A Demonstration of Wavefront Sensing and Mirror Phasing from the Image Domain

1.2 MB·

by  Benjamin Pope

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed30January2014 (MNLATEXstylefilev2.2)
A Demonstration of Wavefront Sensing and Mirror Phasing from
the Image Domain
Benjamin Pope1,2(cid:63), Nick Cvetojevic1,3,4, Anthony Cheetham1, Frantz Martinache5,
Barnaby Norris1, Peter Tuthill1
4 1SydneyInstituteforAstronomy,SchoolofPhysics,UniversityofSydney,NSW2006,Australia
1 2Astrophysics,UniversityofOxford,DenysWilkinsonBuilding,KebleRd,OxfordOX13RH,UK.
0 3CentreforUltrahighbandwidthDevicesforOpticalSystems(CUDOS),InstituteofPhotonicsandOpticalScience(IPOS),
2 SchoolofPhysics,UniversityofSydney,Sydney,NSW2006,Australia
4AustralianAstronomicalObservatory,NSW2121,Australia
n 5LaboratoireLagrange,CNRSUMR7293,ObservatoiredelaCôted’Azur,Bddel’Observatoire,06304Nice,France
a
J

9
30January2014
2

]
M ABSTRACT
Inastronomyandmicroscopy,distortionsinthewavefrontaffectthedynamicrangeofahigh
I
. contrastimagingsystem.Theseaberrationsareeitherimposedbyaturbulentmediumsuchas
h
theatmosphere,bystaticorthermalaberrationsintheopticalpath,orbyimperfectlyphased
p
subapertures in a segmented mirror. Active and adaptive optics (AO), consisting of a wave-
-
o front sensor and a deformable mirror, are employed to address this problem. Nevertheless,
r thenon-common-pathbetweenthewavefrontsensorandthesciencecameraleadstopersis-
t
s tentquasi-staticspecklesthataredifficulttocalibrateandwhichimposeafloorontheimage
a contrast. In this paper we present the first experimental demonstration of a novel wavefront
[
sensorrequiringonlyaminorasymmetricobscurationofthepupil,usingthesciencecamera

1 itselftodetecthighorderwavefronterrorsfromthespecklepatternproduced.Weapplythisto
v correcterrorsimposedonadeformablemicroelectromechanical(MEMS)segmentedmirror
6 inaclosedloop,restoringahighqualitypointspreadfunction(PSF)andresidualwavefront
6 errorsoforder∼10nmusing1600nmlight,fromastartingpointof∼300nminpistonand
5 ∼ 0.3mradintip-tilt.Werecommendthisasamethodformeasuringthenon-common-path
7 error in AO-equipped ground based telescopes, as well as as an approach to phasing diffi-
.
1 cultsegmentedmirrorssuchasontheJamesWebbSpaceTelescopeprimaryandasafuture
0 directionforextremeadaptiveoptics.
4
1 Key words: techniques: interferometric — techniques: image processing — techniques:
: adaptiveoptics
v
i
X
r
a
1 INTRODUCTION withadeformablemirror(DM)inafeedbacklooptomeasureand
correctfordistortionsinthephaseoftheincominglight(Davies&
A major goal in present-day astronomy is the direct detection of
Kasper2012).
planets and other faint companions to stars, a task which simul-
The application of AO is not strictly limited to astronomy,
taneously requires high contrast and high resolution. This task is
however. In optical microscopy it is also the case that the speci-
principallylimitedbythediffractionoflightfromtheparentstar,
men under study introduces aberrations into the optical path that
duetostaticaberrationsinthetelescopeoptics,quasi-staticerrors
limit the ability of a microscope to resolve detail deep below the
thatvarywithtelescopepointingandenvironmentalconditions,and
surfaceofanotherwisetransparentspecimen(Booth2007).
thedynamiceffectsofatmosphericturbulence.
Thereisathird,related,probleminopticsforwhichawave-
Ofthese,byfarthemostseveredifficultyiswithatmospheric
front sensor is necessary. The primary mirrors of large telescope
turbulence,or‘seeing’.IthasbeenapparentatleastsinceNewton’s
areconstructedoutofsegments.Currentexamplesincludethetwo
timesthatatmosphericturbulencelimitstheresolutionandsensi-
10-meterW.M.KeckTelescopesandtheJamesWebbSpaceTele-
tivityofastronomicalobservations.Thisproblemcanbesubstan-
scope.FutureExtremelyLargeTelescopes(ELTS)willfollowthis
tiallyovercomebytheuseofadaptiveoptics(AO),firstproposed
design:theprimarymirrorsoftheEuropeanELT,theThirtyMeter
byBabcock(1953).Inthisapproach,awavefrontsensorispaired
Telescope(TMT)andtheGiantMagellanTelescope(TMT)areall
tobemadeofmultipleelements.Discontinuitiesinthepupiland
(cid:63) E-mail:[email protected] sharp segment edges however come with difficulties that are not
(cid:13)c 0000RAS
2 B.J.S.Popeetal.
welladdressedbytraditionalwavefrontsensorswhosereconstruc- achievetherequireddiversity,whichisnecessarilytime-consuming
torsassumecontinuityofthewavefront.Itisthereforeofprimeim- andthereforeachallengeforbusyobservingschedules.
portancetoensurethatallsegmentsarepositionedcorrectlyrelative Thealternativetothisisinterferometriccalibration,asisal-
tooneanotherwithoutintroducingsignificantwavefronterrors. ready done with sparse aperture masking with AO (Tuthill et al.
