Vlasov analysis of microbunching instability for magnetized beams

For a high-brightness electron beam with high bunch charge traversing a recirculation beam line, coherent synchrotron radiation and space charge effects may result in microbunching instability (MBI). Both tracking simulation and Vlasov analysis for an early design of a circulator cooler ring (CCR) for the Jefferson Lab Electron Ion Collider (JLEIC) reveal significant MBI [Ya. Derbenev and Y. Zhang, Proceedings of the Workshop on Beam Cooling and Related Topics, COOL’09, Lanzhou, China, 2009 (2009), FRM2MCCO01]. It is envisioned that the MBI could be substantially suppressed by using a magnetized beam. In this paper we have generalized the existing Vlasov analysis, originally developed for a nonmagnetized beam (or transversely uncoupled beam), to the description of transport of a magnetized beam including relevant collective effects. The new formulation is then employed to confirm prediction of microbunching suppression for a magnetized beam transport in the recirculation arc of a recent JLEIC energy recovery linac (ERL) based cooler design for electron cooling. It is found that the smearing effect in the longitudinal beam phase space originates from the large transverse beam size as a nature of the magnetized beams and becomes effective through the $x-z$ correlation when the correlated distance is larger than the microbunched scale. As a comparison, MBI analysis of the early design of JLEIC CCR is also presented in this paper.

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Figure 1

Schematic layout of the early CCR design for the JLEIC [44].

Figure 2

CSR gain functions $G(s)$ for MEIC CCR lattice: (red) $\lambda =100\mu \mathrm{m}$, (green) $\lambda =350\mu \mathrm{m}$, (blue) $\lambda =1000\mu \mathrm{m}$.

Figure 3

CSR gain spectra as a function of initial modulation wavelengths for the JLEIC CCR lattice. $G_f$ is evaluated as one-turn microbunching gain. The red curve is obtained by the formulation developed in this paper, while the blue curve is from our previously developed semianalytical Vlasov solver [43] for nonmagnetized beams.

Figure 4

Longitudinal phase space distributions for the JLEIC CCR: left: initial quiet beam; right: when the beam circulates one turn. Note that $z>0$ is for bunch head.

Figure 5

Schematic layout of the JLEIC ERL cooler design [45].

Figure 6

(a) Bunch decompression along the arc; (b) microbunching gain function $G(s)$ for $\lambda =300\mu \mathrm{m}$; (c) gain spectrum; (d) Derbenev ratio as a function of $s$. Red dots represent elegant tracking with steady-state (ss) CSR effect.

Figure 7

Microbunching gain spectra for (a) density-to-density; (b) density-to-energy; (c) energy-to-density; and (d) energy-to-energy modulation. Note that in the figures the resultant modulations are evaluated in units of initial modulations.

Figure 8

Initial current dependence of the maximal microbunching gains for the example arc. In the simulation we have included CSR and LSC effects.

Figure 9

Steady-state CSR gain functions $G(\mathrm{s})$ for the JLEIC CCR lattice. Note here that $\lambda =300\mu \mathrm{m}$ for both the semianalytical solution and elegant tracking. In elegant tracking we impose an initial density modulation amplitude 0.2% on a flattop density distribution.

Figure 10

Steady-state CSR gain spectrum as a function of the initial modulation wavelength for the JLEIC CCR lattice. Here we fix the initial density modulation amplitudes to be 0.2% for various modulation wavelengths to obtain the final gain spectrum.