# Direct detection and characterization of exoplanets using imaging Fourier
transform spectroscopy

Jingwen Zhang(张婧雯) NASA FINESST Fellow  Institute for Astronomy, University of
Hawaii at Manoa
2680 Woodlawn Dr.
Honolulu, HI 96822, USA  Michael Bottom  Institute for Astronomy, University
of Hawaii at Manoa
640 N. Aohoku Pl,
Hilo, HI 96720, USA  Eugene Serabyn  Jet Propulsion Lab, California Institute
of Technology
4800 Oak Grove Dr, Pasadena, CA 91109, USA

(Received xxx; Revised xxx; Accepted xxx)

###### Abstract

Space-based direct imaging provides prospects for detection and spectral
characterization of exoplanets at optical and near-infrared wavelengths.
Integral field spectrographs (IFS) have been historically baselined for these
mission concepts. However, multiple studies have revealed that detector noise
is a serious obstacle for such instruments when observing extremely faint
targets such as Earth-like planets. Imaging Fourier transform spectrographs
(iFTS) are generally less sensitive to detector noise, and have several other
compelling features such as simultaneous imaging and spectroscopy, smaller-
format detector requirements, and variable spectral resolution. To date, they
have not been studied as options for such missions.

In this work, we compare the capabilities of integral field spectrographs and
imaging Fourier transform spectrographs to directly obtain spectra from an
Earth-like planet using analytic and numerical models. Specifically, we
compare the required exposure time to achieve the same signal-to-noise ratio
of the two architectures over a range of detector and optical system
parameters. We find that for a 6-meter telescope, an IFS outperforms an iFTS
at optical wavelengths. In the near-IR, the relative efficiency of an IFS and
iFTS depends on the instrument design and detector noise. An iFTS will be more
efficient than an IFS if the readout noise of near-IR detector is above ~$2-3\
e^{-}\ pix^{-1}\ \rm{frame}^{-1}(t_{\rm{frm}}=1000s)$, which correspond to
half to one-third of detector noise of the state-of-art. However, if the
readout noise is further reduced to below this threshold, the performance of
an IFS will experience a substantial improvement and become more efficient.
These results motivate consideration of an iFTS as an alternative option for
future direct imaging space missions in the near-IR.

astrobiology – planets and satellites: atmospheres – planets and satellites:
detection – planets and satellites: gaseous planets – planets and satellites:
terrestrial planets – techniques: spectroscopic

##  1 Introduction

Spectroscopic observations of exoplanet atmospheres are a powerful tool to
measure chemical compositions and abundances, and provide key insights into
the formation, evolutionary history and habitability of exoplanets. In the
fortuitous case of an exoplanet transit, transmission spectroscopy provides
one avenue for measuring the absorption features imprinted by the planetary
atmosphere on the stellar spectrum (Charbonneau et al., 2002). Direct imaging
offers another way to obtain the reflection or thermal emission spectrum of an
exoplanet by suppressing the host star’s light using a coronagraph or a
starshade. Although simple in principle, direct imaging is challenging due to
the large flux ratio and small separation between a planet and its host star.
For example, a Jupiter twin orbiting a Sun-like star at 10 pc has a planet-
star flux ratio on the order of $10^{\text{-}9}$ in optical at a separation of
$0.52^{\arcsec}$. Using large ground-based telescopes, thermal emission
spectra of several young gas-giant exoplanets at wide orbits have been
obtained, for which planet-star flux ratios are on the order of
$10^{\text{-}4}$ to $10^{\text{-}6}$. Such systems include 51 Eri b (Macintosh
et al., 2015) and the four planets around HR 8799 (Marois et al., 2008; Barman
et al., 2011; Konopacky et al., 2013).

Space-based direct imaging telescopes are expected to bring a breakthrough in
sensitivity by avoiding the limitations imposed by the Earth’s atmosphere. An
important development in this direction is the coronagraph instrument on the
Nancy Grace Roman Space Telescope (Roman-CGI), which will achieve optical
contrast ratios on the order of $10^{\text{-}9}$ (Gersh-Range et al.,
2022).Roman-CGI is expected to detect and characterize gas giant exoplanets
around nearby stars. NASA has also studied mission concepts such as the
Habitable Exoplanet Observatory (HabEx) (Gaudi et al., 2018) and Large UV
Optical Infrared Surveyor (LUVOIR) (The LUVOIR Team, 2019). The Decadal Survey
on Astronomy and Astrophysics 2020 (Astro 2020, National Academies of
Sciences, Engineering, and Medicine 2021) recommended a 6 meter high-contrast
direct imaging telescope as the highest priority mission, currently referred
to as the Habitable Worlds Observatory (HWO), with the goal of detecting and
characterizing around $25$ terrestrial planets around nearby sun-like stars.

Spectroscopy at ultraviolet (UV), visible and near-infrared (NIR) wavelengths
is key to atmospheric characterization of these planets, especially to measure
abundances of biosignature gases such as $\rm{O_{2}}$, $\rm{O_{3}}$,
$\rm{CH_{4}}$. Integral field spectrographs (IFS) were baselined for HabEx and
LUVOIR, and were the choice for Roman-CGI until descoped (Stark et al., 2014;
The LUVOIR Team, 2019; Gong et al., 2019; Martin et al., 2019; Gaudi et al.,
2020). By placing an lenslet array in the focal plane, an IFS can transfer the
light at each individual location to prisms or gratings, which disperse the
target photons into different detector pixels to retrieve the spectral content
(Courtes, 1982). Therefore, one can use an IFS to obtain spatially resolved
spectra from a two dimensional image. However, IFS performance is severely
compromised by detector noise when observing faint targets like terrestrial
exoplanets. Robinson et al. (2016) and Lacy et al. (2019) presented detailed
simulations of space-based coronagraphs equipped with integral field
spectrographs, and both studies concluded that detector noise dominates the
overall error budget, followed by photon noise of the target, residual
starlight, and zodiacal light. For example, for an Earth twin at 5 pc observed
with a 2 m telescope, Robinson et al. (2016) compute that the planet signal
rate of a spectral element at 546-554 nm is
$\raisebox{2.15277pt}{\texttildelow}2\ \rm{e^{\text{-}}\ hr^{\text{-}1}}$,
whereas the dark current is $\raisebox{2.15277pt}{\texttildelow}240\
\rm{e^{\text{-}}\ hr^{\text{-}1}}$, requiring roughly 5000 hrs to achieve a
SNR of 5. Similarly, Lacy et al. (2019) find that for the Roman-CGI, the dark
current is between 1.5 and 3 orders of magnitude higher than the planet photon
rate from a Jupiter twin at 10 pc for all wavelengths between 550 and 1000 nm.
This sensitivity to detector noise is exacerbated by an IFS dispersing photons
into a relatively large number of pixels per spectral element.

An imaging Fourier transform spectrograph (iFTS) provides an alternative way
to obtain spatially resolved spectra by modulating the light intensity in the
time domain through interference using a translatable mirror. As such, it has
a fundamental difference from an integral field spectrograph as it does not
spatially disperse photons. This brings a penalty of higher photon noise and
causes worse performance when observing bright targets in the photon limited
case. But due to the lack of dispersion, an iFTS has the advantage of fewer
pixels and being less sensitive to detector noise when observations are
detector limited. An iFTS also requires a smaller detector than an IFS, with
the detector size matched to the required field of view, and independent of
spectral resolution. Indeed, such instruments can operate with variable
spectral resolution from unity (eg, as an imaging camera) to very high,
depending on the distance the moving mirror is translated (see Appendix C.2
for details). There is also increased complexity in data acquisition, analysis
and engineering to build such instruments, such as the mechanisms to translate
the mirror with sufficient accuracy. In spite of these caveats, the
aforementioned features make it an intriguing possibility for characterizing
faint exoplanets. Given that detector noise presents a significant hindrance
to the direct characterization of explanatory spectra, it is important to
investigate the potential of an iFTS, which exhibits reduced sensitivity to
such noise.

First introduced by Peter Fellgett in 1949, Fourier transform spectrographs
have been successfully applied to the field of planetary exploration on the
Mariner, Voyager, and Cassini spacecrafts (Hanel et al., 1972, 1979; Flasar et
al., 2004) and on ground-based telescopes (Weisstein, 1996). They have also
found use in cosmology, such as in the FIRAS instrument of the Cosmic
Background Explorer satellite (Mather et al., 1990). An imaging Fourier
transform spectrograph was also considered for the James Webb Space Telescope
(Graham et al., 1998), but was not selected for the final instrument suite as
improvements in detector performance, particularly the advent of the HAWAII
arrays (Hodapp et al., 1996), obviated the advantages of the iFTS for the
typical targets observed by Webb. Recent and successful ground-based
implementations can be found in the solar Fourier Transform Spectrometer on
Kitt Peak National Observatory and the SITELLE instrument on the Canada-
France-Hawaii Telescope (CFHT, Grandmont et al. 2012). Unlike IFS’s which
routinely obtain exoplanet spectra on ground based telescopes, IFTS’s have not
yet been demonstrated to work with high contrast imaging systems, though there
is at least one pathfinder instrument, SPIDERS, planned for deployment at
Subaru telescope in the summer of 2024 (Marois et al., 2022). However, there
is no study to date of using an iFTS for space-based exoplanet spectroscopy
and comparing its performance with that of an IFS.

In this paper, we compare the performance of an iFTS to an IFS in the same
spirit as the work of Robinson et al. (2016) and Lacy et al. (2019), using
analytic models and detailed numerical simulations over a range of telescope
and instrument parameters. Specifically, we use the required exposure times to
achieve a signal-to-noise ratio of 5 at continuum as a metric to compare the
capabilities of an iFTS and an IFS in identifying key planetary molecular
bands and atmospheric features at visible and near-infrared (NIR) wavelengths.

The outline of the paper is as follows: in Section 2, we provide a brief
review of the iFTS concept. Section 3 describes the analytical model and
numerical simulations, while Appendices present the detailed derivation and
calculation of our models. Section 4 outline the instrument parameters used in
our simulations. Section 5 presents our results of comparing the performance
of an iFTS and an IFS. Section 6 and Section 7 discusses further avenues for
research and summarizes our conclusions.

##  2 iFTS Concept

Most spectrographs are hardware implementations of the Wiener-Khinchin
theorem, which states that the Fourier transform of the autocorrelation of a
signal yields its power spectral density, ie, its spectrum (Kauppinen &
Partanen, 2001). In a grating spectrograph, for example, the autocorrelation
is computed spatially through the displacement of the signal by the successive
grooves of the grating.

An imaging Fourier transform spectrograph (iFTS) computes the autocorrelation
temporally, by moving a mirror (Kauppinen & Partanen, 2001). In its classic
one-port implementation, an iFTS is essentially a Michelson interferometer,
consisting of a beamsplitter, fixed mirror, moving mirror, and imaging optics
( Figure 1). The incident beam is divided by the beamsplitter into two equal
parts that go toward two mirrors and are reflected back. The path length of
each beam is twice of the length of each arm, and a translation of the moving
mirror by $\frac{\delta}{2}$ creates an optical path difference of $\delta$
between the two beams when they recombine and interfere. Therefore, by moving
the position of one mirror, the optical path difference between the two beams
changes, leading to a modulation of their interfered intensity at the focal
plane. Retrieving spectral information involves scanning the optical path
difference in discrete steps and recording the interfered intensity on the
focal plane at every step, then taking a Fourier transform of the result.

To examine this in detail, consider two electromagnetic waves with electric
fields $E_{1}=A_{1}e^{i(kz-\omega t)}$ and $E_{2}=A_{2}e^{i(kz-\omega
t+\phi)}$ propagating in the same direction $z$. Here $A_{i}$ is the amplitude
of the wave, $k$ is the wave vector, $z$ is the position, $\omega$ is the
angular frequency, $t$ is time, and $\phi$ is the phase difference between two
waves. If $I_{1}$ and $I_{2}$ are the intensities of the two waves, the
intensity of the interfered wave is
$I=I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos\phi$. For the case of a classic one-port
imaging Fourier transform spectrograph, if the input light intensity is
$I_{0}$ and with an ideal beamsplitter ($50\%$ transmissivity and $50\%$
reflectivity), the output will be the inference of two waves with equal
intensity $I_{1}=I_{2}=\frac{I_{0}}{4}$ as input light passes through the
beamsplitter twice. Thus intensity of resulting wave can be expressed:

$I=\frac{I_{0}}{2}(1+\cos\phi)$ (1)

Assuming the spectrum of input light is continuous, it can be described as a
function of wavenumber $B(\nu)$, where $\nu=\frac{1}{\lambda}$ is the
wavenumber, with units of waves per unit distance. It is convenient to present
the spectrum with respect to wavenumber rather than wavelength, with
$B(\lambda)=\nu^{2}B(\nu)$.

