Collimator Wakefields

Collimator Wakefields

Igor Zagorodnov

BDGM, DESY

25.09.06

Codes

• ECHO• CST MS• ABCI• MAFIA• GdfidL• VORPAL• PBCI• Tau3P

• K. Yokoya, CERN-SL/90-88, 1990• F.-J.Decker et al., SLAC-PUB-7261, 1996• G.V. Stupakov, SLAC-PUB-8857, 2001• P. Tenenbaum et al, PAC’01, 2001• B. Podobedov, S. Krinsky, EPAC’06, 2006• I. Zagorodnov, K.L.F. Bane, EPAC‘06, 2006• K.L.F. Bane, I.A. Zagorodnov, SLAC-PUB-

11388, 2006• I. Zagorodnov et al, PAC‘03, 2003• and others

References

Used by me

Outline

• Round collimators– Inductive regime– Diffractive regime– Near wall wakefields– Resistive wakefields

• 3D collimators (rectangular, elliptical)– Diffractive regime– Inductive regime

• Simulation of SLAC experiments• XFEL collimators

– Effect of tapering and form optimization– Kick dependence on collimator length

Effect of the kick

d

1b2b

d

1b2b

Figure 1: Top half of a symmetric collimator.

Round collimator. Regimes

Inductive

11 2 1b

Diffractive

1 1

2/ bd

1b2b

d

1b2b

Figure 1: Top half of a symmetric collimator.


Inductive

11 2 1b

Diffractive

1 1

2/ b

d

1b2b

d

1b2b

|| 0k

Inductive 11 2 1b

0

2 1

2 1 1

4

Z ck

b b

202

iZ f dz

c

2

1 2i fZ dz

c f

0 4Z c

2

k Ob

Round collimator. Inductive

d

1b2b

d

1b2b

|| 0k

Inductive 11 2 1b

0

2 1

2 1 1

4

Z ck

b b

202

iZ f dz

c

2

1 2i fZ dz

c f

0 4Z c

2

k Ob


d

1b2b

d

1b2b

[mrad]

a [mm] b [mm] l [mm] Q [nC]

[mm]

