Perspectives on relativistic electron–positron pair plasma experiments of astrophysical relevance using high-power lasers

Relativistic electron–positron (⁠ $e − e +$⁠) pair plasmas are abundant in high-energy astrophysical systems, such as those associated with neutron star and black hole environments.^1–8 The interactions of $γ$-photons with each other and with strong magnetic fields—which exceed the quantum critical field $B c = 4.4 × 10 13$ G in some systems^9,10—lead to prolific pair creation via electromagnetic cascades.^4,11–16 These pair plasmas are relativistically hot (with temperature $T ± >$ MeV) and can be accelerated to high speeds (with Lorentz factors reaching up to $γ ± ∼ 10 4$⁠) in the relativistic winds or jets associated with these compact astrophysical objects.^4,17 Important examples are pulsar magnetospheres,^14,18–20 relativistic jets from active galactic nuclei (AGN),^21–24 and gamma-ray bursts (GRBs),^7,25 all believed to be associated with relativistic pair plasmas. Collective plasma processes, such as streaming instabilities,^26–28 magnetic reconnection,^29–31 collisionless shocks,^32,33 and turbulence,^34,35 shape the magnetic field dynamics, energy partition, and radiation emission in these systems. The plasma behavior in such extreme, relativistic, and often radiative, environments can depart significantly from that of traditional non-relativistic plasmas studied in the laboratory and in space. Despite significant advances in theoretical and numerical studies of relativistic pair plasmas, our understanding of the plasma dynamics at these exotic, but rather important, regimes is still limited.

The fundamental interest and astrophysical importance of relativistic pair plasmas has motivated efforts to produce and study them in the laboratory.^36–40 While the full range of conditions associated with compact astrophysical objects is unattainable in terrestrial laboratories, there is a significant value in the development of experimental platforms that would enable studies of the basic collective processes associated with pair plasmas in a controlled environment. Such experiments can offer a unique avenue for testing theoretical and numerical models and developing a deeper understanding of the plasma physics that shapes high-energy astrophysical systems.

Advances in high-power lasers are enabling the generation of $e − e +$ beams in high atomic number (high-Z) metal targets, using either direct irradiation of the target by an intense laser pulse^38,41–44 or by a relativistic electron beam produced by laser wakefield acceleration (LWFA).^40,45–48 In the past two decades, significant progress has been made in controlling and characterizing pair production with lasers. Experiments using direct laser irradiation of a gold target have now produced $∼ 10 12$ relativistic pairs,⁴² and experiments based on LWFA have reached pair beam densities of $∼ 10 15$ cm⁻³.⁴⁰ There is now a better understanding of how the number of pairs and their density scales with laser energy, and how to control the pair beam duration, divergence, energy, and charge neutrality by tuning both laser and target parameters.

At the same time, the continued increase in laser intensity is starting to open new avenues to explore pair production in ultra-strong fields that can exceed the quantum critical (or Schwinger) electric field $E c = 1.3 × 10 16$ V/cm⁴⁹ leading to prolific pair production in vacuum. Under such conditions, the photon mean free path to one-photon pair decay can become small and lead to the onset of a quantum electrodynamic (QED) cascade with high multiplicity.⁵⁰ This is a tantalizing prospect that would enable the generation of very dense relativistic pair plasmas in the laboratory and enable controlled studies of the interplay between strong-field QED and collective plasma processes.^50–54 

In this paper, we discuss the current status of relativistic pair production with intense lasers and give our perspective on future developments and opportunities for experimental laboratory studies of relativistic pair plasmas and their relevance to astrophysical systems. We note that other schemes not involving powerful lasers have been proposed, and are being explored, for the generation of pair plasmas. These include trapping of positrons from radioactive sources,³⁷ nuclear reactors,^39,55–57 and particle accelerators.^36,58,59 While these schemes also hold promise for the generation of pair beams and plasmas, they will not be discussed in this paper, which is focused on laser-based configurations.

This paper is organized as follows. In Sec. II, we start with a general discussion of the astrophysical and fundamental plasma physics motivation to study relativistic pair plasmas and the requirements for laboratory experiments to access the basic processes associated with these plasmas. We review the main mechanisms for pair production in experiments with intense lasers in Sec. III. In Sec. IV, we discuss the current status of relativistic pair generation and characterization using intense lasers. In Sec. V, we provide our perspectives for future laboratory studies of relativistic pair plasmas based on the current status and expected advances. Finally, in Sec. VI, we present our conclusions.

In the past decade, extraordinary discoveries associated with extreme astrophysical plasmas have excited scientists and the public alike—from the first images of the plasma orbiting a black hole^60,61 to the high-energy cosmic rays and radiation produced by relativistic jets and gamma-ray bursts^62,63 to fast radio bursts from galactic neutron stars.⁶⁴ It has long been recognized that the plasmas at the core of these extreme environments are relativistic and often electron–positron pair dominated.^1–4 Pair-driven plasma processes shape the dynamics and energy partition in these systems over a wide range of scales and conditions—from highly magnetized regions near the compact objects to the interaction of very dilute relativistic pair beams with the intergalactic medium at 100 Mpc distances—in ways that are not well understood. This is schematically illustrated in Fig. 1.

Relativistic pair plasmas can be produced around neutron stars and black holes by interactions of $γ$-ray photons with each other or with strong magnetic fields in the magnetospheres of these compact objects [Fig. 1(a)]. For example, particles accelerated near the neutron star surface emit energetic $γ$-rays that propagate across a magnetic field of near quantum critical strength (⁠ $B ∼ B c$⁠) and decay into an $e − e +$ pair. The freshly produced pairs undergo acceleration and also emit $γ$-rays, triggering an avalanche of pair production.^4,11–16 Newly born pairs start screening the accelerating electric field, thus preventing further particle acceleration. Similar electromagnetic (EM) cascades occur near the event horizon of supermassive black holes. These EM cascades fill the magnetospheres of compact objects with pair plasma and might comprise a significant part of the observed flux of cosmic rays (CR) high-energy positrons. They are expected to fill relativistic jets launched by compact objects with pair plasma.¹⁷ The pair collective plasma motions in the discharge process can excite coherent EM waves that provide critical observational signatures of these extreme environments.⁶⁵ The most intriguing class of neutron stars are magnetars, whose magnetic field strength can reach $∼ 10 15$ G.^9,10 This field strength significantly exceeds the QED critical field $B c$⁠. Above this value, fascinating QED phenomena arise, such as photon splitting due to nonlinear QED vacuum birefringence.^66,67 The equivalent Schwinger critical electric field $E c$ is capable to transfer the electron rest-mass energy, $m e c 2$⁠, over the reduced Compton wavelength, $λ ¯ c$⁠, giving rise to prolific pair production in vacuum, a phenomena that is exponentially suppressed at lower field strength $N ± ∝ e − E c / E$⁠.^49,68 Understanding pair-plasma generation in these extreme fields and the interplay between QED and collective plasma processes is a frontier topic in plasma physics, with important implications to the understanding of the radiative signatures of compact astrophysical objects.⁵² 

A common observational feature of high-energy astrophysical plasmas is the broad non-thermal radiation spectrum, which is thought to be produced by energetic particles accelerated at relativistic shocks, turbulence, or during magnetic reconnection (or by a combination of these). In the highly magnetized regions of relativistic jets and winds, it is thought that magnetic reconnection and turbulence [Fig. 1(b)], which may be triggered by large-scale magnetohydrodynamic (MHD) plasma instabilities,^69–72 can be responsible for the efficient conversion of the magnetic energy into high-energy non-thermal particles and radiation.^{29,34,71,73,74} In lower magnetization regions, radiation mediated shocks^33,75 and internal collisionless shocks,^76,77 both of which are shaped by pair-plasma instabilities, will further drive gamma-ray emission [Fig. 1(c)]. Photon–photon induced cascades will also load the upstream region of external shocks in GRBs with relativistic pairs^78,79 potentially impacting their magnetic field amplification via streaming instabilities,^28,80,81 particle acceleration,^32,82 and afterglow emission in GRBs^83,84 [Fig. 1(d)]. In all of these cases, relativistic pair plasma processes have an important impact in particle acceleration and radiation emission.

