Spin and polarization effects on the nonlinear Breit-Wheeler pair production in laser-plasma interaction
Huai-Hang Song, Wei-Min Wang, Yan-Fei Li, Bing-Jun Li, Yu-Tong Li, Zheng-Ming Sheng, Li-Ming Chen, Jie Zhang
SSpin and polarization effects on the nonlinearBreit-Wheeler pair production in laser-plasmainteraction
Huai-Hang Song , , Wei-Min Wang , , , Yan-Fei Li , Bing-JunLi , Yu-Tong Li , , , , Zheng-Ming Sheng , , , , Li-MingChen , , , and Jie Zhang , , Beijing National Laboratory for Condensed Matter Physics, Institute of Physics,Chinese Academy of Sciences, Beijing 100190, China Department of Physics and Beijing Key Laboratory of Opto-electronic FunctionalMaterials and Micro-nano Devices, Renmin University of China, Beijing 100872,China Department of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an710049, China Key Laboratory for Laser Plasmas (MoE) and School of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai 200240, China SUPA, Department of Physics, University of Strathclyde, Glasgow G4 0NG, UnitedKingdom School of Physical Sciences, University of Chinese Academy of Sciences, Beijing100049, China CAS Center for Excellence in Ultra-intense Laser Science, Shanghai 201800, China Collaborative Innovation Center of IFSA, Shanghai Jiao Tong University, Shanghai200240, China Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, ChinaE-mail: [email protected] and [email protected]
Abstract.
The spin effect of electrons/positrons ( e − / e + ) and polarization effectof γ photons are investigated in the interaction of two counter-propagating linearlypolarized 10-PW-class laser pulses with a thin foil target. The processes of nonlinearCompton scattering and nonlinear Breit-Wheeler pair production based on spin-and polarization-resolved probabilities are implemented into the particle-in-cell (PIC)algorithm by Monte Carlo methods. It is found from PIC simulations that the averagedegree of linear polarization of emitted γ photons can exceed 50%. This polarizationeffect leads to reduced positron yield by about 10%. At some medium positronenergies, the reduction can reach 20%. Furthermore, we also observe that the local spinpolarization of e − / e + leads to a slight decrease of the positron yield about 2% and someanomalous phenomena about the positron spectrum and photon polarization at thehigh-energy range, due to spin-dependent photon emissions. Our results indicate thatspin and polarization effects should be considered in calculating the pair productionand laser-plasma interaction with the laser power of 10-PW class. a r X i v : . [ phy s i c s . p l a s m - ph ] F e b
1. Introduction
Over the past decades, the intensity of lasers has increased rapidly [1, 2] with the lasertechnical progress based on chirped pulse amplification [3]. Several multi-petawatt(PW) [4] and 10-PW-class [5, 6] femtosecond laser systems have been built, which areexpected to achieve an unprecedented peak power density up to the order of 10 − W/cm with tightly focusing. Such ultraintense lasers enable laser plasma interactionsto enter the quantum electrodynamics (QED) regime [7–9]. Electrons experiencingthe ultra-intense transverse field can stochastically radiate γ photons by nonlinearCompton scattering and lose a considerable amount of energy if the quantum parameter χ e = ( e ¯ h/m e c ) | F µν p ν | ∼ F µν is the field tensor, p ν is the electron four-momentum, and the constants ¯ h , m e , e and c are the reduced Planck constant, theelectron mass and charge, and the speed of light, respectively. As another cross channelof the same reaction, γ photons traveling through the ultraintense field possibly furtherdecay into electron-positron ( e − e + ) pairs by nonlinear Breit-Wheeler process [13] withanother characteristic parameter χ γ = ( e ¯ h /m e c ) | F µν k ν | , where ¯ hk ν is the photonfour-momentum. This sort of pair production by light-by-light scattering was firstdemonstrated by the famous SLAC E-144 experiment [14] in 1990s, where only aboutone hundred positrons was detected due to the limitation of the laser intensity at thattime. The laser pulse and electron beam collisions utilizing today’s high-intensity laserfacilities in all-optical setups are also studied recently [15, 16].With upcoming 10-PW-class lasers [6], abundant e − e + pairs can even be producedin laser-plasma interactions without the need to pre-accelerate electrons to GeV energies.When such an ultraintense laser irradiates plasmas, electrons would be acceleratedto ultrarelativistic energies and deflected by laser or strong self-generated fields inplasmas to gain a Lorentz boosted field strength in the electron’s moving frame toachieve χ e ∼
