1. Introduction

With the development of multi-PW laser facilities such as Extreme Light Infrastructure (ELI) [1] and SG-II laser facilities [2], people are expected to generate femtosecond optical pulses of intensity up to 10²³⁻²⁴ W/cm², which opens up wide possibilities for theoretical and experimental investigation of strong-field quantum electrodynamics (QED) effects. It is predicted that the QED vacuum is unstable [3] when exposed to ultra-intense laser light with electric field approaching the Schwinger limit [4] (E_S = 1.3 × 10¹⁸ V/m), which corresponds to a light intensity up to 10²⁹ W/cm². Finally, e⁻e⁺ pairs can be spontaneously created from vacuum. However, the required laser intensity is at least 6 orders of magnitude higher than the current achievable, making it unlikely to demonstrate the pairs creation directly under terrestrial laboratory conditions. Fortunately, recent studies showed that the laser-plasma interaction in the near-QED regime can significantly reduce the threshold of the laser intensity for dense pair creation. With the projected PW laser facilities, this offers a very promising approach to creating dense e⁻e⁺ pairs by ultra-intense laser-plasma interactions.

At the intensity above 10²³ W/cm², the laser-matter interaction becomes strongly dissipative [5] and enters into the near QED regime. Efficient laser energy transform into high energy γ-rays occurs, which enables copious e⁻e⁺ pair production via various mechanisms, such as trident process, Bethe-Heitler (B-H) process [6] and Breit-Wheeler (B-W) process [7]. For example, irradiating an ultra-thin foil by an ultra-intense laser pulse with intensity ~ 10²³ W/cm², the multi-photon B-W process typically has a much larger cross-section of the pair creation and therefore dominates the pair production over the trident process and B-H process [8, 9].

High energy γ-ray flashes and dense pair plasmas are ubiquitous in the high energy astrophysical environment, such as γ-ray bursts, black hole and pulsar wind nebulae [5, 10]. The possibility of probing the process of laboratory astrophysics and strong-field QED has motivated the pursuit of the generation of bright γ-ray bursts and high-flux relativistic pair jets [11–17]. Recently, a large amount of effort has been dedicated to acquiring bright γ-rays and dense positrons in the near-QED regime. For example, Ridgers et al. [12] numerically demonstrated the production of pure e⁻e⁺ pair plasmas by an ultra-intense laser striking solid foils. During the hole-boring process, energetic electrons radiate high energy γ-photons via the so-called skin-depth emission, which is able to trigger the B-W process so that dense relativistic e⁻e⁺ pairs are produced in the laser fields. Luo et al. [13] and Chang et al. [14] proposed an alternative method by irradiating the target with two ultra-intense colliding laser pulses. Due to the symmetric compression of the target, the positron production is significantly enhanced. Recently, Liu et al. [15] and Zhu et al. [16, 17] reported, for the first time, the bright γ-ray source and dense positron beam generation in the near-critical-density (NCD) plasmas. By using double cone targets, the required laser intensity for the radiation trapping effect is reduced by two orders of magnitude, so that the B-W process is triggered at current laser intensities. It is also shown that a dense positron bunch could be generated in gas plasmas driven by two counter-propagating laser pulses [8]. In this scenario, the first laser is responsible for accelerating the electron bunches to GeV energies in the laser wakefield regime [18, 19], while the second is used for triggering the nonlinear Compton scattering and B-W process. In spite of these different concepts, the proposed schemes seem to be complicated, which depend on either an engineering-shaped cone structure or staged acceleration. Meanwhile, the obtained γ-photons are highly localized and the resulting positrons merger into a single bunch. This poses a great challenge in the beam manipulating and processing for potential applications where special structure of photon beam and positron bunch are required.

In this paper, we propose a novel scheme for ultra-bright γ-ray flashes and high-energy-density positron bunches generation. By irradiating an ultra-intense laser pulse with intensity of 10²³ W/cm² onto a micro-wire target, the electrons near the wire surface are quickly dragged-out by the laser transverse fields, forming attosecond bunches in space, and are effectively accelerated to several GeV energies by the laser ponderomotive force. These attosecond electron bunches behave like “relativistic flying mirrors (RFMs)” [5], which subsequently backscatter the probe laser photons to high energy γ-photons along the electron motion direction. The emitted γ-photons are bunched in space, forming ultra-high brightness γ-ray flashes characterized by a spatial gap of about one laser wavelength. As soon as they collide with the probe pulse, the multi-photon B-W process is triggered and dense attosecond e⁻e⁺ pair bunches are produced. Our multi-dimensional simulations indicate that the final positron energy density is as high as 1 × 10¹⁷ J/m³ and the positron number is up to 10⁹ per shot. The produced ultra-bright ultra-short γ-ray flashes and attosecond positron bunches with high energy and density may have diverse applications in fundamental physics, applied science and laboratory astrophysics.

