Transcript LINAC-I, II

LINAC-II
ILC School
Chicago, Oct. 22, 2008
T. Higo, KEK
1
Contents of LINAC-II
• Beam quality preservation
–
–
–
–
–
–
–
–
–
Luminosity
Linac optics
Perturbations
Wake field
Wake field suppression
Alignment to beam
Breakdown rate
Dark current
Summary of LINAC-II
2
Energy and luminosity
Plinac  RF   Beam
e Ec  Ea Llinac 
N N b Frep
L0 N b Frep
L
1 L0


 RF   Beam
Plinac e Ec N N b Frep /  RF   Beam e Ec N
3
Luminosity
4
Luminosity related parameters
• Important parameters relevant in this lecture are
– Beam transverse size
• Main theme of this lecture
– Number of bunches in a train
• Long-range wake field
– Number of particles in a bunch
2
L
f rep nb N H D
4  
*
x
*
y
• Short range wake field
– Repetition frequency
• Wall plug power
5
Requirements for linear collider
• Beam quality from DR and BC should be preserved.
• Phase space
– Longitudinal
• Energy acceptance of the final focus
– Transverse
• Emittance preservation to keep beam size small
• Wake field
– Single bunch:
• Short range (intra-bunch) wake field
• Dispersive effect in the bunch
– Multi-bunch:
• Long range (inter-bunch) wake field among bunches
• Bunch to bunch dispersive effect
6
Linac example parameters
ILC and CLIC
Item
units
ILC(RDR)
CLIC(500)
ELinac
GeV
25 / 250
/ 250
Acceleration gradient
Ea
MV/m
31.5
80
Beam current
Ib
A
0.009
2.2
Peak RF power / cavity
Pin
MW
0.294
74
Initial / final horizontal emittance
ex
mm
8.4 / 9.4
2 /3
Initial / final vertical emittance
ey
nm
24 / 34
10 / 40
RF pulse width
Tp
ms
1565
242
Repetition rate
Frep
Hz
5
50
Number of particles in a bunch
N
109
20
6.8
Number of bunches / train
Nb
2625
354
Bunch spacing
Tb
360
0.5
468
6
Injection / final linac energy
Bunch spacing per RF cycle
Tb/ TRF
ns
7
Linac example parameters
ILC and CLIC
Item
RF frequency
Beam phase w.r.t. RF
F
units
ILC(RDR)
CLIC(500)
GHz
1.3
12
5
15
SW
TW
degrees
EM mode in cavity
Number of cells / cavity
Nc
9
19
Cavity beam aperture
a/l
0.152
0.145
Bunch length
z
0.3
0.044
mm
ILC parameters are taken from Reference Design Report of ILC for 500GeV.
CLIC500 parameters are taken from the talk by A. Grudief, 3rd. ACE, CLIC Advisory Committee,
CERN, Sep. 2008, http://indico.cern.ch/conferenceDisplay.py?confId=30172.
8
Bunch pattern
ILC
2625 bunches / pulse
z=300mm
468 rf / separation
CLIC
Many bunches / train
Many rf cycles (~103) till next bunch
z/l~0.0013
354 bunches / pulse
z=44mm
6 rf / separation
Only 6 rf cycles till next bunch
z/l~0.0018  actual z 10 times smaller
9
Bunch profile and power spectrum
1 
P( )  Exp[ ( z ) 2  2 ]
2 c
z2
f ( z )  Exp[
]
2
2 z
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
1.0
0.5
0.5
1.0
0
mm
100
200
300
400
500
GHz
10
Emittance
dilution / preservation
11
Emittance preservation
Emittance preservation along the linac is one of
the key issues of the main linac of LC.
Emittance in 6 dimensions;
Single bunch emittance vs multi-bunch emittance
Longitudinal emittance vs transverse emittance in X and y
Error sources for dilution should be minimized.
Better alignment, wake field suppression, etc.
But if diluted, we try to correct
by realignment, corrector fast magnets, etc.
Coherent dilution can be corrected but it phase space volume is
fully filled in an incoherent manner, it cannot be corrected.
Multi-bunch dilution can be corrected bunch by bunch correction.
12
Very basic optics of linac
L
Transverse motion of eq.
d2
x  k ( z) x  0
dz 2
cg
[m 2 ]
E/e
  k Lr   k LQ ,   2LQ / L
k ( z) 

 Cos
MF 
  k Sin 

1


Sin  
 Cosh
k
, M D 
  k Sinh 
Cos 

1

Sinh 
k

Cosh 
M FDF  M F / 2 M drift M D M drift M F / 2
Twiss parameter at QF;
 Sinm
 Cos   Sinm


M FDF  
Cos   Sinm 
   Sinm
In a thin lens approximation
 2 2
m 
Lr
Lr
Transfer matrix per period;
c g L2
Cosm 1 
, Sin 

 (1  )
2
2 2 8E /e
LQ
LQ
Betatron oscillation along the linac
is the very basic of the system.
x( z )  A  ( s) Sin ( ( s)   ) ,
Assume (s) constant
x( z )  Sin (k s   ) , k  
d
1

