Guiding of relativistic electron beams in dense matter by longitudinally imposed strong magnetic fields
M.Bailly-Grandvaux, J.J.Santos, C.Bellei, P.Forestier-Colleoni, S.Fujioka, L.Giuffrida, J.J.Honrubia, D.Batani, R.Bouillaud, M.Chevrot, J.E.Cross, R.Crowston, S.Dorard, J.-L.Dubois, M.Ehret, G.Gregori, S.Hulin, S.Kojima, E.Loyez, J.-R.Marques, A.Morace, Ph.Nicolai, M.Roth, S.Sakata, G.Schaumann, F.Serres, J.Servel, V.T.Tikhonchuk, N.Woolsey, Z.Zhang
GGuiding of relativistic electron beams in dense matter by longitudinally imposedstrong magnetic fields
M. Bailly-Grandvaux, J.J. Santos, ∗ C. Bellei, P. Forestier-Colleoni, S. Fujioka, L. Giuffrida, J.J. Honrubia, D. Batani, R. Bouillaud, M. Chevrot, J.E. Cross, R. Crowston, S. Dorard, J.-L. Dubois, M. Ehret,
1, 7
G. Gregori, S. Hulin, S. Kojima, E. Loyez, J.-R. Marqu`es, A. Morace, Ph. Nicola¨ı, M. Roth, S. Sakata, G. Schaumann, F. Serres, J. Servel, V.T. Tikhonchuk, N. Woolsey, and Z. Zhang Univ. Bordeaux, CNRS, CEA, CELIA (Centre Lasers Intenses et Applications), UMR 5107, F-33405 Talence, France Institute of Laser Engineering, Osaka University,2-6 Yamada-oka, Suita, Osaka, 565-0871, Japan ETSI Aeron´autica y del Espacio, Universidad Polit´ecnica de Madrid, Madrid, Spain LULI-CNRS, ´Ecole Polytechnique, CEA: Universit´e Paris-Saclay; UPMCUniversit´e Paris 06: Sorbonne Universit´es, F-91128 Palaiseau cedex, France Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK Department of Physics, Heslington, University of York, YO10 5DD, UK Institut f¨ur Kernphysik, Tech. Univ. Darmstadt, Germany (Dated: November 6, 2018)High-energy-density flows through dense matter are needed for effective progress in the produc-tion of laser-driven intense sources of energetic particles and radiation, in driving matter to extremetemperatures creating state regimes relevant for planetary or stellar science as yet inaccessible atthe laboratory scale, or in achieving high-gain laser-driven thermonuclear fusion. When interact-ing at the surface of dense (opaque) targets, intense lasers accelerate relativistic electron beamswhich transport a significant fraction of the laser energy into the target depth. However, the overalllaser-to-target coupling efficiency is impaired by the large divergence of the electron beam, intrinsicto the laser-plasma interaction. By imposing a longitudinal 600 T laser-driven magnetic-field, ourexperimental results show guided ≥
10 MA-current of MeV-electrons in solid matter. Due to the ap-plied magnetic field, the transported energy-density and the peak background electron temperatureat the 60 µ m-thick targets rear surface rise by factors ≈
5, resulting from unprecedentedly efficientguiding of relativistic electron currents.
Production of high-energy-density (HED) flowsthrough solid-density or denser matter is a majorchallenge for improving laser-driven sources of energeticparticles and radiation [1], or for optimizing the isochoricheating of dense matter of major relevance in high-gaininertial confinement fusion (ICF) [2, 3], in the study ofstellar opacities [4] or of warm dense matter states [5].When interacting with dense targets, intense laserpulses drive high-current relativistic electron beams(REB), which can transport a significant fraction of thelaser energy into the targets’s depth [6–8]. However,the energy-density flux degrades rapidly against thepenetration depth due to resistive and collisional energylosses [9–12] and mostly to the intrinsically large diver-gence of the REB [13–15], as a result of the laser-plasmainteraction and the development of electromagneticinstabilities at the target surface [16, 17]. Devisingmeans of controlling the REB transverse spread andconfine its propagation within a small radius would begreatly beneficial in the aforementioned research fieldsand applications. For example, in the framework of theFast Ignition (FI) scheme for ICF [18, 19], imposed axialmagnetic-fields (B-fields) in the 1 −
10 kT range shouldbe able to guide GA currents of MeV electrons over100 µ m distances from the laser-absorption region, i.e.,up to the dense core of nuclear fuel [20, 21]. This wouldenhance the electrons kinetic energy coupling to the core and could significantly reduce the ignitor-laser energyneeded to initiate nuclear-fusion reactions, potentiallyleading to high-gain fusion energy release.Radially-confined REB transport has been experimen-tally reported due to self-generated resistive B-fields byusing specific laser irradiation schemes and/or targetstructures [22–25]. The common principle is that col-limating B-fields are induced by REB intense currents inresistive media, due either by the beam inhomogeneity orby radially-converging gradients of resistivity along theREB propagation axis [26, 27]. Nonetheless, many elec-trons are not magnetically-trapped, maintain their initialdivergence and are scattered through collisions. In bothcases, the number of guided electrons remains under 50%.Moreover, proposed improvements involve sophisticatedtarget structures [28–30], which efficiency is unproven inthe harsh conditions of an ICF target.Here we present an alternative and apply for the firsttime an external B-field enough strong to guide MeV-electrons aligned with the REB propagation axis in solidtargets. The 600 T B-field was produced by an all-optical technique using laser-driven coil targets [31–35](see Methods). This technique creates a stable magneticfield that is of sufficiently long duration to fully magne-tize the transport target prior to REB generation. Ourresults clearly show efficient REB-guiding and increasein energy-density-flux at the rear surface of solid-density a r X i v : . [ phy s i c s . p l a s m - ph ] A ug a) B CTRREB
CH/Cutarget
Coil-target driven byLP-laser
Shield
Intense Laser(SP) yx z xx H offset = -70 µm V offset = 120 µm Configuration i) H offset = 0 µm V offset = 50 µm Configuration ii) b) c)d) e) -100 -70 -4070 z [µm] y [ µ m ] -30 z [µm] y [ µ m ] z = -70 µm z = 0 x = 0 x = 0 [T] z B [T] y B FIG. 1.
