Towards laser based improved experimental schemes for multiphoton e+ e- pair production from vacuum
aa r X i v : . [ h e p - ph ] J u l Towards laser based improved experimental schemes for multiphoton e + e − pairproduction from vacuum. I. Ploumistakis † , S. D. Moustaizis † , and I. Tsohantjis ‡ , Technical University of Crete, Department of Sciences, † Institute of Matter Structure and Laser Physics, ‡ Division of MathematicsChania GR-73100, Crete, Greece (Dated: November 1, 2018)Numerical estimates for pair production from vacuum in the presence of strong electromagneticfields are derived, for two experimental schemes : the first concerns a laser based X-FEL andthe other imitates the E144 experiment. The approximation adopted in this work is that of twolevel multiphoton on resonance. Utilizing achievable values of laser beam parameters, an enhancedproduction efficiency of up to 10 and 10 pairs can be obtained, for the two schemes respectively. PACS numbers: 12.20.-m, 42.50.Hz, 42.55.-fKeywords: Pair production, Multiphoton processes, High intensity lasers
I. INTRODUCTION
Electron-positron pair production from vacuum breakdown in the presence of strong electromagnetic fields is oneof the most interesting non-linear QED phenomena and at the same time a rich area of theoretical and experimentalinvestigations[1, 2, 3, 4, 5, 6, 7, 8]. Recently the possibility for an experimental verification of the phenomenon isplausible due to the rapid development of ultra-intense laser facilities [11, 12, 13]. The first steps on the theoreticaltreatment of the phenomenon can be found [1], but Schwinger [2] was the first that examined it thoroughly. By theimplementation of the proper time method Schwinger obtained the following conditions for pair creation to occur: The invariant quantities F = F µν F µν = − (cid:16) ~ E − c ~ B (cid:17) , G = F µν ˜ F µν = c~ E · ~ B , where F µν and ˜ F µν = ǫ µναβ F αβ are the electromagnetic field tensor and its dual respectively, must be such that neither F =0 , G =0 nor F > G =0. These restrictions are for example satisfied close to the antinodes of a standing wave or at the region ofa focused laser beam. In order to have sizable effects the electric field strength must exceed the critical value E c = mc eλ c ≃ . × V /m which corresponds to focal laser intensities of the order of 10 W/cm and above. Suchlaser intensities can be reach in the very near future from facilities such as [14], [29], [30] and XFEL [18]. Brezin andItzykson [3] and soon after Popov [6] used multiphoton atom ionization technics such as the imaginary time method[6, 9, 10] showed that the probability of pair creation over a 4- Compton volume can be obtained as w P = X n>n w n (1)With the parameter of the theoretical treatments being γ = mcω/e E = ~ ω E c /mc E , which is the equivalent of Keldyshparameter, the two important areas γ ≪ γ ≫ N ( τ ) [6] : N ( τ ) = 2 − / n ( E / E c ) exp( − π E c E (1 − (cid:16) n EE c (cid:17) ))( ωτ / π ) , γ ≪ N ( τ ) ≈ πn / (cid:18) γe (cid:19) − n ( ωτ / π ) , γ ≫ τ being the pulse duration and n = 2 mc / ~ ω . The first experimental verification of e − e + pair production tookplace at SLAC ( E-144 experiment)[18]. Highly energetic electrons (maximum 46.6GeV) underwent nonlinear Comptonscattering with n laser photons (of λ =527nm, ω = 2 . eV and laser pulses of energy 500mJ ), e + nω → e ′ + ω ′ ,producing backscattered photons ω ′ of 27 − GeV which then collide with the laser photons to produce e − , e + pairs, via the multiphoton Breit -Wheeler mechanism, ω ′ + nω → e − e + . The number of positrons measured in 21962laser pulses was 175 ±
13 for a n = 5 . ± . statistical ) +0 . − . ( systematic ) multiphoton order process, a result that is invery good agreement with the theory. These results have been shown in [20] to be obtained also from Popov’s theory.In a recent paper Avetissian et al [19] treated e − e + pair production in a standing wave A = 2 A cos ωt formed byoppositely directed laser beams of plane transverse linearly polarized electromagnetic waves of frequency ω, using atwo level on resonance multiphoton approximation [19, 20, 21]. The main difference between this approach and theprevious one is the condition of resonance, which as shown in the detailed investigation presented in [21, 22, 23, 24],can lead to very high production efficiency. The probability density of pair creation for the case of oscillating electricfield, is given in relation (33) in [19] . This approximation will be considered in the rest of the paper. Following [19, 21],taking the momentum of the created electrons(positrons) to be p = ( p x = p sin θ, p y = p cos θ, θ is anglebetween A (lying on the Oy axis) and p , the probability density w n and the number of pairs created N at angle θ = 0 (where both maximize) for an n th order process are respectively given by [21, 22, 23] w n = nω π f n (cid:0) n ω − m (cid:1) (2) N = w n V τ = 14 π V τV e q p q − m c f n (3)In deriving (2) and (3) the resonance condition n = 2 E/ ~ ω = 2 qmc / ~ ω, q ≥ f n = E p y (1 − p E ) n ~ ωJ n (4 ξ mpEω ) .