Super-ponderomotive electron acceleration in blowout plasma heated by multi-picosecond relativistic intensity laser pulse
Sadaoki Kojima, Masayasu Hata, Natsumi Iwata, Yasunobu Arikawa, Alessio Morace, Shouhei Sakata, Seungho Lee, Kazuki Matsuo, King Fai Farley Law, Hiroki Morita, Yugo Ochiai, Akifumi Yogo, Hideo Nagatomo, Tetsuo Ozaki, Tomoyuki Johzaki, Atsushi Sunahara, Hitoshi Sakagami, Zhe Zhang, Shota Tosaki, Yuki Abe, Junji Kawanaka, Shigeki Tokita, Mitsuo Nakai, Hiroaki Nishimura, Hiroyuki Shiraga, Hiroshi Azechi, Yasuhiko Sentoku, Shinsuke Fujioka
SSuper-ponderomotive electron acceleration in blowout plasma heated by multi-picosecondrelativistic intensity laser pulse
Sadaoki Kojima, ∗ Masayasu Hata, Natsumi Iwata, Yasunobu Arikawa, Alessio Morace, Shouhei Sakata, Seungho Lee, Kazuki Matsuo, King Fai Farley Law, Hiroki Morita, Yugo Ochiai, Akifumi Yogo, Hideo Nagatomo, Tetsuo Ozaki, Tomoyuki Johzaki, Atsushi Sunahara, Hitoshi Sakagami, Zhe Zhang, Shota Tosaki, Yuki Abe, Junji Kawanaka, ShigekiTokita, Mitsuo Nakai, Hiroaki Nishimura, Hiroyuki Shiraga, Hiroshi Azechi, Yasuhiko Sentoku, and Shinsuke Fujioka
1, † Institute of Laser Engineering, Osaka University, 2-6 Yamada-Oka, Suita, Osaka, 565-0871 Japan. National Institute for Fusion Science, National Institutes of Natural Sciences, 322-6 Oroshi, Toki, Gifu, 509-5292, Japan. Department of Mechanical Systems Engineering, Hiroshima University, Higashi-Hiroshima, Hiroshima, 739-8527, Japan. Institute for Laser Technology, 1-8-4 Utsubo-honmachi, Nishi-ku Osaka, Osaka, 550-0004, Japan. Beijing National Laboratory of Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China.
The dependence of the mean kinetic energy of laser-accelerated electrons on the laser intensity, so-called pon-deromotive scaling, was derived theoretically with consideration of the motion of a single electron in oscillatinglaser fields. This scaling explains well the experimental results obtained with high-intensity pulses and dura-tions shorter than a picosecond; however, this scaling is no longer applicable to the multi-picosecond (multi-ps)facility experiments. Here, we experimentally clarified the generation of the super-ponderomotive-relativisticelectrons (SP-REs) through multi-ps relativistic laser-plasma interactions using prepulse-free LFEX laser pulsesthat were realized using a plasma mirror (PM). The SP-REs are produced with direct laser acceleration assistedby the self-generated quasi-static electric field and with loop-injected direct acceleration by the self-generatedquasi-static magnetic field, which grow in a blowout plasma heated by a multi-ps laser pulse. Finally, we the-oretically derive the threshold pulse duration to boost the acceleration of REs, which provides an importantinsight into the determination of laser pulse duration at kilojoule- petawatt laser facilities.
