Exoplanets Sciences with Nulling Interferometers and a Single-Mode Fiber-Fed Spectrograph

Understanding the atmospheres of exoplanets is a milestone to decipher their formation history and potential habitability. High-contrast imaging and spectroscopy of exoplanets is the major pathway towards the goal. Directly imaging of an exoplanet requires high spatial resolution. Interferometry has proven to be an effective way of improving spatial resolution. However, means of combining interferometry, high-contrast imaging, and high-resolution spectroscopy have been rarely explored. To fill in the gap, we present the dual-aperture fiber nuller (FN) for current-generation 8-10 meter telescopes, which provides the necessary spatial and spectral resolution to (1) conduct follow-up spectroscopy of known exoplanets; and (2) detect planets in debris-disk systems. The concept of feeding a FN to a high-resolution spectrograph can also be used for future space and ground-based missions. We present a case study of using the dual-aperture FN to search for biosignatures in rocky planets around M stars for a future space interferometry mission. Moreover, we discuss how a FN can be equipped on future extremely large telescopes by using the Giant Magellan Telescope (GMT) as an example.


INTRODUCTION
Direct imaging and spectroscopy of exoplanets provides a wealth of data sets to understand planet orbital dynamics and atmospheric compositions. Current-generation instruments can detect planets that are ∼ 10 6 times fainter than the host star at sub-arcsec separation (Macintosh et al. 2015;Keppler et al. 2018). In parallel, direct spectroscopy of substellar companions with high-resolution spectro-graphs (R>20,000) becomes an emerging field, which opens the window for probing atmospheric circulation (Snellen et al. 2010), surface inhomogeneity (Crossfield et al. 2014), and planet rotation (Schwarz et al. 2016;Bryan et al. 2018).
Combining high-contrast imaging and highresolution spectroscopy is logically the next step to improve sensitivity and broaden the science scope of direct imaging and spectroscopy. We use the term high dispersion coronagraphy (HDC) for the combination of the two techniques (Wang et al. 2017;Mawet et al. 2017).
HDC invokes multiple stages to suppress stellar light and extract the planet's signal. Specifically, high-contrast imaging suppresses stellar light and spatially separates the planet from its host star. A single-mode fiber injection system filters out stellar noise at the planet location since the electric field of a stellar speckles does not couple to the fundamental mode of a singlemode fiber. High-resolution spectroscopy further distinguishes the planet signal from stellar signal by its unique spectral features such as absorption lines and radial velocity. Using this three-pronged starlight suppression, HDC can achieve the high sensitivity to study terrestrial planets in the habitable zone (Kawahara et al. 2014;Lovis et al. 2017;Wang et al. 2017;Mawet et al. 2017;Wang et al. 2018).
Alternatively, an interferometer can be used to effectively suppress starlight. In contrast to a co-axial beam combiner, which is used for the Keck Interferometer (Millan-Gabet et al. 2011;Mennesson et al. 2014) and the Large Binocular Telescope Interferometer (Ertel et al. , 2020, a multi-axial beam combiner maximizes the spatial resolution, e.g., the Palomar fiber nuller (PFN, Haguenauer & Serabyn 2006;Mennesson et al. 2011;Serabyn et al. 2019) and the Fizeau imaging mode at LBT (Spalding et al. 2018). We will focus on the multi-axial interferometry because of its enhanced spatial resolution and its potential of feeding a high resolution spectrograph with a single-mode fiber.
A more recent development of HDC is the vortex fiber nuller (VFN, Ruane et al. 2018). VFN provides a unique solution for coronagraphy and high-resolution spectroscopy at sub λ/D angular resolution for next generation groundbased extremely large telescopes (ELTs, Ruane et al. 2019), and the concept has been demonstrated .
While ELTs with full capability of HDC are a decade away, we present in this paper a FN concept that can be applied to current-generation 8-10 meter telescopes, namely the dual-aperture FN. The concept-combining interferometry with high-resolution spectroscopy-has the potential to expedite the science goal of direct spectroscopy of exoplanets at tens of mas separations.
The dual-aperture FN can also be a choice for future space missions in search for habitable planets and biosignatures in their atmospheres, especially for planets around M stars. Spatial resolution is the major limiting factor that prevents space missions from pursuing direct spectroscopy of habitable planets around M stars. The dual-aperture FN permits (1) sufficient angular resolution with long-base line interferometry; and (2) searching for biosignatures in near infrared where their spectral features are abundant.
We will introduce a dual-aperture FN in §2 and evaluate its performance in §3. Science cases that are enabled by the dual-aperture FN are discussed in §4. A comparison between FN and VFN is given in §5. Our findings are summarized in §6.

