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Asymmetric magnetic reconnection drivenby ultraintense femtosecond lasers
Cite as: Phys. Plasmas 26, 122110 (2019); doi: 10.1063/1.5130512Submitted: 4 October 2019 . Accepted: 27 November 2019 .Published Online: 12 December 2019
Yongli Ping,1,a) Jiayong Zhong,1,b) Xiaogang Wang,2 and Gang Zhao3
AFFILIATIONS1Department of Astronomy, Beijing Normal University, Beijing 100875, China2Department of Physics, Harbin Institute of Technology, Harbin 150001, China3Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences (CAS),Beijing 100012, China
a)[email protected])[email protected]
ABSTRACT
Three-dimensional asymmetric magnetic reconnection (AMR) driven by ultraintense femtosecond (fs) lasers is investigated by relativisticparticle-in-cell (PIC) simulation. The reconnection rate is found to be only one-third of that in the previous symmetric reconnection PICsimulations. Similar to the case of dayside reconnection at geomagnetopause, magnetic X- and velocity stagnation points are not colocated,with the X-point at the lower field side and the stagnation point at the higher field side. Moreover, the moving direction of the X-point asreconnection evolving with the laser irradiation is determined by dBH=dBL, and the moving of stagnation point is dominated byneHBL=neLBH , where dB and ne are the magnetic field disturbance and the electron density with the subscripts “H” for the higher field sideand “L” for the lower field side, respectively. Then, the hosing instability triggered by AMR and the merging of two parallel currents resultingin the tilt of the electron beam generated by the weak laser are also investigated.
Published under license by AIP Publishing. https://doi.org/10.1063/1.5130512
I. INTRODUCTION
Magnetic reconnection (MR) is a key mechanism of fastenergy transfer and release in space, astrophysical, and laboratoryplasmas, where the magnetic energy is converted to thermal andkinetic energies. Most previous studies on MR are in symmetricconfigurations where the physical variables are either symmetric(e.g., the plasma density and temperature) or antisymmetric (e.g.,the magnetic field) about the separatrix.1,2 However, in most cases,the symmetry is only an ideal approximation. In real space andastronomical events, reconnection is, in fact, mostly asymmetric,broadly observed at the dayside magnetopause,3–6 solar coronas,7
CME/flare,8,9 and tokamak plasmas.10 Thus, asymmetric magneticreconnection (AMR) has gradually attracted attention in recentyears. Particularly, AMR scaling laws were obtained by Cassak andShay11 in a Sweet-Parker type analysis on AMR for outflow speed,downstream density, and reconnection rate. They showed that themagnetic field X-point and the flow stagnation point (S-point)were not collocated, which led to an across X-point flow.Furthermore, using particle-in-cell (PIC) simulations for antipar-allel AMR, Malakit et al.12 presented a comprehensive test to verify
the scaling laws11 and to apply them for predicting the S-point ofthe flow pattern in the diffusion region.
In recent years, the dynamic evolution of the magnetic fielddriven by ultraintense lasers, particularly magnetic interaction13 andreconnection,14 has been intensively studied in laser-generated plas-mas due to the development of a high power laser technology. Thus,strongly driven fast reconnection processes can be realized in labora-tory using high power laser pulses to simulate impulsive astrophysicalevents.15–19 Due to having the features of small scale, short pulse, andstrong field, such laser pulses provide a unique opportunity to studyboth global structure and local physics of MR. Nanosecond laser driv-ing experiments have been applied to study self-generated fast recon-nection of magnetic field structures between two colliding laser-produced plasma bubbles, mostly driven by the Biermann batteryeffect. Using the three-dimensional (3D) fully kinetic simulations,Matteucci et al.20 studied both the self-consistent initial field genera-tion by Biermann battery effect and the reconnection process of such afield. Generation processes of ultrahigh magnetic fields of hundreds ofMega-Gausses (MGs) by relativistic intense lasers have been demon-strated numerically21–23 and theoretically24,25 in plasmas at a moderate
Phys. Plasmas 26, 122110 (2019); doi: 10.1063/1.5130512 26, 122110-1
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density. An MR process in self-generated magnetic fields produced bytwo parallel incident laser beams shooting into a near-critical plasmavolume was also recently investigated numerically in a 3D geometry.26
It was found that, instead of the electromagnetic turbulence effect, theelectrostatic turbulence contribution to the reconnecting electric fieldplayed an essential role in the process. Then, electron acceleration wascorrespondingly studied in this MR process. It was presented thatalthough the electron energy distribution was still in a power-law spec-trum, the electron pickup rings were nevertheless observed in themomentum phase space.27
On the laser driven AMR, Rosenberg et al.28 studied experimen-tally by two laser beams incident on a foil with a delay between them,leading to asymmetry in magnetic flux, temperature, density, and pres-sure. In the experiment, the annihilation rate of the magnetic flux wasinsensitive to the initial asymmetry. In addition, the reconnection ratewas hardly affected by the out-of-plane magnetic field from asymme-try in the 3D plasma environment either.
