D'Amico, T E

Guignard, Gilbert

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 External bending magnetExternal bending magnetCentre bending magnetρ φ1 1 ρ φ2 2 ρ φ1 1 External bendingmagnetDefocusingquadrupoleof strength k HalfCentre bendingmagnetFocusingquadrupoleof strength k L1 1L22L3A NEW FAMIL Y OF ISOCHR ONOUS AR CSG.GuignardandE.T.d’Amico,CERN, 1211Geneva 23,SwitzerlandFor theCompact LinearCollider(CLIC),thebunchtimestructureshouldbe preserved intheinjectorcom-plex,especiallyintherecirculationarcsandaftertheflnalbunch compressionstageup tothemain linacinjection.Atthesametime,becausethetransversemittancesaresotiny,theirgrowth,essentiallyduetosynchrotronradiation,shouldbekeptaslow aspossible.Inotherprojects,everalisochronousarcshave beendesignednumericallytomeettheserequirementsfora particularclayout.Thesede-signscannotbe easilyadaptedtodifierentconflgurations.The purposeofthisstudyisto obtainanalyticallythemainparametersofa new classofisochronousarcswhichcanbe quicklytailoredtospecialapplications.Some ofthesearepresentedand theyemphasizethesmalltrans-verseemittancegrowth achievableeven atlargeinjectionenergywhilekeepingthearcradiusina reasonablerange.Becauselocallytheflrst-orderanisochronicity isfullycan-celled,higher-ordercontributionsarelessimportant thaninotherdesigns.I.INTRODUCTIONIn theCompact LinearCollider(CLIC)many consid-erations(wake-fleldefiects,highluminosity)requirethatthebunch timestructureshouldbepreservedafterthelastbunch compressionhastakenplace.Thisconditioni gen-eralcannotbe fulfllledwhen thebeam passesthroughade°ectingsystembecauseofthedifierenceinlengthbe-tweentheindividualorbitsduetotheenergyspreadandto thedifierent initialconditions.The systemiscalledisochronouswhenitdoesnotchangethebunch timestruc-ture.Itcanbeproved[1]thatinthelinearapproximationsuch a systemshouldbenondispersiveandsuch that:Z S2S1D (s)‰(s)ds= 0 (1)whereD (s)isthehorizontaldispersion,‰(s)theradiusofcurvatureandS1,S2 arethepositionsofthebeginningandendoftheinsertion.The relation(1)shows thatcontributionstotheinte-gralcome onlyfromde°ectingmagnetsand ofi-centredquadrupoles.Severalschemesofisochronousarcshavebeendeveloped[2],[3].They arebasedon latticesncompassingseveralde°ectingmagnetswheretheintegral(1)isminimizednu-mericallyoverthewholearc.Thepurposeofthistudywastoinvestigateanalyticallyanisochronousmodulewiththeminimum numberofde°ectingmagnets.Thejuxtapositionofidenticalmodulesallows thebuildingup ofa wholefam-ilyofisochronousarcsdependingupon some parameterswhich canbe adjustedtomeetspecialdesignconstraints,such asminimizationftheemittancegrowth duetosyn-chrotronradiation.Itcanbe proved [1]thattheminimum numberofde-°ectingmagnetsinan isochronousmoduleisthree.Forreasonsofsimplicity we have chosena symmetricmoduleaboutthemid-planeofthecentralde°ectingmagnet.II.ISOCHR ONICITY CONDITIONLetusconsideranisochronousinsertionwiththreebend-ingmagnets(seeFig.1),whereweneglectforthemomentthepresenceofothermagneticelementsassumedtobeper-fectlycentred.To simplifythealgebrathebendingmag-netswillbe treatedassectormagnetsofthesamelengthbutofdifierent curvatureradii‰1 and ‰2,thede°ectionanglesbeingrespectively` 1 and`2.Figure1:Isochronousinsertion:bendingmagnetconflguration.Assumingthatthedispersionanditsderivativearezeroattheentranceoftheflrstmagnet,itiseasytoshow thattheisochronicity