11institutetext: Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
22institutetext: School of Physics and Astronomy, Monash University, VIC 3800,
Australia

# Spectroscopic and interferometric signatures of magnetospheric accretion in
young stars

B. Tessore 11 A. Soulain 11 G. Pantolmos 11 J. Bouvier 11 C. Pinte 1122 and K.
Perraut 11

(xx/xx/xx; yy/yy/yy)

###### Abstract

Aims. We aim to assess the complementarity between spectroscopic and
interferometric observations in the characterisation of the inner star-disc
interaction region of young stars.

Methods. We use the code MCFOST to solve the non-LTE problem of line formation
in non-axisymmetric accreting magnetospheres. We compute the Br$\gamma$ line
profile originating from accretion columns for models with different magnetic
obliquities. We also derive monochromatic synthetic images of the Br$\gamma$
line emitting region across the line profile. This spectral line is a prime
diagnostics of magnetospheric accretion in young stars and is accessible with
the long baseline near-infrared interferometer GRAVITY installed at the ESO
Very Large Telescope Interferometer.

Results. We derive Br$\gamma$ line profiles as a function of rotational phase
and compute interferometric observables, visibilities and phases, from
synthetic images. The line profile shape is modulated along the rotational
cycle, exhibiting inverse P Cygni profiles at the time the accretion shock
faces the observer. The size of the line’s emission region decreases as the
magnetic obliquity increases, which is reflected in a lower line flux. We
apply interferometric models to the synthetic visibilities in order to derive
the size of the line-emitting region. We find the derived interferometric size
to be more compact than the actual size of the magnetosphere, ranging from 50
to 90% of the truncation radius. Additionally, we show that the rotation of
the non-axisymmetric magnetosphere is recovered from the rotational modulation
of the Br$\gamma$-to-continuum photo-centre shifts, as measured by the
differential phase of interferometric visibilities.

Conclusions.  Based on the radiative transfer modelling of non-axisymmetric
accreting magnetospheres, we show that simultaneous spectroscopic and
interferometric measurements provide a unique diagnostics to determine the
origin of the Br$\gamma$ line emitted by young stellar objects and are ideal
tools to probe the structure and dynamics of the star-disc interaction region.

###### Key Words.:

Radiative transfer – Line: profiles – Stars: variables: T Tauri, Herbig Ae/Be
– Accretion, accretion disks

##  1 Introduction

The early evolution of low mass stars ($M_{\ast}$ ¡ 2 $M_{\odot}$) during the
classical T Tauri (CTT) phase depends on the interaction between the star and
its accretion disc, on a distance of a few stellar radii. At the truncation
radius, matter from the disc surface is channelled onto the stellar surface
following the magnetic field lines and forming an accretion funnel or column
(Ghosh et al., 1977; Zanni & Ferreira, 2009; Romanova & Owocki, 2016;
Pantolmos et al., 2020). The star-disc interaction is responsible for
accretion and ejection phenomena that have a strong impact on spectral lines
formed in the close vicinity of the star’s surface.

Ghosh et al. (1977) developed an analytical model of magnetospheric accretion
around a rotating neutron star with a dipolar magnetic field. Hartmann et al.
(1994) applied this magnetospheric accretion model to the formation of
emission lines in the spectrum of T Tauri stars. This fundamental paper sets
the general theoretical framework for the density and temperature
distributions in aligned axisymmetric magnetospheres. The coupling between
this representation of magnetospheric accretion in T Tauri systems with
radiative transfer calculations has provided a crucial tool to interpret
spectroscopic, photometric, and interferometric observations. The sensitivity
of hydrogen lines to the parameters of the magnetospheric models was studied
in detail by Muzerolle et al. (2001), improving the earlier calculations by
Hartmann et al. (1994).

Near-infrared observations of the Brackett $\gamma$ (Br$\gamma$) line with the
Very Large Telescope Interferometer (VLTI) GRAVITY instrument (Gravity
Collaboration et al., 2017) also probe the inner part of the star-disc
interaction region (Gravity Collaboration et al., 2020; Bouvier et al.,
2020a). However, it is still difficult to associate the characteristic sizes
derived from interferometry with the actual size of the magnetospheric
accretion region, a key parameter in our understanding of the star-disc
interaction.

In this paper, we aim at studying the formation of the Br$\gamma$ line and
compute its spectroscopic and interferometric signatures for non-axisymmetric
models of the inner star-disc interaction region, akin to state-of-art MHD
simulations (Romanova & Owocki, 2016). In particular, we want to clarify the
meaning of the sizes inferred through near-infrared interferometric
observations and how they compare with the overall size of the magnetospheric
accretion region.

In sections §2 and §3, we describe the model used to compute the line
formation in accreting magnetospheres. We discuss spectroscopic and
interferometric signatures in sections §4 and §5, respectively.

##  2 Radiative transfer framework

We use the code MCFOST 111https://github.com/cpinte/mcfost (Pinte et al.,
2006, 2009; Tessore et al., 2021) to compute emergent line fluxes from
multidimensional models of magnetospheres for a 20-level hydrogen atom. The
atomic model, with 19 bound levels and the ground state of HII, consists of
171 bound-bound transitions (atomic lines) and 19 bound-free transitions
(continua). We focus here on the Br$\gamma$ line at 2.1661 µm although, the
Balmer lines H$\alpha$ and H$\beta$ and the Paschen $\beta$ line (Pa$\beta$)
are modelled as well. These specific hydrogen lines are commonly used to
characterise accretion and ejection phenomena in young systems (Folha &
Emerson, 2001; Alencar et al., 2012; Bouvier et al., 2020a; Pouilly et al.,
2020; Sousa et al., 2021). The method to solve for the non-LTE populations of
hydrogen and the microphysics are the same as in Tessore et al. (2021). The
updated version of the code we use now simultaneously solves the charge
equation and the statistical equilibrium equations, which has been proven to
increase the convergence in chromospheric conditions (Leenaarts et al., 2007).
We tested our code for different magnetospheric models taken as benchmarks in
Muzerolle et al. (2001) and Kurosawa et al. (2006). The results of this
comparison are presented and discussed in Appendix A.

##  3 Magnetospheric accretion model

Matter from the circumstellar disc is channelled onto the stellar surface
along the dipolar magnetic field lines. The stellar magnetic field truncates
the disc at a distance $R_{t}$ from the star, the truncation radius. In
practice, the interaction between the stellar magnetic field and the disc
takes place over a small region between $R_{t}$ and $R_{t}+\delta r$. Both
$R_{t}$ and $\delta r$ are used to define the size of the disc region
magnetically connected to the star. As the gas approaches the stellar surface,
it decelerates in a shock and is heated at coronal temperatures. Theoretical
models of accretion shocks by Calvet & Gullbring (1998) show that the
optically thin emission of the pre/post-shock dominates below the Balmer jump
and that the optically thick emission of the heated photosphere contributes to
the total continuum emission at larger wavelengths. In the following, we only
consider the contribution of the heated photosphere to the shock radiation.
The shock222We assume that the shock region is unresolved and is part of the
stellar surface. temperature is computed from the energy of the gas infalling
onto the stellar surface following the prescription of Romanova et al. (2004)
unless specified. This approach assumes energy conservation and that the shock
radiates as a black body, meaning that its temperature is determined by the
specific kinetic energy and enthalpy of the gas deposited at the stellar
surface. The shock temperature hence derived is of the order of 4500 K - 6000
K.

###  3.1 The stellar surface

The stellar surface is considered as the inner boundary of the model and emits
as a blackbody whose temperature is determined by the stellar parameters.
Throughout the paper, the stellar parameters are $T_{\ast}=4,000$ K,
$M_{\ast}=0.5$ $M_{\odot}$, and $R_{\ast}=2$ $R_{\odot}$. We set the distance
to the star at 140 pc, which is typical of the nearest star forming regions
such as Upper Scorpius ($\approx$ 146 $pc$ Galli et al., 2018a) or Taurus
($\approx$ 130 $pc$ Galli et al., 2018b).

###  3.2 Geometry of the accretion funnels

We consider 3D non-axisymmetric models of the magnetospheric accretion region.
These models are parametrised by the same set of parameters as the
axisymmetric magnetospheric model of Hartmann et al. (1994) (see also
Muzerolle et al., 1998, 2001; Kurosawa et al., 2006; Lima et al., 2010;
Kurosawa et al., 2011; Dmitriev et al., 2019).The density and the velocity
fields of the accretion columns are fully described with a set of independent
parameters: the mass accretion rate $\dot{M}$, the rotation period $P_{rot}$,
$R_{t}$, and $\delta_{r}$.

Figure 1: Density distribution of a non-axisymmetric model with an obliquity
of 10∘. The rotation axis of the star $\Omega$ is shown with a white arrow and
the dipole axis, $\mu$, with a red arrow. The density is computed from Eqs.
(1) and (2). The colour map scales with the density.

For our study, the value of $\dot{M}$, $R_{t}$ and $\delta_{r}$, and of the
temperature of the magnetosphere are fixed. The impact of these parameters on
the line formation has been discussed thoroughly in Muzerolle et al. (1998,
2001, see also App. A). The line’s response to the mass accretion rate and to
the temperature is an essential proxy for understanding the physics of the
star-disc interaction region. We use a mass accretion rate $\dot{M}=10^{-8}$
$M_{\odot}\,{\rm yr^{-1}}$, a truncation radius $R_{t}=4\,R_{\ast}$, and
$\delta_{r}=1\,R_{\ast}$. The value of the rotation period is deduced from the
maximum truncation radius ($R_{t}+\delta_{r}$), imposing that stable accretion
occurs at 90% of the corotation radius, consistent with the work of Blinova et
al. (2016). The rotation period is therefore fixed at $P_{rot}=6$ days,
corresponding to slowly rotating T Tauri stars (see Herbst et al., 2007;
Bouvier et al., 2014, for a review). The rotational velocity for that period
is thus of the order of 80 ${\rm km\,s^{-1}}$ at the outer edge of the
magnetosphere.

When the magnetic field axis ($\mu$) is misaligned with respect to the
rotational axis ($\Omega$) of the star, the geometry of the accretion flow
changes dramatically. The equations for the magnetic field components of a
non-axisymmetric dipole, i.e. with a non-zero obliquity, are provided in
Mahdavi & Kenyon (1998). The parameter $\beta_{ma}$ describes the angle
between the dipole moment and the star’s rotational axis, the magnetic
obliquity.