2. Two-dimensional simulations and results

We first carry out two-dimensional (2D) particle-in-cell (PIC) simulations using the open source code EPOCH [20] to demonstrate feasibility of the scheme. The simulation box size is x × y = 14λ_L × 12λ_L, which is sampled by 5600 × 3600 cells with 64 pseudo-electrons and 36 pseudo-carbons in each cell. Here λ_L = cT₀ = 1 μm is the laser wavelength and T₀ is the laser period. The left driving pulse is linearly polarized along the y direction, which is grazing incident on the wire target from the left to the right, as schematically shown in Fig. 1. The laser profile can be expressed as $a=a_0{e}^{-{\left(y/{\sigma }_L\right)}^2}{e}^{-{\left(t/{\tau }_L\right)}^2}$, where a₀ = eE₀/m_ecω₀ is the dimensionless amplitude of the laser electric field with e and m_e the charge and mass of electron, c the speed of light in vacuum, and E₀ the laser peak electric field strength, σ_L = 4.5λ_L is the focus spot radius and τ_L = 4T₀ is the pulse duration. The probe pulse is polarized along the same direction and propagates from the right to the left. Its profile is given by $a^{\prime }=a_0{e}^{-{\left(y/{\sigma }_L\right)}^2}$ sin⁴(πt/τ_L′), with τ_L′ = 3T₀. Both laser pulses have the same peak intensity of 10²³ W/cm². The wire target is located between x₀ = 1 μm and x₁ = 7 μm, with radius of 0.2 μm. The wire material is graphite with realistic density of 1.2 g/cm³, corresponding to a number density of n_e = 360n_c, where n_c = 1.1 × 10²¹ cm⁻³ is the critical density for λ_L = 1 μm laser pulse. Since the laser intensity under consideration is extremely high, we assume the target to be completely ionized (C⁶⁺) and its initial temperature is 1 keV. Here the skin-depth of the plasma $\delta =({\lambda }_L/2\pi )\sqrt{n_c/n_e}$ can be resolved, while the initial Debye length is extremely small. However, due to the abrupt increase of the plasma temperature, the Debye length becomes much larger soon so that it can be resolved in only 1T₀ of the laser-wire target interaction.

 

Fig. 1 Schematic view of ultra-bright γ-ray flashes emission and attosecond e⁻e⁺ bunches generation by two ultra-intense colliding lasers irradiating a micro-wire target.

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2.1. Attosecond electron bunches generation

High-energy-density electrons are vital to stimulate the QED effects in ultra-intense laser fields because they determine the cross-section of the electron-photon collision. During the laser-wire interaction under discussion, the amplitude of the driving laser’s electric field is sufficiently strong to immediately break the Langmuir oscillation of electrons on the wire surface [21], so that some electrons in the skin layer could be spontaneously extracted into vacuum. For a linearly-polarized laser pulse, the surface electrons are mainly dragged out by negative transverse laser electric field (E_y < 0), forming a series of sawtooth-like spikes as shown in Fig. 2(a). Due to the non-uniform distribution of the laser electric field in the transverse direction, the laser ponderomotive force acting on the attosecond electrons closer to the wire surface is much stronger than that in the margin region. Therefore, these electrons experience both stronger transverse laser electric field and transverse ponderomotive force, making them flee the wire surface and resulting in an arrow structure in space. From the simulation results above, one sees that the longitudinal thickness of each bunch is merely 0.2 μm, which is much less than the wavelength of the laser pulse. This indicates that the duration of each bunch in time is exactly in the attosecond domain, i.e., 660 as at t = 8T₀, which is at the same level as in [22, 23]. On the other hand, we find the electron number density in each spike is up to 100n_c, with charge of about 5 nC. The obtained peak electron density in our scheme is several orders of magnitude higher than in the gas scenario [19] and at least 3 times larger than those obtained in the NCD plasmas [17].

 

Fig. 2 Density distribution of attosecond electron bunches at t = 8T₀ (a). Phase-space distribution of electrons (x, p_x) at t = 8T₀ (b). Evolution of the electron energy spectrum (c). Distribution of the electron divergence angle at t = 8T₀ and 12T₀ (d).

