Multipoles of the accelerating field and the beam distortion in TBTS

Alexej Grudiev29/05/2013

CLIC RF Structure Development Meeting

MeshTD24_vg1p8Ntetr = 1188991; dxyz ~ 0.5 mm near axis

Electric field

Ln

rnn

L

kickzkick

L

cvz

kickzkickcvkickzkick

zcj

kick

zcj

kick

dzFnunurc

rp

dzHuZEcedz

vFrp

HuZEeBvEeF

eHHeEE

z

z

0

)(1)(

00

0

0

)sin()cos(1),(

),(

;

)(1)(

0

)sin()cos(),(

1:where

~for;),,(),(

naccr

nn

r

tjL

acc

Vnununrjerp

ru

ru

eEzrEdzjerp

Multipole expansion of Ez

n

nnacc

n

innnaccacc

Lnacc

nacc

L

accacc

zcj

zacc

nrVerVrV

dzzEVdzzrErV

ezrEzrE

)cos(),(

)(;),,(),(

),,(),,(

)()(

0

)()(

0

Accelerating gradient:

Accelerating voltage:

Multipole expansion in vacuum only:

Skew components = 0 due to the symmetry

Panofsky-Wenzel (PW) theorem:

Gives an expression for multipolar RF kicks:

Lorenz Force (LF):Gives an expression for kick directly from the RF EM fields:

Which can be decomposed into multipoles:

Equating the RF and magnetic kicks, RF kick strength can be expressed in magnetic units:

]/[1

]/[1

1

0

)(

0

)()()(

1)()()(

nL

nacc

Lnnn

nnacc

nn

mTmVnjdzFec

dzBb

mTEnjFec

B

TD24_vg1p8: multipoles of Eacc at Vz=1V;

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-10

0

10on crest

{E

acc

(0)

} [V

/m] @

1V

r = 2 mmr = 1 mmr = 0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-10

0

10

{E

acc

(1)

} [V

/m2 ] @

1V

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-1

0

1x 10

4

{E

acc

(2)

} [V

/m3 ] @

1V

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-2

0

2x 10

6

{E

acc

(3)

} [V

/m4 ] @

1V

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-2

0

2x 10

9

{E

acc

(4)

} [V

/m5 ] @

1V

z [m]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-5

0

590o off crest

{E

acc

(0)

} [V

/m] @

1V

r = 2 mmr = 1 mmr = 0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-20

0

20

{E

acc

(1)

} [V

/m2 ] @

1V

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-1

0

1x 10

4

{E

acc

(2)

} [V

/m3 ] @

1V

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-5

0

5x 10

6

{E

acc

(3)

} [V

/m4 ] @

1V

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-5

0

5x 10

9

{E

acc

(4)

} [V

/m5 ] @

1V

z [m]

•Quadrupolar kick strength Fx and corresponding multipole of Eacc

(2) have very different dependence along the beam axis but the integrals are equal.

TD24_vg1p8: quadrupolar kick; LF versus PW

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-3

-2

-1

0

1

2

3x 10

-6

z [m]

Qia

drup

olar

kic

k in

[T/m

] @ 1

V

on crest

F(2)x /ec

j2/w*Eacc(2)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-3

-2

-1

0

1

2

3

4

5x 10

-6

z [m]

Qia

drup

olar

kic

k in

[T/m

] @ 1

V

90o off crest

F(2)x /ec

j2/w*Eacc(2)

Comparison b(2) @Vz=1VLF: 0.10 - 0.91i [nTm/m2]PW: 0.02 - 0.65i [nTm/m2]

TD24_vg1p8: octupolar kick; LF versus PW

Octupolar kick is maximum for particle on zero crossing.

Comparison b(4) @Vx=1VLF: 0.17 +3.23i [mTm/m2]PW: 0.22 +3.22i [mTm/m2]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

z [m]

Oct

upol

ar k

ick

in [T

/m3 ] @

1V

on crest

F(4)x /ec

j4/w*Eacc(4)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

z [m]

Oct

upol

ar k

ick

in [T

/m3 ] @

1V

90o off crest

F(4)x /ec

j4/w*Eacc(4)

Summary table for Vz = 22.8 MV; Pin = 46.5 MW

TD24_vg1p8f [GHz] 11.994Vz(x=0) [MV] 22.8 +0iVx [MV] 0b(2) [mTm/m] 0 - 15ib(3) [Tm/m2 ] 0b(4) [kTm/m3] -4.6 +73.4i

NB: the b(n)‘s B-field : By(n)(y=0,x=x0) = b(n)x0

n-1. This is not MAD convention for multipolar strength.

sjnr

ns

n ebnunuerrp )(1)( )sin()cos(),,(

There is the following dependences of the multipolar kick on the RF phase, where δφs is the deviation of the (macro)particle RF phase from the crest

ΔVy@Δx=2mm/structure

Δx after 5m for 180 MeV beam

18 V

176000 V ~5 mm

Beam spot distortion due to OctupoleBeam spot: in the structure on the screen

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

x [mm]

y [m

m]

Vz = 0 MeV

Wilfred Farabolini

Beam spot distortion due to OctupoleBeam spot: in the structure on the screen

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

x [mm]

y [m

m]

Vz = 0.5 MeV

Wilfred Farabolini

Beam spot distortion due to OctupoleBeam spot: in the structure on the screen

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

x [mm]

y [m

m]

Vz = 1 MeV

Beam spot distortion due to OctupoleBeam spot: in the structure on the screen

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

x [mm]

y [m

m]

Vz = 1.5 MeV

Beam spot distortion due to OctupoleBeam spot: in the structure on the screen

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

x [mm]

y [m

m]

Vz = 2 MeV

Beam spot distortion due to OctupoleBeam spot: in the structure on the screen

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

x [mm]

y [m

m]

Vz = 2.5 MeV

Beam spot distortion due to OctupoleBeam spot: in the structure on the screen

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

x [mm]

y [m

m]

Vz = 3 MeV

Beam spot distortion due to OctupoleBeam spot: in the structure on the screen

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

x [mm]

y [m

m]

Vz = 3.5 MeV

Beam spot distortion due to OctupoleBeam spot: in the structure on the screen

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

x [mm]

y [m

m]

Vz = 4 MeV

Beam spot distortion due to OctupoleBeam spot: in the structure on the screen

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

x [mm]

y [m

m]

Vz = 4.5 MeV

Beam spot distortion due to OctupoleBeam spot: in the structure on the screen

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

x [mm]

y [m

m]

Vz = 5 MeV

Beam spot distortion due to OctupoleBeam spot: in the structure on the screen

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

-4 -3 -2 -1 0 1 2 3 4

x 10-3

-4

-3

-2

-1

0

1

2

3

4x 10

-3

x [mm]

y [m

m]

Vz = 6 MeV

Wilfred Farabolini

Thanks for your attention

• Probe beam distortion in TBTS is due to octupolar component of the 12 GHz accelerating field

• RF octupole is 90 degree out of phase with respect to the accelerating field. Maximum octupolar kick at 0-crossing of the main RF

• 8-star shape of the beam near the on crest acceleration (0-crossing for RF octupole) is probably due to multi-bunch RF phase spread