In this paper, we employ a 3D relativistic PIC code to simulateAMR processes with two femtosecond (fs) ultraintense lasers driving aplasma target. The asymmetry is introduced by an intensity differencebetween the two lasers. Then, after reconnection occurring, the out-of-plane magnetic field is also asymmetric. The effect of asymmetry on theMR rate is also studied, and the features of X- and S-points are investi-gated. Then, the AMR excited hosing instability, resulting in the tilt ofan electron beam generated by the weaker laser, is also considered. Thispaper is organized as follows. In Sec. II, the PIC simulation setup andthe theoretical scaling laws are presented. In Sec. III, the structure of theasymmetric magnetic reconnection and the reconnection rate are inves-tigated. In Sec. IV, the tilt of an electron beam generated by the weakerlaser is analyzed. The conclusion is then given in the last section.
II. SIMULATION SETUP AND THEORETICAL SCALINGLAWS
To investigate AMR driven by ultraintense lasers, we carried outsimulations by a fully relativistic particle-in-cell (PIC) code(KLAP).29,30 A configuration similar to the previous symmetric mag-netic reconnection (SMR) driven by ultraintense fs lasers26 is thenapplied. The simulation box is Lx0 � Ly0 � Lz0 ¼ 24� 24� 50lm3,with 20 grids per micrometer and 8 particles per grid averagely. Theplasma density is set as follows. For z ¼ 5–10 lm, the density is line-arly increased to the critical value of nc ¼ mex2
0=4pe2 ¼ 1:15
� 1021cm�3, whereme is the electron rest mass, x0 is the incident laserfrequency, and e is the element charge; for z ¼ 10–15 lm, the plasmadensity remains at the critical density nc, and for z ¼ 15– 35 lm, theplasma density is linearly decreased as nðzÞ ¼ ð1� ðz � z0Þ=L0Þnc,where L0 ¼ 20lm and z0 ¼ 15 lm.
In this numerical simulation, two circularly polarized lasers shootat the target and then propagate parallel (along the z-direction) intothe plasma with their axes separated by a distance of 8 lm, and themagnetic field is then generated by the interaction between lasers andplasma. Each laser has a spot diameter of 3 lm, and a wavelength ofk0 ¼ 1lm, corresponding to a period of T0 ¼ k0=c ¼ 3:33 fs. Theincident laser pulse has a rising front of 2 laser-cycles followed by aflat top in the z-direction and a Gaussian form in the x and they-directions. In order to construct an AMR configuration, however,the two lasers are set with different peak intensities. The higher peak is5� 1020W=cm2, corresponding to a normalized laser vector potential
of a0 ¼ 13.5. The lower peak is set differently with three various runs,as 2� 1020 W=cm2 in case I, 3� 1020 W=cm2 in case II, and4� 1020 W=cm2 in case III. The self-generated magnetic field due tothe laser-plasma interaction is proportional to the laser peak intensityto create the asymmetry. Therefore, the ratio of the antiparallel mag-netic field on the two sides should be 0.4, 0.6, and 0.8 in the three sim-ulation runs corresponding to the intensity ratio. The typical AMR isthe dayside reconnection at the magnetopause, where the magneto-sheath (with a magnetic field of 20–30nT and a plasma density of20–30 cm�3) reconnects with the magnetosphere (with a magneticfield of 50–60 nT and a plasma density of 0.3–0.5 cm�3).31,32 The mag-netic field ratio of the magnetosheath to the magnetosphere is thenabout 0.33–0.6, falling in our simulation range.
