; TeX output 1996.03.25:22207gxg91x2YB(z)f`h( D) 5=3+d 5=3 @ (z)6] 1 =02ljT 'D6+Td (z)Xtj\8ʟ*5=3l#~j21>=02lj lT *sD6Td (z)[j_A*5=3!=f`i(:;|(4)where-Cnx2n(z)-istheturbulencestrengthasafunctionofaltitude;k0V=UR2n9=;whereisthewravelengthUofopSeration;Distheaperturediameter;andTd d(z)isthevrectordisplacementofthetrwopaths.The sumsofthetermsinbracrkets almostcancel,thruscausingdicultiesifonetriestoevXaluatethisinrtegralnumericallyV.8Thetermsintheabsolute-vXaluesignareequaltojUTT vD6Td (z)@jC՟*5=3T3=URf`h6( DA)#Ӎ*2+>92 DSd(z)cosR(M')';+d 2(z)f`i*5=6;|(5)3y7gxgwhere'BistheanglebSetrween'BT o7and'BTd xQ(z)'B.Thisexpressioncanbesimpli edandthenrumerical[dicultiescanbSeeliminatedbryusingGegenbauerpSolynomials.28Theirgeneratingfunctionis"Íf`12ax+a 2Ff`L*[=;1 ZX ҁURp=0y)Cp!x'(x)a pC#:|(6)These%functionsaresometimesreferredtoasultrasphericalfunctionsbSecausetheyarea[generalizationoftheLegendrepSolynomialsPnP(t),whosegeneratingfunctionisf`12ax+a 2Ff`L*1=2cm=;1 ZX ҁURp=0y)Pp](x)a p=:|(7)ThenGegenrbauerpSolynomialswiththecosineofavXariableastheargumentaregiveninEq.(8.934#2)ofRef.8andcanbSerewrittenasCp /jx[Ncos)(.5f'):y]A~=UQ bDp MX ҁURm=0ō[C+m&].![C+pm;no]@cosRmM[U(p2m)'><]ßQmfeRQ 0W2@m!(pm)!([C]!J)%ܱ*2oG;|(8)where9-[Cx]nisthegammafunction.%A9particularGegenrbauerpSolynomialthatisrequiredisC2 (x5=6#[&Hcos6u(')Fs]MrQ=J25N=06f`h z1j21>=03lcos*Ÿ 21(6$')t÷f`i{|:|(9)FVor2 DB>d(z),^thetermsinthestructurefunctioncanbSeexpandedinGegenrbauerpSolynomials. UWThezeroth-andallodd-ordertermscancel. UWWhenthesummationindexiscrhangedbythesubstitutionpUR!2ptheresultisDUV( DS)UR=2(2:91)k0 pW2㈍U^1xz㇫Z0&dzCn>x2YB(z)8 < : d 5=3 @ (z)( DS) 5=31 X ҁ@ p=1 cC2p1Ox5=6I[L[cos^(c@s')o]v5f\"ō}пd(z)}пQmfe<  Mx D@f\#*2p9 = ;L::V(10)$%It`isthiscancelingofthe rsttrwo`termsofthepSorwer`seriesthatwrouldcausenumericaldiculties.8De neadistancemomenrtasdm ZUR2:91k0 pW2㈍U^1xz㇫Z0&dzCn>x2YB(z)d m(z)(11)$%andaphasevXarianceas' 2=URd5=3 @ :(12)4!ܠ7gxgUnlikrethecalculationforStrehlratioforuncorrectedturbulenceandforcorrectedturbulence[withwtiltjitter,;NanexactanalyticalsolutioncannotbSefoundforanisoplanatism.FVortunately,foradaptivre-opticssystems,StheStrehlratioshouldbSefairlyhighbydesign,SwhichrequiresthestructurefunctiontobSesmall.*GThisassumptionallorwsonetoretainonlythe rsttermoftheGegenrbauerexpansiontogive!kPDUV( DS)UR=2' 2"R2x;(13)wherexUR=d2f`h x1j21>=03lcos*Ÿ 21(6$')_f`ir:} 5st=06|K( DS) 1=3܄:(14)WVejustifythissingle-termapprorximationbSelowbyshowingthatitproSducesaresultclose[totheexactresult..T..TheStrehlratiowiththesixtermapprorximationis!kPSRōexp|( '3Ɵ227)Qmfe7  u2?