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Production of electron–positron pairs by intense laser pulses in an overdense plasma
Physics Letters A 360 (2007) 624–628 www.elsevier.com/locate/pla
Production of electron–positron pairs by intense laser pulses in an overdense plasma V.I. Berezhiani a,b , D.P. Garuchava b , P.K. Shukla c,∗ a Graduate School of Frontier Sciences, University of Tokyo, 5-1-5-Kashivanoha, Kashiwa-shi, Chiba 277-8561, Japan b Institute of Physics, Georgian Academy of Sciences, 6 Tamarashvili, 0177 Tbilisi, Georgia c Institut für Theoretische Physik IV and Centre for Plasma Science and Astrophysics, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum,
D-44780 Bochum, Germany Received 8 August 2006; accepted 16 August 2006 Available online 28 August 2006 Communicated by V.M. Agranovich
Abstract We consider the production of electron–positron (e–p) pairs by circularly polarized intense laser pulses impinging onto a slightly overdense electron–ion plasma. For this purpose, we numerically solve one-dimensional system of Maxwell and relativistic two fluid equations. The production of e–p pairs is possible due to the regular motion of relativistic electrons in the field of plasma ions (trident process). The highest rate of e–p pair production is observed at the early stage of laser–plasma interactions when the ion plasma density modification is small. However, effects related to the ion motion reduces the efficiency of the pair creation. Furthermore, it is found that, in contrast to the case of a strongly overdense plasma, in a slightly over-critical plasma double-sided illumination of a thin plasma target by intense laser beams does not lead to an effective creation of e–p pairs. © 2006 Elsevier B.V. All rights reserved. PACS: 52.38.-r; 52.35.Mw; 52.27.Ny
Compact terra/petawatt sub-picosecond pulsed high-intensity laser systems have now been developed in a number of laboratories worldwide [1]. By using modern focusing devices, it is now possible to focus laser beams at a very small spatial region with focal-spot dimensions that are order of diffraction limit. The focal intensities beyond I = 1020 W/cm2 [2] can now be achieved and the promise of even higher intensity (exceeding I = 1024 W/cm2 ) pulses [1,3,4]. The nonlinear response of electromagnetically active media could then be drastically modified in the presence of intense fields of such laser pulses. In particular, the electron quiver energy in such high laser fields is fully relativistic, and leads to the generation of high brightness beams of energetic gamma rays, protons, neutrons and heavy ions. An interesting by-product of the interaction of ultrarelativistic laser pulses with a plasma could be an intensive
* Corresponding author.
E-mail address: [email protected] (P.K. Shukla). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.08.048
creation of e–p pairs. It is believed that very high intensity laser beams can produce relativistic super-thermal electrons. If the jitter energy of the electrons is larger than 2me c2 , collisions of the electrons with ions of charge Z in the plasma will produce e–p pairs via Bremsstrahlung photons, or by colliding with a nucleus (the trident process) [5,8]. A significant fraction of super-thermal electrons and newly produced pairs could be confined and re-accelerated to relativistic energies. In Ref. [5] (see also [6]) it has been suggested that this can be realized by using double-sided laser illumination so that super-thermal electrons are confined by the laser ponderomotive pressure in the front and back and by strong magnetic fields on sides. Moreover, the double-sided illumination of the target could lead to plasma densities much higher than solid density. Since the pair creation rate is proportional to the plasma density, the expected number of created pairs will be increased significantly. The above described scheme of e–p pair creation is efficient in a highly overdense plasma. However, due to relativistic selfinduced transparency in a slightly overdense plasma, part of the
V.I. Berezhiani et al. / Physics Letters A 360 (2007) 624–628
incident laser radiation penetrates certain depth far exceeding the plasma skin depth [9]. As a consequence, the efficiency of the plasma compression is reduced. On the other hand, when a circularly polarized laser pulse is normally incident onto an overdense plasma the efficiency of super-thermal electron production is reduced considerably. Indeed, the laser field has no longitudinal component across the vacuum–plasma boundary while the ponderomotive force has no high-frequency component. However, in this case, which is the case of our main interest here, the production of e–p pairs is possible by the regular motion of relativistic electrons that are directly driven by the laser field. This case has been investigated in the past, see for instance Refs. [10–12]. In these publications, it was shown that a 2 laser pulse with intensity larger than Ith = 2 × 1019 λ−2 0 W/cm (where the laser wavelength is measured in micrometers) is capable of accelerating the plasma electrons to relativistic speeds with γe 3, where γe = (1 + pe2 /m2e c2 )1/2 is the relativistic gamma factor of the electrons oscillatory motion, and as a consequence it can readily produce e–p pairs by the trident process. There are estimates for the number of e–p pairs created when a laser pulse impinges on either an overdense ([11]) or an underdense plasma ([12]). These estimates, however, are rather crude because a detailed dynamics of laser–plasma interactions has been ignored. For instance, the reflected radiation as well as the strong plasma density modifications (in the high field regions) were not taken into account. These effects could severely suppress the number of e–p pairs that is produced via the trident process. On the other hand, there could be considerable enhancement in pair production because of the long-time confinement of relativistic strong radiation in an overdense plasma region. Indeed, the authors of Ref. [13] have presented a comprehensive investigation of nonlinear interactions of intense laser pulses with an overdense electron plasma, by numerically solving the system of Maxwell and collisionless relativistic hydrodynamic equations. For ultra-relativistic pulses, it has been shown that under certain conditions a major part of the penetrated laser energy is trapped in a nonstationary layer near the plasma boundary for a long time. The plasma electron distribution consists of a sequence of over-critical density spikes separated by deep density wells where a part of the electromagnetic (EM) radiation is trapped. An additional consequences of the long-time confinement of relativistic strong radiation in an overdense plasma region is that the intensive pair production, driven by the motion of plasma electrons, can take place. However, if the characteristic time of the radiation field confinement exceeds the characteristic time of the ion motion, then for an adequate description of the pair production process one should account for ion motion as well [14]. In this Letter, we consider the e–p pair production by a circularly polarized laser pulse impinging onto a slightly overdense electron–ion plasma, by numerically solving the system of Maxwell and relativistic two fluid equations in one-dimensional geometry. Neglecting the motion of the newly created e–p pairs, the rate equation governing the pair production reads [15] ∂n+ = σT ne ni vrel , ∂t
(1)
625
where n+ is the density of created e–p pairs, ne and ni are the electron and ion number densities, respectively, and σT is the total cross-section of the trident e–p pair production process [16]; vrel = (ve − vi )2 − (ve × vi )2 , where ve,i are 2 )1/2 . the velocities of electrons and ions |ve,i | = c(1 − 1/γe,i Here we remark that unless the laser intensity is not extremely strong (I ≈ 1024 W/cm2 ) the electron velocity acquired in the laser field is much larger than the ion velocity, ve vi (≈ ve (γe me /mi )). In what follows, the intensity of laser radiation is assumed to be I < 1021 W/cm2 and consequently the terms related to the ion velocity can be neglected (i.e. vrel = |ve |). For the total cross-section of the trident process, we use approximate formulas first time suggested in Ref. [12] (see also Refs. [7,8]). Namely, for γe < 14 we apply the formula established in Ref. [11] σT = 9.6 × 10−4 (Zr0 /137)2 (γe − 3)3.6 ,
(2)
while for larger γe , we use the expression [16] σT = (28/27π)(Zr0 /137)2 (ln γe )3 ,
(3)
where r0 = 2.8 × 10−13 cm is the classical electron radius and Z is the ion nuclear charge. Since the pair creation requires the electron relativistic factor γe > 3, it could happen only in regions where the EM field maxima are localized. In the region of the laser pulse penetration in an overdense plasma, wherever the field intensity is high, the electron plasma density will be obviously reduced considerably due to the action of the ponderomotive force of the laser field. In the relevant region for pair production there are two opposing tendencies at play: the cross-section for the trident process increases with γe favoring pair production, and the density decrease in the field localization region suppresses the process. Moreover, after the characteristic time ti ≈ ωi−1 (where ωi is the ion plasma frequency) the ions begin to move due to the action of the charge separation field, which causes strong modification of ion and electron density profiles. The net result will be necessarily detail dependent and to arrive at a believable estimate, we solve Eq. (1) along with the system of Maxwell and twofluid relativistic equations. Since the density of created pairs is usually considerably smaller than the plasma density, we can safely neglect the contribution of the newly created e–p pairs on the overall dynamics. The electron and ion thermal velocities are assumed to be much smaller than their quiver velocities and the plasma can be treated as a cold electron and ion fluids. The Maxwell equations will be written in terms of the vector and scalar potentials, viz. A and φ. The final system for the one-dimensional problem pertinent to a laser pulse normally incident onto an overdense plasma (semi-infinite or a finite thickness layer) reads ne n i Z 2 ∂ 2 A⊥ ∂ 2 A⊥ − + + A⊥ = 0, (4) γe γi ∂t 2 ∂z2 ∂pe,i ∂φ ∂γe,i (5) + = qe,i , ∂t ∂z ∂z
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∂ne,i ∂ ne,i pe,i = 0, + ∂t ∂z γe,i
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(6)
and ∂ 2φ = n e − ni . ∂z2
(7)
