Introduction—Fast radio bursts (FRBs) are the most extreme coherent electromagnetic radiation observed to date from the universe [1]–[4]. Observational data are rapidly accumulating [5]–[9], yet the physical origins of these bursts remain unclear. Extensive theoretical studies have explored their emission mechanisms and the propagation effects shape the observed signals, including their temporal, spectral and polarization properties [4], [10]–[14]. FRBs have also been leveraged as powerful cosmological probes [15]–[18], with an estimated all-sky rate exceeding \(10^4\) per day [4]. Their multi-wavelength and multi-messenger counterparts, including X-rays [19], gamma-rays [20], neutrinos [21], and gravitational waves [22], have been observed, providing crucial insights into their origins [23]–[25]. To the best of our knowledge, there has been no discussion on the possible connection between FRBs and cosmic rays (CRs).

CRs represent another extreme cosmic phenomena. In particular, the origin of ultra-high-energy cosmic rays (UHECRs) with energies extending beyond 100 EeV remains a mystery. Diffusive shock acceleration (DSA) is the most common shock acceleration to explain UHECRs, notable for naturally producing a power-law energy spectrum similar with observations [26], [27]. However, relativistic DSA is incapable of accelerating proton to 100 EeV [28]. Non-relativistic DSA can potentially achieve ultra-high energy particle acceleration, but only under special shock modes and plasma conditions [28], [29]. Research on UHECRs remains challenging due to the difficulty in detecting the proposed sources. Multi-wavelength and multi-messenger observations, including very-high energy (VHE) neutrinos and ultra-high energy (UHE) gamma rays, have provided deeper insights into the underlying mechanisms, but these messengers are also constrained by the very low statistics of detectable events [30]–[32].

Thanks to their extremely high luminosities, FRBs are unique high-field coherent electromagnetic waves near their sources [4], [33], [34], far surpassing the intensity of the lasers currently available in laboratories. Over the past three decades, extensive research in laser-plasma laboratories has investigated particle acceleration driven by ultra-intense laser pulses. Charged particles can be efficiently accelerated either directly by a laser pulse with specific configurations [35]–[39] or indirectly through high-amplitude electron plasma wakefields excited by the driver pulses [40]–[42]. In astrophysical contexts, the electromagnetic radiation from pulsars can directly accelerate single-particle to ultra-high energy [43]. It has also been proposed that relativistic Alfvenic wave pulses emitted from the accretion disk around a black hole could drive wakefield acceleration [44]–[46].

In this Letter, we theoretically demonstrate the remarkable acceleration capabilities of FRBs near their sources, suggesting their potential as cosmic-ray accelerators, even capable of producing UHECRs. In the ultra-relativistic regime, where the normalized field strength \(a_0 = {eE_0}/{m_ec\omega} > 1000\), two distinct regimes of particle acceleration are identified as the FRB front is eroded in plasma to form a shock-like front, as shown schematically in Fig.\(\,\)1. In the wakefield regime, ion acceleration is dominated by the space charge field of the plasma. At higher field strengths or plasma densities, ions are directly accelerated by the Lorentz force of the FRBs in the so-called piston regime. These regimes exhibit different energy scalings for accelerated ions.

FRBs as ultra-relativistic electromagnetic waves near their sources—The observed FRBs isotropic electromagnetic energy \(W_{iso} = 10^{35} \sim 10^{43}\) erg, duration \(T \approx 1\) ms, frequency \(f = \omega/2\pi\) ranges from 0.1 to 10 GHz [4]. An FRB pulse appears with extremely high electric field strength near its source, with \(E_0 = \sqrt{W_{iso} / cTR^2}\), where \(c\) the light speed, \(R\) the distance from its source. At the frequency of \(f = 1\) GHz, the normalized \(E_0\) can be calculated as \[a_0 = \frac{eE_0}{m_ec\omega} \approx 5.1\times{10}^{13} \left( \frac{W_{iso}}{10^{40}\mathrm{erg}} \right) ^{1/2} \left( \frac{R}{1\mathrm{cm}} \right) ^{-1}, \label{equ:equ1}\tag{1}\] with \(e\) and \(m_e\) the unit charge and the rest mass of electrons, respectively. Generally, the size of the FRB source should be smaller than \(cT \approx 3 \times 10^7\) cm [4], giving the maximum \(\{a_0\}_{max} > 1.7 \times 10^6 \sqrt{W_{iso} / 10^{40}\mathrm{erg}}\). The normalized field strength should below the Schwinger limit \(a_S \approx 1.2 \times 10^{11}\). The presence of GeV photons [47] or highly magnetized particles [48], [49] that may collide with FRB and trigger quantum-electrodynamic (QED) cascades to dampen the radio pulse, further constrain the upper limit of field strength.

