A single-laser scheme for observation of linear Breit–Wheeler electron–positron pair creation

We show that a single laser pulse, traveling through a dense plasma, produces a population of MeV photons of sufficient density to generate a large number of electron–positron pairs via the linear Breit–Wheeler process. While it may be expected that the photons are emitted predominantly in the forward direction, parallel to the laser propagation, we find that a longitudinal plasma electric field drives the emission of photons in the backwards direction. This enables the collision of oppositely directed, MeV-level photons necessary to overcome the mass threshold for the linear Breit–Wheeler process. Our calculations predict the production of 107 electron–positron pairs, per shot, by a laser with peak intensity of just 3 × 1022 W cm−2. By using only a single laser pulse, the scheme sidesteps the practical difficulties associated with the multiple-laser schemes previously investigated.


Introduction
With the development of modern laser technology, the intensity of state-of-the-art lasers has surpassed the level of 10 22 W cm −2 [1][2][3]. Under electromagnetic fields of such magnitude, which are able to accelerate particles to ultrarelativistic energies, effects of quantum electrodynamics (QED) are expected to become important [4]. This capability has led to wide interest in using ultra-intense lasers to study strong-field QED phenomena, the theory of which has been developed for at least sixty years [5][6][7]. Experimental investigation of the strong-field regime is at an early stage [8,9], but is expected to develop significantly as the next generation of high-intensity laser facilities are commissioned.
It is of particular interest to investigate the annihilation of two photons into an electron-positron pair, or the linear Breit-Wheeler process (γ + γ → e − + e + ) [10], using the most advanced laser facilities. The linear Breit-Wheeler process, besides being a fundamental prediction of the theory of QED, also has important applications in astrophysics: for example, in understanding the opacity of the Universe [11] and in studying the pair cascade in pulsar magnetospheres [12][13][14][15]. Despite its importance, however, it has never been observed in a laboratory with real photons. The experimental difficulty comes from its small cross section and its MeV center-of-mass energy threshold. To overcome these difficulties, a system with colliding dense multi-MeV photons is required. Various geometries have been proposed, including the combination of GeV bremsstrahlung photons with keV blackbody photons produced in a hohlraum [16] or with an intense x-ray laser pulse [17]; the combination of multi-MeV photon beams each produced by a separate laser-irradiated foil or low-density plasma targets [18][19][20][21][22]; and the combination of multi-MeV photon beams produced by two colliding laser pulses within a structured target [23].
However, in these works, multiple laser pulses are required in one experiment, which limits the choice of facility where the proposed experiments can be conducted. Moreover, if the photon beams are generated in separate sources, then they must travel over a certain distance to reach the collision point. This leads to two

Emission of forward-directed collimated gamma-rays
It is now well-recognized that a high-intensity laser pulse propagating through a relativistically-transparent plasma can efficiently generate a collimated beam of energetic gamma-rays [22,24,[28][29][30][31]. In this section, we review the main physics elements of this process. A comprehensive analysis is available in [22,32].
The emission of collimated energetic gamma-rays is closely associated with the presence of a strong quasi-static azimuthal magnetic field inside the laser-irradiated plasma. The field is generated and maintained by a volumetrically distributed longitudinal electron current driven by the propagating laser pulse. Figure 1 illustrates this phenomenon for a 25 fs, 800 nm laser beam with a peak intensity of I 0 ∼ 3 × 10 22 W cm −2 propagating through a structured plastic target with a pre-filled channel. The beam has the parameters similar to those expected at the ELI-NP laser facility [1]. Detailed laser and target parameters and the information regarding the simulation setup are given in the appendix. Figure 1(a) shows stable propagation of the laser beam through the channel and figure 1(c) shows a time-averaged magnetic field (the averaging is performed over one laser period) generated as a result of this propagation.
