1 Linac Design for FELs, Lecture Tu 5 Velocity bunching & more on magnetic-chicane bunch compression MV last revised 15-June
1
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
• Beam energy change through rf structure– Ultrarelativistic approx.
• Setting of RF phase to control the beam energy chirp
• Concept of magnetic-chicane bunch compressor– Orbit path-length dependence on particle energy– Correlation between particle energy and its position along the bunch (energy
chirp)
• Momentum compaction R56 for 4-bend C-shaped chicane
• Compression factor
• Linearization of the longitudinal phase-space using harmonic rf cavities
2
Outline
1. Compressing low-energy bunches – Ballistic compression
– Velocity bunching
2. More on magnetic-chicane compression – Sensitivity of beam jitters to RF parameters
– One-stage vs. multiple-stage compression.
3
Compression at low energy: velocity bunching
• High beam energy (v~c): different energies -> no difference in velocity (c). – Bends provide the dispersion needed to establish a dependence of the path-length on
beam energy that can be exploited to do compression
• Low beam energy (v<c): different energies -> difference in velcity can be significant– Exploit different times of arrival by particles with different velocities to do compression
in straight non-dispersive channels.
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• In some cases low-energy compression is desirable– Typically when the e-source generates beams with too low a current. – Desirable feature compared to magnetic compression (no CSR effects).
But it is more vulnerable to space-charge effects
• Note on use of terms:– Velocity-bunching compression can be done during acceleration – The term “Ballistic compression” is often used when compression is done without accelerating– “Velocity bunching” is the more generic term– “RF compression” also used
Conceptual picture of ballistic compression
• For compression, particles in the tail should have larger energy (velocity) – 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 “zero-crossing”
head tail
tail
head
Before entering cavity Right after cavity Downstream cavity
Example of Ballistic Compression in action: Studies for the NGLS injector
6
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)
“Buncher”Single 1.3 GHz RF cavity
(Normal Conducting)
Solenoids (emittance compensation)
“Booster”1.3 GHz Cavities (Tesla style)(Super Conducting)
Beam half-way through gun accelerating gap
7
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. Qiang𝑠 (mm)
𝑠 (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 flattened…
…some space-chargeinduced chirpCurrent drops
slightly
1.4mm
head
𝑠 (mm)𝑠 (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
𝑠 (mm)𝑠 (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
𝑠 (mm) 𝑠 (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
𝑠 (mm) 𝑠 (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 × 𝟒
head
𝑠 (mm)𝑠 (mm)
2nd order chirp from non-linearities
Ballistic compression: expression for compression factor (linear approximation)
• Expand equations about reference orbit 𝑡 = 𝑡𝑟 + Δ𝑡 , 𝛾 = 𝛾𝑟 + Δ𝛾
• Find the equations for Δ𝑡 and Δ𝛾 (first order)
• Define time so that 𝑡 = 0 when the ref. particle crosses the cavity
• Set rf phase so that the reference particle goes through the cavity at “zero crossing” : cos 𝜑rf = 0 (i.e. energy of reference particle doesn’t change) -> 𝜑rf = ±𝜋/2
• Choose 𝜑rf = −𝜋/2. (To have the right energy chirp - higher energy particles in the tail.)
• Compression factor:
13
𝑑𝛾
𝑑𝑠= −
𝑒𝐸𝑠0 𝑠
𝑚𝑐2cos 𝜔rf𝑡 + 𝜑rf
𝑑𝑡
𝑑𝑠=
𝛾
𝑐 𝛾2 − 1
𝐶 ≡Δ𝑡(𝑠)
Δ𝑡 𝑠 = 0= 1 −
𝑒𝑉0𝑚𝑐2
𝑠𝑘rf
𝛾2 − 132
−1
Dirac 𝛿 function
Standing wave RF cavityRelativistic equation
For longitudinal motion(with s as the independent
variable)
cos 𝜔rfΔ𝑡 − 𝜋/2
Δ𝑡
tail
Drift
Rf-cavity operated at “zero-crossing”
Drift
• Assume kick approximation for cavity (0-length cavity length at 𝑠 = 0).
−𝑒𝐸𝑠0 𝑠
𝑚𝑐2=
𝑒𝑉0𝑚𝑐2
𝛿(𝑠)
𝒔
Work-out detailsas an exercise
Velocity bunching: compress while accelerating
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• 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)
• Compression takes place over longer, multi-cavity structure • Dynamics of compression is best illustrated in travelling-wave structures
t=0 t>0
𝑠𝑠
Δ𝐸
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?
• Because of collective effects (space charge) max. amount of velocity bunching compression at low energy is limited by desire to preserve beam quality – Magnetic compression is still needed (usually done at sufficiently high energy to limit adverse
impact of collective effects)
• Magnetic compression: is it better to do all the compression at once, or break it up through two or more chicanes?
