overview of optics tuning results and tolerances€¦ · hor. dispersion along the ring:...

Overview of Optics Tuning Results and Tolerances for FCC-ee

Sandra Aumon, CERNOn behalf of the FCC-ee Lattice Design team

10/18/2017 FCC Week 2017 - Berlin 1

10/18/2017 FCC Week 2017 - Berlin

Beta function in the IR:

Hor. Dispersion along the ring:

bety_max=5220mbetx_max=1886m

Phase advance per cell: 90/90 deg.

2

IP

FCC-ee Optics

10/18/2017 FCC Week 2017 - Berlin

Z W H tt

Beam energy [GeV] 45.5 80 120 175

Beam current [mA] 1450 152 30 6.6

Bunches / beam 91500 5260 780 81

Bunch population [1011] 0.33 0.6 0.8 1.7

Transverse emittance ε

- Horizontal [nm]

- Vertical [nm]

0.09

0.001

0.26

0.001

0.61

0.0012

1.3

0.0025

Momentum comp. [10-5] 0.7 0.7 0.7 0.7

Betatron function at IP β*

- Horizontal [mm]

- Vertical [mm]

1000

2

1000

21000

2

1000

2

Energy loss / turn [GeV] 0.03 0.33 1.67 7.55

Total RF voltage [GV] 0.2 0.8 3 10

3

Challenges and constraints (1)

Small emittance ratio 0.2% requires• Coupling correction• Small vert. dispersion

Small beta functions• make lattice sensitive towards

FF misalignments• require strong sextupoles

(coupling)

10/18/2017 4

Emittance tuning for electron machine

FFODO =1

2siny

5+3cosy

1- cosy

L

lB

L: cell lengthlB: dipole length

: phase advance/cell

• Horizontal emittance:

FCC Week 2017 - Berlin

• Vertical emittance:

• Source of vertical emittance growth

- vertical dispersion Dy (quad-sext. displacement and tilt)- betatron coupling (hor. sextupole misalignments)- opening angle here negligible ~

10/18/2017 FCC Week 2017 - Berlin

• Challenging and strict emittance budget (1.3nm, 2.5pm) and coupling ratio (0.2%) at high energy.

• Very sensitive machine to alignment errors.• Study strategy:

study each error separately Establish appropriate correction scheme to get convergence on a maximum seed

number (MADX fails easily to find closed orbit) Merge step by step the different errors together, keeping the Final Focus Doublets

for the end. Unless mentioned, FF quadrupoles are left perfectly aligned. Emittance calculations (depends only on the beam energy):

Fully tapered machine: every magnet strength follows beam energy loss

5


10/18/2017 FCC Week 2017 - Berlin

• Orbit correction with MICADO & SVD from MADX Hor. corrector at each QF, Vert. corrector at each QD BPM at each quadrupole

• Vertical dispersion and orbit:Orbit Dispersion Free Steering (DFS)

• Linear coupling:Linear Coupling resonant driving terms (RDT) 1 skew at each sextupole

• Beta beating correction & Hor dispersion via Response Matrix:Rematching of the phase advance at the BPMs 1 trim quadrupole at each sextupole

6

Correction methods

• Build numerically a matrix for vertical orbit (u) & dispersion (Du) response under a corrector kick (al)

Dispersion Free Steering: Principle

10/18/2017 7

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

0.20

Vert. DispersionVert. Dispersion

)p (2m

0 5 10 15 20 25

)1/2

(m

1/2

bD

/

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

Norm. Vert. DispersionNorm. Vert. Dispersion

Figure 2: Vert ical dispersion in the SPS ring due to a kick of 0.1 mrad at corrector

MDV.10307. The solid line is the MADX predict ion, the points correspond to the analyt ical

expression of Eq. 20 evaluated at the locat ion of the BPMs.

st rength, 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 equat ion for the quadrupoles by

−K 2Dx , see for example Ref. [4] for a rigorous treatment of the dispersion response.

