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OPTICSOF CHARGED PARTICLES OPTICSOF CHARGED PARTICLES SecondEdition HERMANNWOLLNIK
NewMexicoStateUniversity,LasCruces, NM,UnitedStates
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Forewordtosecondeditionxi Forewordxiii
1.Gaussianopticsandtransfermatrices1
1.1 Themethodoftransfermatrices1
1.1.1 Thedescriptionofstraightraysinopticalsystems1
1.1.2 Propertiesofthinlenses3
1.2 Anopticalsystemthatcontainsonethinlens4
1.2.1 Theformationofarealimageorofavirtualimage6
1.3 Generalopticalsystems8
1.3.1 Thesignificanceofthedisappearanceofelementsof transfermatrices10
1.4 Transfermatricesoflensmultiplets11
1.4.1 Thetransfermatrixofadoubletoftwothinlenses12
1.5 Roundlensesforchargedparticles18 References19
2.Generalrelationsforthemotionofchargedparticlesin electromagneticfields21
2.1 Energy,velocity,andmassofacceleratedchargedparticles21
2.2 Forcesonchargedparticlesinelectromagneticfields22
2.2.1 Themagneticrigidity25
2.2.2 Theelectrostaticrigidity26
2.3 Thedescriptionofabundleofparticlesofdifferentkinetic energiesandmasses28
2.4 Therefractiveindexoftheelectromagneticfield29
2.5 TheEuler Lagrangeequations32 Appendix33 Hamilton’svariationalprincipleandtheLagrangeequations33 Themotionofchargedparticlesintime-independent magneticandelectrostaticfields35 References37
3.Quadrupolelenses39
3.1 Theelectricandmagneticfieldsinquadrupolelenses40
3.1.1 Fieldsinelectricquadrupolelenses41
3.1.2 Fieldsinmagneticquadrupolelenses42
3.2 Particletrajectoriesinquadrupolelenses43
3.2.1 Theequationsofmotion43
3.2.2 Focusingpropertiesofsinglequadrupolelenses46
3.3 Thedesignofquadrupolemultiplets49
3.3.1 Quadrupoledoublets50
3.3.2 Quadrupolemultiplets56
3.4 Thin-lensapproximationsforquadrupolemultiplets59
3.4.1 Doublypoint-to-parallelfocusingthin-lensquadrupole multiplets60
3.4.2 Stigmaticfocusingthin-lensquadrupolemultiplets63
3.4.3 Thinlensquadrupolemultipletsforwhich ðx jaÞ and ðbjbÞ vanishsimultaneously65
3.4.4 A “beamrotator” 65 References66
4.Sectorfields67
4.1 Homogeneousmagneticsectorfields68
4.1.1 Particletrajectoriesinhomogeneousmagneticsectorfields68
4.1.2 Focusinganddispersingpropertiesofhomogeneousmagnetic sectorfields72
4.1.3 Obliqueentranceandexitofparticlebeamsinsectormagnets81
4.2 Inhomogeneousmagneticsectorfieldsformedbyinclinedplanarpole faces(wedgemagnets)86
4.2.1 Particletrajectoriesinwedgemagnets87
4.2.2 Focusinganddispersingpropertiesofwedgemagnets90
4.3 Radiallyinhomogeneoussectorfieldsformedbyconicalpolefacesor toroidalelectrodes92
4.3.1 Theequationsofmotioninradiallyinhomogeneoussectorfields93
4.3.2 Particletrajectoriesinradiallyinhomogeneousmagneticor electrostaticsectorfields98
4.3.3 Focusinganddispersingpropertiesofradiallyinhomogeneous sectorfields100
4.3.4 Examplesofradiallyinhomogeneoussectorfields104
4.4 Particleflighttimesinradiallyinhomogeneoussectorfields, quadrupoles,andfield-freeregions106 Appendix109 Paraxialtrajectoriesinwedgemagnets109 References112
5.Chargedparticlebeamsinphasespace113
5.1 Liouville’stheoremandfirst-ordertransfermatrices114
5.2 Phase-spaceareasofparticlebeamspassingthroughopticalsystems115
5.2.1 Phase-spaceareasofparticlebeamsinfield-freeregions116
5.2.2 Phase-spaceareasofparticlebeamsinimage-formingsystems117
5.2.3 Thevirtualthinobjectlens118
5.3 Beamenvelopes120
5.3.1 Envelopesofbeamswithparallelogram-likephase-spaceareas122
5.3.2 Envelopesofbeamswithoctagon-likephase-spaceareas123
5.3.3 Envelopesofbeamswithellipticalphase-spaceareas123
5.4 Positionsandsizesofenvelopeminima129
5.4.1 Imagesandpupils129
5.4.2 Beamwaists130
5.4.3 Pupilsandwaistsforpoint-to-parallelfocusingsystems131
5.5 Minimalbeamenvelopesatpostulatedlocations131
5.6 Liouville’stheoremanditsapplicationtowide-anglebeams134
5.6.1 Abbe’ s “SineLaw” foratwo-dimensionalphasespace135
5.6.2 Thecurrentdensity136
5.7 Beamswithspacecharge137
5.7.1 Thecompensationofspacechargeinparticlebeams138
5.7.2 Thecalculationoftransporteffectsinconstantdiameter ionbeams139
5.7.3 Thedesignofbeamlinesforhigh-intensitybeams141 References143
6.Particlebeamsinperiodicstructures145 6.1 Single-particletrajectoriesandbeamenvelopes146
6.1.1 Systemswithpostulatedopticalproperties149
6.1.2 Opticalsystemsthathaveidenticalentranceandexitbeams151
6.2 Ringsofunitcells158
6.2.1 Lateralbeamdeviations158
6.2.2 Longitudinalbeamdeviations159 References160
