linear imperfections equations of motion with imperfections: smooth approximation orbit correction...
DESCRIPTION
Skew Multipoles: Example Skew Quadrupole normal quadrupole: clockwise rotation by 45 o skew quadrupole 3 p+p+ p+p+TRANSCRIPT
Linear Imperfections
equations of motion with imperfections: smooth approximation
orbit correction for the un-coupled case
transfer matrices with coupling: element and one-turn
what we have left out (coupling)
perturbation treatment: driven oscillators and resonances
1
sources for linear imperfections
CAS, Darmstadt 2009 Oliver Brüning CERN BE-ABP
sources for linear imperfections:
ygB
xxgB
x
y
⋅−=
Δ+⋅−= )~(xxx Δ+=~
2
Sources for Linear Field Errors
-magnetic field errors: b0, b1, a0, a1
-powering errors for dipole and quadrupole magnets
-energy errors in the particles change in normalized strength
-feed-down errors from quadrupole and sextupole magnets
example: feed down from a quadrupole field
dipole + quadrupole field component
-roll errors for dipole and quadrupole magnets
p+
Skew Multipoles: Example Skew Quadrupolenormal quadrupole: clockwise rotation by 45o skew quadrupole
€
By = g ⋅ x ⇒ Fx = −q ⋅v ⋅g ⋅ xBx = g ⋅ y ⇒ Fy = +q ⋅v ⋅g ⋅ y xNvqFxNB
yNvqFyNB
yx
xy
⋅⋅⋅−=⇒⋅−=
⋅⋅⋅−=⇒⋅+=
xB
g y
∂∂
= ⎟⎟⎠⎞
⎜⎜⎝⎛
∂∂
−∂∂
=xB
yB
N xy
21
3
p+ p+
Sources for Linear Field Errors
-magnet positioning in the tunnel transverse position +/- 0.1 mmroll error +/- 0.5 mrad
-tunnel movements:slow driftscivilizationmoonseasonscivil engineering
-closed orbit errors beam offset inside magnetic elements
-energy error: dispersion orbit
sources for feed down and roll errors:
4
Equation of Motion ISmooth approximation for Hills equation:
perturbation of Hills equation:
in the following the force term will be the Lorenz force of a charged particle in a magnetic field:
)/()),(),(()()( 202
2 pvssysxFswswdsd ⋅=⋅+ω
0)()()(22 =⋅+ sωsKsω
dsd
)sin()( 00 φω +⋅⋅= sAsω
(constant -function and phase advance along the storage ring)
0)()( 202
2 =⋅+ sωsωdsd ω
K(s) = const
LQ /2 00 ⋅= πω(Q is the number of oscillations during one revolution)
BvqFrr
×⋅=5
yxw ,=
Equation of Motion Iperturbation for dipole field errors:
perturbation of Hills equation:⎪⎩
⎪⎨⎧
⋅−⋅−
−=⋅+
)()()()(
1
1
02
022
sysxk
ksxsx
dsd
κω
pB
pvF yΔ
−=⋅
6
perturbation for quadrupole field errors:
ypg
pvF
xpg
pvF
y
x
⋅Δ
+=⋅
⋅Δ
−=⋅
normalized multipole gradients:
]/[][3.00 cGeVp
TBk Δ⋅=
]/[]/[3.01 cGeVp
mTN⋅=κ
]/[]/[3.01 cGeVp
mTgk ⋅=
Coupling I: Identical Coupled Oscillators
π mode:
ω mode:
kk 20 +=πω0k=ωω
fundamental modes for identical coupled oscillators:
0)()( 22
22
2 =⋅+ tqtqdtd
πω0)()( 12
12
2 =⋅+ tqtqdtd
ωω
yxtq +=)(1 yxtq −=)(2
7
weak coupling (k << k0):
description of motion in unperturbed ‘x’ and ‘y’ coordinates
degenerate mode frequencies
k0k
0kk
0k
0k
solution by decomposition into ‘Eigenmodes’:
distributed coupling: )()()( 12
22 sysxsx xds
d ⋅−=⋅+ κω
)()()( 12
22 sxsysy yds
d ⋅−=⋅+ κω
8
Coupling II: Equation of Motion in Accelerator
0)()( 22222
2 =⋅+ sqsqdtd ω0)()( 1
2112
2 =⋅+ sqsqdtd ω
ybxasq ⋅+⋅=)(1 ydxcsq ⋅+⋅=)(2
With orthogonal condition: 0=⋅+⋅ dca
take second derivative of q1 and q2:
2
1
22
1
222
