Abstract
The onset and evolution of magnetic fields in laboratory and astrophysical plasmas is determined by several mechanisms^{1}, including instabilities^{2,3}, dynamo effects^{4,5} and ultrahighenergy particle flows through gas, plasma^{6,7} and interstellar media^{8,9}. These processes are relevant over a wide range of conditions, from cosmic ray acceleration and gamma ray bursts to nuclear fusion in stars. The disparate temporal and spatial scales where each process operates can be reconciled by scaling parameters that enable one to emulate astrophysical conditions in the laboratory. Here we unveil a new mechanism by which the flow of ultraenergetic particles in a laserwakefield accelerator strongly magnetizes the boundary between plasma and nonionized gas. We demonstrate, from timeresolved largescale magneticfield measurements and fullscale particleincell simulations, the generation of strong magnetic fields up to 10–100 tesla (corresponding to nT in astrophysical conditions). These results open new paths for the exploration and modelling of ultrahighenergy particledriven magneticfield generation in the laboratory.
Main
Strong cosmic magnetization requires ultrahighenergy particle flows. These nonthermal particle streams can be produced^{10} by statistical acceleration processes, such as shock^{11,12} and Fermi acceleration^{13}, and by direct particle acceleration mechanisms in strong wave fields, such as those found in pulsars^{14}. In addition to direct astronomical observations, the physical processes occurring when these mechanisms take place may also be explored in the laboratory. For instance, the conditions for the onset of statistical acceleration mechanisms through collisionless shocks in the laboratory have been investigated theoretically^{15}. Direct cosmic acceleration can also be explored in the laboratory through laserdriven plasma wakefields^{16,17,18}. Here we show that the nonthermal particle flows produced in a laserwakefield accelerator (LWFA; refs 19, 20, 21) can strongly magnetize the plasma and the plasma–neutral gas boundary. This observation is also the first showing that the strong magnetization occurring at the flow of energetic particles from ionized to nonionized interstellar material can be reproduced in the laboratory.
A LWFA uses short and intense laser pulses to drive largeamplitude plasma waves. In the LWFA scheme, the laser ponderomotive force excites ultrarelativistic, largeamplitude plasma waves where electrons can be trapped and accelerated to high energies.
In our LWFA experiment the plasma is created in a helium gas jet produced in vacuum by a nozzle of 400 μm in diameter^{22,23}. The gas leak reaches, at its centre, a neutral density of n_{A} = 3.5 × 10^{19} atoms cm^{−3}, which corresponds, when fully ionized, to an electronic density of n_{0} = 4% n_{c} with n_{c} = (2πc)^{2}m_{e}ɛ_{0}/e^{2}λ^{2} the critical density at λ = 800 nm, where c is the speed of light in vacuum, e and m_{e}, respectively, the electron charge and mass, and ɛ_{0} the vacuum permittivity. A driver laser pulse of 30 fs in duration is focused perpendicularly to the gas jet direction by an offaxis parabolic reflector (Fig. 1a) to a transverse spot size of 8 μm fullwidth at halfmaximum (FWHM), reaching a peak intensity of I_{0} = 3 × 10^{19} W cm^{−2}. Because the laser power exceeds the critical power for relativistic selffocusing^{24}, the peak laser intensity in the plasma is further increased, which leads to the excitation of strongly nonlinear plasma waves that, at this density, are above the wavebreaking threshold. Moreover, the driver laser pulse duration is very close to the plasma period, hence the plasma waves are resonantly excited, which enhances the onset of wave breaking.
Wave breaking occurs at the transition between laminar and turbulent electron flows, when the plasma wave amplitude exceeds a given threshold. As the wave breaks, electrons thermalize while accelerating to relativistic energies, damping the amplitude of the plasma wave. Thus, wave breaking leads to nonthermal particle flows at relativistic energies that expand radially away from the laser axis, as they also move at relativistic velocities in the forward direction.
These nonthermal electrons expand radially, eventually reaching the plasma–neutral gas boundary, located where laser fields are not sufficiently high to ionize the background gas. As they propagate through the ionized medium, return currents are set up to balance the hot electron flow, preventing effective magneticfield generation. These return currents appear only inside the plasma, where the gas is ionized. In the neutral gas region, however, there are no free charges that are capable of generating return currents to compensate for hot electron currents (impact ionization of neutral gas due to nonthermal electron propagation is negligible). Thus, hot electrons crossing the plasma–neutral gas boundary are able to generate strong fields that magnetize the neutral gas and the peripheral regions of the plasma cylinder. Although direct measurement of wave breaking is extremely challenging, the measured largescale magnetic field is indirect evidence for its onset, because of the key role wavebreakingproduced electrons play in the field generation.