The theoretical scaling laws and the positions of the X- and theS-points predicated in Ref. 11 for AMR are as follows:
E � BHBL
BH þ BL
voutc
2dLe; (1)
dXHdXL� BH
BL; (2)
dSHdSL� qHBL
qLBH; (3)
where v2out � ½BHBL=4pq�; q ¼ mini þmene, and mi(me) and ni(ne)are the mass and the density of ion (electron), respectively. Also, BH isthe higher magnetic field driven by the stronger laser and BL is thelower magnetic field driven by the weaker laser, d is the half-width,and Le is the half-length of the dissipation region. dX is the distancefrom the dissipation region edge to the X-point, and dS is the distancefrom the dissipation region edge to the S-point, with the other sub-scripts “H” for the higher field side and “L” for the lower field side.
III. STRUCTURE OF ASYMMETRIC MAGNETICRECONNECTION AND RECONNECTION RATE
The detailed structure of AMR at t¼ 50T0 in case I is presented inFig. 1. Figure 1(a) shows the asymmetry of two ring-like distributions
FIG. 1. (a) The in-plane magnetic field Bin, (b) the out-of-plane magnetic fields Bz,(c) out-flow current jy, and (d) out-of-plane current jz in the x-y plane withz ¼ 28lm at t ¼ 50T0. Here, the magnetic fields are normalized by the initial laserfield 1:45� 105T .
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Phys. Plasmas 26, 122110 (2019); doi: 10.1063/1.5130512 26, 122110-2
Published under license by AIP Publishing
for the in-plane magnetic field due to different intensities of drivinglasers and a magnetic hole around the X-point in the reconnectionregion between two magnetic rings. The X-point is located in the lowerfield side. The out-of-plane magnetic field with an asymmetric quadru-polar structure is presented in Fig. 1(b). As be evolving, the quadrupo-lar field is distorted in the reconnection region and gets more obviousand widespread in the lower field side. Figures 1(c) and 1(d) show thein-plane (y component) and the out-of-plane currents in the x-y crosssection at z ¼ 28 lm. The direction of the in-plane (y component) cur-rent is opposite to the outflow velocity because it is mainly the electroncarried.27
Figure 2 illustrates the reconnection electric field at t ¼ 50T0 incase I with I2 ¼ 2� 1020 W=cm2, case II with I2 ¼ 3� 1020 W=cm2,and case III with I2 ¼ 4� 1020 W=cm2. We can find that the electricfield Ez is stronger at the X-point region in Figs. 2(a)–2(c). Then, thenormalized reconnection rate Ez=VAeB0 is presented at 50T0 for case I(the red line), case II (the green line), and case III (the blue line) inFig. 2(d), with the magnetic field B0 and the electron Alfv�en speed VAe
being at the maximum magnetic field B0 generated by the primarylaser at the same time. Here, VAe � c=
ffiffi
�p
and � ¼ 1þ 2ðxpe=XceÞ2,where Xce ¼ eB=mec.
