LINAC-I
ILC School
Chicago, Oct. 21, 2008
T. Higo, KEK
Contents of LINAC-I
• Acceleration for energy
–
–
–
–
–
–
–
–
–
T. Higo
Requirement on acceleration to high energy
Maxwell’s eq. to describe microwave transmission
Cavity as a unit base of acceleration
Coupled cavity system in a SW regime
Pillbox cavity as a simple base to represent practical
cavities
Circuit modeling of cavity
SW versus TW
TW linac
Summary of LINAC-I
LC School, Chicago, 2008, LINAC-I
2
Energy and luminosity
Plinac  RF   Beam
e Ec  e 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
T. Higo
LC School, Chicago, 2008, LINAC-I
3
Acceleration for energy
T. Higo
LC School, Chicago, 2008, LINAC-I
4
Requirements for high energy machine
•
•
•
•
•
High gradient
High efficiency
Emittance preservation
Stable operation
Low cost
– Construction, operation,
– Electric consumption, cooling,
T. Higo
LC School, Chicago, 2008, LINAC-I
5
Evolution of gradient and energy
Type
Beam Energy
Acceleration
scheme
Cockcroft Walton
1MeV
1MV/m
DC Rectify
Van de Graaff
10MeV
10MV/m
DC Charging
Cyclotron
100MeV
1MV/Dee
Cyclic
Synchrotron
100GeV
100MV/ring
Cyclic
Linear accelerator
1TeV
100MV/m
Periodic
Plasma accelerator
10TeV?
>10GV/m
Plasma
T. Higo
LC School, Chicago, 2008, LINAC-I
6
Early accelerator developments
after pioneering experiment with electrons or protons
• 1896 Thomson vacuum tube with thermal electrons
– Cathode ray tube
• 1911 Rutherford Radioactive material
– Alpha particle scattering by nucleus
• 1928 Wideröe
– Linear accelerator idea
• 1931 Sloan & Laurence
– Linear accelerator experiment
• 1932 Cockcroft & Walton ~1MeV
– High voltage by rectifier
• 1933 Van de Graaff ~a few MeV
– High voltage by carrying charge by belt
T. Higo
LC School, Chicago, 2008, LINAC-I
7
Wideröe
Linear accelerators
Sloan Lawrence
Independent driven electrodes
Driven electrodes
Alvarez
Resonant cavity with drift
tube without
acceleration/deceleration
T. Higo
LC School, Chicago, 2008, LINAC-I
8
Development for higher energy
• 1931 Lawrence Several to higher than 10MeV, proton
– Cyclotron acceleration
• 1945 McMillan and Veksler higher and higher energy
– Synchrotron acceleration for proton
– 1952 Courant, Snyder: Strong focus AGS (alternating gradient synchrotron)
• 1950~1960’s Stanford electron linear accelerators GeV electron
– ~1955: Ginzton, Hansen, Chodorow: Mark-II~III
• Microwave technology the legacy of world war II
– 1967: Panofsky: 2-mile accelerator 20GeV  ~10MV/m
• Late 1980’s Richter: Stanford Linear Collider
– Energy doubler by pulse compression technique
T. Higo
LC School, Chicago, 2008, LINAC-I
9
For higher energy machine
TW DLS for electron high energy machine for years
SW Side-coupled cavity for
proton high energy machine
Disk Loaded Structure, 6MeV
Stanford Univ. 1947
SLAC:
Targeting highest
energy with
electron
T. Higo
LC School, Chicago, 2008, LINAC-I
LANL SCS:
Side coupled structure
10
Storage ring, collider to linear collider
• Storage ring in e+e- colliding mode
– PEP / PETRA ~17.5X2 GeV
– TRISTAN 30X2 GeV
– LEP 100X2 GeV
• Linear collider plans
– 1980’s VLEPP 14GHz, 100MV/m
– 1990’s TESLA 1.3GHz, 23.4MV/m  500Gev  higher?
– 1990’s GLC/NLC 11.4GHz, 50MV/m  1TeV
• ILC  500GeV
• CLIC  3TeV
• Further higher energy machine plasma, laser, etc……..
T. Higo
LC School, Chicago, 2008, LINAC-I
11
For efficiency toward high energy LEP
and for stability toward high current KEKB
LEP cavity
with storage cavity
for efficiency improvement
T. Higo
ARES for KEKB
with storage cavity for beam stability
LC School, Chicago, 2008, LINAC-I
12
Higher and higher energy for
lepton linear collider
14GHz DLS VLEPP single-bunch, high rep-rate
3GHz DLS S-band DESY multi-bunch
1.3GHz 9-cell SCC cavity TESLA DESY
11.4GHz DLS SLAC/KEK DDS (weakly damped & detuned)
30GHz DLS CLIC (Heavily damped)
1.3GHz 9-cell SCC cavity ILC developing
12GHz DLS in study
Only two types; TW DLS and SW 9-cell shaped cavity.
Simple and low cost.
T. Higo
LC School, Chicago, 2008, LINAC-I
13
Higher electron energy being developed
With care against wake field in various manners.
Continue with TW DLS for higherenergy electron linear machine
Super-conducting cavity for
higher-energy electron machine
SW SCC cavity
TESLA TDR 9-cell cavity
with HOM damping port
CLIC Quadrant-type DLS
T. Higo
DLS: Medium-damped
detuned structure
LC School, Chicago, 2008, LINAC-I
14
Livingston
plot
LHC
ILC/CLIC
2010
T. Higo
LC School, Chicago, 2008, LINAC-I
2020
2030
15
Acceleration scheme
• Electric field to accelerate
• Voltage across electrodes
• DC: once
• RF in ring: many times
– n/freq=circumference/v
• RF in line: once
– Synchronization along a line
• It is important to focus electric field along
beam axis to effectively accelerate beam.
T. Higo
LC School, Chicago, 2008, LINAC-I
16
How to reach higher energy
• Energy  gradient X length
• DC: Van de Graaf, CockCroft Walton
• RF based
– Sloan Wideroe Alvarez
– SW and TW
– Independent RF source or Two beam acceleration
• Plasma accelerator
T. Higo
LC School, Chicago, 2008, LINAC-I
17
Limiting factors against ultimate
gradient
•
•
•
•
•
Peak power available
Breakdown in structure
Quench
Mechanical stability
He cooling
• Thermal / mechanical
• Dark current loading
• Phase coherency along a long line
T. Higo
LC School, Chicago, 2008, LINAC-I
18
In this lecture
• RF acceleration is the only technology with
which we can reach TeV range accelerator in
very near future
• We focus here, as the examples, on the RF
acceleration at microwave range
– L-band (1.3GHz) superconducting cavity
– X-band (11-12GHz) normal conducting cavity
T. Higo
LC School, Chicago, 2008, LINAC-I
19
Wave length / Frequency / Band
Wave length
Frequency
60cm 20cm 10cm
1.3GHz
2.6cm 1cm
3mm
11GHz
Band
From Microwave Tubes by A. S. Gilmour, Jr.
T. Higo
LC School, Chicago, 2008, LINAC-I
20
Two types of linear accelerators
• I try to overview two types of acceleration
scheme as an introduction of linear accelerator.
– Super-conducting / Normal conducting
– Lower frequency / Higher frequency
– Standing wave / Travelling wave
• These happen to be two candidates for linear
collider which I believe we can explore in near
future.
– ILC / CLIC
T. Higo
LC School, Chicago, 2008, LINAC-I
21
Linac example parameters
ILC and CLIC
Parameters
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
T. Higo
ns
Tb/ TRF
LC School, Chicago, 2008, LINAC-I
22
Linac example parameters
ILC and CLIC
Parameters
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
sz
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.
T. Higo
LC School, Chicago, 2008, LINAC-I
23
Maxwell’s eq. to describe
microwave transmission
T. Higo
LC School, Chicago, 2008, LINAC-I
24
Maxwell’s equation and wave propagation
(1)   D  
Then, Maxwell’s eq. becomes;
  E   jm H
  H  (s  je ) E
(2)   B  0
B
(3)   E  
t
D
(4)  H  j 
t
If all quantities vary time
harmonically;
E  E0 e
jt
H  H 0 e jt
This has a solution of a propagation
along the z-direction;
E  E0 ei t   0 z
H  H 0 ei t   0 z
where
 0   0  j  0    2 (e  js /  ) m

