1. Longitudinal Space Charge Instability C. Limborg-Déprey, Z. Huang, J
Longitudinal Space Charge Instability C.Limborg-Déprey, Z.Huang, J.Wu, SLAC T.Shaftan, BNL September 24, 2003
LSC observed experimentally
Theory
Application to LCLS
Comparison theory with Simulations
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

2. DUV FEL TTF W.S. Graves, et al., PAC 2001, p. 2860
M. Huning et al., NIM A 475 (2001) p. 348
TTF
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

3. Motivations Experimental Theory TTF observations
M. Huning et al., NIM A 475 (2001) p. 348
DUVFEL observations
W.S. Graves, et al., PAC 2001, p. 2860
T.Shaftan et al. experiments at the DUVFEL
T.Shaftan et al., PAC03, “Micro-bunching and beam break-up in the DUV FEL accelerator”, also in FEL03
with modeling of mechanism discussed in March at SLAC
Z.Huang, T.Shaftan, SLAC-PUB-97, May 2003
Theory
Saldin , Schneidmiller, Yurkov, “Longitudinal Space Charge Driven MicroBunching Instability in TTF2 Linac” TESLA-FEL ,May 2003
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

4. The DUVFEL Accelerator
RF gun
4.5 MeV
Tank 1
Tank 2
Tank 3
Tank 4
35 MeV
75 MeV
Spectrometer
dipole
Chicane
Monitor
Beam
Dump
Drive
Laser
to FEL
Positive
RF slope
Accelerate bunch at
RF zero-crossing of tank 4
Spectrometer
dipole
Zero RF
slope
Negative
Size of the
chirped beam
image at the
monitor 14 is
proportional
to the
bunch length !
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

5. The DUVFEL Accelerator
Structure with a Large Number Spikes
“Zero-phasing” image of uncompressed bunch with a large number of sharp spikes
Energy spectrum, derived from the image,
horizontal axis is scaled in picoseconds
Time or Energy?
Frequency spectrum of upper plot. Spectrum
shows modulation with harmonics in THz range. Harmonics  sharpness of the spikes.
Time, ps
Frequency spectrum (THz)
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

6. Modulation dynamics with Compression (~300 pC)
Dynamics at “high” (>200 pC) charge differs
from the dynamics at “low” charge
Uncompressed bunch profile is smooth
Experiment on modulation dynamics:
Keep chicane constant and increase chirping
tank phase (0-13-19-25 degrees)
Modulation shows up during compression
Process
Compression: a} decreases modulation period
(C times); b) increases bunch peak current (C times)
100 0°
1.42 ps 50 2 4 6 8
100
-13°
1.24 ps 50 1 2 3 4 5
200
-19°
0.93 ps
100 1 2 3 4
-25°
0.43 ps
200
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C.Limborg,Z.Huang,T.Shaftan,J.Wu

7. Isolines of constant bunch length
This is not CSR!
Isolines of constant bunch length
CSR-related effect ?  should be sensitive to the bending radius in the chicane magnets
Experiment:
final bunch length maintained constant
while
chicane strength changed
Compression factor is 1-hR56  there is always a combination of h and R56 that will maintain a constant compression ratio
Result of the experiment:
in general the modulation is insensitive to the chicane settings 5 10 15 20 25 30 35 40 50 60 70
Tank 2 Phase (Degree)
Chicane Current (A)
0.1
0.3
0.6 1
1.2
1.5
1.7
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8. Plasma Oscillation in a Coasting Beam
Current Density
Energy
The self-consistent solution is the space charge oscillation
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

9. Longitudinal Space Charge Impedance
For a round, parallel electron beams with a uniform transverse cross section of radius rb, the on-axis longitudinal space charge impedance is (cgs units)
Free space approximation is good when
 /(2) << beam pipe radius
Off-axis LSC is smaller and can increase the energy spread
Longitudinal Space Charge Impedance
(pancake beam)
(pencil beam)
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10. Integral Equation
Bunching parameter b at modulation wavelength l = 2p/k
Linear evolution of b(k;s) can be described by an integral equation (Heifets, Stupakov, Krinsky; Huang & Kim)
For arbitrary initial condition (density and/or energy modulation), this determines the final microbunching
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

11. Including acceleration
beam energy r(s) increases in the linac. Generalize the momentum compaction R56’(! s) as the path length change at s due to a small change in  (not ) at :
for r À 1
The integral equation for LSC microbunching in the linac is
In a drift,
 Space charge oscillation
When ! 1, R56’=0, b(k,s)=b0(k,s), beam is “frozen”
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

12. Space Charge Oscillation – 2 regimes
Density and energy modulation in a drift at distance s
At a very large , plasma phase advance (s/c) ¿ 1,
beam is “frozen,” energy modulation gets accumulated
(Saldin/Schneidmiller/Yurkov, TESLA-FEL )
LSC acts like a normal impedance at high energies
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

