11institutetext: Department of Particle and Astrophysics, Graduate School of
Science, Nagoya University Furocho, Chikusa-ku, Nagoya, Aichi 466-8601, Japan
11email<EMAIL_ADDRESS>22institutetext: ETH Zurich, Institute
for Particle Physics & Astrophysics, Wolfgang-Pauli-Str. 27, 8093 Zurich,
Switzerland 33institutetext: Institute of Astronomy, KU Leuven,
Celestijnenlaan 200D, 3001, Leuven, Belgium
44institutetext: Department of astronomy, Faculty of mathematics, University
of Belgrade Studentski trg 16, Belgrade 11000 Serbia

# Large Interferometer For Exoplanets (LIFE):

XI. Phase-space synthesis decomposition for planet detection and
characterization

Taro Matsuo Felix Dannert Romain Laugier Sascha P. Quanz Andjelka B. Kovačević
1122332244 LIFE collaboration

(Received ; accepted )

###### Abstract

Context. A mid-infrared nulling-space interferometer is a promising way to
characterize thermal light from habitable planet candidates around Sun-like
stars. However, one of the main challenges for achieving this ambitious goal
is a high-precision stability of the optical path difference (OPD) and
amplitude over a few days for planet detection and up to a few weeks for in-
depth characterization (depending on mission parameters such as aperture size,
number of apertures and total instrument throughput).

Aims. Here we propose a new method called phase-space synthesis decomposition
(PSSD) to shorten the stability requirement to minutes, significantly relaxing
the technological challenges of the mission.

Methods. Focusing on what exactly modulates the planet signal in the presence
of the stellar leak and systematic error, PSSD prioritizes the modulation of
the signals along the wavelength domain rather than baseline rotation.
Modulation along the wavelength domain allows us to extract source positions
in parallel to the baseline vector for each exposure. The sum of the one-
dimensional data converts into two-dimensional information. Based on the
reconstructed image, we construct a continuous equation and extract the
spectra through the singular value decomposition (SVD) while efficiently
separating them from a long-term systematic stellar leak.

Results. We performed numerical simulations to investigate the feasibility of
PSSD for the Large Interferometer For Exoplanets (LIFE) mission concept. We
confirm that multiple terrestrial planets in the habitable zone around a Sun-
like star at 10 $pc$ can be detected and characterized despite high levels and
long durations of systematic noise. We also find that PSSD is more robust
against a sparse sampling of the array rotation compared to purely rotation-
based signal extraction. Using PSSD as signal extraction method significantly
relaxes the technical requirements on signal stability and further increases
the feasibility of the LIFE mission.

Conclusions.

###### Key Words.:

Methods: data analysis – Techniques: interferometric – Techniques: high
angular resolution – Planets and satellites: terrestrial planets – Planets and
satellites: atmospheres –

##  1 Introduction

Since the Michelson stellar interferometer was mounted on the Hooker telescope
and successfully measured the diameter of Betelgeuse in 1920 (Michelson &
Pease 1921), ground-based interferometry has been widely used for optical,
infrared, and radio astronomy (Beckers et al. 1990; Colavita & Wizinowich
2000; ten Brummelaar et al. 2005; ALMA Partnership et al. 2015). An image of
the sky was also reconstructed based on the Van Cittert-Zernike theorem (Born
& Wolf 1999). The Fourier transform of the filled U-V plane provides a two-
dimensional image by rotating the baseline and changing its length.

Bracewell (1978) introduced the concept of nulling interferometry to search
for exoplanet around nearby stars by introducing a $\pi$ phase shift to one of
the beams of a 2-beam interferometer. When the observed sky consists of a host
star and multiple planets, this concept generates a
$\sin^{2}(\frac{\pi}{\lambda}\@vec{B}\cdot\@vec{\theta})$ fringe pattern that
can null the host star at the centre of the field-of-view and transmit light
from an off-axis point source, such as a planet, where $\lambda$ is the
observing wavelength, $\@vec{B}$ is the baseline vector, and $\@vec{\theta}$
is the position vector on the sky. Rotating the baseline of the interferometer
also modulates the signal of the off-axis source as a function of time, which
can be leveraged for signal extraction purposes. Followed by the proposal of
mid-infrared nulling interferometry, Angel et al. (1986) noticed that the mid-
infrared wavelength range is useful for the characterization of temperate
Earth-like planets. This is because of the relatively low contrast between the
planet and its host star compared to that observed in the visible wavelength
range. In addition, $CH_{4}$ and $O_{3}$ are atmospheric biosignatures that
have strong absorption bands in the same wavelength range (e.g., Des Marais et
al. 2002; Fujii et al. 2018). A combination of these two studies led to the
construction of a concept for remotely measuring the activity of primitive
life on distant planets through detecting a variety of $CH_{4}$ and $O_{3}$,
named $Darwin$ (Léger et al. 1996). Darwin was an ESA-led concept and similar
parallel activities were going on on the US side in the context of Terrestrail
Planet Finder-Interferometer (TPF-I; Lawson et al. 2008).

Angel & Woolf (1997) used a cross-correlation technique to efficiently find
the modulated signal while rotating the baseline, leveraging Bracewell’s idea.
A nuller consisting of four apertures was also introduced to obtain a fourth-
order null of the host star. Furthermore, Mennesson & Mariotti (1997) proposed
five collectors to suppress modulation of the exozodiacal light during
baseline rotation, keeping the fourth-order null. Instead, Velusamy et al.
(2003) mentioned the advantage of a dual Bracewell interferometer consisting
of two equivalent second-order nullers, which overcomes the ambiguity of the
planet positions with a phase chop. We note that a phase shifter ($\pi/2$ for
nulling interferometers) is introduced to one of the two beams, and the two
states are formed by inserting or removing the phase shifter. In addition, the
subtraction of the two chopped states can separate symmetric components,
including stellar leakage and background light, from the off-axis point
sources. Finally, both the TPF-I and Darwin mission concepts favored a dual
Bracewell interferometer (Cockell et al. 2009). Although TPF-I and Darwin were
anticipated to detect and characterize the thermal emissions from Earth-like
planets for the first time, they were postponed indefinitely due to the
technical difficulties.

However, stellar leak is more sensitive to the optical path difference (OPD)
and low-order aberrations in the second-order null than in the fourth-order
null (e.g., Hansen et al. 2022). Lay (2004) quantified the systematic errors
generated from the fluctuation of the null depth, which could obscure the
modulated planet signal over the baseline rotation. The ideal phase chop
technique can successfully remove most of the systematic noise errors, and
only the first-order phase error and the cross-term of phase and amplitude
errors remain in the demodulated signal. To identify an Earth-like planet
around a Sun-like star at 10 $\mu m$, the OPD and amplitude perturbations need
to be stabilized to 1.5 $nm$ and 0.1%, respectively (Lay 2004). Because
observing such a planet with a signal-to-noise ratio of seven requires an
integration time of a few days, according to preliminary analyses for the LIFE
mission presented in Dannert et al. (2022), the 1.5-$nm$ OPD stability
requirement holds for the same period. This imposes strict requirements on the
formation flight and the optical beam transport and combination system. Lay
(2006) proposed to stretch the aspect ratio of the rectangular four collector
array of the Double Bracewell interferometer to remove instrumental noise
induced by the systematic effects from the data to mitigate the requirements
by a factor of 10. The method utilized different behaviors between the planet
and instability noise achieved by stretching the ratio between the nulling and
imaging baseline to 1:6. While this relaxes the requirements on the null
stability, the streching of the baselines requires a more fuel consumption
compared to baseline rotation.

Instead of using the modulated signals of off-axis objects during baseline
rotation, Matsuo et al. (2011) proposed a method for estimating the positions
of off-axis objects and obtaining their spectra from a few baselines, focusing
on the modulation of the signal along the wavelength domain. This method
requires only relative stability among the wavelength channel across the
observed spectrum instead of stability of the null depth during the baseline
rotation. In addition, when the number of baselines is larger than that of the
detectable objects in the field of view of the interferometer, one can
effectively separate planet signals from long-term fluctuations caused by
systematic effects. The extended method optimizes for continuously rotating
the baseline instead of fixing baselines, which could further mitigate the
stability requirements of a space-based nulling interferometer. The present
study is complementary to developments of the formation flying interferometry
(e.g., Hansen et al. 2022; Matsuo et al. 2022) and ground-based nulling
experiments (e.g., Ertel et al. 2020; Ranganathan et al. 2022). These efforts
provide a support for the Large Interferometer for Exoplanets (LIFE), which
represents a science theme that was recognized as one out of three potential
science themes for a future L-class mission in the Voyage 2050 of the European
Space Agency. Based on the heritage of Darwin and TPF-I, but leveraging most
recent scientific and technological developments, LIFE will directly detect
and characterize the thermal light from habitable planet candidates. LIFE
could detect 25 - 45 terrestrial planets in the habitable zone around nearby
F-, G-, K-, and M-type stars under conservative assumptions (Quanz et al.
2022; Kammerer et al. 2022).

In the following we propose the phase-space synthesis decomposition (PSSD)
method for extracting the planet signal. This method could mitigate the
rigorous requirements imposed on the nulling space interferometer, which is
complementary to ongoing technological demonstrations for LIFE. Section 2
provides a brief overview of PSSD and its mathematical explanation. We perform
a numerical simulation to investigate the feasibility of the method using the
LIFE simulator (Dannert et al. 2022) in Section 3. Section 4 discusses the
limitation of PSSD and its advantages and disadvantages. We conclude with our
main findings in Section 5.

##  2 Concept

This section provides an overview of PSSD and then introduces its procedures
from planet detection to spectral characterization. The analytical equations
constructed for PSSD explain the two processes, planet detection and
characterization, in detail.