The magnitude of the quasi-static magnetic field in the considered example is very high-it is 3 GG or 20% of the strength of the oscillating laser magnetic field (in the absence of the target). Such a strength is achieved by using a channel with an electron density that is higher than the classical cutoff density n c = πmc 2 /(eλ 0 ) 2 for a laser with a vacuum wavelength of λ 0 = 800 nm, where c is the speed of light and e and m are the electron charge and mass. In our example, the normalized laser amplitude, defined as a 0 = 0.85I 1/2 0 [10 18 W cm −2 ]λ 0 [μm], is a 0 = 120. The fact that a 0 1 means that the electrons in the irradiated plasma become relativistic. This, in turn, raises the cutoff density to approximately a 0 n c = 120n c , making the plasma in the channel, whose density is n e ≈ 2.8n c , transparent. The increased plasma transparency is a manifestation of a more general phenomenon referred to as relativistically induced transparency [33][34][35][36]. The channel is used in our simulation to stabilize the laser propagation [28,37], with the dense walls (n e ≈ 28n c ) providing optical guiding. The mechanism described here and the resulting electron acceleration and photon emission discussed in this section have been shown not to be sensitive to the bulk density. For example, similar simulations have been performed in [22] with a bulk electron density of 100n c . Structured targets with foam-filled channels have been successfully fabricated by general atomics and used for an experimental campaign at the Texas Petawatt laser facility [24].
The plasma magnetic field has two important functions: to enhance laser-driven electron acceleration and to induce emission of energetic photons by the accelerated electrons. At a 0 1, DLA produces forward moving electrons due to a forward push by the Lorentz force of the laser magnetic field. Transverse deflections by the plasma magnetic field can keep the transverse electron velocity anti-parallel to the transverse laser electric field and thus enable the accelerated electrons to continue gaining energy despite the oscillations of the laser electric field [32]. The deflections are particularly effective when the frequency of transverse electron oscillations caused by the magnetic field is comparable to the Doppler shifted frequency of the laser [38,39]. Such a condition can be realized only in a plasma with a sufficiently high current density [32]. In our simulation, the electrons reach a maximum energy of 700 MeV through the described mechanism. The deflections by the magnetic field not only enable the electrons to gain more energy from the laser field, but they also lead to energy losses via synchrotron emission of electromagnetic radiation. The rate of the energy gain typically greatly exceeds the rate of energy losses associated with the emission, so the laser accelerated electrons can efficiently convert the laser energy carried by optical photons into gamma-rays.
The photon emission process by an electron with a velocity v is characterized by a dimensionless parameter with E being an effective transverse field strength: Here E and B are the electric and magnetic fields acting on the electron, γ e = 1/ 1 − v 2 /c 2 is the relativistic factor, and B crit ≈ 4.4 × 10 13 G is the magnetic equivalent of the well-known Schwinger (or critical) electric field [40]. The characteristic energy of emitted photons, ε γ , is given by [41] ε γ = 0.44γ e χ e mc 2 , These photons are emitted in the direction of the electron momentum into a narrow cone whose opening angle is roughly 1/γ e . As a result, forward moving electrons generate a forward-directed beam of photons.
In the case of ultra-relativistic forward-moving electrons, like those generated by the laser in the presence of the plasma magnetic field, the contributions from the laser electric and magnetic fields to the effective field E nearly cancel each other out. Therefore, one can estimate E by simply using the plasma magnetic field B pl , with E ≈ B pl , which yields the following approximate expression for χ e : It follows from this estimate that we have χ e ≈ 0.020 for an electron with γ e ≈ 300 that is deflected by B pl ≈ 3 GG ≈0.19B 0 , which is close to the maximum amplitude of the quasi-static field shown in figure 1(c). The characteristic photon energy is ε γ ≈ 1.4 MeV. This example illustrates the important role played by the plasma magnetic field in generating energetic gamma-rays. The spectrum of forward-emitted photons in our simulation is shown in figure 2(a). It is indeed well-collimated in the forward direction, in agreement with our expectations. The photon spectrum is broad because the synchrotron spectrum is itself broadband and because electrons with a wide range of energies contribute to the emission. In our simulation, the photon emission is modeled using a Monte Carlo module for quantum synchrotron radiation [42]. The module computes χ e at each time-step for each charged macro-particle. This value is then used to determine the energy and number of emitted photons. The finite width of the emission cone is neglected, so the photons are emitted along the momentum of the emitting particle. The module also self-consistently accounts for the recoil experienced by the particle when emitting individual photons, as described in [42,43].