We approximate the density, $\rho$, along the non-axisymmetric magnetic field
lines with,

$\rho=\alpha\dfrac{B}{\varv}=\alpha B\dfrac{\rho_{axi}}{B_{axi}}\text{,}$ (1)

where $\alpha$ is a constant along a given field line and $B$ the analytic
misaligned dipolar field. $\varv$, $\rho_{axi}$, and $B_{axi}$ denote the
velocity field, density, and dipolar magnetic field, respectively, and they
are taken from the axisymmetric model of Hartmann et al. (1994). In other
words, the 3D density structure is computed from Eq. (1) under the assumption
that the infalling gas has a velocity field on the poloidal plane. The value
of $\alpha$ is computed from the numerical integration 333For this 3D
magnetospheric accretion model, an explicit formula for the shock area does
not exist (see also Mahdavi & Kenyon, 1998) of the mass flux over the shock
area,

$\dot{M}=\displaystyle\int\rho\mathbf{v}\cdot\mathbf{dS}\text{,}$ (2)

where $dS$ is the surface element and $\mathbf{v}$ the velocity field. In our
model, the value of $\dot{M}$ is an input parameter and is held constant.
Therefore, $\alpha$ is obtained to ensure consistency between Eqs. (1) and
(2). We compute five models with an obliquity $\beta_{ma}$ ranging from five
to forty degrees in step of ten degrees, representative of what has been
measured for T Tauri stars with spectroscopy (McGinnis et al., 2020) and
spectropolarimetry (Donati et al., 2008, 2010, 2013; Johnstone et al., 2014;
Pouilly et al., 2020). For these non-axisymmetric models, the shortest field
lines – defining the main accretion columns444Geometrically, the shortest
field lines obey the following criterion $\cos\phi^{\prime}\times z>0$ where
$\phi^{\prime}$ is the azimuth in the frame aligned with the dipole axis and
$z$ the coordinate parallel to the rotation axis. – carry most of the gas
density. We remove the longest field lines – the secondary columns – in our
modelling as in Esau et al. (2014). This yields models with one crescent-
shaped accretion spot per stellar hemisphere reminiscent of numerical
simulations of misaligned dipoles (Romanova et al., 2003).

Figure 1 shows the density of a non-axisymmetric magnetosphere with an
obliquity of 10∘.

###  3.3 Temperature of the funnels

The temperature of the magnetospheric accretion region is not well
constrained. The determination of the temperature by Martin (1996) from first
principles was not able to reproduce the observations. A self-consistent
calculation of the temperature of the magnetosphere is beyond the scope of
this paper. Instead, we adopt here the temperature profile of Hartmann et al.
(1994), which has been extensively used in the past to model line fluxes from
accreting T Tauri stars. The temperature is computed using a volumetric
heating rate ($\propto r^{-3}$) and balancing the energy input with the
radiative cooling rates of Hartmann et al. (1982). The exact balance between
the heating and cooling mechanisms is unknown. Instead, the temperature
profile is normalised to a free parameter, $T_{max}$, that sets the value of
the maximum temperature in the funnel flow. In the following, we have set the
temperature maximum to $T_{max}=8,000$ K.

##  4 Spectroscopic signatures

Thanks to the Doppler shift of the funnel flow, it is possible to reconstruct
the origin of the emission line by looking at the brightness maps in various
velocity channels. Figure 2 shows the contribution of the different parts of
the magnetosphere to the total integrated Br$\gamma$ line flux at a given
velocity for an inclination of 30∘, matching the model illustrated in Fig. 1.
At those density and temperature, the continuum emission comes from the
stellar surface (${\rm I_{surf}/I_{mag}>100}$). Locally, the continuum
emission from the shock is three times larger than the emission from the star.
Overall, given the small covering area of the accretion shock (around 1%), the
total continuum emission at the frequency of the Br$\gamma$ line is dominated
by the star’s radiation, ${\rm F_{shock}/F_{\ast}=3\%}$.

Figure 2: Origin of the emission seen across the Brackett $\gamma$ line. The
contribution of individual images to the total line flux is indicated on the
central image showing the line profile. The brightness maps are in units of
the maximum emission. The emission of the stellar surface is saturated. Orange
to red colours indicates the regions of maximum emission. The system is seen
at an inclination of 30∘ and an rotational phase of $\sim$0.25, similar to
Fig. 1.

The low-velocity components (¡ 50 ${\rm km\,s^{-1}}$) of the line form in the
regions where the projected velocity along the line-of-sight is close to zero
and near the disc. The geometry of the non-axisymmetric model, defined in §3,
is responsible for a rotational modulation of the integrated line flux. Many
classical T Tauri stars shows modulated photometric variability (e.g Cody et
al., 2014) and, more directly related to the magnetospheric region, many also
show rotational modulation of the longitudinal component of the stellar
magnetic field (e.g Donati et al., 2020). Indeed, the periodic variability of
optical and emission line profiles has been reported in various systems (for
instance Sousa et al., 2016; Alencar et al., 2018; Bouvier et al., 2020a),
which indicates that the emission region is stable on a timescale of several
rotation periods.

Figure 3 shows the variability of the Br$\gamma$ line at different phases of
rotation at an inclination of $60^{\circ}$ for different obliquities. The
origin of the rotational phase is defined such that at phase 0.5, the
accretion shock is facing the observer. The red-shifted absorption seen for
the Br$\gamma$ line at phases 0.250, 0.47 and 0.69, results from a lower
source function of the gas above the shock (see App. A). From observations,
red-shifted absorption in the Pa$\beta$ and Br$\gamma$ lines are seen in less
than 34% and 20% of the line profiles, respectively (Folha & Emerson, 2001).
The inverse P Cygni profile disappears when the shock, or a significant
fraction of it, is hidden on the opposite side of the star. The line, with
either a double-peaked profile or a moderate red-shifted absorption, is
reminiscent of Reipurth et al. (1996) cases II and IV. While the profiles with
redshifted absorption agree with observations, those that display an M-shape
are usually not observed in young stellar objects. This suggests that
magnetospheric accretion is not the only contribution to the profile, which
can also be impacted by various types of outflows (e.g., stellar, interface,
and disk winds Lima et al., 2010; Kurosawa et al., 2011). The optically thick
accretion disc is not included in our models. The effect of the disc emission
and absorption on the spectroscopic and interferometric observables will be
discussed in a subsequent paper.

We also observe a decrease of the line flux as the obliquity increases. Figure
4 shows the radius encompassing 90% of the total line flux, $R_{90}$, at an
inclination of 60∘ for non-axisymmetric models with different obliquities for
the H$\alpha$, H$\beta$, Pa$\beta$ and Br$\gamma$ lines.

Figure 3: Brackett $\gamma$ line variability along the rotational cycle. Each
column corresponds to a specific rotational phase. At phase 0, the shock area
is unseen on the stellar surface, while phase of 0.5, the shock is fully seen
on the visible hemisphere. The colours correspond to different values of the
obliquity. All fluxes are computed with an inclination of 60∘. Figure 4:
Radius encompassing 90% of the total flux ($R_{90}$) for each line as a
function of the obliquity, $\beta_{ma}$. Hydrogen lines are labelled with
different colours.

As $\beta_{ma}$ increases, the volume of the magnetospheric accretion region
decreases because the arc length of the accreting field lines shortens.
Therefore, the total flux, for all lines, decreases accordingly, independently
of the viewing angle of the system. However, we also note a dependence of
$R_{90}$ with the line. The value of $R_{90}$ represents the size of the
emitting region in a given line, which is a function of density and
temperature, and of the viewing angle.

##  5 Interferometric signatures

In this section, we compute the size of the Br$\gamma$ line-emitting region
inferred from interferometric observations, and we compare it to model flux
radii (see §4).

###  5.1 Interferometric observables

The interferometric observables are derived from the radiative transfer (RT)
model using the ASPRO2555Available at https://www.jmmc.fr software developed
by the Jean-Marie Mariotti Center (JMMC). These observables represent what
would be observed with GRAVITY in the near-infrared. We consider the
configuration obtained with the Very Large Telescope (i.e. 4x8m telescopes),
encompassing a range of baselines from 35 to 135 m. With a typical night of 8
hours, we compute one observing point per hour for the six baselines of the
VLTI to increase the Fourier sampling, namely u-v coverage, which is crucial
for the fitting part of our approach. As described in Bourgès & Duvert (2016),
we derive the observables from the RT images (see Fig. 2) by computing the
complex visibility in each spectral channel around the Br$\gamma$ line and
interpolating them to match GRAVITY’s spectral resolution (R = 4000).
Specifically, we simulate a total of 37 spectral channels (from 2.161 to 2.171
µm with a step of $2.8\,10^{-4}$ µm) for the six projected baselines repeated
eight times. Within this range, 31 spectral channels are used to measure the
K-band continuum and six channels sample the Br$\gamma$ line emitting region.

Figure 11 illustrates the resulting u-v plane projected on-sky for a typical
object observed at the VLTI with a declination of -34∘ (e.g. TW Hydrae).

Figure 5 shows the interferometric observables along the rotational cycle for
a model with an inclination of 60∘ and an obliquity of 10∘. The two main
observables are: the modulus of the complex visibility – the visibility
amplitude – and the differential phase – its argument – dispersed in
wavelength. The phase is normalised to zero in the continuum. The visibility
amplitude can then be used to estimate the object’s size, while the phase
measures the photo-centre shifts between the line-emitting region and the
continuum. The phase can only be used as a relative measurement (e.g. between
the line and the continuum), the absolute phase being lost due to a
combination of atmospheric and instrumental effects. We repeat the simulated
observations and compute nine datasets over a rotational cycle sampled every
40 degrees ( 0.11 in phase). In this study, we are interested in the line’s
emitting region only. Therefore, we use pure line quantities, instead of total
visibilities and phases, to remove the contribution from the stellar surface
(see appendix B for the derivation of the pure line interferometric
quantities).

Figure 5: Synthetic interferometric measurements and modelling across the
rotational phase of the system. Top: integrated images over the Br$\gamma$
line. Green (uniform disc) and yellow (Gaussian disc) ellipses are the
characteristic sizes measured with a GRAVITY-like instrument. Middle-top: Pure
line visibility amplitude observables associated with the corresponding
models. The visibility variation (for a given baseline) as the u-v plane
rotates is the specific signature of an elongated object. Middle-bottom: Pure
phase visibility across the line profile for the six baselines of the VLTI.
Colours encode the observing time. Bottom: Velocity map of the radiative
transfer model. The coloured dots represent the measurement of the photo-
centre derived from the phase visibility in each available spectral channel
(see appendix B). In each figure, the magnetosphere and the stellar surface
have been normalised independently, for display purpose.