1. Many theoretical proposals for producing dense e − e + pairs or evenavalanche-like cascades have been put forward, such as through the laser collisionconfiguration seeded by electrons/positrons ( e − / e + ) [17–19] or plasmas [20], and directlylaser-solid interactions [21–24]. Generating copious positrons or dense e − e + plasmas inthe laboratory is of great importance in astrophysics [25–27], nuclear physics [28], andmaterials science [29].Moreover, the e − e + spin effect and γ -photon polarization effect have arousedinterest in the strong-field QED regime [30–32]. An ultrarelativistic electron beamis found to be transversely spin-polarized by a single-shot collision of an ellipticallypolarized laser pulse [33] or a two-color laser pulse [34, 35] due to hard γ photonemissions, analogous to the Sokolov-Ternov effect [36, 37] in the magnetic field. Similarspin polarization processes for newly created e − e + pairs are also investigated [38–40].The general view is that constructing an asymmetric laser field is the key to realize spin-polarized electrons or positrons. The emitted γ photons could be polarized via nonlinearCompton scattering [41–43], whose polarization strongly depends on the initial spin ofelectrons [42]. It is found that only linearly polarized γ photons can be generated byunpolarized or transversely polarized electrons [30, 42].A more sophisticated description for e − e + pair production by taking into accountthe e − e + spin and γ -photon polarization has been discussed [30, 44, 45], based on single-particle model analyses or simulations. The photon polarization is shown to significantlyreduce the pair yield by a factor of over 10% in the collision of ultraintense laser pulseand electron beam [44]. In the rotating electric fields, the growth rate of the e − e + cascade is also found to be suppressed [45]. However, it still demands to be studied thatin the laser-plasma interaction to what extent the e − e + spin and γ -photon polarizationimpact on the positron yield.In this paper, we study the e − e + pair production in the laser-plasma interaction,by taking into account the spin of e − /e + and the polarization of γ photons. The spin-and polarization-resolved probabilities of nonlinear Compton scattering and nonlinearBreit-Wheeler pair production are both implemented into the widely employed QEDparticle-in-cell (PIC) algorithm, which can self-consistently capture the collective plasmadynamics and QED related processes. Here, we focus on the pair production inthe interaction of two counter-propagating linearly polarized laser pulses of the samefrequency and intensity with a thin foil target. This is a particularly advantageousconfiguration under the current laser intensity for triggering the QED pair production,due to the formation of the linearly-polarized electromagnetic standing wave (EMSW)[19, 46]. Our simulation results show that the positron yield is reduced by about 10%with the spin and polarization effects included. This significant difference is primarilycaused by the polarized intermediate γ photons with an average linear-polarizationdegree of more than 50%. In addition, we also observe a decrease of positron number byabout 2% and some anomalous phenomena for high-energy particles due to local spinpolarization of e − /e + . This work indicates that the previously widely adopted spin- andpolarization-averaged probabilities implemented in QED-PIC codes cannot accuratelycalculate the positron yield in the laser-plasma interaction for 10-PW-class lasers, andspin and polarization effects should be considered.
