LINAC-II

ILC SchoolChicago, Oct. 22, 2008

T. Higo, KEK

1

Contents of LINAC-II• Beam quality preservation

– Luminosity– Linac optics– Perturbations– Wake field– Wake field suppression– Alignment to beam– Breakdown rate– Dark current– Summary of LINAC-II

2

Energy and luminosity

3

BeamRFcBeamRFrepbc

repb

linac

repb

BeamRFlinaclinacac

NL

EeFNNEeFNL

PL

FNNPLEEe

00 1/

Luminosity

4

Luminosity related parameters• Important parameters relevant in this lecture are

– Beam transverse size• Main theme of this lecture

– Number of bunches in a train• Long-range wake field

– Number of particles in a bunch• Short range wake field

– Repetition frequency• Wall plug power

5

**

2

4 yx

Dbrep HNnfL

Requirements for linear collider • Beam quality from DR and BC should be preserved.• Phase space

– Longitudinal• Energy acceptance of the final focus

– Transverse • Emittance preservation to keep beam size small

• Wake field– Single bunch:

• Short range (intra-bunch) wake field• Dispersive effect in the bunch

– Multi-bunch: • Long range (inter-bunch) wake field among bunches• Bunch to bunch dispersive effect

6

Linac example parametersILC and CLIC

7

Item units ILC(RDR) CLIC(500)Injection / final linac energy ELinac GeV 25 / 250 / 250Acceleration gradient Ea MV/m 31.5 80Beam current Ib A 0.009 2.2Peak RF power / cavity Pin MW 0.294 74Initial / final horizontal emittance ex mm 8.4 / 9.4 2 / 3Initial / final vertical emittance ey nm 24 / 34 10 / 40RF pulse width Tp ms 1565 242Repetition rate Frep Hz 5 50Number of particles in a bunch N 109 20 6.8Number of bunches / train Nb 2625 354Bunch spacing Tb ns 360 0.5Bunch spacing per RF cycle Tb/ TRF 468 6

Linac example parametersILC and CLIC

8

Item units ILC(RDR) CLIC(500)RF frequency F GHz 1.3 12Beam phase w.r.t. RF degrees 5 15EM mode in cavity SW TWNumber of cells / cavity Nc 9 19Cavity beam aperture a/l 0.152 0.145Bunch length z mm 0.3 0.044

ILC parameters are taken from Reference Design Report of ILC for 500GeV.

CLIC500 parameters are taken from the talk by A. Grudief, 3rd. ACE, CLIC Advisory Committee, CERN, Sep. 2008, http://indico.cern.ch/conferenceDisplay.py?confId=30172.

Bunch pattern

9

Many bunches / trainMany rf cycles (~103) till next bunchz/l~0.0013

Only 6 rf cycles till next bunchz/l~0.0018 actual z 10 times smaller

ILC 2625 bunches / pulse

468 rf / separation

z=300mm

CLIC 354 bunches / pulse

6 rf / separation

z=44mm

Bunch profile and power spectrum

10

]2

[)( 2

2

z

zExpzf

])(21[)( 22

cExpP z

0 10 0 2 0 0 30 0 4 0 0 50 0

0 .2

0 .4

0 .6

0 .8

1 .0

1.0 0.5 0 .5 1 .0

0 .2

0 .4

0 .6

0 .8

1 .0

mm GHz

Emittance dilution / preservation

11

Emittance preservation

12

Emittance preservation along the linac is one of the key issues of the main linac of LC.

Emittance in 6 dimensions; Single bunch emittance vs multi-bunch emittance Longitudinal emittance vs transverse emittance in X and y

Error sources for dilution should be minimized. Better alignment, wake field suppression, etc.But if diluted, we try to correct by realignment, corrector fast magnets, etc.

Coherent dilution can be corrected but it phase space volume is fully filled in an incoherent manner, it cannot be corrected.

Multi-bunch dilution can be corrected bunch by bunch correction.

Very basic optics of linac

13

LQ LQ

L

LrLr

0)(2

2

xzkxdzd

][/

)( 2 meEgczk

LLLkLk QQr /2,

In a thin lens approximation

)1(/4

2 ml

eELgc

Lk

Transverse motion of eq.

CoshSinhk

Sinhk

CoshM

CosSink

Sink

CosM DF

1,

1

Transfer matrix per period;

2/2/ FdriftDdriftFFDF MMMMMM Twiss parameter at QF;

mmmmSinCosSin

SinSinCosM FDF

)1(/822

,2

1222

mm eELgcSinCos

Betatron oscillation along the linac is the very basic of the system.

)(1,))(()()(sds

dsSinsAzx

Assume (s) constant

l

2)(

1,)()( s

kskSinzx

In the left FODO system,

Transverse emittance

14

Multi bunch

x

X’

Phase space

volume

x

X’ Single bunchCorrectable

CoherentCorrectable

x

X’ Single bunchIncorrectable

IncoherentIncorrectable

x

X’ Single bunchCorrectable

CoherentCorrectableSing

le bu

nch

x

X’ Multi bunchbest

CoherentCorrectable

x

X’ Multi bunchCorrectable

if Tb large

Bunch-by-bunchCorrectable

x

X’ Multi bunchCorrectable

if Tb large

Difficult toCorrect

If before filling the whole phase space, the emittance can be controlled.