###  5.2 Physical characteristic and sizes

Once the interferometric observables are computed, we apply standard modelling
methods to interpret the data (Berger, 2003). Firstly, we average the
visibility amplitude of the six spectral channels available within the
Br$\gamma$ line666Five and four spectral channels only were used at phase 0.25
and 0.47, respectively, due to a limited line-to-continuum ratio (see Appendix
B for details).. We use the average visibilities to recover the global size of
the Br$\gamma$ emitting region, where the different velocities probe specific
parts of the moving material within the magnetosphere. Then, we fit the
averaged visibility amplitude using elongated Gaussian or uniform disc models.
Such models are typically used in interferometry to estimate the system’s
characteristic size and on-sky orientation. The source’s brightness
distribution is defined by its half-flux radius in the case of a Gaussian disc
or its radius for the uniform disc model, and an elongation factor and a
position angle. In the following, we adopt the definition of ”radius” for both
models, which corresponds to the half-flux semi-major axis for the Gaussian
model and the semi-major axis for the uniform disc model. The recovered sizes
and orientations are represented in the top panel of Fig. 5. While neither
model can fully account for the size of the magnetosphere, the uniform disc
probes a larger area of the magnetosphere, while the Gaussian disc seems
limited to the most luminous parts. We note that the fit of the visibility is
equally good for both models and, thus, does not allow us to discriminate
between the models from the synthetic visibilities only (middle-top, Fig. 5).

In order to quantify the physical meaning of the interferometric measurements,
we compare the interferometric sizes with reference flux radii of the RT
models. We set these radii to represent 50, 80, 90, and 99$\%$ of the total
flux emitted by the magnetospheric accretion region. Figure 6 compares the
sizes derived with interferometry to the characteristic radii of the RT
models.

We find that the size derived from the uniform disc model is modulated around
an average value of 3.5 R∗ corresponding to 90$\%$ of the Br$\gamma$ emitting
region. The size obtained by interferometry appears to be modulated by the
position of the funnel flows close to the star, with a minimum located around
phase 0.8. The Gaussian model exhibits the same modulation but with a lower
amplitude ($2.1\pm 0.4$ R∗) and appears sensitive to the magnetosphere’s
innermost region, close to the 50$\%$ flux radius. The size derived from the
uniform disc model emerges as being the most appropriate to recover the
reference model size, accounting for at least 80$\%$ of the total flux emitted
by the magnetosphere.

Figure 6: Interferometric radii as a function of the rotational phase. Uniform
and Gaussian disc models are shown with green and yellow markers,
respectively. Blue lines correspond to the radii encompassing 50, 80, 90 and
99$\%$ of the total RT model’s flux. The blue shaded areas represent the
standard deviation of these radii across the rotational phase.

The red shaded area indicates the inner ($R_{t}$) and outer radius
($R_{t}+\delta r$) of the RT model.

The derived orientations obtained from interferometry seem to be particularly
representative of the position of the accretion funnel flow and the on-sky
orientation of the Br$\gamma$ emitting region (Fig. 5). The measured position
angle agrees with the magnetosphere’s orientation, particularly when the shock
faces the observer (phase = 0.5). Nevertheless, it appears somewhat hazardous
to decipher the shape and orientation of the emitting region across the
rotational cycle from this observable only, as different magnetospheric
configurations can be described by a very similar interferometric model (e.g.
phases 0.03 and 0.25). A stronger constraint on the orientation of the funnel
flows arises from differential phase measurements.

###  5.3 Differential phases and photo-centre shifts

From the differential phases, we can derive the photo-centre shift between the
continuum and the Br$\gamma$ line emitting region. In the regime of marginally
resolved sources, there is a direct relationship between the projected photo-
centre displacement vector (P) and the phase along each baseline (Lachaume,
2003):

$\phi_{i}=-2\pi\frac{B_{i}}{\lambda}\textbf{P},$ (3)

where $\phi_{i}$ is the differential phase measured for the $i$th baseline,
$B_{i}$ is the length of the corresponding baseline, and $\lambda$ is the
effective wavelength of the spectral channel. A four telescope beam-combiner
like GRAVITY gives access to six projected baselines that enable us to
accurately retrieve the value and orientation of the photo-centre shifts in
each spectral channel (Le Bouquin et al., 2009; Waisberg et al., 2017). Such a
measurement results in a position-velocity plot of the displacement of the
photo-centre across the Br$\gamma$ line relative to the continuum. This is
illustrated in the bottom panels of Figure 5.

The photo-centre shifts trace the accretion funnel flow’s direction and follow
the stellar rotation. For instance, when the northern accretion shock
(N-shock) is located behind the star (phase = 0), the accreting material falls
onto the stellar surface in the direction of the observer. Accordingly, the
photo-centre measured in the blue-shifted part of the line profile ($\simeq$
-75 ${\rm km\,s^{-1}}$) lies on the blue-shifted part of the velocity map,
corresponding to the approaching funnel flow. Equivalently, the photo-centre
measured in positive velocity channels of the line profile ($\simeq$ +75 ${\rm
km\,s^{-1}}$) is shifted towards the receding funnel flow. In contrast, when
the shock faces the observer (phase = 0.5), the velocity map goes from blue to
red in the east-west direction, and the photo-centre shifts recover this trend
as demonstrated at phase 0.47.

We can thus identify three privileged directions and shapes of the photo-
centre shifts: – linear north-south at phase $\simeq 0$ (N-shock behind), –
S-shape at phase 0.25 and 0.69 and – linear east-west at phase $\simeq 0.5$
(N-shock in front). The differential phase is, therefore, a key ingredient to
recover the geometry and orientation of the line-emitting region, tracing the
moving material along a rotational cycle.

###  5.4 Signal-to-noise considerations

As a proof-of-concept, the results presented above assume infinite signal-to-
noise ratio. The goal is to predict the spectroscopic and interferometric
signatures of the magnetospheric accretion process. Thus, the models predict
typical visibility amplitudes ranging from 1 down to 0.97 (see Fig. 5). Such a
modest interferometric signal requires a measurement accuracy of about 1$\%$
to be securely detected. Similarly, the models predict a deviation of the
differential phases by 1 to 2 degrees (Fig. 5), which requires an accuracy of
order of a fraction of a degree to yield a robust detection. Recent
interferometric studies performed with VLTI/GRAVITY in the K-band demonstrate
that these levels of accuracy can be routinely obtained indeed with reasonable
exposure times on young stellar objects (e.g. Bouvier et al., 2020b; Gravity
Collaboration et al., 2020, 2022), or active galactic nuclei (Gravity
Collaboration et al., 2018).

##  6 Summary and conclusion

We presented non-LTE radiative transfer modelling of the Brackett $\gamma$
line emission for non-axisymmetric models of accreting magnetospheres. We used
the equations of a misaligned dipolar magnetic field to derive the geometry of
the magnetospheric accretion region for different obliquities of the magnetic
dipole. We used MCFOST to compute radiative signatures of the Br$\gamma$ line
along a full stellar rotational cycle. Further, we derived near-infrared
interferometric observables for the line, comparable to what the GRAVITY
instrument has already measured for T Tauri stars.

The main conclusions of this study are the following:

  * 1)

The total flux in the line, and the line-to-continuum ratio, depends on the
obliquity of the dipole. As the obliquity increases, the size of the emitting
region decreases, leading to a lower integrated flux. Also, projection effects
make the emission region of lines forming close to the stellar surface
appearing narrower.

  * 2)

The Br$\gamma$ line total flux varies with the rotational phase due to the
non-axisymmetry of the models induced by the magnetic obliquity. The line
profiles exhibit a red-shifted absorption, that is an inverse P Cygni profile,
when a significant fraction of the accretion shock is aligned with the
observer’s line of sight. When the shock is hidden on the opposite side of the
star, the line profiles exhibit a double-peaked shape, reminiscent of the
lines formed in rotating envelope. The latter is due to the relatively large
rotational velocity of the magnetospheric model ($\sim$80 ${\rm km\,s^{-1}}$).

  * 3)

Near-infrared interferometric observations in the Br$\gamma$ line directly
probe the size of the magnetospheric accretion region. The Gaussian disc model
is sensitive to the brightest parts of the magnetosphere, up to 50% of the
truncation radius, while a uniform disc model grasps 90$\%$ of the
magnetosphere. It is of prime importance to consider this aspect when
estimating magnetospheric radius from interferometric measurements. In both
cases, the measured radius varies with the rotational phase (due to the non-
axisymmetry of the dipole). A robust interferometric estimate of the
magnetospheric radius therefore requires monitoring the system over a full
rotational cycle.

  * 4)

The combined knowledge of the differential phase and the associated photo-
centre shifts gives hints on the object orientation and geometry. More
specifically, the relative direction of the photo-centre shifts indicates the
changing orientation of the accreting material along the rotational cycle in
the non-axisymmetric case.

Near-infrared interferometry of the Brackett $\gamma$ line is used to
characterise the inner star-disc interaction region, and offers a good estima
te of the size of the line’s forming region, at sub-au precision. Comparing
this size with reference model radii, such as the truncation radius, allows us
to distinguish between multiple origins of the Br$\gamma$ line, within or
beyond these radii (e.g. magnetosphere, stellar and disc winds, jets).
Further, simultaneous spectroscopic and interferometric observations along a
rotational cycle, have the potential to unveil the geometry and orientation of
the line’s emitting region. The variability of the line associated with the
photo-centre shifts, provides a unique and unambiguous proxy of the physical
processes occurring in the magnetosphere of young accreting systems, within a
few hundredths of an astronomical unit around the central star.