Experimental configuration for the study of REB-transport with imposed B-field. a)
Sketch of theexperimental setup: the REB is generated by the intense SP laser, focused parallel to the coil axis and at normal incidenceonto the centre of the front surface of a neighboring solid 50 µ m-CH / 10 µ m-Cu thick target of 200 µ m diameter. The Nicoil-target (coil radius of 250 µ m) is previously driven by the ns LP laser. REB-patterns were investigated by imaging the CTRemitted from the transport targets’ rear surface. b) to e) B-field distribution in vacuum at its peak value, 1 ns after LP-laserdriving (origin of the spatial coordinates at the coil center), as experimentally and numerically characterized in [34]: b) , d) Amplitude of the B-field longitudinal component averaged over the 60 µ m target thickness, B z , at the two explored positions ofthe transport target. The dashed circles represent the position of the transport target in the perpendicular plane: the coil axisand the SP laser axis are respectively represented by the cross-signs and the center of the pink circles, which radius correspondsto the REB initial radius, r , in the REB-transport simulations. c) , e) Absolute value of the B-field vertical component | B y | (color scale) and arrow representation of the B-field lines over the target x = 0 -slice, for the two target positions. The plotscorrespond also to the B-field embedded into the targets as initial conditions for the REB-transport simulations in magnetizedconditions, in agreement with predictions of the B-field resistive diffusion. targets of 60 µ m thickness.The experiments were conducted at the LULI pico 2000laser facility with a 1 . µ m wavelength (1 ω ) dual laserbeam configuration: i) a high-energy long-pulse beam[LP: 1 ns, 500 ±
30 J, (1 . ± . × W / cm ] focusedinto Ni coil-targets produced a B-field of several hundredsTesla and duration of a few ns [34], ii) at different delays∆ t with respect to the LP, a high-intensity short-pulsebeam [SP: 1 ps, 46 ± . × W / cm ] focusedat normal incidence generated a REB in solid plastic tar-gets, located at the coil vicinity. The setup at the coilvicinity is sketched in Fig. 1-a). Further details on thecoil-targets geometry and laser irradiation are given inMethods.The REB transport targets were 200 µ m-diameter and50 µ m-thick plastic (CH) cylinders with a 10 µ m-thickCu-coating on the rear side. The cylinder’s axis wasinvariably parallel to the coil axis, and for the two ex-perimental runs, we explored successively positioning thetarget i) shifted from the coil plane (with horizontal andvertical offsets of the target centre with respect to thecoil centre of H offset = − µ m, V offset = 120 µ m), and ii)at the coil plane ( H offset = 0 µ m, V offset = 50 µ m). Thisenabled us to explore two different 3D spatial distribu-tions for the B-field imposed to the transport targets, asseen in Fig. 1-b) and c) for configuration i) and Fig. 1-d)and e) for configuration ii). For each of the two configu-rations, the choice of ∆ t controlled the time allowed for B-field diffusion in the transport targets prior to REBinjection, testing REB-transport in different conditionsof target magnetization.The evolution of the transport-target magnetizationhas been predicted by simulations of the B-field resistivediffusion inside the target as the B-field rises up to itspeak-value (rise-time of ≈ ≈ τ diff = µ L /η ≈ L = 50 µ m of the target CH-layer assum-ing a constant resistivity of η = 10 − Ω m (expected forCH at 1 eV).The REB transverse pattern after crossing the targetthickness was investigated by imaging the Coherent Tran-sition Radiation (CTR) emission from the rear surface attwice the SP-laser frequency, 2 ω . The emitting surfacewas imaged at a 22 . ◦ horizontal angle from the targetnormal into an optical streak camera used with a wideslit aperture as a fast gated 2D frame grabber [13]. Sam-ple results of the CTR signals are shown in the first rowof Fig. 2, for target position configurations i) on the left,and ii) on the right, with and without imposing an ex-ternal B-field, as labelled. The aspect ratio of the signalshas been corrected from the observation angle. For the a.u.a.u.