Practical errors to be considered• Optical error and misalignment

– Field stability in Q– Misalignment of Q, structures and BPM

• RF error– Phase, amplitude jitter– Pulse to pulse, within pulse,– Asymmetry in cavity to time dependent transverse kick

• Wake field– Long range and short range– Longitudinal and transverse

15

Possible cares and corrections• Optical error and misalignment

– Alignment with using beam information– BPM information, on- /off-energy, on- /off- Q setting

• RF error– Feedback with cavity field– Mechanical precision of cavity, offset, tilt, – Suppression of field asymmetry

• Wake field– Cavity HOM damping and cancelling– Multi-bunch energy compensation with injection timing,

ramping pattern, etc

16

Wake field and impedance

17

Wake fields driven by relativistic particle

18

Lorentz contracted fieldIn free space

1/

EM field associated to the particle is scattered by periphery shape.

Wake field

19

Driving bunch: Unit charge bunch at offset radius rWitness bunch: trailing at z=ct behind

Wake field W(s) is the kick received by the witness bunch.

)(

),,(),,(1),(

21

/)(11

11

sWqqp

tzrBzctzrEdzq

srWczst

r1

s=ctr2

q2 q1

czstzL tzrEdz

qsrW /)(1

11 ),,(1),(

czstT

T tzrBzctzrEdzq

srW/)(,

111

1 ),,(),,(1),(

Panofsky-Wenzel theorem

20

)(

)(

zz

zz

EEz

etBe

tBeEe

tczdzd

1

zTzTT Edzq

BeEz

dzq

zyxWz 11

11),,(

HEz

Think about TM11 field

rz Bt

E

LTT WWs

s

LTT syxWdssyxWor )',,('),,(

Cylindrical symmetric systemand multipole expansion

21

WWWs

W TTL

,

0

)(),',(),,',(m

m mCossrrWsrrW

From Panofsky Wenzel theorem,

For axisymmetric environments, W can be expanded into multi-poles,

From Maxwell’s equation, form of Wm can be found,

mmmm rrsFsrrW ')(),',(

Wake functions are expressed now as,

)]()([')(),',(

)(')(),',(

1)(

)(

mSinmCosrrrsFmsrrW

mCosrrsFs

srrW

mmm

mT

mmm

mL

Near axis, only m=0 for longitudinal wake and m=1 for transverse wake.Longitudinal: constant over radiusTransverse: linear over drive bunch offset but constant on witness bunch position.

Impedance

22

Impedance: Fourier transform of the wake function

c

sj

TT esyxWcsdjyxZ

),,()(),,(

c

sj

LL esyxWcsdyxZ

),,()(),,(

),,(),,( yxZyxZc LTT

Panofsky Wenzel theorem becomes,

Wake is real then)}(Im{)}(Im{,)}(Re{)}(Re{ LLLL ZZZZ

From causality,

)()}(Im{)()}(Re{

00)}(

csSinZd

csCosZd

sallforsW

LL

L

Then,

)}]()(Im{)()}(Re{[{21

)(21)(

csSinZ

csCosZd

eZdsW

LL

csj

LL

Actual impedance shape

23

0 2 4 6 8 1 00 .0

0 .2

0 .4

0 .6

0 .8

1 .0

Re(Z||()

Leaking to beam pipe

Trapped modes

)()}(Re{{1)(

csCosZdsW LL

Finally, wake function is calculated from only the real part of the impedance, Actual real part of the

impedance is illustrated as shown in the left figure.

Low frequency trapped modes and higher frequency component with escaping into the beam pipe.

)(1)(

0

0

Qj

RZL

Foe each resonance,

Loss parameter

24

czj

nzn

n

tjnz

n

n

erEdzV

erEtrE

)(

})(Re{),(

,

0

Define loss parameter;

n

nn U

Vk

4

2

For delta-function-like driving particle;

)(2)(0 c

sCosksW nn

nL

Self wake is half of the wake behind the bunch,Energy lost in mode n is

21qkU nn

Electric field is expressed as,

For finite bunch length,

}2

)(exp{21)(

),(),(

2

20

1

z

sss

ctzqtr

l

l

Actual wake is a convolution,

)'()'(')(0

sWssdssW LLl

Loss to fundamental mode;

Loss to higher modes;

})(21exp{ 22

0 zfund ckk

1

22 })(21exp{2

nz

nnHOM ckk

n

nn u

Ek

4

2

SW

TW

Longitudinal wake function in DLS

25

Summation of resonant modes up to a certain frequency,

Higher than the frequency, and in high energy limit >>a/c, optical resonator model predicts

dCosAWAddk

ma

2/302/3

0 )(2)(,

Total wake field calculated for the SLAC disk loaded structure became as shown in right figure (P. Wilson Lecture)

Fundamental mode dominates for long-range wake, with some high Q modes superposed.

Much higher than 400th mode contributes in very short range wake field.

)(2)(0 c

sCosksW n

N

nnL

Transverse wake function in DLS

26

We follow a paper* by Zotter and Bane on transverse wake field calculation on disk-loaded structure.

* B. Zotter and K. Bane, “Transverse Resonances of Periodically widened Cylindrical Tube with Circular Crosssection”, PEP-Note-308, SLAC, 1979.