##  Appendix A benchmark

We present here the comparison between line profiles obtained with MCFOST and
previous studies. The magnetospheric model corresponds to the axisymmetric and
compact configuration of Muzerolle et al. (2001) with a fixed shock
temperature at 8 000 K, a rotation period of 5 days and the following
canonical T Tauri parameters: $M_{\ast}=0.8\,M_{\odot}$,
$R_{\ast}=2\,R_{\odot}$ and $T_{\ast}=4\,000\,K$. The inclination of the
system is 60 degrees. The continuum emission of the stellar surface (shock and
photosphere) is constant for all models. Figures 7, 8, 9, and 10 show the
H$\alpha$, H$\beta$, Pa$\beta$ and Br$\gamma$ lines profiles for different
values of $T_{max}$ and $\dot{M}$. An inverse P Cygni profile, with a red-
shifted absorption, is seen for all lines although it is dependent on the
value of the mass accretion rate and of the maximum temperature. For a given
mass accretion rate, an increase of the maximum temperature results in a
higher line emission peak and a shallower red-shifted absorption. As the
temperature increases, the line source function increases, which is the cause
of a higher emission above the continuum emission. The appearance of the red-
shifted absorption component is caused by absorption from the gas above the
stellar surface. It is controlled by the ratio between the source function of
the line in the accretion funnel and that of the underlying continuum from the
stellar surface, especially at low mass accretion rates and temperatures.
Eventually, for the highest mass accretion rate and temperature, the lines
become so optically thick that the red-shifted absorption is washed out by the
large wings of the line. The red-shifted absorption is more pronounced for
lines forming closer to the accretion shock like the H$\beta$ line. At a
temperature larger than 8,000 K and a mass accretion rate above 10-8
$M_{\odot}\,{\rm yr^{-1}}$, the continuum emission from the magnetosphere
becomes important and the line-to-continuum ratio decreases. This effect is
seen for instance in the H$\alpha$ line (see Fig. 7). When the mass accretion
rate increases for a given temperature, the density of the magnetosphere
increases. As a consequence, the line source function increases. At high
temperature and high density, the background continuum emission of the
magnetosphere dominates for certain wavelengths, and absorption occurs. The
latter effect is seen in the Pa$\beta$ (Fig. 9) and Br$\gamma$ (Fig. 10) lines
where the strong continuum contribution at the disc surface leads to
absorption at low velocities, where the lines source function is small. These
results are consistent with the previous studies and demonstrate the
robustness of our code for modelling the close environment of T Tauri stars
(Tessore et al., 2021).

Figure 7: Dependence of H$\alpha$ line flux with mass accretion rates
$\dot{M}$ and maximum temperatures $T_{max}$. The inclination of the system is
$60^{\circ}$. Figure 8: Same as Fig. 7 for H$\beta$ Figure 9: Same as Fig. 7
for Pa$\beta$. Figure 10: Same as Fig. 7 for Br$\gamma$.

##  Appendix B Derivation of the interferometric pure-line phase and
visibility

We focus on the magnetospheric emission probed by the Br$\gamma$ line and,
therefore, aim to remove any additional contributions (stellar photosphere,
the accretion shocks, dusty disc, etc…). Following Kraus et al. (2008);
Bouvier et al. (2020b), we compute the continuum-subtracted observables, the
so-called pure line visibility and phase, by using the emission line profiles
computed in Sect. 3. This is of prime importance in the case of Br$\gamma$
line as the magnetospheric emission is quite faint in the infrared ($\approx
1.3$ excess flux compared to the continuum, Fig. 3). The derivation of the
pure line quantities is only possible if the source is marginally resolved
(i.e, size $<\lambda/2B$).

In this case, the pure line visibility $V_{line}$ and phase $\Phi_{line}$ are
given by:

$V_{Line}=\frac{F_{L/C}V_{Tot}-V_{Cont}}{F_{L/C}-1},$ (4)
$\Phi_{Line}=arcsin\left(\frac{F_{L/C}}{F_{L/C}-1}\frac{V_{Tot}}{V_{Line}}\sin{\Phi_{Tot}}\right).$
(5)

Where $F_{L/C}$ denotes the line-to-continuum flux ratio as taken from the
normalised spectrum (Fig. 3), $V_{Cont}$ is the visibility computed in the
continuum (star+shock only), and $V_{Tot}$, $\Phi_{Tot}$ are the total complex
quantities measured by GRAVITY. In Eq. (4), we note that in the case when
$F_{L/C}$ is close to one, the derived $V_{Line}$ cannot exist (converges to
infinity). Such non-ideal profiles appear if the red absorption becomes too
important. Therefore, we assume to discard the affected spectral channels for
phases 0.25 and 0.47, where $F_{L/C}$ is too close to one: – one point ($v=53$
${\rm km\,s^{-1}}$) at phases 0.25 and – two points ($v=15$, 53 ${\rm
km\,s^{-1}}$) at phase 0.47.

Figure 11: Fourier sampling (u-v coverage) of the simulated data. The
different colours correspond to the six different baselines of the VLTI. The
eight points per baseline represent a typical observational sequence with one
data point per hour.

###### Acknowledgements.

The authors thank Claudio Zanni, Lucas Labadie, Catherine Dougados, and
Alexander Wojtczak for fruitful discussions. This project has received funding
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme (grant agreement No 742095; SPIDI:
Star-Planets-Inner Disk-Interactions, http://www.spidi-eu.org). B. Tessore
thanks the french minister of Europe and foreign affairs and the minister of
superior education, research and innovation (MEAE and MESRI) for research
funding through FASIC partnership. C. Pinte acknowledges funding from the
Australian Research Council via FT170100040 and DP180104235. The numerical
simulations presented in this paper were performed with the Dahu supercomputer
of the GRICAD infrastructure (https://gricad.univ-grenoble-alpes.fr), which is
supported by Grenoble research communities.

## References

  * Alencar et al. (2018) Alencar, S. H. P., Bouvier, J., Donati, J. F., et al. 2018, A&A, 620, A195
  * Alencar et al. (2012) Alencar, S. H. P., Bouvier, J., Walter, F. M., et al. 2012, A&A, 541, A116
  * Berger (2003) Berger, J. P. 2003, in EAS Publications Series, Vol. 6, EAS Publications Series, ed. G. Perrin & F. Malbet, 23
  * Blinova et al. (2016) Blinova, A. A., Romanova, M. M., & Lovelace, R. V. E. 2016, MNRAS, 459, 2354
  * Bourgès & Duvert (2016) Bourgès, L. & Duvert, G. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9907, Optical and Infrared Interferometry and Imaging V, ed. F. Malbet, M. J. Creech-Eakman, & P. G. Tuthill, 990711
  * Bouvier et al. (2020a) Bouvier, J., Alecian, E., Alencar, S. H. P., et al. 2020a, A&A, 643, A99
  * Bouvier et al. (2014) Bouvier, J., Matt, S. P., Mohanty, S., et al. 2014, in Protostars and Planets VI, ed. H. Beuther, R. S. Klessen, C. P. Dullemond, & T. Henning, 433
  * Bouvier et al. (2020b) Bouvier, J., Perraut, K., Le Bouquin, J. B., et al. 2020b, A&A, 636, A108
  * Calvet & Gullbring (1998) Calvet, N. & Gullbring, E. 1998, ApJ, 509, 802
  * Cody et al. (2014) Cody, A. M., Stauffer, J., Baglin, A., et al. 2014, AJ, 147, 82
  * Dmitriev et al. (2019) Dmitriev, D. V., Grinin, V. P., & Katysheva, N. A. 2019, Astronomy Letters, 45, 371
  * Donati et al. (2020) Donati, J. F., Bouvier, J., Alencar, S. H., et al. 2020, MNRAS, 491, 5660
  * Donati et al. (2013) Donati, J. F., Gregory, S. G., Alencar, S. H. P., et al. 2013, MNRAS, 436, 881
  * Donati et al. (2008) Donati, J. F., Jardine, M. M., Gregory, S. G., et al. 2008, MNRAS, 386, 1234
  * Donati et al. (2010) Donati, J. F., Skelly, M. B., Bouvier, J., et al. 2010, MNRAS, 409, 1347
  * Esau et al. (2014) Esau, C. F., Harries, T. J., & Bouvier, J. 2014, MNRAS, 443, 1022
  * Folha & Emerson (2001) Folha, D. F. M. & Emerson, J. P. 2001, A&A, 365, 90
  * Galli et al. (2018a) Galli, P. A. B., Joncour, I., & Moraux, E. 2018a, MNRAS, 477, L50
  * Galli et al. (2018b) Galli, P. A. B., Loinard, L., Ortiz-Léon, G. N., et al. 2018b, ApJ, 859, 33
  * Ghosh et al. (1977) Ghosh, P., Lamb, F. K., & Pethick, C. J. 1977, ApJ, 217, 578
  * Gravity Collaboration et al. (2017) Gravity Collaboration, Abuter, R., Accardo, M., et al. 2017, A&A, 602, A94
  * Gravity Collaboration et al. (2020) Gravity Collaboration, Garcia Lopez, R., Natta, A., et al. 2020, Nature, 584, 547
  * Gravity Collaboration et al. (2018) Gravity Collaboration, Sturm, E., Dexter, J., et al. 2018, Nature, 563, 657
  * Gravity Collaboration et al. (2022) Gravity Collaboration, Wojtczak, J. A., Labadie, L., et al. 2022, arXiv e-prints, arXiv:2210.13095
  * Hartmann et al. (1982) Hartmann, L., Avrett, E., & Edwards, S. 1982, ApJ, 261, 279
  * Hartmann et al. (1994) Hartmann, L., Hewett, R., & Calvet, N. 1994, ApJ, 426, 669
  * Herbst et al. (2007) Herbst, W., Eislöffel, J., Mundt, R., & Scholz, A. 2007, in Protostars and Planets V, ed. B. Reipurth, D. Jewitt, & K. Keil, 297
  * Johnstone et al. (2014) Johnstone, C. P., Jardine, M., Gregory, S. G., Donati, J. F., & Hussain, G. 2014, MNRAS, 437, 3202
  * Kraus et al. (2008) Kraus, S., Hofmann, K. H., Benisty, M., et al. 2008, A&A, 489, 1157
  * Kurosawa et al. (2006) Kurosawa, R., Harries, T. J., & Symington, N. H. 2006, MNRAS, 370, 580
  * Kurosawa et al. (2011) Kurosawa, R., Romanova, M. M., & Harries, T. J. 2011, MNRAS, 416, 2623
  * Lachaume (2003) Lachaume, R. 2003, A&A, 400, 795
  * Le Bouquin et al. (2009) Le Bouquin, J. B., Absil, O., Benisty, M., et al. 2009, A&A, 498, L41
  * Leenaarts et al. (2007) Leenaarts, J., Carlsson, M., Hansteen, V., & Rutten, R. J. 2007, A&A, 473, 625
  * Lima et al. (2010) Lima, G. H. R. A., Alencar, S. H. P., Calvet, N., Hartmann, L., & Muzerolle, J. 2010, A&A, 522, A104
  * Mahdavi & Kenyon (1998) Mahdavi, A. & Kenyon, S. J. 1998, ApJ, 497, 342
  * Martin (1996) Martin, S. C. 1996, ApJ, 470, 537
  * McGinnis et al. (2020) McGinnis, P., Bouvier, J., & Gallet, F. 2020, MNRAS, 497, 2142
  * Muzerolle et al. (1998) Muzerolle, J., Calvet, N., & Hartmann, L. 1998, ApJ, 492, 743
  * Muzerolle et al. (2001) Muzerolle, J., Calvet, N., & Hartmann, L. 2001, ApJ, 550, 944
  * Pantolmos et al. (2020) Pantolmos, G., Zanni, C., & Bouvier, J. 2020, A&A, 643, A129
  * Pinte et al. (2009) Pinte, C., Harries, T. J., Min, M., et al. 2009, A&A, 498, 967
  * Pinte et al. (2006) Pinte, C., Ménard, F., Duchêne, G., & Bastien, P. 2006, A&A, 459, 797
  * Pouilly et al. (2020) Pouilly, K., Bouvier, J., Alecian, E., et al. 2020, A&A, 642, A99
  * Reipurth et al. (1996) Reipurth, B., Pedrosa, A., & Lago, M. T. V. T. 1996, A&AS, 120, 229
  * Romanova & Owocki (2016) Romanova, M. M. & Owocki, S. P. 2016, Accretion, Outflows, and Winds of Magnetized Stars, Vol. 54, 347
  * Romanova et al. (2004) Romanova, M. M., Ustyugova, G. V., Koldoba, A. V., & Lovelace, R. V. E. 2004, ApJ, 610, 920
  * Romanova et al. (2003) Romanova, M. M., Ustyugova, G. V., Koldoba, A. V., Wick, J. V., & Lovelace, R. V. E. 2003, ApJ, 595, 1009
  * Sousa et al. (2016) Sousa, A. P., Alencar, S. H. P., Bouvier, J., et al. 2016, A&A, 586, A47
  * Sousa et al. (2021) Sousa, A. P., Bouvier, J., Alencar, S. H. P., et al. 2021, A&A, 649, A68
  * Tessore et al. (2021) Tessore, B., Pinte, C., Bouvier, J., & Ménard, F. 2021, A&A, 647, A27
  * Waisberg et al. (2017) Waisberg, I., Dexter, J., Pfuhl, O., et al. 2017, ApJ, 844, 72
  * Zanni & Ferreira (2009) Zanni, C. & Ferreira, J. 2009, A&A, 508, 1117