A Dual-Aperture FN
The dual-aperture FN serves as a bridge between current 8-10 meter telescopes and future ELTs. For example, Keck telescopes, the Large Binocular Telescope, and the Very Large Telescopes Interferometer (VLTI) are all capable of dual-aperture interferometry, providing spatial resolutions that are comparable or even better than those from ELTs. However, ELTs offer superior photon collecting power to existing facilities.
Our dual-aperture FN concept is illustrated in Fig. 1. The dual-aperture is similar to the large binocular telescope interferometer (LBTI, Hinz et al. 2016), with a baseline-aperture ratio of 22.8 m / 8.4 m = 2.71. The concept can be generalized to any dual-aperture interferometers such as Keck and VLTI.
Because of the highly non-gaussian point spread function (PSF), a beam shaping device can be used to improve the coupling efficiency into a single-mode fiber. We calculate the coupling efficiency using the following equation: where η is the fiber coupling efficiency, E(r, θ) is the electromagnetic (EM) field and Ψ(r, θ) is the mode profile of a single-mode fiber. The resulting coupling efficiency map is shown in Panel (f) in Fig. 1. This is for an on-axis nuller for which the phases of the two sub-apertures are offset by π. On-axis light is completely canceled.

Coupling Efficiency
The coupling efficiency peaks at 35.3% at 0.28 λ/D or 0.76 λ/B, where λ is wavelength, D is the sub-aperture size, and B is the edge-toedge baseline. Coupling efficiency as a function of angular separation is shown in Fig. 2. The region with at least half of the peak efficiency goes from 0.36 to 1.12 λ/B.
According to Eq. 1, the coupling efficiency is the overlapping integral of the EM field and the mode profile of a single-mode fiber. To increase the coupling efficiency, the above two profiles need to be matched. We assume an optical device (e.g., a pair of cylindrical lenses) to change the aspect ratio of the interferometer PSF (shown in panel (d) in Fig. 1) to ∼1:1. This is equivalent to feed the EM field to an elongated two-dimensional gaussian beam (shown in panel (c) in Fig. 1), which is the "footprint" of the fundamental mode of a single- Figure 2. Left: throughput vs. angular separation. Middle: maximum throughput vs. core size of a single-mode fiber. Right: peak throughput vs. central obscuration. R 1 and R 2 are the radius of primary and secondary mirror, respectively. mode fiber after the beam-shaping device. The aspect ratio of the gaussian beam is determined by the baseline-aperture ratio, for which we adopt the LBTI value of 2.71. The maximum throughput as a function of the gaussian σ (along the elongation direction) is shown in Fig. 2.
We also show in Fig. 2 how the peak efficiency is affected by central obscuration of the secondary mirror. For LBT, the radius ratio between the secondary (R 2 ) and the primary mirror (R 1 ) is 0.45 / 4.20 = 0.11. The peak efficiency is reduced by less than 1%. In addition, the central obscuration does not affect the onaxis starlight suppression as long as the primary and the secondary mirrors are well aligned.

Coupling Efficiency Loss Due to Unmatched PSF
The point spread function (PSF) of the interferogram is non-gaussian as shown in panel (d) in Fig. 1. Consequently, the beam shape Ψ(r, θ) that maximizes coupling efficiency η should not be Gaussian. This does not fit the Gaussian fundamental mode of a single-mode fiber. Therefore, using a single-mode fiber, while not compromising the on-axis starlight suppression, will suffer from a peak coupling efficiency loss. Indeed, our simulation shows that the maximum throughput decreases by 2.3 times if no beamshaping device is used.

Gaps in Planet Searching Area
The area with high coupling efficiency is no longer azimuthally symmetric for the dualaperture case when compared to the singleaperture case . The implication is that the search time will increase in order to cover all phase angles for a given angular separation. However, for planets with known position angles, follow-up observations can be optimized using the right parallactic angle.
The key advantage of the dual-aperture case is the spatial resolution that rivals the spatial resolution of ELTs that are coming online in the next decade. We will discuss how we quantify the performance of the dual-aperture FN system in §3 and lay out science cases that are enabled in §4 by using LBTI as an example. The same concept can also be applied to other dual-aperture interferometers.