Synchronous space harmonic component of the n-th TW mode, axial electric field is expressed as

)}/({)()(0 cztCosmCosarEE n

mnzn

Where E0n is the field at r=a, iris opening radius. Loss parameter is

2220 )(,

4q

ar

kuanduEk mq

nnn

nn rq is drive bunch position

qkar

E nmq

n )(20

Therefore,

)()()()(2czCosmCos

ar

arqkE n

mqmnzn

With Panofsky Wenzel theorem;)()( )/()( cmf

zTcmf

TT EcjBcE

Transverse wake field (cont.)

27

Finally for the transverse wake field,

)()()(2)(

nq

n

nTn Sin

ar

ackW

Summation gives total from resonant-like modes,

)()()(2)(

nn n

nqT Sin

ack

ar

W

Over maximum frequency m, integration gives wake field using

2/31

A

dkd

The calculated wake field for SLAC DLS are shown in three time ranges;

Dipole wake field parametrization

28

})/()/1(1{4)( 00400 ssExpss

ascZsWT

17.1

38.079.1

00 169.0,377LgasZ

402)(acZsW

s T

Initial slope;

Transverse wake field for NLC structure;

10 .0 0 .2 0 .4 0.6 0 .8 1 .0

50

1 0 0

1 5 0

2 0 0

a=3mm

a=4mm

a=5mm

s [mm]

WT

[V/p

C/m

m/m

]

Transverse wake field for NLC.

00

100

200

Linear slope andstrong a-dependence!

Single-bunch beam dynamics and cures

29

Beam dynamics under shot range transverse wake field

30

Force and momentum change in transverse direction;

dsdx

dsdcm

dtdxm

dtd

dtdpF x

x 200

Under the force due to the transverse wake field, the equation of motion becomes

z

yzywysxdyNeezsxkcmzsxdsd

dsdcm )()(),(),(),( 22

02

0 l

The second term in left hand side is the betatron oscillation term in a linac optics.Right hand side is the force due to wake field inside the bunch at the position z due to the offset of the precedent part of the same bunch at y with offset value x(s,y).

Consider a bunch at s along the linac with the transverse position x(s,z) within the bunch with its charge distribution l(y).

20

2

0

02

41

)()(),(4),(),(1

cmerwhere

yzywysxdyrNzsxkzsxdsd

dsd

e

ze

e

l

e

This becomes

Two particle model view

31

If no acceleration case;

z

e yzywysxdyrNzsxkzsxdsd )()(),(4),(),( 02

2

2

l

e

Bunch is divided into two part, head and tail

Each charge Ne/2, separated by 2z and

Head bunch behaves as

skCosxsx 11 )( The force experienced by the tail particle due to the wake field driven at the head particle

)2(2 12 zwxeNeF

skjze exCxwrNxkxdsd

e

110

22

22

2

2)2(4

Then the equation of motion of the tail particle becomes

This is the forced oscillation with the same oscillation frequency as the drive field.Find a solution of the form

skjesyx )(2

Growth estimation in two particle model

32

1''' )(2)( xsykjsy

This has a solution

skxjsy2

)( 1

Then,

skjeskxjsx

2)( 1

2

This result states that 1. Amplitude increases linearly 2. Phase is 90 degrees delayed

x1

x2

More general but constant energy case

33

z

e yzywysxdyrNzsxkzsxdsd )()(),(4),(),( 02

2

2

l

e

Again in no acceleration case;

s

yz Bunch frame

Position in linac

Assume initial offset without slope

0,),0(0

0

ss

xxzx

And assume wake field is small

2kwrN e

Let us find a solution of the formskjezsazsx ),(),(

Substitute into the above equation

ze yzywysadyk

rNjsa )()(),(

24 0 le

Linear slope wake and square bunch

34

blzzww

2/1)(

'

l

And define a parameter

b

e

lkwrNr

e

24 '

0

Then,

zysazydyrj

sa ),()(

This has an asymptotic solution,

])2(4

3[)2(6

3/122/3

6/120 zsrExpzsrxa

Derived by A. W. Chao, B. Richter and C. Y. Yao, NIM, 178, p1, 1980

2rsz2

])2(4

3[)2( 3/122/3

6/12 zsrExpzsr

Take energy gain into account

35

z

e yzywysxdyrNzsxkzsxdsd

dsd )()(),(4),(),(1 02 l

e

Back to equation

Assume slow acceleration

k/'And assume no wake first, then

0' 22

2

xkxdsdx

dsd

Let us design asskjezsazsx ),(),(

The equation on a becomes

0'2 asa

This has a solution,

)(/)(),( 00 szazsa

It states the adiabatic damping

Now let the wake ON,

With linear slope wake and energy gain

36

ze yzywysadyrNakj

sakj )()(),(4'2 0 l

e

)(/),(),( szsbzsa

This time let us define

ze yzywysbdy

skrNj

sb )()(),(

)(24 0 l

e

Then the equation becomes

This is exactly the same as a(s,z) case but with varying (s)

Let us define coordinate S

)(/ sdsdS Then

ze yzywysbdyk

rNjSb )()(),(

24 0 le

This form is also exactly the same as that of a(s,z) before.This has an asymptotic form as a.