In obtaining Eqs. (4)–(7) we have assumed that the laser pulse propagates in the z direction and all dynamical variables are independent of x and y (∂x = ∂y = 0). The following dimensionless quantities are introduced now: ne,i = ne,i /n0 , z = (ωe /c)z, t = ωe t , φ = (e/me c2 )φ, A⊥ = (e/me c2 )A⊥ . The z component of the electron and ion momenta is normalized as pe,i = pe,i /me c and the relativistic factors for the electrons and ions are, respectively, γe = (1 + A2⊥ + (pe )2 )1/2 and γi = (δ 2 + Z 2 A2⊥ + (pi )2 )1/2 ; δ = mi /me is the ratio of ion and electron mass; qe = −1 and qi = Z are the electron and ion charges. The principal results of this Letter are obtained from a numerical integration of Eqs. (1)–(7) for a circularly polarized laser pulse. For the incident laser pulse (the vacuum solution of Eq. (4)), we choose the form A⊥ (z, t) = A0 exp −0.5(z − z0 − t)β /azβ cos[ω0 (z − z0 − t)] × (8) , sin[ω0 (z − z0 − t)] where A0 is a measure of the pulse amplitude, az is the characteristic width of the Gaussian envelope of the pulse and ω0 is the frequency. In dimensional units, the pulse amplitude is related to the peak intensity (I ) and the vacuum wavelength (λ0 = 2πc/ω0 ) by the relation I [W/cm2 ]λ20 [µm] = 2.74 × 1018 A20 [17]. The equilibrium plasma density is modeled by the function ne = ni = (n0 /2)[1 + tanh(z/zw )], where n0 is the density at the plasma pedestal, and zw is the characteristic width of the slope. To solve the system of equations, a second order finite difference algorithm has been applied. Details of the numerical method can be found in Ref. [13]. Accuracy of the numerical scheme is controlled by the integrals of motion. We insure the conservations of the energy
2 ∂A⊥ 2 ∂A⊥ 2 ∂φ 1 dz + + E= 2 ∂t ∂z ∂z + 2ne (γe − 1) + 2ni (γi − δ) , (9) and the total momentum and demand charge neutrality (at each step). We solve Eqs. (1)–(7) when an ultra-relativistic laser pulse impinges on a semi-infinite overdense deuterium (δ = 3680, Z = 1) plasma. The parameters of incident laser pulse (8) at t = 0 are taken to be A0 = 5, az = 126, β = 4, z0 = −300. The frequency of radiation is ω0 = 0.8. For the laser pulse with the vacuum wavelength λ0 = 1 µm, these parameters imply that in physical units the peak laser intensity I 7 × 1019 W/cm2 and the pulse duration (full width at half maximum) Tp 100 fs, respectively. The plasma den-
Fig. 1. The transverse field |A⊥ | (red line), the plasma electron density n (black line) and ion (green line) spatial profiles are shown at t = 200. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
sity is n0 ≈ 1.8 × 1021 cm−3 . To better appreciate the results of forthcoming simulations, we remark that 100 units of dimensionless t and z correspond, respectively, to 42 fs temporal and 13 µm spatial intervals. In Fig. 1 the spatio-temporal dynamics of the process is demonstrated when there exists a sharp density slope at the plasma boundary zw = 0.3 (= 0.04λ0 ). One sees that in the first stage of nonlinear interactions (see Fig. 1) the strong exertion of the radiation pressure causes a pile up of the electrons in front of the pulse creating a strongly over-critical but a thin plasma “wall”. A part of the pulse pushes its way past the first wall, creating deep electron density cavity and begins creating (in front of it) another plasma wall. This process goes on until the laser energy penetrates to a certain depth. The plasma electron distribution consists of a sequence of over-critical density spikes separated by deep density wells. Since initially penetration occurs on a smooth ion background, in the region of depleted electrons strong electrostatic fields due to the charge separation are produced. After certain time (at t ≈ 400), the ions start to produce an inhomogeneity in the form of an ion layers. However, in the next stage of interactions these layers exhibit complex (stochastic) oscillatory motion. A detailed study of the dynamics of this process is beyond the scope of this Letter. Near the plasma boundary, the fraction of electrons are ejected backward. With the increase of the laser pulse intensity, augmented by radiation reflected from deeper plasma layers, the density of ejected electrons increases. Strong charge separation field of the electrons drags a fraction of plasma ions. After certain time, these ions localize around the ejected electron layer. Thus, the formation of backward moving plasma bunch take place. This process can be seen in Fig. 2(a), (b) where the electron and the ion density distributions are exhibited in the (t, z) plane. A contour plot of laser field strength (|A⊥ |) is plotted in Fig. 3. A bulk part of the laser energy (up to 80% of the incident pulse energy) is reflected back from the sharp vacuum plasma boundary at the first stage of interactions (0 < t < 550), while the rest part of radiation is partially locked in a “cavitating” space between the bunch and the bulk part of the plasma during the entire interval (0 < t < 2100) (see Fig. 3). This field pushes the plasma bunch leading to the expansion of
V.I. Berezhiani et al. / Physics Letters A 360 (2007) 624–628
627
Fig. 4. The pair density n+ versus z at t = 2100.
Fig. 2. The t –z diagram of the plasma electron (a) and ion (b) density distributions.
Fig. 3. The t –z diagram of |A⊥ | shows the trapping of the part of the laser pulse in an expanding cavitation region.
the cavitating region. The bunch quickly obtains the high velocity and after a certain transient time its motion becomes almost ballistic. Similar effect has been observed in Ref. [18] for the
Fig. 5. The accumulated-total number of pairs per area N + versus t for (a) deuterium plasma; (b) hydrogen plasma; (c) double-sided illuminated deuterium plasma slab.
