Nonlinear plasma wavelength scalings in a laser wakefield accelerator

H. Ding, A. Döpp, M. Gilljohann, J. Götzfried, S. Schindler, L. Wildgruber, G. Cheung, S. M. Hooker, and S. Karsch

Phys. Rev. E 101, 023209 – Published 24 February 2020

Abstract

Laser wakefield acceleration relies on the excitation of a plasma wave due to the ponderomotive force of an intense laser pulse. However, plasma wave trains in the wake of the laser have scarcely been studied directly in experiments. Here we use few-cycle shadowgraphy in conjunction with interferometry to quantify plasma waves excited by the laser within the density range of GeV-scale accelerators, i.e., a few $10^{18} {cm}^{- 3}$ . While analytical models suggest a clear dependency between the nonlinear plasma wavelength and the peak potential $a_{0}$ , our study shows that the analytical models are only accurate for driver strength $a_{0} ≲ 1$ . Experimental data and systematic particle-in-cell simulations reveal that nonlinear lengthening of the plasma wave train depends not solely on the laser peak intensity but also on the waist of the focal spot.

Received 26 November 2019
Accepted 24 January 2020

DOI:https://doi.org/10.1103/PhysRevE.101.023209

Physics Subject Headings (PhySH)

Plasma waves

Laser wakefield acceleration

Accelerators & Beams

Authors & Affiliations

H. Ding ^1,3, A. Döpp^1,3,*, M. Gilljohann^1,3, J. Götzfried¹, S. Schindler¹, L. Wildgruber¹, G. Cheung², S. M. Hooker², and S. Karsch^1,3,†

¹Ludwig-Maximilians-Universität München, Am Coulombwall 1, D-85748 Garching, Germany
²John Adams Institute & Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
³Max Planck Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany

^*andreas.doepp@physik.uni-muenchen.de
^†stefan.karsch@physik.uni-muenchen.de

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Vol. 101, Iss. 2 — February 2020

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Images

Figure 1
(Top) Plasma waves excited by lasers with a Gaussian envelope and different peak potential $a_{0}$ according to one-dimensional fluid theory, Eq. (3). The transition from sinusoidal at low intensity ( $a_{0} < 1$ ) to increasingly nonlinear density profiles ( $a_{0} > 1$ ) is clearly visible. (Bottom) Comparison of wavelength intensity dependence among different models. Note that the analytical expression for rectangular pulse, Eq. (4), has two disconnected regions of validity. The dashed segment of the red curve is to guide the eye.
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Figure 2
Schematic representation of the experimental setup. (Insets) (a) An example few-cycle shadowgram of a nonlinear laser-driven plasma wave with a 50- $μ m$ scale bar. (b) A raw image recorded with the Nomarski interferometer. (c) Phase shift caused by the plasma, deduced from (b). (d) The transverse electron density profile retrieved from Abel inversion at the position marked by the white line in (c). Note that the density bumps at the shoulders and feet of the profile are a retrieval artifact. (e) The longitudinal electron density profile at $x = 0$ in (c) together with the density profile used for simulations (cf. Figs. 4 and 5). Note that the coordinate position $z$ of the measured profile is shifted by $- 0.45$ mm compared to (c), and the plasma density of simulation input is scaled to match the measurement. (f) Retrieved FROG trace of the probe beam. (g) Far field profile of the probe beam measured with a CCD camera.
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Figure 3
(Left) Representative shadowgrams of laser-driven plasma waves in the plasma density range of $n_{e} = 2 - 4 \times 10^{18} {cm}^{- 3}$ . (a) A nonlinear plasma wave driven by a 70-TW pulse. (b) A strongly nonlinear plasma wave driven by a 70-TW pulse with a weaker secondary wave above it. Note that the secondary wave starts at the same position as the main wave, but its modulation at the front is poorly visible due to its overlap with a diffraction feature in the probe's near-field profile. (c) A quasilinear plasma wave driven by a 13-TW pulse. (Right) The wavelength of plasma oscillation as a function of electron density. The nonlinear wavelengths (orange dots) are obtained from the main waves whereas the linear wavelengths (blue diamonds) are deduced from the filaments [cf. (b)]. The low power shots (green triangles) are taken at 13 TW [cf. (c)]. Each data point is an average of two to nine shots. The vertical error bars represent the standard error of mean (s.e.m.) of each run. The horizontal error bars are the estimated uncertainties in the density retrieval from interferometry. A least-square fit to the nonlinear wavelengths (dashed red line) yields the elongation factor $λ_{p, n l} / λ_{p}$ of $α = 1.13$ .
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Figure 4
Snapshots of a quasi-3D simulation of a 70-TW, 30-fs (FWHM) pulse propagating in a 3-mm-long hydrogen gas jet with a nominal electron density of $n_{e} = 3 \times 10^{18} {cm}^{- 3}$ : Upper panels are for the beginning of the jet and lower panels are for the center of the jet where the experimental data are taken (cf. Fig. 3). From left to right as follows: (a) and (b) the intensity distribution (false color) together with the E-field envelope of the laser pulse in transverse and longitudinal direction (red lines), normalized by $m_{e} c ω_{0} / e$ , with $ω_{0}$ the laser carrier frequency; (c) and (d) the electron density distribution; (e) and (f) line-by-line Fourier transform of the electron density with the abscissa converted from wave number to wavelength and the intensity corrected by the Jacobian (false color), and the position of the intensity maximum at each transverse coordinate $x$ (the dashed line) [note that the wiggles in (f) are a numerical artifact due to the weak density modulation outside the drive laser]; (g) the evolution of the peak laser potential (red solid line) and the beam waist (green dashed line) (the horizontal lines indicate the matched condition from Lu et al. [20]); (h) the evolution of the elongation factor (blue line), which shows good agreement with the measurement (orange dot). The vertical error bar of the measured dot indicates the 95% confidence interval of the elongation estimate and the horizontal error bar is the sum in quadrature of the length of the visible wave train and the uncertainty in determining the length of the gas jet up-ramp.
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Figure 5
Comparison of plasma wave train formation in the wake of a tightly focused spot (right) and a 3 times as wide spot (left) at various laser peak potentials ( $a_{0} = [1.0, 2.0, 3.0, 4.0]$ from top to bottom). Colored lines show the trajectories of electrons with different initial position in radial coordinates. The normalized E-field strength of the laser is shown as grayscale contour plot and the transverse field gradient is indicated with an overlaid color map. The transverse gradient of the tightly focused laser leads to stronger transverse electron motion and thus prevents them from experiencing the peak laser potential. This further leads to cavitation and suppresses the elongation of the wave train. In contrast, the simulations for a wide focal spot are comparable to laminar models, with a characteristic horseshoelike shape, until wave breaking sets in for $a_{0} ≳ 3$ . All simulations are performed for a plasma density $n_{e} = 3 \times 10^{18} {cm}^{- 3}$ and using an FWHM pulse duration of $τ = 30$ fs.
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Figure 6
Plasma wave elongation according to PIC simulations with different aspect ratios $w_{0} / c τ$ of the laser pulse (dots). The fit function (colored dashed lines) agrees well with the simulations, while approaching the 1D nonlinear model for $w_{0} / c τ ≫ 1$ and $a_{0} ≲ 2$ .
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