DRAFT: Main Linac Beam Dynamics
89
DRAFT: Main Linac Beam Dynamics Kiyoshi KUBO Draft 2006.04.24
description
DRAFT: Main Linac Beam Dynamics. Kiyoshi KUBO Draft 2006.04.24. ILC Main Linac Beam Dynamics. Introduction Lattice design Beam quality preservation Longitudinal Transverse Wakefield Single bunch Multi-bunch Dispersive effect Errors and corrections Initial alignment, fixed errors - PowerPoint PPT Presentation
Transcript of DRAFT: Main Linac Beam Dynamics
DRAFT:Main Linac Beam Dynamics
Kiyoshi KUBO
Draft 2006.04.24
ILC Main Linac Beam Dynamics
• Introduction • Lattice design• Beam quality preservation
– Longitudinal – Transverse
• Wakefield– Single bunch– Multi-bunch
• Dispersive effect
– Errors and corrections• Initial alignment, fixed errors• Vibration, ground motion, jitters, etc.
Note
• Very little will be discussed anything which are irrelevant to ILC.
• Basics will not be lectured here.– Beam optics: What are Emittance, Beta-function, Dispersion
function, etc.. – Wakefield: Definition of wake-function, etc..
• Numbers quoted here and simulation results shown here may be preliminary.
• Corrections, questions, and any comments will be welcome.
Introduction• Main Linac is very simple, compare with most of
other part of LC. – Basically many iteration of a simple unit.
• Analytical treatment is very limited. And most studies are based on simulations.– Even in simulations, approximations are
necessary for reasonable calculation time.• e.g., 2E10 particles cannot be simulated individually.
Detailed treatment of edge fields of magnets and cavities may not be necessary. Space charge can be ignored for high energy beams.
• “What can be ignored?” is important in simulation.
Note on Simulation CodesA lot of codes exist.Probably, two kinds:
1. Track mocro particles, each have 6 parameters, [x, y, z(or t), px, py, E(or pz)]
2. Track “slices”, each have 14 parameters, [x, y, z(or t), px, py, E(or pz), <xx>, <yy>, <xy>, <xpx>, <ypy>, <xpy>, <ypx> <pxpx>, <pxpy>, <pypy>]
Some codes cannot change z of particles in tracking.(This makes wakefield calculation significantly fast.)
“How accurate” and “How fast” may be correlated. (?)Code bench marking is being performed.
Parameter of IL Main Linac(ECM=500 GeV)
Beam energy 15 GeV to 250 GeV
Acc. gradient 31.5 MV/m
Bunch Population 1 ~ 2 x1010 /bunch
Number of bunches <= 5640 /pulse
Total particles <=5.64 x1013 /pulse
Bunch spacing >= 150 ns
Bunch Length 0.15 ~ 0.3 mm
Emittance x
Emittance y
1 x10-5 m-rad
3~8 x10-8 m-rad
Lattice design
• Basic layout– One quad per four cryomodules
• Or one quad/three
– Simple FODO cell
x/y = 75/65 degree phase advance/cell
Or, configuration may change along linac• Higher the energy, larger the beta-function• FOFODODO, FOFOFODODODO, etc., in high energy re
gion
– Vertically curved, following the earth curvature
Unit of main linacabout 240 units/linac
Cryomodule with magnet package
Cryomodules without magnet package
9-cell SC cavity
Magnet package
BPM
Quadrupole SC magnet
DipoleSC magnet
Beta-function
0
50
100
150
200
0 2 103 4 103 6 103 8 103 1 104
x (m)
y (m)
x,y (
m)
s (m)
Baseline designConstant phase advance
Beam size
0
5 10-5
0.0001
0.00015
0.0002
0
5 10-6
1 10-5
1.5 10-5
2 10-5
0 2000 4000 6000 8000 10000
x (=1E-5 m)
y(=2E-8 m)
x (m
) y (m
)
s (m)
Vertical:5~10 m at 15 GeV1.5 ~ 2.5 m at 250 GeV
Beta-function Possible alternative design
0
50
100
150
200
250
300
350
0 2 103 4 103 6 103 8 103 1 104
x (m)
y (m)
x,y (
m)
s (m)
Possible alternative designPhase advance ~ sqrt(E)
0
5 10-5
0.0001
0.00015
0.0002
0
5 10-6
1 10-5
1.5 10-5
2 10-5
0 2000 4000 6000 8000 10000
x (m) (=1E-5 m)
y (m) (=2E-8 m)
x (m
) y (m
)
s (m)
Beam size
BPM-Quad-Dipole corrector packageAlignment line
Designed Beam Orbit
Alignment and Beam Orbit in Curved Linac,Following earth curvature
(Vertical scale is extremely exaggerated)
cryomodule
-0.002
-0.0015
-0.001
-0.0005
0
0 50 100 150
Alignment lineDesign Orbit
y (m
)
s (m)
Alignment and Beam Orbit in Curved Linac,Following earth curvature
Design orbit and dispersion
-5 10-5
-4 10-5
-3 10-5
-2 10-5
-1 10-5
0
0
0.0005
0.001
0.0015
0.002
0.0025
0 50 100 150 200 250 300
Design Orbit and Dispersion
y (m) y (m)
y (m
)
y (m)
s(m)
Injection orbit and dispersion are non-zero, and should be matched to the optics.
