11institutetext:  LESIA, Observatoire de Paris, PSL, CNRS, Sorbonne
Université, Université de Paris, 5 place Janssen, 92195 Meudon, France
22institutetext: European Southern Observatory, Karl-Schwarzschild-Straße 2,
85748 Garching, Germany 33institutetext: Kavli Institute for Cosmology,
University of Cambridge, Madingley Road, Cambridge CB3 0HA, United-Kingdom
44institutetext: Institute of Astronomy, University of Cambridge, Madingley
Road, Cambridge CB3 0HA, United-Kingdom

# Exoplanets in reflected starlight with dual-field interferometry

A case for shorter wavelengths and a fifth Unit Telescope at VLTI/Paranal

S. Lacour Exoplanets in reflected starlight with dual-field
interferometryExoplanets in reflected starlight with dual-field
interferometryExoplanets in reflected starlight with dual-field
interferometryExoplanets in reflected starlight with dual-field interferometry
Ó. Carrión-González Exoplanets in reflected starlight with dual-field
interferometryExoplanets in reflected starlight with dual-field interferometry
M. Nowak Exoplanets in reflected starlight with dual-field
interferometryExoplanets in reflected starlight with dual-field
interferometryExoplanets in reflected starlight with dual-field
interferometryExoplanets in reflected starlight with dual-field interferometry

(Received September 15, 1996; accepted March 16, 1997)

###### Abstract

Context. The direct observation of cold and temperate planets within 1 to 10
AU would be extremely valuable for uncovering their atmospheric compositions
but remains a formidable challenge with current astronomical methods. Ground-
based optical interferometry, capable of high angular-resolution imaging,
offers a promising avenue for studying these exoplanets, complementing space-
based observations.

Aims. Our objective is to explore the fundamental limits of dual-field
interferometry and assess its potential for characterizing exoplanets in
reflected light using the Very Large Telescope Interferometer (VLTI).

Methods. We developed analytical expressions to describe the performance of
dual-field interferometry and integrated these with simulations of atmospheric
wavefronts corrected by extreme Adaptive Optics. An analytical solution for
optimal phase apodization was formulated to enhance starlight rejection when
injected into a single-mode fibre. This framework was applied to determine the
detectability of known exoplanets in reflected light across various wavelength
bands for both the current VLTI and a proposed extended version.

Results. Our results indicate that employing shorter wavelengths improves
detectability, enabling at least seven Jupiter-mass exoplanets to be observed
in the J band with current VLTI’s baselines. Adding new baselines with lengths
beyond 200 meters significantly enhances VLTI’s capabilities, increasing the
number of detectable exoplanets and revealing potential habitable zone
candidates such as $\tau$ Ceti e and Proxima Centauri b.

Conclusions. To substantially improve the VLTI’s exoplanet characterization
capabilities, we recommend developing instrumentation at wavelengths shorter
than 1$\,\mu$m, as well as the addition of a fifth Unit Telescope (UT5).

###### Key Words.:

Planets and satellites: fundamental parameters — planets and satellites:
terrestrial planets — planets and satellites: gaseous planets — Astronomical
instrumentation, methods and techniques — Techniques: high angular resolution
— Techniques: interferometers

##  1 Introduction

Over the past two decades, direct-imaging has become a key method for the
characterization of exoplanets. Its development is now a priority within the
community, as illustrated by key community reports and space agencies (Astro
2020 Decadal Survey, and ESA’s Voyage 2050 Senior Committee report, for
example). Direct-imaging has a particular role to play as it can overcome some
of the main limitations of other methods. For example, observing molecular
(bio)signatures via transit spectroscopy can be made extremely challenging by
the fact that these molecules are often concentrated in lower atmospheric
layers (Lustig-Yaeger et al., 2019). Consequently, several direct-imaging
space observatories have been proposed, and are under study. The Coronagraph
Instrument on the Nancy Grace Roman Space Telescope (Spergel et al., 2015) is
poised to be the first to directly image exoplanets in reflected light, from
space. Future flagship missions, like the Habitable Worlds Observatory, are
being conceptualized based on the LUVOIR (The LUVOIR Team, 2018) and HabEx
(Mennesson et al., 2016; Gaudi et al., 2018) studies. In Europe, momentum is
building for the LIFE project (Quanz et al., 2021), a mid-infrared optical
interferometer targeting thermal emission.

Compared to their space-based counterparts, ground-based observatories offer
distinct advantages, with usually much larger apertures and higher angular
resolution, but also specific challenges, such as an important thermal
background and atmospheric aberrations. Several young massive exoplanets have
already been successfully discovered and directly imaged from the ground.
These are usually hot planets, such as $\beta$ Pictoris b (approximately 1700
K) and 51 Eri b (around 800 K) which are still cooling off by gradually
dissipating the gravitational energy accumulated during their formation. Their
thermal emission is comparatively bright against their host stars, with
contrast ratios on the order of a few $10^{-5}$. But the vast majority of
planets are older and colder than these young giants, with a temperature set
by the radiative equilibrium with their stellar hosts. Such bodies primarily
emit in the mid-infrared wavelength range. To be detected in the near-
infrared, these objects must be observed in reflected light, typically at a
contrast on the order of $10^{-7}$ or lower, depending on their radius, semi-
major axis, phase angle and albedo.

Given the challenges associated with achieving sufficient contrast levels, the
question arises: can exoplanets be effectively characterized in reflected
light from ground-based observatories? Guyon (2005) provides a comprehensive
overview of the challenges involved. The first major challenge is atmospheric
perturbations, which can be mitigated in single-dish telescopes using Adaptive
Optics (AO) systems, such as the Spectro-Polarimetric High-contrast Exoplanet
REsearch (SPHERE) extreme AO (Beuzit et al., 2019). The second challenge
involves the diffraction pattern, which generates photon noise at the location
of the planet; this issue can be addressed with the use of a coronagraph
(Rouan et al., 2000). The third, and perhaps most significant challenge,
involves non-common path quasi-static aberrations that produce slowly moving
speckles, difficult to calibrate. These speckles, with intensities around
$10^{-5}$, require stabilization to within 1%, necessitating a wavefront
stability of 4 pm (Guyon, 2005).

These quasi-static speckles can be subtracted using advanced post-processing
techniques such as Spectral Differential Imaging (SDI Racine et al., 1999) and
Angular Differential Imaging (ADI Marois et al., 2006), or in real time
through focal plane wavefront sensing and correction methods like the dark-
hole method (Potier et al., 2022). However, these methods require that the
wavefront remains stable throughout the observation period; otherwise,
shifting speckles cannot be accurately removed. This stringent requirement for
temporal stability has spurred the development of new techniques that utilize
high-resolution spectral signatures detectable through cross-correlation
techniques (Snellen et al., 2013, 2015). Such techniques are seen as powerful
tools for the first generation of Extremely Large Telescope (ELT) instruments,
including MICADO, HARMONI, and ANDES, which leverage the substantial photon
counts captured by the ELT (Palle et al., 2023). Unfortunately, these
techniques are limited to detecting targets with strong molecular absorption
bands and are ineffective at seeing smooth emission, thus failing to probe
continuum albedos or detect exoplanets lacking a dense atmosphere.

In this context, optical interferometry offers a complementary alternative, as
its post-processing does not rely on any spectral feature. Severak high-
contrast interferometric designs exist, generally classified into two main
types: single-field and dual-field interferometers. Single-field
interferometry differentiates the coherent signals of stars and exoplanets
using techniques such as closure phase (Gardner et al., 2018), nulling
(Bracewell, 1978), or a combination of these methods, such as Kernel nulling
(Martinache & Ireland, 2018; Chingaipe et al., 2023). Dual-field
interferometry, as summarized in Fig. 1, operates by simultaneously observing
two narrow fields of view — typically the size of the telescope’s point spread
function — situated within a larger field of view that can include both a star
and an exoplanet. This configuration allows one interferometer to focus on the
star while the other targets the exoplanet, minimizing the contamination of
the exoplanet signal by stellar photon noise due to the narrowness of each
telescope’s diffraction pattern.

The GRAVITY instrument (Gravity Collaboration et al., 2017) at the Very Large
Telescope Interferometer (VLTI) has pioneered dual-field interferometry, and
was incidently the first interferometer to directly detect an exoplanet
(GRAVITY Collaboration et al., 2019). It has also provided the first direct
detection of two short-separation radial-velocities exoplanets: $\beta$
Pictoris c (Nowak et al., 2020) and HD 206893 c (Hinkley et al., 2023). These
two planets have not been directly observed by any other instrument to date,
suggesting that dual-field interferometry is performing better than imaging at
such small separations. The commissioning of GRAVITY+ will bring extreme AO to
all the Unit Telescopes (UTs) which, coupled with dual-field interferometry
(Gravity+ Collaboration et al., 2022), will further increase this capability
and optimize the VLTI for high-contrast (Pourré et al., 2024).

These developments prompt an inquiry into the fundamental limits of dual-field
interferometry and its potential to detect exoplanets using reflected light
from the ground. In Section 2, we introduce a detailed analytical model to
assess the performance and limitations of dual-field interferometry using the
existing VLTI. Because photon noise is a prime limitation, Section 2.5
presents a novel, semi-analytical method for determining the optimal phase
apodizer necessary for starlight suppression when channeling light into a
single-mode fibre. Section 3 suggests a possible upgrade of the VLTI, in which
a new 8 m Unit Telescope (UT5) would be added to optimize the VLTI for
exoplanet detection, by creating four additional baselines in the 200 m regime
to the North-West direction. In Section 4, we estimate the number of known
Radial Velocity (RV) exoplanets that could potentially be characterized by the
VLTI, including UT5 or not. Finally, Section 5 highlights the main results of
this paper.

Figure 1: Schematic diagram of a dual field optical interferometer using two
telescopes. A beam splitter creates two images of the focal plane, where two
fibres are positioned—one on the star to feed Interferometer 1 (serving as the
phase reference) and the other on the exoplanet, feeding Interferometer 2. The
exoplanet is detected as coherent flux in Interferometer 2, with a phase
offset relative to the stellar coherent flux due to an optical path delay
$\delta\text{OPD}$, corresponding to the scalar product of the narrow angle
baseline vector $\bm{B}$ and the angular separation between the star and
exoplanet $\bm{\alpha}$: $\delta\text{OPD}=\bm{\alpha}\cdot\bm{B}$.

##  2 Simulation of Contrast Range

###  2.1 The Detection Principle

A concise overview of the working principle of a dual-field optical
interferometer used for high-contrast exoplanet observations is given in Fig.
1. This setup involves splitting light from two telescopes into four focal
planes, in which single-mode fibres are placed. Two fibres, which feed a first
interferometer, are centred on the geometrical image of the star, whereas the
other two fibres, which feed a second interferometer, are centred on the
geometrical image of the planet. A dedicated metrology system monitors the
non-common optical-path differences between the two interferometers.

The detection principle relies on the premise that the exoplanet’s location is
already known, with a precision well within the field of view of the single
mode fibres ($\approx 60\,$mas for an 8 m telescope in the K band). This
implies that this method is more suited for characterizing exoplanets (for
example, measuring their albedo) rather than discovering new ones.

Given the typical planet to star contrast ratio (at least $10^{-4}$, and
usually much smaller), the first interferometer $I_{1}$ is always dominated by
photons from the star, while the second interferometer $I_{2}$ receives both
the photons from the planet, and some non-negligible amount of residual
starlight. The light from the planet and the light from the star being fully
incoherent, this results in two superimposed systems of fringes observed in
the second interferometer.

Separating these two systems of fringes observed on interferometer $I_{2}$ to
recover the planet coherent flux is one of the main concern of high-contrast
dual-field interferometry. For this, two strategies are used in combination:
(1) minimizing the amount of starlight entering the fibres feeding
interferometer $I_{2}$ in the first place (a process thereafter referred to as
”apodization”); (2) separating the two fringe systems in post-processing, by
taking advantage of the fact that the on-axis starlight and the off-axis
planet light creates fringes centred at different Optical Path Difference
(OPD) values.

###  2.2 Spelling out the electric fields

In the next sections of this paper, we will use the symbol $\star$ in
subscript to denote parameters associated with the star, and we will use
$\medbullet$ for those associated with the planet. We denote $\bm{\alpha}$ the
vector representing the angular separation between the star and the exoplanet:
$\bm{\alpha}=\bm{\alpha}_{\medbullet}-\bm{\alpha}_{\star}$, where
$\bm{\alpha}_{\medbullet}$ points towards the planet, and
$\bm{\alpha}_{\star}$ points towards the star.

We model the interferometric setup using two independent 8 m telescopes, with
pupil shapes akin to those of a UT of the VLT. We assume that these telescopes
are equipped with an AO system which corrects for the atmospheric turbulence,
and we denote $\Phi(\bm{u})$ the phase residuals downstream of the AO on the
pupil of a given telescope. A term $\Psi(\bm{u})$ will also be added at the
beginning of section 2.5 to include a pupil phase apodizer. The two-
dimensional vector $\bm{u}$ is the X-Y coordinate vector within the
telescope’s pupil. The narrow-angle baseline, the vector between the two
telescopes (the vector delimited by the metrology, as outlined by Woillez &
Lacour, 2013), is denoted as $\bm{B}$.

With this setup, a monochromatic point source of wavevector $k=2\pi/\lambda$
located at a vector position $\bm{\alpha}$ on the sky and radiating on the
telescope with a flux $F=A^{2}$, generates an electric field within the pupil
of a given telescope which can be described by:

$\mathsf{W}(\bm{u})=Ae^{-ik\bm{\alpha}\cdot{}\bm{u}+i\Phi(\bm{u})}.$ (1)

The flux entering a single-mode fibre is determined by projecting the electric
field onto the fibre’s fundamental mode. This can be calculated either in the
focal plane or in the pupil plane, leveraging the Parseval-Plancherel theorem
and assuming the far-field Fraunhofer condition. Using the pupil-plane
approach, the electric field from Eq. (1) couples into a single-mode fibre as
follows:

$E=\iint_{D}Ae^{-ik\bm{\alpha}\cdot{}\bm{u}+i\Phi(\bm{u})}\mathsf{M}^{\ast}(\bm{u})\
d\bm{u},$ (2)

where the integral covers the pupil area with diameter $D$. The fibre’s
fundamental mode, represented in the pupil plane as $\mathsf{M}(\bm{u})$, can
be approximated by a Gaussian-shaped mode with beam waist $w$:

$\mathsf{M}(\bm{u})\propto{\displaystyle\exp\left(-\frac{|\bm{u}|^{2}}{2w^{2}}\right)}\
,$ (3)

where a normalization factor must be added to ensure the conservation of
energy. For a circular aperture telescope without central obstruction, optimal
injection is achieved with $w=\sqrt{0.1}\,D$ and $\mathsf{M}(\bm{u})$ can be
exactly spelled as:

$\mathsf{M}(\bm{u})=\frac{2}{\displaystyle\sqrt{0.1}\pi
D^{2}}{\displaystyle\exp\left(-\frac{|\bm{u}|^{2}}{0.2D^{2}}\right)}\ ,$ (4)

leading to the optimal throughput in the absence of phase aberrations:
$\left\lVert\iint_{D}\mathsf{M}(\bm{u})\ d\bm{u}\right\rVert^{2}=0.81.$

This expression of $\mathsf{M}(\bm{u})$ is valid for a fibre located at the
focal point of the telescope (i.e. ”on-axis”). If the fibre is moved at the
location of the geometrical image on an object situated at the on-sky vector
position $\bm{\beta}$, the fibre mode $\mathsf{M}$ in the pupil plane becomes:

$\mathsf{M}_{\beta}(\bm{u})=\mathsf{M}(\bm{u})e^{-ik\bm{\beta}\cdot{}\bm{u}},$
(5)

and the coupled electric field becomes:

$\displaystyle E$
$\displaystyle=\iint_{D}Ae^{i\Phi(\bm{u})}\mathsf{M}(\bm{u})e^{ik\left(\bm{\beta}-\bm{\alpha}\right)\cdot{}\bm{u}}\
d\bm{u}.$ (6)

At this point, it is useful to note that since $\mathsf{M}$ is a positive non-
zero function, we can define a scalar product $\langle\cdot\,|\,\cdot\rangle$
using:

$\langle\mathsf{G}\,|\,\mathsf{H}\rangle=\iint_{D}\mathsf{G}(\bm{u})\,\mathsf{M}(\bm{u})\,\mathsf{H}^{\ast}(\bm{u})\,d\bm{u},$
(7)

as this expression has all the required properties (linearity, conjugate
symmetry, and definite-positiveness). It is defined such that $\langle
1\,|\,1\rangle=\sqrt{0.81}$ in the case where $\mathsf{M}(\bm{u})$ is given as
in Eq. (4).

