Nonlinear properties of the FCC/TLEP final focus with respect to L *

Nonlinear properties of the FCC/TLEP final focus

with respect to L*

A. Bogomyagkov, E. Levichev, P.PiminovBudker Institute of Nuclear Physics

Novosibirsk

Seminar at CERN, March 24th 2014 2

Outline

• Introduction (goals,assumptions, tools)• FF optical blocks• Comparison of nonlinear sources of FF

(theoretical)• Simulation results• Chromatic properties of the telescope• Conclusions


Goals, assumptions, tools

• Estimate nonlinear features of TLEP final focus as a function of L* and *. Assuming domination of the vertical plane.

• Design several lattices of FF (from IP to beginning of the arc) for several L*.

• Close the ring with linear matrix providing tunes good for luminosity. Find DA, detuning. Optimize DA, chromatical aberrations.


Optics blocksCRABX2X1Y2Y1

FFT YXCCS CRAB

-I -I

IP: K

inem

atic

term

s: e

xtre

mel

y lo

w IP

bet

a

Chromatic section:strong sextupoles, large beta

First quad fringes: large strength and beta


Final focus

D0=0.7 mQ0: L=4.6 m, G=95 T/m @ 175 GeVD1=0.4 mQ1: L=2 m, G=96 T/m @ 175 GeV

IP

L*(D0) QD0


Nonlinearity: figure of meritFinal goal is large dynamic aperture. However, DA can’t be calculated for part of the ring. Therefore we took nonlinear detuning coefficient as FF nonlinearity figure of merit.

Advantages• Perturbations are 3rd order (octupole like) and detuning is

calculated by 1st order of perturbation theory:

• is additive for different sources


Detuning and DA scalingNaive approach:

Jmnr 0/

~A where 0/ mn

Octupole resonance fixed point:

2cos22 JhJJH h

A~

zero Fourier harmonich resonance driving Fourier harmonic (we believe that both are of the same magnitude order)


Final quad QD0 and chromaticityDefining the QD0 focusing requirements as o = –I one can find

*012L

LK QD

The beta and its derivative in the end of L* are given by

*

2*

Le

*

*

Le

FF vertical chromaticity(half of FF): sssy LKL

2*

*

21

QD0 natural chromaticity Correction by sextupoles

Note: For FF ’ corresponds to the chromatic function excitation introduced by B.Montague (LEP Note 165, 1979)

2A

y

ABAW 22


KinematicsFor the extremely low * and large transverse momentum the first order correction of non-paraxiality is given by

2222 81

yx ppH

dssykyy )(

163 2

yxyxxxx JJ yyyxxyy JJ

dsss yxkxy )()(

81

dssykyy )(

163 2

The main contribution comes from the IP and the first drift:

where 2L is the distance between 2 QD0 quads around the IP and

*

*

2*

*

832

163

yy

kyy

L


QD0 Fringe FieldsQuadrupole fringe field nonlinearity is defined by

24/62/)( 2241

21 yxykypxskH y

and the vertical detuning coefficient is given by

221110161

yyyyfyy k

E.Levichev, P.Piminov, arXiv: 0903.3028A.V.Bogomyagkov et al. IPAC13, WEPEA049, 2615

Or, with above assumptions (k10 is the central strength):

2**102*

3*

10 41

41

LkLk

y

fyy


Octupole error in QD0An octupole field error (or corrector) in QD0:

24/6)( 222432 yyxxskH

02

32

33 161)()(

161

QDyyoyy Lkdssskk

If a field quality at the quad aperture radius ra )()(

aQ

ao

rBrB

q

One can rewrite the vertical octupole detuning (for 2 QD0) as

2**22*

3*

22

0102 23

23

43

L

rqL

rqLk

rq

aaQD

a

oyy


Chromatic sextupolesVertical chromatic sextupole pair separated by –I transformer gives the following coordinate transformation in the first order*)

*) A.Bogomyagkov, S.Glykhov, E.Levichev, P.Piminov http://arxiv.org/abs/0909.4872

0yy 02

030

22

0 6yxy

LLKpp ssyy

0yy

020

30

30 3

6yxy

LKpp yy

Pair of sextupoles Octupole

By analogy to the octupole and using the expression for the FF chromaticity we found for the vertical detuning (2 pairs)

2*2

2

*

*

222

2 41

41

161

s

s

s

syss

spyy

LLLLLK

http://arxiv.org/abs/0909.4872




-test for different lattices

1) CDR2) K.Oide, FCC Kick-off Meeting, Geneva, 14 Feb 20143) T.M.Taylor, PAC 19854) H.G. Morales, TLEP Meeting, CERN, 18 Nov 2013

Note: Different lattice versions may have different parameters.