I. INTRODUCTION
When a high-intensity laser pulse is irradiated on a mate-rial, its surface is instantaneously ionized, and the electrons inthe ionized material, i.e., plasma, are then accelerated closeto the speed of light by the ponderomotive force of the laserlight. These energetic electrons are often called relativisticelectrons (REs). The energy distribution of REs is approx-imated by a Maxwell-Boltzmann distribution function withslope temperature T RE as dN / dE ∝ exp ( − E / T RE ) where N and E denote the number and energy, respectively. The scal-ing laws of T RE on laser intensity have been investigated ex-perimentally [1–3], theoretically, and computationally [4–7].These scaling laws are useful to determine laser parametersfor high-intensity short pulse laser experiments and to designapplications. The effect of pulse duration on T RE is not con-sidered explicitly in the reported scaling laws; however, re-cent computational and theoretical studies [8, 9] have revealedthat T RE generated by multi-picosecond (multi-ps) laser pulsescould be several times higher than that predicted by the re-ported scaling laws. With the development of kilojoule-classhigh-power lasers such as LFEX [10], NIF-ARC [11], LMJ-PETAL [12], and OMEGA-EP [13], it has become possible toirradiate relativistic laser pulses continuously over multi-ps.In this study, we have clarified the generation of super-ponderomotive RE (SP-RE) in multi-ps laser-plasma interac-tion using ultra-high-contrast LFEX laser pulses realized us-ing a plasma mirror (PM). The slope temperature of REs was ∗ [email protected] † [email protected] increased more than twice by extending the laser pulse dura-tion from 1.2 ps to 4.0 ps. The following two accelerationmechanisms were identified as essential for the generation ofSP-REs in multi-ps laser-plasma interaction with the help ofparticle-in-cell (PIC) simulations.One mechanism is the generation of SP-REs by the combi-nation of a laser field and a quasi-static electric field reportedby Sorokovikova et al. [9]. In a laser-heated plasma, a quasi-static electric field is generated spontaneously by charge sep-aration at the forward edge of the plasma expansion, and thedirection of this field is generally parallel to the direction oflaser propagation. Such a quasi-static electric field is able topush electrons along the laser propagation direction; there-fore, electrons can stay in the acceleration phase longer thanthat without a quasi-static electric field, i.e., electrons undergohigher energy gain.The other mechanism is multiple electron injection in theregion where the laser field and quasi-static electric field co-exist due to the cyclotron motion of REs in a self-generatedquasi-static azimuthal magnetic field [14, 15]. This distinc-tive injection mechanism is referred to as loop-injected directacceleration (LIDA). A tens of megagauss (MG) quasi-staticmagnetic field also develops within the expanding plasma inmulti-ps laser-plasma interaction and LIDA plays a signifi-cant role in the generation of SP-REs in multi-ps laser-plasmainteraction. The LIDA is triggered by the transition fromthe hole boring phase to the blowout phase in a laser-heatedplasma. Here, we obtained the equation of transition timingfor arbitrary laser pulses. a r X i v : . [ phy s i c s . p l a s m - ph ] M a r II. EXPERIMENTAL OBSERVATION OFSUPER-PONDEROMOTIVE ELECTRONS
We have experimentally investigated the dependence of REenergy distributions on the pulse durations under conditionsfree from pre-plasma formation. The experiment was con-ducted using the LFEX laser system at the Institute of LaserEngineering, Osaka University. The LFEX laser consists offour beams, where the spot diameter of the spatially over-lapped LFEX beams on a target was 70 µ m of the full widthat half maximum (FWHM), and 30% of the laser energy wascontained in this spot. One LFEX beam delivered 300 Jof 1.053 µ m wavelength laser light with a 1.2 ps duration(FWHM), and the peak intensity of one beam was 2.5 × W / cm .It is well known that SP-REs can be accelerated in a long-scale-length pre-plasma; therefore, a PM [16] was imple-mented to realize the pre-plasma-free condition to excludethe other known mechanisms from this experiment. The con-trast ratio of the LFEX laser pulse was improved by two or-ders of magnitude through implementation of the PM [17]down to 10 at 150 ps before the main pulse (as shown inFig. 