Emission of backward-directed gamma-rays
In section 2, we reviewed how a high-intensity laser pulse propagating through a relativistically-transparent plasma can efficiently generate a collimated beam of energetic gamma-rays. In what follows, we discuss the source of backward-directed gamma-rays that are also present, but frequently overlooked, in the considered setup. Figure 2(c) shows that indeed, in addition to forward-directed gamma-rays, the plasma electrons in our simulation emit energetic backward-directed photons. These photons, which have an almost uniform angular distribution, originate from electrons that are moving backwards over at least some segments of their trajectories. The backward-emitting electrons are concentrated primarily at the leading part of the  laser pulse: this may be seen in figure 1(f), which shows the density of electrons that have p x < 0, γ e > 15 and χ e > 0.03. In order to identify their origin, we track five randomly selected electrons that are initially located inside the channel prior to the arrival of the laser pulse. The electrons were selected, at t = 17 fs, from the region defined by |y| < 0.5 μm and 18.75 μm < x < 19.25 μm. As seen in figure 3, these electrons reverse the direction of their longitudinal motion after being swept up by the leading edge of the laser pulse.
The backward acceleration of the electrons inside the channel is caused by a longitudinal electric field, E x , which is shown in figure 1(b) at t = 67 fs. This field is concentrated at the leading edge of the laser pulse. It is formed by charge separation as the laser enters the plasma channel, displacing electrons ahead of the much heavier and therefore slower moving ions, and moves forward with the leading edge of the laser pulse. This aspect is shown in figure 4, where figure 4(a) shows the time evolution of E x in the region close to the axis and figure 4(b) shows the time evolution of the time-dependent magnetic field B z , associated primarily with the laser, in the same region. At each location along x, both fields are a spatial average over the region |y| < 1.5 μm. We find that the E x structure moves with a constant speed of 0.82c and that its amplitude is slowly varying after the initial ramp-up. The field structure disappears by about x ≈ 70 μm following the laser depletion, which again confirms that it is driven by the laser pulse itself.
In contrast to the case of the forward-moving electrons, the effective field E for the backwardpropagating electrons is primarily determined by the fields of the laser. The reason for this difference is the lack of compensation between E and B of the laser. These fields are much stronger than the fields created by the plasma, so, in the absence of their compensation, they dominate the expression for E given by equation (2). Therefore we neglect the plasma fields, set E laser = B laser , and assume that the electron is moving purely backwards to find that the effective field acting on the backward-moving electrons is E ≈ 2B laser . This effective field is stronger than the effective field for the forward moving electrons by a factor of 2B laser /B pl ≈ 13 [in our case max(B laser ) ≈ 20 GG]. Thus, while a relativistic factor of γ e ≈ 300 would be necessary for a forward-moving electron to have a typical photon energy of 1.4 MeV, only γ e ≈ 80 is necessary for a backward-moving electron to emit photons with the same energy.

Yield of electron-positron pairs via the linear Breit-Wheeler process
As discussed in sections 2 and 3, the laser-plasma interaction under consideration generates a gamma-ray population with both backward and forward moving photons. Figures 2(b) and (c) show that the energies of these photons exceed 100 keV and reach 10 MeV, even for the backward emission. This energy range is favorable for the electron-positron pair creation via binary photon collisions, i.e. the linear Breit-Wheeler process, because it can overcome the center-of-mass energy threshold: where ε f γ and ε b γ are the energies of the forward and backward-directed colliding photons. In this section, we show that, in the interaction, binary collisions between photons of the right energy range occur frequently enough to generate an appreciable yield of electron-positron pairs.