ξ ∼ γ is the relativistic invariant parameter given by ξ = mc E ~ ω E c , (5) E is the amplitude of one incident wave and ξ . V e = 7 . × − m s is the4-Compton volume of the electron, V ∼ σ l is the interaction volume, τ is the pulse duration, σ is the cross sectionradius and l ≪ λ the electromagnetic wavelength [19, 21]. Finally since in the Bessel function J n ( x ), x ≃ n , theapproximation J n ( n sec hα ) = (1 / √ πn tanh α ) ( a = sech − ( ξq (cid:16) − q (cid:17) )) has to be used giving, f n = 14 (cid:0) q − (cid:1) − n ~ ω exp ( n tanh a − na ) √ πn tanh a (6)As shown in [21], relation (3) leads to very good estimates for the theory both when higher harmonics are im-plemented and when the range of its applicability is examined. For example for a Nd-Yag laser with ω = 1 . eV , λ = 1 µm , intensity 1 . × W/m , τ ∼ − s , σ ∼ − m , l = 0 . λ, the fifth harmonic can create approximatelyup to 10 pairs for n ∼ , provided that the electric field strength does not exceed the order of 10 V /m .In section two that follows we present the use of a table top laser based X-ray Free Electron Laser scheme for paircreation and in section three a scheme analogous to the one used in the E-144 experiment. The aim of this paper is toemphasize the advantages of using such schemes for the experimental observation of the on resonance approximation[19, 21], as they can be realized in well equipped laboratory and can provide satisfactory results which we will showon our numerical estimations that will follow. One should mention that the following two setups allow the increaseof electric field strength E with a simultaneous increase of ω, so that ξ remains .
1, as it is essential for the validityof the multiphoton on resonance approximation.
II. PAIR CREATION USING LASER BASED X -FEL SYSTEM
In this section we investigate the use of what one could call a table top laser based X-ray Free Electron Laser forelectron positron pair creation. Such system is similar to a standard X -FEL system [17] with the main differencebetween the two to be the electron beam creation and acceleration mechanism. The table top XFEL utilizes arelativistic electron beam produced and accelerated by a high intensity ( ∼ − W/cm ) ultrashort laser beam[25,26, 27, 28]. Then the electron beam propagates through a wiggler system and X- ray beam is created. Generation ofsuch relativistic electron beam from a laser source can be achieved by interaction with a solid or gas target and is awide field of research. For our purposes we consider a laser system with intensity of 10 W/cm and pulse duration20 f s - 30 f s. Recent experiments [25, 26, 27, 28] have confirmed that the accelerated electron beam can reach energies E ebeam close to 1 GeV and charge close to 1 nCb , but is expected that in the near future this charge will becomeavailable. This beam enters an undulator with period λ u and after the interaction with the magnetic field in it, X-rayphotons of wavelength λ = λ u γ L (7)will be produced, where γ L is the relativistic factor ( γ L = E beam m e c ). The undulator period for our scheme is λ u = 5 mm but it can be adjusted depending on the X ray photon wavelength we wish to produce.Using (7) for E ebeam = 1 GeV we obtain X-ray beam wavelength λ = 0 . nm ,which corresponds to photon energy ω = 1 . KeV . Two opposite directed X-ray beams form a standing wave to a circular spot area of radius σ = 100 nm .The interaction 4-volume in which pair creation takes place is taken as V = σ (0 . λ ) τ , where τ = 100 f s . We alsoassume that the conversion efficiency of the electron beam energy to X-ray photons is about 10% and given the powerof the electron beam P ebeam = 1 × W , we obtain the energy of the X-ray beam E b = 0 . J . As in [21],with theabove parameter values, we firstly investigate, using (3) the dependence of pair number N to q (electron rest massunits). The result is presented in Figure 1, where the envelope of N o ( q ) is plotted, for ξ = 0 . ξ (leading to high production efficiency), is dictated by energy conservation between the X-ray beam energy E b and the energy E = 2 qmc N of created pairs: one can numerically solve the energy conservation equation E b = 0 . J = 2 qmc N ( ξ, q ) for appropriate range of values of the q parameter (see [21] for a detailed analysis on thechoice of ξ ). ´ ´ ´ ´ No FIG. 1: Envelope of created e + e − pair number N o vs rest mass unit q for ξ = 0 . . keV. The most probable pair number is the one that corresponds to q = 1 .
