Laser field absorption in self-generated electron-positron pair plasma
E.N. Nerush, I.Yu.Kostyukov, A.M. Fedotov, N.B. Narozhny, N.V. Elkina, H. Ruhl
aa r X i v : . [ phy s i c s . p l a s m - ph ] N ov Laser field absorption in self-generated electron-positron pair plasma
E. N. Nerush, I. Yu. Kostyukov ∗ Institute of Applied Physics, Russian Academy of Sciences, 603950 Nizhny Novgorod, Russia
A. M. Fedotov, N. B. Narozhny,
National Research Nuclear University MEPhI, Moscow, 115409, Russia
N. V. Elkina, H. Ruhl
Ludwig-Maximillians Universit¨at M¨unchen, 80539, Germany
Recently much attention has being attracted to the problem of limitations on the attainableintensity of high power lasers [A.M. Fedotov et al.
Phys. Rev. Lett. , 080402 (2010)]. The laserenergy can be absorbed by electron-positron pair plasma produced from a seed by strong laser fieldvia development of the electromagnetic cascades. The numerical model for self-consistent study ofelectron-positron pair plasma dynamics is developed. Strong absorption of the laser energy in self-generated overdense electron-positron pair plasma is demonstrated. It is shown that the absorptionbecomes important for not extremely high laser intensity I ∼ W/cm achievable in the nearestfuture. PACS numbers: 12.20.Ds,41.75.Jv,42.50.Ct
Due to an impressive progress in laser technology, laserpulses with peak intensity of nearly 2 × W/cm arenow available in the laboratory [1]. When the matteris irradiated by so intense laser pulses ultrarelativisticdense plasma can be produced. Besides of fundamen-tal interest, such plasma is an efficient source of particlesand radiation with extreme parameters that opens brightperspectives in development of advanced particle acceler-ators [2], next generation of radiation sources [3, 4], labo-ratory modeling of astrophysics phenomena [5], etc. Evenhigher laser intensities can be achieved with the cominglarge laser facilities like ELI (Extreme Light Infrastruc-ture) [6] or HiPER (High Power laser Energy Researchfacility) [7]. At such intensity the radiation reaction andquantum electrodynamics (QED) effects become impor-tant [8–13].One of the QED effects, which has recently attractedmuch attention, is the electron-positron pair plasma(EPPP) creation in a strong laser field [11, 12]. Theplasma can be produced via avalanche-like electromag-netic cascades: the seed charged particles are acceler-ated in the laser field, then they emit energetic pho-tons, the photons by turn decay in the laser field andcreate electron-positron pairs. The arising electrons andpositrons are accelerated in the laser field and producenew generation of the photons and pairs. It is predicted[12] that an essential part of the laser energy is spent onEPPP production and heating. This can limit the attain-able intensity of high power lasers. That prediction wasderived using simple estimates, therefore self-consistenttreatment based on the first principles is needed.The collective dynamics of EPPP in strong laser fieldis a very complex phenomenon and numerical modelingbecomes important to explore EPPP. Up to now the nu-merical models for collective QED effects in strong laser field have been not self-consistent. One approach in nu-merical modeling is focused on plasma dynamics and ne-glects the QED processes like pair production in the laserfield. It is typically based on particle-in-cell (PIC) meth-ods and uses equation for particle motion with radia-tion reaction forces taken into account [13]. The secondone is based on Monte Carlo (MC) algorithm for photonemission and electron-positron pair production. This ap-proach has been used to study the dynamics of electro-magnetic cascades [14]. However, it completely ignoresthe self-generated fields of EPPP and the reverse effect ofEPPP on the external field. The latter effect is especiallyimportant to determine the limitations on the intensityof high power lasers [12, 15].Quantum effects in strong electromagnetic fields canbe characterized by the dimensionless invariants [16, 17] χ e = e ~ / ( m c ) | F µν p ν | ≈ γ ( F ⊥ /eE cr ) and χ γ ≈ ( ~ ω/mc )( F ⊥ /eE cr ), where F µν is the field-strength ten-sor, p µ is the particle four-momentum, ~ ω is the pho-ton energy, γ is the electron gamma-factor, F ⊥ is thecomponent of Lorentz force, which is perpendicular tothe electron velocity, E cr = m c / ( e ~ ) = 10 V/cm isthe so-called QED characteristic field, ~ is the Planckconstant. χ e determines photon emission by relativisticelectron while χ γ determines interaction of hard photonswith electromagnetic field. QED effects are importantwhen χ e & χ γ &