T. Higo
LC School, Chicago, 2008, LINAC-I
25
to wave equation
0 attenuation and 0 wave number
along the propagation direction
Using Z0 and 0 to rewrite
Maxwell’s eq.;
In vacuum, s=0  velocity
  E    0 (Z 0 H )
v 1 / em
  (Z 0 H )   0 E
In a plane wave, Ex and Hy
From these, the wave
equation becomes;
Wave impedance becomes;
m
Z 0  Ex / H y 
e  js /
T. Higo
LC School, Chicago, 2008, LINAC-I
 E  0 E  0
2
 H  0 H  0
2
26
Wave propagation in uniform medium
E  E0 ei t   0 z
H  H 0 e i t   0 z
where  0   0  j  0    2 (e  js /  ) m
Insulator / vacuum
Poor conductor
s 0
 0  j em
v 1 / em
Velocity in
free space
s /   e
1
2
0  s
m
 j e m
e
Good conductor
s /   e
 0  (1  j )
Z 0  (1  j )
Lossy
propagation
sm
2
m
2s
E and H at 45
degrees
s (Cu)  5.8 107 [1 /( m)]
e / s 108 1 at 11.4GHz
T. Higo
LC School, Chicago, 2008, LINAC-I
27
Reflection from good conductor
and surface resistance
Ex / H y  Z 0  (1  j )
m
2s
This means phase lag between E and H.
Better conductor makes E smaller and smaller.
Assume plane wave incident in z-direction,
Wave equation for the transmitted wave into
material becomes
 Et   0
2
x
e0, m0
e, m, s
z
2
Et  ( 2  j  m s ) Et  0
z
Et  Es e   z
  ( j  m s )1/ 2 
s 
T. Higo
2
sm
1 j
s
Exponential decaying filed into material.
Skin depth = e-folding depth.
~ 0.6micron in copper at 12GHz.
LC School, Chicago, 2008, LINAC-I
28
Surface resistance
Corresponding magnetic field in medium is
Ht 
1

  Et 
E s e  z
 j m
j m
Then wave impedance in the medium (Cu case) is
Zm 
j  m 1 j

 40m  Z 0 

s s
m0
 377
e0
From boundary condition at the surface, reflection coefficient becomes

Zm  Z0
2 Zm
, T 1   
1
Zm  Z0
Zm  Z0
Almost full reflection
Surface current
s Es
[ A / m]
0

Magnetic field is terminated by the surface current Js within the thickness
s. Loss occurs in the volume current with equivalent surface resistance

H s  J s   s E dz 
T. Higo
LC School, Chicago, 2008, LINAC-I
29
EM wave along a uniform guide
E  E0 ei t   z
 E0  (   ) E0  0
2
t
2
2
0
H  H 0 ei t   z
In H field, also the same story.
This equation can be solved with proper cutoff propagation
constant to satisfy boundary condition;
 c2   2   02
In non conducting case,
 2   c2   2 em
Then, no propagation at low frequency;
  c  c / em
T. Higo
LC School, Chicago, 2008, LINAC-I
30
Propagation field along a uniform guide
E  (kt Et  k z E z ) ei t   z
H  ( k t H t  k z H z ) e i t   z
Where Et, Ht are transverse component vector, while Ez, Hz are both scalar and
2 Ez  c2 E z  0 and 2 H z  c2 H z  0
Them, Maxwell’s equation gives
0
( k z  Z 0 H z )
2
2
c
c


Z 0 H t  2  Z 0 H z  02 (k z  E z )
c
c
Et 

 Ez 
A function Ez, Hz, which satisfy the wave equation, make the transverse component.
T. Higo
LC School, Chicago, 2008, LINAC-I
31
TE (H) wave / TM (E) wave
We can choose either Ez=0 or Hz=0, making
TE : H t 
TM : Et 


2
c


2
c
 Hz
 Ez
These are classified into two modes;
a pure TE (no longitudinal E field) or
a pure TM (no longitudinal H field)
If the waveguide has a modulation along z direction,
pure TE nor pure TM can exist.
This is the reality and we call it HEM, hybrid mode.
T. Higo
LC School, Chicago, 2008, LINAC-I
32
Transverse field pattern in rectangular waveguide
Solving wave equation with satisfying boundary condition
nx
my  jz
Cos
e
a
b

