1. Introduction

The laser technology is developing quickly in recent years. For example, by upgrading HERCULES’s laser power to 300 TW, laser intensity of ∼ 2 × 10²² W/cm² has been firstly demonstrated in experiment in 2008 [1]. A stronger electromagnetic field with E > 10¹⁵V/m can be supplied by an intense laser such as the ELI facilities in the near future [2]. Exposed in such a strong laser field, laser-matter interactions enter into the near quantum electrodynamics (QED) regime [3, 4]. It is predicted that the laser can breakdown the pure vacuum into electron-positron pairs when the Schwinger field, E_s = 1.3×10¹⁸ V/m is approached, which corresponds to a light intensity up to 10²⁹ W/cm² [5]. Even in the case of a single real electron in the laser focus, the required laser intensity for pairs is still of order of 10²⁵ W/cm² [6]. For a lower laser intensity, it is also possible to generate positrons by laser driven electron-nuclei interaction or direct photon-photon collisions [7–22]. Up to now, several schemes have been proposed to produce dense positrons under current laser conditions. For early investigations, the attention has been focused on the trident process, which relies on the colliding of laser-accelerated electrons with a high-Z nuclei. However, the process is very inefficient due to the log dependence of the pair production cross section σ_ei on the Lorentz factor γ_e [9, 10]. The Bethe-Heitler (BH) process [23] also attracted great attention, which makes use of the bremsstrahlung radiation [11, 12] by decelerated electrons in a thick target. The corresponding cross section of pairs production is usually 10³ times larger than that with the trident process, making it very attractive for high-number e⁻e⁺ pairs production [13, 17].

With the increase of laser intensities in recent years, the multi-photon Breit-Wheeler (BW) process has been revisited, because it is capable to enhance the positron production efficiency significantly [15–22]. This process can be simply described by two steps: (1) e⁻ + mγ_l → e⁻ + γ_h, where γ_l refers to the laser photons and γ_h represents the γ photons; (2) γ_h + nγ_l → e⁻ + e⁺. Burke et al., have observed a signal of 106 ± 14 positrons above background in experiment and the results are the first laboratory evidence for inelastic light-by-light scatting involving only real photons [21]. Recently, Pike et al., proposed an experimental scheme for a pure photon-photon collider, suggesting that pure light can be transformed into electron-positron pairs through the BW process [22]. In theory, the positron production probabilities are determined by the parameter [16] η = γ_e |E_⊥ + β_e × cB|/E_s, where E_⊥ and B are the components of the laser electromagnetic field and β_e is electron velocity normalized by the light speed c. It is found that, for a laser propagating parallel to the γ photons emission, η ∼ 0, which implies few positrons generation. In order to increase the cross section of γ photons colliding with the laser photons, a counterpropagating laser pulse is usually required, with the first laser generating γ photons and the second providing low energy photons. In this case, the pair production becomes more efficient because of η ≥ 1. An alternative scheme is to substitute the second laser by a reflected laser from a solid target [20]. However, the γ photons emission relies significantly on the foil surface dynamics and the required laser intensity is above 4 × 10²³ W/cm², an order of magnitude higher than avaliable in current laboratories.

Here, we propose an all-optical scheme for bright γ-rays and dense positrons generation by irradiating a currently approachable laser onto near-critical-density plasmas. Our 2D PIC simulations with QED effects incorporated show that, the radiation damping force becomes large enough to compensate for the Lorentz force, causing radiation reaction trapping of a dense electron bunch in the laser field. The γ photons are emitted in two ways: (1) nonlinear Compton scattering due to the oscillation of electrons in the laser fields, and (2) Compton backwardscattering resulting from the bunch colliding with the reflected laser. As a result, abundant e⁻e⁺ pairs are generated via the multi-photon BW process with a maximal positron density of 10²⁷m⁻³. The scheme is further demonstrated by full 3D PIC simulations and the positron number is up to 2 × 10⁹ with an average energy > 100 MeV. This compact γ-rays and pair source may have potential applications in many fields, e.g., laboratory studies of astrophysics and nuclear physics [24–27].