7.Fringefields161 7.1 Particletrajectoriesinthefringefieldsofdipolemagnets161
7.1.1 Positionsofeffectivefieldboundariesindipolemagnets163
7.1.2 Fringe-fieldshuntsfordipolemagnets165
7.1.3 Fringe-fieldshuntsforhomogeneousandinhomogeneous dipolemagnets169
7.1.4 y-Focusinginthefringefieldsofdipolemagnets169
7.1.5 Transfermatricesofrealisticmagneticsectorfields174
7.2 Particletrajectoriesinfringefieldsofelectrostaticdeflectors174
7.2.1 Electrostaticdeflectorsbiasedsymmetricallytoground175
7.2.2 Electrostaticdeflectorsbiasedasymmetricallytoground176
7.2.3 Transfermatricesofrealisticelectrostaticsectorfields178
7.3 Particletrajectoriesinthefringefieldsofquadrupolelenses179 References182
8.Imageaberrations183
8.1 Systematicsofimageaberrations183
8.2 Originofimageaberrations190
8.2.1 Geometricaberrations191
8.2.2 Chromaticaberrations196
8.3 RelationsbetweencoefficientsofEq.(8.2)duetotheconditionof symplecticity201
8.3.1 Canonicaltransformationswithtimebeingtheindependentvariable201
8.3.2 Canonicaltransformationswithpositionbeingthe independentvariable202
8.3.3 Theconditionofsymplecticity204
8.4 Imageaberrationsof nth order211
8.4.1 Overallimageaberrations213
8.4.2 Theaberrationdrivingtermsofallorders217
8.4.3 Imageaberrationsduetofringefields217 Appendix222
Coefficientsofimageaberrationsof nthorder222
Equationsofmotionintime-independentfields223
Explicitparticletrajectoriesinradiallyinhomogeneouselectromagnetic sectorfields224 References228
9.Designofparticlespectrometersandbeamguidelines229
9.1 Single-sectormagneticspectrometers229
9.1.1 Ionbeamsfrompoint-likeandslit-likeionsources232
9.1.2 Asectormagnetionanalyzerassistedbyonequadrupolelens233
9.1.3 Aphase-spaceadapterforionbeamsinsectormagnets234
9.2 Aqualityfactorforparticlespectrometers235
9.2.1 Arigidity-or QΔ -value236
9.2.2 Anenergy QK -valueandamass Qm -value240
9.2.3 A Q-valuefor N cascadedspectrometers241
9.2.4 Achromaticsystems242
9.3 Time-of-flightparticlespectrometers245
9.3.1 A Qt -valuefortime-of-flightmassanalyzers247
9.3.2 Isochronousopticalsystems248
9.3.3 Angleandenergy-focusingisochronousopticalsystems249
9.4 Thealignmentofanopticalsystemandthecorrectionofits aberrations251
9.4.1 Thecorrectionofthesecond-orderapertureaberrationsin magneticsystems253
9.4.2 Thecorrectionofimageaberrationsinelectrostaticsystems263 References265
10.Time-of-flightmassspectrographs267
10.1 Time-of-flightmassspectrographsthatusesectorfields269
10.1.1 Time-of-flightmassspectrographsthatusemagnetic sectorfields269
10.1.2 Acceleratorstorageringsusedastime-of-flightmass spectrographs270
10.1.3 Determiningthepassagetimeforhigh-energyions271
10.2 Time-of-flightmassspectrographsthatuseelectrostaticsectorfields272
10.3 Low-energyionbeamsintime-of-flightmassspectrographs275
10.3.1 Ioncoolinginaquadrupoleiontrap275
10.3.2 Coolingionsinandextractingthemoutofaflattrap277
10.3.3 RF-carpetstointroduceionsintocoolers277
10.4 Acceleratingandbunchingofionsinpulsedelectricfields278
10.4.1 Acceleratingandbunchingbyasinglepulsedfield279
10.4.2 Ionbunchingtoadesiredposition281
10.4.3 Delayedionextraction283
10.5 Energy-isochronoustime-of-flightmassspectrographsthatuse ionmirrors285
10.5.1 Energy-isochronoustime-of-flightmassspectrographsthat usegriddedionmirrors285
10.5.2 Energy-isochronoustime-of-flightmassspectrographsthat usegrid-freeionmirrors287
10.5.3 Energy-isochronoustime-of-flightmassspectrographsthat useseveralionmirrors287
10.6 Amultireflectiontime-of-flightmassspectrograph288
10.6.1 MovingionsintoandextractingthemoutofaMRTOF-MS290
10.6.2 Determiningthenumberoflapsthationshaveperformed inamultireflectiontime-of-flightmassspectrograph291
10.6.3 ThedesignofaMRTOF-MSforprecisemassdeterminations292
10.6.4 Achievablemassresolvingpowers293
10.6.5 ORBITRAP:asmallhighfrequencymultireflectiontime-of-flight massspectrograph296
References297 Index 301
Forewordtosecondedition Thesecondeditionof OpticsofChargedParticles followsthefirstedition aftermorethan30yearsandattemptstooutlineagainhowchargedparticlesmoveinmagneticandelectricfields.Inthissecondedition,newly arisenquestionsareaddressedas,forinstance,thoseofnewtime-of-flight measurements,andoldexplanationsareimprovedinsimplicityandcompletenessdescribinghowchargedparticlesmovethroughthemainand thefringefieldsofopticalsystemsandthroughindividualopticalelements. Also,itusesthesametypeofformulationandnomenclaturethroughout thebook.