n
nx
my  jz
H x   j 2nm Anm
Sin
Cos
e
 c ,nm
a
a
b
H z  Anm Cos
y
b
Hy  j
(x,y)
a
x
 nm
m
nx
my  jz
A
Cos
Sin
e
nm
 c2,nm
b
a
b
E x   Z h ,nm j
 nm
m
nx
my  jz
A
Cos
Sin
e
nm
 c2,nm
b
a
b
E y   Z h ,nm j
 nm
n
nx
my  jz
A
Sin
Cos
e
nm
2
 c ,nm
a
a
b
where
lg
0
Z h ,nm 
Z0  Z0
 nm
l0
T. Higo
LC School, Chicago, 2008, LINAC-I
33
Typical field patterns
T. Higo
E-mode / TM-mode
E. Marcuvitz ed.,
Microwave Handbook
H-mode / TE-mode
LC School, Chicago, 2008, LINAC-I
34
Transverse field pattern in cylindrical waveguide
Solving wave equation with satisfying boundary condition
TM mode case
m
c
m
E  j 2m
c
Er   j
Sin (m )
J m' (  c r )
me
Sin (m )
e  j z
1
J m (  c r ) e  j z
r
Cos(m ) J m (  c r )
Ez 
Hr   j
Cos(m )
e  j z
(r,)
r=a
1
J m (  c r ) e  j z
r
 c2
e
H   j
Cos(m ) J m' (  c r )
c
e  j z
Hz 0
T. Higo
LC School, Chicago, 2008, LINAC-I
35
Typical field patterns
T. Higo
E-mode / TM-mode
E. Marcuvitz ed.,
Microwave Handbook
H-mode / TE-mode
LC School, Chicago, 2008, LINAC-I
36
Dispersion relation
 2   c2   2 em    z2
In TM case with m=0,
Then the propagation of the form
e
j ( t   z z )
E z  E0 J 0 (  c r ) e
j ( t  z )
Er  j E0 Z 0 (1  (c /  ) 2 ) J1 (  c r ) e j (t  z )
H  j E0 J1 (  c r ) e j (t  z )
No acceleration in z-direction with TE
mode because Ez=0.

freq
No acceleration in z-direction with
TM mode because phase velocity is
not c, making phase slip.
30
c
  c2   z2
25
20
15
Therefore, it cannot accelerate
electrons in a long distance.
10
5
400
T. Higo
200
LC School, Chicago, 2008, LINAC-I
0
200
400
betaz
37
Reducing phase velocity to meet with
beam velocity
Add periodical perturbation with its
period=d.
If d=half wavelength, then reflection
from each obstacle add coherently,
making large reflection, resulting in
a stop band.
ｄ
Then wave component with
harmonics z=2/d suffer from
significant reflections, making a
stop band.
T. Higo
LC School, Chicago, 2008, LINAC-I
38
Expansion to space harmonics

Ez 
4
n 
a
n  
n
J 0 (k rn r ) e
j (  t  n z )
3
where
n  0  2 n / d
k k 
2
rn
5
2
2
m=0
1
2
n
m=-1
6
4
2/d
This is equivalent to the Floquet’s theorem.
2
/d
0
0
2
/d
m=1
4
2/d

6
Now it can be tuned to have a phase velocity of light.
This is required for high energy linac structure.
The accelerating field contains infinite number of
space harmonics, driven at frequency .
There are stop bands. No propagation mode exists.
T. Higo
LC School, Chicago, 2008, LINAC-I
39
Uniform waveguide to isolated cavity
ｄ
If the perturbation becomes large,
reflection from the obstacle is so
large that each cell becomes almost
isolated cavity.
ｄ
ｄ
Power propagation only through a
very small aperture.
In this extreme, the system can
better be analyzed by a weakly
coupled cavity chain model.
Now let us start from isolated
cavities.
T. Higo
LC School, Chicago, 2008, LINAC-I
d
d
40
Cavity as a unit for
acceleration
T. Higo
LC School, Chicago, 2008, LINAC-I
41
The extreme of isolated cavities
• Isolated cavities
– Should be synchronized with beam
• With external reference
• With phasing among cavities
• External control
– Need many input circuits along linac
– Space factor is not high.
• Coupling between cavities
– The only way to practically apply for very long linac as
linear collider
– Two ways, TW and SW.
T. Higo
LC School, Chicago, 2008, LINAC-I
42
Inside a cavity
• Frequency
– Field phase should be synchronous to beam
• Acceleration field on axis
– Focusing Ez to beam axis
– R/Q=V2/2U focusing the field within stored energy
• R shunt impedance
– R=V2/Pwall keep field by feeding power
• Loss factor
– beam cavity interaction
– kL= V2/4U beam loading energy  kq2
– beam loading voltage -kqCos
T. Higo
LC School, Chicago, 2008, LINAC-I
43
Most basic parameter: Frequency
Resonant frequency in electric circuit
1
Freq 
2 L C
Cavity frequency can be tuned by changing L and/or C by
perturbing magnetic field and/or electric field.
Slater’s perturbation theory states;
 2  02
  ( H 2  E 2 ) dV
2
0
V
2
H
 dV 1,
Cavity
SCC cavity tuning
Blue nominal freq
Freq up green
Freq down red
2
E
 dV 1
Cavity
Actual cavity tuning can be done by deforming cell shape, local
dimple tuning, inserting rod, etc.
Four dimple tuning per cell in NCC.
T. Higo
LC School, Chicago, 2008, LINAC-I
44
Acceleration related parameters
Basic acceleration-related parameters. In a cavity or in a unit length.
V   E z ( z , t ) dz
R
2
V
Pc
V
2 U
U G
Q

Pc
Rs
Q
Wall loss by surface integral
Stored energy by volume integral
Pc 
U
T. Higo
Rs
2
m
2


2
Eacc
R/ L
( Pc / L)
2
Eacc
( R / L) / Q 
( Pc / L)
2
R/Q 
Eacc  V / L
 (U / L)
( Pc / L)
G  m
H
H
H 2 dS
H 2 dV 
e
2

E 2 dV
Rs 
m
2s
LC School, Chicago, 2008, LINAC-I
2
dV
2
dS
Geometrical factor
due to geometry.
Surface
resistance due to
surface loss
mechanism.
45
Efficient acceleration R/Q
How to concentrate the Ez field on axis to make
an efficient acceleration?  Increase R/Q.
V2
R/Q 
2 U
For higher R/Q
 Smaller beam aperture  smaller cell-to-cell coupling.
 Nose cone  same as above  need other coupling mechanism
ILC super-conducting cavity
 smooth, polish with liquid, high pressure rinse, etc.
 with circle-ellipsoid smooth connection,
nose cone is difficult
 less effort on higher R/Q, simply decreasing beam hole aperture
because storing large energy with longer period is possible
Choke mode cavity needs field at choke area to establish imaginary short
 sacrifice several % loss in R/Q
Shaped disk-loaded structure
 only change R/Q by beam hole aperture
T. Higo
LC School, Chicago, 2008, LINAC-I
46
Loss factor
Loss factor KL described later
kL 
R
4Q
The energy left after a bunch, with change
q, passes a cavity is
U m  k L, m q 2
Larger R/Q makes bigger energy left in the cavity.
It may cause various problems;
Phase rotation of accelerating mode
Transverse kick field
Heating beam pipe
In a ring application, such as storage ring and DR, sometimes
R/Q should be reduced.
In the linac application, it usually tuned to be maximized to get a
better acceleration efficiency.
T. Higo
LC School, Chicago, 2008, LINAC-I
47
Acceleration: Transit time factor
Assume TM010 mode in a pillbox of length L
E z ( z , t )  E0 e j  t
In  mode cavity
Maximum acceleration occurs if the electric field is
maximum when the beam passes the center of the cavity.
Then transit time factor becomes
In case of thin cavity, where L<< c / f,
Run 
T 
2
0
V
, V0  E0 L
P
Sin ( / 2) 2
  0.64
 /2

