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Collimator Wakefields. Igor Zagorodnov BDGM, DESY 25.09.06. 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 - PowerPoint PPT Presentation
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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.
Round collimator. Regimes
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
Round collimator. Inductive
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
Round collimator. 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
Round collimator. Inductive
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
Round collimator. Diffractive
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).
Round collimator. Diffractive
|| 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. Regimes
|| 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. 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
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
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
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
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
3D collimators. Inductive Regime
ECHO
GdfidL
3D collimators. Inductive Regime
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 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
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
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.
Table 3:Kick factor [V/pC/mm]; = 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]
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.
Table 3:Kick factor [V/pC/mm]; = 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
2.43.028568
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.
SLAC experiment 2006
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
XFEL collimators. Tapering
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. 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?
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.