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In such a geometry, the electrons on the wire surface will be dragged out with a large transverse momentum. Meanwhile, the J × B force turns the transverse motion of electrons into the direction of laser’s propagation so that the electrons can be further accelerated by the longitudinal ponderomotive force [24]. The laser ponderomotive acceleration or so-called direct laser acceleration is quite efficient for electron acceleration, which is usually dominant in the context of ultra-intense laser-plasma interaction. In this regime, the ponderomotive force is proportional to the laser intensity and reciprocal to the laser duration, i.e., ∼ $\pi a_0^2/2{\tau }_0$, so that an ultra-short ultra-intense laser pulse is capable of accelerating electrons to energies up to $0.255a_0^2\text{MeV}$ [25]. Thus, in a few laser cycles the electrons can be immediately accelerated to relativistic velocities and therefore remain in the acceleration phase for a long time. However, the compact electron structure will be deteriorated due to the space charge effect as well as the ponderomotive force scattering. Both effects result in an increasing beam dispersion at a later stage. In order to maintain the compact structure meanwhile accelerate the electrons to sufficiently high energy for the γ-ray emission, the electron bunches are expected to collide with the probe pulse as soon as they flee the right wire tip. Finally, the attosecond electron bunches can acquire sufficient energy in a short time with the maximum energy up to 10 GeV, as shown in Fig. 2(c). At t = 9T₀, their collisions with the probe pulse take place and the radiation loss of electrons becomes significant after the collisions, e.g., at t = 13T₀. Figure 2(d) shows the electron divergence before and after the collisions. We see the attosecond electron bunches have a high collimation before the collisions (∼ 10°), which is very favorable for the γ-ray emission at the following stage.

2.2. Ultra-bright ultra-short γ-ray flashes emission

In the nonlinear QED regime, two significant quantum effects dominate the laser-plasma interaction at ultra-high laser intensities. Firstly, ultra-relativistic electrons quiver violently in the laser fields and concomitantly radiate a large amount of photons in the synchrotronlike way. Alternatively, as the electrons counter-propagate with the laser pulse, they are able to scatter off the incident laser photons to high energy γ-photons in the strong laser fields. When the photon energy in the electron rest frame is comparable to the electron rest mass, multi-photon scattering occurs, which is called multi-photon nonlinear Compton scattering (NCS) [26, 27], i.e., (e⁻ + mω_l → e⁻ + ω_γ), where ω_l and ω_γ denote the laser photon and γ-photon, respectively. Secondly, the emitted high energy γ-photons can further decay into e⁻e⁺ pairs via the B-W process, and therefore generate copious e⁻e⁺ pairs. Under the quasi-stationarity and weak field approximation [28], the cross-section of these processes can be assumed to be only dependent on the following two crucial gauge-invariant parameters:

Figure 3(a) and 3(c) show the photon number density distribution during the γ-ray emission process, which can be divided into two independent stages (I) and (II). In the initial stage (I), the collision between the attosecond electron bunches and the probe pulse has not yet occurred and the emission results from the electron acceleration in the strong laser fields. This is actually the synchrotronlike radiation. It is interesting to see that several ultra-short γ-ray flashes are emitted along the motion direction of the attosecond electron bunches. These denser γ-photons are generated and mainly concentrated at the tail of each electron bunch. By contrast, there are only a few photons emitted on the laser axis. We can interpret it from the electron energy and dynamical motion at this stage. Here, the radiation power is proportional to the fourth power of electron energy, i.e., $P_{\gamma }=e^{2}c{\gamma }_e^4/6\pi {\epsilon }_0{\rho }^2$, where ρ is the radius of the trajectory and ε₀ is the permittivity of vacuum [30]. For each attosecond electron bunch, the electrons located farther away from the micro-wire surface are extracted much earlier and therefore obtain more energies from the laser pulses as compared to those near the wire surface. Meanwhile, the radius of the trajectory is also much smaller for these high energy electrons. Thus these electrons with higher energy and smaller curvature radius emit more photons. However, near the wire surface the electrons are just extracted by the laser fields and have a small energy so that we do not see any dense photon emission here. On the other hand, these photons have a large divergence angle and fly away from the micro-wire surface as soon as they are emitted by the electrons. Finally we see higher power and brighter γ-ray emission at the tail of each electron bunch, as is seen in Fig. 3(a). Figure 3(b) presents the angular distribution of electron bunches at 9T₀. As expected, the photon emission is concentrated in a cone along the direction of ±45° with respect to the wire axis. The corresponding cut-off energy is up to 4 GeV, as shown in Fig. 3(e).