electron–positron plasma. In our simulations, the ion density in the depleted regions continuously reduces and at t = 2100 (which corresponds to t ≈ 1 ps) is dropped to nmin ≈ 0.05n0 (which, in dimensions, means nmin ≈ 9 × 1019 cm−3 ), implying that the complete cavitation (i.e. nmin = 0) of plasma does not take place. Though in this very region the relativistically strong radiation is trapped, the plasma ion density in this cavitating region is reduced by two orders of magnitude. The plasma electrons in this regain acquire the relativistic factor that exceeds the pair creation threshold, and as a consequence the pair creation takes place even in initial “vacuum” region (z < 0). However, dominant region where the pair creation takes place is the thin layer near the plasma boundary 0 < z < 150 (in dimension, the corresponding width of this layer is 20 µm). This can be seen in Fig. 4, where the pair density n+ is plotted versus z at t = 2100. The maximum of the e–p pair density is n+ ≈ 1.3 × 1011 cm−3 . The total number of pairs per area N + = n+ dz versus t is displayed in Fig. 5(a). One observes that N + ≈ 5 × 107 pairs are created within one picosecond. The highest rate of the e–p pair creation is observed at the early stage of interactions (0 < t < 400) when the ion plasma density modification
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Fig. 6. The t –z diagram of the plasma density distribution for the plasma layer of thickness zs = 100.
is small. At the later stage, due to a strong ion density modification, the pair generation rate reduces nearly 10 times. Thus, effects related to the ion motion reduces efficiency of pair creation. In Fig. 5(b) we plot the results of simulations conducted for the case of a hydrogen (δ = 1840, Z = 1) plasma, while all other parameters of the plasma and the laser pulse remain same. One sees that the number of generated e–p pairs as well as the rate of pair creation are reduced, which is obviously related to the fact that for the lighter ion case the laser pressure gives rise to a stronger ion density modification than for the heavy ion case. It is interesting to test now what happens during doublesided illumination of a plasma of finite thickness. One could expect that for a slightly overdense plasma slab with the thickness shorter the laser pulse penetration depth, the double-sided illumination could lead to even stronger plasma density modification as compared to a single pulse shot. As a consequence, the efficiency of pair creation will be reduced. Indeed in Fig. 6 we present the simulation results for a finite plasma layer of thickness zs = 100 (zs = 13 µm) symmetrically embedded in the field of counter-propagating identical laser pulses. All other parameters are assumed to be same as in Fig. 2. We see that strong plasma density modification in the form of multi-layer cavitating structures takes place. In Fig. 5 the curve (c) corresponds to the number of created pairs during this process. Furthermore, one sees that after initial stage of interactions (when t > 500) the creation of e–p pairs does not take place.
The total number of e–p pairs is reduced few times as compared to the one sided illumination case. Thus, in contrast to the case of a strongly overdense plasma, in the slightly over-critical plasma the double-sided illumination of thin plasma target does not lead to an effective creation of e–p pairs. To summarize, we have investigated the electron–positron pair creation by the driven motion of plasma electrons in the field of relativistic strong laser pulses impinging onto a slightly overdense plasma. By solving numerically the full 1D-system of Maxwell and collisionless relativistic hydrodynamic equations of our electron–ion plasma, it is shown that for the circularly polarized laser pulses the intensive pair production takes place due to the trident process. Efficiency of the pair creation is increased for heavier ion plasma cases. It is also shown that in a slightly over-critical thin plasma slab its double-sided illumination leads to the strong plasma density modification, which reduces the efficiency of the pair production. Acknowledgements The authors thank Dr. S. Mikeladze for valuable discussions. This work was partially supported by ISTC grant G1366. References [1] G.A. Mourou, T. Tajima, S.V. Bulanov, Rev. Mod. Phys. 78 (2006) 309. [2] S.P. Hatchett, et al., Phys. Plasmas 7 (2000) 2076; M.H. Key, et al., Phys. Plasmas 5 (1998) 1966. [3] D. Umstadter, J. Phys. D: Appl. Phys. 36 (2003) R151. [4] M. Marklund, P.K. Shukla, Rev. Mod. Phys. 78 (2) (2006) 591. [5] E.P. Liang, S.C. Wilks, M. Tabak, Phys. Rev. Lett. 81 (1998) 4887. [6] B. Shen, J. Meyer-ter-Vehn, Phys. Rev. E 65 (2001) 016405. [7] C. Gahn, et al., Phys. Plasmas 9 (2002) 987. [8] D.A. Gryaznykh, Y.Z. Kandiev, V.A. Lykov, JETP Lett. 4 (1998) 257. [9] P.K. Kaw, J.M. Dawson, Phys. Fluids 13 (1970) 472. [10] F.V. Bunkin, A.E. Kazakov, Dokl. Akad. Nauk SSSR 193 (1970) 1274, Sov. Phys. Dokl. 15 (1971) 758. [11] J.W. Shearer, et al., Phys. Rev. A 8 (1973) 1582. [12] V.I. Berezhiani, D.D. Tskhakaya, P.K. Shukla, Phys. Rev. A 46 (1992) 6608. [13] V.I. Berezhiani, et al., Phys. Plasmas 12 (2005) 062308. [14] M. Tushentsov, et al., Phys. Rev. Lett. 87 (2001) 275002. [15] L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields, Pergamon/ Addison–Wesley, Oxford/Reading, 1987. [16] W. Heitler, The Quantum Theory of Radiation, Clarendon, Oxford, 1954. [17] E. Esarey, et al., IEEE Trans. Plasma Sci. 24 (1996) 252. [18] M. Ashour-Abdalla, et al., Phys. Rev. A 23 (1981) 1906.