Solution of growth

37

Assume uniform acceleration

Lss ifi /)()(

Then,

ifiif

sLsLS

)(ln)(ln

Therefore, as the case with a,

b

e

lkwrNrwherezSrExpzSrxb

e 2

'4])2(4

3[)2(6

03/122/3

6/120

Since Sfinal is

i

ffeffeff

i

f

ffinal whereLLS

ln//ln

From these formula, we can estimate the growth along the bunch

b

e

lkwrNrwherezSrExpzSrx

sa

e 2

'4])2(4

3[)2(6)(

1 03/122/3

6/120

BNS damping

38

Equation of motion in the two particle model,

The tail particle resonant growth is suppressed. In FODO lattice,

2tan21 m

m

Ldd

Lddk

)1(/82

2

m

eELgcSin )1(

/8221

2

2

mm

eE

LgcddCos

22 m

m Tan

dd

We make the energy variation within a bunch to introduce variation of k

This suppression is called BNS damping.

skjze exCxwrNxkxdsd

e

110

22

22

2

2)2(4

skjze exCxwrN

xkkxdsd

e

110

222

22

2

2)2(4

)(

Varying the tail particle oscillation frequency from that of the head particle,

&

In practice; Energy tapering can be produced by setting the bunch in RF slope. The longitudinal wake function help decreased the energy toward the tail.

Autophasing

39

z

e yzywysxdyrNzsxkzsxkzsxdsd )()(),(4)},(),({),( 022

2

2

l

e

If we can vary the k as

z

e yzywdyrNzsk )()(4)},( 02 l

e

The solution becomes simply )cos()(),( 0 skxsxzsx

This is stable and x does not depend on z, which means the both head and tail stays as in the right figure.

x

X’ Single bunchCorrectable

CoherentCorrectable

This suppression scheme is autophasing.The slope on k is produced by energy profile inside the bunch with amplitude of the order of

'02 4 WrNk ze

e

The big energy slope should be compensated at the downstream of linac.

Start with the equation of motion;

Long range wake field

40

Fundamental theorem of beam loading0: Assume a cavity field in phasor diagram with one dominant mode

tjeVtV )(

1: Point particle passed an empty cavity, leaving a field, wake field,

Vc=t2: When the second bunch comes in, superposition applies;

e

Vb

Ve Vreference

21

1

b

be

VU

VfqVqE

Beam energy loss

Cavity stored energy

e

Vb2+

Vb1+

Vb1++Vb

2+

=t

CosV

VeVU

eVV

b

bj

b

jbb

1(2

)(2

2212

1

Fundamental theorem of beam loading cont.While loss of the second bunch;

)(2 e CosVqVqE be

Since particle energy loss = cavity stored energy;

221 UEETherefore,

0)2()(2 ee CosSinqCosVCosqVfq bb

This should always true for any , then

21,

2,0 fqVb

e

When a bunch passes a cavity, it excites the cavity with the field in a decelerating direction, or it remains a deceleration wake field in the cavity. The bunch feels half of this excited field.

Long range wake field in a cavity

43

2/,

,))()((2)(

022

0

Q

csSin

csCoseksW c

s

LL

Longitudinal wake excited in a cavity is expressed as

In a cavity with very high Q value,

Longitudinal wake field behaves cosine-like. The bunch is decelerated and excite the field in the cavity. Point-like bunch suffer from the wake, deceleration.

Transverse wake behaves sine-like. It increases linearly in time at very short time, usually within the bunch.

)(2)(

)(2

,)(2)(

12

2

1

csSineksW

QRkwhere

csCoseksW

cs

QTT

LL

LLcs

QLL

L

Calculation of impedance in SW or TW

44

energyStoredUc

kdzezEVU

VQR L zkj

zL

,,)(,][

2 0

2

For longitudinal mode;

For transverse mode;

energyStoredUc

kdzerzErV

kUrV

QR L zkjz

T

,,)(/,][1

2/

02

2'

Then for longitudinal wake;

)(2]//[1)( 2,0

sc

CosekmCVdzEqL

sW n

n

cs

QnL

L cmfzL

n

n

Calculation of kL by SW field solver

45

For SW cavity;

LQRL

UVk LSWL /)(

2/

4

2

,

For TW cavity;

uEk TWL 4

20

,

where only n=0 space harmonics contributes;

)(0 cstCosEEz

When we consider SW field, SW = FW + BW;

})()({),( 0 cztCos

cztCosEtzE SW

z

The field calculated by SW mode is the case with t=0 in the above equation;

)(2),( 0 czCosEtzE SW

z

Loss parameter formula

46

When we consider the coupling of beam to TW mode, the relevant energy is only the forward wave, which is half of USW

SW

SW

SW

SWTWL U

VLUVk

2/

)2/(4

22

,

LQRk

LTWL /,

Then, loss parameter is

SWLTWL kk ,, 2

LVz

cCoszE

LE

L

L

SWz )()(1 2/

2/0

Therefore, E0 can be calculated as

Transverse wake calculation

47

TBvE )(

Transverse wake field excited by a bunch with charge q passing a cavity with transverse position offset of r,

The wake field due to this field is expressed

)(/

2][)(1)( 2,

0 2 sc

Sinec

kmCVdzBvE

rqLsW nc

sQ

n

nTL cmfTT

n

n

This can be explained for both SW and TW as follows .

cmfzT

cmfT E

cjBvE /

)(

If we apply this to the above eq.

rV

cj

rqLsW

T

/1)(0

(#)

For SW case, from Panofsky Wenzel,

Transverse wake calculation (cont.)

