From Bunch Wakes to Delta Wakes

1
From Bunch Wakes to Delta Wakes

• Adriana Bungau / Roger Barlow
• COLSIM meeting
• CERN, 1st March 2007

2
The question

• Tracking Programs (Placet, Merlin) need Delta
  Wakes the effect on one particle of a preceding
  particle
• EM codes (GDFIDL, ECHO) give Bunch Wakes the
  effect on one particle of the preceding part of
  the (Gaussian) bunch
• To get bunch wakes from delta wakes, just
  integrate
• How do you get delta wakes from bunch wakes?

3
Why do we want to know?

• To handle non-Gaussian bunches
• validate the formulae in the literature, with
  their different regions of validity
• obtain numerical interpolation tables for delta
  wakes of collimator shapes with no formula in the
  literature

4
How not to do it

• Simulate point charge delta function as Gaussian
  bunch with very very small ?
• Why not? Because EM simulations need cell size
  ltlt ?
• And computation time ? (cell size)-2 at least

5
Alternative approach

• Bunch wake is convolution of delta wake with
  Gaussian bunch shape
• FT of convolution is product of FTs
• Fourier Transform Bunch wake
• Divide by FT of Gaussian (also Gaussian)
• Transform back to time domain

6
Example

• Take beam pipe radius 19 mm
• Taper in to 2 mm over 50 mm
• Taper out again

Not to scale!
7
Analytic answer

• Wm(s)2(1/1.9 2m-1./0.22m)e-ms/0.2??(s)
• Zotter Kheifets
• and elsewhere
• Modal decomposition

8
Bunch wake simulation

• Simulated using Echo-2D (Igor Zagorodnov)
• Gaussian beam, ?0.1 cm
• Need to follow for 200 mesh points, not the
  default 52

9
Fourier Deconvolution

• Wbunch(s,m)Wdelta(s,m)??Gaussian

Take FT of ECHO result (here mode1) and FT of
Gaussian (red and blue are sine and cosine
parts) Divide to obtain FT of delta
wake Back-transform.Horrible! (Look at y axis
scale) But mathematically correct combined with
Gaussian reproduces original Due to noise in
spectra at high frequency. Well known problem
10
Apply simple inverse filter

• FTdw(k)FTbw(k)/FTg(k)
• Cap factor 1./FTg(k)
• at some value ?
• ? 100 seems reasonable
• Lower values lose structure
• Higher values gain
• noise

11
Reconstructed delta wakes

• Compare with analytic formula qualitative
  agreement on increase in size and decrease in
  width for higher modes
• Overall scale factor not understood yet
• Positive excursions not reproduced by formula
• At least one of them is wrong

12
EM simulation different bunches

• Bunch wakes for different Gaussian beams
• ?0.1 cm
• ?0.2 cm
• ?0.05 cm
• Oscillation in green curve (s0.05cm) due to
  ECHO2D grid size 0.01 cm

13
Delta wakes Consistency check

• Give the same delta wakes
• Use FT to extract delta wakes from the different
  bunch wakes
• Agreement reasonable method validated
• Green oscillation artefact of ECHO2D, not of
  Fourier extraction

14
Next steps

• Use more sophisticated filter, incorporating
  causality (W(s)0 for slt0)
• Compare simulations and formulae and establish
  conditions for validity
• Use Delta wakes extracted from simulations in
  Merlin/Placet through numerical tables, for
  collimators where analytical formulae not known
• Extend to non-axial collimators.