R  Run T 2  0.4 Run
The acceleration felt by the beam decays as time,
Ez ( z, t )  E0 Cos( t ),
1
L
2f
c
z ct
Voltage acquired by beam is then
V ( L)  
L/2
L / 2
( E0 Cos  t ) d (c t ) 
2 c E0

Sin (
L
2c
)
Transit time factor:
1.0
2c
L
Sin ( x)
L
T  V ( L) / V0 
Sin (
)
, where x 
L
2c
x
2c
0.8
x=
0.6
0.4
0.2
1
2
3
4
5
6
0.2
T. Higo
LC School, Chicago, 2008, LINAC-I
48
7
Surface loss and Q0
G
Q 
Rs
Super conductor, Nb case:
1 f 2  17T.67
RBCS ()  2 10
( ) e
T 1. 5
f (GHz ), T ( K )
4
at 1.3GHz, T=2K << 9K  BCS=11n
Normal conductor:
Equivalent surface current in thin skin
depth s with surface resistance Rs.
Rs depend on mostly choice of material.
Higher freq  larger BCS loss.
Possible to increase geometrical
factor, G by shaping. It reduces
cryogenic power consumption.
Rs 
m
2s
sCu=5.8X107(1/)  Rs~28m
Actually, Rs = RBCS + Rresidual
Need to keep smaller Rs by making
proper material surface.
Higher Rs makes larger pulse surface
heating during short pulse.
Suppressing multipacting and field
emission loading.
T. Higo
LC School, Chicago, 2008, LINAC-I
49
How to increase Q0
ILC SCC
 TESLA to LL shape
 expanding cell outer area  reduce H field  against quench
 eventually increase Q0 by larger G
Heavily damped cavity
 Loss in choke mode cavity,
to establish imaginary short by storing power at choke
Loss in heavily damped cavity with damping waveguide
 opening toward damping waveguide, magnetic field gets higher
For disk-loaded structure
 good to have higher Q0 to reduce wall power, higher transfer efficiency
 near round cell shape, make it close to sphere
T. Higo
LC School, Chicago, 2008, LINAC-I
50
Suppression of local field enhancement
Electric field; Ep / Eacc
Peak surface electric field
 Field emission source
 Breakdown in NCC
Magnetic field; Hp / Eacc
 Surface temperature rise within a pulse in NCC
 Quenching of superconductor above magnetic field threshold
How to decrease these ratios?
 Shaping global cell shape
 Make it smooth locally
Need care on SCC EWB welding quality
NCC remove burrs and sharp corners
EBD: Electron beam welding
T. Higo
LC School, Chicago, 2008, LINAC-I
51
Cares on local field enhancement
Care on the opening edge to
damping waveguide.
Care on EBW bead shape in SCC cavity.
Bump above
design contour
0.2mm
Smooth
opening to
reduce Hp
Smooth
shape to
suppress
Ep
Bump over EBW line
Enhancement at small edge on opening
Hmax/Hwall
10
w=5mm
w=3mm
w=2mm
w=1mm
w=0.3
1
1
10
100
r [mm]
1000
2D calculation of cylinder with radial opening channel
T. Higo
Hs
enhancement
over a small
sphere sitting
on a spherical
cavity.
Height / radius < 0.001
for Hs/Hs<a few %
LC School, Chicago, 2008, LINAC-I
52
Shaping of accelerator cell profile examples
Damped cavity for storage ring
With nose cone.
SCC
Typical cavity
SCC
Reentrant cavity
SW TM010
SCC  mode
Smooth
More Ez on axis
Less Hs at outer
Single cell
Damped cavity
TW DLS
TM010
5/6-mode
(figure shows  field)
HDDS for GLC/NLC
T. Higo
LC School, Chicago, 2008, LINAC-I
DAW cavity
TM020-like /2
Floating washer
Coupling mode in
addition to
accelerating mode.
High Q, high R
53
Coupled cavity system in
a SW regime
T. Higo
LC School, Chicago, 2008, LINAC-I
54
Cell-to-cell coupling
Ep/Eacc increases, but easy to
confine field to increase T if
normalized in the field-existing area.
Coupled cavity needs coupling
between cells through some
mechanism other than beam aperture.
Weakly coupled-cell through beam aperture
SCC cavity for ILC
Transit time factor cannot be improved,
similar to that of pillbox or less.
T. Higo
LC School, Chicago, 2008, LINAC-I
55
Weakly coupled resonators
• Each resonator has
–
–
–
–
Internal freedom
Eigen modes in the cavity in an almost closed surface
Excited resonant modes couple to beam
Acceleration, deceleration, transverse kick, etc.
• Total system
– Weak coupling usually to adjacent cavity through some apertures
– Total system is described as coupled resonator system
• Mathematically equivalent to
– Mechanically coupled oscillator model
– Electrically coupled resonant circuit model
T. Higo
LC School, Chicago, 2008, LINAC-I
56
Coupled resonator model to
describe the total system
Assume each cavity is represented by
a resonant circuit. (described later)
I 0  X 0 (1 
0
2
 02 )  k X 1
j Q 
0
02 k
I n  X n (1 
 )  ( X n 1  X n 1 )
j Q 2 2
0
02
I N  X N (1 
 )  k X N 1
j Q 2
where 02  2 L C
Xn 
/9-mode
2 L in
Q  2 0 L / R
0.5
2
If Q>>1,
X nq  const Cos(
with
T. Higo
q2 
-mode
 qn
N

)e
j q t
4
6
8
Dispersion
1.10
0.5
1.05
1.0
1.00
2
0
1  k Cos(
q
N
0.95
)
LC School, Chicago, 2008, LINAC-I
0.90
0
2
4
6
8
57
10
Perturbation analysis
 X0 
X 
X  1 ,
 
 
X N 
k
0
0 
 1
k / 2 1
k/2 0
0 



k /2 1
k /2 0
M 





 
 
 0
0
k / 2 1 k / 2


0
0
k
1 
 0
Equation:
0 2
) X 0

Frequency errors
M X (
02n'  02  02n
Solution:
1


 Cos(q / N ) 




X q 

Cos
(

qn
/
N
)






 Cosq 
2
 00
0
 ( 2 )

q


012
2

0
( 2 )
M X q  02 X q  
q

q

 

0
 0

First order equation
T. Higo




0
0  q
X


0 
02N 
0  ( 2 )
q 
0
LC School, Chicago, 2008, LINAC-I
0
58
Perturbation analysis (cont.)
For  mode
N
X n / X n   e~p (1  k ) Cos