33 The maximum reconnection rate Ez=Ve0B0 isfound 0.2422, 0.3609, and 0.5059 for case I, case II, and case III,respectively, corresponding to the peaks of the red, green, and bluecurves in Fig. 2(d). It can be seen that the reconnection rate increasesas the asymmetry of the configuration weakens, which means a stron-ger asymmetry leading to a lower reconnection rate. It is a typical fea-ture of AMR observed by satellites during subsolar magnetopausecrossing where reconnection is in general asymmetric.34
It has to be pointed out that, in our simulation, the rate of AMRdriven by lasers is Ez=VAeB0 ¼ 0:2422 for the asymmetric field ratioof 0.4, much higher than the previous AMR results of 0.01 with thesame magnetic field ratio.11,35 In other words, the AMR rate in ourcase I is 1/3 of that for the SMR rate26 while in the previous 2D PICsimulation, the AMR rate is only 1/10 of the SMR rate.36 The satellite
observation results for magnetosphere plasmas, on the other hand, are0:02 for magnetopause AMR,37,38 and 0:1 for magnetotail SMR,39
giving a 1/5 ratio of AMR to SMR rates.One of the reasons for the difference is that the MR driven by
ultraintense lasers is a strong driven process with a time-dependentinward driving flow. It has been shown that in forced reconnection(i.e., driven reconnection), the reconnection electric field is propor-tional to n3=20 in the resistive MHD case, where n0 is the boundarydriving strength40,41 and proportional to the boundary driving in thesteady collisionless reconnection.42
On the other hand, the prediction of Eq. (1) for reconnection elec-tric fields Ez ¼ 3:12; 5:766, and 7:236� 1011V=m for case I, case II,and case III, respectively, a much lower than the above simulatedEz ¼ 2:1253; 3:115, and 4:44� 1012 V=m, correspondingly, althoughthe variation tendency of the prediction is consistent with the simulatedresult. Then, a question is raised: if the difference comes from the self-generated electric field due to the laser-plasma interactions. Thus, wecarry out the other 2 simulation runs for a single laser driving target,with the intensities of I ¼ 5� 1020 W=cm2 and I ¼ 2� 1020 W=cm2,respectively. At the same position where the reconnection electric fieldreaches its maximum in case I, the total electric fields generated in bothnew runs are 0:998� 1011 V=m, an order of magnitude lower thanthose in the two laser driven runs. Clearly, the reconnection electricfield Ez in the two laser driven reconnections includes only a small por-tion of the self-generated electric field by the laser-plasma interactions.Therefore, Eq. (1) introduced in Ref. 11 is not proper for a strongdriven reconnection where the significant magnetic flux compressionduring the reconnection process should be taken into consideration.
IV. X-POINT AND STAGNATION POINT
The X-point is defined as the place where the antiparallel magneticfield lines merge and annihilate to a null in magnetic reconnection. Inour simulation configuration, By is the antiparallel magnetic field.Therefore, the X-point can be obtained from By shown in Figs. 3(a)–3(c)for case I, case II, and case III. Clearly, the X-point is located at the lowerfield side and approaches the symmetric plane (x ¼ 12 lm) in themiddle of the two lasers as the asymmetry weakening. The antiparallel
FIG. 2. The normalized reconnection (out-of-plane) electric field Ez in the x-y planeof z ¼ 28 lm at t ¼ 50T0 for (a) case I of I2 ¼ 2� 1020 W=cm2, (b) case II ofI2 ¼ 3� 1020 W=cm2, and (c) case III of I2 ¼ 4� 1020 W=cm2; here, the electricfield is normalized by the initial laser field of 4:34� 1013 V=m. (d) Along the x-axisfor case I (red line), case II (green line), and case III (blue line), where the electricfield is the electric field normalized by Ez=VAeB0.
FIG. 3. At t ¼ 50T0 ðredÞ; 55T0 ðgreenÞ, and 60T0 ðblue lineÞ along the x-axis withz ¼ 28 lm; y ¼ 12 lm, the antiparallel magnetic fields By and inflow current jx for(a) and (d) case I, (b) and (e) case II, and (c) and (f) case III, respectively.
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magnetic fields are presented at moments of t ¼ 50T0 (red line),55T0 (green line), and 60T0 (blue line). In Figs. 3(a) and 3(b), the X-point moves toward the higher field side direction as time increases.In Fig. 3(c), the movement of X-point is toward the higher fieldfrom 50T0 to 55T0. It reaches the symmetric plane of x ¼ 12lm att ¼ 55T0 and then bounces back from 55T0 to 55T0 due to the weakasymmetry.
The S-point is defined by the flow reversal spot in reconnection.In SMR, it overlaps with the X-point due to symmetry, while in AMR,it separates with the latter. However, during the magnetic reconnec-tion process driven by ultraintense fs lasers, the Hall effect is muchmore significant, with the electrons accelerated much faster than theions.27 Therefore, the quasineutral plasma flow is hardly defined in theconcerned region. We then instead define the S-point by the current jxreversal, along the inflow direction. The inflow currents jx in case I,case II, and case III in Figs. 3(d)–3(f) indicate that the S-point islocated at the higher field side and approaches the middle plane(x ¼ 12lm) as the asymmetry is getting weaker. They are then givenat t ¼ 50T0 (red), 55T0 (green), and 60T0 (blue), by different colorcurves. For the stronger asymmetry cases of I and II, as shown inFigs. 3(d) and 3(e), the S-point moves away from the middle plane astime increases, though the entire tendency is toward the middle planeas the asymmetry becomes weaker. For the weakest case III, as shownin Fig. 3(f), the S-point indeed moves toward the middle plane from50T0 to 55T0, but also bounces back similar to the X-point case.
