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Frantz Martinache

Publications and source records attributed to Frantz Martinache.

At least 55 records · Page 3Linked to original sources

Calibration of the instrumental polarization effects of SCExAO-CHARIS' spectropolarimetric mode

SCExAO at the Subaru telescope is a visible and near-infrared high-contrast imaging instrument employing extreme adaptive optics and coronagraphy. The instrument feeds the near-infrared light (JHK) to the integral field spectrograph CHARIS. Recently, a Wollaston prism was added to CHARIS' optical path, giving CHARIS a spectropolarimetric capability that is unique among high-contrast imaging instruments. We present a detailed Mueller matrix model describing the instrumental polarization effects of the complete optical path, thus the telescope and instrument. The 22 wavelength bins of CHARIS provide a unique opportunity to investigate in detail the wavelength dependence of the instrumental polarization effects. From measurements with the internal light source, we find that the image derotator (K-mirror) produces strong wavelength-dependent crosstalk, in the worst case converting ~95% of the incident linear polarization to circularly polarized light that cannot be measured. Theoretical calculations show that the magnitude of the instrumental polarization of the telescope varies with wavelength between approximately 0.5% and 0.7%, and that its angle is exactly equal to the altitude angle of the telescope. We plan to more accurately determine the instrumental polarization of the telescope with observations of a polarization standard star, and fit more comprehensive physical models to all experimental data. In addition, we plan to integrate the complete Mueller matrix model into the existing CHARIS post-processing pipeline, with the aim to achieve a polarimetric accuracy of <0.1% in the degree of linear polarization. Our calibrations of CHARIS' spectropolarimetric mode will enable unique quantitative polarimetric studies of circumstellar disks and planetary and brown dwarf companions.

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SCExAO/CHARIS Direct Imaging Discovery of a 20 au Separation, Low-Mass Ratio Brown Dwarf Companion to an Accelerating Sun-like Star

We present the direct imaging discovery of a substellar companion to the nearby Sun-like star, HD 33632 Aa, at a projected separation of $\sim$ 20 au, obtained with SCExAO/CHARIS integral field spectroscopy complemented by Keck/NIRC2 thermal infrared imaging. The companion, HD 33632 Ab, induces a 10.5$σ$ astrometric acceleration on the star as detected with the $Gaia$ and $Hipparcos$ satellites. SCExAO/CHARIS $JHK$ (1.1--2.4 $μ$m) spectra and Keck/NIRC2 $L_{\rm p}$ (3.78 $μ$m) photometry are best matched by a field L/T transition object: an older, higher gravity, and less dusty counterpart to HR 8799 cde. Combining our astrometry with $Gaia/Hipparcos$ data and archival Lick Observatory radial-velocities, we measure a dynamical mass of 46.4 $\pm$ 8 $M_{\rm J}$ and an eccentricity of $e$ $<$0.46 at 95\% confidence. HD 33632 Ab's mass and mass ratio (4.0\% $\pm$ 0.7\%) are comparable to the low-mass brown dwarf GJ 758 B and intermediate between the more massive brown dwarf HD 19467 B and the (near-)planet mass companions to HR 2562 and GJ 504. Using $Gaia$ to select for direct imaging observations with the newest extreme adaptive optics systems can reveal substellar or even planet-mass companions on solar system-like scales at an increased frequency compared to blind surveys.

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The MKID Exoplanet Camera for Subaru SCExAO

We present the MKID Exoplanet Camera (MEC), a z through J band (800 - 1400 nm) integral field spectrograph located behind The Subaru Coronagraphic Extreme Adaptive Optics (SCExAO) at the Subaru Telescope on Maunakea that utilizes Microwave Kinetic Inductance Detectors (MKIDs) as the enabling technology for high contrast imaging. MEC is the first permanently deployed near-infrared MKID instrument and is designed to operate both as an IFU, and as a focal plane wavefront sensor in a multi-kHz feedback loop with SCExAO. The read noise free, fast time domain information attainable by MKIDs allows for the direct probing of fast speckle fluctuations that currently limit the performance of most high contrast imaging systems on the ground and will help MEC achieve its ultimate goal of reaching contrasts of $10^{-7}$ at 2$λ/ D$. Here we outline the instrument details of MEC including the hardware, firmware, and data reduction and analysis pipeline. We then discuss MEC's current on-sky performance and end with future upgrades and plans.