Photon collisions are possible only if the backward-moving photons are emitted ahead of forward-moving photons. As these photons are emitted by backward and forward-moving electrons, respectively, we examine the spatial distribution of these two components of the electron population, selecting only those electrons which have γ e > 15 and χ e > 0.03. We apply this selection criterion because, according to equation (3), the characteristic energy of the photons emitted by such electrons is ε γ > 100 keV, which is within the favorable energy range for the linear Breit-Wheeler process. Snapshots of the density for backward and forward-moving electrons, selected according to the described criterion, are shown in figures 1(e) and (f), respectively. The forward-moving and thus forward-emitting electrons are spread throughout entire region occupied by the laser pulse (see figure 1(a)). The backward emitting electrons, by contrast, are primarily concentrated at the front of the laser pulse. This is in good agreement with the mechanism of their generation by E x at the leading edge of the laser pulse (see figure 1(b)).
The difference in the spatial localization of the emitting electrons translates into photon emission profiles that are favorable for the photon-photon collisions. Figure 5(b) shows snapshots of ∂ 2 N γ /∂x∂t for forward and backward-directed photons, where N γ is the number of photons with energy between 100 keV and 10 MeV emitted in a thin vertical slice. We can see from the two curves that the backward-moving photons are emitted ahead of the forward-moving photons. For completeness, figure 6 shows the energy distribution of the forward and backward directed electrons in our simulation. The snapshots in figures 6 and 5 are taken at t = 67 fs. Figure 1(d) shows that our setup produces an extremely dense population of gamma-rays with energies greater than 100 keV. What is also remarkable is that this population consists of overlapping forward and backward-moving photons whose densities are shown in figures 1(g) and (h), which is a result of the longitudinal offset in the photon emission. As the two populations collide, they produce electron-positron pairs, with the yield being enhanced due to the high photon density.
We first estimate the yield using the density of photons from our simulation. In our system, this density of photons with energy ε γ > 100 keV is n γ ≈ 10n c , as shown in figure 1(d), where n c ≈ 1.7 × 10 27 m −3 is the critical density for λ 0 = 0.8 μm. A useful figure of merit is an approximate macroscopic cross-section for the photons in the MeV energy range: Σ γγ = σ γγ n γ , where n γ is their density and σ γγ is the microscopic cross-section for the linear Breit-Wheeler process. We set σ γγ ≈ 1.7 × 10 −29 m −2 which is close to its maximum value (a head-on collision of two 700 keV photons). The number of photons with ε γ > 700 keV is about 40% of the number of photons with ε γ > 100 keV. We therefore re-scale the density by setting n γ ≈ 4n c and use this value to find that Σ γγ ≈ 1.2 × 10 −1 m −1 . The total number of photons with ε γ > 700 keV is N γ ≈ 4 × 10 13 , where we have assumed that the size of the photon cloud in the third dimension is equal to the channel width d ch = 5 μm (our simulation is two-dimensional). It is worth pointing out that in the energy range between 0.1 and 10 MeV the number of backward-moving photons (4.4 × 10 13 ) is comparable to the number of forward-moving photons (5.9 × 10 13 ). The backward-emitted photons are not collimated, so we assume that they leave the cloud after traveling a distance roughly equal to the channel radius, d ch /2. This gives the following estimate for the pair yield via the linear Breit-Wheeler process: N BW lin ≈ Σ γγ d ch N γ ≈ 1.2 × 10 7 . In order to perform a quantitative assessment of the pair yield, we use the post-processing algorithm developed in [23]. The algorithm uses the photon data provided by the PIC simulation, which includes, in addition to the photon energy, the emission time, location, and the macro-particle weight. The photons are grouped into collimated mono-energetic beamlets. Each longitudinal position along the x-axis emits  multiple such beamlets. They are assumed to be uniform along the y-axis (the assumption is motivated by the PIC data) with a width equal to the width of the channel.
