Monte Carlo calculations of pair production in high-intensity laser-plasma interactions
aa r X i v : . [ h e p - ph ] O c t Monte Carlo calculations of pair production inhigh-intensity laser-plasma interactions
R Duclous , J G Kirk & A R Bell , Clarendon Laboratory, University of Oxford, Parks Road, Oxford UK OX1 3PU Max-Planck-Institut f¨ur Kernphysik, Postfach 10 39 80, 69029 Heidelberg, Germany STFC Rutherford Appleton Laboratory, Didcot, Oxfordshire UK OX11 0QXE-mail: [email protected]
Abstract.
Gamma-ray and electron-positron pair production will figure prominently in laser-plasmaexperiments with next generation lasers. Using a Monte Carlo approach we show that straggling e ff ects arising from the finite recoil an electron experiences when it emits a highenergy photon, increase the number of pairs produced on further interaction with the laserfields.
1. Introduction
Next generation lasers such as ELI [1, 2] and VULCAN [3, 4] are expected to achieveintensities of 10 W cm − or greater, opening up rich opportunities to study QED processesin the strong field regime[5, 6, 7]. One of these processes is pair production in counter-propagating beams, which is a significant e ff ect at 10 W cm − [8], and, at 10 W cm − issu ffi ciently strong that a cascade may develop in which each electron produces a pair that inturn produces further pairs, to initiate an avalanche.In intense laser beams, the dominant route to pair production is a two-stage process.Firstly, an electron with Lorentz factor γ interacts with the laser fields ( E , B ) to produce agamma-ray with high energy h ν . Secondly, the gamma-ray interacts with the same fields asit propagates to produce an electron-positron pair [8, 9]. The crucial parameters for the two-stage process are (i) η ≈ γ | E ⊥ + v × B | / E crit ‡ where E crit = . × Vm − is the Schwingerfield, and (ii) χ = h ν | E ⊥ + ( c k / k ) × B | / m e c E crit where ~ k is the gamma-ray momentumand E ⊥ is the component of the electric field perpendicular to the direction of motion ofthe electron or photon, as appropriate. Strong field QED e ff ects such as pair productionbecome important when these parameters reach unity. The energy of the gamma-ray photonis typically h ν ≈ . ηγ mc , implying χ ≈ . η , and the cross-section for pair productionis proportional to exp( − / χ ) for χ ≪
1. Because of the exponential cut-o ff to the cross-section, the process is very sensitive to the value of η reached by the electron, when theregime of strong QED is approached. This leads to an abrupt laser-intensity threshold for ‡ This expression is generally accurate for γ ≫
1, a precise definition is given in (2) onte Carlo calculations of pair production in high-power laser-plasma interactions η can lead to a largeincrease in pair production.In [8], [9] and [11], a semi-classical treatment was used in which electrons were subjectto a continuous loss of energy through radiation of gamma-rays. In this continuous model,the radiation reaction force is approximated, to the lowest order in 1 /γ , by the expression f rad = − α f m c ~ η g ( η ) p p , g ( η ) = √ πη Z ∞ F ( η, χ ) d χ , (1)where g ( η ) ∈ [0 ,
1] accounts for the reduction of the total power radiated by the electron. F ( η, χ ) is the quantum synchrotron emissivity, as defined in Appendix A of [9]. However,in reality, gamma-ray emission is a random quantum process [12, 13]. As η approachesunity, the energy of the emitted gamma-ray becomes a substantial fraction, typically 0 . η ,of the kinetic energy of the electron before emission, and the resulting electron trajectorystarts to fluctuate substantially away from the semi-classical average. This enhances pairproduction in two ways. Firstly, if an accelerating electron propagates by chance over anunusually large distance before emitting a gamma-ray, it may reach a large Lorentz factor γ and emit a gamma-ray with an unusually high energy. This, in turn, has a much increasedprobability of producing a pair, as compared to the photons emitted by an electron movingon a classical trajectory in the same field. Such events cumulate, and we show in 3.1 thatthis straggling e ff ect [12] enhances pair production. Secondly, the deviations in the trajectorycaused by straggling lead the electron to sample the laser fields at locations other than thoseon the classical path. In [9], it was shown that, under the influence of continuous radiationlosses, electrons naturally migrate towards points in the laser field at which they emit fewhigh-energy gamma-rays. In the case of counter-propagating circularly polarised waves,these are the nodes in the electric field. Once an electron settles at an E = ff ects on the electron trajectory. This is doneusing a Monte-Carlo method that is described in section 2. In section 3, we compare thecontinuous with the more realistic discontinuous loss case, and show that the latter produces alarger number of pairs at laser intensities around 10 W cm − . Pair production at 10 W cm − is relatively little a ff ected by straggling since, at this intensity, even semi-classical, continuouselectron trajectories lead to gamma-rays that are well over the threshold for pair production.
