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

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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

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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

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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)

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Canceling the Bunch Off-Set

Off-set vanishes if

k

- + -

2

2

2 cos sin 0

0.36 or 1.524 4

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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

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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

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SLAC S-Band Deflector (LOLA II & III, LOLITA I)

3/2 mode traveling wave

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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

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HEM, Cont’ued

Off-set vanishes when:

0sincos2

1

2a

Or when:

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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

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Applications of Deflecting Cavities

Particle separation Temporal beam diagnostics Crab-crossing in colliders X-ray pulse compression Emittance exchange

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Deflecting Cavity for Longitudinal Phase space diagnostics

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Crabbing to Enhance Collider Luminosity Crab angle

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X-ray Pulse Compression

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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

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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

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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)

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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