Beam Modulation due to

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Beam Modulation due to Longitudinal Space Charge
Zhirong Huang, SLAC Berlin S2E
Workshop 8/18/2003
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Introduction

• 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)

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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?) ltlt beam pipe radius

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Space Charge Oscillation

• If there is a density modulation, space charge
  pushes particles from high density to low
  density, creating energy modulation in the process

I
E

• Energy modulation converts back to density
  modulation to complete space charge oscillation
  with frequency

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Space Charge Oscillation II

• Density and energy modulation in a drift at
  distance s

• At a very large g, plasma phase advance (Ws/c)
  ltlt 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)

ignore in the linac
LSC
In 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 g, R560, b(k,s)b0(k,s), beam
  is frozen

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Comparison with Parmela

• Energy Modulation

• 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)

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(courtesy of J.H. Wu)
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,

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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

Parmela simulations (C. Limborg)

• LSC induced energy modulation in the LCLS
  injector is small at shorter wavelengths (lt250
  ?m), where the downstream gain is the highest
• Density modulation at these wavelengths is also
  reduced

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SDL microbunching experiment
(W. Graves, T. Shaftan et al.)
65 MeV Energy spectrometer
X (E) profile
E
E
E
E
z
z
z
z
z
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Long. Phase Space Distortion

• rf zero phasing energy spectrum is sensitive to
  beam energy
• modulation

• 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)

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Enhancement of horizontal modulation
Energy deviation chirp sinusoidal modulation
Total charge
Energy profile
or
magnification
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• Define gain x modulation amplitude/current
  modulation

I0300 A, ?130, rb600 ?m ? Gm gtgt 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 Its Effect on the modulation
Beam size in the zero-phasing linac is varied
(courtesy of T. Shaftan)
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IR measurements
Bolometer signal, uVs
(T. Shaftan)
gt40 um
Filters
gt100 um
gt160 um
Wavelength, um
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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)