Aswiththefirstedition,thesecondeditiondoesnotrequirethe readertohaveanyspecialpreknowledgeofchargedparticleoptics. However,itrequiresthatthereaderhassomebasicunderstandingofthe physicsandmathematicsequivalenttoanundergraduateeducation.The intentionofthesecondeditionisalsonottoleadthereadertoindividual mountaintopsofscientificfindings,butrathertoahighplateauofunderstandingfromwherethemountaintopscanbeseenandreachedbyhisor herownstrengthandskill.
Ihavetriedtosupplyacomprehensivesetofreferences,whichinclude theearliestpublicationsaswellasthemostimportantones.However,not allpossiblereferencesarelisted,asthiswouldhaveexceededtheavailable space.
Atthispoint,Iwouldliketoacknowledgethemanyfruitfuldiscussions andinsightsabouttheobjectofchargedparticleopticswithmycolleagues M.Berz,K.Brown,S.Dodonov,K.Halbach,A.Kalimov,H.Matsuda, T.Matsuo,S.Schepunov,P.Schury,D.Vieira,andM.Wada.Ialso extendmythankstomywifeAnnettewhoconstantlysupportedmyefforts oversomanyyears.
SantaFe,03-05-2021 HermannWollnik
Foreword Someintroductorybooksaswellasseveralarticlesinscientificjournals andresearchreportshavebeenpublishedonthesubjectofchargedparticleoptics.However,exceptforbasicaspects,itisverydifficulttoobtaina generaloverviewfromthesesources,asmostconcentrateonisolatedproblemsandnormallyusedifferentmathematicalformalisms.Furthermore, astheresultsareappliedtoquitedifferentproblems,thesolutions obtainedareoftenincompatible.Thisbookunifiessuchapproaches, resultinginadescriptionofhowchargedparticlesmoveinthemainand fringingfieldsofmagneticorelectrostaticdipoles,quadrupoles,hexapoles, etc.,usingthesametypeofformulationandconsistentnomenclature throughout.Besidesthedescriptionofparticletrajectoriesandbeam shapes,guidelinesaregivenfordesigningparticleopticalinstruments.
Thisbookdoesnotrequirethereadertohaveanyknowledgeof chargedparticleoptics;however,theequivalentofanundergraduateeducationinphysicsandmathematicsisneeded.Itiswrittenneithertocarry everyonetothemountaintopsofscientificfindingsnortostopatthe foothillsofthemountainrange.Rather,itshouldleadtheinterested readertoahighplateauofunderstandingfromwhichhecanreachthe mountaintopsbyhisownstrength.
Ihavetriedtosupplyacomprehensivesetofreferenceswitheach chapter,normallyquotinganearlyandarecentpublicationforanygiven problem.However,notallpossiblereferencesaregivensincethiswould haveexceededtheavailablespace.
Atthispoint,IwouldliketoacknowledgethemanyfruitfuldiscussionsIhavehadwithM.Berz,H.Matsuda,T.Matsuo,andH.Nestle. IamalsogreatlyindebtedtoR.Kosempelfortheskilledandcareful drawingofthemanyfiguresinthisbookandtoM.Gowansfortheexperiencedandpatienttypingandretypingofthemanuscript.
Gaussianopticsandtransfer matrices Chargedparticleopticsisverysimilartolightoptics.Therefore,inthis chapter,wemainlydescribetheeffectspresentinlight-opticalsystems. Theresultsobtainedcanlaterbeappliedtothediscussionofoptical systemsforchargedparticles.