Super C-Tau1)

NovosibirskSuperB V.161)

ItalySuperKEKB2)

JapanLEP3)

CERNTLEP4)

CERN

103 *(m) 0.8 0.27 0.27 10 1

L*(m) 0.6 0.32 0.77 3.5 3.5

-* 1500 2400 5600 700 7000

-K1(m-2) 1.3 5.4 5.1 0.11 0.19

LQD0(m) 0.2 0.5 0.33 2 2.2

10-6 f (m-1) 0.14 0.79 2.4 0.015 2.6

10-6 k (m-1) 0.22 1.1 9.8 0.008 0.84


Our design, different L** = 1 mm, K1 = 0.16 m-1, Ls = 0.5 m, s= 5 cm

K1(QD0)const

This estimation is very approximate and just shows the trend. We did not take into account realistic beta and dispersion behavior, magnets other but QD0, etc. All these issues are included in simulation.

L*(m) 0.7 1 2 3

-* 1400 2000 4000 6000

10-6 k (m-1) 0.08 0.12 0.24 0.36 L*

10-6 f (m-1) 0.004 0.013 0.1 0.34 L*3

10-6 sp (m-1) -2 -4 -16 -36 L*2


Theoretical conclusions

• FF nonlinearities may increase as L* in high power.• Major part of the vertical nonlinearity for the extra-low beta IP

comes from chromatic sextupoles due to the finite length effect.• The finite length effect in the –I sextupole pair can be improved

by additional (low-strength) sextupole correctors.• Nonlinear errors in the quads with high beta may be a problem.

Correction coils (for instance, the octupole one) can help.• Third order aberrations including the fringe field and kinematics

can be mitigated by a set of octupole magnets located in proper beta and phase.


SimulationScenario:

• We use the following lattice and tracking codes: MAD8 and Acceleraticum1) (BINP home made).

• The FF structure is closed optically by a linear matrix .• The tunes are fixed at (0.53, 0.57) to get large luminosity .• Dynamic aperture and other nonlinear characteristics are

defined from tracking.• DA is increased by additional sextupole and octupole correctors

properly installed.

1) D.Einfeld, Comparison of lattice codes, 2nd NL Beam Dynamics Workshop, Diamond Light Source, 2009


comparison for L*=0.7 mKin Fringe Sextupole pair

Simulationxx (cm-1) 0.6 11 -23xy =yx (cm-1) 3.8 153 -712yy (cm-1) 755 1137 -1.8105

Estimationyy (cm-1) 844 43 -0.2105

Simulation considers all quads fringes (included those strong in the Y chromatic section) and realistic beta behavior


Dynamic apertureW/o correctors With correctors

Black: L* = 0.7 mRed: L* = 1 m (aperture )Green: L* = 1.5 m (aperture )Blue: L* = 2 m (SURPRISE! APERTURE )x=3.24 10-5m, y=6.52 10-8m, *x=0.5m, *x=0.001m


ExplanationY chromaticity correction section is a main source of nonlinear perturbation. Produced aberration is proportional to (K2 )2

L*=0.7 m, K2=-12 m-3,y=5255 m

L*=2 m, K2=-14 m-3 ,y= 5149 m

L*=1.5 m, K2=-14 m-3 ,y= 7707 m


Conclusion• The major source of the DA limitation is the –I Y chromatic

correction section through the sextupole length effect.• Simple calculation of yy confirms it well, however for details

computer simulation is necessary.• DA dependence on L* is more complicated than we expected

in the beginning of this study. Increase of L* will emphasize the fringes (~L*3) compare to the sextupoles (~L*2) (we saw it definitely in SuperKEKB) but for which L*?

• Sextupole and octupole corrections works well.• For L*=2 m we have Ax > 100x and Ay > 700y which seems

quite enough to start.


Guiding principles for FF• IR must provide small beta functions at IP.• Chromaticity of FF doublet must be compensated semi-locally.• Geometrical aberrations from the strong sextupoles pairs, FF

quadrupole fringes must be cancelled in IR.• CRAB sextupole should be as close as possible to IP, at the

position with low chromaticity of beta functions and phase advances to minimize its influence on dynamic aperture.

• Accelerator operates at different energies therefore it is better to have no longitudinal field integral over the FF lenses.

• It is better if integral of the detector longitudinal field should be compensated before each final focus lens.

• FF band width should be more than ±2%.• Synchrotron radiation background should be minimized.


