ILC Beam Dynamics - Main Linac and Bunch Compressor

Kiyoshi KUBO2006

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 of

other 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 most studies are based on simulations.

Parameter of ILC Main Linac(ECM=500 GeV)

Beam energy 13~15 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 (at DR exit/IP)

Emittance y (at DR exit/IP)

8/10~12 x10-6 m-rad2/3~8 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 high

energy region• Vertically curved, following the earth curvature

Unit of main linacabout 320 units/linac

Cryomodule with magnet package

Cryomodules without magnet package

9-cell SC cavity

Magnet packageBPM

Quadrupole SC magnetDipole SC magnet

Beta-function

0

50

100

150

200

0 2 103 4 103 6 103 8 103 1 104

βx (m) β

y (m)

β 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

σx (γε=1E-5 m) σ

y(γε=2E-8 m)

σ x (m) σ

y (m)

s (m)

Vertical:5~10 μm

at 15 GeV1.5 ~ 2.5 μm

at 250 GeV

Strength of quad magnet is proportional to beam energy.

Tolerances depend on optics

β

β

1offset Cavity Acc. of Tolerance

rationsmagnet vib Quad of Tolerance

∝

∝

(See later discussions.)

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 w.r.t. the reference line and dispersion

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

-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

y (m) ηy (m)

y (m

) ηy (m

)

s(m)

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.( beam-beam interaction)

High quality is generated in Damping Ring. Preservation of the quality 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 less

severe than in bunch compressors.

Main Linac does nothing to the timing and bunch length.

( Bunch Compressor)

Single bunch: 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

Cha

rge

Den

sity

(C/m

)

z (m)

E(z) = dz'−∞z∫ λ(z' )W (z − z' )

Charge density

Deceleration by wakefield

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 by

wak

efie

ld (V

/m)

z (m)calculated from TESLA-TDR formula

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 fie

ld (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

σ E (e

V)

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

σ E/E

E (eV)

Undulator for e+ source

Bunch by bunch energy difference

• Should be (much) smaller than single bunch energy 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

Common error of all klystrons

00phase00amplitude sincos φφ eVeVE Δ−Δ≈Δ

( )00phase00amplitudekly

sincos1 φδφδδ 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

−

−

−

−

≈<Δ

≈<Δ

×≈<

×≈<

<

φσ

σ

φσδ

σδ

σ

E

E

En

En

E

E

E

E

E

⎟⎟⎠

⎞⎜⎜⎝

⎛phasecrest -off :

voltage total:

0

0

φV

Fluctuations faster than 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 emittance

preservation 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 σy beam size σy

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−=Δ

2

41exp)0()(

y

yLyLσ

Offset between two beams should be (much ?) smaller than the beam size 0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

L/L

0

Δy/σy

beam-beam force will change this significantly

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 222222 kNaEEkay fqquad

fiiiiiff ββϕββ ≈>≈< ∑

iiffiiif EEkay ϕββ sin/−=ΔFinal 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 : 1

i

i

kEiEifkia

i

fi

ifi

-ii

ϕ

ββ

square value-k of average: on,betafuncti of average:

quads, ofnumber : offset, of rms:2k

Na q

β

Estimation of beam position change due to magnet strength change

error.strength quad theis , where,by pagepreviousin the Replace

i

iiiik

kakaδ

δFinal position change due to strength change of i-th quad:

error.strength dipole theis , where

,by page previous in the Replace

,0

,0

i

iii

k

kka

δ

δFinal 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.

Estimation of beam position change due to accelerating voltage jitter in tilted cavities

angle.t cavity til theis andenergy beam the error, voltage theis where ,2/by page previous in the Replace

iii

iiiiiθEV

)θ/EV(ekaδ

δδ

Final position change due to strength change of i-th cavity:

See lecture note.

Stronger focus optics make quad vibration tolerance tighter

βεβε

ββ

σ y

q

fy

fq

y

f NakNay~

222=≈

><

Beam position jitter / beam size:

proportional to:quad vibrationsquare root of number of quads

inversely proportional to:square root of average beta-functionsquare root of emittance

Beam quality - Emittance• Our goal is high luminosity• For (nearly) Gaussian distribution, emittance is a

good 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 of quality dilution in the main linac, anyways.– Usually OK.– No other good parameter.

