Proton Acceleration with Relativistic Electromagnetic Shock

Abstract

Understanding the mechanisms behind the extreme energies of cosmic rays is crucial for unraveling fundamental physical processes in astrophysical environments. This study proposes a novel mechanism for accelerating cosmic‐ray protons. By examining a high‐velocity collision between an astrophysical object and static magnetic fields, the generation of an intense transverse electric field capable of trapping and accelerating protons are find to relativistic energies. Through Hamiltonian analysis, a scaling law that correlates the proton energy is derived to the minimum longitudinal thickness of the relativistic electromagnetic shock required for acceleration. One‐dimensional (1D) Particle‐In‐Cell (PIC) simulations show that an electromagnetic shock driver with a given intensity can accelerate protons from 4.7 MeV to 13 GeV, driven by the transverse electric field induce by the compressed static magnetic field. These results suggest that this mechanism can be experimentally realized in magnetized laser‐plasma systems, offering a novel approach for studying astrophysical phenomena in controlled laboratory experiments.

New Ion Acceleration Mechanism! A novel cosmic‐ray proton acceleration mechanism is proposed, where relativistic electromagnetic shocks accelerate protons transversely. Hamiltonian analysis yields a scaling law linking the proton energy to the minimum longitudinal thickness of the shock. One‐dimensional Particle‐In‐Cell simulations demonstrate that a shock driver of fixed intensity can accelerate protons from 4.7 MeV to 13 GeV.

1. Introduction

The study of high‐energy cosmic rays is of significant interest to scientists, as it offers unique insights into the universe's most extreme and energetic phenomena. The energies carried by high‐energy cosmic rays far exceed those achievable in terrestrial laboratories, enabling the exploration of particle physics in a regime that may reveal physics beyond the Standard Model.^{[ 1} , ² , ³ , ^{4 ]} Recent advancements in observation techniques have greatly enhanced astrophysics and space science,^{[ 5} , ⁶ , ⁷ , ^{8 ]} providing critical information about the composition, structure, and evolution of astrophysical systems.^{[ 9} , ¹⁰ , ¹¹ , ^{12 ]}

Numerous theoretical models have been developed to explain the origin of high‐energy cosmic rays in the universe.^{[ 13} , ¹⁴ , ¹⁵ , ¹⁶ , ¹⁷ , ^{18 ]} In extreme environments such as supernova remnants (SNRs), active galactic nuclei (AGNs), black holes, and pulsars, strong magnetic fields and shock waves are believed to play pivotal roles in particle acceleration.^{[ 19} , ²⁰ , ²¹ , ²² , ²³ , ²⁴ , ²⁵ , ^{26 ]} Observations of SNRs, for example, indicate that charged particles can be accelerated to energies exceeding 100 TeV, consistent with relativistic shock models.^{[ 27 ]} The first‐order Fermi acceleration mechanism, driven by interactions between Alfven waves and converging plasma flows, is widely considered a primary explanation for particle acceleration in SNR shocks, both within the Milky Way and in extragalactic sources.^{[ 28 ]} On larger scales, such as in galaxy clusters and AGNs, the magnetic field is thought to be the primary driver of high‐energy particle acceleration.^{[ 29 ]} The cyclotron auto‐resonance acceleration mechanism has been proposed to explain proton acceleration when the magnetic and radiant fields are strictly parallel in binary systems merging or in supernova explosions.^{[ 30 ]} The compressible turbulence generated by galaxy cluster mergers can re‐accelerate electrons to high energies, triggering observable non‐thermal radiation.^{[ 31 ]} In interstellar matters, quasi‐perpendicular shock waves are employed to explain proton acceleration.^{[ 32} , ³³ , ^{34 ]} Moreover, the diffusive shock acceleration mechanism in galaxy clusters operates on larger spatial scales and produces higher particle energies than SNRs.^{[ 35 ]}

Previous studies demonstrate that the synergy between magnetic fields and shock waves is the key factor leading to particle acceleration in astrophysical environments and is essential for understanding the origin and energy distribution of cosmic rays.^{[ 36} , ³⁷ , ³⁸ , ^{39 ]} Understanding how protons attain such high energies provides insights into the fundamental physical processes occurring in these extreme environments, facilitating the exploration of the fundamental forces of nature, particle interactions, and the behavior of matter under extreme conditions.^{[ 40} , ⁴¹ , ⁴² , ^{43 ]} The interaction of high‐energy protons with interstellar matter can provide valuable insights into the structure of galaxies, cosmic magnetic fields, and cosmic dynamics.^{[ 44} , ⁴⁵ , ⁴⁶ , ^{47 ]} Therefore, understanding proton acceleration mechanisms is crucial for exploring the origins of cosmic rays and unraveling the mysteries of high‐energy phenomena in the universe.^{[ 48} , ⁴⁹ , ⁵⁰ , ^{51 ]}

