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Beam Modulation due toLongitudinal Space Charge
Zhirong Huang, SLAC
Berlin S2E Workshop
8/18/2003
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• SDL microbunching observations through rf zero-phasing
• LSC driven microbunching instability (TESLA-FEL-2003-02)
• Injector modulation studies
Important to know beam modulation induced by LSC • Discuss methods to evaluate current and energy modulation in the linac
• Discuss its impact on rf zero-phasing measurements
• Do not discuss gain in bunch compressors (until Thursday)
Introduction
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LSC Impedance• For a round, parallel electron beams with a uniform transverse cross section of radius rb, the longitudinal space charge impedance on axis is (cgs units)
• Off-axis LSC is smaller and can increase the energy spread
• Free space approximation is good when /(2) << beam pipe radius
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Space Charge Oscillation
• Energy modulation converts back to density modulation to complete space charge oscillation with frequency
• If there is a density modulation, space charge pushes particles from high density to low density, creating energy modulation in the processI E
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Space Charge Oscillation II• 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-2003-02)
• LSC acts like a normal impedance at high energies
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Non-rigid beam• At lower energies (in the injector…), beam is not rigid • Space charge simulations may be time-consuming and noisy at high frequencies
• Linear evolution of high-frequency beam modulations can be described by the same integral equation for CSR microbunching (Heifets et al., PRSTAB-064401; Huang/Kim, PRSTAB-074401)
dampingLandau
...),exp(...)),(()()()(),( kernel
));((),());(());((
56
00
kZIIsRsiksK
kbsKdsskbsskb
A
s
ignore in the linacLSCIn a drift
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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 :
• The integral equation for LSC microbunching in the linac is
• In a drift,
Space charge oscillation
• For very large , R56’=0, b(k,s)=b0(k,s), beam is “frozen”
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Comparison with Parmela
• Parmela simulations (C. Limborg) of a 3-m drift at 6 and 12 MeV (beam size changes due to optics and transverse SC)• Theory-1D: integral equation using average LSC impedance• Theory-3D takes into account transverse variations of LSC (J.H. Wu)
• Energy Modulation
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LSC 3-D Model• LSC impedance is r-dependant, which leads to decoherence
• We have
• Impedance at arbitrary radial coordinate r from a -ring with unit charge and radial coordinate a is
• Convolution with a Parabolic distribution,
)()()()(2)(
)()(2)(),(
0201
002
RKIIIR
RIKRikRZ ringLSC
);;( and );;( rskrskb
2
2
2 31
32)(
rr
aan
(courtesy of J.H. Wu)
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Comparison with Elegant• Borland implemented 1-D LSC impedance in elegant
• Current modulation at different accelerating gradients
Elegant tracking (M. Borland) Analytical calculation
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Injector Modulation Studies• Assume 10% initial density modulation at gun exit at 5.7 MeV
• After 67 cm drift + 2 accelerating structures (150 MeV in 7 m), LSC induced energy modulation
• LSC induced energy modulation in the LCLS injector is small at shorter wavelengths (<250 m), where the downstream gain is the highest• Density modulation at these wavelengths is also reduced
Parmela simulations (C. Limborg)
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SDL microbunching experiment
E
z
E
z
E
z z
E
z
65 MeVEnergy
spectrometer
(W. Graves, T. Shaftan et al.)
X (E) profile
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Long. Phase Space Distortion
• Small modulation gets projected to large modulation• Energy modulation can be induced by LSC in the zero-phasing section if c/ » L (length of the section, ~15 m)
• rf zero phasing energy spectrum is sensitive to beam energymodulation
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Energy deviation = chirp + sinusoidal modulation
or
magnification
Total charge
Enhancement of horizontal modulation
Energy profile
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• Define “gain” = x modulation amplitude/current modulation
I0=300 A, =130, rb=600 m Gm >> l
(Z. Huang, T. Shaftan, SLAC-PUB-9788, 2003)
• zero-phasing images are dominated by effects of energy modulation instead of current modulation
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Beam size and It’s Effect on the modulationBeam size and It’s Effect on the modulation
0 50 100 150 200 250 300 350 4000
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1402
0 50 100 150 200 250 300 350 4000
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Beam size in the zero-phasinglinac is varied (courtesy of T. Shaftan)
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0 50 100 150 2000
50
100
150
200
250 a
IR measurements1 2
Wavelength, um
Bolometer signal, uVs
>40 um >100 um >160 umFilters:
(T. Shaftan)
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Summary• LSC induced modulation in the linac can be described by a modified integral equation that includes acceleration
• Comparable energy modulation with Parmela simulations
• Initial studies suggest that accumulated energy modulation at the end of the injector is small at the most dangerous modulation wavelengths for LCLS
• Density modulation is reduced in the injector, but can be amplified by downstream bunch compressors…
• Energy spectrum of a chirped beam is sensitive to beam energy modulation, which could be induced by LSC in the SDL linac ( means to measure energy modulation)