Coherent-radiation-induced longitudinal single-pass beam breakup instability of a steady-state microbunch train in an undulator

It has been known that a high-brightness electron beam emits broadband synchrotron radiation when traversing a curved orbit. The radiation reaction at wavelengths comparable to the bunch length or to the wavelength of a phase space modulated beam may lead to collective instabilities. In this paper, we investigate a potential single-pass instability mechanism of coherent-radiation-induced longitudinal multibunch beam breakup (BBU) instability in the presence of a closely spaced microbunch train in an undulator, particularly when the microbunch spacing is close to the resonant wavelength of the undulator. This problem is formulated based on the macroparticle model together with the slippage constraint on the beam-wave interaction. The set of coupled differential equations for individual microbunches can be solved analytically for the first few microbunches with linearization of the coherent radiation wakefield, and numerically in general nonlinear cases for unequal spacing or nonuniform filling charges. The underlying mechanisms, including the slippage effect, the potential-well effect leading to extra focusing, dependence of microbunch spacing and filling patterns, are discussed. The analysis is then applied to the recently proposed steady-state microbunching (SSMB) mechanism with two examples serving for the high average coherent radiation power sources for extreme ultraviolet (EUV) and infrared wavelength regions. For the specific scenario considered in this paper, it is found that when the microbunch spacing is close to the fundamental resonant wavelength, the coherent radiation can provide extra longitudinal focusing for the individual microbunches, leading to more stable multibunch oscillations. For the preliminary nominal SSMB designs with the average beam current of 1 A, our studies show that the single-pass longitudinal BBU instability should not be a severe issue.

Figure 1

The undulator radiation impedances (a) and the corresponding coherent radiation wake functions (b) for the first four individual harmonics and for the total superposition of them (black curve). Here $z>0$ indicates the bunch head or in the forward direction. The relevant parameters are $K=4.2$, ${\lambda }_w=2\mathrm{cm}$, $\gamma =\frac{400}{0.511}$, $k_r\approx 3.92\times {10}^7\mathrm{rad}/\mathrm{m}$.

Figure 2

Schematic layout of the modulator system. The system consists of an undulator, a microbunch train and the external seed laser. The external laser modulates the beam and provides longitudinal focusing to the individual microbunches. The undulator is designed to match the resonance condition with the external laser and the bunch spacing is set close to the undulator resonant wavelength.

Figure 3

Solutions to the first macroparticle. (a) Without potential-well effect, the red curve from Eq. (21) with $k_s=k_{s0}$ and the blue dashed curve from Eq. (27). (b) With potential-well effect, the red curve from the same Eq. (21) with $k_s$ given by Eq. (22) and the blue dashed curve from Eq. (26). For reference purposes, the gray curve in (b) is identical to the blue dashed curve in (a). The slippage effect is included in all the cases. Here ${\widehat{z}}_1(0)=3\mathrm{nm}$, $K=4.2$, ${\lambda }_w=2\mathrm{cm}$, $k_{s0}=4\pi$, $k_s\approx 4.6\pi$, $N_{\mathrm{pb}}=1.875\times 10^7$, $Q=3\mathrm{pC}$, $\gamma \approx 783$, $\mathrm{\Delta }L=\frac{2\pi }{k_r}\approx 0.16\mu \mathrm{m}$, $H_{\max }=1$.

Figure 4

Evolution of the most leading ($n=3$) macroparticle along $s$ based on three different cases. The slippage effect is included. The red curve is obtained from Eq. (21) with $k_s=k_{s0}$; the green curve from Eqs. (21) with $k_s$ given in Eq. (22); the blue curve from Eq. (17), where the potential-well effect is intrinsically included. The inset shows a zoom for the oscillation in $0<s<2\mathrm{m}$. Here the initial microbunch offsets assume ${\widehat{z}}_n(0)=(-1{)}^{n+1}\times 3\mathrm{nm}$ $(n=0\sim 3)$, and the remaining parameters are the same as those assumed in Fig. 3.

Figure 5

Evolution of the most leading ($n=49$) macroparticle along $s$ for a bunch train with 50 microbunches. The blue and red curves are obtained from Eq. (17) without and with the slippage condition, respectively. The parameters are the same as those used in Fig. 4.

Figure 6

Evolution of the longitudinal centroid offsets along $s$ based on Eq. (17). Here, we assume the initial offsets ${\widehat{z}}_n(0)=0$ for all ten microbunches $(n=0-9)$.

Figure 7

Evolution of the longitudinal centroid offsets along $s$ based on Eq. (21) with $k_s=k_{s0}$, i.e., without potential-well term. The initial parameters are the same as those in Fig. 6.

Figure 8

Macroparticle phase space evolution for individual microbunches (a) at $s=2\mathrm{m}$ based on the nonlinear model [Eq. (17)]; (b) at $s=2\mathrm{m}$ based on the linear model [Eq. (21)] and neglecting potential-well term; (c) at $s=10\mathrm{m}$ based on the nonlinear model. Here $H_{\max }=13$. For (c), the modulator voltage is adjusted to retain the same laser field as that for (a).

Figure 9

Evolution of the bunching spectrum along the undulator as a function of the normalized wavenumber. (a,c) from the nonlinear model; (b,d) from the linear model without potential-well term. From (c), the sideband appears due to large-amplitude synchrotron oscillation. See the context for more discussions.

Figure 10

Dependence of the bunching factors $|\mathcal{B}|$ on the number of electrons per microbunch for different radiation harmonics. The bunching factor is evaluated at $s=2\mathrm{m}$.

Figure 11

(a,b) Macroparticle phase space evolution for individual microbunches at $s=10\mathrm{m}$ with different microbunch charges; (c,d) Evolution of the bunching spectrum along the undulator as a function of the normalized wavenumber. Starting from the $n=0$ microbunch, the individual microbunch charges for the left column assume an increasing sequence $(1,2,3,\dots ,10)\times 1.2\times {10}^5$, while for the right column a decreasing sequence $(10,9,8,\dots ,1)\times 1.2\times {10}^5$. See the context for more discussions.

Figure 12

(a,b,c) Macroparticle phase space evolution for individual microbunches at $s=10\mathrm{m}$ with different bunch spacings; (d,e,f) Evolution of the bunching spectrum along the undulator as a function of the normalized wavenumber. The microbunch spacing for the left column assumes $\mathrm{\Delta }L=0.99{\lambda }_r$, the central column $\mathrm{\Delta }L={\lambda }_r$, and the right column $\mathrm{\Delta }L=1.01{\lambda }_r$. For the three cases, the bunch spacing is uniform for all microbunches. See the context for more discussions.

Figure 13

Illustration of the coherent undulator radiation wake function (dashed line), its derivative (solid line), and a few microbunches for different bunch spacings. See the context for more discussions. The wake function is scaled to help visualization.

Figure 14

Macroparticle phase space evolution for individual microbunches at $s=2\mathrm{m}$; (b) Evolution of the bunching spectrum along the undulator. Here $H_{\max }=15$ and $z_n(0)=0$ for all macroparticles.

Figure 15

Dependence of the bunching factors $|\mathcal{B}|$ on the number of electrons per microbunch including different radiation harmonics. The bunching factor is evaluated at $s=2.1\mathrm{m}$.