PERFORMANCE METRICS FOR A FN SYSTEM
We use a merit system that is based on required exposure time to evaluate the performance of a VFN system (Ruane et al. 2018). The total amount of exposure time is a summa-tion of required exposure times to overcome a variety of noise sources: where τ L is the exposure time to overcome leaked stellar noise due to low-order aberration, τ Θ for leaked stellar noise due to the finite size of a star, τ bg for background noise, τ dc for detector dark current, and τ rd for readout noise. All these terms are discussed in detailed in Ruane et al. (2018), so we refer readers to §3 in that paper. We note in Equation 2, τ L absorbs the term τ tt in Equation 15 in Ruane et al. (2018) because we consider tip-tilt as low-order aberration.

Sensitivity to Low-Order Aberrations
We pay special attention to τ L because it is system-specific. We adopt the same parameterization as Ruane et al. (2018) except that we change the fixed power-law dependence of 2 to a variable γ: where η p is the coupling efficiency at the planet location or planet throughput, η s is the on-axis throughput or the starlight suppression level, τ 0 will be given in the next equation. The term η s can be approximated as the summation of contribution of all Zernike modes. Subscript i is Zernike mode number, b i is the coefficient describing how starlight suppression depends on Zernike aberration, and ω is the RMS wavefront error. The term τ 0 is given in Equation 4 in Ruane et al. (2018), which describes the dependence of exposure time on parameters other than planet and star throughput, η p and η s : where R is spectral resolution, λ is central wavelength, S/N is the desired signal to noise ratio (S/N), is planet-star flux ratio, Φ s is star flux in the unit of photons per unit area per unit time per unit wavelength at the primary mirror, A is aperture size, q is the quantum efficiency of the detector, and T is the instrument throughput that affects the star and the planet equally.
We conduct simulation to numerically quantify b i . We use a functional form η s = (b i ω) γ , where γ = 2 or 4, to fit the numerical points for each Zernike mode. We set one sub-aperture to have zero wavefront error and add aberration to the other sub-aperture. In the case in which both sub-apertures have comparable aberrations, the wavefront error increases by a factor of √ 2.  Table 1.

Compared to a Single-Aperture VFN
The coefficients of sensitivity to low-order aberrations are shown in Fig. 3 and summarized in Table 1. Overall, the functional form provides a good approximation. Except for piston, all other even-numbered Zernike modes have a power=4 dependence on aberrations. This is different from the power=2 dependence for VFN (Ruane et al. 2018).
The implication is that small aberrations (e.g., RMS wavefront error < λ/100) in these Zernike modes contribute negligibly to starlight leakage, and the contribution becomes significant at large aberrations (e.g., RMS wavefront error ∼ λ/10). Because we are interested in the FN performance at small aberrations, these evennumbered Zernike modes can be omitted in τ L calculation (Eq. 3).
In contrast to the single-aperture VFN (Ruane et al. 2018), the dual-aperture FN is not azimuthally symmetric, so it loses the advantage of the single-aperture case, i.e., the singleaperture VFN is only sensitive to Z ±1 n (see Table  1 for the GMT case and §5.4 for more details).
In addition, it is also shown in Fig. 3 that the dual-aperture FN is sensitive to piston aberration. This is because changing piston for one sub-aperture while maintaining piston for the other sub-aperture would shift the interferogram along the baseline direction. The effect is similar to the impact of tip-tilt in a singleaperture VFN.

