Caustic-based approach to understanding bunching dynamics and current spike formation in particle bunches

Current modulations, current spikes, and current horns, are observed in a range of accelerator physics applications including strong bunch compression in Free Electron Lasers and linear colliders, trains of microbunching for terahertz radiation, microbunching instability and many others. This paper considers the fundamental mechanism that drives intense current modulations in dispersive regions, beyond the common explanation of nonlinear and higher-order effects. Under certain conditions, neighboring electron trajectories merge to form caustics, and often result in characteristic current spikes. Caustic lines and surfaces are regions of maximum electron density, and are witnessed in accelerator physics as folds in phase space of accelerated bunches. We identify the caustic phenomenon resulting in cusplike current profiles and derive an expression which describes the conditions needed for particle-bunch caustic formation in dispersive regions. The caustic expression not only reveals the conditions necessary for caustics to form but also where in longitudinal space the caustics will form. Particle-tracking simulations are used to verify these findings. We discuss the broader implications of this work including how to utilize the caustic expression for manipulation of the longitudinal phase space to achieve a desired current profile shape.

Figure 1

Optical caustics, which are analogous to electron–trajectory caustics found in accelerator physics. (a) image of caustic lines appearing in a coffee cup. (b) illustration of light rays forming the caustic (red line), and (c) intensity of the rays in the vicinity of the caustic.

Figure 2

Illustrations of current spikes induced by caustics in a variety of applications.

Figure 3

Electron trajectories through a chicane in $s$–$z$ plane, showing how the electron trajectories overlap in a caustic at the extreme values of $z$ (i.e., at the head and the tail of the bunch). (b) Typical phase space distribution and corresponding current profile seen at the end of a strong bunch compressor [equivalent to $s=9.5\mathrm{m}$ in (a)].

Figure 4

(a) elegant distribution taken before bunch compressor BC2, with a fit (thin, red curve), $\delta =81.06z_i+5929.08z_i^2+1.30\times 10^8{z}_i^3$. (b) Derivative of polynomial fit emphasizing the nonlinearity of the distribution.

Figure 5

Two electron trajectories coalescing at the caustic point $P$.

Figure 6

Caustic expression [Eq. (8)] shown in red overlayed on electron trajectories, showing where in $z$ current spikes can be anticipated for a given $R_{56}$ value, for a 4 dipole chicane where $T_{566}=-3/2R_{56}$ and $U_{5666}=2R_{56}$.

Figure 7

Electron trajectories caustics seen forming in 3 scenarios differentiated by the different values of the $R_{56}$, $T_{566}$, and $U_{5666}$ encountered. Column (a) shows the double–horned current structure produced at the end of a dispersive region where $R_{56}=-10.78\mathrm{mm}$, $T_{566}=16.35\mathrm{mm}$ and $U_{5666}=-11.38\mathrm{mm}$. Column (b) shows a single-horn current profile produced with $R_{56}=-10.82\mathrm{mm}$, $T_{566}=-41.07\mathrm{mm}$ and $U_{5666}=0.40\mathrm{m}$, and column (c) shows a current profile produced with $R_{56}=-11.76\mathrm{mm}$, $T_{566}=16.10\mathrm{mm}$ and $U_{5666}=2.60\mathrm{m}$. The top row shows trajectories (blue, thin) with the caustic expressions [Eq. (8)] (red, thick). The second row of images shows histograms of electron density, calculated at the value of $R_{56}$ indicated by the gray vertical line in the top row of images. The third and fourth rows of images were created using elegant, showing the phase space distribution and current profiles, respectively, where the head of the bunch is on the left-hand side of these figures.

Figure 8

Electron trajectories (in $s$–$z$ space) through a bunch compressor. (b) shows a close up of the trajectories through the fourth dipole [i.e., region outlined with a red box in (a)]. Caustics can be seen forming at head and tail of the bunch in (b).

Figure 9

Electron trajectories passing through a dispersive region. Labels $a$ and $b$ are used to mark the longitudinal distance separating the trajectories at $s=0$ and $s=s_2$ respectively.

Figure 10

Local compression ratio, $C_l$ varying with the initial longitudinal position, $z_i$. Where $C_l$ is negative, this indicates that the electrons with those initial $z_i$ values are in fact spreading out rather than compressing.

Figure 11

The local compression ratio mapped to the final longitudinal position values, $z$. The three colors shown distinguish the electrons from each of the 3 sections of the bunch visible in Fig. 3—the core of the bunch and the two edges which fold around on themselves in longitudinal phase space.

Figure 12

Current profile calculated with the parametric equations, Eqs. (16) and (17).

Figure 13

Linearly ramped current profile calculated from Eq. (17) and obtained through optimizing $R_{56}$, $T_{566}$, and $U_{5666}$. Note the head of the bunch is on the left-hand side, at negative values of $z$.

Figure 14

Electron trajectories in $z$–$R_{56}$ space, and caustic expression [Eq. (8)] shown in red. The gray vertical line represents the value of $R_{56}$ where a linearly ramped current profile (along the $z$ direction) can be obtained.

Figure 15

Electron trajectories $s$–$z$ space, where the gray vertical line indicates the end of the compressor where the current profile is evaluated.

Figure 16

Histogram of the density of electron trajectories at the end of the compressor, corresponding to the gray vertical line in Fig. 15.

Figure 17

Longitudinal phase space distribution at the entrance to the beam line (blue) and at the exit (orange), where the final bunch displays a linearly ramped current profile shown in Fig. 13.