The Palomar Kernel Phase Experiment: Testing Kernel Phase Interferometry for Ground-based Astronomical Observations
Benjamin Pope, Peter Tuthill, Sasha Hinkley, Michael J. Ireland, Alexandra Greenbaum, Alexey Latyshev, John D. Monnier, Frantz Martinache
aa r X i v : . [ a s t r o - ph . I M ] O c t Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 2 October 2018 (MN L A TEX style file v2.2)
The Palomar Kernel Phase Experiment: Testing Kernel PhaseInterferometry for Ground-based Astronomical Observations
Benjamin Pope ⋆ , Peter Tuthill , Sasha Hinkley , Michael J. Ireland ,Alexandra Greenbaum , Alexey Latyshev , John D. Monnier , Frantz Martinache Oxford Astrophysics, Denys Wilkinson Building, Keble Rd, University of Oxford, Oxford OX1 3RH, UK Sydney Institute for Astronomy, School of Physics, University of Sydney, NSW 2006, Australia School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, UK Research School of Astronomy & Astrophysics, Australian National University, Canberra, ACT 2611, Australia Department of Physics and Astronomy, The Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218 Department of Astronomy, University of Michigan, 1085 S University Ave, Ann Arbor, MI 48109-1090, USA Laboratoire Lagrange, CNRS UMR 7293, Observatoire de la Côte d’Azur, Bd de l’Observatoire, 06304 Nice, France
ABSTRACT
At present, the principal limitation on the resolution and contrast of astronomical imaginginstruments comes from aberrations in the optical path, which may be imposed by the Earth’sturbulent atmosphere or by variations in the alignment and shape of the telescope optics. Theseerrors can be corrected physically, with active and adaptive optics, and in post-processing ofthe resulting image. A recently-developed adaptive optics post-processing technique, calledkernel phase interferometry, uses linear combinations of phases that are self-calibrating withrespect to small errors, with the goal of constructing observables that are robust against theresidual optical aberrations in otherwise well-corrected imaging systems. Here we present adirect comparison between kernel phase and the more established competing techniques, aper-ture masking interferometry, point spread function (PSF) fitting and bispectral analysis. Weresolve the α Ophiuchi binary system near periastron, using the Palomar 200-Inch Telescope.This is the first case in which kernel phase has been used with a full aperture to resolve asystem close to the diffraction limit with ground-based extreme adaptive optics observations.Excellent agreement in astrometric quantities is found between kernel phase and masking,and kernel phase significantly outperforms PSF fitting and bispectral analysis, demonstrat-ing its viability as an alternative to conventional non-redundant masking under appropriateconditions.
Key words: techniques: interferometric — techniques: image processing — instrumentation:adaptive optics — instrumentation: high angular resolution
Kernel phase interferometry (Martinache 2010) is a powerful tech-nique for image analysis, applicable to any observations whichboth Nyquist-sample all spatial frequencies, and have appropri-ately small wavefront errors. The method is based on a linear ap-proximation, introduced first in Martinache (2010), considering atransfer matrix propagating small phase errors in a pupil or re-dundant array into the corresponding space of u, v baselines. Thekernel of this matrix generates ‘kernel phases’ that are a general-ization of the self-calibrating closure phase quantity well-knownin interferometry (Jennison 1958). These linear combinations ofphases have the property that small phase errors cancel, meaning ⋆ E-mail: [email protected] for example that the residual aberrations after adaptive optics donot propagate at first order into the kernel phase measurements.The kernel phase technique holds promise for detecting objects athigh contrast at or just inside the diffraction limit ∼ λ/D of spacetelescopes, or of ground-based telescopes assisted with extremeadaptive optics (AO). Such instruments are now present on manyof the world’s largest optical