• Favoring multi-stage magnetic compression:– CSR effects on transverse emittance (2nd and further stage compression done at higher
energy to reduce CSR effects) – Reduced sensitivity to rf jitters
• Favoring single-stage magnetic compression:– Control of microbunching– Reduced system complexity (not critical)
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16
Examples of velocity-bunching compression experiments
-105 -100 -95 -90 -85 -80 -75 -700
2
4
6
8
10
12
14
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
• Fluctuations of rf structure parameters (voltage, phase) around set values are unavoidable.
• They cause undesirable “jitters” in– Final electron beam energy – Beam arrival time – Beam peak current
• E.g. for Superconducting Structures (typically easier to stabilize) aggressive but not unreasonable targets for max. rffluctuations are– 0.01deg (rf phases) – 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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𝐶 =1
1 + ℎ𝑅56ℎ = −
𝑒𝑉𝐿1krfsin𝜑𝐿1𝐸𝐵𝐶1
Δ𝐶
𝐶=
1
C
𝜕𝐶
𝜕ℎΔℎ = −
1
𝐶𝐶2𝑅56Δℎ = −𝐶𝑅56ℎ
Δℎ
ℎ= −𝐶(𝐶−1 − 1)
Δℎ
ℎ= (𝐶 − 1)
Δℎ
ℎ
Δℎ
ℎ=
1
ℎ
𝜕ℎ
𝜕𝜑𝐿1Δ𝜑𝐿1 = −
𝐸𝐵𝐶1
𝑒𝑉𝐿1krfsin𝜑𝐿1(−
𝑒𝑉𝐿1krfcos𝜑𝐿1
𝐸𝐵𝐶1)Δ𝜑𝐿1 =
Δ𝜑𝐿1
tan 𝜑𝐿1
𝜟𝑪
𝑪= (𝑪 − 𝟏)
𝚫𝝋𝑳𝟏
𝐭𝐚𝐧𝝋𝑳𝟏
Proportional to 𝐶 − 1 ≃ 𝐶
singular at 0-phase (for fixed compression 𝐶)
L1L0 BC1
𝝋𝑳𝟏
𝑉𝐿1
𝜑𝐿0 = 0
𝑉𝐿0
Beam enters herew/o energy chirp
Study sensitivity of compression factorto L1 rf phase errors
𝐸0 𝐸𝐵𝐶1
Compression factor: Linear chirp after L1:
Example 2: Sensitivity of compression factor to RF phase. Double-stage magnetic compression
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𝑎1 ≡ −𝑒𝑉𝐿1krfsin𝜑𝐿1
𝐸𝐵𝐶1
𝑎2 ≡ −𝑒𝑉𝐿2krfsin𝜑𝐿2
𝐸𝐵𝐶2
ℎ1 = 𝑎1
ℎ2 = 𝐶1𝑎1𝐸𝐵𝐶1𝐸𝐵𝐶2
+ 𝑎2
𝐶 = 𝐶1𝐶2 ≃1
1 + 𝑎1𝑅56×
1
1 + 𝑎2𝑟56
Suppose 𝐶1𝑎1𝐸𝐵𝐶1
𝐸𝐵𝐶2≪ 𝑎2 therefore ℎ2 ≃ 𝑎2 , and 𝐶2 ≃
1
1+𝑎2𝑟56
Δ𝐶
𝐶= −
1
𝐶𝐶12𝑅56Δ𝑎1𝐶2 + 𝐶1𝐶2
2𝑟56Δ𝑎2
BC1 BC2L1 L2
𝝋𝑳𝟐
𝑉𝐿2
𝜑𝐿1𝑉𝐿1
𝐶1 =1
1 + ℎ1𝑹𝟓𝟔𝐶2 =
1
1 + ℎ2𝒓𝟓𝟔
Totalcompression
𝐸0 𝐸𝐵𝐶1 𝐸𝐵𝐶2
chirp contributedby L1
chirp contributedby L2
Example 2: Sensitivity of compression factor to RF phase. Double-stage magnetic compression (cont’d)
• Conclusion: in two-stage compression there is the potential to make sensitivity to rfphase jitter smaller– The condition for the most benefit is 𝐶1𝑎1
𝐸𝐵𝐶1
𝐸𝐵𝐶2≪ 𝑎2 . (Although it may be difficult to