ds  ( s)
1
2

 (s) l
In the left FODO system,
k 
2
l

m
L

cgL
 (1  )
4E/e
13
X’
Phase
space
volume
Transverse emittance
If before filling the whole phase space, the emittance can be controlled.
x
X’
X’
Single bunch
Correctable
Single bunch
Correctable
X’
Multi bunch
best
x
Coherent
Correctable
Incoherent
Incorrectable
X’
X’
Multi bunch
Correctable
if Tb large
Multi bunch
Correctable
if Tb large
x
x
Coherent
Correctable
Single bunch
Incorrectable
x
x
Coherent
Correctable
X’
Bunch-by-bunch
Correctable
x
Difficult to
Correct
14
Practical errors to be considered
• Optical error and misalignment
– Field stability in Q
– Misalignment of Q, structures and BPM
• RF error
– Phase, amplitude jitter
– Pulse to pulse, within pulse,
– Asymmetry in cavity to time dependent transverse kick
• Wake field
– Long range and short range
– Longitudinal and transverse
15
Possible cares and corrections
• Optical error and misalignment
– Alignment with using beam information
– BPM information, on- /off-energy, on- /off- Q setting
• RF error
– Feedback with cavity field
– Mechanical precision of cavity, offset, tilt,
– Suppression of field asymmetry
• Wake field
– Cavity HOM damping and cancelling
– Multi-bunch energy compensation with injection timing,
ramping pattern, etc
16
Wake field and
impedance
17
Wake fields driven by relativistic particle
1/
Lorentz contracted field
In free space
EM field associated to the particle is
scattered by periphery shape.
18
Wake field
Driving bunch: Unit charge bunch at
offset radius r
Witness bunch: trailing at z=ct behind
r2
q2
q1
r1
s=ct
Wake field W(s) is the kick received
by the witness bunch.
1
W (r1 , s ) 
q1
 dz E (r , z, t )  c z  B(r , z, t )

1
1

t ( s  z ) / c
p  q1 q2 W ( s )
1
WL (r1 , s ) 
q1
1
WT (r1 , s ) 
q1

 dz E
z
(r1 , z , t )t ( s  z ) / c

 dz E (r , z, t )  c z  B(r , z, t )

1

1
T , t ( s  z ) / c
19
Panofsky-Wenzel theorem
B
ez    E  ez  ( 
)
t
B

 ez
 e ( E    E z )
t
z
d
 1 


dz z c t

1
WT ( x, y, z ) 
z
q1

or
H
Ez
Think about TM11 field

  Ez  
Br
t
1


 dz  z ET  ez  B    q1 T




dz Ez

WT   T WL
s
WT ( x, y, s)   T

s

ds'WL ( x, y, s' )
20
Cylindrical symmetric system
and multipole expansion
From Panofsky Wenzel theorem,
WL  

W , WT  T W
s
For axisymmetric environments, W can be expanded into multi-poles,

W ( r , r ' ,  , s )   Wm (r , r ' , s ) Cos(m )
m 0
From Maxwell’s equation, form of Wm can be found,
Wm (r , r ' , s )  Fm ( s ) r m r 'm
Wake functions are expressed now as,
WL( m ) (r , r ' , s ) 

Fm ( s ) r m r 'm Cos( m )
s
WT( m ) (r , r ' , s )  m Fm ( s ) r m 1 r 'm [r Cos(m )   Sin ( m )]
Near axis, only m=0 for longitudinal wake and m=1 for transverse wake.
Longitudinal: constant over radius
Transverse: linear over drive bunch offset but constant on witness bunch position.
21
Impedance
Impedance: Fourier transform of the wake function
s
 j
s
Z L ( x, y,  )   d ( ) WL ( x, y, s) e c

c
s
 j

s
ZT ( x, y,  )   j  d ( ) WT ( x, y, s) e c

c

Panofsky Wenzel theorem becomes,

c
Z T ( x, y ,  )   T Z L ( x, y ,  )
Wake is real  then
Re{Z L ()}  Re{Z L ()},
Then,
1
WL ( s ) 
2
1

2
From causality,



d Z L ( ) e
Im{ Z L ()}   Im{ Z L ()}
j
s
c

s
s
d

[{
Re{
Z
(

)}
Cos
(

)

Im{
Z
(

)
Sin
(

)}]
L
L

c
c
WL ( s)}  0 for all s  0


s
s
  d Re{Z L ( )}Cos( )    d Im{ Z L ( )}Sin ( )


c
c
22
Actual impedance shape
Finally, wake function is calculated from only the
real part of the impedance,
WL ( s ) 
1

s
d

{
Re{
Z
(

)}
Cos
(

)
L




c
Low frequency
trapped modes and
higher frequency
component with
escaping into the
beam pipe.
Re(Z||()
1.0
Trapped
modes
0.8
0.6
Leaking to
beam pipe
0.4
0.2
0.0
0
2
4
6

8
Actual real part of the
impedance is
illustrated as shown
in the left figure.
10
Foe each resonance,
R
Z L ( ) 
 0
1 j Q (

)
0

23
Loss parameter
Electric field is expressed as,

E z (r , t )  Re{ En (r ) e j n t }
n 0

Vn   dz E z ,n (r ) e
z
j n
c

For finite bunch length,
 (r , t )  q1 l ( z  ct ),
l (s) 
Define loss parameter;
kn 
kn 
2
Vn
4U n
En
Actual wake is a convolution,
SW
TW
For delta-function-like driving particle;
Loss to fundamental mode;
k fund  k0 exp{

s
WL ( s )   2 k n Cos(n )
c
n 0


WL (s)   ds' l ( s  s' ) WL (s' )
0
2
4 un
( s  s0 ) 2
1
exp{ 
}
2
2

2
z
Self wake is half of the wake behind the bunch,
Energy lost in mode n is
1  2 2
( ) z}
2 c
Loss to higher modes;

k HOM   2 k n exp{
n 1
1 n 2 2
( ) z}
2 c
U n  k n q12
24
Longitudinal wake function in DLS
Summation of resonant modes up to a certain frequency,
N
s
WL ( s )   2 kn Cos(n )
c
n0

