6. betatron coupling sources: skew quadrupoles, solenoid fields concerns: reduction in dynamic (&...
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![Page 1: 6. betatron coupling sources: skew quadrupoles, solenoid fields concerns: reduction in dynamic (& effective physical) aperture; increase of intrinsic &](https://reader030.fdocuments.net/reader030/viewer/2022032805/56649ef45503460f94c07a6a/html5/thumbnails/1.jpg)
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
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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
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
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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,…)
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two new eigenmodes of coupled betatron oscillations;beam is tilted in x-y plane, e.g.,
tilt angle varies along beam line
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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
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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)
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(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
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(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
|_|
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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
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(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
|_|
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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
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(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
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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
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(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
|_|
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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)
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(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
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(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)
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(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
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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
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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
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)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)
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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
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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
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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