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Abstract
Bright $\gamma$-ray sources are of great significance for fundamental research, medicine, and industry. However, $\gamma$-ray production by bremsstrahlung, Compton scattering, and synchrotron radiation is often subjected to large divergence, wide size, and/or low efficiency, making it difficult to achieve high-brightness $\gamma$-ray sources. Here, we have found an efficient mechanism to generate $\gamma$-rays with, to our knowledge, unprecedented brilliance by use of an ultrarelativistic electron beam with nC charge incident onto a solid surface at grazing incidence. With this interaction configuration, extreme high quasistatic magnetic fields up to the gigagauss level or effective electric fields up to ${{10}^{14}}\;\text{V}/\text{m}$ are induced by the background electron backflows at the target surface. Subsequently, the electron beam is strongly focused by such fields by over an order of magnitude to submicrometer diameter, and its density is increased beyond the solid density; the induced effective fields are high enough to trigger quantum electrodynamics (QED) effects. These result in the production of extremely brilliant dense $\gamma$-ray beams with photon energy reaching multi-GeV, where the electron-to-photon energy conversion efficiency can exceed 60%. This offers a promising route to push the beam–target interaction to the QED regime, leading to ultrabright $\gamma$-ray sources for various applications.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. INTRODUCTION
Bright high-energy photon beams may pave the way to new frontiers of research in strong-field quantum electrodynamics (QED) physics [1,2], photon colliders [3], relativistic laboratory astrophysics [4,5], photonuclear physics [6,7], and high-resolution imaging [8,9]. For example, collimated high-density photon beams with energies in the MeV to GeV range can greatly enhance the possibility of photon–photon collisions and luminosity, making it possible to investigate matterless photon–photon physics such as light–light scattering, the linear Breit–Wheeler process, and other QED processes [1–3,10–12]. In addition, high-density energetic photon beams may allow one to access some important unexplored physics regimes [13–15] and to explore in the laboratory various extreme astrophysical phenomena from energetic dense lepton-dominated jets to gamma-ray bursts [16–19].
So far, bright high-energy radiation sources mostly rely on large-scale and expensive facilities, such as synchrotrons and ${ x}$-ray free-electron lasers (XFELs) [20,21], which are typically limited to photon energy ranging from a few keV to hundreds of keV. New ${ x}/\gamma$-ray sources based upon relativistic laser–plasma interactions [22–26] exhibit attractive properties, such as ultrashort duration and relatively compact size; however, their brilliance and radiation efficiency are very limited. In recent years, there has been growing interest in developing brilliant $\gamma$-ray sources and increasing photon energy, density, and flux, and various schemes have been proposed, such as ultra-intense laser–plasma interactions [27–35], multiple colliding laser pulses [36,37], electron–solid collisions [38,39], and laser–electron collisions [40,41]. Furthermore, one of the primary applications for upcoming multi-PW laser facilities is to produce high-flux $\gamma$-ray pulses through the inverse Compton scattering technique [42,43], which is expected to obtain photon energy up to 19.5 MeV and high brilliance in the range of ${{10}^{20}}\! -\! {{10}^{23}}\;\text{photons}\;{\text{s}^{- 1}}\;{\text{mm}^{- 2}}\;{\text{mrad}^{- 2}}\;\text{per}\;{0.1}\%$ bandwidth [44]. Nevertheless, they are still well below the requirements of many cutting-edge applications in terms of photon energy and brilliance. The efficiency of $\gamma$-rays produced with currently available lasers is roughly a few percent level. Due to laser ponderomotive scattering, disruption, and/or plasma instabilities, the resulting photon beams typically have wide size, large divergence, and low density. Therefore, it is very challenging to attain collimated brilliant $\gamma$-ray sources with high density and efficiency.
In this work, we present a scheme to tackle this challenge based upon the interaction of an ultrarelativistic electron beam with two solid slabs, which involves new physics of effective electron beam pinching and intense photon emission in the QED regime. As a result, a collimated brilliant $\gamma$-ray beam can be stably obtained with photon energies reaching multi-GeV and high efficiency, making a major step towards the development of future compact bright $\gamma$-ray sources.