The quiver motion of an ion with mass number \(A\) and charge number \(Z\) is highly relativistic in an electromagnetic wave with \(a_0 \approx {Am_p} / Z{m_e}\), where \(m_p\) is the proton rest mass. The electromagnetic waves with \(a_0 \geq m_p/m_e \sim 1000\) are called ultra-relativistic. The interaction of an FRB pulse with charge particles are well in the ultra-relativistic regime before the burst propagates to a radius \(R_c \approx 5.1\times{10}^{10} \sqrt{W_{iso} / 10^{40}\mathrm{erg}}\) cm. In the following, we investigate the particle acceleration by FRB pulses within the region \(r < R_c\), where \(a_0\) typically ranges from \(\sim a_S/10\) to \(10^3\). At larger radii with \(a_0 \gtrsim 1\) the acceleration of particles in the electron-wakefield wave driven by FRBs may be inefficient [34]. Our model does not account for potential background magnetic fields in the local environments of FRBs. These fields provide weak confinement for accelerating particles if the electron gyration frequency (\(\omega_c = eB_0/m_ec\)) is lower than the FRB frequency, typically when \(B_0 < 100\)G, or if the radiation pressure of the burst exceeds the magnetic pressure [50]. This condition applies to environments such as supernova remnant (SNR, \(\sim 100\mu\)G) [51], pulsar wind nebulae (PWN, \(\sim 10\) \(\mu\)G) [52] and interstellar medium (ISM, \(\sim \mu\)G) [53].

The quasi-static and quasi-periodic plasma waves—We start by considering the propagation of an FRB pulse in plasma and the produced wakefield. As the transverse spatial scale of an FRB pulse is extremely large, it can be described by a one-dimensional (1D) model within certain propagation distance with the normalized vector potential \(\boldsymbol{a}(x,t) = e\boldsymbol{A}(x,t)/m_ec^2\), which depending solely on the longitudinal coordinate \(x\) and time \(t\). In such condition, the responses of plasma electrons and ions to an FRB pulse obey the 1D relativistic cold-fluid equations. Introducing a frame co-moving with the pulse \(\xi = x-c\beta_ct\), \(\tau = t\), with \(\beta_c = v_c/c\) the co-moving velocity [54], and taking the quasi-static approximation (\(\partial/\partial\tau=0\)) [55], one obtains the normalized scalar potential equation in the limit \(\beta_c \rightarrow 1\) as \[k_p^{-2}\frac{d^2\phi}{d\xi^2} = \frac{1+a^2}{2\left(1+\phi\right)^2} - \frac{1+\left(a/\sigma_i\right)^2}{2\left(1-\phi/\sigma_i\right)^2}. \label{equ:equ8}\tag{2}\] where \(\phi(\xi) = e\Phi(\xi)/m_ec^2\), \(\sigma_i = Am_p/Zm_e\). \(k_p = 2\pi/\lambda_p = \left(4\pi e^2n_cN_0 / m_ec^2\right) ^{1/2}\) is the plasma wavenumber, \(N_0 = n_0/n_c\) is the normalized ambient electron density with \(n_c = m_e\omega^2 / 4\pi e^2 \approx 1.2\times10^{10} \mathrm{cm}^{-3}\) the critical density. Once the scale potential is found from Eq.\(\,\)(2 ), other quantities can be obtained. When the value of \(a\) is a constant with \(a\left(\xi\right) = a_0\) for the circularly polarized pulse and for \(\sigma_i/a_0^2 \ll 1\), an analytical solution of Eq.\(\,\)(2 ) can be found, which approximates a triangular wave with amplitude \(\phi_{max} \approx \sigma_i\) and period \(\Lambda_p \approx \sigma_i\lambda_p / \pi a_0 \propto \left(a_0\sqrt{N_0}\right)^{-1}\). See Appendix A for the detailed derivation.

