Collimator Wakefields

Igor ZagorodnovBDGM, 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

Usedby 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

d1b

2bα d

1b2b

α

Figure 1: Top half of a symmetric collimator.

Round collimator. Regimes

Inductive1

1 2 1bρ α σ −≡Diffractive

1 1ρ

2/ bσd1b

2bα d

1b2b

α

Figure 1: Top half of a symmetric collimator.

Round collimator. Regimes

Inductive1

1 2 1bρ α σ −≡Diffractive

1 1ρ

2/ bσ

d1b

2bα d

1b2b

α

|| 0k =

Inductive 11 2 1bρ α σ −≡

0

2 1

2 1 14Z c

kb b

αππσ⊥

⎛ ⎞= −⎜ ⎟

⎝ ⎠

( )202

iZ f dzcω ∞

−∞′= Θ ∫

21 2i fZ dz

c f∞

⊥ −∞

′ ⎞⎛= Θ ⎟⎜

⎝ ⎠∫

0 4Z c πΘ =

2k O

bα

⊥⎛ ⎞

= ⎜ ⎟⎝ ⎠

Round collimator. Inductive

d1b

2bα d

1b2b

α

|| 0k =

Inductive 11 2 1bρ α σ −≡

0

2 1

2 1 14Z c

kb b

αππσ⊥

⎛ ⎞= −⎜ ⎟

⎝ ⎠

( )202

iZ f dzcω ∞

−∞′= Θ ∫

21 2i fZ dz

c f∞

⊥ −∞

′ ⎞⎛= Θ ⎟⎜

⎝ ⎠∫

0 4Z c πΘ =

2k O

bα

⊥⎛ ⎞

= ⎜ ⎟⎝ ⎠

Round collimator. Inductive

d1b

2bα d

1b2b

α

α[mrad]

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

σ [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

20 0

/W

V pC

-5 0 5-1

0

1

2

3

4

analytical

/s σ

1

/ /W

V pC mm⊥

/ 5hσ =/ 10hσ =/ 20hσ =

analytical

/ 5hσ =/ 10hσ =/ 20hσ =

/s σ-5 0 5-20

-10

0

10

20 0

/W

V pC

-5 0 5-1

0

1

2

3

4

analytical

/s σ

1

/ /W

V pC mm⊥

/ 5hσ =/ 10hσ =/ 20hσ =

analytical

/ 5hσ =/ 10hσ =/ 20hσ =

/s σ

Inductive

Round collimator. Inductive

0 20 40 60

50

100

150

200

20 40 600

100

200

300

400

0

/L

V p C

/ 5 ( )h ABCIσ =analytical

/ gradα / gradα

( )1

2 20

/

W

V pC

Δ

/ 10 ( )h ABCIσ =/ 20 ( )h ABCIσ =

/ 5 ( )h ECHOσ =

ABCI

/ 5 ( )h ECHOσ =

0 20 40 60

50

100

150

200

20 40 600

100

200

300

400

0

/L

V p C

/ 5 ( )h ABCIσ =analytical

/ gradα / gradα

( )1

2 20

/

W

V pC

Δ

/ 10 ( )h ABCIσ =/ 20 ( )h ABCIσ =

/ 5 ( )h ECHOσ =

ABCI

/ 5 ( )h ECHOσ =

0 20 40 60 800

20

40

60

80

20 40 60 800

10

20

30

401

/ /L

V pC mm⊥ ( )

12 21

/ /

W

V pC mm⊥Δ analytical

/ gradα

/ 5 ( )h ECHOσ =

analytical

/ gradα

/ 5 ( )h ECHOσ =

0 20 40 60 800

20

40

60

80

20 40 60 800

10

20

30

401

/ /L

V pC mm⊥ ( )

