Photon orbital angular momentum in astronomy

40 Pages·2008·0.97 MB·English

A&A492,883–922(2008) Astronomy
DOI:10.1051/0004-6361:200809791 &
(cid:2)c ESO2008 Astrophysics
Photon orbital angular momentum in astronomy
N.M.EliasII1,2
1 ZentrumfürAstronomiederUniversitätHeidelberg,Landessternwarte,Königstuhl12,69117Heidelberg,Germany
2 Max-Planck-InstitutfürAstronomie;Königstuhl17,69117Heidelberg,Germany
e-mail:[email protected]
Received17March2008/Accepted7October2008
ABSTRACT
Context.Photonorbitalangularmomentum(POAM)hasbeencreatedinthelaboratory,yetitisstillrelativelyunknown.Howdoes
POAMmanifestitselfinastronomy?ArethereanyapplicationsformeasuringastrophysicalPOAM?
Aims.Inthispaper,I1)explainPOAMinanastronomicalcontext;2)definethePOAMobservablesforastronomy;3)creategeneric
systems-based calculi thatdescribe how POAMpropagates fromcelestial sphere todetector; 4) usethecalculi withseveral astro-
nomicalinstrumentsasexamplesoftheirutility;5)demonstrateanapplicationforastrophysicalPOAMmeasurements;and6)relate
POAMtoexistingastronomicalinstrumentsandconcepts.
Methods.ElectricfieldsareexpandedintoazimuthalFouriercomponents,andtheintensitiesareexpandedintocorrelationsorran-
cors.Thesourceelectricfieldsarespatiallyincoherent.Inthesystems-basedcalculi,theinputsarelocatedonthecelestialsphere,the
systemisrepresentedbypropagationthroughfreespaceandinstrument,andtheoutputsarelocatedinaspecificplane.Thediffrac-
tionandpoint-spreadfunctionexpansionsareverygenericandcanbeusedwithanytypeofinstrument.Iemploytheseexamplesto
demonstratethecalculi(inorderofincreasingdifficulty):freespace,singletelescopes,interferometers,coronagraphs,andrancorime-
ters.
Results.TheazimuthalFouriercomponentsoftheelectricfieldcorrespondtoPOAMvortexstates.Rancorscontainlessinformation
thancorrelations,yettheyareeasiertomeasureandcanbeusedinmanyapplications.Propagationthroughanaberratedtelescope
appliesexternaltorque,whichmaybeexpressedintermsofZernikepolynomials.Iprovethatasectoredphasemaskinafocal-plane
coronagraphappliestorquetothelow-orderstates,producinganull.Also,IprovethataMichelsoninterferometerisinherentlyca-
pableoffilteringPOAM;e.g.,tracking180◦ fromthecentralfringeeliminatesevenstates,producinganull.Alimitedrancorimeter
canbecreatedbyplacingafocal-planewedgemaskinacoronagraph.Theresultingrancorscanbeusedtoperformsuper-Rayleigh
observationsofunresolvedunresolvedobjects,suchasbinarystars.TherearethreetypesofsourcePOAM:intrinsic,structure,and
pointing.InstrumentalPOAM,whichmustbecalibrated,includesopticalaberrationsandatmosphericturbulence.
Conclusions.Thispaperrepresentsthestartingpointforfutureresearch:1)makingaprioripredictionsabouttheintrinsicPOAMof
astronomicalsources;2)designingground-andspace-basedPOAM-measuringinstruments;3)understandingexistinginstrumentsin
termsofPOAM;4)minimizingtheeffectsofrandomnoiseonPOAM;and5)calibratingalltypesofinstrumentalPOAM.
Keywords.instrumentation:miscellaneous–methods:analytical–methods:observational–techniques:miscellaneous
1. Introduction
Electromagneticfieldsmaybedescribedintermsofintensity,wavelength,coherence,andangularmomentum.Angularmomentum
consistsoftwodistinctparts,spinandorbital(Marcuse1980;Heitler1984).Spinangularmomentumisclassicallyrepresentedby
photonsrotatingabouttheir own axis. Orbitalangularmomentum,on the otherhand,is somewhatof a misnomerbecause under
normalcircumstancesphotonsdonotorbitabouta fixedpointin space. Thecanonicalviewof orbitalangularmomentumshows
the photon Poynting vector precessing about the nominal propagation direction, i.e., spiral wavefronts. Spin and orbital angular
momentum are different in another fundamental way, namely that the former consists of two orthogonal states while the latter
consistsofaninfinitenumberoforthogonalstates.
1.1.History
Becausetheconceptofphotonorbitalangularmomentum(POAM)isdifficulttovisualizeandunderstand,itismerelymentioned
(Jackson1975;Mandel&Wolf 1995)andevenneglected(Reitzetal. 1980;Griffiths1981)inelementaryelectromagneticstext-
books. Furthermore, a search of the scientific literature shows that the number of POAM articles is smaller than the number of
photonspinangularmomentum(PSAM,morecommonlyknownaspolarization)articles.Inrecentyears,however,theamountof
theoreticalandappliedPOAMresearchhasbeenincreasingsteadily.
Allenetal. (1992)recognizedthatlightbeamswithahelicalphasedependencecarryquantizedPOAMthatisindependentof
PSAM.Tamm&Weiss(1990)havedirectlygeneratedLaguerre-Gaussian(LG)laserbeams,whichexhibitthesehelicalwavefronts.
POAMmaybegeneratedwithspatiallightmodulators(SLMs),whicharepixelatedliquid-crystaldevices(Bazhenovetal. 1990).
Leachetal. (2002)havedevelopedatechnique,employingamodifiedMach-Zehnderinterferometer,tomeasurethePOAMstateof
ArticlepublishedbyEDPSciences
884 N.M.EliasII:Photonorbitalangularmomentuminastronomy
individualphotons.AllofthisprogresshasbeensummarizedbyPadgettetal. (2004).Molina-Terrizaetal. (2007)alsopublished
ageneralreviewofvariousaspectsofPOAM.
Harwit (2003) described astrophysical processes that generate POAM, including light scattering off inhomogeneities in the
environmentssurroundingenergeticsources(e.g.masers,pulsars,andquasars),andphotonsscatteringoffrotatingblackholes.In
addition,hementionstheadvantagesofPOAMforSETI,namelymultiple-bitencodingperphoton,entanglement,andthelackof
naturalhigh-POAMsources.Sjöholm & Palmer (2007)andThidé et al. (2007)have also conductedPOAM simulationsatradio
wavelengths,whichmayberelevantforastronomicalapplications.
