Transcript Lecture 4 - Bunch compressors - International Linear Collider

Bunch compressors
ILC Accelerator School
May 20 2006
Eun-San Kim
Kyungpook National University
Korea
1
Locations of bunch compressors in ILC
 BCs locates between e- (e+) damping rings and main linacs,
and
make bunch length reduce from 6 mm rms to 0.15 mm rms.
1st stage ILC : 500 GeV
2nd stage ILC : 1 TeV
- extension of main linac
- moving of SR and BC
2
Why we need bunch compressors
 Beams in damping rings has bunch length of 6 mm rms.
- Such beams with long bunch length tend to reduce effects of
beam instabilities in damping rings.
- Thus, beams are compressed after the damping rings.
 Main linac and IP in ILC require very short beams:
- to prevent large energy spread in the linac due to the curvature of the rf.
- to reduce the disruption parameter ( ~ sz) :
(ratio of bunch length to strength of mutual focusing between colliding beams)
 Thus, bunches between DRs and main linacs are shortened.
- Required bunch length in ILC is 0.15 mm rms.
3
Main issues in bunch compressors
 How can we produce such a beam with short bunch
length?
 How can we keep low emittance (ex/ey= 8mm / 20nm)
and high charge (~3.2 nC) of the e- and e+ beams in
bunch compression?
 How large is the effects of incoherent and coherent
synchrotron radiation in bunch compression?
4
How to do bunch compression
 Beam compression can be achieved:
(1) by introducing an energy-position correlation along the bunch with
an RF section at zero-crossing of voltage
(2) and passing beam through a region where path length is energy dependent
: this is generated by bending magnets to create dispersive regions.
DE/E
-z
Tail
lower energy trajectory
(advance)
Head (delay)
center energy trajectory
higher energy trajectory
 To compress a bunch longitudinally, trajectory in dispersive region must be
shorter for tail of the bunch than it is for the head.
5
Consideration factors
in bunch compressor design
 The compressor must reduce bunch from damping
ring to appropriate size with acceptable emittance
growth.
 The system may perform a 90 degree longitudinal
phase space rotation so that damping ring extracted
phase errors do not translate into linac phase errors
which can produce large final beam energy deviations.
 The system should include tuning elements for
corrections.
 The compressor should be as short and error tolerant
as possible.
6
Beam parameters in
bunch compressors for ILC
 beam energy
: 5 GeV
 rms initial horizontal emittance : 8 mm
 rms initial vertical emittance
: 20 nm
 rms initial bunch length
 rms final bunch length
 compression ratio
: 6 mm
: 0.15 mm
: 40
 rms initial energy spread
 charge / bunch
: 0.15 %
: 3.2 nC (N=2x1010)
7
Different types of bunch compressor
Chicane
Double chicane
Chicanes as a Wiggler
Arc as a FODO-compressor
8
Different types of bunch compressor
 Chicane : Simplest type with a 4-bending magnets for bunch
compression.
 Double chicane : Second chicane is weaker to compress
higher charge density in order to minimize emittance growth due to
synchrotron radiation.
 Wiggler type : This type can be used when a large R56 is required,
as in linear collider. It is also possible to locate quadrupole
magnets between dipoles where dispersion passes through zero,
allowing continuous focusing across the long systems.
 Arc type : R56 can be adjusted by varying betatron phase advance
per cell. The systems introduce large beamline geometry and
need many well aligned components.
9
Path length in chicane
 A path length difference for particles with a
relative energy deviation d is given by:
Dz = hd = R56d + T566 d2 + U5666 d3 ……
h
: longitudinal dispersion
d
: relative energy deviation (= DE/E)
R56 : linear longitudinal dispersion
(leading term for bunch compression)
T566 : second - order longitudinal dispersion
U5666 : third - order longitudinal dispersion
10
Longitudinal particle motion
in bunch compressor
 Longitudinal coordinates
z : longitudinal position of a particle with respect to bunch center
Positive z means that particle is ahead of reference particle (z=0).
d : relative energy deviation
When a beam passes through a RF cavity on the zero crossing
of the voltage (i.e. without acceleration, frf =  /2 )
z1 = z0
d1 = d 0 +
eVRF


cos  k RF z0 
E0
2

krf = 2 frf/c
11
Longitudinal particle motion
in bunch compressor
 When reference particle crosses at some frf,
reference energy of the beam is changed from Eo to E1.
Initial (Ei) and final (Ef) energies of a given particle are
Ei = Eo (1 + d 0 )
E f = E1 (1 + d1 ) = Ei + eVrf cos(frf  k rf zo )
E1 = Eo + eVrf cos(frf )
Then,
d1 =
Eo (1 + d 0 ) + eVrf cos(f rf  krf zo )
E0 + eVrf cos(frf )
1
12
Longitudinal particle motion
in bunch compressor
To first order in eVrf/Eo << 1,
z1 = z0
 eVrf cos(f rf )  eVrf
 +
d1 = d 0  1 
cos(f rf  k rf zo )  cos(f rf )
E0