20 µm20 µm 20 µm
20 µm
20 µm a.u.
20 µm 20 µm a.u.
20 µm 20 µm
No applied B-field No applied B-field E x p . C T R S y n t h e t i c C T R With applied B-field With applied B-field∆t = 0.5 ns ∆t = 1 ns ∆t = 1 ns Configuration i) Configuration ii) a) b) c) d) e)f) g) h) i)
FIG. 2.
Results of the experimental CTR imaging (first row) and of the synthetic CTR calculated from 3D-PIChybrid simulations of fast electron transport (second row) , for the two configurations i) target out of the coil planeand ii) target at the coil plane (see Fig. 1), with and without imposed B-field. The contour lines correspond to the half-heightof the signals. The crossed dashed lines indicate the position of REB-injection at the targets’ front surface. two data sets, the average SP laser energy and intensitywere respectively i) 47 ± . ± . × W / cm ,ii) 49 ± . ± . × W / cm . The differencein laser intensity is mainly due to different focal spots inthe two independent experimental runs.Without externally imposed B-field [Fig. 2-a) and d)],we obtained rather large ( ≈ ± µ m half-width-half-maximum, HWHM) and fairly-symmetric CTR patterns.When imposing the longitudinal B-field for the targetposition i) and varying the delay of REB-injection, at∆ t = 0 . t = 1 ns we have obtained CTR patterns sig-nificantly different than the case without B-field, forboth target positions i) and ii) [respectively Fig. 2-c)and e)]. The CTR-patterns are clearly narrower hori-zontally. Vertically, the signal is also narrower for con-figuration ii), while it is elongated for configuration i).These correspond to half-height areas of equivalent ra-dius ≈ µ m for configuration i) and ≈ µ m for config-uration ii). The CTR yield decreased for configuration i)and increased for configuration ii) relative to the corre-sponding signals without B-field. As discussed above, thedelay ∆ t = 1 ns corresponds to REB transport in mag-netized targets. The differences in the patterns shapeand yield between Fig. 2-c) and e) are then related to thedifferent B-field distributions inside the targets, as rep-resented in Fig. 1-c) and e): beam electrons are trapped and follow B-field lines if their Larmor radius becomescomparable or smaller than the beam radius. A 600 Tfield [see values of B z around the REB axis in Fig. 1-b) and d)] is strong enough to confine MeV-electrons. It isworth noting that the B y -component in config. i) [Fig. 1-c)] deviates downwards the REB propagation axis. Theconsequent inclined REB path in the transport targetresults in an observed yield drop on the detected CTR.This behaviour of the CTR is reproduced and will befurther explained in the following part of the article, bymeans of REB transport simulations. In comparison, forconfig. ii) with optimum imposed B-field symmetry, thenarrowing of the signal e) compared to d) is significantand is a signature of a radially-pinched REB. Here theREB axis is not deviated. As a result, an increase inthe REB density occurs and is directly observed as anenhancement of the CTR yield by a factor 6.Details of the REB propagation and energy transportwere unfolded by simulations using a 3D PIC-hybrid codefor the electron transport [36], accounting for fast elec-tron collisions with the background material and REBself-generated fields. We simulated REB transport bothwith and without imposed B-fields. For the former case,we assumed full target magnetization (corresponding tothe experimental ∆ t = 1 ns) imposing as initial condi-tions the 3D B-field distributions illustrated in Fig. 1-b)-e).The initial REB total kinetic energy was set to 30% ofthe on target SP-laser energy, and injected at the frontsurface over a region of r ≈ µ m-radius HWHM, cor-responding to empiric factors 4 or 3 of the SP-laser focalspot size HWHM, respectively for config. i) or ii). Theinjected electron kinetic energy spectra [dashed lines inFig. 3-a) and b)] were characterized by power laws for thelow energy part ∝ ( E k ) − . and exponential laws for the R e a r - s i d e R E B d e n s i t y R e a r - s i d e e l e c t r o n t e m p e r a t u r e No applied B-field With applied B-field
Configuration i)
40 µm 40 µm40 µm
40 µm
No applied B-field With applied B-field
Configuration ii) e) f)i) j) T e [eV]n h [10 cm -3 ]
40 µm 40 µm
40 µm 40 µm c) d)g) h)
Rearside a) Rearside b) FIG. 3.
REB energy spectra (first row), rear-side REB density (second row) and background electron-temperature (third row) unfolded from 3D PIC-hybrid transport simulations.