2. Theoretical model
In order to determine the spin of electron after the photon emission and also thepolarization of the emitted γ photon, the spin- and polarization-resolved photon emissionprobability is employed, which is derived in the Baier-Katkov QED operator method[12, 42], d W rad dudt = αm c √ π ¯ hε e (cid:40) u − u + 21 − u K ( y ) − Int K ( y ) − uK ( y )( S i · e )+ (cid:104) K ( y ) − Int K ( y ) (cid:105) ( S i · S f ) − u − u K ( y )( S f · e )+ u − u (cid:104) K ( y ) − Int K ( y ) (cid:105) ( S i · e v )( S f · e v )+ u − u K ( y )( S i · e ) ξ + (cid:34) u − u − u K ( y ) − u Int K ( y ) (cid:35) ( S i · e v ) ξ + (cid:20) K ( y ) − u − u K ( y )( S i · e ) (cid:21) ξ (cid:27) , (1)where K ν ( y ) is the second-kind modified Bessel function of the order of ν , Int K ( y ) ≡ (cid:82) ∞ y K / ( x ) dx , y = 2 u/ [3(1 − u ) χ e ], u = ε γ /ε e , ε e the electron energy before the photonemission, ε γ the emitted photon energy, and α ≈ /
137 the fine structure constant. e v is the unit vector along the electron velocity, e is the unit vector along the electrontransverse acceleration, and e = e v × e . S i and S f are the spin vectors of an electronbefore and after photon emission, respectively, with | S i,f | = 1. The photon polarizationis represented by Stokes parameters ξ = ( ξ , ξ , ξ ), defined with respect to the basisvector ( e , e , e v ). The case ξ = ξ = 0, ξ = 1( −
1) means the photon is linearlypolarized along e ( e ), and the case ξ = ξ = 0, ξ = ± S i and summing up over S f and ξ , the widely employedspin- and polarization-averaged photon emission probability is obtained [18, 47]. Notethat the above description can also be applied to the positron.For convenience, we define high-energy photons with u/ (1 − u ) ≥ u/ (1 − u ) (cid:28)
1. We ignore the correlation terms involving both S f and ξ in equation (1), because the final electron spin and photon polarization are calculatedseparately in our Monte-Carlo method [48].Utilizing the similar method, the probability of the pair production with the photonpolarization included can be written as [12, 40, 44], d W pairs dε + dt = αm c √ π ¯ hε γ (cid:40) ε + ε − ε + ε − K ( y ) + Int K ( y ) − ξ (cid:48) K ( y ) (cid:41) , (2)where y = 2 ε γ / (3 χ γ ε + ε − ), ε − and ε + are the energies of the produced electron andpositron, respectively. The last term containing ξ (cid:48) in equation (2) accounts for thephoton polarization effect on the pair production. It is apparent that if ξ (cid:48) is a positivevalue, the pair production probability d W pairs / ( dε + dt ) is reduced and consequentlythe positron yield decreases. The decrease can be up to 30% for medium-energypositrons [44]. The Stokes parameters need to be transformed from the photon emissionframe ( e , e , e v ) to the pair production frame ( e (cid:48) , e (cid:48) , e v ) to obtain a new set of Stokesparameters ( ξ (cid:48) , ξ (cid:48) , ξ (cid:48) ) that required in equation (2), through the matrix rotation [49] ξ (cid:48) = ξ cos(2 θ ) − ξ sin(2 θ ) ,ξ (cid:48) = ξ , (3) ξ (cid:48) = ξ sin(2 θ ) + ξ cos(2 θ ) , where e (cid:48) is the unit vector along E + e v × B − e v · ( e v · E ), e (cid:48) = e (cid:48) × e v , and θ is theangle between e and e (cid:48) .The semiclassical formulas of photon emission probability in equation (1) andpair production probability in equation (2) are derived based on the local constantfield approximation [50, 51], which is justified at an ultraintense laser intensity of a = | e | E L /mcω (cid:29)
1, where ω is the laser frequency. The stochastic photon emissionby an electron or a positron and pair production by a γ photon are calculated using thestandard QED Monte-Carlo algorithm [47, 52–55] but with the spin- and polarization-resolved probabilities. The e − e + dynamics in the external electromagnetic field isdescribed by classical Newton-Lorentz equations, and their spin dynamics are calculatedaccording to the Thomas-Bargmann-Michel-Telegdi equation [57, 58]. Detailed Monte-Carlo methods for numerical modeling of spin and polarization we employ can be foundin Refs. [33, 40, 42, 48]. In our simulations, spin vectors of newly created pairs are alsoincluded, which however has a relatively weak influence on the pair production in thelinearly-polarized EMSW.