telescopes, including PALM-3000 atPalomar (Bouchez et al. 2008), Subaru Coronagraphic Extreme AO(SCExAO) (Martinache et al. 2009), VLT-SPHERE (Beuzit et al.2010), and the Gemini Planet Imager (GPI) (Macintosh et al. 2008,2014). Moreover, the upcoming James Webb Space Telescope(Gardner et al. 2006) will be capable of nearly-diffraction-limitedresolution at very high sensitivity, and it will be important to es-tablish and benchmark the performance of kernel phase and other c (cid:13) B. J. S. Pope et al. image analysis approaches to best exploit the high image quality itwill provide.Pope et al. (2013) first applied kernel phase interferometry toa sample of brown dwarf systems imaged by Reid et al. (2006) andReid et al. (2008) using the
HST -NICMOS NIC1 camera. Wherethe original papers found a total of ten binary systems out of 72studied, Pope et al. (2013) recovered all of these and found five ad-ditional systems with high confidence, as well as four other newmore marginal candidates at lower confidence or higher contrast.Kernel phase interferometry has not previously been used toresolve a close system with ground-based full-aperture extremeadaptive optics observations, and a detailed discussion of kernelphase reduction compared to other competing techniques has notbeen published. Martinache (2011) reported the extraction of ker-nel phases from a Keck II NIRC image, but did not report the de-tection of any companion. Ireland & Kraus (2014) have presenteda kernel-phase image reconstruction of the LkCa15 system in an Mfilter. In this paper, we discuss kernel phase and NRM observationsof the close binary system α Ophiuchi ( α Oph, or Rasalhague) un-der identical observing conditions on the same night, with a viewto using this as a benchmark for comparing the two methods. Thisis the first time such a simultaneous comparison has been made. Apreliminary analysis of these observations was presented in confer-ence proceedings by Martinache (2013); Hinkley et al. (2015).Hinkley et al. (2011) carried out an adaptive optics non-redundant masking (NRM) study of this system, a nearby binarywith an A5 III primary, in order to characterise its orbital param-eters. Hinkley et al. (2011) find a contrast ratio of . ± . inthe K band from resolved Palomar-PHARO (Hayward et al. 2001)imaging. The primary is known to be rotating at ∼ of its pre-dicted breakup velocity (Zhao et al. 2009), and asterosesimic anal-ysis with MOST shows rotationally-modulated g -modes that probethe conditions of the interior (Monnier et al. 2010). Establishing itsmass with precision is therefore valuable for constraining modelsof its rotational dynamics. Hinkley et al. (2011) predicted the com-panion would pass periastron at ∼ April 2012, and for this rea-son there was an observing campaign in 2012 to track its orbit at itsapparent closest point, which is particularly critical in deliveringa fully-constrained dynamical orbit. While α Oph is ordinarily awell-separated binary, at periastron the companion is buried withinthe PSF of the primary.
Observations were made on 26 June 2012, two months after peri-astron, using the PHARO camera on the 5.1 m Hale Telescope atMt. Palomar Observatory. Data were obtained using a 9-hole aper-ture mask, an 18-hole aperture mask and with the full aperture (withno mask) in CH and K s bands, whose filters are centred at 1.57 µ m and 2.145 µ m respectively with bandpasses of 0.1 and 0.310 µ m. The 9-hole mask contained projected baselines ranging from0.75 to 4.15 m with a projected hole diameter of 0.5 m, while for the18-hole mask the projected baselines ranged from 0.37 to 4.81 mwith a projected hole diameter of 0.25 m.In addition to α Oph, PSF reference stars ǫ Oph and ǫ Herwere observed as calibrators. These data were then reduced to astandard FITS cube form using existing masking code. Regrettably,the full aperture observations in the CH band suffered from poorAO performance and detector saturation and were excluded fromthe present study. The seeing varied between 1.5 and 2 arcsecondsduring the observations. By modelling the PSF, we determined the Figure 1.
