Probing Ultrafast Magnetic-Field Generation by Current Filamentation Instability in Femtosecond Relativistic Laser-Matter Interactions

TL;DR

This study demonstrates the ultrafast generation of strong magnetic fields during femtosecond laser interactions with solid targets, using relativistic electron probes and simulations to reveal their origin from a Weibel instability.

Contribution

It provides the first experimental measurement of femtosecond-scale magnetic fields generated by current filamentation instability in laser-matter interactions.

Findings

Measured a line-integrated magnetic field of 2.70 ± 0.39 kT μm.

Simulations show the fields originate from a Weibel-type filamentation instability.

Fields grow rapidly, affecting electron beam angular distribution.

Abstract

We present experimental measurements of the femtosecond time-scale generation of strong magnetic-field fluctuations during the interaction of ultrashort, moderately relativistic laser pulses with solid targets. These fields were probed using low-emittance, highly relativistic electron bunches from a laser wakefield accelerator, and a line-integrated $B$-field of $2.70 \pm 0.39\,\rm kT\,\mu m$ was measured. Three-dimensional, fully relativistic particle-in-cell simulations indicate that such fluctuations originate from a Weibel-type current filamentation instability developing at submicron scales around the irradiated target surface, and that they grow to amplitudes strong enough to broaden the angular distribution of the probe electron bunch a few tens of femtoseconds after the laser pulse maximum. Our results highlight the potential of wakefield-accelerated electron beams for ultrafast probing of relativistic laser-driven phenomena.

Equations

$$B_{x,\mathrm{int}}=\sqrt{{⟨{\left(\int B_{x}dz\right)}^2⟩}_{n_b}},$$

Full text

The Weibel-type current filamentation instability (CFI) Weibel, E. S. (1959); Fried (1959) has been extensively investigated in past decades owing to its recognized importance in an increasing variety of plasma environments. Induced by temperature anisotropies or relative drifts between the plasma constituents Davidson et al. (1972); Bret et al. (2010); Achterberg and Wiersma (2007); Huang et al. (2019), it gives rise to kinetic-scale, current filaments surrounded by toroidal magnetic fields, through which the charged particles are progressively isotropized Davidson et al. (1972); Achterberg et al. (2007); Ruyer et al. (2015). This instability is widely thought to underpin the physics of relativistic outflows in powerful astrophysical objects (e.g. gamma-ray bursts, pulsar winds, active galactic nuclei), especially as the source of the collisionless shock waves held responsible for generating nonthermal high-energy particles and radiations Medvedev and Loeb (1999); Spitkovsky (2008); Kato and Takabe (2008); Marcowith et al. (2016); Lemoine et al. (2019). Moreover, it is expected to operate in magnetic reconnection scenarios Swisdak et al. (2008), and has been invoked as a possible generation mechanism for cosmological magnetic fields Schlickeiser and Shukla (2003).

On the laboratory side, the CFI stands as a key process in intense laser-plasma interactions. In the case of overdense plasmas irradiated at relativistic laser intensities ($I_{0}\lambda_{0}^{2}\gtrsim 10^{18}\,\mathrm{W\>cm}^{-2}\>\mu\mathrm{m}^{2}$, where $I_{0}$ and $\lambda_{0}$ are the laser intensity and wavelength, respectively), it arises from the counterstreaming of the forward-directed, laser-accelerated fast electrons and the current-neutralizing, cold plasma electrons Sentoku et al. (2000); Silva et al. (2002); Ren, C. and Tzoufras, M. and Tonge, J. and Mori, W. B. and Tsung, F. S. and Fiore, M. and Fonseca, R. A. and Silva, L. O. and Adam, J.-C. and Heron, A. (2006); Robinson et al. (2014). The resulting magnetic fluctuations may grow fast enough to cause significant scattering and deceleration of the fast electrons Sentoku et al. (2003); Chrisman, B. and Sentoku, Y. and Kemp, A. J. (2008); Adam et al. (2006); Debayle et al. (2010). These effects are generally considered detrimental to fast-electron-based applications, e.g. the fast ignition approach to inertial confinement fusion Robinson et al. (2014) or target normal sheath ion acceleration Fuchs et al. (2003); Göde et al. (2017); Scott et al. (2017). Still, they can also be triggered purposefully in laboratory astrophysics experiments addressing the physics of collisionless shocks, whether involving relativistic laser-solid interactions Fiuza et al. (2012); Ruyer et al. (2015), laser-driven interpenetrating plasma flows Fox et al. (2013); Huntington et al. (2015), or electron beam-plasma interactions Allen et al. (2012); Benedetti et al. (2018).