We demonstrate that this mechanism can produce strong magnetic fields scaling as 32η_{hot}(n_{0} [10^{16} cm^{−3}])^{1/2} T in the laboratory, where η_{hot} is the fraction of hot electrons to background plasma density n_{0}. Thus, under astrophysical conditions, fields created by the same mechanism can reach amplitudes scaling as 320 η_{hot}(n_{0} [cm^{−3}])^{1/2} nT. Hot electrons here play the role of suprathermal beams, as opposed to thermal beams, which are responsible for nonthermal radiation spectra and to strong magnetization in astrophysical scenarios.
Our experiment provides the first timeresolved measurements of the spatial distribution of a magnetic field in a laserplasma accelerator for the whole plasma volume and with a high temporal resolution. As the laser propagates through the gas target we observed strikingly complex magnetic structures, with several inversions of the field orientation at the plasma core. These observations are in excellent agreement with threedimensional (3D) onetoone particleincell (PIC) simulations in OSIRIS (refs 25, 26) capturing the global plasma dynamics and magneticfield evolution over the entire gas jet, each taking several hundreds of thousands of CPUhours (see Methods).
The laser interaction with the gas jet is thoroughly scanned at high temporal and spatial resolutions for electron density (via phase recording) and magneticfield mapping. The plasma is probed by a single laser pulse of t = 30 fs in duration at λ = 400 nm. During its propagation in the plasma cylinder, polarization rotation and absolute dephasing are integrated and recorded on three separate chargecoupled device (CCD) cameras, as shown in Fig. 1a. The 3D density and magnetic maps are then reconstructed from the recorded information (see Methods). The delay between the driver and the probe pulses can be varied (with an accuracy of ∼2 fs) to explore in time the propagation of the driver pulse and the evolution of the plasma. This pump–probe experiment permits us to freeze the plasma state to a highresolution snapshot lasting for only 30 fs and to follow its evolution. The validity of the magneticfield measurement rests on the assumption that B ⋅∇ n_{0} = 0 in the plasma cylinder, which is confirmed by simulations.
Snapshots of the spatial distribution of the magnetic field at selected times are shown in Fig. 2. Each image represents the symmetrized radial map of (n_{0}B)_{θ}(r, x), the product between the polar component of the magnetic field and the local electron density, as reconstructed from the probe polarization maps. The laser pulse propagates from left to right. A polar magnetic field is observed in the trail of the laser pulse soon after entering the gas jet (t = 0.7 ps). From the density measurements we can infer a magneticfield magnitude reaching 100 T. This field is positive in the plasma core (r < 50 μm) and changes in sign close to the radial border of the plasma (r ∼ 50 μm). As we will see from simulations below, the inversion of the field direction at the boundary is a distinctive feature indicating an electron current passing through the plasma/gas boundary. These electrons (ultraenergetic particles in astrophysical scenarios) are relativistic and, although accelerated from the wave breaking, they are not trapped by the wakefield.
Figure 2 shows another striking feature, consisting of a strong magnetization of the plasma core. A large magnetic field is indeed expected close to the laser propagation axis, in the wakefield region. This field is created by the very high longitudinal wakefield current, which is not screened by the plasma. At a distance corresponding to the limit of the laser beam waist (that is, a few micrometres), a weak magnetic field survives (the bow wave magnetic field^{27}). Farther away from the axis, plasma return currents typically screen hot (nonthermal) electrons that expand radially from the wakefield (note that the main velocity component of these electrons is along x), thus no significant field is expected to survive up to the vicinity of the boundary. The experimental results suggest that in the plasma core the screening of hot electron current coming from the wakefield is not as efficient as expected.
The magneticfield structure remains alike until t = 1.1 ps: starting from t = 1.3 ps, an island of reversed field starts building up close to the gas jet density peak (x = 530 μm), relatively far from the laser waist. At t = 1.5 ps (Fig. 2), this island has expanded longitudinally and radially, and a second island appears closer to the laser axis (x = 580 μm), at a larger radius. Simulations suggest that these islands are caused by the filamentation of the wings of the laser pulse—that is, the laser energy surrounding the central spot.