2. Description of the Monte Carlo algorithm
In 4-vector notation, the quantities η and χ are defined as η = e ~ m c (cid:12)(cid:12)(cid:12) F µν p ν (cid:12)(cid:12)(cid:12) , χ = e ~ m c (cid:12)(cid:12)(cid:12) F µν k ν (cid:12)(cid:12)(cid:12) , (2) onte Carlo calculations of pair production in high-power laser-plasma interactions p ν and ~ k ν , respectively, with F µν theelectromagnetic field tensor. In the following, we make use of the approach presented in [9],that retains the weak, quasi-stationary field approximation. The probabilities for gamma-rayphoton emission and the subsequent pair production are in this approximation functions of η and χ only. An equivalent system is then chosen, for the same given χ and η , in which theelectron moves in a plane perpendicular to a uniform, static B field. This allows us to use thetransition probabilities as formulated in [14].The radiation emitted by the particle due to its acceleration in the laser fields — herecalled synchrotron radiation — is assumed to be a random walk process [12, 13]. Theprobability of emission of a gamma-ray photon is governed by an optical depth τ e . At thestart of the calculation, and immediately following the emission of a photon, the currentoptical depth τ e is set to zero and a randomly chosen ‘final’ optical depth τ (f)e is assigned to theelectron. The current optical depth τ e increases as the electron propagates until it reaches τ (f)e ,at which point a photon is emitted. At each computational timestep the electron’s position,momentum and current optical depth are updated according to the equationsd p d t = − e ( E + v ∧ B ) , (3)d x d t = p m e γ , (4)d τ e d t = Z η/ d N d χ d t ( η, χ ) d χ , (5)where d N d χ d t ( η, χ ) = √ πτ C α f ηγ F ( η, χ ) χ (6)is the di ff erential rate of production of photons of parameter χ by an electron of parameter η and F ( η, χ ) is the quantum synchrotron emissivity, expressed in the weak, quasi-stationaryfield approximation [14]. An expression for F ( η, χ ) is given in Appendix A of [9]. TheCompton time is τ C = Ż C / c , where Ż C = ~ / m e c is the Compton wavelength, and α f is the finestructure constant.When τ e = τ (f)e is reached, a photon is emitted, and its energy is randomly assigned asfollows. First, the parameter χ (f) is found from the relation ξ = R χ (f) d N d χ d t d χ R η/ d N d χ d t d χ = R χ (f) F ( η,χ ) χ d χ R η/ F ( η,χ ) χ d χ , (7)where ξ is a uniformly distributed random number in [0,1], and the right hand side is amonotonic, increasing function of χ (f) , which is tabulated in both the χ (f) and η directions. Thevariables χ (f) and η , in equation (7), depend on the electromagnetic field at the point of photonemission. Then, assuming that the photon is emitted parallel to the electron momentum, itsenergy h ν , is determined by the relation h ν = m e c χ (f) γη . (8) onte Carlo calculations of pair production in high-power laser-plasma interactions η and χ and the assumption that the electron isrelativistic, γ ≫
1. For computational e ffi ciency, we ignore the rare photons emitted in the lowenergy part of the spectrum, χ (f) < χ (f)min , where χ (f)min is chosen such that the neglected photonscarry only 10 − of the total energy and are incapable of producing pairs. The computationalprocedure was successfully tested by its ability to reproduce the correct synchrotron spectrum.This is shown in Fig.1, for a large number (10 ) of emitted photons in a regime where η = . χ ( f ) parameter, which represents the photon energy, is assigned to each photon at theemission point, according to the sampling relation (7). The occurrence (in percent of the totalnumber of the 10 emission events) of emissions in each energy interval gives a distributionalong the χ ( f ) axis, that scales with the quantum synchrotron emissivity F ( η, χ ( f ) ). S t a t i s t i c a l w e i gh t ( i n % ) log ( χ (f) )C ste xF( η , χ (f) )Sampling of 10 photons Figure 1.