Higher than the frequency, and in high energy limit >>a/c,
optical resonator model predicts
 Cos( )
A
dk
 30/ 2 ,   Wa ( )  2 A0 
d
m
d 
 3/ 2
Total wake field calculated for the
SLAC disk loaded structure became
as shown in right figure (P. Wilson
Lecture)
Fundamental mode dominates for long-range
wake, with some high Q modes superposed.
Much higher than 400th mode contributes in very
short range wake field.
25
Transverse wake function in DLS
We follow a paper* by Zotter and Bane on transverse wake field
calculation on disk-loaded structure.
Synchronous space harmonic component of the n-th TW mode, axial electric field is
expressed as
r
E zn  E0 n ( ) m Cos(m ) Cos{n (t  z / c)}
a
Where E0n is the field at r=a, iris opening radius. Loss parameter is
rq
E02n
kn 
, and un  k n ( ) 2 m q 2
4 un
a
rq
 E0 n   2 ( ) m k n q
a
rq is drive bunch position
Therefore,
rq
r
z
 Ezn   2 k n q ( ) m ( ) m Cos(m ) Cos(n )
a
a
c
With Panofsky Wenzel theorem;
( ET  c BT )( cmf )  j (c /  ) T Ez( cmf )
* B. Zotter and K. Bane, “Transverse Resonances of Periodically widened Cylindrical Tube with
Circular Crosssection”, PEP-Note-308, SLAC, 1979.
26
Transverse wake field (cont.)
Finally for the transverse wake field,
WTn ( )  2 (
k n c rq
) ( ) Sin (n  )
n a a
Summation gives total from resonant-like modes,
WT ( )  2 (
rq
a
) (
n
kn c
) Sin (n  )
n a
Over maximum frequency m, integration gives wake field using
dk
A
 31/ 2
d 
The calculated wake field for SLAC DLS are shown in three time ranges;
27
Dipole wake field parametrization
Transverse wake field for NLC structure;
WT ( s ) 
4Z 0 c s0
{1  (1  s / s0 ) Exp( s / s0 ) }
4
a
a1.79 g 0.38
Z 0  377 , s0  0.169
L1.17
Transverse wake field for NLC.
200
2Z c

WT ( s )  0 4
s
a
Linear slope and
strong a-dependence!
WT [V/pC/mm/m]
Initial slope;
200
a=3mm
150
a=4mm
100
100
a=5mm
50
0
0.0
0
0.2
0.4
s [mm]
0.6
0.8
1.0
1
28
Single-bunch beam
dynamics and cures
29
Beam dynamics under shot range
transverse wake field
Force and momentum change in transverse direction;
Fx 
dp x d
dx
d dx
  m0
 m0 c 2 
dt dt
dt
ds ds
Consider a bunch at s along the linac with the transverse position x(s,z)
within the bunch with its charge distribution l(y).
Under the force due to the transverse wake field, the equation of motion becomes
m0 c 2

d d
 x( s, z )  m0 c 2 k 2 x( s, z )  e Ne  dy x( s, y ) w( y  z ) l ( y )
z
ds ds
The second term in left hand side is the betatron oscillation term in a linac optics.
Right hand side is the force due to wake field inside the bunch at the position z due to
the offset of the precedent part of the same bunch at y with offset value x(s,y).
This becomes
N r 4 e 0
1 d d
 x( s, z )  k 2 x( s, z )  e
 ds ds



z
dy x( s, y ) w( y  z ) l ( y )
e2
where re 
4  e 0 m0 c 2
1
30
Two particle model view
If no acceleration case;
N re 4  e 0
d2
2
x
(
s
,
z
)

k
x
(
s
,
z
)


ds 2



z
dy x( s, y) w( y  z ) l ( y)
Bunch is divided into two part, head and tail
Each charge Ne/2, separated by 2z and
Head bunch behaves as

x1 ( s)  x1 Cos k  s
The force experienced by the tail particle due to the wake field driven at the head particle
F2  e
Ne
x1 w (2 z )
2
Then the equation of motion of the tail particle becomes
N re 4  e 0 w (2 z ) x1
d2
 j k s
2
x

k
x




C
x
2
 2
1e
ds 2
2
This is the forced oscillation with the same oscillation frequency as the drive field.
Find a solution of the form
x2  y ( s) e
j k s
31
Growth estimation
in two particle model

y '' (s)  2 j k y ' (s)  x1
This has a solution

j x1
y( s)  
s
2 k
x1
Then,

j x1 j k s
x2 ( s)  
se
2 k
x2
This result states that
1. Amplitude increases linearly
2. Phase is 90 degrees delayed
32
More general but constant energy case
Again in no acceleration case;
N re 4  e 0
d2
2
x
(
s
,
z
)

k
x
(
s
,
z
)


ds 2



z
dy x( s, y) w( y  z ) l ( y)
Assume initial offset without slope
x(0, z )  x0 ,
x
0
s s 0
And assume wake field is small
N re w

z
 k 2
Let us find a solution of the form
x ( s, z )  a ( s, z ) e
Position in
linac
y
Bunch frame
s
j k s
Substitute into the above equation
N r 4 e 0
a
 j e
s
2 k 


z
dy a( s, y) w( y  z ) l ( y)
33
Linear slope wake and square bunch
w  w' z
l ( z ) 1 / 2lb
2 1/ 6
(2 r s z )
And define a parameter
33 / 2
Exp[
(2 r s z 2 )1/ 3 ]
4
N re 4  e 0 w'
r
2 k   lb
Then,

a
  j r  dy ( y  z ) a( s, y )
z
s
This has an asymptotic solution,
a
3/ 2
x0
3
(2 r s z 2 ) 1/ 6 Exp[
(2 r s z 2 )1/ 3 ]
4
6
2rsz2
Derived by A. W. Chao, B. Richter and C. Y. Yao, NIM, 178, p1, 1980
34
Take energy gain into account
Back to equation
N r 4 e 0
1 d d
 x( s, z )  k 2 x( s, z )  e
 ds ds



z
dy x( s, y) w( y  z ) l ( y)
Assume slow acceleration
 ' /   k 
And assume no wake first, then
d2
' d
x

x  k 2 x  0
2
ds
 ds
Let us design as
x ( s, z )  a ( s, z ) e
This has a solution,
a( s, z )  a0 ( z )  0 /  ( s)
j k s
It states the adiabatic damping
The equation on a becomes
a  '
2
 a 0
s 
Now let the wake ON,
35
With linear slope wake and energy gain
N r 4 e 0
a  '
2 j k
 j k a  e
s 



z
dy a( s, y) w( y  z ) l ( y)
This time let us define
a(s, z)  b(s, z) /  (s)
Then the equation becomes
N r 4 e 0
b
 j e
s
2 k   ( s)


