## Abstract

Ultrahigh electromagnetic fields (≥~10^{23} W cm^{−2}) are necessary for the study of strong-field quantum electrodynamics (QED). In this study, for the first time, we propose the compression of a pre-seeding static magnetic field with a relativistic flying mirror to generate a high electromagnetic field. The produced field intensity can be further amplified to be 5 × 10^{23} W cm^{−2} owing to the multiple reflections between the flying mirror and a stationary solid target; this produced field intensity is approximately four orders of magnitude larger than that of the seeding field and far exceeds that of the driver laser field (9.6 × 10^{22} W cm^{−2}). Therefore, the ultrahigh electromagnetic field can significantly facilitate strong-field QED effects such as high-energy gamma photon emission. An analytical theory is developed to self-consistently describe the motion of the flying mirror and the field amplification. The predications from the theory are well demonstrated by numerical simulations. The scheme of producing high-intensity electromagnetic fields proposed in this letter provides a new, powerful means to study strong-field QED with a relatively low laser intensity.

© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

The invention of chirp pulse amplification (CPA) [1] technology enabled the production of relativistic lasers ($> 10^{18}$ W cm$^{-2}$) in the laboratory for the first time. Owing to the continuous advances in laser technologies, the highest intensity in the laboratory has reached $10^{22}$ W cm$^{-2}$[2,3].

However, much higher intensity lasers are still required to explore strong-field quantum electrodynamics (QED) under extreme conditions [4–7]. For example, QED predicts an interesting and fascinating phenomenon wherein electron–positron pairs spontaneously appear in vacuum [8,9] when the Schwinger field $E_s = m_e^2 c^3/e\hbar = 1.32 \times 10^{18}$ V m$^{-1}$ is applied [10]. The corresponding intensity is $I_s = \epsilon _0 c E_s^2/2 = 2.3 \times 10^{29}$ W cm$^{-2}$, which is approximately seven orders of magnitude higher than the current laser intensity. Moreover, some strong-field QED phenomena such as gamma photon emission [6,11], radiation reaction effect [12–14], and vacuum birefringence effect [4] will occur when laser intensities exceed $10^{23}$ W cm$^{-2}$.

Although generating an ultraintense beam of $\geq 10^{23}$ W cm$^{-2}$ in the near future [4,15] is possible, further enhancing laser powers and intensities is extremely difficult due to laser-induced damage for solid-state optical components [16,17]. In addition, to reach the Schwinger intensity $I_s$, the energy contained in a femtosecond laser pulse would be tens of MJ [18], which is considered a technologically hopeless path in the near future.

Herein, we report a new, efficient scheme to achieve more intense electromagnetic fields than the present record based on a relativistic flying mirror (RFM) [19–24]. An RFM is formed naturally when a relativistic circularly polarized laser pulse interacts with a sub-$\mu$m solid target. Owing to the continuous acceleration driven by the radiation pressure of the laser pulse, the flying mirror accelerates to 0.93c (c denotes the speed of light) quickly in our case and compresses an applied seed magnetic field to be an intense electromagnetic field. The magnetic field gain is approximately twenty-five-fold. If another stationary target exists, the generated field intensity can be further amplified to be $5\times 10^{23}$ W cm$^{-2}$ owing to the multiple reflections between the mirror and the target, which is larger than the driver laser intensity of $9.6\times 10^{22}$ W cm$^{-2}$. With respect to the initial magnetic field, the final field gain was approximately two orders of magnitude. An analytic theory is derived to successfully explain the magnetic field compression. The amplified ultrahigh field holds promise in terms of opening up new research frontiers in the field of strong-field QED, such as energetic gamma photon emission.