APPLICATIONS
Direct spectroscopy of exoplanets is an alternative way of studying exoplanet atmospheres to transit spectroscopy. Since only less than 10% of planets transit their host stars, direct spectroscopy in principle makes it more accessible to probe exoplanet atmospheres, especially for the most nearby exoplanets that are detected by the radial velocity technique and do not transit.
Moreover, direct spectroscopy builds up signal-to-noise ratio (SNR) more easily than transit spectroscopy because there is no constraint for waiting for the next transit.
Together with planet mass and metallicity as inferred from radial velocity data, and possibly age from asteroseismology due to their proximity, and planet chemical composition measurements provided by direct spectroscopy, this information can be used as bench marks to test and improve planet atmospheric modeling. This science case is discussed in §4.1.
Dual-aperture FN offers excellent starlight suppression at a spatial resolution that is comparable to that of ELTs ( §3). Aided by high resolution spectroscopy, the effective starlight suppression level can be improved by another few orders of magnitude (Wang et al. 2017). This allows us to improve inner working angle (IWA) to observe lower-mass planets that are intrinsically more frequent than gas giant planets that can be currently detected (Bowler 2016;Fernandes et al. 2019).
Moreover, to alleviate the the large sample size ( 100) that is usually required to directly image a couple of exoplanets, we can conduct the search for planets around dusty systems, whose long-period planet occurrence rate is boosted by a factor of ∼10 compared to systems without such a constraint (Meshkat et al. 2017). We will discuss this science case in §4.2.
Direct imaging and spectroscopy of rocky planets in the habitable zone is the major science driver for ELTs and future space missions. Space missions such as HabEx and LUVOIR are limited by spatial resolution λ/D. Increasing aperture size D will significantly increase the cost. Another cost driver is the cooling systems that are required to reach mid-and thermal-infrared wavelengths, which the above space mission avoids. However, avoiding long wavelengths in infrared will limit these space missions' ability in searching for biosignatures, which usually have more much abundant spectral lines in infrared than at shorter wavelengths. Space interferometry creates a niche in high-spatial-resolution infrared spectroscopy for temperate planets around nearby M stars, which are traditionally in the reign of groundbased ELTs. This science case will be discussed in §4.3.

Follow-up Observations of Exoplanets
Detected by other techniques We use 4152 exoplanets from the NASA Exoplanet Archive (NEA) service 1 . We put these planets on a separation -planet-star contrast plot as shown in Fig. 4 in order to select amenable targets. Targets with contrast lower than 5×10 −7 and angular separation larger than 15 mas are given in Table 2. We focus on K and L band, which are a trade-off between thermal background and wavefront aberration. As an example, we use the LBTI to present the following two science cases. Below we detail how separation and planet-star contrast are calculated based on information available from NEA.
We calculate the planet-star separation based on their reported distance and semi-major axis. When the latter is not available, we calculate it using orbital period and stellar mass. Planetstar contrast is calculated using the following equation: where, R p is planet radius, a is semi-major axis, and A g is albedo, which assume to be 0.3. When planet radius is not available, we use the massradius relation in (Chen & Kipping 2017) to calculate radius based on mass.
The code for target selection and exposure time calculation is available through a Python notebook on GitHub 2 . Here we present two examples to illustrate how to interpret the calculated exposure times given in Table 2. 4.1.1. 55 Cnc c in K-band LBTI observation 55 Cnc c (McArthur et al. 2004) is among the most challenging exoplanets on our list in terms angular separation (19.2 mas) and planet-star contrast (7.18 × 10 −7 ). The angular separation corresponds to 1.04 and 0.59 λ/B in K and L band assuming a 22.8-m baseline.
Low-order aberration is the major noise source that limits the total required exposure time from the breakdown of exposure times. The 4659-hour exposure time can be greatly reduced by using the cross-correlation technique, which boosts the SNR by ∼40 in K band (see Fig. 7 and more details in Ruane et al. 2018). Since exposure time changes inversely to the square of SNR (Eq. 4), the required exposure time for the cross-correlation technique is reduced to 4659/40 2 = 2.91 hours for a 5-σ detection.

55 Cnc c in L-band LBTI observation
Planet throughput is assumed to be 31.0% for a 0.59 λ/B angular separation (see Fig. 2). Star suppression level is again 4.81×10 −4 if assuming λ/100 wavefront error. However, L-band wavefront quality is better than that of K band, so η s should be lower. A full description of the parameters used in the simulation are given in Table. 6.
We have the following numbers for required exposure times to overcome various noise sources ( §3): τ L = 1.83 × 10 7 s, τ Φ = 3.69 × 10 5 s, τ bg = 7.26 × 10 8 s (assuming a L band thermal background of 2.0 mag per square arcsec), τ dc = 1.73 × 10 5 s, and τ rd = 1.12 × 10 3 s. The summation of the above terms is 7.45 × 10 8 s, or 207121 hours. The required exposure time is therefore limited by the L-band thermal background noise. Accounting for the boost factor of ∼35 that is brought by the cross-correlation technique, the required exposure time is reduced to 169 hours.