㇫ZKdT K( )u:Nf\ }$1+x+ōx22۟Qmfe mV  ƭ2) +ōx23۟Qmfe mV  ƭ6+ō,x24۟Qmfe  24{+ō*x25۟Qmfe  120Ⱏf\!Z:(15)Ifjustthe rstterminthelastparenrtheticalexpressionisretained,theresultisequivXalent[tojtheextendedMarrsechaljapproximation.vItisshownbSelowthatthesix-termapproximationisOmbSestforaperturesizesnormallyencounrtered.g/Theangleintegralforthenthterm,hafteruseofthebinomialtheorem,ispropSortionalto'X(n)UR=ō1Qmfe  2㈍"B{2xn㇫Zۊ0+td'f`h1j 1>=035GcosEk 2L+(Pc')]f`ia*nj#=ō1Qmfe  2Pn X ҁ7m=0fV)u0)uB)uBfi)u@ANTnZ4vnmfVWʻ1WʻCWʻCfiWʻAb3 m㈍Z&2xş㇫Z]0$Nd'6cosF. 2mT(YW')g;(16)where%lfV0BBfi@8nZ nmfV.F1.FC.FCfi.FA:ʙ=ōXvn!Qmfe=  (nm$ ))!m!C:(17)(Equation(4.641#4)inGradshrteynandRyzhik28is% I{=28덍㇫Zܩ0+d'*dcos: 2mHd(L')\=ōn9(2m1)!!Qmfe;tٟ  2(2m)!!A0;(18)5.p7gxgwhere#U(2m1)!!7=D*(2m1)(2m3):::ʜ(3)(1);(19)[(2m)!!7=D*(2m)(2m2):::ʜ(4)(2):(20)Withtheserelations,theangleinrtegralisequalto)](n)UR=1 G n dX ҁm=1fV 0 B Bfi @/DnZ"nmfVE,1E,CE,CfiE,AP+3 mōo(2m1)!!oQmfe4\T  ER(2m)!!Ec:(21)* ThevXaluesofinrteresttousare(0)=1,2(1)=0:8333,(2)=0:7083,(3)=0:6134,[(4)UR=0:5404,-andf(5)=0:4836.TheapSertureinrtegrationforthenthtermisproportionalto#UYp(n)UR=㈍>1xX㇫Z|t0\d "):  1n=3cKܞ( ):(22)%}ThisisageneralizationoftheinrtegralevXaluatedbyTVatarskiinSec.55,Eq.(22)ofRef.9.ItsvXalueis' Yn( .2n)=ō#]8QmfeC1  (2n=3)hp hzKEfV266fi4 n=6+Fu3۟콉fe@'2Z n=6+3fVA«3A7A7fiA5jMPfor~V$nUR<6:(23)The5vXaluesofinrteresttousareYn( .20)>=L1,*Yn( .21)=1:402,*Yn( .22)=2:087,*Yn( .23)=3:396,Yn( .24)=6:419, andYn( .25)=16:94. WiththesevXaluesfortheinrtegral, theStrehlratioapprorximationis#USRUR(1+0:9736E^+0:5133E 2+0:2009E 3+0:0697E 4+0:02744E 5CO:)exp(' 2);(24)whereE i=ō 6dj2QmfeU  DS1=3:(25).T..[ThereisanerrormadeinusingthisapprorximationforthecentralpartoftheapSerturethatincreaseswitheacrhtermintheapproximation.uOnehastodeterminewhetherthiserroris6:7gxgless]"thanorgreaterthantheincreasedaccuracyacrhieved]"overtheremainderoftheapSerture[brykEusingadditionalseriesterms.jTVoresolvetheseuncertainties,Ik$comparedtheStrehlratio,usingvXariousnrumbSersofterms,withexactcalculations.IcalculatedtheStrehlrationrumericallyforthecaseinwhichthedisplacementdoSesnotvXarywithpropagationdistance.'WInFig.1areplottedtheexactStrehlratiovrersusdisplace-menrtEfortheHufnagel{VValley21(HV-21)moSdelofturbulence210{12andtheStrehlratiofromrelation*(24)forDS=ro =t1,ywithonlytheunitryterminparenthesis(extendedMarechalapprorximation)andwithdi erentnumbSersoftermsintheparenthesis..T...€3.DISPLAtCEMENT2ANISOPLANA\TISM&9NInthesimplestcaseofdisplacemenrtanisoplanatism,;2whichwastreatedinSection2,;2thedisplacemenrtjaisconstantalongthepropagationdirection. Thetermstouseto ndtheStrehlratioare#[d(z)=% d;(26)[ dj2=% 2:91k g2ڍ0'0d 2;(27) E=% 6:88f\ ōd ۟Qmfe   Dff\! P*2'ßf^ō/jD/jQmfe   ro:褟f^BE=5=3Q";(28)$h n92ڍ'=% 2:91k g2ڍ0'0d 5=3^=UR6:88f\ ōd ۟Qmfe K  ro8Yf\! #*5=3/c:(29)TheStrehlratiosareplottedinFigs.2andP3..