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### Bunch compressors

ILC Accelerator School

May 20 2006

Eun-San Kim

Kyungpook National University

Korea

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

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.

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?

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 isenergy dependent

: this is generated by bending magnets to create dispersive regions.

DE/E

-z

Tail

(advance)

lower energy trajectory

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.

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.

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 : 6 mm
- rms final bunch length : 0.15 mm
- compression ratio : 40
- rms initial energy spread : 0.15 %
- charge / bunch : 3.2 nC (N=2x1010)

Different types of bunch compressor

Chicane

Double chicane

Chicanes as a Wiggler

Arc as a FODO-compressor

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.

Path length in chicane

- A path length difference for particles with a relative energy deviationd is given by:
- Dz = hd = R56d + T566 d2 + U5666d3……
- 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

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= p/2)

krf = 2p frf/c

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

Then,

Longitudinal particle motion in bunch compressor

To first order in eVrf/Eo << 1,

In a linear approximation for RF,

Longitudinal particle motion in bunch compressor

In a wiggler (or chicane),

In a linear approximation R56 >> T566d1,

Total transformation

For frf= p/2, R66=1, the transformation matrix is sympletic,

which means that longitudinal emittance is a conserved quantitiy.

A simple case of4-bending magnet chicane

- Zeuthen Chicane : a benchmark layout used for CSR calculation comparisons at 2002 ICFA beam dynamics workshop

B2

B3

qo

B1

B4

DL

LB

DL

DLc

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 mm

Relations among R56, T566 and U5666 in Chicane

q

b

a

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

Relations among R56, T566 and U5666 in Chicane

Difference in path length due to relative energy error is

By performing a Taylor expansion about d = 0

For large d, d2 and d3 terms may cause non-linear deformations of the

phase space during compression.

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

- From the conservation of longitudinal emittance,
final bunch lengthis

RF phase angle

- Energy-position correlation from an rf section is

- 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).

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.

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.
- CSR longitudinal wake function is

lr

sz

Lo

R

Coherent radiation forlr > sz

q

R=Lo/q

Overtaking length : Lo (24 sz R2)1/3

Coherent synchrotron radiation

- CSR-induced relative energy spread per dipole for a Gaussian bunch is
- 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 (psz√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.

Incoherent Synchrotron Radiation

When an electron emits a photon of energy u, the change in the betatron action

is given by

H=bxh'2+2axhh'+gxh2

- Transverse emittance growth is

- Increase of energy spread is

- The increase in energy spread is given by:

- Beamenergy loss is

Cq=3.84x10-13m

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

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)

A wiggler based on a chicane between each pair of quadrupoles

Each chicane contains 8 bend magnets (12 chicanes total).

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.

- 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.

- Show that emittance growth and increase of energy spread due to incoherent synchrotron radiation are given by
1)

2)

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