When all contribut ions are combined the dispersion response at monitor i due to the kick

from corrector j becomes

B i j = {

quad

l

K lL lβl

4sin(πQ)2cos(|µi − µl | − πQ) cos(|µl − µj | − πQ)

−

sext

m

K 2,mDx,mLmβm

4sin(πQ)2cos(|µi − µm | − πQ) cos(|µm − µj | − πQ)

−cos(|µi − µj | − πQ)

sin(πQ)} βlβj (20)

where the sums run over all quadrupoles (i ) and sextupoles (m). The last term is the direct

effect from the corrector kick itself. Eq. 20 is valid for the horizontal plane. For the vert ical

6

• Orbit response

• Dispersion response

“Emittance optimization with dispersion free steering at LEP”R. Assmann et al. Phys. Rev. ST Accel. Beams 3, 121001

• SVD analysis to solve the system and find a solution


10/18/2017 8

Resonance driving terms (RDT)Coupling RDT f1001-f1010 are related to the coupling parameter via:

References:-Vertical emittance reduction and preservation in electron storage rings via resonance driving terms correction, A. Franchi et al, PRSTAB 14, 034002

f1001-f1010 can be computed via analytical formulas, or via a matrix formalism with the coupling matrix:

Figure 1: Beta functions (upper figure) in the IR and in the

arcs and horizontal dispersion (lower figure).

FCC-hh layout

A. Bogomyagkov (BINP) FCC-ee crab waist IR 8 / 25

Figure2: Racetrack layout withchromaticty correctionin

the arcs [1].

tially taperedoption- or sectorwiseversion- themachine

providesataperingtothedipolesonly, leavingthereforea

remaining horizontal orbit asshown in Fig. 4 [5].

Therefore, with targetedemittancesof theorder of nm

andpm, FCC-eeisacollider withforeseenperformancesof

light sources (ESRF, SLS).

AMPLIFICATION FACTOR BY ERRORTYPE

Inthissection, amplification factorsontheorbit and/or

thevertical dispersion arecomputed by errorstype. For

emittancetuningpurposes, anysourceof vertical dispersion

andcouplinghastobeidentifiedandshouldbecorrectedas

13/10/16 LowEmittanceWorkshop2015 6

~12 m

30 mrad

9 m“Middle straight”

~1570 m

FCC-hh

Common

RF

Common

RF

“90/ 270 straight”

~4.7 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 =Lcell

2

´ 1+

1

2sin

cell

2

¸

¸ sin2 cell

2

¸

= p

p

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

SB

QQBB BSQ S

S

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

Katsunobu’slattice–fullytapered(dipole,quadrupole,sextupole)

BastianRacetrack–taperingsectorwisedipoleonly

FCC-eeRacetrackLayout

ToleranceconsiderationforAnton’sIRlayoutwillbecoverbySergey(nexttalk)

Figure3: Racetracklayoutwithlocal chromatictycorrection

at the IR [2].

13/10/16 19

Opticsfunctionswithsynchrotronradiation

FCCweek2016-Rome

S(m)

X(m)

Issueswithsector-wisetapering(Bastian’slattice+AndreasTapering)DFS+localverticaldispersioncorrectionattheIpsbringstheverticaldispersionfrom1.0e-2mdownto1.0e-5m->Forafullytaperedmachine(Katsunoburacetrack),thiswon’tbeanissue.

Dyrms(m)

Nbofiterations

X(m)

Figure4: Ingreen, typical horizontal orbit remaindingafter

correction without synchrotron radiation, in blue [5]

muchaspossible. Let consider themost important errorsto

consider.

A vertical offset ∆y inthequadrupoleprovidesadipolar

kick since [6],

Bx = k(y + ∆y) = ky + k∆y (2)

with k thenormalisedquadrupolestrength. Theconstant

termk∆y providesavertical dipolecomponentandtherefore

vertical dispersion. Sextupoleoffsetsproducecouplingand

vertical dipole kick since,

Bx = kxy + kx∆y

By = k(x2 − y2) − 2k∆y − (∆y2)(3)

Quadrupoleroll anglesproduceaskewstrength, generating

betatroncoupling andtransferinghorizontal emittanceto

vertical emittance. Theresultingvertical dispersionchange

due to askew strength componant is

∆Dy = −(∆Jw )Dx

pβy βy0

2sin(⇡Q)cos(⇡Q − |φy0 − φy |) (4)

where Jw is theskew strength, Dx is thehorizontal dis-

persion, βy and βy0 arerespectivelyvertical betafunction

For FCC-ee, I build a RDT and vertical dispersion response matrix with skew quadrupole kick

Jc are the skew strength

Emity[pm]

FullDFScorrectionscheme

Status,sofar:-quadrupolesmisalignementstolerancehasbeenimprovedfrom5micromto20microm-1/2oftheseedsproducetoohighverticalemittance.-BPMerrorsfrom5to20microm–worsecasescenariosofar

Forasectorwisetaperingmachine

Figure9: RMSvertical dispersionfor several iterationsof

DispersionFreeSteeringfirst without sextupoleandthen

with sextupoles.

Figure10: Vertical, horizontal emittanceandcouplingratio

as a function of the errors in the BPMs.