1.1Themethodoftransfermatrices Forgeometriclightoptics,ithasbeencustomarysincethetimeof Newtontouseanalgebraicformulationforallequationsinvolved;however,inrecentyearsthismethodhasatleastpartiallybeenamendedby theuseof transfermatrices.Forsimpleopticalsystems,theuseoftransfer matriceshasnoparticularadvantage.Forcomplexsystems,ontheother hand,itoffersanunexcelledsimplicityandclarity,ashasbeenshown forparticleopticsby Cotte(1938), Penner(1961), Brownetal.(1964), and Wollnik(1967) aswellasby Herzberger(1958) or Halbach(1964) forlightoptics.
1.1.1Thedescriptionofstraightraysinopticalsystems Assumethe z-axisofaCartesiancoordinatesystemtorepresenttheoptic axisofabundleofraysinwhichcasethedeviationfromtheoptic axis-the z-axis-ofanystraightrayofthisbundlecanbedescribed (see Fig.1.1)by
OpticsofChargedParticles
DOI: https://doi.org/10.1016/B978-0-12-818652-7.00004-7
with l 5 z2 z1 .Thoughnotimportanthere,itisgenerallyusefulto describenotonlythedeviations xzðÞ; yðzÞ butalsotheanglesof 1
optic axis z
profile-plane-1 at profile-plane-2 at
Figure1.1 Thedeviationofastraightrayfromtheopticaxispassingobliquely throughtwoprofile-planeslocatedat z1 and z2 from x1 ; y1 to x2 ; y2 .
inclinations α z ðÞ; β ðzÞ ofarayrelativetotheopticaxis.Inaregionofa constantrefractiveindextheseinclinationsstayconstant
tanα z2 ðÞ 5 tanα z1 ðÞ orabbreviatedtanα2 5 tanα1 (1.1c) tanβ z2 ðÞ 5 tanβ z1 ðÞ orabbreviatedtanβ 2 5 tanβ 1 (1.1d)
The Eqs.(1.1) canalsobewritteninamatrixnotationandsodescribe astraightrayasitmovesthrougharegionoflength l 5 z2 z1 ofconstantrefractiveindex:
Herein a tanα and b tanβ denoteinclinationsofrayswithrespectto theopticaxis,whereintheseinclinationsareonlyusedasabbreviationsfor tanα andtanβ .Onlyin Section1.3,moredetailsareoutlined.
Thematrixnotationsasusedin Eqs.(1.2) provideagoodoverview overthedetailsofanopticalsystemandprovideacompactdescription.In detailthefactisexpressedthatthereare
Position-vectorsX1 (x1, a1)and Y1 (y1, b1)inprofile-plane1, thatis,atposition z1,aretransformedby x-and y-transfermatricestonewposition-vectors X2 (x2,a2)andY2 (y2, b2)inprofile-plane2,thatis,atposition z2.
asisillustratedin Fig.1.1.Incaseofrotationallysymmetricsystems,the x-and y-transfermatricesareidentical.Asinthischapter,onlyrotationally symmetricsystemsaredescribed,itissufficienttodescribeonlytheprojectionofraysontothe xz-plane,asisdoneherein Eq.(1.2a) oronto the yz-planeasisdoneherein Eq.(1.2b)
1.1.2Propertiesofthinlenses Athinlensisdefinedasaninfinitesimallythindevicethatcausesabundleofparallelrays,thatis,abundleoflightrays,tobefocusedtoasocalledfocalpoint(see Fig.1.2 ).Indetail,araythatarrivesatthe profile-plane-2at z2 justupstreamofathinlensadistance x2 5 xðz2 Þ awayfromtheopticaxiswithaninclination a2 5 az2 ðÞ tanα2 and movestotheprofile-plane-3at z3 ,itkeepsthepositionx 2 ,butisbent byanangle x2 =f towardtheopticaxisincaseofafocusinglens(see Fig.1.2 ),sothat
Herein f istheso-calledfocallengthofthethinlens. Eq.(1.3) canalsobe writteninamatrixnotationas
Figure1.2 Athinlensthatfocusesabundleofparallelstraightraystoapointis shown.Incasetheincomingraysareaxis-parallel,theraysarefocusedtoapointon theopticaxis,theso-calledfocalpoint.Incasetheincomingraysareobliquetothe opticaxis,thispointisoffthe z-axisandlocatedapproximatelyinthefocalplane, whichisperpendiculartotheopticaxisandincludesthefocalpoint.
Thetransfermatrixof Eq.(1.4) describestheactionofathinlens, whentheinitialraysareparalleltothe z-axis.However,thistransfer matrixisalsocorrectforthecasethattheinitialraysareinclinedrelative totheopticaxis.Inthiscasethebend Δα issimplyaddedtotheinitial inclination a2 asisshowninthesecondpartof Fig.1.2
Noteherethatthetransfermatrixof Eq.(1.4) expressesallpropertiesof athinlenstofirstorderin x andin a andthatthefocallength f orbetter therefractivepower1=f istheonlyquantitythatcharacterizesthepropertiesofthethinlens.Notealsothataray,thatpassesthroughthemiddleof thethinlens,passesunbentsothatfor x2 5 0 Eq.(1.4) yields a3 5 a2 .