By applying the described algorithm, we found that the total number of pairs is N BW lin ≈ 1.1 × 10 7 , which is close to our earlier estimate. The number of generated pairs is three orders of magnitude larger than the number of pairs produced by colliding two beams consisting of the collimated forward-emitted photons 250 μm away from each laser-irradiated structured target [22], where each beam is generated by a separate target as described earlier in section 2. The significant increase is due to the increased density of the colliding photons in our configuration. It must be noted that adding a counter-propagating laser to this configuration can boost the yield by an order of magnitude [23], but this makes the corresponding experimental setup much more challenging.   Figure 7 shows the yield of the linear Breit-Wheeler process in a collision of forward and backward-directed photons as a function of their energies, where the superscripts f and b indicate the direction of the photon motion. As anticipated, the majority of the pairs are produced by photons within the energy range 100 keV <ε f,b γ < 10 MeV, where ε γ is the photon energy. This is a consequence of the broad photon spectrum. Figure 8 shows the momentum distribution of the generated positrons. We distribute the pairs generated in each photon-photon collision isotropically in the corresponding center of mass frame. The result can be refined by taking into account the differential cross-section [20]. The 'initial' distribution shown in figure 8 is likely to evolve because the positrons inside the channel experience a superposition of strong laser and plasma fields.

Test-particle model for backward electron acceleration and photon emission
In this section, we use a reduced model motivated by the PIC simulation results to examine the backward electron acceleration that leads to the emission of the backward-directed gamma-rays. The purpose is to assess the impact of the plasma electric and magnetic fields on the electron dynamics.
The model is similar to that developed in [32] for analyzing DLA of electrons in the presence of a static azimuthal plasma magnetic field. The electron is treated as a test particle, which means that the laser and plasma fields are externally prescribed. There are two modifications that we have made to adapt the model to our problem: we have added a longitudinal plasma field E x , which moves with the laser pulse, and the force of radiation friction f RF caused by the emission of photons. The force of radiation friction, in the Landau-Lifshitz prescription [44], is directed anti-parallel to the electron momentum p: 2 3 where E is the effective field strength given by equation (2), r e ≡ e 2 /m e c 2 ≈ 2.8 × 10 −13 cm is the classical electron radius, ω is the laser frequency, and λ 0 = 2πc/ω is the vacuum wavelength. The equations that describe the electron dynamics are dp where p x and p y are the components of the electron momentum, parallel and perpendicular to the laser propagation respectively, and x and y are the electron coordinates. The fields are a superposition of the laser and plasma fields: where is the phase variable, with ξ = 0 at the initial location of the electron in our calculations (x = x init at t = t init ). We neglect the slight superluminosity of the wave fronts (caused by the plasma and the finite transverse size of the channel) by assuming that the phase velocity is equal to c. More importantly, the longitudinal field in this model is also moving with the speed of light for the sake of simplicity. The assumption here is that the difference between the actual speed, 0.82c, and the speed of light is inconsequential, because the electrons are moving in the opposite to the direction of the laser propagation. It is more important that the envelopes of the laser pulse and the longitudinal electric field should move forward at the same velocity. We perform our calculations for where for ξ Δ y and E env (ξ) = 0 otherwise. Here Δ y = 3π and σ y = 57.53, and a 0 = |e|E 0 /m e cω = 120.4. The values of these parameters are chosen such that the field given by equation (15) reproduces the snapshot of the laser electric field shown in figure 9(a). The longitudinal plasma field in our calculations is for ξ Δ x and E x (ξ) = 0 otherwise. Here Δ x = 2π, σ x = 5.5, μ = 20.12, and |e|E max x /m e cω = 16.34. The values of these parameters are chosen such that the field given by equation (17) reproduces the snapshot of the laser electric field shown in figure 9(b). The plasma magnetic field is The uniform current density is set at j 0 = 10J A /πλ 2 0 to reproduce the transverse linear profile of the magnetic field in our PIC simulation, where J A = m e c 3 /|e| is the non-relativistic Alfvén current. . Laser and plasma electric fields in the PIC simulation (blue) and in the test-particle model (red). The PIC simulation electric fields are the snapshots taken at t = 150 fs and averaged over a region with |y| < 1.5 μm. The analytical form of the test-particle fields is given by equations (15) and (17). The phase variable is given by equation (14), with x init = 20 μm and t init = 150 fs.  