48

The energy of the cavity excited by charge q with offset r is equal to the energy loss if the bunch interacting with the field excited by itself,

LrqkrrVqrU SW

T2)(

21)(

whereL

QRkk T

SWT /)'(

22

ckW T

T /20

and thinking

The equation (#) is found proven. (This is for SW case)In the TW case, as in the longitudinal wake,

SWTTWT kk ,, 2Therefore,

LQRkk

TTWT /

'2

,

Frequency scaling

49

12 3

22

ll

UV

QR

L

Per cavity;

112

/2

2'

kU

rVQR

T

Per unit length;

L

QR

L

/

2/

L

QRkW

LLL

L

QR

T /'

4'

2 /

L

QRkk TT

3

TT

kW

L

T

[]

[]

Dipole mode field to calculate R/Q

50

dlaKwhereH

zCosrKJCosK

jH

zCosrKJr

SinK

jH

zCosrKJCosE

zSinrKJr

SinK

E

zSinrKJCosK

E

znc

z

zcc

zcc

r

zcz

zcc

z

zcc

zr

/,/0

)()()(

)()(1)(

)()()(

)()(1)(

)()()(

1

'1

12

1

12

'1

e

e

Let us take the most typical dipole mode, TM1nl, in a pillbox cavity

Dipole mode field to calculate R/Q (cont.)

51

)()(

)2

()3(212

/

12'

1222

22

2

2'

nc

c

T

JK

kad

dkSinkK

kUrV

QR

em

Then for a cavity

Per a cavity

2/

0

2/

0

2/

2/)(3)(5.12)(/

L

c

L

c

L

L

zkjz dzkzCosKdzkzCosKdzerzErV

)2

(3/ LkSinkKrV c

crc

cccc

z KrKrKJrKJKrKJ

rrzE 5.1])()(2[)()(

0

101

The slope of Ez in r-direction on =0 plane at the beam axis

Comments on dipole mode

52

Firstly note that TE mode cannot couple to beam because of no Ez field!

Secondly, there are two polarization in dipole modes with the same field pattern.

Therefore, it is necessary to separate in frequency for these modes to be stable unless the two polarizations can be divided by geometry condition.

Actual higher order modesexamples

53

Actual modes in 9-cell SCC cavityMeasurement setup

NA Al shorting plate

Inner side Outer side

Transmission measurement 1.5~3GHz

-100

-80

-60

-40

-20

0

1.5 2 2.5 3

ICHIRO #1 Cavity S21 measurement

90-90 [dB]

90-0 [dB]

S21

[dB

]

Freq GHz

050509

Rotational symmetry identification.

Passband nature.

Comparison with pillbox modes.

Leakage to beam pipe.

High Q modes.

Beam excitation of modes

56

• Beam interacts mostly near vp=c line.• Everyband can be excited by beam.• Most concern is the lowest dipole modes.

ACC

vp=c

1st HOM

1000

1500

2000

2500

3000

3500

0 20 40 60 80 100 120

ICHIRO #1 Cavity Frequency Measurement050509 & 050512

S21

S11_in

S11_out

S21

, S11

_in,

S11

_out

[M

Hz]

Mode Numbering

M

D

D

M

frequ

ency

Phase advance

X-band detuned structure

57

(a,t) distribution along structure

Dipole mode distribution in frequency and kick factor

Contour of dipole mode frequency vs (a,t)

Beam excited modes in detuned structure

58

Trapped modes

Position-modal frequency

dependence

Beam excitation

UP

Middle

Down

vp=c

Excited modes

Should be damped

Can be used as SBPM

Actual modes in X-band structure

59

-120

-100

-80

-60

-40

-20

10 12 14 16 18 20

6-disk Spectrum #97--102

S12 [dB]

Freq [GHz]

Extract a part along structure and stack 6 identical cells. Measure dispersion characteristics to confirm the HOM.

Ways to deal with coupled cavity system on

HOM estimation

60

Various ways of treatment• Equivalent circuit model• Matching

– Mode matching– S-parameter – Open mode expansion model

• Mesh based Numerical– Finite element model HFSS, 2– Finite difference model MAFIA, GdFidl,

61

Mode matching techniques

62

Propagating mode

Resonant modeField matching

Propagating mode

Field matching

Resonant SW modeField matching

Mode matching

Scattering matrix

Open mode expansion

Field matching

Expansion modes in cylindrical waveguide

SW & TR field matching

S-matrix for 3D cal can be used

Resonators with small coupling between cells

An example: Open mode expansion technique

63

Open modes for expansion base

Actual open modes used for calculation

Base_1

Base_8

Base_2

Calculated field for 150-cell cavity

64

1 21 41

61 81 101

121 141 161

Base_1

Base_8

Base_2

Cell-1

Cell-150

sN

k

openj

nj eazrE

1

8

1

),(

aaX 2

Na

aaa

a

8

3

2

1

Calculation result

65

Kick factorWake field

N

nnnTT tSinktW

1, )(2)(

Finite element or finite difference

66

• 3, MAFIA, GdfidL, HFSS

• Parallel PC arrays today can deal with the whole cavity in 3D as a whole.

