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1 ILC Bunch compressor Damping ring ILC Summer School August Eun-San Kim KNU

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2 Contents Bunch compressor Damping ring - Transverse motion - Longitudinal motion We present the basic concepts than detailed present issues in ILC. ( Accelerator Physics Lecture in 3 rd year class )

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3 Locations of bunch compressors BCs locate between e - (e + ) damping rings and main linacs, and make bunch length reduce from 9 mm rms to 0.3 mm rms. Beam diagnostics Bunch compressor Bunch compressor make bunch length short.

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4 Why we need bunch compressors Beams in damping rings has bunch length of 9 mm rms. - Beams with long bunch length tend to reduce effects of beam instabilities in damping rings. - Thus, beams should be compressed after the damping rings. Main linac and IP require very short beams: - to prevent large energy spread in the linac due to the curvature of the rf. - to reduce the disruption parameter ( ~ z ) : (ratio of bunch length to strength of mutual focusing between colliding beams) Thus, bunches between DRs and main linacs are shortened. - ~ 0.3 mm rms.

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5 Main issues in bunch compressors How can we produce such a beam with short bunch length? How can we keep low emittance ( x / y = 8 m / 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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6 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. -z E/E lower energy trajectory higher energy trajectory center 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. Tail (advance) Head (delay)

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7 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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8 Initial beam parameters in bunch compressors Initial beam energy : 5 GeV rms initial horizontal emittance : 8 m rms initial vertical emittance : 20 nm rms initial bunch length : 9 mm rms final bunch length : 0.3 mm compression ratio : 30 rms initial energy spread : 0.15 % charge / bunch : 3.2 nC (N=2x10 10 )

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9 Different types of bunch compressor Chicane Double chicane Chicanes as a Wiggler Arc as a FODO-compressor

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10 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 R 56 is required. It is also possible to locate quadrupole magnets between dipoles where dispersion passes through zero, allowing continuous focusing across the long systems. Arc type : R 56 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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11 Path length in chicane A path length difference for particles with a relative energy deviation is given by z R 56 …… : longitudinal dispersion : relative energy deviation (= E/E) R 56 : linear longitudinal dispersion (leading term for bunch compression)

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12 Momentum compaction The momentum compaction R 56 of a chicane made up of rectangular bend magnets is negative (for bunch head at z<0). First-order path length dependence is From the conservation of longitudinal emittance, final bunch length is

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13 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 ~ N 2 ISR and CSR may greatly increase beam emittance in bunch compressors with shorter bunch length than the damping rings.

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14 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 r R R=L o / zz Coherent radiation for r > z LoLo

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15 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 : baseline design A chicane type that produces necessary momentum compaction with a chicane made of 4 bending magnets : alternative design

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16 Baseline design BC1 RF BC1 Wiggler BC2 RF BC1 Wiggler

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17 Alternative design

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18 Initial Energy Spread [%]0.15 Initial Bunch Length [mm] Initial Emittance [ m] / 0.02 BC1 Voltage [MV]348 BC1 Phase [°]-93 BC1 R 56 [mm] End BC1 Bunch Length [mm]1.45 End BC1 Energy [GeV]4.98 End BC1 Energy Spread [%]1.63 BC2 Voltage [MV]14,080 BC2 Phase [°]-36 BC2 R 56 [mm]-50.8 End BC2 Bunch Length [mm] End BC2 Emittance [ m] / 0.02 End BC2 Energy [GeV]16.1 End BC2 Energy Spread [%]2.4 Alternative design

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19 With optics and orbit corrections, emittance growth due to 6 machine errors shows 4% in vertical only. Machine errors 300 m vertical misalign of Q 300 m horizontal misalign of Q 300 rad rotation error of Q 300 m vertical misalign of B 300 m horizontal misalign of B 300 rad rotation error of B Tolerance of bunch compressor Alternative design

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20 Damping ring Betatron motion

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21 Betatron motion Mid-plane symmetry: magnetic field in the horizontal plane is perpendicular to the plane (s) = local radius of curvature Particles are kept on a nearly circular trajectory by bending and focusing magnetic fields. The reference trajectory is the equilibrium closed orbit for a particle of momentum p 0. Quadrupoles act as focusing systems which produce small betatron oscillations around the reference trajectory y

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22 Motion of equation The linearized betatron motion is governed by Hill’s equation. x” + K x (s) x = 0 where K x = 1/ 2 - (∂B y /∂x) /B y” + K y (s) y = 0 and K y = (∂B y /∂x) /B The focusing functions are periodic: K x, y (s+L) = K x,y (s)

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23 Transfer matrices Let y(s)= (y(s),y’(s)) be the “position vector” y(s) = M(s|s 0 ) y(s 0 ) where M(s|s 0 ) is the betatron transfer matrix. The passage through a magnetic element can be described by a 2x2 matrix, which transforms the "position vector" of a particle.

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24 Solution with constant K z” + K (s) z = 0 ( z = x or y ) z(s) = a cos(√Ks +b) K > 0 focusing quad z(s) = as + b K = 0 drift space z(s) = a cosh(√-Ks +b) K < 0 defocusing quad M x = cos sin /√|K| K > 0 focusing quad -√|K| sin cos = s √|K| M y = cosh sinh /√|K| K < 0 defocusing quad - √|K| sinh cosh 1 L K = L drift space of length

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25 Quadrupole Strenght K x = (∂B y /∂x) /B K y = -(∂B y /∂x) /B Field B x = (∂B y /∂x) y B y = (∂B y /∂x) x A quadrupole is always focusing in one plane and defocusing in the other one

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26 One turn matrix

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27 One turn matrix

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28 Twiss parameters

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29 Twiss parameters

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30 Dispersion function

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31 Momentum compaction

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32 Chromaticity

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33 Chromaticity

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34 Damping ring Synchrotron motion

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35 Radiated power

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36 Energy loss per turn

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37 Synchrotron oscillation

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38 Synchrotron oscillation

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39 Summary EDR Activity in Korea - Damping rings (low emittance) : Ecloud and Ion beam instabilities - Bunch compressor (short bunch) : Beam dynamics - RF cavity (beam acceleration) : cavity design / processing - Beam instrumentation (beam diagnostics) : BPM design / processing Welcome to ILC accelerator R&Ds ! Damping rings (e -,e+) RF cavity Beam diagnostics Bunch compressor Bunch compressor Beam diagnostic

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