Maximum Entropy model-independent image reconstruction us-ing BSMEM for K s band non-redundant masking data. median Strehl across the exposures to have been ∼ . in the K s band.In the full-pupil imaging, a neutral-density filter was used todiminish the brightness of the star, as is necessary to avoid satu-rating the science camera. This introduces a ‘ghost’ (a reflectionartefact), which severely limits the maximum size of the windowthat can be used in kernel phase analysis, as discussed in Section 3.In future, it would be beneficial to choose filters in such a way asto avoid ghosts wherever possible, or to avoid using such filters al-together, for example by using very short exposure times or simplyobserving fainter targets.Non-redundant 18-hole aperture masking observations witha Br γ filter were processed to obtain the arguments of themean bispectrum, i.e. bispectral-amplitude-weighted average clo-sure phases. We used a Markov Chain Monte Carlo (MCMC)method to fit a binary model to these closure phases, recover-ing a companion at . ± . mas separation, position angle . ± . ◦ and . ± . contrast. As we were not able to obtainBr γ observations with a filled aperture, these are useful only forcomparison with the kernel phase data. Non-redundant 9-hole ob-servations with a K s filter found the same companion at . ± . mas separation, . ± . ◦ position angle and . ± . con-trast. This contrast ratio is consistent with the . ± . in theK band reported in (Hayward et al. 2001). In addition to this, weperformed a maximum entropy model-independent image recon-struction using BSMEM (Figure 1), in which it is apparent thatthe parametric model accurately captures the information about thesource intensity distribution. These aperture masking observationsset the standard with which the kernel phase-based analyses mustbe compared. c (cid:13) , 000–000 alomar Kernel Phase Experiment In Sections 2.1 and 2.2 we discuss the methods used to analysedata in this paper. In the interests of reproducibility and open sci-ence, we have made public our code and data: .
FITS files for ourraw data are stored on Figshare , together with the IPython note-book which was used to analyse these observations; and the PYSCO
PYthon Self-Calibrating Observables package, a Python modulefor extracting and analyse kernel phase data, is available in a pub-lic GitHub repository . All other packages used in this analysisare publicly distributed elsewhere. We welcome efforts by other re-searchers to apply this body of analysis software to these or otherdata, with appropriate citation.In all full-frame images, the peak of the PSF pushed the detec-tor into its nonlinear response regime. Kernel phase requires strictlinearity, being a Fourier technique, and so each frame thereforehad to be calibrated with a nonlinear gain curve map in order torestore linearity. The core of ǫ Oph was fully saturated in K s band,and therefore ǫ Her was used as the sole calibrator.
As discussed in Section 1, kernel phases are self-calibrating lin-ear combinations of phases, which are robust with respect to smallresidual wavefront errors. We calculate a matrix to generate these a priori based on an assumed discrete model of the pupil.We obtained direct images of the PHARO ‘medium cross’pupil, and found the ratio of the outer radius to the central obscu-ration, and of this radius to the thickness of the spiders, to differ toa small extent from the nominal values reported in Hayward et al.(2001). This is very important to establish carefully, as informationfrom longer baselines than exist in the telescope consists purely ofnoise and will corrupt any kernel phases obtained. In order to estab-lish a precise pupil model as is necessary for kernel phase, we usedthe visibility amplitudes extracted from the point-source calibrator, ǫ Her, to constrain the overall scale of the pupil. This may differfrom the published values because of an error in the measured pro-jected pupil size, or an offset in the effective filter bandpass, whichwe model as the nominal K s band centre of . µ m, but maydiffer from this nominal value due for example to a slope in thestellar spectrum.The absolute magnitude of the Fourier transform of the imageof a point source, in the case of a flat wavefront, is a map of theoptical modulation transfer function. This is itself found as the au-tocorrelation of the pupil, whose magnitude is approximately givenby the redundancy of each baseline in a discrete pupil model. Wetherefore varied this overall outer scale in the vicinity of the valuereported in Hayward et al. (2001), which lists a projected radiusof 2.32 m. In order to be as sensitive as possible to the outer ra-dius, we perform a least-squares fit by brute force between the log-arithms of the redundancy matrix elements and the magnitude ofthe stacked Fourier transforms of all 100 observed frames. The fitis best-conditioned by the low visibilities, at the edge of the pupil,and there is some discrepancy at intermediate visibilities (low spa-tial frequencies), where we are sensitive to the faint binary-like sig-nal of the ND filter ghost and residual low-order aberrations. Ide-ally, we would model this pupil cojointly with the binary model, http://figshare.com/s/4e69f7b2b30411e4bf4a06ec4bbcf141 https://github.com/benjaminpope/pysco and marginalize over uncertainties, which in the present circum-stances we are unable to do due to the prohibitively long computa-tional times. The best fit is found with an outer projected radius