As most FRBs are linearly polarized, we can express their vector potential as \(a_y\left(\xi\right) = a_0 \sin\left(k\xi\right)\) for \(-cT \le \xi \le 0\) with \(k\) the wavenumber, where \(cT \approx 10^6\lambda_0\). Equation\(\,\)(2 ) can be obtained by numerically with the Runge-Kutta method based on the Dormand-Prince (4, 5) pair [56]. A typical solution shows in Figs.\(\,\)2 (a1) - 2 (a4) for \(a_0={10}^3\). The electrons with lower inertia are pushed by the intense longitudinal Lorentz force \(-e{\beta_y}{B}\) of the pulse, piling up to form energetic electron peaks as shown in Fig.\(\,\)2 (a2). In the meanwhile, the background ions are pulled forward by the strong Coulomb force of these electron sheets, which also form energetic ion sheets. There exists quasi-periodic electron - ion sheets (Fig.\(\,\)2 (a2)) and the electrostatic field (Fig.\(\,\)2 (a4)) between them, which co-moves with the pulse as a wake-wave. The leading electron sheet at the pulse front (\(\xi \sim 0\)) is named as the front electron sheet (FES) in the following. Particles at density peaks possess extremely high Lorentz factors (Fig.\(\,\)2 (a3)). Here, the ion motion is dominated by the space-charge force, placing them in the wakefield regime. When the electromagnetic pulse takes an even higher amplitude, ion motion is governed by the pulsed Lorentz force, with \(\beta_yB \propto a_yB \propto \sin(2k\xi)\), within the piston regime. Figures\(\,\)2 (b1) - 2 (b4) are the results with \(a_0=10^5\). Electrons and ions simultaneously accumulate under the Lorentz force to form a second-harmonic density distribution, which is futher shaped by the electrostatic field of the plasma wave, as shown in Fig.\(\,\)2 (b2). The plasma wave intensifies as the peak densities, Lorentz factors, and electrostatic field strength increase, while its period reduces to less than \(\lambda_0/2\) (Fig.\(\,\)2 (b4)).

Ion acceleration in the wakefeld regime—The above solutions only give the quasi-static structures based upon the fluid equations, which exclude the effects of the pulse energy loss and the plasma wave-breaking. These effects can be found from particle-in-cell (PIC) simulation with the code EPOCH [57]. The simulated FRB pulse has a normalized electric field \(e_y\left(\xi\right) = a_0\tanh\left(-\xi/\Delta\xi_{up}\right)\sin\left(k\xi\right)\) for \(\xi=x-ct\le0\), where \(\Delta\xi_{up}\) represents the length of the FRB rising edge and interacts with cold electron-proton plasma. Detailed simulation instructions are provided in Appendix B.

Figure\(\,\)3 illustrates the temporal evolution of both the FRB pulse front and the energetic sheets under the condition where \(a_0={10}^3\). The FRB pulse front undergoes continuous erosion by the FES, leading to the formation of a sharp front [58]. This is evident in Figs.\(\,\)3 (a) and 3 (c), where the pulse front moves backward within the moving window, while the FES remains at the first half-cycle of the pulse. The speed of the pulse front is significantly lower than the group velocity estimated from the dispersion relation. The proton sheet located behind the FES is continuously accelerated by the Coulomb force from the FES (Fig.\(\,\)3 (b)) until this proton sheet catches up with the FES. The speed of the proton sheet is higher than the propagation velocity of the pulse, so that it moves ahead of the pulse and form a stable energetic plasma sheet with charge neutrality together with the FES. Meanwhile, the new FES is formed and the acceleration process we just discussed happens all over again. Such a process continues with the formation of multiple plasma sheets, as shown in Figs.\(\,\)3 (c) and 3 (d). These quasi-periodically relativistic plasma sheets diffuse outward and may be eventually detected as CRs.