In this article, we propose a new mechanism for proton acceleration driven by high‐velocity collisions between astrophysical objects and static magnetic fields. Due to the conservation of magnetic flux,^{[ 52} , ^{53 ]} the magnetic field is compressed by the induced plasma current from the object. In astrophysical environments, the scenario of objects with transverse sizes of 10⁵ to 10²⁰ meters and speeds exceeding 0.9c compressing magnetic fields with strength from 10⁻¹⁰ to 10⁻⁵ T could take place in gamma‐ray bursts afterglow shocks^{[ 54} , ⁵⁵ , ⁵⁶ , ^{57 ]} or relativistic jets exploded from AGNs.^{[ 26} , ⁵⁸ , ⁵⁹ , ^{60 ]} Here, c is the speed of light in the vacuum. The strong transverse electric field is induced as the strong magnetic field can move at a velocity close to the speed of light. This electric field, in conjunction with the magnetic field, can trap and accelerate the injected protons to relativistic energies. Using Hamiltonian analysis,^{[ 61} , ^{62 ]} we derive a scaling law that connects the maximum energy E _max of the accelerated protons to the relativistic factors γ _p associated with the drift motion of the astrophysical object and the initial relativistic factor γ₀ of the accelerated protons. Additionally, we derive the criterion for the minimum longitudinal thickness d of the electromagnetic shock required to achieve proton trapping.

This mechanism provides a method for achieving high‐energy proton beams in laboratory laser‐plasma experiments. This mechanism has excellent scalability and promises to accelerate protons up to multi‐GeV. Given that the propagation direction of the driving laser pulse is perpendicular to the proton acceleration direction, this mechanism is ideal for coupling with multiple stages to achieve cascade proton acceleration, where the long‐distance transverse drifting motion could be realized by employing a transverse flying focus configuration.^{[ 63 ]}

2. Results and Discussion

2.1. Theoretical Model

When an astrophysical object moves with velocity v_p (normalized to the speed of light c) along the x‐direction and collides with a static magnetic field B _z0 in the perpendicular direction along z axis, the magnetic field is compressed due to the conservation of magnetic flux.^{[ 52} , ^{53 ]} If the velocity of the object is relativistic, the relationship between the compressed magnetic field B_z and the static magnetic field B _z0 is given by $B_z=2{\gamma }_p^2{B}_{z0},$ ^{[ 64 ]} where the relativistic factor is ${\gamma }_p=1/\sqrt{1-v_p^2}$. The fast variation in the magnetic field generates a strong transverse electric field E_y , which is given by  E_y =  B_zv_p .^{[ 64 ]} Protons can be accelerated (or decelerated) transversely by this electric field E_y (in pink‐shadowed region, transverse electric field assumed to be uniform in the x‐direction), as shown in Figure  1 .

Figure 1.

Schematic of Proton Acceleration. The astrophysical object colliding with magnetic fields compresses the field due to flux conservation, generating a transverse electric field that accelerates protons transversely. The colored circles show the trajectory of the accelerated proton.

We present a theoretical model of proton motion in the electromagnetic field. The coordinate in the co‐moving frame of the astrophysical object can be written as ξ  =  x − v_pt. During the motion, the proton is transversely accelerated in the y‐direction by the electric field, while the magnetic field deflects it in the longitudinal direction. We have A_x =   − yB_z and φ  =   − yE_y , where  E_y =  B_zv_p , with v_p being the velocity of the astrophysical object (normalized to the speed of light c). Since the magnetic field only alters the direction of the proton's motion, the longitudinal canonical momentum is conserved, P_cx = P _cx0 , where P_{cx 0} is the proton's initial longitudinal canonical momentum. Assuming the proton's initial transverse position is y ₀ =  0 (as marked in Figure 1), P_cx = P _cx0  = p _x0 , where p _x0 is the proton's initial momentum in the x‐direction. The proton's momentum in the x‐direction is p_x = p _x0  + a_xyβ. β  = m_e  / m_i =  1/1836 and m_e (m_i ) is electron (proton) mass, β refers to the ratio of electron mass to the proton mass. For convenience, we normalize the electric field E_y as a_y = E_y   e/(m_e ω₀ c), with e being the proton charge, and the magnetic field B_z as a_x = B_z   e/(m_e ω₀), where ω₀ =  2πc/λ₀ is the angular frequency, with λ₀ =  0.8 µm being the laser wavelength in our simulation. The coordinate y is normalized as k ₀ = λ₀ /(2π).

