velocity bunching & more on magnetic-chicane bunch
TRANSCRIPT
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Linac Design for FELs, Lecture Tu 5
Velocity bunching & more on magnetic-chicane bunch compression
MVlast revised 15-June
Brief summary of what we learned yesterday
β’ Beam energy change through rf structureβ Ultrarelativistic approx.
β’ Setting of RF phase to control the beam energy chirp
β’ Concept of magnetic-chicane bunch compressorβ Orbit path-length dependence on particle energyβ Correlation between particle energy and its position along the bunch (energy
chirp)
β’ Momentum compaction R56 for 4-bend C-shaped chicane
β’ Compression factor
β’ Linearization of the longitudinal phase-space using harmonic rf cavities
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Outline
1. Compressing low-energy bunches β Ballistic compression
β Velocity bunching
2. More on magnetic-chicane compression β Sensitivity of beam jitters to RF parameters
β One-stage vs. multiple-stage compression.
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Compression at low energy: velocity bunching
β’ High beam energy (v~c): different energies -> no difference in velocity (c). β Bends provide the dispersion needed to establish a dependence of the path-length on
beam energy that can be exploited to do compression
β’ Low beam energy (v<c): different energies -> difference in velcity can be significantβ Exploit different times of arrival by particles with different velocities to do compression
in straight non-dispersive channels.
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β’ In some cases low-energy compression is desirableβ Typically when the e-source generates beams with too low a current. β Desirable feature compared to magnetic compression (no CSR effects).
But it is more vulnerable to space-charge effects
β’ Note on use of terms:β Velocity-bunching compression can be done during acceleration β The term βBallistic compressionβ is often used when compression is done without acceleratingβ βVelocity bunchingβ is the more generic termβ βRF compressionβ also used
Conceptual picture of ballistic compression
β’ For compression, particles in the tail should have larger energy (velocity) β Same sign of energy chirp as for magnetic compression in 4-bend C-Shape chicane
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z
DE
z
DE
z
DE
sz0 szsz0
Drift Drift
Rf-cavity operated at βzero-crossingβ
head tail
tail
head
Before entering cavity Right after cavity Downstream cavity
Example of Ballistic Compression in action: Studies for the NGLS injector
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Fig courtesy of C. Papadopoulos
186 MHz Normal Conducting RF-GunAccelerating electrons to 1.25 MeV energy (~750keV kinetic energy)Accelerating gap: ~4cm (19MV/m field at cathode) (APEX- prototype presently under commissioning)
βBuncherβSingle 1.3 GHz RF cavity
(Normal Conducting)
Solenoids (emittance compensation)
βBoosterβ1.3 GHz Cavities (Tesla style)(Super Conducting)
Beam half-way through gun accelerating gap
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Current profile
14 16 18 20 22 24 260.75
0.80
0.85
0.90
0.95
1.00
z mm
EM
eV
s 2cmEnergy profile
head
Beam snap-shot in time*
head
tailtail
Particles in the head have higher energy;
they have been accelerated for a
longer time
Relatively low-current
1.2mm
*IMPACT simulations by J. Qiangπ (mm)
π (mm)
Beam at exit of gun accelerating gap
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Fig. from C. Papadopoulos
65 70 751.235
1.240
1.245
1.250
1.255
z mm
EM
eV
s 7cm
head
Current profile Energy profile
Largely flattenedβ¦
β¦some space-chargeinduced chirpCurrent drops
slightly
1.4mm
head
π (mm)π (mm)
Beam at entrance of buncher
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Fig. from C. Papadopoulos
695 700 7051.23
1.24
1.25
1.26
z mm
EM
eV
s 70cm
head
Current profile Energy profile
Current drops further
1.8mm
π (mm)π (mm)
Larger Chirp(space charge)
Beam at exit of buncher
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Fig. from C. Papadopoulos
995 1000 1005
1.30
1.32
1.34
1.36
z mm
EM
eV
s 1m
head
1.8mm
Current profile Energy profile
Current is about the same
Chirp reversed by buncher
head
π (mm) π (mm)
Beam just downstream the buncher
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Fig. from C. Papadopoulos
1392 1394 1396 1398 1400 1402 1404 1406
1.30
1.31
1.32
1.33
1.34
1.35
1.36
1.37
z mm
EM
eV
s 1.4m
head
Energy profile Current profile Current starts to increase
1.6mmhead
tail
π (mm) π (mm)
Beam 1.2m downstream the buncher
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Fig. from C. Papadopoulos
2186 2188 2190 21921.3151.3201.3251.3301.3351.3401.3451.350
z mm
EM
eV
s 2.19m
head
Energy profile
8.5mm
Current profile Peak current
up by Γ π
head
π (mm)π (mm)
2nd order chirp from non-linearities
Ballistic compression: expression for compression factor (linear approximation)
β’ Expand equations about reference orbit π‘ = π‘π + Ξπ‘ , πΎ = πΎπ + ΞπΎ
β’ Find the equations for Ξπ‘ and ΞπΎ (first order)
β’ Define time so that π‘ = 0 when the ref. particle crosses the cavity
β’ Set rf phase so that the reference particle goes through the cavity at βzero crossingβ : cos πrf = 0 (i.e. energy of reference particle doesnβt change) -> πrf = Β±π/2
β’ Choose πrf = βπ/2. (To have the right energy chirp - higher energy particles in the tail.)