1(a)). These clean intense laser pulses create the idealsituation where the REs are accelerated predominantly in theinherent plasma formed by the main laser pulse itself duringthe picosecond time range. The density scale length of thepreformed plasma was calculated to be 1.5 µ m at 10 ps be-fore the intensity peak from a 2D radiation hydrodynamicssimulation with the PINOCO-2D code [18].These “clean” pulses were focused on a 1 mm gold cube.The thickness of the gold cube is also an important parameterto investigate RE acceleration by multi-ps laser-plasma inter-actions. The REs generate a sheath electric field at the rearsurface of the target. This sheath field refluxes especially lowenergy REs and the refluxed REs are re-injected to the accel-eration region. This recirculation process also generates SP-REs, which was investigated by Yogo and Iwata et al. [19, 20].One cycle of the recirculation process takes at least 6.7 ps inthe 1 mm-thick gold cube, which is longer than the pulse dura-tions (1.2 or 4.0 ps) in this experiment; therefore, the recircu-lation process can be eliminated from the SP-RE mechanismsin this study.LFEX laser pulses can be stacked temporally with arbi-trary delays between the beams, as shown in Fig. 1(b). In thisstudy, a single beam (case A: 1.2 ps FWHM pulse durationand peak intensity of 2 . × W / cm ) was used and twotypes of four-stacked beams (case B: 4.0 ps FWHM pulse en-velope and peak intensity of 3 . × W / cm , and case C:1.2 ps FWHM pulse duration and peak intensity of 1 . × W/cm ). We emphasize here that the leading edge of thestacked pulse remains similar to that of the single beam. Ifthe pulse duration is extended by adjusting the pulse compres-sor of the laser system, the leading edge would inevitably bemodified into a more gradual shape.The energy distribution of REs emanated from the target tothe vacuum was measured with an electron energy analyzerlocated 20.9 ◦ from the incident axis of the LFEX laser. Fig-ure 2(a) shows the experimental results of the time-integrated energy distribution. The slope temperatures were 0.65 MeVfor case A (red circles) and 1.7 MeV for case B (green trian-gles). The slope temperature for case B was more than twicethat for case A, even though the peak intensities were veryclose. The energy distributions of REs obtained for case B(green triangles) and case C (blue squires) were almost identi-cal, even though the peak intensities were different by a factorof four. These slope temperatures cannot be explained us-ing the reported scaling laws, whereby the dependence of theslope temperature on the pulse duration is not considered. III. TWO-DIMENSIONAL (2D) PIC SIMULATIONS WITHEXPERIMENTAL CONDITIONSA. Electron acceleration dynamics in multi-picosecondlaser-plasma interaction
The experimental results were compared with those com-puted using the 2D PIC simulation code (PICLS-2D [21]).Calculations were performed with temporal and spatial scalesthat were comparable to the experimental scales. The goldcube was replaced with a 20 µ m planar plasma with a peakdensity of 40 n c , where n c = 1.0 × cm − is the criticalelectron density for 1.053 µ m wavelength light. The bulkplasma has an exponential density profile from 0.1 to 40 n c and a scale length of 1 µ m. Due to computational limita-tions, the ionization degree was fixed to be +40 in the PICLS-2D simulation, which was determined based on the result ofa one-dimensional PICLS simulation with the dynamic ion-ization model of gold described by field ionization [22] anda fast electron collisional ionization [23]. The ionization de-gree rose from +10 (given by the radiation hydrodynamic codePINOCO-2D) to around +40 for first several picoseconds. Inthe 2D-PIC simulation with dynamic ionization, it was re-ported that ionizing defocusing counteracting laser filamen-tation and self-focusing occurs.However, it does not signifi-cantly affect the short-scale-length pre-plasma in the order ofthe laser wavelength.The slope temperatures of the REs in the simulation were0.7, 2.0, and 2.0 MeV for cases A, B, and C, respectively.Thus, the PIC simulation reproduces well the experimentallyobserved dependence of the slope temperature on the laser in-tensity and pulse duration [24], as shown in Fig. 2.Figures 2 (c)–(e) show a comparison of the pulse shapes(red lines) and the temporal evolution of maximum energy ofREs (blue lines between circles) for cases A, B, and C. Thetemporal evolution of the maximum energy of the REs is sim-ilar to the laser pulse shapes for the cases of 1.2 ps pulse du-ration (cases A and C). In contrast, the situation for the 4.0ps pulse duration (case B) is completely different. For caseB, the maximum energy increases, even after the timing whenthe laser intensity reaches the plateau at 2.0 ps. The most ener-getic REs were produced near the end of the intensity plateau(5.5 ps). The time-integrated energy distributions of the REsfor cases B and C seem to be identical; however, the temporalbehavior of RE acceleration in case B is completely differentfrom that in case C. FIG. 1. (Color online) (a) Experimental setup. The geometrical positions of the target, PM and the diagnostics instruments, and the ray traceare illustrated. (b) Temporal intensity profiles of LFEX laser pulses. Pulses temporally stacked to generate various pulse shapes.FIG. 2. (Color online) RE energy distributions measured experimentally and computationally by changing the intensity and duration of laserpulses. (a) Comparison of experimental data for cases A (red circles) and B (green triangles), and computational data for cases A (grey line)and B (black line). (b) Comparison of experimental data for cases A (red circles) and C (blue squares), and computational data for cases A(grey line) and C (black line). Comparison between laser pulse shapes (red lines) and temporal evolution of the maximum energy of REs (linesbetween circles) for cases (c) A, (d) B, and (e) C.
