Magnetic pinching of relativistic particle beams: a new approach to strong-field QED physics

In general, collimation and focusing of energetic electron beams are achieved by use of conventional magnetic devices. However, this technique has several drawbacks such as low field gradients, large sizes, slow switching times, and transverse asymmetry. These make it unsuitable for focusing high-current electron beams with large divergence and small size. Laser-wakefield acceleration from plasma [32] is a promising candidate for next-generation compact accelerators [33] and light sources [34], where the resulting electron beams typically have a few microns in size, several milliradians in divergence and hundred-kA-class currents, making it difficult to focus or collimate and transport them during applications. To address this problem, considerable efforts have been dedicated to developing novel types of focusing elements (e.g., plasma lenses) [35–38], which can provide the field gradients several orders of magnitude larger than quadrupole magnets. Many applications can benefit greatly from a well-focused relativistic electron beam, for instance, multistage plasma-based accelerators [39, 40], compact free-electron lasers [41–43], synchrotron radiation sources [44–46], brilliant betatron x-rays [47], and effective laser-Thomson scattering [48, 49]. Despite significant progress made in the last few decades, there is no applicable way to focus relativistic high-current electron beams down to the submicron scale.

To tackle the challenges of high-current beam focusing and high-field QED testing mentioned above, we propose a completely different approach by use of a properly designed hollow cone target; see figure 1(a). We demonstrate the scheme using three-dimensional (3D) particle-in-cell (PIC) simulations (see section 2.2 for details). As the high-flux electron beam traverses the cone target, ultra-intense return currents are formed to create gigagauss-scale magnetic fields that radially focus the beam inwards, so that the beam diameter is reduced by about an order of magnitude to the submicron scale, as shown in figures 1(b) and (c). Accordingly, the beam density can be increased hundreds of times, approaching the solid density. Such dense beam may enable a novel regime of radiation-dominated beam–plasma interaction and pave the way for future compact ultrabright γ-ray sources.

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To demonstrate the proposed scheme, 3D simulations with the QED-PIC code EPOCH [50] have been carried out. In addition to nonlinear QED effects, Coulomb collisions, pair creation and bremsstrahlung radiation are also taken into account in the code, but these have little effect on beam focusing and subsequent γ-ray emission. The size of the simulation window is $10\,\mu {\text{m}}\left( x \right)\, \times \,6\,\mu {\text{m}}\left( y \right)\, \times \,6\,\mu {\text{m}}\left( z \right)$ with grid cells of $400\; \times \;240\; \times \;240$, where the macro-particles in each cell for the beam electrons, target electrons and ions are 27, 8 and 8, respectively. The simulation window moves at the speed of light along the x-axis, where the absorbing boundary conditions are employed for both particles and fields.

We take an example of a 5 GeV electron beam with about 3.5 nC charge, 5% energy spread, 4 mm-mrad normalized emittance, and Gaussian distribution of ${\text{exp}}\left[ { - {r^2}/\sigma _ \bot ^2 - {{\left( {x - vt} \right)}^2}/\sigma _\parallel ^2} \right]$, where ${\sigma _\parallel } = {\sigma _ \bot } = 2\,\mu {\text{m}}$ and velocity $v$ along $x$. Comparable beam parameters are expected by using advanced accelerator techniques [51, 52] and/or laser–plasma accelerators [53, 54]. The electron beam current is much less than the Alfvén limit ${I_b}/{I_A}\sim {10^{ - 3}}$ for the considered parameters, where ${I_A} = {m_e}{\gamma _b}{c^3}/e\, \approx 17{\gamma _b}{\text{kA}}$, and ${\gamma _b}$ is the electron Lorentz factor. The beam and its self-fields are initialized in vacuum, which can be self-consistently resolved in the code. A hollow cone target is used as the focuser, which has a $190\;\mu {\text{m}}$ axially-longitudinal length, with a left aperture diameter of ${D_0}\; = \;4\;\mu {\text{m}}$ and a right aperture diameter of ${d_0}\; = \;0.6\;\mu {\text{m}}$. In order to induce the focusing fields quickly and to save computational resources, the entrance aperture of the cone is preferably taken to be comparable to the radial size of the electron beam. It is shown that the target can be made of a variety of materials such as aluminum or carbon, and the results obtained are similar. The target is initialized as fully ionized aluminum plasmas with a density of ${n_p}\; = \;1\; \times \;{10^{29}}{{\text{m}}^{ - 3}}$, where its initial temperature is 100 eV. Such cone targets are readily attainable with state-of-the-art target technologies [55], which have been applied in a variety of laser-based studies such as laser fusion [56, 57], high-energy-density physics [58], radiation reaction effects [59, 60], and laser microtube implosions [61]. However, the direct interaction of ultra-relativistic electron beam with cone target and the magnetic pinching have not been explored.