12 21

/ /

W

V pC mm⊥Δ analytical

/ gradα

/ 5 ( )h ECHOσ =

analytical

/ gradα

/ 5 ( )h ECHOσ =

Round collimator. Inductive

0.5 longshortk k⊥ ⊥≈

1.5 1 1|| 0 1 20.5 log( )k cZ b bπ σ− − −=

02 2

2 1

1 12

long Z ck

b bπ⊥⎛ ⎞

= −⎜ ⎟⎜ ⎟⎝ ⎠

20 2

2 42 1

14

short Z c bkb bπ⊥

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠

|| 2(log( ))k O b= 22

1k Ob⊥

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠

Round collimator. Diffractive

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

d1b

2bα d

1b2b

α

0.5 longshortk k⊥ ⊥≈

1.5 1 1|| 0 1 20.5 log( )k cZ b bπ σ− − −=

02 2

2 1

1 12

long Z ck

b bπ⊥⎛ ⎞

= −⎜ ⎟⎜ ⎟⎝ ⎠

20 2

2 42 1

14

short Z c bkb bπ⊥

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠

|| 2(log( ))k O b= 22

1k Ob⊥

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠

Round collimator. Diffractive

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

d1b

2bα d

1b2b

α

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

=0.5mmσ=0.3mmσ

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

=0.5mmσ=0.3mmσ

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

=0.5mmσ=0.3mmσ

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 (σ = 0.3 mm, right).

Round collimator. Diffractive

|| 2(log( ))k O b=

22

1k Ob⊥

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠

|| 0k =

2k O

bα

⊥⎛ ⎞

= ⎜ ⎟⎝ ⎠

d1b

2bα d

1b2b

α

Inductive1

1 2 1bρ α σ −≡Diffractive

1 1ρ

Round collimator. Regimes

|| 2(log( ))k O b=

22

1k Ob⊥

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠

|| 0k =

2k O

bα

⊥⎛ ⎞

= ⎜ ⎟⎝ ⎠

d1b

2bα d

1b2b

α

Inductive1

1 2 1bρ α σ −≡Diffractive

1 1ρ

Round collimator. Regimes

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⊥

( )( )( )

22 1

22

log 1( )a

r bL r L b

r b⊥ ⊥

−= − ⋅ ⋅

1313 2 1

1

( ) m m

mL 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⊥

( )( )( )

22 1

22

log 1( )a

r bL r L b

r b⊥ ⊥

−= − ⋅ ⋅

1313 2 1

1

( ) m m

mL 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 geometricand 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)4Z c

