Long Range Wake Potential of BPM in Undulator Section

Igor Zagorodnov and Martin Dohlus

Beam Dynamics Group Meeting

29.01.08

BPM Geometry (Dirk Lipka)

|| 2 cosis

ci i

i

sw s k e

c

2

2 22

20.5

|| || 2( ) ( ) ( ) 2 cos ,

i i iss

c ci i i i

i

sW s w s s s ds k e e s

c c

2

221( )

2

s

s e

( )||( ) 2 cos ,

is

cii

i

sW s k e s

c

2 22 22 2 2

10.5 1 0.5

4( )i i

ic Q ci iik k e k e

(1) (2)*(1) (2),

4ii

V r V rk r r

U

2i

iiQ

Longitudinal wake

( ) sin ,is

ci

ii

R sw s c e s

Q c

( , ) ( )w s r r w s

For structure with symmetry group of rotations C4

2 i

i

kR

Q

*

24i

i

V r V rk

U r

(1) (2) (1) (2), , (0,0) ( ) ( ) (2)QDw s r r w W s r W s r O ��

( )

( ) sinis

ci

ii

R sW s c e

Q c

2

22

( ) 0.5 ic

i i

R Re

Q Q

Transverse wake

FD - Frequency DomainTD - Time Domain

MWS - CST Microwave StudioPS - CST Particle Studio

ECHO – time-domain wakefield codeProny – Prony-Pisarenko method

0 10 20 30 400

0.2

0.4

0.6

0.8

65 70 75 80 85-3

-2

-1

0

1

2

3

f, GHz k, V/pC

4.3744 0.4696

11.0772 0.6294

15.7603 0.3466

21.7338 0.0577

2.4048 ~ 23GHz2cutc

fa

f, GHz k, V/pC

4.3713 0.4721

11.066 0.6505

15.7501 0.36574

21.677 0.0566

ECHO + Prony

CST PS + Prony

||V

pCW

cms

GHzf

V

pCk

Reference Cavity (PEC)

f,

GHz

k,

V/(pC*m*m)

6.9594 14.1565e3

12.8695 34.5164e3

17.1830 40.8118e3

1.841 ~ 17.6GHz2cutc

fa

5 10 15 20 250

10

20

30

40

50

65 70 75 80 85-600

-400

-200

0

200

400

600

Reference Cavity (PEC)V

pC*mW

cms GHzf

V

pC*m*mmk

ECHO + Prony

f, GHz

(MWS, FD)

k, V/pC

(MWS, FD)

k, V/pC

(PS, TD)

k, V/pC

(ECHO, TD)

Q

(MWS, FD)

Q

(PS, TD)

sec-1

4.37 0.46 0.47 0.47 636 602 2.2e7

11.06 0.60 0.65 0.63 995 890 3.5e7

15.75 0.33 0.36 0.35 1113 1075 4.4e7

21.72 0.16 0.07 0.06 1712 1126 4.0e7

f,

GHz

(MWS, FD)

k, V/(pC*m*m)(MWS, FD)

k, V/(pC*m*m)

(PS, TD)

k, V/(pC*m*m)(ECHO, TD)

R/Q,

/mm2

Q

(MWS, FD)

Q

(PS, TD)

sec-1

6.96 13e3 15e3 14e3 3.74 797 1989 2.7e7

12.86 31e3 32e3 34e3 4.82 1000 1129 4.0e7

17.22 39e3 48e3 41e3 4.53 1117 608 4.8e7

24.43 25e3 32e3 10e3 2.05 1586 2353 4.8e7

transverse (dipole)

Reference Cavity (Lossy)

Stainless Steel 304

conductivity1,40E+06longitudinal (monopole)

longitudinal

transverse

f, GHz

(MWS, FD)

K, 1e12

(MWS, FD)

K, 1e12

(PS, TD)

Q

(MWS, FD)

sec-1

3.51 0.12 0.11 288 3.8e7

8.56 0.23 0.19 428 6.3e7

13.44 0.22 0.19 543 7.8e7

18.31 0.16 0.17 683 8.4e7

f, GHz

(MWS, FD)

k, V/(pC*m*m(MWS, FD)

k,

V/(pC*m*m

(PS, TD)

Q

(MWS, FD)

R/Q,

/mm2

sec-1

4.46 5.1e3 6.5e3 367 2.3 3.8e7

7.67 2.0e3 2.7e3 469 0.5 5.1e7

9.3 2.3e3 2.7e3 470 0.5 6.2e7

10.46 6.1e3 6.1e3 549 1.2 6.0e7

14.07 11.1e3 12.5e3 607 1.6 7.3e7

15.16 2.4e3 2.5e3 719 0.3 6.6e7

19.17 13.5e3 17e3 715 1.4 8.4e7

20.4 17.7e3 4.9e3 902 1.7 7.1e7

Resonator Cavity (Lossy)Stainless Steel 304

conductivity1,40E+06

( )( )|| ||

1 1

( )

( ) ( ) 2 cos

2 2

1 1

ij s

cii

j j i

s

c cii

i ic c

j sW s W j s k e

c

e ek k

e e

min i

( ) ( )

1

c

iic

e RW s c

Qe

Asymptotic multi-bunch loss

Asymptotic multi-bunch kick

1c

1|| 2.2 7 [sec ]e

2 7 [sec]ec

||

||

0.01

1

c

c

e

e

( )||

kV( ) 0.01 2 0.05

nCii

W s k

12.7 7 [sec ]e

( ) kV( ) 0.005 0.0066

nC×mmii

RW s c

Q

Multi-bunch effect

- bunch spacing

0.005

1

c

c

e

e

( )||

kV( ) 0.01 2 0.05

nCii

W s k

( ) kV( ) 0.005 0.0066

nC×mmii

RW s c

Q

22

225μm||

kV16.6 2

nC

ic

ii

k k e

0.503 2.5

2 3Kick ( )

4

Z cg O

a

0.502.5

1Loss ( )

44

Z c gO

a

25μm kV0.045 ( )

nC×mm iii

Rk c D

Q

Long Range Wakes

Short Range Wakes

Long Range vs. Short Range

( )||

kV( ) 0.05

nCW s ( ) kV

( ) 0.0066nC×mm

W s

Comparison with beam parameters in the undulator

( ) rad0.0004

mm

r eQW

r E

( )

|| 0.05keVE eQ W

0

50eV3 9

17.5GeV

Ee

E

1.4mm*mrad1 rad

34247*45mn

x

0 17.5GeVE

4 4

mmx

r e

r

Geometry scaling

,sii

aQ F i

d

(see R.E.Collin „Foundations for microwave engineering“)

If we scale the cavity by factor as

, ,a d a d then

ii

i iQ Q

1.5i

i

|| ||1

i ik k

21

i i

R R

Q Q

2a

d

2s

i

31

i ik k

1 2 3 4 50

5

10

15

Geometry scaling

If we scale only the cavity by factor as

, ,a d a d then

1.5

( )||

(0.01)( , ) 2 i

i

W s k

1.5

( )2

(0.005)( , )

ii

RW s c

Q

ii

( )||

( )||

( , )

( )

W s

W s

( )

( )

( , )

( )

W s

W s

2a

d

1.5i

i

1i ik k