Not zero
Beam Quality Preservation
Beam Quality
• Longitudinal particle distribution– Energy and arriving time
• E and t or z
• Transverse particle distribution– Horizontal and vertical position and angle
• x, x’, y, y’ (or x, px, y, py)
Generally, “Stable and small distributions at IP”is important.
Longitudinal Beam Quality
Longitudinal beam quality• Beam energy stability• Small energy spread.These require RF amplitude and phase stability,
which rely on RF control.The stability requirement in main linacs is less s
evere than in bunch compressors.
Main Linac does (almost) nothing to the timing of particles.
Longitudinal short range wakefield
0
5 1012
1 1013
1.5 1013
2 1013
2.5 1013
3 1013
3.5 1013
4 1013
0 0.001 0.002 0.003 0.004 0.005
WL
(V
/C/m
)
s (m)
Wake-function
0
1 10-6
2 10-6
3 10-6
4 10-6
5 10-6
-0.001 -0.0005 0 0.0005 0.001
Ch
arge
Den
sity
(C
/m)
z (m)
E(z) dz'z (z' )W (z z' )
Deceleration by wakefield
Charge density
0
1 104
2 104
3 104
4 104
5 104
6 104
7 104
8 104
-0.001 -0.0005 0 0.0005 0.001
Dec
eler
atio
n b
y w
akef
ield
(V
/m)
z (m)
Total acceleration (RF off-crest phase 4.6 deg. minimizing energy spread.)
3.125 107
3.13 107
3.135 107
3.14 107
3.145 107
0
5 104
1 105
1.5 105
2 105
-0.001 -0.0005 0 0.0005 0.001
RF fieldTotal acceleration
wakefield Acc
eler
atin
g fi
eld
(V/m
)
wakefield
(V/m
)
z (m)
Energy spread along linac. Initial spread is dominant.
1.5 108
1.52 108
1.54 108
1.56 108
1.58 108
1.6 108
0 5 1010 1 1011 1.5 1011 2 1011 2.5 1011
Positron
E (
eV)
E (eV)
0
0.002
0.004
0.006
0.008
0.01
0.012
0 5 1010 1 1011 1.5 1011 2 1011 2.5 1011
Positron
Electron
E/E
E (eV)
Undulator for e+ source
Energy Fluctuation, Required RF Stability
Common error of all klystrons
00phase00amplitude sincos eVeVE
00phase00amplitudekly
sincos1 eVeVn
E
Independent, random fluctuations of each klystron.
rad10sin1.0
101.0
rad102sin1.0
1021.0
error,each from 1.0stability For
30phase
4amplitude
20klyphase
3klyamplitude
E
E
En
En
E
E
E
E
E
phasecrest -off :
voltage total:
0
0
V
Transverse Motion
Transverse beam quality• Beam position at IP
– Offset between two beams should be (much) smaller than the beam size
• Beam size at IP, Emittance– From consideration of beam-beam interaction, ‘flat’ beam is
desirable.• vertical size is much smaller than horizontal size
– Beam size at IP is limited by emittance• Hour glass effect and Oide limit
– Then, vertical emittance need to be very small.• Vertical position stability and vertical emittance preservation ar
e considered
Beam position jitter
• There will be position feedback in Main Linac, BDS, then, at IP.
• In Main Linac, only fast jitter (faster than the feedback) should be important, unless it causes emittance dilution.
• The dominant source can be:– vibration of quadrupole magnets.– Strength jitter of quadrupole and dipole mognets (i
nstability of power supplies)
Estimation of beam position change due to quad offset change
Final beam position is sum of all quads’ contribution. Assuming random, independent offset, expected beam position offset is:
4/sin 22222 kNaEEky fqquad
fiiiiff
iiffiiif EEkay sin/
Final position change due to offset of i-th quad:
final toquadth - from cephaseadvan:
onbetafuncti final : quad,th -at on betafuncti :
value,-k:energy beam final : quad,th -at energy beam :
quad,th - of value-k: quad,th - ofoffset :
i
i
kEiE
ikia
i
fi
ifi
ii
square value-k of average: on,betafuncti of average:
quads, ofnumber : offset, of rms:
2k
Na q
Please confirm or confute expressions in this page.
Estimation of beam position change due to magnet strength change
error.strength quad theis , where,by page previous in the Replace iiiii kkaka Final position change due to strength change of i-th quad:
error.strength dipole theis , where,by page previous in the Replace ,0,0 iiii kkka Final position change due to strength change of i-th dipole:
Strength fluctuation is important especially for curved linac, because of non-zero designed dipole kicks, even without alignment errors.
Note on Beam quality - Emittance
• Our goal is high luminosity• For (nearly) Gaussian distribution, emittance is a good
measure of luminosity– We are usually interested in this case.
