Tolerance study in FCC-ee: Status of ver7cal dispersion ... · Status of ver7cal dispersion and coupling correc7on at 175GeV. ... On behalf of the FCC-ee Lattice Design team ... •
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TolerancestudyinFCC-ee:Statusofver7caldispersionandcouplingcorrec7onat175GeV.SandraAumonCERNOnbehalfoftheFCC-eeLatticeDesignteam.eeFACT16Workshop,Daresbury
TableofParameters Emittancetuninginelectronmachine Beamphysicsoflatticeerrors FCC-eeRacetrackandAmpliIicationfactorbyerrors. Toolsandcorrectionmethods BPMerrortoleranceinFCC-ee Resultsandstatus:
DispersionFreeSteeringinFCC-ee(Quadrupolesmisalignments)RollanglesinquadrupolesBetatronCouplingcorrectionwithresonancedrivingterms
Conclusions:statusoutlooksimmediateandfuturework.
25/10/16 eeFACT16 2
Outline
25/10/16 3
Z W H tt Beamenergy[GeV] 45.5 80 120 175 Beamcurrent[mA] 1450 152 30 6.6 Bunches/beam 91500 5260 780 81 Bunchpopulation[1011] 0.33 0.6 0.8 1.7 Transverseemittance- Horizontal[nm]- Vertical[nm]
0.090.001
0.260.001
0.610.0012
1.3
0.0025
Momentumcomp.[10-5] 0.7 0.7 0.7 0.7
BetatronfunctionatIP*- Horizontal[mm]- Vertical[mm]
10002
10002
10002
10002
Energyloss/turn[GeV] 0.03 0.33 1.67 7.55
TotalRFvoltage[GV] 0.2 0.8 3 10
Numberstoremember- 8GeVenergyloss/turn
dp/p~1%- beta_x*=1m,beta_y*=2mm- e_x=1nm,ey=2pm- Couplingratio
V.emit/H.emit~2/1000
FCC-eeBeamParameters
P =cC3E4
2RP175/P45 200
eeFACT16Daresbury
qx/qy=0.07/0.14
25/10/16 4
x
=C
g
Jx
23F FFODO =1
2sin5+3cos1 cos
LlB
L:celllengthlB:dipolelength:phaseadvance/cell
Alignmenterrorsandx/ycouplingspoiltheverticalemittanceandcompromisethecouplingratioof2/1000->betatroncouplingandDyshouldbeundercontrol.
Emittancetuningforelectronmachine
y =
dp
p
2(D2 + 2DD0 + D02)
Horizontalemittance:phaseadvanceinthearcs
Verticalemittance
D:dispersionfunction
Vert.emittancegrowth-verticaldispersion(Dy)-betatroncoupling
eeFACT16Daresbury
Quadrupolesoff-set:dipolarkick* Sextupolesoff-set* Quadrupoleroll:skewquad,couplingoftheplanes Dipoleroll:verticaldipole
25/10/16 5
Bx
= k(y +y) = ky + ky
ConstanttermVerticaldipole->verticaldispersion
Feed misalignments Tuning algorithm
Quadrupole vertical off-set (QV)
Quadrupole roll (QR)
Dipole roll (DR)
Sextupole vertical off-set (SV)
= + = + constant term (vertical dipole)
= cos = cos + sin skew quadrupole
= cos 0 = cos vertical dipole
= + = + = + =k( - ) ( )
skew quadrupole
ortogonal quad +
ortogonal quad +
horizontal dipole +
ortogonal sextupole +
Mainly emittance grows through: Betatron coupling Directly generated and vertical
non-zero closed orbit [through sexts]
Vertical dispersion Directly generated and vertical
non-zero closed orbit [through quads]
1st Workshop on Low Emittance Lattice Design J. Alabau-Gonzalvo
+ vertical dipole Skewquad(coupling)+verticaldipole
Sourceemittancegrowth*SYLeeAcceleratorPhysics,JavierBarcelonaPresentation
Latticeerrorsforthetolerancestudy
eeFACT16Daresbury
25/10/16 LowEmittanceWorkshop2015 6
12 m30 mrad
9 mMiddle straight
1570 m
FCC-hh
CommonRF
CommonRF
90/270 straight4.7 km
IP
IP
A conceptual layout of FCC-ee
0.8 m
As the separation of 3(4) rings is within 15 m,one wide tunnel may be possible around the IR.
FCC-eeRacetrackLayout
2RFsections,2IPsImportantsawtootheffect
Needtotaperthemachine
Twooptions:-K.Oideslattice->fullytapered(dipoles,quadrupoles,sextupoles)-Sectorwise(dipolesonlybygroup),lessexpensive.
FCC-eeRacetrackLayout
25/10/16 FCCweek2016-Rome 7
Interaction Region
The optics in the interaction region are asymmetric. The synchrotron radiation from the upstream dipoles are suppressed below 100 keV up to 450
m from the IP. The crab sextuples are integrated in the local chromaticity correction in the vertical plane.
RFRF
IPBeam
Local chromaticity correction + crab waist sextupoles
Local chromaticity correction
+ crab waist sextupoles
K.Oidelatticeoption(K.Oidestalk)2IPs
Ring Optics
Above are the optics for tt, *x/y = 1 m / 2 mm. 2 IP/ring. The optics for straight sections except for the IR are tentative, customizable for infection/
extraction/collimation, etc.