# Simultaneous Transmitting and Reflecting-Reconfigurable Intelligent Surface
in 6G: Design Guidelines and Future Perspectives

Waqas Khalid, Zeeshan Kaleem, Rehmat Ullah, Trinh Van Chien, Song Noh and
Heejung Yu Waqas Khalid and Heejung Yu (corresponding author) are with Korea
University, Sejong, South Korea; Zeeshan Kaleem is with COMSATS University
Islamabad, Wah Campus, Pakistan; Rehmat Ullah is with Cardiff Metropolitan
University, Cardiff, UK; Trinh Van Chien is with Hanoi University of Science
and Technology, Vietnam; Song Noh (corresponding author) is with Incheon
National University, Incheon, South Korea.

###### Abstract

Reconfigurable intelligent surfaces (RISs) have been considered as a promising
technology for the sixth-generation (6G) wireless networks that can control
wireless channels in a desirable way and significantly enhance the network
performance. Simultaneous transmitting and reflecting-RISs (STAR-RISs) can
overcome limitation of reflecting-only RISs by leveraging the higher design
flexibility and full-space coverage. Despite the benefits, the modeling and
analysis of STAR-RISs are complicated because of various control parameters
for both transmission and reflection links. In this article, a general
framework to facilitate the design and implementation of STAR-RISs in 6G
scenarios and network topologies is presented. We provide a systematic
introduction to the STAR-RIS operating protocols for different communication
modes and discuss recent efforts to identify the research progress and
combination solutions. Finally, we provide the design concepts, research
challenges, potential solutions, and future directions related to the channel
modeling, channel estimation, hardware implementations, modeling and
limitations, and optimization.

##  I Introduction

The conventional transmission strategy focuses on the design of transceivers
by leveraging multiple antennas, effective encoding and decoding procedures,
and advanced communication protocols. In future sixth generation (6G)
networks, reconfigurable intelligent surfaces (RISs) are envisioned to control
wireless channels and provide solutions to radio-frequency (RF) impairment and
signal propagation. An RIS ensures wireless connectivity via non-line-of-sight
(nLoS) links under harsh propagation conditions, e.g., LoS link blockage or
shadowing, and extends the communication distance via controlled scattering
and multipath components. An RIS enables solutions for better coverage and
transmission reliability without sophisticated signal processing or RF
operations. However, signal manipulation is widely investigated for
reflecting-only RISs [1]. The topological constraint, i.e., the same side
deployment of the transmitter and receiver, for reflecting-only RISs causes
performance degradation. A reflecting-only RIS reflects the signals within
front half-space and blocks the signals to the users behind the RIS, which
results in degraded performance for blocked users. The location constraint for
the nodes limits the service coverage to $180^{o}$, restricts the
effectiveness of the RIS, and hinders 6G small-cell implementations. To
overcome such limitations, simultaneous transmitting and reflecting-RISs
(STAR-RISs) has been introduced [2].

A STAR-RIS offers flexible implementation to provide both transmission (T) and
reflection (R) signals. The STAR-RIS establishes a full-space (i.e.,
$360^{o}$) wireless coverage via serving users located on both sides. The
location of a user determines that its signal is reflected or transmitted.
Thus, the STAR-RIS provides a higher design flexibility with more degrees of
freedom for signal manipulation. The STAR-RIS introduces more adjustable
parameters and both T and R communication links. Therefore, the previous
results of conventional RISs do not directly apply to STAR-RISs. The modeling
and analysis of STAR-RISs are still in an early stage. Motivated by the above
discussion, we present a general framework to facilitate the design and
implementation of STAR-RISs in future 6G deployments and discuss the effective
STAR-RIS designs under variable network topologies and application scenarios.
The contributions of this article are summarized as follows:

  * •

A systematic overview on the STAR-RISs is presented by discussing the
differences with other variants of RISs. The benefits and drawbacks of the
operating protocols of STAR-RIS are examined and their performance is
evaluated. We also present a research review to discuss the recent efforts and
identify the research gaps. Furthermore, the promising solutions for STAR-RISs
to achieve diverse communication objectives in 6G scenarios are discussed.

  * •

Key ideas, design issues and potential solutions of STAR-RISs are
investigated. Based on the investigation, valuable insights and future
directions are provided with respect to the channel modeling, channel
estimation, beamforming and element reconfiguration, hardware implementations,
modeling and limitations, and optimization.

The article is organized as follows. Section II presents a systematic overview
and research review. Section III discusses the research challenges, potential
solutions, and future directions from the perspective of the design and
implementation of 6G STAR-RISs. Section IV concludes the article.

Figure 1: Concept of full-space service coverage by STAR-RIS.

##  II STAR-RIS: A Systematic Overview and Research Review

A STAR-RIS is composed of many subwavelength reconfigurable elements. As
compared to other variants, i.e., holographic MIMO surfaces and RISs, the
STAR-RIS facilitates simultaneous or consecutive T and R signals with coupled
or independent controls. Both STAR-RIS and intelligent omni-surface (IOS) are
innovative metasurface implementations to serve users in full-space coverage.
However, IOS and STAR-RIS are different in terms of the architectural design
and communication protocols. An IOS splits the incident signal into T and R
signals with different amplitudes and phases, and provides the simultaneous T
and R mode. A STAR-RIS supports both electric and magnetic currents to produce
T and R signals, as shown in Fig. 1. The three operating protocols for STAR-
RISs, i.e., energy splitting (ES) [2], mode selection (MS) [3], and time
splitting (TS) [4], unlock different communication modes, including T, R, and
simultaneous T and R modes. The important aspects of the operating protocols
are shown in Fig. 2(a), and are highlighted as follows [2, 3, 4, 5]:

  * •

ES protocol: All elements enable simultaneous T and R mode. The amplitude and
phase variations can be jointly optimized in element-wise T and R
coefficients.

  * •

MS protocol: The elements are partitioned to operate separately in T and R
modes. Thus, the number of selected elements and the corresponding phase shift
variations can be jointly optimized.

  * •

TS protocol: The elements are periodically switched in the time domain for T
and R modes. Thus, the orthogonal time slots for both modes can be optimized.

(a) (b)

Figure 2: Comparison between the STAR-RIS operating protocols and other RIS types: (a) Details and (b) Achievable rate performance. Table I: Recent efforts toward the realization of STAR-RISs. Ref. |  System model  ————– Operating protocol (network) |  —————- Performance metric |  Design and optimization ——————- Design variables |  ————– Optimization methodology |  Insight remarks |  CSI estimation |  ——— Channel model |  Modeling and limitation ————– Hardware model |  ————— Hardware limitation
---|---|---|---|---|---|---|---|---|---
[3] |  ES, TS, MS (MIMO) |  Sum rates |  Precoding matrices and STAR-RIS coefficients |  BCD (analytical) |  Design the beamforming matrices in a sub-optimal manner |  $\times$ |  Rayleigh fading |  Phase shift |  Ideal architecture
[4] |  ES, MS (MIMO) |  Coverage probability |  STAR-RIS coefficients |  —- |  Incorporate the statistical CSI for T and R users |  $\times$ |  Correlated Rayleigh fading |  Phase shift |  Discrete amplitude and phase shifts
[5] |  ES, TS (SISO) |  Channel estimation error |  Training pattern matrices |  Sub-optimal methods (analytical) |  TS outperforms ES in terms of uplink channel estimation |  $\checkmark$ |  Rayleigh fading |  Phase shift |  Coupled phase shifts
[8] |  ES (SISO-NOMA-CoMP) |  Achievable rates |  STAR-RIS coefficients |  —- |  Eliminate the inter-cell interference and boost the desired signal |  $\times$ |  Rayleigh fading |  Phase shift |  Ideal architecture
[9] |  ES (MISO-NOMA) |  EE |  Precoding matrices and STAR-RIS coefficients |  DDPG (ML) |  Solve the fractional objective function with non-convex constraints |  $\times$ |  Rayleigh fading |  Phase shift |  Ideal architecture
[10] |  ES (SISO-NOMA) |  Outage probability |  Transmit SNR and a target SNR at receiver |  —- |  Incorporate the coupled phase shifts of T and R users |  $\times$ |  Rician fading |  Load impedance |  Coupled phase shifts

Achievable rates in Fig. 2(b) are extended from the previous work [6] for the
ES, MS and TS operating protocols and provide a comparison with a reflecting-
only RIS. We consider correlated Rayleigh fading channels, multicast
transmission (i.e., the same information is sent to for T and R users), and
power splitting in simultaneous wireless information and power transfer
(SWIPT) design. The detail parameters used for simulations are given in [6].
The numerical results show that the optimal selection and superiority of the
operating protocols depend on the communication objective. By altering the
parameters of a STAR-RIS (e.g., an energy splitting ratio in ES protocol, the
number of elements in MS protocol, and the duration of orthogonal time slots
in TS protocol), the STAR- RIS can satisfy the demands of both T and R users
and can also demonstrate a simple RIS scenario (to serve either T or R user).
The selection strategy determines the hardware and design complexity of STAR-
RISs as well as performance gain.