of2.392 m, which we therefore adopt as fixed in the following anal-ysis. The discretized pupil generated with this model is shown inFigure 2, containing 1128 elements, and generating 3256 baselines.Using a singular value decomposition, we find this model togenerate 2692 kernel phases. We centre each image in real spaceand recentre it to sub-pixel precision by subtracting a phase slopein its Fourier transform, and apply this matrix to phases extractedfrom the corresponding 3256 baselines in this Fourier transform inorder to obtain the kernel phases.There are several differences between the application of kernelphase to this dataset and to the previously-published HST samplein Pope et al. (2013). In particular, each observation consists of adatacube of 100 frames, yielding excellent experimental diversityso that the statistics on each kernel phase can be readily recovered,as opposed to the case with the
HST snapshot data where an en-semble average over many different targets was required. Kernelphases are therefore extracted separately from each frame of data,and then combined such that in the following Sections we take asour data the ensemble mean of each kernel phase over the set offrames, and the statistical uncertainties are taken to be the standarderrors of the mean (SEM).As is standard practice in NRM interferometry, the PSF ref-erence stars ǫ Oph and ǫ Her were processed in the same way. Bysubtracting the kernel phases measured on these point sources, it istherefore possible to calibrate systematic offsets in the instrumentalkernel phase measurements. The uncertainties on each of the cal-ibrator’s kernel phases, again taken to be the standard error of themean, are added in quadrature to the uncertainties on the sciencetarget’s kernel phases.In addition to this, we also add in quadrature a second errorterm of . ◦ to account for uncalibrated systematic errors. We fita parametric binary model to the data as described in Section 2.2,and iteratively adjust the magnitude of this additional error term sothat the reduced χ of the best-fitting parameters is approximatelyunity. The next steep is to fit a parametric model to these kernel phasedata, defined by the binary parameters separation (mas), posi-tion angle (deg) and contrast, proceeding in a similar fashionto Pope et al. (2013). We estimate these parameters using twoBayesian inference algorithms, namely M
ULTI N EST (Feroz et al.2009), an implementation of multi-modal nested sampling, and
EMCEE (Foreman-Mackey et al. 2013), an affine-invariant ensem-ble Markov Chain Monte Carlo sampler. We used both approachesfirstly as a check for consistency, but also because they have com-plementary strengths (Allison & Dunkley 2014): on the one hand,M
ULTI N EST efficiently and reliably converges on the global peakof a posterior distribution without significant sensitivity to an initialguess and avoids being trapped in local likelihood maxima. On theother hand,
EMCEE is more effective at exploring and characteriz-ing potentially non-Gaussian, curving degeneracies in the shape ofthe posterior mode; as noted in Pope et al. (2013), there is typicallysignificant degeneracy between separation and contrast in kernelphase fits to systems at close to the diffraction limit, and it is im-portant to explore the shape of this curve.We began by running M
ULTI N EST , obtaining the parameterestimates listed in Table 1. The corresponding correlation diagram c (cid:13) , 000–000 B. J. S. Pope et al. [h]
Figure 2.
Hale Telescope medium-cross pupil model. Red dots representpupil sampling points; note that they avoid the spiders, which on the HaleTelescope are vertical and horizontal with respect to the detector axes. is displayed in Figure 3. After our first attempt with no additionaluncertainty added in quadrature, we iteratively re-ran the MCMCadding an additional error term in quadrature until the fit of theposterior mean achieved a reduced χ of approximately unity. Thisterm was found to be ∼ . ◦ in the kernel phase case, and . ◦ inthe case of the bispectral phases.As discussed above, interferometric determinations of binaryparameters at close to the diffraction limit often suffer from de-generacy between contrast and separation. As a result, we used theM ULTI N EST output to initialize an
EMCEE run with 100 walkersand 200 burn-in steps and recorded 1000 subsequent steps to sam-ple from the posterior. From this, it is apparent that there is only asmall degree of covariance between these parameters, and we findgood agreement between the M
ULTI N EST and
EMCEE estimates ofthe posterior mean and standard deviation.In order to test whether the kernel phase processing itself in-troduces a bias into the contrast and separation estimates, we sim-ulated binaries with the same parameters each as the best fit to thekernel phase full aperture observations and to the non-redundant 9hole closure phase measurements. These simulations use no atmo-sphere, but include a realistic Palomar ‘medium cross’ pupil modelidentical to that used to derive the kernel phase relations. Modelfitting is performed with M
ULTI N EST as in Section 2.2, with un-certainties on each kernel phase taken to be the same as from thereal observations. For an input model with the parameters of thekernel phase model (129 mas, 83.6 degrees, 34.2 contrast) we re-trieve . ± . mas separation, . ± . deg position angle andcontrast . ± . ; and for the aperture masking parameters (129.6mas, 83.5, 28.7 contrast), we retrieve . ± . mas, . ± . Table 1.