Hithertothemostcommontechnologyforwavefrontsensing 2006).AtthecostofFouriercoverageandthroughput,itispossi-
has been the Shack-Hartmann wavefront sensor. It uses an array bletoextractclosurephaseswhichareself-calibratingwithrespect
oflensletsplacedinare-imagedpupil,suchthatthepositionoff- to phase errors in the pupil. Because they are measured from the
setofthespotproducedbyeachlensletencodesinformationabout science camera image, closure-phase are robust against residual
thelocalslopeofthewavefront.Thishasbeenregularlyappliedto aberrations,NCP-errorotherwise.Theideaofclosurephasescan
generalwavefrontsensingapplicationssinceLane&Tallon(1992) begeneralizedto‘kernelphases’forarbitrarypupilsinthelimitof
andisnowubiquitousinastronomicalAO(Davies&Kasper2012). awell-correctedwavefront(Martinache2010).Thisapproachhas
HoweverasnotedbyGuyon(2005),theShack-Hartmanntechnol- provenfruitfulindetectingfaintcompanionstobrowndwarfsinthe
ogyisintrinsicallyvulnerabletophotonnoiseandrequiresalarge ‘superresolution’regime,beyondtheformaldiffractionlimit,and
fraction of the incoming light be diverted for the sole purpose of withthecalibrationschemesproposedbyIreland(2013)cancom-
wavefrontsensing.Thisisasignificantdrawbackintheregimeof plementADIandsimilartechniquesatsmallangularseparations.
fainttargetsorhigh-contraststudies. Kernel-phaseusesamatrixpseudoinverseapproachandtherefore
Other technologies such as the pyramid wavefront sensor hasformalfeaturesincommonwithelectricfieldconjugation.The
(Ragazzoni1996),curvaturewavefrontsensor(Roddier1988)and responsematrixishowevernotempiricallydetermined,andrelies
phasecontrastmethod(Zernike1934)arepossible.Theyhowever insteadonasimplemodelofthepupilgeometry,whileassuming
alsosufferfromsimilardrawbacks. thatamplitudeerrorsarenegligible.Thishastheadvantageofnot
Allaforementionedapproachessufferfromthenon-common- requiringcalibrationofthesameextentorduration.
path (NCP) error problem, as aberrations occurring downstream Inthispaperwepresentthefirstlaboratorydemonstrationof
fromthewavefrontsensorarenotdirectlymeasured.Theseresid- theasymmetricpupilFourierwavefrontsensorproposedbyMarti-
ualerrorsgiverisetoquasi-staticspecklesthatimposeadetection nache(2013).Thisemergesnaturallyasadualtothekernelphase
floorinhigh-contrastimaging,suchaswiththeP1640integralfield imageanalysismethodputforwardbyMartinache(2010).Bycon-
spectrograph(Hinkleyetal.2008;Creppetal.2011;Hinkleyetal. sideringtheproblemofPSFformationfromaninterferometricper-
2011). spective,itispossibletorecoverwithpost-processingamapofthe
Solutionstothisproblemrequiresomeformofphaseretrieval wavefrontgivingrisetothePSFofanyinversion-asymmetricpupil
usingthesciencecameraitself.Possibleimplementationsfallinto subjecttoweakaberrations.Thisincursamodesthardwarecostof
twocategories:thefirstreliesonanactivedifferentialprocess.Di- aminorasymmetricobscurationofthepupil.Forsegmentedmir-
versityinthepointspreadfunction(PSF)isintroducedbyanactive rors,thismaybeachievedbysegmenttiltingorbyajudiciousar-
elementtodistinguishcoherent,variablediffractionfeaturesfrom rangementofamaskorspiders.Theadvantagehereisthatthisisin
incoherent,fixedcelestialobjects.Theotherisapassivedifferential principleasingle-step,requiringnophasediversityormodulation.
process. One standard approach representative of this category is Usingsimulations,Martinache(2013)showsthatthemethod
angulardifferentialimaging(ADI)(Maroisetal.2006;Lafrenière isparticularlyadaptedtothedetermination(andthereforecorrec-
etal.2007),wherethepupilisallowedtorotatewithrespecttothe tion)oftheNCP-errorinanAOsystem.Here,weprovideexperi-
sky(asin,forinstance,analt-azimuthmountedtelescope).Ifthis mentalresultsshowingthatitisequallysuitedtothephasingofa
rotationoccursonatimescaleshorterthanthatofthevariationof segmentedmirror.
thequasi-staticspeckles,itispossibletodistinguishfaintcompan-
ionsfromspecklesinthePSF,inpost-processing.