Direct Spectroscopy of Exoplanets
Embedded in Systems with Disks . Planet-star contrast vs. angular separation. Colored data points have angular separation larger than 15 mas and planet-star contrast higher than 5 × 10 −7 and therefore amenable for direct spectroscopy (see also Table 2). Red pluses are targets in the north and blue crosses are targets in the south. Blue and red dashed lines mark 1 λ/B for K and L band for a baseline of 22.8 meter.
We select targets using the Catalog of Circulstellar Disks 3 . There are 48 debris disk systems with r magnitudes brighter than 8th and inclinations lower than 60 degree, i.e., more faceon systems (Table 3). The magnitude cut is to ensure optimal AO performance and the inclination cut is to minimize the extinction due to the increasing viewing angle. Below we use HD 104860, the faintest debris-disk system in our sample with R = 8.0, as an example to demonstrate the feasibility of using LBTI and the dualaperture FN to search for planets in debris-disk systems. We provide a Python notebook to compute required exposure time to achieve the sensitivity for a given planet-star contrast.

HD 104860
HD 104860 (Morales et al. 2013, and references therein) represents the worst-case scenario among all targets because it is the faintest debris-disk system in our sample with R = 8.0. We convert R-band magnitude into K or L-band magnitude using the updated Table 5 in Pecaut & Mamajek (2013) for a given effective temperature. In calculating exposure times, we set planet-star contrast to 10 −6 . While the contrast is comparable to the stateof-the-art performance, the greatest gain is the IWA of the FN, which brings the IWA to ∼20 mas ( Table 7). The breakdown of the required exposure times to overcome various noise sources ( §3) are as follows: τ L = 2.31 × 10 8 s, τ Φ = 1.11 × 10 6 s, τ bg = 3.97 × 10 6 s, τ dc = 2.22 × 10 7 s, and τ rd = 1.41 × 10 4 s. The total exposure time is 2.59 × 10 8 s, or 71963 hours for K-band observation. Accounting for the boost factor of ∼40 that is brought by the cross-correlation technique, the required exposure time is reduced to 44.9 hours.
We note that the final exposure time is very sensitive to planet-star contrast.
Relaxing the targeted planet-star contrast by two times would reduce the exposure time by a factor of 4 (Eq. 4). The exposure time is also sensitive to planet throughput and wavefront error to the second power (Eq. 3). Therefore, improving wavefront quality and planet throughput is the key to increase the efficiency of planet search.

Background induced by disk brightness
Below we will show that the background noise due to the emissivity of sky and instrument is almost always higher than the background due to the disk brightness. We can therefore only consider the sky and instrument background when calculating τ bg . Using HD 191089 (Soummer et al. 2014) as an example, the system has a bright debris disk that has a flux of 1 mJy per square arcsec (Ren et al. 2019) in H band. In comparison, the star is 3750 mJy in H band. Given the extent of the disk at ∼1 square arcsec, the ratio between the integrated disk flux and the star flux is 2.6 × 10 −4 or a delta magnitude of 8.9 mag. Since scattered light is the major component in near infrared, it is reasonable to assume that the ratio is similar in K and L band. In the case of HD 191089, the background induced by disk brightness is ∼14 mag per square arcsec, lower than the assumed thermal background in our calculation, i.e., 12.2 mag per square arcsec in K band and 2.0 mag per square arcsec in L band.

Characterizing Rocky Planets Around M Stars with Space Interferometric Array
The small angular separations (<20 mas) of habitable planets around M stars are formidable for space direct-imaging missions due to limited aperture sizes. Moreover, searching for multiple tracers of biosignatures (e.g., water, oxygen, and methane), which reduces the likelihood of false positives (Domagal-Goldman et al. 2014;Harman et al. 2015), requires infrared observations. Observing at infrared wavelengths further decreases the spatial resolution for space missions.
Infrared interferometry provides a solution to the above issue (e.g., Kammerer & Quanz 2018). In addition, the dual-aperture FN concept alleviates many of the technical challenges towards a space interferometry mission (Monnier et al. 2019).
We again start from 4152 exoplanets from the NASA Exoplanet Archive service. Following the angular separation and planet-star contrast calculations that are detailed in §4.1, we select planets with (1) contrasts lower than 1 × 10 −7 ; (2) angular separations larger than 5 mas; and (3) radii smaller than 0.2 R Jupiter . Table 4 lists and Fig. 5 shows the 23 potential rocky planets that meet the above criteria. We provide two examples below for the purpose of feasibility demonstration. Calculations for other planets are available through a Python notebook that is available on GitHub 4 .