The interference signal from the whole spectral band is

$I=\int_{0}^{\infty}\frac{B(\nu)}{2}(1+cos\phi)d\nu$ (2)

The phase difference $\phi$ is determined by the optical path difference
$\delta$ between two beams $\phi=\frac{2\pi\delta}{\lambda}=2\pi\nu\delta$,
thus the interfered intensity as a function of optical path difference is

$I(\delta)=\int_{0}^{\infty}\frac{B(\nu)}{2}(1+cos(2\pi\nu\delta))d\nu$ (3)

Equation (3) is for a classical one-port iFTS which detects half of the
incident light, on average, as the other half goes back to the source after
the second pass through the beamsplitter. Modern implementations use a more
efficient two-port iFTS (see Figure 2), which directs the output beam to two
cameras (Davis et al., 2001). In this design, every photon that enters the
instrument is detected in one of these two output ports, excluding efficiency
losses. A dual-port iFTS theoretically doubles the amount of photons the
instrument collects with respect to a single-port interferometer. But the
detector noise is also increased since the system requires two cameras. The
recorded intensities at two outputs are anti-phased:

$I_{\pm}(\delta)=\int_{0}^{\infty}\frac{B(\nu)}{2}(1\pm\cos(2\pi\nu\delta))d\nu$
(4)

where the $\pm$ signs correspond to the two output ports. The sum and
difference of the two interference signals at two ports are

$I_{\sum}(\delta)=I_{+}(\delta)+I_{-}(\delta)=\int_{0}^{\infty}B(\nu)d\nu$ (5)

$I_{\Delta}(\delta)=I_{+}(\delta)-I_{-}(\delta)=\int_{0}^{\infty}B(\nu)cos(2\pi\nu\delta)d\nu$
(6)

The sum of the two port signals $I_{\sum}(\delta)$is the integral of the full
band intensity, which is equivalent to the broadband (i.e. defined by any
upstream filter) image of the targets. The difference signal
$I_{\Delta}(\delta)$ is called the interferogram, and can be used to obtain
the spectrum. Therefore, one gets a concurrent image (by summing the two port
signals) and spectrum (by subtracting one port signal from another) with a
two-port iFTS.

In practice, spectra are only obtained in the positive wavenumber range.
However, the operation of the Fourier transform integrates the spectrum over
all wavenumbers from $-\infty$ to $\infty$. Thus, it is useful to construct a
symmetric spectrum $B_{e}(\nu)$ from the asymmetric spectrum $B(\nu)$ (see
Figure 3):

$B_{e}(\nu)=\frac{1}{2}[B(-\nu)+B(\nu)].$ (7)

The integral of $B(\nu)$ over positive wavenumbers is equivalent to the
integral of $B_{e}(\nu)$ over negative and positive wavenumbers. We can modify
Eqn. 6 to

$I_{\Delta}(\delta)=\int_{-\infty}^{\infty}B_{e}(\nu)cos(2\pi\nu\delta)d\nu$
(8)

$I_{\Delta}(\delta)$ and $B_{e}(\nu)$ form a Fourier transform pair.
111Technically, Eq.7 $\&$ 8 present a cosine transform pair. However, due to
interferogram being a real-valued signal and the nearly negligible imaginary
component of the spectrum when the interferogram exhibits symmetry, we refer
it to as a Fourier transform pair. Fourier transforming the interferogram
$I_{\Delta}(\delta)$ and taking the real part yields the spectral information:

$B_{e}(\nu)=\int_{-\infty}^{\infty}I_{\Delta}(\delta)\cos(-2\pi\nu\delta)d\delta$
(9)

Figure 1: The data acquisition of an iFTS. The core optical configuration of
an iFTS is a Michelson interferometer with a fixed mirror and a movable
mirror. By moving the position of one mirror and taking images at every step,
one can get the interferogram for every pixel in the image as a function of
optical path difference. The measured interferogram contains all spectral
information and can be Fourier transformed to obtain the spectrum of the input
light. Panel (a) is adapted from Drissen et al. (2014).

The Fourier transform of interferogram distributes the intensity into both
positive and negative spectral domain. The output, symmetric spectrum
$B_{e}(\nu)$ has an amplitude half that of the original input spectrum
$B(\nu)$.

The above derivations hold for the interferograms of each pixel in the frame.
If imaging optics are used to create focal planes at the two output ports,
each pixel will measure an interferogram, with independent spectral
information corresponding to the different points on the sky. In this way, an
iFTS is able to retrieve spatially resolved spectral information from a two-
dimensional focal plane.

Practical operation of an iFTS involves choices about the scan length, mirror
step size, and time spent per step, which affect the resolution and signal-to-
noise ratio of the resulting spectrum. We direct the reader to the appendix C
for a more detailed discussion of these issues.

Figure 2: Schematic optical layout of a two-port iFTS with an ideal
beamsplitter ($50\%$ transmissivity and $50\%$ reflectivity). The light passes
through the beamsplitter twice: the input light first splits into two equal
beams that go toward two mirrors. After being reflected by the mirrors, the
two beams pass through the beamsplitter again and each of them is split again.
The input light all goes to one of two detectors, ignoring the light loss from
optical elements. Figure 3: Construction of a symmetric spectrum $B_{e}(\nu)$
from an asymmetric spectrum $B(\nu)$ for Fourier transform.

##  3 Methodology

###  3.1 Detector noise model

Detector noise is of primary importance in space-based exoplanet imaging and
especially spectroscopy. In this work, we consider three types of detector
noise: dark current, readout noise and clock-induced charge, ignoring some
more complex effects such as charge trapping, cosmic ray amplification, and
dynamic range compression. Dark current has a Poisson distribution with a
variance per hour as $\sigma_{DC}^{2}=DC$, where $DC$ is the dark current rate
in number of electrons per pixel per hour. Read noise is Gaussian with a
variance given by the single-frame readout noise $RN^{2}$ (in electrons per
pixel per frame) multiplied by the number of frames,
$\sigma^{2}_{RN}=RN^{2}N_{\rm{frm}}$ following the fact that
$var(\sum_{N}X_{i})=Nvar(X_{i})$. The number of frames $N_{\rm{frm}}$ per hour
can be approximated by $1/t_{\rm{frm}}$ where $t_{\rm{frm}}$ is the
integration time per frame in hours. Therefore, the read noise variance per
hour is $\sigma^{2}_{RN}=\frac{RN^{2}}{\rm{t_{frm}}}$. Similarly, clock-
induced charge has a Poisson distribution with a variance per hour equal to
the clock-induced charge $CIC$ (in electrons/pixel/frame) multiplied by the
number of frames per hour, $\sigma^{2}_{CIC}=\frac{CIC}{t_{\rm{frm}}}$.
Combining these terms, the detector noise variance per pixel per hour can be
expressed as

$\sigma^{2}_{d}=DC+\frac{RN^{2}}{t_{\rm{frm}}}+\frac{CIC}{t_{\rm{frm}}}$ (10)

Unfortunately, it is not possible to eliminate the effects of read noise and
clock-induced charge by taking very long frames, due to cosmic ray saturation.
The above computation of the “variance rate” has been used in previous studies
(Robinson et al., 2016; Lacy et al., 2019; The LUVOIR Team, 2019) for IFS
instruments, and we adopt it for the iFTS as well. We use $t_{\rm{frm}}$ of
100 and 1000 seconds in our simulation for both the IFS and iFTS, which are
approximately the maximum exposure times baselined for the Roman-CGI and JWST.

The number of pixels involved in the spectral observations is another factor
determining the noise. As an IFS and iFTS employ different operational
principles, they have different numbers of pixels contributing to an
observation. For an IFS, spatially resolved spectroscopic observations are
obtained by placing a grid of lenslets in the focal plane which transfer the
light to dispersive elements (e.g. prisms or gratings). The light from each
lenslet is spectrally dispersed and recorded on different pixels. The number
of contributing pixels per spectral element is the product of the number of
lenslets $n_{\rm{len}}$ involved and the number of pixels per spectral element
per lenslet $n_{\rm{pix}}$.

Assuming the lenslet array is Nyquist sampled at the wavelength $\lambda_{0}$
(see Figure 4),the angular diameter of the lenslet $\theta_{\rm{len}}$ is

$\theta_{\rm{len}}=\frac{\lambda_{0}}{2D}$ (11)

where $D$ is the telescope diameter. We follow the optical design in LUVOIR
report to Nyquist sample the lenslet array at the shortest wavelength of
visible and NIR bands ($\lambda_{0}=540nm$ in visible and $\lambda_{0}=1000nm$
in NIR). At wavelength $\lambda$, the angular diameter of the PSF is
$\rm{\theta_{PSF}=\lambda/D}$. The number of lenslets inside the point-spread
function (PSF) at wavelength $\lambda$ is proportional to the square of the
wavelength:

$n_{\rm{len}}=\left(\frac{\theta_{\rm{PSF}}}{\theta_{\rm{len}}}\right)^{2}=4\left(\frac{\lambda}{\lambda_{0}}\right)^{2}$
(12)

Therefore, at the Nyquist wavelength $\lambda_{0}$, there are four lenslets
within the PSF, and this number increases at longer wavelengths.

The prisms or gratings disperse photons into different pixels. The number of
pixels contributing to one spectral element for each lenslet $n_{pix}$ is the
product of pixels in horizontal direction $n_{h}$ and vertical direction
$n_{v}$. Thus the number of pixels contributing to one spectral element at
wavelength $\lambda$ is

$N_{\rm{IFS}}=n_{\rm{pix}}n_{\rm{len}}=4n_{\rm{pix}}\left(\frac{\lambda}{\lambda_{0}}\right)^{2}$
(13)

Then the detector noise variance per spectral element per hour at wavelength
$\lambda$ is

$\sigma^{2}_{\rm{IFS}}=\sigma^{2}_{d}\times
N_{\rm{IFS}}=4\sigma^{2}_{d}n_{pix}\left(\frac{\lambda}{\lambda_{0}}\right)^{2}$
(14)

On the other hand, an iFTS does not disperse photons, so the number of
contributing pixels is determined by the size of PSF aperture. If a pixel has
an angular diameter $\theta_{\rm{pix}}$ and the detector is Nyquist sampled at
wavelength $\lambda_{0}$, then

$\theta_{\rm{pix}}=\frac{\lambda_{0}}{2D}$ (15)

where we use the same Nyquist wavelength $\lambda_{0}$ as the IFS. Because a
two-port iFTS uses two cameras, it requires double the contributing pixels:

$N_{\rm{iFTS}}=2\left(\frac{{\rm{\theta_{PSF}}}}{\rm{\theta_{pix}}}\right)^{2}=8\left(\frac{\lambda}{\lambda_{0}}\right)^{2}$
(16)

The detector noise in this case is introduced at the interferogram. The
detector noise variance per hour at wavelength $\lambda$ is

$\sigma^{2}_{\rm{iFTS}}=\sigma^{2}_{d}\times
N_{\rm{FTS}}=8\sigma^{2}_{d}\left(\frac{\lambda}{\lambda_{0}}\right)^{2}$ (17)

Figure 4: Schematic illustration showing the number of pixels contributing to
one spectral element of an IFS and an iFTS. (a) An IFS works in the way of
placing a lenslet array in the focal plane, which transfer the light from
individual location to the prims or gratings. The prims or gratings disperse
light into different pixels to get spectral information. The number of pixel
contributing to an spectral element equals to the number of lenslets inside
PSF area $N_{len}$ multiplied by the number of pixels per spectral element per
lenslet $n_{pix}$. (c) An iFTS does not disperse photons so that the number of
pixels contributing to each spectral elements is those inside the PSF area.
Because an two port iFTS uses two cameras, we multiply the pixel number by
two.

###  3.2 iFTS Simulation

In section 2, we consider the spectrum and interferogram as continuous
function and the Fourier transform is expressed as integrals from $-\infty$ to
$\infty$. In practice, the measurement of a signal usually gives us a finite
number of data measured at discrete points. Consequently, the integral from
$-\infty$ to $\infty$ are replaced by a sum of discrete points from $-N$ to
$N$, which is called the discrete Fourier transform. Here, we will present our
numerical simulations of these digital Fourier transforms and discuss the
signal-to noise ratio model.