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High-contrast H$α$ imaging with Subaru/SCExAO+VAMPIRES

We present current status of H$α$ high-contrast imaging observations with Subaru/SCExAO+VAMPIRES. Our adaptive optics correction at optical wavelengths in combination with (double) spectral differential imaging (SDI) and angular differential imaging (ADI) was capable of detecting a ring-like feature around omi Cet and the H$α$ counterpart of jet around RY Tau. We tested the post-processing by changing the order of ADI and SDI and both of the contrast limits achieved $\sim10^{-3}-5\times10^{-4}$ at $0.3^{\prime\prime}$, which is comparable to other H$α$ high-contrast imaging instruments in the southern hemisphere such as VLT/SPHERE, VLT/MUSE, and MagAO. Subaru/VAMPIRES provides great opportunities for H$α$ high-contrast imaging for northern hemisphere targets.

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Kernel nullers for an arbitrary number of apertures

The use of interferometric nulling for the direct detection of extrasolar planets is in part limited by the extreme sensitivity of the instrumental response to tiny optical path differences between apertures. The recently proposed kernel-nuller architecture attempts to alleviate this effect with an all-in-one combiner design that enables the production of observables inherently robust to residual optical path differences (<< lambda). Until now, a unique kernel nuller design has been proposed ad hoc for a four-beam combiner. We examine the properties of this original design and generalize them for an arbitrary number of apertures. We introduce a convenient graphical representation of the complex combiner matrices that model the kernel nuller and highlight the symmetry properties that enable the formation of kernel nulls. The analytical description of the nulled outputs we provide demonstrates the properties of a kernel nuller. Our description helps outline a systematic way to build a kernel nuller for an arbitrary number of apertures. The designs for 3- and 6-input combiners are presented along with the original 4-input concept. Combiners grow in complexity with the square of the number of apertures. While one can mitigate this complexity by multiplexing nullers working independently over a smaller number of sub-apertures, an all-in-one kernel nuller recombining a large number of apertures appears as the most efficient way to characterize a high-contrast complex astrophysical scene. One can design kernel nullers for an arbitrary number of apertures that produce observable quantities robust to residual perturbations. The designs we recommend are lossless and take full advantage of all the available interferometric baselines. They are complete, result in as many kernel nulls as the theoretically expected number of closure-phases, and are optimized to require as few outputs as possible.

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SCExAO/CHARIS Near-IR Integral Field Spectroscopy of the HD 15115 Debris Disk

We present new, near-infrared ($1.1 - 2.4$ $μm$) high-contrast imaging of the debris disk around HD 15115 with the Subaru Coronagraphic Extreme Adaptive Optics system (SCExAO) coupled with the Coronagraphic High Angular Resolution Imaging Spectrograph (CHARIS). SCExAO/CHARIS resolves the disk down to $ρ\sim 0.2''$ ($\rm{r_{proj}} \sim 10$ $\rm{au}$), a factor of $\sim 3-5$ smaller than previous recent studies. We derive a disk position angle of $\rm{PA}$ $\sim 279.4^\circ - 280.5^\circ$ and an inclination of $\rm{i}$ $\sim 85.3^\circ - 86.2^\circ$. While recent SPHERE/IRDIS imagery of the system could suggest a significantly misaligned two ring disk geometry, CHARIS imagery does not reveal conclusive evidence for this hypothesis. Moreover, optimizing models of both one and two ring geometries using differential evolution, we find that a single ring having a Hong-like scattering phase function matches the data equally well within the CHARIS field of view ($ρ\lesssim 1''$). The disk's asymmetry, well-evidenced at larger separations, is also recovered; the west side of the disk appears on average around 0.4 magnitudes brighter across the CHARIS bandpass between $0.25''$ and $1''$. Comparing STIS/50CCD optical photometry ($2000-10500$ $Å$) with CHARIS NIR photometry, we find a red (STIS/50CCD$-$CHARIS broadband) color for both sides of the disk throughout the $0.4'' - 1''$ region of overlap, in contrast to the blue color reported at similar wavelengths for regions exterior to $\sim 2''$. Further, this color may suggest a smaller minimum grain size than previously estimated at larger separations. Finally, we provide constraints on planetary companions, and discuss possible mechanisms for the observed inner disk flux asymmetry and color.