• If the distribution is far from Gaussian, correlation between emittance and luminosity is not necessarily good.
• For accurate evaluation, beam-beam interaction at IP should be considered.
• We use emittance dilution as a measure of quality dilution in the main linac, anyways.
),,(),,( zyxzyxvvdxdydzL
Luminosity, general definition
beam lectron)positron(e ofdensity particle :
beam lectron)positron(e of velocity :
)(
)(
ρ
v
Instantaneous luminosity:
)(int tdtLL
Integrated Luminosity
Luminosity per bunch crossing for Gaussian beam head on collision, no de-formation due to beam-beam force
2,
2,
2,
2,2 yyxx
NN
zz
yxyx
ctzyx
),(
2
),()(),(
22
)(),( 2))((
exp,2
)(exp
yyxxzzbc dydxNcNdzdtL ,,,,,,2
zyxNzyx ),,(
If e+ and e- beams are the same size, luminosity is proportional to inverse of beam’s cross section.
2E10 particlesy=10 my’=1 rad
Luminosity: L0
Emittance: 1E-11 m
2E10 – 2E4 particlesy=10 my’=1 rad
17 mm
2E4 particles
Luminosity: 0.999999 L0
Emittance: 2E-11 m
1/1E6 of haloAlmost the same luminosityEmittance increase by factor 2
Example where emittance does not well correlated with luminosity
NOTE
• We use emittance as a measure of quality in the main linac, anyways.
• Halos, tails far from core should be ignored in calculating the emittance.– Effects of halos or tails should be considered in
other context.
• It is usually OK, though not always appropriate.
Dominant Sources oftransverse Emittance dilution
• Wakefield (transverse) of accelerating cavities– Electromagnetic fields induced by head
particles affect following bunches.
• Dispersive effect– Different energy particles change different
angles by electromagnetic fields (designed or not-designed).
Effects of Transverse Wakefield
Transverse Wakefield ofAccelerating cavities
• Short range - Single bunch effect– Wakefunction is monotonic function of distance– Not seriously important for ILC
• Long range - Multi-bunch effect– Many higher order mode– For ILC
• Need to be damped• Frequency spread is needed.• Need to be careful for x-y coupling
• BBU (injection error)• Effect of cavity misalignment
Rough estimation of BBU (Bunch break up) (by two particle model)
In perfectly aligned linac.The first particle oscillates with injection error.Wake of the first particle excites oscillation of the following particle.
entrance thefrom distance:
constant) (assumegradient acc.:
atenergy andenergy initial:,
constant) (assumefunction -beta:
constant) (assumeon wakefuncti:
particlefirst theof cahnge:
0
s
g
sEE
W
q
2
1/
)0 (initialn oscillatio particle 2nd of Amplitude
/
) (initial oscillaion particlefirst of Amplitude
0
001
0000
00
sWqea
gEEWqeaa
EEaa
a
1/,1 Requiring 001 EEeqgWaa
Rough estimated requirement from BBU
1/,1 Requiring 001 EEeqgWaa
213 V/C/m103
170 m, 100 nC, 3 MeV/m, 30
W
E/E qg For ILC,
This is barely satisfied for short range wake of TESLA design.
Short range wake of warm accelerating structure is much larger.And special cure is necessary, such as BNS damping, or auto-phasing technique: Introduce z-correlated energy spread (tail has lower energy), Avoid ‘resonance’ between head and tail.
BNS, auto-phasing is good for BBU due to wake and necessary for warm LC.But introducing energy spread causes dispersive effect and will not be used in ILC.
Explanation of auto-phasing by 2 particle model
Continuous, constant focusing optics. No acceleration.
strength, focusing const.: function, wake:
particle leading of charge: particle, following and leading ofenergy :
)()()('' :particle Following
)()('' :particle Leading
01,0
01
01
11
00
0
KW
qE
syE
Weqsy
E
Ksy
syE
Ksy
)/( ,/)sin()0(')cos()0()(: , ofSolution 00000 EKkkksyksysyy
Amplitude of the 3rd term grow linearly with distance
No growth of induced oscillation
kksE
Weqskksyksysy
EE
/)sin(2
/)sin()0(')cos()0()(
, If
0
0011
01
particle. following theofsolution a is )()(
, If
01
01
0
1
sysy
E
K
E
Weq
E
K
Continuous, constant focusing optics.
ntacc.gradie const.: strength, focusing const.:
)0()( :Energy
)()(
)(')(
)()('' :particle Second
)(')(
)()('' :particle Leading
00
01
01
111
00
00
gK
gsEsE
sysE
Weqsy
sE
gsKysy
sysE
gsKysy
kkssykssysEEsy
kKEgy
/)sin()(')cos()()(/)0()(
/ assuming ofSolution
00000
2200
If you have time, consider auto-phasing by 2 particle model, with acceleration.