The results are plotted for target-position configurations i) on the left and ii) on the right, without and with B-field. a) , b) REB energy spectra at the targets’front-side (source, dashed lines) and rear-side (full lines). c) - f ) REB density at the target rear surface (time-integrated). g) - j) Final background electron-temperature at the target rear surface. In c) - j) the contour lines correspond to the half-heightof the signals and the crossed dashed lines indicate the position of REB-injection at the targets’ front surface. high energy part ∝ exp ( − E k /T h ) with T i) h = 2 . T ii) h = 1 . ° mean radial angle and a 55 ° dispersion angle as de-fined in [30, 38] (see Methods). All the above geometricand energy REB source parameters are consistent withour previous characterization in the same laser facilityand equivalent laser parameters [38, 40].For a direct comparison with the experimental data, wedeveloped a synthetic CTR-emission post-processor ap-plied to the transport code output. CTR is reconstructedby the coherently added transition radiation fields pro-duced by each simulated macro-particle. The yield, spa-tial and angular distributions of CTR are non-linearlydependent on the density of fast electrons crossing thetarget-vacuum boundary, on their momentum distribu-tion (norm and angle relative to the target normal) andon the observation angle relative to the symmetry axis of the momentum distribution [39], as illustrated in theMethods. Further details on the parameters and as-sumptions of REB-transport simulations and CTR post-processing are found in the Methods.Synthetic CTR-signals are presented in Fig. 2-f), g),h) and i), reproducing fairly well the experimental CTRpatterns as well as the relative signal yield change whenimposing the B-field. In more details, the simulationsreproduce with 15 ±
2% relative errors the ratio of CTR-yield (with B-field / without B-field) for both target po-sitions. By applying the B-field, the synthetic CTR emis-sion is pinched when the target is placed at the coil plane[Fig. 2 i)] and the signal is vertically elongated when thetarget is placed outside the coil plane [Fig. 2 g)]. The ex-perimental CTR patterns’ radius (azimuthally averaged)are reproduced with 15 ±
5% relative errors for config.i) and ii), except when the target is placed outside thecoil plane with applied B-field: the elliptic shape of theexperimental signal is not exactly reproduced and its av-eraged radius is reproduced with a ≈
33 % relative error.As for the REB-transport, Figure 3 shows the corre-sponding simulation results at the 60 µ m-thick targets’rear side surface. Figure 3 a) and b) show the time-integrated REB kinetic energy spectra E k for simula-tions with (full red) and without (full blue) B-field, com-pared to the spectrum at the front surface (dashed black).Beam electrons with energy E k <
100 keV are absorbedor scattered out of the simulation box before crossing thefull target thickness as expected from the direct collisionswith the background material and by resistive field effectlinked to the neutralizing return current of thermal elec-trons [38, 40]. In config. ii) the symmetric B-field slightlymitigates energy losses for E k up to ≈ ≈
15 and ≈
20 and decreasing the beam mean radius by factor ≈ ≈
2, respectively for configurations i) and ii).An other encouraging outcome is that the imposed B-field smoothes the REB-filaments compared to simula-tions without B-field, for both target positions. Yet, itis noticeable in simulations that the B-field asymmetryin configuration i) deviates vertically the REB from itsinjection axis, exiting the target with a vertical shift of ≈ µ m with respect to its injection axis and an angle of ≈ ◦ . On one hand, this vertical deviation broads bothtime-of-flight and momentum-angle of the beam electronsat the target rear side. This contributes to the coherenceloss of the CTR [Fig. 2-c), g)] compared to the unmagne-tized transport case [Fig. 2-a), f)]. On the other hand, thedeviation points the beam away from the CTR collectinglens. Calculations of the CTR yield variation as a func-tion of the collecting lens position are given in Methods.The chosen lens position in our setup accounts for halfof the total CTR signal drop observed in config i). Theremaining part of the CTR drop is due to the additionalbroadening of the electron bunches and consequent CTRloss of coherence.The substantially smoother, narrower and denserbeams in Fig. 3-d) and f) correspond to unprecedentedefficient guiding and improved energy-density flux. Theimpact is clearly seen in the reached peak backgroundelectron temperature [Figure 3 g) - j)], higher with B-field by a factor ≈ . x , p x ) is plotted in Fig. 4: in a) without B-field, the in-clined shape of high ellipticity is characteristic of sym-metric correlated transverse momentum and position ofa regularly diverging beam; in b) with B-field, the phase-space map is significantly narrower than in a) but only forthe spatial-coordinates. The electrons cyclotron move-ment de-correlates positions and momenta and their ra- Depth [µm] N o r m a li z e d R M S e m i tt a n ce [ mm . m r ad ] T r a n s po r t e d k i n e t i ce n e r g y [ J ] c) x [µm] x [µm] − −
50 0 50 100 − −
50 0 50 100 − − p x [ β γ ] With B b) Without B a) (cid:31) x (cid:31) y E k Without B (cid:31) x (cid:31) y E k E k - r With B E k - r FIG. 4.