1
22
1
22
21
2;
21
2 ⎟⎟⎠⎞
⎜⎜⎝⎛ −
+−−
=⎟⎟⎠⎞
⎜⎜⎝⎛ −
++−
=κωω
κωω
κωω
κωω yxyxyxyx ca
9
Coupling II: Equation of Motion in Accelerator
0)()( 22222
2 =⋅+ sqsqdtd ω0)()( 1
2112
2 =⋅+ sqsqdtd ω
use Orthogonal condition for calculating a,b,c,d (set b=1=d)
expressions for ω1 and ω2 as functions of a, b, c, d, ωx, ωy
with:
€
ω1,22 = 1
2 ⋅ ωx2 +ωy
2( ) ± Ω
2222
21 ⎟⎟⎠⎞
⎜⎜⎝⎛ −
+=Ω yx ωωκ
yields:
very different unperturbed frequencies:
10
Coupled Oscillators Case Study: Case 1
expansion of the square root:
12
2
1
22
>>⎟⎟⎠⎞
⎜⎜⎝⎛ −
κωω yx
xyx
x ωωω
κωω ≈−
+= 22
21
1 yyx
y ωωω
κωω ≈−
−= 22
21
2
‘nearly’ uncoupled oscillators
€
ω1,22 = 1
2 ⋅ ωx2 +ωy
2( ) ± 12 ⋅ ωx
2 −ωy2( ) ⋅
2κ1
ωx2 −ωy
2( )
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
+1
εε 2111 +≈+
1;1;1;1 =−≈=≈ dca
almost equal frequencies:
11
Coupled Oscillators Case Study: Case 2
Ω±= ~02,1 ωω
keep only linear terms in Δ
Δ−= 21
0ωωyΔ+= 21
0ωωx
20
221
20
22,1 ωκωω ⋅Δ+±=
expansion of the square rootfor small coupling and Δ:
20
22 2ωωω ≈+ yx Δ≈− 022 2ωωω yx
20
2
40
21
02,1 1ωω
κωω Δ+±⋅=
2
20
21
21~ Δ+⋅=Ω
ωκ
with:
( )yx ωωω +=21
0
measurement of coupling strength:
12
Coupled Oscillators Case Study: Case 2
measure the difference in the Eigenmode frequencies while bringing the unperturbed tunes together:
220
21
02,1 21
Δ+⋅±=ωκωω
the minimum separation yields the coupling strength!!
ω1,2
ωx
ωy
0ωκ
Δ
Coupled Oscillators Case Study: Case 2
with
initial oscillation only in horizontal plane:
€
x(s) = A ⋅cos ˜ Ω ⋅s( ) ⋅cos 12 ω1 +ω2[ ] ⋅ s+ 1
2 [φ1 + φ2]( )
yxtq −=)(1
yxtq +=)(2
13
0)0(;0)0(;0)0(;)0( =′==′= yyxAx
)sin( 111 φω +⋅⋅= sAq )sin( 222 φω +⋅⋅= sAqand
and
€
y(s) = −A ⋅sin ˜ Ω ⋅s( ) ⋅sin 12 ω1 +ω2[ ] ⋅ s+ 1
2 [φ1 + φ2]( )
€
ω1,2 = 12 ⋅ ωx +ωy( ) ± ˜ Ω
modulation of the amplitudes
sum rules for sin and cos functions:
Beating of the Transverse Motion: Case I
modulation of the oscillation amplitude:
frequencies can not be distinguished and energy can be exchanged between the two oscillators
two almost identical harmonic oscillators with weak coupling:
π-mode and ω=mode frequencies are approximately identical!
x
s
14Ω~
( )yx ωωω +=21
0
Driven Oscillators
large number of driving frequencies!
equation of motion driven un-damped oscillators:
)(
..
21 2
2
2 )()()( klmLyx smlsk
mlkklmwds
dwds
d eWswswQsw φωω π
ωω +⋅⋅+⋅+⋅⋅− ∑=++
15
Perturbation treatment:substitute the solutions of the homogeneous equation of motion:
into the right-hand side of the perturbed Hills equation andexpress the ‘s’ dependence of the multipole terms by their Fourierseries (the perturbations must be periodic with one revolution!)
)sin()( 00 φω +⋅⋅= sAsω
Driven Oscillators
general solution: )()()( swswsw sttr +=
)cos()()()()( 02
01
02
2 φωωω +⋅⋅=⋅+⋅⋅+ − ssΩsωsωQsω dsd
dsd
16
single resonance approximation:
consider only one perturbation frequency (choose ):
without damping the transient solution is just the HO solution
)sin()( 00 φω +⋅⋅= sasωtr
0ωω ≈
Lyx mlk πωωω 2++=
Driven Oscillators
resonance condition:
0ωω =n
17
)](cos[)()( 20
ωaωωω
−⋅⋅= sΩsωststationary solution:
where ‘ω’ is the driving angular frequency!and W(ω) can become large for certain frequencies!