With this notation, the electric field of Eq. (6) is simply the scalar product
of the pupil-plane electric field and a complex exponential representing the
offset between the source and the fibre:

$\displaystyle E$ $\displaystyle=A\langle
e^{i\Phi(\bm{u})}\,|\,e^{ik\left(\bm{\alpha}-\bm{\beta}\right)\cdot\bm{u}}\rangle.$
(8)

This formulation simplifies the expression of the fields injected into the
fibres of the dual-field interferometer. These fields are summarized in Table
1 for the telescopes $\mathbb{A}$ and $\mathbb{B}$. The first set of fields
considers a star radiating with a flux $F_{\star}=A_{\star}^{2}$ located on
the optical axis. The second set involves a planet with a flux
$F_{\medbullet}=A_{\medbullet}^{2}$ positioned at $\bm{\alpha}$. For the first
interferometer, we assume $\bm{\beta}=\bm{0}$. For the second interferometer,
we assume $\bm{\beta}=\bm{\alpha}$.

Table 1: The eight electric fields inside the four fibres of a two-telescope
dual-field interferometer.

Interferometer 1 | Interferometer 2
---|---
(on-axis; $\bm{\beta}=\bm{0}$) | (off-axis; $\bm{\beta}=\bm{\alpha}$)
$E_{\mathbb{A},1,\star}=A_{\star}\langle e^{i\Phi_{\mathbb{A}}(\bm{u})}\,|\,1\rangle$ | $E_{\mathbb{A},2,\star}=A_{\star}\langle e^{i\Phi_{\mathbb{A}}(\bm{u})}\,|\,e^{-ik\bm{\alpha}\cdot{}\bm{u}}\rangle$
$E_{\mathbb{A},1,\medbullet}\approx 0$ | $E_{\mathbb{A},2,\medbullet}=A_{\medbullet}\langle e^{i\Phi_{\mathbb{A}}(\bm{u})}\,|\,1\rangle$
$E_{\mathbb{B},1,\star}=A_{\star}\langle e^{i\Phi_{\mathbb{B}}(\bm{u})}\,|\,1\rangle$ | $E_{\mathbb{B},2,\star}=A_{\star}\langle e^{i\Phi_{\mathbb{B}}(\bm{u})}\,|\,e^{-ik\bm{\alpha}\cdot{}\bm{u}}\rangle$
$E_{\mathbb{B},1,\medbullet}\approx 0$ | $E_{\mathbb{B},2,\medbullet}=A_{\medbullet}e^{-ik\bm{\alpha}\cdot{}\bm{B}}\langle e^{i\Phi_{\mathbb{B}}(\bm{u})}\,|\,1\rangle$
111In this table $\Phi_{\rm A}(\bm{u})$ and $\Phi_{\rm B}(\bm{u})$ are the
phase aberrations of, respectively, telescopes $\mathbb{A}$ and $\mathbb{B}$.

The two quantities of interest which are derived from the observations on the
two interferometers are the coherent energies, denoted $\Gamma$, and given by:

$\displaystyle\Gamma_{1}$
$\displaystyle=E_{\mathbb{A},1,\star}E_{\mathbb{B},1,\star}^{\ast},$ (9)
$\displaystyle\Gamma_{2}$
$\displaystyle=E_{\mathbb{A},2,\star}E_{\mathbb{B},2,\star}^{\ast}+E_{\mathbb{A},2,\medbullet}E_{\mathbb{B},2,\medbullet}^{\ast},$
(10)

where we implicitly neglect the exoplanet light in the first interferometer.
The exoplanet signal
$\Gamma_{2,\medbullet}=E_{\mathbb{A},2,\medbullet}E_{\mathbb{B},2,\medbullet}^{\ast}$
is observable in the second interferometer, blended with some amount of
residual stellar coherent energy
$\Gamma_{2,\star}=E_{\mathbb{A},2,\star}E_{\mathbb{B},2,\star}^{\ast}$. In the
rest of the paper, we usually refer to $\Gamma_{2,\star}$ as ”coherent
speckle”, by analogy with the speckles observed by classical imaging.

In practice, though, the coherent energies $\Gamma$ are never observed
directly, but are rather derived from measurements of the interference
patterns as a function of optical delay $x$, described by:

$\displaystyle I_{1}(x)=$
$\displaystyle|E_{\mathbb{A},1,\star}|^{2}+|E_{\mathbb{B},1,\star}|^{2}+2\Re\left[\Gamma_{1}e^{ix}\right],$
(11) $\displaystyle I_{2}(x)=$
$\displaystyle|E_{\mathbb{A},2,\star}|^{2}+|E_{\mathbb{B},2,\star}|^{2}+|E_{\mathbb{A},2,\medbullet}|^{2}+|E_{\mathbb{B},2,\medbullet}|^{2}+2\Re\left[\Gamma_{2}e^{ix}\right].$
(12)

A common beam combination pattern is the ABCD scheme, for which the detectors
of the interferometers record $I_{1}(x)$ and $I_{2}(x)$ as a function of four
well-calibrated values of $x$: $0,\pi/2,\pi,3\pi/2$. The coherent energies
$\Gamma_{1}$ and $\Gamma_{2}$ are then reconstructed by a proper linear
combination of these measurements.

###  2.3 Signal to Noise

In the numerator of the S/N, the signal can have multiple definitions:

$\displaystyle S_{\rm entrance}$ $\displaystyle=$
$\displaystyle\gamma\,F_{\medbullet}$ (13) $\displaystyle S_{\rm interfero}$
$\displaystyle=$
$\displaystyle\gamma\,\Gamma_{2,\medbullet}=\gamma\,F_{\medbullet}\,\eta_{\medbullet}^{\mathrm{inj}}$
(14) $\displaystyle S_{\rm processed}$ $\displaystyle=$
$\displaystyle\gamma\,\Gamma_{2,\medbullet}\,\eta_{\medbullet}^{\mathrm{proc}}=\gamma\,F_{\medbullet}\,\eta_{\medbullet}^{\mathrm{inj}}\,\eta_{\medbullet}^{\mathrm{proc}}\,.$
(15)

The definition of $S_{\rm entrance}$ in Eq. (13) corresponds to the number of
photons of exoplanetary origin entering the interferometer. It depends solely
on the flux of the exoplanet $F_{\medbullet}=A_{\medbullet}^{2}$ with a
proportionality factor $\gamma$ that encompasses the observation parameters.
$S_{\rm entrance}$ represents an upper limit of the signal that we can expect.

The definition of $S_{\rm interfero}$ used in Eq. (14) corresponds to the
number of coherent exoplanetary photons that we can observe. To simplify the
study, we assume a visibility of one, meaning that the exoplanet flux is fully
coherent. Compared to $S_{\rm entrance}$, $S_{\rm interfero}$ is therefore
reduced by the injection losses of the light into the single-mode fibre. These
losses are denoted as $\eta_{\medbullet}^{\mathrm{inj}}$ and correspond to the
product of two scalar products as in Eq. (8) averaged over time. Its value is
always below 1 and is formally discussed in Section 2.5.

The definition of $S_{\rm processed}$ in Eq. (15) includes an additional term,
$\eta_{\medbullet}^{\mathrm{proc}}$, accounting for the post-processing of the
data that reduces the amount of detectable signal
($\eta_{\medbullet}^{\mathrm{proc}}<1$). This term depends on the algorithm
used for post-processing and is described in Section 2.6.

The term $\gamma$ is a proportionality factor that converts the flux into the
signal in units of photons. This conversion factor incorporates several terms:
the total observation duration ($\Delta t$), the telescope’s collecting area
($\pi D^{2}/4$), the spectral bandwidth ($\Delta\lambda/\lambda$), the
transmission efficiency ($T_{r}$), and Planck’s constant ($h$):

$\gamma=\rho\times\frac{1}{h}\times\frac{\Delta\lambda}{\lambda}\times\frac{\pi
D^{2}}{4}\times T_{r}\times\Delta t\,.$ (16)

The coefficient $\rho$ corresponds to how the flux of a telescope is
dispatched between the different interferometric baselines. Under the
assumption that all baselines are recombined, it is equal to the number of
telescopes divided by the maximum number of baselines:

$\rho=N_{\mathrm{tel}}\times\left(\cfrac{N_{\mathrm{tel}}(N_{\mathrm{tel}}-1)}{2}\right)^{-1}=\frac{2N_{\mathrm{tel}}}{N_{\mathrm{tel}}^{2}-N_{\mathrm{tel}}}\,.$
(17)

Thus, we have $\rho=2$ for a two-telescope interferometer and $\rho=2/3$ for
the recombination of the 4 UTs with the GRAVITY instrument. The signal is then
defined for a specific baseline, while the signals from multiple baselines can
be combined to obtain the final S/N in Section 3.

In the denominator of the S/N, the noise can be broken into its individual and
independent components:

$N=\sqrt{N_{\mathrm{readout}}^{2}+N_{\mathrm{thermal}}^{2}+N_{\mathrm{shot}}^{2}+N_{\mathrm{speckle}}^{2}},$
(18)

where the individual noise elements account for detector readout noise,
thermal background photon noise, shot noise, and speckle noise. These elements
are discussed in the following subsections.

Combining all sources of noise, and using $S_{\rm processed}$ for the signal,
the S/N ratio takes the form:

$\mathrm{S/N}=\frac{\gamma
F_{\medbullet}\,\eta_{\medbullet}^{\mathrm{inj}}\,\eta_{\medbullet}^{\mathrm{proc}}}{\sqrt{N_{\mathrm{readout}}^{2}+N_{\mathrm{thermal}}^{2}+N_{\mathrm{shot}}^{2}+N_{\mathrm{speckle}}^{2}}}.$
(19)

###  2.4 Readout and thermal background noise

The detector readout noise, $N_{\mathrm{readout}}$, depends on the
characteristics of the detector, on the number of frames and on the number of
used pixels:

$N_{\mathrm{readout}}=\sigma_{e^{-}}\sqrt{4\times\frac{3\Delta\lambda}{\delta\lambda}\times\frac{\Delta
t}{\delta t}}$ (20)

where $\sigma_{e^{-}}$ represents the read noise per pixel per frame. For
modern Teledyne H2RG detectors using a Fowler-16 detection readout scheme,
$\sigma_{e^{-}}$ is typically around 5 electrons. The term under the square
root quantifies the total number of reads during the entire observation. The
factor of 4 accounts for the ABCD recombination scheme. The spectral bandwidth
is expressed as $\Delta\lambda$ and the spectral resolution is denoted by
$\delta\lambda$. With an oversampling factor of 3, the total number of pixels
across the band is $3\Delta\lambda/\delta\lambda$. The number of frames is
given by the ratio of the total integration time to the detector integration
time, $\Delta t/\delta t$.

The thermal background noise, $N_{\mathrm{thermal}}$, can be estimated by
assuming that blackbody emission from the laboratory at temperature $T_{\rm
lab}$ is the dominant source of background. This assumption is reasonable
given the low throughput ($T_{r}<10\%$) of the optical chain, i.e. sky
background can be neglected. The background emission is analytically estimated
using the limited acceptance angle of a single-mode fibre:

$N_{\mathrm{thermal}}=\rho\times\sqrt{(1-T_{r})\times\frac{2c\,\Delta\lambda\,\Delta
t}{\lambda^{2}\,(\exp(\frac{hc}{k\rm T_{\rm lab}\lambda})-1)}}\,,$ (21)

which can be simplified by using the Planck’s law for a blackbody radiation
$B_{\lambda}(\rm T_{\rm lab},\lambda)$:

$N_{\mathrm{thermal}}=\sqrt{\gamma\times\frac{4}{\pi
D^{2}\,T_{r}}\times(1-T_{r})\times\lambda^{2}\times B_{\lambda}(\rm T_{\rm
lab},\lambda)}\,.$ (22)

The term $\lambda^{2}$ represents the étendue of the single-mode fibre. The
factor $\gamma$ is expressed by Eq. (16) and denotes the conversion factor
between an incoming flux and the number of detected photons in our
instrumental system.

Figure 2: Contrast limit at the VLTI as a function of the total transmission
of the interferometer. The coloured curves represent theoretical contrast
limits set by the thermal background of the telescope and interferometric
laboratory (300 K). The dashed curves indicate the contrast limit from
detector readout noise. The simulation parameters are detailed in Table 2.

To better understand the contrasts accessible with these readout and thermal
background noise sources, we can convert these noise expressions into first-
order contrast limits, by writing the planet raw signal as a function of the
stellar flux:

$S_{\rm entrance}=\gamma\,CF_{\star}\,,$ (23)

where $C$ is the planet to star contrast ratio: $C=F_{\medbullet}/F_{\star}$.

Using a threshold of $\mathrm{S/N}=3$, this gives two contrast limits set
respectively by the amount of readout and thermal background noise:

$\displaystyle C_{\mathrm{readout}}=3\frac{\sigma_{e^{-}}}{\gamma
F_{\star}}\sqrt{4\frac{3\Delta\lambda}{\delta\lambda}\frac{\Delta t}{\delta
t}}\,,$ (24) $\displaystyle
C_{\mathrm{thermal}}=3\frac{1}{\sqrt{\gamma}F_{\star}}\sqrt{\frac{4(1-T_{r})}{\pi
D^{2}T_{r}}\lambda^{2}B_{\lambda}(300\,\mathrm{K},\lambda)}\,.$ (25)

Both $C_{\text{readout}}$ and $C_{\text{thermal}}$ are depicted in Fig. 2 for
a two telescopes interferometer with parameters as in Table 2. For comparison,
Figure 2 also shows the contrast ratios in reflected light expected for two
known planets: Gliese 876 b and Proxima Centauri b. This plot demonstrates the
importance of high transmission, especially given the inherent low brightness
of exoplanets. At a transmission of 2%, typical of the VLTI, the readout noise
begins to limit detection slightly above $10^{-9}$. The thermal emission
equals readout noise near the H band. At longer wavelengths, near 3 microns,
the 300 K emission of the laboratory strongly limits the achievable contrast
and impedes the observation of reflected light from Gliese 876 b and Proxima
Centauri b.

Table 2: Simulation parameters

| Fig. 2 | Fig. 4 | Fig. 5 | Fig. 6
---|---|---|---|---
$F_{\star}$ | 10 Jy | 10 Jy |  | 10 Jy
$\delta t$ |  $100$ s |  $100$ s |  |  $100$ s
$\Delta t$ | 3 h | 3 h |  | 3 h
$T_{r}$ | 0-10% | 5% |  | 5%
$\rm T_{\rm lab}$ | 300 K | 300 K |  | 300 K
$D$ | 8 | 8 | 8 | 8
$N_{\rm tel}$ | 2 | 2 | 2 | 2
${\Delta\lambda}/{\lambda}$ | 0.23 | 0.23 | 0.23 | 0.23
${\lambda}/{\delta\lambda}$ | 100 | 100 | 100 | 100
$\lambda_{B}$ | J/H/K/L | I/K | K | I/J/H/K/L
Strehl |  | 98 | 98 | 88/98
$|\bm{B}|$ |  |  | 100/200 m | 100/200 m
$n_{\rm poly}$ |  |  | 3-7-11 | 11
222 $F_{\star}$ is flux of the stellar host. $\delta t$ is the integration
time of individual frames while $\Delta t$ the total integration time. $T_{r}$
is the total transmission of the instrument, $D$ is the diameter of the
telescopes and $N_{\rm tel}$ is the number of telescopes.
${\Delta\lambda}/{\lambda}$ the spectral bandwidth and
${\lambda}/{\delta\lambda}$ is the spectral resolution. $\lambda_{B}$ is the
name of the spectral band. $|\bm{B}|$ is the projected baseline length, while
$n_{\rm poly}$ is the polynomial order used for stellar light post processing.