Production of electron–positron pairs by intense laser pulses in an overdense plasma V.I. Berezhiani a,b , D.P. Garuchava b , P.K. Shukla c,∗ a Graduate School of Frontier Sciences, University of Tokyo, 5-1-5-Kashivanoha, Kashiwa-shi, Chiba 277-8561, Japan b Institute of Physics, Georgian Academy of Sciences, 6 Tamarashvili, 0177 Tbilisi, Georgia c Institut für Theoretische Physik IV and Centre for Plasma Science and Astrophysics, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum,
D-44780 Bochum, Germany Received 8 August 2006; accepted 16 August 2006 Available online 28 August 2006 Communicated by V.M. Agranovich
Abstract We consider the production of electron–positron (e–p) pairs by circularly polarized intense laser pulses impinging onto a slightly overdense electron–ion plasma. For this purpose, we numerically solve one-dimensional system of Maxwell and relativistic two fluid equations. The production of e–p pairs is possible due to the regular motion of relativistic electrons in the field of plasma ions (trident process). The highest rate of e–p pair production is observed at the early stage of laser–plasma interactions when the ion plasma density modification is small. However, effects related to the ion motion reduces the efficiency of the pair creation. Furthermore, it is found that, in contrast to the case of a strongly overdense plasma, in a slightly over-critical plasma double-sided illumination of a thin plasma target by intense laser beams does not lead to an effective creation of e–p pairs. © 2006 Elsevier B.V. All rights reserved. PACS: 52.38.-r; 52.35.Mw; 52.27.Ny
Compact terra/petawatt sub-picosecond pulsed high-intensity laser systems have now been developed in a number of laboratories worldwide [1]. By using modern focusing devices, it is now possible to focus laser beams at a very small spatial region with focal-spot dimensions that are order of diffraction limit. The focal intensities beyond I = 1020 W/cm2 [2] can now be achieved and the promise of even higher intensity (exceeding I = 1024 W/cm2 ) pulses [1,3,4]. The nonlinear response of electromagnetically active media could then be drastically modified in the presence of intense fields of such laser pulses. In particular, the electron quiver energy in such high laser fields is fully relativistic, and leads to the generation of high brightness beams of energetic gamma rays, protons, neutrons and heavy ions. An interesting by-product of the interaction of ultrarelativistic laser pulses with a plasma could be an intensive
* Corresponding author.
E-mail address: [email protected] (P.K. Shukla). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.08.048
creation of e–p pairs. It is believed that very high intensity laser beams can produce relativistic super-thermal electrons. If the jitter energy of the electrons is larger than 2me c2 , collisions of the electrons with ions of charge Z in the plasma will produce e–p pairs via Bremsstrahlung photons, or by colliding with a nucleus (the trident process) [5,8]. A significant fraction of super-thermal electrons and newly produced pairs could be confined and re-accelerated to relativistic energies. In Ref. [5] (see also [6]) it has been suggested that this can be realized by using double-sided laser illumination so that super-thermal electrons are confined by the laser ponderomotive pressure in the front and back and by strong magnetic fields on sides. Moreover, the double-sided illumination of the target could lead to plasma densities much higher than solid density. Since the pair creation rate is proportional to the plasma density, the expected number of created pairs will be increased significantly. The above described scheme of e–p pair creation is efficient in a highly overdense plasma. However, due to relativistic selfinduced transparency in a slightly overdense plasma, part of the
V.I. Berezhiani et al. / Physics Letters A 360 (2007) 624–628
incident laser radiation penetrates certain depth far exceeding the plasma skin depth [9]. As a consequence, the efficiency of the plasma compression is reduced. On the other hand, when a circularly polarized laser pulse is normally incident onto an overdense plasma the efficiency of super-thermal electron production is reduced considerably. Indeed, the laser field has no longitudinal component across the vacuum–plasma boundary while the ponderomotive force has no high-frequency component. However, in this case, which is the case of our main interest here, the production of e–p pairs is possible by the regular motion of relativistic electrons that are directly driven by the laser field. This case has been investigated in the past, see for instance Refs. [10–12]. In these publications, it was shown that a 2 laser pulse with intensity larger than Ith = 2 × 1019 λ−2 0 W/cm (where the laser wavelength is measured in micrometers) is capable of accelerating the plasma electrons to relativistic speeds with γe 3, where γe = (1 + pe2 /m2e c2 )1/2 is the relativistic gamma factor of the electrons oscillatory motion, and as a consequence it can readily produce e–p pairs by the trident process. There are estimates for the number of e–p pairs created when a laser pulse impinges on either an overdense ([11]) or an underdense plasma ([12]). These estimates, however, are rather crude because a detailed dynamics of laser–plasma