Binary Parameter Estimates for α Ophiuchi at JD2456104.847025.Mode Separation Position Angle Contrast(mas) (deg) ( K s )Kernel Phase 129.3 ± ± ± ± ± ± ± ± ± ± ± ± Figure 3.
Correlation diagram for α Ophiuchi kernel phases in K s band.We plot model kernel phases on the x -axis and the observed signal on the y -axis, such that for a good fit we expect the data to lie on a straight line ofgradient unity (overplotted green line). deg and . ± . contrast. It is evident from these fits that kernelphase fitting itself introduces no bias towards lower contrasts. Using the same full-aperture observations analysed with kernelphase in Section 2.2, we also performed an identical analysis us-ing the arguments of the bispectrum, i.e. closure phases. The clo-sure phases are the arguments of the bispectrum or ‘triple prod-uct’ of three complex visibilities around a closing triangle, andwe therefore call these closure phases or bispectral phases inter-changeably. Roddier (1986) recognized bispectral analysis as be-ing equivalent to the existing triple-correlation method of specklemasking (Labeyrie 1970; Weigelt 1977), and it provides a more ro-bust observable than raw phases even on partially redundant pupils(Haniff & Buscher 1992).For a redundant pupil, there are a combinatorically large num-ber of baselines, and for reasons of hardware memory we are notable to use a pupil model as dense as in the above kernel phaseanalysis. We instead use a pupil model with the same dimensionsbut only 508 pupil samples. Using this coarser model we conducta kernel phase fit, finding good agreement with the denser model,with a separation of ± . mas, a position angle of . ± . degrees, and a contrast ratio of . ± . . We see that this is inreasonable agreement with the denser model and aperture maskingobservations.We then find all possible combinations of triangles and testfor closure, finding 378662 closing triangles in our redundant pupil c (cid:13) , 000–000 alomar Kernel Phase Experiment model. As there are only 1456 independent u, v baselines, the infor-mation in the raw bispectrum is extremely redundant, and unless wemodel our data in a reduced-dimensionality representation, we willboth encounter unnecessary computational cost, and underestimateour uncertainties. We therefore first construct the × matrix containing the full set of closure relations, and find a rank-reduced operator with the same range using a sparse SVD. Usingthis, we find that the space of closure phases is spanned by theexpected N baseline − orthonormal vectors, and usethese as our bispectral observables. These are therefore linearly-independent closure phases (Sallum et al. 2015), but we do not havea sufficiently large number of observations to re-diagonalize theseas statistically-independent closure phases as in Kraus & Ireland(2012); Ireland (2013a). It is important to note that in the generalnon-redundant cases these orthonormal closure phases do not spanthe same space as kernel phases, and that only in the case of a non-redundant pupil are these two spaces of observables expected tobe the same. Ideally, we would for each triangle average the com-plex bispectrum across all frames, extract the phase of the resultingmean complex bispectrum, and then project these onto the minimalspanning set of orthonormal closure phases. Due to the combinator-ically large number of triangles this is not possible, and we insteadaverage the orthonormal closure phases themselves.Data are processed as for the kernel phases in Section 2.1, ex-cept using this orthonormalized matrix of closure phase relationsinstead of the kernel phase matrix. We recover the binary at a sep-aration of 140.9 ± ± ± ∼ masin separation, and the best-fitting contrast is much lower, at 15.4 ± ∼ times lower than the best kernel phase or maskingestimate and similar to that from PSF fitting. This is consistent withthe effects of a speckle introduced by a phase aberration perturbingall the full-frame images, while kernel phase and aperture maskingare by construction resilient against this form of aberration.While the difference in sampling density means that we donot compare the kernel phase and bispectral methods on a levelplaying field, we note that finding the full set of triangles generatedby the denser model was not possible due to memory constraints,and therefore the bispectral method is inherently more limited thankernel phase in its applicability to very dense, redundant pupils. Weare therefore restricting our comparison to methods of equivalentcomputational resource usage. In this situation we lack the diversity of calibration images touse the most advanced PSF analysis techniques, such as LOCI(Lafrenière et al. 2007) or KLIP (Soummer et al. 2012), but the 100frames of our single calibrator are sufficiently diverse to permit PSFfitting. We adopt a maximum-likelihood approach, finding the best-fitting incoherent sum of two shifted calibrator PSFs that minimizea χ objective function for each individual frame of α Oph. Theoffset and scaling between these two calibrator PSFs is then takento be the model parameters of the binary system. In Table 1 we re-port the mean and standard error of the mean of all such fits with a χ within a factor of 2 of the median χ across all frames, to allowfor the possibility of failing to retrieve the binary in some frames.We note the low contrast found in the results of PSF fitting, notfound in the kernel phase analysis. We suggest that a quasi-static Figure 4.