 E0


In a linear approximation for RF,
 z1   1

d 


 1   R65
0   z0 





R66   d 0 

R65 =
eVrf
E0
R66 = 1 
 
sin frf krf
eVrf
E0
 
cos frf
13
Longitudinal particle motion
in bunch compressor
In a wiggler (or chicane),
z2 = z1 + R56d1 + T566d12 + U 5666d13 
d 2 = d1
In a linear approximation
R56 >> T566 d1,
 z 2   1 R56   z1 
   
   
 d 2   0 1   d1 
Total transformation
z 
 z2 
   M   0 
d2 
d0 
1 + R65 R56
M = 
 R65
R56 R66 

R66 
For frf =  /2, R66=1, the transformation matrix is sympletic,
which means that longitudinal emittance is a conserved quantitiy.
e = s2z sd2  s z2d , s2z = z 2  sd2 = d2  s zd = zd 
14
A simple case of
4-bending magnet chicane

Zeuthen Chicane : a benchmark layout used for CSR calculation
comparisons at 2002 ICFA beam dynamics workshop
B2
•
•
•
•
•
•
•
•
•
B1
qo
LB
DL
B3
B4
DLc
DL
LB
Bend magnet length
: LB = 0.5m
Drift length B1-B2 and B3-B4(projected) : DL = 5 m
Drift length B2-B3
: DLc = 1 m
Bend radius
: r = 10.3 m
Effective total chicane length
: (LT-DLc) = 12 m
Bending angle
: qo = 2.77 deg
Bunch charge
: q = 1nC
Momentum compaction : R56 = -25 mm
Electron energy : E = 5 GeV
2nd order momentum compaction : T566 = 38 mm Initial bunch length : 0.2 mm
Total projected length of chicane : LT = 13 m
Final bunch length : 0.02 mm15
Relations among R56, T566 and U5666 in Chicane
q
a
b
a
If a particle at reference energy is bent by qo, a particle with
relative energy error d is bent by q = qo / 1+d.
Path length from first to final bending magnets is
1
  q 
2a
 q 
s=
+ b = 2a cos  o   + b  2a + a o  + b
cos( q
 1+ d 
  1 + d 
R56
ds
=
 2aqo
dd d = 0
2
16
Relations among R56, T566 and U5666 in Chicane
Difference in path length due to relative energy error is
1 
1 
 qo 


Ds = s(d  s(d =   a
  aqo = R56 1 
2 
2  (1 + d) 
 1+ d 
2
By performing a Taylor expansion about d = 0
Ds  R56 d 
3
R56 d  + 2 R56 d  ......
2
T566  
3
R56
2
U 5666  2R56
For large d, d and d terms may cause non-linear deformations of the
phase space during compression.
17
Momentum compaction
 The momentum compaction R56 of a chicane made up of
rectangular bend magnets is negative (for bunch head at z<0).
 The required R56 is determined from the desired compression,
energy spread and rf phase.
First-order path length dependence is
dz
h
= R56 =  ds
dd
r
 From the conservation of longitudinal emittance,
final bunch length is
sdf szf = sdi szi
szf  R56sdi
18
RF phase angle
 Energy-position correlation from an rf section is
Vrf krf sin(frf )
dd
= R65 =
dz
Eo + Vrf cos(frf )
 In general case that beam passes through RF away zero-
crossing of voltage, that is R66 = 1, there is some damping
(or antidamping) of the longitudinal phase space,
associated with acceleration (or deceleration).
19
Synchrotron Radiation
 Incoherent synchrotron radiation (ISR) is the result of individual
electrons that randomly emit photons.
Radiation power P ~ N
(N : number of electrons in a bunch)
 Coherent synchrotron radiation (CSR) is produced when a group
of electrons collectively emit photons in phase. This can occur
when bunch length is shorter than radiation wavelength.
Radiation power P ~ N2
 ISR and CSR may increase beam emittance in bunch
compressors with shorter bunch length than the damping rings.
20
Coherent synchrotron radiation

Opposite to the well known collective effects where the
wake-fields produced by head particles act on the particles
behind, radiation fields generated at tail overtake the head of
the bunch when bunch moves along a curved trajectory.
lr
sz
Lo
Coherent radiation for lr > sz
R
q

R=Lo/q
Overtaking length : Lo  (24 sz R2)1/3
CSR longitudinal wake function is
Q
W =
//
e o (2)3 / 2 (3s 4 r2 )1/ 3
z
21
Coherent synchrotron radiation

CSR-induced relative energy spread per dipole for a Gaussian
bunch is
re NL
 DE 
 0.22 2 / 3 4 / 3


E
R s z

rm s

This is valid for a dipole magnet where radiation shielding of a
conducting vacuum chamber is not significant, that is, for a full
vacuum chamber height h which satisfies
h  (sz√R)2/3  hc.