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SCExAO/CHARIS High-Contrast Imaging of Spirals and Darkening Features in the HD 34700 A Protoplanetary Disk

We present Subaru/SCExAO+CHARIS broadband ($JHK$-band) integral field spectroscopy of HD 34700 A. CHARIS data recover HD 34700 A's disk ring and confirm multiple spirals discovered in Monnier et al. (2019). We set limits on substellar companions of $\sim12\ M_{\rm Jup}$ at $0\farcs3$ (in the ring gap) and $\sim5\ M_{\rm Jup}$ at $0\farcs75$ (outside the ring). The data reveal darkening effects on the ring and spiral, although we do not identify the origin of each feature such as shadows or physical features related to the outer spirals. Geometric albedoes converted from the surface brightness suggests a higher scale height and/or prominently abundant sub-micron dust at position angle between $\sim45^\circ$ and $90^\circ$. Spiral fitting resulted in very large pitch angles ($\sim30-50^\circ$) and a stellar flyby of HD 34700 B or infall from a possible envelope is perhaps a reasonable scenario to explain the large pitch angles.

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On-sky verification of Fast and Furious focal-plane wavefront sensing: Moving forward toward controlling the island effect at Subaru/SCExAO

High-contrast imaging (HCI) observations of exoplanets can be limited by the island effect (IE). On the current generation of telescopes, the IE becomes a severe problem when the ground wind speed is below a few meters per second. This is referred to as the low-wind effect (LWE). The LWE severely distorts the point spread function (PSF), significantly lowering the Strehl ratio and degrading the contrast. In this article, we aim to show that the focal-plane wavefront sensing (FPWFS) algorithm, Fast and Furious (F&F), can be used to measure and correct the IE/LWE. We deployed the algorithm on the SCExAO HCI instrument at the Subaru Telescope using the internal near-infrared camera in H-band. We tested F&F with the internal source, and it was deployed on-sky to test its performance with the full end-to-end system and atmospheric turbulence. The performance of the algorithm was evaluated by two metrics based on the PSF quality: 1) the Strehl ratio approximation ($SRA$), and 2) variance of the normalized first Airy ring ($VAR$). Random LWE phase screens with a peak-to-valley wavefront error between 0.4 $μ$m and 2 $μ$m were all corrected to a $SRA$ $>$90\% and an $VAR\lessapprox0.05$. Furthermore, the on-sky results show that F&F is able to improve the PSF quality during very challenging atmospheric conditions (1.3-1.4'' seeing at 500 nm). Closed-loop tests show that F&F is able to improve the $VAR$ from 0.27 to 0.03 and therefore significantly improve the symmetry of the PSF. Simultaneous observations of the PSF in the {optical} ($λ= $ 750 nm, $Δλ=$ 50 nm) show that during these tests we were correcting aberrations common to the optical and NIR paths within SCExAO. Going forward, the algorithm is suitable for incorporation into observing modes, which will enable PSFs of higher quality and stability during science observations.

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Angular differential kernel phases

To reach its optimal performance, Fizeau interferometry requires that we work to resolve instrumental biases through calibration. One common technique used in high contrast imaging is angular differential imaging, which calibrates the point spread function and flux leakage using a rotation in the focal plane. Our aim is to experimentally demonstrate and validate the efficacy of an angular differential kernel-phase approach, a new method for self-calibrating interferometric observables that operates similarly to angular differential imaging, while retaining their statistical properties. We used linear algebra to construct new observables that evolve outside of the subspace spanned by static biases. On-sky observations of a binary star with the SCExAO instrument at the Subaru telescope were used to demonstrate the practicality of this technique. We used a classical approach on the same data to compare the effectiveness of this method. The proposed method shows smaller and more Gaussian residuals compared to classical calibration methods, while retaining compatibility with the statistical tools available. We also provide a measurement of the stability of the SCExAO instrument that is relevant to the application of the technique. Angular differential kernel phases provide a reliable method for calibrating biased observables. Although the sensitivity at small separations is reduced for small field rotations, the calibration is effectively improved and the number of subjective choices is reduced.