Even at very large (⁠ $>$ Mpc) distances from the central engine, very high-energy gamma-rays produce highly relativistic and extremely dilute pair flows. It has been proposed that pair-driven plasma instabilities can lead to significant energy loss of the pairs on timescales faster than inverse Compton scattering and potentially explain the absence of enhanced GeV emission that would be expected from inverse Compton cascades in TeV blazar jets.^85,86 This would also have important implications for current constraints on the intergalactic magnetic fields.⁸⁷ While the growth rates associated with plasma micro-instabilities are indeed very fast,^88,89 the nonlinear phase of the instabilities for high Lorentz factors and extremely low density ratios between the pair beam and ambient plasma are not well understood.^28,90,91

The ability to produce relativistic pair plasmas in the laboratory and study the basic processes and instabilities (in particular their nonlinear stages) that control energy partition in the plasma over a wide range of conditions, including different magnetizations, pair Lorentz factors, and radiative regimes, would be highly valuable. This would provide a unique opportunity to advance our understanding of these exotic plasma regimes and benchmark current theoretical models and numerical simulations of the role these plasma processes play in astrophysical environments.

Pair plasmas represent a paradigm for the study of basic plasma science, and many open questions exist regarding these unique systems.³⁶ A central research topic in plasma physics is instability.^27,92–95 Of primary importance to many of these physical systems are the so-called streaming instabilities, which arise in plasmas hosting interpenetrating flows—a common configuration in high-energy astrophysical and laboratory plasma environments. Relativistic pair-driven streaming instabilities develop when the relative velocity between the pair plasma flows and the ambient medium approaches the speed of light and constitute a long-standing problem in fundamental plasma physics, with direct and far-reaching consequences for high-energy astrophysical environments. These instabilities control the energy transfer between relativistic flows, fields, and particles in these environments.^{27,28,32,90–93,96,97} However, because the properties of these relativistic pair plasmas can depart significantly from those of traditional non-relativistic plasmas studied in the laboratory and in space, our understanding of the basic plasma processes in this frontier regime is still very limited. Important open problems include the competition between different instabilities, their saturation level, and nonlinear evolution, the impact of the ambient magnetization, and the coupling with radiative processes in strong-field environments.

In pair plasmas, there is no separation between electron and positron kinetic scales, and this has qualitative implications for the properties of the waves and instabilities that develop in these systems. A very important goal for the study of pair-driven plasma instabilities is, thus, to understand the long-term evolution of micro-instabilities, their role on the electromagnetic field dynamics, particle acceleration, and radiation emission, and how they influence (and are influenced by) the large-scale dynamics of the system.

The study of basic kinetic collective plasma processes—instabilities, shocks, magnetic reconnection, turbulence—in relativistic, pair-dominated environments is, thus, very important to build a systematic knowledge base for the development of theoretical models and for the interpretation of high-energy astrophysical observations. Advances in spacecraft measurements, dedicated laboratory experiments, computational power and numerical tools, and analytical theory have led over the past decade to mature understanding of these basic processes in non-relativistic electron–ion plasmas. However, for relativistic, pair-dominated regimes, this understanding is still scarce. The ability to produce and probe relativistic, pair-dominated plasmas in the laboratory represents a unique opportunity to better understand these processes and to help develop better analytical and numerical models.

In the following, we discuss some of the important requirements for experimental studies of collective processes in relativistic pair plasmas.

To enable the study of collective processes in relativistic pair plasmas, laboratory experiments need to meet a set of basic conditions.

In order to minimize the number of pairs required to study collective behavior, one should maximize the pair plasma density and produce relativistic pairs with low to moderate Lorentz factor (⁠ $γ ± ≲ 10$⁠). For example, for a relativistic pair plasma with density $n ± = 10 14$ cm⁻³ and $γ ± = 10$⁠, the number of pairs that needs to be produced is $N ± ≫ 3 × 10 12$⁠.

To investigate the behavior of pure pair plasmas, these need to be quasi-neutral, meaning that the number of electrons and positrons produced should be comparable, i.e., $n e + / ( n e − + n e + ) ∼ 0.5$⁠. While pair production schemes will typically lead to the same number of secondary electrons and positrons being produced, the primary source of energy for the pair production can be dominated by one of the species. As we discuss below, many laser-driven pair production schemes will first convert the laser energy into relativistic electrons, and the latter will then generate the high-energy $γ$ photons required for secondary pair production. Because of this, electrons are typically the primary source of energy and tend to dominate the resulting number of charged particles. Different pair production schemes will enable different levels of control over the resulting neutrality. We also note that it is of interest to explore the physics of mixed-species plasmas containing electrons, positrons, and ions both from a fundamental perspective and for its relevance to many astrophysical systems of interest, where the exact composition ratio of the different species is not well understood.²³ 

Given that pair production by high-energy photons intrinsically requires a minimum photon energy equal to the rest mass energy of the electron–positron pair (i.e., $> 1$ MeV), the majority of the pairs produced are naturally relativistic. Thus, this enables the generation of relativistic pair plasmas or flows with $γ ± ≫ 1$⁠.

In studies of pair-driven instabilities, the duration of the processes of interest is typically some multiple $δ$ (typically $∼ 10$⁠) of the inverse instability growth rate, i.e., $τ = δ / Γ$⁠, which corresponds to the saturation of the linear phase and transition to nonlinear evolution. The micro-instabilities mentioned above are typically the fastest growing in relativistic pair plasmas and, therefore, are more accessible experimentally. Other instabilities of relevance, including MHD instabilities, will require significantly longer pair plasma duration.

Ultimately, the electron–positron annihilation time needs to exceed the timescales of interest. The annihilation timescale is $τ a ∼ 1 / ( π r e 2 c n ± )$⁠,¹⁰³ where $r e$ is the electron classical radius. For $n ± ∼ 10 20$ cm⁻³, $τ a ∼$ ms, which is far greater than the ps–ns typical laser-driven pair-plasma interaction time. Thus, annihilation is not expected to be an issue for the laser-driven pair-plasma studies considered in this paper.

It will be important to guarantee that the pair plasma or beam does not expand significantly during the timescale of the processes of interest, $τ$⁠. That is, it is important to guarantee that $v th τ < L ±$⁠. As discussed above, the duration of the processes of interest is, for example, a multiple of the inverse growth rate of some pair plasma instability, i.e., $τ = δ / Γ$⁠. This condition can further constrain the requirements in terms of pair density and number to study a given process. As an example, we illustrate here how the requirements are changed for the study of the Weibel instability, considering its fastest growth rate given by Eqs. (5) and (6).

In practice, the minimum number of pairs required is given by the largest of Eqs. (1) and (7) for a pure pair plasma and Eqs. (2) and (8) for a pair beam interaction with an electron–ion plasma. We would like to stress again that Eqs. (7) and (8) illustrate only the case of the fastest growing rate of the Weibel instability. More generally, this condition needs to be evaluated for the duration of the process of interest.

Production of an electron–positron pair requires a minimum energy of $2 m e c 2$⁠. The energy source can be in the form of cosmic rays, fast electrons from earthly events like lightnings,¹⁰⁹ charged particles from terrestrial accelerators,^59,110 neutrons from nuclear reactors,⁵⁵ or light from powerful lasers.³⁸ 

The generation of pairs with lasers typically involves a multi-step process, where first the laser field accelerates electrons to relativistic energies, the electrons then radiate high-energy $γ$ photons, and finally, the $γ$ produce pairs via the interaction with the strong laser field itself, a nuclear field, or other photons.

