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 ... •

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.


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

[email protected]

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]


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.


*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


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


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]


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

eeFACT16Daresbury

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

eeFACT16Daresbury

25/10/16 22

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

eeFACT16Daresbury

ThankyouforyouraDen7on!

InfluencequadrupolesofIPs

25/10/16 25eeFACT16Daresbury

25/10/16 26

MADXhasbeenused. Allthecorrectionsaredonewithoutsynchrotronradiationlossin

energyinthemagnet.Approachcorrectforafullyorsectorwisetaperedmachine.

Thispresentationisfocusontransversedisplacementsandtiltofquadrupoles,BPMerrors.

EmittancecomputationinMADX:

EMITusestheChaoformalismtocomputetheemittances.EMITtakesintoaccounttheenergylossviasynchrotronradiationindipoles,quadrupoles,sextupolesetc..


eeFACT16Daresbury

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

eeFACT16Daresbury

Ver7calOrbit

25/10/16 28

------

eeFACT16Daresbury

25/10/16 29eeFACT16Daresbury

LocalDispersionCorrec7on extremelyefIicient:

itbringstheverticaldispersionatthesameorderofmagnitudeasthearcs.

2%->0.5%inemittanceratio. DFSextraiterationsdonotchangethekickerstrengthsintheIRs


Documents

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 ... •