RFRFRF
IP IP
2Ips
Nolocalchromaticitycorrection
Ring Optics
Above are the optics for tt, *x/y = 1 m / 2 mm. 2 IP/ring. The optics for straight sections except for the IR are tentative, customizable for infection/
extraction/collimation, etc.
RFRFRF
IP IP
RacetrackfollowsofIicialdesign
50600 50700 50800 50900s(m)
500
1000
1500
2000
2500
x,y(m)Beta_y
Beta_x
LEP-likeIR
25/10/16 8
Sectorwisetapering,dipolesonly-AndreasDoblhammerMasterthesiswork
Sector-wisetapering
S(m)
X(m) --Hor.COwithradiationdampingandtapering
--Hor.COwithoutradiationdampingandtapering
eeFACT16Daresbury
25/10/16 9
AmpliIicationfactorbyerrortype
Errortype Effectonvert.orbit EffectonDyQuadV.displacement(ex:2micro)
300(LHC150)(6e-4mRMS)
1.0e6(LHC1.0e4)2mRMS,20m@IPs
Rollquad(10mrad)
25(0.2mmRMS)
SextupoleV.displacement(1microm)
Ilatbeam,ieey
BeamPhysicsChallenges
FCC-eeisacolliderwithbeamparametersofaLightSource(Ilatbeam,extremelowverticalemittance)->Hor~nm,Vert.pm!!
Howtoconserve2pmVert.Emittancewithspuriousdispersion HowtokeepcouplingratioEy/Exof2/1000
withstrongsextupoleIield(x100ink2largerthanLEP)? Howtokeeptheverticaldispersionto1mmRMS
25/10/16 10
Latticeerrorsandchallengesforthetolerancestudy
Algorithm&MethodologyChallenges Gettingalgorithmsoflatticecorrection(orbit-dispersionmatrixresponse,andcouplingcorrection)inspiredfromlightsourceexperiences,LHCorothercolliders
Apply,adaptthemto100kmcolliderwithaemittanceratioof~2/1000 MADXitselfisnotoptimalforcomputationwithRF&radiationdamping
eeFACT16Daresbury
1.Errors(transversedisplacementofquadrupoleand/ortilt)2.Roughorbitcorrections(nosextupoles)3.LocalverticaldispersioncorrectionattheIps(nosextupoles)4.DispersionFreeSteeringwithcorrectors(nosextupoles)Correctverticaldispersion5.Correctionofthechromaticity6.LocalverticaldispersioncorrectionattheIPs(sextupoles)7.DispersionFreeSteeringwithcorrectors(sextupoles)Correctverticaldispersion8.Couplingandverticaldispersioncorrectionwithskewquadrupoles.9.Tapering(fullytaperedorsectorwise)10.Opticsmatching,tunerematched.-----------------------------------------------------11.Emittancecomputation:EMITcommand(Chaoformalism)
25/10/16 11
Opticscorrectionmethodsalsouseinlightsources,LEP,LHC.
Toolsandcorrectionmethods
CorrectionmethodsdoneinPython
MADXwithoutsynchrotronradiationdamping
MADXwithradiationdamping
eeFACT16Daresbury
BPMErrorTolerance
25/10/16 12
RMS Dispx RMS Dispy
0 1 2 3 4 5 6 70.00
0.05
0.10
0.15
BPM Errors RMS @mmD
Disp
ersio
nRM
S@mD
Emitx Emity
0 1 2 3 4 5 6 710-17
10-15
10-13
10-11
10-9
10-7
BPM Errors RMS @mmD
Emitt
ance
s@m.r
adD
KatsunobuRacetracklattice,175GeV,2RF,fullytapered(BothBastiansandKatsunobuslaticehavesimilartolerancewithrespecttoBPMserrors)
Verylowtolerance,emittancegrowthdrivenbyverticaldispersion. PreferabletocorrectthevertDisp.viaaDISPERSIONFREESTEERING(DFS)ratherthanbyaorbitcorrection
~10mmVert.Dispersion0.1-0.2mattheIP
eeFACT16Daresbury
Buildnumericallyamatrixforverticalorbit(u)&dispersion(Du)responseunderacorrectorkick(al)
DispersionFreeSteering:Principle
PRST-AB 3 EMITTANCE OPTIMIZATION WITH DISPERSION FREE 121001 (2000)
(N . M) or under (N , M) constrained. In the for-mer and most frequent case, Eq. (7) cannot be solvedexactly. Instead, an approximate solution must be found,and commonly used least square algorithms minimize thequadratic residual
S ! k "u 1 A "uk2. (8)
Dispersion free steering is based on the extension ofEq. (7) to include the dispersion at the BPMs. The ex-tended linear system is
!1 2 a""u
a "Du
1
!1 2 a"A
aB
"u ! 0 , (9)
where vector "Du (dimension N) represents the dispersionat the BPMs. B is the N 3 M dispersion response matrix,its elements Bij giving the dispersion change at the ithmonitor due to a unit kick from the jth corrector. Theweight factor a is used to shift from a pure orbit (a !0) to a pure dispersion correction (a ! 1). In general,the optimum closed orbit and dispersion rms are not ofthe same magnitude and a must be adjusted for a givenmachine. a can, in principle, be evaluated from the BPMaccuracy and resolution. Applied to Eq. (9), a least squarealgorithm will minimize
S ! !1 2 a"2 k "u 1 A "uk2 1 a2k "Du 1 B "uk2. (10)
Singular response matrices are a well-known problem oforbit corrections. The singularities are related to redundantcorrectors, i.e., areas of the machine where the samplingof the orbit is insufficient. Such situations yield numeri-cally unstable solutions where large kicks are associatedto minor changes in the orbit. A standard cure consists indisabling a subset of correctors and removing the corre-sponding lines from the linear systems of Eqs. (7) and (9).Regularization can also be obtained by extending Eq. (9)to constrain the size of the kicks,0
B@!1 2 a" "u
a "Du"0
1CA 1
0@ !1 2 a"AaB
bI
1A "u ! 0 . (11)
Here "0 is a null vector of dimensionM, I is a unit matrix ofdimension M 3 M, and b is a kick weight. The quadraticresidual now contains the rms strength of the correctorkicks,
S ! !1 2 a"2 k "u 1 Auk2 1 a2k "Du 1 B "uk2
1 b2k "uk2, (12)
and large kicks are suppressed since they receive a penaltywhich can be adjusted with b.Various other constraints can be added to the linear sys-
tem to be solved, for example, to maintain a constant or-bit length or to stabilize the beam at given locations inthe ring. Adequate weight factors can be used to control
the importance of such constraints. It is also possible tocorrect the machine coupling using a similar scheme. Theorbit coupling of horizontal corrector kicks into the ver-tical plane is then minimized using skew quadrupoles ascorrecting elements [10]. To simplify the expressions inthe following sections, vector "d and matrix T are definedas
"d !