Figures 3(a)–(f) show three selected RE trajectories attwo different periods ( t = 3.0–3.5 and 5.0–5.5 ps) overlaidon the electron densities [Figs. 3(a) and (b)], self-generatedazimuthal magnetic fields [Figs. 3(c) and (d)], and self-generated electric fields [Figs. 3(e) and (f)]. Figures 3(a) and (b) are colored using the lookup table of electron density log-arithm normalized with the critical density ( n c ). When a high-intensity laser is irradiated on a target, quasi-static electric andmagnetic fields are spontaneously generated on the target sur-face. The quasi-static term indicates that the time variation ofthe fields is slower than that of the laser field. The electricfield is formed with plasma expansion and its direction is per-pendicular to the target. The magnetic field is in the azimuthaldirection of the laser axis. These self-generated electric andmagnetic fields assist RE acceleration as discussed below.In the earlier period (the top panels of Fig. 3), the REsmove around the near-critical density region. The energeticRE source is initially accelerated to 3–4 MeV by the reflectedlaser field in the near-critical density region. 3–4 MeV is closeto the kinetic energy (3.5 MeV) of a RE obtained by the pon-deromotive force from the reflected laser field ( a = . I = . × W / cm ) without absorption of the incident laserfield. The electron travels outwardly (the opposite directionof laser propagation) through the magnetic and electric fieldsthat are generated by the Biermann battery effect and chargeseparation. In this period, the self-generated magnetic fieldstrength is not sufficient to change the RE motion. The self-generated electric field decelerates the outwardly moving RE,and the RE eventually stops and is then accelerated again in-wardly. The effect of the quasi-static electric field not onlydirectly imparts additional energy to the electrons but alsoreduces the dephasing rate of the RE from the accelerationphase of the laser field [9, 25–30]. The RE continues to rideon the acceleration phase, whereby the RE gains energy fromthe laser field. i.e., the RE obtains more energy when it is ac-celerated by the incident laser field. In this simulation, the REis accelerated up to 15 MeV by the combination of the quasi-static electric field and the laser field, as shown in Fig. 3(g).In the later period (bottom panels of Fig. 3), the self-generated magnetic field is sufficiently strong that some ofthe REs (blue and green trajectories) are reflected outwardlyby the vvv × BBB force and they are re-injected to the region whereboth the self-generated electric field and laser field coexist(Fig. 3(b), loop(ii)). In loop (ii), the turning point of the RE isfarther from the near-critical density region than that in loop(i) because the RE receives more kinetic energy in loop (i).After loop (ii), the kinetic energy of the REs reaches beyond15 MeV, as shown in Fig. 3(h). This re-injection mechanismis the LIDA [14].The solid lines in Figs. 4(a) and (b) show energy distribu-tions of REs accelerated in the two periods. The histogramsshow the ratio of the RE numbers between the two groups:one group (red bars) consists of REs that experienced sin-gle loop-injection and the other (green bars) consists of REsthat experienced multiple loop-injection. The correlation be-tween multiple loop-injections and energetic RE generationis clearly evident; namely, the highest energy component ofREs in Fig. 4(b) above 20 MeV is generated predominantlyby multiple loop-injection.