In this section, we briefly describe the main pair production processes relevant to laser-driven experiments, comparing their likelihood (quantified in the cross sections or pair production rates). We then discuss the range of laser intensities associated with the different regimes of pair production.

Pair production near atomic nuclei is dominated by the so-called “Bethe–Heitler” (BH) process,¹¹¹ named after the two authors who first described the process in their seminal paper published in 1935. This is a two-step process, $e → γ$ and $γ → e − + e +$⁠.

We note that although pair production can also occur in the Coulomb field of electrons, the corresponding cross section scales as $σ γ → e − e + ∝ α ( r e ) 2 Z$⁠, which is at least an order of magnitude smaller for high-Z targets, and therefore, this process typically does not play an important role.

This mechanism is characterized by a very weak probability, making it difficult to produce a large number of pairs. Recent experiments at the Relativistic Heavy Ion Collider produced $≳ 6 × 10 3$ pairs via linear BW.¹¹⁰ Using high-power lasers, different experimental configurations have been suggested in the last few years to study this process. The main idea is that a high-intensity laser is used to produce a collimated $γ$-ray source via bremsstrahlung or Compton scattering, and this $γ$-ray source then collides with either a similar $γ$-ray beam (⁠ $γ − γ$ collider configuration)^120–122 or with a bath of x-rays produced by multiple high-power lasers in a gold hohlraum.¹²³ Theoretical and numerical calculations predict that the number of pairs that would be produced using state-of-the-art laser systems is $10 5 − 10 7$⁠,¹²⁰ which is relatively low for pair plasma studies, but could enable the first experimental demonstration of this fundamental process.

Figure 3 compares the pair production rates for the two dominant processes associated with the interaction of a high-energy photon with (1) a gold target via the BH process and (2) a laser with intensity of $5 × 10 22$ W/cm². The actual total number of pairs produced is dependent on the total number of high-energy photons and the duration of the interaction. Needless to say that only the interacting particles with energies above the pair creation threshold contribute. For laboratory studies, although highly relativistic electron beams can be driven by conventional accelerators as in earlier and more recent experiments,^130,131 there is now also significant interest in driving them using more compact high-power laser systems as discussed in Sec. III D.

The high-energy electrons and/or photons required for the different pair production processes discussed above can be provided by lasers when the focal brightness (commonly called laser intensity, which is used hereafter) is large enough. The electrons oscillating in the electromagnetic field of a laser acquire $v osc = a 0 c / γ$⁠, where $γ = [ 1 − ( v osc / c ) 2 ] − 1 / 2$ is the Lorentz factor of the electrons oscillating with quiver velocity $v osc$ and $a 0 ∼ 8.55 × 10 − 10 [ I ( W / cm 2 ) λ 0 2 ( μ m ) ] 1 / 2$ is the normalized vector potential for a linearly polarized laser. For example, for laser intensities $I λ 2 > 10 18$ W/cm $2 μ$m², $a 0 > 1$⁠, and it is possible to accelerate electrons to velocities near the speed of light. Above such laser intensities, the electron energy gain can significantly exceed MeV allowing the generation of the high-energy photons needed to produce electron–positron pairs and for significant energy conversion efficiencies to be reached between the laser and electron–positron pairs.

Broadly speaking, three main regimes can be identified for the dominant laser-driven pair production mechanisms when the electron acceleration is produced by the laser itself, e.g., via the interaction with a plasma. These regimes are defined according to the laser intensity, as shown in Fig. 4 for a typical 1 $μ$m laser wavelength, and discussed below.

In regime #1, for laser intensity $10 18 < I < 5 × 10 22$ W/cm², pair production is expected to be dominated by the BH process, where the pairs are produced near the nuclear field of a high-Z target by above threshold photons that are produced by laser accelerated electrons. We estimate this upper intensity limit by considering a photon energy $ϵ γ ∼ ϵ e ∼ a 0 m e c 2$⁠. By comparing the maximum pair-production rate of the different processes outlined above, we find that the BH process is dominant for $a 0 < 150$ (see Fig. 3), corresponding to a threshold laser intensity $I ∼ 5 × 10 22$ W/cm².

In regime #2, for laser intensity $5 × 10 22 ≤ I < 10 24$ W/cm², pair production through NBW is expected to be dominant. Similarly to the case of photons, electrons interacting with the laser field experience a quantum field strength $χ ≃ ( γ / E c ) | E ⊥ + v × B | ≃ γ ϑ E L / E c$⁠, where $v$ is the electron velocity, $E ⊥$ is the component of the laser electric field perpendicular to it, and the parameter $ϑ ∈ [ 0 , 2 ]$ is related to the geometry of the interaction. In the case of an electron beam counter-propagating with a laser, $ϑ ∼ 2$⁠. In the case of electrons directly accelerated by the laser either in a configuration of two counter-propagating lasers in a low-density (underdense) plasma or in the interaction of a laser with a solid-density plasma that reflects a large fraction of the laser light, $ϑ ∼ 1$⁠. In the latter case, considering that electrons acquire $γ ∼ a 0$ in the laser interaction, we can estimate $χ ∼ 2 I ( 10 24 W / cm 2 ) λ 0 ( μ m )$⁠. For the laser intensities in this regime, as $χ$ approaches unity, the probabilities for high-energy photon emission via nonlinear Compton scattering and pair production by NBW in the strong laser field rise steeply. The upper intensity limit for this regime is associated with two effects. In the case of the interaction of the laser with a solid-density plasma, at $I ∼ 10 24$ W/cm², electrons get so energetic that the plasma becomes transparent due to the relativistic mass correction. At this point, accelerated electrons move primarily in the same direction as the laser and the $χ$ experienced drops significantly leading to a strong reduction in pair production. In the case of two counter-propagating lasers in an underdense plasma (or in the presence of a seed electron population), $χ$ continues to increase with the laser intensity, and as $χ$ exceeds unity, it can give rise to the onset of QED cascades.⁵⁰ 

Regime #3, for laser intensities $I ≥ 10 24$ W/cm², is dominated by QED cascades. Above this threshold, the probability for pair production by NBW in the strong laser field is no longer exponentially suppressed. Emitted pairs are accelerated in the laser field and emit more $γ$ photons, which quickly decay into pairs, giving rise to a QED pair cascade. It is predicted that at $I ∼ 10 24$ W/cm², the number of secondary pairs becomes comparable to the number of primary particles,⁵⁰ and for $I > 10 24$ W/cm², the number of pairs can significantly exceed the number of seed particles, potentially enabling the generation of very dense relativistic pair plasmas.^132,133

The highest laser intensity currently achieved in the laboratory¹³⁴ is $I = 1.1 × 10 23$ W/cm². Such record intensities start to enable significant pair production by the BW process (regime 2 in Fig. 4). However, for the majority of currently available lasers, the operating intensities are in the range $I = 10 18 − 10 22$ W/cm² (regime 1 in Fig. 4). At these intensities, as discussed above, hot electrons can be accelerated up to tens of MeV, and the energy conversion efficiency from fast electrons to the above threshold (⁠ $> 1.02$ MeV) bremsstrahlung photons can reach over 50% for thick targets.¹¹⁶ Combined with the high cross section for pair production in high-Z targets, the BH process underlines the primary pair-production process for current experiments using high-power lasers.

We note that other processes could be explored to convert the relativistic electrons into the necessary $γ$ source for pair production. These include betatron radiation^135–137 and Compton scattering.^138–140 However, for current operating laser intensities, the efficiency of these processes can be significantly lower than bremsstrahlung, and therefore, they have not been widely explored for pair production. This could change at higher intensities, where, in particular, nonlinear Compton scattering^141–143 and synchroton emission^144–146 can become efficient as will be discussed in Secs. V B–V D.

