AstronomicalObservingTechniques
Lecture6:EverythingYouAlwaysWantedtoKnowAboutOpAcs
Outline1. GeometricalOp<cs
– WavesandRays– ImagesandPupils– Aberra<ons
2. PhysicalOp<cs– Diffrac<on– TransferFunc<ons– ImageMetrics
3. HighContrastImaging
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SphericalandPlaneWaves• lightsource:collec<onofsourcesofsphericalwaves
• astronomicalsources:almostexclusivelyincoherent
• lasers,masers:coherentsources
• sphericalwaveorigina<ngatverylargedistancecanbeapproximatedbyplanewave
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IdealOpAcs
• idealop<cs:sphericalwavesfromanypointinobjectspaceare• imagedintopointsinimagespace• correspondingpointsarecalledconjugatepoints• focalpoint:centerofconvergingordivergingsphericalwavefront• objectspaceandimagespacearereversible7-3-2016 Astronomical Observing Techniques 2016: Optics 4
IdealOpAcalSystem
idealop<calsystemtransformsplanewavefrontintospherical,convergingwavefront
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AzimuthalSymmetry
• mostop<calsystemsareazimuthallysymmetric• axisofsymmetryisop<calaxis
Optical Axis
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LocallyFlatWavefronts
• raysnormaltolocalwave(loca<onsofconstantphase)
• localwavearoundraysisassumedtobeplanewave7-3-2016 Astronomical Observing Techniques 2016: Optics 7
Rays
• geomtricalop<csworkswithraysonly• raysreflectedandrefractedaccordingtoFresnelequ.• phaseisneglected(incoherentsum)7-3-2016 Astronomical Observing Techniques 2016: Optics 8
FiniteObjectDistance
• objectmayalsobeatfinitedistance• alsoinastronomy:reimagingwithininstrumentsandtelescopes
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GeometricalOpAcsExample:SPEX
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SPEXonNASAER-2plane
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picture by Ken Ulbrich, NASA
• Aperturestop:determinesdiameteroflightconefromaxialpointonobject.
• Fieldstop:determinesthefieldofviewofthesystem.
ApertureandFieldStops
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Images
• everyobjectpointcomestofocusinimageplane• lightinimagepointcomesfromallpupilposi<ons• objectinforma<onencodedinposi<on,notangle
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Pupils
• allobjectraysaresmearedoutovercompleteaperture• lightinonepupilpointcomesfromdifferentobjectposi<ons
• objectinforma<onisencodedinangle,notinposi<on
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Speedofop<calsystemdescribedbynumericalaperture(NA)orFnumber:
NA)(21 and sinNA =≡⋅=
DfFn θ
f
DΘ
• fastop<cs(largeNA)• slowop<cs(smallNA)
n
Speed/F-Number/NumericalAperture
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AberraAonsAberra<onsaredeparturesoftheperformanceofanop<calsystemfromtheidealop<calsystem.1. On-axisaberra<ons:aberra<onsthatcanbeseen
everywhereintheimage,alsoontheop<calaxis(centeroftheimage)
2. Off-axisaberra<ons:aberra<onsthatareabsentontheop<calaxis(centeroftheimage)a) Aberra<onsthatdegradetheimageb) Aberra<onsthataltertheimageposi<on
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• ReferencesphereSwithradiusRforoff-axispointP’andaberratedwavefrontW
• “Aberrated”rayfromobjectintersectsimageplaneatP”
• Rayaberra<onisP’P”
• Waveaberra<onisn·QQ ( )
rrW
nRri
i ∂∂=SmallFOV,radiallysymmetricwavefrontW(r)
WaveandRayAberraAons
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22
NA1
22 ⎟
⎠⎞⎜
⎝⎛== λλδ F
defocusedimage
Usuallyreferstoop<calpathdifferenceofλ/4.
Depthoffocus:
focusedimage
Defocus(OutofFocus)
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Raysfurtherfromtheop<calaxishaveadifferentfocalpointthanraysclosertotheop<calaxis.
SphericalAberraAon
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Made with Touch Optical Design
HubbleTrouble
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• Nullcorrectorformeasuringmirrorshapewasincorrectlyassembled(onelensmisplacedby1.3mm).
• Amanagementproblem:Mirrormanufacturerhadanalyzedsurfacewithothernullcorrectors,whichindicatedtheproblem,buttestresultswereignoredbecausetheywerebelievedtobelessaccurate.
• Nullcorrectorcancelsnon-sphericalpor<onofasphericmirrorfigure.WhencorrectmirrorisviewedfrompointA,combina<onlookspreciselyspherical.
HSTPrimaryMirrorSphericalAberraAon
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ComaVaria<onofmagnifica<onacrossentrancepupil.Pointsourceswillshowacometarytail.Comaisaninherentpropertyoftelescopesusingparabolicmirrors.
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Fromoff-axispointAlensdoesnotappearsymmetricalbutshortenedinplaneofincidence(tangen<alplane).Emergentwavewillhaveasmallerradiusofcurvaturefortangen<alplanethanforplanenormaltoit(sagifalplane)andformanimageclosertothelens.