3. Simulation results and analysis
We implement the spin- and polarization-resolved probabilities in equations (1) and(2) into the two-dimensional QED-PIC code by the Monte-Carlo method as describedin section 2, to self-consistently study the e − e + spin and γ photon polarization effectson the pair production in the laser-plasma interaction. The standard QED moduleshave been benchmarked following Refs. [24, 55], and benchmarks about the spin andpolarization modules are presented in Appendix A. In the following simulations, wecan artificially switch on or off these two modules for better identifying their impactsby comparison. In our simulation setups, two counter-propagating laser pulses withthe same profile of a L = a sin ( πt/τ ) × exp( − r /r ) within 0 < t ≤ τ are normallyincident from the left and right boundaries, respectively, and they are both linearlypolarized along the y axis. We take the laser normalized peak strength a = 800 (peakintensity I ≈ . × W/cm ), spot size r = 2 λ , and pulse duration τ = 10 T ,where λ = 1 µ m is the laser wavelength and T = 2 π/ω ≈ .
33 fs is the laser period. A1 µ m-thickness fully ionized foil target, composed of electrons and carbon ions, is initiallyplaced in the laser overlapping center of 3 . λ < x < . λ with an electron density of n e = 50 n c , where n c = m e ω / πe is the critical density. The computational domainhas a size of 8 λ × λ in x × y directions with 384 ×
720 cells. Each cell contains 100macro electrons and 16 macro carbon ions. Absorbing boundary conditions are used forboth particles and fields in any direction.In the first simulation case, labeled as case (i) , the spin and polarization effects areboth incorporated. As reference cases, we have also performed another three simulationsunder the same physical parameters as those in case (i) except that: in case (ii) , thespin and polarization effects are not included, which is the widely adopted method in thecurrent QED-PIC codes [47, 52, 54]; in case (iii) , only spin effect is included, where thepolarization-dependent terms are summed up over in equation (1) and averaged over inequation (2), while retaining spin-dependent terms in these two probability equations; in case (iv) , we switch off the photon annihilation and pair production processes but stillinclude the spin and polarization effects, to check the original polarization characteristics x/λ y / λ (a) − E y /E c n + /n c . . . x/λ . . . . . . t / T (b) - - - - - - - - - - - - P rad (a . u . ) 3 . . . x/λ . . . . . . t / T (c) - - - - - - - - - - - - − S z Figure 1.