P 1
pn
p
/ k (1  Cos )
N
N
For /2 mode
 /2
 /2
X n / X n
1 N ~
pn
n
p
  e p ( Sin
Sin / Sin )
k P 1
N
2
N
~
pn
Where e p is Fourier component of frequency perturbation of Cos
N
Both perturbation scales as 1/k.
As for number of cells,  mode scales as N^2, while  /2 mode linearly on N.
For longer structure,  mode becomes difficult.
This is related to no energy exchange in  mode because of zero group velocity.
For energy transfer, we need other mode than  mode, which destroys  mode
itself.
T. Higo
LC School, Chicago, 2008, LINAC-I
59
-mode and /2 mode
0
Most basic but no net acceleration
/2
/3
Stable cavity system but half acceleratin
Good compromize in TW linac
Most efficient but weak against
perturbation
/2
/2
T. Higo
Electrically /2 but acceleration
efficiency ~ -mode
Actual -mode acceleration with /2
coupling element outside accelerating cell.
LC School, Chicago, 2008, LINAC-I
60
SCC 9-cell cavity example
•
•
•
•
•
•
Cell to cell coupling k~2%
Dispersion curve f0/(1+ k Cosf)1/2
Band width BW ~ k f0
Small mode separation f - f8/9 ~ 0.06*BW
Tuning of SCC 9-cell cavity see next page
Shunt Impedance of Total system
– Rtotal = Rsingle X 9 if flat field and right frequency
T. Higo
LC School, Chicago, 2008, LINAC-I
61
Practical issues in SCC being analyzed
with coupled resonator model
• SCC 9-cell cavity is basically expressed as
– a single chain of coupled resonators.
• Field flatness consideration and tuning
– Frequency of cells
– Lorentz force detuning
– EP deformation of cell shape
• Coupling between cells
– It makes coupling coefficient between resonators.
– It represent robustness of field flatness against perturbations
– It gives spacing to the nearest resonance, 8/9 mode.
• Some other system such as super-structure
– Also can be described by a weekly coupled two 9-cell systems.
T. Higo
LC School, Chicago, 2008, LINAC-I
62
Frequency error and field distribution
M 0  x0  l0 x0
( M 0   )  ( x0  x)  (l0  l) ( x0  x)
   x0  M 0  x  l  x0  l0  x
because l / l0  x / x0
then   x0  (l0  M 0 )  x
we know design values : x0 , l0 , M 0 and measured x
so that we get   2  ( / 0 )
T. Higo
LC School, Chicago, 2008, LINAC-I
63
Frequency error estimation from measured field
delf MHz
Field relative amplitude
1.02
0.04
1.01
0.02
1
0
-0.02
0.99
-0.04
0.98
2
3
4
5
6
7
8
9
2
Frequency error
3
4
5
6
7
8
Tuning:
Mechanical
deformation
9
Field uniformity
Design field & Error field X 10
Freq. error estimated
0.4
0.04
0.2
0.02
0
0
-0.2
-0.4
2
4
6
8
10
Field measurement
Deviation from  mode
T. Higo
Eq.
circuit
-0.02
-0.04
2
LC School, Chicago, 2008, LINAC-I
4
6
8
10
64
Pillbox cavity as a simple
base to represent
practical cavities
T. Higo
LC School, Chicago, 2008, LINAC-I
65
SW Cavity example: pillboxd
In a cylindrical waveguide, two propagation
modes exist;
a
ei ( t   ) z and ei ( t   ) z
Forward wave and Backward wave
For satisfying the
boundary condition at
both end plates, the
solution with the
superposition of
these two counterpropagating modes in
proper phase and
amplitude becomes
SW in a pillbox cavity.
T. Higo
Er  
E 
z
Cos( m )
Kc
m z
K c2
Sin ( m )
Cos( m )
Ez 
Hr   j
J m' ( K c r )
Sin (  z z )
1
J m ( K c r ) Sin (  z z )
r
J m (Kc r)
Cos(  z z )
me
1
Sin
(
m

)
J m ( K c r ) Cos(  z z )
K c2
r
H   j
e
Kc
Cos( m )
J m' ( K c r )
Cos(  z z )
Hz 0
where K c   mn / a,  z  l  / d
LC School, Chicago, 2008, LINAC-I
66
Bessel’s functional form representing
pillbox field to satisfy boundary condition
1.0
1.0
0.8
0.8
0.6
0.6
1.0
0.4
0.4
0.2
0.2
0.5
0.0
0.0
0.0
0.5
1.0
1.5
TM01
Ez and Hr
2.0
0
2
3
0.0
0.5
0
Acceleration
Max at center and 0 at r=a
T. Higo
1
1
2
3
4
5
TM02
Ez and Hr
Acceleration
More energy storage for
a given acceleration
LC School, Chicago, 2008, LINAC-I
TM11
Ez and Hr
Transverse kick
Two polarizations
Zero at center and linearly
increase as r increases.
Big kick at center H field.
67
Mode frequency in a pillbox cavity
Modes are classified as TM and TE mode.
Frequencies are determined to satisfy
boundary condition at two end surface,
mnl
 mn
l 2
(
) (
) (
)
c
a
d
2
100
80
2
where mnl is Bessel’s functions zero for
TM or derivative of Bessel’s function
becomes zero for TE.
Modal density increases as higher
frequency region.
60
40
TM mode frequencies.
20
0
0
10
20
30
40
Accelerating mode is usually TM010 mode.
If we want more stored energy with the
same acceleration, TM011, TM020 or
others can be used.
T. Higo
LC School, Chicago, 2008, LINAC-I
68
Q value of modes in pillbox cavity
U
m
2

H
2
dV 
e
2

2
E dV ,
P
Rs
2

H
2
dS ,
Q
U
P
For TM modes;
Volume integral;
H
2
dV 

 ( H  H ) r dr d dz
A d  [ H ( z  0)  H  ( z  0) ] r dr d
2
r
2
2
r
2
where A  1 for   0 and A  1 / 2 for   0
for m  0
H
2
dV  A  d (
 A d (
e
)2  [ (
Kc
e
Kc
 A d (
m 2 2
) J m ( K c r )  J m' 2 ( K c r ) ] r dr
Kc r
) 2 I (r  a)
e
Kc
)2
a 2 '2
J m (  mn )
2
for m  0 similar result
I  [ (
m 2 2
r2
2
m 2
) J m ( K c r )  J m' 2 ( K c r ) ] r dr 
[ J m' 2 ( K c r ) 
J m' ( K c r ) J m ( K c r )  {1  (
) } J m2 ( K c r )]
Kc r
2
Kc r
Kc r
T. Higo
LC School, Chicago, 2008, LINAC-I
69
Q (cont.)
Finally we get,
H
2
e
a 2 '2
dV  A B  d (
)
J m (  mn )
Kc
2
2
where B  2 for m  0 and B  1 for m  0
Surface integral;