In the ideal case, the electron beam is incident along the target axis; if the beam is injected away from the axis, the off-axis distance should be much less than the exit radius of the cone. Otherwise the beam will lose a lot of energy because of the direct interaction of the beam with the cone wall in the later stage. For example, if the off-axis distance is within one third of the cone exit radius, the beam energy loss can be less than 20%. However, if the off-axis distance is as large as the exit radius of the cone, the beam energy loss reaches over 50% due to the significant interaction of the beam with the cone wall. In order to reduce the beam energy loss, an appropriately large cone outlet can be utilized.

When the beam propagates in the hollow cone, a large backflow current is rapidly formed because the free electrons in the cone wall move towards the opposite direction relative to the beam velocity to compensate for the magnetic field of the beam current. This occurs within the charge relaxation time [62] ${\tau _e}$, where ${\tau _e} = 1/4\pi \sigma$ is of the order ${10^{ - 18}}{\text{s}}$ and $\sigma \sim {10^{17}}{{\text{s}}^{ - 1}}$ is the electrical conductivity for usual conductors. In our scenario, since ${\tau _e}\; \ll \;{\tau _b}\sim 10{\text{fs}}$, beam propagation is accompanied by self-generation of intense quasi-static magnetic fields, resulting in beam pinching.

After passing through the cone, the beam can be pinched from its initial diameter of $4\;\mu {\text{m}}$ to approximately $0.4\;\mu {\text{m}}$, as seen in figure 2(a). To elucidate the underlying physics of beam pinching, it is necessary to analyze the role of electric and magnetic fields. It is well known that when a relativistic electron beam propagates in vacuum, the self-generated electromagnetic fields experienced by the beam are balanced since the electric term ${{\mathbf{E}}_ \bot }$ is almost exactly offset by the magnetic term ${\boldsymbol{\unicode{x03B2}}} \times {\mathbf{B}}$. With the presence of a cone, the two fields are no longer cancelled at the inner cone surface (compare figures 2(b) and (c)), where a strong spontaneous azimuthal magnetic field can be rapidly formed with amplitude comparable to the beam self-magnetic field and the beam's space-charge field is shielded at the target surface. This is because when the beam propagates within the cone target, ultra-intense return currents are formed by the target electrons flowing back along the cone walls (figure 2(d)). The generated magnetic field can be described by the electron magnetohydrodynamic equations [63, 64], which can be written as ${\mathbf{B}} = \frac{c}{e}\nabla \times {{\mathbf{p}}_{{\text{ret}}}}$, where ${{\mathbf{p}}_{{\text{ret}}}}$ is the background electron momentum in the cone target. For an electron beam with a length of ${l_b}\; \gg \;{v_b}/{\omega _p} \cong c/{\omega _p}$, where ${\omega _p} = {\left( {4\pi {n_p}{e^2}/{m_e}} \right)^{1/2}}$, the displacement current term ${c^{ - 1}}\partial {\mathbf{E}}/\partial t$ is about the order of magnitude ${\left( {{v_b}/{l_b}{\omega _p}} \right)^2}\; \ll \;1$ compared to the plasma electron current term, so it can be ignored. Furthermore, the magnetic field also satisfies $\nabla \times {\mathbf{B}} = \frac{{4\pi }}{c}{{\mathbf{J}}_{{\text{ret}}}}$. Here ${{\mathbf{J}}_{{\text{ret}}}} = - e{n_p}{{\mathbf{v}}_{{\text{ret}}}}$ is the background return current density with typical velocity $\left| {{{\mathbf{v}}_{{\text{ret}}}}/c} \right|\; \ll \;1$, ${n_p}$ is assumed to be constant, and the plasma electron momentum can be approximately given by ${{\mathbf{p}}_{{\text{ret}}}} \approx {m_e}{{\mathbf{v}}_{{\text{ret}}}}$. Then one can obtain ${\nabla ^2}{\mathbf{B}} - k_p^2{\mathbf{B}} = 0$, where ${k_p} = \frac{{{\omega _p}}}{c}$. In the cylindrical coordinates, ${\mathbf{B}} = {B_p}\hat \theta$ meets $\frac{{{\partial ^2}{B_p}}}{{\partial {r^2}}} + \frac{1}{r}\frac{{\partial {B_p}}}{{\partial r}} - k_p^2{B_p} = 0$. Therefore, one has ${B_p}\left( r \right) = \alpha {K_0}\left( {{k_p}r} \right)$, where $\alpha$ is a constant, and ${K_0}$ is the zeroth-order modified Bessel function. At the interface $r = {r_0}$, ${B_p}\left( {{r_0}} \right) = {B_b}\left( {{r_0}} \right) \propto - 2\pi e{n_{b0}}\sigma _ \bot ^2/{r_0}$, thus $\alpha = - 2\pi e{n_{b0}}\sigma _ \bot ^2r_0^{ - 1}\left[ {1 - \exp \left( { - \frac{{r_0^2}}{{\sigma _ \bot ^2}}} \right)} \right]/{K_0}\left( {{k_p}{r_0}} \right)$, where ${B_b}\left( r \right)$ is the beam self-magnetic field, and ${n_{b0}}$ is the initial density. In consequence, the magnetic force term can exceed the electric term of the Lorentz force near the inner cone surface, producing a net focusing force on the beam. In the central region of the hollow cone, the electric repulsive force of the beam is almost completely compensated by its magnetic field pinching. Hence, the beam can be confined and focused magnetically within the cone. As a whole, the beam undergoes a large pinching force ${{{\boldsymbol{f}}}_ \bot } = - e{{\mathbf{E}}_{{\text{eff}}}}$ (figure 2(e)) as it travels through the cone with a decreasing diameter, where ${{\mathbf{E}}_{{\text{eff}}}} = {{\mathbf{E}}_ \bot } + {\boldsymbol{\unicode{x03B2}}} \times {\mathbf{B}}$ is the effective transverse field. Figure 2(f) shows the distribution of the self-generated magnetic fields, which agrees with our simulations. The electron beam energy spectrum is broadened a bit due to photon emission during the beam pinching, as presented in figure 2(g). However, since the unique interaction configuration where the electron beam is mainly located near the central axis and the effective field exists mainly on the inner surface of the cone wall, the beam energy loss during focusing can be limited to a few precent while its total charge is almost unchanged.