The formation of neutral plasma sheet requires further dynamics on top of the quasi-static structure. As shown in Fig.\(\,\)2 (a4), the scalar potential between the FES and the adjacent proton sheet approximately follows \(\phi = \alpha_1\sigma _i\left( -\xi/\Delta x_0 \right)^{\alpha_2}\), where \(\Delta x_0\) is the interlayer spacing with \(\Delta x_0 / \lambda_0 = \alpha_3\left( {a_0}\sqrt{N_0} \right)^{\alpha_4}\). Through parameter scanning of Eq.\(\,\)(2 ) across the range of \(a_0\) from \(10^3\) to \(10^5\) and \(N_0\) from \(10^{-3}\) to \(10^2\), the constants \(\alpha_1=0.74\), \(\alpha_2=1.2\), \(\alpha_3=94\), and \(\alpha_4=-0.56\) are obtained. The electrostatic field \(E_x = m_ec^2/e \times d\phi/d\xi \approx m_ec^2\alpha_1\sigma_i /e\Delta {x_0}\). The electromagnetic energy of the pulse within \(\left[\xi - d\xi, \xi \right]\), given by \(\left[E_y\left(\xi\right)\right]^2d\xi/4\pi\), is converted into the kinetic energy of the FES and completely depleted after the erosion time \(\Delta t_{er}\). The work done on FES within \(\Delta t_{er}\) is \(N_\xi F_\xi L_\xi\). The electron numbers \(N_\xi=n_e(\xi)d\xi\), the force \(F_\xi=-eE_y(\xi)\), the displacement \(L_\xi=v_{ey}(\xi)\Delta t_{er}(\xi)\). With the help of the fluid model for the electron density and the relativistic factor, we obtain \(\Delta t_{er} = (1+\phi)\left[1+\cos(k\xi)\right]\tau_0 / 2\pi N_0\). The duration required for the depletion of electromagnetic energy within the first half-cycle is roughly \(\Delta t_{er}(\xi=-\lambda_0/4)\). The front of the FRB pulse erodes with a speed \(v_{er} = \lambda_0/2\Delta t_{er}\). Ultimately, the energy gained by the ion sheet during erosion is \[\mathcal{E}_{wf} = ZecE_{x1}\Delta{x_0}/v_{er} \approx 3.0\times10^8\left(a_0/N_0\right)^{0.67} \mathrm{eV}. \label{equ:equ9}\tag{3}\] As shown in Fig.\(\,\)4 (a), the energy spectrum of accelerated protons exhibits a single peak with energy about 50 GeV. Three-dimensional (3D) simulations of localized particle acceleration by FRB pulses yield results consistent with those of 1D simulations, as detailed in Appendix B.

Ion acceleration in the piston regime—As the amplitude of the FRB pulse \(a_0\) increases to \(10^5\), as shown in Fig.\(\,\)6 given in the End Matter, the immense field strength pushes all the particles along its path to accumulate within the first half-cycle of the pulse, resulting in intense pulse front dynamics. Far behind the pulse front, almost no particle left and thus no electrostatic field is formed, as shown in Figs.\(\,\)6 (b) - 6 (d). Compared with the wakefield regime, both the electrons and protons are accelerated simultaneously by the Lorentz force of the pulse at the leading edge of the pulse with \(\Delta{x_0} / \lambda_0 < 1/2\). The acceleration is governed by momentum conservation in the pulse front co-moving frame [59]–[61]. See Appendix C for the detailed derivation. For ultra-relativistic FRBs with \(a_0^2/N_0 \gg \sigma_i\), the kinetic energy of ions in the plasma sheet is \[\mathcal{E}_{pis} \approx 1.5\times{10}^7 A^{0.5} Z^{0.5} a_0 N_0^{-0.5} \mathrm{eV}. \label{equ:equ12}\tag{4}\] The proton energy spectrum still exhibits a quasi-monoenergetic peak with higher energy, as shown in Fig.\(\,\)4 (b).