Let $S(\nu_{i})$ denote the discrete input spectrum coming into the iFTS,
where the wavenumber $\nu_{i}=i\Delta\nu$, for $i=0,1,2...N$. As discussed in
section 2, it is convenient to use the symmetric spectrum
$S_{e}(\nu_{i})=\frac{1}{2}[S(\nu_{i})+S(-\nu_{i})]$, where
$i=-N,-N+1...-1,0,1,...N$. The units for $S(\nu_{i})$ and $S_{e}(\nu_{i})$ are
photon numbers per wavenumber per hour.

Based on Eqn. 8, the interferogram is obtained from the difference between
intensities at two cameras:

Figure 5: Flow chart of the numerical simulation.

$\begin{split}F_{\Delta}(\delta_{j})&=F_{+}(\delta_{j})-F_{-}(\delta_{j})\\\
&=T(\delta_{j})\sum_{i=-N}^{N}{S_{e}(\nu_{i})\cos{(2\pi\nu_{i}\delta_{j})}}\Delta\nu\\\
\delta_{j}&=j\Delta\delta,j=-N,-N+1..,-1,0,1...N-1,N\end{split}$ (18)

where $\delta_{j}$ is the optical path difference between the two beams (with
a unit of nm) and $T(\delta_{j})$ is the exposure time at every scanning step
(with a unit of hour). The simulated interferogram $F_{\Delta}(\delta_{j})$
has a unit of photon numbers. The interferogram is double-sided and symmetric
about the zero optical path difference. We also add photon noise and detector
noise to the intensity at two cameras at every step. The difference of
intensity at two cameras has the same noise variance as the sum of intensity,
which is the the total intensity of the incident light during the exposure
time (photon noise) and the detector noise.

$\begin{split}\sigma^{2}[F_{\Delta}(\delta_{j})]&=\sigma^{2}[F_{\sum}(\delta_{j})]\\\
&=T(\delta_{j})\sum_{i=-N}^{N}{S_{e}(\nu_{i})\Delta\nu+\sigma^{2}_{\rm{iFTS}}}T(\delta_{j})\end{split}$
(19)

where $\sigma^{2}[X]$ represents the noise variance of the variable X.

We then apply the Fourier transform to the simulated interferogram
$F_{\Delta}(\delta_{j})$ and take the real part as the measured output
spectrum. Based on Eqn.7 and 8, the output spectrum will be a symmetric
spectrum in both negative and positive frequencies. 222The imaginary part is
nearly zero because the interferogram is double sided and symmetric.

$O_{\rm{FTS}}(\nu_{i})=\sum_{j=-N}^{N}F_{\Delta}(\delta_{j})cos(-2\pi\nu_{i}\delta_{j})\Delta\delta$
(20)

The scans only cover a finite range of the interferogram instead of from
$-\infty$ to $\infty$. The truncation of the interferogram in a range of
$(-L,L)$ leads to the convolution of the original spectrum with an Instrument
Line Shape (ILS) function. If we assign a total exposure time
$T_{tot}=\sum_{-N}^{N}T(\delta_{j})$ evenly at every step, the truncated
interferogram can be seen as the product of a infinite interferogram with a
boxcar function (see details in Appendix C.2). Thus, the output spectrum is
the convolution of the $S_{e}(\nu_{i})$ with a sinc function times the total
exposure time

$O_{\rm{FTS}}(\nu_{i})=T_{tot}*[S_{e}(\nu_{i})\circledast 2Lsinc(2L\nu_{i})]$
(21)

If the full width at half maximum (FWHM) of the ILS is $\Delta\nu_{\rm{ILS}}$,
the intensity of output spectrum will be

$O_{\rm{FTS}}(\nu_{i})=S_{e}(\nu_{i})\Delta\nu_{\rm{ILS}}T_{\rm{tot}}$ (22)

where the output spectrum has a unit of photon numbers per spectral element
(in total time). Based on the Rayleigh energy theorem, the total noise
variance in spectral domain equals to the total noise variance in the
interferogram domain. But as we only take real part of the Fourier transform
output and discard the imaginary part, the variance of the output spectrum is
only half of the total variance of the interferogram (for details see chapter
8 in Davis et al. 2001).

$\frac{1}{2N}\sum_{-N}^{N}\sigma^{2}[O_{\rm{FTS}}(\nu_{i})]=\frac{1}{2}\sum_{-N}^{N}\sigma^{2}[F_{\Delta}(\delta_{j})]$
(23)

As the spectral noise is white and independent of the wavenumbers, the sum
sign and $2N$ in the denominator in the left term of Eqn. 23 can be canceled
out and we have

$\begin{split}\sigma^{2}[O_{\rm{FTS}}(\nu_{i})]&=\frac{1}{2}\sum_{-N}^{N}\sigma^{2}[F_{\Delta}(\delta_{j})]\\\
&=\frac{T_{\rm{tot}}\sum_{-N}^{N}{S_{e}(\nu)\Delta\nu+\sigma^{2}_{\rm{iFTS}}}T_{\rm{tot}}}{2}\end{split}$
(24)

In the numerator of Eqn. 24, the first term represents the total photon count
entering the iFTS during the entire exposure (photon noise), while the second
term accounts for the total detector noise variance. Combining Eq 22 and Eq
24, the signal-to-noise ratio of the output spectra can be described by

$\rm{SNR_{FTS}(\nu_{i})}=\frac{S_{e}(\nu_{i})\Delta\nu_{\rm{ILS}}T_{\rm{tot}}}{\sqrt{\frac{1}{2}[T_{\rm{tot}}\sum_{-N}^{N}{S_{e}(\nu_{i})\Delta\nu+\sigma^{2}_{\rm{iFTS}}}T_{\rm{tot}}]}}$
(25)

The above expression is a symmetric function from negative to positive
wavenumbers. If we go back to the convention of the input spectrum
$S(\nu_{i})$ only spanning in positive frequencies, we then obtain

$\rm{SNR_{FTS}(\nu_{i})}=\frac{S(\nu_{i})\Delta\nu_{\rm{ILS}}T_{\rm{tot}}}{\sqrt{2[T_{\rm{tot}}\sum_{0}^{N}{S(\nu_{i})\Delta\nu+\sigma^{2}_{\rm{iFTS}}}T_{\rm{tot}}]}}$
(26)

In above equations, $S(\nu_{i})$ denotes the input signal fed into the iFTS,
which in these simulations consists of the planet spectrum $c_{pl}(\nu_{i})$
and Solar System zodiacal light $c_{z}(\nu_{i})$, exozodiacal light
$c_{ez}(\nu_{i})$ and residual starlight $c_{sp}(\nu_{i})$ (details see
Appendix A). We assume stellar flux is suppressed by a factor $C$ at the image
plane after a coronagraph, e.g. $C$ is $10^{-10}$ for the Habitable World
Observatory. We also consider light loss due to the finite throughput of
coronagraphs $\tau_{core}$, spectrograph optics $\tau_{optical}$ and the
quantum efficiency of the detectors $\tau_{\rm{QE}}$ (details see Appendix B).
We use the same optical throughput for an iFTS and an IFS for consistency.
Figure 5 shows a flow chart to illustrate the procedure and workflow of our
numerical simulations.

Practical observations will involve background subtraction of some sort, to
remove systematic errors from speckles, zodi, exozodi, and detector noise. In
this case, we assume the observation uses the technique of roll deconvolution
or angular differential imaging (ADI), where the target star is observed at
two roll angles, allowing for background subtraction with the planet at two
different positions (we assume smooth zodi and exozodi backgrounds). The
effect of this subtraction is to increase the background variance by a factor
of two:333 Other background subtraction techniques using reference stars or
models can result in smaller noise increases, but these involve assumptions
about the targets and the ability to model systematic errors (Lacy et al.,
2019).

$\rm{SNR_{FTS}^{{}^{\prime}}(\nu_{i})}=\frac{c_{pl}(\nu_{i})\Delta\nu_{\rm{ILS}}T_{\rm{tot}}}{\sqrt{2T_{\rm{tot}}[\sum_{0}^{N}c_{pl}(\nu_{i})\Delta\nu+2\sum_{0}^{N}c_{b}(\nu_{i})\Delta\nu+2\sigma^{2}_{\rm{iFTS}}}]}$
(27)

where we use $c_{b}(\nu_{i})=c_{z}(\nu_{i})+c_{ez}(\nu_{i})+c_{sp}(\nu_{i})$
to denote the total astrophysical backgrounds.

Theoretical derivations of SNRs in Fourier transform spectrographs are
provided in Treffers (1977); Davis et al. (2001); Graham et al. (1998) and
Bennett (1999), but the latter two are inconsistent with the former two,
claiming a better SNR by a factor of $\sqrt{2}$. To address this discrepancy,
we carried out Monte Carlo simulations of the FTS data acquisition and
transformation process, resulting in the posterior distribution of intensity
for each spectral element. The numerical SNR is computed as the mean of the
posterior distribution divided by its standard deviation. Fig. 6 exhibits two
such sample simulations. The simulated SNRs are found to be in very good
agreement with the theoretical predictions of Treffers (1977) and Davis et al.
(2001), but do not yield the extra $\sqrt{2}$ noise reduction predicted by
Graham et al. (1998) and Bennett (1999).

|  Name | Earth twin
---|---|---
$R_{p}$ |  planet radius | $1\ R_{\oplus}$
$a$ |  planet orbit distance | 1 AU
$d$ |  system distance from Earth | 10 pc
$\alpha$ |  planet orbit phase angle | $60^{\circ}$
$T_{\star}$ |  temperture of the host star | 5770 K
$F_{o,V}$ |  standard zero-magnitude V-band specific flux | $3.63\times 10^{-11}\rm{W\ m^{-2}\ nm^{-1}}$
$F_{\odot,V}(1AU)$ |  solar flux density at 1 AU in V band | $1.86\times 10^{3}\rm{W\ m^{-2}\ nm^{-1}}$
$M_{z,V}$ |  V-band zodiacal light surface brightness | $23\ \rm{mag\ arcsec^{-2}}$
$M_{ez,V}$ |  V-band exozodiacal light surface brightness | $22\ \rm{mag\ arcsec^{-2}}$
$N_{ez,V}$ |  number of exozodis in exoplanetary disk | 1

###  3.3 IFS Simulation

For simulating the IFS performance, it is more convenient to work in units of
wavelength rather than wavenumber, hence $S(\lambda)$ denotes the input
spectrum. We divide the input spectrum into several spectral elements and the
width of each element depends on the resolving power $R$:
$\Delta\lambda_{\rm{eff}}=\frac{\lambda}{R}$. We then integrate the spectrum
in each element and add the detector noise and multiply them with the total
exposure time $T_{\rm{tot}}$. The signal is:

$\begin{split}O_{\rm{IFS}}(\lambda)&=T_{tot}\int_{\lambda}^{\lambda+\Delta\lambda_{\rm{eff}}}S(\lambda)d\lambda\\\
&\approx S(\lambda)\Delta\lambda_{\rm{eff}}T_{tot}\end{split}$ (28)

The noise variance of each element is the sum of photon counts and detector
noise

$\sigma^{2}[O_{\rm{IFS}}]=S(\lambda)\Delta\lambda_{\rm{eff}}T_{tot}+\sigma^{2}_{\rm{IFS}}T_{\rm{tot}}$
(29)

Therefore, the SNR for an IFS is

$\rm{SNR}_{\rm{IFS}}=\frac{S(\lambda)\Delta\lambda_{\rm{eff}}T_{tot}}{\sqrt{S(\lambda)\Delta\lambda_{\rm{eff}}T_{tot}+\sigma^{2}_{\rm{IFS}}T_{\rm{tot}}}}$
(30)

When considering using ADI for the background subtraction, the SNR is:

$\rm{SNR}_{\rm{IFS}}^{\prime}=\frac{c_{pl}(\lambda)\Delta\lambda_{\rm{eff}}T_{tot}}{\sqrt{c_{pl}(\lambda)\Delta\lambda_{\rm{eff}}T_{tot}+2c_{b}(\lambda)\Delta\lambda_{eff}T_{tot}+2\sigma^{2}_{\rm{IFS}}T_{\rm{tot}}}}$
(31)