1.2.Applications
So,whataretheapplicationsforPOAMinastronomy?
The most obvious application is the characterization of POAM and how it relates to underlying physical processes. Harwit
(2003)showed that exotic astronomicalsourcesgenerateelectric fields with non-zeroPOAM. Do commonastronomicalsources
alsogeneratenon-zeroPOAM?Ifso,whatproducesit?Canitbeexploitedtoobtainvaluableinformation?
I found these questions difficult to answer because laboratory and astrophysical POAM differ from each other in funda-
mental ways. Laboratory POAM sources typically consist of individual vortices and are created by lasers and dedicated optics.
AstrophysicalPOAMsources,ontheotherhand,couldcontainindividualvortices,butitismorelikelythattheycontainmultiple
vorticesduetolarge-scalestructure.Inaddition,theelectricfieldsproducedbylaboratorysourcesaregenerallyspatiallycoherent,
while the electric fields arising from the celestial sphere are generally spatially incoherent. These distinctions are important and
mustbetakenintoaccount.
The focus of this paper has shifted significantly with time. I had originally wanted to investigate POAM generated by astro-
nomical sources. I do invest some effort on that topic, but I decided to leave these detailed analyses for the future. I found that
understandingPOAM observablesand how they propagate is a prerequisite for understandingastrophysicalPOAM mechanisms
aswellasadvancedtopicslikeinstrumentdesign,randomerrorminimization,andsystematicerrorcalibration.Tothatend,Icre-
ate new observablesand propagationcalculi and present several importantexamples of how they are used, including a practical
application.
1.3.Overview
In Sect. 2, I describe astronomical POAM in terms of a semi-classical/semi-quantum model that is suitable for astronomy. The
electric fields are analogous to quantum wavefunctions, while the intensities are analogous to probabilities. Vortices are helical
wavefrontswithquantizedPOAMthatareindependentofcoordinatesystemorigin.
In Sect. 3, I expand arbitrary spatially incoherent electric fields on the celestial sphere in terms of azimuthal Fourier series.
EachcomponentrepresentsasinglePOAMstate.Theintensitycanbeexpandedintermsofcorrelationsofthestates.Correlations
betweendifferentstatesarecalledorbitalpolarization,analogoustoPSAMspinpolarization.Theintensitycanalsobeexpandedin
termsofitsownazimuthalFourierseries.Eachtermiscalledarancor,whichisequivalenttothesumofcorrelatedstateswiththe
samerancorindexandrelatedtoorbitalpolarization.Also,IderivethePOAMexpectationvalueintermsofintegratedprobabilities.
InSect.4,Icreatetwocalculi,startingwiththeelectricfieldandintensityoverlapintegrals,thatcanbeusedtomodelPOAM
propagationfromcelestialspherethroughfreespaceandinstrumentto detector.Inthe followingfive sections,Iuse these calculi
in conjunction with astronomically relevant examples (in order of increasing difficulty) to demonstrate their utility: free space
(Sect.5),singletelescope(Sect.6),focal-planecoronagraph(Sect.7),Michelsoninterferometer(Sect.8),andlimitedrancorimeter
(Sect.9).Propagationthroughfreespaceandanunaberratedtelescopeappliesnoexternaltorquetotheelectricfield.Coronagraphs
applytorqueto(modulate)thezero-orderstateofthecentralstar,producingacentralnull.Michelsoninterferometersfilterstates
accordingtothebaselineanddelay,e.g.,single-Bracewellinterferometerstrack180◦fromthecentralfringetoeliminateevenstates
and produce a null. I convertthe coronagraphto a limited rancorimeterby replacing the focal-plane phase mask with a rotating
opaquewedgemask.Itmeasuresonlyrancors,yetitcanperformtaskssuchassuper-Rayleighmodelingofanunresolvedbinary
star.
InSect.10,Idefineanaturalzero-POAMcalibrationsource:anunresolvedorpartiallyresolvedstaratthefield-of-view(FOV)
center. Several simple POAM sources are also discussed, which are used to describe instrinsic, structure, and pointing POAM.
Structure and pointing POAM are related to each other. In Sect. 11, I derive general expressions for the responses of aberrated
systemsthatlead toinstrumentalPOAM,andshowthatZernikepolynomialsmaybe writtenintermsofPOAM expansions(and
viceversa).InstrumentalPOAM(duetoopticalaberrationsandatmosphericturbulence)isanalogoustoinstrumentalPSAMbecause
theybothmustbecalibratedformeaningfulscientificresults.
2. SimplePOAMmodel
In astronomy, treating POAM with pure quantum mechanics is not necessary most of the time. Gori et al. (1998) presented a
simplifiedsemi-classical/semi-quantumPOAMmodel,drawingonformalismsfromboththeclassicalandquantumregimes.Inthis
section,Igeneralizetheirapproach.
Usingthefirst-orderparaxialapproximation,thePOAM expectationvaluedependsontheazimuthalderivativeofthe electric
field as well as the intensity in the plane of interest. Vortex electric fields represent pure POAM states with a constant POAM
expectationvaluethatisindependentofcoordinatesystemorigin.Themathematicsinothersectionsofthispaperusetheformulae
→
inthissection,replacingthevectorHwithvectorsfromotherplanes.
N.M.EliasII:Photonorbitalangularmomentuminastronomy 885
2.1.Thesemi-classical/semi-quantumregime
ConsideranensembleofphotonstravelingperpendicularlythroughaplanealongtheZ axis.Theplanecanrepresentthecelestial
sphere(locally,aplane),theobservationplane(containingtheentrancepupiloftheinstrument),theinstrumentimageplane,etc.
I assume that all of the photons have the same polarization, so the PSAM nature of the radiation can be ignored. The scalar
spatial/temporalwavefunctionontheplaneis
(cid:2) (cid:2)
(cid:2) (cid:2)
→ (cid:2) → (cid:2) →
Ψ(H;t) = (cid:2)(cid:2)Ψ(H;t)(cid:2)(cid:2) ejΦ(H;t), (1a)
→
whereH=(H ,H )=(Hcos(cid:2),Hsin(cid:2))isthevectorintheplane,H andH aretheCartesiancoordinates,Histheradialcoordinate
x y x y
intheplane,(cid:2) istheazimuthalcoordinateintheplane,andtisthetime.Itcanbedividedintoamodulusfunction
(cid:3)
(cid:4) (cid:5) (cid:4) (cid:5)
→ → →
|Ψ(H;t)| = Re2 Ψ(H;t) + Im2 Ψ(H;t) (1b)
andanaberrationfunction
⎡ ⎤
Φ(H→;t) = tan−1⎢⎢⎢⎢⎢⎢⎣ImΨ(H→→;t)⎥⎥⎥⎥⎥⎥⎦. (1c)
ReΨ(H;t)
Thiswavefunctionmaybeemployedtorepresentoneormorephotons.