Experimental evidence for the development of the CFI in relativistic laser-driven plasmas has been mainly provided through characterization of the spatial profiles of the fast electron Tatarakis et al. (2003); Wei et al. (2004); Jung et al. (2005); Manclossi et al. (2006) or ion Fuchs et al. (2003); Metzkes et al. (2014); Göde et al. (2017); Scott et al. (2017); King et al. (2019) beams exiting the target. In situ measurements of the magnetic-field fluctuations at the irradiated target surface have been performed using optical polarimetry Mondal et al. (2012); Chatterjee et al. (2017), yet this technique cannot access the volumetric distribution of the fields, and the data obtained so far could not capture their femtosecond time-scale dynamics.

In this Letter, we demonstrate a novel method for diagnosing the kT-level, electromagnetic fluctuations induced in femtosecond laser-solid interactions using an ultrarelativistic probe electron bunch with energies above 100 MeV, produced by a laser wakefield accelerator (LWFA) Tajima and Dawson (1979); Faure et al. (2004); Geddes et al. (2004); Mangles et al. (2004); Gonsalves et al. (2019). In our experimental setup, the laser pulse driving the LWFA is the same one that induces the electromagnetic fluctuations in a neighboring foil target (see Fig. 1). This ensures a well-controlled time delay between the electron bunch and the laser pump, and therefore probing of the fluctuations a few tens of femtoseconds only after the on-target laser pulse maximum. Their line-integrated field strength is then inferred from the angular broadening induced upon the electron bunch. Our measurements are supported by 3D fully relativistic particle-in-cell (PIC) simulations, which indicate that the field fluctuations indeed result from a Weibel-type CFI excited at the target surface. Note that ultrafast probing of plasma electromagnetic fields by a LWFA-driven electron beam was previously exploited to image plasma wakefields in LFWAs Zhang et al. (2017) or large-scale, radially expanding magnetic fields in relativistic laser-solid interactions Schumaker et al. (2013).

The experiment was performed at Laboratoire d’Optique Appliquée with the ‘Salle Jaune’ Ti:Sapphire laser system, delivering laser pulses with 30 fs full width at half maximum (FWHM) duration and up to $1.5\,\rm J$ energy on target. The laser pulse had a $810\,\rm nm$ central wavelength and was linearly polarized along the horizontal $x$-axis. Corrected using adaptive optics, it was focused at the entrance of a 3-mm exit diameter gas jet target by a $f/16$ off-axis parabola to a $20\,\rm\mu m$ FWHM spot size in vacuum, yielding a normalized peak vector potential of $a_{0}\simeq 1.5$ when accounting for the experimental intensity distribution in the focal plane. The supersonic gas jet used for the LWFA consisted of a mixture of 99% hydrogen and 1% nitrogen, enabling well-controlled electron acceleration through ionization injection Rowlands-Rees et al. (2008); Pak et al. (2010); McGuffey et al. (2010); Doepp et al. (2017). Due to relativistic self-focusing and self-steepening in the LWFA stage, the laser field strength is expected to be enhanced to $a_{0}\gtrsim 3$ Corde et al. (2013). After exiting the gas jet, the laser pulse and the electron beam impinged on a thin Mylar or aluminium foil, located at a variable position along the propagation axis. The electron beam transmitted through the foil was characterized by an electron spectrometer comprising a 10-cm-long, 1.0 T dipole magnet deflecting electrons depending on their energy along the horizontal $x$-axis, and a scintillating screen imaged onto a 16-bit camera [see Fig. 1(a)]. The spectrometer also recorded angular information along the nondispersive vertical $y$-axis (perpendicular to laser polarization), but the large distance (about 35 cm) between the foil and the scintillating screen prevented sub-micron-scale structures of the beam profile close to the target from being resolved.

The LWFA was operated in the highly non-linear regime Esarey et al. (2009), and the electrons from the inner shells of the nitrogen dopant were ionized within the blowout cavity by the high-intensity part of the laser pulse. This resulted in continuous injection as the laser propagated through the gas, and therefore in electron beams with a broad energy spectrum extending beyond 200 MeV [Fig. 2(a) (top)], a 50-100 pC charge (above 100 MeV) and a 2-4 mrad FWHM divergence. The longitudinal separation between the electron beam and the laser pulse was on the order of the plasma wavelength ($\sim 10\,\mu$m for an electron plasma density of $\sim 10^{19}\,\rm cm^{-3}$). After exiting the gas jet, the peak intensity of the diffracting laser pulse decreased with the propagation distance.