0@ !1 2 a" "ua "Du
"0
1A, T !
0@ !1 2 a"AaB
bI
1A , (13)
with
"d 1 T "u ! 0 . (14)
A. Singular value decomposition (SVD) and orbiteigenvectors
Dispersion free steering is particularly interesting inconjunction with the singular value decomposition (SVD)algorithm [11,12], because it allows a simultaneous limi-tation of the corrector kick strength. The SVD algorithmis a powerful tool to handle singular systems and to solvethem in the least square sense. For M $ N the singularvalue decomposition of matrix T has the form
T ! UWVt ! U
0BB@
w1 0 00 w2
00 0 wM
1CCAVt, (15)
where W is a diagonal M 3 M matrix with non-negativediagonal elements and Vt is the transpose of the M 3 Morthogonal matrix V,
VVt ! VtV ! I , (16)
while U is an N 3 M column-orthogonal matrix
UtU ! I . (17)
The vector "q !i", corresponding to the ith column ofmatrix V,
"q !i" !
0BB@
V1iV2i VMi
1CCA , (18)
is an eigenvector with eigenvalue w2i $ 0 of the M 3 Msymmetric matrix TtT [12,13],
TtT "q !i" ! w2i "q!i". (19)
It follows from Eq. (16) that the M vectors "q !i" form anorthonormal base of the corrector space since
! "q !i" ? "q !j"" ! "q !i"t "q !j" ! dij . (20)
121001-3 121001-3
25/10/16 13
Ai,j =
pij
2 sin(Qy)cos(|i j | Qy)
s (m)0 1000 2000 3000 4000 5000 6000
D (m
)
-0.20
-0.15
-0.10
-0.05
-0.00
0.05
0.10
0.15
0.20Vert. DispersionVert. Dispersion
) (20 5 10 15 20 25
)1/
2 (m
1/2
D
/
-0.020
-0.015
-0.010
-0.005
0.000
0.005
0.010
0.015
0.020Norm. Vert. DispersionNorm. Vert. Dispersion
Figure 2: Vertical dispersion in the SPS ring due to a kick of 0.1 mrad at correctorMDV.10307. The solid line is the MADX prediction, the points correspond to the analyticalexpression of Eq. 20 evaluated at the location of the BPMs.
strength, the horizontal dispersion at the sextupole and to the orbit offset in the sextupole.The sextupole term is obtained by replacing the K in the equation for the quadrupoles byK2Dx, see for example Ref. [4] for a rigorous treatment of the dispersion response.
When all contributions are combined the dispersion response at monitor i due to the kickfrom corrector j becomes
Bij = {quad
l
KlLll4 sin(Q)2
cos(|i l| Q) cos(|l j| Q)
sext
m
K2,mDx,mLmm4 sin(Q)2
cos(|i m| Q) cos(|m j| Q)
cos(|i j | Q)
sin(Q)}
lj (20)
where the sums run over all quadrupoles (i) and sextupoles (m). The last term is the directeffect from the corrector kick itself. Eq. 20 is valid for the horizontal plane. For the vertical
6
Orbitresponse
Dispersionresponse
EmittanceoptimizationwithdispersionfreesteeringatLEPR.Assmannetal.Phys.Rev.STAccel.Beams3,121001DispersionFreeSteeringforYASPanddispersionforTI8,J.Wenninger,LHC-Performance-Note-005.