z
dy b(s, y) w( y  z ) l ( y)
This is exactly the same as a(s,z) case but with varying (s)
Let us define coordinate S
dS  ds /  ( s )
Then
N re 4  e 0
b
 j
S
2 k


z
dy b( s, y) w( y  z ) l ( y)
This form is also exactly the same as that of a(s,z) before.
This has an asymptotic form as a.
36
Solution of growth
Assume uniform acceleration
 ( s)   i  s( f   i ) / L
Then,
S
L
 ( s) L  ( s)
ln

ln
 f  i
i  f
i
Therefore, as the case with a,
x0
N r 4  e 0 w'
33 / 2
2 1 / 6
b
(2 r S z ) Exp[
(2 r S z 2 )1/ 3 ] where r  e
4
2 k  lb
6
Since Sfinal is
S final 
L
f
ln
f
f
 L /  eff where  eff   f / ln
i
i
From these formula, we can estimate the growth along the bunch
1
a
 (s)
x0
33 / 2
2 1 / 6
(2 r S z ) Exp[
(2 r S z 2 )1/ 3 ]
4
6
where r 
N re 4  e 0 w'
2 k  lb
37
BNS damping
Equation of motion in the two particle model,
N re 4  e 0 w (2 z ) x1
d2
 j k s
2
x

k
x




C
x
2
 2
1e
ds 2
2
Varying the tail particle oscillation frequency from that of the head particle,
N re 4  e 0 w (2 z ) x1
d2
 jk s
2
2
x

(
k


k
)
x


  C x1 e 


2
2 2
ds
2
The tail particle resonant growth is suppressed. In FODO lattice,
m
c g L2
Sin 
 (1   ) &
2 8E /e
1
m dm c g L2
Cos

 (1   )
2
2 d 8 E 2 / e
dm
m
  2 Tan
d
2
We make the energy variation within a bunch to introduce variation of k
dk 
d

1 dm
2
m
  tan
L d
L
2
This suppression is called BNS damping.
In practice;
Energy tapering can be produced by setting the bunch in RF slope.
The longitudinal wake function help decreased the energy toward the tail.
38
Autophasing
Start with the equation of motion;
N re 4  e 0
d2
2
2
x
(
s
,
z
)

{
k
x
(
s
,
z
)


k
x
(
s
,
z
)}



ds 2



z
dy x( s, y) w( y  z ) l ( y)
If we can vary the k as
k 2 ( s, z )} 
N re 4  e 0



z
dy w( y  z ) l ( y )
X’
Single bunch
Correctable
The solution becomes simply
x( s, z )  x( s)  x0 cos( k  s)
This is stable and x does not depend on z,
which means the both head and tail stays as
in the right figure.
x
Coherent
Correctable
This suppression scheme is autophasing.
The slope on k is produced by energy profile inside the bunch with
amplitude of the order of
N r 4 e 0  z
k 2  e
W'

The big energy slope should be compensated at the downstream of linac.
39
Long range wake field
40
Fundamental theorem of beam loading
0: Assume a cavity field in phasor
diagram with one dominant mode
=t Vc
V (t ) V e jt
2: When the second bunch comes in,
superposition applies;
e
1: Point particle passed an empty cavity,
leaving a field, wake field,
Ve
Vreference
e
Vb1++Vb2+
Vb1  Vb e j
Vb
E1  q Ve  q f Vb
U1   Vb2
=t
Vb2+
Vb1+
U 2   (Vb1 e j  Vb 2 ) 2
Beam energy loss
Cavity stored energy
 2  Vb2 (1  Cos
Fundamental theorem of beam loading cont.
While loss of the second bunch;
E2  q Ve  q Vb Cos(e   )
Since particle energy loss = cavity stored energy;
E1  E2  U 2
Therefore,
2 (q f   Vb )  (q Cose  2 Vb ) Cos  q Sine Cos  0
This should always true for any , then
 e  0,
q
Vb 
,
2
1
f
2
When a bunch passes a cavity,
it excites the cavity with the field in a decelerating direction,
or it remains a deceleration wake field in the cavity.
The bunch feels half of this excited field.
Long range wake field in a cavity
Longitudinal wake excited in a cavity is expressed as
WL ( s)  2 k L e

s
c
s 
s
( Cos( )  Sin ( ) ) ,
c 
c
  02   2 , Q  0 / 2
In a cavity with very high Q value,
WL ( s)  2 k L e
WT ( s)  2 kT e

L s

1 s
2Q c
2Q c
s
 R
Cos(L ) , where k L  L ( ) L
c
2 Q
s
Sin (1 )
c
Longitudinal wake field behaves cosine-like. The bunch is decelerated
and excite the field in the cavity. Point-like bunch suffer from the wake,
deceleration.
Transverse wake behaves sine-like. It increases linearly in time at very
short time, usually within the bunch.
43
Calculation of impedance in SW or TW
For longitudinal mode;
L
R
V2

  
[] , V   E z ( z ) e j k z dz , k  , U  Stored energy
0
c
 Q L 2 U
For transverse mode;
2
L E ( z )
R

V / r  1

jk z
z
  T 
[

]
,

V
/

r

e
dz
,
k

,U  Stored energy
2

0
2 U k
r
c
Q
'
Then for longitudinal wake;
1
WL ( s)  
Lq

L
0
Ezcmf dz [V / C / m] 
 2 k L, n e
n

n s
2 Qn c
Cos(
n
c
s)
44
Calculation of kL by SW field solver
For SW cavity;
V2
 R
k L , SW 
/ L  ( )L / L
4U
2 Q
For TW cavity;
2
E
k L ,TW  0
4u
where only n=0 space harmonics contributes;
s
E z  E0 Cos  (t  )
c
When we consider SW field, SW = FW + BW;
z
z
E zSW ( z, t )  E0 {Cos  (t  )  Cos  (t  ) }
c
c
The field calculated by SW mode is the case with t=0 in the above equation;
z
E zSW ( z, t )  2 E0 Cos  ( )
c
45
Loss parameter formula
Therefore, E0 can be calculated as
1 L / 2 SW