• This is a final confirmation but these are becoming to even the tool in early design stage.

Example: SCC 9-cell cavity simulation with 3D FEM (Z. Li, SLAC)

Cures against multi-bunch emittance growth

by suppression of wakefield

67

Cures from structure design

68

extwallL PP

UQ

LQt

e 2

deftW tj)()(

)(2)( 2, tSineksW n

n

tQ

nTTn

n

There are two ways of suppressing wake field.One is to align beam to axis, while the other is to suppress wake field.

Transverse wake field is expressed as

Damping the mode with low Q

Effectively lower Q by frequency spread

Intrinsic Q0 is too large.Both ILC and CLIC need external damping.

X-band approach utilizes this.

The relativ

e ratio

depends on

design.

110

6

110468

/2

4

10

CLICfor

ILCforQ

tfQt

L

b

L

b

External coupling ILC and CLIC

69

It is not easy to make Q very low by external coupling.This results in the application of frequency spread for effective damping (cancellation) of wake field in addition to low Q.

53 mmQuadrant-type heavily damped structure

Q ~105

Q ~101

Detuning to make frequency spread

Frequency distribution

Fourier transform

Wake field

i

ii tSinKtW )()()(

Cancellation of wake field

As of excitation by beam

Fi=it

Calculation with equivalent circuit

71

Geometry parameters along a structure is distributed to make

the coupling to beam (kick factor) as gaussian like.

Analyse the whole system with coupled resonator equivalent circuit model.

pioneered by K. Band and R. Gluckstern and explored by R. Jones.

Moderate damping byextraction of dipole modes into manifold

12000

14000

16000

18000

20000

0 30 60 90 120 150 180

RDDS1 DispersionFd1(97-102)Fd2(97-102)Mahifold(97-102)

Phase shift / cell

2nd dipole

Manifold

1st dipole

Example of middle cells of RDDS1

Avoided crossing due to the coupling of cavity dipole mode and manifold mode.

Example cell shape

Manifold Cell

Distribution of (a, t) and introduction of damping

• Faster damping need larger width

• Truncation makes tail up in wake field

• DS detuned only DDS damped detuned

• Recurrent due to finite number of distribution points

• Interleaving makes longer recurrent

73

Result of equivalent circuit model calculation and estimate of tolerances

74

Wake field calculation based on frequency error info from

fabrication.Wake field simulation with more frequency errors to investigate

frequency tolerances.

Actual design (RDDS1) and typical cells

75

Input / output waveguide

HOM damping waveguide

Detuned cells

Manifold to carry HOM to outside

Frequency control of actual structure

76

-3

-2

-1

0

1

2

3

0 50 100 150 200

Accelerating mode frequency with feed forward in '2b' and integrated phase

2b offset [micron]Freq. meas. & estim. [MHz]Integ. phase

2b offset [micron]

Frequency error [MHz]

Integrated phase from cell #6-->#199

Disk number

Check each cell frequency and feedforward to the later cell fabrication.

RDDS1 dispersion

-120

-100

-80

-60

-40

-20

10 12 14 16 18 20

6-disk spectrum #194-199

S12 [dB]

Freq [GHz]

-120

-100

-80

-60

-40

-20

10 12 14 16 18 20

6-disk Spectrum #97--102

S12 [dB]

Freq [GHz]

-120

-100

-80

-60

-40

-20

10 12 14 16 18 20

6-disk Spectrum #6-11 S12 [dB]

S12 [dB]

Frequency [GHz]

12000

13000

14000

15000

16000

17000

18000

19000

20000

0 30 60 90 120 150 180

First Dipole Mode Pseudo Dispersion Curves

Fd1(6-11)Fd1(40-45)Fd1(74-79)Fd1(91-96)Fd1(109-114)Fd1(133-138)Fd1(166-171)Fd1(184-189)Fd1(194-199)

First Dipole Mode [MHz]

Phase shift / cell

[MHz]

12000

13000

14000

15000

16000

17000

18000

19000

20000

0 30 60 90 120 150 180

Second Dipole Mode Pseudo Dispersion Curves

Fd2(6-11)Fd2(40-45)Fd2(74-79)Fd2(91-96)Fd2(109-114)Fd2(133-138)Fd2(166-171)Fd2(184-189)Fd2(194-199)

Phase shift / cell

[MHz]

12000

13000

14000

15000

16000

17000

18000

19000

20000

0 30 60 90 120 150 180

Manifold Mode Pseudo Dispersion Curves

Mahifold(6-11)Mahifold(40-45)manifold(74-79)Mahifold(91-96)Mahifold(109-114)Mahifold(133-138)manifold(166-171)manifold(184-189)manifold(194-199)

Phase shift / cell

[MHz]

Proof of wake field in RDDS1

78

10-2

10-1

100

101

102

0 2 4 6 8 10 12 14 16

Wak

e A

mpl

itude

(V/p

C/m

/mm

)

SQRT[Time(ns)]

RDDS1 Wake Data (Wx = , Wy = ·) and Prediction (Line)

Cures by improving alignment

79

One-to-one steering

80

If every BPM is aligned perfectly to the magnetic center of each quadrupole magnet, it is easy to adjust the beam with respect to those quad’s. Just align the beam to zero the BPM reading.