2. PHYSICAL SCHEME
As well known, high-energy photon emission in intense electromagnetic fields represents one of the primary QED effects [1,2], which can be determined by the radiation parameter ${\chi _e}$. For ultrarelativistic electrons with energy ${\varepsilon _b} = {\gamma _b}{m_e}{c^2}$ (${\gamma _b} \gg 1$), this parameter can be written as
To overcome these challenges, we introduce a simple yet very efficient approach by utilizing a single electron beam interacting directly with solid slab targets, as shown schematically in Fig. 1, where one of the slabs (called the interacting plate) is slightly tilted relative to another (called the baffle plate) so that the electron beam is incident with a grazing angle towards the interacting plate. The interacting plate serves as a focuser to pinch the injected electron beam, while the baffle plate placed parallel to the beam propagation direction mainly acts to enhance beam focusing and to prevent the beam from diverging (see Supplement 1 for more details). When the electron beam approaches the interacting plate, a huge return current is induced at the target surface. This gives rise to rapid formation of gigagauss-class self-generated magnetic fields, which can confine and focus the injected electron beam. More importantly, the electron beam can be pinched along the interacting plate, where beam density and its self-fields increase significantly. Subsequently, it expels background electrons out of the target surface, such that ultra-intense static fields of up to ${{10}^{14}}\;\text{V}/\text{m}$ are excited in the transverse direction. This leads to both strong beam focusing and high-energy photon emission, producing ultrabright GeV $\gamma$-rays.
Fig. 1. Schematic illustration. A high-flux ultrarelativistic electron beam propagates between two solid slabs, where the slab at the top parallel to the propagation direction acts as a baffle to enhance beam focusing, while the tilted slab at the bottom acts as the interacting target to pinch the electron beam by self-generated magnetic fields and to produce collimated $\gamma$-rays. The red arrows indicate the orientations of the transverse focusing forces experienced by the beam.
3. ELECTRON BEAM PINCHING DYNAMICS AND GAMMA-RAY EMISSION
To quantitatively demonstrate the proposed scheme, we carry out three-dimensional (3D) particle-in-cell (PIC) simulations with the QED-PIC code EPOCH [46]. The simulation window has a size of $10\;{\unicode{x00B5}\text{m}}(x) \times 8\;{\unicode{x00B5}\text{m}}(y) \times 8\;{\unicode{x00B5}\text{m}}(z)$ with $400 \times 320 \times 320$ grid cells, which moves along the $x$ direction with velocity $c$, where the macro-particles per cell for the beam electrons, target electrons, and ions are 27, eight, and eight, respectively. The absorbing boundary conditions are utilized for both fields and particles. An example of the injected electron beam has about 5nC charge, 5 GeV mean energy with Gaussian momentum distribution, energy spread of 5% full width at half maximum (FWHM), 4 mm mrad normalized emittance, Gaussian spatial distribution of ${n_{b0}}\text{exp}({- \frac{{{r^2}}}{{2\sigma _ \bot ^2}} - \frac{{{{({x - {x_0} - vt})}^2}}}{{2\sigma _\parallel ^2}}})$ with ${n_{b0}} = {10^{27}}\;{\text{m}^{- 3}}$, ${\sigma _\parallel} = {\sigma _ \bot} = 2\;{\unicode{x00B5}\text{m}}$, ${x_0} = - {\sigma _\parallel}$, and velocity $v$ along the $x$ axis. For the parameters considered, the electron beam self-current is much less than the Alfvén current limit ${I_b}/{I_A}\sim{10^{- 3}}$, where ${I_A} = {m_e}{\gamma _b}{c^3}/e\; \approx 17{\gamma _b}\text{kA}$. Such beams could be generated either by conventional accelerators with a plasma lens [47,48] or laser plasma accelerators [49]. As an example, we take the targets initialized as fully ionized carbon plasmas, which in practice correspond to a carbon or plastic target. Targets made of other materials can also be used. Here, two solid slabs are used as the baffle and interacting targets, which have a 200 μm longitudinal length and number density of ${n_p} = {10^{29}}\;{\text{m}^{- 3}}$. The baffle target is parallel to the beam propagation direction, while the interacting slab is tilted with an angle of ${\theta _0} \approx 0.6^\circ$. Additional simulation results indicate that the effects of Coulomb collisions, bremsstrahlung radiation, and pair production on electron beam pinching and high-energy photon emission can be neglected, as detailed in Supplement 1. In experiments, one may use a movable target tape to attain efficient generation of brilliant $\gamma$-ray pulses. For example, thin solid foils can be wound onto a reel and positioned at the beam focus by a tape target delivery system, which allows operation with high repetition rates [50,51].
