Howdowedetectcompanions? PSFreconstruction

High-contrast Imaging:Below the Diffraction Limit with Kernel Phase

Jens Kammerer, Mike Ireland & Frantz Martinache

RESEARCH SCHOOL OFASTRONOMY & ASTROPHYSICS

We extract kernel phases, which are interferometric measureables independent of pupil plane phase noise to second order, from archivalNACO mid-infrared cube mode data. Since high Strehl is essential for this technique we perform lucky imaging first and calibrate ourmeasureables (which are often dominated by systematic errors) against those of well-known point sources. Finally, we fit analyticalmodels directly to the kernel phases. We demonstrate the direct detection of a low-contrast companion below the classical diffraction-limit and the capabilities towards smaller separations and contrasts.

Whatisthekernelphasetechnique?

Howdoweextractthekernelphases?

Sparseaperturemasking:• LookatFouriertransformofdetectorimage• Fourierplanephasecanbesplitinto

eigenphase andkernelphase(~closurephase)• Kernelphaseisindependentofpupilplane

phasenoise(whichisdominatedbyphasepistonsoneachhole,seeeigenphase)tosecondorder[1]

Fullpupilkernelphaseimaging:• Assumethatfullpupilishighlyredundantarray

ofindividualsub-apertures(leftpanel)• NotransmissionlossesandNyquistorbetter

Fourierplanesampling(rightpanel)• Inhigh-Strehl regime,thereisalinear

relationshipΦ = 𝑅$%𝐴𝜑 +Φ)*+,-. betweenFourierplanephaseΦ andpupilplanephase𝜑,where𝑅$%𝐴 isdeterminedbypupilmodel[2]

VLT pupil model (left panel) and its Fourier planecoverage (right panel). The axes represent size inmeters.

Fullpupilkernelphaseimaging:• Multiplicationwithleftkernel𝐾 of𝑅$%𝐴 yields

𝐾Φ = 𝐾𝑅$%𝐴𝜑 + 𝐾Φ)*+,-. = 𝐾Φ)*+,-.• Hence,inhigh-Strehl regime,kernelphase𝐾Φ

isindependentofnoiseinpupilplanephase𝜑• Suchnoisecanoriginatefromatmospheric

seeingoropticalaberrationsininstrumentitself

Keck JHKs-band multicolorimage of HR 8799 after ADIpost-processing revealingthe sub-stellar companionsHR 8799 b/c/d. Theinnermost 10-15 𝜆/𝐷 aredominated by speckleswhich are caused by pupilplane phase noise and formthe limit for classical high-contrast imaging (fromMarois et al. 2008 [3]).

Fullyautomaticdatacleaningpipeline:• Detectorlinearization,bias/flat/background/jittersubtraction• Read-noiseestimationfrombiasframes• Badpixelcorrection/cosmicrayrejectionusingFouriertechniques• Luckyimagingtopickhigh-Strehl frames• Finalcontrastofindividualframes~500:1andfinalcontrastofmedian

frameoverfulldatacubes>1000:1

Cross-section of the PSF of HIP 11484 before (upper left panel) andafter application of our reconstruction algorithm (upper rightpanel). The x-axes represent pixels and the y-axes representdetector counts. The PSF core is saturated and therefore marked asbad pixels (lower left panel) and our iterative reconstructionalgorithm also identifies additional bad pixels based on their Fourierpower (lower right panel).

• Reconstructionofsaturatedpointspreadfunctions(PSFs)fromthePSFwingsbyminimizingtheFourierpoweroutsidetheregionofsupportpermittedbythepupilgeometry

• ThisisimportantforFourierplaneimagingtechniquesbecausesaturatedpixelscauseFourierplanephasenoise

Fourierplanephaseandkernelphaseextraction:• FourierplanephaseΦ isobtainedbyanalyticalFourier

transformofeachindividualcubemodeframe• KernelphaseKΦ isobtainedbymatrixmultiplication

foreachindividualcubemodeframe,thenacovarianceweightedmeankernelphaseiscomputedforeachdatacube(seefiguretotheleft)

Median of raw data cube (left panel) of HIP 50156, median ofbias/flat/background subtracted data cube (middle panel) and median ofdata cube after subtracting the median cleaned data cube of a jitteredimage (right panel).

Fourier plane phase (upper rightpanel) and covariance betweenthe 192 kernel phases extractedfrom the data of HIP 50156 (lowerleft panel). The median Fourierplane phases (over all frames of adata cube) at the Fourier planepositions of our pupil model areshown in the lower right panel(blue curve), but the maximum(green curve) and minimum(orange curve) reveal spikes whichramp up to pi. This must beprevented by improving the pupilmodel and is work in progress!

Modelfitting:• Fitanalyticalbinarymodel𝑙% + 𝑙5𝑒$578(:;<=>:?@AB) directly

tokernelphasesbymultiplyingmodelphaseswithleftkernel𝐾 of𝑅$%𝐴,where𝑈 and𝑉 representFourierplanecoordinatesnormalizedbyobservingwavelength

• Firstperformgridsearchforfixedsmallcontrast(e.g.𝑙5/𝑙% =0.001),thenoptimizecontrastforbestfitgridpositioninordertofindpriorforleastsquaresfit

• Leastsquaresfitoptimizesseparation,positionangleandcontrastsimultaneously

Measured kernel phases of HIP 50156 and model kernel phasesfor a grid search fit (upper left panel) and a least squares fit (upperright panel). The lower right panel gives the chi2 if varyingseparation, position angle or contrast around the best fit from theleast squares routine and illustrates that it found the minimum.

Datacalibration:• Kernelphasesareindependentofpupilplanphasenoiseto

secondorder,butarestillaffectedbysystematicsatthirdorderinphase[4]

• Subtractingoffkernelphasesofwell-knownpointsourcesisessentialinhigh-contrastregime

• WeuseKarhunen-Loeve projection𝑇 → 𝑇 −∑ < 𝑇|𝑍 > 𝑍�RSTUV ,where𝑍 followsfrom

eigendecomposition ofcovariancematrixofcalibratorkernelphases

• Thisallowsustogetridofstatisticallymostsignificantcalibratorkernelphases

Measured kernel phases of HIP 50156 (src), measured meankernel phases of three different calibrator stars (cal) andKarhunen-Loeve projected (i.e. calibrated) kernel phases of HIP50156 (src_poise). This will only become relevant in the high-contrast regime.

Companionwithseparation=74mas(~0.75𝜆/𝐷),

positionangle=63deg andrawcontrast=0.22

sep =74maspa=63degcon=0.22

74-20mas63-20deg0.22-0.1

74+20mas63+20deg0.22+0.1

Bowleretal.2015reportcompanionat~90mas,1.8magKs-bandcontrast,substantialorbitalmotion[5]

References: [1] Ireland 2016, ASSL, 439, 43I; [2] Martinache 2010,ApJ, 724, 464M; [3] Marois et al. 2008, Sci, 322, 1348M [4] Ireland2013, MNRAS, 433, 1718I; [5] Bowler et al. 2015, ApJS, 216, 7B