A Geometric View of Closure Phases in Interferometry
Nithyanandan Thyagarajan, Christopher L. Carilli
备注：
Accepted in PASA. 18 pages including 3 appendices, 11 captioned figures. Abstract abridged for arXiv. Removed some previous appendices. Parallels of closure phases in crystallography, interferometric synthetic aperture radar, seismic imaging, gravitational wave time-delay interferometry, and Bargmann invariants in quantum mechanics, described in appendices of arXiv:2012.05254v3
发表信息：
Publications of the Astronomical Society of Australia, 39, e014 (2022)
Closure phase is the phase of a closed-loop product of correlations in a $\ge 3$-element interferometer array. Its invariance to element-based phase corruption makes it invaluable for interferometric applications that otherwise require high-accuracy phase calibration. However, its understanding has remained mainly mathematical and limited to the aperture plane (Fourier dual of image plane). Here, we lay the foundations for a geometrical insight. we show that closure phase and its invariance to element-based corruption and to translation are intricately related to the conserved properties (shape, orientation, and size, or SOS) of the principal triangle enclosed by the three fringes formed by a closed triad of array elements, which is referred herein as the "SOS conservation principle". When element-based amplitude calibration is not needed, as is typical in optical interferometry, the 3-element interference image formed from phase-uncalibrated correlations is a true and uncorrupted representation of the source object's morphology, except for a possible shift. Based on this SOS conservation principle, we present two geometric methods to measure the closure phase directly from a 3-element interference image (without requiring an aperture-plane view): (i) the closure phase is directly measurable from any one of the triangle's heights, and (ii) the squared closure phase is proportional to the product of the areas enclosed by the triad of array elements and the principal triangle in the aperture and image planes, respectively. We validate this geometric understanding across a wide range range of interferometric conditions using data from the Very Large Array and the Event Horizon Telescope. This geometric insight can be potentially valuable to other interferometric applications such as optical interferometry. These geometric relationships are generalised for an $N$-element interferometer.
For short-wavelength VLBI observations, it is difficult to measure the phase of the visibility function accurately. The closure phases are reliable measurements under this situation, though it is not sufficient to retrieve all of the phase information. We propose a new method, Phase Retrieval from Closure Phase (PRECL). PRECL estimates all the visibility phases only from the closure phases. Combining PRECL with a sparse modeling method we have already proposed, imaging process of VLBI does not rely on dirty image nor self-calibration. The proposed method is tested numerically and the results a…
An $N$-element interferometer measures correlations among pairs of array elements. Closure invariants associated with closed loops among array elements are immune to multiplicative, element-based ("local") corruptions that occur in these measurements. Till recently, it has been unclear how a complete set of independent invariants can be analytically determined. We view the local, element-based corruptions in co-polar correlations as gauge tranformations belonging to the gauge group $\textrm{GL}(1,\mathbb{C})$. Closure quantities are then naturally gauge invariant. We use this to provide a simp…
We explore the sensitivity and completeness of long baseline interferometric observations for detecting unknown, faint companions around bright unresolved stars. We derive a linear expression for the closure phase signature of a faint companion in the high contrast regime (<0.1), and provide a quantitative estimation of the detection efficiency for the currently offered four-telescope configurations at the Very Large Telescope Interferometer. The results are compared to the performances provided by linear and Y-shaped interferometric configurations in order to identify the ideal array. We find…
The discovery of magnetic fields close to the M87 black hole using Very Long Baseline Interferometry (VLBI) by the Event Horizon Telescope collaboration utilized the novel concept of "closure traces", that are immune to element-based aberrations. We take a fundamentally new approach to this promising tool of polarimetric VLBI, using ideas from the geometric phase and gauge theories. The multiplicative distortion of polarized signals at the individual elements are represented as gauge transformations by general $2\times 2$ complex matrices, so the closure traces now appear as gauge-invariant qu…
Astronomical optical interferometers (OI) sample the Fourier transform of the intensity distribution of a source at the observation wavelength. Because of rapid atmospheric perturbations, the phases of the complex Fourier samples (visibilities) cannot be directly exploited , and instead linear relationships between the phases are used (phase closures and differential phases). Consequently, specific image reconstruction methods have been devised in the last few decades. Modern polychromatic OI instruments are now paving the way to multiwavelength imaging. This paper presents the derivation of a…