REB transverse phase-space ( x , p x ) at thetarget’s rear surface and REB emittance and totalkinetic energy as a function of its propagation depth,for configuration ii) with transport targets at the coilplane: cases with (red) and without (blue) imposedB-field . a) REB phase-space ( x , p x ) at the target rear-surface, for the case without B-field. b) Idem for the casewith B-field. c) REB normalized RMS emittance in horizon-tal (cid:15) x , and vertical (cid:15) y , transverse directions (left-hand sideordinates); E k and E k − r respectively the total transportedkinetic energy (diamonds) and its fraction within the initialREB radius r centred in target axis (stars), as a function ofthe propagation depth (right-hand side ordinates). dial spread is limited as they are trapped and flow along(rotating around) the B-field lines, as expected from theguiding criterium where the ≈ µ m-Larmor radius, cal-culated for the REB-source mean kinetic energy ≈ r ≈ µ m. Yet the B-field does not really affectthe electrons divergence as the width of the transversemomenta distribution is maintained.Figure 4-c) summarizes the evolution with targetdepth of the REB normalized root-mean-square (RMS)emittance in both horizontal ( (cid:15) x ) and vertical ( (cid:15) y )transverse coordinates (left-hand side ordinates; forfurther details on (cid:15) calculation, see Methods). The plotclearly shows the effect of the imposed B-field: withoutB-field the electron beam emittance rises considerablydue to collisional scattering (full blue lines), an effectconsiderably mitigated in case of magnetic guidingof the REB (full red lines).The emittance evolutionover the first ∼ µ m is particularly steep for theunmagnetized case, this is related to the high diffusivityof the lower-energy electrons in the REB. For largerdepths, the two slopes are less steep, as the low energyelectrons are previously lost by collisional scattering andOhmic heating [12, 38, 40].To understand the energy transport, we plot inthe same graph (right-hand side ordinates) the evolutionof the time-integrated REB total energy ( E k , diamondsconnected by dashed lines) and REB energy encircledover the surface corresponding to the initial REB-source, πr ( r is the initial REB radius) kept centred with theinjection axis ( E k − r , stars connected by dotted lines).The E k -loss rate against target depth is comparablefor the two cases without (blue) and with (red) B-field,except for the first ∼ µ m where the B-field efficientlyconfines electrons and also smoothes REB filamentation.About 45% more energy is transported to the target rearin the magnetized case due to the magnetic confinementmitigating the high diffusivity of low-energy particles.Much more importantly, as a consequence of REBguiding in the magnetized case, the r -encircled energyaround the injection axis at the target rear contains ≈
66% of the total transported energy, against only ≈
18% for the unmagnetized case. As a consequence,the REB energy-density flux after crossing the targetthickness increases by ≈ . × by applying the B-field.In conclusion, we have succeeded to efficiently guidea laser-accelerated MeV electron beam through solid-density matter by, for the first time, imposing a 600 TB-field parallel to the electron beam propagation axis.The B-field was generated by an all-optical process us-ing a coil-target driven by a high-energy ns laser inter-action [34]. This B-field was driven 1 ns before the REBacceleration, providing a sufficient time for the magne-tization of the transport target. In our best setup con-figuration, we found at the rear surface of a 60 µ m tar-get that the energy density transported by the fast elec-trons and the peak background electron temperature in-crease respectively by factors of ≈ . ≈ . METHODSProduction of strong magnetic-fields driven by laser
The first part of our experimental campaign wasdevoted to producing and characterizing strong B-fields [34]. We used a LP laser with 1 . µ m wavelength,500 ±
30 J energy, 1 ns flat-top duration ( ≈
100 ps risetime), focused into loop-shaped targets. These weremade out of 50 µ m-thick Ni-foil, laser-cut and bent toform two parallel disks, connected by a coil-shaped wireof 500 µ m-diameter [see Fig. 5-a)]. The laser was incidentalong the surface normal of one disk, passing throughthe hole of the other, yielding (1 . ± . × W / cm of focused intensity. The target is charged by the laser-generated supra-thermal electrons escaping its potentialbarrier. We believe also that some of them are capturedon the opposite holed disk. The resulting discharge cur-rent through the wire loop closes the circuit producing aquasi-static (time-scale of a few ns), dipole-like B-field inthe coil region. The laser-charging and discharge throughthe wire process during the laser irradiation, after this thetarget discharges like an RL-circuit. Figure 5-b) showsresults for the B-field strength at the coil center, B ,as a function of time, inferred from induction measure-ments at 7 cm from the coil, approximately along thecoil axis, using 2 . I and to the coil radius a , accordingto B ≈ µ I/ a . µ is the vacuum permeability. Theinferred results show reproducible charging-time, consis-tent with the driver LP laser duration, and peak valueof ≈
600 T ( ± -scale volume. The spatial-integratedenergy of the B-field at peak-time corresponds to ≈ . Simulations of the magnetic-field diffusion over thetransport targets
The transport-target magnetization has been predictedby simulations of the B-field resistive diffusion inside thetarget [42], as the B-field rises with time up to its peak-value. The model describes the penetration of externalB-fields in the target material, according to the diffusionequation ∂ t (cid:126)B = η ∇ B/µ , which is valid for small mag- -2000200400600800-2 0 2 4 6 8 10 B [ T ] time [ns] Faradayrotationprotonsprobing
B-dot probing a) b) B (cid:31)(cid:30) I + ++++ ---- - FIG. 5.