interactions has been ignored. For instance, the reflected radiation as well as the strong plasma density modifications (in the high field regions) were not taken into account. These effects could severely suppress the number of e–p pairs that is produced via the trident process. On the other hand, there could be considerable enhancement in pair production because of the long-time confinement of relativistic strong radiation in an overdense plasma region. Indeed, the authors of Ref. [13] have presented a comprehensive investigation of nonlinear interactions of intense laser pulses with an overdense electron plasma, by numerically solving the system of Maxwell and collisionless relativistic hydrodynamic equations. For ultra-relativistic pulses, it has been shown that under certain conditions a major part of the penetrated laser energy is trapped in a nonstationary layer near the plasma boundary for a long time. The plasma electron distribution consists of a sequence of over-critical density spikes separated by deep density wells where a part of the electromagnetic (EM) radiation is trapped. An additional consequences of the long-time confinement of relativistic strong radiation in an overdense plasma region is that the intensive pair production, driven by the motion of plasma electrons, can take place. However, if the characteristic time of the radiation field confinement exceeds the characteristic time of the ion motion, then for an adequate description of the pair production process one should account for ion motion as well [14]. In this Letter, we consider the e–p pair production by a circularly polarized laser pulse impinging onto a slightly overdense electron–ion plasma, by numerically solving the system of Maxwell and relativistic two fluid equations in one-dimensional geometry. Neglecting the motion of the newly created e–p pairs, the rate equation governing the pair production reads [15] ∂n+ = σT ne ni vrel , ∂t
(1)
625
where n+ is the density of created e–p pairs, ne and ni are the electron and ion number densities, respectively, and σT is the total cross-section of the trident e–p pair production process [16]; vrel = (ve − vi )2 − (ve × vi )2 , where ve,i are 2 )1/2 . the velocities of electrons and ions |ve,i | = c(1 − 1/γe,i Here we remark that unless the laser intensity is not extremely strong (I ≈ 1024 W/cm2 ) the electron velocity acquired in the laser field is much larger than the ion velocity, ve vi (≈ ve (γe me /mi )). In what follows, the intensity of laser radiation is assumed to be I < 1021 W/cm2 and consequently the terms related to the ion velocity can be neglected (i.e. vrel = |ve |). For the total cross-section of the trident process, we use approximate formulas first time suggested in Ref. [12] (see also Refs. [7,8]). Namely, for γe < 14 we apply the formula established in Ref. [11] σT = 9.6 × 10−4 (Zr0 /137)2 (γe − 3)3.6 ,
(2)
while for larger γe , we use the expression [16] σT = (28/27π)(Zr0 /137)2 (ln γe )3 ,
(3)
where r0 = 2.8 × 10−13 cm is the classical electron radius and Z is the ion nuclear charge. Since the pair creation requires the electron relativistic factor γe > 3, it could happen only in regions where the EM field maxima are localized. In the region of the laser pulse penetration in an overdense plasma, wherever the field intensity is high, the electron plasma density will be obviously reduced considerably due to the action of the ponderomotive force of the laser field. In the relevant region for pair production there are two opposing tendencies at play: the cross-section for the trident process increases with γe favoring pair production, and the density decrease in the field localization region suppresses the process. Moreover, after the characteristic time ti ≈ ωi−1 (where ωi is the ion plasma frequency) the ions begin to move due to the action of the charge separation field, which causes strong modification of ion and electron density profiles. The net result will be necessarily detail dependent and to arrive at a believable estimate, we solve Eq. (1) along with the system of Maxwell and twofluid relativistic equations. Since the density of created pairs is usually considerably smaller than the plasma density, we can safely neglect the contribution of the newly created e–p pairs on the overall dynamics. The electron and ion thermal velocities are assumed to be much smaller than their quiver velocities and the plasma can be treated as a cold electron and ion fluids. The Maxwell equations will be written in terms of the vector and scalar potentials, viz. A and φ. The final system for the one-dimensional problem pertinent to a laser pulse normally incident onto an overdense plasma (semi-infinite or a finite thickness layer) reads ne n i Z 2 ∂ 2 A⊥ ∂ 2 A⊥ − + + A⊥ = 0, (4) γe γi ∂t 2 ∂z2 ∂pe,i ∂φ ∂γe,i (5) + = qe,i , ∂t ∂z ∂z
626
∂ne,i ∂ ne,i pe,i = 0, + ∂t ∂z γe,i
V.I. Berezhiani et al. / Physics Letters A 360 (2007) 624–628
(6)
and ∂ 2φ = n e − ni . ∂z2
(7)