-15
-10
-5
0
5
10
15
0 50 100 150 200 250 300
0.1Eauto
-15
-10
-5
0
5
10
15
0 50 100 150 200 250 300
Eauto
-15
-10
-5
0
5
10
15
0 50 100 150 200 250 300
1.5Eauto
Oscillation piriod of the leeding bunch
-30
-20
-10
0
10
20
30
0 50 100 150 200 250 300
E=0
-15
-10
-5
0
5
10
15
0 50 100 150 200 250 300
0.5Eauto
Oscillation of second particle.Two particle model.No acceleration
Unit of the amplitude of the leading particle .
No energy difference: Amplitude grows linearly.
Auto-phasing:Oscillate as same as the leading particle.
More energy difference:BNS Damping
Note on auto-phasing, BNS damping
• It is impossible to maintain exact auto-phasing condition for all particles. But,
• Exact condition is not necessary to suppress BBU. Correlated energy difference close to the exact condition will work.
• Useful for single bunch BBU– Wakefunction is monotonic with distance– Longitudinally correlated energy difference can be
introduced by controlling RF off-crest phase.– Warm IL must use this technique– Not strictly necessary for ILC
• Difficult to apply to multibunch BBU
Effect of misalignment of cavities
Assume Beam offset << typical misalignment
Induced wake almost only depend on misalignment,not on beam offset.
ignore the beam offset, which is much smaller than misalignment of cavities
Effect of misalignment of cavities-continued
iiffiijii
f EEWaey sin/1,
fcav
fjf
iiifij
fiifj gEL
EEWaeW
EE
eay
2
)/log(sin 0
22222
,
222
,
)( :wake"- sum" , kjjk
ikij zzWqW
Position change at linac end due to i-th cavity:
Expected Position change square:
gradient, acc.: length,cavity :
particel,th -k of charge:
final, cavity toth -i from advance phase:
beta, final andcavity th -iat beta:,
energy, final andcavity th -iat energy :,
cavity,th -i ofoffset :
gL
q
EE
a
cavi
k
i
ii
fi
i
iiffiijifi EEWeay sin/1,,
Position change at linac end:
Please confirm or confute expressions in this page.
Due to low RF frequency of ILC, using superconducting cavity, wakefield is much less serious, compare with warm LC (X-band or higher frequency).
length scale by factor 1/a
frequency change by factor a
WaW
t
WaW
taWatW
4
3
3
)0 effect,bunch single( beginning at the Slope : wakerangeShort
effect)bunch -(multi wakerange long of Amplitude
sinsin
mode e)(transvers dipoleevery ofon Wakefuncti
Explanation of Wakefunction Scaling with size
length scaled by factor 1/a
Look at one dipole mode (justified by principle of superposition, for many modes)
)/(ion wakefunctse transverand 2/
on,preservatienergy from then, voltage, theof half one a feels'' change drive The
:gradient Average , :Voltage , :Energy
y,offset at cavity ugh the went thro charge drive aby fielld Exited :cavityLeft
0000
000
LyqV WqVU
EVU
q
yq
y/aq’
0311
01
20
10
3
001
03
)'/(function wakee transversand
'2/'
on,preservatienergy From
:gradient Average ,:Voltage , :Energy
scaled)(length on distributi field same the
gradient, same theexited offset ugh went thro' charge driveA :cavityRight
WaLaqyaV a
qaqVaqUa
E V aUa
y/aq
-
-
cavity length Lcavity length L/a
Please confirm or confute expressions in this page.
Example of Short range wakefunctionFrom TESLA-TDR
0
10
20
30
40
50
60
0 0.0005 0.001 0.0015 0.002
Transverse-short-wake(from TESLA-TDR formula)
WT (
V/p
C/m
2 )
s (m)
Example of simulation of short range wake effect-1
Single bunch BBU, with injection offset in perfectly aligned linac.Emittance along linac, injection offset 1 of beam size. monochromatic beam.
y-z, y-’z, y’-y distribution at the end of linac
2 10-8
2.005 10-8
2.01 10-8
2.015 10-8
2.02 10-8
2.025 10-8
0 2000 4000 6000 8000 10000
pro
ject
ed
s (m)
8 10-7
1 10-6
1.2 10-6
1.4 10-6
1.6 10-6
1.8 10-6
-8 10-4 -6 10-4 -4 10-4 -2 10-4 0 100 2 10-4 4 10-4 6 10-4 8 10-4
injection 1 sigma, monochro
y (m
)
z (m)
-5 10-9
0
5 10-9
1 10-8
1.5 10-8
2 10-8
2.5 10-8
-8 10-4 -6 10-4 -4 10-4 -2 10-4 0 100 2 10-4 4 10-4 6 10-4 8 10-4
injection 1 sogma, monochro
y'
z (m)
-5 10-9
0
5 10-9
1 10-8
1.5 10-8
2 10-8
2.5 10-8
8 10-7 1 10-6 1.2 10-6 1.4 10-6 1.6 10-6 1.8 10-6
y'
y (m)
Example of simulation of short range wake effect-2
Single bunch, random misalignment of cavities, sigma=0.5 mm.Emittance along linac. One linac and average of 100 seeds. monochromatic beam.