Experiments showed several inversions of the magneticfield orientation, as illustrated in Fig. 3. Several shots at different delays are used to compose a picture of the local evolution in time of the magnetic field. The temporal resolution in this figure is 200 fs. On entry into the gas (Fig. 3a) the laser propagation is accompanied by the formation of a positive poloidal magnetic field in the plasma core, which changes sign in the plasma boundary. The inner field component is consistent, in sign, with a negative current propagating with the laser pulse and remains stable for approximatively 1.2 ps. Figure 3b shows the evolution in the region where the first island appears, x = 480 μm. At t ∼ 1.1 ps the field sign is reversed in the plasma core, corresponding to the formation of the first island. This inversion lasts up to t ∼ 1.5 ps. When the laser exits from the gas profile (Fig. 3c) a poloidal field is again formed, comparable to Fig. 3a. At this time, however, the plasma core magnetization is opposite in sign to what was observed at the entrance.
Our experimental findings are confirmed by 3D PIC simulations run with parameters closely matching experimental laser and plasma conditions (see Methods). As it enters the gas, the laser ionizes the gas up to a radius of 100 μm away from the axis and excites weakly nonlinear plasma waves. Relativistic pulse selffocusing enhances the wakefield amplitude beyond the wavebreaking threshold after 400 μm of propagation. When wave breaking occurs, a fraction of the resulting hot electrons expand radially through the plasma.
The laserdriven plasma waves lead to complex longitudinal electron current structures in the ionized volume. Figure 4a shows the current structures at the end of the simulation, at t = 2.67 ps, where it is possible to distinguish between backward electron currents (blue) and forward electron currents (red). As electrons cross the plasma–neutral gas boundary an inner return current is set up, located at the boundary itself, at a variable radius in the range r ≲ 60–100 μm. These currents, Fig. 4a, are the origin of the largescale poloidal magnetic fields observed in Fig. 4b. Simulations also show that a small fraction of hot electrons at the plasma entrance (x < 500 μm) flow away from the laser in the backward direction. These are indicated by the red structures surrounding a blue core (r ≳ 60 μm) in Fig. 4a I. For x > 500 μm most of the hot electrons are accelerated in the forward direction, leading to the outer forward current structures (blue) in Fig. 4a II and III.
Return currents electrons propagate forward at the leading edge of the gas profile (Fig. 4a I) and backward for the remainder of the gas jet length (Fig. 4a II, III). Because hot electrons flow in opposite directions for x < 500 μm (Fig. 4b I) and for x > 500 μm (Fig. 4b III), the sign of the magnetic field changes in these regions. The transition at x ≃ 500 μm is shown in Fig. 4b II, which illustrates the poloidal magnetic field corresponding to the longitudinal currents shown in Fig. 4a. This transition to a richer current structure around x ≃ 500 μm is confirmed by experimental observation (for example, Figs 2 and 3b).
The largescale magnetic field that surrounds the plasma around y ≃ 60–100 μm reaches amplitudes that are in fair agreement with experimental measurements (n_{0}B_{z} ≃ 10^{21} T cm^{−3}).
An estimate of the poloidal magneticfield amplitude can then be determined considering the return currents at the plasma–gas interface. Using Ampère’s law to estimate the amplitude of the resulting azimuthal magnetic field (B_{θ}) in cylindrical symmetry gives
where r is the distance to the axis, dl and dS are the line and surface elements for the integrations and μ_{0} is the magnetic permeability of the vacuum. For relativistic hot electrons moving at velocity c longitudinally, j ≃ ec(η_{hot}n_{0}) e_{z}, where e is the elementary charge and e_{z} is the unit vector pointing in the z direction. When r is greater than the plasma radius r_{p}, the return current density flux is η_{hot}en_{0}4π^{2}[r_{p}^{2} − (r_{p} − Δ)^{2}] ≃ 4π^{2}c e n_{0} η_{hot}r_{p}Δ, where Δ ≃ c/ω_{p} ≪ r is the thickness of the plasma the return currents are set up in, hence B_{θ} ≃ 4πe n_{0} c η_{hot}(r_{p}/r)(c/ω_{p}). The typical generated magnetic fields are of the order of for r ≃ r_{p}. Considering η_{hot} ≃ 0.05 (taken from simulations) and n_{0} = 7 × 10^{19} cm^{−3}, we obtain B_{θ} ≃ 80 T, which is consistent with the simulation results. In astrophysical scenarios, the amplitude of the cosmic magnetic fields generated by this mechanism can be extrapolated to nT. This is consistent with the magnetization in the interstellar and extragalactic media, both of the order of 0.1–1 nT for densities of ∼1 particle cm^{−3} (interstellar medium) and 10^{−2}–10^{−4} particles cm^{−3} (extragalactic medium)^{28} considering η_{hot} between η_{hot} ∼ 0.01 (as for the conditions of the experiment and simulations) and η_{hot} ∼ 1 (which may be obtained by using a smaller plasma radius and higher laser intensities).
