Observation of Coherent Oscillations

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Observation of Coherent Oscillations

• Susumu KAMADA / KEK

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How and what can we learn from the data taken by
a turn-by-turn monitor?

• Cooking method of data
• Related beam physics
• Experiments, analytic approaches and simulations

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Cooking method of data(1)
Attention! Figures are not consistent each other.
(1) Raw data
(3a) Amplitude decay
turns
(2) Fourier transformation
(3b) Tune decay
tune
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Cooking method of data(2)

• From the sampled signal of a turn-by-turn
  position monitor, the instantaneous spectrum is
  calculated with a series of data which
  corresponds to N256 turns, for example, and is
  applied the Hanning window as a weight function.
• The time variation of the spectrum is obtained by
  shifting the data for Fourier transformation by
  128 turns, for example.
• In order to find the frequency and amplitude of
  the peak precisely, parabolic fitting is used.
  Where the true continuous spectrum around the
  peak is approximated by a parabola fitted to the
  largest three points.
• The time variations of the tune and the amplitude
  are given for the transverse collective
  oscillation of the bunch.

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Related beam physics

• Radiation damping negligibly slow in many cases
• Head-tail effect damping
• Amplitude dependent tune shift measure lattice
  nonlinearity
• Nonlinear smear (filamentation) Landau damping
• Nonlinear Head-tail effect Suppression of
  damping

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Head tail effect

• Data taken at the interleaved TRISTAN optic by
  horizontal kick
• Damping rate is proportional to bunch current
• Oscillation amplitude decays exponentially

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Head-tail damping rate

• Data taken at the non-interleaved TRISTAN optic
• Head-tail damping rate is proportional to
  chromaticity times bunch current

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Amplitude dependent tune shiftexperiment(1)

• Data taken at the interleaved TRISTAN optic
• Amplitude dependent tune shift is positive and
  fairy large
• Power line ripple is seen at 50Hz

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Amplitude dependent tune shiftexperiment (1) and
SAD simulation

• Data taken at the interleaved TRISTAN optic
• Particle tracking by SAD agrees very well with
  the measurement.

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Amplitude dependent tune shift Amplitude vs Tune

• Data taken at the KEKB under fairly scattered
  conditions
• Be aware the real tune is above the half integer
  so that the figure must be turned over.
• Suggests the parabolic positive dependence of
  tune on the amplitude.

Tune
Amplitude
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Nonlinear smearAppears at small bunch current

• Data taken at the non-interleaved TRISTAN optic
• Amplitude decays in exponential at large bunch
  current and in square exponential at small bunch
  current

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Nonlinear smear oscillation damping rate

• Data taken at the non-interleaved TRISTAN optic
• rate of nonlinear smear is proportional to kick
  amplitude
• nD head-tail damping radiation damping
• nNF nonlinear smear

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Nonlinear smeardamping rate analytic expression

• The lowest order nonlinear-coefficients of
  lattice, Hamiltonian and tune

• Nonlinear damping rate nNF is given by the
  following when coherent kick is so large that
  horizontal and longitudinal motion can be
  negligible.

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Nonlinear smearAction distribution of Gaussian
beam

• Before coherent kickexponential
• After small amplitude coherent kickmixture
• After large amplitude coherent kickGauss

Before kick
Jm
After large kick
After small kick
Jm
Jm
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Nonlinear smear Tune distribution

• (a) tune distribution and Gaussian fit after
  small amplitude kick
• (b) tune distribution and Gaussian fit after
  large amplitude kick

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Nonlinear smear Nonlinear coefficients for the
simulation
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Vert. emittance vs nonlinear damping rate kick
amplitude and initial emittance (a)

• The interleaved TRISTAN optic
• Vertical emittance measurement seems possible by
  this kind of experiment.
• BUT, Nonlinear Head-tail effect will affect it.

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Vert. emittance vs nonlinear damping rate kick
amplitude and initial emittance (b)

• The non-interleaved TRISTAN optic
• Agreement is quite poor even between tracking and
  analytic calculation, higher order nonlinear-term
  must be considered.
• Experimental observation is highly affected by
  errors, such as spurious nonlinearity.

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Vert. emittance vs nonlinear damping rate kick
amplitude and initial emittance (c)

• Simulation for KEKB-LER, Non-interleaved optic
• Non-sextupole contribution to nonlinear
  coefficients is large, Fringe of IR Qs
• Nonlinear Head-tail effect will affect

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Nonlinear head-tail effect 2 particle
modelevolution of X1 and X2

• (a) amplitude-dependent tune shift Positive
• (b) amplitude-dependent tune shift Negative

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Nonlinear head-tail effect multi-particle
tracking

• five macro-particles out of 1000 or 2000
  macro-particles
• (a) amplitude-dependent tune shift Positive
• (b) amplitude-dependent tune shift Negative

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Nonlinear head-tail effect multi-particle
tracking

• phase space distribution of particles after 500
  turns

• (a) amplitude-dependent tune shift Positive

• (b) amplitude-dependent tune shift Negative

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Nonlinear head-tail effectSimulation and
Experiment

• Simulation
• (a) amplitude-dependent tune shift Positive
• (b) amplitude-dependent tune shift Negative
• Experiment at KEK-PF where octupoles can change
  the sign of tune shift
• (c) amplitude-dependent tune shift Positive
• (d) amplitude-dependent tune shift Negative
• Good agreement including echo effect

SIMULATION
EXPERIMENT
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References

• S.Kamada, Proc of the Workshop on Nonlinear
  Dynamics in Particle Accelerators, Arcidosso,
  Italy,1994, AIP Conf. Proc. No.344,1(1995)
• N.Akasaka and S.Kamada, Proc. of EPAC96, 1141
  (1996)
• S.Kamada, N.Akasaka and K.Ohmi, Proc. of
  Advanced ICFA Workshop on Beam Dynamics Issues
  for ee- Factories,INFN, Frascati(Rome),
  Oct.20-25, 1997 (1998)
• K.Ohmi and Y.Kobayashi, Phys. Rev. E Vol.59 No.1
  1167(1999)