1. If χ e & ~ ω ∼ γmc andthe quantum recoil imposed on the electron by the emit-ted photon is strong. The probability rate of emissionof a photon with energy ~ ω by relativistic electron withgamma-factor γ can be written in the form [17–19] dW em ( ξ ) = αmc √ π ~ γ (cid:20)(cid:18) − ξ + 11 − ξ (cid:19) K / ( δ ) − Z ∞ δ K / ( s ) ds (cid:21) dξ, (1)where ~ ω is the photon energy, m is the electron mass, c is the speed of light, δ = 2 ξ/ [3(1 − ξ ) χ e ] and ξ = ~ ω/ ( γmc ). ~ ωdW em can be considered as the energydistribution of the electron radiation power. For electronradiation in constant magnetic field B perpendicular tothe electron velocity it reduces to the synchrotron radia-tion spectrum in the classical limit χ e ≪ ~ ω is [17–19] dW pair ( η − ) = αm c √ π ~ ω (cid:20)(cid:18) η + η − + η − η + (cid:19) K / ( δ ) − Z ∞ δ K / ( s ) ds (cid:21) dη, (2)where δ = 2 / (3 χ γ η − η + ), η − = γmc / ( ~ ω ) and η + =1 − η − are the normalized electron and positron energies,respectively. It follows from Eq. (2) that in the classicallimit χ γ ≪ ~ ω ≪ mc ) while the en-ergy of the photons emitted by accelerated electrons andpositrons is very high ( ~ ω ≫ mc ). The emitted photonsare hard and can be treated as particles. Conversely, theevolution of the laser and plasma fields is calculated bynumerical solution of the Maxwell equations. Therefore,the dynamics of electrons, positrons and hard photons aswell as the evolution of the plasma and laser fields are cal-culated by PIC technique while emission of hard photonsand pair production are calculated by MC method.The photon emission is modeled as follows. On everytime step for each electron and positron we sample a pho-ton emission by a probability distribution which approx-imates Eq. (1) with the accuracy within 5%. The newphoton is included in the simulation region. The coordi-nates of a new photon are equal to the electron (positron)coordinates at the emission instance. The photon mo-mentum is parallel to the electron (positron) momentum.The electron (positron) momentum value is decreased bythe value of the photon momentum. Similar algorithmis used for modeling of pair production by photons. Thenew electron and positron are added in the simulationregion while the photon that produced a pair is removed. The sum of the electron and positron energy is equal tothe photon energy. The pair velocity is directed alongthe photon velocity at the instance of creation.The MC part of our numerical model has been bench-marked to simulations performed by other MC codes.We simulated the electromagnetic showers in a static ho-mogeneous magnetic field, the interaction of relativisticelectron beam with a strong laser pulse, and the devel-opment of electromagnetic cascades in circularly polar-ized laser pulses. The obtained results are in reasonablygood agreement with those published by other authorsand are discussed in Ref. [19]. The particle motion andevolution of the electromagnetic field are calculated withstandard PIC technique [23]. The PIC part of the modelis two-dimensional version of the model used in Ref. [24].In order to prevent memory overflow during simulationbecause of the exponential growth of particle numberin a cascade, the method of particles merging is used[22]. If the number of the particles becomes too large therandomly selected particles are deleted while the charge,mass, and the energy of the rest particles are increasedby the charge, mass, and energy of the deleted particles,respectively.We use our numerical model to study productionand dynamics of EPPP in the field of two collidinglinearly polarized laser pulses. The laser pulses haveGaussian envelopes and propagate along the x -axis. Thecomponents of the laser field at t = 0 are E y , B z = a exp( − y /σ r ) sin ζ h e − ( x + x ) /σ x ± e − ( x − x ) /σ x i ,where the field strengths are normalized to mcω L / | e | ,the coordinates are normalized to c/ω L , time is nor-malized to 1 /ω L , a = | e | E / ( mcω L ), E is the electricfield amplitude of a single laser pulse, ω L is the laserpulse cyclic frequency, 2 x is the initial distance be-tween the laser pulses, ζ = x − φ , and φ is the phaseshift. The parameters of our simulations are φ = 0 . π , a = 1 . · , σ x = 125, σ r = 40, x = σ x / λ = 2 πc/ω L = 0 . µ m correspond to theintensity 3 · W/cm , pulse duration 100 fs, the focalspot size 10 µ m at 1 /e intensity level. The cascade isinitiated by a single electron located at x = y = 0 withzero initial momentum for t = 0 when the laser pulsesapproach to each other (the distance between the pulsecenters is σ x ).The later stage ( t = 25 . λ/c ) of the cascade develop-ment is shown in Fig. 1, where the electron and photondensity distributions and the laser intensity distributionare presented. The laser pulses passed through each otherby this time instance and the distance between the pulsecenters becomes about 1 . σ x . It is seen from Figs. 1 thatthe micron-size cluster of overdense EPPP is producedand the laser energy at the backs of the incident laserpulses is spent on EPPP production and heating. Theplasma density exceeds the relativistic critical density a n cr in about 2 times, where n cr = mω / (cid:0) πe (cid:1) is the − −
20 0 20 40 x/λ -505 y / λ c 01 ρ l -505 y / λ a 01 ρ e -505 y / λ b 06 ρ γ FIG. 1: The normalized electron density ρ e = n e / ( a n cr )(a), the normalized photon density ρ γ = n γ / ( a n cr ) (b) andthe laser intensity normalized to the maximum of the initialintensity ρ l (c) during the collision of two linearly-polarizedlaser pulses at t = 25 . λ/c .