Thespatialprobabilitydensityisproportionaltothetimeaverageofthesquaredmagnitudeofthewavefunction,
(cid:12) (cid:2) (cid:2) (cid:13)
(cid:2) (cid:2)
→ 1(cid:2) → (cid:2)2
p(H) ∝ (cid:2)(cid:2)Ψ(H;t)(cid:2)(cid:2) , (2a)
2
where(cid:6)·(cid:7)denotesthetimeaverage2.Itshouldbenormalizedsuchthattheintegralovertherelevantregionintheplaneisunity.The
wavefunctionisproportionaltotheelectricfield
(cid:2) (cid:2)
(cid:2) (cid:2)
→ → (cid:2) → (cid:2) →
Ψ(H;t) ∝ E(H;t) = (cid:2)(cid:2)E(H;t)(cid:2)(cid:2) ejΦ(H;t), (2b)
whichmeansthatthespatialprobabilitydensityisproportionaltotheintensity
(cid:12) (cid:2) (cid:2) (cid:13)
(cid:2) (cid:2)
→ → 1(cid:2) → (cid:2)2
p(H) ∝ I(H) = (cid:2)(cid:2)E(H;t)(cid:2)(cid:2) . (2c)
2
Therefore,thenormalizedprobabilitydensityistheintensitydividedbytheintensityintegratedoveragivenregionintheplane,or
→ →
→ I(H) I(H)
p(H) = (cid:14) = , (2d)
→ Iplane
d2HI(H)
whered2H =dH dH =dHH d(cid:2)isthedifferentialareaelement.
x y
Usingthefirst-orderparaxialapproximation,thetimeaveragePOAMperphotonabouttheZaxisis
⎡ ⎤
LZ(H→) = (cid:15)LZ(H→;t)(cid:16) = h (cid:4)HxκHy(H→) − HyκHx(H→)(cid:5) = (cid:2) ⎢⎢⎢⎢⎢⎢⎣Hx∂Φ∂H(H→) − Hy∂Φ∂H(H→)⎥⎥⎥⎥⎥⎥⎦ = (cid:2)∂Φ∂((cid:2)H→), (3a)
y x
wherehisthePlanckconstant,(cid:2)=h/2πisthereducedPlanckconstant,
(cid:15) (cid:16) → (cid:12) → (cid:13)
→ → 1 ∂Φ(H) 1 ∂Φ(H;t)
κ (H) = κ (H;t) = = (3b)
Hx Hx 2π ∂H 2π ∂H
x x
and
(cid:15) (cid:16) → (cid:12) → (cid:13)
→ → 1 ∂Φ(H) 1 ∂Φ(H;t)
κ (H) = κ (H;t) = = (3c)
Hy Hy 2π ∂H 2π ∂H
y y
arethetransversewavenumbers,and
→ (cid:12) → (cid:13)
∂Φ(H) 1 1 → ∂E∗(H;t)
= − Im E(H;t) (3d)
∂(cid:2) → 2 ∂(cid:2)
I(H)
2 Ileavethelengthofthetimeaveragearbitrary,becauseitdependsonthephotonrate,theamountofatmosphericturbulence,detectorstatistics,
andthespecificapplication.
886 N.M.EliasII:Photonorbitalangularmomentuminastronomy
is theazimuthalderivativeofthe aberrationfunctionderivedinAppendixA.Equation(3a)canbeconvertedto itsclassical form
when h κ → p and h κ → p (the linear momenta). The transverse wavenumbers are proportional to the slopes of the
Hx Hx Hy Hy
aberration function and are constant only across uniform wavefronts. The POAM expectation value per photon is the weighted
averageovertheplane,or
(cid:17) (cid:17) →
→ → → ∂Φ(H)
Lˆplane = d2Hp(H)L (H) = (cid:2) d2Hp(H) · (4)
Z Z ∂(cid:2)
Inrealsystems,integrationsoveradditionalvariablesmayberequired(Sects.4and8).
2.2.Vortices
Ingeneral,Lˆplane dependsonthecoordinatesystemorigin.InEq.(4),POAMismeasuredwithrespectto(H ,H )=(0,0).IfIshift
Z x y
thereferencepointby(ΔH ,ΔH ),thePOAMintheplanebecomes
x y
(cid:17) (cid:4) (cid:18) (cid:19) (cid:5) (cid:18) (cid:19)
Lˆplane,(cid:9) = h d2H (H + ΔH ) κ (H→) − H + ΔH κ (H→) = Lˆplane + h ΔH κˆplane − ΔH κˆplane , (5a)
Z x x Hy y y Hx Z x Hy y Hx
where
(cid:17)
→ →
κˆplane = d2Hp(H)κ (H) (5b)
Hx Hx
and
(cid:17)
→ →
κˆplane = d2Hp(H)κ (H) (5c)
Hy Hy
arethetransversewavenumberexpectationvalues.AspatiallyinvariantPOAMexpectationvalue,orLˆplane,(cid:9)=Lˆplane,requiresκˆplane=
Z Z Hx
κˆplane =0fornon-zeroΔH andΔH .