###  2.1 Overview

PSSD is divided into two processes: (1) search for the planet signal, and (2)
measurement of the planet spectrum. Both steps require a continuous baseline
rotation and the same operation as the previous method that extracts the
modulated signal through the cross-correlation or the maximum likelihood of
the data obtained while rotating the baseline (e.g., Angel & Woolf 1997;
Dannert et al. 2022). However, the previous method and PSSD differ in how the
planetary positions are reconstructed. The previous process transforms
interferometric signals collected by spinning the baseline into the planet
position by fitting the modulation in both the azimuth and wavelength domains
simultaneously. However, focusing on the fact that the spectrum of a G-type
star is smoothly distributed at mid-infrared wavelengths (e.g., Husser et al.
2013), a one-dimensional image in parallel to baseline is first formed by
correlation of the interferometric signal only along the wavelength domain,
which basically has the same characteristics as Fourier transform of the
signal in the same direction (Matsuo et al. 2011). Because the wavelength
dependence of the stellar leak is largely different from that of the planet
signal, the partial correlation only along the wavelength domain can decompose
the stellar leak and planet signal. After spinning the baseline, summing over
one-dimensional images, instead of a cross-correlation of the signal in the
azimuth domain, transforms a set of one-dimensional positional information
into two-dimensional positional information. Thanks to the partial correlation
only in the wavelength domain, an impact of a long-term systematic error on
image reconstruction can be avoided.

PSSD receives both the advantages of the two previous methods, the cross-
correlation method (Angel & Woolf 1997) and Fourier transform of the signal
along the wavelength domain (Matsuo et al. 2011). While the corss-correlation
method increases the signal-to-noise ratio as much as possible, the latter
efficiently decomposes the stellar leak and planet signal. PSSD combines the
two methods by employing a local cross-correlation of the signal only along
the wavelength domain, instead of the full cross-correlation in both the
wavelength- and time-domains. Generally, local (or segmented) cross-
correlation is used when cross-correlating data in smaller segments. We can
directly compare small sections of two arrays of data by cross-correlating
corresponding segments, allowing for a more localised analysis. This method is
very useful when analyzing complex astronomical phenomena with fluctuations or
patterns in various portions of compared signals (e.g., Kovačević et al.
2018).

Thanks to the combination of the two previous methods, PSSD provides three
advantages in terms of planet detection. First, PSSD could shorten the
required stability duration from a few days to a few minutes. Second, PSSD
could also mitigate the impact of the limited number of baselines on search
for the planet signal. Third, PSSD could have robustness against a larger OPD
fluctuation. Utilizing the advantages of the planet detection process, PSSD
also develops a method for extracting the planet spectrum embedded in the
stellar leak.

Regarding the first advantage, the required stability duration could be
shortened from a few days to a few minutes because we do not use the
correlation of the planet signals collected while rotating the baseline. This
is equivalent to the period for obtaining the two-phase chop states. We note
that the period of switching between the two-phase chop states is determined
such that the slow change of the background can be fully sampled (Absil et al.
2003). PSSD only requires relative stability along the wavelength (i.e. among
the spectral data), rather than the stability of signals received while
turning the baseline. The continuous and wide wavelength range obtained from
space, such as 4 to 18.5 $\mu m$ for the LIFE observatory, realizes this
alternative approach. We note that what type of object (e.g. Jovian planet or
terrestrial planet) orbits the host star is unclear in the planet detection
phase because the light is integrated over the entire wavelength range (4 to
18 $\mu m$ for the LIFE mission) in this phase. Regarding the second
advantage, because PSSD reconstructs a one-dimensional image from one imaging
baseline, two-dimensional positionalal information can be extracted from fewer
baselines. In other words, PSSD has more resistance against a limited number
of data collected during baseline rotation than the previous cross-correlation
technique. In terms of the last advantage, a fluctuation of OPD during
baseline rotation does not correlate with a modulation of the planet signal
along the wavelength domain. Instead, the OPD error contributes to the
observing data as a noise because the stellar leak is inversely proportional
to approximately the fourth power of wavelength. PSSD can detect the planet
light unless the modulation of the planet signal is embedded in the stellar
leak. Thus, PSSD is more robust against a large OPD error in terms of planet
detection.

Next, the planetary spectra are derived based on the positional information of
the planets. Because we calculate the modulation of planet light while
rotating the baseline based on the information of the estimated planet
position, the planet light for each spectral channel can be extracted by
fitting the data through the singular value decomposition (SVD) method.
However, because the modulation of the planet signal during baseline rotation
is used for the reconstruction of the planet spectrum, the reconstructed
spectrum is more affected by a long-term systematic error compared to planet
detection. As a result, the large contrast between the stellar leak and a
temperate planet at short wavelengths prevents us from precisely
reconstructing the planet spectrum in the same wavelength range (Section 4.1).
Before applying the data to the SVD method, the stellar leak has to be
subtracted from the data if the stellar leak is much brighter than the planet
light. On the other hand, it is difficult to measure the OPD change because
the number of available photons is very limited at the nulled output.

Here we find that the stellar leak induced by the systematic OPD error could
be measured from the data at short wavelenths. Because warm and temperate
planets are much fainter than the stellar leak in the short wavelength range,
only the stellar leak mainly contributes to the data. Since there is no a
strong chromatic aberration in the optical system, thanks to reflectors
constructing the optical system of LIFE, the stellar leak may be able to be
expressed as a function of wavelength. For example, when the stellar leak is
induced by the OPD error, the stellar leak is inversely proportional to
approximately the fourth-power of wavelength. Since there are a large volume
of the data collected during baseline rotation, we could estimate the
wavelength dependence of the stellar leak using a simple model, such as an
exponential function. We note that modeling the wavelength dependence of the
stellar leak is already performed in the data reduction pipeline of GRAVITY
(e.g., GRAVITY Collaboration et al. 2020). Once the wavelength dependence is
derived, the modeled stellar leak could be extrapolated to a longer wavelength
range. The modeled stellar leak is subtracted from the data and is not, in
principle, contaminated in the reconstructed spectra. Because the stellar leak
is much weaker at the wavelengths longer than 10 $\mu m$, the planet spectrum
is less affected even if there exists a chromatic aberration.

We note, however, that if a Jovian planet close to its host star exists in the
observing object, the planet is brighter than the stellar leak even at short
wavelengths. Subtracting the bright planet from the data is required for
modeling the wavelength dependence of the stellar leak at short wavelengths.
Thanks to the long imaging baseline, an inner planet can be spatially resolved
from the host star. Because both the position and spectrum of the inner planet
are obtained through PSSD, we can estimate how the planet signal is modulated
during baseline rotation and subtract it from the data.

We also need to emphasize that PSSD is validated only for objects having
smoothed spectra such as a Planck function. If the spectra have sine
components, the positions of the objects are shifted from the true positions.
The reconstructed spectra are systematically affected by the wrongly estimated
positions. Since the low-dispersion spectra of the atmospheres in exoplanets
are close to a Planck function, the systematic shifts are not considered in
this study. We note that the reflected light of Europa from the Sun in optical
does not affect the position on the reconstructed image in spite of the
spectrum including a number of lines (Matsuo et al. 2022).

Here we overview the concrete data reduction of this method. The process of
the planet detection consists of the following five steps.

  * 1

Subtract the two chop states for each baseline to obtain the sine component of
the complex visibility.

  * 2

Extract the modulated signals of the off-axis point sources along the
wavelength (see Section 2.2).

  * 3

Repeat processes [1] and [2] during rotation of the baseline. This process is
done for each set of the two-phase chop states.

  * 4

Transform a set of reconstructed one-dimensional images into a two-dimensional
image (i.e. phase-space synthesis).

  * 5

Search for planet light in the reconstructed two-dimensional image and measure
the planet position if it exists.

The characterization process is as follows:

  * 6

Perform procedure [1] with longer integration time.

  * 7

Model the wavelength dependence of the stellar leak from the collected data at
short wavelengths (e.g., 4 to 6 $\mu m$) if the stellar leak is much brighter
than the planet light due to a large OPD error.

  * 8

Subtract the stellar leak from the collected data after extrapolating the
stellar leak model constrcuted in process [7] to the long wavelength range.

  * 9

Construct a matrix equation of the following form from the set of collected
data: $O=RI$, where $O$ is the observable vector, $R$ is the response
function, and $I$ is the vector of the input sky.

  * 10

Solve the matrix equation using the SVD method to extract the planet spectrum
(i.e. phase-space decomposition).

The phase information is summed for planet detection through processes [1] to
[5]. In contrast, the phase information is decomposed in the planet
characterization phase through procedures [6] to [10]. If the stellar leak is
not much brighter than the planet light at short wavelengths, processes [7]
and [8] can be skipped. As shown in Section 3.2, when the systematic OPD RMS
error is 0.75 $nm$, corresponding to the standard requirement of LIFE (Dannert
et al. 2022), the planet spectra can be precisely extracted without processes
[7] and [8]. Although processes [1] and [6] are equal in data reduction, the
required integration time is different. Because the planet light is integrated
over the entire wavelength range in the reconstructed image through processes
[1] – [5], the required integration time for planet detection is much shorter
than that for planet spectrum obtained through processes [6] - [10].

We explain the planet detection and characterization in Sections 2.2 and 2.3,
respectively, constructing equations for PSSD.

