6. betatron coupling

sources: skew quadrupoles, solenoid fields

concerns: reduction in dynamic (& effective physical) aperture;increase of intrinsic & projected y emittance in e- storage rings; degraded tuning performance; increased spot size at collision point

two new eigenmodes, no longer purely x or y

xyQd

yd

yxQd

xd

202

2

202

2

2

y-x v,

2 yx

u

0

0

202

2

202

2

vQd

vd

uQd

ud

20

2

20

2

QQ

QQ

v

u

222 vu QQ

2 coupled linear oscillators

: coupling

normal-mode coordinates:

decoupled equations new eigen-frequencies

frequency split:measure of strength of coupling

in a real storage ring, the coupling is not constant, but variesaround the ring (localized sources) 2 global parametersdriving terms for sum and difference resonance

sources of coupling:skew quadrupole field errors, vertical orbit offset in sextupole

Nyx

Nxy

NxyH

~''

~''

'~''

'~''

)(

Ryx

Rxy

xpypRH yx

solenoid fields (detector field, solenoids against e-cloud,…)

two new eigenmodes of coupled betatron oscillations;beam is tilted in x-y plane, e.g.,

tilt angle varies along beam line

two linear resonances in Hamiltonian

yxyx

yxyxyyxx

IIq

IIqIQIQH

)cos(

)cos(

0

000

sum resonance

difference resonance

uncoupled linear motion

0 qQQ yx

)/2)()()(()()()(

2

1 LsqQQssi

yxsyxyxessskds

resonance driving terms:

ks(s): normalized gradient of skew quadrupoleL: circumference

minimizing the driving term improved beam lifetimeincreased dynamic aperturesmaller emittance

electron storage ringvertical emittance due to weak betatron coupling:

dsQQ

sWsW

Q

sWH

ds

Cx

qy

sinsin

)()(Re2

sin

)(1

16

**

2

2

31

3

2

qQQQ

DDDDH

C

yx

x

q

22

13

''2

m 1084.3

)())()(())()(()()()()( yxyxyx QQzzssiLs

s

yxs ezzzdzksW

driving term ‘including all Fourier components’

where

on resonance: 2)(sW

(Raubenheimer)

(A) first turn analysis

difference orbits

kick

identify coupling sourceand devise correction

one can fit large number of orbits & BPM data to determine skew component of each magnet

measuring betatron coupling

(B) kick response over many turns

envelopes ofhorizontal andvertical oscillationsexhibit beating

plane of kick

orthogonalplane

beating period

2max

2min

x

xS

brevTf

S1

define

one can show that ! exampleATF

|_|

frequency spectrum of horizontal pick up viewed on a spectrum analyzer

monitoring betatron coupling at the ATF Damping Ring

evolution of the peak signal in the frequency spectrum vs. time,on an oscilloscope; the slowvariation reflects synchrotronmotion; the fast period is dueto transverse coupling;the amplitude and period of themodulation can be used todetermine the driving term |_|,in this case |_|~0.02

(C) closest tune approach

near the difference resonance 0qQQ yx

22_, 2

1 qQQqQQQ yxyxIII

the tunes of the two eigenmodes, in the vertical plane, are

uncoupled tunes

tunes can approach each otheronly up to distance |_|

correction strategy;use two skew quadrupoles(ideally with x-y)~/2) tominimize |_|, namelythe distance of closest tune approach

|_|

closest tune approach in the PEP-II HER before final correction; shown arethe measured fractional tunes as a function of the horizontal tune knob; the minimum tune distance is equal to the driving term |_| of the differenceresonance

(D) compensating the sum resonance

near difference resonance, energy exchange x y

near sum resonance, motion is unstable

is solution for

(note: thesephases arenot exactly thesame asbefore but

0

...~~

sin~

...~~

sin~

yx

yxyxy

y

yxyxx

x

II

IIH

I

IIH

I

0~~

,0 ,0

...~~

sin~

...~~

sin~

yxyx

yxyx

y

y

yxyx

x

x

II

IIH

I

IIH

I

qQQ yx yxyxyx Q ,,,

~

transforming intoresonance basis)

resonance stop band

in principle, |+| could be compensated by adjusting two skew quadrupolesso as to minimize the stopband width,ideally at locations separated by

21 nyx

minimum number of skew quadrupoles for global correction in a ring:

2 for ||2 for |+|2 for Dy

6: minimum number for independentcorrection of 6 global effects and emittanceoptimization

this does not yet correct the local coupling effects, which may alsocontribute to emittance growth, especially in lepton machines