TESLA 20 17.5 0.4 20 1. 0.3

NLC 20 17.5 0.2 20 1. 0.1

-5 0 5-20

-10

0

10

200

/

W

V pC

-5 0 5-1

0

1

2

3

4

analytical

/s

1

/ /

W

V pC mm

/ 5h / 10h / 20h

analytical

/ 5h / 10h / 20h

/s -5 0 5-20

-10

0

10

200

/

W

V pC

-5 0 5-1

0

1

2

3

4

analytical

/s

1

/ /

W

V pC mm

/ 5h / 10h / 20h

analytical

/ 5h / 10h / 20h

/s

Inductive


0 2 0 4 0 6 0

5 0

1 0 0

1 5 0

2 0 0

2 0 4 0 6 00

1 0 0

2 0 0

3 0 0

4 0 0

0

/L

V p C

/ 5 ( )h A B C I a n a l y t i c a l

/ g r a d / g r a d

1

2 20

/

W

V p C

/ 1 0 ( )h A B C I / 2 0 ( )h A B C I

/ 5 ( )h E C H O

A B C I

/ 5 ( )h E C H O

0 2 0 4 0 6 0

5 0

1 0 0

1 5 0

2 0 0

2 0 4 0 6 00

1 0 0

2 0 0

3 0 0

4 0 0

0

/L

V p C

/ 5 ( )h A B C I a n a l y t i c a l

/ g r a d / g r a d

1

2 20

/

W

V p C

/ 1 0 ( )h A B C I / 2 0 ( )h A B C I

/ 5 ( )h E C H O

A B C I

/ 5 ( )h E C H O

0 2 0 4 0 6 0 8 00

2 0

4 0

6 0

8 0

2 0 4 0 6 0 8 00

1 0

2 0

3 0

4 01

/ /

L

V p C m m

12 21

/ /

W

V p C m m

a n a l y t i c a l

/ g r a d

/ 5 ( )h E C H O

a n a l y t i c a l

/ g r a d

/ 5 ( )h E C H O

0 2 0 4 0 6 0 8 00

2 0

4 0

6 0

8 0

2 0 4 0 6 0 8 00

1 0

2 0

3 0

4 01

/ /

L

V p C m m

12 21

/ /

W

V p C m m

a n a l y t i c a l

/ g r a d

/ 5 ( )h E C H O

a n a l y t i c a l

/ g r a d

/ 5 ( )h E C H O


0.5 longshortk k

1.5 1 1|| 0 1 20.5 log( )k cZ b b

02 2

2 1

1 1

2long Z c

kb b

20 2

2 42 1

1

4short Z c b

kb b

|| 2(log( ))k O b2

2

1k O

b

Round collimator. Diffractive

I. Zagorodnov, K.L.F. Bane, EPAC‘06, 2006

d

1b2b

d

1b2b

0.5 longshortk k

1.5 1 1|| 0 1 20.5 log( )k cZ b b

02 2

2 1

1 1

2long Z c

kb b

20 2

2 42 1

1

4short Z c b

kb b

|| 2(log( ))k O b2

2

1k O

b


I. Zagorodnov, K.L.F. Bane, EPAC‘06, 2006

d

1b2b

d

1b2b

0.01 0.1 1 10

2

3

4

5[V/pC/mm]k

[cm]d

short

long

0.01 0.1 1 10

2.5

3

3.5

4

4.5

5

σ=0.1mm

=0.5mm=0.3mm

long

in+out

[V/pC/mm]k

[cm]d

short

0.01 0.1 1 10

2

3

4

5[V/pC/mm]k

[cm]d

short

long

0.01 0.1 1 10

2

3

4

5[V/pC/mm]k

[cm]d

short

long

0.01 0.1 1 10

2.5

3

3.5

4

4.5

5

σ=0.1mm

=0.5mm=0.3mm

long

in+out

[V/pC/mm]k

[cm]d

short

0.01 0.1 1 10

2.5

3

3.5

4

4.5

5

σ=0.1mm

=0.5mm=0.3mm

long

in+out

[V/pC/mm]k

[cm]d

short

Figure 2:Kick factor vs. collimator length. A round collimator(left), a square or rectangular collimator ( = 0.3 mm, right).


|| 2(log( ))k O b

22

1k O

b

|| 0k

2

k Ob

d

1b2b

d

1b2b

Inductive

11 2 1b

Diffractive

1 1


|| 2(log( ))k O b

22

1k O

b

|| 0k

2

k Ob

d

1b2b

d

1b2b

Inductive

11 2 1b

Diffractive

1 1


Round collimator

Round collimator. Near-wall wakes

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

/r mm

/V pC

( )aL r

1 ( )L r

13( )L r

2

2 12

2

log 1( )a

r bL r L b

r b

1313 2 1

1

( ) m m

m

L r L r

References1. F.-J.Decker et al.,

SLAC-PUB-7261, Aug 1996.

2. I. Zagorodnov et al.,TESLA 2003-23, 2003.

Round collimator. Near-wall wakes

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

/r mm

/V pC

( )aL r

1 ( )L r

13( )L r

2

2 12

2

log 1( )a

r bL r L b

r b

1313 2 1

1

( ) m m

m

L r L r

References1. F.-J.Decker et al.,

SLAC-PUB-7261, Aug 1996.

2. I. Zagorodnov et al.,TESLA 2003-23, 2003.

Round collimator. Resistive wall wakes

Analytical estimations are available. To be studied.

Can the total wake be treated as the direct sum of the geometric and the resistive wakes? Numerical modeling is required.

Discrepancy between analytical estimations and measurements.Transverse geometric kick for long collimator is approx. two times larger as for the short one.