Production of strong magnetic-fields drivenby laser. a)
Illustration of the B-field production mecha-nism with laser-driven coil-targets. b) Experimental resultsfor the B-field strength at the center of the coil as a func-tion of time, inferred from measurements by B-dot probes(green curves), Faraday rotation (black square) and proton-deflectometry (red circle). H offset from coil plane [µm] B z [T] CHtarget vacuum -‐120 -‐80 -‐40 0 H offset from coil plane [µm] t = 1 ns CHtarget vacuum V o ff s e t f r o m c o i l a x i s [ µ m ] t = 0.5 ns FIG. 6.
Axis-symmetric simulations of the B-field re-sistive diffusion over the transport targets.
The B-fieldspatial distribution is given at ∆ τ = 0 . τ = 1 ns (right panel) for the target positioning config-uration with H offset = 70 µ m and V offset = 120 µ m. From thebreaking on the contour lines at the target-vacuum interface,we conclude that there is a discontinuity of the B-field for∆ τ = 0 . τ = 1 ns, assumed as the time of full target magnetization. netic Reynolds number. At each time iteration, Amp`ere’slaw calculates the induced current density, from whichohmic heating is then computed for the temperature mapof the target along with a new resistivity map. Figure 6presents sample results for the target position configu-ration i), at t = 0 . t = 1 ns (rightpanel). At t ≈ PIC-hybrid simulations of electron beam transport
PIC-hybrid simulations allow to describe fast electronbeam transport in dense matter, where the injected beamcurrent is modeled kinetically by a particle-in-cell (PIC)method and the neutralizing return current of back-ground thermal electrons is described by Ohm’s law asan inertialess fluid [36, 44, 45]. The hybrid method ne-glects high frequency effects, hence enabling a simplifi-cation of Maxwell equations by neglecting the Poissonequation and the displacement current in the Maxwell-Ampere equation.Our simulation box corresponded to the transport-target dimensions, reproducing its CH-Cu structure interms of background density and resistivity behaviouras a function of the evolving background electron tem-perature due to REB-deposited energy. Both collisionaland ionization processes were taken into account for com-puting the evolution of the background resistivity, com-puted using the classical Drude model according to theEidmann-Chimier model [46, 47] and as described in[38]. The background electron temperature is initiatedat 0 . E k distributions between 8 keVand 10 MeV, characterized by power laws for the lowenergy part ∝ ( E k ) − . and exponential laws for thehigh energy part ∝ exp ( − E k /T h ) with T i) h = 2 . T ii) h = 1 . T h = m e c ( (cid:112) a − a is the normalized laserintensity. The corresponding fast electron angular distri-bution was approximated with the following form [30]: f h ( θ, r ) ∝ exp (cid:2) − ( θ − θ r ) / ∆ θ (cid:3) with ∆ θ = 55 ° thedispersion angle, and θ r = arctan (tan (Θ) r/r ), whereΘ = 30 ° is the mean radial angle at initial REB ra-dius r ≈ µ m in configuration i) and r ≈ µ min configuration ii). These geometric and energy REBsource parameters are consistent with an experimentalcharacterization made previously in the same facilitywith equivalent laser parameters and are supported bybenchmarked simulations [38]. Approximately 50 mil-lions macro-particles were used to simulate the propaga-tion of ∼ − × electrons through the target. Thetotal simulation time was set to 3 . µ m, respectively.Given the ps-time scale of the REB-transport, veryfast if compared to the ns-scale evolution of the B-fieldstrength or of its diffusion in the target, we assumed thatthe B-field distribution is constant over each simulationrun-time. For the cases with applied external B-field,we only considered fully-magnetized targets (∆ t = 1 ns).The B-field spatial distribution inside the target is calcu-lated as in vacuum by a 3D magnetostatic code [48], con-sistently with the experimental characterization of theB-field space-time evolution obtained with laser-drivencoil-targets [34]. Coherent Transition Radiation as a diagnostic oflaser-accelerated relativistic electron beams
Coherent Transition Radiation (CTR) is produced bythe REB crossing the target-vacuum boundary [49, 50].Its time-scale, of the order of a few ps, follows that of thefast electron flux envelope. Yet, the pulsed character ofrelativistic laser-acceleration mechanisms modulates lon-gitudinally the REB-current as a comb of periodic micro-bunches. The coherent interference of the transition ra-diation produced by the electron-comb crossing the rearsurface yields peak emissions at the spectral harmonicsof the bunch frequency.In the present experiment, the CTR imaging system ofthe transport targets’ rear surface was composed by twodoublet-lenses with an optical aperture of f /
9, and wasaligned on the equator plane looking at the target rearCu-surface with a 22 . ° -angle with respect to its normal.The optical system produced images with a magnifica-tion of ≈