Figure 6: Examples of simulation result for an Earth twin at 10 pc in a
visible (710-800 nm, R=140, top) and NIR bandpass (1050-1250 nm, R = 70,
bottom). (a)$\&$(d): simulated interferogram of a two-port iFTS, with a Boxcar
apodization function. Red points are the difference of intensity at two
outputs at each scanning step, with error bars from photon noise and detector
noise. (b)$\&$(e) present simulated spectra for an iFTS (red) and an IFS
(blue). (c)$\&$(f) present the analytical and numerical signal-to-noise ratios
of an iFTS (red) and an IFS (blue). Here, we use a 6 m sized telescope with
dark current of $0.77\ \rm{e^{-}/hr/pix}$ for the visible camera and readout
noise of $5.9\ \rm{e^{-}/frame/pix}$ for the NIR camera with a frame time of
1000s. The SNRs are computed for a 500 hrs exposure time analytically and
numerically. Other instrument parameters see in Table 3.4 Figure 7: Comparison
of plant count rate and major noise rates for the fiducial planet-star system
using the following configurations: (a) IFS within the visible band (700-820
nm); (b) iFTS within the visible band (700-820 nm); (c) IFS within the NIR
band (1050-1200 nm); (d) iFTS within the NIR band (1050-1200 nm). We assume
the fiducial target to be an Earth twin located at a distance of 10 pc (
details see Table LABEL:tab:rvs). The solid lines depict the rate of planetary
signal and photon noise from the planet, residual starlight, Solar System
Zodiacal light, and exo-zodiacal light. The dashed lines represent the dark
currents of the Roman EMCCD ($0.77\ \rm{e^{-}/hr/pix}$) in the visible band
and the readout noise of HxRG on JWST ($5.9\ \rm{e^{-}/frame/pix}$) in the NIR
band. In the case of IFS, we present the detector noise rate for various
numbers of pixels assigned to each spectral element. Other instrument
parameters see in Table 3.4

We compared our IFS simulations to those in Robinson et al. (2016) and Lacy et
al. (2019) and obtained values for the planet photon flux and noise count rate
matching theirs within $5\%$. Figure 6 also shows that our IFS numerical
simulations agree with the analytical models.

###  3.4 Signal and noise comparison between iFTS and IFS

Figure 7 illustrates the signal rate from an Earth twin at 10 pc away and
primary noise count rates using a 6-meter telescope with an IFS and an iFTS,
respectively. In the visible, the dominant contributor to noise for the IFS is
the dark current, which surpasses the planet signal by nearly twofold when
adopting the EMCCD dark current value from the Roman-CGI. In contrast, the
iFTS has higher photon noise in the visible band due to the integrated photon
noise across the entire bandpass.

In the NIR, the readout noise is the most prominent noise source for both an
IFS and an iFTS, exceeding the planet signal and photon noise by nearly one
order of magnitude when using the effective noise in 1000s of the HxRG
detector in JWST. Combining Eq.13, Eq.16, Eq.23 and Eq.27 reveals that the
ratio of detector noise variance between an iFTS and an IFS is determined by
the number of pixels allocated to an individual spectral element of the IFS,
denoted as $n_{\rm{pix}}$:

$\frac{2\sigma^{2}_{\rm{IFS}}T_{\rm{tot}}}{4\sigma^{2}_{\rm{iFTS}}T_{\rm{tot}}}=\frac{n_{pix}}{4}$
(32)

The above expression implies that if an IFS uses more than 4 pixels per
spectral element per lenslet, its detector noise will be higher than that of
an iFTS.

If the two instruments have the same resolution, then the required exposure
time ratio between an IFS and an iFTS to achieve the same $\rm{SNR}$ is given
by

$\frac{T_{\rm{IFS}}}{T_{\rm{iFTS}}}=\frac{c_{pl}(\lambda)\Delta\lambda_{\rm{eff}}+2c_{b}(\lambda)\Delta\lambda_{\rm{eff}}+2\sigma^{2}_{\rm{IFS}}}{2[\sum
c_{pl}(\nu)\Delta\nu+2\sum c_{b}(\nu)\Delta\nu+2\sigma^{2}_{\rm{iFTS}}]}$ (33)

The required exposure time ratio is determined by total noise variance of the
two spectrographs.

Simulation1 Fig.6$\&$7 |  Example | $10\%$ |  $\rm{VIS}:24\%$ $\rm{NIR}:\ 32\%$ |  $\rm{VIS}:6$ $\rm{NIR}:\ 4$ |  $\rm{VIS}:140$ $\rm{NIR}:\ 70$ |  $\rm{i_{d}:0.77e^{-}pix^{-1}hr^{-1}}$ $i_{r}:0$ $\rm{q_{cic}:0.01e^{-}pix^{-1}frame^{-1}}$ $(t_{\rm{frm}}=100s)$ |  $\rm{i_{d}:0}$ $\rm{i_{r}:5.9e^{-}pix^{-1}frame^{-1}}$ ($t_{\rm{frm}}=1000s)$ $\rm{q_{cic}:0}$
---|---|---|---|---|---|---|---
Simulation2a Fig.8 |  General in VIS | $10\%$ |  $\rm{VIS}:24\%$ |  $\rm{VIS}:6$ |  $\rm{VIS}:140$ |  $\rm{i_{d}:0.1\sim 3.4\ e^{-}pix^{-1}hr^{-1}}$ $i_{r}:0$ $\rm{q_{cic}:0.01e^{-}pix^{-1}frame^{-1}}$ ($t_{\rm{frm}}=100s)$ | –
Simulation2b Fig.9 |  General in VIS | $3\%-40\%$ |  $\rm{VIS}:24\%$ |  $\rm{VIS}:2-10$ |  $\rm{VIS}:140$ |  $\rm{i_{d}:0.1\sim 3.4\ e^{-}pix^{-1}hr^{-1}}$ $i_{r}:0$ $\rm{q_{cic}:0.01e^{-}pix^{-1}frame^{-1}}$ $(t_{\rm{frm}}=100s)$ | –
Simulation3a Fig.10 |  General in NIR | $10\%$ |  $\rm{NIR}:\ 32\%$ |  $\rm{NIR}:\ 6$ |  $\rm{NIR}:\ 70$ | – |  $i_{d}:0$ $\rm{i_{r}:0.1\sim 10e^{-}pix^{-1}frame^{-1}}$ $(t_{\rm{frm}}=1000s)$ $\rm{q_{cic}:0}$
Simulation3b Fig.11 |  General in NIR | $3\%-40\%$ |  $\rm{NIR}:\ 32\%$ |  $\rm{NIR}:\ 2\sim 10$ |  $\rm{NIR}:\ 70$ | – |  $\rm{i_{d}:0}$ $\rm{i_{r}:0.1\sim 10e^{-}pix^{-1}frame^{-1}}$ $(t_{\rm{frm}}=1000s)$ $\rm{q_{cic}:0}$
Simulation4a Fig.12 |  General $+$ varied noise | $10\%$ |  $\rm{VIS}:24\%$ $\rm{NIR}:\ 32\%$ |  $\rm{VIS}:6$ $\rm{NIR}:\ 6$ |  $\rm{VIS}:140$ $\rm{NIR}:\ 70$ |  $\rm{i_{d}:0.1\sim 3.4\ e^{-}pix^{-1}hr^{-1}}$ $i_{r}:0$ $\rm{q_{cic}:0.01\ e^{-}pix^{-1}frame^{-1}}$ $(t_{\rm{frm}}=100s)$ |  $\rm{i_{d}:0}$ $\rm{i_{r}:0.1\sim 10\ e^{-}pix^{-1}frame^{-1}}$ $(t_{\rm{frm}}=1000s)$ $\rm{q_{cic}:0}$
Simulation4a Fig.13 |  HWO \+ current detector noise | $10\%$ |  $\rm{VIS}:24\%$ $\rm{NIR}:\ 32\%$ |  $\rm{VIS}:6$ $\rm{NIR}:\ 6$ |  $\rm{VIS}:140$ $\rm{NIR}:\ 70$ |  $\rm{i_{d}:0.77\ e^{-}pix^{-1}hr^{-1}}$ $i_{r}:0$ $\rm{q_{cic}:0.01\ e^{-}pix^{-1}frame^{-1}}$ $(t_{\rm{frm}}=100s)$ |  $\rm{i_{d}:0}$ $\rm{i_{r}:5.9\ e^{-}pix^{-1}frame^{-1}}$ $(t_{\rm{frm}}=1000s)$ $\rm{q_{cic}:0}$
Simulation4b Fig.13 |  HWO \+ future detector noise | $10\%$ |  $\rm{VIS}:24\%$ $\rm{NIR}:\ 32\%$ |  $\rm{VIS}:6$ $\rm{NIR}:\ 6$ |  $\rm{VIS}:140$ $\rm{NIR}:\ 70$ |  $\rm{i_{d}:0.1\ e^{-}pix^{-1}hr^{-1}}$ $i_{r}:0$ $\rm{q_{cic}:0.0013\ e^{-}pix^{-1}frame^{-1}}$ ($t_{\rm{frm}}=100s)$ |  $\rm{i_{d}:0}$ $\rm{i_{r}:0.5\ e^{-}pix^{-1}frame^{-1}}$ ($t_{\rm{frm}}=1000s)$ $\rm{q_{cic}:0}$

11footnotetext: Astrophysical parameters of targets are listed in table
LABEL:tab:rvs

22footnotetext: $\tau_{core}$ is the core throughput of coronagraph or
starshade, not including optical throughput due to other optical opponents and
detector quantum efficiency.

33footnotetext: $\tau_{\rm{optical}}$ is the throughput of optical component
of the system.

44footnotetext: $n_{\rm{pix}}$ is the number pf pixels contributing to one
spectral element in IFS simulation.

55footnotetext: $i_{d}$: dark current count rate, $i_{c}$: readout noise count
rate, $q_{cic}$: clock induced charge, $\rm{t_{frm}}$: integration time per
frame.

66footnotetext: Other parameters: coronagraph contrast $10^{-10}$; Nyquist
wavelength: VIS 540 nm, NIR 1000 nm; Telescope diameter: 6m

##  4 Simulation Setup

###  4.1 General Simulation Setup

Table LABEL:tab:rvs lists the astrophysical properties of our fiducial target
system. We consider the target as an Earth-like planet around a sun-like star
at 1AU at 10 pc away.

As shown by the models presented in Appendix A, the signal-to-noise ratio is
determined by instrument parameters including the telescope size, the detector
noise rate, the throughput of the coronagraph. To understand their effect on
the relative sensitivity of an IFS to an iFTS to detecting an Earth-like
planet around a Sun-like star, we first simulate a general space-based mission
with varying instrument parameters including coronagraph throughput
$\tau_{core}$ and the pixel number assigned to each IFS spectral element
$n_{pix}$. Simulations $2\&3$ in Table 3.4 present two such examples in a
visible band (720-800 nm) and an NIR band (1050-1200 nm), respectively. We
consider $10\%$ bandwidth as it becomes increasingly challenging for a
coronagraph to maintain high-contrast across broader bandpasses. We assume a
telescope diameter of 6 meter and coronagraph contrast of $C=10^{-10}$. We
consider the system throughput as the product of optical throughput
$\tau_{optical}$, coronagraph core throughput $\tau_{core}$ and detector
quantum efficiency $\tau_{\rm{QE}}$ (details see Appendix B). We use the
average optical throughput of $\tau_{optical}=23.8\%$ at visible wavelengths
and $\tau_{optical}=32.4\%$ in the NIR wavelengths from the end-to-end studies
of LUVOIR (The LUVOIR Team, 2019). We also adopt the spectral resolution
values from LUVOIR (The LUVOIR Team, 2019), which are 140 in the optical and
70 in the infrared. The coronagraph core throughput ranges between $3\%$ and
$40\%$, while the IFS $n_{pix}$ varies from 2 to 10.

