Batch Bayesian optimization of attosecond betatron pulses from laser wakefield acceleration

Abstract

Laser wakefield acceleration can generate a femtosecond-scale broadband X-ray betatron radiation pulse from electrons accelerated by an intense laser pulse in a plasma. The micrometer-scale of the source makes wakefield betatron radiation well-suited for advanced imaging techniques, including diffraction and phase-contrast imaging. Recent progress in laser technology can expand these capabilities into the attosecond regime, where the practical applications would significantly benefit from the increased energy contained within the pulse. Here we use numerical simulations combined with batch Bayesian optimization to enhance the radiation produced by an attosecond betatron source. The method enables an efficient exploration of a multi-parameter space and identifies a regime in which a plasma density spike triggers the generation of a high-charge electron beam. This results in an improvement of more than one order of magnitude in the on-axis time-averaged power within the central time containing half of the radiated energy, compared to the reference case without the density spike.

Figures (11)

• Figure 1: An illustration of the proposed setup for betatron radiation enhancement. The plasma density profile (gray thick line) varies along the longitudinal coordinate $x$, while remaining constant in the $y$ and $z$ coordinates. The laser pulse first enters a gradient of length $L_\mathrm{g}=40~{\mathrm{\upmu m}}$, providing the injection of the electron ($e^-$) beam into the bubble. The $e^-$ beam produces a betatron pulse that is enhanced by a density spike introduced further along $x$. In the simulations, the length of the density spike $d_{\mathrm{s}}$, the length of the uniform region with density $n_0$ before the spike $d_{\mathrm{u}}$ and the maximum density value of the density spike $n_{\mathrm{p}}$ were adjusted in the optimization process. The total plasma length is equal to $L_\mathrm{acc}=256~{\mathrm{\upmu m}}.$
• Figure 2: Convergence and wall-clock time of BBO for different batch sizes. a) Evolution of the best-performing normalised cost value $C/C^{\mathrm{ref}}$ achieved up to corresponding iteration for different batch sizes: $N=1$ (solid blue), $N=4$ (dashed yellow), $N=8$ (dotted green). The dash-dotted magenta line marks the reference baseline, i.e., the case with $C/C^{\mathrm{ref}} = 1$. b) Accumulated computational wall-clock time versus iteration for different batch sizes.
• Figure 3: Values of the normalised cost function around the best-performing point found by BBO. The values are shown for a) varying $d_{\mathrm{u}}$, fixed $d_\mathrm{s}=d_\mathrm{s}^\mathrm{opt}$, $n_\mathrm{p}=n_\mathrm{p}^\mathrm{opt}$, b) varying $d_{\mathrm{s}}$, fixed $d_\mathrm{u}=d_\mathrm{u}^\mathrm{opt}$, $n_\mathrm{p}=n_\mathrm{p}^\mathrm{opt}$, and c) varying $n_\mathrm{p}/n_0$, fixed $d_\mathrm{u}=d_\mathrm{u}^\mathrm{opt}$, $d_\mathrm{s}=d_\mathrm{s}^\mathrm{opt}$. The normalised cost function $C/C^{\mathrm{ref}}$ is shown with orange lines and circular markers at the evaluated points. The normalised model prediction $\hat{C}/C^{\mathrm{ref}}$ for $N=8$ from the final (8th) iteration, is shown with blue lines with crosses at evaluated points. Colored shaded regions indicate the area within predicted standard deviation $\sigma_{\hat{C}}$. The large black circle marks the best-performing case.
• Figure 4: Betatron radiation characteristics for the reference and best-performing cases. a) On-axis radiated energy $W$ per time $t$ per solid angle $\Omega$ (temporal profile) as a function of the observer's time $t$ for the best-performing optimized (red dotted) and reference (blue solid) cases. The first two insets show zoomed-in profiles of the betatron peaks in b) the best-performing and c) reference cases, corresponding to the shaded areas in a). d) On-axis radiated energy $W$ per photon frequency $\omega$ per solid angle $\Omega$ of the betatron pulses as a function of the photon energy $E_{\mathrm{ph}}$. Critical energies for the reference case $E_c^{\mathrm{ref}}$ and the best-performing case $E_c^{\mathrm{opt}}$ are marked with a blue cross and a red star, respectively. The energy spectra are calculated from the whole temporal profile depicted in a).
• Figure 5: Plasma density evolution for the best-performing case. a) Early stage of propagation shortly after the first electron injection. b) Propagation through an increasing gradient of the density spike. c) A second injection is triggered while the laser pulse propagates through the decreasing gradient. d) End of the acceleration and degradation of the bubble structure. The density profiles are shown at the centre of the $z$-axis, $z=25.6~\mathrm{\upmu m}$, and $x_{\mathrm{mw}}$ is the $x$-coordinate comoving with the simulation window. For better visualization, $n/n_0$ is transformed using a power-law function with exponent $0.35$, $\left((n/n_0)/(n/n_0)_{\max}\right)^{0.35}$, where $(n/n_0)_{\max}$ corresponds to the maximum value of the normalised density $n/n_0$.