multipoles of the accelerating field and the beam distortion in tbts
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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
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