Table 3.4 also presents the detector noise rate in the simulations. We
consider two levels of detector noise, referred to as “current” and “future”
levels. The current level refers to state of the art (SoA) values from
detectors used in operating or soon-to-launch space missions. For the future
detector noise, we adopt HabEx and LUVOIR requirements. In the visible bands
(500-1000nm), we consider the dark current and clock-induced charge (CIC) of
the electron multiplying charge-coupled device (EMCCD). For the current level,
we adopt values from the coronagraph instrument on Roman, which uses the e2v
CCD201-20 EMCCD for both imaging and spectrograph cameras. There are several
reference values of its dark current444Recently, it was found that operating
the EMCCD in inverted mode led to a remarkable decrease in dark current to
$0.05-0.1\ \rm{e^{-}\ hr^{-1}\ pix^{-1}}$. (P. Morrissey, private
communication). This may become the flight baseline. However, our detector
numbers are based on the most recent published work, Morrissey et al. (2023).
If the dark current remains this low, CIC will be the dominant contributing
noise, but there will be no change to the conclusions of this paper regarding
the unsuitability for an iFTS in the optical. (Morrissey et al., 2023):
current best estimate (CBE, $0.77\ \rm{e^{-}\ hr^{-1}\ pix^{-1}}$), beginning
of life requirement ($\rm{BOL\\_REQ}$, $1.65\ \rm{e^{-}\ hr^{-1}\ pix^{-1}}$)
and end of life requirement ($\rm{EOL\\_REQ}$, $3\ \rm{e^{-}\ hr^{-1}\
pix^{-1}}$). Similarly, the CIC at beginning of life is 0.01 e-/pixel/frame.
In this case, the dark current is the dominant noise source as it is between
0.77-3 electron/pix/hr, whereas CIC is only 0.36 electron/pix/hr assuming 100s
exposure time per frame. In addition, we adopt a constant CIC as the photon
number per pixel per frame from a Earth like planet and the backgrounds are
$\ll 1\rm{e^{-}\ \rm{frame}^{-1}\ pix^{-1}}$ for both an IFS and an iFTS. For
the future level, we adopt the dark current values required by LUVIOR and
HabEx as $0.1\ \rm{e^{-}\ \rm{frame}^{-1}\ pix^{-1}}$ (Gaudi et al., 2020; The
LUVOIR Team, 2019).

In the near-IR (1000-2000 nm), we consider the HAWAII H2RG as the state-of-
the-art. The total effective noise of H2RG detectors on JWST is $5.9\
\rm{e^{\text{-}}\ frame^{\text{-}1}\ pix^{\text{-}1}}$ with an integration
time per frame of 1000s and $10\ \rm{e^{\text{-}}\ frame^{\text{-}1}\
pix^{\text{-}1}}$ with an integration time per frame of 100s (Birkmann et al.,
2022). Because readout noise is the dominant noise source of NIR detector, we
use a readout noise of $5.9\ \rm{e^{\text{-}}\ frame^{\text{-}1}\
pix^{\text{-}1}}$ with an integration time per frame of 1000s and ignore the
dark current. HabEx baselined (and LUVOIR listed as backup) HgCdTe linear mode
avalanche photodiode array detectors (LMAPD)for the NIR channels. 555Both the
HabEx and LUVOIR report appear to suggest that current LMAPDs like the small-
format SAPHIRA arrays can achieve dark currents of 1.5e-3 electrons per pixel
per second simultaneously with sub-electron read noise, and thus development
of larger arrays is all that is needed. This is false. When the effective read
noise is reduced to sub-electron levels through avalanching, the dark current
increases by approximately three orders of magnitude. There are no space-
qualified infrared arrays that can simultaneously achieve low dark current and
read noise(Atkinson et al., 2017; Goebel et al., 2018), though there are
development efforts underway (Claveau et al., 2022) HabEx required
$<0.002\rm{e^{\text{-}}\ s^{\text{-}1}\ pix^{\text{-}1}}$ dark current and
$\ll 1\ \rm{e^{\text{-}}\ \rm{frame}^{\text{-}1}\ pix^{\text{-}1}}$ read
noise. LUVOIR required $3\ \rm{e^{\text{-}}\ frame^{\text{-}1}\
pix^{\text{-}1}}$ readout noise with $0.001\rm{e^{\text{-}}\ hr^{\text{-}1}\
pix^{\text{-}1}}$. Here, we adopt a readout noise of $0.5\ \rm{e^{\text{-}}\
frame^{\text{-}1}\ pix^{\text{-}1}}$ as the future level NIR detector noise.

###  4.2 Simulation setup for HWO in various bands

Simulation $4a\&$b in Table 3.4 present the instrument parameters of
simulations for a 6-meter direct imaging telescope. We use the baseline
designs and requirements from LUVOIR and HabEx (The LUVOIR Team, 2019; Gaudi
et al., 2020) as references for other parameters, with a coronagraph
throughput of $10\%$. We use filter specifications and optical throughput of
LUVOIR because the optical design is incomplete for the Habitable Worlds
Observatory.

##  5 Results

###  5.1 General simulation results in Visible

Figure 8: (a): the required exposure time to achieve an SNR of 5 using an IFS
as the function of EMCCD dark current in a visible band (720-800 nm, R=140);
(b) the same as (a), but for an iFTS; (c) the required exposure time ratio
between an IFS and an iFTS $\rm{T_{IFS}/{T_{iFTS}}}$ as the function of EMCCD
dark current. We use fixed instrument parameters including diameter$D=6m$,
coronagraph throughput $\tau=10\%$, and IFS $n_{pix}=6$. The target is an
Earth-twin at 10 pc way. The normalized planet spectrum is superimposed on the
three panels. The horizontal dashed lines correspond to several reference
values for detector noise, including the state-of-art dark current from EMCCD
on Roman space telescope (best and worst scenarios) and detector noise
requirements of future missions.  Figure 9: For the same band in figure 8, we
further compare the IFS and iFTS required time at the continuum ($\lambda=750\
nm$) with various instrument parameters. (a)-(c): the required exposure time
to achieve an SNR of 5 using an IFS, an iFTS and the time ratio between an IFS
and iFTS as the functions of EMCCD dark current and coronagraph throughput.
(d)-(f): the required exposure time to achieve an SNR of 5 using an IFS, an
iFTS and the time ratio between an IFS and iFTS as the functions of EMCCD dark
current and IFS detector sampling number $n_{IFS}$. Solid lines mark where the
time ratio equals to one. The horizontal lines correspond to several reference
values for detector noise, including the state-of-art dark current from EMCCD
on Roman space telescope (best and worst scenarios) and future detector noise
required by future missions. Figure 10: (a): the required exposure time to
achieve an SNR of 5 using an IFS as the function of readout noise in a NIR
band (1050-1200 nm, R=70); (b) the same as (a), but for an iFTS; (c) the
required exposure time ratio between an IFS and an iFTS
$\rm{T_{IFS}/{T_{iFTS}}}$ as the function of readout noise. We use fixed
instrument parameters including diameter$D=6m$, coronagraph throughput
$\tau=10\%$, and IFS $n_{\text{pix}}=4$. The target is an Earth-twin at 10 pc
way. The normalized planet spectrum is superimposed on the three panels. The
horizontal dashed lines correspond to several reference values for detector
noise, including the state-of-art dark current from HxRG on JWST and future
detector noise required by future missions. Figure 11: For the same band in
figure 10, we further compare the IFS and iFTS required time at the continuum
($\lambda=1050\ nm$) with various instrument parameters. (a)-(c): the required
exposure time to achieve an SNR of 5 using an IFS, an iFTS and the time ratio
between an IFS and iFTS as the functions of readout noise and coronagraph
throughput. (d)-(f): the required exposure time to achieve an SNR of 5 using
an IFS, an iFTS and the time ratio between an IFS and iFTS as the functions of
readout noise and IFS detector sampling number $n_{IFS}$. Solid lines mark
where the time ratio equals to one. The horizontal dashed lines correspond to
several reference values for detector noise, including the state-of-art dark
current from HxRG on JWST and future detector noise required by future
missions.

Figure 8 (a) $\&$ (b) display the required exposure times of an IFS and an
iFTS to achieve the SNR of 5 as the functions of detector noise and wavelength
in a visible bandpass. We vary the dark current rate for the visible camera
from state of the art value (EMCCD on Roman-CGI) to the future value required
by LUVOIR/HabEx (detector noise details see Table 3.4). We focus on the band
from 720 nm-800 nm due to the strong $\rm{O_{2}}$ line at 760 nm. Both the IFS
and iFTS need significantly longer exposure times in the absorption feature
compared to the continuum. For the continuum, the IFS exposure time exhibits a
strong dependency on the dark current, decreasing from
$\raisebox{2.15277pt}{\texttildelow}2000$ hours with Roman-CGI’s worst
detector noise ($3\ \rm{e^{\text{-}}\ hr^{\text{-}1}\ pix^{\text{-}1}}$) to
$\raisebox{2.15277pt}{\texttildelow}500$ hours with Roman’s best detector
noise ($0.77\ \rm{e^{\text{-}}\ hr^{\text{-}1}\ pix^{\text{-}1}}$). If the
detector noise is further reduced to $0.1\ \rm{e^{\text{-}}\ hr^{\text{-}1}\
pix^{\text{-}1}}$, as required by LUVIOR and HabEx, the IFS exposure time at
the continuum will be $\raisebox{2.15277pt}{\texttildelow}100$ hours.

In contrast, the iFTS exhibits lower sensitivity to the dark current,
requiring $>1000$ hours of exposure time across varying dark current values.
As elaborated in section 3.4, the iFTS enjoys the benefit of reduced
sensitivity to detector noise, yet it is more susceptible to the effects of
photon noise. In optical wavelengths, the impact of photon noise outweighs the
advantages of reduced detector noise for the iFTS, limiting its performance.
Figure 8 (c) shows the required exposure time ratio between an IFS and an iFTS
in the same bandpass as Figure 8 (a)$\&$(b). It is evident that an IFS
outperforms an iFTS in most cases. The time ratio exhibits a weak dependence
on the wavelength because the IFS detector noise increases at longer
wavelengths.

Figure 9 (a)-(c) compare the performance of an IFS and an iFTS with varied
detector noise and coronagraph throughput for a 6 meter telescope. The
simulation is conducted within the same bandpass as depicted in Figure 8, and
the exposure time considered is that of the continuum. For reference, Roman-
CGI has a low coronagraph throughput of
$\raisebox{2.15277pt}{\texttildelow}3\%$, whereas studies on coronagraphs for
future missions are underway to achieve higher throughput. When the
coronagraph throughput $\tau_{core}$ is below $5\%$, both the IFS and iFTS
required exposure times are over 5000 hours, rendering the detection
unfeasible. As the coronagraph throughput increases, the IFS and iFTS
performance both improve. Nonetheless, in cases of higher throughput, the iFTS
experiences a higher penalty from photon noise compared to the IFS. This is
evident from Figure 9(c), which illustrates that the IFS would surpass the
iFTS in performance when the throughput exceeds around $10\%$.

Figure 12: The required exposure time to achieve a SNR of 5 as the function of
the detector noise for an iFTS (dashed line), solid lines are for an IFS with
varied $n_{pix}$ (solid lines). The left is for the visible band (720-800 nm)
same as figure 8 and figure 9, the required time is calculated at the
continuum. The right is for the NIR band (1050-1200 nm) same as figure 10 and
figure 11, the required time is calculated at the continuum. The dashed lines
correspond to the state-of-art detector noise from EMCCD on Roman-CGI (best
scenario) and the readout noise for HxRH detector on JWST.

Figure 9 (d)-(f) investigate how the IFS detector sampling influences the
relative performance of an iFTS and an IFS for a 6 meter telescope. As an IFS
retrieves spectral information through dispersing photons into different
pixels, the total detector noise depends on the number of pixels per spectral
element per lenslet $n_{pix}$. As shown in section 3.1, $n_{pix}$ is the
product of pixel numbers in vertical and horizontal directions. HabEx and
LUVOIR both use 6 pixels ($2\times 3$) and 4 pixels ($2\times 2$) per spectral
element per lenslet in the optical and NIR bandpasses for coronagraph mode,
respectively (The LUVOIR Team, 2019; Gaudi et al., 2020). We varied $n_{pix}$
from 2 to 10 in our simulations. In the visible bandpass, an IFS always
outperforms an iFTS if the IFS uses less than 6 pixels per spectral elements
per lenslet. But the IFS performance degrades as $n_{pix}$ increases, being
less effective than an iFTS if $n_{pix}>6$ and dark current rate $>2\
\rm{e^{\text{-}}\ hr^{\text{-}1}\ pix^{\text{-}1}}$.

In summary, these simulations reveal that an iFTS is less effective than an
IFS at optical wavelengths for a 6 meter telescope, because an iFTS is
fundamentally limited by photon noise.