Two main experimental approaches have been used in laser experiments of pair production (illustrated in Fig. 5). One is the “direct” approach, where the laser accelerates electrons and produces pairs on the same target, and the other is the “indirect” approach, where the electrons are first accelerated by the laser in a gas target and these electrons then bombard a second, high-Z solid target to produce the pairs. Note that the indirect approach follows the same process of positron generation used in electron–positron colliders where a high-Z target (tungsten, for example) is bombarded by GeV electrons from the accelerator.¹³⁰ In experiments using either approach, the laser energy is first transferred to electrons then to bremsstrahlung photons and finally to pairs.

The electron source can be directly measured in the indirect method, while it can only be inferred in the direct method as only electrons that escape the target are measured. Although in principle bremsstrahlung photons are measurable, it is difficult to completely characterize their spectrum as the photons have large flux (⁠ $> 10 20$ ph/s) and a wide spectral range from 10 keV up to hundreds MeV (with the upper limit being the maximum energy of the source electrons). The resulting pair number and energy distributions from experiments using both methods are directly observable using magnetic spectrometers^147,148 at a given solid angle. The spectrometer generally uses either film-like static detectors, such as image plates, or scintillating materials coupled to photon detectors, such as CCD.

A summary of representative state-of-art experimental results is given in Table I. As mentioned earlier, the direct measurements are on the electron and positron distributions. The number, density, and other parameters are derived from the spectra together with the laser pulse parameters.

In general, the laser-driven electron source for pair production experiments can be described by a Maxwell–Boltzmann type distribution $f e ( ϵ ) ∝ exp ( − ϵ / T e )$⁠, where $T e$ is the electron temperature, and $ϵ$ its energy. This distribution is typically a good approximation of the electron distribution from both the direct approach and the indirect approach when using processes like self-modulated LWFA (SM-LWFA)^157–159 or Direct-Laser-Acceleration.^160–162 However, in some cases, more complex expressions are needed, as, for example, in two-temperature distributions originating from different acceleration processes.^163,164 For electron beams from LWFA, a mixture of a broader Maxwell–Boltzmann distribution and a narrower (quasi-monoenergetic) distribution is often needed.^{137,165–168}

In experiments using the direct approach, the resulting positron spectrum can also be well-described by a Maxwell–Boltzmann distribution in general but at a different temperature from that of the electrons.³⁸ The positron spectrum from the indirect approach can be more complex due to its multi-distribution source function. For example, for a mono-energetic electron source, the indirect approach produces positrons (and electrons) with non-Maxwell–Boltzmann distribution, as illustrated in Fig. 6.

Examples of experimentally measured electron and positron spectra are shown in Fig. 7 for both approaches: (a) direct approach using a 100 J laser at intensity of about $5 × 10 19$ W/cm² on a 1 mm thick gold target with broad electron spectrum between 1 and 50 MeV (upper red) and a positron distribution between 3 and 30 MeV (lower red); (b) indirect approach using a laser with about 14 J energy and a peak intensity of about $3 × 10 19$ W/cm² on 5–40 mm thick lead targets,⁴⁰ with electron (upper blue) energy from 0.1 to 2 GeV and broad positron distribution (lower blue) measured from 200 to 400 MeV.

An interesting difference in the positron distribution from the two approaches is the additional energy “boost” observed in the direct approach. This near uniform energy shift of the positron energy distribution (e.g., Fig. 1 in Ref. 149) is due to the target normal sheath electric field¹⁶⁴ and is largely absent in the indirect approach. This may be caused by the combination of relatively less strong sheath fields and the wider positron energy distributions leading to a less pronounced change in the positron energy distribution.

In the following, we will discuss the deduction of the pair numbers and densities in the experiments.

Experimentally, each energy spectrum of electrons or positrons $f ( ϵ , θ )$ is measured at a given angle theta relative to the laser propagation (or target normal) direction. By integrating over the energy (⁠ $ϵ$⁠) and angle (⁠ $θ$⁠), one can calculate the total number of electrons (⁠ $N −$⁠) or positrons (⁠ $N +$⁠) as $N ± = ∫ ∫ f ( ϵ , θ ) dϵdθ$⁠. This means that to infer the total pair yield from an experiment, both the energy and the angular distribution are needed. The energy distributions are normally obtained by using magnetic spectrometers that can disperse the charged particles according to their kinetic energy and the angular distributions is either obtained experimentally^43,149,152 by placing the magnetic spectrometer at various angles or obtained via calculations.^{43,46,47,156,170,171}

So far, the highest pair number reported per laser shot is about $10 12$ from the direct method using 10 ps, 1 kJ laser pulse,^42,150 and $10 9$ from the indirect method using 0.04 ps, 14 J laser pulse.⁴⁰ The large difference in pair number from the two approaches is not surprising given the difference in laser energies used in the experiment: the maximum number of pairs produced is directly related to the number of laser accelerated MeV electrons, which, in turn, is proportional to the laser energy and intensity.

To further infer the pair density (n_±), the volume over which the pairs are created is needed. Inside the dense target, where positrons are produced, it is not currently possible to measure their volume directly. Positron characterization is typically done outside the target, but direct measurements using existing diagnostic techniques (such as interferometry or shadowgraphy) can also be challenging due to the relatively low pair density (⁠ $< 10 16$ cm³), long optical depth and short duration (< ns). Instead, the volume is often estimated as a frustum, as illustrated in Fig. 8: (a) the smaller base of the frustum is the source's transverse size, (b) the “slant height” is the pair spatial length, given as the product of pair beam duration and the pair velocity, and (c) the frustum's larger base is given by the pair's angular divergence together with (a) and (b). The accuracy of pair volume derivation is dependent on the accuracy of the measurement of (a), (b), and (c), or their estimates based on the physics processes that dominate them.

The positron source size can be measured using standard techniques, such as “pepper-pot” in 2D, or “knife-edge” in 1D. These techniques rely on the penumbra of the particles from the edge of the hole (in pepper-pot) or edge (in knife-edge). Although used in MeV positrons from the direct method,¹⁵² they are not applicable to high-energy (hundreds MeV) positrons where the source sizes are either calculated or assumed as a point source.

The pair beam duration is a convolution of electron source duration, the BH pair production rate, and transport time in the target. While the electron source duration can be approximated as the laser pulse duration due to the fact that electrons are accelerated directly in the laser field, this is not necessarily the case for positrons. The positron beam duration can be affected by small-angle scattering in the dense target. For example, calculations for a 1.5 mm thick gold target show that for 10 MeV positrons, the increase in beam duration due to scattering is about 190 fs.⁴⁷ This would be larger for thicker targets and, therefore, should be taken into account into the pair density evaluation.¹⁵⁶ In short, the increase in the positron beam duration over the laser pulse duration due to the transport is negligible for the direct method using multi-ps lasers,¹⁴⁹ but it can be comparable to that of the laser pulse length for much shorter laser pulse duration (tens of fs) used for the indirect approach and therefore needs to be considered^47,156 in positron volume evaluation.

The positrons originating from the BH process have a spatial distribution that depends on the source of electrons and the target material and thickness. The pair angular distribution can be measured as part of beam emittance characterization.¹⁵² Divergence measurements based on the direct method show that the sheath electric field at the target rear surface can reduce the angular divergence of positrons substantially.^152,172 In cases where the divergence is not measured, it is common to estimate it based on Monte Carlo simulations, although their accuracy depends on the physics included. For example, straggling from small-angle deflections has been shown to have a significant effect.^47,156 In addition, electromagnetic fields produced in the plasma can further affect the positron angular distribution,¹⁷² although these are not included in the Monte Carlo simulations.

