Electromagnetic elds of charged beams in
gradually tapering waveguides
David A Burton
(with DC Christie
1
,JDA Smith & RW Tucker)
Department of Physics,Lancaster University
and the Cockcroft Institute of Accelerator Science and Technology,UK
MOPNET 3
HeriotWatt University
2022 September 2010
1
University of the Highlands and Islands,Scotland
Motivation
Examples of nonuniform metallic structures in accelerators:
I
collimators
Motivation
Examples of nonuniform metallic structures in accelerators:
I
collimators
Motivation
Examples of nonuniform metallic structures in accelerators:
I
transition between subsystems,e.g.beam pipe in a small gap
undulator
Max Radius:
20mm
Length: 1.4m
Charge: v=c
Min Radius:
2mm
Challenge
I
Rapid changes in spatial prole tend to have undesirable
consequences
I
induces instabilities in the particle beam,and destroys its
useful properties
I
employ a structure with a gradually varying prole
I
Slow variation =) computationally intensive
I
BUT slow variation =) amenable to analytical methods
I
(practical limitations:size of tunnel and resistance of the
structure)
Challenge
I
Rapid changes in spatial prole tend to have undesirable
consequences
I
induces instabilities in the particle beam,and destroys its
useful properties
I
employ a structure with a gradually varying prole
I
Slow variation =) computationally intensive
I
BUT slow variation =) amenable to analytical methods
I
(practical limitations:size of tunnel and resistance of the
structure)
Challenge
I
Rapid changes in spatial prole tend to have undesirable
consequences
I
induces instabilities in the particle beam,and destroys its
useful properties
I
employ a structure with a gradually varying prole
I
Slow variation =) computationally intensive
I
BUT slow variation =) amenable to analytical methods
I
(practical limitations:size of tunnel and resistance of the
structure)
Challenge
I
Rapid changes in spatial prole tend to have undesirable
consequences
I
induces instabilities in the particle beam,and destroys its
useful properties
I
employ a structure with a gradually varying prole
I
Slow variation =) computationally intensive
I
BUT slow variation =) amenable to analytical methods
I
(practical limitations:size of tunnel and resistance of the
structure)
Electromagnetic 2form
I
Electromagnetic 2form F is related to E and B as follows:
F = dt ^(E
x
dx +E
y
dy +E
z
dz)
B
x
dy ^dz B
y
dz ^dx B
z
dx ^dy (1)
I
Perfect conductor boundary condition at S = 0
dS ^F = 0
S=0
(2)
is equivalent to n E = 0 and n B = 0 where n is normal to
S
Maxwell's equations
I
Maxwell's equations
dF = 0;"
0
d?F = %?
e
V (3)
where %V is the electric 4current of the particle beam
Maxwell's equations
I
Maxwell's equations
dF = 0;"
0
d?F = %?
e
V (3)
where %V is the electric 4current of the particle beam
I
for an unperturbed beam
V =
1
c
@
@t
+
@
@z
(4)
assuming that it is travelling close to the speed of light c
Potential decomposition
I
Exploit cylindrical symmetry of waveguide:
F =
@
u
H
+@
u
H
B
+@
H
B
2@
2
u
W @
2
W
d ^du
+du ^
h
d
?
@
u
W +@
W H
B
+@
u
#
?
d
?
X
i
+d ^
h
d
?
H
B
@
W
+#
?
d
?
(@
X H
'
)
i
+
@
H
'
+@
u
H
+H
b
2@
2
u
X @
2
X
#
?
1 (5)
where = z,u = z ct,d
?
is the exterior derivative acting
in the (x;y) subspace (transverse subspace) and#
?
is the
Hodge map on the transverse subspace
I
Follows from Hodge decomposition
1
(D) = d
?
F
d
(D) #
?
d
?
F(D) on the transverse
crosssection D of the waveguide
Field equations
Maxwell's equations may be reduced to:
?
d
?
H
B
= 0;(6)
d
?
H
b
=#
?
d
?
@
u
H
B
;(7)
d
?
H
'
=#
?
d
?
H
;(8)
?
d
?
W 2@
2
u
W @
2
W
+@
u
H
+@
u
H
B
+@
H
B
= P(r;;u);(9)
?
d
?
X 2@
2
u
X @
2
X +@
H
'
+@
u
H
'
+H
b
= 0 (10)
where
?
d
?
is the transverse Laplacian and
@
u
P(r;;u) = (r;;u) (11)
Boundary conditions
Cylindrical symmetry with S as r = R():
W = 0j
r=R()
;(12)
@
r
X = 0j
r=R()
;(13)
H
B
= @
Wj
r=R()
;(14)
H
= R
0
()
1
r
@
Xj
r=R()
(15)
Asymptotic approximation scheme
I
Introduce\slow variation"in :
R() = R() (16)
where 1 > 0
I
Introduce\slow coordinate"s = and
=
1
X
n=0
n
n
;(17)
where
2
n
W;
X;
H
B
;
H
b
;
H
;
H
'
o
(18)
N.B.Assume potentials depend on through s only.
Yields a hierarchy of Poisson and Laplace equations
Asymptotic approximation scheme
I
Introduce\slow variation"in :
R() = R() (16)
where 1 > 0
I
Introduce\slow coordinate"s = and
=
1
X
n=0
n
n
;(17)
where
2
n
W;
X;
H
B
;
H
b
;
H
;
H
'
o
(18)
N.B.Assume potentials depend on through s only.