Figure 13: The required integration time to achieve $\rm{S/N=5}$ for a 6 m
telescope equipped with an iFTS (red) and an IFS (dark blue) at different
spectroscopy bands in the optical (top panel) and NIR (bottom panel). The
filter bands are shown as colorful shaded area and every point corresponds to
the central wavelength of every band. In the visible wavelengths, we use Roman
current best estimate dark current (solid lines) and the required dark current
by future mission (dashed lines). In the NIR, there are also two types of
readout noise: JWST current best estimate readout noise ( solid lines) and the
future level of readout noise (dashed lines). The target is a 1 $R_{\oplus}$
planet orbiting a Sun-like star at 1 AU from 10 pc away. Instrument parameters
are listed in 3 (a) and (b) in Table 3.4

###  5.2 General simulation results in NIR

Figure 10 compares the require exposure times of an IFS and an iFTS in a NIR
band containing a $\rm{H_{2}O}$ absorption feature from 1050 nm to 1200 nm.
For the NIR detectors, the readout noise is the dominant noise source for
which we vary from the state of the art HxRG detectors on JWST to future
LMAPD-like level, in both cases with a frame time of 1000 s (detector noise
details see Table 3.4). LUVOIR and HabEx reports use an $n_{\rm{pix}}=4$
design for the IFS in the NIR. As detailed in section 3.4, when
$n_{\rm{pix}}=4$, an iFTS will have equivalent detector noise to an IFS but
higher photon noise. Consequently, an iFTS is always less effective than an
IFS with $n_{\rm{pix}}=4$. Hence, in our simulations, we employ an IFS
$n_{\rm{pix}}=6$ to investigate the relative sensitivity within this setup.

Figure 10 reveals that the exposure times needed for the IFS and iFTS in the
NIR band are longer than those in the optical. This is because NIR detectors
have higher detector noise, primary the readout noise. Specifically, at the
continuum of the $\rm{H_{2}O}$ absorption band, if the readout noise is above
$3\ \rm{e^{\text{-}}\ frame^{\text{-}1}\ pix^{\text{-}1}}$, both the IFS and
iFTS demand exposure times over 1000 hours, with the iFTS displaying slightly
better efficiency. However, as the readout noise decreases to levels below
around $3\ \rm{e^{\text{-}}\ frame^{\text{-}1}\ pix^{\text{-}1}}$, the IFS
surpasses the iFTS in performance, requiring only half the exposure time of
the iFTS.

Figure 11 is similar to Figure 9, but specifically for the NIR band ranging
from 1050 nm to 1200 nm. It is evident that both IFS and iFTS benefit from
increased coronagraph throughput and reduced readout noise, resulting in
decreased required time. Considering the readout noise of the HxRG detector on
JWST, the time required by IFS and iFTS will diminish to less than 1000 hours
when throughput surpasses $\raisebox{2.15277pt}{\texttildelow}25\%$.In these
scenarios, iFTS demonstrates $\raisebox{2.15277pt}{\texttildelow}20\%$ better
efficiency. Should the readout noise adhere to the specifications outlined in
the LUVOIR report ($3\ \rm{e^{\text{-}}\ frame^{\text{-}1}\
pix^{\text{-}1}}$), an iFTS requires less time if the coronagraph throughput
is below $\raisebox{2.15277pt}{\texttildelow}10\%$. However, the opposite
occurs when the throughput is larger than
$\raisebox{2.15277pt}{\texttildelow}10\%$. As the readout noise is further
reduced below 1 e-/pix/frame an IFS emerges as the more efficient choice.

Figure 11 (d)-(f) shows that the performance comparison between an iFTS and an
IFS also depend on the IFS detector sampling design. Consistent with the
argument in section 3.4, if an IFS is assigned $\leq 4$ pixels per spectral
element per lenslet, it always wins over an iFTS. (Note that the plot limit is
set at 2 pixels per spectral element per lenslet, which corresponds to a
lenslet being imaged to a single row of pixels on the detector with spectral
sampling at the Nyquist limit.) However, if the IFS $n_{pix}>4$, the IFS
performance will deteriorate. When IFS $n_{pix}>4$ and readout noise is above
$2\ \rm{e^{\text{-}}\ frame^{\text{-}1}\ pix^{\text{-}1}}$, an iFTS would be
more efficient by $\raisebox{2.15277pt}{\texttildelow}30\%$.

In summary, the comparison of the iFTS and IFS performance in the near-IR
channel depends on the instrument parameters, especially the number of pixels
per spectral element per lenslet used by the IFS. Furthermore, we have
implemented a frame integration time of 1000 seconds for NIR bandpass in our
simulations. A longer integration period helps to reduce readout noise, while
resulting in a higher number of cosmic ray hits. While a comprehensive study
to determine the optimal integration time per frame is necessary, this task
falls outside the scope of this paper.

###  5.3 Exposure times with decreased detector noise

Figure 12 presents the required exposure time to achieve a SNR of 5 at
continuum by an iFTS and an IFS as the function of detector noise. As
discussed in section 5.1, an iFTS is limited by the photo noise in the
visible, always requiring $>1000$ hours exposure time. For an IFS, to reduce
the exposure time to the order of $100$ hours, the dark current needs to be
reduced by a factor of 5 from the current published level (Morrissey et al.,
2023). In the NIR, if the readout noise remains $>2-3\ \rm{e^{\text{-}}\
frame^{\text{-}1}\ pix^{\text{-}1}}$ and IFS $n_{pix}>4$, an iFTS will use at
least $\raisebox{2.15277pt}{\texttildelow}20\%$ time although the absolute
time scale is above 1000 hrs for the two spectrographs in the case. The
required exposure time by an IFS can be decreased to the order of $100$ hours
if readout noise is reduce to below $1\ \rm{e^{\text{-}}\ frame^{\text{-}1}\
pix^{\text{-}1}}$ with a frame time of 1000s.

###  5.4 Results for HWO in various bands

Figure 13 presents the required exposure time for an IFS and an iFTS to
achieve a SNR of 5 in different filters in the optical and NIR. We compare the
required time with current and future level detector noise, respectively. In
the optical, the required integration time rises sharply at channels longer
than 800 nm, because the detector quantum efficiency drops significantly at
the red wavelengths (see Appendix B.3). Thus, we mainly focus on the channels
shorter than 800 nm. In the optical range, photon noise is the dominant source
of noise in an iFTS, so reducing detector noise does not significantly improve
performance. The iFTS required exposure time is
$\raisebox{2.15277pt}{\texttildelow}1100$ hrs in channels shorter than 820 nm
with current dark currents, and $\raisebox{2.15277pt}{\texttildelow}1000$ hrs
with future dark currents. In contrast, an IFS currently requires
approximately $\raisebox{2.15277pt}{\texttildelow}700$ hrs for operation, but
a substantial reduction in dark current greatly enhances its performance,
reducing the required time to $\raisebox{2.15277pt}{\texttildelow}100$ hrs
with a future level of detector noise.

In the NIR, the required exposure time is longer at bands with wide absorption
features, mostly from $\rm{H_{2}O}$. As the wavelength increases, the PSF
aperture also expands, prompting the involvement of a greater number of pixels
in both the IFS and iFTS methods. This augmented pixel utilization contributes
to higher detector noise, consequently increasing the required exposure time.
The most promising band for detection is from 1050nm to 1200nm with a
$\rm{H_{2}O}$ absorption feature.

With an IFS $n_{pix}$ of 6, an IFS requires longer exposure times than an iFTS
to achieve a SNR of 5 with current-level readout noise. However, the absolute
time scale is beyond a few thousand hours and even higher than that in the
longer wavelengths, making the detection unfeasible. If the readout noise is
reduced to the $0.5\ \rm{e^{\text{-}}\ frame^{\text{-}1}\ pix^{\text{-}1}}\
(t_{\rm{frm}}=1000\ s)$, an IFS will be more efficient. At the band from
1050-1200nm with a $\rm{H_{2}O}$ absorption line, an IFS will require $<100$
hrs to achieve a SNR of 5. The required time increases at longer wavelengths.
It takes at least a few hundred hours in the bands from 1300-1500 nm and even
a few thousand hours in he bands from 1800 nm to 2000nm, both of which
encompass broad absorption features attributed to $\rm{H_{2}O}$.

##  6 Discussion

###  6.1 Practical operational concerns

Our results have been derived from a purely “point-source, photon budget”
perspective, keeping as many instrument parameters between the two
spectrograph architectures as equal as possible. For example, iFTS advantages
in throughput were reduced to IFS values, and iFTS-specific issues like field-
dependent modulation efficiency, beamsplitter chromaticity, polarization
effects, etc were ignored. We have also not discussed challenges and
opportunities with operating an iFTS as part of a space-based high-contrast
imaging system. Here we briefly discuss such practical considerations,
focusing on wavefront control, sensitivity to space radiation effects, and the
flexibility offered by the instrument.

To properly understand the feasibility of integrating an iFTS with an
exoplanet imaging system, simulated two-dimensional images of a coronagraph,
wavefront control system, and iFTS would be needed, which are missing in this
work. This paper is a first step in exploring the potential use of an iFTS for
direct imaging exoplanets, with an analysis based on point photometry and the
assumption that an iFTS works with upstream instruments. Future work is to add
two-dimensional simulations, including realistic tolerances, of how electric
fields propagate through coronagraph and wavefront control systems before they
are fed into the spectrograph.

####  6.1.1 Wavefront Control

Operation of a coronagraph in space requires nanometer to picometer level
control of wavefront error. To achieve this, space coronagraphs use the
science camera in conjunction with the deformable mirrors to iteratively sense
and correct the electric field at the detector to achieve a zone of very high
contrast, known as “digging a dark hole”. Once the dark hole is established,
it needs to remain stable.

The situation is more complicated with an iFTS due to the moving mirror. While
a dark hole could be established at the zero-path difference of the
interferometer, an open question is whether the contrast would be stable
during a mirror scan. Rough arguments suggest that it would. First, the main
optical errors corrected by the dark hole are upstream of the coronagraph,
with downstream optics playing a secondary role. Second, the tolerances
already demonstrated by iFTS instruments on mirror stability and position are
close to those required for high contrast imaging–for example, SITELLE
achieves sub-nm accuracy over a 1 cm scan using a 15 cm flat mirror (Grandmont
et al., 2012). Finally, phase-to-amplitude wave mixing happens on distances
comparable to the Talbot length, which is ~meters in a high contrast imaging
system, much longer than any mirror scan distance (Mazoyer et al., 2017). This
suggests the dark hole contrast should not be meaningfully affected.

The need for wavefront control also negates one of the principal advantages of
an iFTS, which is its broad bandwidth, limited by reflectance properties of
the mirrors and detector quantum efficiency. However, wavefront control
algorithms struggle with broadband light, and are not expected to operate with
more than 10-20% bandwidth (The LUVOIR Team, 2019; Gaudi et al., 2020). Thus
upstream filters must limit the bandwidth through the iFTS.

####  6.1.2 Detector Effects

Space radiation effects on detectors can play a dominant role in instrument
performance. We briefly comment on the effects of cosmic rays and accumulated
radiation damage on iFTS performance.

The focal-plane image in an iFTS is orders of magnitude smaller and brighter
than in an IFS since an iFTS does not disperse photons. The chance of a cosmic
ray hit on the planet signal is thus proportionally reduced, though suffers a
minor penalty from the two detectors. On the other hand, while a cosmic ray in
an IFS only affects the the section of the 2-d spectrum it hits, in an iFTS, a
cosmic ray affects the entire spectrum, due to the effect of Fourier
transforming the interferogram. Special care would need to be taken to clean
the interferogram of cosmic rays before extracting the spectrum.

Accumulated damage from cosmic rays causes degradation in detector parameters
like dark current. For example, the dark current in the least noisy detector
of the wide-field planetary camera of the Hubble Space Telescope quadrupled
between 1995 and 1998, with rates of increase closely agreeing with
independent measurements of galactic and solar cosmic rays (McMaster et al.,
2008). Such increases in dark current would substantially affect IFS
sensitivity, while an iFTS would only have a modest decrease in performance,
as it is mainly limited by photon noise (see Figures 9).

####  6.1.3 Variable Resolution of iFTS

There is additional flexibility in the iFTS for spectroscopic exoplanet
surveys beyond that afforded by a camera and IFS.