Figure 10(a) shows a trajectory of an electron computed using the test-particle model. The electron is located at x init = 20 μm at t init = 150 fs. In our fully self-consistent PIC simulation, the electrons in the channel typically acquire some negative longitudinal momentum before they encounter the laser beam and the longitudinal electric field structure driven by its front. In order to capture this feature in our test-particle calculations, we set p x = −0.3m e c at t = t init . After encountering the laser pulse, the electrons gets pushed in the forward direction. As it slips further into the laser pulse, the longitudinal plasma field, associated with the leading edge of the laser pulse, ramps up at the electron location. The increase in E x causes the electron to reverse the direction of its longitudinal motion while gaining energy. In order to make the correlation evident, we use black round markers to show the beginning and the end of that segment of the trajectory where E x 0.1E max x . The calculation confirms the role of the plasma E x in generating relativistic (γ e ≈ 100) backward-moving electrons.
The plasma magnetic field also has a profound impact on the electron trajectory. As seen in figure 10(c), the electron moves backwards while performing transverse oscillations after its interaction with E x . This trajectory is obtained by setting j 0 = 0 in the electron equations of motion, so that there is no plasma magnetic field. The energy gain is comparable to that seen in figure 10(a). However, the magnetic field in figure 10(a) causes appreciable transverse electron deflection that leads to electron loss once the electron reaches the wall of the channel. The deflection is in qualitative agreement with the magnetic field orientation. We terminate the calculation once the electron cross the initial channel boundary shown with the thick lines.  Figure 11 provides additional information regarding the electron dynamics, including the characteristic energy of emitted photons ε γ . In order to clearly distinguish the backward electron motion, we use the red color to indicate p x < 0. The region between the vertical dashed lines corresponds to that part of the electron trajectory where E x 0.1E max x . Figure 11(e1) shows that ε γ reaches the 0.5 MeV level toward the end of the electron interaction with E x . More importantly, the emission occurs in the backward direction. The electron continues to emit in the backward direction even after the interaction with E x , which indicates the long-lasting effect of the plasma electric field. Our result confirms that E x at the leading edge of laser pulse can indeed lead to emission of energetic backward-directed photons suitable for the pair production. Figure 11 also provides details of the electron dynamics without radiation friction and without the plasma magnetic field. By comparing figures 11(e1) and (e3), we find that ε γ is essentially the same in the absence of the plasma magnetic field. The backward emission is prolonged because the electron remains within the channel. The middle column in figure 11 shows the electron dynamics in the absence of the radiation friction. Even though the electron trajectory in the absence of the radiation friction remains relatively unaffected (compare figures 10(a) and (b)), the energy of the emitted photons changes appreciably. As seen from comparing figures 11(e1) and (e2), the difference occurs after the interaction with E x . The conclusion then is that the photon emission should be calculated self-consistently, with radiation friction taken into account in real-time.

Summary and discussion
We have shown that electron-positron pair creation by the linear Breit-Wheeler process is possible when a single laser travels through a dense plasma target. In particular, using a laser pulse with parameters similar to those at the ELI-NP laser facility, it is possible to produce 10 7 linear Breit-Wheeler pairs in a single shot with a peak laser intensity of just 3 × 10 22 W cm −2 .