13. Klystron Amplitfication mechanism
Saldin, Schneidmiller, Yurkov
R56 ri
Z(k) Dg rf
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14. Gain is high enough to amplify the shot noise bunching
LCLS Set-UP
3 keV intrinsic energy spread is too small for the high-frequency LSC
Gain is high enough to amplify the shot noise bunching
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

15. Solution = Wiggler before BC2
Only CSR
Solution = Wiggler before BC2
CSR + LSC
Peak gain increased by a factor of 3
Solution = Adding Energy Spread on beam before BC1
“laser heater”
3 keV-> 30 keV before BC1
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

16. Even More Landau Damping
Current design has energy spread 1x10-4 for the FEL
Since FEL  ~ 5x10-4, small increase in energy spread is allowed, say 1.7x10-4
Laser heater increases energy spread 3 keV  50 keV
Wiggler increases energy spread to 5x10-5 at 4.5 GeV
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

17. Real danger is in fact energy spread >  = 5.10-4
Wiggler
 
Laser
 
Elegant (from M.Borland)
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C.Limborg,Z.Huang,T.Shaftan,J.Wu

18. All the simulations are based on theory just described
Let’s compare this theory with results from space charge simulation codes
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

19. Simulations Simulations description +/- 5% 40k/200k particles
Distribution generated using the Halton sequence of numbers
Longitudinal distribution
2.65 m of drift
With 3 cases studied
6MeV, 1nC
6 MeV , 2nC
12 MeV, 1nC
+/- 5%
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

20. Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

21. Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

22. Half a plasma period, 12 MeV, 1nC,  = 250 m
Oscillation of density characterized by
“ modulation amplitude”
current density  energy
But,
1- Residual energy modulation after half a period
2- Line density modulation amplitude strongly increased
Two parameters have also evolved along the beamline :
- uncorrelated energy spread
- transverse beam size
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

23. 6 MeV, 2nC,  = 100 m
30 cm
65 cm
110 cm
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24. 150 cm 190 cm 230 cm 265 cm Shift + broadening
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25. Evolution with energy 6 MeV , +/- 5% 12 MeV , +/- 5%
½ plasma period shorter for shorter wavelengths
line density modulation  with shorter wavelengths
Energy modulation  with shorter wavelengths
Amplitude Energy modulation gets larger than for 6 MeV
Uncorrelated energy spread gets larger by 60%
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

26. Comparison with theory
Transverse beam size evolution along beamline taken into account
(Radial variation of green’s function for 2D )
Evolution of peak current NOT taken into account yet
Absence of dip in 6MeV curve :
“Coasting beam “ against “bunched beam” with edge effects
Intrinsic energy spread
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

27. Evolution transverse profile
6 MeV , 1nC, +/- 5%
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28. 3D model – impedance with r dependency
1-D model: transverse uniform pancake beam with longitudinal modulation
where, rb is the radius of the pancake beam.
need to find realistic effective rb
radius dependence of impedance increases energy spread and so damping
define
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

29. LSC in accelerating structures
Initial beam
6MeV
No energy spread,
Current modulation M
drift 67 cm
Linac 3m
linac 3m
drift 1m
Summary PARMELA simulations
Summary Theory [Huang,Wu]
Courtesy of Z.Huang
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

30. LSC in gun Modulation introduced on the laser (as in real life)
Difficulties to study the effect for the LCLS pulse
Study of  = 50 m , dz = 5 m
z ~ 5 mm
Nz x Nr ~ 1000 x 20
200k particles
Per longitudinal bin ~ 200 particles , 1/200 ~7%
 For initial modulation of less than 7% modulation of current density at gun exit is in noise
Case studied :
 = 50 m, 100 m, 250 m, 500 m, 1000 m
For +/-10% and +/- 20% initial modulation amplitude
Other cases under study
 < 50 m but for a shorter pulse than the LCLS pulse
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31. +/-5% , = 50 m C.Limborg,Z.Huang,T.Shaftan,J.Wu
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32. +/-5% , = 100 m +/-5% , = 250 m C.Limborg,Z.Huang,T.Shaftan,J.Wu
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33. +/-5% , = 500 m
+/-5% , = 1000 m
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34. +/-10% +/-20% C.Limborg,Z.Huang,T.Shaftan,J.Wu
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35. Noise Level = 4% C.Limborg,Z.Huang,T.Shaftan,J.Wu
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C.Limborg,Z.Huang,T.Shaftan,J.Wu

36. LSC along the LCLS
Modulation introduced on the laser (as in real life)
Difficulties to study the effect for the LCLS pulse
Study of  = 50 m , dz = 5 m
z ~ 5 mm
Nz x Nr ~ 1000 x 20
200k particles
Per longitudinal bin ~ 200 particles , 1/200 ~7%
 For initial modulation of less than 7% , modulation in noise
Case studied :
 = 50 m, 100 m, 250 m, 500 m, 1000 m
For +/-10% and +/- 20% initial modulation amplitude
Other cases under study
 < 100 m but for a shorter pulse than the LCLS pulse
Theory Institute, ANL , 24 September 2003
C.Limborg,Z.Huang,T.Shaftan,J.Wu