2. 2D PIC simulations and results

In the scheme, a plasma filled cone target is used because such targets have several features, such as efficient electron guiding and laser focusing. It can be attributed to the fact that the laser radiation tends to bend towards the area with a larger dielectric ε determined by [28]

We first carry out 2D simulations using QED-PIC code EPOCH [20,32]. The simulation box size is X × Y = 50λ₀ × 16λ₀ with λ₀ = 1μm being the laser wavelength, which is sampled by 2000 × 400 cells with 20 particles in each cell. In order to reduce the required laser intensity, we use a plasma-filled cone target, as schematically shown in Fig. 1, which is initially located between 5λ₀ ∼ 42λ₀, with an electron density of n_e=780n_c and thickness of 2μm. Instead of the high-Z materials like gold, here we employ an aluminum cone so that the multiple-photon BW process is dominant over the trident process and BH process. The left cone opening radius is 6λ₀, while the right tip radius is 3λ₀. Meanwhile, hydrogen near-critical-density (NCD) plasma with a uniform density of 2n_c is filled into the cone, which can increase the laser absorption and provide more background electrons for γ photons emission. A linearly polarized Gaussian laser pulse, with a transverse profile $a=a_0\mathit{exp}(-y^2/{\sigma }_0^2)$, is incident from the middle of the left boundary and strikes on the right cone tip eventually, where σ₀ =5λ₀ is the laser focal spot radius and a₀=150 is the dimensionless laser electric field amplitude, corresponding to an achievable laser intensity 3 × 10²² W/cm² [1]. The laser pulse has a platform profile in time and the duration is τ₀=10T₀.

 

Fig. 1 Schematic view of an intense linearly polarized laser striking a near-critical-density plasmas filled Al cone.

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As shown in Fig. 2(a), the conical target has a strong focusing effect on the laser pulse so that the laser peak intensity is enhanced from 3 × 10²² W/cm² to 1.3 × 10²³ W/cm², corresponding to an enhanced transversely electric field ∼ 10¹⁵V/m at t=42T₀. Exposed in such a strong laser field, hydrogen atoms are fully ionized, forming NCD plasmas. Simultaneously, electrons are rapidly accelerated to relativistic velocities with the Lorentz factor of the order of several hundreds. They emit abundant photons outwards and suffer from strong inward recoil forces by the photons. As a result, the radiation reaction effect becomes significant and the damping force gets comparable with the Lorentz force [33,34]. Figures 2(c) and 2(d) present the electron density and energy distribution in space at t = 42T₀. We see a dense electron beam is trapped in the cone along the laser axial direction with a peak density of 7n_c and energy up to 2.4 GeV at t=42T₀. At t=44T₀, the laser pulse penetrates through the plasma with its front arriving at the cone tip. A curved surface of the tip is thus formed by the strong laser pressure [35] and the laser wave is soon reflected, as illustrated in Fig. 2(b). Though the depletion of the laser pulse in the NCD plasmas is serious, we find the reflected laser electric field amplitude is still comparable with its original strength due to the focusing effect of the cone. This is very beneficial for the photon emission and the BW process in the following stage.

 

Fig. 2 The transverse electric field evolution at t=42T₀ (a) and 54T₀ (b). Here, E₀ = m_ecw₀/e = 3.2 × 10¹² V/m. The electron density (c) and energy distributions (d) at t=42T₀.

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Figures 3(a)–3(d) demonstrate a bright γ-rays emission in the cone, which is produced with two independent processes: (1) nonlinear Compton scatting due to the oscillation of trapped electrons in the laser field, and (2) Compton backwardscattering resulting from the trapped electron bunch colliding with the reflected laser by the cone tip. Before the laser reflection at t=44T₀, the trapped electrons have a high density and large transverse oscillation amplitude as shown in Figs. 2(c) and 2(d). This is actually the nonlinear Compton scattering, which promises high energy γ photons emission, as seen in Figs. 3(a) and 3(c). After the laser reflection, the co-moving electrons back-scatter the reflected laser photons to high energy γ photons. The spectrum of γ photons from Compton back-scatterings is broad and the maximum energy can be estimated as $E_{\gamma ,\mathit{max}}=4E_e^2\overline{h}{\omega }_0/\left(4E_e\overline{h}{\omega }_0+{\left(m_e{c}^2\right)}^2\right)$, where E_e is the electron energy, and h̄ω₀ is the laser photon energy. As seen in Fig. 2(d), E_e is up to 2.4 GeV at t=42T₀ and will get higher later. For example, E_e ≈ 2.8 GeV at t=44T₀, which leads to E_γ,max ≈ 141 MeV. This is in excellent agreement with the simulation results as shown in Fig. 3(a). It is noticed that the photon energy density gets larger, while the photon energy decreases significantly at t=48T₀. This can be interpreted as follows. First, the photon energy decreasing results from the initiation of the BW process, which consumes the high energy part of γ photons to generate positrons. Figures 3(e) and 3(f) exhibit the positron density distribution at t=44T₀ and 48T₀. Only few positrons are produced before the laser reflection because the controlling parameter, η ∼ 0. By comparison, at t=48T₀ a large number of positrons are observed, leading to the sharp decrease of the photon energy. Second, a standing wave is set up when the incident laser overlaps the reflected one. It further enhances the electric potential. On the other hand, the reflected laser by the curved surface of the tip is focused so that the subsequent Compton backscattering process gets efficient, producing high energy density photons. This interprets the increase of the photon energy density at t=48T₀. In practice, we have to take into account of the pre-plasma before the cone tip, which may influence the laser reflection and alter the γ rays emission. However, the influences should be limited because of the thin thickness of the tip and sharp front of the laser pulse after its penetrating the NCD plasma.