Example where emittance does not well correlated with luminosity

So called “banana effect”

Same projected emittance.Different luminosity

Beam-beam interaction

z-y correlated

no correlation

Collision with angle

NOTE• We use emittance as a measure of quality in

the main linac.• Halos, tails far from core should be ignored in

calculating the emittance.– Effects of halos or tails should be considered in

other context.• Luminosity may not be well correlated to emittance, in

some cases, because of Beam-beam force at IP

Dominant Sources oftransverse Emittance dilution

• Wakefield (transverse) of accelerating cavities– Electromagnetic fields induced by head

particles 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.

entrancethefromdistance:constant) (assumegradient acc.:

atenergy andenergy initial:, constant) (assumefunction -beta: constant) (assumeon wakefuncti:

particlefirst theofcahnge:

0

sg

sEE

Wq

β

( )⎩⎨⎧

=>

≈

≈

)0(2)0(2)0()(log

)0 (initialn oscillatio particle 2nd of Amplitude/

)(initialoscillaionparticlefirst of Amplitude

0

01

0000

00

gsWqeaggEsEWqea

a

EEaa

a

ββ

Rough estimated BBU

15.001 ≈⇒ aa

213 V/C/m103

,170 m, 100 nC,3 MeV/m, 30

×<

≈≈≈≈

W

E/Eqg βFor ILC,

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 may not be used in ILC.

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

WeqsyEKsy

syEKsy

+−=

−=

)/( ,/)sin()0(')cos()0()(: , ofSolution 00000 EKkkksyksysyy =+=

Amplitude of the 3rd term grows linearly with distance

No growth of induced oscillation

kksEWeqskksyksysy

EE

/)sin(2

/)sin()0(')cos()0()(

, If

0

0011

01

++=

=

particle. following theofsolution a is )()(

, If

01

01

0

1

sysyEK

EWeq

EK

=

−=+−

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

-15

-10

-5

0

5

10

15

0 50 100 150 200 250 300

0.1ΔEauto

-15

-10

-5

0

5

10

15

0 50 100 150 200 250 300

ΔEauto

-15

-10

-5

0

5

10

15

0 50 100 150 200 250 300

1.5ΔEauto

Oscillation piriod of the leeding bunch

-30

-20

-10

0

10

20

30

0 50 100 150 200 250 300

ΔE=0

-15

-10

-5

0

5

10

15

0 50 100 150 200 250 300

0.5ΔEauto

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

• 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 help

banana-effect)• Difficult to apply to multibunch BBU

– Complicated wake function need to introduce complicated bunch-by-bunch energy difference

Note on auto-phasing, BNS damping

Effect of misalignment of cavitiesAssume

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

∑−=i

iiffiijifj EEWaey ϕββ sin/1,,

f

fjfcav

iiifij

fi

caviifj gE

EEWLaeW

EELeay

2)/log(

sin 0222

22,

2222

,ββ

ϕββ ≈><>=< ∑

)( :wake"- sum" , kjjk

ikij zzWqW −= ∑<

Position change of j-th particle at linac end due to i-th cavity:

Expected Position change square of j-th particle:

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 -iofoffset :

gLq

EEa

cavi

k

i

ii

fi

i

ϕββ

iiffiijifi EEWeay ϕββ sin/1,, −=

Position change at linac end:

assume all cavities have the same wakefunction

caviL

caviL

Effect of misalignment of cavities-continued

Stronger focus optics (smaller beta-function) make Effects of wakefield weaker

( )⎩⎨⎧

=>

≈

≈

)0(2)0(2log

)0 (initialn oscillatio particle 2nd of Amplitude/

)(initial oscillaionparticlefirst ofAmplitude

0

001

0000

00

gsWqeaggEEWqea

a

EEaa

a

ββ

BBU:

Misalignment:

f

fjfcav

iiifij

fiifj gE

EEWLaeW

EEeay

2)/log(

sin 0222

22,

222

,ββ

ϕββ ≈><>=< ∑

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 σ of beam size. monochromatic beam.

Only 1% emittance increase.

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

proj

ecte

d γε

(μ)

s (m)

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

/pC

/m2 )

s (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. 7% emittance increase.

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

proj

ecte

d γε

(m

)

s (m)

Day 2 (Aug.31)

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

/pC

/m2 )

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.

Damping of Higher Order Mode Wakefield

Two HOM Couplers at both sides of a cavity

TESLA-TDR

TESLA-TDR

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

Detuning of wakefield

10 offactor by reduction Wakefield60bunch)next (next ns 500~for effectivefully becomes Detuning

MHz2~ 0.1% spread 2GHz,~frequency HOM Typical Linac,Main ILCIn

.2/ of Amplitude

,/2for , spread has If, of Amplitude

spread, no has If

sin

function,-betatocomparablelength in cavities

effective

effective

1effective

→≈→

→≈

≈

>≈

≈ ∑=

nt

WnW

tWnW

tWW

n

i

i

n

ii

ω

ωω

δ

δπδω

ω

ω

In X-band warm LC, carefully designed damped-detuned structures were necessary. Not for ILC.

