ILC Beam Dynamics - Main Linac
and Bunch Compressor
Kiyoshi KUBO
2006

ILC Main Linac Beam Dynamics• Introduction• Beam dynamics related design• Beam quality preservation– Longitudinal– Transverse• Wakefield– Single bunch, Multi-bunch– BBU, Cavity misalignment• Dispersive effect– Errors and corrections• Initial alignment, fixed errors• Vibration, ground motion, jitters, etc.

Comment• Main Linac is very simple, compare with most ofother part of LC.
– Basically many iteration of a simple unit.
• Some statistical calculations are useful because of
the large number of identical components.
• Still, analytical treatment is very limited. And moststudies are based on simulations.

Parameter of ILC Main Linac(ECM=500 GeV)
Beam energy

13~15 GeV to 250 GeV

Acc. gradient31.5 MV/m

Bunch Population

1 ~ 2 x1010 /bunch

Number of bunches<= 5640 /pulse

Total particles<=5.64 x1013 /pulse

Bunch spacing>= 150 ns

Bunch Length0.15 ~ 0.3 mm

Emittance x (at DR exit/IP)
Emittance y (at DR exit/IP)

8/10~12 x10-6 m-rad
2/3~8 x10-8 m-radLattice design• Basic layout– One quad per three (four) cryomodules• Changed from one quad/four modules– Simple FODO cell
x/y = 75/65 degree phase advance/cellOther possible configurations: change along linac• Higher the energy, larger the beta-function• FOFODODO, FOFOFODODODO, etc., in highenergy region• Vertically curved, following the earth curvature

Unit of main linac
about 320 units/linac
Cryomodules without magnet package

Cryomodule with magnet package
9-cell SC cavity

Magnet package
BPM

Dipole SC magnet
Quadrupole SC magnet

Beta-function
200

βx,y (m)Baseline design
Constant phase advance

Strength of quad magnet
is proportional to beamenergy.

βx (m)βy (m)

2 103

4 103

150

100

50

0

0

6 103

8 103

1 104

s (m)

Beam size

σx (m)σx (γε=1E-5 m)-5σy(γε=2E-8 m)

0.00015

1.5 100.0001

1 10

5 10-5

5 10

0

0

0

2000

4000

6000

s (m)

8000

10000

-5

-6

-5

σy (m)

Vertical:
5~10 μm
at 15 GeV
1.5 ~ 2.5 μm
at 250 GeV

2 10

0.0002

Tolerances depend on optics
Tolerance of Quad magnet vibrations ∝ βTolerance of Acc. Cavity offset ∝ 1(See later discussions.)βAlignment and Beam Orbit in Curved Linac,
Following earth curvature
Alignment line

BPM-Quad-Dipole corrector package

cryomodule

Designed Beam Orbit(Vertical scale is extremely exaggerated)

Alignment and Beam Orbit in Curved Linac,
Following earth curvature
0

y (m)

-0.0005

Alignment line
Design Orbit-0.001-0.0015-0.002

0

50

100

s (m)

150

Design orbit w.r.t. the reference line
and dispersion

-1 10

-5

0.002

-2 10

-5

0.0015

-3 10

-5

0.001

-4 10

-5

y

y (m)

0.0025

η (m)

0

η (m)

y (m)
-5 10

0.0005

y

-5

0
0

50

100

150

s(m)

200

250

300

Injection orbit and dispersion are non-zero, andshould be matched to the optics.

Beam Quality Preservation

Beam Quality• Longitudinal particle distribution– Energy and arriving time, or longitudinal position• 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 preferable.
Exception: x and z distribution at IP should not be too small.
(Æ beam-beam interaction)
High quality is generated in Damping Ring. Preservation of thequality is the issue from DR to IP.

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 lesssevere than in bunch compressors.
Main Linac does nothing to the timing andbunch length.
(Æ Bunch Compressor)

Single bunch:Longitudinal short range wakefield
4 1013

Deceleration by wakefield

Wake-function

3.5 1013

WL (V/C/m)

3 1013z

E(z) = ∫−∞ dz'λ (z')W (z − z')

2.5 1013
2 1013
1.5 1013

8 104

1 1013

0
0

0.001

0.002

0.003

0.004

0.005

s (m)
5 10-6Charge density
Charge Density (C/m)

4 10

-6

3 10-62 10-6Deceleration by wakefield (V/m)