B. Generation of a giant quasi-static magnetic field duringmulti-ps laser-plasma interaction
The PIC simulation shows that the quasi-static magneticfield is generated by three different mechanisms in case B,which are dependent on the time during the multi-ps laser-plasma interaction. At the leading edge of the 4 ps flat-top pulse ( < γ n c ) into the overdense region, andthe heated underdense plasma expands into the vacuum. Anelectric field is generated at the outer boundary of the expand-ing plasma (which is referred to as the first electric field.).An azimuthal magnetic field is generated in the overdenseplasma due to the ∇ n × ∇ I effect [4, 31–33], where n and I arethe plasma electron density and laser intensity, respectively.When the laser intensity reaches the plateau at 2.0 ps, plasmaevacuation by the laser field is eventually halted by the chargeseparation due to depletion of the local electron density. The ∇ n × ∇ I mechanism becomes relatively small, whereas the ∇ T × ∇ n (Biermann battery) effect [34–39] becomes the dom-inant mechanism for generation of the magnetic field. Here, T is the plasma electron temperature. Along with a change of thegeneration mechanism, the generation region also moves fromthe overdense region to the underdense region. The strongestmagnetic field is generated at the edge of the laser spot inthe underdense plasma (Fig. 5(f)). This magnetic field influ-ences the motion of REs around the near-critical density re-gion. Some of the REs are moved transversely from the laserspot by the EEE × BBB drift. The drift current heats the surface ofthe bulk plasma via the two-stream instability. Enhancementof the energy transfer to the transverse direction due to the sur-face magnetic field is discussed in Refs. [40–42]. The electricfield that contributes to the
EEE × BBB drift is a weak electric fieldgenerated in a limited region near the critical density surface.The heated bulk plasma begins to expand at the edge of thelaser spot, while the expansion is suppressed at the inside ofthe laser spot by the laser ponderomotive pressure. The heatedbulk plasma surface, which has been flat so far, deforms into abow shape (which is referred to as a bow-shaped bulk plasmasurface). The first electric field is carried out by the plasmaexpansion far away from the critical density surface and nolonger contributes to the drift.When the thermal pressure of the heated bulk plasma ex-ceeds the ponderomotive pressure of the incident laser at 3.8ps, the bulk plasma begins to expand at the inside of the laserspot, and the strong quasi-static electric field (the second elec-tric field) is then generated at the near-critical density region.Figure 5(a) shows the electric fields in the longitudinal direc-tion ( E x ) of the two regions. The second electric field is gen-erated at the expansion front of the heated bulk plasma at theinside of the laser spot. The newly generated strong electricfield contributes to the EEE × BBB drift by combination with themagnetic field (Fig. 5(b)). REs move along the bow-shapedbulk plasma surface by the
EEE × BBB drift. When the REs flowin the plasma, the return-current is driven to maintain currentneutrality in the plasma. Figure 5(c) shows the RE drift cur-rent in the lower density region and the return-current flow inthe higher density region. The current loop produced by thespatial separation between the RE drift current and the returncurrent generates a magnetic field along the outer edge of thebow-shaped bulk plasma surface (Fig. 5(c)). This third mag-netic field (30–50 MG) is stronger than the magnetic field gen-erated by the ∇ T × ∇ n effect ( <
10 MG). In the plasma regionwhere RE current terminates, the electric field is enhanced by
FIG. 3. Three examples of RE trajectories at two different periods ( t = 3.0–3.5 and 5.0–5.5 ps) overlaid on the electron densities [(a) and (b)],self-generated azimuthal magnetic fields [(c) and (d)], and self-generated electric fields [(e) and (f)]. The electron density maps [(a) and (b)]are colored using the lookup table of electron density logarithm normalized according to the critical density ( n cr ). (g,h) Kinetic energies ofREs along the longitudinal position for the two different periods. the inflow of electrons (Fig. 5(d)).The positive feedback between the growth of the fields andthe field-driven drift current results in the rapid growth of thequasi-static electric and magnetic fields with time (Figs. 5(c)–(h)). The maximum energy of the REs increases from 3.5 psuntil 5.5 ps, which corresponds to the timing of rapid growthof the self-generated fields. The SP-RE are accelerated bya laser field under a quasi-static self-generated electric field.In addition, when positive feedback starts, the strength of theself-generated magnetic field grows by several tens of MGapproximately several picoseconds after the beginning of thelaser-plasma interaction, and the strong magnetic field be-gins the LIDA. Thus, SP-RE acceleration is not a processthat gradually progresses with time but a process that pro-ceeds in a threshold manner. This has not been pointed out inprevious studies on REs acceleration by multi-ps laser pulse.[8, 9, 19, 20, 43] C. Transition timing to super-ponderomotive electronacceleration