TomeasuretheNCP-errorfromthesciencedetector,thestan- 2 PHASETRANSFERALGORITHM
dardoptionisthephasediversitytechnique.Multipleimagestaken
2.1 Theory
in and out of focus, make it possible to determine uniquely the
staticaberrationsacrosstheentirewavefront(Kendricketal.1994; Here we will summarize and explain the theory of the wavefront
Campbell et al. 2004; Sauvage et al. 2012). This NCP-error esti- sensingapproachproposedbyMartinache(2013).Inthefollowing
matecanbeusedtooffsetthewavefrontsensorzero-positionfor discussionitwillbeimportanttodistinguishbetweenthreeplanes
close-loopoperation. intheopticalpath:thepupilplane,definedatthetelescopeaper-
TheDMitselfcanbeusedtointroducetheknownphaseper- ture;theimageplane,asrecorded(forinstance)byacamera;and
turbationinordertocalibrateorsuppressspeckles.Thelargenum- theFourierplane,theFouriertransformoftheimage.Theimage
ber of degrees of freedom offered by a DM leads to variety of ofapointsource(orequivalentlythepointspreadfunctionofthe
control algorithms: from random perturbations of the DM while imagingsystem)isgivenbythesquaredFouriertransformofthe
monitoring a metric function of the PSF (Ren et al. 2012); iter- electricfieldinthepupil,andthereforethecomplexvisibilitiesin
ative speckle nulling loop compatible with coronagraphic mode theFourierplanearegivenbythefield’sautocorrelation.Forarbi-
(Martinacheetal.2012);toamorecompletedeterminationofthe traryaberrations,itisnotingeneralpossibletouniquelymapfrom
wavefront with a finite number of DM modulations (Keller et al. informationintheimageplaneonlytothewavefrontinthepupil
2012). A comparable method compatible with a coronagraphic plane, because the autocorrelation of an arbitrary function is not
imagingmodeshouldrelyontheelectricfieldconjugationframe- uniquelyinvertible.
work (Give’on et al. 2007; Thomas et al. 2010; Give’On et al. If the aberrations are small (ϕ (cid:46) 1 rad), it is however pos-
2011),toconstructacomplextransfermatrixfortherealandimag- sible to treat this mapping as a linear operation. Using a discrete
inarypartsoftheelectricfieldbetweenpupilandfocalplanes,for modelofthepupilitisthenstraightforwardtocomputethephase
direct speckle nulling within a finite region of the field of view. transfermatrixassociatedwiththisoperator.Inthispaper,willcon-
These approaches all rely on sequential motions of the DM to sidermodelsofthehexagonalsegmentedMEMSmirrorusedfor
(cid:13)c 0000RAS,MNRAS000,000–000
WavefrontSensingfromtheImageDomain 3
[h!] structedfromtheFourierplane(Martinache2013).Inthefollow-
ing,wewillbrieflyoutlinetheinversionmethod.
Itiscrucialinthiscasethatthepupilitselfnotbecentrosym-
metric:thatis,thatunderthetransformation(x,y) → (−x,−y)
itdoesnotmapontoitself.Thisisbecauseotherwiseitisnotpos-
sible to distinguish between pupil modes of odd and even parity
underinversion.
Withasymmetricpupil,onlyoddmodesofthepupilappear
in the row space and can be sensed when constructing the pseu-
doinverse. Introducing a pupil plane asymmetry, even of a very
small character, breaks this degeneracy, and the pseudoinverse of
thistransfermatrixwillaccordinglymaptothefullspaceofpupil
modes.
ThecolumnvectorsinUprovideasetoforthonormalmodes
for describing phases in the pupil. The transfer matrix A maps
these one-to-one onto a corresponding orthonormal basis of row
vectors in V, normal modes for the Fourier plane. These can be
Figure1.AmodeloftheDragonflysegmentedmirror.Bluepointsrepresent
thoughtofasbeingsimilartoasymmetry-adaptedsetofZernike-
pointstiltedon-axis;redpoints,anequivalentsamplingtoillustratemirrors
like modes for an arbitrary pupil. The Moore-Penrose pseudoin-
tiltedawayfromtheopticalaxis.
verseA+mapsthevectorsbackintheoppositedirection:
theDragonflypupilremappinginterferometer(Kotanietal.2009),
A+ =V·Σ+·UT (4)
whichisascalemodeloftheJWST pupil.Afilledpupilisshown k
inFigure1. whereΣ+ denotesthediagonalmatrixwhoseentriesarethe
k
Martinache(2010)showedthatthesingularvaluedecompo- reciprocalsofthefirstkdiagonalentriesofΣ.Examplesofthese
sition(SVD)ofthetransfermatrixisseparableintomappingsbe- modesaredisplayedinFigure2.
tweenpairsoforthonormalphasevectorsinthepupilandFourier
planes,andakernelspaceofvectorsintheFourierplanetowhich
nosmallperturbationsinthepupilplanecanbemapped.