1.2Anopticalsystemthatcontainsonethinlens Asimpleopticalsystemconsistsofonethinlens(see Eq.1.5)of focallength f anddriftregions(see Eqs.1.2) oflengths l1 upstreamand l2 downstreamofthethinlens.Suchanopticalsystemissketchedin Fig.1.3 characterizedby6position-vectors:
1. X1 ðx1 ; a1 Þ at z1 ; thatis,adistance w1 upstreamofthefocal-plane1
2. X2 ðx2 ; a2 Þ at z2 ,thatis,atthepositionofthefocal-plane1
3. X3 ðx3 ; a3 Þ at z3 ; thatis,ontheupstreamsideofthethinlens
4. X4 x4 ; a4 ðÞ at z4 ; thatis,onthedownstreamsideofthethinlens
5. X5 x5 ; a5 ðÞ at z5 ; thatis,atthepositionofthefocal-plane2
6. X6 x6 ; a6 ðÞ at z6 ; thatis,adistance w2 downstreamofthefocal-plane2
profile-plane 1
profile-plane 2 =focal-plane 1
profile-plane 6
optic axis z
profile-plane 5 =focal-plane 2
3 profile-plane 4
Figure1.3 Sketchedisanarbitraryraythatpassesthrougharegionthatincludesa thinlens.Shownarealsotheray’s x -positionsand a-inclinationsatdifferent z-positionsalongtheopticaxis.
Theseposition-vectors X1 ; X2 ; X3 ; X4 ; X5 ; X6 describeanarbitraryray (see Fig.1.3)byusingrepeatedly Eqs.(1.2a) and (1.4):
Asknownfromclassicalmatrixalgebra,itispossibletoobtainthe positionvector X6 x6 ; a6 byapplyingtheposition-vector X1 x1 ; a1 to theproductofthelisted5transfermatricessothat:
orexplicitly
Advantageously,themiddlethreetransfermatricesarecombinedtoa singlematrixthatdescribestheionmotionfromthefocal-plane1, upstreamofthethinlens,tothefocal-plane2,downstreamofthethin lens,(see Fig.1.3):
Usingthetransfermatrix Tff Eq.(1.5) canberewrittenas x6 a6 5 x
For w1 w2 5 f 2 thematrixelement ðxjaÞ vanishesandallraysthatstart fromapoint x1 5 xðz1 Þ underdifferentinclinations a1 meetagaininone point x6 5 xjx ðÞx1 .Inthiscaseonespeaksofan “object-image relationbetweentheprofileplanesat z1 and z6 ”.Thisrelation,the so-called “lensequation” isknownsincealongtimeandhasbeen reportedalreadyinNewton’sPRINCIPIAin1685.
Noteherethatthedeterminantsofallsofarshowntransfermatrices havethemagnitudeof1,thatis
orexplicitly xjx ðÞ aja ðÞ xja ðÞ ajx ðÞ 5 1.Thispropertyholdsverygenerallyfor anytransfermatrixthatdescribestrajectoriesofchargedparticlesmoving throughanopticalsystem,inwhichtherefractiveindexforlightopticalsystemsortheelectricpotentialforsystems ofchargedparticlesisidenticalatthe beginningandattheend.ThereasonforthispropertyisthatparticletrajectoriesarealwaysdeterminedfromHamiltonianequationsofmotion,inwhich casetheobtainedsolutionsforparticlemotionsarecalledtobesymplectic.
1.2.1Theformationofarealimageorofavirtualimage Fromthecondition xja ðÞ 5 0in Eq.(1.7),whichestablishesanobjectimagerelationbetweentheprofileplanesat z1 and z6 in Eq.(1.7) or z1 and z2 in Fig.1.4,theso-called lensequation: w1 w2 5 f 2 or 1 l1 1 1 l2 5 1 f (1.9) isderived,forwhich w1 w2 isalsoreplacedby(l1 f Þðl2 f)asonemay takefrom Fig.1.4
2
Figure1.4 Theformationoftherealimageofanobjectbyathinlensoffocal length f .Notethatthecenterrayofonegroupofrayspassesunbentthrough thecenterofthethinlensandthatthecenterrayofanothergrouppasses throughthefirstfocalpointandthenisparalleltotheopticaxisdownstreamof thethinlens.
Forthecasethatthe Eq.(1.9) arevalidthe Eqs.(1.5) and (1.7) transformto
describingthattheimage x6 is xjx ðÞ timeslargerthantheobjectofsize x1 .Noteherethat xjx ðÞ 52 w2 =f 5 1 l2 =f isthesocalledlateralmagnification Mx ,whichisalwaysnegativeaslongas l2 . f .Inotherwords, theimage x6 isalwaysupsidedownwithrespecttotheobject x1 asis shownin Fig.1.4
Incaseofafocusingthinlenswith f . 0,onecanreadfrom Eq.(1.10b) that,whentheobjectislocatedat l1 with 1. l1 , f avirtualmagnifieduprightimageisformed,locatedat l2 . f Incaseoflightrayssuchopticalsystemsarecalled “loups” or “magnifyingglasses.”