Figure 2(a) displays typical electron energy-angle spectra recorded during the experiment. The top panel shows the reference spectrum from the LWFA (no solid target). When a 13-$\rm\mu m$-thick Mylar foil is placed 0.42 mm from the gas jet exit, the beam divergence is significantly increased [Fig. 2(a), middle]. This effect is strongly reduced when the jet-foil distance is increased to 2.61 mm [Fig. 2(a), bottom], corresponding to a decrease in the laser intensity on the solid target surface. Multiple scattering of beam electrons in the foil due to elastic collisions cannot account for this behavior since it should cause a negligible increase in the divergence in 13-$\rm\mu m$-thick Mylar (scattering angle of 0.38 mrad for 150 MeV electrons) and be independent of the foil position.

Figure 2(b) plots the variations of the electron beam divergence with the jet-foil distance (in the range from 0.25 to 3.2 mm) as measured with different targets (13-$\mu$m-thick Mylar and 8 to 60-$\mu$m-thick Al). For each target type, the beam divergence is seen to decrease monotonically with the jet-foil distance. Increasing the Al foil thickness only entails detectable changes at large distances ($\gtrsim 1.5$ mm) due to stronger multiple scattering. These results indicate that the angular broadening of the electron beam takes place in the vicinity of the irradiated target surface.

To assess the possible effect of the foil-reflected laser pulse on the probe electrons, the foil was tilted at $45^{\circ}$ with respect to the laser axis, so that the reflected laser and the electron beam no longer overlapped. This yielded negligible differences in the observed electron beam divergence, thus proving that the reflected pulse does not account for the increase in the electron beam divergence, and that the source of the latter hardly depends on the laser incidence angle on the target. The only possible scenario, therefore, is that of beam scattering by laser-driven electromagnetic fields within a thin layer behind the target surface.

As a result, our data provide a direct measurement of the integrated Lorentz force experienced by the beam in the solid foil, expressed as an equivalent line-integrated magnetic field,

[TABLE]

where the average is weighed by the transverse profile of the electron beam. This field induces a spread $\sigma_{p_{y}}=eB_{x,\mathrm{int}}$ in the transverse momentum distribution of the electron beam, and therefore contributes to a total divergence $\theta_{y}^{2}=\theta_{y,\mathrm{ref}}^{2}+\theta_{y,\mathrm{sc}}^{2}+\theta_{y,\mathrm{B}}^{2}$. Here $\theta_{y,\mathrm{ref}}$ is the original divergence of the LWFA-generated beam, $\theta_{y,\mathrm{sc}}$ is the contribution from the multiple scattering and $\theta_{y,\mathrm{B}}\simeq\sigma_{p_{y}}/p_{z}=ecB_{x,\mathrm{int}}/E$ is the contribution from the integrated equivalent magnetic field, with $E$ the electron energy. From the experimentally measured divergence, $\theta_{y}=13.23\pm 1.31$ mrad (FWHM), of the 150 MeV energy electrons passing through the 13-$\mu$m-thick Mylar foil at a 0.42-mm distance, one infers an integrated equivalent magnetic field of $B_{x,\mathrm{int}}=2.70\pm 0.39$ kT$\>\mu\mathrm{m}$.

In order to identify the physical mechanism behind the electromagnetic field generation around the target surface, 2D and 3D PIC simulations have been performed using the code calder Lefebvre et al. (2003); Nuter et al. (2011); Pérez et al. (2012); Lobet et al. (2016). These fully relativistic simulations describe both the laser-foil and subsequent beam-plasma interactions, including the effects of binary Coulomb collisions, impact ionization and field ionization. The laser is modeled as a planar wave with a Gaussian temporal profile and a 20 fs FWHM pulse duration. Its field strength on target is estimated to drop from $a_{0}=2.3$ to 0.7 when the jet-foil distance is increased from 0.5 to 2.7 mm (assuming 1 Joule of laser energy and 15 $\mu$m FWHM spot size at the gas jet exit). The electron beam is initialized with an energy of 150 MeV, a 1 $\mu$m root-mean-square (RMS) transverse size, a 1.6 $\mu$m RMS bunch length, a 50 pC total charge, and a 11 $\mu$m peak-to-peak separation with the laser pulse. The target consists of a 8-$\mu$m-thick, solid-density, neutral plasma of $e^{-}$ and Al3+ ions. On its front side is added a linearly ramped preplasma of $0.8\,\rm\mu m$ length to take account of an imperfect laser contrast (see Supplemental Material See Supplemental Material for further details at for a discussion of the weak effect of the preplasma length on the resulting integrated $B$-field). The 3D domain size is $L_{x}\times L_{y}\times L_{z}=2.1\times 2.1\times 45\,\rm\mu m^{3}$ with a cell size in each direction of $\Delta x=\Delta y=\Delta z=\lambda_{0}/64$, while for the 2D simulations, the domain size is $L_{x}\times L_{z}=2.1\times 45\,\rm\mu m^{2}$ with a cell size $\Delta x=\Delta z=\lambda_{0}/64$. 50 macro-particles per cell for each species are used in all simulations.