Correlation diagram for bispectral phases for α Ophiuchi in K s band. We plot model phases on the x -axis and the observed signal on the y -axis, such that for a good fit we expect the data to lie on a straight line ofgradient unity. speckle very close to the position of the binary companion, or avariable incoherent second ghost at the same location, could causethis effect, which is corrected in the kernel phase approach. Thisgood kernel phase correction is consistent with this being a phase-aberration induced speckle; in the case of an amplitude-inducedspeckle, it would be desirable to find an equivalent ‘kernel ampli-tude’ which would be self-calibrating with respect to such an aber-ration, but such a quantity is not presently known.We note that this PSF fitting approach is enabled by the diver-sity over both many calibrators and many frames on each calibrator,whereas in principle any single image can be analysed with kernelphase given appropriate single images of calibrators. This is an ad-vantage for kernel phase in campaigns where many short exposuresare not feasible, or in space telescope snapshot campaigns, wherePSF diversity may otherwise be lacking. For our benchmark α Oph dataset, the kernel phase analysisstrongly favours a binary model over a point source, and success-fully obtains the same system configuration as found with aperturemasking. The results are displayed in Table 1. The spatial astro-metric components agree remarkably well, to much better than σ ,and in contrast the kernel phase is slightly higher, at 34.2 vs 28.7.As the uncertainties on each are ∼ . and ∼ . respectively, wesee these estimates do not entirely agree, though the higher con-trast value derived from kernel phase may represent a systematicerror which may require further improvements to the method. Asnoted below, the uncertainties on the kernel phase-derived contrastare probably underestimated here, though with present software im-plementations and the limited number of frames available in theseobservations, this cannot be resolved at present. Nevertheless, it isclear that kernel phase presents a realistic alternative to aperturemasking for telescopes with extreme AO, with the potential for sig-nificant advantages in throughput and Fourier coverage.The science target was observed in the middle of the detector,near the corner of the four CMOS chips which tile the focal plane.This leads to lines of noise running through the sides of the PSF.On the other hand, the calibrators were observed in the middle of c (cid:13) , 000–000 B. J. S. Pope et al. each chip (‘dithered’), as is standard practice. The subtraction ofsystematic errors from the science data was therefore imperfect,and in future it will be important to dither the science source andcalibrators identically.The binary astrometry from kernel phase is reasonably close( ∼ . − σ masking ) to that obtained by aperture masking; this isremarkably good agreement given that the observations were per-formed under far from ideal conditions: as noted in Section 1.1, onthe night these observations were made, the seeing was . − arcseconds, which is substantially worse than median for Palomar.This translated to the relatively low Strehl of 0.5 in K s , somewhatlow for extreme-AO and toward the lower end of the kernel phaseregime. We therefore note that we may expect improved perfor-mance under better AO conditions in future.The retrieved binary parameters are remarkably insensitive tothe size of the super-Gaussian window used in preprocessing. Bythe convolution theorem, windowing an image is equivalent to con-volving its Fourier transform with a kernel whose dimension isinversely proportional to that of the window. A narrower windowtherefore has a wider convolution kernel, which blurs Fourier phaseinformation. This results in both a blurring in the Fourier plane,which is especially significant at high spatial frequencies wherethis mixes real signals from inside the support of the modulationtransfer function with noise from outside its support.In addition to the above instrumental errors, we noted in Sec-tion 2.4 that systematic optical aberrations are likely to remain evenafter calibration. By using an ensemble of calibrators as describedin Kraus & Ireland (2012), it is possible to substantially improvethe correction of systematic errors. Residual uncalibrated system-atics enter at first order in phase, and third order in kernel phase,and can introduce statistical correlations between kernel phase re-lations which are algebraically orthogonal. It