###  2.2 Search for planet signal (phase-space synthesis)

When a dual-Bracewell nulling interferometer with a $\frac{\pi}{2}$ phase chop
observes the sky, the observed two-chop states in the unit of photoelectrons
are as follows (e.g., Beichman & Velusamy 1999; Matsuo et al. 2011)

$\displaystyle O_{\pm}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\int d^{2}\theta
I(\lambda,\@vec{\theta})\sin^{2}\left(\frac{\pi}{\lambda}\@vec{b}\cdot\@vec{\theta}+\delta
l_{n}\right)$ $\displaystyle\times$
$\displaystyle\left\\{1\pm\sin\left(\frac{2\pi}{\lambda}\@vec{B}\cdot\@vec{\theta}+\delta
l_{i}\right)\right\\},$

where $\@vec{\theta}$ is the position vector in the sky, $\@vec{b}$ and
$\@vec{B}$ are the nulling and imaging baseline vectors,
$I(\lambda,\@vec{\theta})$ is the signal without the effect of sky
transmission caused by the dual-Bracewell nulling interferometer at wavelength
of $\lambda$, and $\delta l_{i}$ and $\delta l_{n}$ are the optical path
differences of the imaging and nulling baselines, respectively. The $(+)$ and
$(-)$ notations indicate the two chop states. In Equation 2.2, we assumed that
the two nulling baselines for the dual-Bracewell nulling interferometer have
the same optical path difference error, $\delta l_{n}$, for simplicity.

The planetary system is the sum of the host star
$I_{*}(\lambda,\@vec{\theta})$, multiple planets $N_{p}$,
$\Sigma_{k}^{n}I_{p,k}$, local zodiacal light, $I_{lz}(\lambda)$, and the
exozodiacal light $I_{ez}(\lambda,\@vec{\theta})$. The spectrally resolved
signal for the planetary system is written as

$\displaystyle O_{\pm}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\int_{\Omega_{*}}d^{2}\theta
I_{*}(\lambda,\@vec{\theta})\sin^{2}\left(\frac{\pi}{\lambda}\@vec{b}\cdot\@vec{\theta}+\delta
l_{n}\right)$ $\displaystyle\times$
$\displaystyle\left\\{1\pm\sin\left(\frac{2\pi}{\lambda}\@vec{B}\cdot\@vec{\theta}+\delta
l_{i}\right)\right\\}$ $\displaystyle+$
$\displaystyle\frac{1}{2}\Sigma_{k}^{N_{p}}I_{p,k}\left(\lambda,\@vec{\theta}_{p,k}\right)\Omega_{p,k}\sin^{2}\left(\frac{\pi}{\lambda}\@vec{b}\cdot\@vec{\theta}_{p,k}+\delta
l_{n}\right)$ $\displaystyle\times$
$\displaystyle\left\\{1\pm\sin\left(\frac{2\pi}{\lambda}\@vec{B}\cdot\@vec{\theta}_{p,k}\right)+\delta
l_{i}\right\\}$ $\displaystyle+$
$\displaystyle\frac{1}{2}\int\int_{\Omega_{fov}}d^{2}\theta\left(I_{lz}(\lambda)+I_{ez}(\lambda,\@vec{\theta})\right)\sin^{2}\left(\frac{\pi}{\lambda}\@vec{b}\cdot\@vec{\theta}+\delta
l_{n}\right)$ $\displaystyle\times$
$\displaystyle\left\\{1\pm\sin\left(\frac{2\pi}{\lambda}\@vec{B}\cdot\@vec{\theta}+\delta
l_{i}\right)\right\\},$

where $\Omega_{*}$ and $\Omega_{p,k}$ are the solid angles of the host star
and the $k$-th planet, respectively, $\Omega_{fov}$ is the field of view of
the interferometer, and $\theta_{p,k}$ is the position vector of the $k$-th
planet. As shown above in step [1], the demolutated signal is given by

$\displaystyle O(\lambda)$ $\displaystyle=$ $\displaystyle
O_{+}(\lambda)-O_{-}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\int_{\Omega_{*}}d^{2}\theta
I_{*}(\lambda,\@vec{\theta})\sin^{2}\left(\frac{\pi}{\lambda}\@vec{b}\cdot\@vec{\theta}+\delta
l_{n}\right)$ $\displaystyle\times$
$\displaystyle\sin\left(\frac{2\pi}{\lambda}\@vec{B}\cdot\@vec{\theta}+\delta
l_{i}\right)$ $\displaystyle+$
$\displaystyle\frac{1}{2}\Sigma_{k}^{N_{p}}I_{p,k}\left(\lambda,\@vec{\theta}_{p,k}\right)\Omega_{p,k}\sin^{2}\left(\frac{\pi}{\lambda}\@vec{b}\cdot\@vec{\theta}_{p,k}+\delta
l_{n}\right)$ $\displaystyle\times$
$\displaystyle\sin\left(\frac{2\pi}{\lambda}\@vec{B}\cdot\@vec{\theta}_{p,k}+\delta
l_{i}\right)$ $\displaystyle+$
$\displaystyle\frac{1}{2}\int\int_{\Omega_{fov}}d^{2}\theta
I_{ez}(\lambda,\@vec{\theta})\sin^{2}\left(\frac{\pi}{\lambda}\@vec{b}\cdot\@vec{\theta}+\delta
l_{n}\right)$ $\displaystyle\times$
$\displaystyle\sin\left(\frac{2\pi}{\lambda}\@vec{B}\cdot\@vec{\theta}+\delta
l_{i}\right),$

where the local zodiacal light was assumed to be removed from the demodulated
signal because of its symmetrical structure. If the host star is perfectly
positioned at the center of the field of view (FOV), the stellar leak
disappears in Eq. 2.2 and contributes only as shot noise. Now we move to step
[2].

There are two approaches for step [2]: the cross-correlation method (Angel &
Woolf 1997) and the Fourier transform (Matsuo et al. 2011). While the former
focuses on the modulated signal while rotating the baseline, the latter uses
the modulated one along the wavelength domain for each baseline. Here, we
combine the advantages of the two methods. We use the correlation method
(Angel & Woolf 1997) to extract the signal correlated to the modulation of the
planet along the wavelength and derive the planet position for each baseline.
After rotation of the baseline, the positions of the planets are obtained. We
employ a rectangular array configuration with a baseline ratio of 6:1 based on
the baseline of the LIFE mission concept (see Figure 1). The parameters of the
configuration are the same as those used for the numerical simulations in
Section 3. The configuration is optimized to maximize the throughput of the
habitable zone around a G-type star at 10 $pc$.

Given that the position vector of the correlated signal is
$\@vec{\alpha}_{corr}$ (i.e. the two-dimensional position of the signal), the
positional information reconstructed from the $j$-th imaging baseline vector,
$\@vec{B}_{j}$, is

$M_{corr,j}(\alpha_{j})=\Sigma_{i}^{N_{i}}O(\lambda_{i})\sin\left(\frac{2\pi}{\lambda_{i}}\@vec{B}_{j}\cdot\@vec{\alpha}_{corr}\right)\sin^{2}\left(\frac{\pi}{\lambda_{i}}\@vec{b}_{j}\cdot\@vec{\alpha}_{corr}\right),$
(4)

where $\lambda_{i}$ is the $i$-th spectral element, and $N_{i}$ is the number
of elements. $M_{corr,j}$ tells us about the position of the correlated
signal, $\alpha_{j}$, projected to the baseline vector, $\@vec{B}_{j}$. As
shown in Equation 2.2, $O(\lambda_{i})$ is equal to the sum of the stellar
leak, planet signals, and background components. Each component has a
different spectrum energy distribution (see Figure 3(b)).

In order to explain how PSSD works, we perform a simulation under a simple
condition that an Earth-like planet is positioned at 1 $AU$ parallel to the
$x$-axis (panel (a) of Figure 2). When the azimuths of the imaging baseline
are 0 and 45∘ (panel Fig. 2 (b)), two one-dimensional images parallel to the
imaging baseline vector are generated (panels (c) and (d) of Figure 2). When
the azimuth of the imaging baseline is 0∘, the planet light is nulled for the
rectangular array because the nulling baseline is parallel to the $x$-axis
(panel Fig. 2(c)). In fact, the peak value of the planet is almost 0. In
contrast, for the azimuth angle of 45∘, the planet light is extracted at 1
$AU$ of the $x$ axis (panel Fig. 2 (d)). The peak value is a much larger than
that of the nulled planet. The data obtained by the rectangular array also has
information on the planet position in the direction perpendicular to the
imaging baseline, thanks to the nulling baseline. The planet position along
the nulling baseline is weakly constrained for each angle of the imaging
baseline. Thus, both the imaging baseline and the nulling baseline can be
utilized for planet detection.

Because the stellar leak caused by the OPD error is inversely proportional to
approximately $\lambda^{-4}$, the first term of $O(\lambda_{i})$ in Equation
2.2 does not correlate with
$\sin\left(\frac{2\pi}{\lambda_{i}}\@vec{B_{j}}\cdot\@vec{\alpha}_{corr}\right)\sin^{2}\left(\frac{\pi}{\lambda_{i}}\@vec{b_{j}}\cdot\@vec{\alpha}_{corr}\right)$
along the wavelength domain. The OPD error does not have less influence on the
reconstruction of one-dimensional images, compared to the previous method that
extracts the modulated signal through the cross-correlation of the data
collected while rotaing the baseline. In other words, PSSD has robustness
against a large OPD fluctuation, which is the third advantage of PSSD, as
discussed in Section 2.1.

The two-dimensional positional information is obtained by summing the one-
dimensional images collected while rotating the baseline:

$M_{corr}(\@vec{\alpha}_{corr})=\Sigma_{j}^{N_{j}}M_{corr,j}(\alpha_{j}),$ (5)

where $N_{j}$ is the number of collected baselines. Panel (e) shows the two-
dimensional image reconstructed through Equation 5. The pixel value at the
planet position on the two-dimensional image corresponds to the sum of the
pixel values at the same position on the one-dimensional images. Because PSSD
focuses on the signal modulated by the wavelength for each baseline instead of
one modulated by the rotation of the baseline, PSSD is less affected by the
long-term systematic noise than the purely rotation-based signal extraction,
which is the first advantage of PSSD discussed in Section 2.1. The required
stability duration could be shortened to a few minutes, corresponding to the
period for obtaining the two-phase chop states.