Hy x y
WhatkindofelectricfieldyieldsaspatiallyinvariantPOAMexpectationvalue?Consider
(cid:20) (cid:21)
→
E(H;t) → Em(H;t)ejm(cid:2) = |Em(H;t)| ejΘm(H;t) ejm(cid:2), (6a)
whichiscalleda vortexbecauseofitshelicalformforeach H (Indebetouw1993).Theparameterm mustbeanintegerquantum
number,otherwisethehelixwillbediscontinuousin(cid:2).Theintensityofthisvortexis
(cid:12) (cid:2) (cid:2) (cid:13) (cid:12) (cid:13)
(cid:2) (cid:2)
→ 1(cid:2) → (cid:2)2 1
I(H) = (cid:2)(cid:2)E(H;t)(cid:2)(cid:2) → Im,m(H) = |Em(H;t)|2 , (6b)
2 2
whichisindependentof(cid:2).Themodulusandaberrationfunctionsare
(cid:2) (cid:2)
(cid:2) (cid:2)
(cid:2) → (cid:2)
(cid:2)(cid:2)Ψ(H;t)(cid:2)(cid:2) ∝ |Em(H;t)| (7a)
and
→
Φ(H;t) = Θ (H;t) + m(cid:2). (7b)
m
The modulus is also independentof (cid:2), and for a given H and t the aberration function is a linear function of (cid:2). This aberration
functionleadstoκˆplane =κˆplane =0aswellasthefundamentalequation
Hx Hy
Lˆplane = m(cid:2). (8)
Z
Note that Lˆplane is independent of coordinate system origin, which means that vortices must be pure POAM states. Laboratory
Z
studies of POAM typically employ LG laser modes, which uniquelyset the radial dependencesof the E (H;t) (Tamm & Weiss
m
1990).Fornaturalastronomicalsources,however,thereareaninfinitenumberofpossibleradialdependences,sothegenericform
oftheE (H;t)ispreferredhere.
m
N.M.EliasII:Photonorbitalangularmomentuminastronomy 887
3. ArbitraryPOAMexpansions
In this section, I expand the electric field into an azimuthal Fourier series, where each component represents a POAM state. I
expresstheintensityintermsofcorrelationsofthesestates.Correlationsbetweendifferentstatesarecalledorbitalpolarization.The
intensitycanalsobeexpandedintoitsownazimuthalFourierseries,whereeachrancorcomponentisequivalenttoaninfinitesum
ofcorrelations.Igiveestimatesforthermselectricfieldamplitudesandphases.Last,IdefinethePOAMexpectationvalueinterms
ofaninfinitesumofintegratedprobabilities.
Astronomical electric fields generally do not consist of a single vortex, so how can they be expressed within the POAM
paradigm?Since(cid:2) isperiodicin2π,anyelectricfieldmaybeexpandedintermsofanazimuthalFourierseries,
(cid:17)
→ (cid:22)∞ F 1 2π →
E(H;t) = E (H;t)ejm(cid:2) ⇔ E (H;t) = d(cid:2)e−jm(cid:2)E(H;t). (9a)
m m
2π
m=−∞ 0
→
I employ the same vector H as in Sect. 2. Yao et al. (2006) used constants and Kennedy et al. (2002) used LG expansions for
theFouriercomponents,butheretheyarearbitraryfunctionsofH andt.Thisexpansionisjustthemacroscopicdescriptionofthe
POAMstatespectrumintheplane.ItisanalogoustotheJonesvector,
⎧
→ → → → J ⎪⎪⎨E (H→;t) = Hˆ ·→E(H→;t)
E(H;t) = EHx(H;t)Hˆx + EHy(H;t)Hˆy ⇔ ⎪⎪⎩EHx(H→;t) = Hˆx·→E(H→;t) , (9b)
Hy y
whichisthemacroscopicdescriptionforPSAM.
Theintensity,expressedintermsofthestatesinEq.(9a),is
(cid:22)∞ (cid:22)∞ (cid:22)∞ (cid:22)∞ (cid:22)∞
→
I(H) = I (H)ej(m−n)(cid:2) = I (H) + 2Re I (H)ej(m−n)(cid:2), (10a)
m,n m,m m,n
m=−∞n=−∞ m=−∞ m=−∞n=m+1
wherethe
(cid:12) (cid:13)
1
I (H) = I∗ (H) = E (H;t)E∗(H;t) (10b)
m,n n,m 2 m n
arethecomponentsofthecorrelationspectrum.Thesumsconsistoftwotypesofterms,autocorrelated(m=n)andcross-correlated
(m(cid:2)n).Thecross-correlatedtermscontainorbitalpolarization,orrancor2.ThesequantitiesareanalogoustoPSAMspinpolariza-
tion.Ifallofthecrosscorrelationsarezero,theradiationisunrancored.
Thermsstatesofanysourcecanbedetermineddirectlyfromitscorrelations.Thermsmagnitudeofeachstatemcomesfrom
the(m,m)autocorrelation
(cid:27)
(cid:12) (cid:13)
(cid:28)
1
E¯ (H) = |E (H;t)|2 = I (H)· (11a)
m m m,m
2
The rmsphasesarezeroforautocorrelations.Usingthen = 0state as a reference,the rmsphaseofeachstate m comesfromthe
(m,0)crosscorrelation,
(cid:29) (cid:30)
ΔΦ¯ (H) = tan−1 ImIm,0(H) = tan−1 Im (cid:29)12Em(H;t)E0∗(H;t)(cid:30) · (11b)
m,0 ReIm,0(H) Re 12Em(H;t)E0∗(H;t)
Theothercrosscorrelationscontainredundantrmsphases,andtheyserveasausefulcheck.
An ensemble of limited measurementscan often be used to gain valuable insights about astronomicalsources. For example,
thediameterofastarortheorbitalparametersofabinarystarcanbeextractedfromafinitenumberofinterferometricvisibilities.
ForthePOAMcase,significantresultscanbeobtainedwithoutreferringtotheelectricfieldsatall.ConsidertheazimuthalFourier
seriesoftheintensity
(cid:22)∞
→
I(H) = I (H)ejm(cid:2), (12a)
m
m=−∞
wherethe
(cid:17)
1 2π → (cid:22)∞
Im(H) = d(cid:2)e−jm(cid:2)I(H) = Ik,k−m(H) (12b)
2π
0 k=−∞
arethecomponentsoftherancorspectrum.ThePOAMinformationcontainedintherancorsisnotcomplete,butitmaybesufficient
forcertaintypesofmodeling(Sect.9.2).Asamatteroffact,ifrancorsaremeasuredwithsufficientspatialresolution,itispossible
2 Ichosetheword“rancor”becauseitisconvenient andasynonymforpolarization.Relatedexpressionsnaturallyfollow,e.g.,rancorimetry,
rancorimeter,(un)rancored,and(de)rancorizer.
888 N.M.EliasII:Photonorbitalangularmomentuminastronomy
toreconstructanimageusingonlythisFourierseries.Notethateachrancorisaninfinitesumofcorrelationswiththesamerancor
indexm(AppendixB).