€4.ANGULAR2ANISOPLANA\TISM&9NWhenUthepropagationbSeamiso setbryaconstantanglefromthedirectionalongwhich[turbulenceUismeasured,sethee ectiscalledangularanisoplanatism.24ItarisesnaturallywhenoneqistracrkingasatellitetargetanddirectingalaserbSeamatit.Becauseofthe nitespeed7D^7gxgoflighrt,thelaserbSeamhastoleadthetrackingdirection,resultinginanangulardi erence[bSetrweenbYthedirectionalongwhicrhthetargetistrackedandtheonealongwhichthelaserbSeamdisdirected.Thiserrorcanbeeliminatedifthetargethasare ectorforthebeaconthatextendsasuitabledistanceinthepSoinrt-aheaddirection.+FVorthecaseofanangularerror#[d(z)=% Sz;(30)[ dj2=% 2:91k g2ڍ0'2S 2;(31) E=% 6:88ō31231Qmfe ǟ  08)f\ ōyVQmfe   D&Bf\!.*25Hf^ō=ؗD=ؗQmfe   roI џf^P}E=5=3_O;(32)& n92ڍ'=% 2:91k g2ڍ0'S 5=3㈍LLx㇫Z~0vdzCn>x2YB(z)QRz 5=3A=UR(S=o)";*5=31{;(33)%DKwheretheisoplanaticangleisde nedbryso=URf` N62:91k g2ڍ0'5=3Af`G*3=5]v:(34)....€5.TIME2DELA\Y&9NIfq#thereisatimedelarybSetweenwhenturbulenceismeasuredandwhenacorrectionis[appliedtothedeformablemirror, thereisadegradationinpSerformance.27Thise ectisnotoftenthoughrtofasananisoplanatice ect;however,itcanbSetreatedassuch.8...d(z)=% vn9(z);(35) ad2=% 2:91k g2ڍ0㈍ Lx K㇫Z +0ndzCn>x2YB(z)vn9 2.=(z)W 2 ls=UR(W=2)"*2)d;(36)%Ӎ E=ō1W22&>Qmfe"  ߍW2g2!DS1=3I;(37)%ns n92ڍ'=% 2:91k g2ڍ0㈍ Lx K㇫Z +0ndzCn>x2YB(z)vn9 5=3 E(z)W 5=3{=URf` N6W=5=3'f`-ן*5=3=2;(38)%DKwherethetempSoralmomenrtisde nedas8 Mt7gV1= W5=3ڍm{=UR2:91k g2ڍ0㈍ Lx K㇫Z +0ndzCn>x2YB(z)vn9 mr(z):(39)">.T../Ӎ6.CHRtOMA\TIC2ANISOPLANATISM%IfIthebSeaconbeamthatsensestheturbulencehasawravelengthIdi erentfromthatofthe[laser^bSeamthatissenrtout,thenthetwobSeamswillfollowdi erentpathsthroughtheatmospherebSecauseofthedispersivrepropertiesoftheatmosphere.%TheanalysisgivrenhereparallelsthatgivrenbyBelsherandFVried.21.T..ThecrhangeofrefractiveindexwithwavelengthhasbSeengivenbyAllen216 asn0V=URf` N6 2ڍ1j 2ڍ2/.1f`7'f\"ōe֬29498:1>/Qmfewѐ  (146ߍ2g2j16W)<(Az146ߍ2g1j1s?)+ō,255:4۟Qmfel  (41ߍ2g2j10w )7(;41ߍ2g1j1g)4%6f\#;10 6 \|:(40)The)atmosphericdensitryversusaltitudeisgivenbyCole.217 )Theratioofthe...G.Thus)thebSeamdisplacemenrtalongthepathis$'d< dd c (z)UR=ō33T sin( ts).kn033QmfeDKi  ocosOş2(s)fZJ˫2 J4㈍\czxRҟ㇫ZU0`'+dz 0W (Qz 0q)+ōz۟Qmfe  L㈍ȇLx3㇫Z0Wydz 0 (Qz 0q)fZ'3 '5:(41)#De netheinrtegraloftheairdensityasI(Ez)h=㈍>zx㇫Z0 dz 0W (Qz 0q):(42)EvXaluatingtheinrtegraland.T..[Themomenrtsofthisdisplacementaredm Z=URf\"ō ]sin(\s)+n0 ]ܟQmfe7  ?cosP2R($Ms)Cff\#I;p*mS@2Tm;(43)whereTm Z=UR2:91k g2ڍ0'sec;W(s)㈍3Hx,!㇫Z.0;.^dhCn[x2_(h)f\" SI(h)ōhsec6(s)۟Qmfe(  YxL-I(L)p^f\#v*mw:(44)H=isthealtitudeofthetarget.%ThelastterminbracrketsgoSestozeroastherangebecomes[in nite.8.T..FVorthein niterange,thisreducesto9 XB7gVTm Z=UR2:91k g2ڍ0'sec;W(s)㈍3Hx,!㇫Z.0;.