(E) emittance near difference resonance for leptons

near the difference resonance

14

12

2

2

0

Q

Qxx

22,)( IIIyx QqQQQ

IIIIII QQQ ,

2,

2_

0 21

IIIxx Q

where

measured tunedifference

combining the above relations yields

(Guignard)

recipe: infer ex from synchrotron light monitor for different values of QI,II; then determine x0 and |_| by nonlinear fit

|_|

Horizontal emittance as a function of the tune separation QI,II at the ATF Damping Ring; the measured data and the result of a nonlinear fit are shown; fit gives x0~2.44 nm, |_|~0.037 (closesttune approach measured at the same time yielded |_|~0.042)

(F) emittance near sum resonance

qQQQ

Q

Q

Q

yxIII

III

IIIxx

III

xy

,,

2,,

2

2,,

2

0

2,,

2

2

0

where

5

3

5

2

|+|

near the sum resonance

(derived from Guignard’sexpressions)

alternative theoretical formula from T. Raubenheimer;simulation results from MAD (Chao formalism. probably notapplicable for vicinity of sum resonance);simulation result from SAD (Ohmi-Oide-Hirata formalism);

caution!

4 different answers! experiments at ATF unclear

personal preference for SAD

(G) local coupling correction

minimizing vertical closed-orbit response to horizontal steering (at KEK ATF DR); by measuring cross-planeresponse matrix for all dipole correctors and all BPMs, and computing skew-quad correction based on optics model

(J. Urakawa,2000)

(H) coupling transfer function

excite beam in x detect coherent y motion

used for continuous monitoring of coupling at the CERN ISR in the 1970s;is considered for LHC coupling control

r

i

ir

c

c

ccA

arctg

22

amplitude and phaseof vertical response;complex value of _

ISRcouplingtransferfunction

Nn

mMT

1VUVT

B0

0AU

IC

CIV

,1121

1222

2221

1211

CC

CC

CC

CCCC 12 C

mathematically exact formulation of coupling

4x4 one-turn matrix

Edwards-Teng factorization

new matrix U is block-diagonal;A and B are of the same formas for the uncoupled case

factorization matrix V describesthe coupling

symplectic conjugate of C

aaaaa

aaaaa

sincossin

sinsincosA

bbbbb

bbbbb

sincossin

sinsincosB

ba

ba

ba

ba

ba

,

,

,

,

,

01

G

b

a

G0

0GG

block-diagonal matricesfor eigenmodes are of theCourtant-Snyder type

2x2 matrices for normalization of A, B

IC

CIGVGV

1

4x4 normalizationmatrix

normalized coupling matrix

)sincos(

cos

1222 aab

aaa

CCA

y

x

bb

bbbb

CCA

y

x

cos

)sincos( 1211

if mode a is excited

if mode b is excited

D. Sagan & D. Rubin, PRST-AB 2, 074001 (1999)

1

0

1

0

1

0

1

0

2sin2

2cos2

2sin2

2cos2

N

n yn

na

py

y

N

n xn

na

py

y

N

n xn

na

px

x

N

n xn

na

px

x

p

ynQ

NS

S

y

ynQ

NC

C

p

xnQ

NS

S

p

xnQ

NC

C

the complete coupling matrix can be determined by harmonic analysis,e.g., excite beam at eigenmode frequency a, measure responsein both planes over N turns and form 8 sums:

the px is obtained bycombining information fromtwo nearby BPMs

1

1121

1222

pxpx

xx

pypy

yy

SC

SC

SC

SC

YX

X

CC

CC

exciting also the eigenmode b can serve as a test & each mode measurement gives more precise answer for half of the Cij

E. Perevedentsev, 2000

pxpx

xx

SC

SCX

pypy

yy

SC

SCY

flat versus round beams for e+e- colliders

yx

brevb nfNL

4

2

00

y0

1 ,

1 xxx

x

yyxxx y ,

)(2 ,

)(2 yxy

yeby

yxx

xebx

rNrN

yx

1

y

x

y

x

y

revb

e

bnfN

rL

1

2

1

luminosity

emittances could be varied by coupling:

naturally flat due tosynchrotron radiation

beam sizes at collision point

beam-beam tune shift

one wants to maximize both:constraint

round beams give 2x higher luminosity, but requires ! yx

Summarytune measurements

FFT with interpolation, Lob periodogrambeam transfer functionsphase locked loopmultibunch spectrum

function measurementsKphase advancecorrector excitationsymmetry pointR matrix from trajectory fit

phase advance measurementsmulti-turn BPMs & harmonic analysis

gradient errors1st turn, or closed-orbit distortionphase advance

bumpsmultiknobs

beam response to kick excitationcoherent dampingfilamentationchromaticity

betatron coupling various measurement techniques