5 1012

7 104
6 104
5 104
4 104
3 104
2 104
1 104

1 10-60-0.0010-0.001-0.00050z (m)0.0005-0.0005

0

z (m)

0.001

calculated from TESLA-TDR formula

0.00050.001

Total acceleration(RF off-crest phase 4.6 deg. minimizing energy spread.)
3.145 107

2 105

3.14 107

1.5 1053.135 107

1 105

wakefield

3.13 107

3.125 107
-0.001-0.000500.0005

wakefield (V/m)

Accelerating field (V/m)

RF field
Total acceleration

5 104

0
0.001

z (m)

Energy spread along linac. Initial spread is dominant.
1.6 108

0.012

0.01

1.58 108

PositronPositronElectron

Undulator for e+ source

σE (eV)

σE/E

0.008

0.006

1.56 108

1.54 108

0.004
1.52 108

0.002

0
0

5 10101 1011

1.5 1011

E (eV)

2 1011

2.5 1011

1.5 1080

5 10101 1011

1.5 1011

E (eV)

2 1011

2.5 1011

Bunch by bunch energy difference• Should be (much) smaller than single bunchenergy spread• Accurate RF control will be essential– Compensation of beam loading– Compensation of Lorentz detuning• Longitudinal higher order mode wakefield will
not be a problem, assuming moderate“detuning of wakefield.”Energy Fluctuation, Required RF Stability
Independent, random fluctuations of each klystron.

1(δ amplitudeeV0 cosφ0 + δ phaseeV0 sin φ0 )
δE ≈
nkly
Common error of all klystrons

ΔE ≈ Δ amplitudeeV0 cos φ0 − Δ phase eV0 sin φ0⎛V0 : total voltage ⎞⎟⎟⎜⎜⎝ φ0 : off - crest phase ⎠For stability < 0.1σ E from each error,δ amplitude < 0.1 nklyσ E E ≈ 2 × 10 −3δ phase< 0.1 nkly σ E E sin φ0 ≈ 2 × 10 − 2 rad

Δ amplitude < 0.1σ E E ≈ 10 − 4
Δ phase< 0.1σ E E sin φ0 ≈ 10 −3 rad

Fluctuations fasterthan feedback are
relevant.

Transverse Motion

Transverse beam quality• Stable Beam Position at IP– Offset between two beams should be (much) smaller
than the beam size
• Small Beam Size at IP, Small 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• Hourglass effect and Oide limit– Then, vertical emittance need to be very small.
• Vertical position stability and vertical emittancepreservation are consideredBeam position jitter• There will be position feedback in Main Linac,
BDS, then, at IP, and feed-forward in RTML
(turnaround).
• 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 magnets(instability of power supplies)– RF strength jitter (with tilted cavities)

Beam position offset vs. luminosity- estimation without beam-beam force
offset Δybeam size σybeam size σy

1

0.8

0

0.6

L/L

2⎤
⎡ ⎛
⎞
1 Δy ⎟ ⎥L(Δy ) = L(0) exp ⎢− ⎜⎢ 4 ⎜⎝ σ y ⎟⎠ ⎥⎣⎦

Æ Offset between two beams
should be (much ?) smaller thanthe beam size

0.4

0.2

0
0

0.5

1

1.5

2

2.5

3

3.5

4

Δy/σ

y

beam-beam force will change this significantly

Estimation of beam position change
due to quad offset change
Final position change due to offset of i-th quad:Δy f = − ai ki Ei / E f β f β i sin ϕ iai : offset of i - th quad,ki : k - value ( f -1 ) of i - th quad,

Ei : beam energy at i - th quad,

E f : final beam energyki : k - value,

β i : betafunction at i - th quad,β f : final betafunction

ϕ i : phaseadvance from i - th quad to final
Final beam position is sum of all quads’ contribution. Assuming
random, independent offset, expected beam position offset is:< y 2f >≈ β f ∑ ai2 β i ki2 Ei sin 2 ϕi / E f ≈ a 2 N q β f β k 2 4quad

a : rms of offset, N q : number of quads,

β : average of betafunction, k 2 : average of k - value square

Estimation of beam position change due
to magnet strength change
Final position change due to strength change of i-th quad:

Replace ai ki in the previous page by aiδ ki ,where δ ki , is the quad strength error.
Final position change due to strength change of i-th dipole:

Replace ai ki in the previous page by δ k 0,i ,where δ k 0,i , is the dipole strength error.

Strength fluctuation is important especially for curved linac,
because of non-zero designed dipole kicks, even withoutalignment errors.