First, at the zero-path difference of the instrument, the iFTS functions as a
conventional camera, with all the light going to one detector, with minimal
efficiency loss due to the extra surfaces and no noise penalty over a
conventional camera. Even when scanning, broadband imaging data is
simultaneously available by adding the two output port images, though this
increases the detector noise in the imaging data by $\sqrt{2}$. This allows
for dynamic planet identification and localization during long observations.
The increase in image brightness in an iFTS compared to an IFS is advantageous
for this task, as is the higher spatial resolution over lenslets. The
simultaneous imaging and spectroscopy allows for simultaneous planet search
and characterization, which is qualitatively different than search strategies
with a camera and IFS. However, the time advantage conferred may be minor as
these missions are unlikely to rely on blind searches.

The flexible resolution is also potentially advantageous. As the mirror scan
distance increases, the resolution increases, allowing for “dynamical” vetting
of interesting sources at low resolution. The resolution may be adjusted to
observe Earth-like planets with low resolution mode and bright young giant
plants with medium/high resolution mode. Similarly, if it is found that more
resolution is a science need on a particular target, the mirror scan may
simply be restarted from the last position.

The previous two advantages suggest a high-level analysis of observations with
an iFTS and their effects on survey yield (eg, Stark et al. (2014); Savransky
et al. (2017)), are needed to understand how and if this flexibility can be
leveraged.

##  7 Conclusion

In this work, we have compared the performance of an imaging Fourier transform
spectrograph to an integral field spectrograph for the challenging task of
obtaining spectra from an Earth-like planet. To do this, we created analytical
and numerical models to simulate the iFTS and IFS observations of exoplanet
reflective spectra from visible to near-infrared wavelengths using an future
space direct imaging telescopes with varied instrument parameters. While the
relative performance of the two instrument concepts depends on telescope,
instrument, detector, and planetary system parameters, several general
conclusions can be drawn.

First, in the optical, an IFS outperforms an iFTS to characterize Earth-like
planets around nearby stars, since the latter is limited by photon noise.
Reductions in dark current by a factor of 5 from current published values
(Morrissey et al., 2023) will improve the performance of an IFS, allowing it
to obtain the spectra from Earth-like planets within a reasonable time (below
$\raisebox{2.15277pt}{\texttildelow}100$ hrs) in the bluest wavelengths,
whereas an iFTS has a more minor gain in performance.

On the other hand, in the near-IR, the relative sensitivity between an IFS and
an iFTS depends on the instrument design. An IFS will always be more efficient
if it uses 4 or less pixels per spectral element per lenslet. If using more
than 4 pixels per spectral element per lenslet, the IFS performance is less
efficient by at least $\raisebox{2.15277pt}{\texttildelow}20\%$ than an iFTS
when the readout noise is higher than $2-3\ e^{-}\ pix^{-1}\ \rm{frame}^{-1}\
(\rm{t_{frm}=1000s})$, though the absolute sensitivity depends strongly on the
readout noise level. But if the effective readout noise levels is below 2
electrons, the required time by an IFS becomes much shorter than an
iFTS,reduced to 200-1000 hrs at the shortest near-IR bandpass (1050-1200 nm).
The readout noise need to be reduced to sub-photo level ($<1\ e^{-}\ pix^{-1}\
\rm{frame}^{-1}$) to achieve a SNR of 5 within 100 hrs.

Given the high priority of a space high contrast imaging mission, it is
encouraging to see that at least one other instrument concept exists that can
provide comparable sensitivity to an IFS at infrared wavelengths, where most
of the deep biosignature features exist. Other aspects of the iFTS, like
simultaneous imaging and spectroscopy, flexible resolution, and the ability to
use orders-of-magnitude smaller-format detectors, may present new
opportunities for such a mission, though these features were not explored in
this work. We hope the results of this paper motivate consideration of an iFTS
in future design trades and yield simulations of this important future
mission.

## Acknowledgements

We gratefully acknowledge

  * •

Laurie Rousseau-Nepton and Simon Prunet, for their help with understanding
imaging Fourier transform spectrographs

  * •

Tyler Robinson, for his help in understanding the various noise terms in the
direct imaging error budget

  * •

Matthew Bolcar and Stefan Martin, for their help with understanding the LUVOIR
and HabEx mission optical designs,

  * •

Bertrand Mennesson, for his help in understanding the exozodiacal flux levels
and coronagraph performance for space-based exoplanet imaging.

This work was supported by NASA FINESST grant #20-ASTRO20-0172/Jingwen Zhang.
MB would also like to particularly thank his undergraduate research advisor,
Amber Miller, for introducing him to the fascinating world of Fourier
transform spectroscopy. Part of this work was carried out at the Jet
Propulsion Lab, California Institute of Technology, under contract with NASA
(80NM00018D0004).

##  Appendix A Planet light and Astrophysical Background

We derived analytic expressions for the SNRs of integral field spectrographs
and imaging Fourier transform spectrographs in section 3.2 and section 3.3.
These expressions include the planet light, background (local zodiacal light,
exzodiacal light, speckle light). In this section, we will present the
detailed analysis of each term. The analysis in this section follows the work
in Robinson et al. (2016), Lacy et al. (2019), and Stark et al. (2019). For
the IFS cases, we checked these results against their published work, finding
excellent agreement in computed count rates. Here, we present the spectra as
the function of wavelength, and they can be transformed to the format of
wavenumber for iFTS.

###  A.1 Planet count rate, $c_{p}$

In the optical and near-infrared, thermal emission from mature terrestrial and
gas giant exoplanets is negligible. The flux from the planet is due to
reflection of stellar light. We assume the host star has a radius $R_{\star}$
and an effective temperature of $T_{\rm{eff}}$. The stellar specific flux
density at the planet’s orbit is

$F_{\star,\lambda}(a)=\pi
B_{\lambda}(T_{\rm{eff}})\left(\frac{R_{\star}}{a}\right)^{2}$ (A1)

where $B_{\lambda}(T_{\rm{eff}})$ is the blackbody spectrum, $a$ is the
orbital distance of the planet. More realistic stellar spectra, either from
observations or models, can also be used for $F_{\star,\lambda}(a)$.

The fraction of stellar light at wavelength $\lambda$ reflected by a planet
toward an observer is determined by the planet’s radius $R_{p}$, geometric
albedo $A_{g}(\lambda)$, and phase function $\Phi(\alpha)$. The albedo depends
on the wavelength-dependent scattering properties of the atmosphere and
planetary surface and is of primary scientific interest. The phase function
accounts for the variation of reflectivity over the planet’s orbit, with
$\alpha$ being the exoplanet-centric angle between the star and the observer
(Madhusudhan & Burrows, 2012). While the phase function may also depend weakly
on wavelength (Mayorga et al., 2016), we ignore such effects and use a
Lambertian form:

$\displaystyle\Phi(\alpha)$
$\displaystyle=\frac{\sin\alpha+(\pi-\alpha)\cos\alpha}{\pi}$ (A2)

Combining these expressions, we see the specific flux density of a planet at a
distance of $d$ from Earth is

$\displaystyle F_{p}(\lambda)$
$\displaystyle=A_{g}(\lambda)\Phi(\alpha)F_{\star,\lambda}(a)\left(\frac{R_{p}}{d}\right)^{2}$
(A3) $\displaystyle=\pi
B_{\lambda}(T_{\rm{eff}})A_{g}(\lambda)\Phi(\alpha)\left(\frac{R_{\star}}{a}\right)^{2}\left(\frac{R_{p}}{d}\right)^{2}$

We can convert the planet specific flux density in the unit of $W\ m^{-2}\
nm^{-1}$ into photon density in the unit of $e^{-}\ m^{-2}\ nm^{-1}$ by
multiplying by $\frac{\lambda}{hc}$. The actual detected photons from the
planet also depends on the effective collecting area $A_{\rm{PM}}$ and total
system throughput $\tau_{tot}$. Thus, the planetary photons at wavelength
$\lambda$ measured by the detector per unit time is

$\displaystyle c_{p}(\lambda)$
$\displaystyle=A_{\rm{PM}}\tau_{tot}F_{p}(\lambda)\frac{\lambda}{hc}$ (A4)
$\displaystyle=\pi A_{\rm{PM}}\tau_{tot}$

###  A.2 Zodiacal and exozodiacal background light

Zodiacal and exozodiacal light consists of scattered starlight from dust in
the solar system and extrasolar systems, respectively, and is a primary source
of astrophysical noise at optical and near infrared wavelengths ($0.5-2\ \mu
m$). Stark et al. (2014) demonstrated that a uniform V-band surface brightness
of $M_{\rm{z,V}}=23\ \rm{mag\ arcsec^{\text{-}2}}$ is a reasonable
representation of solar system zodiacal light, ignoring the variation with
ecliptic latitude and longitude given by Levasseur-Regourd & Dumont (1980).
The solar system zodiacal light flux density is given by

$F_{z}(\lambda)=\Omega\frac{F_{\rm{\odot,\lambda}}(1\rm{AU)}}{F_{\rm{\odot,V}}(1\rm{AU)}}F_{\rm{0,V}}10^{\text{-}M_{\rm{z,V}}/2.5}$
(A5)

where $F_{\rm{\odot,\lambda}}$ is the wavelength-dependent specific solar flux
density, $F_{\rm{\odot,V}}$ is the solar flux density in the V-band, and
$F_{\rm{0,V}}$ is the standard zero-magnitude V-band specific flux density.
$\Omega$ is the photometric aperture in units of $\rm{arcsec^{2}}$. Following
Robinson et al. (2016), we use a square aperture area
$\Omega=\left(\frac{\lambda}{D}\right)^{2}$ to represent the PSF region of the
planet. Thus, the solar system zodiacal light photons at wavelength $\lambda$
received by the detector per unit time is

$\displaystyle c_{z}(\lambda)$
$\displaystyle=A_{\rm{PM}}\tau_{tot}F_{z}(\lambda)\frac{\lambda}{hc}$ (A6)
$\displaystyle=A_{\rm{PM}}\tau_{tot}\Omega\frac{F_{\rm{\odot,\lambda}}(1\rm{AU)}}{F_{\rm{\odot,V}}(1\rm{AU)}}F_{\rm{0,V}}10^{\text{-}M_{\rm{z,V}}/2.5}\frac{\lambda}{hc}$

Following Stark et al. (2014), a “zodi” is defined as the surface brightness
of an exozodiacal disk at $1\ \rm{AU}$ from a solar twin. The surface
brightness in V band $M_{\rm{ex,V}}$ is 22 $\rm{mag\ arcsec^{\text{-}2}}$,
which is roughly twice the local zodi level, since the observer receives
reflected light of dust from both above and below the mid-plane of the system.
Therefore, the exozodiacal light flux density from a system with $N_{ex}$
zodis of dust around a star with effective temperature $T_{\rm{eff}}$ at the
planet orbit is given by

$\displaystyle F_{ez}(\lambda)$
$\displaystyle=N_{ex}\Omega\frac{F_{\star,\lambda}(a)}{F_{\star,V}(1\rm{AU})}\frac{F_{\star,V}(1\rm{AU})}{F_{\odot,V}(1\rm{AU})}$
(A7) $\displaystyle\times F_{\rm{0,V}}10^{\text{-}M_{\rm{ez,V}}/2.5}$
$\displaystyle=N_{ex}\Omega\frac{\pi
B_{\lambda}(T_{\rm{eff}})}{F_{\odot,V}(1\rm{AU})}\left(\frac{R_{\star}}{a}\right)^{2}$
$\displaystyle\times F_{\rm{0,V}}10^{\text{-}M_{\rm{ez,V}}/2.5}$

where $a$ is the planet’s orbital distance. The term
$\frac{F_{\star,V}}{F_{\odot,V}}$ accounts for the exozodical light scaling
with the brightness of the host star. Notice the exozodical light decreases
with the increasing orbital distance as $1/a^{2}$.

The exozodiacal light photons received at wavelength $\lambda$ by the detector
per unit time is

$\displaystyle c_{ez}(\lambda)$
$\displaystyle=A_{\rm{PM}}\tau_{tot}F_{ez}(\lambda)\frac{\lambda}{hc}$ (A8)
$\displaystyle=A_{\rm{PM}}\tau_{tot}N_{ex}\Omega\frac{\pi
B_{\lambda}(T_{\rm{eff}})}{F_{\odot,V}(1\rm{AU})}\left(\frac{R_{\star}}{a}\right)^{2}$
$\displaystyle\times
F_{\rm{0,V}}10^{\text{-}M_{\rm{ez,V}}/2.5}\frac{\lambda}{hc}$

The above expressions reveal that the solar system zodiacal and exozodiacal
light signals are independent of the telescope diameter. But the ratio of the
planetary signal to the solar system zodiacal and exozodiacal light signals
scale with telescope size as $c_{p}/(c_{z}+c_{ez})\propto D^{2}$. Thus, a
larger telescope helps to distinguish the planet light from these sources of
background noise.