H
2
dS  2
[ H
2
r
( z  0)  H 2 ( z  0) ] r dr d   [ H 2 ( r  a ) ] a d dz
e
a 2 '2
e 2
 2 B (
)
J m (  nm )  A B  d (
) a J m' 2 (  nm )
Kc
2
Kc
 B (
e
Kc
2
) 2 a ( a  Ad ) J m' 2 (  nm )
Therefore, for TM modes,
Q
1
Where s is skin depth,
a
s 1 a
s 
Ad
2
sm
Similarly we obtain for TE modes;
Q
T. Higo
1
 s  '4
mn
a 2
'2
) (  ) 2 ] (  mn
 m2 )
d
a 3
a
2a
'
 2 ( ) (   mn
) 2  ( ) 2 (1 
) ( m) 2
d
d
d
'2
a [  mn
(
LC School, Chicago, 2008, LINAC-I
70
Acceleration in a single pillbox cavity
1 d / 2c
Re [ E z ( z , t ) ] dt
d  d / 2 c
1 d / 2c
 
Re [ E z ( z ) beam e j ( t f ) ] dt
d d / 2c
1 d /2
 
Re [ E z ( z ) { Cos(  z z  f )  j Sin (  z z  f )} ] dz
d d / 2
( for synchronou s beam  t   z z  0)
Eacc 

1
d

d /2
d / 2
E z ( z ) Cos(  z z  f ) dz
For an even function, such as the case of TM010, the f=0 to maximize
acceleration. This is the case of on-crest acceleration.
T. Higo
LC School, Chicago, 2008, LINAC-I
71
R/Q value of modes in pillbox cavity

Normalization:
2
E dV  1   U 
e
2
Definition of field integral:
Eacc 
E0 
1
d
1
d

d /2
d / 2

d /2
d / 2
E ( z , t ) Cos(t ) dz
E ( z , t ) dz
Transit time:
T  Eacc / E0
Impedance:
2
Eacc
2 d Z0
R 1 ( Eacc d ) 2
2



Eacc
Q d 2 U
2  (U / d ) 2  / c
T. Higo
LC School, Chicago, 2008, LINAC-I
72
R/Q for TM0nl field
In a previously obtained field expression of pillbox mode,

2
E dV  Anl
An 0   d a 2 J12 (  n 0 ) for l  0
2
Anl   d a 2 J12 (  nl )
2
1
l a 2
{1  (
) } for l  0
2
d  nl
Then, to make the normalization,

2
E dV  1
1
Ez ( z) 
Cos( m ) J m ( K c r ) Cos(  z z )
Anl
For TM010,
1
E010, z ( z ) 
J 0 ( r  0)
A10
Finally,
J 0 (r  0) d / 2

Eacc 
T
A10 d

d / 2
Cos(
c
z  fs ) dz 
J0 2 c
d
Sin (
) Cosfs
A10 d 
2c
2c
d
Sin (
) Cosfs
d
2c
2 c J0 2
R 2 d Z0
d

(
) Sin 2 (
) Cos 2fs
Q
 / c A10 d 
2c
T. Higo
LC School, Chicago, 2008, LINAC-I
73
For TM0nl mode
For TM0nl, with
Eacc 
0
J0
Anl d

d /2
d / 2
Cos(


z ) Cos( z  fs ) dz
d
c
 d
  d
)
Sin ( 
)
J 0 Cosfs
c
d 2 
c
d 2]

[
 
 
Anl d


c
d
c
d
2  J 0 Cosfs
1

B
 2
 2 l
Anl d c
( ) (
)
c
d

d
where Bl  Cos(  ) Sin (
) for   even
2
2c

d
 Sin (  ) Cos(
) for   odd
2
2c
Sin (
T. Higo


LC School, Chicago, 2008, LINAC-I
74
Circuit modeling of cavity
T. Higo
LC School, Chicago, 2008, LINAC-I
75
Cavity to equivalent circuit
Ez
Ez
Hf
Hf
Current flow
Power loss into wall
Resistive loss
Magnetic field
Inductive energy
storage
This system can intuitively be
expressed with series resonant
circuit in electric circuit
PS
T. Higo
Electric field
Capacitive energy
storage
The differential equiation is mathematically
equivalent and the system can also be
presented by parallel resonant circuit.
PS
LC School, Chicago, 2008, LINAC-I
76
Circuit model of cavity
From P. Wilson Lecture Note.
RF power source, transmission line and cavity can be
described as an equivalent circuitry.
T. Higo
LC School, Chicago, 2008, LINAC-I
77
Cavity response and beam
y: Cavity tuning angle
tan y   2 QL 
y
vb
y
vc
vbr
y
vb
  (  0 ) / 0
vg
If frequency changes to the
order of Q band width, y
becomes as big as 45deg!
vg/vgr = 0.7 (reduced much)
vgr
f 
va
Phasor diagram
-ib
To save power, need smaller
 <<1/QL
Need vgr to keep f>0.
T. Higo
LC School, Chicago, 2008, LINAC-I
78
Transient field around cavity
Emitted field from cavity stored field
Eout  Ee   Ein
dU c
Pin  Pout  Pc 
dt
  1
tc 
tc
T. Higo
2QL