Insofar, the reported maximum pair densities outside the target range from $n ± = 10 9 − 10 14$ cm⁻³ with^153,154 and without^43,149 magnetic collimation for the direct method using 100–1000 J lasers, and $n ± = 10 14 − 10 16$ cm⁻³ for the indirect method based on laser wakefield accelerated electrons.^40,46,173

We note that the maximum pair density can quickly drop with distance from the target, depending on the pair divergence. This can affect the pair plasma behavior or the instabilities produced by the interaction of the pair jet with an ambient plasma (as discussed in §II C) and as such must be considered carefully in experimental studies.

It is important to have control over the charge neutrality of the electron–positron beams produced both for studies of pair-driven instabilities in a background plasma or pair plasma studies. The level of charge neutrality of the electron–positron pair beams can be calculated^43,170,174 based on the individual electron and positron densities, which are estimated separately using the same method as described above.

While the number of electrons and positrons produced via the pair production mechanisms discussed above, including the BH (or Trident) process, is symmetric, the presence of the primary electrons will lead to an excess of negative charges in the resulting pair beams. Different strategies are being explored to control the level of charge neutrality. One possibility is to use thicker targets than can range out the majority of the primary electrons.^40,43,59 This is illustrated in the calculated data in Fig. 9. The ratio of total number of electrons and positrons outside the back of the target is calculated for four different thicknesses of a gold target at a range of pencil-type electron beams. A better than $50 %$ charge neutrality can be achieved for thick targets (10 and 50 mm) when using primary electron energies above 100 MeV. For a small (e.g., $2 °$⁠) collecting angle relative to the target normal direction, the electron and positron ratio can reach about $80 % − 90 %$⁠, as shown in the inset figure. We note that while using a thicker target to improve the charge neutrality can be advantageous at high electron energy (⁠ $> 100$ MeV), at lower energies, an important drawback is the reduced conversion efficiency. This is shown in Fig. 10.

A promising possibility for improving the charge neutrality with a lower energy electron source is to apply an external axial magnetic field that is designed to collimate only a narrow energy band of the electrons and positrons, resulting in a quasi-charge neutral electron–positron jet.^153,154 As discussed earlier, by using an externally applied magnetic field to the target, one should be able to remove the excess electrons from the pair beams and effectively achieve the charge neutrality without compromising the positron yield. For example, experiments have shown that the ratio of positrons to electrons in the pair beam increased from 0.01 without magnetic lens to 0.3¹⁵³ and 1¹⁵⁴ with the lens applied.

To project the pair yield (Y) for near future lasers, we now discuss the relationship between the BH pair production, the source electron temperature, and the target material without going into the detailed derivations, which have been discussed elsewhere.¹¹⁶ The pair yield is strongly dependent of the target Z (⁠ $Y ∝ Z 4$⁠) as each of the cross sections of the two-step process are proportional to $Z 2$⁠.^114–116 The pair yield further depends on the laser intensity via the source electron distribution n_e(E) produced by the laser and target plasma interactions. As will be discussed separately for the direct and indirect methods, the primary electron source ultimately determines the pair yield.

In the direct pair production configuration, the dependence of primary electron source distribution $f e ( ϵ )$ on the driving laser intensity has been studied for many years for applications, such as electron fast-ignition and secondary source (gamma, proton, and ion as well as neutron) generations. Basically, depending on the laser and target parameters, scalings, such as “ponderomotive”¹⁶⁴ and “Pukhov-scaling,”¹⁶⁰ have been used to describe the MeV electron-distribution for laser–solid interactions: $T e$ (MeV) ≈ $( 0.5 − 1.5 ) × [ I ( 10 18 W / cm 2 ) λ 0 2 ( μ m ) ] 1 / 2$⁠.^160,164,172 Together with the electron re-circulation in the target, this leads to a quadratic scaling of the positron yield with the laser energy $N ± ∝ ϵ laser 2$⁠, as demonstrated experimentally and shown in Fig. 11. This scaling was supported by analytical analysis and by particle-in-cell simulations.^42,172

Recently, a higher electron temperature scaling (superponderomotive) was found for multi-ps kilojoule (KJ) laser–plasma interactions: $T e$ (MeV) ≈ $5 [ I ( 10 18 W / cm 2 ) λ 0 2 ( μ m ) ] 1 / 2$⁠.^175,176 Although the scaling of this long-pulse regime is more favorable for the pair yield,¹⁵¹ current multi-ps lasers [such as NIF Advanced Radiographic Capability (ARC)¹⁷⁷] operate at relatively low laser intensity (⁠ $I ≈ 10 18$ W/cm²) for which the expected pair yield is lower than that obtained at sub-ps lasers operating at higher laser intensities (⁠ $I > 5 × 10 19$ W/cm²).

In the indirect pair production configuration, primarily electrons are typically produced by laser wakefield acceleration. The laser converts its energy into a plasma wake with some efficiency over a dephasing length, and then the electrons absorb the wake energy with a separate efficiency. The electron scaling is more complex because the acceleration mechanisms differ for different experimental setups for a wide range of laser intensities (10¹⁸⁻²⁰ W/cm² and plasma target with density from under to over critical density). Like that in laser–solid interactions, the laser interactions with low-density plasmas have been studied extensively for applications, such as advanced accelerator concepts and low-emittance photon sources. These applications require different electron distributions than that needed for pair plasma applications. As discussed in Sec. II B, to produce large enough relativistic pair plasmas (in skin depth units) for studying the underlying collective processes, we want to maximize $N ± / γ ±$⁠. This is naturally proportional to the charge and inversely proportional to the energy of the primary electron beam. So we are interested in maximizing the charge without significantly increasing the electron energy. The scaling of the beam charge and electron distribution with laser energy depends on the specific regime of LWFA considered.

We first consider the blowout regime of nonlinear LWFA, where the laser is matched to the blowout radius, and the pulse length is a fraction of it, as derived by Lu et al.¹⁶⁵ The accelerated beam charge scales with $( ϵ L a 0 ) 1 / 3$ and the matched laser duration scales with $( ϵ L / a 0 2 ) 1 / 3$⁠, where $ϵ L$ is the laser energy. In order to maximize the beam charge, we would like to maximize $a 0$⁠, but this can also put significant constrains on the required laser duration. In order to understand the optimal scaling for the duration currently envisioned for near future high-intensity laser systems, we consider a laser duration of 20 fs, which seems a reasonable lower limit.

For a fixed laser duration, the matched spot size is also fixed, and we have that $a 0$ scales with $ϵ L 1 / 2$⁠. This yields a beam charge scaling with $ϵ L 1 / 2$⁠. For this fixed laser duration, the accelerated electron beam energy would also be fixed, meaning that $N ± / γ ±$ would also scale with $ϵ L 1 / 2$⁠. For current 1–10 J laser systems, the produced beam charge at this laser duration is $∼ 0.1 − 1$ nC. For future laser systems, such as the recently proposed EP-OPAL, with 1 kJ, 20 fs, it could be possible to produce a 400 MeV electron beam with a maximum charge of $∼ 10$ nC.

The nonlinear bubble regime¹⁷⁸ can potentially lead to the injection of more charge than the blowout regime. However, in the case of the bubble regime, both the charge and the electron beam energy scale with $ϵ L 1 / 2$⁠. Therefore, in this regime, higher energy lasers do not bring a significant advantage in terms of the $N ± / γ ±$ scaling.

The linear LWFA regime, in particular, the SM-LWFA regime, can be more advantageous in terms of producing very high charge by using a laser pulse that is considerably longer than the plasma wake wavelength.^157–159 In this case, the electron beam spectrum will have a large energy spread in contrast with the nearly mono-energetic spectrum associated with the blowout and bubble regimes of LWFA. While the characterization of the energy conversion efficiency between the laser and the beam in SM-LWFA has not been thoroughly established, it seems reasonable to consider values of $∼ 10 %$ based on recent studies¹⁷⁹ that have demonstrated $11 %$ conversion efficiency in the generation of about 200 MeV, 0.7 $μ$C, electron beams using the kJ-class OMEGA EP laser. For 10 kJ-class laser systems, such as the NIF-ARC, this would potentially enable the generation of 10 $μ$C electron beams with similar energy.