Electrons in electromagnetic fields are accelerated by the Lorentz force but decelerated by radiating photons. If the radiation reaction force \(\boldsymbol{F}_{\boldsymbol{RR}} \approx -{2e^4}/{3m_e^2c^4}E^2\gamma^2\left(1-\beta_x\right)^2\boldsymbol{\beta}\) approaches the Lorentz force \(\boldsymbol{F_L}=-e\left(\boldsymbol{E}+\boldsymbol{\beta}\times\boldsymbol{B}\right)\), the classical radiation reaction should be considered [62]. For an electron under ultra-relativistic pulse with \(\beta_y = a / \gamma\), \(\beta_x \approx 1-a^2/2\gamma^2\) [62], \(\boldsymbol{F}_{\boldsymbol{RR}} \sim \boldsymbol{F_L}\) gives a critical condition: \({a^4}/{\gamma} \sim {3m_ec^3}/{e^2\omega}\). In the piston regime, the acceleration process occurs at the pulse front with \(a(\xi\approx0.01\lambda_0)\approx0.002a_0\) and the Lorentz factor for Eq.\(\,\)(4 ). This gives a threshold \(a_{th} \sim 3.70\times{10}^7N^{-1/6}\) as shown in Fig.\(\,\)5. The radiation loss of electron is introduced into PIC simulation using the Monte Carlo method [63]. As shown in Fig.\(\,\)4, the 1D-PIC simulation including radiation reaction obtained a proton energy spectrum similar to that without radiation reaction. Strong-field QED is significant as the quantum nonlinearity parameter \(\chi_e = {F^\ast}/{F_{cr}} \gtrsim 1\), which is the ratio of the field strength in the instantaneous rest frame of the electron to the Schwinger field [64]. When electrons move at relativistic speeds in the same direction as the electromagnetic field, the parameter \(\chi_e \ll 1\). The quantum effect is suppressed during acceleration.

As shown in Figs.\(\,\)5 (a) - 5 (c), these two acceleration models describe the PIC simulation results very well. When \(a_0=10^3\), the separation between neighboring sheets \(\Delta{x_0}\) is sufficiently large, placing them in the wakefield regime. At \(a_0=10^4\), an increase in \(N_0\) results in a transition from the wakefield regime to the piston regime, with the boundary at \(N_0 \approx 1.5\). \(a_0=10^5\) is within the piston regime. Figure\(\,\)5 (d) shows the proton energies from Eqs.\(\,\)(3 ) and (4 ). Ultra-high-intensity bursts can accelerate particles to energies exceeding EeV in tenuous plasma, reaching the energy upper limit of observed CRs.

The diffusion process— The diffusion process sets a lower limit on the plasma density required for significant acceleration. Considering an FRB pulse spreads out in uniform electron – proton plasma with density \(N_0\), its field strength decreases as \(a_0 = 10^{10}\left(R/1\mathrm{cm}\right)^{-1}\). The plasma is accelerated in the piston regime within radius \(R_t \approx 8.7\times10^5N_0^{0.5}\) cm and in the wakefield regime within the range \(R_t \sim R_c=10^7\mathrm{cm}\). It takes time \(t_{pis}\) to obtain a plasma sheet with \(m_i\gamma_{pis}c = Zm_ec{\omega}a_0t_{pis}\) in the piston regime. The time required for a plasma sheet to accelerate in wakefield regime is \(t_{wf} = \Delta{x_0}/v_{er}\). The number of plasma sheets obtained through acceleration is \(\mathcal{N} = \int_{1}^{R_t} dR/ct_{pis} + \int_{R_t}^{R_c} dR/ct_{wf}\). Substantial particle acceleration requires \(\mathcal{N} \ge 1\), which sets a lower limit for the plasma density at \(7.8\times10^{-5}n_c\). On the other hand, the FES continuously erodes the burst, with the total erosion length given by \(l_{er} = \int_{1}^{R_t}(1-\beta_{FRB})dR + \int_{R_t}^{R_c}v_{er}dR/c\). The FRB pulse is severely eroded when \(l_{er} \sim cT\), making it difficult for the FRB pulse and their accelerated particles to escape the source simultaneously. This sets an upper limit on the plasma density at \(556n_c\).