V
E0   E z ( z ) Cos( z ) 
L L / 2
c
L
When we consider the coupling of beam to TW mode, the relevant
energy is only the forward wave, which is half of USW
2
k L ,TW
2
VSW
V

/ L  SW
4 (U SW / 2)
2 U SW
R
 k L ,TW     / L
 Q L
Then, loss parameter is
k L ,TW  2 k L , SW
46
Transverse wake calculation
Transverse wake field excited by a bunch with charge q passing a cavity
with transverse position offset of r,
( E  v  B)T
The wake field due to this field is expressed
1
WT ( s) 
L q r


L
0
( E  v  B)Tcmf
2kT , n  2Qnn c
n
V
dz [
]

e
Sin
(
s)
n  / c
C m2
c
s
(#)
This can be explained for both SW and TW as follows .
For SW case, from Panofsky Wenzel,
( E  v  B)Tcmf 
j
T E zcmf
/c
If we apply this to the above eq.
WT0 ( s ) 
1
j V
L q r  / c r
47
Transverse wake calculation (cont.)
The energy of the cavity excited by charge q with offset r is equal to the
energy loss if the bunch interacting with the field excited by itself,
1 V
U (r )  q
r  kTSW (q r ) 2 L
2 r
where

R
kTSW  k 2 ( )'T / L
2
Q
and thinking
2k
WT0  T
/c
The equation (#) is found proven. (This is for SW case)
In the TW case, as in the longitudinal wake,
kT ,TW  2 kT , SW
Therefore,
'
R
kT ,TW   k 2   / L
 Q T
48
Frequency scaling
Per cavity;
L
Per unit length;
R
V2
l2
  

1
3
 Q L 2 U  l
[]
R
  / L  
 Q L
R
WL  k L     / L   2
 Q L
'
2
R

V / r  1
  T 
1
2
Q
2

U
k
 
'
T
[]
R
  T / L  
Q
'
R
kT   k 2   T / L   4
Q
WT 
kT

 3
49
Dipole mode field to calculate R/Q
Let us take the most typical dipole mode, TM1nl,
in a pillbox cavity
Er  
E 
z
Cos( )
Kc
z
Sin ( )
K c2
e
H   j
K c2
e
Kc
Sin (  z z )
1
J 1 ( K c r ) Sin (  z z )
r
Cos( )
Ez 
Hr   j
J1' ( K c r )
Sin ( )
Cos( )
J1 ( K c r )
Cos(  z z )
1
J 1 ( K c r ) Cos(  z z )
r
J 1' ( K c r )
Cos(  z z )
Hz 0
where K c  1n / a,  z  l  / d
50
Dipole mode field to calculate R/Q (cont.)
The slope of Ez in r-direction on =0 plane at the beam axis
J (K r)
Ez ( z ) 

J1 ( K c r )  K c [2 J 0 ( K c r )  1 c ] 1.5K c
r
r
K c r r 0
Per a cavity
L/2
L/2
E z ( z ) j k z
e dz  2  1.5 K c Cos(kz) dz  3 K c 
Cos(kz) dz
L / 2
0
0
r
V / r  
 V / r 
L/2
3 Kc
kL
Sin ( )
k
2
Then for a cavity
3 Kc 2
kd
) Sin 2 ( )
k
2
R
V / r  1 
  T 
2  U k 2  m  d a 2 k 2 ( e )2 J '2 (  )
Q
1
1n
Kc
'
2
2(
51
Comments on dipole mode
Firstly note that TE mode cannot couple to beam
because of no Ez field!
Secondly, there are two polarization in dipole modes
with the same field pattern.
Therefore, it is necessary to separate in frequency
for these modes to be stable unless the two
polarizations can be divided by geometry condition.
52
Actual higher order modes
examples
53
Actual modes in 9-cell SCC cavity
Measurement setup
Outer side
Inner side
NA
Al shorting plate
Transmission measurement 1.5~3GHz
050509
ICHIRO #1 Cavity S21 measurement
Rotational symmetry
identification.
0
90-90 [dB]
90-0 [dB]
S21 [dB]
-20
Passband nature.
Comparison with
pillbox modes.
-40
-60
Leakage to beam
pipe.
-80
High Q modes.
-100
1.5
2
2.5
Freq GHz
3
Beam excitation of modes
ICHIRO #1 Cavity Frequency Measurement
050509 & 050512
frequency
S21, S11_in, S11_out [MHz]
3500
1st HOM
3000
2500
ACC
M
D
2000
D
S21
1500
S11_in
vp=c
S11_out
M
1000
0
20
40
60
80
100
120
Mode Numbering
Phase
advance
• Beam interacts mostly near vp=c line.
• Everyband can be excited by beam.
• Most concern is the lowest dipole modes.
56
X-band detuned structure
Contour of dipole mode
frequency vs (a,t)
(a,t) distribution
along structure
Dipole mode
distribution in
frequency and
kick factor
57
Beam excited modes in detuned
structure
Beam excitation
Down
Excited modes
Middle
Trapped modes
UP
Should be
damped
vp=c
Position-modal
frequency
dependence
Can be used
as SBPM
58
Actual modes in X-band structure
Extract a part along structure and stack 6 identical cells.
Measure dispersion characteristics to confirm the HOM.
6-disk Spectrum
#97--102
-20
S12 [dB]
-40
-60
-80
-100
-120
10
12
14
16
Freq [GHz]
18
20
59
Ways to deal with
coupled cavity system on
HOM estimation
60
Various ways of treatment
• Equivalent circuit model
• Matching
– Mode matching
– S-parameter
– Open mode expansion model
• Mesh based Numerical
– Finite element model HFSS, 2
– Finite difference model MAFIA, GdFidl,
61
Mode matching techniques
Resonant mode
Field
Field matching matching
Propagating mode
SW & TR field
matching
Propagating mode
Scattering
matrix
S-matrix for 3D
cal can be used
Field matching
Mode
matching
Expansion modes
in cylindrical
waveguide
Resonant SW mode
Field matching
Open mode
expansion
Resonators
with small
coupling
between
cells
62
An example:
Open mode expansion technique
Open
modes for
expansio
n base
Base_8
Base_1
Base_2
Actual open modes used for calculation
63
Calculated field for 150-cell cavity
Base_8
1
21
41
Ns
E (r , z )    a j eopen
j
Base_2
Base_1
Cell-1
8
k 1 n 1
61
81
121
141
101
Cell-150
161
 a1 