The presented magnetic field increases with the background plasma density and the fraction of hot (suprathermal) particles as η_{hot} n_{0}^{1/2}: this is a key scaling parameter which relates these results to astrophysical conditions. The ratio η_{hot} can be controlled by changing the volume of the plasma, which plays the role of the ionized interstellar (or extragalactic) region, and by the laser intensity, which plays the role of a central engine accelerating suprathermal particles. Hence, by controlling the plasma and the parameters that define the magneticfield amplitude it will be possible to quantitatively explore the magnetization of the interstellar/extragalactic magnetic fields in scaled experiments.
Methods
Highdensity micrometric target.
The target is a pulsed, highpressure gas jet system which can drive a submillimetric gas nozzle to atomic densities in the range 10^{19}–10^{21} atoms cm^{−3} (ref. 22).
In our experiment we used a transonic nozzle with an output diameter of 400 μm, producing an expanding flow with a Mach number M = 1.3. The radial atomic density profile at a distance of 200 μm from the nozzle level (laser propagation axis) is fitted using n_{A} = n_{max} exp[− (r/r_{0})^{(2.1)}], where n_{max} = 3.5 × 10^{19} atoms cm^{−3} and r_{0} = 170 μm is the jet radius.
The peak density in the jet decreases exponentially along the vertical direction with a characteristic scale length of L = 263 μm.
Probing of the plasma.
The plasma is probed by a linearly polarized τ_{p} = 30 fs, λ = 400 nm probe pulse synchronized to the main (pump) beam. The probe beam is split into three separate optical setups, for simultaneous measurement of density and polar magnetic field.
The radial density map is extracted from the integrated phase of the probe pulse, by means of a Nomarski interferometer on the first transmitted copy of the probe beam. The integrated phase is retrieved by 2D wavelet analysis of the interferogram. The phase map is then normalized by subtraction of a reference phasemap (accounting for aberrations in the laser transport and in the imaging system) and inverted by Hankel–Fourier implementation of the Abel transform.
Magnetic field is retrieved from the polarization rotation φ_{rot} produced on the probe beam by the Faraday effect as
where n_{c} = (2πc)^{2}m_{e}ɛ_{0}/e^{2}λ^{2} is the electron plasma critical density at a wavelength of λ and dz is the line element along the probe beam propagation (see ref. 29).
To obtain a 2D map of the polarization status in the probe beam profile, the pulse is split into two separate diagnostic lines, each equipped with an analyser and a high dynamic range CCD camera. A total of four images, two with pump and two without, are used for each snapshot of the magnetic field to eliminate effects due to laser intensity fluctuation and systematic optical deformation. The correct superposition of images from the two separate polarization measurement lines is ensured by spatial markers and automated numerical pattern recognition methods.
The map of the polarization rotation φ_{rot}(x, y) is obtained from the ratio between the images recorded by the two ‘Faraday rotation’ CCDs (Fig. 1). Let I_{p1}(x, y) (resp. I_{p2}(x, y)) be the plasma shadowgraphy at a polarization angle θ_{p1} (resp. θ_{p2}).
The map of the ratio between the two images R(x, y) = I_{p1}/I_{p2}(x, y) can be written as
where θ_{p{1,2}} are the analysers’ angles and φ_{rot}(x, y) the local polarization rotation. From simple trigonometric operations, having defined θ_{1} = π/2 + θ_{p1} + φ_{rot} and Δ_{p} = θ_{p2} − θ_{p1}, φ_{rot}(x, y) is obtained through
where the coefficients are given by
To retrieve the 2D map of the quantity (n_{0} B_{θ})(r, x) it is necessary to eliminate the scalar projection in the integral kernel of equation (1). This operation is accomplished via the integral back transform developed in ref. 30:
Owing to the sensitivity of the algorithm, φ_{rot} maps are antisymmetrized before performing the back transform.