The SP-RE acceleration is started when the plasma ther-mal pressure exceeds the laser ponderomotive pressure. Fig-ure 6(a) shows the evolution of an initially exponential plasmaprofile during the interaction with a high-intensity laser pulse.The color map shows the electron density (log ( n e / n c ) ) andthe red solid line shows the temporal intensity profile of thelaser. At t = x i ( t ) = x c ( ) + l s ln (cid:20) + c l s (cid:115) R cos θ ( + R ) Zm e M i (cid:90) tt (cid:18) γ ( t ) − γ ( t ) (cid:19) / dt (cid:21) . (1)Here, I ( t ) / c = m e c n c a ( t ) / t is the time when the nor- FIG. 4. (a,b) Energy distributions (solid lines) of REs accelerated in the two periods (3.0–3.5 and 5.0–5.5 ps). The histograms show the ratioof the RE numbers between the two groups, where one group (red bars) consists of REs that experienced single loop-injection and anothergroup (green bars) consists of REs that experience multiple loop-injections due to LIDA. A correlation between multiple loop-injections andenergetic electron generation is clearly evident.FIG. 5. Spatial maps of longitudinal electric field E x [((a), (d), and (g)], transverse current density J y [(c) and (f)], and azimuthal magneticfield B θ [(b), (e), and (h)] at 3.8, 4.0, and 4.5 ps. Loop current by the REs and return current rapidly enhance the strength of the electric andmagnetic fields. malized laser amplitude a reaches 1. Note that the position ofthe interface x c should vary with time. However, here the ini-tial position of the critical density, i.e., x c = x c ( ) = constant,was substituted considering that the temporal profiles of real-istic lasers increase from 0 to the peak intensity.The transition timing can be obtained by coupling Eq. (1)with the hole boring limit density, which is derived from themomentum transfer equation for the stationary state of the in- terface [45]: n s n c = ε a (cid:20) + R − ( − R ) β − h α − (cid:21) , (2)where ε is the polarization factor ( ε =1 and √ n h ) and bulk electrons ( n b ) as n e = n h + n b , and the momentum flux of the bulk electron compo-nent is negligible compared to that of the RE component (i.e., n e T e c β e ≈ n h T h c β h ). β h is the ratio of the drift velocity of REs( v h ) to the speed of light, c . α ≡ ir / r = r = i =
1, 2, or 3 repre-sents the dimension of the momentum distribution. When a1D relativistic Maxwell distribution α = β h =
1, and linear po-larization ε =
1, Eq. (2) reduces to n s / n c = Ra .By substituting a = .
79 and R = .
7, the electron densitythreshold n s for the experimental condition of case B in Fig. 6is obtained as n s = . n c . This density is almost identical tothe electron density threshold in which the plasma compres-sion terminates in the PIC simulation. Substituting Eq. (2) andthe initial electron density profile ( n e ( x ) = n c exp [( x − x c ) / l s ] )into Eq. (1) yields8 ε a (cid:20) + R − ( − R ) β − h α − (cid:21) = exp (cid:40) x c + l s ln (cid:20) + c l s (cid:113) R cos θ ( + R ) Zm e M i (cid:82) t s t (cid:113) γ ( t ) − γ ( t ) dt (cid:21) − x c l s (cid:41) , (3)where a is the normalized laser intensity. When the laser intensity is constant in time, the transition timing is then ob-tained as t s = F c (cid:20) l s c (cid:26)(cid:113) ε a [ + R − ( − R ) β − h α − ] − (cid:27)(cid:115) ( + R ) R cos θ α m p Z ∗ Zm e γγ − (cid:21) + t , (4)where M i = α ∗ m p Z ∗ represents the ion mass, m p is the pro-ton mass, Z ∗ is the ion charge number for the fully ionizedstate, and α ∗ = α ∗ = F c is added to take the laser pulse pro-file into account. For cases where the laser intensity is con-stant in time, F c =
1. The lines in Fig. 6(b) show the transitiontiming calculated from Eq. (4) with F c = Z ∗ = l s = µ m in the chargestate of Z =
40 interacts with a linearly polarized laser ( ε = α = a or reflectivity R increases, the transition timingis delayed. Low reflectivity reduces the effective laser inten-sity at the interface and reduces the hole boring limit density.When the normalized laser intensity is a =1.79 (intensity is I λ = . × W µ m / cm ), the transition timing is esti-mated to be t s = F c =
