Pragmatic Method of Deducing a Wake Function for a General 3D Structure

A key quantity in simulating collective beam instabilities is the wake potential of a bunch of particles whose charge distribution is continuously evolving in time. However, obtaining such wake potential is only possible if a wake excited by a single particle in the surrounding environment is known. A practical self-consistent approach was developed to obtain an effective wake function from a numerical wake potential computed for a finite length bunch. The wake potential is processed to a numerical impedance which is decomposed into a set of well-known analytical wake functions. The decomposed impedance is then transformed back into time domain and, thus, converted into an effective wake function which is by nature physical and most consistent with the numerical wake potential. Though the method is limited by the initial numerical impedance data and the choice of impedance decomposition, the retrieved wake function can be used in instability simulations with a bunch whose length is comparable to that used in the electromagnetic field solver. We show that the method can be applied to a general 3D structure, which allows finding effective wake functions of realistic vacuum chambers.