Rigorous formulation of space charge wake function and impedance by solving the three-dimensional Poisson equation

Shobuda, Yoshihiro   ; Chin, Y. H.*

In typical numerical simulations, the space-charge force is calculated by slicing a beam into manylongitudinal segments and by solving the two-dimensional Poisson equation in each segment. Thismethod neglects longitudinal leakage of the space-charge force to nearby segments owing to itslongitudinal spread over 1/Lorentz-. By contrast, the space-charge impedance, which is the Fourier transformof the wake function, is typically calculated directly in the frequency-domain. So long as we followthese approaches, the longitudinal leakage effect of the wake function will remain to be unclear. In thepresent report, the space-charge wake function is calculated directly in the time domain by solving thethree-dimensional Poisson equation for a longitudinally Gaussian beam. We find that the leakage effectis insignificant for a bunch that is considerably longer than the chamber radius so long as the segmentlength satisfies a certain condition. We present a criterion for how finely a bunch should be sliced so thatthe two-dimensional slicing approach can provide a good approximation of the three-dimensional exactsolution.