enforce)– 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
𝐶1 =1
1 + 𝑎1𝑅56𝐶2 ≃
1
1 + 𝑎2𝑟56
= −𝐶1 𝐶1−1 − 1
Δ𝑎1
𝑎1− 𝐶2 𝐶2
−1 − 1Δ𝑎2
𝑎2
Δ𝐶
𝐶= −
1
𝐶𝐶12𝑅56Δ𝑎1𝐶2 + 𝐶1𝐶2
2𝑟56Δ𝑎2 = −𝐶1𝑅56Δ𝑎1 − 𝐶2𝑟56Δ𝑎2
= −𝐶1𝑅56𝑎1Δ𝑎1𝑎1
− 𝐶2 𝑟56𝑎2Δ𝑎2𝑎2
𝚫𝑪
𝑪= 𝑪𝟏 − 𝟏
𝚫𝒂𝟏𝒂𝟏
+ 𝑪𝟐 − 𝟏𝚫𝒂𝟐𝒂𝟐
~ 𝟐 𝑪𝚫𝝋
𝐭𝐚𝐧 𝝋
For comparison, for single-stage we found Δ𝐶
𝐶= 𝐶 − 1
Δℎ
ℎ~ 𝑪
𝜟𝝋
𝐭𝐚𝐧 𝝋
Assuming 𝐶1~𝐶2 ≫ 1,𝐶 = 𝐶1𝐶2
𝛥𝜑𝐿1~𝛥𝜑𝐿2~𝛥𝜑𝜑𝐿1~𝜑𝐿2~𝜑
Summary
• RF compression is one more tool in the tool-box for beam manipulations– It has to be done at low energy (++ and --)
• Multi-stage magnetic compression is usually preferred as a way to reduce certain collective effects (CSR impact on transverse emittance)– It can make the beam more sensitive to other collective effects (microbunching
instability).
• Sensitivity to rf parameter errors is smaller in multi-stage compression – E.g. compression (peak current) jitter dependence on rf phase errors
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𝜟𝑪
𝑪= (𝑪 − 𝟏)
𝚫𝝋𝑳𝟏
𝐭𝐚𝐧𝝋𝑳𝟏
Bonus material23
Velocity bunching: analytical model
• Travelling wave structure – 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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𝑑𝛾
𝑑𝑠= −
𝑒𝐸𝑠0 𝑠
𝑚𝑐2cos 𝑘rf𝑠 − 𝜔rf𝑡 + 𝜑0
𝑑𝑡
𝑑𝑠=
𝛾
𝑐 𝛾2 − 1
𝑑𝜙
𝑑𝑠= 𝑘rf 1 −
𝛾
𝛾2 − 1
𝑑𝛾
𝑑𝑠= −
𝑒𝐸𝑠0 𝑠
𝑚𝑐2cos 𝜙
𝜑 = 𝑘rf𝑠 − 𝜔rf𝑡(𝑠) + 𝜑0
Chang dynamical Coordinate from 𝑡 to 𝜑
New equations
𝑑𝜑
𝑑𝑠= 𝑘rf − 𝜔rf
𝑑𝑡
𝑑𝑠=
= 𝑘rf 1 −1
𝛽= 𝑘rf 1 −
𝛾
𝛾2 − 1
Derivation of invariant
• H is independent of 𝑠 𝐻 is a constant of motion
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𝑑𝜙
𝑑𝑠= 𝑘rf 1 −
𝛾
𝛾2 − 1≡𝜕𝐻
𝜕𝛾
𝑑𝛾
𝑑𝑠= 𝛼 cos 𝜙 ≡ −
𝜕𝐻
𝜕𝜑
System of equationscan be thought of as a canonical system witheffective Hamiltonian H:
𝑯 = 𝒌𝐫𝐟 𝜸 − 𝜸𝟐 − 𝟏 − 𝜶𝒌𝐫𝐟 𝐬𝐢𝐧𝝋𝛼 = −
𝑒𝐸𝑠0 𝑠
𝑚𝑐2
Beam injectedat zero crossing
Phase space
Beams movesto higherenergy,is compressed
𝚫𝝋𝟎
𝚫𝝋𝒇
Compression factor in the linear approximation
• Use invariance of Hamiltonian:
• Assume for simplicity that 𝛾 − 𝛾2 − 1 ~0 at exit of Compressing structure
• Compression factor
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𝐻0𝑟 = 𝑘rf 𝛾0𝑟 − 𝛾0𝑟2 − 1 − 𝛼𝑘rf sin𝜑0𝑟
𝑘rf 𝛾0 − 𝛾02 − 1 − 𝛼𝑘rf sin𝜑0 = sin𝜑1 ≡
𝐶 =|Δ𝑡0|
|Δ𝑡1|=|Δ𝜑0|
|Δ𝜙1|≃cos𝜑𝑟0cos𝜑𝑟1
Max. compression Is when phase
𝜑𝑟1 =𝜋
2
Invariant @injectionFor reference partice