## 2. PIC simulation setup

To verify our scheme, we performed 1D and 2D PIC simulations using the fully relativistic code epoch [25]. A circularly polarized laser pulse was injected into the simulation box from the left boundary and propagated along the x-axis. The intensity of the pulse is $I_0 \approx 9.6\times 10^{22}$ W/cm$^{2}$, corresponding to a normalized laser amplitude of $a_0 \equiv eE_0/m_\textrm{e}c\omega _0 = 150$, where $E_0$ is the electric field amplitude of the laser, $\omega _0 = 2\pi c/\lambda _0$ is the angular frequency, and $\lambda _0 = 800~\textrm {nm}$ is the wavelength. The thin solid target is located at $x_0 = 8~\rm {\mu m}$ ($10 \lambda _0$) and modeled using a hydrogen plasma with a thickness $d = 240~\textrm {nm}$ and electron density $n_0 = 200n_\textrm{c}$, where $n_\textrm{c} = \varepsilon _\textrm{0} m_\textrm{e}\omega _0^2/ e^2 \approx 1.7\times 10^{21}$ cm$^{-3}$ is the critical density, $m_\textrm{e}$ is the electron mass, $- e$ is the electron charge, and $\varepsilon _\textrm{0}$ is the permittivity of vacuum. The applied magnetic field, which is along the y-axis and in the domain of $x > 10 \lambda _0$, satisfies $e\mathbf {B}_0/m_\textrm{e}\omega _0=3\mathbf {e}_y$. The dimensions of the 1D simulations are 300$\lambda _0$, and the cell size is $\Delta x = \lambda _0/200$; the dimensions are $90 \times 60\lambda _0^2$ in the 2D x-y plane, and the corresponding cell sizes are $\Delta x=\lambda _0/80$ and $\Delta y=\lambda _0/40$. In 2D simulations, the incident laser pulse has a Gaussian profile with a duration of $16\rm {T}_0$ (full width at half maximum), where $\rm {T}_0$ is the laser cycle, and a transverse spatial distribution of a fourth-order super-Gaussian, $a \propto \exp [-(r/w_0)^4]$, with $w_0 = 20\lambda _0$. We used 200 (1D) and 8 (2D) macroparticles per cell per species and open boundary conditions in the $\pm x$ and $\pm y$ directions. To speed up the simulation, moving window technology was used in 2D simulations.

## 3. Results and discussions

We begin by performing a 2D PIC simulation to test our scheme. Figure 1(a) shows the schematic of magnetic field compression. A high-intensity circularly polarized laser pulse interacting with a thin plasma target exerts a large force on the irradiated area owing to the radiation pressure of the pulse. The volume of irradiated plasma can be constantly accelerated and detached from the thin target. The pre-seeding magnetic field [Fig. 1(a)] was compressed by the flying mirror.

The simulation shows that, at the beginning of compression, the generated magnetic field is small and then gradually increases with time. Figure 1(b) shows the spatial and on-axis distribution of the compressed magnetic field. The maximum value of the field has increased approximately twenty-five-fold. As the velocity of the mirror and the value of the compressed magnetic field have the same dependence on time, they are proportional to each other. A detailed analysis is performed using the following theoretical model. The corresponding electric field distribution $E_z$ is shown in Fig. 1(c), which has almost the same normalized magnitude as the magnetic field. A comparison between Fig. 1(b) and (c) indicates that an electromagnetic field was formed.

Owing to the transverse super-Gaussian profile of the laser pulse, the resulting flying mirror also has the same profile in the transverse space. This usage of the super-Gaussian laser helps maintain the structure of the mirror, thus leading to a longer acceleration time and a larger velocity. At this time of $t=130 {\rm T_0}$, the velocity of the mirror is approximately $0.93c$ shortly after the laser interactions with the mirror are complete.

Subsequently, we show that the intensity of the electromagnetic field in Fig. 1 can be further amplified to $5\times 10^{23}$ W cm$^{-2}$, far exceeding the value corresponding to the driver laser. QED effects cannot be ignored at this intensity. QED effects such as pair production, synchrotron emission and radiation reaction are included in the simulations. To this end, in addition to using the same input parameters as those in Fig. 1, another plasma slab was placed at $x=130 \lambda _0$ (white dashed line in Fig. 2(a)). The simulation results for the two-target geometry are shown in Fig. 2(a)–(f), where the QED effects were included in the simulations.

The field amplification by the flying mirror in Fig. 1 is viewed as the first stage of magnetic field compression. The produced field and flying mirror propagate forward together. However, the field cannot penetrate the second stationary target owing to its high density. Then, the produced field can be further compressed and amplified owing to the multiple reflections between the mirror and the second target, which is viewed as the second stage. When QED effects are included, the maximum value of $eB/m_e\omega _0$ is 343 ($5\times 10^{23}$ W cm$^{-2}$), which is slightly smaller than the case without the QED considered, as shown in Fig. 2(a) and (b). Figure 2(c) shows that the electric field from the first stage cannot be further amplified efficiently in the second stage owing to the boundary effect of the stationary target. Including the QED module in the simulations has little effect on the maximum electric fields, as shown in Fig. 2(c)–(d), and the normalized electric field values are approximately one-third the value of the normalized magnetic field. The amplitude of the produced field is enhanced by two orders of magnitude with respect to the initial seed magnetic field and is approximately $2.5$ times higher than that of the laser field.

Compared with that in Fig. 2(b), the decrease in the magnetic field in Fig. 2(a) may be related to the generation of the gamma photon emission. According to the QED theory [5], when energetic electrons interact with electromagnetic fields, the energy of the field can be absorbed, and several energetic gamma photons are emitted, or even prolific electron–positron pair production is possible [8,26]. Although we did not observe a large number of positrons in our simulations, the gamma photon emission is significantly different when a seed magnetic field is present, as shown in Fig. 2(e) and (f). Energetic gamma photon emission occurs during the second stage (see $x=130\lambda _0$ in Fig. 2(f)) when a magnetic field is applied, and the maximum energy of the photons is approximately $400$ MeV. Before RFM impinges on the stationary target, the gamma photon energies produced are mostly tens of MeV, and there is no significat energy peak in the spatial distribution of photon energy.