Estimation of beam position change due toaccelerating voltage jitter in tilted cavities
Final position change due to strength change of i-th cavity:
Replace aiδ ki in the previous page by (eδVi /Ei )θi / 2,
where δVi is the voltage error, Ei the beam energy and θi is the cavity tilt angle.

See lecture note.

Stronger focus optics make quad
vibration tolerance tighter
Beam position jitter / beam size:< y 2f >σy≈a Nqβ f β k 2 2

ε yβ f

=~

a Nq

ε yβ

proportional to:quad vibrationsquare root of number of quads
inversely proportional to:square root of average beta-function
square root of emittanceBeam quality - Emittance• Our goal is high luminosity• For (nearly) Gaussian distribution, emittance is agood measure of luminosity• But, In some cases, emittance is not necessarily
correlated with luminosity., e.g. Banana effect,
etc. . . . .
• We use emittance dilution as a measure ofquality dilution in the main linac, anyways.
– Usually OK.
– No other good parameter.

Example where emittance does not wellcorrelated with luminosity
So called “banana effect”

z-y correlatedSame projected
emittance.
Different luminosity

no correlation
Æ Beam-beam interactionCollision with angle

NOTE• We use emittance as a measure of quality inthe main linac.
• Halos, tails far from core should be ignored in
calculating the emittance.
– Effects of halos or tails should be considered inother context.
• Luminosity may not be well correlated to emittance, insome cases, because of Beam-beam force at IP

Dominant Sources oftransverse Emittance dilution• Wakefield (transverse) of accelerating cavities– Electromagnetic fields induced by headparticles affect following bunches.
• z-correlated orbit difference• Dispersive effect– Different energy particles change different
angles by electromagnetic fields (designed or
not-designed).
• energy correlated orbit differenceEffects of Transverse WakefieldTransverse Wakefield of
Accelerating 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 oscillating mode
– For ILC
• Need to be damped• Frequency spread is needed. (may naturally exist ?)• Need to be careful for x-y coupling
BBU [Beam Break Up] (injection error)
Effect of cavity misalignment

Rough estimation of BBU (Beam 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.

Amplitude of first particle oscillaion (initial a00 ) q : cahnge of the first particle
a0 ≈ a00 E0 / E

W : wakefunction (assume constant)
Amplitude of 2nd particle oscillation (initial 0) β : beta - function (assume constant)⎧a0 e q W β log( E ( s ) E (0) ) 2 g ( g > 0) E0 , E : initial energy and energy at s
a1 ≈ ⎨
g : acc. gradient (assume constant)( g = 0)
⎩ a0 e q W β s 2
s : distance from the entranceRough estimated BBU
For ILC,g ≈ 30 MeV/m, q ≈ 3 nC, β ≈ 100 m, E/E 0 ≈ 17,

W < 3 × 1013 V/C/m 2⇒ a1 a0 ≈ 0.15
Short range wake of warm accelerating structure is much larger.
And special cure is necessary, such as BNS damping, or autophasing 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 forwarm LC.
But introducing energy spread causes dispersive effect and may
not be used in ILC.

Explanation of auto-phasing by 2 particle modelContinuous, constant focusing optics. No acceleration.

Leading particle :y0 ' ' ( s ) = −Following particle : y1 ' ' ( s ) = −K
y0 ( s )E0K
eq W
y1 ( s ) + 0 y0 ( s )E1
E1

E0,1 : energy of leading and following particle, q0 : charge of leading particle
W : wake function, K : const. focusing strength,
Solution of y0 , : y0 ( s ) = y0 (0) cos(ks) + y0 ' (0) sin( ks) / k , (k = K / E0 )If

If

E1 = E0 ,Amplitude of the 3rd
eq W
y1 ( s ) = y1 (0) cos(ks) + y0 ' (0) sin( ks) / k + s 0 sin( ks) / k term grows linearly
2 E0with distance−

K eq0W
K
+
=− ,
E0
E1
E1

y1 ( s ) = y0 ( s ) is a solution of the following particle.

No growth of
induced oscillation

30

ΔE=0

20
10
0

Oscillation of second particle.
Two particle model.
No acceleration

-10
-20
-30

0

50

100

150

200

250

300

100

150

200

250

300

100

150

200

250

300

100

150

200

250

300

100

150

200

250

300

15
10

0.1ΔEauto

5

Unit of the amplitude of theleading particle .

