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

Heriot-Watt University

20-22 September 2010

1

University of the Highlands and Islands,Scotland

Motivation

Examples of non-uniform metallic structures in accelerators:

I

collimators

Motivation

Examples of non-uniform metallic structures in accelerators:

I

collimators

Motivation

Examples of non-uniform 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 2-form

I

Electromagnetic 2-form 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 4-current of the particle beam

Maxwell's equations

I

Maxwell's equations

dF = 0;"

0

d?F = %?

e

V (3)

where %V is the electric 4-current 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

cross-section 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 on-axis 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:on-axis harmonic charge density

Z

k

on-axis

(!) = Z

k

1 on-axis

+Z

k

2 on-axis

+Z

k

4 on-axis

+Z

k

6 on-axis

+:::

(22)

where

Z

k

1 on-axis

=

1

2"

0

c

ln

R

1

R

2

;(23)

Z

k

2 on-axis

=

i!

4"

0

c

2

Z

1

1

R

02

dz (24)

Z

k

4 on-axis

=

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:on-axis harmonic charge density

4"

0

cZ

k

6 on-axis

=

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

non-circular cross-sections

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

non-circular cross-sections

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

non-circular cross-sections

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

non-circular cross-sections

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

non-circular cross-sections

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

non-circular cross-sections

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.acc-ph]

(submitted)

We thank the Cockcroft Institute for support.

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