###  2.5 Starlight Suppression

####  2.5.1 Attenuation Formulas

At short wavelengths, thermal background noise and readout noise limit the
achievable contrast to values of $\sim 10^{-9}$. Unfortunately, the shot noise
caused by stellar speckles can be several orders of magnitude above this
limit. Therefore, it is necessary to remove as much of this parasitic light as
possible, a process known as starlight suppression.

For high-contrast interferometry, we propose leveraging the efficient spatial
filtering provided by single-mode fibres. This filtering can be combined with
pupil-plane phase apodization, which can achieve near-perfect rejection of
stellar light into the fibre. The effect of the apodization can be modeled
with two coupling coefficients: one affecting the stellar light
($\eta_{\star}^{\mathrm{inj}}$) and the other the planetary light
($\eta_{\medbullet}^{\rm inj}$). The shot noise caused by the stellar flux
then becomes:

$N_{\rm shot}=\sqrt{\gamma F_{\star}\eta_{\star}^{\rm inj}},$ (26)

while the exoplanet signal is given by:

$S_{\rm interfero}=\gamma{F_{\medbullet}\eta_{\medbullet}^{\rm inj}}.$ (27)

Phase apodization works by adding an extra phase term $\Psi(\bm{u})$ in the
pupil-plane of the telescope, for example using an offset to the deformable
mirror of the AO. With this additional phase term in the pupil-plane, the
coupling formulas for the electric field in the second interferometer (see
Table 1) are given by:

$\displaystyle E_{2,\star}=A_{\star}\langle
e^{i\Phi(\bm{u})+i\Psi(\bm{u})}\,|\,e^{-ik\bm{\alpha}\cdot{}\bm{u}}\rangle\,$
(28) $\displaystyle E_{2,\medbullet}=A_{\medbullet}\langle
e^{i\Phi(\bm{u})+i\Psi(\bm{u})}\,|\,1\rangle\,$ (29)

and the corresponding apodization coefficients are:

$\displaystyle\eta_{\star}^{\rm
inj}=\frac{\left|E_{2,\star}\right|^{2}}{F_{\star}}=\left|\langle
e^{i\Phi(\bm{u})+i\Psi(\bm{u})}\,|\,e^{-ik\bm{\alpha}\cdot\bm{u}}\rangle\right|^{2}$
(30) $\displaystyle\eta_{\medbullet}^{\rm
inj}=\frac{\left|E_{2,\medbullet}\right|^{2}}{F_{\medbullet}}=\left|\langle
e^{i\Phi(\bm{u})+i\Psi(\bm{u})}\,|\,1\rangle\right|^{2}.$ (31)

These expressions are valid for a single value of the wave-vector $k$. For a
polychromatic source, these expressions must be integrated over the waveband
to obtain appropriate values for the apodization coefficients.

####  2.5.2 Designing a Phase Apodizer

The overall objective of a good apodization scheme is of course to minimize
the value of $\eta_{\star}^{\rm inj}$ while maximizing the value of
$\eta_{\medbullet}^{\rm inj}$. There are various strategies for doing so, as
multiple solutions exist. State-of-the-art techniques, which include the
solutions proposed by Por (2020) or Haffert et al. (2020), leverage from an
iterative optimization algorithm (Por, 2017).

In this study, we adopt a more qualitative approach, which takes advantage of
the fact that we are only interested in minimizing the flux at a particular
position, corresponding to the location of the fibre. Therefore, we know in
advance for which value of $\bm{\alpha}$ our apodization function
$\Psi_{\bm{\alpha}}$ should be designed. We use a two-step process: the first
step involves a slight simplification of the apodization problem to derive a
simple analytical solution, which is then used as a starting point for a
numerical optimization to derive the final apodization function. This second
step is ultimately intended to be performed in real-time using a dedicated
control-loop in the instrument, the details of which are outside the scope of
the present study.

To derive an analytical solution, we first simplify the problem by ignoring
the impact of atmospheric fluctuations, setting $\Phi(\bm{u})=0$ in Eq. (30)
and (31). We also simplify the apodization function by enforcing a set of
symmetries. Specifically, we look for a solution that is a function of
$\bm{\alpha}\cdot{}\bm{u}$ (axially symmetric) and anti-centrosymmetric333A
similar derivation can also be done without the anti-centrosymmetry
requirement, in the case where we want to cancel imaginary speckles:

$\displaystyle\Psi_{\bm{\alpha}}(\bm{u})=\Psi_{\bm{\alpha}}(\bm{u^{\prime}})\quad\text{if}\quad\bm{\alpha}\cdot\bm{u}=\bm{\alpha}\cdot\bm{u^{\prime}},$
(32) $\displaystyle\Psi_{\bm{\alpha}}(\bm{u})=-\Psi_{\bm{\alpha}}(-\bm{u}).$
(33)

We also assume that the apodization function is small, i.e. that
$\Psi_{\bm{\alpha}}(\bm{u})\ll 1$. This is consistent with the requirement
that $\eta_{\medbullet}^{\rm inj}$ should be as close to 1 as possible. This
can be seen by using a linear development of Eq. (31) to the second order:

$\eta_{\medbullet}^{\rm inj}=\left|\langle
e^{i\Psi_{\bm{\alpha}}(\bm{u})}\,|\,1\rangle\right|^{2}\simeq{}\left|\langle
1\,|\,1\rangle-\frac{1}{2}\langle\Psi_{\bm{\alpha}}(\bm{u})\,|\,\Psi_{\bm{\alpha}}(\bm{u})\rangle\right|^{2}\,.$
(34)

The first order has canceled out due to the central symmetry of
$\Psi_{\bm{\alpha}}$: $\langle\Psi_{\bm{\alpha}}\,|\,1\rangle=0$. Also, we
made use the fact that $\Psi$ is a real function, which implies
$\langle\Psi_{\bm{\alpha}}^{2}\,|\,1\rangle=\langle\Psi_{\bm{\alpha}}\,|\,\Psi_{\bm{\alpha}}\rangle$
from the definition of $\langle\cdot\,|\,\cdot\rangle$ given in Eq. (7). With
such a form, $\eta_{\medbullet}^{\rm inj}$ can be expected to be close to
$\langle 1\,|\,1\rangle=0.81$ if $\Psi_{\bm{\alpha}}$ is small.

The first order development of Eq. (30) for the starlight apodization,
however, gives:

$\displaystyle\eta_{\star}^{\rm inj}$ $\displaystyle=\left|\langle
e^{i\Psi(\bm{u})}\,|\,e^{-ik\bm{\alpha}\cdot\bm{u}}\rangle\right|^{2}$ (35)
$\displaystyle\simeq{}\left|\langle
1\,|\,e^{-ik\bm{\alpha}\cdot{}\bm{u}}\rangle+\langle
i\Psi_{\bm{\alpha}}(\bm{u})\,|\,e^{-ik\bm{\alpha}\cdot{}\bm{u}}\rangle\right|^{2}$
(36) $\displaystyle\simeq{}\left|\langle
1\,|\,\cos\left(k\bm{\alpha}\cdot{}\bm{u}\right)\rangle-\langle\Psi_{\bm{\alpha}}(\bm{u})\,|\,\sin\left(k\bm{\alpha}\cdot{}\bm{u}\right)\rangle\right|^{2},$
(37)

where the last line makes use of the fact that $\langle
1\,|\,\sin\left(k\bm{\alpha}\cdot{}\bm{u}\right)\rangle$ and
$\langle\Psi_{\bm{\alpha}}(\bm{u})\,|\,\cos\left(k\bm{\alpha}\cdot{}\bm{u}\right)\rangle=0$
due to the definition of $\langle\cdot\,|\,\cdot\rangle$ and the symmetries of
the functions involved. This equation shows that the injected starlight can be
minimized by setting:

$\langle\Psi_{\bm{\alpha}}(\bm{u})\,|\,\sin\left(k\bm{\alpha}\cdot{}\bm{u}\right)\rangle=\langle
1\,|\,\cos\left(k\bm{\alpha}\cdot{}\bm{u}\right)\rangle.$ (38)

This guarantees that the injected starlight is 0 at the particular wavevector
$k$, but does not guarantee that the injected light is minimum over the entire
waveband. If the waveband is small enough, minimization over the entire band
can be ensured by setting the successive derivatives of $\eta_{\star}^{\rm
inj}$ with respect to $k$ to 0. The first and second derivates are given by:

$\displaystyle\frac{\mathrm{d}\eta_{\star}^{\rm inj}}{\mathrm{d}k}$
$\displaystyle=2\langle
e^{i\Psi_{\bm{\alpha}}(\bm{u})}\,|\,e^{-ik\bm{\alpha}\cdot{}\bm{u}}\rangle\Re{\left(\langle
e^{i\Psi_{\bm{\alpha}}(u)}\,|\,\left[-i\bm{\alpha}\cdot{}\bm{u}\right]e^{-ik\bm{\alpha}\cdot{}\bm{u}}\rangle\right)},$
(39) $\displaystyle\frac{\mathrm{d^{2}}\eta_{\star}^{\rm
inj}}{\mathrm{d}k^{2}}$ $\displaystyle=2\langle
e^{i\Psi_{\bm{\alpha}}(\bm{u})}\,|\,\left[-i\bm{\alpha}\cdot{}\bm{u}\right]e^{-ik\bm{\alpha}\cdot{}\bm{u}}\rangle\Re{\left(\langle
e^{i\Psi_{\bm{\alpha}}(u)}\,|\,\left[-i\bm{\alpha}\cdot{}\bm{u}\right]e^{-ik\bm{\alpha}\cdot{}\bm{u}}\rangle\right)}$
$\displaystyle\quad+2\langle
e^{i\Psi_{\bm{\alpha}}(\bm{u})}\,|\,e^{-ik\bm{\alpha}\cdot{}\bm{u}}\rangle\frac{\mathrm{d}\Re{\left(\langle
e^{i\Psi_{\bm{\alpha}}(u)}\,|\,\left[-i\bm{\alpha}\cdot{}\bm{u}\right]e^{-ik\bm{\alpha}\cdot{}\bm{u}}\rangle\right)}}{\mathrm{d}k}.$
(40)

Given our previous Eq. (38), which guarantees that $\eta_{\star}^{\rm
inj}=\left|\langle
e^{i\Psi_{\bm{\alpha}}(\bm{u})}\,|\,e^{-ik\bm{\alpha}\cdot{}\bm{u}}\rangle\right|^{2}=0$,
the first derivative is necessary 0, and the second derivative can be
simplified to:

$\frac{\mathrm{d^{2}}\eta_{\star}^{\rm inj}}{\mathrm{d}k^{2}}=2\langle
e^{i\Psi_{\bm{\alpha}}(\bm{u})}\,|\,\left[i\bm{\alpha}\cdot{}\bm{u}\right]e^{-ik\bm{\alpha}\cdot{}\bm{u}}\rangle\Re{\left(\langle
e^{i\Psi_{\bm{\alpha}}(u)}\,|\,\left[i\bm{\alpha}\cdot{}\bm{u}\right]e^{-ik\bm{\alpha}\cdot{}\bm{u}}\rangle\right)}.$
(41)

This last expression shows that canceling the second derivative of
$\eta_{\star}^{\rm inj}$ with respect to $k$ requires canceling out the real
part of $\langle
e^{i\Psi_{\bm{\alpha}}(u)}\,|\,\left[ik\bm{\alpha}\cdot{}\bm{u}\right]e^{-ik\bm{\alpha}\cdot{}\bm{u}}\rangle$,
where we took advantage of the linearity of the scalar product to add a factor
$k$ in the second term and keep all elements dimensionless.

By using again the first-order development of $e^{i\Psi_{\bm{\alpha}}}$ and
the different symmetries, we find:

$\displaystyle\Re{\left(\langle
e^{i\Psi_{\bm{\alpha}}(u)}\,|\,\left[ik\bm{\alpha}\cdot{}\bm{u}\right]e^{-ik\bm{\alpha}\cdot{}\bm{u}}\rangle\right)}$
$\displaystyle\simeq{}\langle
1\,|\,\left[k\bm{\alpha}\cdot{}\bm{u}\right]\sin(k\bm{\alpha}\cdot{}\bm{u})\rangle$
$\displaystyle\quad+\langle\Psi_{\bm{\alpha}}(\bm{u})\,|\,\left[k\bm{\alpha}\cdot{}\bm{u}\right]\cos(k\bm{\alpha}\cdot{}\bm{u})\rangle,$
(42)

which shows that in order to cancel out this real part, we need:

$\displaystyle\langle\Psi_{\bm{\alpha}}(\bm{u})\,|\,\left[k\bm{\alpha}\cdot{}\bm{u}\right]\cos(k\bm{\alpha}\cdot{}\bm{u})\rangle=-\langle
1\,|\,\left[k\bm{\alpha}\cdot{}\bm{u}\right]\sin(k\bm{\alpha}\cdot{}\bm{u})\rangle.$
(43)

Putting everything together, to solve our apodization problem, we are
therefore looking for a function $\Psi_{\bm{\alpha}}$ such that:

$\displaystyle\langle\Psi_{\alpha}(\bm{u})\,|\,\Psi_{\alpha}(\bm{u})\rangle=\left\lVert\Psi_{\bm{\alpha}}\right\rVert^{2}\text{
is as small as possible,}$ (44)
$\displaystyle\langle\Psi_{\bm{\alpha}}(\bm{u})\,|\,H_{1}(u)\rangle=z_{1},$
(45)
$\displaystyle\langle\Psi_{\bm{\alpha}}(\bm{u})\,|\,H_{2}(u)\rangle=z_{2},$
(46)

where:

$\displaystyle H_{1}(u)=\sin\left(k\bm{\alpha}\cdot{}\bm{u}\right),$ (47)
$\displaystyle z_{1}=\langle
1\,|\,\cos\left(k\bm{\alpha}\cdot{}\bm{u}\right)\rangle,$ (48) $\displaystyle
H_{2}(u)=\left[k\bm{\alpha}\cdot{}\bm{u}\right]\cos\left(k\bm{\alpha}\cdot{}\bm{u}\right),$
(49) $\displaystyle z_{2}=-\langle
1\,|\,\left[k\bm{\alpha}\cdot{}\bm{u}\right]\sin\left(k\bm{\alpha}\cdot{}\bm{u}\right)\rangle.$
(50)

This system is readily solved by noting that in order to minimize its norm in
Eq. (44), while fulfilling the two conditions given by Eq. (45) and (46),
$\Psi_{\bm{\alpha}}$ needs to be in the plane defined by $(H_{1},H_{2})$:

$\Psi_{\bm{\alpha}}=aH_{1}+bH_{2}\,.$ (51)

Then, solving the linear system resulting from Eq. (45) and (46) gives the
following closed-form solution for $\Psi_{\bm{\alpha}}$:

$\Psi_{\bm{\alpha}}=\frac{z_{1}\left\lVert H_{2}\right\rVert^{2}-z_{2}\langle
H_{1}\,|\,H_{2}\rangle}{\left\lVert H_{1}\right\rVert^{2}\left\lVert
H_{2}\right\rVert^{2}-{\langle
H_{1}\,|\,H_{2}\rangle}^{2}}H_{1}+\frac{z_{2}\left\lVert
H_{1}\right\rVert^{2}-z_{1}\langle H_{1}\,|\,H_{2}\rangle}{\left\lVert
H_{1}\right\rVert^{2}\left\lVert H_{2}\right\rVert^{2}-{\langle
H_{1}\,|\,H_{2}\rangle}^{2}}H_{2}.$ (52)

This closed-form solution for the apodization is valid under the conditions
that $\Psi_{\bm{\alpha}}$ remains small and in the absence of atmospheric
fluctuations. It also ignores the true dependency of the injected flux as a
function of the wavelength, as we have only tried to cancel the flux at a
particular wavenumber in Eqs. (45) and (46). Moreover, in practice, the
optical path delays is applied by a deformable mirror and it is therefore in
units of length. So our strategy is to use the solution given in Eq. (52) as
an initial guess in an optimization process.