2 10-8
2.02 10-8
2.04 10-8
2.06 10-8
2.08 10-8
2.1 10-8
2.12 10-8
2.14 10-8
0 2000 4000 6000 8000 10000
average of 100 seedsexample, from one seed
pro
ject
ed
(m
)
s (m)
Example of Long range wakefunction
Sum of 14 HOMs from TESLA-TDR
-200
-150
-100
-50
0
50
100
150
200
0 50 100 150 200
TESLA-TDR
WT (
V/p
C/m
2 )
time (ns)
Mitigation of long range transverse wakefield effect
• Damping– Extract higher order mode energy from
cavities through HOM couplers.
• Detuning– Cavity by cavity frequency spread.
• Designed spread or• Random spread (due to errors)
• In LC, both will be necessary.
-200
-100
0
100
200
300
0 50 100 150 200 250 300 350 400
no dampingwith damping
sum
-Wak
e (V
/C/m
2 )
Bunch number
“Sum-Wake” from 14 HOM (from TESLA-TDR) with/without damping
Note: Bunch spacing is set as 219/650E6 s.The result strongly depend on the spacing.
)(
jk
kjj zzWW
With damping, sum-wakes are almost the same for most of the bunches, except in the beginning of the beam pulse.-->Orbit changes due to cavity misalignment are the same for most of the bunches.
Please consider implication of this.
Damping of Higher Order Mode Wakefield
Two HOM Couplers at both sides of a cavity
Special shapes:Accelerating mode should be stopped.HOM should go through. - trapped mode may cause problem.
TESLA-TDR
TESLA-TDR
Detuning of wakefield
7 offactor by reduction Wakefield50
bunch)next (next ns 500~for effectivefully becomes Detuning
MHz2~ 0.1% spread 2GHz,~frequency HOM Typical
Linac,Main ILCIn
. of Amplitude
,/2for , spread has If
, of Amplitude
spread, no has If
sin
function,-beta tocomparablelength in cavities
effective
effective
1effective
n
t
WnW
t
WnW
tWW
n
i
i
n
ii
For BBU: effective wake is from sum of cavities within length comparable to beta-function
In x-band warm LC, carefully designed damped-detuned structures were necessary. Not for ILC.
Wakefunction envelope from HOMs (from TESLA-TDR) with/without random detuning (50 cavities) and damping
No detuning
Random detuningf/f=0.1%
1
10
100
0 1 103 2 103 3 103 4 103 5 103 6 103
f=0, no damp
f=0, damp
Wen
velo
pe (
V/C
/m2 )
time (ns)
1
10
100
0 1 103 2 103 3 103 4 103 5 103 6 103
fsf=0.1%, no damp
f=0.1%, damp
Wen
velo
pe (
V/C
/m2 )
time (ns)
Example of simulation, long range wake effect-1Multibunch BBU. Injection offset in perfectly aligned linac.
y vs. bunch number at the end of linac,
injection offset 1 sigma of beam size.
w/wo damping. w/wo frequency spread 0.1%.
1 10-6
1.5 10-6
2 10-6
2.5 10-6
0 50 100 150 200 250 300
df=0.1%, no dampdf=0.1% damp
y (m
)
Bunch number
-2 10-5
-1 10-5
0
1 10-5
2 10-5
0 50 100 150 200
df=0, no dampdf=0, damp
y (m
)Bunch number
(vertical scales are different by more than 1 order)Detuning is very effective for BBU.
Example of simulation, long range wake effect-2Multibunch, random misalignment of cavities
y vs. bunch number at the end of linac,
Misalignment of cavities, sigma=0.5 mm.. w/wo damping. w/wo frequency spread 0.1%.
-3 10-5
-2 10-5
-1 10-5
0
1 10-5
2 10-5
3 10-5
0 50 100 150 200
df=0, no dampdf=0, damp
y (m
)
Bunch number
-3 10-5
-2 10-5
-1 10-5
0
1 10-5
2 10-5
3 10-5
0 50 100 150 200 250 300
df=0.1%, no dampdf=0.1%, damp
(vertical scales are the same)Detuning for random misalignment is not so effective as for BBU.
Possible vertical orbit induced by long range wakefield excited by horizontal orbit
x-y coupling of wakefield mode
• Horizontal beam orbit will be less stable than vertical. (Horizontal emittance is much larger than vertical, more than factor of 400 at the DR extraction.)
• Some dipole modes of wakefield may be x-y coupled
• Horizontal orbit may induce vertical orbit.
x-y coupling of long range wakefield
Consider dipole wakefield.
If cavity is perfectly x-y symmetric, two polarization modes has the same frequency (perfectly degenerated.)Induced field by particles with horizontal offset kicks following particles only horizontally.