By using Hamiltonian analysis, the derivation of the proton's momentum in the y‐direction as

$$p_y=\sqrt{{\left({\gamma }_0+a_{y}y\beta \right)}^2-1-p_x^2}$$ (1)

which is the function of transverse acceleration distance y. γ₀ refers to the initial Lorentz factor of the proton. The momentum of the proton is correspondingly expressed as $p=\sqrt{{({\gamma }_0+a_{y}y\beta )}^2-1}$. We can derive the normalized quantities from $a_x=2{\gamma }_p^2{a}_{x0}$ and a_y = a_x   v_p . Here, a _x0 is the initial magnetic field normalized by a _x0 = B _z0  e/(m_e ω₀), where B _z0 is the initial magnetic field.

As the proton can be deflected with the strong magnetic field induced by conservation of magnetic flux, the proton's transverse acceleration distance changed by both Lorentz force and strong transverse electric field. When p_y =  0 in Equation (1), the acceleration process terminates, and the proton reaches the position (x ₂, y ₂). Given that the proton's initial position is y ₀ =  0, the maximum acceleration distance y _max can be determined. By normalizing the maximum acceleration distance y _max in meters (m) and formulating the magnetic field B _z0 in Tesla (T), the relationship between the maximum acceleration distance and the magnetic field strength is given by

$$y_{\max }\left(\mathrm{m}\right)=\frac{v_p{\gamma }_0-p_{x0}+\sqrt{{\left(v_p{\gamma }_0-p_{x0}\right)}^2+p_{y0}^2/{\gamma }_p^2}}{B_{z0}\left(\mathrm{T}\right)}\frac{m_{e}c}{2\beta e}$$ (2)

as shown in Figure  2 . If the velocity of the astrophysical object v_p /c is close to unit, the maximum acceleration distance is approximated as $y_{\max }\approx \frac{{\gamma }_0-p_{x0}}{a_{x0}\beta }$. From this, we can obtain the energy at point 2, which represents the maximum energy E _max reached after acceleration:

$$E_{\max }\approx {\gamma }_0+2{\gamma }_p^2\left({\gamma }_0-p_{x0}\right)$$ (3)

Figure 2.

The relationship between the proton's maximum acceleration distance and the initial magnetic field strength with the maximum energy. The lines show the relationship between the proton's maximum acceleration distance y _max and the magnetic field strength B _z0, with the velocity of the astrophysical object v _p = 0.925. The color area shows the representative magnetic field strengths and distance ranges for several astrophysical objects such as interstellar matter (ISM), pulsar wind nebulae (PWN), Gamma‐ray burst (GRB), active galactic nucleus (AGN), white dwarf stars (WD), and neutron stars (NS). The scale size of the magnetic fields of these objects is beyond the maximum acceleration distance y _max.

In the scenario that proton's initial proton momentum is extremely large like astrophysical object, we can describe the maximum energy mainly by $2{\gamma }_p^2{\gamma }_0$. The proton's maximum energy E _max is related to the initial longitudinal momentum p _x0 not initial transverse momentum p _y0 from Equation (3), the initial proton momentum in the x‐direction is denoted as p _x0, while the initial proton momentum in the y‐direction p _y0 are all zero. The initial energies of the proton are calculated as ${\gamma }_0=\sqrt{1+p_{x0}^2}$.