Calculating Lˆplane in termsof correlationsis relativelystraightforward.Accordingto AppendixA, the azimuthalderivativeof
Z
theaberrationfunctionis
⎡ ⎤
∂Φ(H→) = 1 Re (cid:22)∞ (cid:22)∞ nI (H)ej(m−n)(cid:2) = 1 ⎢⎢⎢⎢⎢⎣ (cid:22)∞ mI (H) + 2Re (cid:22)∞ (cid:22)∞ nI (H)ej(m−n)(cid:2)⎥⎥⎥⎥⎥⎦. (13)
∂(cid:2) → m,n → m,m m,n
I(H) m=−∞n=−∞ I(H) m=−∞ m=−∞n=m+1
SubstitutingthisequationintoEq.(4),IobtainthePOAMexpectationvalue
(cid:22)∞
Lˆplane = pplanem(cid:2), (14a)
Z m,m
m=−∞
where
(cid:14)
∞ (cid:17)
pplane = Impl,amne = 2π 0(cid:14) dHHIm,m(H) = 2π ∞dHHp (H) (14b)
m,m Iplane → m,m
d2HI(H) 0
is the probability of a single photon being in state m, p (H) is the corresponding probability density, Iplane is the integrated
m,m m,m
correlation(m,m),Iplane istheintensityintegratedovertheplane,
(cid:17) (cid:17)
(cid:22)∞ Iplane ∞ I (H) ∞
pplane = 0 = 2π dHH 0 = 2π dHHp (H) = 1, (14c)
m,m Iplane Iplane 0
m=−∞ 0 0
Iplane istheintegratedunrancoredterm,andp (H)istheunrancoredprobabilitydensity.Therancoredprobabilitydensitiescontain
0 0
phaseinfofrmation,buttheydonotcontributetoLˆplane becausetheej(m−n)(cid:2) areperiodicin(cid:2).
Z
4. POAMpropagationthroughagenericsystem
InSects.2and3,IpresentedthePOAMexpansionsofelectricfieldsandintensities.Thoseexpressionscanbeappliedtoanyplane
→
by simply changingthe spatial coordinateH. In this section, I employ a mathematicalsystems approach using overlap integrals
todescribehowthePOAMexpansionsofelectricfieldsandintensitiespropagatethroughagenericsystem,fromcelestialsphere
→ → → →
(input; H → Ω) to a specific plane (output; H → N). These generic expansions can be applied to any optical system; I present
importantandverydifferentexamplesinSects.5–9todemonstratetheirutility.
4.1.Electricfields
Theexpressionforthepropagationofelectricfieldsisgivenby
(cid:17)
→ → → → → →
E(N;a,t) = d2ΩD(N,Ω;a)E(Ω;t), (15)
→
whereN=(N ,N )=(Ncosν,Nsinν)isthevectorinthespecificplane, N and N aretheCartestiancoordinatesinthespecific
x y x y
→
plane,Nistheradialcoordinateinthespecificplane,νistheazimuthalcoordinateinthespecificplane,Ω=(α,δ)=(ρcosφ,ρsinφ)
isthevectoronthecelestialsphere,αandδaretheCartestiancoordinatesonthecelestialsphere,ρistheradialcoordinateonthe
→
celestialsphere,φisazimuthalcoordinateonthecelestialsphere,and a isanoptionallistofparameters.Thisequationisanoverlap
integral, linking the input electric fields on the celestial sphere to the output electric fields in the specific plane via the system
diffractionfunction(DF).
Equation(15)isvalidforbothspatiallycoherentandincoherentelectric fieldsonthecelestialsphere,butin thispaperI deal
onlywiththelatter.TheDF,whichdescribesthepropagationbehaviorthroughfreespaceandinstruments,couldbeafunctionof
timeinthepresenceoftheatmosphereandthermomechanicalinstrumentstresses.Inthemathematicalsystemsapproach,theDFis
alsoknownastheelectric-fieldimpulseresponse.
WhentheelectricfieldsandDFofEq.(15)areexpandedinazimuthalFourierseries,POAMpropagationbecomesclear.There
are three different generic expansions, all displayed in Table 1. The formulae for the input sensitivities, output sensitivities, and
input/outputgainsoftheDFarelocatedinTable2.IderivetheseresultsinAppendixC.
Thefirstelectric-fieldexpansionusestheinputsensitivityoftheDF.Eachtermindicateshowtheinputstatesfromthecelestial
spherearemodifiedbytheinstrumentbeforetheyarriveatthespecificplane.Thisexpansionisoflimitedusebecauseeachcompo-
nentisnotaspecific-planestate,butIincludeitforthesakeofcompleteness.Eachterminthesecondexpansionisaspecific-plane
state,writtenintermsoftheoutputsensitivityandthetotalinputelectricfield.Thisformisemployedinsituationswheretheexact
form of the input POAM spectrum is not specified. The third expansionis a combinationof the first two, using the input/output
gainsandtheinputstates.Thegainsaredirectindicatorsofhowmuchsourcestatemisconvertedintospecific-planestate p.This
formmaybeusedforanumberofdifferentpurposes,includingopticalsystemanalysis.
N.M.EliasII:Photonorbitalangularmomentuminastronomy 889
→ → → → → →
Table1.ThePOAMexpansionsofE(N;a,t)intermsofthePOAMexpansionsofD(N,Ω;a)(Table2)andE(Ω;t).
POAMexpansiontype Expression
Input E(N→;→a,t)=(cid:31)∞m=−∞Eˆm((cid:14)N→;→a,t)
whereEˆ (N→;→a,t)=2π ∞dρρD−m(N→,ρ;→a)E (ρ;t)
Output E(N→;→a,tm)=(cid:31)∞p=−∞E(cid:14)p(N0;→a,t)ejpν m
→ → → →
whereE (N;a,t)= d2ΩD (N,Ω;a)E(Ω;t)
p p
Input/Output wE(hN→er;e→aE,t)(=N;(cid:31)→a∞p,=t)−∞=E(cid:31)p(∞N;→a2,tπ)(cid:14)ej∞pνdρρD−m(N,ρ;→a)E (ρ;t)
p m=−∞ 0 p m
→ → →
Table2.ThePOAMexpansionsoftheD(N,Ω;a).