Simulation results of case (i) . (a) Spatial distribution of the density ofproduced positrons n + , together with the electric field E y of two counter-propagatinglaser pulses at t = 12 T , where E c is mcω / | e | . The dotted lines outline the initialboundaries of the foil target. Space-time evolution of (b) the radiation power P rad of photons with energies above 100 MeV and (c) the average electron spin degree S z along y = 7 . λ . The laser magnetic field B z normalized by mcω / | e | is also shown bycontour lines to outline the magnetic region of the EMSW both in (b) and (c). of emitted γ photons. For case (i) , figure 1(a) illustrates the spatial distribution of the positron density n + at the end of the laser-foil interaction at t = 12 T when two laser pulses have passedthrough each other. We can see that dense and spatially modulated [59] positrons areproduced, with a maximum density of n + = 60 n c , already exceeding the initial electrondensity of the foil target. A transient linearly EMSW responsible for the abundantpositron production is constructed by two counter-propagating linearly polarized laserpulses in the time interval 7 T < t < T , covering the entire foil plasma zone. Thespace-time evolution of magnetic field component B z of EMSW along y = 7 . λ is shownby contour lines both in figures 1(b) and 1(c). The formed EMSW is divided into electricregion (maximize at magnetic nodes x = mλ / t = nT / T /
4) and magneticregion (maximize at magnetic antinodes x = mλ / λ / t = nT / m and n are integers. These two distinct regions are shifted by λ / T / π/ y direction in the electric region, while emitting photonsand simultaneously losing energies primarily in the magnetic region [22]. The photonradiation power P rad shown in figure 1(b) verifies that more high-energy photons areemitted in the early stage after entering the magnetic region. The previous study [46]has presented that this type of field configuration is favorable for improving quantumparameter χ e with 10-PW-class laser facilities, and consequently are the photon emission t/T . . . . . . N + ( a . u . ) × (a) (i)(ii)(iii) . . . . . . ε + (GeV)02468 d N + / d ε + ( a . u . ) × (b) 1050-5-10-15-20 δ N i − ii + ( % ) . . . . . . ε γ (GeV)0-2-4-6-8 δ N iii − ii γ ( % ) (c) Figure 2. (a) The time evolution of total positron number N + in three simulationcases: case (i) with spin and polarization effects (blue solid line), case (ii) withoutspin and polarization effects (red dashed line), and case (iii) with only spin effect(green dotted line). (b) Positron energy spectrum dN + /dε + under the three casesat the end of the laser interaction. The relative deviation δN i − ii+ = ( dN i+ /dε + − dN ii+ dε + ) /dN ii+ dε + between case (i) and case (ii) is also presented by the cyan line.(c) The relative deviation δN iii − ii γ = ( dN iii γ /dε γ − dN ii γ /dε γ ) /dN ii γ dε γ of photon numberversus photon energy ε γ between case (iii) and case (ii) . and pair production. The quantum parameter χ e can reach a maximum of 3 in our case.At the end of the simulation, about 30% laser energies are absorbed, among which, 24%are transformed into photons, 5 .
5% into electrons and positrons, and less than 0 .
5% intoions.The time evolution of the total positron number N + of cases (i)-(iii) is illustratedin figure 2(a). During the existence of EMSW in the time interval of 7 T < t < T , N + increases dramatically. The stair-step-like growth with a period of 0 . T isattributed to the fact that electrons strongly emit photons mostly in magnetic regionsas already shown in figure 1(b). The most important feature is that when the spinand polarization effects are fully considered in case (i) , the total number of positronsis reduced by 12% compared with the case (ii) that excluding the two effects, i.e.∆ N i − ii+ = ( N i+ − N ii+ ) /N ii+ ≈ − δ i − ii+ also depends on the positron energy ε + shown in figure 2(b), exhibitinga maximum difference as large as −
20% at ε + = 200 MeV. The difference of positronyield is mainly attributed to the linear polarization of emitted γ photons, which will bedetailed in the next subsection.Then, we analyze the spin dynamics of electrons in the linearly-polarized EMSW.As emitting a high-energy photon, the electron spin more probably flips to the directionantiparallel to the magnetic field in the electron’s rest frame, according to equation (1)[see figure 4(b) in reference [34]]. In the magnetic region where photon emissions areconcentrated on, the spin-flip trend of the electron is determined by the direction ofmagnetic field B z . More specifically, the electron spin is more likely antiparallel to x/λ y / λ (a) 010002000 n γ /n c x/λ y / λ (b) − . − . . . . ξ . . . . . . ε γ (GeV) − . − . − . . . . . . ξ (c) (i)(iv) Figure 3.