Comparison between the distribution in χ ( f ) resulting from the sampling relation (7)and the quantum synchrotron emissivity F ( η, χ ( f ) ). After the electron has emitted a photon, which is assumed to occur instantaneously, itcontinues its trajectory beginning at the same point but with a di ff erent energy γ (f) mc andmomentum p (f) . To find p (f) , we again use the approximation that the photon is emitted parallelto the pre-emission electron momentum, so that, by conservation of momentum, p (f) = − h ν cp ! p . (9)In reality, the photon may be emitted over a cone of opening angle approximately 1 /γ , but weneglect this small angular spread here. Equation (9) implies that a small amount of energy isextracted from the laser fields during photon emission, which we also neglect. This neglectis unimportant in the present test-particle model in which the laser fields are prescribed, sincethe fraction of the electron energy transferred to the laser field is small ∼ / ( γγ (f) ), as shownin Appendix A. However this point may need more careful treatment in a more completedynamic treatment of the development of an electron-positron cascade and its inclusion in aparticle-in-cell code [15, 16]. onte Carlo calculations of pair production in high-power laser-plasma interactions
3. Straggling e ff ects As in [9], we adopt the configuration of two counter-streaming, circularly polarized laserbeams, which can qualitatively be considered equivalent to the interaction of an incident andreflected beam from a solid [8]. We restrict the treatment to pair creation by the two-stepprocess described in section 2. Pairs can also be produced directly by an electron interactingwith the laser field, without the need to produce an intermediate real photon. This processis analogous to trident pair production by an electron in the Coulomb field of a nucleus.However, although this direct, single-step process may be important at lower laser intensity,it is dominated by the indirect two-step process at laser intensities above 10 Wcm − , and wedo not include it in the results presented here.In the two-step process, the emitted gamma-rays travel through the plasma and mayproduce pairs according to the cross-section which depends on χ . In many ways, the processis comparable with that of gamma-ray production by electrons. The photon trajectory could befollowed through the electromagnetic fields and the optical depth integrated as for electrons.However, in order to make a direct comparison with the rate computed assuming continuouslosses, we here follow the much simpler approximate approach that was used in [9]. Theprobability of pair production is calculated for a gamma-ray with χ = χ (f) , that propagatesthrough electromagnetic fields that are assumed to equal those at the point where the gamma-ray was emitted. We follow [9] and make the assumption that conversion into pairs can occuruntil the photon has propagated a distance of one laser wavelength. Here the approximation isthat the beams have a transverse structure with cylindrical geometry, whose radius is onewavelength, representative of beams which are tightly focussed at the di ff raction limit toachieve maximum laser intensity. The real photons only produce pairs while they propagatethrough the region of large laser field. In contrast to electrons, whose trajectory is determinedby the electromagnetic fields, the photons are undeflected. This means that all along its paththrough the laser fields, the value of the parameter χ remains unchanged at χ = χ (f) . Furtherdetails can be found in section 3.3 of [9]. The di ff erential pair production rate via real photonsis the product of the di ff erential rate for real photon emission, d N d χ d t ( η, χ ), and an e ff ectiveconversion probability into pairs, (cid:16) − exp( − D τ p E ) (cid:17) , where D τ p E is an e ff ective optical depthto pair conversion. The rate of pair production via real photons per electron is then addressedwith the same rate equation as in [9]d N real dt = Z η/ d χ d N d χ d t ( η, χ ) (cid:16) − exp( − D τ p E ) (cid:17) . (10) ff ects of the straggling at a B = node In this section, we select a particular trajectory at a B = onte Carlo calculations of pair production in high-power laser-plasma interactions z -axis, each with an intensity I × W cm − , with I = .