From Figure 2, The upper‐left portion represents the proton acceleration process in astrophysical environments. The lower‐right portion represents the same process in laser plasmas with higher magnetic field but extremely small distance range. The color area in Figure 2 shows the representative magnetic field strengths and distance ranges for several astrophysical objects as interstellar matter (ISM),^{[ 65 ]} pulsar wind nebulae (PWN),^{[ 66 ]} Gamma‐ray burst (GRB),^{[ 54 ]} active galactic nucleus (AGN),^{[ 67 ]} white dwarf stars (WD),^{[ 68 ]} and neutron star (NS).^{[ 69 ]} In our mechanism, the maximum proton energy can be approximated as $E_{\max }\sim 2{\gamma }_p^2{\gamma }_0$, where γ _p is the flying mirror's Lorentz factor and γ₀ is the proton's initial Lorentz factor. Different strength ranges of the magnetic field correspond to different astrophysical environments, such as interstellar matter (ISM), pulsar wind nebulae (PWN), Gamma‐ray burst (GRB), active galactic nucleus (AGN), white dwarf stars (WD), and neutron star (NS). Most of them, in principle, could take place the proposed electromagnetic shock acceleration to generate protons with the energy of $2{\gamma }_p^2{\gamma }_0$.

The scale size of magnetic fields of these objects is beyond the maximum acceleration distance y _max required to attain the maximum energy E _max in our mechanism. In both scenarios, the energy gained through acceleration is nearly identical, indicating that the astrophysical acceleration mechanism can be simulated based on laboratory laser‐plasma experiments. It is difficult for us to simulate the real large spatial acceleration range to reach the maximum energy for the astrophysical environment. However, when the plasma kinetic energy density ε _k =  n(γ _p − 1)m_ec ² significantly exceeds the magnetic energy density ε _B = B ² /(2µ₀), the magnetic field can be fully compressed (as shown in Figure S1 in Supporting Information), giving rise to a transverse electric field. Here, n represents the plasma density, γ _p is the plasma Lorentz Factor, m_e denotes the electron mass, B is the magnetic field strength, and µ₀ stands for the vacuum magnetic permeability. With the magnetization ratio σ  =  ε _B /ε _k ∼ 10⁻² ≪ 1, this parameter serves as a scaling factor bridging astrophysical environments and laboratory laser‐plasma interactions for our acceleration mechanism. On the other hand, we can consider a short acceleration range to get a lower energy under the same magnetic field, as the dashed lines shown in Figure 2. The range of parameters that can be simulated by the laser plasma is around on the order of ≈µm and ≈10 kT, shown as the blue line ($\mathrm{lo}{\mathrm{g}}_{10}(2{\gamma }_p^2{\gamma }_0)=1.1$) in Figure 2. The laser‐plasma interaction shows that the proton energy can approach up to 13 GeV. The acceleration distance corresponding to the magnetic field strengths of different astrophysical environments on the blue line is much smaller than that provided by the astrophysical environment, so we can simulate the proton acceleration process corresponding to the astrophysical environmental parameters on this line in the laboratory. For $\mathrm{lo}{\mathrm{g}}_{10}(2{\gamma }_p^2{\gamma }_0)=2,3,4$ (shown as the light blue, green and yellow line in Figure 2, respectively), such acceleration lengths are expected to be achieved in laboratory by cascade acceleration in the future.

For protons to be trapped by the magnetic field and continuously accelerated by the transverse electric field, they must undergo longitudinal reflection. This implies that the electromagnetic shock has sufficient longitudinal thickness. Assume the proton enters the electric field at t ₀ =  0, y ₀ =  0, and x ₀ =  0, and is reflected at point 1. In the object's co‐moving frame, where ξ  =  x − v_pt, when the proton's velocity in the x‐direction is v _x1 = v_p  , the position at the reflection point 1 is $y_1=\frac{{\gamma }_0{v}_p-p_{x0}}{2a_{x0}\beta }$; From p_x = p _x0  + a_xyβ and Equation (1), the relationship between the proton's position ξ and y can be derived as $\xi =\frac{1}{2a_{x0}\beta {\gamma }_p^2}(\sqrt{p_{y0}^2}-\sqrt{-4{\beta }^2{a}_{x0}^2{\gamma }_p^2{y}^2+4\beta ({\gamma }_0{v}_p-p_{x0})a_{x0}{\gamma }_p^{2}y+p_{y0}^2})$. Then we can get the corresponding position at the reflection point as ${\xi }_1=\frac{1}{2a_{x0}\beta {\gamma }_p^2}(\sqrt{p_{y0}^2}-\sqrt{{\gamma }_p^2{({\gamma }_0{v}_p-p_{x0})}^2+p_{y0}^2})$. Finally, by taking the negative of ξ₁ and normalized in the micron (µm), we obtain the minimum longitudinal thickness d of the relativistic electromagnetic shock required for the proton to undergo reflection and continue the acceleration process:

$$d\left(\mu \mathrm{m}\right)=\frac{0.2}{\pi a_{x0}\beta {\gamma }_p^2}\left(\sqrt{{\gamma }_p^2{\left(v_p{\gamma }_0-p_{x0}\right)}^2+p_{y0}^2}-\sqrt{p_{y0}^2}\right)$$ (4)

From Figure 3b, a longitudinal thickness of approximately 50 µm for the electromagnetic shock to accelerate protons with this mechanism can be realized in laboratory experiments.