 

Fig. 3 Density distribution of high energy γ-photons at t = 9T₀ (a) and 11T₀ (c). The red dashed lines mark the micro-wire locations. Angle-resolved photon energy spectrum at t = 9T₀ (b) and 11T₀ (d). Here the cone angle φ = 12° and θ = 20°. Evolution of the photon energy spectra (e), total photon number, and photon average energy (f). Here (I) and (II) mark different stages of the photon emission during the laser-wire interaction.

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In the second stage (II), the attosecond electron bunches start leaving the wire tip and subsequently collide with the probe pulse. As a result, a large amount of γ-photons are emitted via the multi-photon NCS process. It is clear to see that the resulting γ-ray flashes have a similar structure to the attosecond electron bunches. The duration of each pulse can be approximately estimated as ${\tau }_{\gamma }={\tau }_e+{\tau }_L/4{\gamma }_e^2$ with τ_e the electron bunch duration. For ultra-relativistic electrons with γ_e ≫ 1, the second term above in r.h.s. is negligible and the γ-ray source has almost the same duration of the attosecond bunch. On the other hand, these photons have a high collimation and are located almost at the same positions of the attosecond electron bunches. Note that these electrons are ultra-relativistic here, which can follow the photons at this stage. Figure 3(d) presents the photon angular distribution after the collision. We see these photons propagate along the laser axial with a smaller cone angle of 20° and source size of 0.2 × 2 × 0.8 μm³. Compared to the emission at the stage (I), the photon number increases significantly and the average photon energy is also boosted sharply as seen in Fig. 3(f). For example, at t = 11T₀, the photon cut-off energy is as high as 8 GeV and the average photon energy is about 24 MeV with an equivalent total photon number up to 1.7 × 10¹³. This indicates that nearly 10% of the total laser energy is transferred to the γ-photons.

Since ultra-intense laser pulses are employed in our scheme, the laser-wire target interaction is highly nonlinear and high-order multi-photon scattering occurs. The number of the absorbed laser photons is a random value with its own distribution law, but the first order of scattering has the largest cross-section which occupies the major portion of the scattering spectrum [31]. Finally, we may estimate the energy of the scattered photons on assumption of a head-on collision of electrons and photons by virtue of the formula below [32].

2.3. High-energy-density attosecond positron bunches production

It is well-known that a plane EM wave of arbitrary intensity and frequency is impossible to create e⁻e⁺ pairs from vacuum [37]. In our configuration, copious e⁻e⁺ pairs are generated via the multi-photon B-W process in the interaction zone of the two colliding focused laser pulses. The collision between ultra-relativistic electron bunches and the probe pulse begins at t = 9T₀, while the pair creation is gradually enhanced from the collision onward. Here, the electron energy and density determine the final γ-photon emission via the NCS process, while the pair creation is highly dependent on the γ-photon’s distribution. Finally, the electron bunches are able to transfer the feature of the ultra-short duration to first the photons and then positrons, so that we can obtain bright γ-ray flashes and compact attosecond positron bunches, as is seen in Fig. 4(a). Once generated, these positrons are also accelerated by the laser pressure of the driving pulse, like the attosecond electrons, and propagate forward along the laser axial direction. Figures 4(b) and 4(c) present the positron energy spectrum and divergence angle distribution. We see the positron emission angle is relatively small, with a cone angle of 10° at t = 11T₀, which is beneficial to the further applications in practice. Meanwhile, the positron bunches have a broad energy spectrum ranging from several MeV to GeV. The corresponding mean energy is up to 338 MeV at t = 11T₀, which is even higher than the parent γ-photon’s energy (several tens of MeV) due to the laser acceleration of positrons. The equivalent final positron number in 2D is about 1×10¹⁰ and the corresponding laser energy conversion efficiency to the positrons is about 0.07%, which is at least 3 times larger than in the B-H experiments [11] and is much higher than that in the laser wakefield scenario [8, 18]. In theory, we can estimate the final positron number by [17]

 

Fig. 4 Energy distribution (a) and energy spectra (b) of positrons. Positron divergence angle distribution at t = 9T₀, 10T₀, and 11T₀ (c).