0
-5
-10
-15 0

50

15

No energy difference:
Amplitude grows linearly.

105

0.5ΔEauto

0
-5
-10
-15 0

Auto-phasing:Oscillate as same as theleading particle.

15
10

50

ΔEauto

5
0
-5
-10
-15 0

More energy difference:
BNS Damping

50

15
10

1.5ΔEauto

5
0
-5
-10
-15 0

50

Oscillation piriod of the leeding bunch

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 LC must use this technique– Not strictly necessary for ILC (though it will helpbanana-effect)• Difficult to apply to multibunch BBU– Complicated wake function Æ need to introduce
complicated bunch-by-bunch energy differenceEffect of misalignment of cavities
AssumeBeam offset << typical misalignment
Induced wake almost only depend on misalignment,
not on beam offset.

ignore the beam offset,
which is much smaller
than misalignment ofcavitiesEffect of misalignment of cavities-continued
Position change of j-th particle at linac end due to i-th cavity:
Lcavi

yi , f = −eaiW j ,i 1 / Ei E fβ f β i sin ϕ i ai : offset of i - th cavity,"sum - wake": W j ,i = ∑ qkWi ( z j − zk )k< jPosition change at linac end:
Lcavi

y j , f = −e∑ aiW j ,i 1 / Ei E f
i

Ei , E f : energy at i - th cavity and final energy,

βi , βi : beta at i - th cavity and final beta,
ϕi : phase advance from i - th cavity to final,
qk : charge of k - th particel,β f β i sin ϕ i Lcavi : cavity length, g : acc. gradient,

Expected Position change square of j-th particle:
assume all cavities have the same wakefunction

2 2
2
2 2
log( E f / E0 )
e
a
L
W
β
β
e
L
cav
f
j
2
2
2
2
cavi
W j ,i β f β i sin ϕ i ≈< y j , f >= ∑ < ai >2 gE fEi E fi

Stronger focus optics (smaller beta-function)
make Effects of wakefield weaker
BBU:
Amplitude of first particle oscillaion (initial a00 )a0 ≈ a00 E0 / E
Amplitude of 2nd particle oscillation (initial 0)⎧a0 e q W β log( E E0 ) 2 g ( g > 0)
a1 ≈ ⎨( g = 0)
⎩ a0 e q W β s 2

Misalignment:
2 2
2
2
β
β
e
a
L
W
e
cav f
j log( E f / E0 )22
2
2
< y j , f >= ∑ < ai >W j ,i β f β i sin ϕ i ≈Ei E f
2 gE fi

Example of simulation of short range
wake effect-1
60

Transverse-short-wake(from TESLA-TDR formula)40

2

W (V/pC/m )

50

T

30

20

10

0
0

0.00050.001

0.0015

0.002

s (m)

Single bunch BBU, with injection offset in perfectly aligned linac.
Emittance along linac,
2.025 10
injection offset 1 σ of beam size. 2.02 10
monochromatic beam.
2.015 10
-8

Only 1% emittance increase.

projected γε (μ)

-8

-8

-8

2.01 10-82.005 10

-8

2 10

0

2000

4000

6000

s (m)

8000

10000

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. 7% emittance increase.
-8

2.14 10average of 100 seeds
example, from one seed

-8

projected γε (m)2.12 10-8

2.1 10-8

2.08 10-82.06 10-82.04 10-8

2.02 10

-8

2 10

0

2000

4000

6000

s (m)

8000

10000

Day 2 (Aug.31)

Example ofLong range wakefunction
Sum of 14 HOMs from TESLA-TDR
200
150

TESLA-TDR

T

2

W (V/pC/m )

100
50
0
-50
-100
-150
-200
0

50

100

time (ns)

150

200

Mitigation of long range transverse
wakefield effect• Damping– Extract higher order mode energy fromcavities through HOM couplers.

• Detuning– Cavity by cavity frequency spread.
• Designed spread or• Random spread (due to errors)• In LC, both will be necessary.