 a2 
a   a3 





a 
 8N 
X a  2 a
64
Calculation result
Wake field
Kick factor
N
WT (t )   2 kT ,n Sin (n t )
n 1
65
Finite element or finite difference
Example: SCC 9-cell cavity simulation with 3D FEM (Z. Li, SLAC)
• 3, MAFIA, GdfidL, HFSS
• Parallel PC arrays today can deal with the whole cavity
in 3D as a whole.
• This is a final confirmation but these are becoming to
even the tool in early design stage.
66
Cures against multibunch emittance growth
by suppression of
wakefield
67
Cures from structure design
Transverse wake field is expressed as
WT ( s ) 
2k

T ,n
e
n
2 Qn
t
Sin (nt )
 tb
2 QL

f tb
468
 10 for ILC 1
QL /  10

n
6
for CLIC 1
10 4
There are two ways of suppressing wake field.
One is to align beam to axis, while the other is to suppress wake field.
Damping the mode with low Q

e
t
QL 
2 QL
U
Pwall  Pext
Effectively lower Q by frequency spread
Intrinsic Q0 is too
large.
Both ILC and CLIC
need external
damping.
W (t )   f ( ) e j  t d
X-band approach
utilizes this.
68
External coupling ILC and CLIC
Q ~105
Q ~101
Quadrant-type heavily
damped structure
It is not easy to make Q very low by external coupling.
This results in the application of frequency spread for effective damping
(cancellation) of wake field in addition to low Q.
69
Detuning to make frequency spread
Wake
field
Fourier
transform
Frequency
distribution
W (t )   K (i ) Sin (i t )
i
Fi=it
As of
excitation by
beam
Cancellation
of wake field
Calculation with equivalent circuit
Geometry parameters along a
structure is distributed to make
the coupling to beam (kick
factor) as gaussian like.
Analyse the whole system with coupled
resonator equivalent circuit model.
pioneered by K. Band and R. Gluckstern
and explored by R. Jones.
71
Moderate damping by
extraction of dipole modes into manifold
Example of middle cells of RDDS1
RDDS1 Dispersion
2nd
dipole
Fd1(97-102)
20000
Fd2(97-102)
Mahifold(97-102)
Manifold
18000
Manifold
1st dipole
16000
Cell
14000
12000
0
30
60
90
120
Phase shift / cell
Example cell shape
150
180
Avoided crossing
due to the coupling
of cavity dipole
mode and manifold
mode.
Distribution of (a, t) and
introduction of damping
• Faster damping need larger
width
• Truncation makes tail up in wake
field
• DS detuned only  DDS damped
detuned
• Recurrent due to finite number
of distribution points
• Interleaving makes longer
recurrent
73
Result of equivalent circuit model
calculation and estimate of tolerances
Wake field calculation based
on frequency error info from
fabrication.
Wake field simulation with more
frequency errors to investigate
frequency tolerances.
74
Actual design (RDDS1) and typical cells
Input / output
waveguide
HOM damping
waveguide
Manifold to carry
HOM to outside
Detuned cells
75
Frequency control of actual structure
Accelerating mode frequency with
feed forward in '2b' and integrated phase
2b offset [micron]
Frequency error [MHz]
Integrated phase from cell #6-->#199
3
2b offset [micron]
Freq. meas. & estim. [MHz]
Integ. phase
2
1
0
-1
-2
-3
0
50
100
150
200
Disk number
Check each cell frequency
and feedforward to the later
cell fabrication.
76
6-disk Spectrum
S12 [dB]
#6-11
RDDS1 dispersion
-20
-40
S12 [dB]
-60
Second Dipole Mode
Pseudo Dispersion Curves
[MHz]
20000
-80
19000
18000
-100
17000
-120
10
12
14
16
Frequency [GHz]
18
20
16000
15000
6-disk Spectrum
#97--102
Fd2(6-11)
Fd2(40-45)
Fd2(74-79)
Fd2(91-96)
Fd2(109-114)
Fd2(133-138)
Fd2(166-171)
Fd2(184-189)
Fd2(194-199)
14000
-20
13000
12000
-40
0
30
60
90
120
-60
-80
[MHz]
-100
19000
-120
10
12
14
16
18
Freq [GHz]
6-disk spectrum #194-199
20
Manifold Mode
Pseudo Dispersion Curves
[MHz]
20000
-20
19000
18000
-40
S12 [dB]
First Dipole Mode
Pseudo Dispersion Curves
20000
First Dipole Mode [MHz]
S12 [dB]
Phase shift / cell
-60
12
14
16
Freq [GHz]
18
20
15000
14000
12000
0
Mahifold(6-11)
Mahifold(40-45)
manifold(74-79)
Mahifold(91-96)
Mahifold(109-114)
Mahifold(133-138)
manifold(166-171)
manifold(184-189)
manifold(194-199)
12000
10
16000
16000
13000
-120
17000
13000
14000
-100
18000
17000
15000
-80
Fd1(6-11)
Fd1(40-45)
Fd1(74-79)
Fd1(91-96)
Fd1(109-114)
Fd1(133-138)
Fd1(166-171)
Fd1(184-189)
Fd1(194-199)
0
30
60
90
120
Phase shift / cell
150
180
30
60
90
120
Phase shift / cell
150
180
150
180
Proof of wake field in RDDS1
RDDS1 Wake Data (Wx = , Wy = ·) and Prediction (Line)
2
Wake Amplitude (V/pC/m/mm)
10
1
10
0
10
-1
10
-2
10
0
2
4
6
8
10
12
14
16
SQRT[Time(ns)]
78
Cures by improving
alignment
79
One-to-one steering
If every BPM is aligned perfectly to the magnetic center of
each quadrupole magnet, it is easy to adjust the beam
with respect to those quad’s. Just align the beam to zero
the BPM reading.
Changing the Q strength  transverse kick measured at
downstream BPM’s to know the beam position w.r.t. Q
center and BPM calibration.
By
x
It is straightforward way but suffers from errors in BPM
reading, BPM misalignment w.r.t. Q magnet, etc.
This is the local correction, but there is better way of
correcting more globally, DF or WF correction scheme.
80
DF and WF correction
The equation of motion in transverse plane in high energy linac;
1 d
d
 ( s) x( s; z,  )  (1   ) K [ x( s; z,  )  xq ]
 ( s) ds
ds