 

Fig. 3 The photon energy distribution [(a),(b)], the photon energy-density distribution [(c),(d)], and the positron density distribution [(e),(f)] at t=44T₀ and 48T₀, respectively.

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Figure 4(a) shows the positron energy spectra. As expected, only few positrons are generated at the very beginning. When the laser pulse is reflected, both the number and energy of positrons are dramatically increased. Here, the average Lorentz factor of the positrons generated inside the cone can be roughly determined by [36]

 

Fig. 4 The positron energy spectra (a) and the divergence angle distributions (b) at t=44T₀, 46T₀ and 48T₀.

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3. The robustness of the scheme

In the process of the 2D simulations, we also consider the influences of the cone tip thicknesses and density of NCD plasmas on the pair production. As shown in Fig. 5(a), a 2μm thick tip is optimal for the positron generation under our laser conditions. This is easy to understand: a 1μm thick cone tip is so thin as to cause relativistic transparency of the tip for the laser pulse, making the laser reflection weaken and the BW process inefficient. By comparison, a 3μm tip is a litter thick, which absorbs considerable laser energy so that the efficiency of the BW process becomes low.

 

Fig. 5 The positron number evolutions with different cone tip thicknesses (a) and NCD plasma densities (b).

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Meanwhile, the NCD plasma density also plays an important role for the positron generation. As seen in Fig. 5(b), we consider three different NCD plasmas: 0.8n_c, 2n_c, and 4n_c. It is found that 2n_c plasma is optimal for the positron generation in our scheme. When 0.8n_c plasmas is filled in the cone, only few electrons are trapped so that the photon emission becomes trivial. On the contrary, using a 4n_c plasma is also disadvantageous, because most laser energy is consumed by the NCD plasmas and only a small part laser wave is reflected by the cone tip. Therefor, an optimal value of the NCD plasma density exsits, e.g., 2n_c in our case.

Finally, we conduct full 3D simulations to demonstrate the robustness of the scheme, as shown in Fig. 6. Here, the box size is X × Y × Z = 50λ₀ × 16λ₀ × 16λ₀, sampled by 600 × 128 × 128 cells and 8 pseudoions per cell. All other parameters are the same as those in the 2D simulations. It is obvious to see that a dense relativistic electron bunch is trapped along the laser axis with a density of tens n_c. These electrons oscillate in the lase fields and emit γ-rays forward. Finally, we obtain a positron beam with a peak density of 1n_c and a number > 2 × 10⁹. Meanwhile, the average positron energy is up to 107 MeV when laser front has just been reflected at t=48T₀, which is smaller than that in the 2D case. As we know, in a more realistic 3D case, the focusing effect of the cone on the laser pulse becomes weakened due to the another dimension along the Z direction as compared with the 2D case. This results in a smaller laser electric field in the cone (not shown here). Finally, the positron energy gets smaller according to the Eq. (2).

 

Fig. 6 3D simulation results: (a) the electron density distribution at t=57T₀, (b) the positron density distribution at t=57T₀, and the positron energy spectrum evolution.

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4. Conclusion

In conclusion, we propose an all-optical scheme for bright γ-rays emission and dense e⁻e⁺ pair production by using the QED-PIC simulations. It is shown that the radiation reaction leads to the electron trapping in the laser field. These electrons oscillate transversely and emit high-energy γ photons through the nonlinear Compton scattering and Compton backwardscattering processes. Finally, the multi-photon BW process is initiated and dense e⁻e⁺ pairs are produced with a density of ∼ 10²⁷m⁻³. Full 3D PIC simulations indicate the final positron number is up to 10⁹, which may have many potential applications in fundamental science and astrophysics.