GJ 1061 b
While GJ 1061 b (Dreizler et al. 2020) has a favorable planet-star contrast at 1.47 × 10 −6 , its angular separation (5.7 mas) and faintness (K=6.6) pose challenges for direct spectroscopy. In the following calculation, we assume a subaperture diameter of 4 meter and a baseline of 50 meter. This corresponds to an angular separation of 0.69 λ/B for K band at the 50meter baseline. Wavefront RMS error is λ/100 and this translates to a star suppression level of 4.81 × 10 −4 . For thermal background, we as-sume a level that is comparable to JWST thermal background at 0.2 MJy/SR, which is 20.4 mag per square arcsec 5 . Full parameters in simulation are given in Table 8.
The breakdown of the required exposure times to overcome various noise sources ( §3) are as follows: τ L = 1.06 × 10 8 s, τ Φ = 1.10 × 10 7 s, τ bg = 6.22 × 10 2 s, τ dc = 3.10 × 10 7 s, and τ rd = 6.47 × 10 3 s. Adding up these terms leads to a total exposure time of 1.48×10 8 s, or 41199 hours for a K-band observation. Accounting for the boost factor of ∼40 that is brought by the cross-correlation technique, the required exposure time is reduced to 25.7 hours.
The total exposure time has three major contributions within the same order of magnitude: low-order aberration, finite-size of the host star, and dark current. The low-order aberration component can be improved by reducing wavefront error. However, ∼20 nm RMS error (λ/100) is at the level of JWST wavefront error (Aronstein et al. 2016) and may take significant effort to improve. The finite-size component becomes significant because of the 50meter baseline resolution (9 mas) approaches within two orders of magnitude to the angular diameter of the star (0.4 mas). The dark current component becomes significant because of the increasing ratio between dark current to stellar flux due to the decreasing aperture size (see Equation 13 in Ruane et al. (2018)).

Proxima Cen b
Proxima Cen b (Anglada-Escudé et al. 2016) is the closest planetary system to the solar system and therefore presents a compelling case for direct spectroscopy. Here we discuss a case for an L-band observation with space-based dualaperture FN (Table 9).
There are two game changers for the spacebased observation. First, the thermal background, which is the major limitation for ground-based observations, is significantly reduced. We assume a JWST thermal background level at 0.2 MJy/SR, or 19.5 mag per square acrsec. Second, space interferometry achieves a superior spatial resolution to any previous space missions that allows Proxima Cen b to be observed at 1.03 λ/B at a 20-meter baseline in L band.
The breakdown of the required exposure times to overcome various noise sources ( §3) are as follows: τ L = 6.64 × 10 8 s, τ Φ = 2.21 × 10 7 s, τ bg = 8.78 × 10 3 s, τ dc = 3.93 × 10 7 s, and τ rd = 4.01 × 10 4 s. Adding up these terms leads to a total exposure time of 7.25×10 8 s, or 210579 hours for L-band observation. Accounting for the boost factor of ∼35 that is brought by the cross-correlation technique, the required exposure time is reduced to 164.5 hours.
The major limiting factor for the total required exposure time is the low-order aberration. Note that the exposure time is comparable to numbers from HDC simulations for ELT ground-based observations (Wang et al. 2017).

The Connection
Both a FN and a VFN are a nuller, i.e., a device that suppresses on-axis starlight by manipulating the phase of an EM field. A FN achieves the phase manipulation by changing piston. A VFN achieves the phase manipulation with a vortex plate. For an azimuthally-changing EM, a FN and a VFN deliver a similar performance in terms of starlight suppression and IWA.
We use the Giant Magellan Telescope (GMT, Johns et al. 2012) configuration to illustrate the similarity between a FN and a VFN. For the VFN setup, we use a vortex plate with charge=1. The resulting coupling map is shown in Fig. 6. For the FN setup, we block the cen- Figure 5. Planet-star contrast vs. angular separation. Colored data points have angular separation larger than 5 mas and planet-star contrast higher than 1 × 10 −7 and therefore amenable for direct spectroscopy with a space interferometric mission (see also Table 4). Red pluses are targets in the north and blue crosses are targets in the south. Dotted and dashed lines mark 1 λ/B at 1 µm for a baseline of 30 and 10 meter.
tral sub-aperture and use only the outer six subapertures. To achieve a similar performance to the GMT VFN, we change piston for the six sub-apertures so that their phases change from 0 to 5π/6 with an increment of π/6. This effectively create a charge=1 phase ramp. The resulting coupling map of the GMT FN setup is similar to that of the GMT VFN setup (Fig.  7).
Note that the very same idea can be applied to other ELTs. In addition to changing piston and using a vortex plate, phase manipulation can also be achieved with a deformable mirror.