The mechanism involves a strong longitudinal electric field driven by the laser pulse. As the laser pulse travels through the plasma channel, it induces a longitudinal electric field at the front of the laser pulse due to charge separation. The electric field pulls the channel electrons back when they encounter the laser pulse, causing them to travel opposite to the direction of laser propagation. The backward-moving electrons experience strong acceleration from the laser pulse, causing them to emit energetic backward-directed photons. These photons are in the energy range suitable for the linear Breit-Wheeler process, and therefore when they collide with forward moving photons that are typically associated with the considered setup, electron-positron pairs are created. The high pair yield is due to the fact that the photon collisions occur locally, i.e. close to the point of emission, and thus at high density. The practical advantage of the scheme presented is that only a single laser pulse is required and therefore the difficulty of spatially and temporally overlapping multiple laser pulses is avoided.
The number of positrons produced by the linear Breit-Wheeler process in our setup is 10 7 even for a 0 = 120, equivalent to I 0 ≈ 3 × 10 21 W cm −2 , which is within reach of today's high-intensity laser facilities. The uniqueness of our setup compared to the setup that involves two lasers [23] is that it produces virtually no pairs via the nonlinear Breit-Wheeler process, i.e. no pairs were produced in our simulation by the module that computes the corresponding yield. For this process to become significant for a single laser pulse propagating in a plasma channel, we would need intensities 10 24 W cm −2 [45], or if the laser pulse is subsequently reflected from a high-density mirror, 10 23 W cm −2 [46]. The linear Breit-Wheeler process then only needs to compete with the Bethe-Heitler process. In this process, a γ-ray with energy greater than 2m e c 2 creates an electron-positron pair by interacting with the Coulomb field of an atomic nucleus. Using a simple post-processing algorithm detailed in [23], we determine that the Bethe-Heitler process generates 5.7 × 10 5 pairs inside the channel. This number is roughly 20 times lower than the yield from the linear Breit-Wheeler process, so we conclude that the linear Breit-Wheeler process is the primary source of positrons in our setup. Figure 8 shows the momentum distribution of the positrons right after they were created, but this 'initial' distribution is likely to change before the positrons leave the plasma. The positrons created inside the channel are subject to the influence of the laser and plasma fields. One would need to develop a PIC code that can produce the linear Breit-Wheeler pairs in order to reliably assess the impact of these fields on the positron momentum. The plasma magnetic field is likely to have a confining effect. The transverse confinement however causes charged particles to slide along the magnetic filament [32], with the direction dependent on the polarity of the field and the particle charge. In the magnetic field configuration shown in figure 1(c), the electrons slide forward, which enables the increased energy gain from the laser [32], so we anticipate that the positrons are likely to slide back toward the channel opening.
The backward emission of energetic photons by a similar mechanism related to the charge separation at the leading edge of the laser pulse was previously discussed in [47], albeit in the case of a much higher laser intensity. This regime was then further investigated in [48]. The strength of the longitudinal electric field goes up with the laser intensity or, equivalently, with a 0 , which means that the energies of backward accelerated electrons and the energies of the photons they emit also increase with a 0 . Even though the nonlinear Breit-Wheeler process is inefficient for the value of a 0 considered in our work, it can become an import source of electron-positron pairs at much higher a 0 due to the discussed increase of the photon energies [47,48]. It must be pointed out that references [47,48] only considered the nonlinear Breit-Wheeler process, so further research is needed to determine at what value of a 0 the nonlinear Breit-Wheeler process becomes the dominant mechanism. The plastic structured target contains a pre-filled channel whose axis is aligned with the axis of the laser beam. The target is initialized as a fully ionized plasma. The density of the carbon ions is set to be equal to the density of the protons in the bulk and inside the channel. The values of the electron density in the bulk and inside the channel are listed in table 1.