Finally, we note that other schemes for electron acceleration are under investigation. In particular, recent experiments using Direct-Laser-Acceleration^160,161 demonstrated the generation of $≈$ 0.2 $μ$C electron beams.¹⁶² While this beam charge is slightly lower than that obtained from SM-LWFA under similar laser conditions, this could also be a promising source for electron–positron generation that should continue to be explored in the future.

The number and density of pairs produced by previous experiments is still below the requirements to produce a large enough pair plasma that supports collective behavior, as we will discuss in more detail in Sec. V E. However, this situation is expected to change over the coming years. As the energy and power of lasers increase, it is expected that not only will the number and density of pairs produced by the BH process significantly increase but also it will become possible to access different regimes of pair production associated with higher laser intensities. In this section, we will first estimate the pair beam parameters that could be produced with these near future laser systems and discuss their application to laboratory relativistic pair plasma studies. Finally, we will comment on the diagnostic development needed for the experiments.

For clarity and simplicity, we consider as relevant laser parameters for the BH pair-production regime laser energy of up to 10 kJ, pulse duration between 10 fs and 10 ps, and maximum laser intensity of $5 × 10 22$ W/cm², which corresponds to the transition to the BW dominated pair regime. For laser intensities below this threshold, the electron temperature is less than 500 MeV assuming an optimal superponderomotive coupling of five times the ponderomotive electron temperature.

For the pair generation target, we considered a 1 mm thick gold slab target in most cases, although the results of using thicker (up to 10 mm) target are discussed too. The goal is to provide a baseline picture for the discussion. The pair yield can be affected by the details of the target, such as target material and transverse size or shape. See Refs. 47, 116, and 172 for a more detailed discussion of how the pair yield depends on the target and laser parameters.

Depending on the scheme of pair-production considered, we assume the source electron distribution to be either (1) Maxwell–Boltzmann or (2) quasi mono-energetic. For the direct method based on laser–solid target interactions, we assume a $30 %$ energy coupling efficiency between the laser and electrons based on previous experimental results.¹⁷² We note that higher coupling factors, up to 90%, are possible if an under-critical density plasma is present at the laser–target interface.^176,180 In the case of the indirect method based on SM-LWFA, we assumed a $10 %$ energy conversion based on previous studies.¹⁷⁹ Finally, for the indirect method based on LWFA, we assume a quasi-monoenergetic electron distribution with a maximum 10 nC charge as discussed in Sec. IV C.

For simplicity, we consider a pulse duration of 10 ps for the electron source produced by laser pulses with similar duration in both direct and indirect methods based on SF-LWFA. This consideration is based on the fact this is currently the longest pulse duration for relativistic laser intensities and one for which there is a large number of experimental results.¹⁷² For the indirect method based on LWFA, we consider a pulse duration of $0.01 − 0.1$ ps for the quasi mono-energetic source.

For electron sources with Maxwell–Boltzmann distribution, the number of pairs produced as a function of input electron temperature has been well documented analytically and found to agree with experiments.^42,116,172 Based on previous studies, we assume that the fraction of pairs that escape the solid target is $∼ 10 %$ of the total number of pairs created.¹⁷² In Fig. 12, we show the estimated pair yield as a function of electron temperature from $1 − 500$ MeV for a 10 kJ laser and 1 mm thick gold target. Under such conditions, the predicted maximum positron yield is $N ± = 10 14$⁠. This is about two orders of magnitude higher than the current maximum positron yields achieved with kJ class lasers.⁴² 

For the indirect pair production approach based on the SM-LWFA scheme, we estimate a maximum positron yield of $N ± = 10 13$ for the same 10 kJ laser system and 1 mm thick gold target. For the LWFA scheme, we estimate the pair yield based on a monoenergetic electron beam interacting with either a 1 or 10 mm thick gold target.

A maximum positron yield of $N ± = 10 11$ could be achieved from a 10 nC electron beam driven by a kJ laser. This is about three orders of magnitude higher than the maximum currently achieved yield based on LWFA.⁴⁰ 

These estimates are based on the current understanding of the scaling of the different schemes. We note that the exact yields can depend on the details of the laser–target parameters, including target material,⁴⁷ size,⁴³ and shape.¹⁵⁰ 

The angular spread of the pair beam depends on the laser energy and target conditions. For the direct method, we consider a range of $10 − 40 °$ full width at half maximum (FWHM) angular spread based on previous measurements.^38,152,172 For the indirect methods, previous studies show divergence angles varying from a few mrad up to rad depending on the positron energy.^{46,47,173,181} We considered a range of $10 − 100$ mrad in our estimates.

Based on the previous discussion for the different pair parameters, we can estimate the pair density. The key to derive pair density is to estimate their volume, which can be described as frustum with the source size for smaller areas. The height is the pair length (duration) and pair source divergence determines the shape of the frustum.

The resulting pair density is shown in Fig. 13. For the direct pair production method, the range of densities estimated are shown in the red band. The upper and lower bounds represent the ranges covered by laser pulse duration of $1 − 10$ ps and positron divergence angle of $10 − 40 °$ (FWHM).

For the indirect laser pair production using monoenergetic electrons based on LWFA, the green and purple bands represent the range of pulse duration (⁠ $10 − 100$ fs) and positron divergence (⁠ $10 − 100$ mrad) considered for either a 1 mm (green) or 10 mm (purple) thick gold target. For pair production based on SM-LWFA, the estimated pair density is shown in the blue band for a range of pulse durations between 1 and 10 ps, a positron divergence of 10 mrad, and a 1 mm thick gold target.

Up now experimental studies have been limited to the laser intensity regime $I < 5 × 10 22$ W/cm², and the dominant pair production mechanism has been BH. Exciting recent developments in laser systems promise to open a new window for pair production studies at higher intensities in the near future like that achieved on Ref. 134. This has motivated important theoretical and numerical studies of pair production at ultrahigh intensities.^51,182–188 While there is no experimental data yet for this regime, we shall discuss the current understanding of the properties of pair beams and plasmas that could be produced based on theoretical arguments and existing numerical studies.

In the interaction of an ultra-intense laser with a solid target, while the target is overdense, the reflected laser light will create strong-field conditions in the counter-propagating laser field region that can lead to significant gamma-ray emission by Compton scattering of laser-accelerated electrons and pair production.^51,182,187 The condition that the plasma remains relativistically overdense limits the laser intensity to $∼ 10 24$ W/cm² for typical solid targets, such as aluminum. In this regime, pair production can be achieved via two main processes: BH and BW.

In the case of BH pair production, previous numerical studies considered a $∼ 100$ J laser with $I ∼ 10 23$ W/cm², $τ = 25$ fs interacting with a solid-density 50 $μ$m thick gold target.¹⁸⁷ It was shown that $10 13$ pairs, with a density of $∼ 3 × 10 19$ cm⁻³ and an average energy of 6 MeV, can be produced, which would enable the study of collective effects in relativistic pair plasmas.

In the case of BW, it was shown that a 300 J, 10 PW, 30 fs laser can produce $N ± = 10 10$⁠.^51,182 This is a negligible fraction of the number of particles of the target in the same volume. (Inside the target, the resulting plasma is not pair dominated.) However, as the relativistic electron–positron pairs escape from the rear side, they form a pair dominated jet with density $∼ 10 20$ cm⁻³. The pairs have an average energy of 200 MeV per particle, which can be of interest to study relativistic pair-driven instability from the propagation of this jet in an ambient electron–proton plasma.