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Kernel-phase analysis: aperture modeling prescriptions that minimize calibration errors

Kernel-phase is a data analysis method based on a generalization of the notion of closure-phase invented in the context of interferometry, but that applies to well corrected diffraction dominated images produced by an arbitrary aperture. The linear model upon which it relies theoretically leads to the formation of observable quantities robust against residual aberrations. In practice, detection limits reported thus far seem to be dominated by systematic errors induced by calibration biases not sufficiently filtered out by the kernel projection operator. This paper focuses on the impact the initial modeling of the aperture has on these errors and introduces a strategy to mitigate them, using a more accurate aperture transmission model. The paper first uses idealized monochromatic simulations of a non trivial aperture to illustrate the impact modeling choices have on calibration errors. It then applies the outlined prescription to two distinct data-sets of images whose analysis has previously been published. The use of a transmission model to describe the aperture results in a significant improvement over the previous type of analysis. The thus reprocessed data-sets generally lead to more accurate results, less affected by systematic errors. As kernel-phase observing programs are becoming more ambitious, accuracy in the aperture description is becoming paramount to avoid situations where contrast detection limits are dominated by systematic errors. Prescriptions outlined in this paper will benefit any attempt at exploiting kernel-phase for high-contrast detection.

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High-Resolution Near-Infrared Polarimetry and Sub-Millimeter Imaging of FS Tau A: Possible Streamers in Misaligned Circumbinary Disk System

We analyzed the young (2.8-Myr-old) binary system FS Tau A using near-infrared (H-band) high-contrast polarimetry data from Subaru/HiCIAO and sub-millimeter CO (J=2-1) line emission data from ALMA. Both the near-infrared and sub-millimeter observations reveal several clear structures extending to $\sim$240 AU from the stars. Based on these observations at different wavelengths, we report the following discoveries. One arm-like structure detected in the near-infrared band initially extends from the south of the binary with a subsequent turn to the northeast, corresponding to two bar-like structures detected in ALMA observations with an LSRK velocity of 1.19-5.64 km/s. Another feature detected in the near-infrared band extends initially from the north of the binary, relating to an arm-like structure detected in ALMA observations with an LSRK velocity of 8.17-16.43 km/s. From their shapes and velocities, we suggest that these structures can mostly be explained by two streamers that connect the outer circumbinary disk and the central binary components. These discoveries will be helpful for understanding the evolution of streamers and circumstellar disks in young binary systems.

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Overview of focal plane wavefront sensors to correct for the Low Wind Effect on SUBARU/SCExAO

The Low Wind Effect (LWE) refers to a phenomenon that occurs when the wind speed inside a telescope dome drops below $3$m/s creating a temperature gradient near the telescope spider. This produces phase discontinuities in the pupil plane that are not detected by traditional Adaptive Optics (AO) systems such as the pyramid wavefront sensor or the Shack-Hartmann. Considering the pupil as divided in 4 quadrants by regular spiders, the phase discontinuities correspond to piston, tip and tilt aberrations in each quadrant of the pupil. Uncorrected, it strongly decreases the ability of high contrast imaging instruments utilizing coronagraphy to detect exoplanets at small angular separations. Multiple focal plane wavefront sensors are currently being developed and tested on the Subaru Coronagraphic Extreme Adaptive Optics (SCExAO) instrument at Subaru Telescope: Among them, the Zernike Asymmetric Pupil (ZAP) wavefront sensor already showed on-sky that it could measure the LWE induced aberrations in focal plane images. The Fast and Furious algorithm, using previous deformable mirror commands as temporal phase diversity, showed in simulations its efficiency to improve the wavefront quality in the presence of LWE. A Neural Network algorithm trained with SCExAO telemetry showed promising PSF prediction on-sky. The Linearized Analytic Phase Diversity (LAPD) algorithm is a solution for multi-aperture cophasing and is studied to correct for the LWE aberrations by considering the Subaru Telescope as a 4 sub-aperture instrument. We present the different algorithms, show the latest results and compare their implementation on SCExAO/SUBARU as real-time wavefront sensors for the LWE compensation.