Figure 3.

The trapped conditions and proton energy. a) The minimum longitudinal thickness d (shown by the color code) of the relativistic electromagnetic shock corresponding to different injected proton's initial momenta p_x0 and p _y0. b) The maximum proton energy E _max (shown by the color code) corresponding to different injected proton's initial momenta p _x0 and p _y0. The initial proton momentum with thickness of 10 µm (blue dashed line) and 50 µm (red dashed line) is marked on the plot, respectively.

2.2. Simulation Validation

To understand this mechanism, we scale the parameters to a range suitable for laboratory simulation and perform the One‐dimensional (1D) Particle‐In‐Cell (PIC) simulations.

In the relativistic flying mirror's^{[ 64 ]} co‐moving frame, where ξ  =  x − v_pt, v_p ≈ 0.925 is the speed of the relativistic flying mirror (obtained from the simulation results and normalized to the speed of light c). Figure  4 shows that when the strong laser interacts with the plasma, the static magnetic field B _z0 =  40000 T ( a _x0 = B _z0  e/(m_e ω₀) ≈ 3) is compressed, generating a maximum transverse electric field E_ye/(m_e ω₀ c) of nearly 60, as shown by the red solid line. The relationship between the electric field E_y and the initial static magnetic field B _z0 is expressed as $E_y=2{\gamma }_p^2{B}_{z0}c/v_p$. The red dotted line represents the theoretically achievable transverse electric field E_y .

Figure 4.

The relativistic electromagnetic shock induced by compressing a static magnetic field through interaction with a strong laser (a ₀ = 150) at 295T₀.

The selected protons from the trapped high‐energy proton group in x (ξ) coordinate system are shown following the motion trajectories depicted in Figure  5b,c. At the start of the simulation (0T ₀), a proton beam with an initial momentum p _x,in =  0.1 (normalized with respect to the proton's momentum m_ic) and p _y,in =  0 is injected into the region 250λ₀ < x < 300λ₀, and its initial energy is approximately 4.7 MeV. After approximately 295T ₀ (about half cycle of the proton's motion in the 40000 T magnetic field), the proton beam is deflected by the initial static magnetic field B _z0, altering its momentum to p _x0 =   − 0.1, p _y0 =  0. It then enters the relativistic electromagnetic shock, where it is accelerated by the transverse electric field. At t  =  770T ₀, the proton energy spectrum is shown in Figure 5a. The high‐energy portion of the spectrum results from the trapping and acceleration of protons by the intense transverse electric field E_y . The inset shows the energy distribution of the high‐energy portion, where the red line represents the proton energy spectrum, and the color code represents the proton density distribution. This demonstrates that this process can accelerate protons from 4.7 MeV to 13 GeV. Preliminary 2D simulations (as shown in Figure S2 in Supporting Information) indicate that the flying mirror can last for 1 fs and accelerate protons.

Figure 5.

The acceleration result and process of the trapped protons. a) The energy distribution of protons at 770T₀ after acceleration. The acceleration process of the trapped protons, with the color code representing the simulation time. b, c) The motion trajectories of the high‐energy protons in the x and ξ coordinate systems, separately. d) The energy source of the high‐energy protons.

The protons enter the transverse electric field E_y at 295T ₀, traveling considerable distances in the transverse direction due to the acceleration (Figure 5b,c). The energy gained by the protons primarily originates from E_y (Figure 5d), the transverse electric field generated by the compression of the static magnetic field due to the interaction between the strong laser and the plasma. After 770T ₀, the protons, influenced by the Lorentz force, acquire negative transverse momentum p_y , leading to their deceleration and energy loss. As a result, the instantaneous power values fall below the blue line y  =  x in Figure 5d. The simulated acceleration length is y _max =  107λ₀, where λ₀ =  0.8 µm is the laser wavelength. The simulated proton energy is E _max =  13 GeV. Comparing the simulation result with the theoretically predicted value of y _max =  100λ₀ and E _k2 =  15.3 GeV, we find that the simulation results basically agree well with the predictions of the Hamiltonian analysis.