If symmetry is broken, two polarization have different frequency and their axis can be slant.Induced field by particles with horizontal offset consists of two slant modes and can kick following particles vertically too.
x-y coupling of long range wakefield
x
y
axis of mode-1
axis of mode-2
charge ttWxeq
tWtWtW
tWtWtW
tWtW
tWtW
eqWW
WxeqW
y
x
cos)2/sin(2sin
cos)(sin)()(
sin)(cos)()(
)2/sin()0()(
)2/sin()0()(
sin)0(
cos)0(
21
21
22
11
2
1
-1.2
-0.8
-0.4
0
0.4
0.8
1.2
0 20 40 60 80 100 120
WxWy
W
t
Cures of x-y coupling due to long range wakefield
• Extremely good cylindrical symmetry of cavities. 0 : difficult? and/or cost?
• Stronger damping : difficult?• Intentionally broken symmetry. 0 : cost?• x-y tune difference (Different phase advance
per FODO cell). Suppress the effect of the coupling.
• etc. ???
Dispersive effect
Dispersive effect• Dominant source of emittance dilution in ILC
Main Linac• Depend on initial energy spread• Important errors:
– Quad misalignment– Cavity tilt
• Need rather sophisticated corrections– DFS (Dispersion Free Steering)– Kick Minimum– etc.
Note: Correction of Linear Dispersion -1
• Energy-position correlation will be measured after the main linac. And linear dispersion will be well corrected, (we assume).
• “Linear dispersion corrected emittance” should be looked, not projected emittance. (see appendix - 1)
• There is designed dispersion in curved linac. Even without any errors, projected emittance is significantly larger than linear dispersion corrected emittance.
Note: Correction of Linear Dispersion -2
• In principle, correction of non-linear dispersion is possible. But practically, it will be very difficult. Only 1st order dispersion can be measured and corrected (practically).
• Even if 1st order dispersion is corrected at the end of linac, there can be large higher order dispersion remained.
• Transverse E-M fields at zero dispersion will induce linear (1st order) dispersion. And transverse, position dependent E-M fields (quad magnet) at non-zero n-th order dispersion will induce (n+1)th order dispersion.
• 1st order dispersion should be kept small everywhere in the linac to suppress higher order dispersions, then for preservation of low emittance.
Emittances in curved linac without errors
2 10-8
2.5 10-8
3 10-8
3.5 10-8
4 10-8
0 2 103 4 103 6 103 8 103 1 104
Projected Emittance (No Error)
pro
jrct
ed (
m)
s (m)
2 10-8
2.002 10-8
2.004 10-8
2.006 10-8
2.008 10-8
2.01 10-8
0 2 103 4 103 6 103 8 103 1 104
Linear Dispersion Corrected Emittance (No Error)
Lin
ear
Dis
per
sin
Cor
rect
ed (
m)
s (m)
This is getting small as the relative energy spread becomes small.
The emittace increases by 0.1% of nominal.Initial dispersion should be matched
Dispersive effect in perfect linac “Filamentation” with injection error
y'sq
rt(
y/y)
y/sqrt(y
y)
y'sq
rt(
y/y)
y/sqrt(y
y)
y'sq
rt(
y/y)
y/sqrt(y
y)
y'sq
rt(
y/y)
y/sqrt(y
y)
Different phase advance for different energy particles.
Dispersive effect from quad misalignment
• Charged particle goes through quad magnet with offset is kicked.• The momentum change is proportional to the offset.• The angle change proportional to inverse of energy of the particle.• Many of such angle differences induce non-linear dispersion,
which cannot corrected later.Simulation result - Emittance along linac (straight linac)Random offset, 1um..
2 10-8
2.2 10-8
2.4 10-8
2.6 10-8
2.8 10-8
3 10-8
3.2 10-8
0 2000 4000 6000 8000 10000
projected (m) average of 50 seedsprojected (m), one example corrected (m) average of 50 seeds corrected (m) one example
(
m)
s (m)
Quad misalignment 1 m
Dispersive effect of cavity tilt• Charged particle is transversely kicked by the tilted cavity.• The momentum change is about Vc*tilt angle / 2 (see next slide).• The angle change proportional to inverse of energy of the particle. • Many of such angle change differences and quad magnet fields ind
uce non-linear dispersion, which cannot corrected later.Simulation result - Orbit and Emittance along linac (straight linac)Rndom tilt, 10 rad.
2 10-8
2.1 10-8
2.2 10-8
2.3 10-8
2.4 10-8
2.5 10-8
2.6 10-8
0 2000 4000 6000 8000 10000
cavity tilt 10 rad
projected (m) average of 50 seedsprojected (m) one example corrected (m) average of 50 seeds corrected (m) one example
(
m)
s (m)
Note: Edge focus reduce the effect of cavity tilt
beam
Transverse kick in the cavity: pt = eEL
Acc. field E, length L, tilt angle
offset: y0+L/2 offset: y0-L/2
entranceexit
Transverse kick at the entrance: pt = -eE (y0+ L/2)/2Transverse kick at the exit: pt = eE (y0- L/2)/2
Edge (de)focus [see appendix]
Total transverse kick by the cavity: pt = eEL/2
Static corrections(transverse motion)
Necessity of Beam based corrections
• Without corrections, required alignment accuracy to keep emittance small will be, roughly:– ~0.1 um for quad offset– ~1 urad for cavity tilt
which will not be achieved.