Through intelligent placement and effective beamforming designs in a full
space smart radio environment offered by the STAR-RISs, 6G wireless paradigm
can deliver its full potential in terms of signal boosting for millimeter-wave
(mmWave) and terahertz (THz) signals, power consumption reduction, throughput
enhancement, interference suppression, secure transmission (via physical layer
security), non-orthogonal multiple access (NOMA), and SWIPT. Inspired by the
aforementioned benefits, STAR-RISs can enable novel application scenarios,
e.g., STAR-RISs can be deployed on the existing structures (such as facades,
walls and ceilings of the buildings) to provide cascaded indirect links to
both indoor and outdoor users. However, research efforts from industry and
academia have mainly focused on the design and optimization of the
conventional RISs. In this context, unmanned aerial vehicle (UAV)-RIS can
alter the wireless propagation through maneuver control and signal
manipulation in full-space. However, UAV-RIS poses crucial challenges,
including complex channel estimation and modeling, and more vulnerability to
the security attacks [7]. There have been initial studies that covered STAR-
RIS-aided systems and discussed their potential benefits and combination
solutions [3, 4, 5, 8, 9, 10, 11]. Table I summarizes the relevant research
contributions.

STAR-RISs have been studied in next-generation wireless systems [3, 4, 5]. In
[3], a conventional (sub-optimal) block coordinate descent (BCD) algorithm was
employed for solving the sum rate optimization problem in STAR-RIS-aided MIMO
networks for unicast and broadcast communications. In [4], the authors derived
a coverage probability expression for a STAR-RIS-aided massive MIMO system
under correlated fading and phase-shift errors. Moreover, [5] presented the
uplink channel state information (CSI) acquisition techniques for the TS and
ES protocols. Several studies have shown that STAR-RIS provides a natural fit
for NOMA to meet stringent future 6G requirements [8, 9, 10]. In particular,
NOMA improves spectral efficiency and connectivity, and STAR-RIS extends the
full-space coverage and facilitates the basic NOMA principle. In particular,
the channel gains can be adjusted via ES and MS protocols to provide flexible
decoding orders according to the requirements of T and R users. In [8], the
transmission design of a STAR-RIS-NOMA-coordinated multi-point (CoMP) system
was addressed to simultaneously eliminate inter-cell interference and boost
the desired signal. In [9], the energy-efficiency (EE) maximization problem
for a STAR-RIS-NOMA network was solved using machine learning (ML).
Specifically, a deep reinforcement learning approach, i.e., a deep
deterministic policy gradient (DDPG), was considered. Moreover, correlated T
and R phase shifts were proposed for a STAR-RIS-NOMA system and the outage
probability and power scaling law were derived [10].

Despite the aforementioned research efforts, the study and evaluation of 6G
STAR-RISs are in an early stage. The design and implementation of STAR-RISs
under variable network topologies and application scenarios require further
investigation. Specifically, the synergy between different communication modes
and the inherent complexity with more adjustable parameters have not been
studied extensively.

##  III Research Challenges, Potential Solutions, and Future Directions

Table II:  Details of channel models and approximations for STAR-RISs.  Methods |  Physics-based models |  Details
---|---|---
Exact |  Ray-tracing-based ————— Huygens– Fresnel–based |  – Compatible with the phase shift and load impedance models – Consider the principles of geometrical optics for ray-element interaction – Incorporate the physical factors, such as environment geometry, frequency, and locations of nodes – Applicable to the far-field region under certain conditions – Simple signal power calculation ——————————————- – Applicable to the near-field region – Analyze the wavefront transformation function of the STAR-RIS – Limited to LoS and free-space scenarios – Challenge to select a proper wave-front to characterize T and R coefficients
Approximate |  Central limit ————— M-fold convolution ————— Curve-fitting |  – Tractable and accurate expressions for the performance metrics – Limited to a large number of elements and a low-medium SNR region – Not suitable for the diversity gains ——————————————- – Diversity analysis to evaluate the performance under high SNR – Low-complex asymptotic analysis ——————————————- – Wider applications (e.g., fewer elements of STAR-RIS and 6G multi-cell scenarios) – Low-complex tractable derivation – Adjustable curve-fitting functions with a challenge of a suitable selection – Not suitable for a high SNR region

###  III-A Channel Modeling for STAR-RISs

The channel models for the STAR-RISs integrate the hardware implementations
with the communication theory and find out the received signal power for T and
R users. Table II emphasizes the details of the channel modeling. The channel
models are required to incorporate the necessary propagation statistics, e.g.,
path loss, shadowing, and small-scale fading [12]. The previously reported
accurate but tractable path loss models (for RISs) can be extended for STAR-
RISs. The analytical framework can be formulated to compute T and R
communications incorporating the impact and scaling laws of design parameters
(e.g., the physical size, and the transmission distances). We present the
novel physics-based approaches that can be used to develop the channel models
for the STAR-RIS from perspective of the far-field and near-field regions
(based on the changing behavior of power with distance for the specific
configuration). In this context, ray-tracing technique can be adopted for far-
field channel models. For near-field users, the wave-front transformation
function of STAR-RISs can be studied by Huygens–Fresnel principle, and
Fresnel–Kirchhoff diffraction formula can be extended to calculate the channel
gains. For approximation analysis, the central-limit, M-fold convolution, and
curve-fitting models can be adopted. The central-limit model is a mathematical
tool to derive the channel model as a summation of random variables. The
M-fold convolution method can derive the diversity order (i.e., an asymptotic
reliability metric) to evaluate a high SNR performance. Furthermore, different
curve-fitting functions can be exploited. In addition, gamma distribution can
be considered for the approximation of the received signal.

####  III-A1 Open Issues and Future Directions

In the literature, design and physical properties are generally discussed.
However, there is a lack of physical implementations-compliant and
mathematical-tractable channel models for 6G STAR-RISs.

  * •

Numerically reproducible channel models to calculate T and R coefficients and
derive the physical-layer performance with the adjustable parameters for the
propagation environment and geometric setting (e.g., size and placement plane
of the STAR-RIS) are still in their infancy.

  * •

Channel models are required for different network scales (e.g., single- and
multi-user), deployment scenarios (e.g., indoor and small cell), channel
conditions (e.g., static and time-varying), and antenna settings (e.g., single
antenna and massive MIMO).

  * •

Outdoor application scenarios are generally formulated. Near-field propagation
(e.g., indoor scenarios) and wireless solutions must be discussed extensively
considering implementations with a large aperture (in terms of wavelengths) or
using higher carrier frequency (e.g., mmWave and THz).

  * •

Since the exact fading distribution to characterize the system design and
performance optimization will be non-trivial in complex environment, the
derivation of appropriate mathematical approximations will be required.

###  III-B Channel Estimation Designs for STAR-RISs

Timely and accurate CSI acquisition is indispensable for a substantial
performance gain in conventional RIS- and STAR-RIS-aided networks. However, in
practical terms, it is a nontrivial task to acquire perfect CSI with
negligible training overhead, complexity, and on-board signal processing, even
for a conventional RIS. This is because of many reflecting elements with
hardware constraints and non-linear characteristics. The passive RIS lacks RF
chains to transmit or receive pilot signals to estimate unknown channel
coefficients. Most of the literature studies for the conventional RIS assume
the CSI availability (at the transceiver, and RIS). The channel estimation
methods for both cascaded and separate channels have been investigated under
different system and channel models. The multidimensional structure of the
received signals using parallel factor (PARAFAC) tensor decomposition can be
leveraged to estimate the high dimensional RIS channels with low complexity,
and high accuracy and stability in various scenarios [13]. The channel
features, e.g., high-dimensional but quasi-static (base station-RIS channels),
mobile but low-dimensional (RIS-user channels), and low-rank and block
sparsity (mmWave channels) can also be exploited. In addition, deep learning
(DL) approaches can estimate multiple large RIS channels for the general
systems or complicated channel conditions. However, the disadvantages of
offline trained networks are the higher hardware complexity, power
consumption, and a mismatch with the real-world conditions. The channel
estimation methods for RIS cannot be directly applied to STAR-RIS because both
T and R patterns need to be designed for STAR-RIS in contrast to designing a
single T (or R) pattern for reflecting-only (or transmission-only) RIS. The
channel estimation for STAR-RISs revolves around hardware designs and
operating protocols. The estimation performance in terms of the training
overhead, accuracy and computational complexity is generally influenced by the
number of T and R users, elements, and operating protocols. The channel
estimation designs for STAR-RISs are still in an early stage and are presented
under the top-level taxonomy of model-based (e.g., signal-processing), and
data-driven-based (e.g., ML) methods.