1. The color mapsin Figs. 6(c) and (d) show the dependence of the correctionfactor F c on the normalized laser intensity a and the halfwidth at half maximum (HWHM) of the Gaussian leadingedge for the low reflectivity case ( R =0.3) and high-reflectivitycase ( R =0.7). As the normalized laser intensity or HWHM in-creases, a larger correction factor is required. In the presentrange of 1 ≤ a ≤ ≤ HWHM ≤ F c = IV. SUMMARY
In summary, with the development of kilojoule-class high-power lasers, it has become possible to continuously irradi-ate relativistic laser pulses on matter over multi-ps. Althoughelectron acceleration using a conventional sub-ps laser pulsehas been explained theoretically as the interaction of a sin-gle electron with a laser field, it is necessary to consider thecollective effect of electrons when the pulse duration reachesthe multi-ps range. Energetic RE generation was experimen-tally clarified with an average energy far beyond the pondero-motive scaling using the prepulse-free LFEX laser. Duringthe multi-ps interaction, a quasi-static electric field is gener-ated by plasma expansion. In addition, a quasi-static magneticfield is gradually generated due to three different mechanismsof the ∇ n × ∇ I effect, the ∇ T × ∇ n effect, and a loop currentdriven by the E × B drift. The third mechanism of the currentloop rapidly amplifies the magnetic field strength by the posi-tive feedback between the electric and magnetic fields and thefield-driven drift current. Under the quasi-static electric field,REs are accelerated efficiently above ponderomotive scalingby the laser field because the dephasing rate of the REs fromthe laser field is reduced. Furthermore, when the quasi-staticmagnetic field becomes sufficiently strong to reflect REs backto the laser-plasma interaction region, the reflected REs gain FIG. 6. (a) Evolution of the initially exponential plasma profile during interaction with a laser pulse having a normalized laser intensity of a = .
79 ( I λ = . × W µ m / cm ). The motion of the interface calculated from Eq. (1) (red dotted line) is in good agreement with themotion obtained by the PIC simulation until t = (cid:96) s = µ m in the charge state of Z =
40. Correction factor of the pulsetemporal profile F c for cases of (c) low reflectivity R = . R = .
7. The normalized laser intensity a or half width athalf maximum (HWHM) of the Gaussian leading edge increases; therefore, a larger correction factor is required. further additional energy. The boosting timing of electron ac-celeration by the LIDA mechanism is related to the transitiontiming of the laser-plasma interaction state from the hole bor-ing phase to the blowout phase. The equation for transitiontiming can be derived from the equations for the motion ofa relativistic critical density interface and the equations forthe electron density where the hole boring terminates. In thisstudy, the equation for transition timing was extended to alaser pulse with an arbitrary intensity temporal profile. Thetheoretical result was then compared with the result of PICsimulation. The mechanism for the generation of SP-REs inthe multi-ps laser-plasma interaction reported here is usefulfor various applications. For instance, Yogo et al. reportedthat the maximum proton energy is enhanced more than twiceby extending the pulse duration to the multi-ps regime, dueto the electron temperature evolution beyond the ponderomo- tive energy in the over picoseconds interaction. The accelera-tion mechanism of REs investigated here is also important forfast-ignition inertial confinement fusion and laboratory astro-physics using high-intensity multi-ps laser pulses. ACKNOWLEDGMENTS
The authors thank the technical support staff of the In-stitute of Laser Engineering (ILE) at Osaka University andthose of the Plasma Simulator at the National Institute forFusion Science (NIFS) for assistance with laser operation,target fabrication, plasma diagnostics, and computer simu-lations. We also acknowledge A. Sagisaka, K. Ogura, A.S. Pirozhkov, M. Nishikino, and K. Kondo of the KansaiPhoton Science Institute, National Institutes for Quantumand Radiological Science and Technology for valuable dis-cussions on intensity contrast improvement using the PM.This work was supported by the Collaboration Research Pro-gram between the NIFS and ILE at Osaka University, theILE Collaboration Research Program, and by the JapaneseMinistry of Education, Culture, Sports, Science and Tech-nology (MEXT) through Grants-in-Aid for Scientific Re-search (Nos. 24684044, 24686103, 70724326, 15K17798,25630419, 16K13918, and 16H02245), the Bilateral Programfor Supporting International Joint Research of the Japan So-ciety for the Promotion of Science (JSPS), and Grants-in-Aid for Fellows from JSPS (Nos. 14J06592, 17J07212 and15J00850).