(cavity offset ~ a few 100 um may not be serious problem.)
Beam based static corrections• Corrections using information from beam
measurement will be necessary– 1 to 1 correction
• non-invasive, but will not be enough– Kick minimization
• non-invasive, but cannot correct for cavity tilt and need additional correction
– DFS (Dispersion Free Steering) • invasive, seems promising
– Ballistic Steering• invasive. Not yet studied if it is good or not for curved linac ?
– etc. • These are “Local” corrections. Beam quality is to be
corrected everywhere in the linac.
Quad shunting for finding Quad - BMP center offset
• Some correction methods need to know BPM - Quad center offset accurately. (BPM is attached to quad.)
• Quad shunting (change strength) and measuring beam will probably be the best way.
• Changing strength of superconducting magnet cannot be so fast. The procedure will take time. (Possible? How long?)
• The accuracy depend of BPM resolution, how much strength changed and stability of field center (for different strengths and also long term).
BPMaBPMb
quad(strength change)
Orbit change
change)strength (Q
change) readBPM(-read)BPM(offset) BPM(Q b
aa
One to one correctionMake BPM readings zero, or designed readings
2 10-8
2.5 10-8
3 10-8
3.5 10-8
4 10-8
4.5 10-8
5 10-8
5.5 10-8
6 10-8
0 100 2 103 4 103 6 103 8 103 1 104
1to1, quad offset 300 m
average of 100 seedsone example
co
rrec
ted
(m)
s (m)
-0.001
-0.0005
0
0.0005
0.001
0 100 2 103 4 103 6 103 8 103 1 104
1to1, quad offset 300 m
Quad offsetBeam Orbit
y (m
)
s (m)
Simulation result:Quad random offset 300 m, no other errorsExample of quad offset and beam orbit
Emittance along linac
One to one correction will not be enough.
Kick Minimization (KM)
• Basically:– Steer beam to minimize kick angle at every quadrupole-dipol
e magnet pair. Or minimize deviation from designed kick angle. (Requiring Quad and dipole magnets are attached or very close each other.)
– See appendix for a little more details
• Non-invasive correction• Accurate information of Quad - BPM offset is importa
nt.• Can correct quad misalignment but not effective for c
avity tilt.
KM, example of simulation resultSimulation result:Quad random offset 300 m, no other errorsExample of quad offset and beam orbit
Emittance along linac
-0.001
-0.0005
0
0.0005
0.001
0 2000 4000 6000 8000 10000
KM Quad offset 300 m
Quad offsetBeam orbit
y (m
)s (m)
2 10-8
2.02 10-8
2.04 10-8
2.06 10-8
2.08 10-8
2.1 10-8
0 2000 4000 6000 8000 10000
KM Quad offset 300 mAverage of 100 seedsOne example
co
rrec
ted
e
s (m)
KM, example of simulation result - 2
Sensitivity to errors.Emittance at the end of linac. Average of 100 random seeds.
2 10-8
2.1 10-8
2.2 10-8
2.3 10-8
2.4 10-8
2.5 10-8
2.6 10-8
2.7 10-8
2.8 10-8
0 200 400 600 800 1000
co
rrec
ted
(m)
Quad offset (m)
2 10-8
2.1 10-8
2.2 10-8
2.3 10-8
2.4 10-8
2.5 10-8
2.6 10-8
2.7 10-8
2.8 10-8
0 100 200 300 400 500
co
rrec
ted
(m)
Cavity Tilt (rad)
DFS (Dispersion Free Sttering)• Basically:
– Change beam energy and measure beam orbit. – Steer beam to minimize orbit difference. Or, minimize
deviation from designed orbit difference in the case of the curved linac.
– See appendix for an example of algorithms (not necessarily the best one)
• Results seem to depend on some details of algorithms. (?)
• Need to change accelerating voltage to change beam energy. ~ 10%– How accurately it can be, practically???
• BPM resolution is important but information of Quad - BPM offset is not so important.– Because “difference of orbit” is looked at.
DFS, example of simulation result
-0.001
-0.0005
0
0.0005
0.001
0 100 2 103 4 103 6 103 8 103 1 104
DFS, Quad offset 300 m
Quad offset y (m)Beam orbit (m)
Qu
ad o
ffse
t y
(m)
s (m)
2 10-8
2.1 10-8
2.2 10-8
2.3 10-8
2.4 10-8
2.5 10-8
2.6 10-8
2.7 10-8
0 100 2 103 4 103 6 103 8 103 1 104
DFS, Quad offset 300 m
average of 100 seedsone example
co
rrec
ted
(m)
s (m)
Simulation result:Quad random offset 300 m, no other errorsExample of quad offset and beam orbit
Emittance along linac
This particular algorithm may not be optimum.Do not quote these figures.