PSSD is not also less impacted by a sparse U-V sampling. We discuss how the
limited number of collected baselines affects the reconstructed two-
dimensional image in Section 4.2.

We also note the relationship between the correlation and Fourier tranform
methods. The Fourier transformation of the demodulated signal along the
wavelength gives a one-dimensional image of the sky in parallel to the $j$-th
imaging baseline vector:

$M_{FT,j}(\alpha_{FT,j})=\int
d\left(\frac{1}{\lambda}\right)O(\lambda)\sin\left(\frac{2\pi}{\lambda}|\@vec{B}_{j}|\alpha_{FT,j}\right)\sin^{2}\left(\frac{\pi}{\lambda}|\@vec{b}_{j}|\alpha_{FT,j}\right),$
(6)

where $\alpha_{FT}$ is the one-dimensional coordinate system in parallel to
the $j$-th imaging baseline vector. The origin of the coordinate system is the
center of the field of view. Comparing Equation 6 with Equation 5, we found
that both approaches are analytically equal if the sky consists of multiple
point sources. A continous source can be reconstructed only through Fourier
transform of the interferometric signal (i.e. complex visibility). However,
focusing on the fact that the transmission pattern of the sky induced by the
nulling baseline can be utilized to increase the signal-to-noise ratio of the
planet detection, the Fourier transform method requires the condition that the
imaging and nulling baselines are aligned for better planet detection. In
contrast, the correlation method can be applied to any telescope
configuration. Thus, the correlation method would be more utilized for planet
detection, compared to the Fourier transform method (Matsuo et al. 2011).

Figure 1: Rectangular array used for this study. Each filled circle represents
a 2$m$-diameter telescope. The ratio of the imaging to nulling baselines is
6:1.

Figure 2: Procedure of image reconstruction. Panel (a) shows the planet
position for this simulation. The planet is positioned at 1 $AU$ in parallel
to the $x$ axis, where $(x,y)$ is the coordinate system of the object plane in
the unit of $AU$.

The planetary system at 10 $pc$ consists of only one Earth-radius planet with
an effective temperature of 285 $K$ (without its host star). Panel (b) shows
the directions of the imaging baseline for the two reconstructed one-
dimensional images (panels (c) and (d)). Panel (e) shows a two-dimensional
image converted from the set of the one-dimensional images. The vertical axis
of panels (c) and (d) and the color bar of panel (e) represent the number of
photoelectrons. The parameters of the telescope and instrument are compiled in
Table 2.

###  2.3 Extraction of planet spectrum (phase-space decomposition)

Once the planet light is successfully detected in the reconstructed two-
dimensional image, we can estimate the planet spectrum. We calculate how the
planet light is modulated while rotating the baseline based on the two-
dimensional planet positional information. The planet signal for each spectral
channel can be extracted from the data collected during baseline rotation.
However, because the long-term fluctuation of OPD correlates with the
modulation of the planet signal during baseline rotation, the reconstruction
of the plaent spectrum is more easily affected by a long-term OPD error. In
other words, the characterization of the planet light is more challenging than
planet detection. Therefore, if the stellar leak is much brighter than planet
signal, the bright stellar leak has to be subtracted before extracting the
planet signal from the data.

Here, as introduced in Section 2.1, how the stellar leak changes while
rotating the baseline could be measured from the data at short wavelengths.
This is because the signals of warm and temperate planets except for a hot
Jupiter are negligible compared to the stellar leak at short wavelengths. If
we confirm from the reconstructed image that the stellar leak mainly
contributes to the data at short wavelengths, the wavelength dependence of the
stellar leak can be modeled in the same wavelength range. After the stellar
leak model is extrapolated to the longer wavelength range, the stellar leak is
subtracted from the demodulated signal shown in Equation 2.2.

There are mainly two systematic error terms, the first-order phase error and
the cross-term of phase and amplitude, in the demodulated signal. Because the
demodulated signal of the stellar leak in Equation 2.2 is characterized by
$\sin^{2}\left(\frac{\pi}{\lambda}\@vec{b}\cdot\@vec{\theta}+\delta
l_{n}\right)\sin\left(\frac{2\pi}{\lambda}\@vec{B}\cdot\@vec{\theta}+\delta
l_{i}\right)$, the wavelength dependence of the stellar leak could be simply
expressed by a power law of wavelength under the condition that the systematic
errors are much smaller than wavelength:

$I_{leak}(\lambda,t)=\left\\{a\left(\frac{\lambda}{\lambda_{0}}\right)^{\alpha}+b\left(\frac{\lambda}{\lambda_{0}}\right)^{\beta}\right\\}I_{leak}(\lambda_{0},t),$
(7)

where $\lambda_{0}$ is the reference wavelength, and $I_{leak}(\lambda,t)$ is
the stellar leak model in the demodulated signal at the wavelength $\lambda$
as a function of time, $t$. We note, however, that different wavelength
dependencies may exist in the long wavelength range because a coating
dispersion error or a pupil shear error could impact at long wavelengths.
Because the coating dispersion and pupil shear errors drastically decreases in
the longer wavelength regime, the subtraction of the estimated stellar leak
model from the demodulated signal would less impact the reconstructed planet
spectrum at long wavelengths. We also note that thanks to the bright stellar
leak at short wavelengths, the systematic aberration could be modeled from the
intensity and wavelength dependence of the stellar leak, which is a similar
work as measuring coronagraphic low-order aberrations (Guyon et al. 2009).

After subtracting the stellar leak from the demodulated signal shown in
Equation 2.2, the planet spectra are reconstructed from the residual data
through the SVD method. The matrix equation for the $i$-th spectral element
can be written as

$O_{i}=R_{i}I_{i},$ (8)

where $O$ is the vector composed of the observed data, $R$ is the matrix of
the response function of objects (i.e. sky transmission of object), and $I$ is
the vector of the input sky. In order to reconstruct the spectra of the
planets, we solve the matrix equation for each spectral channel.

After the observed data is subtracted from the averaged value over the
baseline rotation, the $O$ vector of the $i$-th spectral element is

$O_{i}=\left(\begin{array}[]{cccc}O_{i,1}\\\ O_{i,2}\\\ \vdots\\\
O_{i,N_{j}}\end{array}\right),$ (9)

where $O_{i,j}$ is the observation data of the $j$-th azimuth angle for the
$i$-th spectral element. The number of the elements for the $O$ vector is
$N_{j}$, corresponding to the number of the collected data during baseline
rotation. The response matrix for the $i$-th spectral element, $R_{i}$, is
written as

$R_{i}=\left(\begin{array}[]{ccc}R_{1,1}&\ldots&R_{1,N_{p}}\\\
\vdots&&\vdots\\\ R_{N_{j},1}&\ldots&R_{N_{j},N_{p}}\end{array}\right),$ (10)

where we assumed that the continuum component is removed from the observed
vector, $O_{i}$, by the subtraction of the two chop states. The $R$ matrix is
a $N_{j}\times N_{p}$ matrix. Each component is as follows:

$\displaystyle R_{1,1}$ $\displaystyle=$
$\displaystyle\sin^{2}\left(\frac{\pi}{\lambda_{i}}\@vec{b}_{1}\cdot\@vec{\theta}_{p,1}\right)\sin\left(\frac{2\pi}{\lambda_{i}}\@vec{B}_{1}\cdot\@vec{\theta}_{p,1}\right)$
(11) $\displaystyle R_{1,N_{p}}$ $\displaystyle=$
$\displaystyle\sin^{2}\left(\frac{\pi}{\lambda_{i}}\@vec{b}_{1}\cdot\@vec{\theta}_{p,N_{p}}\right)\sin\left(\frac{2\pi}{\lambda_{i}}\@vec{B}_{1}\cdot\@vec{\theta}_{p,N_{p}}\right)$
$\displaystyle R_{N_{j},1}$ $\displaystyle=$
$\displaystyle\sin^{2}\left(\frac{\pi}{\lambda_{i}}\@vec{b}_{N_{j}}\cdot\@vec{\theta}_{p,1}\right)\sin\left(\frac{2\pi}{\lambda_{i}}\@vec{B}_{N_{j}}\cdot\@vec{\theta}_{p,1}\right)$

The vector of the input sky for the $i$-th spectral element is

$I_{i}=\left(\begin{array}[]{cccc}I_{p,1}(\lambda_{i},\@vec{\theta}_{p,1})\\\
I_{p,2}(\lambda_{i},\@vec{\theta}_{p,2})\\\ \vdots\\\
I_{p,N_{p}}(\lambda_{i},\@vec{\theta}_{p,N_{p}})\end{array}\right).$ (12)

The number of elements for the $I$ vector is $N_{p}$. When the number of the
observed data is much larger than that of the elements for the input matrix,
the $N_{p}$ planet signals for each spectral element can be decomposed by the
SVD method.

Finally, we need to emphasize that there are several ways to decompose the
planet signals and unknown stellar leaks under the condition that the planets
are successfully detected. This decomposition could be solved using modified
orthogonal projections, or kernels, such as used in (Laugier et al. 2020).
They preserve important properties of the covariance matrix of errors, and
this decomposition therefore is well suited for further data whitening
approaches.