2.2 SoftwareImplementation
Inparticular,wecanwritethisexpressionas.
In this experiment we have used a Python implementation of the
asymmetric pupil Fourier wavefront sensing algorithm, sharing
Φ=A·ϕ+Φ (1)
0 severalcommonfeatureswiththepysco‘PYthonSelf-Calibrating
for a transfer matrix A, pupil phases ϕ, observed Fourier Observables’ kernel-phase analysis code. We calculate the pseu-
phasesΦand‘true’sourceFourierphasesΦ .Asimplemethod doinversefromtheSVDofthetransfermatrixA,includingonly
0
forobtainingthematrixAisdescribedinMartinache(2010).We thefirst200singularvaluesandcorrespondingmodesoutofatotal
thenobtaintheSVDofA, numberof∼1000.Thiscutoffat200modeswaschoseninorderto
helpsmooththeresultantwavefrontestimate,asthemirrorphasing
problemrequirespredominantlylow-orderinformation
A=U·Σ·VT (2) WhileAisingeneralsparse,wetakecarenottonaïvelyapply
sparsematrixalgebrapackages,andinsteadtreatthefullmatrixin
Theleft-annihilatorofA,K,isthenspannedbythecolumns
all operations. We do this in order to preserve small components
ofUcorrespondingtosingularvaluesofzero.Itisthenpossibleto
ofthenormalmodes,whichwefoundweretruncatedinthesparse
extractkernelphasessuchthat:
treatmentinsuchawayastopreventtheconvergenceofthealgo-
rithm.
K·Φ=K·A·ϕ+K·Φ Thealgorithmproceedsasfollows.Theimageisloaded,bias-
0
∴K·Φ=0+K·Φ . (3) subtracted,re-centred,theFouriertransformtaken,andsampledat
0
the baselines generated by the pupil model. We then operate on
By extracting these kernel phases K · Φ from high-Strehl theseuv phaseswiththetransfermatrixpseudoinverse,toobtain
astronomicalobservations,suchasfromspacetelescopesorwith a pupil wavefront map. Next, we fit and subtract an overall tip-
assistancefromextremeadaptiveoptics,itispossibletodramati- tiltfromthismap,toaccountforimperfectionsintherecentring.
callyenhancethesignal-to-noiseofphaseinformationcontributed Finally, we obtain the piston, tip and tilt on each segment from
by real asymmetries in an astronomical source. Even in the case the mean, x gradient and y gradient of the wavefront across the
ofanominallydiffraction-limitedspacetelescope,low-levelther- samplingpoints.
malandvibrationalmodesofthetelescopeopticsneverthelesscon- Inallpracticalcases,theseoperationsareveryfast.Forapupil
tributetothedegradationofthewavefrontqualityandthereforethe samplingasdenseasinFigure1,thistakesoforderseveralseconds
PSF.Usingkernelphaseanalysis,Popeetal.(2013)wereableto on a laptop with Intel(R) Core(TM) i7-2760QM CPU running at
obtainhighresolution,high-contrastinformationatandbeyondthe 2.40GHz. Likewise, the entire process of extracting a wavefront
formaldiffractionlimitoftheHubbleSpaceTelescope. fromafreshimageframecantakeaslittleastwosecondsforsuch
TheremainingphasevectorsintheSVDcanbeusedtocon- asampling.
structaMoore-Penrosepseudoinverseofthephasetransfermatrix, We have not attempted to optimize the computational speed
sothatsmallaberrationsinthewavefrontcanbeuniquelyrecon- of this algorithm, which is implemented in pure Python. As the
(cid:13)c 0000RAS,MNRAS000,000–000
4 B.J.S.Popeetal.
Figure2.Examplesofpupilmodepatternscalculatedforthefirstnon-redundantmaskgeometryasshowninFigure4,interpolatedontotheconvexhullof
thepupilsampleset.Colourscaleisarbitrary.
densityofapupilsamplingincreases,thenumberofuvbaselines
growsrapidly,andalthoughrequiredonlyonce,theSVDmaybe
very slow, or fail to converge. Precalculation of these matrices is
thereforeessential.