Damping of Higher Order Mode Wakefield
Two HOM Couplers at both sides of a cavity

TESLA-TDR

Special shapes:
Accelerating mode should be stopped.
HOM should go through.
- trapped mode may cause problem.
TESLA-TDR

Detuning of wakefield
For BBU: effective wake is from sum of cavities within lengthcomparable to beta-functionn cavities in length comparable to beta - function,n

Weffective ≈ ∑ W sin ωi t
i =1

ωi: HOM frequency of i-th cavity

If ωi has no spread,
Amplitude of Weffective ≈ n W ,If ωi has spread δ ω , for t > 2π / δ ω ,Amplitude of Weffective ≈ n / 2 W .
In ILC Main Linac,
Typical HOMfrequency ~ 2GHz, spread ≈ 0.1% → δ ω ~ 2MHz
→ Detuning becomes fully effective for t ~ 500 ns (next next bunch)n ≈ 60 → Wakefield reduction by factor of 10

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(V/pC/m )No detuning
2

100

W

envelope10

σ =0, no dampf
σ =0, damp

1

f

0

1 10

3

3

3

2 10

3

3 10

3

4 10

3

5 10

6 10

fs =0.1%, no dampf
σ =0.1%, damp

100

f

10

W

envelope

2

Random detuning
σf/f=0.1%(V/pC/m )time (ns)

1

0

1 10

3

3

2 10

3

3 10

time (ns)

3

4 10

3

5 10

3

6 10

Dispersive effectDispersive effect• Dominant source of emittance dilution in
ILC Main Linac• Important errors:– Quad misalignment– Cavity tilt (rotation around x-axis)• Need rather sophisticated corrections

Note: Correction of Linear Dispersion• Energy-position correlation will be measured
after the main linac. And linear dispersion will
be well corrected for fixed or slow changing
errors.
• “Linear dispersion corrected emittance”
should be looked, not “projected emittance”,
for fixed errors.
• “Projected emittance” should be looked, not“Linear dispersion corrected emittance”, forfast varying errors.
(see lecture note.)

Emittances in curved linac without errors
Projected Emittance (No Error)
4 10-8This is getting small as
the relative energy
spread becomes small.

projrcted γε (m)3.5 10-8

3 10-82.5 10-8

2 10-8
0

2 103

4 103

6 103

8 103

1 104

s (m)

Linear Dispersion Corrected Emittance (No Error)
Linear Dispersin Corrected γε (m)2.01 10-8

The emittace increases
by 0.1% of nominal.
Initial dispersion should
be matched2.008 10-8

2.006 10-8

2.004 10-8

2.002 10-8

2 10-8
0

2 103

4 103

6 103

s (m)

8 103

1 104

Dispersive effect in perfect linac“Filamentation” with injection error

y

y

y

y'sqrt(β /ε )yy'sqrt(β /ε )Different phase advance for different energy particles.

y/sqrt(β ε )y/sqrt(β ε )

y y

y

y

y

y'sqrt(β /ε )yy'sqrt(β /ε )y yy/sqrt(β ε )y yy/sqrt(β ε )y yDispersive effect from quad misalignment
Simulation result - Emittance along linac(straight linac)
Random offset, σ 1um..
-8

3.2 10projected γε (m) average of 50 seeds
projected γε (m), one example
η corrected γε (m) average of 50 seeds
η corrected γε (m) one example

-8

3 10

-8

γε (m)

2.8 10-8

Quad misalignment 1 μm

2.6 10-8

2.4 10-8

2.2 10

-8

2 10

0

2000

4000

s (m)

6000

8000

10000

Quad offset 1 micron RMS causes• 45% projected emittance increase– Fast 0.1 micron quad vibration causes
0.45% emittance increase• 5% dispersion corrected emittance– 100 micron quad misalignment causes
50000% dispersion corrected emittance• Need correction
Emittqance increase ∝ (Misalignment) 2

Dispersive effect of cavity tilt
Simulation result - Orbit and Emittancealong linac (straight linac)
Random tilt, σ 10 μrad.
-8

2.6 10projected γε (m) average of 50 seeds
projected γε (m) one example
η corrected γε (m) average of 50 seeds
η corrected γε (m) one example-8

2.5 10-8

γε (m)

2.4 10-8

2.3 10cavity tilt 10 μrad-8

2.2 10-8

2.1 10

-8

2 10

0

2000

4000

s (m)

6000

8000

10000

Cavity tilt 10 micro-radian RMS causes• 25% projected emittance increase– Fast 1 micro-rad cavity vibration causes
0.25% emittance increase• 5% dispersion corrected emittance– 100 micro-rad cavity tilt causes 500%
dispersion corrected emittance• Need correction

Cavity tilt and Edge focus
Acc. field E, length L, tilt angle θ
beamTransverse kick in the cavity: Δpt = θ eEL/c
Edge (de)focus [see appendix]
entrance

offset: y0+Lθ/2

exit

offset: y0-Lθ/2

Transverse kick at the entrance: Δpt = -eE (y0+θ L/2)/2c
Transverse kick at the exit:
Δpt = eE (y0-θ L/2)/2cÆTotal transverse kick by the cavity: Δpt = θ eEL/2c

Static corrections(transverse motion)

Necessity of Beam based corrections• Without corrections, required alignment
accuracy to keep emittance small will be,
roughly:– 0.1~1 μm for quad offset– 1~10 μrad 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 beammeasurement will be necessary– 1 to 1 correction– Kick minimization– DFS (Dispersion Free Steering)– Ballistic Steering– etc.