We could also combine PSSD with wavelet-based signal reconstruction methods
(e.g., del Ser et al. 2018), either by suppressing wavelet coefficients at
different levels, corresponding to unwanted signals, or convolving them with
specially design convolution kernels. It is expected that while the low-
frequency signal is mainly caused by the systematic OPD residual, the mid- to
high-frequency signals are caused by off-axis point sources and stochastic
noises. Knowing the models of the systematic errors and stochastic noises
makes it possible to suppress sections of the wavelet decomposition connected
to these signals and then, using the inverse wavelet transform, rebuild the
planet signal. Reconstruction of the signal will depend on the cadence and its
level with respect to noise. With advanced techniques we could expect signal
recovery even its contribution is up to $\sim 10-30\%$.

##  3 Simulations

We performed numerical simulations to check the feasibility of PSSD under the
LIFE baseline scenario. First, we briefly explain the simulation setup
regarding the target system and instrument. Next, we show the results
generated by PSSD under the ideal condition where only the astronomical noise
contributes to the data as shot noise. Finally, we include a long-term
systematic OPD error in the simulations and show its impact on PSSD.

###  3.1 Setup

The distance of the considered target is 10 $pc$. The target system consists
of three Earth-sized ($R_{\oplus}$) planets, a Sun-like star with a Sun radius
($R_{\odot}$) and an effective temperature of 5778 $K$, and an exozodiacal
dust disk. The semi-major axes of the three planets are 1, 0.73, and 1.5 $AU$,
which are the same as those of Earth, Venus, and Mars. Given that the
effective temperatures of the three planets are simply proportional to the
inverse square root of the semi-major axis, the temperatures of planets P1,
P2, and P3 were set to 285, 330, and 232 $K$, respectively. All of the target
objects were assumed to emit blackbody radiation. The orbital phases of the
three planets were set to 0, -45, and 90∘. The phase angle of 0∘ points along
the positive $x$-axis, where $(x,y)$ is the coordinate system of the sky in
the unit of $AU$. The arrangement of the three planets is shown in Figure
3(a). The exozodiacal light is equal to three times that of the solar system
(Ertel et al. 2020). The surface brightness of the exozodiacal light is
generated based on the previous model (Kennedy et al. 2015), which is applied
to the software tool, LIFEsim (Dannert et al. 2022). Table 1 compiles all the
parameters of the target system.

We employ a dual-Bracewell nulling interferometer with imaging and nulling
baselines of 87.3 $m$ and 14.55 $m$ (see Figure 1) so that the maximum of
transmission is achieved for the center of the habitable zone around a Sun-
like star at 10 $pc$ at a wavelength of 15 $\mu m$ (Quanz et al. 2022). The
diameter of each telescope is 2 $m$, and the imaging and nulling baselines are
perpendicular to each other. Although the observing wavelength ranges from 4
to 18.5 $\mu m$, the same as the LIFE baseline, we limited the wavelength
range to larger than 8 $\mu m$ in the planet detection phase. The reason is
that the bright stellar leak is more than 100 times brighter than the light of
the planets we consider at short wavelengths and deteriorates the performance
of PSSD. We note that the shorter wavelengths are effective in looking for
inner planets because of the combination of higher spatial resolution and
brighter planets in that wavelength range. PSSD needs to optimize the
wavelength range used for planet detection based on what type of planets we
find.

We set the resolving power of the spectrum to 50 for both planet detection and
their characterization. The minimum resolving power for planet detection is
determined by the required field of view. When the resolving power is 50, the
field of view is 1.14 arcsecond at 10 $\mu m$. And for the characterization a
spectral resolution of 30-50 was suggested in Konrad et al. (2022) in order to
detect the various molecules in an Earth-twin atmosphere. The total throughput
was set to 0.035, given that the instrument throughput and quantum efficiency
are 0.05 and 0.7, respectively. The integration time was set to 55 $h$ for
planet detection and 75 days for planet characterization, respectively.

We do not assume a continuous rotation but a discrete rotation in steps of one
degree, where the spacecraft come to a halt before rotating again by one
degree, because of computational cost. We note that the continuous rotation
provides a better reconstruction thanks to the continuous U-V coverage
compared to the discrete rotation. Assuming that the baseline rotates by 360∘
at a one-degree interval, the integration time for each baseline is 550 and
180,000 seconds for planet detection and characterization phases,
respectively. In addition to the ideal observing case, we studied the
feasibility of PSSD under systematic OPD error. Although there are mainly the
first-order phase and the phase-amplitude cross-term in the demodulated signal
(Lay 2004), only the former was considered in this simulation. We note,
however, that this simulation on the feasibility of PSSD is not largely
impacted by the phase-amplitude cross-term because the two systematic
components have a similar frequency dependence of $\frac{1}{f}$, called ”pink
noise”, where $f$ is the frequency. The root-mean-square (RMS) of the OPD
error was set to 0.75 $nm$, corresponding to the LIFE baseline scenario for
the case of only phase error (Dannert et al. 2022). The baseline value is
larger than that for the case of both phase and amplitude errors. We also
consider 5, 10, and 15 times the baseline values, 3.8, 7.5, and 11.3 $nm$ RMS
errors, to investigate the limitation of PSSD in Section 4. Table 2 compiles
all the instrumental parameters.

Figure 3(b) shows the astronomical signals obtained by the Bracewell nulling
interferometer under the above-observed conditions. The stellar leak and
background, such as the local zodiacal and exozodiacal light, cover the
modulations of the three planets. When there is no OPD error, all astronomical
signals contribute to the data as the shot noise. We perform the numerical
simulations under the ideal condition in Section 3.2.1 and then consider the
fluctuation of the stellar leak due to the systematic OPD error in Section
3.2.2.

Figure 3: Target planetary system. (Left) Configuration of three planets.
(Right) Signals of planet P1 (black), planet P2 (blue), planet P3 (green), the
nulled host star (red), local zodiacal light (brown), and exozodiacal light
(purple) with a resolving power of 50 per a unit of time. Target system and
instrument parameters are compiled in Tables 1 and 2.

###  3.2 Results

####  3.2.1 Ideal condition

We first show the feasibility of PSSD under the ideal condition, in which only
shot noise exists due to astronomical sources. After collecting data while
rotating the baseline by 360∘ in steps of 1 degree, we generated a two-
dimensional image through processes [1] to [4]. Figure 4(a) shows the
reconstructed two-dimensional images for an integration time of 55 $h$. The
signal-to-noise ratios for the detection of planets under the ideal condition
are compiled in Case 1 of Table 3. We successfully detected signals of planets
P1 and P2 with signal-to-noise ratios of 10.8 and 14.6, respectively. The
higher temperature of planet P2 allows us to obtain higher signal-to-noise
ratio compared to that of planet P1. In contrast, the signal-to-noise ratio of
planet P3 is only 3.6 because of its lower temperature. We require a longer
integration time to achieve the signal-to-noise ratio of five for planet P3.

The signal-to-noise ratio is defined as the ratio of the planet signal to the
starndard deviation at the same angular distance as its planet. In order to
calculate the standard deviation, a two-dimensional image without the planet
signals is generated and divided into annular rings. The noise floor is
calculated as the standard deviation for each annular ring. We note that the
absolute value of the signal-to-noise ratio cannot be directly compared with
that calculated by Dannert et al. (2022) because PSSD reconstructs both the
signal and noise in a different way.

Next, we reconstructed spectra of the three planets through processes of [6],
[9] and [10], assuming that the planet positions are correctly obtained.
Figures 4(b), (c), and (d) show the reconstructed spectra of the three
planets. The spectra of planet P1 and planet P2 are consistent with the input
spectra (solid gray lines). We also derived for each data-point the average
and standard deviation by performing the numerical simulations 100 times. The
signal-to-noise ratio of the spectrum for planet P1 agrees with that of the
previous study (Konrad et al. 2022). LIFE could detect the methane and ozone
absorption bands at 7.6 and 9.6 $\mu m$ with signal-to-noise ratios of
approximately 5 and 15, respectively. This combination of simultaneously
detected absorption features is thought to be a good indicator of a non-
equilibrium atmosphere caused by biological activity on the planet (Kasting et
al. 2014).

On the other hand, the signal-to-noise ratio worsens at shorter wavelengths
than 7.5 $\mu m$. This is because the stellar leak drastically increases due
to the narrower null pattern on the sky in the shorter wavelength range. The
planet signals rapidly decrease at the same time (see Figure 3(b)). In
addition, PSSD also obtained the entire spectrum of planet P3 but at a signal-
to-noise ratio lower than three except for wavelengths longer than 10 $\mu m$
due to its faintness compared to the other planets. Table 4 compiles the
signal-to-noise ratios of the reconstructed spectra at wavelengths of 5, 7.5,
10, and 15 $\mu m$.

Although planet P3 was not detected for an integration time of 55 $h$, the
signal of planet P3 could be obtained while integrating the data in the
characterization phase, and its position would be well determined. Planets P1
and P2 were also detected with higher signal-to-noise ratios compared to those
in the planet detection phase, which can reduce the systematic errors of the
reconstructed spectra due to the estimation errors of the planet positions.

Thus, we confirmed that PSSD could detect the planet signals and characterize
their atmospheres under the ideal condition, in which the data is affected
only by the shot noise due to the stellar leak, background, and planets.

Figure 4: Image and spectrum reconstruction under photon-noise limited
condition. (a) Reconstructed two-dimensional image. The white arrows denote
positions of planet P1, planet P2, planet P3, respectively. The integration
time was set to 55 $h$. The unit of the color bar is the number of
photoelectrons. Reconstructed spectra of three planets (b) P1, (c) P2, and (d)
P3 for an integration time of 75 days. The grey line and grey vertical bar of
each panel show the input model and the standard deviation of each data point
derived through 100 numerical simulations, respectively.