c 

Pout
Pc
2Qc
(1   c ) 
dEe
2 c
 Ee 
Ein
dt
1 c
Ee 
2 c
Ein (1  e t / tc )
1 c
Emitted (out-going, reflected) field
from cavity
Eout  Ee  Ein 
2 c
Ein (1  e t / tc )  Ein
1 c
Cavity filled voltage
2 c
R
V (t )  (1  e t / tc ) ( )  tc Pin
Q
1 c
LC School, Chicago, 2008, LINAC-I
79
Cavity transient response
Reflected field
2.0
1.4
Input power
1.2
1.5
1.0
1.0
0.8
0.5
0.6
0.4
0.0
0.2
0.5
1
1
0
1
2
3
4
5
2
3
4
5
1.0
Cavity field
Reflected power
2.0
0.0012
c=5
c=2
0.0010
0.0008
c=1
0.0006
c=0.5
0.0004
1.5
1.0
0.5
0.0002
0.0
0
0.0000
0
T. Higo
1
2
3
4
1
2
3
4
5
5
LC School, Chicago, 2008, LINAC-I
80
SW cavity beam loading
From energy conservation,
Pin  Pout  Pc 
dU c
 I b Va
dt
With proper timing t=tr, we can make
dU c
 I b Va  0
dt
Then, we can make the beam loading compensation to keep
the voltage gain the same within the bunch train.
This is realized in a proper timing, feeding power and beam
current to adjust to the transient behavior of the cavity.
T. Higo
LC School, Chicago, 2008, LINAC-I
81
SW versus TW
T. Higo
LC School, Chicago, 2008, LINAC-I
82
SW and TW
SW : e j  t Sin (kz) , e j  t Cos(kz)
 H 
Superposition of Cos + j Sin
Example pillbox TM010 mode
Ez and Hf is 90 degrees out of phase
Hf
D
t
Ez
Forward or backward wave F+B or F-B
Er and Hf in phase to make Poynting vector
TW : e j (t  kz ) , e j (t  kz )
T. Higo
LC School, Chicago, 2008, LINAC-I
83
SW field  TW real field
Short b.c.
Cosine
E
T. Higo
TW
Open b.c.
Sine
E
SW
Short
 jE
SW
Open
LC School, Chicago, 2008, LINAC-I
84
SW   TW
• R/QTW = R/QSW X 2
– The space harmonics of SW propagating against beam cannot contribute to
the net acceleration, while the reverse-direction power is needed to
establish the SW field.
• SCC cavity operated in TW mode
– Resonant ring like: return the outgoing power from cavity to feed again into
the upstream of the cavity. But need sophisticated system outside.
• Many NCC at high gradient use TW
– High field need high impedance.
– Power not used for beam acceleration nor wall loss need be absorbed by
outside RF load.
• High gradient in SW cavity
– Probably resistive against damage from arcing due to easy detuning.
– Stability requires short cavity need more feeding points
T. Higo
LC School, Chicago, 2008, LINAC-I
85
TW linac
T. Higo
LC School, Chicago, 2008, LINAC-I
86
TW basic idea different than standing wave
• TW
– Travelling wave = microwave power flow in one direction
– Beam to be coupled to the field associated to this power flow
• Power flow
– Acceleration field decreases along the structure
• Due to wall loss and energy transfer to beam
– Attenuation parameter is a key parameter
– Extracted power is absorbed by load or re-inserted into the structure
• Shaping of attenuation along a structure
– Accelerating mode  CZ, CG
– Higher mode consideration  Detuned, …
• Synchronization to beam
– Control of frequencies of cells to make the phase velocity right
• Input matching with mode conversion
– Matching to TM01-type mode in the periodic chain of cavities
– From TE10 in waveguide
T. Higo
LC School, Chicago, 2008, LINAC-I
87
Acceleration field in TW linac
E z (r , , z, t ) 


n 
 E p (r , , z ) e
j t   z
5
4
n  
n 
3
 E (r , ) e
n  
j (  t  n z )
n
 z
e
2
where
n  0  2 n / d
j ( 2 n / d ) z
1 zd
En ( r ,  )  
Ed ( r ,  , z ) e
d z
E z (r ,  , z, t ) 
m=0
1
m=-1
6
n 
a
n  
n
4
2/d
J 0 (k rn r ) e
2
/d
0
0
2
/d
m=1

4
2/d
j (  t  n z )
where  n   0  2  n / d and k rn2  k 2   n2
T. Higo
LC School, Chicago, 2008, LINAC-I
88
6
Structure parameters
in a uniform structure (CZ)
SW cavity

TW linac
Acceleration voltage in a cavity  Acceleration gradient
Stored energy in a cavity  Stored energy in a unit length
Power loss in a cavity  Power attenuation in a unit length
Ea2
r
dP / dz
Q
u
Ea2
r /Q 
u
dP / dz
Group velocity and attenuation parameter
P  v g  u with v g  d / d
dP
  2 P
dz
dEa
   Ea
dz
E a  E 0 e  z

Pa  P0 e  2 z

2 vg Q
At the end of a structure, of length L
EL  E0 e  L
T. Higo
PL  P0 e  2 L    L 
LC School, Chicago, 2008, LINAC-I
L
2 vg Q
89
1.1
DLS mode
Frequency
1.05
Cal. With SW field patterns
1
0.95
0.9
0
30
60
90
120
150
180
Phase advance
0 mode
T. Higo
/2 mode
2/3 mode
LC School, Chicago, 2008, LINAC-I
5/6 mode
 mode
90
Shunt impedance vs phase advance
Actual acceleration field on axis can be
decomposed into space harmonics.
n=number of disks per
wavelength.
Net acceleration for long distance is
realized by only synchronous one.
Thinner disk makes larger
Gross peak at 2/3 mode
T. Higo
LC School, Chicago, 2008, LINAC-I
91
DLS basic design with changing (a,t)
Simple geometry for DLS design
d
“b” to tune frequency
t
a
b
t
a
“Q” varies as (a,b,t) changes
T. Higo
“vg” varies as (a,b,t) changes
LC School, Chicago, 2008, LINAC-I
92
DLS Parameter variation with (a,t)
“r” varies as (a,t) changes
“Ep/Eacc” varies as (a,t) changes
“r/Q” varies as (a,t) changes
T. Higo
LC School, Chicago, 2008, LINAC-I
Hp/Eacc
93
CG: Constant gradient structure
c
vgvgvs
a
12
Assume CG case is realized.
By neglecting weak variation of r,
10
8
6
dP / dz  const
4
P( z )

 1  ( z / L) (1  e 2 )
P0
2
3.0
3.5
4.0
4.5
r M Ohm m
Group velocity should be varied linearly as
5.0
5.5
6.0
10
r vs a
12
10
vg ( z ) 
L 1  ( z / L) (1  e 2 )
1  e 2
Q
Filling time becomes
8
6
4
2
Tf 

L
0
dz
2Q

vg ( z )

3.0
4.0
4.5
5.0
5.5
6.0
2/3 mode in X-band
disk-loaded structure
This is the same as CZ case
T. Higo
3.5
LC School, Chicago, 2008, LINAC-I
94
ENL and ELD actual and CG characteristics
GLC structure
H60VG4S17
Basically changing beam
aperture “a” to make it roughly
CG
8
a, t [mm], vg/c [%]
7
6
a [mm]
5
Roughly constant gradient by
linearly tapered vg
4
t [mm]
3
2
Reduce field near input coupler
with initial taper
vg/c [%]
1
0
0
10
20
30
40
50
Cell number
T. Higo
LC School, Chicago, 2008, LINAC-I
95
Acceleration with/without beam
Field with beam is
superposition of externally-driven field + beam-induced field
Steady state case
Acceleration along a structure without beam
CZ : V0  (r L P0 )1/ 2 [( 2 /  )1/ 2 (1  e  )]
CG : V0  (r L P0 )1/ 2 (1  e 2 )1/ 2
Beam induced field (beam loading) in an empty structure
dP
 I 0 Eb  2  P
dz
Since
Eb : Beam induced field
I0 : DC current of beam
P : Power flow of beam induced field
Eb2  2  r P , then
dEb
E d
 I 0  r   Eb  b
dz
2  dz
This becomes simple when we consider CZ or CG structure,
dEb
 I 0  r   Eb
dz
dEb
CG case :
 I0  r
dz
CZ case :
T. Higo
LC School, Chicago, 2008, LINAC-I
96
Beam loading voltage
Beam loaded field along a structure is described as,
CZ : Eb ( z )  I 0 r (1  e  z )
Beam loading in CZ
I0 r
1  e  2
CG : Eb ( z )  
ln (1 
z)
2
L
40
30
20
By integrating along a structure
(1  e  )
CZ : Vb  I 0 r L [1 
]
10
0.2