kb b

αππσ⊥

⎛ ⎞= −⎜ ⎟

⎝ ⎠0

2 22 1

1 1 (2)4Z chk

b bπασ π⊥

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠

0.5 1.5 12 02.7 (4 ) (3)k A b Z cσ α π− − −

⊥ =

2 1 12 2 1h bρ α σ− −≡Inductive

Intermediate 1 1,ρ 22ρ π≥

11 2 1bρ α σ−≡

round rectangular

1 1,ρ

Diffractive 1 1ρ >>1 1ρ >>

0

2 1

2 1 1 (1)4Z c

kb b

αππσ⊥

⎛ ⎞= −⎜ ⎟

⎝ ⎠0

2 22 1

1 1 (2)4Z chk

b bπασ π⊥

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠

0.5 1.5 12 02.7 (4 ) (3)k A b Z cσ α π− − −

⊥ =

2 1 12 2 1h bρ α σ− −≡Inductive

Intermediate 1 1,ρ 22ρ π≥

11 2 1bρ α σ−≡

round rectangular

1 1,ρ

Diffractive 1 1ρ >>1 1ρ >>

3D Collimators. Diffractive Regime

1 2

2 21 22

0

1 (5)eZ ds dsQ Z

ϕ ϕΩ Ω

⎛ ⎞⎜ ⎟= ∇ − ∇⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫

1 2

212

0

1 (6)eZ dsQ Z

ϕΩ −Ω

⎛ ⎞⎜ ⎟= ∇⎜ ⎟⎜ ⎟⎝ ⎠∫

long short

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

0 0( ) ( )i x Z Q x xϕ δΔ = −r r ri ix ∈ Ωr

( ) 0i xϕ =ri ix ∈ ∂Ωr 1, 2i =

1Ω

2Ω

1 2

2 21 22

0

1 (5)eZ ds dsQ Z

ϕ ϕΩ Ω

⎛ ⎞⎜ ⎟= ∇ − ∇⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫

1 2

212

0

1 (6)eZ dsQ Z

ϕΩ −Ω

⎛ ⎞⎜ ⎟= ∇⎜ ⎟⎜ ⎟⎝ ⎠∫

long short

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

0 0( ) ( )i x Z Q x xϕ δΔ = −r r ri ix ∈ Ωr

( ) 0i xϕ =ri ix ∈ ∂Ωr 1, 2i =

1Ω

2Ω

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

0 0( ) ( )i x Z Q x xϕ δΔ = −r r ri ix ∈ Ωr

( ) 0i xϕ =ri ix ∈ ∂Ωr 1, 2i =

1Ω

2Ω

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

=0.5mmσ=0.3mmσ

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

=0.5mmσ=0.3mmσ

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

=0.5mmσ=0.3mmσ

long

in+out

[V/pC/mm]k⊥

[cm]d

short

4.251.997874square6.112.437256rect.5.012.507878roundlongshortlongshort

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

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

Table1: Loss and kick factors as estimated by 2D electrostatic calculation. The bunch length σ = 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

=0.5mmσ=0.3mmσ

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

=0.5mmσ=0.3mmσ

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

=0.5mmσ=0.3mmσ

long

in+out

[V/pC/mm]k⊥

[cm]d

short

4.251.997874square6.112.437256rect.5.012.507878roundlongshortlongshort

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

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

Table1: Loss and kick factors as estimated by 2D electrostatic calculation. The bunch length σ = 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

3D collimators. Inductive Regime

−5 0 5 10 15 20 25 30 35 40 45−50

−40

−30

−20

−10

0

10

20

s [cm]

Wy [V

/pC

/m]

Dipolar Wakes

IZ, roundIZ, ellipticalBP, roundBP, elliptical

ECHO

GdfidL

3D collimators. Inductive Regime

−5 0 5 10 15 20 25 30 35 40 45−25

−20

−15

−10

−5

0

5

s [cm]

WQ

[V/p

C/m

]

Quadrupolar Wakes

IZ, dipolar roundIZ, ellipticalBP, ellipticalBP dipolar round

ECHO

GdfidL

3D collimators. Inductive Regime

-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

3D collimators. Inductive Regime

-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 dxF s

W x dx

σ

σ

⊥−

⊥−

= →∫

∫

,

5

,

5

( )( ) 0.52

( )

sq el

qs

d round

W x dxF s

W x dx

σ

σ

⊥−

⊥−

= →∫

∫

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 σ-3 -2 -1 0 1 2 3 4 5-4

-3

-2

-1

0

/s σ

[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 σ-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 σ-3 -2 -1 0 1 2 3 4 5-4

-3

-2

-1

0

/s σ

[V/pC/mm]W⊥

square(#2)

rect.(#3)

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

-3

-2

-1

0

/s σ

[V/pC/mm]W⊥

square(#2)

rect.(#3)

round

298335335168α [mrad]3.81.91.91.9b2 [mm]

rect.rect.squarerect.Type4321Coll. #

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

Table 2: Geometry of SLAC collimators of 2001

1 / 2 19 mmb h= =

Rectangular 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 σ-3 -2 -1 0 1 2 3 4 5-4

-3

-2

-1

0

/s σ

[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 σ-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 σ-3 -2 -1 0 1 2 3 4 5-4

-3

-2

-1

0

/s σ

[V/pC/mm]W⊥

square(#2)

rect.(#3)

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

-3

-2

-1

0

/s σ

[V/pC/mm]W⊥

square(#2)

rect.(#3)

round

298335335168α [mrad]3.81.91.91.9b2 [mm]

rect.rect.squarerect.Type4321Coll. #

Figire 3: Transverse wake of Gaussian bunch, with σ = 0.65 mm (left) and σ=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]σ

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]σ0 .1 0.3 0.5 0.7 0.9 1.11

1.5

2

2.5[V/pC/mm]k⊥

[mm]σ

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]σ

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]σ0.2 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]σ

0.622.52.5Eq. 121.13.32.4Eq. 31.02.51.24Eq. 1

0.54(0.05)1.4(0.1)1.4(0.1)1.2(0.1)meas.0.501.721.751.28simul.

