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**ILC Accelerator School Kyungpook National University**

Bunch compressors ILC Accelerator School May Eun-San Kim Kyungpook National University Korea

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**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 Sz=6 mm rms 2nd stage ILC : 1 TeV - extension of main linac - moving of SR and BC

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**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.

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**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?

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**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 Head 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.

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**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.

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**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 : mm compression ratio : 40 rms initial energy spread : % charge / bunch : 3.2 nC (N=2x1010)

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**Different types of bunch compressor**

Chicane Double chicane Chicanes as a Wiggler Chicane Double chicane Chicane as a Wiggler Arc as a FODO-compressor Arc as a FODO-compressor

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**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.

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**Path length in chicane h : longitudinal dispersion**

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

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**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 ) Let us consider effect of passing a bunch through a RF cavity on the zero crossing of the voltage (i.e. without acceleration) krf = 2p frf/c

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**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,

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**Longitudinal particle motion in bunch compressor**

To first order in eVrf/Eo << 1, In a linear approximation for RF,

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**Longitudinal particle motion in bunch compressor**

In a wiggler (or chicane), In a linear approximation R56 >> T566 d1, Total transformation For frf = p/2, R66=1, the transformation matrix is sympletic, which means that longitudinal emittance is a conserved quantitiy.

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**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 B3 qo B1 B4 LB DL DLc DL LB Bend magnet length : LB = 0.5m Drift length B1-B2 and B3-B4(projected) : DL = 5 m Drift length B2-B : 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

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**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

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**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.

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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 length is

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**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).

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**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.

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**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 for lr > sz q R=Lo/q Overtaking length : Lo (24 sz R2)1/3

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**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.

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**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 Cq=3.84x10-13m The increase in energy spread is given by: Beam energy loss is Cq=3.84x10-13m

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**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

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**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)

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**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).

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**Baseline design for ILC BC**

BC2 RF BC1 RF BC1 Wiggler BC1 Wiggler

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**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 cell RF cavities arranged in 57 cryomodules. - A wiggler has optics identical to the wiggler in the first BC, but with weaker wiggler.

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**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] 8.2 / 0.02 End BC2 Energy [GeV] 11.7 End BC2 Energy Spread [%] 2.73

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**Alternative design for ILC BC**

Main linac Matching Chicane 1 Quadrupoles Chicane 2 RF section

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**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 [%] BC2 Voltage [MV] 11,800 BC2 Phase [°] -45 BC2 R56 [mm] -50.8 End BC2 Bunch Length [mm] End BC2 Emittance [mm] 8.3 / 0.02 End BC2 Energy [GeV] 13.26 End BC2 Energy Spread [%] 2.2

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**Bunch compressors for ILC**

Alternative Baseline Chicane length 68.4 m 480 m Matching 4 m 310 m Number of RF cavity 452 488 Total length 680 m 1400 m Alternative Baseline Required bunch length achieved System length shorter longer Tolerence of emittance acceptable comparable GDE Requirement correction of vertical dispersion shorten system length

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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.

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Problems Show that emittance growth and increase of energy spread due to incoherent synchrotron radiation are given by 1) 2)

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