4010BPMs2006V.correctors2004H.correctors1corr/quads
SVDanalysistosolvethesystemandIindasolution
eeFACT16Daresbury
25/10/16 14
FullDFScorrectionscheme
1. Errors(quadrupolesdisplcements)2. IPLocaldispersioncorrection
(LDC)3. Verticaldispersioncorrectionwith
DFSnosextupoles+LDC4. Chromaticitycorrection5. DFSwithsextupoles+LDC6. Sectorwisetapering7. Emittancecomputation
ForoneSEED
RomeFCCweek:toleranceat5microm.sincethen..IPstreatedseparatelyfromtherestofthemachine+DFSToleranceimprovedfrom5to30microm
eeFACT16Daresbury
Emity[pm]
FullDFScorrectionscheme
Status,sofarinsummarymaximumeffortonverticaldispersioncorrection-quadrupolesmisalignementstolerancehasbeenimprovedfrom5micromto20microm.-1/2oftheseedsproducetoohighverticalemittance.- BPMerrorsfrom5to20/30microm- LocalcorrectionattheIPscanbestillimproved.
Forasectorwisetaperingmachine
eeFACT16Daresbury
25/10/16 16
FullDFScorrec7onscheme
S(m)
X(m)
Issueswithsector-wisetapering(LEP-likeIRlattice+sectorwiseTapering)DFS+localverticaldispersioncorrectionattheIPsbringstheverticaldispersionfrom1.0e-2mdownto1.0e-5mQuestion:howdoesthesectorwisetaperinginIluencethecomputationoftheemittance?->extrasourceofdispersionwhendampingisinthesimulation?->Forafullytaperedmachine(Katsunoburacetrack),thiswontbeanissue.
Dyrms(m)
Nbofiterations
Nodamping
eeFACT16Daresbury
25/10/16 17
MADXhasbeenused. Allthecorrectionsaredonewithoutsynchrotronradiationlossinenergy
inthemagnet. EmittancecomputationinMADXbasedontheA.Chaoformalism*IfthemachinecontainsatleastoneRFcavity,andifsynchrotronradiationisenabledwithBEAM,RADIATE=true;,theEMITcommandcomputestheequilibriumemittancesandotherelectronbeamparametersusingthemethodinA.Chaopaper.Inthiscalculationtheeffectsofquadrupoles,sextupolesandoctupolesalongtheclosedorbitarealsoconsidered.EMITtakesintoaccounttheenergylossviasynchrotronradiationindipoles,quadrupoles,sextupolesetc..BUT!Opticsfunctionslikebeta-functionanddispersionfunctionarenotcorrect(TWISSmoduleis4D),onlyclosedorbitiscorrect.
Toolsandcorrectionmethods
*A.Chao.Evaluationofbeamdistributionparametersinanelectronstoragering.JournalofAppliedPhysics,50:595598,1979.
eeFACT16Daresbury
RollangleampliIicationfactor~25notbigissue
Status:20/30micromquadrupoles,50microradisacceptableastiltinquadrupoles.
25/10/16 18
Rollanglesforquadrupoles
eeFACT16Daresbury
1skewevery6FODOcells,i.e.dmux=540deg.Anddmuy=360(90/60degphaseadvance/per)
272skewsinstalledinthelatticeinthearcs(onlydispersiveplaces) BothusesforbetatroncouplingcorrectionandverticaldispersionThisschemewillbeimprovedwithlocalbetatroncouplingattheIPs.Tocorrectthebetatroncoupling,thelinearcouplingresonancedrivingtermsaremitigatedwitharesponsematrix(LHC-ESRF)
25/10/16 19
Betatroncouplingcorrectionscheme
References:-ImprovedcontrolofthebetatroncouplingintheLargeHadronCollider,T.Perssonetal,PRSTAB17,051004-Verticalemittancereductionandpreservationinelectronstorageringsviaresonancedrivingtermscorrection,A.Franchietal,PRSTAB14,034002
eeFACT16Daresbury
25/10/16 20
Resonancedrivingterms(RDT)
injection oscillations. The coupling is reconstructed locallyby the 500 TbT BPMs. Finally, we will describe thedesign of a feedback to control the coupling beyond 2015.In Sec. II we start with describing a more precise equationrelating the f1001 to the C. We continue in Sec. III withdemonstrating the benefit of selecting two BPMs close to 2when measuring the coupling. The measurement resolutionis also increased using a singular value decomposition(SVD) cleaning, which is described in Sec. IV. Theautomatic coupling correction approach based on injectionoscillations, which was used in normal operation in 2012, isdemonstrated in Sec. V. The coupling is reconstructed fromall BPMs and the paring algorithm ensures that the phaseadvance is close to the optimal. The method to use theinjection oscillations, however, only provides measure-ments for injection energy. The resolution of the BPMsis not good enough to measure the coupling from accept-able excitations during normal operation with high inten-sity beams. The diode orbit and oscillation (DOROS) [21]are being developed at CERN and will provide very precisephase and amplitude measurements. The location of theBPMs equipped with DOROS electronics are, however, notoptimized for coupling measurements, since their mainpurposes are to provide very precise orbit and phasemeasurements close to the interaction points (IPs). As aconsequence the phase advance is far from the optimal 2. InSec. VI a feedback based on the combined informationfrom all the BPMs equipped with DOROS electronics ispresented.