22
001
1)(
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⋅=
+−ω
ω
ω
ωω
Qnn
nΩΩ
justification for single resonance approximation:
all perturbation terms with: 0ωω ≠n de-phase with the transient
no net energy transfer from perturbation to oscillation (averaging)!
Resonances and Perturbation Treatment
nQLn LQ =⏐⏐⏐⏐ →⏐⋅= ⋅=0
/20
00/2 πωπω
example single dipole perturbation:
resonance condition:
∑∞
−∞=
⋅⋅⋅−=⋅+ ⋅n
Lsnlkswsw Ldsd )/2cos()()( 1
02
022 πω
)()(00 ssk
pvsF
L −⋅−=⋅
d
avoid integer tunes!
(see general CAS school for more details) 18€
ΔCO(s) =β (s)
2sin(πQ)⋅ Δk0(t) ⋅ β (t) ⋅cos(|φ(t) −φ(s) | −πQ)∫ dt
Fourier series of periodic d-function
Resonances and Perturbation Treatment integer resonance for dipole perturbations:
19 dipole perturbations add up on consecutive turns! Instability
assume:
Q = integer
Resonances and Perturbation Treatment integer resonance for dipole perturbations:
20
dipole perturbations compensate on consecutive turns! stability
assume:
Q = integer/2
Resonances and Perturbation Treatment Orbit ‘kink’ for single perturbation (SPS with 90o Q = n.62):
21
)|)()(cos(|)sin(2)()(
)( 000 Qsskl
Qss
sCO πφφπ
−−⋅Δ⋅=Δ
)()( 00 εε −′Δ=+′Δ sOCsOC
Resonances and Perturbation Treatment
2/22 0
/20
00nQLn LQ =⏐⏐⏐⏐ →⏐⋅=⋅ ⋅= πωπω
example single quadrupole perturbation:
resonance condition:
avoid half integer tunes!
(see general CAS school for more details) 22
xkpvsF
⋅−=⋅ 1)( )cos()( 0,00 φω +⋅⋅= sAsω xwith:
∑∞
−∞=
±⋅±⋅⋅−=⋅+n
xx sLnAswsw Llk
dsd )]/2cos([)()( 0,0
120,2
2
2 φωπω
dtQstttkQs
s∫ −−⋅⋅Δ⋅=
Δ)2|)()(|2cos()()(
)2sin(21
)()(
10
πφφπ
Resonances and Perturbation Treatment half integer resonance for quadrupole perturbations:
23
quadrupole perturbations add up on consecutive turns! Instability
assume:Q = integer + 0.5
feed down error:
ybvqFybB yx ⋅⋅⋅+=⇒⋅= 11
Resonances and Perturbation Treatment
nQQLn yxLQ
yx =±⏐⏐⏐ →⏐⋅=± ⋅= /20,0, /2 πωπωω
example single skew quadrupole perturbation:
resonance condition:
avoid sum and difference resonances!
24
ypvsFx ⋅−=
⋅ 1)( κ )cos()( 0,00 φω +⋅⋅= sAsy ywith:
∑∞
−∞=
±⋅±⋅⋅−=⋅+n
yx sLnAsxsx Ll
dsd )]/2cos([)()( 0,0
120,2
2
2 φωπω κ
difference resonance stable with energy exchange sum resonance instability as for externally driven dipole
coupling with:
25
drive and response oscillation de-phase quickly no energy transfer between motion in ‘x’ and ‘y’ plane
small amplitude of ‘stationary’ solution:
no damping of oscillation in ‘x’ plane due to coupling
coupling is weak tune measurement in one plane will show both tunes in both planes but
with unequal amplitudes
tune measurement is possible for both planes
yxyx QQorQQ <<>>
2)(2])(1[
1
00
20)(
ωω
ωω
ω
Q
ΩΩ+−
⋅=
Resonances and Perturbation Treatment: Case 1
coupling with:
26
drive and response oscillation remain in phase and energy can be exchanged between motion in ‘x’ and ‘y’ plane:
large amplitude of ‘stationary’ solution:
damping of oscillation in ‘x’ plane and growth of oscillation amplitude in ‘y’ plane
‘x’ and ‘y’ motion exchange role of driving force!
each plane oscillates on average with:
Impossible to separate tune in ‘x’ and ‘y’ plane!