andsymmetryconditionsyieldthefollow-ingexpressionsforthedispersionanditsderivative attheentranceofthecentremagnet[4]:D j = ‰2£D 0jctn(` 2=2)+ 1⁄D 0j = ¡‰1‰2µ32`1 ¡ sin` 1¶: (2)III.INSERTION DESIGNTo transportthebeam throughtheinsertiondescribedinFig.1,wehavetoaddquadrupolesbetweenthebendingmagnets.The simplestconflgurationisa FODO, asshowninFig.2 whereonlya half-insertionisdrawn.Figure2:Layoutofhalfisochronousinsertion.The threespacesL1;L2;L3 and thetwo quadrupolestrengthsk1;k2 havetobechoseninorderfortheexpres-sions(2)tobe satisfled.Aftersomemanipulationfthetransfermatrices(seeAppendixA ofreference[4])thefol-lowingexpressionsforthethreedriftlengthsasfunctions          Table1:Permittedrangesofk1;k2;¢ L3q2 < q1 ¡ d and¢ L3 > Maxfd¡ D j=D 0j;¡ q2gk1 • Minfk(1)1 ;kmax g q2 > q1 ¡ d andMaxfd¡ D j=D 0j;¡ q2g < ¢ L3 < ¢ L(1)3k(1)1 < k1 • Minfk(2)1 ;kmax gpk2 > Max‰acosh(Max f1;C⁄2 g)lq;qk(1)2¾andMaxfd¡ D j=D 0j;¢ L(2)3 g < ¢ L3 < ¢ L(1)3k(1)1 < k1 • Minfk(2)1 ;k(3)1 ;kmax gpk2 <acosh(C ⁄2 )lqandMaxfd¡ D j=D 0j;¢ L(2)3 g < ¢ L3 < ¢ L(1)3k(2)1 < k1 < Minfk(3)1 ;kmax gpk2 < Min‰qk(1)2 ;acosh(C ⁄2 )L q¾andMaxfd¡ D j=D 0j;¢ L(2)3 g < ¢ L3 < ¢ L(1)3k(1)1 < k1 < kmax q2 < q1 ¡ d and¢ L3 > Maxfd¡ D j=D 0j;¢ L(2)3 gwherek(1)1 ;k(2)1 ;k(3)1 ;k(1)2 arethesolutionsofthefollowingtranscendentalequations:(l+ d)qk(1)1 tan(Lqqk(1)1 )= 1;qk(2)1 =cos(L qpk(2)1 )(l+2 d)+r hlcos(L qpk(2)1 )i2+4 d(l+ d)2sin(L qpk(2)1 )d(l+ d)(l+ d)qk(3)1 sin(Lqqk(3)1 )¡ cos(Lqqk(3)1 )= a;qk(1)2 sinh(lqqk(1)2 )¡pk⁄2flflflflcosh(lqqk(1)2 )¡ C ⁄2flflflfl= 0andC ⁄2;k⁄2;kmax ;¢ L(1)3 ;¢ L(2)3 ,aregivenby theexpressionsC ⁄2 =aq1C 1(l¡ q1+ d);k⁄2 =1hd¡ q1‡1+C ⁄2aC 1·i2 ;kmax = …4L 2q ; ¢ L(1)3 =bd+ q2¡ q1 ¡ q2;¢ L(2)3 =q2C 1aq1C 2(l¡ q1 + d)¡ q2ofk1;k2 and ofthefreeparameter¢ L3 = L3 ¡ D j=D 0j,may beobtained:L1 = aC2q1C1q2(¢L3 + q2)¡ l+ q1L2 = q1 ¡ q2 +b¢ L3 + q2L3 = D j=D 0j + ¢ L3 (3)wherel = ‰1 tan(` 1=2); a = ¡D 0j=sin(` 1);b = q2C 2‡q2C 2+q1aC 1·; qi =CiSipki;C1 = cos(Lqpk1); S1 = sin(Lqpk1);C2 = cosh(Lqpk2); S2 = sinh(Lqpk2)(4)Lq beingthequadrupolelength.Table1 givesa subsetoftherangesofk1;k2;¢ L3 forwhich thethreedriftlengthsarelargerthana givenvalued,when1q1•1l+ d+1Lq=2+ d:Thiscanbe shown tobe thecaseformostoftheusualhardwareconflgurations.The fullsetofconditionsmay befoundinAppendixC of[4].IV.AR C DESIGNTobuildupanarcwehavetoconnectasmany insertionsasarenecessarytoobtainthedesiredde°ection.To avoidlargexcursionsofthebetatronfunctions,theeasiestwayistotakeadvantageoftheinsertionsymmetryandtoensurethatthevaluesoftheTwissparametersatbothendsofamodulecomposedofan insertionasdescribed above andofa matchingsectionarethesame.Itiseasytoshow thatthisispossibleonlywhen thebetatronfunctionand itsderivativeatbothendsofsuch a modulearerespectively:fl0 =p1¡ m 2=jm 21j and fl00 = 0 (5)wherem = m 11 = m 22 and m 21 aretheelementsofthetransfermatrixforthemodule.Itisverydi–cultodowithoutthematchingsectionwhilesatisfyingthesecon-straintsinbothplanes.W e have preferredtochooseasamatchingsectionhalfatripleta bothendsoftheinsertiontoobtaina modulewith¡1 < m < 1 inbothplanes.TheTwissparametersattheendofthetransferlineinjectinginthearcshouldthenbe matchedtothevaluesgivenbytheexpressions(5).Inordertoreducetoa minimum thecontributionfmagneticerrorsand thesextupoleefiectswe addtheconditionthatthephaseadvanceovera smallnumberofmodulesshouldbe an integerm ultipleof… inbothplanes.Aftersomemanipulationsitispossibletoshow