Changing the Q strength transverse kick measured at downstream BPM’s to know the beam position w.r.t. Q center and BPM calibration.

By

x

It is straightforward way but suffers from errors in BPM reading, BPM misalignment w.r.t. Q magnet, etc.

This is the local correction, but there is better way of correcting more globally, DF or WF correction scheme.

DF and WF correction

81

The equation of motion in transverse plane in high energy linac;

z ae

q

xzsxzzsWzddzrNs

G

xzsxKzsxdsds

dsd

s

])',';([)';()','('')(

1)1(

]),;([)1(),;()()(

1

0

where = E/E, G(s) steering field, K(s) = Q magnetic field, Wt = transverse wake field, N = number of particles in a bunch, re = classical electron radius, (z,d) = charge distribution, xq = Quad misalignment, xa structure misaligment

yy B

cpesG

dxdB

cpesK

00

)(,)(

),()0,( zzd xxx Correlated energy spread

Uncorrelated energy spread

),()0,( zzw xxx

zz

z

HeadTail

zz

Equation of motion and force term

82

The equation for the difference are in the first order approximation

])0,;()([)()((

),;()(),;()()(

1

zq

zdzd

sxsxsKsG

sxsKsxdsds

dsd

s

})()0,;(({)2,()(2

})0,;()({)()(

),;()(),;()()(

1

0

sxsxsWsrN

sxsxsKsG

sxsKsxdsds

dsd

s

azze

zq

zwzw

Steering Q mag

No wake field

With wake field

Steering

Q mag

Wake field

Oscillation

Oscillation

zz

zz

z

HeadTail

Dispersion free (DF) correction

83

In the first order approximation, the solutions for these are written as the transferred position originated from the kick at s’ upstream

s

zqzd sxsxsKsGssRdssx0 12 }])0,;'()'([)'()'({)',('),;(

No wake field

])0,;'()'()]'()'()'([

])0,;'()'([)'()'(

zq

zq

sxsKsxsKsG

sxsxsKsG

Minimize xd is equivalent to locally minimize the following value small

In reality, from the i’th BPM reading mi and its difference mi with their predicted values, xi and xi, minimization does dispersion free correction;

BPMN

i prec

i

BPMprec

i xx1

2

2

22

2

min

Q misalignment Local correction

Wakefield free (WF) correction

84

In the first order approximation, the solutions for these are written as the transferred position originated from the kick at s’ upstream

})'()0,;'({)2,'()'(2

])0,;'()'({)'()'([)',('),;(

0

0 12

sxsxsWsrN

sxsxsKsGssRdssx

azze

s

zqzw

With wake field

Minimize xw is equivalent to locally minimize the following value small

)'()2,'()'(2

)0,;'(})2,'()'(2

)'({)]'()'()'([

})'()0,;'({)2,'()'(2

])0,;'()'({)'()'([

00

0

sxsWsrNsxsW

srNsKsxsKsG

sxsxsWsrNsxsxsKsG

aze

zze

q

azze

zq

Wake field correction Structure misalignment

Q misalignment

Wake field term cannot be cancelled out by term because of constant Wt while alternating in K(s’). Taking QF only or QD only makes the correction of Wt.

Wakefield free (WF) correction

85

In reality, from the i’th BPM reading mi and its difference mi with their predicted values, xi and xi, minimization does wake field free correction;

BPMN

i prec

QDi

prec

QFi

BPMprec

i xxx1

2

2

2

2

22

2

22min

Where the xiQF and xi

QD are those difference orbit due to the variation of only QF and QD.

Method eyTrajectory rms

1-to-1 23 ey0 72 mm

DF 9 ey0 55 mm

WF 1 ey0 44 mm

Correction example; NLC case from T. Raubenheimer, NIM A306, p63,1991.

Alignment with using excited field in the actual structure

86

If we measure dipole mode in a structure, we can estimate the position of beam which excites the mode. The coupling is linear as offset.

If the modal frequency depends on the position of the mode, it can distinguish the position there by frequency filtering.

In such cavity as ILC, it can be done with using power from HOM couplers.

In such cavity CLIC, it can be done with extracted power from manifold or damping waveguide.

Both directions x and y are measured with distinguishing two modes in different polarizations, almost degenerate but with some frequency difference.

Alignment measurement in situ as SBPM

87

0

100

200

300

400

500

-400 -300 -200 -100 0 100 200 300 400

15 GHz Dipole Power Scan

Parabolic Fit

Measured Power

Sign

al P

ower

(au)

Beam Y Position (Microns)

Excited power

0

50

100

150

200

-400 -300 -200 -100 0 100 200 300 400

15 GHz Dipole Phase Scan

Measured Phase

Arctan(Y/27)

Phas

e (d

egre

es)

Beam Y Position (Microns)

Phase of excited field

Position measured from frequency filtered signal compared to mechanical measurement

Power from manifold

Frequency filtered

Phase and amplitude

Frequency-to-position

Straightness measure

Dark current issue

88

Dark current

89

What is dark current? Dark current is a stable emission of electrons under high field.