To do so, we keep $\Psi_{\bm{\alpha}}$ as a linear combination of two
functions $H_{1}$ and $H_{2}$ defined for a central wavenumber $k_{0}$, and we
directly integrate the square root of the scalar product over all wavenumbers
$k$:

$\eta_{\star}^{\rm inj}(a,b)=\int_{k_{0}-\Delta k/2}^{k_{0}+\Delta
k/2}\left|\langle
e^{i\frac{k}{k_{0}}\left[a\sin(k_{0}\bm{\alpha}\cdot\bm{u})+bk_{0}\bm{\alpha}\cdot\bm{u}\cos(k_{0}\bm{\alpha}\cdot\bm{u})\right]}\,|\,e^{-ik\bm{\alpha}\cdot{}\bm{u}}\rangle\right|^{2}\frac{\mathrm{d}k}{\Delta
k}\,.$ (53)

This apodization coefficient can then be numerically minimized to find the
coefficients $a$ and $b$:

$(a,b)=\mathrm{argmin}\left\\{\eta_{\star}^{\rm inj}(a,b)\right\\}\,,$ (54)

which will give a set of values $(a,b)$ for any value of the angular
separation $\bm{\alpha}$.

In Figure 3, we show the values of the coefficients $a$ and $b$ obtained both
by the closed-form solution of the approximated apodization problem from Eq.
(52), and after the optimization process from Eq. (54), as a function of the
separation $\bm{\alpha}$. Also shown are the apodization functions
$aH_{1}+bH_{2}$ for the two specific cases of $\bm{\alpha}=\lambda/D$ and
$\bm{\alpha}=2\lambda/D$.

Figure 3: Upper panel: Coefficients $(a,b)$ used to compute the phase apodizer
as $\Psi_{\bm{\alpha}}=aH_{1}+bH_{2}$. The solid lines show the optimized
coefficient used in the final definition of the apodizer, i.e. the solution of
Eq. (54). The dashed lines show the initial guess calculated from Eq. (52).
The simulation has been made for K band observations ($k_{0}=2\pi/2.2\,\mu$m)
and the shape of the VLT pupil. Lower panels : The apodizer function,
$\Psi_{\bm{\alpha}}(\bm{u})$, for a target at an angular separation of
$100\,$mas (left panel) and $200\,$mas (right panel).

####  2.5.3 Photon noise after pupil apodization

Figure 4:  Apodization due to the spatial filtering caused by single-mode
fibres as a function of the angular separation $\bm{\alpha}$ between the star
and the exoplanet, in the K band (upper panel) and in the I band (lower
panel). The green curves show $\eta_{\medbullet}^{\rm inj}$, the injection of
the exoplanet flux into interferometer 2, while the blue and orange curves
represent the injection of the stellar flux ($\eta_{\star}^{\rm inj}$). The
blue curves are without a phase apodization ($\Psi_{\bm{\alpha}}=0$), while
the orange curves are with phase apodization
($\Psi_{\bm{\alpha}}=aH_{1}+bH_{2}$). The solid curves include the residuals
of atmospheric correction by an extreme AO system, giving a Strehl ratio of
98% in the K band and 72% in the I band. The dotted curves are obtained
assuming no atmospheric residuals: $\Phi(\bm{u})=0$, hence a Strehl ratio of
100%. The red curves correspond to the contrast limit, $C_{\rm shot}$ in Eq
(57), that would be achieved when limited by photon noise, for an observation
with parameters as stipulated in Table 2. The phase apodizer is extremely
efficient at decreasing the flux injected into the fibre in the K band.
However, at shorter wavelengths, the lower Strehl ratio makes it less
effective, with the injected flux mostly due to the atmospheric halo.

To calculate the amount of photon noise introduced by the starlight, we have
to account for the atmospheric perturbations. To do so, we simulated the
atmospheric perturbations using HCIPy, a Python framework for high-contrast
imaging simulation (Por et al., 2018). The atmospheric residuals are obtained
using a closed-loop adaptive optics system similar to the one used in the
SPHERE instrument, which employs a Shack-Hartmann Wavefront Sensor (WFS). The
simulations include WFS Photon Noise and the Time Lag of the AO system. The
aberrations have been adjusted to the lab measurements of the SAXO optimal
performance, giving 98% in K band, and 72% in I band (Fusco et al., 2016).

The phase masks $\Phi(u)$ generated by the atmospheric residuals are then
added to the phase mask $\Psi(u)$ of the apodizer to calculate the two broad
band attenuations coefficients:

$\eta_{\star}^{\rm inj}=\int_{k_{0}-\Delta k/2}^{k_{0}+\Delta
k/2}\left|\langle
e^{i\Phi(\bm{u})+i\Psi(\bm{u})}\,|\,e^{-ik\bm{\alpha}\cdot\bm{u}}\rangle\right|^{2}\frac{\mathrm{d}k}{\Delta
k}$ (55)

and

$\eta_{\medbullet}^{\rm inj}=\int_{k_{0}-\Delta k/2}^{k_{0}+\Delta
k/2}\left|\langle
e^{i\Phi(\bm{u})+i\Psi(\bm{u})}\,|\,1\rangle\right|^{2}\frac{\mathrm{d}k}{\Delta
k}\,.$ (56)

The two panels of Figure 4 display simulations under these atmospheric
conditions in the two bands. For reference, the dotted curves in both panels
represent a system without atmosphere (Strehl ratio of 100%). These
simulations use parameters representative of the VLTI, as listed in Table 2.

Figure 4 also shows the contrast limit obtained assuming a threshold of 3 on
the S/N, using $S_{\rm interfero}$ for the signal as defined in Eq. (27), and
$N_{\rm shot}$ for the noise as defined by Eq. (26). According to this
threshold, the contrast is defined as follows:

$C_{\rm shot}=3\frac{1}{\sqrt{\gamma F_{\star}}}\frac{\sqrt{\eta_{\star}^{\rm
inj}}}{\eta_{\medbullet}^{\rm inj}}\,.$ (57)

These curves indicate that the photon noise contrast limit is just sufficient
to detect the exoplanet Gliese 876b in the K band. Proxima Cen b, on the other
hand, is only observable at smaller angular separations and can only be
detected at shorter wavelengths.

Achieving a high stellar extinction $1/\eta_{\star}^{\rm inj}$ of at least
$10^{3}$ is crucial for observing exoplanets with contrast ratios around
$10^{-8}$. This can potentially be achieved at long wavelengths with very high
Strehl and excellent apodization, or by taking advantage of the smaller
diffraction spot at short wavelengths, which somewhat loosens the requirements
on the Strehl ratio and apodization.

###  2.6 Coherent Speckle Removal

####  2.6.1 Principle of post-processing

Due to the chromatic behaviour of stellar light and due to residual optical
aberrations, an apodizer alone cannot completely eliminate the coherent
stellar light. However, the residual coherent speckles can be removed by a
dedicated post-processing algorithm. This is similar to what happens in
direct-imaging, where the use of a coronagraph is always complemented by a
post-processing technique (for example ADI or SDI) to reach the highest levels
of contrast.

In dual-field interferometry, the residual coherent speckles can be removed by
taking advantage of the simultaneous observation of the central-star by the
first interferometer. A coherent flux $\Gamma_{1,\star}$ is measured on the
first interferometer (see Figure 1), which can be used to phase-reference the
stellar coherent speckles on the second interferometer, thanks to the internal
monitoring of the non-common optical path by the metrology.

In practice, the two coherent fluxes $\Gamma_{1,\star}$ and $\Gamma_{2,\star}$
are not equal, as they can have vastly different amplitudes. But they remain
similar in the sense that neither their phase, nor their spectral shape are
significantly different. Experience shows that these differences are well
captured by a low-order polynomial in $k$, the wavenumber:

$\frac{\Gamma_{2,\star}}{\Gamma_{1,\star}}=\sum_{n=0}^{n_{\rm
poly}}a_{n}k^{n},$ (58)

where the coefficients $a_{n}$ are complex-coefficients which represent the
chromatic difference between the on-axis stellar light $\Gamma_{1,\star}$ and
the off-axis stellar light $\Gamma_{2,\star}$. The order $n_{\rm poly}$ is
typically 5 to 15 for a good approximation444We also tested spline-based
linear models, similar to those used by Ruffio et al. (2019, 2023) to model
the continuum, but did not observe any improvement for an identical number of
degrees of freedom..

Another advantage of using a metrology system to reference one interferometer
to the other is the ability to phase reference the planet light as well. It
guarantees that the phase of the coherent flux observed in the second
interferometer will follow the analytical formula for a binary companion at a
given separation ${\bm{\alpha}}$:

$\frac{\Gamma_{2,\medbullet}}{\Gamma_{1,\star}}=\left(\frac{\mathsf{F}_{\medbullet}}{\mathsf{F}_{\star}}\right)e^{-ik\left(\bm{\alpha}\cdot\bm{B}\right)}=Ce^{-ik\left(\bm{\alpha}\cdot\bm{B}\right)}\,,$
(59)

ignoring the spectral differences between the planet and the star by reducing
the contrast $F_{\medbullet}/F_{\star}$ to a single parameter $C$.

Hence, there is a structural difference between the phase of the low-order
polynomial-like off-axis coherent flux, $\arg(\Gamma_{2,\star})$, and the
phase of planet coherent flux, $\arg(\Gamma_{2,\star})$. It is this difference
which is used to filter out $\Gamma_{2,\star}$ from the combined coherent
signal $\Gamma_{2}=\Gamma_{2,\star}+\Gamma_{2,\medbullet}$ without
significantly impacting the second. This idea is at the heart of the
ExoGRAVITY Collaboration’s post-processing technique, pioneered in GRAVITY
Collaboration et al. (2019) and detailed in GRAVITY Collaboration et al.
(2020a). In the rest of this Section, we use the algorithms detailed in these
publications to estimate the residual stellar contamination $N_{\rm speckle}$
in Eq. (19).

####  2.6.2 Mathematical formulation

As outlined by GRAVITY Collaboration et al. (2020a) and Nowak et al. (2020),
the formalism of the ExoGRAVITY method involves considering all wavelength-
dependant quantities as column-vectors. In this formalism, the coherent flux
$\Gamma$ is a column-vector
$\left[\Gamma(k_{1}),\Gamma(k_{2}),\dots,\Gamma(k_{n})\right]^{T}$. Retrieving
the value of $C$ involves two steps: (1) projecting the coherent flux vector
$\Gamma_{2}$ onto a subspace orthogonal to that of the coherent speckle model
of $\Gamma_{2,\star}$; (2) projecting the theoretical planet coherent flux
from Eq. (59) onto this same subspace to retrieve the contrast $C$.

In the first step, the subspace orthogonal to the coherent speckle is
calculated from the series of vectors $J_{n}$ and $Q_{n}$ defined as:

$\displaystyle J_{n}$ $\displaystyle=k^{n}\,\Gamma_{1,\star}$ (60)
$\displaystyle Q_{n}$ $\displaystyle=ik^{n}\,\Gamma_{1,\star}.$ (61)

The projector $P_{\perp}$ on the subspace orthogonal to all the $J_{n}$ and
$Q_{n}$, $n\in\\{0,\dots,n_{\rm poly}\\}$ is the matrix $P_{\perp}$ defined
by:

$\displaystyle P_{\perp}=I_{n_{\rm poly}}-\sum_{n=0}^{n_{\rm
poly}}\left[\frac{J_{n}^{T}}{J_{n}^{T}\cdot{}J_{n}}+\frac{Q_{n}^{T}}{Q_{n}^{T}\cdot{}Q_{n}}\right].$
(62)

The first step therefore consists in multiplying the coherent flux by
$P_{\perp}$:

$P_{\perp}\cdot\Gamma_{2}=P_{\perp}\cdot\Gamma_{2,\star}+P_{\perp}\cdot\Gamma_{2,\medbullet},$
(63)

with the hope that the starlight residual, $P_{\perp}\cdot{}\Gamma_{2,\star}$,
will be close to 0.

In the second step, we calculate the vector:

$K_{\bm{\alpha}}=P_{\perp}\cdot\left[\Gamma_{1,\star}e^{-ik\left(\bm{\alpha}\cdot\bm{B}\right)}\right],$
(64)

which serves as a model for the coherent flux of the exoplanet
$\Gamma_{2,\medbullet}$, as expected according to Eqs. (59), projected in the
same subspace as the data. Assuming $P_{\perp}\cdot\Gamma_{2,\star}=0$, we now
have:

$P_{\perp}\cdot\Gamma_{2}=CK_{\bm{\alpha}},$ (65)

which gives the solution:

$C=\frac{K_{\bm{\alpha}}^{T}\cdot P_{\perp}}{K_{\bm{\alpha}}^{T}\cdot
K_{\bm{\alpha}}}\cdot\Gamma_{2}.$ (66)

This product post-processing is required to extract the planet signal from the
stellar speckles, but it also attenuates the signal by reducing
$\Gamma_{2,\medbullet}$.

Figure 5: Attenuation of the exoplanet signal ($\eta_{\medbullet}^{\rm
inj}\eta_{\medbullet}^{\rm proc}$, in dashed lines) and the corresponding
contrast limit ($C_{\rm speckle}=3\eta_{\star}^{\rm inj}\eta_{\star}^{\rm
proc}/\eta_{\medbullet}^{\rm inj}\eta_{\medbullet}^{\rm proc}$, in solid
lines) after post-processing of the stellar speckle, as a function of angular
separation. The three colour curves represent three levels of polynomial
fitting strength ($n_{\rm poly}$ of 3, 7, and 11) used for filtering the
coherent speckles. The dotted curves show the attenuation of the exoplanet
signal, illustrating that a lower polynomial order yield a smaller inner
working angle. The solid curves indicate the contrast limit after post-
processing of the stellar speckle, demonstrating that higher polynomial orders
result in a better contrast limit. The upper plot represents observations with
a 100 m baseline, while the lower plot corresponds to a 200 m interferometer
baseline.

####  2.6.3 Simulation of Post-Processing Attenuations

To calculate the level of attenuation of the coherent star and exoplanetary
light, we must account for atmospheric perturbations. We again used HCIPy to
simulate the wavefront residuals on two telescopes, each equipped with a
SPHERE-like AO system with similar performance. As in Section 2.5.3, we then
added a phase apodizer to obtain two sets of phase masks
$\Phi_{\mathbb{A}}(\bm{u})$ for telescope ${\mathbb{A}}$ and
$\Phi_{\mathbb{B}}(\bm{u})$ for telescope ${\mathbb{B}}$. Both
$\Phi_{\mathbb{A}}$ and $\Phi_{\mathbb{B}}$ depend on $k$ and time.

The total attenuation can be calculated from the electric field expressed in
Table 1:

$\eta_{\medbullet}^{\rm inj}\eta_{\medbullet}^{\rm
proc}=\frac{K_{\bm{\alpha}}^{T}\cdot P_{\perp}}{K_{\bm{\alpha}}^{T}\cdot
K_{\bm{\alpha}}}\cdot\begin{bmatrix}\langle
e^{i\Phi_{\mathbb{A}}(\bm{u})}\,|\,1\rangle\langle
e^{i\Phi_{\mathbb{B}}(\bm{u})}\,|\,1\rangle^{\ast}e^{-ik_{1}\bm{\alpha}\cdot{}\bm{B}}\\\
\vdots\\\ \langle e^{i\Phi_{\mathbb{A}}(\bm{u})}\,|\,1\rangle\langle
e^{i\Phi_{\mathbb{B}}(\bm{u})}\,|\,1\rangle^{\ast}e^{-ik_{n_{\lambda}}\bm{\alpha}\cdot{}\bm{B}}\end{bmatrix}.$
(67)

Similarly, the attenuation of the coherent stellar speckle can be written as:

$\eta_{\star}^{\rm inj}\eta_{\star}^{\rm proc}=\frac{K_{\bm{\alpha}}^{T}\cdot
P_{\perp}}{K_{\bm{\alpha}}^{T}\cdot
K_{\bm{\alpha}}}\cdot\begin{bmatrix}\langle
e^{i\Phi_{\mathbb{A}}(\bm{u})}\,|\,e^{-ik_{1}\bm{\alpha}\cdot{}\bm{u}}\rangle\langle
e^{i\Phi_{\mathbb{B}}(\bm{u})}\,|\,e^{-ik_{1}\bm{\alpha}\cdot{}\bm{u}}\rangle^{\ast}\\\
\vdots\\\ \langle
e^{i\Phi_{\mathbb{A}}(\bm{u})}\,|\,e^{-ik_{n_{\lambda}}\bm{\alpha}\cdot{}\bm{u}}\rangle\langle
e^{i\Phi_{\mathbb{B}}(\bm{u})}\,|\,e^{-ik_{n_{\lambda}}\bm{\alpha}\cdot{}\bm{u}}\rangle^{\ast}\end{bmatrix}.$
(68)

The attenuations due to apodization and post-processing are combined in our
analysis because it is difficult to calculate the attenuation of post-
processing without including the spatial filtering of the single-mode fibre.