II. MORE PRECISE EQUATION FOR C
The closest the horizontal and vertical tune can approacheach other is termed Qmin and is equal to the jCj. TheRDT f1001 is a local property related to the Hamiltonianterm h1001. A relation of the f1001 to the jCj close to thedifference resonance, was described as [22]
Qmin jCj 41
N
XN
i1jf1001ij; 1
where Qmin is the closest the tunes can approach eachother, N is the number of BPMs and is the fractional tunesplit. A more precise relation was published in [23] butnever applied to data. The nomenclature used in this articleis different and we therefore derive the relation in theAppendix for clarity. The relation is described as
Qmin jCj !!!!42R
Idsf1001eixyis=R
!!!!; 2
where R is the radius of the machine, x is the horizontalphase, y is the vertical phase, and s is the longitudinaldistance. The integral extends over the entire ring but inpractice it will only be evaluated at the locations of the
BPMs. In Fig. 1 the jCj is calculated from Eqs. (1) and (2).TheQmin is retrieved by trying to match the tunes as closeas possible to each other in methodical accelerator design(MAD) [24]. We observe that the two formulas give almostidentical and correct results close to the resonance butEq. (1) deviates more when the fractional tune splitincreases. We also observe that the term is=R has anegligible effect on the calculated jCj. This also holds truefor the European Synchrotron Radiation Facility (ESRF)booster [25]. The main differences between the formulascan then be interpreted as Eq. (1) is the average of thejf1001j while Eq. (2) is the absolute value of the aver-age f1001.
III. OPTIMAL PARING OF BPMs
We reconstruct the f1001 and the f1010 terms from TbTdata [26] using the Courant-Snyder variable [27]
hx; x ipx; 3
where x is the normalized horizontal position and px is thehorizontal transverse momentum. The momentum is not adirectly measurable quantity with a BPM but needs to bereconstructed using two BPMs. The momentum at the ithBPM can be written as [22]
pxi xi1 xi cosx
sinx; 4
where x is the horizontal phase advance between the ithand i 1th BPM under the assumption that the regionbetween the two BPMs is free of coupling sources andnonlinearities contributing to the main and the couplingline. Equation (4) indicates that a phase advance of 2 is the
FIG. 1. A comparison of Eqs. (1) and (2) to calculate the Qminfrom f1001. Qy was kept constant at 59.31 while Qx was variedbetween 64.22 and 64.40. Injection optics for the LHC was usedin the simulation.
T. PERSSON AND R. TOMS Phys. Rev. ST Accel. Beams 17, 051004 (2014)
051004-2
CouplingRDTf1001-f1010arerelatedtothecouplingparametervia:
A. RMS apparent emittances
As far as the apparent emittances of Eq. (2) are con-cerned, the following relations apply (a dependence on shas to be assumed in all quantities, bar the two eigenemit-tances Eu;v):
Ex C2Eu S2$ S2 $ 2S$S cosq q$'Ev; (4)
Ey C2Ev S2$ S2 $ 2S$S cosq $ q$'Eu; (5)
The following definitions (all s dependent) apply:
C cosh2P ; (6)
S $ sinh2P
Pjf1001j; (7)
S sinh2P
Pjf1010j; (8)
P ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi$jf1001j2 jf1010j2
q; (9)
f10011010
PW
w Jw;1ffiffiffiffiffiffiffiffiffiffiffiffiffi!wx!
wy
pei!"w;x(!"w;y
41$ e2#iQu(Qv; (10)
q$ argff1001g; q argff1010g: (11)Jw, w 1; 2; 3 . . . , W are the skew quadrupole integratedstrengths present in the ring and originated by quadrupoletilts, sextupole misalignments, insertion devices, and cor-rector skew quadrupoles already powered. Qu;v are theeigentunes, which are equal to the measured tunes up tothe first order in strengths, Qu;v Qx;y OJ2w;1. !wrdenotes the Twiss parameter corresponding to the locationof the skew quadrupole kick, whereas !"w;r is its phaseadvance with respect to the position where the RDTs f1001and f1010 are either measured or computed. Both!r and"rrefer to the ideal, uncoupled lattice. Even though all quan-tities in Eqs. (6)(10) are complex numbers, the followingrelations hold:
1 C2 S2$ $ S2; C2;S2$;S2;SS$ 2 224; (32)
where "c is the beta function at the location of the skew
corrector c, and !#cw is the phase advance between thesame corrector and the BPM w. By inverting via SVD thelinear system of Eq. (30) the strengths for the correctormagnets that best reduce the coupling RDTs are derived.By itself coupling correction implies thatC2 1,S2+ 0,
and hence that %y Ey Ev. This, however, is not suffi-cient, as the eigenemittance Ev is minimized only after afurther correction of vertical dispersion, i.e., after minimiz-ingH y, see Eq. (27). Skewquadrupolesmay still be used tothis end. Indeed, Eq. (30) may be generalized as follows:
a1 ~f1001
a1 ~f1010
a2 ~Dy
0BBB@
1CCCA
meas
"M ~Jc; (33)
whereM is now a 224 ( 2 224 ) 32matrix. The genericelement of the additional 224) 32 block reads
mw;c !Dwy
!Jc1; (34)
where!Dwy is the vertical dispersion distortion at the BPM
number w induced by the skew corrector strength !Jc1 .These terms need to be computed by means of optics codes,as they depend on the error lattice model. The weightsa2 1" a1 are introduced in order to determine the bestcompromise between correction of dispersion and deterio-ration of coupling. Their determination is empirical and inthe case of the ESRF storage ring the best correction isfound for a2 0:7. Note that the system of Eq. (33) isanalogous to the one already proposed and successfullyimplemented in Ref. [14], with the difference of havingthe RDTs instead of the vertical orbit distortion to be
20
30
40
50
60
70
80
90
vert
ical
app
aren
t em
ittan
ce [p
m]
from AT modelfrom in-air x-ray monitor
C5 C10 C11 C14 C18 C21 C25 C26 C29 C31
FIG. 3. Example of comparison between the apparent emittan-ces Ey (before coupling correction) measured at ten available in-air x-ray detectors (blue) and the predictions of AT (red) aftercreating the lattice error model from the ORM measurement ofJanuary 16, 2010.