After this, processing time for a typical single frame in this
experimentissetbytheFastFourierTransform(FFT)ofa1024x
1024image,samplingatthe1872uniquebaselinepointsandmul-
tiplyingthisvectorbya1874x284matrix.Thistook∼2sonour
operatinglaptop.TheFFTsteptakes76%ofthecalculationtime
inthiscase,whichwesuggestcouldinAOapplicationsbesignif-
icantlyimprovedwithmoreefficientalgorithmsandhardware.A
high order AO system such as PALM-3000 or SCExAO on a 5-
10mtelescopeneedstooperateat∼kHzrates(Davies&Kasper
2012)anditisnotunreasonabletoexpectthistobeachievable.
Future applications should rely on a library of pupil models
pre-computed with high-performance methods. With such high-
performanceresources,thisapproachtowavefrontsensingwillbe
compatible for a very fast closed AO loop. For this experiment,
Figure3.Experimentallayoutusedfortestingthesegmentphasingalgo-
suchalinkwasmaintainedbetweencomputerscontrollingthecam-
rithm.AsinglemodelasersourcewascollimatedontotheMEMS,passing
era and running analysis software using a file-sharing client. In
throughamasktoremoveunwantedstraylightfromtheperiphery.Each
future implementations, we recommend both operations be con- segmentwascontrolledintip,tilt,andpiston.Thelightwasthenfocused
ductedonthesamecomputer. ontoanInGaAsNIRdetectorusinganachromaticdoubletlensformingthe
image.InsetimagesofMEMSfromHelmbrechtetal.(2011).
3 APPARATUS
Camera), focused using a 200 mm focal length NIR achromatic
TheexperimentalsetupusedtotestthealgorithmisshowninFig- doublet lens (Thorlabs AC254-200-C-ML). A number of silver
ure 3. A Micro-Electro-Mechanical Segmented Mirror (MEMS) coatedmirrorswithtip-tiltadjustmentwereusedintheopticalpath
served as an analogue of a typical telescope segmented primary tosteerthebeam,ensuringon-centreincidenceforallkeyoptical
mirror.TheMEMSconsistsof37hexagonal,gold-platedmirrors components.
arrangedina4-ringhexagonalclose-packedconfiguration,which As mentioned above, the MEMS acted as a scale model for
canbeadjustedelectronicallyinpiston,tip,andtilt,toaprecision theJWST pupil.However,becausetheMEMShasanadditional
ofafewnm(IrisAOPT-111DMHelmbrechtetal.(2011)).Each outer ring of segments when compared to the JWST primary, a
hexagonalsegmentis700µmcorner-to-corner(or606.2µmflat- maskwasplacedatadistanceof< 1mminfrontoftheMEMS
to-flat),withthewholeMEMSactiveareameasuring∼ 4.2mm which blocked out the unwanted segments. To further match the
in diameter. The MEMS was illuminated by a narrowband laser JWST pupil,atiltwasappliedtothecentralsegmentsuchthatit
source(TunicsTunableC-bandlaser)atawavelengthof1600nm, didnotcontributeanylighttotheimageplane,steeringthebeam
which was injected into a single-mode optical fibre (SMF-28), away from the centre of the detector. The ability to tilt away un-
and collimated using a reflective fibre collimator (a 90◦ off-axis wantedMEMSsegmentsallowedforanarbitrarylayoutforvarious
parabolic mirror). The reflected light from the MEMS formed an pupil-shapestobecreatedfortestingthealgorithm’sperformance.
image on an InGaAs detector array (Xenics Xeva 1.6-640 NIR ExamplescanbeseeninFigure4.
(cid:13)c 0000RAS,MNRAS000,000–000
WavefrontSensingfromtheImageDomain 5
Figure4.SchematicsofthevariousMEMSpupilpatternsusedtotestthealgorithm.Foralltests,theouterringofsegmentswasblockedbyapupilmask(gray
segments)anddidnotcontributelighttothefinalimage.Thesegmentswhichwereunwantedweresteeredawaybytiltingthesegmentstotheirmaximum
travelsuchthattheywerenotfocusedontotheNIRdetector(pinksegments).Theremainingsegments(blue)wereusedtoformtheimage.Thistechnique
enablesthecreationofarbitrarysegmentpatternsinthepupiltoassesstheimpactofsymmetry-breakingrequiredforconvergence,aswellasapproximating
spiderlayoutsinthepupil.