^dhCn[x2_(h)I mG(h):(45).T../b#7.COMBINED2DISPLAtCEMENT%Ifttherearesevreralanisoplanatice ectspresent,IwitheachnotdecreasingtheStrehlratio[mruch,?it?isacommonpracticetomrultiplytheStrehlratiosfortheindividuale ectstogettacomrbinedStrehlratio.KEThevXalidityofthisassumptionisnowexamined.KEThetotaldisplacemenrtENthatisduetoatranslation,[anangularo set,atimedelaryV,andacrhromatico setisd< dd t (z)UR=&drdUFd +T ʥz3+n|vvv&(z)+n|ddd 1c `-(z);(46)wherecrhromaticdisplacementisgiveninEq.(50).ThetwotermsnecessaryforcalculatingtheStrehlratioarePE =ō#@dj2QmfeU  DS1=36AK;(47)[ n92ڍ' =dj5=3;(48)wheredm Z=UR2:91k g2ڍ0㈍ 1x '㇫Z nC0'-dzCn>x2YB(z)jURdt(z) #j#y1*m-}:(49).T...T..Trylerm3@ cmti12etal.218utoSokadvXantageofthevectornatureofthedisplacementalmosttoeliminateKthee ectofcrhromaticanisoplanatismonanadaptive-opticssystembychoSosinganoptimalo setangleofabSeaconfromthepropagationdirection./b#8.SUMMAR\Y%AnrFapprorximateexpressionfortheStrehlratiothatiseasilyevXaluatedforanyturbulencedistribution3wrasderived.ItappliesforvXariousanisoplanatice ects.Thisexpressionwras 10 d7gxgshorwnCtogivemuchbSetteragreementwiththeexactanswerthantheextendedMarechal[approrximation. ~ThekzenithdepSendenceisincludedintheformula. ~Thisapproximationwrasappliedtoparallelpathdisplacements,$Rangularo sets,time-delaryinducedo sets,ando sets6orwingtorefractivee ectsthatvXarywithwavelength. ExamplesforeachtypSeofanisoplanatismatvXariouszenithangleswrereevaluated.The"kStrehlratiointhepresenceofsevrerale ectswasexamined.*Itwasshownthat,depSendingonthedirectionoftherelativredisplacements,onecangetacancellationoranenhancemenrt ofthee ectofthedisplacements.!WThereforeitispSossiblefortheretobelittlereductionintheStrehlratioifthereislittlenetpathdisplacemenrt.Ifthedisplacementsareinthesamedirection,theStrehlratioislessthantheproSductoftheStrehlratiosoftheindividualterms..€AtCKNOWLEDGMENTS&9NThisUresearcrhwasspSonsoredbytheStrategicDefenseInitiativeOrganizationthroughtheU.S.DepartmenrtoftheAirFVorce. 11 mI7gxgREFERENCES$[1:~J.BelsherandD.FVried,\Chromaticrefractioninducedpseudoanisoplanatism,"tOSC[~Rep.TR-433(OpticalSciencesCo.,Placenrtia,Calif.,1981).[2:~B.;CL.EllerbroSekandPV.H.Roberts,i\TVurbulenceinducedangularseparationerrors;~expSectedόvXaluesfortheSOR-2experimenrt,"tOSCυRep.TR-613(OpticalSciencesCo.,~Placenrtia,Calif.,1984).3:~D.tL.FVried,%\Di erenrtialangleofarrivXal:theory,%evaluation,andtmeasuremenrtfeasibil-~itryV,"RadioSci.5N cmbx1210,71-76(1975).4:~D.FVried,\Anisoplanatisminadaptivreoptics,"J.Opt.SoSc.Am.72,52-61(1982).5:~D.‹Kor ,ʑG.Druden,andR.PV.Learvitt,\Isoplanicity:$the‹translationinrvXarianceofthe~atmosphericGreen'sfunction,"J.Opt.SoSc.Am.65,1321-1330(1975).6:~J.\H.Shapiro,yH\Proint-ahead\limitationonreciproScitrytracking,"yHJ.Opt.SoSc.Am.65,~65-68(1975).7:~G.eEA.Tryler,\TVurbulence-inducedadaptive-opticspSerformancedegradation:.evXaluation~inthetimedomain,"J.Opt.SoSc.Am.A1,251-262(1984).8:~I.S.GradshrteynandI.M.Ryzhik,TableofInteffgrals, ?Series,andPrffoducts(Academic,~NewYVork,1980).9:~V.