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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

• 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

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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.

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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

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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)

“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

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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. 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:

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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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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