yx QQ ≈
( )yx QQ +21
2)(2])(1[
1
00
20)(
ωω
ωω
ω
Q
ΩΩ+−
⋅=
Resonances and Perturbation Treatment: Case 2
Exact Solution for Transport in Skew Quadrupolecoupled equation of motion: 01 =⋅+′′ yx κ
01 =⋅+′′ xy κand
27
can be solved by linear combinations of ‘x’ and ‘y’:
0)()( 1 =+⋅+′′+ yxyx κ 0)()( 1 =−⋅−′′− yxyx κand
solution as for focusing and defocusing quadrupole
transport matrix for ‘x-y’ and ‘x+y’ coordinates for κ1 > 0:
iniend yxyx
ll
ll
yxyx
⎟⎟⎠⎞
⎜⎜⎝⎛
′−′−
⋅⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⋅
=⎟⎟⎠⎞
⎜⎜⎝⎛
′−′−
)cos()sin(
)sin()cos(
111
1
11
κκκκκ
κ
iniend yxyx
ll
ll
yxyx
⎟⎟⎠⎞
⎜⎜⎝⎛
′+′+
⋅⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⋅
=⎟⎟⎠⎞
⎜⎜⎝⎛
′+′+
)cosh()sinh(
)sinh()cosh(
111
1
11
κκκκκ
κ
Transport Map with Couplingtransport map for skew quadrupole:
transport map for linear elements without coupling:
inisqend zMz rr⋅=
with:
28
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
′
′=
yyxx
zr
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−
−−=
adcbbadccbaddcba
M sq
11
11
κκ
κκand
inilend zMz rr⋅=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=
4443
3433
2221
1211
0000
0000
mmmm
mmmm
M lwith
Transport Map with Couplingcoefficients for the transport map for skew quadrupole:
with:
29
NsNsNsNsN
NsNsNsNsN
dcba
2/)]sinh()[sin(2/)]cosh()[cos(
2/)]sinh()[sin(2/)]cosh()[cos(
−−
++
====
One-Turn Map with Couplingone-turn map around the whole ring:
)()()( 000 szsTLsz rr⋅=+
30
∏=i
iMT
⎟⎟⎠⎞
⎜⎜⎝⎛
=NmnM
T
T is a symplectic 4x4 matrix
with:
with: being 2x2 matrices 16 parameters in total
nmNM ,,,
notation:
STSTt =⋅⋅ with: ⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−
−=
0100100000010010
S
determines n*(n-1)/2 parameters for a n x n matrix
Parametrization of One-Turn Map with Couplinguncoupled system: parameterization by Courant-Snyder variables
31
)sin()cos( μμ ⋅+⋅= JIT
⎟⎟⎠⎞
⎜⎜⎝⎛
=1001
I ⎟⎟⎠⎞
⎜⎜⎝⎛
−−=
aγa
J
with:
aγ
21+=
T is a 2 x 2 matrix 4 parameters
T is symplectic determines 1 parameter
3 independent parameters
Parametrization of One-Turn Map with Couplingrotated coordinate system:
using a linear combination of the horizontal and vertical position vectors the matrix can be put in ‘symplectic rotation’ form
or:
32
⎟⎟⎠⎞
⎜⎜⎝⎛ −⋅⎟⎟⎠⎞
⎜⎜⎝⎛⋅⎟⎟⎠⎞
⎜⎜⎝⎛−
=−−
)cos()sin()sin()cos(
00
)cos()sin()sin()cos( 1
2
11
φφφφ
φφφφ
IDDI
AA
IDDIT
1−⋅⋅= RURT with:
is a symplectic 2x2 matrice 3 independent parameters
total of 10 independent parameters for the One-Turn map
D
2,1);sin()cos( =⋅+⋅= iJIA iiiii μμ
One-Turn Map with Couplingrotated coordinate system:
33
x
y
x~y~
rotated coordinate system:
new Twiss functions and phase advances for the rotated coordinates
)sin()cos( iiii JIA μμ ⋅+⋅= ⎟⎟⎠⎞
⎜⎜⎝⎛
−−=
ii
iiiJ αγ
βα
€
cos(μ1) − cos(μ2) = 12 Tr M −N( )[ ]
2+ det(m + n+)
with:
( ))(
det)2tan(
21 NMTr
nm−+−
=+
φ
⎟⎟⎠⎞
⎜⎜⎝⎛−
−⏐⏐⏐⏐ →⏐⎟⎟⎠
⎞⎜⎜⎝⎛ −
+
acbd
dcba matrixadjoint
Summary One-Turn Map with Coupling
coupling changes the orientation of the beam ellipse along the ring
34
a global coupling correction is required for a restoration of the uncoupled tune values (can not be done by QF and QD adjustments)
coupling changes the Twiss functions and tune values in thehorizontal and vertical planes
a local coupling correction is required for a restoration of the uncoupled oscillation planes ( mixing of horizontal and vertical kicker elements and correction dipoles)