&I.TVatarski,sThe$ E effctsOfTheTurbulentA2tmosphereOnWavePropagation&(U.S.~DepartmenrtofCommerce,WVashington,D.C.,1971).10:~R.7E.Hufnagel,ȚOpticffaliPropagationthroughTurbulence7(OpticalSoScietryofAmerica,~WVashington,D.C.,1974).11:~J.LL.Bufton,,PV.O.Minott,M.W.Fitzmaurice,andPV.J.Titterton,\Measuremenrtsof~turbulencepro lesinthetropSosphere,"J.Opt.Soc.Am.62,1068-1070(1972). 12 rh7gxg12:~G.C.VValley,z\Isoplanaticdegradationoftiltcorrectionandshort-termimagingsystem,"[~Appl.Opt.19,574-577(1980).[13:~M.vG.MillerandPV.L.Zieskre,)\Turbulenceenrvironmentalvcharacterization,")RADC-~TR-79-131(RomeAirDevrelopmentCenter,GrissAirFVorceBase,N.Y.,1979).14:~D. PV.GreenrwoSod,;\Bandwidth speci cationsforadaptivreopticssystems,";J.Opt.Soc.~Am.67,390-393(1977).15:~D.{L.FVried,Q\Time-delary-inducedmean-squareerrorinadaptiveoptics,"QJ.Opt.SoSc.~Am.A7,1224-1225(1990).16:~C.W.Allen,Astrffophysical35Quantities(Arthlone,London,1963).17:~A.xE.Cole,qA.Court,andA.J.Kanrtor,Handbffook4ofGeffophysicsandSpaceEnviron-~ments,S.L.VValley,ed.(McGrarw-Hill,NewYVork,1965).18:~G.Tryler,̂J.BelsherandD.FVried,\Ameliorationofcrhromaticrefractioninducedpseu-~doanisoplanatism,"tOSCRep.TR-465(OpticalSciencesCo.,Placenrtia,Calif.,1982). 13{7gxgrFIGURES,Fig.{1.!ComparisonoftheMar"Dec!halandthetwo-tosix-termapproximationswiththeexact[vdDaluefoftheStrellratio,forananisoplanaticdisplacemen!t,for( b> 3 cmmi10DM=rz0fjequalto1.;VэFig.{2.!ComparisonoftheMar"Dec!halandthetwo-tosix-termapproximationswiththeexact[vdDaluefoftheStrellratio,forananisoplanaticdisplacemen!t,forDM=rz0fjequalto5.Fig.{3.!ComparisonoftheMar"Dec!halandthetwo-tosix-termapproximationswiththeexact[vdDaluefoftheStrellratio,forananisoplanaticdisplacemen!t,forDM=rz0fjequalto10.SV,JFig.)4.Strehlratioforangularanisoplanaticerroratzenith,forvdDariousturbulencemoMdels,[v!ersusseparationanglefora0.6-msystem.UppMer-altitudeturbulencehasastronge ectontheStrehlfratio./[Fig.f5.5Strehlratioforangularanisoplanatismat30fjfora0.6-msystem.Fig.f6.5Strehlratiov!ersustimedelayatzenithfora0.6-msystem.:̍Fig.47.5Strehlratiov!ersustimedelayfora0.6-msystemat30P8zenithangle.wStrehlratioat30[forfa0.6-msystem.Fig.f8.5Di erence(106)inrefractiv!eindexbMetween0:51mandotherwavelengths. 14F7gxgTVABLES,Teable1.,ValuesUofTz2 andT:u5=3toSolv!efortheChromaticDisplacementforVeariousTurbulence[MoMdelsfforaWea!velengthfof0.5mwffTMoMdel%XTz2#fcmti8aLT:u5=3 @ b lffSLC-Da!y2:71B1060W2:00B107ZHV-216:16B1060W3:60B107HV-543:40B1050W1:87B106HV-725:95B1050W3:25B106 lff B 5aThefunitsofTz2fjarem1=3 @ .[ 5bT:u5=3risfdimensionless. 15];TDF cmmib10 3 cmmi10'K`y 3 cmr10#fcmti8K cmsy82cmmi8|{Ycmr8u cmex10