Spatial distribution of (a) photon density n γ and (b) average linear-polarization degree ξ at T = 12 T in case (i) . (c) ξ versus photon energy ε γ obtained from case (i) and case (iv) , respectively, where we switch off the pairproduction module while include spin and polarization effects in case (iv) . the B z direction after the photon emission (and positron spin is more likely parallel tothat). This is evidenced by the time-space evolution of the average spin component S z of electrons shown in figure 1(c). It shows that S z is temporally oscillating with a laserfrequency ω , but the total degree of spin still remains nearly zero due to the symmetryof the field, which is a kind of local spin polarization.By comparing the positron number in case (ii) and case (iii) in figure 2(a), onealso notices that the positron yield is reduced by about 2 .
5% due to the pure spin effect.This reduction is caused by the local spin polarization of electrons in the EMSW. Whenelectrons just enter the magnetic region, their initial spins are mainly parallel to thedirection of magnetic field (spin-up, defined by S i · e >
0) [see figure 1(c)], owing tothe radiative spin polarization in the previous magnetic region. The spin-dependentphoton emission probability is therefore reduced, especially for high-energy photons [seethe third term − uK ( y )( S i · e ) at the right hand side of equation (1) and figure 4(a) inRef. [34]]. Then, the electron spin gradually flips to be anti-parallel to the magnetic fielddirection (spin-down, defined by S i · e < B z is reversed. The electron spin direction becomes parallelto the B z direction again, and consequently the same process arises with slightly weakerphoton emission. The relative deviation of emitted photon number δ iii − ii γ is plottedin figure 2(c). One can see the absolute value of δ iii − ii γ increases with the increase ofphoton energy ε γ when ε γ < . γ photons more likely decayinto e + e − pairs, leading to a slightly smaller positron yield in figure 2(a). The spineffect discussed above is five times weaker than the photon polarization effect in termsof positron yield, and therefore we will mainly focus on the latter one. However, wenote that the polarization of high-energy emitted photons highly relies on the spin ofemitting electrons, hence the local spin polarization could have a prominent impact onthe high-energy positron yield [see the next subsection]. . . . . . . u − . − . . . . S i · e (a) − . − . . . . ξ . . . u . . . . ξ (b) χ e = 1 χ e = 5 χ e = 10 Figure 4.
Theoretical calculations according to equation (4). (a) ξ versus emittedphoton energy ratio u and the initial electron spin vector S i · e for χ e = 1. (b) ξ versus u at S i · e = 0 for χ e =1, 5, and 10. The remarkable difference of the positron yield between case (i) and case (ii) [seefigures 2(a) and 2(b)] mostly originates in the highly polarized γ photons. Figures 3(a)and 3(b) present spatial distributions of the photon density n γ and average photonpolarization ξ at t = 12 T in case (i) , in which both spin and polarization effectsare included. The other two polarization components ξ and ξ are nearly zero, so notshown here. ξ is rather uniform in space with a positive average value of about 0.58,indicating that photons are emitted predominantly with a linear polarization orientedalong e , i.e. always in the x - y plane.The spin- and polarization-resolved probabilities in equations (1) and (2) can besimplified in the interaction between the considered linearly-polarized EMSW and un-prepolarized electrons. It is appropriate to sum up over S f terms and neglect S i · e and S i · e v terms in equation (1) since electrons can only be spin-polarized along e ,i.e. B z direction. Therefore, the average Stokes parameters of emitted photons can beapproximately as ξ ≈ ,ξ ≈ , (4) ξ ≈ K ( y ) − u − u K ( y )( S i · e ) u − u +21 − u K ( y ) − Int K ( y ) − uK ( y )( S i · e ) . Equation (4) indicates that the emitted photons cannot be circularly polarized since ξ = 0. According to equation (4), the theoretical average polarization ξ as a functionof u and S i · e is shown in figure 4(a). For the low-energy photon, ξ is always positivewith a value of about 0 .