####  III-B1 Model-based Methods

Least square (LS), linear minimum mean-squared-error (LMMSE), compressed
sensing, and matrix factorization, can be adopted to estimate STAR-RIS
channels. Specifically, the closed-form analytical expressions of the pilot
sequences and training patterns of the T and R users can be constructed.
Different strategies, e.g., grouping of elements, (with a common T or R
coefficient), low-rank channel covariance, and use of common channel (e.g.,
BS-STAR-RIS), can be applied to provide a flexible trade-off between training
overhead and beamforming performance. The simple/low-complex pilot-based
channel estimation methods, including LS/LMMSE, are applicable to
overdetermined linear signal models, and a wider range of system
configurations (e.g., single-/multi-user) and antenna configurations (e.g.,
SISO/SIMO/MISO/MIMO). However, LS/LMMSE-based estimation suffers from
performance degradation under non-linear and non-stationary channels.
Compressed sensing exploits the sparse and low-rank mmWave and THz STAR-RIS
channels, reducing the training overhead and performance error, and is
applicable to under-determined linear signal models. Furthermore, matrix
factorization methods decompose the high-dimensional aggregate STAR-RIS
channels (e.g., MIMO and multi-user) into a series of lower-dimensional sub-
channels with a lower training overhead. The compressed sensing and matrix
factorization offer a higher computational complexity than LS/LMMSE. The
channel estimation of the STAR-RIS operating protocols can be generally
discussed as:

  * •

ES protocol: Simultaneous estimation of T and R cascaded channels requires a
lesser pilot overhead at the expense of a higher design complexity. The
intricately coupled T and R channel coefficients per element cause performance
degradation. To solve a challenging CSI acquisition for the ES protocol, a
solution based on independent arbitrary T and R channel coefficients can be
first obtained which can then be utilized to develop a nearly-optimal solution
with a coupled phase-shift model.

  * •

MS protocol: The MS protocol requires lesser pilot overhead and computational
complexity because the correlation relationship of the T and R coefficients is
not required with the separate estimation of the channels.

  * •

TS protocol: The separate estimation of T and R channels (in different time
slots) results in a higher accuracy at the expense of a higher pilot overhead.

####  III-B2 Data-driven-based Methods

Data-driven approaches cover different wireless areas, such as channel
estimation, channel tracking and prediction, and beamforming. Specifically,
the feature extraction and nonlinear mapping capabilities can be applied for
channel estimation. ML-based approaches can learn the STAR-RIS channels
efficiently for any network configuration and signal model. The conventional
approach relies on the channel model, which can be complex and costly to
estimate, and can only provide an approximation to reality. On the other hand,
ML-based approaches can provide efficient estimation solutions with low
training overhead and high accuracy without explicit channel models. ML-based
approaches can demonstrate better estimation performance in the real-time
under complex conditions, such as statistical correlation and non-stationary
channels.

####  III-B3 Open Issues and Future Directions

  * •

The key aspects of the model-based and data-driven channel estimation
approaches for the STAR-RIS are to derive the explicit channel models and
determine the accurate but low complex learning algorithms, respectively.

  * •

The channel estimation methods with the optimal performance-overhead tradeoff
will be crucial for 6G STAR-RIS operating protocols.

  * •

ES protocol can be deployed in scenarios with the low mobility and channel
variation to tackle the channel estimation for the intricately coupled T and R
signals.

  * •

The joint design of the training signal and channel estimation algorithm is
nontrivial. The optimization of a training signal under a chosen estimation
algorithm can be an efficient approach.

Figure 3: Conceptual comparison of hardware implementations and modeling for
STAR-RISs.

###  III-C Hardware Implementations and Modeling

The conceptual comparison and compatibility of physical implementations and
hardware models of the STAR-RISs are illustrated in Fig. 3.

####  III-C1 PIN Diode- and Metasurface-based Implementations

From a manufacturing perspective, the full-space manipulation of T and R
signals demands different hardware implementations for STAR-RIS. Depending on
the tuning mechanism, hardware implementations can be divided into positive
intrinsic negative (PIN) diode- and metasurface-based implementations. PIN
diode-based implementation relies on the PIN diodes distribution over two
symmetrical layers. More precisely, the amplitude and phase variations of the
T and R signals can be adjusted by the ON/OFF states of PIN diodes, which are
controlled via DC voltage signals (biasing) [14]. Some other detail is as
follow:

  * •

Operating frequency: Generally used for lower frequencies (10 kHz up to 1
GHz).

  * •

Advantage: Most common hardware realization owing to its reduced cost, size,
weight, and power consumption, and its voltage-controlled operating mechanism.

  * •

Disadvantage: Challenge to achieve the independent T and R signals with a
limited number of PIN diodes.

Metasurface-based implementation uses a material whose EM properties (e.g.,
permittivity or conductivity) can be modified for effective electric and
magnetic responses. In this context, the prototype dynamic metasurface
designed by NTT Docomo Inc. can dynamically control 28-GHz 5G radio signals. A
prototype metasurface can successfully manipulate T and R signals in a highly
transparent and compact package; therefore, it is suitable for unobtrusive use
in buildings and vehicles. In addition, graphene has excellent EM properties
in terms of carrier mobility and electrical conductivity, thereby enabling
different potential applications in photonic devices. The conductivity of
graphene can be dynamically modulated (electrically or magnetically).
Therefore, tunable R and T coefficients can be achieved. Graphene-based STAR-
RISs have excellent tunable response and low fabrication cost.

####  III-C2 Modeling for Hardware Implementations

STAR-RISs differ from conventional RISs in terms of hardware design, physics
principles, and communication system design. The development of a unified
model for analyzing the reconfigurability of elements in different hardware
implementations is still in its infancy. The physics-compliant hardware
modeling to reflect the arbitrary EM field interaction with a STAR-RIS and
characterize the T and R coefficients per element is an intrinsically complex
problem; however, it is required to facilitate research on 6G STAR-RISs. In
this regard, we present phase shift, load impedance, and generalized sheet
transition conditions (GSTC) models to discuss the necessity and prerequisites
of hardware characteristics and study the boundary conditions of the EM field
at STAR-RISs. The phase shift model characterizes the response of the EM wave
in terms of phase-shift values. In the load impedance model, the radiated T
and R fields are characterized by electric and magnetic impedances. The GSTC
model provides an analytical (algebraic) formulation of the electric and
magnetic polarization densities as a function of macroscopic fields and dyadic
surface susceptibilities. The aforementioned hardware models are based on
different assumptions and approximations; however, they can enable a suitable
insightful analysis.

####  III-C3 Open Issues and Future Directions

There is an inherent tradeoff between independent controls and design
complexity for STAR-RISs. Therefore, novel hardware implementation and
manufacturing solutions are required to make STAR-RISs more scalable and
affordable while maintaining their tuning capabilities and independent phase
and amplitude controls.

###  III-D Practical Hardware Limitations for STAR-RISs

Because of the complex nature of the interaction between STAR-RISs and the
environment, ideal hardware realizations and oversimplified hardware models
have been initially adopted in research studies. This limits the design and
performance optimization in practice. The development of 6G STAR-RISs while
addressing practical hardware limitations will be a challenge; therefore,
further research is required. The practical hardware constraints, including
the maximum number of elements, coupled phase shifts, phase-dependent
amplitude, discrete (quantized) amplitude and phase shifts, and correlated
channels, need to be considered.

####  III-D1 Coupled Phase Shifts

The amplitude and phase shift variations for T and R users are generally
chosen as constrained (law of energy conservation) and independent,
respectively. However, independent T and R coefficients are non-trivial to
realize for purely passive lossless STAR-RISs. We next list some details about
coupled phase shifts:

  * •

In the ES protocol, the coupled phase control is required in general. For
semi-passive or active STAR-RISs, however, the independent phase shift is
possible at the expense of increased manufacturing cost and reduced
scalability.

  * •

In the TS and MS protocols, only the independent phase control is required due
to separate T and R modes. However, such operations entail a higher hardware
complexity due to frequently updating T and R coefficients.

####  III-D2 Phase-Dependent Amplitude

The amplitude response for conventional RISs is generally assumed to be
independent of the phase-shift response. However, such ideal control is only
possible for circuit implementations with zero energy dissipation. Similar to
conventional RISs, the amplitude adjustments for T and R users introduced by a
STAR-RIS will also be described in terms of non-uniform functions of the
respective phase shifts. Specifically, each amplitude exhibits a minimum value
for the phase shift near zero and a maximum value for the phase shift around
$-\pi$ or $\pi$. It is important to characterize the intertwined relationship
between the amplitude and phase-shift responses and capture the phase-
dependent amplitude variations in element-wise T and R coefficients.

####  III-D3 Discrete Amplitude and Phase Shift

The discrete phase shift and amplitude selection limits the ability of STAR-
RISs to adjust the wireless environment and degrades the system performance.
However, the continuous responses provide an upper-bound (and simpler)
analysis at the expense of a high implementation cost owing to the high-
resolution phase shift and amplitude controllers. The selection of the minimum
control bits for quantized phase shift and amplitude levels per element to
provide a cost-effective yet nearly-continuous shifting-based performance is
vital.

####  III-D4 Correlated STAR-RIS Channels

The spatial correlation cannot be avoided in an isotropic scattering
environment. Although independent and identically distributed (i.i.d.)
Rayleigh model is not appropriate for sub-wavelength-sized elements, it can be
used in design methodologies to provide an upper-bound, simpler, and tractable
performance analysis.

####  III-D5 Open Issues and Future Directions

The aforementioned hardware limitations complicate design and analysis. The
development of 6G STAR-RISs can be summarized as follows:

  * •

Assuming coupled phase shifts, phase-dependent amplitudes, and discrete
amplitudes and phase shifts for T and R users, the investigation of the STAR-
RIS designs as well as the characterization of performance degradation will be
challenging.

  * •

Spatially correlated Rayleigh fading provides a valid model and accurate
insights under isotropic scattering while deriving performance metrics. The
research on STAR-RISs can be first established using the simpler i.i.d. model
for later extension to spatially correlated channels.