We first investigate the electron beam pinching process, as illustrated in Fig. 2. For a relativistic electron beam propagating in vacuum, the self-generated electromagnetic fields experienced by the beam are greatly suppressed since the electric term ${{\bf E}_ \bot}$ is almost exactly offset by the magnetic term ${\boldsymbol \beta} \times {\bf B}$, leading to the radiation parameter ${\chi _e}\sim 0$ and thus ineffective photon emission. When the energetic electron beam propagates between two solid slabs, the electric and magnetic fields cannot be cancelled out [see Figs. 2(d) and 2(e)]. This is attributed to the formation of ultra-intense background return currents [Fig. 2(f)], which gives rise to strong self-generated magnetic fields exceeding gigagauss. Consequently, the magnetic term can exceed the electric term associated with the electron beam, leading to magnetic focusing of the beam and subsequent radiation emission. In addition, the electric field formed by the charge separation at the target surface is conducive to attracting and confining the beam to the plasma target, thus further enhancing beam pinching. During propagation, the beam diameter can be focused by more than an order of magnitude to a submicrometer scale [see Fig. 2(a)]. Accordingly, the beam density and its self-fields increase drastically, such that the plasma electrons are expelled out of the target surface, as shown in Figs. 2(b) and 2(c). This will excite extremely high static fields of more than ${{10}^{14}}\;\text{V}/\text{m}$ [see Fig. 2(g)], where ${\chi _e}\sim 1$, so that QED effects become prominent and the beam–target interaction enters the QED regime. In this way, the electron beam is not only drastically shrunk to submicrometer size, but also most of its energy can be efficiently transferred to intense high-energy photon emission.
Fig. 2. (a) Evolution of the electron trajectory and corresponding energy in 3D geometry. Spatial distributions of (b) plasma density (${n_p}$), (c) beam density (${n_b}$), (d) electric field (${E_y}$), (e) magnetic field (${B_z}$), (f) longitudinal electric current density (${J_x}$), and (g) effective transverse field (${E_{\text{eff}}}$) at $x = 160\;{\unicode{x00B5}\text{m}}$ are given in the ($\xi ,y$) plane, where $\xi = x - ct$. The inset in (g) shows distribution of the effective transverse field in the ($\xi ,z$) plane.
In the following, we explain the underlying physics of magnetic field formation responsible for beam pinching. As the electron beam travels through the target, a balancing electron flow or the return current from the plasma target is induced immediately with the average velocity ${v_p}$ moving backward along the target surface. By using electron magnetohydrodynamic equations [52,53], the self-generated magnetic fields in plasmas satisfy
To illustrate details for the beam pinching and photon emission processes, we show the evolution of the beam electron density, plasma density, transverse electric field, and photon density in Fig. 3. It is evident that the electron beam traveling between two slab targets can be well pinched by the induced transverse fields, as seen in Figs. 3(a) and 3(b). As the beam shrinks, its density increases dramatically since ${n_b}\sim{n_{b0}}r_{b0}^2/r_b^2$, where ${n_{b0}}$ is the initial beam density, and ${r_{b0}}$ and ${r_b}$ are the initial and present beam radii, respectively. Therefore, the space-charge force generated by the electron beam is large enough to push the plasma electrons away from the target. The electron beam is attracted to the tilted target by the ion Coulomb force and pinched by the self-generated magnetic fields. As a consequence of these strong surface fields, most of the beam electrons can be confined and focused to a very thin region near the target surface. The beam radius can be reduced to about 0.2 µm or about a tenth of its initial size for this case [Fig. 3(a1) compared with Fig. 3(a3)], and subsequently its density is increased by more than two orders of magnitude to over ${{10}^{29}}\;{\text{m}^{- 3}}$. This induces super-strong static fields beyond ${{10}^{14}}\;\text{V}/\text{m}$, such that strong-field QED effects can be triggered, producing a collimated $\gamma$-ray beam with submicrometer width and extreme high density. Figure 3(c) shows distributions of the emitted photon density, where the peak density can be as large as a few times of ${{10}^{29}}\;{\text{m}^{- 3}}$ after a propagation distance of 200 µm.