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As the electron beam shrinks, its number density increases and its transverse radius decreases, which are two important characteristics of beam pinching. To clarify these two features, we investigate the maximum number density and the minimum radius of the focused electron beam. It is shown that an appropriately small cone angle is essential for beam pinching (figure 3(a)), which agrees with our analytical model (see methods). For example, when the cone open angle is reduced to about 0.5°, the minimum radius can be less than $0.2\;\mu {\text{m}}$, and the maximum density can reach $1.3 \times {10^{29}}{{\text{m}}^{ - 3}}$, which is more than 100 times the initial peak density. Furthermore, they also depend on the aperture size of the cone tip, as discussed below.

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On the other hand, the outlet aperture size of the cone target also plays an important role in the focusing of the electron beam. It is shown that when the cone target has an appropriately small open angle (e.g., less than 0.9°), a relatively small outlet aperture is of great benefit to electron beam focusing. As the aperture diameter of the cone tip decreases, the electron beam can be focused along the cone to a smaller size, leading to a higher number density (figure 3(b)). Theoretically, the electron beam may be focused down to a spot size of order $\sim c/{\omega _p}$. However, when the number density of the focused electron beam increases close to the target density, it is dense enough to expel the plasma electrons from the target surface, such that the beam self-fields cannot be fully shielded at the target surface, and the magnetic pinch effect is reduced. As a result, the focusing of the electron beam will reach a limit, yielding a spot size of about 0.1 μm. Additionally, if the cone tip aperture is reduced to zero (i.e. a closed cone), the electron beam will directly interact with the cone wall, which may cause substantial energy loss and hinder the beam pinching.

After interaction with the cone target, the beam diameter is reduced by about an order of magnitude and correspondingly its peak density is enhanced by two orders of magnitude from 10²⁷ m⁻³ to over 10²⁹ m⁻³, as presented in figure 4. In this stage, the beam mostly resides in the central region of the hollow cone that is the lowest field strength region (figures 4(d) and (e)), such that the electrons undergo moderate photon emission and retain more than 90% of the beam energy. This is very important for the next stage to trigger high-field QED effects, as well as other applications as mentioned earlier.