The energy of the accelerated particles decreases as \(a_0\) decreases, \(\mathcal{E}_{pis} \propto a_0 \propto R^{-1}\), \(\mathcal{E}_{wf} \propto a_0^{0.67} \propto R^{-0.67}\). The number of accelerated particles \(\propto R^2\). As a result, the entire diffusion process naturally yields a power-law energy spectrum for the two regimes, giving by \(\mathcal{E}_{pis}^{-2}\) and \(\mathcal{E}_{wf}^{-2.99}\), respectively. The index closely matches the CR spectrum (index (i.e. -3) [30]) and is similar to the DSA (index (i.e. -2) [28]). When an ultra-intense FRB has a long rising edge, under the erosion, the acceleration will also transition from wakefield to piston regime. In most cases, the piston regime accelerates higher-energy particles than the wakefield regime, but its energy spectrum is flatter. This leads to an ankle in the total spectrum.

Conclusion—FRBs as ultra-relativistic electromagnetic waves, can efficiently accelerate particles to ultra-high energies near their sources, making them potential CR-accelerators. It is found that the particle energy increases with the field amplitude \(a_0\) and decreases with the plasma density \(N_0\). Two distinct ion acceleration regimes have been identified, wakefield and piston, each exhibiting different energy scalings. These acceleration mechanisms are also beneficial for understanding the interactions between ultra-relativistic electromagnetic pulses and plasmas in the universe [46] and future exawatt laser facilities [65]. The energies of accelerated particles can cover the entire cosmic ray spectrum (\(10^9 \sim 10^{20}\) eV) and a power-law distribution with an index near -3 naturally emerges as the burst diffuses outward. Additionally, the particle energy show a power-law dependence on both the mass number and the charge number according to Eq.\(\,\)(4 ). This acceleration is more robust than DSA [30] in two aspects: particles can be accelerated directly by the FRB pulse from rest to relativistic speeds, i.e., no injection problem. Once accelerated, the plasma sheets propagate ahead of the burst and no longer influence subsequent acceleration.

FRBs are the first potential directly observable messenger-type accelerators. Active repeating sources produce bursts with similar luminosity [6], [66] and can also repeatedly emit CRs with similar energy spectra and composition. These repeating bursts offer an opportunity to study CRs, particularly UHECRs, and provide insights into the local environments of FRBs. Joint observation of FRBs, CRs and their possible counterparts, such as gamma rays, neutrinos, and gravitational waves, may help us understand these high-energy astrophysical phenomena. This multi-messenger approach could also serve as a unique tool for probing cosmic structures, including interstellar magnetic fields, dark matter, dark energy, and for testing fundamental physics like the weak equivalence principle, Lorentz invariance, and physics beyond the standard model [4], [67].

1 End Matter↩︎

Appendix A: The wakefield equations—The Lagrangians for electron and ion fluids in 1D geometry, \(L=-{mc^2}/{\gamma}-q{\Phi}+{q\left(\boldsymbol{{v}\cdot{A_y}}\right)}/{c}\), does not depend on the transverse coordinates, leading to the conservation of transverse canonical momentum, \(u_y=-{qA_y}/{mc^2}\), \(u_z=-{qA_z}/{mc^2}\). The longitudinal fluid momentum equations can be written as \[\begin{align} \frac{1}{c}\frac{\partial u_{ex}}{\partial t}+\frac{\partial\gamma_e}{\partial x}-\frac{\partial\phi}{\partial x} &= 0, \tag{5} \\ \frac{1}{c}\frac{\partial u_{ix}}{\partial t}+\frac{\partial\gamma_i}{\partial x}+\frac{1}{\sigma_i}\frac{\partial\phi}{\partial x} &= 0. \tag{6} \end{align}\] where \(\boldsymbol{u} = \gamma \boldsymbol{\beta}\) is the normalized momentum. The subscript \(e\) represents electron fluid, and \(i\) represents the ion fluid. The thermal effects are neglected because the quiver energy of particles in the ultra-relativistic pulses is much greater than their thermal energy.