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SUBARU Near-Infrared Imaging Polarimetry of Misaligned Disks Around The SR24 Hierarchical Triple System

The SR24 multi-star system hosts both circumprimary and circumsecondary disks, which are strongly misaligned from each other. The circumsecondary disk is circumbinary in nature. Interestingly, both disks are interacting, and they possibly rotate in opposite directions. To investigate the nature of this unique twin disk system, we present 0.''1 resolution near-infrared polarized intensity images of the circumstellar structures around SR24, obtained with HiCIAO mounted on the Subaru 8.2 m telescope. Both the circumprimary disk and the circumsecondary disk are resolved and have elongated features. While the position angle of the major axis and radius of the NIR polarization disk around SR24S are 55$^{\circ}$ and 137 au, respectively, those around SR24N are 110$^{\circ}$ and 34 au, respectively. With regard to overall morphology, the circumprimary disk around SR24S shows strong asymmetry, whereas the circumsecondary disk around SR24N shows relatively strong symmetry. Our NIR observations confirm the previous claim that the circumprimary and circumsecondary disks are misaligned from each other. Both the circumprimary and circumsecondary disks show similar structures in $^{12}$CO observations in terms of its size and elongation direction. This consistency is because both NIR and $^{12}$CO are tracing surface layers of the flared disks. As the radius of the polarization disk around SR24N is roughly consistent with the size of the outer Roche lobe, it is natural to interpret the polarization disk around SR24N as a circumbinary disk surrounding the SR24Nb-Nc system.

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Atmospheric Characterization and Further Orbital Modeling of $κ$ And b

We present $κ$ Andromeda b's photometry and astrometry taken with Subaru/SCExAO+HiCIAO and Keck/NIRC2, combined with recently published SCExAO/CHARIS low-resolution spectroscopy and published thermal infrared photometry to further constrain the companion's atmospheric properties and orbit. $κ$ And b's Y/Y-K colors are redder than field dwarfs, consistent with its youth and lower gravity. Empirical comparisons of its Y-band photometry and CHARIS spectrum to a large spectral library of isolated field dwarfs reaffirm the conclusion from Currie et al. (2018) that it likely has a low gravity but admit a wider range of most plausible spectral types (L0-L2). Our gravitational classification also suggests that the best-fit objects for $κ$ And b may have lower gravity than those previously reported. Atmospheric models lacking dust/clouds fail to reproduce its entire 1--4.7 $μm$ spectral energy distribution, cloudy atmosphere models with temperatures of $\sim$ 1700--2000 $K$ better match $κ$ And b data. Most well-fitting model comparisons favor 1700--1900 $K$, a surface gravity of log(g) $\sim$ 4--4.5, and a radius of 1.3--1.6\,$R_{\rm Jup}$; the best-fit model (DRIFT-Phoenix) yields the coolest and lowest-gravity values: $T_{\rm eff}$=1700 K and $\log g$=4.0. An update to $κ$ And b's orbit with ExoSOFT using new astrometry spanning seven years reaffirms its high eccentricity ($0.77\pm0.08$). We consider a scenario where unseen companions are responsible for scattering $κ$ And b to a wide separation and high eccentricity. If three planets, including $κ$ And b, were born with coplanar orbits and one of them was ejected by gravitational scattering, a potential inner companion with mass $\gtrsim10M_{\rm Jup}$ could be located at $\lesssim$ 25 au.

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Performance and Early Science with the Subaru Coronagraphic Extreme Adaptive Optics Project

We describe the current performance of the Subaru Coronagraphic Extreme Adaptive Optics (SCExAO) instrument on the Subaru telescope on Maunakea, Hawaii and present early science results for SCExAO coupled with the CHARIS integral field spectrograph. SCExAO now delivers H band Strehl ratios up to $\sim$ 0.9 or better, extreme AO corrections for optically faint stars, and planet-to-star contrasts rivaling that of GPI and SPHERE. CHARIS yield high signal-to-noise detections and 1.1--2.4 $μm$ spectra of benchmark directly-imaged companions like HR 8799 cde and kappa And b that clarify their atmospheric properties. We also show how recently published as well as unpublished observations of LkCa 15 lead to a re-evaluation of its claimed protoplanets. Finally, we briefly describe plans for a SCExAO-focused direct imaging campaign to directly image and characterize young exoplanets, planet-forming disks, and (later) mature planets in reflected light.