2. 2f . l1 . f arealmagnifiedupside-downimageisformed,locatedat l2 . 2f
3. l1 . 2f arealdemagnifiedupside-downimageisformed,locatedat 2f . l2 . f (see Fig.1.5).
2
Figure1.5 Theformationofavirtualimageofanobjectbyalightopticthinlensof focallength f isshown.Noteherethatanobliqueraythatstartsfromx1 canpass unbentthroughthecenterofthethinlensandthatanaxis-parallelraythatstarts fromx1 isbentbythethinlenssothatitpassesthroughthedownstreamfocal point.Inthiscaseallraysseemtooriginatefromanenlargedvirtualimagelocated inbackoftheobject.
1.3Generalopticalsystems Therelationbetweenpositionvectors Xm1n and Xm inprofile planesat zm1n and zm canbewrittenforanarbitraryopticalsystemas
1n
(1.11) with xm and xm1n describinghowmucharayisapartfromtheopticaxis andfurthermorewhatinclinations am and am1n ithasrelativetotheoptic axis.Suchatransfermatrixalsodescribestheactionofagenerallens system,inwhichraysarebenttowardtheopticaxisincaseoffocusing lensesorawayfromitincaseofdefocusinglenses.Whenspeakingof inclinations,itisirrelevantforallfirst-orderdiscussions,whetheronetalks ofinclinations a; b oroftanα; tanβ with α; β beinganglesofinclinations. However,itisgoodtounderstandthattheinclinationsaredefinedas ratiosofcomponentsofparticlevelocitiesorincaseoflightraysofvelocitiesofshortpulsesoflight:
Herein v isthevelocityofaparticleorofashortlightpulseand vx ; vy ; vz arecomponentsthereof.Explicitlytheserelationscanbewrittenas
Also,itissufficienttoconsiderthatlensesdeflectparticletrajectories orlightraysproportionallytothedistancetheyhavefromtheopticaxis atthepositionofthelens,thatis,theproportionalityconstantto x is1. However,verygenerallythisproportionalitycouldalsobenonlinear.
Definingittobe k2 wouldyield:
wherein kðzÞ canalsovarywith z.Thisgeneralpropagationequationis namedaftertheastronomerGeorgeHill1 asHill’sequation.Assuming that kðzÞ ispiecewiseconstant,andthatthedifferential Eq.(1.14) issolved foreachsectionof z withconstant k2 . 0,itisfoundthat xzðÞ 5 c ðÞcos kz ðÞ 1 ðd Þsin kz ðÞ
Introducingtheinitialconditions xz1 ðÞ 5 x1 and az1 ðÞ 5 a1 intothese relationsthecoefficients c and d aredeterminedandthe Eqs.(1.15) can bewrittenfor k2 . 0with k 5 kðzÞ: xðzÞ aðzÞ 5 cosðkzÞ ksinðkzÞ k 1 sinðkzÞ cosðkzÞ x1 a1 (1.16)
Noteherethatfor k2 , 0thefunctionssinðkzÞ andcosðkzÞ arereplaced bysinh kz ðÞ andcosh kz ðÞ
Thetransfermatricesof Eq.(1.16) aresolutionsofHill’sequationund thusthemostgeneralfirst-orderdescriptionsoftrajectoriesofcharged
1 Throughoutthisbookanequation d 2 x=dz2 52 k2 ðzÞx willbecalledHill’sequation,thoughin mostpublicationsHill’snameisonlyusedif kðzÞ isperiodicwith z
particlespassingthroughvaryingpotentialsaslongasthepotentialsat z1 and z2 areidentical.
1.3.1Thesignificanceofthedisappearanceofelementsof transfermatrices
Tounderstandthemeaningofthedifferentmatrixelementsforageneral opticalsysteminmoredetail,itisusefultodiscuss,whattheirindividual disappearanceimplies(Halbach,1964).Noteherethatthedeterminantof anytransfermatrixequalsunity(see Eq.1.8)sothatnevermorethantwo diagonalmatrixelementscanvanishsimultaneously,thatis,either xjx ðÞ 5 aja ðÞ 5 0oralternatively xja ðÞ 5 ajx ðÞ 5 0.
1.3.1.1Theconditions x ja ðÞ 5 0and/or ðajx Þ 5 0inatransfermatrix
For xja ðÞmn 5 0 thefirstrowof Eq.(1.11) reads xm1n 5 xjx ðÞmn xm describing that xm1n doesnotdependon am .Consequently,allraysthatstartedunder differentinclinations am fromonepointontheprofileplaneat zm ,meet againatonepointontheprofileplaneat zm1n .Thismeansthatthereisan object-imagerelationbetweenprofileplanesat zm and zm1n withalateral magnification Mx 52 xjx ðÞmn .Thissituationisillustratedin Fig.1.6A,where thestartandtheendpointsaresituatedonthe z-axis.
the case ( | ) =0
(B) the case ( | ) =0
the case ( | ) =0
(D)the case ( | ) =0
Figure1.6 Schematicrepresentationsofopticalsystems,whicharereferredtoas(A) beingparallel-to-pointfocusing,when x jx ðÞmn 5 0,(B)beingpoint-to-pointfocusing, when x ja ðÞmn 5 0,(C)havingnofocusingpower,when ajx ðÞmn 5 0,(D)beingpointto-parallelfocusing,when aja ðÞmn 5 0.