is possible to diago-nalize kernel phases and closure phases in a Karhunen-Loève ba-sis which properly takes into account their statistical covariance(Ireland 2013b); such a calibration is improved significantly withgreater calibrator diversity than is available in this work. In future,it will be valuable to include more calibrator sources, and observethese and the science target with a range of pupil orientations tomaximize the calibration diversity. We expect that this may reducesystematic effects, and also increase the parameter uncertaintiessomewhat, given that presently we assume data to be independentwhich are in fact correlated.The issue of pupil modelling remains an outstanding problemfor kernel phase, in that at present it is not feasible to marginalizeover uncertainties in pupil scale, or to model the pupil as denselyas may be preferred for high-performance applications. There aretherefore systematic errors associated with any mismatch betweenthe discrete pupil model and the real effective model of the tele-scope, as well as any amplitude aberrations or spatial variations intransmission. Resolving this difficulty is beyond the scope of thispaper, but will be important for future work. The recovery of the α Oph binary system illustrates both the po-tential of kernel phase in conjunction with extreme AO, and thepotential for improvements in future observations. It is clear thatkernel phase recovers the binary parameters with remarkable pre-cision, and it will be a valuable tool in probing systems than cannotbe observed with aperture masking. In this test case, under idealconditions for aperture masking and more challenging conditions for kernel phase, nevertheless kernel phase delivers comparableresults. The performance of kernel phase in this low to moderateStrehl, single calibrator regime is not expected to be representativeof higher Strehls and multiple calibrators - an analysis of whichwould be important future work.We have also demonstrated the benefits of using kernel phasesover more standard PSF fitting and bispectrum (closure phase) ap-proaches in parameter estimation. Deconvolving structure from anAO-corrected PSF is significantly enhanced by the use of kernelphase, and we expect this will enable new science to be done at andnear the diffraction limit.As noted in Pope et al. (2013), wavelength diversity acrossseveral filters can help alleviate the degeneracy between separationand contrast, by jointly fitting to kernel phases extracted in severalbands and enforcing the condition that the position of a companionmust be fixed, while its flux can vary. This is a promising optionfor future kernel phase work, as the extreme AO systems SPHEREand GPI, as well as the P1640 instrument on Palomar, are equippedwith integral field spectrographs which can obtain images in manywavelength channels simultaneously.Given these encouraging results, we see that the best currentadaptive optics systems are already able to make use of kernelphase for high contrast imaging. In particular, we have shown thatin the extreme-AO regime, kernel phase obtains comparable resultsto those using non-redundant masking. Where for hardware reasonsor due to throughput considerations it is not possible to use a mask,or where very dense Fourier coverage is desired for imaging, thekernel phase approach may be much more effective than standardalternatives, opening up new parameter space for high-resolutionimaging of faint companions and circumstellar environments. Wehave also discussed observing strategies, and in particular the im-portance of calibrator sources and wavelength diversity, which willbe of use in planning future kernel phase work from the ground.
ACKNOWLEDGEMENTS
This research has made use of NASA’s Astrophysics DataSystem. This research has also made use of Astropy, acommunity-developed core P
YTHON package for Astronomy(Astropy Collaboration et al. 2013), the IP
YTHON package(Pérez & Granger 2007), and
MATPLOTLIB , a P
YTHON library forpublication quality graphics (Hunter 2007).BP thanks Balliol College and the Clarendon Fund for theirfinancial support for this work.This work was performed in part under contract with the Cali-fornia Institute of Technology, funded by NASA through the SaganFellowship Program.We are grateful to Anthony Cheetham, Rupert Allison,Michael Bottom, Matthew Kenworthy, Daniel Foreman-Mackey,Patrick Roche and Suzanne Aigrain for their helpful comments. Wewould especially like to thank our referee, Chris Haniff, for his ad-vice in improving this paper.