IWA for a FN with a phase ramp
IWA for a VFN (in the unit of λ/D) is determined by the charge number of a vortex plate (Ruane et al. 2018), see also Fig. 8. For a FN, the IWA as a function of charge number follows the same trend. Three examples are given in Fig. 9 to illustrate the effect of charge number on IWA.  As the effective charge number increases, the IWA of a FN is pushed outward. This is quantitatively similar to a VFN on GMT (Fig. 8 Left). However, one noticeable difference is the searching area (bottom rows of Fig. 9). Unlike a VFN, the FNs with charge number of 2 and 3 have a partial coverage at a given λ/D.
Since the IWA of a VFN or a FN with an effective phase ramp is determined by the charge number, there is a trade-ff between the system complexity and a full coverage at a given angular separation. For a VFN, the system is more complicated with the addition of a vortex plate, but the coverage is continuous for a given λ/D. In contrast, a FN system is simpler since phase is controlled by the piston of each sub-aperture, but there are insensitive planetsearch areas along an annulus. However, this drawback of a FN can be compensated by the simplicity/flexibility of the system: by combining different configurations, e.g., adding up coupling maps for charge 1, 2, and 3 as shown in Fig. 9, will result in a complete coverage from the IWA for charge=1 to the IWA for charge=3.

IWA for a FN with mirror symmetry
The LBT FN is one variation of FNs with mirror symmetry, and we refer to it as a dualaperture FN. The IWA of this type of FN is determined by the baseline of the interferometer. To make a connection between a dual-aperture FN and a dual-aperture VFN, we add a vortex plate in the optical system to investigate if the charge number would affect the IWA of a dual-aperture system. Equivalently, the dualaperture FN corresponds to a charge=0 VFN.
As shown in Fig. 8 Right, the angular separation with the highest throughput does not Figure 9. Left Column: top panel is the phase ramp of a FN for GMT. Phases of the six outer sub-apertures change from 0 to 5π/6 with an increment of π/6. This correspond to charge = 1. Bottom panel is the corresponding coupling map. Middle Column: the same as the left column except charge = 2, i.e., phases of the six outer subapertures change from 0 to 5π/3 with an increment of π/3. Right Column: the same as the left column except charge = 3. move out as charge increases. An analogy to help understand the dependence is a traditional dual-aperture interferometer: the locations of the first null and/or the first constructive interferogram remain the same as long as the baseline remains the same, regardless of the diameter of the sub-aperture. The net effect of increasing the charge number in a dual-aperture system is to reduce the peak throughput without adding the benefit of relaxing the sensitivity to low-order aberrations.