DC field emission is the tunneling feature of electron migration near surface. It is studied by Fowler-Northeim. Here work function and field enhancement factor play important roles.Actually the field E0 is replaced by the local field E0.

RF field emission is estimated to be the superposition of DC field emission. It is calculated by J. Wang and G. Loew. It should exist in ant RF field, whether or not in SW and TW or in NCC and SCC .

Tracking of FE electrons are studied by various authors. Nowadays, numerical tracking is usual one. Simple analytical estimate gives minimum threshold field for capture.

5.19152.462

0

20, 1053.6,101054.1],[)(

5.0

GFmAEGExpEFi DCFE

5.1975.152.4122

0

5.20, 1053.6,10107.5],[)(

5.0

GFmAEGExpEFi RFFE

Acceleration in linear accelerator

dz

cmEednzSinkJ

dzd

ckdnkkk

rkJEjzktjE

p

nn

rn

n

nn

zrn

dnzjrn

n

nnzz

)11(

/

)2()(

/2/)/2(

exp)()(exp

20

0

22

/20

e

el

Appk

dzdandSin

dzd

p

)1

()(2

00 e

e

AppkCos

p

)1

()(2

0 e

Finally, by combining these equations,

If, p~, then is slowly varying function. Therefore, only n=0 is dominant.

Plot the contour of this equation with A in the next page.

Acceleration field is expressed as the summation of all space harmonics in the periodic structure.

R. Helm and R. Miller, in Linear Accelerators, ed. by P. M. Lapostolle and A. L. Septier, North-Holland Publishing Co., 1970 90

Separatrics describing longitudinal motion

momentumNormalizedcm

p 12

0

particleby seen phase RF

p

5.0p

p

trap

ped

untr

appe

d1p

p

p

91

Capture threshold

Minimum field e0 for being tramped, eq. to being max in left-hand side,

le

cmE thresholdacc

20 61 MV/m at 11.424GHz

7 MV/m at 1.3GHz

l = 26.242mm at 11.424GHzl = 230.6 mm at 1.3GHz

AppkCos

p

)1

()(2

0 e

For p=1, A=Cos(m) when approaching at pinfinity

)1()()( 2

0

ppkCosCos m e

2)()( mCosCos

For zero-energy electron, p=0 to be captured, the minimum field becomes

92

Tolerable breakdowns or quenches

93

To keep the beam energy under RF failure

94

It is important to make the integrated luminosity high by keeping the instantaneous luminosity high.

Once some failure happens in some cavity, the power feeding the cavity or the bunch of cavities is shut off.

In such a system as CLIC two beam scheme, the off-timing operation cannot be applied and gradual power recovery is needed.

During this recovery period, other cavities than nominal should be used to keep the beam energy. Therefore, linac needs extra acceleration capability than the nominal one.

Then, the power or pulse width will be recovered taking some pulses starting from a little lower power level. During this period, the cavities in recovering mode are off in timing from acceleration for the linac if powered by independent power supply.

Extra spare cavities or extra power/gradient capability is required.

Simple estimation of tolerable failure rate

95

Rfail failure rate for a cavity (1 trip / N pulses)Trec recovery time in the unit of pulses

Number of failures during the recovery time for a cavity

recfailstrunit TRNN

It should be less than the number of spare units

unitrecfailstrunit NTRNN

Then the tolerable failure rate is

strrefail NT

Rcov

01.05060

83000

HzsT

NN

rec

str

unit

7104 failR

In this example,

An example;Order estimation

Nunit Nstr Nunit Nstr

Nominal Spare

Compensation with spare cavitires:

Reduce failure rate or more margin in accelerator gradient

• Cares on SCC system– Margin of accelerator gradient

• Increase quench field due to Hs

• Suppress FE• Increase Q0

• Variable feeding system– Mechanical long life for tuner

• Reduce breakdown rate in NCC– Possible trigger source of breakdown

• Surface quality chemically and physically• Reduce micro protrusions • Reduce pulse temperature rise?

96

Pulse temperature rise

Pritzkaw Thesis, p99, SLAC-Report 577.

d

pss c

THRT

e

2

21 2

97

Surface heating and heat diffusion into body.

degC in rise Temperature

0

10

20

30 0

200

400

600

800

1000

0

100

200

0

10

20

30

Time

Depth

Tem

pera

ture

V. Dolgashev and L. Laurent, AAS08

Pulse heated surface

There may trigger breakdowns.

p

w

ctxPtxutxu

t),(),(),( 2

　　　　 CuRs=27.85mHs=1MA/mTp=400ns=4*10^-7=8.93 10^3 kg/m^3Ce=380 J/kg/Kd=k/ce= m^2/sk=401 W/m/K

T=270degC

Beam Physics Seminar 98050113

Care in complicated shape formation

We need to avoid additional local field enhancement due to non-smoothness especially at red areas.

Special care is taken at points (2,3,4) where smooth junction is difficult due to the junction between milling and turning

1

23

4

Esurface HsurfaceTpulse

Summary of LINAC-II• By keeping the emittance growth within a tolerable level, the

luminosity will be kept.

• Various sources, especially wake-field origin, were discussed.• Various cures on HOM origin are discussed.• Cares on alignment to suppress single-bunch wake field was

discussed, using structure BPM and BBA.

• These perturbations and cures are almost similar to both warm and cold linac.

99