This residual starlight, which is neither rejected by the fibre and the
apodization nor by the post-processing algorithm, corresponds to the speckle
noise term in Eq. (19):

$N_{\rm speckle}=\gamma F_{\star}\eta_{\star}^{\rm inj}\eta_{\star}^{\rm
proc}.$ (69)

In Figure 5, we show the evolution of $\eta_{\medbullet}^{\rm
inj}\eta_{\medbullet}^{\rm proc}$ as a function of the angular separation
$\bm{\alpha}$, for three different levels of aggressiveness in deconvolution,
corresponding to three different orders for the polynomial model of the
coherent residual starlight ($n_{\rm poly}=3$, 7, and 11), and two baseline
lengths: $B=100\leavevmode\nobreak\ \mathrm{m}$ and 200 m. This graph shows
how, as the planet gets closer to the central star ($\alpha\to 0$), its signal
becomes too similar to the coherent flux of the central star and gets removed
by the post-processing step ($\eta_{\medbullet}^{\rm
inj}\eta_{\medbullet}^{\rm proc}\to 0$).

We also plot a contrast limit assuming a threshold of 3 on the S/N, using
$S_{\rm processed}$ for the signal as defined in Eq. (15), and $N_{\rm
speckle}$ for the noise as defined in Eq. (69). According to this threshold,
the contrast is defined as follows:

$C_{\rm speckle}=3\ \frac{\eta_{\star}^{\rm inj}\eta_{\star}^{\rm
proc}}{\eta_{\medbullet}^{\rm inj}\eta_{\medbullet}^{\rm proc}}\,.$ (70)

###  2.7 Final Contrast Curve

Figure 6: Contrast limit as a function of angular separation, with simulation
parameters detailed in Table 2. The upper panel illustrates the contrast
dependence on waveband for a 100 m baseline. The lower panel displays the
contrast limits for a 200 m baseline. The shaded area represents a range of K
band Strehl ratios, from 88% (worse contrast limit) to 98% (best contrast
limit). At short wavelength, the contrast range is limited by the Strehl. In
the K and L bands, the contrast range is predominantly limited by thermal
background noise. The inner working angles are constrained by both the
telescope’s point spread function and the interferometric resolution, both
linear function of wavelength.

Putting everything together, the contrast limit of a dual-field interferometer
on the VLTI can be calculated using the formula presented in Eq. (19),
replacing $F_{\medbullet}$ by $C\,F_{\star}$ to convert to contrast, and using
an arbitrary threshold of $\mathrm{S/N}=3$ to define a detection limit:

$C=3\ \frac{\sqrt{N_{\rm readout}^{2}+N_{\rm background}^{2}+\gamma
F_{\star}\eta_{\star}^{\rm inj}+(\gamma F_{\star}\eta_{\star}^{\rm
inj}\eta_{\star}^{\rm proc})^{2}}}{\gamma F_{\star}\eta_{\medbullet}^{\rm
inj}\eta_{\medbullet}^{\rm proc}}.$ (71)

The resulting contrast curve is given in Figure 6.

This accounts for the cumulative effects of readout noise, thermal background
noise, stellar photon noise, and residual speckle noise after post-processing,
and provides critical insights into the system’s performance. As shown in the
upper panel of Fig. 6, a contrast of $10^{-6}$ can be achieved at 25 mas in
the I band, but this increases to 70 mas in the K band. Reducing the baseline
length also results in an inversely proportional increase of the detection
limit at low separations, highlighting the importance of baseline extension
for reducing this angle beyond the current capabilities of the VLTI.

Another important observation relates to the Strehl ratio, which strongly
influences the contrast that can be achieved, impacting both the signal
injected into the fibre and the effectiveness of stellar flux apodization, as
illustrated in Fig. 4. As anticipated, the Strehl ratio - and consequently,
the contrast - is more adversely affected at shorter wavelengths. Two distinct
regimes are observed: in a regime of good correction, above a Strehl ratio of
$\simeq{}50\,\%$ our simulations show a preference for shorter wavelengths due
to the beneficial increase in angular resolution that compensates for the
slight decrease in Strehl ratio. In the regime where the Strehl ratio is low,
the correction quality is insufficient for proper light injection into the
single-mode fibre, and the advantages of a narrower point spread function no
longer offset the losses.

Furthermore, the figure underscores the difficulty of detecting reflected
light from exoplanets beyond 2 microns, attributed to the escalation of
thermal background noise. At these wavelenght, exploring thermal emissions
from exoplanets emerges as a viable alternative, albeit likely requiring the
colder operational conditions provided by space-based observatories.

##  3 An additional Unit Telescope in the context of exoplanet detection

###  3.1 The Effect of the 2D Geometry of the VLTI

Figure 6 shows how the contrast limit changes as a function of separation for
baseline lengths of 100 and 200 m. However, this does not account for the
geometry of the VLTI, which has six baselines. In practice, the geometry of
the array is crucial, as the relevant quantity when determining the ”baseline
length” is the projected baseline vector,
$\bm{\alpha}\cdot\bm{B}/|\bm{\alpha}|$, projected both in the plane of the sky
and along the position angle of the exoplanet.

The projection along the plane of the sky depends on the position of the star.
In the upper panel of Fig. 7, the UV coverage is shown for Proxima Cen when it
passes the meridian, representing almost optimal conditions. The frequency
domain covered by the VLTI (red points) is mostly constrained in the North-
East direction due to the intrinsically elongated geometry of the VLTI along
the UT1-UT4 baseline.

The second projection is along the position angle (PA) of the exoplanet (the
argument of $\bm{\alpha}$): if the exoplanet PA aligns with the direction of
the maximal baseline, the contrast will be optimal. However, if a baseline is
orthogonal to the planet’s position, the baseline becomes useless. Thus, the
simultaneous usage of all baselines is crucial to cover all possible PAs of
the exoplanet.

To illustrate this effect, Figure 8 shows the contrast limit using all
different baselines of the array as a function of the 2D vector $\bm{\alpha}$
shown in RA-DEC. To compute the contrast limit, we calculated the
$S_{ij}/N_{ij}$ on each projected baseline and summed it as:

$S/N=\sqrt{\sum_{ij}(S_{ij}/N_{ij})^{2}}\,.$ (72)

Then, we estimated the final contrast using a requirement of 3 on the S/N.
This clearly shows that the performance of the VLTI is fundamentally under-
performing in the North-West direction.

###  3.2 Limitations of the Inner Working Angle

Figure 7:  Upper panel: The frequency coverage of the VLTI when observing
Proxima Cen passing the meridian is shown in red. The coverage of the current
VLTI is confined within an oval oriented towards the North-East direction. The
proposed upgrade, with the addition of a new 8 m telescope (highlighted in
green), would create four additional baselines extending into the North-West
direction. Lower panel: Inner Working Angle (IWA) as a function of the PA-
averaged projected baseline length
$\bm{\alpha}\cdot\bm{B}/|\bm{\alpha}|=|\bm{B}|/\sqrt{2}$. For very long
baseline lengths, the IWA is limited by telescope diffraction, approaching one
$\lambda/D$. However, for most of the VLTI baselines, the IWA is typically
limited to several $\lambda/D$. The addition of a new telescope would create
four new baselines with an IWA only limited by the 8-meter telescope diameter.
Figure 8: Contrast limit of a dual-field interferometer using the current
four-UT configuration of the VLTI, compared to the proposed five-UT
configuration shown in Figure 9. Figure 9: Aerial view of the proposed
extenstion of the VLTI. The new UT5 telescope would be positioned to the
southeast of the platform, on the opposite side of the access road. The beam
would be propagated to the VLTI tunnel through the J arm of the existing
VLTI/AT interferometer, creating four new baselines with north-south
orientations, each approximately 200 meters in length. With a length of 220
meters and an orientation at 154 degrees to the South-East, the UT3-UT4
baseline would be almost orthogonal to the existing long UT1-UT4 baseline
giving an optimal UV coverage. The long delay line could be obtained by
serializing the two remaining, presently unused, VLTI delay lines.

As detailed in Eq. (73) of Section 4, the reflected light flux from an
exoplanet decreases with the square of its distance from the host star.
Consequently, the detectability of exoplanets heavily depends on the
instrument’s ability to observe companions close to their star. This
capability is quantified by the Inner Working Angle (IWA), a critical metric
for assessing the effectiveness of astronomical instruments in detecting
companions at short separation.

The IWA showed in Fig. 8 is influenced by two factors which both reduces the
exoplanet signal: $\eta_{\medbullet}^{\mathrm{inj}}$ and
$\eta_{\medbullet}^{\mathrm{proc}}$. The first parameter,
$\eta_{\medbullet}^{\mathrm{inj}}$, arises from the phase apodization used to
reduce the stellar flux. Its impact gets bigger as $\bm{\alpha}$ becomes close
to $\lambda/D$ (as shown in Figure 4), meaning that the extent of its effect
is governed by the diameter $D$ of the individual telescopes of the array.

Conversely, the influence of $\eta_{\medbullet}^{\mathrm{proc}}$, depicted in
Figure 5, becomes significant when the separation is too small for the
oscillatory term $e^{ik\bm{\alpha}\cdot{}\bm{B}}$ to effectively separate the
exoplanet signal from the residual starlight. This starts to occur when the
$\bm{\alpha}$ becomes close to a few $\lambda/\Delta\lambda$ of
$\lambda/\bm{B}$, where $\bm{B}$ is the projected baseline.

Hence, the IWA depends both on the diameter of the telescopes and the baseline
length, whichever limit comes first. This is illustrated in the lower panel of
Fig. 7, which plots the IWA as defined by the angular separation at which the
quantity $\eta_{\medbullet}^{\mathrm{inj}}\eta_{\medbullet}^{\mathrm{proc}}$
equals 1/2. This figure shows the IWA’s noticeable increase at shorter
baselines (baseline-limited regime) and its stabilization at longer baselines
(telescope diameter-limited regime). The red vertical lines represent the
lengths of the six existing baselines of the VLTI averaged over all possible
orientations. They highlight that the current VLTI IWA is limited by its
baselines, thus not fully capitalizing on the individual telescopes’ size.
Four additional baselines with projected lengths above 100 meters would enable
the observation of exoplanets with a minimal IWA.

###  3.3 Physical implementation of the Fifth UT

Given the need to both enhance the UV coverage in the North-West direction and
increase the baseline lengths for maximizing the Inner Working Angle (IWA), we
have devised a proposal for integrating a fifth UT that addresses these
challenges without extensive change to the VLTI architecture.

This new telescope could be constructed to the south of UT4, across the road.
It would be aligned with the AT J band arm, facilitating the propagation of
the beam toward the delay line tunnels. The required delay line for a baseline
of approximately 200 meters could be achieved by serially utilizing the two
remaining unused delay lines of the VLTI. An aerial view of this proposed
setup is presented in Figure 9.

In terms of UV coverage, the four new baselines would ideally complement the
existing frequency coverage, as shown by Fig. 7. With these four new
baselines, the IWA would also reach optimal levels, approaching the
theoretical limit of $\lambda/D$. Additionally, this configuration would
enhance the contrast limit at many position angles, improving the
detectability of exoplanets. As shown in the right panel of Fig. 8, this setup
would enable the detection of an exoplanet at a 1 AU distance from a star
located 10 parsecs away across all position angles. This setup would also
allow for the detection of Proxima Centauri b at its most favorable position
angle. The enhancements provided by these additional capabilities are
quantified in the subsequent section.

##  4 Detection rate of reflected light exoplanets

Figure 10: Detectable exoplanets with the current four-UT configuration with a
100 m baseline (left) at the J band and the proposed five-UT with 200 m
baselines (right) at the I band. Blue and purple circles and histograms show
the detectable planets in 30 h of integration time in each case, and green
ones correspond to planets detectable in 3 h. Empty circles indicate planets
that could be detectable by such an instrument but are out of the
$\rm{DEC}\in\\{-90^{\circ},+30^{\circ}\\}$ interval observable from Paranal,
and thus not included in the tally of detectable planets. Grey dots and
histograms show the ensemble of known exoplanets within 30 pc. For non-
detectable exoplanets, we plot either $M_{p}$ or $M_{p}\,sin\,i$ as reported
in the NASA Archive. Horizontal lines indicate the masses of Neptune (blue),
Earth (green) and Mars (red).