3. The electron is initially located at a B = γ =
10, with itsmomentum oriented along the positive y -axis. The electron motion is tracked over seven laserperiods during which it remains at the magnetic node. In Fig. 2(a), the discontinuous MonteCarlo calculations lead to the straggling of the Lorentz factor for the particle (green curve).This is compared with the Lorentz factor of a particle that undergoes continuous radiationloss, computed as in [9] (red curve) for the same initial conditions. A feature of the stragglingin this regime is a ∼
30% fluctuation about the Lorentz factor derived from the continuousdescription. The particle motion, as shown in Fig. 2(b) is characterized by excursions tohigher and lower η values than those reached by an electron undergoing continuous losses.The discrepancies in the particle y position, between the two descriptions, is also shown inFig. 2(b). Finally, the comparison of Fig. 3, and Fig. 2(b), shows that the high η excursions arevery e ff ective at producing pairs. Moreover, the accumulation of pairs, produced by the high η excursions, leads to an enhancement of the overall pair production, in this example by about40% after only seven laser periods. At this time, the average number of pairs produces by anelectron reaches unity, which can be considered an approximate threshold for the initiation ofan electron-positron avalanche. E l ec t r on L o r e n t z f ac t o r γ time (laser phase) Continuous radiation lossDiscontinuous radiation loss -3.7-3.2-2.7-2.2-1.7-1.2-0.7-0.2 0.3 0.8 1.3 1.8 0 5 10 15 20 25 30 35 40 45 y , l og ( η ) time (laser phase) Continuous log ( η )Discontinuous log ( η )Continuous yDiscontinuous y ( a ) ( b ) Figure 2.
Case of an electron at a B = η -parameter & spatial coordinate y . onte Carlo calculations of pair production in high-power laser-plasma interactions N r ea l time (laser phase) Continuous N real
Discontinuous N real
Figure 3.
Number of pairs produced against time, by the continuous-loss trajectory (red curve)and the discontinuous one (green curve). ff -B = nodes In [9], the authors showed that the electron trajectory at a B = E = E = E = I = .
3, except that the initial electron position is now z = . z =
0, which is thelocation of the B = electrons is followed until the time t max =
20. Because of random fluctuations, each of the 10 trajectories is di ff erent, resultingin a di ff erent number of pairs finally produced. In Fig. 4(a), the z -position for two sampletrajectories is tracked, and compared with the z -position of an equivalent electron startingfrom the same point but subject to a continuous radiation-reaction force. Two di ff ering kindsof behaviour are highlighted. In one case (the blue curve), an electron trajectory describedby discontinuous radiation losses reaches the stable E = E = E = z = π/
2, averaged over 10 discontinuous trajectories, i.e., D ( z − π/ E . For comparison,the green curve shows the case of continuous radiation loss. The longer migration time,together with the excursions to large η discussed above, combine to enhance pair production onte Carlo calculations of pair production in high-power laser-plasma interactions trajectories. The N real axis isdivided into 100 equally spaced intervals in the logarithmic range log ( N real ) ∈ [ − , z = .
1, with I = . ( N real ) ≃ − .
01, obtained with continuous radiation loss. The figure shows that 77 . ( N real ) ≃ I = .
1. The results are presented in Fig. 5(a), where the N real axis is again divided into 100 equally spaced intervals, this time in the logarithmic rangelog ( N real ) ∈ [ − , ( N real ) ≃ − .
19 obtained with the continuous radiation loss. Electron energystraggling and longer migration times prove to have a significant impact on the number ofpairs produced. For the top 1% of the trajectories, one order of magnitude more pairs areproduced than are predicted using continuous radiation losses.In contrast, at higher laser intensities the spread of the distribution is quite narrow, andis centred close to the continuous value. This is illustrated in Fig. 5(b), where we presentresults for I =
1. In this figure, the N real axis is divided into 100 equally spaced intervalsin the logarithmic range log ( N real ) ∈ [ − , ( N real ) ≃ .
55. At high laser intensity, straggling has less e ff ect, because the process isabove threshold and less dependent on statistically rare excursions to large η .In order to obtain a more straightforward figure of merit, in terms of produced pairs, wecompute an e ff ective value for the number of pairs produced per electron whose motion isfollowed: h N real i = X i ∈ I N real ( i ) P ( i ) , where I stands for the complete set of equally spaced intervals that discretize the N real axis.The index i refers to one of these intervals, in which the probability is P ( i ). This procedureamounts to summing over histograms such as those in Fig. 4(c), 5(a) & 5(b), for the totalexpected number of pairs produced per electron starting from an initial position z = . I = .