Typically the value of h required to shield CSR effects (to cutoff low
frequency components of the radiated field) is too small to allow an
adequate beam aperture
(for R  2.5 m, h « 10 mm will shield a 190 mm bunch.)

With very small apertures, resistive wakefields can also generate
emittance dilution.
22
Incoherent Synchrotron Radiation
When an electron emits a photon of energy u, the change in the betatron action
2
is given by
 u 
 H
DJ = 
H=bxh'+axhh'+xh
 Eo 
 Transverse emittance growth is
De  = 23 Cq re 6 I5
H
I5 = 
r
3
ds
 Increase of energy spread is
Ds d = Cq re I3
2
5
4
3
I3 = 
1
r
ds
3
• The increase in energy spread is given by:

Beam energy loss is
U0 =
C
2
E04 I 2
I2 = 
1
r
2
ds
Cq=3.84x10-13m
23
Bunch compressors for ILC
 Two-stages of bunch compression were adopted to
achieve σz = 0.15 mm.
 Compared to single-stage BC, two-stage system
provides reduced emittance growth.
 The two-stage BC is used :
(1) to limit the maximum energy spread in the beam
(2) to get large transverse tolerances
(3) to reduce coherent synchrotron radiation
that is produced
24
Designed types of
bunch compressors for ILC
A wiggler type that has a wiggler section made
up of 12 periods each with 8 bending magnets
and 2 quadrupoles at each zero crossing of the
dispersion function : baseline design (SLAC)
A chicane type that produces necessary
momentum compaction with a chicane made of
4 bending magnets : alternative design (E.-S. Kim)
25
Baseline design for ILC BC
A wiggler based on a chicane between each pair of quadrupoles
Each chicane contains 8 bend magnets (12 chicanes total).
26
Baseline design for ILC BC
BC2 RF
BC1
RF
BC1
Wiggler
BC1
Wiggler
27
Baseline design for ILC BC
 First stage BC
- contains 24 9-cell RF cavities arranged in 3 cryomodules.
- Because the bunch is long, relatively strong focusing is used to
limit emittance growth from transverse wakefields.
 Second stage BC
- contains 456 9-cell RF cavities arranged in 57 cryomodules.
- A wiggler has optics identical to the wiggler in the first BC,
but with weaker wiggler.
28
Parameters of baseline design
Initial Energy Spread [%]
0.15
Initial Bunch Length [mm]
Initial Emittance [mm]
6.0
8 / 0.02
BC1 Voltage [MV]
253
BC1 Phase [°]
-100
BC1 R56 [mm]
-750
End BC1 Bunch Length [mm]
1.14
End BC1 Energy [GeV]
4.96
End BC1 Energy Spread [%]
0.82
BC2 Voltage [MV]
12,750
BC2 Phase [°]
-58
BC2 R56 [mm]
-41
End BC2 Bunch Length [mm]
End BC2 Emittance [mm]
0.15
8.2 / 0.02
End BC2 Energy [GeV]
11.7
End BC2 Energy Spread [%]
2.73
29
Alternative design for ILC BC
 Main linac
Matching Chicane 1
Quadrupoles Chicane 2
RF section
30
Parameters of alternative design
Initial Energy Spread [%]
0.15
Initial Bunch Length [mm]
Initial Emittance [mm]
6.0
8 / 0.02
BC1 Voltage [MV]
348
BC1 Phase [°]
-114
BC1 R56 [mm]
-474.2
End BC1 Bunch Length [mm]
1.1
End BC1 Energy [GeV]
4.86
End BC1 Energy Spread [%]
1.1
BC2 Voltage [MV]
11,800
BC2 Phase [°]
-45
BC2 R56 [mm]
-50.8
End BC2 Bunch Length [mm]
End BC2 Emittance [mm]
End BC2 Energy [GeV]
End BC2 Energy Spread [%]
0.15
8.3 / 0.02
13.26
2.2
31
Bunch compressors for ILC
Alternative
Chicane length
Baseline
68.4 m
480 m
Matching
4m
310 m
Number of RF cavity
452
488
Total length
680 m
1400 m
Alternative
Baseline
Required bunch length
achieved
achieved
System length
shorter
longer
Tolerence of emittance
acceptable
comparable
GDE
Requirement
correction of
vertical dispersion
acceptable
comparable
shorten
system length
32
Summary
 Compared to single-stage BC, two-stage BC system
provides reduced emittance growth at σz = 0.15 mm.
 Two stage system can be tuned to ease transverse
tolerances.
 Two stage system is longer than one-stage system.
– A shorter 2-stage may be also possible.
33
Problems
 Show that emittance growth and increase of
energy spread due to incoherent synchrotron
radiation are given by
1)
2)
De  = 23 Cq re 6 I5
Ds d2 = 43 Cq re 5 I3
I5 = 
I3 = 
H
r
3
ds
1
r
3
ds
34