ROLLSIN QUADRUPOLESANDCOUPLING CORRECTION

Current skewquadrupolecorrectorsschemeforFCC-ee

In order to correct the betatron coupling, one skew

quadrupolehasbeen installed every 6FODOcells, with

ahorizontal andvertical phaseadvanceof ∆φx = 540and

∆φy = 360degrees, sincethelatticehasa90/60degrees

phaseadvanceper cell. Thereforethetotal amount of skews

inthemachineis272, installed indispersiveplaces. Cur-

rently, theyareusedtocorrect bothbetatroncouplingand

verticadispersion. Nolocal correctionof thecouplingat the

IPsisperformed,but isforeseenasnext stepinorder tocom-

pensatethecouplinggeneratedby theroll anglesof thefinal

focusdoublets. Tocorrect thebetatroncoupling, thecou-

plingresonancedrivingterms, socalled f1001 for difference

resonanceand f1010 for thesumresonance, aremitigated,

as successfully applied in LHC and at the ESRF [7] [8].

Theclosest tuneapproachisrelatedtothecomplexcou-

plingparameter,C− -herethedifferencecouplingparameter

- whichisdirectlyafunctionof thecouplingresonancedriv-

ing terms(RDT) as [7] [8] [9]

∆Qmi n = |C− | = |4∆

2⇡R

I

dsf1001e− i (φx −φy )+i s∆/ R | (7)

Theresonant drivingterms f1001 and f1010 canbecomputed

from several ways, here theanalytical formula

f 10011010 =

Pw Jw

qβw

x βwy ei (∆φw , x − / +∆φw , y )

4(1− e2⇡ i (Qu − / +Q v ))(8)

whereJaretheskewstrength, βwx β

wy arethehorizontal and

vertical betafunctionat thelocationof theskewstrength,

∆φw,x ,∆φw,y arethephaseadvancebetweentheobserva-

tion point and theskew componant.

Usingthematrixformalism,aresponsematrixof theRDT

using the quadrupoleskewsof the lattice can becomputed,

( ~f1001)meas = −M ~J (9)

whereJ arethevector of theskew, ~f1001 arethecomplex

coupling RDT at theBPMs, M istheresponsematrix of

theRDT toskewquadrupolekicks. ~f1010 isneglected to

thedistanceof theworkingpoint withrespect tothesum

coupling resonance.

Couplingcorrectionfor alatticewithroll angles

in thequadrupoles

Theroll quadrupoletolerancesaremuchlesstighcom-

paredtothetransversedisplacement withanamplification

factor of 25onthevertical dispersion. Let usconsider the

FCC-eelatticeat 175GeV with50µradroll anglegaussian

distributedcut at 2sigma. Sincenoother error isconsid-

eredinthissimulation, thecouplingmainlycomesthetilted

quadrupoles.

ThecouplingRDTat theBPMsarecomputedandcor-

rectedwiththecorrespingresponsematrixafter aSVD, and

skewquadrupolesstrengtharethenapplied. Theresulting

RDTarecomparedtotheinitial RDT.Thesuccessivecor-

rections allows to correct by a factor 10 theRDT ~f1001.

Thiscorrectioncanbecombinedwitharesponsematrix

of the vertical dispersion to theskew quadrupoles:

( ~Dy) = −M ~J (10)

where ~Dy isthevertical dispersionmeasuredat theBPMs,M

istheresponsematrixof theRDTtotheskews,Jaretheskew

strength. Whileintroducingroll anglesinthequadrupoles

of thelattice, dispersionistransferredfromthehorizontal

planetothevertical oneEq.4. Thecorrectionof thevertical


• Errors , no strength in sextupoles• X-y orbits correction• Pure coupling correction• Rematch the horizontal dispersion• 1 step Dispersion Free Steering wo sextupole (Dy correction)

+1 step coupling correction (kicker strength change the coupling configuration)

• Save x,x’,y,y’ at the beginning of the machine

• Switch on sextupoles to +10% of their design current- orbit corrections- coupling correction, tune matching- beta beat correction, Dx correction- coupling + Dy correction- increase by 10% the sextupole strength

• Emittance computation

10/18/2017 FCC Week 2017 - Berlin 9

This avoid the tunes run of to resonance and maximize the number of seeds

Sextupole and Quad. transverse displacements

StrategyQuadrupole

Sextupole

10/18/2017 FCC Week 2017 - Berlin 10

Orbit correction+DFS (no sextupole)