In obtaining Eqs. (4)–(7) we have assumed that the laser pulse propagates in the z direction and all dynamical variables are independent of x and y (∂x = ∂y = 0). The following dimensionless quantities are introduced now: ne,i = ne,i /n0 , z = (ωe /c)z, t = ωe t , φ = (e/me c2 )φ, A⊥ = (e/me c2 )A⊥ . The z component of the electron and ion momenta is normalized as pe,i = pe,i /me c and the relativistic factors for the electrons and ions are, respectively, γe = (1 + A2⊥ + (pe )2 )1/2 and γi = (δ 2 + Z 2 A2⊥ + (pi )2 )1/2 ; δ = mi /me is the ratio of ion and electron mass; qe = −1 and qi = Z are the electron and ion charges. The principal results of this Letter are obtained from a numerical integration of Eqs. (1)–(7) for a circularly polarized laser pulse. For the incident laser pulse (the vacuum solution of Eq. (4)), we choose the form A⊥ (z, t) = A0 exp −0.5(z − z0 − t)β /azβ cos[ω0 (z − z0 − t)] × (8) , sin[ω0 (z − z0 − t)] where A0 is a measure of the pulse amplitude, az is the characteristic width of the Gaussian envelope of the pulse and ω0 is the frequency. In dimensional units, the pulse amplitude is related to the peak intensity (I ) and the vacuum wavelength (λ0 = 2πc/ω0 ) by the relation I [W/cm2 ]λ20 [µm] = 2.74 × 1018 A20 [17]. The equilibrium plasma density is modeled by the function ne = ni = (n0 /2)[1 + tanh(z/zw )], where n0 is the density at the plasma pedestal, and zw is the characteristic width of the slope. To solve the system of equations, a second order finite difference algorithm has been applied. Details of the numerical method can be found in Ref. [13]. Accuracy of the numerical scheme is controlled by the integrals of motion. We insure the conservations of the energy
2 ∂A⊥ 2 ∂A⊥ 2 ∂φ 1 dz + + E= 2 ∂t ∂z ∂z + 2ne (γe − 1) + 2ni (γi − δ) , (9) and the total momentum and demand charge neutrality (at each step). We solve Eqs. (1)–(7) when an ultra-relativistic laser pulse impinges on a semi-infinite overdense deuterium (δ = 3680, Z = 1) plasma. The parameters of incident laser pulse (8) at t = 0 are taken to be A0 = 5, az = 126, β = 4, z0 = −300. The frequency of radiation is ω0 = 0.8. For the laser pulse with the vacuum wavelength λ0 = 1 µm, these parameters imply that in physical units the peak laser intensity I 7 × 1019 W/cm2 and the pulse duration (full width at half maximum) Tp 100 fs, respectively. The plasma den-
Fig. 1. The transverse field |A⊥ | (red line), the plasma electron density n (black line) and ion (green line) spatial profiles are shown at t = 200. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
sity is n0 ≈ 1.8 × 1021 cm−3 . To better appreciate the results of forthcoming simulations, we remark that 100 units of dimensionless t and z correspond, respectively, to 42 fs temporal and 13 µm spatial intervals. In Fig. 1 the spatio-temporal dynamics of the process is demonstrated when there exists a sharp density slope at the plasma boundary zw = 0.3 (= 0.04λ0 ). One sees that in the first stage of nonlinear interactions (see Fig. 1) the strong exertion of the radiation pressure causes a pile up of the electrons in front of the pulse creating a strongly over-critical but a thin plasma “wall”. A part of the pulse pushes its way past the first wall, creating deep electron density cavity and begins creating (in front of it) another plasma wall. This process goes on until the laser energy penetrates to a certain depth. The plasma electron distribution consists of a sequence of over-critical density spikes separated by deep density wells. Since initially penetration occurs on a smooth ion background, in the region of depleted electrons strong electrostatic fields due to the charge separation are produced. After certain time (at t ≈ 400), the ions start to produce an inhomogeneity in the form of an ion layers. However, in the next stage of interactions these layers exhibit complex (stochastic) oscillatory motion. A detailed study of the dynamics of this process is beyond the scope of this Letter. Near the plasma boundary, the fraction of electrons are ejected backward. With the increase of the laser pulse intensity, augmented by radiation reflected from deeper plasma layers, the density of ejected electrons increases. Strong charge separation field of the electrons drags a fraction of plasma ions. After certain time, these ions localize around the ejected electron layer. Thus, the formation of backward moving plasma bunch take place. This process can be seen in Fig. 2(a), (b) where the electron and the ion density distributions are exhibited in the (t, z) plane. A contour plot of laser field strength (|A⊥ |) is plotted in Fig. 3. A bulk part of the laser energy (up to 80% of the incident pulse energy) is reflected back from the sharp vacuum plasma boundary at the first stage of interactions (0 < t < 550), while the rest part of radiation is partially locked in a “cavitating” space between the bunch and the bulk part of the plasma during the entire interval (0 < t < 2100) (see Fig. 3). This field pushes the plasma bunch leading to the expansion of
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Fig. 4. The pair density n+ versus z at t = 2100.
Fig. 2. The t –z diagram of the plasma electron (a) and ion (b) density distributions.
Fig. 3. The t –z diagram of |A⊥ | shows the trapping of the part of the laser pulse in an expanding cavitation region.
the cavitating region. The bunch quickly obtains the high velocity and after a certain transient time its motion becomes almost ballistic. Similar effect has been observed in Ref. [18] for the
Fig. 5. The accumulated-total number of pairs per area N + versus t for (a) deuterium plasma; (b) hydrogen plasma; (c) double-sided illuminated deuterium plasma slab.