(C)
(A)
For ajx ðÞmn 5 0 thesecondrowof Eq.(1.11) reads am1n 5 aja ðÞmn am describingthat am1n doesnotdependon xm .Consequently,abundleof parallelrays,whichattheprofileplaneat zm areallinclinedbytheangle am relativetothe z-axis,aretransformedtoabundleofparallelraysat theprofileplaneat zm1n withallraysbeinginclinedbyanangle am1n 5 aja ðÞmn am orinotherwordsmagnifiedbyanangularmagnification Ma 52 aja ðÞmn .Thissituationisillustratedin Fig.1.6B,inwhichinitial andfinalbundlesofparallelraysareshown.
1.3.1.2Theconditions x jx ðÞ 5 0 and/or ðajaÞ 5 0 inatransfermatrix
For xjx ðÞmn 5 0 thefirstrowof Eq.(1.11) reads xm1n 5 xja ðÞmn am describingthat xm1n doesnotdependon xm .Consequently,abundleofparallel rays,whichattheprofileplaneat zm formsabundleofparallelraysareall inclinedbythesameangle am relativetothe z-axisandwillbefocusedto thesamepointintheprofileplaneat zm1n : Thismeansthattheprofile planeat zm1n isthesecondfocalplaneoftheopticalsystem.Thissituation isillustratedin Fig.1.6C,forthecasethattheinitialraysareallparallelto the z-axis,sothatthefinalfocusissituatedonthe z-axis.
For aja ðÞmn 5 0 thesecondrowof Eq.(1.11) reads am1n 5 ajx ðÞmn xm describingthefactthat am1n doesnotdependon am .Consequentlyallrays, thatstartedfromonepointunderdifferentanglesofinclination am fromthe profileplaneat zm ; willbeparallelafteradistance f downstreamoftheprofile planeat zm1n .Thissituationmayalsobeunderstoodastheprofileplaneat zm beingthefirstfocalplaneofthecorrespondingopticalsystem.Thissituationis illustratedin Fig.1.6D forthecasethatallraysstartedfromonepointonthe z-axis,sothattheraysofthefinalbundleareallparalleltothe z-axis.
1.4Transfermatricesoflensmultiplets Lensmultipletscanconsistoftwo,three,four....individuallenses,and itspropertiescanbedeterminedfromtheproductofthetransfermatricesof allsubparts.Thefirststepininvestigatingsuchlensmultipletscanbetoestablishatransfermatrixthatconnectsthemultiplet’sentrancefocal-plane-1toits exitfocal-plane-2(seealso Fig.1.7)as xjx ðÞff ajx ðÞff xja ðÞff aja ðÞff 5 1 0
principal-plane 1 principal-plane 2
focal-plane 1
2
Figure1.7 Apoint-to-pointfocusinglensmultipletisshown,inwhichthedistance betweenthetwofocalplanesis2f 1 dpp ,with dpp beingthedistancebetweenthe twoprincipleplanesofalensmultiplet.
Hereinthecentermatrixdescribesthepropertiesofthelensmultipletandthe driftlengths r1 ; r2 mustbechosensothat xjx ðÞff 5 aja ðÞff 5 0.Thisresultsin
(1.18a)
Experimentally r1 ; r2 couldbeobtainedforanexistinglensmultiplet bysendingaparallelbeamthroughthemultipletfromtheleftandfrom therightsideandobservethepositionsofthefocuspointsontheopposite sideofthelensmultiplet.
1.4.1Thetransfermatrixofadoubletoftwothinlenses Incaseofathinlensdoublet(see Fig.1.8),theoveralltransfermatrix fromanobjecttoanimageisdescribedby Eq.(1.17),whereinthemiddle transfermatrixof Eq.(1.17) is
Introducingtheelementsofthistransfermatrixintothe Eqs.(1.18) onefindsthefocallength fd ofthequadrupoledoubletandthedistances r1 ; r2 betweenthedoubletfocallengthand(see Fig.1.8)thepositionsof thefocalplanesoflens-1andoflens-2
with d 5 f1 1 f2 1 dff asonemaytakefrom Fig.1.8.Noteherethat fd can bevariedbyvaryingthedistance d betweenthelenses.Notefurtherthatfor fd tobepositiveoneorallthreeofthequantities dff ; f1 ; f2 mustbenegative.
planes-lens-1 principle-planes of doublet focal-planes of
planes-lens-2
Figure1.8 Alensdoubletconsistingoftwofocusingthinlenses.