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Now we consider an example of applications with such dense electron beams to generate collimated bright γ-rays via beam–solid interaction by placing a thin solid foil with a thickness of $20\;\mu {\text{m}}$ and the same density as the cone behind the cone tip as a γ-ray converter. The tightly-focused beam is dense enough (${n_b} > {n_p}$) to expel nearly all plasma electrons from the beam volume, leaving the massive ions there. Because of their low velocities, the plasma electrons mainly experience the Coulomb repulsive force from the beam electrons. An almost electron-evacuated channel is formed as a result of the balance between the repulsive force of the beam electrons and the space charge force of the plasma ions. The static fields generated in the plasma channel are dramatically raised to above 10¹⁴V m⁻¹ (figure 4(f)), where the electron beam is surrounded by the huge fields and undergoes stronger effective fields as it traverses the converter compared to in the hollow cone. This brings the quantum parameter up to ${\chi _e} \sim 1$, leading to a significant enhancement of high-energy photon emission. Finally, a highly-collimated γ-ray beam is produced with small size of ∼0.15 μm full width at half maximum and ultrahigh photon density on the order of 10³⁰ m⁻³ (figure 4(g)). This results in unprecedented high brilliance of γ-rays, exceeding ${10^{28}}$ photons s⁻¹ mm⁻² mrad⁻² per 0.1% bandwidth, as detailed in appendix A.

In the following, we give a simple estimation on the cone open angle required for effective beam pinching. The trajectory of the electron beam at its edge in the self-generated magnetic fields can be described as

$$\begin{equation}\frac{{{d^2}r}}{{d{t^2}}} = \frac{e}{{{m_e}{\gamma _b}}}\beta {B_\theta },\end{equation} \tag{ 2 }$$

where ${B_\theta } \approx 2{I_b}/cr$ is the maximum field strength at the beam edge. For simplicity, the beam's current is assumed to be ${I_b} = - \pi e{n_{b0}}c\sigma _ \bot ^2$. Equation (2) can be integrated once to

$$\begin{equation}{\left( {\frac{{dr}}{{dt}}} \right)^2} = A - f\sigma _ \bot ^2\frac{{\omega _{b0}^2}}{{2{\gamma _b}}}\left( {\ln r} \right),\end{equation} \tag{ 3 }$$

where $A$ is a constant, $\omega _{b0}^2 = 4\pi {e^2}{n_{b0}}/{m_e}$, and $f < 1$ is a reduction factor due to the decrease in the pinching force. At the entrance of the cone, one has $r = \frac{{{D_0}}}{2}$ and $\frac{{dr}}{{dt}} = 0$, thus $A = f\sigma _ \bot ^2\frac{{\omega _{b0}^2}}{{2{\gamma _b}}}\left( {\ln \frac{{{D_0}}}{2}} \right)$. Substituting $A$ into equation (3), one obtains ${\left( {\frac{{dr}}{{dt}}} \right)^2} = f\sigma _ \bot ^2\frac{{\omega _{b0}^2}}{{2{\gamma _b}}}\left( {\ln \frac{{{D_0}}}{{2r}}} \right)$, which can be rewritten as $\frac{{dr}}{{dx}} = - {\left[ {f\sigma _ \bot ^2\frac{{\omega _{b0}^2}}{{2{c^2}{\gamma _b}}}\left( {\ln \frac{{{D_0}}}{{2r}}} \right)} \right]^{1/2}} =$ $- {\left[ {2f\frac{{\left| {{I_b}} \right|}}{{{I_A}}}\left( {\ln \frac{{{D_0}}}{{2r}}} \right)} \right]^{1/2}}$ using $dt = dx/c$, where $r < {D_0}/2$, and ${I_A} = {m_e}{\gamma _b}{c^3}/e$ is the Alfvén current. Assuming the final diameter of the focused beam is ${d_f}$ and let

$$\begin{equation}{\theta _f} = {\left[ {2f\frac{{\left| {{I_b}} \right|}}{{{I_A}}}\left( {\ln \frac{{{D_0}}}{{{d_f}}}} \right)} \right]^{1/2}},\end{equation} \tag{ 4 }$$

which represents the beam focusing angle. In order to focus the electron beam effectively, the cone angle should satisfy the relation ${\theta _0} < {\theta _f}$. If taking ${\gamma _b} = 9785$, ${D_0} = 4\;\mu {\text{m}}$, ${n_{b0}} = {10^{27}}{{\text{m}}^{ - 3}}$, ${d_f} = 0.4\;\mu {\text{m}}$ and $f = 0.1$ in one of our simulations, for instance, one has ${\theta _0} < 2^\circ$. This is consistent with our numerical observations in figure 3(a). From equation (4), one notes that ${\theta _f}$ depends strongly on the beam current ${I_b}$, but relatively weakly depends upon the beam diameter ratio ${D_0}/{d_f}$. We wish to emphasize that the beam focusing scheme can also be applied to focus other charged particles, e.g., positrons, as illustrated in appendix B.