OurapparatusdiffersfromJWST’sgeometryinanotherim- metricsamplingbutafullpupil,wefailtoachieveanysignificant
portant respect: while we have tilted away the central segment restorationofthePSF.
whichwouldbeblockedbythesecondarymirroroftherealtele- Unfortunately,theNIRcamerausedintheseexperimentshad
scope,wehavenotattemptedtoincludethespiderswhichsupport afarlowerdynamicrange,andfarhighernoisefloor,thanthatof
thissecondarymirror.Oneshouldnotethatwiththreespiders,the state-of-the-artNIRdetectorsusedonmoderntelescopes.Assuch,
JWST pupil is already asymmetric, and it will be useful in fu- visualising the faint airy rings without saturating the central spot
turetoestablishwhetherthisinitselfprovidesenoughasymmetry wasnotpossible.Thisisfurthercomplicatedbyhighdetectornon-
forourwavefrontsensingscheme.Thisremainsbeyondthescope linearity at both the low, and high, ADU ranges. While we have
of the present work. This may raise the question as to whether correctedforthese,thisprocesswillhaveinducedsmallextraerrors
thepresenceofspidersmayactuallyharmtheperformanceofthis inourmeasurement.
method.Sincethespiderscontributetothepupilstructureonlyat
veryhighspatialfrequencies,wedonotexpectthemtocontribute
detrimentally to wavefront sensing at the low spatial frequencies 4.1 InitialTests
requiredinthemirrorphasingproblem.
Inordertoestablishtheeffectivenessofthewavefrontsensor,we
initially tested its accuracy using the simplest possible cases. As
mentioned in Section 2.1, the sensor only functions if the pupil
symmetryisbrokeninsomemanner.Hence,forourinitialexper-
4 RESULTS
imentsweusedtheSectorAsymmetryPattern#1(seeFig4.)by
In the following Subsections 4.2-4.4 we will discuss the results tiltingawayascalenetriangleofsegments,providingthegreatest
ofthedifferentwavefrontsensingexperimentsperformed:inSec- possiblepupilasymmetry.
tion4.1wedescribeinitialtestsestablishingtheeffectivenessofthe With the pupil pattern established, we set all remaining
technique;inSection4.2,wephaseanentiremirrortiltingawaya MEMS segments to their mechanical zero positions, and recon-
scalenearrayofthreesegmentseachtime;inSection4.3,removing structedawavefrontfromanexposurehere.Wethentookthisas
awholequadrantofthefullmirror,thenwithonlyasinglemirror ourarbitraryzerowavefrontforallsegments.Itisimportanttonote
missingattheedge;andinSection4.4,withoutalteringthemirror thatwhiletheMEMSisabletozerothepistonsofthesegmentsto
structure and simulating the asymmetry in the pupil sampling. In anaccuracyof10nm,therearefurtherwavefronterrorsandglobal
allcaseswithrealpupilasymmetrywerecoverahigh-qualityPSF tiltsinducedbydownstreamoptics,forinstancemirrors,lens,de-
fromaninitially-heavily-degradedmap;andinthecasewithasym- tector window, so that this zero-point wavefront map is not per-
(cid:13)c 0000RAS,MNRAS000,000–000
6 B.J.S.Popeetal.
tonmeasurementforthethreesegmentsthatwerepistoned.Ascan
beseeninFig.5,thereconstructedpistononthecorrectsegment
tracks the input values very closely (within the measurement er-
rorof10%)between−135nm&+300nmofpiston,outsideof
which (< −135 nm) it varies more or less wildly. This is in ac-
cordance with our expectation that for input errors larger than a
thresholdvalueof1radthereconstructionshouldfail,asthecubic
termdominatesoverthelinearterm.Wenotethatthisinpractice
occurredforrelativelysmallnegativevalues,butmaintaineditsac-
curacyforpositivevaluesforasfaraswetested;thismaybeowing
owingtophaseoffsetsfromthezero-phasereferencelevel,which
wasbiasedsuchthatthe‘errorcentre’tookapositivevalue.Also
shown in Fig. 5 is the mean residual measured pistons from the
remainingunpistonedsegments.
We note that the wavefront recorded for the other segments
variesinproportiontothepistonedsegment.Thisgivesrisetoan
RMS wavefront error across the pupil that grows monotonically
with any local error. We conjecture that the reason for this is the
choice of basis. The phase is projected onto a truncated basis of
modessupportedonalimited,discretesetofpoints.Accordinglya
largephaseerrornearananti-nodeofoneofthesemodesisliable
tobiasthereconstructionacrossthewholepupil.Becausethisbias
ismonotonicwiththeinput,whenanegativefeedbackloopisap-
plied,asinthefollowingexperiments,itwillingeneralbeironed
outastheiterationsprogressandwillnotsignificantlyaffectcon-
vergenceofthealgorithm.
4.2 Non-RedundantTripletAsymmetries
As noted in Section 2.1, the singular value decomposition con-
structstwobasissetstospanthepupilandFourierplanes.While
themethodonlyrequiresthatthepupilpossessnoinversionsym-
metry,itisapparentfromsimpleinspectionofthegeneratedpupil
modesthattheyarebyconstructionsymmetry-adaptedtoanyother
symmetries present in the pupil, most notably lines of reflection
symmetry.