Then, transform the space - time coordinates from the laboratory frame \((x, t)\) to the co-moving frame (\(\xi=x-c\beta_ct\), \(\tau=t\)) and taking the quasi-static approximation with \({\partial}/{\partial\tau}=0\). In the limit \(\beta_c \rightarrow 1\), the Poisson’s equation \(\partial^2\phi / \partial{x^2} = k_p^2(N_e-ZN_i)/N_0\) can be written as Eq.\(\,\)(2 ). The continuity equation \(\partial{N}/\partial{t} + c\partial{(N\beta_x)} = 0\) become \[\begin{align} \frac{N_e}{N_0} &= \frac{1+a^2}{2\left(1+\phi\right)^2}+\frac{1}{2}, \tag{7} \\ \frac{ZN_i}{N_0} &= \frac{1+\left(a/\sigma_i\right)^2}{2\left(1-\phi/\sigma_i\right)^2}+\frac{1}{2}. \tag{8} \end{align}\] The Lorentz factors are \[\begin{align} \gamma_e &= \frac{1+a^2}{2\left(1+\phi\right)}+\frac{1+\phi}{2}, \tag{9} \\ \gamma_i &= \frac{1+\left(a/\sigma_i\right)^2}{2\left(1-\phi/\sigma_i\right)}+\frac{1-\phi/\sigma_i}{2}. \tag{10} \end{align}\]

Equation (2 ) has an approximate analytic solution when \(a(\xi)=a_0\). For natural boundary conditions \(\phi = \partial\phi/\partial\xi = 0\) at the pulse front \(\xi=0\), Eq.\(\,\)(2 ) can be rewritten as \[-2\sqrt{\frac{a_0^{2}+\sigma_{i}}{\sigma_{i}(\sigma_{i}+1)}}k_{p}\xi = \int_{-1}^{\sin\theta} \sqrt{\frac{1-\kappa^{2}(x-\alpha)^{2}}{1-x^{2}}}dx, \label{equ:a7}\tag{11}\] where \(\theta = \arcsin \left\{ 2(a_0^{2}+\sigma_{i})\phi / [a_0^{2}(\sigma_{i}-1)] - 1 \right\}\), \(\kappa={a_0^{2}(\sigma_{i}-1)}/[(a_0^{2}+\sigma_{i})(\sigma_{i}+1)]\), and \(\alpha = \sigma_{i}/a_0^{2}\). For ultra-relativistic electromagnetic waves with \(\alpha \sim 0\), the right side of Eq.\(\,\)(11 ) can be rewritten as \(\mathbb{E}(\kappa) + \mathbb{E}(\kappa, \theta)\), which is the sum of the second kind complete and incomplete elliptic integrals.

Differentiating Eq.\(\,\)(11 ) with respect to \(\xi\) yields \(d\phi/d\xi = -k_pa_0\). From \(\max(\sin\theta) = 1\), we obtain \(\phi_{max}={a_0^{2}(\sigma_{i}-1)}/{a_0^{2}+\sigma_{i}} \approx \sigma_{i}\). So, the electrostatic field satisfies \(E_{x}=m_{e}c^{2}/e \times (2\phi_{max})/\Lambda_p\), where the period \(\Lambda_p = {a_0(\sigma_{i}-1)\lambda_p}/[\pi(a_0^{2}+\sigma_{i})] \approx {\sigma_{i}\lambda_p}/{\pi a_0}\). The scalar potential can be approximated as a triangular wave \[\phi(\xi) = k_pa_0 \begin{cases} -\xi-n\Lambda_p & \xi \in -[n, n+1/2]\Lambda_p, \\ \xi+(n+1)\Lambda_p & \xi \in -[n+1/2, n+1]\Lambda_p, \end{cases} \label{equ:a8}\tag{12}\] where \(n=0,1,2,...\).

Appendix B: The PIC simulations—In 1D-PIC simulations, particles respond to FRB pulses in three velocity dimensions, but their coordinates are updated only in the x-dimension. This means that the FRB pulses and plasmas are considered infinitely flat in the y and z directions. The 1D configuration is adequate for studying the localized particle acceleration processes induced by FRB pulses. The simulation window moves at the speed of light \(c\) to observe the acceleration at the FRB pulse front. The background plasma is initially at rest. The spatial resolution of the simulation grid is \(\lambda_0/1000 = 0.3\) mm. Each grid contains one macro-particle.