Next, we give an explanation for the peak occuring in Fig. 2(f). In the QED regime, the probability for photon emission can be characterized by the quantum invariant [5]

which is the ratio of the electromagnetic field in the electron’s rest frame to the Schwiger field $E_s$, where $\gamma _e$ is the relativistic factor of the electron, $\bf {E}_{\bot }$ is the electric field perpendicular to the direction of motion of the electron, and $\textbf{P}/m_e c$ is the normalized momentum of electrons. When $\eta \sim 1$, the QED process of gamma photon emission must be considered. And the energy of radiated photons can be estimated as [8] When RFM impings on the stationay target at $t= 151 {\rm T_0}$, the electrons in the RFM have ultrahigh energies with the maximum value of relativistic factors approaching to $1400$, as shown in Fig. 3. As mentioned before, the laser pulse has been completely reflected at $t=130\rm {T}_0$, so the electrons in the RFM only interact with the produced electromagnetic field at this time. The quantum invariant can be reduced as $\eta = \frac {1}{E_s}\sqrt {(\gamma _e E_z + P_x B_y/m_e)^2 + (P_z B_y/m_e)^2}$ for our case. If taking $\gamma _e = 1400$, $eE_z/m_e \omega _0 c= -120$, $e B_y/m_e \omega _0 = 350$, $P_x/m_e c=1200$, and $P_z/m_e c = -1000$, then $\eta = 1.31$ and the maximum energy of radiated photons $W=412$ MeV, which is in agreement with the simulation results. However, note that because $\gamma _e E_z$ and $P_x B_y$ have different signs, so $(\gamma _e E_z + P_x B_y/m_e)^2 < (|\gamma _e E_z| + |P_x B_y/m_e|)^2$. When only considering the magnetic field component, then $\eta =\frac {1}{E_s}\sqrt {(P_x B_y/m_e)^2 + (P_z B_y/m_e)^2} = 1.65$, and $W=519$ MeV, which is larger than the simulation results. In the case of only electric field $E_z$, $\eta = \frac {1}{E_s}|\gamma _e E_z |= 0.51$ and $W=160$ MeV, which can’t explain the peak in Fig. 2(f). Therefore, we can conclude that the electromagnetic fields $E_z$ and $B_y$ produced in the interaction of RFM and the stationary target both have an effect on the emission process of gamma photons, and that the enhancement of the gamma photon emission at $x = 130\lambda _0$ is mainly related to the generation of an ultrahigh magnetic field. A complete exploration of the QED effects in this process is beyond the scope of this letter, and this issue will be addressed in future work.We now illustrate how to form the flying mirror with 1D PIC simulation and derive a theory to describe the magnetic field compression in the first stage. We traced the trajectories of typical particles (50 electrons and 50 protons), as shown in Fig. 4. It can be inferred that electrons and protons leave the target together as an electrically neutral plasma sheet at time $\Delta t = 2.4 \rm {T}_0$, as shown by the dashed cross lines in Fig. 4. The slope of the trajectories represents the velocity of the flying mirror and increases rapidly with time, indicating that the mirror rapidly accelerates under the effect of light pressure, which is extremely beneficial for generating a high electromagnetic field by compressing the magnetic field behind the target. The 1D PIC simulation shows that the velocity of mirror reaches $0.9c$ at $t=90\rm {T}_0$.

The light pressure drives the acceleration of the mirror that then compresses the applied magnetic field. Figure 5(a) shows typical magnetic field distribution at two different moments. At $t=120\rm {T}_0$, the laser-plasma interaction was terminated for a while, while the magnetic field of the laser pulse was still in the box, as shown by the sinusoid-like curve on the left of the field distribution. The distribution of the magnetic field at $t=250\rm {T}_0$ shows that the field magnitude on the left of the peak is zero, indicating the conservation of magnetic flux during magnetic field compression. The compressed magnetic field gains a maximum value of $eB/m_e \omega _0 = 60$, which is 20 times higher than the initial value. At the end of the laser–plasma interaction, the maximum value of the field remains the same, as shown in Fig. 5(a). The width of the compressed magnetic field increases gradually, as indicated by the arrow in Fig. 5(a). This indicates that the magnetic field produced at the previous moment propagates forward at the speed of light. Figure 5(b) and (c) clearly show that the static magnetic field is compressed into an electromagnetic field.