ωi: HOM frequency of i-th cavity

For BBU: effective wake is from sum of cavities within length comparable to beta-function

Wakefunction envelope from HOMs (from TESLA-TDR) with/without random detuning (50 cavities) and damping

No detuning

Random detuningσf/f=0.1%

1

10

100

0 1 103 2 103 3 103 4 103 5 103 6 103

σf=0, no damp

σf=0, dampW

enve

lope

(V/p

C/m

2 )

time (ns)

1

10

100

0 1 103 2 103 3 103 4 103 5 103 6 103

fsf=0.1%, no damp

σf=0.1%, damp

Wen

velo

pe (V

/pC

/m2 )

time (ns)

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”, for fast varying errors.

(see lecture note.)

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)

proj

rcte

d γε

(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

pers

in C

orre

cted

γε (m

)

s (m)

The emittace increases by 0.1% of nominal.Initial dispersion should be matched

This is getting small as the relative energy spread becomes small.

Dispersive effect in perfect linac “Filamentation” with injection error

y'sq

rt(β

y/εy)

y/sqrt(βyε

y)

y'sq

rt(β

y/εy)

y/sqrt(βyε

y)

y'sq

rt(β

y/εy)

y/sqrt(βyε

y)

y'sq

rt(β

y/εy)

y/sqrt(βyε

y)

Different phase advance for different energy particles.

Dispersive effect from quad misalignment

Simulation result - Emittance along linac (straight linac)

Random offset, σ 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 γε (m) average of 50 seedsprojected γε (m), one exampleη corrected γε (m) average of 50 seedsη corrected γε (m) one example

γε (m

)

s (m)

Quad misalignment 1 μm

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

2ent)(Misalignm increase Emittqance ∝

Dispersive effect of cavity tilt

Simulation result - Orbit and Emittance along linac (straight linac)

Random tilt, σ 10 μrad.

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 μrad

projected γε (m) average of 50 seedsprojected γε (m) one exampleη corrected γε (m) average of 50 seedsη corrected γε (m) one example

γε (m

)

s (m)

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

beam

Transverse kick in the cavity: Δpt = θ eEL/c

offset: y0+Lθ/2 offset: y0-Lθ/2

entrance exitEdge (de)focus [see appendix]

Transverse kick at the entrance: Δpt = -eE (y0+θ L/2)/2cTransverse kick at the exit: Δpt = eE (y0-θ L/2)/2c

Total transverse kick by the cavity: Δpt = θ eEL/2c

Cavity tilt and Edge focusAcc. field E, length L, tilt angle θ

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 tiltwhich 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– 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 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 μm

average of 100 seedsone example

η co

rrec

ted

γε (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 μm

Quad offsetBeam Orbit

y (m

)

s (m)

Simulation result:Quad random offset σ300 μm, no other errorsExample of quad offset and beam orbit

Emittance along linac

One to one correction will not be enough.

• Basically:– Steer beam to minimize kick angle at every

quadrupole-dipole magnet pair. (Requiring Quad and dipole magnets are attached or very close each other.)

• Accurate information of Quad - BPM offset is important.

• Can correct quad misalignment but not effective for cavity tilt.

Kick Minimization (KM)

Simulation result:Quad random offset σ300 μm, 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 μm

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 μmAverage of 100 seedsOne example

η co

rrec

ted

e

s (m)

KM, example of simulation result

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

η c

orre

cted

γε

(m)

Quad offset (μm)

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

η c

orre

cted

γε

(m)

Cavity Tilt (μrad)

KM, example of simulation result - 2

DFS (Dispersion Free Steering)• Basically:

– Change beam energy and measure beam orbit.

– Steer beam to minimize orbit difference. • Need to change accelerating voltage to

change beam energy. ~ 10%• BPM resolution is important.

– Because “difference of orbit” is looked at.

DFS, example of simulation resultSimulation result:Quad random offset σ 300 μm, no other errorsExample of quad offset and beam orbit

Emittance along linac

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

-0.001

-0.0005

0

0.0005

0.001

0 2 103 4 103 6 103 8 103 1 104

Quad offset (m)

Vertical orbit (m)

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

0 2 103 4 103 6 103 8 103 1 104

Average of 40 seeds

η co

rrec

ted

γε (m

)

s (m)

DFS, example of simulation result - 2Sensitivity 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.

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

η c

orre

cted

γε

(m)

Quad (BPM attached) offset error ( μm)

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

η c

orre

cted

γε

(m)

Cavity Tilt err300 400 500

or (μrad)

“Global” corrections

• In addition to “Local” corrections.• Scan knobs, measuring beam at the end of

the linac, 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

advance 90 deg. apart. – The length of the section should not be so long.