Figure A.1:  The specific flux density of planet, zodiacal light, exozodiacal
light, and speckle light at the pupil of a telescope for an Earth twin
orbiting a Sun-like star 10 pc away. Here, the zodiacal and exozodiacal light
are 23 and 22 $\rm{mag\ arcsec^{-2}}$, respectively. The simulated telescope
has a 6 m primary mirror and achieves a contrast of $10^{-10}$.

###  A.3 Speckle Light

Another astrophysical noise source arises from the interaction of stellar
light with unsensed or uncorrected optical aberrations in the telescope and
coronagraph, causing a relatively bright “speckle” halo surrounding the image
of the star in the focal plane. The speckle flux density is the product of
stellar specific flux density and the coronagraph raw contrast $C_{\rm{raw}}$:

$\displaystyle F_{sp}(\lambda)$
$\displaystyle=C_{\rm{raw}}F_{\star,\lambda}(d)$ (A9)
$\displaystyle=C_{\rm{raw}}\pi
B_{\lambda}(T_{\rm{eff}})\left(\frac{R_{\star}}{d}\right)^{2}$

Thus, speckle light photons at wavelength $\lambda$ received by the detector
per unit time is

$\displaystyle c_{sp}(\lambda)$
$\displaystyle=A_{\rm{PM}}\tau_{tot}F_{\rm{sp}}(\lambda)\frac{\lambda}{hc}$
(A10) $\displaystyle=A_{\rm{PM}}\tau_{tot}C_{\rm{raw}}\pi
B_{\lambda}(T_{\rm{eff}})\left(\frac{R_{\star}}{d}\right)^{2}\frac{\lambda}{hc}$

Figure A.1 shows the specific flux density of each term at the telescope pupil
from 450-2000 nm for a Earth twin orbiting a Sun-like star at 1 AU from 10 pc
away.

##  Appendix B System Throughput

The system throughput $\tau_{tot}$ is the product of optical throughput
$\tau_{\rm{optical}}$, coronagraph core throughput $\tau_{\rm{core}}$ and
detector quantum efficiency $\tau_{\rm{QE}}$. Here, we briefly describe these
parameters.

###  B.1 Optical throughput

The optical throughput of the instrument depends on the various reflectivities
of the surfaces, transmissive and coupling losses (such as from lenslets),
etc. Generally, optical throughput is a weak function of separation between
the star and planet, but can be somewhat dependent on wavelength. In our
simulations, we use the optical throughput in optical ($24\%$) and near-IR
($32\%$) from the LUVOIR report (The LUVOIR Team, 2019). We use the same the
optical throughput of the IFS and iFTS in the simulations.

Figure B.1: Azimuthal mean raw contrast and coronagraph core throughput
$\tau_{core}$ of Roman-SPC, LUVOIR A-APLC, LUVOIR B-DMVC and HabEx as a
function of separation. LUVOIR A 1,2,3 correspond to three adopoted masks
desinged for LUVOIR A. Sources: Roman-SPC (Lacy et al., 2019), LUVOIR (Stark
et al., 2019), Habex (Ruane et al., 2018)

###  B.2 Coronagraph core throughput

The intrinsic faintness of the signals involved in exoplanet imaging leads to
the conclusion that it is not realistic to expect to detect photons from the
planet outside the central core of the planet’s PSF. This leads to the concept
of “core throughput”, which captures both the diffraction of the light outside
the center of the planet PSF due to the coronagraph as well as any throughput
losses from light being blocked by the coronagraph optics (for example, an
opaque hard-edge Lyot coronagraph will heavily attenuate point sources located
behind the focal plane mask). Core throughput is a weak function of
wavelength, but strongly dependent on the separation between the planet and
the star. Figure B.1 shows the contrast and core throughput of different
coronagraphs as a function of separation.

###  B.3 Detector quantum efficiency

Detector quantum efficiency, or QE, is the final component of throughput,
which depends on the detector substrate and coatings. QE can be a strong
function of wavelength. In the optical, silicon detectors are typically used,
which have excellent QE in visible wavelengths but become increasingly
transparent at infrared wavelengths. To capture this effect, we used the
quantum efficiency for the visible detector from Lacy et al. (2019). In the
infrared, HgCdTe has become the substrate of choice. While the QE wavelength
cutoffs may be adjusted by changing the relative abundances of the elements,
the QE at near-infrared wavelengths is well approximated by a constant 80%
from 1-2 $\mu$m, which is the functional form we adopt.

$\tau_{\rm{QE}}=\begin{cases}0.9,&\lambda\leq 0.7\mu m\\\
0.9(1-\frac{\lambda-0.7}{0.3}),&0.7<\lambda\leq 1\mu m\\\ 0.8,&1<\lambda\leq
2\mu m\end{cases}$ (B1)

##  Appendix C Practical observation of spectra using an iFTS

###  C.1 Aliasing

A truncated, discrete interferogram inevitably leads to some spectral
distortions compared to an infinitely long, continuous signal. One distortion
due to the discrete sampling is aliasing. In section 2 and section 3.2, we
have discussed that the output spectrum from the Fourier transform to the
interferogram is a symmetric mirror spectrum along the zero point. However, we
can see Eqn. 20 is not only valid for indices $i$ from $-N$ to $N$, but also
for all other integers. If we replace $\nu_{i}$ with
$\nu_{i}+{n}/{\Delta\delta}$, where $n$ can be any non-zero integer, we see
that

$\displaystyle O_{FTS}\left(\nu_{i}+\frac{n}{\Delta\delta}\right)$
$\displaystyle=\Delta\delta\sum_{-N}^{N}F_{\Delta}(\delta_{j})\cos\left(-2\pi\left(\nu_{i}+\frac{n}{\Delta\delta}\right)\Delta\delta\right)$
(C1)
$\displaystyle=\Delta\delta\sum_{-N}^{N}F_{\Delta}(\delta_{j})\cos(-2\pi\nu_{i}\Delta\delta)$
$\displaystyle=O_{FTS}(\nu_{i})$

Figure C.1: The output spectra from Fourier transform repeat at an interval of
$\frac{1}{\Delta\delta}$.

which means that the mirror-symmetrical spectrum computed from an
interferogram sampled with the interval $\Delta\delta$ is periodic with the
period $\frac{1}{\Delta\delta}$ as shown in Figure C.1. The replication of the
original spectrum and its mirror images on the wavenumber axis is called
aliasing. It is clear that the original spectrum can only be obtained if it
does not overlap with the replications. Thus, the critical sampling interval
of an interferogram is

$(\Delta\delta)_{\rm{Nyquist}}=\frac{1}{2\nu_{\rm{max}}}$ (C2)

where $\nu_{\rm{max}}$ is the maximum wave number (ie, smallest wavelength) of
the spectral band. For example, for a spectrum with a minimum wavelength of
400 nm, the optical path difference would need to be 200 nm or less, ie, a
mirror step of 100 nm. If the interferogram is sampled at exactly the critical
sampling interval, the spectrum and its mirror image replicate with the period
of $2\nu_{max}$. If the sampling interval is smaller than this critical
interval $\Delta\delta<\frac{1}{2\nu_{\rm{max}}}$, then the spectral orders of
the sampled signal do not overlap and the original spectrum can be faithfully
recreated. In contrast, if the sampling interval
$\Delta\delta>\frac{1}{2\nu_{\rm{max}}}$, the period of the replications is
smaller than $2\nu_{\rm{max}}$. Then the portion of the spectrum with
$|\nu|>\nu_{\rm{max}}$ is aliased into the basic period
$[-\nu_{\rm{max}},\nu_{\rm{max}}]$ and thus overlaps with the spectral
information originally located in this interval. In this case, we obtain a
distorted spectrum.

In summary, the interferogram sampling interval needs to be smaller than the
Nyquist interval $\frac{1}{2\nu_{\rm{max}}}$ to avoid the aliasing.

###  C.2 Spectral Resolution

The finite mirror scan distance in the range $(-L,L)$, where
$L=N\Delta\delta$, leads to a truncation of the interferogram. This results in
the convolution of the true spectrum with an instrument profile, similarly to
how a dispersive spectrograph convolves the true spectrum with a line-spread
function that depends on the resolution. This can be derived as follows: the
truncated interferogram $I_{L}(\delta)$ can be written as the product of the
“infinite” interferogram $I_{\Delta}(\delta)$ and a boxcar function:

$I_{L}(\delta)=\Pi(\delta)I_{\Delta}(\delta)$ (C3)

where $\Pi(\delta)$ is the boxcar function:

$\Pi(\delta)=\begin{cases}1,&|x|\leq L,\\\ 0,&|x|>L\end{cases}$ (C4)

Figure C.2: The truncation effect of the interferogram. The measured spectrum
is the convolution of the original spectrum with the instrument profile, which
is the Fourier transform of a boxcar function.Figure is adapted from Drissen
et al. (2011)

Based on the properties of the Fourier transform, we can see that the spectrum
computed from the truncated interferogram $B_{L}(\nu)$ is the convolution of
true spectrum $B(\nu)$ and the inverse Fourier transform of the boxcar
function

$\begin{split}B_{L}(\nu)&=\mathcal{F}^{\text{-}1}\\{\Pi(\delta)I_{\Delta}(\delta)\\}=\mathcal{F}^{\text{-}1}\\{\Pi(\delta)\\}\ast\mathcal{F}^{\text{-}1}\\{I_{\Delta}(\delta)\\}\\\
&=\mathcal{F}^{\text{-}1}\\{\Pi(\delta)\\}\ast B(\nu)\end{split}$ (C5)

The inverse Fourier transform of the boxcar function is a sinc function,
$2L\rm{sinc}(2\pi\nu L)$. Thus, the effect of truncation of the interferogram
is that the true spectrum is convolved by a sinc function, also called the
instrumental line shape. For example, if we measured a monochromatic beam, the
computed spectrum using truncated interferogram would be a sinc functions
instead of a ideal delta function (see Figure C.2).

The full width at half maximum (FWHM) of the instrumental line shape is
$\Delta\nu=1.207/(2L)$. Thus, the resolution can be given by

$R=\frac{\lambda}{\Delta\lambda}=\frac{\lambda}{\lambda^{2}\Delta\nu}=\frac{2L}{1.207\lambda}=\frac{1.65L}{\lambda}$
(C6)

Thus, one can modify the resolution of an iFTS by changing the maximum scan
length.666There is a small subtlety here, in that an iFTS delivers a fixed
resolution in wavenumber, which is slightly different than saying the quantity
$\lambda/\Delta\lambda$ is fixed. Comparing the performance of an iFTS and IFS
necessarily means matching the resolution at the center of the filter. The
variation in resolution along the filter is usually about 5-10%.

Figure C.3: Boxcar, triangular and hamming apodization functions and their
instrument profiles.

###  C.3 Apodization

For a basic boxcar scan pattern (equal time per step) the corresponding
instrument profile is a sinc function shown in the Figure C.3(a). We can see
besides a main lobe centered at 0, there are numerous additional peaks called
side lobes which cause a leakage of the spectral intensity. For the sinc
function, the amplitude of the largest side lobe is $22\%$ of that of the main
lobe. Because side lobes do not contribute to the measured spectral
information but represent the instrument pattern due to the truncation of
signal, it is desirable to reduce their amplitude.

The method to attenuate the side lobes is called apodization. The easiest way
to do this is by modifying the time spent per step in mirror position. Figure
C.3 (b) and (c) present a triangular apodization function and a Hamming
apodization function, respectively. Compared to the boxcar function, the
instrument profiles of these functions have a wider main lobe for the same
scan length, leading to higher signal-to-noise ratio and lower resolution. The
inverse Fourier transform of the triangular function is $Lsinc^{2}(\pi\nu L)$,
which has a FWHM of $\frac{1.77}{2L}$, ie, about 30% wider. Another option is
the Hamming function, $0.54-0.45\cos(2\pi x/L)$ with a FWHM of
$\frac{1.812}{2L}$. There has been significant research into optimal
apodization functions (Naylor & Tahic, 2007), and the choice can depend on the
science goals, and on the strength and number of the lines.

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