Numerical growth of emittance in simulations of laser-wakefield acceleration

Transverse emittance is a crucial feature of laser-wakefield accelerators, yet accurately reproducing its value in numerical simulations remains challenging. It is shown here that, when the charge of the bunch exceeds a few tens of picocoulombs, particle-in-cell (PIC) simulations erroneously overestimate the emittance. This is mostly due the interaction of spurious Cherenkov radiation with the bunch, which leads to a steady growth of emittance during the simulation. A new computational scheme is proposed, which is free of spurious Cherenkov radiation. It can be easily implemented in existing PIC codes and leads to a substantial reduction of the emittance growth.

Figure 1

Top: energy spectrum after $300\mu \mathrm{m}$ of acceleration (from the position when injection started). The grayed area represents the sub-bunch for which the emittance was computed. Bottom: transverse emittance of this sub-bunch in the $y$ (parallel to the laser polarization) and $z$ (orthogonal to the laser polarization) direction, as a function of the acceleration distance.

Figure 2

Snapshot of the transverse field $E_y-c{B}_z$ in the $x\mathrm{\text{−}}y$ plane, at the back of the bubble and after $300\mu \mathrm{m}$ of acceleration. Shaded regions correspond to zones of high electron density (in this case the sheath of the bubble and the accelerated bunch). While the laser pulse is clearly apparent on the right-hand side of the figure, one also notices higher-frequency radiation around the bunch.

Figure 3

Physical phase velocity along $x$ ($v_{\varphi ,x}$, as deduced from ${\omega }^2=c^2{\overrightarrow{k}}^2$) as a function of $k_x$ and $k_y$ for electromagnetic modes propagating in the $x\mathrm{\text{−}}y$ plane ($k_z=0$). The blue horizontal plane corresponds to $v_{\varphi ,x}=\beta c$ with $\beta =1-9\times {10}^{-6}$ (which is representative of particles having an energy of 120 MeV).

Figure 4

Numerical phase velocity along $x$ [$v_{\varphi ,x}$, as deduced from Eq. (2)] as a function of $k_x$ and $k_y$ for electromagnetic modes propagating in the $x\mathrm{\text{−}}y$ plane ($k_z=0$). The blue horizontal plane corresponds to $v_{\varphi ,x}=\beta c$ with $\beta =1-9\times {10}^{-6}$ (which is representative of particles having an energy of 120 MeV).

Figure 5

Spatial Fourier transform of the field $E_y-c{B}_z$ from Fig. 2. The two red spots correspond to the laser field ($k_x\Delta x/\pi =\pm k_{\mathrm{laser}}\Delta x/\pi =\pm 0.08$). The structures observed near $k_x=0$ correspond to the fields of the bubble (their broad extent along $k_y$ in Fourier space corresponds to the sharp edges of the bubble in real space). The dashed line materializes the position of the modes satisfying Eq. (3) (with $\beta =1–9\times {10}^{-6}$) and thus corresponds to the intersection of the two theoretical surfaces of Fig. 4.

Figure 6

Position of the nodes used to compute ${\nabla }_x^{*}F{|}_{i+1/2,j,k}^n$ in our modified scheme.

Figure 7

Numerical phase velocity along $x$ [$v_{\varphi ,x}$, as deduced from Eq. (8)] as a function of $k_x$ and $k_y$ for electromagnetic modes propagating in the $x\mathrm{\text{−}}y$ plane ($k_z=0$). The blue horizontal plane corresponds to $v_{\varphi ,x}=\beta c$, and there is no intersection between these two surfaces. Although not shown on this figure, there is no intersection either for the other modes ($k_z\ne 0$).

Figure 8

Snapshot of the transverse field $E_y-c{B}_z$ in the $x\mathrm{\text{−}}y$ plane, at the back of the bubble and after $300\mu \mathrm{m}$ of acceleration. The top panel corresponds to the simulation ran with the Yee scheme, while the bottom panel corresponds to the one run with our modified scheme. Both snapshots correspond to the same iteration, so that the bubble has a slight advance in the bottom panel, due to the alteration of the velocity of the laser mentioned in Sec. 4b.

Figure 9

Transverse emittance of the sub-bunch considered in Fig. 1, in the $y$ (parallel to the laser polarization) and $z$ (orthogonal to the laser polarization) direction, as a function of the acceleration distance.

Figure 10

Energy spectrum after $300\mu \mathrm{m}$ of acceleration (from the position when injection started) for the two numerical schemes considered here.