Yields a hierarchy of Poisson and Laplace equations
Longitudinal wake potential
W
k
(r;;u) =
1
q
0
Z
1
1
E
z
ct=zu
dz (19)
For simplicity,assume that the source is onaxis and gaussian in
u = z ct
Longitudinal wake potential:I
r = R(z) = 20 18 exp[z
2
=(8 10
5
l
2
)]; = 1=
p
8 10
5
(20)
l = 1mm and z;r are measured in mm
Longitudinal wake potential:II
r = R(z) = 20 18sech(0:01z=l ); = 0:01 (21)
l = 1mm and z;r are measured in mm
Impedance:onaxis harmonic charge density
Z
k
onaxis
(!) = Z
k
1 onaxis
+Z
k
2 onaxis
+Z
k
4 onaxis
+Z
k
6 onaxis
+:::
(22)
where
Z
k
1 onaxis
=
1
2"
0
c
ln
R
1
R
2
;(23)
Z
k
2 onaxis
=
i!
4"
0
c
2
Z
1
1
R
02
dz (24)
Z
k
4 onaxis
=
i!
96"
0
c
Z
1
1
5R
04
+3
R R
00
2
2
!
2
c
2
R
2
R
00
2
dz
(25)
Impedance:onaxis harmonic charge density
4"
0
cZ
k
6 onaxis
=
i!
c
Z
1
1
3
16
(R
00
R
0
R)
2
+
11
120
R
06
+
1
48
(R
2
R
000
)
2
dz
+
i!
3
c
3
Z
1
1
11
256
R
3
R
000
2
1
6
R
2
R
0
R
00
2
73
768
R
5
R
003
dz
+
i!
5
c
5
Z
1
1
19
160
(R
3
R
0
R
00
)
2
+
73
1920
R
7
R
003
19
1920
(R
000
R
4
)
2
dz
(26)
I
Similar integrals obtained for Z
?
(!)
Optimal geometry
Stationary variations of an impedance
Z[R] =
Z
z
2
z
1
(R;R
0
;R
00
;R
000
) dz (27)
yield a BVP for an optimal geometry.
0 = Z =
Z
z
2
z
1
@
@R
d
dz
@
@R
0
+
d
2
@
dz
2
@R
00
d
3
@
dz
3
@R
000
R dz
+
@
@R
00
d
dz
@
@R
000
R
0
z
2
z
1
+
@
@R
000
R
00
z
2
z
1
(28)
Optimal geometry
Hence,solve the ODE
@
@R
d
dz
@
@R
0
+
d
2
@
dz
2
@R
00
d
3
@
dz
3
@R
000
= 0;(29)
subject to the natural BCs
@
@R
00
d
dz
@
@R
000
z
1
;z
2
= 0;(30)
@
@R
000
z
1
;z
2
= 0 (31)
Optimal geometry
An optimal geometry (w.r.t.longitudinal impedance) has a linear
prole,R(z) = az +b,independent of the source frequency!
I
2nd order unaected by 4th order and 6th order corrections
I
Does this hold to all orders?
I
(optimal geometry w.r.t.transverse impedance is frequency
dependent)
Optimal geometry
An optimal geometry (w.r.t.longitudinal impedance) has a linear
prole,R(z) = az +b,independent of the source frequency!
I
2nd order unaected by 4th order and 6th order corrections
I
Does this hold to all orders?
I
(optimal geometry w.r.t.transverse impedance is frequency
dependent)
Optimal geometry
An optimal geometry (w.r.t.longitudinal impedance) has a linear
prole,R(z) = az +b,independent of the source frequency!
I
2nd order unaected by 4th order and 6th order corrections
I
Does this hold to all orders?
I
(optimal geometry w.r.t.transverse impedance is frequency
dependent)
Optimal geometry
An optimal geometry (w.r.t.longitudinal impedance) has a linear
prole,R(z) = az +b,independent of the source frequency!
I
2nd order unaected by 4th order and 6th order corrections
I
Does this hold to all orders?
I
(optimal geometry w.r.t.transverse impedance is frequency
dependent)
Further directions
I
Resistive waveguide
I
noncircular crosssections
I
Propagating modes missing
I
Fully inductive:here,a source with compact support in
u = z ct yields elds with compact support in u = z ct
I
But sharp corners lead to wave emission,propagation and
diraction
I
=) modal analysis
Further directions
I
Resistive waveguide
I
noncircular crosssections
I
Propagating modes missing
I
Fully inductive:here,a source with compact support in
u = z ct yields elds with compact support in u = z ct
I
But sharp corners lead to wave emission,propagation and
diraction
I
=) modal analysis
Further directions
I
Resistive waveguide
I
noncircular crosssections
I
Propagating modes missing
I
Fully inductive:here,a source with compact support in
u = z ct yields elds with compact support in u = z ct
I
But sharp corners lead to wave emission,propagation and
diraction
I
=) modal analysis
Further directions
I
Resistive waveguide
I
noncircular crosssections
I
Propagating modes missing
I
Fully inductive:here,a source with compact support in
u = z ct yields elds with compact support in u = z ct
I
But sharp corners lead to wave emission,propagation and
diraction
I
=) modal analysis
Further directions
I
Resistive waveguide
I
noncircular crosssections
I
Propagating modes missing
I
Fully inductive:here,a source with compact support in
u = z ct yields elds with compact support in u = z ct
I
But sharp corners lead to wave emission,propagation and
diraction
I
=) modal analysis
Further directions
I
Resistive waveguide
I
noncircular crosssections
I
Propagating modes missing
I
Fully inductive:here,a source with compact support in
u = z ct yields elds with compact support in u = z ct
I
But sharp corners lead to wave emission,propagation and
diraction
I
=) modal analysis
Reference
I
\Wake potentials and impedances of charged beams in
gradually tapering structures"
DAB,DC Christie,JDA Smith,RW Tucker.
arXiv:0906.0948 [physics.accph]
(submitted)
We thank the Cockcroft Institute for support.
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