thatthegrowth ofthenormalizedhorizontalemittance¢ °† x isingood approximationinverselyproportionaltothefourthpowerofthenumberofmodulesrequiredtoassembleanarc[4].The diameterofa full-circlearcisofcoursepro-portionaltothenumberofmodules.Clearlya compromisem ustbe foundbetweenthesetwo veryimportant designparameters.To flnd itwe have writtena simpleinterac-tive programasan Excelspreadsheetwhich permitsonetoquicklyobtainthemainfeaturesofa 2… arcaccordingtodifierentchoicesofthenumberofrequiredmodules,oftheratiobetweentheradiiofcurvatureoftheexternalandcentralbendingmagnets,and ofthegradientsofthetwoquadrupolesandofthedistance¢ L3.V. APPLICA TIONSIneach branch ofCLIC,two 360-degreearcsareneededtoguidetheparticlesinthereversedirection,eat3GeVforthedrive beam and theotherat9 GeV forthemainbeam. Thesearcsshouldnotperturbthebunch length,which iscarefullychosenforoptimum performanceattheflnalinteractionregioninthemain linacand forpowertransfer–ciencyinthedrivelinac.Thustheyhavetobeisochronous.A preliminarystudyofthemhasbeencarriedattheflrstorderusingthetoolsdescribedintheprevioussection.The resultsaresummarizedinTable2 andFigs.3and4.The lesstringentconstraintonthehorizontalemittancegrowth forthedrive beam allows onetoobtaina smallerarcradiusthancouldbe expectedfromtheenergyscal-ingalone.Thuslargerhorizontalemittancegrowth wouldbe acceptablebutdi–cultoachieve duetolimitationsinopticsmatching.On thecontraryforthemainbeam thefractionalhor-izontalemittancegrowth (» 7:4%) cannotbe furthere-laxedtoobtainasmallerarcradiusbecauseitwouldinducea signiflcantlossofluminosity.VI.DISCUSSIONThisreportshows theexistenceofa parametricfam-ilyofisochronousarcsandanalyticalprocedurestodesignthem.Simpleinteractiveprogrammingtoolshavebeende-velopedtoimplementtheseprocedureswhich speedup thesearch ofnearoptimizedisochronousarcs.The flrst-orderanisochronicity isfullyeliminatedandthelow valuesofthedispersioncontributeto thesecond-orderfiectsas wellastolimitingthehorizontalemittancegrowth. On theotherhand,thismakesthecorrectionfthechromatic-ity withsextupolesmoredi–cultbecausetheycannotbeplacedwherethedispersionissu–cientlyhigh.This,how-ever,becomesa severeproblemwhen thearcispartofaringthroughwhich thebeam passeseveraltimes.Fur-therinvestigationswillbe aimedatlimitingtheseefiectsand studyingtheenergyspreadacceptanceofsuch arcs.Trackingshouldprovideresultson thebehaviourofthisfamilyofisochronousarcsathigherorders.Table2:Parametersofthe360-degreeisochronousarcsParameter 3 GeV arc 9 GeV arcNum berofinsertions 3 48Lengthofbendingmagnet 1.8m 1 mQuadrupolelength 0.3m 0.5mGradientofthefocusingQuad 55 T/m 60 T/mGradientofthedefocusingQuad 55 T/m 60 T/mL1 1.366m 2.068mL2 0.227m 0.925mL3 1.164m 0.310mOverallarcdiameter 15 m 214mHorizontalphaseadvance …/2 …/2Verticalphaseadvance …/3 …/3Nominal°† x (m¢rad) 5£ 10¡ 4 2:5£ 10¡ 6¢ °† x (m¢rad) 8:16£ 10¡ 6 1:84£ 10¡ 7Figure3:Opticsfunctionsofthe3 GeV isochronousmodule.Figure4:Opticsfunctionsofthe9 GeV isochronousmodule.VII.REFERENCES[1]K.G. Stefien,HighEnergyBeam Optics,New York:Interscience,1965.[2]G. Guignard,A Latticewithno Transitiona d LargeDynamicApertureinProc.13th ParticleAcceleratorCon-ference,Chicago,1989.[3]CEBAF DesignReport,Ma y 1986.[4]E.T.d’Amicoand G. Guignard,CLIC note(inprepa-ration).

A new family of isochronous arcs

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