β’ Compression factor:
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ππΎ
ππ = β
ππΈπ 0 π
ππ2cos πrfπ‘ + πrf
ππ‘
ππ =
πΎ
π πΎ2 β 1
πΆ β‘Ξπ‘(π )
Ξπ‘ π = 0= 1 β
ππ0ππ2
π πrf
πΎ2 β 132
β1
Dirac πΏ function
Standing wave RF cavityRelativistic equation
For longitudinal motion(with s as the independent
variable)
cos πrfΞπ‘ β π/2
Ξπ‘
tail
Drift
Rf-cavity operated at βzero-crossingβ
Drift
β’ Assume kick approximation for cavity (0-length cavity length at π = 0).
βππΈπ 0 π
ππ2=
ππ0ππ2
πΏ(π )
π
Work-out detailsas an exercise
Velocity bunching: compress while accelerating
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β’ If current out-of gun is already high further compression is best done while also accelerating (to reduce effect of space-charge forces; ease emittance compensation)
β’ Compression takes place over longer, multi-cavity structure β’ Dynamics of compression is best illustrated in travelling-wave structures
t=0 t>0
π π
ΞπΈ
tail
head
Tail particle gainsmore energy and gets closerto ref particle
Reference particle (and whole beam)gain energy
Forward moving traveling wave
Compression continues until
beam reaches the rf crest and v<c
Traveling wave slips forward
compared to bunch(phase velocity > beam
velocity)
Single-stage or multi-stage compression?
β’ Because of collective effects (space charge) max. amount of velocity bunching compression at low energy is limited by desire to preserve beam quality β Magnetic compression is still needed (usually done at sufficiently high energy to limit adverse
impact of collective effects)
β’ Magnetic compression: is it better to do all the compression at once, or break it up through two or more chicanes?
β’ Favoring multi-stage magnetic compression:β CSR effects on transverse emittance (2nd and further stage compression done at higher
energy to reduce CSR effects) β Reduced sensitivity to rf jitters
β’ Favoring single-stage magnetic compression:β Control of microbunchingβ Reduced system complexity (not critical)
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Examples of velocity-bunching compression experiments
-105 -100 -95 -90 -85 -80 -75 -700
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4
6
8
10
12
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Phase (deg)
Com
pre
ssio
n facto
r
+Exp. Measurements
-- Parmela simulations
SPARC
Ferrario et al. Phys. Rev. Lett. 104, 054801 (2010)
Pio
t et
al.
Ph
ys. R
ev. S
T A
ccel
. Bea
ms
6, 0
33
50
3 (
20
03
)
DUV-FELBNL
More on Magnetic Compression17
Multi-stage magnetic compression is more robust against various sources of jitters
β’ Fluctuations of rf structure parameters (voltage, phase) around set values are unavoidable.
β’ They cause undesirable βjittersβ inβ Final electron beam energy β Beam arrival time β Beam peak current
β’ E.g. for Superconducting Structures (typically easier to stabilize) aggressive but not unreasonable targets for max. rffluctuations areβ 0.01deg (rf phases) β 0.01% (rf voltages)
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Example 1: Sensitivity of compression factor to RF phase. Single-stage magnetic compression
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πΆ =1
1 + βπ 56β = β
πππΏ1krfsinππΏ1πΈπ΅πΆ1
ΞπΆ
πΆ=
1
C
ππΆ
πβΞβ = β
1
πΆπΆ2π 56Ξβ = βπΆπ 56β
Ξβ
β= βπΆ(πΆβ1 β 1)
Ξβ
β= (πΆ β 1)
Ξβ
β
Ξβ
β=
1
β
πβ
πππΏ1ΞππΏ1 = β
πΈπ΅πΆ1
πππΏ1krfsinππΏ1(β
πππΏ1krfcosππΏ1
πΈπ΅πΆ1)ΞππΏ1 =
ΞππΏ1
tan ππΏ1
ππͺ
πͺ= (πͺ β π)
π«ππ³π
πππ§ππ³π
Proportional to πΆ β 1 β πΆ
singular at 0-phase (for fixed compression πΆ)
L1L0 BC1
ππ³π
ππΏ1
ππΏ0 = 0
ππΏ0
Beam enters herew/o energy chirp
Study sensitivity of compression factorto L1 rf phase errors
