Conditions for coherent-synchrotron-radiation-induced microbunching suppression in multibend beam transport or recirculation arcs

The coherent synchrotron radiation (CSR) of a high-brightness electron beam traversing a series of dipoles, such as transport or recirculation arcs, may result in beam phase space degradation. On one hand, CSR can perturb electron transverse motion in dispersive regions along the beam line and possibly cause emittance growth. On the other hand, the CSR effect on the longitudinal beam dynamics could result in microbunching instability. For transport arcs, several schemes have been proposed to suppress the CSR-induced emittance growth. Correspondingly, a few scenarios have been introduced to suppress CSR-induced microbunching instability, which however mostly aim for linac-based machines. In this paper we provide sufficient conditions for suppression of CSR-induced microbunching instability along transport or recirculation arcs. Examples are presented with the relevant microbunching analyses carried out by our developed semianalytical Vlasov solver [C.-Y. Tsai, D. Douglas, R. Li, and C. Tennant, Linear microbunching analysis for recirculation machines, Phys. Rev. ST Accel. Beams 19, (2016)]. The example lattices include low-energy ($\sim 100\mathrm{MeV}$) and high-energy ($\sim 1\mathrm{GeV}$) recirculation arcs, and medium-energy compressor arcs. Our studies show that lattices satisfying the proposed conditions indeed have microbunching gain suppressed. Beam current dependences of maximal CSR microbunching gains are also demonstrated, which should help outline a beam line design for different scales of nominal currents. We expect this analysis can shed light on the lattice design approach that aims to control the CSR-induced microbunching.

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Figure 1

Illustration of a two-dipole system. The in-between section can be a general transport section (see context for definition of notations).

Figure 2

$|R_{56}^{s_1\to s_2}|$ pattern for an achromatic and isochronous unit, $R_{16}=R_{26}=R_{56}=0$. (Top) $R_{56}^{s_1\to s_2}(\alpha ,\psi )$ for (a) small and (b) large $\beta$ functions; (bottom) $R_{56}^{s_1\to s_2}(\beta ,\psi )$ for (c) small and (d) large $\alpha$ functions. The color bar is measured in meters.

Figure 3

Twiss functions and $R_{56}(\mathrm{s})$ for HERA v1 [(a) and (c)] and v2 [(b) and (d)].

Figure 4

Relative momentum compaction function $|R_{56}^{s\prime \to s}|$ for HERA v1 (left) and v2 (right). Note that $s^{\prime }$ is denoted as the source position and $s$ the observation or test position. The function is evaluated by Eq. (5). We quantify $\mathrm{\Delta }L$ as 40 m and 20 m for HERA v1 and v2. The double-headed arrows indicate the effective length $\mathrm{\Delta }L$.

Figure 5

Microbunching gain spectra for HERA v1 (left) and v2 (right) including steady-state (s.s.) only, steady-state plus entrance (tr.) and drift (drif.) transients, and the integrated (CSR-plus-LSC) effects. The dots are taken from particle tracking simulation by elegant for steady-state CSR. For HERA v1, the dots are taken as the average over a set of data with initial modulation amplitude 0.05% and total of 70-million macroparticles are used. For HERA v2, the initial modulation amplitude is set 0.8% and the same number of macroparticles as HERA v1 is used. Detailed information of the impedance models of entrance transient and drift CSR and LSC effects can be found in Ref. [18].

Figure 6

Twiss $\alpha$ functions at dipoles and betatron phase differences ${\psi }_{21}$ (in unit of $\pi$) for HERA v1 [(a) and (c)] and v2 [(b) and (d)]. The dashed lines in (c) and (d) only help visualize the dispersion within dipoles and do not represent exact behavior of dispersion functions.

Figure 7

Dependence of maximal CSR gains of HERA v1 (left) and v2 (right) on initial (peak) bunch current.

Figure 8

$R_{56}(\mathrm{s})$ for LERA v1 (left) and v2 (right).

Figure 9

Relative momentum compaction function $|R_{56}^{s\prime \to s}|$ for LERA v1 (left) and v2 (right). Note that $s^{\prime }$ is denoted as the source position and $s$ the observation or test position. We quantify $\mathrm{\Delta }L$ as 4 m and 2 m for LERA v1 and v2. The double-headed arrows indicate the effective length $\mathrm{\Delta }L$.