Init DY After CO-correctionFactor 2e4

DFSfactor 50

Dispersion correction during the coupling correction (coupling due to sextupole)

Factor ~10

Transverse displacement of arc quadrupoles

Switch on sextupoles

10/18/2017 FCC Week 2017 - Berlin 11

Coupling matrix

10/18/2017 FCC Week 2017 - Berlin 12

• 700 valid seeds under 1000, gaussian distributed cut at 2.5 sigma

Rolls and transverse displacements

emit 100 μrad

εy (pm)

Z energy 45 GeVW energy 90 GeVttbar energy 175 GeV

0.016+/-0.0020.067+/-0.010.25+/-0.03

εy/εx 0.0002

Those values can be reduced by a factor 10 by installing dedicated skews at the IP.

• Δy=10 μm RMS gaussian distributed truncated at 2.5 sigmaNo correction

10/18/2017 FCC Week 2017 - Berlin 13

Sextupole transverse displacements, no roll

2.5 pm vertical emittance design value!

• After correction:

(pm)

10/18/2017 FCC Week 2017 - Berlin 14


Vertical Δβ/β after beta beat correction (initial up to 50%)

Horizontal Δβ/β after beta beat correction below 1%

Vertical dispersion during correction with sextupole misalignments

10/18/2017 FCC Week 2017 - Berlin 15

RMS values

Dy_rms = 1.25 mmx_rms = 28 microm, px_rms = 3.71e-06y_rms= 27 microm, py_rms = 3.74e-06

Sextupoles

10/18/2017 FCC Week 2017 - Berlin 16

Kicker strength

example of hor. kicker str, comparable to vertical kicker str.vkick= 1.43e-7 rad ~ @45 GeV 2.5e-5 T.mhkick= 2e-8 rad

normalised skew strengthk1s = 5.8e-6 -> 9e-4 T.m

• No horizontal displacement of the arc quadrupoles100 μm in vertical BPM errors (worst case scenario, work on going)

10/18/2017 FCC Week 2017 - Berlin 17

emit 100μm

εy (pm) 0.2 +/-0.1

εx(nm) 1.25+/-0.008

εy/εx 0.0002

175GeV (79 seeds out of 100)

εy (BPM error)~20 εy (no BPM errors)

First considerations on BPM reading errors

10/18/2017 FCC Week 2017 - Berlin

• Tool box for dispersion, optics correction in FCC-ee.

• Vertical dispersion and coupling are successfully corrected.

• Arc quadrupolesSextupoles do not need special care, so far.

• The sextupole misalignments contribute the most to the vertical emittance budget Introduction of misalignments in the FF quadrupoles:

The treatment of the FF roll angle need a special treatment with a local correction, a factor 10 reduction in vertical emittance with local correction for coupling @ the IP.

Work on going for more statistics for H&V displacement of the FF quadrupoles.

• Working on going:

Treatment of the FF quadrupoles, BPM errors.

18

Conclusions –

10/18/2017 FCC Week 2017 - Berlin 19

10/18/2017 FCC Week 2017 - Berlin 20

Quadrupoles of the arcs (no FF)

Sextupoles

εy = 0.36 pmεx= 1.25 nmεy/εx= 0.0003

Sextupole misalignments dominate the source of vertical emittance

Sextupole and arc quadrupole displacements

10/18/2017 FCC Week 2017 - Berlin 21

Example:

εy=32pmεx=1.79nm

with the following damping partition numbers

jx=0.987, jy=0.897, js=2.16

Sextupole and arc quadrupole displacements

jx

jy

js

• Large non corrected chromaticities Q’x~-500 Q’y~-1000

• Special chromaticity correction scheme with local chromaticity correction at the IPs (K. Oide’s talk)-> strong sextupole fields in the IR

• Working point Qx=0.08 & Qy=0.14 (for polarization purpose) 1/Sin(pi Q) amplification in orbit, dispersion, coupling response due to errors

10/18/2017 FCC Week 2017 - Berlin 22

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

0.20

Vert. DispersionVert. Dispersion

)p (2m

0 5 10 15 20 25

)1/2

(m

1/2

bD

/

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

Norm. Vert. DispersionNorm. Vert. Dispersion

Figure 2: Vert ical dispersion in the SPS ring due to a kick of 0.1 mrad at corrector

MDV.10307. The solid line is the MADX predict ion, the points correspond to the analyt ical

expression of Eq. 20 evaluated at the locat ion of the BPMs.

st rength, 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 equat ion for the quadrupoles by

−K 2Dx , see for example Ref. [4] for a rigorous treatment of the dispersion response.