POAMexpansion Expression
Inputsensitivity:
(cid:14)
Integralform(forward) D−m(N→,ρ;→a)= 1 2πdφejmφD(N→,Ω→;→a)
Sumform(reverse) D(N→,Ω→;→a)=(cid:31)2∞mπ=−∞0 D−m(N→,ρ;→a)e−jmφ
Outputsensitivity:
(cid:14)
Integralform(forward) D (N,Ω→;→a)= 1 2πdνe−jpνD(N→,Ω→;→a)
Sumform(reverse) D(pN→,Ω→;→a)=(cid:31)2∞pπ=−0∞Dp(N,Ω→;→a)ejpν
Input/Outputgain:
(cid:14) (cid:14)
Integralform(forward) D−m(N,ρ;→a)= 1 2πdφejmφ 1 2π dνe−jpνD(N→,Ω→;→a)
Sumform(reverse) D(pN→,Ω→;→a)=(cid:31)2∞pπ=−∞0 (cid:31)∞m=−∞D2−pπm(N0,ρ;→a)e−jmφejpν
4.2.Intensities
Forspatiallyincoherentelectricfieldsonthecelestialsphere,theexpressionforthepropagationofintensitieshasthesameoverlap-
integralformasEq.(15),
(cid:12) (cid:2) (cid:2) (cid:13) (cid:17) (cid:17) (cid:2) (cid:2) (cid:12) (cid:2) (cid:2) (cid:13)
(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)
→ → 1(cid:2) → → (cid:2)2 → → → → (cid:2) → → → (cid:2)2 1(cid:2) → (cid:2)2
I(N;a) = (cid:2)(cid:2)E(N;a,t)(cid:2)(cid:2) = d2ΩP(N,Ω;a)I(Ω) = d2Ω (cid:2)(cid:2)D(N,Ω;a)(cid:2)(cid:2) (cid:2)(cid:2)E(Ω;t)(cid:2)(cid:2) . (16)
2 2
I derive this equation in Appendix D. The intensity impulse response of the mathematical systems approach is more commonly
knownasthepoint-spreadfunction(PSF)inastronomy.
WhentheinputelectricfieldsandDFofEq.(15)areexpandedinazimuthalFourierseriesandpluggedintoEq.(16),POAM
propagationbecomesclear. There are eight differentgeneric expansionsinvolvingboth correlationsand rancors, all displayed in
Table 3. The formulae for the input sensitivities, output sensitivities, and input/outputgains of the PSF are located in Table 4. I
derivetheseresultsinAppendixC.
ThefirsttwoexpansionsusethecorrelatedandrancoredinputsensitivitiesofthePSF.Eachtermshowshowtheinputcorrela-
tionsorrancorsaremodifiedbytheinstrumentbeforetheyarriveatthespecificplane.Theseexpansionsareoflimitedusebecause
eachcomponentisnotaspecific-planecorrelationorrancor,butIincludethemforthesakeofcompleteness.Eachterminthenext
twoexpansionsiseitheracorrelationorrancor,writtenintermsofthetotalinputintensity.Theseformsareemployedinsituations
where the exact forms of the input correlation or rancor spectra are not specified. The last four expansions are combinations of
the first two, using the input/outputgains and the input correlations and rancors. The gains are a direct indication of how much
inputcorrelation(m,n)(orrancorm)isconvertedintospecific-planecorrelation(p,q)(orrancor p).Theseformsmaybeusedfor
anumberofdifferentpurposes,includingopticalsystemanalysis.
5. POAMpropagationthroughfreespace
→ → → →
Spatiallyincoherentelectricfieldsonthecelestialsphere(H→Ω)travelthroughfreespacetotheobservationplane(H→R),which
containstheentrancepupiloftheinstrument.Inthissection,IformtheDF,PSF,andthePOAMexpansionsforthissimplesystem
andprovethat1)eachstateonthecelestialspheregivesrisetoonlythesamestateintheobservationplane;2)onlytheunrancored
correlationsintheobservationarenon-zero;and3)theintegratedprobabilitiesandPOAMexpectationvalueareconserved,which
meansthatnoexternaltorqueisappliedtothewavefronts.
5.1.Classicalforms
TheDFforpropagationthroughfreespaceis
D(N→,Ω→;→a) → D(R→,Ω→) = ej2πκR→·sˆ(Ω→) = ej2πκR→·sˆxy(Ω→) ≈ ej2πκRρcos(ψ−φ), (17a)
890 N.M.EliasII:Photonorbitalangularmomentuminastronomy
→ → → → → →
Table3.ThePOAMexpansionsofI(N;a),foraspatiallyincoherentsource,intermsofthePOAMexpansionsofP(N,Ω;a)(Table4)andI(Ω).
POAMexpansiontype Expression
Input I(N→;→a)=(cid:31)∞m=−∞(cid:31)∞n=−(cid:14)∞Iˆm,n(N→;→a)
(correlated) whereIˆ (N→;→a)=2π ∞dρρP−m+n(N→,ρ;→a)I (ρ)
Input I(N→;→a)m=,n(cid:31)∞m=−∞Iˆm(N→(cid:14);0→a) m,n
(rancored) whereIˆ (N→;→a)=2π ∞dρρP−m(N→,ρ;→a)I (ρ)
Output I(N→;→a)=m(cid:31)∞p=−∞(cid:31)(cid:14)∞q=−∞0Ip,q(N;→a)ej(p−q)ν m
→ → → →
(correlated/unexpanded) whereI (N;a)= d2ΩP (N,Ω;a)I(Ω)
Output I(N→;→a)p=,q(cid:31)∞p=−∞I(cid:14)p(N;→a)epj,pqν
→ → → →
(rancored/unexpanded) whereI (N;a)= d2ΩP (N,Ω;a)I(Ω)
p p
I(cnoprurte/Olautetdp/uctorrelated) wI(hN→e;r→ae)I=((cid:31)N∞p;=→a−∞)=(cid:31)(cid:31)∞q=−∞∞Ip,q(cid:31)(N∞;→a)2eπj((cid:14)p−∞q)νdρρP−m+n(N,ρ;→a)I (ρ)
I(cnoprurte/Olautetdp/urtancored) wI(hN→e;r→ae)Ip=,q((cid:31)N∞p;=→a−∞)=(cid:31)(cid:31)∞q=−m∞∞=−I∞p,q2(πNn=(cid:14);−∞→a∞)deρj(ρp0−Pq)−νm(N,ρp;,q→a)I (ρ) m,n
I(nrapnucto/Oreudt/pcuotrrelated) wI(hN→e;r→ae)Ip=,q((cid:31)N∞;p=→a−)∞=I(cid:31)p(∞Nm=;−→a∞)(cid:31)ej∞pν0 2π(cid:14)∞dp,ρqρP−m+n(Nm,ρ;→a)I (ρ)
I(nrapnucto/Oreudt/pruantcored) wI(hN→e;r→ae)I=p((cid:31)N∞;p=→a−)∞=I(cid:31)p(m∞N=;−→a∞)2eπjnp=(cid:14)ν−∞∞dρρ0P−m(N,ρp;→a)I (ρ) m,n
p m=−∞ 0 p m
→ → →
Table4.ThePOAMexpansionsofP(N,Ω;a).