Closure phases are critical in astronomical interferometry. However, their uncertainties are difficult to compute numerically. We provide a method to efficiently compute interferometric closure phase distributions in terms of an approximate distribution that is valid in the low signal-to-noise ratio regime. This is done by first showing that the true phase distribution is well approximated by the von Mises distribution, then performing a convolution of three von Mises distributions. The resulting approximation is superior than the normal distribution for all signal-to-noise ratios and, being f…
Closure phases and closure amplitudes have proven critical to modern radio interferometry due to their insensitivity to the uncertain station gains. We present the first set of closure quantities constructed from parallel-hand and cross-hand visibilities that are insensitive to both station gains and to polarimetric leakage. These complex "closure traces" are a natural extension of closure amplitudes and closure phases, are independent of all station-based linear corruptions of the polarized visibilities, and are complete in the sense that they contain all remaining information present in the …
Imaging the direct light signal from a faint exoplanet against the overwhelming glare of its host star presents one of the fundamental challenges to modern astronomical instrumentation. Achieving sufficient signal-to-noise for detection by direct imaging is limited by three basic physical processes: aberration of the wavefronts (both instrumental and atmospheric), photon noise, and detector noise. In this paper, we advance a novel optical setup which synthesizes the advantages of two different techniques: nulling interferometry to mitigate photon noise, and closure phase to combat optical aber…
In long-baseline interferometry, bandwidth smearing of an extended source occurs at finite bandwidth when its different components produce interference packets that only partially overlap. In this case, traditional model fitting or image reconstruction using standard formulas and tools lead to biased results. We propose and implement a method to overcome this effect by calculating analytically a corrective term for the conventional interferometric observables: the visibility amplitude and closure phase. For that purpose, we model the interferogram taking into account the finite bandwidth and t…
We present the first high angular resolution observation of the B[e] star/X-ray transient object CI Cam, performed with the two-telescope Infrared Optical Telescope Array (IOTA), its upgraded three-telescope version (IOTA3T) and the Palomar Testbed Interferometer (PTI). Visibilities and closure phases were obtained using the IONIC-3 integrated optics beam combiner. CI Cam was observed in the near-infrared H and K spectral bands, wavelengths well suited to measure the size and study the geometry of the hot dust surrounding CI Cam. The analysis of the visibility data over an 8 year period from s…
Kernel-phase is a recently developed paradigm that tackles the classical problem of image deconvolution, based on an interferometric point of view of image formation. Kernel-phase inherits and borrows from the notion of closure-phase, especially as it is used in the context of non-redundant Fizeau interferometry, but extends its application to pupils of arbitrary shape, for diffraction limited images. The additional calibration brought by kernel-phase boosts the resolution of conventional images and enables the detection of otherwise hidden faint features at the resolution limit and beyond, a …
We present here three recipes for getting better images with optical interferometers. Two of them, Low- Frequencies Filling and Brute-Force Monte Carlo were used in our participation to the Interferometry Beauty Contest this year and can be applied to classical imaging using V 2 and closure phases. These two addition to image reconstruction provide a way of having more reliable images. The last recipe is similar in its principle as the self-calibration technique used in radio-interferometry. We call it also self-calibration, but it uses the wavelength-differential phase as a proxy of the objec…
Latest-generation spectro-interferometric instruments combine a milliarcsecond angular resolution with spectral capabilities, resulting in an immensely increased information content. Here, I present methodological work and results that illustrate the fundamentally new scientific insights provided by spectro-interferometry with very high spectral dispersion or in multiple line transitions (Brackett and Pfund lines). In addition, I discuss some pitfalls in the interpretation of spectro-interferometric data. In the context of our recent studies on the classical Be stars β CMi and ζ Tau, I present…