In order to verify the scaling predictions of Eqs. (2) and (3),certain parameters from simulation results are listed in Table I. Whenthe reconnection rate is strongest at t ¼ 50T0, we get BH=BL > 1, i.e.,dXH > dXL, which implies that the X-point is located at the lower fieldside. At the same time, nHBL=nLBH < 1, i.e., dSH < dSL, which impliesthat the S-point is located at the higher field side. The spatial separa-tion between the S-point and the X-point implies there being a plasmabulk flow across the X-point region. Table I shows that the drift of theS-point is determined by neHBL=neLBH not niHBL=niLBH , due to theelectrons being accelerated by MR more significantly than the ionsin our simulations. However, we have to point out that, as shown inFigs. 3(a)–3(c), the moving position of the X-point is not dominantlydetermined by BH=BL at different moments. It is clearly due to the
ultraintense laser driven that induces turbulent, not steady, motions ofthe X- and the S-points. But from Table I, we can find that the X-pointwill drift toward the strong field when dBH=dBL > 1; if not, then thedrift direction should be a reversal.
V. TILT OF ELECTRON BEAM ACCELERATED BY WEAKLASER
When the relativistic laser is transmitted through the near-criticalplasma, lots of instability are triggered, such as the hosing instabilityexcited by the transverse asymmetric laser intensities or perturbationsof the plasma density.43–49 Then, a tilt of the electron beam acceleratedby an ultraintense laser pulse is caused to affect the quality of the elec-tron beams and related radiation sources in the process.
Figure 4 shows the electron density ne, out-flow current densityjy, and antiparallel magnetic field By in the x-z plane with y ¼ 12 lmat t ¼ 50T0. In Figs. 4(a-4)–4(c-4), there is a plasma channel formedwhen a laser pulse of 2� 1020 W=cm2 propagates into a near-criticaldensity plasma nonuniformly along the incident direction, and anelectron beam is shown generated around the center of the channel.The electron beam, the current, and the plasma channel are approxi-mately straight in the single laser case with a magnetic field symmetricaround the laser, as shown in Fig. 4(c-4).
At the same time, in Figs. 4(a-1)–4(a-3), we can find that the elec-tron beam, thus the current, around the center of the weaker laser pulsebecomes tilted and slopes to the stronger laser side in the two lasercases. It can also be found in Figs. 4(a-1)–4(a-3) and 4(b-1)–4(b-3) thatthe more asymmetric, the worse tilt of the electron beam and the cur-rent is. There are two reasons why the electron beam is titled. One isthat in AMR, the magnetic field generated by the weaker laser isreconnected faster than that by the stronger laser, leading to a struc-ture drifting toward the higher field side with the X-point. Therefore,the hosing instability is then triggered. The other is the attractionforce F, estimated by Eq. (4), between two parallel currents generatedby the lasers. If the two currents are IH generated by the strongerlaser and IL generated by the weaker laser, then we can write in CGSunits as
F � 2c2IHILd; (4)
where c is the light speed and d is the distance between the two cur-rents. According to Newton’s second law, the acceleration of the elec-tron beam under the attraction is
aH ¼FMH
; (5)
aL ¼FML
: (6)
Here, MH and ML are the mass of the two beams generated by thestronger and weaker lasers, respectively. In Figs. 4(a-1)–4(a-3), onecan find that the density of the electron beam accelerated by the stron-ger laser is higher than that accelerated by the weak laser, leading tothe mass of the electron beam around the stronger laser being largerthan that around the weaker laser, i.e., MH > ML. Based on Eqs. (5)and (6), we have aH < aL. Therefore, the weak electron beam movesmore obviously and slopes to the strong side.