Itthereforeseemedmostpromisingforaninitialtesttophase
a pupil with no symmetries at all. One choice is to remove first
onescalenetriangleofsegments,andthenbringthesebackinand
Figure5.Measuredsegmentpistonsusingthewavefrontsensorforseg- removeacomplementarytripletnon-redundantwithrespecttothe
ment #3 (top), #8 (middle), & #19 (bottom). The RMS piston values of first.Inthismanner,allFouriercomponentsandallmirrorsegments
theremainingunpistonedsegmentsarealsoshown.Thehighlightedorange aresensed,andtherearenospatialsymmetriesinthepupilsused
sectionofthegraphshowstheregionwhichisoutsidethe1radthreshold forsensing.
(dominatedbythecubictermandnolongerlinear),wherethereconstruc- Theremainingmirrorsweresettotheirfiducialflatconfigura-
tionhasfailed. tions.Then,randompistonsandtip-tiltsweredrawnfromnormal
distributionswithvariancesof300nmand0.3mradrespectively,
andappliedtotheMEMScontrols.Theseerrorswerechosensuch
fectlyflat.However,becauseintheexperimentspresentedinthis thatatthe1600nmwavelengththeRMSerrorwouldbejustbe-
subsectionwemeasurearelativepistoncomparedtoanarbitrary yondoftheexpected∼λ/2π =250nmlimitforthelinearphase
startingposition,andnotanabsolutepiston,thisdifferentialtech- regime.
niqueisadequatetodemonstratethewavefrontsensor’sapplicabil- Startingtheloopfromthispoint,weranthePSFrestoration
ity. loop first for the first non-redundant pattern shown in Figure 4,
MEMSsegments3,8,&19(seeFig.4.)wereindependently whichremovedsegments3,15,&19.AsshowninFigures6,the
pistonedfrom−300nmto+300nmin20nmincrements.Ateach PSFwasquicklyrestored.
step an image was taken of the resulting PSF and the wavefront Inordertophasetheentiremirrorincludingtheinactiveseg-
reconstructedbyouralgorithm.Afterthephase-mapwasnormal- ments,wetiltedthesebacktotheirfiducialzeropositionsandtilted
izedtothezeroreferencemap,pistonofallactivesegmentswere awaythreephasedsegmentsasshowninPattern2ofFigure4.We
extracted.Animportantsubtletyisthatthemeasuredwavefrontpis- repeatedtheprocedurehere,andsimilarlyobtainedahighquality
tonisactuallytwicetheMEMSsegmentpiston,becausethelight PSF.
reflectingoffthesegmentacquiresanopticalpathlengthdifference Bythenrestoringthesettingsofthesegmentsremovedinthe
inbothdirections. firstpattern,wewerethenabletophasetheentiremirror.Unfortu-
Figure 5 shows the accuracy of the wavefront sensor’s pis- nately,thefinalphasedmirrorconfigurationledtosaturationofthe
(cid:13)c 0000RAS,MNRAS000,000–000
WavefrontSensingfromtheImageDomain 7
centralpixels,whichwasnoticedonlyinsubsequentanalysis.Nev- 5 CONCLUSIONS
ertheless,itisvisuallyapparentfrominspectionofthePSFthatthe
This paper provides experimental evidence that the focal-plane
diffraction pattern agrees extremely well with that calculated for
wavefront sensing technique proposed by Martinache (2013) is
thediffractionlimitofaflatpupil.
sound:evenforasegmentedaperture,itispossibletodirectlysense
aberrationsfromtheanalysisofasingleaberratedPSF,ifonein-
troducessomeasymmetryinthepupil.Fromtheaboveresults,it
isclearthatlargelyindependentofthedegreeorstructureofasym-
4.3 WedgeandSingleMirrorAsymmetries metry, mirror phasing was rapid and effective. It is not possible
basedonlyontheseresultstosuggestanoptimalasymmetricmask
We repeated the previous experiment for two other asymmetries:
strategy;ontheotherhand,itisclearthatthealgorithmisveryfor-
first,withthewedgeasymmetryasshowninFigure4,andsecond,
givingwithrespecttothepupilgeometry.Itmayaccordinglybea
withthesingle-mirrorasymmetry.Inbothcasesthealgorithmcon-
fruitfuldirectionforfuturesimulationstooptimizethepupilmask
vergedtoahigh-qualityPSFaswiththenon-redundantpattern.
geometryandanyeffectthattelescopespidersmayhave.
