Beam-beam interaction in a dielectric laser accelerator electron-positron collider

We examine through numerical calculation the collision of counter-propagating trains of optically spaced electron or positron microbunches in a 1 TeV collider scenario for a dielectric laser accelerator. A time-dependent envelope equation is derived for arbitrary number of bunches in the classical limit, with inclusion of the radiation reaction force. Example parameters are examined based on a constrained luminosity relation that takes into account the bunch charge for optimal efficiency, material damage limits, and power constraints. We find that for initially identical counter-propagating Gaussian bunch trains, the periodic temporal structure leads to a peak in luminosity with number of bunches. For longer bunch trains, the enhancement then decreases inversely with number of bunches. The corresponding fractional energy loss of the beam is found to be of order 1.75%, which is reduced to 0.35% when the nonlinear radial dependence of the transverse force is included, with an average beamstrahlung parameter of 0.075, an important result considering that beamstrahlung losses are a critical concern for future TeV colliders.

Figure 1

Constrained parameters from Eq. (9) as function of number $M$ of bunches, showing (a) macrobunch population (black) and optimal efficiency (gray); (b) corresponding field intensity parameters; and (c) corresponding laser repetition rate (black) and wallplug power (gray). All points on the curves are matched to ${\overline{\mathcal{L}}}_{\mathrm{ref}}=2\times {10}^{34}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$. Vertical blue dashed lines mark the value $M=100$ corresponding to the parameters of Table 3.

Figure 2

Schematic of the $e^{+}{e}^{-}$ collision process with $M=3$ microbunches per train. At $t=0$ the first microbunches of each train collide. For an arbitrary $M$, the full duration of the interactions is $\mathrm{\Delta }t=(M-1)\lambda /c$ with a total of $M^2$ microbunch crossings.

Figure 3

Frames (a) and (b) show the normalized spot size ${\overline{\sigma }}_i={\sigma }_i/{\sigma }_0$ and energy ${\overline{\gamma }}_i={\gamma }_i/{\gamma }_0$ respectively from solving Eq. (33) for three representative microbunches $i=150100$ in a train of $M=100$ bunches. The calculations are performed in the time range $0\le t\le (M-1)\lambda /c$. The parameters correspond to those of Table 3 with ${\mathrm{\Omega }}^2=0.042$ and ${\mathrm{\Theta }}^2=2.7\times {10}^{-4}$. Part (c) shows the partial luminosity enhancement factor ${\mathcal{H}}_{ij}$ along each trajectory.

Figure 4

Computed solutions for (a) luminosity enhancement, (b) beamstrahlung parameter, and (c) fractional energy loss of the entire bunch train for various number of bunches $M$ for the case ${\mathrm{\Omega }}^2=0.042$, ${\mathrm{\Theta }}^2=0.00027$. Note that the values of Table 3 correspond to $M=100$ on these plots (vertical dashed blue markers). In part (c), numerical solutions of fractional energy loss (black dots) are shown for every fifth $M$ value to aid visualization. Including nonlinearity of the transverse force yields approximately a factor of four reduction in energy loss (blue curve).

Figure 5

Coordinate system for single particle of velocity $v$ colliding with the charge density function $\rho$ with velocity $-v$.