10 15 ct/λ . W W Σ W las W e + e − W γ FIG. 2: The electron and positron energy (solid line), the pho-ton energy (dotted line), the laser energy (dashed line) andthe total energy of the system (dash-dotted line) as functionsof time. All the energies are normalized to the initial energyof the system. nonrelativistic critical density for the electron-positronplasma. The evolution of the particle and laser energyis shown in Fig. 2. It is seen from Fig. 2 that abouta half of the laser energy is absorbed by self-generatedEPPP and then mostly reradiated in ultrashort pulse ofgamma-quanta. The total energy of the particles in the n e / ( a n c r ) N e ct/λ FIG. 3: The number of the electrons produced in the cascade(line 1) and the EPPP density normalized to the relativisticcritical density (line 2) as functions of time. n e / ( a n c r ) − − x/λ FIG. 4: The profile of the electron density along x -axis at y = 0 for initial stage ct = 6 . λ (line 1) and for the laterstage ct = 16 . λ (line 2) of the cascade development. Theelectron density for ct = 16 . λ is normalized to a n cr andthat for ct = 6 . λ is normalized to 3 . × − a n cr . cascade and the electromagnetic field is conserved withaccuracy about 1% during our simulation.At initial stage of the cascade development the num-ber of created particles is growing exponentially N ∼ e Γ t [12], where Γ is the multiplication rate. It follows fromthe energy conservation law that the number of parti-cles that can be created is limited by the laser pulsesenergy. Thus, at some instant the exponential growthis replaced by much slower growth. Equating the initialenergy of laser pulses to the overall particles energy af-ter the pulse collision we get N ∼ a σ x σ r N / ¯ γ , wherewe assume N e ∼ N p ∼ N ph ∼ N , N e , N p and N ph arethe number of electrons, positrons and photons producedby the cascade, respectively, mc ¯ γ is the average parti-cle energy, N = n cr ( c/ω ) = λ/ (16 πr e ), r e = e / ( mc ). ct/λ χ FIG. 5: The dependence of χ e for the primary electron ontime. The multiplication rate decreases when the field strengthgoes down, that, by turn, occurs if the plasma densityreaches the value a n cr . This is in good agreement withthe numerical results shown in Fig. 3, where the mul-tiplication rate drops dramatically and EPPP densityreaches the value about a n cr at the same instant oftime t s ≈ λ/c . The value of t s can be estimated as t s ≈ Γ − ln N . It follows from Fig. 3 that Γ ≈ . ω L for t < t s . The typical lifetime t em for electrons andpositrons with respect to hard photon emission can beestimated as 1 / Γ > /ω L [12]. Thus, for the parametersof numerical simulation ¯ γ can be estimated as ¯ γ ∼ a hence N ∼ a σ x σ r N ∼ · and ct s /λ ∼ B -node of linearly polarized standing electromagneticwave so far as χ e and χ γ are less than unity. However,our numerical simulations show that the cascade quasi-periodically develops between B and E nodes of such awave. This is because under such conditions the electronmotion becomes complicated and is not confined to thedirection of polarization on the temporal scales aboutthe laser period. It turns out that there occur the timeintervals of duration of the order of ω − L with χ e > N e ( t ) and of EPPP density along the x -axis at the initial stage of the cascade development areseen in Fig. 3 and 4, respectively. At the later stages thespatial modulation of the density is strongly smoothedout due to EPPP expansion (see Fig. 4, line 2).In conclusion we develop the numerical model which al-lows us to study EPPP dynamics in strong laser field self-consistently. We have demonstrated efficient productionof EPPP at the cost of the energy of the laser pulses. Weshow that even not extremely high intensity laser pulses( I ∼ W/cm with duration ∼
100 fs) can produce overdense EPPP so that the QED effects can be experi-mentally studied with near coming laser facilities like ELI[6] and HiPER [7]. The simulations and estimates showthat for intensity
I > W/cm the overdense EPPPcan be produced during a single laser period. In suchhigh-intensity regime few-cycle laser pulses can be used inexperiments. High-energy photons or electron-positronpair can be also used as a seed to initiate cascade insteadof an electron. Photon-initiated cascade can be moresuitable for experimental study in low intensity regime( I ∼ W/cm ) because the laser intensity thresholdfor pair creation in vacuum is about ∼ W/cm [25].This work was supported in parts by the Russian Foun-dation for Basic Research, the Ministry of Science andEducation of the Russian Federation, the Russian FederalProgram “Scientific and scientific-pedagogical personnelof innovative Russia”, the grant DFG RU 633/1-1, andthe Cluster-of-Excellence ’Munich-Centre for AdvancedPhotonics’ (MAP). ∗ Electronic address: [email protected][1] V. Yanovsky et al.
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