Figure 3 shows results from the 3D PIC simulation for a $0.5\,\rm mm$ jet-foil distance. While several mechanisms may give rise to strong electromagnetic fluctuations in the vicinity of the foil surface (e.g. parametric decay of laser-driven surface oscillations or Rayleigh-Taylor-like instability Macchi et al. (2002); Kluge, T. and Metzkes, J. and Zeil, K. and Bussmann, M. and Schramm, U. and Cowan, T. E. (2015)), the Weibel-type CFI appears to be the dominant process under our experimental conditions (see Supplemental Material See Supplemental Material for further details at ). The resulting fluctuations, of mainly magnetic nature, exhibit a characteristic filamentary pattern with a $\sim 0.4\,\rm\mu m$ transverse periodicity, and extending to a $\sim 1\,\rm\mu m$ depth [Fig. 3(b)]. The time evolution of the $z$-integrated magnetic field during and after the laser irradiation is presented in Fig. 3(e), showing that the beam electrons experience fully-grown magnetic fields as soon as they enter the target. Their $(\theta_{x},\theta_{y})$ angular distribution after transiting through the target is displayed in Fig. 3(d): the beam divergence along the vertical ($y$) direction is measured to be $\theta_{y}\simeq 10\,\rm mrad$ (FWHM), much larger than its initial value ($\simeq 0.1\,\rm mrad$) in the simulation. Moreover, these magnetic deflections translate into strong transverse modulations in the beam profile [compare Fig. 3(a) and Fig. 3(c)]. The asymmetry between the horizontal ($x$) and vertical ($y$) divergences originates from the stronger laser-induced electron heating along the laser polarization axis ($x$); this excites current modulations preferentially along the cold ($y$) axis, hence leading to $B_{x,\rm int}>B_{y,\rm int}$ and to a larger vertical divergence.

To further compare the simulation results with the experimental observations, we plot in Fig. 4 the results of a parametric scan using 3D PIC simulations where the jet-foil distance is varied from 0.5 to $2.7\,\rm mm$, corresponding to an estimated $a_{0}$ ranging from 2.3 to 0.7. The integrated magnetic field experienced by the electron beam and the resulting angular broadening are observed to monotonically decrease as the foil is moved away from the gas jet exit, in good agreement with the experimental measurements [Fig. 2]. This shows the sensitivity of the CFI-induced magnetic fluctuations to the intensity of the femtosecond laser drive pulse.

To conclude, we have demonstrated, both experimentally and numerically, the potential of low-emittance, LWFA-generated electron beams to probe the submicron-scale magnetic fields induced by a Weibel-like current filamentation instability developing during ultraintense laser-solid interactions. Supported by 3D PIC simulations, our measurements have allowed us to infer the generation of an integrated magnetic field strength $B_{x,\rm int}=2.70\pm 0.39\,\rm kT\,\mu m$ around the front surface of a solid target irradiated by a $\sim$ 20 fs, $\sim 10^{19}$ W cm*-2* laser pulse. These results pave the way for time-resolved $B$-field measurements at femtosecond time scales, by generating the probe electron beam from an auxiliary laser pulse with controlled delay. Our results are also of prime interest for staged plasma-based accelerators Steinke et al. (2016), including novel hybrid schemes, which aim to miniaturize beam-driven plasma wakefield accelerators (PWFA) Chen et al. (1985); Litos et al. (2014) and achieve unprecedented beam quality by using relativistic electron drive beams from a LWFA Hidding, B. and Königstein, T. and Osterholz, J. and Karsch, S. and Willi, O. and Pretzler, G. (2010); Chou et al. (2016); Ferri et al. (2018); Gilljohann et al. (2019); Martinez de la Ossa et al. (2019), and separating the LWFA and PWFA by a thin foil. The present study highlights the need to mitigate the CFI (e.g. by depleting the laser pulse energy before it hits the solid target) so as to avoid degrading the quality of the electron beam driving the subsequent acceleration stage.