Figure 10

CSR microbunching gain spectrum for LERA v1 (left) and v2 (right). Only steady-state CSR is included. The dots are taken from particle tracking simulation by elegant, with a total of 40-million (with 0.1% initial modulation amplitude) and 10-million (with 0.5% initial modulation amplitude) macroparticles used for LERA v1 and v2, respectively.

Figure 11

Twiss $\alpha$ functions at dipoles and betatron phase differences ${\psi }_{21}$ (in unit of $\pi$) for LERA v1 [(a) and (c)] and v2 [(b) and (d)]. The dashed lines in (c) and (d) only help visualize the dispersion within dipoles and do not represent exact behavior of dispersion functions.

Figure 12

Dependence of maximal CSR gains on (peak) bunch current for LERA v1 (left) and v2 (right).

Figure 13

$R_{56}(\mathrm{s})$ for FODO compressor arc (left) and modulated compressor arc (right).

Figure 14

Relative momentum compaction function $|R_{56}^{s\prime \to s}|$ for FODO compressor arc (left) and modulated compressor arc (right). Note that $s^{\prime }$ is denoted as the source position and $s$ the observation or test position. $\mathrm{\Delta }L$ is approximately quantified to be 2.7 and 2.4 m for FODO and modulated arcs, though the periodicity of the quilt patterns is not obvious here.

Figure 15

Twiss $\alpha$ functions at dipole locations and betatron phase differences ${\psi }_{21}$ (in unit of $\pi$) for FODO compressor arc [(a) and (c)] and modulated compressor arc [(b) and (d)]. The dashed lines in (c) and (d) only help visualize the dispersion within dipoles and do not represent exact behavior of dispersion functions.

Figure 16

CSR microbunching gain spectrum for FODO compressor arc (left) and modulated compressor arc (right). Note that only steady-state CSR is included. The dots in the right figure are taken from elegant by tracking a total of 60-million macroparticles with the initial modulation amplitude set 1%.

Figure 17

$|R_{56}^{s_1\to s_2}|$ pattern for a dispersive but isochronous unit, $R_{16}=1\mathrm{m}$, $R_{26}=0$, $R_{56}=0$. (Top)$R_{56}^{s_1\to s_2}(\alpha ,\psi )$ for (a) small and (b) large $\beta$ functions; (bottom) $R_{56}^{s_1\to s_2}(\beta ,\psi )$ for (c) small and (d) large $\alpha$ functions.

Figure 18

$|R_{56}^{s_1\to s_2}|$ pattern for an achromatic but nonisochronous unit, $R_{16}=0$, $R_{26}=0$, $R_{56}=0.05\mathrm{m}$. (Top)$R_{56}^{s_1\to s_2}(\alpha ,\psi )$ for (a) small and (b) large $\beta$ functions; (bottom) $R_{56}^{s_1\to s_2}(\beta ,\psi )$ for (c) small and (d) large $\alpha$ functions.

Figure 19

$|R_{56}^{s_1\to s_2}|$ pattern for a dispersive and nonisochronous unit (small $R_{26}$), $R_{16}=0$, $R_{26}=0.4$, $R_{56}=0.05\mathrm{m}$. (Top)$R_{56}^{s_1\to s_2}(\alpha ,\psi )$ for (a) small and (b) large $\beta$ functions; (bottom) $R_{56}^{s_1\to s_2}(\beta ,\psi )$ for (c) small and (d) large $\alpha$ functions.

Figure 20

$|R_{56}^{s_1\to s_2}|$ pattern for a dispersive and nonisochronous unit (small $R_{16}$ and $R_{26}$), $R_{16}=0.4\mathrm{m}$, $R_{26}=0.2$, $R_{56}=0.1\mathrm{m}$. (Top) $R_{56}^{s_1\to s_2}(\alpha ,\psi )$ for (a) small and (b) large $\beta$ functions; (bottom) $R_{56}^{s_1\to s_2}(\beta ,\psi )$ for (c) small and (d) large $\alpha$ functions.

Figure 21

$|R_{56}^{s_1\to s_2}|$ pattern for a dispersive and nonisochronous unit (large $R_{16}$ and $R_{26}$), $R_{16}=4\mathrm{m}$, $R_{26}=2$, $R_{56}=0.1\mathrm{m}$. (Top) $R_{56}^{s_1\to s_2}(\alpha ,\psi )$ for (a) small and (b) large $\beta$ functions; (bottom) $R_{56}^{s_1\to s_2}(\beta ,\psi )$ for (c) small and (d) large $\alpha$ functions.