An analytical optimization of plasma density profiles for downramp injection in laser wake-field acceleration

Abstract

We propose and detail a multi-step analytical procedure, based on an improved fully relativistic plane model for Laser Wake Field Acceleration, to tailor the initial density of a cold diluted plasma to the laser pulse profile, so as to control and optimize the partial wave-breaking of the plasma wave and maximize the acceleration of small bunches of electrons self-injected by the first wave-breaking at the density down-ramp, at least in the first stages of their acceleration phase. We find an excellent agreement with the results of Particle In Cell simulations obtained with the same input data.

Figures (15)

• Figure 1: Schematic illustration of the direct and inverse problems
• Figure 2: a) As every particle travels slower than light, $\tilde{\xi}(t)=ct\!-\!z(t)$ grows strictly, and $\xi=ct\!-\!z$ can replace $t$ as the independent parameter along its worldline (WL) $\lambda$ in Minkowski space and in its eq. of motion Fio18JPA. $\tilde{\xi}\to\xi_f<\infty$ as $t\to\infty$ implies $\dot z\to c$, see e.g. WL $\lambda'$. While the instants of intersection with the front and the end of a plane EM wave (\ref{['EBfields0']}) (whose support we have painted in pink) depend on the particular WL (we have pinpointed the ones $t_{1i},t_{1f}$ for $\lambda_1$), the corresponding light-like coordinates are the same for all WLs: $\xi_{1i}=\xi_{2i}=0$, $\xi_{1f}=\xi_{2f}=l$: the pulse acts as a "clock" for "time" $\xi$. b) We label the "particles" (i.e. elements) of the electron fluid by their initial positions ${\bm X}$; the hydrodynamic regime (HR) holds as long as their WLs do not intersect, i.e. the maps ${\bm X}\mapsto {\bm x}$ are one-to-one.We denote Eulerian, Lagrangian observables as follows: Eulerian observables${\color{red}{f}}(t,{\bm x})={\color{red}{\check f}}({\color{blue}{\xi}},{\bm x})={\color{magenta}{\tilde{f}}}(t,{\bm X})={\color{magenta}{\hat{f}}}({\color{blue}{\xi}},{\bm X})$Lagrangian observables.
• Figure 3: a) An initial plasma density of the type (\ref{['n_0bounds']}) with an up-ramp, a down-ramp, and a plateau. b) An essentially short, slowly modulated quasi-monochromatic (SMM) laser pulse.
• Figure 4: (a) Normalized gaussian pulse of FWHM $l'\!=\!10.5\lambda$, linear polarization, peak amplitude $a_0\!\equiv\!\lambda eE^{ \perp}_{ M}/2\pi mc^2\!=\!2$, as in section III.B of BraEtAl08. We consider $l\!=\!40\lambda$ and cut the tails outside $|\xi\!-\!l/2|\!<\!l/2$. (b) Corresponding solution of (\ref{['heq1']}-\ref{['heq2']}) if $\widetilde{n_0}(z)\!=\!\bar{n}^j\!\equiv\! n_{cr}/268.8$ ($n_{cr}\!=\! \pi mc^2/e^2\lambda^2$ is the critical density); as a result, $E/mc^2\equiv h\!=\!1.28$. Adopting $\bar{n}=\bar{n}^j$ as the density plateau maximizes the LWFA of test electrons, see section \ref{['WFA']}.B. A wavelength $\lambda=0.8\mu$m leads to a peak intensity $I\!=\!1.7\!\times\!10^{19}$W/cm$^2$ and $\bar{n}^j\!=\!6.5\times\! 10^{18}$cm$^{-3}$. (c) Corresponding phase portrait (at $\xi>l$). (d) Corresponding electron density at $t=100\lambda/c$.
• Figure 5: Assuming the same input data as in fig. \ref{['graphsb']}, we plot: in the first row, $\check{\bm U}^{ \perp}(\xi,\xi_{{ +}})\equiv e\check {\bm A}\!^{ \perp}(\xi,\xi_{{ +}})/mc^2$ and $\hat{\bf u}^{ \perp}(\xi)\equiv e{\bm \alpha}^{ \perp}(\xi)/mc^2$ vs. $\xi$ resp. for $\xi_{{ +}}/\lambda=200,400,800$; in the second row, $V\equiv \check{\bm U}^{ \perp}{}^2(\xi,\xi_{{ +}})$ and $v\equiv \hat{\bf u}^{ \perp}{}^2(\xi)$ vs. $\xi$ resp. for $\xi_{{ +}}/\lambda=200,400,800$; in in the third row, the averages $V_a(\xi,\xi_{{ +}}),v_a(\xi)$ vs. $\xi$ resp. for $\xi_{{ +}}/\lambda=200,400,800$.

Theorems & Definitions (2)

• Proposition 1
• Proposition 2