It is important to note that the driving laser pulse must have an ultrahigh temporal contrast to ensure interaction with a solid target. However, the significant improvement in plasma mirror [27] technology has allowed to achieve a contrst of $3 \times 10^{11}$, which is great for accelerating ultrathin foils as a rigid body. In order to show the scheme of generating ultra-high electromagnetic fields, a laser pulse with super-Gaussian intensity distribution in transverse direction is used as the driving laser in the paper. However, when the intensity of the laser pulse has a Gaussian distribution in transverse direction, the scheme of generating fields of QED intensity is still efficient. The produced fields in Fig. 1 also have a Gaussian distribution in transverse direction.

## 4. Theory

The high magnetic field generation associated with the motion of the mirror in the first stage can be calculated as follows: First, as mentioned above, the flux is conserved before and after magnetic compression. The flux before compression is $\Phi _1=B_0\beta \Delta t$, and the flux after compression is $\Phi _2 = B(1-\beta ) \Delta t$, where $\beta$ is the velocity of the mirror, normalized to $c$. According to the conservation of magnetic flux, the compressed field can be expressed as follows:

When the velocity of the flying mirror approaches $c$, Eq. (3) can be simplified as where $\gamma = \sqrt {1/(1-\beta ^2)}$ is the relativity factor of the mirror.The behavior of the flying mirror was subject to two forces. Owing to the high density of the plasma target, the incident laser pulse is reflected off the surface of the mirror, which exerts a force $\mathbf {F}_1$ to the mirror, named radiation pressure. This force $\mathbf {F}_1= (1+R)\varepsilon _0 E^2 \frac {1-\beta }{1+\beta } \mathbf {e}_x$ was first given by Einstein [28], where $R$ is the reflection coefficient and is often used to describe the radiation pressure acceleration of ions [29–32]. In addition, because the produced magnetic field propagates at velocity $c$, it exerts a recoil force to inhibit the movement of the flying mirror. Thus, the second force can be expressed as $\mathbf {F}_2 = -\frac {B^2}{\mu _0} \frac {1-\beta }{\beta }\mathbf {e}_x$. Combining $\mathbf {F}_1$ and $\mathbf {F}_2$ provides a precise description of the motion of the mirror under the effect of the laser and the produced electromagnetic field, which can be written as

Equations (3)–(5) give a self-consistent solution to the magnetic compression scheme presented herein. To examine the correctness of the theoretical model, a comparison between the theory and PIC simulations is shown in Fig. 6, where Eqs. (3) and (5) are numerically solved. From $t=10-115\rm {T}_0$, the flying mirror gradually accelerates under the combined effect of the radiation pressure from the incident pulse and the recoil force from the produced electromagnetic field, and the $\gamma$ approaches 2.96, indicating that the velocity of the flying mirror is $0.94 c$.

As the light pressure is much greater than the recoil force, the acceleration does not stop until the laser is completely consumed. Subsequently, the flying mirror starts to slow down owing to the recoil force, and the produced magnetic field begins to decrease. Therefore, the profile of the pulse magnetic field is as shown in Fig. 6(b). Equation (4) clearly shows the numerical relation between the produced magnetic field and the $\gamma$ factor. For example, it implies that $B$ increases to 18 times the initial value ($\rm {B}_0$) for the maximum $\gamma$, which is in good agreement with Fig. 6(b). Another case is shown in Fig. 6(c) and (d), where a sinusoidal laser is used. It can be seen that the analytic theory agrees well with the PIC simulation results.

## 5. Conclusion

In conclusion, we have presented a new scheme for generating high-intensity electromagnetic field. The static magnetic field can be compressed by an RFM into an electromagnetic field, and the generated field is enhanced approximately twenty-five-fold with respect to its original amplitude. Owing to the multiple reflections between the flying mirror and the second target, the produced field intensity can be further amplified to a QED intensity $5\times 10^{23}$ W cm$^{-2}$, far exceeding the driver laser intensity $9.6\times 10^{22}$ W cm$^{-2}$, and is approximately four orders of magnitude larger than the applied magnetic field intensity. Therefore, the ultrahigh field can significantly facilitate strong-field QED effects such as high-energy gamma photon emission. This new phenomenon opens up new opportunities to explore strong-field QED with a relatively low laser intensity as well as paves the way for laboratory astrophysics [33–35] and high-energy density physics associated with high magnetic fields [36–45].

Although a magnetic field of $3\times 10^{4}$ T is used to generate an ultrahigh electromagnetic field in this paper, the principle of magnetic field compression applies to magnetic fields of any value due to the normalized representation of magnetic fields. Currently, technology to generate strong magnetic fields based on high-power lasers is developing rapidly, and can already generate fields of kilotesla order in experiments [46]. In theory, a new way of generating megatesla magnetic fields has been proposed [45]. With the development of magnetic field generation technology, such a strong field ($\sim 10^{4}$ T) may be produced in the near future.

## Funding

Ministry of Science and Technology of the People's Republic of China (2016YFA0401102, 2018YFA0404803); National Natural Science Foundation of China (11922515, 11935008, 12005138).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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