####  3.2.2 Systematic error

We consider instrumental noise and investigate its negative impact on planet
detection and characterization. In order to evaluate it, we included an OPD
error in the numerical simulations as the instrumental noise. We note,
however, that the phase-amplitude cross-term also constributes to the long-
term stellar fluctuation in the demodulated signal under the existence of the
amplitude error (Lay 2004). Because both systematic components have the same
spectrum in terms of the time domain (e.g., Dannert et al. 2022), the phase-
amplitude cross-term would not significantly impact the results. We assumed
that the systematic noises have a dependency of $\frac{1}{f}$, where $f$ shows
the frequency. According to Dannert et al. (2022), when the OPD RMS error is
larger than 0.75 $nm$, the instrumental noise is dominant over the statistical
noise (i.e. fundamental noise) from the astronomical objects at the shortest
wavelength. Because the systematic OPD error is much smaller than the
observing wavelength, the amount of stellar leak is proportional to the OPD
error for each spectral element. The OPD error impacts the null depth and
leaves the stellar leak in the subtraction of the two chop states (Equation
2.2). While the former contributes to the data as the Poisson noise, the
latter affects the planet signal during baseline rotation, which prevents us
from reconstructing the planet spectrum. In our simulations, we added the same
OPD error to the imaging and nulling baselines to reduce the calculation cost.

Figure 5(a) compares the spectra of the three planets with the stellar leak
left after subtracting the two chop states in the entire wavelength range
(i.e. 4 - 18.5 $\mu m$). The stellar leak drastically increases at the shorter
wavelengths because of the shallower null depth and the planet signal drops
instead. It is more challenging to perform planet detection and
characterization at shorter wavelengths than longer ones. Panels (b), (c), and
(d) of Figure 5 compare the modulated signal of planet P1 with the stellar
leak at 4, 8, and 12 $\mu m$ as a function of the azimuth of the imaging
baseline. Although the systematic OPD RMS error of 0.75 $nm$ does not affect
the planet signal at 12 $\mu m$, the stellar leak covers the planet signal at
4 $\mu m$, which is consistent with Figure 5(a).

Figure 6(a) shows a reconstructed two-dimensional image under the condition
that the systematic OPD RMS error is 0.75 $nm$. Thanks to robustness of PSSD
against a long-term OPD error, we obtained the same signal-to-noise ratios for
detecting the three planets as those for the ideal state.

Finally, we reconstructed the planet spectra over the entire wavelength based
on processes of [6], [9], and [10] without subtraction of the stellar leak
from the demodulated signal in processes [7] and [8]. Panels (b), (c), and (d)
of Figure 6 compare the reconstructed spectra of the three planets with the
input models. The reconstructed spectra are consistent with the models over
the entire wavelength range, except for the shorter range. Comparing the
reconstructed spectra with the ideal case, we found that the shot noise limits
the performance of PSSD in the wavelength range longer than 7.5 $\mu m$. In
addition, even though the systematic stellar leak is a few times brighter than
the planet signals (Figure 5(a)), the signal-to-noise ratios are almost the
same as those for the reconstructed spectra under the ideal condition (Case 2
of Table 4). This is because the SVD method efficiently separates the mid- to
high-frequency componets induced by planets from the low-frequency component
due to the systematic stellar leak. Thus, the SVD method could reconstruct the
spectra of the three planets from the modulations of the planet signals during
baseline rotation while efficiently separating them from the long-term
systematic stellar leak.

Figure 5: Systematic stellar noise. Panel (a) shows comparison of signals of
the three planets P1 (black), P2 (blue), and P3 (green) with the nulled
stellar leaks left in the subtraction of the two chop states over the entire
wavelength range. The red, brown, gray, and light gray lines represent the
stellar leaks for systematic OPD RMS errors of 0.75, 3.8, 7.5, and 11.3 $nm$.
Panels (b), (c), and (d) show fluctuation of the stellar leak due to an OPD
RMS error of 0.75 $nm$ (red) and the demodulated signal of planet P1 (black)
at wavelengths of 4, 8, and 12 $\mu m$, respectively. The integration time of
each data point is 550 $s$. Figure 6: Same as Figure 4 except for a systematic
OPD RMS error of 0.75 $nm$.

##  4 Discussion

We have confirmed thus far that PSSD detects planet light and extracts planet
spectra under the existence of OPD fluctuation, which follows $\frac{1}{f}$.
In this section we investigate how much noise amplitude PSSD can endure
(Section 4.1) and compare PSSD with the previous method (Section 4.2).

###  4.1 Robustness against a large OPD error

We set the systematic OPD RMS error to 3.8, 7.5, and 11.3 $nm$, which are
equal to the baseline values multiplied by factors of 5, 10, and 15,
respectively. Figures 7(b), (c), and (d) show the reconstructed images for
these three different OPD errors. As the OPD error increases, systematic
patterns are brighter in the centre region of each image. In contrast, the
noise floor is limited by the shot noise at semi-major axes larger than 1.0
$AU$ (panel Fig. 7(d)). As discussed in Section 3.2.1, the noise floor was
calculated for a reconstructed two-dimensional image without the planet
signals.

We derived the signal-to-noise ratios of the three planets for each OPD RMS
error. Although the signal-to-noise ratio is gradually worsened as the OPD
error increases, the signal-to-noise ratios of the planets are higher than 5
except for planet 3 (Cases 2 - 4 of Table 3). Thus, PSSD could successfully
detect the inner two planets even under systematic OPD errors 15 times larger
than the LIFE baseline requirement. In addition, because the shot noise limits
the detection of planet 3, PSSD could also detect planet 3 with a longer
integration time.

Next, we reconstructed the spectra of the three planets for an OPD RMS error
of 7.5 $nm$ by solving the matrix equation with the SVD method (process [9];
see left panels of Figure 8). However, the OPD RMS error of 7.5 $nm$ deforms
the spectra of planet P1 and planet P2 at the shorter wavelengths than 10 $\mu
m$ and the spectrum of planet P2 over the entire wavelength range. This is
because the planet spectra are reconstructed from the long-term modulation of
the signal while rotating the data, which correlates with the systematic OPD
error. Thus, characterizing the planet atmosphere is much more affected by the
long-term OPD error, compared to planet detection.

Here, we utilize the advantages of PSSD in terms of planet detection. Because
a two-dimensional image can be reconstructed even for a large OPD error, we
can investigate what kind of objects orbits the host star. Once we confirm
that the stellar leak is dominant over the planet signals at short
wavelengths, we could measure the long-term fluctuation of the stellar leak
from the data at short wavelengths (process [7]) and subtract the stellar leak
from the demodulated signal (process [8]). There are several steps to
reconstruct the planet spectrum. First, we apply a low pass filter to the
demodulated signal to decrease the impact of statistical noise on the data.
Second, the wavelength dependence of the stellar leak is estimated from a
large number of the data points collected during baseline rotation. Finally,
based on the estimated wavelength dependence, we extrapolate the stellar leak
model to the longer wavelength range and subtract it from the demodulated
signal shown in Equation 2.2 over the entire wavelength. The planet spectra
are reconstructed through applying the subtracted data to the SVD process.
Panels (b), (d), and (f) of Figures 8 show the reconstructed spectra of the
three planets through the above process for a large OPD RMS error of 7.5 $nm$,
corresponding to ten times larger than the baseline requirement of LIFE. In
this simulation, the wavelength range applied to estimation of the wavelength
dependence was 4 - 5.5 $\mu m$, in which the stellar leak is more than one-
hundred times larger than planet signals for an OPD RMS error of 7.5 $nm$
(Figure 5(a)). The signal-to-noise ratios for the reconstructed spectra (Case
4 of Table 4) are almost the same as those under the ideal condition (Case 1
of Table 4). Thus, if we confirm from the reconstruted image that only the
stellar leak mainly contributes to the demodulated signal at short
wavelengths, the planet spectra could be reconstructed.

Figure 7: Image reconstruction under large systematic noises. Panels (a), (b),
and (c) show reconstructed images for systematic OPD RMS errors of 3.8, 7.5,
and 11.3 $nm$, respectively. Panel (d) shows the noise floors for systematic
OPD RMS errors of 3.8 (black), 7.5 (yellow), and 11.3 $nm$ (brown) were
compared with the planet signals (star symbol). The noise floor for each OPD
error was calculated for a reconstructed two-dimensional image without the
planet signals. The gray line shows the noise floor for the ideal case (i.e.,
only the shot noise) as a reference. Because the reconstructed planet signals
are slightly affected by the OPD error, the black, yellow, and brown star
symbols represent the planet signals for OPD RMS errors of 3.8, 7.5, and 11.3
$nm$, respectively. Figure 8: Spectrum reconstruction under large systematic
noise. Panels (a), (c), and (e) show reconstructed spectra of planet P1,
planet P2, and planet P3 only through the SVD process. Panels (b), (d), and
(f) show the spectra of planet P1, planet P2, and planet P3 reconstructed
through subtracting the stellar leak from the demodulated signal before the
SVD method. The OPD RMS error was set to 7.5 $nm$ for both the two cases.

###  4.2 Comparison with previous method

We compare PSSD with the previous signal extraction through the cross-
correlation or the maximum likelihood of the data obtained while rotating the
baseline (e.g., Angel & Woolf 1997; Dannert et al. 2022). The main difference
between the two methods is whether a one-dimensional image from each baseline
is first reconstructed or a two-dimensional image is reconstructed at one
time. PSSD performs the correlation of planet signal among the obtained
spectrum and converts one-dimensional information into a two-dimensional
image. In contrast, the previous method simultaneously finds the modulation of
the planet signal in both the wavelength- and time-domains. As discussed in
Section 2.1, the advantages of PSSD are robustness against a large OPD error
and a limited number of baselines.