πΈ0 πΈπ΅πΆ1
Compression factor: Linear chirp after L1:
Example 2: Sensitivity of compression factor to RF phase. Double-stage magnetic compression
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π1 β‘ βπππΏ1krfsinππΏ1
πΈπ΅πΆ1
π2 β‘ βπππΏ2krfsinππΏ2
πΈπ΅πΆ2
β1 = π1
β2 = πΆ1π1πΈπ΅πΆ1πΈπ΅πΆ2
+ π2
πΆ = πΆ1πΆ2 β1
1 + π1π 56Γ
1
1 + π2π56
Suppose πΆ1π1πΈπ΅πΆ1
πΈπ΅πΆ2βͺ π2 therefore β2 β π2 , and πΆ2 β
1
1+π2π56
ΞπΆ
πΆ= β
1
πΆπΆ12π 56Ξπ1πΆ2 + πΆ1πΆ2
2π56Ξπ2
BC1 BC2L1 L2
ππ³π
ππΏ2
ππΏ1ππΏ1
πΆ1 =1
1 + β1πΉπππΆ2 =
1
1 + β2πππ
Totalcompression
πΈ0 πΈπ΅πΆ1 πΈπ΅πΆ2
chirp contributedby L1
chirp contributedby L2
Example 2: Sensitivity of compression factor to RF phase. Double-stage magnetic compression (contβd)
β’ Conclusion: in two-stage compression there is the potential to make sensitivity to rfphase jitter smallerβ The condition for the most benefit is πΆ1π1
πΈπ΅πΆ1
πΈπ΅πΆ2βͺ π2 . (Although it may be difficult to
enforce)β In practice, compression sensitivity to phase jitters will be higher than predicted by the
formula above but still reduced compared to one-stage compression 21
πΆ1 =1
1 + π1π 56πΆ2 β
1
1 + π2π56
= βπΆ1 πΆ1β1 β 1
Ξπ1
π1β πΆ2 πΆ2
β1 β 1Ξπ2
π2
ΞπΆ
πΆ= β
1
πΆπΆ12π 56Ξπ1πΆ2 + πΆ1πΆ2
2π56Ξπ2 = βπΆ1π 56Ξπ1 β πΆ2π56Ξπ2
= βπΆ1π 56π1Ξπ1π1
β πΆ2 π56π2Ξπ2π2
π«πͺ
πͺ= πͺπ β π
π«ππππ
+ πͺπ β ππ«ππππ
~ π πͺπ«π
πππ§ π
For comparison, for single-stage we found ΞπΆ
πΆ= πΆ β 1
Ξβ
β~ πͺ
ππ
πππ§ π
Assuming πΆ1~πΆ2 β« 1,πΆ = πΆ1πΆ2
π₯ππΏ1~π₯ππΏ2~π₯πππΏ1~ππΏ2~π
Summary
β’ RF compression is one more tool in the tool-box for beam manipulationsβ It has to be done at low energy (++ and --)
β’ Multi-stage magnetic compression is usually preferred as a way to reduce certain collective effects (CSR impact on transverse emittance)β It can make the beam more sensitive to other collective effects (microbunching
instability).
β’ Sensitivity to rf parameter errors is smaller in multi-stage compression β E.g. compression (peak current) jitter dependence on rf phase errors
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ππͺ
πͺ= (πͺ β π)
π«ππ³π
πππ§ππ³π
Bonus material23
Velocity bunching: analytical model
β’ Travelling wave structure β Straightforward extension of formalism to standing-wave structures (Decompose
travelling wave into forward + backward travelling waves; effect of backward wave will average out)
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ππΎ
ππ = β
ππΈπ 0 π
ππ2cos πrfπ β πrfπ‘ + π0
ππ‘
ππ =
πΎ
π πΎ2 β 1
ππ
ππ = πrf 1 β
πΎ
πΎ2 β 1
ππΎ
ππ = β
ππΈπ 0 π
ππ2cos π
π = πrfπ β πrfπ‘(π ) + π0
Chang dynamical Coordinate from π‘ to π
New equations
ππ
ππ = πrf β πrf
ππ‘
ππ =
= πrf 1 β1
π½= πrf 1 β
πΎ
πΎ2 β 1
Derivation of invariant
β’ H is independent of π π» is a constant of motion
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ππ
ππ = πrf 1 β
πΎ
πΎ2 β 1β‘ππ»
ππΎ
ππΎ
ππ = πΌ cos π β‘ β
ππ»
ππ
System of equationscan be thought of as a canonical system witheffective Hamiltonian H:
π― = ππ«π πΈ β πΈπ β π β πΆππ«π π¬π’π§ππΌ = β
ππΈπ 0 π
ππ2
Beam injectedat zero crossing
Phase space
Beams movesto higherenergy,is compressed
π«ππ
π«ππ
Compression factor in the linear approximation
β’ Use invariance of Hamiltonian:
β’ Assume for simplicity that πΎ β πΎ2 β 1 ~0 at exit of Compressing structure
β’ Compression factor
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π»0π = πrf πΎ0π β πΎ0π2 β 1 β πΌπrf sinπ0π
πrf πΎ0 β πΎ02 β 1 β πΌπrf sinπ0 = sinπ1 β‘
πΆ =|Ξπ‘0|
|Ξπ‘1|=|Ξπ0|
|Ξπ1|βcosππ0cosππ1
Max. compression Is when phase
ππ1 =π
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Invariant @injectionFor reference partice