Simulations.
Threedimensional simulations were performed using the fully relativistic particleincell code Osiris^{25,26}, which is routinely used to model laserwakefield acceleration and astrophysical scenarios.
The particleincell technique (PIC) is a firstprinciples approach which performs almost no physical approximations as long as quantum effects can be neglected. These simulations therefore include a fully kinetic description of the plasma dynamics, and the selfconsistent laser–plasma interaction in relativistic regimes including field ionization of the neutral gas in three dimensions. The initial transverse and longitudinal laser profiles, frequency and focal point, were set to closely follow measured experimental profiles. The initial gas jet density profile measured in experiments was also accurately reproduced in the simulations. The simulations make the following assumptions, which have no influence on the findings: binary collisions are not modelled (a valid assumption in our scenario, which describes collisionless plasmas), impact ionization has not been included (impact ionization is negligible because the probability of hot electron beam electrons ionizing a neutral helium atom is lower than 5 × 10^{−3}), ions remain immobile (similar simulations including ion motion, but without ionization, showed no differences in results and conclusions).
The simulation window is 1,272 × 300 × 300 (μm)^{3}, divided into 33,000 × 1,000 × 1,000 cells with 2 × 1 × 1 particles per cell, giving a total of 6.6 × 10^{10} simulation particles and a total simulation time between 200,000 and 400,000 CPUh. Simulations considered an initial helium gas jet density profile given by n_{0} = 3.5 × 10^{19} cm^{−3} exp[(x − 700 μm/340 μm)^{2.1}], x being the longitudinal position in microns, which reproduces the experimental conditions. Ionization was modelled using ADK tunnel ionization rates. The longitudinal profile of the laser electric field is symmetric and given by 10τ^{3} − 15τ^{4} + 6τ^{5}, with , where τ_{FWHM} is the FWHM duration of the laser pulse. The transverse laser profile is a fit to a transverse line taken from the experimental laser intensity profile using higherorder Hermite–Gaussian modes. Given the distance to the propagation axis, each mode electric field profile is defined as:
where E_{0} is the peak electric field, k_{0} = ω_{0}/c, ω_{0} = 2.34 × 10^{15} radian s^{−1} is the laser frequency corresponding to a central laser wavelength of 800 nm, ζ_{p}(x) = (m + n + 1) tan^{−1}(x, Z_{r}) is the Gouy phase shift, Z_{r} = k_{0}W_{0}^{2}/2 is the Rayleigh length, W(x)^{2} = W_{0}^{2}(1 + x^{2}/Z_{r}^{2}), and H_{n} the nthorder Hermite polynomial. Table 1 shows the laser peak normalized vector potentials (a_{0}) of the five higherorder Gaussian beams associated with the fit to the experimental laser profile. A value of W_{0} = 9.8 μm was used.
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Acknowledgements
The authors acknowledge the support of OSEO project n.I0901001WSAPHIR, the support of the European Research Council through the PARIS ERC project (contract 226424) and the national research grants BLAN081380251 (GOSPEL) and IS0400201 (ILA). A.F. acknowledges collaboration with T. Vinci (LULI, École Polytechnique). The work of J.V. and L.O.S. is partially supported by the European Research Council through the Accelerates ERC project (contract ERC2010AdG267841) and by FCT, Portugal (contract EXPL/FIZPLA/0834/1012). We acknowledge PRACE for access to resources on SuperMUC (Leibniz Research Center).
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S.K., F.S., M.V. and A.F. conceived, designed and carried out the experimental measurements, A.F. conceived, designed and realized the analysis tools and performed the data analysis, A.L., J.V. and L.O.S. carried out the numerical simulations, A.F., J.V. and L.O.S. wrote the manuscript, V.M. provided overall supervision.
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Flacco, A., Vieira, J., Lifschitz, A. et al. Persistence of magnetic field driven by relativistic electrons in a plasma. Nature Phys 11, 409–413 (2015). https://doi.org/10.1038/nphys3303
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DOI: https://doi.org/10.1038/nphys3303
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