We report results from 5-day VLBI observations of the well-known quasar 3C 279 at 1.3 mm (230 GHz) in 2011. The measured nonzero closure phases on triangles including stations in Arizona, California and Hawaii indicate that the source structure is spatially resolved. We find an unusual inner jet direction at scales of $\sim$1 parsec extending along the northwest-southeast direction (PA = $127^{\circ}\pm3^{\circ}$), as opposed to other (previously) reported measurements on scales of a few parsecs showing inner jet direction extending to the southwest. The 1.3 mm structure corresponds closely wi…
Astronomers usually need the highest angular resolution possible, but the blurring effect of diffraction imposes a fundamental limit on the image quality from any single telescope. Interferometry allows light collected at widely-separated telescopes to be combined in order to synthesize an aperture much larger than an individual telescope thereby improving angular resolution by orders of magnitude. Radio and millimeter wave astronomers depend on interferometry to achieve image quality on par with conventional visible and infrared telescopes. Interferometers at visible and infrared wavelengths …
Université de Nice Sophia Antipolis（尼斯索菲亚·安提波利斯大学）；CNRS（法国国家科学研究中心）；Observatoire de la Côte d’Azur（蔚蓝海岸天文台）；Université de Lyon（里昂大学）；Université Lyon 1（里昂第一大学）；Observatoire de Lyon（里昂天文台）；Centre de Recherche Astrophysique de Lyon（里昂天体物理研究中心）；École Normale Supérieure de Lyon（里昂高等师范学院）
Astronomical optical interferometers sample the Fourier transform of the intensity distribution of a source at the observation wavelength. Because of rapid perturbations caused by atmospheric turbulence, the phases of the complex Fourier samples (visibilities) cannot be directly exploited. Consequently, specific image reconstruction methods have been devised in the last few decades. Modern polychromatic optical interferometric instruments are now paving the way to multiwavelength imaging. This paper is devoted to the derivation of a spatio-spectral (3D) image reconstruction algorithm, coined P…
A General Response Theory for Closure Phases through Visibility Nulls and Image Structure
University of Waterloo（滑铁卢大学）；Perimeter Institute for Theoretical Physics（珀因特理论物理研究所）；Waterloo Centre for Astrophysics, University of Waterloo（滑铁卢大学天体物理中心）
New massively redundant low frequency arrays allow for a novel investigation of closure relations in interferometry. We employ commissioning data from the Hydrogen Epoch of Reionization Array to investigate closure quantities in this densely packed grid array of 14m antennas operating at 100 MHz to 200 MHz. We investigate techniques that utilize closure phase spectra for redundant triads to estimate departures from redundancy for redundant baseline visibilities. We find a median absolute deviation from redundancy in closure phase across the observed frequency range of about 4.5deg. This value …
We develop a new process of image plane self-calibration for interferometric imaging data. The process is based on Shape-Orientation-Size (SOS) conservation for the principal triangle in an image generated from the three fringes made from a triad of receiving elements, in situations where interferometric phase errors can be factorized into element-based terms. The basis of the SOS conservation principle is that, for a 3-element array, the only possible image corruption due to an element-based phase screen is a tilt of the aperture plane, leading to a shift in the image plane. Thus, an image ma…
In this paper we compare the performance of multi and single-mode interferometry for the estimation of the phase of the complex visibility. We provide a theoretical description of the interferometric signal which enables to derive the phase error in presence of detector, photon and atmospheric noises, for both multi and single-mode cases. We show that, despite the loss of flux occurring when injecting the light in the single-mode component (i.e. single-mode fibers, integrated optics), the spatial filtering properties of such single-mode devices often enable higher performance than multimode co…
Filling out the dearth of detections between direct imaging and radial velocity surveys will test theories of planet formation and (sub)stellar binarity across the full range of semi-major axes, connecting formation of close to wide separation gas giants and substellar companions. Direct detection of close-in companions is notoriously difficult: coronagraphs and point spread function (PSF) subtraction techniques fail near the $λ/D$ diffraction limit. We present a new faint companion detection pipeline called Argus which analyzes kernel phases, an interferometric observable analogous to closure…
The new operational mode of aperture masking interferometry has been added to the CONICA camera which lies downstream of the Adaptive Optics (AO) corrected focus provided by NAOS on the VLT-UT4 telescope. Masking has been shown to deliver superior PSF calibration, rejection of atmospheric noise and robust recovery of phase information through the use of closure phases. Over the resolution range from about half to several resolution elements, masking interferometry is presently unsurpassed in delivering high fidelity imaging and direct detection of faint companions. Here we present results from…