Telescope: optical functions

2

2266226222

21662160

2612612602116611611

UTRUT

UTUTRR

322111

116126

111

1266

111

1262

O

RTT

RU

RT

n

32

1

2126

111661121161116111

2112 22

O

TURTTRR

32

1

2212662261262161162166111

1

221261216112

O

RUTTTTUR

RTTR

Transport matrixKarl L. Brown, SLAC-PUB-4159

Convenient for matching routines


Telescope: second and third order

0

211

2116 )(2sin)cos(

21

ddKKT x

0

212

21126 )(sin)cos( ddKKT x

0

212

21216 )(cos)cos(

ddKKT x

01662

221

0211262

0

2321

2211226

)(sin)cos(

)(sin)sin(

2)(sin)cos(

dTK

ddKKT

ddK

dKKU

x

xx

From equations of motionKarl L. Brown, SLAC-PUB-4159


Telescope: optical functions

0

212

2

)(2sin1

ddKK

dd

x

0

21)(2exp ddKKiQ x

0

212

111

1262 )(sin

ddKK

RT

dd

x

2

111

116126

111

126622

2

2RTT

RU

dd

0

212 )(2cos

ddKKdd

x22

1

baW

ddb

dd

dda

One vector describeschromaticities of and


Telescope: conclusions

• Simplicity of the matching and understanding chromatical properties.

• One vector describes chromaticities of and . Other elements of the FF region have their influence.

• Sextupoles should be almost in phase with both FF quadrupoles.• It is difficult to cancel 1st order chromaticities of and

simultaneously with 1st order chromaticity of phase advance.• Once tuned it does not change even if * changes.• Additional sextupoles provide knobs for higher order chromaticity

tuning.


Example with L*=2m


Example with L*=2mNonlinear aberrations optimized Nonlinear chromaticity optimized

L*=2 m, K2=-14 m-3 , y= 5149 m, dx=0.077 m,K2 y=72000 m-2

L*=2 m, K2=-7 m-3 , y= 7711 m , dx=0.108 m,K2 y=54000 m-2


Example with L*=2m

Wx=32Wy=15

x=0 x=-6.4e-3y=0 y=-4.3e-5Wx=32 Wx=3Wy=15 Wy=4


Example with L*=2m

x=-6.4e-3 y=-4.3e-5Additional sextupole OFF

x=-6.4e-3 y=-4.3e-5Additional sextupole ON𝐵𝑋= 1

𝛽𝑥

𝑑 𝛽𝑥

𝛿


Example with L*=2m

QX0= 4 QY0= 3QX1= -3.53 QY1= -1.63QX2= -176 QY2= -37QX3= -1636 QY3= -36502

QX0= 4 QY0= 3QX1= -3.69 QY1= -1.63QX2= -175 QY2= -37QX3= -2048 QY3= -1826

x=-6.4e-3 y=-4.3e-5Additional sextupole OFF

x=-6.4e-3 y=-4.3e-5Additional sextupole ON


Why additional sextupoles work

22

2

2

2

0,

2

2

2

022

ddd

ddd

ddd

ddLK

dd xx

x


Trajectories L*=2m


Trajectories L*=1.5m


Trajectories L*=1m


Trajectories L*=0.7m


Summary about FF quadrupoles

L*, m SEQ0G(T/m)

SEQ0L(m)

SEQ0R, (m)@20x

SEQ1G(T/m)

SEQ1L(m)

SEQ1R, (m)@20x

0.7 95 4.6 0.013 96 2 0.021 97 4.3 0.013 98 2 0.021.5 97 3.9 0.013 99 2 0.0192 94 3.6 0.012 93 2 0.019

E=175 GeV, x=2.1 nm

E.Paoloni for SuperBItalyP.V

obly

for C

tau

Nov

osib

irsk


Conclusions• Presented study is independent of head-on or CW collsions.• The major source of the DA limitation is the –I Y chromatic correction

section through the sextupole length effect.• Sextupole and octupole corrections works well.• For L*=2 m we have Ax > 100x and Ay > 700y, band width > 2% which

seems quite enough to start.• Telescopic transformations provide simplicity of the matching and

understanding chromatical properties.• Sextupoles should be almost in phase with both FF quadrupoles.• Additional sextupoles provide knobs for higher order chromaticity tuning.• To define L* feedback from detector is required.• Key questions is prototype of the final focus quadrupole (CERN

superconductive group ?!).• Include realistic arcs and detector field.• Energy acceptance due to beamstrahlung should be as large as possible.

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Nonlinear properties of the FCC/TLEP final focus with respect to L *

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