Comparing LBT FN and GMT VFN
An LBT FN (B = 22.8 m) can provide similar spatial resolution to that of GMT (B = 25.2 m). Although the light collecting power of LBT is (7/2) = 3.5 lower than GMT, a FN can have 2 higher planet throughput and lower sensitivity to low-order aberrations than a VFN. These factors would significantly reduce the exposure time for the LBT FN (Eq. 3). In order to understand the trade-off between light collecting power, the planet throughput, and the sensitivity to low-order aberrations, we study two specific cases: an LBT FN and a GMT VFN. In the comparison, we assume everything is the same except for the light collecting area, planet throughput, and the sensitivity to loworder aberrations.
The sensitivity to lower-order aberrations for the GMT VFN is calculated the same way as described in §3 (see also Table 1) with one exception: wavefront errors are applied across a pupil that consists of 7 sub-apertures rather than one sub-aperture. The numerical results are in agreement with (Ruane et al. 2018). Fig. 10 shows the dependence on RMS wavefront error per Zernike mode. Only Zernike modes with l = ±1, i.e., with Zernike indices of 1, 2, 7, and 8 are coupled into a single-mode fiber. This is also confirmed in Ruane et al. (2018Ruane et al. ( , 2019. Given the non-zero numerical values for other Zernike modes that should not be theoretically decoupled from a single-mode fiber, we conclude that the uncertainties of b i coefficients that are reported in Table  §1 are ∼0.01. Quantitative relationships between η s and wavefront error are given in Table 1. Using 55 Cnc c as an example, the total exposure time to reach a 5-σ detection for a K-band LBT FN is 2.9 hours (Table 5) vs. 11.4 hour for a VFN at GMT (Table 10). Despite a factor of ∼4 lower light collecting power, the LBT FN needs only a factor of ∼4 shorter exposure time to reach the same detection significance as a VFN on GMT. The loss in effective aperture size is out-weighted by the increase of planet throughput and the decrease in low-order aberrations (Eq. 3). By comparing Table 5 and  Table 10, the difference in planet throughput is 27.0% for the LBT FN vs. 15.8% for the GMT VFN. A factor of 1.7 translates into a difference of 2.9 in exposure time based on Eq. 3. In addition, the starlight suppression level for the LBT is 6.4 times better than that for the GMT VFN. This further reduces the LBT FN exposure time by a factor of 6.4 times. Together, these factors explain why the LBT FN outperforms the GMT VFN by a factor of 4.
The above comparison of performance in K holds as long as the total exposure time is dominated by the exposure time to overcome the loworder aberrations. However, this is no longer true in L band, in which case the dominating noise source is the thermal background. Although the exposure time to overcome the thermal background τ bg is sensitive to the planet throughput, i.e., ∝ η −2 p , (see Eq. 12 in Ruane et al. 2018), τ bg is also proportional to the solid angle subtended by the fiber, which is a factor of ∼3 smaller for the GMT case than the LBT case. Therefore, the L-band exposure time ratio between the LBT FN and the GMT VFN for 55 Cnc c is 2.3 (Table 6 and Table 11). While the ratio is not as promising as the K-band case, it is nonetheless better than 3.5, which is from simply scaling the effective aperture size.

SUMMARY
We present a concept of combining nuling interferometry with single-mode fiber-fed highresolution spectroscopy, i.e., the dual-aperture FN, which can be applied to current-generation 8-10 meter telescopes. The dual-aperture FN provides spatial resolution that is comparable to that of future ELTs, and therefore enables several unique science cases for 8-10 m telescopes before the era of ELTs in 2030s and future space interferometric missions.
We conduct numerical simulations as a proof of concept in §2. We quantify planet throughput as a function of angular separation, single-mode fiber core size, and central obscuration due to a secondary mirror (Fig. 2). We use a merit system, which is based on the required exposure time, to evaluate the performance of the dualaperture FN in §3. In particular, we quantify the sensitivity of starlight leakage to low-order aberrations as expressed by Zernike modes (Fig.  3 and Table 1).
Because of the superior spatial resolution brought by interferometry and the planet sensitivity brought by single-mode fiber-fed highresolution spectroscopy, a number of science cases are enabled by the dual-aperture FN, including (1) follow-up spectroscopic observations on exoplanet systems that are detected by the radial velocity technique ( §4.1); (2) searching for planets in debris-disk system ( §4.2); and (3) direct spectroscopy for biosignatures in rocky planets around nearby M stars ( §4.3).
Targets for each case are given in Table 2,  Table 3, and Table 4. Specific examples are discussed and general exposure time calculators are provided on GitHub 6 . In all cases, we find that the dual-aperture FN is a viable pathway to achieve the science goals with reasonable telescope time investments.
We compare FN and VFN in §5. The two concepts are connected by pupil-plane phase manipulation in order to achieve starlight suppression. We use GMT as an example to illustrate the connection and the synergy between the two concepts. We also compare the performance of a GMT VFN and an LBT FN. The comparison showcases that the LBT FN is indeed bridging the gaps between 8-10 telescopes and future ELTs because of its spatial resolution, planet throughput and sensitivity. Acknowledgements We would like to thank Elodie Choquet, Karl Stapelfeldt, and Bin Ren for helpful discusions on the debris-disk science case, Dan Echeverri, Garreth Ruane, and Dimitri Mawet for insights into VFN, Steve Ertel, Jordan Stone, and Amali Vaz for useful information on LBT and LBTI, and Bertrand Mennesson for discussing fiber nullers.