Duringthefirstattemptwiththewedgeasymmetry,wenoted
This new wavefront sensing method is immediately applica-
thatonemirror’spistonpositionasrecordedbythecontrollerwas
bletoseveralexistingandnear-futuresystemswhereahigh-quality
approximately800nmawayfromtheothers,thatis,halfthewave-
wavefrontisdegradedbyquasi-staticwavefronterrorsarisingfrom
lengthofthe1600nmlaserusedasthelightsource.Thissuggested
smallopticalaberrations.Asthisexperimenthasshown,theprob-
that while the wavefront appeared flat under this monochromatic
lem of phasing a segmented mirror such as the upcoming James
source,thiswasbecausetheFourierwavefrontsensingapproachis
WebbSpaceTelescopeiseasilysolvedbythismethod.Whilethe
insensitivetoa2π wrappinginphase.Wethereforemanuallyad-
initialopticalpatherrorfollowingunfoldingislikelytobeofor-
justedthismirror800nmdown,neartothesettingsfortheother
der∼tensofµm,andothermethodswillberequiredtoachieve
mirrors,andfoundthatthealgorithmquicklyconvergedagaintoa
coarse phasing, our approach is suitable as a ‘tweeter’ on top of
diffraction-limitedPSF.
thisbywhichthemirrorshapecanbefine-tunedandmaintained.
TheresultsofbothexperimentsaredisplayedinFigure6.
Inthisway,itwillbepossibletohaveanequivalentof‘activeop-
Inthiscase,forbothpupilconfigurationsnosuchsaturation
tics’tomaintainauniformlyhighqualitywavefrontbyadjusting
occurred, and we were able to estimate the final Strehl ratio. In
theJWST segments,requiringonlythatitbrieflyobserveabright
ordertoavoidabiasfromresidualscatteredlight,wedidnotdi-
pointsource.
rectlycalculatetheoverlapintegralwithadiffraction-limitedPSF,
Asecondexampleinwhichthismethodcanbeappliedisin
butrathercalculatedradialencircledenergyprofilesformeasured
correctingstaticNCP-erroronground-basedtelescopeswithAO.
and theoretical images. These were calculated for the diffraction
ThisNCP-errorisinmanyhighcontrastimagingapplicationsthe
limit,andfora99.0%StrehlratioPSFsimulatedforthemanufac-
limitingerrorsource(e.g.inCreppetal.(2011))andavarietyof
turer’squotedpositionaccuracylimitfortheMEMS,withsegment
solutionshavebeenproposed(Thomasetal.2012;N’Diayeetal.
positionsdrawnfromauniformdistributionbetween±10nmin
2013).Fewofthese,withtheexceptionofRenetal.(2012),avoid
pistonand±0.05mradintip-tilt.SimulatedPSFswith97%Strehl
the necessity of introducing substantial additional hardware into
ratiowereapoorfittotheexperimentalprofiles.Wethereforecon-
thebeampath,withtheattendantcostsoftime,expenseandresid-
cludethatthefinalStrehlS wasintherange0.97 < S < 0.99.
ual optical errors. Ren et al. (2012) present a novel and effective
Nevertheless, for all images there were departures at large radial
algorithm for attaining a particular desired PSF, but the method
distances, which we suggest are due to contamination with stray
demonstratedherehastheadvantageofproducingaflatwavefront
light.Fromthesecalculationswearguethatintheinnerregionsof
quickly,foranarbitrarypupil,withoutmodel-dependentdistortions
thePSFweattainperformancelimitedprimarilybythetoleranceof
andsuitableforgeneralobserving.
theMEMSpositioning,andcloselyapproachthediffractionlimit.
In applications where the readout speed of a camera can be
mademuchfasterthantheatmosphericcoherencetimet ,thetech-
0
nique demonstrated here would appear an ideal method of high-
orderwavefrontsensing.Thiswillbethecaseinanysituationin
4.4 NoAsymmetry whichspeckleinterferometrymightordinarilybedone,andisac-
cordinglyrestrictedonlytobrighttargets.
InSection2.1,itwasstatedthatthematrixpseudoinversedoesnot
spanthewholerangeofsymmetricandantisymmetricmodesofthe
pupilunlessthepupilmodelitselfisasymmetric.Fromthisitisnot
immediatelyapparentthatanasymmetricsamplingofasymmetric ACKNOWLEDGEMENTS
pupilwouldnotpermitthesensingofallmodes.
ThisresearchhasmadeuseofNASA’sAstrophysicsDataSystem.
Inordertotestthis,wetiltedallmirrorson-axisapartfromthe
This research was supported by the Australian Research
centrepanel,anddeletedonepointfromthepupilsamplinginorder
CouncilCentreofExcellenceforUltrahighbandwidthDevicesfor
tomakeitasymmetricandobtainamatrixwithmodesspanningall
OpticalSystems(projectnumberCE110001018).
therequiredmodes.Afterrandomizingtheon-axismirrorsettings
PartofthisworkwasperformedattheOptoFabnodeofthe
as previously, we attempted to phase the mirror as before. After
AustralianNationalFabricationFacility(ANFF).
seventeeniterationstherewasnosignofimprovementatallinthe
PSF,andweconcludedthatthemethodwasineffectiveinthiscase.
Thisistobeexpectedontheoreticalgrounds,asforaneven
mode of an even pupil, the phase of the autocorrelation vanishes
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