20 with a spatial resolution of ≈ µ m − σ .The streak camera was used with an open slit ( ≈ . ω -light was selected by a 10 nm FWHM-bandwidth interferometric filter centred at 532 nm. Asan extra precaution for reducing the noise level due toany spurious light from the coil-target, the first lens pro-duced an intermediate image where mechanical filteringallowed to select only the central region of the imagedfield, corresponding to the REB-transport target surface.The CTR data was used as benchmarking reference forthe electron transport simulations coupled to a syntheticCTR-emission post-processor. As the hybrid transportcode continuously injects particles during the laser pulseduration, and as such does not simulate the REB lon-gitudinal (or temporal) modulations, we assumed thatthe periodic electron micro-bunches produced through-out the duration of the laser-plasma interaction are iden-tical and that the CTR spatial pattern and angular distri-bution are not dependent on the number of bunches. TheCTR is therefore calculated for a single electron bunch,yielding intensities at one given wavelength in arbitraryunits.In general, the coherence of the radiation can be de-composed into a temporal and a spatial component [11,39]. For the temporal coherence, the phase difference,Φ i − Φ j of the transition radiation fields emitted at thetarget rear side is calculated after considering the time of flight of each individual macro-particle, i.e. we record thetime difference between injection and time of arrival atthe target rear side. For the spatial coherence, we assumeno phase shift between macro-particles belonging to thesame cell at the target rear side. This is justified by not-ing that the cell size in the simulation, of 1 µ m × µ m, iscomparable to the observed wavelength and smaller thanthe spatial resolution of the experimental imaging sys-tem. The finite spatial resolution is taken into accountby convoluting the CTR out of the simulation with a3 µ m standard deviation Gaussian function.The coherent addition of the fields (cid:126)E i emitted by eachmacro-particle is therefore given by: I T R ∝ (cid:88) i | (cid:126)E i | + (cid:88) i (cid:88) j, j (cid:54) = i | (cid:126)E i || (cid:126)E j | exp ( i (Φ i − Φ j )) . (1)The final synthetic images to be compared to the ex-perimental CTR data take into account the imaging sys-tem solid angle, angle of observation and magnificationfactor. Figure 7 shows results for the synthetic CTR forthe four experimental situations analysed in the article:with and without B-field, with in config. i) the trans-port target outside the coil plane and in config. ii) thetransport target at the coil plane. The time and spaceintegrated CTR yields are plotted as a function of theimaging diagnostic orientation with respect to the targetnormal, by varying both latitude (dashed lines) and lon-gitude (full lines). The figure also shows sample syntheticCTR images, corresponding to the lens coordinates indi-cated in the graphs. The four images corresponding tothe experimental setup (longitude 22 . ◦ , latitude 0 ◦ ) areidentified by the thicker box frames and by the dots onthe graphs. The size of the dots are representative of thesolid angle of the collecting lens in the experiment. TheCTR yield calculation obtained either by considering thefull solid angle of the lens or its center point differs byless than 1%, justifying the point-like lens approximationmade for the calculations of Figure 7.In the case without applied B-field (bluecurves/frames), the diverging REB propagation issymmetric around the injection axis. The result is asymmetric CTR emission pattern. As expected [39],peak emission in the CTR image occurs at an angle closeto 1 /γ , where γ is the relativistic factor correspondingto the mean energy of the REB. With B-field (redcurves/frames), the REB propagation is confined to asmall radius and revolves at the cyclotron frequencyaround its symmetry axis. In configuration i), the REBaxis deviates downwards due to the inclination in B-fieldlines, explaining the asymmetric angular dependenceof the CTR emission. The peak emission in latitudecorresponds to the direction angle of the REB exitingat the target rear. In configuration ii) the angulardistribution is closer to that of a beam symmetric tothe target normal, yet due to the cyclotron movement, Configuration i)Configuration ii)
20 µm
20 µm
20 µm -22.5°
20 µm -45°
20 µm -22.5°
20 µm
20 µm -22.5°
20 µm i)i) ii) iii) iv)v) vi) vii) viii)
20 µm
20 µm
20 µm -22.5°
20 µm -45°
20 µm -22.5°
20 µm
20 µm -22.5°
20 µm i) ii) iii) iv)v) vi) vii) viii)+i)+ vii)+ viii)+ii) iii)iii)vi) iv)+ii)+ +iv)+v)+ v)+vii) viii)+ +vi) a.u.a.u.
FIG. 7.
Synthetic CTR calculated from the benchmarked simulations of REB transport in the experimentalconditions.
The graphs on the left plot the total CTR yield as a function of the lens position, in terms of varying longitude(full lines) or varying latitude (dashed lines). The symbols identify the coordinates corresponding to the sample syntheticCTR images on the right. Blue curves and boxes correspond to REB transport without B-field, red curves and boxes to REBtransport with imposed B-field. The four experimental configurations analysed in the article are identified by the thicker framesand by the dot-symbols in the graphs. the exiting position and angle are related to the targetthickness.