Panels (a), (b), and (c) of Figure 9 show the reconstructed two-dimensional
images through the previous cross-correlation method. The images were formed
based oh the following Equation:

$M_{corr}(\@vec{\alpha}_{corr})=\Sigma_{j}^{N_{j}}\Sigma_{i}^{N_{i}}O(\lambda_{i})\sin\left(\frac{2\pi}{\lambda_{i}}\@vec{B}_{j}\cdot\@vec{\alpha}_{corr}\right)\sin^{2}\left(\frac{\pi}{\lambda_{i}}\@vec{b}_{j}\cdot\@vec{\alpha}_{corr}\right).$
(13)

Compared with Figure 7, the OPD error induces brighter systematic patterns at
semi-major axes smaller than 1.0 $AU$, which prevent us from detecting the two
inner planets. As shown in panel (d) of Figure 9, the intensity of the
systematic pattern is roughly proportional to the OPD RMS error in the inner
region. We note that the noise floor under the baseline requirement of LIFE is
almost equal to that for a systematic OPD RMS error of 0.75 $nm$, which is
consistent to the previous study (Dannert et al. 2022). The signal-to-noise
ratios for the detection of planets with PSSD under an OPD RMS error of 7.5
$nm$ (Case 2 of Table 3) are almost the same as those for the cross-
correlation method under the ideal case (Case 5 of Table 3). Therefore, there
exists a large difference between the robustness against a large OPD error.

We also compare PSSD with the cross-correlation method in terms of the impact
of a limited number of baselines on planet detection. The left panels of
Figure 10 show the PSSD reconstructed images with a limited number of
baselines and under the existence of only long-term systematic error (without
shot noise). In other words, the reconstructed image is not influenced by the
integration time. We randomly selected available baselines, which are more
sparsely distributed over 360∘ as the number of baselines decreases. The
systematic pattern is slightly brighter as the number of baselines decreases.
However, the three planets could be detected even though the fraction of the
available baselines is limited to only 8 %. In contrast, images reconstructed
using the previous method were largely affected by the limited number of
baselines because the modulation in the time domain was lost (the right panels
of Figure 10). In addition, the artificial pattern fully covers the three
planets if the number of baselines is limited to 8 %, and the limited azimuth
coverage elongates the point sources, which means that the long-term
systematic error modulates the data collected while rotating the baseline.

Thus, PSSD is much less impacted by both a large systematic error and a
limited number of available baselines compared to the previous method, which
could relax the requirement of LIFE.

Figure 9: Same as Figure 7 except for using the cross-correlation method. The
noise floor derived under the LIFE baseline requirement (dashed line) is added
as a reference. Figure 10: Images reconstructed through PSSD (left panels) and
the previous method (right panels) under the condition that the number of
baselines is limited by (a)(b) 100 %, (c)(d) 28 %, and (e)(f) 8 %. Only a
long-term systematic error was included in the simulations; shot noise was not
considered while investigating how the limited number of the baselines affects
the image reconstruction. The OPD RMS error was set to 3.8 $nm$.

##  5 Conclusion

We proposed a method for planet detection and characterization with future
nulling space interferometers, such as large interferometer for exoplanets
(LIFE). The proposed method is named ”phase-space synthesis decomposition
(PSSD).” PSSD focuses on the correlation of the planet signal over the entire
wavelength range instead of that along the baseline rotation. Because a one-
dimensional image parallel to the baseline can be derived for each baseline, a
large number of one-dimensional images are collected after rotating the
baseline. A two-dimensional image can be reconstructed by summing over the
one-dimensional images. Once the two-dimensional image is obtained, a
continuous equation is constructed based on the planet position information,
and its solution through singular value decomposition (SVD) allows us to
extract the planet spectra embedded in the stellar fluctuation. As long as the
modulation of the planet signal has a different frequency from the stellar
fluctuation, the SVD method efficiently decomposes the stellar leak and planet
signal. PSSD provides three advantages in planet detection compared to
previous methods that find a modulation of the planet signal during baseline
rotation in both the wavelength- and time-domains. One is robustness against a
large systematic OPD error. Because the stellar leak has a different
wavelength dependence from the planet signal, only the correlation of the
signal efficiently decomposes the stellar leak and the planet signal. The
second is that PSSD does not correlate with a long-term fluctuation of the
stellar leak because a two-dimensional image is formed by summing over one-
dimensional images that require only two-phase chop states. The third
advantage is robustness against a limited number of baselines.

We performed numerical simulations to investigate the feasibility of PSSD
under various conditions. We put three terrestrial planets with semi-major
axes of 0.73, 1, and 1.5 $AU$, corresponding to those of Venus, Earth, and
Mars, respectively, around a Sun-like star at 10 $pc$. The simulation included
both statistical and systematic noises. PSSD successfully detected the three
planets and reconstructed their spectra for an OPD RMS error of 0.75 $nm$,
which is the same as the baseline requirement of LIFE.

We also confirmed that PSSD has robustness against a large systematic OPD
error. We increased the amplitude of the systematic OPD error by a factor of
5, 10, and 15. PSSD successfully detected the three planets almost without
being affected by the stellar leak, even under the largest systematic OPD
error of 11.3 $nm$. This is because the stellar leak is inversely proportional
to approximately the fourth power of wavelength, which is largely different
from the modulation of the planet signal in the wavelength domain.

In contrast, the reconstructed spectra were more affected by the long-term
systematic noise than planet detection. This is because PSSD uses the
modulation of the planet signal during baseline rotation to reconstruct the
spectra of the three planets. The long-term systematic OPD error more easily
correlates with the modulation of the planet signal. The spectra could not be
accurately extracted under an OPD RMS error of 7.5 $nm$, corresponding to ten
times the baseline requirement of LIFE. The signal-to-noise ratio
significantly decreases in the wavelength range shorter than 7.5 $\mu m$.

Here, focusing on the fact that the planet signals are much smaller than the
stellar leak at shorter wavelengths than 6 $\mu m$, we can measure the
fluctuation of the stellar leak and its wavelength dependence for the data at
short wavelengths. Because PSSD can successfully obtain the planet signals
even under a large systematic noise, PSSD utilizes the planet position
information and tells us what type of sources contribute to the signal at
short wavelengths. If we confirm from the reconstructed two-dimensional image
that the stellar leak is dominant over the planet signals at short
wavelengths, the wavelength dependence of the stellar leak can be modeled from
the data in the short wavelength range. After extrapolating the estimated
stellar leak model to the longer wavelength range, the stellar leak could be
subtracted from the data over the entire wavelength range. The spectra of the
three planets were successfully reconstructed by applying the subtracted data
to the SVD process. The signal-to-noise ratios were almost the same as those
for the ideal condition.

Finally, we compared PSSD with a previous method that reconstructs a two-
dimensional image by simultaneously fitting the modulation of the planet in
the time- and wavelength domains after the baseline is rotated. Because the
long-term noise is correlated with the planet signal in the time domain,
systematic patterns were formed in the reconstructed image and covered the
planet signals under systematic OPD errors larger than 3.8 $nm$. The signal-
to-noise ratio for planet detection significantly decreased for large OPD
errors compared to PSSD. This is because the noise floor increased in the
inner region due to the systematic OPD error. In addition, limited azimuth
coverage in the U-V plane impacted planet detection because the modulation in
the time domain was lost. In contrast, PSSD can reconstruct a two-dimensional
image from fewer baselines. Even in the case where the azimuth coverage of the
baseline is limited to 8 $\%$, the three planets could be discovered by PSSD.

Thus, PSSD is more robust against a large OPD error and a limited number of
baselines, which could relax the requirements of LIFE regarding the OPD error
and the stability duration. However, this numerical simulation was performed
as the first step under an ideal case that makes detection of terrestrial
planets easier. We will investigate various impacts not considered in this
study, such as asymmetric exozodiacal structure (Defrère et al. 2010), and the
other systematic errors, on planet detection and characterization as the next
step.

###### Acknowledgements.

We express our sincere gratitude to Dr. Lacour for many valuable comments and
suggestions on this study. Part of this work was supported by JST FOREST
Program, Grant Number JPMJFR202W. Part of this work has been carried out
within the framework of the National Centre of Competence in Research PlanetS
supported by the Swiss National Science Foundation under grants
51NF40$\\_$182901 and 51NF40$\\_$205606\. FD and SPQ acknowledge the financial
support of the SNSF. Part of this work was carried out within the project
SCIFY, which has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation program (grant
agreement CoG - 866070). A.B.K. acknowledges funding provided by University of
Belgrade-Faculty of Mathematics (the contract 451-03-47/2023-01/200104),
through the grants by the Ministry of Science, Technological Development and
Innovation of the Republic of Serbia.