3D collimators. Regimes

3D collimators. Regimes

0

2 1

2 1 1(1)

4

Z ck

b b

0

2 22 1

1 1(2)

4

Z chk

b b

0.5 1.5 12 02.7 (4 ) (3)k A b Z c

2 1 12 2 1h b Inductive

Intermediate1 1, 2

2

11 2 1b

round rectangular

1 1,

Diffractive 1 1 1 1

0

2 1

2 1 1(1)

4

Z ck

b b

0

2 22 1

1 1(2)

4

Z chk

b b

0.5 1.5 12 02.7 (4 ) (3)k A b Z c

2 1 12 2 1h b Inductive

Intermediate1 1, 2

2

11 2 1b

round rectangular

1 1,

Diffractive 1 1 1 1

3D Collimators. Diffractive Regime

1 2

2 21 22

0

1(5)eZ ds ds

Q Z

1 2

212

0

1(6)eZ ds

Q Z

long short

Diffractive Regime 0.5 longshortk k || 2 (4)eZ Z

0 0( ) ( )i x Z Q x x

i ix

( ) 0i x

i ix 1, 2i

1

2

1 2

2 21 22

0

1(5)eZ ds ds

Q Z

1 2

212

0

1(6)eZ ds

Q Z

long short


0 0( ) ( )i x Z Q x x

i ix

( ) 0i x

i ix 1, 2i

1

2


0 0( ) ( )i x Z Q x x

i ix

( ) 0i x

i ix 1, 2i

1

2

S.Heifets and S.Kheifets, Rev Mod Phys 63,631 (1991)I. Zagorodnov, K.L.F. Bane, EPAC‘06, 2006

3D collimators. Diffractive Regime

0.01 0.1 1 10

2

3

4

5[V/pC/mm]k

[cm]d

short

long

0.01 0.1 1 10

2.5

3

3.5

4

4.5

5

σ=0.1mm

=0.5mm=0.3mm

long

in+out

[V/pC/mm]k

[cm]d

short

0.01 0.1 1 10

2

3

4

5[V/pC/mm]k

[cm]d

short

long

0.01 0.1 1 10

2

3

4

5[V/pC/mm]k

[cm]d

short

long

0.01 0.1 1 10

2.5

3

3.5

4

4.5

5

σ=0.1mm

=0.5mm=0.3mm

long

in+out

[V/pC/mm]k

[cm]d

short

0.01 0.1 1 10

2.5

3

3.5

4

4.5

5

σ=0.1mm

=0.5mm=0.3mm

long

in+out

[V/pC/mm]k

[cm]d

short

4.251.997874square

6.112.437256rect.

5.012.507878round

longshortlongshort

ktr [V/pC/mm]k|| [V/pC]Type

Figure 2:Kick factor vs. collimator length. A round collimator(left), a square or rectangular collimator (= 0.3 mm, right).

Table1: Loss and kick factors as estimated by 2D electrostatic calculation. The bunch length 0.3 mm. ``Short'' means using Eq. 6, ``long'' Eq. 5

The good agreement we have found between direct time-domain calculation [1] and the approximations (5, 6), suggests that the latter method can be used to approximate short-bunch wakes for a large class of 3D collimators.

0.01 0.1 1 10

2

3

4

5[V/pC/mm]k

[cm]d

short

long

0.01 0.1 1 10

2.5

3

3.5

4

4.5

5

σ=0.1mm

=0.5mm=0.3mm

long

in+out

[V/pC/mm]k

[cm]d

short

0.01 0.1 1 10

2

3

4

5[V/pC/mm]k

[cm]d

short

long

0.01 0.1 1 10

2

3

4

5[V/pC/mm]k

[cm]d

short

long

0.01 0.1 1 10

2.5

3

3.5

4

4.5

5

σ=0.1mm

=0.5mm=0.3mm

long

in+out

[V/pC/mm]k

[cm]d

short

0.01 0.1 1 10

2.5

3

3.5

4

4.5

5

σ=0.1mm

=0.5mm=0.3mm

long

in+out

[V/pC/mm]k

[cm]d

short

4.251.997874square

6.112.437256rect.

5.012.507878round

longshortlongshort

ktr [V/pC/mm]k|| [V/pC]Type

Figure 2:Kick factor vs. collimator length. A round collimator(left), a square or rectangular collimator (= 0.3 mm, right).

