Structure resonances due to space charge in periodic focusing channels

The Vlasov-Poisson model is one of the most effective methods to study the space charge dominated beam evolution self-consistently in a periodic focusing channel. Since the approach to get the solution with this model is not trivial, previous studies are limited in degenerated conditions, either in smoothed channel (constant focusing) [I. Hofmann, Phys. Rev. E 57, 4713 (1998)] or in alternating gradient focusing channel with equal initial beam emittance condition in the degrees of freedom [I. Hofmann et al., Part. Accel. 13, 145 (1983); Chao Li et al., THOBA02, IPAC2016]. To establish a basis, we intentionally limit this article to the study of the pure transverse periodic focusing lattice with arbitrary initial beam condition, and the same lattice structure in both degrees of freedom, but with possibility of different focusing strengths. This will show the extension of the existing work. The full Hamiltonian is invoked for a pure transverse focusing lattice in various initial beam conditions, revealing different mode structure and additional modes beyond those of the degenerated cases. Application of the extended method to realistic lattices (including longitudinal accelerating elements) and further details will then reveal many new insights, and will be presented in later work.

Figure 1

Locations of possible eigenvalues for map $M(S)$, where $S$ is period length.

Figure 2

Eigenvalues $Abs|\lambda |$ and eigenphases $Arg|\lambda |$ of the 2nd even mode, (a) and (b), and odd mode, (c) and (d)$vs$ tune depression ${\sigma }_y/{\sigma }_{y0}$, for a FODO channel with the parameter ${\sigma }_{x0}={\sigma }_{y0}=100°$, emittance ratio $\mathrm{\Gamma }=1/2$, filling factor $\eta =0.5$.

Figure 3

Eigenvalues (a) and eigenphases (b) of the 3rd even mode $vs$ tune depression ${\sigma }_y/{\sigma }_{y0}$ for FODO channel (filling factor $\eta =0.5$, emittance ratio $\mathrm{\Gamma }=1/2$) with the parameter ${\sigma }_{x0}={\sigma }_{y0}=80°$.

Figure 4

Eigenvalues and eigenphases of the 4th order even, (a) and (b), and odd mode, (c) and (d), $vs$ tune depression ${\sigma }_y/{\sigma }_{y0}$ for FODO channel (filling factor $\eta =0.5$, emittance ratio $\mathrm{\Gamma }=1/2$) with the parameter ${\sigma }_{x0}={\sigma }_{y0}=80°$.

Figure 5

The 3rd, (a) and (b), and the 4th, (c) and (d), order even modes eigenvalues and eigenphases as function of tune depression, when the external focusing is smoothed. The tune ratio is ${\sigma }_y/{\sigma }_x=0.48$, rms beam size ratio is $a/b=1.54$. It gives the same result as Fig. 5 and Fig. 7 shown in Ref. [6].

Figure 6

Eigenvalues and eigenphase evolution of the 3rd order even mode, (a) and (b), and the 4th order even mode, (c) and (d) $vs$ the tune depression in a smoothed constant focusing channel where fill factor $\eta =0.5$, emittance ratio $\mathrm{\Gamma }=1/2$, phase advance ${\sigma }_{x0}={\sigma }_{y0}=80°$.