1 
 (1   ) G 
N re  dz '  d '  ( z ' ,  ' ) W ( s; z  z ' ) [ x( s; z ' ,  ' )  xa ]
z

 0 ( s)
where
e dBy
e
K ( s) 
, G( s) 
By
 = E/E,
p
c
dx
p
c
0
0
G(s) steering field, K(s) = Q magnetic field,
Wt = transverse wake field, N = number of particles in a bunch,
re = classical electron radius,
(z,d) = charge distribution,

xq = Quad misalignment, xa structure misaligment
Correlated energy spread 
xd  x( z ,0)  x( z ,  )
Uncorrelated energy spread 
xw  x( z ,0)  x( z ,  )
z


z
z
z
Tail

z
Head
81
Equation of motion and force term
The equation for the difference are in the first order approximation
No wake field
Oscillation

1 d
d
 ( s) xd ( s; z ,  )  K ( s) xd ( s; z ,  )
 ( s) ds
ds
  (G ( s)  K ( s) [ xq ( s)  x( s;  z ,0) ]
Steering
With wake field
z

Q mag
Oscillation
1 d
d
 ( s ) xw ( s;  z ,  )  K ( s ) xw ( s;  z ,  )
 ( s ) ds
ds
  G ( s )   K ( s ) {xq ( s )  x( s;  z ,0) }
Steering
z

N re
W ( s,2 z ) { ( x( s;  z ,0)  xa ( s ) }
2  0 (s)

Q mag
z
z
Tail

z
Head
Wake field
82
Dispersion free (DF) correction
In the first order approximation, the solutions for these are written as the
transferred position originated from the kick at s’ upstream
No wake field
xd (s; z ,  ) 

s
0
ds' R12 (s, s' )  {G(s' )  K (s' ) [ xq (s' )  x(s' ; z ,0) ]}
Minimize xd is equivalent to locally minimize the following value small
G( s' )  K ( s' ) [ xq ( s' )  x( s' ; z ,0) ]
 [G( s' )  K ( s' ) xq ( s' )]  K ( s' ) x( s' ; z ,0) ]
Q misalignment
Local correction
In reality, from the i’th BPM reading mi and its difference mi with their predicted
values, xi and xi, minimization does dispersion free correction;
2
2
 N BPM

x

x
i
i 

min  2
 2
 i 1    2


prec
BPM
prec


83
Wakefield free (WF) correction
In the first order approximation, the solutions for these are written as the
transferred position originated from the kick at s’ upstream
With wake field
xw ( s; z ,  ) 

s
0
ds' R12 ( s, s' )  [G( s' )  K ( s' ) { xq ( s' )  x( s' ; z ,0) ]

N re
W ( s' ,2 z ) {x( s' ; z ,0)  xa ( s' ) }
2  0 ( s' )
Minimize xw is equivalent to locally minimize the following value small
 [G ( s ' )  K ( s ' ) { xq ( s' )  x( s' ;  z ,0) ] 
  [G ( s ' )  K ( s ' ) xq ( s ' )]  { K ( s ' ) 
Q misalignment
N re
W ( s ' ,2 z ) {x( s ' ;  z ,0)  xa ( s ' ) }
2  0 ( s' )
N re
N re
W ( s ' ,2 z ) } x( s ' ;  z ,0) 
W ( s ' ,2 z ) xa ( s ' )
2  0 (s' )
2  0 ( s' )
Wake field correction
Structure misalignment
Wake field term cannot be cancelled out by  term because of constant Wt while alternating in
K(s’). Taking QF only or QD only makes the correction of Wt.
84
Wakefield free (WF) correction
In reality, from the i’th BPM reading mi and its difference mi with their predicted
values, xi and xi, minimization does wake field free correction;
2
QF 2
QD 2 
 N BPM
x
x
x
min   2 i 2  i 2  i 2 
 i 1  prec   BPM 2  prec 2  prec 