The localized particle acceleration is also examined in 3D-PIC simulations. The simulated domain extends \(5\lambda_0\) in both \(y\) and \(z\) directions, with periodic boundary conditions applied. The FRB pulse and plasma are uniformly distributed in these directions. The spatial resolutions are \(\lambda_0/50\) in the \(x\) direction and \(\lambda_0/10\) in both \(y\) and \(z\) directions. Fig.\(\,\)7 shows the 3D simulation results with the same parameters as Fig.\(\,\)3. The acceleration process is identical to the 1D simulation, with no multi-dimensional effects observed that could alter the acceleration. The proton energy spectrum is shown in Fig.\(\,\)4.

For FRB pulses with finite width, the acceleration requires that the particles being accelerated do not escape the pulse front transversely before the acceleration process is complete. The transverse displacement of particles satisfies \(D_{wf} < ct_{wf} \approx 19 \lambda_0 {a_0}^{0.04}{N_0}^{-1}\) in the wakefield regime, and \(D_{pis} < ct_{pis} \approx 4.7\lambda_0 {N_0}^{-0.5}\) in the piston regime. These displacements are far smaller than the spatial scale of FRB pulses, so they have minimal impact on the acceleration processes.

Appendix C: The piston model—The piston model has been developed to study the ion acceleration by intense circularly polarized laser pulses in laser interaction with thin solid foils. The front plasma sheet pushed by a circularly polarized pulses is invariant, attributed to the invariant time-domain light intensity distribution of the pulses. This plasma sheet can perfectly reflect the incident pulses. The intensity of linearly polarized electromagnetic wave oscillates as a sine function, this temporal fluctuation induces instability of the front plasma sheet. Under high \(a_0\) or \(N_0\), as shown in Fig.\(\,\)2 (b) and Fig.\(\,\)6, both the FES and subsequent ion sheet exist within the first half-cycle of the FRB pulse. Before their merging, these sheets can be treated as invariant, making the piston model effective.

Considering the momentum flux in the moving frame of FEL. The photon momentum flux of the electromagnetic wave is \(P_{FRB}^\prime={I^\prime}/{c}\), \(I^\prime=I{\left(1-\beta_{FRB}\right)}/{\left(1+\beta_{FRB}\right)}\) represents the light intensity in the moving frame, following the relativistic Doppler shift and the conservation of photon number. \(c\beta_{FRB}\) is the propagation velocity of the burst. The intensity of linearly polarized electromagnetic wave is \(I={a_0^2n_cm_ec^3}/{2}\). The momentum flux of background plasma is \(P_\mathrm{bg}^\prime=-cn_0^\prime\left(m_e^\prime+{m_i^\prime}/{Z}\right)\beta_{FRB} = -c^2\gamma_{g}^2n_0\left(m_e+{m_i}/{Z}\right)\beta_{FRB}^2\), where \(\gamma_g={1}/{\sqrt{1-\beta_{FRB}^2}}\). The reflectivity of electromagnetic wave in the moving frame is \(R^\prime=R_0\left[{\left(1+\beta_{FRB}\right)}/{\left(1-\beta_{FRB}\right)}\right]^2\), where \(R_0\) is the reflectivity in laboratory frame. The electromagnetic wave is perfectly reflected (\(R^\prime\approx1\)) when \(\beta_{FRB} \sim 1\). The momentum conservation law is written as \(P_{FRB}^\prime=-P_{bg}^\prime\), so \[\beta_{FRB}=1-\frac{1}{1+\sqrt{{a_0^2}/{[2N_0\left(1+\sigma_i\right)]}}}. \label{equ:b1}\tag{13}\] The velocity of the plasma sheet in laboratory frame is \(\beta_{pis}={2\beta_{FRB}}/{\left(1+\beta_{FRB}^2\right)}\), the kinetic energy of of protons in the plasma sheet is \[\frac{\mathcal{E}_{pis}}{Am_pc^2} = \frac{a_0^2/[N_0(1+\sigma_i)]}{1+\sqrt{2a_0^2/[N_0(1+\sigma_i)]}}. \label{equ:b2}\tag{14}\] which reduces to Eq.\(\,\)(4 ) as long as \(a_0^2/N_0 \gg \sigma_i\).

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