In passing, we wish to point out that the instabilities normally found in long-time beam–plasma interactions such as the two-stream instablity, Weibel instability, and hosing instability do not develop significantly in our case. This is mainly due to ultrashort beam length of a few microns and that the beam propagates mainly inside the hollow cone target with a 100 μm-scale length. Therefore, these instabilities have weak effects on the propagation and focusing of the beam during the interaction.

When a high-energy electron traverses intense electromagnetic fields, the rate of high-energy photon emission by the electron with energy ${\varepsilon _b} = {\gamma _b}{m_e}{c^2}$ is given by [67] $\frac{{{d^2}{N_\gamma }}}{{d{\chi _\gamma }dt}} = \left( {\sqrt 3 {\alpha _f}{m_e}{c^2}{\chi _e}/2\pi \hbar {\gamma _b}} \right)$ $F\left( {{\chi _e},{\chi _\gamma }} \right)/{\chi _\gamma }$, where ${\alpha _f} = \frac{{{e^2}}}{{\hbar c}}$ is the fine structure constant, and $F\left( {{\chi _e},{\chi _\gamma }} \right)$ is the quantum-corrected synchrotron function. In the ultrarelativistic limit, the quantum parameter associated with strong-field QED can be characterized by

$$\begin{equation}{\chi _e} = \left( {\frac{{e\hbar }}{{m_e^3{c^5}}}} \right){\varepsilon _b}\left| {{{\mathbf{E}}_{{\text{eff}}}}} \right|,\end{equation} \tag{ 5 }$$

where ${{\mathbf{E}}_{{\text{eff}}}} = \left( {{{\mathbf{E}}_ \bot } + {\boldsymbol{\unicode{x03B2}}} \times {\mathbf{B}}} \right)$ is the effective transverse field experienced by the electron, and ${{\mathbf{E}}_ \bot }$ is the electric field term perpendicular to the electron motion. The radiation power emitted by the electron can be written as

$$\begin{equation}{P_\gamma } = \left( {\frac{{2m_e^2{c^4}}}{{3\hbar }}} \right){\alpha _f}\chi _e^2{\text{g}}\left( {{\chi _e}} \right),\end{equation} \tag{ 6 }$$

where ${\text{g}}\left( {{\chi _e}} \right) \approx {\left[ {1 + 4.8\left( {1 + {\chi _e}} \right)\ln \left( {1 + 1.7{\chi _e}} \right) + 2.44\chi _e^2} \right]^{ - \frac{2}{3}}} \in \left( {0,\,1} \right)$ is the quantum-corrected function [67]. By substituting equation (5) into equation (6), the radiation power can be estimated by ${P_\gamma } \approx \left( {\frac{{2{\alpha _f}\hbar {e^2}}}{{3m_e^4{c^6}}}} \right)\varepsilon _b^2E_{{\text{eff}}}^2{\text{g}}\left( {{\chi _e}} \right)$. Therefore, both QED effects and high-energy photon emission are mainly determined by the electron energy ${\varepsilon _b}$ and the effective field ${E_{{\text{eff}}}}$. When the electron energy ${\varepsilon _b}$ is fixed, the key to increase the radiation emission is to have extremely high ${E_{{\text{eff}}}}$. Many previous studies have proposed to achieve this by use of high-power intense lasers focused on the order of 10²³W cm⁻², allowing the paremater ${\chi _e}$ to be increased to 1 via the head-on collisions of focused laser fields with multi-GeV electrons or much higher intensity laser–plasma interactions. Technically achieving such extreme laser fields and stable laser-electron-beam head-on collisions are still challenging. Moreover, there are inevitable physical restrictions due to laser ponderomotive scattering and nonlinear effects, which cause large divergence and transverse size. These make it challenging to produce collimated high-brilliance γ-ray sources via electron beam collision with ultra-intense lasers. In this work, it is shown that such high effective fields can be realized via direct electron beam-target interactions without laser pulses, as described above in section 2.