Fig. 3. Distributions of (a) plasma density (${n_p}$) and beam density (${n_b}$), (b) effective transverse field (${E_{\text{eff}}}$), and (c) photon density (${n_\gamma}$) at $x = 10\;{\unicode{x00B5}\text{m}}$, $100\;{\unicode{x00B5}\text{m}}$, and $200\;{\unicode{x00B5}\text{m}}$. Transverse distributions of electron beam density at the beam center at these three positions are shown in (a1)–(a3). In the right column in (c), the inset displays the transverse distribution of the $\gamma$-photon beam along the green dashed line.
Figure 4 shows the energy spectrum of the electron beam and $\gamma$-rays and their energy efficiency evolution. After interacting with two slab targets, the conversion efficiency of the electron beam to an ultra-intense $\gamma$-ray pulse is as high as 58%, where maximum photon energy can reach nearly 5 GeV or close to the initial electron energy. In this scenario, QED correction plays an important role in radiation emission, which has an upper bound in the energy of the emitted photons, that is, the photon energy cannot exceed the kinetic energy of the electrons. The emitted $\gamma$-rays are highly collimated with only a $4 \times 2$ mrad FWHM divergence. Assuming a $0.2 \times 0.5\;{\unicode{x00B5}\text{m}}$ source size as shown in Fig. 3(c), a 6 fs long duration, and $1.2 \times {10^{11}}$ MeV photons, we can estimate the $\gamma$-ray peak brilliance at about $2.5 \times {10^{28}}\;\text{photons}\;{\text{s}^{- 1}}\;{\text{mm}^{- 2}}\;{\text{mrad}^{- 2}}\;\text{per}\;{0.1}\%$ bandwidth at 1 MeV. For 1 GeV $\gamma$-rays with $3 \times {10^7}$ photons within the 0.1% energy bandwidth, one can still obtain ultrahigh brilliance of about $6 \times {10^{27}}$ (usual units). Such energetic $\gamma$-rays are inaccessible to traditional radiation sources such as synchrotrons and XFELs.
Fig. 4. (a) Energy conversion efficiency ($\eta$) of the electron beam to $\gamma$-rays as a function of the interaction distance. The inset displays the angular distribution of the emitted $\gamma$-rays. (b) Final $\gamma$-photon energy spectrum after the interaction distance of 200 µm. The inset displays the initial and final energy distribution of the electron beam.
To fulfill intense high-energy $\gamma$-ray emission, ${\chi _e} \gt 0.1$ is required, where ${\chi _e} = ({{\varepsilon _b}/{m_e}{c^2}}){E_{\text{eff}}}/{E_s}$, and ${E_s} = m_e^2{c^3}/e \hbar\, \approx 1.3 \times {10^{18}}\;\text{V}/\text{m}$ is the critical field of QED. In our configuration, the induced magnetic field provides the dominant contribution to the effective field $| {{{\bf E}_{\text{eff}}}} | = | {{{\bf E}_ \bot} + {\boldsymbol \beta} \times {\bf B}} | \approx | {{\boldsymbol \beta} \times {\bf B}} |$, where the amplitude of the magnetic field can be estimated roughly by $B \approx 2{I_b}/c{r_b}$. Therefore, the effective field can be estimated by ${E_{\text{eff}}}[{\text{V}/\text{m}}] \approx 6 \times {10^7}{I_b}[\text{A}]/{r_b}[{{\unicode{x00B5}\text{m}}}]$. For a given beam profile with duration of about 13 fs in our case, for example, one can further get ${E_{\text{eff}}}[{\text{V}/\text{m}}] \approx 4.6 \times {10^{12}}{Q_b}[{\text{nC}}]/{r_b}[{{\unicode{x00B5}\text{m}}}]$, where ${Q_b}$ is the total beam charge. Finally, the QED parameter can be rewritten as
For example, if the minimum radius of the beam can be focused to ${r_b} \approx 0.1\;{\unicode{x00B5}\text{m}}$ after optimizations, the threshold of the total beam charge should be ${Q_b} \gt 0.3\;\text{nC}$ for 5 GeV electrons to satisfy ${\chi _e} \gt 0.1$. Note that for the simulation parameters given above with ${Q_b} = 5\;\text{nC}$, ${\varepsilon _b} = 5\;\text{GeV} $, ${r_b} \approx 0.2\;{\unicode{x00B5}\text{m}}$, one has ${\chi _e} \approx 0.9$ according to Eq. (5), indicating that this scheme is in the QED dominant regime.