In the interaction of ultra-intense counterstreaming laser-pulses, it is possible to accelerate and trap a population of seed electrons that can initiate a QED pair cascade. Electrons accelerated in the laser field will emit high-energy photons that will decay in $e − e +$ pairs. It was predicted that the threshold laser intensity for the development of a QED cascade is $I ∼ 10 24$ W/cm².⁵⁰ This exciting prospect has motivated a large number of theoretical and numerical studies.^{50,132,133,189–198}

For two counterstreaming lasers with 1 kJ each, such as the L4 laser at ELI,¹⁹⁹ a maximum number $N ± , max = 3.4 × 10 12$ pairs could be produced. This is roughly an order of magnitude larger than the largest number of pairs currently produced using the BH process at similar laser energy.⁴² 

The electrons and positrons produced during the cascade will be heated in the transverse direction by the laser fields, leading to comparable pair transverse and longitudinal momenta. Thus, the divergence angle of the resulting pair plasma is $θ = tan − 1 ( p ⊥ / p ∥ ) ∼ 45 °$⁠, and the pair density will drop due to plasma expansion following the cascade.

We note that these laser–laser cascades need to be seeded and that in order reach saturation of the QED cascade within the pulse interaction, it will be important to start already with a large number of electrons. This could be achieved, for example, by having the two lasers focused on a $μ$m-scale liquid hydrogen jet with density $∼ 5 × 10 22$ cm⁻³.²⁰⁰ For a laser intensity $I > 10 24$ W/cm², the plasma will be relativistically transparent. However, it will be important to address the role of plasma effects on the laser focusing and in establishing an optimal QED cascade.¹⁹⁶ This, as well as the control of the initial seed plasma density profile, could be achieved by using a thinner (sub $μ$m) liquid jet that is pre-expanded by the laser pre-pulse (or by a separate laser).²⁰¹ Additionally, the production of dense pair plasma from QED cascades will also require precise transverse alignment of the laser beams and adjustment between the laser duration and transverse spot size.²⁰² 

The interaction of high-energy electron beams and intense lasers can also lead to prolific pair production.^{54,141,142,203–205} The electron beams can be produced either by conventional accelerators, as in the SLAC E-144 experiment,¹²⁵ or by an intense laser. Recent numerical studies have shown that by colliding a GeV electron beam with a laser of intensity $I ≳ 5 × 10 23$ W/cm², it is possible to convert a significant fraction (⁠ $≳ 10 %$⁠) of the number of primary electrons into pairs.¹⁴² For typical electron beams with nC charge, the resulting number of pairs $N ± ≃ 10 9$ could still be relatively small for collective pair plasma studies. However, at higher electron energies, as $χ ≫ 1$⁠, the number of pairs could increase significantly via the onset of QED cascades, as shown recently.^54,205

The high-energy electron beam enables a strong boost of the laser field strength experienced by electrons in their rest frame $χ ≃ 0.8 ϵ b ( GeV ) [ I ( 10 22 W / cm 2 ) ] 1 / 2$⁠. For electron beams from current linear accelerators with $ϵ b ≳ 10$ GeV and currently available laser intensities, it is possible to reach $χ ≫ 1$⁠, opening an interesting route for the study of QED cascades and associated pair plasmas. In this scheme, the electron beam also provides the dense seeding population for the cascade, with $n ± ≃ χ n b$⁠. For highly compressed electrons beams, such as those being explored at the FACET-II facility at SLAC,²⁰⁶ beam densities $n b ≳ 10 20$ cm⁻³ could be produced in the near future, enabling the generation of pair plasmas with densities potentially approaching the relativistic critical density. The energy of the resulting pair is primarily dictated by the laser intensity with $γ ± ≃ a 0$⁠, and similarly to the laser–laser cascades its divergence angle is $θ ∼ 45 °$⁠.

Recent 3D particle-in-cell (PIC) simulations of the head-on collision of a 3 PW laser pulse (50 fs duration, 2.5 $μ$m waist, $3 × 10 22$ W cm⁻² intensity) with a 30 GeV, 1 nC, $4 × 10 20$ cm⁻³ electron beam indeed showed the possibility to produce a pair plasma with $N ± ∼ 10 11$⁠, $n ± ∼ 5 × 10 21$ cm⁻³, $γ ± ∼ 100$⁠, and $θ ∼ 45 °$⁠. As discussed in Ref. 205, such a configuration would open the possibility for the study of collective effects associated with the dense relativistic pair plasmas.

As in the case of laser–laser configurations, the production of dense pair plasma from laser–electron-beam QED cascades will also require precise transverse alignment and adjustment of the duration between the laser pulse and the electron beam.

The scalings for pair production with high-power lasers discussed for the different regimes are summarized in Figs. 14 and 15. We can see in Fig. 14 that while the number and density of pairs produced by previous experiments is still below the requirements to produce a large enough pair plasma that supports collective behavior, the availability of high-intensity kJ-class lasers is expected to enable the generation of pair plasmas with a size of up to $10 c / ω p$⁠. This would allow, for example, studies of the nonlinear regime of the Weibel instability in a relativistic pair plasma, following Eq. (7). Different approaches can be explored.

For 10 kJ, multi-ns lasers, such as NIF-ARC¹⁷⁷ and PETAL,²⁰⁷ that are now becoming available, both the direct method and the indirect SM-LWFA regime could enable the generation of the required pair plasmas with densities $n ± = 10 16 − 10 18$ cm⁻³ via the BH process. In parallel, upcoming short pulse (⁠ $≲ 30$ fs), ultra-intense lasers, such as the ones becoming available at ELI,¹⁹⁹ APOLLON,²⁰⁸ OMEGA-MTW/OPAL,^209,210 and Michigan-ZEUS,²¹¹ offer exciting opportunities to explore different regimes.

For example, at 0.1 kJ and $10 23$ W/cm², relativistic pair plasma jets with density $∼ 10 19$ cm⁻³ could be produced via BH in the laser interaction with high-Z solid targets.¹⁸⁷ In the collision between two kJ, $10 24$ W/cm² lasers, or between a 0.2 kJ, $3 × 10 22$ W/cm² laser and a 30 GeV electron beam, QED cascades can be produced in the laboratory giving rise to the generation of relativistic and ultradense pair plasmas. This is a tantalizing prospect, where under the action of strong fields, the coupling of QED processes and collective plasma effects becomes important in the pair plasma dynamics. This constitutes a new strong-field QED-plasma regime that is largely unexplored and is relevant to the understanding of the dynamics and radiative signatures of compact astrophysical objects,^{4,13–16,30,31} such as the environments of black hole and neutron stars—in particular, magnetars.^9,10

In Fig. 15, we observe that the relativistic pair beams currently generated in high-Z targets via the BH process (either based on the direct or indirect method) can already enable the study of pair-driven plasma instabilities associated with the propagation of these beams in an ambient electron–ion plasma. As an example, we consider the case of the relativistic Weibel, or current-filamentation, instability, in which growth rate, in the cold limit, is given by Eq. (5) and shown in Fig. 2. The pair number, density, and beam divergence produced in previous experiments (see Table I) puts current studies already in the linear regime.²¹² However, based on the experimental and simulation results, and associated scalings discussed in this section, we see that near future experiments could probe the saturation and nonlinear dynamics of these type of plasma instabilities, where we have assumed that the saturation time corresponds to ten e-foldings of the instability (⁠ $δ = 10$⁠) in Eq. (6).