POAMexpansion Expression
Inputsensitivity(separate):
Integralform(forward) P−m,−n(N→,ρ;→a)=D−m(N→,ρ;→a)D−n,∗(N→,ρ;→a)
Sumform(reverse) P(N→,Ω→;→a)=(cid:31)∞m=−∞(cid:31)∞n=−∞P−m,−n(N→,ρ;→a)e−j(m−n)φ
Outputsensitivity(separate):
Integralform(forward) P (N,Ω→;→a)=D (N,Ω→;→a)D∗(N,Ω→;→a)
Sumform(reverse) P(pN→,q,Ω→;→a)=(cid:31)∞p=−p∞(cid:31)∞q=−∞Ppq,q(N,Ω→;→a)ej(p−q)ν
Input/Outputgain(separate):
Integralform(forward) P−m,−n(N,ρ;→a)=D−m(N,ρ;→a)D−n,∗(N,ρ;→a)
Sumform(reverse) P(pN→,q,Ω→;→a)=(cid:31)∞p=−∞p(cid:31)∞q=−∞(cid:31)∞m=−q∞(cid:31)∞n=−∞P−p,mq,−n(N,ρ;→a)e−j(m−n)φej(p−q)ν
Inputsensitivity(combined):
SInutmegrfaolrmfor(mrev(eforsrew)ard) PP(−N→m(,N→Ω→,;ρ→a;)→a=)=(cid:31)∞m21π=−(cid:14)∞02πPd−φm(eN→jm,φρP;→(aN→),eΩ→−j;m→aφ)=(cid:31)∞k=−∞P−k,−k+m(N→,ρ;→a)
Outputsensitivity(combined):
SInutmegrfaolrmfor(mrev(eforsrew)ard) PP(pN→(N,Ω→,Ω→;→;a→a))==(cid:31)21∞pπ=(cid:14)−0∞2πPdpν(eN−,jΩ→pν;P→a()N→e,jΩ→pν;→a)=(cid:31)∞l=−∞Pl,l−p(N,Ω→;→a)
Input/Outputgain(combined#1):
SInutmegrfaolrmfor(mrev(eforsrew)ard) PP(−pN→m(,NΩ→,;ρ→a;)→a=)=(cid:31)∞2p1π=−(cid:14)∞02π(cid:31)d∞mφ=−e∞jmPφ−p21πm((cid:14)N02,πρd;ν→ae)−ejp−νjmPφ(N→ej,pΩν→;→a)=(cid:31)∞k=−∞(cid:31)∞l=−∞Pl−,lk−,−pk+m(N,ρ;→a)
Input/Outputgain(combined#2):
Integralform(forward) P−m(N,ρ;→a)= 1 (cid:14)2πdφejmφP (N,Ω→;→a)=(cid:31)∞ P−k,−k+m(N,ρ;→a)
Sumform(reverse) P(pN→,q,Ω→;→a)=(cid:31)∞2pπ=−∞0(cid:31)∞q=−∞(cid:31)∞m=p−,q∞P−p,mq(N,ρ;→a)k=e−−∞jmφpe,qj(p−q)ν
Input/Outputgain(combined#3):
SInutmegrfaolrmfor(mrev(eforsrew)ard) PP(−pN→m,,−Ω→n(;N→a,)ρ)==(cid:31)2∞1pπ=−(cid:14)0∞2π(cid:31)d∞mν=e−−∞jp(cid:31)νP∞n=−−m∞,−nP(N→−pm,,ρ−n;(→aN),=ρ;(cid:31)→a∞l)=−e∞−j(Pml−−,lmn−),φp−ne(jNpν,ρ;→a)
→
whereR= (X,Y)=(Rcosψ,Rsinψ) isthevectorintheobservationplane, X andY aretheCartesiancoordinates,R istheradial
coordinate,ψisazimuthalcoordinate,κ=1/λisthewavenumber,
→ → →
sˆ(Ω) = sinρ cosφXˆ + sinρ sinφYˆ + cosρZˆ = s (Ω) + cosρZˆ ≈ ρ cosφXˆ + ρ sinφYˆ + 1Zˆ (17b)
xy
N.M.EliasII:Photonorbitalangularmomentuminastronomy 891
istheunitvectorpointingtowardthecelestialsphere,andXˆ,Yˆ,andZˆ aretheunitvectorsoriginatingfromtheobservationplane.
→ →
Onlysˆ (Ω)isrequired,sinceRandZˆ areperpendicular.TheapproximationinEq.(17b)arisesbecausethedistancebetweenthe
xy
celestialsphereandtheobservationplaneisextremelylarge.ItleadstotheapproximationinEq.(17a),whoseexponentialkernelis
theFraunhoferpropagator.Theelectricfieldintheobservationplanethenbecomes
(cid:17)
E(N→;→a,t) → E(R→;t) = d2Ωej2πκRρcos(ψ−φ)E(Ω→;t). (17c)
Also,thisDFleadstoaPSFofunity,whichmeansthattheintensitybecomes
(cid:17)
→ → → →
I(N;a) → I(R) = d2ΩI(Ω) = Ics = constant. (18a)
Aconstantintensityovertheentireobservationplanesuggeststhatthesourceemitsaninfiniteamountofenergy,whichisimpossi-
ble.Inreality,theintensityisconstantoveraverylargespherecenteredonthesource.Locally,eachpointonthespheremaybewell
approximatedbyanobservationplanethatis largerthanthe entrancepupilofanypracticalastronomicaltelescope,whichmeans
thattheintensityintegralovertheentrancepupil
(cid:17)
→
Ient = d2RI(R) = A Ics = constant (18b)
ent
ent
willbefinite,whereA istheareaoftheentrancepupil.
ent
5.2.POAMexpansions
Eachstateintheobservationplanehastheform
(cid:17) (cid:17)
∞ ∞
E (R;t) = 2π dρρD−m(R,ρ)E (ρ;t) = jm2π dρρJ (2πκRρ)E (ρ;t) = jmH [E (ρ;t);R], (19)
m m m m m m m
0 0
where H [E (ρ;t);R]is themth-orderHankeltransform(Bracewell1986),and J (·)isthemth-orderBesselfunctionofthe first
m m m
kind(AppendixE).Thisequationshowsthateachstateonthecelestialspheregivesriseonlytothesamestateintheobservation
plane, which meansthat propagationthroughfree space doesnotapply externaltorque.A small uniformm (cid:2) 0 disk at the FOV
centerproducesdecayingringswithlargeradiiintheobservationplane.Asthediskradiusincreases,theringradiidecrease.