2.3. Discussion

The proposed electromagnetic shock acceleration requires an external magnetic field with a strength of 40000 T, which could be achievable in the near future. Laser‐plasma experiments have already realized kT‐level magnetic field,^{[ 70 ]} while there is some different opinion.^{[ 71 ]} The PIC simulations have generated magnetic fields as strong as megatesla.^{[ 72 ]} The relevant details of producing such a 40000 T magnetic field in the laboratory still need to be further studied. Considering the compression ratio for the magnetic field is much higher for the flying mirror with higher velocity, it is possible to realize the proposed electromagnetic shock acceleration based on the current magnetic field of kT‐level by applying a more intense laser pulse. Alternatively, the magnetic and electric fields can be further enhanced by using the double‐flying‐mirror scheme, where the magnetic field can be further compressed by another flying mirror in the opposite direction.

To achieve longer acceleration distances and higher proton energies, multiple laser‐plasma modules can be used for cascaded acceleration, as shown in Figure  6 . In addition, using a transverse flying focus laser configuration^{[ 63 ]} can further optimize the acceleration effect. Electron acceleration can also be achieved with this method. It is worth emphasizing that the transverse flying focus technique can help reduce the required laser power to realize the extended transverse acceleration distance. Therefore, the proposed mechanism of electromagnetic shock proton acceleration is expected to be realized within the state‐of‐the‐art multi‐PW laser systems.

Figure 6.

Schematic diagram of relativistic electromagnetic shock cascaded acceleration. The direction of the magnetic field needs to change according to the direction of motion of the proton so that the transverse electric field is always in the y‐direction.

In addition, the acceleration mechanism of relativistic electromagnetic shock indicates that the maximum proton energy can approach to $E_{\max }\sim 2{\gamma }_p^2{\gamma }_0{m}_i{c}^2$. The observed proton energy of cosmic rays up to 10¹⁸ eV corresponds to $2{\gamma }_p^2{\gamma }_0\sim {10}^9$. In Figure 2, the orange line of $\mathrm{lo}{\mathrm{g}}_{10}(2{\gamma }_p^2{\gamma }_0)=10$ illustrates that the scaled parameters could be relevant to astrophysical scenarios of interstellar matter (ISM), pulsar wind nebulae (PWN), white dwarf stars (WD), neutron star (NS), and in the region of Gamma‐ray burst (GRB) and active galactic nucleus (AGN). For the magnetic field strength of the above astrophysical scenarios, the distance required for the proton's energy (the order of 10¹⁸ eV) is shorter than the scale length of these scenarios. Therefore, this mechanism could, in principle, take place in the above astrophysical scenarios considering the spatial scale and strength of their magnetic fields to realize the acceleration of ultra‐high‐energy cosmic‐ray particles with energies greater than 10¹⁸ eV.

3. Conclusion

In summary, this work shows a new cosmic‐ray protons acceleration mechanism in astrophysical environments, where the astrophysical object colliding with magnetic fields (plasma is not required) compresses the field due to flux conservation, generating a relativistic electromagnetic shock that accelerates protons transversely. Using Hamiltonian analysis, we analytically derived a scaling law that relates the energy of protons to the minimum longitudinal distance of the relativistic electromagnetic shock required for acceleration. 1D PIC simulations, using a laser pulse with an intensity of 9.6 × 10²² W/cm² to interact with plasma and compress a 40000 T static magnetic field, demonstrate that the transverse electric field generated by this process can accelerate protons from 4.7 MeV to 13 GeV. Moreover, cascaded acceleration can increase the acceleration distance and reach higher proton energies. This acceleration mechanism not only provides a novel method for proton acceleration but also enables the simulation of high‐energy physical phenomena in astrophysical environments, thereby advancing laboratory astrophysics.

4. Experimental Section

Hamiltonian Analysis

The Hamiltonian for proton motion in the electromagnetic field was derived as

$$H=\sqrt{m_i^2{c}^4+{\left(P_{cx}-e{A}_x\left(y\right)\right)}^2{c}^2+p_y^2{c}^2}+e\phi \left(y\right)$$ (5)

m_i was the proton mass, c was the speed of light, P_cx was the canonical momentum of the proton in the x‐direction, where P_cx = p_x   + eA_x , with e being the proton charge and A_x the vector potential in the x‐direction. p_x and p_y were the proton momenta in the x‐ and y‐directions, respectively, and φ was the scalar potential.