Considering the propagation of a pair beam on a cold and denser background electron–ion plasma, the magnetic field produced by the Weibel instability at saturation (via magnetic trapping^91,213) has amplitude $B ( T ) ≃ 6.4 n ± ( 10 16 cm − 3 ) / [ n 0 ( 10 18 cm − 3 ) ] 1 / 2$ and the magnetic wavelength $λ ( μ m ) ≃ 33.4 / [ n 0 ( 10 18 cm − 3 ) ] 1 / 2$⁠, where we have considered the most unstable mode $k max = ω p 0 / c$ with $ω p 0$ the background plasma frequency. See Ref. 91 for a detailed discussion of the dependence of the field structure on the beam and plasma parameters. If $n ± / n 0 ≪ 1$⁠, the resulting background density modulation will be very small, $δ n / n 0 ≪ 1$⁠. However, in future studies, using high-energy lasers, the pair beam density can be comparable to the background density. For a pair beam of $n ± ∼ n 0 ∼ 10 18 cm − 3$⁠, the resulting B-field has $B ≃ 640$ T, and the background plasma electron density will develop strong modulations $δ n / n 0 ∼ 1$⁠.

Finally, we comment on the possibility of future experiments probing highly magnetized pair plasma regimes with $σ B ≥ 1$⁠. Given the dependence $B 0 ( T ) ∝ γ ± n ±$ from Eq. (9), magnetized studies favor the use of moderate-intensity (⁠ $I ≳ 10 19$ W/cm²) ps laser drivers producing moderate pair energy $γ ± ∼ 10$ and densities $n ± ∼ 10 16 − 10 18 cm − 3$⁠. For such conditions, the required external magnetic field is $B 0 = 10 2 − 10 3$ T. While challenging, recent advances in both pulsed-power and laser-driven magnetic delivery devices could produce the required magnetic field strengths over mm–cm scales. Current experiments have demonstrated the capability of producing 10–60 T magnetic fields using pulsed power systems^105,106,108 and tens to hundreds of T using laser-driven capacitor coils.^214–216 Moreover, new experimental schemes have been proposed to produce $≈$ kT field using the spinning angular momentum of intense lasers.²¹⁷ 

Future laboratory experimental studies of relativistic pair plasmas, be it from basic plasma modes to instabilities, particle acceleration, or radiative processes, will require advanced diagnostics to observe the signatures of the relevant plasma processes, including particle energy distributions, plasma density, magnetic field, and emitted radiation. This will require advances in current instruments in terms of sensitivity, precision, coverage, and better spatial and temporal resolving power, which we summarize below.

For example, to measure a 10% change in the energy distribution of electrons, positrons, or ions as a result of pair plasma driven instabilities or shocks,^32,218–220 one needs the particle energy spectrometer to measure with similar accuracy both the particle numbers as well as its energy. Although both the dispersion^147,221 and dose¹⁴⁸ have achieved better than $10 %$ accuracy, the noise/background of the measurement will need to be counted for the needed overall measurement accuracy. At the present, the detection threshold for electrons is about $5 − 10 × 10 7$ (sr MeV) $− 1$¹⁷⁴ for high-power laser experiments. Obviously, if the pair numbers are smaller, improvements must be made in the background reduction in order to reach the lower detection threshold. This can be done through experimental and diagnostic hardware design. For example, in addition to selecting the detectors with higher sensitivities to pairs but not to background photons, one can also increase the signal-to-noise ratio by adding shielding and reducing the secondary radiation from the spectrometer housing onto the detectors.

Another importance signature from the pair plasma interactions discussed above are the generation of magnetic fields (few to hundreds of Tesla with modulations on scales of few to tens of $μ$m). Because of its prominence in high-power laser–plasma interactions, diagnostics for electromagnetic fields have been studied extensively with techniques, including charged particle deflection,^222–225 Faraday rotation of the polarized light,^107,226 and atomic spectroscopy.^227,228 While the atomic spectroscopy is still being developed experimentally, and the Faraday rotation technique is not applicable for pure pair plasma,²²⁹ the short pulse driven charged-particle radiography technique^224,230,231 has been developed for over a decade, and it is widely used and remains to be a powerful electromagnetic field diagnostics. This is due to its high sensitivity, and $μ$m spatial and sub-ps temporal resolving capabilities. Recent developments in high-energy density plasma studies have established methods for extracting path integrated two-dimensional data, distinguishing between electric and magnetic fields, and interpreting complex field structures.^{222,232–236} Riding along the continuing diagnostic development, it is expected that charged-particle radiography will play an important role in the measurement of electromagnetic field from the pair plasma experiments.

As to the density modulations from the pair plasma interactions, which is estimated to be of the order of 10%–100% of background plasma densities of 10¹⁶–10²⁰ cm⁻³ with spatial size of $1 − 10 μ$m, the measurement techniques will strongly depend on the density level.

For a small volume (≤100 $μ$m³), low-density (≤10¹⁸ cm⁻³) pair plasma, either refractive index or scattering based density measurement techniques, such as interferometry, reflectometry, or coherent and incoherent scattering,¹⁰⁷ would have difficulties in obtaining sufficient signals over background. At a higher density (≥10¹⁹ cm⁻³), optical interferometry, shadowgraphy, or optical to x-ray Thompson scattering could be used.^107,237 Although not yet demonstrated, fluorescent spectroscopy might be of use to inform the local plasma density if tracer elements are used as dopant in the interaction regions.

Finally, the measurement of MeV to GeV radiation signatures of the above-mentioned pair plasma processes will be challenging too. Conventional single-counting, pulse-height based solid detectors are likely to have pileup issues due to the high photon flux that is typical to the radiation source produced in short pulse, high-power laser experiment. To use such a technique would require a large array (⁠ $≥$ 1000 $×$ 1000) of sensors to obtain sufficient statistics in single-counting mode. Low efficiencies in both photon dispersion techniques based on Bragg reflection/transmission and in detection due to the low scatting and stopping cross section of the photons in materials also need to be considered. Moreover, MeV–GeV photons from the laser- or beam–plasma interaction contribute as a background in the measurements.

Despite these difficulties, and due to its broad application in high-energy physics and high-power laser-driven high-energy density physics experiments, a large effort has been dedicated to improved MeV–GeV radiation diagnostics. This includes step-filters,^238,239 scintillating arrays,²⁴⁰ magnetic electron spectrometers that detect electrons from forward Compton scattering of MeV photons,^241–243 gas Cherenkov detectors,^244,245 or nuclear activation excited by MeV gammas.²⁴⁶ All of these works will benefit future pair plasma experiments.

Given the recent developments in diagnostics and new techniques now being proposed that may be applicable in the near future, we are optimistic that the diagnostics needed can be fielded in experiments to observe the signatures of the pair plasma interaction processes. This, of course, will require careful evaluation of the different signatures of the processes of interest and ensuring the necessary data quality and quantity due to the challenges discussed above.

We have discussed the current status of laser-driven generation of relativistic electron–positron pairs and progress toward the study of basic pair plasma processes of relevance to high-energy astrophysical systems. Based on current experimental and numerical studies and the understanding of the scaling of pair production using different laser-driven configurations, we presented our perspectives for future experimental studies in this exciting frontier topic. In particular, we discussed the possibility of producing relativistic pair plasmas and study their collective modes over a wide range of conditions, from different degrees of magnetization to the QED-plasma regime, where the interplay between quantum and collective processes becomes important. We have also addressed the prospects for studying streaming instabilities associated with the propagation of relativistic pair beams on ambient electron–proton plasmas. This is a fast developing field, motivated by exciting observational discoveries, theoretical and computational developments, and technological advances in light sources and diagnostics that are pushing the limits of our understanding of relativistic pair plasmas.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DEAC5207NA27344 and by SLAC under Contract No. DE-AC02-76SF00515. F.F. acknowledges the support by the U.S. DOE Early Career Research Program under FWP 100331, and H.C. the funding by LLNL LDRD (Nos. 17-ERD-010 and 20-LW-021).

The authors have no conflicts to disclose.

Hui Chen: Conceptualization (equal); Data curation (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Frederico Fiuza: Conceptualization (equal); Data curation (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.