1  e  2
CG : Vb  I 0 r L [ 
]
2 1  e 2
60
50
0.4
0.6
0.8
z
1.0
Beam loading in
CG
40
The net acceleration voltage then becomes,
CZ , CG : VLD V0 Cos  Vb
30
20
10
0.2
Where  is the off crest angle w.r.t. RF on-crest phase.
T. Higo
LC School, Chicago, 2008, LINAC-I
0.4
0.6
0.8
1.0
z
97
Beam loaded field with beam
80
E0 (0A)
Estimation with CG case
60
ELD(0.8A)
40
ELD(1.6A)
20
ELD(2.4A)
0.0
0.1
0.2
0.3
GLC structure
0.4
0.5
0.6
H60VG4S17
100
Acceleration field [MV/m]
r=60 M/m
=0.6
L=0.6 m
E0=65 MV/m
I=0 - 0.8 - 1.6 - 2.4 A
80
Recursive calculation at
Ib=0.9A with actual
parameters (a,b,t)  r, Q
along the structure  Field
ENL
60
ELD
40
Beam loading voltage build
up toward downstream end
20
0
0
T. Higo
10
20
30
Cell number
40
50
LC School, Chicago, 2008, LINAC-I
98
Full loading in CTF3
In a steady state
regime, beam can
fully absorb stored
energy of the
structure. This is
fully loaded
condition.
P. Urschütz et al., “Efficient Long-Pulse Fullｙ
Loaded CTF3 Linac Operation”, LINAC06.
T. Higo
LC School, Chicago, 2008, LINAC-I
99
Transient property of travelling
wave propagation
•
Wave propagation suffers
from the dispersive effect
of the periodic structure.
t
E0 (t )  Re  e j  t h(t )
0
t
Eq (t )  Re  e j  ( t  ) Gq ( ) d
0
 Z q (t ) Cos( t  q   )
•
•
/2 mode  f=0, only
amplitude modulation
Accelerated particle sees
as function of time.
Rising front of pulse
transmission at a certain
position q down the
structure. (Linear Accelerators)
Solid line: Field amplitude
along the structure.
Dotted line: Field a
particle sees. (Linear
Accelerators)
The same property is calculated straightforwardly based on the equivalent circuit
model by T. Shintake. (Frontiers in Accelerator Technology, 1996)
T. Higo
LC School, Chicago, 2008, LINAC-Is
100
TW accelerator structure in practice
• Design
• Cell fabrication
– Frequency control
• Bonding
• Tuning
– Matching
– Phase tuning
– Minimize small reflection
– HOM
T. Higo
LC School, Chicago, 2008, LINAC-I
101
Typical TW structure
made of stacked disks
T. Higo
LC School, Chicago, 2008, LINAC-I
102
Small reflection and tuning
Express wave propagation
Vk  1e  k ( j  )  R e 2 m ( j  ) e k ( j  ) for k  m
T e k ( j  )
Frequency
perturbation
Forward
Transmit
for k  m
From continuity at m-th cell,
m
T  1 R
Reflect
From coupled resonator model,
(m2  m2   2  j
m
1
) Vm  m2 (k m 1/ 2 Vm 1  k m 1/ 2 Vm 1 )
Qm
2
Explicitly written as,
(m2  m2   2  j
m
1
) (T e  m ( j    ) )  m2 k m 1/ 2 (e ( m 1)( j    )  e  2 m ( j    ) R e ( m1)( j    ) )
Qm
2
1
 m2 k m 1/ 2 T e ( m1)( j    )
2
Tuning condition,
m2  m2   2  j
T. Higo
m
Qm
1
 m2 (k m 1/ 2 e ( m 1)( j    )  k m 1/ 2 e ( m 1)( j    ) )
2
LC School, Chicago, 2008, LINAC-I
103
Reflection (cont.)
Finally reflection from cell m becomes,
R
m2 / m2
m2 / m2  km1/ 2 (Cos  j Sin )
Assume; loss-less TW with 2/3 mode, k~0.02, 2/2 ~ 10-4
m2 / m2
10 4
R

10 2
km1/ 2 Sin 0.02 ( 3 / 2)
If coherent error from 10 cells, those cohere so that R becomes as much as 0.1.
Such systematic error is not allowed.
In contrast, random error makes much smaller and such amount of error is
allowed.
T. Higo
LC School, Chicago, 2008, LINAC-I
104
Phase error
Dispersion relation;

0
1  k Cos(f )
Group velocity formula;

1
 vg   0 k Sin (f )
f
2
Frequency error to phase advance error
  f d 

vg
d
d /c
l ( / 2 ) / c
 
 
 
vg / c
vg / c
vg / c 
At X-band, 1MHz gives phase advance error of 0.6degree/cell.
In 20 cell structure such as CLIC,
Systematic error of this order is not good
but random error should be OK, as long as acceleration mode.
T. Higo
LC School, Chicago, 2008, LINAC-I
105
Summary of LINAC-I in comparison
of SCC and NCC
• How the linac design can be determined from
gaining energy point of view?
• Energy mode
– SW or TW
• Confinement mechanism
– SCC or NCC
• Frequency choice
– 1GHz-10cm or 10GHz-1cm?
T. Higo
LC School, Chicago, 2008, LINAC-I
106
Choice of material for EM field
confinement
• SCC
• Nb
•  Geometrical factor
• Intrinsic limit Hs
• Mechanical strength
• Quench
• Cryogenic power
T. Higo
• NCC
• Cu
• R Shunt impedance
• Thermal
• Breakdown
• RF generation
LC School, Chicago, 2008, LINAC-I
107
Choice of frequency
• 1GHz
• 10cm
• Drawing or
hydroforming
• Longer pulse ~1ms
• Longer power
T. Higo
• 10GHz
• 1cm
• High precision turning /
milling
• Shorter pulse ~100ns
• High peak power
LC School, Chicago, 2008, LINAC-I
108
Choice of EM mode & Efficiency
• SW
• SCC
• Power loss
– Cavity wall  cryogenic
– Reflection from cavity
T. Higo
• TW
• NCC
• Power loss
– Cavity wall
– Transmitted to RF load
LC School, Chicago, 2008, LINAC-I
109
Potential for higher energy
• Es/Ea
• Hs/Ea
• Multi pactor
• Field emission
• Lorentz detuning
• Quench
• Q0  cryogenic power
T. Higo
• Es/Ea
• Hs/Ea
• Surface temperature rise
in a pulse
• Fatigue
• Breakdown
• R/Q * Q  efficiency
LC School, Chicago, 2008, LINAC-I
110