## References

  * Absil et al. (2003) Absil, O., Karlsson, A., & Kaltenegger, L. 2003, in Interferometry in Space, ed. M. Shao, Vol. 4852, International Society for Optics and Photonics (SPIE), 431 – 442
  * ALMA Partnership et al. (2015) ALMA Partnership, Fomalont, E. B., Vlahakis, C., et al. 2015, ApJ, 808, L1
  * Angel et al. (1986) Angel, J. R. P., Cheng, A. Y. S., & Woolf, N. J. 1986, Nature, 322, 341
  * Angel & Woolf (1997) Angel, J. R. P. & Woolf, N. J. 1997, ApJ, 475, 373
  * Beckers et al. (1990) Beckers, J. M., Enard, D., Faucherre, M., et al. 1990, in Advanced Technology Optical Telescopes IV, ed. L. D. Barr, Vol. 1236, International Society for Optics and Photonics (SPIE), 108 – 124
  * Beichman & Velusamy (1999) Beichman, C. A. & Velusamy, T. 1999, in Astronomical Society of the Pacific Conference Series, Vol. 194, Working on the Fringe: Optical and IR Interferometry from Ground and Space, ed. S. Unwin & R. Stachnik, 408
  * Born & Wolf (1999) Born, M. & Wolf, E. 1999, Principles of Optics
  * Bracewell (1978) Bracewell, R. N. 1978, Nature, 274, 780
  * Cockell et al. (2009) Cockell, C. S., Herbst, T., Léger, A., et al. 2009, Experimental Astronomy, 23, 435
  * Colavita & Wizinowich (2000) Colavita, M. M. & Wizinowich, P. L. 2000, in Interferometry in Optical Astronomy, ed. P. J. Lena & A. Quirrenbach, Vol. 4006, International Society for Optics and Photonics (SPIE), 310 – 320
  * Dannert et al. (2022) Dannert, F. A., Ottiger, M., Quanz, S. P., et al. 2022, A&A, 664, A22
  * Defrère et al. (2010) Defrère, D., Absil, O., den Hartog, R., Hanot, C., & Stark, C. 2010, A&A, 509, A9
  * del Ser et al. (2018) del Ser, D., Fors, O., & Núñez, J. 2018, A&A, 619, A86
  * Des Marais et al. (2002) Des Marais, D. J., Harwit, M. O., Jucks, K. W., et al. 2002, Astrobiology, 2, 153
  * Ertel et al. (2020) Ertel, S., Defrère, D., Hinz, P., et al. 2020, AJ, 159, 177
  * Fujii et al. (2018) Fujii, Y., Angerhausen, D., Deitrick, R., et al. 2018, Astrobiology, 18, 739
  * GRAVITY Collaboration et al. (2020) GRAVITY Collaboration, Nowak, M., Lacour, S., et al. 2020, A&A, 633, A110
  * Guyon et al. (2009) Guyon, O., Matsuo, T., & Angel, R. 2009, ApJ, 693, 75
  * Hansen et al. (2022) Hansen, J. T., Ireland, M. J., Laugier, R., & the LIFE collaboration. 2022, arXiv e-prints, arXiv:2204.12291
  * Hansen et al. (2022) Hansen, J. T., Ireland, M. J., Travouillon, T., et al. 2022, in Optical and Infrared Interferometry and Imaging VIII, ed. A. Mérand, S. Sallum, & J. Sanchez-Bermudez, Vol. 12183, International Society for Optics and Photonics (SPIE), 121831B
  * Husser et al. (2013) Husser, T. O., Wende-von Berg, S., Dreizler, S., et al. 2013, A&A, 553, A6
  * Kammerer et al. (2022) Kammerer, J., Quanz, S. P., Dannert, F., & LIFE Collaboration. 2022, A&A, 668, A52
  * Kasting et al. (2014) Kasting, J. F., Kopparapu, R., Ramirez, R. M., & Harman, C. E. 2014, Proceedings of the National Academy of Science, 111, 12641
  * Kennedy et al. (2015) Kennedy, G. M., Wyatt, M. C., Bailey, V., et al. 2015, ApJS, 216, 23
  * Konrad et al. (2022) Konrad, B. S., Alei, E., Quanz, S. P., et al. 2022, A&A, 664, A23
  * Kovačević et al. (2018) Kovačević, A. B., Pérez-Hernández, E., Popović, L. Č., et al. 2018, MNRAS, 475, 2051
  * Laugier et al. (2020) Laugier, R., Martinache, F., Cvetojevic, N., et al. 2020, A&A, 636, A21
  * Lawson et al. (2008) Lawson, P. R., Lay, O. P., Martin, S. R., et al. 2008, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7013, Optical and Infrared Interferometry, ed. M. Schöller, W. C. Danchi, & F. Delplancke, 70132N
  * Lay (2004) Lay, O. P. 2004, Appl. Opt., 43, 6100
  * Lay (2006) Lay, O. P. 2006, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6268, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. J. D. Monnier, M. Schöller, & W. C. Danchi, 62681A
  * Léger et al. (1996) Léger, A., Mariotti, J. M., Mennesson, B., et al. 1996, Icarus, 123, 249
  * Matsuo et al. (2022) Matsuo, T., Ikari, S., Kondo, H., et al. 2022, Journal of Astronomical Telescopes, Instruments, and Systems, 8, 015001
  * Matsuo et al. (2011) Matsuo, T., Traub, W. A., Hattori, M., & Tamura, M. 2011, ApJ, 729, 50
  * Mennesson & Mariotti (1997) Mennesson, B. & Mariotti, J. M. 1997, Icarus, 128, 202
  * Michelson & Pease (1921) Michelson, A. A. & Pease, F. G. 1921, ApJ, 53, 249
  * Quanz et al. (2022) Quanz, S. P., Ottiger, M., Fontanet, E., et al. 2022, A&A, 664, A21
  * Ranganathan et al. (2022) Ranganathan, M., Glauser, A. M., Birbacher, T., Gheorghe, A. A., & Quanz, S. P. 2022, in Optical and Infrared Interferometry and Imaging VIII, ed. A. Mérand, S. Sallum, & J. Sanchez-Bermudez, Vol. 12183, International Society for Optics and Photonics (SPIE), 121830L
  * ten Brummelaar et al. (2005) ten Brummelaar, T. A., McAlister, H. A., Ridgway, S. T., et al. 2005, ApJ, 628, 453
  * Velusamy et al. (2003) Velusamy, T., Angel, R. P., Eatchel, A., Tenerelli, D., & Woolf, N. J. 2003, in ESA Special Publication, Vol. 539, Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, ed. M. Fridlund, T. Henning, & H. Lacoste, 631–636

Table 1: Parameters of the target system. | Host star | Planet 1 | Planet 2 | Planet 3
---|---|---|---|---
Distance ($pc$) | 10 $pc$ | - | - | -
Radius | 1 $R_{\odot}$ | 1 $R_{\oplus}$ | 1 $R_{\oplus}$ | 1 $R_{\oplus}$
Temperature ($K$) | 5778 | 285 | 330 | 232
Semi-major axis ($AU$) | - | 1 | 0.73 | 1.5
Orbital phase (∘) | - | 0 | -45 | 90
Table 2: Parameters of the LIFE instrument used for planet search and
characterization.

| Planet search phase | Planet characterization phase
---|---|---
Telescope diameter ($m$) | 2 | 2
Configuration | Dual-Bracewell interferometer | Dual-Bracewell interferometer
Imaging baseline ($m$) | 87.3 | 87.3
Nulling baseline ($m$) | 14.55 | 14.55
Instrument throughput | 0.05 | 0.05
Quantum efficiency | 0.7 | 0.7
Wavelength range ($\mu m$) | 8 - 18.5 | 4 - 18.5
Resolving power | 50 | 50
Integration time | 55 hours | 75 days
Field of view ($\frac{\lambda}{D}$) | 1 | 1
Systematic OPD RMS error ($nm$) | 0.75, 3.8, 7.5, 11.3 | 0.75, 3.8, 7.5, 11.3

Quanz et al. (2022); Dannert et al. (2022); Konrad et al. (2022)

Table 3: Signal-to-noise ratios for planet detection with the previous method and PSSD under several OPD values. | Method | OPD error ($nm$) | Planet 1 | Planet 2 | Planet 3
---|---|---|---|---|---
Case 1 | PSSD | 0 (only shot noise) | 10.8 | 14.6 | 3.6
Case 2 | PSSD | 3.8 | 10.9 | 13.3 | 3.6
Case 3 | PSSD | 7.5 | 10.6 | 8.4 | 2.9
Case 4 | PSSD | 11.3 | 6.1 | 6.9 | 3.9
Case 5 | cross-correlation | 0 (only shot noise) | 9.5 | 11.7 | 3.6
Case 6 | cross-correlation | 3.8 | 4.8 | 3.1 | 4.6
Case 7 | cross-correlation | 7.5 | 3.3 | 1.5 | 2.2
Case 8 | cross-correlation | 11.3 | 2.1 | 0.4 | 2.1

Table 4: Signal-to-noise ratios for reconstructed spectra of three planets. | Wavelength ($\mu m$) | OPD error ($nm$) | Subtraction of stellar leak | Planet 1 | Planet 2 | Planet 3
---|---|---|---|---|---|---
Case 1 | 5 | 0 (only shot noise) | No | 0.24 | 0.79 | 0.02
| 7.5 | 0 (only shot noise) | No | 3.2 | 9.1 | 0.76
| 10 | 0 (only shot noise) | No | 9.5 | 21.9 | 2.0
| 15 | 0 (only shot noise) | No | 19.0 | 16.5 | 7.5
Case 2 | 5 | 0.75 | No | 0.16 | 0.43 | 0.01
| 7.5 | 0.75 | No | 3.0 | 11.1 | 0.7
| 10 | 0.75 | No | 8.9 | 21.4 | 2.2
| 15 | 0.75 | No | 16.3 | 16.0 | 7.4
Case 3 | 5 | 7.5 | No | 0.02 | 0.05 | 0.001
| 7.5 | 7.5 | No | 0.74 | 3.2 | 0.2
| 10 | 7.5 | No | 8.0 | 13.4 | 1.6
| 15 | 7.5 | No | 15.9 | 19.2 | 7.5
Case 4 | 5 | 7.5 | Yes | 0.18 | 0.77 | 0.01
| 7.5 | 7.5 | Yes | 3.1 | 8.0 | 0.5
| 10 | 7.5 | Yes | 10.5 | 22.8 | 2.6
| 15 | 7.5 | Yes | 17.2 | 23.2 | 6.7