From Bunch Wakes to Delta Wakes Adriana Bungau / Roger Barlow COLSIM meeting CERN, 1 st 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 W m (s)=2(1/1.9 2m -1./0.2 2m )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 Deconvolution W bunch (s,m)=W delta (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)=FT bw (k)/FT g (k) Cap factor |1./FT g (k)| at some value   =100 seems reasonable Lower values lose structure Higher 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 cm Oscillation 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.