Thedistances l1fd and l2fd betweenthefocalplanesofthedoubletand thepositionsofthefirstandthesecond thinlenses,respectively,arefoundas
andthedistances l1pd and l2pd betweentheprincipleplanesofthethin lensdoubletandthefocalplanesarefoundas
Importantisalsothatthedistance dppd betweenthetwoprinciple planesofthedoubletis
1.4.1.1Thetelescopeaspeciallensdoublet
Awell-knownlensdoubletisatelescope,whichbecameimportant alreadyseveral100yearsagotoobserveobjectsthatarelongdistances away.Suchdevicesformasmallrealimageofadistantobject,whichis
observedthroughaloupeormagnifyingglass(see Fig.1.5).Thus,the twolens-systemachievesanobject-imagerelationbetweenthedistant objectandthevirtualimage,whichrequires xja ðÞff 5 0forthetransfer matrixofthelensdoublet.Also,asisshownin Fig.1.9,theincoming lightraysthatoriginatedfromonepointofthedistantobjectforma quasi-parallelbeam,whichshouldbetransformedbythetelescopeintoan outgoingparallelbeam,sothattheincominglightiscompletelypassedto theobserver.Suchasituationwasalreadyshownin Fig.1.6B and discussedasbeingcausedby ajx ðÞff 5 0.
axis
Figure1.9 Twotelescopesareshowneachoneofwhichconsistsoftwolenses,a firstso-calledobjectivelensandasecondsocalledeyepieceorocular.Thereisa Keplertelescope,whichhasanegativeangularmagnification a2 =a1 andforwhich bothlensesarefocusing,andaGalileantelescope,whichhasapositiveangularmagnification a2 =a1 andforwhichoneofthelensesisfocusingwhiletheotherisdefocusing.NotethatforhighperformanceapplicationstheKeplertelescopeispreferred whileGalileantelescopesareused,whencompactdevicesaredemanded.
Applyingtheconditionthattheoutgoingraysformaparallelbeam, whentheincomingbeamisaparallelbeamto Eq.(1.19) resultsin dff 5 0,whichisthesameasstatingthatthesecondfocalplaneofthe objectlensandthefirstfocalplaneoftheeye-pieceorocularcoincide. Forasystemtoobservefarawayobjectsthetelescopemustenlargethe
inclination a1 oftheincomingbeamtoamagnifiedinclination aja ðÞa1 of theoutgoingbeamwiththeratioofthetwoinclinationsbeing
Toachieveahighangularmagnification, f1 mustbelargeand f2 mustbesmall. Noteherethatthediameteroflens-1,theobjectlens,limitsthe diameter D1 oftheincomingparallelbeamandthatthetelescopeshould bedesignedsothatthediameter D2 ofthefinalparallelbeamissmaller thanthediameteroftheobserver’seyepupil.Notefurtherthatthe amountoflightthatiscollectedbytheeyeoftheobserverisincreasedby thetelescope’ s ðD1 =D2 Þ2 ,whichmakesitadvantageoustochooselarge diametersfortheobjectlens.
Therearetwosolutionsforatelescope.Thereistheso-calledastronomicalorKeplertelescopeforwhich f1 . 0and f2 . 0and Ma isnegative,thatis,theobjectisseenupsidedown.ThereisalsotheDutchor Galileantelescope,forwhich f1 . 0and f2 , 0and Ma ispositive,thatis, theobjectisseenupright.Inbothcasestheusedlensesshouldbewell designedlensmultiplets,sothattheimageaberrationsofthelensesdonot limittheresolvingpowerofthetelescope.
1.4.1.2Lensesoflongandshortfocallengthsforphotographic cameras
Forsomeapplicationsoflensdoubletsonedefocusingandonefocusing lensarecombined,sothatthefocallength fd ofthelensdoubletispositive onlyfor dff , 0accordingto Eq.(1.20a).Consequently,thetwolenses arerelativelyclosetoeachotherandaccordingto Eq.(1.21c) theprincipleplanesareshiftedtowardthesideofthepositivelensasisshownin Fig.1.10.Thispropertyofadoubletcanbeusedtoconstructlensesfor whichtheprincipalandthefocalplanesareshiftedtosuitablepositions relativetothephysicaldimensionsofthelenses.Oneexampleofsucha designisatele-lensthatcanbeusedforaphotographiccamera.Insucha cameratheimageofaverydistantobject,thatis, w1fd N,issituated ataposition w2fd 5 f 2 d =w1fd closebyanddownstreamofthefocal-plane-2 ofthecameralens.Theimageofadistantobjectislocatedatthesecond focalpaneofthedoubletanditssizeisthemagnitudeoftheobject multipliedbythelateralmagnification Mx 5 fd =w1fd ofthedoublet. Tokeepthismagnificationwithinlimitsandstillallowtoobservedetails oftheimage,itisadvantageoustousealensoflongfocallength fd .On theotherhand,suchalensislargeandinconvenienttocarry.Asolution