DFS, example of simulation result - 2
2 10-8
2.2 10-8
2.4 10-8
2.6 10-8
2.8 10-8
3 10-8
0 1 2 3 4 5
co
rrec
ted
(m)
BPM resolution (m)
2 10-8
2.2 10-8
2.4 10-8
2.6 10-8
2.8 10-8
3 10-8
0 0.2 0.4 0.6 0.8 1 1.2
co
rrec
ted
(m)
Cavity tilt (mrad)
2 10-8
2.2 10-8
2.4 10-8
2.6 10-8
2.8 10-8
3 10-8
0 0.1 0.2 0.3 0.4 0.5
co
rrec
ted
(m)
Quad offset (mm)
Sensitivity to errors.Emittance at the end of linac. Average of 100 random seeds.
This particular algorithm may not be optimum.Do not quote these figures.
“Global” corrections
• In addition to “Local” corrections.• Scan upstream knobs, measuring beam at th
e end of the linac or certain linac section, and finding the best setting of the knobs.
• For ILC, we are considering:– Knobs: Orbit bumps– Measurement: emittance or beam size– Correct dispersive effect: “Dispersion bumps”– Correct wakefield effect: “Wakefield bumps”
Dispersion Bumps and Wakefield Bumps
• Energy-dependent kick (dispersion) or/and z-dependent kick (wakefield) in the section is to be compensated by the bumps.– Possible (in principle) by two bumps, phase advan
ce 90 deg. apart. – The length of the section should not be so long. In
duced dispersion or z-x/y correlation (by wakefield) should be corrected before significant filamentation.
bump -> tail is kicked Monitortail is kicked by error
Cancelled
phase advance ~90o
Dynamic corrections (Feedback)
Dynamic errors
• Mechanical motion– Motion induced by the machine itself (motors for
cooling, bobbles in pipes, etc.)– Cultural noise (nearby traffic, etc.)– Ground motion (slow movement, earth quake)
• Strength– Field strength of magnets– Accelerating field, amplitude and phase
• EM field from outside
Typical speed of fluctuations and correctionsfor transverse motion
Speed (Hz) Possible source Effective feedback For
>106 DR kickers, ? None -
10 ~ 106 Machine, cultural noise, Power supply, ground motion
Intra pulse feedback at IP
Position
0.001 ~ 1 Temperature change, ground motion,
What else?
Simple orbit feedback (in BDS and Linac)
Position
? ~ 0.01 More sophisticated orbit FB. (If necessary)
Emittance
0 ~ 0.0001? “static” corrections Emittance
Dynamic correction in ILC Main Linac
Studies have not well matured and there is no commonly agreed scheme, but probably:
1. Orbit correction at several locations.2. Energy feed back. One or two loops (?)3. One-to-one orbit correction (using all or most
correctors and BPM, slower than1)
Need to be careful from Machine protection point of view.
IF feedbacks change machine parameter too much, the beam may hit some part of machine.
LAST SLIDE
• Coming Later
Appendix - 1
Definition ofProjected emittance andLinear Dispersion Corrected emittance
Projected emittance
( y2 y 2 )( y'2 y'2 ) ( yy' y y')2
y : Vertical offset, y': Verticale angle
: Relative energy deviation
( y y ) /(2 2 ), '( y' y' ) /(2 2 )
: Average over all macro - particles
Linear Dispersion Corrected emittance
( (y )2 y 2 )( (y' ')2 y' ' 2 ) ( (y )(y' ') y y' ' )2
Appendix - 2 Edge (de)focus of cavity
beam
Deep inside cavity,No transverse E-field
offset r
Out of cavity,No E-field
Apply Gauss’ Law to the cylinder, radius=r.Then, relation between integrated transverse force,felt by the beam, and accelerating field strength will be given.
Cylindrical symmetry
Appendix- 3, Example of KM Algorithm
Every quad should have a BPM and a dipole corrector attached. Divide linac into sections, and in each section, from upstream to down stream,Minimize additional offset and additional kick” at quads.
3
22
10= ratio. Weight :
quadth - theof value-K :
quadth -at orbit designed fromOffset :
quadth -at steering of
kick) designed tol(additiona anglekick Additional :
, Minimize
r
ik
iy
i
ykyr
i
i
i
iiii
ii
Appendix- 4, Example of DFS Algorithm
One-to-one orbit correction (BPM reading zeroed)Divide linac into sections, and in each section:(1) Measure orbit with nominal beam energy. (y0,i at i-th BPM)(2) Reduce initial beam energy and accelerating gradient from the linac entrance to the end of previous section by a common factor (e.g. 10% or = -0.1).(3) For the second section or downstream, orbit adjusted at the two BPMs in the previous section to make the position at the BPM y = y0 + (y0 is the position with nominal energy, the dispersion at BPM.)(4) Measure orbit. (y,i at i-th BPM)(5) Set dipole correctors in the section to minimize
w(y,i - y0,i - ycal,i)2 + y0,i - ycal,i)2 (ycal,i is the calculated orbit difference, ycal,I the calculated orbit, without errors, at I-th BPM. w is the weight factor, w=5000.).(6) Iterate from (1) to (5).(7) Go to next section.
