Multipoles of the accelerating field and the beam distortion in TBTS
description
Transcript of Multipoles of the accelerating field and the beam distortion in TBTS
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