REFERENCES
Allison, R. & Dunkley, J. 2014, MNRAS, 437, 3918Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013,A&A, 558, A33Beuzit, J.-L., Feldt, M., Mouillet, D., et al. 2010, in In the Spiritof Lyot 2010 c (cid:13) , 000–000 alomar Kernel Phase Experiment Bouchez, A. H., Dekany, R. G., Angione, J. R., et al. 2008, in So-ciety of Photo-Optical Instrumentation Engineers (SPIE) Con-ference Series, Vol. 7015, Society of Photo-Optical Instrumenta-tion Engineers (SPIE) Conference SeriesFeroz, F., Hobson, M. P., & Bridges, M. 2009, MNRAS, 398, 1601Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J.2013, PASP, 125, 306Foreman-Mackey, D., Price-Whelan, A., Ryan, G., et al. 2014Gardner, J. P., Mather, J. C., Clampin, M., et al. 2006, Space Sci.Rev., 123, 485Haniff, C. A. & Buscher, D. F. 1992, Journal of the Optical Soci-ety of America A, 9, 203Hayward, T. L., Brandl, B., Pirger, B., et al. 2001, PASP, 113, 105Hinkley, S., Monnier, J. D., Oppenheimer, B. R., et al. 2011, ApJ,726, 104Hinkley, S., Pope, B., Martinache, F., et al. 2015, in American As-tronomical Society Meeting Abstracts, Vol. 225, American As-tronomical Society Meeting Abstracts, 313.02Hunter, J. D. 2007, Computing In Science & Engineering, 9, 90Ireland, M. J. 2013a, MNRAS, 433, 1718Ireland, M. J. 2013b, ArXiv e-printsIreland, M. J. & Kraus, A. L. 2014, in IAU Symposium, Vol. 299,IAU Symposium, ed. M. Booth, B. C. Matthews, & J. R. Gra-ham, 199–203Jennison, R. C. 1958, MNRAS, 118, 276Kraus, A. L. & Ireland, M. J. 2012, ApJ, 745, 5Labeyrie, A. 1970, A&A, 6, 85Lafrenière, D., Marois, C., Doyon, R., Nadeau, D., & Artigau, É.2007, ApJ, 660, 770Macintosh, B., Graham, J. R., Ingraham, P., et al. 2014, Proceed-ings of the National Academy of Science, 111, 12661Macintosh, B. A., Graham, J. R., Palmer, D. W., et al. 2008, inSociety of Photo-Optical Instrumentation Engineers (SPIE) Con-ference Series, Vol. 7015, Society of Photo-Optical Instrumenta-tion Engineers (SPIE) Conference SeriesMartinache, F. 2010, ApJ, 724, 464Martinache, F. 2011, in Society of Photo-Optical Instrumenta-tion Engineers (SPIE) Conference Series, Vol. 8151, Society ofPhoto-Optical Instrumentation Engineers (SPIE) Conference Se-riesMartinache, F. 2013, in Proceedings of the Third AO4ELT Con-ference, ed. S. Esposito & L. Fini, 6Martinache, F., Guyon, O., Lozi, J., et al. 2009, ArXiv e-printsMonnier, J. D., Townsend, R. H. D., Che, X., et al. 2010, ApJ,725, 1192Pérez, F. & Granger, B. E. 2007, Computing in Science and Engi-neering, 9, 21Pope, B., Martinache, F., & Tuthill, P. 2013, ApJ, 767, 110Reid, I. N., Cruz, K. L., Burgasser, A. J., & Liu, M. C. 2008, AJ,135, 580Reid, I. N., Lewitus, E., Allen, P. R., Cruz, K. L., & Burgasser,A. J. 2006, AJ, 132, 891Roddier, F. 1986, Optics Communications, 60, 145Sallum, S., Eisner, J. A., Close, L. M., et al. 2015, ApJ, 801, 85Skilling, J. & Bryan, R. K. 1984, MNRAS, 211, 111Soummer, R., Pueyo, L., & Larkin, J. 2012, ApJ, 755, L28Weigelt, G. P. 1977, Optics Communications, 21, 55Zhao, M., Monnier, J. D., Pedretti, E., et al. 2009, ApJ, 701, 209 c (cid:13)000