• These are “Local” corrections. Beam quality
is to be corrected everywhere in the linac.

One to one correction
Make BPM readings zero, or designed readings
Simulation result:

Example of quad offset
and beam orbit

Quad offset
Beam Orbit

0.0005y (m)Quad random offset σ
300 μm, no other errors

0.001

0

-0.0005-0.0010
0 10

1to1, quad offset 300 μm
3

2 10

3

3

4 10

6 10

3

1 10

3

1 10

8 10

4

s (m)
-8

average of 100 seeds
one example-8η corrected γε (m)Emittance along linac

6 10
5.5 10

-8

5 10

-8

4.5 10

-8

4 10

-8

3.5 10-8

3 10

1to1, quad offset 300 μm-82.5 10

-8

2 10

0

0 10

3

2 10

3

3

4 10

6 10

s (m)

One to one correction will not be enough.

8 10

4

Kick Minimization (KM)
• Basically:– Steer beam to minimize kick angle at every
quadrupole-dipole magnet pair. (Requiring
Quad and dipole magnets are attached orvery close each other.)• Accurate information of Quad - BPM
offset is important.
• Can correct quad misalignment but noteffective for cavity tilt.

KM, example of simulation result
0.001

Simulation result:

0.0005y (m)Quad random offset σ
300 μm, no other errors

Quad offset
Beam orbit

Example of quad offset
and beam orbit

0-0.0005

KM Quad offset 300 μm-0.001
0

2000

4000

6000

8000

10000

s (m)
-8

2.1 10Emittance along linac

Average of 100 seeds
One example-8η corrected e

2.08 10KM Quad offset 300 μm-82.06 10-82.04 10-8

2.02 10

-8

2 10

0

2000

4000

6000

s (m)

8000

10000

KM, example of simulation result - 2
Sensitivity to errors.
Emittance at the end of linac. Average of 100 random seeds.
-8-8

2.8 10-8

2.7 10-8

2.6 102.8 10-8

2.7 10-8

η corrected γε (m)η corrected γε (m)2.6 10-8

2.5 10-8

2.4 10-8

2.3 10-8

2.5 10-8

2.4 10-8

2.3 10-8-8

2.2 10-8

2.1 10-82 10

2.2 10-8

2.1 10

-8

2 10

0

200

400

600

800

Quad offset (μm)

1000

0

100

200

300

400

Cavity Tilt (μrad)500

DFS (Dispersion Free Steering)• Basically:– Change beam energy and measure beam
orbit.
– Steer beam to minimize orbit difference.

• Need to change accelerating voltage tochange beam energy. ~ 10%• BPM resolution is important.
– Because “difference of orbit” is looked at.

DFS, example of simulation result
0.001

Simulation result:

Example of quad offset
and beam orbit

Vertical orbit (m)0.0005y (m)Quad random offset σ 300
μm, no other errors

Quad offset (m)0-0.0005-0.001

Emittance along linac

This particular algorithm may
not be optimum.
Do not quote these figures.

η corrected γε (m)0
2.6 10-8

2.5 10-8

2.4 10-8

2.3 10-8

2.2 10-8

2.1 10

-8

2 10

-8

2 10

3

4 10

3

6 10

3

s (m)

8 10

3

1 10

4

Average of 40 seeds

0

2 10

3

4 10

3

6 10

s (m)

3

8 10

3

1 10

4

DFS, example of simulation result - 2
2.8 10-8

2.8 10-8

2.7 10-8

2.7 10-8

2.6 10-8

2.6 10-8

η corrected γε (m)η corrected γε (m)Sensitivity to errors.
Emittance at the end of linac. Average of 100 random seeds.