37. At end LCLS injector beamline:
Current density modulation
strongly attenuated
residual energy oscillation has amplitude between 2 keV and 4 keV for wavelengths [50 m, 500 m]
Impedance defined by
Theory Institute, ANL , 24 September 2003
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38. Conclusion
Good agreement Simulations / Theory for drift and Acceleration
Discrepancies are under investigation
Intrinsic energy spread increase
Edge effects for bunched beam
More comparisons simulations /theory in high energy regime
Difficulties to do simulations for small modulation amplitude
Noise Problem
Shorter wavelengths, not enough particles
More comparisons experiments / simulations
“Attenuation” in gun might make situation not as critical as first thought
But not enough attenuation : extrapolation from 10% case
for wavelengths >100 m : attenuation line density modulation by factor of 3
for wavelengths <100 m : attenuation line density modulation by factor of 4
Beam Heater is under discussion for the LCLS beamline
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39. DISCUSSION ON EXPERIMENTAL RESULTS AT DUVFEL from T.Shaftan et al.
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40. Modulation analysis
An example of the chirped beam image. One of the interesting features here is evolution of the modulation wavelength along the bunch, corresponding to nonlinear chirp. Interpretation of double peaks on the left: “overmodulated” periods. Every couple of double spikes in this region represents a single modulation period. Tail of the bunch is folded back over, introducing bright region on the right side of the image (space charge).
“Analysis of space charge driven modulation in electron bunch energy spectra”,
T. Shaftan and L.H. Yu, BNL preprint.

41. “Overmodulation” effect
Measuring distance E between “double spikes” we can determine amplitude of the energy modulation.
where
is the energy modulation,  is the
modulation frequency
“Analysis of space charge driven modulation in electron bunch energy spectra”,
T. Shaftan and L.H. Yu, BNL preprint.

42. Sensitivity to the Transverse Beam Size
large linac beam (1 mm)
Space charge force is a function of rb
Change the beam size of the compressed beam along the accelerator  effect on modulation ?
200 A
40 A
small linac beam (250  m) 20 40 60 80
100 5 10 15 25 30 35 G a
200 A
40 A
Analytical calculations of “gain”
in the modulation by Z. Huang
3 different lattice solutions  3 different
RMS beam sizes along the accelerator

43. Results of the Experiment
“Zero-phasing” profiles of the beam (300 pC) for different lattice solutions:
Average RMS beam sizes along the accelerator: 0.25 mm, 0.5 mm, 1 mm

44. IR Radiation Measurements
Does modulation enhance any bunching in the bunch longitudinal density ?
Experiment:
Change the beam size  “modulated” and “non-modulated” bunch profiles
We measured CTR from metallic mirror, using IR detector and low-pass IR filters (cut-off of 40 µm, 100 µm, 160 µm)
Modulation wavelength = 90 µm (from “zero-phasing”)  expect enhancement of the coherent IR power in this spectral region if bunching
Result of the experiment:
No difference is found between“modulated” and “non-modulated” beam conditions
“Modulated” bunch profile
“Non-modulated” bunch profile
Bolometer signal, uVs
Filters:
>40 um
>100 um
>160 um
Wavelength, um

45. Dependence on Energy 60 MeV 80 MeV 110 MeV
Space charge force is a function of 
Change the energy of the compressed beam (300 pC) along the accelerator   effect on modulation ?
Vary tank 3 energy, maintaining the
same all other beam parameters (bunch
length, transverse beam size, charge)
80 MeV
110 MeV

46. (I) Results of the experiment
Number of modulation periods and modulation wavelength for different energies is different !
Product is constant: bunch length is the same for different energies
Why ?
“Experimental investigation of a space charge induced modulation in high-brightness electron beam”, T. Shaftan and Z. Huang, BNL JA

47. Model of “Modulation versus Energy” experiment
3 m long linac section
Space Charge “kicks”
“Initial phase space”
Beam: 70 MeV and other
parameters of the experiment
Slippage “kicks”
“Final phase space”, spectrometer
Assumptions: 1) Do not take into consideration bunch prehistory before chicane. We did not change anything but energy using tank after chicane.
2) Initial conditions at the end of the chicane  no initial energy modulation, certain amount of initial density modulation (spectral shape is unknown)

48. (II) Results of the experiment
50 MeV
Simulated normalized spectra of energy and density modulations for 50 MeV and 110 MeV
With energy increase average spectral frequency of energy modulation shifts toward higher freq. range
Therefore modulation wavelength decreases and number of modulation periods increases for higher energy
Another words:
Plasma oscillations at different frequencies accumulate different phase advance while the bunch travels down to the accelerator.
Initial density modulation
Final energy modulation
Final density modulation
Initial density modulation
Final energy modulation
Final density modulation
“Experimental investigation of a space charge induced modulation in high-brightness electron beam”, T. Shaftan and Z. Huang,
BNL JA
110 MeV