40 and is of the order of 10 pairs. Themultiphoton order of the process as obtained by (4) is n = 750.Secondly one can also investigate the dependence of N o from ξ (equivalently E ). Using (3) and the data from Table1 we present in Figure 2, the dependence of pair number N from ξ for three E ebeam values of 1 GeV , 400
M eV and200
M eV , the last two being considered in order to estimate and compare pair creation efficiencies for energies thatcan be easily created with the recent technology in large laser facilities. Note that in order to obtain each of the threeplots in Figure 2 the values of ( q , n ) are respectively ( 1.41, 750), (1.40, 4742), (1.40,18833).Each one of the curves has a different ending point, because of previous mentioned energy conservation consideration.The choice of plotting vs. parameter ξ is justifiable, since it depends both from the electric field strength E o and photonenergy and thus allowing to investigate the dependence of N on both of these physical quantities. By adjusting thephoton energy and/or the electric field one can estimate the number of produced pairs for the corresponding value of ξ from figures like Figure 2. Observe the very sensitivity of N from ξ which is a result of resonance approximation(thisis due to the presence of the Bessel function in rel (3)). Ξ No FIG. 2: Log- plot of the number of created pairs N vs ξ for electron beam energy 1 GeV (top curve), 400
MeV (middle curve)and 200
MeV (bottom)
Electron beam Wavelength of X ray X-ray photon Maximum Electric fieldenergy E ebeam laser beam ( nm ) energy ( eV ) Strength ( V /m )1 GeV . × M eV . × M eV . × Table 1. Typical parameters for obtaining Figure 2The electron beam charges considered for these energies case are respectively 1nCb for 1GeV , 0.4 nCb for 400MeVand 0.2 nCb for 200MeV. The estimates of Figure 2 show that even if we choose to lower the electron beam energy,we can achieve efficient electron positron pair production ( of the order of 10 for 1GeV , 10 for 400 and 200 MeV)and strongly suggest the possible use of this scheme for experimental observation of resonance approximation. III. PAIR CREATION USING A CONFIGURATION ANALOGOUS TO THE E144 EXPERIMENT.
The second scheme proposed is based on the E144 experiment at SLAC [18].It consists of two steps. On the firststep a relativistic electron beam is produced and accelerated by an ultrashort laser beam of photon energy 1 eV . Onthe second step the relativistic electron beam interacts with the main high intensity (10 W/cm ) highly focused laserbeam. On the electron’s frame the electric field intensity of the laser beam is thus increased due to the relativisticfactor while the focusing guarantees that the condition E − H > , for pair creation is satisfied. On the electronsreference frame the electric field E ∗ and the photon energy E ∗ photons are given by E ∗ = γ L E laserlab (8) E ∗ photons = γ L E photons (9)where γ L = E ebeam m e c , E ebeam being the electron beam energy.For E ebeam = 1 GeV, we obtain from (9) E ∗ photons = 1 . keV , which corresponds to λ = 0 . nm and E ∗ ∼ V /m .We will apply the same analysis as in the section two , to estimate the efficiency of resonance approximation for thisscheme. Using (3) where the interaction 4-Volume V = σ (0 . λ ) τ , λ = 0 . nm, σ = 50 µm and τ = 10 − s ,we presentin Figure 3 the envelope of the created pair number N as a function of q for ξ = 0 . ξ is obtained with sameconsiderations as before).The most probable process corresponds to q = 1 .
41 and can produce up to the order of 10 pairs. The multiphotonorder of this process is n = 735. We also examine the efficiency of pair creation for two more energies of the electronbeam, 400 M eV and 200
M eV.