Table1: Loss and kick factors as estimated by 2D electrostatic calculation. The bunch length 0.3 mm. ``Short'' means using Eq. 6, ``long'' Eq. 5

The good agreement we have found between direct time-domain calculation [1] and the approximations (5, 6), suggests that the latter method can be used to approximate short-bunch wakes for a large class of 3D collimators.

3D collimators. Inductive Regime


ECHO

GdfidL


ECHO

GdfidL


-4 -2 0 2 4 6 8 10 12 14 16

-1000

-500

0

500

1000

-5 0 5 10 15

-20

-15

-10

-5

0

Round: 2D vs.3D

[ ]s cm [ ]s cm

|| [ / / / ]dW V pC m m [ / / ]dW V pC m

3 2

2

( ( ) ( ))100% 5%

( )

D D

D

W s W s ds

W s ds

Blue-3DGreen-2D accurate


-5 0 5 10 15 20 25 30 35 40 450.46

0.47

0.48

0.49

0.5

0.51

0.52

0.53

0.54

0.55

0.56

-5 0 5 10 15 20 25 30 35 40 451

1.5

2

2.5

3

3.5

4

4.5

[ ]s cm

( )dF s ( )qF s

,

5

,

5

( )

( ) 3

( )

sd el

ds

d round

W x dx

F s

W x dx

,

5

,

5

( )

( ) 0.52

( )

sq el

qs

d round

W x dx

F s

W x dx

Dipolar Quadrupolar

[ ]s cm

1dx dy dz mm

5offset mm

I estimate that error in this numbers is about 5%

Simulations of SLAC experiment 2001Rectangular Collimators

SLAC experiment 2001

-3 -2 -1 0 1 2 3 4 5

-3

-2

-1

0

1[V/pC/mm]W

round(3D/2D)

square(#2)

rect.(#3)

/s -3 -2 -1 0 1 2 3 4 5

-4

-3

-2

-1

0

/s

[V/pC/mm]W

square(#2)

rect.(#3)

round

-3 -2 -1 0 1 2 3 4 5

-3

-2

-1

0

1[V/pC/mm]W

round(3D/2D)

square(#2)

rect.(#3)

/s -3 -2 -1 0 1 2 3 4 5

-3

-2

-1

0

1[V/pC/mm]W

round(3D/2D)

square(#2)

rect.(#3)

/s -3 -2 -1 0 1 2 3 4 5

-4

-3

-2

-1

0

/s

[V/pC/mm]W

square(#2)

rect.(#3)

round

-3 -2 -1 0 1 2 3 4 5-4

-3

-2

-1

0

/s

[V/pC/mm]W

square(#2)

rect.(#3)

round

298335335168mrad]

3.81.91.91.9b2 [mm]

rect.rect.squarerect.Type

4321Coll. #

Figire 3: Transverse wake of Gaussian bunch, with = 0.65 mm (left) and =0.3 mm (right).

Table 2: Geometry of SLAC collimators of 2001

1 / 2 19 mmb h

Rectangular Collimators


-3 -2 -1 0 1 2 3 4 5

-3

-2

-1

0

1[V/pC/mm]W

round(3D/2D)

square(#2)

rect.(#3)

/s -3 -2 -1 0 1 2 3 4 5

-4

-3

-2

-1

0

/s

[V/pC/mm]W

square(#2)

rect.(#3)

round

-3 -2 -1 0 1 2 3 4 5

-3

-2

-1

0

1[V/pC/mm]W

round(3D/2D)

square(#2)

rect.(#3)

/s -3 -2 -1 0 1 2 3 4 5

-3

-2

-1

0

1[V/pC/mm]W

round(3D/2D)

square(#2)

rect.(#3)

/s -3 -2 -1 0 1 2 3 4 5

-4

-3

-2

-1

0

/s

[V/pC/mm]W

square(#2)

rect.(#3)

round

-3 -2 -1 0 1 2 3 4 5-4

-3

-2

-1

0

/s

[V/pC/mm]W

square(#2)

rect.(#3)

round

298335335168mrad]

3.81.91.91.9b2 [mm]

rect.rect.squarerect.Type

4321Coll. #

Figire 3: Transverse wake of Gaussian bunch, with = 0.65 mm (left) and =0.3 mm (right).