###  4.1 Simulated orbital realizations

Table 3: Detectability at each spectral band of the exoplanets within 30 pc that are potential targets ($P_{detect}$¿25%) in at least one band with 3 h of integration time. Both the configurations assuming 4 or 5 telescopes are shown. In all cases, we adopted a value of $A_{g}$=0.3 for bands I and J, and $A_{g}$=0.1 for H and K. The corresponding targets in 30 h of integration time are displayed in Tables 4 and 5. |  |  |  |  | I | J | H | K
---|---|---|---|---|---|---|---|---
| Planet | $d$ | $a$ | $M_{p}$ | $P_{detect}$ | $\alpha_{obs}$ | $P_{detect}$ | $\alpha_{obs}$ | $P_{detect}$ | $\alpha_{obs}$ | $P_{detect}$ | $\alpha_{obs}$
|  | [pc] | [AU] |  [$M_{J}$] | [%] | [deg] | [%] | [deg] | [%] | [deg] | [%] | [deg]
VLTI | HD 62509 b | 10.34 | 1.62 | 2.74 | 100.00 |  [31${}^{+26}_{-18}$,127${}^{+2}_{-5}$] | 100.00 |  [31${}^{+26}_{-14}$,123${}^{+1}_{-2}$] | 99.90 |  [32${}^{+25}_{-2}$,101${}^{+2}_{-2}$] | 97.80 |  [44${}^{+11}_{-2}$,86${}^{+2}_{-3}$]
eps Eri b | 3.20 | 3.53 | 0.68 | 100.00 |  [14${}^{+12}_{-5}$,115${}^{+1}_{-1}$] | 100.00 |  [14${}^{+12}_{-6}$,100${}^{+6}_{-2}$] | 100.00 |  [24${}^{+14}_{-3}$,71${}^{+9}_{-5}$] | 0.00 | $-$
alf Tau b | 20.44 | 1.47 | 7.44 | 100.00 |  [28${}^{+29}_{-6}$,118${}^{+1}_{-1}$] | 100.00 |  [37${}^{+21}_{-6}$,115${}^{+1}_{-1}$] | 49.70 |  [61${}^{+6}_{-4}$,81${}^{+3}_{-3}$] | 0.00 | $-$
alf Ari b | 20.21 | 1.20 | 2.13 | 100.00 |  [31${}^{+28}_{-2}$,105${}^{+1}_{-1}$] | 100.00 |  [39${}^{+20}_{-1}$,98${}^{+1}_{-2}$] | 0.00 | $-$ | 0.00 | $-$
GJ 876 b | 4.68 | 0.21 | 2.28 | 100.00 |  [41${}^{+1}_{-1}$,97${}^{+1}_{-1}$] | 100.00 |  [53${}^{+0}_{-0}$,94${}^{+1}_{-1}$] | 0.00 | $-$ | 0.00 | $-$
GJ 896 A b | 6.26 | 0.64 | 2.23 | 71.60 |  [53${}^{+4}_{-3}$,73${}^{+3}_{-5}$] | 99.90 |  [47${}^{+2}_{-2}$,80${}^{+2}_{-3}$] | 0.00 | $-$ | 0.00 | $-$
GJ 876 c | 4.68 | 0.13 | 0.71 | 100.00 |  [46${}^{+0}_{-0}$,92${}^{+1}_{-1}$] | 0.00 | $-$ | 0.00 | $-$ | 0.00 | $-$
GJ 9066 c | 4.47 | 0.88 | 0.24 | 12.70 |  [56${}^{+5}_{-5}$,66${}^{+5}_{-4}$] | 47.60 |  [54${}^{+15}_{-10}$,87${}^{+2}_{-8}$] | 0.00 | $-$ | 0.00 | $-$
Extended VLTI | HD 62509 b | 10.34 | 1.62 | 2.74 | 100.00 |  [31${}^{+26}_{-21}$,127${}^{+2}_{-5}$] | 100.00 |  [31${}^{+26}_{-17}$,123${}^{+2}_{-2}$] | 99.90 |  [31${}^{+26}_{-8}$,102${}^{+3}_{-3}$] | 99.30 |  [33${}^{+23}_{-2}$,91${}^{+3}_{-2}$]
alf Tau b | 20.44 | 1.47 | 7.44 | 100.00 |  [28${}^{+29}_{-10}$,119${}^{+1}_{-2}$] | 100.00 |  [29${}^{+28}_{-5}$,119${}^{+1}_{-1}$] | 99.70 |  [44${}^{+13}_{-5}$,96${}^{+2}_{-3}$] | 43.50 |  [62${}^{+6}_{-4}$,82${}^{+2}_{-3}$]
eps Eri b | 3.20 | 3.53 | 0.68 | 100.00 |  [14${}^{+12}_{-5}$,113${}^{+1}_{-2}$] | 100.00 |  [14${}^{+12}_{-6}$,108${}^{+5}_{-5}$] | 100.00 |  [24${}^{+7}_{-3}$,71${}^{+6}_{-4}$] | 0.00 | $-$
alf Ari b | 20.21 | 1.20 | 2.13 | 100.00 |  [31${}^{+28}_{-10}$,109${}^{+1}_{-1}$] | 100.00 |  [31${}^{+28}_{-4}$,110${}^{+1}_{-1}$] | 0.00 | $-$ | 0.00 | $-$
GJ 876 c | 4.68 | 0.13 | 0.71 | 100.00 |  [34${}^{+0}_{-0}$,110${}^{+0}_{-1}$] | 100.00 |  [47${}^{+0}_{-0}$,100${}^{+1}_{-0}$] | 0.00 | $-$ | 0.00 | $-$
GJ 876 b | 4.68 | 0.21 | 2.28 | 100.00 |  [31${}^{+0}_{-0}$,103${}^{+1}_{-1}$] | 100.00 |  [38${}^{+0}_{-0}$,111${}^{+1}_{-1}$] | 0.00 | $-$ | 0.00 | $-$
GJ 896 A b | 6.26 | 0.64 | 2.23 | 67.00 |  [45${}^{+4}_{-3}$,79${}^{+4}_{-9}$] | 100.00 |  [39${}^{+2}_{-4}$,88${}^{+3}_{-3}$] | 0.00 | $-$ | 0.00 | $-$
HD 27442 b | 18.27 | 1.27 | 1.81 | 7.20 |  [56${}^{+2}_{-1}$,63${}^{+2}_{-2}$] | 71.30 |  [46${}^{+3}_{-2}$,66${}^{+4}_{-5}$] | 0.00 | $-$ | 0.00 | $-$
nu Oct A b | 21.15 | 1.25 | 2.34 | 30.60 |  [55${}^{+5}_{-3}$,70${}^{+3}_{-4}$] | 47.10 |  [53${}^{+5}_{-4}$,68${}^{+4}_{-4}$] | 0.00 | $-$ | 0.00 | $-$
61 Vir d | 8.50 | 0.48 | 0.08 | 41.20 |  [55${}^{+15}_{-8}$,84${}^{+16}_{-14}$] | 26.60 |  [62${}^{+12}_{-7}$,86${}^{+17}_{-14}$] | 0.00 | $-$ | 0.00 | $-$
GJ 9066 c | 4.47 | 0.88 | 0.24 | 4.90 |  [55${}^{+4}_{-8}$,67${}^{+5}_{-4}$] | 54.50 |  [51${}^{+17}_{-10}$,87${}^{+3}_{-11}$] | 0.00 | $-$ | 0.00 | $-$
HD 26965 b | 5.04 | 0.22 | 0.03 | 29.70 |  [61${}^{+13}_{-11}$,89${}^{+13}_{-19}$] | 21.80 |  [64${}^{+13}_{-6}$,94${}^{+8}_{-16}$] | 1.20 |  [84${}^{+2}_{-2}$,94${}^{+1}_{-3}$] | 0.00 | $-$
HD 147513 b | 12.90 | 1.32 | 1.39 | 50.30 |  [54${}^{+3}_{-2}$,63${}^{+3}_{-3}$] | 0.00 | $-$ | 0.00 | $-$ | 0.00 | $-$
HD 3651 b | 11.13 | 0.30 | 0.27 | 42.70 |  [86${}^{+4}_{-5}$,100${}^{+2}_{-3}$] | 0.00 | $-$ | 0.00 | $-$ | 0.00 | $-$

We used the Planetary System database of the NASA Exoplanet
Archive555https://exoplanetarchive.ipac.caltech.edu (Akeson et al., 2013) as
our main source of information on the confirmed exoplanets. As of January 25,
2024, it contained a total of 5572 exoplanets. We focused in this study on the
451 known exoplanets within 30 pc.

We applied the methodology of Carrión-González et al. (2021a) and computed
1000 orbital realizations for each of these planets (we refer to Sects. 3 and
4 therein for a more detailed description). Each orbital realization was
discretized into 360 orbital positions, at which we computed the planet-star
angular separation ($\Delta\theta$) and the brightness of the planet in
reflected starlight, which is given by the planet-to-star contrast ratio
($F_{\medbullet}/F_{\star}$). For each orbital position,
$F_{\medbullet}/F_{\star}$ is computed at a given wavelength ($\lambda$) as:

$\frac{F_{\medbullet}}{F_{\star}}=\left(\frac{R_{\medbullet}}{r(t)}\right)^{2}A_{g}(\lambda)\,\Phi(\alpha,\lambda),$
(73)

with $R_{\medbullet}$ being the planetary radius, $r$ the distance between the
planet and the star, and $\alpha$ the corresponding star-planet-observer phase
angle. We assumed for all planets a Lambertian scattering phase law ($\Phi$),
and carried out our analysis for geometric albedo ($A_{g}$) values of 0.3 and
0.1 (see Sect. 4.2).

When a planet lacked a radius measurement, we computed it using either the
mass-radius relationships from Otegi et al. (2020) for rocky and for volatile-
rich exoplanets, or the one from Hatzes & Rauer (2015) for giant planets. In
case of a missing value for the orbital inclination ($i$) or the argument of
periastron of the planet ($\omega_{p}$), we assumed an isotropic distribution
of orbital orientations and drew random values from $\cos(i)\in[-1,1]$ and
$\omega_{p}\in[0,2\pi]$ at each orbital realization. Similarly, we drew values
for the orbital eccentricity ($e$) from $e\in[0,1)$ if it had not been
measured. We assumed that the NASA Exoplanet Archive reports for each planet
the argument of periastron of the host star ($\omega_{\star}$), following the
radial-velocity convention. Sect. 4.1.1 in Carrión-González et al. (2021a)
provides details on the lack of homogeneous criteria to report the planetary
argument of periastron on catalogues, and its impact for exoplanet
detectability studies.

If no value was available for the stellar mass, radius or temperature
($M_{\star}$, $R_{\star}$, $T_{\star}$), we used those in the Planetary
Systems Composite Data database of the NASA Exoplanet Archive. Also, if the
stellar magnitude at bands I, J, H or K was not reported in the NASA Exoplanet
Archive, we obtained it from the SIMBAD database (Wenger et al., 2000).

This affects in particular the I magnitude, which is unavailable for 198 of
the planets within 30 pc. If also unavailable in SIMBAD, we adopted the value
of the J magnitude as an approximation – this however happened for none of the
detectable-exoplanet hosts in Table 3.

With the S/N analytical model from Sect. 2, assuming the VLTI parameters used
to make Fig. 8 with a projection along the best possible orientation, and
adapting in each case for the stellar luminosity, we computed the
detectability of the orbital realizations simulated for each exoplanet. A
planet is considered detectable at a given orbital position of a given orbital
realization if an $\mathrm{S/N}=3$ is achievable in less than 3 h of
integration time. The probability of a planet to be detectable ($P_{\rm
detect}$) is given by the number of detectable orbits – those with at least
one orbital position detectable – divided by the total number of orbits
simulated. As described below, we also explored the number of planets
detectable if the integration time is increased to 30 h.

###  4.2 Detectable exoplanets

We applied our methods to the 451 confirmed exoplanets within 30 pc, and
obtained a list of detectable planets at each spectral band. For each planet,
the value of $A_{g}$ will strongly depend on the structure and composition of
its atmosphere. In the case of the exoplanets under study here, this
information will be a priori completely unknown. For reference, Solar-System
bodies have geometrical albedos at spectral bands I, J, H and K ranging from
about 0.0 to 0.5, significantly impacted by the presence of clouds and by
methane absorption bands in the case of gaseous planets (e.g. Fink & Larson,
1979; Madden & Kaltenegger, 2018). In agreement with values from Solar System
giants, we adopted in Table 3 an albedo of 0.3 at bands I and J, and
$A_{g}=0.1$ for bands H and K.

In order to build a target list that is statistically robust, we only
considered as suitable targets those planets meeting $P_{\rm detect}>25\%$.
Similarly, we discarded any planet with declination out of the interval
$\rm{DEC}\in\\{-90^{\circ},+30^{\circ}\\}$ and thus not observable from
Paranal.

Table 3 contains the list of targets and their detectability conditions, and
Fig. 12 shows the simulated orbits for each planet compared to the sensitivity
curves of the instrument. For integration times of 3 h, we find 8 exoplanets
that are detectable in at least one of the four bands under study with the
current VLTI configuration. For the extended VLTI configuration (ie., the
current VLTI plus UT5), this count rises to 14 detectable planets. We find
that the majority of the detectable targets are long-period planets in the
mass range of Jupiter, although the new UT5 can allow to detect planets in the
Neptune-mass regime (Fig. 10). These targets, and in particular the lower-mass
ones, are detectable mostly at the I and J spectral bands (Table 3), while the
expected low-albedos at H and K bands makes them undetectable at such longer
wavelengths. Extending the integration time to 30 h increases the tally of
detectable planets to 40 with the VLTI and 50 with the extended VLTI (Tables 4
and 5). With the new UTs, low-mass planets down to the Earth-like range (e.g.
Proxima Cen b) become detectable (see Sect. 4.2.1).

We find wide ranges of observable phase angles ($\alpha_{obs}$) for several of
the exoplanets detectable even in 3 h of integration (Table 3), reaching
phases both before and after quadrature ($\alpha=90^{\circ}$). These are prime
targets for atmospheric characterization, as combining the information of
multi-phase observations has been found to significantly improve the
atmospheric retrievals in direct imaging (Damiano et al., 2020; Carrión-
González et al., 2021b). For 30 h of integration, this list of targets
detectable at multiple phase angles increases significantly, including several
low-mass planets (Tables 4 and 5). In addition, several multi-planet systems
have more than one detectable planet offering a good opportunity to study the
architecture of planetary systems, and the evolution of the planets therein
and their atmospheres.

Notably, two promising targets for this instrument in 3 h of integration (HD
62509 b and $\epsilon$ Eri b) are potentially detectable as well with the
Nancy Grace Roman Space Telescope in reflected starlight (Carrión-González et
al., 2021a). Roman will be the first space-based facility capable of directly
imaging cold and temperate exoplanets in reflected starlight. It will be
launched in late 2026 or 2027, with a mission duration of five years and a
potential extension of five more. With the spectral range of Roman’s
coronagraph instrument limited to within 575 and 825 nm (Akeson et al., 2019;
Zellem et al., 2022), this showcases synergies between both facilities for
complementary –and potentially simultaneous– observations in a wider spectral
range.

Figure 11: Detectability conditions for Proxima Cen b with the extended VLTI
configuration. Left panel show the simulated orbits (black lines) assuming
$A_{g}=0.3$, as well as the mean orbital realization (as shown in Figs. 4-6)
in yellow. Colored lines show the detectability limits of the interferometer
at each spectral band with the VLTI (solid lines) and the extended VLIT (Fig.
9, dotted lines) configurations. In this case, we show the detectability
limits for a $\rm{S/N}=3$ in 30 h of integration time, instead of 3 h. The
histograms on the right panel show the range of $\alpha_{obs}$ at each band.

####  4.2.1 Pushing the instrumental limits: the case of Proxima Cen b

The adopted detectability criteria ($\mathrm{S/N}=3$ in less than 3 h of
integration time, $P_{\rm detect}$¿25%) ensure the robustness of the target
list in Table 3. However, tens of additional known exoplanets that are not
included in Table 3 are at the limits of our detectability criteria. These
planets could be considered as potential targets for high-risk high-gain
observations by increasing the integration time beyond 3 h. Tables 4 and 5
showcase the detectable exoplanets ($P_{detect}$¿25%) for an integration time
of 30 h.

Several temperate low-mass exoplanets are among these, including the
habitable-zone super Earths $\tau$ Cet e and f, as well as Proxima Cen b, a
potentially rocky planet at the habitable zone of our nearest stellar
neighbour. We find that for a detection threshold of $\mathrm{S/N}=3$, 30 h of
integration would be enough to detect Proxima Cen b with the upgraded VLTI
configuration at bands I and J (Fig. 11). The a priori unknown atmospheric
properties of the planet will also modify the detectability, and we note that
if its albedo is for instance similar to that of Venus (0.45 at band I, 0.4 at
band J, Madden & Kaltenegger, 2018) the prospects for detecting the planet
will be more favourable than reported here.

##  5 Conclusion and perspectives

In this paper, we have demonstrated that:

  1. 1.

Dual-field interferometry shows significant potential for detecting exoplanets
in reflected light, positioning it as a competitive technique alongside cross-
correlation methods used with the ELT or space-based coronagraphy.

  2. 2.

The primary limitation of this technique is shot noise from stellar light,
which can be effectively mitigated through the combined use of phase
apodization and a high-performance AO system.

  3. 3.

Sensitivity is enhanced at shorter wavelengths, provided the AO system
performs well (Strehl ratio above 50%). Furthermore, planets are expected to
exhibit higher geometrical albedo at these wavelengths, further improving
detectability.

  4. 4.

The inner working angle is constrained by both the telescope diameter and the
baseline length, with optimal performance achieved when the projected baseline
length is $\geq 10$ times the telescope diameter.

The proposed addition of a new Unit Telescope to the South of the VLTI
platform would greatly enhance the VLTI’s efficiency. This upgrade would not
only increase the number of baselines ¿ 100 meters by five but it would also
double the angular resolution and expand the UV coverage to include the North-
West direction. Such improvements would significantly boost the VLTI’s
capability, offering the best opportunity to detect signals from temperate,
Earth-mass exoplanets like Proxima Centauri b in reflected light.

As of now, no direct-imaging observations of exoplanets in reflected starlight
have been performed, as they require sensitivity levels out of reach for
current facilities. The proposed extension of the VLTI will join a handful of
other missions and concepts capable of reaching this sensitivity in the near-
and mid-term future, such as the PCS instrument envisioned for the ELT (Kasper
et al., 2021), the Roman Space Telescope’s Coronagraph Instrument technology
demonstrator (Bailey et al., 2023), and the future Habitable Worlds
Observatory. This will enable the characterization of the poorly-studied
population of cold and temperate long-period exoplanets, including in the case
of an extended VLTI several low-mass temperate targets.

Additionally, the increased resolution to the VLTI would also help numerous
other science cases that are fundamental to the astronomical community. Thanks
to a better astrometry, it would be possible to detect exomoons, to measure
the masses of galactic stellar black holes (Gaia Collaboration et al., 2024),
as well as high redshit super-massive black holes (Abuter et al., 2024). The
extended UV coverage would also bring increased imaging capabilities to the
UTs, that could be fundamental to image faint objects, like protoplanets (Wang
et al., 2021) and AGN’s nuclei (GRAVITY Collaboration et al., 2020b).