5, insensitive to the initial spin vector S i of the emitting electron.While for the high-energy photon, ξ strongly depends on S i · e . Spin-up electrons areprone to emit photons linearly polarized along e axis ( z axis in our case) with ξ < ξ >
0. In general, ξ value decreases with the increase of photon energy ε γ . This trend is supported by0the curve for case (i) shown in figure 3(c), that ξ ≈ . ε γ = 0 . ξ ≈ . ε γ = 0 . ξ , ξ , ξ ) to ( ξ (cid:48) , ξ (cid:48) , ξ (cid:48) ) can alsobe simplified in our case with the linearly-polarized EMSW. We can obtain thetransformation angle θ ≈ π for the fact that the acceleration direction andvelocity direction of electrons are both well confined in the x - y plane (laser polarizationplane). Consequently, ξ with respect to the emitted frame ( e , e , e v ) can be directlysubstituted into equation (2), i.e. ξ ≈ ξ (cid:48) . Hence, the positive values of ξ directly leadsto the reduction of the positron yield.The photon polarization and positron yield at their higher energy range exhibitsome anomalous phenomena compared with those at their lower energy range, dueto the local spin polarization of electrons. As outlined above, the most intense photonemissions occur in the early stage of each magnetic region, in which the spin-up emittingelectrons dominate. The resulting decrease in the photon yield has been observed infigure 2(c). Besides the photon yield, the polarization of high-energy photons is alsostrongly affected by these locally spin-polarized electrons. Figure 3(c) for case (iv) indicates the original emitted photons with ε γ > . ξ . Itis coincident with the theoretical analysis shown in figure 4(a) that spin-up electronsare in favor of emitting high-energy photons of negative-value ξ . This eventually leadsto an increase of positron yield at positron energies higher than 0.5 GeV, rather thandecreasing like that at lower energies in figure 2(b), due to the opposite signs of ξ between these two energy ranges. Moreover, when we switch on the pair productionprocess and include the γ -photon annihilation in case (i) , the polarization ξ of high-energy photons increases as compared to those in case (iv) , shown in figure 3(c). Thisis because photons of negative ξ are easier to annihilate into e − e + pairs, according tothe polarization-dependent pair production probability in equation (2). In figure 5(a), we investigate the difference of positron yield between with and withoutspin and polarization effects under various laser strengths a in order to find the laserintensity at which these two effects need to be taken into account. For the caseof initial plasma density n = 50 n c , the absolute relative deviation | ∆ N + | of totalpositron number first increases with laser intensity a and reaches a maximum 12% at a = 800, then it gradually decreases to 7% at a = 1200. For a higher-density plasmaof n = 200 n c , | ∆ N + | always decreases with a in the scanned parameter range, i.e.decreasing from 11% at a = 500 to 6% at a = 1200. On the whole, | ∆ N + | is larger inthe case of n = 50 n c than that of n = 200 n c case at the same a . These results can beexplained according to figure 4(b), in which the average polarization ξ as a function of u under different quantum parameter χ e is plotted according to equation (4). Here, weassume S i · e = 0, which is approximately valid since electrons or positrons cannot gaina net degree of spin polarization in the linearly EMSW and the influence of local spin1 | ∆ N + | ( % ) (a) n = 50 n c n = 200 n c
600 800 1000 1200 a | ∆ η | ( % ) (b) Figure 5.
The absolute relative deviation of (a) positron number | ∆ N + | = | ( N inc+ − N exc+ ) /N exc+ | and (b) total laser absorption | ∆ η | = | ( η inc − η exc ) /η exc | betweenincluding and excluding spin and polarization effects, under various laser strengths a = 500-1200 and initial plasma densities n /n c = 50 and 200. polarization on the positron yield is smaller compared with the photon polarization one.One can see that the average polarization ξ is smaller for a higher χ e (corresponding toa higher laser intensity), hence a smaller difference between positron yields can expected.Figure 5(b) shows the spin and polarization effects on the total laser absorption η with laser strength a . The absolute relative deviation | ∆ η | can reach a maximumof 6% for n = 50 n c and 2% for n = 200 n c , respectively. Below a = 800, | ∆ η | decreases as reducing a and only ∆ η < .