VERTICAL EMITTANCE REDUCTION AND PRESERVATION . . . Phys. Rev. ST Accel. Beams 14, 034002 (2011)
034002-7
Jcaretheskewstrength
References:-ImprovedcontrolofthebetatroncouplingintheLargeHadronCollider,T.Perssonetal,PRSTAB17,051004-Verticalemittancereductionandpreservationinelectronstorageringsviaresonancedrivingtermscorrection,A.Franchietal,PRSTAB14,034002
Figure 1: Beta functions (upper figure) in the IR and in thearcs and horizontal dispersion (lower figure).
FCC-hh layout
A. Bogomyagkov (BINP) FCC-ee crab waist IR 8 / 25
Figure 2: Racetrack layout with chromaticty correction inthe arcs [1].
tially tapered option - or sectorwise version- the machineprovides a tapering to the dipoles only, leaving therefore aremaining horizontal orbit as shown in Fig. 4 [5].
Therefore, with targeted emittances of the order of nmand pm, FCC-ee is a collider with foreseen performances oflight sources (ESRF, SLS).
AMPLIFICATION FACTOR BY ERRORTYPE
In this section, amplification factors on the orbit and/orthe vertical dispersion are computed by errors type. Foremittance tuning purposes, any source of vertical dispersionand coupling has to be identified and should be corrected as
13/10/16 LowEmittanceWorkshop2015 6
12 m30 mrad
9 mMiddle straight
1570 m
FCC-hh
CommonRF
CommonRF
90/270 straight4.7 km
IP
IP
A conceptual layout of FCC-ee
0.8 m
As the separation of 3(4) rings is within 15 m,one wide tunnel may be possible around the IR.
0.0 100. 200. 300. 400. 500. 600. 700. 800. s (m)
FCC-ee DS MAD-X 5.02.03 20/03/15 11.17.17
10.
20.
30.
40.
50.
60.
70.
80.
90.
100.
x(m
), y
(m)
x
FCC-ee: The Storage Ring Bastian Haerer
D = Lcell2
1+ 1
2sin cell
2
sin2
cell2
= pp
2
D2 + 2 D D + D 2( )
Optics & Cell design in the arcs determine the emittance
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 1 2 3 4 5 6 7 8 9
Dx
in m
s in km
Basic Cell: 900/600 (at present), l = 50m, optimised for 175 GeV
S BQ QBB BSQ SS
B = bending magnet, Q = quadrupole, S = sextupole
50 m FODO cell
10.5 m 1.5 m0.5 m 0.65 m
50 m
0.55 m0.65 m
Katsunobuslatticefullytapered(dipole,quadrupole,sextupole)
BastianRacetracktaperingsectorwisedipoleonly
FCC-eeRacetrackLayout
ToleranceconsiderationforAntonsIRlayoutwillbecoverbySergey(nexttalk)Figure 3: Racetrack layout with local chromaticty correctionat the IR [2].
13/10/16 19
Opticsfunctionswithsynchrotronradiation
FCCweek2016-Rome
S(m)
X(m)
Issueswithsector-wisetapering(Bastianslattice+AndreasTapering)
DFS+localverticaldispersioncorrectionattheIpsbringstheverticaldispersion
from1.0e-2mdownto1.0e-5m
->Forafullytaperedmachine(Katsunoburacetrack),thiswontbeanissue.
Dyrms(m)
Nbofiterations
X(m)
Figure 4: In green, typical horizontal orbit remainding aftercorrection without synchrotron radiation, in blue [5]
much as possible. Let consider the most important errors toconsider.
A vertical oset y in the quadrupole provides a dipolarkick since [6],
B
x
= k (y + y) = ky + ky (2)
with k the normalised quadrupole strength. The constantterm ky provides a vertical dipole component and thereforevertical dispersion. Sextupole osets produce coupling andvertical dipole kick since,
B
x
= k xy + k xyB
y
= k (x2 y2) 2ky (y2)(3)
Quadrupole roll angles produce a skew strength, generatingbetatron coupling and transfering horizontal emittance tovertical emittance. The resulting vertical dispersion changedue to a skew strength componant is
Dy
= (Jw
)Dx
py
y0
2sin(Q)cos(Q |
y0 y |) (4)
where Jw
is the skew strength, Dx
is the horizontal dis-persion,
y
and y0 are respectively vertical beta function
Emity[pm]
FullDFScorrectionscheme
Status,sofar:-quadrupolesmisalignementstolerancehasbeenimprovedfrom5micromto20microm-1/2oftheseedsproducetoohighverticalemittance.-BPMerrorsfrom5to20micromworsecasescenariosofar
Forasectorwisetaperingmachine
Figure 9: RMS vertical dispersion for several iterations ofDispersion Free Steering first without sextupole and thenwith sextupoles.
Figure 10: Vertical, horizontal emittance and coupling ratioas a function of the errors in the BPMs.