electron–positron plasma. In our simulations, the ion density in the depleted regions continuously reduces and at t = 2100 (which corresponds to t ≈ 1 ps) is dropped to nmin ≈ 0.05n0 (which, in dimensions, means nmin ≈ 9 × 1019 cm−3 ), implying that the complete cavitation (i.e. nmin = 0) of plasma does not take place. Though in this very region the relativistically strong radiation is trapped, the plasma ion density in this cavitating region is reduced by two orders of magnitude. The plasma electrons in this regain acquire the relativistic factor that exceeds the pair creation threshold, and as a consequence the pair creation takes place even in initial “vacuum” region (z < 0). However, dominant region where the pair creation takes place is the thin layer near the plasma boundary 0 < z < 150 (in dimension, the corresponding width of this layer is 20 µm). This can be seen in Fig. 4, where the pair density n+ is plotted versus z at t = 2100. The maximum of the e–p pair density is n+ ≈ 1.3 × 1011 cm−3 . The total number of pairs per area N + = n+ dz versus t is displayed in Fig. 5(a). One observes that N + ≈ 5 × 107 pairs are created within one picosecond. The highest rate of the e–p pair creation is observed at the early stage of interactions (0 < t < 400) when the ion plasma density modification
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Fig. 6. The t –z diagram of the plasma density distribution for the plasma layer of thickness zs = 100.
is small. At the later stage, due to a strong ion density modification, the pair generation rate reduces nearly 10 times. Thus, effects related to the ion motion reduces efficiency of pair creation. In Fig. 5(b) we plot the results of simulations conducted for the case of a hydrogen (δ = 1840, Z = 1) plasma, while all other parameters of the plasma and the laser pulse remain same. One sees that the number of generated e–p pairs as well as the rate of pair creation are reduced, which is obviously related to the fact that for the lighter ion case the laser pressure gives rise to a stronger ion density modification than for the heavy ion case. It is interesting to test now what happens during doublesided illumination of a plasma of finite thickness. One could expect that for a slightly overdense plasma slab with the thickness shorter the laser pulse penetration depth, the double-sided illumination could lead to even stronger plasma density modification as compared to a single pulse shot. As a consequence, the efficiency of pair creation will be reduced. Indeed in Fig. 6 we present the simulation results for a finite plasma layer of thickness zs = 100 (zs = 13 µm) symmetrically embedded in the field of counter-propagating identical laser pulses. All other parameters are assumed to be same as in Fig. 2. We see that strong plasma density modification in the form of multi-layer cavitating structures takes place. In Fig. 5 the curve (c) corresponds to the number of created pairs during this process. Furthermore, one sees that after initial stage of interactions (when t > 500) the creation of e–p pairs does not take place.
The total number of e–p pairs is reduced few times as compared to the one sided illumination case. Thus, in contrast to the case of a strongly overdense plasma, in the slightly over-critical plasma the double-sided illumination of thin plasma target does not lead to an effective creation of e–p pairs. To summarize, we have investigated the electron–positron pair creation by the driven motion of plasma electrons in the field of relativistic strong laser pulses impinging onto a slightly overdense plasma. By solving numerically the full 1D-system of Maxwell and collisionless relativistic hydrodynamic equations of our electron–ion plasma, it is shown that for the circularly polarized laser pulses the intensive pair production takes place due to the trident process. Efficiency of the pair creation is increased for heavier ion plasma cases. It is also shown that in a slightly over-critical thin plasma slab its double-sided illumination leads to the strong plasma density modification, which reduces the efficiency of the pair production. Acknowledgements The authors thank Dr. S. Mikeladze for valuable discussions. This work was partially supported by ISTC grant G1366. References [1] G.A. Mourou, T. Tajima, S.V. Bulanov, Rev. Mod. Phys. 78 (2006) 309. [2] S.P. Hatchett, et al., Phys. Plasmas 7 (2000) 2076; M.H. Key, et al., Phys. Plasmas 5 (1998) 1966. [3] D. Umstadter, J. Phys. D: Appl. Phys. 36 (2003) R151. [4] M. Marklund, P.K. Shukla, Rev. Mod. Phys. 78 (2) (2006) 591. [5] E.P. Liang, S.C. Wilks, M. Tabak, Phys. Rev. Lett. 81 (1998) 4887. [6] B. Shen, J. Meyer-ter-Vehn, Phys. Rev. E 65 (2001) 016405. [7] C. Gahn, et al., Phys. Plasmas 9 (2002) 987. [8] D.A. Gryaznykh, Y.Z. Kandiev, V.A. Lykov, JETP Lett. 4 (1998) 257. [9] P.K. Kaw, J.M. Dawson, Phys. Fluids 13 (1970) 472. [10] F.V. Bunkin, A.E. Kazakov, Dokl. Akad. Nauk SSSR 193 (1970) 1274, Sov. Phys. Dokl. 15 (1971) 758. [11] J.W. Shearer, et al., Phys. Rev. A 8 (1973) 1582. [12] V.I. Berezhiani, D.D. Tskhakaya, P.K. Shukla, Phys. Rev. A 46 (1992) 6608. [13] V.I. Berezhiani, et al., Phys. Plasmas 12 (2005) 062308. [14] M. Tushentsov, et al., Phys. Rev. Lett. 87 (2001) 275002. [15] L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields, Pergamon/ Addison–Wesley, Oxford/Reading, 1987. [16] W. Heitler, The Quantum Theory of Radiation, Clarendon, Oxford, 1954. [17] E. Esarey, et al., IEEE Trans. Plasma Sci. 24 (1996) 252. [18] M. Ashour-Abdalla, et al., Phys. Rev. A 23 (1981) 1906.