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Focal-plane wavefront sensing with the vector Apodizing Phase Plate

In this article we show that the vector-Apodizing Phase Plate (vAPP) coronagraph can be designed such that the coronagraphic point spread functions (PSFs) can act as a wavefront sensor to measure and correct the (quasi-)static aberrations, without dedicated wavefront sensing holograms nor modulation by the deformable mirror. The absolute wavefront retrieval is performed with a non-linear algorithm. The focal-plane wavefront sensing (FPWFS) performance of the vAPP and the algorithm are evaluated with numerical simulations, to test various photon and read noise levels, the sensitivity to the 100 lowest Zernike modes and the maximum wavefront error (WFE) that can be accurately estimated in one iteration. We apply these methods to the vAPP within SCExAO, first with the internal source and subsequently on-sky. In idealised simulations we show that for $10^7$ photons the root-mean-square (RMS) WFE can be reduced to $\simλ/1000$, which is 1 nm RMS in the context of the SCExAO system. We find that the maximum WFE that can be corrected in one iteration is $\simλ/8$ RMS or $\sim$200 nm RMS (SCExAO). Furthermore, we demonstrate the SCExAO vAPP capabilities by measuring and controlling the lowest 30 Zernike modes with the internal source and on-sky. On-sky, we report a raw contrast improvement of a factor $\sim$2 between 2 and 4 $λ/D$ after 5 iterations of closed-loop correction. When artificially introducing 150 nm RMS WFE, the algorithm corrects it within 5 iterations of closed-loop operation. FPWFS with the vAPP's coronagraphic PSFs is a powerful technique since it integrates coronagraphy and wavefront sensing, eliminating the need for additional probes and thus resulting in a $100\%$ science duty cycle and maximum throughput for the target.

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Enabling the next generation of scientific discoveries by embracing photonic technologies

The fields of Astronomy and Astrophysics are technology limited, where the advent and application of new technologies to astronomy usher in a flood of discoveries altering our understanding of the Universe (e.g., recent cases include LIGO and the GRAVITY instrument at the VLTI). Currently, the field of astronomical spectroscopy is rapidly approaching an impasse: the size and cost of instruments, especially multi-object and integral field spectrographs for extremely large telescopes (ELTs), are pushing the limits of what is feasible, requiring optical components at the very edge of achievable size and performance. For these reasons, astronomers are increasingly looking for innovative solutions like photonic technologies that promote instrument miniaturization and simplification, while providing superior performance. Astronomers have long been aware of the potential of photonic technologies. The goal of this white paper is to draw attention to key photonic technologies and developments over the past two decades and demonstrate there is new momentum in this arena. We outline where the most critical efforts should be focused over the coming decade in order to move towards realizing a fully photonic instrument. A relatively small investment in this technology will advance astronomical photonics to a level where it can reliably be used to solve challenging instrument design limitations. For the benefit of both ground and space borne instruments alike, an endorsement from the National Academy of Sciences decadal survey will ensure that such solutions are set on a path to their full scientific exploitation, which may one day address a broad range of science cases outlined in the KSPs.

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Kernel-phase Detection Limits : Hypothesis Testing and the Example of JWST NIRISS Full Pupil Images

The James Webb Space Telescope will offer high-angular resolution observing capability in the near-infrared with masking interferometry on NIRISS, and coronagraphic imaging on NIRCam & MIRI. Full aperture kernel-phase based interferometry complements these observing modes, probing for companions at small separations while preserving the telescope throughput. Our goal is to derive both theoretical and operational contrast detection limits for the kernel-phase analysis of JWST NIRISS full-pupil observations by using tools from hypothesis testing theory, applied to observations of faint brown dwarfs with this instrument, but the tools and methods introduced here are applicable in a wide variety of contexts. We construct a statistically independent set of observables from aberration-robust kernel phases. Three detection tests based on these observable quantities are designed and analysed, all guaranteeing a constant false alarm rate for small phase aberrations. One of these tests, the Likelihood Ratio or Neyman-Pearson test, provides a theoretical performance bound for any detection test. The operational detection method considered here is shown to exhibit only marginal power loss with respect to the theoretical bound. In principle, for the test set to a false alarm probability of 1%, companion at contrasts reaching 10^3 at separations of 200 mas around objects of magnitude 14.1 are detectable. With JWST NIRISS, contrasts of up to 10^4 at separations of 200 mas could be ultimately achieved, barring significant wavefront drift. The proposed detection method is close to the ultimate bound and offers guarantees over the probability of making a false detection for binaries, as well as over the error bars for the estimated parameters of the binaries detectable by JWST NIRISS. This method is not only applicable to JWST NIRISS but to any imaging system with adequate sampling.

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