Table 2: Geometry of SLAC collimators of 2001

1 / 2 19 mmb h

Simulations of SLAC experiment 2001

0 .1 0.3 0.5 0.7 0.9 1.11

1.5

2

2.5[V/pC/mm]k

[mm]

square(#2)

round

exper.(#2)

0.2 0.4 0.6 0.8 1 1.2

0.5

1

1.5

2

[V/pC/mm]k

rect.(#3)

rect.(#1)

rect.(#4)

[mm]0 .1 0.3 0.5 0.7 0.9 1.1

1

1.5

2

2.5[V/pC/mm]k

[mm]

square(#2)

round

exper.(#2)

0 .1 0.3 0.5 0.7 0.9 1.11

1.5

2

2.5[V/pC/mm]k

[mm]

square(#2)

round

exper.(#2)

0.2 0.4 0.6 0.8 1 1.2

0.5

1

1.5

2

[V/pC/mm]k

rect.(#3)

rect.(#1)

rect.(#4)

[mm]0.2 0.4 0.6 0.8 1 1.2

0.5

1

1.5

2

[V/pC/mm]k

rect.(#3)

rect.(#1)

rect.(#4)

[mm]

0.622.52.5Eq. 12

1.13.32.4Eq. 3

1.02.51.24Eq. 1

0.54(0.05)1.4(0.1)1.4(0.1)1.2(0.1)meas.

0.501.721.751.28simul.

1.7/431/981.00.5/50

4321Coll. #

0.62.52.5Eq. 12

0.82.41.7Eq. 3

0.51.30.7Eq. 1

0.49(0.15)1.08(0.1)1.2(0.1)0.78(0.13)meas.

0.411.301.350.90simul.

0.9/240.5/530.50.3/27

4321Coll. #

Table 4:Kick factor [V/pC/mm]; = 1.2 mm. Measurement errors are given in parentheses.


We note good agreement for rectangular collimators #1 and #4. On the other hand the short bunch results for collimators #2 and #3 agree very well with the calculations of the previous section, but they disagree with the experimental data (by 20 %).

0 .1 0.3 0.5 0.7 0.9 1.11

1.5

2

2.5[V/pC/mm]k

[mm]

square(#2)

round

exper.(#2)

0.2 0.4 0.6 0.8 1 1.2

0.5

1

1.5

2

[V/pC/mm]k

rect.(#3)

rect.(#1)

rect.(#4)

[mm]0 .1 0.3 0.5 0.7 0.9 1.1

1

1.5

2

2.5[V/pC/mm]k

[mm]

square(#2)

round

exper.(#2)

0 .1 0.3 0.5 0.7 0.9 1.11

1.5

2

2.5[V/pC/mm]k

[mm]

square(#2)

round

exper.(#2)

0.2 0.4 0.6 0.8 1 1.2

0.5

1

1.5

2

[V/pC/mm]k

rect.(#3)

rect.(#1)

rect.(#4)

[mm]0.2 0.4 0.6 0.8 1 1.2

0.5

1

1.5

2

[V/pC/mm]k

rect.(#3)

rect.(#1)

rect.(#4)

[mm]

0.622.52.5Eq. 12

1.13.32.4Eq. 3

1.02.51.24Eq. 1

0.54(0.05)1.4(0.1)1.4(0.1)1.2(0.1)meas.

0.501.721.751.28simul.

1.7/431/981.00.5/50

4321Coll. #

0.62.52.5Eq. 12

0.82.41.7Eq. 3

0.51.30.7Eq. 1

0.49(0.15)1.08(0.1)1.2(0.1)0.78(0.13)meas.

0.411.301.350.90simul.

0.9/240.5/530.50.3/27

4321Coll. #



We note good agreement for rectangular collimators #1 and #4. On the other hand the short bunch results for collimators #2 and #3 agree very well with the calculations of the previous section, but they disagree with the experimental data (by 20 %).


