From Bunch Wakes to Delta Wakes

Adriana Bungau / Roger Barlow

COLSIM meeting

CERN, 1st March 2007

Slide 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?

Slide 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

Slide 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 <<

And computation time (cell size)-2 – at least

Slide 5

Alternative approach

Bunch wake is convolution of delta wake with Gaussian bunch shape

FT of convolution is product of FTs

1. Fourier Transform Bunch wake

2. Divide by FT of Gaussian (also Gaussian)

3. Transform back to time domain

Slide 6

Example

• Take beam pipe radius 19 mm

• Taper in to 2 mm over 50 mm

• Taper out again

Not to scale!

Slide 7

Analytic answer

Wm(s)=2(1/1.9 2m-1./0.22m)e-ms/0.2(s)

Zotter & Kheifets

and elsewhere

Modal decomposition

Slide 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

Slide 9

Fourier DeconvolutionWbunch(s,m)=Wdelta(s,m)Gaussian Take FT of ECHO result (here mode=1) 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

Slide 10

Apply simple inverse filter

FT(k)=FTbw(k)/FTg(k)

Cap factor |1./FTg(k)| at some value =100 seems

reasonable Lower values lose

structureHigher values gain

noise

Slide 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’

Slide 12

EM simulation: different bunches

Bunch wakes for

different Gaussian beams:

=0.1 cm =0.2 cm =0.05 cmOscillation in green curve

(=0.05cm) due to ECHO2D grid size 0.01 cm

Slide 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

Slide 14

Next steps

• Use more sophisticated filter, incorporating causality (W(s)=0 for s<0)

• 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.