ROLLS IN QUADRUPOLES ANDCOUPLING CORRECTION
Current skew quadrupole correctors scheme for
FCC-ee
In order to correct the betatron coupling, one skewquadrupole has been installed every 6 FODO cells, witha horizontal and vertical phase advance of
x
= 540 and
y
= 360 degrees, since the lattice has a 90/60 degreesphase advance per cell. Therefore the total amount of skewsin the machine is 272, installed in dispersive places. Cur-rently, they are used to correct both betatron coupling andvertica dispersion. No local correction of the coupling at theIPs is performed, but is foreseen as next step in order to com-pensate the coupling generated by the roll angles of the final
focus doublets. To correct the betatron coupling, the cou-pling resonance driving terms, so called f1001 for dierenceresonance and f1010 for the sum resonance, are mitigated,as successfully applied in LHC and at the ESRF [7] [8].
The closest tune approach is related to the complex cou-pling parameter, C - here the dierence coupling parameter- which is directly a function of the coupling resonance driv-ing terms (RDT) as [7] [8] [9]
Qmin
= |C | = | 42R
Ids f1001e
i (x
y
)+is/R | (7)
The resonant driving terms f1001 and f1010 can be computedfrom several ways, here the analytical formula
f
10011010 =
Pw
J
w
qwx
wy
e
i (w,x/+w,y )
4(1 e2i (Qu/+Qv ) ) (8)
where J are the skew strength, wx
wy
are the horizontal andvertical beta function at the location of the skew strength,
w,x ,w,y are the phase advance between the observa-tion point and the skew componant.
Using the matrix formalism, a response matrix of the RDTusing the quadrupole skews of the lattice can be computed,
( ~f1001)meas = M ~J (9)
where J are the vector of the skew, ~f1001 are the complexcoupling RDT at the BPMs, M is the response matrix ofthe RDT to skew quadrupole kicks. ~f1010 is neglected tothe distance of the working point with respect to the sumcoupling resonance.
Coupling correction for a lattice with roll angles
in the quadrupoles
The roll quadrupole tolerances are much less tigh com-pared to the transverse displacement with an amplificationfactor of 25 on the vertical dispersion. Let us consider theFCC-ee lattice at 175 GeV with 50 rad roll angle gaussiandistributed cut at 2 sigma. Since no other error is consid-ered in this simulation, the coupling mainly comes the tiltedquadrupoles.
The coupling RDT at the BPMs are computed and cor-rected with the corresping response matrix after a SVD, andskew quadrupoles strength are then applied. The resultingRDT are compared to the initial RDT. The successive cor-rections allows to correct by a factor 10 the RDT ~f1001.
This correction can be combined with a response matrixof the vertical dispersion to the skew quadrupoles:
( ~Dy) = M ~J (10)
where ~Dy
is the vertical dispersion measured at the BPMs, Mis the response matrix of the RDT to the skews, J are the skewstrength. While introducing roll angles in the quadrupolesof the lattice, dispersion is transferred from the horizontalplane to the vertical one Eq. 4. The correction of the vertical
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25/10/16 21
ConclusionsCouplingresonancedrivingtermscorrection
Example:50microradtiltinquadrupoles,noothererrors,sector-wisetapering. Couplingisintroducedbytherollsquads. Couplingcorrectionreducesf1001byafactor10(0.010RMS->0.001rms)
Dispersioncorrectionswithskews:Dyrms=3.5mm->0.5mm
--Beforecouplingcorrection--Aftercouplingcorrection--Aftercouplingcorrectionwithsynchrotronlightlosses
Re(f1001)
BPMnb
Gaininemittancebyafactor2,insteadofmore.0.5pm->0.25pm
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ConclusionsCouplingresonancedrivingtermscorrection
Example:50microradtiltinquadrupoles,noothererrors,sector-wisetapering.
Hor.Orbitaftersectorwisetapering
Fromwherethisextracouplingcomesfrom,whensynchrotronlightistakenintoaccountintothesimulations?- Couplingx&yorbitthroughsextupoles- sector-wisetaperingstillallows3.5e-5mHor.Orbit- FullytaperedmachinefromKatsunobuwillanswerthequestion.
eeFACT16Daresbury
Statusofthetolerance:-quadrupoledisplacement20microm(gainalmostafactor4)-quadrupoletilt50microrad-BPMverticaldisplacement:20microm(factor4)
TheimprovementiscomingfromtheverticaldispersioncorrectionwiththecombinationoftheDFS+IPlocalcorrection.
VerysuccessfulDFSmethod->mto1.0e-5minDyNextSteps:
LocalCouplingcorrectionattheIPstocompensatethecouplingfromtherollsofFFquads.
MergingDFS+RDTcorrectionongoing..Moretricky. Gobacktoafullytaperedlattice.
Katsunobuslattice
IssueswithMADXwithsector-wisetaperingforthecouplingcorrection->WouldnotbeaproblemwithfullytaperedKatsunobuslattice.
ComparisonbetweenSADandMADX(benchmarkemittance,corrections..) Sextupolemisalignments
25/10/16 23
Conclusions:Outlook,futurework
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ThankyouforyouraDen7on!
InfluencequadrupolesofIPs
25/10/16 25eeFACT16Daresbury
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MADXhasbeenused. Allthecorrectionsaredonewithoutsynchrotronradiationlossin
energyinthemagnet.Approachcorrectforafullyorsectorwisetaperedmachine.