4. EFFECTS OF TARGET AND BEAM PARAMETERS
We now discuss the robustness of this scheme in terms of target and beam parameters. One of the key parameters is the inclination angle of the interacting target. It is found that both electron beam focusing and subsequent radiation would benefit from the interacting target aligned at a small angle. As long as the target angle ${\theta _0} \lt 1^\circ$, collimated brilliant $\gamma$-ray flashes can be efficiently generated, with up to 60% electron-to-photon conversion efficiency and extraordinary brilliance reaching $3 \times {10^{28}}\;\text{photons}\;{\text{s}^{- 1}}\;{\text{mm}^{- 2}}\;{\text{mrad}^{- 2}}\;\text{per}\;{0.1}\%$ bandwidth at 1 MeV [Fig. 5(a)]. It is also noted that, when the angle of the tilted target is reduced to below 0.5°, the resulting photon emission becomes saturated. Physically, it implies that beam pinching has achieved its limit. Beam pinching may be analyzed qualitatively as follows, assuming a cylindrically symmetric geometry for simplicity. Beam focusing in the generated magnetic fields can be described by estimating the electron trajectory with
where the azimuthal magnetic field $B \approx 2{I_b}/cr$ is the maximum field strength at the beam edge. We assume the beam density is uniform for simplicity, that is, ${I_b} = - \pi e{n_{b0}}c\sigma _ \bot ^2$. Since the self-generated magnetic field is characterized by an exponential decay along the transverse direction, its average field strength can be assumed to be $\bar B \approx B \exp (-1)$. As the beam is focused to a smaller diameter, the particles initially at the beam edge are no longer located at the edge, and the magnetic fields experienced by them are reduced, so there should be a reduction factor $f \lt 1$ for electron beam pinching. Operating on Eq. (6) with $dr/dt$, its integral givesFig. 5. Effects of (a) target inclination angle (${\theta _0}$) and (b) initial electron beam length (${\sigma _\parallel}$) on radiation efficiency and peak brilliance ($\text{photons}\;{\text{s}^{- 1}}\;{\text{mm}^{- 2}}\;{\text{mrad}^{- 2}}\;\text{per}\;{0.1}\%$ bandwidth).
The effect of electron beam length on photon emission is illustrated in Fig. 5(b). It is found that a relatively long electron beam is beneficial for efficient generation of bright $\gamma$-rays. When the beam pulse is longer than 2 µm, the electron-to-photon energy conversion efficiency is quite high, up to 61.2%, and the emitted $\gamma$-rays are extremely brilliant. However, when the initial electron beam length is reduced to 1 µm, the radiation efficiency and brilliance of $\gamma$-rays decrease drastically. Because electron beam pinching takes time to develop, actually the beam head is not well focused. This is different from the scheme of $\gamma$-ray generation based upon beam–foil collision [39], which requires ultrashort electron beams with submicrometer length.
5. CONCLUSION
In conclusion, we have presented an efficient and simple scheme to generate brilliant GeV $\gamma$-rays by grazing a single relativistic electron beam onto a solid target. In this scheme, the electron beam is first focused transversely over an order of magnitude to submicrometer size and pinched to solid density, which is difficult to achieve with other methods, such as laser-based methods. With this highly focused beam, ultrahigh surface static fields are excited and the beam–target interaction is driven in the QED regime. This remarkably enhances the electron-to-photon energy conversion efficiency to over 60%. Due to the high photon flux, submicrometer size, and milliradian divergence, the produced $\gamma$-ray beams are extremely dense and bright, reaching unprecedented levels. This opens up a new way for the development of future compact $\gamma$-ray sources for both fundamental and applied research, such as exploring laserless strong-field QED physics, photonuclear physics, and extreme astrophysical systems relevant to $\gamma$-ray emission.
Funding
National Natural Science Foundation of China (12205186, 12135009, 11991074, 11975154); National Postdoctoral Program for Innovative Talents of China (BX20220206); Strategic Priority Research Program of Chinese Academy of Sciences (XDA25050100); Science Challenge Project (TZ2018005).
Acknowledgment
The authors thank the Supercomputer Center at SJTU for providing computing resources.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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