The Hamiltonian describing the proton's motion during the acceleration process was given by $H=\sqrt{m_i^2{c}^4+{(p_{x0}+ey{B}_z)}^2{c}^2+p_y^2{c}^2}-ey{E}_y$. After normalization, the expression becomes:

$$H=\sqrt{1+{\left(p_{x0}+a_{x}y\beta \right)}^2+p_y^2}-a_{y}y\beta$$ (6)

p _x0 was the initial x‐direction momentum of the proton, and p_y was its y‐direction momentum, both normalized with respect to the proton's momentum m_ic. The longitudinal vector potential a_x was used to describe the transverse magnetic field B_z , where a_x = B_z   e/(m_e ω₀) was the normalized magnetic field strength. The proton was initially located at y ₀ =  0. The initial Hamiltonian h ₀ of the proton was derived as

$$h_0=\sqrt{1+{p_{x0}}^2+p_{y0}^2}-a_y{y}_0\beta ={\gamma }_0$$ (7)

here p _y0 was the initial y‐direction momentum of the proton. By using the conservation of the Hamiltonian during the acceleration process, Equation (6) and (7) were thus equivalent. This leads to the derivation of the proton's momentum in the y‐direction.

PIC Simulation

The 1D PIC simulations using the completely relativistic code EPOCH was performed. A circularly polarized laser pulse was injected from the left boundary of the simulation window and propagates along the x‐axis. The pulse intensity was I ₀ ≈ 9.6 × 10²² W/cm², corresponding to normalized amplitude of the laser a ₀ = E ₀ e/(m_e ω₀ c)  =  150, where E ₀ was the laser's electric field amplitude, ω₀ =  2πc/λ₀ was the angular frequency, and λ₀ =  0.8 µm was the wavelength. The laser pulse has a trapezoidal temporal profile: 3T ₀ − 10T ₀ − 3T ₀, where T ₀ = λ₀/c  was the laser cycle. A thin solid target was located at x_m =  10λ₀, and it was modeled as a hydrogen plasma with a thickness of d_m =  240 nm and electron density n_e =  200 n_c , where $n_c\approx {\epsilon }_0{m}_e{\omega }_0^2/e^2\approx 1.7\times {10}^{21}\mathrm{c}{\mathrm{m}}^{-3}$ was the critical density, ε₀ was the vacuum permittivity, m_e was the electron mass, and e was the proton charge. The sharp plasma density boundary utilized here does not impact the effect of the proposed mechanism. In addition, the plasma here can be considered as collisionless ideal plasma, so magnetic diffusion can be negligible. A magnetic field with 40000 T along the z‐axis in the region x > 10λ₀. The 1D simulations have a size of 300λ₀, with a grid size of Δx  = λ₀/200 . 200 (50) macro‐particles were used per real particle for target (injected protons) in each grid, with open boundary conditions applied in the ± x‐direction. To speed up the simulation, the moving window technique was applied. It was worth emphasizing that in this 1D PIC simulation, the spatial grid was 1D, but electromagnetic fields (E and B) and particle velocities retain three components. This was crucial for capturing phenomena like magnetic field effects or particle acceleration in perpendicular directions. In addition, the normalized magnetic field B ≈ 4m_e ω₀/e and electron density n_e =  200n_c . For the scenario of laser‐plasma interaction, the typical laser frequency was ω₀ =  2πc/λ₀ where λ₀ =  1 µm. For the situation of the ISM irradiated by astrophysical GRB outflows, if one chooses λ₀ =  4 × 10⁵ m, the magnetic field strength can be estimated as B ≈ 4m_e ω₀/e ∼ 10⁻⁷ T and plasma density was approximately n_e =  200n_c −1.2 cm⁻³.

Conflict of Interest

The authors declare no conflict of interest.

Supporting information

Supporting Information

Xiao T., Zhang X., Kong F., Zheng X., Gong Z., Shen B., Proton Acceleration with Relativistic Electromagnetic Shock. Adv. Sci. 2025, 12, e03538. 10.1002/advs.202503538

Data Availability Statement

All data needed to evaluate the conclusions of the paper are present in the paper.