TABLE I. Theoretical values of BH=BL; dBH=dBL; neHBL=neLBH , and niHBL=niLBLat times 50T0; 55T0; 60T0 in case I, case II, and case III.
tðT0ÞBH
BL
dBH
dBL
neHBL
neLBH
niHBL
niLBH
Case I 50 2.8118 … 0.3205 0.347955 4.4117 1.7286 0.2250 0.220160 22.28 1.6613 0.0687 0.0507
Case II 50 2.4965 … 0.4675 0.430655 2.9731 2.1498 0.3882 0.372060 7.8824 1.2713 0.1966 0.1995
Case III 50 1.3542 … 0.8652 0.750255 1.2060 1.4568 0.9305 0.750360 3.6672 0.5901 0.313 0.4018
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Phys. Plasmas 26, 122110 (2019); doi: 10.1063/1.5130512 26, 122110-4
Published under license by AIP Publishing
VI. CONCLUSION
In conclusion, AMR is investigated in ultraintense fs lasers drivingnear-critical plasmas by fully kinetic PIC simulations. The asymmetricenvironment of the plasma and the magnetic field created by laserdriving with the intensity ratio of 0:4–0:8 is similar to that of themagnetopause plasma with the ratio of the magnetosheath to themagnetosphere.31,32 The out-of-plane quadrupolar magnetic field isdistorted asymmetrically, also similar to the nonrelativistic AMR at thegeomagnetopause. The AMR rate in our cases is found to be slowerthan that in the previous AMR simulations, also similar to the nonrela-tivistic cases. However, instead of 1/10 of the SMR rate in nonrelativisticsimulations, and 1/5 in magnetospheric observations, our AMR rate is1/3 of the previous SMR runs due to the strong driving. The X-point isfound at the lower magnetic field side and the S-point at the highermagnetic field side, again similar to the nonrelativistic cases. Moreover,the X-point moves toward the higher magnetic field as reconnectionevolving with laser irradiation, a direction determined by dBH=dBL.And the moving of S-point is dominated by neHBL=neLBH . Then, thehosing instability excited by AMR and the attraction of two parallel cur-rents lead to the tilt of the electron beam generated by the weaker laser.
The laser driving MR experiments extensively carried out inmany labs were mostly symmetric without significant asymmetricfeatures, such as the distorted out-of-plane magnetic field, thedecoupled X-point and S-point, and so on. Even in the AMRexperiment,28 the magnetic field of the large and the small bubbleshad almost the same strength of B ¼ 506 20T , with the density ofthe large bubble qlb ¼ 2:5–4:9� 10�4gcm�3 and that of the small
bubble qsb ¼ 0:6–1:8� 10�4gcm�3. In the SMR experiment of thesame paper, the density of both bubbles was qs ¼ 1:6–2:8� 10�4gcm�3. According to Eqs. (1)–(3), we can find that theAMR rate should be almost the same as the SMR rate in theirexperiment, with dXH=dXL ¼ 1 but dSH=dSL ¼ 4:1667. Therefore,in that experiment, only the S-point is deviated by the asymmetricdensity. Thus, further studies for other asymmetric characteristicsshould be carried out.
In the nanosecond laser driving solid target experiments, theplasma beta b > 1, which is a bit high to significantly affect the recon-nection process, while MR is more effective in b < 1 plasma circum-stance where the magnetic pressure is stronger than the thermalpressure. Therefore, future AMR experiments would be carried out inthe regime of lower plasma beta with b < 1, which can be reached byusing a double-turn Helmholtz capacitor-coil target.50,51 Ultraintensefs/ps lasers driving SMR has been performed in laboratory;52 therefore,the corresponding AMR should be able to be utilized in similarconfigurations.
ACKNOWLEDGMENTS
This work was supported by Science Challenge Project No.TZ2016005, the National Basic Research Program of China (No.2013CBA01500), the National Natural Science Foundation of ChinaNos. 11622323 and U1930108, and the Fundamental ResearchFunds for the Central Universities. Computer runs were performedon the Laohu high performance computer cluster of the NationalAstronomical Observatories, Chinese Academy of Sciences (NAOC).
FIG. 4. (a) The electron density normalized by nc, (b) the out-flow current density jy, and (c) the reconnection magnetic field By in the x-z plane with y ¼ 12 lm at t ¼ 50T0,with (a-1), (b-1), and (c-1) for case I, (a-2), (b-2), and (c-2) for case II, (a-3), (b-3), and (c-3) for case III and (a-4), (b-4), and (c-4) for the single laser run withI ¼ 2� 1020 W=cm2.
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