AsAgmaAsm
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Onlyobjectsclosetoop<calaxiswillbeinfocusonflatimageplane.Off-axisobjectswillhavedifferentfocalpoints.
FieldCurvature
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DistorAon
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Straightlineonskybecomescurvedlineinfocalplanebecausemagnifica<ondependsondistancetoop<calaxis.1. Outerpartshavelargermagnifica<onàpincushion2. Outerpartshavesmallermagnifica<on àbarrel
AberraAonsSummary
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Refrac<veindexvaria<onwithwavelengthn(λ)resultsinfocallengthoflensf(λ)todependonwavelength;differentwavelengthshavedifferentfoci
ChromaAcAberraAon
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Fresneldiffrac<on=near-fielddiffrac<on:Whenawavepassesthroughanapertureanddiffractsinthenearfielditcausestheobserveddiffrac<onpaferntodifferinsizeandshapefordifferentdistances.ForFraunhoferdiffrac<onatinfinity(far-field)thewavebecomesplanar.
1
1
2
2
<<⋅
=
≥⋅
=
λ
λ
drF
drFFresnel:
Fraunhofer:F=Fresnelnumber,r=aperturesizeandd=distancetoscreen
FresnelandFraunhoferDiffracAon
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FraunhoferDiffracAon• ElectricfieldinimageplaneisFouriertransformofelectricfieldinaperture
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E x, y, z( ) = A u,v( )eiϕ u ,v( )e− i2π
λzxu+yv( )
dudv∫∫
Whenthecircularpupilisilluminatedbyapointsourcethentheresul<ngPSFisdescribedbya1storderBesselfunc<on:ThisisalsocalledtheAiryfunc<on.Theradiusofthefirstdarkring(minimum)isat:
Point Spread Function (1)
( ) ( ) 2
0
011 / 2
/ 22⎟⎟⎠
⎞⎜⎜⎝
⎛=
λθπλθπθ
rrJI
DfrFr λαλ 22.1or 22.1 1
11 ===
ThePSFisojensimplycharacterizedbythehalfpowerbeamwidth(HPBW)orfullwidthhalfmaximum(FWHM)inangularunits.
cωθ 2
1=Δ
AccordingtotheNyquist-ShannonsamplingtheoremI(Θ)(oritsFWHM)shallbesampledwitharateofatleast:
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PointSpreadFuncAons
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OpAcal/ModulaAonTransferFuncAonRayleighcriterion:twosourcescanberesolvedifthepeakofthesecondsourceisnocloserthanthe1stdarkAiryringofthefirstsource.
Abefermeasureoftheresolu<onthatthesystemiscapableofistheop<caltransferfunc<on(OTF):
Dλ22.1sin =Θ
(a) Input (b) output
( ) ( )minmax
minmax
0
re whe,IIIIC
CfCfMTF
+−==
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OpAcal/ModulaAonTransferFuncAon(2)Op<calTransferFunc<on(OTF)describesspa<alsignalvaria<onasafunc<onofspa<alfrequency.Withspa<alfrequencies(ξ,η)Modula<onTransferFunc<on(MTF)describesitsmagnitude,andthePhaseTransferFunc<on(PTF)thephase.
( ) ( ) ( )( ) ( )( ) ( )ηξπληξ
ηξηξηξηξηξ
,2,
,,,,,
iePTF
OTFMTFPTFMTFOTF
−=
=⋅=
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• Strehlra<oisconvenientmeasureofop<calquality.• Strehlra<o(SR)isra<oofobservedpeakintensityofPSFcomparedtotheore<calmaximumpeakintensityofpointsourceseenwithperfectimagingsystem
• Withwavenumberk=2π/λ,RMSwavefronterrorω
• Examples:• SR>80%considereddiffrac<on-limitedàWFE~λ/14• typicaladap<veop<cssystemdeliversSR~10-80%• seeing-limitedPSFon8mtelescopehasaSR~0.1-0.01%
22122
ωω keSR k −≈= −
StrehlRaAo
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Q: What is the maximum concentration of light within a small area? The fraction of the total PSF intensity within a certain radius is given by the encircled energy (EE):
Encircled Energy
( ) ⎟⎠⎞⎜
⎝⎛−⎟
⎠⎞⎜
⎝⎛−=
FrJ
FrJrEE
λπ
λπ 2
1201
Note that the EE depends strongly on the central obscuration ε of the telescope:
F is the f/# number
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SPHEREatVLT
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AdapAveOpAcs(lecture13)
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Student-BuiltLeidenAO
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Coronagraph
• goal:minimizediffractedlightclosetostar• canintroduceop<csinpupiland/orfocalplanes
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ApodizingPupilPhaseCoronagraph
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Gilles Otten Frans Snik Matt Kenworthy Christoph Keller UofA MagAO team NCSU OLEG group
simulation on-sky close binary companion
360°vAPPon-skyatMagAO
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theory on-sky
Otten et al. 2015
gvAPPonMagAO/Clio2
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