5% is observed at a = 500, indicating thatthe spin and polarization effects have a negligible impact on the laser absorption atlaser intensity below I min = 3 . × W/cm , although the difference of positron yieldis more obvious at relatively low laser intensities. This is because the pair productionprobability d W pairs / ( dε + dt ) in equation (2) is exponentially small for χ γ (cid:28)
1, so thatonly a small number of positrons is produced. At I < I min , the energy conversionefficiency from laser to positron is less than 1%, hence the impact of positron yieldreduction induced by spin and polarization effects on the laser-plasma interaction canbe neglected.
4. Conclusion
In conclusion, we have investigated the pair production in the interaction of two counter-propagating 10-PW-class laser pulses with a thin foil target using QED-PIC simulations,with e − e + spin and γ -photon polarization effects included. The two effects result in thedecrease of total positron yield by about 10%, and the relative difference can even reach20% for the medium-energy positrons. In other words, the spin- and polarization-averaged probabilities widely implemented in QED-PIC methods overestimate the2 t/T − . − . . S x (QED-PIC) S y (QED-PIC) S z (QED-PIC) t/T . . . . ξ (QED-PIC) ξ (QED-PIC) ξ (QED-PIC) t/T − . − . . . . t/T . . . . Figure 6.
Comparison of our 2D QED-PIC results (thick colored lines) with those bythe code in reference [40] (thin black lines) in the process of radiative spin polarizationof initially (a)(b) unpolarized electrons and (c)(d) longitudinally polarized electrons,respectively. (a)(c) Average spin degree of electrons S x , S y and S z . (b)(d) Averagepolarization degree of all emitted photons ξ , ξ and ξ with respect to the emittedframe. positron yield, which can also affect the laser plasma interaction at laser intensities above3 . × W/cm in our laser-foil interaction. The decrease of positron yield mainlycomes from the linear polarization up to 50% of emitted γ photons. In addition, we alsoobserve several anomalous phenomena about positron yield and photon polarization ofhigh-energy particles caused by the local spin polarization of e − e + . For example, withspin and polarization effects included, the positron yield is increased in the high-energyrange rather than decreased like that in the low-energy range. Acknowledgments
This work was supported by the National Key R&D Program of China (GrantNo. 2018YFA0404801), National Natural Science Foundation of China (Grant Nos.11775302, 11721091, 11991073, and 12075187), the Strategic Priority Research Programof Chinese Academy of Sciences (Grant Nos. XDA25050300, XDA25010300, andXDB16010200), Science Challenge Project of China (Grant Nos. TZ2016005 andTZ2018005), and the Fundamental Research Funds for the Central Universities, theResearch Funds of Renmin University of China (20XNLG01).3
Appendix A. Spin and polarization benchmarks
In our QED-PIC code, the e − e + radiative spin polarization and photon polarizationdetermination modules follow the Refs. [40, 42, 48], in which the three-dimensional spindynamics can be resolved. Here, we perform two additional 2D QED-PIC simulations tobenchmark against the one-particle code used in reference [40], where ultrarelativisticelectrons with 1 GeV move along + x axis in the x - y plane under a perpendicularlystatic external magnetic field of B = 100 mcω / | e | , where ω = 1 µ m. We takethe computational domain of 8 λ × λ in x × y directions with 128 ×
128 cells,and 16 macro electrons per cell. The electron density is low enough to avoid theinfluence of self-generated electromagnetic field, and periodic boundaries are employed.Figures 6(a) and 6(b) show the time evolution of average spin degree S of electronsand average polarization degree ξ of emitted photons for the initially unpolarized (orrandomly polarized) electrons, respectively, and figures 6(c) and 6(d) are those forinitially longitudinally polarized electrons. Our QED-PIC simulation results are in goodagreement with the single-particle model simulation results of the code in reference [40]. References [1] Cartlidge E 2018
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