We present an in-situ antenna characterization method and results for a "low-frequency" radio astronomy engineering prototype array, characterized over the 75-300 MHz frequency range. The presence of multiple cosmic radio sources, particularly the dominant Galactic noise, makes in-situ characterization at these frequencies challenging; however, it will be shown that high quality measurement is possible via radio interferometry techniques. This method is well-known in the radio astronomy community but seems less so in antenna measurement and wireless communications communities, although the mea…
Using kilometric arrays of air Cherenkov telescopes, intensity interferometry may increase the spatial resolution in optical astronomy by an order of magnitude, enabling images of rapidly rotating stars with structures in their circumstellar disks and winds, or mapping out patterns of nonradial pulsations across stellar surfaces. Intensity interferometry (pioneered by Hanbury Brown and Twiss) connects telescopes only electronically, and is practically insensitive to atmospheric turbulence and optical imperfections, permitting observations over long baselines and through large airmasses, also a…
Assessing the quality of aperture synthesis maps is relevant for benchmarking image reconstruction algorithms, for the scientific exploitation of data from optical long-baseline interferometers, and for the design/upgrade of new/existing interferometric imaging facilities. Although metrics have been proposed in these contexts, no systematic study has been conducted on the selection of a robust metric for quality assessment. This article addresses the question: what is the best metric to assess the quality of a reconstructed image? It starts by considering several metrics, and selecting a few b…
The study of binary stars is critical to apprehend many of the most interesting classes of stars. Moreover, quite often, the study of stars in binary systems is our only mean to constrain stellar properties, such as masses and radii. Unfortunately, a great fraction of the most interesting binaries are so compact that they can only be apprehended by high-resolution techniques, mostly by interferometry. I present some results highlighting the use of interferometry in the study of binary stars, from finding companions and deriving orbits, determining the mass and radius of stars, to studying mass…
Context. Theoretical predictions from models of red giant branch stars are a valuable tool for various applications in astrophysics ranging from galactic chemical evolution to studies of exoplanetary systems. Aims. We use the radiative transfer code OPTIM3D and realistic 3D radiative-hydrodynamical (RHD) surface convection simulations of red giants to explore the impact of granulation on interferometric observables. Methods. We compute intensity maps for the 3D simulation snapshots in two filters: in the optical at 5000 \pm 300 Å and in the K band 2.14 $\pm$ 0.26 μm FLUOR filter, corresponding…
Long baseline radio interferometers can provide some interesting opportunities for future SETI searches. Known advantages (compared to single dishes or beam-formed arrays), include the large reduction in false-positives due to the interferometer's natural suppression of RFI. This paper presents other advantages - the presence of multiple interferometer baselines in an array provide an important level of redundancy and additional confidence (verification) in the detection of faint and potentially transient signals. The SETI requirement for high time and frequency resolution is well matched to w…
3D mapping of matter distribution in the universe through the 21 cm radio emission of atomic hydrogen HI is a complementary approach to optical surveys for the study of the Large Scale Structures, in particular for measuring the BAO (Baryon Acoustic Oscillation) scale up to redshifts z < 3, and therefore constraining dark energy parameters. We propose a novel method to map the HI mass distribution in three dimensions in radio, without detecting or identifying individual compact sources. This method would require an instrument with a large instantaneous bandwidth (> 100 MHz) and high sensitivit…
The Large Binocular Telescope Interferometer (LBTI) has the longest baseline in the world, 22.7 m, for performing astronomical interferometry in Fizeau mode, which involves beam combination in a focal plane and preserves a wide field-of-view. LBTI can operate in this mode at wavelengths of 1.2 to 5 and 8 to 12 μm, making it a unique platform for carrying out high-resolution imaging of circumstellar disks, evolved stars, solar system objects, and possibly searches for planets, in the thermal infrared. Over the past five years, LBTI has carried out a considerable number of interferometric observ…
We used a new realistic 3D radiative-hydrodynamical model atmosphere of Procyon generated with the Stagger Code and synthetic spectra computed with the radiative transfer code Optim3D to re-analyze interferometric and spectroscopic data from the optical to the infrared of Procyon. We compute intensity maps in two optical filters centered at 500 and 800 nm (MARK III) and one infrared filter centered at 2200 nm (VINCI). We constructed stellar disk images accounting for the center-to-limb variations and used them to derive visibility amplitudes and closure phases. We provide 3D limb-darkening coe…