Induced dispersion or z-x/y correlation (by wakefield) should be corrected before significant filamentation.

bump -> tail is kicked Monitortail is kicked by errorCancelled

phase advance ~90o bumps may be just before the monitor

Dynamic corrections (Feedback)

Dynamic (time dependent) 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 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 correctionsfor transverse motion

Speed Possible source Effective correction For

~100 ns DR kickers,what else ?Machine, cultural noise, Power supply, Ground motion (fast)+ Temperature change ,

etc.

1000 s ~ + Ground motion (slow)

“static” corrections

Feed forward (in turnaround)

Emittance

1μs ~ 0.1 s Intra pulse feedback at IP

Position

Position

Position1 s ~ 10 s Simple orbit feedback (in BDS and Linac)

10 s ~ 100 s More sophisticated orbit FB(, e.g. KM, If necessary).

Emittance

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 some studies have been done.

I (we?) expect the design performance will be achieved 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 not well done yet.

– Quantitative discussions are very limited.

Bunch should be short at IP

Hourglass effectbeam size = sqrt(βε)

β=β∗+s2/β∗

Ignoring beam-beam force,Ignoring horizontal size change for max. luminosity β∗∼ 0.26σz

Small betaSmall betaSmall beta

Large beta

zL σ1∝

Bunch should be compressed just after Damping Ring

• Damping Ring cannot produce short bunch, low emittance beam – Corrective effect

• Compress very high energy beam is not realistic

• Shorter bunch reduces transverse wakefield effect in the main linac

Principle of bunch compress

δz

δ

z

δ

z

RF fieldChicane

),,,,,(),,',,',( 654321 wwwwwwzyyxx ≡δ

LL+++= ∑∑∑ ljkl

kjijkljk

kjijkj

jiji wwwUwwTwRw '

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡δδz

Rz

101

''

65

z-dependent acceleration (deceleration): Slope of RF field

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ +=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡δδδz

RRRRz

RRz

11

101

101

''''

65

566556

65

56

Generally

And E-dependent orbit length

>≡< jiij wwσ

⎥⎦

⎤⎢⎣

⎡ +⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ +=⎥

⎦

⎤⎢⎣

⎡1

11

1''''

56

656556

6665

5655

65

566556

6665

5655R

RRRR

RRRσσσσ

σσσσ

255 zσσ ≡,2

66 δσσ ≡

22265

265

56δσσ

σ+

−=

z

zR

RR

22265

222'

δ

δ

σσσσσ

+=

z

zz

R

22265

2' δδ σσσ += zR

For up-right distribution )0''( 65655656 ==== σσσσ

)invariant. is space phase in the area theorem,s(Liuvile'

unchanged.is :Note δσσ z

R65 and higher order in RF field

L+++=

−+

36555

265565

0

00 cos)/cos(

zUzTzR

EeVczeV φωφ

00

65 cosφωcE

eVR =

0

2

0655 sin

21 φω

⎟⎠⎞

⎜⎝⎛=

cEeVT

0

3

06555 cos

61 φω

⎟⎠⎞

⎜⎝⎛−=

cEeVU

φ

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

R56 and higher order in a simple chicane

2cos24 DD LLl ++= θρθ )sin( θρ=BL

2056 3

22 θ⎟⎠⎞

⎜⎝⎛ +−≈ BD LLR

56566 23 RT −≈

565666 2RU ≈

LB LD LB LBLB LDLD2

Assuming small θ.

θ

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 Acceleration 2nd compressionAcceleration makes relative energy spread small.(Absolute spread does not change.)

Rotation in longitudinal phase space

δz

δ

z

δ

z

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

Simple 2-stage compression: About 180 degree.energy energy, position position

Beam from DR: Stable energy but fluctuating position.Requirement after Compressor :

Tight position stability, loose energy stability.

Requirement of Longitudinal stability after Bunch Compressor

Tight position stability:Directly affect arrival time at IP.

Deviation of colliding point Hourglass effect reduces luminosity

Loose energy stability:Energy deviation at the beginning of Main Linacwill be smeared out after acceleration.

Total 90 degree rotation is desirablee.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 increase in bunch compressor - Designed

∫= dsssR)()(

56 ρη

•For bunch compression, there must be dispersion in bending regions.

•Dispersion in bending section causes radiation excitation of transverse emittance.•Design horizontal emittance >> vertical emittance

• 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, where bunch is short. (No CSR of wave length shorter than bunch is significant.)

•Total horizontal emittance increase: a few %

Transverse emittance increase in 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 with

reasonable 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