As in section two, using formula (3) and the data from Table 2 we present in Figure 4,the dependence of pair number N from ξ for three E ebeam values of 1 GeV,
M eV and 200
M eV with correspondingvalues ( q , n ) for the later two energies as (1.41, 1844) and (1.40, 3686) ´ ´ ´ ´ ´ ´ No FIG. 3: Envelope of created e + e − pair number N o vs rest mass unit q for ξ = 0 . . keV. Ξ No FIG. 4: Log- plot of the number of created pairs N vs ξ for electron beam energy 1 GeV (top curve), 400
MeV (middle curve)and 200
MeV (bottom)
Electron beam X-ray photon Electric fieldenergy E e − beam energy ( eV ) Strength ( V /m )1 GeV . × M eV
782 7 . × M eV
391 1 . × Table 2. Typical parameters for obtaining Figure 4The estimates of Figure 4 of this scheme strongly suggest the possible use of this scheme for experimentalobservation of resonance approximation ( of the order of 10 for 1GeV , 10 for 400 and 200 MeV). IV. CONCLUSION
The above analysis aims to support the experimental verification of e + e − pair creation using the on resonancemultiphoton approximation by presenting two possible schemes. One is based on using a table-top X-FEL laser systemand the second is analogous to the E144 experiment. The numerical estimates presented that a sufficient number ofpairs can be produced by both schemes that can reach up to the order of 10 pairs for the table top X-FEL and 10 for the E144 like scheme. An important fact in both proposed schemes is that they use laser based modern technologyfor production and acceleration of electron beam and consequently they do not need any large acceleration facilities.Moreover, since the existing laser laboratories have this kind of technology, the required experimental values of theparameters needed for pair production can be achieved. Our choice to carry out our numerical estimates with E ebeam no greater than 1GeV is thus justified, as electron beams with energies like 400 MeV or 200 MeV can be created quiteeasily. Besides the above advantages of the two schemes, the laser beam parameters can be easily adjusted to changethe electron beam energy, in order to operate on resonance. Moreover the criterion for the choices of the values ofthe parameters used, is that they should be close to typical values rather than extreme ones, in order to maintainour estimates as realistic as possible. Consequently our analysis, suggests that both of the schemes presented arepromising for a future experimental verification of the on resonance pair production, especially when new powerfullaser systems like X-FEL[17], HIPER[29] and ELI[30] are under development. [1] O. Klein, Z. Phys., , 157 (1929); F. Sauter, Z. Phys. , 742 (1931); W. Heisenberg, H. Euler, Z. Phys. , 718 (1936).[2] J. W. Schwinger, Phys. Rev., , 664 (1951).[3] E. Brezin and C. Itzykson, Phys. Rev. D , 1191 (1970).[4] G.J. Troup and H.S. Perlman, Phys. Rev. D , 2299 (1972).[5] W. Greiner, B. Muller, J. Rafelski, ‘Quantum Electrodynamics of Strong Fields’, Springer – Verlag, Berlin, 1985.[6] V.S. Popov, JETP Lett. , 185 (1971); Sov. Phys. JETP , 709 (1972); Sov. Phys. JETP , 659 (1972); V.S. Popovand M. S. Marinov, Sov. J. Nucl. Phys. , 449 (1973) ; JETP Lett. , 255 (1974); Sov. J. Nucl. Phys., , 584 (1974).[7] A. I. Nikishov, Nucl. Phys. B , 346 (1970).[8] N.B. Narozhnyi and A. I. Nikishov, Sov. J. Nucl. Phys. , 596 (1970); Sov. Phys. JETP, , 427 (1974).[9] A. Ringwald, Phys. Let. B , 107 (2001).[10] V.S. Popov, Phys. Let. A , 83 (2002).[11] M. Perry and G. Mourou, Science , 917 (1994)[12] P. Chen and C. Pellegrini, in Quantum Aspects of Beam Physics, Proc.15th Advanced ICFA Beam Dynamics Workshop,Monterey, Cal., 4-9 Jan 1998 (World Scientific, Singapore, 1998) p. 571.[13] P. Chen and T. Tajima, Phys. Rev. Lett. , 256 (1999).[14] T. Tajima and G. Mourou, Phys. Rev. STAccel.Beams 5, 031301 (2002)[15] S. S. Bulanov,N.B. Narozhny,V.D. Mur, and V.S. Popov, Phys. Lett. A 330, 1 (2004)[16] D. B. Blaschke, A.V. Prozorkevich, C. D. Roberts, S. M. Schmidt and S. A. Smolyansky, Phys. Rev. Lett, 96, 140402(2006).[17] XFEL-Technical Disign Report (2006) Publisher: DESY XFEL Project Group, European XFEL Project Team, DeutschesElektronen-Synchrotron, Member of the Helmholtz Association, Notkestrasse 85, 22607 Hamburg, Germany ISBN 3-935702-17-5[18] D.L. Burke et. al., Phys. Rev. Let., , 1626 (1997).[19] H. K. Avetissian, A. K. Avetissian, G. F. Mkrtchian and Kh. V. Sedrakian, Phys. Rev. E66