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3. Robustness of the scheme

At ultra-high laser intensities, nonlinear effects during the QED processes become remarkable and the electron dynamics is strongly modified by the radiation reaction [30, 38]. With the increase of the laser intensity, the cross-section of collisions between electron-photon and between photon-photon increases so that both the number of pairs and the laser energy conversion efficiency increase significantly. Figure 5(a) shows the effective temperature of the attosecond electron bunches against the laser dimensionless parameter a₀. In these simulations, we keep all other parameters unchanged but only vary a₀. We see the electron temperature tends to be saturated after a₀ = 300. This nonlinear phenomenon results from the modification of radiation reaction [20, 39]. At a relatively low laser intensity, the electron acceleration is more important and the radiation loss of electrons is not serious. Once the radiation damping force becomes comparable with the Lorenz force [30], $F_R/F_L=(2/3)k_0{r}_e{\gamma }_e^2{\beta }_{e^{-}}{a}_0~1$, the electrons dynamics changes and the electrons act as an energy converter, so that more laser energy is transferred to the γ-photons. Here, k₀ = 2π/λ₀ and r_e is the classical electron radius. This corresponds to a threshold a₀ ∼ 320, assuming an average relativistic factor $\overline{{\gamma }_e}=500$, which is in excellent agreement with the simulation result above. Figure 5(b) presents the final yield of secondary particles and the corresponding laser energy conversion efficiency against the parameter a₀. As the laser intensity increases, both the production of γ-photons and positrons and the corresponding laser energy conversion increase. Especially, once a₀ > 450 the positron production becomes highly efficient.

 

Fig. 5 Effective temperature of attosecond electron bunches against the laser dimensionless parameter a₀ (a). Final yield of secondary particles and the corresponding laser energy conversion efficiency against the parameter a₀ (b). Here, the total number of photons and positrons, and the conversion efficiencies are all normalized by the corresponding maximal value in the case with a₀ = 750. Number density distribution of γ-photons in full 3D simulations at t = 11T₀ (c). The xz panel shows the projection of the photon density. Energy distribution of positron bunches in space at t = 11T₀ (d). The insets in (c) and (d) show the corresponding energy spectra.

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Finally, we carry out a 3D simulation to demonstrate the feasibility of the scheme. Here, the box size is x × y × z = 14λ_L × 12λ_L × 12λ_L, sampled by 1400 × 480 × 480 cells and 12 pseudo-electrons and 6 pseudo-carbons in each cell. All parameters are the same as in the 2D cases except that the z direction is considered. Figures 5(c) and 5(d) present the density and energy distribution of γ-photons and positrons, respectively. It is shown that the peak brightness of the γ-ray flashes is up to 1.8 × 10²⁴ photons s⁻¹mm⁻²mrad⁻² per 0.1%BW at 24 MeV. These γ-photons can subsequently decay into high-energy-density positron bunches via the B-W process. The total positron number is about 1 × 10⁹ per shot with a clear spatial attosecond structure. As compared to the 2D simulations above, both the yield and the number density of γ-photons and positrons decrease, since the expansion of the produced attosecond electron bunches along the z direction should be taken into account in the 3D configuration. However, the resulting γ-ray flashes and the attosecond structure of positrons are obvious. At t = 11T₀, the positron energy achieved is up to 600 MeV and the photon density is as high as 300n_c. This demonstrates that the underlying physics during the laser-wire interaction in the full 3D configuration does not vary. With the progress of projected PW laser facilities in the near future, the resulting ultra-bright ultra-short γ-ray flashes and dense attosecond positron bunches may be tested soon, which might have diverse applications in various fields.

4. Conclusion

In summary, we propose a novel scheme to generate ultra-bright ultra-short γ-ray flashes and high-energy-density attosecond positron bunches by using multi-dimensional QED-PIC simulations. It is shown that the attosecond electron bunches are extracted from the micro-wire by the laser transverse fields and are accelerated forward to ultra-relativistic energies by the laser ponderomotive force. As soon as the attosecond electron bunches flee the wire tip and collide with the probe pulse, ultra-brilliant γ-ray flashes are emitted with ultra-high brightness of 1.8 × 10²⁴ photons s⁻¹mm⁻²mrad⁻² per 0.1%BW at 24 MeV. These γ-photons can subsequently decay into high-energy-density positron bunches via the B-W process. It is shown that the total positron number in full 3D simulations is about 1 × 10⁹ per shot with a clear spatial attosecond structure and over critical density. These unique properties are beneficial for further applications in laboratory astrophysics and high energy physics in the future.