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Deflecting Cavities for Advanced Accelerator Applications
Kwang-Je Kim ANL
ICFA Mini-Workshop Deflecting/Crabbing Cavity Applications in Accelerators
April 23-25, 2008
SSRF, Shanghai
2Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
Contents
EM Fields of a Deflecting Mode
Applications of Deflecting Cavities
3Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
Deflecting Mode in a Pill-Box Cavity
TM110
Standing wave:
Deflecting force:
1
01
0 1
2 ( ) cos
0
2( )sin
2 ( )cos
i tz o
r z
i tr
i t
E E J kr e
E E cB
iEcB J kr e
kr
cB iE J kr e
cos , sini t i te t ie t 2
2 20 (1 (3 ) )sin
8x z y
kF ev B eE x y t
20
2 2sin
eEd xt
dz mc
R
3.8c
R
Aberration terms
4Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
Multi-Cell Pillbox, -Mode
kz t sin sin( ) sin cos cos sint kz kz kz
-3λ/4
-λ/2 -λ/4 λ/4 3λ/4λ/2
z = 0
-ℓ ℓ
Θ(z) = +1 Θ(z) = -1 Θ(z) = +1 Θ(z) = -1 Θ(z) = +1
-λ λ
2
2 21 ( )sinoed x
z tdz mc
02
sin e
x z kz dzmc
02 0
2 sin cos
ek z z dz
mc
cos kz
5Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
Multi-Cell Pillbox, -Mode (cont’d)
2 1 : 2 1 full-cells4
n n
2 : 2 1 full-cells 2 half-cells4
n n
0 00
22 cos 2 kz dz
2 sin 1 aberration
eVx
mc
V┴: kick voltage Transit time factor
6Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
Multi-Cell Pillbox: Off-Set of Bunch Center
Independently by J. Power (simulation) & D. Edwards (calculation)
Large for low energy application such as 15 MeV emittance exchange experiment
Probably not important especially when + & - deflections are added
2
2
2
0 for 0
for2 4
d xx
dzeV
mc
(integer)
7Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
Canceling the Bunch Off-Set
Off-set vanishes if
k
- + -
2
2
2 cos sin 0
0.36 or 1.524 4
8Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
Multi-Cell Pillbox with Holes: HEM Mode
Solution of Maxwell Equation of the form
B.C. on beam hole:
( ) (up to quadratic polynomials in x & y)i kz ckte
0 @E r a
B
y
z
x
E-field due to a hole
---
+ + +
2 2 2 ( )
4i kz kct
x
ika x y e
( )
2i kz kct
y
ikx y e
( )i kz kctz x e
2x
ikc x y
2
2 2 ( )1 ( )1
4 4i kz kct
y
ka ikc x y e
ik
( )i kz kctz
kc y e
k
BB B
9Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
Traveling Wave Solution
Take real part of the complex solution
No bunch off-set The quadratic terms cancel out No aberration! SLAC S-band deflector in 60’s. (G. Lowe, …)
2 2 2( ) sin4
sin2cos
x
y
z
kE a x y
kE xy
E x
2 2 2
sin2
( ) sin4
cos
x
y
y
kcB xy
kcB a x y
cB y
(H. Hahn, Y. Garault. referred by G. Lowe)
1( ) sinx x yF e E cB
k
10Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
SLAC S-Band Deflector (LOLA II & III, LOLITA I)
3/2 mode traveling wave
11Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
HEM Standing Wave (Most Cases) Complex solution Construct
( , )iE k ck
No-vanishing bunch off-set → cancel by adjusting the end-cell length, or radius
Aberration terms do not cancel
Re ( , ) ( , )i iE k ck E k ck
2 2 2 sin cos ,4
sin cos ,2
cos cos ,
x
y
z
kE a x y z t
kE xy kz t
E x kz t
2 2 2 2
cos sin ,2
1 ( ) ( )1 cos sin ,
4 4
sin sin .
x
y
z
kcB xy kz t
ka k x ycB kz t
k
cB y kz t
22 2 2
2
1( ) sin ( ) cos sin
4
d x ka x y z t kz t
d z k
kz t
12Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
HEM, Cont’ued
Off-set vanishes when:
0sincos2
1
2a
Or when:
13Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
Simulation of Bunch Offset in Real Cavity (John Power) Confirms bunch off-set & vanishing of off-set for
some ℓ, but not for a
Le
gmge
r 2r
C.L.
ge
a abendbmiddlebend
Le
14Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
Applications of Deflecting Cavities
Particle separation Temporal beam diagnostics Crab-crossing in colliders X-ray pulse compression Emittance exchange
15Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
Deflecting Cavity for Longitudinal Phase space diagnostics
16Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
Crabbing to Enhance Collider Luminosity Crab angle
17Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
X-ray Pulse Compression
18Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
19Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
Emittance Exchange Emittance Exchange (EEX): Complete exchange of x- and z-phase spaces: (x,z) (z,x)
Applications
To improve performance of high-gain FEL… Reduce transverse emittance at the expense of longitudinal
To obviate electron damping ring in linear collider Photocathode gun flat beam transformation EEX
(x, y, z)=(1, 1, 8)(50,0.02, 8)(8, 0.02,50) m
20Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
Construction of an Exact Exchange Optics
1k
1000
10
0010
00L0
MD
Thin Trasverse Cavity
Dog-Leg for generating dispersion
Mc
C
0 1 0 0
0 1 k 0M
0 0 1 0
k 0 0 0
B
B
B
B
1k
0 0 / /
0 0 1/ /
0 / 0 0
0 / 0 0
L L
L
L
B Exact Exchange (KJK)
Approximate Exchange (M.Cornacchia & P. Emma)
0 2 / /
0 2 1/ /
0 / 0 0
0 / 0 0
L L L
L
L
21Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
EEX Demonstration Experiment at FNAL
(x, y, z)=(4, 4,130) m (130, 3,3) m
3.9 GHz, 5-cell, LiN-cooled copper cavity (adopted from CKM)
22Deflecting Cavities for Ad. Acc. Appl., Shanghai, 4/23-25/08
Gun Linac
ICT1 GVGV
TQ1,TQ2,TQ3
YAG1 YAG2 Emittance Exchanger
Demonstration Experiment for EEX at Argonne AWA
initialx=9 mz= 4 m final
x=4 mz= 9 m
• AWA- and APS-ANL, NIU, and Tsinghua U collaboration• L-band, TM110, (1/2-1-1/2)-cell cavity from Tsinghua U
TM110
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