1.7/431/981.00.5/50ρ1/ ρ2

4321Coll. #

0.62.52.5Eq. 120.82.41.7Eq. 30.51.30.7Eq. 1

0.49(0.15)1.08(0.1)1.2(0.1)0.78(0.13)meas.0.411.301.350.90simul.

0.9/240.5/530.50.3/27ρ1/ ρ2

4321Coll. #

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

Table 3:Kick factor [V/pC/mm]; σ = 0.65 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]σ

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]σ0 .1 0.3 0.5 0.7 0.9 1.11

1.5

2

2.5[V/pC/mm]k⊥

[mm]σ

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]σ

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]σ0.2 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]σ

0.622.52.5Eq. 121.13.32.4Eq. 31.02.51.24Eq. 1

0.54(0.05)1.4(0.1)1.4(0.1)1.2(0.1)meas.0.501.721.751.28simul.

1.7/431/981.00.5/50ρ1/ ρ2

4321Coll. #

0.62.52.5Eq. 120.82.41.7Eq. 30.51.30.7Eq. 1

0.49(0.15)1.08(0.1)1.2(0.1)0.78(0.13)meas.0.411.301.350.90simul.

0.9/240.5/530.50.3/27ρ1/ ρ2

4321Coll. #

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

Table 3:Kick factor [V/pC/mm]; σ = 0.65 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 %).

SLAC experiment 2006

2.43.0285682.73.1346072.32.9245166.87.1478150.770.74244045.16.1336333.13.6336021.71.928501

σ=0.5mmσ=0.3mmσ=0.5mmσ=0.3mmktr [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.

SLAC experiment 2006

2.43.0285682.73.1346072.32.9245166.87.1478150.770.74244045.16.1336333.13.6336021.71.928501

σ=0.5mmσ=0.3mmσ=0.5mmσ=0.3mmktr [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

d1b

2bα d

1b2b

αInductive

[deg]α

2[ ]b mm

When a short bunch passes by an out-transition, a significan reduction in thewake will not happen until the taperedwalls cut into the cone of radiation, i.euntil

2tan /z bα σ0.025 mmzσ =

Collimator geometry optimization.Optimum d ~ 4.5mm

200 mm

17 mm2 mm

100 mmd

200 mm

17 mm2 mm

100 mmd

0 5 10 15 20100150200250300350400450

( )1

2 20WΔ

0L

/d mm

/V p C

0 5 10 15 20

1

2

3

4

5

1L ⊥

( )1

2 21W⊥Δ

/ /V pC mm

/d mm0 5 10 15 20100

150200250300350400450

( )1

2 20WΔ

0L

/d mm

/V p C

0 5 10 15 20

1

2

3

4

5

1L ⊥

( )1

2 21W⊥Δ

/ /V pC mm

/d mm

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

XFEL collimators. Tapering

d mm

10 / 20 mm

4.5 mm3.4 mm

25mkmσ =

50 (45%)

38

110

Loss,V/pC

29 (67%)

42

43

Spread,V/pC

-83taper 20mm

-82 (53%)

taper 20mm +step

-156step

Peak,V/pC

-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. 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 12

long Z ck

b bπ⊥⎛ ⎞

= −⎜ ⎟⎜ ⎟⎝ ⎠

20 2

2 42 1

14

short Z c bkb bπ⊥

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠

10-1 100 101 102 1030.8

1

1.2

1.4

1.6

1.8

2

[ ]d mm

[ / / ]KickV 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 12

long Z ck

b bπ⊥⎛ ⎞

= −⎜ ⎟⎜ ⎟⎝ ⎠

20 2

2 42 1

14

short Z c bkb bπ⊥

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠

10-1 100 101 102 1030.8

1

1.2

1.4

1.6

1.8

2

[ ]d mm

[ / / ]KickV pC mm

Form optimization?Exponetial tapering?

Acknowledgements to K. Bane,

M. Dohlus,B. PodobedovG. Stupakov, T. Weiland

for helpful discussions on wakes and impedances, and to

CST GmbH for letting me use CST MICROWAVE

STUDIO for the meshing.