Table III:  Details of the optimization methods for STAR-RISs. Problem description |  Divisions |  Optimization methods |  Benefits |  Limitations
---|---|---|---|---
– Non-convex (e.g., mixed-integer or NP-hard) problems due to the coupled variables (e.g., precoding and T and R coefficient matrices) and theoretic fundamental constraints (e.g., energy conservation and coupled phase-shift constraints in the ES protocol, and unit modulus constraints in the TS and MS protocols) |  Analytical approach |  Iterative/search/alternating-based methods (e.g., AO, BCD, SCA, and SDR) |  – Acceptable sub-optimal or approximate solutions – Flexible performance-complexity trade-off |  – Rely on the mathematical models – Long execution time and high computational complexity for large-scale systems; not suitable for resource-limited and delay-sensitive applications – Perfectly-known environment (e.g., availability of CSI) is required – Unstable sometimes as the performance depends on the designed strategy – Not applicable to the practical scenarios due to simple analysis assumptions, e.g., static users – Challenge to cope with a dynamic and heterogeneous environment
– 6G networks also generate high-dimensional /nonlinear/multi-objective problems |  ML approach |  – Supervised learning (e.g., regression and classification) – Unsupervised learning (e.g., clustering and association analysis) – Federated learning – DL (e.g., deep neural network) – DRL (e.g., Q-learning, DDPG and policy gradient) |  – Do not completely rely on the mathematical models – Can manage the lack of the environment statistical information – Applicable under stringent computational time constraints due to high-speed graphical processing units – Suitable for the large-scale networks as well as dynamic and uncertain environment |  – Challenge to design an approach with appropriate training data set in terms of availability, quality, and sufficiency – Challenge to design an approach with fast convergence rate (learning speed) – Often presents high computational complexity – Selection of an approach applicable to large state and action spaces – Selection of an approach suitable for joint discrete and continuous state spaces (as both discrete and continuous parameters exist) – Selection of a relatively stable approach (i.e., insensitive to parameter setting)

###  III-E Optimization Designs for STAR-RISs

The T and R coefficients design under theoretical fundamental constraints of
STAR-RISs constitutes a different mathematical problem to conventional RISs
having a single coefficient under a unit modulus constraint. The intractable
objective function with complex constraints turns an optimization problem of
STAR-RIS into a non-trivial task. More precisely, the problems become non-
convex because of the coupled variables, e.g., precoding and STAR-RIS
coefficient matrices, and the theoretical fundamental constraints per element.
Furthermore, complex topologies and non-linear components in 6G also give rise
to high-dimensional problems. Therefore, the optimization designs of
conventional RISs need to be extended, and effective solving methods are
required for 6G STAR-RISs. Exact methods (e.g., linear programming, dynamic
programming, and exhaustive search) can provide an optimal solution in a
limited time for simpler problems only. However, 6G STAR-RIS optimization
problems are diverse and complex. The solution and execution time via exact
methods are unacceptable for large problems. As shown in Table III, the top-
level taxonomy of optimization theory (analytical) and ML covers the
optimization techniques for 6G STAR-RISs.

Generally, analytical approaches adopt iterative-based methods. Specifically,
originally complicated problems are approximated into simpler sub-problems
with convex relaxations to find a sub-optimal solution. The transformed sub-
problems become tractable with decoupled optimization variables and can be
solved with the guarantee of convergence to a stationary point of the original
problem. In this context, iterative algorithms and alternative optimization
techniques, including the penalty method, successive convex approximation
(SCA), semidefinite relaxation (SDR), alternative optimization (AO), BCD, can
be exploited. The second optimization approach implies the use of ML, which
transforms wireless communications through its learning capability and large
search space. By exploring the training data, ML-based approaches can directly
learn solutions without explicit channel models while efficiently solving
complex optimization problems of 6G STAR-RIS systems. For example, the coupled
T and R signals for an ES protocol requires a high-dimensional continuous and
discrete control policy which can be developed using deep reinforcement
learning (DRL) algorithms [15]. Furthermore, DRL algorithms can also handle
the multi (conflicting) objective problems, e.g., coverage and capacity
optimization.

####  III-E1 Open Issues and Future Directions

The future optimization design aspects of 6G STAR-RISs are as follows:

  * •

Performance tradeoff on both sides of a STAR-RIS is an open research issue.
The beamforming and deployment (location) optimization problems with users in
the T and R half-spaced regions are more challenging, especially for realistic
multi-user scenarios.

  * •

The challenge for analytical approaches is to manage a highly dynamic wireless
propagation environment, whereas for ML approaches it is to introduce a low-
complex, accurate, and stable learning algorithm. With certain advantages and
disadvantages, the selection of an appropriate approach depends on the design
scenarios and communication objectives.

##  IV Conclusions

This article aims to facilitate STAR-RIS implementations in 6G era. We study
STAR-RIS designs for the simultaneous or consecutive T and R modes with
coupled or independent controls. Specifically, the CSI acquisition, i.e.,
channel estimation, methods are presented. The comparison and compatibility of
the physical implementations and hardware modeling, and the near- and far-
field channel modeling are described. Finally, the high-dimensional
optimization problems due to the STAR-RIS constraints and complex 6G networks
are discussed. The future directions include the validation of proposed design
solutions and insights with the hardware platforms and experimental
measurements, and their extension in complex situations, e.g., mmWave/THz
communications, hardware impairments, multilayer/multicell networks, and
randomness of multiple users.

## Acknowledgement

This research was supported by the Basic Science Research Program through the
National Research Foundation of Korea (NRF) funded by the Ministry of
Education(MOE) (NRF-2022R1I1A1A01071807, 2021R1I1A3041887), by the Institute
of Information & communications Technology Planning & Evaluation (IITP) grant
funded by the Korea government (Ministry of Science & ICT (MSIT))
(2022-0-00704, Development of 3D-NET Core Technology for High-Mobility
Vehicular Service), and by Korea University Grant.

## References

  * [1] W. Khalid et. al., “RIS-Aided Physical Layer Security with Full-Duplex Jamming in Underlay D2D Networks,” IEEE Access, vol. 9, Jul. 2021, pp. 99667–99679.
  * [2] X. Mu et. al., “Simultaneously Transmitting and Reflecting (STAR) RIS Aided Wireless Communications,” IEEE Trans. Wireless Commun., vol. 21, no. 5, May 2022, pp. 3083–3098.
  * [3] H. Niu et al., “Weighted Sum Rate Optimization for STAR-RIS-Assisted MIMO System,” IEEE Trans. Veh. Technol., vol. 71, no. 2, Feb. 2022, pp. 2122–2127.
  * [4] A. Papazafeiropoulos et al., “Coverage Probability of STAR-RIS Assisted Massive MIMO Systems with Correlation and Phase Errors,” IEEE Wireless Commun. Lett., vol. 11, no. 8, Aug. 2022, pp. 1738–1742.
  * [5] C. Wu et al., “Channel Estimation for STAR-RIS-Aided Wireless Communication,” IEEE Commun. Lett., vol. 26, no. 3, Mar. 2022, pp. 652–656.
  * [6] W. Khalid et. al., “Rate-Energy Tradeoff Analysis in RIS-SWIPT Systems with Hardware Impairments and Phase-Based Amplitude Response,” IEEE Access, vol. 10, Mar. 2022, pp. 31821–31835.
  * [7] H. Lu et al., “Aerial Intelligent Reflecting Surface: Joint Placement and Passive Beamforming Design With 3D Beam Flattening,” IEEE Trans. Wireless Commun., vol. 20, no. 7, Jul. 2021, pp. 4128–4143.
  * [8] T. Hou et al., “A Joint Design for STAR-RIS Enhanced NOMA-CoMP Networks: A Simultaneous-Signal-Enhancement-And-Cancellation-Based (SSECB) Design,” IEEE Trans. Veh. Technol., vol. 71, no. 1, Jan. 2022, pp. 1043–1048.
  * [9] Y. Guo et al., “Energy-Efficient Design for a NOMA Assisted STAR-RIS Network with Deep Reinforcement Learning,” Nov. 2021. [Online]. Available at: https://arxiv.org/abs/2111.15464.
  * [10] J. Xu et al., “STAR-RISs: Simultaneous Transmitting and Reflecting Reconfigurable Intelligent Surfaces,” IEEE Commun. Lett., vol. 25, no. 9, Sep. 2021, pp. 3134–3138.
  * [11] G. M. Minopoulos et al., “Exploitation of Emerging Technologies and Advanced Networks for a Smart Healthcare System,” Appl. Sci., vol. 12, no. 12, Jun. 2022, p. 5859.
  * [12] S. Noh et al., “Channel Estimation Techniques for RIS-Assisted Communication: Millimeter-Wave and Sub-THz Systems,” IEEE Veh. Technol. Mag., vol. 17, no. 2, Jun. 2022, pp. 64–73.
  * [13] L. Wei et al., “Channel Estimation for RIS-Empowered Multi-User MISO Wireless Communications,” IEEE Trans. Commun., vol. 69, no. 6, Jun. 2021, pp. 4144–4157.
  * [14] H. Zhang et al., “Intelligent Omni-Surfaces for Full-Dimensional Wireless Communications: Principles, Technology, and Implementation,” IEEE Commun. Mag., vol. 60, no. 2, Feb. 2022, pp. 39–45.
  * [15] R. Zhong et al., “Hybrid Reinforcement Learning for STAR-RISs: A Coupled Phase-Shift Model Based Beamformer,” IEEE J. Sel. Areas Commun., vol. 40, no. 9, Sep. 2022, pp. 2556–2569.

##  V Biographies

Waqas Khalid received the Ph.D. degree in information and communication
engineering from Yeungnam University, Gyeongsan, South Korea, in 2019. He is
currently a Research Professor with the Institute of Industrial Technology,
Korea University, Sejong, South Korea.
---
Zeeshan Kaleem (Senior Member, IEEE) received the Ph.D. degree in electronics
engineering from Inha University, South Korea, in 2016. He is currently an
Assistant Professor with the Electrical and Computer Engineering Department,
COMSATS University Islamabad, Wah Campus.
---
Rehmat Ullah received the Ph.D. degree in electronics and computer engineering
from Hongik University, South Korea, in 2020. He is currently a Lecturer
(Assistant Professor) at the Cardiff School of Technologies, Cardiff
Metropolitan University, UK.
---
Trinh Van Chien (Member, IEEE) received the Ph.D. degree in communication
systems from Linköping University, Sweden, in 2020. He is now a lecturer at
the School of Information and Communication Technology, Hanoi University of
Science and Technology, Vietnam.
---
Song Noh (Member, IEEE) received the Ph.D. degree in electrical and computer
engineering from Purdue University, West Lafayette, IN, USA, in 2015. Since
September 2018, he has been with the Department of Information and
Telecommunication Engineering, Incheon National University, Incheon, Korea,
where he is now an Associate Professor.
---
Heejung Yu (Senior Member, IEEE) received the Ph.D. degree in electrical
engineering from the Korea Advanced Institute of Science and Technology
(KAIST), Daejeon, South Korea, in 2011. He is currently a Professor with the
Department of Electronics and Information Engineering, Korea University,
Sejong, South Korea.
---