###### Acknowledgements.

This research made use of HCIPy, an open-source object-oriented framework
written in Python for performing end-to-end simulations of high-contrast
imaging instruments (Por et al., 2018). SL acknowledges the support of the
French ”Agence Nationale de la Recherche” (ANR), under grants ANR-21-CE31-0017
(project ExoVLTI) and ANR-22-EXOR-0005 (PEPR Origins). The authors thank Dr.
A. Kellerer and Dr. G. Bourdarot for their thorough proofreading and extensive
review of the mathematical formulations.

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##  Appendix A Simulated orbits and detectability of known exoplanets

Figure 12: Detectability conditions for the targets with $P_{detect}$¿25% in 3
h of integration. Left panels show the simulated orbits of each planet (black
lines) assuming $A_{g}$=0.3. Colored lines show the detectability limits of
the interferometer at each spectral band with the current VLTI (solid lines)
and the extended VLTI (dotted lines) baselines, for a $S/N$=3 in 3 h of
integration. The histograms on the right panels show the range of
$\alpha_{obs}$ at each band with the extended VLTI configuration.

##  Appendix B Detactable targets in 30 h of integration time

Table 4: List of detectable targets as in Table 3, but for an integration time
of 30 h with the current VLTI configuration.

|  |  |  |  | I | J | H | K
---|---|---|---|---|---|---|---|---
| Planet | $d$ | $a$ | $M_{p}$ | $P_{detect}$ | $\alpha_{obs}$ | $P_{detect}$ | $\alpha_{obs}$ | $P_{detect}$ | $\alpha_{obs}$ | $P_{detect}$ | $\alpha_{obs}$
|  | [pc] | [AU] |  [$M_{J}$] | [%] | [deg] | [%] | [deg] | [%] | [deg] | [%] | [deg]
VLTI | HD 62509 b | 10.34 | 1.62 | 2.74 | 100.00 |  [31${}^{+26}_{-20}$,140${}^{+1}_{-17}$] | 100.00 |  [31${}^{+26}_{-17}$,137${}^{+1}_{-14}$] | 100.00 |  [31${}^{+26}_{-11}$,123${}^{+1}_{-1}$] | 100.00 |  [31${}^{+26}_{-5}$,115${}^{+1}_{-1}$]
eps Eri b | 3.20 | 3.53 | 0.68 | 100.00 |  [14${}^{+12}_{-9}$,130${}^{+2}_{-2}$] | 100.00 |  [14${}^{+12}_{-9}$,128${}^{+2}_{-9}$] | 100.00 |  [14${}^{+12}_{-5}$,92${}^{+16}_{-3}$] | 97.30 |  [16${}^{+9}_{-2}$,47${}^{+5}_{-11}$]
alf Tau b | 20.44 | 1.47 | 7.44 | 100.00 |  [28${}^{+29}_{-8}$,133${}^{+1}_{-10}$] | 100.00 |  [34${}^{+24}_{-6}$,130${}^{+1}_{-8}$] | 100.00 |  [47${}^{+11}_{-7}$,112${}^{+2}_{-4}$] | 62.70 |  [64${}^{+10}_{-7}$,96${}^{+2}_{-3}$]
alf Ari b | 20.21 | 1.20 | 2.13 | 100.00 |  [31${}^{+28}_{-11}$,124${}^{+1}_{-3}$] | 100.00 |  [31${}^{+28}_{-2}$,117${}^{+1}_{-1}$] | 99.50 |  [50${}^{+9}_{-2}$,84${}^{+2}_{-2}$] | 0.00 | $-$
GJ 896 A b | 6.26 | 0.64 | 2.23 | 100.00 |  [24${}^{+14}_{-1}$,111${}^{+1}_{-1}$] | 100.00 |  [26${}^{+12}_{-3}$,113${}^{+1}_{-1}$] | 87.10 |  [54${}^{+3}_{-3}$,71${}^{+4}_{-5}$] | 0.00 | $-$
GJ 9066 c | 4.47 | 0.88 | 0.24 | 98.30 |  [38${}^{+22}_{-9}$,99${}^{+7}_{-10}$] | 100.00 |  [35${}^{+24}_{-8}$,106${}^{+8}_{-8}$] | 40.70 |  [57${}^{+12}_{-10}$,84${}^{+2}_{-4}$] | 0.00 | $-$
nu Oct A b | 21.15 | 1.25 | 2.34 | 100.00 |  [34${}^{+22}_{-3}$,106${}^{+2}_{-2}$] | 100.00 |  [43${}^{+14}_{-4}$,101${}^{+2}_{-3}$] | 0.00 | $-$ | 0.00 | $-$
bet Pic c | 19.74 | 2.72 | 8.20 | 100.00 |  [30${}^{+2}_{-2}$,98${}^{+3}_{-3}$] | 100.00 |  [43${}^{+4}_{-3}$,84${}^{+5}_{-6}$] | 0.00 | $-$ | 0.00 | $-$
HD 27442 b | 18.27 | 1.27 | 1.81 | 100.00 |  [31${}^{+25}_{-1}$,106${}^{+1}_{-1}$] | 100.00 |  [36${}^{+19}_{-1}$,105${}^{+1}_{-1}$] | 0.00 | $-$ | 0.00 | $-$
HD 160691 b | 15.60 | 1.52 | 4.30 | 100.00 |  [30${}^{+1}_{-0}$,99${}^{+2}_{-2}$] | 100.00 |  [40${}^{+2}_{-1}$,88${}^{+2}_{-5}$] | 0.00 | $-$ | 0.00 | $-$
HD 147513 b | 12.90 | 1.32 | 1.39 | 100.00 |  [29${}^{+28}_{-3}$,106${}^{+1}_{-1}$] | 100.00 |  [38${}^{+19}_{-1}$,100${}^{+1}_{-1}$] | 0.00 | $-$ | 0.00 | $-$
GJ 876 c | 4.68 | 0.13 | 0.71 | 100.00 |  [40${}^{+0}_{-0}$,114${}^{+0}_{-0}$] | 100.00 |  [68${}^{+1}_{-1}$,81${}^{+1}_{-1}$] | 0.00 | $-$ | 0.00 | $-$
GJ 876 b | 4.68 | 0.21 | 2.28 | 100.00 |  [31${}^{+0}_{-0}$,120${}^{+0}_{-0}$] | 100.00 |  [45${}^{+0}_{-0}$,117${}^{+0}_{-0}$] | 0.00 | $-$ | 0.00 | $-$
GJ 832 b | 4.96 | 3.70 | 0.98 | 100.00 |  [39${}^{+2}_{-2}$,82${}^{+2}_{-2}$] | 100.00 |  [39${}^{+2}_{-2}$,93${}^{+1}_{-3}$] | 0.00 | $-$ | 0.00 | $-$
7 CMa c | 19.81 | 2.12 | 0.87 | 100.00 |  [25${}^{+0}_{-0}$,106${}^{+1}_{-1}$] | 100.00 |  [30${}^{+1}_{-1}$,105${}^{+1}_{-1}$] | 0.00 | $-$ | 0.00 | $-$
7 CMa b | 19.80 | 1.77 | 1.85 | 100.00 |  [26${}^{+1}_{-1}$,105${}^{+1}_{-1}$] | 100.00 |  [32${}^{+1}_{-1}$,104${}^{+1}_{-1}$] | 0.00 | $-$ | 0.00 | $-$
HD 160691 e | 15.60 | 0.93 | 7.00 | 99.70 |  [38${}^{+13}_{-3}$,97${}^{+2}_{-2}$] | 93.90 |  [54${}^{+10}_{-2}$,86${}^{+2}_{-2}$] | 0.00 | $-$ | 0.00 | $-$
tau Cet e | 3.60 | 0.54 | 0.01 | 94.70 |  [41${}^{+18}_{-9}$,95${}^{+13}_{-17}$] | 87.00 |  [44${}^{+16}_{-8}$,94${}^{+12}_{-13}$] | 8.00 |  [74${}^{+8}_{-7}$,94${}^{+6}_{-15}$] | 2.80 |  [82${}^{+3}_{-4}$,93${}^{+4}_{-9}$]
HD 114613 b | 20.28 | 5.33 | 0.41 | 87.20 |  [37${}^{+12}_{-2}$,79${}^{+9}_{-12}$] | 96.90 |  [45${}^{+11}_{-2}$,77${}^{+3}_{-5}$] | 0.00 | $-$ | 0.00 | $-$
61 Vir d | 8.50 | 0.48 | 0.08 | 98.40 |  [46${}^{+13}_{-6}$,98${}^{+10}_{-12}$] | 78.10 |  [59${}^{+9}_{-6}$,96${}^{+11}_{-12}$] | 3.50 |  [83${}^{+4}_{-4}$,100${}^{+1}_{-4}$] | 0.00 | $-$
HD 190360 b | 16.01 | 3.90 | 1.80 | 100.00 |  [34${}^{+1}_{-1}$,76${}^{+2}_{-3}$] | 56.30 |  [61${}^{+4}_{-8}$,75${}^{+4}_{-11}$] | 0.00 | $-$ | 0.00 | $-$
HD 26965 b | 5.04 | 0.22 | 0.03 | 97.40 |  [48${}^{+12}_{-7}$,96${}^{+11}_{-11}$] | 45.20 |  [62${}^{+9}_{-5}$,91${}^{+13}_{-13}$] | 0.70 |  [86${}^{+3}_{-1}$,93${}^{+1}_{-3}$] | 0.00 | $-$
GJ 649 b | 10.38 | 1.11 | 0.30 | 62.20 |  [45${}^{+15}_{-5}$,82${}^{+3}_{-4}$] | 72.20 |  [45${}^{+15}_{-6}$,87${}^{+2}_{-5}$] | 0.00 | $-$ | 0.00 | $-$
55 Cnc f | 12.59 | 0.77 | 0.17 | 88.00 |  [47${}^{+12}_{-7}$,92${}^{+14}_{-12}$] | 34.00 |  [60${}^{+8}_{-6}$,91${}^{+9}_{-13}$] | 0.00 | $-$ | 0.00 | $-$
GJ 887 c | 3.29 | 0.12 | 0.03 | 76.90 |  [53${}^{+14}_{-9}$,93${}^{+11}_{-12}$] | 42.60 |  [63${}^{+15}_{-11}$,90${}^{+12}_{-13}$] | 1.70 |  [83${}^{+8}_{-6}$,90${}^{+5}_{-7}$] | 0.10 |  [89${}^{+0}_{-0}$,89${}^{+0}_{-0}$]
HD 10647 b | 17.34 | 2.02 | 1.11 | 99.00 |  [37${}^{+22}_{-1}$,88${}^{+1}_{-2}$] | 19.70 |  [58${}^{+3}_{-3}$,66${}^{+2}_{-3}$] | 0.00 | $-$ | 0.00 | $-$
HD 192310 c | 8.80 | 1.18 | 0.09 | 61.30 |  [51${}^{+14}_{-10}$,90${}^{+14}_{-14}$] | 51.00 |  [53${}^{+14}_{-10}$,90${}^{+15}_{-14}$] | 5.00 |  [77${}^{+4}_{-6}$,92${}^{+2}_{-11}$] | 0.00 | $-$
HD 39091 b | 18.27 | 3.29 | 12.16 | 99.20 |  [49${}^{+3}_{-2}$,75${}^{+3}_{-4}$] | 0.00 | $-$ | 0.00 | $-$ | 0.00 | $-$
HR 810 b | 17.32 | 0.92 | 2.69 | 99.10 |  [43${}^{+14}_{-2}$,91${}^{+2}_{-2}$] | 0.00 | $-$ | 0.00 | $-$ | 0.00 | $-$
70 Vir b | 17.90 | 0.48 | 8.70 | 98.90 |  [54${}^{+8}_{-2}$,94${}^{+1}_{-2}$] | 0.00 | $-$ | 0.00 | $-$ | 0.00 | $-$
HD 19994 b | 22.52 | 1.31 | 1.58 | 93.10 |  [52${}^{+3}_{-2}$,77${}^{+2}_{-3}$] | 0.00 | $-$ | 0.00 | $-$ | 0.00 | $-$
HD 140901 c | 15.25 | 12.61 | 1.79 | 35.20 |  [43${}^{+10}_{-4}$,62${}^{+11}_{-12}$] | 56.40 |  [57${}^{+5}_{-5}$,65${}^{+5}_{-6}$] | 0.00 | $-$ | 0.00 | $-$
tau Cet h | 3.60 | 0.24 | 0.01 | 60.30 |  [55${}^{+8}_{-6}$,78${}^{+21}_{-8}$] | 19.10 |  [64${}^{+11}_{-8}$,93${}^{+9}_{-14}$] | 1.40 |  [83${}^{+2}_{-1}$,92${}^{+3}_{-5}$] | 0.20 |  [88${}^{+0}_{-0}$,92${}^{+0}_{-0}$]
GJ 849 b | 8.80 | 2.31 | 1.03 | 0.00 | $-$ | 75.70 |  [38${}^{+1}_{-1}$,51${}^{+4}_{-4}$] | 0.00 | $-$ | 0.00 | $-$
HD 60532 c | 25.98 | 1.60 | 2.91 | 70.10 |  [55${}^{+3}_{-2}$,69${}^{+3}_{-3}$] | 0.00 | $-$ | 0.00 | $-$ | 0.00 | $-$
HD 160691 c | 15.60 | 5.10 | 4.40 | 34.20 |  [47${}^{+9}_{-15}$,97${}^{+3}_{-19}$] | 35.90 |  [46${}^{+14}_{-5}$,85${}^{+4}_{-7}$] | 0.00 | $-$ | 0.00 | $-$
GJ 229 b | 5.76 | 1.09 | 0.05 | 30.10 |  [59${}^{+13}_{-10}$,82${}^{+14}_{-13}$] | 32.70 |  [57${}^{+13}_{-9}$,86${}^{+15}_{-15}$] | 4.20 |  [77${}^{+3}_{-5}$,90${}^{+8}_{-10}$] | 0.00 | $-$
HD 102365 b | 9.29 | 0.46 | 0.06 | 42.60 |  [56${}^{+15}_{-9}$,89${}^{+13}_{-13}$] | 17.30 |  [67${}^{+10}_{-8}$,88${}^{+7}_{-13}$] | 0.00 | $-$ | 0.00 | $-$
HD 3651 b | 11.13 | 0.30 | 0.27 | 52.80 |  [92${}^{+4}_{-3}$,110${}^{+4}_{-7}$] | 1.10 |  [94${}^{+4}_{-2}$,96${}^{+3}_{-3}$] | 0.00 | $-$ | 0.00 | $-$
| 61 Vir c | 8.50 | 0.22 | 0.07 | 40.00 |  [67${}^{+6}_{-5}$,92${}^{+12}_{-12}$] | 0.00 | $-$ | 0.00 | $-$ | 0.00 | $-$

