.5$
and $\sigma^{in}=+$. For different distances from the metasurface to the WS2
monolayer: (a) $h=30\,nm$, (b) $h=60\,nm$, and (c) $h=90\,nm$.

In Fig. 3, we represent $A_{T}^{+}$ for three different distances between the
metasurface and the WS2 monolayer, $h$. The SPPs parabolic branches are
excited at lower incident energies as $h$ increases. At the same time, we
observe the appearance of new parabolas for larger $h$. These are the emerging
guided modes.

Figure 4: Total absorptance, considering $\epsilon_{I}=1$, $\epsilon_{d}=2.5$,
$h=90\,nm$ and $\sigma^{in}=+$. For the different structures above each
contour plot: (a) With the WS2 monolayer (same as Fig. 3(c)), (b) Without the
WS2 monolayer, and (c) With the WS2 monolayer and $\epsilon_{I}=2.5$.

In order to understand the origin of these guided modes, Fig. 4 renders
$A_{T}^{+}$ for three different situations, sketched above the plots. Fig.
4(a) is the same as Fig. 3(c). Fig. 4(b) is computed by removing the WS2 from
the top of the dielectric spacer. Fig. 4(c) considers the TMD, but with the
region above the WS2 also having dielectric constant $\epsilon_{I}=2.5$. These
modes already appear in Fig. 4(b), when no WS2 monolayer is present. Besides,
only the SPPs are excited if the semi-infinite superstrate is considered to
have the same dielectric constant of the spacer, even if we consider the
presence of the WS2 layer. Thus, the guided modes are due to the presence of
the spacer and not to the TMD (although the 2D material slightly affects the
required energy for their excitation).

Figure 5: Total absorptance, considering $h=90\,nm$ and $\epsilon_{d}=2.5$.
For incident polarization: (a) $\sigma^{in}=+$, (b) $\sigma^{in}=-$, (c)
$\sigma^{in}=p\equiv TM$, and (d) $\sigma^{in}=s\equiv TE$.

These spacer-induced guided modes are TE-polarised. To show how this changes
the polarisation response of the considered structure, we represent in Fig. 5
the total absorptance of the system $A_{T}$ for four incident polarisations.
Figs. 5(a,b) also show that the SML holds for the guided modes. However, for
linearly polarised light with $\sigma^{in}=TM$ (see Fig. 5(c)), absorptance is
enhanced at the central plasmonic parabola but vanishes for the guided mode.
Complementarily, for $\sigma^{in}=TE$ (see Fig. 5(d)), we observe just the
opposite: an enhancement for the central parabola of the guided mode and a
substantial reduction for the corresponding SPP.

Note that the modified SML rules, developed in [39] to address the resonant
excitation of plasmons, also apply to these guided modes when slightly
modified to incorporate their TE nature properly. Scattering under resonant
excitation can be understood as a three-step scattering process. First, the
incoming plane wave picks up grating momentum to excite the SPP/guided mode,
ending with polarisation dictated by the SML rules. In the second step, this
polarisation is projected onto TM/TE polarisation states due to the linear
polarization of the confined modes Lastly, the SPP/guided mode outcouples via
another SML process. Both in- and out-coupling processes can be “normal”
(maintaining the polarisation) or “chiral” (which reduces or increases the
plane wave spin when the geometric momentum $k_{g}$, associated with the
rotation of the dimples, is added or subtracted).

Figure 6: (a) Circular dichroism. (b) Normalised circular dichroism. We have
considered $h=90\,nm$ and $\epsilon_{d}=2.5$.

To conclude, in Fig. 6, we represent the $CD$ and $CD_{norm}$ considering
$h=90\,nm$, $\epsilon_{I}=1$ and $\epsilon_{d}=2.5$. The TE-guided modes also
induce circular dichroism in the WS2 monolayer. We notice that $|CD|\simeq
0.04$ around the A-exciton band and $|CD_{norm}|\simeq 0.3$ when we excite one
of these modes. These magnitudes are not as large as those Figs. 2(e,f) but
demonstrate inducing chiral behaviour in the WS2 monolayer through the TE-
guided modes, and the possibility to use the spacer layer to enhance this
phenomenon further.

##  5 Conclusion

We have presented a way to obtain circular dichroism in transition metal
dichalcogenides using chiral plasmonic metasurfaces. Additionally, a
transverse electric waveguide mode has been observed and demonstrated to cause
$CD$ in TMDs. This work is a proof of concept in a non-optimized structure;
$CD$ values are expected to be substantially higher after optimisation over
the whole range of parameters. In particular, after optimisation, the
transverse electric character of the guided modes is expected to couple them
better than SPPs to the TMDs.

Our characterisation of the 2D material is simple, only based on its
permittivity tensor. By changing this tensor into the circularly polarised
basis, we can easily split the full TMD optical behaviour into the $K$ and
$K^{\prime}$ valleys components. This allows us to study the different valley
phenomena with our simple formalism. However, magnitudes such as the valley
contrast or the valley-selective circular dichroism would still require
considering the inter-valley scattering to embrace the exciton dynamics, which
has been proved to be essential in the analysis of TMDs [46, 47, 32].

The findings presented here pave the way for further exploration in the field
of chiral optics and valleytronics. Indeed, the induced circular dichroism in
TMDs may have applications in areas such as chiral sensing, imaging, and
quantum information processing [15, 48, 6, 49, 50, 51].

Funding We acknowledge Project PID2020-115221GB-C41 was financed by
MCIN/AEI/10.13039/501100011033 and the Aragon Government through Project
Q-MAD.

This work is part of the Interdisciplinary Thematic Institute QMat of the
University of Strasbourg, CNRS, and Inserm. It was supported by the following
programs: IdEx Unistra (ANR-10-IDEX-0002), SFRI STRATUS project
(ANR-20-SFRI-0012), and USIAS (ANR-10-IDEX-0002-02), under the framework of
the French Investments for the Future Program.

Acknowledgments FL and LMM also acknowledge the Aragón government project
Q-MAD.

Disclosures The authors have no conflicts to disclose.

Data Availability Statement Data underlying the results presented in this
paper are not publicly available but may be obtained from the authors upon
reasonable request.

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