1, thenumber of pairs h N real i associated with discontinuous-loss trajectories is about 5 times largerthan the number associated with continuous-loss trajectories. This is due to the importanceof stragglers in this regime, and has a strong dependence on the η -parameter. Because of thelarge range in N real (eight orders of magnitude) this e ff ect is more easily seen in Fig 6(b),which plots the ratio between these two numbers, showing that h N real i / N real shrinks as thelaser intensity increases, until it becomes very close to unity, at I =
1. The strength of the onte Carlo calculations of pair production in high-power laser-plasma interactions z po s iti on time (laser phase) Continuous z positionDiscontinuous z positionDiscontinuous z position -7-6-5-4-3-2-1 0 1 2 0 5 10 15 20 l og ( < ( z - π / ) > ) time (laser phase) Discontinuous log (<(z- π /2) >)Continuous log ((z- π /2) ) ( a ) ( b ) S t a t i s t i c a l w e i gh t ( i n % ) log (N real ) Distribution in log (N real ) (%)Continuous log (N real ) ( c ) Figure 4.
The discontinuous and continuous descriptions are compared at a laser intensity I = .
3, for initialization at z = .
1, away from the B = z = z isshown as a function of time, in (b) the square of the distance to the stable E = z = π/ N real produced by atrajectory. straggling process is found to be very sensitive to the laser intensity for about I < .
4. Conclusion
In this paper, we report on calculations of the number of electron positron pairs produced asan electron moves in the field of a laser with intensity in the range range 10 − W cm − .In this regime, the dominant route to pair production involves the emission of gamma-rayphotons via synchrotron radiation, and their further conversion into electron-positron pairs onte Carlo calculations of pair production in high-power laser-plasma interactions S t a t i s t i c a l w e i gh t ( i n % ) log (N real ) Distribution in log (N real ) (%)Continuous log (N real ) 0 5 10 15 20 25 30 35 -2 -1 0 1 2 3 4 S t a t i s t i c a l w e i gh t ( i n % ) log (N real ) Distribution in log (N real ) (%)Continuous log (N real ) ( a ) ( b ) Figure 5.
The statistical distribution (red bars) of a set of 10 electrons identically initializedat z = . B = z =
0) for (a) I = .
1, and (b) I =
1. The number ofpairs produced by an electron undergoing continuous losses is shown for comparison (greenlines). -8-6-4-2 0 2 4-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 l og ( N r ea l ) log (I ) Continuous radiation lossDiscontinuous radiation loss 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 < N r ea l > / N r ea l I Fraction
Fit with rational function ( a ) ( b ) Figure 6.
The continuous- and discontinuous-loss descriptions are compared in terms of thenumber of pairs produced, with respect to the laser intensity parameter I , for two counter-streaming, circularly polarized pulses, the particle being initialized at z = .
1, close to the B = z =
0. The curves are fits with rational functions. onte Carlo calculations of pair production in high-power laser-plasma interactions straggling . Our main result is that straggling increases thenumber of pairs produced for given initial conditions, and for a given laser intensity. Thisbehaviour is more pronounced for laser intensities less than 0 . × Wcm − where thetransition probabilities are strongly dependent on η and χ ; in this range we observe up to afivefold increase in the number of pairs produced. Straggling e ff ects are less important athigher intensities, where most gamma-rays are above threshold for pair production and thenumber of electron-positron pairs produced is large.Two e ff ects contribute to the increase in pair production due to straggling. Firstly, thevalue of the electron’s η -parameter, which controls the energy of the emitted gamma-ray,varies considerably about its mean. Occasional excursions to large η result in the emissionof higher energy gamma-rays with a correspondingly larger probability of producing pairs.Secondly, straggling increases, on average, the time taken for electrons to congregate in aregion where few pairs can be created (a node with E = Acknowledgments
We thank the UK Engineering and Physical Sciences Research Council for support undergrant number EP / G055165 /
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FermiSymposium, Washington D.C., Nov. 2-5 onte Carlo calculations of pair production in high-power laser-plasma interactions Appendix A. Estimation for the energy transferred to the laser field by a single particle
From the momentum change expression at the photon emission time p (f) = p − ~ k , we estimatethe error, in the Monte Carlo algorithm, related to the energy transferred to the laser field bya single particle. The expression for the relative error reads ∆ γγ ≡ γ (cid:12)(cid:12)(cid:12) γ (f) − γ + h ν/ ( m e c ) (cid:12)(cid:12)(cid:12) In the limit where the 1 /γ and 1 /γ (f) parameters are small, it can be approximated as ∆ γγ ≃
12 1 γ γ (f) − γ ! , which is of the order 1 / ( γγ (f)(f)