AUTHOR CONTRIBUTIONS
S. K. and S. F. are the principal investigators who proposedand organized the experiment. M. H. performed the PIC sim-ulations in the collaboration with T. J. and H. S.. N. I., S. K.,M. H. and Y. S. developed the theoretical model. Y. A. and A.M. designed and constructed the large size plasma mirror. H.M. and Y. O. carried out the theoretical analysis. H. N. andA. S. performed the radiation hydrodynamic simulations. K.M., S. S., S. L., K. F. F. L. and Y. A. measured electron energydistribution of high energy component with some help fromT. O.. S. T. measured electron energy distribution of low en-ergy component with some help from Z. Z. and A. Y.. S. T.and J. K. are in charge of LFEX laser facility development inILE. M. N., H. N., H. S. and H. A. supervised the project andprovided overall guidance. All authors participated in the dis-cussions and contributed to the preparation of the manuscript.
V. METHODSA. PM implementation
The LFEX parabola cannot be focused at an offset posi-tion far from the target chamber center due to its mechanicallimitations; therefore, a spherical PM was used that allowsthe original focal pattern to be relayed at an offset positionwith respect to the target chamber center. A spherical con-cave mirror (2-inch diameter and 202 mm curvature length)with a 1.053 µ m anti-reflection coating on both surfaces wasplaced after the focus point, as shown in Fig. 1(a). The LFEXwas focused at 3 mm above the target chamber center (off-set position). The image at the offset position is relayed tothe target chamber center with an image magnification of 1 bythe spherical mirror. According to ray-trace code calculations,the deterioration of the image due to spherical aberration and astigmatism of the spherical mirror is negligible compared tothe 70 µ m diameter of the LFEX spot. The laser energy flu-ence on the PM surface was optimized to be 90 J/cm to obtainacceptable reflectivity (50%) and spatial uniformity of the re-flected pulse [46]. B. Model for the hydrodynamics of the critical surfaceirradiated by multi-ps laser pulse
We have previously derived the transition timing from thehole boring to the blowout phase t s , under constant laser irra-diation (Eq. (8) in Ref. [45]). Here, the equation of transitiontiming is extended to a laser pulse with an arbitrary intensitytemporal profile and the theoretical result is compared withthat obtained by PIC simulation.At the interface, plasma is pushed into a high-density re-gion by the laser ponderomotive pressure caused by hole bor-ing. The hole boring velocity is conventionally derived withassumption of the initial and terminal states of the plasmacomponents (ions, bulk electrons) and REs [44, 47–50]. Itis assumed that the laser is reflected at the interface, which ismoving with velocity v p , and that electrons and ions are ini-tially immobile. The flow velocity of REs is assumed to be v h .In a frame moving with the interface at velocity v p , ions andbulk electrons drift at − v p toward the interface, where they arereflected. 1 − f e is the fraction of electrons reflected elasticallyto the velocity + v p , at the interface. The remaining fraction f e of electrons are heated by the laser and accelerated to rela-tivistic velocities p h / γ m e . All ions are assumed to be reflectedelastically to + v p , which is the same assumption as that madeby Vincenti et al. [49]. The equations of the momentumflux conservation and the energy flux conservation are givenby ( + R ) I ( t ) / c cos θ ≈ M i n i ( t ) v p + f e n e ( t ) v h ( t ) p h ( t ) and ( − R ) I ( t ) cos θ = f e n e ( t ) v h ( t ) p h ( t ) c , respectively. Here, R is the reflectivity of the incident laser on the plasma and θ isthe laser incident angle. M i and n i ( m e and n e ) are the ion(electron) mass and number density, respectively. Reflectionand density steepening occur at the relativistic critical electrondensity γ ( t ) n c with γ ( t ) = (cid:113) + ( + R ) a ( t ) /
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