When all contribut ions are combined the dispersion response at monitor i due to the kick

from corrector j becomes

B i j = {

quad

l

K lL lβl

4sin(πQ)2cos(|µi − µl | − πQ) cos(|µl − µj | − πQ)

−

sext

m

K 2,mDx,mLmβm

4sin(πQ)2cos(|µi − µm | − πQ) cos(|µm − µj | − πQ)

−cos(|µi − µj | − πQ)

sin(πQ)} βlβj (20)

where the sums run over all quadrupoles (i ) and sextupoles (m). The last term is the direct

effect from the corrector kick itself. Eq. 20 is valid for the horizontal plane. For the vert ical

6


Sextupole displacement: final vertical dispersion & Orbits

10/18/2017 FCC Week 2017 - Berlin 23

Beta Beat Correction• Used also for Hor. Dispersion correction.

• Similar to the skews, quadrupolar component installed at every sextupoles -> number/scheme could be optimized later.

• Correction based on a response matrix of phase advances, tunes and hor. dispersion to a gradient k1

**PROCEDURES AND ACCURACY ESTIMATES FOR BETA-BEAT CORRECTION IN THE LHC, R. Toma s et. Al EPAC 2006

10/18/2017 24FCC Week 2017 - Berlin

10/18/2017 25

1. Roll angles in arc quadrupoles

2. Vertical and horizontal displacements in arc quadrupoles

3. Roll, vertical and horizontal displacements in arc quadrupoles

4. Vertical and horizontal displacements in sextupoles

5. Transverse displacement in sextupoles and arc quadrupoles

6. The case of the IP quadrupoles

7. BPM errors

Results of the emittance tolerance error by error


10/18/2017 FCC Week 2017 - Berlin 26


Horizontal Dispersion during correction

10/18/2017 FCC Week 2017 - Berlin 27

Sextupole displacements: Beta beat

<1% after beta beat correction

Correction methods for sextupole only

10/18/2017 FCC Week 2017 - Berlin 28

Correction method

• Misalignment errors in sextupoles

• Switch on sextupoles to +10% of their design current- orbit corrections- coupling correction, tune matching- beta beat correction, Dx correction- coupling + Dy correction- increase by 10% the sextupole strength

• Emittance computation

• Quadrupoles off-set: dipolar kick *

• Sextupoles off-set *

• Quadrupole roll: skew quad, coupling of the planes

10/18/2017 29

Skew quad (coupling) + vertical dipole

Source emittance growth* SY Lee “Accelerator Physics”, Javier Barcelona Presentation

Source of vertical dispersion & coupling


Constant termVertical dipole -> vertical dispersion

• Δy=Δx=100μm RMS displacement in arc quadrupoles, errors gaussiandistributed truncated at 2.5sigma (No sextupole)

• Response of vertical dispersion Dy to Qy (nominal working point qx=0.08 qy=0.14)

• Δy=Δx=100μm RMS displacement in arc quadrupoles (No sextupole, Qy=0.14)

10/18/2017 FCC Week 2017 - Berlin

(m )

No correction

30

Example of FCC-ee lattice sensitivity to errors

Misalignments of elements (E =175 GeV; After CO & SQ correction)

Element

dx,y,u

m

Tilt, m

rad

CO

be

fore

co

rr.

Dy , m

m

Qx

Qy

εx , n

m*ra

d

εy

/ εx , %

ARC BPM 100 0 + 5.5 0.003 0.041 1.40 0.66

FF BPM 25 0 + 3.2 0.000 0.002 1.39 0.20

CCS_XY BPM 50 0 + 7.8 0.002 0.043 1.40 0.72

Dipole - 0.2 + 2.6 0.000 0.000 1.39 0.14

Quads 100 0 - 3.3 0.003 0.063 1.46 0.12

FF Quads 25 0 - 4.0 0.000 0.000 1.40 0.38

Quads 0 0.1 + 1.5 0.000 0.000 1.40 0.05

CCS_XY Quad 100 0 - 1.8 0.002 0.024 1.40 0.07

Sextupole 100 0 + 3.1 0.005 0.009 1.42 0.26

CCS_XY Sext 100 0 + 1.8 0.004 0.082 1.40 0.28

Total * * - 7.0 0.011 0.048 1.46 1.44

* - all misalignments included.

Work from Sergey Sinyatkinfor Anton Bogomyagkov’sFCC-ee layout opticslink

https://eventbooking.stfc.ac.uk/news-events/eefact2016-349?agenda=1

overview of optics tuning results and tolerances€¦ · hor. dispersion along the ring:...

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