Table 5: List of detectable targets as in Table 3, but for an integration time of 30 h with the extended VLTI configuration. |  |  |  |  | I | J | H | K
---|---|---|---|---|---|---|---|---
| Planet | $d$ | $a$ | $M_{p}$ | $P_{detect}$ | $\alpha_{obs}$ | $P_{detect}$ | $\alpha_{obs}$ | $P_{detect}$ | $\alpha_{obs}$ | $P_{detect}$ | $\alpha_{obs}$
|  | [pc] | [AU] |  [$M_{J}$] | [%] | [deg] | [%] | [deg] | [%] | [deg] | [%] | [deg]
Extended VLTI | alf Tau b | 20.44 | 1.47 | 7.44 | 100.00 |  [28${}^{+29}_{-13}$,134${}^{+1}_{-12}$] | 100.00 |  [28${}^{+29}_{-8}$,134${}^{+1}_{-11}$] | 100.00 |  [36${}^{+22}_{-5}$,120${}^{+1}_{-2}$] | 100.00 |  [49${}^{+10}_{-7}$,111${}^{+2}_{-4}$]
HD 62509 b | 10.34 | 1.62 | 2.74 | 100.00 |  [31${}^{+26}_{-21}$,140${}^{+1}_{-17}$] | 100.00 |  [31${}^{+26}_{-20}$,138${}^{+1}_{-15}$] | 100.00 |  [31${}^{+26}_{-15}$,124${}^{+1}_{-2}$] | 100.00 |  [31${}^{+26}_{-10}$,118${}^{+1}_{-2}$]
eps Eri b | 3.20 | 3.53 | 0.68 | 100.00 |  [14${}^{+12}_{-9}$,126${}^{+5}_{-2}$] | 100.00 |  [14${}^{+12}_{-9}$,126${}^{+2}_{-2}$] | 100.00 |  [14${}^{+12}_{-5}$,100${}^{+4}_{-2}$] | 99.00 |  [17${}^{+9}_{-2}$,49${}^{+5}_{-13}$]
alf Ari b | 20.21 | 1.20 | 2.13 | 100.00 |  [31${}^{+28}_{-16}$,128${}^{+1}_{-7}$] | 100.00 |  [31${}^{+28}_{-10}$,126${}^{+1}_{-5}$] | 100.00 |  [36${}^{+23}_{-1}$,104${}^{+1}_{-2}$] | 52.90 |  [62${}^{+5}_{-4}$,75${}^{+3}_{-4}$]
GJ 896 A b | 6.26 | 0.64 | 2.23 | 100.00 |  [21${}^{+16}_{-3}$,113${}^{+1}_{-2}$] | 100.00 |  [22${}^{+16}_{-4}$,116${}^{+2}_{-1}$] | 100.00 |  [39${}^{+4}_{-2}$,86${}^{+5}_{-3}$] | 0.00 | $-$
GJ 876 b | 4.68 | 0.21 | 2.28 | 100.00 |  [31${}^{+0}_{-0}$,125${}^{+1}_{-1}$] | 100.00 |  [31${}^{+0}_{-0}$,126${}^{+0}_{-0}$] | 100.00 |  [52${}^{+0}_{-0}$,101${}^{+1}_{-1}$] | 0.00 | $-$
HD 27442 b | 18.27 | 1.27 | 1.81 | 100.00 |  [30${}^{+26}_{-8}$,108${}^{+1}_{-1}$] | 100.00 |  [30${}^{+26}_{-3}$,109${}^{+1}_{-1}$] | 85.00 |  [55${}^{+3}_{-2}$,72${}^{+3}_{-4}$] | 0.00 | $-$
GJ 9066 c | 4.47 | 0.88 | 0.24 | 97.80 |  [37${}^{+24}_{-10}$,99${}^{+7}_{-11}$] | 100.00 |  [33${}^{+27}_{-11}$,106${}^{+7}_{-8}$] | 52.20 |  [53${}^{+15}_{-12}$,88${}^{+2}_{-11}$] | 0.00 | $-$
61 Vir d | 8.50 | 0.48 | 0.08 | 100.00 |  [36${}^{+22}_{-5}$,103${}^{+8}_{-9}$] | 100.00 |  [46${}^{+12}_{-5}$,101${}^{+10}_{-9}$] | 13.00 |  [74${}^{+4}_{-6}$,95${}^{+7}_{-10}$] | 0.00 | $-$
HD 26965 b | 5.04 | 0.22 | 0.03 | 100.00 |  [35${}^{+25}_{-4}$,101${}^{+10}_{-9}$] | 100.00 |  [48${}^{+12}_{-5}$,100${}^{+9}_{-9}$] | 8.20 |  [76${}^{+7}_{-6}$,95${}^{+4}_{-11}$] | 0.10 |  [87${}^{+0}_{-0}$,89${}^{+0}_{-0}$]
nu Oct A b | 21.15 | 1.25 | 2.34 | 100.00 |  [30${}^{+26}_{-7}$,109${}^{+2}_{-2}$] | 100.00 |  [34${}^{+23}_{-4}$,109${}^{+2}_{-2}$] | 1.90 |  [61${}^{+1}_{-2}$,68${}^{+2}_{-2}$] | 0.00 | $-$
bet Pic c | 19.74 | 2.72 | 8.20 | 100.00 |  [25${}^{+3}_{-3}$,97${}^{+3}_{-3}$] | 100.00 |  [33${}^{+5}_{-2}$,87${}^{+4}_{-5}$] | 0.00 | $-$ | 0.00 | $-$
HD 160691 b | 15.60 | 1.52 | 4.30 | 100.00 |  [30${}^{+0}_{-0}$,103${}^{+4}_{-6}$] | 100.00 |  [32${}^{+3}_{-1}$,94${}^{+4}_{-3}$] | 0.00 | $-$ | 0.00 | $-$
HD 147513 b | 12.90 | 1.32 | 1.39 | 100.00 |  [29${}^{+28}_{-10}$,106${}^{+1}_{-1}$] | 100.00 |  [29${}^{+28}_{-2}$,102${}^{+1}_{-1}$] | 0.00 | $-$ | 0.00 | $-$
GJ 876 c | 4.68 | 0.13 | 0.71 | 100.00 |  [31${}^{+0}_{-0}$,124${}^{+0}_{-0}$] | 100.00 |  [39${}^{+0}_{-0}$,113${}^{+0}_{-0}$] | 0.00 | $-$ | 0.00 | $-$
GJ 832 b | 4.96 | 3.70 | 0.98 | 100.00 |  [39${}^{+2}_{-2}$,70${}^{+8}_{-4}$] | 100.00 |  [39${}^{+2}_{-2}$,89${}^{+2}_{-2}$] | 0.00 | $-$ | 0.00 | $-$
7 CMa c | 19.81 | 2.12 | 0.87 | 100.00 |  [20${}^{+1}_{-1}$,106${}^{+1}_{-1}$] | 100.00 |  [22${}^{+0}_{-0}$,107${}^{+1}_{-1}$] | 0.00 | $-$ | 0.00 | $-$
7 CMa b | 19.80 | 1.77 | 1.85 | 100.00 |  [20${}^{+1}_{-1}$,105${}^{+1}_{-1}$] | 100.00 |  [24${}^{+0}_{-0}$,107${}^{+1}_{-1}$] | 0.00 | $-$ | 0.00 | $-$
HD 160691 e | 15.60 | 0.93 | 7.00 | 100.00 |  [30${}^{+14}_{-0}$,103${}^{+4}_{-8}$] | 99.70 |  [40${}^{+15}_{-4}$,92${}^{+6}_{-2}$] | 0.00 | $-$ | 0.00 | $-$
70 Vir b | 17.90 | 0.48 | 8.70 | 100.00 |  [41${}^{+16}_{-2}$,105${}^{+1}_{-2}$] | 97.00 |  [63${}^{+7}_{-3}$,92${}^{+2}_{-2}$] | 0.00 | $-$ | 0.00 | $-$
HR 810 b | 17.32 | 0.92 | 2.69 | 99.60 |  [32${}^{+25}_{-3}$,96${}^{+1}_{-1}$] | 96.90 |  [46${}^{+10}_{-3}$,86${}^{+2}_{-2}$] | 0.00 | $-$ | 0.00 | $-$
tau Cet e | 3.60 | 0.54 | 0.01 | 94.40 |  [42${}^{+18}_{-10}$,94${}^{+13}_{-17}$] | 88.00 |  [42${}^{+18}_{-9}$,94${}^{+12}_{-14}$] | 9.70 |  [72${}^{+9}_{-8}$,93${}^{+7}_{-16}$] | 3.80 |  [80${}^{+4}_{-7}$,95${}^{+3}_{-7}$]
GJ 887 c | 3.29 | 0.12 | 0.03 | 94.10 |  [43${}^{+20}_{-9}$,101${}^{+10}_{-10}$] | 87.00 |  [53${}^{+17}_{-10}$,99${}^{+9}_{-12}$] | 8.60 |  [76${}^{+12}_{-11}$,90${}^{+8}_{-13}$] | 0.70 |  [85${}^{+4}_{-2}$,88${}^{+5}_{-1}$]
HD 10647 b | 17.34 | 2.02 | 1.11 | 98.50 |  [34${}^{+25}_{-2}$,88${}^{+2}_{-2}$] | 86.80 |  [48${}^{+4}_{-2}$,74${}^{+3}_{-3}$] | 0.00 | $-$ | 0.00 | $-$
HD 19994 b | 22.52 | 1.31 | 1.58 | 97.20 |  [37${}^{+18}_{-2}$,84${}^{+2}_{-2}$] | 86.80 |  [51${}^{+3}_{-2}$,71${}^{+3}_{-4}$] | 0.00 | $-$ | 0.00 | $-$
tau Cet h | 3.60 | 0.24 | 0.01 | 95.80 |  [47${}^{+12}_{-6}$,82${}^{+13}_{-8}$] | 83.40 |  [51${}^{+10}_{-5}$,79${}^{+16}_{-9}$] | 3.50 |  [79${}^{+3}_{-5}$,94${}^{+4}_{-14}$] | 0.80 |  [85${}^{+2}_{-2}$,94${}^{+3}_{-5}$]
HD 114613 b | 20.28 | 5.33 | 0.41 | 86.70 |  [37${}^{+11}_{-2}$,79${}^{+9}_{-12}$] | 96.00 |  [42${}^{+14}_{-3}$,76${}^{+4}_{-6}$] | 0.00 | $-$ | 0.00 | $-$
55 Cnc f | 12.59 | 0.77 | 0.17 | 96.50 |  [39${}^{+18}_{-9}$,95${}^{+12}_{-13}$] | 75.60 |  [47${}^{+12}_{-6}$,86${}^{+17}_{-15}$] | 0.00 | $-$ | 0.00 | $-$
GJ 649 b | 10.38 | 1.11 | 0.30 | 61.00 |  [43${}^{+16}_{-6}$,81${}^{+3}_{-4}$] | 79.80 |  [43${}^{+15}_{-9}$,91${}^{+2}_{-9}$] | 0.00 | $-$ | 0.00 | $-$
HD 190360 b | 16.01 | 3.90 | 1.80 | 100.00 |  [34${}^{+1}_{-1}$,77${}^{+2}_{-3}$] | 36.40 |  [62${}^{+4}_{-8}$,74${}^{+4}_{-11}$] | 0.00 | $-$ | 0.00 | $-$
tau Cet g | 3.60 | 0.13 | 0.01 | 100.00 |  [45${}^{+13}_{-3}$,86${}^{+9}_{-6}$] | 33.00 |  [65${}^{+7}_{-5}$,85${}^{+15}_{-8}$] | 0.80 |  [85${}^{+1}_{-0}$,93${}^{+1}_{-1}$] | 0.10 |  [90${}^{+0}_{-0}$,91${}^{+0}_{-0}$]
Proxima Cen b | 1.30 | 0.05 | 0.00 | 27.80 |  [59${}^{+14}_{-6}$,77${}^{+15}_{-10}$] | 99.30 |  [61${}^{+5}_{-5}$,85${}^{+9}_{-5}$] | 0.70 |  [85${}^{+2}_{-1}$,93${}^{+1}_{-4}$] | 0.00 | $-$
HD 3651 b | 11.13 | 0.30 | 0.27 | 100.00 |  [71${}^{+6}_{-5}$,116${}^{+5}_{-7}$] | 26.10 |  [98${}^{+5}_{-4}$,112${}^{+3}_{-6}$] | 0.00 | $-$ | 0.00 | $-$
61 Vir c | 8.50 | 0.22 | 0.07 | 99.30 |  [50${}^{+10}_{-5}$,101${}^{+9}_{-10}$] | 19.90 |  [73${}^{+5}_{-5}$,98${}^{+6}_{-8}$] | 0.00 | $-$ | 0.00 | $-$
HD 192310 c | 8.80 | 1.18 | 0.09 | 59.30 |  [52${}^{+14}_{-10}$,90${}^{+14}_{-14}$] | 53.20 |  [51${}^{+16}_{-11}$,91${}^{+15}_{-15}$] | 6.10 |  [76${}^{+5}_{-5}$,97${}^{+2}_{-16}$] | 0.00 | $-$
HD 102365 b | 9.29 | 0.46 | 0.06 | 60.80 |  [51${}^{+16}_{-11}$,92${}^{+13}_{-15}$] | 40.80 |  [56${}^{+15}_{-10}$,89${}^{+13}_{-13}$] | 3.40 |  [79${}^{+4}_{-4}$,88${}^{+5}_{-8}$] | 0.00 | $-$
HD 39091 b | 18.27 | 3.29 | 12.16 | 100.00 |  [42${}^{+2}_{-1}$,82${}^{+2}_{-2}$] | 0.00 | $-$ | 0.00 | $-$ | 0.00 | $-$
HD 60532 b | 25.98 | 0.77 | 1.21 | 99.90 |  [43${}^{+15}_{-1}$,89${}^{+2}_{-2}$] | 0.00 | $-$ | 0.00 | $-$ | 0.00 | $-$
HD 60532 c | 25.98 | 1.60 | 2.91 | 95.60 |  [46${}^{+6}_{-2}$,78${}^{+2}_{-3}$] | 0.00 | $-$ | 0.00 | $-$ | 0.00 | $-$
GJ 3021 b | 17.56 | 0.49 | 3.95 | 84.80 |  [73${}^{+5}_{-3}$,93${}^{+1}_{-2}$] | 0.00 | $-$ | 0.00 | $-$ | 0.00 | $-$
HD 140901 c | 15.25 | 12.61 | 1.79 | 34.50 |  [43${}^{+10}_{-4}$,62${}^{+11}_{-12}$] | 40.40 |  [57${}^{+5}_{-10}$,64${}^{+5}_{-5}$] | 0.00 | $-$ | 0.00 | $-$
GJ 849 b | 8.80 | 2.31 | 1.03 | 0.00 | $-$ | 74.70 |  [39${}^{+1}_{-1}$,50${}^{+3}_{-4}$] | 0.00 | $-$ | 0.00 | $-$
HD 160691 c | 15.60 | 5.10 | 4.40 | 36.50 |  [48${}^{+9}_{-15}$,97${}^{+8}_{-20}$] | 36.00 |  [45${}^{+11}_{-10}$,85${}^{+11}_{-7}$] | 0.00 | $-$ | 0.00 | $-$
GJ 229 b | 5.76 | 1.09 | 0.05 | 28.60 |  [60${}^{+13}_{-11}$,82${}^{+14}_{-13}$] | 37.30 |  [54${}^{+16}_{-9}$,83${}^{+18}_{-15}$] | 5.30 |  [76${}^{+4}_{-6}$,89${}^{+8}_{-10}$] | 0.50 |  [79${}^{+2}_{-2}$,81${}^{+5}_{-2}$]
55 Cnc c | 12.59 | 0.24 | 0.18 | 70.70 |  [62${}^{+8}_{-5}$,96${}^{+10}_{-8}$] | 0.00 | $-$ | 0.00 | $-$ | 0.00 | $-$
HD 192310 b | 8.80 | 0.32 | 0.05 | 43.10 |  [48${}^{+9}_{-6}$,78${}^{+8}_{-8}$] | 7.30 |  [65${}^{+3}_{-3}$,77${}^{+4}_{-4}$] | 0.00 | $-$ | 0.00 | $-$
HD 20794 e | 6.00 | 0.51 | 0.02 | 27.10 |  [59${}^{+12}_{-11}$,87${}^{+13}_{-14}$] | 20.60 |  [63${}^{+11}_{-10}$,87${}^{+13}_{-16}$] | 1.30 |  [82${}^{+3}_{-4}$,93${}^{+2}_{-6}$] | 0.50 |  [86${}^{+1}_{-0}$,91${}^{+4}_{-1}$]
HD 180617 b | 5.91 | 0.34 | 0.05 | 16.50 |  [67${}^{+10}_{-9}$,89${}^{+11}_{-14}$] | 27.30 |  [60${}^{+14}_{-8}$,86${}^{+14}_{-14}$] | 1.90 |  [82${}^{+4}_{-4}$,94${}^{+3}_{-9}$] | 0.00 | $-$
HD 210277 b | 21.30 | 1.13 | 1.49 | 26.30 |  [59${}^{+5}_{-4}$,70${}^{+2}_{-3}$] | 0.00 | $-$ | 0.00 | $-$ | 0.00 | $-$
| GJ 887 b | 3.29 | 0.07 | 0.01 | 25.20 |  [66${}^{+10}_{-5}$,96${}^{+7}_{-11}$] | 0.80 |  [85${}^{+2}_{-2}$,93${}^{+1}_{-2}$] | 0.00 | $-$ | 0.00 | $-$