2.43.028568

2.73.134607

2.32.924516

6.87.147815

0.770.7424404

5.16.133633

3.13.633602

1.71.928501

0.5mm0.3mm0.5mm0.3mm

ktr [V/pC/mm]k|| [V/pC]Coll. #

Table 5: Loss and kick factors for new set of collimators.

Perhaps the most interesting result in Table 5 is a noticeable difference in the kicks for collimators #2 and #3. In agreement with the analytic models discussed in the paper, a long collimator length results in a kick factor increase by a factor ~2.


2.43.028568

2.73.134607

2.32.924516

6.87.147815

0.770.7424404

5.16.133633

3.13.633602

1.71.928501

0.5mm0.3mm0.5mm0.3mm

ktr [V/pC/mm]k|| [V/pC]Coll. #

Table 5: Loss and kick factors for new set of collimators.

Perhaps the most interesting result in Table 5 is a noticeable difference in the kicks for collimators #2 and #3. In agreement with the analytic models discussed in the paper, a long collimator length results in a kick factor increase by a factor ~2.

XFEL collimators. Tapering

2 3 4 50.2

0.4

0.6

0.8

1

d

1b2b

d

1b2b

Inductive

[deg]

2[ ]b mm

When a short bunch passes by an out-transition, a significan reduction in the wake will not happen until the tapered walls cut into the cone of radiation, i.e until

2tan /z b

0.025 mmz

Collimator geometry optimization.Optimum d ~ 4.5mm

200 mm

17 mm

2 mm

100 mm

d

200 mm

17 mm

2 mm

100 mm

d

0 5 10 15 20100

150

200

250

300

350

400

450

1

2 20W

0L

/d mm

/V p C

0 5 10 15 20

1

2

3

4

5

1L

1

2 21W

/ /V pC mm

/d mm

0 5 10 15 20100

150

200

250

300

350

400

450

1

2 20W

0L

/d mm

/V p C

0 5 10 15 20

1

2

3

4

5

1L

1

2 21W

/ /V pC mm

/d mm

Geometry of the “step+taper” collimator for TTF20.05 mmz


d mm

10 / 20 mm

4.5 mm3.4 mm

25mkm

Loss,

V/pC

Spread,

V/pC

Peak,

V/pC

step 110 43 -156

taper 20mm 38 42 -83

taper 20mm +step

50 (45%)

29 (67%)

-82 (53%)

-0.01 -0.005 0 0.005 0.01

-150

-100

-50

0

50

taper 20 mm

step (d=3.4mm)

taper (d=4.5mm)

step+taper (d=4.1mm)

|| [ / ]W V pC

[ ]s cm

Geometry of the “step+taper” absorber for XFEL


XFEL collimators. Kick vs. length

1 25mmb

d

2 3mmb

1.041

1.1410

1.3550

1.4890

1.73200

1.9685long

1.98500

0.998short

Kick,

V/pC/mm

d,

mm

02 2

2 1

1 1

2long Z c

kb b

20 2

2 42 1

1

4short Z c b

kb b

10-1

100

101

102

103

0.8

1

1.2

1.4

1.6

1.8

2

[ ]d mm

[ / / ]

Kick

V pC mm

Form optimization?Exponetial tapering?

XFEL collimators. Kick vs. length

1 25mmb

d

2 3mmb

1.041

1.1410

1.3550

1.4890

1.73200

1.9685long

1.98500

0.998short

Kick,

V/pC/mm

d,

mm

02 2

2 1

1 1

2long Z c

kb b

20 2

2 42 1

1

4short Z c b

kb b

10-1

100

101

102

103

0.8

1

1.2

1.4

1.6

1.8

2

[ ]d mm

[ / / ]

Kick

V pC mm

Form optimization?Exponetial tapering?

Acknowledgements to K. Bane,

M. Dohlus,B. Podobedov G. Stupakov, T. Weiland

for helpful discussions on wakes and impedances, and to

CST GmbH for letting me use CST MICROWAVE

STUDIO for the meshing.

Documents

Collimator Wakefields