Where the xiQF and xiQD are those difference orbit due to the variation of only
QF and QD.
Correction example; NLC case from T. Raubenheimer, NIM A306, p63,1991.
Method
ey
Trajectory rms
1-to-1
23 ey0
72 mm
DF
9 ey0
55 mm
WF
1 ey0
44 mm
85
Alignment with using excited field
in the actual structure
If we measure dipole mode in a structure, we can estimate the position
of beam which excites the mode. The coupling is linear as offset.
If the modal frequency depends on the position of the mode, it
can distinguish the position there by frequency filtering.
In such cavity as ILC, it can be done with using power from
HOM couplers.
In such cavity CLIC, it can be done with extracted power from
manifold or damping waveguide.
Both directions x and y are measured with distinguishing two modes in
different polarizations, almost degenerate but with some frequency difference.
86
Alignment measurement in situ as SBPM
Power from manifold
Excited power
15 GHz Dipole Power Scan
Frequency filtered
500
Phase and amplitude
Signal Power (au)
400
Parabolic Fit
300
Frequency-to-position
Measured Power
200
Straightness measure
100
0
-400
-300
-200
-100
0
100
200
300
400
Beam Y Position (Microns)
Phase of excited field
15 GHz Dipole Phase Scan
Position measured from frequency
filtered signal compared to
mechanical measurement
200
Phase (degrees)
Measured Phase
150
Arctan(Y/27)
100
50
0
-400
-300
-200
-100
0
100
200
300
400
Beam Y Position (Microns)
87
Dark current issue
88
Dark current
What is dark current?
Dark current is a stable emission of electrons under high field.
DC field emission is the tunneling feature of electron migration near surface.
It is studied by Fowler-Northeim.
Here work function and field enhancement factor play important roles.
Actually the field E0 is replaced by the local field E0.
iFE , DC  F E02 Exp(
G
) [ A m 2 ],
E0
F 1.54 10 6 10 4.52
0.5
 1 , G  6.53109  1.5
RF field emission is estimated to be the superposition of DC field emission.
It is calculated by J. Wang and G. Loew.
It should exist in ant RF field, whether or not in SW and TW or in NCC and SCC .
iFE , RF  F E02.5 Exp(
G
) [ A m  2 ],
E0
F  5.7 10 12 10 4.52
0.5
 1.75 , G  6.53109  1.5
Tracking of FE electrons are studied by various authors.
Nowadays, numerical tracking is usual one.
Simple analytical estimate gives minimum threshold field for capture.
89
Acceleration in linear accelerator
Acceleration field is expressed as the summation of
all space harmonics in the periodic structure.
E z  exp j (t  k z z )
n 
  jE
n  
n
J 0 (k rn r ) exp  2 jnz / d
k rn  k 2  (k z  2n / d ) 2
k   / c  2 / l
n 
d
2nz
   e n J 0 (k rn ) Sin ( 
)
dz
d
n  
e n  e En / m0c 2
 (
1
p

1

) dz
R. Helm and R. Miller, in Linear Accelerators, ed. by
P. M. Lapostolle and A. L. Septier, North-Holland
Publishing Co., 1970
If, p~, then  is slowly varying function.
Therefore, only n=0 is dominant.
d
  e 0 Sin ( )
dz
and
p 2 1
d k
 (
 p)  A
dz e 0
p
Finally, by combining these equations,
Cos( ) 
k
e0
(
p 2 1
p
 p)  A
Plot the contour of this equation with A
in the next page.
90
Separatrics
describing longitudinal motion
  
untrapped
p
 p
 p  0. 5
p
  2 1
m0 c

Normalized momentum
trapped
p
 p
 p 1

 RF phase seen by particle
91
Capture threshold
Cos( ) 
k
e0
(
p 2 1
p
 p)  A
For p=1, A=Cos(m) when approaching at pinfinity
Cos( )  Cos( m ) 
k
e0
( p 2  1  p)
Minimum field e0 for being tramped, eq. to being max in left-hand side,
Cos( )  Cos( m )  2
For zero-energy electron, p=0 to be captured,
the minimum field becomes
2

m
c
threshold
0
Eacc

el
l = 26.242mm at 11.424GHz
l = 230.6 mm at 1.3GHz
61 MV/m at 11.424GHz
7 MV/m at 1.3GHz
92
Tolerable breakdowns or
quenches
93
To keep the beam energy under RF failure
It is important to make the integrated luminosity high by keeping
the instantaneous luminosity high.
Once some failure happens in some cavity, the power feeding the
cavity or the bunch of cavities is shut off.
Then, the power or pulse width will be recovered taking some
pulses starting from a little lower power level. During this period,
the cavities in recovering mode are off in timing from acceleration
for the linac if powered by independent power supply.
In such a system as CLIC two beam scheme, the off-timing
operation cannot be applied and gradual power recovery is needed.
During this recovery period, other cavities than nominal should be
used to keep the beam energy. Therefore, linac needs extra
acceleration capability than the nominal one.
Extra spare cavities or extra power/gradient capability is required.
94
Simple estimation of tolerable failure rate
Compensation with spare cavitires:
Nunit Nstr
Nominal
 Nunit Nstr
Spare
Rfail failure rate for a cavity (1 trip / N pulses)
Trec recovery time in the unit of pulses
Number of failures during the recovery time for a cavity
N unit N str R fail Trec
It should be less than the number of spare units
N unit N str R fail Trec   N unit
Then the tolerable failure rate is
R fail 

Trecov N str
An example;
Order estimation
N unit  3000
N str  8
Trec  60 s  50 Hz
  0.01
In this example,
R fail  4107
95
Reduce failure rate or more margin
in accelerator gradient
• Cares on SCC system
– Margin of accelerator gradient
•
•
•
•
Increase quench field due to Hs
Suppress FE
Increase Q0
Variable feeding system
– Mechanical long life for tuner
• Reduce breakdown rate in NCC
– Possible trigger source of breakdown
• Surface quality chemically and physically
• Reduce micro protrusions
• Reduce pulse temperature rise?
96
Pulse temperature rise
Surface heating and heat diffusion into body.
Pw ( x, t )

2
u ( x, t )   u ( x, t ) 
t
cp 
Pulse heated
surface
Temperature
degC in rise Temperature
200
1000
100
800
0
0
600
400
10
200
20
30
0
2 Tp
1
2
T  Rs H s
2
 ce   d
Pritzkaw Thesis, p99, SLAC-Report 577.
Cu
Rs=27.85m
Hs=1MA/m
Tp=400ns=4*10^-7
=8.93 10^3 kg/m^3
Ce=380 J/kg/K
d=k/ce= m^2/s
k=401 W/m/K
V. Dolgashev and L.
Laurent, AAS08
There may trigger
breakdowns.
T=270degC
97
Care in complicated shape formation
4
2
3
1
HsurfaceTpulse
Esurface
We need to avoid additional local field enhancement
due to non-smoothness especially at red areas.
Special care is taken at points (2,3,4) where smooth junction is difficult
due to the junction between milling and turning
050113
Beam Physics Seminar
98
Summary of LINAC-II
• By keeping the emittance growth within a tolerable level, the
luminosity will be kept.
• Various sources, especially wake-field origin, were discussed.
• Various cures on HOM origin are discussed.
• Cares on alignment to suppress single-bunch wake field was
discussed, using structure BPM and BBA.
• These perturbations and cures are almost similar to both
warm and cold linac.
99