Perturbation-minimized triangular bunch for high-transformer ratio using a double dogleg emittance exchange beam line

The longitudinal shape, i.e., the current profile, of an electron bunch determines the transformer ratio in a collinear wakefield accelerator and thus methods are sought to control the longitudinal bunch shape. The emittance exchange (EEX) appears to be promising for creating a precisely controlled longitudinal bunch shapes. The longitudinal shape is perturbed by two sources: higher-order terms in the beam line optics and collective effects and these perturbations can lead to a significant drop of the transformer ratio. In this paper, we analytically and numerically investigate the perturbation to an ideal triangular longitudinal bunch shape and propose methods to minimize it.

Figure 1

The longitudinal bunch shaping (LBS) beam line is based on the $\mathrm{mask}+\mathrm{EEX}$ method.

Figure 2

The mask is used to produce the initial x-distribution after the mask (left) and the ideal EEX beam line linearly transforms this initial x-distribution to produce the final z-distribution (right). In the four examples shown, the coefficient ($C_0$) in Eq. (8) $x$ is equal to 0.5 (i.e. $z_f=0.5x_i$) so that the separation ($\mathrm{\Delta }$) between the particles (top) and beamlets (second from top) is reduced by a factor of 2 ($\mathrm{\Delta }/2$).

Figure 3

The transformer ratio of the triangular current profile as a function of the Courant-Snyder parameters at the entrance of an emittance exchange beam line calculated by the linear matrix beam transport (left) and GPT simulation (right).

Figure 4

Density plot of the transverse Gaussian distribution. Red curves on each axis show the projected profile that is Gaussian, and the cyan-dashed curve in the density plot shows the mask shape for the triangular profile.

Figure 5

The general perturbation of a slice (left). The $j$th slice of the ideal longitudinal bunch shape has $n_0(z_j^i)$ particles at the ideal position ($z_j^i$). In general, $P(\zeta )$ shifts particles in the slice to new locations by a spread of distances. The total perturbed bunch shape, $n(z)$, is due to the sum of all the slice perturbations (right).

Figure 6

The four key characteristics of the triangular profile.

Figure 7

The perturbed longitudinal bunch shapes for: $x_0^2$-series with $C_1<0$ (a), $z_0^2$-series with $C_4>0$ (b) and $C_4<0$ (c), $x_0{z}_0$-series with $C_7\ne 0$ (d), $y_0^2$-series with $C_{11}>0$ (e) and $C_{11}<0$ (f), SC effect (g), and CSR effect (h). The color corresponds to the ratio of the maximum deviation ${\zeta }_{\max }$ to the ideal bunch length ($l_b$): ${\zeta }_{\max }/l_b$.

Figure 8

The relative change of the transformer ratio versus the ratio of the maximum deviation (${\zeta }_{\max }$) to the ideal bunch length ($l_b$) for each of the perturbation cases.

Figure 9

Suppression of second-order effects with the slope-control method. The final longitudinal bunch shape, $n(z)$, from GPT simulation (left column) and corresponding normalized change in the transformer ratio, $\mathrm{\Delta }R/R$ (right column). The top row shows the effect of the horizontal slope, the middle row shows the effect of the longitudinal slope (i.e. chirp), the bottom row shows the effect of the vertical slope on $n(z)$ (left) and $\mathrm{\Delta }R/R$ (right).

Figure 10

The CSR effect. Longitudinal bunch shape at the end of EEX beam line with different charges after the mask (top), and corresponding transformer ratio drop (bottom). As we move from left to right (a)–(e), ${\alpha }_2=({20}^0,{18}^0,{16}^0,{14}^0,{12}^0)$ and $L_{D2}=(2.00\mathrm{m},2.31\mathrm{m},2.68\mathrm{m},3.16\mathrm{m},3.79\mathrm{m})$. Corresponding ${\xi }_2$ for each case is (0.29 m, 0.27 m, 0.24 m, 0.21 m, 0.18 m). Note that the beam energy is 50 MeV to generate strong CSR but weak SC.

Figure 11

The space charge effect. Current profiles at the end of the EEX beam line with different charges after the mask (top), and corresponding transformer ratio drop (bottom). As we move from left to right (a)–(e), ${\alpha }_2=({20}^0,{18}^0,{16}^0,{14}^0,{12}^0)$ and $L_{D2}=(0.85\mathrm{m},1.00\mathrm{m},1.19\mathrm{m},1.43\mathrm{m},1.74\mathrm{m})$). Corresponding ${\xi }_2$ for each case is (0.14 m, 0.13 m, 0.11 m, 0.10 m, 0.09 m). The beam energy is 15 MeV that generates a weak CSR effect but a strong space-charge effect.

Figure 12

Final longitudinal bunch shape (blue) and corresponding wakefield (red) for the CSR dominant case (a) and the SC dominant case (b). In both cases, we used a 12-degree bending angle with a charge after the mask of 3.5 nC.

Figure 13

The $x_0^2$-series perturbation. Bottom: The ideal $n_0(z_f^i)$ profile. Middle: The perturbation function $P(\zeta )$ at various positions along the bunch. Top: Slices of the ideal profile $n_0(z_f^i)$ (blue) and the perturbed profile $n_0(z_f^i)$ (red). The shift, $\zeta$ depends on $z_f^i$. Particles move to the back of the bunch and its distance is proportional to the square of the distance from the center of the bunch.

Figure 14

The $z_0^2$-series perturbation for the case of $C_4<0$. Bottom: The ideal $n_0(z_f^i)$ profile. Middle: The perturbation function $P(\zeta )$. Top: Slices of the ideal profile $n_0(z_f^i)$ (blue) and the perturbed profile $n(z_f^p)$ (red). The shift is a skewed Gaussian towards the back of the bunch.

Figure 15

The $x_0{z}_0$-series perturbation. Bottom: The ideal $n_0(z_f^i)$ profile. Middle: The perturbation function $P(\zeta )$. Top: Slices of the ideal profile $n_0(z_f^i)$ (blue) and the perturbed profile $n(z_f^p)$ (red). The Gaussian spread of the slices gets stronger the farther out from the center.

Figure 16

The $y_0^2$-series perturbation for the case of $C_{11}<0$. Bottom: The ideal $n_0(z_f^i)$ profile. Middle: The perturbation function, $P(\zeta )$, for the $y_0^2$-series with $C_{11}<0$. Top: Slices of the ideal profile $n_0(z_f^i)$ (blue) and the perturbed profile $n(z_f^p)$ (red). The skewed Gaussian spread of the slices gets stronger the farther away from the head.

Figure 17

The SC perturbation. Bottom: The ideal $n_0(z_f^i)$ profile. Middle: The perturbation function $P(\zeta )$. Top: Slices of the ideal profile $n_0(z_f^i)$ (blue) and the perturbed profile $n(z_f^i)$ (red). The SC perturbation has a symmetric Gaussian-like spread with spikes at each end, and there is no shift in $z_f^i$.

Figure 18

The CSR perturbation. Bottom: The ideal $n_0(z_f^i)$ profile. Middle: The perturbation function $P(\zeta )$. Top: Slices of the ideal profile $n_0(z_f^i)$ (blue) and the perturbed profile $n(z_f^p)$ (red). The CSR perturbation has skewed-Gaussian-like spread with spikes at each end, and it is the same for all slices.