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.