Collimator Wake fields : Formulae and Simulations From Bunch Wakes to Delta Wakes

1. + + + + Roger Barlow, Adriana Bungau, and Roger Jones University of Manchester and the Cockcroft Institute, ILC collimators Small apertures: Effects of high (angular) order transverse short range wakes need to be studied Bunch wakes Particle affected by all leading particles Given by EM solvers (GdfidL, ECHO, MAFIA) + + + + Delta wakes Particle affected by single leading particle Needed by tracking simulations (Merlin, PLACET) Delta wakes Need to know Wm(s) for various spoiler/absorber shapes and materials Some formulae exist, but only for special cases Bunch Wakes from Delta wakes Integrate or sum: Wb(s)=(s’) W(s’-s) ds’ Delta wakes from Bunch wakes through Deconvolution Bunch = Delta  Gaussian Delta = Bunch  Gaussian Take FT of bunch. Divide by FT of Gaussian (another Gaussian). Back-transform Try on bunch wakes from ECHO2D for simple tapered collimator (a=19 mm, b=2 mm) Crazy result (though mathematically correct) Fluctuations in denominator of division at high frequencies Smooth by simple inverse filter: cap multiplication by  =10 gives sensible-looking results (less loses detail, more gives noise) Extract first 4 transverse modes Similar to but different from commonly used formula 2(1/a2m- 1/b2m )exp(-ms/a) (s) But still some structure which is clearly an artefact of the transformations – has non-physical non-zero values for s<0 Collimator Wake fields : Formulae and Simulations From Bunch Wakes to Delta Wakes Imposing Causality General Fourier expansion f(x)=b0+ ak sin(kx) +  bk cos(kx) Ensure f(x)=0 for all points x=r/L by b0=b1-b2+b3-b4… ak=N/2 (b0+  sin (kj/L) cos(jm/L) bm) For any set of cosine coefficients bj, fixing other coefficients in this way uniquely guarantees a causal function Determine bj though 2 fit to the data and use Tikhonov (k=2) regularisation for smoothing Result still has some artefacts (work in progress) – but generally confirms result from simple damping. There is a lot of structure – simple formulae not fully adequate Can be done for any axially symmetric collimator Results can be written to table usable by particle tracking codes m=1