2.5 10-8
2.4 10-8

2.3 10-8

2.2 10-8

2.1 10-8
2 10

2.5 10-8
2.4 10-8

2.3 10-8

2.2 10-8

2.1 10-8

-8

2 10
0

100

200

300

400

500

Quad (BPM attached) offset error ( μm)

-8

0

100

200

300

400

Cavity Tilt error (μrad)

This particular algorithm may
not be optimum.
Do not quote these figures.

500

“Global” corrections• In addition to “Local” corrections.
• Scan knobs, measuring beam at the end of
the linac, and finding the best setting of theknobs.
• 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 zdependent kick (wakefield) in the section is tobe compensated by the bumps.
– Possible (in principle) by two bumps, phaseadvance 90 deg. apart.
– The length of the section should not be so long.
Induced dispersion or z-x/y correlation (bywakefield) should be corrected before significant
filamentation.
Cancelled
bump -> tail is kicked

phase advance ~90otail is kicked by error

Monitorbumps may be just before the monitor

Dynamic corrections(Feedback)Dynamic (time dependent) errors• Mechanical motion– Motion induced by the machine itself (motors forcooling, bobbles in pipes, etc.)– Cultural noise (nearby traffic, etc.)– Ground motion (slow movement, earth quake)• Strength fluctuation– Field strength of magnets– Accelerating field, amplitude and phase• EM field fluctuation from outside– no problem for high energy beamTypical speed of fluctuations and corrections
for transverse motion
Speed

Possible source

Effective correction For~100 ns

DR kickers,
what else ?

Feed forward(in turnaround)

Position

1μs ~ 0.1 s

Machine, cultural
Intra pulse
noise, Power supply, feedback at IP
Ground motion (fast)

Position

1 s ~ 10 s

+ Temperaturechange ,Simple orbit
feedback (in BDS
and Linac)

Position

More sophisticatedorbit FB(, e.g. KM,
If necessary).

Emittance“static” corrections

Emittance

10 s ~ 100 s etc.

1000 s ~+ Ground motion(slow)

Status of ILC-ML beam dynamics study• “Static” simulations have been well performed.
• “Dynamic” simulations are less developed.
• Integrated studies from Damping Ring to IP.
More simulations will be needed though somestudies have been done.
I (we?) expect the design performance will beachieved with reasonable tolerances of errors.
Still, a lot of things to do for confirmation of the
expected performance.

Bunch Compressor• Why we need Bunch Compressor?
• Basics of Bunch Compressor• Some comments– Study for ILC Bunch Compressor has notwell done yet.
– Quantitative discussions are very limited.

Bunch should be short at IP
Hourglass effect

Small beta

beam size = sqrt(βε)β=β∗+s2/β∗Large beta
Ignoring beam-beam force,
Ignoring horizontal size change
for max. luminosityβ∗∼ 0.26σz

L ∝1 σ zBunch should be compressedjust after Damping Ring
• Damping Ring cannot produce shortbunch, low emittance beam
– Corrective effect• Compress very high energy beam is notrealistic• Shorter bunch reduces transverse
wakefield effect in the main linac

δ

δ

δ

Principle of bunch compresszz

ChicaneRF fieldzGenerally( x, x' , y, y ' , z , δ ) ≡ ( w1 , w2 , w3 , w4 , w5 , w6 )wi ' = ∑ Rij w j + ∑ Tijk w j wk + ∑ U ijkl w j wk wl + LL
j

jk

jkl

z-dependent acceleration (deceleration): Slope of RF field⎡ z ' ⎤ ⎡1 0 ⎤ ⎡ z ⎤⎥⎢ ⎥⎢δ '⎥ = ⎢ R⎣ ⎦ ⎣ 65 1⎦ ⎣δ ⎦And E-dependent orbit length⎡ z ' ' ⎤ ⎡1 R56 ⎤ ⎡1 0 ⎤ ⎡ z ⎤ ⎡1 + R56 R65=⎢⎥⎢δ ' '⎥ = ⎢0 1 ⎥ ⎢ R
⎢
⎥
⎣ ⎦ ⎣
⎦ ⎣ 65 1⎦ ⎣δ ⎦ ⎣ R65

R56 ⎤ ⎡ z ⎤1 ⎥⎦ ⎢⎣δ ⎥⎦σ ij ≡< wi w j >σ 66 ≡ σ δ2 ,σ 55 ≡ σ z2

⎡σ 55 ' σ 56 '⎤ ⎡1 + R56 R65⎢σ ' σ '⎥ = ⎢ R⎣ 65
66 ⎦ ⎣ 65

R56 ⎤ ⎡σ 55 σ 56 ⎤ ⎡1 + R56 R65
1 ⎥⎦ ⎢⎣σ 65 σ 66 ⎥⎦ ⎣⎢ R56

R65 ⎤
1 ⎥⎦

For up-right distribution (σ 56 = σ 56 = σ '65 = σ '65 = 0)
R56 =− R65σ z2
2 2
R65
σ z + σ δ2

σ '2z =

σ z2σ δ2
2 2
R65
σ z + σ δ2

2 2
σ 'δ2 = R65
σ z + σ δ2Note : σ zσ δ is unchanged.
(Liuvile' s theorem,
area in the phase space is invariant.)

