The project of a high-power FEL at KAERI

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Optimal Beamlines for Beams with Space Charge
Effect
S.V.Miginsky Budker Institute of Nuclear
Physics, Novosibirsk, Russia
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Basics and previously published results
An optimizer for high-current beamlines. ICAP04
-gt NIM A 558 (2006). Minimization of Space Charge
Effect. SR'06 -gt NIM ???
Kapchinsky Vladimirsky rms values
Space charge can be neglected only if
AND!
If space charge is significant and charge density
is not homogeneous, emittance dilution occurs due
to nonlinearity of the effect.
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Basic space charge effects
Effect of longitudinal inhomogeneity each slice
is uniformly charged and independent from others.
Slices have different currents and states.
Effect of transverse inhomogeneity nonlinear
radial force
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Basic assumptions

• Longitudinal
• circular symmetric beam
• slices are charged uniformly
• no thermal emittance
• slices move independently
• independent on transverse inhomogeneity.

• Transverse
• circular symmetric beam
• no thermal emittance
• laminar motion trajectories don't cross each
  other
• independent on longitudinal inhomogeneity.

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Emittance compensation
Longitudinal
Small (linear!) vibration around main phase
trajectory, x dx x(1d)
Transverse
If the initial beam emittance is zero and
vibration is perfectly linear, phase portraits of
all the slices coincide twice per a period of
vibration and the emittance is zeroed at these
points.
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Basic results

• Emittance dilution in an optimized space charge
  affected beamline is many times less than in
  arbitrary one due to emittance compensation
  effect.
• Effect of longitudinal inhomogeneity is typically
  the strongest.
• Effect of transverse inhomogeneity is usually
  weaker, but comparable to the previous one.
• Combined effect causes greater emittance
  dilution. Parameters of the optimal beamline in
  this case are compromise between the two above
  cases.
• The best emittance compensation occurs when the
  charge vibration phase is about 2p. The emittance
  is significantly better in 2np minima, than in
  (2n1)p ones.
• The obtained analytical formulae improved by
  numerical simulation provide good estimations of
  emittance dilution in optimized beamlines in the
  mentioned cases. They also gives approximate
  parameters of optimized beamlines.
• The differences between uniformly focusing
  beamlines and non-uniform ones are moderate, so
  the obtained estimates can be applied well to
  arbitrary beamlines.

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Basic scaling
I peak current r rms beam size n f
2np ? a coefficient depending on the
considered effect, the type of the beamline, etc.
810-3510-2

• Valid for longitudinal, transverse and combined
  effect for uniform and nonuniform beamlines.
• For optimal beamlines only. An optimal beamline
  is to provide matched focusing and f 2np.
• Parameters of optimal beamlines were estimated.
• All these parameters are considered as a good
  initial approximation for a designed beamline.

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Bunching basic equations
Longitudinal effect. Persistent energy. Matched continuous focusing x const.
Basic equation
Hamiltonian
Charge vibration phase taper
Adiabatic damping
Valid only if
?
Analytical estimate
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Bunching simulation

• Effect of longitudinal inhomogeneity.
• 2p-minimum emittance (triangles red solid line
  0.0215?1/3) .
• Optimal focusing (blue dashed line).
• Optimal length (green dash-dot line),
  15.7?-1/3.

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Bunching simulation

• Effect of longitudinal inhomogeneity.
• No bunching, ? 1.
• Emittance vs. length L and focusing g .

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Bunching simulation

• Effect of longitudinal inhomogeneity.
• ? 2 bunching.
• Emittance vs. length L and focusing g .

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Bunching simulation

• Effect of longitudinal inhomogeneity.
• ? 5 bunching.
• Emittance vs. length L and focusing g .

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Bunching simulation

• Effect of longitudinal inhomogeneity.
• ? 10 bunching.
• Emittance vs. length L and focusing g .

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Bunching simulation

• Effect of longitudinal inhomogeneity.
• ? 20 bunching.
• Emittance vs. length L and focusing g .

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Bunching simulation

• Effect of combined inhomogeneity.
• 2p-minimum emittance (triangles red solid line
  0.0349?0.28).
• Optimal focusing (blue dashed line).
• Optimal length (green dash-dot line),
  12.7?-0.28.

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Bunching simulation

• Effect of combined inhomogeneity.
• No bunching, ? 1.
• Emittance vs. length L and focusing g .

17
Bunching simulation

• Effect of combined inhomogeneity.
• ? 2 bunching.
• Emittance vs. length L and focusing g .

18
Bunching simulation

• Effect of combined inhomogeneity.
• ? 5 bunching.
• Emittance vs. length L and focusing g .

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Bunching simulation

• Effect of combined inhomogeneity.
• ? 10 bunching.
• Emittance vs. length L and focusing g .

20
Bunching simulation

• Effect of combined inhomogeneity.
• ? 20 bunching.
• Emittance vs. length L and focusing g .

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Accelerating basic equations
Longitudinal effect. Persistent current. Matched continuous focusing x const.
Basic equation
Hamiltonian
Charge vibration phase taper
Adiabatic damping
Valid only if
a
Analytical estimate
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Accelerating simulation

• Effect of longitudinal inhomogeneity.
• 2p-minimum emittance (triangles red solid line
  0.0220a-0.136).
• Optimal focusing (blue dashed line).
• Optimal length (green dash-dot line), 11.96
  6.05a.

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Accelerating simulation

• Effect of longitudinal inhomogeneity.
• No accelerating, a 1.
• Emittance vs. length L and focusing g .

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Accelerating simulation

• Effect of longitudinal inhomogeneity.
• a 2 accelerating.
• Emittance vs. length L and focusing g .

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Accelerating simulation

• Effect of longitudinal inhomogeneity.
• a 5 accelerating.
• Emittance vs. length L and focusing g .

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Accelerating simulation

• Effect of longitudinal inhomogeneity.
• a 10 accelerating.
• Emittance vs. length L and focusing g .

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Accelerating simulation

• Effect of longitudinal inhomogeneity.
• a 20 accelerating.
• Emittance vs. length L and focusing g .

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Accelerating simulation

• Effect of combined inhomogeneity.
• 2p-minimum emittance (red solid line).
• Optimal focusing (blue dashed line),
  0.115a0.227.
• Optimal length (green dash-dot line), 10.89
  5.03a.

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Accelerating simulation

• Effect of combined inhomogeneity.
• No accelerating, a 1.
• Emittance vs. length L and focusing g .

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Accelerating simulation

• Effect of combined inhomogeneity.
• a 2 accelerating.
• Emittance vs. length L and focusing g .

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Accelerating simulation

• Effect of combined inhomogeneity.
• a 5 accelerating.
• Emittance vs. length L and focusing g .

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Accelerating simulation

• Effect of combined inhomogeneity.
• a 10 accelerating.
• Emittance vs. length L and focusing g .

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Accelerating simulation

• Effect of combined inhomogeneity.
• a 20 accelerating.
• Emittance vs. length L and focusing g .

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Summary formula coeffs
Beamline Beam ?
Bunching, ? Iend/Istart Transverse uniform longitudinal Gaussian 0.0215?1/3
Totally Gaussian 0.0349?0.28
Accelerating, a (ß?)1/(ß?)0 Transverse uniform longitudinal Gaussian 0.0220a-0.136
Totally Gaussian 0.0350