Thispresentationisfocusontransversedisplacementsandtiltofquadrupoles,BPMerrors.
EmittancecomputationinMADX:
EMITusestheChaoformalismtocomputetheemittances.EMITtakesintoaccounttheenergylossviasynchrotronradiationindipoles,quadrupoles,sextupolesetc..
Toolsandcorrectionmethods
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DFSoptimizationwithsingularvaluedecomposition
25/10/16 27
WithoutSVD,thecorrectiondoesnotworkmandatorystep. SVDprinciple
Wisadiagonalmatrixwithw_iarethesingularvalues. Establishacutofftoeliminatethesingularvaluetooptimizethe
correctionCompromisebetweennoiseinthecorrectionandefIiciencyMoresingularvalue->morelocalcorrection,morenoiseLesssingularvalue->globalcorrection,IP
PRST-AB 3 EMITTANCE OPTIMIZATION WITH DISPERSION FREE 121001 (2000)
(N . M) or under (N , M) constrained. In the for-mer and most frequent case, Eq. (7) cannot be solvedexactly. Instead, an approximate solution must be found,and commonly used least square algorithms minimize thequadratic residual
S ! k "u 1 A "uk2. (8)
Dispersion free steering is based on the extension ofEq. (7) to include the dispersion at the BPMs. The ex-tended linear system is
!1 2 a""u
a "Du
1
!1 2 a"A
aB
"u ! 0 , (9)
where vector "Du (dimension N) represents the dispersionat the BPMs. B is the N 3 M dispersion response matrix,its elements Bij giving the dispersion change at the ithmonitor due to a unit kick from the jth corrector. Theweight factor a is used to shift from a pure orbit (a !0) to a pure dispersion correction (a ! 1). In general,the optimum closed orbit and dispersion rms are not ofthe same magnitude and a must be adjusted for a givenmachine. a can, in principle, be evaluated from the BPMaccuracy and resolution. Applied to Eq. (9), a least squarealgorithm will minimize
S ! !1 2 a"2 k "u 1 A "uk2 1 a2k "Du 1 B "uk2. (10)
Singular response matrices are a well-known problem oforbit corrections. The singularities are related to redundantcorrectors, i.e., areas of the machine where the samplingof the orbit is insufficient. Such situations yield numeri-cally unstable solutions where large kicks are associatedto minor changes in the orbit. A standard cure consists indisabling a subset of correctors and removing the corre-sponding lines from the linear systems of Eqs. (7) and (9).Regularization can also be obtained by extending Eq. (9)to constrain the size of the kicks,0
B@!1 2 a" "u
a "Du"0
1CA 1
0@ !1 2 a"AaB
bI
1A "u ! 0 . (11)
Here "0 is a null vector of dimensionM, I is a unit matrix ofdimension M 3 M, and b is a kick weight. The quadraticresidual now contains the rms strength of the correctorkicks,
S ! !1 2 a"2 k "u 1 Auk2 1 a2k "Du 1 B "uk2
1 b2k "uk2, (12)
and large kicks are suppressed since they receive a penaltywhich can be adjusted with b.Various other constraints can be added to the linear sys-
tem to be solved, for example, to maintain a constant or-bit length or to stabilize the beam at given locations inthe ring. Adequate weight factors can be used to control
the importance of such constraints. It is also possible tocorrect the machine coupling using a similar scheme. Theorbit coupling of horizontal corrector kicks into the ver-tical plane is then minimized using skew quadrupoles ascorrecting elements [10]. To simplify the expressions inthe following sections, vector "d and matrix T are definedas
"d !
0@ !1 2 a" "ua "Du
"0
1A, T !
0@ !1 2 a"AaB
bI
1A , (13)
with
"d 1 T "u ! 0 . (14)
A. Singular value decomposition (SVD) and orbiteigenvectors
Dispersion free steering is particularly interesting inconjunction with the singular value decomposition (SVD)algorithm [11,12], because it allows a simultaneous limi-tation of the corrector kick strength. The SVD algorithmis a powerful tool to handle singular systems and to solvethem in the least square sense. For M $ N the singularvalue decomposition of matrix T has the form
T ! UWVt ! U
0BB@
w1 0 00 w2
00 0 wM
1CCAVt, (15)
where W is a diagonal M 3 M matrix with non-negativediagonal elements and Vt is the transpose of the M 3 Morthogonal matrix V,
VVt ! VtV ! I , (16)
while U is an N 3 M column-orthogonal matrix
UtU ! I . (17)
The vector "q !i", corresponding to the ith column ofmatrix V,
"q !i" !
0BB@
V1iV2i VMi
1CCA , (18)
is an eigenvector with eigenvalue w2i $ 0 of the M 3 Msymmetric matrix TtT [12,13],
TtT "q !i" ! w2i "q!i". (19)
It follows from Eq. (16) that the M vectors "q !i" form anorthonormal base of the corrector space since
! "q !i" ? "q !j"" ! "q !i"t "q !j" ! dij . (20)
121001-3 121001-3
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Ver7calOrbit
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------
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25/10/16 LowEmittanceWorkshop2015 30
LocalDispersionCorrec7on extremelyefIicient:
itbringstheverticaldispersionatthesameorderofmagnitudeasthearcs.
2%->0.5%inemittanceratio. DFSextraiterationsdonotchangethekickerstrengthsintheIRs
25/10/16 LowEmittanceWorkshop2015 31
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