Calculation of REB-emittance
Emittance is a quantity of area or volume in phasespace of particles. It is usually used as a property to de-scribe a beam along its propagation when the motion ofparticles in transverse and longitudinal planes are weaklycoupled. As in conventional accelerators, fast electronbeams accelerated by high intensity lasers have momen-tum mostly directed in the longitudinal direction. Onthe canonical phase-space, a statistical definition of thenormalized Root Mean Square (RMS) emittance in the( x, p x ) plane is given by : ε x,n,RMS = (cid:113) (cid:104) x (cid:105) (cid:104) p x (cid:105) − (cid:104) x p x (cid:105) It is convenient to use momenta in dimensionless units of βγ (relativistic parameters). (cid:10) x (cid:11) defines the second cen- tral moment of the particle distribution x . For macropar-ticules distributions extracted from PIC simulations, theweights w i of each macroparticle have to be taken intoaccount : (cid:10) x (cid:11) = (cid:80) i (cid:0) w i ( x i − ¯ x ) (cid:1)(cid:80) i w i and (cid:104) x p x (cid:105) = (cid:80) i ( w i ( x i − ¯ x )( p x i − ¯ p x )) (cid:80) i w i where ¯ x and ¯ p x are respectively the weighted-average of x and p x :¯ x = (cid:88) i ( w i x i ) / (cid:88) i w i and ¯ p x = (cid:88) i ( w i p x i ) / (cid:88) i w i . The definitions are similar for the emittance calculationin the ( y, p y ) plan.0 ACKNOWLEDGMENTS
We gratefully acknowledge the support of the LULIpico 2000 staff during the experimental run. J.J.S. grate-fully acknowledges fruitful discussions with L. Gremil-let. This work was performed through funding from theFrench National Agency for Research (ANR) and thecompetitiveness cluster Alpha - Route des Lasers, projectnumber TERRE ANR-2011-BS04-014. The authors alsoacknowledge support from the COST Action MP1208”Developing the physics and the scientific communityfor Inertial Fusion” through three STSM visit grants.The research was carried out within the framework ofthe ”Investments for the future” program IdEx BordeauxLAPHIA (ANR-10-IDEX-03-02) and of the EUROfusionConsortium and has received funding from the Euro-pean Union’s Horizon 2020 research and innovation pro-gram, grant 633053. The views and opinions expressedherein do not necessarily reflect those of the EuropeanCommission. The Japanese collaborators were supportedthe Japanese Ministry of Education, Science, Sports,and Culture through Grants-in-Aid for Young Scientists(Grants No. 24684044), Grants-in-Aid for Fellows byJSPS (Grant No. 14J06592), and the program for pro-moting the enhancement of research universities. Sim-ulation work has been partially supported by the Span-ish Ministry of Economy and Competitiveness (grant No.ENE2014-54960-R) and used HPC resources and techni-cal assistance from CeSViMa Centre of the UPM.
AUTHOR CONTRIBUTIONS
J.J.S. designed and executed the experiment as princi-pal investigator, with the help from M.B.-G., C.B., P.F.-C., S.F., L.G., J.E.C., R.C., J.-L.D., M.E., S.K., J.-R.M.,A.M., S.S., J.S. and Z.Z. and the engineering supportfrom R.B., M.C., S.D., E.L. and F.S.; data analysis wasperformed by M.B-G., with contributions from P.F-C.,L.G., M.E., S.K., S.S. and Z.Z. and under the supervisionof J.S.S., D.B. and S.F.; B-field diffusion modelling wasperformed by J.J.H., who also developed the PIC-hybridcode; the CTR post-processor was developed by C.B.;PIC-hybrid simulations were run and analysed by M.B.-G. with supervision of J.J.S, C.B. and J.J.H.; targetswere manufactured by G.S.; G.G., S.H., Ph.N., M.R.,V.T.T. and N.W. contributed to the discussion of the re-sults; M.B.-G. and J.J.S. led the manuscript writing; allthe figures were prepared by M.B.-G. ∗ [email protected][1] Ledingham, K.W.D., Galster, W. Laser-driven particleand photon beams and some applications. New Journal of Physics , 045005 (2010).[2] Robinson, A.P.L. et al . Theory of fast electron transportfor fast ignition. Nucl. Fusion , 054003 (2014).[3] Norreys, P. et al . Fast electron energy transport in soliddensity and compressed plasma. Nucl. Fusion , 054004(2014).[4] Hoarty, D.J. et al . Observations of the Effect ofIonization-Potential Depression in Hot Dense Plasma. Phys. Rev. Lett. , 265003 (2013).[5] P´erez, F. et al . Enhanced Isochoric Heating fromFast Electrons Produced by High-Contrast, Relativistic-Intensity Laser Pulses.
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