R65 and higher order in RF fieldeV cos(φ0 + ωz / c) − eV cos φ0
E0φ= R65 z + T655 z 2 + U 6555 z 3 + L
R65 =

eV ω
cos φ0E0 c
21 eV ⎛ ω ⎞T655 =⎜ ⎟ sin φ0
2 E0 ⎝ c ⎠3

Small frequency reduce higher order effect,
but require higher voltage.

1 eV ⎛ ω ⎞U 6555 = −⎜ ⎟ cos φ0
6 E0 ⎝ c ⎠R56 and higher order in a simple chicane

θ

LB

LD

LB

l = 4 ρθ + 2 LD cosθ + LD 22 ⎞⎛R56 ≈ −2⎜ LD + LB ⎟θ 02
3 ⎠⎝3
T566 ≈ − R562

U 5666 ≈ 2 R56

LD2

LB

LD

( LB = ρ sin θ )Assuming small θ.

LB2-stage compression• Bunch length from DR: > 6 mm
• Required bunch length in ML: < 0.3 mm or 0.15 mm• Then, Compression ratio: > 20 or 40– Energy spread becomes > 20 or 40 times– Longitudinal: higher order effects become large– Transverse: Dispersive effects become large(emittance growth)

2-stasge compression
1st compression Æ Acceleration Æ 2nd compression
Acceleration makes relative energy spread small.
(Absolute spread does not change.)

δ

δ

δ

Rotation in longitudinal phase space

z

z

z

Simple 1-stage compression: About 90 degree.
energy deviation Æ position deviation, position Æ energy
Simple 2-stage compression: About 180 degree.
energy Æ energy, position Æ position

Beam from DR: Stable energy but fluctuating position.
Requirement after Compressor :Tight position stability, loose energy stability.

Requirement of Longitudinalstability after Bunch Compressor
Tight position stability:
Directly affect arrival time at IP.
Æ Deviation of colliding pointÆ Hourglass effect reduces luminosity
Loose energy stability:
Energy deviation at the beginning of Main Linac
will be smeared out after acceleration.

Total 90 degree rotation is desirable
e.g. 90 degree + 0 degree
Example of (about) 0 degree rotation bunch compression

RF Voltage and Phase stability
requirement in Bunch Compressor
Most severe effect is longitudinal bunch
position stability (Luminosity reduction
due to collision position deviation. )• Voltage: < 0.1%
• Phase: <0.1 degree (1.3 GHz)
Not easy, but possible.

Transverse emittance increasein bunch compressor - Designed•For bunch compression, there must be dispersion in bending
regions.
η ( s)
R56 = ∫ds
ρ ( s)•Dispersion in bending section causes radiation excitation oftransverse emittance.
•Design horizontal emittance >> vertical emittance•Æ Bending and dispersion should be in horizontal plane•CSR (Coherent Synchrotron Radiation) is as important as Incoherent
radiation.
•CSR is important near the end of 2nd bunch compressor, wherebunch is short. (No CSR of wave length shorter than bunch issignificant.)•Total horizontal emittance increase: a few %Transverse emittance increasein bunch compressor - Due to errors• Vertical will be important• Similar in the RF cavity sections• There are not satisfactory studies yet.
– Much less advanced than studies for Main
Linac

Status of ILC Main Linac beam dynamics study• “Static” simulations have been well performed.
••••“Dynamic” simulations are less developed.
Integrated studies from Damping Ring to IP. More simulations
will be needed though some studies have been done.
I expect the design performance will be achieved withreasonable tolerances of errors.
Still, thereare things to do for confirmation of the expected
performance.

Status of ILC Bunch Compressor beam dynamics study•••

No fatal problems have been found.
But studies so far are not satisfactory yet.
– Much less advanced than studies for Main Linac
A lot of things to do.

END