Thecorrelationsintheobservationplaneare
(cid:17) (cid:17)
∞ ∞
Im,n(R) = 2π dρρP−mm,n,−n(R,ρ)Im−n(ρ) = jm−n2π dρρJm(2πκRρ)Jn(2πκRρ)Im−n(ρ). (20)
0 0
Onlytheunrancoredcorrelationsintheobservationplanearenon-zero,whichmeansthat
(cid:17)
→ (cid:22)∞ (cid:22)∞ ∞ (cid:22)∞
I(R) = I (R) = I (R) = 2π dρρI (ρ) = Ics , (21a)
0 m,m m,m m,m
m=−∞ m=−∞ 0 m=−∞
wherethesecondlastequalitycomesfromAppendixE,andthelastequalitycomesfromEq.(14b).Notonlyisthesumconstant
(Eq.(18a)),buteachtermisconstantaswell.UsingEq.(14b)again,Ifindthat
(cid:14)
2π RentdRRI (R) AentIcs
pent = 0 m,m = m,m = pcs , (21b)
m,m Ient AentIcs m,m
whereR istheradiusoftheentrancepupil.Theintegratedprobabilityintheentrancepupilisidenticaltotheintegratedprobability
ent
on the celestial sphere.Therefore,the POAM expectationvalueis conserved(Lˆent = Lˆcs) and noexternaltorqueis appliedto the
Z Z
wavefronts,asshownpreviously.
6. POAMpropagationthroughasingletelescope
→ → →
Electricfieldsintheobservationplane(H→R),whichoriginatefromaspatiallyincoherentsourceonthecelestialsphere(H→
→ → →
Ω) and propagatethrough free space, must be sent to an instrument for analysis (H → r). In this section, I employ a telescope
for thistask (Fig. 1).Thetelescope hasanexitpupil,whichrepresentsthe imageof the entrancepupilplusaberrations.The exit
pupilisincludedsothatPOAMdevices(e.g.,mask,hologram,sorter,etc.)ornon-POAMdevices(e.g.,deformablemirror)maybe
introducedintotheopticalsystem.IformtheDF, PSF, andthePOAM expansionsforanunaberratedtelescopeandprovethat1)
theinputsensitivities,outputsensitivities,andinput/outputgainsoftheDFcanbeexpressedintermsthesameprincipalDFPOAM
functions;2)thesensitivities,outputsensitivities,andinput/outputgainsofthePSFcanbeexpressedintermsofthesameprincipal
PSFfunctions;3)theintegratedprobabilitiesandPOAMexpectationvaluesareconserved,whichmeansthatnoexternaltorqueis
appliedtothewavefronts.
892 N.M.EliasII:Photonorbitalangularmomentuminastronomy
Fig.1.Theschematicdiagramofpropagationfromasourceonthecelestialspherethroughfreespaceandtelescopetotheimageplane.
6.1.Classicalforms
→
When the integral over Ω in Eq. (17c) is removed, the result is the electric field in the observation plane due to a point source
(impulse)onthecelestialsphere,or
E(R→,Ω→;t) = ej2πκR→·→sxy(Ω→)E(Ω→;t). (22a)
Theelectricfieldintheexit-pupilplaneisamodifiedversionofEq.(22a)
E(R→,Ω→;t) = ej2πκM→R·→sxy(Ω→)D(R→,Ω→)E(Ω→;t), (22b)
→ →
whereR=R/M =(X,Y)=(Rcosχ,Rsinχ)isthevectorinthetheexit-pupilplane,XandYaretheCartesiancoordinates,Ris
→ →
theradialcoordinate,χistheazimuthalcoordinate,D(R,Ω)isanamplitude/phaseaberrationfunctionoftheinstrumentfront-end
projected into the exit-pupilplane, and M is the amount of beam compression between the entrance- and exit-pupilplanes. The
electricfieldintheimageplaneduetoapointsourceonthecelestialsphereis
(cid:17)
E(→r,Ω→;t) = 1 d2Re−j2πκ→rf·→RE(R→,Ω→;t) = D(→r,Ω→)E(Ω→;t), (23a)
A
ep
where
(cid:17)
D(→r,Ω→) = 1 d2Re−j2πκM[→Θr−→sxy(Ω→)]·→RD(R→,Ω→) (23b)
A
ep
is the telescope DF, A = πR2 is the area of the exit pupil, R is the radius of the exit pupil, Θ = Mf is the plate scale (units
ep ep ep
→
are length per angle, e.g. μm arcsec−1), f is the focal length between the exit-pupil plane and the image plane, r = (x,y) =
(rcosξ,rsinξ)isthevector,xandyaretheCartesiancoordinates,ristheradialcoordinate,andξistheazimuthalcoordinate. The
overlapintegrals,
(cid:17)
→ → → →
E(r;t) = d2ΩD(r,Ω)E(Ω;t) (24a)
and
I(→r) = (cid:17) d2ΩP(→r,Ω→)I(Ω→) = (cid:17) d2Ω (cid:2)(cid:2)(cid:2)(cid:2)(cid:2)D(→r,Ω→)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)2 (cid:12)1(cid:2)(cid:2)(cid:2)(cid:2)E(→r;t)(cid:2)(cid:2)(cid:2)(cid:2)2(cid:13), (24b)
2
→ → → →
areexpressedintermsoftheDFandPSFandhavethesameformsasinSect.4(N→ r and a → 0).
→ →
ConsideranunaberratedtelescopewithD(R,Ω)→D =realconstant(thephaseisarbitrary,soIchoosezero).TheDFthen
0
becomes
(cid:4) (cid:5)
→ → → → → →
D(r,Ω) → D(r,Ω) = D J 2πκR Γ(r,Ω) , (25a)
0 1 tel