What We Have Left Out
dispersion beat:
-beat: skew quadrupole perturbations generate -beatsimilar to normal quadrupole perturbations
35
integer tune split and super symmetry
skew quadrupole perturbations generate vertical dispersion
the (1,-1) coupling resonance in storage rings with supersymmetry can be strongly suppressed by an integer tune split
general definition of the coupling coefficients:
dsesss srssiyxr
Lyx ))()((1,
2
)()()(21 ⋅+±
± ⋅⋅⋅= ∫πφφκ
πκ⏐⏐ →⏐= = ω
ωκκ /11
Orbit Correctiondeflection angle:
36
)(]/[][3.0
iy
i sXcGeVp
lTB′Δ=
⋅Δ⋅−=q
))(sin()()( iii sssZ φφq −⋅⋅⋅=Δtrajectory response:
))(cos()(/)( iii sssZ φφq −⋅⋅=′Δ
closed orbit bump
compensate the trajectory response with additional dipolefields further down-stream ‘closure’ of the perturbationwithin one turn
Orbit Correction3 corrector bump:
37
12
113
23
133 )cos(
)tan()sin( q
φφφq ⋅⋅⎟⎟⎠
⎞⎜⎜⎝⎛
Δ−ΔΔ
= −−
−
closure
sensitive to BPM errors; large number of correctors
123
13
2
12 )sin(
)sin( qφφ
q ⋅ΔΔ
⋅−=−
−
limits
SVD Algorithm Ilinear relation between corrector setting and BPM reading:
38
global correction:
),...,,( 21 mcccCOR = vector of corrector strengths
CORABPM ⋅=
problem A is normally not invertible (it is normally not even a square matrix)!
),...,,( 21 nbbbBPM = vector of all BPM data
A being a n x m matrix
BPMACOR ⋅= −1
solution minimize the norm: CORABPM ⋅−
SVD Algorithm IIsolution:
39
singular value decomposition (SVD):
find a matrix B such that
pm
i
pixx
/1
⎟⎠⎞⎜
⎝⎛
= ∑
any matrix can be written as:
where O1 and O2 are orthogonal matrices and D is diagonal
CORBABPM ⋅⋅−
attains a minimum with B being a m x n matrix and:
21 ODOA ⋅⋅=
tOO =−1
SVD Algorithm IIIdiagonal form:
40
define a pseudo inverse matrix:
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛==⋅
10
011 O)kDD
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=
0000
00
0
00
0
0 22
11
LMLL
MLL
LL
O
L
LOM
kk
D
σ
σσ
),min( mnk ≤
⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
=
0
0/10
0
0
00
0/10
0
00
0/1
11
11
11
M
M
LML
O
LMLL
M
M)σ
σσ
D
1k being the k x k unit matrix
SVD Algorithm IVcorrection matrix:
41
main properties:
define the ‘correction’ matrix:
( ) ( ) ktt ODOODOBA 11221 =⋅⋅⋅⋅⋅=⋅
)
SVD allows you to adjust k corrector magnets
tt ODOB 12 ⋅⋅=)
if k = m = n one obtains a zero orbit (using all correctors)
for m = n SVD minimizes the norm (using all correctors)
the algorithm is not stable if D has small Eigenvalues can be used to find redundant correctors!
),min( mnk ≤
Harmonic FilteringUnperturbed solution (smooth approximation):
42
orbit perturbation
spectrum peaks around Q = n small number of relevant terms!
periodicity:
022 =⋅⋅+′′ xQx Lπ sQi LeAsx ⋅⋅⋅⋅=
π2
)(
)(22 sFxQx L =⋅⋅+′′ π
sni
nn
LefsF ⋅⋅⋅⋅=∑π2
)( sni
nn
LedsCO ⋅⋅⋅⋅=∑π2
)(
( ) ( )2222 QQfd
LL
nn ππ −
=
Most Effective Correctorthe orbit error is dominated by a few large perturbations:
43
brut force:
minimize the norm: CORBABPM ⋅⋅−
using only a small set of corrector magnets
select all possible corrector combinations time consuming but god result
selective: use one corrector at the time + keep most effective much faster but has a finite chance to miss best solution and can generate π bumps
MICADO: selective + cross correlation between orbitresidues and remaining correcotr magnets
Example for Measured & Corrected Orbit Data LEP:
44